Performance evaluation of ANN and ANFIS models for estimating velocity and pressure in water distribution networks

Challenges faced by water distribution systems (WDSs), including system deterioration, leakage, pipeline disruptions, insufficient capacity to meet demand, unreliability, and mismanagement have emphasized the necessity of replacing conventional techniques with precise and efficient computer software and methods for designing WDSs (Vairavamoorthy et al. 2001; Longe et al. 2010). The modeling of WDSs has emerged as a critical aspect, facilitating hydraulic assessments to ensure that these systems can effectively meet demand and quality standards. The water distribution network (WDN) is modeled for velocity and pressure to predict the optimum diameter required in a WDN. Based on the optimum diameter the total cost of the network is decided. Several other factors also play a vital role while modeling WDN such as elevation, demand, and the height of the Elevated Service Reservoir (ESR). Recognizing the importance of modeling WDSs, a multidisciplinary team comprising professionals, researchers, scholars, engineers, and programmers joined forces to develop software specifically designed for the design and modeling of WDSs. These advanced hydraulic simulation and modeling software tools enable thorough analysis of water behavior (Sonaje & Joshi 2015).

In recent years, artificial neural networks (ANNs) have emerged as a promising and efficient technique for modeling and forecasting. Its applications are across various engineering fields. One notable early example of using backpropagation ANNs was demonstrated by (Crommelynck et al. 1992), who applied ANN to model daily and hourly water demand forecasts in select communities of Paris, France and compared the performance of ANN models against statistical models, and their findings indicated that the ANN models perform better than the statistical models. To accurately model water demand forecasts, (Miaou 1990) various types of data can be categorized into two main classes: socio-economic variables and climatic variables. Socio-economic variables, including population, income, water price, and housing characteristics, primarily influence the long-term patterns of water demand. On the other hand, climatic variables such as rainfall and maximum air temperature play a significant role in short-term seasonal fluctuations in water demand.

Jain et al. (2001) utilized ANNs to model short-term water demand at the Indian Institute of Technology (IIT) in Kanpur, India. They developed and compared six neural network models, five regression models, and two-time series models. The results indicated that the neural network models consistently outperformed the other models in terms of performance. (Bougadis et al. 2005) conducted comparable analyses to predict both peak daily demand and weekly water demand. Their findings revealed that ANN models outperformed time series and regression models, offering superior results. Furthermore, numerous other researchers, such as Chang & Makkeasorn (2006), Zhang et al. (2004) and Lui et al. (2003), among others, have also acknowledged and reported the success of ANN in water demand prediction and forecasting.

Almheiri et al. (2020) predict the average time to failure using three techniques: ANN, ridge regression and decision trees (DT). After applying these methods to a real case study, the authors recommend DT because of their simplicity and computational efficiency. Spatial clustering is commonly used to identify regions with high-failure rates (De Oliveira et al. 2011). This technique usually serves as a support to other predictive models, providing additional input information. Giraldo & Rodriguez (2020) used k-means clustering to create groups of pipes with similar characteristics and then estimate the total number of failures of each group using three regression models: linear regression, Poisson regression and evolutionary polynomial regression (EPR) where Poisson regression shows a superior accuracy compared to the other two models. Chen & Guikema (2020) merge spatial clustering and regression models to predict the number of pipe breaks in a real water network of the USA. Experts in the field have expressed their commitment to improve water network databases, mainly aided by advances in GIS (Barton et al. 2022).

Limited applications of a comparative study is performed using ANN and ANFIS in modeling pipeline failure trend exist, unlike in many hydrological applications. Tabesh et al. (2009) employed ANN and ANFIS to model pipe failure rate, considering five input parameters. They compared the results with a multivariate regression approach. Jafar et al. (2010) demonstrated an application of ANN in modeling the failure rate of 4,862 urban mains using a 14-year database from a city in northern France. Their study also involved estimating the optimal replacement time for individual pipes in an urban WDS. Ansari et al. (2020) employed a combination of the ANFIS model with GA and PSO optimization algorithms to enhance the accuracy of predicting influential variables in wastewater treatment plants. In addition to the use of ANN models, FIS models are also extensively employed in water science. Researchers have investigated the risk of water quality failure, pipe failure in the WDN, and the potential for leakage within the network using FIS models (Sadiq et al. 2007; Fares & Zayed 2010; Islam et al. 2011; Valis 2013; Zangenehmadar & Moselhi 2016; Pandey et al. 2020).

Numerous studies have been conducted to explore the application of statistical and soft models in various areas, including leak detection, calibration of pipe roughness coefficient, water quality prediction, and prediction of network pipe failure rate. The findings from these investigations demonstrate that hybrid models can achieve highly accurate predictions of diverse phenomena (Kapelan et al. 2003; Tu et al. 2005; Berardi et al. 2008; Xu et al. 2011, 2013; Soltani & Tabari 2012; Farmani et al. 2017). Despite the previous studies, very few studies performed use of data-driven methods (ANN and ANFIS) to predict velocity and pressure in a WDN by using ANN and ANFIS.

In this paper, advanced artificial intelligence (AI) techniques are used, including ANFIS, ANN-LM, ANN-BR, and ANN-SCG, to predict velocity and pressure. For case 1, Pipe diameter and flow rate are selected as independent variables, and velocity as dependent variable. For case 2, Node demand and node elevation are selected as independent variables, and pressure as dependent variable. 80% of the data is used to train, test, and validate the ANN and ANFIS models, while the remaining 20% is used to evaluate the models. The best model is selected based on statistical performance such as root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R²). Sensitivity analysis is performed using the best model obtained after statistical performance to assess the influence of each independent variable on velocity and pressure.

The pipe and node details of Gadhra WDN (DMA-5) are given in Supplementary material, Appendix A (Tables A1 and A2).

The second set of constraints is based on the law of energy conservation, which states that the sum of friction loss and the minor loss due to valves minus the grade difference between the two points of known energy minus the energy added to the liquid by the pump must equal zero.

The function “v” is used to indicate the weights connected to each input and the total bias, where the weights for each connection are w₁, w₂,…, and w_n. The input vector is given by (X = x₁, x₂, …, x_n), the weight vector is given by (W′ = w₁, w₂,…, w_n), and “b” stands for the bias as shown in Equation (7). The activation function is represented by “f”.

The process of training the model involves two steps in order to achieve the desired output by minimizing the error. The first step entails the forward computation of input weights, while the second step involves the backward process of updating weights based on the obtained error (Shaik et al. 2020). The fundamental structure of the Feedforward network comprises an input layer, a hidden layer, an output layer, and their connections, which are governed by adjustable synaptic weights. Each layer consists of a specific number of neurons or nodes that facilitate information exchange and decision-making. Each node receives inputs from its preceding nodes, calculates the weighted sum of the inputs along with the added bias, and then passes the result through an activation function at each layer to generate the intended prediction. The process is iteratively performed using randomly generated weights until the desired output is achieved. Due to its nature as a black box model (Ibnelouad et al. 2021), this technique effectively discovers the intricate and valuable relationship between the input and output variables in order to attain the target output.

The Levenberg–Marquardt (LM) algorithm was specifically developed to achieve second-order training speed without the need for calculating the Hessian matrix. In cases where the performance function can be expressed as a sum of squares, an approximation of the Hessian matrix can be utilized, and the gradient can be computed following the approach described in (Hagan & Menhaj 1994; Kisi & Uncuoghlu 2005) as shown in Equations (9) and (10).

The Bayesian Regularization (BR) training algorithm incorporates the principles of LM optimization (MacKay 1992; Foresee & Hagan 1997) to update the weights and bias values. It aims to minimize a combination of squared errors and weights, seeking the optimal combination that leads to a well-generalizing network (Pan et al. 2013). Additionally, BR introduces network weights into the training objective function, referred to as F(ω) in Equation (12).

In the BR framework, represents the sum of the network weights squared, while represents the sum of network errors. Both α and β serve as parameters for the objective function. Within this framework, the network weights are perceived as random variables, and the distribution of the weights and training set follows a Gaussian distribution. The α and β factors are defined using Bayes’ theorem, which establishes a relationship between two variables or events, A and B. This relationship is based on their prior or marginal probabilities and posterior or conditional probabilities, as described in Equation (13) (Li & Shi 2012).

To determine the optimal weight space, it is necessary to minimize objective function (12), which is equivalent to probability function denoted in Equation (14).

Here, P(A|B) represents the posterior probability of A given B, P(B|A) represents the prior probability of B given A, and P(B) represents the non-zero prior probability of event B, serving as a normalizing constant.

The fundamental backpropagation algorithm modifies the weights by moving in the direction of steepest descent, which corresponds to the negative gradient. In most CG algorithms, the step size is adapted during each iteration. The search is conducted along the conjugate gradient direction to find the step size that minimizes the performance function along that particular path. Initially, all CG algorithms begin by searching in the direction of steepest descent in the first iteration as shown in Equation (15). Frequently, CG algorithms employ line search techniques, approximating the step size without calculating the Hessian matrix to determine the optimal distance for movement along the current search direction as shown in Equation (16). Subsequently, the subsequent search direction is chosen to be conjugate to the previous search direction as shown in Equation (17). The general procedure for determining the new search direction involves combining the new steepest descent direction with the previous search direction (Hagan et al. 1996).

Figure 1 illustrates the layout structure of ANFIS, comprising five distinct layers. Layer 1 is responsible for identifying the input and output variables and determining their descriptors. Layer 2 defines the membership functions for each input and output variable. Layer 3 constructs the rule base. Layer 4 performs Rule Evaluation, and the final layer, Layer 5, conducts Defuzzification.

Fuzzy MFs are of different shapes such as Gaussian, triangular, trapezoidal.

The two-way analysis of variance (ANOVA) normally scrutinizes the influence of two independent variables on their outcome. In the present study, the effect of two independent variables (i.e., diameter and flow rate) on a dependent variable (velocity) for case 1 and the effect of two independent variables (i.e., elevation and demand) on a dependent variable (pressure) for case 2 has been determined by two-way ANOVA test at 5% significant level (α) as shown in Table 1.

In the present study, the F-value obtained from the ANOVA test is greater than the F-critical and the p-value is less than 0.05 at the significance level is 5% indicating that the null hypothesis is rejected.

In WaterGEMS, the simulated output, denoted as represents the results obtained from hydraulic simulation. The mean of the simulated output, represented as ⁠. In MATLAB, the target output, denoted as⁠, represents the desired results after optimization. The mean of the target output, represented as ⁠. The values of n represent the number of data points used in the hydraulic simulation for pipe links and nodes, which are used to calculate velocity and pressure, respectively.

A neural network was used to predict velocity and pressure using the backpropagation algorithm, includingLM, BR, and SCG. The parameters used in the model were collected from hydraulic design and distribution network data, and are shown in Table 2. The model has three layers: independent, hidden, and dependent. Neurons in the hidden layer use weights (w) and biases (b) to compute the results of a neural network. The input layer provides the hidden layer with data, and the hidden layer then passes its output to the output layer as shown in Figure 6.

The neural network was trained using the training data. The results were then adapted according to any changes identified in the model during ongoing training. To obtain the best possible ANN predictive model for velocity and pressure, we tested different values of the number of neurons in the hidden layer and learning algorithms. The results for training, test and all are shown in Tables 3 and 4 for (FFBN-ANN) considering LM, BR and SCG algorithms.

Prior to running ANFIS on the dataset, the 74 pipe/link data and 73 node/junction data are divided into training and checking datasets. This ensured that the range of input and output data in the training dataset is representative of the range of input and output data in the checking dataset. This is important because it ensured that the checking samples could accurately represent the population and generate generalized ANFIS velocity and pressure models that could accurately predict velocity and pressure in the WDN.

The model generation process began after the dataset was classified. In the current scenario, for case 1, 74 data points are divided into 41 training samples, 10 testing samples, and 10 checking samples which are 80% of the total pipe data. For case 2, 73 data points are divided into 41 training, 10 testing samples, and 10 checking which are 80% of total node data. To generate the ANFIS model, training dataset is used and to assess the validity of the model and calculate the error, checking dataset is used in velocity and pressure prediction for case 1 and case 2, respectively.

During the sensitivity analysis of an independent variable, all other independent variables are set to their mean values as indicated in Tables 8 and 9. The slope of the graph that shows the relationship between the independent variable and the observed velocity is used to assess the individual effect of a one-unit change in the independent variable on velocity and pressure when all other variables are held constant. The results reveal that flowrate exhibits a positive correlation with velocity, while diameter demonstrates a negative correlation. Similarly, pressure shows a positive correlation with demand and a negative correlation with elevation. Tables 10 and 11 provide a comprehensive summary of the magnitude of change in velocity and pressure corresponding to a unit change in the value of each independent variable. Of all data-driven models, the ANN-LM perform best for velocity and pressure in terms of statistical performance therefore sensitivity analysis is carried out for ANN-LM.

Figures 18 and 19 show that there is a positive correlation of 0.016 m/s per cumec with flow rate and a negative correlation of (−) 0.01 m/s per 0.1 m with diameter when used as independent variable while predicting velocity. Figures 20 and 21 show that there is a positive correlation of 0.03 m per 0.0001 MLD with demand and a negative correlation of (−)0.51 m per m with elevation when used as independent variable while predicting pressure.

The effective management of WDSs plays a crucial role in enhancing water use efficiency in residential areas. Hence, this study introduces a seldom approach to assess the statistical performance of WDSs using data-driven models such as ANN-LM, ANN-BR, ANN-SCG, and ANFIS. The Gadhra WDN was employed to train and validate these models, and the data were collected from the DW&S, Jharkhand Government.

The results of this study can be summarized as follows:

• (1)

  In the ANN modeling, the ANN-LM model outperforms the ANN-BR and ANN-SCG models in predicting velocity and pressure. The number of hidden layer neurons and the type of transfer functions used in the hidden and output layers significantly impact the performance of the ANN. The ANN-LM method exhibits the best prediction accuracy for estimating velocity (RMSE = 0.0189, MAE = 0.0122, R² = 0.9568) and pressure (RMSE = 0.3244, MAE = 0.2176, R² = 0.9773) when considering both training and testing data.

• (2)

  The ANFIS model also demonstrates satisfactory performance in predicting velocity and pressure. By utilizing the Gaussian MF instead of the triangle and trapezoidal functions and increasing the number of membership functions in the univariate output mode, the model's performance improves. The ANFIS method yields reliable predictions for velocity (RMSE = 0.0201, MAE = 0.0215, R² = 0.8745) and pressure (RMSE = 0.5520, MAE = 0.4472, R² = 0.9336) when considering both training and testing data.

• (3)

  The findings of the sensitivity analysis are significant as they identify potential parameters that influence velocity and pressure and provide numerical estimates of the magnitude of each independent variable's impact on these quantities. The sensitivity analysis reveals the following relationships for parameters (flow rate, diameter, demand, and elevation): ▴V/▴Q = 0.016 m/s per cumec, ▴V/▴d = (−)0.01 m/s per 0.01 m, ▴P/▴D = 0.03 m per 0.0001 MLD, and ▴P/▴Elev. = (−) .51 m per m. Utilizing this prediction model will aid in adjusting the values of these variables, facilitating effective management of velocity and pressure.

The prediction model obtained for velocity and pressure has a good statistical performance by using four independent parameters (i.e., diameter, flow rate, elevation, and demand). The performance of the model could have been enhanced if the model is also predicted for head loss including velocity and pressure.

This study demonstrates the reliability and practicality of ANN and ANFIS models for evaluating the performance of WDSs. While various data-driven models, including ANN-LM, ANN-BR, ANN-SCG, and ANFIS, performed well, ANN-LM exhibited lower errors and higher accuracy than all the other models. Therefore, it is advisable to prioritize the use of the ANN-LM model in future studies. The findings of this study serve as a valuable guide for selecting an appropriate model to assess the performance of WDNs. Consequently, instead of relying on time-consuming and complex conventional methods, it is recommended to employ ANN-LM, ANN-BR, ANN-SCG, and ANFIS models for evaluating the performance of WDNs.

The authors are thankful to the reviewers for their valuable suggestion which has enhanced the quality of this manuscript The authors are thankful to the *DW&S of the Jharkhand government for providing the details of Gadhra Water Distribution Networks.

All authors contributed to the study conception and design. Data collection was performed by A.R. and S.K. Analysis and optimization were performed by A.R. and S.K. The first draft of the manuscript was written by A.R. All authors read and approved the final manuscript.

Funding not received from any agency.

All analyses were made by licensed software WaterGEMS and MATLAB.

Authors gave their permission.

Authors gave their permission.

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

The authors declare there is no conflict.