Optimal Design of a SVC Controller Using a Small Population Based PSO T. K. Das, Student Member, IEEE, S. R. Jetti, Student Member, IEEE and G. K. Venayagamoorthy, Senior Member, IEEE Real-Time Power and Intelligent Systems Laboratory Department of Electrical and Computer Engineering University of Missouri-Rolla, MO 65409-0249 USA
[email protected]
Abstract—Static Var Compensator (SVC) is a shunt Flexible AC Transmission System (FACTS) device. The primary purpose of a SVC is voltage control. In this paper, Particle Swarm Optimization (PSO) algorithm with a small population is used for the design of an optimal SVC controller that can provide better voltage control. Optimal SVC PI controller parameters are determined for a two-area power system subjected to small and large disturbances. Results are presented to show that the SPPSO algorithm based SVC PI controller minimizes the voltage deviations at the point of common coupling from the set reference level during dynamic and transient conditions.
P
I. INTRODUCTION
ower electronic devices have had a revolutionary impact on the electric power systems around the world. The availability and application of thyristors has resulted in a new breed of thyristors-based fast operating devices called Flexible AC Transmission System (FACTS) devices used for control and switching operations. These controllers are fast and increase the stability operating limits of the transmission systems when their controllers are properly tuned. A Static Var Compensator (SVC) is a shunt FACTS device employed primarily for voltage stability by providing appropriate reactive compensation. The conventional control of the SVC is of the Proportional plus Integral type (PI). The conventional method of obtaining best performance of the SVC is by tuning the parameters of the PI control. Since power systems are highly nonlinear systems, with configurations and parameters that change with time, the Conventional SVC design based on a linearized model of the power system cannot guarantee its performance in a practical operating environment. Thus, it is important to determine the parameters of the SVC and similar controllers using power system simulation models and tools where the nonlinear behavior of the power system is realizable but this becomes a challenge as size of the system studied becomes larger. Several SVC design techniques are reported in literature, few are listed here [1]-[3]. The conventional method for SVC AC voltage control design is based on a linearised system model which ensures system stability and satisfactory control performance around the operating condition where the linearised model is obtained [1]. In [2], Ju, Handschin and Reyer apply a genetic algorithm (GA) to aid SVC design which handles SVC continuous and discrete actions. To accommodate for the nonlinear nature of power systems and
SVC, the H∞ method was used by Parniani and Iravani in [3] for an SVC design with guaranteed robustness to the variations of power system operating conditions. In [4] and [5], Particle Swarm Optimization (PSO) is applied to tune the parameters of SVC external damping controller but based on some linearised mathematical models of power systems. A Small Population based Particle Swarm Optimization (SPPSO) algorithm for the determination of optimal PI controller parameters for a nonlinear system is presented in this paper. PSO has been shown to have great potential for single and multi-objective optimization [6]. It is a population based algorithm which does not cause individuals/particles to reproduce over generations but it simply evolves better solutions through the collective interaction of all the individuals. PSO has flexible and well balanced mechanism to carry out local and global search [7]. PSO, GA and other similar population based algorithms require a lot of computations to determine optimal parameters. Thus, they are only used in offline modes. The SPPSO is suitable for online implementations. The major contributions of this paper are two-fold: • The SPPSO algorithm is applied to determine the optimal parameters of a SVC internal PI controller. • To the knowledge of the authors, the authors are the first to report the implementation of a population based optimization algorithm in PSCAD/EMTDC power system simulation software. The paper is organized as follows: Section II describes multimachine power system; Section III describes the PSO algorithm. Section IV describes how SPPSO cost functions are formulated and used in determining the optimal controller parameters of the SVC. Section V presents simulation results obtained using the SPPSO algorithm. Finally, conclusions are given in Section VI. II. MULTI-MACHINE POWER SYSTEM For the study in this paper, the two area multi-machine power system [9], [10] is simulated in the PSCAD/EMTDC environment [8]. The two area power system is shown in Fig. 1, consists of two fully symmetrical areas linked together by two transmission lines. Each area is equipped with two identical synchronous generators rated 20 kV/900 MVA. All the generators are equipped with identical speed governors and turbines, exciters and Automatic Voltage Regulators
(AVRs), and Power System Stabilizers (PSSs). The loads in the two areas are such that area 1 is exporting 413MW to area 2. These loads can represent motor loads. This power network is specifically designed to study low frequency electromechanical oscillations in large interconnected power systems. Despite the small size of this power network, it mimics very closely the behavior of typical systems in actual operation [9]. 5
G1 1
6
7
8
9
10
11
~
3 G3 ~
SVC
4 ~ G4 AREA 2
2 G2
~
AREA 1
Fig. 1. Two area multi-machine power system with a SVC.
A SVC is connected at bus 8. A typical block diagram of a SVC controller is shown in Fig. 2. The difference of the actual voltage at the point of common coupling Vpcc (i.e., bus 8 where SVC is connected) and the reference value is given as input to the PI control block. The PI controller consists of gain constant, Kp, and integral constant Ti. These are the parameters that need to be optimally selected for the SVC to ensure optimal system performance under a wide range of operating conditions and disturbances. B, susceptance value of the SVC is the output of the PI control block. Bmax and Bmin are the limits for B. Bmax
Vref +
∑ -
Kp
B
1+sTi
Bmin
Vpcc
Fig. 2. Block diagram of SVC PI controller.
III. SPPSO ALGORITHM Particle swarm optimization is a form of evolutionary computation technique (a search method based on natural systems) developed by Kennedy and Eberhart [11], [12]. The system initially has a population of random solutions. Each potential solution, called particle, is given a random velocity and is flown through the problem space. The particles have memory and each particle keeps track of previous best position and corresponding fitness. The previous best value is called the pbest of the particle and represented as pid. Thus, pid is related only to a particular particle i. The best value of all the particles’ pbests in the swarm is called the gbest and is represented as pgd. Each particle is accelerated towards its pid and the pgd locations at each time step. The amount of acceleration with respect to both pid and pgd locations is given random weighting. The SPPSO algorithm given in this paper consists of
mainly two features. The first one is the use of a small population of particles, few as five or lesser; calling this algorithm the SPPSO. This idea is synonymous to the Micro GA (μGA) algorithm [14]. The second feature is regeneration concept where new particles are randomly created every N iterations to replace all but the gbest particle in the swarm This means that the new particles inherit the previous generation’s best characteristics but have new positions and velocities to explore the search space. In addition to keeping the gbest’s particle parameters, the population pbest attributes are also transition from one set of population to the next every N iterations. The concept of regeneration is incorporated to make the convergence faster like it would with a large population of PSO. Randomize the positions and velocities of the particles helps the particles move out of local minima and find the global optimum. Fig. 3 illustrates briefly the concept of PSO which is also applicable to SPPSO, where xi is current position, xi+1 is modified position, vini is initial velocity, vmod is modified velocity, vpid is velocity considering pid and vpgd is velocity considering pgd. The following steps explain the procedure in the SPPSO algorithm. (i) Initialize a small population of particles with random positions and velocities in d dimensions of the problem space. (ii) For each particle, evaluate the desired optimization fitness function. (iii) Compare every particle’s fitness evaluation with its pbest value, pid. If current value is better than pid, then set pid value equal to the current value and the pid location equal to the current location in ddimensional space. (iv) Compare the updated pbest values with the population’s previous gbest value. If any of pbest values is better than pgd, then update pgd and its parameters. (v) Compute the new velocities and positions of the particles according to (1) and (2) respectively. vid and xid represent the velocity and position of ith particle in dth dimension respectively and, rand1 and rand2 are two uniform random functions.
vid = w × vid + c1 × rand 1 × ( p id − xid ) (1) + c 2 × rand 2 × ( p gd − xid )
xid = vid + xid (vi)
(2)
After N iterations randomly create new particles to replace all except the gbest and pbest of the particles. Repeat from step 2 until a specified terminal condition is met, usually a sufficiently good fitness or a maximum number of iterations.
Y
J1 =
xi+1 vmod vini
(4)
fault = 1 fault
vpid X
t =t
Fig. 3. Movement of a PSO particle in two dimensions from one instant k to another instant k+1.
The parameters in (1) are: w is called the inertia weight, which controls the exploration and exploitation of the search space. Local minima are avoided by small local neighborhood, but faster convergence is obtained by larger global neighborhood. Synchronous updates are more costly than the asynchronous updates. The velocity is restricted to a certain dynamic range. vmax is the maximum allowable velocity for the particles i.e. in case the velocity of the particle exceeds vmax then it is reduced to vmax. Thus, resolution and fitness of search depends on vmax. If vmax is too high, then particles will move beyond good solution and if vmax is too low, then particles will be trapped in local minima. c1 and c2 termed as cognition and social components respectively are the acceleration constants which changes the velocity of a particle towards pid and pgd (generally somewhere between pid and pgd). Velocity determines the tension in the system. A swarm of particles can be used locally or globally in a search space. In the local version of the PSO, the pid is replaced by the lid and the entire procedure is same. IV. OPTIMAL DESIGN OF A SVC CONTROLLER The SVC aids in maintaining the voltage magnitude setpoint at the point of common coupling by providing appropriate reactive power compensation. Moreover the objective of the optimization is for the voltage control which means minimizing voltage deviations under small and large disturbances. Therefore, a fitness function is formulated as a function of the area under voltage (at point of common coupling, Vpcc) time response curve during a transient. SPPSO is applied to determine PI parameters that results in the least fitness value subject to the constraints in (3) for various faults and disturbances. K min ≤ K p ≤ K max
Tmin ≤ Ti ≤T max
fault
is given in Where s is the number of faults applied and J (5). t Δt 2 fault J = ∑ ( ΔV pcc (t)) × (A × (t - t ) 2 × Δt ) (5)
vpgd
xi
s
∑J
(3)
The SPPSO algorithm minimizes the following cost function given by J 1 given in (4).
0
0
Where ΔVpcc is the voltage deviation at the point of common coupling, A is weighting factor, t0 is the time the fault is cleared, t2– t0 is transient period time considered for cost function calculation, ∆t is the voltage signal sampling period and t = simulation time in seconds. In this study, two faults are applied; these are a three phase short circuit (J1) and a transmission line outage (J2). The SPPSO minimizes J given by (6). J = J1 + J 2 (6) The original parameters of the SVC controller are obtained by trial and error using the cost function given in (4) and the best parameters found are: Kp = 1.3, Ti = 0.001s. These parameters are used to compare the effectiveness of SPPSO determined parameters in Section V. V. SIMULATION RESULTS The entire power system and SPPSO simulation is carried out in the PSCAD/EMTDC/FORTRAN environment. Each particle is a two area power system with SVC in PSCAD. The number of particles in the SPPSO is five, thus five PSCAD cases. The multiple run feature in PSCAD is used to carry out a set of SPPSO iterations. The parameters w, c1 and c2 in (1) are taken to be 0.8, 2.0 and 2.0 respectively. SPPSO being a stochastic optimization technique, a statistical evaluation is provided. 20 trials have been carried out for 40 SPPSO iterations to show the consistency of the optimization. Regeneration is carried out on 16th and 32nd iteration. Table I shows the worst and best fitness values and the corresponding optimized PI parameters. TABLE I FITNESS VALUES AND THE CORRESPONDING OPTIMIZED PI PARAMETERS Fitness
Kp
Ti
Worst26.825
1.0987
0.00147
Best25.9423
2.9496
0.00114
The average fitness for 20 trials is found to be 26.360± 0.6222. Three tests are carried out using the above mentioned best fitness parameters to compare the
performance of the PI parameters optimized by SPPSO and that without optimization.
1.01 1
Voltage (pu)
0.99 0.98
1.4 1.2 1 0.8 0.6 0.4 0.2 0 3
3.5
4
4.5 Time (secs)
5
5.5
6
Fig. 6. Corresponding susceptance (B) response of SVC for Fig. 4.
B. Test 2 A transmission line outage is carried out at 10 seconds between buses 7 and 8 in (Fig. 1) and this outage is not restored immediately. Fig. 7 shows the Vpcc response for the line outage. It can be seen that the SPPSO optimized parameters is able to keep the system stable. Fig. 8 shows the corresponding reactive power responses at the SVC bus. Fig. 9 shows the susceptance, B of SVC.
0.97 0.96
0.95 Vref step change PI parameters without optimization PI parameters optimized by SPPSO 3
3.5
4
4.5 Time (secs)
5
5.5
6
Fig. 4. Vpcc response for a step change in reference level of the PI controller of SVC.
0.945 Voltage (pu)
0.94
0.94 0.935 0.93 0.925
0
PI parameters without optimization PI parameters optimized by SPPSO
-20
0.92 0.915
-40 -60
0.91 10
-80
10.5
11
11.5
12
12.5 13 Time (secs)
13.5
14
14.5
Fig. 7. Vpcc response for transmission line outage between buses 7 and 8.
-100 -120 -140 -160 -180 -200 -220
PI parameters without optimization PI parameters optimized by SPPSO
0.955
0.95
Reactive Power (MVAR)
PI parameters without optimization PI parameters optimized by SPPSO
1.6
Susceptance (pu)
A. Test 1 Step changes in the SVC PI controller voltage reference (Vref) are applied to test the performances of the SPPSO optimized and unoptimized PI controllers. Fig. 4 shows the Vpcc response to the change in Vref. At 3 seconds, the Vref is changed from 0.94 p.u to 1.0 p.u and at 5 seconds, another step change in Vref from 1.0 p. u to 0.97 p.u is applied. It can be seen that SPPSO optimized parameters provides good control of voltage as per the required reference level. This test shows that the SVC with the optimized controller parameters provides better voltage regulation which is the primary purpose of the SVC. Fig. 5 shows the corresponding reactive power for reference voltage changes at the SVC bus. Fig. 6 shows the PI control output, i.e., susceptance, B of SVC.
1.8
3
3.5
4
4.5 Time (secs)
5
5.5
Fig. 5. Corresponding reactive power response at SVC bus for Fig. 4.
6
15
-20
PI parameters without optimization PI parameters optimized by SPPSO
-40 -60
0.96
-80 -100
Voltage (pu)
Reactive power (MVAR)
PI parameters without optimization PI parameters optimized by SPPSO
0.97
-120 -140
0.94
0.93
-160 -180 -200 10
0.95
0.92 10.5
11
11.5
12 12.5 Time (secs)
13
13.5
14
0.91 11
11.5
12
Fig. 8. Corresponding reactive power response at SVC bus for Fig. 8.
13
13.5
14
Fig. 10. Vpcc response for a 3 phase 100 ms short circuit applied at bus 7 followed by transmission line outage between buses 7 and 8.
PI parameters without optimization PI parameters optimized by SPPSO
1.2
12.5 Time (secs)
PI parameters without optimization PI parameters optimized by SPPSO
-20
1
-60 Reactive power (MVAR)
Susceptance (p.u)
-40 0.8
0.6
0.4
0.2
0 10
-80 -100 -120 -140 -160
10.5
11
11.5
12 12.5 Time (secs)
13
13.5
14
-180
Fig. 9. Corresponding susceptance (B) response of SVC for Fig. 7
11.5
12
12.5
13 13.5 Time (secs)
14
14.5
15
Fig. 11. Corresponding reactive power response at SVC bus for Fig. 10.
1
0.8
Susceptance (pu)
C. Test 3 This test consists of double faults. A 100 ms three phase short circuit fault is applied at bus 7 for the system followed by a transmission line outage between buses 7 and 8 in Fig. 1. This test is carried out to show the robustness of the SPPSO optimized PI controller. It can be seen in Fig. 10 that SVC parameters optimized by SPPSO algorithm is able to survive the double disturbances. Fig. 11 and 12 shows the corresponding reactive power responses at the SVC bus and the susceptance, B of SVC respectively.
-200 11
0.6
0.4
0.2
0
-0.2 10.5
PI parameters without optimization PI parameters optimized by SPPSO 11
11.5
12
12.5 13 Time (secs)
13.5
14
14.5
Fig. 12. Corresponding susceptance (B) response of SVC for Fig. 10.
15
VI. CONCLUSION The optimal design of a SVC controller is presented using a small population based particle swarm optimization algorithm (SPPSO). The SPPSO algorithm is used to optimize the SVC PI controller parameters. These parameters provide better voltage stability than those parameters obtained by trial and error given the same cost function as that of SPPSO. The SPPSO algorithm has been implemented in a commercial power system simulation tool with detailed nonlinear models of the power system elements. The potential of the SPPSO algorithm is valuable for large power systems with multiple SVCs and other controllers where optimization of controller parameters is necessary to avoid any adverse effects. In such cases, linearized model based controller designs will not perform as desirable. In addition, the SPPSO algorithm is feasible for online implementations due to its lesser computational demand. VII. REFERENCES [1] [2] [3] [4]
[5]
[6]
[7] [8] [9] [10] [11] [12] [13] [14]
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