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Design of Robust Current Controller for Two-Level 12-Pulse VSC-based STATCOM

Design of Robust Current Controller for Two-Level 12-Pulse VSC-based STATCOM Hindawi Publishing Corporation Advances in Power Electronics Volume 2011, Article ID 912749, 7 pages doi:10.1155/2011/912749 Research Article Design of Robust Current Controller for Two-Level 12-Pulse VSC-based STATCOM 1 1 2 M. Janaki, R. Thirumalaivasan, and Nagesh Prabhu School of Electrical Engineering, VIT University, Vellore 632014, India Department of Electrical Engineering, Canara Engineering College, Mangalore 574219, India Correspondence should be addressed to M. Janaki, janaki.m@vit.ac.in Received 2 November 2010; Revised 22 March 2011; Accepted 6 April 2011 Academic Editor: Jose Pomilio Copyright © 2011 M. Janaki et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The static synchronous compensator (STATCOM) is a shunt connected voltage source converter (VSC) based FACTS controller using GTOs employed for reactive power control. A typical application of a STATCOM is for voltage regulation at the midpoint of a long transmission line for the enhancement of power transfer capability and/or reactive power control at the load centre. The PI controller-based reactive current controller can cause oscillatory instability in inductive mode of operation of STATCOM and can be overcome by the nonlinear feedback controller. The transient response of the STATCOM depends on the controller parameters selected. This paper presents a systematic method for controller parameter optimization based on genetic algorithm (GA). The performance of the designed controller is evaluated by transient simulation. It is observed that the STATCOM with optimized controller parameters shows excellent transient response for the step change in the reactive current reference. While the eigenvalue analysis and controller design are based on D-Q model, the transient simulation is based on both D-Q and 3-phase models of STATCOM (which considers switching action of VSC). 1. Introduction (1) a significant reduction in size can be achieved because of the reduced number of passive elements and their The concept of flexible AC transmission system (FACTS) smaller size, envisages the use of advances in power electronics technology (2) the STATCOM can supply required reactive current to achieve flexibility of system operation together with fast even at low values of bus voltages while the reactive and reliable control [1]. Fast control over the reactive power current capability of the FC-TCR at its limit varies can allow secure loading of transmission lines nearer to their linearly with the voltage. thermal limits, regulate the voltage, and improve system damping. The availability of high-power gate turn off (GTO) In this paper a 2-level, 12-pulse voltage source converter thyristors has led to the development of STATCOM. (VSC) is considered for STATCOM configuration with type- The STATCOM is a VSC based FACTS device used for 2 controller. The type-2 controller regulates the reactive cur- shunt reactive power compensation making use of gate turn rent output of STATCOM by adjusting the phase angle of the off (GTO) power semiconductor devices. The VSC is con- converter output voltage relative to the bus voltage [1–3, 6]. nected to the system bus through an interfacing reactance, The PI controller-based reactive power control with feed- which is the leakage reactance of the coupling transformer back (from the reactive current) can destabilize the system [2, 3]. The STATCOM is connected at the midpoint of while operating in inductive mode [6]. Schauder and Mehta long transmission line to regulate the voltage and enhance [2] propose a non linear feedback controller to overcome the the power transfer capability. The major advantages of problem of instability in the inductive mode of operation. the STATCOM over the fixed capacitor-thyristor controller However, the response of PI controller with state variable reactor- (FC-TCR-) type SVC are [4, 5]: feedback mainly depends on the controller parameters that 2 Advances in Power Electronics θ π V s I R 4 I − Rref Σ k p Σ R , X s s 4 π V k s s i Figure 2: Type-2 controller for 2-level VSC based STATCOM. θ + α where θ is thephase angleof bus voltage. α is the angle by dc which the fundamental component of converter output voltageleads theSTATCOM busvoltage V . Schematic Equivalent circuit s k is the modulation index (constant for a two-level con- Figure 1: STATCOM shunt FACTS controller. verter with fundamental frequency modulation and 180 conduction mode) and is given by k = 2 6/π for a 12-pulse are selected. The objective of this paper is to present converter. a systematic method for controller parameter optimization The following equations in the D-Q variables can be giv- based on genetic algorithm (GA), while ensuring stability en for describing STATCOM: and improvement of transient response. This paper is organized as follows. Section 2 describes the dI R ω ω sD s B B =− I − ω I + V − V , modeling of STATCOM. The design of reactive current con- sD o sQ sD sD dt X X s s troller is explained in Section 3. The GA-based optimization dI of controller parameters is described in Section 4, Section 5 R ω ω sQ s B B = ω I − I + V − V , o sD s sQ (2) Q sQ gives the conclusions. dt X X s s dV ω ω dc B B 2. Modeling of STATCOM =− I − V , dc dc dt b b R c c p The schematic of STATCOM is shown in Figure 1.The STAT- where COM is connected to the bus (with voltage V )through a coupling transformer with resistance and reactance of R and X , respectively. In the power circuit of a STATCOM, i =− k sin(θ + α)I + k cos(θ + α)I . (3) s dc s sD s sQ the converter has either a multipulse and/or a multilevel configuration. Here the STATCOM is realized by 12-pulse I , I D-Q components of STATCOM current. sD sQ two-level configuration. The real and reactive currents are defined as The detailed three-phase model of a STATCOM is devel- I = I sin(θ ) + I cos(θ ), oped by modeling the converter operation by switching P sD s sQ s (4) functions, [6–8]. The modeling of two-level VSC based on I =−I cos(θ ) + I sin(θ ). R sD s sQ s switching functions is discussed in detail in [6] and it is not repeated here. Values of I and I result in positive values when the STAT- P R COM is absorbing real and reactive power. 2.1. Mathematical Model of STATCOM in D-Q Frame of Ref- erence [7, 9]. When switching functions are approximated by their fundamental frequency components neglecting 2.2. STATCOM Current Control (Two-Level VSC). With a 2- harmonics, STATCOM can be modeled by transforming the level VSC, the reactive current control can be achieved by three-phase voltages and currents to D-Q variables using varying α alone (refer Figure 2). When STATCOM regulates Kron’s transformation [10]. The STATCOM can be repre- theBus voltage, thereactivecurrent reference i in Figure 2 Rref sented functionally, as shown in Figure 1. The magnitude is obtained as the output of the bus voltage controller. control of converter output voltage V is the function of DC However, in the present study i is kept constant. s Rref voltage V for type-2 converters [2, 6]. The converter output dc In this controller, the modulation index k is constant. voltage can be represented in the D-Q frame of reference as The capacitor voltage is not regulated but depends upon the phase difference between the converter output voltage 2 2 i i i and the bus voltage. The reactive current control is effected V = V + V , s sD sQ by converter output voltage magnitude (which is a function i (1) of DC voltage) and achieved by phase angle control [2, 6]. V = kV sin(θ + α), dc s sD This causes the variation of capacitor voltage over a small range with change in operating point. V = kV cos(θ + α), dc s sQ Advances in Power Electronics 3 Table 1: Eigenvalues with PI controller. Capacitive region i =−1Inductiveregion i = 1 R R Inductive region −834.58 −775.36 1500 −81.819 ± j1429.812.969± j1485.8 k = 0 −10 −10 p −9.9137 −9.9137 3. Reactive Current Controller Design Capacitive region 3.1. PI Controller without Nonlinear State Variable Feedback. 1350 In this section we investigate the design of the reactive current controller. The reactive current of the STATCOM depends on the difference between the bus voltage and the −100 −80 −60 −40 −20 0 20 40 converter output voltage. To simplify the design procedure Real part we design the reactive controller assuming the voltage at STATCOM bus is constant (neglecting the dynamics in the Figure 3: Movement of the critical eigenvalues with PI controller for k = 0–10 and k /k = 10. transmission network). The eigenvalue analysis is performed p i p by linearizing the system represented by Figure 1 and de- scribed by (1)–(4) for capacitive and inductive mode of 2.5 operation of STATCOM and are given in Table 1.It is tobe noted that the system is unstable in the inductive mode of Inductive region operation of STATCOM. The movement of the critical eigenvalues for operating 1.5 points in the inductive region and capacitive region when a PI controller (refer Figure 2.) is used is shown in Figure 3. Referring to Figure 3 it is observed that in the inductive 0.5 region the poles move towards the imaginary axis resulting in oscillatory instability. The instability in the inductive mode of operation of a STATCOM is also reported in the literature [2, 6]. −0.5 As detailed in [3], the condition for stability can be −1 expressed as Capacitive region −1.5 ωC 0 0.5 1 1.5 i < V . (5) R dc Time (s) Figure 4: Step response of STATCOM with PI controller. The above condition is always satisfied when i < 0 (when the converter is operating in the capacitive mode). Thus, the problem of instability can arise only while operating in the inductive mode and is also verified in the controller is used is also brought out by simulation as shown present study. in Figure 4. The important point is that the variation of locations of zeros with the operating point, the angle of departure for this complex mode varies from 0 to −180 as the operating point 3.2. PI Controller with Nonlinear State Variable Feedback. is changed [3]. Thus design of a compensator in cascade with described and Mehta [2] propose a nonlinear feedback con- the PI controller which is suitable for all operating points is troller to overcome the problem of instability in the inductive also difficult. mode of operation of STATCOM. The block diagram of the It is to be noted that the system is unstable in the induc- controller is shown in Figure 5. tive mode of operation. To validate the linearized analysis It is to be noted that the nonlinear controller is active (which neglects harmonics in the switching functions) and only when i > (ωC/k)V and also only during a transient. R dc to check the performance for large deviations from an In steady state, (when V is a constant), the output of the dc operating point, the transient simulation of a 12-pulse 3- multiplier in nonlinear feedback is zero. phase STATCOM is carried out using MATLAB-SIMULINK. The root locus of the critical eigenvalues with nonlinear The action of the converter is modeled using the switching feedback controller for inductive and capacitive mode of op- functions (switching instants are obtained from θ and the eration is shown in Figure 6. controller output α)[3, 6, 11]. The instability predicted It is observed from the root locus plot that the critical by linearized analysis in the inductive region when a PI modes are stable. Imaginary part R 4 Advances in Power Electronics 1.5 sT 1+ sT Inductive region dc b − 0.5 0 π + ∑ Rref − −0.5 i s −1 Capacitive region −1.5 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Figure 5: Nonlinear feedback controller. Time (s) Figure 7: Phase “a” current of STATCOM with nonlinear feedback controller. k = 0 1.5 Inductive region k = 0 1450 p 0.5 Inductive region Capacitive region −0.5 −200 −150 −100 −50 0 −1 Real part Capacitive region Figure 6: Root locus with nonlinear feedback for k = 0–10 and −1.5 k /k = 10. 0 0.5 1 1.5 i p Time (s) Table 2: Eigenvalues with nonlinear feedback. Figure 8: Response of STATCOM with nonlinear feedback con- troller. Capacitive region i =−1Inductiveregion i = 1 R R −834.58 −1173.2 current. It is to be noted that, although the system is stable −81.819 ± j1429.8 −102.73 ± j1400.7 during inductive operation, the transient response of the −10 −7.3766 ± j4.3491 STATCOM is slow. −9.9137 Hence there is a need to optimize the controller parame- ters to improve the transient response of the STATCOM. The optimization of reactive current controller parameters based The eigenvalues of the system with nonlinear feedback on GA to improve the transient response will be discussed in and suboptimal controller parameters [6]are given in the section to follow. Table 2. The suboptimal controller parameters [6]are k = 0.33, k = 3.33, g = 2, and T = 0.1. It is to be noted that, i W the eigenvalues are stable with good stability margin for both 4. Application of Genetic Algorithm for inductive and capacitive mode of operation of STATCOM. Controller Parameters Optimization The transient simulation of the 3 phase model of the STATCOM for step change in the reactive current is shown 4.1. Introduction. GA has been used as optimizing the pa- in Figures 7 and 8. rameters of control system that are complex and difficult to It can be seen that, transition from capacitive to inductive solve by conventional optimization methods [12]. mode of operation of STATCOM is slow and steady state It maintains a set of candidate solutions called popu- is reached after 0.2 sec following the step change in reactive lation and repeatedly modifies them. Each member of the Imaginary part sa Advances in Power Electronics 5 population is evaluated using a fitness function. The popula- tion undergoes reproduction in a number of iterations. One or more parents are chosen stochastically, but strings with ζ = 10% D-contour higher fitness values have higher probability of contributing the offspring Genetic operators, such as crossover and mu- tation are applied to parents to produce offspring. The offspring are inserted into the population and the process is repeated. α =−0.5 Given a random initial population, GA operates in cycles −5 called generations as follows. The basic steps involved in GA are as follows. −10 Step 1: Begin with a randomly generated population of chro- −15 mosome-encoded “solutions” to a given problem. −20 −5 −4 −3 −2 −10 1 Step 2: Calculate the fitness of each chromosome, where Real fitness is a measure of how well a member of the population performs at solving the problem. Figure 9: D-contour with α =−0.5and ξ = 10%. Step 3: Retain only the fittest members and discard the least The D-contour in Figure 9 can be mathematically de- fit members. scribed as Step 4: Generate a new population of chromosomes from f (z) = Re(z) − min[− Im(z), α] = 0, (6) the remaining members of the old population by applying the operations reproduction, crossover, and where zεC is a point on D-contour and C represents the com- mutation. plex plane. Step 5: Calculate the fitness of these new members of the Define J as population, retain the fittest, discard the least fit, and J = max Re(λ ) − min −ζ|Im(z)|, α i = 1, 2, 3...n, reiterate the process. (7) The GA-based optimization guarantees the system stabil- ity under varying operating conditions. where n is the number of eigenvalues. λ is the ith eigenvalue of the system at an operating point. A negative value of J implies that all the eigenvalues lie on the left of the D- 4.2. Objective Function. For damping oscillations, the damp- contour. Similarly some or all eigenvalues will lie on the right ing factor ζ of around 10% to 20% is considered to be ade- of the D contour if J is positive. quate. A damping factor of 10% would be acceptable to most On the basis of these facts, objective function E is defined utilities and can be adopted as the minimum requirement. as Further, having the real part of eigenvalue restricted to be less than a value, say α, guarantees a minimum decay rate 2 Sum Squared Error (E) = e,(8) α.A value α =−0.5 is to be considered adequate for an acceptable settling time. The closed loop mode location where should simultaneously satisfy these two constraints for an e = i − i,(9) acceptable small disturbance response of the controlled Rref R system. where i and i are the Reactive current reference input Rref R If all the closed loop poles are located to the left of the and reactive current output of the STATCOM, respectively. contour shown in Figure 9, then the constraints on the Hence the optimization problem can be stated as follows: damping factor and the real part of eigenvalues are satisfied and a well-damped small disturbance response is guaranteed. Minimize E, This contourisreferred to as the D-contour [3, 13]. (10) The system issaid tobeD-stableifitisstablewith respect Subjected to J ≤ 0. to this D-contour, that is, all its poles lie on the left of this In addition to this, the boundaries of optimal parameters are contour. This property is referred to as generalized stability in control literature. This generates a neat specification, g ≤ g ≤ g T ≤ T ≤ T , min max W W W min max the closed loop system should be robustly D-stable, that (11) is, D-stable, for the entire range of operating and system K ≤ K ≤ K K ≤ K ≤ K . p p p i p min max min i max conditions. Hence a system is said to be “robust,” if, in spite of changes in system and operating conditions, the closed loop Genetic algorithm is adopted to obtain the optimal poles remain on the left of the D-contour for the specified parameters of PI controller with nonlinear feedback loop range of system and operating conditions. [13, 14]. Imaginary 6 Advances in Power Electronics 1.5 Table 3: Parameters used for optimization with genetic algorithm. Parameter Value/type Inductive region Maximum generations 25 Population size 200 0.5 Type of selection Normal geometric [0 0.08] Type of crossover Arithmetic [4] Type of mutation Nonuniform [4 20 3] Termination method Maximum generation −0.5 −1 Table 4: Eigenvalues with optimal controller parameters based on Capacitive region GA. −1.5 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Capacitive region i =−1Inductiveregion i = 1 R R −1856.7 −2194.6 Time (s) −83.949 ± j1373.6 −58.401 ± j1398.3 Figure 10: Phase “a” current of STATCOM with optimal controller −55.556 −32.538 ± j4.9879 parameters. −23.842 1.5 Inductive region The parameters used with GA are given in Table 3.The optimized parameters as obtained by GA are k = 0.69, k = p i 0.5 16.45, g = 2.54, and T = 0.018. The parameters used with GA are given in Table 3. Eigenvalues of the system with nonlinear feedback and optimized controller parameters are shown in Table 4.Com- paring with the eigenvalue results given in Table 2,itis −0.5 to be noted that although the damping of critical mode is decreased (in inductive region), the damping of other −1 modes is increased. The GA-based optimization ensures that the system is robustly D-stable for various operating points Capacitive region −1.5 under consideration. 0 0.5 1 1.5 It can be seen from Figures 10 and 11 that the transition Time (s) from capacitive to inductive mode of operation of STATCOM is very fast and takes less time, about 0.04 sec, to reach steady Figure 11: Step response with 3Φ model of STATCOM with state. optimal controller parameters. It is to be noted that the transient response of the STAT- COM is significantly improved. The steady state oscillations Appendix in thereactivecurrent of Figure 11 are due to harmonics in the converter output voltage. Thus with optimal controller STATCOM data in p.u: parameters the transient response of the STATCOM is improved. R = 0.01, X = 0.15, R = 78.7, b = 1.136. s s p c (A.1) The suboptimal controller parameters without nonlinear 5. Conclusion feedback [6]are k = 0.33 and k = 3.33, p i The suboptimal controller parameters with nonlinear In this paper we have presented a systematic method for con- troller parameter optimization based on genetic algorithm feedback [6]are k = 0.33, k = 3.33, g = 2, and T = 0.1. p i W The optimized parameters as obtained by GA are k = (GA) for the design of STATCOM reactive current controller. p 0.69, k = 16.45, g = 2.54, and T = 0.018. The GA-based optimization ensures that the system is i w robustly D-stable in the entire range of operation and system conditions. The performance of the designed controller is References evaluated and it was observed that the STATCOM with optimized controller parameters provides excellent transient [1] N. G. Hingorani, “Flexible AC transmission,” IEEE Spectrum, response. vol. 30, no. 4, pp. 40–45, 1993. R i sa Advances in Power Electronics 7 [2] C. Schauder and H. Mehta, “Vector analysis and control of advanced static VAR compensators,” IEE Proceedings C,vol. 140, no. 4, pp. 299–306, 1993. [3] K. R. Padiyar, FACTS Controllers in Power Transmission and Distribution, New age International (P), New Delhi, India, [4] Y.Sumi, Y. Harumoto,T.Hasegawa, M. Yano,K.Ikeda,and T. Matsuura, “New static VAR control using force-commutated inverters,” IEEE Transactions on Power Apparatus and Systems, vol. 100, no. 9, pp. 4216–4224, 1981. [5] L. Gyugyi, “Reactive power generation and control by thyristor circuits,” IEEE Transactions on Industry Applications, vol. 15, no. 5, pp. 521–532, 1979. [6] K. R. Padiyar and A. M. Kulkarni, “Design of reactive current and voltage controller of static condenser,” International Jour- nal of Electrical Power and Energy Systems, vol.19, no.6,pp. 397–410, 1997. [7] K. R. Padiyar and N. Prabhu, “Design and performance evaluation of subsynchronous damping controller with STAT- COM,” IEEE Transactions on Power Delivery,vol. 21, no. 3,pp. 1398–1405, 2006. [8] K. R. Padiyar and V. Swayam Prakash, “Tuning and perfor- mance evaluation of damping controller for a STATCOM,” International Journal of Electrical Power and Energy Systems, vol. 25, no. 2, pp. 155–166, 2003. [9] N. Prabhu, Analysis of sub synchronous resonance with voltage source converter based FACTS and HVDC controllers,Ph.D. dissertation, IISc Bangalore, 2004. [10] K. R. Padiyar, Power System Dynamics: Stability and Control, B.S. Publications, Hyderabad, India, 2nd edition, 2000. [11] K. R. Padiyar and N. Prabhu, “Analysis of subsynchronous resonance with three level twelve-pulse VSC based SSSC,” in Proceedings of the IEEE Conference on Convergent Technologies for Asia-Pacific Region (TENCON ’03),vol.1,pp. 76–80, October 2003. [12] D. E. Goldberg, Genetic Algorithm in search, Optimization and Machine Learning, Addison Wesley, Reading, Mass, USA, 1989. [13] R. Singh, A novel approach for tuning of power system stabilizer using genetic algorithm, M.Sc. dissertation, Indian Institute of Science, Bangalore, India, July 2004. [14] A. L. B. do Bomfim, G. N. Taranto, and D. M. Falcao ˆ , “Simul- taneous tuning of power system damping controllers using genetic algorithms,” IEEE Transactions on Power Systems,vol. 15, no. 1, pp. 163–169, 2000. 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Design of Robust Current Controller for Two-Level 12-Pulse VSC-based STATCOM

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Hindawi Publishing Corporation Advances in Power Electronics Volume 2011, Article ID 912749, 7 pages doi:10.1155/2011/912749 Research Article Design of Robust Current Controller for Two-Level 12-Pulse VSC-based STATCOM 1 1 2 M. Janaki, R. Thirumalaivasan, and Nagesh Prabhu School of Electrical Engineering, VIT University, Vellore 632014, India Department of Electrical Engineering, Canara Engineering College, Mangalore 574219, India Correspondence should be addressed to M. Janaki, janaki.m@vit.ac.in Received 2 November 2010; Revised 22 March 2011; Accepted 6 April 2011 Academic Editor: Jose Pomilio Copyright © 2011 M. Janaki et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The static synchronous compensator (STATCOM) is a shunt connected voltage source converter (VSC) based FACTS controller using GTOs employed for reactive power control. A typical application of a STATCOM is for voltage regulation at the midpoint of a long transmission line for the enhancement of power transfer capability and/or reactive power control at the load centre. The PI controller-based reactive current controller can cause oscillatory instability in inductive mode of operation of STATCOM and can be overcome by the nonlinear feedback controller. The transient response of the STATCOM depends on the controller parameters selected. This paper presents a systematic method for controller parameter optimization based on genetic algorithm (GA). The performance of the designed controller is evaluated by transient simulation. It is observed that the STATCOM with optimized controller parameters shows excellent transient response for the step change in the reactive current reference. While the eigenvalue analysis and controller design are based on D-Q model, the transient simulation is based on both D-Q and 3-phase models of STATCOM (which considers switching action of VSC). 1. Introduction (1) a significant reduction in size can be achieved because of the reduced number of passive elements and their The concept of flexible AC transmission system (FACTS) smaller size, envisages the use of advances in power electronics technology (2) the STATCOM can supply required reactive current to achieve flexibility of system operation together with fast even at low values of bus voltages while the reactive and reliable control [1]. Fast control over the reactive power current capability of the FC-TCR at its limit varies can allow secure loading of transmission lines nearer to their linearly with the voltage. thermal limits, regulate the voltage, and improve system damping. The availability of high-power gate turn off (GTO) In this paper a 2-level, 12-pulse voltage source converter thyristors has led to the development of STATCOM. (VSC) is considered for STATCOM configuration with type- The STATCOM is a VSC based FACTS device used for 2 controller. The type-2 controller regulates the reactive cur- shunt reactive power compensation making use of gate turn rent output of STATCOM by adjusting the phase angle of the off (GTO) power semiconductor devices. The VSC is con- converter output voltage relative to the bus voltage [1–3, 6]. nected to the system bus through an interfacing reactance, The PI controller-based reactive power control with feed- which is the leakage reactance of the coupling transformer back (from the reactive current) can destabilize the system [2, 3]. The STATCOM is connected at the midpoint of while operating in inductive mode [6]. Schauder and Mehta long transmission line to regulate the voltage and enhance [2] propose a non linear feedback controller to overcome the the power transfer capability. The major advantages of problem of instability in the inductive mode of operation. the STATCOM over the fixed capacitor-thyristor controller However, the response of PI controller with state variable reactor- (FC-TCR-) type SVC are [4, 5]: feedback mainly depends on the controller parameters that 2 Advances in Power Electronics θ π V s I R 4 I − Rref Σ k p Σ R , X s s 4 π V k s s i Figure 2: Type-2 controller for 2-level VSC based STATCOM. θ + α where θ is thephase angleof bus voltage. α is the angle by dc which the fundamental component of converter output voltageleads theSTATCOM busvoltage V . Schematic Equivalent circuit s k is the modulation index (constant for a two-level con- Figure 1: STATCOM shunt FACTS controller. verter with fundamental frequency modulation and 180 conduction mode) and is given by k = 2 6/π for a 12-pulse are selected. The objective of this paper is to present converter. a systematic method for controller parameter optimization The following equations in the D-Q variables can be giv- based on genetic algorithm (GA), while ensuring stability en for describing STATCOM: and improvement of transient response. This paper is organized as follows. Section 2 describes the dI R ω ω sD s B B =− I − ω I + V − V , modeling of STATCOM. The design of reactive current con- sD o sQ sD sD dt X X s s troller is explained in Section 3. The GA-based optimization dI of controller parameters is described in Section 4, Section 5 R ω ω sQ s B B = ω I − I + V − V , o sD s sQ (2) Q sQ gives the conclusions. dt X X s s dV ω ω dc B B 2. Modeling of STATCOM =− I − V , dc dc dt b b R c c p The schematic of STATCOM is shown in Figure 1.The STAT- where COM is connected to the bus (with voltage V )through a coupling transformer with resistance and reactance of R and X , respectively. In the power circuit of a STATCOM, i =− k sin(θ + α)I + k cos(θ + α)I . (3) s dc s sD s sQ the converter has either a multipulse and/or a multilevel configuration. Here the STATCOM is realized by 12-pulse I , I D-Q components of STATCOM current. sD sQ two-level configuration. The real and reactive currents are defined as The detailed three-phase model of a STATCOM is devel- I = I sin(θ ) + I cos(θ ), oped by modeling the converter operation by switching P sD s sQ s (4) functions, [6–8]. The modeling of two-level VSC based on I =−I cos(θ ) + I sin(θ ). R sD s sQ s switching functions is discussed in detail in [6] and it is not repeated here. Values of I and I result in positive values when the STAT- P R COM is absorbing real and reactive power. 2.1. Mathematical Model of STATCOM in D-Q Frame of Ref- erence [7, 9]. When switching functions are approximated by their fundamental frequency components neglecting 2.2. STATCOM Current Control (Two-Level VSC). With a 2- harmonics, STATCOM can be modeled by transforming the level VSC, the reactive current control can be achieved by three-phase voltages and currents to D-Q variables using varying α alone (refer Figure 2). When STATCOM regulates Kron’s transformation [10]. The STATCOM can be repre- theBus voltage, thereactivecurrent reference i in Figure 2 Rref sented functionally, as shown in Figure 1. The magnitude is obtained as the output of the bus voltage controller. control of converter output voltage V is the function of DC However, in the present study i is kept constant. s Rref voltage V for type-2 converters [2, 6]. The converter output dc In this controller, the modulation index k is constant. voltage can be represented in the D-Q frame of reference as The capacitor voltage is not regulated but depends upon the phase difference between the converter output voltage 2 2 i i i and the bus voltage. The reactive current control is effected V = V + V , s sD sQ by converter output voltage magnitude (which is a function i (1) of DC voltage) and achieved by phase angle control [2, 6]. V = kV sin(θ + α), dc s sD This causes the variation of capacitor voltage over a small range with change in operating point. V = kV cos(θ + α), dc s sQ Advances in Power Electronics 3 Table 1: Eigenvalues with PI controller. Capacitive region i =−1Inductiveregion i = 1 R R Inductive region −834.58 −775.36 1500 −81.819 ± j1429.812.969± j1485.8 k = 0 −10 −10 p −9.9137 −9.9137 3. Reactive Current Controller Design Capacitive region 3.1. PI Controller without Nonlinear State Variable Feedback. 1350 In this section we investigate the design of the reactive current controller. The reactive current of the STATCOM depends on the difference between the bus voltage and the −100 −80 −60 −40 −20 0 20 40 converter output voltage. To simplify the design procedure Real part we design the reactive controller assuming the voltage at STATCOM bus is constant (neglecting the dynamics in the Figure 3: Movement of the critical eigenvalues with PI controller for k = 0–10 and k /k = 10. transmission network). The eigenvalue analysis is performed p i p by linearizing the system represented by Figure 1 and de- scribed by (1)–(4) for capacitive and inductive mode of 2.5 operation of STATCOM and are given in Table 1.It is tobe noted that the system is unstable in the inductive mode of Inductive region operation of STATCOM. The movement of the critical eigenvalues for operating 1.5 points in the inductive region and capacitive region when a PI controller (refer Figure 2.) is used is shown in Figure 3. Referring to Figure 3 it is observed that in the inductive 0.5 region the poles move towards the imaginary axis resulting in oscillatory instability. The instability in the inductive mode of operation of a STATCOM is also reported in the literature [2, 6]. −0.5 As detailed in [3], the condition for stability can be −1 expressed as Capacitive region −1.5 ωC 0 0.5 1 1.5 i < V . (5) R dc Time (s) Figure 4: Step response of STATCOM with PI controller. The above condition is always satisfied when i < 0 (when the converter is operating in the capacitive mode). Thus, the problem of instability can arise only while operating in the inductive mode and is also verified in the controller is used is also brought out by simulation as shown present study. in Figure 4. The important point is that the variation of locations of zeros with the operating point, the angle of departure for this complex mode varies from 0 to −180 as the operating point 3.2. PI Controller with Nonlinear State Variable Feedback. is changed [3]. Thus design of a compensator in cascade with described and Mehta [2] propose a nonlinear feedback con- the PI controller which is suitable for all operating points is troller to overcome the problem of instability in the inductive also difficult. mode of operation of STATCOM. The block diagram of the It is to be noted that the system is unstable in the induc- controller is shown in Figure 5. tive mode of operation. To validate the linearized analysis It is to be noted that the nonlinear controller is active (which neglects harmonics in the switching functions) and only when i > (ωC/k)V and also only during a transient. R dc to check the performance for large deviations from an In steady state, (when V is a constant), the output of the dc operating point, the transient simulation of a 12-pulse 3- multiplier in nonlinear feedback is zero. phase STATCOM is carried out using MATLAB-SIMULINK. The root locus of the critical eigenvalues with nonlinear The action of the converter is modeled using the switching feedback controller for inductive and capacitive mode of op- functions (switching instants are obtained from θ and the eration is shown in Figure 6. controller output α)[3, 6, 11]. The instability predicted It is observed from the root locus plot that the critical by linearized analysis in the inductive region when a PI modes are stable. Imaginary part R 4 Advances in Power Electronics 1.5 sT 1+ sT Inductive region dc b − 0.5 0 π + ∑ Rref − −0.5 i s −1 Capacitive region −1.5 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Figure 5: Nonlinear feedback controller. Time (s) Figure 7: Phase “a” current of STATCOM with nonlinear feedback controller. k = 0 1.5 Inductive region k = 0 1450 p 0.5 Inductive region Capacitive region −0.5 −200 −150 −100 −50 0 −1 Real part Capacitive region Figure 6: Root locus with nonlinear feedback for k = 0–10 and −1.5 k /k = 10. 0 0.5 1 1.5 i p Time (s) Table 2: Eigenvalues with nonlinear feedback. Figure 8: Response of STATCOM with nonlinear feedback con- troller. Capacitive region i =−1Inductiveregion i = 1 R R −834.58 −1173.2 current. It is to be noted that, although the system is stable −81.819 ± j1429.8 −102.73 ± j1400.7 during inductive operation, the transient response of the −10 −7.3766 ± j4.3491 STATCOM is slow. −9.9137 Hence there is a need to optimize the controller parame- ters to improve the transient response of the STATCOM. The optimization of reactive current controller parameters based The eigenvalues of the system with nonlinear feedback on GA to improve the transient response will be discussed in and suboptimal controller parameters [6]are given in the section to follow. Table 2. The suboptimal controller parameters [6]are k = 0.33, k = 3.33, g = 2, and T = 0.1. It is to be noted that, i W the eigenvalues are stable with good stability margin for both 4. Application of Genetic Algorithm for inductive and capacitive mode of operation of STATCOM. Controller Parameters Optimization The transient simulation of the 3 phase model of the STATCOM for step change in the reactive current is shown 4.1. Introduction. GA has been used as optimizing the pa- in Figures 7 and 8. rameters of control system that are complex and difficult to It can be seen that, transition from capacitive to inductive solve by conventional optimization methods [12]. mode of operation of STATCOM is slow and steady state It maintains a set of candidate solutions called popu- is reached after 0.2 sec following the step change in reactive lation and repeatedly modifies them. Each member of the Imaginary part sa Advances in Power Electronics 5 population is evaluated using a fitness function. The popula- tion undergoes reproduction in a number of iterations. One or more parents are chosen stochastically, but strings with ζ = 10% D-contour higher fitness values have higher probability of contributing the offspring Genetic operators, such as crossover and mu- tation are applied to parents to produce offspring. The offspring are inserted into the population and the process is repeated. α =−0.5 Given a random initial population, GA operates in cycles −5 called generations as follows. The basic steps involved in GA are as follows. −10 Step 1: Begin with a randomly generated population of chro- −15 mosome-encoded “solutions” to a given problem. −20 −5 −4 −3 −2 −10 1 Step 2: Calculate the fitness of each chromosome, where Real fitness is a measure of how well a member of the population performs at solving the problem. Figure 9: D-contour with α =−0.5and ξ = 10%. Step 3: Retain only the fittest members and discard the least The D-contour in Figure 9 can be mathematically de- fit members. scribed as Step 4: Generate a new population of chromosomes from f (z) = Re(z) − min[− Im(z), α] = 0, (6) the remaining members of the old population by applying the operations reproduction, crossover, and where zεC is a point on D-contour and C represents the com- mutation. plex plane. Step 5: Calculate the fitness of these new members of the Define J as population, retain the fittest, discard the least fit, and J = max Re(λ ) − min −ζ|Im(z)|, α i = 1, 2, 3...n, reiterate the process. (7) The GA-based optimization guarantees the system stabil- ity under varying operating conditions. where n is the number of eigenvalues. λ is the ith eigenvalue of the system at an operating point. A negative value of J implies that all the eigenvalues lie on the left of the D- 4.2. Objective Function. For damping oscillations, the damp- contour. Similarly some or all eigenvalues will lie on the right ing factor ζ of around 10% to 20% is considered to be ade- of the D contour if J is positive. quate. A damping factor of 10% would be acceptable to most On the basis of these facts, objective function E is defined utilities and can be adopted as the minimum requirement. as Further, having the real part of eigenvalue restricted to be less than a value, say α, guarantees a minimum decay rate 2 Sum Squared Error (E) = e,(8) α.A value α =−0.5 is to be considered adequate for an acceptable settling time. The closed loop mode location where should simultaneously satisfy these two constraints for an e = i − i,(9) acceptable small disturbance response of the controlled Rref R system. where i and i are the Reactive current reference input Rref R If all the closed loop poles are located to the left of the and reactive current output of the STATCOM, respectively. contour shown in Figure 9, then the constraints on the Hence the optimization problem can be stated as follows: damping factor and the real part of eigenvalues are satisfied and a well-damped small disturbance response is guaranteed. Minimize E, This contourisreferred to as the D-contour [3, 13]. (10) The system issaid tobeD-stableifitisstablewith respect Subjected to J ≤ 0. to this D-contour, that is, all its poles lie on the left of this In addition to this, the boundaries of optimal parameters are contour. This property is referred to as generalized stability in control literature. This generates a neat specification, g ≤ g ≤ g T ≤ T ≤ T , min max W W W min max the closed loop system should be robustly D-stable, that (11) is, D-stable, for the entire range of operating and system K ≤ K ≤ K K ≤ K ≤ K . p p p i p min max min i max conditions. Hence a system is said to be “robust,” if, in spite of changes in system and operating conditions, the closed loop Genetic algorithm is adopted to obtain the optimal poles remain on the left of the D-contour for the specified parameters of PI controller with nonlinear feedback loop range of system and operating conditions. [13, 14]. Imaginary 6 Advances in Power Electronics 1.5 Table 3: Parameters used for optimization with genetic algorithm. Parameter Value/type Inductive region Maximum generations 25 Population size 200 0.5 Type of selection Normal geometric [0 0.08] Type of crossover Arithmetic [4] Type of mutation Nonuniform [4 20 3] Termination method Maximum generation −0.5 −1 Table 4: Eigenvalues with optimal controller parameters based on Capacitive region GA. −1.5 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Capacitive region i =−1Inductiveregion i = 1 R R −1856.7 −2194.6 Time (s) −83.949 ± j1373.6 −58.401 ± j1398.3 Figure 10: Phase “a” current of STATCOM with optimal controller −55.556 −32.538 ± j4.9879 parameters. −23.842 1.5 Inductive region The parameters used with GA are given in Table 3.The optimized parameters as obtained by GA are k = 0.69, k = p i 0.5 16.45, g = 2.54, and T = 0.018. The parameters used with GA are given in Table 3. Eigenvalues of the system with nonlinear feedback and optimized controller parameters are shown in Table 4.Com- paring with the eigenvalue results given in Table 2,itis −0.5 to be noted that although the damping of critical mode is decreased (in inductive region), the damping of other −1 modes is increased. The GA-based optimization ensures that the system is robustly D-stable for various operating points Capacitive region −1.5 under consideration. 0 0.5 1 1.5 It can be seen from Figures 10 and 11 that the transition Time (s) from capacitive to inductive mode of operation of STATCOM is very fast and takes less time, about 0.04 sec, to reach steady Figure 11: Step response with 3Φ model of STATCOM with state. optimal controller parameters. It is to be noted that the transient response of the STAT- COM is significantly improved. The steady state oscillations Appendix in thereactivecurrent of Figure 11 are due to harmonics in the converter output voltage. Thus with optimal controller STATCOM data in p.u: parameters the transient response of the STATCOM is improved. R = 0.01, X = 0.15, R = 78.7, b = 1.136. s s p c (A.1) The suboptimal controller parameters without nonlinear 5. Conclusion feedback [6]are k = 0.33 and k = 3.33, p i The suboptimal controller parameters with nonlinear In this paper we have presented a systematic method for con- troller parameter optimization based on genetic algorithm feedback [6]are k = 0.33, k = 3.33, g = 2, and T = 0.1. p i W The optimized parameters as obtained by GA are k = (GA) for the design of STATCOM reactive current controller. p 0.69, k = 16.45, g = 2.54, and T = 0.018. The GA-based optimization ensures that the system is i w robustly D-stable in the entire range of operation and system conditions. The performance of the designed controller is References evaluated and it was observed that the STATCOM with optimized controller parameters provides excellent transient [1] N. G. Hingorani, “Flexible AC transmission,” IEEE Spectrum, response. vol. 30, no. 4, pp. 40–45, 1993. R i sa Advances in Power Electronics 7 [2] C. Schauder and H. Mehta, “Vector analysis and control of advanced static VAR compensators,” IEE Proceedings C,vol. 140, no. 4, pp. 299–306, 1993. [3] K. R. Padiyar, FACTS Controllers in Power Transmission and Distribution, New age International (P), New Delhi, India, [4] Y.Sumi, Y. Harumoto,T.Hasegawa, M. Yano,K.Ikeda,and T. Matsuura, “New static VAR control using force-commutated inverters,” IEEE Transactions on Power Apparatus and Systems, vol. 100, no. 9, pp. 4216–4224, 1981. [5] L. Gyugyi, “Reactive power generation and control by thyristor circuits,” IEEE Transactions on Industry Applications, vol. 15, no. 5, pp. 521–532, 1979. [6] K. R. Padiyar and A. M. Kulkarni, “Design of reactive current and voltage controller of static condenser,” International Jour- nal of Electrical Power and Energy Systems, vol.19, no.6,pp. 397–410, 1997. [7] K. R. Padiyar and N. Prabhu, “Design and performance evaluation of subsynchronous damping controller with STAT- COM,” IEEE Transactions on Power Delivery,vol. 21, no. 3,pp. 1398–1405, 2006. [8] K. R. Padiyar and V. Swayam Prakash, “Tuning and perfor- mance evaluation of damping controller for a STATCOM,” International Journal of Electrical Power and Energy Systems, vol. 25, no. 2, pp. 155–166, 2003. [9] N. Prabhu, Analysis of sub synchronous resonance with voltage source converter based FACTS and HVDC controllers,Ph.D. dissertation, IISc Bangalore, 2004. [10] K. R. Padiyar, Power System Dynamics: Stability and Control, B.S. Publications, Hyderabad, India, 2nd edition, 2000. [11] K. R. Padiyar and N. Prabhu, “Analysis of subsynchronous resonance with three level twelve-pulse VSC based SSSC,” in Proceedings of the IEEE Conference on Convergent Technologies for Asia-Pacific Region (TENCON ’03),vol.1,pp. 76–80, October 2003. [12] D. E. Goldberg, Genetic Algorithm in search, Optimization and Machine Learning, Addison Wesley, Reading, Mass, USA, 1989. [13] R. Singh, A novel approach for tuning of power system stabilizer using genetic algorithm, M.Sc. dissertation, Indian Institute of Science, Bangalore, India, July 2004. [14] A. L. B. do Bomfim, G. N. Taranto, and D. M. Falcao ˆ , “Simul- taneous tuning of power system damping controllers using genetic algorithms,” IEEE Transactions on Power Systems,vol. 15, no. 1, pp. 163–169, 2000. 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