Forecasting operational demand for an urban water supply zone

Journal of Hydrology 259 (2002) 189±202

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Forecasting operational demand for an urban water supply zone S.L. Zhou a, T.A. McMahon a,*, A. Walton b, J. Lewis b a

Department of Civil and Environmental Engineering, Centre for Environmental Applied Hydrology, The University of Melbourne, Parkville, Vic. 3052, Australia b Water Transfer, Melbourne Water Corporation, 68 Ricketts Road, Mt Waverley, Vic. 3149, Australia Received 30 January 2001; revised 28 September 2001; accepted 16 November 2001

Abstract A time series forecasting model of hourly water consumption 24 h in advance for an urban zone within the Melbourne (Australia) water supply system is developed. The model comprises two modulesÐdaily and hourly. The daily module is formulated as a set of equations representing the effects of three factors on water use namely seasonality, climatic correlation, and autocorrelation. The hourly module is developed to disaggregate the estimated daily consumption into hourly consumption. The models were calibrated using hourly and daily data for a 6 year period, and independently validated over an additional seven month period. Over this latter period, the hourly forecast model accounted for 66% of the variance in the peak hourly water consumption with a standard error of 162 l/p/d. q 2002 Elsevier Science B.V. All rights reserved. Keywords: Statistics; Time series; Water consumption; Forecasting

1. Introduction The general objective of an urban water supply authority is to match supply and demand at a level of service acceptable to consumers. This requires very frequent adjustments in response to demand variations in order to minimise costs. Relative demand variation is diurnal, by day of week, and by month and season. It is modi®ed by weather, by weekend and holiday patterns, as well as by the regular domestic and industrial activities of consumers. Prediction, taking account of the earlier factors, is necessary for system optimisation in successive control periods. Often system operators do this from experience. They use such information as day of the week, hour of the day, weather, special events (holiday, a major * Corresponding author. Fax: 161-3-9344-6215. E-mail address: [email protected] (T.A. McMahon).

sporting event), previous day and previous hour consumptions. However, it is only very recently that statistical analysis of water use and climate data monitored by the authority has begun to be used for prediction purposes. The water supply system for Melbourne's three million consumers was wholly managed by Melbourne Water until 1995, when as a result of organisational reform, the business was disaggregated into three Retail Water Companies and the Melbourne Water Corporation remained as a wholesaler providing bulk supply. The water supply system is hydraulically simulated by computer models of the bulk supply system (Melbourne Water Corporation) and the network systems (Retail Water Companies), that comprise reservoirs, pipes, pumps and control valves. To model effectively the system operation, future water demandsÐboth long- and short-termÐmust be predicted. This paper deals with forecasting demands 24 h in advance.

0022-1694/02/$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S 0022-169 4(01)00582-0

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Water demand forecasting, say 24 h ahead, can be performed by a computer-based mathematical model that relies on past demand data and other information, such as weather forecasts. The objective of this study is to investigate a methodology for forecasting water demand at hourly interval 24 h in advance within a pressure zone using monitored records of water consumption and climatic information. This paper summarises the development of the water demand prediction model for the Chelsea distribution zone that is currently managed by the Retail Water Company, South East Water Limited and supplied by the Melbourne Water Corporation. 2. Literature review Urban water supply systems provide water to serve uses ranging from human consumption to ®re control, and from garden irrigation to industrial processes. Water demands are highly variable, depending on such factors as size of city, characteristics of the population, the nature and size of commercial and industrial establishments, climatic conditions, and cost of supply. A brief survey of the literature reveals an increase in past decades or so in the development of statistical models, typically multiple regression and time series, for predicting urban water use. Time series of annual urban water consumption have been related to population, household income, water price, rainfall, air temperature, and evaporation by Wong (1972), Young (1973), and Willsie and Pratt (1974). More speci®c accounting of the effect of climatic variables was achieved by separating the year into two seasons, with the winter season characterising indoor or domestic water use and the summer season having, in addition, outdoor or sprinkler water use (Howe and Linaweaver, 1967; Carver and Boland, 1980). Sometimes seasonality was modelled using a regression function for each month (Morgan and Smolen, 1976; Yamauchi and Huang, 1977; Cassuto and Ryan, 1979). Yamauchi and Huang (1977) ®tted two models, an additive model and an exponentially multiplicative model formed by replacing daily water consumption by its natural logarithm as the dependent variable. They consider that the multiplicative model is more appropriate than an additive

model because in it the seasonal variation of water use changes from year to year along with the trend in the mean, while in the additive model the seasonal variation is constant from year to year. Although the time structure of water use may be indirectly accounted for by special devices used in the multiple regression, time series analysis methods are more appropriate to account for that structure. The earliest comprehensive study of modelling monthly water use as a time series is reported by Salas and Yevjevich (1972). They investigated municipal, agricultural and hydropower water uses and developed mathematical models for trends and periodicities in monthly means and standard deviations. They also investigated the coherence and cross-correlation functions of water use, precipitation and air temperature to determine the interrelationships among these three variables. Maidment and Parzen (1984a) developed a cascade model which was used to analyse monthly time series of municipal water use by detrending, deseasonalising, autoregressive ®ltering, and ®nally developing a multiple regression. Maidment et al. (1985) developed a transfer function model to forecast daily water use for the city of Austin, TX, as a combination of both long and shortmemory components. Maidment and Miaou (1986) applied the methodology to daily water use data from nine cities, three cities each from Florida, Pennsylvania, and Texas. They concluded that, as a proportion of the mean annual use, the seasonal use averaged for the three cities in each state was 23% in Texas, 15% in Florida, and 5% in Pennsylvania. The dynamic response of water use to rainfall and air temperature was found to be similar across the cities within each state. There was little impact of city size on the response functions. The response of water use to rainfall depended ®rstly on the occurrence of rainfall and secondly on its magnitude. In Austin, a spatially averaged rainfall series showed a clearer relationship with water use than did rainfall data from a single gauge, and there was a non-linear response of water use to air temperature changes. They also concluded that for small cities, such as College Station, TX, there is a relatively higher inherent randomness in the daily water use data than in larger cities, so smaller cities are harder to model than larger cities. Viswanathan (1985) investigated the effect of restrictions on daily water consumption levels in

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191

Fig. 1. Average weekly pro®le of water consumption for the Chelsea zone (based on available hourly data from November 1989 to November 1995).

Newcastle, NSW using a linear model in which the independent variables were rainfall depth and the concurrent and previous day's daily maximum air temperature. In his model each coef®cient was a sinusoidally varying function of time over the year and the parameters were estimated using recursive least squares with a different parameter set estimated for each year of data. Many variables have been utilised in models to account for the impact of climate on water use. Temperature and precipitation are two of the most common, and possibly the most effective, meteorological variables. Morgan and Smolen (1976) found these two variables most signi®cant when compared with potential evapotranspiration minus precipitation and monthly binary seasonal variables in a crosssectional study for 33 cities in Southern California. Weeks and McMahon (1973) found that in Australia the number of rainy days per annum was the most signi®cant climatic variable affecting annual per capita water use, but that weekly pan evaporation and average maximum daily temperature were more signi®cant explanatory variables than rainfall in a multiple linear regression describing maximum weekly demands. Maidment and Parzen (1984b) utilised precipitation, temperature and pan evaporation as climatic variables in a study of six Texas cities. Franklin and Maidment (1986) used the Maidment and Parzen procedure (the Cascade Model) to model weekly and monthly water use in Deer®eld Beach, Florida. In summary, various types of time series models

have been developed to forecast urban water use. Total water use is generally separated into base use and seasonal use. The base water use is characterised by the water use during the winter months. The methods used to estimate the base use include regressions based on population served by the water system, number of connections to the distribution system, water price, and household income as independent variables, or against time, e.g. a polynomial function of time. Often seasonal use is divided into two components, one which varies smoothly over the year and another which represents the short-term variations. The long-term cycle of the seasonal use can be expressed as a Fourier series, while the short-term variations can be simulated by climatic regression and autoregression. 3. Methodology The combined effect of large numbers of consumers gives rise to overall consumption pro®les with certain well de®ned features. These are shown in Fig. 1 which shows the average weekly pro®le of water consumption for the Chelsea zone which is a residential area and has a population of 35 000 persons. In general, this indicates a daily periodic pattern with a slowly varying component and added random variations. Based on the literature review and taking into account the requirements of Melbourne Water, the methodology adopted to forecast water consumption

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Fig. 2. Water demand simulation model for zones in water supply system of Melbourne.

24 h in advance is based on the techniques of time series analysis. The demand simulation model comprises hourly and daily modules. For the Chelsea zone, total water consumption is divided into base and seasonal consumption, and the consumption is characterised on a monthly basis separately for weekdays and for weekends (including public holidays). The daily module simulates seasonal water consumption using the techniques of time series analysis incorporating climate data. The hourly module consists of base consumption and seasonal use disaggregated from daily seasonal use. The model structure is shown schematically in Fig. 2. 3.1. Base consumption Base water consumption represents the weatherinsensitive portion of the total use. Furthermore, it is less sensitive to a change in consumption behaviour, and a reduction will be brought about mainly by appliance redesign. Such a change can only occur slowly, as more ef®cient appliances are installed by consumers. Therefore, within the period for which consumption data were available for this study, base water consumption is assumed to be a ®xed percentile of the hourly ¯ow records for weekdays and weekends separately in each month. The daily base consumption is then derived by integrating the hourly base consumption to calculate daily base consumption by month. The procedure of deriving hourly base consumption curves is as follows: (1) select all hourly consumption data for weekdays and weekends (including public

holidays) separately in each month, and (2) calculate the lower ten percentile values for each month (weekday and weekend separately) of the selected series. The 10% value was adopted as base consumption on advice from Melbourne Water staff. Thus, 24 curves of hourly base consumption can be obtained for each zone. The daily base consumption is then derived by integrating the hourly base consumption to calculate daily seasonal consumption. 3.2. Daily seasonal consumption Seasonal water consumption represents the weather-sensitive portion of the total use. Based on the earlier literature review, the time series of daily seasonal consumption Cjx is represented by a structure consisting of three additive components: C^ xj ˆ C^ Sj 1 C^ Cj 1 C^ Pj ;

…1†

where C represents daily per capita water use; the superscripts S, C and P indicate the seasonal cycle, climate related consumption and persistence components, respectively. j is a daily time index. A carat, ^, indicates estimated values. 3.2.1. Seasonal cycle Various methods are available for estimating monthly components. One could form the twelfth differences of data (Box and Jenkins, 1976), or use an autoregressive model of CjS in which the coef®cients re¯ect seasonality (twelfth differencing is a special case of this method). Alternatively one could regress CjS against a seasonal variable such as

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193

maximum air temperature. However, a straightforward approach is to use the arithmetic or Fouriersmoothed daily means of CjS : In this study we adopted the latter approach and represented the seasonal cycle in water use by the following Fourier series:  K  X 2pk 2pk C^ St ˆ a0 1 ak cos t 1 bk sin t ; 365 365 kˆ1 …2†

been included in the model. Past techniques for climatic explanation of consumption tended to rely on the antecedent effect of precipitation. This quite logical approach has been incorporated into the model in the form of antecedent precipitation Index (API) and also as the number of days since previous precipitation (de®ned as being greater than or equal to 0.2 mm per day). Therefore, the basic structure of the climatic component is written as:

t ˆ 1; 2; ¼; 365;

C^ Cj ˆ h0 1 h1 f1 …MT† 1 h2 f2 …P† 1 h3 f3 …API†

in which a0 is the mean value of the seasonal cycle, ak and bk the Fourier coef®cients, and K is the number of signi®cant harmonics. In this example only two were found to be signi®cant at 5% level of signi®cance (Yevjevich, 1972). The Fourier coef®cients, ak and bk, of Eq. (2) are given by ak ˆ

365 2 X 2pt k; Smt cos 365 tˆ1 365

…3†

365 2 X 2pt k; Sm sin 365 tˆ1 t 365

…4†

and bk ˆ

in which Smt is the sample estimate of the seasonal mean, and it is computed by Smt ˆ

n 1 X C ; n yˆ1 y;t

t ˆ 1; 2; ¼; 365;

…5†

where n is the number of years of record and Cy,t is the total consumption value less base consumption. For simplicity, it is assumed that no periodic components (seasonal cycles) exist in standard deviation and high order statistics, although this may not be true in practice. 3.2.2. Climatic component Weather affects urban water consumption on a daily, weekly and monthly basis. These effects have been utilised to develop detailed relationships between weather and water consumption so that short-term consumption trends could be pre-determined using known weather and weather forecasts. Three climatic variablesÐmaximum temperature, precipitation and Class A pan evaporationÐhave

1 h4 f4 …ND† 1 h5 f5 …E†;

…6†

where h0 ; ¼; h5 are model parameters, f1 ; ¼; f5 the variable functions, MT the daily maximum temperature in 8C, P is the daily precipitation in mm, ND the number of days since a rainfall of at least 0.2 mm, and E represents the daily pan evaporation in mm. API is calculated by APIj ˆ kAPIj21 1 Pj21 ; where Pj21 is the precipitation in mm on day j 2 1. The value of k is dependent on the potential loss of moisture and varies seasonally usually between 0.85 and 0.98, but in this study a constant value of 0.85 was adopted as being representative of the previous day's rainfall (Bruce and Clark, 1966). 3.2.3. Persistence component In this study the persistence component represents a short-memory process of the system. An autoregressive procedure is ®tted to the residual time series (after the base, the seasonal and the climatic components have been removed) to account for the dependence of water use on its own past values. It is written as: C^ Pj ˆ f0 1 f1 C 0j21 1 f2 C 0j22 1 ¼ 1 fp C 0j2p ;

…7†

where C 0 is the residual consumption, p the order of the autoregressive procedure, AR(p), f 0, f 1, f 2,¼, and f n are the coef®cients of the AR(p). 3.3. Hourly seasonal consumption Both detailed design and short-term operation of a water supply and distribution system require hourly estimates of water consumption. For design purposes, only the expected range of short-term demand patterns is required. However, for operational control purposes, it is necessary to be more speci®c, and to be

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able to predict the expected hourly demands for the forecast period. The combined effect of large numbers of consumers gives rise to overall consumption pro®les with certain well de®ned features. In general, this indicates a daily periodic pro®le with a slowly varying component and added random variations. It is noted that although the consumption pattern is different from day to day, for modelling, daily periodic pro®les in a week were only categorised into two patternsÐone for weekdays and other for weekend days (including public holidays). An empirical procedure is used to disaggregate the daily seasonal consumption into hourly consumption in consideration of hourly peak consumption. The ®rst step is to build a relationship between hourly peak seasonal consumption and daily seasonal consumption for weekdays and for weekend days in each month. The relation can be expressed as pk x Cw;m ˆ aw;m 1 bw;m Cw;m ;

…8†

where C represents per capita water use, the superscripts pk and x represent the hourly peak and daily water use less base use, respectively, the subscript w indicates weekday or weekend, and m is a monthly time index from 1 to 12. The coef®cients a and b are calibrated from historical data. The next step is to identify the appropriate hourly pk pattern given the estimated peak hourly value, Cw;m : The appropriate pattern is chosen from one of ®ve pk patterns based on Cw;m : As an example, Table 4 shows the relationship between the hourly peak values and the appropriate patterns for weekdays in January. Tables were prepared for the other 23 relationships. An example of developing an appropriate pattern is given in Section 4.3. Finally, the resulting hourly pattern is adjusted so that the summation of the hourly values equals the daily estimate. 4. Model application to Chelsea zone 4.1. Data sets Chelsea is a single ¯ow monitored zone categorised as a predominantly residential area within the Melbourne water supply system. Hourly water consumption data from 25 November 1989 to 28

November 1995 were available for this study. These data were automatically measured by ¯ow metering. The use of automated measurement techniques can provide more consistent and continuous data on which to base re®ned analyses. However, corruption of telemetered data can occur which may be caused by data transmission error or equipment failure. A ®rst requirement in the data analysis was to determine speci®c quanti®able features of hourly consumption patterns for the area. Hourly consumption patterns derived from the data were displayed as boxplots. After the analysis of the data series by coarse screening, gaps or missing data were ®lled using the following criteria: 1. The average of the last and the next valid data is used up to ®ve hours between 0:00 and 5:00. 2. The average of last and next record is used up to two hours between 10:00 and 12:00. 3. For missing single hours the average of records each side is used from 5:00 to 7:00, 12:00 to 18:00, and 20:00 to 0:00 next day. 4. Gaps are not ®lled from 8:00 to 10:00, and 18:00 to 20:00. There are 86.8% of data (1905 days) available for the period after in®lling of missing data using the above criteria. The data in Ml/d were converted to per capita water consumption (l/p/d) using population data for the Chelsea zone obtained from the Victorian Department of Infrastructure, Australia. Concurrent data sets of maximum temperature, daily precipitation and Class A pan evaporation recorded at Melbourne Regional Of®ce (Station #86071), Australian Bureau of Meteorology were used as climatic variables. 4.2. Daily model calibration 4.2.1. Model ef®ciency criteria Methods commonly used to compare estimated and measured water consumption can be classi®ed into two groups: graphical plots and statistical parameters including description of the characteristics of the time series and dimensionless coef®cients. The choice of appropriate graphical plots is important as they can enhance or discredit the model simulations. The statistical parameters of describing time series, e.g. mean

S.L. Zhou et al. / Journal of Hydrology 259 (2002) 189±202

195

Fig. 3. Hourly base consumption.

and standard deviation, can be used to compare the estimated consumption with the measured consumption. Dimensionless coef®cients are very useful indicator in assessing model adequacy. The classical measure of ®t in regression analysis is the coef®cient of multiple determination, R 2, which is estimated as P ^ 2 …C 2 C† R2 ˆ 1 2 P  2; …C 2 C†

…9†

P ^ 2 is the sum-of-squares error (SSE) where …C 2 C† ^ predicbetween theP measured (C) and estimated …C† 2  tands, and …C 2 C† is the sum-of-squares total  R 2 falls exclusively (SST) of C about its mean …C†: in the range 0±1, with the former being analogous to using the predictand mean as the best predictor of C

and the latter being a perfect ®t to each observation of C. 4.2.2. Calibration results Based on Melbourne Water's recommendation, the values of hourly base consumption for weekdays and weekends of each month were calculated for the Chelsea zone, assuming that hourly base use is the ten percentile value of hourly total water use. The results plotted for summer months of December, January and February in Fig. 3 show that, for this zone, the average diurnal patterns of base consumption for weekdays are signi®cantly different from those for weekends. This observation holds true for the other nine months of the year. The average daily base consumption by month was derived by aggregating the hourly base consumption as shown graphically in Fig. 4. No long-term

Fig. 4. Daily base consumption.

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S.L. Zhou et al. / Journal of Hydrology 259 (2002) 189±202

Fig. 5. Seasonal cycle of water consumption.

trend was evident in the base consumption data. Time series of daily seasonal consumption was obtained by subtracting the base consumption de®ned in Fig. 4 from the daily total consumption. A regular seasonal cycle is evident in the Chelsea zone as shown in Fig. 5, with higher water use normally occurring in summer months and lower water use normally occurring in winter months. It should be noted that to emphasise the variability of summer water consumption, the time in the ®gure is rearranged from 1 July in a year to 30 June of next year. The seasonal cycle can be represented as a Fourier series (Eq. (2)); the coef®cients ak and bk are summarised in Table 1, where the time unit is Julian days from 1 January to 31 December in a year. In this application, it is suf®cient to consider only the ®rst two harmonics, which are statistically signi®cant. The variations in seasonal water use arise from the collective action of many people responding to heat and wetness conditions in their decisions regarding Table 1 Seasonal component expressed as Fourier coef®cients for the Chelsea zone Number of harmonics k

Coef®cients ak

bk

Variance accounted by the kth harmonics ck (%)

1 2

75.09 19.55

18.12 12.72

69.19 6.31

Coef®cient a0 (the mean) ˆ 84.8

watering their lawns and gardens. The use of mathematical equations to describe the collective response is an approximation of a very complex process that involves a lot of individual decisions. Therefore the variables used in describing this process can be regarded only as indicator or explanatory variables. In modelling the seasonal variation it is assumed that there is a linear relationship between water consumption and concurrent maximum temperature and the two previous day's maximum temperature. A regular pattern of seasonal variation, which is interrupted by the occurrence of precipitation, is evident particularly over the summer. Each time there is rainfall, the water consumption is reduced immediately then it gradually resumes its regular seasonal pattern. This effect is modelled by precipitation variables, API and ND. The pan evaporation rate is also considered in this investigation. The climatic component is of a multiple linear regression form: C^ Cj ˆ c0 1 c1 MT 0j21 1 c2 MT 0j22 1 c3 Pj21 1c4 APIj21 1 c5 NDj21 1 c6 Ej21 ;

…10†

where c0 ; ¼; c6 are the regression coef®cients, MT 0 j21 and MT 0 j22 the seasonal residuals of maximum temperature, which is the value in 8C of the difference between actual and seasonal average daily values, on day j 2 1 and j 2 2, respectively, Pj21 the precipitation in mm measured on day j 2 1, APIj21 represents the value on day j 2 1, NDj21 the number of days since a rainfall of at least 0.2 mm on day j 2 1, and

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197

Table 2 Coef®cients in Eq. (12) for climate dependent consumption of Model #3 Independent variables

Coef®cients

Standard error

t Stat

P-value

Intercept MT 0 j MT 0 j21 MT 0 j22 Pj APIj NDj Ej21

7.40 8.61 22.22 21.16 21.57 22.05 5.11 4.82

2.85 0.35 0.40 0.34 0.23 0.11 0.46 0.70

2.60 24.93 25.58 23.44 26.92 218.25 11.02 6.87

0.009 0.000 0.000 0.001 0.000 0.000 0.000 0.000

R2 ˆ 52%; Standard error ˆ 54

Ej21 is the pan evaporation rate in mm on day j 2 1. The overall ef®ciency, R 2, of this model known as Model #1 which includes the base, seasonal cycle, climatic and persistence components is 75% with a standard error of 54 l/p/d. To account for the immediate effects of air temperature on water consumption, the climatic component is modi®ed as follows (and is known as Model #2): C^ Cj ˆ c0 1 c1 MT 0j 1 c2 MT 0j21 1 c3 MT 0j22 1 c4 Pj21 1 c5 APIj21 1 c6 NDj21 1 c7 Ej21 ;

(11)

where MT 0 j is the seasonal residual of maximum temperature concurrently with daily water consumption on day j. Other variables are the same as those in Eq. (10). The overall ef®ciency R 2 of Model #2 is 82% with standard error 46 l/p/d. It indicates signi®cant improvement over Model #1; nevertheless, daily maximum temperature must be predicted to forecast daily consumption in operating Model #2. To further account for the immediate effects of wetness on water consumption, a third model (Model #3) was developed that includes the following climatic component: C^ Cj ˆ c0 1 c1 MT 0j 1 c2 MT 0j21 1 c3 MT 0j22 1 c4 Pj 1 c5 APIj 1 c6 NDj 1 c7 Ej21 ;

(12)

where Pj and NDj are the values concurrently with daily water consumption on day j. MT 0 j, MT 0 j21 and MT 0 j22 are the same as those in Eq. (11). In Model #3, the lag effect of evaporation on consumption was

considered by APIj and Ej21. Table 2 contains the regression coef®cients which are consistent for MT 0 j, Pj, APIj, NDj, and Ej21 : However, the negative coef®cients for MT 0 j21 and MT 0 j22 are assumed to be due to the correlation among the explanatory variables. The overall ef®ciency R 2 of Model #3 is 84% with standard error 43 l/p/d. It indicates improvement over Model #2, and is adopted herein. However, in operating Model #3 both daily maximum temperature and precipitation must be ®rst predicted to forecast daily consumption. To account for the dependence of water use on its own past values, the following autoregressive equation was ®tted to the residual time series in the three models: C^ Pj ˆ f0 1 f1 C 0j21 1 f2 C 0j22 1 f3 C 0j27 :

…13†

The presence of the f 3 term indicates an additive signi®cant day of the week cyclic effect. The persistence component was determined by individual regression after the climatic consumption was removed from the consumption series. No consideration was given to component interactions, although it is accepted that they are present. For model #3, the overall ef®ciency R 2 is 84% which is comprised of base and seasonal cycle 50%, climate component 26%, and persistence 8%. 4.2.3. Consumption errors analysis In calibrating water demand models, there exist four sources of uncertainty: (1) the natural randomness, (2) the model error due to the form of the model, (3) parameters estimated for the model, and (4) errors

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S.L. Zhou et al. / Journal of Hydrology 259 (2002) 189±202

Fig. 6. Histograms of relative errors of daily consumption modelling: (a) all months; (b) months in December, January and February.

in model output (dependent variable) and inputs (independent variables). In modelling past water demand, the residuals incorporate the ®rst three sources of uncertainty as measured from lack of ®t. The dependent and independent variables are assumed to have been measured without errors and therefore have no uncertainty. Relative errors of consumption estimates for Model #3 are shown in Fig. 6. Fig. 6(a) is the histogram for total months, and indicates that about 68% of model predictions fall within the relative error band ^13% and over 95% within the range ^26%. Fig. 6(b) is the histogram for summer months in December, January and February showing a slightly larger error bound. 4.2.4. Cross-validation There are many possible ways to validate

numerical models of the form used here. The most common approach is to withhold a portion of the data from the calibration exercise and use it to test the validity of model estimates. The withheld data typically comprise a contiguous record, which is traditionally called the veri®cation period. However, it is perfectly acceptable to withhold randomly selected observations as well. In order to evaluate the performance of the adopted model (Model #3), two split-data tests were carried out: 1. The model was calibrated using the data from years 1 (25 November 1989±31 December 1990), 3 (1 January 1992±31 December 1992), 5 (1 January 1994±31 December 1994), and veri®ed using the data from years 2 (1 January 1991±31 December

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cast daily rainfall classi®cations were used to specify input of appropriate values in the model:

Table 3 Model test results Test case

1 2

199

Calibration

Veri®cation

R 2 (%)

SE (l/p/d)

R 2 (%)

SE (l/p/d)

85 83

45 41

82 85

43 45

1991), 4 (1 January 1993±31 December 1993) and 6 (1 January 1995±28 November 1995); 2. The model was calibrated using the data from years 2, 4 and 6 and veri®ed using the data from years 1, 3, and 5. P In the present work, the initial variance, …C 2  2 ; is obtained for the test periods as the sum of C† squares of deviation from the mean of the calibration period, C c : Thus, the present de®nition of R 2 represents the proportion of the sum of squares of error associated with mean of the calibrated period. The results of model calibration and veri®cation for these two tests are shown in Table 3. It indicates that the model performs consistently between the calibration and veri®cation periods. 4.2.5. Model tests Model #3 developed earlier was tested using an independent data set for the period from 1 December 1996 to 30 June 1997, using separately observed and forecasted weather information. The following fore-

Classi®cation

Values (mm)

No rain Light (drizzle) Showers Rain Heavy rain Very heavy rain

0.0 0.1±0.9 1.0±2.9 3.0±9.9 10.0±24.9 $ 25.0

In using model #3 to forecast daily water consumption, rainfall forecasts from the Australian Bureau of Meteorology were amended using the above classi®cation. The model was operated as follows: #3.1: Water consumption on the current day was forecasted using forecasted daily maximum temperature and rainfall for the current day, and measured pan evaporation and water consumption on previous days. #3.2: Water consumption on the current day was estimated using observed weather information, and water consumption on previous days. For model #3.1 (forecasting), the ef®ciency R 2 is 83% with a standard error of 57 l/p/d during the test period from 1 December 1996 to 30 June 1997. For model #3.2 (simulation), R 2 is 88% with a standard error of 49 l/p/d. In general, both models perform satisfactorily. The comparison of measured, simulated and forecasted water consumption is shown in Fig. 7.

Fig. 7. Comparison of measured, simulated and forecasted daily water consumption for the Chelsea zone using independent test data.

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Table 4 Relationship between hourly peak consumption and hourly patterns on weekdays in January Pattern

Hourly peak value less base water pk † use (l/p/d) …Cw;m

1 2 3 4 5

, 400 400±800 800±1200 1200±1600 . 1600

4.3. Hourly consumption For the purpose of forecasting hourly water consumption 24 h ahead, the daily modi®ed consumption (de®ned as total consumption less base consumption) estimated by the daily module is disaggregated into hourly values. Thus, to do this we need to determine speci®c diurnal pro®les of water consumption. In general, the consumption patterns vary with peak hourly consumption, and are distinguished between days of week for each month. For this study, the diurnal patterns derived for the Chelsea zone vary with peak hourly consumption, and were separately considered for weekdays and weekends in each month. In selecting the patterns, the ®rst step was to determine the number of consumption patterns. For example, for January weekdays ®ve patterns were chosen as shown in Table 4 and for each a typical

pattern was adopted as shown in Fig. 8. As the characteristics of water use are different from month to month, the number of patterns varied for weekdays and weekends in each month. A linear regression between hourly peak modi®ed consumption and daily modi®ed consumption for weekdays or weekend in each month was developed to estimate peak consumption and hence hourly pattern. Twenty-four regressions between peak and daily modi®ed consumption were derived from the historical data. As an example, Fig. 9 shows graphically the relationship between peak and daily modi®ed consumption on weekdays in January. In operation, the model integrates base consumption and modi®ed water use on an hourly time interval. The base consumption is determined according to the day of the week and of the month. Daily modi®ed consumption is estimated according to weather conditions and other components of the daily module, and then disaggregated into hourly values using the patterns derived before. As an example of the performance of this part of the model, comparison between the measured and forecasted hourly water consumption in December 1996 is shown in Fig. 10, where Model #3.1 was used to estimate the daily consumption. Over the complete test period from 1 December 1996 to 30 June 1997, the model accounted for 66% of the variance in the peak hourly water consumption with a standard error of 162 l/p/d.

Fig. 8. Patterns of hourly water consumption on weekdays in January.

S.L. Zhou et al. / Journal of Hydrology 259 (2002) 189±202

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Fig. 9. Relationship between hourly peak consumption and daily modi®ed consumption on weekdays in January.

5. Conclusions The demand simulation model developed in this study comprises an hourly and a daily module. Total water consumption is split into two components: base (weather-insensitive) and seasonal use (weather-sensitive). For the Chelsea pressure zone, average base water consumption in each month was determined as 10% of the hourly water usage averaged for weekdays and weekends separately. Daily seasonal water consumption was modelled using a time series approach, which included the seasonal cycle, a climatic component and a persistence component. The adopted daily model depended climatically on daily maximum temperature, daily precipitation, antecedent precipitation index, the number of days since a rainfall of at least 0.2 mm, and

pan evaporation. Hourly water consumptions are produced by adding base hourly consumption to disaggregated modi®ed daily consumption. The former is derived from hourly patterns of base consumption and the latter from the predicted daily modi®ed value. For the independent test period model #3.1 accounted for 83% of the variance in daily water consumption with a standard error of 57 l/p/d and 66% of the variance in the peak hourly water consumption with a standard error of 162 l/p/d. Acknowledgements K. Gan, N. Dzadey and B.G. Rhodes of Melbourne Water Corporation were involved at an early stage of

Fig. 10. Comparison between measured and forecasted hourly consumption in December 1996 using independent test data.

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