FORECAST MODEL OF HEAT DEMAND

Transcription

1 FORECAST MODEL OF HEAT DEMAND Absrac: The paper deals wih he uilizaion of ime series predicion for conrol of echnological process in real ime. An improvemen of echnological process conrol level can be achieved by ime series analysis in order o predicion of heir fuure behaviour. We can find an applicaion of his predicion also by he conrol in he Cenralized Hea Supply Sysem (CHSS), especially for he conrol of ho waer piping hea oupu. Due o he large operaional coss involved, efficien operaion conrol of he producion sources and producion unis in a disric heaing sysem is desirable. Knowledge of hea demand is he base for inpu daa for operaion preparaion of CHSS. Term hea demand is insananeous required hea oupu or insananeous consumed hea oupu by consumers. The course of hea demand can be demonsraed by means of hea demand diagrams. Mos imporan is daily diagram of hea demand (DDHD), which demonsraes he course of requisie hea oupu during he day. This diagram is mos imporan for echnical and economic consideraion. Therefore forecas of his diagram course is significan for shor-erm and long-erm planning of hea producion. In his paper we propose he forecas model of DDHD based on he Box-Jenkins mehodology. The model is based on he assumpion ha he course of DDHD can be described sufficienly well as a funcion of he oudoor emperaure and he weaher independen componen (social componens). Time of he day affecs he social componens. The ime dependence of he load reflecs he exisence of a daily hea demand paern, which may vary for differen week days and seasons. Forecas of social componen is realized by means of Box-Jenkins mehodology. The complee forecas algorihm wih inclusion of oudoor emperaure is described. Finally is presened some compuaional resuls and conclusions. Key words: Predicion, Disric Heaing Conrol, Box-Jenkins, Conrol algorihms, Time series analysis. 1 Inroducion Improvemen of echnological process conrol level can be achieved by he ime series analysis in order o predic heir fuure behaviour. We can find applicaions of his predicion also in he conrol of he Cenralized Hea Supply Sysem (CHSS), especially for he conrol of ho waer piping hea oupu. Knowledge of hea demand is he base for inpu daa for he operaion preparaion of CHSS. The erm hea demand means an insananeous hea oupu demanded or insananeous hea oupu consumed by consumers. The erm hea demand relaes o he erm hea consumpion. I expresses hea energy which is supplied o he cusomer in a specific ime inerval (generally a day or a year). The course of hea demand and hea consumpion can be demonsraed by means of hea demand diagrams. The mos imporan one is he Daily Diagram of Hea Demand (DDHD) which demonsraes he course of requisie hea oupu during he day (See Fig. 1). These hea demand diagrams are of essenial imporance for echnical and economic consideraions. Therefore forecas of hese diagrams course is significan for shor-erm and long-erm planning of hea producion. I is possible o judge he quesion of peak sources and paricularly he quesion of opimal load disribuion beween he cooperaive producion sources and producion unis inside hese sources according o he ime course of hea demand. The forecas of DDHD is used in his case (Balá, 1982).

2 Hea demand [/hod] Dae Fig. 1: DDHD from he concree localiy 2 Forecas mehod for concree ime series of hea demand I is possible o use differen soluion mehods for he calculaion of ime series forecas. (For example: soluion by means of linear models, soluion by means of non-linear models, specral analysis mehod, neural neworks ec.). In he pas, los of works were creaed which solved he predicion of DDHD and is use for conrolling he Disric Heaing Sysem (DHS). Mos of hese works are based on mass daa processing. Neverheless, hese mehods have a big disadvanage ha may resul in ou-of-dae real daa. From his poin of view i is suiable o use he forecas mehods according o he Box Jenkins mehodology (Box and Jenkins, 1976). As his mehod achieves very good resuls in pracice, i was chosen for he calculaion of DDHD forecas. This paper is dealing wih he idenificaion of a model of concree ime series of he DDHD and hen he inclusion of oudoor emperaure influence in calculaion of predicion of DDHD is presened. 3 Idenificaion of a model of DDHD Idenificaion of ime series model parameers is he mos imporan and he mos difficul phase in he ime series analysis. We have paricularly focused on deerminaion of difference degree as well as on obaining a suiable order of auoregressive process and order of moving average process. Differencing he ime series consiss in reducion of nonsaionary ime series o saionary ime series and also i means o deermine a degree of seasonal differencing in he case of periodic behaviour of ime series. A number of possibiliies for deerminaion of difference degree exis. 1. I is possible o use a plo of he ime series, for visual inspecion of is saionariy. In case of doubs, he plo of he firs or second differencing of ime series is drawn. Then we review saionariy of hese series. 2. Invesigaion of esimaed auocorrelaion funcion (ACF) of ime series is a more objecive mehod. 3. Anderson (Anderson, 1976) prefers o use he behaviour of he variances of successive differenced series as a crierion for aking a decision on he difference degree required. Many observed non-saionary ime series exhibi cerain homogeneiy and can be accouned for by a simple modificaion of he ARMA model, he auoregressive inegraed moving average (ARIMA) model. Deerminaion of a degree of differencing d is he main problem of ARIMA model building. In

3 pracice, i seldom appears necessary o difference more han wice. Tha means ha saionary ime series are produced by means of he firs or second differencing. In pracice, many ime series have seasonal componens. These series exhibi periodic behaviour wih a period s. Therefore i is necessary o deermine a degree of seasonal differencing - D. The seasonal differencing is marked by D s. In seasonal models, necessiy of differencing more han once occurs very seldom. Tha means D= or D=1. An example of he deerminaion of he difference degree for our ime series of DDHD is shown in his par of paper. The course of ime series of DDHD conains wo periodic componens (daily and weekly periods). The daily period presens increase and decrease in hea demand during he day. Hea demand drop a he weekend forms he weekly period of DDHD. The general model according o Box- Jenkins enables o describe only one periodic componen. We can propose wo evenual approaches o calculaion of forecas o describe boh periodic componens (Dosál, 1986). The mehod ha uses he model wih double filraion The mehod superposiion of models For ime series analysis we will furher consider only ime series of he DDHD wihou Saurday and Sunday values. This ime series exhibis an eviden non-saionariy (see Fig. 2). I is necessary o difference. Fig. 2: The course of DDHD wihou Saurday and Sunday values The course of ime series of he firs differencing is shown in Fig. 3. The differenced series looks saionary now. This fac is also confirmed by he sample ACF (see Fig. 4 and Fig. 5) and by he esimaed variance of non-differenced and differenced series.

4 Number of value Fig. 3: Time series of hea demand afer firs differencing 1 Sample Auocorrelaion Funcion (ACF) Sample Auocorrelaion Lag Fig. 4: The course of esimaed ACF of DDHD As our ime series of he DDHD exhibis an obvious seasonal paern, i is necessary o make an analysis of he seasonal differencing of our ime series, as well. The course of he ACF sample evidences he seasonal paern (see Fig. 4 or Fig. 5). These funcions have heir local maxima a lags 48, 96, ec. Tha represens a seasonal period of 24 hours by a sampling period of 3 minues. On he basis of he execued analysis, i is necessary o make he firs differencing and also he firs seasonal differencing of he DDHD in he form (1). 48 = z z 1 z 48 + z 49 z (1)

5 1 Sample Auocorrelaion Funcion (ACF) Sample Auocorrelaion Lag Fig. 5: The course of esimaed ACF of he DDHD afer he firs differencing 3.1 Deerminaion of AR process order and MA process order Afer differencing he ime series, we have o idenify he order of auoregressive process AR(p) and order of moving average process MA(q). The radiional mehod consiss in comparing he observed paerns of he sample auocorrelaion and parial auocorrelaion funcions wih he heoreical auocorrelaion and parial auocorrelaion funcion paerns. The order of model is usually difficul o deermine on he basis of he ACF and PACF. This mehod of idenificaion requires a lo of experience in building up models. From his poin of view i is more suiable o use he objecive mehods for he ess of he model order. A number of procedures and mehods exis for esing he model order (Cromwell, 1994). These mehods are based on he comparison of he residuals of various models by means of special saisics. In our case, he Akaike Informaion Crierion (AIC), Bayesian Informaion Crierion (BIC) and Schwarz es are used for esing. Adequacy of he suggesed model is examined by means of Pormaneau es. 4 Inclusion of oudoor emperaure influence Above menioned mehods do no describe sudden flucuaion of meeorological influences. In his case we have o include hese influences in calculaion of predicion. Previous works on hea load forecasing [Arvasson, 21], show ha he oudoor emperaure has he greaes influence on DDHD (wih respec o meeorological influences). Oher weaher condiions like wind, sunshine and so on have less effec and hey are pars of sochasic componen. For inclusion of oudoor emperaure influence in calculaion of predicion of DDHD was proposed his general plan: a) The influence of oudoor emperaure filer off from ime series of DDHD by means of heaing characerisic (funcion ha describes he emperaure-dependen par of hea consumpion) b) Predicion of DDHD by means of Box-Jenkins mehod for his filered ime series c) Filraion of prediced values for he reason of inclusion of oudoor emperaure influence (on he base of weaher forecas) From he previous plan is eviden ha he principal aim is o derive an explici expression for he emperaure-dependen par of he hea load. I is obvious ha he emperaure dependence is non-

6 linear. For relaively high oudoor emperaures, he emperaure has less influence. For example, he load will almos be he same for 25 C and 27 C. A corresponding conclusion is also rue for relaively low emperaures, e.g. wheher he oudoor emperaure is -28 C or -3 C does no maer, he producion unis will produce a heir maximum rae anyway. Regarding o previous consideraion we can used he emperaure-dependen par of hea consumpion in he form (2): z 3 = x1 T x2 T (2) where: z is correcion value of hea consumpion in ime including oudoor emperaure influence, T is real value of oudoor emperaure in ime, x 1, x 2 are consans The course of heaing characerisic for consans x 1 =.2, x 2 = 3.5 is shown in he Fig. 6. correcion value of hea consumpion [/hod] 6 Heaing characerisic Oudoor emperaure [ O C] -4-6 Fig. 6: The sample of heaing characerisic (cubic funcion) The emperaure dependen par can assumed o vary as a piecewise linear funcion, see he illusraing example in Fig. 7 (Dozauer, 22). Here a funcion wih five segmens is used, bu he number of segmens can of course be chosen arbirarily. Given he number of segmens as a N s. and he emperaure levels as τ i, i = 1,..., Ns + 1. Now we can consider he emperaure-dependen par of hea consumpion in he form (3): z = α T + β (3) i i where: z is correcion value of hea consumpion in ime including oudoor emperaure influence, T is real value of oudoor emperaure in ime, i is he slope of i-h segmen, i is absolue equaion erm of i-h segmen. Consans (x 1, x 2 and, I, i ) have o be deermined for concree localiy empirically. Filraion ime series of DDHD ha inpus in predicion model is defined in he form (4). i z filr = z z (4) where: z filr z is hea consumpion in ime wih filering off he influence of oudoor emperaure, is correcion value of hea consumpion in ime including oudoor emperaure influence, z is real value of hea consumpion in ime

7 correcion value of hea consumpion [/hod] 6 Heaing Characerisic Oudoor Temperaure [ O C] -4-6 Fig. 7: The sample of heaing characerisic (piecewise linear funcion) Afer predicion calculaion of filering off ime series is necessary o filrae he prediced values for he reason of inclusion of oudoor emperaure influence (on he base of weaher forecas). We can define his operaion in he form (5). filr + + z = z + z (5) where: filr + z is prediced value of filer off ime series of hea consumpion in ime, + correcion value of hea consumpion in ime including oudoor emperaure influence, z is prediced value of hea consumpion in ime. The value z is obained by applicaion of he equaion (2) or (3) for his operaion. We use weaher forecas (emperaure forecas). 5 Concree Resuls of DDHD predicion We use real daa measured in concree localiy (in our case in he ciy Olomouc and Zlin) for calculaion. In his case we realized he calculaion of DDHD predicion for dae In his ime he sudden flucuaion of oudoor emperaure was observed (see Fig. 8). The samples of resuls of DDHD predicion are shown in he Fig. 9 and Fig. 1. z is Fig. 8: The course of oudoor emperaure

8 Fig. 9: The course of DDHD predicion wihou inclusion of oudoor emperaure influence 6 Conclusion Fig. 1: The course of DDHD wih inclusion of oudoor emperaure influence This paper presens he mehod for building up he model of ime series of DDHD and he possibiliy of improvemen of his forecas model wih help of inclusion of oudoor emperaure influence. The modelling is based on he Box-Jenkins mehodology. The ime series analysis was made for he DDHD from he concree localiy. This predicion of DDHD is necessary for he conrol in he Cenralized Hea Supply Sysem (CHSS), especially for he qualiaive-quaniaive conrol mehod of ho-waer piping hea oupu he Balá Sysem (Balá, 1982).