Short-Term Load Forecasting for Electric Power Systems Using the PSO-SVR and FCM Clustering Techniques

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1 Energes 20, 4, 73-84; do:0.3390/en40073 Artce OPEN ACCESS energes ISSN Short-Term Load Forecastng for Eectrc Power Systems Usng the PSO-SVR and FCM Custerng Technques Pan Duan, *, Kagu Xe 2, Tngtng Guo 2 and Xaogang Huang State Key Laboratory of Power Transmsson Equpment & System Securty and New Technoogy, Chongqng Unversty, Chongqng, , Chna Schoo of Automaton Engneerng, Chongqng Unversty, , Chna; E-Mas: kaguxe@yahoo.cn (K.X.); guottcqu@yahoo.cn (T.G.) Chongqng Tongnan Eectrc Power Company, Chongqng, , Chna; E-Ma: huangxgcqu@yahoo.cn * Author to whom correspondence shoud be addressed; E-Ma: duanpancqu@gma.com; Te.: Receved: 4 November 200; n revsed form: 3 December 200 / Accepted: 5 January 20 / Pubshed: 20 January 20 Abstract: Ths paper presents a new combned method for the short-term oad forecastng of eectrc power systems based on the Fuzzy c-means (FCM) custerng, partce swarm optmzaton (PSO) and support vector regresson (SVR) technques. The tranng sampes used n ths method are of the same data type as the earnng sampes n the forecastng process and seected by a fuzzy custerng technque accordng to the degree of smarty of the nput sampes consderng the perodc characterstcs of the oad. PSO s apped to optmze the mode parameters. The compcated nonnear reatonshps between the factors nfuencng the oad and the oad forecastng can be regressed usng the SVR. The practca oad data from a cty n Chongqng was used to ustrate the proposed method, and the resuts ndcate that the proposed method can obtan hgher accuracy compared wth the tradtona method, and s effectve for forecastng the short-term oad of power systems. Keywords: oad forecastng; short-tme oad; PSO

2 Energes 20, Introducton Short-term oad forecastng (STLF) ams at predctng the system oad over a short tme perod ke a day or a week, and pays an mportant roe n power system operaton. Many oad forecastng approaches have been reported n the ast decade, such as conventona smoothng technque, regresson method, statstca anayss, tme seres anayss, autoregressve movng average mode, and expert system [,2]. Athough these technques and modes are reabe and usefu, they are not sutabe for anayzng unusua weather condtons and vared hoday actvtes, whch usuay resut n a day oad wth hghy non-near characterstcs. Load forecastng for any gven day s a dffcut task, because t depends not ony on the oad of the prevous days, but aso on that on the same day n the prevous weeks and even the prevous years [3]. Furthermore, t s dffcut to mode the reatonshp between the oad and the mentoned externa factors nfuencng the oad demand, such as weather varatons, hoday actvtes, etc. These are the major factors makng the modeng process more compcated. Because the mode parameters are decded accordng to the hstorca data, some errors may be ntroduced [4]. The tme seres methods treat the oad pattern as a tme seres sgna wth known perod, and offer a rough predcton of the oad for the gven season. The technques used n tme seres methods are as foows [5,6]: (a) Kaman Fter method; (b) Box Jenkns method; (c) Regresson processes; (d) Spectra expanson technque. The spectra expanson technque utzes the Fourer seres. Generay, these technques use a arge number of compex reatonshps and requre ong computaton tmes. An ntegent system exhbts ntegence n capturng and processng nformaton, and empoys artfca ntegence technques to fuf some or a of ts computatona requrements. Artfca Neura Networks (ANNs) are capabe of performng non-near modeng and adaptaton. In addton, they do not requre a functona reatonshp between oad and weather varabes n advance. An expert system has the abty to act as a knowedge expert, therefore the oad-forecast mode can be but usng the knowedge about the oad-forecast doman provded by an expert n the fed [7 9]. Recenty methods oad forecastng based on the oad smarty were reported, whch forecast the oad curve by usng the oad nformaton of the days wth weather condtons smar to the objectve day. In these methods severa smar days are seected to mprove the accuracy of oad forecastng, whch can not ony dea wth the non-near reatonshp of the oads, but aso the nfuence caused by weekends or speca days. However the weghts of dfferent nfuencng factor cannot refect the actua degree of nfuence due to the ack of adequate cognton of these factors. In order to sove the probems mentoned above, ths paper presents a nove preprocessng procedure based on the PSO-SVR and the FCM custerng technques, whch can mprove the precson of day oad forecastng. Practca oad data from a cty n Chongqng, Chna, was used to demonstrate the proposed technques, and the resuts ndcate that the proposed method s effectve for forecastng the short-term power system oad. The remanng parts of the paper are arranged as foows: Secton 2 ntroduces the support vector regresson based on PSO. Secton 3 descrbes the short-term oad forecastng process usng the PSO-SVR and FCM custerng technques, and an anayss of a practca case study foows n Secton 4. Fnay the concusons w come at Secton 5.

3 Energes 20, Support Vector Regresson Base on PSO Partce swarm optmzaton (PSO) was ntroduced by Eberhart and Kennedy n 995 [0]. A PSO has two prmary operators: a veocty update operator and a poston update operator. Durng the convergence process, each partce s acceerated toward the partces wth prevous best poston and the goba best poston. A new veocty for each partce s cacuated n an teraton based on ts current veocty and the dstances from ts prevous best poston and the goba best poston. The new veocty s then used to cacuate the next poston of the partce n the search space. Ths process s repeated unt a pre-specfed mnmum error s met. The updated veocty and poston are gven by: k+ k k k vd = w vd + c r ( pd xd ) + c2 r2 ( pgd xd ) k+ k+ k xd = vd + xd () Support vector regresson (SVR) [] s a regresson method based on the SVM method. Dfferent from SVM, SVR tres to fnd a hyper-pane whch can accuratey predct the dstrbuton of nformaton, but not the pane on whch to cassfy the data. However, the prerequste for SVR to acheve the better resuts s to fnd three approprate parameters: the kerne functon K, the baance factor C and the nsenstve oss functon ε, whch are crtca for achevng hgh accuracy and generazaton of the regresson mode. For a tranng sampe set: T = {( x, y )} x R m s the nput vector, y R s the correspondng objectve output and s the number of tranng sampes. Usng a nonnear mappng φ ( x) R m R M, the SVR maps a ow dmensona nput space to a hgh dmensona feature space, then t regresses neary n a hgh dmensona feature space, whch can demonstrate the nfuence of nonnear regresson n orgna nput space. The genera functon of a decson near regresson s: = f ( x) = ( w φ( x)) + b (2) where w s a weght vector, and b s a threshod vaue. The parameters of the optmum regresson n Equaton (2) are soved usng (3) to mnmze the structura rsk R: 2 mn R = w + CR emp (3) 2 where: R = ( L ( x, y f( x ))) (4) emp e = L ( x, y f( x)) = max{0, y f( x) ε} (5) e The tem w 2 /2 n (3) s the expressng abty of the compexty of the decson functon; the tem R emp s the tranng error, and C s the baance factor whch ndcates the degree of matchng of the two parts above. Experence rsk R emp s measured by the oss functon, whch usuay adopts the nsenstve oss functon L ( x, y f( x)). e

4 Energes 20, 4 76 By ntroducng the sack varabes ξ and ξ ( =,2,..., ), Equaton (3) can be rewrtten as: C mn ( ) (6) 2 w + ξ + ξ 2 = ( w φ( x)) + b y ε + ξ St : y ( w φ( x)) b ε + ξ ε, ε 0( =,2,..., ) Equaton (6) s caed the prmtve probem of SVR. The prmtve probem s actuay a probem wth near constrants. Introducng the Lagrange mutpers to the Lagrange functon, the parta dervatve of the Lagrange functon wth respect to w,b, ξ, ξ can be obtaned, and ts dua probem s gven by: mn ( α α )( α α ) (, ) ε ( α α ) ( α α ) (7) j j K x xj + + y 2 = = = ( α α ) 0 = = St : C 0 α, α (,2,..., ) = b = yj ( α )( ) α x, xj + ε = C f α j (0, ) St : b = yj ( α α )( x xj) ε, = C f α j (0, ) where K( x, xj) = ( φ( x) φ( xj)) s a kerne functon. The kerne functon s used to obtan the decson-makng regressve equaton wthout knowng the defnte form of nonnear mappng φ ( x) from the ow dmensona nput space to hgh dmensona feature space. Assumng that α = ( α, α,..., α, α) T s the souton of dua probem, there s at east a zero n a par of ( α, α ) accordng to the sparse characterstc of the SVR. Decson functon s ony determned by support vectors and has nothng to do wth nonsupport vectors. Decson regresson functon (2) can be wrtten as: f ( x) = ( α α ) ( ) K x, x + b (8) = 3. Short-Term Load Forecastng Usng the PSO-SVR and FCM Custerng Technque Ths secton presents the optmum FCM custerng anayss and a new short-term oad forecastng method usng the PSO-SVR and FCM custerng technque. Fuzzy c-means (FCM) s a custerng method whch aows a arge number of data to be cassfed nto two or more custers. Ths method (deveoped by Dunn n 973 [2] and mproved by Bezdek n

5 Energes 20, [3]) s wdey used n pattern recognton. The FCM s formuated as a mode of mnmzng the foowng objectve functon: N C m 2 Jm = uj x cj, m< (9) j where m s any rea number greater than, u j s the degree of membershp of x beongng to custer j, x s the th of d-dmensona measured data, c j s a d-dmenson center of the custer, and * s any norm expressng the smarty between any measured data and the center. Fuzzy parttonng s carred out through an teratve optmzaton of the objectve functon shown above, wth the updates of membershp u j and the custer centers c j by: u j = C x cj x c k = k 2 m (0) c j = N m uj x = N m uj = () The teraton s stopped when ( k+ ) ( k) max j{ j j } u u < ε, where ε s a pre-specfed error crteron and k s the teraton step. Ths procedure converges to a oca mnmum or a sadde pont of J m. The man dea for the short-term oad forecastng s as foows: () Use the FCM custerng agorthm to obtan the optmum oad pattern cassfcaton sampes consderng the change rues of the hstorca day oad n a power system. (2) Use the pattern recognton method to extract the optma tranng sampes. (3) Construct the oad forecastng mode usng the PSO-SVR regresson agorthm. Ths method can reduce the number of tranng sampes and mprove the predcton accuracy of support vector machnes. The foowng are the detas for ths method: () Pre-process the hstorca oad data and normaze the basc data. A basc sampe for each predcted tme can be obtaned based on the nput-output oad data, and the forecastng nput sampe xt ( t =,2,, N) s formed. (2) Set the kerne functon and mode parameters, and seect the weght of the FCM custer anayss, here m =.3, the argest possbe number of custers c max = n, where n s the oad sampe number of the dataset used n custerng anayss, then set the pre-specfed teratve convergence 6 error of FCM custerng agorthm, here ε = 0. The basc sampe set for each predct tme n the objectve day s anayzed usng optma FCM custer method, then the cassfed nformaton of the basc sampe set wth the best custers number can be coected. The dstance between the forecast tme perod of the nput sampes x t s cacuated, and the mnmum dstance correspondng to the sub-cass as the PSO-SVR forecastng methods sampe set w be set. By usng the sampe set at each forecastng tme and the seected custerng parameters, the PSO-SVR oad forecastng mode s constructed, and the parameters for support vector machne regresson

6 Energes 20, 4 78 agorthm are obtaned. The objectve functon as Equaton (7) can be formed and soved. Fnay, by substtutng the optma souton of SVR modeα andb to Equaton (8), the optma regresson functon at each tme perod can be 0. (3) Use the forecast nput sampes and the optma regresson functon at each tme perod to predct the oad n the future. The deta of the oad forecastng process usng the PSO-SVR and FCM custerng technques s shown n Fgure. Fgure. Fow chart of the oad forecastng process. 4. Smuaton Resuts and Dscusson In order to verfy the feasbty of the proposed method, a case study on the oad forecastng from a cty n Chongqng, Chna, was used. The tradtona methods are frst used to forecast the oad, and the method n ths paper s aso used to forecast the short-term oad, and comparsons were made to anayze the characterstcs of the dfferent methods. The hstorca oad data s shown n Tabe. Frsty the tme seres method s used to forecast the short-term oad, and Tabe 2 shows the oad forecastng resuts usng the tme seres method. It can be concuded that the maxmum forecastng error s 9.362%, whch s a tte arge for practca engneerng. The average error of the tme seres method s.36%.

7 Energes 20, 4 79 Tabe. Hstorca oad data. Month Tme(hour) : : : : : : : : : : : : : : : : : : : : : : : : Tme (hour) Actua Load (MW) Tabe 2. Load forecastng usng the tme seres method. Forecastng Load (MW) Forecastng Tme (hour) Actua Load (MW) Forecastng Load (MW) Forecastng 0: : : : : : : : : : : : : : : : : : : : : : : :

8 Energes 20, 4 80 Then the method based on PSO-SVR and method n ths paper were used, respectvey, to forecast the oad on December th, and the resuts are shown n Tabe 3. It can be concuded that the average error for the PSO-SVR method s.443%, and the average error for the method presented n ths paper s.066%. The maxmum errors for the dfferent methods are 3.25% and 2.798%, therefore the method n ths paper mproves the accuracy of the short-term oad forecastng. Tabe 3. Comparson of oad forecastng on December th between the method based on PSO-SVM and method n ths paper. PSO-SVR Mode FCM Based PSO-SVR Mode Tme Actua Load Forecasted Forecastng Forecasted Forecastng (hour) (MW) Load (MW) Load (MW) 0: : : : : : : : : : : : : : : : : : : : : : : : Average Error (Absoute Vaue ) Maxmum Error (Absoute Vaue ) To make a cearer comparson between the tme seres method, the method based on PSO-SVM, and the method n ths paper, the comparson between these methods and the rea oad s made as shown n Fgure 2. It can be concuded that the method n ths paper s mosty cose to the rea oad. The houry forecastng error curves of the three oad forecastng modes are shown n Fgure 3, whch shows the oad at dfferent hours on December, t aso ndcates that the error between the rea oad and the method n ths paper s the owest.

9 Energes 20, 4 8 Fgure 2. Comparson of forecastng oads usng dfferent methods oad actua tme seres PSO-SVR PSO-SVR based on the FCM 2600 oads/mw hours/h 5 Fgure 3. Comparson of oad forecastng errors usng dfferent methods. forecast Proportona error percent 0-5 tme seres propotrona error PSO-SVR propotrona error PSO-SVR based on FCM proportona error hours/h For further study, the oad on the weekend s forecasted by the tme seres method, the PSO-SVR mode and method n ths paper, respectvey, and the resut s shown n Fgure 4. The same concuson can be reached.

10 Energes 20, 4 82 The above forecastng s regardng the oad at dfferent ponts n a day. In addton, the oad forecastng on the dfferent day n one week s carred out wth the dfferent methods, and the resuts are shown n Tabe 4. It can be concuded from the Tabe that the technque proposed n ths paper has a hgher accuracy compared wth the conventona mode. Tabe 4 and Tabe 5 aso show that the proposed mode mproves accuracy of the houry oad forecastng sgnfcanty. Fgure 4. Comparson of the oad forecastng vaue on the weekend actua oad tme seres PSO-SVR PSO-SVR based on FCM oad/mw Date hours/h Tabe 4. Comparson of oad forecastng usng dfferent methods. Average Absoute Tme Seres Maxmum PSO-SVR Mode Average Absoute Maxmum FCM Based PSO-SVR Mode Average Maxmum Absoute Average n a week

11 Energes 20, Concusons Tabe 5. Load forecastng errors n a week. Date Average Maxmum Error > 3% Error > 4% An mproved method for the short-term oad forecastng s presented to mprove the forecastng accuracy of day oads, especay n the scenaro where the day oad s affected by weather and date factors serousy. Consderng the perodca feature of oad, the optma FCM custerng was used to obtan the optma data pattern dvson of hstory oad sampes and the best tranng sampe set. The data reguarty of nput-output functon reaton n the tranng sampes can be enhanced, the consstency of data characterstcs can be ensured. The effectve ntegraton of PSO-SVR and the optma FCM custerng agorthm s acheved. The oad forecastng resuts of a cty n Chongqng ustrate the proposed hybrd method s effectve, and not ony reduces the number of tranng sampes, but aso mproves the oad forecastng accuracy. Acknowedgement The authors woud ke to acknowedge the grant from the Fundamenta Research Funds for the Centra Unverstes (Project No. CDJXS50022). References. L, Y.B.; Zhang, N.; L, C.B. Support vector machne forecastng method mproved by chaotc partce swarm optmzaton and ts appcaton. J. Cent. South Unv.Techno. 2009, 6, Wang, B.; Ta, N.L.; Zha, H.Q.; Ye, J.; Zhu, J.D.; Q, L.B. Hybrd optmzaton method based on evoutonary agorthm and partce swarm optmzaton for short-term oad forecastng. Proc. CSU-EPSA 2008, 20, Wang, L.J.; Lu, C. Short-term prce forecastng based on PSO tran BP neura network. Eectr. Power Sc. Eng. 2008, 24, L, X.M.; Gong, D.; L, L.; Sun, C.Y. Next day oad forecastng usng SVM. Lect. Notes Comput. Sc. 2005, 3498, Mahaanabs, A.K.; Kothar, D.P.; Ahson, S.I. Computer Aded Power System Anayss and Contro; Tata McGraw-H Pubshng Company mted: New Deh, Inda, Box, G.E.; Jenkns, G.M. Tme Seres Anayss Forecastng and Contro; Hoden-Day: San Franssco, CA, USA, 976.

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