DIFFERENTIAL EVOLUTION APPROACH FOR OPTIMAL POWER FLOW SOLUTION

Transcription

1 DIFFERENIAL EOLUION AROACH FOR OIMAL OWER FLOW SOLUION K.asakh, L.R.Srnvas rofessor, Department of Electrcal Engneerng, Andhra Unversty, sakhapatnam, A, Inda Assoc. rof., Department of Electrcal Engneerng, SRKR Engneerng College, Bhmavaram, W.G.Dst., A, Inda E-mal: vasakh_k@yahoo.co.n ABSRAC hs paper presents an algorthm for solvng optmal power flow problem through the applcaton of Dfferental Evoluton (DE. he obectve s to mnmze the total fuel cost of thermal generatng unts havng quadratc cost characterstcs subected to lmts on generator real and reactve power outputs, bus voltages, transformer taps and power flow of transmsson lnes. he proposed method has been tested under smulated condtons on IEEE 30-bus system.he optmal power flow results obtaned usng DE are compared wth other evolutonary methods. It s shown that DE total generaton fuel cost s less expensve than those of evolutonary programmng, tabu search, hybrd tabu search, and smulated annealng. Keywords: Optmal ower Flow; Dfferental evoluton; Lne flow constrants. INRODUCION Evolutonary Algorthms (EAs are optmzaton technques based on the concept of a populaton of ndvduals that evolve and mprove ther ftness through probablstc operators lke recombnaton and mutaton. hese ndvduals are evaluated and those that perform better are selected to compose the populaton n the next generaton. After several generatons these ndvduals mprove ther ftness as they explore the soluton space for optmal value. he feld of evolutonary computaton has experenced sgnfcant growth n the optmzaton area. hese algorthms are capable of solvng complex optmzaton problems such as those wth a non-contnuous, non-convex and hghly nonlnear soluton space. In addton, they can solve problem that feature dscrete or bnary varables, whch are extremely dffcult. Several algorthms have been developed wthn the feld of Evolutonary Computaton (EC beng the most studed Genetc Algorthms were frst conceved n the 90 s when Evolutonary Computaton started to get attenton. Recently, the success acheved by EAs n the soluton of complex problems and the mprovement made n computaton such as parallel computaton have stmulated the development of new algorthms lke Dfferental Evoluton (DE, artcle Swarm Optmzaton (SO, Ant Colony Optmzaton (ACO and scatter search present great convergence characterstcs and capablty of determnng global optma. Evolutonary algorthms have been successfully appled to many optmzaton problems wthn the power systems area and to the economc dspatch problem n partcular [-8].. OERIEW OF DIFFERENIAL EOLUION One extremely powerful algorthm from evolutonary computaton due to t s excellent convergence characterstcs and few control parameters s dfferental evoluton. Dfferental evoluton solves real valued problems based on the prncples of natural evoluton [-] usng a populaton of Np floatng pont-encoded ndvduals ( that evolve over G generatons to reach an optmal soluton. In dfferental Evoluton, the populaton sze remans constant throughout the optmzaton process. Each ndvdual or canddate soluton s a vector that contans as many parameters ( as the problem

2 decson varables D. he basc strategy employs the dfference of two randomly selected parameter vectors as the source of random varatons for a thrd parameter vector. In the followng, we present a more rgorous descrpton of ths new optmzaton method. = [ Y... YNp ] ( Y = [ X, X,..., X D ] =,, Np ( Extractng dstance and drecton nformaton from the populaton to generate random devatons result n an adaptve scheme wth excellent convergence propertes. Dfferental Evoluton creates new offsprngs by generatng a nosy replca of each ndvdual of the populaton. he ndvdual that performs better from the parent vector (target and replca (tral vector advances to the next generaton. hs optmzaton process s carred out wth three basc operatons: Mutaton Cross over Selecton Frst, the mutaton operaton creates mutant vectors by perturbng each target vector wth the weghted dfference of the two other ndvduals selected randomly. hen, the cross over operaton generates tral vectors by mxng the parameters of the mutant vectors wth the target vectors, accordng to a selected probablty dstrbuton. Fnally, the selecton operator forms the next generaton populaton by selectng between the tral vector and the correspondng target vectors those that ft better the obectve functon. 3. DE OIMIZAION ROCESS A. Intalzaton he frst step n the DE optmzaton process s to create an ntal populaton of canddate solutons by assgnng random values to each decson parameter of each ndvdual of the populaton. Such values must le nsde the feasble bounds of the decson varable and can be generated by Eq. (3. In case a prelmnary soluton s avalable, addng normally dstrbuted random devatons to the nomnal soluton often generates the ntal populaton. Y mn max mn = Y + η ( Y Y (3 (0, =,,. Np, =,,. D mn max Where Y and Y are respectvely, the lower and upper bound of the th decson parameter and η s a unformly dstrbuted random number wthn [0,] generated anew for each value of. B. Mutaton After the populaton s ntalzed, ths evolves through the operators of mutaton, cross over and selecton. For crossover and mutaton dfferent types of strateges are n use. Basc scheme s explaned here elaborately. he mutaton operator s ncharge of ntroducng new parameters nto the populaton. o acheve ths, the mutaton operator creates mutant vectors by perturbng a randomly selected vector ( Y a wth the dfference of two other randomly selected vectors ( Yb andy c accordng Eq. (. All of these vectors must be dfferent from each other, requrng the populaton to be of at least four ndvduals to satsfy ths condton. o control the perturbaton and mprove convergence, the dfference vector s scaled by a user defned constant n the range [0,.]. hs constant s commonly known as the scalng constant ( S. ' Y = Y + S( Y Y a b =,,. Np ( WhereY, Y, Y, are randomly chosen vectors a {,,...Np } b c and a b c Y a, Yb, Yc are generated anew for each parent vector, S s the scalng constant. For certan problems, t s consdered a =. C. Crossover he crossover operator creates the tral vectors, whch are used n the selecton process. A tral vector s a combnaton of a mutant vector and a parent (target vector based on dfferent dstrbutons lke unform dstrbuton, bnomal dstrbuton, exponental dstrbuton s generated n the range [0, ] and compared aganst a user defned constant referred to as the crossover constant. If the value of the random number s less or equal than the value of the crossover constant, the parameter wll come from the mutant vector, otherwse the parameter comes from the parent vector as gven n Eq. (. c

3 he crossover operaton mantans dversty n the populaton, preventng local mnma convergence. he crossover constant (CR must be n the range of [0, ]. A crossover constant of one means the tral vector wll be composed entrely of mutant vector parameters. A crossover constant near zero results n more probablty of havng parameters from the target vector n the tral vector. A randomly chosen parameter from the mutant vector s always selected to ensure that the tral vector gets at least one parameter from the mutant vector even f the crossover constant s set to zero. ' ' X f C or = q G, η ''( R X, = X, otherwse ( Where =,, Np =,, D q s a randomly chosen ndex {,,...D} that guarantees that the tral vector gets at least one parameter from the mutant ' vector; η s a unformly dstrbuted random number wthn [0, generated anew for each value of. X s the parent (target vector, '( G,, X the mutant vector and X the tral ", vector. Another type of crossover scheme s mentoned n []. ' X for = n n + G,,,..., ''( D D X, = X, otherwse ( Where the acute brackets denote the D modulo functon wth modulus D. he startng ndex n s a randomly chosen nteger from the nterval [0, D-]. he nteger L s drawn from nterval [0, D-] wth the probablty r (L=v = (CR v. CR [0,] s the crossover probablty and consttutes a control varable for the DE scheme. he random decsons for both n and L are made anew for each tral vector. D. Selecton he selecton operator chooses the vectors that are gong to compose the populaton n the next generaton. hs operator compares the ftness of the tral vector and ftness of the correspondng target vector, and selects the one that performs better as mentoned n Eq. (. " " Y f f Y f Y + ( G ( ( Y = Y otherwse (7 =, Np he selecton process s repeated for each par of target/ tral vector untl the populaton for the next generaton s complete.. ALICAION OF DE O OF Dfferental Evoluton has been appled to problems from several areas. Some power engneerng problems have been solved wth DE ncludng: Dstrbuton systems capactors placement, harmoncs voltage dstrbuton reducton and passve shunt harmonc flter plannng. DE has also been used n the desgn of flters, neural network learnng, fuzzy logc applcaton, and optmal control problems, among others. he obectve functon of OF Ng F = ( a g + b g + F = c $/Hr COS = (8 Subected to the constrants g( x, u = 0, (9 h( x, u 0. n + where L g s the equalty constrants and D represent typcal load flow equatons. h s the system operatng constrants E. Dependent arables X s the vector of dependent varables consstng of slack bus power G, load bus voltages L, generator reactve power outputs Q G, and transmsson lne loadngs S l. Hence, X can be expressed as X =[ G, L, QG, Sl ] (0.e., X =,,..., Q,... Q, S,... S ] [ G L L Npq G G Ng l l Nl Npq Ng, Nl where, are number of load buses, number of generators, and number of transmsson lnes, respectvely. 3

4 F. Independent arables U s the vector of ndependent varables consstng of generator voltages G, generator real power outputs G, except at the slack bus G, and transformer tap settngs. Hence, U can be expressed as U = [ G, G, ] (.e., u = [ G,... G Ng, G... G Ng,,... Nt ] where Nt s the number of the regulatng transformers. G. Intalzaton he frst step n ths algorthm s to create an ntal populaton. All the ndependent varables [ G, G, ] have to be generated accordng to formula (3, where each ndependent parameter of each ndvdual n the populaton s assgned a value nsde the gven feasble regon of the generator. hs creates parent vectors of ndependent varables for the frst generaton. As they have created wthn ther lmts, they readly satsfy the correspondng nequalty constrants. o fnd dependent varables X = [ G, L, QG, Sl ] correspondng to each ndvdual, Newton- Raphson power flow soluton s mplemented. After gettng all vectors correspondng to dependent varables, constrant-handlng method of penalty functons s appled to handle the nequalty constrants related to dependent varables. enalty factors correspondng to each dependent varable of each ndvdual n populaton have to be calculated. If they volate a lmt whether lower or upper, dfference of that value and correspondng lmt volated was taken as penalty ndex and t s multpled wth a constant so as to match wth basc obectve functon.e., fuel cost. he penalty functons for slack bus power, voltages of load buses, lne flows and reactve power generatons are consdered to calculate ftness of each populaton member. Ftness ncludes fuel cost functon and also penaltes correspondng to dependent varables. Incluson of these penaltes n ftness gves us a great opportunty to assgn better ftness to that partcular populaton member whose control parameters are wthn the operatonal lmts n addton to mnmum fuel cost. Ft = Ng Npq Nl F where COS + ( k* Spf Slack bus penalty Lne flows penalty Q G enalty oltage penalty + ( k* = Qgpf, + ( k3* pf, + ( k* LFpf Spf Lfpf Qgpf pf =. DE IMLEMENAION RESULS ( he sutablty of the proposed method has been tested for IEEE-30 bus shown n Fg.. It s chosen as t s a benchmark system, has more control varables and provdes results for comparson of the proposed method. he approach can be generalzed and easly extended to large-scale systems. he IEEE-30 bus system conssts of sx generators, four transformers, lnes, and two shunt reactors. In DE soluton for OF, the total control varables are : sx unt actve power outputs, sx generator bus voltage magntudes, and four transformers tap settngs and are gven n able. All generator actve power, and generator bus voltages and transformer tap settng are consdered as contnuous for smplcty. he generators cost coeffcents of the IEEE 30-bus test system are gven n the able.he lmts of varables for the IEEE-30 bus system s gven n able 3. In ths secton, the DE soluton of the OF s evaluated usng the test system IEEE-30 bus system [7]. he results, whch follow, are the best soluton over the ten runs. he results are compared wth E and other methods. ABLE I SYSEM DESCRIION OF CASE SUDY Sl.No. arables 30-bus system 3 Buses Branches Generators Generator buses Shunts reactors ap-changng transformers 30 = p,

5 ABLE II GENERAOR COS COEFFICIENS OF IEEE 30-BUS SYSEM Bus No Real ower Output lmt (MW Cost Coeffcents Mn Max a b c ABLE III LIMIS OF ARIABLES FOR IEEE 30-BUS SYSEM No. Descrpton Unts Lower Lmts oltage Qbus u 0.9 oltage bus u ransformer taps u 0.90 Upper Lmts ABLE I DE ARAMEERS FOR BES RESULS OF OIMAL OWER FLOW FOR IEEE 30-BUS SYSEM Sl.No. 7 8 arameters of Dfferental evoluton arameters alues opulaton 0 Generatons 00 enalty factors of ftness functon Slack bus generaton penalty factor Reactve power penalty factor Load bus voltage penalty factor Lne flows penalty factor 0, Fgure : IEEE 30-bus system ABLE OIMAL ACIE AND REACIE OWER GENERAION LEELS FOR 30-BUS SYSEM Unt No. 3 Sl. No. 3 Bus No 8 3 Generator unt real and reactve power control Unt real power [MW] Unt reactve power [MAR] ABLE I CONROL ARIABLES FOR HE 30-BUS SYSEM I. Generator voltages II. ower generaton III. ransformer taps Gen voltage alue g alue ransf. ap alue G g G g G g g8 G8.0.3 g G G3 g3

6 ABLE II COMARISON OF HE OAL GENERAOR FUEL COSS OF DE WIH S, S/SA, IS, E, AND IE Cost ($/hr Algorthm S S/SA IS E IE DE Best cost Average cost Worst cost he DE parameters used for the optmal power flow soluton are gven n able. hey are treated as contnuous controls. able shows the optmal settng of the generator bus actve power and correspondng reactve generaton for DE. able shows the optmal control varables obtaned for the optmal power flow of the IEEE- 30 bus system. able 7 shows the comparson of the cost of generaton for the IEEE-30 bus system for the above cases wth other soft computng methods Generat ons Fgure : Cost s Generatons.. Fgure shows the convergence of DE for the optmal power flow problem. he operatng costs of the best soluton n the normal operaton acheved by the DE and E are, respectvely, $80.30 and $ per hour. It can be observed from Fg. that the convergence of DE s faster whle obtanng a better soluton n lesser computatonal tme. Fgure shows the bus voltage profles of the 30-bus system acheved by the DE and E.. CONCLUSIONS hs paper presents a DE soluton to the optmal power flow problem and s appled to an IEEE 30- bus power system. he man advantage of DE over other modern heurstcs s modelng flexblty, sure and fast convergence, less computatonal tme than other heurstc methods. And t can be easly coded to work on parallel computers. he man dsadvantage of DE s that t s heurstc algorthms, and t does not provde the guarantee of optmal soluton for the OF problem. he DE approach s useful for obtanng hgh-qualty soluton n a very less tme compared to other methods. he future work n ths area conssts of the applcablty of DE solutons to large-scale OFF problems of systems wth several thousands of nodes, utlzng the strength of parallel computers. oltage(pu Bus Number Fgure : Bus voltage profles DE E 7. REFERENCES [] R.Gnanadas,.enkatesh, Narayana rasad adhy, Evolutonary rogrammng Based Optmal ower Flow For Unts Wth Non- Smooth Fuel Cost Functons, Electrc ower Components and Systems, ol.33, 00, pp. -0. [] Jason Yuryevch, Kt o Wong, Evolutonary rogrammng Based Optmal ower Flow Algorthm, IEEE ransactons on ower

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