Resonance Damping in a Higher Order Filter (LCL) in an Active Front End Operation

Transcription

1 Resonance Damping in a Higher Order Filter (LCL) in an Active Front End Operation A Project Report Submitted for partial fulfillment of degree Master of Engineering In Electrical Engineering By NILANJAN MUKHERJEE June 2009

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3 i Acknowledgements I express my profound gratitude to my guide Dr. Vinod John for suggesting me the topic and I wish to thank him for giving me opportunity to work with such a fascinating project of current interest. I am again thanking him for providing the course Dynamics of Linear System that helped me a lot in my present work and also for providing me freedom that I have enjoyed throughout my work. I cannot forget those ours long discussion with him about the various things related to power electronics. He was really active in discussion with me in each and every problem that I have faced in my work. I sincerely thank Dr. G. Narayanan for his course PWM Power converter and its application that helped me to expose such fascinating research area. I am also grateful to him for provided me an opportunity to deal with a highly prospectus mini-project related to my main work that nobody has touched so far. I am grateful to Prof. V.T. Ranganathan, Prof. V. Ramanarayanan for their courses namely, Electric Drives, Switch Mode Power Conversion as those courses also helped me in my project. I consider myself fortunate to be a part of this Power Electronics Group. I have extreme gratitude & affection for this institute, as I have received every opportunity for self-improvement. I gratefully acknowledge the financial support from the institute for being a Master student. I have all along been enjoying the company of a very good set of friends and seniors here at PEG, especially Dipankar who was there with me all the time as a friend and sometime like an elder brother with all the happenings in these two years. I cherish the wonderful time I have had with him that I cannot forget in my life. It was a great pleasure working with Anirban in the same lab. I got a lot of advice with the discussion with him in the lab. Over all it was a great pleasure to work with this Power Electronics Group. I also had a great time with all my friends here at IISc. I thank Mrs. Silvi Jose for her help in purchase of components etc. I sincerely thank Mr. D.M.Channe Gowda and his colleagues at the EE Office who have always been very helpful.

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5 Abstract The use of Distributed or Disperse Generation is rapidly increasing in the modern distribution networks because of their potential advantages. They need DC/AC converter in order to interface to the grid.these type of active rectifiers/inverters are used more frequently in regenerative systems and distributed power systems. The switching frequency of these converters is generally between 5 khz and 20 khz and causes high order harmonics that can disturb other EMI sensitive loads/equipment on the grid side. Choosing a high value for the line-side inductance can solve this problem, but this makes the system expensive and bulky. On the contrary, to adopt an LCL- filter configuration allows to use reduced values of the inductances (preserving dynamic performance) and to reduce the switching frequency pollution emitted in the grid. The main goal is to ensure a reduction of the switching frequency ripple at a reasonable cost and, at the same time, to obtain a high performance active rectifier. Usually the converter side reactor is bigger than the grid side one because it is responsible for the attenuation of most of the switching ripple. The ac capacitor is limited in order not to reduce too much the reactive power drawn and the grid side reactor is chosen in order to properly tune the cut-off frequency of the LCL-filter. As a drawback these LCL filters have very high gain at the filter cutoff frequency, so naturally if that frequency get excited then system will oscillate. As a result system becomes highly sensitive to outside disturbances. One way of reducing the resonance oscillation in current & voltage of the system is by adding a passive damping circuit to the filter. This damping circuit can be purely resistive or more complex solution consisting of a combination of resistors, capacitors and inductors. A more attractive option is the use of active damping where the output voltage from the converter is used (suitable control action) to damp out the resonance oscillations. The greater emphasis is given to active damping in current work. Some of the techniques are available in the literature based on current control strategy of the converter. But in the current work state-space based method of control is adopted in order to get more flexibility & optimized the energy during control action.

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7 Contents Acknowledgements Abstract List of Figures Nomenclature i iii ix xiii 1. Introduction Organization of the project.2 2. Active Front End Operation with LCL filter and Problem of Resonance 2.1 Introduction Active Front End Converter (AFEC) Grid connected operation with L filter Grid connected operation with LCL filter Problem of resonance AFEC Conclusion.7 3. Introduction to passive damping in LCL filter 3.1 Introduction Different passive damping topologies Solution Solution Solution Conclusion Introduction to Active damping 4.1 Introduction Active damping based on traditional approach..20

8 vi Contents 4.3 Active damping by means of State-space based method Filter modeling in state-space Pole-placement of the system Physical realization of Active Damping (Concepts of Virtual resistance) Active damping loop realization Conclusion Grid synchronization & Introduction to Phase Locked Loop 5.1 Introduction PLL algorithm Conclusion Control Scheme for Standalone and Grid Interactive Mode with LCL Filter 6.1 Introduction Model for control design Over view of the control loop consisting of three states of the system Current control strategy Analysis of controller performance Inclusion of inner most state-space based damping loop.; Operation in standalone mode with LCL filter Control in standalone mode with LCL filter Sensor less operation Conclusion Simulation Results 7.1 Introduction System parameters Standalone mode results..53

9 Contents vii 7.4 Grid connected mode results Conclusion Experimental Results 8.1 Introduction Standalone Mode of operation grid connected mode of operation Conclusion...79 A. Further Investigation about damping A.1 Introduction.81 A.2 Comparison of Active & Passive damping.81 A.3 Optimal damping (Advanced Active Damping).83 A.3.1 Introduction.. 83 A.3.2 Optimal gain matrix calculation 83 A.4 Conclusion 86 B. Hardware Implementation B.1 Introduction..87 B.2 Experimental Setup..87 B.2.1 Diode bridge rectifier...87 B.2.2 Pre-charging Autotransformer.89 B.2.3 IGBT based inverter 89 B.2.4 FPGA Controller.89 B.3 Implementation stages..92 B.4 Conclusion...92 References....94

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11 List of Figures 2.1 Grid connected operation with the simple L filter Grid connected operation with the third order LCL filter PWM waveform applied to the filter LCL filter is being excited by unit impulse frequency response of the capacitor current in LCL filter series damping in LCL filter Frequency response of capacitor current after damping bypassing the damping resistance by L f Frequency response with bypassing inductance R-C parallel damping for the LCL filter damping by changing C/C f ratio...15 il2 3.8 Root locus analysis of transfer function in damped case...16 Uinv 3.9 change of closed loop poles with the variation of a Comparison of transient response with different a Variation of Q with the a overall comparison of Loss, attenuation, Damping with C/C f ratio Typical scheme for Active damping Pole placement to LHS of s-plane Active damping by weight age capacitor current feedback Approximate Circuit representation for Active damping comparison of different damping factor in Active Damping State-space based active damping loop (general case) Comparison of virtual resistance based damping & actual resistance il2 based damping of transfer function.29 Uinv 5.1 Basic PLL structure Reference frame Simplified block diagram of SRF PLL Theta is synchronized with grid voltage...33 ' 5.5 sin( θ ) and sin( θ ) synchronized with each other Active rectifier with LCL filter Grid connection with LCL filter Conventional three loop control strategy for LCL filter 39

12 x List of Figures 6.4 Two loop control strategy for LCL filter Single loop control strategy for LCL filter Modified two loop control strategy for LCL filter Two loop control strategy for LCL filter 42 * il2 6.8 Root locus of with K p = il2 * il2 6.9 Root locus of with K p = 1 44 il2 * il Root locus of with K p = il2 * il Bode plot of with different value of K c 45 il Current control with State-space based damping loop Standalone mode of operation Vector control of standalone mode Vector control in grid parallel mode d-axis load side current in standalone when the reference 0.05 sec q-axis load side current in standalone when the reference 0.05 sec R-phase load side current in standalone when the reference 0.05 sec capacitor current in standalone when the reference 0.05 sec Capacitor voltage in standalone when the reference 0.05 sec d-axis load side current in standalone when the damping loop is being 0.02 sec q-axis load side current in standalone when the damping loop is being 0.02 sec Capacitor current in standalone when the damping loop is being 0.02 sec Capacitor voltage in standalone when the damping loop is being 0.02 sec R-phase load side current in standalone when the damping loop is being 0.02 sec..59

13 List of Figures xi 7.11 d-axis grid side current in no-load where the reactive power reference 0.02 sec q-axis grid side current in no-load grid side current in no-load DC-link voltage profile in no load Fundamental of R-phase inverter voltage& grid voltage of R-phase d-axis current of grid side with load q-axis current of grid side with load 0.08sec line side current of R-phase with load 0.08sec DC-link voltage profile when the load is being 1.5sec Line side current & line voltage with UPF operation Line side current when active damping 0.1 sec capacitor voltage when active damping 0.1 sec capacitor current when active damping 0.1 sec q-axis load side current with change of reference q-axis load side current with change of reference load side current with change of reference Un-damped load current & its spectra in standalone mode Damped load current & its spectra in standalone mode Active damping test of load current with lesser state-weight age Active damping test of load current with higher state-weight age Un-damped capacitor current & its spectra Active damping test of capacitor current Active damping test of q-axis capacitor current DC bus control test (voltage rise) DC bus control test (voltage falling) Distorted current from grid (full of resonance) Less distorted current grid (state-weight age K=10) smooth current from grid (state-weight age 25) and its FFT grid side current dynamics when sudden change in DC-bus grid side current dynamics when sudden change load Line side current when active damping is being enabled mid-way Active damping loop is being enabled mid-way with BW of 1.2 KHz Distortion in the utility voltage and smoothing out by active damping..78

14 xii List of Figures il1 A.1 Frequency response of for un-damped, actively damped & Uinv passively damped system 82 il2 A.2 Frequency response of for un-damped, actively damped & Uinv passively damped system 82 A.3 Root loci of the original system & optimally damped system 85 il1 A.4 Frequency response of in the original system & optimally damped Uinv system.85 il2 A.5 Frequency response of in the original system & optimally damped Uinv system 86 B.1 Complete Hardware Setup.88 B.2 Block diagram of the FPGA board 90

15 Nomenclatures Symbols Definition V dc : DC-bus voltage. U : Inverter output voltage. inv U g : V c : Grid voltage Capacitor voltage in LCL filter Q : Quality factor / state weight age matrix w r : Resonance frequency in rad/s A : System matrix in LCL filter LCL B : Input matrix in LCL filter LCL C : Output matrix in LCL filter LCL X LCL : State vector ξ : Damping ratio i L1 : Inverter side current i L2 : Grid side current/line side current i c : Capacitor current in LCL filter R d : L f : Damping resistance in LCL filter Bypassing inductance in damping circuit

16 xiv Nomenclatures L 1 : Converter side inductance L 2 : C : C f : Grid side inductance Capacitance of the LCL filter Capacitance of the parallel branch in LCL filter a : - V : C f C Voltage space vector V α, V β : α β axis components of - V K p : Proportional part of PI-controller K PWM : Gain of the inverter T c : Integral time constant of PI-controller R : Input weight age matrix J : Cost function AFEC : Active Front End Converter BW : Bandwidth

17 Chapter 1 Introduction Three-phase grid connected PWM rectifiers (Active rectifier) are often used in regenerative system & adjustable drives system when regenerative braking is required. Apart from power generation, they offer control of power factor as well as dc link voltage while injecting lower current harmonics to the grid than passive diode rectifier bridges. Traditionally, LC filter is used for an inverter power supply like in UPS system. A grid-interconnected inverter used in such an application however, has some unique requirements that an LC filter may not be sufficient. (Based on IEEE ) A PWM converter with higher switching frequency will result in smaller LC filter size. However, switching frequency is generally limited in high power applications. As an alternative solution, LCL filter is more attractive for two reasons: First, they offer advantages in terms of costs & dynamics second; it has better attenuation than LC filter given the size. LCL filter also provides inductive output at the grid interconnection point to prevent inrush current compared to LC filter. One drawback LCL filter based Active rectifier system is that the filter can oscillate with the resonance frequency. Active damping is generally more attractive than passive damping as of it is being loss-less & more flexible however, the control bandwidth is quite limited in high power converters due to their limitation in switching frequency. Here in the present work the design part of the filter is not considered. Concentration is given to the damping part of LCL filter. First part of the work is concentrated on the passive damping & the next part is focused on active damping.

18 2 Chapter 1. Introduction 1.1 Organization of the report Chapter 2: Active front end operation with LCL filter and problem of LCL resonance. Chapter 3: Introduction to passive damping in LCL filter Chapter 4: Introduction to Active damping Chapter 5: Grid synchronization & Introduction to Phase Locked Loop Chapter 6: Control Scheme for Standalone and Grid Interactive Mode with LCL Filter Chapter 7: Simulation Results Chapter 8: Experimental results.

19 Chapter 2 Active Front End Operation with LCL filter and Problem of Resonance 2.1 Introduction Active rectifier / active front end converters have been used in drives as well as distributed generation system & now becoming more and more popular because of its ability to control the line side power factor and load voltage at a time. This type of converters connected between load and the grid /utility in order to supply fine quality of power to the load. Now AFEC can be connected to the grid through L filter & LCL filter The major advantage of the L type of filter are, design is simple and implementation is bit easy, but at the same time disadvantage of the L filter are large magnitude required to provide required attenuation, poor dynamic response due to large time lag. System will be bulky and cost as well as inefficient in application like 100 s of KW or MW range. On the other hand LC filter has better attenuation compared to L filter but there is always a problem of inrush current due to grid voltage fluctuation. Now LCL filter is better than these filters because of giving better decoupling between filter and grid (reducing the dependence on grid parameters.) as well as excellent attenuation of -60dB/decade to the switching frequency harmonics.

20 4 Chapter 2. Active Front End Operation with LCL filter and Problem of Resonance 2.2 Active Front End Converter (AFEC) The converter consists of a three-phase bridge, a high capacitance on the dc side and a three-phase filter in the line side. The voltage at the mid point of a leg or the pole voltage Vi is pulse width modulated (PWM) in nature. The pole voltage consists of a fundamental component (at line frequency) besides harmonic components around the switching frequency of the converter. Being at high frequencies, these harmonic components are well filtered by the high inductances (L) or some higher order line filter (LCL). Hence the current is near sinusoidal. The fundamental component of Vi controls the flow of real and reactive power. It is well known that the active power flows from the leading voltage to the lagging voltage and the reactive power flows from the higher voltage to the lower voltage. Therefore, controlling the phase and magnitude of the converter voltage fundamental component with respect to the grid voltage can control both active and reactive power. As the grid voltage leads the converter pole voltage, real power flows from the ac side to the dc side, while the reactive power flows from the converter to the grid Apart from control of real and reactive power flow, an FEC should also have a fast dynamic response. Operation of FEC with the first order L filter is well reported in the literature but operation of this type active rectifier with LCL filter has now started drawing attention [12],[2]. The present work chooses LCL filter based Active Front End converter as its control platform

21 2.3 Grid connected operation with L filter Grid connected operation with L filter V dc i dc Active rectifier L O A D +Vdc/2 +Vdc/2 O R Y B L L L i L Grid Fig 2.1 Grid connected operation with the simple L filter 2.4 Grid connected operation with LCL filter i Load L O A D V dc O i dc R Y B L1 Active Rectifier i i L1 L 2 V c L1 L1 C C C L2 L2 L2 Grid Grid Grid N Neutral Fig 2.2 Grid connected operation with the third order LCL filter

22 6 Chapter 2. Active Front End Operation with LCL filter And Problem of Resonance Simplified Power circuit L1 il1 i L 2 Vc L2 C i c Grid U g U inv Fig 2.3 PWM waveform applied to the filter 2.5 Problem of LCL resonance in AFEC Normally grid impedance reflected back to the converter side is generally very less so, if the resonance is excited, the oscillation of that can continue for ever and it can make the entire system very vulnerable. Actually when the PWM converter is switched on, the filter (LCL) encounter a sudden pulse at the input as a result filter starts to oscillate in its cutoff frequency. Here is an attempt to show how the resonance is excited by PWM converter (AFEC) itself. For that LCL filter is modeled inside the FPGA based controller & fed from a very narrow single pulse from the same controller. After exciting from the pulse, filter starts to oscillate at resonant frequency. Pulse is being generated in FPGA by means of switch de-bouncing logic

23 2.6 Conclusion 7 Fig 2.4 LCL filter is being excited by unit impulse 2.6 Conclusion In the grid-connected operation with LCL filter, like Distribution Generation case or in Front End operation, damping carries a significant part of designs if we want to utilize the advantages of higher order filter. There are two ways to damp the resonance, Passive Damping and Active damping. First one can damp resonance in all condition but it is of loss full process & second one can act only when the power converter is switching. Firstly Passive damping is taken to describe & then active damping is analyzed.

24 8 Chapter 2. Active Front End Operation with LCL filter and Problem of Resonance

25 Chapter 3 Introduction to passive damping in LCL filter 3.1 Introduction Damping is essential in LCL filter based grid connected system or in the case, when it is connected to light load (standalone mode of operation). This resonance effect can cause instability in the output, especially if some harmonic voltage/current is near the resonant frequency. Damping by control algorithm (Active damping) is most attractive however; the control bandwidth is quite limited in high power converters due to their low switching frequency. Therefore passive damping (by addition of passive circuit elements) is considered in most of the cases. Here in this chapter different passive damping topologies are described & compared. The criteria for the following comparison are effective resonance suppression without deteriorating the attenuation at switching frequency. The Bode plots are used to analyze the performance of damping filters. The five different damping topologies are described. There is always a tradeoff exits between losses & damping in the passive damping, so this also give limitation of passive damping in different cases. Those issues are compared; solution is given depending on power level, damping requirement & efficiency (or losses).

26 10 Introduction to passive damping in LCL filter 3.2 Different passive damping topologies Solution 1 The LCL filter transfer function of line side current & inverter input voltage in grid-connected mode of operation is given below. i 1/ (L + L ) = U s(1+s (L L.C)) L inv 1 2 (1) From the transfer function it is clear that, at the frequency of 1 it has high gain (infinite Q ). The simplest solution may be 2 π (L 1 L 2).C the addition of series resistance with the capacitor to reduce the Q as the capacitor current is most responsible for resonance in LCL filter. ic L1 sc =. 2 (2) U L + L (1+s (L L ).C) inv It is also clear from the frequency response of capacitor current. It carries basically resonant component & very less fundamental as well as switching component.

27 Different passive damping topologies 11 Fig 3.1 frequency response of the capacitor current in LCL filter L 1 L 2 i L1 C i L2 U inv i c R d U g Fig 3.2 series damping in LCL filter Fig 3.2 shows the first & the simplest kind of passive damping topology. The bode-plot is given for the capacitor current vs. inverter voltage & line side current Vs inverter output voltage. It is clear from those that larger series resistance can give better damping or lower Q as clear from the transfer function after damping.

28 12 Introduction to passive damping in LCL filter i L sc =. U L + L (1+ CR s+ s (L L )C) c 2 2 inv 1 2 d 1 2 (3) i U 1+sCR L2 d = 3 2 inv s (L1L 2C) +s (L 1+ L 2)CR d+s(l 1+ L 2)..(4) So, here damping factor is proportional to R d. But on the other hand, larger resistance tends to reduce the attenuation above the resonant frequency. It is undesirable from the harmonic filtering point of view. Moreover higher Rd can also increase the losses at low frequency. So, there is a trade off exits between losses & damping in this case as a result this method cannot be used for higher power rating like KW or MW level. Fig 3.3 Frequency response of capacitor current after damping

29 Solution Solution 2 Second solution is a slight modification over the first one. In the series damping method the disadvantage was losses at fundamental & that restricts us to use that type of damping method for higher power ratings. Here one more inductance is inserted parallel to damping resistance. As a result current at fundamental will be bypassed through L & loss will be considerably saved. f L 1 L 2 i L1 C il2 U inv R d L f U g Fig 3.4 bypassing the damping resistance by In this type of damping process, if we see the transfer characteristic the attenuation at switching frequency is improved but at the same time damping is bit affected. But the major advantage of this type damping loss at fundamental frequency is considerably improved. L f

30 14 Introduction to passive damping in LCL filter Solution 3 Fig 3.5 Frequency response with bypassing inductance L 1 Vc L 2 U inv il1 Cf R d C i L2 U g Fig 3.6 R-C parallel damping for the LCL filter

31 Solution 3 15 In this type of damping is much more preferred compared to simple series damping as here damping is not only depends on resistance but also on the a ( C f / C ) ratio.it is shown by following frequency plots: - Fig 3.7 damping by changing C f / C ratio

32 16 Introduction to passive damping in LCL filter il2 Fig 3.8 Root locus analysis of U inv transfer function in damped case Fig 3.9 change of closed loop poles with the variation of a

33 Solution 3 17 Fig 3.10 Comparison of transient response with different a Fig 3.11 Variation of Q with the a

34 18 Introduction to passive damping in LCL filter Now if we see the transfer function, il2 1+ scf R d = Uinv s L1L 2CCf R d + s L1L 2(C+ C f ) + s Cf R f (L 1+ L 2) + s(l 1+ L 2) (5) The loss, attenuation & damping ( Q factor) with variation of C / C (a) are analyzed below: - f Fig 3.12 overall comparison of Loss, attenuation, damping with C / C ratio f

35 Design of a 19 The Design of a : - To find the Q-factor of this damping circuit derive Q = V j0.5 w C R c r d d So, = U a inv jw rcdr d 1+ a the expression for Q as follows:- V U c lim = 0.5 ω 0 substituting for C d in terms of C we get inv Q = a 1+ jw rcr d 1+ a 2.. (7) a jw CR (1+ a) r d 2 The Q-factor is also affected by the choice of R d. It is taken equal to the characteristic impedance of the LCL circuit. L R = Or a = 1. C d This gives the lowest Q for the damping circuit. The comparison for different ratings: - Unit 1KVA 10KVA 100KVA V b V I b A Z b Ω L mh C µ F L 1 = L 2 mh C d = C µ F R d Ω

36 19 Introduction to passive damping in LCL filter 3.3 Conclusion In this chapter several passive damping filters [12] as part of the overall output filter have been discussed. Passive damping is the simplest way of damping resonance excited by this type of filter. But there is trade off between losses and damping in passive damping topologies. Passive damping can be used where the efficiency is not so much of importance or where damping required (Q factor) is not so high. The results of passive damping are given in chapter 7.

37 Chapter 4 Introduction to Active Damping 4.1 Introduction In case of passive damping, damping provided by physical element like resistors. But this process is associated with losses and in the high power application with process cannot be afforded. To reduce losses and improve the performance inductors, capacitors are provided along with resistors. In case of active damping, damping is being provided by means of control algorithm, this process is not a lossy process so this process is much more attractive. But there is a limitation of active damping, such as this control technique depends on the switching of power converter, so this is effective only when power converter is switching. On the other hand switching frequency of the power converter is limited hence the control BW of the active damping is also limited. There are broadly two methods of active damping can be thought one is based on traditional PI-controller and the other is based on generalized statespace approach. In this chapter we will focus on a method of active damping based on State-space (arbitrary pole placement) 4.2 Active damping based on traditional approach The traditional approach is based on different current control strategy such as conventional PI-controller based [3] (in rotating frame) combined with lead compensator or a resonant controller as a main compensator in α β domain. In these approaches BW of the system or settling time cannot be arbitrarily fixed as these based upon main current controller BW. In other words placement of the closed loop poles is determined by the current controller design.

38 21 Introduction to Active Damping 4.3 Active damping by means of State-space based method This approach is more generalized than traditional PI-controller based method because of flexibility. This method gives us the freedom of arbitrary pole placements or in other words BW can be independently fixed without depending upon the current controller BW. More over the energy required for damping can be optimized by means of state-space based method. So, state-space based method offers good stability margin and robustness to parameter uncertainty in the grid impedance at the same time implementation is bit easier in case of state-space based method. Now for the stable under-damped system pole of the any system should be on left half of s-plane. Now in our present case providing damping is equivalent to shifting the closed loop poles in the left half of s-plane. So all we have to do is to shift the closed loop poles in the left half of s-plane by means of suitable gain (or gain matrix). Shifting of the poles can be determined by required settling time of the closed loop system. LQ regulator (Appendix A) allows us to choose different state-weight for different states and the inputs. So, apart from damping, the transient response can also be controlled for the different states. The results of optimal damping are given in Appendix A. 4.4 Filter modeling in state-space There are two inputs to the system 1. PWM output of the power converter and 2.Grid voltage. First one is the active input (that can be controlled) and second one is the disturbance input (passive input). The states of the system are two inductor currents and one capacitor voltage dv dt c 0 1/ C 1/ C V 0 0 c dil 1 = 1/ L1 0 0 il 1 + 1/ L1 Uinv + 0 U g dt 1/ L2 0 0 i L2 0 1/ L 2 di L2 dt

39 4.5 Pole-placement of the system 22 So in more compact form,. x = A x + B u + B ug y = C x LCL LCL 1 inv 2 LCL LCL. (1).. (2) Where the matrices are following, A LCL 0 1/ C 1/ C = 1/ L / L2 0 0 B 0 = 1/ L B 2 0 = 0 1/ L 2 C LCL = So, the eigen values of system are + λ = 0, j 1 ( L L ). C 1 2 Now position of the poles are on imaginary axis hence the system is oscillatory and highly sensitive to outside disturbances. 4.5 Pole-placement of the system Typical scheme (in brief) for active damping control can be visualized by the following figure.

40 23 Introduction to Active Damping PWM delay and the digital controller delays correspond to phase errors in the requirements of the active damping control. Fig 4.1 Typical scheme for Active damping As mentioned earlier that poles of the closed loop system is on imaginary axis for un-damped system. Now the poles can be moved to the left half of the s-plane properly choosing the BW or settling time of the entire system. Fig 4.2 Pole placement to LHS of s-plane

41 Per unitization 24 Let the imaginary axis pole is to be shifted to ξ + 2 ξω jω 1 ξ Where the is the damping factor of the system. Here only task to choose this damping factor for designing this system. ξ is taken as 0.6 (sufficient to provide the damping) r r For assigning above pole in the system we use control law U inv = -KX LCL Or in other words inv 1 c 2 L1 3 L2 U = -K V - K i - K i.. (3) The task is to find the gain matrix for the system. For that system is per unitized. Per unitization:- For 10KVA inverter and 440V grid voltage we can per unitized the system. L1=L2 = 3mH and C = L1 = L2 = 0.05p.u C = 0.09p.u 16µ F can be per unitized as dv dt di c Vc 0 0 = i U + 0 U L1 L1 inv g dt i L di L2 dt.. (4) Now the required pole j16.32 Hence the required gain matrix is K= [ ].

42 25 Introduction to Active Damping The complete system model can be written in state-space as:-. X = (A- Bk) X + Br.. (5) Where r is reference input, which is determined by the overall control This gives the damping loop description, which is based on statespace based method. * Uinv K PWM Uinv 1 sl 1 il1 U g Vc 1 1 sc sl2 i L2 V c K i c LCL Filter Fig 4.3 Active damping by weight age capacitor current feedback

43 4.6 Physical realization of Active Damping (concepts of Virtual resistance) Physical realization of Active Damping (concepts of Virtual resistance) The concepts of active damping can be realized from the equation (5). After splitting the state-space form we get, dv dt dil1 L 1 = -(1+ K1) Vc - K2 il1- K3 i L2+ Uinv dt dil2 L 2 = Vc - Ug dt c C = il1-i L So, if we try to give the circuit form of the above equation then it approximately looks like:- L1 Rv L (1 + K1) V 2 c V c U inv C i i L 2 L1 U g R p Fig 4.4 Approximate Circuit representation for Active damping

44 27 Introduction to Active Damping R v is the series and R p is the parallel virtual resistance. Now from circuit representation it is clear that these two resistances are providing the damping to the LCL resonance though these resistances do not exits in practical. These are coming just because of control action which is used to damp the resonance that is why these are called Virtual Resistances. [5] It is quite clear from the following transfer functions:- il2 1 = U K inv sl L C(s + s + w ) r L1 i And U 2 L1 2 inv 1+ s L C =. K sl L C(s + s + w ) r L1 Fig 4.5 comparison of different damping factor in Active Damping K So the damping factor D = which is proportional to K. The 2L 1 w r results of in depth comparison of active and passive damping is given at Appendix A

45 4.8 Active damping loop realization Active damping loop realization The following figure shows the general practical approach of active damping loop. It consists of three feedbacks like two inductor currents and one capacitor voltage though it is shown that there is no need of feeding backs the capacitor voltage for arbitrary pole placements. But in the case of optimal damping (Appendix A) all the three feedback signals are needed. L1 i L 1 i L 2 Vc L2 INVERTER Current sense voltage sense Gate pulses for IGBT PWM Pulse ZC generation C Current sense U g Processing Fig 4.6 State-space based active damping loop (general case)

46 29 Introduction to Active Damping Fig 4.7 Comparison of virtual resistance based damping and actual i U resistance based damping of L2 inv transfer function 4.9 Conclusion Active damping by means of state-space based method is much more robust compared to other ones. The implementation of the statespace based method is much simpler than complex control topology. The active damping implemented in MATLAB/SIMULINK and also in practically. The results are given in chapter 7 and 8.

47 Chapter 5 Grid synchronization & Introduction to Phase Locked Loop 5.1 Introduction A phase locked loop is used in applications where tracking of phase and frequency of the incoming signal is necessary. For grid connected applications of power converters such as distributed generation and power quality improvement, a PLL is used to generate unit sine and cosine signals synchronized to system frequency from utility voltage. Closed loop control of power converters in synchronous reference frame method needs these unit sine and cosine signals to compute feedback and modulating signals. Here SRF based (Synchronous frame based) algorithm is described. It is robust algorithm that also works under abnormal grid condition. 5.2 PLL algorithm Fig 5.1 Basic PLL structure

48 31 Grid synchronization & Introduction to Phase Locked Loop Operation of a three phase PLL can be schematically shown in the Fig. 5.1 to obtain the phase information the three phase voltage signals ( V a, V b & V c ) are transferred into stationary two phase system (V α,v β ). Where, Va = Vmcos( ωt) Vb = Vmcos( ωt 2 π / 3) V = V cos( ωt 4 π / 3) c m Now phase angle (θ ) can be obtained by either synchronizing the voltage space vector V, along q axis or along d axis of synchronously rotating reference frame. The voltage space vector is to be aligned with q axis (Fig. 5.2). Then position of d axis (θ ) is related to ω t by θ ωt π /2 =. Fig 5.2 Reference frame ' ' θ gets estimated as θ by integrating estimated frequency ( ω ) which is the (1 ) summation of output of PI controller (. + sτ K p ) and the feed-forward sτ frequency ( ω ).The voltage vectors in synchronously rotating reference ff ' frame can be found out using θ from following equation (1) & (2):-

49 5.1 PLL algorithm 32 3 Vd = Vmsin( θ θ '). (1) 2 3 Vq = Vmcos( θ θ ') (2) 2 Now if we can make θ θ ' then V will be along the q- axis & grid will be synchronized with the system. The voltage vector along synchronously 3 rotating d-axis (eqn 3) can be expressed Vd = Vm( θ θ ') (3) 2 So, overall system in Fig. 5.1 can be simplified to that shown in Fig 5.3. Fig 5.3 Simplified block diagram of SRF PLL The analysis & design of the PI- controller is given in the reference [1].

50 33 Grid synchronization & Introduction to Phase Locked Loop Fig 5.4 Theta is synchronized with grid voltage Fig 5.5 sin( θ ) and ' sin( θ ) synchronized with each other

51 5.3 Conclusion Conclusion A three-phase PLL system is presented here for utility interface applications. Operation is suitable especially under distorted utility conditions such as line harmonics and frequency disturbances but in the present case balanced grid is assumed. The PLL is completely implemented in inside FPGA based controller as well as in practically without the use of any hardware filters. All analytical results were experimentally verified.

52 Conclusion

53 Chapter 6 Control Scheme for Standalone and Grid Interactive Mode with LCL Filter 6.1 Introduction Here in this chapter dq-based control strategy [1] is adopted. The control scheme for LCL filter based system is quite different as well as complicated from those of simple L filter based grid-connected system. The number of sensors is also more in case of LCL filter. Control scheme that is described, can be applicable to both in standalone and grid parallel mode. Initially current controllers, voltage controller and then at last State-space based damping loop is described. 6.2 Model for control design For control design, the grid is modeled as an ideal sinusoidal three phase voltage source without line impedances, although, in reality there are line impedances and distortions like line harmonics and unbalances. The space notation is used. Three phase values are transformed into the dqreference frame that rotates synchronously with the line voltage space vector. From control point of view it is advantageous to control dc values since PI controller can achieve reference tracking without steady state errors. Modeling of the LCL filter in the dq-reference frame without frequency dependences of the inductances is performed here.

54 37 Control Scheme for Standalone and Grid Interactive Mode with LCL Filter i Load L O A D V dc O i dc R Y B L1 Active Rectifier i i L1 L 2 V c L1 L1 C C C L2 L2 L2 Grid Grid Grid N Neutral Fig 6.1 Active rectifier with LCL filter Simplified model for Grid connected mode L 1 Vc L 2 U inv i L1 i c C i L2 U g Fig 6.2 Grid connection with LCL filter

55 Simplified model for Grid connected mode 38 The differential equations are written in space vector domain - d i - - L1 1 c inv L = V - U...(1) dt - d i - - L2 2 g c L = U - V...(2) dt - d V - - c C = il 2-i L1...(3) dt Here for simplicity of the analysis the leakage resistance of the inductors is neglected at the same time ESR of the capacitance is being neglected. After transforming to dq-domain we get the differential equations:- dil1q L 1 = Vcq - Uinvq - wl1i L1d...(4) dt dil1d L 1 = Vcd - U invd + wl1i L1q...(5) dt dil 2q L 2 = Ugq - Vcq - wl2i L 2d...(6) dt dil 2d L 2 = Ugd - V cd + wl2i L 2q...(7) dt dv cq C = il 2q - i L1q...(8) dt dvcd C = il 2d -i L1d...(9) dt

56 39 Control Scheme for Standalone and Grid Interactive Mode with LCL Filter Now if we include the dynamics of the DC-bus voltage of the power converter we get, dv 3 i U C = i -i = -i...(10) dc L2q gq dc dc load load dt 2 Vdc Now the control will contain simple decoupling terms in order to decouple the d- and q-current dynamics. No perfect dynamic decoupling can be achieved due to delays in the loop and filter resonance. 6.3 Over view of control loop consisting of three states of system * i L2 * V c * i L1 * U inv i L2 V i c L 1 Conventional three loop control strategy for LCL filter Fig 6.3 Reduction of Controller complexity But here for this type of filter the capacitor voltage can be indirectly controlled so there is no need of capacitor voltage controller for LCL filter.

57 Two loop control strategy for LCL filter 40 by the i -i = i and the V c = c The L2 L1 c V c. i L1 and L 2 1 i dt, hence if we can control C i separately then that itself control the i c followed Two Loop current control Strategy for LCL filter Now the control loop may be reduced to following fashion:- * i L2 * i L1 * U inv i i L2 L 1 Two loop control strategy for LCL filter Fig 6.4 So here the output of the line side current controller is becoming the reference of converter side current. Now here line side current and converter side current are almost equal in magnitude and phase in fundamental as capacitor size is limited in LCL filter because of reactive power burden. Hence further more simplification is possible. The converter side current controller can also be omitted and only line side current controller is fair enough to control the current. The output of line side current controller will become inverter input reference.

58 41 Control Scheme for Standalone and Grid Interactive Mode with LCL Filter * i L2 * U inv i L2 Single loop control strategy for LCL filter Fig 6.5 Limitation of Single loop control strategy for LCL filter Single grid current loop controller is not sufficient for stability of the overall system. The resonance of the filter can make the system unstable as here we are only concentrating on the fundamental current where LCL filter has significant amount of resonance frequency super imposed over the fundamental. So we need to consider the resonance carefully. Higher-level control [10] loops are required to provide fast dynamic compensation for the system disturbances and improve stability. 6.4 Current control strategy Conventional PI controller [7] in innermost loops is not reasonable from point of view of speed and complexity. Basically in order to control the resonance in the filter inner most current loop should be very fast (ideally instantaneous). So, for this type of filters inner most current loop ( i L1 ) can be designed by a proportional controller. This controller or this inner loop makes the system dynamic response very good by limiting the unwanted

59 6.4 Current control strategy 42 resonance in the system or in other words it shifts the closed loop poles in LHS of s-plane quite a bit. The results are given in chapter 7 & 8. * i L2 * i L1 Kp * U inv i i L2 L 1 Modified two loop control strategy for LCL filter Fig 6.6 The total current control loop structure becomes like following where outer loop is line side current and inner loop is the converter side current and converter current feedback is used for stability [7] (damping oscillation). * i L2 1+ st * i L1 c Kp Kc st c KPWM U inv 1 sl 1 il1 U g Vc 1 1 sc sl2 i L2 i L2 i L1 V c Fig 6.7 Two-loop control strategy for LCL filter Now we can see the transfer function of the closed loop current controller with the inner loop, it is a 4 th order system. In the outer loop PI-controller

60 43 Control Scheme for Standalone and Grid Interactive Mode with LCL Filter used as usual and proportional controller is used in inner loop. The closed loop transfer function is given below, is used to analyze the stability. * i (KpKcK L2 PWM )s+ KpKcK PWM / Tc = i (L L C)s + (K K L C)s + (L + L )s + (K K K + K K )s+ K K K / T L2 1 2 c PWM p c PWM c PWM p c PWM c 6.5 Analysis of controller performance A. Analysis of the outer loop... (11) The outer loop is a PI-controller, where the value of K p should be quite high value to track the reference but at the same time Kp should not be as large as wish, which can be seen from the following analysis. As it grows bigger, the poles will shift towards the right side of s-plane. Fig 6.8 Root locus of i i * L2 L2 with K p = 0.5

61 6.5 Analysis of controller performance 44 Fig 6.9 Root locus of i i * L2 L2 with K p = 1 Fig 6.10 Root locus of i i * L2 L2 with K p = 2

62 45 Control Scheme for Standalone and Grid Interactive Mode with LCL Filter B. Analysis of the inner loop The inner loop basically improves the stability of the system and increases the robustness. In other words more important role of the inner loop is to damp the resonance peak but at the same time very high of Kccan make system unstable also. So, the value of Kchas to be limited and we cannot depend the value of Kcto damp the oscillation. Fig 6.11 Bode plot of i i * L2 L2 with different value of K c

63 6.6 Inclusion of inner most state-space based damping loop Inclusion of innermost state-space based damping loop As mentioned earlier that in the converter current loops the value of K c is limited from the point of view of stability. So, when high damped system is desired this method of damping not so much preferable. The state-space based method offer more flexibility of choosing the controller parameter and at the same time robustness. The total current control loop with the damping loop is shown below Control part Power converter LCL filter * i L2 1+ st U * inv K c Kp PWM st c Uinv 1 sl 1 Vc 1 1 sc sl2 i L2 V c il1 U g i L2 K Fig 6.12 Current control with State-space based damping loop

64 47 Control Scheme for Standalone and Grid Interactive Mode with LCL Filter 6.7 Operation in standalone mode with LCL filter Standalone mode operation is very useful especially in distributed generation case. This is the operation when grid outage occurs. here the power converter supplies a load which is connected in grid side. The power converter supplies the power to the load according it demands. Voltage across the load has to be maintain constant according to the demand. Instead of sensing the voltage here we can set the current reference from the active or reactive power demand of the load. L1 Vc L2 DC-bus PWM Inverter i C L1 i L 2 LOAD Fig 6.13 Standalone mode of operation

65 6.7 Control in standalone mode with LCL filter Control in standalone mode with LCL filter Here in standalone mode voltage loop is neglected as current reference is being generated from the active power reference. The complete vector control of the system is shown below which include active damping loop also in that. The results are given in chapter 7,8 and also in Appendix C * i L 2 d sensed i L 2 abc i L 2 d abc-dq i cabc K 1+ stc p stc Decoupling & dq-abc * U inv U invabc K KPWM PWM i L 1 abc sensed U gabc i L 2 q K 1+ stc p stc K sensed PLL Algorithm θ * i L 2 q Fig 6.14 Vector control of standalone mode Here though the grid is not connected but the unit vector is generated from the grid only by SRF based PLL algorithm. Single grid current control loop with the state-space damping loop is included to damp the un-wanted resonance of LCL filter. 6.8 Control in grid parallel mode with LCL filter In grid connected mode load is connected across the DC-bus, which is to be supplied from the grid with good power quality (FEC mode)[16]. So,

66 49 Control Scheme for Standalone and Grid Interactive Mode with LCL Filter naturally the load voltage has to be maintaining constant. Here DC-link voltage controller is must for this kind of operation. Reactive Power reference * i L 2 d * V dc i L 2 d K 1+ stdc dc stdc K 1+ stc p stc Decoupling & dq-abc * U invabc U invabc K KPWM PWM V dc sensed * i L 2 q Active Power reference i L 2 q K 1+ stc p stc K i cabc U gabc sensed PLL Algorithm θ Fig 6.15 Vector control in grid parallel mode 6.9 Sensor less operation In the control of LCL filter based system (grid connected or standalone mode) consists of several loops cascaded. Naturally while doing the implementation in practice it needs quite a few sensors. These sensors, specially the LEM current sensors are costly hence it is better to use minimal number of sensors.

67 6.10 Conclusion 50 So, in the LCL filter based system the idea of minimizing sensors really makes sense. The way out is to estimate the corresponding quantities like voltages or currents. There are two ways to eliminate sensors, one way to run a parallel process in the controller and then calculate quantities and use for control, the second way to design the reduced order observer to estimate the states Conclusion The control strategies that is described in this chapter equally applicable for standalone and grid parallel mode with very little difference. The above control strategy is simulated and experimentally verified and results are given in chapter 7 and 8.

68 Conclusion

69 Chapter 7 Simulation Results 7.1 Introduction The entire system is simulated in standalone and grid connected mode. The system parameters are specified in the following table. The simulation is carried out in MATLAB/SIMULINK software. At first the standalone mode results are presented followed by grid connected mode results. 7.2 System parameters Symbols Values used U g V (L-n) V dc V L 1 3mH L 2 3mH C 16 µ F f sw 5-10KHz f 50Hz C dc 3300 µ F Load (grid connected mode) ohm Load (in standalone mode) 10-20ohm

70 53 Simulation Results 7.3 Standalone mode results These are the results that are simulated in standalone mode (Power circuit in chapter 6). Actually this mode of operation is essential from point of proving the controller performance as in this case we can test the each loop separately (current control loop, damping loop).this flexibility is not available in grid parallel mode where all control has to be worked in cascaded manner. Hence this mode of operation is essential before going to grid interactive mode. - Actually the current control loop & damping loops are decoupled in the sense, their BW is well apart. So for current loop sees the damping loop instantaneous similar reasons applicable for voltage loop & current loop. So, we can test these loops separately & verify. System has simulated in all condition like with damping, without damping, with load & without load & functions of all the three loops are shown separately. In standalone mode the all the results are with 20ohms load in the grid side Figs 7.1 to 7.5 are the results of standalone mode without damping. Figs 7.6 to 7.10 are the results of standalone mode with damping. 7.4 Grid connected mode results In this mode all the three loops like voltage (DC-bus loop), current (grid current loop) & active damping loop are cascaded one after other. DCbus loop is the outer most loops so naturally much slower compared to other two. Hence while designing the DC-bus loop the other loop are taken as one. Similarly for the current loop, damping loop is taken as one.

71 7.4 Grid connected mode results 54 Figs 7.11 to 7.15 are the results of grid connected mode with damping and with no-load Figs 7.16 to 7.20 are the results of grid connected mode with damping and with load Figs 7.21 to Fig 7.23 are the results of Active damping in grid parallel mode

72 55 Simulation Results Fig 7.1 d-axis load side current in standalone when the reference 0.05 sec Fig 7.2 q-axis load side current in standalone when the reference 0.05 sec

73 7.3 Standalone mode results Fig 7.3 R-phase load side current in standalone when the reference 0.05 sec Fig 7.4 capacitor current in standalone when the reference 0.05 sec

74 57 Simulation results Fig 7.5 Capacitor voltage in standalone when the reference 0.05 sec Fig 7.6 d-axis load side current in standalone when the damping loop is being 0.02 sec

75 7.4 Standalone mode results Fig 7.7 q-axis load side current in standalone when the damping loop is being 0.02 sec Fig 7.8 Capacitor current in standalone when the damping loop is being 0.02 sec

76 59 Simulation results Fig 7.9 Capacitor voltage in standalone when the damping loop is being 0.02 sec Fig 7.10 R-phase load side current in standalone when the damping loop is being 0.02 sec

77 7.4 Grid connected mode results Fig 7.11 d-axis grid side current in no-load where the reactive power reference 0.02 sec Fig 7.12 q-axis grid side current in no-load

78 61 Simulation results Fig 7.13 grid side current in no-load Fig 7.14 DC-link voltage profile in no load

79 7.4 Grid connected mode results Fig 7.15 Fundamental of R-phase inverter voltage& grid voltage of R- phase Fig 7.16 d-axis current of grid side with load

80 63 Simulation results Fig 7.17 q-axis current of grid side with load 0.08sec Fig 7.18 line side current of R-phase with load 0.08sec

81 7.4 Grid connected mode results Fig 7.19 DC-link voltage profile when the load is being 1.5sec Fig 7.20 Line side current & line voltage with UPF operation

82 65 Simulation results Fig 7.21 Line side current when active damping 0.1 sec Fig 7.22 capacitor voltage when active damping 0.1 sec

83 7.5 Conclusion Fig 7.23 capacitor current when active damping 0.1 sec 7.5 Conclusion Here in this chapter the entire system is simulated in two modes Standalone and Grid connected mode. The test of controller can be only be done in standalone only. Hence the controller test results are given in standalone mode and test of active damping results are given in FEC mode as well as in standalone mode.

84 Conclusion

85 Chapter 8 Experimental Results 8.1 Introduction The system has experimentally verified on 10KVA power converter which is operated in standalone mode & FEC mode with LCL filter in Lab. Parameters are same as in simulation. 8.2 Standalone Mode of operation Figs 8.1 to 8.3 are the results of standalone mode without damping. Figs 8.4 to 8.10 are the results of standalone mode with damping. 8.3 grid parallel mode of operation Figs 8.11 to Fig 8.20 are the results of grid parallel mode including damping loops.

86 69 Experimental Results Fig 8.1 q-axis load side current with change of reference Fig 8.2 q-axis load side current with change of reference

87 8.2 Standalone mode operation 70 Fig 8.3 load side current with change of reference Fig 8.4 Un-damped load current & its spectra in standalone mode

88 71 Experimental Results Fig 8.5 Damped load current & its spectra in standalone mode Fig 8.6 Active damping test of load current with lesser state-weight age

89 8.2 Standalone mode results 72 Fig 8.7 Active damping test of load current with higher state-weight age Fig 8.8 Un-damped capacitor current & its spectra

90 73 Experimental Results Fig 8.9 Active damping test of capacitor current Fig 8.10 Active damping test of q-axis capacitor current

91 8.3 Grid connected mode results 74 Fig 8.11 DC bus control test (voltage rise) Fig 8.12 DC bus control test (voltage falling)

92 75 Experimental results Fig 8.13 Distorted current from grid (full of resonance) Fig8.14 Less distorted current grid (state-weight age K=10)

93 8.3 Grid connected mode results 76 Fig 8.15 smooth current from grid (state-weight age 25) and its FFT Fig 8.16 grid side current dynamics when sudden change in DC-bus

94 77 Experimental results Fig 8.17 grid side current dynamics when sudden change load Fig 8.18 Line side current when active damping is being enabled mid-way

95 8.3 Experimental results 78 Fig 8.19 Active damping loop is being enabled mid-way with BW of 1.2 KHz Fig 8.20 Distortion in the utility voltage and smoothing out by active damping

96 79 Experimental results 8.3 Conclusions The standalone mode results are given in the chapter and those results are enough to prove concepts of controls. The grid connected mode results are very similar to above results with very little difference. It is also clear how state-feedback approach can be very effective for analysis and practical implementation. The analysis is given in chapter 6.

97 Conclusion

98 Appendix A Further Investigation about damping A.1 Introduction In this chapter some more results are presented to show the comparison between active & passive damping. The active damping that is presented in this report is based on the State-space based (Pole-assignment) where pole of the system are placed in the LHS of s-plane based on the requirement in the transient response. The only disadvantage of this approach is that this approach is not energy optimized or in other words energy required in this method is not in the designer s hand. So, more is the damping, more is the energy required. The way out of this problem or in other words to balance the energy required & damping is to go for optimal pole assignment. In this approach energy required is minimized at the same time very good damping is achieved. This chapter is also focuses on the Optimal damping in LCL filter. A.2 Comparison of Active & Passive damping The comparison of active & passive damping is given below by the i U L1 help of the transfer function inv transfer function.

99 82 Further Investigation about damping Fig A.1 Frequency response of passively damped system. i U L1 inv for un-damped, actively damped & Fig A.2 Frequency response of passively damped system. i U L2 inv for un-damped, actively damped &

100 A.3 Optimal Damping (Advanced Active Damping) 83 A.3 Optimal damping (Advanced Active Damping) A.3.1 Introduction The optimal pole is done by solving Riccati s Equation. Here we have complete balance between energy & damping. Linear quadratic (LQ) optimal control gives that freedom to the designer to place pole of the system optimally. In the control law Uinv = KX the gain matrix can be found from the Riccati s Equation. In the LQ based control we can get good stability margin and robustness. In this method we determine the gain matrix K after minimizing the cost function. 1 J = 2 T T (X(t) QX(t) + U (t) RU(t)) dt 0 Here the Q state weight age matrix and R is the input weight age matrix. The ith diagonal of Q determine direct weight age and off diagonal element determines the coupling between the states, for the linear system generally these off diagonal terms are not much of importance and are neglected in the current study. LCL A.3.2 Optimal gain matrix calculation There are many ways to select these Q and R matrices. Here in the present case selection of R is pretty simple as grid is not a controllable input hence R can be taken as one. But the selection of Q is bit controversial. In this design the Q is selected by keeping in mind the following criterion.

101 84 Further Investigation about damping (a) Heavy weight is putted on the L1 so as to get fast and well damped transient response. (b) Impedance of the capacitor is fairly and very little current bypasses through it also; thus almost similar weight can be putted on thei L2. (c) Vc should be given least weight (or may zero weight) as it should be directed by the grid voltage in order avoid converter saturation By the above criterion Q = and R = Solving the equation, T 1 T A P + PA PBR B P + Q = P = Now system is stable as P is positive definite matrix. Now for this P, optimal gain matrix [7 11 2]. The optimal poles are at i K = λ = 165, T R B P + j which is Here we can see that the gain matrix that is coming is consists of nonzero weight age to all the states unlike in arbitrary pole placement where capacitor weight age was zero.

102 A.3 Optimal Damping (Advanced Active Damping) 85 Fig A.3 Root loci of the original system & optimally damped system. Fig A.4 Frequency response of optimally damped system. i U L1 inv in the original system &

103 86 Further Investigation about damping Fig A.5 Frequency response of optimally damped system. i U L2 inv in the original system & A.4 Conclusion In this chapter the few more comparison of among the different damping procedure is done. The optimal damping seems to be most perfect method of damping as in this method the energy & damping both are balanced but it needs one more sensing of capacitor voltage. Hence this procedure of damping can be applied suitably for sensor less system otherwise system may be complicated & costly also.