Paralleled three-phase inverters

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1 Paralleled three-phase inerters Hoff, E., Skjellnes, T. & Norum, L. Department of Electrical Power Engineering, Norwegian Uniersity of Science and Technology, NTNU 749 Trondheim, NORWAY Phone (+47) Fax (+47) erikhoff@elkraft.ntnu.no Abstract Paralleling of inerters are useful for a number of reasons, where modularity and redundancy are some of the key points. Load sharing and oltage control may be done using different methods. The three-phase oltage is controlled by either using fixed frame, αβ-coordinates, or synchronized frame, dqcoordinates. Models for the dynamic behaior of paralleled inerters will be inestigated through Matlab Simulink simulations. =/= =/= Renewable energy sources Critical load Noncritical load Grid I. INTRODUCTION AND BACKGROUND The use of oltage source inerters is increasing. They are used both for feeding power from distributed generators to the transmission grid, and for draining power to arious types of electronic loads. This paper will focus on inerters as an interface between renewable energy sources and the transmission grid. There has been much focus on anti-islanding methods for distributed generators [3]. This is due to security issues regarding grid failures and maintenance. Howeer, the use of automatic communication may change this. In case of transmission grid failure, it may be desirable to let an isolated part of the grid work in islanding mode [2]. This will increase power reliability in these isolated parts of the grid (microgrid), as shown in Figure. A renewable energy based microgrid will need a high degree of local adaption in the design. This is true for both geographical ariations, where different energy sources are aailable, as well as for daily generationand load ariations. This argues for the use of seeral paralleled inerters, simplifying the design process. Reliability is a major concern, especially from the customer point of iew. Paralleling will in fact increase reliability without oersizing the single inerters. A Simulink model of two paralleled inerters, feeding a diode rectifier load, was deeloped (Figure 2). The load inductances were deliberately chosen to be different, giing a normal parameter ariation, in order to show the robustness of the load sharing. Voltage control and load sharing was implemented using a irtual resistance [8]. Inerter A Inerter B Paralleled inerters Figure : Microgrid consisting of seeral parallel inerters.3mh 65µF mh 65µF mh Figure 2: The system implemented in Simulink. II. THREE PHASE TRANSFORMATIONS Diode rectifier load The load sharing is a rather slow process. Howeer, a rapidly responding oltage regulation is needed. This oltage regulator has to compensate for both load changes and harmonics caused by for instance a diode rectifier load. This will otherwise excite the LC filter resonant frequency, which may be close to the Nyquist frequency of the regulator. Dampening this resonant frequency is a demanding task for the oltage regulator. The typical three phase system without a zero conductor can be regulated efficiently using spaceector modulation. The system then has to be transformed to a two-axis coordinate system. Two possible coordinate frames exist: αβ-coordinates (fixed), and dq-coordinates (synchronized).

2 The transformation from phase oltages to power inariant αβ-coordinates is gien by [4]. Neglecting the zero sequence component, this simplifies to: a () α b = β c 2 2 The dq-transformation is a frequently used method for regulating AC currents for instance in motors. The relationship with αβ-coordinates is gien below: d cos( θ) sin( θ) α = (2) q sin( θ) cos( θ) β Two alternaties of irtual resistance locations are shown in Figure 4. Resistor R is the most common placement, while R2 is the position used in this paper. A common method to dampen oscillations is using feedback from the filter capacitor current, R in Figure 4. It is efficient, but it requires one extra current measurement. Aoiding this measurement is important from a cost perspectie. An alternatie to capacitor current feedback is inductor current feedback, R2 in Figure 5. This current is already measured, in order to secure the transistors against oer currents. This is in fact state feedback control. It has preiously been done in [7]. Load unbalance may cause deiations from the model, as it excites the negatie rotating system. Because of this, the system is sometimes split into one positiely rotating and one negatiely rotating system. This result in more calculation time needed, compared to αβ-coordinates. III. STATE FEEDBACK USING αβ-coordinates VE R2 7 L C 65uF VS A simple oltage control strategy is implemented by using two separate sinusoidal references and regulators for the α-axis and the β-axis respectiely. This may be done due to the fact that the 5Hz oltage reference will not change significantly during one sampling period. The inerter is feeding an LC-filter. The major role of the oltage controller is to dampen oscillations close to the filter resonance frequency (Figure 3). The concept of irtual resistance can then be used []. Load sharing can also be accomplished by using these irtual resistors [8]. A irtual resistance is implemented as a negatie current feedback in the software. The circuit will then behae as if a resistor actually was present, but no power will be dissipated. Figure 4: Two possible locations of a irtual resistor for oscillation dampening. VE R2 7 I L,αβ L C 65uF R V C,αβ VS Figure 5: The implemented irtual resistance. Figure 3: LC-filter oltage transfer function.

3 IV. QUADRATIC OPTIMAL VOLTAGE REGULATOR To design a discrete oltage regulator, a continuous model was first deeloped. Because the fixed frame model does not include axis coupling, it is sufficient to model one axis at a time: x = Ax + Bu C, αβ C C, αβ (3) E, αβ i = + L, αβ R i L L, αβ L L L Here, L =, C = 65µF, and R =.Ω. Inserting these alues, gies the numerical expression of (3): 6 C, 65 αβ C, αβ (4) E, αβ i = L, αβ i + L, αβ the inerter oltage. Without going through the method of calculating quadratic optimal state feedback, the result is presented. The discrete state feedback matrix E was calculated equal to [-3-7]. The resulting closedloop transfer function is shown in Figure 6. The strong current feedback gain (-7), gies seere oltage droop when load current is high. This gies the same oltage drop as a 7Ω series connected resistor. This is a disadantage for heay load conditions. The problem can be corrected using high-pass-filtering of the current measurement. To aoid aliasing, a low-pass filter is added. A compromise of these two requirements is a band pass filter, with center frequency equal to the LC resonant frequency. This gies zero phase and unity gain at the frequency of interest. The amplitude has a drop of 2dB per decade both for higher and lower frequencies. It has a bode plot as shown in Figure 7. To build a discrete regulator, discretization of the continuous model was done [6]. A sampling frequency of khz gies T s equal to.ms. A discretization of matrix A from (3) is shown in equation (5): A d = I + ( AT ) n= n s n= n! Ts =.. + Ts ( ) Ts ( ) Ad (5) A similar discretization of matrix B is shown in equation (6): Ts n= n ( AT ) s Bd = B I + dt n= n! (6) B Ts = Ts L Bd Figure 6: Closed-loop transfer function for the LC-filter with oltage regulator. The oltage regulator was designed using statefeedback. Two identical regulators was constructed for the two separate axes (α and β). The method used, was Quadratic Optimal Regulation (p296, []). It tries to minimize the cost function J using state feedback: T T J = x Qx+ u Ru (7) The weight functions were chosen so that Q = I and R =, arbitrarily chosen. This gies equal high penalty on oltage and current, but less restrictions on Figure 7: A passie band pass filter used for filtering of inductor current measurement

4 It can be seen that the current feedback gain is dampened by a ratio of -4.8dB at 5Hz. In other words, the effectie current gain for a resistie load will be 8% of its original alue, resulting in 3Ω at 5Hz. V. LOAD SHARING The irtual series resistor used for oscillation dampening, also seres a second role. It gies oltage droop for actie load P sharing. The reactie power Q sharing, on the other hand, is connected to the frequency and phase-locked loop. A reactie load (positie Q) will lead to increasing phase angle θ, and thereby increasing frequency for a short period of time. As shown with a dashed line in Figure 8, the power flow control will be different if a irtual resistor (b) is used instead of an inductance (a) for load sharing. Using an inductance, actie power will depend on the phase angle θ. On the other hand using a irtual resistor, actie power will now depend on oltage amplitude difference. The load-sharing oltage ector presented here is therefore 9 degrees dephased compared to the load sharing methods found in the literature [5]. a) I s I s,p I s,q Figure 9: Simulink calculation of P and Q VI. RESULTS OF SIMULATIONS The load sharing preents any oscillations between the two inerters, een when a diode rectifier load is applied. The load currents of the two inerters are shown in Figure. The irtual resistor showed satisfactory power load sharing capability. This is shown in Figure and Figure 2. Both reactie load sharing and actie load sharing are satisfactory. Due to the intentional difference in the load inductances, a 2% difference of actie power was howeer obsered. V s V E I s j X I s,q j X I s,p j X b) I s,p I s,p R I s I s,q V s I s R I s,q R V E Figure 8: Power dissipated by an inerter connected to the grid (V S ) through a) an inductance (X), and b) a irtual resistor (R). Dashed line gies a certain leel of actie power flow. The inerter is here producing P and Q (reactie load). Power flow is defined into the inerter (V E ). To make this load sharing possible, calculation of P and Q is done in the αβ-coordinates. P is defined as the dot product of the ectors V and I. Equation (8) and (9) are also shown in Figure 9. P = V I = Vα Iα + Vβ Iβ (8) Figure : Inerter load currents. Thick lines are inerter A currents, and thin lines are inerter B currents. Q can be defined as the cross product of the ectors I and V: Q = I V = Iα Vβ Iβ Vα (9)

5 Figure : Power flow of inerter A. Thick line is actie power P. Thin line is reactie power Q. Figure 3: Connecting a diode rectifier load at t=.2, band pass filtering of current is enabled. Figure 2: Power flow of inerter B. Thick line is actie power P. Thin line is reactie power Q. Figure 4: Connecting a diode rectifier load at t=.2, band pass filtering of current is disabled. The irtual series resistors show current-limiting properties when a diode rectifier load is connected. It will then reduce the oltage amplitude, which in turn will reduce the inflow currents to the diode rectifier. This is shown in Figure 3. To illustrate the adantage of band pass filtering, a simulation was run without using the filter, as shown in Figure 4. This shows a significantly higher oltage droop, compared to the simulations where the band pass filter is enabled.

6 VII. DISCUSSION The simulations were performed using a three phase continuous LC-filter model. The controller and inerter were discrete, giing a satisfactory model approximation. Including switching ripple current in would make the simulations more close to real measurements. The goal of this paper was to show the principles of paralleled inerters, and switching ripple was therefore not necessary in this context. Without a band pass filter, the optimal current feedback was too high for the oltage regulator to work efficiently. To reach a limit of ±% oltage ariation a band pass filtering of the current measurement is needed. When implementing the simulation model in the lab, parameters can be adjusted to refine the regulators. Howeer, cooling requirements, electromagnetic noise, and cost will put limits on component sizing. Increasing the filter capacitor size is one method to allow a lower current feedback gain, while still maintaining stability. This will increase maximum load. Results from calculations of quadratic optimal state feedback for seeral alues of filter inductances and capacitances, indicates this. For future work, it would therefore be adantageous to hae a relatiely high output filter capacitance compared to the inductance. The placement of the irtual resistance in series with the load gies as an extra bonus in addition to LC-filter oscillation dampening. The resistance also contributes to an een actie load sharing. The band pass filter gies the irtual resistance a phase angle at 5Hz, unintentionally making it act like a irtual inductance. This did not, howeer, cause any problems in the simulations. VIII. CONCLUSION Simulations of a αβ-based oltage regulation strategy gae a satisfactory response, een in the presence of a diode rectifier load. Current feedback was used for LC-filter oscillation dampening. In order to make it most sensitie to the LC resonant frequency, a passie band pass filter proed efficient. If the current feedback gain must be further decreased, the output capacitor should be increased. The inductor current feedback made a oltage droop sufficient to control the actie load sharing. The reactie load sharing was implemented with the phase locked loop. It increased the phase and frequency if the load appeared reactie. These simulations indicate that a simple approach for controlling paralleled three phase inerters may be possible. The solutions presented may lead to a lower inerter cost and simplified oltage regulators. In future work this will be inestigated by laboratory tests. REFERENCES [] Dahono, P. A.: A Control Method to Damp Oscillation in the Input LC Filter of AC-DC PWM Conerters: IEEE, Bandung, Indonesia, 22 [2] Illindala, M. S., Piagi, P., Zhang, H., Venkataramanan, G., Lasseter, R. H.: Hardware Deelopment of a Laboratory-Scale Microgrid Phase 2: Operation and Control of a two-inerter Microgrid. Winsconsin Power Electronics Research Center, Madison, Wisconsin, 24. [3] Kern, G.A., Bonn, R. H., Ginn, & J., Gonzalez, S.: Results of Sandia National Laboratories Grid-Tied Inerter Testing. 2nd World Conference and Exhibition on Photooltaic Solar Energy Conersion, Vienna, Austria, 6- July 998. [4] Lindgren, M.: Modeling and Control of Voltage Source Conerters Connected to the Grid. Technical Report No. 35, Chalmers Uniersity of Technology, Göteborg, Sweden, 998. [5] Peças Lopes, J.A.: Management of MicroGrids. JIEEC23, Bilbao, October 23. [6] Satina, M., Stubberud, A. R. & Hosetter, G. H.: Digital Control System Design. Second edition, Saunders College Publishing, Florida, 994 [7] Sato, Y. & Kataoka, T.: State Feedback of Current-Type PWM AC-to-DC Conerters. IEEE Transactions on Industry Applications, Vol 29, no. 6, 993. [8] Skjellnes, T. & Norum, L. E.: Load sharing for parallel inerters without communication. NORPIE, Stockholm, Sweden, 22