Short Time Price Forecasting for Electricity Market Based on Hybrid Fuzzy Wavelet Transform and Bacteria Foraging Algorithm

Transcription

1 Shor Time Price Forecasing for Elecriciy Marke Based on Hybrid Fuzzy Wavele Transform and Baceria Foraging Algorihm Keivan Borna* Deparmen of Compuer Science, Faculy of Mahemaics and Compuer Science, Kharazmi Universiy, Tehran, Iran Sepideh Palizdar Deparmen of Compuer Engineering, Faculy of Engineering, Kharazmi Universiy, Tehran, Iran Received: 3/Jan/6 Revised: /Mar/6 Acceped: /Jun/6 Absrac Predicing he price of elecriciy is very imporan because elecriciy can no be sored. To his end, parallel mehods and adapive regression have been used in he pas. Bu because dependence on he ambien emperaure, here was no good resul. In his sudy, linear predicion mehods and neural neworks and fuzzy logic have been sudied and emulaed. An opimized fuzzy-wavele predicion mehod is proposed o predic he price of elecriciy. In his mehod, in order o have a beer predicion, he membership funcions of he fuzzy regression along wih he ype of he wavele ransform filer have been opimized using he E.Coli Bacerial Foraging Opimizaion Algorihm. Then, o beer compare his opimal mehod wih oher predicion mehods including convenional linear predicion and neural nework mehods, hey were analyzed wih he same elecriciy price daa. In fac, our fuzzy-wavele mehod has a more desirable soluion han previous mehods. More precisely by choosing a suiable filer and a muliresoluion processing mehod, he maximum error has improved by 3.6%, and he mean squared error has improved abou 7.9%. In comparison wih he fuzzy predicion mehod, our proposed mehod has a higher compuaional volume due o he use of wavele ransform as well as double use of fuzzy predicion. Due o he large number of layers and neurons used in i, he neural nework mehod has a much higher compuaional volume han our fuzzy-wavele mehod. Keywords: Price Predicion; Wavele Transform; Fuzzy Logic; Baceria Foraging Algorihm; Elecriciy Marke.. Inroducion In he las couple of decades, he elecric power indusry, in mos counries of he world, has undergone radical and fundamenal changes which are referred o by differen names such as deregulaion, resrucuring, law revision, ec.. Also in Iran, changes in he elecric power indusry, began wih he launch of Iran's elecriciy marke in November 3. Wih Iran Grid Managemen Company being esablished in 4, Iran's elecriciy marke ook a more serious shape. Also, considering he approval of he implemening regulaions relaed o paragraph (b) of Aricle 25 of he Third Developmen Plan law by he Council of Minisers, he approval of he law relaed o he independence of disribuion companies by he Islamic Consulaive Assembly, and he inerpreaion of Aricle 44 of he consiuion by he supreme leader, Iran's elecriciy indusry will undergo fundamenal changes in he coming years []. A lo of work has been done wih respec o price predicion. In 2, a paper was presened in which an auoregressive inegraed moving average (ARIMA) model has been used o predic he prices in he elecriciy marke. Since he price in he elecriciy marke depends on various facors, by comparing he resuls of his sudy wih fuure sudies, i can be seen ha his mehod has no had an appropriae soluion. This mehod does no have a desirable soluion especially where he price changes are big. In 9, a paper was presened in which a hybrid model, which combined an auoregressive inegraed moving average (ARIMA) model and generalized auoregressive condiional heeroskedasiciy (GARCH) model, has been used o predic he average daily price in Iran's elecriciy marke. This mehod had a medium compuaional volume, and according o he resuls obained, i can be seen ha his forecasing mehod has no had a desirable soluion where he speed of price changes were high. [] Since he ime series models did no have a desirable soluion in drasic changes in he desired signal, hus a paper was presened in 8, in which he wavele ransform and a neural nework have been used. One of he capabiliies of he wavele ransform is ha, where drasic changes occur, he lengh of he window, in which he signal is evaluaed, becomes smaller and calculaions will be done more carefully, and among he disadvanages of his mehod an increased compuaional volume can be menioned. [5] Reference [7] has presened a mehod in which a parallel mehod has been used o reduce predicion errors, * Corresponding Auhor

2 Journal of Informaion Sysems and Telecommunicaion, Vol. 4, No. 4, Ocober-December 6 2 which has been used o predic he nex day price of he elecriciy marke, and in which he daa prediced in he previous period are used as daa for he nex period. In 9, a mehod was presened based on adapive regression. According o he aricle, oher predicion mehods can a mos predic up o nex 3 or 4 seps, bu his mehod can predic up o nex seps. A disadvanage of his mehod is ha i requires he ambien emperaure daa simulaneously wih he elecriciy price daa for predicion. [8]. In he nex secion we consider linear predicion. price of marke elecriciy from he previous ime period. The figure below shows how o choose a neural nework. 2. Linear Predicion In his secion we will explain linear predicion mehod. A linear predicion model represens he ime series of he signal samples during a given ime period. Is usual linear model is as follows [8]: T a. y a. 2 y T a. y m T () y m. where a, a 2,..., a m are linear predicion coefficiens, m is he model order, T is he sampling ime, y(+t) is he fuure sample, and y(), y(-t),..., y(-mt) are he presen and pas observaions. In his relaion, he funcion oupu is a linear combinaion of he presen and pas samples, hus his funcion is called a linear predicion. Two sages have o be done o achieve a linear predicion of prices in he elecriciy marke using equaion (). According o his equaion, firs he model order has o be chosen carefully, and hen he coefficiens a, a 2,..., a m have o be calculaed from he modeling window. And hen he obained model can be used o predic he price of elecriciy marke in he ime seps ahead. A leas-squares error mehod can be used o esimae he coefficiens a, a 2,..., a m in equaion (). This error is measured beween he esimaed value and acual value. In use of leas-squares mehod, he energy in he error signal, falls o is minimum. A soluion o his problem is vecor B which esimaes he unknown vecor of parameer β. The leas-squares soluion is as follows: T T X. X. X. y ˆ b (2) In he nex secion predicion using he neural nework mehod is sudied. 3. Predicion Using he Neural Nework Mehod In his secion we will explain predicion using he neural nework mehod and he ype of neural nework ha we use is considered hroughly. Due o heir remarkable abiliies o infer resuls from complex and ambiguous daa, neural neworks can be used o exrac paerns and idenify various rends, which are very difficul o idenify for humans and compuers. In his sudy, a back-propagaion neural nework is used, wih 5 layers and 5 inpus, all of which are he daa of he Fig.. The srucure of a BP neural nework for price predicion in he elecriciy marke 4. Predicion Using he Fuzzy Regression Mehod In he nex secion we sudy he hisory of fuzzy logic and fuzzy regression. Dr. Lofizadeh inroduced he heory of fuzzy sysems in 965. In such an amosphere where he scieniss of engineering sciences were seeking for mahemaical mehods o defea more difficul problems, he fuzzy heory ook seps oward anoher kind of modeling. In convenional fuzzy sysems, he number and ype of membership funcions are deermined by rial and error. Bu wha is obvious is ha, a larger number of membership funcions are needed for more complicaed sysems. On he oher hand, as he number of membership funcions increases, he number of fuzzy rules usually increases, which ulimaely leads o he complexiy of implemenaion. Membership funcions can have differen shapes, such as riangular, Gaussian, bell, rapezoidal, ec.. In he linear regression, he goal is o find he fuzzy coefficiens of he polynomial below. In oher words, he goal is o express he linear predicion using he fuzzy coefficiens. Is usual linear model is as follows [8]: 2 m y T A. y A. y T A. y m. T (3) Where; ( )s are fuzzy coefficiens which are expressed by fuzzy membership funcions. Here he goal is o find ( )s in a way ha he predicion error is minimized. 5. Our Propsed Mehod: Fuzzy-Wavele Predicion In his secion he hisory of wavele and our proposed mehod which is combinaion of fuzzy predicion mehod and wavele is presened. The consan resoluion problem in he shor-ime Fourier ransform, has is roos in Heisenberg's uncerainy principle. According o his principle, a imefrequency descripion of a signal canno be achieved exacly; ha is, i canno be found ou exacly ha a a given signal, wha frequency componens are available a wha ime inervals, bu we can only found ou ha wha

3 22 Borna & Palizdar, Shor Time Price Forecasing for Elecriciy Marke Based on Hybrid Fuzzy frequency bands are available a wha ime inervals. This principle direcly reurns o he concep of resoluion. Alhough he ime and frequency resoluion problems are resuls of a physical phenomenon (heheisenberg uncerainy principle) and exis regardless of he ransform used, i is possible o analyze anysignal by using an alernaive approach called he muliresoluion analysis (MRA). The wavele ransform is a kind of windowing echnique wih variable- sized windows. The wavele analysis gives us he possibiliy o achieve our goal boh in a long duraion where we require high accuracy a low frequency daa, and in shorer duraions where we need high-frequency daa. The wavele ransform does no conver ino a ime-frequency region, bu raher ino a ime-scale region. Using he wavele ransform, he price signal of elecriciy marke can be divided ino daa wih big changes (deails) and daa wih lile changes (generaliies). Figure 2 shows an overview of his conversion. Fig. 2. The srucure of he wavele ransform for price predicion in he elecriciy marke. Since in order o increase he accuracy of predicors when many changes occur, i is necessary o increase he number and/or degrees of membership funcions, which increases he volume of calculaions, anoher soluion is o use he muliresoluion analysis, in his way ha; high frequency daa (deails) are esimaed by a predicor and low frequency daa (generaliies) by anoher predicor. By doing his, he desired accuracy and less compuaional volume can be achieved. This paper uses a combinaion of wavele ransform and fuzzy regression o predic he price of elecriciy marke. Fig. 3. shows an overview of his sysem. Figure 3. The general srucure of he proposed fuzzywavele ransform, for price predicion in he elecriciy marke. To compare he predicion mehods presened in his paper, some indicaors are defined as follows. The mean absolue percenage error (MAPE): This error is defined as follows: n MAPE PE (4) n Where; n is he number of daa, and PE represens a relaive error and is defined as follows: y F y PE (5) If y is he acual observaion for duraion, and F a forecas for he same duraion, hen he error will be defined as in equaion (6): e y F (6) The maximum forecas error: is he greaes value of error in he predicion for he es daa se which is defined as in equaion (4-): ME max(price Price ) (7) acual prediced The maximum forecas percenage error is defined as follows: Priceacual Price prediced MPE max( ) (8) Price acual Also he mean squared error (MSE) is defined as follows: MSE N 2 Priceacual Price prediced (8) k,..., N In he nex secion he simulaion resuls and comparisions are presened.

4 Journal of Informaion Sysems and Telecommunicaion, Vol. 4, No. 4, Ocober-December Simulaion and Resuls In order o deermine he bes model in quaniaive erms, he hree measures of predicion errors: MSE, mean absolue error (MAE), and mean absolue percenage error (MAPE), were used o evaluae and compare he models. For a beer evaluaion, he resuls of hese forecasing mehods as well as heir insananeous errors are presened in Figures 4 o. By aking hem ino consideraion, i can definiely be concluded ha he values resuling from he wavelefuzzy forecasing mehod has a beer soluion han he previous forecasing mehods. Price, Euro/ kw Price Original Prediced Price by LP Fig. 4. The resuls obained from he linear predicion, for price predicion in he elecriciy marke Error or prediced price, Euro/kW Fig. 8. The insananeous error for price predicion in he elecriciy marke, using he linear predicion mehod Error or prediced price, Euro/kW Number of sampels (every 3 Min.) Fig. 9. The insananeous error for price predicion in he elecriciy marke, using he neural nework predicio mehod Price, Euro/kW Price Original Prediced Price by ANN Fig.. The insananeous error for price predicion in he elecriciy marke, using he fuzzy regression predicion mehod Fig. 5. The resuls obained from he neural nework predicion, for price predicion in he elecriciy marke Fig. 6. The resuls obained from fuzzy predicion, for price predicion in he elecriciy marke Error or prediced price, Euro/ kw - - Fig.. The insananeous error for price predicion in he elecriciy marke, using our fuzzy-wavele predicion mehod Table. Comparison of errors among he predicion mehods presened [3] Mehod MAPE (%) Maximum (%) MSE Volume of calculaions Linear predicion[3] 2.59% High Neural nework predicion[3] 6.2% Medium Fuzzy predicion[3] 9.4% Medium Fuzzy-wavele predicion[our mehod] % High Fig. 7. The resuls obained from our fuzzy-wavele predicion, for price predicion in he elecriciy marke

5 24 Borna & Palizdar, Shor Time Price Forecasing for Elecriciy Marke Based on Hybrid Fuzzy 7. Conclusion By evaluaing he resuls obained for price predicion in Queensland elecriciy marke, i can be seen ha use of fuzzy logic-wavele forecasing mehod resuled in an improved performance, compared wih ha of fuzzy logic forecasing mehod. Also choosing wo differen ypes of filers; low-pass and high-pass, in he wavele ransform, increased he efficiency of he predicor in he fuzzy predicion mehod. To furher invesigae he presened mehods, he resuls of hese mehods have been colleced in Table. As can be seen, he fuzzy-wavele mehod has a more desirable soluion han he oher presened mehods have, also by choosing a suiable filer and a muliresoluion processing mehod, he maximum error has improved by 3.6%, and he mean squared error has improved abou 7.9%. Bu in comparison wih he fuzzy predicion mehod, he proposed mehod has a higher compuaional volume due o use of wavele ransform as well as double usage of fuzzy predicion. Due o he large number of layers and neurons used in i, he neural nework mehod has a much higher compuaional volume han our fuzzy-wavele mehod has, bu his mehod, depending on he daa used for raining, has a greaer maximum error han he proposed mehod has. References [] Inernaional Energy Oulook 5, Energy Informaion Adminisraion; hp:// [2] Pael, M.R., Elecriciy price and solar power sysems, CRC Press LLC, 999. [3] Global power source, Global elecriciy price energy council, hp:// [4] Coa, A., A review on he young hisory of elecriciy price power shor erm predicion, Journal on Renewable Energy Review, vol. 2, Issue 6, pp , 8. [5] Sideraos, G. Haziargyriou, N.D. An Advanced Saisical Mehod for Elecriciy price Power Forecasing, IEEE Transacion on power sysems, Na. Tech. Univ. of Ahens, vol. 22, Issue, pp , Feb 7. [6] Hui, L., Hong-Qi, T., Chao, C., Yan-fei, L. A Hybrid Saisical o Predic Elecriciy price Speed and Elecriciy price Power, Renewable energy, Science direc, December 9. [7] Monfared, M., Rasegar, H., Kojabadi, H. M. On Comparing Three Arificial Neural Neworks for Elecriciy price Speed Forecasing, Applied energy, Science direc, January. [8] MATLAB, Mahemaical Foundaions of Muliple Linear Regressions, R7a. [9] Sahin, A. D., Zekai, S., Firs-order Markov chain approach o elecriciy price speed modeling, Journal of wind Engineering and Indusrial Aerodynamics 89 () [] Shamshad, A., Bawadi, M.A., Wan Hussin, W.M.A., Majid, T.A., Sanusi, S.A.M., Firs and second order Markov chain models for synheic generaion of elecriciy price speed ime series, Energy 3 (5) [] Kennedy, J., and Eberhar, R., Paricle Swarm Opimizaion, IEEE Inernaional Conference on Neural Neworks, pp , 995. [2] M. Monfared, H. Rasegar, H. M. Kojabadi A New Sraegy for Elecriciy price Speed Forecasing Using Arificial Inelligen Mehods, Renewable energy, Science direc, Vol. 34, Issue 3, pp , March 9. [3] A. Moamedi, H. Zareipour, W.D. Rosehar, "Elecriciy Price and Demand Forecasing in Smar Grids", IEEE Transacions on Smar Grid, vol. 3, pp , 2. Keivan Borna joined he Deparmen of Compuer Science a he Faculy of Mahemaics and Compuer Science of Kharazmi Universiy as an Assisan Professor in 8. He earned his Ph.D. in Mahemaics from he Deparmen of Mahemaics, Saisics and Compuer Science of he Universiy of Tehran. His research ineress include Compuer Algebra, Crypography, Approximaion Algorihms, Evoluionary Compuaions and Compuaional Geomery. He is he auhor of he "Advanced Programming in JAVA" (in Persian) and is a life member of "Elie Naional Foundaion of Iran". Sepideh Palizdar received her M.Sc. degree from Faculy of Engineering a Kharazmi Universiy, Tehran, Iran a 5. Her research ineress include evoluionary compuaions.