Forecast of Next Day Clearing Price in Deregulated Electricity Market

Transcription

1 Proceedngs of the 009 IEEE Interntonl Conference on Systems, Mn, nd Cybernetcs Sn Antono, TX, USA - October 009 Forecst of Next Dy Clerng Prce n Deregulted Electrcty Mrket Hu Zhou, Xnhu Wu, We Wng School of electrcl engneerng Bejng Jotong Unversty Bejng, Chn hzhou@bjtu.edu.cn Lpng Chen Northern chn power grd Ltd. compny Stte Power Grd Compny Bejng, Chn Lpng.chen@ncpg.com.cn Abstrct the dly clerng prce curve n electrcty mrket vred wth mult-perod nd strong fluctuton chrcterstc. When grey GM (, ) model s used n forecst, the forecst error exceeded the permtted precson. Ths s becuse GM (, ) model s nvldted only f the prce seres dd not follow the rule of exponentl growth. In ths cse, grey model wth perod resdul modfcton s proposed, whch nherts the dvntges of grey model nd mkes the forecstng prce curve fluctuted. Menwhle, seres of technology s used, such s smooth processng to orgnl dt, mprovement of ntl condton nd perod resdul modfcton. Thus the fttng curve s closer to orgnl dt nd the forecstng precson s mproved. Smulton results verfed the fesblty of the proposed pproch. Keywords deregulted electrcty mrket; forecst of next dy clerng prce; Grey Model of GM (, ); perod resdul modfcton; qudrc exponentl smoothng omponent; I. INTRODUCTION Wth compettve mrket structure hvng been reconstructed from prevous monopolzton mngement n electrcty ndustry, electrcty prce s the most mportnt prmeter concerned by every mrket prtcpnt nd regulton commttee of electrcty mrket [-]. Ths s becuse the clerng prce, sometmes clled equlbrum prce, would nfluence ther economc beneft n the trnscton of electrc power energy. Even n the worse cse, fluctuted prce wth bnorml vrton n frequency nd mpltude, whch my be resulted from the mnpulton by genertor-sde unon, would dsturb the mrket order. n, equlbrum prce s formed when supply nd demnd utomtclly reched on n equlbrum pont under mrket competton condton. Therefore, clerng prce forecstng s becomng one of focus ssues for domestc nd bord reserchers. n, forecst of next dy clerng prce s consdered s one of chllengng problems. In the pst, deep reserch on lod forecst hs been done for power system. Mny mture lgorthms nd forecst prctces hve been reported. In some sense, prce forecst s ble to refer to those lgorthms of lod forecst prtly. Approch used n clerng prce forecstng ncludes tme sequence nlyss [3-4], rtfcl neurl net [5-7], nd combnton predcton [8]. n, combnton pproch s weghted model, whch s generted by group of predcton result of conventonl forecstng technology, so t could bstrct useful nformton from every sngle model. The essence of the pproch mentoned bove s to fnd the vrton rule of clerng prce bsed on mss of dt, then to construct the forecst model. More or less, there re some shortcomngs. For exmple, some models requre long clculton tme or enough smples; some models requre dt to be conformed to those clsscl probblty dstrbuton. For grey system model, unque dvntge hs been demonstrted [9], such s no requrement on probblty requrement to orgnl dt, poor smples, smple clculton processng etc. Actully, n electrcty mrket, equlbrum prce could be consdered s grey vrble. Ths s becuse form mechnsm of equlbrum prce s very complcted [0]. Menwhle, equlbrum prce s usully nfluenced by mny fctors, whch re dffcult to dentfy respectvely. So clerng prce s synthetc vrble nd mples the synthetc effect of mny fctors. As the detled nformton of every nfluencng fctor s ncomplete nd ther effect s mostly uncertn, whch s ccordnce wth chrcterstc of grey vrble. Consequently, forecst of clerng prce s dpted to be solved wth uncertn theory, such s grey system theory [-]. As we known, model lke GM (, ) s constructed by oneorder or second-order generted sequences, nd s especlly used n the sequence wth pproxmte exponentl vrton. In ths cse, hgher fttng precson s gurnteed. If ths condton s not be stsfed, the forecstng error wll be ncresed gretly. Wth explorng the dtum of clerng prce from Clforn electrcty mrket, Amerc n 000, whch s n open dtbse nd s freely shred by ll cdemc reserchers, we found tht clerng prce s fluctuton sequence wth certn cycle,.e. dly clerng prce curve hs smlr chrcterstc wthn perod of tme, such s the curve shpe, the occurrence ntervl of mxml clerng prce, etc. Then we ttempted to use GM (, ) wth perod resdul modfcton [3-4] to hndle the dscussed problem. Wth numercl clculton, we found tht the pproch s effectve nd ts predcton s more ccurte thn tht of GM (, ), nd predcton precson s mproved /09/$ IEEE /09/$ IEEE 4507

2 II. QUADRIC EXPONENTIAL SMOOTHING APPLIED If the orgnl sequence hs some chrcterstc of multperod, frequent vrton nd strong fluctuton, greter error occurred when we drectly use t to modelng [5]. Therefore, qudrc exponentl smoothng technology ws ppled to orgnl prce sequence nd generted new sequence. And the smoothng formul s denoted s followng: S k = X k + S k S ( k) = αs ( k) + ( α) S ( k ) k=,n The menng of prmeter n the equton s nterpreted s below. The prmeter s the smoothng prmeter. X (k) s orgnl sequence of clerng prce. S (k) s new sequence generted by X (k) wth lner exponentl smoothng nd clled s lner exponentl smoothng sequence. S (k) s nother sequence, whch s generted by S (k) nd clled s qudrc exponentl smoothng sequence. The frst dt or verge of nteror dtum of X (k) cn be worked s S. The mgntude of smoothng prmeter s set before clculton, whch s usully determned s followng prncples: The greter X (k) fluctutes, the smller s; vce vers. S (k), s the orgnl sequence, s nputted nto GM (, ) model. When fnshed modelng, the forecsted sequence S ( k) should be reduced twce to obtn X ( k) ccordng to the equton (). k=,n () Compred wth X (k), rndomness of S (k) becomes weker. Although the mthemtcl expecttons E(S (k)) stll remned constnt, ts vrnce VAR(S (k)) decresed, whch tells us the vrton extent of S (k) s wekened. Usully, stedy seres s helpful to ncresng forecstng precson when t s used n modelng. III. ( ) α ( ) ( α) ( ) S ( k) = [ S ( k) ( α) S ( k )]/ α X ( k) = [ S ( k) ( α) S ( k )]/ α IMPROVED GM (, ) FORECASTING MODEL A. Orgnl GM (, ) Model Grey system theory founded n 98, hs been used n mny felds wdely [6]. The reson s becuse t hs remrkble dvntges such s poor dt, smple clculton, excellent fttng precson, bck-test crteron etc. wheren, GM (, ) s the model wth the most extensve pplcton. GM (, ) s frst-order dfferentl equton nd s used to del wth sngle vrble problem. The steps of GM (, ) modelng re lsted s followng: Step : Genertng frst-order ccumultve sequence. Supposng tht hstorcl dt of clerng prce ws wrtten s x, whch s group of dt vred wth the tme, nd the length of sequence s n. x s denoted s below: Then, we generted the frst-order ccumultve sequence: Step: Estblshng dfferentl equton bsed on ccumultve sequence. As we known, when x (k) vred pproxmtely ccordng to exponentl growth rule, ts expresson s the sme s the soluton of frst-order dfferentl equton. Therefore, the new sequence x (k) s consdered s meetng the frstorder dfferentl equton: d x ( t) + x () t = u (5) dt In the equton (5), the menng of prmeter s ntercepted s below. The prmeter, s clled s the development prmeter of model, stndng for development tendency of x s well s orgnl sequence. The prmeter u, s the coordnton prmeter, tells us trnsformton reltons of these seres. Wrtten A s A= [, u] T, every element of mtrx A re determned by lest squre pproch. Detled clculton of mtrx A s shown s equton (6). x = x, x (),, x ( n) x = x x x n, (),, ( ) k x ( k) = x ( ) = T T A = ( B B) B Y x () [ x + x ()] x (3) Y = B = [ x () + x (3)] x ( n) [ ( ) x n + x ( n )] Step3: Estblshng grey forecstng model. We put nd u nto equton (5) nd got the underlyng forecstng expresson: u u x ( k+ ) = [ x ]e k + (3) (4) (6) k=, n (7) 4508

3 The equton (7) s clled s functon of tme response for GM (, ). Wth ccumultve reducton clculton, the forecstng model of x s descrbed s followng: u k x ( k+ ) = x ( k+ ) x ( k) = ( e )[ x ]e k=, n (8) n, x ( k+ ) the (k+) th ntervl. B. Improvement of Intl Vlue Condton mens forecstng sequence of x t Accordng to the equton (7), x ( k+ ), the soluton of dfferentl equton, would be nfluenced drectly by ntl vlue x. Actully, x s not the most optml selecton when t s worked s ntl vlue [7]. Ths s becuse relton between x nd x ( k+ ) s not closely correltve, n ths cse, the soluton precson of the dfferentl equton mentoned bove would be nfluenced f x wth no specl proceedng s employed. Now, some modfcton nto the ntl vlue of smple dt, nd modfed formul s set s underlyng expresson. (9), s the modfcton tem to x, nd then the forecstng equton becomes new one: (0) () Equton (0) nd () cn be trnsformed nto expresson () nd (3): (3) () Checked () (3), we found tht the ddtonl modfcton tem,.e., the tem e -k or the tem (- e )e - k, presented modfcton to the conventonl forecst expresson. When equls to zero, expresson () nd (3) re reduced to () nd (). The prmeter s solved ccordng to the sme pproch s ntroduced bove, whch mens tht the sum of error squre between orgnl sequence nd forecstng sequence rrves to the mnmum. The clculton of s shown s followng: = + x x σ u u x ( k+ ) = [ x + σ ]e k + u x ( k+ ) = ( e )[ x + σ ]e x k + = x k + + σ ( ) ( ) e k x k x k σ n mn x ( k+ ) x ( k+ ) k = σ = p u [ x ] q k ( + ) = ( + ) + ( e )e k (4) (5) n p = x ( k)e k = ( k ) e n ( e ) e q = IV. GM (, ) MODEL WITH PERIOD RESIDUAL MODIFICATION Consderng tht clerng prce curve vred frequently, extensvely, perodclly, when we constructed fttng model by GM (, ), the postve nd negtve sgns of resdul sequence wll pper lterntely nd hs rregulr vrton perod. The precson of model s decresed when GM (, ) s dopted, nd the model s dffcult to reflect the fluctunt vrton of prce sequence. In ths cse, we nlyzed the perod of resdul sequence n the grey model nd found tht f we dvded the resdul sequence nto few sectons, of whch perod nd mpltude s dfferent, then used sne (or cosne) curve to ft the resdul sequence, the mproved model re ble to pproxmte the specl vrton of curve. The resdul modfed tem n every secton could be clculted ccordng to (6), referred to reference [3]. In ths cse, we nlyzed the perod of resdul sequence n the grey model nd found tht f we dvded the resdul sequence nto few sectons, of whch perod nd mpltude s dfferent, then used sne (or cosne) curve to ft the resdul t ˆ( ) sn Et = A (6) T The prmeter Eˆ ( t ) s the modfed tem t the t th ntervl n the th perod. A nd T stnd for the mxmum mpltude nd the length t the t th ntervl n the th perod respectvely. In order to smplfy clculton, mpltude of every secton cn be tken s verge of resdul bsolute vlue, lsted s follows: M ε ( j) j= A = M j=, M M s the number of concerned resdul sequence. The length of dvded secton s determned by the ntervl of sgn lterton of resdul sequence. Generlly, the resdul sequence contns few of sectons, some re referred to the segment wth postve sgn, nd others re referred to the segment negtve sgn. Of course, the dvson s ble to do some djustment ccordng to requrement from ctul stuton. Then, every resdul modfed tem s dded to the correspondng reduced expresson t the sme ntervl, shown s equton (7). xˆ ( t+ ) = x ( t+ ) + E( t+ ) ˆ (7) 4509

4 After the dsposl s dscussed bove, the resdul s decresed. Ths mens fttng curve s closer to the orgnl curve; therefore, precson of forecstng model s ncresed. For clerng prce n next dy, the chrcterstc of perodcl vrton s suggested to be smlr to the known sequence, the mpltude n every dvded secton s ssumed to be wthn the permtted rnge. Once these prmeters re determned, the resdul modfcton tem t every ntervl n future s clculted nd s dded to the forecstng expresson. V. ANALYSIS OF EXAMPLES In our reserch, the dt of clerng prce n Clforn mrket durng the 5 th - th, Mrch, 000 re tken s orgnl dt. Conventonl grey model nd modfed grey model re estblshed respectvely, nd these models re used to forecst the clerng prce on the th, 3 th, nd 4 th, Mrch, 000. Forecstng result s evluted ccordng to the followng ndexes: percentge of reltve error APE nd verge of reltve error MAPE, whch re wrtten s equton (8). x x δape = 00% x δ MAPE T x x = T x = =,T (8) T s the number of forecstng vlue. x represents the hstory clerng prce; x represents the forecsted clerng prce. In tble, we lsted out the ctul clerng prce, forecsted result of two models s well s ther error on the 4 th, Mrch, 000. After completng the forecstng clculton on the 4 th, Mrch, 000, we drew out the curves of the ctul clerng prce curve nd the forecsted curve of two models, whch s dsplyed n fgure. As fgure shown, the forecstng precson hs ncresed by the GM (, ) model wth perod resdul modfcton. Tble showed the forecstng error wth mproved GM (, ) model nd GM (, ) model on the th, 3 th, nd 4 th Mrch, 000. We notced tht the error of model wth perod resdul modfcton hs decresed, when compred wth generl grey model. The verge percentge of reltve error n three dys s 7.38%, whch reched the requrement of engneerng. In ddton, we use the forecst result on the sme dy to compre the modfed GM (, ) wth ARIMA, whch s one of commonly lgorthms used n tme seres forecst. Both of the models hve represented reltvely pproxmte forecst blty. Referred to the mxml reltve error for sngle ntervl n forecsted dy, the former s.80%, slghtly lower thn 3.85% of the ltter. As for the verge of reltve error, the former s 7.87%, slghtly hgher thn 7.69% of the ltter. TABLE I. Prce /(USD/MWh) COMPARISON OF CLEARING PRICE BETWEEN TWO MODELS UNIT: USD/MWH,% Intervl Hstory GM(,) generl model Improved GM(,) modfed model ctul vlue t/h Error of Improved GM(,) 0: : : : : : : : : : : : : : : : : : : : : : : : Fgure. Comprson between two models n forecstng prce TABLE II. ERROR PERCENTAGE OF TWO MODELS UNIT:% Dt Error of GM(,) Error of Modfed GM(,) 9.96% 7.53% 3.6% 8.8% 450

5 Dt Error of GM(,) Error of Modfed GM(,) % 5.8% VI. CONCLUSIONS In ths pper, we proposed new pproch of clerng prce forecstng, tht s grey model wth perod resdul modfcton. Frstly, we nlyzed the chrcterstc of the clerng prce,.e. t s wth uncompleted nd uncertn nformton. Therefore, clerng prce s tken s grey vrble. Menwhle, we mde used of dvntge of grey system theory, whch mnfested n poor smples, smple clculton nd forecst result beng tested. Secondry, qudrc smoothng technology ws ppled to orgnl prce sequence, nd then generted new sequence, whch decrese the fluctuton of sequence. Consderng the ntl vlue of grey model would nfluence the soluton of dfferentl equton, new pproch of ntl vlue desgn s dopted. Addng tem wth perod resdul modfcton nto GM (, ) mkes forecst curve fluctuted, whch would trce the tendency of orgnl sequence. Wth the bove dsposls, the forecst precson hs been mproved compred wth GM (, ). Fnlly, n exmple of Clforn electrcty mrket hs verfed tht when pplyng proposed pproch to the ctul system, the forecst result s stsfctory. And the forecst precson of modfed GM (, ) s compred wth tht of ARIMA, both s reltvely close. Deep reserch n grey model s helpful to mprove the forecst precson of clerng prce, whch would gve grey system model more extensve pplcton. REFERENCES [] Du Songhu, Wen Fushung, L Yng,et.l, Operton of electrc power system n deregulted mrket----predcton, pln, rsk mngement, Bejng: Chn Electrc Power Press,005. [] Wng Xfn, Zhng Xn, "Revew of the short-term electrcty prce forecstng," Automton of Electrc Power Systems, Vol.30, pp.9-0, Mrch 006. [3] Jver C, Rosro E, Frncsco J N, et l, "ARIMA models to predct next-dy electrcty prces, " IEEE Trns on Power Systems, Vol.8, pp.04-00, Mrch, 003 [4] Zhou Mng, Ne Ynl, L Gengyn, et l, "Wvelet nlyss bsed ARIMA hourly electrcty prces forecstng pproch, "Power System Technology, Vol. 9, pp.50-55, Sept.005. [5] Szkut B R, Snbr L A, Dllon T S, "Electrcty prce short-term forecstng usng rtfcl neurl networks, "IEEE Trns on Power Systems, Vol.4, pp , Mrch,999. [6] Zho Qngbo, Zhou Yunbn, Zeng Mng et l, "Applcton of fuzzy neurl network n power system mrgnl prce forecstng, "Power System Technology, Vol.8, pp.45-48,july, 004. [7] Wu Xnghu, Zhou Hu, "Short-term electrcty prce forecstng bsed on substrctve clusterng nd dptve neuro-fuzzy nference system," Power System Technology, Vol.3, pp.69-73, Sept [8] Ln qyou, Chen Xngyng, Wng Zhwe, "Applcton of dt mnng n electrcty prce forecstng," Power System Technology, Vol.30, pp.83-87, Dec.006. [9] Lu Sfeng, Dng Yoguo, Fng Zhgen et l, Grey system theory nd ts pplcton, Bejng,: Scence Press, 004. [0] Lu Gungjn, Hu Sngo, D Junlng, "The chotc property of system mrgnl prce nd ts forecstng," Proceedngs of the CSEE, Vol.3, pp.6-8, My, 003. [] Su Jun, Du Songhu, "The GM (, ) short-term spot prce forecstng grey model," Rely, Vol.36, pp.46-49, Jun.006. [] Cheng Xoxn, Zhou Yuhu, "Reserch on electrcty prce forecstng bsed on mproved grey model, Journl of North Chn Electrc Power Unversty," Vol.33, pp.47-50, Jun.006. [3] Zhng Xnwen, Wng Xuemeng, Ne Hongsheng, Anlyss of rurl economcs grey system, Bejng: Acdemc Journl Press, 989. [4] Ho Yunhong, Hung Dengyu, Zhng Wenzhong, et l, "Perod resdul modfcton of GM (, ) modelng nd ts pplcton n predctng the sprng dschrge," Mthemtcs n Prctce nd theory, Vol.33, pp.35-37, Sept.003. [5] J.L.Deng, Introducton to Grey System Theory, Interntonl Journl of Grey System, 987, Vol., 3, PP-4 [6] Zho Xoyn, Lu Tnjo, Zhou Bo, et l, "The smoothng mprovement nd the pplcton of grey model GM (, ), Journl of Northest Dnl Unversty (Nturl Scence Edton), Vol.6, pp.63-66, Aprl 006. [7] Zhng Hu, Hu Shgeng, "Anlyss of boundry condton for GM (, ) model, Journl of Huzhong unversty of Scence nd Technology, Vol.9, pp.0-, Aprl, 00 45