Seasonal models of peak electric load demand

Technological Forecasting & Social Change 72 (2005) 609–622

Seasonal models of peak electric load demand

Shamsuddin Ahmed

College of Business and Economics, United Arab Emirates University, P.O. Box 17555, Al Ain, United Arab Emirates

Edith Cowan University, School of Engineering and Mathematics, Perth, WA, Australia

Received 21 November 2003; received in revised form 17 February 2004; accepted 19 February 2004

Abstract

Energy consumption in a pilgrim city belonging to a Gulf Cooperation Council (GCC) country exhibits strong

seasonal pattern due to higher demand in summer season and additional load during the pilgrimage months. The

pilgrimage month’s timing is not fixed in the Gregorian calendar. The event varies according to the lunar calendar

called the Hegira calendar, which lags behind the former by approximately 14 days in a year. Ten seasonal demand

models are developed to model energy estimate for a GCC pilgrimage city. Among the long-range forecast models,

three trigonometric models, a multiplicative model, and a multivariate model using categorical variables are

considered. Further, a composite nonlinear model whose coefficients are nonlinear is suggested. This model

combines the seasonality extracted from a multivariate regression model and a model that represents the peak

electric load pattern. Adopting least square fit of a chi-square error function expanded by parabolic expansion, the

parameters of the nonlinear model are identified. Moreover, smoothing-based techniques, such as moving average,

double exponential smoothing, Winter’s, and a multiplicative seasonal model, are suggested. The peak electric load

model on lunar and solar calendars is closely related, and the deference in fitting error can be attributed to the

magnitude of data. Computational results and statistical tests are presented to analyze the models. It is observed

that the multiplicative model performs better to predict the peak electric load demand.

D 2004 Elsevier Inc. All rights reserved.

Keywords: Forecasting; Seasonal model; Peak electric load; Nonlinear model; Decomposition

1. Introduction

Factors that influence the peak electric load pattern are identified as seasonal variation,

monthly cycles, major religious events, and some unexpected random events [1–4]. This study

focuses on seasonal peak electric load demand models for a pilgrim city in a Gulf Cooperation

0040-1625/$ - see front matter D 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.techfore.2004.02.003

E-mail address: [email protected].

S. Ahmed / Technological Forecasting & Social Change 72 (2005) 609–622610

Council (GCC) country. The pilgrimage months require an additional demand of electricity due to

millions of pilgrims converging for a particular religious event called the Hajj. It is one of the

important duties of a pilgrim, who is supposed to perform the obligation once in his/her lifetime.

This Hajj event is of significant importance so far as the logistics and demand of electricity are

concerned. The city hosts millions of pilgrims for more than a month every year according to

the lunar calendar called the Hegira calendar. This event does not fall on a fixed month in the

Gregorian calendar, as it lags by 14 days. The pilgrimage months have extra peak demand apart

from the average daily peak demand and cannot be attributed to a particular month in the

Gregorian calendar. This characteristic is unique, as it influences the peak load pattern. In such a

model, one of the contributing factors of seasonality changes according to the lunar calendar.

The pilgrim host city in the GCC country gets its power demand through the Saudi Consolidated

Electric in the western region, SCECO-WEST. Models of peak electric load demand forecast based on

effectiveness have been proposed in Ref. [5]. New techniques with neural networks and fuzzy logic to do

short and long-range forecast have been proposed in Refs. [6–10]. These models require extensive

computations to develop forecast. The models introduced in this article are simpler in structure, with

both short- and long-term features.

A plot of monthly demand of electric load (in megawatts) in 12-month cycles is shown in Fig. 1.

This shows that there is a strong seasonality with slight upward trend. The consumption of the

electric load shows a seasonal pattern due to the extreme weather conditions of the long summer

months, mainly May through October, followed by a gradual decrease in electric load demand in

other months. The winter months in the western region are moderate and do not require room

heating, and the electric load demand during winter months is minimal for domestic consumption.

The objective of this paper is to develop different peak electric load forecast models that explain the

additional demand due to pilgrimage months.

Fig. 1. Monthly peak electric load.

S. Ahmed / Technological Forecasting & Social Change 72 (2005) 609–622 611

2. Energy forecast models

Table 1 depicts 10 different seasonal models to estimate the peak electric load for the pilgrim city.

Models 1–5 are of a regression type. Model 9 can be described as a multiplicative decomposition

method. Models 6, 7 and 8 are smoothing-based.

Many different seasonal characteristics can be modeled using trigonometric functions involving sine

and cosine terms. This can be achieved by controlling three characteristics of the sine wave, such as

the amplitude, phase angle, and period. Thus, x=b sin xt defines a sine wave with amplitude b and

origin at t=0. A 12-period season is obtained by setting x=2p/12, while x is in radians. In some

seasonal models, the amplitude of the seasonal cycle is proportional to the trend. Model 1 captures

these characteristics. A seasonal pattern that exhibits more irregular behavior can be constructed by

including a second sine–cosine pair to introduce a higher-frequency harmonic into the theoretical

formulation. This process is represented by Model 2. A formulation containing four harmonic

frequencies can describe almost any periodic process and is represented by Model 3. It contains an

irregular 12-point periodic function. One may refer to the documents in Refs. [11,12] for more

information.

Expression 4 is a multiple regression model. The seasonality is modeled by introducing

categorical variables for the 12-month periods. Model 5 is a multiplicative model based on Model

4. A moving average process is expressed by Model 6, while Model 7 is related to the double

exponential smoothing method. A Winter-type model is described by Model 8. A multiplicative

decomposition technique is represented by Model 9. This model first fits a trend component on the

early seasonal data and then corrects it with the computed seasonal ratio. Model 10 is nonlinear in

nature, where a Gausion function is embedded with Model 4 to account for the peak load with

quadratic background for more accurate data modeling. Parameter estimations for Models 1–9 are

straightforward. The methods of estimating the parameters can be found in Ref. [12]. Inasmuch as

the parameters of Model 10 are nonlinear, they are estimated by methods described in the

following section. The proposed models have the characteristics to estimate seasonal peak electric

load demand. Multivariate regression techniques and the moving average process model the

nonlinearity.

3. Functions with nonlinear parameters

Consider fitting a nonlinear seasonal Model 10 with a set of m parameters aj, j=1,. . ., m. In the

proposed models, aj is expressed as d0, d1, b1,b2,. . ., b11, X1, X2, and X3. This type of model is classified

as nonlinear in coefficients. One way to estimate the parameters is to adopt a goodness of fit criterion.

The measure of goodness of fit can be defined by the following chi-square (v2) expression:

v2 ¼Xnt¼1

1

r2xt �

Xmj¼1

ajxtt

!2

where r2 is the uncertainties in the data point xt at time t, and there are n data points indexed as

t=1,2,3,. . ., n.

Table 1

Seasonal models

Xt ¼ d0 þ d1t þ ½b1 þ b2t�sin2Pt

12

� �þ ½c1 þ c2t�cos

2Pt

12

� �þ et ð1Þ

Xt ¼ d0 þ d1t þX2j¼1

bjsin2Pjt

12

� �þ cjcos

2Pjt

12

� �� þ et ð2Þ

Xt ¼ d0 þ d1t þX4j¼1

bjsin2Pjt

12

� �þ cjcos

2Pjt

12

� �� þ et ð3Þ

Xt ¼ d0 þ d1t þX12�1

j¼1

bjMj þ et; ð4Þ

where et=Error term at time t; Xt=predicted value at time t

M1 ¼1 If Month 1

0 If Month 2; 3; 4; . . . ; 12

8><>:

M2 ¼1 If Month 2

0 If Month 1; 3; 4; . . . ; 12

8<:

M11 ¼1 If Month 11

0 If Month 1; 2; . . . ; 10; 12; ðConsider a season of 12 monthsÞ

8<:

xt ¼ e

d0þd1tþX12�1

j¼1

bjMj

!þ et

ð5Þ

Xt ¼Xnprdi¼1

Yt�i

nprdþ et; ð6Þ

where nprd=n�point moving average; Yt�i=response at time t�i

Xt ¼ mYt þ ð1� mÞðXt þ Tt�1Þ þ et Tt ¼ kðXt � Xt�1Þ þ ð1� kÞTt�1 ð7Þ

where Yt=response at time t ; Xt=predicted value at time t; Tt=trend component ; 1zm, kz0

Xt ¼ ðU0 þ U1tÞwt þ et ð8Þ

where U0=base signal; U1=linear trend component; wt=multiplicative seasonal factor

Xt ¼d0 þ d1t

12

� � Xnyrsj¼1

Yjk

!=nyrs

" #

X12j¼1

Xnyrsi¼1

Yij

!=nyrs

" #( )=12

þ et ; k ¼ 1; 2; 3; . . . ; 12 ð9Þ


Table 1 (continued)

Xt ¼ d0 þ d1t þX12�1

j¼1

bjMj þ X3e�1

2ft�X1

X2g2

h iþ et ð10Þ

where

M1 ¼1 If Month 1

0 If Month 2; 3; 4; . . . ; 12

ðRegression model with Gaussian PeakÞ

8><>:

M2 ¼1 If Month 2

0 If Month 1; 3; 4; . . . ; 12ðet ¼ Error term at time t; Xt ¼ Predicted value at time tÞ

8<:

M11 ¼1 If Month 11

0 If Month 1; 2; . . . ; 10; 12; ðConsider a season of 12 monthsÞ

8<:

Where (d0 and d1), (X1, X2,. . .,Xk) (b0, b1, b2,. . ., bL), (c1, c2, c3,. . ., cM), are coefficients, and the values of index K, L, andM are

dependent on the model. nyrs=data for ‘nyrs’ number of years, (t=1,2,3,. . ., nyrs*12).


Three sources of errors that contribute to the size of v2 are

(i) the fluctuations of v2 about the expected value of v, which may be statistically greater or less than the

expected theoretical uncertainty in aj;(ii) v2 is a continuous function with all parameters of aj; and(iii) the choice of the approximate function as a true analytical function will influence the value of v2.

The contribution of the errors can be minimized in two steps. First, the parameter aj can be estimated by the

least square method, which minimizes v2 from (ii). Second, an iterative procedure can be used to test

several different functions with an arbitrary choice of aj, such that an appropriate function is determined.

The estimation of aj is not straightforward. A modified form of Gauss Newton method was suggested

in Ref. [13] to solve such nonlinear problem. For several years, it was the only alternative to the steepest

descent method. The method behaves unsatisfactorily when some of the residuals are large at the

solution space. Nazareth [14] and McKeown [15] explain the demerits of this method. Marquaradt [16]

has put forth an elegant method related to an earlier suggestion of Levenberg [17] for varying smoothly

between the extremes of the inverse Hessian method and the steepest descent method. Marquardt’s [16]

first observation about the Hessian matrix was that it provides no precise information about the search

direction but gives information about the order of magnitude of the scale of the problems. The second

observation was the selection of step size so that the v2 function is minimized. This method is somewhat

more dependable than the modified Gauss Newton method but frequently fails to converge, as claimed in

Refs. [18,19]. The v2 function is of a quasiconvex type. It is shown in Refs. [20,21] that the function

contains several local minima. The computational experience reveals that gradient-based search process

faces difficulty, as the Hessian matrix tends to be ill conditioned. It could be verified by the Eigenvalues

of the Hessian matrix. It is claimed in Ref. [22] that this type of problem would not perform better with


an algorithm that requires derivative information. He suggested a derivative-free algorithm. The code

implementing a second-order Newton-type method can be found in Ref. [23]. Another method, which

classifies derivative-free algorithm, is the simplex method outlined in Ref. [24], and an improved Nelder

and Mead method [24] is shown in Ref. [25]. The proposed method employs a procedure that

approximates the v2 function and uses gradient information to search the optimized parameters. A

numerical approximation method is used to represent the derivative of the v2 function. An algorithm in

Section 5 is outlined to estimate the parameters for the seasonal Model 10.

4. Searching parameters in Em dimensional space

One of the difficult computational steps in searching parameters aj by minimizing v2 function is that

there are several local minima within the reasonable range of aj. The v2 function can be considered a

continuous function of m parameters in aj, describing a hyper surface in Em dimensional space. By

expanding v2 function into a second-order Taylor series containing aj parameters, the v2 surface can be

approximated with a parabolic hyper surface. Such expansion of the function makes it easy to search the

parameters. The search scheme explores the approximate function with initial estimates determined at

the beginning of the computation. If the estimates are close enough to the minimum trajectory, the

parabolic approximation of the v2 hyper surface will be close to the original function, and therefore, it

would provide an appropriate estimate of aj.

5. An algorithm to estimate the model parameters

The first step in estimating the parameters aj with j=1,. . .,m is to define a v2 merit function. The best

fit parameters are determined by an iterative minimization procedure. Supplying initially the trial values

for the parameters, the search procedure improves the trial solution. Section 5.1 shows the steps of the

nonlinear forecast (NLF) algorithm.

Table 2

Peak load data

Initialization Validation

1 2 3 4 5 . 11 12

1 774987 13 844948.5 25 858405.5 37 995240 49 1093810 . 121 1441844 133 1649371

2 802550 14 895417.5 26 957958 38 1063817 50 1151488 . 122 1545458 134 1547936

3 821357 15 940760.5 27 984996.5 39 1105389 51 1197457 . 123 1471433 135 1698465

4 815251 16 927171 28 996470.5 40 1041177 52 1171060 . 124 1542287 136 1531830

5 798545 17 823056 29 905901.6 41 976566 53 1032625 . 125 1337216 137 1571064

6 589409 18 649692 30 658394 42 808272 54 783523 . 126 1059582 138 1175476

7 482309 19 522538 31 493505.5 43 616155 55 668767 . 127 755114 139 905122

8 495076 20 464687 32 480823 44 531927.5 56 475088.5 . 128 792202 140 761702

9 369670 21 451027.5 33 536153 45 504449 57 432578 . 129 814115 141 961613

10 575153 22 615095.5 34 575929 46 685147 58 634057.5 . 130 808330 142 878376

11 634944 23 704391 35 711404 47 864022.5 59 916419 . 131 1084498 143 1157074

12 847835 24 792831 36 930286.5 48 1035545 60 1092229 . 132 1288743 144 1393310

Table 3

Parameters’ values

Coefficient Model 1 Model 2 Model 3 Model 4 Model 5 Model 9 Model 10

d0 643514.09 644831.05 645780.49 848948.63 13.62 7500152.8 158180.14

d1 3880.31 3850.87 3836.59 3831.96 0.0043 551802.2 440.43

b1 238109.94 282289.61 282236.33 44330.85 0.0394 512123.39

b2 39753.36 125276.78 125291.06 48705.17 0.0459 �45142.78

b3 �40043.31 19306.18 0.0173 555362.64

b4 30746.44 �118354.0 �.1132 304864.07

b5 �324301.6 �.3591 �49133.04

b6 �470104.7 �.5646 �2128.63

b7 �536147.9 �.6672 203312.81

b8 �509895.3 �.6448 �117108.02

b9 �400752.8 �.4479 473984.93

b10 �184645.4 �.1967 80627.21

b11 �2461.86 �.0118 419415.44

X1 230.80

X2 186.57

X3 717202.82

c1 �1284.9 �40018.58 �3696.08

c2 687.90 30732.16 26898.71

c3 �8186.55


5.1. The NLF algorithm

(1) Provide initial estimastes of aj and a small incremental step size aj, (i.e., daj) select m.(2) Set j=j+1

(3) aj=aj+daj

c4 5867.08

Table 4

Statistical parameters

Statistics Model 1 Model 2 Model 3 Model 4 Model 5 Model 10 Model 6 Model 7 Model 8 Model 9

No. of

coefficients

6 10 6 13 13 16

df 132 132 132 132 132 132 129 132 132 132

MSE1010 1.086 1.106 1.115 1.1269 0.8625 7.9629

SSE1010 136.95 135.00 140.54 134.105 108.673 1051.1 718.410 222.09 141.305 123.481

r21010 1.086 1.106 1.115 1.1269 0.8625 7.9628

r e105 1.043 1.05194 1.0561 1.06157 9.5425 3.01 2.3593 1.2966 1.003 0.9672

R2 0.86689 0.868794 .86341 .86966 .8943 .873

R̄2 .86161 .859115 .85799 .85652 .8837 .8560

F test 112.91 89.7598 159.305 66.1708 83.976 52.988

D–W test 0.7350 0.5885 0.6462 0.5671 0.6106 2.152 .4329 .6960 .5431 .5979

S e105 0 0 0 0 7.019 .3443 6.665 �4.588 �35.3654 0

Relative

percentage e�1.6322 �1.214 �1.218 �1.212 �0.723 �2.048 �4.6255 1.625 �4.148 �1.4195

MAE105 0.7842 0.7775 0.7744 0.7739 0.7364 2.22676 1.9923 1.0531 0.7265 0.7209

MAPE 9.491 9.727 9.691 9.696 8.968 26.382 25.012 13.018 9.149 8.909

Fig. 2. Electric load forecast: initialization period (Actual, and Models 1,2, and 3).


(4) Minimize v2, using v2 expansion error function.

(5) Is v2 minimized? If yes go to Step 6, otherwise select new daj and go to Step 3.

(6) Select aj when v2 is minimum and call it ajmin; select two estimates of ajmin {ajl, ajr} in the neighborhoods of

ajmin, such that ajmin is minimum between {ajl, ajr}.(7) Use these three points to construct a parabolic approximation of v2 hyper surface.(8) Search for the value of aj that minimizes the new approximate function of v2, and set it equal to aj.(9) If the new minimum improves reduction in error function, go to Step 10, otherwise go to Step 3.

(10) If j=m, stop, otherwise go to Step 2.

The algorithm performs well with user-supplied initial estimates of the parameters. The suggested

NLF algorithm builds up information during the search process. It has been coded to solve the problem.

6. Performance of the models

The seasonal data available are divided into two sets, as shown in Table 2. The first data set contains

132 data points and is called initialization or fitting data or test set. The second data set contains 12 data

Fig. 3. Electric load forecast: initialization period (Actual, and Models 4, 5, and 10).

Fig. 4. Electric load forecast: initialization period (Actual, and Models 6 and 7).


points defined as validation data. The first set of data is used to find the best fitted models based on

statistical criteria, such as F test, adjusted R-square (R̄2), mean absolute percentage error (MAPE), and

D–W test. The second set of data is used to test the forecast models. The MAPE, relative percentage

error, and absolute error criteria are used to check the accuracy of the seasonal models.

Table 3 lists the values of the parameters of Models 1–5, 9, and 10, using initialization data.

Parameters of the smoothing-based methods are determined by simulation adopting square error criteria.

Model 8 is related to Winter’s method that contains three parameters a, b, and c, with values 0.02, 0.3,

and 0.02, respectively. The double-exponential-type Model 7 selects both the parameters as 0.089.

Comprehensive statistical analysis of the models is presented in Table 4. Figs. 2–6 show the

performance of the 10 models on initialization data set.

7. Analysis of results

The analysis of the models based on several statistical criteria could be categorized into two groups.

Models 1–5 are regression-based and hence provide long-range forecast, while Models 6, 7, 8, and 9 are

smoothing-based, and are therefore good for short-range forecast. The models are tested with the test

data and the validation data. Different test statistics, as shown in Table 4, are used to rank the models.

F test indicates that Models 1–5 are highly significant. The regressors explain a significant amount of

variability more than 86% in the change of monthly peak electric load. Adjusted R2 values are also

Fig. 5. Electric load forecast: initialization period (Actual, and Models 8 and 9).

Fig. 6. Peak electric load forecast in validation period.


highly significant in all the models. Model 5 appears to be best explained by the R2 statistics, with a

value of .8943. Model 10 has a close value of R2=.873. However, it should be noted that R2 alone does

not increase the appropriateness of a model [11]. Root mean square error at 86248 with Model 5 implies

that one can expect to predict 95% of the time a monthly peak demand accurate to within 172496 MWof

its true value. It is slightly higher for other models. Model 4 performs better than Model 10 with MAPE

and mean absolute error (MAE) statistics at 9.696, 77390 and 26.382, 222676, respectively.

The statistics based on the values of MAPE and MAE suggest that the Model 5 is more appropriate

for the application. D–W statistics generally provides information as to whether or not the model has

been correctly fitted. A value of 2 would imply that the fitting errors are random. All models exhibit D–

W values of less than 2, except for Model 10. It appears that there exists a positive autocorrelation, and

Table 5

Validation results

Forecast error: (actual forecast)

1 5 8 9 4 10

Error Error Error Error Error Error

1 142,551.71 202,219.91 149,632.77 235,101.05 290,771.75 807,482.98

2 �7,537.86 36,195.04 �15,806.84 70,753.58 141,173.93 191,734.29

3 200,131.34 170,422.97 120,448.63 210,563.16 283,496.66 897,352.33

4 180,619.17 40,468.54 �8,024.82 77,324.99 142,428.68 128,048.07

5 416,643.92 256,534.71 210,600.66 291,371.54 315,490.90 415,630.24

6 213,548.26 143,183.95 105,143.18 159,796.31 122,018.59 371,902.50

7 78,207.51 60,968.92 33,179.80 74,891.46 �6,364.32 52,420.96

8 �25,630.15 �3,407.45 �24,960.86 12,734.67 �87,573.04 �298,549.79

9 106,078.98 175,809.64 145,467.24 173,347.80 82,253.37 219,686.71

10 �136,449.26 �82,472.20 �108,093.39 �57,461.55 �113,958.04 �456,724.49

11 �66,636.21 �83,494.61 �116,089.91 �66,058.48 �55,199.45 213,264.22

12 �33,748.67 �105,598.25 �139,796.27 �72,804.45 �4,978.95 108,659.35

Sum error 1,067,778.74 810,831.19 351,700.20 1,109,560.08 1,109,560.08 2,650,907.37

Absolute error 1,607,783.04 1,360,776.18 1,177,244.37 1,502,209.04 1,645,707.69 4,161,455.93

MAPE 1.24 1.05 0.95 1.14 1.22 3.27

Relative percentage error �0.69 �0.56 �0.20 �0.79 �0.61 �1.44

S.D. 153,581.45 122,091.97 116,736.21 123,295.50 149,999.20 386,347.35

Fig. 7. Electric load forecast in validation period (Actual, and Models 1 and 4).


hence, a relatively smooth pattern cannot be ruled out. Model 10 provides a D–W value of 2.152, and

therefore, it is preferred over Models 1–5.

Among the smoothing-based forecast models, the Model 9, has the smallest MAPE with a value of

8.909, followed by Model 8 with a value of 9.149. MAE for Model 9 is 72090, which is lower than

Model 8 with a value 72650. In case of D–W statistics, Model 9 slightly outperforms Model 8 with

values of 0.5979 and 0.5431, respectively. Model 8 can be ignored due to its limitation to forecast in

limited period only.

In light of the above information, Models 4, 5, and 10, belonging to the regression-based model, and

Models 8 and 9 from smoothing-based methods are used as candidates for forecasting. The forecast

performance is evaluated based on the test data set. The forecast values are compared with the actual values

to identify the best model. Referring to Table 5, Model 10 can be dropped due to higher values of MAPE

andMSE. Using the lowest values of MAPE, relative percentage error, and absolute error as criteria, it can

be concluded that Model 5 performs best, followed by Models 4 and 9. Forecasts in validation period by

Models 4, 5, 8, 9, and 10 are shown in Figs. 7–9. Finally, it can be concluded from the above analysis that

Model 5 seems superior to all in predicting the peak electric load demand for the GCC city.

8. Reliability of models with solar and lunar calendars

The trigonometric model, Model 1, is further tested with lunar and solar calendars. The lunar calendar

lags behind the solar calendar by approximately 14 days, and this lag is used to define a factor for

Fig. 8. Electric load forecast in validation period (Actual, and Models 8 and 9).

Table 6

Solar and lunar calendars’ peak load profiling

ANOVA df SS MS F Significance F

Peak load estimation on solar calendar

Regression 5 8.39437E+12 1.68E+12 111.6247 1.4714E�44

Residual 126 1.89508E+12 1.5E+10

Total 131 1.02895E+13

Peak load estimation on lunar calendar

Regression 5 8.92E+12 1.78E+12 164.1254 2.09E�53

Residual 126 1.37E+12 1.09E+10

Total 131 1.03E+13


periodic cycle. The ratio (14/30c0.5) between the lag time of 14 days in the lunar calendar and the

average number of days taken as 30 in a month is added to model the cyclic pattern. Model 1 is now

fitted with this factor, and the results are listed in Tables 6 and 7. The fitted results are superimposed over

the actual data for comparison, as shown in Fig. 9.

The R-square values in Table 7 suggest that the lunar calendar peak electric load data fits better than

that of the solar calendar. The standard error is comparatively smaller with the lunar calendar. The

ANOVA in Table 6 suggests that the model fits well with 95% reliability. The F statistics is also

significant at this level.

The trigonometric series can be used to estimate and forecast all the components of the peak electric

load. The trend component is estimated by d0+d1t, the irregular pattern is estimated with the mean and

variance of the resulting residuals, while the sine and cosine terms are used to estimate the cyclical and

seasonal pattern. Such model does not require operating parameter values to determine the best fit.

Trigonometric models forecast all forecast components simultaneously, such as trend, seasonality, and

cycles. Although computational complexity is a drawback, it is objectively easy to use. The casual

prediction model has the advantage to forecast power load with simple prediction models. Often, the

relationship between power load and factors controlling the power load is nonlinear; it is difficult to

identify the nonlinear nature with the casual models.

Fig. 9. Comparison of load estimation with lunar and solar calendars.

Table 7

Regression statistics with solar and lunar calendar

Solar calendar Lunar calendar

Multiple R 0.903229041 0.931072

R Square 0.815822701 0.866896

Adjusted R Square 0.808514078 0.861614

Standard Error 122639.1261 104257.4

Observations 132 132


Long-term forecast for utilities are important for capacity expansion, capital investment, cash flow

analysis, project feasibility analysis, and cost estimation. The uncertainty in long-term forecast due to

unforeseen events makes it difficult to arrive at a reliable estimate. Hence, an accurate long-term load

estimation is crucial for future planning and long-term strategic investment policies. The proposed

models are easy to incorporate for strategic planning with little extra efforts in computations. For capital

extensive project, such as power plant constructions and operations, it is crucial to have a very reliable

long-term estimate where all possible nonlinearities are taken into account for policy formulations.

9. Conclusions

The study shows 10 long- and short-range forecast models for peak electric load forecast. There are

five multivariate regressions: one of a multiplicative decomposition type, three smoothing, and one

nonlinear model. Software is developed for all the peak electric load forecast models. It is observed that

Model 5 is suitable to forecast peak electric load demand for the GCC city, followed by Models 4 and 9.

Model 10 outperforms others in the D–W statistics. Note that Model 9 is simple in concept, economical

to maintain, and could further be improved for long-term predictions. It is noticed that the modes of peak

load in the lunar and solar calendars are closely related and show little deference in magnitude of error.

This could be attributed to the magnitude of data. The extra pilgrimage electric load demand contributed

to the irregular components. It should be noted that NLF algorithm requires several experiments to find

the best estimates of the statistical parameters. A modified simulation strategy may improve the local

minimum and would hence provide a more reasonable estimate of the parameters. The results are

reported with initial estimates for all aj=100 and daj=0.01.

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Shamsuddin Ahmed is an engineer by training. He holds a PhD degree from Edith Cowan University, Australia, in Neural

Computations. He started his professional career in manufacturing and utility industries. During his service, he designed and

commissioned a production industry, which is fully functional now. Later, he moved to academic institutions and is currently a

faculty member at the College of Business and Economics at the United Arab Emirates University, which is an AACSB

International-accredited institution. His research interests include ANN, industrial simulation, service industry management,

multivariate statistical analysis, and business intelligence.

Documents

Seasonal models of peak electric load demand