PARTICLE SWARM OPTIMIZATION BASED OPTIMAL POWER FLOW FOR VOLTVAR CONTROL


 Christopher Webb
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1 ARPN Journal of Engneerng and Appled Scences Asan Research Publshng Networ (ARPN). All rghts reserved. PARTICLE SWARM OPTIMIZATION BASED OPTIMAL POWER FLOW FOR VOLTVAR CONTROL M. Balasubba Reddy 1, Y. P. Obulesh 2 and S. Svanaga Raju 3 1 Praasam Engneerng College, Kanduur, Andhra Pradesh, Inda 2 L.B.R. College of Engneerng, Mylavaram, Andhra Pradesh, Inda 3 J.N.T.U. College of Engneerng, Kanada, Andhra Pradesh, Inda Emal: ABSTRACT Evolutonary computaton (EC) technques such as genetc algorthms (GAs), utlze multple searchng ponts n the soluton space le PSO. Whereas GAs can treat combnatoral optmzaton problems, PSO was amed to treat nonlnear optmzaton problems wth contnuous varables orgnally. Moreover, PSO has been expanded to handle combnatoral optmzaton problems and both dscrete and contnuous varables as well. Effcent treatment of mxednteger nonlnear optmzaton problems (MINLPs) s one of the most dffcult problems n practcal optmzaton. Moreover, unle other EC technques, PSO can be realzed wth only a small program; namely, PSO can handle MINLPs wth only a small program. Ths feature of PSO s one of ts advantages compared wth other optmzaton technques. In ths paper, the basc PSO method s combned wth Newton s method, and nteror pont method for the optmal power flow/voltvar optmzaton. The results obtaned on IEEE 30bus system showed that the hybrd method based on gves the best results compared to the other methods. It has been demonstrated that the proposed method can be easly appled to large systems. Keywords: optmal power flow, PSO, VAR control, voltage control. 1. INTRODUCTION Natural creatures sometmes behave as a swarm. One of the man streams of artfcal lfe research s to examne how natural creatures behave as a swarm and reconfgure the swarm models nsde a computer. Reynolds developed bod as a swarm model wth smple rules and generated complcated swarm behavor by computer graphc anmaton [1]. Boyd and Rcherson examned the decson process of human bengs and developed the concept of ndvdual learnng and cultural transmsson [2]. Accordng to ther examnaton, human bengs mae decsons usng ther own experences and other persons experences. A new optmzaton technque usng an analogy of swarm behavor of natural creatures was started n the begnnng of the 1990s. Dorgo developed ant colony optmzaton (ACO) based manly on the socal nsect, especally ant, metaphor [3]. Each ndvdual exchanges nformaton through pheromones mplctly n ACO. Eberhart and Kennedy developed partcle swarm optmzaton (PSO) based on the analogy of swarms of brds and fsh schoolng [4]. Each ndvdual exchanges prevous experences n PSO. These research efforts are called swarm ntellgence [5, 6]. Ths paper focuses on PSO as one of the swarm ntellgence technques. 2. BASIC PARTICAL SWARM OPTIMIZATION Swarm behavor can be modeled wth a few smple rules. Schools of fshes and swarms of brds can be modeled wth such smple models. Namely, even f the behavor rules of each ndvdual (agent) are smple, the behavor of the swarm can be complcated. Reynolds utlzed the followng three vectors as smple rules n the researches on bod. Step away from the nearest agent Go towards the destnaton Go to the center of the swarm The behavor of each agent nsde the swarm can be modeled wth smple vectors. The research results are one of the basc bacgrounds of PSO. Boyd and Rcherson examned the decson process of humans and developed the concept of ndvdual learnng and cultural transmsson [2]. Accordng to ther examnaton, people utlze two mportant nds of nformaton n decson process. The frst one s ther own experence; that s, they have tred the choces and now whch state has been better so far, and they now how good t was. The second one s other people s experences,.e., they have nowledge of how the other agents around them have performed. Namely, they now whch choces ther neghbors have found most postve so far and how postve the best pattern of choces was. Each agent decdes ts decson usng ts own experences and the experences of others. The research results are also one of the basc bacground elements of PSO. Accordng to the above bacground of PSO, Kennedy and Eberhart developed PSO through smulaton of brd flocng n a twodmensonal space. The poston of each agent s represented by ts x, y axs poston and also ts velocty s expressed by vx (the velocty of x axs) and vy (the velocty of y axs). Modfcaton of the agent poston s realzed by the poston and velocty nformaton. Brd flocng optmzes a certan objectve functon. Each agent nows ts best value so far (pbest) and ts x, y poston. Ths nformaton s an analogy of the personal experences of each agent. Moreover, each agent 20
2 ARPN Journal of Engneerng and Appled Scences Asan Research Publshng Networ (ARPN). All rghts reserved. nows the best value so far n the group (gbest) among pbests. Ths nformaton s an analogy of the nowledge of how the other agents around them have performed. Each agent tres to modfy ts poston usng the followng nformaton: The current postons (x, y), The current veloctes (vx, vy), The dstance between the current poston and pbest The dstance between the current poston and gbest Ths modfcaton can be represented by the concept of velocty (modfed value for the current postons). Velocty of each agent can be modfed by the followng equaton: 1 v + = wv + c rand ( pbest s ) + c rand *( gbest s ) (1) 1 1 * 2 2 where v s velocty of agent at teraton, w s weghtng functon, c 1 and c 2 are weghtng factors, rand 1 and rand 2 are random numbers between 0 and 1, s s current poston of agent at teraton, pbest s the pbest of agent, and gbest s gbest of the group. Namely, velocty of an agent can be changed usng three vectors such le bod. The velocty s usually lmted to a certan mum value. PSO usng (1) s called the Gbest model. The followng weghtng functon s usually utlzed n (1): w = w w w ) /( ter )) * ter (2) (( mn Where w s the ntal weght, w mn s the fnal weght, ter s mum teraton number and ter s current teraton number. The meanngs of the rghthand sde (RHS) of (1) can be explaned as follows [7]. The RHS of (1) conssts of three terms (vectors). The frst term s the prevous velocty of the agent. The second and thrd terms are utlzed to change the velocty of the agent. Wthout the second and thrd terms, the agent wll eep on flyng n the same drecton untl t hts the boundary. Namely, t tres to explore new areas and, therefore, the frst term corresponds wth dversfcaton n the search procedure. On the other hand, wthout the frst term, the velocty of the flyng agent s only determned by usng ts current poston and ts best postons n hstory. Namely, the agents wll try to converge to ther pbests and/or gbest and, therefore, the terms correspond wth ntensfcaton n the search procedure. As shown below, for example, w and w mn are set to 0.9 and 0.4. Therefore, at the begnnng of the search procedure, dversfcaton s heavly weghted, whle ntensfcaton s heavly weghted at the end of the search procedure such le smulated annealng (SA). Namely, a certan velocty, whch gradually gets close to pbests and gbest, can be calculated. PSO usng (1) (2) s called nerta weghts approach (IWA). Fgure1. Concept of modfcatons of a Searchng pont by PSO. s = current searchng pont +1 s = modfed searchng pont v = current velocty +1 v = modfed velocty v = velocty based on pbest pbest v gbest = velocty based on gbest The current poston (searchng pont n the soluton space) can be modfed by the followng equaton: s = s + v (3) Fgure1 shows a concept of modfcaton of a searchng pont by PSO, and Fgure1 shows a searchng concept wth agents n a soluton space. Each agent changes ts current poston usng the ntegraton of vectors as shown n Fgure1. PSO algorthm Step 1: Generaton of ntal condton of each agent. Intal searchng ponts ( s ) and veloctes ( v ) of each 0 agent are usually generated randomly wthn the allowable range. The current searchng pont s set to pbest for each agent. The best evaluated value of pbest s set to gbest, and the agent number wth the best value s stored. Step 2: Evaluaton of searchng pont of each agent. The objectve functon value s calculated for each agent. If the value s better than the current pbest of the agent, the pbest value s replaced by the current value. If the best value of pbest s better than the current gbest, gbest s replaced by the best value and the agent number wth the best value s stored. Step 3: Modfcaton of each searchng pont. The current searchng pont of each agent s changed usng (1), (2), and (3). Step 4: checng the ext condton. The current teraton number reaches the predetermned mum teraton number, then exts. Otherwse, the process proceeds to step 2. The features of the searchng procedure of PSO can be summarzed as follows: As shown n (1), (2), and (3), PSO can essentally handle contnuous optmzaton problems. 0 21
3 ARPN Journal of Engneerng and Appled Scences Asan Research Publshng Networ (ARPN). All rghts reserved. PSO utlzes several searchng ponts, and the searchng ponts gradually get close to the optmal pont usng ther pbests and the gbest. The frst term of the RHS of (1) corresponds wth dversfcaton n the search procedure. The second and thrd terms correspond wth ntensfcaton n the search procedure. Namely, the method has a wellbalanced mechansm to utlze dversfcaton and ntensfcaton n the search procedure effcently. The above concept s explaned usng only the x, y axs (twodmensonal space). However, the method can be easly appled to ndmensonal problems. Namely, PSO can handle contnuous optmzaton problems wth contnuous state varables n an ndmensonal soluton space. Sh and Eberhart tred to examne the parameter selecton of the above parameters [7, 8]. Accordng to ther examnaton, the followng parameters are approprate and the values do not depend on problems: c = 2.0, w = 0.9, w mn = 0.4, The values are also proved to be approprate for power system problems [9, 10]. The basc PSO has been appled to a learnng problem of neural networs and Schaffer f6, a famous benchmar functon for GA, and the effcency of the method has been observed [4]. 3. MATHEMATICAL MODEL OF OPF PROBLEM The OPF problem s to optmze the steady state performance of a power system n terms of an objectve functon whle satsfyng several equalty and nequalty constrants. Mathematcally, the OPF problem can be formulated as gven: Mn F ( x, u) (4) Subject to g ( x, u) = 0 (5) h ( x, u) 0 (6) where x s a vector of dependent varables consstng of slac bus power P G 1, load bus voltages V L, generator reactve power outputs Q G, and the transmsson lne loadngs S, Hence, x can be expressed as gven: l T x = P, V... V, Q... Q, S... S ] (7) [ G1 L1 LNL G1 GNG l lnl where NL,NG and nl are number of load buses, number of generators and number of transmsson lne, respectvely. u s the vector of ndependent varables consstng of generator voltages V G, generator real power outputs P G except at the slac bus P G 1, transformer tap settngs T, and shunt VAR compensatons Q C. Hence u can be expressed as gven: T u = V... V, P... P, T... T, Q... Q ] (8) [ G1 GNG G2 GNG 1 NT C1 CNC Where NT and NC are the number of the regulatng transformers and shunt compensators, respectvely. F s the objectve functon to be mnmzed. g s the equalty constrants that represents typcal load flow equatons and h s the system operatng constrants. Objectve functons In ths paper, the objectve(s) (J) s the objectve functon to be mnmzed, whch s one of the followng: () Objectve functon1 (Fuel cost mnmzaton) It sees to fnd the optmal actve power outputs of the generaton plants so as to mnmze the total fuel cost. Ths can be expressed as: NG J = f ($ / h) (9) where f s the fuel cost curve of the th generator and t s assumed here to be represented by the followng quadratc functon: f 2 = a P + b P c ($/ h) (10) G G + where a, b, and c are the cost coeffcents of the th generator. () Objectve functon2 (Voltage profle mprovement) Voltage profle s one of the qualty measures for power system. It can be mproved by mnmzng the load bus voltage devatons from 1.0 per unt. The objectve functon can be expressed as J = 1 (11) V NL () Objectve functon3 (Voltage stablty enhancement) Voltage profle mprovement does not necessary mples a voltage secure system. Voltage nstablty problems have been experenced n systems where voltage profle was acceptable [11]. Voltage secure system can be assured by enhancng the voltage stablty profle throughout the whole power system. An ndcator Lndex s used n ths study to evaluate the voltage stablty at each bus of the system. The ndcator value vares between 0 (no load case) and 1 (voltage collapse) [1214]. One of the best features of the Lndex s that the computaton speed s very fast and so can be used for onlne montorng of power system. Enhancng the voltage stablty and movng the system far from voltage collapse pont can be acheved by mnmzng the followng objectve functon: J = L (12) where L s the mum value of Lndex as: 22
4 ARPN Journal of Engneerng and Appled Scences Asan Research Publshng Networ (ARPN). All rghts reserved. { L, K 1 NL} L = K,..., (13) = (v) Objectve functon4 (Real power loss mnmzaton) The optmal reactve power flow problem to mnmze actve losses can be formulated as: mn J = f ( Ζ) s. t g( Ζ) = 0 (14) Ζ mn Ζ Ζ Where f ( ) s the objectve functon for actve losses. g ( ) Nonlnear vectors functon representng power flow equatons. Ζ = [ x u] T Vector of decson varables whose components are the vector of state varables x (voltage phase angles and magntudes, etc.) and the vector of dscrete control varables u (generator termnal voltages, tap poston of OLTC transformers, number of connected shunt compensaton devces etc.). Ζ mn and Ζ vectors modelng operatonal lmts on state and control varables 4. SIMULATION RESULTS The smulaton results of the proposed basc PSO method for dfferent objectve functons (.e., fuel cost mnmzaton, voltage profle mprovement, voltage stablty enhancement, and real power loss mnmzaton) have been appled to IEEE30 bus system wth NRload flow, NewtonOPF and Interor Pont Method. It s chosen as t s a benchmar system, have more control varables and provde results for comparson of the proposed methods. The approach can be generalzed and easly extended to largescale systems. The IEEE30 bus system conssts of sx generators, four transformers, 41 lnes, and nne shunt capactors. In all these dfferent PSO methods, the total control varables are 25: sx unt actve power outputs, sx generator bus voltage magntudes, four transformer tap settngs, and nne bus shunt admttances. The basc PSO methods have been run for 20populatons and for 150 teratons. To test the ablty of the proposed hybrd algorthm for solvng optmal power flow problem to reduce specfed objectve functon, t was appled on selected bus system. Four objectve functons are consdered for the mnmzaton usng the proposed hybrd algorthm namely cost of generaton, voltage profle mprovement, voltage stablty enhancement and real power loss mnmzaton. Fgures 2 to 5 show the convergence characterstcs of the three OPF methods under the selected objectve functon. It can be observed that IPM PSO converges to a mnmum value than PSONewton and methods. Objectve functon1 (cost) Objectve functon2 (V.D) Objectve functon3 (Lndex) Objectve functon4 (loss) Number of teratons Fgure2. Convergence characterstcs of objectve functon Number of teratons Fgure3. Convergence characterstcs of objectve functon Number of teratons Fgure4. Convergence characterstcs of objectve functon Number of teratons Fgure5. Convergence characterstcs of objectve functon4. 23
5 ARPN Journal of Engneerng and Appled Scences Asan Research Publshng Networ (ARPN). All rghts reserved. The best results for dfferent PSO methods combned wth NRload flow, NewtonOPF, and Interor Pont method are compared and results are tabulated n Table1. In ths table, the optmal settngs of the control varables and varous performance parameters wth four objectve functons are presented. From ths table, t was found that all the state varables satsfy lower and upper lmts. From the results t s evdent that proposed IPMPSO hybrd method outperforms n achevng mnmum of the specfed objectve when compared wth other optmzaton methods. Control Varables P 1 P 2 P 5 P 8 P 11 P 13 V 1 V 2 V 5 V 8 V 11 V 13 T 11 T 12 T 15 T 36 Q C10 Q C12 Q C15 Q C17 Q C20 Q C21 Q C23 Q C24 Q C29 Cost ($/h) V.D L Index Ploss (pu) Table1. Optmal settngs of control varables of IEEE 30bus system. Objectve functon1 (cost) Objectve functon2 (V.D) Objectve functon3 (Lndex) Objectve functon4 (loss) CONCLUSIONS In ths paper, the basc PSO method s combned wth Newton s method, and nteror pont method for the optmal power flow/voltvar optmzaton. The results obtaned on IEEE 30bus system showed that the hybrd method based on gves the best results compared to the other methods proposed n ths paper. It has been demonstrated that the proposed method can be easly appled large systems. ACKNOWLEDGEMENTS The authors would le to than Dr. Kancharla Ramaah, correspondent and secretary of Praasam Engneerng College, Kanduur, Inda for hs encouragement and support and provdng us wth the faclty for completng ths paper. 24
6 ARPN Journal of Engneerng and Appled Scences Asan Research Publshng Networ (ARPN). All rghts reserved. REFERENCES [1] Reynolds C Flocs, herds, and schools: A dstrbuted behavoral model. Computer Graphcs. 21(4): [2] Boyd R. and Rcherson P.J Culture and the evolutonary process. Chcago: Unversty of Chcago Press, USA. [13] Tuan T.Q., et al., Emergency load sheddng to avod rss of voltage nstablty usng ndcator. IEEE Trans. On Power Systems. 9(1): [14] Kessel P. and Glavtsch H Estmatng the voltage stablty of a power system. IEEE Trans. on Power Delvery. 1(3): [3] Colorn A, Dorgo M and Manezzo V Dstrbuted optmzaton by ant colones. Proceedngs of 1 st European Conference on Artfcal Lfe. Cambrdge, MA: MIT Press. pp [4] Kennedy J. and Eberhart R Partcle swarm optmzaton. Proceedngs of IEEE Internatonal Conference on Neural Networs (ICNN 95) Perth, Australa: IEEE Press. IV: [5] Bonabeau E, Dorgo M and Theraulaz G Swarm ntellgence: From natural to artfcal systems. Oxford: Oxford Unversty Press. [6] Kennedy J and Eberhart R Swarm ntellgence. San Mateo, CA: Morgan Kaufmann. [7] Sh Y and Eberhart R A modfed partcle swarm optmzer. Proceedngs of IEEE Internatonal Conference on Evolutonary Computaton (ICEC 98). Anchorage: IEEE Press. pp [8] Sh Y and Eberhart R Parameter selecton n partcle swarm optmzaton. Proceedngs of the 1998 Annual Conference on Evolutonary Programmng. San Dego: MIT Press. [9] Fuuyama Y. et al., A partcle swarm optmzaton for reactve power and voltage control consderng voltage securty assessment. IEEE Trans Power Systems. 15(4): [10] Naa S, Genj T, Yura T and Fuuyama Y A hybrd partcle swarm optmzaton for dstrbuton state estmaton. IEEE Trans Power Systems. 18(1): [11] Overbye T. and DeMarco C Voltage Securty Enhancement Usng Energy Based Senstvtes. IEEE Trans. On Power Systems. 6(3): [12] Km S. and Song T., et al., Development of Voltage Stablty Constraned Optmal Power Flow (VSCOPF). Power Engneerng Socety Summer Meetngs, 2001 IEEE. 3: July, pp