Le DONG 1, Shuang GAO 1, Xaozhong LIAO 1, Yang GAO 2 Beng Insttute of echnology (1), Shenyang Insttute of Engneerng (2) Short-erm Wnd Power Forecastng wth Combned Predcton Based on Chaotc Analyss Abstract. Wth the ntegraton of wnd energy nto electrcty grds, t s becomng ncreasngly mportant to obtan accurate wnd power forecasts. In ths paper, models for short-term wnd power predcton n large wnd farms are dscussed. he analyss of modelng wth low dmensons nonlnear dynamcs ndcates that wnd power tme seres have chaotc characterstcs and wnd power can be predcted n the short-term. he wnd power predcton models are bult wth phase space reconstructon method and the combnaton model wth dfferent embeddng dmensons s tested. Streszczene. Opsano metodę krótkotermnowego prognozowana mocy elektrown watrowych bazuącą na chaotycznym charakterze watru w krótkch odstępach czasu. (Krótkotermnowa prognoza mocy elektrown watrowych bazuąca na teor chaosu). Keywords: Wnd Power Generaton, Short erm, Chaotc Characterstc, Phase Space Reconstructon. Słowa kluczowe: sec elektrown watrowych, prognozowane. Introducton Wnd power generaton has rapdly developed over the last decade and global nstalled wnd power contnues to grow at around 28% per year [1]. By the year 2020 wnd power generaton wll supply about 12% of the total world electrcty demands [2]. Because of the wnd power fluctuatons n drecton and speed, wnd power generaton ntegrated n electrcal power system may cause several problems, these problems nclude; power qualty, stablty and especally power dspatchng. he problems ncrease as the penetraton of wnd energy ncreases [3-6]. Predcton of wnd power, along wth load forecastng, permts schedulng the connecton or dsconnecton of wnd turbnes or conventonal generators, thus achevng low spnnng reserve and optmal operatng cost. he horzon of predcton may vary: long-term horzon, one year; shortterm horzon, 36hours, 10 hours, and 1hour. Dfferent horzons correspond to dfferent obectves. here are two man state-of-art approaches, one based on physcal determnstc modellng [3,7-9] and a second one based on statstcal or tme seres modellng [2,4-6]. Physcal models are usually used for long-term predcton and physcal consderaton has a great effect on the accuracy. Snce wnd power s a functon of wnd speed, predctons of wnd power are generally derved from predctons of wnd speed [4,5]. he statstcal models are drect transformatons of the nput varables to wnd power. Advanced statstcal analyss methods are ARMA [10], Kalman Flters [11], artfcal neural network (ANN) [12-14], fuzzy logc [15], support vector machne [16] and so on. hs paper s organzed as follows. Frstly, the method of dynamcal phase space reconstructon s gven. he chaos ANN predcton model s presented. hen, the lnear combnaton and ANN combnaton predcton method are presented. he expermental results of comparng the algorthm proposed n ths paper wth other algorthms are also presented. Fnally, our work of ths paper s summarzed n the last secton. Reconstructon of dynamcal phase space he wnd power generaton system s a complex dynamcal system. he reconstructon of dynamcs from the tme seres of wnd power generaton should be done before dscussng the chaotc behavor of tme seres of wnd power generaton, whch s bult based on akens embeddng theorem. hs theorem provdes the mathematcal bass of the dynamc reconstructon problem. It states that model reconstructon of a nonlnear dynamcal system, usng ust one observaton of the system, should succeed to a certan extent and the reconstructon s ndependent of whch sgnal component s used [17]. Wth a chaotc tme seres, consder a D-dmensonal compact manfold M, akens embeddng theorem says wth m 2D+1, the map Φ:M R m defnes a correspondng traectory, the traectory of the reconstructed space s dffeomorphsm wth orgnal dynamcal system, where m s the embeddng dmenson [17]. ypcally, a wnd power tme seres x 1, x 2,, x n-1, x n s a seres of record values of observatons of a dynamcal system at regular, dscrete tme ntervals (. Assume the embeddng dmenson s m, and the tme delay s τ=kt, then the reconstructed phase space becomes (1) Y ( ) [ ), ), 2 ),..., ( m1) )], 1,2,..., N, N n ( m 1) Embeddng dmenson m and tme delay τ are two mportant parameters to perform the dynamc reconstructon. hey wll mpact the aspect of the optmal predcton of a reconstructed dynamcal model. Embeddng dmenson m s the component number of each pont n the reconstructed phase space, and t can be the nput number of ANN predcton model. me delay τ s the tme nterval between two nearby components of each pont n the reconstructed phase space and t can be the nputs tme nterval of ANN predcton model,.e. the nput values of ANN represent a phase pont n the reconstruct phase space. A nonlnear approach s used to measure tme delay τ called the mutual nformaton I. As far as the qualty of predcton wth neural networks s concerned, t s best to choose the value of τ somewhere n between 1 and the frst local mnmum of the mutual nformaton functon I(τ) [18]. he mutual nformaton of a tme seres as follows I( Q, S) Q S) S) S, (2) P( q )logp( q ) P( s )logp( s ) P( s, q )logp( s, q ) where: Q, S two dscrete varables, H(*) nformaton entropy, the more H(*)s, the stronger uncertanty s. H(S, unted nformaton entropy, H(Q S) condtonal entropy, whch ndcates the uncertanty of Q n case that S s known. P(q ) appearance probablty of q event, P(q, s ) smultaneous appearance probablty. he value of I(Q,S) ndcates the relevance level between Q and S. he smaller I(Q,S) s, the weaker the relevance between Q and S. PRZEGLĄD ELEKROECHNICZNY (Electrcal Revew), ISSN 0033-2097, R. 88 NR 5b/2012 35
Suppose the tme seres comes from an attractor, E 1( wll stop changng when m s greater than some value m0. hen the m 0 +1 s the mnmum embeddng dmenson. o nvestgate the wnd power tme seres of Fgure 1, E 1( s calculated and s shown n Fgure 3. When m s greater than 9 the E 1( stops changng and then the embeddng dmenson of wnd power generaton tme seres n Fgure 1 s 10. Fg.1. Measured wnd power tme seres from Fun wnd farm (2006. 1. 1 to 2007. 12. 31) Fgure 1 shows orgnal wnd power tme seres from 2006. 1. 1 to 2007. 12. 31, whch are from the wnd farm located n the Fun Chna. he wnd power tme seres from 2006. 1. 1 to 2006. 12. 31 n Fun Wnd Farm are used for tranng the predcton model and the wnd power tme seres from 2007. 1. 1 to 2007. 12. 31 used to test the predcton results. Assume S=k) and Q=k+τ), the I(Q,S) s shown n Fgure 2. When the tme delay τ changes from 1 to 50, the frst local mnmum of the mutual nformaton functon I(τ) s 12, so we choose 12 as tme delay of the wnd power tme seres. 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 I(Q,S) 0.2 0 10 20 30 40 50 Fg.2. Mutual nformaton versus tme delay Paper [15] presents a method to determne the mnmum embeddng dmenson. Consder a wnd power tme seres x 1, x 2,, x n-1, x n. he tme delay has been determned above, so the reconstructed phase space s shown as equaton (1). Defne Ym1( ) Ym1( n(, ) (3) a(, Ym ( ) Ym ( n(, ) 1 N (4) m a(, N m 1 m 1) (5) E1( where: 1,2,, N m. Y m 1( n(, ) s the nearest neghbour of Y m 1( ), and Y m ( n(, ) s the nearest neghbour of Y m (). E ( s a mddle functon, whch can smplfy the process to access embeddng dmenson m. If m s qualfed as an embeddng dmenson by the embeddng theorem, then any two ponts whch stay close n the m-dmensonal reconstructed space wll be stll close n the (m+1)-dmensonal reconstructed space [15]. Fg.3. he values E1( for wnd power tme seres It has to be confrmed that the wnd power generaton system s chaotc before applyng chaos theory to predct the wnd power, to satsfy ths need, chaotc characterstcs of wnd power must be determned. In practce, t can be defned that a bounded determnstc dynamcal system wth at least one postve Lyapunov exponent s a chaotc system [17]. he determnstc and bounded propertes are obvous of the wnd power generaton system n terms of fnte nstallaton power capacty and fnte attractor dmenson. hen the crteron of at least one postve Lyapunov exponent plays a key role n the characterzaton of a process as chaotc, because the Lyapunov exponents not only show qualtatvely the senstve dependence on ntal condtons, but also gve a quanttatve measure of the average rate of separaton or attracton of nearby traectores on the attractor. An algorthm developed by Wolf et al [19], whch mplements the theory n a very smple and drect fashon, can be used to calculate the largest Lyapunov exponent. As mentoned above, the ntal pont of the reconstructed phase space of wnd power tme seres n equaton (1) s Y ( t 0 ). Assume the nearest neghbour of the ntal pont s Y 0 ( t 0 ). Let L 0 denote the Eucldean dstance between them. Next, we have to terate both ponts forward for a fxed evoluton tme, whch should be of the same order of magntude as the tme delay τ, and fnally they evolve to two ponts, Y ( t 1 ) and Y 0 ( t 1 ), along wth ther traectores separately. he dstance of the two new ponts s L 0 Y ( t1) Y0 ( t1), where >0. hen another nearest neghbour Y 1 ( t 1 ) of the pont Y ( t 1 ) should be found, whch has the mnmum dstance between the par of ponts, L 1 Y ( t1) Y1 ( t1). hs procedure s repeated untl the ntal pont Y ( t 0 ) reaches the end of the tme seres, shown n Fgure 4 [19]. Fnally, the largest Lyapunov exponent s calculated accordng to the equaton 1 M L (6) max ln t M t 0 0 L 36 PRZEGLĄD ELEKROECHNICZNY (Electrcal Revew), ISSN 0033-2097, R. 88 NR 5b/2012
where: M the total number of replacement steps. max ndcates the average separaton rate of nearby traectores n the reconstructed phase space, and t determnes the future predcton tme. L 1 L 2 used. he NMAE of 2007 s 5.96%, and the NRMSE of 2007 s 9.35%. From comparsons, although the pure chaos-based ANN model does not play well as the persstence model from the NMAE, the NRMSE s smaller than persstence model. And the predcton accuracy can be mproved by buldng a combned predcton model on the bass of chaos-based ANN model. L 0 L 2 ) L 0 L 1 t 3 ) t t1 0 Fg.4. Wolf method for largest Lyapunov exponent calculaton t 2 2 ) P out By usng equaton (6), the largest Lyapunov exponent for the attractor presented n Fgure 1 can be calculated. he largest Lyapunov exponent converges very well to б max =0.0638. hs s a frm proof for the chaotc behavor of the wnd power generaton system. ANN model s sutable for wnd power predcton but the ANN model s dffcult to be confgured. here are 3 steps for ANN model desgn, 1) dentfy the approprate nput parameters, 2) set up approprate network structure, and 3) desgn a tranng algorthm that yelds the best tranng performance. Shuhu L, et al [7] presents an ANN model wth 4 nput data, whch are wnd speed and drecton from two meteorologcal towers. Rohrg, K, et al [20] presents another ANN model wth 8 nput data n terms of 3 wnd speed values, 3 drecton values, and 2 values related to the tme seres. Unfortunately, these patterns are lmted by the meteorologcal towers and cannot gve out the hub heght wnd speed for each turbne. hs ssue can be consdered wth the applcaton of the chaos-based ANN model. An ANN serves as the reconstructed phase space model of the wnd power generaton system. he nput data are the Y()=[), +τ), +2τ),, +(m-1)τ)], =1,2,, N, N=n-(m-1)τ), and the number of nput parameters s the embeddng dmenson m. he predcted value of the tme seres pont +(m-1)τ)+1,. e. +(m-1)τ+1), s set as the output of the chaos-based ANN model. he nonlnear DE functon ˆ : M R wll be obtaned by tranng the ANN. hs ANN model s dffeomorphsm wth orgnal dynamcal system of wnd power generaton system, and has locally (.e. short-ter predctable ablty. he confguraton of ANN for wnd power predcton shows n the Fgure 5 [14]. he predcton accuracy of the models wth monthly normalzed mean absolute error (NMAE) and normalzed root mean squared error (NRMSE) are expressed n equatons (7) (8). 1 1 N (7) NMAE x ( ) ) 100% Pcap N 1 1 1 N 2 (8) NRMSE ( x ( ) )) 100% Pcap N 1 where: x () actual value, x ( ) predcted value, N the number of predcton sample, P cap the rated capacty of the wnd turbne. he chaos ANN model presented n ths paper performs well, NMAE of 2007 s 6.17%,and NRMSE of 2007 s 9.22%. Persstence s a smple method, whch consders that the wnd power producton remans the same n all lookahead tmes [15]. he same data as the chaos method are ( m 1) ) Fg.5. Mult-layer neural network for wnd power generaton predcton Combned predcton model Wnd power predcton of large wnd farm usng only a sngle model brngs the rsk of hgh and costly errors, partcularly n the case of extreme events where ndvdual models can go wrong. he choce of the embeddng dmenson of reconstructed dynamcal model wll be not accurate because the nflexon of E 1( curve s very dffcult to be determned. In order to dmnsh the effect of reconstructed parameters of predcton of chaotc system, a combned model for wnd power predcton based on multdmenson embeddng s proposed. We ntroduce the lnear predcton model frst. (t=1,2,, n)s the measured wnd power generaton at tme t. Assume there are p dfferent embeddng dmensons around the determned one n the above-mentoned. And the predcted values of respectve ANN models wth dfferent embeddng dmensons are 1(, 2(,, p (. he lnear combnaton model can be wrtten as (9) x ˆ ( l1 1( l2 2 ( l p p ( where: L ( l1, l2,, l p ) the weghts of each model and (10) l1 l2 l p 1 he predcton error of the model at tme t s (11) et (, 1,2,, p, t 1,2,, n hen the error matrx shows as (12) E [( e t ) pn ][( et ) pn ] he quadratc sum of error of the lnear combnaton model s n n p 2 2 (13) J [ ( ] ( let ) L EL t 1 t 1 1 he mnmum of the quadratc sum of error s selected as obectve functon, then combnaton weghts can be obtaned by calculatng the optmal functon (14) (14) mn J L EL s. t. R L 1, R (1,1,,1) Accordng to the lnear combnaton model, the weghts are fxed, so t can not be sutable for changng weather stuaton. Choosng a combnaton of the best weghts for the specfc weather stuaton, leads to a sgnfcantly mproved wnd power predcton. If the predcted x ˆ( satsfed (15) ( ( 1(, 2(,..., p ( ) PRZEGLĄD ELEKROECHNICZNY (Electrcal Revew), ISSN 0033-2097, R. 88 NR 5b/2012 37
where: a nonlnear functon. hs combnaton model s nonlnear predcton model.e. the combnaton weghts are changng. Because the ANN can approxmate any nonlnear functon, the ANN combnaton model wll be a good choce for wnd power predcton. he ANN combnaton model can also regulate the weghts of the dfferent models after tranng of the ANN, so more hstory nformaton can be used for future predcton. Expermental result he wnd power tme seres from 2006. 1. 1 to 2006. 12. 31 n Fun Wnd Farm are used for tranng the predcton model and the wnd power tme seres from 2007. 1. 1 to 2007. 12. 31 used to test the predcton results. hree models based on phase space reconstructon are bult. he dmensons are m 1 =9, m 2 =10, m 3 =11, respectvely. By usng quadprog functon n MALAB optmzaton toolbox, the optmal combnaton weghts of the three predcton models are obtaned by calculatng the optmal functon (14), w 1 =0.1866, w 2 =0.5667, w 3 =0.267. he NMAE of lnear combnaton model of 2007 s 6.16%, whle the values for the ndvdual predctons of 2007 are 6.21%, 6.17% and 6.20%. And the NRMSE of lnear combnaton model n 2007 s 9.21% whch s smaller than persstence model. o fnd an optmal combnaton of dfferent phase space reconstructon model wth m 1 =9, m 2 =10, m 3 =11, the ANN s used for regulatng weghts of each model, whch can synthesze nformaton and fuse predcton devaton n dfferent embeddng dmensons, resultng n the predcton accuracy s mproved. he predcton results of the three embeddng dmensons are used as three nputs of ANN and the actual wnd power s the output. hen the weghts of the three predcton methods n ANN combnaton model can be obtaned by self-learnng. he node number of hdden layer s determned by cut-and-try method, and the network structure of ANN combnaton model s 3-2-1. he NMAE of ANN combnaton model n 2007 s 6.10% and NRMSE s 9.21%, whch s also smaller than persstence model. he NRMSE of dfferent predcton models are shown n Fgure 6. From the comparsons n Fgure 6, although the errors are very smlar, the NRMSE of persstence model are the bggest for all months of 2007. he lnear combnaton model and ANN combnaton model have good performance, and they perform a lttle better than the pure ANN model. Fg.6. NRMSE of dfferent predcton models he errors of dfferent models of 2007 are shown n able 1. From the comparsons n able 1, t can be concluded that: (1) When the sngle embeddng dmenson s used, m=10 s the best and the weght n lnear combnaton model s the largest; (2) he combnaton model s better than ANN model wth the sngle embeddng dmenson; (3) he NRMSE of combnaton models are smaller than persstence model. able 1. Errors of Dfferent Models. Models NMAE [%] NRMSE [%] ANN model (m=9) 6.21 9.25 ANN model (m=10) 6.17 9.22 ANN model (m=11) 6.20 9.24 Lnear combnaton model 6.16 9.21 ANN combnaton model 6.10 9.21 Persstence model 5.96 9.35 Conclusons he tme seres of wnd power generaton are analyzed n ths paper. he wnd power system ndcates typcal chaotc characterstcs. Accordng to the chaotc behavors of wnd power generaton, the ANN model based on phase space reconstructon s used for wnd power predcton. he desgn method of ANN model s dscussed n ths paper. Predcton accuracy s affected by dfferent embeddng dmenson whch s very dffcult to choose n phase space reconstructon. In order to dmnsh the effect of reconstructed parameters of predcton of chaotc system, a combned model for wnd power predcton based on multdmenson embeddng s proposed. he lnear combnaton model and the ANN combnaton model show a lttle better performance of wnd power predcton when compared wth the pure ANN model. he NRMSE of combnaton models are smaller than persstence model. hs study was supported by Chna Power Investment Corporaton scence & technology proect (2009064ZDGKJX). REFERENCES [1] Analyss and Prospects of Global Renewable Energy Development Stuaton n 2008 [Onlne].Avalable:http://www.chnapower.com.cn [2] El-Fouly HM, El-Saadany EF, Salama MMA. One day ahead predcton of wnd speed usng annual trends. Proceedngs of the Power Engneerng Socety General Meetng 2006 [3] Ferrera LM. Evaluaton of Short-erm Wnd Predctablty. IEEE ransactons on Energy Converson, 7 (1992) 409-417 [4] Alexads MC, Dokopoulos PS, Sahsamanoglou HS. 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