PSO Based Location and Parameter setting of ...

PSO Based Location and Parameter setting of Advance SVC Controller with Comparison to GA in Mitigating Small Signal Oscillations D. Mondal, A. Chakrabarti and A. Sengupta, Member, IEEE

Abstract-- This paper aims to select the optimal location and setting parameters of Static VAR Compensator (SVC) controller using Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) to mitigate small signal oscillations in a multimachine power system. Though Power System Stabilizers (PSSs) are prime choice in this issue, its performance gets affected by changes in network configurations, load variations etc. Hence installation of FACTS device, SVC has been suggested here in order to achieve appreciable damping of oscillations. However, performance of any FACTS devices highly depends upon its parameters and suitable location in the power network. In this paper PSO as well as GA based techniques are used to investigate this problem. An attempt has also been made to compare the performance of the PSO based SVC controller with its GA based design. The validity of the proposed techniques is simulated in a multimachine system following two common disturbances. It has been revealed that the PSO based SVC controller is more effective than GA based controller even during critical loading condition.

Index Terms-- Flexible AC Transmission System, Genetic Algorithm, Particle Swarm Optimization, Small Signal Stability, Static VAR compensator

I. INTRODUCTION OW frequency (0.2-2.5 Hz) power oscillations are the challenging problem in interconnected power systems. Conventionally, additional damping in power system is introduced by the application of PSS [1]-[2]. The development of FACTS [3] has generated much attention of the researchers in this issue. In [4] it has been reported the better effectiveness of a SVC than PSS in damping power system oscillations. An optimal power flow (OPF) and transmission loss minimization model with SVC has been developed in [5] to improve the system stability and security of a practical power network. The optimal placement of FACTS controller in power system networks has been reported in literatures based on different aspects. A residue factor was proposed in [6] based on the relative participation of the parameters of SVC controller to the critical mode in order to find the optimal

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D. Mondal is with the Department of Electronics and Instrumentation Engineering, Haldia Institute of Technology, Purba Medinipur, Haldia721657, India (e-mail: [email protected] ). A. Chakrabarti and A. Sengupta are with the Department of Electrical Engineering, Bengal Engineering and Science University, Howrah-711103, India (e-mail: [email protected] and [email protected] ).

978-1-4673-0136-7/11/$26.00 ©2011 IEEE

location of the SVC controllers. An evolutionary algorithm for optimally locating FACTS controllers has been presented in [7] to maximize the power system loadability within security limit. The optimal allocation of SVC using GA has been reported in [8] to achieve the optimal power flow (OPF) with lowest cost generation in power system. But the optimal allocations of SVC using PSO to investigate the small signal oscillations have not been discussed in existing literature. In this paper this fact has been taken into consideration as well as PSO based technique has been proposed to place SVC controller in a multimachine system in order to damp small signal oscillations. It is a well known fact that optimal parameter tuning of power system analysis controller is a complex exercise. The conventional techniques [9]-[10] are time consuming, require heavy computation burden and they have slow convergence rate too. Many stochastic search methods have been utilized for global optimization problems in power systems, such as simulated annealing, artificial neural network and evolutionary programming [11]-[12]. The application of a GA based fuzzy hybrid SVC controller has been proposed in [13] for improving damping inter-area modes of oscillations. Recently, Particle Swarm Optimization (PSO) method [14] has appeared as a promising algorithm for handling the optimization problems. PSO can generate high-quality solutions and has more stable convergence characteristics than other stochastic methods. PSO has no complex evolution operators such as crossover and mutation like GA. Though several attempts have been made in literatures [15]-[17] for the design of optimal FACTS controllers using PSO, the applications are mostly limited to the case of single machine infinite bus system. In this paper, both PSO and the GA are used to search the best location and the parameters of SVC controller and the application is extended to study the small signal oscillation problem in case of a multimachine power system. To the best of the author’s knowledge this work has not been reported yet. The paper is organized as follows: section II describes the small signal modeling of the multimachine system with SVC controller. The desired objective function and parameter optimization algorithms have been formulated in section III. In section IV, the SVC controller parameters and its optimal location are identified separately by both algorithms and subsequently PSO and the GA based SVC controllers are applied in a standard test system. Finally, comparisons between PSO and the GA based results have been discussed in section V.

II. SYSTEM MODELING A. Modeling of SVC The most popular configuration of this type of shunt connected device is a parallel combination of fixed capacitor C with a thyristor controlled reactor (TCR) (Fig. 1(a)). The block diagram of a basic SVC incorporating an auxiliary controller [18] has been shown in Fig. 1(b). The voltage input, ΔVsvc of the SVC controller is measured from the SVC bus. The machine speed, Δν ( = Δω / ω s ) is taken as the control input to the auxiliary controller. The firing angle ( α ) of the thyristor determines how much susceptance is included in the network. The SVC equivalent susceptance, Bsvc at fundamental frequency is given by [19] X X L − C (2(π − α) + sin (2α)) π B svc = − (1) XC X L while its profile as a function of firing angle for X C = 1.1708 pu and X L = 0.4925 pu has been shown in Fig. 2.

K 1 Δ Vs = − Δ V s + svc ωs T2

⎛ 1 ⎜ ⎜T ⎝ 2

⎞ K ⎟ Δ ω + svc ⎟ ωs ⎠

⎛ T1 ⎜ ⎜T ⎝ 2

⎞ ⎟ Δ ω ⎟ ⎠

Δα = − K I ΔV s + K I ΔV svc − K I ΔVref ΔB svc = −

1 T svc

Δα −

1 T svc

(2)

(3) (4)

ΔB svc

where T svc is the time delay of the SVC module and σ = (π − α) is the conduction angle of the thyristor. K svc , T1 and T2 are the gain, lead and lag time constant of the auxiliary controller respectively. B. Multimachine small signal model with SVC The small signal modeling of a multimachine system with IEEE-Type I exciter has been described in [20]. The statespace equations representing the dynamic model of the system for eigenvalue analysis are given by ΔX = A1ΔX + B1ΔI g + B 2 ΔV g + E1ΔU (5) 0 = C 1 Δ X + D 1 Δ I g + D 2 Δ Vg

(6)

0 = C 2 ΔX + D3 ΔI g + D4 ΔV g + D5 ΔVl

(7)

0 = D6 ΔV g + D7 ΔVl (8) Here (5) and (6) represent the linearized differential equations and linearized stator algebraic equations of the machine while (7) and (8) correspond to the linearized network equations pertaining to the generator buses and the load buses. The inclusion of a SVC controller in this multimachine model results in addition of state variables,

Fig. 1(a). Advance SVC module

Δx svc = [ΔV s Δα ΔB svc ]T corresponding to the SVC controller in (5)-(6) and the SVC reactive power output equation in the network equation (8). The SVC linearized reactive power equation at node n can be obtained from the following equation ⎡Δθ n ⎤ Δ Q n = ⎡ 0 − 2V n B svc 2V n2 (1 − cos 2 α )/X L ⎤ ⎢⎢ Δ V n ⎥⎥ ⎢⎣ ⎥⎦ ⎢⎣ Δ α ⎥⎦

where, Fig. 1(b). Block diagram of SVC controller

Qn = − B svcV n2 .

Eliminating

ΔI g from

(9)

the

respective equations, the overall system matrix with SVC controller for an m-machine system can be obtained as [Asvc ](7 m +3)×(7 m +3) = [A ′] − [B ′][D ′]−1 [C ′] (10) where A′ = A1 − B1 D1−1C1 , B′ = [B2 − B1D1−1D2 0] , C ′ = [K 2 0]T −1 −1 ⎡K D ⎤ D ′ = ⎢ 1 5 ⎥ with K1 = [D4 − D3D1 D2] and K2 = [C2 − D3D1 C1] ⎣ D6 D7 ⎦

III. PROBLEM FORMULATION

Fig. 2.

Bsvc as function of firing angle α

Setting K P is zero, the linearized state equations of the SVC controller can be represented as

A. Objective Function and Optimization Problem The optimization problem represented here is to search the optimal location and the optimal parameter set of the SVC controller using PSO and GA algorithms. It is worth mentioning that the SVC controller is designed to minimize the power system small signal oscillations after a disturbance

so as to improve the stability. This results in minimization of the critical damping index (CDI) given by: CDI = J = (1 − ζ i ) (11) Here, ζ i = −σ i / σ i 2 + ωi 2 is the damping ratio of the ith critical swing mode. The objective of the optimization is to maximize the damping ratio (ζ ) as much as possible. There are four tuning parameters of the SVC controller; the controller gain (Ksvc), lead time constant (T1), lag time constant (T2) and the location number (Nloc). These parameters are to be optimized by minimizing the objective function J given by (13). With the change of locations and parameters of the TCSC controller the damping ratio (ζ ) as well as J varies. The problem constraints are the bounds on the possible locations and parameters of the SVC controller. The optimization problem can then be formulated as: Minimize J (12) S.T min max K svc ≤ K svc ≤ K svc ; T1 min ≤ T1 ≤ T1 max

T2 min ≤ T2 ≤ T2 max ; N loc min ≤ N loc ≤ N loc max

is kept within a typical range. The particle configuration corresponding to the SVC controller is represented in Fig 3. Here first string corresponds to the SVC controller gain, second and third strings for lead and lag time constants and fourth contain the number of transmission line where the SVC is to be located. All the load buses (bus #6, 7, 8, 9, 10, 11, 12, 13 and 14) of the test system are proposed here for possible locations of the SVC and therefore, bus #6 and bus #14 are min max and N loc (Fig. 3) respectively. The considered as N loc

computational flow chart of the implemented PSO has been shown in Fig. 4.

Fig. 3. Particle configuration for SVC controller

B. Particle Swarm Optimization (PSO) Particle Swarm Optimization was first developed in 1995 by Eberhart and Kennedy [14]. The algorithm begins by initializing a random swarm of M particles, each having R unknown parameters to be optimized. In each iteration, the fitness of each particle is evaluated according to the selected fitness function. The algorithm stores and progressively replaces the best fit parameters of each particle (pbesti, i=1, 2, 3, . . . , M) as well as a single most fit particle (gbest) among all the particles in the group. The trajectory of each particle is influenced in a direction determined by the previous velocity and the location of gbest and pbesti. Each particle’s previous position (pbesti) and the swarm’s overall best position (gbest) are meant to represent the notion of individual experience memory and group knowledge of a “leader” respectively. The parameters of each particle (pi) in the swarm are updated in each iteration (n) according to the following equations: veli (n) = w × veli (n −1) + acc1 × rand1 × (gbest − pi (n −1)) (13) + acc2 × rand2 × (pbesti − pi (n − 1)) p i (n) = p i (n − 1) + vel i (n) (14)

where, veli (n) is the velocity vector of particle i. acc1, acc2 are the acceleration coefficients that pull each particle towards gbest and pbesti positions respectively and are often set to be 2.0. w is the inertia weight of values ∈ (0,1) . rand 1 and rand 2 are two uniformly distributed random numbers in the ranges [0, 1]. 1) Algorithms for Implemented PSO: To optimize (12), routines from PSO toolbox [21] are used. The objective function corresponding to each particle is evaluated by the eigenvalue analysis program [18] of the proposed test system shown in Appendix (Fig. 8). The particle is defined as a vector which contains the SVC controller parameters and the location number: Ksvc, T1, T2 and Nloc. The initial population is generated randomly for each particle and

Fig. 4. Flow chart of the implemented PSO

C. Overview of Genetic Algorithm (GA) Genetic algorithms (GAs) [22] are essentially global search algorithms based on the mechanisms of natural selection and genetics. It has been used for optimizing the parameters of the control system that are complex and difficult to solve by conventional optimisation methods. GA maintains a set of candidate solutions called population and repeatedly modifies them. At each step, the GA selects individuals from the current population to be parents and uses them to produce the children for the next generation. Candidate solutions are usually represented as strings of fixed length, called

chromosomes. A fitness or objective function is used to reflect the goodness of each member of the population. The GAs start with random generation of initial population and then the selection, crossover and mutation are preceded until the maximum generation is reached. 1) Algorithms for implemented GA: The objective is to find the optimal locations and parameters for the SVC controller applying GA. The individuals for the SVC device have been configured following similar procedure as described for particle configuration in PSO. Each individual is encoded by four parameters: the controller gain (KSVC), lead and lag time constants (T1, T2) and the SVC location number (Nloc). It is to be noted that the range of minimum and maximum values of these parameters has been kept identical with the particle configuration for PSO (Fig. 3). The entire initial population of size Nind has been calculated by repeating the individuals for Nind times and shown in Fig. 5.

for two operating scenarios: (i) real and reactive load increased at a particular bus # 9 (15 % more than nominal case) (ii) outage of a transmission line (# 4-13). The swing modes of the system without SVC are listed in Table I. It has been observed that the mode # 4 is the critical one as the damping ratio of this mode is smallest compared to other modes. Therefore, stabilization of this mode is essential in order to improve small signal performance of the system. Both PSO and the GA algorithms separately generate the best set of parameters as well as the best location (Table II) corresponding to the SVC controller by minimizing the desired objective function J. The damping ratio of the critical swing mode # 4 with the application of these PSO and the GA based SVC controllers in their respective optimal locations has been represented in Table III. TABLE I SWING MODES WITHOUT SVC

# 1 2 3 4

Load increased at bus # 9 (PL=0.339 pu, QL=0.190 pu) Damping Swing modes ratio -1.5446 ± j7.5274 0.2010 -1.4244 ± j6.5313 0.2130 -1.1590 ± j6.1460 0.1853

Transmission line (# 4 - 13) outage Damping Swing modes ratio -1.5482 ± j7.5222 0.2015 -1.4291 ± j6.5339 0.2136 -1.1501 ± j6.1659 0.1833

-0.8831 ± j5.8324

-0.8845 ± j5.8336

0.1497

0.1499

TABLE II SVC CONTROLLER PARAMETERS AND LOCATIONS Obtained controllers PSO based GA based

Lead Time (T1) 1.00 0.8892

Lag Time (T2) 0.15 0.014

SVC Location (Nloc ) Bus # 10 Bus # 9

TABLE III APPLICATION OF SVC CONTROLLER

Fig. 5. Calculation of the entire population

The algorithms of the implemented GA have been described here in following steps; Step 1: Specify parameters for GA: population size, generation limit, number of variables etc. Step 2: Generate initial population for SVC controller parameters: Ksvc, T1, T2 and Nloc. Step 3: Run small signal stability and eigenvalue analysis program for the proposed test system. Step 4: Evaluate objective function (J ) for each individual in the current population. Step 5: Determine and store best individual which minimizes the objective function. Step 6: Check whether the generation exceeds maximum limit/stall generation limit. Step 7: If generation < max. limit, update population for next generation by crossover and mutation and repeat from step 3. Step 8: If generation > max. limit, stop program and produce output.

SVC gain (Ksvc) 20.0 11.97

Applied disturbance

with PSO based SVC Critical swing Damping mode # 4 ratio

with GA based SVC Critical swing Damping mode # 4 ratio

Load increase (15 %)

-0.98121 ± j6.0070

0.16121

-0.88107 ± j5.6195

0.1549

Line outage (# 4-13)

-0.98224 ± j6.0568

0.16008

-0.88313 ± j5.614

0.1554

The convergence rate of the objective function J towards best solutions with population size 15 and number of generations 200 has been shown in Fig. 6(a) and 6(b).

IV. RESULTS AND PERFORMANCE STUDY A. Application of PSO and GA in the Test System The validity of the proposed PSO and the GA based algorithms has been tested here on an IEEE-14 bus system. This system has also been used widely in the literature [23] for small signal stability analysis. In order to study the small signal performance of the system the simulation is carried out

Fig. 6 (a). Convergence of the objective function with PSO

controller (Table VI). This implies that PSO based SVC controller can mitigate the small signal oscillations problem more efficiently than GA based controller. Again, the plots of convergence rate of the objective function (Fig. 6 (a) and Fig. 6(b)) indicate that PSO method has more fast and stable convergence characteristics than GA.

Fig. 6 (b). Convergence of the objective function with GA

B. Implication of SVC on Critical Loading In order to study the effect of loading on system stability, the real load (constant power) at bus # 9 is increased form its nominal value PL=0.295 pu, QL=0.166 pu in steps. In each case eigenvalues of the system matrix are checked for stability. It has been observed that at load PL=2.60 pu, QL=0.166 pu Hopf bifurcation [20] takes place for the critical swing mode # 4 (Table IV) and led to low frequency oscillatory instability of the system. When the PSO and the GA based SVCs are installed at bus #10 and bus # 9 separately there is no Hopf bifurcation of swing modes (Table V). This implies that the SVC controllers so obtained by PSO and GA methods can put off the Hopf bifurcation until further increase of load levels. The angular speed ( Δω ) response of machine # 1 has been plotted in Fig. 7 for both type of disturbances. It has been observed that the application of PSO based SVC controller imparted better settling time compared to the GA based SVC controller. TABLE IV EFFECT OF CRITICAL LOADING WITHOUT SVC

# 1 2 3 4

Nominal load at bus # 9 (PL=0.295 pu, QL=0.166 pu) Damping Swing modes ratio -1.6071 ± j7.5211 0.2089 -1.4987 ± j6.5328 0.2236 -1.2074 ± j6.1633 0.1922 -0.9461 ± j5.8552 0.1595

Hopf bifurcation load at bus # 9 (PL=2.60 pu, QL=0.166 pu ) Damping Swing modes ratio -1.1190 ± j7.7098 0.1436 -1.6357 ± j5.9069 0.2668 -0.9230± j2.5144 0.3446 0.0072 ± j4.6175 -0.0015

TABLE V APPLICATION OF SVC WITH HOPF BIFURCATION LOAD

# 1 2 3 4

PSO based SVC at bus #10 Damping Swing modes ratio -1.2337 ± j7.5753 0.16074 -1.4921 ± j6.0160 0.24072 -1.776 ± j2.7131 0.54768 -0.9712 ± j3.4139 0.27363

GA based SVC at bus # 9 Damping Swing modes ratio -1.1871 ± j 7.7211 0.15196 -1.5778 + 6.059 0.25201 -0.77007 + 2.6005 0.28393 -0.20113 + 4.7623 0.04219

V. COMPARISON BETWEEN PSO AND GA The performance comparisons between the PSO and the GA based results have been drawn in this section. It is evident that both PSO and GA handle the proposed optimization problem efficiently and generate satisfactory results. But the PSO based SVC controller imparted reasonably more damping to the critical swing mode # 4 compared to the GA based SVC

Fig. 7 (a). Speed deviation response of machine #1 for load increase

Fig. 7 (b). Speed deviation response of machine #1 for line outage TABLE VI COMPARISON BETWEEN PSO AND GA BASED RESULTS Damping ratio of critical swing mode # 4 Line outage Critical load (# 4-13) at bus # 9

Application of SVC Controllers

Load at bus # 9 (15 % more than nominal )

PSO based GA based

0.16121 0.15490

0.16008 0.15540

0.27363 0.04219

VI. CONCLUSIONS In this paper a novel stochastic method, PSO has been implemented for optimal parameter setting and identification of optimal site of the SVC controller in a standard multimachine power system in order to mitigate the small signal oscillation problem. The performance of the PSO and the GA based SVC controller has been compared against power system disturbances e.g. varying load and transmission line outage. The nature of critical eigenvalue and time response analysis reveal that the PSO based SVC controller is more superior than the GA based SVC controller even during critical loading. The present approach of PSO based optimization technique seems to have good accuracy, faster convergence rate and is free from computational complexity than GA based technique.

[8]

VII. APPENDIX A A.1 Proposed Study System

[9] [10]

[11] [12]

[13] [14] [15] [16]

[17] Fig. 8. IEEE-14 bus system with the application of SVC

A.I1 Parameter Constants of Implemented PSO and GA

[18]

TABLE VII

[19]

PARAMETER AND CONSTANT SETTINGS FOR PSO AND GA PSO Parameters PSO type Max. generations(epoch) Dimension of inputs Population size acc1, acc2 Minimum error gradient terminates run Epochs before error gradient criterion terminates run

Value Common ‘0’ 200 4 15 2, 2

GA Parameters Max. generations Number of variables Population size Elite count

Value Double vector 200 4 15 2

1× e-8

Population creation

uniform

100

Stall generation limit

50

Population type

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[5]

[6] [7]

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IX. BIOGRAPHIES Debasish Mondal received his degree of engineering and Master of Engineering in 1998 and 2000, respectively. He has total 11 years of industrial and teaching experience. He holds a permanent post of Assistant Professor at the Haldia Institute of Technology, India. His research interests on the areas like power systems stability, soft computing and robust control. He is a member of the Institution of Engineers (India). Abhijit Chakrabarti received B.E., M.Tech. and Ph.D. (Tech) degrees in 1978, 1987 and 1991 respectively. He is a professor at the Department of Electrical engineering, Bengal Engineering and Science University, India. He has 30 years of research and teaching experience and has around 120 research papers in National and International journal and conferences. He has active interest on the areas like power systems, power electronics and circuit theory. He is a Fellow of the Institution of Engineers (India). Aparajita Sengupta (M’41432862) received her B.E., M.Tech. and Ph.D. degrees in 1992, 1994 and 1997, respectively. She is employed in teaching and research at the Department of Electrical Engineering, Bengal Engineering and Science University, India, for 14 years. Her areas of interest are power system, robust control and nonlinear control and state estimation methods.