AN ANALYSIS OF MONTHLY WATER DEMAND
IN PHOENIX, ARIZONA
Pawel Jerzy Mizgalewicz, M.Sc.,
THESIS (Selected Parts) Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Degree of Master of Public Affairs and Master of Science in Engineering
THE UNIVERSITY OF TEXAS AT AUSTIN , December, 1991
Supervisor: David R. Maidment, David J. Eaton
Partial support and cooperation: William Mee and Jeff DeWitt, TheWater and Wastewater Department in Phoenix, Arizona.
Cooperation: Ben Dziegielewski of Southern Illinois University, Carbondale, Illinois
In 1990 the Water and Wastewater Department and the Water Rates Advisory Committee of the City of Phoenix, Arizona, developed a new water rate structure simpler than the one previously in use. This new water rate structure encourages water conservation. Preliminary calculations done by city staff confirmed the advantages of the new billing system (City of Phoenix, 1990). However, a specific water forecast model is required for a more precise estimation of the influence of the new rate structure on water consumption and on revenue collection. Such a model will allow ttesting of different rate structures and different conditions affecting water use and therefore revenue (water price, conservation programs, weather conditions).
The water forecasting system is very powerful decision support tool. The precise estimation of the future water consumption is essential for determining the water management policy including the efficient water use and for the water purchase planning. The revenue projections are necessary for budget preparation. The knowledge of the future water production is indispensable for utility planning and management.
The scope of this work is determined by the local requirements of the Water and Wastewater Department in Phoenix, Arizona.
Taking into consideration specific environment of the City of Phoenix, Arizona, the forecasting model distinguishes between three water user categories: two residential categories--single-family and multi-family--and a single non-residential user category. The non-residential category includes: commercial, industrial and institutional consumers. Some of the factors supporting such differentiation between the water consumers are variations in water consumption per account, seasonal variability, distribution of meter sizes, sensitivity to the weather conditions, price elasticity, and response to the conservation programs.
The model also takes into account the spatial division of the City service area which results from different rights to the water stored and developed by the Salt River Project (more information about Salt River Project is presented in Section 1.3).
Some factors influencing water consumption such as: weather conditions, water conservation programs, or water price changes, are not utilized in the present model as they demand more detailed study of additional data which are unavailable at the moment. Such a study was performed at Southern Illinois University in parallel with this study.
Based on the water use forecast and the prediction of the number of accounts the model presented here estimates monthly revenues generated by the volume and base charges in accordance with the water rate structure. Since some of the customers do not pay their bills, the bad debt is also incorporated into revenue projections.
The Water and Wastewater Department also needs the predictions of the revenue from the "Water Resource Acquisition Fees" and the "Development and Occupational Fees" which are designed to cover some of the costs of providing necessary water resources to a new development.
One of the applications of the model is in the process of budget preparations for next fiscal year and corrections of the budget within the current fiscal year. Therefore, the model must have a procedure which updates the fiscal year predictions, utilizing the values observed in elapsed months of the current fiscal year.
The Phoenix Department of Water and Wastewater estimates future water use by the application of a forecasting model named IWR-MAIN (Institute for Water Resources - Municipal And Industrial Needs). The IWR-MAIN model predicts the water consumption in 5 or 10 year time steps taking into consideration the factors which determine water demand, for instance: population, number of users, market value and type of housing units in the residential sector, employment in various service and manufacturing industries, water and wastewater rates, weather conditions, personal income, and conservation programs (City of Phoenix, 1989, Dziegielewski and Boland 1989). The present study is intended to supplement IWR-MAIN by making a more detailed forecast on monthly time steps within the next 5-10 years.
The essence of the model is based on the methodology of water use projections published in the professional journals, however, two new procedures have been developed in this research. One of them is a method of updating the projections of annual water use, the other is a method for the estimation of the unaccounted-for-water from time-shifted data of production and consumption.
1.3 OUTLINE OF THE THESIS
The following Section 1.4 introduces the spatial division of the water and wastewater service area. Section 1.5 describes the features of the water rate structure recently proposed by the Water and Wastewater Department. Section 2 reviews selected literature on the forecasting of water use. Sections 3.1 to 3.7 are devoted to a detailed mathematical description of the model components. Sections 4.1 to 4.6 give the real estimation of the parameters of the model described mathematically in Section 3. Section 5 presents summary, general conclusions and recomendations for further study
1.4 SPATIAL DIVISION OF THE SERVICE AREA
The Salt River Project (SRP) is an organization originating from the Salt River Valley Water Users' Association (SRVWUA) formed in 1903. In 1904 the members of the Association negotiated a contract promising to repay the federal government the cost of building Roosevelt Dam. They gave their lands as security for a loan to finance the dam construction (Luckingham, 1989). Although the debt was paid back 40 years ago, the owners of the lands within the boundaries of the original SRP are still entitled to the water from the dams and reservoirs built at the turn of the century. These regions are called the "on-project" water service area.
There are some sectors within on-project area which are referred to as "non-member" service area. The owners of this territories declined the original SRVWUA membership for two reasons: their property had little agricultural value or it received enough water from the Salt and Verde Rivers (City of Phoenix 1989). In 1910, the Kent Decree confirmed a preestablished system of water rights based on the principle of prior appropriation--the SRVWUA (and later the SRP) was to supervise the distribution of project water (Luckingham 1989).
The City of Phoenix provides water to all users within the City who are not served by SRP, and it also serves the non-member areas within the SRP jurisdiction. In this thesis, "off-project" means any area not supplied with SRP water. This term labels the regions which lie outside the original SRP water rights boundaries. The members of both non-member and off-project service areas are not entitled to the water of the SRP. Therefore, these two spatial categories are grouped together and considered as a single off-project category.
1.5 BILLING SYSTEM IN PHOENIX, ARIZONA
Since the model developed in this work answers the requirements of the newly introduced water rate structure, it is pertinent to describe this water billing system. The current Section makes this presentation, based on (City of Phoenix, 1990). The old rate structure was a complex system: each consumer group had different price of water and the charges during the summer season were related to the amount of water consumed during winter months.
The new water rate structure assumes the same price of water for all the consumers: single-family users, multi-family users and nonresidantial consumers. It defines a certain monthly amount of water called a Lifeline--10 ccf (ccf is one hundred cubic feet of water) for summer months and 6 ccf for the rest of the year. The cost of the water consumed within the Lifeline is included into the base charges. This provides an inexpensive, stable monthly water bill for low water use consumers. The amount of water above the Lifeline is sold at a constant unit price.
The new water rate structure introduces different charges per unit of water for three seasons, which are the same for all the consumer groups. The lowest price of water is in "Low" months, December to March. The highest water rates are in "High" months, June to September. The remining months of the year: April, May, October, and November have "Medium" rates.
There are additional fees which are assessed for new meters in service area. The Water Resource Acquisition Fee is designated to assist in paying capital costs of new projects. It is different for the three areas: on-project, off project South of Jomax Road and off-project North of Jomax Road. The fee varies by meter size for single-family and non-residential categories within each area. For multi-family dwellings or apartments the fee depends on the number of the units in the structure.
The development and occupation fees are designated to offset the cost of new development. They are composed of the water fee and the sewer fee, and it is constant for the whole service area. Each unit from the single-family category and each unit from the multifamily category have a specified fee ($/unit) which do not depend on the meter size. The non-residential fee varies with the meter size.
2 LITERATURE REVIEW
The models of water use which are discussed in professional journals are based on the cross-sectional data, time-series data or pooled cross-sectional and time-series data. The cross-sectional regression models do not really examine water use patterns over time, and therefore their application for forecasting purposes is questionable. Cross-sectional-time-series models are complex models. They are usually focused on the modelling of the relationship between water use and variables which are able to explain the variations in water consumption. For example, average and marginal price of water, personal or household income, evapotranspiration, rainfall depth (Morgan and Smolen, 1976, Agthe and Billings, 1980, Carver and Boland, 1980), temperature, lot value, household size (Danielson, 1979) are variables which are customarily considered as primary determinants of water use. Some authors attempt to explain variations in water use by such factors as: daylight hours (Hansen and Narayanan, 1981), number of weekend days in month, or average elevation of the census tract (Cassuto and Ryan, 1979). Some studies have more than 15 variables which are regarded as influencial factors on water consumption (Billing, 1987).
Time-series models seem to be more applicable for the complex Phoenix forecasting system. The general time-series model was introduced in 1972 by Salas-LaCruz and Yevjevich. They separated three components of the water use: trend (linear or quadratic function of time), seasonality (approximated by Fourier series), and stochastic part (modeled by a autoregressive stochastic model). In modeling the trend and periodicity, they used both the monthly means and the standard deviations. They also studied cross-correlation functions of water use, air temperature, and precipitation. They found that trend accounts for 10-69 percent of the variance in water use, whereas seasonality accounts for 24-79 percent of the variance.
Oh and Yamauchi (1974) estimated the trend and the cyclical component of monthly water consumption in Honolulu, Hawaii, by the application of the twelve-month centered moving average. They expressed the seasonal and irregular components of analyzed time series by the ratio of the observed water use to the moving average. The seasonal component was separated from the irregular part by the averaging ratios for corresponding months of the year. Oh and Yamauchi attempted to explain some of the seasonal variations by the rainfall.
Maidment and Parzen (1984a) give a comprehensive description of modeling water consumption by the "cascade model". The model is composed of four components:
- long-term trend and cyclical changes resulting from city development, increasing service area, and changes in socioeconomic factors.
- seasonal variations resulting from the natural variations in climatic factors and related human behavior including commercial/industrial activity.
- irregular components which are resulting from the past events (autocorrelation).
- irregular components related to the factors affecting water use e.g. weather conditions.
This methodology was applied to model the water use in six Texas cities (Maidment and Parzen, 1984b). The regression of observed annual water consumption against population, number of water connections, household income, and water price determined the trend in mean annual water use. Household income and the water price were found not significant for all six cities. Since the population and the number of connections are highly correlated, in each city only one of them were found significant. The seasonal component was estimated by fitting a Fourier series to the monthly means of the detrended time series. After having removed the trend and the seasonal components from the observed water consumption, the residuals were filtered by the auto-regression method. Such "prewhitened" data were regressed against the detrended, deseasonalized, and autoregressive filtered time series of precipitation, evaporation, and maximum air temperature. The cascade models explained 80-87% variance of the water use. Each city exhibited different distribution of the percentage of the variation in water consumption explained by particular model component: trend explained 0-63% of variation, seasonality 11-76%, autocorrelation 2-23%, and climate correlation 1-11%. On average (for six Texas cities) the trend and seasonality accounted for 70% of total variability, the auto-correlation and climate correlation 15%, and residual error 15%.
Franklin and Maidment (1986) applied the cascade modelling to the weekly and monthly municipal water use in Deerfield Beech, Florida. They used a polynomial function of population to model the trend. To increase the applicability of the model for risk analysis, they added a rainfall component, so that the distribution of the possible future water consumption could be determined based on the distribution of historical rainfall. They found that the inclusion of the auto-correlation term considerably improves the accuracy of the forecasts made on the weekly basis but it does not improve the precision of the monthly projections.
The cascade modelling methodology is very flexible. It allows one to increase the precision of the model by relatively simple model extension (addition of required components). Therefore the model can be developed by the "step by step" method--a very convenient procedure for the development of the complex water forecasting system for Phoenix, Arizona. The base of this system should be constructed from the regression lines since the regression is considered in literature as a superior to the autoregressive-moving average methods and to the exponential smoothing method for fitting actual consumption and forecasting future consumption (Weber, 1989).
The wide application of the Phoenix forecasting system, the required precision of predictions, and the character of the available data requires development of new procedures, which constitute the author's contribution to water demand modelling. These methods are:
- application of Bayesian methods for updating predicted annual water consumption, revenue and water production from new data.
- disaggregated modelling of water consumption, water production, and city revenue divided by two areas, three user types, and seven meter diameter categories.
- estimation of the unaccounted for water from two shifted in time data sets: the production data and the consumption data.
5 SUMMARY, CONCLUSIONS, AND RECOMENDATIONS FOR FURTHER STUDY.
In this thesis a model of water consumption, water production and the revenue generated from water delivery for the City of Phoenix, Arizona is developed. The traditional modelling of water use called "cascade modelling" has been applied. Two new procedures have been developed in this research: one of them, based on Bayesian theory, has been used in the process of the updating predictions after new data are available. The second procedure, based on Fourier series, enables estimation of the unaccounted for water when the production observations are not compatible. The production readings are time shifted in relation to the consumption measurements.
The model considers the water use in two planning areas: on-project and off-project. Three types of users are distinguished for each planning area: single-family, multi-family, and nonresidential. Each planning area/type of user category is further subdivided into seven meter size classes: 5/8", 1", 1&1/2", 2", 3", 4", and 6".
Six values represent the average-month water consumption per account. The water use per account for two categories: on-project/single-family category and off-project/multi-family category showed small but significant trend. Since the factors causing these trends are not clear at this stage of analysis, a constant values of monthly water use per account [in ccf/acct/month] have been assumed as follows: for on-project planning area and for single-family consumers 17.92, for multi-family consumers 86.20, and for non-residential consumers 108.40. For off-project service area these values are slightly higher and they are respectively: 19.67, 128.29, and 152.41 [ccf/acct/month].
The seasonal pattern of the water consumption for the residential consumers is similar for the two planning areas. The single-family monthly water use varies from 153% of average month in July (maximum) to 64% of average month in December (minimum). The highest multi-family consumption is in July--128% while the lowest multi-family consumption is in February--79% of an average month. For the non-residential category the seasonal variation of water use in off-project area is significantly higher than the one for the on-project area: For this category the water use in on-project area varies from 74% (February) to to 130% (July) whereas in off-project area it varies from 57% (February) to 140% (July).
The growth of the city is represented by the six regression lines in numbers of water accounts, each for a given type of user/planning area category (a month is assumed as a time unit). The number of accounts in Phoenix, Arizona was increasing but at a declining rate during the study period (July, 1986 to December, 1990). For the on-project/single family category the growth rate estimated in the fiscal year 1986/87 was 120 accounts per month, whereas for the same category the growth rate in 1989 and 1990 years was 27 accounts per month. The respective growth rates for the multi-family category were 8 and -12 accounts per month (where a negative number indicates decrease in number of accounts). For the non-residential category, the respective growth rates were 18 and 7 accounts per month. The off-project service area showed even higher decreases in growth rates (For a single-family user category, the growth rate changed from 503 accounts per month in fiscal year 1986/87 to average 240 accounts per month in calendar years 1989-1990. For the multi-family user category, the growth rate changed from 19 account per account in fiscal year 1986/87 to about 3 accounts per month in 1989-90. For the non-residential category, the growth rate decreased from 35 accounts per month in fiscal year 1986/87 to 19 accounts per month in years 1989-1990). The regression lines estimated from the 1989-1990 data are assumed as most reliable for predicting water use in the immediate future. Yet the analysis of the growth rates in different periods showed considerable uncertainty about predictions of number of accounts, especially those which are made for 5-10 year horizon.
For each planning area/type of user category the proportions of monthly water consumption below the two lifelines--6 ccf and 10 ccf--were determined in order to estimate the revenue generated by the volume charges. The analysis showed that there is a linear relationship between the proportions of water used below the lifeline 10 ccf and the proportions of water used below the lifeline 6 ccf. This feature can be used to derive additional sets of proportions from the base one for different lifelines.
To estimate the revenue generated from the base charges, the number of accounts in each meter size category must be modelled. This is achieved by the multiplication of estimated number of accounts by meter size fractions, where the meter size fractions are equal to the number of accounts falling into a given meter size category divided by the total number of accounts. For all categories (planning-area/user/meter size), the meter size fractions have shown trends. Therefore the monthly fractions used in estimating revenue have been calculated from most recent observations available, from July, 1990 to December,1990. This approach precisely estimates the number of accounts falling into categories of small meter size: 5/8", 1", 1&1/2", and 2", whereas the estimates of the number of accounts falling into categories of large meters: 3", 4", and 6" are questionable by this approach. Since the accounts with 5/8" - 2" meters constitute more than 99.5% of the total number of accounts (5/8"-90%, 1"-4.6%, 1&1/2"-2%, and 2"-3%), it has been concluded that the errors in modelling trends in accounts with 3", 4", and 6" meters have no significant influence on the projected revenue produced from base charges
To predict water production, the relationship between total consumption and production for the city as a whole for data from 1982 and 1985-88 has been analyzed. It has been demonstrated that on average in these years, recorded consumption lags recorded production by about 16 days and that unaccounted for water averages 9.2% of production annually, varying in a smooth fashion from a high of 15% in the summer months to a low of 3% in the winter months. Using these figures, projections of monthly total production can be computed from projections of monthly total consumption.
Since the parameters of the water forecasting model have been determined from the five year data set, the maximum reasonable time horizon for the projections of the water use, water production, and the revenues generated by the volume and base charges is about five years.
The conclusions reached as a result of the presented in this thesis research are presented below.
The research shows that the disaggregated cascade model introduced here is applicable for forecasting water consumption, water production, and revenue associated with water delivery system.
Subdivision of the total service area into different subareas and into separate water consumer groups allows detailed modeling of the dynamics and patterns of city development. Traditional methods of demand forecasting are only able to model the growth of city as a unit. The analysis of the Phoenix water delivery system shows that although the subdivision of water consumers into three user categories (single-family residential, multi-family residential and nonresidential) is sufficient for water forecasting purposes, it might be useful to recognize more than two service subareas. Detailed modelling of water consumption in specified city regions is important for the water distribution network and water production facilities planning and maintenance. Further study of the proper degree of city disaggregation needs to be performed.
In the case of Phoenix, the growth in the total water consumption is related exclusively to the growth of the number of accounts. The consumption per account does not show significant trendsin time over the years 1986 to 1990. Seasonal patterns in water consumption are practically independent of the number of accounts.
Whereas revenue generated by volume charges depends on the precision of water use estimation, the revenue generated by base charges depends on the precise modelling of the number of accounts with meters of a given size. Since the model is more precise at this level of detail, dissagregation of the water system into meter size categories allows better estimation of the number of accounts and therefore a better forecast of revenue from base charges.
Although the subdivision of the water delivery system into spatial and consumer type categories advances the process of modelling water consumption, water production, and related revenues, further improvements may be introduced by the application of models in which the correlation between water use in adjacent regions and different user types is modelled explicitly instead of each series being treated separately. This requires a further study of correlation between the water use in different city regions and different water consumers.
TABLE OF CONTENTS
1 INTRODUCTION. . . 1
1.1 PURPOSE . 1
1.2 SCOPE . . . 1
1.3 OUTLINE OF THE THESIS . . . . . . . . . . . 3
1.4 SPATIAL DIVISION OF THE SERVICE AREA . . . . . . 4
1.5 BILLING SYSTEM IN PHOENIX, ARIZONA . . . . . . . . 5
2 LITERATURE REVIEW. . . . . . . . . . . 7
3 MATHEMATICAL DESCRIPTION OF THE FORECASTING SYSTEM . . . . . 11
3.1 PROJECTING WATER USE PER ACCOUNT . . 13
3.1.1 Monthly water consumption. . . . . . . . . . 13
3.1.2 Seasonal variations. . . . . 14
3.1.3 Average deseasonalized monthly water use. . . . . . . . . . 18
3.1.4 Normal weather predictions. . . . . . . . . . 21
3.1.5 Demand adjustment factors. . . . . . . . . . 22
3.2 PROJECTING NUMBER OF ACCOUNTS. . . . . 25
3.2.1 Average monthly number of accounts. . 25
3.2.2 Meter size fractions. . . . . 26
3.3 PROJECTING TOTAL WATER CONSUMPTION. . . . . 28
3.4 CONSUMPTION ABOVE THE LIFELINE. . . . . 29
3.5 PROJECTING REVENUE. . . . . . 30
3.5.1 Revenue from volume charge. . . . . . . . . 31
3.5.2 Deviations of revenue projections from normal values . . 33
3.5.3 Revenue from base charges. . . . . . . . . . . 33
3.6 FORECASTING WATER CONSUMPTION WITHIN THE
CURRENT YEAR. 35
3.6.1 Updating projections of annual water consumption. . . 36
3.6.2 Correction of monthly revenue from volume charges. . 40
3.6.3 Update of revenue from base charge. . . . 41
3.7 PROJECTING WATER PRODUCTION. . . . . . . . 42
3.7.1 Time-shift coefficient. . . . 43
3.7.2 Time adjustment of water consumption record . . . . . . . 47
3.7.3 Estimation of water production from consumption. . . . 48
4 STATISTICAL ESTIMATION OF MODEL PARAMETERS. . . . . . . . 49
4.1. DATA AND COMPUTER SOFTWARE DESCRIPTION. 49
4.2 MODELLING WATER CONSUMPTION PER ACCOUNT. . . . . 52
4.2.1 Seasonal Indices. . . . . . . 53
4.2.2 Weather Patterns. . . . . . . 56
4.2.3 Normalization of Core Models. . . 58
4.2.4 Demand adjustment factors. . . . . . . . . . 64
4.3 FORECASTING NUMBER OF ACCOUNTS. . . . 67
4.3.1 Distribution of number of accounts. . . . . 67
4.3.2 Analysis of meter size fractions. . . . . . . 70
4.3.3 Trends in number of accounts by meter size category . . 76
4.3.4 Comparison of selected models. . 78
4.4 LIFELINE PROPORTIONS OF WATER CONSUMPTION. . . . . 82
4.5 REVENUE UPDATE WITHIN CURRENT FISCAL YEAR. . . . 86
4.6 UNACCOUNTED FOR WATER AND WATER PRODUCTION. . 92
4.6.1 Data availability. . . . . . . . 92
4.6.1 Determination of time-shift coefficient. . . . . . . . 92
4.6.2 Unaccounted for water as a function of month of the year. . . . . 96
4.6.4 Forecasting water production . . . . . . . . 100
5 SUMMARY, CONCLUSIONS, AND RECOMENDATIONS FOR
FURTHER STUDY. . . 102
APPENDIX 1: SELECTED DATA APPLIED IN THE ANALYSIS. . . . . 107
APPENDIX 2: ESTIMATION OF MODEL PARAMETERS. . . . 110
APPENDIX 3: ANALYSIS OF WEATHER CONDITIONS. . . . . 145
LIST OF TABLES
CHAPTER 4: ESTIMATION OF THE MODEL PARAMETERS
Tab. 4.1 Seasonal indices applied in modelling seasonal variations of water consumption per account.
Tab. 4.2 Parameters of the normalized deseasonalized water consumption per account (Eq. 3.6).
Tab. 4.3 Distribution of the number of accounts falling into particular meter size classes within spatial area/type of use categories--average values for the calendar year 1990.
Tab. 4.4 Growth rates of the number of accounts by planning area/user category [#acct/month].
Tab. 4.5 Comparison of predicted number of accounts for December 1995 by application both: individual meter size regression models, and core account models and meter size fractions.
Tab. 4.6 Results of the regression analysis of the linear relationship between the proportions specific for the Lifeline 6 ccf and the proportions specific for the Lifeline 10 ccf (Eq. 4.1).
Tab. 4.7 Estimated values of the coefficient ß(k).
Tab. 4.8 The Fourier series coefficients of the monthly proportions of the unaccounted for water.
APPENDIX 1: SELECTED DATA APPLIED IN THE ANALYSIS
Tab. A1.1 Monthly water production in Phoenix, Arizona from the report City of Phoenix Arizona, Water and Wastewater Department, Water Production and Consumption in Millions of Gallons (for various fiscal years).
Tab. A1.2 Monthly water supply in Phoenix, Arizona from the report City of Phoenix Arizona, Water and Wastewater Department, Water Production and Consumption in Millions of Gallons (for various fiscal years).
Tab. A1.3 Monthly water consumption in Phoenix, Arizona from the WCIS 825.1 Report.
Tab. A1.4 Inconsistencies in the data from Report WCS 865.2.
APPENDIX 2: ESTIMATION OF MODEL PARAMETERS
Tab. A2.1 Water consumption per account calculated from Report WCS 865.2.
Tab. A2.2 Analysis of the differences between sets of seasonal indices.
Tab. A2.3 Deseasonalized water use per account.
Tab. A2.4 Results of the stepwise regression analysis of the deseasonalized water consumption per account.
Tab. A2.5 Demand adjustment factors for July, 1986 to December, 1990.
Tab. A2.6 Percentage distribution of the number of accounts in planning area, type of water use, and meter size categories.
Tab. A2.7 Six-month average proportions (Jan.-Jun. and Jul.-Dec.) of accounts with given meter size to the total number of accounts by planning area/type of use category.
Tab. A2.8 Coefficients (constant and slope) of the trend lines in the number of accounts estimated by the least squares method for three time intervals.
Tab. A2.9 Comparison of the annual change in number of accounts modelled by the individual regression lines and modelled by the core regression lines and the meter size fractions for differen time intervals.
Tab. A2.10 Water consumption below the Lifeline as a proportion of the total water use.
Tab. A2.11 Regression models of water supply (analysis of the coefficient ß).
Tab. A2.12 Comparison of the water supply: observed and predicted by the regression models.
Tab. A2.13 Distribution of average monthly fractions of water supply determined for different time intervals.
Tab. A2.14 Cumulative distribution of average monthly fractions of water supply determined for different time intervals.
Tab. A2.15 Cumulative distribution of recorded water supply.
Tab. A2.16 Predictions of fiscal year water supply Wd(y,k) from the cumulative water supply (Table A2.15) divided by the cumulative monthly fractions (Table A2.14).
Tab. A2.17 Differences between observed W(y) and predicted Wd(y,k) water supply.
Tab. A2.18 Production and consumption data applied in modelling of unaccounted for water , corrected water consumption, estimated unaccounted for water, and proportions of the unaccounted for water to the corrected water consumption.
Tab. A2.19 Proportions of unaccounted for water to annual corrected fiscal year water consumption.
Tab. A2.20 Proportions of unaccouted for water to water consumption and to water production by month as estimated from the Fourier series model.
APPENDIX 3: ANALYSIS OF WEATHER CONDITIONS
Tab. A3.1 Average daily maximum temperatures.
Tab. A3.1.1 Regression analysis of the fiscal year average daily maximum temperatures.
Tab. A3.1.2 Regression analysis of the winter average daily maximum temperatures.
Tab. A3.1.3 Regression analysis of the summer average daily maximum temperatures.
Tab. A3.2 Average daily minimum temperatures.
Tab. A3.2.1 Regression analysis of the fiscal year mean daily minimum temperatures.
Tab. A3.2.2 Regression analysis of the winter average daily minimum temperatures.
Tab. A3.2.3 Regression analysis of the summer average daily minimum temperatures.
Tab. A3.3 Average daily mean temperatures.
Tab. A3.3.1 Regression analysis of the fiscal year average daily mean temperatures.
Tab. A3.3.2 Regression analysis of the winter average daily mean temperatures.
Tab. A3.3.3 Regression analysis of the summer average daily mean temperatures.
Tab. A3.4 Total precipitation depth.
Tab. A3.4.1 Regression analysis of the fiscal year total precipitation depth.
Tab. A3.4.2 Regression analysis of the winter depth of the precipitation.
Tab. A3.4.3 Regression analysis of the summer depth of the precipitation.
Tab. A3.5 Total evaporation depth.
Tab. A3.5.1 Regression analysis of the fiscal year total evaporation depth.
Tab. A3.5.2 Regression analysis of the winter evaporation depth.
Tab. A3.5.3 Regression analysis of the summer evaporation depth.
Tab. A3.6 Monthly mean maximum temperature.
Tab. A3.7 Monthly precipitation depth.
Tab. A3.8 Monthly number of days with total precipitation greater than 0.01 inch.
Tab. A3.9 Differences between observed monthly values of mean maximum temperature, total precipitation depth, and number of days with the total precipitation greater than 0.01 inch, and the respective averages determined for the study period (July, 1986 to December, 1990).
LIST OF FIGURES
CHAPTER 3: MATHEMATICAL DESCRIPTION OF THE FORECASTING SYSTEM
Fig. 3.1 Selected components of the Phoenix forecasting system.
Fig. 3.2 Categories of water demand used in the modelling process.
Fig. 3.3 Calculation of the centered twelve-month moving average.
Fig. 3.4 Example of seasonal factors (single-family user category).
Fig. 3.5 Example of the observed and the deseasonalized water use per account (Off-project/single-family category).
Fig. 3.6 Example of the demand adjustment factors (Off-project/single-family category).
Fig. 3.7 Example of the meter size fractions (Off-project/single-family category,size of meter: 1&1/2").
Fig. 3.8 Water production and consumption time series before time-shifting of water consumption (Phoenix, Arizona).
CHAPTER 4: ESTIMATION OF THE MODEL PARAMETERS
Fig. 4.1 Seasonal indices of the water consumption per account for different user categories and planning areas: (a) single-family, (b) multi-family, and (c) non-residential.
Fig. 4.2 Deseasonalized water consumption per account and normal weather average for on-project planning area and users: (a) single-family, (b) multi-family, and (c) non-residential.
Fig. 4.3 Deseasonalized water consumption per account and normal weather average for off-project planning area and users: (a) single-family, (b) multi-family, and (c) non-residential.
Fig. 4.4 Demand adjustment factors for on-project planning area and users: (a) single-family, (b) multi-family, and (c) non-residential.
Fig. 4.5 Demand adjustment factors for off-project planning area and users: (a) single-family, (b) multi-family, and (c) non-residential.
Fig. 4.6 Distribution of average number of accounts in each planning area/type of water use category recorded in 1990.
Fig. 4.7 Distribution of average number of accounts in the calendar year 1990.
Fig. 4.8 Proportions of accounts given meter size to the total number of accounts (meter size fractions) for single-family category averaged in six month intervals.
Fig. 4.9 Proportions of accounts given meter size to total number of accounts (meter size fractions) for multi-family category averaged in six month intervals (proportions of meter size categories: 5", 6", and 7" are not shown).
Fig. 4.10 Proportions of accounts given meter size to total number of accounts (meter size fractions) for on-project/non-residential category averaged in six month intervals.
Fig. 4.11 Proportions of accounts given meter size to total number of accounts (meter size fractions) for off-project/non-residential category averaged in six month intervals.
Fig. 4.12 Proportions of water consumed below the Lifeline 6 [ccf] to total water consumption for users: (a) single-family, (b) multi-family, and (c) non-residential.
Fig. 4.13 Proportions of water consumed below the Lifeline 10 [ccf] to total water consumption for users: (a) single-family, (b) multi-family, and (c) non-residential.
Fig. 4.14 Example of the relationship between the proportions for the Lifelines 6 [ccf] and 10 [ccf] (off-project/single-family category).
Fig. 4.15 Weight coefficient ß(k).
Fig. 4.16 Monthly water production and monthly water consumption in Phoenix, Arizona during fiscal years 1986, 1987, and 1988.
Fig. 4.17 Comparison of the Fourier series approximation of production proportions, water consumption proportions, and corrected water consumption proportions.
Fig. 4.18 Proportions of annual unaccounted for water to annual water use.
Fig. 4.19 The Fourier series approximation of the monthly proportions of the unaccounted for water to the corrected water consumption.
Fig. 4.20 Plot of the recorded water production and the water production estimated from the water consumption record.
APPENDIX 2: ESTIMATION OF MODEL PARAMETERS
Fig. A2.1 Number of accounts for on-project planning area and for different user/meter size categories: (a) single-family/meter size 5/8", (b) single-family/meter size 1", and (c) single-family/meter size 1 1/2".
Fig. A2.2 Number of accounts for on-project planning area and for different user/meter size categories: (a) single-family/meter size 2", (b) single-family/all accounts, and (c) multi-family/meter size 5/8".
Fig. A2.3 Number of accounts for on-project planning area and for different user/meter size categories: (a) multi-family/meter size 1", (b) multi-family/meter size 1&1/2", and (c) multi-family/meter size 2".
Fig. A2.4 Number of accounts for on-project planning area and for different user/meter size categories: (a) multi-family/all accounts, (b) non-residential/meter size 5/8", and (c) non-residential/meter size 1".
Fig. A2.5 Number of accounts for on-project planning area and for different user/meter size categories: (a) non-residential/meter size 1 1/2", (b) non-residential/meter size 2", and (c) non-residential/meter size 3".
Fig. A2.6 Number of accounts for on-project planning area and for different user/meter size categories: (a) non-residential/meter size 4", (b) non-residential/meter size 6", and (c) non-residential/all accounts.
Fig. A2.7 Number of accounts for off-project planning area and for different user/meter size categories: (a) single-family/meter size 5/8", (b) single-family/meter size 1", and (c) single-family/meter size 1 1/2".
Fig. A2.8 Number of accounts for off-project planning area and for different user/meter size categories: (a) single-family/meter size 2", (b) single-family/all accounts, and (c) multi-family/meter size 5/8".
Fig. A2.9 Number of accounts for off-project planning area and for different user/meter size categories: (a) multi-family/meter size 1", (b) multi-family/meter size 1&1/2", and (c) multi-family/meter size 2".
Fig. A2.10 Number of accounts for off-project planning area and for different user/meter size categories: (a) multi-family/all accounts, (b) non-residential/meter size 5/8", and (c) non-residential/meter size 1".
Fig. A2.11 Number of accounts for off-project planning area and for different user/meter size categories: (a) non-residential/meter size 1 1/2", (b) non-residential/meter size 2", and (c) non-residential/meter size 3".
Fig. A2.12 Number of accounts for off-project planning area and for different user/meter size categories: (a) non-residential/meter size 4", (b) non-residential/meter size 6", and (c) non-residential/all accounts.
APPENDIX 3: ANALYSIS OF WEATHER CONDITIONS
Fig. A3.1.1 Fiscal year average daily maximum temperatures.
Fig. A3.1.2 Winter average daily maximum temperatures.
Fig. A3.1.3 Summer average daily maximum temperatures.
Fig. A3.2.1 Fiscal year average daily minimum temperatures.
Fig. A3.2.2 Winter average daily minimum temperatures.
Fig. A3.2.3 Summer average daily minimum temperatures.
Fig. A3.3.1 Fiscal year average daily mean temperatures.
Fig. A3.3.2 Winter average daily mean temperatures.
Fig. A3.3.3 Summer average daily mean temperatures.
Fig. A3.4.1 Fiscal year depth of precipitation
Fig. A3.4.2 Winter depth of precipitation
Fig. A3.4.3 Summer depth of precipitation
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