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Short-term electric load forecasting based on Kalman filtering

Short-term electric load forecasting based on Kalman ?ltering algorithm with mo v ing window weather and load model

H.M.Al-Hamadi,S.A.Soliman *

Power System Research Group,Electrical Engineering Department,College of Engineering,Uni v ersity of Qatar,P.O.Box 2713,Doha,Qatar

Recei v ed 11February 2003;recei v ed in re v ised form 24April 2003;accepted 4May 2003

Abstract

This paper presents a no v el time-v arying weather and load model for sol v ing the short-term electric load-forecasting problem.The model utilizes mo v ing window of current v alues of weather data as well as recent past history of load and weather data.The load forecasting is based on state space and Kalman filter approach.Time-v arying state space model is used to model the load demand on hourly basis.Kalman filter is used recursi v ely to estimate the optimal load forecast parameters for each hour of the day.The results indicate that the new forecasting model produces robust and accurate load forecasts compared to other approaches.Better results are obtained compared to other techniques published earlier in the literature.#2003Else v ier B.V.All rights reserved.

Keywords:Short-term electric load;Kalman ?ltering algorithm;Mo v ing window

1.Introduction

Short-term electric load forecasting is v ital for power generation and operation.It is fundamental in many applications such as pro v iding economic generation,system security,and management and planning.Fore-cast of the total load demand is necessary for unit commitment and economic dispatch of generating units.Other applications of short-term load forecasting that are essential for system security and management are monitoring and controlling,on hourly basis,bus load,system reacti v e power,load flow,and spinning reser v e allocation.

Careful study of the electric load demand re v eals that the load demand is a non-stationary process.It dom-inantly depends on two factors:first,human social acti v ities and habits during day and night,and second on the weather conditions.Short-term forecast deals with predicting load demand from few minutes up to few weeks ahead [1].

Fig.1illustrates the three steps used for load forecasting.The first step is model identification.It is the most crucial step in the load-forecasting technique.It entails deciding on the model structure,model order,and the influential past load and weather data to be used in the model.The second step is model parameters estimation.Database of past load and weather data is used to estimate those parameters.The estimation technique produces optimal parameters that best relate the load at the k th instant of time to the pre v ious load v alues before the k th instant,as well as weather data at and before the k th instant.The last step is load prediction.The optimal parameters at the k th instant are used to predict the future load for next hours,days or weeks.Due to its importance to power system planning and operation,short-term load forecasting has gained researchers’attention during the past few decades.Se v eral re v iews of the forecast methods are published [1,2].

Load-forecasting methods can be categorized as parametric-based methods and artificial-intelligence-based methods.The later method uses neural networks as load models,where the former method formulates load models as mathematical function exploiting the relationship that relates the load to dominant factors affecting this function [3á7].

*Corresponding author.E-mail addresses:helal@https://www.sodocs.net/doc/e414224272.html,.qa (H.M.Al-Hamadi),

solimans@https://www.sodocs.net/doc/e414224272.html,.qa (S.A.Soliman).

Electric Power Systems Research 68(2004)47á

The accuracy of load prediction depends mainly on the model used.Load models are di v ided into three groups [8]:(1)non-weather sensiti v e models that depend on pre v ious v alues of the load,(2)weather-sensiti v e models that depend on weather v ariables,and (3)hybrid models.The time series models are the most popular prediction models used for load-forecasting techniques and they are still used today by many power utilities.All types of time series models [1,8],autoregressi v e models (ARs),mo v ing-a v erage models (MAs),autoregressi v e mo v ing-a v erage models (ARMAs),and autoregressi v e-integrated mo v ing-a v erage (ARIMA)are of the non-weather-sensiti v e models.Another non-weather-sensi-ti v e load model is presented in [9],which use only one pre v ious v alue of the load.Weather-sensiti v e models used in [10]utilize temperature de v iations and wind chill factor for winter day’s model and temperature de v ia-tions and humidity factor for summer day’s model.Many of the models proposed in the literature in conjunction with Kalman filter estimation consider either the dependence of the load on the weather [11,12]or on the pre v ious load as a time series AR [13].In [14,15],the load model consists of two compo-nents,periodic load component that is an expansion of pure sinusoid function of time,and a residual load component that is an ARMA of load and temperature as input.

In this paper,hybrid models are used to express the load as a combination of past loads and dominant weather v ariables to predict future load.Moreo v er,Kalman filter is used to estimate the load model parameters.A detailed re v iew of state space modeling and Kalman filter parameter estimation is found in [1].The rest of the paper is organized as follows:In Section 2,the load model is described.Section 3presents state space model and Kalman filter formulation for parameter estimation.Se v eral examples with illustrati v e results are presented in Section 4.Finally,Section 5is reser v ed for the conclusion.

2.Load-forecasting model

One of the most difficult tasks in load forecasting is the load model identification step.Setting up an appropriate load model has a great impact on the prediction performance.Model formation entails to decide on the model order and to postulate the v ariables that ha v e an effect on the load.In this paper,we will use a parametric model,with parameters that depend on past load beha v ior and weather conditions.

Pre v alent weather patterns ha v e a significant impact on the nature of the load profile.It shows a v ery strong dependence of the load on time.Typical load profiles pre v ail v ery strong correlation at certain periodic time inter v als.For example,the daily load profile for a specific day retains the same shape more or less as the next day.Moreo v er,there is a strong correlation in

load

Fig. 3.The correlation between load cur v es of the same hour of pre v ious days (hours 1á7,January

1995).

Fig.4.The correlation between load demand and weather v ariables o v er 1day (Thursday,January

1995).

Fig.1.Schematic diagram for the load prediction steps.

H.M.Al-Hamadi,S.A.Soliman /Electric Power Systems Research 68(2004)47á59

of successi v e hours,which cannot be o v erlooked in designing a load model.Figs.2á4show these effects. These smooth fluctuations in load are mainly due to the smooth weather changes.The effect of these changes on the same hour on successi v e days and the changes on the successi v e hours of the same day has to be impeded into the model to maximize prediction accuracy.

Fig.2displays the load fluctuations o v er24h of7 days of the first week of January1995for the Canadian weather data.Each cur v e represents the load for1day. As noticed,the cur v es almost ha v e the same shape,with peaks of loads around10:00 A.M.and07:00 A.M.For load modeling,this fact strongly suggests expressing the load as a function of the load v alue of the same hour and/or adjacent hours of the pre v ious day.Fig.3 displays the load for1h o v er the whole month of January1995.Fi v e hours,from01:00to05:00A.M.,are plotted one cur v e per hour.The chart shows the strong correlation of these cur v es and the strong dependence of the load of a certain hour on the pre v ious hour(s)of the same day.Fig.4shows three cur v es:load,temperature, and wind chill factor o v er24h for Thursday,January5, 1995(the load and wind cur v es ha v e been scaled to display the three cur v es on one chart).The impact of weather,particularly temperature,on load v ariation is ob v ious.As temperature peaks low,the load peaks high and v ice v ersa.Similarly,the wind chill factor has an effect on the load demand v ariation but with lesser amount than the temperature.As the chill factor increases at about07:00 A.M.,the load starts rising at about09:00A.M.

Based on the abo v e facts of strong dependence of load on weather and time,the load model is expressed as a linear function of past v alues of load and dominant weather v ariables.The winter model is expressed next.

2.1.A winter model

The dynamic v ariation of load with respect to and weather factors is expressed as time-v arying linear model with v ariables belong to mo v ing windows of recent past load and weather v ariables.The general load model at any discrete time instant k,k01,2, (24)

corresponding to24h of1day,can be expressed as:

y(k)0a

0(k)'a

(k)y(k(k

)'ááá'a

(k)y(k(k

)

0(k)t

(k)'b

(k)t(k(l

)'ááá

n2(k)t(k(l

)'c

(k)w

(k)'c

(k)w(k(m

)

'ááá'c

n3(k)w(k(m

)(1a)

where,at any instant time k,y(k)is the load,t(k)the temperature and w(k)the wind.y(k(k i)is the pre v ious load v alues associated with a window at time instant k where i01,...,n1,t(k(l i)the pre v ious temperature v alues associated with a window at time instant k where i01,...,n2and w(k(m i)the pre v ious wind v alues associated with a window at time instant k where i0 1,...,n3.a0(k)is the base load at time instant k,a i(k) the load coefficients at time instant k where i01,...,n1, b i(k)the temperature coefficients at time instant k where i01,...,n2and c i(k)the wind coefficients at time instant k where i01,...,n3.

The model assumes that the load,temperature,wind and their coefficients are constant o v er each discrete time instant k,k01,...,24,of the24h of the day.Next, without loss of generality,we define a3)3window of the load at time instant k.Temperature and wind windows are treated in exactly the same manner.The load data are arranged in a matrix of24columns with rows holding24v alues for1day.The load window associated with the k th time instant is defined in Fig.5. It illustrates the positions of the load v alues relati v e to the k th time instant in the data matrix.

The indices are labeled as follow:0represents the current v alue,y(k);1represents the pre v ious hour v alue,y(k(1);2represents the hour of the pre v ious day v alue,y(k(24);3represents1h before the hour of the pre v ious day v alue,y(k(25);4represents1h after the hour of the pre v ious day v alue,y(k(23);5 represents the hour of2days back v alue,y(k(48);6 represents1h before the hour of2days back v alue, y(k(49);and7represents1h after the hour of2days back v alue,y(k(47).

As shown,although the model sets a3)3window of v alues for only2days back and only1h before and1h after the current hour,it can be easily generalized by stretching the window to include more v alues.The results,as will be illustrated in the following section, showed that the size of window considered in Fig.5is sufficient for parameter estimation using Kalman filter. The load model associated with the window defined in Fig.5at time instant k is gi v en

y(k)0a

0(k)'a

(k)y(k(1)'a

(k)y(k(24)

3(k)y(k(25)'a

(k)y(k(23)

5(k)y(k(48)'a

(k)y(k(49)

7(k)y(k(47)'b

(k)t

(k)'b

(k)t(k(1)

2(k)t(k(24)'b

(k)t(k(25)'b

(k)t(k(23)

5(k)t(k(48)'b

(k)t(k(49)'b

(k)t(k(47)

0(k)w

(k)'c

(k)w(k(1)'c

(k)w(k(24)

3(k)w(k(25)'c

(k)w(k(23)

5(k)w(k(48)'c

(k)w(k(49)

(k)w(k(47)(1b) Since coefficients in the model are assumed constant o v er1h time instant,parameter estimation is carried out for each of the24discrete instances in a day. Accordingly,24sets of coefficients are required to be estimated for1day.The estimated coefficients can be plugged into the model to predict hourly loads for the next day.

3.Kalman?lter parameter estimation

3.1.Basic Kalman?lter

The detailed deri v ation of Kalman filtering can be found in[16,17].In this section,only the necessary equation for the de v elopment of the basic recursi v e discrete Kalman filter will be addressed.Gi v en the discrete state equations:

x(k'1)0A(k)x(k)'w(k);

z(k)0C(k)x(k)'v(k)

(2)

where x(k)is n)1system states,A(k)is n)n time-v arying state transition matrix,z(k)is m)1measure-ment v ector,C(k)is m)n time-v arying output matrix, w(k)is n)1system error and v(k)is m)1measure-ment error.

The noise v ectors w(k)and v(k)are uncorrected white noises that ha v e:

Zero means:E[w(k)]0E[v(k)]00(3) No time correlation:E[w(i)w T(j)]0E[v(i)v T(j)]00

for i0j

(4) Known co v ariance matrices(noise le v els):

E[w(k)w T(k)]0Q

1;E[v(k)v T(k)]0Q

(5)

where Q1and Q2are positi v e semi-definite and positi v e definite matrices,respecti v ely.The basic discrete-time Kalman filter algorithm is gi v en by the following set of recursi v e equations.Gi v en as priori estimates of the

state v ector x?(0)0x?

0and its error co v ariance matrix,

P(0)0P0,set k00then recursi v ely computer:Kalman gain:

K(k)0[A(k)P(k)C T(k)][C(k)P(k)C T(k)'Q

](1

(6) New state estimate:

x?(k'1)0A(k)x?(k)'K(k)[z(k)(C(k)x?(k)]

(7) Error covariance update:

P(k'1)0[A(k)(K(k)C(k)]p(k)[A(k)(K(k)C(k)]T

'K(k)Q

K T(k)

(8)

An intelligent choice of the priori estimate of the state

and its co v ariance error P0enhances the con v ergence characteristics of the Kalman filter.Few samples of the output wa v eform z(k)can be used to get a weighted

least-squares as an initial v alues for x?

and P0:

0[H T Q(1

H](1H T Q(1

;

0[H T Q(1

H](1

(9)

where z0is(mm1))1v ector of m1measured samples and H is(mm1))n matrix.

z(1)

z(2)

z(m

)

5and H0

C(1)

C(2)

C(m

)

5(10)

3.2.Prediction Kalman?lter model

In this section,the weatheráload model is used to form a time-v arying discrete dynamic system suited for Kalman filter.The dynamic system of Eq.(2)is used with the following definitions:

1)State transition matrix,A(k),is a constant identity

matrix.

2)The error co v ariance matrices,Q1and Q2,are

constant matrices.Q1and Q2v alues are based on some knowledge of the actual characteristics of the process and measurement noises,respecti v ely.Q1 and Q2are chosen to be identity matrices for this simulation;Q1would be assigned better v alue if more knowledge was obtained on the sensor accu-racy.

3)The state v ector,x(k),consists of N parameters.

From Eq.(1a),N0n1'n2'n3'1.

4)C(k)is N elements time-v arying row v ector,which

relates the measured load and weather data to the state v ector(refer to Eq.(11)).

5)The obser v ation v ector,z(k),for this application,is

a scalar representing the load at time instant k(refer

to Eq.(11)).

The obser v ation equation z(k)0C(k)x(k)has the following form:

H.M.Al-Hamadi,S.A.Soliman/Electric Power Systems Research68(2004)47á59 50

y (k )0[1y 1áááy n 1

t 0ááát n 2w 0áááw n 3]

?a 0a 1n a n 1b 1n b n 2c 1n c n 3

266666666666666437777777

77777775(11)

where the parameters and load áweather v alues are defined in Eqs.(1a)and (1b),with k representing the time instant of the 24discrete hours of the day (k 01,...,24).

4.Examples and results

To v erify the effecti v eness of the proposed load model parameters estimation and forecasting demand load,we used a load of one of a largest utility company in Canada as well as the weather data for the years 1994á1995for the same utility company.

The steps taken to predict the next day load are as follows:

Step 1.Initial condition:For all examples,the mo v ing window of the load and weather data is selected as in Fig.5.The model order for all examples is taken to be 10.The initial condition of the parameter v ector is fixed arbitrarily to one in all examples.The system noise and measurement

error

Fig.6.Kalman ?lter con v ergence of some of the estimated parameters.

H.M.Al-Hamadi,S.A.Soliman /Electric Power Systems Research 68(2004)47á5951

co v ariance matrices Q1and Q2of Eq.(5)are set to unity matrices with the appropriate dimension.

Step2.Run Kalman filter for the first hour of the day and sa v e the resulting coefficient to be used later for predicting the first hour of any day need to be predicted.

https://www.sodocs.net/doc/e414224272.html,e the estimated parameters of the pre v ious hour as initial condition for estimating the next hour coefficient using Kalman filter.Run Kalman filter and sa v e the estimated parameters.

Step4.Repeat Step3for all24h of the day.

https://www.sodocs.net/doc/e414224272.html,ing the mo v ing windows as described in Step1,the24sets of estimated parameters,one set per hour,are used to predict the load24h ahead. To measure the effecti v eness of load prediction,two measures are used:M and P.M is the Euclidean norm error between the actual and predicted loads o v er the24 h of a day,and P is the mean absolute percentage error with respect to the actual load.For24h,M and P are gi v en by the following equations:

???????????????????????????

'ááá'd2

(12)

P0 ?d

'ááá'

a24

)

100

(13)

where d k0y p k(y a k,k01,...,24;y p k and y a k are the predicted and actual loads at instant k,respecti v ely. The execution time for estimating10parameters, using Kalman filter with169iterations for each para-meter on a desktop500MHz CPU speed was0.075of a second,and that for estimating the24h of1day was 1.743s.

4.1.One-hour prediction

In this section,the results of predicting1h(01:00 A.M.,February28,1995)is discussed.The load and

weather data of the first hour of the day(01:00A.M.)of the pre v ious2months(January and February1995)are used to estimate the load model parameters.Moreo v er, two extra points between e v ery two actual v alues of load as well as weather are generated using cubic interpola-tion to boost up Kalman filter iterations.The model of Eq.(11)is used with load,temperature and wind data windows of sizes n104,n202,and n301,respec-ti v ely.This entails the use of a window of four load v alues(y1,y2,y3,and y4),a window of three temperature v alues(t0,t1,and t2),and a window of two wind v alues (w0and w1)as defined in Fig.5.The combined load and weather model is then written as:y(k)0[1y

]

(14) Fig.6presents sample of the estimated model coefficients of Kalman filter iterations.As illustrated, all estimated parameters con v erge to their steady-state v alue after some transient fluctuations.The first para-meter,a0,requires more iteration to con v erge;in fact,it has continued its fluctuation in the second hour then con v erged in the rest of the hours.Fig.7shows

the Fig.9.Difference between actual and estimated loads.

H.M.Al-Hamadi,S.A.Soliman/Electric Power Systems Research68(2004)47á59 52

con v ergence of the10parameters for the last hour of the same day.A comparison between load resulting from the estimated parameters and the actual load is shown in Fig.8.The results show how closely the estimated model matches with the actual load.Fig.9displays the difference between the estimated and actual loads.

4.2.Twenty-four hour prediction

The1-h prediction is repeated for24h of1day (February28,1995).Table1presents24sets;each set consists of10v alues of estimated parameters for1day. The estimated parameters are used to predict demand load for1day.The results are shown in Table2and Figs.10á13.The results show that the load percent error,P,as gi v en by Eq.(13),is less than1%for all24 predicted loads except for the second hour,which is due to the transient beha v ior of a0parameter.The v ariation of each coefficient o v er24h is illustrated in Fig.14. The dependency of the load model on the load and weather history is e v ident in the results depicted in Table 1.For example,the first hour,h00,parameters depend mainly on y1,y2loads,the three temperature v alues,and the two wind factors.The effects of the load v alues y1, y2,y3,and the three temperature v alues dominate the predicted load in h12hour.The wind effect has less effect on the predicted load at that hour.The results necessitate the use of a combined load and weather model.

4.3.Weekdays and weekends pro?les

Weekdays and weekends load demand profiles pos-sess different patterns.Accordingly,treating them

Table2

The difference and error percentage of the predicted to the actual load Hour Actual Predicted Difference Percent 0990.4990.9(0.48(0.048 1915.4882.233.17 3.623 2883.2875.57.730.876 3873.8866.17.700.881 4875.9873.2 2.650.303 5890.9888.2 2.720.305 6930.4926.0 4.360.469 71046.41041.1 5.340.511 81205.81202.9 2.860.237 91301.11300.0 1.100.084 101356.11357.9(1.76(0.130 111364.81370.8(6.01(0.440 121384.61388.5(3.87(0.280 131338.41345.6(7.18(0.536 141312.11312.8(0.69(0.053 151277.41280.4(2.99(0.234 161279.81281.6(1.80(0.141 171342.01342.6(0.56(0.042 181356.81362.9(6.06(0.447 191365.71368.2(2.46(0.180 201347.91350.2(2.31(0.172 211306.51308.0(1.52(0.116 221252.11254.3(2.15(0.172 231157.41158.0(0.65(

0.056Fig.11.Difference between actual and estimated loads o v er24

Fig.12.Actual and predicted loads o v er24

separately is expected to produce better results than combining all days as if they ha v e the same profile patterns.The actual and predicted forecasted loads for2 months,4months,weekdays,and weekends of4months are pro v ided in Table3.It can be noted from the results in Table3that the Euclidean norm error,M,and the mean absolute error,P,are less for the separated weekdays than for the combined days.The M and P for the weekends is slightly more than that for the all days’prediction because of the number of data used for weekend case which is less than the number of data used for the former three cases.To increase the accuracy,the number of data must be increased.It can be achie v ed by two possible ways.First,by increasing the actual past data pro v ided for Kalman filter estimation.But this option would take the weather and load profiles to another season,which would be the summer in this case. Most weather and load profiles experience major changes with season’s change,which has great effect on the pre v ailing predicting model.The second way is

generate extra points between actual weather and load data using interpolation technique.Tables4and5and Fig.14depict the effect of introducing points using cubic interpolation technique.As clearly seen,the prediction error is decreased by increasing total number of points up to about10points.After10points,the results are indeterminately fluctuating,which indicate that increasing the number of interpolation point has little or no significance in impro v ing the prediction accuracy.

To illustrate the effect of load demand pattern on the parameter estimation,only Sunday’s data are taken for winter months(No v ember1994áFebruary1995).Since there are only17Sundays in the4months,there is no sufficient data for Kalman filter to produce adequate con v ergence of the parameters.Extra points ha v e been generated using cubic interpolation technique.The parameter con v ergence as well as the24-h load predic-tion has been impro v ed as shown in Table3.With one interpolation point,the total number of points used was 32as illustrated in Table3.The prediction error is14.1% because Kalman filter did not get enough iteration for v igorous parameter con v ergence.This error decreases as the number of interpolation points had been increased (Fig.15).5.Conclusion

This paper presents a no v el algorithm-forecasting model that depends on a mo v ing window of pre v ious load and weather influential factors.The load at a certain instant of time is expressed as a linear combina-tion of pre v ious load and weather data with time-v arying coefficients.Kalman filter-based estimation is used to estimate the model parameters using pre v ious history of load and weather data.The prediction technique has shown indisputable superior results o v er other techniques.The results were presented and com-pared with results based on the ordinary least-squares-based techniques.The result showed a less than1% prediction error for24h demonstrating superior per-formance of the presented model and technique.It has been shown,from the predicted load,that the proposed model with Kalman filtering algorithm produces better and accurate results compared to those obtained using different algorithms published earlier in the literature such as those mentioned in Ref.[10].

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Table5

Effects of interpolation points

Weekend days(No v ember1994áFebruary1995)

N N points M P 16289.3 1.31 29457.60.93 312648.60.74 415841.60.60 519036.00.49 622231.60.42 725428.60.37 828629.60.39 931827.40.34 1035024.70.33 1138229.00.38 1241420.70.27 1344623.60.29 1447815.20.17 1551019.70.26 1654248.10.32 1757433.20.46 1860634.70.35 1963878.40.45 2067026.90.40 2170228.80.25 2273463.20.37 2376612.40.15 2479816.90.23 2583012.8

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