A short-term, pattern-based model for water-demand forecasting

Transcription

1 39 Q IWA Publishing 2007 Journal of Hydroinforatics A short-ter, pattern-based odel for water-deand forecasting Stefano Alvisi, Marco Franchini and Alberto Marinelli ABSTRACT The short-ter, deand-forecasting odel described in this paper fors the third constituent part of the POWADIMA research project which, taken together, address the issue of real-tie, near-optial control of water-distribution networks. Since the intention is to treat water distribution as a feed-forward control syste, operational decisions have to be based on the expected future deands for water, rather than just the present known requireents. Accordingly, it was necessary to develop a short-ter, deand-forecasting procedure. To that end, onitoring facilities were installed to easure short-ter fluctuations in deands for a sall experiental network, which enabled a thorough investigation of trends and periodicities that can usually be found in this type of tie-series. On the basis of these data, a short-ter, deand-forecasting odel was forulated. The odel reproduces the periodic patterns observed at annual, weekly and daily levels prior to fine-tuning the estiated values of future deands through the inclusion of persistence effects. Having validated the odel, the deand forecasts were subjected to an analysis of the sensitivity to possible errors in the various coponents of the odel. Its application to uch larger case studies is described in the following two papers. Key words deand forecasting, short-ter, POWADIMA, water distribution Stefano Alvisi (corresponding author) Marco Franchini Dipartiento di Ingegneria, Università degli Studi di Ferrara, Ferrara 44100, Italy Tel.: Fax: E-ail: salvisi@ing.unife.it Alberto Marinelli DISTART, Università degli Studi di Bologna, Bologna 40136, Italy INTRODUCTION Need for deand forecasting In the context of operational control, the deand for water coprises not only that required for custoers but also leakage fro the distribution network, since it is the cobined aount which is put into supply. Therefore, deands are norally estiated by eans of district etering. For the efficient operation of water-distribution networks, soe for of deand forecasting is usually needed in order to prograe the puping arrangeents over the next 24 h, to take advantage of the electricity tariff structure. For exaple, in the case of pup-scheduling (Sterling & Coulbeck 1975; Zessler & Shair 1989; Jowitt & Geranopoulos 1992; etc.), a short-ter deand forecast is required at the beginning of each orning for the following 24 h. More often than not, these are based on averaged deand profiles for the particular day of the week, which ay vary according to the season or onth of the year. If for any reason, there is a significant difference between the deand profile assued and that which aterializes as the day progresses, it ay be necessary to re-run the pupscheduling software with the revised data but even then, further assuptions need to be ade about the future deands for the reaining portion of the 24 h. Real-tie control requireents The alternative to using a pre-defined deand profile is to develop an adaptive deand-forecasting process which can be continually updated to account for any deviations between that forecasted and reality. For real-tie, doi: /hydro

2 40 S. Alvisi et al. POWADIMA: short-ter deand forecasting Journal of Hydroinforatics near-optial control of water-distribution networks, this alternative approach is the ore appropriate since the pup and valve control settings need to be re-optiized at short, regular intervals, in response to the highly variable deands. Accordingly, this particular paper describes the deand-forecasting process used in the POWADIMA (Potable Water Distribution Manageent) research project (Jaieson et al. 2007). As such, it fors the third constituent part of the DRAGA-ANN (Dynaic, Real-tie, Adaptive, Genetic Algorith Artificial Neural Network) control syste, the other two being the eans of predicting the consequences of different control settings (Rao & Alvarruiz 2007) and the search engine to find the best cobination of control settings (Rao & Saloons 2007). The ai was to develop a odel for real-tie deand forecasting, consistent with the needs of the DRAGA-ANN control process. As such, the spatial variation in deands has been incorporated by having a separate forecast for each of the district etering areas (DMA) that coprise the distribution network. MONITORING CAMPAIGN AND DATA ANALYSES Choice of case study Prior to the outset of the POWADIMA research project, the University of Ferrara was already involved with the Municipality of Castelfranco Eilia in easuring shortter fluctuations of water deands. Therefore, rather than installing a copletely new onitoring schee at a different location, it seeed logical to ake use of the existing facilities by extending the to eet the revised requireents. Castelfranco Eilia is situated in Eilia Roagna, Italy, between Modena and Bologna. Its waterdistribution network is greatly influenced by the town s developent pattern, which extends radially fro the centre towards the ore sparsely inhabited rural areas. As a result, there is a dense, grid-like network of pipes in the town centre, which is replaced by a ore open, ribbon-like structure in the outlying areas (Figure 1). The overall length of pipes foring the network aounts to soe 160 k, the network itself being supplied by a local well field via a storage tank near the town centre. Previous exaples of deand-forecasting odels Any cursory exaination of the scientific literature will confir that a considerable aount of effort has been expended on water-deand forecasting. In the case of operational control, the interest is restricted to short to ediu tiescales (hourly, daily and onthly), rather than annual which largely relates to longer-ter water-resources planning. Even then, there are still a large nuber of papers detailing various ethodologies for hourly forecasts (for exaple, Shvartser et al. 1993; Zhou et al. 2002; etc.) as well as daily/onthly tiescales (Maident & Parzen 1984a,b; Maident et al. 1985; Franklin & Maident 1986; Sith 1988; Miaou 1990). Without exception, all these papers refer to the recurring patterns and periodicities that exist in water-deand data, at different levels of teporal aggregation. For this reason, work began with an extensive onitoring capaign for a sall, urban water-distribution network in order to understand and, later, predict the observed variability in deands, before progressing to the uch larger networks of Haifa-A (see Saloons et al. 2007) and Valencia (Martinez et al. 2007). Monitoring schee The distribution network is onitored for both flow and pressure at 14 stations, strategically located so that data could be gathered for different types of custoers (residential, coercial, etc.) at different levels of spatial as well as teporal aggregation, thereby enabling an in-depth analysis of how space tie correlation characteristics vary according to the level of aggregation (Alvisi et al. 2003). Since the overall ai is to develop a water-deand forecasting odel suitable for large, coplex distribution networks, as in the case of Haifa-A and Valencia, special attention was focused on data acquisition for Station 9, which is located near to the storage tank serving the whole network. This easured the supply to the entire population of soe 23,000 inhabitants, which roughly equates to the size of a DMA found in ajor cities. Water-deand patterns As expected, an analysis of the observed hourly and daily water-deand tie-series revealed the existence of patterns

3 41 S. Alvisi et al. POWADIMA: short-ter deand forecasting Journal of Hydroinforatics Figure 1 The ain water distribution network of Castelfranco Eilia (Province of Modena, Italy) and location of the 14 onitoring stations installed. Station 9 was used in this study. in which it is possible to identify seasonal and weekly periodicities in daily water deands as well as daily periodicities in hourly water deands. Figure 2 shows the pattern in daily water deands for a whole year, with deand rising over the suer period and during the week fro Sunday to Saturday. Likewise, the hourly water deands show a variable diurnal behaviour over the day, with different patterns depending whether it is a weekday or weekend and, to a lesser extent, the season (Figure 3). Analysis of the observed daily and hourly tie-series data indicated that the deand pattern during public holidays was very siilar to that observed during weekends. Therefore, public holidays have been treated as weekends for the purposes of deand forecasting. PATTERN-BASED WATER-DEMAND FORECASTING MODEL Structure of odel The forecasting odel proposed is based on the patterns iplicit in the water-deand tie-series data and for this reason, it is referred to as the Pattern-based Water Deand Forecasting (Patt_WDF) odel. In the first of its two odules, whose purpose is to estiate the average daily water deand on the day (or days) covered by the rolling 24 h forecasting window, the seasonal and weekly patterns are taken into account, together with the ediu-ter persistence. In the second odule, these daily aounts are then cobined with the daily deand patterns and shortter persistence to provide hourly forecasts over the following 24 h period (Figure 4). The daily water-deand odule (DM) In this approach, the average daily water deand Q d;for for the Julian date is estiated by eans of the following relationship: Q d;for ¼ Q d;s where Q d;s þ D d;w i;j þ d d ð1þ is the long-ter average daily water deand representing the seasonal periodic coponent, D d;w i;j is a correction representing the weekly periodic coponent and

4 42 S. Alvisi et al. POWADIMA: short-ter deand forecasting Journal of Hydroinforatics Figure 2 Average daily water deand Q d,obs easured at the outlet of the storage tank in Castelfranco Eilia during the year Seasonal and weekly periodicities are shown. d d is a deviation representing the ediu-ter persistence coponent. The long-ter average daily water deand Q d;s (Figure 2) is odelled using a Fourier series: calculated as: a k ¼ X365 ¼1 Q d;obs cos 2p k; ð3þ 365 Q d;s ¼ a 0 þ XK k¼1 ¼ 1; 2; ; 365 a k cos 2pk 365 þ b k sin 2pk 365 ; ð2þ b k ¼ X365 ¼1 Q d;obs sin 2p 365 k ð4þ where Q d;obs are the observed average daily water deands. The weekly correction factor D d;w i;j is defined as: D d;w i;j ¼ Q d i;j 2 Q w j ð5þ where a 0 is the ean value of the seasonal cycle, a k and b k are Fourier coefficients and K is the nuber of haronics considered. The Fourier coefficients a k and b k are where Q d i;j is the ean value of the average daily water deand observed on day i of the week (i ¼ 1,,7, Monday,, Sunday), j the season ( j ¼ 1,,4, winter, spring, suer, autun) and Q w j is the ean value of the average

5 43 S. Alvisi et al. POWADIMA: short-ter deand forecasting Journal of Hydroinforatics Figure 3 Daily patterns in hourly water deand for the seven days of the week: (a) in winter, (b) in spring, (c) in suer and (d) in autun. weekly water deand in season j. Finally, d d, which represents the deviation between the average daily water deand Q d and the ean value estiated solely on the basis of the periodic coponents Q d;s and D d;w i;j, is odelled using an autoregressive process AR(1) (Box et al. 1994). For forecasting purposes, this ay be written as: d d ¼ F 1 d d 21 whose paraeter F 1 is calibrated on the basis of the observed deviations: d d;obs ¼ Q d;obs 2 Q d;s þ D d;w i;j ð6þ ; ð7þ Figure 5 shows the pattern of the daily residuals observed and odelled by eans of Equation (7), with reference to a year of easured data at Station 9. The hourly deand-forecasting odule (HM) The hourly odule, like the daily odule, is coposed of two parts: a periodic coponent and a persistence coponent. Here, the hourly water deand Q h;for tþk forecasted at the tth hour for k hours ahead is given by Q h;for tþk ¼ Qd;for þ D h n;i;j þ 1 tþk ð8þ Figure 4 Structure of the two odules aking up the Patt_WDF odel.

6 44 S. Alvisi et al. POWADIMA: short-ter deand forecasting Journal of Hydroinforatics Figure 5 Daily residuals d d observed (unbroken line) and odelled via the AR(1) odel (dotted line). where Q d;for is the average daily water deand forecast by eans of the DM using the inforation available up to Julian date -1 (see Equations (1) and (6)), which is replaced with Q d;for þ1 when t þ k leads to the definition of an hour falling in the following day (see Figure 6). D h n;i;j is the hourly deviation representing the daily pattern defined by D h n;i;j ¼ Q h n;i;j 2 Q d i;j where Q h n;i;j represents the ean value of the average hourly water deands observed in hour n (n ¼ 1,,24, the hour of the day) of day i in season j, and Q d i;j is the ean value of the average daily water deands observed on day i in season j (see Figure 7). Finally e tþk represents the hourly persistence coponent, which is odelled using a regression on the errors e tþk-1 and e tþk-24 as follows: 1 tþk ¼ c 1 1 tþk21 þ c 24 1 tþk224 : ð10þ The coefficients c 1 and c 24 depend on the hour of the day t þ k ; n ¼ 1,2,,24 corresponding to the hour t þ k considered (it should be noted that the hour t þ k is counted starting fro the beginning of the year) and are calibrated on the basis of the observed errors 1 obs t where 1 obs t ¼ Q h;obs t 2 Q d;obs þ D h n;i;j : ð11þ ð9þ ahead and so forth up to 24 h ahead, independently of the tie serving as the starting point of the forecast. Then followed an analysis of how the odel s perforance varied according to the hour in which the forecast was ade. These assessents were ade in reference to both the year of data on which the calibration was based and the year of data utilised for validation. Forecasting accuracy was easured by the explained variance (EV), the root ean square error (RMSE) and the ean absolute percentage error (MAE%) as defined by EV ¼ 1 2 P n i¼1 ðe i 2 e Þ 2 2 ð12þ 2 obs P n i¼1 xobs i vffiffiffiffiffiffiffiffiffiffi u1 RMSE ¼ t Xn e 2 i n MAE% ¼ 1 n Xn i¼1 e i i¼1 obs ð13þ ð14þ 100 where n is the nuber of observed data (which in this case is the nuber of hours in the year), e ¼ x obs 2 x for the errors, x obs the observed values, x for the forecast values, e the ean error and obs the ean of the observed values. ANALYSES OF RESULTS Accuracy of forecast The accuracy of the forecast was initially assessed by separately considering the forecast for 1 h ahead, 2 h Figure 6 Forecast average hourly water deands Q h;for obtained by cobining the tþk forecast average daily water deand Q d;for, the daily pattern D h n;i;j and an estiated error e t þ k.

7 45 S. Alvisi et al. POWADIMA: short-ter deand forecasting Journal of Hydroinforatics Figure 7 The daily patterns D h n;i;j associated with a weekday (Friday), Saturday and Sunday in the winter and suer seasons. The daily pattern is characterized on the basis of the ean deviations in hourly flow copared to the average daily value. Figure 8 shows the three indexes obtained considering the water-deand forecast for 1, 2,, 24 h ahead, independently of the forecast starting point, for both the calibration and validation phases. In the calibration phase, the explained variance for the one-hour-ahead forecast is in excess of 0.97 and close to 0.96 for forecasts over the longer tie horizons. The corresponding EV values in the validation phase are 0.97 for the one-hour-ahead forecast and 0.94 for forecasts over the longer tie horizons. As ight have been expected, in both cases the best forecast was obtained for the first hour ahead. This is understandable bearing in ind that the correlation, and hence the persistence, rapidly declines as the nuber of hours ahead increases. The other observation that can be ade relates to the quality of the forecast, which does not vary significantly between the calibration and validation phases. The RMSE and MAE% display a siilar pattern to the one described for EV. In particular, it can be observed that in the calibration phase the RMSE is less than 3.5 l/s for the 1-hour-ahead forecast and less than 4 l/s for the 24-hourahead forecast, for an average hourly water deand of 54 l/s. This corresponds to a MAE% in the range of 4 6%. Siilarly, in the validation phase the RMSE is less than 4.5 l/s for the 1-hour-ahead forecast and less than 8 l/s for the 24-hour-ahead forecast, for an average hourly water deand of 68 l/s, corresponding to a ean absolute percentage error of between 5 9%. Again, the quality of the forecast is better for the first hour ahead and does not change significantly between calibration and validation phases. This is deonstrated in Figure 9, where the 1- and 24-hour-ahead water-deand forecasts for the calibration and validation phases are copared to the observed data. In all four cases, the points are near the 458 line, which represents an exact fit, but in the case of the 1-hour-ahead forecast (Figure 9(a) for calibration and Figure 9(c) for validation) the cluster shows less dispersion copared to the 24-hour-ahead forecast (Figure 9(b) for calibration and Figure 9(d) for validation). Moreover, the 1- hour-ahead forecast for the calibration phase (Figure 9(a)) shows a siilar degree of dispersion to the 1-hour-ahead forecast for the validation phase (Figure 9(c)). The statistics entioned so far, which relate to different lead ties independent of the forecasting start point, furnish a good estiate of the odel s perforance. However, they do not provide an adequate basis for analysing the errors incurred in relation to the different hours of the day to which the forecast refers. Indeed, the water deands observed during the day not only show different average values but also different standard deviations. The latter are largest in the early orning and

8 46 S. Alvisi et al. POWADIMA: short-ter deand forecasting Journal of Hydroinforatics Figure 8 EV, RMSE and MAE% between the observed hourly water deands and those forecast with the Patt_WDF odel, based on the periodic coponent alone and on the coplete odel, 1,2,,24 h ahead in the calibration (cal.) and validation (val.) phases. evening hours due to the fact that consuption varies significantly at these ties, resulting in a wider dispersion. Larger or saller standard deviations in water deand at different ties of the day will result in higher or lower forecasting precision. Figure 10 shows the RMSE and corresponding MAE% obtained in the calibration phase when the tie of forecasting was fixed and the errors incurred in the predictions for the next 24 h were calculated. For exaple, in Figure 10(a), all the forecasts ade at hour 1 for hours 2, 3,, 24 of the sae day and hour 1 of the next day are considered. Looking at the graphs in Figures 10(a), 10(b), 10(c) and 10(d), it can be seen that the RMSE pattern is siilar but the curve shifts laterally according to the tie of forecasting. The patterns only differ for the first or second hour following the tie the forecast was ade, revealing a slight decrease in the RMSE copared to when the water deands of the sae hour of the day are forecast any hours in advance. This is due to the effect of short-ter persistence, which leads to iproveents in the forecasts for the hours that iediately follow the tie of forecasting. Beyond this, the RMSE behaviour is solely a function of the hour of the day considered which is deterined by the representation of the hourly water-deand patterns present in the hourly forecasting odule. For exaple, the forecast of the water deand for hour 7 shows an RMSE equivalent to 6 l/s, regardless of whether the forecast was ade at hour 1 of the sae day (6 h ahead), at hour 12 of the previous day (19 h ahead), or at hour 18 of the previous day (12 h ahead), as indicated in Figures 10(a), 10(c) and 10(d). The RMSE is

9 47 S. Alvisi et al. POWADIMA: short-ter deand forecasting Journal of Hydroinforatics Figure 9 Water deand as forecast (a) 1 h ahead in the calibration phase, (b) 24 h ahead in the calibration phase, (c) 1 h ahead in the validation phase and (d) 24 h ahead in the validation phase, copared to the observed data. slightly lower (5.2 l/s) only when the forecast is ade 1 h ahead of tie, that is to say at hour 6 (Figure 10(b)). Siilar behaviour can be seen in the MAE% (Figures 10(e), 10(f), 10(g), 10(h)). In particular, it is evident that there is an increase in the percentage errors during the night-tie hours which is due to the low values of the average water deand used to noralize the errors. The above considerations find further confiration in Figure 11. In this case, the fixed paraeter was the hour of the day pertaining to the forecast. Here, the RMSE and MAE% relating to the forecasts for a specific hour, ade at different ties of the day, were calculated during the calibration phase. For exaple, Figure 11(a) shows the RMSE of the forecast for hour 1 ade at hours 24, 23,., 2. Again, the RMSE follows a siilar pattern to that shown in Fig. 8, except in this case there are 24 different series associated with the 24 forecasted hours of the day. The curves obtained also confir that the RMSE of the water-deand forecasts for a given hour do not depend on when the forecast was ade, apart fro the two or three hours iediately before the forecasted hour. On the other hand, RMSE values vary according to the hour pertaining to the forecast. For exaple, forecasts for hour 12 are associated with a lower RMSE than forecasts for hour 6, since the forer are characterized by a lower degree of variability. Siilar behaviour ay be observed for MAE%, though higher values are associated with the night-tie hours. This is due to the low values of the average water deand during the night which were used to noralize the errors and hence calculate the MAE%. Sensitivity analysis The sensitivity of the forecasting odel to its pattern and persistence coponents has been assessed on the basis of both its daily and hourly water-deand forecasts, taking into Figure 10 Patterns in RMSE and MAE% obtained by fixing the hour in which the forecast was ade and calculating the errors in the forecasts for each of the next 24 h.

10 48 S. Alvisi et al. POWADIMA: short-ter deand forecasting Journal of Hydroinforatics Figure 11 Patterns in RMSE and MAE% obtained by fixing the hour the forecast was for and calculating the errors in the forecasts ade starting fro each of the previous 24 h. account, firstly, the periodic coponents alone and, secondly, the periodic and persistence coponents together. To that end, the forecasts were copared with the actual water deands observed in order to analyse the errors incurred for each case. More specifically, the EV and RMSE were calculated for the daily water-deand forecasts based on the patterns alone (the seasonal pattern Q d;s and weekly pattern D d;w i;j ), in addition to the forecasts based on both the periodic and persistence coponents (i.e. also taking Table 1 Explained variance EV and RMSE of daily and hourly water deand as forecast on the basis of patterns alone and on the basis of both the periodic and persistence coponents of the Patt_WDF odel in the calibration (average observed water deand 54 l/s) and validation (average observed water deand 68 l/s) phases Calibration Validation Daily Hourly Daily Hourly EV periodic cop EV periodic þ persistence cop RMSE periodic cop RMSE periodic þ persistence cop account of the residual d d, odelled using the autoregressive process AR(1)). Siilarly, the EV and RMSE were calculated for the hourly water deand forecasts based on the patterns alone ( Q d;s, D d;w i;j and D h n;i;j ) and the forecasts based on both the periodic and persistence coponents (i.e. also taking into account the daily residual d d as well as the hourly residual 1 tþk, odelled using a regression equation). Table 1 shows the values obtained in the four cases described for both the calibration and validation phases. In both instances, inclusion of the persistence coponent has the effect of fine-tuning the forecast. This can also be seen fro Figure 8, where the EV, RMSE and MAE% relating to the hourly forecasts using the coplete Patt_WDF odel, are copared to the values obtained when only the periodic coponents of the odel were considered. When the persistence coponents are absent, the EV takes on a constant value of about 0.94, regardless of lead tie, in coparison with an EV of 0.97 for a lead tie of 1 and an EV of 0.96 for all other lead ties, using the coplete odel. In ters of ean absolute percentage error, oitting the persistence coponent would cause the MAE% to increase fro 4.3% for a lead tie of 1 and 5.4 for all other lead ties, to 6.4%. Whilst the contribution of the persistence coponent for the calibration period is odest, it increases

11 49 S. Alvisi et al. POWADIMA: short-ter deand forecasting Journal of Hydroinforatics arkedly when the focus shifts to the validation period (see Table 1). In this latter case, the variance explained by the periodic coponent alone decreases significantly but this is copensated by the increased contribution of the persistence coponent. For exaple, when using the coplete odel at the hourly level, both the calibration and validation phases have an EV of 0.97 whereas for the periodic coponent alone, the EV deteriorates fro 0.94 in the calibration phase to 0.89 in the validation phase. Siilar trends can be seen in Figure 8 where, in the case of the coplete odel, the MAE% is 5.0% for a lead tie of 1 and 7.7% for lead ties greater than 1, whilst that for the pattern coponent alone is 19.3%. The explanation of why the contribution of the persistence coponent increased in the validation phase is likely to be a result of the fact that the patterns used for validation were the sae as those defined in the calibration year. Whilst using such patterns akes it possible to represent periodicities in water deand effectively, soe changes ay well occur fro one year to the next. For exaple, although the observed water-deand patterns in Castelfranco Eilia have been very siilar over the past few years, average consuption has undergone a slight increase, thereby increasing the size of the deviations. However, it would see fro this analysis that the persistence coponent is able to copensate for these systeatic discrepancies. CONCLUSIONS Methodology developed Prior to selecting this odular, pattern-based approach to short-ter water-deand forecasting, consideration was given to using an artificial neural network (Stark et al. 1999) to serve the sae end, which perhaps was ore in keeping with the nature of the overall research project. Whilst the latter provided equally good results for a particular day, difficulties were encountered with the transition fro one day to the next, especially between weekdays and weekends. By adopting a pattern-based approach, which contains seasonal, weekly and daily cycles, these difficulties could be avoided. However, reliance on the cyclic patterns alone also has its own shortcoings since aongst other things, no attention is paid to short-ter exogenous variables such as the prevailing weather which can significantly influence deands on a day to day basis. Nevertheless, by including a persistence coponent as well as a periodic coponent, these liitations can, to a large extent, be overcoe since such factors are iplicitly taken into account by the hourly updating of the persistence coponent. Therefore, the resulting deand-forecasting odel developed for the POWADIMA research project coprised two odules, each containing a periodic coponent and a persistence coponent. Whereas the ai of the first odule was to forecast the daily water deands, that for the second odule was to superipose the hourly fluctuations. By this eans, the observed seasonal, weekly and daily cycles can be preserved as well as the persistence effects. Attributes of the odel developed An analysis of the results obtained shows that the odel is capable of delivering an accurate and robust forecast of future water deands on an hourly basis. Whilst the periodic coponent represents the ainstay of the odel, the persistence coponents enables the forecast to be finetuned. This is of particular relevance to real-tie control where the forecast is updated at short, regular intervals by grounding any discrepancies between the actual deand at the next update and that forecast at the previous tiestep. In this way, rather than just scaling the existing 24 h forecast in accordance with the actual deand, a new forecast can be generated at the next tie-step, taking the internal structure of the deand patterns into account. However, it is iportant to stress that, in the light of the results obtained fro the calibration and validation phases of the Castelfranco Eilia s water-deand forecasting odel, consideration should be given to re-estiating the periodic and persistence paraeters whenever there is a distinct increase in the nuber of custoers served or a noticeable change in the per capita consuption. ACKNOWLEDGEMENTS This water-deand forecasting odel was developed as a constituent part of the POWADIMA research project,

12 50 S. Alvisi et al. POWADIMA: short-ter deand forecasting Journal of Hydroinforatics which was funded by the European Coission under its Vth Fraework theatic prograe on Energy, Environent and Sustainable Developent (Contract Nuber EVK1-CT ). Mebers of the POWADIMA consortiu would like to take this opportunity to thank the Coission and its project officers for their support. We also wish to acknowledge the assistance of Castelfranco Eilia Municipality and Meta Spa which acted as subcontractors to the Università degli Studi di Ferrara for this particular aspect of the project. REFERENCES Alvisi, S., Franchini, M. & Marinelli, A A stochastic odel for representing drinking water deand at residential level. Wat. Res. Mngnt. 17 (3), Box, G. E. P., Jenkins, G. M. & Reinsel, G. C Tie Series Analysis Forecasting and Control, 3rd edn. Prentice-Hall Englewood Cliffs, NJ. Franklin, S. L. & Maident, D. R An evaluation of weekly and onthly tie series forecasts of unicipal water use. Wat. Res. Bull. 22 (4), Jaieson, D. G., Shair, U., Martinez, F. & Franchini, M Conceptual design of a generic, real-tie, near-optial control syste for water-distribution networks. J. Hydroinforatics 9 (1), Jowitt, P. W. & Geranopoulos, G Optial pup scheduling in water-supply networks. J. Wat. Res. Plann. Mngnt., ASCE 118 (4), Maident, D. R. & Parzen, E. 1984a Tie patterns of water uses in six Texas cities. J. Wat. Res. Plann. Mngnt., ASCE 110 (1), Maident, D. R. & Parzen, E. 1984b Cascade odel of onthly unicipal water use. Wat. Res. Res. 20 (1), Maident, D. R., Miaou, S. P. & Crawford, M. M Transfer function odels of daily urban water use. Wat. Res. Res. 21 (4), Martinez, F., Hernandez, V., Alonso, J. M., Rao, Z. & Alvisi, S Optiizing the operation of the Valencia water-distribution network. J. Hydroinforatics 9 (1), Miaou, S. P A class of tie series urban water deand odels with non-linear cliatic effects. Wat. Res. Res. 26 (2), Rao, Z. & Alvarruiz, F Use of an artificial neural network to capture the doain knowledge of a conventional hydraulic siulation odel. J. Hydroinforatics 9 (1), Rao, Z. & Saloons, E Developent of a real-tie, nearoptial control process for water-distribution networks. J. Hydroinforatics 9 (1), Saloons, E., Goryashko, A., Shair, U., Rao, Z. & Alvisi, A Optiizing the operation of the Haifa-A water-distribution network. J. Hydroinforatics 9 (1), Shvartser, L., Shair, U. & Feldan, M Forecasting hourly water deands by pattern recognition approach. J. Wat. Res. Plann. Mngnt., ASCE 119 (6), Sith, J. A A odel of daily unicipal water use for shortter forecasting. Wat. Res. Res. 24 (2), Stark, H.L., Stanley, S.J. and Buchanan, I.L Water deand forecasting using artificial neural network., OWWA Conference paper, Ottawa. Sterling, M. J. H. & Coulbeck, B A dynaic prograing solution to the optiisation of puping costs. Proc. ICE 59 (2), Zessler, U. & Shair, U Optial operation of water distribution systes. J. Wat. Res. Plann. Mngnt., ASCE 115 (6), Zhou, S. L., McMahon, T. A., Walton, A. & Lewis, J Forecasting operational deand for an urban water supply zone. J. Hydrol. 259,