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Frequency analysis of water consumption for metropolitan area of Melbourne
Journal of Hydrology 247 (2001) 72±84
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Frequency analysis of water consumption for metropolitan area of Melbourne S.L. Zhou, T.A. McMahon*, Q.J. Wang 1 Department of Civil and Environmental Engineering, Centre for Environmental Applied Hydrology, The University of Melbourne, Victoria 3010, Australia Received 8 June 1999; revised 30 January 2001; accepted 22 February 2001
Abstract In this paper, a procedure is developed to estimate the average recurrence interval of maximum daily (and 2, 3 and 5 consecutive days) water consumption for Melbourne, Australia. The method consists of three steps: calibration of a daily water demand simulation model for high consumption months, estimation of a water consumption time-series and then calculation of the average recurrence interval of the extreme events. The deterministic/probabilistic approach of deriving the frequency curves for the Melbourne area can be used to improve water supply strategies that depend on demand estimates. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Statistics; Time series; Water consumption; Frequency
1. Introduction The operation of water supply and distribution requires very frequent adjustments in response to demand variation in order to minimise costs of water delivery. Relevant demand variation will be diurnal, by day of the week, and by month and season, affected by weather and by holiday patterns as well as by the regular domestic and industrial activities of consumers. Melbourne Water Corporation is responsible for the day to day transport of high quality water from large headwater storage reservoirs to a series of junctions where retail companies distribute the water to consumers. As part of the operation of a reliable * Corresponding author. 1 Present address: Institute of Sustainable Irrigated Agriculture, Fergson Road, Tatura, Vic. 3616, Australia. Fax: 161-3-93446215. E-mail address: [email protected] (T.A. McMahon).
water supply, the frequency of occurrence of speci®c water demands for various durations needs to be known. There is a large number of research papers dealing with modelling annual or monthly urban water use. Few address daily water use. The basic approach in modelling daily urban water use is to divide total water use into base use which is weather-insensitive and seasonal use which is weather-sensitive (Maidment and Miaou, 1986; Maidment et al., 1985; Zhou et al., 2000). The weather-dependent water use is statistically related to weather information, such as temperature and rainfall. This paper presents the results of estimating the average recurrence interval of extreme water consumptions for one, two, three and ®ve consecutive days for Melbourne, Australia, a city of three million persons. The method developed consists of three steps: calibration of a water demand simulation model for high consumption months, estimation of a
0022-1694/01/$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0022-169 4(01)00357-2
S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
73
Table 1 Statistics of water consumption and weather conditions for Melbourne metropolitan area Year (November± February)
1989/90 1990/91 1991/92 1992/93 1993/94 1994/95 1995/96 1996/97 Average
Average daily consumption (l/ p/d)
602 603 532 444 454 510 457 560 520
Daily maximum temperature in 8C (no. of days) .30
.32
.34
.36
23 27 11 23 19 29 13 29 22
15 18 9 14 12 24 5 22 15
8 13 7 11 4 14 2 17 10
3 9 2 7 4 8 2 11 6
water consumption time-series and determination of the frequency of extreme consumptions.
2. Development of the water demand model 2.1. Model structure and parameters This section outlines brie¯y the model structure and parameter estimation. A more detailed discussion is given in Zhou et al. (2000). Using daily water consumption (as estimated by Melbourne Water Corporation), maximum daily temperature and daily precipitation data for seven years from 1 July 1990 to 31 March 1997, a water demand simulation model was developed for this study. The unit of water consumption is litre per person per day (l/p/d). Population data for Melbourne
Average maximum temperature (8C)
No. of rain days
Average precipitation (mm/d)
24.5 25.2 23.0 24.9 23.6 25.5 23.1 25.4 24.4
37 29 50 60 46 37 35 31 41
1.7 1.7 1.8 3.3 3.4 1.6 2.5 0.8 2.1
metropolitan area were obtained from the Victorian Department of Infrastructure, Australia. As the focus of this study is the magnitude of high consumption, after the base usage had been removed from the total consumption, the weather-dependent consumption for the four summer months in the period of November to February was estimated. As noted below, the data for 1989/90 period was initially included but was later excluded because the data revealed characteristics dissimilar to that displayed by the following seven years, as shown in Table 1 and Fig. 1. Table 1 shows that the same amount of water was consumed in the summer months of 1989/90 with lower maximum temperature and more rain days as that consumed in the summer months in 1990/91, and more water was consumed in the summer months of 1989/90 (with lower maximum temperature and more rainfall) than that consumed in the summer months in
Fig. 1. Daily water consumption for Melbourne metropolitan area.
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S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
Fig. 2. Scatterplot of total consumption less base consumption against daily maximum temperature for the summer periods of November± February.
1994/95 and 1996/97. Advice from Melbourne Water Corporation suggested this anomalous usage was partly due to changes in the Corporation's boundaries and in water pricing and the data should not be used in the analysis. Monthly base water consumption can be identi®ed from the lowest monthly water use (Maidment et al., 1985; Zhou et al., 2000). In this study, using daily per capita data, base consumption was removed by applying a third order polynomial to the winter months (June and July) from 1 July 1989 to 31 March 1997. This function accounts for 81% of the variance of the winter-month consumption. This allowed the weather-insensitive water consumption to be separated from the weather-sensitive consumption. The daily equivalent of the polynomial function of time is expressed as
consumption (C C) was based on data after removal of the base consumption (Eq. (1)). Fig. 2 shows a scatterplot between the climate dependent consumption and daily maximum temperature in the summer months from November to February with different symbols for each water year in the period of 1 July 1989 to 31 March 1997. The scatterplot shows that the data for 1989/90 period are dissimilar to that displayed by the following seven years. As noted earlier, the data for 1989/90 period were excluded during further calibration of the model. Next, the effect of climate on consumption in summer between November and February was identi®ed using multiple regression where the independent variables were daily maximum temperatures of the current day and the following day, and antecedent precipitation index (API). The API is calculated by
C^ Bt 392:61 2 1:57 £ 1022 t 2 8:76 £ 1026 t2 1 3:62
APIj k´APIj21 1 Pj ;
£ 1029 t3 ; 1 where t 1; 2; ¼ and t 1 corresponds to 1 July 1989, the ®rst daily observation. The magnitude of base consumption varies from 392 l/p/d on 1 July 1989 to 356 l/p/d on 15 July 1996 (Fig. 1). High water consumptions are caused by a combination of extreme conditions. Peak demands occur in summer months, usually associated with hot, dry periods. Weather-sensitive or climate dependent
2
where Pj is the precipitation in mm on day j. Following a trial and error analysis, the constant k was set equal to 0.85, adopted as being representative of the previous day's rainfall (Bruce and Clark, 1966). A variety of relationships between the climate dependent consumption and climate variables was investigated in this study. Table 2 lists the correlation coef®cients between total water consumption less base consumption and maximum temperatures and API. Fig. 3 suggests a non-linear relationship between water consumption less base consumption and
S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84 Table 2 Correlation coef®cients between total consumption less base consumption and climate variables for Melbourne metropolitan area (data used from November to February in the period of 1 July 1990±31 March 1997)
CjC MTj11 MTj APIj
CjC
MTj11
MTj
1.00 0.46 0.58 2 0.61
1.00 0.48 2 0.21
1.00 2 0.20
APIj
1.00
maximum temperature. There appears to be a structural change in the relationship at about the point of maximum temperature MT 398C. The two straight lines in Fig. 3 can be represented by C^ Cj c0 1 c1 MTj ;
MT # 39
and
C^ Cj c0 2 c2 £ 39 1 c1 1 c2 MTj ;
MT . 39; 3
where MTj is the maximum temperature in 8C on day
75
j. In this case, the intercepts and slopes of the two lines are different, subject to the condition that the two lines intersect at MT 398C: Eq. (3) can be written as C^ Cj c0 1 c1 MTj 1 c2 MTpj with a new variable ( 0; p MT MT 2 39;
4
MT # 39; MT . 39:
The term MTpj introduced in Eq. (4) gives additional weight to the extreme high temperature days. The least squares estimators for the parameters c0, c1 and c2 in Eq. (4) are given in Table 3. The ®tted lines are shown in Fig. 3. The scatterplot in Fig. 4 suggests a linear relationship between today's consumption residuals (after removing the effect of MTj using Eq. (4)) and tomorrow's temperature MTj11. By introducing the variable to the regression function, the equation becomes C^ Cj c0 1 c1 MTj 1 c2 MTpj 1 MTj11 :
5
Fig. 3. Scatterplot of total consumption less base consumption against daily maximum temperature for the summer periods of November± February from 1 July 1990 to 31 March 1997. Table 3 Regression coef®cients of climate dependent consumption in Eq. (4) Independent variables
Coef®cients
Standard error
t Statistics
P-value
Intercept MTj MTpj
2158.45 12.38 69.01
13.67 0.55 15.53
211.6 22.4 4.4
0.000 0.000 0.000
R2 44:3% Standard error 88:3
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S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
Fig. 4. Scatterplot of consumption residuals against tomorrow's maximum temperature.
The least squares estimators for the parameters c0, c1, c2 and c3 in Eq. (5) are given in Table 4. The R 2 increased from 44.3 to 51.5% compared with Eq. (4). After removing the estimated consumption using Eq. (5), several transformations were applied which showed that a non-linear relationship between the consumption residuals and the cubic root of API
was the most appropriate (Fig. 5). It was found that the relationship between water consumption and API remains almost constant when cubic root of API is greater than 3; in other words, beyond a certain level water consumption is not sensitive to wetness. From Fig. 5, it would appear that a broken-line function of API transformation Z API1=3 with a break point at Z 3 provides a satisfactory ®t to the data.
Table 4 Regression coef®cients of climate dependent consumption in Eq. (5). Independent variables
Coef®cients
Standard error
t Statistics
P-value
Intercept MTj MTpj MTj11
2231.18 9.08 80.24 6.27
14.34 0.59 14.54 0.56
216.1 15.3 5.5 11.1
0.000 0.000 0.000 0.000
R2 51:5% Standard error 82:4
Fig. 5. Scatterplot of consumption residuals against cubic root of API.
S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
77
Table 5 Regression coef®cients of climate dependent consumption in Eq. (6)
Intercept MTj MTpj MTj11 Zj Zjp
Coef®cients
Standard error
t Statistics
P-value
75.08 7.58 61.37 4.25 2103.41 93.60
13.36 0.39 9.59 0.38 3.48 11.85
5.6 19.2 6.4 11.3 229.7 7.9
0.000 0.000 0.000 0.000 0.000 0.000
R2 79:1% Standard error 54:2
Therefore, the ®nal equation is written as C^ Cj c0 1 c1 MTj 1 c2 MTpj 1 c3 MTj11 1 c4 Zj 1 c5 Zjp ;
6
where MTj, MTj11 is the maximum temperature in 8C on day j, j 1 1, respectively. MT p is de®ned in Eq. (4), and the term Z p is de®ned as ( 0; Z # 3; p Z 7 Z 2 3; Z . 3: The least squares estimators for the parameters c0, c1, c2, c3, c4 and c5 in Eq. (6) are given in Table 5. The R 2 increased from 51.5 to 79.1% compared with Eq. (5). All ®ve terms in Eq. (6) are highly signi®cant as indicated by the P-values in Table 5. Note that c4 and c5 are opposite in sign and have only a small difference in absolute value. This simulates the very small dependence of water consumption on the increases of API for API 1/3 . 3 as mentioned above. The scatterplot of the consumption error (calculated
as total daily consumption less base consumption and less the climate dependent consumption) against the climate dependent consumption in Fig. 6 shows heteroscedasticity where the scatter of points is larger at larger values of simulated climate dependent consumption, Cà C. In this case, the ordinary least squares estimators are still unbiased but no longer gives minimum variance. More importantly, the estimates of the standard errors for the coef®cients may be understated, and hence test statistics may be too large. The ®nal step in the calibration was to determine a model for the error component. The consumption errors were divided into sub-classes based on simuà C. The standlated climate dependent consumption, C ard deviations of the consumption errors from zero for each of the sub-classes, as shown in Fig. 7, are calculated by v u X u1 n SE t Cer2i ; n i1
Fig. 6. Consumption errors against simulated climate dependent consumption.
8
78
S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
Fig. 7. Standard deviations of consumption errors against simulated climate dependent consumption.
where Ceri is the consumption error on day i, and n is the number of days for each of the sub-classes. Based on these standard deviations an error model was formulated which is shown in Fig. 7 and can be approximately described by the following equation: 8 > 40; C^ C , 250; > < s er C^ C 0:060C^ C 1 43; 250 # C^ C # 450 ; 9 > > : 70; C^ C . 450: A ®nal assumption was made that the consumption errors are normally distributed and independent, Ã C) is de®ned by Eq. Cer , N 0; s er2 C^ C where s er(C (9). Fig. 8 shows a normal plot for the standardised consumption errors (Cer/s er), where the plotting position formula is de®ned as P Cer=s er m 2 3=8= N 1 1=4 (Blom, 1958; Cunnane, 1978) and m is the rank of error value (Cer/s er) and N is the total number of error values. It is noticed that there are deviations from the normal line for large positive
standardised errors. The normality of the standardised errors is assessed by using the Shapiro±Wilk statistic giving P-value equal to 0.08 (Shapiro and Wilk, 1965). The result shows that errors are not signi®cantly different from normal at 5% level of signi®cance. The normality of the error series may be partially weakened by the approximation of Eq. (9) to calculated standard deviations of the consumption errors, particularly in the range of simulated climate dependent consumption from 150 to 350 l/p/ d as depicted in Fig. 7. Considering that the climate model accounts for 79.1% of the total variance, the use of a normal distribution for the consumption error is considered adequate. 2.2. Model validation The model developed above was tested using an independent data set during the summer periods from 1 November 1997 to 31 January 2000. The data series of daily maximum temperature observed
Fig. 8. Plot of standardised consumption errors on Normal probability scale.
S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
79
Fig. 9. Comparison of measured and modelled water consumption series for the summer periods of November±February from 1 November 1997 to 31 January 2000.
and API calculated from daily rainfall were used as model (Eq. (6)) inputs. The ef®ciency R 2 is 76.0% for the test period compared with 79.1% for model calibration. Fig. 9 shows the measured and modelled water consumption series. It should be noted that the unit of measured water consumption is converted from megalitres per day, available from Melbourne Water Corporation, to litre per person per day by a constant population of 2,818,000, which is the same ®gure used on 31 January 1997. The modelled water consumption is calculated from the model output plus a constant base consumption of 356 l/p/d, adopted as at 15 July 1996. From the above independent test, the model was considered satisfactory. 2.3. Adopted model The model ®nally adopted to generate simulated daily water consumptions, knowing daily maximum temperatures and daily rainfall, contains the summation of two components Ð a deterministic part
and a random part. For a given day, the deterministic component, Eq. (6), is calculated and then a normally distributed random variate with a mean of zero and a variance equal to s er2 (Eq. (9)) is added. The above methodology was checked in Fig. 10 by comparing the simulated water consumption using the above model with the 842 estimated consumptions (November±February, July 1990±March 1997) after the base consumptions were removed. Excellent agreement is achieved. This con®rms the adequacy of the climate and error models.
3. Simulation of water consumption series Daily maximum temperature and daily rainfall data at Melbourne Regional Of®ce (Station No. 86071) have been recorded since 1855. Therefore, 142 years of consumption data could be generated using the demand simulation model as de®ned in Section 2.2.
Fig. 10. Comparison of total consumption less base consumption against simulated climate dependent plus random error consumption.
80
S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
Fig. 11. Double mass curve of annual rainfall of Melbourne Regional Of®ce (86071) against Yan Yean for the period from 1877 to 1995.
3.1. Homogeneity analysis of rainfall
3.2. Annual maximum temperature series
To check the homogeneity of the historical record for Melbourne, we carried out a double mass analysis. Fig. 11 shows the double mass curve of annual rainfall of Melbourne Regional Of®ce against Yan Yean which according to Lavery et al. (1992) is homogenous. Therefore, from Fig. 11 it is reasonable to assume that the Melbourne Regional Of®ce data are essentially homogeneous, although there are 16 years of missing data for Yan Yean; these were not included in the ®gure. Hence, for the purpose of the frequency analysis of consumption data, we assume that the rainfall data recorded at the Melbourne Regional Of®ce can be directly used for simulating water consumption without data correction.
Table 6 shows the annual maximum temperature series (selected from November of previous year to February of current year) in the period of 1855/56± 1996/97, and Fig. 12 shows the decadal average values of maximum temperature. It was concluded that the temperature data do not exhibit any longterm non-stationary trends. 3.3. Simulation of water consumption series The synthetic consumption series were obtained sequentially as follows: ² Using the observed daily temperature and rainfall
Table 6 Annual maximum temperature (8C) series recorded at Melbourne Regional Of®ce (86071), for data selected from November of previous year to February of current year Year
0
1
2
3
4
5
6
7
8
9
1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990
43.9 42.7 41.3 39.7 41.6 40.2 40.5 39.7 39.8 38.7 40.6 38.2 40.4 36.6
36.9 36.7 39.9 39.4 43.1 35.9 41.9 37.1 39.8 40.1 41.7 38.7 41.8 41.6
44.0 41.1 43.6 38.4 39.9 41.3 40.3 42.7 40.9 43.1 40.0 37.2 43.3 39.4
41.7 39.1 40.5 39.5 40.5 40.7 41.1 37.3 41.9 37.3 39.8 40.5 43.2 41.3
35.9 38.4 38.1 39.1 35.9 41.1 40.1 39.1 40.2 42.1 39.6 36.3 38.7 41.0
39.7 43.5 38.7 40.9 42.5 39.9 40.1 39.8 38.4 40.6 41.6 39.6 42.2 40.2
36.9 42.3 39.3 40.1 42.2 43.1 40.1 40.0 40.9 40.9 38.3 39.3 40.6 38.3 39.2
39.5 42.4 43.7 40.5 39.0 39.9 37.9 40.9 37.3 40.0 39.4 40.7 40.3 40.0 41.2
42.9 40.3 37.3 40.0 42.9 44.2 38.3 40.0 39.4 40.2 38.7 43.7 37.9 40.4
40.0 43.3 41.1 38.9 43.0 38.6 41.4 38.3 45.6 37.7 42.8 38.7 41.3 38.8
S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
81
Fig. 12. Decadal average values of annual maximum temperature.
data for the months of November±February from 1855 to 1997 calculate the weather dependent component (Eq. (6)). ² Generate the consumption error series using a Normal distribution with the standard deviations of error as de®ned in Eq. (9), and add to the weather dependent consumption. ² Finally, the daily consumption series for the four summer months is obtained by adding the weatherdependent and stochastic generated components to the constant base consumption (adopted as at 15 July 1996, 356 l/p/d). In order to estimate the error in the frequency diagram, which is discussed in the next section, 1000 replicates each of length 142 years of four months of daily consumptions were synthetised as set out above.
4. Consumption frequency analysis To estimate the average recurrence interval (years) a partial series plot was developed from the simulated data. The method to do this for each 142 years sequence was as follows: ² Scan the 142 years of four months of daily consumption and identify the largest 142 values (that is, the same as the length of the sequence years). In order to ensure the independence of the largest values, each value selected was separated by at least four days from other chosen values. Four days were adopted following inspection of a plot of daily consumptions and weather information which showed that adoption of a four day separation would ensure independence of the data incorporated in the partial series.
Fig. 13. Comparison of maximum consumption derived, respectively, from 142 and 7 years of partial data series using standard deviations of consumption errors de®ned as Eq. (9) (calibration: November±February, 1/7/1990±31/3/1997, simulation: 1855±1997, base consumption adopted as at 15 July 1996, 356 l/p/d).
82
S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
Fig. 14. Comparison of three-day maximum consumption from partial series of different replicates.
² Arrange the 142 consumptions in descending order of magnitude. ² Calculate the average recurrence interval T (years) for each consumption, C, according to the plotting position formula (Cunnane, 1978) T C
N 1 0:2 ; m 2 0:4
10
where m is the rank of C and N is the number of years in the time series (142). ² Plot C versus average recurrence interval using the N values similar to Fig. 13. ² The individual points shown in Fig. 13 are the median values of the 1000 replicates of 142 years of simulated data with the 5 and 95% con®dence band based on the same data.
Fig. 13 shows the comparison of maximum consumption with con®dence limits for Melbourne metropolitan area derived from 142 and 7 years of partial data series, respectively. It is unrealistic to assume that a variable is measured without error. The presence of measurement errors in water consumption, maximum temperature and precipitation would have an impact on the results of the model calibration. The con®dence bands are based on the error term in the model. In reality they will be wider because of ¯uctuations in temperature and precipitation. (In view of the time constraints in this project, these were not estimated.) The symbol £ in Fig. 13 represents the seven largest measured values, each value selected was separated by at least four days from other chosen values, between November and February from 1 November,
Fig. 15. Comparison of maximum consumption (median values) from partial and annual series.
S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
83
Fig. 16. Maximum water consumption from partial series.
1990 to 28 February, 1997. Analysis showed that there is no signi®cant difference using 2 or 4 consecutive days for dependence checks when selecting the partial series for three-day maximum consumption. Fig. 14 shows that the three-day maximum consumption derived from 10,000 replicates has the same pattern as that derived from 1,000 replicates. Furthermore, the results for 1 and 3 consecutive days derived from annual series show similar patterns at long recurrence intervals, as depicted in Fig. 15. Fig. 16 shows for the Melbourne metropolitan area the average recurrence interval in years with con®dence limits for 1, 2, 3 and 5 consecutive days consumption. It is noticed that there are kinks for long recurrence intervals for 3 and 5 consecutive days consumption. They result from sampling, because the climate dependent consumptions were simulated using a ®xed 142 years series (length of the historical data) and because smoothed curves through the data points are used rather than computing a theoretical curve to ®t the data. A similar feature is exhibited in Fig. 13 for the one-day maximum consumption derived from seven years of partial data series. 5. Conclusions The frequency curves with con®dence limits for 1, 2, 3 and 5 consecutive days consumption for Melbourne were derived by a deterministic/probabilistic approach.
This approach includes the calibration of a water demand simulation model for high consumption months in seven years, estimation of water consumption series for a 142 years period, and frequency analysis of the consumption series. The frequency curves can be used to improve water supply strategies that depend on demand estimates. It should be noted that weather variables are essential in explaining differences in water use from year to year. The daily consumption series was obtained by assuming the base consumption of 356 l/ p/d (adopted as at 15 July 1996). Acknowledgements The advice and assistance of A. Walton, Melbourne Water Corporation in carrying out the research are gratefully acknowledged. The daily water consumption data were provided by Melbourne Water Corporation and the 142 years of daily maximum temperature and rainfall data were obtained from the Bureau of Meteorology, Australia. The research was supported by funds provided by Melbourne Water Corporation. References Blom, G., 1958. Statistical Estimates and Transformed Beta Variables. Wiley, New York, NY. Bruce, J.P., Clark, R.H., 1966. Introduction to Hydrometeorology. Pergamon Press, Oxford, UK, p. 319. Cunnane, C., 1978. Unbiased plotting positions Ð a review. Journal of Hydrology 37, 205±222. Lavery, B., Kariko, A., Nicholls, N., 1992. A historical rainfall data set for Australia. Australian Meterological Magazine 40, 33±39.
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Maidment, D.R., Miaou, S.P., 1986. Daily water use in nine cities. Water Resources Research 22 (6), 845±851. Maidment, D.R., Miaou, S.P., Crawford, M.M., 1985. Transfer function models of daily urban water use. Water Resources Research 21 (4), 425±432.
Shapiro, S.S., Wilk, M.B., 1965. An analysis of variance test for normality (complete samples). Biometrika 52 (3±4), 591±611. Zhou, S.L., McMahon, T.A., Walton, A., Lewis, J., 2000. Forecasting daily urban water demand: a case study of Melbourne. Journal of Hydrology 236, 153±164.
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Frequency analysis of water consumption for metropolitan area of Melbourne S.L. Zhou, T.A. McMahon*, Q.J. Wang 1 Department of Civil and Environmental Engineering, Centre for Environmental Applied Hydrology, The University of Melbourne, Victoria 3010, Australia Received 8 June 1999; revised 30 January 2001; accepted 22 February 2001
Abstract In this paper, a procedure is developed to estimate the average recurrence interval of maximum daily (and 2, 3 and 5 consecutive days) water consumption for Melbourne, Australia. The method consists of three steps: calibration of a daily water demand simulation model for high consumption months, estimation of a water consumption time-series and then calculation of the average recurrence interval of the extreme events. The deterministic/probabilistic approach of deriving the frequency curves for the Melbourne area can be used to improve water supply strategies that depend on demand estimates. q 2001 Elsevier Science B.V. All rights reserved. Keywords: Statistics; Time series; Water consumption; Frequency
1. Introduction The operation of water supply and distribution requires very frequent adjustments in response to demand variation in order to minimise costs of water delivery. Relevant demand variation will be diurnal, by day of the week, and by month and season, affected by weather and by holiday patterns as well as by the regular domestic and industrial activities of consumers. Melbourne Water Corporation is responsible for the day to day transport of high quality water from large headwater storage reservoirs to a series of junctions where retail companies distribute the water to consumers. As part of the operation of a reliable * Corresponding author. 1 Present address: Institute of Sustainable Irrigated Agriculture, Fergson Road, Tatura, Vic. 3616, Australia. Fax: 161-3-93446215. E-mail address: [email protected] (T.A. McMahon).
water supply, the frequency of occurrence of speci®c water demands for various durations needs to be known. There is a large number of research papers dealing with modelling annual or monthly urban water use. Few address daily water use. The basic approach in modelling daily urban water use is to divide total water use into base use which is weather-insensitive and seasonal use which is weather-sensitive (Maidment and Miaou, 1986; Maidment et al., 1985; Zhou et al., 2000). The weather-dependent water use is statistically related to weather information, such as temperature and rainfall. This paper presents the results of estimating the average recurrence interval of extreme water consumptions for one, two, three and ®ve consecutive days for Melbourne, Australia, a city of three million persons. The method developed consists of three steps: calibration of a water demand simulation model for high consumption months, estimation of a
0022-1694/01/$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S 0022-169 4(01)00357-2
S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
73
Table 1 Statistics of water consumption and weather conditions for Melbourne metropolitan area Year (November± February)
1989/90 1990/91 1991/92 1992/93 1993/94 1994/95 1995/96 1996/97 Average
Average daily consumption (l/ p/d)
602 603 532 444 454 510 457 560 520
Daily maximum temperature in 8C (no. of days) .30
.32
.34
.36
23 27 11 23 19 29 13 29 22
15 18 9 14 12 24 5 22 15
8 13 7 11 4 14 2 17 10
3 9 2 7 4 8 2 11 6
water consumption time-series and determination of the frequency of extreme consumptions.
2. Development of the water demand model 2.1. Model structure and parameters This section outlines brie¯y the model structure and parameter estimation. A more detailed discussion is given in Zhou et al. (2000). Using daily water consumption (as estimated by Melbourne Water Corporation), maximum daily temperature and daily precipitation data for seven years from 1 July 1990 to 31 March 1997, a water demand simulation model was developed for this study. The unit of water consumption is litre per person per day (l/p/d). Population data for Melbourne
Average maximum temperature (8C)
No. of rain days
Average precipitation (mm/d)
24.5 25.2 23.0 24.9 23.6 25.5 23.1 25.4 24.4
37 29 50 60 46 37 35 31 41
1.7 1.7 1.8 3.3 3.4 1.6 2.5 0.8 2.1
metropolitan area were obtained from the Victorian Department of Infrastructure, Australia. As the focus of this study is the magnitude of high consumption, after the base usage had been removed from the total consumption, the weather-dependent consumption for the four summer months in the period of November to February was estimated. As noted below, the data for 1989/90 period was initially included but was later excluded because the data revealed characteristics dissimilar to that displayed by the following seven years, as shown in Table 1 and Fig. 1. Table 1 shows that the same amount of water was consumed in the summer months of 1989/90 with lower maximum temperature and more rain days as that consumed in the summer months in 1990/91, and more water was consumed in the summer months of 1989/90 (with lower maximum temperature and more rainfall) than that consumed in the summer months in
Fig. 1. Daily water consumption for Melbourne metropolitan area.
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S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
Fig. 2. Scatterplot of total consumption less base consumption against daily maximum temperature for the summer periods of November± February.
1994/95 and 1996/97. Advice from Melbourne Water Corporation suggested this anomalous usage was partly due to changes in the Corporation's boundaries and in water pricing and the data should not be used in the analysis. Monthly base water consumption can be identi®ed from the lowest monthly water use (Maidment et al., 1985; Zhou et al., 2000). In this study, using daily per capita data, base consumption was removed by applying a third order polynomial to the winter months (June and July) from 1 July 1989 to 31 March 1997. This function accounts for 81% of the variance of the winter-month consumption. This allowed the weather-insensitive water consumption to be separated from the weather-sensitive consumption. The daily equivalent of the polynomial function of time is expressed as
consumption (C C) was based on data after removal of the base consumption (Eq. (1)). Fig. 2 shows a scatterplot between the climate dependent consumption and daily maximum temperature in the summer months from November to February with different symbols for each water year in the period of 1 July 1989 to 31 March 1997. The scatterplot shows that the data for 1989/90 period are dissimilar to that displayed by the following seven years. As noted earlier, the data for 1989/90 period were excluded during further calibration of the model. Next, the effect of climate on consumption in summer between November and February was identi®ed using multiple regression where the independent variables were daily maximum temperatures of the current day and the following day, and antecedent precipitation index (API). The API is calculated by
C^ Bt 392:61 2 1:57 £ 1022 t 2 8:76 £ 1026 t2 1 3:62
APIj k´APIj21 1 Pj ;
£ 1029 t3 ; 1 where t 1; 2; ¼ and t 1 corresponds to 1 July 1989, the ®rst daily observation. The magnitude of base consumption varies from 392 l/p/d on 1 July 1989 to 356 l/p/d on 15 July 1996 (Fig. 1). High water consumptions are caused by a combination of extreme conditions. Peak demands occur in summer months, usually associated with hot, dry periods. Weather-sensitive or climate dependent
2
where Pj is the precipitation in mm on day j. Following a trial and error analysis, the constant k was set equal to 0.85, adopted as being representative of the previous day's rainfall (Bruce and Clark, 1966). A variety of relationships between the climate dependent consumption and climate variables was investigated in this study. Table 2 lists the correlation coef®cients between total water consumption less base consumption and maximum temperatures and API. Fig. 3 suggests a non-linear relationship between water consumption less base consumption and
S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84 Table 2 Correlation coef®cients between total consumption less base consumption and climate variables for Melbourne metropolitan area (data used from November to February in the period of 1 July 1990±31 March 1997)
CjC MTj11 MTj APIj
CjC
MTj11
MTj
1.00 0.46 0.58 2 0.61
1.00 0.48 2 0.21
1.00 2 0.20
APIj
1.00
maximum temperature. There appears to be a structural change in the relationship at about the point of maximum temperature MT 398C. The two straight lines in Fig. 3 can be represented by C^ Cj c0 1 c1 MTj ;
MT # 39
and
C^ Cj c0 2 c2 £ 39 1 c1 1 c2 MTj ;
MT . 39; 3
where MTj is the maximum temperature in 8C on day
75
j. In this case, the intercepts and slopes of the two lines are different, subject to the condition that the two lines intersect at MT 398C: Eq. (3) can be written as C^ Cj c0 1 c1 MTj 1 c2 MTpj with a new variable ( 0; p MT MT 2 39;
4
MT # 39; MT . 39:
The term MTpj introduced in Eq. (4) gives additional weight to the extreme high temperature days. The least squares estimators for the parameters c0, c1 and c2 in Eq. (4) are given in Table 3. The ®tted lines are shown in Fig. 3. The scatterplot in Fig. 4 suggests a linear relationship between today's consumption residuals (after removing the effect of MTj using Eq. (4)) and tomorrow's temperature MTj11. By introducing the variable to the regression function, the equation becomes C^ Cj c0 1 c1 MTj 1 c2 MTpj 1 MTj11 :
5
Fig. 3. Scatterplot of total consumption less base consumption against daily maximum temperature for the summer periods of November± February from 1 July 1990 to 31 March 1997. Table 3 Regression coef®cients of climate dependent consumption in Eq. (4) Independent variables
Coef®cients
Standard error
t Statistics
P-value
Intercept MTj MTpj
2158.45 12.38 69.01
13.67 0.55 15.53
211.6 22.4 4.4
0.000 0.000 0.000
R2 44:3% Standard error 88:3
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S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
Fig. 4. Scatterplot of consumption residuals against tomorrow's maximum temperature.
The least squares estimators for the parameters c0, c1, c2 and c3 in Eq. (5) are given in Table 4. The R 2 increased from 44.3 to 51.5% compared with Eq. (4). After removing the estimated consumption using Eq. (5), several transformations were applied which showed that a non-linear relationship between the consumption residuals and the cubic root of API
was the most appropriate (Fig. 5). It was found that the relationship between water consumption and API remains almost constant when cubic root of API is greater than 3; in other words, beyond a certain level water consumption is not sensitive to wetness. From Fig. 5, it would appear that a broken-line function of API transformation Z API1=3 with a break point at Z 3 provides a satisfactory ®t to the data.
Table 4 Regression coef®cients of climate dependent consumption in Eq. (5). Independent variables
Coef®cients
Standard error
t Statistics
P-value
Intercept MTj MTpj MTj11
2231.18 9.08 80.24 6.27
14.34 0.59 14.54 0.56
216.1 15.3 5.5 11.1
0.000 0.000 0.000 0.000
R2 51:5% Standard error 82:4
Fig. 5. Scatterplot of consumption residuals against cubic root of API.
S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
77
Table 5 Regression coef®cients of climate dependent consumption in Eq. (6)
Intercept MTj MTpj MTj11 Zj Zjp
Coef®cients
Standard error
t Statistics
P-value
75.08 7.58 61.37 4.25 2103.41 93.60
13.36 0.39 9.59 0.38 3.48 11.85
5.6 19.2 6.4 11.3 229.7 7.9
0.000 0.000 0.000 0.000 0.000 0.000
R2 79:1% Standard error 54:2
Therefore, the ®nal equation is written as C^ Cj c0 1 c1 MTj 1 c2 MTpj 1 c3 MTj11 1 c4 Zj 1 c5 Zjp ;
6
where MTj, MTj11 is the maximum temperature in 8C on day j, j 1 1, respectively. MT p is de®ned in Eq. (4), and the term Z p is de®ned as ( 0; Z # 3; p Z 7 Z 2 3; Z . 3: The least squares estimators for the parameters c0, c1, c2, c3, c4 and c5 in Eq. (6) are given in Table 5. The R 2 increased from 51.5 to 79.1% compared with Eq. (5). All ®ve terms in Eq. (6) are highly signi®cant as indicated by the P-values in Table 5. Note that c4 and c5 are opposite in sign and have only a small difference in absolute value. This simulates the very small dependence of water consumption on the increases of API for API 1/3 . 3 as mentioned above. The scatterplot of the consumption error (calculated
as total daily consumption less base consumption and less the climate dependent consumption) against the climate dependent consumption in Fig. 6 shows heteroscedasticity where the scatter of points is larger at larger values of simulated climate dependent consumption, Cà C. In this case, the ordinary least squares estimators are still unbiased but no longer gives minimum variance. More importantly, the estimates of the standard errors for the coef®cients may be understated, and hence test statistics may be too large. The ®nal step in the calibration was to determine a model for the error component. The consumption errors were divided into sub-classes based on simuà C. The standlated climate dependent consumption, C ard deviations of the consumption errors from zero for each of the sub-classes, as shown in Fig. 7, are calculated by v u X u1 n SE t Cer2i ; n i1
Fig. 6. Consumption errors against simulated climate dependent consumption.
8
78
S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
Fig. 7. Standard deviations of consumption errors against simulated climate dependent consumption.
where Ceri is the consumption error on day i, and n is the number of days for each of the sub-classes. Based on these standard deviations an error model was formulated which is shown in Fig. 7 and can be approximately described by the following equation: 8 > 40; C^ C , 250; > < s er C^ C 0:060C^ C 1 43; 250 # C^ C # 450 ; 9 > > : 70; C^ C . 450: A ®nal assumption was made that the consumption errors are normally distributed and independent, Ã C) is de®ned by Eq. Cer , N 0; s er2 C^ C where s er(C (9). Fig. 8 shows a normal plot for the standardised consumption errors (Cer/s er), where the plotting position formula is de®ned as P Cer=s er m 2 3=8= N 1 1=4 (Blom, 1958; Cunnane, 1978) and m is the rank of error value (Cer/s er) and N is the total number of error values. It is noticed that there are deviations from the normal line for large positive
standardised errors. The normality of the standardised errors is assessed by using the Shapiro±Wilk statistic giving P-value equal to 0.08 (Shapiro and Wilk, 1965). The result shows that errors are not signi®cantly different from normal at 5% level of signi®cance. The normality of the error series may be partially weakened by the approximation of Eq. (9) to calculated standard deviations of the consumption errors, particularly in the range of simulated climate dependent consumption from 150 to 350 l/p/ d as depicted in Fig. 7. Considering that the climate model accounts for 79.1% of the total variance, the use of a normal distribution for the consumption error is considered adequate. 2.2. Model validation The model developed above was tested using an independent data set during the summer periods from 1 November 1997 to 31 January 2000. The data series of daily maximum temperature observed
Fig. 8. Plot of standardised consumption errors on Normal probability scale.
S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
79
Fig. 9. Comparison of measured and modelled water consumption series for the summer periods of November±February from 1 November 1997 to 31 January 2000.
and API calculated from daily rainfall were used as model (Eq. (6)) inputs. The ef®ciency R 2 is 76.0% for the test period compared with 79.1% for model calibration. Fig. 9 shows the measured and modelled water consumption series. It should be noted that the unit of measured water consumption is converted from megalitres per day, available from Melbourne Water Corporation, to litre per person per day by a constant population of 2,818,000, which is the same ®gure used on 31 January 1997. The modelled water consumption is calculated from the model output plus a constant base consumption of 356 l/p/d, adopted as at 15 July 1996. From the above independent test, the model was considered satisfactory. 2.3. Adopted model The model ®nally adopted to generate simulated daily water consumptions, knowing daily maximum temperatures and daily rainfall, contains the summation of two components Ð a deterministic part
and a random part. For a given day, the deterministic component, Eq. (6), is calculated and then a normally distributed random variate with a mean of zero and a variance equal to s er2 (Eq. (9)) is added. The above methodology was checked in Fig. 10 by comparing the simulated water consumption using the above model with the 842 estimated consumptions (November±February, July 1990±March 1997) after the base consumptions were removed. Excellent agreement is achieved. This con®rms the adequacy of the climate and error models.
3. Simulation of water consumption series Daily maximum temperature and daily rainfall data at Melbourne Regional Of®ce (Station No. 86071) have been recorded since 1855. Therefore, 142 years of consumption data could be generated using the demand simulation model as de®ned in Section 2.2.
Fig. 10. Comparison of total consumption less base consumption against simulated climate dependent plus random error consumption.
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Fig. 11. Double mass curve of annual rainfall of Melbourne Regional Of®ce (86071) against Yan Yean for the period from 1877 to 1995.
3.1. Homogeneity analysis of rainfall
3.2. Annual maximum temperature series
To check the homogeneity of the historical record for Melbourne, we carried out a double mass analysis. Fig. 11 shows the double mass curve of annual rainfall of Melbourne Regional Of®ce against Yan Yean which according to Lavery et al. (1992) is homogenous. Therefore, from Fig. 11 it is reasonable to assume that the Melbourne Regional Of®ce data are essentially homogeneous, although there are 16 years of missing data for Yan Yean; these were not included in the ®gure. Hence, for the purpose of the frequency analysis of consumption data, we assume that the rainfall data recorded at the Melbourne Regional Of®ce can be directly used for simulating water consumption without data correction.
Table 6 shows the annual maximum temperature series (selected from November of previous year to February of current year) in the period of 1855/56± 1996/97, and Fig. 12 shows the decadal average values of maximum temperature. It was concluded that the temperature data do not exhibit any longterm non-stationary trends. 3.3. Simulation of water consumption series The synthetic consumption series were obtained sequentially as follows: ² Using the observed daily temperature and rainfall
Table 6 Annual maximum temperature (8C) series recorded at Melbourne Regional Of®ce (86071), for data selected from November of previous year to February of current year Year
0
1
2
3
4
5
6
7
8
9
1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990
43.9 42.7 41.3 39.7 41.6 40.2 40.5 39.7 39.8 38.7 40.6 38.2 40.4 36.6
36.9 36.7 39.9 39.4 43.1 35.9 41.9 37.1 39.8 40.1 41.7 38.7 41.8 41.6
44.0 41.1 43.6 38.4 39.9 41.3 40.3 42.7 40.9 43.1 40.0 37.2 43.3 39.4
41.7 39.1 40.5 39.5 40.5 40.7 41.1 37.3 41.9 37.3 39.8 40.5 43.2 41.3
35.9 38.4 38.1 39.1 35.9 41.1 40.1 39.1 40.2 42.1 39.6 36.3 38.7 41.0
39.7 43.5 38.7 40.9 42.5 39.9 40.1 39.8 38.4 40.6 41.6 39.6 42.2 40.2
36.9 42.3 39.3 40.1 42.2 43.1 40.1 40.0 40.9 40.9 38.3 39.3 40.6 38.3 39.2
39.5 42.4 43.7 40.5 39.0 39.9 37.9 40.9 37.3 40.0 39.4 40.7 40.3 40.0 41.2
42.9 40.3 37.3 40.0 42.9 44.2 38.3 40.0 39.4 40.2 38.7 43.7 37.9 40.4
40.0 43.3 41.1 38.9 43.0 38.6 41.4 38.3 45.6 37.7 42.8 38.7 41.3 38.8
S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
81
Fig. 12. Decadal average values of annual maximum temperature.
data for the months of November±February from 1855 to 1997 calculate the weather dependent component (Eq. (6)). ² Generate the consumption error series using a Normal distribution with the standard deviations of error as de®ned in Eq. (9), and add to the weather dependent consumption. ² Finally, the daily consumption series for the four summer months is obtained by adding the weatherdependent and stochastic generated components to the constant base consumption (adopted as at 15 July 1996, 356 l/p/d). In order to estimate the error in the frequency diagram, which is discussed in the next section, 1000 replicates each of length 142 years of four months of daily consumptions were synthetised as set out above.
4. Consumption frequency analysis To estimate the average recurrence interval (years) a partial series plot was developed from the simulated data. The method to do this for each 142 years sequence was as follows: ² Scan the 142 years of four months of daily consumption and identify the largest 142 values (that is, the same as the length of the sequence years). In order to ensure the independence of the largest values, each value selected was separated by at least four days from other chosen values. Four days were adopted following inspection of a plot of daily consumptions and weather information which showed that adoption of a four day separation would ensure independence of the data incorporated in the partial series.
Fig. 13. Comparison of maximum consumption derived, respectively, from 142 and 7 years of partial data series using standard deviations of consumption errors de®ned as Eq. (9) (calibration: November±February, 1/7/1990±31/3/1997, simulation: 1855±1997, base consumption adopted as at 15 July 1996, 356 l/p/d).
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S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
Fig. 14. Comparison of three-day maximum consumption from partial series of different replicates.
² Arrange the 142 consumptions in descending order of magnitude. ² Calculate the average recurrence interval T (years) for each consumption, C, according to the plotting position formula (Cunnane, 1978) T C
N 1 0:2 ; m 2 0:4
10
where m is the rank of C and N is the number of years in the time series (142). ² Plot C versus average recurrence interval using the N values similar to Fig. 13. ² The individual points shown in Fig. 13 are the median values of the 1000 replicates of 142 years of simulated data with the 5 and 95% con®dence band based on the same data.
Fig. 13 shows the comparison of maximum consumption with con®dence limits for Melbourne metropolitan area derived from 142 and 7 years of partial data series, respectively. It is unrealistic to assume that a variable is measured without error. The presence of measurement errors in water consumption, maximum temperature and precipitation would have an impact on the results of the model calibration. The con®dence bands are based on the error term in the model. In reality they will be wider because of ¯uctuations in temperature and precipitation. (In view of the time constraints in this project, these were not estimated.) The symbol £ in Fig. 13 represents the seven largest measured values, each value selected was separated by at least four days from other chosen values, between November and February from 1 November,
Fig. 15. Comparison of maximum consumption (median values) from partial and annual series.
S.L. Zhou et al. / Journal of Hydrology 247 (2001) 72±84
83
Fig. 16. Maximum water consumption from partial series.
1990 to 28 February, 1997. Analysis showed that there is no signi®cant difference using 2 or 4 consecutive days for dependence checks when selecting the partial series for three-day maximum consumption. Fig. 14 shows that the three-day maximum consumption derived from 10,000 replicates has the same pattern as that derived from 1,000 replicates. Furthermore, the results for 1 and 3 consecutive days derived from annual series show similar patterns at long recurrence intervals, as depicted in Fig. 15. Fig. 16 shows for the Melbourne metropolitan area the average recurrence interval in years with con®dence limits for 1, 2, 3 and 5 consecutive days consumption. It is noticed that there are kinks for long recurrence intervals for 3 and 5 consecutive days consumption. They result from sampling, because the climate dependent consumptions were simulated using a ®xed 142 years series (length of the historical data) and because smoothed curves through the data points are used rather than computing a theoretical curve to ®t the data. A similar feature is exhibited in Fig. 13 for the one-day maximum consumption derived from seven years of partial data series. 5. Conclusions The frequency curves with con®dence limits for 1, 2, 3 and 5 consecutive days consumption for Melbourne were derived by a deterministic/probabilistic approach.
This approach includes the calibration of a water demand simulation model for high consumption months in seven years, estimation of water consumption series for a 142 years period, and frequency analysis of the consumption series. The frequency curves can be used to improve water supply strategies that depend on demand estimates. It should be noted that weather variables are essential in explaining differences in water use from year to year. The daily consumption series was obtained by assuming the base consumption of 356 l/ p/d (adopted as at 15 July 1996). Acknowledgements The advice and assistance of A. Walton, Melbourne Water Corporation in carrying out the research are gratefully acknowledged. The daily water consumption data were provided by Melbourne Water Corporation and the 142 years of daily maximum temperature and rainfall data were obtained from the Bureau of Meteorology, Australia. The research was supported by funds provided by Melbourne Water Corporation. References Blom, G., 1958. Statistical Estimates and Transformed Beta Variables. Wiley, New York, NY. Bruce, J.P., Clark, R.H., 1966. Introduction to Hydrometeorology. Pergamon Press, Oxford, UK, p. 319. Cunnane, C., 1978. Unbiased plotting positions Ð a review. Journal of Hydrology 37, 205±222. Lavery, B., Kariko, A., Nicholls, N., 1992. A historical rainfall data set for Australia. Australian Meterological Magazine 40, 33±39.
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Maidment, D.R., Miaou, S.P., 1986. Daily water use in nine cities. Water Resources Research 22 (6), 845±851. Maidment, D.R., Miaou, S.P., Crawford, M.M., 1985. Transfer function models of daily urban water use. Water Resources Research 21 (4), 425±432.
Shapiro, S.S., Wilk, M.B., 1965. An analysis of variance test for normality (complete samples). Biometrika 52 (3±4), 591±611. Zhou, S.L., McMahon, T.A., Walton, A., Lewis, J., 2000. Forecasting daily urban water demand: a case study of Melbourne. Journal of Hydrology 236, 153±164.