Urban Water Demand: Statistical Optimization Approach to Modeling Daily Demand

One of the primary objectives of this work was to develop a predictive water demand model that requires minimal data for calibration and validation, both for use in simple computational platforms as well as for broad applicability among municipalities. This model requires four daily time-series data sets for calibration: historical water demand, population, average daily temperature, and daily precipitation. For the model calibration in this paper, 5 years (2006–2010) of daily time-series data were used for temperature and precipitation, and an additional preceding year (2005) was required for water demand and population (i.e., 2005–2010, a total of 6 years of daily data) to calculate the 365-day trailing average for the first model calibration year (2006). The model then was validated with the following 5 years of data (2011–2015).

The model uses known meteorological and social factors to represent the forces that affect water demand trends based on historical water demand data decomposed into three constituents: long-term trends, seasonal patterns, and noise (Dokumentov and Hyndman 2015). The goal of the model was to create sets of fitting parameters that could explain these constituents to be combined to model the daily water demand. From the observed water demand, population, daily average temperature, and daily precipitation data, an equation was developed to synthesize the three constituents into mathematical terms. The daily water demand is calculated using Eq. (1), where per-capita unit consumption, $U$, and population, $P$, describe the trend; monthly, day of the week, and holiday scaling coefficients, $C_M$, $C_{\mathrm{DotW}}$, and $C_H$, respectively, describe seasonal patterns; and temperature and precipitation functions, $f_T$ and $f_P$, respectively, describe the noise due to daily meteorological conditions. The weighting factors, $W_1$ and $W_2$, attenuate the amplitude of daily fluctuationswhere $Q_{(i,m,d,c)}$ = predicted water demand ($\mathrm{L}/\mathrm{day}$) for $i$th day of model, where $m$ = month, $d$ = day of week, and $c$ = calendar day; $W_1/W_2$ = weighting factor; $U_{i-1}=365$-day average unit consumption lagged by one day ($\mathrm{L}/\mathrm{day}/\mathrm{person}$); $P$ = population for $i$th model day (persons); $C_{M(m)}$ = monthly constant for month $m$; $C_{\mathrm{DotW}(m,d)}$ = weekly constant for month $m$ and day of week $d$; $C_{H(c)}$ = holiday constant for calendar day $c$; $f_{\mathrm{Temp}}$ ($T$) = temperature factor, where $T$ = average daily temperature (°C); $f_{\mathrm{precip}}$ ($\overline{p}$) = precipitation factor, where $\overline{p}=3$-day average precipitation of $i$th day of model and two previous days (cm); and $\overline{Q}$ = average predicted water demand for previous 3 days ($\mathrm{L}/\mathrm{day}$).

The model was calibrated and validated using a common split sample approach (e.g., Bakker et al. 2014) and model performance evaluation. An optimization-based approach was implemented to improve model calibration by minimizing the RMSE of daily water demand using the generalized reduced gradient (GRG) nonlinear solver in Microsoft Excel on a 5-year calibration period (2006–2010). Subsequently, the model was validated against the next 5 years (2011–2015) of water demand data. This combination of calibration and validation periods was intended to prevent overfitting by calibrating the model with data within the calibration period and observing the accuracy of the model during the validation period.

The modeling framework proposed in this study was applied to the city of El Paso, Texas, the 6th largest city in Texas and the 22nd largest city in the US, with a population of approximately 682,000 in 2019 (US Census Bureau 2019). The city is located at the far west corner of Texas where it borders the state of New Mexico and the country of Mexico. El Paso is sustained by three primary sources of water, a river that borders the city, named the Rio Grande (or Rio Bravo in Mexico), and two aquifers that lie on either side of the Franklin Mountain Range: the Mesilla Bolson (west) and Hueco Bolson (east). Based on temperature data from 1994 to 2015 (NOAA 2020), the area is classified as a BWh dry arid desert according to the revised Köppen–Geiger system (Peel et al. 2007). The average daily temperature for this period was 19°C, and El Paso had average daily temperatures below $-12°\mathrm{C}$ and as high as 37°C. Since the mid-1900s, drought conditions, increased farming activity (USDA 2012), and national and international agreements have caused El Paso to rely heavily on the aquifers, especially during low flow in the river. Fortunately, since the 1980s, El Paso leaders have strategically advanced regional water sustainability by promoting residential water conservation, reclamation of treated wastewater effluent (purple pipe) for irrigation of municipal parks, indirect potable reuse of treated municipal wastewater through aquifer recharge from the Fred Hervey Water Reclamation Plant, and brackish groundwater desalination at the Kay Bailey Hutchison Desalination Plant. These factors have made El Paso an interesting case to study for water modeling and forecasting, because effective water management is essential to the sustainability of this region.

For the selected study area and time period, the four temperature fitting constants ($C_{T1}$, $C_{T2}$, $C_{T3}$, and $C_{T4}$) were determined to be 1.204, 0.377, 28.009°C, and 9.791°C, respectively, so that the values of the temperature factor, $f_T$, ranged from a minimum of 0.827 in cold weather to a maximum of 1.58 in impracticably hot weather [Fig. 4(a)]. At an average daily temperature of 22.1°C, the temperature factor evaluated to unity and had no effect in Eq. (1). Over the study period, the minimum temperature factor value of 0.827 was observed each year, and the greatest temperature factor value observed was 1.435. An important feature of using this particular method to describe behaviors based on temperature is that for many regions, wintertime consumption is not temperature sensitive, but certain locations are temperature sensitive (Berkes et al. 1998; Donkor et al. 2014), and this hyperbolic tangent function can accommodate both conditions. The El Paso fit in Fig. 4 had little temperature sensitivity for average daily temperatures less than 15°C.

The two precipitation fitting constants, $C_{P1}$ and $C_{P2}$, were determined to be 0.24 and $2.12\mathrm{day}/\mathrm{cm}$ respectively, such that for increasing precipitation in the previous 3 days, the values for the precipitation factor, $f_p$, range from a value of 1 at no precipitation in the previous 3 days to the minimum value asymptote of 0.75 for an average precipitation greater than $2.5\mathrm{cm}/\mathrm{day}$ in the preceding 3 days [Fig. 4(b)]. The lowest observed precipitation factor value within the model was 0.753.

The monthly $C_M$ coefficient was used to capture the general annual cycle, and varied from 0.86 in the December to 1.12 in May and June [Fig. 5(a)]. Furthermore, weekly watering restrictions in El Paso [Fig. 5(b)] affected the day of the week constant [shown in Fig. 5(c) for 4 months that represent the four seasons]. The value of $C_{\mathrm{DotW}}$ ranged from 0.854 to 1.138 and had greater amplitude during the spring and summer (greater lawn irrigation) than the winter. Day of the week values for all 12 months are listed in Table 1.

Selected values of the holiday constant, $C_H$, are listed in Table 2, and holidays with values farthest from unity include regional holidays such as Cinco de Mayo. The unconstrained average of this factor was unity, which indicates that because the holiday constant was determined after the other factors and constants, no significant bias was corrected or introduced by the $C_H$ constant.

Following the meteorological and seasonal regressions, the weighting factors, $W_1$ and $W_2$, were determined to be 0.592 and 0.408, respectively.

Fig. 6 displays known daily water demand values and predicted demand values, and model performance evaluation results are listed in Table 3. To verify the results of a regression model, one method is to check for normality of relative errors, which provided no visual indication of nonnormal distributions. Following the rule of thumb by Bulmer (1979), skewness of errors between 0.5 and $-0.5$ can be interpreted as being approximately symmetric, and an excess kurtosis between $-2$ and 2 is acceptable (George and Mallery 2009). From visual inspection of the distribution of errors, the errors were assumed to be normally distributed, and the skewness and excess kurtosis values (0.12 and 0.94, respectively) gave no reason to reject this assumption. Fig. 7 shows the nonexceedance curve for errors for the validation period of 2011–2015; more than 90% of the errors were less than 10%.

During the validation period, the model demonstrated acceptable performance compared with the goodness-of-fit factors in the literature for similar modeling applications. Commonly reported $R^2$ values within recent literature ranged between 0.75 and 0.85, and some models produced $R^2$ values greater than 0.9 (most of which employ ANN or SVM). When using ANN or SVM models, units may not be conserved because they might use exponential beta values, log functions, or other methods that result in the loss of units, whereas the model presented here conserves units by identifying and solving for scalar terms and using unit consumption and population to derive the units of megaliters per day. Another difference is that ANN or SVM methods might output only the desired forecasted values, whereas the parameters used in this model provide insight into the behavior of the system. For example, the temperature and precipitation response curves used in this model might be used for other applications and decision-making policies, such as considering a new building code standard for smart sprinkler systems that automatically will decrease irrigation based on precipitation. If landscape irrigation represents a significant fraction of total demand in that community, and this model shows that the demand historically has been insensitive to precipitation events, then enforcing such a building code likely would help reduce demand.

One of the key features of the presented model is its ability to predict annual peak-day demand. Other studies primarily focused on average model error, but other key pieces of information are important to water utility management. The annual peak-day demand dictates the minimum capacity of system water treatment and storage infrastructure. For prediction of the peak-day demand during the validation period, the mean absolute-value peak-day relative error was 2.38% and the maximum absolute-value peak-day relative error was 5.98%. The statistical modeling approach to estimate the annual peak-day demand advances the capability to quantify this critical factor, which currently is represented at some water utilities using a simple average or rule-of-thumb approach (e.g., City of Oxnard 2017; Vallecitos Water District 2018). For a municipal water utility that calibrates this model with their data, theoretically that water utility could use population growth projections to estimate peak-day demand to help with their capital improvements planning for plant expansions and distribution storage capacity.

Furthermore, a municipal water utility that calibrates this model with their data theoretically could use short-term weather forecast data to optimize of the timing of water production. For example, if cooler weather was expected in short-term weather forecasts, a water utility could reduce their electrical consumption by shifting more production to shoulder and off-peak hours.

All models are sensitive to source data accuracy; thus, some amount of uncertainty is introduced to this model, accordingly. One such limitation is the accuracy and resolution of population data for the given study area. The US Census Bureau is one of the most reliable sources for population counts, but it releases data only in the form of annual values. Therefore, interpolation of the data from 1 year to the next was necessary, which introduces uncertainty about what is occurring from day to day. For example, winter snowbirds or rapid changes in Fort Bliss are not necessarily captured in US Census Bureau data. Increasing the resolution of data has made models increasingly more accurate, but public data availability of population figures seems to have lagged in this aspect. In a similar manner, lack of inclusion of spatial modeling methods such as those in use with WaterGAP (Eicker et al. 2014) also can contribute a certain amount of uncertainty. The model described herein treats the El Paso water demand system as a single entity despite the fact that multiple water treatment facilities serve different parts of the city. Spatial modeling methods could reduce uncertainty by parsing sections of the city that use water in different ways.