A chaotic particle swarm optimization (cpso) algorithm for solving optimal reactive power dispatch problem

A chaotic particle swarm optimization (cpso) algorithm for solving optimal reactive power dispatch problem

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  Industrial Engineering Letterswww.iiste.org ISSN 2224-6096 (Paper) ISSN 2225-0581 (online) Vol.4, No.3, 2014 11 A Chaotic Particle Swarm Optimization (CPSO) Algorithm for Solving Optimal Reactive Power Dispatch Problem K. Lenin1* , Dr.B.Ravindranath Reddy2 , Dr.M.Surya Kalavathi3 1.Research Scholar, Jawaharlal Nehru Technological University Kukatpally, Hyderabad 500 085, India 2.Deputy Executive Engineer, Jawaharlal Nehru Technological University Kukatpally, Hyderabad 500 085, India 3.Professor of Electrical and Electronics Engineering, Jawaharlal Nehru Technological University Kukatpally,Hyderabad 500 085, India * E-mail of the corresponding author: gklenin@gmail.com Abstract This paper presents a chaotic particle swarm algorithm for solving the multi-objective reactive power dispatch problem. To deal with reactive power optimization problem, a chaotic particle swarm optimization (CPSO) is presented to avoid the premature convergence. By fusing with the ergodic and stochastic chaos, the novel algorithm explores the global optimum with the comprehensive learning strategy. The chaotic searching region can be adjusted adaptively. In order to evaluate the proposed algorithm, it has been tested on IEEE 30 bus system and simulation results show that (CPSO) is more efficient than other algorithms in reducing the real power loss and maximization of voltage stability index. Keywords:chaotic particle swarm optimization, Optimization, Swarm Intelligence, optimal reactive power, Transmission loss. 1.Introduction One of the major problems faced by power system operators is the reactive power dispatch imposed on electric power utilities for a continuous and reliable supply of energy. Major power loads require a significant amount of reactive power that has to be supplied while maintaining load bus voltages within their permissible operating limits. In order to maintain desired levels of voltages and reactive flows under various operating conditions and system configurations, power system operators may utilize a number of control tools such as switching var sources, changing generator voltages, and by adjusting transformer tap settings. By an optimal adjustment of these controls, the redistribution of the reactive power would minimize transmission losses.Various mathematical techniques have been adopted to solve this optimal reactive power dispatch problem. These include the gradient method (O.Alsac et al.1973; Lee K Yet al.1985), Newton method (A.Monticelli et al.1987) and linear programming (Deeb Net al.1990; E. Hobson1980; K.Y Lee et al.1985; M.K. Mangoli 1993) .The gradient and Newton methods suffer from the difficulty in handling inequality constraints. To apply linear programming, the input- output function is to be expressed as a set of linear functions which may lead to loss of accuracy. Recently Global Optimization techniques such as genetic algorithms have been proposed to solve the reactive power flow problem (S.R.Paranjothi et al 2002;D. Devaraj et al 2005) . In recent years, the problem of voltage stability and voltage collapse has become a major concern in power system planning and operation. To enhance the voltage stability, voltage magnitudes alone will not be a reliable indicator of how far an operating point is from the collapse point (C.A. Canizareset al.1996). The reactive power support and voltage problems are intrinsically related. Hence, this paper formulates the reactive power dispatch as a multi-objective optimization problem with loss minimization and maximization of static voltage stability margin (SVSM) as the objectives. Voltage stability evaluation using modal analysis is used as the indicator of voltage stability. In recent years, several new optimization techniques have emerged. The evolutionary algorithms (EAs) for reactive power optimization problem have been extensively studied. Several global optimization algorithms such as differential evolution (DE) (Dib.N et al .2010; Lin. C et al.2010), genetic algorithm (GA) (Zhang et al.2009 ; Vaitheeswaran.S et al.2008) , simulated annealing (SA) (Ferreira, J. A et al.1997 ), Ant colony optimization (ACO) (Hosseini.S. A et al 2008 ), particle swarm optimization (PSO) (Perez Lopez et al.2009; Liu, D et al.2009; Li, W.-T et al.2010 ; Goudos, S et al.2010 ; Shavit, R et al.2005) are used for reactive power optimization problem. However, these methods present certain drawbacks with the possibility of premature convergence to a local optimum. In this paper, a novel chaotic PSO algorithm (CPSO) is proposed. Based on the ergodicity, regularity and pseudo-randomness of the Chaotic variable, chaotic search is used to explore better solutions. The performance of (CPSO) has been evaluated in standard IEEE 30 bus test system and the results analysis shows that our proposed approach outperforms all approaches investigated in this paper. 2. Voltage Stability Evaluation 2.1 Modal analysis for voltage stability evaluation Modal analysis is one of the methods for voltage stability enhancement in power systems. The linearized steady state system power flow equations are given by.
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  Industrial Engineering Letterswww.iiste.org ISSN 2224-6096 (Paper) ISSN 2225-0581 (online) Vol.4, No.3, 2014 12 ∆P ∆Q = J J J J (1) Where ∆P = Incremental change in bus real power. ∆Q = Incremental change in bus reactive Power injection ∆θ = incremental change in bus voltage angle. ∆V = Incremental change in bus voltage Magnitude Jpθ , J PV , J Qθ , J QV jacobian matrix are the sub-matrixes of the System voltage stability is affected by both P and Q. However at each operating point we keep P constant and evaluate voltage stability by considering incremental relationship between Q and V. To reduce (1), let ∆P = 0 , then. ∆Q = J − J J J ∆V = J ∆V (2) ∆V = J − ∆Q (3) Where J = J − J J JPV (4) J is called the reduced Jacobian matrix of the system. 2.2 Modes of Voltage instability Voltage Stability characteristics of the system can be identified by computing the eigen values and eigen vectors Let J = ξ˄η (5) Where, ξ = right eigenvector matrix of JR η = left eigenvector matrix of JR ∧ = diagonal eigenvalue matrix of JR and J = ξ˄ η (6) From (3) and (6), we have ∆V = ξ˄ η∆Q (7) or ∆V = ∑ ! "# ∆Q (8) Where ξi is the ith column right eigenvector and η the ith row left eigenvector of JR. λi is the ith eigen value of JR. The ith modal reactive power variation is, ∆Q$% = K%ξ% (9) where, K% = ∑ ξ%'(' − 1 (10) Where ξji is the jth element of ξi The corresponding ith modal voltage variation is ∆V$% = *1 λ%⁄ -∆Q$% (11) It is seen that, when the reactive power variation is along the direction of ξi the corresponding voltage variation is also along the same direction and magnitude is amplified by a factor which is equal to the magnitude of the inverse of the ith eigenvalue. In this sense, the magnitude of each eigenvalue λi determines the weakness of the corresponding modal voltage. The smaller the magnitude of λi, the weaker will be the corresponding modal voltage. If | λi | =0 the ith modal voltage will collapse because any change in that modal reactive power will cause infinite modal voltage variation. In (8), let ∆Q = ek where ek has all its elements zero except the kth one being 1. Then, ∆V = ∑ ƞ . ξ λ% (12) ƞ / k th element of ƞ V –Q sensitivity at bus k 0 1 0 1 = ∑ ƞ . ξ λ% = ∑ . λ% (13) 3.Problem Formulation The objectives of the reactive power dispatch problem considered here is to minimize the system real power loss and maximize the static voltage stability margins (SVSM).
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  Industrial Engineering Letterswww.iiste.org ISSN 2224-6096 (Paper) ISSN 2225-0581 (online) Vol.4, No.3, 2014 13 3.1Minimization of Real Power Loss It is aimed in this objective that minimizing of the real power loss (Ploss) in transmission lines of a power system. This is mathematically stated as follows. P23445 ∑ g/( ( 8 9 ( : 9 ;<= θ 9 ) ? /5 /5(%,') (14) Where n is the number of transmission lines, gk is the conductance of branch k, Vi and Vj are voltage magnitude at bus i and bus j, and θij is the voltage angle difference between bus i and bus j. 3.2 Minimization of Voltage Deviation It is aimed in this objective that minimizing of the Deviations in voltage magnitudes (VD) at load buses. This is mathematically stated as follows. Minimize VD = ∑ |V/ − 1.0|?2 /5 (15) Where nl is the number of load busses and Vk is the voltage magnitude at bus k. 3.3 System Constraints Objective functions are subjected to these constraints shown below. Load flow equality constraints: DEF – DHF − IF ∑ JK LM KN OFP cos TFP +VFP sin TFP = 0, Y = 1,2 … . , ] (16) ^EF − ^HF − IF ∑ JK LM KN OFP cos TFP +VFP sin TFP = 0, Y = 1,2 … . , ] (17) where, nb is the number of buses, PG and QG are the real and reactive power of the generator, PD and QD are the real and reactive load of the generator, and Gij and Bij are the mutual conductance and susceptance between bus i and bus j.Generator bus voltage (VGi) inequality constraint: IEF _F` ≤ IEF ≤ IEF _bc , Y ∈ e (18) Load bus voltage (VLi) inequality constraint: IfF _F` ≤ IfF ≤ IfF _bc , Y ∈ g (19) Switchable reactive power compensations (QCi) inequality constraint: ^hF _F` ≤ ^hF ≤ ^hF _bc , Y ∈ i (20) Reactive power generation (QGi) inequality constraint: ^EF _F` ≤ ^EF ≤ ^EF _bc , Y ∈ e (21) Transformers tap setting (Ti) inequality constraint: jF _F` ≤ jF ≤ jF _bc , Y ∈ k (22) Transmission line flow (SLi) inequality constraint: lfF _F` ≤ lfF _bc , Y ∈ g (23) Where, nc, ng and nt are numbers of the switchable reactive power sources, generators and transformers. 4. Principle of CPSO 4.1Principle of CPSO Inspired by the social behaviours of animal, bird flocking and fishing, PSO was developed by (Kennedy et al . 1995) . The particle is endowed with two factors: velocity and position which can be regarded as the potential solution in the D dimension problem space. In basic PSO, they can be updated by following formulas: mFn(k + 1) = omFn(k) + i p n DFn(k) − qFn(k) + i:p:n rDsn(k) − qFn(k)t (24) qFn(k + 1) = qFn(k) + mFn(k + 1) (25) Where i = 1,….., N, d = 1,…,D, N is the number of particles. o is the inertia weight factor to control the exploration and exploitation. r1d and r2d are two random numbers within the range [0, 1]. vid(t) and xid(t) are the velocity and position of the current particle i at time step t in the dth-dimensional search space respectively. When vid(t) and xid(t) are beyond the boundary, the solution may be illegal. So, the treatment of boundaries in the PSO method is important in order to prevent the swarm from explosion (Xu.S et al .2007 ). In many practical problems, the search range xid is in [Xmin;Xmax]D. vid should be clamped to a maximum magnitude Vmax. pi is the previous best position of particle I which is also called “personal best”, and its dth-dimensional part is pid. The global best" pg is the best position found in the whole particles, and its dth-dimensional part is pgd. c1, c2 are the acceleration constants which change the velocity of a particle towards the pi and pg. 4.2Modification Techniques in CPSO The basic PSO uses pg as neighbourhood topology. Each particle learns from its pi and pg. Restricting the social learning part to pg can make basic PSO converge quickly. However, because all particles in the swarm learn
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  Industrial Engineering Letterswww.iiste.org ISSN 2224-6096 (Paper) ISSN 2225-0581 (online) Vol.4, No.3, 2014 14 from the pg even if the current pg is far from the global optimum, particles may easily be attracted to the area and trapped in a local optimum. Furthermore, the fitness value of a particle is determined by all dimensions. A particle that has discovered the region corresponding to the global optimum in some dimensions may have a low fitness value because of the poor solutions in other dimensions ( Liang, J.-Jet al 2006 ) . In order to acquire more beneficial information from the entire swarm, we define pc as “comprehensive best position”. uv = w ∑ xy z yN { , ∑ xy( z yN { , . . , ∑ xy| z yN { , . . , ∑ xy} z yN { ~ (26) Where i = 1,..,N. Thus Equation (1) is modified as mFn(k + 1) = omFn(k) + i p n DFn(k) − qFn(k) + i:p:n Dvn(k) − qFn(k) (27) where pcd is the dth-dimensional part of pc. By using pc instead of pg, all particles' pi can potentially be used as the exemplars to guide their flying direction. The comprehensive learning strategy yields a larger potential search space than that of the basic PSO. On the other hand, a particle can learn from pg, as well as its personal best and the other particles' best, so that the particle can learn from particle itself, the elite and other particles. The strategy can increase the initial diversity and enable the swarm to overcome premature convergence problem. Basic PSO has shown some important advances by providing high speed of convergence in specific problems. However it does exhibit some shortages (Modares.H et al.2010). During the process of evolution, sometimes particles lose their abilities of exploration and will be stagnated. When some particles' velocity is be close to zero, other particles will quickly fly into the region near the inactive particles position that guided by pi and pg. Because of the particles randomicity in initialization and evolution process, the updating sometimes looks aimless. As a result, when pg is trapped in a local optimum, the whole swarm becomes premature convergence, and the exploration performance will not be improved. Optimization algorithms based on the chaos theory are stochastic search methodologies that differ from any of the existing evolutionary algorithms. Due to the non- repetition of chaos, it can carry out overall exploration at higher velocities than stochastic and ergodic searches that depend on probabilities (Coelho, L. Det al.2009) . Chaotic PSO can be divided into two types. In the first type, chaos is embedded into the velocity updating equation of PSO. In (Modares.H et al.2010) , c1 and c2 are generated from the iterations of a chaotic map instead of using the rand function. In (Wang, Y., et al. 2010 ), a chaotic map is used to determine the value of o during iterations. In the second type, chaotic search is fused with the procedures of PSO. This type is a kind of multi-phase optimization technique that chaotic optimization and PSO can switch to each other according to certain conditions (Wu.Q 2011 ) . Therefore, this paper provides a new strategy, which not only introduces chaotic mapping with certainty, ergodicity and stochastic property into PSO algorithm, but also proposes multi-phase optimization integrated by chaotic search and PSO evolution. The multi-phase optimization of chaotic PSO includes: vid and xid are updated by basic PSO with comprehensive learning strategy. If the swarm is stagnated, chaotic disturbance would be introduced. Here, variance •: demonstrates the converge degree of all particles. •: = ∑ €F − €b•s €⁄ :{ F5 (28) € = ‚ƒq „1, ‚ƒq…†€F − €b•s†‡ˆ (29) Where fi is the fittness of the ith particle; favg is the average fitness value; f is the factor of fitness value. The bigger •: is the broader ith particles will spread. Otherwise, they will almost converge. The chaotic sequence can be generated by the logistic map introduced by Robert May in 1976. It is often cited as an example of how complex behaviour can arise from a simple dynamic system without any stochastic disturbance (He, Y.-Y., et al. 2009) The equation is the following ‰Fn(k + 1) = Š‰Fn(k) 1 − ‰Fn(k) (30) Where ‰Fn(k) ∈ (0,1), Y = 1, . . , ‹ , Œ = 1, . . , •. Š is usually set to 4 obtain ergodicity of ‰Fn(k + 1) ŽYkℎY (0,1). When the initial value ‰Fn(0) ∈ •0.25.0.5.0.75“ using equation (30) we can obtain chaotic sequences. In order to increase the population diversity and prevent premature convergence, we add adaptively chaotic disturbance Dv at the time of stagnation. Thus, Dv Y” ‚•ŒY€Y–Œ ƒ” Dv ′ . uvn — (k + 1) = uvn(k) + ˜Fn 2™Fn(k) − 1 (31) ˜Fn = š|uvn(k) − uFn(k)| (32) Where š is the region scale factor. Because‰Fn ∈ (0,1), the second part of Equation (31) is in the range of (−|˜Fn|, |˜Fn|) that would restrict the searching area around pc. In addition, the searching range can be adaptively adjusted by the distance between pi and pc. If pc is surrounded with the previous best positions pi, it means that a good region may have been found, and it is reasonable to search elaborately in a small area. On the contrary, if pi is far from pc, this probably suggests that a good area has not yet been found. For better solution,
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  Industrial Engineering Letterswww.iiste.org ISSN 2224-6096 (Paper) ISSN 2225-0581 (online) Vol.4, No.3, 2014 15 searching region should be enlarged (Lin, C.et al.2007). Thus in CPSO, the new position can be expressed as mFn(k + 1) = omFn(k) + i p n uFn(k) − qFn(k) + i:p:n uvn — (k) − qFn(k) (33) Different from Equation (24), pg is replaced by uv — at the time of stagnation when •: is less than the stagnation factor ›. Chaotic search is restricted into a small range to obtain high performance in local exploration. Additionally, the algorithm keeps a dynamic balance between global and local searches due to its adaptive mechanism. With the new updating rule, different exemplars are used in different dimensions to explore a larger search space than the basic PSO. In addition, chaotic disturbance is embedded in different dimensions to maintain the diversity which plays an important role in avoiding early convergence. 5.Simulation Results The validity of the proposed Algorithm technique is demonstrated on IEEE-30 bus system. The IEEE-30 bus system has 6 generator buses, 24 load buses and 41 transmission lines of which four branches are (6-9), (6-10) , (4-12) and (28-27) - are with the tap setting transformers. The lower voltage magnitude limits at all buses are 0.95 p.u. and the upper limits are 1.1 for all the PV buses and 1.05 p.u. for all the PQ buses and the reference bus. Table 1. Voltage Stability under Contingency State Sl.No Contigency ORPD Setting Vscrpd Setting 1 28-27 0.1400 0.1422 2 4-12 0.1658 0.1662 3 1-3 0.1784 0.1754 4 2-4 0.2012 0.2032 Table 2. Limit Violation Checking Of State Variables State variables limits ORPD VSCRPD Lower upper Q1 -20 152 1.3422 -1.3269 Q2 -20 61 8.9900 9.8232 Q5 -15 49.92 25.920 26.001 Q8 -10 63.52 38.8200 40.802 Q11 -15 42 2.9300 5.002 Q13 -15 48 8.1025 6.033 V3 0.95 1.05 1.0372 1.0392 V4 0.95 1.05 1.0307 1.0328 V6 0.95 1.05 1.0282 1.0298 V7 0.95 1.05 1.0101 1.0152 V9 0.95 1.05 1.0462 1.0412 V10 0.95 1.05 1.0482 1.0498 V12 0.95 1.05 1.0400 1.0466 V14 0.95 1.05 1.0474 1.0443 V15 0.95 1.05 1.0457 1.0413 V16 0.95 1.05 1.0426 1.0405 V17 0.95 1.05 1.0382 1.0396 V18 0.95 1.05 1.0392 1.0400 V19 0.95 1.05 1.0381 1.0394 V20 0.95 1.05 1.0112 1.0194 V21 0.95 1.05 1.0435 1.0243 V22 0.95 1.05 1.0448 1.0396 V23 0.95 1.05 1.0472 1.0372 V24 0.95 1.05 1.0484 1.0372 V25 0.95 1.05 1.0142 1.0192 V26 0.95 1.05 1.0494 1.0422 V27 0.95 1.05 1.0472 1.0452 V28 0.95 1.05 1.0243 1.0283 V29 0.95 1.05 1.0439 1.0419 V30 0.95 1.05 1.0418 1.0397 Table 3. Comparison of Real Power Loss
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  Industrial Engineering Letterswww.iiste.org ISSN 2224-6096 (Paper) ISSN 2225-0581 (online) Vol.4, No.3, 2014 16 Method Minimum loss Evolutionary programming[11] 5.0159 Genetic algorithm[12] 4.665 Real coded GA with Lindex as SVSM[13] 4.568 Real coded genetic algorithm[14] 4.5015 Proposed CPSO method 4.2031 6. Conclusion A large-scale power system should supply power to the customers in a reliable and economic way while keeping system voltages in permissible limits. The purpose of the optimal reactive dispatch problem is to improve system voltage profile by minimizing power system losses. In this research paper a chaotic particle swarm has been utilized to sole the optimal reactive power dispatch problem. The performance of the proposed algorithm has been tested in standard IEEE test system and simulation results shows the best performance of the proposed system and the real power loss has been considerably reduced . References O.Alsac,and B. Scott, “Optimal load flow with steady state security”,IEEE Transaction. PAS -1973, pp. 745-751. Lee K Y ,Paru Y M , Oritz J L –A united approach to optimal real and reactive power dispatch , IEEE Transactions on power Apparatus and systems 1985: PAS-104 : 1147-1153 A.Monticelli , M .V.F Pereira ,and S. Granville , “Security constrained optimal power flow with post contingency corrective rescheduling” , IEEE Transactions on Power Systems :PWRS-2, No. 1, pp.175-182.,1987. Deeb N ,Shahidehpur S.M ,Linear reactive power optimization in a large power network using the decomposition approach. IEEE Transactions on power system 1990: 5(2) : 428-435 E. Hobson ,’Network consrained reactive power control using linear programming, ‘ IEEE Transactions on power systems PAS -99 (4) ,pp 868-877, 1980 K.Y Lee ,Y.M Park , and J.L Oritz, “Fuel –cost optimization for both real and reactive power dispatches” , IEE Proc; 131C,(3), pp.85-93 1985. M.K. Mangoli, and K.Y. Lee, “Optimal real and reactive power control using linear programming” , Electr.Power Syst.Res, Vol.26, pp.1-10,1993. S.R.Paranjothi ,and K.Anburaja, “Optimal power flow using refined genetic algorithm”, Electr.Power Compon.Syst , Vol. 30, 1055-1063,2002. D. Devaraj, and B. Yeganarayana, “Genetic algorithm based optimal power flow for security enhancement”, IEE proc-Generation.Transmission and. Distribution; 152, 6 November 2005. C.A. Canizares , A.C.Z.de Souza and V.H. Quintana , “ Comparison of performance indices for detection of proximity to voltage collapse ,’’ vol. 11. no.3 , pp.1441-1450, Aug 1996 . Wu Q H, Ma J T. Power system optimal reactive power dispatch using evolutionary programming. IEEE Transactions on power systems 1995; 10(3): 1243-1248 . S.Durairaj, D.Devaraj, P.S.Kannan ,’ Genetic algorithm applications to optimal reactive power dispatch with voltage stability enhancement’ , IE(I) Journal-EL Vol 87,September 2006. D.Devaraj ,’ Improved genetic algorithm for multi – objective reactive power dispatch problem’ European Transactions on electrical power 2007 ; 17: 569-581. P. Aruna Jeyanthy and Dr. D. Devaraj “Optimal Reactive Power Dispatch for Voltage Stability Enhancement Using Real Coded Genetic Algorithm” International Journal of Computer and Electrical Engineering, Vol. 2, No. 4, August, 2010 1793-8163. Dib, N. I., S. K. Goudos, and H. Muhsen, Application of Taguchi's optimization method and self-adaptive differential evolution to the synthesis of linear antenna arrays," Progress In Electromagnetics Research, Vol. 102, 159-180, 2010. Lanza Diego, M., J. R. Perez Lopez, and J. Basterrechea, Synthesis of planar arrays using a modi¯ed particle swarm optimization algorithm by introducing a selection operator and elitism," Progress In Electromagnetics Research, Vol. 93, 145{160,2009. Qu, Y., G. Liao, S.-Q. Zhu, and X.-Y. Liu, Pattern synthesis of planar antenna array via convex optimization for airborne forward looking radar," Progress In Electromagnetics Research, Vol. 84, 1- 10, 2008. Zhang, S., S.-X. Gong, Y. Guan, P.-F. Zhang, and Q. Gong, A novel IGA-EDSPSO hybrid algorithm for the synthesis of sparse arrays," Progress In Electromagnetics Research, Vol. 89, 121-134,2009. Oliveri, G. and L. Poli, Synthesis of monopulse sub-arrayed linear and planar array antennas with optimized sidelobes," Progress In Electromagnetics Research, Vol. 99, 109-129, 2009. Vaitheeswaran, S. M., Dual beam synthesis using element position perturbations and the G3-GA algorithm," Progress In Electromagnetics Research, Vol. 88, 43-61, 2008.
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