Optimal Reactive Power Dispatch Using Efficient Particle Swarm Optimization Algorithm

Transcription

1 Optmal Reactve Power Dspatch Usng Effcent Partcle Swarm Optmzaton Algorthm MESSAOUDI Abdelmoumene *, BELKACEMI Mohamed ** *Electrcal engneerng Department, Delfa Unversty, Algera ** Electrcal engneerng Department, Batna Unversty, Algera Emal : messaoud3@yahoo.fr Emal : belacem_m@hotmal.com Abstract-hs paper presents an effcent partcle swarm optmzaton (EPSO algorthm for the soluton of the optmal reactve power flow (ORPF. he obectve s to mnmze the total actve power loss wth optmal settng of control varables wthout volatng nequalty constrants and satsfyng equalty constrant. Control varables are both contnuous and dscrete. he contnuous control varables are generator bus voltage magntudes, whle the dscrete varables are transformer tap settngs and reactve power of shunt compensators. he PSO algorthm soluton has been tested on the standard IEEE 30-Bus test system wth both contnuous and dscrete control varable. he results have been compared to genetc algorthm method. Key words- actve power Loss mnmzaton, partcle swarm Optmzaton (PSO, load flow (Lf I-Introducton he man obectve of optmal reactve power dspatch (ORPD of electrc power system s to mnmze an actve power loss va the optmal adustment of the power system control varables, whle at the same tme satsfyng varous equalty and nequalty constrants. he equalty constrants are the power flow balance equatons, whle the nequalty constrants are the lmts on the control varables and the operatng lmts of the power system dependent varables. he problem control varables nclude the generator bus voltages, the transformer tap settngs, and the reactve power of shunt compensator, whle the problem dependent varables nclude the load bus voltages, the generator reactve powers, and the power lne flows. Generally, the ORPD problem s a largescale hghly constraned nonlnear non convex and multmodal optmzaton problem. o solve the ORPD problems, the optmzaton methods are classfed nto classcal and heurstc optmzaton methods. Classcal optmzaton methods, such as gradent based optmzaton algorthm [,], quadratc programmng, nteror pont method [3], non lnear programmng[5]. Recently, due to the basc effcency of nteror pont method, wtch offer fast convergence and convenence n handlng nequalty constrants n comparson wth other methods, nteror pont method has been wdely used to solve the ORPD problem of large scale power systems. It converts the nequalty constrants to equaltes by the ntroducton of nonnegatve slac varables. A logarthmc barrer functon of the slac varables s then added to the obectve functon and multpled by a barrer parameter, whch s gradually reduced to zero durng the soluton process. Lnear programmng methods are fast and relable but ther man dsadvantage s assocated wth the pecewse lnear approxmaton. Nonlnear programmng method s nown to suffer from the complex algorthms. Most of these methods are based on the combnaton of the obectve functon and the constrants by Lagrange formulaton, Kuhn ucer condton, and applyng sensty analyss and gradent-based optmzaton algorthm [4]. Heurstc methods such as genetc algorthm (GA, evolutonary programmng algorthm, and partcle swarm optmzaton (PSO have been recently proposed for solvng the ORPD problem. hese algorthms have recently found extensve applcatons n solvng global optmzaton searchng problems, when the closed-form optmzaton technque cannot be appled. GA s parallel and global search technque emulatng natural genetc operators such as, selecton, crossover and mutaton. A GA based optmzaton method s more lely to converge toward the global soluton because t, smultaneously, evaluates many ponts n the parameter space. It does not need to assume that the search space s dfferentable or contnuous. he PSO algorthm s also a global search method whch explores search space to get to the global optmum. he PSO s a stochastc, populatonbased computer algorthm modeled on swarm ntellgence. PSO fnds the global mnmum of a

2 multdmensonal, multmodal functon wth best optmum. In reference [8], a standard genetc algorthm method has been proposed to mnmze the total actve power loss. In the present paper an effcent PSO algorthm method s used to mprove the qualty of soluton, leadng to the near global optmum, and gets the best soluton wth both contnuous and dscrete control varables. he contnuous control varables are generator bus voltage magntudes, whle the dscrete varables are transformer tap settngs and reactve power of shunt compensators. hs method has been tested on the IEEE 30-bus standard system wth two cases, one problem contans two AR shunt compensators; the other contans three AR shunt compensators. he results are compared wth the standard GA. II-Problem Formulaton he OPF problem s consdered as a general mnmzaton problem wth constrants, and can be wrtten n the followng form: Mnmze f ( x, u ( subect to g ( x, u = 0 ( and: h ( x, u 0 (3 Where f ( x, u s the obectve functon, g( x, u and h ( x, u are respectvely the set of equalty and nequalty constrants. x s the vector of state varables, and u s the vector of control varables. he state varables are the load buses voltages, angles, the generator reactve powers and the slac actve generator power: x = ( Pg, θ,.. θ N, L,.. LNL, Qg,.. Qgng (4 he control varables are the generator bus voltages, the shunt capactors/reactors and the transformers tap-settngs: U (,, = g Q c (5 or : U =,...,,..., Q,.. Q (6 ( g gng Nt c cnc Where Ng, N, NC are the number of generators, number of tap transformers and the number of shunt compensators respectvely. A-Obectve Functon he obectve of the reactve power dspatch s to mnmze the actve power loss n the transmsson networ, whch can be descrbed as follows: F = g ( or F Nbr + cosθ Ng = Pg Pd = Pgslac + Pg Ng slac P d (7 (8 Where g s the conductance of branch between nodes and, Nbr s the number of transmsson lnes. P d s the total actve power demand, P g s s the generator actve power of unt, and P gsalc s the generator actve power of slac bus. B-Equalty Constrant he equalty constrant g(x,u of the ORPD problem s represented by the power balance equaton, where the total power generaton must cover the total power demand and the power losses: P = P + P (9 G D L hs equaton s solved by runnng Newton Raphson load flow method, by calculatng the actve power of slac bus to determne actve power loss. C-Inequalty Constrants he nequalty constrants h(x,u reflect the lmts on physcal devces n the power system as well as the lmts created to ensure system securty: Upper and lower bounds on the actve power of slac bus, and reactve power of generators: P Q mn gslac P P (0 gslac gslac Q Q, N ( g g mn g g

3 Upper and lower bounds on the bus voltage magntudes: mn N ( Upper and lower bounds on the transformers tap ratos: mn N (3 Upper and lower bounds on the compensators reactve powers: Qc mn Qc Qc N c (4 Where N s the number of buses, N s the number of ransformers, N c s the number of shunt reactve compensators, In PSO search algorthm all control varables stand n there lmts except actve power n slac bus By addng the nequalty constrants to the obectve functon, the augmented ftness functon to be mnmzed becomes: F = F + λ ( + λ S NL : = lm lm ( P P gsalac gslac (5 Where λ and λ s are the penalty factors, and both penalty factors are large postve constants; NL s a number of load buses (PQ buses. F s the total actve power loss gven by (8. and lm P are defned as: lm gslac mn mn < lm f = (6 f > mn mn P < lm gslac f Pgslac Pgslac P gslac = (7 Pgslac f Pgslac > Pgslac he equalty constrant and generators reactve power nequalty constrants are handlng by Newton Raphson load flow calculaton method. Generally, for the reactve optmzaton problem, the range of varatons of P gslac s small then we set λ = 0. s III- Overvew of PSO Partcle Swarm Optmzaton was ntroduced by R, Eberhart and J, Kennedy n 995 [0], nspred by socal behavor of brd flocng or fsh schoolng. It s a part of modern heurstc optmzaton algorthm, t wor on populaton or group n wtch ndvduals called partcles move to reach the optmal soluton n the multdmensonal search space. It wors wth drect real - valued numbers, whch elmnates the need to do bnary converson of a classcal canoncal genetc algorthm. he number of partcles n the group s Np. he ntal populaton of a PSO algorthm s randomly generated wthn the control varables bounds. Each partcle adusts ts poston through ts present velocty, prevous postons and the postons of ts neghbors. In d dmensonal search space the poston and velocty of partcle are represented as the vectors: X = (x,...,xd and t = (,...d respectvely where Np, and d s the number of elements n the partcle. It represents n general the number of control varables n the obectve functon. best best Let Xpbest = (x,...xd the best prevous poston gbest gbest of partcle, and Xgbest = (x,...xd the best partcle among all the partcles n the swarm. he updated velocty of partcle s modfed under the followng equaton: + = ω + c rand + c rand (Xpbest (Xgbest K X X where K t : velocty of partcle at teraton ; ω : nerta weght factor ; c,c : acceleraton constant; : current teraton; rand, rand : random numbers between 0 and ; Xpbest :best poston of partcle untl teraton ; K (8 Xgbest : best poston of the swarm untl teraton ; Each partcle changes ts current poston to the new poston by addng the modfed velocty (8 usng the followng equaton: X = X + t, =,,..Np (9 + K + In general nerta weght factor ω decreases lnearly from ω to ωmn accordng to the followng equaton: ω ωmn ω = ω (0

4 where s the mum number of teratons. I- mplementaton of the ORPD PSO algorthm I- Intalzaton Intal value of each partcle s generated randomly between [ u mn,u ] X = (x,,...,x, then x 0 mn, = random(u,,u,. Also ntals values of velocty of each partcle s generated randomly between[ t mn,t ] 0 mn = random(,,,, mn, = t, = (u, u /Nv Where Nv s an nteger value, representng the number of ntervals. mn Where =,..N P, =,..d and u,u are mum and mnmum values of control varables respectvely. I- Algorthm OF ORPD-EPSO Step : gve the PSO parameters Np; ω mn, ω,, c,c, d=dmenson of vector of control arables U, set =. Step : Intalze at random Np partcles wthn ther lmts. Step 3: Calculate ftness functon of each ntal 0 partcle X usng obectve functon F (5. 0 Step 4: set Xpbest = X as a prevous X and Xgbest to the best partcle have the best ftness among all partcles Xpbest Step 5: set teraton K=; Step 6: update velocty of each partcle usng equaton (8, If < then =, or f, >, then mn,, mn, =, mn,, Step 7: adusts the poston of each partcle usng equaton (9 f the element of vector of partcle X exceeds ts lmts, enforce t wthn ts lmts. Step 8: calculate new ftness functon of each partcles X usng obectve functon F (5. Step 9: f the ftness value of each partcle s better than prevous Xpbest, the current s set to be Xpbest f the best partcle of all Xpbest s better than Xgbest, the current s set to Xgbest. Step 0: f < K set K=K+ and go to step 5, otherwse go to step. Step : tae U best = Xgbest and runnng load flow to calculate real slac power, actve power loss, and other elements of state varables. o calculate the ftness functon of each partcle X set the vector of control varables U = X, and runnng load flow to calculate real slac power, actve power loss usng(8, and ftness functon usng (5. I-3- Handlng of dscrete arables he dscrete control varables are adustng by 0.0 step sze. hen each transformer tap settng s rounded to ts nearest decmal nteger value of 0.0, by utlzng the roundng operator. he same prncple apples to the dscrete reactve power necton of shunt compensators. he roundng operator s only performed n evaluatng the ftness functon. - Numercal Results he PSO algorthm has been tested on the IEEE 30- bus, 4 branch system [6]. It has a total of 3 control varables as follows: 6 generator-bus voltage magntudes, 4 transformer-tap settngs, and 3 bus shunt reactve compensators. he consdered securty constrants are the voltage magntudes of all buses, the reactve power lmts of the shunt AR compensators and the transformers tap settngs lmts. he varables lmts are lsted n able. he transformer taps and the reactve power source nstallaton are dscrete wth the changes step of 0.0. g mn g ABLE ARIABLES LIMIS IN PU L mn L mn SHUN AR COMPENSAOR LIMIS IN PU Q c mn Q c he power lmts generators buses are represented n able. Generators buses are: P buses,5,8,,3 and slac bus s.the others are PQ-buses. he total power demand s 83.4 Mw.

5 he PSO populaton sze s taen equal to 30. he mum number of generatons s 00, acceleraton factors C =C =, mum and, mnmum nerta factors are ω =0.9,ω mn =0., the penalty factor n (5 s chosen as λ v = 500. he complete algorthm has been mplemented n Delph orented obect programmng. 0 runs have been performed for two cases of AR shunt compensators, and the results whch follow are the best soluton of these 0 runs. ABLE Generators power Lmts n Mw and Mvar Bus N 0 P g P gmn P g Q gmn Q g Case: wo AR shunt compensators In ths case reactve power of shunt compensator at bus 3 s equal to zero. he optmal settngs of the control varables are gven n table 3 case of PSO. he total actve power loss was ntally 5.8 Mw and, t has been reduced by the proposed PSO to Mw. hs soluton s mproved than the optmal actve power loss obtaned by the other heurstc method reported n the lterature wth both contnuous and dscrete control varables such as standard genetc algorthm SGA[8] wth 4.98 Mw. he real power of slac bus s Mw. Case: hree AR shunt compensators In he case, reactve power of shunt compensator at bus 3 s consdered, the optmal settngs of the control varables are gven n table 3 case of PSO. he total actve power loss has been reduced by the proposed PSO to Mw. hs soluton s mproved than the optmal actve power loss of case. he real power of slac bus s Mw. It s clear that all control varables are wthn ther boundary lmts. System of voltage profle of case and case for all buses s shown n Fgure. All voltage magntudes of buses are wthn ther lmts, n PQ buses (load buses voltage magntude does not exceed ts lmts (.05 and 0.95 except voltage magntude at bus 3 of case s pu. It s reduced n case to.04997pu. he proposed approach succeeds n eepng the dependent varables wthn ther lmts. By addng the AR shunt compensator at bus 3, actve power loss s reduced. able 3 summarzes the results of the optmal settngs of control varables obtaned by PSO and GA methods. hese results show that the optmal dspatch soluton determned by the PSO lead to lower actve power loss than that found by GA method; wtch confrms that PSO s well capable of determnng the global or near global optmum dspatch soluton. oltage Magntude n pu arables n p.u , 6,9 6,0 8,7 Q 3 Q 0 Q case case N bus Fgure: oltage Profle Dagram ABLE 3 ALUES OF CONROL ARIABLES AFER OPIMIZAION AND ACIE LOSS Intal PSO case PSO case SGA Loss (Mw

6 Generator reactve powers are gven n table 4. hese values are wthn ther lmts. ABLE 4 Generator Reactve Powers IN Mvar Case Case Q g Q g Q g Q g Q g Q g I- Concluson In ths paper, a PSO soluton to the OPF problem has been presented for determnaton of the global or near-global optmum soluton for optmal reactve power dspatch. he man advantages of the PSO to the ORPD problem are optmzaton of convex or non-convex obectve functon, real coded of both contnuous and dscrete control varables, and easly handlng nonlnear constrants. he proposed algorthm has been tested on the IEEE 30-bus system to mnmze the actve power loss. he optmal settng of control varables are obtaned n both contnuous and dscrete value. he results were compared wth the other heurstc method such as SGA algorthm reported n the lterature and demonstrated ts effectveness and robustness. II. References [] Lee K, Par Y, Ortz J. A unted approach to optmal real and reactve power dspatch IEEE rans Pwr Appar Syst 985; 04(5: [] Y. Y.Hong, D.I. Sun, S. Y. Ln and C. J.Ln Multyear Mult-case optmal AR plannng IEEE rans. Power Syst.,vol.5, no.4, pp.94-30,nov.990. [3] J. A. Momoh, S. X. Guo, E.C. Ogbuobr, and R. Adapa, he quadratc nteror pont method solvng power system optmzaton problems IEEE rans. Power Syst. vol. 9, no. 3, pp ,Aug.994. [4] H.W.Dommel, W.F.nney, Optmal Power Flow Solutons, IEEE, rans. On power Apparatus and Systems, OL. PAS-87,octobre 968,pp [5] Y.C.Wu, A. S. Debs, and R.E. Marsten A Drect nonlnear predctor-corrector prmal-dual nteror pont algorthm for optmal power flows IEEE ransacyons on power systems ol. 9, no., pp , may 994. [6] M.A. Abdo. Optmal Power Flow Usng Partcle Swarm Optmzaton, Elec.Power Energy Syst.,4 (00 pp [7] L.L.La,J..Ma, R. Yooma, M. Zhao Improved genetc algorthms for optmal power flow under both normal and contngent operaton states, Electrcal Power & Energy System, ol. 9, No. 5, p. 87-9, 997. [8] Q. H. Wu, Y.J. Cao, and J.Y. Wen, Optmal Reactve Power dspatch Usng an Adaptve Genetc Algorthm Int. J. Elect. Power Energy Syst., vol. 0, pp , Aug [9].H.Quntana, G.L.orres, and J.Medna-Palomo, Interor-pont methods and ther applcatons to power systems: A classfcaton of publcatons and software codes IEEE trans. On Power systems,ol 5,No.,pp 70-76,Feb.000. [0] J. Kennedy and R.Eberhart, Partcle Swarm Optmzaton n Proc. IEEE Int. Conf. Neural Netw., vol. 4,Nov.995, pp [] B. Zhao, C. X. Guo, and Y.J. Cao Multagent-Based Partcle Swarm Optmzaton approach for Optmal Reactve Power Dspatch IEEE rans. Power Syst. ol. 0, no., pp , May 005. [] J. G. lachoganns, K.Y. Lee, A Comparatve Study on Partcle Swarm Optmzaton for Optmal Steady-State Performance of Power Sytems IEEE trans. on Power Syst., vol., no. 4, pp , Nov [3] M. aradaraan,k. S. Swarup, Networ Loss Mnmzaton wth oltage Securty Usng Dfferental Evoluton Electrc Power Sys. Research 78(008, pp