SHORT TERM LOAD FORECASTING USING A MULTILAYER NEURAL NETWORK

8/7/2019 SHORT TERM LOAD FORECASTING USING A MULTILAYER NEURAL NETWORK

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Tmnsactions on Power Systems, Vol. 7,No. 1, February 1992 . 141SHOR T TERM LOAD FORECASTING USING A MULTILAYER N EURA L NETWORKWITH AN ADAPTIVE LEARNING ALGORITHM

Kun-Long Ho, Student Member IEEE Yuan-Yih Hsu, Senior Member IEE E, Chlsn-fl:uea YangDepartme nt of Electrical EngineeringNational Taiwan UniversityTaipei, Taiw an, ROC

Abstract - A multilayer neural netwo rk with an adaptivelearning algorithm is designed for short term load forecast-ing. Extensive studies have been performed on the effectof various factors such as learning rate, momentum, thenumber of presentations in an iteration, etc. on th eefficiency and accuracy of the backpropagation-momentumlearning metliod which is employed in the training ofartificial neural n etworks. To speed up the training process,a new learning algorithm for the adaptive training of neuralnetworks is presented. The effectiveness of the neuralnetwork with tlie proposed adaptive learning algorithm isdemonstrated by short term load forccasting of Taiwanpower system. It is found that. once trained by the pro-posed learning algorithm, the neural network can yield thedesired ho urly load forecast very efficiently and accurately.Moreover, the proposed adaptive learning algorithm con-verges much faster than the conventional backpropagation-momentum learning method.Keyw ords: load forecasting, artificial neur al netw orks,artificial intelligence, machine learning systems.

1. INTRODUCTIONThe main objective of short term load forecasting isto predict the hourly loads, one day or even one weekbeforehand,w hich are necessary for th e operational planningof a power system [ 1 I . Since the electric load demand isa :unction of weather variables and hum an social activities,both statistical techniques [2-71 and expe rt systemapproaches [8-10] have been proposed. Usually, th estatistical approaches are effective for the forecasting ofnormal days but fail to yield good results for those dayswith special events, e.g., the superbowl day. To take these

special events into account, operators experience andheuristic rules must be employed . This motivated thedevelopment of rule-based ex per t systems for load forecast-ing [8-101.In this paper, a new approach using artificial neuralnetworks (ANN) is proposed for short-term load fore-casting. Am ong the various artificial neural netw orkspresented so far, the multilayer feedforward neural network[ 11-17] is employe d. To determine the connectionweights between neurons, tlie conventional backpropa-gat ion mo me ntu m learning technique with constant learningrate q and momentum fl [ 1 1] is first employed in the91 3.1 45C-7 P: b A paper reco,:neiueci inc ansroveclb27 th e I K G /ewer S ,stem Cnrineerinc COP l i t t e e o fthe I& Power Lnzinecrin, Socic t y f o r 1 r e s e q ta t io na t t he IZ%/?LS 1591 Sunmer lee ti nC, San Diego,Ca l i fo rn ia , July 28 - Au;dst 1, 1991. anu scr ip5submitted July 9 , 1950: na!e av ai ls bl e f o r : x i n t i n ;Jiine 19, 1991.

training process. Th e effec t of different learning ratesand momenta on the convergence property of the learningprocess is extensively studied. In addition, the numbe rof presentat ions in an i terat ion and the number of input/output patterns in a presentation are also varied to see theireffect on the convergence rate. T o speed up the conver-gence rate of the learning process, ari adaptive learningalgorithm in which the momentum is automatically adaptedin the training process is presented. Results from thepresent study indicate that the proposed adaptive learningalgorithm converges mu ch faster than th e conventionalbackpropagation-momlentuin technique.To demonstrate the effectiveness of the artificial.neural network, short-term load forecasting is perfqmedon Taiwan power system. The neural netw ork has 46input nodes, 60 hidden nodes, and one output node. Inthe training process, the weather variables and load datain the past few days are used. Once trained, th e neuralnetw ork is capable of forecasting the hou rly loads with amean absolute error (MAE) of 0.7%.In the next section, the problem of load forecastingis briefly described. This is followed by an introduc tionof the artificial neural network and the backpropagation-mom entum learning algorithm. Then , we present a newadaptive learning algorithm for the neural network. Anexample is then given to demonstrate the effectivenessof the proposed neural network.2. PROBLEM FORMULATION

The objective of short-term load forecasting is topredict the 24 hourly loads L(i) , i = 1, 2, ... , 24, of thenext day. There are fou r main steps involved in the fore-casting process [ 101.Step 1.Step 2.

Step 3.

where

Read in the day to be forecasted and tlie fore-casted weather variables.Identify day ty pe and get th e 24 normalized hourlyloads L n(i) , i = 1, 2, ... ,24 ,fo r that particular daytype. These normalized hourly loads have beenstored in the data base to represent the loadpattern of each day type.Compute the forecasted daily peak load and v?lleyload using the following equations.( 1 )(2)

I

L p = gp * T p + h pL t = gt * Tt + h tLp = pcak load of the dayLt = valley load of the dayT p = equivalent forecasted high temperatureT t = cquivalent forecasted low temperature o f

of the system on the daythe system on the day

U-M-I 0885-8950/92$03.00(91992 IEEE

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gp , h p , g t , and lit = coefficients determinedby least-square-error technique using theload and weather data in the data base.

Step 4. Compute the forecasted hourly loads L(i) asfollows.L(i) = Ln(i) (L p - L t) + Lt

i = 1 , 2 , ... , 24 (3 )Details of the aforem entione d procedures are given in[ l o 1 *It has been noted during the course of our previouswork [ 101 that the forecasted hourly loads depend heavilyon the forecasted peak and valley loads. In fact, the meanabsolute error (MAE) can be reduced to a great exte ntif the actual peak and valley loads are employed to replacethe forecasted peak and valley loads. This implies tha t therule-based expert system developed in our previous work[ 101 is good a t identifying day type and providing th e24normalized hourly loads Ln(i), i = 1, 2 , ... ,24 , for each daytype . However, t o have even bette r forecast results, th eapproach for peak and valley load forecasting employed inthat work needs further refinement. Thus, the presentstudy will be focused on th e application of neural networksto forecast the peak load Lp and valley load Lt of Taiwanpowe r system. These peak load and valley load values,together with the normalized hourly loads Ln(i) obtainedin our previous work [ I O ] , can be used to forecast thehourly loads L(i). Of course, the proposed neural networkcan also be employed to achieve the normalized hourlyloads L(i). But ou r main effor t in this stud y will be placedon peak and valley load forecasting.

signals form external sources and the nodes in the ou tputlayer provide the desired output signals. Since only onehidden layer is employed in the present study, it is calleda three-layer neural network according to Rumelhart 'sdefinition [ 1 11 . However, since the input nodes simplypass the inputs to the nodes in the hidden layer and do notperform the activation function and output function thatnormal neurons are supposed to carry o ut, some peopleregard this kind of network as a two-layer neural network[ 181. The importan t thing is to no te th at on ly the nodesin th e hidden layer and outp ut layer are, regarded asordinary neurons which perform activation function andoutp ut function. It should also be noted that, in theemployed feedforward netw ork str uctu re, signal propaga-tion is allowed only from the input layer to the hiddenlayer and from the hidden layer to the outp ut layer. Con-nections between nodes within the same layer or fromthe input layer directly to the output layer are notpermitted.For each neuron k in the hidden layer and neuron !2in the o utpu t layer, the net inputs are given by

NJnetk = j=21 wk j Oj k = 1 , ... ,NK

an dNK

netQ = k=;C1 W Q k O k II = 1 , ... , N L

respectively. The neuron outp uts are given by3 . MULTILAYER NEURAL NETWORKS O j = n e t j

Consider the multilayer feedforward neural network asshown in Fig. 1 [ 11 1 . 1

(4)

1

Oulput Patterns

t t t fInternalRepresentationUnits

1-(netQ + 0, )op = = f Q ( n et Q,eQ) (8)l t e

where net. is the inpu t signal from the external sources tonode j in the inp ut layer.Let t he conne ction weights Wkj and WQk be updatedafter each presentation. In other words, they are changedafter NQ inpu t/ou tput patterns (pairs) have been presented.Then for a presentation p, the sum of squared errors to beminimized is given by

Input Patterns

Fig. 1 A multilayer neural networkwhere tpqQ and 0 qQ are the target output value andcomputed output va%e at output node i! fo r the q th inpu t /outp ut pair in presentation p , respectively.By minimizing the error Ep using the technique ofgradient descent, the connection weights between hidden

The nodes in the neural network can be divided int o tiueelayers: the inpu t layer , the ou tpu t layer and one o r morehidden layers. The node s in the input layer receive inpu tunit k and ou tpu t un i t Q can be updated by using theequat ions [ l

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where

a n d o r q k IS. , the ou tp u t o f h idden un i t k fo r the q th inpu t /ou tpu pair in presentation p. Note that the leaming rate qand the momentum a give the relative weights for thepresent error and the error in the previous presentation.These factors affect the convergence rate of the learningprocess t o a great exten t. Details on the selection of q an da will be addressed later. The con nection weights betwe eninput unit j and hidden unit k can b e updated using similarequations

whereNL

'pqk = fi( (netpqk) Qzl'pqP WQkNote that the threshold 0 is also updated in the learn-The overall training (learning) process is summarizedIn applying the above learning rule, there are several

ing process using an equatio n similar to eq. ( 1 2) .by the flow chart in Fig. 2 .issues which sho uld be ad dressed.What are the optimal values of learning rate (q) an dmomentum (a)? Most works on feedforward neuralnets use constant values of 77 an d a. Rumelhart [ 11 ]recoinmendcd that a combination of q = 0.25 an d

a=0.9 can yield good results for most problems.But there is still no consensus as to what values of qand a should be used in the learning process. In fact,the optimal values of q an d Q may be problem-depend ent. In the present work, extensive studieshave been carried out on the effect of different valuesof q and a on the convergence rate of the learningmethod. It is found that th e combination of ~ ~ 2 . 5an d CY= 0.9 is a good choice for the load forecastingproblem. To further speed up the convergence rate,an adaptive algorithm as described in next section isdeveloped to adapt the momen tum during the learningprocess.What are the optimum values for NP (number ofpresentations in an iteration) and NQ (number ofinput/output patterns in a presentation) given a fixedvalue of NR (number of input/output patterns in aniteration)? If any inp ut/o utpu t pattern is allowed toappear in only one presentation, then(14)

It is obvious that we will have less presentations inan iteration if we use more patterns per presentation.In the present study, the convergence properties forvarious co mbinations of NP an d NQ are investigated.We also examine the case i n which an inpu t /ou tpu t

NR = NP * NQ

START[ E(o)=O I

ITER= 1

DIVIDE THE TOTAL INPUT/OUTPUT PATTERNS(NR) INTO NP GROUPS (PRESENTATIONS)

41t

I SET INITIAL WEIGHTS w &(o) and wM(o) II 1rAPPLY EQS. (4)-(8) TO COMPUTE1 , I , NEURON INPUTS AN D OUTPUTS J

P=P+l 11'yI APPLY EQS. (10);13) TO UPDATE 1kjWEIGHTS W~ and w1

lYESUSE EQ.(9) TO COMPUTETH E SU M OF SQUARED ERRORFOR THE PRESENT ITERATION

1 2 0 0E (ITER) -E (ITER - 1)

wIlk and w

(*)Fig. 2 The learning procedure

pattern is allowed to appear in two presentation.In this case, we defineNS = the number of inpu t /ou tpu t pat terns in thecurrent presentation that will appear in nex tpresentation

Then , we haveNR = N P . N Q - ( N P - I ) . N S (1 5 )

The effect of different values of NS on convergencerate is also examined .



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(31 In the learning procedure described in Fig. 2, con-vergence is checked at the end of each iteration whenthe connect weights have been updated NP times.It may be argued that convergence can also be checkedafter each presentation. Indeed, there is no reasonwhy we can not check convergence after each presenta-tion in order to save some presentations (or eveniteration s). However, we made tlie decision to checkconvergence only at the time when an iteration iscompleted because we found that the comp utation ofthe sum of squared error, E, is very time consumingand we d o not want to spend a lot of CPU time incomp uting E. In othe r words, we believe that therequired CPU time per iteration is equally importantas the required number of iterations since the totalrequired CPU time in the learning process is theprod uct of th e two figures. Thus , in presentingdifferent learning algorithms in our load-forecastexample, com parisons will be made based on therequired CPU times as well as the num ber of iteratio ns.4. AN ADAPTIVE LEARNING ALGORITHMTO make the learning process converge more rapidlythan the conventional method i n which both tlie learningrate and momentum are kept constant during the learningprocess, a new adaptive training algorithm is developed toada pt momentu m in the training process. The proposed

adaptation rule is as follows.1.01 c u b ) , A E n > Oi ,0 .9 9 a b ) , otherwise (1 6 )where a(n ) i s momentum at i teration n , and AEn = E(n-1)- E(n) with E(n) being the sum of squared errors at theend of n th iteration. Here, the basic idea is to increaseawhen AEn is positive and decrease 'a when AEn is negative.Note that AEn is positive when the error is decreasing(E(n) < E(n-l)), which implies that the coiinection weightsare updated to t he correct direction. In this case, i t isreasonable to maintain this update direction in nextiteration. The way we achieve this is to increase themom entum in next iteration. On the other hand, if theconnection weights are moved to t he wrong direction,causing the error to increase, we should try to ignorethis direction in next iteration by reducing the momentumterm.

u ( n + l ) =

5 . EXAMPLETo demonstrate the effectiveness of the proposedadaptive learning algorithm, load forecasting is performedon Taiwan power system using the neural network withthe conventional algorithm and the adaptive algorithm.Table 1gives the forecast results for a typical day (August19, 19 87) using the rule-based expe rt system in [ 101.It is observed tha t the MAE is 1.79% and tlie errors in peakload and valley load are 19.42 hf W (1.78%) and 171.63 MW(1.57%), respectively.To further improve the accuracy in the forecastedhourly loads, it is desirable to reduce the errors in peak loadand valley load forecasting. In the present work, we propos eto use neural networks to forecast peak a nd valley load.The inputs to the neural net contain the forecasted high

(low) temperatures in the three areas of Taiwan for the dayload forecast is being conducte d, the recorded area high (low)tempe ratures in the previous da y, and the recorded areahigh (low) temperatu res and peak (valley) load in the past tendays with tlie same load pattern as the forecast day. T hus, wehave 46 inpu t nodes in all. The neural netwo rk has 60

Table 1. Load forecas t results using the rule-basedexpert system (August 19, 1987). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - - - - _ - - _ - _ - _ - _ - - - - - - _ _ - -

4CTUAL FORECASTEDHOUR I,OADS(MW) L OADS(MW) ERRO R(MW) X ERROR_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - - - - - _ - - - _ - - - _ - - - - - - - - - - - -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - - _ _ - - _ _ - - _ _ _ - - _ - - - - - - _1 7fi19.00 7524.28 94.72 8.663-032 7326.00 7218.27 107.73 9.84E-033 7074.00 6988.41 85.59 7.823-034 6895.00 6862.72 32.28 2.95E-035 67 91 .0 0 fi683.49 107.51 9.82E-036 6735.00 6593.15 141.85 1.30E-027 6747.00 6563.37 183.63 1.68E-028 7395.00 7142.06 252.94 2.31E-029 9538.00 9235.29 302.71 2.773-0210 10201.00 9937.47 263.53 2.41E-02

1 1 10485.00 10267.33 217.67 1.993-0212 10616.00 10372.71 243.29 2.22E-0213 3539.00 9279.03 259.97 2.383-0214 10823.00 10588.48 234.52 2.14E-0215 10944.00 10749.54 194.42 1.783-0216 10785.00 10575.12 209.88 1.92E-0217 10558.00 10373.14 184.86 1.693-0218 9715.00 9531.42 183.58 1.68E-0219 9732.00 9479.65 252.35 2.31E-0220 9977.00 9797.35 179.65 1.643-0221 9721.00 9560.17 160.83 1.473-0222 9352.00 9080.00 272.00 2.491-0223 8994.00 8752.46 241.54 2.21E-0224 8522.00 8234.76 287.24 2 . 6 2 3 - 0 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ - - - _ _ - - - - - - - - - - - - - -** MAE = 1.79 % ** RM S = 1.90 %** FORECASTED PEAK = 10749.54 MW** ACTUAL PEAK = 10944 M W ERROR = 194.42 MW(1.7890** FORECASTED VALLEY = 6563.37 MU** ACTUAL VALLEY = 6735 MW ERROR= 171.63 MW(1.57%)

======1==1====:11:1=-=:-==l=E=-=:===-I===============:=========

hidden units and one output nodes which gives the peak(valley) load .In the training process, 30 input/output patternsselected from the data base of Taiwan Power Companyare employed to determine the weights for the neuralnetwork. The convergence criterion used for trainingis to have a root mean square error (RMSE) of 0 .01 orless ( E = 0.01 in Fig. 2 ) .The efficiency and accuracy of the neural networkusing the conventional learning algorithm with fixedlearning rate and momentum are compared in Tables 2an d 3 for different combinations of q ,a,NS , NP , an d NQ.Note that the learning efficiency is evaluated by th e numberof iterations and the required CPU time in the learningprocess. Once the netwo rk has been trained, th e accuracyof the neural net can be evaluated by testing the neuralnet with 7 load forecasts and record the maximum errorand root-mean-square e rror of th e forecasts. Table 2 givesthe results for the case in which each presentation usesd if feren t inpu t /ou tp u t pat terns (N S = 0) while Table 3 givesthe results for the case in which a certain num ber of input/output patterns in the current presentation will appear innext presentation (NS # 0). We use a set of 30 i n p u t /output training patterns in the achieving the results inTable 2 and Table 3 (case 1) . To see how the input/ou tputpattcrns affect the final results, training is also performedusing another set of 30 inp ut/o utp ut patterns. The resultsfor this case (case 2 ) are summarized in Tables 4 and 5.The results obta ined by using the adaptivc learningalgorithm are given in Tablc 6.



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r ) = 0 . 2 5 a = 0 . 9learning efficiency test results

NQ NP number of CPU time maximum root-mean-error square error7iterations (sec)

r ) = 2 . 5 a = 0 . 9learning efficiency test results

number of CPU time maximum root-mean-iterations (sec) error square error_ _15

IO1530I I I I I I I I , 1

9154 1.947% 1.405% diverge3 0 5 6 66 429 3920 1.419% 0.835% I53 1420 1.197% 0.841%3 8 8 8 7272 1.403% 0.827% 119 977 I .044% 0.533%2 1376 10870 1.299% 0.745% 14 1 1116 1.063% 0.585%1 2794 21303 1.307% 0.751% 292 2286 1.058% 0.564%

q = O . 2 5 a = 0 . 9learning efficiency test results

NS NP number of CPU time maximum root-mean-iterations (sec) error square error8 11 67 4 15906 0.586% 0.362%6 6 47 7 6695 1.347% 0.751%5 5 572 6928 1.332% 0.760%0 3 88 8 7272 1.403% 0.827%

1030

q Z 2 . 5 a = 0 . 9learning efficiency test results

num ber of CPU time maximum root-mean-square erroriterations3 3 7 8 4 0.8 10% 0.429%82 1151 0.371% 0.266%78 950 1.125% 0.675%1I 9 9 7 7 1.044% 0.533%

(sec) error

Table 4. The results fo r case 2 using conventional learning algorithm (NS=O)!r ) = 0 . 2 5 ~2 = 0 . 9 r) = 2 . 5 a ~ 0 . 9

r ) = O. 2 5 a = 0 . 9learning efficiency I test results

learning efficiencyiterations (sec)

493 4532997 81801493 117903007 23170

q = 2.5 a = 0 .9learning efficiency 1 test results

test resultsmaximum root-mean-error

2.741% 1.214%2.166% 1.004%2.218% I .008%2.196%2.225%

number ofiterations29 246 854 399 7

learning efficiency(sec)iterations

diverge

363 2992436 3415

CPU time maximum root-mean- number of CPU time maximum(sec) error square error iterations (sec) error6942 1.812% 1.126% 35 808 2 .196%6570 2.068% 0.977% 362 5081 1.812%6575 2.198% 1.003% 165 1998 1.652%8180 2.2 18% 1.008% 158 1329 1.779%

test results

1.93 1% 0.870%1.779% 0.871%1.687% 0.818%1.958% 0.879%

algorithm

conventionaladaptive

Table 5 . The results for case 2 using conventional learning algorithm (N Q=lO )

r) = 0.25 r) = 2.5test resultslearning efficiency

- - ~ _ _ _ _ _ _learning efficiency test results

~ ____number of CPU time maximum root-mean- number of CPU time maximum root-mean-iterations (sec) error square error iterations (sec) error square error888 7272 1.403% 0.827% 1 I 9 9 7 7 1.044% 0.533%84 68 8 2.176% 1.519% 81 665 1.229% 0.608%

53

root-mean-

0.926%0.885%0.809%0.871%



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From the results in Tables 2-6, the following observa-tions can be made:(1) Fo r the conventional learning algorithm with NS=O(Table 2) the choice of ~ ~ 2 . 5and a=0 . 9 can yieldgood results in terms of training efficiency andaccuracy. In the training process, the numb er ofiterations and the required CP U t ime on a VA X11 /780 computer are in the o rder o f 100 and 1000seconds, respectively. However, both the num ber ofiterations and CPU time will increase drastically(eight times as great) if the learning rate is reducedto 0 . 2 5 .(2) From the results in Table 2, i t seems that, in ours tudy , the combinat ion o f NP= 3 a n d N Q = 10(77-2 .5) can achieve most efficient learning if eachinpu t/outp ut pair is used in only one presentation.(3 ) By comp aring the results in Table 2 and Table 3 , i tcan be concluded that more efficient learning can beachieved by allowing some input/o utpu t pattern s t oappear in more than one presentation. To be specific,the required CPU time can be reduced from 977seconds in Table 2 t o 784 seconds in Table 3. It isalso noted the convergence rate will be affected bythe number o f inpu t /ou tpu t pat terns in the cu rren tpresentation that will appear in next presentation(NS).(4) As evidenced by the results in Tables 2-3 and Tables4-5,the previous observations also hold for case 2.Thus, i t is concluded that the above observationsremain valid regardless of the set of training patternsused. In fact, many other sets of inpu t/outp utpatterns and different values of q an d a have beentested. The results are no t presented du e to limitedspace.( 5 ) Table 6 compares the convergence rates of the conven-tional learning algorithm and the adaptive learningalgorithm. It is observed that both th e required CPUtime and the number of iterations can be significantlyreduced by the proposed adaptive learning algorithm.Thus, the adaptive algorithm converges much fasterthan the conventional algorithm. One characteristicfeature of the adaptive learning algorithm is that,no matte r what value of r) is used, the adaptive learningalgorithm can make the Iearning process very efficient.For example, even a value of 0 .25 is picked, therequired CPU time can b e reduced from 7 2 7 2 secondsto 688 seconds by the adaptive algorithm. Thus, i t isconcluded that, with the conventional learningalgorithm, th e values of leaming rate and mom entuminust be carefully selected i n order to achieve efficientlearning. However, if the adaptive learning algorithmis employed, the sclection of learning rate and initialvalue for m omen tum will have only a mino r effecton the convergence ra te . Thus, we d o no t have toworry about the selection of 17 an d cy , if we use theadaptive algorithm.(6 ) To see how the momentum is changed by the adaptivealgorithm through out t he learning process,Fig.3depictsthe value of a as a funct ion o f i terat ion number fo rthe adaptive algorithm with 17 = 2 .5 .(7) To examine the convergence characteristics of theconventional algorithm and the adaptive algorithm,the root mean square (RMS) errors in the learningprocess are depicted in Figs. 4 -6 . Fig. 4 shows theRM S errors for the conventional algorithm withNSZ 0 an d q = 2 . 5 . Figs. 5an d 6 compares the RM Serrors obtained by using the conventional algorithman d the adaptive algorithm for r) = 0 .25 an d r) = 2.5,respectively. The excellent convergent characteristicof th c ntlaptive alg orithni can be easily obscrved.

(8) It was mentioned earlier that the error is computedon ly a t the end o f one i terat ion to save the computa-tional time. Since there would be more than onepresentation in an iteration and the connection weightsare updated after each presentation, i t is possible thatthe RMS erro r a t on e p resen tat ion wi th in an i terationis smaller than that at the end of the iteration. Fig. 7depicts the situation for the conventional algorithm

1 .o

0.9

NUMBER O F ITERATIONFig. 3 The momentum as a funct ion of i tera t ion numbe r

for the adaptive algorithm ( ~ ~ 2 . 5 )LT0nwn40

z4w

t0n

NUM9ER O F ITERATION

Fig. 4 The RM S erro r as B funct ion of i teration numberfor the conventional algorithm (Table 3 ,NQ = l o , ? = 2 .5 )

K(LII:Yw$ VI

adaptiveConventional--_-_

NUM9ER O F ITERATION

Fig. 5 Comparison of the RM S errors as a fun ction ofiteration number (77 = 0 . 25)



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n0nww60v)zwIF00U

a

conventional_ _ _ _ _ _

0.5 , , , , , , , , , I , , , , , , ,0.

0.

0.

J

50 7 a0 150 zooNUMBER O F ITERATION presentation number

Fig. 6 Comparison of the RM S error as a function ofiteration num ber (q = 2.5) Fig. 8 Thc con nection weight as a function of th epresentation number

l o o - , , , , l , , , , l , , , , , , , , , , , , , ,0

the RMS error evaluated ateach presentationthe RMS error evaluated ateach iteration -

1

I-

Now, the neural network has been trained and tested,it can be applied to forecast the required peak load. Aseparate neural network is also trained for valley loadforecasting. Table 7 lists th e forecasted peak load, valleyload, and ho urly loads. By comparing the results inTable 1and Table 7, it is observed that the error in peak load canbe reduced from 1.78% to 0.41% while the error in valleyload is reduced from 1.57% to 0.56%. The resultant MAEfor hourly load forecasting is reduced from 1.79% to0.7%.To see how the neural network performs in the otherseasons of the year, short term load forecasting is alsoperformed on othe r days of the year. Due to limited space,only the results for one day in spring (May 7, 1987) arepresented in Table 8.

Table 7. Load forecast results using the neural network(August 19 , 1987)The RMS errors at each presentation and eachiteration = : 3 : = = : = ; = = : : = = : = = = = = = = : = = 1 1 - 1 - - = = = = = = = = = = =NUMSER O F lTERATlON

Fig. 7ACIIJAI, FORECASTEDHOUR L O A D S 1 M W I LOAI)S(MW) ERROR( H W ) % ERROR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 7619.00 7fi43.48 -24.48 -2 -2 43 -0 3with N S = O , N Q = 10, an d q = 2. 5 (Table 2 ) . It is 2 7326.00 7334.56 -8.56 -7,823-043 7074.00 7102.50 -28.50 -2.603-035 6791.00 6794.67 -3.67 -3.36E-04tion. However, we do not employ this policy for fi 6735. 00 6703 .47 31.53 2.881-03

73.59 6.723-03the reason mentioned above. 7 6747.00 6673.41obvious that the required number of iterations can bereduced if we evaluate the error after each presenta- -80.61 -7.373-034 6895.00 6975.60

(9) The connec tion weights are updated after each presen-tation. Fig. 8 shows how one of the connectionweights is varied at each presentation for the conven-tional algorithm w ith N S = 0, N P = 3 an d 7 7 ~ 2 . 5 .(10) Test results in Table 6 reveal that the neural networkis capable of yielding very accurate results (with anRMS error of 0 .608%) once it is trained by the pro-posed adaptive algorithm. The neural netw ork trainedby the conventional algorithm also gives very goodaccuracy (with an RMS error of 0 .533%). It is notsurprising that the tw o algorithm s give different RMSerrors since they will converge to different points inthe weight space. However, the differenc e in the twoRMS errors is insignificant.(11) The initial values for the connection weights havebeen generated by th e random num ber generator.

8 7395.00 7257.61 137.39 1.263-029 9538. 00 9370.83 167.17 1.53E-0210 10201.00 10079.71 121.29 1.llE-021 1 10485.00 10412.72 72.28 6.60E-0312 10616.00 10519.12 96.88 8.85E-0313 9539.00 9414.99 124.01 1.13E-0214 10823.00 10736.94 86 .06 7.86E-0315 10!)44.00 10899.58 44.42 4.06E-0316 10785.00 10723.46 61.54 5.623-0317 10558.00 10519.55 38.45 3.513-0318 9715.00 9669.79 45.21 4.131-03114.47 1.05E-0219 9732.00 9617.5320 9977.00 9938.25 38.75 3.543-0322.18 2.03E-0321 9721.00 9698.8222 9352.00 9214.06 137.94 1.26E-0223 8994.00 8883.39 110.61 1.01E-0224 8522.00 8360.75 161.25 1.473-02===I==Il=5:=D=* * MAE = 0.70 % ** RM S = 0 . 8 2 %

** FORECASTED PEAK = 10899.58 MW* * ACTUAL PEAK 10944 MW ERROR = 44.42 M W (0.41%)** FORECASTED VALLEY = 6673.41 MW** ACT UAL VAL L E Y = 6735 MW ERROR = 61.59 MW(0.56%)===================:==::D=I=IPE



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148Table 8. Load forecast results for May 7, 19 87

8.11E-039.166-01-4.126-03-8 .486-03-11.591-03-1.516-02-2.286-02-1.816-02-1.796-02- 6 . 6 1 6 - 0 3-1.726-022.196-02-2.556-02-1.27F-021.56E-032.416-031.708-021.OIE-02-1.528-022.16E-028.596-039.62E-03-9.511433.21E-03

From the results in Table 8, it is observed th at, byusing the artificial neural network, the error in peak loadis reduced from 1.56% to 1.19% while th e error in valleyload is reduced from 0.68% to 0.25%. The resul tant MAEfor hourly load forecasting is reduced from 1.01% to0.92%.By co mparing the effectiveness of the neural netwo rkin differen t seasons (see Tab!t.s 7-10), it is foundthat the improvement in peak load and valley load forecastis more pronounce d in summer than in spring. This isdue to the fact that , in a summer peak system such asTaiwan power system, peak load is more dependent ontemperature in summer than in other seasons. Thc neuralnetwork improves the forecasted peak load by simulatingthe complicated re lationship between temperature andpeak load using the ne urons and their associated weights.It is also observed from Table 8 that the neuralnetwork yields slightly better forecast results than thesystem operator . Moreover, the neural network approachgives smaller forecast errors than the rule-based expertsystem by using a more sophisticated algorithm to reducethe error in both the forecasted peak load and the fore-casted valley load. Note that th e two approach es use th esame hourly load pattern, Lll(i), and it is the peak load,Lp , and the val ley load, Lt , that causes the difference inforecast error.

6 . NOMENCLATUREsubscripts j, k, f! = any node in thc input layer, hidden layer,NJ, NK, NL = number of nodes in the inp ut layer , hiddenNP = number of presentations in a n iterationNQ = number of inpu t/outp ut pat terns in a presentat ionNR = number of input/output pat tems in an i terat ionNS = number of input/o utpu t pat terns in the current presen-tation that will appear in next presentation8 = thresholdw = connection weight between neuronst = target (desired) outp ut valueo = computed outp ut value using the neural netE = sum of squared error77 = learning ratea = momentum

and o utp ut laye r, respectively.layer, and o utp ut layer, respectively.

5813 .315681.815570.975 5 0 0 . 0 25496.065621.525773.066208.807152.518309.378593.868649.357250.408330.748496.618486.618418.908021.188113.018191.997850.667319.206889.326411.34

rllOUI1 OPERATOR I RULE-BASED EXPERTSYSTEM NEIlIlAI. NI~:'I'Wlllth I-i %E1111011ACTUAI ORECAS'TBILOAD1 MW IOIlECAST61LOA01MY )LI i 5916515856:115 5 7 05 5 5 56 1 i Z D5180626218848 3 7 18 6 9 11311841986368631861882188261824979151364fig116428

8730

58405750561056505630516059806420804084208880850076Vb8590857086108500813084008060184072807 0 0 06400

4.896-038.73E-031.226-036.75E-U31.69E-047,951-046.09E-03I . 516-027.06E-031.566-02I. 62E-021.45E-021 . T OE - 0 21.60E-021.656-022.28E-022.256-021.086-026.538-031.37E-035,136-033.17E-031.916-031.56%

8.70~-035973.015111.115655.1555Rll.555708.315867.666325.867949.148534.708833.868781.057421.158551.188131.608721.088681.438231.668 3 9 1 . 3 18411.278052.357493.507041.466538.848833.86

551u.311 - 9 .I IB - l l :II9

IO1 11213I415161 7181920

- I . I l JG lJ 2-5.008-03-6.956-03- 1 .031i-02- I .5GE-U3-1.42E-02

- I . I O B - O Z

- I ,436-02-1.2lE-022 12 2I ;:

8730-5555

88805630 0.68% 5576.38

1.22% I 1.01% 0.92% II . 0 l i X1.40% 1.19%

Ta'J!e 9 . Load {orecast rcsult; f r x Feb. 25 , 1987____________________-------.----------.---------____________________--------.-.---.-..----------I 5501.00 5538.70 -31.70 -5.768-032 5366.00 5365.80 0.20 3.738-053 5255.00 5262.10 -1.10 -1.358-034 5196.00 5197.20 -1.20 -2.318-045 5198.00 5207.80 -9.80 -1.89E-036 5339.00 5330.10 8.90 1.6'78-037 5615.00 5543.20 71.80 1.28E-028 6028.00 5940.10 87.90 1.46E-029 7515.00 1446.40 68.60 9.13E-0310 7924.00 1849.90 14.10 9.356-03

1 1 8148.00 8090.40 51.60 1.078-0312 8065.00 8092.40 -21.40 -3.40E-0313 6112.00 6782.70 -10.10 -1.O5E-0214 7794.00 1761.60 32.40 4.16E-0315 7891.00 7910.70 -19.70 -2.50E-0316 1864.00 1902.10 -38.10 -4.92E-0317 7927.00 1986.10 -59.70 -1.53E-0318 7613.00 1655.00 -42.00 -5.528-0319 1697.00 1865.80 -168.80 -2.19E-0220 7506.00 1645.10 -139.10 -1.86E-0221 7218.00 1326.60 -108.60 -1.50E-0222 6160.00 6842.90 -82.90 -1.231-0223 6329.00 6460.70 -131.70 -2.08E-0224 5838.00 6003.00 -165.00 -2.83E-02t* MAE= 0.91 X ** RUS error 1.18 X** FORECASTED PEAK = 8092.4** ACTUAL PEAK = 8065 HW ERROR = 2P.4UW (0.34%)** PORECASTED VALLEY = 5197.2 UY** ACTUAL VALLEY = 5196.0 MY ERROR =I.ZUk(O.OZXl

______-_--______________________________-________________________________________--------

--______________________________________----------______________________________________-7. CONCLUSIONSA multilayer feedforward neural network is proposedfor short-term load forecasting. The effect of learning rateand momentum on the efficiency of conventional learningalgorithm, in which the learning rate and momentum areboth fixed, is extensively studied. It is foun d that theconvergence rate of the conventional algorithm is signifi-cantly affected by thc learning rate and momentum. Tospeed up the learning process, an adaptive learningalgorithm, in wich the m ome ntum is adapted in the learningprocess, is developed. It is found that the adaptivealgorithm convcrges much faster than th e conventionalalgorithm. Moreover, the convergence property of theadaptive learning algorithm will not be affccted by t hclearning rate and initial value of momen tum. This makesthe adaptive algorithm more convenient than the convcn-tional algorithm. Thc proposed ncural netwo rk with the

adaptive algorithm ha s been applied to load forecasting ofTaiwan power systcin. I t is found that accurate loadforecasting results can be achieved b y th e neural network ina very efficient way.

Table 16. Load forecast results fc r >!ov. 27 , 1987-----._-________..-____.__._______________----.--____.-_______-------___.YO -9.6PE-031 6094.00 t i 152 5934.00 5926.50 1.50 1.268-03

3 5194.00 5118.30 15.10 2.71E-034 5667.00 5681.60 -14.60 -2.58E-035 5690.00 5681.00 9.00 1.58E-036 5817.00 5808.50 8 . 5 0 1.46E-037 6021.00 5957.10 63.90 1.06E-028 6469.00 6411.80 51.20 1.91E-039 8114.00 8221.10 -41.10 -5.768-0310 8113.00 8792.80 -19.80 - 2 . 2 6 1 - 0 311 9067.00 9168.50 -101.50 -1.12E-021 2 9038.00 9112.30 -134.30 -1.491-02

13 1724.00 7833.60 -1OY.60 -1.421-0214 8816.00 8961.30 -85.30 - 9 . 6 1E - 0315 9037.00 9120.00 -83.00 -9.18E-0316 9065.00 9096.90 -31.90 -3.52E-0311 9161.00 9120.00 41.00 5.13E-0319 8860.00 8884.20 -24.20 -2.73E-0320 8597.00 8573 .30 23.10 2.76E-0321 8189.00 8196.40 -7.40 -9.048-0422 1616.00 1629.30 -3.30 -1.15E-0323 1126.00 7168.00 -42.00 -5.898-0324 6542.00 6641.90 -99.90 -1.53E-02-------____--__-._______________________--..--~-___-__--_.__.__________________________--..* * FORECASTED PEAK = 9112.3** ACTUAL PEAK = 9167 UW ERROR = 5.3 MY (0.05%)

18 9051.00 8859.10 197.90 2.1 9~- 02

** UAE: 0.69 x *+ nus error 0.88 x


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149

8. ACKNOWLEDGEMENTSThe authors would like to thank the reviewers fortheir valuable comments. Sincere gratitude is also given toMessrs. C.C. Liang, K.K. Chen, T.S. h i , an d B.S. Chang ofthe System Operation Department of Taiwan PowerCompany for providing the valuable system data. Financialsupport given to this work by the National Science Councilof ROC under contract nuniber NSC-78-0404 -E002-30is appreciated.

propagation trained multiperception, Proc. 1989Internatio nal Symposium o n Circuits and Systems,. .pp. 486-489.[ 171 M.E. Aggoun e, L.E. Atlas, D.A. Colin, M.J. Damborg,M.A. El-Sharkawi, and R.J. Marks 11, Artificialneural networks for power system static securityassessment, Proc. 1989 Internatio nal Symposiumon circuits and Systems, pp. 490-494 .[ 181 R.P. Lippmann, An introduction to computingwith neural nets, IEEE ASSP Magazine, pp. 4-22,1 Q R 71,.9. REFERENCES

G. Gross and F.D . Galiana, Short term load fore-casting, Proc. IEEE, Vol. 75, No. 12 , p p . 1558-1573 , 1987 .W.R. Christianse, Short-term load forecasting usinggeneral expon ential smo othin g, IEEE Trans. PAS,J. Davey, J.J . Soach s, G.W. Cunningham and K.W.Priest, Practical applica tion of weathe r sensitiveload forecasting to system planning, IEEE Trans.PAS, Vol. 91 , pp. 91 7-977 , 19 72.R.P. Thom pson, Weather sensitive dema nd andenergy analysis on a large geographically diversepowe r system-application to short-tern1 hourlyelectric deman d forecasting, IEEE Trans. PAS,A.D. Papalexopoulos and T.C. Hesterberg, Aregression-based approach to short-term system loadforecasting, 1989 PICA Conference, pp. 41 4-4 23.F. Meslier, New advances in short term load fore-casting using Box and Jenkins Approaches, PaperA78 051-5, presented at the IEEElPES 1 978 WinterMeeting.G.D. Irisarri, S.E. Widergren, and P.D. Yehsakul,On-line load forecasting fo r encrgy control centerapplica tion, IEEE Trans. PAS, Vol. 101, pp. 71-78 ,1982 .S. Ranm an and R. Bhatnagar, An exp ert systembased algorithm fo r shor t term load forecast, IEEETrans. PWRS, Vol. 3 , pp. 392-399, 1988.K. Jabbour, J.F.V. Riveros, D. Landsbergen, and W.Meyer, AL FA: automa ted load forecastingassistant, IEEE Trans. PWRS, Vol. 3, pp. 908- 914,

Vol. 90, pp. 900-911 , 197 1 .

Vol. 95 , pp. 384-393, 1976.

1988.[ l o ] K.L. Ho, Y.Y. Hsu, C.F. Chen, T.E. Lee, C.C. Liang,T.S. Lai, and K.K. Chen , Short term load fore-casting of Taiwan power system using a knowledge-based expert system, Paper 90 WM 259-2 PiVRS,presented a t th e IEEE/PES 199 0 Winter Meeting.[ 1 I ] D.E. Rumelhart, G.E. Hinton, and R.J. Williams,Leaming internal representations by error propaga-tion, in D.E. Rumelhart and J.L. McClelland,Parallel Distributed Processing, Vol. 1 , Ch. 8.[ 121 D.J. Sobajic and Y.H. Pao , Artificial neural-netbased dynamic security assessment for electric powersystem s, IEEE Trans. PWKS, Vol. 4 , pp. 220-22 8,1989 .[ 13 1 M. Aggoune, M.A. El-Sharkaw i, D.C. Park, M.J. Dam-borg, and R.J. Marks 11, Preliminary results on usingartificial neural .networks for security assessment,1989 PICA Conference, pp . 252-258.1141 M.I. Santoso and O.T. Tan , Neural-net based real-time cortrol of capacitors installed on distributionsystems, Pa er 8 9 SM 768-3 PWRD, presen ted atthe IEEE JPEJ 1989 Summer Meet ing .

[ 151 E.H.P. Chan, Application of neural-network coni-puting i n intelligent alarm processing, 1989 PlCAConference, pp. 2 46-25 1.[1 6 ] R. Fischl, M. Kam, J.C. Chow and S. Ricciardi,Screening power system conti!igencies using a back-

Kun-Long Ho was born in 1964.He received his B.S . degree in elec-trical engineering from Chung-YuanUniversity, Chungli, Taiwan. He hasbeen working toward his Ph.D. degreein the Electrical Engineering Depart-ment of National Taiwan University,Taipei, Taiwan.At present, his research interestsinclude security control, load forecast-ing, and th e application of expert systems t o power systemproblems.

Yuan-Yih Hsu was born in Tai-wa n on Jun e 19, 1955 . He reccivedhis B. Sc., M. Sc., and Ph.D. degrees,all in electrical engineering fromNational Taiwan University, Taipei,Taiwan.Since 197 7, he has been withNational Taiwan University, where heis now a professor. He worked at theUniversity of Calgary, Alberta, Canada,as a postodctoral research fellow andinst ructo r f rom 1982 to 1983 . F rom 1988 to 1989 , he wasa visiting scholar at the University of California, Berkeley.At present, his research interests include power systemdynamics, stability analysis, reliability analysis and theapplication of expert systems t o pow er system problems.He is a senior member of IEEE.

SHORT TERM LOAD FORECASTING USING A MULTILAYER NEURAL NETWORK

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