Medium Term Load Forecasting for Jordan Electric Power System Using Particle Swarm Optimization Algorithm Based on Least Square Regression Methods

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1 Journal of Power and Energy Engneerng, 2017, 5, ISSN Onlne: ISSN Prnt: X Medum Term Load Forecastng for Jordan Electrc Power System Usng Partcle Swarm Optmzaton Algorthm Based on Least Square Regresson Methods Mohammed Hattab 1, Mohammed Ma tah 2, Tha er Swedan 2*, Mohammed Rfa 1, Mohammad Moman 1 1 Electrcal Power Engneerng Department, Yarmouk Unversty, Irbd, Jordan 2 Electrcal Engneerng Department, The Hashemte Unversty, Zarqa, Jordan How to cte ths paper: Hattab, M., Ma tah, M., Swedan, T., Rfa, M. and Moman, M. (2017) Medum Term Load Forecastng for Jordan Electrc Power System Usng Partcle Swarm Optmzaton Algorthm Based on Least Square Regresson Methods. Journal of Power and Energy Engneerng, 5, Receved: January 25, 2017 Accepted: February 20, 2017 Publshed: February 23, 2017 Copyrght 2017 by authors and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY 4.0). Open Access Abstract Ths paper presents a technque for Medum Term Load Forecastng (MTLF) usng Partcle Swarm Optmzaton (PSO) algorthm based on Least Squares Regresson Methods to forecast the electrc loads of the Jordanan grd for year of Lnear, quadratc and exponental forecast models have been examned to perform ths study and compared wth the Auto Regressve (AR) model. MTLF models were nfluenced by the weather whch should be consdered when predctng the future peak load demand n terms of months and weeks. The man contrbuton for ths paper s the conducton of MTLF study for Jordan on weekly and monthly bass usng real data obtaned from Natonal Electrc Power Company NEPCO. Ths study s amed to develop practcal models and algorthm technques for MTLF to be used by the operators of Jordan power grd. The results are compared wth the actual peak load data to attan mnmum percentage error. The value of the forecasted weekly and monthly peak loads obtaned from these models s examned usng Least Square Error (LSE). Actual reported data from NEPCO are used to analyze the performance of the proposed approach and the results are reported and compared wth the results obtaned from PSO algorthm and AR model. Keywords Medum Term Load Forecastng, Partcle Swarm Optmzaton, Least Square Regresson Methods DOI: /jpee February 23, 2017

2 1. Introducton MTLF s extremely mportant for energy supplers and other partcpants n electrc energy generaton, transmsson, dstrbuton and markets. It helps make decsons, ncludng decsons on purchasng and generatng electrc power system utltes. The man role of electrc load forecastng n the electrc system s to help the energy companes to plant the purchasng and generatng of electrc power needed. Load forecastng studes can be categorzed based on forecastng perod nto three categores: short term load forecast (STLF) from one hour to a week, medum term load forecast (MTLF) from one week to a year and long term load forecast (LTLF) longer than one year [1]. These categores of forecasts are dfferent as well, for example for a partcular regon t s possble to predct the next three days peak load wth accuracy (1% - 2%), but t s mpossble to predct the next year peak load wth smlar accuracy because we do not have weather observatons [2]. MTLF s one of the most dffcult problems n dstrbuton power system plannng and analyss [3]. There are many factors affectng the load forecastng such as hstorcal load data, populaton growth and economc development. MTLF s not easy due to: frstly, because the load seres s complex and shows vacllatng behavor [4]; secondly, there are many mportant varables that must be consdered, ncludng weather data. To determne the accuracy of MTLF, a comparson between the actual load taken by NEPCO and the approxmaton load calculated by LSRM and PSO algorthm must be acheved. To acheve a good forecastng predcton many approaches have to deal wth programmed power network [5]. Accurate trackng of weekly demand and monthly peak demands s very mportant for the operaton of any power system. MTLF s bascally used to decde whether an extra power generaton should be provded to meet the demand or not. The demand can be met by ether ncreasng the generaton, nstallng new generaton unts nto servce or by power exchange from neghborng countres. On the other hand, MTLF can also be used to decde whether the output of the runnng generaton unts should be decreased or stopped. In order to predct the electrc load demand of a power system, t s mportant to nvestgate the load pattern, ts response and the factors effect on the demand [4]. Two man challenges have a drect mpact on the Jordan power operator center, the frst one: obtanng optmal economc dspatch for electrcal utltes and the second one s determnng medum term unt commtment n order to mantan the system relablty. Therefore, there s a necessty to make a robust MTLF models as a frst step for power system operaton and plannng based on LSRM and PSO optmzaton. The exstng forecastng predctons of MTLF employed by Natonal Electrc Power Company (NEPCO) n Jordan are based on the educated guess assumptons whch depend on gatherng the electrcty consumpton of domestc, commercal, ndustral, and publc lghtng sectors. The average error obtaned by NEPCO was n the range from 8.2% to 12.8% n 2015, whch s hgh. Therefore, t s necessary to have relable model to predct the load for medum term perods [5]. 76

3 2. Forecastng Procedure The MTLF procedure for the models can be vewed n Fgure Data Source Input varable selecton, ncludng: month type, peak load, average electrcal demand, humdty and temperature data, and weather nfluences of prevous tme Hstorcal Data The monthly or weekly peak load demand data recorded from NEPCO for the years ( ) takng nto account external varables lke holdays, weather and populaton growth Data Pre-Processng It may be nevtable to have mproperly recorded data and observaton error. Fgure 1. MTLF procedure. 77

4 Therefore the monthly and weekly reported data from NEPCO used to ntalze the smulaton results Smulaton In ths part, the peak load forecastng output s smulated usng Matlab Convergence Crtera The stoppng crteron s met when the parameters of PSO are achevng a global forecastng error wthn an effcent computaton tme. The convergence error must be less than 0.01% to make a suffcently good ftness value Post Processng The LSRM and PSO coeffcents requre calculatons to prompt the desred forecasted load results Error Analyss As characterstcs of load changes, error observatons become more sgnfcant for the forecastng process. LSE s used to mprove the accuracy of these models. 3. Forecastng Methodology Regresson analyss s wdely used n the analyss of data for any desgn. Regresson models one of the most commonly used statstcal analyss technques n any research [6]. Typcally, regresson analyss s used to dscover the relatonshps between a dependent varable and a set of ndependent varables based on a sample from nput data [7]. We wll study the method n the context of a regresson problem, where the varaton n one varable, called the response varable Y, can be partly explaned by the varaton n the other varables, called co-varables X. For example, varaton n exam results Y are manly caused by varaton n abltes X of the students [8]. The least squares estmates used to mnmze the error sum of squares: ^ ( ) 2 n LSE = Y Y (1) = 1 ^ where: Y : Actual load value n MW for week or month, Y : Predcted load value n MW for week or month, n : Number of samples (weeks or months) Least Square Regresson Methods Regresson analyss s the study of the acton of the tme seres or process n the past and t s mathematcal model, therefore the future behavor can be expected from t. In the forecastng process of medum term peak load, least squares regresson methods are used by dfferent relatons between the nput and output [9] Lnear Regresson The medum term load forecastng of many busness seres such as, sales exports and producton usually approxmates a straght lne. The smple lnear regres- 78

5 son method LRM model s desgned to study the relatonshp between a par of varables that appear n a data set. It s a model based on the lnear relatonshp between the total demand y and month x as shown n Equatons (2)-(4) [10]. y = ax + b (2) where: a : The slope, b : The ntercepton pont at y axs. The least squares crteron s used to generate the lne y = ax + b that fts a set of n data ponts. By usng the least square error approach [10], a and b coeffcents can be gven by: n 2 n n ( x )( y) ( 1 1 xy )( x = = = 1 ) a = (3) n 2 2 n n x x ( = 1 ) ( = 1 ) n n n ( 1 ) = ( = 1 )( = 1 ) n 2 2 n n( x 1 ) ( x = = 1 ) n xy x y b = where: n : The number of months whch the forecastng s based on, y : The total load demand for all perod for forecastng, x : The total sum of months. When a and b coeffcents are obtaned, the load forecastng s performed by Equaton (2.2) Quadratc Regresson In ths approach the parabolc functon whch s gven n Equaton (5) s used (4) 2 y = ax + bx + c. (5) After applyng least square error we can fnd a, b and c parameters n matrx form [10] x x x a xy 3 2 x x x b = xy 2 x x n c y When a, b and c coeffcents obtaned the load forecastng s performed by Equaton (5) Exponental Regresson In ths method the exponental functon s obtaned through Equatons (8)-(15) to get Equaton (7). x y = ab (7) By wrtng the equaton n logarthmc form, the equaton becomes: (6) x log y = lo g ab. (8) The propertes of algorthms gve log y = log a+ xlog b. (9) Ths expresses log y as a lnear functon of x wth slope. Slope = log b= m (10) 79

6 Intercept = log a = A (11) Therefore, f we fnd the best lne usng log y as A functon the slope and ntercept wll be gves as lnear regresson, so that the coeffcents m and A derves as lnear equaton. (13). m = n A = 2 ( x )( y ) ( xy )( x) 2 2 n( x ) ( x) ( xy ) ( x)( y ) 2 2 n( x ) ( x) (12) (13) After lnearzaton, a and b coeffcents are shown n Equatons (12) and a = 10 A (14) b = 10 m (15) When a and b coeffcents are obtaned the load forecastng s performed by Equaton (7) [10] Partcle Swarm Optmzaton A Partcle swarm optmzaton PSO technque s used to fnd the optmal parameters for dfferent forecastng methods. Ths algorthm s used to solve a wde class of complex optmzaton problems n engneerng and scence. Both lnear and nonlnear models wll be used n the system and the results wll be obtaned usng PSO. Through the mplementaton of PSO all partcles are kept as members of the populaton. The basc dea of the PSO s the mathematcal modelng and smulaton of the food searchng actvtes of a swarm of brds n the multdmensonal space where the optmal soluton s sought. Each partcle n the swarm s moved towards a pont where t obtans optmal soluton by the nfluence of ts velocty. The velocty of a partcle s affected by three factors; nertal momentum, cogntve and socal [11]. The goal of PSO s to fnd the optmal varable values for a certan functon. Each partcle knows ts optmal value ( p best ) and ts velocty and poston. Also, each partcle knows the optmal value n the group ( g best ) among pbests. Each partcle seeks to adjust ts poston usng the current velocty and the dstance obtaned from the p best and g best. Based on the above dscusson, the mathematcal model for PSO s represented as velocty update equaton gven by Equatons (16)-(18). ( ) ( ) v + = wv + c r pbest x + c r gbest x (16) k 1 k k k k k x = x + v (17) k+ 1 k k+ 1 where: v : The velocty of partcle. x : The current poston of partcle. c 1 and c 2 are postve constants, used to pull each partcle to g. best p best and 80

7 r 1 and r 2 are two randomly generated numbers wth a range [0 1]. w s the nerta weght and t keeps balance between exploraton and explotaton. wmax wmn wk ( ) = wmax k Max.Iter. w mn : The ntal weght. w max : The fnal weght. pbest : The best partcle poston acheved. gbest : The best poston of all partcles acheved. k : The teraton ndex. In ths work, PSO s employed to mnmze the LSE between the real values and predcton values. To evaluate the forecastng process for each model, LSE error can be used Auto Regressve (AR) Model The AR model was developed by Box and Jenkns n 1970 to analyze hstorcal data that had relatons wthn t. In ths study, the parameters were obtaned from NEPCO. The AR process utlzes the least squares (LS) method, and t s an analogous way to ft a model by mnmzng the sum of square errors for estmatng parameters. The LSE uses the normal equatons to mplement the polynomal system. The parameters can be solved by Matlab. The purpose of ths study was to mplement the dscounted least squares method wth drect smoothng for estmatng autoregressve model parameters [12]. The AR model structure s gven by Equaton (19) (18) Aq ( ) yt ( ) = et ( ). (19) Aq ( ) : The parameters that are estmated usng varants of the least-squares method. y( t ) : ddata object that contans the tme-seres data (one output channel). et ( ) : Random Error. The parameters of Aq ( ) can be estmated by Equaton (20). ( ) 1 A q = 1 + aq + = 1, 2,, q (20) = 1 a : The coeffcents for each order. q : Scalar that specfes the order of the model you want to estmate (the number of A parameters n the AR model). : Random error. 4. Results and Dscussons Real peak loads are used n ths study, so the electrc peak loads n the years [ ] have been founded for Jordan country. The data used are monthly and weekly peak loads recorded n the years [ ]. Ths system of equaton s solved usng the proposed PSO algorthm to fnd the optmal coeffcents 81

8 for dfferent forecastng models. Lnear, quadratc, exponental and AR models are used n the system and the results obtaned usng PSO are compared wth those of LSRM. A Matlab code was generated to execute the PSO Algorthm and LSRM Usng the peak data of NEPCO grd. For PSO, all partcles start at a random poston n the range [0, 1] for each dmensons. The swarm sze was lmted to 250 partcles. The selectons of some parameters to carry out the procedures of the work successfully has great effect on the model, these parameters are maxmum speed, nerta weght and acceleraton constants. The most sutable values for maxmum speed s set to be 2, w max and w mn are 0.9 and 0.4, C 1 and C 2 are 2. Key parameters of PSO algorthm used n ths paper are presented n Table Case One: MTLF Based on Peak Load Data Peak Loads of NEPCO recorded n the years [ ] are used to estmate the coeffcents of lnear, quadratc and exponental models for MTLF. The nputs of these models are the number of weeks or months and the peak loads recorded n the years [ ], whereas the output s the monthly or weekly peak loads predcted for the year Monthly Forecastng PSO and LSRM technques are used to estmate models parameters. Horzon and the computed parameters are tabulated n Table 2. Snce the forecastng n ths work s carred out on monthly bases, monthly least square error s performed and calculated by Equaton (21). The equaton used s gven by: Forecasted peak loads LSE = 1 100% Actual peak loads (21) Table 1. PSO parameters. Parameter Populaton Stop crteron Value 250 partcles 500 teratons Velocty V max = 2, V mn = 0 Acceleraton constants C 1 = 2, C 2 = 2 Inerta weght w max = 0.9, w mn = 0.4 Table 2. Monthly estmated coeffcents based on LSRM and PSO. Coeffcents Lnear model Quadratc model Exponental model LSRM PSO LSRM PSO LSRM PSO a b c

9 The monthly forecasted peak loads based on the parameters of lnear, quadratc and exponental models and monthly least square error are shown n Table 3. It can be concluded from the tables that the results computed by PSO are more close to the peak load and have less error. In both approaches, the monthly peak load s ncreased contnuously from January tll December when usng lnear or exponental models. In quadratc model, the peak load s decreased contnuously from January tll June and ncreased from June tll December. The forecasted monthly peak load usng dfferent models are shown n Fgure 2. It can be seen from the fgure that the PSO and LSRM are close to each other. In each model the dfference between LSRM and PSO s arranged from [10] [11] [12] MW. Ths dfference make LSRM very close to the real peak load n Aprl, May and October, otherwse the PSO acheves better estmaton for the predcted peak load. The results show that the PSO model s more accurate than LSRM and moreover, t s closer to the real peak load data for the year The monthly error performed by LSRM and PSO algorthm s shown n Fgure 3. From Fgure 3, t can be observed that the Error for 2015 wth LSRM and PSO s arranged from 0.04% to 16.85%. The Error less than 10% for nne months n LSRM and PSO approaches. The average error n LSRM for lnear model s 6.64%, for quadratc model s 6.40%, and for exponental s 7.41%. So t can be seen that the best represented model between the months and peak load n LSRM s the quadratc regresson model. The average error n PSO for lnear model s 6.47%, for quadratc model s 6.18%, and for exponental s 7.09%. Table 3. Monthly mornng peak loads for the year 2015 compared wth the actual readngs. Month Peak load LSRM Lnear model Quadratc model Exponental model PSO Error (%) LSRM PSO Error (%) LSRM PSO Error (%) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Average error (%)

10 Fgure 2. Monthly forecasted peak load usng LSRM and PSO. Fgure 3. Monthly error assocated wth LSRM and PSO. Therefore the best represented model between the months and peak load n PSO s the quadratc regresson model Weekly Forecastng Weekly real peak demands recorded n the years [ ] are used n ths secton. The data set s used to establsh an over determned system of equatons. Ths system of equatons s LSRM and PSO technque. The weekly real peak loads are used to fnd the coeffcents for lnear, quadratc and exponental models. PSO and LSRM technques are used to estmate models parameters for the same tme horzon and the computed parameters are tabulated n Table 4. The forecasted loads based on the parameters of lnear, quadratc and exponental models and weekly least square error are shown n Table 5. It can be concluded from the tables that the error computed by PSO are less than LSRM. In lnear and exponental models, the weekly predcton load s ncreased from the frst week tll last week of the year In quadratc model, the weekly predcton load has vertex pont [mnmum peak value] occurs at week number 26. The peak load demand expected usng LSRM and PSO shown n Fgure 4. 84

11 Table 4. Weekly estmated coeffcents based on PSO and LSRM. Coeffcents Lnear model Quadratc model Exponental model LSRM PSO LSRM PSO LSRM PSO a b c Table 5. Weekly peak loads for the year 2015 compared wth the actual readngs. Week Peak load LSRM Lnear model Quadratc model Exponental model PSO Error (%) LSRM PSO Error (%) LSRM PSO Error (%) Average error (%) Fgure 4. Weekly forecasted peak load usng LSRM and PSO. 85

12 From Fgure 4, t can be concluded that the PSO technque gves more accurate results than LSRM. In LSRM model, the predcton of the weekly peak load data gves results close to the real value n the weeks number 20, 24 and 44, otherwse the PSO technque represent the best model for all weeks n the year The weekly error performed by LSRM and PSO algorthm s shown n Fgure 5. From Fgure 5, t can be seen that the error has mnmum values for 20 weeks and maxmum values for 10 weeks arranged from 0.0% to 15.99%. From average error pont of vew t s found that PSO method has produced better estmates than the LSRM and the quadratc model has the least error. Therefore, the best represented model between the weeks and peak load n LSRM and PSO s the quadratc regresson model Case Two: MTLF Based on Weather Effect Weather s the most mportant ndependent varable for MTLF. In ths secton, MTLF models used weather nfluence to predct the future peak load demand n terms of month and week. Varous weather varables could be consdered for MTLF. Temperature s the most commonly used for load predctors. The result of the prevous secton shows that there s a hgh postve correlaton between temperature and peak load durng summer and there s a negatve correlaton between temperature and peak load durng wnter. For these postve and negatve correlatons, LSRM used to predct the peak load n the hot and cold days. Because the relaton between temperature and peak load s very complcated n nature and cannot be analyzed wth ordnary mathematcal models, the quadratc model used for data obtaned n summer and wnter seasons Monthly Forecastng Peak loads of NEPCO are used to estmate the coeffcents of quadratc model for MTLF n summer and wnter season. Summer season extends from June tll September; wnter season extends from December tll March. Long studes nvestgated that the changng of the temperature affects the peak load. The researches focused on the effect of the hgher and lower temperatures on electrcty Fgure 5. Weekly error assocated wth LSRM and PSO. 86

13 consumpton usng peak load data and temperature nfluence. The studes ndcate that the mpact of a one-degree n temperature hgher than 25 C, the peak load predcted wll ncrease by 8 MW and a one-degree n temperature lower than 15 C, the peak load predcted wll ncrease by 6 MW [13]. The nputs of ths model s the number of months per each season and the peak loads recorded n the years [ ], whereas the output s the monthly peak loads predcted for the year 2015 after takng the temperature effect by each season. PSO and LSRM technques are used to estmate quadratc model parameters for wnter and summer season. The quadratc parameters are tabulated n Table 6. Table 7 represents the adjusted forecasted loads based on the parameters of quadratc model after takng the temperature effect by each season and monthly least square error. It can be concluded from the table that n summer season, the peak load s ncreased contnuously from June tll August and decreased from August tll Table 6. Monthly estmated coeffcents for LSRM and PSO based on temperature effect. Coeffcents Summer Wnter LSRM PSO LSRM PSO a b c Table 7. Monthly forecasted peak load by quadratc model based on temperature effect. Month Peak load Quadratc MODEL LSRM PSO Error (%) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Average error (%)

14 September, whle n wnter season the peak load s ncreased contnuously from December tll January and decreased contnuously from January tll March. The forecasted monthly peak load usng quadratc model and PSO algorthm based on temperature effect s shown n Fgure 6. It can be observed from Fgure 6 that the results obtaned by the PSO are very close to the real values and more accurate than the results obtaned by LSRM. The predcton load obtaned by LSRM s very close to the real peak load n January, Aprl and October, otherwse the PSO made better estmaton for the predcted peak load. Fgure 7 s shown the monthly least square error based on temperature effect. From Fgure 7, t can be notced that the error for 2015 wth quadratc model and PSO algorthm s less than 5% for every month. The mnmum error 0.0% s happened n November, whle the maxmum error 4.68% s happened n September. The average error n Quadratc model s 2.02% and n PSO algorthm s 1.73%. Therefore, the PSO approach gves the best represented model between the months and peak load based on temperature effect Weekly Forecastng The hstorcal and weather data for the perod [ ] are used for estmaton the models parameters for wnter and summer seasons. Data for the year Fgure 6. Monthly peak load usng quadratc model based on temperature effect. Fgure 7. Monthly forecastng error usng quadratc model based on temperature effect. 88

15 2015 are used for testng LSRM and PSO models. The nputs of ths model s the number of weeks per each season, whereas the output s weekly peak loads predcted for the year 2015 after takng the temperature effect by each season. The quadratc parameters are tabulated n Table 8. The weekly forecasted peak load usng quadratc model and PSO algorthm based on temperature effect s shown n Fgure 8. From Fgure 8, t can be notced that the PSO results are very close to the real peak load data for the most weeks n the year The maxmum peak loads n 2015 are happened n the weeks representng the summer and wnter seasons, therefore the dfference n values between real and predcton load data has the maxmum n these weeks. The adjusted forecasted loads based on the parameters of quadratc model and weekly least square error are shown n Table 9. It can be concluded from the table that n summer season, the peak load s ncreased contnuously from the 22 nd week tll the 35 th week and decreased from the 36 th week tll the 39 th week (the weeks representng summer semester), whle n wnter season the peak load s ncreased contnuously from the 48 th week tll the 3 rd week and decreased contnuously from the 4 th week tll the 12 th week (the weeks representng the wnter semester). The weekly least square error depends on the temperature effect s shown n Fgure 9. From Fgure 9, t can be observed that the error has mnmum values for 45 weeks and maxmum values for 7 weeks arranged from 0.02% to 2.25%. The average error n Quadratc model s 2.20% and n PSO algorthm s 1.55%. From Table 8. Weekly estmated coeffcents for LSRM and PSO based on temperature effect. Coeffcents Summer Wnter LSRM PSO LSRM PSO a b c Fgure 8. Weekly peak load usng quadratc model based on temperature effect. 89

16 Fgure 9. Weekly forecastng error usng quadratc model based on temperature effect. Table 9. Weekly forecasted peak load by quadratc model based on temperature effect. Week Peak load Quadratc model LSRM PSO Error (%) Average error (%) average error pont of vew t s found that PSO method has produced better estmaton than the LSRM MTLF Usng AR Model In ths paper, the autoregressve data were generated wth order equal 13. Therefore, AR (13) was nvestgated and the forecasts of ths model were started from the month [84-95] for monthly predcton and from the week [ ] for weekly predcton (The months and weeks whch are represented the year A Matlab code was used to generate the data and check the AR property. The results were evaluated by LSE to compare computng peak loads and accuracy of the two forecast bass, respectvely. 90

17 Monthly Forecastng Usng AR Model Monthly real peak demands recorded n the years [ ] are used to mplement AR model and the real data for 2015 s used to test ths model. The monthly AR model parameters whch are estmated usng varants of the leastsquares method are gven by the followng equaton ( ) = A q q q q q q q q q q q q q q (22) The monthly forecasted loads based on the parameters of AR model and monthly least square error are shown n Table 10. It can be concluded from the table that the error computed by AR model s hgh compared wth other technques. In February, March and May the AR model gves good results, otherwse the results gves unacceptable predcton of peak load data. By usng Equaton (19) n Matlab, The monthly forecasted peak load usng AR model can be shown n Fgure 10. From Fgure 10, t can be observed that the dfference between the results obtaned by AR model for monthly peak load and the real data s relatvely hgh. Ths dfference makes an average error hgh compared wth the LSRM and PSO technques used n prevous secton. The monthly least square error usng AR model s shown n Fgure 11. From Fgure 11, t can be observed that the error s very hgh shown n 6 months for the year The AR model gves the maxmum error 17.96% n January whle gves the mnmum error 2.35% n March wth an average error equal to 8.88% for ths model. Table 10. Monthly forecasted peak load usng AR model. Month Peak load AR Error (%) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Average error (%)

18 Fgure 10. Monthly peak load usng AR model. Fgure 11. Monthly forecastng error usng AR model Weekly Forecastng Usng AR Model Weekly real peak demands recorded n the years [ ] are used n ths secton. The data set s used to establsh AR model. The weekly AR model parameters whch are estmated usng varants of the least-squares method are gven by the followng equaton ( ) = Aq q q q q q q q q q q q q q (23) The weekly forecasted loads usng AR model and weekly least square error are shown n Table 11. It can be seen from the table that the results computed by AR have the least accuracy compared wth LSRM and PSO. In AR model, the weekly predcton load has maxmum value n the week number 6 for the year 2015, but t stll has 92

19 a bg dfference compared wth real value shown n many weeks for ths year. By usng Equaton (19) n Matlab, The weekly forecasted peak load usng AR model can be shown n Fgure 12. From Fgure 12, t can be notced that the results obtaned by AR have a bg dfference wth the real peak load data for the most weeks n the year 2015, therefore the dfference n values between real and predcton load data has maxmum for ths model. The weekly least square error usng AR model s shown n Fgure 13. Table 11. Weekly forecasted peak load usng AR model. Week Peak load AR Error (%) Average error (%) 8.12 Fgure 12. Weekly peak load usng AR model. 93

20 Fgure 13. Weekly forecastng error usng AR model. From Fgure 13, t can be observed that the error has mnmum values for 10 weeks and maxmum values for 25 weeks arranged from 0.2% to 18.99%. From average error pont of vew t s found that AR model has produced least estmates than the LSRM and PSO wth an average error equal to 8.12%. 5. Concluson The partcle swarm optmzaton PSO, least square regressve LSR and auto regressve AR methods are presented as MTLF technques for Jordan electrc power systems. Ths paper has presented approaches used for MTLF of electrc loads: LSRM and PSO algorthm. The predcton s made ether weekly or monthly based on hstorcal peak load data and weather nfluence. Least square error (LSE) s ntroduced to evaluate the performance of the two models then compared these models wth AR model and the educated guess assumptons currently used n NEPCO. The comparson between LSRM, PSO and AR models s made by usng an average error and depcted by usng tables and fgures. A PSO algorthm s presented for optmal parameter estmaton of MTLF n power system. The soluton s mplemented and tested usng actual recorded data obtaned from NEPCO. Real peak load data from NEPCO are used to valdate the performance of these approaches; three dfferent models for LSRM and PSO algorthm based on the peak load data and weather nfluence are used; the quadratc model provdes the least errors for both monthly and weekly peak load compared wth the lnear and exponental models. The forecasted peak load resulted by usng the PSO algorthm has been compared wth that obtaned wth LSRM and AR models. From average error pont of vew, t s found that LSRM and PSO algorthm have produced better estmaton than the AR model and current predcton assumptons used n NEPCO. The results are shown the average error for each model used. Two cases were obtaned; forecastng depends on peak load data and forecastng depends on peak load data nfluenced by the weather 94

21 Table 12. The monthly and weekly average error for each model covered n ths work (%). Forecastng type Forecastng depends on peak load Forecastng depends on peak load adjusted consderng weather effects Lnear model Quadratc model Exponental model Quadratc model LSRM PSO LSRM PSO LSRM PSO LSRM PSO Forecastng by usng AR model Monthly Weekly effects. The average error for LSRM and PSO for the three models and forecastng usng AR model are represented n Table 12. References [1] Alfares, H. and Nazeeruddn, M. (2002) Electrc Load Forecastng: Lterature Survey and Classfcaton of Methods. Internatonal Journal of Systems Scence, 33, [2] Chow, J., Wu, F. and Momoh, J. (2005) Appled Mathematcs for Restructured Electrc Power Systems. Sprnger, New York. [3] Hahn, H., Meyer-Neberg, S. and Pckl, S. (2009) Electrc Load Forecastng Methods: Tools for Decson Makng. European Journal of Operatonal Research, 199, [4] Espnoza, M., Suykens, J., Belmans, R. and De Moor, B. (2007) Electrc Load Forecastng-Usng Kernel Based Modelng for Nonlnear System Identfcaton. IEEE Control Systems Magazne Specal Issue on Applcatons of System Identfcaton, 27, [5] NEPCO (2016) Prvate Communcaton. [6] Schachter, J. and Mancarella, P. (2014) A Short-Term Load Forecastng Model for Demand Response Applcatons. 11th Internatonal Conference on the European Energy Market, Krakow, May 2014, [7] Dng, C.S. (2006) Usng Regresson Mxture Analyss n Educatonal Research. Practcal Assessment Research & Evaluaton, 11, [8] Van De Geer, S.A. (2005) Least Squares Estmaton. Encyclopeda of Statstcs n Behavoral Scence, 2, [9] Wlls, H. and Northcote-Green, J.D. (1984) Comparson Tests of Fourteen Dstrbuton Load Forecastng Methods. IEEE Transactons on Power Apparatus and Systems, 103, [10] Canale, P. (2006) Numercal Methods for Engneers. 6th Edton, McGraw-Hll Scence, New York, [11] Tomar, S.K. and Prasad, R. (2009) Conventonal and PSO Based Approaches for Model Order Reducton of SISO Dscrete Systems. Internatonal Journal of Electrcal and Electroncs Engneerng, 2, [12] Jantana, P. and Sudasna-Na-Ayudthya, P. (2002) Least Squares and Dscounted Least Squares n Autoregressve Process. Slpakorn Unversty Open Journal Sys- 95

22 tems, 2, [13] Moman, M., Yatm, B. and Al, M. (2009) The Impact of the Daylght Savng Tme on Electrcty Consumpton A Case Study from Jordan. Energy Polcy, 37, Submt or recommend next manuscrpt to SCIRP and we wll provde best servce for you: Acceptng pre-submsson nqures through Emal, Facebook, LnkedIn, Twtter, etc. A wde selecton of journals (nclusve of 9 subjects, more than 200 journals) Provdng 24-hour hgh-qualty servce User-frendly onlne submsson system Far and swft peer-revew system Effcent typesettng and proofreadng procedure Dsplay of the result of downloads and vsts, as well as the number of cted artcles Maxmum dssemnaton of your research work Submt your manuscrpt at: Or contact jpee@scrp.org 96

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