OPTIMAL POWER FLOW USING PARTICLE SWARM OPTIMIZATION

Transcription

1 Internatonal Journal of Engneerng Scences & Emergng Technologes, Dec OPTIMAL POWER FLOW USING PARTICLE SWARM OPTIMIZATION M. Lakshmkantha Reddy 1, M. Ramprasad Reddy 2, V. C. Veera Reddy 3 1&2 Research Scholar, Department of Electrcal Engneerng, SV Unversty, Trupath, Inda 3 Professor, Department of Electrcal Engneerng, SV Unversty, Trupath, Inda lkreddy.eee@gmal.com, ramprasadreddy.madthat@gmal.com, veerareddy_vc@yahoo.com ABSTRACT The Optmal Power Flow (OPF) s an mportant crteron n today s power system operaton and control due to scarcty of energy resources, ncreasng power generaton cost and ever growng demand for electrc energy. As the sze of the power system ncreases, load may be varyng. The generators should share the total demand plus losses among themselves. The sharng should be based on the fuel cost of the total generaton wth respect to some securty constrants. Conventonal optmzaton methods that make use of dervatves and gradents are, n general, not able to locate or dentfy the global optmum. Heurstc algorthms such as genetc algorthms (GA) and evolutonary programmng have been recently proposed for solvng the OPF problem. Unfortunately, recent research has dentfed some defcences n GA performance. Recently, a new evolutonary computaton technque, called Partcle Swarm Optmzaton (PSO), has been proposed and ntroduced. Ths technque combnes socal psychology prncples n soco-cognton human agents and evolutonary computatons. In ths paper, a novel PSO based approach s presented to solve Optmal Power Flow problem. KEYWORDS: Partcle Swarm Optmzaton, Optmal Power Flow, Soft Computng Technques, Evolutonary Computaton Technques. I. INTRODUCTION The fundamental msson of a power system s to provde consumers wth sustaned, relable and costeffcent electrcal energy. In order to acheve ths goal, system operators need to constantly adjust varous controls such as generaton outputs, transformer tap ratos, etc., to assure the contnuous economc and secure system operatons. Ths s a dffcult task that reles hghly on optmal power flow (OPF) functon at power system control centers, the OPF procedure conssts of usng mathematcal methodology to fnd the optmal operaton of a power system under feasblty and securty constrants. It has been consder as basc tool for determnng secure and economc operatng condtons of power systems. The optmal power flow problem can be traced back early as early as 192 s when economc allocaton of generaton was the only concern. The economc operaton of power system was acheved by dvdng loads among avalable generator unts such that ther ncremental generaton costs are equal. Ths was a rather smple problem where only operatng lmts on real power generaton were consdered and the effect of system losses was ether neglected or approxmated by penalty factors calculated from loss formula or load flow Jacoban matrx. As power system became ncreasngly large and complex, the securty became an mportant ssue, whch requres more detaled system models. On the other hand, the evoluton of dgtal computers made such detaled modellng become possble. In 1962, Carpenter for the frst tme establshed the Optmal Power Flow (OPF) problem on a rgorous mathematcal base. He formulated t as a constraned nonlnear programmng problem and derved ts optmalty condtons usng Kuhn-Tucker theorem. In hs formulaton, the OPF problem s expressed n terms of all control and state varables, 116

2 Internatonal Journal of Engneerng Scences & Emergng Technologes, Dec wth both network and securty constrants. The objectve functon can be total generaton cost or transmsson losses, dependng on a specfc applcaton. The frst known Interor Pont (IP) s usually attrbuted to Frsch n 1955, whch s a logarthmc barrer method that was later extensvely studed by Facco and McCormck to solve nonlnearly nequalty constraned problem n 196. In 1979 Khachyan presented an ellpsod method that would solve an LP problem n polynomal tme. The greatest breakthrough n the Interor pont method research feld took place n 1984 by Karmarkar s method. After 1984, several varants of Karmarkar s Interor Pont (IP) method have been proposed and mplemented. 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 make 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 199s. 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]. S. M. Kumar, G. Pryanka and M. Sydulu et al.,[11] comparson of Genetc Algorthms and Partcle Swarm Optmzaton for Optmal Power Flow Includng FACTS devces are descrbed. GA and hybrd partcle swarm optmzaton s used for dstrbuton state estmaton [1]. In [12,13] developed a method for Solvng mult-objectve optmal power flow usng dfferental evoluton algorthm. Optmal Power Flow for Steady State Securty Enhancement usng Enhanced Genetc Algorthm wth FACTS Devces are proposed n [14,15]. The research efforts for loss mnmzaton usng Optmal Power Flow Based on Swarm Intellgences are gven n [16,17]. Modfed dfferental evoluton algorthm for optmal power flow wth non-smooth cost functons[18]. Ths paper focuses on PSO as one of the swarm ntellgence technques. Other evolutonary computaton (EC) technques such as genetc algorthms (GAs), utlze multple searchng ponts n the soluton space lke 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 mxed-nteger nonlnear optmzaton problems (MINLPs) s one of the most dffcult problems n practcal optmzaton. Moreover, unlke 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. II. BASIC PARTICLE 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 behavour 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 toward the destnaton Go to the center of the swarm The behavour of each agent nsde the swarm can be modelled wth smple vectors. The research results are one of the basc backgrounds of PSO. Boyd and Rchardson 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 knds of nformaton n decson process. The frst one s ther own experence; that s, they have tred the choces and know whch state has been better so far, and they know how good t was. The second one s other people s experences,.e., they have knowledge of how the other agents 117

3 Internatonal Journal of Engneerng Scences & Emergng Technologes, Dec around them have performed. Namely, they know 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 background elements of PSO. Accordng to the above background of PSO, Kennedy and Eberhart developed PSO through smulaton of brd flockng n a two-dmensonal 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 flockng optmzes a certan objectve functon. Each agent knows 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 knows the best value so far n the group (gbest) among pbests. Ths nformaton s an analogy of the knowledge 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: k 1 k k k v wv c rand *( pbest s ) c rand *( gbest s ) (1) k where v s velocty of agent at teraton k, w s weghtng functon, c 1 and c 2 are weghtng factors, k rand 1 and rand 2 are random numbers between and 1, s s current poston of agent at teraton k, 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 lke bod. The velocty s usually lmted to a certan mum value. PSO usng eqn. (1) s called the Gbest model. The followng weghtng functon s usually utlzed n eqn. (1): w w (( w w ) / ( ter ))* ter (2) Where mn w s the ntal weght, mn w s the fnal weght, ter s mum teraton number and ter s current teraton number. The meanngs of the rght-hand sde (RHS) of eqn. (1) can be explaned as follows [7]. The RHS of eqn. (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 keep 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 mn w are set to.9 and.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 lke smulated annealng (SA). Namely, a certan velocty, whch gradually gets close to pbests and gbest, can be calculated. PSO usng eqns.(1) & (2) s called nerta weghts approach (IWA). 118

4 Internatonal Journal of Engneerng Scences & Emergng Technologes, Dec Fg.1 Concept of modfcatons of a searchng pont by PSO. k k 1 k k 1 s : current searchng pont s : modfed searchng pont v : current velocty v : modfed velocty v pbest : velocty based on pbest v gbest : velocty based on gbest The current poston (searchng pont n the soluton space) can be modfed by the followng equaton (3): k 1 k k 1 s s v (3) Fgure 1.shows a concept of modfcaton of a searchng pont by PSO, and t shows a searchng concept wth agents n a soluton space. Each agent changes ts current poston usng the ntegraton of vectors. The features of the searchng procedure of PSO can be summarzed as follows: As shown n eqns. (1), (2), and (3), PSO can essentally handle contnuous optmzaton problems. 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 eqn. (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 well-balanced mechansm to utlze dversfcaton and ntensfcaton n the search procedure effcently. The above concept s explaned usng only the x, y axs (two-dmensonal space). However, the method can be easly appled to n-dmensonal problems. Namely, PSO can handle contnuous optmzaton problems wth contnuous state varables n an n-dmensonal 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., w =.9, w mn =.4, The values are also proved to be approprate for power system problems [9, 1]. The basc PSO has been appled to a learnng problem of neural networks and Schaffer [6] a famous benchmark functon for GA, and the effcency of the method has been observed [4]. III. MODELING OF OPTIMAL-POWER FLOW PROBLEM The conventonal formulaton of the optmal-power-flow (OPF) problem determnes the optmal settngs of control varables such as real power generatons, generator termnal voltages, transformer tap settngs and phase-shfter angles whle mnmzng the objectve functon such as fuel cost as gven n eqn.(4). NG 2 Mn (Fuel cost) = mn ( ( a PG b PG C ) (4) 1 Where NG: No. of Generators P G : Actve Power produced by generator. a, b, c : Fuel cost coeffcents of generator. 119

5 Objectve functon (cost) Internatonal Journal of Engneerng Scences & Emergng Technologes, Dec The mnmzaton problem of the objectve functon s subjected to the satsfacton of constrants from eqns. (5-9) () Load-flow constrants: P V N b j 1 V 1,2,..., N Q V NB j 1 1,2,..., N () Voltage constrants: mn V V V j b j ( G pq j, s V ( G j cos j j cos B N b B j j cos ) sn ) j j (5) (6) (7) () Unt constrants mn P P P g g g N g (8) Q mn g Q g Q g N g (9) IV. RESULTS AND DISCUSSIONS Ths secton presents the detals of the study carred out on IEEE-3 bus and IEEE-14 bus test systems for testng the OPF methodology. The proposed algorthm was mplemented n MATLAB computng envronment wth Pentum-IV, 2.66 GHz computer wth 512 MB RAM. the proposed PSO based algorthm was appled to obtan the optmal-control varables n the IEEE 3 & 14 bus systems under base load condtons EP PSO Number of teratons Fg.2 Convergence of generaton cost for 3-bus system 12

6 S h u n Transfor mer tap Generator voltages (p.u) Real power generaton (p.u) Objectve functon (cost) Internatonal Journal of Engneerng Scences & Emergng Technologes, Dec EP PSO Number of teratons Fg.3 Convergence of generaton cost for 14-bus system The upper and lower voltage lmts at all the bus bars except slack were taken as 1.1 and.95 respectvely. The slack bus bar voltage was fxed to a value 1.7p.u. Here the contngences are not consdered and the proposed PSO based algorthm was appled to fnd the optmal schedulng of the power system for the base case loadng condton. The objectve functon n ths case s mnmzaton of total fuel cost. Generator actve-power outputs, generator bus bar termnal voltages, transformer tap settngs and shunt reactve power compensatng elements were taken as optmzaton varables. The optmzaton varables are represented as floatng pont numbers n the populaton. The optmal values of control varables along wth the real power generaton of the slack bus bar generator are gven n Table 1 & 2 for IEEE-3 & 14 bus systems respectvely. The mnmum cost obtaned wth the proposed PSO algorthm for IEEE-3 bus system s $8.966/h, whch s less than the mnmum generaton cost of $ /h obtaned wth nteror pont method. Also, t was found that all the state varables satsfy the lower and upper lmts. For comparson, the OPF problem was solved usng an evolutonary programmng method wth the populaton sze of 2 and 25 generatons. All the solutons satsfy the constrants on reactve power generaton lmts and lne flow lmts. The convergence of generaton cost s shown n Fg.2 & 3 for IEEE 3 & 14 bus systems respectvely. From Fg. 2 & 3, t can be observed that the PSO took approxmately 6 generatons to reach the same producton cost reached by EP. Ths shows that the proposed PSO algorthm occupes less computer space and takes less tme to reach the optmal soluton. Table 1. Soluton for IEEE 3-bus system Control Varables IPM EP PSO P G P G P G P G P G P G6 V G1 V G2 V G3 V G4 V G5 V G6 Tap-1 Tap-2 Tap-3 Tap Q SVC

7 Internatonal Journal of Engneerng Scences & Emergng Technologes, Dec Q SVC2 Q SVC3 Q SVC4 Q SVC5 Q SVC6 Q SVC7 Q SVC8 Q SVC Cost ($/hr) V. CONCLUSIONS Table 2. Soluton for IEEE 14-bus system Control Varables EP PSO P G1 P G2 P G3 P G P G5 V G1 V G2 V G3 V G4 V G5 Tap-1 Tap-2 Tap-3 Q SVC1 Q SVC2 Q SVC3 Q SVC Q SVC Cost ($/hr) In ths paper PSO based OPF algorthm has been valdated wth EP-OPF method usng MATLAB software. It has been observed that optmal soluton obtaned by PSO-OPF s very close to that obtaned by classcal methods and t s clear that t s better than EP-OPF. So the proposed OPF methods are most sutable and vald for ncorporatng new objectve functons and constrants. The algorthm s capable of determnng the global optmum soluton to the OPF problem n the presence of multple local optma. Ths provdes the opportunty to better model power system operatons and therefore determne a more accurate operatng state. The performance of the developed OPF algorthms has been demonstrated by ts applcaton to the modfed IEEE 3-bus and 14-bus test systems. The algorthms were accurately and relably converged to the global optmum soluton n each case. The PSO-algorthm s also capable of producng more favourable voltage profle whle stll mantanng a compettve cost. REFERENCES [1] Reynolds C. Flocks, herds, and schools: A dstrbuted behavoural model. Computer Graphcs 1987; 21(4): [2] Boyd R, Rcherson PJ. Culture and the evolutonary process. Chcago: Unversty of Chcago Press; [3] Colorn A, Dorgo M, Manezzo V. Dstrbuted optmzaton by ant colones. Proceedngs of Frst European Conference on Artfcal Lfe. Cambrdge, MA: MIT Press; pp [4] Kennedy J, Eberhart R. Partcle swarm optmzaton. Proceedngs of IEEE Internatonal Conference on Neural Networks (ICNN 95) Perth, Australa: IEEE Press; Vol. IV.pp

8 Internatonal Journal of Engneerng Scences & Emergng Technologes, Dec [5] Bonabeau E, Dorgo M, Theraulaz G. Swarm ntellgence: From natural to artfcal systems. Oxford: Oxford Unversty Press; [6] Kennedy J, Eberhart R. Swarm ntellgence. San Mateo, CA: Morgan Kaufmann; 21. [7] Sh Y, Eberhart R. A modfed partcle swarm optmzer. Proceedngs of IEEE Internatonal Conference on Evolutonary Computaton (ICEC 98). Anchorage: IEEE Press; pp [8] Sh Y, Eberhart R. Parameter selecton n partcle swarm optmzaton. Proceedngs of the 1998 Annual Conference on Evolutonary Programmng. San Dego: MIT Press; [9] Fukuyama Y. et al. A partcle swarm optmzaton for reactve power and voltage control consderng voltage securty assessment. IEEE Trans Power Systems 2; 15(4). pp [1] Naka S, Genj T, Yura T, Fukuyama Y. A hybrd partcle swarm optmzaton for dstrbuton state estmaton. IEEE Trans Power Systems 23; 18(1),pp [11] S. M. Kumar, G. Pryanka and M. Sydulu,Comparson of Genetc Algorthms and Partcle Swarm Optmzaton for Optmal Power Flow Includng FACTS devces," IEEE Power Tech27, 1-5 July 27, pp [12] M. Varadarajan, K. S. Swarup, Solvng mult-objectve optmal power flow usng dfferental evoluton," Generaton, Transmsson & Dstrbuton, IET, Vol 2, 28, pp [13] S. B. Warkad, M. K. Khedkar and G. M. Dhole, "A Genetc Algorthm Approach for Solvng AC-DC Optmal Power Flow Problem", Journal of Theoretcal and Appled Informaton Technology, Vol. 6, No , pp [14] Mat Syan and Ad Soeprjanto, "Neural Network Optmal Power Flow (NN-OPF) based on IPSO wth Developed Load Cluster Method", World Academy of Scence, Engneerng and Technology,Vol. 72, 21, pp [15] D. Devaraj and R. Narmatha Banu, "Optmal Power Flow for Steady State Securty Enhancement usng Enhanced Genetc Algorthm wth FACTS Devces", Asan Power Electroncs Journal, Vol. 4, No. 3, Dec 21, pp [16] Vjayakumar Krshnasamy, "Genetc Algorthm for Solvng Optmal Power Flow Problem wth UPFC", Internatonal Journal of Software Engneerng and Its Applcatons, Vol. 5 No. 1, Jan 211, pp [17] Numphetch Snsuphun, Uthen Leeton, Umaporn Kwannetr, Dust Uthtsunthorn, and Thanatcha Kulworawanchpong, "Loss Mnmzaton usng Optmal Power Flow Based on Swarm Intellgences", ECTI Transactons on Electrcal Eng., Electroncs, and Communcatons, Vol. 9, No. 1, Feb 211,pp [18] Samr Sayah, Khaled Zehar Modfed dfferental evoluton algorthm for optmal power flow wth non-smooth cost functons Energy Converson and Management, Vol. 49,28, pp Authors M.Lakshmkantha Reddy has receved hs B.Tech degree n electrcal engneerng from J N T U Hyderabad Unversty at Sree Vdyankethan engneerng college Trupat and M.Tech degree n power systems from Natonal Insttute Of Technology Warangal A.P., n the year21and25 respectvely. He has 1 years of teachng experence. Presently, he s a research scholar n the Department of Electrcal Engneerng, S.V. Unversty, Trupat, A.P., Inda. Hs research areas are power system operaton and power system optmzatons. M. Ramprasad Reddy has receved hs B.Tech degree n electrcal engneerng from J N T U Hyderabad Unversty at RGMCET, Nandyal and M.Tech degree n electrcal power systems from S.V.Unversty, Trupat A.P., n the year 2 and 25 respectvely. He has 11 years of teachng experence. Presently, he s a research scholar n the Department of Electrcal Engneerng, S.V.Unversty, Trupat, A.P., Inda. Hs research s n the area power system operaton and control. 123

9 Internatonal Journal of Engneerng Scences & Emergng Technologes, Dec V. C. Veera Reddy has obtaned hs B.Tech and M.Tech degrees n the year1979 and 1981 respectvely. He has done hs research n the area of power Systems. He has 3 years of teachng and research experence. He s presently a Professor at the Dept of Electrcal Engneerng, S.V.Unversty, Trupat, A.P., Inda. He has guded 4 Ph.D. canddates. He s havng 34 research publcatons n Natonal and Internatonal Conferences and Journals. Hs research areas are FACTS and Sold state Drves. 124