To Predict The Power Load with Time Series Modelling

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TO PREDICT THE POWER LOAD WITH TIME SERIES MODELLING M. X. Xu j)1'/Htrtlllt'1I1

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Abstract. This paper discusses how to predict the power load by using time series modelling. Son;e efforts have been made in sampling, stationar:: handling and modelling, so that both accuracy and velocity of the prediction by computer are satisfactory. We had continuo~sly made the actual prediction for about three months and the results showed that the prediction by computer fully corresponded to the man-made prediction which was made by using rich experiences. Keywords. hodelling; prediction; power control; data dependent systeffi; computer applications. SAh.l:'LIlJG AND STATIONARY HANDLING

In general when power system does not experience the sudden disturbance which constitutes the prin~ry factor to affect it, the tiffie series only presents n ~an value-time-varying, variance-time-varyin~ is small. Of course, it is possible to verify whether the view point is correct after the computation. Therefore Y(t) is capable of stationary handling hy uSing addition model, i.e.

The subject discussed by this paper is to predict the power load required actually at each and every hour on the next day before twelve o'clock every afternoon. Here, we consider the variation of the power load with time as time series for modelling. Thus, if we sample the power load by the hour to form time series, to model and predict, the predicted points will be extended forward over 36. Clearly, accuracy for the prediction is lower. And such time series presents a distinct periodicity and it will make the computation increased greatly in modelling by computer. Therefore we don't adopt such sampling fashion, whereas sample the power load for a certain hour by the day to forn; 24 time series, to nlodel and predict.

Y(t) = A(t) + f(t),

(1)

Here, A(t) is er~od i c stationary stochastic process with ze ro-mean value; f(t) is nonstochastic function of t and represents mean value-time-varying, i.e. the trend of the non-stochastic variation existed in the load variation (we call it secular trend). In order to determine the special form of f(t) for a certain sample, first we express f(t) by the unified form as follows:

Now we describe the tin,e series as Y( t) , its sallipling is y, 'Y2"" ,YN'

( 2)

Before modelling we make the handling for y, 'Y2' ... ,y" as follows: to pick out the unrelia ble po int s; to reduce san,pie variance as small as possible on the premise that the distortion is not caused to sample.

then we estirr.ate "o,""o(.by usinr- linear least-squares estimate based on y, ' Y2 ,"·,Y... and determine whether a certain term ~tA (i=C,1,2) is retainable by using F test (Rao,1955). 2109

M. X. Xu

2110

We note that in general the time series does not present the trend of the periodic variation unless sample size t' is especially large, theref or e ~q . ( 1) does not appear the term of periodic function. On the other hand, we adopt least-squares es timat e to ijentify f(t) which is approx iu2 ted by polynomial whose order does not exceed two. In this regard, sa mple size N must not be too large either; otherwise, on the cont rary, identified 1'( t) may deviate from the actual secular trend where the prediction will be done. We adopt less sample size N on the premise that the statistical rationality is ensured. Thus, the c Olllputation is decreased and the rationalities of Eq . (1), Eq . (2)a re ensu red. I-.WELLING After identifying f(t ) , we obtain sample of A(t): x-/{=YJ\-f(k) , (l ~ k " N) . It is known, stationary series At with zero-mean value can be described by using Aru'~(n , m) model as follows (Box; Jenkins, 1970): (1 - ~ i3 -

1'..," - - . . - cp~n,)At

= ( 1 -9,i3 -9zif- . .. -Il,.,B"" )at

,(3)

Wu; Pandit (197tl) suggest the modelling as follows : first fit the type of ARhA(n,n-l) Illodel , n begins from one and increases one by one until find suitable Aru·tA(n ,n-l ) ll.od el. For a certain n, in order to determine the availability of corresponding l,.m· ...,(n,n-l )n·,odel, we fit Aru·LA(n+ l ,n) mode l, then compare the significance of the difference between the corresponding sums of squares of residuals. Aru'lA(n+l ,n) mo del is /lIore available than ARllA(n,n-l) mode l, if the difference is significant; then fit Affi·~(n+2,n+l) model again and repeat the procedure for Aru'lA(n+2,n+l) and Aru·~(n+l ,n) /lIodels. Conversely, when the difference is not significant, we determine that Aru'~(n, n-l) Ii.od el is available. Thus, autore glessive or de r n is determined. Next we fix n and f it the type of ARl·.A(n,m) model , ~ begins from n-l and decreases one by one , then compare the significance or' the difference between the sums of squares of residuals for the two successive models. Once t he dif ference, between Aru·.A(n ,m) and

AR}.A(n,m-l) models, is significant, we determine that AR}lA(n,m) model is available. Thus, moving avera ge ord e r m is determined too. We note that i f AR}~(n ,m) model (m>Olis used to fit At, being due to nonlinear estimate in modelling, the computation will be increased greatly . Thu~, for our case to predict by 24 mode ls, it is impo ssible to renew sample , to ilIode l I' afresh every day. But Yt(l) , the prediction extended forward 1 steps at t, is:

"xtCl)

+ f (t +l ) ,

(4)

~ In Eq. (4), "Xt(l) is very small , Yt(l ) is determined mainly by f(t +l ) . Therefore it is necessary to renew f(t) ti mely, otherwise the model for the prediction can not real ly mirror the secular trend of t he power load. In this regard, we must r e new sample to model afr esh every day. In orde r to take acc ounts of both accuracy and velocity of the prediction , and we notice that its order is lower when power system does not experience the sudden disturbance which constitutes the primary factory to affect it, this paper adopts AR(n) mode l to fit At . Thus, both the determinations of the form of f(t) and the order n of AR( n) model only refer to linear estimate and significance test. Thus, it is possible to renew sample and to mod el every day. Clearly, the renewal of each san~le i s only: y,;;ryn" (1 ~n~N); YN., is a new datum.

After we determine the form of f ( t ) and the order n of AR(n) mode l, we express the form of the mode l for the prediction a nd estimate the paramaters of the model. ',Ihen f(t)=oio+o(,t+oi.t·, the form of the model is:

In Eq . (5) , a t is lesidua l, as sur;.ed as norrual "'hite noise , at-Ii(e ,~ ) ; where as :

2111

Power Load Prediction with Time Series Modelling

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Now define:

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and they can be determined by this way: sQbstitQting Eq. (9) into Eq. (3) and comparing same power of backward shift operater B, we can obtain the reCQrrence forrr,ulars of solving then"

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can be compQted directly from Rq . (3) by Qsing the rules as follows:

riecause N is sufficiently large, r~n+3 (r: rank of W. If r=n+3, aT a is nonsingular; , , ,j...

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CONClUSION The method has been realized on sn.all-sized computer. All the operating time does not exceed ten minQtes for one ti me prediction. We had continQoQsly made the actQal prediction for aboQt three r..onths . 'l"' ,. resQlts had been carefully compared from the month average of proportional error, the maximum proportional error, the month average of standred diviation verSQS load nQmbers and showed that the prediction by compQter fQlly corresponded to the nlBn-n'. ade prediction which was made by using rich experiences, as showed in Fig. 1.

Box, G. E. P., and G. j, .. Jenkins ( 1 07 C) . Time Series Analysis, Forecasting and Control, Holden-~ay. Rao, C. R. (1965). linear Statistical Inference and Its hpplications , John

Of course, after the paramaters of the lJ,odel arc estilUated, it niQst be tested whether at is norn.al white noise. Pli.EDICTION

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We adopt the linear prediction.: (7) ~

It is k~own \hat when G =G..t"a (~ =O,1 ,2,"' ) , :s [;"tt,(-Xt(l)) =minih.un.. ThQs,

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S. k ., and S. ~. Pandit ( 107 P) . Time Series and Sy stem ,'"nalysis j',odelling and Applications.

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