Water Distribution Network Models Research Articles

Model Reduction and Outer Approximation for Optimizing the Placement of Control Valves in Complex Water Networks

The optimal placement and operation of pressure control valves in water distribution networks is a challenging engineering problem. When formulated in a mathematical optimization framework, this problem results in a nonconvex mixed integer nonlinear program (MINLP), which has combinatorial computational complexity. As a result, the considered MINLP becomes particularly difficult to solve for large-scale looped operational networks. We extend and combine network model reduction techniques with the proposed optimization framework in order to lower the computational burden and enable the optimal placement and operation of control valves in these complex water distribution networks. An outer approximation algorithm is used to solve the considered MINLPs on reduced hydraulic models. We demonstrate that the restriction of the considered optimization problem on a reduced hydraulic model is not equivalent to solving the original larger MINLP, and its solution is therefore sub-optimal. Consequently, we investigate the trade-off between reducing computational complexity and the potential sub-optimality of the solutions that can be controlled with a parameter of the model reduction routine. The efficacy of the proposed method is evaluated using two large scale water distribution network models.

Global Gradient Algorithm Extension to Distributed Pressure Driven Pipe Demand Model

It has been proved that the standard representation of water demand in a Water Distribution Network (WDN) leads to pipe head loss errors as well that the fully satisfied demand regardless water pressure assumption is misleading. This follows that different algorithms have been developed in order to overcome these two drawbacks although separately and independently. Consequently, this paper introduces an alternative formulation of the Global Gradient Algorithm (GGA), referred to as UD-PD, which is able to solve uniformly distributed pressure driven demands along the pipes of a WDN in extended period simulations. In addition, this new scheme is tested against reference solutions and its performances are compared with the standard WDN models. Finally, the UD-PD is applied to a real WDN under pressure deficit conditions. Numerical results show that the hydraulic heads computed with the UD-PD result higher than those simulated with standard demand driven models and that the UD-PD is able both to capture the non linear behavior of the hydraulic head along the network and to correctly compute the flow inversion even in pressure driven conditions.

Economic MPC-LPV Control for the Operational Management of Water Distribution Networks

Abstract This paper presents an Economic Model Predictive Control (EMPC) for the operational management of water distribution networks (WDNs) with periodic operation based on embedding the nonlinearity of the model to the Linear Parameter Varying (LPV) model of WDNs. The performance of the WDN is identified by a set of difference-algebraic equations while the relation of hydraulic head/pressure and flow in connected pipes is nonlinear. In particular, the WDN model consists of two sections of nonlinear algebraic equations for bidirectional and unidirectional flows in pipes, respectively. The proposed algorithm is embedded the nonlinear algebraic equations into the LPV model. The proposed control approach allows the controller to accommodate the scheduling parameters. By computing the prediction of the state variables along a prediction time horizon, the system model can be modified according to the evaluation of the estimated state at each time instant. This iterative approach improves the implementation efficiency and reduces the computational burden compared to the solution of a non-linear optimization problem. Finally, the proposed strategy is applied to a well-known benchmark of the Richmond WDN. The closed-loop simulation results are shown and compared with other approaches by using several key performance indices.

A modeling and distributed MPC approach for water distribution networks

This paper deals with the modeling and control of water distribution networks consisting of multiple pressure zones that are interconnected by fixed-speed pumps. A simplified control-oriented model with linear dynamics and non-convex constraints which considers the pressures and losses in the zones is derived. Based on the model, a periodic distributed model predictive scheme which is tailored to the non-convex constraints and minimizes an economic objective is introduced. The scheme uses event-triggered communication between the subsystems to reduce the communicational load. The proposed model and controller are evaluated using a model of the water distribution network of the city of Kaiserslautern, Germany.

Comparison of Pressure-Driven Formulations for WDN Simulation

This paper presents the comparison of five pressure-driven formulations in the context of water distribution network (WDN) modelling. These formulations, which relate nodal outflow q to users to demands d and nodal pressure heads h, were implemented inside the global gradient algorithm for the snapshot solution of the equations concerning mass and energy conservation at WDN nodes and pipes, respectively. The modelling of leakage nodal outflows as a function of pressure was also considered. The applications concerned two case studies, in which nodal demands were suitably amplified to lower service pressure below the desired values. This was done to stress the effects of the pressure-driven dependence q(h) in the WDN. The results showed that the formulations tend to behave similarly in terms of nodal outflows. Compared to a widely used formulation, which features a q(h) relationship with derivative discontinuities, the other four formulations analyzed tend to guarantee faster algorithm convergence, above all for simple and poorly interconnected WDNs, due to their smooth q(h) relationship. The results in terms of nodal pressure heads can be very different, above all for low values of h.

Economic Model Predictive Control with Nonlinear Constraint Relaxation for the Operational Management of Water Distribution Networks

This paper presents the application of an economic model predictive control (MPC) for the operational management of water distribution networks (WDNs) with periodic operation and nonlinear constraint relaxation. In addition to minimizing operational costs, the proposed approach aims to reduce the computational load and to improve the implementation efficiency associated with the nonlinear nature of the MPC problem. The behavior of the WDN is characterized by a set of difference-algebraic equations, where the relation of hydraulic pressure/head and flow in interconnected pipes is nonlinear. Specifically, the considered WDN model includes two categories of nonlinear algebraic equations for unidirectional and bidirectional flows in pipes, respectively. In this paper, we propose an iterative algorithm to relax these nonlinear algebraic equations into a set of linear inequality constraints that will be implemented in the economic MPC design, which improves the implementation efficiency and meanwhile optimizes the economic performance. Finally, the proposed strategy is applied to a well-known benchmark of the Richmond WDN. The closed-loop simulation results are shown and the proposed strategy is also compared with a nonlinear economic MPC using several key performance indexes.

Topological Taxonomy of Water Distribution Networks

Water Distribution Networks (WDNs) can be regarded as complex networks and modeled as graphs. In this paper, Complex Network Theory is applied to characterize the behavior of WDNs from a topological point of view, reviewing some basic metrics, exploring their fundamental properties and the relationship between them. The crucial aim is to understand and describe the topology of WDNs and their structural organization to provide a novel tool of analysis which could help to find new solutions to several arduous problems of WDNs. The aim is to understand the role of the topological structure in the WDNs functioning. The methodology is applied to 21 existing networks and 13 literature networks. The comparison highlights some topological peculiarities and the possibility to define a set of best design parameters for ex-novo WDNs that could also be used to build hypothetical benchmark networks retaining the typical structure of real WDNs. Two well-known types of network ((a) square grid; and (b) random graph) are used for comparison, aiming at defining a possible mathematical model for WDNs. Finally, the interplay between topology and some performance requirements of WDNs is discussed.

Experimental analysis of pressure-discharge relationship in a private water supply tank

AbstractIn Mediterranean countries, users are often equipped with private tanks, which provide a temporary water storage capacity, able to compensate service interruptions due to either scarcity or irregularity of water supply. In the presence of private water storage, water supply is no longer linked to users' consumption and network-operating conditions can be off-design, therefore specific models have to be introduced in simulation models of water distribution networks. Here, a new mathematical model is proposed that is able to reproduce a tank's emptying/filling cycles. Specifically, by means of experimental analysis, a hyperbolic tangent law was tested to reproduce the filling process for private tanks. The flow rate is calculated by means of the classical Torricelli law, in which the float valve emitter coefficient and the valve area are calculated using a function that takes into account the water level within the private tank. The comparison obtained through the mathematical model and those observed from experiments confirmed the ability of the model to predict the flow rate balance within private tanks. The results show that the model is suitable for any length of float valve branch. The mathematical system can be easily used in a transient model to correctly estimate the supplied demand.

Water demand time series generation for distribution network modeling and water demand forecasting

Knowledge of water consumption can help water distribution network modeling. Consumption has been studied based on its stochastic and spatial random features. Considering the difficulty to have access to real-demand data, this work presents three methods to generate synthetically water demand time series from observed data, thus producing final demands in different ways. Each method generates a time series that considers either temporal consumption trends or jointly temporal trends and climatic influence. A random forest algorithm is applied to obtain the relevance of each climatic variable. This study uses water demand and climatic data of various Brazilian cities to extract temporal patterns. The final synthetically generated data can be used as input data for water network models, to feed the methods used according to the objectives of each study or project.

Wiener model of pressure management for water distribution network

In this paper, the pressure management model of water distribution network (WDN) is established, which relies on the hydraulic law of the whole WDN, not the relationship between the pressure and flow based on the leakage. In view of the specific WDN, the configuration is pumped into a closed system with pressure control. And the pressure management model of WDN is divided into two modules by all the tanks which are extracted from the WDN. The two modules are linear dynamic module and nonlinear static module in series. A Wiener model is proposed to describe the nonlinear dynamic relationship of the pressure management model. Moreover, the theoretical data calculated by the Wiener model are compared with the real data generated by EPANET to show the effectiveness and high accuracy of our proposed modelling method.

Wiener model of pressure management for water distribution network

In this paper, the pressure management model of water distribution network (WDN) is established, which relies on the hydraulic law of the whole WDN, not the relationship between the pressure and flow based on the leakage. In view of the specific WDN, the configuration is pumped into a closed system with pressure control. And the pressure management model of WDN is divided into two modules by all the tanks which are extracted from the WDN. The two modules are linear dynamic module and nonlinear static module in series. A Wiener model is proposed to describe the nonlinear dynamic relationship of the pressure management model. Moreover, the theoretical data calculated by the Wiener model are compared with the real data generated by EPANET to show the effectiveness and high accuracy of our proposed modelling method.

An Integrated Optimal Approach for Solar Powered Rural Water Distribution Systems in the Gambia

In the Gambia and across sub-Saharan Africa, reliable access to clean water and electrical power is constrained. As most rural water supply systems are built, enhanced understanding of efficiencies and optimisation is required. Here, methods of integrating estimations of power outputs from solar photovoltaic arrays into gravity-fed water distribution network modelling are investigated. The effects of powering a rural water distribution system that is replenished with groundwater pumps that use solar power are investigated, along with the effect of this on other network design decisions. The water storage tank and pipework of a rural community with an estimated 2800 people and 28 standpipes from a borehole was selected. EPANET modelling software and genetic algorithms were used to run network optimisation simulations of: water tank location, elevation and volume; pipe diameter and configuration; and optimal system design in terms of cost. Different scenarios were produced, including supply, demand and required water storage curves, which could have practical application for rural water distribution system design. Indicative costs for theoretical water distribution networks will be useful for decision makers and planners.

Limitations of demand- and pressure-driven modeling for large deficient networks

Abstract. The calculation of hydraulic state variables for a network is an important task in managing the distribution of potable water. Over the years the mathematical modeling process has been improved by numerous researchers for utilization in new computer applications and the more realistic modeling of water distribution networks. But, in spite of these continuous advances, there are still a number of physical phenomena that may not be tackled correctly by current models. This paper will take a closer look at the two modeling paradigms given by demand- and pressure-driven modeling. The basic equations are introduced and parallels are drawn with the optimization formulations from electrical engineering. These formulations guarantee the existence and uniqueness of the solution. One of the central questions of the French and German research project ResiWater is the investigation of the network resilience in the case of extreme events or disasters. Under such extraordinary conditions where models are pushed beyond their limits, we talk about deficient network models. Examples of deficient networks are given by highly regulated flow, leakage or pipe bursts and cases where pressure falls below the vapor pressure of water. These examples will be presented and analyzed on the solvability and physical correctness of the solution with respect to demand- and pressure-driven models.

Calibration Model for Water Distribution Network Using Pressures Estimated by Artificial Neural Networks

The success of hydraulic simulation models of water distribution networks is associated with the ability of these models to represent real systems accurately. To achieve this, the calibration phase is essential. Current calibration methods are based on minimizing the error between measured and simulated values of pressure and flow. This minimization is based on a search of parameter values to be calibrated, including pipe roughness, nodal demand, and leakage flow. The resulting hydraulic problem contains several variables. In addition, a limited set of known monitored pressure and flow values creates an indeterminate problem with more variables than equations. Seeking to address the lack of monitored data for the calibration of Water Distribution Networks (WDNs), this paper uses a meta-model based on an Artificial Neural Network (ANN) to estimate pressure on all nodes of a network. The calibration of pipe roughness applies a metaheuristic search method called Particle Swarm Optimization (PSO) to minimize the objective function represented by the difference between simulated and forecasted pressure values. The proposed method is evaluated at steady state and over an extended period for a real District Metering Area (DMA), named Campos do Conde II, and the hypothetical network named C-town, which is used as a benchmark for calibration studies.

On the choice of the demand and hydraulic modeling approach to WDN real‐time simulation

AbstractThis paper aims to analyze two demand modeling approaches, i.e., top‐down deterministic (TDA) and bottom‐up stochastic (BUA), with particular reference to their impact on the hydraulic modeling of water distribution networks (WDNs). In the applications, the hydraulic modeling is carried out through the extended period simulation (EPS) and unsteady flow modeling (UFM). Taking as benchmark the modeling conditions that are closest to the WDN's real operation (UFM + BUA), the analysis showed that the traditional use of EPS + TDA produces large pressure head and water discharge errors, which can be attenuated only when large temporal steps (up to 1 h in the case study) are used inside EPS. The use of EPS + BUA always yields better results. Indeed, EPS + BUA already gives a good approximation of the WDN's real operation when intermediate temporal steps (larger than 2 min in the case study) are used for the simulation. The trade‐off between consistency of results and computational burden makes EPS + BUA the most suitable tool for real‐time WDN simulation, while benefitting from data acquired through smart meters for the parameterization of demand generation models.

Robust Periodic Economic Predictive Control based on Interval Arithmetic for Water Distribution Networks

This paper addresses a robust periodic economic model predictive control (EMPC) based on interval arithmetic with unknown-but-bounded additive disturbances for the management of water distribution networks (WDNs). The system constraints in presence of system disturbances are tightened along the prediction horizon by means of the proposed interval arithmetic and considering that system variables in the WDN model are also subject to some algebraic equations. These algebraic equations should be satisfied when the disturbances have effects on those system variables. The EMPC controller is designed with adding the terminal state constraint. The periodically optimal steady states are obtained by employing a periodic EMPC planner with the nominal model. This periodic steady states are subsequently used to be terminal states. The on-line robust constraint satisfactions are implemented into the robust periodic EMPC controller. Finally, the proposed control strategy is verified using a case study.

An Interval Estimator for Chlorine Monitoring in Drinking Water Distribution Systems Under Uncertain System Dynamics, Inputs and Chlorine Concentration Measurement Errors

Abstract The design of an interval observer for estimation of unmeasured state variables with application to drinking water distribution systems is described. In particular, the design process of such an observer is considered for estimation of the water quality described by the concentration of free chlorine. The interval observer is derived to produce the robust interval bounds on the estimated water quality state variables. The stability and robustness of the interval observer are investigated under uncertainty in system dynamics, inputs, initial conditions and measurement errors. The bounds on the estimated variables are generated by solving two systems of first-order ordinary differential equations. For that reason, despite a large scale of the systems, the numerical efficiency is sufficient for the on-line monitoring of the water quality. Finally, in order to validate the performance of the observer, it is applied to the model of a real water distribution network.

Multilinear Method for Hydraulic Analysis of Pipe Networks

AbstractSteady-state modeling of water distribution networks (WDNs) is the calculation of flow rates in pipes and nodal pressures for a given set of boundary conditions (i.e., water levels, pump cu...

Water Quality Modeling Considering Incomplete Mixing in Extended Periods

Diverse studies have shown that mixing within pipe junctions is neither complete nor instantaneous. However, these studies have focused in understanding mixing phenomena at junctions, without applying these results in water quality modeling of water distribution networks. This paper focuses on the application of both the complete and incomplete mixing models to model water quality in water distribution networks. Hydraulic networks are modelled in EPANET and EPANET-BAM, and the concentrations and their distributions are compared. Results show that there is little difference between the two mixing models, so water quality modeling might be done by using EPANET and its complete mixing model.

Feasibility of Mass Balance Approach to Water Distribution Network Model Calibration

The increase of real water losses in water distribution networks (WDNs) is worrisome for social community and water utilities. Nowadays, enhanced WDN hydraulic models allow simulating pressure-dependent background leakages, thus, being of direct relevance to support water loss reduction actions. Unfortunately, traditional calibration approaches rely on demand-driven analyses and do not account for additional parameters of the pressure-dependent background leakage models. This work proposes a novel framework for calibrating enhanced WDN models which permits to pursue mass-balance (matching of flow observations at water sources) by accounting for pressure-dependent water demand components and demand patterns. At the same time, energy balance (matching of pressure observations) allows to get the realistic prediction of all water demands and background leakages. The novel calibration strategy involves the simultaneous estimation of pipe hydraulic resistances, parameters of the background leakage model and relevant customer demand patterns. The feasibility of the strategy is demonstrated on a real WDN located in Southern Italy.