Long term electric load forecasting based on particle swarm optimization

Applied Energy 87 (2010) 320–326

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Applied Energy

journal homepage: www.elsevier .com/ locate/apenergy

Long term electric load forecasting based on particle swarm optimization

M.R. AlRashidi *, K.M. EL-NaggarElectrical Engineering Department, College of Technological Studies, Shuwaikh, Kuwait

a r t i c l e i n f o

Article history:Received 19 February 2009Received in revised form 2 April 2009Accepted 16 April 2009Available online 19 May 2009

Keywords:ForecastingParticle swarm optimizationLeast errorPeak loadEstimation

0306-2619/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.apenergy.2009.04.024

* Corresponding author. Tel.: +965 2231 4312; fax:E-mail addresses: [email protected] (M

hotmail.com (K.M. EL-Naggar).

a b s t r a c t

This paper presents a new method for annual peak load forecasting in electrical power systems. The prob-lem is formulated as an estimation problem and presented in state space form. A particle swarm optimi-zation is employed to minimize the error associated with the estimated model parameters. Actualrecorded data from Kuwaiti and Egyptian networks are used to perform this study. Results are reportedand compared to those obtained using the well known least error squares estimation technique. The per-formance of the proposed method is examined and evaluated. Finally, estimated model parameters areused in forecasting the annual peak demands of Kuwait network.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

One of the primary tasks of an electric utility is to accuratelypredict load requirements at all times. Results obtained from loadforecasting process are used in planning and operation. For exam-ple, long-term load forecasting, one to ten years ahead monthlyand yearly values, is applied in expansion planning, inter-tie tariffsetting and long-term capital investment return problems. Whileshort-term load forecast results, one day to one month aheadhourly and daily values, are needed in unit commitment, mainte-nance and economic dispatch problems. Therefore the accuracyof load forecasting has significant effect on power system planningand operation. The time horizon for mid and long-term forecastingranges between few weeks and several years. Unfortunately, it isquite difficult to forecast load demand over a planning period ofthis length. This fact is due to the uncertain nature of the forecast-ing process. There are large numbers of influential factors thatcharacterize and directly or indirectly affect the underlying fore-casting process; most of them are uncertain and uncontrollable.

Many classic approaches have been proposed and applied tolong-term load forecasting to estimate model parameters, includ-ing static and dynamic state estimation techniques [1–4]. Whilethe least error square (LES) technique has been the most famousconventional static estimation technique and in use for a long timeas the preferred technique for optimum estimation in general,some limitation and disadvantages are associated with this

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+965 24816568..R. AlRashidi), knaggar60@

approach. For example, when the data set is contaminated withbad measurements, the estimates may be inaccurate unless a largenumber of data points are used. Ref. [5] proposed a static methodbased on noniterative least absolute value technique. This methodhas the advantage of detecting bad data.

Another powerful class of estimation is the stochastic dynamicone. Kalman filtering and the least absolute value filtering algo-rithms are examples of such dynamic approaches. Unlike static ap-proaches, where the whole set of data is used to obtain the optimalsolution, dynamic filters are recursive algorithms. In recursive fil-ters, the estimates are updated using each new measurement. Dy-namic filters are well suited to on-line digital processing as dataare processed recursively. They had been used extensively in esti-mation problems for dynamic systems [6]. Dynamic filters have theadvantage of being able to handle measurements that change withtime. Methods based on artificial intelligence such as artificial neu-ral networks (ANN) and expert systems have been also proposedand shown promising and encouraging results [7,8]. Support vectormachine (SVM) has been an attractive tool for load forecasting re-cently. SVM is a form of machine learning method which is devel-oped from statistical learning theory. Like ANN, the SVM has theproblem of network parameter selection [9]. Heuristic searchmethods like genetic algorithms (GA) were also proposed andimplemented. This method is based on the mechanism of naturalselection and natural genetics [10]. Hybrid methods using ANN,GA, SVM were also proposed in many Refs. [11,12].

This paper presents a new method for long-term load forecast-ing using particle swarm optimization (PSO) technique. PSO is aglobal optimization algorithm that deals with problems in whicha best solution can be represented as a point or surface in

M.R. AlRashidi, K.M. EL-Naggar / Applied Energy 87 (2010) 320–326 321

an n-dimensional space. It has been applied to different areas ofpower systems as shown in Refs. [13,14]. The forecasting problemis presented in state space form. PSO technique is used to estimatethe parameters of different load forecasting models. The proposedmethod is tested using actual recorded data for both of the Egyp-tian and Kuwaiti networks. Forecasting results are obtained andevaluated.

2. Mathematical formulation

Load forecasting models are developed to mathematically rep-resent the relationship between load and influential variables suchas time, weather, etc. Coefficients of the formulated model areidentified and used to predict the future loads by extrapolatingthe relationship to desired lead time. Final accuracy of the forecastprocess depends on the selected model and accuracy of the esti-mated parameters. Reviewers of load forecasting models havefound that techniques almost in use today can be categorized asbeing of multiple regression, general exponential smoothing andstatistical methods [3].

Regression analysis or trend analysis is the study of the behav-ior of a time series or process in the past and its mathematicalmodeling so that future behavior can be extrapolated from it [4].A time variant event such as power system load can be brokendown into five components, basic, trends, seasonal variations, cyc-lic variations and random variations. The last three variations havea long-term zero mean. Regression curves used in power systemload forecasting are: linear, polynomial, exponential and power.In general, a multi-variable regression model can be related ton + 1 independent variables and can be written as follows:

PðtÞ ¼ a0 þXn

i¼1

aiti þ rðtÞ ð1Þ

where P(t) is the peak load demand at time t, ao, ai are the regressioncoefficients relating the load P(t) to time t. The last term, r(t), is theresidual load at year t.

Another type of regression technique involves nonlinear regres-sion models that are nonlinear in terms of the parameters and cannot be made so by any transformation. The following models arenonlinear [4]:

PðtÞ ¼ a0ea1t þ rðtÞPðtÞ ¼ a0 þ a1ea1t þ rðtÞ

PðtÞ ¼ ao

1þ a1ea2tþ rðtÞ

ð2Þ

For many years, generation planners have used regression tech-niques as tools in predicting annual peak system demands. Electric-ity demands are known to be influenced by weather conditions,number and type of consumers and general economic conditions.However, a simple relationship in which demand increases expo-nentially with time is generally found to yield an adequate forecastfor system peak demand. Forecasts are frequently obtained from thefollowing simple relationship:

PðtÞ ¼ eaþbt ð3Þ

This equation can be simply transferred into a linear form by takingthe (ln) of both sides.

In order to identify the most adequate model for forecastingapplication among all available linear and nonlinear regressionmodels, different types of graphs must be examined. Graphical vi-sual inspection of a graph of a given observation against time canoften reveal both obvious and less apparent data characteristics.After estimation process, the resultant residuals are subjected towhiteness test.

2.1. Whiteness test

Whiteness test is used to ensure that a selected model ade-quately describes a given set of data [5]. It can be achieved bythe following two steps:

(1) Examination of the estimated residual graph (exploratoryanalysis); and

(2) Calculation of the residual autocorrelation function (RACF)at different time lags (confirmatory analysis).

Step 1 involves testing the whiteness of the estimated residuals.The most informative approach to check for the whiteness is toexamine the graph of estimated residuals against the estimatedobservation (peak demand in our case). If the estimate residualsform a pattern around its mean, then one can conclude that theresiduals are not white (correlated). This means that an importantindependent variable has been omitted from the selected model.The objective of this step is to explore, but not to confirm, whetheror not the estimated residuals are white (uncorrelated). The objec-tive of step 2 is to confirm whether or not the estimated residualsare white. Therefore this step is called confirmatory analysis and isachieved via calculating the RACF as follows:

RACFk ¼Pm

t¼kþ1etet�kPmt¼1e2

t

ð4Þ

where RACFk is the RACF at time lag k, et is the estimated residual attime t.

Typical values of RACF are in the range of [�1, 1]. If a given va-lue (rather than the first one) is significantly different from zero, itwill fall outside a confidence interval level [5].

In order to build a proper forecasting process, one must con-struct the model, select the forecasting method and finally eval-uate the results. As mentioned before, the regression technique isthe most widely used one mainly because of its simplicity andease of use. Therefore, this technique is considered for modeling.It is very important to emphasize that the primary objective ofthis paper is to present the application of PSO technique to esti-mate the parameters of load forecasting models and evaluate theresults obtained. The objective is not to present a modelcomparison.

Parts of data used in this paper are taken from Refs. [5,10]. Inthese references, the data has been tested and appropriate modelhas been chosen. Since the scope of this paper is focused on theapplication of PSO technique for the long-term load forecasting,the models proposed and tested in [5,10] will be used directly here.Therefore, in this paper two model are considered, i.e. i = 1 and 2.Given the peak load (P) at each year t, an equation just like Eq.(1) can be written for each load. If the data consists of m sets ofyears and peak loads, then there will be (m) equations with (n) un-knowns. This system of equation is an overdetermined system(m > n). Then for m years, a discrete system of equations in statespace form can be written as:

PðtÞ ¼ HðtÞWþ rðtÞ ð5Þ

where P(t) is the load demand vector, W is the parameter vector tobe estimated, r(t) is the error vector associated with P(t), H(t) is arow vector that relates P(t) to W.

In this study two models used are:Model 1 (i = 1)

HðtÞ ¼ ½ T 1 �; T ¼ 1;2; . . . m and W ¼ ½a b �T ð6Þ

Model 2 (i = 2)

HðtÞ ¼ ½ T2 T 1 �; T ¼ 1;2; . . . m and W ¼ ½a b c �T ð7Þ

322 M.R. AlRashidi, K.M. EL-Naggar / Applied Energy 87 (2010) 320–326

Now, the problem is to find an estimate for the parameter vector Wfor any model that minimizes the error vector r(t).

3. Particle swarm optimization

In PSO, a number of particles form a ‘‘swarm” that evolve orfly throughout the feasible hyperspace to search for fruitful re-gions in which optimal solution may exist. Each particle hastwo vectors associated with it, the position (Xi) and velocity(Vi) vectors. In N-dimensional search space, Xi = [xi1, xi2, . . ., xiN]and Vi = [vi1, vi2, . . ., viN] are the two vectors associated with eachparticle i. During their search, members of the swarm interactwith each others in a certain way to optimize their search expe-rience. There are different variants of particle swarm paradigmsbut the most commonly used one is the gbest model where thewhole population is considered as a single neighborhoodthroughout the flying experience [15]. In each iteration, particlewith the best solution shares its position coordinates (gbest)

-5-4-3-2-1012345-20

-15

-10

-5

0

5

10

15

20

A

B

C

X1

4-3-

2-1-

01

23

45

B

C

X1

g

Fig. 1. Information sharing m

information with the rest of the swarm. Information sharingmechanism among swarm members of the gbest model can bebest described using the following graphical illustration. A math-ematical function of two variables with multiple valleys andpeaks is depicted in Fig. 1. The known global solution is foundto be (�4.7119, 4.7116) with minimum objective value of�16.4248. In this gbest model, the entire swarm of four mem-bers, namely A, B, C, and D forms a single neighborhood. A snap-shot of each particle location during their flying experience (initeration k) is shown in Fig. 1. It is clear that particle A is theclosest one to the global solution (i.e. having minimum objectivevalue) in this iteration. Thus, it will send its gbest coordinates tothe rest of the swam members. Each particle updates its coordi-nates based on its own best search experience (pbest) and gbestaccording to the following equations:

vkþ1i ¼ wvk

i þ c1rand1ðpbestki � xk

i Þ þ c2rand2ðgbestk � xki Þ ð8Þ

xkþ1i ¼ xk

i þ vkþ1i ð9Þ

-5-4-3-2-1012345

D

X2

5-

5-4-

3-2-

1-0

12

34

5

A

D

X2

best

2D

3D

echanism in gbest model.

2000

3000

4000

5000

6000

7000

8000

1977 1980 1983 1986 1989 1992

Year

Peak

Loa

d (M

W)

M.R. AlRashidi, K.M. EL-Naggar / Applied Energy 87 (2010) 320–326 323

where c1 and c1 are two positive acceleration constants, they keepbalance between the particle’s individual and social behaviorwhen they are set equal; rand1 and rand2 are two randomly gener-ated numbers with a range of [0,1] added in the model to introducestochastic nature in particle’s movement; and w is the inertiaweight and it keeps a balance between exploration and exploita-tion. In our case, it is a linearly decreasing function of the iterationindex:

wðkÞ ¼ wmax �wmax �wmin

Max:Iter:

� �� k ð10Þ

where k is the iteration index.The most important factor that governs the PSO performance in

its search for optimal solution is to maintain a balance betweenexploration and exploitation. Exploration is the PSO ability to coverand explore different areas in the feasible search space whileexploitation is the ability to concentrate only on promising areasin the search space and to enhance the quality of potential solutionin the fruitful region. Exploration requires bigger step sizes at thebeginning of the optimization process to determine the mostpromising areas then the step size is reduced to focus only on thatarea. This balanced is usually achieved through proper tuning ofPSO key parameters.

Recently, PSO developments and applications have been widelyexplored in engineering and science mainly due to its distinctfavorable characteristics. With regard to its mathematical develop-ment, Refs. [16] is an excellent reference that analyzed and studiedthe PSO promising convergence characteristics. The authors suc-cessfully established some mathematical foundation to explainthe behavior of a simplified PSO model in its search for an optimalsolution. Just like in the case of other evolutionary algorithms, PSOhas many key features that attracted many researchers to employit in different applications in which conventional optimizationalgorithms might fail such as:

� It only requires a fitness function to measure the ‘‘quality” of asolution instead of complex mathematical operations like gradi-ent, Hessian, or matrix inversion. This reduces the computa-tional complexity and relieves some of the restrictions that areusually imposed on the objective function like differentiability,continuity, or convexity.

� It is less sensitive to a good initial solution since it is a popula-tion-based method.

� It can be easily incorporated with other optimization tools toform hybrid ones.

� It has the ability to escape local minima since it follows proba-bilistic transition rules.

More interesting PSO advantages can be emphasized whencompared to other members of evolutionary algorithms like:

� It can be easily programmed and modified with basic mathe-matical and logic operations.

� It is inexpensive in terms of computation time and memory.� It requires less parameter tuning.

Table 1PSO parameters.

Parameter Value

Population 10 particlesStop criterion 1000 iterationsVelocity Vmax = 2.0, Vmin = 0Acceleration constants C1 = 3, C2 = 3Inertia weight Wmax = 0.9, Wmin = 0.4

� It works with direct real valued numbers that eliminates theneed to do binary conversion of classical canonical geneticalgorithm.

Pseudo-code of the proposed approach is described as follows:

Set PSO parameters;For each particle

Randomly initialize the parameter vector;Randomly initialize the velocity vector;

EndMeasure the fitness of each particle;Store pbestStore gbestWhile the stopping criteria is not met

For each particleUpdate the velocity and position vectorsMeasure the fitness of the new position vectorIf the new fitness value is better than the previously stored

oneStore the new position vector as pbestStore the new fitness value

EndEndDetermine the particle with lowest fitness value in the searchhistory and store its position vector as gbest

End

4. Practical application and results

The proposed PSO method is implemented in Matlab computingenvironment which gives easy access to compare its outcomeswith other optimization/estimation methods. Real peak demandsof Egyptian and Kuwaiti Networks are used in this study [17,18].

Fig. 2. Egypt annual peak demands.

Table 2Estimated parameters based on PSO and LES.

Coefficients Linear model Quadratic model

PSO LES PSO LES

a 377.8410 363.1600 �8.2818 �7.1169b 1714.2260 1874.8000 508.4816 491.2700c – – 1455.4930 1469.1000

Table 3Performance of both forecasted models.

Year Actual PD

(MW)Linear model Quadratic model

PSO LES PSO LESForecastedPD (MW)

ForecastedPD (MW)

Forecasted PD

(MW)ForecastedPD (MW)

1977 2284 2092.067 2237.96 1955.69278 1953.25311978 2449 2469.908 2601.12 2439.32892 2423.17241979 2742 2847.749 2964.28 2906.40142 2878.85791980 3161 3225.59 3327.44 3356.91028 3320.30961981 3350 3603.431 3690.6 3790.8555 3747.52751982 3981 3981.272 4053.76 4208.23708 4160.51161983 4672 4359.113 4416.92 4609.05502 4559.26191984 5158 4736.954 4780.08 4993.30932 4943.77841985 5361 5114.795 5143.24 5360.99998 5314.06111986 5803 5492.636 5506.4 5712.127 5670.111987 6152 5870.477 5869.56 6046.69038 6011.92511988 6279 6248.318 6232.72 6364.69012 6339.50641989 6664 6626.159 6595.88 6666.12622 6652.85391990 7004 7004 6959.04 6950.99868 6951.96761991 7215 7381.841 7322.2 7219.3075 7236.84751992 7503 7759.682 7685.36 7471.05268 7507.49361993 7657 8137.523 8048.52 7706.23422 7763.9059

7000

8000

)

324 M.R. AlRashidi, K.M. EL-Naggar / Applied Energy 87 (2010) 320–326

The data set is used to establish an overdetermined system ofequations. This system of equations is solved using the proposedPSO technique to find the optimal parameters for different fore-casting models. A series of experiments were conducted to finetune the proposed PSO. Key parameters of PSO algorithm used inthis study are presented in Table 1. Both linear and nonlinear mod-els are used in each test system and results obtained using PSO arecompared with those of LES method.

4.1. Estimation of model parameters and forecast

4.1.1. Case 1: model parameters estimation of Egyptian networkPeak demands of Egyptian power network during 1977–1993

are used to estimate the parameters of both linear and nonlinearlong-term forecasting models. Fig. 2 shows the annual peak loadsof Egypt network. PSO and LES techniques are used to estimatemodels’ parameters for the same time horizon and the computedparameters are tabulated in Table 2. The corresponding forecasteddemands based on the estimated parameters of both modelsare shown in Table 3. Estimation error analysis is carried out in

Table 4Error associated with PSO and LES.

Year Actual PD (MW) Linear model Quadratic model

PSO LES PSO LESError (%) Error (%) Error (%) Error (%)

1977 2284 8.4034 2.0158 14.3742 14.48101978 2449 0.8537 6.2115 0.3949 1.05461979 2742 3.8566 8.1065 5.9957 4.99121980 3161 2.0433 5.2654 6.1977 5.03981981 3350 7.5651 10.1672 13.1599 11.86651982 3981 0.0068 1.8277 5.7080 4.50921983 4672 6.6971 5.4598 1.3473 2.41311984 5158 8.1630 7.3269 3.1929 4.15321985 5361 4.5925 4.0619 0.0000 0.87561986 5803 5.3483 5.1111 1.5660 2.29001987 6152 4.5761 4.5910 1.7118 2.27691988 6279 0.4886 0.7371 1.3647 0.96361989 6664 0.5678 1.0222 0.0319 0.16731990 7004 0.0000 0.6419 0.7567 0.74291991 7215 2.3124 1.4858 0.0597 0.30281992 7503 3.4211 2.4305 0.4258 0.05991993 7657 6.2756 5.1132 0.6430 1.3962

Average error (%) 3.8336 4.2103 3.3488 3.3873

Table 4 that reveals that PSO on average performed better thatLES in minimizing the error associated with the estimation process.Thus, the parameters estimated using PSO are better correlatedwith the actual measurements recorded.

4.1.2. Case 2: model parameters estimation of Kuwaiti networkPeak loads data of Kuwaiti network during 1992–2005 are used

in this case. Fig. 3 shows the maximum load profiles during thatperiod. Similar analysis with identical PSO parameters used in Case1 is applied on Kuwaiti network with the computed models’ coef-ficients shown in Table 5. Based on these coefficients, Table 6 pre-sents corresponding peak demands of both estimation methods fordifferent forecasting models. From average error point of viewshown in Table 7, PSO estimation generated better results whencompared to the LES. It is noted from both test cases that PSO isconsistent in producing better parameter estimation of differentforecasting models. In both cases, the quadratic model generatedbetter average error which indicates that the nonlinear model isbetter suited to peak load forecasting than the linear model.

4.1.3. Case 3: peak demand forecast of Kuwaiti networkBased on the parameters estimated from Case 2, annual peak

load forecast of Kuwaiti network for years 2006–2010 is presentedin Fig. 4. Both models are used to forecast the peak demands. It isnoted that linear model tends to forecast higher peak demandswhen compared to the quadratic model. As mentioned before,accuracy of the forecast models is not the main concern of this pa-per since it mainly focuses on the parameters estimation part ofthe problem. It is also worth mentioning that Kuwait network, justlike in any other power system, may be influenced by other impor-tant factors that should be incorporated in the forecast model. Therecent rapid growth in the country infrastructure as a direct resultof the skyrocketing oil prices has resulted in greater hunger forelectricity. Therefore, oil price is an important factor in forecastingthe annual peak demands for Kuwaiti network.

3000

4000

5000

6000

1992 1994 1996 1998 2000 2002 2004

Year

Peal

Loa

d (M

W

Fig. 3. Kuwait annual peak demands.

Table 5Estimated parameters based on PSO and LES.

Coefficients Linear model Quadratic model

PSO LES PSO LES

a 344.0348 354.5900 �2.8311 �0.7761b 3353.8610 3287.7000 390.0019 366.2300c – – 3207.6570 3256.6000

Table 6Performance of both forecasted models.

Year Actual PD

(MW)Linear model Quadratic model

PSO LES PSO LESForecastedPD (MW)

ForecastedPD (MW)

Forecasted PD

(MW)ForecastedPD (MW)

1992 3460 3697.8958 3642.29 3594.82779 3622.05391993 4120 4041.9306 3996.88 3976.33636 3985.95561994 4350 4385.9654 4351.47 4352.18271 4348.30511995 4730 4730.0002 4706.06 4722.36684 4709.10241996 5200 5074.035 5060.65 5086.88875 5068.34751997 5360 5418.0698 5415.24 5445.74844 5426.04041998 5800 5762.1046 5769.83 5798.94591 5782.18111999 6160 6106.1394 6124.42 6146.48116 6136.76962000 6450 6450.1742 6479.01 6488.35419 6489.80592001 6750 6794.209 6833.6 6824.565 6841.292002 7250 7138.2438 7188.19 7155.11359 7191.22192003 7480 7482.2786 7542.78 7479.99996 7539.60162004 7750 7826.3134 7897.37 7799.22411 7886.42912005 8400 8170.3482 8251.96 8112.78604 8231.7044

Table 7Error associated with PSO and LES.

Year Actual PD (MW) Linear model Quadratic model

PSO LES PSO LESError (%) Error (%) Error (%) Error (%)

1992 3460 6.8756 5.2685 3.8968 4.68361993 4120 1.8949 2.9883 3.4870 3.25351994 4350 0.8268 0.0338 0.0502 0.03901995 4730 0.0000 0.5061 0.1614 0.44181996 5200 2.4224 2.6798 2.1752 2.53181997 5360 1.0834 1.0306 1.5998 1.23211998 5800 0.6534 0.5202 0.0182 0.30721999 6160 0.8744 0.5776 0.2195 0.37712000 6450 0.0027 0.4498 0.5946 0.61712001 6750 0.6549 1.2385 1.1047 1.35242002 7250 1.5415 0.8526 1.3088 0.81072003 7480 0.0305 0.8393 0.0000 0.79682004 7750 0.9847 1.9015 0.6351 1.76042005 8400 2.7340 1.7624 3.4192 2.0035

Average error (%) 1.4699 1.4749 1.3336 1.4434

8000

8200

8400

8600

8800

9000

9200

9400

9600

9800

10000

2006 2007 2008 2009 2010Year

Peak

Dem

and

(MW

)

Linear

Quadratic

Fig. 4. Forecast of peak demands for Kuwait network.

M.R. AlRashidi, K.M. EL-Naggar / Applied Energy 87 (2010) 320–326 325

4.2. Error analysis

The whiteness test described in Section 2 is applied on the re-sults obtained for Kuwaiti network. This is to insure that the se-lected models are adequate. Samples of the obtained results areshown in Fig. 5 for linear model and Fig. 6 for the nonlinear model.In these figures, the RACF for data used in Case 2 are presented.Examinations of these graphs reveals that the both models areappropriate for forecasting the load of the given data set. The RACFis not significantly different from zero; i.e. the models have esti-mated and removed the pattern of the given data set, and whatis left over is a white noise.

4.3. PSO reliability

To better understand the steady performance of the developedalgorithm, 50 independent runs have been conducted to analyzethe PSO reliability. Results are tabulated in Table 8 which signifiesthe robustness of PSO in estimating the optimal model parameters.

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Time Lag (Years)

RACF

Fig. 5. RACF for linear model of Case 2.

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Time Lag (Years)

RACF

Fig. 6. RACF for quadratic model of Case 2.

Table 8PSO statistical data for Cases 1and 2.

Average error (%) Mean Best Worse

Case 1 Linear model 3.8825 3.8336 3.9225Quadratic model 3.3928 3.3488 3.4275

Case 2 Linear model 1.4813 1.4699 1.5011Quadratic model 1.3492 1.3336 1.3578

326 M.R. AlRashidi, K.M. EL-Naggar / Applied Energy 87 (2010) 320–326

5. Conclusions

This paper presents a new application of PSO algorithm forlong-term load forecasting in power systems. The estimation prob-lem is formulated as an optimization one. The solution frameworkis implemented and tested using actual recorded data. Real net-works data are used to validate the performance of the proposedapproach and test its potential. Two different models are usedand the quadratic model is proven to more suitable for represent-ing the available data in terms of average error. Forecasting usingthe PSO method has been compared with that obtained using theLES method. From total error point of view, it is found that PSOmethod has produced better estimates than the LES method. Thisindicates that the PSO approach is quite promising and deservesserious attention as a new tool for parameter estimation. In futurework, incorporation of the oil price in forecasting the annual peakdemands may be included.

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