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.@en

In this paper, a new mathematical framework is proposed for maximizing the self-cleaning capacity (SCC) of drinking water distribution systems by controlling the diurnal peak flow velocities in the pipes under normal operation. This is achieved through an optimal change of the network connectivity (topology). This paper proposes an efficient algorithm for the network analysis of valve closures, which allows enforcing favorable changes in the flow velocities for maximizing the SCC by determining an optimal set of links to isolate in the forming of a more branched network, while concurrently satisfying the hydraulic and regulatory pressure constraints at the demand nodes. Multiple stochastic demands from an end-use demand model are generated to test the robustness in the improved SCC for the modified network connectivity under changing demand. An operational network model is used to demonstrate the efficacy of the proposed approach.@en

This manuscript investigates the problem of optimal placement of control valves in water supply networks, where the objective is to minimize average zone pressure (AZP). The problem formulation results in a nonconvex mixed integer nonlinear program (MINLP). Due to its complex mathematical structure, previous literature has solved this nonconvex MINLP using heuristics or local optimization methods, which do not provide guarantees on the global optimality of the computed valve configurations. In our approach, we implement a branch and bound method to obtain certified bounds on the optimality gap of the solutions. The algorithm relies on the solution of mixed integer linear programs, whose formulations include linear relaxations of the nonconvex hydraulic constraints. We investigate the implementation and performance of different linear relaxation schemes. In addition, a tailored domain reduction procedure is implemented to tighten the relaxations. The developed methods are evaluated using two benchmark water supply networks and an operational water supply network from the UK. The proposed approaches are shown to outperform state-of-the-art global optimization solvers for the considered benchmark water supply networks. The branch and bound algorithm converges to good quality feasible solutions in most instances, with bounds on the optimality gap that are comparable to the level of parameter uncertainty usually experienced in water supply network models.@en

In this paper, we investigate the application of penalty and relaxation methods to the problem of optimal placement and operation of control valves in water supply networks, where the minimization of average zone pressure is the objective. The optimization framework considers both the location and settings of control valves as decision variables. Hydraulic conservation laws are enforced as nonlinear constraints and binary variables are used to model the placement of control valves, resulting in a mixed-integer nonlinear program. We review and discuss theoretical and algorithmic properties of two solution approaches. These include penalty and relaxation methods that solve a sequence of nonlinear programs whose stationary points converge to a stationary point of the original mixed-integer program. We implement and evaluate the algorithms using a benchmarking water supply network. In addition, the performance of different update strategies for the penalty and relaxation parameters are investigated under multiple initial conditions. Practical recommendations on the numerical implementation are provided.

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In this paper, we consider the proactive control of flow velocities to maximise the self-cleaning capacity of the drinking water distribution systems under normal operations both through a change of the network topology and through an optimal control of pressure reducing valve (PRV) settings. Inspired by line outage flow distribution in electrical networks, we show how a fast network graph analysis of link closures can be used to estimate the potential changes in flow velocities, which are then used to determine the most favourable pipes for closure. Where closing of pipes cannot be used because of other conflicting objectives, we consider the optimal control of PRVs to maximise self-cleaning at peak demand periods. We formulate a novel optimisation problem to maximise the network operations for increased self-cleaning capacity, while satisfying hydraulic and regulatory pressure constraints at demand nodes. A new smooth objective function approximation for cleaning capacity of the network is proposed along with a scalable sequential convex programming method to solve the resulting valve optimization problems. We use a published benchmark network as a case study to show the efficacy of these new approaches.@en

In this paper, we study the multiobjective co-design problem of optimal valve placement and operation in water distribution networks, addressing the minimization of average pressure and pressure variability indices. The presented formulation considers nodal pressures, pipe flows and valve locations as decision variables, where binary variables are used to model the placement of control valves. The resulting optimization problem is a multiobjective mixed integer nonlinear optimization problem. As conflicting objectives, average zone pressure and pressure variability can not be simultaneously optimized. Therefore, we present the concept of Pareto optima sets to investigate the trade-offs between the two conflicting objectives and evaluate the best compromise. We focus on the approximation of the Pareto front, the image of the Pareto optima set through the objective functions, using the weighted sum, normal boundary intersection and normalized normal constraint scalarization techniques. Each of the three methods relies on the solution of a series of single-objective optimization problems, which are mixed integer nonlinear programs (MINLPs) in our case. For the solution of each single-objective optimization problem, we implement a relaxation method that solves a sequence of nonlinear programs (NLPs) whose stationary points converge to a stationary point of the original MINLP. The relaxed NLPs have a sparse structure that come from the sparse water network graph constraints. In solving the large number of relaxed NLPs, sparsity is exploited by tailored techniques to improve the performance of the algorithms further and render the approaches scalable for large scale networks. The features of the proposed scalarization approaches are evaluated using a published benchmarking network model.

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Changes in power generation and supply and changes in water distribution systems are creating new opportunities for water utilities to enhance operational efficiency and income through the use of advanced control and optimisation. First, the increase in renewables penetration into the grid is causing a growth in energy storage schemes. Second, variable speed pumps are now fitted to most new systems and many existing water distribution systems are being retrofitted with variable speed pumps to improve the efficiency of the operation. We study how these trends can be jointly exploited to provide energy storage from a water distribution system. We investigate how variable speed drive pumps can enhance the ability of a water distribution network to provide demand response energy to the grid. We show that demand response provision from water distribution systems can be improved through the use of variable speed pumps. We also demonstrate that a network equipped with variable speed pumps can provide demand response profitably across a wider range of operating scenarios compared to a network equipped with only fixed-speed pumps. The results highlight another potential benefit of variable speed pumps in water distribution systems.@en

In the present manuscript, we {\color{red} investigate and demonstrate the use of outer approximation methods} for simultaneously optimising the placement and operation of control valves in water distribution networks. The problem definition results in a mixed-integer nonlinear program with nonconvex constraints. We simplify the formulation, compared to previous literature, in order to reduce the degree of nonlinearity in the constraints and decrease the total problem size. We then {\color{red} formulate the application of} outer approximation based methods for the generation of good quality local optimal solutions for {\color{red} the considered co-design problem}. Finally, we present the results of applying the developed techniques to two case studies, and also comparing the performances of the outer approximation approaches with those of other local mixed integer nonlinear programming solution methods.@en

For electricity grids with an increasing share of intermittent renewables, the power generation mix can have significant daily variations. This leads to time-dependent emission intensities and volatile electricity prices in the day-ahead and spot market tariffs that can be better utilised by energy intensive industries such as water supply utilities. A multi-objective optimisation method for scheduling the operation of pumps is investigated in this paper for the reduction of both electricity costs and greenhouse gas emissions for a benchmark water distribution system. A set of energy supply scenarios has been formulated based on future projections from National Grid plc (UK) in order to investigate the range of cost savings and emission reductions that could be possibly achieved. Pump scheduling options with fixed time-of-use and day ahead market tariffs are analysed in order to compare potential reduction tradeoffs for both electricity costs and greenhouse gas emissions using Pareto optimality. The presented analysis concludes that the explicit inclusion of greenhouse gas emission reductions in optimising the scheduling of pumps operation in water distribution systems could provide considerable benefits; however, more compelling fiscal and regulatory incentives are needed.@en

Significant changes in the power generation mix are posing new challenges for the balancing systems of the grid. Many of these challenges are in the secondary electricity grid regulation services and could be met through demand response (DR) services. We explore the opportunities for a water distribution system (WDS) to provide balancing services with demand response through pump scheduling and evaluate the associated benefits. Using a benchmark network and demand response mechanisms available in the UK, these benefits are assessed in terms of reduced green house gas (GHG) emissions from the grid due to the displacement of more polluting power sources and additional revenues for water utilities. The optimal pump scheduling problem is formulated as a mixed-integer optimisation problem and solved using a branch and bound algorithm. This new formulation finds the optimal level of power capacity to commit to the provision of demand response for a range of reserve energy provision and frequency response schemes offered in the UK. For the first time we show that DR from WDS can offer financial benefits to WDS operators while providing response energy to the grid with less greenhouse gas emissions than competing reserve energy technologies. Using a Monte Carlo simulation based on data from 2014, we demonstrate that the cost of providing the storage energy is less than the financial compensation available for the equivalent energy supply. The GHG emissions from the demand response provision from a WDS are also shown to be smaller than those of contemporary competing technologies such as open cycle gas turbines. The demand response services considered vary in their response time and duration as well as commitment requirements. The financial viability of a demand response service committed continuously is shown to be strongly dependent on the utilisation of the pumps and the electricity tariffs used by water utilities. Through the analysis of range of water demand scenarios and financial incentives using real market data, we demonstrate how a WDS can participate in a demand response scheme and generate financial gains and environmental benefits.

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In this article, a comprehensive review of existing methods is presented and computationally efficient sparse null space algorithms are proposed for the hydraulic analysis of water distribution networks. The linear systems at each iteration of the Newton method for nonlinear equations are solved using a null space algorithm. The sparsity structure of these linear equations, which arises from the sparse network connectivity, is exploited to reduce computations. A significant fraction of the total flops in the Newton method are spent in computing pipe head losses and matrix-matrix multiplications involving flows. Because most flows converge after a few iterations, a novel partial update of head losses and matrix products is used to further reduce computational complexity. Convergence analyses are also presented for the partialupdate formulas. A new heuristic for reducing the number of pressure head computations of a null space method is proposed. These savings enable fast near-real-time control of large-scale water networks. It is often observed that the linear equations that arise in solving the hydraulic equations become ill-conditioned due to hydraulic solutions with very small and zero flows. The condition numbers of the Newton equations are bounded using a regularization technique with insignificant computational overheads. The convergence properties of all proposed algorithms are analyzed by posing them as an inexact-Newton method. Small-scale to large-scale models of operational water networks are used to evaluate the proposed algorithms.

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Water utilities face a challenge in maintaining a good quality of service under a wide range of operational management and failure conditions. Tools for assessing the resilience of water distribution networks are therefore essential for both operational and maintenance optimization. In this paper, a novel graph-theoretic approach for the assessment of resilience for large scale water distribution networks is presented. This is of great importance for the management of large scale water distribution systems, most models containing up to hundreds of thousands of pipes and nodes. The proposed framework is mainly based on quantifying the redundancy and capacity of all possible routes from demand nodes to their supply sources. This approach works well with large network sizes since it does not rely on precise hydraulic simulations, which require complex calibration processes and computation, while remaining meaningful from a physical and a topological point of view. The proposal is also tailored for the analysis of sectorised networks through a novel multiscale method for analysing connectivity, which is successfully tested in operational utility network models made of more than 100,000 nodes and 110,000 pipes.

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Water utilities can achieve significant savings in operating costs by optimising pump scheduling to improve efficiency and shift electricity consumption to low-tariff periods. Due to the complexity of the optimal scheduling problem, heuristic methods that cannot guarantee global optimality are often applied. This paper investigates formulations of the pump scheduling problem solved using a branch and bound method. Piecewise linear component approximations outperform non-linear approximations within application driven accuracy bounds and demand uncertainties. It is shown that the reduction of symmetry through the grouping of pumps significantly reduces the computational effort, whereas loops in the network have the opposite effect. The computational effort of including convex, non-linear pump operating, and maintenance cost functions is investigated. Using case studies, it is shown that linear and fixed-cost functions can be used to find schedules which, when simulated in a full hydraulic simulation, have performances that are within the solver optimality gap and the uncertainty of demand forecasts.

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Many sequential mathematical optimization methods and simulation-based heuristics for optimal control and design of water distribution networks rely on a large number of hydraulic simulations. In this paper, we propose an efficient inexact subspace Newton method for hydraulic analysis of water distribution networks. By using sparse and well-conditioned fundamental null space bases, we solve the nonlinear system of hydraulic equations in a lower-dimensional kernel space of the network incidence matrix. In the inexact framework, the Newton steps are determined by solving the Newton equations only approximately using an iterative linear solver. Since large water network models are inherently badly scaled, a Jacobian regularization is employed to improve the condition number of these linear systems and guarantee positive definiteness. After presenting a convergence analysis of the regularised inexact Newton method, we use the conjugate gradient (CG) method to solve the sparse reduced Newton linear systems. Since CG is not effective without good preconditioners, we propose tailored constraint preconditioners that are computationally cheap because they are based only on invariant properties of the null space linear systems and do not change with flows and pressures. The preconditioners are shown to improve the distribution of eigenvalues of the linear systems and so enable a more efficient use of the CG solver. Since contiguous Newton iterates can have similar solutions, each CG call is warm-started with the solution for a previous Newton iterate to accelerate its convergence rate. Operational network models are used to show the efficacy of the proposed preconditioners and the warm-starting strategy in reducing computational effort.@en

Increasingly more variable speed drive pumps are installed in water distribution systems worldwide. However, the modeling of variable speed drives in such networks remains difficult, especially in the context of mathematical optimization for pump scheduling. For the problem of energy usage minimization, formulated as a mixed integer program, we propose a new simplified convex relaxation of the hydraulic characteristics of a variable speed pump. By using different model approximations for the power curve, as a function of pump speed and volumetric flow rate, we study the trade-off in the computational complexity and quality of solutions obtained. We use two small benchmark networks and real pump data from three different pumps to calculate operating schedules. We show that these schedules, are computed in a time suitable for operational usage, enable operation with lower operating cost compared to fixed speed pumps, and their sub-optimality can be computed.

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The operation of water distribution networks (WDN) with a dynamic topology is a recently pioneered approach for the advanced management of District Metered Areas (DMAs) that integrates novel developments in hydraulic modeling, monitoring, optimization, and control. A common practice for leakage management is the sectorization of WDNs into small zones, called DMAs, by permanently closing isolation valves. This facilitates water companies to identify bursts and estimate leakage levels by measuring the inlet flow for each DMA. However, by permanently closing valves, a number of problems have been created including reduced resilience to failure and suboptimal pressure management. By introducing a dynamic topology to these zones, these disadvantages can be eliminated while still retaining the DMA structure for leakage monitoring. In this paper, a novel optimization method based on sequential convex programming (SCP) is outlined for the control of a dynamic topology with the objective of reducing average zone pressure (AZP). A key attribute for control optimization is reliable convergence. To achieve this, the SCP method we propose guarantees that each optimization step is strictly feasible, resulting in improved convergence properties. By using a null space algorithm for hydraulic analyses, the computations required are also significantly reduced. The optimized control is actuated on a real WDN operated with a dynamic topology. This unique experimental program incorporates a number of technologies set up with the objective of investigating pioneering developments in WDN management. Preliminary results indicate AZP reductions for a dynamic topology of up to 6.5% over optimally controlled fixed topology DMAs.

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A dynamic topology aggregates zones in water distribution networks (WDNs) for improved pressure management and resilience to failure. Based on a sequential convex programming (SCP) approach, we propose an optimization method for the control of pressure reducing valves (PRV) in WDNs with dynamic topology. By restricting the SCP iterations to the feasible search space, we show that reliable convergence of the method is achieved. Using an experimental study in a large operational network, the optimization of PRV settings with a dynamic topology is shown to result in pressure reductions of 3.7% compared to optimized PRVs in a closed DMA structure.

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Hydraulic resilience can be formulated as a measure of the ability of a water distribution network to maintain a minimum level of service under operational and failure conditions. This paper explores a hybrid approach to bridge the gap between graph-theoretic and hydraulic measures of resilience. We extend the concept of geodesic distance of a pipeline by taking into account energy losses associated with flow. New random-walk algorithms evaluate hydraulically feasible routes and identify nodes with different levels of hydraulic resilience. The nodes with the lowest scores are further analysed by considering the availability and capacity of their supply routes.

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In this paper, we consider the use of an efficient null space algorithm for hydraulic analysis that employs preconditioned conjugate gradient (PCG) methods for solving the Newton linear equations. Since large water network models are inherently badly conditioned, a Jacobian regularization is employed to improve the condition number to some degree, this resulting in an inexact Newton method whose analyses is presented. Based on this analysis, constraint preconditioners are used to improve the condition number further for more efficient use of CG solvers. Operational networks are used to study the computational properties of the various approaches.

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