Forecasting Electricity Consumption: A Comparison of Models for New Zealand

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1 Paper Tile: Forecasing Elecriciy Consumpion: A Comparison of Models for New Zealand Auhors: Zaid Mohamed and Pa Bodger,* Affiliaions:. Mohamed, Z., B.E (Hons), is a Ph.D. suden in he Deparmen of Elecrical and Compuer Engineering a he Universiy of Canerbury, Chrischurch, New Zealand. zmo0@suden.canerbury.ac.nz. Bodger, P.S, B.E (Hons), Ph.D., is Professor of Elecric Power Engineering, Deparmen of Elecrical and Compuer Engineering Universiy of Canerbury, P.O Box 4800, Chrischurch, New Zealand, Phone: (64)364070, Fax: (64) p.bodger@elec.canerbury.ac.nz Presener: Zaid Mohamed * Corresponding auhor

2 Forecasing Elecriciy Consumpion A comparison of models for New Zealand Zaid Mohamed and Pa Bodger Deparmen of Elecrical and Compuer Engineering Universiy of Canerbury Chrischurch, New Zealand ABSTRACT Forecasing elecriciy consumpion is of naional ineres o any counry. Fuure elecriciy forecass are no only required for shor and long erm power planning aciviies bu also in he srucure of he naional economy. This paper proposes six forecasing models developed for elecriciy consumpion in New Zealand. Three of hese models (Logisic, Harvey Logisic and Harvey) are based on growh curves. A furher model uses economic and demographic variables in muliple linear regression o forecas elecriciy consumpion while anoher uses hese facors o esimae fuure sauraion values of he New Zealand elecriciy consumpions and combine he resuls wih a growh curve model. The sixh model makes use of he Box-Jenkins ARIMA modeling echnique. The developed models are compared using goodness of fi, forecasing accuracy and fuure consumpion values. The fuure consumpions are also compared wih he available naional forecass. The comparisons revealed ha he bes overall forecass are given by he Harvey model for boh he Domesic and he Toal elecriciy consumpion of New Zealand while a specific form of he Harvey model, he Harvey Logisic model, is he bes in forecasing Non- Domesic elecriciy consumpion.. Inroducion Forecasing elecriciy consumpion has been applied using many heoreical mehods including growh curves [-6], muliple linear regression mehods ha use economic, social, geographic and demographic facors [7-3], and Box-Jenkins auoregressive inegraed moving average (ARIMA) echniques [4-0]. This paper invesigaes he effeciveness of six forecasing models developed for elecriciy consumpion in New Zealand. Firsly, a Logisic model [, ] based on he growh curve is developed. The fiing of logisic curves o he hisorical daa employs a Fibonacci search echnique o esablish opimum asympoes [, ]. In a second model, he influence of seleced economic and demographic variables on he annual elecriciy consumpion in New Zealand has been invesigaed []. The sudy uses populaion, price of elecriciy and Gross Domesic Produc (GDP) of New Zealand. The resuling Combined Models [] are developed using Corresponding auhor

3 muliple linear regression analysis. The hird models [] use an ARIMA echnique in developing elecriciy forecasing models. Fourhly, wo oher growh curve models, Harvey models and Harvey Logisic models, based on growh curves are developed [3]. Finally, he paper discusses a Variable Asympoe Logisic (VAL) model for elecriciy consumpion in New Zealand [4]. The sauraion levels of he logisic curve are esimaed using he Fibonacci search echnique. Correlaion of he esimaed sauraion level wih populaion, price of elecriciy and gross domesic produc (GDP) is deermined. The VAL model for forecasing elecriciy consumpion in New Zealand is proposed using one or more of he above explaining variables. Muliple linear regression is used in sudying he correlaion beween he explaining variables and elecriciy consumpion. Since ARIMA echniques are well known for predicing economic variables, hey are used in predicing fuure values of populaion, price of elecriciy and GDP. The developed models are compared for goodness of fi o he hisorical daa and forecasing accuracy in he shor, medium and long erm. The fuure forecass of hese models are also compared wih he available naional forecass in New Zealand [5, 6].. Models Theory.. Logisic Model The proposed Logisic model is [, ] of he form: F Y = + exp( C0 + C) () where; Y is he annual consumpion daa in GWh, F is he asympoic value obained by he Fibonacci search echnique in GWh, is ime in years, C 0 and C are consans... Combined Model The proposed Combined model using muliple linear regression is of he form []: Y = a + b X + b X + b3 X 3 + u () where, Y is he elecriciy consumpion (GWh), X is GDP ($NZ millions ), X is elecriciy price (cens/ kwh), X 3 is populaion, u is he error (disurbance erm or whie noise). 3

4 Each of he independen variables X, X and X 3 are hemselves obained from simple linear regression applied o daa ses of hese variables over ime (). X = c0 + c X = c0 + c (3) X 3 = c03 + c3 where, c 0, c, c 0, c, c 03 and c 3 are he consans of he respecive simple linear regressions..3. ARIMA models ARIMA models are generally wrien as ARIMA(p,d,q), where p represens he order of he auoregressive (AR) par, d denoes he degree of firs differencing (I) involved and q denoes he order of he moving average (MA) par. The auoregressive (AR) par of he model wih order p is of he form: Y + = c + φ Y + φy φ py p e (4) The moving average (MA) par of he model consiss of he pas errors as he explanaory variable. A moving average model of order q is of he form: where Y + = c + θ e + θ e θ pe q e (5) Y is he elecriciy consumpion e are he error series. The Box-Jenkins mehodology for modeling ime series consiss of idenificaion, esimaion, esing and forecasing. The resuling ARIMA models for New Zealand are discussed in deail in []..4. Harvey Logisic and Harvey Models The Harvey Logisic model is based on he Logisic model. The proposed Harvey Logisic Model is [3]: ln y + δ + γ + ε, = T (6) = lny where, Y is he elecriciy consumpion a year, y = Y Y, = T ε is a disurbance erm wih zero mean and consan variance, 4

5 δ and γ are consans o be found by regression. The Harvey model is based on general modified exponenials. The proposed Harvey model is [3]: ln y = ρ lny + δ + γ + ε (7) where, k ρ =, k / k δ = ln( kβα γ ), ρ, β and γ are parameers o be esimaed. The value of k deermines he form of he modified exponenial funcion. When k = -, i is Logisic and when k = i is a simple modified exponenial [3]. The only difference beween he wo models is ha he parameer ρ in he Harvey model is significanly differen from in he Harvey Logisic model..5. VAL model In he VAL model, he sauraion level F of he Logisic model (Eq. ) is esimaed using economic and demographic variables (X X n ) and used as a variable asympoe F(X) in Eq.. The proposed VAL model akes he form [4]: Y = F( X i ) + exp( a + a ) (8) 0 where, n 0 + ( ci. X i ) i= F( X ) = c (9) i F(X i ) is he sauraion level expressed as a funcion of n variables, c 0 and c i are he parameers obained from he explaining variables..6. Saisical Measures Mean squared error (MSE) is used o measure he goodness of he fi of each of he fied model o he hisorical daa. I is defined as: n MSE = ( Y i Yˆ i ) (0) n i= 5

6 Mean absolue percenage error (MAPE) is used o compare he forecasing accuracy of he models. I is defined as: MAPE = n n i= Y i Yˆ i Yi 00 () where, Y i is he acual consumpion daa a ime i Yˆ i is he forecased consumpion daa a ime i n is he number of daa poins considered. In addiion, he developed models are esed agains various saisical measures including auocorrelaion analysis, Durbin-Wason (DW) saisic, F-es, -es, residual plos and auocorrelaion (ACF) and parial auocorrelaion (PACF) plos of he daa and residuals [, - 4]. 3. Applicaion o Elecriciy Consumpion The developed models are applied o he Domesic, he Non-Domesic and he Toal elecriciy consumpion daa in New Zealand [7, 8]. Populaion and GDP daa for New Zealand are obained from Saisics New Zealand [9, 30]. Elecriciy price daa are obained from he Minisry of Economic Developmen, New Zealand [8, 3]. Elecriciy consumpion for New Zealand from is shown in Fig.. There is an increase in rend in he consumpion daa for all he secors. However, he rae of consumpion growh is generally very slow in he Domesic secor especially from 975 onwards. The resricions on elecriciy brough by he low lake inflows beween lae 99 o mid 99 can be clearly seen in all he hree daa ses wih a sudden decrease in consumpion in 99. Deails of he developed models for elecriciy consumpion are discussed in [] for he Logisic models and [-4] for he Combined models, ARIMA models, Harvey models and VAL models respecively. In he Logisic models, a Fibonacci search echnique is applied o each of he daa ses and he resuling models are proposed [, ]. The Combined models are proposed using he explaining variables populaion, price of elecriciy and GDP. In he ARIMA modeling, each daa se is reaed independenly for saionariy, idenificaion, esimaion and diagnosic checking of residuals. The Domesic and he Toal elecriciy consumpion daa required firs order differencing while he Non-Domesic daa required second order differencing o achieve saionariy. The proposed models are ARIMA(0,,0) for he Domesic secor, ARIMA(0,,) for he Non-Domesic secor and ARIMA(,,0) for he Toal consumpion []. The Harvey and Harvey Logisic models showed accepable degrees of saisical validiy o he New Zealand elecriciy consumpion daa [3]. In he VAL models, he elecriciy consumpion sauraion levels are bes explained by he populaion and price of elecriciy [4]. The addiion of he exra variable GDP degraded he model. This indicaes ha GDP may no be a useful describing variable in forecasing elecriciy consumpion, an observaion ha is conrary o many forecasing pracices using economerics including he proposed Combined model. The VAL 6

7 model was no suiable o describe he Non-Domesic elecriciy consumpion due o he inconsisencies in he sauraion levels obained due o possible immauriy in he secor [4]. 3.5 x 04 3 Domesic Non Domesic Toal Elecriciy Consumpion (GWh) Year Fig.. Elecriciy consumpion for New Zealand The combined models are applied o he daa ses from while all oher models are applied from The combined models are no accepable when applied o he whole daa ses. The naure of elecriciy consumpion growh in New Zealand was no suiable o apply a muliple linear regression model across he whole daa ses, bu his was adequaely saisfied for he daa ses from In he developmen of he VAL models, sauraion levels are obained using he whole daa ses for he years and he model is fied o he hisorical daa from Tha is, o obain he sauraion level for 984, all daa from is used, similarly for he sauraion level of 999, all daa from is used. More deails are available in [4]. 4. Model Fi and Forecasing Accuracy Forecasing accuracy of he six models is compared using MAPE. Forecasing accuracy is measured from one year ahead hrough o nine years ahead. To calculae he MAPE of he year ahead forecas, he acual elecriciy consumpion daa for 999 is held ou while developing hese models. The forecass obained by he models are hen used in Eq. () along wih he 7

8 acual consumpion daa held ou. In his case n =. Similarly, for 9 years ahead MAPE, acual daa from is held ou in developing he models and n = 9. The MAPE plos of he six models from one year ahead hrough o nine years ahead are shown in Fig., Fig. 3 and Fig. 4 for he Domesic, he Non-Domesic and he Toal elecriciy consumpion respecively. The models are ranked according o heir abiliy of model fi, shor ( o 3 years), medium (4 o 6 years) and long (7 o 9 years) erm forecasing accuracy. Models are ranked from (bes) o 6 (wors) for he Domesic and he Toal consumpion, while (bes) o 5 (wors) for he Non- Domesic consumpion as he VAL model is no applicable (n/a) for his secor. In ranking, he average of he MAPE values over he shor, medium and long erm is calculaed and ranked from he lowes MAPE (bes model) o he highes MAPE (wors model). The overall rankings for he Domesic, he Non-Domesic and he Toal consumpion are calculaed by aking he average of he shor, medium and long erm rankings. Table summarizes he resuls. 4 Domesic Consumpion Forecasing Errors 0 Logisic Model Combined Model ARIMA Model Harvey Logisic Model Harvey Model VAL Model MAPE value year hrough o 9 years ahead forecas Fig.. Domesic elecriciy consumpion forecasing accuracy of he six models 8

9 8 Non Domesic Consumpion Forecasing Errors 6 4 Logisic Model Combined Model ARIMA Model Harvey Logisic Model Harvey Model MAPE value year hrough o 9 years ahead forecas Fig. 3. Non-Domesic elecriciy consumpion forecasing accuracy of he six models 0 Toal Consumpion Forecasing Errors Logisic Model Combined Model ARIMA Model Harvey Logisic Model Harvey Model VAL Model 6 MAPE value year hrough o 9 years ahead forecas Fig. 4. Toal elecriciy consumpion forecasing accuracy of he six models 9

10 Table Ranking of models in erms of model fi and shor, medium and long erm forecasing accuracy ( = bes, 6 = wors). Domesic Non-Domesic Toal Model fi Forecas accuracy Shor medium long Overall MAPE fi Forecas accuracy Shor medium long Overall MAPE fi Forecas accuracy Shor medium long Overall MAPE Logisic Combined ARIMA Harvey Logisic Harvey VAL 5 6 n/a n/a n/a n/a n/a Domesic Secor The bes model in erms of fi is he Combined model while he Logisic is he wors. The shor erm forecass given by all he models excep he Logisic are very comparable. The VAL model is he bes o forecas shor and medium erm Domesic elecriciy, while i is ranked he wors o forecas he long erm. The bes model o forecas long erm Domesic elecriciy is he Harvey model. The wors forecass for he shor and medium erm are given by he Logisic model. Overall, Harvey is he bes model o forecas he Domesic elecriciy followed by he VAL model. The Harvey Logisic and he ARIMA are ranked in he middle. There is a sudden jump in MAPE of he VAL model a year 8. There are wo possible reasons for his. Firsly, in he VAL mehod, he asympoes are esimaed for he years from 984. Thus, in he 8 seps ahead forecas he sauraion levels are iniially calculaed from 984 o 99. This means ha as he number of sep ahead forecas increases, he number of sauraion values calculaed decreases. The decrease in he number of daa poins generally increases he error in he esimae of he coefficiens in he regression analysis. This will lead o an increase in forecasing errors. Secondly, for boh he Domesic and Toal consumpion daa, he decrease in he year 99 due o he elecriciy resricions brough by drough in ha year resuled in an overall increase in he forecasing error. 4.. Non-Domesic Secor The bes model fi is given by he Harvey model. The forecass given by he ARIMA, Harvey Logisic and Harvey models are very similar for he shor erm. The Combined model gave he bes shor erm forecas while he Harvey Logisic gave he bes medium and long erm forecass. The wors forecass are given by he Logisic model for shor erm and by he Harvey model for he medium and long erm forecass. Overall, Harvey Logisic is he bes model o forecas Non- Domesic elecriciy. The ARIMA model coninued o give he second lowes MAPE values up o year 6 wih an overall ranking of. There is a large increase in error a he year 7 for he ARIMA. This corresponds o he forecass made from he year 99. The consumpion is 0

11 significanly low. The ARIMA forecass are very dependen on he laer values of he acual consumpion. Thus, he overall forecas made for 7 years ahead is much lower. This resuled in a significan increase in error ha is refleced in he MAPE plo. The Combined and he Logisic model gave similar and consisen MAPE errors hroughou he compared period wih an overall ranking of and 4 respecively. As discussed before, he VAL model is no applied o he Non- Domesic elecriciy consumpion daa Toal consumpion The bes fi is again given by he Harvey model. The Harvey and ARIMA model gave very low MAPE values from year o year 6. Thus, he ARIMA model was ranked he bes for shor erm forecasing. The sudden decrease in MAPE by he VAL model a year 4 o 6 resuled in ha model being he bes o forecas he medium erm. The Harvey model gave he bes long erm forecas. The Logisic and Combined models are ranked overall he wors forecasing models for Toal elecriciy consumpion. The consisenly low forecasing errors by he ARIMA model excep a year 7 resuled in ha model being he second bes o forecas he Toal elecriciy consumpion. Overall, he Harvey is he bes model o forecas he Toal elecriciy consumpion. 5. Comparison of Fuure Forecass The forecass obained by he six developed models are compared wih each oher as well as wih he naional forecass available in New Zealand. These are he CAE models [30] and he MED models [3]. The CAE forecass are modelled using an annual load growh of.8%. Their sudy has used.8% as he baseline esimae, wih.3% and.3 % growh used for sensiiviy analysis. This paper uses he.8% baseline esimae for comparison purposes. The MED forecass are made by he Minisry of Economic Developmen, New Zealand, using is SADEM energy supply and demand model. The SADEM model is a descripive marke equilibrium model focusing on he enire energy secor. The model deermines equilibrium in he energy marke by projecing demands for a given se of prices and comparing his wih he modelled cos of supplying his level of demand [4]. These demands are re-esimaed if he prices implied by modelling he level of supply are no consisen wih he prices used o deermine he iniial demand. The process of re-esimaion is coninued unil equilibrium is achieved wih demand and supply in balance a marke clearing prices. An inerpolaion echnique is used for updaes of he prices. Thus, he forecass are also inerpolaed beween hese years. The forecass obained by all he models from he year 000 o 05 for he Domesic and he Non-Domesic secors and he Toal consumpion are shown in Fig. 5, Fig. 6 and Fig. 7 respecively. For he Domesic secor, he highes forecass are given by he MED model followed by he CAE model. The forecass by he Combined model and ARIMA models are very similar especially a he long erm forecass. The Harvey model forecass approximaely an average of all he oher models. For he Non-Domesic secor, he ARIMA model prediced he highes consumpion values followed by he Harvey model. CAE and MED forecass are very similar a he early years while CAE and Harvey Logisic model forecass are very similar a he

12 laer years. The forecass of he Combined model are more similar o bu less han he Harvey model forecass. For he Toal elecriciy consumpion, he forecass are generally more comparable. Forecass by he CAE, he MED and he Harvey models are very similar over he enire forecased period. The forecass by he Combined model are lile higher on average han hese hree models. Forecass by he ARIMA models are also very similar o he hree models especially a he early years. In all cases he Logisic model has prediced he lowes consumpion values followed by he Harvey Logisic model excep for he Non-Domesic secor. The VAL model iniially sared wih lower predicions han he Logisic model, bu ulimaely prediced higher consumpions values han he Logisic and Harvey Logisic model for he Domesic secor and Toal consumpion o which i is applied..6 x 04 Domesic Elecriciy Consumpion (GWh) Acual daa Logisic model Combined model ARIMA model Harvey Logisic model Harvey model VAL model CAE model MED model Year Fig. 5. Comparison of he Domesic Elecriciy Consumpion Forecass

13 4 x 04 Non Domesic Elecriciy Consumpion (GWh) Acual daa Logisic model Combined model ARIMA model Harvey Logisic model Harvey model CAE model MED model Year Fig. 6. Comparison of he Non-Domesic Elecriciy Consumpion Forecass 5 x 04 Toal Elecriciy Consumpion (GWh) Acual daa Logisic model Combined model ARIMA model Harvey Logisic model Harvey model VAL model CAE model MED model Year Fig. 7. Comparison of he Toal Elecriciy Consumpion Forecass 3

14 6. Summary This paper has compared six forecasing models developed for elecriciy consumpion in New Zealand. They are he Logisic, Combined, ARIMA, Harvey Logisic, Harvey and he VAL model. One of each model was developed for each of he Domesic and he Non-Domesic secors and he Toal elecriciy consumpion. The models were compared for goodness of fi o he hisorical daa and forecasing accuracy. Forecasing accuracy is measured for shor (-3 years), medium (4-6 years) and long (7-9 years) erm forecass. The VAL model is he bes in he shor and medium erm Domesic consumpion forecasing while he Harvey model is he bes for long erm Domesic consumpion forecasing. The Combined model is he bes for shor erm Non-Domesic consumpion forecasing while he Harvey Logisic model is he bes for boh he medium and long erm Non-Domesic consumpion forecasing. For he Toal consumpion forecass, he bes shor erm forecas is given by he ARIMA model, medium erm by he VAL model and long erm by he Harvey model. Overall, he bes forecass are given by he Harvey models for boh he Domesic and he Toal consumpion and he Harvey Logisic model for he Non-Domesic consumpion. In addiion, he Harvey models gave he bes model fi o he Non- Domesic and he Toal elecriciy consumpion. References [] Mohamed, Z. and Bodger, P.S., Analysis of he Logisic model for Predicing New Zealand Elecriciy Consumpion, presened a he Elecriciy Engineer s Associaion (EEA) New Zealand 003 Conference, Chrischurch, New Zealand, Published in CD-ROM, 0- June 003. [] Bodger, P.S. and Tay, H.S., Logisic and Energy Subsiuion Models for Elecriciy Forecasing: A Comparison Using New Zealand Consumpion Daa, Technological Forecasing and Social Change, vol. 3, pp. 7-48, 987 [3] Skiadas, C.H., Papayannakis L.L., and Mourelaos, A.G, An Aemp o Improve Forecasing Abiliy of Growh Funcions: The Greek Elecric Sysem, Technological Forecasing and Social Change 44, , 993 [4] Sharp, J.A., and Price, H.R., Experience Curve Models in he Elecriciy Supply Indusry, Inernaional Journal of Forecasing 6, , 990. [5] Young, P., Technological Growh Curves: A Compeiion of Forecasing Models, Technological Forecasing and Social Change 44, , 993. [6] Tingyan, X., A Combined Growh Model for Trend Forecass, Technological Forecasing and Social Change 38, 75 86, 990. [7] Egelioglu F, Mohamad AA and Guven H., Economic variables and elecriciy consumpion in Norhern Cyprus. Energy 6, , 00 [8] Harris, J.L. and Liu L., Dynamic srucural analysis and forecasing of residenial elecriciy consumpion. Inernaional Journal of Forecasing 9, , 993 [9] Yan Y.Y., Climae and residenial elecriciy consumpion in Hong Kong. Energy 3(),7-0, 998 [0] Rajan, M. and Jain V.K., Modelling of elecrical energy consumpion in Delhi, Energy 4, 35-36, 999 [] Fung Y.H. and Tummala V.M.R., Forecasing of Elecriciy Consumpion: A Comparaive Analysis of Regression and Arificial Neural Nework Models. IEE nd Inernaional Conference on Advances in Power Sysem Conrol. Operaion and Managemen, Hong Kong, , 993 4

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