## 1. Introduction

[2] The optimization of water distribution systems (WDSs) is a complex problem that usually has a large search space and multiple constraints. Traditionally, the WDS optimization problem has been treated as a single-objective problem with cost as the objective. This is because of the high cost associated with the construction and operation of these systems [*Simpson et al*., 1994]. More recently, it has been recognized that network reliability is also an important criterion in the design and operation of WDSs and should therefore be considered in addition to cost.

[3] In some of the earliest work on the reliability of WDSs, *Gessler and Walski* [1985] used the excess pressure at the worst node in the system as a measure of benefit in a pipe network optimization problem to ensure sufficient water with acceptable pressure is delivered to demand nodes. *Li et al*. [1993] extended network reliability analysis to include a portion of hydraulic reliability—the capacity reliability. This is defined as the probability that the carrying capacity of a network meets the demand. *Schneiter et al*. [1996] applied the concept of capacity reliability to a WDS optimal rehabilitation problem.

[4] References to the multiobjective optimization of WDSs accounting for network reliability can be traced back to the 1980s, when *Walski et al*. [1988] used the WADISO program to solve WDS pipe sizing problems considering both cost and the minimum pressure of the network. In a study by *Halhal et al*. [1997], the network cost and the total benefit (including the improvement in the pressure deficiencies in the network) were maximized. Since then, minimizing the head deficit at demand nodes has been used as a hydraulic capacity reliability measure in a number of multiobjective WDS optimization studies that considered both cost and system reliability [*Atiquzzaman et al*., 2006; *Jourdan et al*., 2005; *Keedwell and Khu*, 2004; *Savic*, 2002]. In 2000, *Todini* [2000] introduced a resilience index approach that was incorporated together with minimization of cost into multiobjective WDS optimization via a heuristic approach. The resilience index was used by *Farmani et al*. [2006] as a hydraulic reliability measure in a multiobjective WDS optimization problem considering cost, reliability, and water quality. Based on the resilience index, *Prasad and Park* [2004] introduced a network resilience measure and applied it to multiobjective genetic algorithm optimization of WDSs. Around the same time, *Tolson et al*. [2004] used a genetic algorithm coupled with the first-order reliability method (FORM) to obtain optimal tradeoffs between the cost and reliability of WDSs represented by the probability of failure. *Kapelan et al*. [2005] applied a multiobjective approach to maximize the robustness of a WDS, which was represented as the possibility that pressure heads at all network nodes are simultaneously equal to or above the minimum required pressure. *Jayaram and Srinivasan* [2008] modified the resilience index introduced by *Todini* [2000] and applied it to the optimal design and rehabilitation of WDSs via a multiobjective genetic algorithm approach. More recently, *Fu and Kapelan* [2011] used a fuzzy random reliability measure, which is defined as the probability that the fuzzy head requirements are satisfied across all network nodes, for the design of WDSs. *Fu et al*. [2012] developed a nodal hydraulic failure index to account for both nodal and tank failure in multiobjective WDS optimization.

[5] In the last 10 years, objectives focused on environmental considerations have started to be included in WDS optimization studies due to increased awareness of climate change, especially global warming. *Dandy et al*. [2006] used a single-objective approach to minimize the material usage, embodied energy, and greenhouse gas (GHG) emissions associated with the manufacture of PVC pipes. *Wu et al*. [2008] first introduced GHG emission minimization as an objective into the multiobjective optimal design of WDSs. Since then, a number of other WDS optimization studies have focused on the incorporation of GHG emissions associated with energy consumption into WDS optimization studies [*Dandy et al*., 2008; *Herstein et al*., 2009; *Wu et al*., 2010a, 2010b, 2012b]. It should be noted that there were a large number of earlier studies that included the minimization of energy consumption as an objective in WDS optimization [see *Ormsbee and Lansey*, 1994; *Pezeshk and Helweg*, 1996; *Nitivattananon et al*., 1996; *Ilich and Simonovic*, 1998; *van Zyl et al*., 2004; *Ulanicki et al.,* 2007]. However, these studies focused on the minimization of energy consumption as a means of cost minimization [*Ghimire and Barkdoll*, 2007], rather than the explicit minimization of environmental impacts.

[6] As can be seen from previous research, economic cost, hydraulic reliability, and environmental impact, especially in terms of GHG emissions, are important design criteria for WDSs. However, while there have been a number of studies that have considered the tradeoffs between cost and reliability and cost and GHG emissions, there have been no studies that have investigated the optimal tradeoffs between all of these three objectives. One reason for this is that traditionally used measures of hydraulic reliability cannot be used as an objective in WDSs where pumping into storages plays a major role. This is because the calculation of the majority of these hydraulic reliability measures relies upon the difference between the required and minimum allowable pressure heads at the outlet of the system, which is generally equal to a constant value of zero in these pumping systems [*Wu et al*., 2011], and can therefore not be used as an objective function. However, it is precisely these systems that are of most interest from a GHG emission minimization perspective, as they often require most pumping energy.

[7] The surplus power factor hydraulic reliability/resilience measure, which was introduced by *Vaabel et al*. [2006], overcomes the shortcoming of existing hydraulic reliability measures outlined above, as it can be used as an objective function in the optimization of water transmission systems (WTSs), which generally include pumping into storages. This is because calculation of the surplus power factor does not require the value of the pressure head at the outlet of the system. Consequently, this resilience measure is the only network hydraulic reliability measure that can be used in conjunction with the objectives of cost and GHG emission minimization for all types of networks, and particularly those that are of most interest from a GHG minimization perspective, such as transmission networks that pump water into storage facilities. As a result, this measure can be used in all studies considering cost, hydraulic reliability, and GHG emissions as objectives. The application of the surplus power factor as a hydraulic reliability measure for WDS optimization can be found in the study by *Wu et al*. [2011].

[8] The tradeoffs between cost minimization, hydraulic reliability maximization, and GHG emission minimization are important. We are now moving into an era where the minimization of GHG emissions from WDSs is becoming increasingly important. Consequently, the main aim of this paper is to gain a good understanding of the generic nature of the tradeoffs between these objectives for systems that involve pumping into storage facilities. The specific objectives include (1) to investigate the shape of the solution space formed by the three objectives for WDSs involving pumping or WTSs, (2) to investigate the location of the Pareto-optimal front in the solution space for a number of case studies, and (3) to elicit generic guidelines that are useful from a practical perspective when optimizing WDSs using cost, hydraulic reliability, and GHG emissions as objective functions. The specific research objectives are achieved via three case studies, including two hypothetical case studies from the literature and one real-world case study.

[9] The remainder of this paper is organized as follows. The formulation of the proposed three-objective optimization problem and the method used to analyze the multiobjective optimal solutions obtained are both introduced in section 2. Then, the three case studies and associated assumptions are presented in section 3, followed by the optimization results and their discussion in section 4. Conclusions are then made.