Short-Term Load Forecasting Using PSO-Based Phase Space Neural Networks

Transcription

1 Proceedngs of he 5h WSEAS In. Conf. on SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, Augus 7-9, 005 (pp78-83) Shor-Term Load Forecasng Usng PSO-Based Phase Space Neural Neworks Jang Chuanwen, Fang nyan, Wang Chengmn Deparmen of Elecrcal Engneerng, Shangha Jaoong Unversy Huashan Road 954, Shangha P.R Chna, Lu Janyu, Wang Lang Eas Chna Grd Company Lmed. Shangha P.R Chna Absrac: - The nonlnear heores of load forecasng, such as he applcaons of neural nework and chaos, have recenly made consderable progress. Generally, s an effecve mehod o combne phase space resrucures heory wh arfcal neural neworks (ANN) model for load forecasng. Bu, hey are no so effecve o forecas aracors wh hgher embedded dmenson. The paper proposes a new dea based on ncdence-degree o deermne he neares pon n phase space. In he mean me, an arfcal neural neworks model based on parcle swarm opmzaon (PSO) learnng algorhm s presened for load forecasng. The proposed mehod has been examned and esed on a praccal power sysem. The es resul shows ha he precson of load forecasng s mproved by means of he new mehod when he embedded dmenson s hgher. Key-Words: -Load forecasng, Parcle Swarm Opmzaon, ANN. Inroducon Shor-erm load forecasng plays an mporan role n power sysem operaon and plannng. Generally, shor-erm load forecasng s always a dffcul ask n pracce, because he load s affeced by a varey of nonlnear facors such as weaher condons, daly, weekly and seasonal perodcy, ec. Nowadays, more and more scholars hnk ha consderng he nonlnear facors n he load forecasng modelng s he key o mprovng he load forecasng level. Recenly, he heory of nonlnear chaos dynamcs whch lnks he deermnaon and randomcy has become he foreland of load forecasng sudy, many scholars do los of useful explore for load forecasng []-[4]. Specally, combnng phase space resrucures heory wh neural neworks s consdered as one of he mos effecve mehods [5][6]. However, he embedded dmenson wll be relavely hgh, because load forecasng s nfluenced by varous nrcae facs. The radonal chaos mehods are proved of hgh precson o forecas me seres wh low-embedded dmenson. Bu, hey are no so effecve o forecas aracors wh hgh-embedded dmenson [7][8]. On he oher hand, s aracve o fnd a fas nework convergen arhmec for aracors wh hgh-embedded dmenson. The paper proposes a new dea based on ncdence-degree o deermne he neares pon n phase space. In he mean me, an arfcal neural neworks model based on parcle swarm opmzaon (PSO) learnng algorhm s presened for load forecasng. The proposed mehod has been examned and esed on a praccal power sysem. The es resul shows ha he precson of load forecasng s mproved by means of he new mehod when he embedded dmenson s hgher. The paper was organzed as follows: secon descrbes a phase space neural neworks forecasng model su for hgher embedded dmenson; an mproved ANN model wh parcle swarm opmzaon learnng algorhm s shown n secon 3; followed by numercal examples n secon 4, and

2 Proceedngs of he 5h WSEAS In. Conf. on SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, Augus 7-9, 005 (pp78-83) concluson n secon 5.. PHASE SPACE NEURAL NETWORKS FORECASTING MODEL SUIT FOR HIGHER EMBEDDED DIMENSION Now, here are several forecas ways based on chaos heory, such as local lnear approxmaon mehod, lnear nerpolaon mehod, arfcal neural neworks mehod and so on. All hese models are based on he phase space resrucures heory proposed by Takens n 98. For a me seres x, x, L, }, we se n a d { 0 x N dmensonal phase space accordng o he phase space resrucures heory: x, x, L, x ) () = ( ( ) τ + ( ) τ + ( d ) τ + To smplfy he presenaon, leτ =, hen: = ( x, x, L, x d+ ) () where s one sae n he d-order sae space. Accordng o he famous Takens Theory [9], when d m + (m s he order of he aracor), here exss a deermnsc mappng wh dmenson beng d: F ( d) d d : R R. I descrbes he evolvng rack of n he sae space and he mappng has he same geomerc srucure and opology wh he orgnal sysem. So: forecasng model combnng phase space resrucures heory and arfcal neural neworks as follows: x x + x d + Fg. ANN Forecasng Model By selecng a suable se of weghs and ransfer funcons, s known ha he ANN can approxmae any smooh, measurable funcon beween he npu and oupu vecors. In hs paper, an oupu mean squared error (MSE) of ANN s consdered and defned as: MSE = N N M = = ( o sk l sk ) s k x (5) where o sk s he expeced oupu, l sk s he predced oupu, M s he number of oupu neurons, and N s he number of ranng se samples. Now, he problem s how o fnd he K he neares pons o ranng neural neworks, he curren chaos forecasng mehods are ofen based on he neares pons mehod. Those mehods are based on Eucld dsance: = ( x x ) + ( x x ) + + ( x d+ x d + ) L (6) ( d ) + = F ( ) (3) and fnd K neares pons o n he sae space o (3) can also be presened as: ~ ( d ) x F ( x, x, L, x ) (4) = + d + ~ ( d ) The sae space descrbed by F s called as he resrucured sae space. d s called as he embedded dmenson, whch s he dmenson of he mnmal sae space ha can compleely conan he aracor ~ ( ) ses comprsed by sae ranson. Because F d s deermnsc, we are only needed o made an esmae on ~ ( d ) F n order o forecas x +. The approach he load evolvng rack. The above algorhm depends n a large degree on he neares pons found accordng o Eucld dsance mehod. If he neares pons are relaed closely o he orgnal sae, he forecasng precson s hgh. Oherwse, s low. When he embedded dmenson d s small, he neares pons found accordng o Eucld dsance mehod can approxmaely reflec he relaonshp wh he orgnal pon. Bu when he dmenson s ncreased, such close relaonshp wll be decreased because he neares

3 Proceedngs of he 5h WSEAS In. Conf. on SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, Augus 7-9, 005 (pp78-83) dsance doesn mean he greaes relaonshp. Ths paper proposed o subsue he Eucld dsance wh he ncdence degree. The ncdence degree can evaluae he relaonshp accordng o he smlary of curves. The larger he ncdence degree s, he beer he fng s. Relang degree s an effcen mehod o deal wh hgh embedded dmenson. Le 0,, beng hree pons n he sae space wh dmenson of d. Defne: mnmnx0( k) x ( k) + ρ maxmaxx0( k) x ( k) k k ξ ( k) = (7) x ( k) x ( k) + ρ maxmaxx ( k) x ( k) 0 ( k =,,..., d : generally ρ =0.5) as he ncdence degree coeffcen beween pons 0 and We call: a he kh elemen. r = d d k = k ξ ( k) (8) as he ncdence degree beween pon 0 and he reference pon 0. The larger he ncdence degree s, he hgher he smlary s. 3. Tranng ANN Usng PSO The above-menoned mehod s very easy and feasble, bu sll has dsadvanage: he neworks wll converge very slowly when usng common ranng arhmec, especally, when he dmenson d becomes enough large, he phenomenon s obvous. Ths paper proposes an mproved parcle swarm algorhm for ranng neural nework. Parcle swarm opmzaon (PSO) was frs nroduced by Kennedy and Eberhar n 995 [0]. Lke evoluonary algorhms, PSO echnque conducs search usng a populaon of parcles, correspondng o ndvduals. Each parcle represens a canddae soluon o he problem a hand. Now, parcle swarm opmzaon has become he focus of research []-[6].Durng he calculaon, he parcle s affeced by hree facors when s movng n space. One of he facors s he parcle s curren velocy V (). Anoher s he opmal pon pbes = ( pbes, pbes, L, pbes ) where he parcle has reached before. The hrd facor s he opmal pon gbes = gbes, gbes, L, gbes ) of ( he communy or he sub-communy. The parcle s velocy s changed owards pbes () and gbes () n every eraon sep. Meanwhle, V pbes () and gbes () are assgned separaely a wegh a random. The velocy and poson s updaed accordng o he formula (9) and (0). v ( ) = w v ( ) + c r ( pbes ( ) x ( )) (9) + c r ( gbes( ) x ( )) x, ( + ) = x ( ) v ( ) (0) ( =,, L n =,, Lm ) Where, c,c are he learnng facors, generally, c = c. w = s he wegh scale operaor. r, r are he randoms whn he nerval of [0,]. s he number of eraon. n s he number of parcles. m s he number of dmensons. I s assumed ha he hree-layered perceprons are chosen for all applcaon cases n hs sudy. W s he connecon wegh marx beween he npu layer and he hdden layer, W s he connecon wegh

4 Proceedngs of he 5h WSEAS In. Conf. on SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, Augus 7-9, 005 (pp78-83) marx beween he hdden layer and he oupu layer. The performance of each ndvdual s measured accordng o a fness funcon. The fness funcon can be calculaed by f W W ) = MSE( W, ) () (, W The procedure of he self-adapve PSO for combned forecasng model wegh opmzaon can be descrbed as follows. Sep Inalzaon: Se =0. Le { W } W, = be a parcle, generae randomly n parcles { 0),, Ln} ( = (se n o 0 n hs paper). All parcles are se beween he lower and upper lms. Smlarly, generae randomly r ( 0), =, Ln, nal veloces of all parcles, { } r ( = m. v r k (0) r r where V 0) { v (0), L, v (0)} generaed by randomly selecng a value wh unform probably over he kh r r max dmenson[ v k v ]. Each parcle n he nal max, k populaon s evaluaed usng he equaon (). For each parcle, se pbes 0) = (0) and f = f, =, L n. f n V ( Le f = mn{ f, L }. Se he parcle assocaed s poson. If f < f, =, L, n, hen pbes ( ) = ( ) f = f Else go o Sep3 Search for he mnmum value If mn f f < hen gbes( ) = mn ( ) f = f mn f mn among f. Else go o Sep3. Sep4 Soppng crera: If one of he soppng crera s sasfed, hen sop. Else go o Sep. 4. Numercal examples Case sudes for he proposed mehod were carred ou for load forecasng usng dfferen hsorcal daa of shangha grd of Eas Chna area n 000. Dmensons of aracors are calculaed by means of G-P algorhm whch s pu up by Grassberger and Procacca, and he mehod whch s pu up by Wolf [7] o calculae Lyapunov exponenal s also ulzed. The correspondng parameers for he load seres: wh f as he global bes gbes (0). Maxmum Lyapunov ndex: λ max = ; Sep Velocy and Poson updang: Le =+. Usng he global bes and ndvdual bes of each parcle, he h parcle velocy and poson n he h dmenson s updaed usng he equaon ()-(3). v ( ) = w v ( ) + c r ( pbes ( ) x + c r ( gbes( ) x ( )) ( )) () Embedded dmenson: D=6; Tme delay: τ =. The 3-dmenson reconsruced sae space of orgnal load seres are shown n Fg., Fg.3 shows he ceran day s predcon resuls produced by he paper s mehod (he real lne represens he real load value, and he dashed lne he forecas value). Table. show he forecas resul of 4 consecuve days. x, ( ) = x ( ) + v ( ) (3) Sep3 Indvdual and global bes updang: Each parcle s evaluaed accordng o s updaed

5 Proceedngs of he 5h WSEAS In. Conf. on SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, Augus 7-9, 005 (pp78-83) n Each algorhm s eraed 0 mes n order o compare he PSO wh BP n erms of he convergence characer and he compuaon speed. Tab. gves he average values for comparson showng ha he PSO s more effcen han BP Table. Performance of PSO and BP n n 0000 Mean erave Mean me (s) BP > PSO < Fg. Three-Dmenson Reconsrucon of Load 5. Concluson The chaoc load seres wh hgh-embedded dmenson Real l oad For ecas l oad s very common n he naure. Therefore s useful o sudy he forecasng mehods of hgh-embedded dmenson. The paper proposes a new dea based on ncdence-degree o deermne he neares pon n phase space. In he mean me, an arfcal neural neworks model based on parcle swarm opmzaon learnng algorhm s presened for load forecasng. The proposed mehod has been examned and esed on a praccal power sysem. The resuls show ha he mehod plays an mporan role o mprove he precson of forecasng of load seres wh hgher embedded dmensons. Fg 3. he ceran day s predcon resuls Table. he 4 consecuve days forecas resuls day Mean References Max day Mean Max [] Iokbe,T., Fumoo,Y., Kanke,M., and SuzukS.: Shor-erm predcon of chaoc me seres by local fuzzy reconsrucon mehod, J.Inell.Fuzzy Sysems 5(997),3- [] James McNames,: Local averagng opmzaon for chaoc me seres predcon, Neurocompung, 48(00), 79-97

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