Global optimality bounds for the placement of control valves in water supply networks
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Abstract
This manuscript investigates the problem of optimal placement of control valves in water supply networks, where the objective is to minimize average zone pressure. 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 stateoftheart 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.
Keywords
Global optimization Mixedinteger nonlinear programming Valve placement Pressure management Water supply networks1 Introduction
The efficient management of hydraulic pressure in pipes results in reduction of leakage (Lambert 2000; Wright et al. 2015) and risk of pipe failure (Lambert and Thornton 2011), and it is therefore one of the main operational challenges in water supply networks (WSNs). Here we consider pressure management using pressure control valves, which regulate pressure at their outlet. We investigate the problem of simultaneously optimizing the placement and operational settings of control valves in WSNs, where the objective is to minimize average zone pressure (AZP). AZP is used as a surrogate measure for leakage. The problem formulation includes flows across network links and hydraulic heads at nodes as continuous decision variables. In addition, binary variables are introduced to model the placement of valves. Mass and energy conservation laws are enforced as optimization constraints, resulting in a nonconvex mixed integer nonlinear program (MINLP). The solution of process network optimization problems frequently relies on the solution of MINLPs. Some examples include synthesis of heat exchanger networks (Zamora and Grossmann 1998), multiperiod blending (Kolodziej et al. 2013), optimal design and operation of gas networks (Pfetsch et al. 2015; Humpola and Fügenschuh 2015), and water supply networks (DAmbrosio et al. 2015). In the framework of WSNs, MINLP formulations are ubiquitous and employed in a variety of applications, ranging from optimal network design (Bragalli et al. 2012; Sherali et al. 1999) to pump scheduling (Menke et al. 2015; Gleixner et al. 2012).
Both heuristic and mathematical optimization methods were applied in previous work to solve the problem of optimal valve placement in water networks. Heuristic approaches based on genetic algorithms (GAs) have been widely used for solving the considered problem—see Reis et al. (1997), Araujo et al. (2006), Nicolini and Zovatto (2009), Liberatore and Sechi (2009), Ali (2015), De Paola et al. (2017). However, they present some limitations. Firstly, they can not guarantee optimality of the computed solutions, not even local optimality. Moreover, the number of objective function evaluations and hydraulic simulations required by these approaches grows rapidly with the size of the network, precluding the application of GAs when large operational water network are considered. Previous work has also investigated the application of mathematical optimization methods for the solution of the problem of optimal valve placement in WSNs—see Hindi and Hamam (1991), Eck and Mevissen (2012), Dai and Li (2014), Pecci et al. (2017a, b). Since the considered problem is nonconvex, approaches implemented in previous literature do not provide theoretical guarantees on the global optimality of the computed valve configuration.
This paper investigates mathematical optimization methods to generate a certified bound on the optimality gap of the computed solutions for the problem of optimal valve placement in WSNs, guaranteeing \(\varepsilon\)suboptimality. We formulate a branch and bound method, which is a common approach in global optimization. To the best of the authors knowledge, global optimization methods have not been previously applied to the problem of optimal valve placement in WSNs. Previous literature has investigated global optimization techniques for optimal design of WSNs (Sherali et al. 1999; Raghunathan 2013). However, when pipe diameters are fixed, hydraulic heads and flow rates are uniquely determined and can be found by solving a strictly convex optimization problem (Raghunathan 2013). As a result, in the case of optimal WSN design, it is sufficient to focus the branch and bound on integer decision variables. On the contrary, when optimal operation of water supply networks is considered, spatial branch and bound is needed (Gleixner et al. 2012). In the present manuscript, we consider the problem of optimal valve placement in WSNs, where locations and operational settings of the control valves need to be simultaneously optimized. Therefore, branching is required on both continuous and integer variables.
The implemented branch and bound algorithm relies on a sequence of lower and upper bounds to the optimal value of the nonconvex MINLP in study—for a general review see Tawarmalani and Sahinidis (2002). Since all convex constraints within the problem formulation for optimal valve placement are linear, it is particularly convenient to generate lower bounds using linear relaxations of the nonconvex constraints (Tawarmalani and Sahinidis 2002, Chapter 4). The nonconvexity of MINLPs arising in the framework of water networks is due to the absolute power functions representing friction energy losses within the system’s conservation laws—see Eq. (2a). Analogous nonconvex expressions have previously been studied in other engineering frameworks, where linear relaxations were formulated—see Humpola and Fügenschuh (2015), Gleixner et al. (2012), Liberti and Pantelides (2003), Tawarmalani and Sahinidis (2002), Udell and Boyd (2015), Vigerske (2012). We define linear relaxations of the nonconvex equality constraints considered here by extending the formulation proposed in Liberti and Pantelides (2003) for monomials of odd degree. Such linear relaxations define an outer approximation of the convex envelopes of the nonconvex equality constraints. We investigate the use of different number of linearizations for the outer approximation. The strength of the linear relaxations depends on the diameter of the decision variables’ domain (Puranik and Sahinidis 2017). Therefore, we implement a domain reduction procedure, based on the solution of a series of linear programs (LPs). The proposed approach takes advantage of the underlying network structure to reduce the number of linear programming solves. Benefits and limitations of the developed methods are evaluated using two benchmark water networks, and a largescale operational network from the UK. Moreover, the numerical results show that the proposed approach outperforms stateoftheart global optimization solvers for the considered benchmark water networks. The branch and bound framework has enabled the convergence to \(\varepsilon\)suboptimal solutions for the problem of optimal valve placement, with bounds on the optimality gap comparable to the order of parameter and data uncertainties inherent in operational network models.
2 Problem formulation
A water supply network with \(n_0\) water sources, \(n_n\) demand nodes and \(n_p\) pipes, is modelled as a directed graph with \(n_n+n_0\) nodes and \(n_p\) edges. The operation of a network is considered under \(n_l\) different demand conditions during the diurnal cycle. The nodal demands are denoted by \(d^t \in {\mathbb {R}}^{n_n}\), while known hydraulic heads at water sources are indicated by \(h_0^t \in {\mathbb {R}}^{n_0}\), for each \(t=1,\ldots ,n_l\). Furthermore, the vector of node elevations is represented by \(\zeta \in {\mathbb {R}}^{n_n}\). Given \(t \in \{1,\ldots ,n_l\}\), we consider hydraulic heads \(h^t \in {\mathbb {R}}^{n_n}\) and flow rates \(q^t \in {\mathbb {R}}^{n_p}\) as continuous decision variables. Moreover, vector \(\eta ^t \in {\mathbb {R}}^{n_p}\) is included to model the unknown head loss introduced by the action of pressure control valves. We introduce auxiliary variables \(\theta ^t \in {\mathbb {R}}^{n_p}\) to isolate the nonconvex terms representing the friction head losses occurring within the pipes of a network. These can be expressed by either the HazenWilliams (HW) or DarcyWeisbach (DW) formulae. Since both friction head loss formulae involve nonsmooth nonconvex terms, it is convenient to use smooth quadratic approximations, computed over a range of flow (Pecci et al. 2017c). When a suitable quadratic approximation has been determined, it can be written as \(\phi _j(q^t_j)=q_j^t(a_jq_j^t+b_j)\), with \(a_j \ge 0\) and \(b_j\ge 0\), for all \(j=1,\ldots ,n_p\) and \(t=1,\ldots ,n_l\)—see also Eq. (28) in Appendix 1.

\(z^+_j=1 \Leftrightarrow\) there is a valve on link j in the assigned positive flow direction

\(z^_j=1 \Leftrightarrow\) there is a valve on link j in the assigned negative flow direction

\(z^+_j=z^_j=0 \Leftrightarrow\) there is no valve on link j

\(z^+_j+z^_j \le 1\) prevents the placement of two valves on the same link.
3 Solution method
The nonconvexity of \(\text {MINLP}(Q)\) is due to the presence of functions \((\phi _j(\cdot ))_{j=1,\ldots ,n_p}\) within equality constraints (2a). Since the other constraints (2b)–(6d) are linear, it is convenient to consider linear relaxations of constraints (2a). Polyhedral relaxations for similar nonconvex expressions have been previously studied by Humpola and Fügenschuh (2015), Gleixner et al. (2012), Liberti and Pantelides (2003), Tawarmalani and Sahinidis (2002), Udell and Boyd (2015), Vigerske (2012). In this paper, linear relaxations developed in Liberti and Pantelides (2003) for monomials of odd degree are extended to the nonconvex equality constraints within the problem formulation for optimal valve placement in WSNs.
We implement a branch and bound method that relies on the generation of a sequence of lower and upper bounds to the optimal value. The present work takes a similar approach to Misener and Floudas (2013, 2014) and compute lower bounds solving Mixed Integer Linear Programming (MILP) relaxations of \(\text {MINLP}(Q)\). As discussed in Smith and Pantelides (1999), MILP relaxations result in tighter lower bounds then relaxed linear programs (LPs) and the algorithm is expected to converge in less iterations. However, they also require an higher computational effort for each branch and bound iteration. Nonetheless, as shown in the numerical results reported in Sect. 4, the implementation of stateoftheart MILP solvers (e.g. Gurobi Optimization 2017) have enabled the application of the considered method to large problem instances.
3.1 Lower bounding MILP
3.2 Generation of upper bounds
The generation of a valid upper bound via the (local) solution of a nonlinear program is often computationally expensive, particularly when large problem instances are considered. However, \(\text {NLP}(Q')\) presents a high level of sparsity, that is retained by constraints (2b) and (2c) from the sparse structure of water supply networks. As a consequence, sparse NLP solvers offer scalable solution approaches for \(\text {NLP}(Q')\)—see the interior point method introduced in Waechter and Biegler (2006).
3.3 Domain reduction
The strength of the MILP relaxations is expected to have a significant impact on the convergence properties of branch and bound schemes (Belotti et al. 2009). This is confirmed by the numerical results reported in Sect. 4. As discussed in Appendix 1, in the case considered here, smaller ranges of flows lead to tighter relaxations. Therefore, we investigate preprocessing strategies to reduce the domains of the flow variables, focusing on Optimization Based Bound Tightening (OBBT), which relies on the solution of a series of optimization problems to tighten upper and lower bounds on selected variables—for a recent review on variable bound tightening approaches see Puranik and Sahinidis (2017).
3.4 Branch and bound algorithm
The computed optimality gaps should be considered within the range of uncertainties that are inherent in modelling of operational water networks. As discussed in Wright et al. (2015), pressure control in operational water networks is subject to multiple sources of data and modelling errors. These include the stochastic nature of customer demand, uncertainty in the hydraulic model parameters, network connectivity, measurements accuracy, and factors affecting the physical operation of control and isolation valves.
Remark 1
If \(Q' \in {\mathscr {Q}}\) is such that \(\text {L}(Q')>\text {UB}\), then the global optimum cannot belong to \(\text {F}(Q')\) and \(\text {MINLP}(Q')\) can be pruned from the branch and bound tree. However, the selection strategy implemented in Algorithm 2 implies that the branch and bound iterations will terminate before \(Q'\) is selected for branching. Therefore, the method does not explicitly implement a pruning procedure.
3.5 Branching strategy
The above branching strategy improves the lower bound, by definition of the linear relaxations given in Appendix 1. In fact, the formulation of \(\text {MILP}(Q^{\text {left}})\) and \(\text {MILP}(Q^{\text {right}})\) requires the inclusion of linear inequalities corresponding to the polyhedral relaxations computed with the new bounds on variable \(q^{l}_{k}\). Such refined linear relaxations are exact at \({\hat{q}}^{l}_{k}\) resulting in tighter MILP relaxations. If \({\hat{q}}^l_k\) is too close to either \((q^l_L)^b_k\) or \((q^l_U)^b_k\), the implemented branching strategy can result in an unbalanced branch and bound tree. However, the relaxation error \({\hat{\theta }}^t_j\phi _j({\hat{q}}^t_j)\) is larger the more distant \({\hat{q}}^t_j\) is from \((q^t_L)^b_j\) and \((q^t_U)^b_j\), see Fig. 11. As a result, in most cases, \({\hat{q}}^l_k\) is expected to be closer to the middle point of the interval \([(q^t_L)^b_k,(q^t_U)^b_k]\) than to its extremes.
4 Results and discussion
Problem size of the 3 case studies
Name  # Cont. var.  # Bin. var.  # Lin. Constr.  # Nonconvex terms 

PescaraNet  365  198  1591  99 
Net25  3192  74  9762  888 
BWFLnet  28,251  2620  96,599  7107 
The second case study is named Net25. This benchmark network model has been used to evaluate solution approaches for optimal valve placement problems in previous literature using heuristics (Reis et al. 1997; Araujo et al. 2006; Liberatore and Sechi 2009; Nicolini and Zovatto 2009; Ali 2015; De Paola et al. 2017) and mathematical optimization methods (Eck and Mevissen 2012; Dai and Li 2014; Pecci et al. 2017a, b). The results presented in this section allow the quantification of the level of suboptimality of valve configurations for Net25 previously computed using heuristics and local optimization methods. The network has 22 nodes, 37 pipes and 3 reservoirs—see Fig. 1b. Details on pipes’ characteristics, nodal demands and reservoirs’ levels are presented in Jowitt and Xu (1990) and Dai and Li (2014). The hydraulic model of Net25 uses the HW formula to model friction head losses. We observe that Net25 has a smaller dimension with respect to PescaraNet; nonetheless, it results in a larger nonconvex MINLP as the problem formulation considers 24 demand conditions, one for each hour of the day—see Table 1. Moreover, we set the maximum allowed velocity in each pipe to 1 (m/s) and the minimum pressure at each node to 30 (m). Analogously to what done for PescaraNet, for each link j, a quadratic approximation of the friction losses \(\phi _j(\cdot )\) is computed as proposed in Pecci et al. (2017c).
We investigate the implementation of algorithms \(\text {dBB}_{N_c}\), with \(N_c \in \{0,1,3,5\}\), for solving the considered optimal valve placement problems in PescaraNet and Net25. Firstly, Algorithm 1 was applied to tighten the flow variable bounds, for each \(N_c \in \{0,1,3,5\}\). When considering PescaraNet, the implementation of Algorithm 1 required the solution of 656 LPs in the case of \(n_v=1\) and \(N_c=0\). In comparison, the domain reduction procedure required the solution of 492 LPs in all the other cases. The average computational time required to run Algorithm 1 in PescaraNet is 18 (s)—see Tables 10, 12, 14, 16 for more details. The application of Algorithm 1 as domain reduction procedure for Net25 required the solution of 5568 LPs for each \(n_v \in \{1,\ldots ,5\}\) and \(N_c \in \{0,1,3,5\}\), and an average computational time of 58 (s)—see Tables 11, 13, 15, 17. Algorithm 2 was then applied to the problem formulations with tightened bounds on the flow variables. The results are summarised in Tables 10, 11, 12, 13, 14, 15, 16, 17. As shown in Fig. 2b, in most instances, the solutions computed by \(\text {dBB}_{N_c}\) are within \(5\%\) of optimality, for all \(N_c \in \{0,1,3,5\}\). Moreover, all the computed feasible solutions for Net25 equal the bestknown valve locations obtained for the considered case study with respect to those reported in Eck and Mevissen (2012), Dai and Li (2014), Pecci et al. (2017a, b). According to these numerical results, the implementations of \(\text {dBB}_{N_c}\) with \(N_c >0\) have better computational performance than \(\text {dBB}_0\).
4.1 Case study 3: BWFLnet
Finally, the proposed global optimization method for optimal valve placement has been applied to BWFLnet, the network model of the Smart Water Network Demonstrator (Field Lab) operated by Bristol Water, InfraSense Labs at Imperial College London and ClaVal (Wright et al. 2014). This water supply network consists of 2310 nodes, 2369 pipes and 2 inlets (with fixed known hydraulic heads); its graph is presented in Fig. 7. The HW formula is used to model friction losses within BWFLnet.
Note that the highest impact in terms of AZP reduction will be achieved by controlling pressure through pipes carrying large quantity of water, hence links with small diameters are not likely to be good candidates for control valve placement. As a consequence, links in BWFLnet whose diameter is smaller than 100 (mm) are discarded from the set of candidate valve locations. The minimum allowed pressure at all demand nodes is set to 18 (m), while this value is relaxed to zero for nodes with no demand. A quadratic approximation of friction losses is computed as discussed in Pecci et al. (2017c). The size of the resulting MINLP is shown in Table 1.
Then, the domain reduction procedure described in Algorithm 1 was applied as preprocessing reoutine to tighten the flow variable bounds within the formulation of \(\text {MINLP}(Q)\) on BWFLnet. In this case, the fraction of links involved in LP solves corresponds to less than the \(10\%\) of the whole set of network links. Such reduction is due to the structure of BWFLnet, a typical water network from the United Kingdom, where a considerable portion of links belongs to the forest. Moreover, hydraulic models of operational networks often present sequences of links connected in series, used to model the existence of different customer connections along network pipes. As a result, the proposed graphtheory based decomposition has significantly reduced the computational cost associated with Algorithm 1, which required the solution of 2532 LPs for each case of \(n_v=1,\ldots ,5\). In the numerical experiments reported here, such LPs were solved sequentially. As a result, Algorithm 1 required roughly one hour of CPU time, in all instance—see Tables 24 and 25. However, as previously observed, the LPs solved at each iteration of Algorithm 1 do not depend on each other. Therefore, in a practical implementation, they can be solved in parallel, exploiting the existence of multiple computational cores.
Algorithm 1 was then applied to \(\text {MINLP}(Q^{\text {tight}})\), with \(N_c \in \{0,3\}\)—see Tables 24 and 25. When \(N_c=0\), the bounds on the absolute optimality gaps obtained for \(n_v\in \{1,2,3\}\) are between 3 and 5 (m)—see also Fig. 9. Optimality bounds of such magnitude are comparable to the order of uncertainty affecting pressure control in BWFLnet (Wright et al. 2015). Hence, the quality of the computed feasible solutions is considered to be acceptable. Again, in the cases of \(n_v \in \{4,5\}\), no feasible solution was found before reaching the time limit. Similar optimality gaps were obtained when \(N_c=3\), but the algorithm failed to compute a feasible solution for \(n_v \in \{3,4,5\}\). These results suggest that the inclusion of different number of linearizations for the outer approximation of the convex envelopes of (2a) did not improve the performance of the branch and bound algorithm for solving the problem of optimal valve placement in BWFLnet.
Next, given optimal valve locations for \(n_v \in \{1,2,3\}\), computed with Algorithm 2, we have formulated a nonconvex nonlinear program to optimize valve operation under the complete set of 96 demand conditions. Such NLP is obtained from the formulation of \(\text {MINLP}(Q)\) by fixing the values of the binary variables corresponding to optimized valve locations, and it is equivalent to the problem of optimizing the actuators operation. Algorithm 1 is applied to tighten the bounds on flow variables, and Algorithm 2 is subsequently implemented to solve the nonconvex NLP with tightened variable bounds. Lower and upper bounds computed at the root of the branch and bound tree are reported in Table 26, for \(n_v \in \{1,2,3\}\). As expected, the upper bounds to the AZP values computed considering the full set of multiple demand conditions are close to those obtained for the restriction to three demand conditions. Moreover, in this case, the domain reduction procedure has resulted in considerably tighter lower bounds, without performing any branching operation. In contrast, the results reported in Table 24 indicate that that inclusion of valve locations as unknowns significantly reduces the ability of Algorithm 1 to compute tight estimates of flow variable domains. This situation has an interpretation from the hydraulic application perspective. In fact, appropriate changes in network topology induced by valves closures can result in increased flow velocities across some pipes (Abraham et al. 2018).
In conclusion, we observe that the average computational cost associated with the solution of a single MILP relaxation for BWFLnet is considerably higher than for Net25 and PescaraNet. In fact, solving the MILP relaxation at the root of the branch and bound tree for BWFLnet required 2–6 orders of magnitude more computational time than what experienced for Net25 and PescaraNet. This behaviour is predictable, as the problem of optimal valve placement and operation in BWFLnet results in a MINLP whose size is one to two orders of magnitude larger than the size of the problem formulation on Net25 and PescaraNet—see Table 1. It is known that computational effort required to solve a mixedinteger program grows combinatorially with the size of the problem. These numerical experiments were conducted on a single computational thread of a desktop machine, using a standard implementation of GUROBI for the solution of the MILP relaxations. The availability of additional computational capability and the use of a tailored MILP solver could speedup the optimization process.
All three case studies show that good quality solutions are often computed at the root node of the branch and bound tree (i.e. iteration 0). This is in accordance with the work by Diamond et al. (2018), presenting a number of examples where good quality solutions to nonconvex optimization problems are recovered from the solution of suitable convex relaxations. The results suggest that Algorithm 2 can be early terminated to generate good quality solutions of \(\text {MINLP}(Q)\), by opportunely setting time limit or maximum number of iterations. Moreover, observe that Algorithm 2 provides more information than local optimization methods for optimal valve placement in WSNs studied in previous work. In fact, the algorithm always generates a certified bound on the optimality gap of the computed solution. Local optimization methods can be implemented before starting Algorithm 2, in order to rapidly generate good quality feasible solutions.
5 Conclusions and future work
In this manuscript, we have investigated the application of branch and bound strategies to compute \(\varepsilon\)suboptimal solutions for the problem of optimal valve placement and operation in water supply networks. The implemented algorithm relies on the solution of MILP relaxations of the original nonconvex MINLP. Furthermore, a tailored domain reduction procedure was implemented to tighten the MILP relaxations. In contrast to previously published solution methods for optimal valve placement in water networks, the presented algorithm terminates with a certified bound on the optimality gap of the computed solution, thus providing additional information to support the design and operation of water supply networks.
The proposed branch and bound method has successfully generated good quality feasible solutions in two benchmark water networks and a large water supply network, after few iterations, with bounds on the optimality gap comparable to the order of uncertainty usually experienced in pressure control of water supply networks. Furthermore, the results suggest that the proposed domain reduction strategy is more effective in improving the convergence properties of the algorithm, than simply increasing the number of linearizations used to define the polyhedral relaxations. Moreover, the results reported in this manuscript show that, for the considered benchmark water networks, the proposed branch and bound algorithms outperform stateoftheart global optimization solvers SCIP and BARON. These results highlight the challenges of applying offtheshelf global solvers to the problem in study.
However, the lower bounds generated by the algorithm experience slow progress, as shown in Fig. 5. They can be improved by performing Algorithm 1 on \(Q^{\text {left}}\) and \(Q^{\text {right}}\), so that the new bounds on the branching variable can be propagated to the remaining flow variables. Recall that, at each iteration of Algorithm 1, the required solutions of the \(2n_l{\mathscr {C}}\) linear programs can be computed in parallel. As a consequence, if enough computational cores are available, the outlined bound propagation strategy can be applied at each branch and bound iteration, without dramatically increasing the overall computational time. Furthermore, future work should investigate the inclusion of additional valid linear inequalities in the formulation of \(\text {MILP}(Q')\). A possible strategy is to use the locally optimal solutions generated at each stage of the branch and bound method, following an approach similar to Humpola et al. (2016). In addition, when large operational water networks are considered, tailored solvers for the relaxed MILPs can be designed using suitable decomposition strategies, following ideas discussed in Vigerske (2012, Chapters 3 and 4).
Although the present work has focused on the minimization of AZP, other convex objective functions can be minimized within the same framework, with little modifications to the discussed formulation. In particular, if the objective function is nonlinear, the generation of lower bounds requires the solution convex MINLPs. Furthermore, optimal design (Bragalli et al. 2012), optimal valve control (Wright et al. 2015), and optimal pump scheduling (Menke et al. 2015) problems result in optimization problems involving nonconvex constraints like (2a). As a consequence, using suitable linear relaxations and NLP subproblems, Algorithms 1 and 2 can be applied to guarantee the optimality of the solution found, or a bound on the optimality gap, for such nonconvex optimization problems.
Notes
Acknowledgements
The authors thank the anonymous reviewers for their helpful comments and suggestions, and for their valuable advice on the formulation of the polyhedral relaxations described in Appendix 1. This work was supported by the NECImperial Smart Water Systems project, and EPSRC (EP/P004229/1, Dynamically Adaptive and Resilient Water Supply Networks for a Sustainable Future).
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