Evaluation of Some Techniques for Forecasting of Electricity Demand in Sri Lanka

Transcription

1 Appeared in Sri Lankan Journal of Applied Statistics (Volume 3) 00 Evaluation of Some echniques for Forecasting of Electricity Demand in Sri Lanka.M.J. A. Cooray and M.Indralingam Department of Mathematics University of Moratuwa

2 Evaluation of Some echniques for Forecasting of Electricity Demand in Sri Lanka.M.J. A. Cooray and M.Indralingam Department of Mathematics University of Moratuwa ABSRAC A review of demand forecasting of electricity in Sri Lanka using five methods are presented.. he widely used five forecasting techniques are used. hese are (i) Box- Jenkins ARIMA Methodology (ii) Autoregression method (iii) Kalman_Filter and State Space Method (iv) Exponential Smoothing method and (v) Bayesian Forecasting technique. A brief discussion of each or these forecasting techniques along with the necessary equations is presented. Algorithm implementing these forecasting techniques have been programmed and applied to the same data set. for direct comparison of these techniques... It must be mentioned here the purpose of this paper is to compare the performances of these five forecasting techniques. It shown that the Bayesian forecasting method can exactly forecast the quarterly demand. he respective mean absolute percentage error of each methods are.89%,.8%,.3%,.99% and.69% A comparative summary of the results is presented to give the reader an understanding of the inherent level of difficulty of each of these techniques and their performances. Keywords: Autoregression, Kalman_Filter and State space Model, Exponential Smoothing method, Bayesian Forecasting Introduction In Sri Lanka, electricity is generated from hydro power and thermal power. While hydro power is the cheapest and environmentally beneficial source of energy, thermal is more expensive and demand for which varies with the changes in the weather pattern and is mostly used for meeting the shortfall in the supply of hydro electricity in the drought periods. he level of energy consumption depends mainly on the activities of domestic, Industrial and commercial sectors, and the pattern of consumption is associated with characteristics to a different sectors. he domestic sector is the largest energy consumer in Sri Lanka and the industrial sector obtains about 5% of its energy requirements from electricity at present. he commercial sector includes activities of shops, supermarkets

3 banks and hotels etc. As result of expansion in service and commercial oriented activities following the economic liberation introduced in November, 977, the commercial demand for electricity has considerably increased. oday electricity has become the main source of energy for commercial activities. In 970, the demand of electricity in Sri Lanka was around 66.7 GWh (Giga Watts Hours) and the rates increase in demand was slow till 977. By the year 977, the demand stood up 04 GWh, and in 000 it went up to 557 GWh. otal energy requirements of the country varies every day. At present, the electricity demand varies from 950 MW (Mega Watts) to 000MW and unit wise, 4.5 million units ( GWh) per day should be generated to meet the present demand. In this study intends to cover the period from 977 to 000. A significant increase in demand for electricity was recorded during this period. Description of Methods As a brief review, the demand series (Chatfield, C,985), y (, is modeled as the output from the linear filter that has a random series input, e(, usually called a white noise as shown in figure. his random input series has a zero mean and unknown fixed variance t σ ( ). White noise x ( Linear Filter Depending on the characteristic of the linear filter, different models can be classified as follows: he Autoregressive (AR) process In the autoregressive process, the current value of the time series x ( is expressed linearly in terms of its previous values x ( t ), t ),... and a random noise e(. he order of this process depends on the oldest previous value at which x ( is regressed on. For an autoregressive process of order p (ie AR(p) ), this model can be written as = α t ) + α t )... α p t p) + e( () 3

4 By introducing the backshift operator = t j) we can write () in the following compact form B j p α Bx ( + α B... α p B t + et φ P ( B ) xt = et = ) where φ( B) = (- σ B α B... α ) p p B he Moving -Average(MA) Process In the moving average model, the current value of the time series x ( is expressed linearly in terms of current and previous values of white noise series e ( t ), e( t ), e( t 3),... his white noise series constructed from the forecast errors or residuals when demand observation become available. he order of this model depends on the oldest white noise value at which x ( is regressed on. For a moving average of order q (ie MA(q) ) this model can be written as = e( β e( t ) β e( t )... β qe( t q) () a similar application of the backshift operator on the white noise series would allow equation (3) to be written as: θ ( B) z( q = where φ(b) = ( β B β B... β ) q q B he Autoregressive Moving-Average (ARMA) Process he mixed Autoregressive moving Average model ARMA (p,q) is a mixture of the previously mentioned two terms and could be formulated as = α t ) + α t ).. α p t p) + e( βe( t ) β e( t ).. β qe( t q) (3) he model in (3) can be represented in the following compact form φ ( B) = φ ( B) e( q q Autoregressive Integrated Moving-Average (ARIMA) Models he time series defined previously as an AR, MA, or as an ARMA is called stationary process (Box, G. E. P, and G. M. Jenkins,970). his means that the mean of the series of 4

5 any of these processes and the covariance among its observations do not change with time. If the process is not stationary, transformation of the series to a stationary process has to be performed first. his can be achieved, for the time series that are non stationary in the mean, by differencing the process. By introducing the operator, the differenced series of order can be written as = t ) = ( B) using the Backshift operator, B. Consequently, an order d differenced time series is written as = ( B). he differenced stationary time series can be model as an AR, MA, or an ARMA to yield an ARIMA time series processes. For series that needs to be differenced d times and has a orders p and q for the AR and MA components (ie ARIMA(p,d,q) ), the model is written as : d φ ( B) = θ ( B) e)( (4) Where φ(b), and θ (B), have been defined earlier. q d d Seasonal ARIMA Models As a result of daily, weekly, quarterly, yearly or any other periodicities, many time series exhibits periodic behaviors in response to one or more of these periodicities, herefore, different class of models which have this property is designated as seasonal processes. Seasonal time series could be model as an AR, MA, ARMA or an ARIMA seasonal process similar to the non seasonal time series discussed previous sections. It has been shown that the general multiplicative model (p,d,q)(p,d,q) s for time series model can be written in the form S d S Φ P ( B ) φ p ( B)( B) = ΘQ ( B ) θ q ( B) e( (5) Autoregression Model Define p x ( = α + φ t j) + e (6) j= j t 5

6 whereφ φ, φ3,... φ are auto-regressive parameters p and e t = ( e e,... e t, t tr )' are rx independent identically distributed zero mean vectors with covariance matrix = E ( e e t t ') and regression matrix B = ( α, φ, φ, φ3... φ p ) r p ) + Estimated co efficient of regression matrix B = x t Z ' t t= p+ t= p + Z Z ' t t (7) Where is the number of observations in given time series, will contain maximum likelihood estimators ˆ α. ˆ φ... ˆ φ Where Z t = (, x' t x' t-.... x' t p he rxr Error sum of product matrix becomes. )' RSP = ( X BZ ˆ )( X Bˆ Z ) t= p+ t t t t (8) his can be used as a basis for identifying the correct order auto-regressive model. hen the residual sum of product matrix computed from lower order process say RSP can be used with the Chi-Squared Approximation. In this case full model having pr + parameters and lower having (p-)r+ parameters where p order of full model and r dimension of series. o check whether model of parameters p is appropriate, then one needs to check the significance of the model by test of hypothesis Ho : model is insignificant (after including p number of lag terms to the fitted model) Vs H : model is significant his leads to test statistic 3 RSP χ = ( rp ) Ln (9) r RSP 6

7 Reject H 0 If r α, r χ χ where α level of significance then add higher order autoregressive terms to the model. Reject H if r α, r χ χ then pth order autoregressive model is appropriate. (Luthkepotu, H 985) mentioned that the sequential testing called PAC procedure has approximate Chisquare distribution with r degrees of freedom. herefore (PAC) Chi-squared values are computed for different order of lags. In addition to that (BIC) values, can be computed to chooses the model of order p, for which BIC value is minimized. RSP Ln( ) BIC( K) = Ln + pr (0) State Space and Kalman Filter Method his is a general forecasting approach that can include the previously mentioned and more such as time varying coefficients model (Kalman, R. E. -960). In this method, the demand is modeled as a state variable using state space formulation which designated by two sets of equations, the system state equations and the measurement equations. hese sets are written for demand model as: State Space Equations x ( t + ) = Φ( + w( Measurement Equations y ( = A( + v( Where x ( = (nx) process vector state vector at time t Φ ( = (nxn) state transition matrix relating to x ( t +) w ( = white noise with a known covariance Q ( y ( = (mx) vector (demand) measurement at time t A ( = (mxn) matrix relating to x ( to y ( without noise 7

8 v ( = (mx) (demand) measurement error which is a white noise with a known covariance R ( he covariance matrices for the vector w ( and v ( are given by E( w(. w( s) E( v(. v( s) Q( ) = 0 R( ) = 0 t = s t s t = s t s he process noise w ( and the measurement noise v ( are assumed uncorrelated and accordingly E ( w(. v( s) ) = 0 for all t and s At any instant t there will be an estimate for the process based on knowledge of the process up to t-. his estimate is called the prior estimate t t-). he associated between the actual and the previous estimates of the process is given as e ( = he error vector has an error covariance matrix expressed by E{ e(. e( } = P( he posterior estimate is obtained by as a linear combination from the prior estimate and the measurement noise as: x ( t = + K( [ y( A( ] where t is an updated estimate and K ( is a blending factor he error associated with the actual and the posterior estimate of the process is e ( t = he covariance matrix of this error vector is expressed by E{ e( t. e( t } = P( t he blending factor K ( is found such that x ( t is optimal such as the minimum-squares error (MSE) criterion. his factor is known as Kalman gain and the procedure for implementing Kalman Filter for demand prediction is as follows.. Find the process prior estimates x ( and the error covariance matrix associated with it, P (. Compute the Kalman gain K( = P( A( { A( P( A( R( 3. Compute the updated estimate error covariance matrix } 8

9 P ( t = { K( A( } P( 4. Compute ahead the prior estimate x ( t + and the covariance matrix P ( t + associated with it. x ( t + = Φ ( X ( t. + P( t + = Φ( P( t Q( Q( 5. Go to the step repeat next steps. It is clear that the State Space method is very attractive for on-line prediction as a result of the recursive property of the Kalman Filter the optimal forecast will be based on the assumed model. herefore the model has to be known prior to using the Kalman Filter. he identification process is the main difficulty of this approach. Exponential Smoothing Method In this method, the demand at time t, y ( is modeled using a fitting function and is expressed in the form (Holt, C. C. (965)). y( = β ( f ( + e( where f ( = fitting function vector for the process β ( = coefficient vector e ( = a white noise he estimates of the coefficients are found using the weighted or discounted mean square error for the recent N sampled intervals N ie to minimize the function w { y( N j) f ( j) β } 0 < w < 0 j= 0 j his minimization gives the estimate vector of the coefficients which has the form ˆ β ( N) = F ( N) h( N) N where F(N) = w f ( j) f ( j) β } j= 0 j N h(n) = w f ( j) y( N j= 0 j j) he forecast of the series of lead time l is found as yˆ ( N + l) = f ( l) ˆ( β N) 9

10 he coefficient estimates and the forecast can be updated respectively using ˆ ) ( ) ( ) ) β N + = L β N + F f (0)[ y( N + ) y( N)] yˆ( N + + l) = f ( l) ˆ( β N + ) Where F = LimN F(N) he L matrix is called the transition matrix and is constructed on the basis that the model will have a fitting function satisfying the relationship: f ( = Lf ( t ) Dynamic Bayesian Models Dynamic models provide the technical components of Bayesian forecasting methods and systems (Harisson, P. J. 965).. he most widely used processes as dynamic linear models, and with associated assumptions of normally distributed components, they also features in various areas of classical time series analysis and control areas state-space model, and econometrics or structural time series models. he approach of Bayesian forecasting and dynamic modeling comprises, fundamentally (Harrison, P. J. and Stevens, C. F. 976)., sequential model definitions for series of observations observed over time, structuring using parametric models with meaningful parameterizations, probabilistic representation of information about all parameters and observable, and hence inference and forecasting derived by summarizing appropriate posterior and predictive probability distributions. he probabilistic view is inherent in the foundation in the Bayesian paradigm, and has important technical and practical consequences. First, the routine analysis of time series data, and the entire process of sequential learning and forecasting, is directly determined by the laws of probability. Joint distributions for model parameters and future values of a time series are manipulated to compute parameter estimates and point forecasts, together with a full range of associated summary measures of uncertainties. Second, and critically in the context of practical forecasting systems that are open to external information sources, subjective interventions to incorporate additional information into data analysis and inference are both explicitly permissible and technically feasible within the Bayesian framework. All probability distributions represent the views of a forecaster, and these may be modified / updated / 0

11 adjusted in the light of any sources of additional information deemed relevant to the development of the time series. he only constraint is that the ways in which such additional information is combined with an existing model form and the historical information abide by the laws of probability. he sequential approach focuses attention on statements about the future development of a time series conditional on existing information. hus, if interest lies in the scalar series Yt, statements made at time t - are based on the existing information set D t-. hese statements are derived from a representation of the relevant information obtained by structuring the beliefs of the forecaster in terms of a parametric model defining the probability density fy ( Yt θ t, Dt ), where θ t is the defining parameter vector at time t. hrough conditioning arguments, the notation explicitly recognizes the dependence of y t on the model parameters and on the available historical information. Parameters represent constructs meaningful in the context of the forecasting problem. For example, θ t may involve terms representing the current level of the Yt series, regression parameters on independent variables, seasonal factors, and so forth. Indexing θ t by t indicates that the parameterization may be dynamic, i.e. varying over time through both deterministic and stochastic mechanisms. In some cases, the parameter may even change in dimension and interpretation. Results and Conclusions ARIMA Methodology he time series approach has been applied to electricity demand data set. he seasonal ARIMA model has been identified and its parameters estimated as using [ eq(5)]: ARIMA(0,,)(0,,) 4 x ( = ( 0.394B B )(.497B B ) e( ) t Autoregression Methodology he Akaike Information criterion [ Eq(9)] and Bayesian Information criterion [Eq(0)] have been applied and computed appropriate model as: X t = X t X t-3

12 Exponential Smoothing Methodology he Exponential smoothing technique has been applied to model and computed parameter are as follows Parameter Alpha(level) Gamma(trend) MAPE MAD MSD Estimated value State Space Kalman Filter he state Space Kalman Filter algorithm has been applied to the demand data set and initial and final estimation of parameters are given below x ( = t ) + w( Q ( = E{ w(. w( } = y ( = + v( R ( = E{ v(. v( } = Bayesian Analysis Bayesian Forecasting technique has been used with aid of software package BAS and following results were obtained. Figure Figure

13 With reference to the figure it is clearly indicated the growth of electricity demand decline at 3 rd Quarter of 987 and 3 rd quarter 0f 996 this because of due to the J.V.P troubles in 988 and drought at 997.herefore those two points are indicated automatically, giving prior values for level, growth analysis were continued. Comparative Summary of results 3 e( Mean absolute percentage error x00 (MAPE) for the forecasting quarterly i= electricity demand in the year 000 and 00 st quarter able Model Period Actual Q: demand Forecast Q: demand MAPE ARIMA (0,,)(0,,) 000/Q % 000/Q /Q Autoregres: method 000/Q3 000/Q4 00/Q % State Space model 000/Q3 000/Q4 00/Q Exp: Smoothing 000/Q3 000/Q4 00/Q Bayesian Model 000/Q3 000/Q4 00/Q % 0.99% 0.69% Conclusions Among the five forecasting techniques Bayesian Forecasting techniques showed minimum percentage error of forecast. One of the attractive features of this method user can intervene with his subjective knowledge of the environment in the forecasting procedure. From the mathematical point of view this technique employs the operators estimate as initial forecasting. hen it combine this initial forecasting with monitoring automatic signals such as increases or decreases of level and intervening outliers to obtain better forecasting. For these reasons Bayesian analysis is better than other methods. his features can be used for the controlling discount for level, growth and seasonality as well as on-line analysis. his method can be easily used by practitioners who are involved system forecasting. References 3

14 Akaike, H (974a). A new look at the statistical model identification, IEEE ransactions on automatic control, AC-9, Box, G. E. P, and G. M. Jenkins (970) ime Series Analysis, Forecasting, and Control San Fransisco: Holdan day. Chatfield, C (985) he ime Series Analysis and Introduction. 4th Edition. London, Chapman and Hall Harisson, P. J. (965). Short-term sales forecasting. Applied Statistics5, Harrison, P. J. and Stevens, C. F. (976). Bayesian Forecasting. J. Roy. Stat. Soc. B. 38: Holt, C. C. (965). Forecasting seasonal and trends by exponentially weighted moving averages. O. N. R. Research Memo. 5, Carnegie Institute of echnology. Kalman, R. E. (960). A new approach of linear filtering and prediction problems, rans. ASME. J, Basic Engrg., Series D, 8, Lutkepohi, H. (985) Comparison of criterion for estimating the order of a vector autoregressive process. J. ime Series Anal. 6: