Improved Power Control of Inverter Sources in Mixed-Source. Microgrids

Transcription

1 Improved Power Control of Inverter Sources in Mixed-Source Microgrids By Micah J. Erickson A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Electrical Engineering) at the UNIVERSITY OF WISCONSIN-MADISON 2013 Date of final oral examination: 12/17/12 The dissertation is approved by the following members of the Final Oral Committee: Thomas M. Jahns, Professor, Electrical Engineering Giri Venkataramanan, Professor, Electrical Engineering Robert D. Lorenz, Professor, Mechanical Engineering Chris L. DeMarco, Professor, Electrical Engineering Bernard Lesieutre, Professor, Electrical Engineering Bulent Sarlioglu, Associate Professor, Electrical Engineering Robert H. Lasseter, Emeritus Professor, Electrical Engineering

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3 i Abstract The Consortium for Electric Reliability Technology Solutions (CERTS) microgrid concept is an established approach for controlling many distributed sources on either an isolated or gridconnected power system that does not need a fast communications network for control. This research investigates the impact of load transients on the operating characteristics of CERTS microgrids including the voltage magnitude, frequency, and power regulation of an islanded power system. For voltage source-based distributed generators, load changes can overload microgrid sources that have low operating margins, causing significant overcurrents and, under worst-case conditions, instability and voltage collapse. This research investigates the dynamic characteristics of inverter microsources in microgrids and determines the impediments to fast load redistribution and rebalancing. Based on these results, control techniques are developed that maintain microgrid stability and rapidly redistribute load changes among the sources to minimize the stresses on sources with limited overload power and energy capabilities. Linear operating point analysis and simulation techniques are used to explore the dynamic characteristics of microgrids under various network parameter sets. Attention is focused on a particularly challenging microgrid configuration consisting of photovoltaic (PV) sources having low amounts of stored energy. The power regulation bandwidth of a photovoltaic inverter must be fast enough to conserve its limited available energy and avoid collapse of its DC bus voltage during load transient conditions. Both grid-forming PV inverter controllers using frequency droop regulation and grid-following inverter controllers using current regulation are developed and compared to evaluate their dynamic power regulation capabilities and their impact on microgrid performance. This comparative analysis leads to the development of techniques for improving the power regulation bandwidth of grid-forming controllers that reduce the risk of instability and faults instigated by load transients. One of the key findings of this research is a

4 ii limit on the maximum frequency droop gain to insure microgrid stability that is directly proportional to the resistance of the electrical lines/cables interconnecting the microgrid sources. Two control techniques are demonstrated for increasing this limit on the maximum frequency droop gain and power regulation bandwidth: 1) by actively increasing the effective line resistance, or 2) by decreasing the effective frequency droop gain. Key results of this work are verified by means of experimental tests using a laboratory CERTS microgrid testbed.

5 iii Contents List of Figures... ix 1 Introduction Background Microgrid Overview Microgrid Sources Combined Heat and Power (CHP) Microgrid Control Techniques Problem Statement Technical Approach Thesis Organization State of the Art Microsource Unit Control Concepts in Modern Publications Isolated Power System Examples Application Considerations for Supervisory Controller Interactions Frequency-Regulated Converter Control Publications System Response Characteristics of a Power-Constrained Droop-Regulated Source in an Islanded Microgrid Introduction Source and System Model Development Definition of System Variables: Inverter Microsource Model:... 41

6 iv Generator-Set Model: Inverter-based Microsource Only System Characteristics Mixed-source Microgrid System Characteristics with Microsource and Synchronous Machine Generator Partial Overloading Response Characteristics Microsource 1 Initially Overloaded, Microsource 2 Tracks Load Microsource 1 Initially Overloaded, Genset2 Tracks Load Change Load Rebalancing Experimental Results Experimental Hardware Transient Experimental Results Conclusions Comparative Analysis of PV Inverter Controller Performance in a Microgrid Introduction PV Microgrid Overview PV Cell Overview Selection of Number of Series Cells to Match Design Voltage Controller Topologies and Implementations Grid-Following Power Controller Grid-Forming Droop Controller Controller Topology Performance Indicators and Results Grid Connected Step Insolation

7 v Maximum Power-Point Tracking While Grid Connected Maximum Power-Point Tracking While Islanded Load Change in Island with Energy Storage (ES) Unit Island When Importing Island When Exporting Island While Exporting, Then Source Shedding Island When Importing, Causes Overload and Frequency Trip of Non-Critical Load Island Alone When Exporting Island Alone when Importing, Causes Overload and Frequency Trip Conclusion Improvement of Dynamic Power Responses of Grid-Forming Sources in a CERTS Microgrid Introduction Voltage Balance Instability Investigated Limit of Combined Frequency and Voltage Gain Experimental Verification of Resistance and Frequency Droop Gain Methods Mitigating Voltage Balance Instability Active Resistance Input (Feedback) Filtering Lead-Lag Compensation

8 vi 5.4 Power Control Bandwidth Improvements Improved Power Response Using Active Resistance Improved Power Response Utilizing Input Filtering Improved Power Response Utilizing Lead-Lag Compensator Dynamic Power Response Improvement Conclusions Stability of Multi-Node Networks Passivity by Bounded Operating Point Eigenvalues Passivity by Tests for Convergence Inductor Stored Energy Decay Passivity by Convergence of P-Q Error Vector Multi-Source System Stability Conclusions Analysis of Microgrid Stability Considering Penetration of Grid-Following versus Grid- Forming Sources Fixed-Power Grid Following Limits Variable-Power Grid Following Limits Frequency Feedback Filtering Grid-Following Penetration Conclusions Scalability of Active Damping Results to Microgrids of Different Power Ratings Reactance and Resistance Values as a Function of Inverter Scale Other System Scaling Considerations System characteristics and frequency droop gain limitations for grid-connected inverters

9 vii 8.3 Conclusions on Applicability of Findings to Scaled Systems Example Microgrid Source-Source Impedance Characteristics Conclusions Contributions Summary Analysis and Simulation of Power-Constrained Droop-Regulated Inverter Microsource in an Islanded Microgrid Performance of PV Source in a CERTS Microgrid Voltage Balance Stability Mode of Frequency-Droop Controlled Sources Analyzed Increased Power Control Bandwidth of Frequency Droop-Regulated Sources Enabled by Avoidance of Voltage Balance Instability Experimental Verification of Active Resistance Damping and Loop-Shaping for Increased Power Control Bandwidth Analysis of Microgrid Stability Considering Penetration of Grid-Following versus Grid-Forming Sources Scalability of Results to Sizes and Voltage Levels of Microgrids Investigated CERTS Inverter Microsource for Passive Characteristics Future Work Specific Points of Continued Research Suggested Future Focus of Research Efforts References Appendix A: Inverter Hardware Description

10 12 Appendix B: Inverter Source Code viii

11 ix List of Figures Figure 1-1 CERTS microgrid Power-frequency and Reactive power-voltage droop Figure 1-2 Power vs. frequency droop operating points Figure 2-1 baseline topology of solid state transformer from [25] Figure 2-2 Operating modes of dual active bridge solid state transformer Figure 2-3 Hybrid AC/DC microgrid from [27] Figure 2-4 Decision tree for intelligent energy controller from [27] Figure 3-1 Worst-case load change drawn from a fixed power source Figure 3-2 Two-source configuration with two microsources Figure 3-3 Inverter microsource 1-wire equivalent circuit model Figure 3-4 Reduced order model of frequency and voltage controller Figure 3-5 Electromechanical model of synchronous machine generator set Figure 3-6 Nominal eigenvalues with inverter-based microsources Figure 3-7 Voltage balance and voltage regulation eigenvalue migration Figure 3-8 Voltage balance and Phi2 eigenvalue migration Figure 3-9 Voltage balance and voltage regulation2 eigenvalue migration Figure 3-10 Power regulation eigenvalue migration Figure 3-11 Power regulation and voltage regulation eigenvalue migration Figure 3-12 Power regulation eigenvalue migration Figure 3-13 Power regulation and load angle eigenvalue migration with increases in KiPMax.. 59 Figure 3-14 Voltage balance eigenvalue migration Figure 3-15 Filter L-C mode (center) and inter-source modes (right) eigenvalue migration Figure 3-16 Power regulation eigenvalue migration and voltage regulation eigenvalue... 63

12 x Figure 3-17 Voltage balance eigenvalue migration Figure 3-18 Inter-source modes eigenvalue migration Figure 3-19 Energy regulation eigenvalue migration and voltage regulation eigenvalue Figure 3-20 Load current eigenvalue migration as a function of increasing the system load Figure 3-21 L-C filter eigenvalue migration as a function of increasing the system load Figure 3-22 Low frequency eigenvalue migrations as a function of increasing the system load. 68 Figure 3-23 Nominal eigenvalues for mixed source microgrid system Figure 3-24 Speed regulation eigenvalue migration Figure 3-25 Torque regulation and Phi1 eigenvalue migration Figure 3-26 Migration of Torque/Speed regulation eigenvalue migration Figure 3-27 Migration of low-frequency eigenvalues in response to increases in KiV Figure 3-28 Voltage balance q-axis damper and Phi1 eigenvalue migration Figure 3-29 Source 2 torque/speed and field voltage eigenvalue migration Figure 3-30 Field voltage and Voltage regulation eigenvalue migration Figure 3-31 Inter-source current and Phi2 eigenvalue migration Figure 3-32 Torque regulation and speed regulation eigenvalue migrations Figure 3-33 Torque regulation and Phi2 eigenvalue migration Figure 3-34 Inter-source current eigenvalue migration while varying L1 and R Figure 3-35 L-C Filter eigenvalue migration as inductance and resistance is varied Figure 3-36 Phi1 and Voltage Regulation eigenvalue migration Figure 3-37 Filter L-C eigenvalue migration when varying L11 and R Figure 3-38 Voltage balance eigenvalue migration when varying L11 and R Figure 3-39 Torque Regulation and Voltage Regulator 1 eigenvalue migrations... 86

13 xi Figure 3-40 Load current eigenvalue migration when varying Rload, varying system load Figure 3-41 Low-frequency eigenvalue characteristic Figure 3-42 L-C filter eigenvalue migration when varying Rload, varying system load Figure 3-43 Inverter microsource system response from a 1pu load to a 2pu load Figure 3-44 Mixed microsource system response from a 1pu load to a 2pu load Figure 3-45 Power system block diagram and instrumentation Figure 3-46 Components of experimental power system Figure 3-47 Battery source inverter and DC capacitors Figure 3-48 Inverter for DC supply source Figure 3-49 Battery pack (APC 196Vx4, 10Ah) Figure 3-50 Magna-Power DC supplies for DC supply microsource 500Vx2, 20A Figure 3-51 Inverter microsource system response from a 0.385pu load to a 0.77pu load Figure 3-52 Simulation transient to match experimental trace in Figure Figure 4-1 Voltage and Current characteristic of a Mitsubishi solar cell Figure 4-2 Maximum power characteristic for various insolation intensities and cell temps Figure 4-3 Direct-tied PV array configuration utilized for the work here Figure 4-4 LCLX filter configuration for PV microgrid inverter Figure 4-5 Power controller block diagram Figure phase PLL Figure 4-7 Complex power calculation and current command generator Figure 4-8 q-axis Command Generator and Current Regulator Figure 4-9 q-axis Capacitor Voltage and Line Current Observer Figure 4-10 Droop controller (usrc) block diagram

14 xii Figure 4-11 Positive sequence filter topology used in droop controller Figure 4-12 Frequency calculator with droop function and power limit controller Figure 4-13 Voltage command regulator Figure 4-14 Power and Reactive Power for step-decrease in insolation Figure 4-15 Array voltage and Inverter Frequency for step-decrease in insolation Figure 4-16 Power and Reactive Power for grid connected step-insolation Figure 4-17 Array Voltage and Inverter Frequency for grid connected step-insolation Figure 4-18 Power and Reactive Power during MPPT while grid connected Figure 4-19 Array Voltage and Inverter Frequency during MPPT while grid connected Figure 4-20 Power and Reactive Power during MPPT in island Figure 4-21 Array Voltage and Inverter Frequency during MPPT in island Figure 4-22 Power and Reactive Power for islanded load increase Figure 4-23 Array Voltage and Inverter Frequency for islanded load increase Figure 4-24 Power and Reactive Power for islanded load decrease Figure 4-25 Array Voltage and Inverter Frequency for islanded load decrease Figure 4-26 Power and Reactive Power during island event with ES source Figure 4-27 Array Voltage and Inverter Frequency during island event with ES source Figure 4-28 Power and Reactive Power during island event with ES source Figure 4-29 Array Voltage and Inverter Frequency during island event with ES source Figure 4-30 Power and Reactive Power during island event Figure 4-31 Array Voltage and Inverter Frequency during island event Figure 4-32 Power and Reactive Power during island event when importing Figure 4-33 Array Voltage and Source Frequency during island event when importing

15 xiii Figure 4-34 Power and Reactive Power during island event Figure 4-35 Array Voltage and Inverter Frequency during island event Figure 4-36 Power and Reactive Power during island event, usrc only, previously importing. 157 Figure 4-37 Array Voltage and Inverter Frequency during island event Figure 5-1 Progressive steps of block diagram simplified restructuring Figure 5-2 Greatest real value of the three eigenvalues Figure 5-3 Limit of system stability based on relationship Figure 5-4 System stability envelope tracking max. real value of eigenvalues Figure 5-5 Experimentally determined limit of system stability Figure 5-6 Active resistance voltage implementation employed in simplified model Figure 5-7 Active resistance implementation on full microsource control diagram Figure 5-8 Frequency response of active resistance contribution to system transients Figure 5-9 Effective pass-through gain for positive sequence filter Figure 5-10 Power signal filtering implementation on full microsource control diagram Figure 5-11 Power signal lead-lag filtering implemented on full microsource control diagram 181 Figure 5-12 Effect of increasing Ra (nominally 0.01) on nominally tuned eigenvalues Figure 5-13 Effect of changing Ra (0.025pu-0.4pu. 0.1pu nominally) on eigenvalue Figure 5-14 Transient load step simulation of an inverter microsource network with Ra Figure 5-15 Transient load step simulation of an inverter microsource network with Ra Figure 5-16 Transient load step simulation of a mixed source microgrid network with Ra Figure 5-17 Neutral stability characteristic defined by the relationship considering Ra Figure 5-18 Individual break-out of the six separate cases defined in Table Figure 5-19 Experimental result of load transient from 0.385pu load to 0.77pu load

16 xiv Figure 5-20 Eigenvalue migration when varying the power measurement filter bandwidth Figure 5-21 Eigenvalue migration when varying the power measurement filter bandwidth Figure 5-22 Eigenvalue migration when varying the power measurement filter bandwidth Figure 5-23 Transient load step simulation of an inverter network with input filtering Figure 5-24 Transient load step simulation of a mixed source network with input filtering Figure 5-25 Comparison of effective frequency droop attenuation versus ideal attenuation Figure 5-26 Experimental result of load transient from 0.385pu load to 0.77pu load Figure 5-27 Effect of increasing Pole2/Zero2 and Zero1/Pole1 around 60Hz Figure 5-28 Effect of increasing Pole2/Zero2 and Zero1/Pole1 around 60Hz Figure 5-29 Transient load step simulation of an inverter microsource microgrid network Figure 5-30 Transient load step simulation of a mixed source microgrid network with Ra Figure 5-31 Experimental result of load transient from 0.385pu load to 0.77pu load Figure 6-1 Gain of Mp at stability limit for grid-connected single source Figure 6-2 Gain of Mp at stability limit for grid-connected single source Figure 6-3 Ratio of realistic frequency droop gain limit against high X/R networks Figure 6-4 Ratio of realistic frequency droop gain limit against high X/R networks Figure 6-5 Maximum real value of eigenvalues for frequency droop gain Figure 6-6 Maximum real value of eigenvalues for frequency droop gain Figure 6-7 Simplified system nominal configuration energy error vector Figure 6-8 Simplified system offset voltage angle configuration energy error vector Figure 6-9 Simplified system offset voltage angle and power set-point energy error vector Figure 6-10 High frequency-droop gain configuration energy error vector Figure 6-11 High frequency-droop gain and voltage angle bias energy error vector

17 xv Figure 6-12 Voltage angle bias configuration energy error vector Figure 6-13 High frequency droop gain and voltage offset config. energy error vector Figure 6-14 Very high frequency droop gain and voltage offset config. energy error vector Figure 6-15 High resistance configuration energy error vector Figure 7-1 Eigenvalue of two-source network with grid-forming and grid following sources Figure 7-2 Eigenvalue migration for changing relative scale grid-following and grid-forming 237 Figure 7-3 Functional diagram of simplified grid-following source paired with grid-forming. 238 Figure 7-4 Minimum values of frequency droop gain to achieve a stable system Figure 7-5 Eigenvalue migration of two-source system with grid-following and grid-forming 241 Figure Reactance vs. resistance characteristics for grid-tied inverters of varying scales Figure 8-2 Grid-tied impedance model used for neutral stability boundary for scaling analysis 248 Figure 8-3 Grid-connected eigenvalues for various systems with nominal droop gain Figure 8-4 Grid-connected eigenvalues for various systems with nominal droop gain Figure 8-5 Grid-connected eigenvalues for various systems with nominal droop gain Figure 8-6 Frequency and voltage droop gains as compared to the theoretical limit values for conditions of neutral stability in (left) full view and (right) zoomed Figure 8-7 Frequency of oscillation as a function of frequency and voltage droop gains Figure 8-8 Source-Source resistance versus nominal frequency droop Figure 11-1 Laboratory inverter hardware used in experiments Figure 11-2 Inverter board block diagram

18 1 1 Introduction Microgrids present an alternative to centralized generation that offers opportunities to achieve higher efficiency and higher power system reliability by increasing the number of sources on the grid. The control of distributed generators present in a microgrid has been a major research objective in the microgrid field. The Consortium of Electric Reliability Technology Solutions (CERTS) Microgrid concept [1] [2] addresses the unit control issues by applying plug-and play concepts that significantly reduce the amount of custom engineering that is required compared to other types of microgrid configurations. In this research program, the focus has been placed on the issue of load re-balancing for sources whose outputs exceed their power capabilities as a result of load transients. The objective of this work is to extend the plug-and-play concept to include power-limit management that promotes stability over the widest possible range of system configurations and transient conditions. 1.1 Background Microgrid Overview Distributed generation offers an alternative to the traditional grid architecture consisting of centralized large generators that originally was economically advantageous given the available generator technology and economies of scale. However, with increasing demands for high local power quality for some customers and the distributed nature of several types of renewableenergy power sources, distributed generation is presenting itself as a desirable alternative to the existing centralized power generation scheme.

19 2 Microgrids collect distributed generators into a single functional unit that can be coordinated internally to collectively act as a good citizen at the point of common coupling (PCC) to the grid. This allows for a greater amount of freedom internally and simplicity when viewed externally. By limiting the power regulation requirements to the single connection point to the grid [3], the internal management of the distributed resources has greater degrees of freedom since there are many solutions to satisfy the interconnecting terminal constraints of power quality [4] [5]. From a customer perspective, microgrids increase power reliability by compounding the reliability of the grid with that of the internal source that can separate itself from the grid (i.e., island itself) during power quality events. This approach requires that both the grid and the microgrid suffer unlikely coincident failures in order to adversely affect the customer. From the perspective of the grid operator, microgrids also promote high power quality by offering added transient suppression capabilities through the adjustable power output from microgrid sources. In addition, microgrids offer the ability to disconnect themselves from the grid supply when power quality becomes compromised on either side of the point of common coupling. Therefore, if a fault event occurs within the microgrid, the microgrid can island itself, thereby isolating the fault. As a result, the grid will be spared the bulk of the disturbance, which increases the power quality for the remainder of the grid. The net result is an increase in power quality and reliability for both parties. The increasing demands for improved power quality are being driven by two major sources. The first originates with utility customers such as hospitals and data centers that require the highest possible grid power reliability. These facilities already implement separate back-up power equipment to enhance the power reliability provided by the grid alone. Secondly, the increasing penetration of intermittent renewable energy causes large power swings on both shorter and

20 3 longer time scales that can seriously stress the transient tolerance of the grid, compromising the stability of the power system as a whole. These parallel trends, both focused on a desire to achieve the highest possible power quality in the presence of large sudden changes in the utility power, encourage the adoption of microgrid technology based on distributed sources as a means of achieving significant improvements in the grid transient ride-through capabilities Microgrid Sources Microgrid sources (referred to henceforth as microsources) can deliver power from a variety of primary energy sources and in a variety of ways, which makes the collection of microsource types significantly diverse. For example, solar energy can be converted directly into DC electricity by means of a photovoltaic panel and the DC electricity can then be converted into AC power using an inverter in order to deliver this energy to an AC microgrid. The same solar energy could be used as a heat source for a Stirling engine, and the resulting rotating mechanical power can then be converted into AC electricity using a rotating synchronous alternator machine. Energy from natural gas has a similar ability to be converted in multiple ways: one way is to feed natural gas to a high-temperature fuel cell, or it can be burned in an internal combustion engine. The common theme in the preceding paragraph is that there are both electrical and mechanical means of converting energy that need to be compatible with the AC microgrid. Some sources include power electronic inverters, while others rely on mechanical means of regulating the power flow (e.g., fuel regulators for internal combustion engines) which is typically much slower. Sources will be classified in two separate categories based on this distinction: rotating machines and inverter-based sources. Two sub-groups within the family of inverter-based sources are those whose controller causes the microsource to behave as a regulated voltage source or a regulated current source. These two sub-groups could also be referred to as grid-

21 4 forming and grid-following, respectively, due to the ability of the voltage-regulated (gridforming) microsources to act as master (or shared master) frequency sources, while the currentregulated (grid-following) microsources require a grid-forming source to be present on the microgrid network in order for the microgrid to operate in island mode, isolated from the grid. A discussion on this topic is presented more extensively in Chapter 4. The three source types mentioned above, as well as their respective prime mover dynamics, each present unique dynamic characteristics during their interactions when connected to each other in a microgrid. Some sources like battery-powered inverters are generally well suited to act as temporary back-up power sources in microgrids due to their general compatibility with high output power slew rates. Other sources like micro-turbines have thermal and kinetic restrictions that limit the rate at which their output power can be changed. These inter-source dynamics are one of the major focal points of the research presented in this thesis. The following subsections ( , and ) summarize the individual microsource types considered during this research program and their respective operational challenges Inverter-based Battery, Ultra-capacitor, and Flywheel Energy Storage Inverters with limited-energy sources supplying their dc link inputs such as batteries, ultracapacitors and flywheels all have similar power characteristics in that they are capable of supporting rates of change in their output power that are fast compared to the other major dynamics in the microgrid. Battery chemistries and configurations can be combined to tune them for a given rated power and rated energy, but they are typically known for having significantly more energy per monetary investment than the competing technologies of ultracapacitors and flywheels.

22 5 A simplifying assumption will be adopted in this research that each storage technology can deliver rated power until the rated limit of the source s stored energy is reached. All of these technologies have reduced power capabilities as they approach the end of their stored energy, but this fact will be omitted unless explicitly stated here for simplicity and general comparison between technologies. These sources are also generally well suited for temporary overload conditions, allowing them to deliver power at levels above their rated values for brief periods provided that the RMS power is below that of the continuous power rating. The main operating constraints for a stored-energy source are the result of its thermal limit and finite energy limit. For short-term energy storage components such as ultra-capacitors, the associated energy storage microsource must shed its load before the available stored energy is fully depleted in order to avoid undesirable uncontrolled reactive power or voltage collapse. This is accomplished by off-loading power delivery responsibilities to other sources on the network or by forcing non-essential loads to disconnect themselves, typically triggered by an under-frequency event Inverter Based Photovoltaic and Fuel-cells Photovoltaic panels or a fuel cell connected to a microgrid via an inverter can operate as a current-regulated or a voltage-regulated source (i.e., grid-following or grid-forming). This topic will be explored further in Chapter 4 for the case of the photovoltaic source, but it will be noted here that photovoltaic power is primarily dependent on the incident solar radiation with very little energy stored in the panel itself or the associated inverter. This lack of stored energy allows for fast changes in system voltage to track the maximum power. However, it also implies that this source cannot be utilized for temporary overload or transient suppression in the way that a stored-energy microsource described in the preceding subsection might be.

23 6 In addition, the desired power response of a photovoltaic system in response to load changes or islanding requires that the PV microsource output power be relatively immune to frequency changes. This is required in order for the PV source to remain operational under all operating condition by appropriately regulating the voltage of the DC bus capacitors in the inverter, the only energy storage components located in the source and converter system. Since the stored energy in the capacitance is relatively small, this requires a relatively fast power-control bandwidth to perform the DC bus voltage regulation. As a benefit, fast regulation of the DC bus voltage also allows faster maximum power point tracking, making it possible to achieve greater utilization of the solar resource under transient shading conditions Rotating Machines Rotating machines represent a dramatically different type of distributed microsource compared to an inverter-based microsource because of the physical inertia of the machine itself that is not present in an inverter. While inverters require no energy to change the frequency of the output voltage, this is not the case for a typical synchronous machine generator. However, the effects of non-negligible mechanical inertia behave as both a detriment and a benefit considering various aspects of microgrid transients. Physical inertia acts as a stored energy source, like a flywheel, but power changes the frequency of the machine, and power is approximately proportional to the load angle (angle of the machine s back-emf voltage phasor with respect to the angle of the machine s terminal voltage phasor). This creates an orthogonally cross-coupled configuration that takes the form of a perfect oscillator that requires damping. This damping represents a direct link between frequency and power which can be provided passively by the machine (e.g., damper windings in a synchronous machine) or actively by the prime mover governor. It should be noted that speed damping from a direct link between power

24 7 and frequency can also be provided by an external source such as an inverter-based microsource which has well-damped power delivery characteristics. The stored energy in the rotating machine is very valuable because, during a load transient on an islanded microgrid, the load will be supplied in part by the stored energy of the rotating mass even if the frequency of the machine is not be regulated properly considering the prime mover dynamics. This fact allows other microsources with limited overload capabilities such as the photovoltaic sources to take advantage of the transient energy-supplying capabilities of a rotating machine regardless of prime mover dynamics. Henceforth, microgrids that utilize different types of microsources with widely varying energy storage capabilities and dynamic characteristics will be referred to as mixed-source microgrids. However, it is the prime mover dynamics that ultimately determine the overall response characteristics of the frequency for the islanded microgrid network. As a result, slow-reacting prime mover governors frequently cause frequency collapse and nuisance trips of frequencysensitive loads in islanded load transient scenarios. Therefore, fast regulation of the prime mover well within the electro-mechanical time constant of the machine is required to provide adequate damping for well-behaved responses, as well as to help limit the maximum frequency deviations resulting from load transients. Unfortunately, in some cases the prime movers are subject to physical limitations that prevent fast torque and power regulation, complicating the microgrid dynamic response. Examples include turbocharger delays, fuel servo/valve actuation delays and thermal delays for compressed gas driven applications (eg. steam boiler, geothermal pump).

25 Micro-Turbines Microturbines are gas turbines that typically have power ratings in the range of tens or hundreds of kw. They operate at such high speeds (20kRPM to 150kRPM) that they typically use a high speed direct-coupled alternator to generate power in place of a directly-coupled synchronous generator which can run at maximum speed of 3600rpm for a 2-pole configuration with 60 Hz output power. The AC electrical output power of the high-speed alternator is typically rectified to serve as a DC source connected to a current-regulated inverter. For the Capstone 30kW turbine [6], the rate at which the power can be changed is limited to 1pu power in 30 seconds which is constrained by thermal and kinetic limits as the turbine speeds up to a higher speed where it is capable of producing its maximum power. For the transient time scales of the microgrid, which are typically on the order of one second, the power from this micro-turbine source cannot actively participate in the microgrid s transient suppression algorithms. The slowness of the micro-turbine s transient response characteristics requires other microsources to compensate for sudden load changes in the short term, but the micro-turbine can ramp its output power so that it can be used to off-load other sources for steady-state operation. For purposes of the research presented in this thesis, the micro-turbine can be considered to be a constant negative load that is effectively ignored in the analysis since the time scale of interest is typically within a second Combined Heat and Power (CHP) In addition to the improvements in power quality and reliability, microgrids can offer higher efficiency by avoiding transmission losses using local generation, as well as by making use of the heat produced by the generation process to offset electricity use that would have otherwise been required for heating and cooling, i.e., cogeneration, and combined heat and power (CHP).

26 9 Approximately 5-10% of losses typically occur in the transmission system between a centralized power station and the customer [7], and these losses can be largely eliminated with distributed generation. In addition, this reduction of power transmission demand contributes to a valuable increased operating margin in the transmission system. The use of heat on-site is an opportunity that can boost efficiencies even higher. Since 60% or more of the energy in a combustion process is typically lost to heat, a large portion of this locally-generated heat can be used for building and industrial heating, or for cooling by means of absorption chillers. By making use of this heat, overall efficiencies up to 80% can be achieved compared to conventional combustion-based electrical generation that typically dissipates the waste heat on-site using cooling towers [8]. From any perspective, localized use of otherwise wasted heat or electricity increases the overall system efficiency, providing one of the major drivers for distributed generation Microgrid Control Techniques The primary challenges facing the development of microgrids are centered on internal management of the distributed generation resources and the interconnect standards that are currently being developed [5]. As previously noted, microgrids have the ability to combine the outputs of local distributed generators and manage their collective actions to abide by power quality standards at the point of common coupling. However, there are many ways of accomplishing this task of generation and load resource management, and solutions range from completely centralized control to completely distributed control. As will be suggested later, the optimal solution should include

27 10 elements of both centralized and distributed control to achieve the greatest benefits of high reliability and efficiency. Centralized control is a popular method that connects many resources together on a fast communications network to balance the power generation and demand [9]. These approaches focus on the optimization of energy production and resource utilization for given market conditions. However, the greatest restriction of these energy managers is the number of nodes that the central controller manages, significantly complicating the optimization strategy when the number of options becomes large. Previous publications on this topic focus more on the market aspects of microgrids and ignore, for the most part, the inter-source dynamics [10] which can be a detriment to power system stability. Distributed, or autonomous, control is an approach adopted by the Consortium for Electric Reliability Technology Solutions (CERTS) microgrid configuration. In the original CERTS Microgrid white paper [3], a method of active power sharing among the distributed sources is introduced using the concept of frequency droop in the 3-phase ac network (Fig. 1-1a). Reactive power sharing is achieved by means of voltage magnitude droop as shown Fig. 1-1b. a) b) Figure 1-1 CERTS microgrid droop characteristics: a) Power versus frequency droop; and b) Reactive power versus voltage droop

28 11 This combination of frequency and voltage droop is critical to achieving improved microgrid dynamic stability using the CERTS algorithm since it does not require a separate communications network. Load changes are automatically divided among all of the microgrid sources on the network by means of the frequency droop characteristics. The adoption of the voltage magnitude droop algorithm reduces any unnecessary reactive power transmission between the distributed sources [1]. These algorithms assume a primarily inductive microgrid network, and this assumption is typically valid because of the presence of electric machine sources and loads, combined with inverters employing filter inductances. In addition, the distribution lines of a microgrid are typically inductive, although the ratio of resistance to reactance can vary significantly. Note that in microgrids, due to the relatively close proximity of the sources and loads, the filter impedance dominates and contributes to the assumption of an overall inductive network. Due to the autonomous control implemented in this approach to microgrid control, power optimization can be performed on a longer time scale by adjusting local set-points for the microsources, taking advantage of the appealing stability characteristics associated with the CERTS microgrid architecture. The second feature that helps to assure stability and relative immunity to the set-points of the network is the incorporation of power limit controllers for the distributed sources. These power limit controllers act to rapidly decrease the source frequency during an overload event on a source-by-source basis (Fig. 1-2), thereby shifting the burden of power output to other sources on the network. However, when all of the sources are simultaneously operating at their maximum

29 12 outputs, the microgrid is in danger of a voltage collapse and black-out regardless of the individual power set-points. Figure 1-2 Power vs. frequency droop operating points for various microgrid conditions, showing the impact of the power limit controller acting at 1 pu active power. Figure 1-2 illustrates the implementation of the power-limit conditions in addition to the baseline linear frequency-droop region. The set-point dictates the steady-state operating power when the microgrid is grid connected, assuming the grid is operating at its nominal frequency. Under these conditions, the power drawn from the grid to supply the on-site loads is determined by the difference between the total of the power set-points of the sources and the total on-site load. However, when a microgrid separates itself from the grid (i.e., islanded operation), the microgrid must supply the entire local load. Immediately following initiation of the islanding event, the incremental power requirements are distributed to all the sources based on their source impedance values, resulting in a communal change in the microgrid frequency that establishes a new frequency operating point for the entire microgrid in a well-damped manner. More recent research in the global microgrid community has focused on development of an agent-based control scheme involving a hierarchy of distributed controllers [11]. The base level

30 13 of this hierarchy is the source controller which often includes some variant of the autonomous CERTS control frequency droop algorithm. The source controller acts as an agent by managing the source voltage, frequency and protection, while giving external access to the power set-point. Since the CERTS-based algorithms maintain a high degree of transient stability regardless of power set-point, it enhances the resistance of the microgrid to cyber-attack. The microgrid energy manager acts as a form of agent-based controller, managing the set-points of the microgrid for energy optimization, while allowing a bulk grid power command from a controller positioned one level in the hierarchy above the microgrid controller. The details for controller above this are relatively unimportant, since the process simply repeats itself until the number of microgrids, microgrid clusters, or regions are of a small enough number to be effectively managed by a regional or grid-wide balancing agent [12], which already performs a power balancing action for the existing grid. 1.2 Problem Statement Frequency droop mechanisms are a proven method to balance load in power systems. However, this process is not instantaneous. In the event that additional loading causes one of the sources to exceed its power limit, the frequency of that overloaded source will change to hold the power at a fixed maximum value in steady-state, transferring the excess load to other sources with additional output capabilities. This condition of unequal load distribution that causes at least one source to operate beyond its maximum power is referred to in this thesis as partial overloading. In cases of partial overloading, the system remains stable and will automatically transfer load in the event of additional load. However, this process of partial overloading and overload transfer response becomes more complex when considering sources that have complicating source-side dynamics which limit the power output response bandwidth. For example, mixed-source

31 14 microgrids that include different types of microsources with widely varying dynamic characteristics and energy storage capabilities present special challenges for both analysis and controller design to provide appropriately responsive dynamic performance under overload conditions. Investigating the complexities of this overloading issue is the primary focus of the research in this thesis. Source overloading is of concern when load is increased in an islanded microgrid or when islanding from a state when power was originally being imported from the grid to support local load. In these cases, all voltage-source type sources (i.e. synchronous machines, voltage-source inverters and induction machines) will undergo an increase in output loading as a function of the interconnecting impedance. The controller of each microsource, whether power regulated or frequency droop regulated, will react to this change in demand once it is sensed. It is particularly important for those sources operating at their maximum output power to maintain adequate stored energy (voltage) in their DC capacitance throughout the load transient events. When the output power exceeds the maximum input power, the difference in power is drawn from whatever supplemental energy storage exists within the source. In rotating machine sources, this would be the physical inertia whereas in a photovoltaic source the only significant stored energy is the DC bus capacitance. In addition, excess power transfer may cause power stage transistors to exceed their safe operational range as heat builds and temperatures rise. Overloading is also worsened as the amplitude of the load change increases with respect to the total microgrid generating capacity. The overload problems are further aggravated when the ratio of fixed-power grid-following sources to the adjustable grid-forming sources increases. The proportion of any added load initially distributed among the connected sources depends on the impedance between the new load and the sources themselves. The analysis of the

32 15 overloading phenomena can be defined in reference to the size of the added load by two assumptions. If it can be assumed: 1) that the interconnecting impedance for each source to any point in the network is dominated by the source s own filter impedance; and 2) that the impedance is sized similarly on a per-unit basis, analysis of the source overloading phenomenon can be restricted to typical network conditions. It will be shown later that the phenomenon of overloading becomes more significant when load is added that is equal to the remainder of adjustable generation headroom and also when the added load is significant with respect to the total generating capacity. The microgrid, as any small power system, typically has a high ratio of load step size to total generation which makes the microgrid a transient-rich environment, often transiently exceeding the microgrid s total steady-state power generation capabilities. As load is added to a network, a shift in current occurs nearly instantaneously and is distributed along the line impedances to each of the sources, preferentially drawing current from sources with low-impedance paths to the load. This condition can cause a temporary power overload in sources that were acting near or at their maximum power limit before the load transient. The overloading exists until the source controller of the overloaded source can act to cooperatively re-distribute power flow in the microgrid so that nominal power operation is restored in the source. If this load rebalancing does not occur before the controller faults, the converter voltage collapses, or a prime mover stalls, then microgrid black-out may occur. One of the popular distributed generation sources is photovoltaic which has many benefits including low maintenance and no fuel costs. However, two compounding issues for photovoltaic sources are: 1) the powerful economic incentive to operate the PV panels at their maximum power available which creates an unstable power characteristic below the maximum

33 16 power point voltage; and 2) the low amount of stored energy in the source that was noted previously. These issues together make it critically important to manage source overloading conditions to properly regulate the DC bus to remain within acceptable levels, avoiding DC bus voltage collapse. 1.3 Technical Approach A key to addressing the overloading problems discussed in the preceding section is the implementation of fast power control techniques that are compatible with the underlying frequency droop algorithm. It has been shown previously [13] [14] that steady-state overloading in a CERTS microgrid is handled well by providing an elevated frequency droop gain at high load levels (Fig. 1-2) and adding an integrated frequency droop loop as well to manage the power limit [1]. However, this process is restricted in bandwidth by the maximum gains that can be used in the power limit controllers, which are limited primarily by the microgrid interconnection impedances, as discussed later in Chapter 5. To increase the rate of power control of each source to the necessary rate without extensive custom engineering, the stability limit of the frequency droop gain in inverter sources needs to be defined more generally and then increased. Furthermore, additional control techniques need to be investigated to determine what improvements can be made to the power control bandwidth while ensuring a stable response under the various conditions that a microgrid will introduce. The topic of transient responses of mixed-source microgrids has been largely unexplored. As there are so many varieties of sources and even more configurations of sources in a microgrid, the possible outcomes are many and analysis is required to understand the underlying mechanisms that define the inter-source power and frequency reactions. The overall goal of this

34 17 area of research is to limit the amount of custom engineering [15], as well as minimizing the energy storage requirement of each source to assure stable and predictable responses in the face of transients. The concept of minimal custom engineering for distributed generation was developed from the CERTS plug-and-play model which utilizes the standardized droop gains to unify the power and reactive power characteristic for each source, relating all values with respect to the power ratings. This concept significantly minimizes one of the major impediments to customer adoption and microgrid implementation. It has been shown that traditional CERTS frequency droop controllers are not capable of regulating power to a specified value as quickly as traditional power-regulated controllers used for grid-tied PV applications [16]. However, the frequency droop controller has been shown to be significantly more stable in a microgrid as it operates as a grid-forming source; one requirement for islanded operation is that at least one source must operate as as a grid-forming source. By having multiple grid-forming sources, the reliability is increased and microgrid configuration flexibility is improved. However, a microgrid could operate with only a photovoltaic source operating in grid-forming mode if the limited amount of stored energy can be managed appropriately in response to load transients, a key point of the work presented here. While the benefits of a grid-forming source are apparent, the power control bandwidth limitations are potentially detrimental to the timely rebalancing of loads and warrant the application of additional control techniques. A more thorough analysis will be conducted to more generally define the stability-constrained maximum power control bandwidth of an inverter microsource. Control system modifications will also be introduced to improve the power control bandwidth, further improving the case for grid-forming control of photovoltaic and other energyconstrained resources in CERTS microgrids.

35 Thesis Organization Initially, the model and characteristics of a microgrid source will be examined with a focus on parameter sensitivities and power control bandwidth. Special attention is focused on photovoltaic power sources since this type of source typically has a very small amount of stored energy compared to its power rating. As a baseline investigation into the issues of using PV sources in microgrid, two possible controller configurations are compared: grid-following and gridforming. The inherent limitations of power regulation using frequency droop control are then identified and methods to mitigate the excitation of destabilizing system modes are presented. Load transients in microgrids can be substantial compared to the power and energy constraints of the local sources, so each source must be designed to offer high power quality in a transient-rich environment. Following this introductory chapter, Chapter 2 provides a thorough state-of-the-art review that examines a wide range of existing technical publications addressing the CERTS microgrid, its suitability for applications with mixed types of microsources, microgrid performance under overloading conditions, and both the steady-state and dynamic performance characteristics of frequency-droop controllers for microgrids and other applications. Chapter 3 investigates system characteristics and step load responses of both inverter-only and mixed-source microgrid configurations. The advantages of achieving a higher power control bandwidth are discussed in the context of applications with low amounts of stored energy such as PV sources. Chapter 4 provides a comparison of PV inverter controllers with a focus on the power control bandwidth and network reliability. This discussion shows how the basic frequency-droop controller compares to the alternative grid-following controller for a variety of microgrid operating conditions.

36 19 Chapter 5 investigates the stability limitations that are exhibited when the controllers are tuned for higher power control bandwidth values. Special control techniques are introduced that make it possible to enhance the microgrid stability by avoiding the excitation of undesirable network modes. Next in Chapter 6, the previous conclusions are compared against microgrids of various scales to investigate their general applicability. For completeness of the controller discussion in Chapter 4, the characteristics and limitations of a microgrid with predominantly grid-following sources are investigated in Chapter 7. Also, nonlinear stability techniques are utilized in Chapter 8 in order to examine if stability can be proven on a full-order system model. Finally, conclusions, key research contributions, and future work are discussed in Chapter 9.

37 20 2 State of the Art Microgrids are a very active area of research by both academic and industrial parties. The three major hot-beds for research are in Japan, Europe, and the USA [17]. A significant piece of the original microgrid research was conducted from 2000 to 2006, published in [3] and [2]. This work encompassed the concept of premium power quality, distributed generation as a mechanism for achieving high power quality, intentional islanding of a distributed resource network and included the droop controls that are used in much of the contemporary research today. Some of the significant technical points generated from this work include selection of filter components, droop gains, power limit control and feeder flow control. In many ways, this pioneering work set up an extensive basis for developing an upper layer of optimization. However, some researchers have chosen to not include the standard frequency-regulated source controller topology in favor of switched mode operation as well as a current regulated variety. As this document will be using the CERTS concepts as a preferred method for agent-like unit control, it is appropriate to consider the perspective of recent publications with respect to the capabilities and characteristics of CERTS microsources. 2.1 Microsource Unit Control Concepts in Modern Publications Many current publications omit the detail of frequency regulation - more attention is paid, if at all, to the dynamics of current regulation. However, those that do consider frequency droop control are addressed later in Section 2.4. The lack of focus on frequency droop as a concept partly implies the well understood concepts of grid-forming frequency regulated source operation, but in some cases the valuable concepts of unit self-regulation abilities are neglected. One recent publication [18] makes a distinction between current and voltage regulated sources, but is more focused on solutions for current harmonic mitigation in a microgrid with non-linear

38 21 loads. While it demonstrates that the same measurement and compensation effects can be used on voltage regulated sources as well as current regulated sources, attention is paid to the mitigation of harmonics using a sliding discrete Fourier transform. The measured voltage error due to current harmonics is compensated using proportional voltage response in the case of the voltage regulated source. Another recent publication [19] utilizes a grid-following variety of frequency droop (presented in detail in Chapters 3 and 7) instead of the more common gridforming droop control. Other recent work regarding unit control includes [20], [21], [22] and [23] which range from optimal sizing of microgrid components to stability-constrained unit control strategies. Another common theme in recent publications is the use of a solid-state transformer in place of a step-down transformer and a thyristor based static switch as presented in previous CERTS literature [3] [2]. The research presented in [24] uses a power-electronic transformer, a fullconversion back-to-back converter, to allow power-limiting functionality and frequency independence to the microgrid. This concept replaces the use of a static switch with a likely expensive full-conversion converter that may or may not have over-voltage protection capabilities. Where the power-transient limitation would typically come from the opening of the static switch, this solution is more effective at interrupting current spikes with fully controlled active switches. Also, it is attractive to replace a large and heavy step-down transformer with a solid state alternative, but the added cost and decreased reliability may not be improved by the supposed benefits like frequency independence and on-demand reconnection to the grid. The work in [25] introduces a dual-active bridge solid-state transformer at the point of connection to provide step-down as well as DC capabilities for a DC section of the microgrid. One interesting point is that it presents four separate operating modes of interconnection/power-

39 22 flow with said solid state transformer (Figure 2-2). Other follow-on examples of hybrid solidstate transformers are presented in [26] that use series compensation techniques on a traditional transformer. Figure 2-1 baseline topology of solid state transformer from [25] Figure 2-2 Operating modes of dual active bridge solid state transformer with accessible DC link from [25] This mode selection is arguably unnecessary as there are still capabilities and desired circumstances for a unified controller approach under the CERTS technology envelope which avoids the need for mode-switching decision engines. Secondarily, there are no proposed or evident benefits from the isolated modes where the DC-link is not used for DC power especially since there are no losses associated with using such a pathway and switching modes may induce undesirable latency. The control theory that is presented also includes mode switching from power-regulated to voltage regulated when islanding event occurs which requires fast

40 23 communication to maintain operation, another application of fast communication that could be solved with CERTS concepts where voltage mode only is used. A hybrid AC/DC network is presented in [27] as a solution to reduce the stages of power conversion and increase the system efficiency. Figure 2-3 Hybrid AC/DC microgrid from [27] The implementation of the 'main converter' is identical to a standard inverter with two sources on its DC bus, separating the AC sources and loads from the DC sources and loads of the system, as shown in Figure 2-3. One point that serves as a detriment to this paper is the two operating modes that the system operates in. For grid connected mode, the 'main converter' has a control objective to regulate the DC link voltage through a power-control method. In an island mode, the 'main converter' switches to a voltage source output that can serve as the master frequency source for the islanded network. The main technical point absent from this paper is the opportunity to define a DC voltage droop curve that generates a frequency reference which could avoid the need for mode switching entirely. Secondarily, the DC-side coordination between a photovoltaic unit and a battery energy storage unit is represented by a flow-chart (Figure 2-4) that requires control of both systems whereas an agent approach with the appropriate power and

41 24 energy limits monitored locally could avoid fast inter-source communication and enable a more plug-and-play friendly implementation. Figure 2-4 Decision tree for intelligent energy controller from [27] The variety of power quality issues that can be solved by the available D-FACTS devices are surveyed in [28] to aid in the function of a microgrid. This approach proposed the use of dynamic voltage restorers and power flow regulators to improve power quality from installations along the distribution feeder. Power source pairing for hybrid-compensation is proposed in [29] where two energy sources are used on a microgrid using CERTS droop algorithms. The proposed architecture utilizes one high-energy source such as a lead-acid battery that is paired with a high-power storage device like an ultra-capacitor. The power set point of each is varied to manage the stored energy in each and ensure appropriate transient suppression capabilities. A mode switching strategy for unit control in a microgrid is proposed in [30] to perform separate functions of load sharing, voltage and frequency stability and resynchronization. While this mode switching may be valuable implemented on a unit basis, these functions would likely be

42 25 better performed by a microgrid manager that could control all of these functions simultaneously without mode switching that may compromise performance in one mode versus another. 2.2 Isolated Power System Examples Isolated small power systems using diesel generators with power-versus-frequency droop predate microgrids and the similarities continue with the inclusion of renewable energy wind turbines. A host of work particularly in the 1980s-1990s investigated this configuration [31] [32] [33] [34] [35] [36] [37] [38]. Efforts in this area aimed particularly at the stability of an isolated system with diesel base-load generators and a varying power source (wind turbines). To confront the issue of limited computation abilities as compared to the modern era of computing resources, reduced-order models were developed through justification and validation of simplifying system assumptions. Linear or simplified non-linear models were enabled by assumptions based on relative scales of time constants [32] such as ignoring the voltage transients following a load change. The dominant characteristics remaining include the diesel genset governor and the pitch control actuator of the wind turbine. Selection of control gains allowed desirable response characteristics in multiple examples [31] [36], indicating that these pairings of near 1:1 of diesel and renewable were possible with proper selection of gains. The development of simplified models that utilize linear analysis techniques is a powerful tool employed in the research presented here as well. However, the model assumptions are important to developing a featurecompatible model that represents the phenomena investigated. Standard models for a range of sources in an islanded power system were developed in [32] which are largely similar to the

43 26 commonly applied models used today. One more recent publication [37] validated a linear model of a nonlinear system by showing general agreement between transient response traces from both systems by utilizing dynamic simulation. This technique, though lacking mathematical robustness, may prove to be a useful tool when more formidable analysis techniques are impractical. Development of control strategies for islanded power systems was also a focal point of research efforts in this era. The manner by which wind power was metered was also a point that varied throughout. In most cases, a maximum power pitch characteristic was used to maximize the wind power generated [34] while others included stipulations for furling at high speeds [31]. Another concept brought forth in this era was the concept of active loads, or prioritized load sets [34], which represents an early form of demand-side management. In the case of [34], the Fair Isle power system included two separate distribution systems, one with high priority lighting and convenience loads, while a lower-priority heating network was disabled when critical conditions occurred. Some papers even included considerations on application of energy storage to isolated systems. One publication noted the inherent benefits of battery energy storage which made transient load responses faster and more damped, but also cautioned against setting gains on other network components that relied on the presence of battery energy storage to remain stable [38]. It can easily be seen how some of these concepts likely influenced the control concepts of a microgrid and it is interesting how some of the issues have carried over. However, the universal application of frequency droop control in the CERTS topology and the increased integration of

44 27 inverter-based distributed generation add a more simplified load balancing strategy and a well damped response characteristic. 2.3 Application Considerations for Supervisory Controller Interactions From the basis set forth by the CERTS source controller, the primary externally controllable quantities are the power set-point and voltage set point. The other main parameters like the power limits, temperature limits, droop gains and power limit gains should not change enough to warrant external assignment particularly for the purposes of cyber security. One of the primary efforts regarding microsource control, though not explicitly stated under the CERTS literature, is that the microgrid system should function regardless of the individual set points. Through power limit control actions, a stable operating point theoretically should be possible so long as the system load is within the power range of the sources on the network. The opinion of the author is that this is a critical aspect to maintain the stability of the network even in the face of cyberattack particularly by limiting the extent to which an external attack can affect the ability of the microgrid to remain functional. However, optimization is only available through sharing of information through the network, and it is important to note that functionality and optimal utilization of the available network resources are typically orthogonal goals. The work presented in [39] represents the latest work in the field of multi-agent systems (MAS) for unit coordination and microgrid optimization. It compares centralized versus distributed algorithm implementations, noting that while the distributed method is more complex as a peerto-peer operation, it is fundamentally more resistant to issues in interconnecting cabling than the centralized communication system. However, this is another condition where the control of the microgrid resources is set in either grid-connected or islanded mode, though the distinction

45 28 seems antiquated, particularly if the grid can be considered as simply another microgrid resource that has selective availability. One approach to microgrid control [40]recently presented optimizes CO 2 emissions by optimizing CHP utilization and suppresses transients with storage elements. Other recent notable work includes other multi-agent [41] [42]and multi-objective managers [43] [44] for microgrid optimization [45] and stability [46]. 2.4 Frequency-Regulated Converter Control Publications Recent literature on inverter control in microgrids is another topic to explore when discussing the larger subject of frequency-regulated sources. Among the contemporary research, the common themes are power balancing strategies and artificial impedance. The concept of artificial impedance is introduced in [13] where the apparent impedance of a line is augmented by controlling the current of the inverter based on the difference in voltage between a measured quantity from the grid and a virtual voltage, where the current commanded is as a function of the difference in voltage and the value of virtual impedance previously selected. The work demonstrated high power quality to non-linear loads and didn t rely on inductive or resistive filters as large as the virtual impedance. With the addition of an inner current control loop in software and filter elements large enough to allow for regulation of current, the proposed approach in this paper effectively can modify its apparent filter impedance and still act as a voltage source (grid forming source). Virtual impedance is also included in [47] where it is used to augment the feeder impedance, combatting the issue of primarily resistive low-voltage microgrid lines. This approach has its drawbacks, however, specifically by its dependence on knowledge of the feeder impedance in a network that may have local loads that are not separated

46 29 from the source by such large impedances. In that case, the application of negative resistance may destabilize the network. Secondarily, this method uses different gains for grid-connected versus islanded configuration and a time-sectioned response, engaging different gains and controllers as a function of fixed delays from a singular load disturbance event in place of high or low pass filters. Researchers focused more on distributed uninterruptable power supplies (UPSs) published [48] which outlines a method for incorporating droop control, marking the presence of distributed generation control concepts well before the term microgrid was introduced. However, it is clear that the methods by which the power was balanced was not well established at this point and it yielded a solution that involves signal injection and line impedance compensation to maintain power sharing. It was reported in this work that previously the operation of a distributed UPS system was not stable in the face of non-linear or inductive loads, which is a conclusion that most recent research fails to find even though the converter topology, interconnecting filter impedance, and droop methodology were similar. One main difference between the more contemporary research and the original CERTS concepts is that the signal injection employed in the referenced work associates variations in frequency and voltage and measured changes in reactive power and power, respectively. This is reverse of modern work that assumes an inductive network including the filter impedance. Also, this previous publication lists a variation in droop gain as a method to incorporate distributed supplies of varying sizes. A similar finding is demonstrated here, but only so far as to maintain similar per-unit droop scaling for a broad range of source sizes. Active or virtual impedance is also used in [49], where two parallel UPS converters are controlled. Also, this work mentions the unit coordination through power setpoint only which incorporates CERTS droop concepts.

47 30 The issue of power and reactive power cross-coupling with respect to simply voltage and frequency droop is presented in [50] and solved with decoupling based on nonlinear equations that omit small angle approximations. Although simplifications of decoupling could have mitigated the majority of this cross-coupling, the solution presented includes nearly complete cross-coupling elimination through this thorough analysis and application. The work presented in [51] combats the issue of under-damped power responses caused by relatively high droop gains considering the relatively low coupling inductance with respect to the bandwidth of the measured power. This method is supposed as decentralized and adaptive. While true, the referenced work does so through the use of sliding mode gain schedules to appropriately tune the response based on the source loading. It utilizes operating point analysis on a multi-loop controller to examine the eigenvalue migration particularly as a function of load changes. The main technical contribution of this work is that it incorporates a variable frequency droop gain to maintain a fixed amount of damping by determining the instantaneous measured power through linear analysis techniques. The issue of X/R ratio is proposed as a poignant parameter of a microgrid in [52] which was again due to the bandwidth of the filtered feedback values. However, the discussion is steered towards the effect of cross coupling with significant amounts of network resistance, a condition rarely found in a microgrid when considering source power filters in the interconnecting impedance. Another solution to poor power response by [53] added a derivative term to the droop gain instead of changing the proportional value, creating a PD controller to add damping to the system in place of adding additional inductance, lowering the droop gain, or modifying the output voltage to mimic increased virtual impedance. This is equivalent to a proportional control on phase angle which would avoid the derivative term. Interestingly this work does not note the

48 31 issue of feedback signal filtering that has a lower bandwidth than the desired power control bandwidth of the source. However, in this case, the work referenced damps the oscillations by adding in a rate term that increases the frequency sensitivity to high frequency changes that were previously omitted previously by a simple low-pass filter. This fact demonstrates the careful selection of feedback signal filter bandwidth with respect to the droop gain and the resulting power regulation rate considering a limit-case scenario of a stiff grid voltage at the output of the inverter filter. In [54], the relationship between droop gain and stability of the converter within the system is considered. This relationship will be investigated later in this document in Chapters 3 and 5 where parameter variations influence stability. However, in the case of the referenced paper, it confronts instability of source power responses as a function of droop gain that is likely due to improperly low power measurement bandwidth. The measurement bandwidth is too close to the power response bandwidth, dictated by the frequency droop gain and the network admittance between sources. This is the same oversight present in [53]. While unfortunate, it also provides an opportunity for a teachable contribution. While [53] does outline the method by which a small signal model can be used to predict system stability, the work presented here performs a more thorough analysis and investigates the underlying cause of the instability. One of the system natural modes, later referred to as the voltage balance mode, is confronted in [55] where an instability can be found at the power frequency when either of the droop gains are sufficiently high. Power-frequency sensitivity is found also with respect to the reactive power droop gain in [56]. While this issue will be discussed in detail in Chapter 5, [55] solves this issue by adding in a supplementary control structure that has a similar structure as three cascaded lead compensators. This referenced document does not attempt to explain why said natural

49 32 mode exists, but does indicate that a negative gain was the most effective, which partially counter-acts the droop gain at high frequencies, a concept that is also investigated further in Chapter 5. A contribution aimed at proving stability of n number of chain connected sources is presented in [57] that illustrated that all eigenvalues were not only negative, but real-valued. However, this paper did use a simplified source model that did not include the transient characteristics of the interconnecting impedances, measurement delays, or individual source impedance. Nevertheless, this contribution presents a methodology for expanding a distributed generation system to many sources generally with a method that ensures response characteristics of any source can be identified as stable within certain stated parameter constraints. More recent inverter controllers include [58] which confronts the issue of inverter control in the face of load unbalance and [59] that uses photovoltaic cell power-voltage gradient to more intelligently find and maintain operation at the maximum power point. Considering the number of recent publications regarding control of droop-controlled inverters for microgrid applications, this demonstrates that there are many significant issues to be solved in relation to this topic. This document will investigate and present system parameter sensitivity analyses to generate a better understanding of the characteristics of inverter microsources. This thesis will also improve upon works noted here by avoiding low frequency power oscillations simply by utilizing power measurements at a bandwidth greater than the intended power regulation bandwidth. Finally, this work will investigate, explain and mitigate power frequency oscillations that have been noted to naturally exist and limit frequency droop gains.

50 33 3 System Response Characteristics of a Power-Constrained Droop-Regulated Source in an Islanded Microgrid As mentioned in previous sections, microgrids are a power system concept that utilizes distributed generation to improve power quality and reliability, being able to vary power flow while grid connected and capable of isolated operation. Power characteristics from sources employed in a microgrid can vary from fixed-power sources such as fuel-cells to variable sources such as battery-based inverters and generator sets. This variety of source types brings into question the multitude of inter-source dynamics that could result, particularly when operating isolated from the grid. In an islanded configuration, the occurrence of one or more sources reaching their maximum power limit and maintaining maximum power in steady state is a regular occurrence which can bring about even more complex system reactions. As load changes distribute the load amongst all the network sources, temporary overload will occur as a result on all sources operating against their maximum power limits. Excluding conditions where the system load exceeds the available generation (complete system overload), conditions of transient source overloading due to positive load transients should still result in a stable response characteristic as the additional load is re-distributed to sources that have remaining power margin. The transient addition of load will cause units previously at their limit to initially exceed the limit due to increased current that will flow from the source to the load as there is a delay between measurements of the current or power error and when corrective action can be taken by the source controller. Any additional load will cause this condition to some degree, but the most severe case is where the load change is equal to the complete range of the adjustable source. This case considers one source at its maximum output and another near its minimum output

51 34 before adding a load equal to the range of the second source. This is the maximum load change possible considering the constraints of loading conditions less than the total capability of the online sources. Figure 3-1 Worst-case load change drawn from a fixed power source, varying the ratio of PV (limited-power) to microsource (variable-power), under the assumption of equal per-unit filter impedances Figure 3-1 shows the impact of worst-case load changes considering the ratio of fixed-power sources to adjustable sources. In the case of equally sized sources in a two-source network, the proportion of load initially drawn from the PV source would be 50% and increases to 100% in the limit case where the PV source represents very little of the power generation. However, this is positive in that even though the additional load may be many times larger than the rating of the fixed-power source, that the worst-case is limited to a value equal to the power rating of the fixed-power source, given the aforementioned assumptions. The two source case is also the simplest case to consider when confronting power balance between sources and it will be verified later that a single source of twice the rating will act as a suitable analog for two sources of equal size in order to simplify the analysis. This chapter will begin by investigating the phenomena of transient source overloading of fixed-power sources in response to positive load transients for a variety of two-source configurations. In these cases, the power output from one source will be temporarily beyond its power rating and the response of the system will demonstrate re-distribution of the added load towards the source with operational power margin.

52 35 The response characteristic will be identified in terms of key system parameters such as settling time, maximum frequency deviation and total overload energy utilized. In consideration of cyber security, the power set-point of each source may be compromised as this is an externally adjustable parameter. Under the assumption that the microgrid should be able to survive additional load up to the rated limits, the reliability of the microgrid should be insensitive to these set-points, to as great of a degree as possible. These simulation and experimental results may also be used as design guidelines for development of microgrid sources that will promote transient stability regardless of unit set-points. 3.1 Introduction Microgrids have been theorized and demonstrated as a solution to increasing power quality and reliability while promoting the integration of distributed renewable energy sources. However, with the increasing variety of source power response characteristics and desired operating configurations, the variety of system transient responses to increases in load should be investigated to define configuration guidelines to ensure stable operation. While sources such as battery-based inverters and generator sets are less sensitive to transient overloads due to significant short-term overload capabilities, other sources such as photovoltaic (PV) inverters, fuel-cells (FC) and micro-turbines (MT), have more limited power capabilities. Section 3.2 will discuss the system configuration and the model of each source. Section 3.3 will present the voltage, frequency and power response characteristics of power re-distribution for various microgrid configurations following positive load transients. Section 3.4 presents experimental results that serve as verification of the results in section 3.3 and validation of the model utilized for analysis in section 3.2.

53 36 A mixture of eigenvalue analysis and transient responses will be presented in this chapter. The eigenvalue analysis will help identify the primary modes of the system while the transient load responses will illustrate the resulting time domain response. For each operating point, a linear system is assumed and for small disturbances around that operating point, the response characteristic is defined in terms of eigenvalues using traditional linear system analysis techniques. Considering a source with small amounts of stored energy, fast power control response is beneficial to maintaining fault-free operation of said source. However, lightly damped modes in the network between the sources components limit the rate at which power balance can be achieved following a disturbance. Frequency and voltage regulation actions of the traditional CERTS microsource controller result in unstable responses with high power control P-I gains in the power limit controller and voltage regulator. Therefore, a tradeoff is defined between the decay rate of lightly damped poles and response rate of the power limit controller. 3.2 Source and System Model Development In all cases presented in this section, the sources will be modeled as their synchronous-frame equivalent circuits using rotating-reference-frame speed-voltages on inductors to satisfy impedance calculations. The models are simplified to neglect friction and magnetic saturation, characteristics that would be present in more representative source models, but these models still give reasonably accurate results in regard to the examined transients. The reduced order models presented here include the basic electrical and mechanical responses necessary for this investigation as they include the basic physical relationships between voltage, current and conservation of power.

54 37 While there are a variety of sources that can be used in a microgrid, there are three main varieties, two of which will be modeled here: voltage source and rotating machines. The third is a fixed-power source with current regulating controller is assumed to perform as a negative load and will not be considered here. Voltage sources include any source with an inverter front-end acting as a grid-forming source. There are two primary operating modes for these sources: first is a nominal range where the output power can freely change, and second is the power limited mode where the DC input has reached a limit (maximum or minimum) and the average AC output power must match the average input power from the DC side. For these sources, the assumption will be made initially that the DC power source coupled to the inverter is ideal and does not have significant power limitations or delays except for a fixed maximum power. This assumption will allow the generalization of voltage-source based sources and by accounting for the overload energy required to survive the transient, each DC source type (battery, PV, etc.) can be evaluated against said energy account. The maximum of the integral of power beyond the maximum input power will determine the overload energy and this value can be compared against the theoretical available energy stored in the source. As mentioned earlier, the energy available from a battery based source is better able to support temporary overloads than a PV source and this methodology allows a single transient simulation to evaluate both cases against the available overload energy of each. On the AC side, these sources will be modeled as ideal voltage sources that change their frequency and magnitudes with respect to other sources based on a CERTS microgrid style droop law. The power sources are inverter-based sources fed by constrained prime movers such as fuel cells and micro-turbines in which thermal considerations limit power variations from each to a fixed

55 38 ramp rate. This type of source is fairly established and will be largely immune to transient dynamics as they are current regulated and have the ability to regulate said current faster than external disturbances can cause significant power changes. Therefore, these types of sources will be considered fixed negative load and neglected from the transient investigations in this section as their dynamics are typically fast and well damped. Lastly, rotating machines include any generator set with a synchronous machine directly tied to the microgrid. These gen-sets are the only components that will include physical inertia. While the physical stored energy in the inertia of the machine allows for immediate supply of power to loads, the prime mover is dynamically limited by the fuel regulator as well as a turbocharger in some cases as equipped. The scenario investigated here involves two sources and an adjustable load which is a simple configuration that can demonstrate the worst case for transient overloading, scaled by a factor displayed in Figure 3-1. In all cases presented in this section, the first and second sources will have power set-points of 1pu and 0pu, respectively and will be of equal size with equal coupling inductances. The first source will always be in a power-limited state, utilizing the power limit controller to maintain the maximum rated power output in steady state of 1pu. As shown previously, this condition can cause an overload up to 100% of the source rating when a large load is added. For proper worst-case consideration, the resulting responses will be discussed with respect to this fact. By setting the adjustable load to output 0pu power before the transient, this creates the largest nominal disparity between pre-transient operating points, maximizing the overload caused on source 1 from positive-load transients. Due to the equal source sizes and equal impedances, this

56 39 causes an initial equal distribution of load increase for each source as seen in Figure 3-1 for a 1:1 source power rating ratio. Figure 3-2 Two-source configuration with two microsources, filter impedances, line impedances, and adjustable load While there are many design variables such as bus capacitance, droop gain, synchronous machine inertia and coupling impedance that could be varied to explore the possibility of the design space, the dimension of the associated operating space becomes overly cumbersome. Therefore, to simplify the analysis to a reasonable set of configurations, this investigation will present nominal source models that are similar to the hardware proven to function well in previous research, specifically those used in the UW Microgrid. The source parameters will be varied one or two at a time to illustrate the effect on the system characteristics from each parameter Definition of System Variables: Mp: the linear frequency droop gain within the power limits of the source. KpPMax and KiPMax: the proportional and integrated gain of the frequency controller to regulate maximum power, respectively. Mq: the linear voltage droop gain within the reactive power limits of the source. KpV and KiV: the proportional and integrated gain of the voltage regulator, respectively. ω: the velocity of the rotating reference frame.

57 40 ω 1 and ω 2: the angular velocity (frequency) of the voltage from Microsource 1 and Microsource 2, respectively. φ 1 and φ 2 : the instantaneous phases (angles) of the voltage from Microsource 1 and Microsource 2 in the arbitrary reference frame, respectively. E set1 and E set2 : the voltage set points of Microsource 1 and Microsource 2 that will serve as the base voltage around which the voltage droop will operate, respectively. P set1 and P set2 : the power set points of Microsource 1 and Microsource 2, respectively, which correspond to the average power output when connected to a 60Hz power system and varies based on system frequency. P max : the maximum power set point of the source. This value is typically defined as the power rating of the converter but may be defined as the maximum input power from the originating power source (PV array, battery, etc.), whichever is smaller. P err1 : the power error between the maximum power and the output power of Microsource 1. E err1 and E err2 : the errors between the commanded terminal voltage considering voltage droop and the measured voltage of Microsource 1 and Microsource 2, respectively. intp err1, inte err1 and inte err2 : the states that exist as integrator values of their respective errors (P err1, E err1 and E err2 ). P 1 and P 2 : the output powers from Microsource 1 and Microsource 2, respectively. Q 1 and Q 2 : the output reactive powers from Microsource 1 and Microsource 2, respectively.

58 41 V mag1 and V mag2 : the calculated voltage magnitudes measured at the microsource point of connection to the microgrid mains for Microsource 1 and Microsource 2, respectively. V q_load and V d_load : the q-axis and d-axis voltages of the load resistor, respectively Inverter Microsource Model: The Inverter-based Microsource model is a voltage source that varies its frequency based on the difference between commanded and measured power and varies its magnitude based on measured reactive power. It includes an L-C-L filter where the initial L-C branch is used for switching harmonic suppression and the coupling inductance is used for passive voltage compliance (impedance required for coupling sources together). Figure 3-3 Inverter microsource 1-wire equivalent circuit model including averaged inverter model and L-C-L filter The frequency changes on a linear droop slope of 0.6Hz for 1pu power error within the nominal power range (Mp=0.01, 1%pu), but once the maximum power is exceeded, the power limit controller engages and regulates output power by the use of an additional proportional term and an integrated term to eliminate steady state error. This gated or limited-domain integration process, shown in Figure 3-4, enforces the power limits following a scenario of operating point d in Figure 1-2. This allows the maintenance of power output at the power limit for frequencies beyond the frequency range defined by the droop slope characteristic and the power limits. The magnitude of the voltage changes on a linear deviation of +/-5% change for +/-1pu reactive power and currently has no provisions regarding the power limitation. From a power quality perspective, the voltage should change no more than 5% under any circumstance, indicating a

59 42 flat voltage characteristic beyond the limits. However, this is impractical from the point of view of the converter considering the inherent current limits of the power electronics and the potential for low-voltage (voltage sag) events that would draw excessive current if the terminal voltage were to be regulated within the 5% band. Therefore, a vertical characteristic is implied for the limit of reactive power, similar to the limit of real power. However, this particular limit will not be tested in the work presented here. The state equations used for the inverter microsource model controller and passive filter elements are presented in equations (3.1) to (3.9): I = 1 V V I R +I L I = 1 V V I R I L (3.2) I = I = (3.1) 1 V V _ I R +I L (3.3) 1 V V _ I R I L (3.4) V = 1 I I +V C (3.5) V = 1 I I V C (3.6) inte = K E (3.7) intp = K Pmax (3.8) φ = ω ω (3.9) Where the state equations utilize the following definitions: V _ =R I +I (3.10) V _ =R I +I (3.11)

60 43 P = V _ I +V _ I (3.12) Q = V _ I V _ I (3.13) E =E _ + _ (3.14) V = + (3.15) Pmax = (3.16) ω = (3.17) ω= (For source 1 only) (3.18) = sin = cos (3.19) (3.20) 2 I_filter 1 V_filter V I I* Conjugate Product 3 P_set Power Re Im Complex to Real-Imag Reactiv e Power Mp KpPMax KiPMax 1 s integrator w/limit Sum w/limit Frequency 5 Omega 1 s Angle -1i e u V_Generator 1 V_inverter 4 V_set Mq V_cmd KpV Vmag_Cmd u Abs Vmag KiV 1 s Figure 3-4 Reduced order model of frequency and voltage controller for inverter-based microsource. The diagram in Figure 3-4 depicts the implementation of these limitations which are enforced through the engagement of integrators at the top of the diagram. They use a commanded input of the power limits and feature zero upper limits for the power maximum integrator. There is also a second proportional gain with the integrators to the aid in the power limiting process. During nominal power operation, the added proportional control has zero gain due to the upper limit on

61 44 the summation. This arrangement of the limiter on the integration process and the composite output creates an anti-wind-up PI regulator. However, for all operating points before and after transients the first source will be in the power limited mode where the power maximum controller has a non-zero output and the second source will be in the nominal range, keeping the power limit controller at a zeroed output. Transiently, the first source may exit the powerlimited mode during power oscillations, but will return back to a limited case as the system reestablishes a steady state power balance. Operating point analyses will assume a system when the power limit controller is engaged for the first source and disengaged for the second in order to appropriately define the system around the operating point as linear systems do not allow conditional statements. The voltage regulator uses a proportional droop characteristic and a cascaded PI regulator to track said characteristic. In this case, the low-voltage ride-through (LVRT) will not be tested and therefore the additional fidelity of a controller capable of managing an overload is not necessary and simplifies the resulting system Generator-Set Model: The generator set model includes physical inertia which is used to change the speed of the rotor proportional to the difference in input and output power. While the output power is defined by the electrical loading from the microgrid and the power contribution of the other source(s), the input power is determined by the fuel governor. The governor is fed a speed error between the commanded speed considering the power-frequency droop characteristic and the actual speed. This error is eliminated in steady state by the PI regulator in the governor if the governor is not limited by realistic fuel delivery characteristics like a fuel servo delay. Also, due to the discontinuous nature of prime-mover torque production, combustion delay ultimately limits the

62 45 torque-control and velocity bandwidth. In this case, the effect of combustion delay and fuel servo delay will be lumped into a single first-order linear delay. Lastly, this source is modeled after a turbo-diesel generator, but the turbocharger model is incorporated into the bulk power response due to the lack of detailed information in [60] to separate the turbocharger characteristic from that of the fuel governor. ψ = ψ ψ + + ψ (3.21) ψ = ψ ψ + ψ (3.22) ψ = ψ ψ (3.23) ψ = ψ ψ (3.24) ψ = ψ ψ + ω = T = (3.25) 1 T T (3.26) 1 T _ T (3.27) inte = K E (3.28) intp = K Pmax (3.29) intω = K ω (3.30) φ = ω ω (3.31) Where the state equations utilize the following definitions in equations : 1 = = ψ = ψ + ψ + ψ (3.32) (3.33) (3.34)

63 46 ψ = ψ + ψ I = ψ ψ I = ψ ψ (3.35) (3.36) (3.37) I = cos + sin (3.38) I = cos sin (3.39) V _ =R I +I (3.40) V _ =R I +I (3.41) P = V _ I +V _ I (3.42) Q = V _ I V _ I (3.43) E =E _ + _ (3.44) V = + (3.45) ω = (3.46) T =V cos Phi I +sin Phi I /Omega (3.47) ω= (For source 1 only) (3.48) = _ = sin = cos (3.49) (3.50) (3.51)

64 47 2 I_filter 1 V_filter V I I* Conjugate Product 3 P_set Power Re Im Complex to Real-Imag Reactiv e Power Mp KpPMax KiPMax 1 s Sum w/limit2 integrator w/limit Frequency Command KpGov KiGov Sum w/limit 1 s 1 0.1s+1 Fuel/Turbo Delay Torque 6 T_elec Torque Sum 4 V_set 1/Jsm Accel 1 s Speed 5 Omega 1 s Angle -1i e u 2 Rotor_Angle 3 Rotor_Speed u Abs Mq Vmag V_cmd KpV KiV 1 s Vmag_Cmd 1 Field Voltage Figure 3-5 Electromechanical model of synchronous machine generator set governor and exciter Inverter-based Microsource Only System Characteristics (a) Figure 3-6 Nominal eigenvalues with inverter-based microsources, one at its power limit and one in nominal operating range. (a) full view, (b) low damping-frequency pole detail Using the state Equations ( ) for each source to form a two inverter-microsource network, the system was evaluated at a nominal operating point with the parameters as in Table 3-3. The baseline eigenvalues are presented at various levels of zoom in Figure 3-6. As this investigation will study only one source against its power limits, this eliminates the second copy of Equation (3.8) leaving 17 states of the two-node system with an equal number of associated eigenvalues. Each can be attributed to a particular characteristic of the network. The participation factors presented in (b)

65 48 Table 3-1 and Table 3-2 associate a given eigenvalue to the various states which are physical (i.e. current) as well as purely mathematical (i.e. integrated power error) characteristics. In Figure 3-6b, the lightly damped complex poles are shown. This investigation will focus primarily on those closest to the real imaginary axis for simplicity as they are the ones that will characterize the majority of the response, particularly for the cases nearing neutral stability. Secondarily, in Figure 3-6b, the low-frequency poles are shown with the power control pole amongst them. The bandwidth of this pole is particularly important when considering the amount of stored energy a source needs to operate nominally in a CERTS microgrid considering the worst-case load transient that the system should reasonably endure. While two of the low-frequency poles are responsible for terminal voltage regulation (reactive power balancing), one eigenvalue resides at the origin which represents the steady-state offset that will exist in the load angle between two sources. This offset will exist during all conditions except for a coincidental case where network load and network impedances cause a lack of an angular difference between the two sources. In the following sections, the sensitivity of the selected eigenvalues to changes in network parameters will be investigated to illustrate the associated effects on stability and system response-time. In later sections, additional control techniques will be applied in an effort to improve the rate of power control response without sacrificing network stability.

66 49 Table Eigenvalue participation factors for Islanded Microsource Network (1 of 2) Eigenvalue State Number Real Imaginary Designator Load Current 1 Load Current 2 Filter L-C 1 Filter L-C 2 Capacitor Voltage 1 Capacitor Voltage 2 Capacitor Voltage 3 Capacitor Voltage 4 Id Iq Vcd Vcq Id Iq inteerr intperr Phi Id Iq Vcd Vcq Id Iq inteerr Phi The participation factors analysis of the nominal operating point eigenvalues gives important insight into the characteristics of the small islanded power system. As expected, the eigenvalue associated with the voltage phase offset angle of one source with respect to the primary source is located at the origin. Secondly, the L-R time constant of the load resistor and filter inductor connected to it (Eigenvalues 1&2, Load Current) is the fastest set of time constants of the network and nearly serves as a step-change in power disruption resulting from changes in load resistance given that it is at a significantly higher frequency than any other eigenvalue. Thirdly, the voltage balance eigenvalues (11&12) have participation factors that include the current in the inductors and not the voltage on the capacitor which implies a mode existing between the two

67 50 sources that is at a frequency that does not interact with the L-C resonant mode of the filter, but instead is affected by the flux balance and resulting current fluctuations between sources. Lastly, it is clear which eigenvalues drive the response of the terminal voltage regulation action. Eigenvalues 14 and 16 are equally balanced between the two sources in the network, noting that cooperation occurs between both voltage regulators to reach a steady-state condition. Table 3-2- Eigenvalue participation factors for Islanded Microsource Network (2 of 2) Eigenvalue State Number Real Imaginary Designator Filter L-C 3 Filter L-C 2 Voltage balance 1 Voltage balance 2 Phi2 Voltage regulation 1 Energy regulation Voltage regulation 2 Phi1 Id Iq Vcd Vcq Id Iq inteerr intperr Phi Id Iq Vcd Vcq Id Iq inteerr Phi

68 51 Table Eigenvalue parameter sensitivity for 2x scaled change in parameter value Eig. Value Designator Voltage balance Filter L-C Modes Integrated Power Control Parameter Nominal Value i i KiV i i Mp i i Mq i i KpPMax i i KiPMax i i L1 (X1) 104E-6 (0.039) i i L11 (X11) 390E-6 (0.147) i i R i i R i i Rload i i The eigenvalue sensitivities table, Table 3-3, shows the response of changes in various eigenvalues calculated as a function of changes in system parameters. This is important to determine the system parameters most responsible for the position of each of the critical network modes. These parameters can be varied to tune system responses, or if the parameters are not easily changed, they can be tracked to determine the response characteristics of the system which will help define control gains to compensate. It should be noted that the parameter sensitivities presented in Table 3-3 do not account for changes in system operating point as a function of change in said system parameter. For parameters that change the operating point, steady state is no longer achieved, and therefore, the results are void. However, in the following section, simulation has been used to achieve steady state for each parameter variation to either validate or supersede the results presented in Table 3-3.

69 Varying KiV: The first parameter sensitivity investigated is the integrated gain on the voltage regulator. The voltage regulator measures the magnitude of the terminal voltage and will adjust the terminal voltage to the set-point using an integrated controller only. While this is generally used to minimize reactive power flow between sources, both reactive power and real power change as a function of increased terminal voltage magnitude. Particularly for the case where resistance of the interconnecting filter impedance is significant, the cross-coupled association between voltage magnitude and power delivered does create a conflict between the frequency regulator and the voltage magnitude regulator, reducing the effectiveness of each controller proportional to the degree of cross-coupling. Secondarily, fast regulation of the terminal voltage represents a steady-state decoupling of the resistive and inductive voltage drop that occurs with power flow. This decoupling of the resistance in particular does act to decrease the damping of the lightly damped poles in particular, which can be seen in Figure 3-7. The migration in eigenvalue in this case corresponds with the reported change calculated in Table 3-3, but this is expected as the value of integrated gain should not change the steady-state operating point, keeping the conditions of Table 3-3 valid. However, the positive direction of the migration indicates this gain as a limiting factor to damping and eventually would jeopardize stability.

70 53 Figure 3-7 Voltage balance and voltage regulation eigenvalue migration as a function of increasing the integrated gain on the terminal voltage regulator As expected, the terminal voltage regulation eigenvalues increase in frequency as a function of increasing integrated gain, as can be seen in Table 3-3. Then nominal parameter values are presented in red and can be seen to correlate to the values in Table 3-3. One discrepancy with Table 3-3 is the change in power regulation eigenvalue, where the table supposed a Hz migration for a 2x change in the voltage regulator integrated gain. The poles for the power regulation eigenvalue can be seen to be grouped over the -0.5Hz region, largely unaffected by the parameter change, as can be seen by the multi-color pole grouping at - 0.5Hz in Figure 3-7. The remaining eigenvalues show no significant movement. Overall, the voltage regulator integrated gain has been shown to increase the voltage regulation bandwidth while removing some damping for lower-frequency voltage balance oscillations that would have come from the resistance present in the inverter filter between the inverter terminals and the connection point of the filter to the microgrid mains.

71 Varying Mp and Mq: The proportional frequency droop gain Mp and voltage droop gain Mq are two main parameters to the CERTS microgrid. While the frequency droop gain defines the change in frequency as a function of differential real power, the voltage droop gain defines the change in terminal voltage set-point as a function of reactive power output. Each gain represents a compliance of the source to changes in system load, whether real or reactive power loads. An increase in frequency droop gain would be similar to decreasing the size of an induction generator (increasing slip gain as a function of power). These gains increase network compliance as compared to the infinite grid which is generally assumed to have very little compliance. However, increasing the source compliance decreases sensitivity to changing network conditions. This insensitivity is beneficial from the perspective of source protection and regulation, but detrimental from the perspective of adequately providing high power quality to the microgrid. Fortunately, most loads are insensitive to the range of frequency variations that the microgrid will encounter (+/- 1%). In the case of the voltage droop gain, the same can be said in terms of increased gain implies increased compliance. The voltage droop gain acts as actively controlled impedance between a fixed voltage source and the microgrid. Typically, this value is set to 0.05pu, which is similar to standard isolation transformer impedance of pu. For exceedingly large voltage droop values, acceptable reactive power draw is met with significant voltage reduction or surge, depending on the polarity of the reactive power flow. Therefore, careful selection of this value is important to allowing enough compliance (active impedance) to limit reactive power flow between sources, while maintaining a nominal voltage range that complies with power quality standards.

72 55 Figure 3-8 Voltage balance and Phi2 eigenvalue migration as a function of increasing the frequency droop (Mp) gain Figure 3-9 Voltage balance and voltage regulation2 eigenvalue migration as a function of increasing the voltage droop (Mq) gain In this section, the two parameters in question were varied individually from 1/4x to 4x their nominal values to explore the sensitivities in both directions by factors of 2. The highest value of Mp generated unstable poles so no steady-state conditions were able to be reached. To determine the eigenvalues of the system with this gain set, the operating point used for the analysis was that of the closest possible stable point which performs a reasonable estimate of the eigenvalues. From Figure 3-8 and Figure 3-9, it can be seen that the voltage balance eigenvalue migrates in a positive real direction for all increases of either parameter. However, the frequency droop eigenvalue has a significantly more pronounced effect as compared to the voltage droop gain.

73 56 This could be supposed to be due to the fact that the voltage regulator has little effect at 60Hz whereas the frequency droop action has a lesser delay on the volt-second balance that drives said eigenvalue. It can be seen plainly that an increase of the frequency droop gain brings the intersource current eigenvalue to a marginally stable or an unstable point when all other system parameters are nominal values. This is concerning and clearly defines a limit of the frequency droop gain in this case. Figure 3-9 shows one of the voltage regulation eigenvalues migrating as a function of increased voltage droop gain. This is particularly interesting because the other voltage regulation eigenvalue is largely unaffected by the change in this gain, which is more likely due to the integrated voltage regulation gain KiV. Figure 3-8 also shows the migration of the load angle (Phi2) eigenvalue as a function of increasing droop gain. The result indicates that the load angle is regulated at a higher rate for higher droop gains in this case, but the result is less important when considering the characteristic in Figure Figure 3-10 Power regulation eigenvalue migration as a function of increasing the frequency droop (Mp) gain

74 57 Figure 3-10 shows the migration of the integrated power error (intperr) eigenvalue as a function of increasing Mp. While it may be thought that the power regulation bandwidth would be increased with an increase in the frequency droop gain, the opposite shows true. Even though the load angle eigenvalue moves significantly left (out of frame of Figure 3-10, shown in Figure 3-8), the integrated power error bandwidth decreases. Considering any system with an increase in proportional gain without an increase in integrated gain, the integrated bandwidth will slow and that is the case here. Later, the power regulator integrated gain will be investigated to illustrate the opposite case. The high frequency L-C modes are largely unaffected by the droop gains which simplifies the number of eigenvalues to consider when varying the droop gains Varying KpPMax and KiPMax: KpPMax and KiPMax are the proportional and integrated gains for the PI controller that regulates the output power of a CERTS microsource to not greater than its maximum output power. These gains enable the load re-balancing that is one of the main considerations of this work. This controller has limiters that maintain zero output when the power of the source is within the maximum and minimum bounds, but will provide both proportional and integrated compensation when power is measured outside the limits. The controller will cease compensating when the frequency command output of the sum of the proportional and integrated output is above zero for the Pmax controller (below zero for the Pmin controller). For all conditions within this chapter, the Pmax controller is engaged for source 1 and power is regulated to the specified maximum power in the steady state. This property is typical for variable input power sources such as wind and solar converters that track the maximum power available. Unfortunately, differences between the output power and input power are buffered

75 58 only by the DC bus capacitor and the output power must be regulated to match the average input power to properly manage the DC bus and avoid voltage collapse. In Figure 3-11, it can be seen that the increase in KpPMax has a similar effect to a previously demonstrated increase in Mp, the frequency droop gain by destabilizing the voltage balance eigenvalues and increasing the load angle (Phi2) regulation. This is expected as KpPMax acts identically to Mp when in the power-limited condition which is valid for source 1 in the configuration investigated in this chapter. The integrated gain KiPMax acts to increase the power control bandwidth as seen in Figure In the highest value cases (4x nominal, KiPMax=120rad/s^2), the second eigenvalue pertaining to source power regulation, load angle, meets the integrated power error eigenvalue and they create a complex pair. This describes a limiting case where further increases in integrated gain, particularly with small proportional gains, decreases damping and eventually jeopardizes stability. This result agrees with the result calculated in Table 3-3 as the integrated gain of the Pmax controller does serve to increase the rate of power convergence from the nominal parameter case. Figure 3-11 Power regulation and voltage regulation eigenvalue migration as a function of increasing the power limit proportional gain (KpPMax) and integrated gain (KiPMax)

76 59 It is interesting how the integrated gain of the Pmax controller does not affect the voltage balance poles in a negative manner. However, the proportional gain is shown to significantly reduce damping and eventually causes unstable poles. As might be expected of control gains, the calculations from Table 3-3 do agree with the results here except that the reported sensitivity to the KpPMax gain was nearly 4x the result which can be attributed in part to local sensitivity that becomes invalid for such a large parameter change. Figure 3-12 Power regulation eigenvalue migration as a function of increasing the frequency droop (Mp) gain Figure 3-13 Power regulation (right) and load angle (left) eigenvalue migration with increases in KiPMax The L-C Filter eigenvalues are largely insensitive to changes in power regulation control gains in this case as well, which limits the focus only to the voltage balance mode eigenvalues.

77 Varying L1 and R1: The filter impedance values are some of the most critical values in the microgrid. As filter impedances typically dominate the interconnection impedance for small power systems, they essentially define the open-loop damping characteristics of the network. While inductance adds a certain level of compliance between voltage sources, a delay that allows for a response time that spans multiple cycles, the resistance adds damping towards steady state while significantly contributing to real power and reactive power cross coupling. One of the basic assumptions of the CERTS microgrid is that the network is primarily inductive in nature which couples power to load angle and reactive power to voltage magnitude differential. If the network is primarily resistive, than the responses phase shift by 90 degrees and the coupling of power and reactive power essentially flip. Typically, the reactance to resistance ratio (X/R) is between 10:1 and 1:1 in most small power systems. While the interconnecting impedances of low voltage networks have been noted to have X/R ratios of less than 1:1, the filter elements are typically much higher particularly depending on the size and quality of the filter components. While strong cross coupling can lead to lightly damped oscillatory network characteristics, the more likely case is where network impedances with a high X/R ratio are present. This condition is concerning as it will be shown that the resistive elements are key to damping transient oscillations and establishing a steady-state condition. This is important as the frequency droop and voltage droop characteristics rely on the power flow relationship that is based on a steady-state condition. With such a wide range of possible configurations in a microgrid it is difficult to select one X/R value as standard. However, the X/R ratio has been chosen to be 10:1 for the purposes of this chapter, which is the most reasonable worst case condition to explore for stability issues.

78 61 Figure 3-14 illustrates how varying two of the network components affect the inter-source current eigenvalue. The figure has a significantly expanded y-axis for readability, so it should be noted that the majority of the movement is along the real axis. To that point, it can be shown that increasing the inductance tends to destabilize this mode, increasing the resistance tends to stabilize this mode and increasing both proportionally has less change but does tend to increase damping. All of these conclusions agree with those from the sensitivity analysis presented in Table 3-3. Figure 3-14 Voltage balance eigenvalue migration as a function of increasing the converter-side filter inductance (L1) and resistance (R1) The voltage balance eigenvalues presented in Figure 3-15 demonstrate an interesting case of eigenvalue interaction where the two Filter L-C modes migrate around the two inter-source modal eigenvalues. The voltage balance eigenvalues are initially quite stable and tend towards the imaginary axis, but then back away from the imaginary axis which is likely due to eigenvalue interactions. This could be due in part to also having identical filter values for both sources under the nominal parameter case (red poles, third in sequence). The immense sensitivity to converter-side inductance and resistance values is quite apparent here especially considering the

79 62 relative insensitivity to previous network parameters investigated. The Filter L-C mode eigenvalues decrease in frequency and increase in damping with an increase in inductance and resistance, respectively. Both results are expected from an RLC resonant circuit, but were not clearly predicted in the parameter sensitivity analysis presented in Table 3-3. However, this is likely due to the complex nature of the apparent response as well as the change in operating point from changes in filter parameters. The inter-source modes appear to be quite sensitive to the frequency of the filter L-C mode, particularly when they coincide, creating a lightly damped mode. It is interesting to note that the value of resistance present in the converter-side impedance has little effect on the position of the eigenvalue in most cases except for the most lightly damped case. Figure 3-15 Filter L-C mode (center) and inter-source modes (right) eigenvalue migration as a function of increasing the converter-side filter inductance (L1) and resistance (R1) A similar insensitivity to changes in network resistance is seen in Figure 3-16 where only the increase in inductance moves the voltage regulation (center) and power regulation (left)

80 63 eigenvalues. The increase in inductance is shown to slow the voltage regulation mode significantly. Figure 3-16 Power regulation eigenvalue migration (left) and voltage regulation eigenvalue (center) as a function of increasing the converter-side filter inductance (L1) and resistance (R1) Varying L11 and R11: The microgrid-side filter inductance (L11) and resistance (R11) is significantly larger than the converter side as it represents the main reactive component that allows for reasonable response characteristics between sources. The x/r ratio of this inductor and resistor series pair was also kept at 10:1 to be consistent with the rest of the network. Also, as these components are significantly larger than those on the converter side, changes to these parameters will result in more movement in some eigenvalues as compared to the previous section. Figure 3-17 shows a characteristic that is very similar to Figure 3-14 where the addition of resistance increases damping of this eigenvalue and the increase of inductance causes, to a lesser degree, a destabilizing effect. In this case, however, the worst case eigenvalue position is much closer to the imaginary axis, but it is fair to mention that this operating point represents a total system X/R close to 25:1, significantly higher than the base case.

81 64 Figure 3-17 Voltage balance eigenvalue migration as a function of increasing the microgrid-side filter inductance (L11) and resistance (R11) The L-C filter eigenvalues presented in Figure 3-18 are particularly interesting as they tend to exhibit the least damping when in the nominal inductance case, where the impedance of the filters of both sources are equal. It appears that setting these impedances and associated resonant modes different than each other assists in avoiding a lightly damped condition, which is a topic that warrants further review as it may become an issue for multiple installations of similar microsources. Again here, the addition of resistance has very little effect as compared to the increase in inductance.

82 65 Figure 3-18 Inter-source modes eigenvalue migration as a function of increasing the microgrid-side filter inductance (L11) and resistance (R11) The energy regulation eigenvalues and the voltage regulation eigenvalues in Figure 3-19 show a characteristic very similar to that of Figure 3-16 except for the pronounced effect on eigenvalue position. These migrations make logical sense as increasing the inductance should increase the compliance of the filter and require more action to actively decouple said inductive voltage drop effects. Secondarily, the increase in power control bandwidth comes from an effect similar to a decrease in the frequency droop gain and/or the Pmax proportional gain that was previously demonstrated to decrease the power regulation bandwidth for increases in these proportional values.

83 66 Figure 3-19 Energy regulation eigenvalue migration (left) and voltage regulation eigenvalue (center) as a function of increasing the microgrid-side filter inductance (L11) and resistance (R11) Varying Rload: Sensitivity to load is an important parameter to investigate. It will help determine which, if any, operating points are more susceptible to instabilities than others. However, the system load was found to have very little effect on the potentially unstable eigenvalues within the acceptable range of system loadings. At twice the rated load, some of the low frequency poles show signs of decreased damping. One interesting result is the migration of the load current eigenvalues that are typically at a very high frequency, presented in Figure These eigenvalues are the ones most affected by the change in load resistance while other system parameters had little effect on these eigenvalues.

84 67 Figure 3-20 Load current eigenvalue migration (center) as a function of increasing the system load (decreasing Rload) The L-C filter eigenvalues, presented in Figure 3-21, are also affected by the change in load. For lower loadings, an increase in load tends to stabilize these eigenvalues, but at the rated power of 1pu for each source, these eigenvalues begin to head towards the inter-source mode eigenvalues which are closer to the imaginary axis. Fortunately, the voltage balance eigenvalues are seemingly unaffected by changes in system load which implies that system loading will not decrease the damping of these lightly-damped poles any further. Figure 3-21 L-C filter eigenvalue migration (center) as a function of increasing the system load (decreasing Rload)

85 68 Figure 3-22 shows the migration of the integrated power error (energy control) eigenvalue merges with the low frequency voltage regulation eigenvalue to create a complex pair at high loadings. This is likely due to the fact that the highest loaded case is 2x the total maximum power of the system, but illustrates how the local linearity of the load-angle versus power transfer breaks down for significant angles. For this level of system loading, the phase angle (load angle) between the source voltages is only 16 degrees, but considering the added phase from the resistance of the network which adds more cross coupling between power regulation and voltage magnitude regulation, it is apparent why a complex pair would develop. Figure 3-22 Low frequency eigenvalue migrations as a function of increasing the system load (decreasing Rload) The remaining eigenvalues have negligible movement through the load range, particularly the voltage balance eigenvalues. The overall conclusion from this section is that stability is not significantly affected by nominal system loading while the voltage and power responses are only affected at system loadings well beyond the nominal system ratings. However, these conclusions assume similar filter impedances. With higher per-unit filter impedance and/or system cabling with significant impedance, the complex eigenvalue pair for the voltage and current response will become less damped at lower levels of system loading.

86 Inverter Microsource System Parameter Sensitivity Conclusions It was shown in this section that each parameter varied at least one eigenvalue significantly. The primary eigenvalues of consequence identified previously were those oscillatory poles close to the imaginary axis as well as the power regulation eigenvalue. While the integrated gain of the frequency regulator (K ipmax ) was the only parameter that showed a dramatic improvement in the power regulation eigenvalue, many other system parameters were identified that modify the location of the lightly damped oscillatory poles (i.e. M p, K ppmax, L 11, R 1 and K iv ). The frequency droop gain M p and the proportional gain of the frequency controller that regulates power to the maximum power limit K ppmax both showed a similarly destabilizing effect on the lightly damped poles with an oscillatory frequency 60Hz. This result is due to the fact that the frequency droop effect increases its voltage vector velocity for decreases in measured power which encourages excitation unbalance in line reactors and transformers, supporting DC-offsets in line currents against the line resistances that cause the offsets to decay. At the limit where the droop effect dominates, this leads to ~60Hz power oscillations in the synchronous frame. It was shown in Figure 3-14 that any positive addition of resistance increased the damping of the aforementioned voltage balance eigenvalues. Unfortunately, resistance added for the purposes of adding damping would introduce losses that hurt the efficiency of power transfer on the microgrid. It was also discovered that increases in microgrid-side inductance (L11) tended to destabilize the lightly damped poles but did not provide evidence of directly causing instability within the explored range, contrary to the case for the proportional frequency gains discussed previously. The integrated gain on the voltage regulator was shown to have a similar effect as a decrease in resistance which is due to the voltage-drop decoupling effect at lower frequencies when the terminal voltage is regulated back to the pre-transient command. Lastly, the change in system

87 70 load was shown to not affect the eigenvalues of interest except by suggesting an under-damped response of the voltage and power particularly for over-loaded cases of 2x and above the rated system loading. The conjecture was also proposed stating that this under-damped response could be achieved at lower loadings with larger system inductances creating weak interconnecting impedances. These results indicate that the nominal operating point of X/R ratio of 10:1 amongst other nominal system parameters is largely stable, but that the system can be brought to an unstable point particularly with increased proportional frequency droop gains and reduced system resistance. Addition of an active controller that either mimicked the effect of resistance or avoided exciting system resonances would prove useful here to increase the stability margin of the system particularly considering the inclusion of other source varieties or sources with X/R ratios higher than 10: Mixed-source Microgrid System Characteristics with Microsource and Synchronous Machine Generator (a) (b) (c) Figure 3-23 Nominal eigenvalues for mixed source microgrid system with inverter microsource and Synchronous Machine Generator (a) full view, (b) Inter-source current oscillatory pole detail, (c) network damping and power regulation detail The next system configuration was constructed with one inverter microsource and a rotating machine generator. It uses state equations from sections and to create a nineteen

88 71 eigenvalue system. The inverter microsource is used as the power limited source in this case and the synchronous machine generator is used as the adjustable source that tracks changes in load. The generator has a delay modeled between the power input from the prime mover and the command for said power. This is in contrast to the previous case with the inverter because the power source side of the inverter was ignored while the overload energy requirements for transient ride-through were monitored. Also, the synchronous machine case here differs in that the energy buffer is the inertia of the machine, and to a lesser degree the stored flux in the machine. When the electrical power exceeds the mechanical input, the machine will slow and this frequency characteristic will be monitored later in this chapter during transient response analysis. The presence of physical inertia imparts slightly different network characteristics and this section is intended to bring forth the pertinent network characteristics before presenting the performance of this network as a result of load transients. The nominal system eigenvalues presented in Figure 3-23 resemble those of the inverter microsource only system presented in section with the addition of six low frequency poles that replace four of the higher frequency oscillatory poles (L-C filter poles) which were quite lightly damped under most cases. The new low frequency poles come from the physical inertia of the synchronous machine, the flux voltages of the synchronous machine, the governor torque regulation and the torque regulation delay element as described in Equation The participation factors for the system in this section are presented in Table 3-4 and Table 3-5.

89 72 Table 3-4 Inverter Microsource and Synchronous Machine Generator Microgrid System Participation Factors (1/2) Eigenvalue State Number Real Imag Designator Filter L- C 1 Filter L-C 2 Filter L- C 3 Filter L-C 4 Load Current 1 Load Current 2 Voltage balance 1 Voltage balance 2 d-axis damper q-axis damper Id Iq Vcd Vcq Id Iq inteerr intperr Phi Psi_ds Psi_qs Psi_dr Psi_qr Psi_fr Omega Tpm intomegaerr inteerr Phi

90 73 Table 3-5 Inverter Microsource and Synchronous Machine Generator Microgrid System Participation Factors (2/2) Eigenvalue State Number Real Imag Designator Src2 Torque Speed 1 Src2 Torque Speed 2 Phi1 Field Reg. 1 Field Reg. 2 Power/Speed Reg. 1 Power/Speed Reg. 2 Voltage Reg. Id Iq Vcd Vcq Id Iq inteerr intperr Phi Psi_ds Psi_qs Psi_dr Psi_qr Psi_fr Omega Tpm intomegaerr inteerr Phi Phi2

91 74 Table 3-6 Inverter Microsource and Synchronous Machine Generator Microgrid Parameter Sensitivities Eigenvalue Designator Parameter Nominal Value Voltage Balance Torque/Speed Power/Speed Reg i i i KiV i i i Mp i i i Mq i i i KpPMax i i i KiPMax i i i KpGov i i i KiGov i i i tau_filt i i i Jsm i i i Varying KpGov and KiGov Two of the main parameters of the generator set prime mover governor are the proportional and integrated gain associated with the mechanical input torque regulation. The bandwidth of this process is inherently limited by the delay element that lumps together the combustion delay, fuel servo delay and turbocharger delay. Therefore, oscillatory poles are expected when the bandwidth approaches the torque regulation pole which exists as one of the pair at -0.26Hz. The eigenvalue migration of Figure 3-24 displays the effect of changing the governor proportional and integrated gains on the machine speed and governor power regulation eigenvalues. It can be seen that the proportional gain (KpGov) is increased to the point where there are diminishing returns on the torque regulation eigenvalues (top and bottom of Figure 3-24). It can also be seen that the integrated gain (KiGov) was increased to a value such that the lowest frequency pole had moved beyond the voltage regulation pole and not so high to cause the torque regulation poles to become lower frequency than the voltage regulation pole at -0.28Hz. This provided a reasonable characteristic considering the limitation of the torque production delay in the governor model.

92 75 Figure 3-24 Speed regulation eigenvalue migration with variation of KpGov only (a) and KiGov only (b) Varying tau_filt, the torque production filter bandwidth Figure 3-25 Torque regulation and Phi1 eigenvalue migration with an increase of the prime mover torque production delay (tau-filt) The torque production delay is one parameter that would ideally be minimized to zero. The delay between torque command and torque production is an undesirable characteristic and the result of an increase of said delay is shown in Figure It can be seen that the torque regulation poles (top and bottom) migrate towards the imaginary axis and even cross it in the most extreme case of tau_filt=0.4 seconds. Secondarily, the load-angle eigenvalue (Phi1) can also be seen to be significantly limited in bandwidth by the torque production delay. The nominal delay value for the governor model was chosen to be 0.1 seconds which represents a 1.6Hz bandwidth by itself, but this value is limited further by other phase delays in the system. While the selection of this value could vary significantly given the performance of the fuel

93 76 regulator and size of the turbocharger, it can be supposed that any reasonable system would not have such a delay as to cause the unstable torque oscillations that can be seen in the limiting case from poles that exist in the right-half plane. Also, the assumed value is close to experimentally determined values for small diesel generators with mechanical fuel regulators and large turbodiesel generators with a dominant turbocharger delay Varying Jsm, gen-set inertia Figure 3-26 Migration of Torque/Speed regulation eigenvalue migration with increasing genset inertia To complete the investigation of new parameters to the mixed microgrid system, the combined inertia of the machine and prime mover is varied to examine its effect on the eigenvalues of the system. As might be expected, the only appreciable effect on the system is in the torque/speed eigenvalues where the oscillatory frequency of the poles decreases with added inertia, but decreasing the real component slightly. This can be explained from the decreased involvement of the damper windings as the generator would be oscillating less often and at a lower frequency. The lower frequency oscillation would excite the damper windings with less current and dissipate less energy, creating a system with slower rate of convergence.

94 Varying KiV: The remainder of the system parameters are those seen in the previous section, but it is important to revisit them in order to examine any significant changes that the new source paring provides. Figure 3-27 Migration of low-frequency eigenvalues (voltage regulation and field voltage regulation poles) in response to increases in KiV The voltage regulation poles move in a similar manner to when the network is only comprised of two inverter microsources, except that the synchronous machine field voltage eigenvalues are a complex pair in this case. It can be seen that the power/speed regulation eigenvalues, as well as all other eigenvalues not shown, are relatively unaffected by the change in the voltage regulation gains. In this manner, the result makes perfect sense and reaffirms the findings of the participation factor analysis that determined the eigenvalue designators in the first place.

95 Varying Mp and Mq: Figure 3-28 Voltage balance q-axis damper and Phi1 eigenvalue migration with increases in frequency droop gain Mp As the frequency droop gain is increased, the voltage balance eigenvalues progress toward the imaginary axis and show a similar trend as the previous case. However, the voltage balance poles originate from a more damped position in this case than the previous. Also, the worst-case gain of 4x the nominal value still leaves the associated eigenvalues a significantly more stable point (negative real value) than the previous case. This result is likely due to both the slowerreacting nature of the synchronous machine, as well as the significant component of resistance associated with the synchronous machine impedance which is closer to an X:R ratio of 3:1 versus the 10:1 as employed for the inverter microsource filter model. It is then expected that this network node is significantly more damped for reasons mentioned previously and particularly investigated in Chapter 5. The q-axis damper eigenvalue can be seen migrating on the left side of Figure 3-28 which is unexpected as the reasons for this link are unclear. Finally, the Phi1 eigenvalue frequency is shown to increase with Mp in the right hand side of Figure 3-28 which follows with the general increase in effective settling gain for this state.

96 79 Figure 3-29 Source 2 torque/speed and field voltage eigenvalue migration with increases in frequency droop gain Mp The torque/speed eigenvalue is the pole most affected by the increase in the frequency droop gain amongst the low frequency poles. It can be seen in Figure 3-29 that the torque/speed response moves into the right-half plane with a 4x increase in the droop gain and nearly goes unstable at 2x the nominal gain. This can be attributed to the low damping of the synchronous machine as compared to the effect of the droop gain on the power regulation actions of the inverter microsource. Figure 3-30 Field voltage and Voltage regulation eigenvalue migration from increasing the voltage droop gain Mq The voltage droop gain can be seen in Figure 3-30 to affect the voltage-associated eigenvalues, but in this case, the field voltage regulation poles are moved to a lower frequency, while the

97 80 inverter voltage regulation eigenvalue increases in frequency. This is likely due to the inherent time delay associated with the magnetizing flux changes in the synchronous machine versus the relatively low latency of the inverter L-C-L filter of the inverter Varying KpPMax and KiPMax: Figure 3-31 Inter-source current and Phi2 eigenvalue migration as a function of increasing the power limit proportional gain (KpPMax) The effect of increasing the power limit proportional gain should have an identical effect to increasing the frequency droop gain as they act on the same variables; however, the KpPMax gain is only in use on Microsource 1, the inverter microsource, which slightly lessens the resulting travel of the eigenvalues. Regardless, the effect seen is similar for the voltage balance, torque/speed eigenvalues and the Phi1 eigenvalue. The voltage balance eigenvalue in this case shows that an increase in the KpPMax gain of 4x will not cause unstable responses and even shows to improve the Phi1 eigenvalue.

98 81 Figure 3-32 Torque regulation and speed regulation eigenvalue migrations as a function of increasing the power limit integrated gain (KiPMax) While the integrated gain of the power limit controller did not have a significant impact on the inter-source current eigenvalues, it can be seen in Figure 3-32 that it causes one of the torque regulation and speed regulation eigenvalues to merge into a complex pair at 2x and 4x the nominal value. It can also be seen that at ¼ the nominal value (in blue), the torque regulation eigenvalues are at a significantly lower frequency with a power regulation bandwidth of approximately 0.1Hz. The nominal case in red appears to give a reasonable response characteristic while still maintaining speed regulation (dominant) eigenvalues with a high damping factor. Figure 3-33 Torque regulation and Phi2 eigenvalue migration as a function of increasing the power limit proportional gain (KpPMax)

99 82 When the power limit proportional gain is increased, the low frequency eigenvalues migrate in a manner opposite to when the integrated gain is increased, a result presented in Figure It can be seen that increases in the KpPMax cause the torque regulation poles to increase in frequency while the speed regulation eigenvalues decrease in frequency. This indicates that increasing the proportional gain any further will not yield any faster of a power response without also increasing the integrated gain. However, it should be noted that the limit of these gains should be tuned to ensure stability in the worst case condition and it appears that the inverteronly system is the less damped case and should define the limit of the power limit gains Varying L1 and R1: Figure 3-34 Inter-source current eigenvalue migration while varying L1 and R1, the inverter side elements of the L- C-L filter The effect seen in Figure 3-34 is very similar to the previous case in section except that the average real value in this case is further from the imaginary axis which is consistent with previous evidence from this system configuration. The same trend remains that addition of inverter-side inductance slightly decreases damping while resistance increases it.

100 83 Figure 3-35 L-C Filter eigenvalue migration as inductance and resistance is varied on the inverter-side of the L-C-L filter The most pronounced effect on the system in response to modifications of R1 and L1 are on the L-C Filter eigenvalues which vary greatly and are presented in Figure This result mimics the previous case, but it is interesting to note that without the more lightly damped pole pairs, previously referred to as the inter-source mode eigenvalues, these parameter variations do not jeopardize stability and are therefore significantly less concerning. Figure 3-36 Phi1 and Voltage Regulation eigenvalue migration as inductance and resistance is varied on the inverter-side of the L-C-L filter Figure 3-36 demonstrates that the resistance of the filter has little effect on low frequency eigenvalues. However, the inductance does decrease the rate of Phi1 convergence and slightly slows the voltage regulation eigenvalue. Both of these responses are due to the increased voltage compliance from the inductance, meaning an increased difference in voltage magnitude and power angles necessary to reach a new operating point after the same change in system load.

101 Varying L11 and R11: The highest frequency eigenvalues of the system, the load current eigenvalues, change significantly with increases in microgrid-side filter inductor size as this forms the primary L-R time constant that defines the load current eigenvalue. R, in this case, is a series combination of the filter resistance and load resistance. It makes sense then why the small changes in filter resistance here have little effect on the load current eigenvalues as they are dominantly defined by the load resistance itself which is significantly larger (~1pu versus 0.02pu). Figure 3-37 Filter L-C eigenvalue migration when varying L11 and R11, the microgrid side inductor components of the L-C-L filter The Filter L-C eigenvalues are also significantly affected by changes in filter inductance as presented in Figure The majority of the change is in the real axis, which indicates that with the increase in inductance, that the stored energy in the L-C circuit is greater compared to the resistive power loss, giving a higher Q value. It is important to note that the Filter L-C eigenvalues do not appear to limit stability (convergence rate) more than other eigenvalues of the network. This result is informative, but it is not particularly concerning for this analysis.

102 85 Figure 3-38 Voltage balance eigenvalue migration when varying L11 and R11, the microgrid side inductor components of the L-C-L filter The voltage balance eigenvalue migration characteristic presented in Figure 3-38 appears nearly identical to all other cases where the inverter filter impedance is modified. This indicates that the voltage balance eigenvalue is insensitive to whether the inductance or resistance is on the inverter side or microgrid side of the source filter. In fact, this eigenvalue has been previously noted to exist as a three-phase flux balancing characteristic where its response is slowed by increased inductance and expedited by increased resistance given the R/L response towards a steady state operating point. The frequency is very similar to the operating frequency as the flux balance is a stationary-frame phenomenon and appears in the synchronous reference frame at the rate of the synchronous reference frame with respect to the stationary frame.

103 86 Figure 3-39 Torque Regulation and Voltage Regulator 1 eigenvalue migrations when varying L11 and R11, the microgrid side inductor components of the L-C-L filter The low frequency eigenvalues migration characteristics in response to changes in filter impedance are presented in Figure The figure indicates that the voltage regulation eigenvalue (right) is slowed while the field regulation eigenvalues (top and bottom) are also slowed with a greater imaginary component as a function of increasing inductance. Therefore, there seems to be a direct link between increasing the line reactance and a slowing of the voltage regulation response, which is due to the increased amount of work required by the regulators to perform the regulation task, requiring a longer time to completion. In this case, it is interesting to note that resistance has no appreciable effect on the low frequency dynamics. The torque regulation eigenvalues (middle right) are driven to a slightly faster but lighter damped condition, but only as a result of eigenvalue interaction at the highest value of inductance.

104 Varying Rload: Figure 3-40 Load current eigenvalue migration when varying Rload, varying system load As might be expected from previous evidence, the load current eigenvalues increase in frequency with an increase in load resistance assuming the inductance of the interconnecting lines are not varied as well. This result is presented in Figure 3-40 and mimics the results of the previous section with only inverter-based microsource network. Figure 3-41 Low-frequency eigenvalue characteristic when varying Rload, changing system load

105 88 More importantly, the low frequency eigenvalues are not significantly affected by changes in load resistance, displayed in Figure This is a positive result in that it simplifies the later analysis to being insensitive of load variations for investigation of power balance convergence and system stability. However, one interesting result can be noted which is in the case of the lowest load resistance, that represents the greatest system loading. It can be seen that in all cases the blue poles (Rload is 1/4 the nominal value, double the nominal system rating) exist as an outlier to the remaining eigenvalue sets for nominal system loadings. This indicates that the network characteristics would change significantly when the product of the loading and line reactance is increased. This represents a voltage difference across the coupling impedance that results in significant reactive power and real power feedback cross-coupling. Both of these results are undesirable which is why the source impedance is suggested to be kept between 10% and 33% on a per-unit basis which preserves the small angle approximation made previously in regard to the frequency droop vs. power relationship. Without this assumption, significant cross coupling exists between power and reactive power in relation to load angle and relative voltage magnitudes as shown in Figure Figure 3-42 L-C filter eigenvalue migration when varying Rload, varying system load

106 89 Lastly, the only other significant change on the system eigenvalues is in the L-C filter poles as a function of changing system load. In this variation of the system load, the final case is very lightly loaded and the majority of the power flow in the system is from Microsource 1 to Microsource 2. The feasibility of a synchronous machine generator acting as a load is ignored in this case, but it is interesting to note that when the nominal case in red is presented, this represents the case of power flow entirely from Microsource 1 to the load, having Microsource 2 at idle. In the more heavily loaded cases (green and blue), the power from Microsource 2 flowing to the load is 1pu for the green case and 3pu for the blue case. It is interesting then to note that the most damped case is when zero power is flowing from Microsource Inverter Microsource and Synchronous Machine Generator System Parameter Sensitivity Conclusions With the use of a synchronous machine generator set in place of the inverter microsource as Microsource 2, the system characteristics changed slightly, but in large part remained similar to the previous case with only inverter microsources. With the loss of an inductor and capacitor from the source impedance of the synchronous machine generator set, the system complexity was reduced, but some complexity was also added with the inertial state of the machine, the flux voltages of the machine model, the governor torque regulator gain and the combined torque production delay element. The resulting change in eigenvalues saw a loss of the lightly damped high frequency poles, but five low-frequency poles were added. Initially, the KpGov and KiGov governor gains were tuned to yield well damped dominant speed regulation poles and reasonably damped torque response poles which was restricted by the torque production delay element that has a bandwidth of 0.5Hz. Secondarily, it was found that the governor response was well behaved over a 2:0.25 range of gains with respect to the nominal

107 90 case which provides confidence in stability of response in the face of changing network parameters like interconnecting impedance and addition of negative loads. Similar to the previous case, the power regulator gains KpPMax and KiPMax were found to decrease damping at high values, particularly with the interaction of the proportional gain against the voltage balance and torque/speed eigenvalues, similar to the effect of increasing the frequency droop gain, Mp. However, only in the case of the torque/speed eigenvalues did the change jeopardize stability. It was also found that varying KpPMax and KiPMax values gave better system response characteristics than the previous case, but because the previous case provided a limit for said gains. Provided that the source is design to act in a plug-and-play manner, its controller must perform well under any reasonable system configuration. Consistent with the previous case, increases in R1 aided in the damping of the voltage balance current eigenvalues, though the resistance of the machine model provided more than enough damping in this case. Lastly, it was noted that while changes in system load affected some eigenvalues in the system, that the low frequency eigenvalues were relatively unaffected, which simplifies analysis in later chapters to ignore the case of loading within the nominal power ratings of the sources. 3.3 Partial Overloading Response Characteristics This section will present the response characteristics of system configurations that include a power-limited inverter microsource as the primary source. The intent in this section is to illustrate the maximum frequency deviations and overload energy required to ride through the transient load change that places the primary source temporarily into an overloaded condition. As the total load is less than the sum total of the source power capabilities on the network, the

108 91 load transient is referred to as partial overloading to indicate that one or more of the sources is initially placed into an overloaded state, initiating an automatic re-balancing of load referred to here as overload transfer. As will be seen later in this work, the inverter microsource has the most preferred transient response properties and represents the fast-acting power response unit that is required in a microgrid to ensure frequency stability within the nominal frequency limits. This is primarily due to the lack of DC-side power restrictions that are neglected here to preserve the generality of the transient analysis. The synchronous machine genset unit is modeled with a torque production delay as presented in section which restricts the prime mover power flow dynamics. However, the gen-set has stored kinetic energy in its inertia which is readily available to aid in the instantaneous support of changing load. Unfortunately, the frequency must deviate significantly to access the stored energy which may cause loads with under-frequency sensors to trip loads offline temporarily if the deviation is significant enough. In this section two network configurations will be investigated: first, the inverter microsource network as presented in section 3.2.4; and second, the inverter microsource and synchronous machine generator mixed system as presented in section The pre-transient system load in both cases is 1pu and the post-transient load is 2pu, leaving no system margin between the load and the combined ratings of the two sources. This will demonstrate how the equal-sharing of added load between voltage sources can cause significant temporary source overloading. Also, it will demonstrate how quickly the load can be re-

109 92 balanced and mitigate the overload energy on the power limited source operating in a gridforming frequency regulated mode. This transient is the largest load transient that can be placed on a system with two sources of equal rating when one source is continuously operating at its maximum power. This illustrates the most severe condition that the microgrid must withstand. The pertinent issue in transient load responses is the amount of load initially transferred to each source and particularly the power beyond the rating. The added load needs to be limited based on the requirements of the DC-side power source characteristics as well as the power converter. The power converter can typically operate beyond its continuous power rating for a fraction of a second or more. However, the source supplying the converter on the DC-side may have more strict requirements on power transients that may be exceeded during the load transients experienced on a microgrid. Therefore, the purpose of this section is to investigate and quantify the amount of overloadenergy required from the power-constrained microsource to continue operation following a large load transient. The configuration that causes the largest per-unit source overloading, assuming a load within the rating of the system, is when the non-power-constrained source is small as compared to the power-constrained sources of the system. This case will cause twice the overload that will be experienced in this chapter where equal source sizing is assumed. The scale factor is dependent on the ratio of the power rating of the power-constrained source and the combined rating of the remainder of the system. The characteristic relating the per-unit overload as a function of relative source size ratings is introduced in the beginning of this chapter in Figure 3-1.

110 Microsource 1 Initially Overloaded, Microsource 2 Tracks Load As mentioned previously, 1pu load will be added to a two-source inverter network with 1pu load pre-transiently. The load transient response in Figure 3-43 indicates that the added load is transferred equally among each source initially after the transient. This causes a 0.5pu overload on Inverter Microsource 1 which is then rejected by decreasing the frequency below that of Inverter Microsource 2 in order to cause load transfer and re-establish operation at a nominal output power. By inspection it can be noted that the load re-balancing takes approximately 0.7 seconds with a time constant of approximately 0.3 seconds which agrees with the projected time constant of 0.31 seconds per the -0.51Hz pole location of the energy regulation eigenvalue. The integration of the overload power yields approximately 0.12 pu-seconds of energy required from Inverter Microsource 1 which is small compared to a source supplied by an ultra-capacitor, flywheel or battery. However, this amount of energy is significantly larger than a photovoltaic source or a static VAR compensator where the vast majority of the stored energy of the source is stored in the DC bus capacitors. The typical storage on the DC bus capacitors of an industrial inverter is close to pu-seconds per Equation This value is calculated considering a 15kW power rating, a 750V nominal DC bus, a 678V minimum DC bus voltage and a 2200uF DC capacitance. From this result it appears that a photovoltaic source, a source likely to exist in a microgrid, would not survive this transient and is therefore the largest point of concern for later investigations. = 1 2 (3.52)

111 94 The inverter voltages can be seen to increase initially, though both values eventually settle back to the point where Inverter Microsource 1 began. Each source will be delivering the same amount of power post transiently in steady state and will require the same inverter voltage to compensate for resistive losses and voltage droop considering reactive power requirements. Due to the identical nature of the source hardware and the equal post-transient source loading, the load angle can be seen to decrease to zero. The rate of convergence is consistent with the differential power between the sources which is expected from the steady-state power-versusfrequency equation that has a direct relationship to load angle for small angles.

112 95 Figure 3-43 Inverter microsource system response from a 1pu load to a 2pu load (system maximum) The load transfer response, particularly the power angle response can be seen as a product of the difference between the operating frequencies between the two sources. The frequency and the reactive power characteristics also illustrate most clearly the voltage balance eigenvalue that exists as 60Hz frequency deviations. For higher droop gains and power regulation gains, the 60Hz oscillations decay slower and at the limit will increase in magnitude, representing an unstable system. In this case, the oscillations decay but indicate the relatively lightly damped response of the associated eigenvalue and the reasoning for limiting the power control gains.

113 Microsource 1 Initially Overloaded, Genset2 Tracks Load Change The next load transient simulation involves the mixed system used in section with a powerlimited inverter microsource as the first source and a synchronous machine as the second source. In this case, the impedance of the synchronous machine is higher than the filter impedance of the inverter microsource so it is expected that a greater proportion of the added load is transferred to the inverter microsource as it represents a lower impedance path to a voltage source. Also, it would then be expected that the overload energy characteristic would initially increase at a faster rate than the previous case. From Figure 3-44 it can be seen that this is the case on both counts. First, the initial load transferred to the Inverter Microsource 1 is approximately 0.7pu versus 0.5pu in the previous case. This result can also be seen by examining the power characteristic of the synchronous machine where the output power only increases to 0.3pu immediately following the load transient. Also, by comparing the overload energy characteristic to the previous case in Figure 3-43, it can be seen that the initial rate of increase for the overload energy is higher, reaching 0.05pu-seconds in only 0.1 seconds where the previous case required 0.15 seconds to reach the same value. Interestingly, the amount of overload energy is nearly identical due to a few key factors regarding the different source characteristics. First, the Inverter Microsource 1 has the ability to change the frequency without energy as it does not have physical inertia. Therefore, the offset in frequency is immediately achieved in response to measured power and the load angle is decreased much more rapidly in this case than in the previous case. Unfortunately, the load angle required for the new steady state operating point is much more of a difference than in the previous case due to the increased source impedance and the individual angles of the output voltages with respect to the load. In fact, it appears that the phase angle between the rotor of

114 97 synchronous machine and the output voltage vector from the inverter is nearly double the original value in the opposite direction, implying nearly three-fold difference in line impedance. The equivalent synchronous machine impedance between the field and the stator terminals is approximately 0.3pu in this case as compared to the 0.24pu impedance of the microsource which doesn t account for such a difference alone, but the synchronous machine has significantly more resistance than the inverter and therefore has a significant amount of cross-coupling that exists between phase angle and reactive power. Figure 3-44 Mixed microsource system response from a 1pu load to a 2pu load (system maximum)

115 98 Regardless, it is interesting to note the significantly more damped response of the 60Hz component in the reactive power characteristic and the more pronounced 7.4Hz component that is effectively the rotor mass and spring time constant for the rotor inertia as the mass and the coupling impedance as the spring. These oscillations persist through the simulation but do show a gradual attenuation which is expected from the pole location. One of the most interesting results is the nearly equal final value of the overload energy. In this case, the presence of coupled inertia acted to allow the inverter microsource to effectively limit its power by transferring load to the synchronous machine even before the prime mover governor could regulate its speed. In a sense, the inverter microsource is afforded the ability to transfer load to the rotating machine simply by virtue of it being a rotating inertia that has dynamic velocity restrictions due to the energy required to adjust the rotor speed. 3.4 Load Rebalancing Experimental Results Experimental Hardware The experimental configuration consists of two inverters and associated filter components connected together to supply a single load. More specifically, a switched load is placed between the two inverter sources on a bus with negligible cable impedance (12 3-conductor 6AWG). This configuration is displayed in Figure 3-44 where the scopecorder utilized is measuring all values of voltage and current in order to calculate the necessary quantities of power, reactive power, voltage magnitude and frequency for each of the three branches. The voltage measurements might at first appear redundant, but closer inspection shows that they are sampled at three different locations including the two inverter AC filter capacitors and the load resistor terminals as shown in Figure For a more complete discussion of the hardware and

116 99 software configuration, the reader is direct to Appendix A. The serial data links are used for monitoring measured parameters and delivering commands to the inverter such as power and voltage set points, hardware over-current fault resets, output contactor enable signal, and controller gains. Figure 3-45 Power system block diagram and instrumentation The first source is a 3-phase inverter supplied by a 20kW DC supply. The supply can operate with voltage or current limits which are externally variable to mimic the steady-state powervoltage characteristics of a PV cell. In this case, the DC supply was set in voltage mode so that the DC supply dynamics did not affect the frequency droop gain limit investigations. The second source is a battery-based inverter source with a rating of 768V and 10Ah. The pack is constructed of 64 series 12V cells, spread amongst the four battery racks. Each battery rack has two parallel strings of batteries that are connected together at the positive and negative of each of the four racks. This unit is only capable of providing approximately 7kW before the voltage on the battery bank falls below the minimum required voltage for 480Vrms output which limits the power rating of this source. Therefore, the power maximum (Pmax) was set to 6,500W (0.433pu) instead of the 15,000W (1pu) inverter rating, to provide test conditions that

117 100 would yield experimental results that can be compared in a meaningful way to the predicted transient results in the preceding section. Figure 3-46 Components of experimental power system Table 3-7 CONVERTER COMPONENT VALUES PER Figure 3-46 Component Parameter Value Converter rating 15kW, 480V (1pu) Primary filter inductor (L1,L2) 1.3mH Primary filter resistance (R1,R2) ~100mΩ Secondary filter inductor (L11,L21) 5.0mH Secondary filter resistance (R11,R21) ~200mΩ Filter capacitor (C1,C2) 30µF Load resistance 2 parallel 10.5Ω 3-phase Y-connected resistors DC bus capacitor 2,200µF (0.0169pu) Power module Fuji 1200V, 100A 3-phase DC power supply 2 series connected 500V, 20A Magna- Power Electronics Battery 4 series connected APC Surt 192 Volt RM Battery Pack, 10Ah

118 101 Figure 3-47 Battery source inverter and DC capacitors The battery inverter board shown in Figure 3-47 shows the custom inverter board, DC capacitance and other various circuitry that is explained in more detail in Appendix A. At the heart of the control system is a DSPic33FJ128MC706A, a 16-bit microcontroller from Microchip with some DSP function sets. It is configured to run at 40MHz, has two parallel ADC peripherals and four PWM channels. The code used to perform the inverter control was written in C and is provided in Appendix B for reference. One of the serial peripherals is connected to a LabView interface that displays the operational data output from the inverter at approx. 50Hz. The interface also has some command buttons to change set-points and gains. It includes highspeed general purpose output compare (PWM) modules to display the value of internal parameters each control cycle through a digital-to-analog R-C filter. Synchronization of the sources is automatic with a synchronization subroutine in the onboard software. If an inverter is powered up but not yet connected via the output contactor, the inverter will match the phase of the microgrid voltage to reduce the initial power flow when connecting to the microgrid. This method also allows for connection at an arbitrary time without waiting for

119 102 a phase synchronization event that periodically occurs from asynchronous sources. Once connected, the frequency of the inverter is defined by the droop algorithm. The inverter board on the DC supply source in Figure 3-48 is functionally identical to the battery inverter, the only difference being that it is a previous revision of the board that has some traces manually repaired to correct for design errors. It also is connected to a LabView interface, utilizing one computer for each inverter. Also shown in Figure 3-48 is an output contactor positioned between the inverter power module terminals and the primary output inductor. Figure 3-48 Inverter for DC supply source The power sources feeding the inverter boards are pictured in Figure 3-49 and Figure The battery is a series connected set of UPS batteries that nominally delivers 768V, though the operational range extends from 680V to 900V depending on the state-of-charge and loading. Below 680V, the source does not have enough DC bus voltage to produce the necessary voltage magnitude even with zero-sequence injection.

120 103 Figure 3-49 Battery pack (APC 196Vx4, 10Ah) The DC power supplies are connected in series with one of them configured as a slave source, tracking the voltage magnitude of the other supply serving as the master source. The transient current response of these sources was measured to have a response time of approximately 3ms, which is slow enough to influence the dynamics of the inverter. However, the software implements a real-time measurement of the DC bus voltage in order to appropriately decouple these voltage variation effects using the PWM modulation index to achieve an accurate output voltage consistent with the inverter command. Figure 3-50 Magna-Power DC supplies for DC supply microsource 500Vx2, 20A

121 Transient Experimental Results The two inverters of the UW-Madison Microgrid described above were employed to benchmark how similar an actual set of microgrid sources will respond to similar transients where one source is near its limit and the second source with significant margin tracks the load. This transient should help validate the microsource model and also help validate the benchmarked amount of overload energy required in the nominally tuned case. Unfortunately, the transient could not be exactly matched due to power capabilities of the sources available for testing as limited by the second source. Resistance decoupling was utilized in the experimental case here to compensate for some of the resistance of the filter components to raise the effective X/R ratio. The remaining uncompensated resistance in this case is approximately half the original value, 0.9Ω versus 1.8Ω per source originally. This yields a lower X/R ratio of 2.55:1 versus 10:1 in the experimental case. Consistent with previous simulated transients, the load is doubled from approximately half the total system rating to nearly the full system rating. To avoid complete overload in the experimental system, the Pmax variable was intentionally set marginally higher than the selected load banks rated at 5750W (0.385pu, ~90% of Pmax) each.

122 Power 1.05 Source Voltages Power [pu] InvMicrosource1 0 InvMicrosource2 Load Time [s] Reactive Power 0.2 V [pu] Time [s] Source Load Angle 0.15 Q [pu] InvMicrosource1 InvMicrosource Time [s] Load Frequencies Load Angle [rad] Time [s] Unbalanced Energy 0.03 Frequency [Hz] Time [s] E [pu-sec] Time [s] Figure 3-51 Inverter microsource system response from a 0.385pu load to a 0.77pu load, Pmax set to 0.433pu The transient in Figure 3-51 shows very similar characteristics to the simulation of the same system in Figure The primary difference between this experimental characteristic and the simulation is the more persistent 60Hz power oscillation which is due to DC offsets in the phase currents. The experimental system also has more uncompensated line resistance than the simulated case, which is surprising as more resistance should result in greater damping. This fact points to the main discrepancies between the simulation transients and the experimental transients. However, it is fair to mention that persisting 60Hz oscillations in power is likely a simple result of a current sensor offset in the inverter. The relationship between line resistance and damping of 60Hz power oscillations will be investigated further in Chapter 5.

123 106 1 Power 1.05 Source Voltages Power [pu] Q [pu] InvMicrosource1 InvMicrosource2 Load Time [s] Reactive Power 0.2 InvMicrosource InvMicrosource2 V [pu] Angle [rad] Time [s] Power Angle Frequency [Hz] Time [s] Source Frequencies 60.2 InvMicrosource InvMicrosource Time [s] E [pu-sec] Time [s] Unbalanced Energy Time [s] Figure 3-52 Simulation transient to match experimental trace in Figure 3-51, Inverter microsource system response from a 0.385pu load to a 0.77pu load, Pmax set to 0.433pu A simulation was run to mimic the hardware transient of this section. Again, the simulated system is more damped, but the resulting unbalanced energy (over-energy) is similar between them. Where the experimental over-energy was 0.028pu-seconds, the simulation yielded 0.032pu-seconds for this transient of 0.385pu. This simulation of the transient shows reasonable agreement with the experimental data and does help to validate the simulation model. It doesn t conclude on the lightly damped response, but Chapter 5 will investigate this further.

124 Conclusions The source models and system load transients presented here demonstrate the need for faster energy regulation to accommodate the use of grid-forming sources with low amounts of stored energy. It was demonstrated that both an inverter microsource and a synchronous machine source offered nearly equal amounts of support in terms of being able to accept the entirety of the new load in a reasonable amount of time. More importantly, the amount of energy required to survive the transient was significantly higher than the 0.005pu-seconds that are typically available in a converter. This alone points to the need for faster power regulation to support sources of energy with more significant constraints on power flow. Investigations into parameter sensitivity earlier demonstrated where an increase in the power control gains would inhibit the stability of the source response characteristic particularly in a network with low line resistance. A more in-depth look at photovoltaic converters follows this chapter in order to more closely examine the use of a photovoltaic converter in a microgrid, including a photovoltaic array model where the dynamics of the DC supply in this chapter had been ignored.

125 108 4 Comparative Analysis of PV Inverter Controller Performance in a Microgrid 4.1 Introduction Photovoltaic (PV) inverters operating in a microgrid environment combine two significant system constraints: a variable hard limit on maximum power and variable frequency and significant load changes that occur from operating in a microgrid. Most previous applications have dealt with only one of these at a time. Most grid-tied PV inverters have been of the sort that injects a prescribed current in phase with a 3-phase voltage vector (grid-following). Other inverter applications in a microgrid have utilized a droop-style controller topology that varies its frequency proportional to output power (grid-forming). The two controllers associated with these inverters represent two distinct approaches to creating a microgrid-compatible PV inverter and the various performance aspects of each will be investigated in the work presented here. Each inverter controller has specific strengths and weaknesses that narrow its application domain. The power-controlling voltage source inverter (VSI) is the most common style of inverter controller for grid-tied and motor control applications due to its ability to regulate current and tightly control power. However, it is susceptible to conditions where the voltage vector, as measured at the output of the inverter filter, changes angle quickly. Also, in network configurations without a grid-forming source present, the phase-lock loop (PLL) used to track the network voltage vector will rapidly accelerate or decelerate and become unstable. This sensitivity limits power-controlling VSI to applications where there are other sources present to serve as the master frequency source of the network, which can be a significant detriment to the

126 109 reliability of the islanded network considering the n-1 stability criterion typically used in power systems reliability estimations. The grid-forming inverter does not suffer this particular disadvantage as it primarily functions as a fixed voltage source with minor variations on frequency around the nominal frequency (e.g. 50 or 60Hz). This model is identical to that presented in where the exact frequency is determined by measuring the output power and comparing that to the power set-point, thereby decreasing (drooping) the frequency for higher output power. This style of control had been implemented in mechanical form in the early power plants controlling the frequency of large synchronous machines and is still in use today. The concept of decreasing frequency for increased loadings leads to a large degree of stability in a microgrid as it typically serves as the primary balancing mechanism. This is particularly due to the equal loading increases or decreases on each source on a per-unit basis in response to transients, which allows for scaleindependent source parings. The same algorithm operates just as well interacting with the grid supply, which makes this topology particularly stable, especially without the physical inertia and the associated second-order response characteristics that have been an issue with large synchronous machine governors. However, this style of inverter has poorer power-regulation characteristics as it does not control power directly, but rather indirectly by causing power flow across a primarily inductive network from angular differences in voltage vectors between sources. Under these conditions, power follows proportionally to the relative phase angle (power angle or load angle) of the 3-phase voltage vector between other sources and inversely proportional to the coupling impedance between the two sources on the network. The development of this angle comes as a function of time as only the frequency is adjusted, developing the angle as an integral of the difference frequency between the two sources.

127 110 Changing the angle too abruptly will cause significant cross coupling of power and reactive power that becomes more of an issue when the reactance to resistance ratio (X/R ratio) becomes large and limits the power-control bandwidth of the grid-forming inverter controller as discussed previously in Chapter 0. Power-control bandwidth is a significant issue that is important to both source controllers as the sources themselves share the same physical inverter topology. The pertinent issue is the lack of any surge-power capability from a photovoltaic panel and limited energy reserve stored locally in the DC bus capacitors. When a positive load transient causes beyond-maximum power to be drawn from the source, the difference power is drawn from the DC-bus capacitor, reducing the DC link voltage. This causes issues primarily by decreasing the voltage margin between the operating voltage and the peak line-line voltage of the system, eventually leading to uncontrolled reactive power or harmonics from voltage over-modulation. Secondarily the PV power characteristic below the maximum power voltage is unstable as can be seen in Figure 4-2. The maximum power available from the PV array decreases with decreased voltage, increasing the difference between the drawn and available power and further accelerating the rate of voltage collapse. Essentially, if the output power for positive-load transients is not reduced below the PV power characteristic before the DC-link voltage reaches its minimum design value, uncontrolled currents and/or DC-link voltage collapse will occur. Allowing the microgrid to island or sustain local changes in load by preventing such a collapse is highly desirable, regardless of inverter controller type. 4.2 PV Microgrid Overview Microgrids, as covered in Chapter 1, are essentially local power systems that can be connected to the grid or operated in island without significant changes in system configuration. The

128 111 Consortium for Electric Reliability Technology Solutions (CERTS) group has developed a set of guidelines for the microgrid configuration outlining the need for plug-and-play integration to reduce integration engineering cost per source which means that all sources should be capable of operating using local information only. This implies that each source will not know if it is in a system that is grid connected or islanded which then requires that each controller topology have a control scheme that can operate in a grid-connected as well as in an islanded network without the ability to make assumptions about the presence of a grid connection. Unplanned microgrid transients include two basic varieties: load transients and islanding/reconnection transients. Each transient results in a phase-angle step-change in the local voltage that re-distributes power flow and is followed by a change in the average frequency for islanded operation. By natural phenomena, the proportion of the load transient absorbed by each source is proportional to the local admittance of each source versus the total admittance of the network from the perspective of the point of the transient in an islanded configuration. However, in a grid-connected configuration, each source is fairly unaffected by load changes as the grid supply serves as an effectively infinite source. The previously mentioned local voltage angle change still occurs when grid connected, but each source is allowed to adjust for the phase angle change in the network though their respective controllers to re-establish power back to the power set-point coincident with grid frequency(~60hz). Coincidently, grid-connected load transients are significantly suppressed by the grid connection and leave very little to cause issues or microgrid responses of significant magnitude. It is for this reason that grid connected load changes are not investigated here. Alternatively, island transients and load transients are similar in that the equivalent increase or decrease in load is precisely what the power was flowing across the static switch prior to the

129 112 island event. This is why there will be little distinction drawn between islanding transients and islanded load transients for the remainder of this section. Lastly, the reconnection or resynchronization event is less dramatic and presents less of an issue to investigate. As the switch is closed at near zero phase angle to re-synchronize the network, a near-zero power transfer results initially due to the small difference angle and power increases or decreases smoothly in proportion to the power set-point error. The resulting source response results in each source settling back to a new steady-state power flow due to the power-frequency droop characteristic with significantly low dynamic content. These transients represent the basic set of events that any source on the microgrid will have to endure. Therefore, these transients, except the resynchronization transient, will define the primary range of tests from the perspective of the microgrid for qualitative comparisons between the two converter controller topologies. 4.3 PV Cell Overview A compounding factor, beyond the transient microgrid environment, that forms the basis for this investigation is the power characteristic of the photovoltaic cell and the lack of any short-term over power capability that is typically found in other sources like generators with physical inertia or chemical batteries. Further, even though short-term over-power is available in the aforementioned non-pv sources, the amount of energy is limited by physical constraints. Typical limitations include flywheel kinetic energy, stored chemical energy or one of the many system thermal constraints like engine exhaust temperature or battery operating temperature. Nevertheless, photovoltaic systems are typically operated at maximum power and are therefore highly susceptible to overload following positive load transients. Unfortunately, the task of regulating the AC output power in coordination with the maximum power of an attached PV

130 113 array cannot be avoided for two reasons: The economic case for photovoltaics is made using the assumption of maximum solar resource utilization. Secondarily, the power converter must be able to ensure the lowest likelihood of a voltage collapse or converter fault scenario by regulating the DC bus voltage to an acceptable range. Consequently, this chapter focuses greatly on the power- regulation capabilities of each controller configuration. The power characteristic of photovoltaic cells is very similar to a voltage and current limited power supply with some source resistance and non-linear rounding particularly around the corner of maximum voltage and maximum current, as can be seen in Figure 4-1. Figure 4-1 Voltage and Current characteristic of a Mitsubishi solar cell for various temperatures and radiant intensities Figure 4-1 also shows how the radiant intensity has a near direct proportionality on maximum current as well as how the equivalent open-circuit voltage is defined primarily by the cell temperature. This data was developed from a model whose terminal characteristics were

131 114 matched against the data in the datasheet of the Mitsubishi 150mm solar cell with good agreement between the two. It is also interesting to note how the effective source resistance, as evidenced by the voltage slope around low currents, is inversely proportional to the radiant intensity of photon radiation (insolation). Figure 4-2 Maximum power characteristic for various insolation intensities and cell temperatures Figure 4-2 shows the power characteristic of the cell under the same conditions as in Figure 4-1 with the inclusion of maximum power point indicators. It is evident that the temperature is the primary driving factor in determining not only the effective open circuit voltage, but the maximum power point voltage as well. In these testing conditions, a variation of maximum power point voltage of 46% was noted in the range of 200W/m 2 to 900W/m 2 and -20 C to 60 C Selection of Number of Series Cells to Match Design Voltage The full range of temperatures investigated occur in places like Canada and Alaska and poses significant design constraints on the voltage rating of the inverter devices versus the minimum operating voltage defined by the system AC voltage. For example, if the inverter is placed on a

132 V network, the DC bus voltage must be at least 5% above the peak line-line voltage to effectively regulate the microgrid voltage to a nominal level. This implies a minimum voltage of 712.8V, which can be defined as the minimum 60 C maximum power point voltage. The highest maximum power point voltage then would be 1040V, which would necessitate 1600V devices in a 2-level inverter to remain in the safe operating region considering bus voltage ringing. Secondarily, if the source is not utilizing all of the input power due to light system loading in an island scenario (source shedding), then measures must be taken to mitigate the lightly loaded DC bus voltage or extend the allowable DC bus voltage range even further. In the investigation here, it is assumed that the DC bus can handle the lightly-loaded voltage. As mentioned in the introduction, operation of the system below the maximum-power-point voltage is an unstable power region and it is the job of the controller to retain stability, a topic explored further in the following sections. Conversely, DC voltage above the maximum power point voltage creates a highly stable operating region due to the inherent voltage limiting characteristic and negative correlation of voltage to power. 4.4 Controller Topologies and Implementations This section will present the controllers mentioned in section 4.1 and explain the controller configurations of both options in detail. The control block diagrams will be presented and the details of each block will be investigated individually. Both inverters are physically configured in an identical manner. They both have the PV array attached directly to the DC bus of the inverter as indicated in Figure 4-3.

133 116 Figure 4-3 Direct-tied PV array configuration utilized for the work here. The PV array is directly tied to the DC connection of the inverter and the AC terminals of the inverter are used to transmit the power to the network through a filter network to reduce the presence of voltage and current harmonics from the inverter switching. The output filters are also identical, configured as an LCLX filter as described in Figure 4-4. Finally, the voltage and current measurements are consistent with Figure 4-4 as well, measuring the voltage on the line and the current from two of the phases of the converter. Only two measured currents are necessary as the current in the third leg can be inferred from the opposite of the two measured currents as there is no ground reference here. In some cases of photovoltaic installations, a ground tie for the negative DC bus is required by safety standards [61]. In such a case, a third current sensor would be required to segregate ground currents from powertransmission current, as well as to sense ground faults. Figure 4-4 LCLX filter configuration for PV microgrid inverter

134 117 The PV array was chosen to be in a directly connected configuration for implementation simplicity. Other options include a boost or buck converter connected to the array, which could effectively decouple the voltage variation to the inverter, but represents a further degree of freedom to be explored in later work. Such a configuration would also eliminate the operation in the unstable region of the PV input to the decoupling inverter, but would not significantly mitigate the occurrence of inverter DC-bus collapse. If it can be assumed that the voltage on the DC bus in response to reasonable system load transients will remain within the region of local maximum on the PV power curve, then this will justify the omission of the decoupling converter here. The LCLX output filter was implemented and proven in earlier microgrid work [1] for its harmonic suppression stage (LC) on the inverter terminals, coupling inductance and isolating transformer. While the primary LC stage provides some coupling inductance, the majority of the impedance between adjacent sources is in the secondary inductor and leakage inductance of the transformer. Though the primary filter is particularly adept at mitigating harmonics, it poses a significant challenge when implementing a current regulator through the filter, which will be explained in Section Minimal sensors were utilized here to reduce implementation cost. Voltage sensors were placed at the output terminals of the filter to adequately measure the line voltage for line voltage regulation/reactive power injection. Also, current sensors were placed at the terminals of the inverter to serve the dual role of measuring power and regulating inverter device current in the event of a fault condition. As will be explained later in this section, additional calculations are required to virtually co-locate the voltage and current measurements by decoupling voltage drops and reactive capacitor currents. These calculations assume constant filter values and are

135 118 therefore sensitive to parameters. However, the scope of the work presented in this section does not address filter parameter sensitivity Grid-Following Power Controller The power controller is, as the name implies, designed primarily to regulate power at the terminals of the inverter. It employs a droop algorithm similar to the original implementation in coal plant steam turbines, where the frequency feedback changes the power input to the system. The output voltage vector is tracked by a PLL and the associated frequency is correlated to an output power through the droop slope adopted by the other sources in the network. However, one primary difference between this implementation and that of a large synchronous machine is the lack of physical inertia. This causes rapid changes in frequency and loss of voltage tracking when the entire system is without inertia such as when operating in an island as the only source and assuming a primarily resistive load. This style of controller employs synchronous-frame proportional-only current regulation loops with state-feedback decoupling on inductor voltages and capacitor currents. The controller was designed in the synchronous frame to easily validate its operation as all of the control quantities are constant in steady-state. Nevertheless, an observer must be employed to obtain an estimate of the line current, the desired control variable, because only the pole current is measured. The same observer is used to develop an estimate of the capacitor voltage, a value used in the command generation block.

136 119 Figure 4-5 Power controller block diagram Figure phase PLL To convert the control variables into a synchronous reference frame, the phase angle of the voltage is tracked so that the q-axis voltage is at an angle of theta (θ). The result is a q-axis voltage that has a magnitude equal to that of the voltage vector and a near-zero value of d-axis voltage, which can also be used as the error signal. Under the assumption of the small angle approximation, the ratio of d-axis voltage to q-axis voltage is the radian value of phase angle error, a fact which is used when selecting PLL gains. The PLL not only develops a phase angle for the reference frame transformation, but the angular velocity of the reference frame as well. This, along with the q-axis voltage, is used to generate the current commands for the current regulation stage.

137 120 Figure 4-7 Complex power calculation and current command generator Using an inverse of the standard droop algorithms outlined in [1], the power can be commanded from the operating frequency and the reactive power can be calculated by the system voltage magnitude, as presented in Figure 4-7 (Kp used in place of Mp elsewhere in the document). In both cases, the inputs are compared to the nominal values. In the case of the frequency, this value is always fixed, but in the case of the voltage set-point, there are some conceivable circumstances where the nominal voltage set-point may be changed. These error signals are related to the associated delta power and delta reactive power commands by the power base divided by the respective droop gains. These so-called delta-power and delta-reactive power values are summed with the respective set-points to generate the raw command. In most cases, the Q_set value is zero, but the P_set value determines the power at nominal system frequency of the grid. The raw commands are filtered through a limiter and the results are divided by the line voltage magnitude to develop the associated current signals. Under heavy loads, the limiters are critical to maintaining safe operation of the inverter. Secondarily, the power limiter includes the job of tracking the maximum power available from the solar panels by way of regulating the DCbus voltage. As will be seen, the process of regulating maximum power in the grid-forming controller is slightly more complex as it requires the use of a PI controller.

138 121 Figure 4-8 q-axis Command Generator and Current Regulator The pole current command generator is a complex component of the power-controller topology, though it should be mentioned that this is not the only method to achieve the desired output. The approach adopted here is the most physically meaningful method. Being that the LCL component of the filter is lightly damped, regulation of the pole current (the current at the output of the inverter) alone would only serve to inject current into filter capacitors during fast transients. This has been shown to result in large, lightly damped, and possibly unstable voltage oscillations at the filter capacitors. This mode was seen previously in Chapter 0 as the L-C Mode. To avoid exciting this mode, the capacitor voltage must be regulated directly in order to develop the proper output current characteristic and achieve the ultimate goal of output power regulation. The command generator is formed as a three-state cascaded proportional controller. Decoupling is included to significantly reduce the necessary gain, increase the linearity of response and reduce steady-state error. In each state-section, separated by color, is a set of three components: an error signal multiplied by a proportional gain, a feedforward value from the adjacent state and a decoupling value that should separate the commanded state from the adjacent state. If the estimated values, designated by the _hat suffix, are exactly correct, then the steady-state error

139 122 should be zero. However, in real systems, the estimates always contain some error, which is reduced by the proportional gain from each stage acting against said error. Primarily, the proportional controllers are used to cause first-order convergence of the respective controlled states. Integrators were not used in each stage as cascading integrators would significantly complicate the dynamics of the command generation process, increase lag, and generally decrease stability. A single integrator was added in to the maximum power-point tracking block to account for estimation errors and ensure the DC-bus voltage would converge to the commanded value. However, this integrator is limited to 0.1pu to avoid unnecessary wind-up for source-shedding conditions when the load of the system was not enough to absorb the maximum PV power available. Figure 4-9 q-axis Capacitor Voltage and Line Current Observer The final component of the power controller is the line current and capacitor voltage observer. To limit the number of sensors required to implement the power control scheme, an observer is employed as a sensor replacement. The model displayed in Figure 4-9 displays the q-axis component of the observer which is effectively a synchronous frame model for the output filter.

140 123 This model primarily relies on a feedforward value from the commanded pole voltage to drive the change of the states. The measured pole current versus the observed pole current is utilized as the main tracking mechanism that corrects for estimation errors in the observed states. Each state is modeled as an integration process that includes synchronous frame decoupling components for resistance as well as frequency-dependent inductive voltages and capacitive currents. As will be explored later, this controller topology demonstrates superior current regulation abilities when tracking of the system voltage vector. However, its applications for islanded operation are restricted as self-definition of operating frequency has led to unstable operation in systems without significant physical inertia to slow the frequency variation dynamics. Therefore, to ensure stable operation this style of controller must have another master frequency source or grid-forming source present Grid-Forming Droop Controller The grid-forming controller, referred to later as the usrc (microsource), is the more traditional solution applied to microgrid inverter controllers. It also represents the dual to the powercontroller topology because in this case the power is the measured value and frequency is the commanded value. Secondarily, the droop controller is simpler because frequency is a variable that an inverter has direct access to without the use of complex cross-coupled control schemes as opposed to the grid-following case. The most complex calculation that this style of controller employs is the calculation of real and reactive power which doesn t require transformation into the synchronous reference frame. By this measure alone, it would seem that this is a preferential method of implementing an inverter controller, but its power control scheme requires a PI

141 124 regulator and is inherently limited in bandwidth by power-frequency oscillations. These drawbacks create issues with properly managing the stored energy on the DC bus. Figure 4-10 Droop controller (usrc) block diagram Figure 4-10 gives an overview of the calculation steps required in the droop controller. After voltage and current is collected, the voltage magnitude, real and reactive power are calculated. Only the real and reactive power values are used to calculate the new values of operating frequency and voltage magnitude. However, the measured voltage magnitude is used to develop a voltage magnitude command that is fed to a PI regulator in the voltage command regulator block. The power calculation from stationary-frame d-q quantities is as follows: =3/2 + (4.1) The reactive power is calculated by: =3/2 (4.2) And the voltage magnitude is calculated as follows: = + (4.3)

142 125 Prior to the calculation of states, the voltages and currents are filtered for their positive sequence component which mitigates the feedback of unbalanced phase loading and voltage excitation. At the heart of the positive sequence filter is a dual-resonant loop that increases each d or q component proportional to the magnitude of the leading quantity and its velocity. Essentially, the filter automatically rotates the voltage vector the expected amount based on the nominal frequency of the system. This value is then compared to the new measured value and the filtered value is modified proportional to this error. Figure 4-11 Positive sequence filter topology used in droop controller The output frequency calculator is a simple function that forms the heart of the power balancing and power limiting functions of the grid-forming controller in an AC microgrid. Under the assumption of a primarily inductive network, this function will change the output frequency of its waveform which will advance on or fall back with respect to the rotational angle of other sources. Advancing on other sources increases its own power output or decreases its absorbed power and vice versa. Since this controller measures the output power and adjusts the frequency accordingly, the rate of advance or retreat is based on the relative power error between the

143 126 sources. The power error here is described as the difference between the measured power and the set-point power and the same definition holds for the other sources. Figure 4-12 Frequency calculator with droop function and power limit controller Power error results in operational frequencies which differ from nominal, which is plainly visible from Figure Since differing power errors will result in different frequencies, the behavior is transient and will cause a converging behavior that settles to a steady-state point where the power errors are not eliminated but equal when scaled by the frequency droop gain. This fact is one of the key components to ensuring a highly stable power system, utilizing equal load sharing where steady state response to changes in load establishes equal per-unit responses on all loads. The caveat to the previous statement is when the power limit for a given source has been reached. The remainder of Figure 4-12 illustrates the simple PI control of the frequency with normally-nulled outputs and anti-wind-up limiters on the integrators that stop continued frequency deviation significantly outside of the allowable frequency operating range. Once the power limit is breached, the frequency will decay at a rate proportional to that overload. In the

144 127 event that another source on the network has not yet reached overload, the decrease in frequency from the overloaded source will cause a relative change in the load angle. This results in more power output from the remote source and a return to the power limit for the previously overloaded source. Secondarily, in the event that multiple sources reach an overloaded condition, the frequency will droop faster in the source that is more overloaded and will equalize the per-unit overloading for each source. This inherent balancing even in the case of complete system overload is a testament to the simple and effective nature of this naturally power-sharing network where even the amount of overloading is shared and the maximum degree of overload is reduced to its minimum possible point automatically. Figure 4-13 Voltage command regulator The remaining component of the grid-forming controller to be explained is the voltage command regulator. This block will adjust the voltage reference by a value proportional to the measured reactive power which is indicative of output voltage magnitude differential in a highly inductive network. The inclusion of droop effectively strikes a compromise between the output voltage and the injected reactive current. In a stiff network, the output voltage will not be affected by this controller, but the resulting reactive power will then change the voltage command to match the system voltage. When two microsources are placed in close proximity to each other, this compromising effect allows the support of reactive power loads without unnecessarily transmitting large amounts of reactive power between sources.

145 Controller Topology Performance Indicators and Results As previously discussed in sections 4.2 and 4.3, island transients and islanded load transients represent the significant transients that test the transient capabilities of a microsource with limited-overload capabilities in a microgrid. For these transient conditions, the change in demand occurs nearly instantaneously, which represents a worst-case rate that load can be applied. Secondarily, fast change of input power from sudden cell shading can occur for various reasons and though this typically occurs on a potentially fast but variable basis, the worst-case condition of a step-change in input power capability will be investigated. The second component in this investigation that creates a potentially unstable point is the lack of surge power available from a PV source. As PV panels are expensive and the utilization of solar energy is free, it makes financial sense to operate PV systems at their maximum power point to generate the greatest economic and ecological impact. However, because the power is at the maximum, any excess power drawn from the source comes from the only remaining source of stored energy, typically the DC-bus capacitors. There are a few main requirements for a PV inverter to operate in a stable fashion in a microgrid. Primary amongst these are the following capabilities that test the range of transient conditions common to a PV source in a microgrid: 1) Tracking maximum power from the PV panels 2) Withstanding quick input power changes 3) Withstanding islanded load changes 4) Withstanding island events and reconnection events 5) Automatically reducing output (source shedding) in an islanded scenario when local load is light 6) Withstanding island into overloaded condition and load shedding event

146 129 The following section further investigates the response to these transients. To adequately compare the performance of each inverter controller, two identical transients were performed on each of the controllers in each result section. First, the transient was performed with the power-controller and next with the droop-controller. The results were overlaid on the same time scale to illustrate the relative response characteristics for comparison. The synchronous frame and scalar quantities are utilized in place of AC waveforms to give more resolution of the characteristics of the events. Aside from power and reactive power, the other main variables to inspect are PV array/dc bus voltage and the instantaneous source frequencies. The PV source will be accompanied by an energy storage unit in most of the following transients presented in this section. The energy storage element has a droop controller identical to the usrc controller presented in section 4.4.2, and has the fixed Pmax and Pmin of 15kW and -5kW respectively. The limits of the energy storage unit will be left unexplored to eliminate unnecessary variables Grid Connected Step Insolation The first transient to explore investigates power-tracking response when the solar insolation changes instantaneously. This situation could occur naturally if a part of a parallel array suffered a blown fuse, but would more likely occur from a sudden shading event like an airplane fly-over or a longer-term shading event from a cloud or tree. As previously mentioned, this baseline case of step-change in input power illustrates a single-worst case scenario and best illustrates the difference in response characteristics between the controllers.

147 Grid Connected Step Insolation from 750 to 500 W/m 2 The first transient will be a step-reduction in solar insolation while grid connected. The stiffness of the grid connection helps illustrate the power-fluctuation response characteristics without the interaction of other controllers. Figure 4-14 illustrates the power response characteristics from the step-decrease in power input. It can be seen that the post-transient power characteristic settles to the same value for each controller topology, indicating that the power available was approximately 11.5kW prior to the transient and 7.5kW post-transiently. Figure 4-14 Power and Reactive Power for step-decrease in insolation from 750W/m 2 to 500W/m 2 The major point of comparison is the rate of response of both controllers. Though they both share similar magnitudes of overshoot in power, the power-controller changes its power to the

148 131 new value in 10ms, whereas the droop controller takes approximately 25ms. Both exhibit similar damped oscillatory responses, but the droop controller exhibits a multi-frequency response. The higher frequency component is at the power frequency (60Hz) and directly illustrates the limiting component to the power-control behavior of the grid-forming controller. This limitation comes inherently from the disparity between the transient and steady-state characteristic of changing a voltage vector on one source versus another source and was referred to previously as the voltage balance mode. Under the assumption of an inductive network in steady-state, real power is proportional to the differential voltage between source voltages that is orthogonal to the source voltage vectors. Also, reactive power is proportional to the difference voltage in-phase with the source voltage vectors. Transiently, this relationship is reversed and damps out in relation to the resistive nature of the interconnecting impedance with respect to the inductive nature of the same impedance. It has been shown that the oscillations take a longer time to damp out as compared to cases with a significant resistance component. Nevertheless, because the frequency of the output voltage and the magnitude of the output voltage do not have a cross-coupled link, there are no opportunities for the droop controller to actively damp out this oscillation, whereas the power controller does this inherently. An overall point to draw from this is that the droop controller must have a closed-loop power control bandwidth less than the settling frequency of the network components. Conveniently, the grid-connected state causes the highest feedback gain, illustrates the fastest rate of response and therefore demonstrates the worst-case power/reactive power oscillations in response to a transient. These gains were kept constant through the entire set of testing, which should be more damped than the grid-tied case. The reactive power illustrates the same oscillation in the case of the droop controller power as seen in the case of the power response for the droop controller. Also, the reactive power

149 132 convergence in the case of the power controller is similar to the power response of the power controller. Both cases differ from each other pre-transiently and post-transiently because the reactive power measurement of the droop controller is measured at the line whereas the powercontroller variety measures the power into the filter from the inverter. Therefore the power controller reactive power measurement includes the reactive power absorbed from the capacitor and is lower than the measured value by the droop controller. Figure 4-15 Array voltage and Inverter Frequency for step-decrease in insolation from 750W/m2 to 500W/m2 Figure 4-15 illustrates the remainder of the transient conditions that resulted in the power response in Figure It can be seen that both the voltage deviation and the frequency deviation for the droop controller are significantly greater than the power controller due to the inherent limit on power-convergence speed. Though the frequency of the droop controller changes within 4ms, the maximum phase angle does not develop until 25ms. This result can be

150 133 noted by the point at which the lowest DC-bus voltage occurs on the DC bus of the grid-forming inverter. It is at this point where the power balance is restored momentarily and the further decrease of the DC-bus is eliminated. However, full bus voltage recovery does not occur until approximately 30ms after the halting of the bus voltage decay. In the case of the power controller, the DC-bus voltage decays only 0.25% versus 1.8% in the case of the droop controller, approximately seven times less resulting disturbance. This fact illustrates the superior power response of the power controller. For a larger value of transient, the DC bus voltage would change proportionally in both cases. With a transient large enough to change the DC bus voltage significantly, the power capability diminishes approximately proportional to the percentage change in voltage. However, it does not appear that this case is likely with realistic load changes as this transient was 0.25pu insolation change and a full 1pu insolation change would only lead to a 1% and 7.2% change respectively, which is not significant enough in either case to suggest eminent voltage collapse Grid Connected Step Insolation from 500 to 750 W/m 2 The counter case to section is presented here, where the insolation is increased. This condition is notably less critical, considering DC-bus surge, not collapse, is the typical result of an increase in insolation. However, the power-control response is still investigated here as the grid-connected scenario is the best for investigating just that. Conversely, while in an islanded condition, the ability to control power to a specific value is sometimes constrained by the power limits of other sources and the dynamics become coupled with that of the other source. Therefore, this case provides a good baseline to investigate the best-case scenario for investigating the power control response.

151 134 Figure 4-16 Power and Reactive Power for grid connected step-insolation from 500W/m 2 to 750W/m 2 It can be seen from Figure 4-16 that both the power and reactive power response seem to be nearly opposite of Figure 4-14 in section However, the primary difference is that the grid-forming controller appears to take measurably longer to settle into its final operating point, 130ms vs. 85ms in the previous case. This is primarily due to the power limit of the source, set nominally at 20kW, being closer to the post-transient steady-state operating point. Once the power-limit is set to its maximum point, the frequency quickly increases to the intersection with the linear droop slope which serves as a frequency limit and increases the DC-bus overshoot and settling time. The power controller exhibits very similar response characteristics in both directions with similar settling times as previously observed in section

152 135 Figure 4-17 Array Voltage and Inverter Frequency for grid connected step-insolation from 500W/m 2 to 750W/m 2 Figure 4-17 shows a significantly higher deviation in DC-bus voltage, ~3% in the case of the droop controller, but an expected 0.25% for the power controller. Also, the chopped frequency characteristic from t=0 to t=90ms illustrates the effect of Pmax hitting the system limit, and the frequency resulting from the linear portion of the droop curve, not the power limiter gains. Under small perturbations where the Pmax value does not hit a limit, the settling time for the grid-forming controller is approximately 60ms in either direction, which is still significantly slower than the ~15ms for the power controller.

153 Maximum Power-Point Tracking While Grid Connected The next capability of a PV inverter to inspect is the ability to track maximum power. Though this trait is not only limited to a microgrid application, the controllers presented here are evaluated on all microgrid configurations that include grid connected and islanded operation. Both controllers have identical maximum power-point tracking (MPPT) controllers which are very basic in topology. Upon initialization, a conservatively high DC-bus voltage command is generated from the MPPT controller. This command is a sub-maximal power point but a stable operating point. The Pmax value is modified slightly higher or lower than the output power to change the output power and regulate the DC-bus. Once the error in the voltage measurement is significantly low for a brief period of time, the MPPT controller assumes that a steady-state power operating point has been reached and a modified voltage command changes the commanded DC-bus voltage. Previous and current measurements of power are compared to determine if the direction of the voltage command perturbation is correct. The system will eventually continue selecting voltages around the maximum power point voltage, indicating that the MPPT controller has found a local maximum in the power characteristic.

154 137 Figure 4-18 Power and Reactive Power during MPPT while grid connected In Figure 4-18, the power and reactive power characteristics are somewhat flat except for the small perturbations every half second or so. Each perturbation results from a change in the DC bus voltage set point from the MPPT algorithm. The droop controller causes slightly longer and flatter deviations in power as compared to the power controller. However, both achieve their goal of changing the power to regulate the DC-bus voltage to the new command. Figure 4-19 indicates the tracking process for both controller topologies. The array voltage can be seen oscillating around the 1.01 pu voltage level for both controller topologies indicating the functionality of the MPPT algorithm in both cases. This demonstrates the functional similarities between the controllers with only the response time and damping coefficient separating them. Interestingly, the response characteristic of the droop controller appears faster due to the

155 138 overshoot, but judging by the frequency of the perturbations, the power controller still regulates the DC bus in less time, as expected. Figure 4-19 Array Voltage and Inverter Frequency during MPPT while grid connected Maximum Power-Point Tracking While Islanded In an islanded condition, the environment may vary from single-source to many-source, the latter of which representing an environment similar to a stiff grid connection. The environment here with a single other source represents a middle-ground where the ratio of source power scale to the rating of the remainder of the network is 1:1. This section demonstrates the MPPT functionality in an islanded configuration which varies only slightly from the grid connection. However, because of the effective compliance of the microgrid as compared to section 4.5.2, the

156 139 frequency stiffness is reduced and will induce a slightly less damped response of the DC-bus voltage regulation. Figure 4-20 Power and Reactive Power during MPPT in island A similar power and reactive power characteristic can be seen in Figure 4-20 as compared to the previously presented Figure 4-18 with the addition of slightly higher reactive power oscillations that result from the weaker connection between the sources and generally less stiff connection. These oscillations foreshadow the inability of the power controller to island alone. The droop controller, on the other hand, appears the same without a distinguishable difference here.

157 140 Figure 4-21 Array Voltage and Inverter Frequency during MPPT in island The figure presented here bares resemblance to Figure 4-19 except that the droop-controller DCbus voltage trace can be seen to exhibit less damping characteristics as compared to the previous case. This is due to the ability of the other source to droop in frequency in proportion to power being drawn from it. However, a high degree of damping and convergence to steady state is demonstrated in both cases. Functionally, the MPPT algorithm can be seen in action here performing well, similar to the previous grid connected case, indicating that the presence of a grid connection only modifies the tracking behavior slightly.

158 Load Change in Island with Energy Storage (ES) Unit In an islanded microgrid, load changes are common and significant with respect to the rated capacity of the system. Load changes are not regulated by any master controller and are an unpredictable inevitability. Therefore, stable responses to positive and negative load changes in an island are instrumental to a reliable microgrid. Particularly in the case of the PV inverter, this scenario is troublesome. Nearly instantaneously following a transient, the change of load is distributed to all of the sources by a factor proportional to the coupling impedance between the load change and each source. This causes drastic changes in output from a primarily fixedpower source that need to be properly counteracted to maintain its own safe operation. However, by maintaining constant power locally, the entirety of additional load is shifted to the other sources in the microgrid. Therefore, the adjustable source power is a smaller percentage of the local power generation mix and effectively serves to destabilize an otherwise reliable and transient-tolerant network. However, it should be noted that the degree to which destabilization has occurred is a function of the size and direction of the load transients on the network as compared to the power margin of adjustable power sources on the network. Essentially, stability in a microgrid is primarily a function of power limits which will not be addressed in this section, but will be investigated in following sections. For sections and load will be changed in an islanded configuration and the PV source will maintain its power output as no other source limits are encountered Load Increased from 11.8kW to 18.5kW In this section, a load increase will be investigated in an islanded configuration. Figure 4-22 shows the power and reactive power traces which appear more harmonic-rich and lightly damped as compared to the single-source power changes presented in section This result was

159 142 foreshadowed in the islanded MPPT characteristics where it was noted that the relative compliance of a microsource as compared to a stiff grid connection increased the feedback gain of angle and power. In fact, because microsources can operate in steady-state deviating from the nominal system frequency, the relationship between angle deviations and power has been decoupled entirely. Therefore, an islanded load transient has both a proportional and integrated component for adjustment of power. Figure 4-22 Power and Reactive Power for islanded load increase from 11.8kW o 18.5kW Nevertheless, each source manages to control power in this scenario, each by a different method. In the case of the power controller, it accounts for the steady-state frequency deviations by adjusting its own synchronous reference frame. The droop controller employs a PI regulator for the maximum and minimum power presented in Figure 4-12 which results in zero steady-state power error for this second-order system.

160 143 Figure 4-23 Array Voltage and Inverter Frequency for islanded load increase from 11.8kW o 18.5kW The DC-bus voltage and source frequency characteristic bare resemblance to the insolation step change presented in section which caused a similar drop in DC-bus voltage in the case of the droop controller. The lower limit on the frequency can be seen affecting the droop controller for the first 60ms following the load transient, indicating correctly that the source was operating in the linear region of the droop characteristic. On the other hand, the power controller frequency deviations are small and therefore do not reach a limiting edge. Interestingly, both controllers exhibit similar increases in settling time, approximately double for each. Also, as might be expected, the damping ratio for all components seems to have been cut in half as well, exhibiting traces with twice as many noticeable oscillations as in the previous grid-connected state. Though, one measure of performance that does not scale in this case is the

161 144 decay in the DC-bus voltage. In each case, the DC-bus voltage dropped a similar value with respect to the load change distributed instantaneously to the PV sources as compared to the change in solar insolation. This indicates a relative insensitivity to islanded or grid-connected configuration, assuming other variable-power sources are connected to the network. As a point of clarification, the voltage change in Figure 4-23 at 185ms is due to the activity of the MPPT algorithm Load Decreased from 18.5kW to 11.8kW The contrary case for islanded load change is presented here. Based on the relative insensitivity to transient polarity, the response characteristics should be similar in time scale to the load increase versus the load decrease presented here. Figure 4-24 Power and Reactive Power for islanded load decrease from 18.5kW o 11.8kW

162 145 Similar to the increase in load case, the power controller exhibits more oscillatory and lighter damped response characteristics. This transient is less critical in terms of time of response due to the swell in DC-bus voltage versus the decay that could potentially cause issues. Also, this transient would not typically exceed the safe operating area of the converter as the maximum power point voltage of the PV array varies due to temperature swings by a greater magnitude than the voltage variations seen here. Figure 4-25 Array Voltage and Inverter Frequency for islanded load decrease from 18.5kW o 11.8kW Figure 4-25 illustrates a swell in magnitude of DC-bus in the case of droop controller that is significantly higher than the power controller. This is an expected result as previous traces have shown similar characteristics. However, it is also interesting to note that in the case of the power

163 146 controller that the deviation in bus voltage is significantly greater in reaction to a voltage perturbation command than any swell that resulted from the decrease in system load Island When Importing Island events represent the other concerning transient especially considering the change in power flow is nearly instantaneous, leaving the difference to be supported by local sources only. This configuration potentially creates overloads, particularly considering a PV source while tracking maximum power previous to the island event. The first island transient explored here is when a PV source and an energy storage (ES) source are connected in the microgrid that was utilizing the grid connection to support some of its onsite load (importing) prior to the island transient. Islanding from an importing state implies that the on-site sources will have to compensate for the loss of the grid power and decrease the system frequency to the new operating point.

164 147 Figure 4-26 Power and Reactive Power during island event with ES source, previously importing In this case, the PV source stays at the same power output and the energy storage source makes up for the difference entirely as the PV source is tracking its own maximum input power. The entirety of the load transient was originally 2kW per source, but is shifted entirely to the ES in 80ms for the droop controller source and nearly instantaneously for the power controller source. The response behavior is similar to the previous transients as well as the frequency and DC-bus voltage characteristic which indicates predictable response behavior under various other conditions.

165 148 Figure 4-27 Array Voltage and Inverter Frequency during island event with ES source, previously importing Island When Exporting The counter case to that presented in section is the exporting case. This simply implies that the nominal power output of the sources while grid connected is greater than the local load and utilizing the grid connection to export power to the grid at large. The amount of power exported in this case is similar to the amount of power imported in section which makes this a fairly good comparison and would highlight any polar characteristics if they were to exist. In fact, the response is nearly identical in response characteristic, just opposite polarity, when compared to the previous case in section This indicates that islanding into conditions without power limitations on the remaining network components is relatively stable and predictable. In later sections, it will be explored how interacting with source limits changes the behavior of an island transient.

166 149 Figure 4-28 Power and Reactive Power during island event with ES source, previously exporting Figure 4-29 Array Voltage and Inverter Frequency during island event with ES source, previously exporting

167 Island While Exporting, Then Source Shedding. A significantly higher degree of exporting can result in an islanded system with less total load than available PV power. In the example presented in this section, the energy storage source is limited in its ability to sink power, acting as a controllable load, to 5kW. Since the total load including the energy storage element totals approximately 6.5kW, the 7.5kW available PV power will have to be decreased by 1kW. This process is referred to as source shedding and will be demonstrated here through an island transient. Figure 4-30 Power and Reactive Power during island event, previously exporting, source shedding The power characteristic illustrates an initial decrease in power from each source, which is expected from a system that was previously importing power. In either case presented here, the power into the energy storage source exceeds the minimum power initially and after the transient is then limited to the maximum power output by its own power limit controller. The total energy

168 151 absorbed by the energy storage element in this case raises the frequency to cause source shedding in the other sources on the network. In this network configuration, power beyond the minimum power (maximum charge power) is required to allow the frequency to change and cause source shedding. Whether this energy could be transiently absorbed by the energy storage element or whether it would need to be dissipated in a braking resistor is defined by the individual requirements of each source. Figure 4-31 Array Voltage and Inverter Frequency during island event, previously exporting, source shedding Although the process towards source shedding begins the moment the energy storage element begins accepting more power than set by the limit controller, said controller will have to integrate this error to completely register a response in frequency that will cause a sourceshedding event from the PV source. In both cases, the entire source shedding event takes approximately 200ms.

169 152 The DC-bus voltage characteristics appear significantly different between the controllers, which comes initially from the power regulation capabilities of each. The power controller initially is able to regulate power and maintain the DC-bus whereas the droop controller of the PV source begins source shedding almost immediately. The act of source shedding disengages the power regulation abilities of the droop controlled PV source as the energy storage element goes in to a power limited mode. This changeover of power control does not occur in the case of the power controller until 150ms after the transient when the frequency sensed by the power controller PV source is above the minimum nominal frequency for the given input power available from the PV panels. The DC-bus voltage of both PV sources begins increasing due to a disparity between input power and output power. The voltage should increase to a value where the input power matches the output power which can be perceived from the PV power characteristics presented in Figure 4-2. The apparent delay of frequency sensing in the case of the power controller configuration indicates a more delayed power response as compared to the droop controller which has until now consistently had the longer response time to transients. However, this is power-frequency tracking and has a bandwidth that is significantly limited by the damping of the PLL. This restriction and further implications will be investigated further in Chapter Island When Importing, Causes Overload and Frequency Trip of Non-Critical Load When the on-site load exceeds the generating capacity of the local sources, overload occurs when islanding. The only stable response of the system is to shed some load so that the total system load is less than the peak power capabilities of the on-site source, a process referred to as load shedding. Because loads could be located anywhere in the microgrid, local measurement of frequency is the only value that can be entirely trusted as it is a parameter that is common to

170 153 every point in the microgrid on average. Therefore, it is assumed here that there is a noncritical remote load with a frequency sensing switch that will isolate itself when a low-frequency level is breached. In this case, the low frequency limit trip is set to 59Hz. Figure 4-32 Power and Reactive Power during island event when importing The power characteristic here for the droop controller here is notably lightly damped which stems from both sources existing in a limited condition. However, the inter-source oscillations do eventually damp out within a second, particularly after the noncritical load is disconnected. On the other hand, the power controller s response is much more damped. Similar amounts of system overload were utilized here of 4kW and a post-load shedding power margin of approximately 5kW. This causes a frequency decay rate which is slightly less than the recovery rate as can be seen in Figure At around 0.5 seconds for each source, the noncritical load is disconnected and the system begins recovering. It is interesting to note that

171 154 the droop controller inverter trips the load first. As the voltage had decayed, less power was available from the PV array and therefore the extra power came from the energy storage source, hastening the frequency decay and trip time. The power controller was able to maintain the DCbus voltage throughout the event, creating a more stable operation as compared to the response of the droop controller. Figure 4-33 Array Voltage and Source Frequency during island event when importing The frequency characteristic of the droop controller source includes the aforementioned lightly damped oscillations on top of the average frequency characteristic that regulated the DC-bus voltage back to the nominal level. Otherwise, all sources decayed their frequencies in proportion to the magnitude of overload. Both configurations recover to a nominal operating point without incident.

172 Island Alone When Exporting Islanding alone is the simplest configuration in a microgrid as there is typically little frequency sensitivity in the load, which makes power drawn from the source deterministic if the voltage magnitude and frequency are kept within a small deviation around the rated values. While the droop controller is quite capable of stable behavior in this condition, the power controller is not. The lack of frequency (or angle) sensitivity to power causes the power controller topology to accelerate or decelerate the synchronous frame uncontrollably. Essentially, the grid-following power controller in an islanded system represents an over-constrained scenario where the load power and the commanded power conflict. The power controller was demonstrated to be able to perform source shedding by measuring the frequency and altering the power command, but was significantly bandwidth limited in doing so. This delay is prohibitive and it is for this reason that the droop controller is the only type of source able to operate in this scenario. Figure 4-34 Power and Reactive Power during island event, usrc alone, previously exporting, source shedding

173 156 The power and reactive power are simple traces which is an expected result from the previous discussion of power determinism on an islanded network. The rate at which the power is changed is only a function of the impedance in the network, which typically lags the power transient by a fraction of a power cycle. Figure 4-35 Array Voltage and Inverter Frequency during island event, usrc alone, previously exporting, source shedding The voltage and frequency traces are also quite simple, illustrating a condition where the on-site load is less than the available power of the source. This results in an increase in frequency to the corresponding value on the linear portion of the droop curve and a DC-bus voltage that increases to its stable point naturally Island Alone when Importing, Causes Overload and Frequency Trip The counter example to section is presented here. Figure 4-36 shows the output power of the microsource and the Pmax value that comes from the system limit, the maximum power point tracker or the lower voltage limit, whichever is lowest. It can be seen that prior to the transient,

174 157 the droop controller source was tracking the maximum power well and that after the transient, the power demanded from the source was approximately 1kW more than the pre-transient value. This overload causes a decrease in DC-bus voltage which lowers the Pmax command to zero by the MPPT algorithm. Secondarily, once the DC-bus voltage gets closer to the lower limit dictated by the AC system voltage constraint, the Pmax value falls further. Figure 4-36 Power and Reactive Power during island event, usrc only, previously importing The increase in the value of Pmax and the decrease in power output indicate that the underfrequency load shedding (UFLS) event has completed. The decrease in load causes the system to return back to a stable operating point where the system load is less than the available power.

175 158 Figure 4-37 Array Voltage and Inverter Frequency during island event, usrc only, previously importing The array voltage indicates the level of overloading and subsequent under-loading by its slope. Upon close inspection, the DC-bus voltage can be seen to have concave-down profile indicating that the overload value increased slightly due to a decrease in the power incoming from the PV panels as the voltage deviated further from the maximum power point. The frequency characteristic illustrates three modes of operation. Initially the system is in a maximum power state until t=0. Immediately following the island event is the decay in frequency as a function of the filtered measured power. As the load is constant power, the overload power is accumulated into a continued decay at a nearly constant slope. When the DCbus gets low enough, the frequency droops even faster to preserve the functionality of the system in the case of potentially critical DC-bus sag. After the load shedding event, the frequency increases proportionally to the change in output power but a further decay in frequency is visible after 200ms because the DC-bus is still low enough to have the Pmax value limited against zero.

176 159 Once the DC-bus voltage increases to a value near nominal, the value of Pmax is increased to encourage a higher power output if there were other sources present to accommodate the change in output power. Lastly, as the value of Pmax sits against the system limit, the frequency establishes a fixed value where the system will remain until the system load is changed. 4.6 Conclusion Considering the two options for PV-based inverter controllers, both appear to be suitable for most conditions in a microgrid. Load transients, island transients, and insolation changes have been investigated here and each controller topology has behaved predictably and in a relatively stable manner in most cases. It has been shown that the typical grid-forming controller suffers from DC-bus voltage decay issues from load transients due to an inherent power-control bandwidth limitation. Conversely, the power controller lacks the ability to operate alone in an island and is limited in its ability to quickly adjust power based on measured frequency. Regardless, both sources are capable of withstanding all of the light transient conditions tested without significant additions of bus capacitance beyond what is typical for an industrial inverter. Note, the transients tested here do not represent the worst case conditions and include an energy manager in the form of a DC-bus regulator that was not included in Chapter 0. Both sources demonstrated the ability to survive insolation changes, track maximum PV power and maintain the maximum power point through islanded load transients. However, the obvious distinction between the two is that the power controller is better at regulating the DC bus. Therefore, the decision of which controller to employ should include whether the PV source will be required to operate alone, which would require the grid-forming controller for added stability.

177 160 5 Improvement of Dynamic Power Responses of Grid- Forming Sources in a CERTS Microgrid It was presented previously that a photovoltaic source requires relatively fast power regulation abilities to be able to track maximum input power from the photovoltaic array by regulating the DC bus of the converter especially in the face of network load transients. Grid-forming photovoltaic sources were shown in the previous chapter to exhibit deficient power regulation characteristics as compared to the current-regulating grid-following variety primarily due to the restrictions on the power limit controller to avoid exciting unstable network modes. Frequency droop algorithms used by CERTS Microgrid inverters adjust power by changing the output frequency. This concept assumes that the interconnecting impedance is primarily inductive. It can also be shown that the reactance-to-resistance (X/R) ratio significantly affects the rate at which steady-state behavior is restored following transient events. Steady state operation is assumed in the power versus frequency relationship outlined in the CERTS concepts, but transiently the characteristic is orthogonal. Secondarily, it will be shown that the magnitude of the droop gain or proportional power control gain counteracts the decay of DC phase current offsets and can cause unstable response characteristics. There are multiple candidate remedies to mitigate the oscillatory mechanisms and maintain stable response characteristics with increased power control gains. Three such candidates will be investigated further. First, the concept of active resistance will be introduced. Second, the effect of synchronous frame filters on measured quantities will be investigated. Lastly, the effectiveness of a lead-lag compensator will be examined. Using a combination of eigenvalue migration analysis, dynamic simulation and experimental verification, it will be demonstrated

178 161 that the power control bandwidth of frequency-regulated grid-forming sources can be significantly increased. It will also be shown that the damping and stability of the system can be maintained concurrently. Finally, these methods will be employed to demonstrate the increased power control bandwidth in an inverter application to reduce the overload energy and more tightly control the output power to limit temporary overloading. 5.1 Introduction The concept of frequency droop algorithms to implement power sharing between sources has been traditionally used in power systems to automatically balance power amongst network sources and has more recently become widely applied in CERTS Microgrids [3]. However, inverter-based sources in a microgrid have no inertia and react significantly faster than a large synchronous machine, which changes the response characteristics of the system significantly. The lightly-damped voltage balance eigenvalue can interact with the ideal first-order power response from the droop controller [51] [54] and create unstable systems when placed in a network with other distributed generation sources. Stable responses under a wide range of system configurations is a critical component which will enable the CERTS microgrid concept to work without adding in fast communication or custom engineering to determine gains or parameters of a specific microgrid installation. One of the primary aims of the CERTS concept is the ability to ensure stability up to an infinite number of sources without significant concern for possible inter-source interactions that have regions of instability. The purpose of this chapter is to identify the mechanisms causing voltage unbalance instabilities, quantify the contributing factors and present stability-promoting control approaches that will enable faster power responses in grid forming inverter-based microsources. Active resistance is a control technique that has been successfully applied in other power electronics applications

179 162 ranging from motor drives [62] to DC-to-DC converters [63], but the benefits of applying these techniques in microgrid power systems have not been explored previously in the literature. The concept of signal measurement filtering, or loop-shaping techniques have been utilized in microgrid research and have been included in publications. However, there has been insufficient focus on the role of signal and command filtering which has led to lost information from the conclusions. 5.2 Voltage Balance Instability Investigated One of the key mechanisms previously identified to jeopardize stability when increasing power control gains was voltage balance mode. To examine this phenomenon more closely and quantify the feedback and controller mechanisms that are responsible for this behavior, a two node network is used to present the concepts in a succinct manner. Each source used in this analysis will be an ideal inverter microsource model with only an inductive filter at the output. In this case, the series inductances and resistances can be lumped into a single R-L model connecting the inverters in absence of a filter capacitor. Secondarily, as it was previously established that the change in system load did not significantly affect the voltage balance eigenvalues in section , the load can also be neglected in this case. Therefore, the resulting pertinent system equations are presented below, describing the power versus frequency droop behavior and the resulting d and q-axis current between the two sources. =1 + (5.1) =1 (5.2) = = + = (5.3)

180 163 If the second source can be assumed to be at a fixed frequency like a grid connection, then it can also be assumed that the reference frame rotates at a similar speed. Then the synchronous frame grid voltage can be assumed a constant value. As a result, the following equations can be used to further define the simplified system. = =0 =1 = sin =cos =1 sin + =1 cos 1 = cos + sin (5.4) (5.5) (5.6) (5.7) (5.8) (5.9) (5.10) (5.11) If it can be assumed that the particular operating point for this analysis is such that φ 1, the voltage angle between two sources, is near zero, the small angle approximation is valid and the state Equations can be expressed by the following equations =1 + =1 1 1 = 1 = (5.12) (5.13) (5.14) The corresponding block diagram from equations is presented in the top of Figure 5-1 and shows a cross-coupled damped oscillatory network with an outer loop that defines the droop. The resonant component is defined by the ωl cross coupling term while the damped portion

181 164 comes from the resistive term R. The outer loop with the frequency droop gain Mp defines a loop that contains three integrators which is the most suspected of causing the instability by introducing as much as 270 degrees of phase shift when the resistive terms are ignored. It can then be seen that the magnitude of the feedback resistance plays a critical role in stabilizing the system. R 1 Noise Mp 1 s Phi1 1/L w*l 1 s Id w*l 1/L R 1 s Iq 1 Iq 2 Noise1 Mp 1 s Phi1 1 L.s+R Transfer Fcn Id -w*l -w*l 1 L.s+R Transfer Fcn1 Iq 2 Iq1 3 Noise2 Mp 1 s Phi1 w*l L^2.s 2+2*L*Rs+R^2+w^2*L^2 Transfer Fcn2 Iq 3 Iq2 4 Noise3 Mp*w*L L^2.s 3+2*L*Rs 2+R^2+w^2*L^2.s+Mp*w*L Transfer Fcn3 Iq 4 Iq3 Figure 5-1 Progressive steps of block diagram simplified restructuring The four-step restructuring and grouping of the system in Figure 5-1 leads to the final system that represents a single transfer function that can be used to evaluate the stability of the simplified system in a closed form. While the symbolic representation of the roots of the transfer function is too cumbersome to list here, some insight can come from the structure of the transfer function itself. For example, the transfer function is third order and shows how the resistive

182 165 terms, which exist only in the middle two terms of the denominator, would contribute to the damping of the resulting poles. Also, it can be seen that the droop gain exists only in the final (0 th order) term of the denominator which indicates that an increase in said term will require an increase from the middle two terms to keep the real components of the poles negative. This then further suggests an interrelation between the droop gain and the resistance in the network. To investigate the interrelation of resistance, reactance and frequency droop gain with system stability, these parameters were varied over an extended range and the maximum real value of the three system eigenvalues was plotted to indicate the damping or instability of the dominant eigenvalue. Figure 5-2 presents the resulting maximum real values as a function of parameter variations and the resulting findings are significant. First, all of the plots within Figure 5-2 have similar characteristics that provide evidence to the consistency of the eigenvalue real values based on parameter variations. Second, each characteristic increases in the real value as a function of reduced inductance and reduced resistance. However, the most important finding is that each of the characteristics has eigenvalues that are positive for the same value of resistance for all values of inductance.

183 166 Unstable Unstable Stable Stable (a) (b) Unstable Stable (c) Figure 5-2 Greatest real value of the three eigenvalues as a function of per-unit resistance and inductance for three separate frequency droop values This result indicates that while the inductance plays a role of sensitivity or amplitude of the eigenvalue real values, it does not change the polarity of the real values. Lastly, it can be seen from each value of frequency droop gain in Figure 5-2 that the point at which the values become positive is at different values of resistance. Therefore the theory can be presented that stability of the simplified system does not depend on the inductance, but a relationship between the resistance and frequency droop gain. To further test this conjecture, the real values of the system eigenvalues were monitored while varying the line resistance for different values of frequency droop gain and the points of neutral

184 167 stability were defined. This procedure was performed for multiple values of inductance yielding no sensitivity to inductance as expected. The result presented in Figure 5-3 shows the result of the sampled points overlaid on the ideal relationship suggested by the data. As it can be plainly seen, and with good agreement from each of the resulting marginal stable points, the result suggests that the system will be unstable for all cases where the per-unit resistance is not more than half the per-unit frequency droop gain. This result is quite significant as it draws a connection between previously observed phenomena and places a closed-form solution to avoiding voltage balance instability. This relationship is also described in Equation (5.15). Stable Unstable Figure 5-3 Limit of system stability based on relationship between per unit resistance and frequency droop gain _ / 2 (5.15) The implications of this stability limit should also be considered not only in light of the nominal frequency droop gain, but also considering the combined frequency droop gain that results from the addition of the proportional gain of the power limit controller. Where a typical value for the nominal droop gain is around 0.01pu, the proportional gain of the power limit controller must be high enough to manage the stored energy of the individual source. For a low stored energy

185 168 source such as a PV inverter, the power limit controller is approximately an order of magnitude greater than the frequency droop gain used for nominal power conditions. This implies that the combined frequency droop gain for power-limited conditions may exist in the unstable region for some sources and may threaten the stability of a microgrid if this limit is not considered in each implementation. Now that the simplified result has been defined, it is important to consider these findings with respect to systems that are more complex than a grid tied inverter with a simplified interconnecting impedance. Considering the conditions explored previously, the two pertinent cases to consider are a two inverter microsource network as well as a mixed system with an inverter and synchronous machine microsource. In these cases, one of the major differences is the effective frequency droop gain, but the inertial response of the synchronous machine does complicate this relationship as well. The effective droop gain between two inverters of equal size is effectively the sum that of each both inverters, which would require proportionally more resistance to maintain stability. However, if the impedance is segmented into two halves, than the original relationship defined in Equation (5.15) holds for each inverter and filter treated separately. Also, systems with more than two nodes will have more than one possible resonant mode and require a more rigorous modal analysis to evaluate the effective margin considering resistance with respect to the frequency droop gain. As a compounding issue, the power regulation controller uses a supplementary droop gain as the proportional component which effectively increases the frequency droop gain when the source is in an at-limit condition. In the case of the mixed-source system investigated previously, the voltage balance eigenvalue had significantly more damping than the system with only inverter microsources primarily due to the increased resistance of the synchronous machine. Here the theory is presented that

186 169 considering the diminished effect of the voltage balance mode on the rotor inertia of the synchronous machine due to the frequency of the oscillation, the effective droop gain at 60Hz frequencies is negligible for the synchronous machine. Assuming a minimum inertia typical of rotating generators, the synchronous machine can then be considered a nearly fixed frequency voltage source. Its characteristic is then similar to a grid connection that has no droop gain and therefore, considering the previous findings, will not be the cause of exciting the voltage unbalance mode. Secondarily, due to the more significant resistances typically found in rotating machinery than in harmonic filters as well as the presence of damper windings, the synchronous machine is even more capable of avoiding excitation of the voltage balance instability. High efficiency synchronous machines and inverter solutions will typically have little resistance and will be more prone to exibiting the voltage balance instability. When considering the myriad of possible microgrid source arrangements the task of ensuring global stability can appear intractible. The principle of passivity can be employed here to suggest that each source resides on the stable side of Equation Therefore, any collection of microgrid sources will be stable and even compensate for the possibility of other sources or loads that may not comply with this criterion. Typically, the frequency droop gain used in the nominal region is around 1% of the system frequency for a 1pu change in power. Therefore, only a 0.005pu resistance would be required to maintain a stable system under the model investigated in this chapter. Since this amount of resistance is significantly smaller than normal, the applicability of the explored range may initially seem unnecessary. However, when the power control bandwidth becomes important for power limit cases, the porportional gain of the power limit controller acts in an identical manner to the nominal frequency droop gain Mp. Since power limit cases must be considered in the

187 170 normal operational range of an inverter microsource, the combined gain of Mp+KpPMax becomes the effective proportional frequency droop gain. This point is explored further in later chapters, but illuminates one practical consideration in implementation of the frequency droop gain limit presented herein Limit of Combined Frequency and Voltage Gain After noting that the frequency droop gain interacted with the differential voltage between two sources orthogonal to the voltage magnitude, a natural analog could be drawn between the differential voltage in-phase with the voltage vectors that is controlled by the voltage droop. Therefore, it is expected that these terms will have a close and well defined interaction. In this system, the rate at which the magnitude of the voltage vectors changes is proportional to both the voltage droop gain Mq and the integrated error voltage droop gain KiV. For the purposes of this section, the effective voltage droop gain equivalent in units to the frequency droop gain will be Mq =Mq*KiV which is in units of (per-unit voltage)/second/(per-unit reactive power). Using the model in Figure 5-1, a second feedback path was added fundamentally in quadrature to the power-frequency droop that measured reactive power (current in Figure 5-1) and altered the q- axis differential voltage. It can be seen from Figure 5-4 that the limit value of stability is defined by the summation of the frequency droop gain and the effective voltage droop gain. Therefore, the results obtained for this one axis naturally extend to the voltage droop gain as well in consideration that the summation of the two are constrained to the limit value of resistance as defined by Equation 5.15.

188 171 Unstable Stable Unstable Figure 5-4 System stability envelope tracking max. real value of eigenvalues vs. droop gains for system frequency (M p ) and voltage (M q ) (left) and plot of stabilty boundaries vs M p and M q droop gains for 3 values of R (right) Experimental Verification of Resistance and Frequency Droop Gain The UW-Madison Microgrid testbed was utilized to test the theoretical relationship in Figure 5-3 and Equation This configuration is identical to that described in Section except that the resistive load between the two inverters was not present in these tests. That is, two inverterbased microsources were connected together with a cable of negligible impedance and in a configuration with no significant load other than magnetizing and series resistive losses. An overview of this configuration can be found in Figure 3-44 and the details of the hardware utilized are included in Appendix A. While the physical line inductance and resistance were not varied, resistance decoupling in software was utilized to augment the effective resistance to achieve different conditions for the characteristic test. The resistance decoupling was achieved by augmenting the output voltage to increase the average output voltage (duty cycle) from the inverter proportional to the instantaneous measured current on the associated phase leg. This accomplished the job of counter-acting the voltage drop of the physical resistance with minimal lag imparted by the 100us delay imparted from the 10kHz control cycle. This delay should provide accurate

189 172 decoupling for transient current frequencies up to 5kHz. The AC resistance of the system was determined by first setting the droop gain of each source to a miniscule value and decoupling an equal amount of resistance per converter until the system reached an unstable point, stepped in 10mΩ increments. The value before the system was noted to go unstable was recorded at 1.85Ω in the experimental system, which is assumed to exist primarily in the power electronic switches and the inverter-side filter inductor by the amount of heat each produced as compared to that of the microgrid-side inductor and AC filter capacitors. This value of measured resistance was used as the basis for decoupling and determining the relationship between frequency droop gain and line resistance. The data points for neutral stability were defined by the point of inverter tripping during the process of increasing the frequency droop gain. Total Resistance [percent pu] Mp vs. R Stability Limit, X=30.5[percent pu] Stable Experimental Predicted Unstable Total frequency droop gain[percent pu] Figure 5-5 Experimentally determined limit of system stability based on relationship between per unit resistance and frequency droop gain Figure 5-5 presents the experimentally gathered data of the frequency droop gain versus the total interconnecting resistance along with the ideal 2:1 relationship between per-unit values. The experimental data shows good agreement between the ideal case for lower values of frequency droop and effective line resistance with significant variability towards conditions with higher

190 173 resistance and higher frequency droop gain. With the X/R ratio dropping from 13:1 in the low resistance case to 1.27:1 in the high resistance case, the system characteristic is expected to diverge from the ideal condition of primarily inductive interconnecting impedance. A more thorough investigation into the effects of system scaling and different X/R ratios is presented later in section Chapter 0. Nevertheless, the practical point can be noted that the system seemed to have a greater region of quasi-stable operation as the resistance and the frequency droop gain increased. The DC offsets of the phase currents were clearly lightly constrained by the resistance of the network particularly near the theoretical limit of stability. For conditions with less resistance, the instability seemed to occur as would be expected from a linear system. The phase currents progressed away from a value of zero-dc offset at an increasing rate and the measured power had a 60Hz ripple where the envelope increased similarly to the phase current DC offset. However, for the conditions with higher values of resistance and frequency droop gain, there was significantly more lightly-damped wandering in the DC offset of the phase currents. Therefore the trip point (inverter over-current protection limit) becomes a more subjective indicator for instability. The effect of this phenomenon can be seen in Figure 5-5 where the results begin to vary to and away from the ideal characteristic. In spite of the less clearly defined limit for stability, the agreement of the experimental data with the ideal characteristic does validate the previously defined relationship between frequency droop gain and total line resistance particularly for cases with X/R ratios of at least 2:1. Secondarily, this validates the simplified model in Figure 5-1 against the more complex system model used elsewhere in this investigation and suggests the use of this model as a valid basis for further exploration of methods for augmenting the defined frequency droop characteristic.

191 Methods Mitigating Voltage Balance Instability To avoid the previously discussed voltage balance instability in efforts to improve the power control bandwidth, the simplest solution may be to add resistance to the impedance of the source. However, this may lead to undesirable loss of efficiency, increased cross-coupling between the droop mechanism feedbacks and would require a physical retrofit to an existing system. In this section, possible solutions to voltage unbalance instability in grid-forming inverter microsources will be presented in the form of control methods that will be presented, tested and discussed. Preliminary analysis is offered in the first half of this chapter in an effort to predict the effectiveness of the proposed control approach on the system. In the later part of this chapter, full system simulations and experimental tests will be performed to validate the preliminary analysis Active Resistance Active resistance is an obvious consideration when applying a control solution to solve a situation that can be solved by adding physical resistance. The obvious benefits are the real losses that can be avoided by controlling the inverter terminal voltage to mimic the voltage drop associated with a physical resistance. Secondarily, to address concerns regarding cross-coupling of droop regulation mechanisms that would still occur even with active resistance, a high-pass filter can be introduced in the path of said active resistance to eliminate a resistive voltage drop in steady state. An example of the sort of pass-through gain of such a filter, centered around 60Hz, is presented in Figure 5-8. In reality, it is best to introduce a band-pass filter that has little effect at low frequencies and also limits the sensitivity to sensor noise at high frequencies. An implementation of this is presented in Figure 5-6 with poles placed at K1 and K2. For cases where sensor noise is not a significant

192 175 consideration, K2 = can be applied to the general form in Figure 5-6. Figure 5-7 also shows how active resistance is applied to the standard microsource controller. Ra R s 1/K2s 2+K1/K2+1.s+K1 1st order Bandpass filter 1 Noise Mp 1 s Phi1 1/L w*l 1 s Id w*l 1/L R 1 s Iq 1 Iq Ra s 1/K2s 2+K1/K2+1.s+K1 1st order Bandpass filter1 Figure 5-6 Active resistance voltage implementation employed in simplified grid-tied inverter model using cascaded single-pole filters to create a band-pass filter for active resistance voltage contribution in reduced-order model Figure 5-7 Active resistance implementation on full microsource control diagram using difference between filtered and instantaneous values of current to generate band-pass filtered quantities that primarily exclude steady-state operation. The resulting currents are multiplied by the active resistance gain Ra to develop a voltage that is added on to the d and q-axis quantities developed from the waveform generator. The active resistance in this case acts very similarly to the actual resistance where the measured currents result in a proportional change to the voltage summation defining the rate of change to the same current. With a broad pass band of the active resistance contribution as indicated in

193 176 Figure 5-8, the effect spans a significant portion of the spectrum represented by the system eigenvalues. This implementation represents a more general addition of damping to any currents not at low frequencies in the synchronous frame. The benefit of this sort of implementation is that it is insensitive to the actual frequency of the lightly damped modes which avoids the need for re-adjustment based on changing network base frequencies. To assess the effectiveness of active resistance implementation in terms discussed previously, the result can be determined most simply if the assumption can be made that the effective gain of the band-pass filter in Figure 5-8 is unity at 60Hz. Then, the effective resistance is the simple summation of the physical resistance and the active resistance and therefore, a microgrid system can become safeguarded from voltage balance instability by active resistance alone. However, it should be mentioned that single-phase loading, representing a 120Hz power oscillation (-60Hz in synchronous frame), would cause voltage fluctuations at the terminals of the inverter that would emulate the effect of series resistance. Therefore, for power quality reasons, the value of active resistance should be limited by power systems considerations and voltage compliance standards such as IEEE-519.

194 Bode Diagram Magnitude (abs) Phase (deg) Frequency (Hz) Figure 5-8 Frequency response of active resistance contribution to system transients, primary oscillations occur at power frequency (60Hz) in the synchronous reference frame One final design trade-off to consider is the voltage regulation loop bandwidth as compared to the active resistance low-pass frequency. If the voltage regulator bandwidth exceeds the lowpass frequency of the active resistance filter, the effectiveness of both controllers will be mitigated at those frequencies as each controller has competing references Input (Feedback) Filtering Another approach to mitigating the gain of the feedback mechanisms is to limit the spectrum of the input signals to avoid passing through the frequencies for which the oscillatory mode exists. One example of this that will be used here is the positive sequence filter, an application of filtering which was developed in [1]. The positive sequence filter is placed on stationary frame d and q-axis voltages and currents to filter out any components not synchronous with the operating frequency of the output voltage. The filter itself has two primary components, a synchronous component that rotates the stationary frame filtered value at synchronous speed, and a low-pass

195 178 filter on the measured values that affect the filtered values, as seen in Equation (5.16) and Equation (5.17). V _ k = V k 1 ωtv _ k 1 M + 1 M V k (5.16) V _ k = V _ k 1 + ωtv _ k 1 M + 1 M V k (5.17) The resulting gain characteristic of the synchronous frame filter is actually quite similar to the active resistance band-pass filter in that the gain at synchronous frequency is near unity and has a 20db/decade roll-off at some frequency away from the target frequency. However, the synchronous frame filter does not contain a similar pass band for the counter-rotating (negative) frequency, which is accomplished by shifting the synchronous frequency to the base-band and using a frequency-symmetric low-pass filter in this case. Secondarily, the filter used here is implemented in-line to the code that determines the output frequency, which then operates from primarily synchronous components. In contrast, the previous case of active resistance was implemented as a damping that acted on non-synchronous components. The primary detriment to the input filter used here is the loss of information due to the filtering process, limiting the rate at which the power calculation converges on the actual value. Therefore, the power regulation controller can only be tuned to a rate significantly slower than the bandwidth of this measurement filter which places a significant restriction on the achievable power control bandwidth, the overall goal of this chapter.

196 179 Figure 5-9 Effective pass-through gain for positive sequence filter as a function of positive sequence filter bandwidth with selected bandwidths (2.5, 5, 10, 20, and 40Hz) and the associated attenuation As can be seen from Figure 5-9, a design trade-off exists between the cut-off frequency of the positive sequence filter and the pass-through gain at the frequency of the voltage unbalance oscillation. For example if a power regulation bandwidth of 5Hz is desired, than a power measurement bandwidth of 15Hz or greater is required, which is a minimum cut-off frequency of log 10 (f c )=2 on the x-axis. In this case, the pass-through gain is only 0.2 which mitigates resistance requirement of the relationship in Equation (5.15) by the reciprocal, 5x, which is significant. Higher measurement bandwidths would result in significantly less effect, but it is important to note that the minimum power control bandwidth is ultimately defined by the amount of stored energy present in the source and varies by application. Figure 5-10 shows the simplistic implementation of power signal filtering on the microsource controller. It should be mentioned that the power signal could also be generated from unfiltered quantities to not interfere with the power filter bandwidth.

197 180 Figure 5-10 Power signal filtering implementation on full microsource control diagram using a low pass filter on the power signal to reduce the high-frequency response of the frequency droop actions Lead-Lag Compensation Output filtering is similar to input filtering discussed previously except that the measured values still can be calculated and resolved at high bandwidths internally. A similar terminal behavior would still result as the filtration would still limit the pass-through gain of the voltage balance mode. However, another variety of filter, or compensator as it is referred to here, can be utilized to specifically mitigate pass-through gain at a specific frequency while limiting the effect on the remainder of the spectrum. This solution is referred to as a lead-lag compensator for its pair of pole-zero functions with one configured as a lead compensator and one configured as a lag compensator. The structure of the lead-lag compensator is described in Equation (5.18). = , > > > (5.18) The resulting effect of the lead-lag compensator is similar to a notch-filter where the gain is near unity for all frequencies except those between the poles. A 20db/decade roll-off occurs between the first pole (P 1 ) and the first zero (Z 1 ), exhibiting flat gain between the two zeroes and then increasing at 20db/decade between the second zero (Z 2 ) and the second pole (P 2 ). To ensure unity gain above the frequency of the compensator, the ratios of the poles and zeroes are kept equal so that the same number of decades exists between the first pole-zero pair and the second.

198 181 The compensator functions by restricting the output voltage activity at the modal frequency and giving a pass-through gain between zero and unity like the positive sequence filtering. Aside from the ability to selectively filter modes, this variety of compensator provides minimal lag for measurement and compensation across the measurement spectrum. This high degree of unity pass-through gain provides the greatest opportunity for retaining the effectiveness of the power regulator in response to step-changes in load which contain many frequencies in the input characteristic. The major disadvantage of this compensator is that separate instances must be implemented for each frequency band or specific frequency that requires mitigation, increasing computational load for systems with multiple lightly damped modes. The implementation of the lead-lag filter on the standard microsource inverter controller is presented in Figure 5-11 as an in-line filter between the power measurement and the signal delivered to the frequency droop calculator. The internal components of the lead-lag filter are consistent with Equation Figure 5-11 Power signal lead-lag filtering implemented on full microsource control diagram using a lead-lag filter on the power signal to reduce the 120Hz response of the frequency droop actions, mitigating the voltage balance mode instability

199 Power Control Bandwidth Improvements Improved Power Response Using Active Resistance The first solution tested was the active resistance implementation as presented previously, except that the upper limit (low-pass element) was not included as it is not yet known at what frequency the noise of the system should be effectively filtered. The effect on a nominally tuned system is presented in Figure 5-12 where it can be seen that the only appreciable movement of the eigenvalues is to the left, if at all. This case presents a nominally tuned active resistance of 0.01pu (1%) which should eliminate the need for physical resistance with the nominal frequency droop gain in each source. With the high pass frequency set to 1Hz, the added poles can be seen to move slightly as a function of the change in active resistance while interacting with the voltage regulation 1 pole (as referenced in Table 3-2). (a) (b) Figure 5-12 Effect of increasing Ra (nominally 0.01) on nominally tuned eigenvalues Most importantly, the lightly damped voltage balance poles can be seen in Figure 5-12 (a) moving significantly to the left even in the case of minimal amounts of added active resistance (varying between 0.25% and 4% in this case).

200 183 Interestingly, the collateral effect of added damping can be seen on the LC Filter poles in Figure 5-12b primarily due to the fact that the active resistance contribution is not band limited to exclude affecting higher frequency poles. While they, for the most part, did not present a stability issue, faster convergence with less side-band activity is a desirable result. In reality, sampling and computational delay may limit the upper bandwidth to approximately 1kHz for a 10kHz control and PWM loop. Therefore, the active resistance gain cannot be assumed to be unity gain at 900Hz as seen in the above figure, but significant gain will still exist enough to mimic part of the effect on the high frequency poles as displayed. The next step is to investigate the effect of active resistance as an enabling agent to allow for stable implementation of higher power control gains to yield an increased power control bandwidth. To test this, the power control gains were increased significantly to where KpPMax=48 and KiPMax=960. These gains were selected as the point where the power control response was still critically damped and represented a significant improvement over the base gains. The actual limits of power control bandwidth are not yet known as they require consideration of the available stored energy, computational delays and noise levels in the system. These items are more concern for experimental implementation which is presented later. Nevertheless, use of fixed power control gains will give a fair comparison of the effectiveness of each control approach in this section against the filtering methods presented in the following sections.

201 184 (a) (b) (c) Figure 5-13 Effect of changing Ra (0.025pu-0.4pu. 0.1pu nominally) on eigenvalues for system with increased power control gains (KpPMax=48, KiPMax=960) With the increased gain set, the energy regulation pole can be seen to be located near 3Hz in Figure 5-13c, a significant improvement over the 0.5Hz in the nominal case. More importantly, the voltage balance eigenvalues can be seen to be drawn into the left half plane by the active resistance actions in Figure 5-13a with no perceivable detriment as the active resistance gain increases even in the case of Ra=0.4pu. Eventually, the active resistance voltage drop will cause a considerable amount of reactive power to flow in response to real power transients, but the transient simulations that follow investigate this more closely. Also note that the high frequency poles experience a similar addition of damping as was seen in the nominally tuned case.

202 185 Figure 5-14 Transient load step simulation of an inverter microsource network with active resistance damping. Ra=0.1, KpPMax=48, KiPMax=960 The transient response result in Figure 5-14 illustrates a drastically improved power regulation bandwidth, particularly when considering the overload energy, which is 31 times less than the base case in section Some other notable differences are the increased excursion in frequency from the first source, dipping to just below 57Hz for part of one cycle. This is a large deviation, but a necessary response considering the rate at which power is regulated and the phase angle that must be traversed so quickly. The peak overload seen on the first source was reduced to 1.3pu versus the 1.5pu experienced in the base case. As a negative point, the voltage balance oscillations are significantly pronounced in the figure above. These oscillations persist because the increased effective droop gain when considering the power controller is high

203 186 (14%pu). This level of frequency droop gain requires at least 0.07pu of line resistance to remain stable, per Equation The eigenvalue plot in Figure 5-13 (a) shows that the voltage balance poles are unstable at Ra=0.05 and stable at Ra=0.10, which agrees with the theoretical value of approximately 0.07pu for the total effective series resistance. Figure 5-15 Transient load step simulation of an inverter microsource network with active resistance damping. Ra=0.2, KpPMax=48, KiPMax=960 To illustrate the effect on the voltage balance mode when increasing the active resistance further, Ra was increased to 0.2pu. As can be seen from Figure 5-15, the power and reactive power characteristics have significantly less 60Hz oscillations, but it is also interesting that the peak reactive power decreased with a higher active resistance. As the active resistance does act like a physical line resistance during transients, it is expected that an increase in resistance would cause

204 187 a greater degree of cross-coupled association between phase angle and reactive power, but the change is hardly perceivable amidst the 60Hz transient response. Figure 5-16 Transient load step simulation of a mixed source microgrid network with active resistance damping. Ra=0.1, KpPMax=48, KiPMax=960 To complete the set of system tests, the same transient, with the same power regulation gains, was run on a mixed source microgrid with a synchronous machine generator as the second source. In this case, presented in Figure 5-16, the power characteristic is improved over the base case significantly which is expected from the previous load transient. However, one surprising result is the increase in overload energy required which is nearly double the inverter microsource microgrid case shown previously. This result is likely due to the higher impedance of the synchronous machine which acts to increase the initial loading on the inverter microsource, but

205 188 also the terminal voltage of the synchronous machine is affected more significantly considering stator resistance and latency of the field voltage regulator. The frequency response, while stable, is lightly damped. However, this characteristic is primarily due to the low-loss damper windings and the latency of the torque (fuel) regulator in the governor Experimental Verification of Active Resistance The concept of active resistance was tested on UW-Madison microgrid test-bed hardware in a manner outlined in section where two inverter sources were connected together with minimal impedance between the inverter filters and power was transferred between them. Both the distribution of the frequency droop gain and the amount of active resistance between each source was varied to create six distinct cases. In theory, the distribution of frequency droop gain applied in one source versus another should not matter, according to the simple system as defined in section 5.2 with two rotating voltages and a single inductor connecting the two. However, the actual system has four series inductors and two shunt capacitors from the two joined L-C-L filters and each section exhibits different X/R ratio. While it may still be a reasonable assumption that the system is physically symmetric, the effect of unequal application of frequency droop gain and active resistance is the key point investigated here in addition to the effectiveness of any and all implementations of active resistance. Figure 5-17 is the result of all cases of application of active resistance, superimposed over the original Figure 5-5. The general trend appears to exhibit a more dispersed clustering of points than the case without active resistance, but more importantly the points of neutral stability for the

206 189 majority of points reside in the stable side of the resistance vs. frequency droop gain characteristic. Table Points of neutral stability for two inverter microgrid with active resistance for cases varying distribution of frequency droop gain and active resistance Case Mp1 [%pu] Mp2 [%pu] R1 [Ω] R2 [Ω] Ra1 [Ω] Ra2 [Ω] Total R [%pu] Total Mp [%pu] Ideal Mp [%pu]

207 190 Total resistance [percent pu] Mp vs. R Stability Limit, X=30.5[percent pu] Stable Unstable Predicted Without Ra Cases with Ra Ignoring Ra Total frequency droop gain [percent pu] Figure 5-17 Neutral stability characteristic defined by the relationship between frequency droop gain and line resistance with active resistance and without active resistance This implies that the majority of cases where active resistance was applied give slightly less damping than those cases where physical resistance was applied. The main difference between the resistance decoupling in the inverter controller to augment the apparent line resistance and the active resistance tested here is the high-pass filter utilized for the active resistance. The 2Hz active resistance high-pass filter for active resistance is nearly unity gain at 60Hz so this is not a contributing factor. Regardless, the characteristic does tend to follow the ideal characteristic more closely than it would without consideration of the active resistance. This is evidenced by the green points in Figure Conversely, the neutrally stable cases well into the previously unstable region above the ideal line redefine the region of stable operating space. The distribution of the points considering the addition of active resistance is considered on a case-by-case basis in Figure 5-18 which correlates to the data presented in Table 5-1. The distribution of the active resistance was tested to be equally distributed in the first condition, 1Ω per source, and 2Ω located only on one source as the second condition. The other variable tested is the distribution of frequency droop gain amongst the sources. A near distribution indicates

208 191 that the majority of the frequency droop gain is located on source 1 versus source 2, with an equal distribution and a far distribution that follow the same naming convention. These two variables make up the six cases investigated. While the results are slightly separate from the ideal case, the individual cases follow a consistent trend, establishing an offset versus the ideal case. Total frequency droop gain [percent pu] Frequency Droop Gain Distribution for Cases with Active Resistance Near dist. Mp, Equal dist. Ra Equal dist. Mp, Equal dist. Ra Far dist. Mp, Equal dist. Ra Near dist. Mp, Near dist. Ra Equal dist. Mp, Near dist. Ra Far dist. Mp, Near dist. Ra Ideal Unstable Stable Total Resistance (Physical and Active) [percent pu] Figure 5-18 Individual break-out of the six separate cases defined in Table 5-1 to explore the effect of frequency droop gain and active resistance distribution between two inverter-based sources In general terms, the equal distribution of Ra seems to offer the greatest amount of effective suppression of the instability as the neutrally stable point is at a measurably higher value of frequency droop gain in those cases. Also, the near distribution of frequency droop gain (Mp) appears to add the most damping, despite the supposed identical nature of the first and third cases. The assumption of identical sources may be invalidated by this finding, but the resulting data could also show a discrepancy based on the temperature-biased resistance of the filter elements as the tests were conducted or from the effective resistance of the power electronic switches in the inverter. The far distribution values were taken last, which does point to

209 192 temperature sensitivity of the filter components which were noticeably elevated after this sequence of testing, but only to approximately 50C. While this raises questions about the symmetry of the sources, the results in other sections track the total amount of resistance consistent with the theoretical relationship. It is also shown from the cases of unequal distribution of active resistance that the least damped case occurs when the active resistance is not co-located with the frequency droop gain. Even if the far distribution case is omitted, the equally distributed case of frequency droop gain for unequal distribution of active resistance also agrees with the previous conclusions. In spite of the more varied results with the active resistance applied as compared to the previous case in section 5.2.2, the result in this section illustrates an important point regarding the effectiveness of active resistance. Inspection of Figure 5-17 clearly indicates how the addition of active resistance can be largely considered in the total effective resistance by its general agreement with the ideal stability boundary. An experimental transient was also performed to demonstrate the time-domain response. The Pmax value of each source was scaled to 0.433pu, so the resulting unbalanced energy is reduced as compared to the simulations presented previously. However, as compared to the baseline transient in Figure 3-51, there is a 28x improvement, yielding only 3.5% as compared to the simulation case which yielded an improvement to just 3.2% of the original overload energy. In both cases, there is a dramatic reduction in overload energy.

210 Power 1.05 Source Voltages Power [pu] InvMicrosource1 0 InvMicrosource2 Load Time [s] Reactive Power 0.15 V [pu] Time [s] Source Load Angle 0.15 Q [pu] InvMicrosource1 InvMicrosource Time [s] Load Frequencies 60.1 Load Angle [rad] Time [s] 1.5 x 10-3 Unbalanced Energy Frequency [Hz] E [pu-sec] Time [s] Time [s] Figure 5-19 Experimental result of load transient from 0.385pu load to 0.77pu load. Pmax set to 0.433pu, constrained by battery power capabilities of source2, Ra=3.07Ω (0.2pu), KpPMax=48 (0.1273pu), KiPMax=960. R1=0.9Ω (0.0586pu), R2=0.9Ω (0.0586pu).

211 Improved Power Response Utilizing Input Filtering Input filtering, as discussed earlier, limits the spectrum of signals that propagate from the feedback to the output of the controller. Proper selection of this spectrum can avoid causing lightly damped natural modes of the system to become unstable through active controller actions. In this case, a first order low-pass filter is implemented on the measured power signal in Figure 5-8 to yield the voltage balance mode pass-through gain consistent with Figure 5-9. (a) (b) Figure 5-20 Eigenvalue migration when varying the power measurement filter bandwidth (2.5Hz-40Hz) on an inverter microsource microgrid, nominally tuned power regulator For the nominally tuned case, the effect of changing the power measurement bandwidth can be seen in Figure 5-20 for the inverter microsource microgrid. The Phi2 eigenvalue begins as a part of a complex pole pair with the power measurement pole (P_filt) which separates above 10Hz power measurement bandwidth. The effect on the remainder of the system in the base case is relatively inconsequential, though it can be seen that the voltage balance poles do begin to migrate at a power measurement bandwidth of 40Hz (magenta case).

212 195 Figure 5-21 Eigenvalue migration when varying the power measurement filter bandwidth (2.5Hz-40Hz) on an inverter microsource microgrid (KpPMax=48, KiPMax=960) By increasing the bandwidth of the power filter in the case with increased power regulator gains, it can be seen that the P_filt and Phi2 remain as a pair (Figure 5-21, center) through the full range of power measurement bandwidth. Secondarily, it can be seen that the voltage balance poles are originally sitting at approximately -2.5Hz as the effective pass-through gain at 60Hz is minimal. As the measurement bandwidth is increased, the system becomes less damped. The last two filter points (20Hz and 40Hz, correlating to 2.1 and 2.4 on the x-axis of Figure 5-9) have passthrough gains that allow the frequency droop to destabilize the voltage balance poles. In the case of the cyan, the effective gain of the frequency droop at the frequency of the voltage balance mode was only 0.25 whereas the magenta case was 0.40, enough to destabilize the response. With a measurement filter bandwidth of 20Hz, the allowable power response bandwidth with favorable damping characteristics can be as high as ~7Hz, or 1/3 the measurement value. Practically, the power control bandwidth can be set to a lower value for better damping, but the lower power control gains would also not necessitate as much attenuation from the measurement filter. However, this is not to say that input filtration cannot be used in concert with the active resistance solution as well to combine the benefits of each, one which mitigates the loop gain of the lightly damped mode and the second that actively damp the oscillations.

213 196 (a) (b) Figure 5-22 Eigenvalue migration when varying the power measurement filter bandwidth (2.5Hz-40Hz) on an mixed-source microgrid (KpPMax=48, KiPMax=960) The case of the mixed system was also investigated and the effect on the eigenvalues can be seen in Figure It shows that an increase in the power measurement bandwidth causes the torque/speed eigenvalues of the synchronous machine to go unstable when the power measurement bandwidth is only 2.5Hz. This is somewhat of a surprising result except that the power regulation bandwidth is intended to be greater than this and a lightly damped response, if not an instability, would be expected from that condition. Secondarily, the power regulation pole (Figure 5-22b) can be seen to increase from 3Hz to approximately 5.5Hz, but exhibits minimal changes beyond the 10Hz power measurement bandwidth. The most positive result here is that there is little to no negative effect seen from the upper values of power measurement bandwidth.

214 197 Figure 5-23 Transient load step simulation of an inverter microsource network with input filtering. Freq_P_meas=20Hz, KpPMax=48, KiPMax=960 The power response of the inverter microsource network with input filtering shows multiple lightly damped modes with a power filter bandwidth of 20Hz, but the oscillations disappear within 150ms after the transient. Also, the overload energy can be seen to peak at a value of 6.7E-3 pu-sec, which is slightly higher than the previous case with active resistance damping, but this is primarily due to the frequency oscillations due to the under-damped power regulation response.

215 198 One interesting point to note is the source-shedding that occurs because of the power response overshoot. The flat-topping of the frequency is evidence of linear frequency droop operation. This fact gives rise to a new discussion point regarding the mitigation of the lightly damped modal response. By only limiting the extent to which the controller excites the already lightlydamped voltage balance mode, the improvement in transient response is inherently limited without the use of active damping that counteracts the propagation of the lightly damped oscillation within the d-q current loop. Nevertheless, it should be noted that the response is stable and generates satisfactory overload management characteristics even without additional damping techniques.

216 199 Figure 5-24 Transient load step simulation of a mixed source microgrid network with input filtering. Freq_P_meas=20Hz, KpPMax=48, KiPMax=960 Again for completeness, the same transient was performed on the mixed system including the synchronous machine generator as the second source. It can be seen that similar characteristics of low-frequency oscillations still exist, though the 60Hz oscillations are unperceivable due to the extra resistance of the synchronous machine. Also, the same flat-topping of the frequency can be seen as well as a minor increase in overload energy, as much as 0.01pu-sec, which is twice the typical value present in an inverter alone.

217 Power Measurement Filtering Hardware Results The UW-Madison Microgrid was again used to validate the analytical and simulation work on power measurement filtering presented previously in this section. Two cases were considered here in the context of a two-inverter microgrid: implementation of power measurement filtering on one source and power measurement filtering on both sources. In all experimental cases here, the uncompensated resistance will be similar to the simulation cases with X/R 13:1. Table Points of neutral stability utilizing power measurement filtering on source 1 to attenuate effective frequency droop gain at higher frequencies, data presented in order taken to illustrate on-time sensitivity Pmeas BW [Hz] R[Ω] Mp1 [%pu] Mp2 [%pu] Total Mp Effective Ideal Limit Table Points of neutral stability utilizing power measurement filtering on both sources to attenuate effective frequency droop gain at higher frequencies Pmeas BW [Hz] R[Ω] Mp1 [%pu] Mp2 [%pu] Total Mp Effective Ideal Limit The results from the two cases are presented in Figure 5-25, illustrating a characteristic that only loosely follows the ideal attenuation trend in both cases. It can be seen that the recorded effective attenuation of the implementation in one source only is less than that of the case with both sources having power measurement filters. Secondarily, it can be seen from the effective attenuation in Table 5-3 that the power measurement filter varied depending on the order in

218 201 which the data was gathered. The data in Table 5-2 is listed chronologically, using the same method outline in section 5.2 where the frequency droop gain was increased until the converter tripped. The result indicates that the converter run-time and the thermal characteristics associated with continuous operation of the experimental setup do affect the results. However, it bears mentioning that this series of tests exhibited the widest range of quasi-stable operation and that the data likely has a significant error band that should be applied. The converter trip point was not defined in consideration of the phenomena measured, where the DC offsets of the currents would waiver to varying magnitude of total offsets which could prematurely trigger an unstable point based on the converter trip point. For all cases presented here, the typical trip point was for filtered power to exceed 3kW input on either source. Regardless, the observed phenomena of large regions of quasi-stability indicate that the loop shaping provided by the power measurement filter does not increase the damping of the system, but simply avoids excitation of a lightly damped mode. This conclusion is mirrored in the findings from Figure 5-20 where the voltage balance mode remains largely stationary at -2.5Hz for all but the case of the highest power filter bandwidth, in which case the mode becomes less damped.

219 202 Frequency Droop Gain Attenuation at 60Hz [unitless] Experimental, on source1 only Experimental, both sources Ideal Unstable Stable Filter Bandwidth [Hz] Figure 5-25 Comparison of effective frequency droop attenuation versus ideal attenuation varying the bandwidth of the low-pass filter used on the power measurement as defined by the resistance and frequency droop gain. Shows the case when power filtering was only used on one source and when it was employed on both sources. The results obtained experimentally in this section illustrate that higher frequency droop gains can be utilized even in lightly damped systems using power measurement filtering. However, the characteristics of a lightly damped real system have been shown to change with only small perturbation in the parameters and that power measurement filtering does not add damping to the system. This statement provides the best argument for a combined approach of power measurement filtering or other loop-shaping techniques and active resistance damping. The time domain experimental transient illustrates the low system damping in Figure This transient was performed consistent with previous experimental transients performed in sections 3.4 and Interestingly, the system required a minimum amount of uncompensated resistance as described in the figure (5.86% per source) with that transient to avoid tripping the converter. Even though the system was tolerant to small disturbances, as seen in Table 5-2, the magnitude of the transient faced in this case acted on such a lightly damped system to cause faults for any resistance lower than the value used. The value of resistance required was

220 203 approximately 2x the value predicted by eigenvalue analysis when considering the maximum allowable power deviation and magnetic saturation of the inductor. 0.8 Power 1.05 Source Voltages Power [pu] Q [pu] InvMicrosource1 0 InvMicrosource2 Load Time [s] Reactive Power 0.3 InvMicrosource InvMicrosource2 V [pu] Load Angle [rad] Time [s] Source Load Angle Frequency [Hz] Time [s] Load Frequencies Time [s] Time [s] 2 x 10-3 Unbalanced Energy E [pu-sec] Time [s] Figure 5-26 Experimental result of load transient from 0.385pu load to 0.77pu load. Pmax set to 0.433pu, constrained by battery power capabilities of source2, Ra=0Ω, KpPMax=48 (0.1273pu), KiPMax=960. R1=0.9Ω (0.0586pu), R2=0.9Ω (0.0586pu), Pfilt_BW=20Hz. Regardless, the application of power signal filtering has been shown to have a significant effect on avoiding the excitation of the 60Hz power oscillation. However, due to practical considerations, it was also shown that the amount of resistance required to have a desirable response in this case was significantly more than the value that would nominally allow stable operation.

221 Improved Power Response Utilizing Lead-Lag Compensator The second approach to limiting the pass-through gain of the controller is to implement the leadlag compensator presented earlier. Firstly, to limit the amount of free variables, the spectral separation between the two zeroes of the system was eliminated, placing them both at 60Hz. This allows for one free variable, the spectral separation between the poles and respective zeros. As previously mentioned, the ratio between each pole-zero pair must be equal to retain unity gain at higher frequencies. Therefore, the value of Pole2/Zero2 and Zero1/Pole1 was varied from 2.5x to 40x. (a) (b) Figure 5-27 Effect of increasing Pole2/Zero2 and Zero1/Pole1 around 60Hz (range x) on nominally tuned system eigenvalues The spectral separation range of the pole-zero pairs determines how many decades occur and therefore the magnitude of attenuation is achieved at the frequency of the voltage balance mode. For reference, the theoretical attenuation is between 8, 20, and 32db for the 2.5x, 10x, and 40x spectral separations, respectively. The actual attenuation of the voltage balance mode is less due to rounding of the gain characteristic that has zeroes located directly at the frequency of the mode. Nevertheless, increasing the spectral separation, effectively increasing the attenuation of the voltage balance pass-through gain, can be seen in Figure 5-27a to cause the voltage balance mode to approach the natural damping frequency of the R-L line impedance. Also, the Phi2

222 205 (relative voltage angle) eigenvalue can be seen moving to the right as a complex pair with the lower frequency pole of the lead-lag compensator. This complex pair still remains at a significantly higher frequency than the power regulation pole in the nominal case (at 0.5Hz). However, it can be seen that the lower Lead-Lag pole and the Phi2 pole may become the dominant characteristics with the power regulation bandwidth at a higher frequency. This would ultimately limit the rate at which the network achieves steady state. (a) (b) Figure 5-28 Effect of increasing Pole2/Zero2 and Zero1/Pole1 around 60Hz (range x) on tuned inverter microsource system (KpPMax=48, KiPMax=960) By examining the tuned case with the increased power regulation gains, the same characteristics appear as a result of the increasing spectral separation between the poles and zeros. It can be seen that the Phi2 and lower Lead-Lag poles are always a complex pair in this case but otherwise move in a very familiar way as compared to the nominally tuned base case. One change here is that the voltage balance poles are unstable for the first two cases of spectral separation and only until the 20db attenuation occurs, at 10x spectral separation, the response becomes stable. The one positive result is the relatively unaffected power regulation pole, presented in Figure 5-28b.

223 206 Figure 5-29 Transient load step simulation of an inverter microsource microgrid network with active resistance damping. Zero1/Pole1=Pole2/Zero2=10, KpPMax=48, KiPMax=960 The transient response characteristic in Figure 5-29 is quite similar to the previous case with only input signal filtering. The 60Hz and lightly damped lower frequency characteristic on the power and reactive power are present as well as a similar level and response characteristic as the previous case. Lastly, the frequency flat-topping can be seen as well, which also agrees with the previous case. All of these similarities make sense considering the pole location in the previous case was at 10Hz and the first pole of the lead-lag controller is at 6Hz, which are similar enough values to generate similar output characteristics. The major difference between the lead-lag compensator case and the previous case is the unity pass-through gain at higher frequencies.

224 207 Figure 5-30 Transient load step simulation of a mixed source microgrid network with active resistance damping. Zero1/Pole1=Pole2/Zero2=10, KpPMax=48, KiPMax=960 The mixed-source transient also shares similar characteristics with the previous case providing further evidence that the lead-lag compensation offers similar network characteristics throughout. 5.5 Dynamic Power Response Improvement Conclusions Voltage balance oscillations were identified to result from a lightly damped mode particularly for conditions in a small power system with low resistance as compared to the inductance. Secondarily, it was presented that the combined frequency droop gain further destabilized the

225 208 network mode, counteracting the damping effect of the physical resistance. More importantly, this relationship between resistance and the combined frequency droop gain with respect to stability was found to be independent of the network inductance. It was noted that with the aim of desired increases in power regulation bandwidth, that the frequency droop gain of the power limit controller would need to be increased, eventually jeopardizing network stability. With the need to mitigate the excitation of this network mode defined, three solutions were introduced and evaluated in the base case investigated earlier, as well as a case with increased power regulation gains. It was found that all solutions that were presented did enable stable responses with the increased power regulation gains, but it was also identified that the last two cases, the input filtering and Lead-Lag compensation, affected only the pass-through gain of the controller at the resonant frequency. Even in cases where the pass-through gain could be effectively eliminated, the mode will never become more damped than it is naturally, which has proven to be a significant issue in the face of large load transients. However, it was shown that the application of active resistance could increase the damping of the network mode beyond its natural state and it was also shown that said active resistance did allow for unaffected steady state operation. As a concluding point on this topic, both active resistance damping and power filtering were employed on a final experimental transient to demonstrate the possibility of using both types of controller techniques. The results are displayed in Figure 5-31 and interestingly demonstrate marginal increase in the overload energy. This is likely due to the rapid response of the power controller being faster than the 20Hz bandwidth of the power measurement filter. Regardless, the transient is well behaved except that the phase current offsets tend to persist for an extended

226 209 period of time, though again this could partially be as a result of phase current measurement offsets. 0.8 Power 1.05 Source Voltages Power [pu] InvMicrosource1 0 InvMicrosource2 Load Time [s] Reactive Power 0.2 V [pu] Time [s] Source Load Angle 0.15 Q [pu] InvMicrosource1 InvMicrosource Time [s] Load Frequencies 60.1 Load Angle [rad] Time [s] 2.5 x 10-3 Unbalanced Energy Frequency [Hz] Time [s] E [pu-sec] Time [s] Figure 5-31 Experimental result of load transient from 0.385pu load to 0.77pu load. Pmax set to 0.433pu, constrained by battery power capabilities of source2, Ra=3.07Ω (0.2pu), KpPMax=48 (0.1273pu), KiPMax=960. R1=0.9Ω (0.0586pu), R2=0.9Ω (0.0586pu), Pfilt_BW=20Hz.

227 210 6 Stability of Multi-Node Networks The stability of multi-node networks is an important consideration as up until this point, only two source networks have been considered and it is assumed that this configuration should be stable when implemented with many sources on the same network. To address this issue, the concept of passivity will be utilized by examining the eigenvalues of a modified design that keeps the voltage magnitude constant for a single grid-connected system. Passivity has the advantage that it can apply to a single network component and will remain stable providing that each component is also proven passive. Two tests for passivity will be investigated in this chapter. The first approach will calculate the stability limit over a range of parameters for steady-state operating points, leaving four freevariables to explore. This method is closer to the previous investigations of operating point stability for a single source, but incorporates the possibility of non-trivial voltage angles between sources. The second will investigate the characteristics of a source settling towards a steady state operating point in the power and reactive-power plane (P-Q plane). The error vector between the steady-state P-Q point and the operating point should be of decreasing magnitude to yield a stable response as evidenced in previous chapters and particularly by inspection from Figure 5-1. The simplifying assumption of constant voltage magnitudes was adopted in this chapter in an effort to reduce the order of the solution. 6.1 Passivity by Bounded Operating Point Eigenvalues Operating point eigenvalues have been investigated previously, yielding insightful results. This section will also utilize this method as well as provide a more thorough investigation into why an explicit solution cannot be defined to the satisfaction of readers looking for insightful results.

228 211 The system employed for this chapter is a third-order system similar to Figure 5-1, except that this model dispenses with the small angle approximation. This is done in order to more accurately investigate the characteristics of the system and investigate where the previously discovered relationships are valid, and where they break down. The three system equations are presented in Equations I = 1 V V I R + I L I = 1 V V I R I L φ = ω ω (6.1) (6.2) (6.3) Where the variables of the above equations can be further defined in terms of system parameters and states by the following equations P = V I + V I (6.4) ω = + (6.5) = sin = cos ω = ω = 2 60 (6.6) (6.7) (6.8) These system equations can be formed into an operating-point specific state-space matrix that is valid for small perturbations around the given operating point.

229 212 cos / = sin / 3 2 sin cos 2 cos + sin (6.9) The eigenvalues for this system are presented in Equation 6.10 and Equation = + + (6.10), = 2 2 ± 3 2 (6.11) The expressions use intermediate variables to simplify the expressions by use of Equations = (6.12) = (6.13) = 4 = 3 + cos + sin 4 (6.14) (6.15) = cos + sin (6.16) = cos + sin (6.17) Although the real value is the only pertinent value for determining stability, the remaining components of the equations still represent a significantly complex expression. Therefore, it is not readily discernible if any simplified dominant characteristics exist at all from this expression. Therefore, the remaining options for investigating a characteristic defining the stability constraints include operating point stability as one of the most useful. One of the greatest benefits of operating point models is that they utilize steady-state operating points to constrain

230 213 the states of the system. Therefore, it is only the parameters that need investigating. In this case with fixed voltage magnitudes, the steady-state power characteristic can be defined with respect to the phase angle, which can then be used to define the phase angle for a given power set-point. Further, the value of steady-state d and q-axis currents can be defined by cross solving equations 6.1 and 6.2 for steady-state conditions in equations 6.18 and _ = (6.18) _ = (6.19) The result relates Pset, φ, I d and I q, combining four of the free variables to one combined steadystate operating point. While this does not change the dimensionality of the system, it does reduce the dimensionality of the steady state points of the operating space by defining a steady state value of, φ, I d and I q for any value of Pset and vice versa. The remaining three parameters of L, R, and Mp can be explored for the nominal operational space. The angle between voltage sources is typically in the single-digit degrees to incite rated power flow, but may be higher for weak or systems with large amounts of line impedance. Therefore, the investigated range of phase will be within 45 on each side. Given that there are a wide variety of X/R ratios possible, as explored in Chapter 8, but in light of the assumption that CERTS networks are predominantly reactive, the X/R ratio will be varied from 20:1 to 1:1. Figure 6-1 and Figure 6-2 show the resulting frequency droop gain when considering the steadystate angle between the source and the grid. This is analogous in a two-source network to the total angle being double the reported value here. In a multi-source network, the result is varied, but this analysis should hold valid for each source connected.

231 214 Figure 6-1 Gain of Mp at stability limit for grid-connected single source as a function of voltage angle at steady state and line resistance. L=0.001pu Figure 6-2 Gain of Mp at stability limit for grid-connected single source as a function of voltage angle at steady state and line resistance. L=0.0001pu

232 215 The sloped characteristic indicates that there still exists the dominant dependence on line resistance affecting the maximum value of frequency droop gain before the system becomes neutrally stable. It can be seen that particularly for X/R ratio near 1:1 and for voltage angles near 45 that the limit increases significantly. In fact, the actual values had to be excluded as they were high enough to consider a negligible limit from a practical standpoint. To compare against the ideal characteristic defined in Chapter 5, the Mp limit data in Figure 6-1 and Figure 6-2 were re-scaled to illustrate the relative agreement with the previously stated characteristic. Figure 6-3 and Figure 6-4 show the resulting agreement and it is clear that there is a high degree of agreement for high values of X/R and slightly negative angles. One important point to consider is the absence of points beneath the 1:1 ratio of actual vs. theoretical limits. This is important because it implies that by using the limit implied by Equation 5.15, the frequency droop gain limit will be accurate in most of the operating space, but conservative in others. This bodes well for the general application of the simplified limit as it is generally useful even when it is not exact.

233 216 Figure 6-3 Ratio of realistic frequency droop gain limit against the ideal value defined for high X/R networks. L=0.001pu. Figure 6-4 Ratio of realistic frequency droop gain limit against the ideal value defined for high X/R networks. L=0.0001pu.

234 217 Another important way to visualize the effect of a conservative gain estimate across the nominal operating space is to examine the maximum real value of eigenvalues that result. Figure 6-5 and Figure 6-6 show that the small-signal model is stable over the explored nominal range. Unstable Stable Figure 6-5 Maximum real value of eigenvalues for frequency droop gain of nominal limitation, L= Unstable Stable Figure 6-6 Maximum real value of eigenvalues for frequency droop gain of nominal limitation, L=0.001

235 218 While this does not prove passivity by itself as it is technically not a robust solution, it gives reasonable confidence that the operating space explored was presented in enough detail to accurately represent the frequency droop gain limit. It also shows how at higher values of resistance the frequency droop gain limit diverges from the linear relationship, agreeing with Figure Passivity by Tests for Convergence The second approach utilizes a gradient method for defining the asymptotic stability criteria of a Lyapunov function. It tracks the convergence of an error vector in the P-Q plane towards the projected steady state point. Two approaches were employed in seeking to define a more general stability relationship for the relatively simple system. The first was to track the stored energy of the system by tracking the stored energy in the line impedance inductor elements. Also, a more general approach was taken to define a composite power and reactive power error vector gradient function as one option for a Lyapunov stability test Inductor Stored Energy Decay Tracking the stored energy in line inductors is one method to concretely define if the energy of the system is decreasing or increasing, possibly towards an unstable point. While some control states can be assigned a representative energy, others are not clearly defined. The approach utilized in this section accounts for only the physically defined energy states of the simple system, those defined from the magnetic field energy stored in the line inductors. This function implies that the steady state point should be at zero current and therefore implies a power setpoint of zero to ensure that the stored energy decreases toward steady state. To monitor this relationship, the stored energy in the d-q representation of the line inductors is summed and the resulting value should be decreasing. The derivation of the expression defining the derivative of

236 219 stored energy, which also defines the condition for passivity, is presented initially in Equation 6.20 and then simplified through Equations until the simplified general form is defined. = < 0 (6.20) + < 0 (6.21) < 0 (6.22) V V I R + I L + V V I R I L < 0 (6.23) I + I + V V + V V < I I I < 0 (6.24) + (6.25) The inequality in Equation 6.25 implies that the resistance has a directly proportional effect on the decay of energy in the simplified system per Figure 5-1 under evaluation here. It also relies heavily on the voltage differential between the sources, implying directly that for positive values of resistance and equal source voltages, that the stored energy would always decay, which makes intuitive sense from the analogy of shorted R-L impedance. Unfortunately this expression does not include consideration of the frequency droop gain M p that has been previously shown to cause system instabilities in the same system. Therefore, the result from this preliminary investigation lacks insight into gain and parameter limitations and highlights the role of the voltage angle in the growth or decay of stored energy in the system. Nevertheless, some special cases can be used to show the stability characteristics of the system. For example, if I d1 =0 and R 1 >0, the expression in Equation 6.26 is valid. V V < (6.26) This relationship further isolates the stability relationship between the controller-defined voltage differential and the resistive voltage drop when the d-axis current is assumed zero.

237 220 Also, for the case when V d1 =V d2 and R 1 >0, Equation 6.27 can be formed, which highlights the ratio of d-axis current to q-axis current, scaled by the line resistance, is an important parameter defining convergence to the steady state resting point. V V < + (6.27) Passivity by Convergence of P-Q Error Vector The alternative to tracking the physical stored energy is to define a representative function that will decay when approaching the supposed steady state operating point. In this section, the concept of a P-Q error vector is used as the representative energy function. Equation 6.28 defines the error vector which is simply defined as the error between the steady state power and reactive power commands and the instantaneous values. =, (6.28) =, = , + (6.29) _ = = + (6.30) = cos + sin + sin cos cos 9 4 sin cos 1 sin cos sin + cos + sin cos sin

238 221 The time derivative of the complex power vector is defined in Equation These two functions are multiplied together in the fashion of a dot product to define the gradient of the power characteristic in Equation 6.30 with the substitutions listed in Equations = 3 2 cos sin (6.31) = cos (6.32) = + + sin (6.33) The variable Qset is removed from Equation 6.30 by cross solving the steady-state current Equations with the power equation in Equation 6.4 and the reactive power equivalent to achieve Equation 6.34 under the precondition of 1pu fixed voltage magnitude. This effectively reduces the free variables from nine to seven (I d1, I q1, φ 1, R 1, L 1, M p1 and P set ) by fixing the voltage magnitude and constraining the steady-state power and reactive power relationship to a single variable. Q = P + ωl R 1 cos φ ωl + R sin φ ω L + R (6.34) Furthermore, it may be useful to define both Pset and Qset by the same approximation of fixed voltage magnitude by defining a new variable that implies the steady-state voltage angle, φ ss. Under the assumption of some steady-state voltage angle used in place of the instantaneous angle, the steady-state currents and voltages can be defined by the same method as described in Equations 6.18 and 6.19 for current and Equations 6.6 and 6.7 for voltages. These steady-state, voltage and current values can then be used to define the Pset and Qset values by Equations 6.35 and P = 3 2 V I + V I (6.35)

239 222 Q = 3 2 V I V I (6.36) Now the same P-Q error vector gradient can be calculated in a more insightful way as the instantaneous voltage angle is used alongside the steady-state voltage angle to simplify the analysis of the system stability by an energy-gradient method. _ = = + (6.37) = cos + sin + sin cos + sin 9 4 cos sin sin 1 cos sin cos + 1 cos + cos sin 9 4 sin 9 4 cos cos + sin Where = 3 2 sin cos + 1 cos + sin (6.38) = + (6.39) = (6.40) = + (6.41) Using the simplified function in Equation 6.37, the gradient characteristic can be visualized by defining system variables such as resistance, inductance, frequency droop gain, initial (instantaneous) voltage angle and steady-state voltage angle. The line currents can be varied over a two dimensional grid to demonstrate the form of the gradient when limited from the original seven-dimensional space down to two. While the use of exploring discrete system

240 223 configurations is not a robust analytical approach, the high dimensionality and complex gradient expression form prohibit drawing more general conclusions without exploring a finite number of discrete operating points to reduce the dimensionality of the system. This analysis will explore the resulting energy gradient function from Equation 6.37 against the gradient determined from dynamic simulation for the same operational points. The consistency between the two will help to validate the energy gradient function and also provide some interpretation of the resulting operating space and parameter sensitivities that will aid in qualitative conclusions. The current was varied from -1pu to 1pu in both the d and q axes which implies a 1.5pu power and/or reactive power from Equations 6.35 and This range was selected to encompass the maximum realistic operating range. However, the extent of the analysis presented here only investigates single or double parameter variations around a nominal case, which does not fully inspect the complexities or corner-cases of the nominal operating space. Nevertheless, it is helpful to initially inspect the system at a nominal steady-state point with only a current disturbance. In this case, the resistance and inductance are set to nominal values, and the frequency droop gain is set to its nominal limit. Also, both the instantaneous and steady-state voltage angles are set to zero, implying that the voltage difference between the sources is initially zero and that the complex power set points (Pset and Qset) are both zero as well. The divergence surface plot in Figure 6-7a shows a symmetrical and fully stable characteristic within the explored area of current, indicated by the negative values of divergence (i.e. convergent). The value at the origin implies that the system is neutrally stable only at the steadystate point and stable elsewhere. The gradient plot in Figure 6-7b shows the anti-clockwise

241 224 inward spiral trajectory towards the indicated steady-state complex power point. This trend is expected from the discussion in Section 5.2 where the simplified system used here was first introduced. There in Figure 5-1, the primary oscillator loop was identified which accounts for the primarily circular trajectory and it was also noted that the resistance encourages convergence to steady state. Here it is shown in Figure 6-7b how that characteristic translates into a natural response gradient function Divergence of P-Q vector from steady-state point. R=0.01 L= Mp= Phi=0 Phi ss =0 Q [pu] Gradient of P-Q plane by simulation. R=0.01 L= Mp= Phi=0 Phi ss =0 Steady state Start point Gradient Id Iq P [pu] Figure 6-7 Simplified system nominal configuration energy error vector divergence and P-Q plane operating point gradient over a range of operating point currents. The first variation of the nominal configuration presented in Figure 6-7 will include a variation of the voltage angle away from the steady state point. This result is presented in Figure 6-8 and clearly indicates how sensitive this convergence function is to the voltage of the system. It can clearly be seen that the energy function tends to grow when the system has negative d-axis current flowing and when the q-axis current is small. The result from Figure 6-8a is mirrored in Figure 6-8b where the gradient clearly centers around the complex power point that would develop if the instantaneous voltage vector was held constant. However, it is clear that for nonzero frequency droop gains that this would not occur and that the voltage angle should approach the origin, even for this case as demonstrated by dynamic simulation. Therefore, this method,

242 225 while it has opportunities for providing a robust solution to determining system stability, doesn t necessarily indicate if a nonlinear system will eventually reach steady state Divergence of P-Q vector from steady-state point. R=0.01 L= Mp= Phi=0.01 Phi ss =0 Q [pu] Gradient of P-Q plane by simulation. R=0.01 L= Mp= Phi=0.01 Phi ss =0 Steady state Start point Gradient Id Iq P [pu] Figure 6-8 Simplified system offset voltage angle configuration energy error vector divergence and P-Q plane operating point gradient over a range of operating point currents. Phi is offset by 0.01 radians from the steady-state voltage angle of 0 radians. Another point to mention in Figure 6-8 is the issue of voltage angle sensitivity to the increase or decrease of the energy function. In section 6.2.1, the correlation of voltage to the increase of the stored energy was noted and here again it is demonstrated in a pseudo energy function that uses power and reactive power instead of energy stored in the inductors. To validate that this relationship is tied to a voltage angle difference between the instantaneous voltage angle and the steady-state voltage angle, the steady-state voltage angle was also modified to match the offset in the instantaneous voltage angle such that they are aligned. Figure 6-9 indicates that the P-Q error vector always converges to the steady-state point in this case. This result implies that a voltage angle differential between the instantaneous and steady-state point causes an energy error vector divergence in some areas of the operating space.

243 Divergence of P-Q vector from steady-state point. R=0.01 L= Mp= Phi= Phi ss = Id Iq Q [pu] Gradient of P-Q plane by simulation. R=0.01 L= Mp= Phi= Phi ss = Steady state Start point Gradient P [pu] Figure 6-9 Simplified system offset voltage angle and power set-point configuration energy error vector divergence and P-Q plane operating point gradient over a range of operating point currents. Phi and Phi_ss are both offset by 0.01 radians from zero. Previously in this document, it was identified that the frequency droop gain had an effect on the stability of the system. However, the sensitivity of the frequency droop gain doesn t seem to have as pronounced of an effect here as Figure It can be seen that for a re-investigation of the nominal case with a 10x increase in frequency droop gain shows that for conditions when the voltage angle and the steady-state voltage angle are both zero, that the error vector will always converge to steady state Divergence of P-Q vector from steady-state point. R=0.01 L= Mp= Phi=0 Phi ss =0 Q [pu] Gradient of P-Q plane by simulation. R=0.01 L= Mp= Phi=0 Phi ss =0 Steady state Start point Gradient Id Iq P [pu] Figure 6-10 High frequency-droop gain configuration energy error vector divergence and P-Q plane operating point gradient over a range of operating point currents. Mp is 10x the nominal limit value as defined in Chapter 5.

244 227 The method diverges from the previous conditions in that this case has the predicate that the voltage vector error is zero and therefore the system should tend towards steady state as defined by the resistance of the system and the inverse of the inductance of the system. Another case was investigated to ensure that the difference between the voltage angle and the steady state voltage angle was responsible for the condition of convergence of the error vector to steady state, presented in Figure Divergence of P-Q vector from steady-state point. R=0.01 L= Mp= Phi= Phi ss = Gradient of P-Q plane by simulation. R=0.01 L= Mp= Phi= Phi ss = Steady state Start point Gradient Q [pu] Id Iq P [pu] Figure 6-11 High frequency-droop gain and voltage angle bias configuration energy error vector divergence and P-Q plane operating point gradient over a range of operating point currents. Mp is 10x the nominal limit value as defined in Chapter 5 and both the voltage angle and steady-state voltage angle are offset by equal amounts. To clarify, the frequency droop instability investigated earlier occurred when the voltage angle was allowed to vary away from the steady-state point by the frequency droop action, which is a mechanism that is not fully captured in this analysis and detracts from the usefulness of this particular energy function. Other energy functions may be formed in the future as a part of continued research in this area that may overcome this issue. Nevertheless, the relationship between frequency droop gain and convergence is somewhat affected by the frequency droop gain, but only at values significantly beyond the theoretical stability limit.

245 Divergence of P-Q vector from steady-state point. R=0.01 L= Mp= Phi=0 Phi ss = Gradient of P-Q plane by simulation. R=0.01 L= Mp= Phi=0 Phi ss = Steady state Start point Gradient Q [pu] Id Iq P [pu] Figure 6-12 Voltage angle bias configuration energy error vector divergence and P-Q plane operating point gradient over a range of operating point currents. The steady-state voltage angle is offset from the voltage angle initially at zero. 500 Divergence of P-Q vector from steady-state point. R=0.01 L= Mp= Phi=0 Phi ss = Gradient of P-Q plane by simulation. R=0.01 L= Mp= Phi=0 Phi ss = Steady state Start point Gradient Q [pu] Id Iq P [pu] Figure 6-13 High frequency droop gain and voltage offset configuration energy error vector divergence and P-Q plane operating point gradient over a range of operating point currents. The frequency droop gain is set to 10x the nominal limit; steady-state voltage angle is offset from the voltage angle initially at zero. A case with the steady-state voltage offset is used to explore the frequency gain sensitivity of the power error divergence function of the single inverter microsource system here. Figure 6-12, Figure 6-13 and Figure 6-14 are associated with the nominal frequency droop gain, 10x nominal, and 100x the nominal frequency droop gain, respectively. Both the 1x and 10x cases demonstrate regions of divergence (increasing energy), but still seem to both have gradient functions with similar points of rotation. While the 1x case in Figure 6-12 centers primarily

246 229 around the origin, the 10x case in Figure 6-13 is slightly offset from the origin towards the steady-state point. The 100x case demonstrates a significant change in the characteristic with significantly higher gradients for the current points furthest from the steady-state operating point. This change in characteristic implies some effect of the frequency droop gain, but only at a significantly higher value than expected, which indicates that the mechanism for instability is tied to the location of the voltage vector as a function of the frequency droop gain, and not necessarily the frequency droop gain itself X: -1 Y: 1 Z: 1059 Divergence of P-Q vector from steady-state point. R=0.01 L= Mp= Phi=0 Phi ss = X: 0 Y: 0 Z: 0 X: 1 Y: 1 Z: X: 1 Y: 0.25 Z: Gradient of P-Q plane by simulation. R=0.01 L= Mp= Phi=0 Phi ss = Steady state Start point Gradient Q [pu] Id Iq P [pu] Figure 6-14 Very high frequency droop gain and voltage offset configuration energy error vector divergence and P- Q plane operating point gradient over a range of operating point currents. The frequency droop gain is set to 100x the nominal limit; steady-state voltage angle is offset from the voltage angle initially at zero. One last case investigated includes sensitivity to the line resistance and the effect on the divergence characteristic as well as the gradient. With the resistance increased 10x from 0.01pu to 0.1pu, the X/R ratio fell to 0.377:1, a significantly low value that is dominated by the resistance of the network. This dominance can be seen in the result presented in Figure 6-15 by the stable characteristic in the face of the offset in voltage angle between the instantaneous value and the steady state value, something previously not found. Secondarily, the gradient plot shows a significantly more damped system than previously, indicating that the P-Q transient

247 230 characteristic would quickly converge to the steady-state value. Lastly, the dominant resistance characteristic can be seen in the significant cross-coupling of the reactive power for the steadystate point which results from an angular differential between two microsource voltage sources. 0 Divergence of P-Q vector from steady-state point. R=0.1 L= Mp= Phi=0 Phi ss = Gradient of P-Q plane by simulation. R=0.1 L= Mp= Phi=0 Phi ss = Steady state Start point Gradient Q [pu] Id Iq P [pu] Figure 6-15 High resistance configuration energy error vector divergence and P-Q plane operating point gradient over a range of operating point currents. The resistance is set to 10x the nominal value. 6.3 Multi-Source System Stability Conclusions This chapter has investigated two main topics in the area of multi-node source stability. Both have approached this issue by examining a single source connected to a stiff grid connection which serves as an analog for a node in a multi-source multi-modal configuration. Essentially, this source configuration was used as a platform to investigate the property of passivity. The first section investigated the operating point stability of a microsource when the power flow angle varied between the microsource and the grid analog. It provided a more complete investigation of the power frequency droop relationship while varying both resistance and load (voltage) angle. The result identified that the previously defined characteristic was valid for high X/R systems, but diverged significantly for configurations with X/R less than 2:1 where a significantly higher frequency droop gain could be used without jeopardizing the stability of the

248 231 system. However, the primary drawback to this approach is that it is based on an operating point model and does not represent the full nonlinear characteristic. The second approach investigated the passivity of the system through the tendency of a power and reactive power vector to tend towards the steady-state value. The use of this pseudo-energy function allowed the incorporation of the frequency droop gain, something the actual stored energy function did not include. The result from this analysis used the simplifying assumption of constant voltage magnitude of 1pu to identify the relationship between the instantaneous voltage angle and the steady-state voltage angle as an indication of the likelihood of a divergent energy characteristic. The frequency droop gain did not have a significant effect by this method until values were used well beyond the supposed limit value, which limits the usefulness of the selected energy function in determining convergence to steady state following a transient as a function of frequency droop gain. Lastly, the line resistance was found to have a positive effect on increasing the rate of convergence of the power and reactive power characteristic to the steady-state point.

249 232 7 Analysis of Microgrid Stability Considering Penetration of Grid-Following versus Grid-Forming Sources As follow-up to the discussion in Chapter 4 investigating the transient capabilities of gridforming versus grid-following sources, the stability of networks primarily composed of gridfollowing sources is addressed here. It was previously stated that only grid-forming sources could operate alone in island due to their inherent shared-master approach to producing a voltage at a given frequency that enabled load balancing between sources. The grid-following variety was noted to demonstrate superior abilities to maintain constant power output, or even changing the power output with respect to internal measured variables. However, the ability of gridfollowing sources to perform source-shedding tasks to track load is investigated here. It will be shown initially that grid-following sources have been used as negative load for some time without issue in PV and fuel-cell applications, but that the added challenge of source shedding can cause significant stability issues. Source shedding is the practice of reducing power output for increases sensed in the measurement of the frequency that occurs when the load is light. This section will investigate these issues and confront the question of what the limit of grid-following source penetration is while maintaining microgrid stability. It has already been established that this number cannot be 100%, but it will be shown that the percentage of grid-forming sources (including rotating motors) required to maintain stability is largely tied to the size of step-load transients that the microgrid will encounter. 7.1 Fixed-Power Grid Following Limits The grid-following source model employed for this chapter is largely similar to that used in chapter 0. It utilizes a PLL to track the voltage vector at the local frequency and synchronous-

250 233 frame current regulators to inject real power in phase with the q-axis voltage vector and reactive power in quadrature to that in the d-axis. The representative source equations for the gridfollowing source are provided below in equations PLL: intv _ = _ V = V (7.1) Phi = = _ (7.2) Phi = Phi Phi (7.3) Omega = Omega Omega (7.4) Current Regulators: I _ = _ I I _ = _ I (7.5) (7.6) inti _ = _ I = I _ (7.7) inti _ = _ I = I _ (7.8) _ = V + _ + _ + + (7.9) _ = V + _ + _ + (7.10) Reference frame transformations = V cos h + V sin h (7.11) = V cos h V sin h (7.12) = I cos h + I sin h (7.13) = I cos h I sin h (7.14)

251 234 _ = V _ cos h V _ sin h (7.15) _ = V _ cos h + V _ sin h (7.16) Supporting reference frame transformations are presented in equations They enable the voltage and current measurements to be processed in the reference frame of the model defined by the PLL. They also translate the output voltage back to the reference frame of the network. The PLL is simply a PI regulator on the sensed d-axis voltage, minimizing the value to zero in order to remain in phase with the q-axis only. The current regulator employs standard voltage, resistance, and inductance decoupling of the source filter elements to achieve a well damped response to changes in power command. To ensure that the response is tuned properly, the current regulator gains were tuned such that the current regulation bandwidth pole and the integrated current regulation bandwidth pole were placed at 770Hz and 51Hz respectively. This spectral separation allows for a well damped response under varied interconnecting impedances that may cause the poles to migrate together. However, the PLL also must be tuned to yield a well-damped response through its own proportional and integrated gains. The interaction of the PLL caused the implemented tuning to have an oscillatory component, but the damping was balanced with closed loop frequency tracking bandwidth to achieve a reasonably damped solution. The process of tuning a current regulator and PLL together is not the focus here as it is well documented [64], but it is critical to functionality of a grid-following source in any grid-tied configuration. The system investigated will pair one grid-forming source with the aforementioned gridfollowing source. Only one source of each type is required in this investigation of relative scale by scaling the impedance of either source to effectively resemble multiple sources in parallel

252 235 [32]. Therefore, the key variable used in this section will be that of relative scale. To further simplify the analysis, only the grid-forming source will be scaled, either increasing or decreasing the effective number of paralleled sources to mimic multiple or fractional (lower power) instances. Figure 7-1 Eigenvalue of two-source network with grid-forming and grid following sources at 1:1 scale The nominal eigenvalues of a 1:1 scaled system are presented in Figure 7-1. Utilizing a participation factor analysis similar to that introduced in section 3.2.4, the eigenvalues can be identified as follows: The load current eigenvalues are at around -900Hz, followed closely by the Id and Iq proportional current regulation poles at -770Hz. The L-C filter eigenvalues for source2 are oscillatory at -22Hz+/-j800Hz so they will be monitored to ensure they do not migrate towards the imaginary axis. The capacitor voltage regulation eigenvalue combined with the PLL pole to create a complex pair at -55Hz+/-J300Hz which will also be monitored. Next, the Phi1 eigenvalue and the integrated current regulation eigenvalues are at -51Hz and -33Hz+/-j28Hz respectively. Finally, the integral PLL error is at -18Hz, the source2 voltage regulation at pole is at -7Hz, and the Phi2 pole sits at the origin as in previous cases. To illustrate the effect of scaling on this system, the relative scale of source1:source2 is varied, where source1 is the gridfollowing source and source2 is the grid-forming source. In Figure 7-2 the scale is varied from

253 236 1:1 to 64:1 by increasing the resistances and inductances, and reducing the capacitance of the filters. The frequency and voltage droop gains are also increased as the second source is scaled down to reflect the frequency change that would become more sensitive to power output. Numerous trends exist that complicate the analysis of the eigenvalue migration, but the reality of system scaling and the increased impedance associated with the decrease in power rating of the grid forming source essentially creates a high impedance path to the reference voltage. This issue plays a significant role in the response of the PLL and the current regulation abilities. It can be seen that near the ratio of 64:1 that the PLL eigenvalues become unstable, defining the limit of stable source pairing between 32:1 and 64:1 for this case, but is realistically defined by the PLL gains. If the PLL was tuned more conservatively or if the current regulators were tuned to be more damped, the 64:1 ratio and higher can be made stable at the expense of current regulation bandwidth. As mentioned previously, grid-following sources are not a new concept for grid-tied applications where they are considered negative load. It has been shown here that significant ratios of nearly 32:1 have been shown to be stable and it has been proposed that higher ratios are capable with appropriate tuning of the PLL and current regulator gains.

254 Eigenvalue migration, varying relative scale from 1:1 to 64:1 4 Eigenvalue migration, varying relative scale from 1:1 to 64: Oscillatory Frequency [Hz] PLL Eigenvalue Settling Frequency [Hz] Settling Frequency [Hz] Figure 7-2 Eigenvalue migration for changing relative scale (ratio) between grid-following and grid-forming sources from 1:1 to 64:1, indicating PLL eigenvalue becoming unstable For this scaling investigation to yield the result of the PLL stability to appear as the weakest link requires careful power balancing. The real and reactive power of the grid forming source was set very closely to that of the load to reduce the power required from the scaled-down second source. With even the case of 32:1, for a small per-unit disparity between the source1 power and load would be amplified 32x for the grid forming source in its own per-unit base. This value may be well beyond the real or reactive power ratings without this careful matching of the load and grid-forming source power. In reality, this power balance would have to be actively managed either through a supervisory control, or through voltage and frequency droop algorithms. 7.2 Variable-Power Grid Following Limits To ensure that load is properly balanced in islanded systems with a significant difference in relative capacity of grid-forming versus grid-following sources, power-versus-frequency and reactive power-versus-voltage magnitude droop can be employed even in the case of the gridfollowing source. Similar to the case in section 4.4.1, the standard form of droop can be implemented here for grid-following sources by reversing the measured and controlled variables.

255 238 Figure 7-3 is an illustrative example diagram for the two-source system investigated here, neglecting the reactive power balancing mechanisms for diagrammatic simplicity. In the top of the figure is a proportional and integrated (Mp2, Jp2) frequency droop that represents the actions of a grid-forming source where the measured value is power and the controlled variable is inverter (or induction machine) frequency. However, the grid-following source uses the PLL to measure the frequency and then commands a new output power directly through the Mp1_inv gain, which is the inverted value of the standard power-versus-frequency droop gain Mp for source 1. Jp2 1 s 0 1 Pset1 P_cmd Pset2 Constant Mp2 2*pi*60 Omega2 1 s Vd_cmd Source angle Vd_errOmega_PLL PI PLL regulator 1 Omega_PLL 2*pi*60 Mp1_inv Omega_nom1 Omega_nom Omega_PLL1 Omega Id2 0 Vd2 Vd2 Terminator1 Fixed voltage2 Vq2 1 Vq2 Iq2 Iq Fixed voltage 0 Fixed Current Id1 Iq1 Vq_load Vd_load Vq Line Impedance L-C-L Model Figure 7-3 Functional diagram of a simplified grid-following source paired with a grid-forming source to illustrate mechanism of frequency measurement feedback on power command The inversion of the frequency droop gain used as a gain in the grid-following system is introduced to bring focus to the fact that that in the case of the grid-following source the inverse of the frequency droop gain can cause stability issues.

256 239 The actual frequency-vs.-power and voltage-vs.-reactive power relationships for the gridfollowing source are provided in Equations and are mathematically identical to the previous case with a reversal of the independent and dependent values. Power and Reactive Power Calculation = + / 1 (7.17) = / 1 (7.18) I _ = 2 3 / I _ = 2 3 / (7.19) (7.20) With the typical values of droop, the addition of this feedback causes a significant instability. This unstable loop is formed through the PLL, the frequency vs. power feedback gain Mp1_inv, and the grid-forming source frequency droop gain Mp2. It can be seen in Figure 7-4 that the minimum value of droop gain in the grid-following source for a stable system is quite high. The value drops with respect to Kp_PLL, confirming that the proportional path through the PLL is the primary path through the PLL upon which the instability propagates. The integrator gain of the PLL appears to have very little effect on the minimum gain requirement, indicating very little involvement.

257 240 Stable Unstable Figure 7-4 Minimum values of frequency droop gain to achieve a stable system utilizing droop characteristics on the grid-following source Regardless, the minimum value for any case in Figure 7-4 is too high for normal operation in a microgrid which is around Mp=3.77, keeping a small value to ensure that frequency excursions are not significant for the full range of power outputs. Therefore, a method should be devised to allow nominal values of frequency droop values that essentially reduce the gain of the unstable loop Frequency Feedback Filtering One method for limiting the gain of this feedback loop at the unstable frequency is to place a low-pass filter on the feedback signal. If the bandwidth of this filter is well below the bandwidth of the current regulators and the PLL, then it can be considered the dominant pole that defines the rate of balancing load via droop algorithms. Figure 7-5 shows the eigenvalue migration that occurs when the frequency measurement filter is added to enable nominal values of frequency droop. It shows a sweep from 0.125Hz to 2Hz and indicates that around 1Hz bandwidth is where the system becomes unstable. This result helps define the maximum rate at which a grid-

258 241 following source can re-balance the system load and is an important quality when considering the conditions when load is added quickly (step-load changes). Figure 7-5 Eigenvalue migration of two-source system with grid-following and grid-forming source of equal sizes. Voltage measurement and frequency measurement filter bandwidths are varied from 0.125Hz to 2Hz With such a restriction on the rate at which power is re-balanced, the duty of load tracking in the interim falls on grid-forming sources. Therefore, the short term power capabilities of the gridforming sources on the network must be greater than any step load, or system fault may occur. Considering the transient environment of microgrids, this practical limitation defines the ratio of grid-following to grid-forming sources required to remain stable is significantly lower ratio than demonstrated previously. This consideration also supports the bias towards grid-forming control when practical to support the microgrid during load transients. 7.3 Grid-Following Penetration Conclusions The limits of penetration of grid-following sources in an islanded microgrid have been defined within two contexts: the source ratio and the load ratio. The source ratio was validated to 32:1 where the combined power rating of the grid-following source significantly outweighed that of the grid-forming (droop controlled) sources power rating. The relative scaling concept outlined in [32] was used to represent multiple machines by scaling them by their impedance and inertial

259 242 values. In this case, the frequency and voltage droop gains were also scaled consistent with the analog to inertia. The findings from the scaling analysis show that when the source ratio grows significantly large, that the PLL tuning and/or current regulator gains have to be de-tuned to accommodate the weak grid reference and maintain stability. However, the point was introduced that the grid-following source power had to be maintained close to that of the load due to the multiplication effect on per-unit loading with high grid-forming to grid-following source ratios. Without this balance, the grid-forming source would easily exceed its maximum power rating and fault or fail. The power-vs.-frequency droop concept was also was applied to the grid-following source and the complicating fact that the minimum required power-vs.-frequency gain was significantly higher than practical microgrids would allow. Finally, a low-pass filter was introduced on the frequency measurement from the PLL to the power command generator, which enabled reasonable frequency-vs.-power gains at the expense of load-balancing bandwidth. In the case presented, a 0.25Hz bandwidth was selected for frequency measurement and 1Hz bandwidth for voltage magnitude measurement. This restriction in power and reactive power re-balancing bandwidth elucidated the fact that all load changes will initially be supported by the grid-forming sources. Considering the transientrich environment of a microgrid, this requirement of grid-forming sources places the most significant practical restriction on the penetration of grid-forming sources.

260 243 8 Scalability of Active Damping Results to Microgrids of Different Power Ratings So far, the results shown have been based on an individual converter rating of 15kW which is on the smaller end of some of the more recent projects [65] that have extended into the MW scale. Many system parameters vary with power rating including voltage, filter inductance, line resistance and frequency. Conveniently, the use of the per-unit system enables a parameter set that varies to a lesser degree. Beyond this, the power rating and voltage rating are removed as free variables in a per-unit scaled system. Therefore, the two remaining parameters of interest are the reactive and resistive impedance characteristics of the lines connecting the sources together and the impedance of the grid connection. As a first step to extending the validation of results to exploring a range of sizes, this section will first investigate the impedance characteristics of a sampling of grid-tied inverter applications. 8.1 Reactance and Resistance Values as a Function of Inverter Scale A publication investigating the stability of wind turbine distributed generation for grid-tied applications [66] provides a good baseline for a range of inverter filter values as a function of size. It also introduces the possibility of a weak grid supply feeder and the presence of an isolation/step-up transformer. Table 8-1 and Table 8-2 include filter impedance values for three power levels of grid-tied inverters. They range from 3kW to 6MW, representing the spectrum of microgrid applications. Table 8-1 also includes the representative impedance for the feeder line for both a stiff and weak connection. In the higher power case of 6MW from [67], it is assumed that the transformer impedance is 5% and that the grid impedance is assumed to be 5% as well.

261 244 The resulting total combined impedance from the output of the filter to the infinite grid is 10%. This analysis adopts the same assumption, with the assumed X/R of 5.5. Table Impedance parameters from [66] for 500kW and 3kW grid-tied inverters including the possibility of a weak feeder Table Megawatt scale inverter (6MW) filter values from [67] For cases of inverter power ratings below 6MW, the transformer impedance is defined by a standard table of NEMA transformers in Table 8-3.

262 245 Table NEMA transformer ratings by size [68] Transformer Rating [KVA] X/R R [%] X [%] Z [%] Nine cases were compiled and their per-unit total impedances were calculated to define the representative operating space for the scaling analysis to follow. The total reactance and resistance values are plotted in Figure 8-1. The low power cases tend to have significantly lower per-unit reactance than the higher power cases, but the lower power cases also exhibit the greatest variability of total system resistance. Where resistance in a higher power case would not only be less efficient, the power dissipated in the lines would pose more of a heat dissipation issue in the interconnecting lines. The trend follows that higher power systems are typically designed to be more efficient, but at the cost of larger inductive elements. This case is evidenced by the 6MW representative large system test case where the resistance is in part due to the appropriately sized feeder to accommodate the full power rating of the source.

263 246 Resistance [pu] Reactance [pu] 6000 kw 500kw 500kw weak 500kw w/xfmr 500kw weak w/xfmr 3kw 3kw weak 3kw w/xfmr 3kw weak w/xfmr Test Cases Figure Reactance vs. resistance characteristics for grid-tied inverters of varying scales Test cases were selected to serve as a representative envelope for both small and large scale sources. The test case operating points are presented in Table 8-4 including the baseline case that has been the subject of this study so far. These operating points form the sub-set of cases that will be investigated in this section to answer the question of applicability of the previous results to microgrid system scaling. Table 8-4 Scaled system source impedances for selected system sizes Representative Sample Reactance [pu] Resistance [pu] X/R C [pu] =L1+L11+Lxfmr+Lfeeder =R1+R11+Rxfmr+Rfeeder Baseline 25.3% =5.9%+18.4%+0.5%+0.5% 1.77% =0.59%+0.184%+0.5%+0.5% Small system, stiff 1% 2% feeder =0.66%+0.33%+0.005%+0.005% =0.1% %+0.9% Small system, weak 11% 10% feeder =0.66%+0.33%+5%+5% =0.1%+0.05%+4.97%+4.97% Large system, medium 45% 3.5%

264 247 voltage =15%+30%+0.5%+0.5% =1.5%+2% Other System Scaling Considerations Interconnection cabling will be ignored in this study as it represents another free variable that would complicate the findings, but affects the system minimally. Cable impedances in [69], have an X/R ratio of approximately 1:1 for 700kcmil 6kV line. This would be the appropriate hook-up wire for a 500kW system. It has a resistance rating of 30mohm/1kft which at 100ft, represents an R=0.0016pu. When this impedance is added to a typical 500kVA transformer perunit resistance of [68] it is only a 15% increase. The resulting total X/R is reduced from 3.85 to 3.48, which is assumed to be a negligible change in this case. Another consideration is of power levels that are well beyond the 6MW limit that will be explored here. As shown in Table 8-5, transformer impedances for higher power systems follow the trend of increasing X/R ratios, with values of 100:1 as standard here. To a limited degree, the results applying to the high power operating point will be useful towards higher power systems in general if the trend is extrapolated. Table 8-5 Transformer impedance values for (a) 180MVA and (b) 20MVA transformers from [70]

265 System characteristics and frequency droop gain limitations for grid-connected inverters One of the two pertinent configurations of microsources in microgrids is the grid-connected mode. In this configuration, additional load transients are largely absorbed by the infinite grid connection. Therefore, the previously investigated load transient has little bearing on this configuration. However, power control bandwidth still competes with stability when power control is desired. The typical droop gains required for power balancing in a microgrid are low and typically do not pose stability issues. Therefore, this work will make the assumption that power control, and the increased proportional power-versus-frequency gain associated with the power limit controller, is the most jeopardizing to source stability. The system configuration investigated in this section follows a single source with both inverter filter and grid-tie feeder impedance consistent with Figure 8-2. The second source in the analysis here will be a fixed-frequency grid analog, while the first source will incorporate a nonzero power vs. frequency droop gain. Figure 8-2 Grid-tied impedance model used for neutral stability boundary for scaling analysis The model used for this analysis is displayed in Figure 8-2 and utilizes a transformer and feeder impedance model to represent a realistic impedance model for grid-tied application. The load for this study is set at 0.9pu (Rload=1.33), for reference, but this is not expected to affect the result considering the findings that noted the insensitivity to loading in section

266 249 To first analyze the system characteristics during normal grid-connected operation in the linear power-versus-frequency droop region, the droop gain was set to the nominal value of 1% (Mp=3.77 for 60Hz system) and the eigenvalues were determined from the resulting steady-state operating point. 1.5 x 104 Eigenvalue migration, varying scale of system Oscillatory Frequency [Hz] Base Case Small system, stiff feeder Small system, weak feeder Large system, medium voltage Settling Frequency [Hz] Figure 8-3 Grid-connected eigenvalues for various scales of systems with nominal frequency droop gain 400 Eigenvalue migration, varying scale of system Oscillatory Frequency [Hz] Base Case Small system, stiff feeder Small system, weak feeder Large system, medium voltage Settling Frequency [Hz] Figure 8-4 Grid-connected eigenvalues for various scales of systems with nominal frequency droop gain, medium frequency zoom

267 250 As might be expected, the system with the highly resistive weak feeder exhibits the most damping for the voltage balance eigenvalues at 60Hz in Figure 8-5. It is also interesting to note the vast similarity between the baseline system and the large system. The green pole at - 3.8Hz+j20Hz and the red pole at -1.3Hz+j7Hz represent the voltage balance poles, the ones that would eventually go unstable with increased voltage and frequency droop gains. The X/R ratio appears to have a significant impact on the damping and oscillatory frequency of these poles where the system with the lowest X/R ratios exhibit the most damping. 60 Eigenvalue migration, varying scale of system Oscillatory Frequency [Hz] Base Case Small system, stiff feeder Small system, weak feeder Large system, medium voltage Settling Frequency [Hz] Figure 8-5 Grid-connected eigenvalues for various scales of systems with nominal frequency droop gain, low frequency zoom To test the limits of voltage and frequency droop gain, the two were varied to find the conditions of neutral stability in the grid-connected configuration. Figure 8-6 shows the result for the four selected cases and indicates that while systems with high X/R ratios have a fairly constant sum of voltage and frequency droop gains, that the systems with low X/R ratios have a significantly different characteristic particularly at low values for either gain. The inductance and resistance play a role here in system characteristic as evidenced by the

268 251 fact that the total resistance of the large system case is significantly higher than the base case, but that the eigenvalues and droop gain limit characteristic remains similar. Earlier it was stated that inductance did not play a role in system stability from the voltage balance mode, but this investigation of scale indicates that systems with low X/R ratios do not follow the expected trend for neutral stability consistent with Equation 5.15 except for conditions where the voltage and frequency droop gains are similar. Frequency droop gain (Mp/Mp_limit) Theoretical Limit Base System Small system, stiff feeder Small system, weak feeder Large system, medium voltage Frequency droop gain (Mp/Mp_limit) Theoretical Limit Base System Small system, stiff feeder Small system, weak feeder Large system, medium voltage Voltage droop gain (Mq*KiV/Mp_limit) Voltage droop gain (Mq*KiV/Mp_limit) Figure 8-6 Frequency and voltage droop gains as compared to the theoretical limit values for conditions of neutral stability in (left) full view and (right) zoomed The value of Mp_limit referenced in Figure 8-6 was determined from the total resistance of the line between the inverter and the grid that is the sum of the inverter-side inductor resistance, the output side inductor resistance, the resistance of the transformer, and the resistance of the grid feeder. One interesting finding is that the distribution of the resistance doesn t seem to affect the results here, as the systems with significantly different X/R ratios in each branch exhibit characteristics based on their sum-total X/R ratios, independent of system load and filter capacitance.

269 252 Oscillatory Frequency [Hz] Base System Small system, stiff feeder Small system, weak feeder Large system, medium voltage Ratio of droop gain (Mp/Mq) Figure 8-7 Frequency of oscillation as a function of ratio of frequency and voltage droop gains Another interesting finding is the frequency of oscillation at the condition of neutral stability for different combinations of voltage and frequency droop gains. Even for the systems with low X/R ratios, the oscillatory frequency is near 60Hz when the voltage and frequency gains are nearly equal. Three of the four systems intersect 60Hz at the 1:1 ratio and only the small system, weak feeder case exhibits slightly higher frequency oscillations. It is important to note as well that the damped oscillations leading to stable systems yielded a lower frequency trend versus the increasing frequency as the instability was more severe. This agrees with physical principles of excited modes, but is interesting to note that the effect is more pronounced for systems with low X/R ratios. 8.3 Conclusions on Applicability of Findings to Scaled Systems Of the four cases examined in this section that represent the full range of power anticipated in microgrid applications, the primary disparities were found with systems that exhibited low X/R ratios. Small systems appeared to have the largest variability in total line resistance, but uniformly had less per-unit inductance than in the case of large systems. This condition is likely driven from voltage compliance standards that are more challenging for lower inverter switching

270 253 frequencies and require significantly more filtering inductance. The base case investigated previously in this document had an impedance characteristic between the high and low power systems, which in part validates the data obtained to generate the various test cases. A nominal operation eigenvalue investigation revealed that systems with low X/R ratios exhibited voltage balance eigenvalues that were significantly more damped and at lower oscillatory frequencies than with systems with higher X/R ratios that mimicked the base case. This was one of the more important findings of the scaling investigation as it implies that conditions with nearly equal voltage and frequency gains are different than when the two types of droop gains that are significantly different from each other. Systems with high X/R ratios were shown to be consistently limited to the previously discovered combined gain limit defined by the interconnecting resistance. However, the systems with low X/R ratios can have gains beyond the limit value for either voltage or frequency droop while maintaining system stability. The most important finding in the context of the scaling investigation here is that the results discovered previously are only applicable to systems with high X/R ratios and for sources that have similar voltage and frequency gains. The capability of low X/R systems to remain stable in the face of higher values of voltage or frequency gain represents an interesting point of future work. Nevertheless, practicality states that efficient power systems will exhibit low resistance and that the work here can still be used as a design principle to avoid unstable operation of frequency-regulated (grid-forming) distributed generator inverters as the conclusions will be, at worst, conservative. 8.4 Example Microgrid Source-Source Impedance Characteristics In this final section, the line resistance versus nominal droop gain is examined for three example Microgrids. The UW Microgrid, the AEP Microgrid and the Santa Rita Jail Microgrid are

271 254 considered as case studies here for practical Microgrid implementations from 15kW base to 1.2MW base. Previously in this work, the stability of a two-source network was investigated to focus on a single-modal analysis. In Microgrids with more than two sources, the number of possible source-source modes become immediately more numerous as described in Equation 8.1 where the infinite grid can also be considered a voltage source. = 1 2 (8.1) In the case of the UW microgrid, only two local sources and the infinite grid are considered, yielding three unique inter-source paths. On these inter-source paths, the total resistance and the combined frequency droop gain are important. Assuming that the resistances are linear, and therefore are available to apply superposition techniques, each inter-source path could be considered independent if the impedance to adjacent sources was large. Unfortunately, the latter assumption of dominant paths with low impedance is not typically the case and therefore invalid, which complicates the analysis of a multi-modal system that is beyond the scope of this work. However, in lieu of avoiding the topic entirely, a few cases can be provided under the aforementioned assumptions that will provide conservative estimates. To elaborate on this point further, any disturbance on a multi-source microgrid will eventually affect the entire network to some degree. If only two sources are considered at a time, than any possibly unstable source pairing can be highlighted. However, this doesn t necessarily mean that the two sources will interact in an unstable manner, providing that adjacent sources can provide sufficient damping through the presence of line resistance and the absence of high frequency droop gain. Therefore, the actual interaction would become more damped than a single unstable

272 255 or neutrally stable pairing resulting from this analysis in the presence of adjacent stable source parings. Table 8-6 UW-Madison Microgrid Inter-Source Impedance Summary Source1 Source2 X/R R [%pu] X [%pu] Effective nominal frequency droop gain Inf. Grid PV Source % 19.3% 1.2% Battery Source PV Source % 35.2% 2.4% Inf. Grid Battery Source % 19.1% 1.2% Table 8-7 AEP Microgrid Inter-Source Impedance Summary Source1 Source2 X/R R [%pu] X [%pu] Effective nominal frequency droop gain Inf. Grid A % 15.4% 1.2% Inf. Grid A % 15.9% 1.2% Inf. Grid B % 19.1% 1.2% A1 A % 27.5% 2.4% A1 B % 30.6% 2.4% A2 B % 31.2% 2.4% Table 8-8 Santa Rita Microgrid Inter-Source Impedance Summary Source1 Source2 X/R R [%pu] X [%pu] Effective nominal frequency droop gain Inf. Grid Batt % 10.8% 1.20% Inf. Grid Gen % 15.0% 1.20% Gen1 Gen % 28.9% 2.40% Gen1 Batt % 24.7% 2.40% The summaries from Table 8-6, Table 8-7 and Table 8-8 indicate a variety of total per-unit impedance figures that are found between sources. However, it can be seen that in all cases presented, that the interconnecting resistance is well above half of the effective nominal frequency droop gain, indicating a stable source paring under nominal conditions. As can be

273 256 seen from Figure 8-8, the Santa Rita Jail Microgrid presents the lightest damped case with all of the unique source parings existing the closest to the stability limit for nominal frequency droop settings. The Santa Rita Jail Microgrid is also the highest power installation and follows previous conclusions drawn regarding the decrease in resistive damping that results from higher power systems and raises the most concern for possibly exceeding the frequency droop gain stability limit. Inter-source Resistance [%pu] 6.0% 5.0% 4.0% 3.0% 2.0% 1.0% Source-Source Resistance Vs. Nominal Power Frequency Droop for Example Microgrids UW Microgrid AEP Santa Rita Jail Stability_Limit 0.0% 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% Frequency Droop [%pu] Figure 8-8 Source-Source resistance versus nominal frequency droop for example microgrids considering only two sources at a time. Nominal droop gain here is 1.2% per source, resulting in 1.2% for interactions with the infinite grid and 2.4% for microsource-to-microsource interactions. It should also be noted that the nominal frequency droop values have been used in this analysis and that for power-limited cases, the effective frequency droop can be significantly higher and would warrant the application of the mitigation approaches outlined in Chapter 5.

274 257 9 Conclusions This research has demonstrated that source overloading resulting from positive load transients can seriously degrade the dynamic performance and stability of islanded microgrids. This degradation of the microgrid s dynamic response has motivated the improvement of power control bandwidth for low-energy storage sources in microgrids. This research has demonstrated that grid-forming sources can jeopardize microgrid stability if the frequency droop and voltage droop gains are raised too high. The deficiencies of grid-following sources for tracking load changes have also been highlighted. However, higher power control gains have been shown to be enabled by both active damping and loop shaping techniques which significantly improve the power control bandwidth, thereby reducing the amount of stored energy required by the microsources to handle large load transients. Toward one of the larger goals of microgrid development that is plug-and-play abilities, the stable implementation of microgrid sources in general, it has been shown that the application of these techniques can allow source operation within the proposed stability region. These findings show that even sources with only the stored energy of the DC bus capacitors can act as grid-forming sources. This allows the network to benefit from the increased reliability provided by multiple microsource units that are individually capable of supporting the microgrid. It also affords sources with more stored energy the flexibility to be located remotely from lowenergy sources in an islanded microgrid. Chapter 3 and Chapter 4 presented a thorough investigation of the characteristics of two types of microgrid sources illustrating the sensitivity of 60Hz eigenvalues to both resistance and droop gains, setting the stage for the primary focus of this work. It utilized participation factor analysis

275 258 and linear operating point models to illustrate the movement of eigenvalues, particularly the voltage-balance mode at 60Hz, to lightly-damped and unstable regions. When the worst-case load transients were applied in the latter part of the chapter, the issue of source overloading and load re-balancing from grid-forming sources was identified as a significant threat to microgrid stability. While the dynamic limits of different types of power sources connected to microsource inverters can vary significantly, it has been shown that designing the power control bandwidth on the AC side to match the requirements on the DC side is important for maintaining microgrid stability. Chapter 4 further explored the dynamic overload issue, highlighting the fact that some sources are not equipped with sufficient stored energy to sustain a transient overload event of the type outlined in Chapter 3. A photovoltaic (PV) source is used as the primary example. This chapter presented a comparison of two major types of source controllers, grid-forming and gridfollowing. Results of this analysis showed that grid-following sources exhibit superior power control abilities, but only grid-forming sources are capable of operating alone in islanded microgrids. Chapter 7 investigated grid-following sources in further detail and highlighted their inability to track loads quickly, which inherently limits their microgrid compatibility where load tracking, in addition to fast power control, is critical to reliable operation. Considering the benefits of grid-forming sources in a microgrid, Chapter 5 was dedicated to addressing their primary drawback in the area of power control characteristics. Analysis showed that the primary restriction to enabling higher power-control gain is a competition between the biased application of voltage that tends to increase the DC current offsets, and the resistive voltage drop that encourages decay towards zero DC offset. A relationship has been derived (Eq. 5.15) showing that the maximum value of the frequency droop gain that avoids instability is

276 259 directly proportional to the resistance of the electrical line/cable interconnected the microsources. The application of active damping and loop shaping techniques utilize this relationship, modifying the effective values of resistance and frequency droop gain. These methods make it possible to achieve higher power control bandwidths that far exceed the baseline value without these techniques. During this investigation, it was determined that the application of active resistance damping allows steady-state conditions to be achieved more quickly than by applying loop shaping alone. It was also noted that the two methods are not mutually exclusive and can be used in combination for maximum effect. However, the bandwidth of any filter applied to the feedback signals inhibits the fast convergence towards steady-state operation. This indicates that the bandwidth of the measurement signal conditioning circuit should be significantly higher than the maximum rate of convergence defined by the droop gains and the filter impedance. With the minimum application of active resistance damping in proper proportion to the level of frequency droop gain, each source can be maintained on the stable side of the stability characteristic presented in this thesis. This concept preserves the plug-and-play abilities of inverter sources as both quantities are software-defined values. The benefits of fast power control for the grid-forming sources make it possible to operate nearly any commercial inverter in a CERTS microgrid. Stored energy requirements were reduced in this analysis to only 0.004pu-sec, which is below the typical value of 0.005pu-sec for commercial inverters. This margin may appear rather small, but this result also does not consider the application of an energy manager. To maintain generality regarding these results, energy management was excluded from the majority of this work. However, the DC-bus regulation employed in Chapter 4 is essentially an energy manager that displays significant

277 260 benefits in recovering the energy lost during load transients, further limiting the amount of energy drawn from an overloaded source. The results related to power control bandwidth improvement were found to be most applicable to microgrid networks that are dominated by inductive impedances between the sources, requiring an X/R ratio of at least 2:1. Large-scale systems fit into this category quite well, having significantly less per-unit resistance than lower-power systems, consistent with the results of analysis in Chapter 8 that investigated the scalability of the previous results. This limitation was further reinforced in Chapter 6 through the operating point stability investigation where it was found that the stability limit relationship breaks down for systems with a low X/R ratio. However, it was noted this stability limit relationship could still be used as a conservative estimate throughout the likely operating space that considered power flow based on non-zero voltage angles between the sources. Two candidate Lyopanov functions were used in this thesis to investigate a simplified system for Lyapunov stability. This effort concluded that a complex power error pseudo energy function was partially insightful, but, in general, revealed a strong sensitivity of the voltage angle to the growth or decay of the pseudo energy error function. This indicates that the mechanism by which the voltage angle is determined is important, but did not provide further insights into the limit of frequency droop gain to ensure stability in the face of significant transients. Using experimental results and dynamic simulation, the gain limits previously defined in the stability limit relationship (Equation 5.15) have been verified for individual cases. However, a generalized form of this stability relationship that is applicable to an arbitrary number of microsources in a CERTS microgrid must await further study. Other candidate Lyopanov functions may exist to provide this more general form for microsource stability in a microgrid.

278 Contributions Summary Analysis and Simulation of Power-Constrained Droop-Regulated Inverter Microsource in an Islanded Microgrid This investigation has used a combination of analysis and simulation to identify and quantify the key factors that influence the dynamic characteristics of islanded microgrids containing sources that have limited overload capabilities and for sources that have power-response bandwidth constraints on the power-source side. Key investigation results can be summarized as follows: Investigation results have clearly demonstrated that source overloading can cause microgrid instability by increasing the transient load on distributed generators beyond their individual ratings even when the total system load is within the combined steadystate power rating of the microgrid system sources. Sensitivity analysis methods have revealed a significant sensitivity to the value of the frequency droop gain, identifying it as a key parameter that influences the system settling time and stability. Line resistance has been identified as a stabilizing agent in small quantities, but it decreases stability when the resistance values increase significantly due to the resulting cross-coupling that reduces the effectiveness of the power and reactive power regulators. Steady-state system loading was found to not have a significant effect on the stability of the system within nominal ranges, contradicting prior conclusions of other authors. Simulation of overload events revealed that the energy required to survive the temporary overloading was more than that contained in typical photovoltaic sources.

279 Performance of PV Source in a CERTS Microgrid This research has provided, for the first time, a direct comparison of the performance characteristics of grid-forming droop-regulated microsource controllers with those of gridfollowing current-regulated controllers, with a particular focus on their dynamic characteristics under overload transient conditions. Key results of this investigation are: The analysis revealed that both types of microsource controller configurations applied to photovoltaic power sources with identical maximum power point tracking algorithms were equally capable of tracking maximum power and providing source-shedding capabilities under light loading. Comparative analysis has shown that, while the current-regulated source achieves higher power-control bandwidth that enhances its transient ride-through capabilities in the face of positive-load transients, only the frequency-regulated controller provides the gridforming ability needed by the microsource to operate alone in an islanded CERTS microgrid. Experimental tests using the 15kW laboratory microgrid have confirmed the key dynamic performance characteristics of the two types of controllers that were revealed in the analyses and simulations Voltage Balance Stability Mode of Frequency-Droop Controlled Sources Analyzed Analysis of the lightly-damped voltage balance mode defined in Chapter 3 yielded valuable insights into the stability characteristics of the microgrid, including identification of the key parameters that determine the network stability characteristics. Key results from this investigation include the following:

280 263 The dynamic characteristics of the voltage balance mode are determined primarily by the line resistance and the frequency droop gain, with little sensitivity to the line inductance. A simple yet powerful system equation has been derived for a simplified microgrid system that determines limiting values of the frequency droop gain as a function of line resistance for stable power response characteristics Increased Power Control Bandwidth of Frequency Droop-Regulated Sources Enabled by Avoidance of Voltage Balance Instability Three approaches to mitigating excitation of the voltage balance mode in order to avoid network instability were presented, analyzed, and demonstrated via simulation for islanded microgrid configurations with increased power control gains. Key results of this investigation include: The approaches have been classified into two categories: loop-shaping and active damping techniques. Each approach has been applied to an example microgrid configuration, and analysis showed that each of the three approaches can achieve a power control bandwidth at least six times the value for the baseline case. A controller solution that combines both categories of techniques has been proposed to maximize damping and power control capabilities while respecting the practical implementation limitations of each individual technique Experimental Verification of Active Resistance Damping and Loop-Shaping for Increased Power Control Bandwidth The UW Microgrid testbed has been used to experimentally demonstrate the dynamic response performance improvement that can be achieved by the introduction of active resistance damping and loop-shaping techniques into frequency droop-controlled microsources in the laboratory

281 264 microgrid. This enhanced controller was implemented in a 15kW microsource. The primary results are as follows: Good agreement between theoretical and experimental data was demonstrated for the stability limit expressed by the relationship between resistance and frequency droop gain in Eq This finding helps validate the simplified model used to develop the theoretical relationship as well as the full-order simulation model used for transient and operating point analysis. Active resistance was employed experimentally and the transient response showed very similar response characteristics to what was expected from the simulations. Power signal filtering was also employed to demonstrate the effective attenuation of the frequency droop gain at the oscillatory frequency, validating the theoretical work regarding loop-shaping techniques Analysis of Microgrid Stability Considering Penetration of Grid-Following versus Grid-Forming Sources An analysis has been conducted regarding the relative scale of grid-forming source ratings with grid-following (current-regulated) sources. The per-unit system was used to simplify and generalize the analysis such that the relative scale between the types of sources could be analyzed. The issue of fixed power applications versus variable power grid-following sources was also investigated. The key results are as follows: It was found that penetration levels up to 97% for the grid-following sources could be achieved in a microgrid, with higher penetrations possible. However, the constraint upon

282 265 which this was made possible was identified by the power balance that needed to be achieved given that the grid-forming sources were treated as fixed-power. The implementation of load tracking via frequency versus power droop allowed for load shedding, but required a low-pass filter on the power signal to avoid oscillations with the PLL for reasonable frequency droop gains. The load tracking abilities of the grid-following sources in this case were demonstrated to be quite slow, so that any load changes will be borne by the grid-forming sources. This defines the practical limit of grid-following penetration by the size of expected load Scalability of Results to Sizes and Voltage Levels of Microgrids An investigation into the per-unit ratings of scaled sources investigated a wide range of microsources that spanned total microgrid power ratings from 3kW to 6MW. Four selected cases were chosen to investigate the limits of stability considering frequency droop. The key results are as follows: Higher power systems tended to have lower resistance and higher reactance, presumably to filter the PWM waveforms that are at lower frequencies. Lower power systems varied from high X/R to very low X/R. High X/R systems were characterized and demonstrated behavior similar to the theory presented previously in this work. In contrast, systems with low X/R were shown to have similar traits only around equal per-unit values of voltage and frequency droop gains Investigated CERTS Inverter Microsource for Passive Characteristics A simplified single inverter source model connected to a grid supply was utilized to examine the passivity characteristics of a single source as a method to test for stability in a multi-source

283 266 network. Two methods were used to discern characteristics of both operating point models and the simplified nonlinear system. Each method had specific restrictions but each provided separate insights: The use of operating point models at steady-state reduced the number of free variables to a manageable number of applicable permutations. This approach showed that the previously defined limit on frequency droop gain was primarily applicable for systems with X/R ratios greater than 2:1. However, the applicability of this method is restricted to small disturbances only. By devising a pseudo energy function tied to the power and reactive power error from steady state, an error divergence characteristic was derived which tested for the increase in time of the error (pseudo energy) function through a Lyapunov-style stability analysis. This method, while more applicable to larger disturbances, identified a sensitivity to the instantaneous voltage angle without giving insight into how the voltage angle trending over time will affect the overall convergence to steady-state conditions. 9.2 Future Work The discussion of future work presented here is broken into two sections. The first of these sections includes research topics that are directly related to the research program that was carried out for this thesis, while the second section consists of research topics in the microgrid field that are inspired by but not directly related to the specific topic of this research program Specific Points of Continued Research Included in this section are future research topics that are directly related to the research that was carried out and presented in this thesis. These include topics in adjacent areas that extend

284 267 beyond the scope of the current research program as well as extensions of topics that were addressed in this investigation but could not be studied because of time limitations Improvements in Load Tracking Abilities of Grid-Following Sources It was shown in Chapter 7 that the load tracking abilities of grid-following sources were limited by the tuning of the PLL and current regulator gains that propagated an unstable mode through the power-frequency droop controller. The theoretical limits of load tracking abilities remains an area that would more fairly define the trade-off when selecting which inverter controller should be employed for an application. This investigation should include the sensitivity of the PLL to determining the output power and present possible methods for increasing the rate at which load can be tracked Characterization of Low X/R Systems with Decoupled Droop Algorithms Systems with low X/R were demonstrated in this work to have differing characteristics than the theory suggests for high X/R systems. The CERTS algorithms utilize the assumption of primarily inductive network impedance, but there are newer methods that utilize decoupled droop algorithms that can be used for systems with lower X/R ratios. The power control response should be investigated to define a need for further work. Secondarily, it should be determined if the decoupled algorithm can be used on low X/R sources on the same system with traditional sources that exhibit higher X/R ratios Limits of Energy Management of Grid-Forming Sources To maintain more general conclusions regarding power control, the results presented here largely ignored the use of an energy manager on top of the power controller. It is proposed that an energy manager reduces the total amount of overload energy and also can act to restore the

285 268 energy balance back to the nominal stored energy value. This work should aim to quantify the further improvement on overload energy that is enabled by an energy manager with respect to the cable impedances and total stored energy being managed Definition of a Candidate Lyopanov Function to Develop a General Frequency Droop Gain Limit Stability of a many-source (multi-node) network is an important consideration, particularly when applying advanced control algorithms. Chapter 6 investigated passivity as an approach to define stability of individual sources that, when placed together, would operate in a stable manner. The energy function associated with the stored energy in the line inductors was used initially to define the energy gradient of a given system configuration to define source stability. Unfortunately, this approach did not yield a useful result which inspired the use of another candidate Lyopanov function that included a dependence on the frequency droop gain, a gain of particular interest in the work presented previously in this document. Alternative forms of an energy function could be devised to define a more insightful stability criterion for inverter microsources that would prove mathematically robust and give greater assurance to the stability of microsources integrated with the grid Utilization of Active Inductance to Shape Transient Response of Sources The concept of active resistance was discussed in this thesis, but the reactive component of active impedance was not discussed. The possible benefits of active inductance include shaping the perceived reactance between each source and the network it is connected to. It has been discussed in Chapter 3 that the magnitude of added load distributed to each grid-forming source is determined by the relative admittance of each source to the added load. With active inductance, this reaction could be actively controlled, giving better control of the power response

286 269 without physically varying the filter inductance of the source. This approach could also reduce the necessary size of reactance for the output filter Suggested Future Focus of Research Efforts Microgrids originated as a solution to provide high power quality to critical loads while taking advantage of local generation. While maintaining these benefits, microgrids will provide the greatest energy savings and added grid stability by leveraging the adjustable sources and smart loads into a controllable grid resource. However, the information required from a microgrid to effectively utilize it as a resource and especially the method by which the information is derived are not well understood. Microgrids do not fall into the traditional roles of energy supply or demand, as they are both concurrently. They also employ smaller generators that are typically capable of faster response times than traditional central coal plants, which allow use in ancillary services and base load support. Lastly, the reliability of microgrid generators and the certainty in microgrid load profiles are currently not rated. These complicating factors point to the development of a standardized approach to determining the stated capabilities for microgrids for use by power aggregators and power system regulators. An example of the required information could include: 1. Basic estimates for use profiles 2. Projected generation availability 3. Gross stored energy capacity (both positive and negative) 4. Confidence level of each of the previously stated quantities 5. Gradient cost per kw for increasing power output or decreasing demand

287 270 The concept of aggregating and conglomerating various levels of information from distributed generators to a central regulating authority is discussed in [71], but the proposed content transmitted is not covered. For the various types of microgrid sources and smart loads, this information can be collected, aggregated and used by a supervisory controller to accurately state the capabilities of a microgrid as a grid resource. The regulating authority can utilize this information to ensure appropriate operating margin, while purchasers in energy markets can put microgrids to work and help offset installation costs. This area of research help would advance microgrids towards a ready-to-act solution for existing distributed generators and incentivize owners of sensitive loads to construct additional microgrids. The net result is a more flexible, adaptable and reliable power system that is accepting of intermittent renewable resources and provides financial compensation to distributed generation owners.

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297 Appendix A: Inverter Hardware Description The inverters used for the experimental verification of presented principles were developed specifically as a multi-functional inverter platform for the Wisconsin Electric Machines and Power Electronics Consortium (WEMPEC) research group. They are single PCB units that incorporate a power stage, measurement and control for simplicity and low signal noise. They utilize a Fuji 1200V, 100A IGBT module and a fully isolated gate drive section for increased noise immunity. Three-phase line reactors are utilized for the L1 and L11 inductive elements. L1 is sized for harmonic suppression along with C (values listed in Table 3-7) for switching frequency harmonic mitigation. The L1 inductor used here was not designed for high frequency and therefore resulted in approximately 100W of losses per inverter while in operation. An isolation step-down transformer is typically used in the UW Microgrid, but in the experimental section here, the transformers were not used and the inverters were connected directly together at the output of their filters as described in Section The experimental hardware is pictured in Figure 11-1 below and shows the inverter board on the lower right, DC bus filter capacitors on the bottom left, AC filter capacitors on the left and the inductor element L1 at the top of the image. Not pictured is the L11 secondary filter inductor or the DC power supplies and storage batteries that were used. Photos of these two sources are included in Section 3.4.

298 Figure 11-1 Laboratory inverter hardware used in power-control bandwidth and stability experiments 281