Short-term water demand prediction using stacking regression–based machine learning

Many cities worldwide experience water scarcity (Greve et al. 2018). More than half the population is expected to live in water-stressed areas by 2030 (Bakkes et al. 2008). For the consumption of water, distribution systems have to be managed sustainably. Water demand prediction is crucial for the same (House-Peters & Chang 2011). Infrastructure decision-makers also consider this to create effective water consumption plans and schedules (Derrible 2016; Pacchin et al. 2019; Pesantez et al. 2020). This is particularly crucial, given the ongoing urbanization of the population and consumer pushback for lower energy and resource consumption. To design, plan, operate, and manage water distribution systems (WDSs), water utilities around the globe rely on urban water demand projections. Water demand approximations are crucial to the optimal functioning of WDS as they provide the basis for efficient scheduling. For example, it helps water enterprises evaluate pricing strategies, pump station operations, pipe network capacity, water distribution, and water production (Herrera et al. 2010). Historical water consumption, operating conditions, socioeconomic factors, and weather patterns frequently influence future water demand (Donkor et al. 2014). Water demand forecasting has been approached from many angles, but it may be broadly divided into learning algorithms and classical procedures. Early articles used time-series and linear regression models, two standard statistical techniques, to address this problem (Zhou et al. 2000; Bougadis et al. 2005; Wong et al. 2010).

Nonlinear approaches are algorithms used for learning. Nonlinear approaches largely hinge on previous data to establish the association between water demand and substantial factors (such as wind speed, relative humidity, and temperature). Artificial intelligence and machine learning (ML) are examples of advanced data analysis approaches that enable learning algorithm models to extract essential knowledge from demand data of water accurately. The characteristics that are selected as model inputs and the prediction algorithms used during model development are the two factors that affect the success of learning algorithm models (Fan et al. 2017).

This research focuses on anticipating water demand in the metropolitan water distribution system of Peroorkada, Trivandrum. According to a study by the Kerala Water Authority, the Peroorkada network consists of 99 nodes (Sankaranarayanan et al. 2017; Sankaranarayanan et al. 2019, 2020). Peroorkada's commercial supply system, with one reservoir, has a water storage facility of about 8 million litres. A total of 13 prediction models were developed, which comprised both mathematical and machine intelligence learning models. Hybrid machine learning prediction situations were discussed. The historical dataset was utilized to train algorithms for predicting future water demand. Their performances were compared to choose the most suited model.

This is how the remaining of the article is structured. Section 2 provides a synopsis of relevant research. The approaches are outlined in Section 3. The Case study, in Section 4, the model test findings are reported in Section 5, and the study's decisions and future research suggestions have been delivered in Section 6.

A prediction model integrates geographic data with economic and social variables to estimate water demand. Explanatory variable selection and model construction are the two primary phases in prediction modelling (Shuang & Zhao 2021). The model characteristics described depend on the demand patterns. Using a procedure to determine the correlation between the chosen characteristics and the prediction target – in this example, water demand is a step in the model-building process.

Water demand predictions have already been made using machine learning prediction methods. Numerous studies have looked at the effectiveness of support vector machine (SVM) regression, a well-liked machine intelligence learning technique that is frequently employed in water demand projections (Braun et al. 2014; Brentan et al. 2017). In addition to SVM, other machine learning approaches that were being exercised to anticipate WDS demand include artificial neural networks (ANNs) (Maier et al. 2010; Romano & Kapelan 2014; Baotić et al. 2015), random forests (RFs, Mouatadid & Adamowski 2016), and extreme learning machines (Li et al. 2017; Lu et al. 2020). In general, machine learning has proved to be promising and is often utilized in estimating water demand.

A prediction model's goal will dictate its prediction periodicity, the amount of time between consecutive forecasts and prediction horizon, or the period of impending demand that should be projected (Bakker et al. 2003). When planning or designing urban WDS, long-term predictions – yearly projections for more than 10 years – are typically employed. To estimate the income or charge of a water supply and enhance the distribution system, medium-period predictions – that is, monthly or yearly forecasts for 1–10 years – are typically employed. The daily routine operations of the water plants are normally grounded on short-term forecasts. WDS smart management is evolving quickly, and there is a need for very accurate short-term predictions, particularly ones that have real-time pipe burst detection and optimal WDS control (Bakker et al. 2013; Hutton & Kapelan 2015).

In order to forecast the monthly water utilization of Austin, Texas, Lu et al. (2020) created a fusion model and noted that the demand was strongly correlated with the city's population, regular mean temperature, and regular mean humidity. Shanghai's yearly water consumption was examined by Li et al. (2017) using prime module analysis regression. The authors discovered that the city's populace and Gross domestic product significantly affect yearly water distribution demand.

For forecasting urban residents' demand for water with variables such as the quality of population, GDP per capita, water cost, the sum of yearly supply and temperature of water from water companies, and weather variations, Zhao & Chen (2014) used a neural network algorithm based on elastic backpropagation. For forecasting the yearly water consumption in the Chinese city of Guangdong with variables such as precipitation, the sum of water resources, GDP, permanent resident information, tertiary factories added value, the added value of the industrial area, added value of agricultural and irrigation area, animal husbandry, forestry, and fishery, Tian & Xue (2017) created a neural network with backpropagation.

Zhi-Guo et al. (2010) developed methods to assess the base demand for water in Chinese cities Beijing and Jinan, taking into account variables including GDP, per capita water resources, added value in elementary, intermediate, and higher sectors, and population. Sun et al. (2017) assessed the maintainable use of Beijing's water resources, taking into account the population, economy, supply of water, demand and resources of land, pollution, and management.

The selection of suitable water demand estimation is quite difficult since such methods are affected by surrounding factors (Fricke 2013). Statistical and machine-learning models are the two main categories of predictive models. Probability theory and quantitative statistics are utilized in statistical models to determine the functional connection among various variables. Regression using statistical techniques, such as least absolute shrinkage and selection operator (Lasso) regression, linear regression, and ridge regression, are often used.

The establishment of links between explanatory variables and dependent variables is not necessary for machine learning models (Mitchell 1997). Rather, they exercise procedures such as RF regression, SVM regression, and decision tree (DT) to identify models in training data and utilize those patterns to forecast future events. A number of statistical models are employed to anticipate water consumption (House-Peters et al. 2010; Kontokosta & Jain 2015; Arbues & Villanua 2016; Ashoori et al. 2016). Finding a single mathematical function that would perform well on a variety of datasets is challenging because statistical models are primarily limited by the need to have a predefined structure (Hastie et al. 2009; Lee & Derrible 2020). Moreover, statistical models frequently fall short in handling intricate data linkages, and as data volume increases, so does the accuracy of their predictions ends to improve (Mu et al. 2020). When working with large and complicated datasets, other techniques have to be utilized (Villarin & Rodriguez-Galiano 2019). For instance, Rozos et al. (Makropoulos et al. 2016) used cellular automata modelling and integrated system dynamics to estimate water demand.

With their outstanding predictive performance in areas including urban infrastructure (Marvuglia & Messineo 2012; Ali et al. 2016; Golshani et al. 2018; Lee et al. 2018), financial risk (Xia et al. 2017; García et al. 2019), energy (Voyant et al. 2017), ecology (Archibald et al. 2009; Darling et al. 2012; Muñoz-Mas et al. 2019), and resource management of water (Rozos 2019), techniques for machine learning are gaining popularity. Depending on how many predictors are used, machine learning techniques may be additionally separated to ensemble algorithms and single machine learning predictors. A single predictor, such as decision tree, SVM, or neural network, consists of just one predictor (or method). Many predictors are aggregated via ensemble algorithms including gradient-boosting tree and RF, which all contribute to the final prediction outcome.

The use of ensemble learning is growing in popularity (Nascimento et al. 2014; Lessmann et al. 2015). It trains several models using the concepts of statistical sampling. A fresh sample is predicted independently using each of these models. The majority voting procedure determines the rate of the concluding forecast constituted for the novel model. Stated differently, ensemble learning combines several hypotheses from individual predictors into a single hypothesis. In order to forecast everyday domestic water consumption in reaction to the housing demand for water, Lee et al. (2018) looked into 12 numerical and ML methods in the field of water supply prediction. With the aim of forecasting the hourly water consumption of 90 accounts, Pesantez et al. (2020) utilized SVMs, RFs, and ANNs to smart-meter data.

Support vector machine regression, a backpropagation ANN, and an excessive understanding machine were utilized by Parisouj et al. (2020) to forecast the daily and monthly streamflow of quaternion catchment reservoirs in the USA. Villarin & Rodriguez-Galiano (2019) developed a multiple variate estimation model for demand of water in Seville, Spain, using regression trees, RFs, and categorization. Sengupta et al. (2018) predicted changes in stream channel morphology using SVM regression, ANNs, and RF regression.

In this study, 13 models were presented to predict water demand:

Numerical methods: ridge regression method, linear regression (LR) method, Lasso regression method, elastic net regression (ENR), and partial least squares (PLS) method.

Machine learning representations: individual predictors: DT, SVM regression; ensemble approaches: DT and RFs. While DT is a serial integration technique, RF is a parallel integration algorithm.

The techniques were implemented using the Python Scikit-Learn module (Pedregosa et al. 2011; Buitinck et al. 2013). For every technique mentioned in Section 3.3, a different prediction model was constructed. The hyperparameter values are used to train the models. The following is a brief description of each algorithm.

Breiman (2001) introduced RF, an ensemble learning method that selects features using the bagging approach. Specifically, substitution sampling is exercised at random to choose a subdivision of factors to build an individual tree in the ensemble, whereas substitution sampling is done to train trees using samples commencing the initial data. To reach the final result, the prediction results given by every trained tree in the ensemble are combined for every novel test sample using the majority voting technique.

Gradient Boosting Tree Regression uses the notion of an ensemble method, which is acquired from the technique called decision trees (Friedman 2001). The decision tree is based on the structure of a tree. The predicted outcome is the target leaf, which commences from the root and divisions of the tree with some conditions before reaching the leaves.

Gaussian processes (GPs) are a nonparametric supervised learning technique for solving regression and probabilistic classification issues. Gaussian process regression (GPR) is a sophisticated and adaptable nonparametric regression approach used in machine learning and statistics (Shuang & Zhao 2021). GPR is a Bayesian technique that can provide some assurance in forecasts, which makes it valuable for numerous functions concerning optimization and time series calculation. GPR is based on the concept of a Gaussian structure, i.e. a set of arbitrary values with a collective Gaussian distribution.

LR uses a linear function to represent the connection among the illustrative value(s) x and the variables of dependencies y (Guo et al. 2018). It uses a linear method to reduce the residual sum of squares amid the observed and projected values of the variable of dependency. The coefficients are calculated via the ordinary least squares (OLS) technique.

Ridge and Lasso are variants of the linear method (Hoerl & Kennard 1970), and ridge is an enhancement to the approach of OLS. It is an added steady and accurate machine learning technique due to the deficiency of impartialness (Khan et al. 2019). To balance variance and bias, the regularization term, i.e., L2 is applied after the sum of squared errors (SSE).

DT is a traditional machine intelligence algorithm. It creates a binary programme tree constructed on the sample characteristics. This method of reaching a prediction outcome is simple for grasping and interpretating from the final decision tree structure, which resembles a flow diagram, with leaf joints representing the predicted variables (Shuang & Zhao 2021).

SVM regression locates a dividing hyperplane and fits y on x with the biggest difference. It employs a function of kernel to translate the initial training group with nonlinear characteristics to a higher dimensional attribute space, in which the values become proportionally distinguishable (Braun et al. 2014). Most common functions that use kernel are polynomial, linear, and radial basis functions.

A variation of linear method that employs the imaginary function is the elastic net regression. It is a superior machine learning technique that comprises components of both Lasso and Ridge (Khan et al. 2019). It is a standardized regression method designed to deliver the issues of multicollinearity and overfitting that are common in big datasets. This technique works by including a penalty term into the normal least-squares objective function.

The PLS method is a strategy that decreases interpreters to a reduced collection of noncorrelated factors and conducts the least-squares method based on these factors, instead of the whole set of variables (Hanrahan et al. 2005).

Stacking regression (Figure 3) is a prominent machine learning strategy of the ensemble method for predicting several nodes with the aim of creating a novel method and enhancing its execution. Stacking permits to train numerous algorithms to address related issues and associates their product to generate a novel algorithm with improved performance. Stacking regression involves stacking the output of distinct predictors and using a machine learning method to obtain the optimal estimate. It permits to use strengths of every distinct estimator by utilizing their product as input to the last estimator (Anbananthen et al. 2021). It employs various meta-algorithms for determination to integrate the best estimates from two or more basic algorithms. Cross-validation and the least squares for values of nonnegative are used to get the stack coefficient. It appears to be more efficient than conventional ML procedures.

A typical aim in machine learning is to investigate and develop methods that are learning and predicting data value (Kohavi 1998). Such methods work by producing data-driven predictions or assessments (Bishop 2006) by developing a mathematical model from incoming data. The input data required to develop the model are often separated into several datasets. Specifically, three datasets are frequently employed at various phases of model development: training, validation, and test sets. Using Python, the input ‘X’ data groups are divided into two parts: training and k-fold validation, with the training group taking up 0.8 of the data groups and the test group taking up the remaining 0.2.

When a model is fitted to the training group and validated using the test group, it is prone to excessive fitting, which implies that it functions excellently on observed data but poorly on unobserved information. To alleviate this problem of excessive fitting, a 10-fold cross-validation (CV) was applied to the training datasets. The k-fold CV spontaneously divides the training data into 10 distinct subgroups, each constituting one-fold of the total. Each model is then trained on the remaining nine-folds and tested on the remaining fold 10 times, with the validation fold changing between iterations. The mean prediction result and standard deviation are calculated using 10-fold scores.

The test data consisted of the remaining data, which made up of 20% of the total dataset. The predicted results from the preceding phase are compared to what the actual outcomes should have been. Several evaluation metrics are created to determine the model's performance. To assess each of the model's performance, three performance metrics were used: mean absolute error (MAE), mean squared error (MSE), and determination coefficient (R²). These measures were evaluated because they are commonly used in studies on demand forecast (Villarin & Rodriguez-Galiano 2019; Lee & Derrible 2020; Mu et al. 2020; Pesantez et al. 2020).

The Peroorkada WDN, which is taken into consideration for this study, is operated on 24-h regular basis. The resource planning and allocation are done for every 24 h substantially to prevent the data managing problem. This measurement of large period calculated for WDS scheduling creates a dynamic model. The inflows and outflows of head and flow of the system are computed and updated for corresponding periods of time. Each consumer node has the outflow from the tank that depends upon the tank head and the inflow from the main water distribution lines. The aforementioned demand dispatch circumstances cannot be accommodated by the traditional steady-state models for WDS as they all take into consideration demand-driven situations, in which the demands are directly related to the WDS.

Forecasting of the states and outcomes, for any estimation approach, requires a numerical model. The WDS implemented in this study operates on a 24-h basis. Minimizing the issue of managing of data on a very large and regular period, the scheduling and resource planning occur every 24 h. The dynamic model of the system is said to be a set of quasi-steady-state difference equations and this is caused by the wide time interval used for scheduling of the WDS. This is a comparatively better depiction of dynamics of WDS. During the specific time period with the use of inflows and outflows, as well as the heads availed, the flow and head values are computed. The resultant is equivalent to the nonlinear comprehensive models over a larger time (Mohamed Hussain et al. 2023). A traditional pressure dependent demand dispatch (PDDD) (Sankaranarayanan et al. 2018) is the significance of this WDS context, in which each civil structure (consumer node) includes outflow from tank and a reservoir tank. These are determined by input from the distribution lines and the head of the tank. Additional modelling of the reservoir at the consumer nodes is used to simulate PDDD as a result due to the unsuitable standard steady-state models for WDS. Direct demand-driven conditions are considered for the demand dispatch settings for all steady states.

The modelling of the WDN used four equivalent sets of equations: (i) the continuity equation (flow balance), (ii) the head balance equation in the WDS, (iii) the flow and head loss correlation, and (iv) the reservoir model and their related points.

They considered WDS is the town's residential sector, with the majority of nodes being residential (individual houses and closely guarded communal complexes) or commercial. The demands are considered as an unknown quantity, and it is highly essential to estimate the proper water management. If end-user demand is directly associated to reservoir tanks excluding an exit valve, the outflow is proportionate to demand and effects tank head. If the consumer's water supply is linked to the tank by a regulating valve, the head of the tank is determined by the working position of the valve. Thus, exit valves are regarded as an important feature that influences the kinetics of the outflow. Since the rate of change of the exit valve co-efficient is almost maintained constant in nominal cases, the working conditions for the valves are often not measured (Sankaranarayanan et al. 2018). The reservoir tank model on demand nodes (non-interacting and interacting) is depicted in figure A1. The structural complexity and the unobservable states lead to the ambiguous idea about the operating condition of the valve coefficients. Furthermore, the diameter of pipes is also considered as an important parameter, since the WDS is vulnerable to the sediment contamination through ages and the pipe leak affects the dynamics of the flow. The considered section of WDS consists of 55 PDDD consumer nodes, out of which 6 nodes are directly connected to the demand points and other 49 are connected through 49 valves. The parameters to be estimated are the 49 valves connected to the reservoir tanks, 6 demand patterns, and the diameter of the 55 pipes. The demand pattern is dynamic in nature, and the considered optimization is a static approach. In order to solve the abovementioned discrepancy, the dynamic demand profile is converted to a static parameter for the estimation.

From the R² test plot, the performance of the models was determined. In an ideal model, the points should be closer to a diagonal line (Kohavi 1998). A total of 13 machine learning models, namely, RF, Gradient Boosting Regression (GBR), GPR, linear regression, Lasso regression, ridge regression, DT, Support Vector machine Regression (SVR), ENR, PLS, Lasso-based stacking regressor, ridge-based stacking regressor, and linear-based stacking regressor, were modelled, and the predicted vs. actual plot (R² plot) has been plotted for each node at the time intervals of 6 h exactly at 0, 6, 12, 18, and 24 h, and the predictions for the nodes 106, 113, 123, 150, 1,041, and 1,057 were plotted. Figures 8 and 9 describe the R² plot for the abovementioned nodes and time intervals. For each node mentioned above, goodness of fit (R²) is higher for Lasso-based stacked regressor (Brentan et al. 2017; Anbananthen et al. 2021) than that of other regression models.

The training R² results for all nodes at 0, 6, 12, 18, and 24 h, with a 6-h interval of comparatively larger standard deviations have been noticed for DT and SVR, which indicates that the R² results were amply distributed around the 10-folds of these methods, implying that the algorithms' execution in every fold suggestively diverged from the mean value. Furthermore, to decrease standard deviations (Shuang & Zhao 2021), R² values of the RF, GBR, as well as the three stacked regressor methods, were in the top five. The test R² findings for all nodes at 0, 6, 12, 18, and 24 h with a 6-h interval showed relatively significant standard deviations for DT and SVR, indicating lower R² values. In addition, higher R² values of test results were observed for RF, Lasso, Linear, and the three stacked regressor models ranked in the top five. The best values of R² were indicated for the Lasso-based stacked regressor (Figure 12) (Lasso SR) with values of 0.789 for nodes at time 0:00 h, 0.98 for nodes at time 6:00 h, 0.967 for nodes at time 12:00 h, 0.881 for nodes at time 18:00 h, and 0.79 for nodes at time 24:00 h.

The training R² results for the nodes 106, 113, 123, 150, 1,041, and 1,057 for SVR had relatively large standard deviations, which indicates that the R² results were amply distributed around the 10-folds of these methods, implying that the algorithms' execution in every fold suggestively diverged from the mean value (Pacchin et al. 2019). Furthermore, to decrease standard deviations, the R² values of the RF, GBR, DT, and three stacked regressor models were in the top six. The test R² results for the nodes 106, 113, 123, 150, 1,041, and 1,057 of relatively large standard deviations were observed for SVR, indicating lower R² values. In addition, higher R² values of test results (Herrera et al. 2010) were observed for GBR, DT, and the three stacked regressor models ranked in the top five. The best values of R² were indicated for Lasso-based stacked regressor (Lasso SR) with values of 0.806 for node 106, 0.978 for node 113, 0.981 for node 123, 0.824 for node 150, 0.61 for node 1,041, and 0.708 for node 1,057. The tables for individual nodes are provided in the Supplementary File.

Tables 1 and 2 show the outcomes attained by the machine learning models over the training set and the test set, respectively, for all nodes at 6-h intervals. The optimal values for each measure are marked. The following table depitcs the mean value of R² values trained and tested for all models. The separate R² values for each node were discussed in Section 5.2.2.

Tables 3 and 4 present the outcomes attained by the machine learning models over the training set and test set, respectively, for the nodes 106, 113, 123, 150, 1,041, and 1,057. The optimal values for each measure are marked. The following table depicts the mean value of R² values trained and tested for all models. The separate R² values for each node were discussed in Section 5.2.2.

Water shortages and wastes are limited or eliminated via the use of demand forecasting and an equal supply of local water resources. The foremost goal of this research study is to use a prediction model to estimate water demand in the Peroorkada urban water distribution system. The elements of predictions included attributes linked to the water demand, such as head, pressure, and base demand. Python is used to create the prediction model. There are 99 nodes in the water distribution network. All of the nodes have had their demand for a 6-h time interval anticipated and displayed, and the sensor placement tools have been used to estimate the susceptible nodes' demand for a 24-h interval.

A total of 13 machine learning algorithms were used in this work, three of which were hybrid/stacked regression methods. Demand prediction is best achieved with the Lasso-based stacking regressor model. Stacking regressor models outperformed single prediction models in terms of success. This is due to the fact that they are able to combine the effectiveness of different individual regressors. These models can improve predictive maintenance performance and handle more complex tasks than single models by managing a variety of data structures.

Two other areas with distinct topographies, climates, and economies were used to further verify the Lasso-based stacked regressor model. In most of the situations, the anticipated accuracies of Lasso-based stacked regressor exceeded 80%. This establishes the model's robustness. Other locations can utilize the same information with the same explanatory qualities to forecast water demand. The study's findings will help municipalities and utilities optimize the association between water distribution and water demand and decrease ambiguous water time arrangement costs by merging their own databases with corporate industrial requests to keep the demand forecast of water. Further segmentation of the demand for water with an examination of the key factors influencing variations in the demand for water is required in the future. Data on water supply should be incorporated into the design of multiscenario water demand forecasts. It is important in the future to take into account extensive information on water demand, including that of various Indian provinces and cities. Since it decreases overall demand, recovered wastewater and polluted water may be factored into the prediction model.

The authors express their thanks, in particular, to the ‘MeitY (Ministry of Electronics and Information Technology), Government of India’ for granting this project work, as well as our institute, ‘National Institute of Technology, Tiruchirappalli (NITT), Tamil Nadu, India, under the Ministry of Human Resources Development (MHRD), Government of India’, for assistance in accomplishing the above work.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.