Application of Gravitational Search Algorithm for Optimal Reactive Power Dispatch Problem

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1 Applcaton of Gravtatonal Search Algorthm for Optmal Reactve Power Dspatch Problem Serhat Duman Department of Electrcal Eucaton, Techncal Eucaton Faculty, Duzce Unversty, Duzce, 8620 TURKEY Yusuf Sonmez Department of Electrcal Technology, Gaz Vocatonal Collage, Gaz Unversty, Ankara, TURKEY Ugur Guvenc Department of Electrcal an Electronc Engneerng, Faculty of Technology, Duzce Unversty, Duzce, 8620 TURKEY Nuran Yorukeren Department of Electrcal Engneerng, Engneerng Faculty, Kocael Unversty, Izmt, 4040 TURKEY Abstract In ths paper, Gravtatonal Search Algorthm (GSA) s apple to solve the optmal reactve power spatch (ORPD) problem. The ORPD problem s formulate as a nonlnear constrane sngle-obectve optmzaton problem where the real power loss an the bus voltage evatons are to be mnmze separately. In orer to evaluate the propose algorthm, t has been teste on IEEE 30 bus system consstng 6 generator an compare other algorthms reporte those before n lterature. Results show that GSA s more effcent than others for soluton of sngle-obectve ORPD problem. Keywors-gravtatonal search algorthm, optmal reactve power spatch, power systems, optmzaton I. INTRODUCTION In recent years the optmal reactve power spatch (ORPD) problem has receve great attenton as a result of the mprovement on economy an securty of power system operaton. Solutons of ORPD problem am to mnmze obect functons such as fuel cost, power system loses, etc. whle satsfyng a number of constrants lke lmts of bus voltages, tp settngs of transformers, reactve an actve power of power resources an transmsson lnes an a number of controllable varables [, 2]. In the lterature, many methos for solvng the ORPD problem have been one up to now. At the begnnng, several classcal methos such as graent base [3], nteror pont [4], lnear programmng [5] an quaratc programmng [6] have been successfully use n orer to solve the ORPD problem. However, these methos have some savantages n the process of solvng the complex ORPD problem. Drawbacks of these algorthms can be eclare nsecure convergence propertes, long executon tme, an algorthmc complexty. Beses, the soluton can be trappe n local mnma [,7]. In orer to overcome these savantages, researches have successfully apple evolutonary an heurstc algorthms such as Genetc Algorthm (GA) [2], Dfferental Evoluton (DE) [8] an Partcle Swarm Optmzaton (PSO) [9]. It s reporte n those that evolutonary or heurstc algorthms are more effcent than classcal algorthms for solvng the RPD problem. Gravtatonal Search Algorthm (GSA) s a new metaheurstc an populaton base search algorthm base on Newton s law of gravty an law of moton an t propose frstly by Rashe et al. n 2009 [0]. GSA has many avantages whch are reporte n [0] such as, aaptve learnng rate, memory-less algorthm an, goo an fast convergence. Beses, n [0] authors have compare the GSA wth PSO, Central Force Optmzaton (CFO) an Real Genetc Algorthm (RGA) usng 23 fferent benchmark functons an they have reporte that GSA s more powerful than other algorthms. Due to all of these avantages, Duman et al. presente gravtatonal search algorthm to solve the economc spatch wth valve pont effects for fferent test systems [4]. In ths paper, GSA s apple for solvng the ORPD problem. In the process of solvng, ORPD problem s formulate as a nonlnear constrane sngle-obectve optmzaton problem where the real power loss an the bus voltage evatons are to be mnmze respectvely. Smulatons have been one usng MATLAB program. The propose algorthm s teste on IEEE 30-bus system for evoluton of effectveness of t. Results obtane from GSA are compare results reporte those n []. Results show that propose algorthm s more effectve an powerful than other algorthms n soluton of ORPD problem. II. FORMULATION OF ORPD PROBLEM The obectve of the ORPD problem s to mnmze one or more obectve functons whle satsfyng a number of constrants such as loa flow, generator bus voltages, loa bus voltages, swtchable reactve power compensatons, reactve power generaton, transformer tap settng an transmsson lne flow. In ths paper two obectve functons are mnmze separately as sngle obectve. Obectve functons mnmze 59

2 n ths paper an constrants are formulate takng from [, ] an shown as follows. A. Mnmzaton of Real Power Loss It s ame n ths obectve that mnmzng of the real power loss (P loss ) n transmsson lnes of a power system. Ths s mathematcally state as follows. Loa bus voltage (V L ) nequalty constrant: V mn L V V, nl (6) L L Swtchable reactve power compensatons ( C ) nequalty constrant: Mnmze P loss = n k (, ) 2 2 g ( V + V 2V V cosθ ) () mn C, nc (7) C C Reactve power generaton ( G ) nequalty constrant: Where n s the number of transmsson lnes, g k s the conuctance of branch k, V an V are voltage magntue at bus an bus, an θ s the voltage angle fference between bus an bus. B. Mnmzaton of Voltage Devaton It s ame n ths obectve that mnmzng of the evatons n voltage magntues (VD) at loa buses. Ths s mathematcally state as follows. mn G, ng (8) G G Transformers tap settng (T ) nequalty constrant: T mn T T, nt (9) Transmsson lne flow (S L ) nequalty constrant: Mnmze VD = nl V k.0 (2) where nl s the number of loa busses an V k s the voltage magntue at bus k. C. System Constrants In the mnmzaton process of obectve functons, some problem constrants whch one s equalty an others are nequalty ha to be met. Obectve functons are subecte to these constrants shown below. Loa flow equalty constrants: P G G P D D V V nb = nb = G cosθ V = 0 + B snθ, =,2,,nb (3) G sn θ V = 0, =,2,,nb (4) + B cos θ where, nb s the number of buses, P G an G are the real an reactve power of the generator, P D an D are the real an reactve loa of the generator, an G an B are the mutual conuctance an susceptance between bus an bus. Generator bus voltage (V G ) nequalty constrant: V mn G V V, ng (5) G G SL S L, nl (0) where, nc, ng an nt are numbers of the swtchable reactve power sources, generators an transformers. Durng the smulaton process, all constrants satsfe as explane below []. The loa flow equalty constrants are satsfe by power flow algorthm. The generator bus voltage (V G ), the transformer tap settng (T ) an the Swtchable reactve power compensatons ( C ) are optmzaton varables an they are self-restrcte between the mnmum an mum value by the GSA algorthm. The lmts on actve power generaton at the slack bus (P Gs ), loa bus voltages (V L ) an reactve power generaton ( G ), transmsson lne flow (S L ) are state varables. They are restrcte by ang a penalty functon to the obectve functons. III. GRAVITATIONAL SEARCH ALGORITHM GSA s the new meta-heurstc optmzaton algorthm motvate by the Newton s laws of gravty an moton. GSA was frstly prouce by Rashe et al. n 2009 [GSA]. Accorng to ths algorthm, agents are consere as obects an ther performance s measure by ther masses. Every obect attracts every other obect wth gravtatonal force. GSA algorthm can be explane followng steps [0, 2, 3]. A. Step : Intalzaton When t s assume that there s a system wth N (menson of the search space) masses, poston of the th mass 520

3 s escrbe as follows. At frst, the postons of masses are fxe ranomly. 2 n X = ( x, x,... x ),,... N where, x = () s the poston of the th mass n th menson. B. Step 2: Ftness Evaluaton of All Agents In ths step, for all agents, best an worst ftness are compute at each epoch escrbe as follows. best( = mn ft (2) {,.., N} worst( = ft (3) {,.., N} where ft ( s the ftness of the th agent of t tme, best( an worst( are best (mnmum) an worst (mum) ftness of all agents. C. Step 3:Compute the Gravtatonal Constant (G() In ths step, the gravtatonal constant at t tme (G() s compute as follows. G G0 t exp( α ) T = (4) where G 0 s the ntal value of the gravtatonal constant chosen ranomly, α s a constant, t s the current epoch an T s the total teraton number. D. Step 4:Upate the Gravtatonal an Inertal Masses In ths step, the gravtatonal an nertal masses are upate as follows. mg ft worst( best( worst( = (5) where ft ( s the ftness of the th agent of t tme. Mg mg = N (6) = mg where Mg ( s the mass of the th agent of t tme. E. Step 5:Calculate the Total Force In ths step, the total force actng on the th agent (F () s calculate as follows. F = ran F ( (7) kbest where ran s a ranom number between nterval [0,] an kbest s the set of frst K agents wth the best ftness value an bggest mass. The force actng on the th mass (M () from the th mass (M () at the specfc t tme s escrbe accorng to the gravtatonal theory as follows. F M M = G( ( x x ) R + ε (8) where R ( s the Euclan stance between th an th agents ( ) an ε s the small constant. X, X 2 F. Step 6:Calculate the Acceleraton an Velocty In ths step, the acceleraton (a () an velocty (v () of the th agent at t tme n th menson are calculate through law of gravty an law of moton as follows. v a F Mg = (9) ( t + ) = ran v a (20) + where ran s the ranom number between nterval [0,]. G. Step 7:Upate the Poston of the Agents In ths step the next poston of the th agents n th (x (t+)) menson are upate as follows. x ( t + ) = x + v ( t + ) (2) H. Step 8:Repeat In ths step, steps from 2 to 7 are repeate untl the teratons reach the crtera. In the fnal teraton, the algorthm returns the value of postons of the corresponng agent at specfe mensons. Ths value s the global soluton of the optmzaton problem also. All these steps explane above escrbes how the GSA works. Beses, the prncple agram of the GSA s llustrate n Fg.. IV. SIMULATION RESULTS Propose approach has been apple to solve ORPD problem. In orer to emonstrate the effcency an robustness of propose GSA approach base on Newtonan physcal law 52

4 of gravty an law of moton whch s teste on stanar IEEE 30-bus test system shown n Fg. 2 an the system ata are gven n [5]. Fgure. The prncple agram of the GSA [0]. The test system has sx generators at the buses, 2, 5, 8, an 3 an four transformers wth off-nomnal tap rato at lnes 6-9, 6-0, 4-2, an an, hence, the number of the optmze control varables s 0 n ths problem. 2, 5, 8,, an 3, an.05 pu for the remanng buses nclung the reference bus. The mnmum an mum lmts of the transformers tappng are 0.9 an. pu respectvely []. The propose approach has been apple to solve ORPD problem for fferent obectve functons. G s set usng n Eq. (4), where G 0 s set to 00 an α s set to 0, an T s the total number of teratons. Maxmum teraton numbers are 200 for all case stues. The optmum control parameter settngs of propose approach are gven n Table. The best power loss an best voltage evatons obtane from propose approach are MW an respectvely. GSA s less by %, % compare to prevously report best results 5.67 MW, respectvely. Fg. 3 an Fg. 4 show the convergence of GSA for mnmum power loss an voltage evatons solutons respectvely. The results obtane from propose algorthm have been compare other methos n the lterature. The results of ths comparson are gven n Table 2 an Table 3. The results n Tables 2 an 3 show that the reactve spatch an voltage evatons solutons specfe by the propose GSA approach lea to lower actve power loss an voltage evatons than that by the ref. [] smulaton results, whch confrms that the propose approach s well capable of specfcaton the optmum solutons. TABLE I. BEST CONTROL VARIABLES SETTINGS FOR DIFFERENT TEST CASES OF PROPOSED APPROACH Control varables settngs Case : Power Loss Case 2: Voltage evatons V G V G V G V G V G V G T T T T Power loss (MW) Voltage evatons TABLE II. COPMARISON OF THE SIMULATION RESULTS FOR POWER LOSS Fgure 2. Sngle lne agram of IEEE 30-bus test system The mnmum voltage magntue lmts at all buses are 0.95 pu an the mum lmts are. pu for generator buses Control varables settngs GSA Invual optmzaton [] Multobectve EA [] As sngle obectve [] V G V G V G V G V G V G T T T T Power loss (MW) Voltage evatons (p.u.)

5 V. CONCULUSION In ths paper, one of the recently evelope stochastc algorthms s the gravtatonal search algorthm has been emonstrate an apple to solve optmal reactve power spatch problem. The problem has been formulate as a constrane optmzaton problem. Dfferent obectve functons have been consere to mnmze real power loss, to enhance the voltage profle. The propose approach s apple to optmal reactve power spatch problem on the IEEE 30- bus power system. The smulaton results ncate the effectveness an robustness of the propose algorthm to solve optmal reactve power spatch problem n test system. The GSA approach can reveal hgher qualty soluton for the fferent obectve functons n ths paper. TABLE III. Control varables settngs Fgure 3. Convergence of GSA for power loss COPMARISON OF THE SIMULATION RESULTS FOR VOLTAGE DEVIATIONS GSA Invual optmzaton [] Multobectve EA[] As sngle obectve [] V G V G V G V G V G V G T T T T Power loss (MW) Voltage evatons (p.u.) Fgure 4. Convergence of GSA for voltage evatons REFERENCES [] M. A. Abo, J. M. Bakhashwan, A novel multobectve evolutonary algorthm for optmal reactve power spatch problem, n proc. Electroncs, Crcuts an Systems conf., vol. 3, pp , [2] W. N. W. Abullah, H. Sabon, A. A. M. Zan, K. L. Lo, Genetc Algorthm for Optmal Reactve Power Dspatch, n proc. Energy Management an Power Delvery conf., vol., pp , 998. [3] K. Y. Lee, Y. M. Park, J. L. Ortz, Fuel-cost mnmsaton for both realan reactve-power spatches, n proc. Generaton, Transmsson an Dstrbuton conf., vol. 3, pp , 984. [4] S. Granvlle, Optmal Reactve Dspatch Trough Interor Pont Methos, IEEE Trans. on Power Systems, vol. 9, pp , 994. [5] N. I. Deeb, S. M. Shahehpour, An Effcent Technque for Reactve Power Dspatch Usng a Revse Lnear Programmng Approach, Electrc Power System Research, vol. 5, pp. 2-34, 988. [6] N. Grunn, Reactve Power Optmzaton Usng Successve uaratc Programmng Metho, IEEE Trans. on Power Systems, vol. 3, pp , 998. [7] M. A. Abo, Optmal Power Flow Usng Partcle Swarm Optmzaton, Electrcal Power an Energy Systems, vol. 24, pp , [8] A. A. Abou El Ela, M. A. Abo, S. R. Spea, Dfferental Evoluton Algorthm for Optmal Reactve Power Dspatch, Electrc Power Systems Research, vol. 8, pp , 20. [9] V. Mrana, N. Fonseca, EPSO-Evolutonary Partcle Swarm Optmzaton, A New Algorthm wth Applcatons n Power Systems, n Proc. of Transmsson an Dstrbuton conf., vol. 2, pp , [0] E. Rashe, H. Nezamaba-pour, S. Saryaz, GSA: A gravtatonal search algorthm, Informaton Scences, vol. 79, pp , [] S. Durara, P. S. Kannan, D. Devara, Applcaton of Genetc Algorthm to Optmal Reactve Power Dspatch Inclung Voltage Stablty Constrant, Journal of Energy & Envronment, vol. 4, pp , [2] E. Rashe, H. Nezamaba-pour, S. Saryaz, Flter moelng usng gravtatonal search algorthm, Engneerng Applcatons of Artfcal Intellgence, vol. 24, pp.7-22, 20. [3] A. Chatteree, G. K. Mahant, Comperatve Performance of Gravtatonal Search Algorthm an Mofe Partcle Swarm Optmzaton Algorthm for Synthess of Thnne Scanne Concentrc Rng Array Antenna, Progress In Electromagnetcs Research B, vol. 25, pp , 200. [4] S S. Duman, U. Güvenç, N. Yörükeren, Gravtatonal Search Algorthm for Economc Dspatch wth Valve-pont Effects, Internatonal Revew of Electrcal Engneerng, vol. 5, no. 6, pp , 200. [5] O. Alsac, B. Stott, Optmal loa flow wth steay-state securty, IEEE Transactons on Power Apparatus an Systems, PAS-93, no. 3, pp ,