Literature on water demand forecasting.
Abstract
Given the increasing trend in water scarcity, which threatens a number of regions worldwide, governments and water distribution system (WDS) operators have sought accurate methods of estimating water demands. While investigators have proposed stochastic and deterministic techniques to model water demands in urban WDS, the performance of soft computing techniques [e.g., Genetic Expression Programming (GEP)] and machine learning methods [e.g., Support Vector Machines (SVM)] in this endeavour remains to be evaluated. The present study proposed a new rationale and a novel technique in forecasting water demand. Phase space reconstruction was used to feed the determinants of water demand with proper lag times, followed by development of GEP and SVM models. The relative accuracy of the three best models was evaluated on the basis of performance indices: coefficient of determination (R2), mean absolute error (MAE), root mean square of error (RMSE), and Nash-Sutcliff coefficient (E). Results showed GEP models were highly sensitive to data classification, genetic operators, and optimum lag time. The SVM model that implemented a Polynomial kernel function slightly outperformed the GEP models. This study showed how phase space reconstruction could potentially improve water demand forecasts using soft computing techniques.
Keywords
- water demand forecasting
- soft computing
- genetic expression programming
- support vector machines
- phase space reconstruction
- lag time
1. Introduction
While water scarcity has become a key concern worldwide, it is particularly so in arid and semiarid regions with limited potable water sources. In designing water distribution systems (WDS), engineers have typically used a “fixture unit” method, which considers the sum of fixture unit demands, facility types, and socioeconomic factors to determine peak demand. However, this overestimates the actual peak demand by as much as 100% [1]. Due to various uncertainties, including those associated with demand, engineers often include large safety factors when designing WDS. Given that WDS rely mainly on regional energy and resources, an overdesigned system can have environmental impacts that will appear in region(s) beyond the jurisdictional boundaries of the system. While short-term demand forecasts are critical to a WDS daily operations [2], long-term forecasts are required for future planning and management of the systems. In providing an accurate estimate of water demand, a robust demand-forecasting model assists managers in designing a more environmentally sustainable WDS and in managing available water resources more efficiently. When coupled with a water demand management strategy, such models can help managers overcome operational problems (e.g., low pressure during peak demands) and issues related to asset management (e.g., nonreplacement of assets or replacement by lower capacity assets reaching the end of their economic life). It has been estimated that a well-predicted monthly average demand might be up to 400% lower than peak demands that cause low pressure; however, a more realistic model can enhance resource management and operating systems. This will eventually lead to significant savings for water and energy (for running pumps, treatment plants, etc.) industries. Considering weather conditions and population, the prime objective of the present study was to develop a predictive model for monthly average water demand. While the present study proposed a generic framework that could be easily adjusted for any specific case, the City of Kelowna (British Columbia, Canada) was employed as a test case.
2. Literature review
Water demand varies greatly both regionally and seasonally. Increasing urbanization and industrialization as well as emerging issues such as shifting weather patterns and population growth have significant impacts on water demand. The main components in demand prediction are the explanatory variables and time scales used. Selecting explanatory variables for a predictive model depend on the desired time scale and the availability of data. Simple models using very few explanatory variables have shown promising accuracy for short-term prediction [3, 4]. In general, the explanatory variables affecting water demand are of two types: weather (e.g., temperature, relative humidity, and rainfall) and socioeconomic (e.g., population and income). Weather conditions affect short-term prediction while their socioeconomic counterparts can affect long-term predictions [5–7]. As has been highlighted by significant worldwide changes in climate, both in terms of weather conditions and global warming, water availability is prone to great uncertainty [8]. Therefore, the impact of evolving weather conditions on long-term water demand predictions should receive greater attention. Furthermore, researchers who have considered weather conditions in short-term water demand prediction have established that it is not feasible to feed online automated WDS with real time weather information [9]. As a result, limited studies have considered weather conditions in their demand forecasting models [10–12]. Table 1 summarizes the relevant literature. Temperature, precipitation, pan evaporation, and number of days since the last rainfall were used in a forecasting model [13]. Another study used temperature, relative humidity, rainfall, wind speed, and air pressure as weather parameters in their hourly water demand model for Sao Paulo, Brazil [12]. Table 1 shows the previous researchers did not consider socioeconomic and weather conditions simultaneously since their effects are highly dependent on the forecast’s time scale. Traditionally, WDS utilities have used historical patterns as explanatory variables in predicting future water demands. Scarce water reserves and the rapid increase in urbanization have raised awareness and led to implementation of statistical approaches. Multiple linear regression (MLR) and time series were the most popular techniques used in the early stages of demand forecasting [6]. While MLR has been widely used to better understand the determinants of water demand [14–18], its major drawback is the fact that it considers linear relationships among variables and water demand, such relationships are nonlinear by nature. Time series have been introduced along with regression as methods for demand forecasting [10, 19]. Due to the common belief that they can deal with complex systems [20], artificial neural networks (ANNs) have been widely applied in water demand forecasting [21–23, 2]. Comparing regression, univariate time series, and ANN models, one study found ANN models drawing on standard rainfall and maximum temperature data could better predict weekly water demand than other models [6]. Similarly, drawing on temperature and rainfall data in their forecasting models, researchers concluded that ANN models provided more reliable forecasts for peak weekly demand than time series and simple and multiple linear regressions [22]. Results of another study showed ANN models performed better for hourly forecasts, whereas regression models were more accurate in forecasting daily demand [23]. To improve the accuracy and robustness of demand forecasting models, hybrid models combining or modifying ANN, MLR, and time series techniques have been tested [24–27]. However, application of nonlinear regression in demand forecasting has remained limited to studies using support vector machines (SVMs) [28–30] and training nonlinear relationships through linear regression models [6, 31]. The present study compares gene expression programming (GEP) and SVM nonlinear approaches. Inspired by Darwin’s theory of evolution, GEP was recently proposed in engineering disciplines to optimize the structure of input variables fed into predictive models [32]. Being a self-learning algorithm, GEP has several advantages over conventional predictive models. GEP defines individual block structures (input variables, response, and function sets) and selects the optimized operating functions and multipliers through the process of learning algorithms. Results of one study indicated GEP models outperformed traditional linear models in the field of hydrology [32]. Since weather information is one of the major determinants of water demand, this research employed GEP to develop a robust and accurate demand-forecasting model.
No. | Reference | Method | Determinant | Time scale |
---|---|---|---|---|
1 | [16] | Linear regression | Seasonal dummies, derivatives of weather and price |
Monthly demand |
2 | [17] | Linear regression | Density, building size, lot size, household size, income, price, temp, rain, drought dummies |
Bimonthly demand |
3 | [18] | Regression using Bayesian moment entropy | Population density | Annual demand |
4 | [13] | Decomposed daily demand followed by composite forecasts |
Daily demand and hourly demand | Daily demand |
5 | [19] | Univariate time series | Y_{t−1} | Annual residential demand |
6 | [22] | Regression and ANN | Temp, rainfall, and lags of peak demand | Peak weakly demand |
7 | [23] | ANN | Temperature, rainfall, and delayed demand | Daily demand |
8 | [2] | Time series | Univariate demand series, temperature in a multivariate model |
Daily, weekly, monthly, annual |
9 | [6] | Time series and ANN | Delayed demands, temperature, and rainfall | Weekly demand |
10 | [24] | Holt-Winters multiplicative smoothing modified regression | Precipitation, temperature, humidity, lagged demand | Weekly (6 days) |
11 | [26] | Weighted average regression and ANN |
Historical demand and time | Annual demand |
12 | [27] | Decomposed annual demand, regression and ANN | GDP, population, temperature, greenery coverage, delayed demand | Annual demand |
13 | [31] | Wavelet-deinoizing and ANN | 7-year long time series of demand | Monthly demand |
14 | [28] | SVM with RBF function is compared with ANN | Delayed demand, population | Daily demand |
15 | [29] | ANN, SVM, Monte Carlo | Rain, demand, wind speed, atmospheric pressure | Hourly demand |
16 | [30] | SVM and adaptive Fourier series | Wind speed, temperature, demand, humidity, and rainfall | Hourly demand |
3. Study area and data collection
This research focused on the City of Kelowna located in the Okanagan Valley (British Columbia, Canada). The City has five water districts including the City of Kelowna District (CKD), Glenmore Ellison Irrigation District (GEID), Black Mountain Irrigation District (BMID), Rutland Water District (RWD), and the South East Kelowna Irrigation District (SEKID). The CKD served as the study area of this research. Using three major pumping stations, the CKD primarily supplies water from the Okanagan Lake. The present study used monthly mean water demand data from 1996 to 2010 (http://www.kelowna.ca/). The population censuses of 1996, 2001, 2006, and 2011, along with the best-fit parabolic equation (with coefficient of determination of
4. Methodology
4.1. Model development
To determine water demand (
where
Classification | Model | Input variables combination* |
---|---|---|
Demand Data Based | ||
Demand + Weather Data Based | ||
Demand + Weather + Population Data Based | ||
Data were used in partitions of 144 samples for training (1996–2007) and 35 samples for validation (2008–2010). The time series of water demand over the time period of 1996–2010 (Figure 1) shows a relatively regular periodic cycle of water demand in CKD that is mainly due to seasonal changes.
4.2. Genetic expression programming (GEP)
Introduced by Ferreira, GEP is an emerging soft computing technique [34]. The strategy used for the learning algorithms was the optimal evolution using the genetic operators. Following Ferreira, this research defined the overall structure of the GEP model by: 30 chromosomes, eight head sizes, and three genes [35]. The selected head size determined how complex each model parameter was. Each of the gene heads underwent a set of different arrangements to model the feeding data. Selecting new random populations was followed by reproduction in order to reach the most suitable model under optimized stopping conditions. Models were developed based on three genes linked together by an addition function. The number of genes per chromosome specified the layers or blocks involved in building the whole model. Although a large gene was useful, dividing the chromosomes into simpler units resulted in a more efficient and manageable learning process. RMSE was used as a fitness function to fit a curve to target values. The stopping condition was a maximum fitness and coefficient of determination (
4.3. Lag time
The literature lists three methods for estimating lag time, AMI, autocorrelation function (ACF), and correlation integral (CI) [36–38]. AMI is considered the best since ACF reflects only linear properties and CI requires a large set of data [39]. Consequently, the present study employed AMI defined as:
where the joint probability of two successive time series,
4.4. Support vector machines (SVM)
For SVM models, in which genetic operators are not used, the input types remained
In this method, the input vectors are considered as supports forming the backbone of the whole model structure through a training process. If
where
where ξ_{k} and
5. Results and discussion
The prime objective of using phase space reconstruction was to find a proper lag time for developing the models in this study. In order to have a comprehensive understanding of model performance, GEP models were defined by all lag times up to the optimum value determined for water demand in the CKD. The AMI calculations of the water demand in the CKD resulted in a lag time of 3 months. Figure 3 shows that the first local minimum point occurs at 3 months, allowing the AMI an optimum lag time for phase space reconstruction (
Figure 4a–c shows the phase space diagrams of water demand for
Prior to analysis with GEP models, a correlation table between the explanatory variables and water demand provided a better understanding of how to define the input factors (Table 3). The correlations were 0.92, 0.84, −0.83, 0.11, and −0.01 for
1.00 | 0.92 | −0.01 | −0.83 | 0.11 | 0.84 | |
0.92 | 1.00 | 0.10 | −0.89 | 0.00 | 0.92 | |
−0.01 | 0.10 | 1.00 | −0.05 | −0.26 | 0.11 | |
−0.83 | −0.89 | −0.05 | 1.00 | 0.02 | −0.84 | |
0.11 | 0.00 | −0.26 | 0.02 | 1.00 | −0.09 | |
0.84 | 0.92 | 0.11 | −0.85 | −0.09 | 1.00 |
Table 4 shows all 27 GEP models developed in the present study. Three superior models were highlighted in each category or classification of determinants. Interestingly, a lag time of 3 months outperformed other combinations in all different classifications which show the importance of using phase space construction in studying complex systems. This shows that an appropriate lag time determined by AMI can significantly improve the performance of the forecasting model. Different genetic operators were also used to understand which mathematical operations better define the nature of these determinants. The first operator {+, −,
Model ID* | Training | Testing | ||||
---|---|---|---|---|---|---|
MAE | RMSE | MAE | RMSE | |||
0.4687 | 0.6974 | 0.6284 | 0.4833 | 0.6067 | 0.6343 | |
0.4718 | 0.6100 | 0.6252 | 0.4849 | 0.6120 | 0.6300 | |
0.4672 | 0.6118 | 0.6235 | 0.4800 | 0.6112 | 0.6281 | |
0.3552 | 0.4721 | 0.7754 | 0.378 | 0.4607 | 0.7892 | |
0.3574 | 0.4721 | 0.7756 | 0.3794 | 0.4608 | 0.7892 | |
0.3008 | 0.4049 | 0.8481 | 0.4188 | 0.5188 | 0.8346 | |
0.2858 | 0.3641 | 0.8691 | 0.3488 | 0.3106 | 0.8452 | |
0.3545 | 0.4647 | 0.7849 | 0.3637 | 0.4548 | 0.8029 | |
0.3777 | 0.4790 | 0.7735 | 0.4529 | 0.5296 | 0.7552 | |
0.3955 | 0.4933 | 0.7560 | 0.4423 | 0.5169 | 0.7546 | |
0.3914 | 0.4893 | 0.7903 | 0.4596 | 0.5488 | 0.7643 | |
0.2463 | 0.3359 | 0.8867 | 0.3015 | 0.3981 | 0.8426 | |
0.3236 | 0.4022 | 0.8438 | 0.3455 | 0.4176 | 0.8473 | |
0.3580 | 0.4450 | 0.8048 | 0.3987 | 0.4798 | 0.8077 | |
0.3619 | 0.4445 | 0.8085 | 0.3893 | 0.4649 | 0.8139 | |
0.3033 | 0.4184 | 0.8502 | 0.3339 | 0.4562 | 0.8260 | |
0.2776 | 0.3810 | 0.8542 | 0.4201 | 0.5869 | 0.7087 | |
0.3474 | 0.4194 | 0.8237 | 0.4154 | 0.5348 | 0.7919 | |
0.2780 | 0.3601 | 0.8861 | 0.3933 | 0.5410 | 0.7714 | |
0.2875 | 0.3694 | 0.8778 | 0.4987 | 0.6332 | 0.6999 | |
0.3514 | 0.4543 | 0.8147 | 0.5694 | 0.6959 | 0.7027 | |
0.3944 | 0.2205 | 0.7827 | 0.5219 | 0.6408 | 0.7401 | |
0.3213 | 0.3961 | 0.8609 | 0.5624 | 0.6556 | 0.6839 | |
0.3907 | 0.4801 | 0.7800 | 0.3655 | 0.4582 | 0.8236 |
The superior GEP models from each classification were compared to SVM models implementing three different kernel functions (RBF, Poly, and Lin). Training and testing performance indices for the SVM models developed with each of the three kernel functions showed
Model ID* | Training | Testing | ||||
---|---|---|---|---|---|---|
RMSE | RMSE | |||||
0.9545 | 0.2123 | 0.9546 | 0.8397 | 0.4051 | 0.8387 | |
0.9856 | 0.1201 | 0.9855 | 0.8701 | 0.3678 | 0.867 | |
0.9416 | 0.2407 | 0.9415 | 0.9258 | 0.3014 | 0.9107 | |
0.9308 | 0.2618 | 0.9309 | 0.8206 | 0.4278 | 0.8201 | |
0.9372 | 0.2497 | 0.9371 | 0.9343 | 0.2593 | 0.9339 | |
0.9428 | 0.239 | 0.9424 | 0.9279 | 0.3002 | 0.9114 | |
0.7864 | 0.4602 | 0.7864 | 0.7945 | 0.4592 | 0.7927 | |
0.8894 | 0.3311 | 0.8894 | 0.8977 | 0.323 | 0.8974 | |
0.9093 | 0.2998 | 0.9004 | 0.9084 | 0.3344 | 0.8901 |
6. Conclusion
In an attempt to improve model prediction accuracy, a wide range of modeling techniques has been proposed by researchers over recent years in the water demand forecasting field. The present research explored a new approach to modeling water demand, namely genetic expression programming along with phase space reconstruction. In this method, input factors are not randomly chosen as in previous studies. Instead, appropriate lag time determinations made by the AMI method defined the structure of the explanatory variables employed in the models. The outcome of this research demonstrated GEP models to be highly sensitive to classification of input factors, proper lag time, and selection of genetic operators. In general, soft computing techniques like GEP should receive more attention in forecasting behaviors of complex systems such as WDS. These models can offer valuable information to WDS operators and designers to deploy optimum determinants in their forecast models. The three best GEP models proposed in this research were compared using different performance indices, however, differentiating between them was difficult due to the similarity in statistical index values. One of three GEP models was selected due to lower cumulative error in predicting demand and less fluctuation in comparison with the other two GEP models. However, these models were slightly outperformed by a SVM model, which showed even better performance indices. This shows that both GEP and SVM can be useful techniques in water demand forecasting and can account for nonlinearity of the input parameters
Acknowledgments
The authors received financial support from the Natural Sciences and Engineering Research Council (NSERC) of Canada. The Okanagan Basin Water Board and the City of Kelowna are thanked for providing water consumption data.
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