Implementation of GCPSO for Multi-objective VAr Planning with SVC and Its Comparison with GA and PSO

Transcription

1 Implementation of GCPSO for Multi-obective VAr Planning with SVC and Its Comparison with GA and PSO Malihe M. Farsang Hossein Nezamabadi-pour and Kwang Y. Lee, Fellow, IEEE Abstract In this paper, Guaranteed Convergence Particle Swarm Optimization (GCPSO) Algorithm is used for VAr planning with the Static Var Compensators (SVC) in a largescale power system. To enhance voltage stability, the planning problem is formulated as a multiobective optimization problem for maximizing fuzzy performance indices. The multi-obective VAr planning problem is solved by the fuzzy GCPSO and the results are compared with those obtained by the Particle Swarm Optimization (PSO) and Genetic Algorithm (GA). Index Terms guaranteed convergence particle swarm optimization, genetic algorithm, particle swarm optimization, SVC, multiobective optimization, fuzzy performance indices. V I. INTRODUCTION OLTAGE collapse and other instability problems can be related to the system s inability to meet VAr demands []. Efforts have been made to find the ways to assure the security of the system in terms of voltage stability. Flexible AC transmission system (FACTS) devices are good choice to improve the voltage profile in a power system, which operates near the steady-state stability limit and may result in voltage instability. Taking advantages of the FACTS devices depends greatly on how these devices are placed in the power system, namely on their location and size. Over the last decades there has been a growing interest in algorithms inspired from the observation of natural phenomena []-[8]. The ability of different algorithms is investigated by the authors in VAr planning by SVC based on single obective and multi-obective functions [9]-[0]. Also, the ability of modal analysis is investigated where this method meets difficulties in placing SVC optimally [9]. The work carried out by the authors in [0] used Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) for multi-obective VAr planning by SVC. It was revealed that both algorithms show the same bus for the placement of SVC but with different MVAr size. In view of this, this paper investigates the applicability of the global best model of the Guaranteed Convergence PSO (GCPSO) in the VAr planning problem with SVC. Also, other obective function, known as cost, is added to the obective function used in [0]. The VAr planning problem is formulated as a multi-obective optimization problem for maximizing fuzzy performance indices, which represent minimizing voltage deviation, RI losses and the cost of installation resulting in the maximum system VAr margin. To validate the results obtained by the GCPSO, the problem is solved by GA. Also, the basic PSO is applied and the results are compared with those obtained by the GCPSO. II. PARTICLE SWARM OPTIMIZATION AND GENETIC ALGORITHM Through cooperation and competition among heuristic methods, the population-based optimization approaches, such as GA and PSO, often can find very good solutions efficiently. The GA is motivated by the hypothesis of evolution while the PSO is motivated from the simulation of social behavior. These optimization approaches update the population of individuals by applying some kinds of operators according to the fitness information obtained from the environment so that the individuals in the population can be expected to move towards better solution areas. A brief explanation of PSO is given below: A. Standard PSO The Particle Swarm Optimizer is a population based optimization method that introduced by Kennedy and Eberhart []. In PSO, each particle moves in the search space with a velocity according to its own previous best solution and its group s previous best solution []. The dimension of the search space can be any positive integer. Each particle updates its position and velocity with the following two equations: X i ( t + ) = X i( + Vi ( t + ) () where X i ( and V i ( are vectors representing the position and velocity of the i-th particle, respectively; and M. M. Farsangi and Hossein Nezamabadi-pour are with the Department of Electrical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran. ( mmaghfoori@mail.uk.ac.ir, nezam@mail.uk.ac.ir). K. Y. Lee is with the Department of Electrical and Computer Engineering, Baylor University, Waco, TX , USA ( Kwang_Y_Lee@baylor.edu). V c r ( t + ) = wv, ( gb X ( + c r ( ), ( pb X ( ) + () 87

2 where,,..., d represents the dimension of the particle; 0 w < is an inertia weight determining how much of the particle's previous velocity is preserved; c and c are two positive acceleration constants;, are two uniform r, r, random sequences sampled from U(0, l); pb i is the personal best position found by the i th particle; and gbis the best position found by the entire swarm so far. The PSO has been proven to be very effective for static and dynamic optimization problems. But in some cases, it converges prematurely without finding even a local optimum. Standard PSO may converge at the early stage: the best particle moves based only on the inertia term since X i = pbi = gb at the time step when it became the best. Later, its position may improve where X i = pbi = gb holds again. Also, its position will worsen where it will be drawn back to pb i = gb by the social component. Therefore, it is possible for the inertia weight to drive all velocities to zero before the swarms manage to reach a local extermum. When all the particles collapse with zero velocity on a given position in the search space, then the swarms have converged, but this does not mean that the algorithm has converged on a local extermum. It merely means that all the particles have converged on the best position discovered so far by the swarm. This phenomenon is referred to as stagnation [3]. Thus; it is possible for the standard PSO to converge prematurely without finding even a local extermum [3]-[4]. B. Guaranteed Convergence PSO (GCPSO) The GCPSO was introduced by Van den Bergh and Engelbrecht [3] to address the issue of premature convergence to solutions that are not guaranteed to be local exterma. The modifications to the standard PSO involve replacing the velocity update () of only the best particle with the following equation: V ( t ) = wv ( X ( + pb + ρ( r + (3) where r is a sequence of uniform random numbers sampled from U(-,) and ρ ( is a scaling factor determined using: ρ ( 0) =.0 ρ( if # successes > sc ρ( t + ) = 0.5ρ( if # failures > f c ρ( otherwise where sc and f c are tunable threshold parameters. Whenever the best particle improves its personal best position, the success count is incremented and the failure count is set to 0 and vice versa. The success and failure counters are both set to 0 whenever the best particle changes. These modifications cause the best particle to perform a directed random search in a non-zero volume around its best position in the search space. C. Genetic Algorithm Genetic algorithm (GA) is a search algorithm based on the mechanism of genetic and natural selection. The GA starts (4) with random generation of initial population and then the selection, crossover and mutation operations are preceded until the fitness function converges to a maximum or the maximal number of generations is reached. A typical simple genetic algorithm is described in detail in [5]. III. PROBLEM FORMULATION The VAr planning problem using SVC can be formulated by considering a number of different obective functions, i.e., multi-obective functions. They include in this paper reduction of voltage deviation, reduction of the active power loss, and reduction of installation cost. A. Multi-obective Functions The goal is that to find the best SVC location and the level of compensation, which would result in the increase of system VAr margin. System VAr margin can be evaluated by stressing the system gradually from an initial operating state until the state of critical voltage stability is reached. This can be done by increasing all loads gradually close to the point of voltage collapse. Increasing system VAr margin could be achieved by placing SVC considering the following obective functions: ) Active power loss. The total power loss to be minimized is as follows: PL = Vi + V VV i cos( δi δ ) Yi cosϕ (5) i where V i and δ i are the magnitude and angle of voltage at bus and Y i and ϕ i are the magnitude and angle of the admittance of the line from bus i to bus. ) Maximum voltage deviation. To have a good voltage performance (to keep the voltage between per un the voltage deviation at each load bus must be made as small as possible. The voltage deviation to be minimized is as follows: f = max V V (6) k Ω k ref k where Ω is the set of all load buses, V is the voltage k magnitude at load bus k and V is the nominal or reference refk voltage at bus k. 3) Cost function of SVC. According to [6], the cost function for SVC in terms of (US$/kVAr) is given by the following equation: C = Q 0.305Q where Q is MVAr size of SVC. 88

3 There are a number of approaches to solve the multiobective optimization problem. Since SVC placement according to the multi-obective functions is difficult with an analytical method, a fuzzy logic technique is proposed in this paper to achieve a trade off between the obective functions. The multi-obective optimization problem is transformed into a fuzzy inference system (FIS), where each obective function is quantified into a set of fuzzy obectives selected by fuzzy membership functions. The FIS is composed of fuzzification, inference engine, knowledge or rule base, and defuzzification. The fuzzification process is an interface between the real world parameters and the fuzzy system. It performs a mapping that transfers the input data into linguistic variables and the range of these variables forms the fuzzy sets. The inference engine uses the rules defined in a rule base and develops fuzzy outputs from the fuzzy inputs. The rule base includes the information given by the expert in the form of linguistic fuzzy rules, or experience gained in the process of experiment [7]-[8]. The defuzzification is a reverse process of the fuzzification. It maps the fuzzy output variables to the real world, or crisp, variables that can be used in controlling a real world system. In this paper, the three obective functions, the voltage deviation ( f ), the power loss ( P L ) and installation cost (C) are inputs to the FIS and the output is an index of satisfaction or fitness achieved. The inputs are fuzzified by the membership functions shown in Figs. -3. The membership function of the output is shown in Fig. 4. The inference engine uses the rules defined in Tables I-III and develops fuzzy outputs from the fuzzy inputs. The fuzzy output is defuzzified by the Center of Gravity (COG) method to yield a crisp value for the level of satisfaction or fitness. Tables I-III show the fuzzy rules for solving the problem. For example in Table I for low cost (C(Low)) if f is good (G) and P L is good (G) therefore the level of satisfaction (fitness) is excellent (Ex). In Tables I-III, G stands for good, M stands for moderate, B stands for bad, V stands for very and Ex stands for excellent. Input ( P L ) Input ( P L ) TABLE II. FUZZY RULES Input ( f ) For G M B C(Med) G VVG M VB M VG B VVB B G VVB VVB TABLE III. FUZZY RULES Input ( f ) For G M B C(High) G VG B VVB M G VB VVB B M VVB VVB Fig.. Membership functions for Input, voltage deviation ( f ). Input ( P L ) TABLE I. FUZZY RULES Input ( f ) For G M B C(Low) G Ex G VB M VVG M VB B VG VB VVB Fig.. Membership functions for Input, active power loss ( P ). L 89

4 initialization is made on the position randomly for each particle. 4 G G 53 G G9 Fig. 3. Membership functions for Input 3, cost function (C) G G 6 63 G G G G G 55 G3 58 G G7 G6 Fig. 5. One-line diagram of a 5-area study system. Fig. 4. Membership functions for output, the level of satisfaction (fitness). IV. STUDY SYSTEM A 5-area-6-machine study system is shown in Fig. 5, which consists of 6 machines and 68 buses. This is a reduced order model of the New England (NE) New York (NY) interconnected system. The first nine machines are the simple representation of the New England system generation. Machines 0 through 3 represent the New York power system. The last three machines are the dynamic equivalents of the three large neighboring areas interconnected to the New York power system. GCPSO, standard PSO and GA incorporating the FIS are used to locate SVC in the power system shown in Fig. 5. The implementation is presented below: ) The use of GCPSO and PSO. Placing of SVC starts from an initial load. All loads are increased gradually near to the point of voltage collapse. In the both version of PSO algorithm, n particles for a population are generated randomly where n is selected to be 00. The goal of the optimization is to find the best location of SVC where the optimization is made on two parameters: its location and size. Therefore, each particle is a d -dimensional vector in which d =. The The number of iteration is considered to be 70, which is the stopping criteria. The parameter in () must be tuned. These parameters control the impact of the previous velocities on the current velocity where, in this paper, c = c = and the weight w is decreasing linearly from 0.9 to 0.. Also, in (4) s c = fc are considered to be 5. Each particle in the population is evaluated by the FIS, searching for the particle associated with the best satisfaction (best fitness). The best previous position of the i th particle is recorded and represented as: pb i = ( pb, pb ) and the best particle among all of the particles in the group is set for the gb. Using the gb and the pb, particle velocity and position is updated. ) The use of GA. The first step in solving an optimization problem using a simple GA is the encoding of the variables. The most usual approach is to represent these variables as binary strings. A collection of such strings is called population. A configuration for the chromosome is considered with two genes. The first gene is related to the location of SVC, and the second gene is related to the size of the SVC. The number of chromosomes for a population is set to be 00. During each generation, the chromosomes are evaluated with some measure of fitness, which is calculated from the FIS. In this paper, one point crossover is applied with the crossover probability p = 0. 9 and the mutation probability is p m c = Also, the weighted roulette wheel method is used for selection. As in the PSO, the number of iteration is considered to be 70. To locate an SVC with GCPSO, PSO and GA, suitable buses are selected based on 0 independent runs under different random seeds. At the end of the 0 independent runs, the following results are observed by the fuzzy GCPSO: 40% of the results show that the SVC should be placed at bus with 546 MVAr size; 30% of the results show that the SVC should be placed at bus 4 with 70 MVAr size and 30% of 90

5 the results show bus 4 with size 544 MVAr. Also, the following results are observed by the fuzzy PSO: 0% of results show that the SVC should be placed at bus with 546 MVAr size and 40% of results show that the SVC should be placed at bus 4 and 50% of results show that the SVC should be placed at buses 4, 36, 37 and 5. But 60% of the obtained results by GA reveal that the SVC should be placed at bus with 546 MVAr size, 0% of results show that the SVC should be placed at bus 4 with 646 MVAr size and 30% of results show that the SVC should be placed at bus 37 with 04 MVAr size. The obtained results are summarized in Table IV. TABLE IV THE OBTAINED RESULTS BY GCPSO, PSO AND GA WITH FUZZIFIED OBJECTIVE FUNCTIONS SVC Placement MV Ar Size Maximum voltage deviation losses Cost fit GCPSO bus * 0 7 bus * 0 7 PSO bus * 0 7 bus * 0 8 GA bus * 0 7 bus * 0 7 The results obtained by three algorithms are averaged over 0 independent runs. The average best-so-far and the mean fitness function of each run are recorded and averaged over 0 independent runs. To have a better clarity, the convergence characteristics in finding the location and size of an SVC is given in Figs. 6-7 for three algorithms. These figure show that the convergence of GCPSO is much better than the PSO. Fig. 7. Convergence characteristics of GCPSO, PSO and GA on the average fitness function in finding the solution, placement of SVC at bus. The voltage profiles when the system is heavily stressed are shown in Figs. 8-9, for before and after placing the SVC. voltage in pu internal bus number Fig. 8. Bus voltage magnitude profile when system is heavily stressed. voltage in pu internal bus number Fig. 9. Bus voltage magnitude profile of the stressed system after placing a 546 MVAr SVC obtained by GCPSO, PSO and GA at bus. Fig. 9 shows that the voltage profile has been improved perfectly. The maximum voltage in Fig. 9 is.05 and the minimum voltage is at bus 8. As it can be seen in table IV (the third column) the other solution found by GCPSO, PSO and GA are not good due to having a voltage deviation for PQ bus more than Fig. 6. Convergence characteristics of GCPSO, PSO and GA on the average best-so-far in finding the solution, placement of SVC at bus. IV. CONCLUSIONS In this paper the ability of GCPSO and PSO with fuzzy obective functions is investigated to place SVC in a power system, where VAr planning is based on the reduction of the 9

6 system losses, reduction of voltage deviations and cost function. Also, to validate the results obtained by the GCPSO, GA is applied. When the population size is 00, the three algorithms find bus. The convergence characteristics of GCPSO, PSO and GA show that GCPSO has a better feature than PSO. Also, the convergence characteristics show that GA performs better; the reason is that PSO are robust in continuous decision space and in this study, one dimension is discrete (location of SVC) but the point is that the GCPSO is capable in finding the solution better than PSO. The ability of binary GCPSO should be investigated and compared with GA which is the future work of the authors. V. REFERENCES [] P. Kundur, Power System Stability and Control, McGraw-Hill: New York, 994. [] K. Y.Lee and M. A. El-Sharkawi (Editors), Tutorial on modern heuristic optimization techniques with applications to power systems IEEE Power Engineering Society, IEEE Catalog Number 0TP60, Piscataway, NJ, 00. [3] K. Y.Lee and M. A. El-Sharkawi (Editors), A tutorial course on evolutionary computation techniques for power system optimization, Proc. IFAC Symposium on Power Plants and Power System Control, Seoul, Korea, 003. [4] K. Y.Lee (Editor), Tutorial on intelligent optimization and control of power systems, Proc. the 3th International Conference on Intelligent Systems Application to Power Systems (ISAP), Arlington, VA, 005. [5] K. Y. Lee, X. Ba and Y. M. Park, Optimization Method for Reactive Power Planning Using a Genetic Algorithm, IEEE Trans. Power Syst., Vol. 0, No. 4, pp , November 995. [6] R. Garduno-Ramirez, J. S. Heo, and K. Y. Lee, Dynamic Multiobective Optimization of Power Plant Using PSO Techniques, Proc. of the IEEE Power Engineering Society General Meeting, in CD, San Francisco, June -7, 005. [7] J. S. Heo, K. Y. Lee, and R. Garduno-Ramirez, Multiobective optimal power plant operation using particle swarm optimization technique, Proc. 005 IFAC Congress, Prague, Czech Republic, July 4-8, 005, Tu-M06-TO/4. [8] J.-B. Park, K.-S. Lee, J.-R. Shin, and K. Y. Lee, A particle swarm optimization for economic dispatch with non-smooth cost functions, IEEE Trans. Power Syst., Vol. 0, No., pp. 34-4, February 005. [9] S. Ebrahim M. M. Farsang H. Nezamabadi-Pour and K. Y. Lee, Optimal Allocation of STATIC VAR COMPENSATORS using Modal analysis, Simulated annealing and Tabu search, Proc. 006 IFAC Symposium on Power Plants and Power Systems, Calgary, Canada, July, 006. [0] M. M. Farsang H. Nezamabadi-Pour and K. Y. Lee, Multi-obective VAr Planning with SVC for a Large Power System Using PSO and GA, Proc. 006 IEEE PES Power Systems Conference and Exposition (PSCE), Atlanta, USA,. 9Oct-Nov, 006. [] J. Kennedy and R. Eberhart, Particle swarm optimization, in Proc. 995 IEEE Int. Conf. Neural Networks (ICNN 95), Vol. IV, pp , 995. [] J. Kennedy, The particle swarm: Social adaptation of knowledge, in Proc. 997 IEEE Int. Conf. Evol. Comput., pp , 997. [3] F. Van den Bergh and A. P. Engelbrecht, A new locally convergent particle swarm optimizer, Proc. IEEE Conference on Systems, Man and Cybernetics. (Hammamet. Tunisia), Vol. 3, 6-9 Oct, 00. [4] E.S Peer, F. Van den Bergh and A.P Engelbrecht, Using neighbourhoods with the guaranteed convergence PSO, in Proc. Swarm Intelligence Symposium, pp. 35 4, 003. [5] D.E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley: New York, 989. [6] L.J. Ca I. Erlich and G. Stamtsis, Optimal choice and allocation of FACTS devices in deregulated electricity market using genetic algorithm, Proc. IEEE PES Power Systems Conference and Exposition (PSCE), pp. 0-07, 0-3 Oct [7] P. Ramaswamy, R. M. Edwards, and K. Y. Lee, An automatic tuning method of a fuzzy logic controller for nuclear reactors, IEEE Transactions on Nuclear Science, Vol. 40, No. 4, pp. 53-6, August 993. [8] Y. M. Park, U. C. Moon, and K. Y. Lee, A self-organizing fuzzy logic controller for dynamic systems using fuzzy auto-regressive moving average (FARMA) model, IEEE Transactions on Fuzzy Systems, Vol. 3, No., pp. 75-8, pp. 75-8, February 995. BIOGRAPHIES M. M. Farsangi is received her B.S. degree in Electrical Engineering from Ferdousi University in 995, and her PhD degree in Electrical Engineering from Brunel Institute of Power Systems, Brunel University in 003. Since 003, she has been with Kerman University, Kerman, Iran, where she is currently an Assistant Professor of Electrical Engineering. Her interests include power system control and stability. Hossein Nezamabadi-pour received his B.S. degree in Electrical Engineering from Kerman University in 998, and his M. Sc and PhD degree in Electrical Engineering from Tarbait Moderres University, in 000 and 004, respectively. Since 004, he has been with Kerman University, Kerman, Iran, where he is currently an Assistant Professor of Electrical Engineering. His interests include pattern recognition, soft computing, evolutionary computation and image processing. Kwang Y. Lee received his B.S. degree in Electrical Engineering from Seoul National University, Korea, in 964, M.S. degree in Electrical Engineering from North Dakota State University, Fargo, in 968, and Ph.D. degree in System Science from Michigan State University, East Lansing, in 97. He has been with Michigan State, Oregon State, Univ. of Houston, the Penn State University, and Baylor University, where he is currently a Professor and Chair of Electrical and Computer Engineering. His interests include power system control, operation, planning, and intelligent system applications to power systems. Dr. Lee is a Fellow of IEEE, Associate Editor of IEEE Transactions on Neural Networks, and Editor of IEEE Transactions on Energy Conversion. He is also a registered Professional Engineer. 9