Lost in Optimisation of Water Distribution Systems? A Literature Review of System Design

Abstract

Optimisation of water distribution system design is a well-established research field, which has been extremely productive since the end of the 1980s. Its primary focus is to minimise the cost of a proposed pipe network infrastructure. This paper reviews in a systematic manner articles published over the past three decades, which are relevant to the design of new water distribution systems, and the strengthening, expansion and rehabilitation of existing water distribution systems, inclusive of design timing, parameter uncertainty, water quality, and operational considerations. It identifies trends and limits in the field, and provides future research directions. Exclusively, this review paper also contains comprehensive information from over one hundred and twenty publications in a tabular form, including optimisation model formulations, solution methodologies used, and other important details.

1. Introduction

Water distribution systems (WDSs) are one of the major infrastructure assets of the society, with new systems being continually developed reflecting the population growth, and existing systems being upgraded and extended due to raising water demands. Designing economically effective WDSs is a complex task, which involves solving a large number of simultaneous nonlinear network equations, and at the same time, optimising sizes, locations, and operational statuses of network components such as pipes, pumps, tanks and valves [1]. This task becomes even more complex when the optimisation problem involves a larger number of requirements for the designed system to comply with (e.g., water quality), includes additional objectives beside a least-cost economic measure (e.g., potential fire damage) and incorporates more real-life aspects (e.g., uncertainty, staging of construction).

The early research related to the design optimisation of WDSs can be dated from the 1890s to 1950s. It was based on the principle of economic velocity [2,3,4], which was gradually reviewed and replaced by establishing the minimum (annual) costs of the system (i.e., least-cost design) [5,6,7]. Due to lack of computational technology in that period, those previous studies involved manual calculations combined with graphical methods, often resulting in practical charts to derive economic pipe diameters. The development of the optimisation of WDS design, therefore, had been an incremental process over time and may have appeared to be “only too true that the design of the transmission and distribution system receives [at that period] little attention in spite of the great sums of money invested in such installations” [8].

A successive period from the 1960s to 1980s displays a more rapid progression, which was initiated by the introduction of digital computers to network analysis in 1957 [9]. The introduction of computers was subsequently followed by the development of iterative methods [10,11] and simulation packages [12,13] to solve simultaneous nonlinear network equations, and eventuated in the application of mathematical deterministic methods to solve WDS design optimisation problems. These methods, including linear programming (LP) [14], nonlinear programming (NLP) [15,16], and others [17], typically minimised the design or capital (and operational) costs of the system, which were combined into one economic measure.

Another significant advancement in the optimisation of WDSs represented an introduction of stochastic methods using principles of biological evolution [18] and natural genetics [19]. Nonetheless, it was not until the 1990s when these methods became more popular [20] due to their ability to solve complex, real-world problems for which deterministic methods incured difficulty or failed to tackle them at all [21,22], and to also control multiple objectives. The popularity of metaheuristics has resulted in a dramatic increase in the application [21,23] to optimal design of WDSs, with “the several hundred research papers written on the subject” by 2001 [24]. Optimisation of WDS design has also progressed from a cost-driven single-objective framework to multi-objective models, when various objectives that continually gain importance (e.g., environmental objectives, community objectives reflecting the level of service provided to customers) can be evaluated on more equal basis [25]. Some of the most recent developments include the use of an engineered (as opposed to a random) initial population to improve the algorithm convergence [26], application of online artificial neural networks (ANNs) to replace network simulations [27], analysis of the algorithm search behaviour [28] in relation to the WDS design problem features [29], and reduction of the search space [30] to increase computational efficiency.

2. Aim, Scope and Structure of the Paper

This paper aims to provide a comprehensive and systematic review of publications since the end of the 1980s to nowadays, which are relevant to the optimisation of WDS design, strengthening (i.e., pipe paralleling), expansion and rehabilitation. The purpose of the review is to enable one’s speedy familiarisation with the scope of the field, insight in the overwhelming amount of publications available and realisation of the future research directions. This paper contributes to and goes beyond the existing review literature for the optimisation of WDS design and rehabilitation [20,21,31,32,33,34,35,36,37,38,39] by not only identifying trends and limitations in the field, but also by providing comprehensive information from over one hundred and twenty publications in a tabular form, including optimisation model formulations, solution methodologies used, and other important details.

The paper consists of two parts: (i) the main review and (ii) an appendix in a tabular form (further referred to as the table), each having a different structure and purpose. The main review is structured according to publications’ design problems and general classification. The design problems cover application areas, such as new system design, existing system strengthening, expansion and rehabilitation, and time, uncertainty and performance considerations. The general classification captures all the main aspects of a design optimisation problem answering the questions: what is optimised (Section 4.1), how is the problem defined (Section 4.2), how is the problem solved (Section 4.3) and what is the application (Section 4.4)? The purpose of the main review is to provide the current status, analysis and synthesis of the current literature, and to suggest future research directions.

A significant portion of this review paper is represented by the table, which refers to over one hundred and twenty publications in a chronological order. Each paper is classified according to an optimisation model (i.e., objective functions, constraints, decision variables), water quality parameter(s), network analysis, optimisation method and test network(s) used. Obtained results as well as other relevant information are also included. The purpose of the table is to provide a representative list of publications on the topic detailing comprehensive information, so that it could be used as a primary reference point to identify one’s papers of interest in a timely manner. Hence, it presents a unique and integral contribution of this review.

The structure of the paper is as follows:

• The main review: Design problems (Section 3), General classification of reviewed publications (Section 4), Future research (Section 5), Summary and conclusion (Section 6), List of terms (Section 7), List of abbreviations.
• The table: Appendix A.

3. Design Problems

Two types of a design problem have been identified based on the field progression as follows: (i) a traditional design (i.e., theoretical or static design) of a WDS with a single construction phase for an entire expected life cycle of the system usually considering fixed loading conditions reflecting maximum (and other) future demands (Section 3.1); (ii) an advanced design (i.e., real-life or dynamic design) of a WDS capturing the system modifications and growth (due to the development of the populated area) over multiple construction phases, including future uncertainties (e.g., in demands, pipe deterioration) and other performance considerations (Section 3.2).

3.1. Application Areas

3.1.1. New Systems: Design

Critical infrastructure, including water, energy and transport systems, is essential in ensuring the survival and wellbeing of populations worldwide. Since the ancient Greek civilisations, WDSs have been an important part of making human settlements sustainable, thus optimising these systems to meet various requirements has over time gained interest of researchers and practitioners alike. Generally, optimisation of WDS design involves determining sizes, locations and operational statuses of network components such as pipes, pumps, tanks and valves, while keeping the system design or capital (and operational) costs at their minimum. The problem scope is primarily dependent on a type of a WDS under consideration, which is either a branched or looped and gravity or pumped system.

A network topology, branched or looped, represents a fundamental distinction in the problem complexity at the network analysis stage due to a way of determining flows in pipes. In branched networks there is a unique flow distribution calculated directly using nodal demands, while in looped systems flows can undertake multiple and alternative paths from a source to a customer [40]. This possible variability results in iterative methods being required to solve pipe flows in looped networks, such as that described in [41].

Regarding gravity WDSs, a basic optimisation model minimises the design cost of the network subject to the nodal pressure requirements, with pipe sizes or diameters being the only decision variables [42,43,44,45,46,47,48]. Popular test networks used to solve such a problem are the two-loop network [14], Hanoi network [49] and Balerma irrigation network [50]. As far as pumped WDSs are concerned, the optimisation problem becomes more complex than in the case of gravity WDSs, because of the presence of pumps and tanks (see Section 3.1.3), which require selecting not only their sizes and locations [14,26,51,52], but also their operational statuses [14,29,53,54], as well as often running an extended period simulation (EPS) for multiple loading conditions. Unlike for gravity WDSs, there does not seem to be any test network that is frequently used by multiple authors for pumped WDSs.

Regarding test networks, nevertheless, study [26] comments that they are limited, in general, to simple transmission networks, so-called benchmark systems, excluding local distribution lines. This exclusion is mainly due to a dramatic increase in the problem dimension, thus computational time, if local pipes were included. A problem of excluding smaller distribution pipes from the optimisation is in oversizing the transmission mains, as local distribution networks provide alternative pathways and display significant capacity to carry when the transmission lines are out of service [26]. The lack of large and complex test networks has recently been addressed by a number of researchers [55,56,57] who developed methodologies for generating synthetic networks of varying sizes and complexity levels. Furthermore, several real-world networks have been used for the design competitions by international research teams working in the area of WDS design, including those that are described by [58,59].

The problem complexity further increases by considering multiple simultaneous objectives. Initially, single-objective optimisation models were used to formulate WDS design problems, in which all objectives are combined into one economic (i.e., least-cost) measure (see, for example, [14,51,60,61,62]). A multi-objective optimisation approach was possibly first applied in the late 1990s (Figure 1), maximising the network benefit on one hand and minimising the system cost (of network rehabilitation) on the other hand [63]. In studies of newly designed WDSs, in addition to the economic measure, the other objectives considered were the pressure deficit [30,62,64,65,66,67] or excess [68,69] at network nodes, the penalty cost for violating the pressure constraint [70], greenhouse gas (GHG) emissions [71,72,73,74,75,76] or emission cost [77], water discolouration risk [68] and water quality [78]. A multi-objective optimisation approach is considered “very appealing for engineers as it provides a tool to investigate interesting trade-offs”, for example, a marginal pressure deficit can be outweighed by a considerable cost reduction [67].

Figure 1. Papers (from Appendix A Table A1) by year and optimisation approach.

The single-objective approach benefits from being able to identify one best solution, which is then easy to analyse and implement. Multi-objective methods, on the other hand, result in a set of tradeoff (Pareto, non-dominated) solutions, which requires an additional step to select only one or a limited number of the promising solutions. Choosing such a reduced number of solutions from a potentially large (or even infinite) non-dominated set is likely to be difficult for any decision maker. This task makes the multi-objective approach less desirable as there is often a requirement to make a clear decision to be implemented. The research question resulting from this challenge is how to select the best solution(s) from the Pareto set, which may involve providing the decision makers with a practical and representative subset of the non-dominated set that is sufficiently small to be tractable [22]. For example, study [79] introduced game-theoretic bargaining models to take into account conflicting requirements and managed to reduce the solution sets to a reasonable size. Further investigation of the methodologies for identifying a handful of useful solutions, such as those where a small improvement in one objective would lead to a large deterioration in at least one other objective, is thus warranted. In addition to game-theoretic models, the approaches that are based on identifying ‘knees’ of the Pareto front or expected marginal utility, maximum convex bulge/distance from hyperplane, hypervolume contribution and local curvature [80] are all promising methods that require a thorough analysis on WDS problems.

3.1.2. Existing Systems

As a consequence of the development/growth and population density increase within urban areas, existing WDSs require to be upgraded to satisfy raising water demands. These upgrades involve system strengthening (i.e., pipe paralleling), rehabilitation (e.g., pipe cleaning and relining) and expansion. Even though these processes often take place within one WDS thus some of the research articles fall under all system strengthening, rehabilitation and expansion, they are divided into separate subsections in order to provide a systematic overview.

Strengthening

System strengthening represents a reinforcement of an existing WDS to meet future demands, through lying duplicated pipes in parallel to the existing water mains. It is also sometimes referred to as parallel network expansion [42] or pipe paralleling. The main objective and decision variables are, similar to the design of new WDSs, the minimisation of the design (or capital) cost and pipe diameters of duplicated pipes, respectively. Publically available test networks involving purely system strengthening include the New York City tunnels [81] and EXNET [82]. In addition, there are test networks considering system strengthening together with other design strategies (e.g., system expansion, rehabilitation), which include the 14-pipe network with two supply sources [20,83] and Anytown network [84]. Of those publically available test networks, the most frequently applied is the New York City tunnels, which was often the only network used to test the proposed methodology. These studies used genetic algorithm (GA) [85,86], combined with ANNs [87], fast messy GA (fmGA) [88] and non-dominated sorting genetic algorithm II (NSGA-II) [89] as a solution algorithm.

The complexity of an optimisation problem involving exclusively system strengthening as a design strategy can be substantially increased by incorporating water quality considerations. Such applications include, apart from pipe sizes as decision variables, also water quality decision variables that can be in a form of disinfectant (i.e., chlorine) dosage rates [27,87]. In order to reduce computational effort of those problems, ANNs were implemented to replace network simulations to a large extent. Further increase in the complexity presents the use of a multi-objective approach, with additional objectives being system robustness [89] (uncertainty and system robustness are contained in Section 3.2.3), the pressure deficit at network nodes [62,65], and the number of demand nodes with pressure deficit [65,90]. In those studies, a conflicting relationship was identified between the economic (i.e., least-cost) objective and pressure deficit/the number of nodes with pressure deficit. Based on such information, the decision maker is able to “quantitatively evaluate the cost of pressure constraints attenuation which implies a reduction in the system service to its consumers.” Optimisation methods used in those studies were NSGA-II [65,89,90], strength Pareto evolutionary algorithm 2 (SPEA2) [65] and cross entropy (CE) [62].

Rehabilitation

Due to aging water infrastructure, which causes a decreased level of service in terms of water quantity as well as quality for customers, increased operation costs and leakage, pipe breaks and other issues, existing WDSs require rehabilitation in a timely manner. Large investments are and will be needed in the future to rehabilitate ever deteriorating pipe networks [91] reaching the end of their lifecycle. Network rehabilitation consists of the replacement of pipes with the same or larger diameter, cleaning, or cleaning and lining of existing pipes; with the main objective to minimise the pipe rehabilitation cost. Within an optimisation model, pipe replacement options can be represented by binary [17] or integer [92] decision variables to identify the pipes selected for replacement, and continuous [17] or integer [92] diameters, respectively, of the replaced pipes. Pipe rehabilitation options are often binary decision variables (i.e., 1 = cleaning/lining, 0 = no action) [17,93]. If a pipe is not scheduled for rehabilitation, it is expected to be subject of break repair over a longer planning horizon. Hence, study [17] added the expected pipe repair costs to the rehabilitation cost of the network. Because a network rehabilitation strategy also has a direct impact on pump operating costs and GHG emissions due to pumping (i.e., they are reduced with an increased quantity of rehabilitated pipes) [94], pump energy costs have been added to the total least-cost objective [17,95].

Some studies consider only a single economic objective to formulate a network rehabilitation problem [17], while other investigations apply a multi-objective optimisation framework in order to incorporate measures affecting the level of service provided to customers (i.e., ‘community objectives’). Accordingly, additional objectives considered, beside the economic measure, include the network benefit [63], pressure violations at network nodes [68,95], velocity violations in pipes [95] causing potential sedimentation problems and subsequent water discolouration, water quality (i.e., disinfectant) deficiencies at network nodes [92], and potential fire damage expressed as lack of available fire flows [92]. To generate multi-objective optimal solutions, those studies use mainly metaheuristics or hyperheuristics, such as structured messy GA (SMGA) [63], NSGA-II [95], non-dominated sorting evolution strategy (NSES) [92], and evolution strategy (ES)/SPEA2 in a hyperheuristic framework with evolved mutation operators [68]. The resulting Pareto fronts can then serve decision makers in selecting a rehabilitation strategy that balances community objectives with a capital expenditure.

Note that publications included in this section belong to the category of static design, which involves a single network rehabilitation intervention for a near planning period, designed based on the current network status. Publications, which are concerned with staged rehabilitation interventions involving their timing over an extended planning horizon, are reviewed in Section 3.2.1.

Expansion

An expansion of a WDS means developing or expanding the existing system beyond its current boundary, with the main objective to minimise the total design (or capital) and operation cost. System expansion can be thought of as the following two interdependent design problems: (i) developing a new network that is connected to the existing one, and simultaneously (ii) strengthening, rehabilitating and upgrading the existing system in order to convey increased water demands. Hence, system expansion is the most complex WDS design problem as it can ultimately contain all aspects of designing new as well as existing systems. A typical example of the optimal network expansion is the Anytown network problem [84]. Essentially, the objective is to determine least-cost design and operation, using locations and sizes of new pipes (including duplicated pipes), pumps and tanks, as well as pipe rehabilitation options (i.e., cleaning and lining) as decision variables. Such extensive problems are often solved by combining a power of optimisation algorithms with “manual calculations and a good deal of engineering judgement” [84].

Although some studies solved the Anytown network problem as initially formulated [84], for example, study [83] by enumeration and [96] using GA, others included new aspects to the (original or modified) problem. Those aspects represent, for example, water quality [97] inclusive of the construction and operation costs of treatment facilities [53], new tank sizing approach (further discussed in Section 3.1.3) [93,98], and additional objectives, such as the network benefit incorporating multiple system performance criteria [93,99] or the hydraulic failure, fire flow deficit, leakage and water age with visual analytics used to explore the tradeoffs between numerous objectives [97]. These studies used SMGA [99], GA [53,93], and ε-NSGA-II [97] to solve the problem. Study [93] combined GA with fuzzy reasoning, where system performance criteria are individually assessed by fuzzy membership functions and combined using fuzzy aggregation operators.

An example of large system expansion represents the battle of the water networks II (BWN-II) optimisation problem, which involves the addition of new and parallel pipes, storage, operational controls for pumps and valves, and sizing of backup power supply, and includes the capital and operational costs, water quality, reliability and environmental considerations as performance measures [58]. This problem was solved by multiple authors within the Water Distribution Systems Analysis (WDSA) conference series [58]. Another example of large and real-world system expansion is presented in [100]. Apart from the decision variables for the BWN-II, it also includes selections of pipe routes, expansions of water treatment plants (WTPs) and configurations of pressure zones. The common approach that is applied to solve both of those optimisation problems was the use of engineering judgement, which led to a reduction in the number and type of decision variables. In the case of the study of [100], some eliminated variables were included in separate optimisation problems. Study [58] demonstrates that “different combinations of engineering experience, computational power and problem formulation can give similar results”.

Despite the advances in optimisation methods developed for new system design, rehabilitation and/or expansion of WDS, most notably over the last three decades, the large, complex systems still represent a significant challenge to solve using a fully automated optimisation procedure. There are several reasons for that, including: (i) complexity resulting from a mixed-discrete, nonlinear optimisation problem with often conflicting and difficult to assess objectives and performance measures; (ii) the large network sizes normally encountered in practice, which translates into large search spaces where a global optimum is almost impossible to find; (iii) the so called No-Free-Lunch theorem [101], which says that not all of the optimisers are well suited to solving all problems, in other words, slow convergence of general population-based optimisation methodologies that do not utilise some form of traditional engineering experience/heuristics; and (iv) the lack of computational efficiency of network simulators required by modern population-based optimisation methods. A number of approaches have been developed to deal with these challenges, mainly aimed at increasing the computational efficiency of the optimisation process. Those improvements often include the division of a design problem into multiple phases [58] that can be solved separately, the involvement of engineering expertise and manual interventions [59] to reduce the search space, or the use of surrogate and meta-modelling to speed up the simulation process [27]. The work that is still needed in the WDS design optimisation area is to understand the link between the performance of an algorithm (and its operators) and certain topological features of a WDS (e.g., existence of pumps/tanks, loops), as indicated in [29].

3.1.3. Problem Elements

Pipes

Unlike other network elements (e.g., pumps, tanks, valves), pipes are always included in the optimisation of WDS design, as the basic model is to determine such pipe sizes (or diameters) for which the design cost of the network is minimal, subject to the nodal pressure requirement. Even though pipe decision variables are incorporated in every optimisation model, they do not seem to have been unified. Assuming a given layout of the pipe network, there are two types of a decision variable, pipe sizes/diameters, and pipe segment lengths of a constant (known) diameter. Pipe sizes/diameters are discrete by nature of the problem, because they are to be selected from a set of commercially available sizes, however both discrete and continuous values are used mainly depending on the optimisation method. Discrete sizes are used mostly for stochastic algorithms (i.e., metaheuristics) [42,70,85,88,102,103,104,105,106,107,108,109], whereas continuous sizes for deterministic methods [16,110,111]. In regards to continuous sizes, the final solution can be modified by splitting a link into two pipes of closest upper- and lower-sized commercially available discrete diameter [16].

WDS design optimisation problems, which use pipe sizes/diameters as decision variables, can be referred to as a single-pipe design [112,113], while problems with pipe segment lengths of a constant (known) diameter as a split-pipe design [112,113]. Pipe segment lengths of a constant (known) diameter are predominantly used in conjunction with deterministic algorithms [14,114,115] or hybrid methods (i.e., combined deterministic and stochastic methods) [113,116,117]. Single-pipe design with discrete decision variables can provide, compared to split-pipe design and continuous diameters solutions, high quality [102], or good quality results without unnecessary restrictions imposed by split-pipe design [42]. Even if only pipe diameters are optimised, the design of WDS is a complex problem that requires a careful selection of decision variables as to minimise the search space. The choice between direct representation of discrete pipe diameters and split-pipe solutions has largely been resolved in favour of the former, but further improvements in decision variable coding might be possible.

In cases of an unspecified network layout (e.g., when designing a new or extending an existing WDS), additional decision variables are required in order to determine or select pipe routes [52,100]. These variables can be formulated, for example, as binary selecting a link which should be included into the pipe route [52]. Pipe routes can also be considered when strengthening an existing WDS, as “parallel pipes do not necessarily have to be laid in the same street”, they “may be laid in a parallel street or right-of-way that may not have existed at previous construction times” [118]. Another possible type of a pipe decision variable are pipe closures/openings to adjust a pressure zone boundary within a WDS [100].

Pumps

There are two main aspects of including pumps into the optimisation of WDS design. First, the pump design or capital cost and second, the pump operating cost due to electricity consumption. Typically, electricity consumption is one of the largest marginal costs for water utilities, with the price of electricity rising globally making it a dominant cost in managing WDSs. Therefore, “the presence of pumps requires that both the design and the operation of the network should be considered in the optimisation” [99]. Accordingly, the minimisation of the pump design or capital cost as well as the pump operating cost to achieve minimal amount of electricity consumed by pumps ought to be included in an optimisation model. Pump operating cost is usually calculated on annual basis using the typical daily demand patterns (i.e., EPS), but a longer period can be considered depending on the planning horizon of a case study, for example, 20 years [17,119], 100 years [72,76,77]. Because this cost occurs at different times in the future, its present value is required to be included in the objective function. This conversion of future economic effects into the current time is undertaken via a present value analysis (PVA), described in detail in [71,72,77], using zero, constant or time varying discount rates.

In the model, pumps are controlled by three types of a decision variable. Firstly, a pump location, which are used when designing a new or extending and upgrading an existing WDS. Possible options to consider are, for example, to predetermine a limited number of potential pump locations [93,120], to evaluate network nodes as potential pump locations (yes/no) via binary variables [52] or to upgrade the current pump stations where new pumps are to be installed in parallel to existing ones [99]. Secondly, a pump size, which can be included as a pump capacity [14,121], pump type [75,76], pumping power [17], pump head/height [52,122], pump operation curve/head-flow [93] or pump size in a combination with the number of pumps [26]. Thirdly, a pump schedule, which describes when the pump is on and off during a scheduling period (e.g., 24 h). It can be specified by a pumping power [53,54] or pump head [123] at each time step, the number of pumps in operation during 24 h [97], binary pump statuses [29], continuous options representing on/off times with a limit imposed on the number of pump switches [76], discrete options representing the time at which a pump is turned on/off using a predefined time step (e.g., 30 min) [75]. All of these decisions impact on the size of the search space and eventually on the computational efficiency of the optimisation algorithm used. A comparative study of various approaches would be useful to help determine what their advantages and disadvantages are and which one to use for a particular situation.

In terms of the model objectives, the pump design or capital and/or operating costs were mostly incorporated together with the costs of other network elements (e.g., pipes, tanks, valves) into one economic function (see, for example, [17,26,51,60,93,95,96,119]). Although a few studies, which considered the design and operating costs as part of separate objectives (e.g., [124]), reported on their conflicting tradeoff, this relationship was not confirmed for a higher-dimensional space when required to balance numerous objectives [97]. Additionally, the pump maintenance cost (see, for example, [61,62,121]) as well as the pump replacement and refurbishment cost [71,72,77] were accounted for. More recently, GHG emission cost or GHG emissions due to the electricity that is consumed by pumps [71,72,73,74,75,76,77] were introduced as an environmental objective. Similar to the pump operating cost, a PVA can be used for the pump maintenance, replacement and refurbishment costs, as well as GHG emissions/cost. Even though there is a significant tradeoff between economic and environmental objectives (i.e., GHG emissions decrease with the increasing costs and vice versa), GHG emissions can be considerably reduced by a reasonable increase in the costs [71,72]. Additional results indicate that the price of carbon has no effect on the tradeoff [77], whereas the discount rates do [72], the use of variable speed pumps (VSPs) (rather than fixed speed pumps (FSPs)) leads to significant savings in both total costs and GHG emissions [74].

The mixed-integer nature of pumps as decision variables and their often significant impact in terms of hydraulic behaviour of the entire system, makes them a difficult element to include and control its impact during an optimisation run. Furthermore, the increased complexity of modelling VSPs and their incorporation into the optimisation problem pose another difficulty that has to be tackled by modern optimisation algorithms. Finally, the formulation of various objectives, including maintenance requirements (i.e., often surrogated by the number of times a pump is switched on during the optimisation period), represents another challenge for including pumps into overall WDS design studies.

Tanks

In spite of having a valuable role in WDSs contributing to their reliability and efficiency [125], storage tanks (further in the text referred simply to as tanks) are not often included in WDS design optimisation problems. Several types of a decision variable have been used in the literature to control tanks in the model, and a few objectives (or objective functions) have been developed to mainly evaluate tank performance. However, the use of those variables as well as objectives seems to vary across studies with no general framework on how to model tanks available. As far as decision variables are concerned, they include tank locations [71,72,96,97,98,99,120], tank volumes [16,53,93,96,98,99], minimum (and maximum) operational levels [93,96,98,99], tank heads [78], tank elevations [14], ratio between diameter and height [98], ratio between emergency volume and total volume [98]. Study [99] compared two approaches to model tanks in terms of operational levels, first of which calculates tank levels analytically during the network analysis, and second of which includes tank levels as independent variables. Although they yielded similar results, the former approach obtained more robust solutions.

In regard to objectives, the most frequently used account for the tank design or capital cost, which is normally part of the total system costs (i.e., pipes, pumps, etc.) [16,53,76,93,96,97,98,99,120]. Furthermore, additional objectives have been introduced evaluating, along with others, the tank performance. These objectives are the network benefit, including storage capacity difference [99], safety and operational volume capacities, and the filling capacity of the tank [93], and system hydraulic failure including tank failure index [97]. A positive relationship was identified between the total cost of the system and network benefit [93,99], whereas a negative relationship exists between the cost and failure index [97]. The effect of changing the tank balancing volume, so called tank reserve size (TRS), on the minimisation of system cost and GHG emissions was also investigated [76]. It was identified that a larger TRS could assist in reducing GHG emissions with no additional cost by modifying pumping schedules.

In addition to pumps, the presence or absence of a tank can also play a significant role in changing hydraulic behaviour of a WDS. This presents a large challenge for any optimisation approach as it creates a discontinuity (i.e., a large change in behaviour with or without a tank at a particular location), which has to be properly managed by the algorithm. Additionally, the setup of the tank (i.e., the link to the system, overflow valve operation, consideration of upper/lower level limits) within a simulation model can also play a significant role in the efficiency of the optimisation run.

Valves

The inclusion of valves in WDS design optimisation problems appears to be rather sporadic and descriptions related to their implementation are often very brief with not many details provided. Studies [14,26] accounted for valves in the overall costs of the system, based on optimal valve locations. The optimisation of a real-life scale WDS incorporating not only transmission pipelines, but also local distribution pipelines, concluded that optimal valve locations are to be affected by the presence of local lines which “provide alternative pathways when the main lines are out of service” [26]. As shutdown of valves used to isolate a portion of the WDS during an emergency (e.g., pipe break or a water quality incident) creates a change in hydraulic behaviour, the valve numbers and locations play part in the overall system design, particularly when the reliability or resilience of the system is considered. For example, study [126] presented a methodology for optimal system design accounting for valve shutdowns. Another application of valves is using their settings to influence the pressure distribution in the network (via pressure reducing valves (PRVs)) [16], or to determine timing of flows and flow rate values (either via flow control valves (FCVs) or PRVs) [127].

Valves were also used to incorporate a simpler model of VSPs into the multi-objective optimisation of WDS design including total economic cost of the system (i.e., design and operation) and GHG emissions [74]. In such an application, a pump power estimation method uses a FCV combined with an upstream reservoir to represent a pump in the system, with the aim to maintain the flows via the FCV into the downstream tanks as close as possible to the required flows. Hence, the determination of the most appropriate FCV setting for calculating pump power is formulated as a single-objective minimisation problem that is subject to multiple flow constraints, which is implemented within a multi-objective GA (MOGA) framework [74].

A combined design of the isolating valve system and the pipe network presents a considerable challenge to optimisation methods. Not only that the number of decisions increases exponentially with the addition of valves, but also the consequences of various valve system designs can only be evaluated by investigating a large number of (probabilistic) scenarios making the whole process computationally inefficient. Furthermore, the location and status of isolating valves can form decision variables also when a WDS is to be divided into manageable subsystems. This is the case with the so-called district metering areas (DMAs), which have been first implemented in the UK primarily for leakage management purposes [128]. Due to the fact that the DMA optimal design is normally performed after a system has been constructed, this problem was deemed beyond the scope of this review paper.

3.2. Time, Uncertainty and Performance Considerations

3.2.1. Staged Design

A staged design represents an optimisation of a WDS over a long planning horizon divided into several construction phases, without considering future uncertainties (e.g., in demands, pipe deterioration). In other words, it is a deterministic dynamic design either over several prefixed time intervals or with timing decisions (i.e., year of action execution). The planning horizon can spread across a number of years to an expected life cycle of the system.

Initial work in the staged design is related to the development of multiquality water resources systems using a single-objective approach, which minimised the costs of water allocation, facilities expansion, water transportation, and losses caused by insufficient supply [129]. It was formulated as a LP optimisation problem, into which nonlinear water quality equations were incorporated using a successive linear approximation iterative scheme. An advantage of using a staged design was demonstrated by realising linkages between certain management processes and variables, and a particular planning period (prefixed time interval).

Concerning WDSs, the staged design is often applied to rehabilitate an existing system as this problem inherently involves the timing of ongoing works over an extended planning horizon. Both single- and multi-objective optimisation models were proposed to solve such problems. Single-objective models included beside the network rehabilitation [130], also network strengthening [131] and expansion [124,132] combined into one least-cost objective, while multi-objective models incorporated the network benefit [131] or the system operating costs [124,132] as additional objectives. Optimisation methods used were GA [130], SMGA [131] and NSGA-II [124,132]. As opposed to the study of [129], these models do not define prefixed time intervals, but include timing decision variables to schedule works, also referred to as event-based coding [124,132]. This coding dramatically reduces the search space, thus the computing and memory requirements, because it eliminates unnecessary zero values of a traditional coding based on a time-interval (e.g., yearly) basis [124]. A further search space reduction can be achieved by so called limited pipe representation introduced by [130], which involves placing an upper bound on the number of pipes considered for rehabilitation. These reductions in the search space and computing requirements are especially important for large size WDSs.

Moreover, the staged design was applied to extend and strengthen existing wastewater, recycled and drinking water systems applying an integrated optimisation scheme within a single-objective framework using GA [127], and to plan a new WDS considering two objectives, the construction costs and network reliability, using NSGA-II [118]. Both of these studies used prefixed time intervals to schedule the construction. In addition, study [118] analysed the effect of the scheduled construction on the network design using a set of scenarios reflecting different lengths of planning horizons (25–100 years), time intervals (25–100 years) and the number of construction phases (1–4). Both studies [118,127] confirmed that for long planning horizons, the staged design is cost effective. The system upgrades guarantee a predefined reliability and there is always opportunity to modify or redesign subsequent upgrades at the later stage, based on new up-to-date predictions of potential future development [118].

By introducing staged design to WDSs, it is obvious that the search space increases almost exponentially to accommodate decisions at various times in the planning horizon. This is one of the key challenges for deterministic staged design, as computational efficiency of optimisation algorithms plays even more significant role than with static design. Another difficulty for achieving the optimised staged design is that even if an optimal solution can be found for each of the intermediate time steps, the algorithm has to ensure that contiguity among the staged decision is maintained, i.e., that the decisions selected in the previous stages are retained in the subsequent ones. An approach by [133] presents one way of obtaining that contiguity of decisions, starting from the solution at one extreme of the Pareto front. However, this issue is still an under-researched area, which requires more investigation. All of the above challenges apply even when the future is assumed to be perfectly known, which is unfortunately not the case.

3.2.2. Flexible Design

A flexible design represents one of the most recent developments in the design optimisation of WDSs. Similar to a staged design, a flexible design represents an optimisation of a WDS over a long planning horizon divided into several construction phases, but with the consideration of future uncertainties (e.g., in demands, pipe deterioration, urban expansion scenarios). Specifically, it is a probabilistic dynamic design over several prefixed time intervals and with the planning horizon ranging from a number of years to an expected life cycle of the system. Such a design allows for flexible and adaptive planning, which is favoured by water organisations that are often encouraged to include risk and uncertainty in their long term plans.

Uncertainties included in the flexible design are related to future demands [122,134,135,136] and future network expansions [137]. They are implemented using either a probabilistic demand assessment [135] or scenario-based approach via demand/expansion scenarios [122,134,136,137]. A decision tree has been introduced to combine the uncertain demands and intervention measures into optional decision paths [135]. Analogously, studies [122,137] have proposed the use of real options (ROs) approach, which is also based on decision trees that reflect future uncertainties. In ROs approach, a decision tree is formed by independent decision paths with assigned probabilities to each of the scenarios. This approach enables flexibility to be incorporated into the decision making process and to subsequently change the investment plan based on new circumstances [122].

The majority of the above studies apply multi-objective optimisation approach, including, besides an economic (least-cost) objective, the system resilience [135], reliability [136] or total pressure violations [137] as another objective. Stochastic optimisation algorithms, such as NSGA-II [135,136], simulated annealing (SA), and multi-objective SA [122,137] have been employed to solve flexible design problems, except for [134] who applied integer LP (ILP) combined with preprocessing methods to reduce the dimensionality of the problem. These preprocessing methods included separating the (branched) network into subnetworks, reducing the number of decision variables (e.g., velocity constraints were used to eliminate unsuitable pipe diameters) and solving each subnetwork separately. As a consequence, the quality of the solution was improved and the proposed methodology [134] can be applied to large size WDSs.

While comparing to a traditional deterministic design, the results indicate that a flexible design has a higher initial cost (i.e., in the first construction phases) [122,136], which enables the system to adapt to various future conditions. However, it outperforms a traditional design in terms of the total cost over the entire planning horizon [122,135].

The application of flexible optimisation methodologies in WDS design that consider long-term uncertainty and management options, is yet to be explored to a larger extent in the literature. One of the key reasons is that it is not clear how various types of uncertainties, i.e., stochastic vs. deep uncertainty or aleatoric vs. epistemic uncertainty, are best represented in the optimisation process. The other possible reason is that the flexible design incurs additional computational costs that affects the overall computational efficiency of the optimisation algorithm. However, as the planning and design exercises are done sporadically, the additional computational burden and costs are often justified. Future uncertainties that might have an impact on WDS design, including climate change, population movements and economic development, make flexible design probably the most promising area of research over the next few decades.

3.2.3. Resilient, Reliable and Robust Design

System resilience, reliability and robustness present performance characteristics of a WDS in relation to current and most importantly future uncertain conditions. Although there is no universally agreed definition of any of these measures, the resilience can be defined in broadest terms as the ability of a WDS to adapt to or recover from a significant disturbance, which can be internal (e.g., pipe failure) or external (e.g., natural disaster) (adapted from [138]). Similarly, the reliability can be defined as the ability of a WDS to provide expected service, and can be expressed as the probability that the system will be in service over a specific period of time (adapted from [139]). The robustness represents the ability of a WDS to maintain its functionality under all circumstances (adapted from [138]), or over everyday fluctuations that have the potential to cause low to moderate (i.e., not catastrophic) loss of performance [89].

A robust design problem of a WDS is primarily concerned with uncertainties in model parameters. These uncertainties are related mainly to future demands [89,110,121,123,140,141], but can also consider pipe roughnesses [89,110,140,141], minimum nodal pressure requirements [110], network expansions [137] and others [142]. Study [89] states that “neglecting uncertainty in the design process may lead to serious underdesign of water distribution networks”.

Several approaches have been proposed in the literature to formulate a robust design problem for WDSs. Firstly, a redundant design approach which adds redundancy to the system to account for the uncertain parameters by assuming that those parameters are larger than expected [140]. Secondly, an integration approach where uncertainties are incorporated into the model formulation via either objective function [89] or constraints [140] sometimes referred to as a chance-constrained model [110]. Both of those approaches assume a probabilistic distribution of uncertain parameters and convert an original stochastic optimisation problem into a deterministic one. Thirdly, a two-phase optimisation approach that initially solves an optimisation problem with deterministic parameters (i.e., no uncertainties), and subsequently uses those obtained solutions as an initial population for a stochastic problem where future demands and pipe roughnesses are considered uncertain variables following a probability density function [141]. Fourthly, a scenario-based approach where the uncertainty is implemented via a set of scenarios, each assigned a probability [121]. Lastly and most recently, a robust counterpart (RC) approach which is non-probabilistic and incorporates the uncertainty through an ellipsoidal uncertainty set constructed according to the user-defined protection level [123].

Despite a number of approaches to incorporate robustness into the design of WDSs, the measure has been defined fairly well and consistently in the literature, and consequently it has been used in optimisation studies. This may be due to the advances in robust optimisation in other fields and/or due to the focus on non-catastrophic loss of performance that is associated with robust operation. However, the other two measures, reliability and most notably resilience, have not been defined consistently in the WDS literature or have been considered seriously only fairly recently. Therefore, this section focuses on robust design of WDSs, with resilience and reliability being outside of the scope of this review paper. This also indicates that future research efforts could be directed toward a consistent and agreed definition of reliability and resilience, with optimisation methods being capable of solving WDS design considering them as objectives/performance measures.

Robust design optimisation problems are mainly solved using stochastic methods, such as GA [140], NSGA-II [89,121], optimised multi-objective GA (OPTIMOGA) [141], PSO [142] and CE [123], except for [110], who solves it as a NLP problem.

3.2.4. Design for Water Quality

In the literature, water quality is incorporated into the WDS design optimisation problems in various ways concerning both an optimisation model and water quality measure used. In terms of optimisation models, single-objective as well as multi-objective exist which include water quality considerations. In the former, water quality related expenditures, such as the cost of disinfection [27,120], cost of water treatment [53] or cost of losses incurred by insufficient quality [129], are combined with the system design/capital (and operation) costs into one objective. Alternatively, water quality is included as a constraint to a single-objective model in a form of minimum (and maximum) disinfectant concentrations at the network nodes [87,143]. In the latter, water quality presents a sole objective, which is either water quality benefit (being maximised) [63,131], water quality deficiencies (being minimised) [92,97,144] or water quality reliability (being maximised) [78]. Regardless of an optimisation model used, study [120] highlighted an importance of incorporating water quality considerations with system design and operation in one optimisation framework, which enables promoting water quality in the design stage, rather than leaving potential water quality issues to be resolved during the system’s operational phase. Indeed, study [78] reports a significant tradeoff between water quality objective based on disinfectant residual and the system capital costs (i.e., the best quality solutions correspond to higher costs and vice versa), and demonstrates the sensitivity of the obtained solutions to a disinfectant dosage rate. Interestingly, there was not tradeoff found between water quality objective based on water age and the cost.

Regarding the water quality measure, it is dependent on the system specifics, its requirements, and also the optimisation model advancements progressively implementing water quality objectives useful to system operators. Basic water quality parameters that are used in optimisation models of drinking WDSs are chlorine [27,87,120,143] and chloramine [120], modelled as non-conservative applying first order decay kinetics, adjusted by a higher decay rate in parts of the system when needed [120]. In contrast, conservative water quality parameters are typically important for regional multiquality WDSs. These parameters are either specified within an optimisation model, such as salinity [129] or unspecified being modelled in conjunction with the operation of treatment facilities [53]. In multi-objective optimisation models, both specific parameters and surrogate measures are used to quantify water quality objectives. Water quality benefit is expressed as a function of the length of renewed and/or lined old pipes, as aged pipes are considered to cause the development of microorganisms and water discolouration [63,131]. Water quality deficiencies can be represented by a performance function on disinfectant residual reflecting governmental regulations [92], water age [97], or the risk of water discolouration due to the potential material after daily conditioning shear stress [144]. Water quality reliability is based on disinfectant residual [145] and/or water age [78].

Optimisation methods used to solve WDS design problems including water quality considerations were LP [129], GA [53,87,100,120,143] and differential evolution (DE) [27] for single-objective models, and SMGA [63,131], NSES [92], ε-NSGA-II [97], NSGA-II and SPEA2 integrated with a heuristic Markov-chain hyper-heuristic (MCHH) [144] and ant colony optimisation (ACO) [78] for multi-objective models. These algorithms were mainly linked with a network simulator EPANET to solve network equations. Because these EPANET simulations, in particular water quality analyses, are very computationally demanding, they were replaced by ANNs [27,87,143] to reduce computational effort.

Unsurprisingly, introduction of water quality considerations increases the complexity of the quest for the optimal design considerably. This increased complexity is caused not only by the more complex simulations required to predict the temporal and spatial distribution of a variety of constituents within a distribution system, but also by the requirement to run shorter time step water quality computations [22]. Furthermore, computational efficiency is affected by the ability to model multiple constituents throughout the WDS via the EPANET Multi-Species Extension, EPANET-MSX [146].

4. General Classification of Reviewed Publications

Based on the selected literature analysis, the following are the four main criteria for the classification of design optimisation for WDSs: application area (Section 4.1), optimisation model (Section 4.2), solution methodology (Section 4.3) and test network (Section 4.4).

4.1. Application Area

As outlined in Section 3, there are four application areas: design of new systems (Section 3.1.1), strengthening, expansion and rehabilitation of existing systems (Section 3.1.2). Numerous publications do not deal only with those design optimisation problems, but also with the operational optimisation (see, for example, [14,26,53,71,120,135]), which is an equally important area if the total cost (i.e., including capital and operation expenditure) is considered. Hence, the system operation has been added to the following analysis. It represents papers optimising (mainly) the pump operation together with the system design, strengthening, expansion and/or rehabilitation. Figure 2 displays distribution of the application areas across the papers analysed and listed in Appendix A Table A1 as follows:

Figure 2. Application areas (of papers from Appendix A Table A1).

• Design of new systems is an application area with the highest representation (41%). Interestingly, an almost identical percentage (43%) totals applications for existing systems (i.e., strengthening, expansion and rehabilitation).
• An application area with the second highest representation (25%) is strengthening of existing systems.
• Expansion and rehabilitation of existing systems are both represented evenly by 9% of applications each.
• Optimisation of the system operation is represented by 16% of applications.

It is not surprising that design (and mostly using pipe diameters as decision variables) dominates the literature, which occurs mostly due to historical reasons. Namely, the sizing of pipes was addressed first in the literature, even before WDS simulation was possible. Other design variants, such as strengthening, expansion and rehabilitation, followed on, but use the same or quite similar performance measures and optimisation tools. The introduction of other network elements, such as pumps, tanks and valves, as well as various performance criteria, including water quality and operational considerations, appeared much later in the literature. Lately, more emphasis was put on robustness, reliability and resilience assessment of WDS design and operation, which seems to be the trend for the future.

4.2. Optimisation Model

An optimisation model is mathematically defined by three key components: objectives, constraints and decision variables. Figure 3 shows how many of these components are included in the optimisation models (of papers analysed in Appendix A Table A1), which indicates the degree of complexity of the formulation. Note that not all of the reviewed papers include mathematical formulations of an optimisation model used. Therefore, our assessment is limited to our interpretation of the provided information in the publications, where explicit formulation was partially presented or missing altogether.

Figure 3. Optimisation models (of papers from Appendix A Table A1) by: (a) number of objectives, (b) number of constraints, (c) number of types of a decision variable, in an optimisation model.

• The number of objectives included in optimisation models ranges from one to six. The majority of models (69%) are single-objective, determining the least-cost design. The second largest proportion (27%) represents two-objective optimisation models. Multi-objective models including more than two objectives (i.e., 3–6 objectives) are very sparse as they represent only 4% of all formulations.
• The number of constraints incorporated in optimisation models ranges from zero to ten. Hydraulic constraints (such as conservation of mass of flow, conservation of energy and conservation of mass of constituent) are normally included as implicit constraints and are forced to be satisfied by a WDS modelling tool, such as EPANET, and thus are not included in these statistics. There are 5% of models with no constraints, which are mainly multi-objective optimisation models where the pressure requirement is defined as an objective rather than a constraint. Almost half models (48%) include only one constraint (mostly the minimum pressure requirement). A quarter of models (25%) incorporate two constraints. The proportion of optimisation models with exactly three or more (i.e., 4–10) constraints is 13% and 9%, respectively.
• The number of types of a decision (i.e., control) variable included in optimisation models ranges from one to 13. The majority of optimisation models (60%) uses one type of a decision variable, being a pipe diameter/size or pipe segment length of a constant (known) diameter. The use of more than one type of a decision variable is considerably less frequent and is represented by 16%, 11% and 13%, respectively, for two, three and more (i.e., 4–13) types of a decision variable.

Inspecting Figure 3, the question arises as to how many optimisation models there are, which include only one objective, one constraint and one type of a decision variable? There are, in total, 129 optimisation models formulated and solved in 124 papers listed in Appendix A Table A1. From those optimisation models, 30% (i.e., 39 models) consist of one objective (mostly design costs), one constraint (mostly the minimum pressure at nodes) and one type of a decision variable (mostly pipe diameters).

As indicated, the prevailing use of single-objective optimisation is probably caused by the preference to arrive at a single solution, which can be implemented by decision makers. On the other hand, the preference for one constraint seems surprising as the number of constraints of the problem depends on the complexity of the system and the number of criteria expressed as constraints rather than objectives. Finally, the number and types of decision variables appearing in the literature is a function of historical developments in the field and the increasing trend is expected in the future. Research questions still remain as how to best formulate the optimisation model for a particular case, and what effect the model formulation has on obtained solution(s) [22,23].

4.2.1. General Optimisation Model

A general multi-objective optimisation model for optimal design of a WDS can be formulated as:

$$\mathrm{Minimise}/\mathrm{maximise} \left(f_1\left(x\right), f_2\left(x\right), \dots , f_n\left(x\right)\right)$$

(1)

subject to:

$$a_i\left(x\right)=0,i\in I=\left\{1, \dots ,m\right\},m\ge 0$$

(2)

$$b_j\left(x\right)\le 0,j\in J=\left\{1, \dots ,n\right\},n\ge 0$$

(3)

$$c_k\left(x\right)\le 0,k\in K=\left\{1, \dots ,p\right\},p\ge 0$$

(4)

where Equation (1) represents objective functions to be minimised (e.g., system capital costs) or maximised (e.g., system reliability), Equations (2)–(4) present three types of a constraint, with $x$ representing decision variables.

Objectives

Objectives of a general optimisation model of WDS design are listed in Table 1. They can be divided into four distinct groups according to their type. The first group represents economic objectives such as capital and rehabilitation costs, and expected operation and maintenance costs of the system. The second group are community objectives, which report on the level of service provided to WDS customers, and which, if inadequate, could eventuate in water supply related issues for those customers. Those objectives include, for example, a benefit function (using various performance criteria), water quality deficiencies, pressure deficit at demand nodes, hydraulic failure of the system and potential fire damages. The third group presents performance objectives, reflecting the operation of a WDS, specifically system robustness, reliability and resilience. These objectives, although ultimately indicating the level of service for WDS customers, have separate classification, due to their primary purpose to report on the performance in relation to a WDS rather than to customers. The fourth group represents environmental objectives, namely GHG emissions, consisting of capital emissions due to manufacturing and installation of network components applicable at the WDS construction phase, and operating emissions due to electricity consumption occurring during the WDS life cycle.

Table 1. Objectives of a general optimisation model.

Objective Type	Objectives	Reference (An Example)
Economic	Capital costs of the system, including purchase, installation and construction of network components (pipes, pumps, tanks, treatment plants, valves, etc.)	[53,74,121]
	Rehabilitation costs of the system, including pipe/pump replacement, pipe cleaning/lining, pipe break repair	[17,124] (pipes), [77] (pumps)
	Expected operation costs of the system, including pump stations, treatment plants and disinfectant dosage	[53] (pump stations and treatment plants), [27] (disinfectant dosage)
	Expected maintenance costs of the system	[121]
Community	Benefit/benefit of the solution (i.e., rehabilitation, expansion and strengthening) using various performance criteria by authors	[131] (welfare index to place greater importance on early improvements), [99] (quantity shortfalls as criteria), [93] (e.g., safety volumes and operational capacities as criteria)
	Water quality (e.g., disinfectant, sedimentation, discolouration) deficiencies or water age at customer demand nodes, water discolouration risk, velocity violations (causing sedimentation/discolouration)	[92,120] (water quality deficiencies), [97] (water age), [144] (water discolouration), [95] (velocity violations)
	Pressure deficit at customer demand nodes (maximum individual or total), or the number of demand nodes with the pressure deficit	[65] (maximum individual deficit), [66,68] (total deficit), [90] (the number of demand nodes)
	Hydraulic failure of the system expressed by the failure index	[97]
	Potential fire damages using either expected conditional fire damages or fire flow deficit	[142] (expected conditional fire damages), [92] (fire flow deficit)
Performance	System robustness using either a redundant design approach, integration approach (via objective function or constraints), two-phase optimisation approach, scenario-based approach or RC approach	[140] (redundant design), [89] (integration via objective function), [110,140] (integration via constraints), [141] (two-phase optimisation), [121] (scenario-based), [123] (RC)
	System reliability	[118]
	System resilience	[135]
Environmental	GHG emissions or emission costs consisting of capital emissions (due to manufacturing and installation of network components) and operating emissions (due to electricity consumption)	[77] (capital and operating GHG emission costs), [73,75] (capital and operating GHG emissions), [132] (operating GHG emission cost)

Constraints

Constraints of a general optimisation model of WDS design are described in Table 2 and divided into three groups as follows. Hydraulic constraints are given by physical laws governing the fluid flow in a pipe network. These constraints are incorporated in an optimisation model either explicitly often in conjunction with deterministic [147] and hybrid optimisation techniques [116,117], or implicitly by a WDS modelling tool (e.g., EPANET) [26] and/or ANNs [27,87] normally in combination with stochastic optimisation algorithms. System constraints arise from limitations and operational requirements of a WDS, and include tank water level bounds, pressure/water quality requirements at demand nodes, etc. The ways to manage these constraints include an integration of EPANET (e.g., tank water levels), the augmented Lagrangian penalty method [17], a penalty function [26], a penalty function with a self-adaptive penalty multiplier [45,88], or a (modified) constraint tournament selection [148,149,150]. Constraints on decision variables, such as pipe diameters being limited to commercially available sizes and others, are handled explicitly by an optimisation algorithm.

Table 2. Constraints of a general optimisation model.

Constraints	Reference (An Example)
Hydraulic constraints given by physical laws of fluid flow in a pipe network: (i) conservation of mass of flow, (ii) conservation of energy, (iii) conservation of mass of constituent	[41]
System constraints given by limitations and operational requirements of a WDS, for example, minimum/maximum pressure at (demand) nodes and flow velocity in pipes, water deficit/surplus at storage tanks at the end of the simulation period, maximum water withdrawals from sources	[54] (limits on nodal pressure, storage tank deficit and water withdrawals from sources), [127] (limits on pipe velocity)
Constraints on decision variables $x$, for example, limits on pipe diameters, pipe segment lengths (so called split-pipe design), pump station capacities	[92] (limits on pipe diameters), [117] (limits on pipe segments), [121] (limits on pump stations)

Decision Variables

Decision variables of a general optimisation model of WDS design are listed in Table 3. They are grouped according to an element or aspect that drives the optimisation (i.e., pipes, pumps, tanks, valves, nodes, water quality and timing). In general, a pipe diameter/size is often the main (or the only) decision variable used in design optimisation of WDSs. Accordingly, a total of 60% optimisation models (of papers listed in Appendix A Table A1) use only one type of a decision variable (see Figure 3c), which is either a pipe diameter/size or the pipe segment length of a constant (known) diameter. As the complexity of an optimisation model increases, so does the number of types of a decision variable. An example of such an optimisation model could be an expansion and rehabilitation of an existing WDS with pumps, tanks and a treatment plant to meet future demands and water quality requirements.

Table 3. Decision variables of a general optimisation model.

Decision Variables	Reference (An Example)
Pipes: pipe diameters/sizes, pipe duplications, pipe rehabilitation options (pipe replacement, pipe cleaning/lining), pipe break repair, pipe segment lengths (so called split-pipe design), future pipe roughnesses, pipe routes, pipe closures/openings (to adjust a pressure zone boundary)	[75] (diameters), [132] (duplications, replacement, lining and break repair), [117] (segments), [141] (roughnesses), [52] (routes), [100] (routes and closures/openings)
Pumps: pump locations, pump sizes (pump capacities, pump types, pumping power, pump head/height or head-flow), the number of pumps, pump schedules (pumping power or pump head at each time step, the number of pumps in operation during 24 h, binary statuses at time steps, on/off times)	[52,99] (locations), [14] (locations and capacities), [75] (types), [17] (power), [52,122] (head/height), [93] (head-flow), [26] (sizes and the number of pumps), [53,123] (power or head at each time step), [97] (the number of pumps in operation), [29] (binary statuses), [75] (on/off times)
Tanks: tank locations, tank sizes/volumes, minimum operational level, ratio between diameter and height, ratio between emergency volume and total volume, tank heads	[98] (locations, sizes/volumes, minimum operational level, ratios), [78] (heads)
Valves: valve locations, valve settings (headlosses or flows)	[14] (locations), [16] (headlosses via a roughness coefficient), [127] (headlosses and flows)
Nodes: flowrates from sources, future nodal demands, threshold demands, hydraulic heads at junctions	[127] (flowrates), [135,141] (demands), [147] (heads)
Water quality: disinfectant dosage rates (at the sources, at the treatment plants, in the tanks), treatment removal ratios, treatment plant capacities	[143] (dosage at the sources), [27] (dosage at the treatment plants), [78] (dosage in the tanks), [53] (removal ratios), [121] (capacities)
Timing: year of action (e.g., network expansion, rehabilitation, pipe replacements) execution	[131] (network expansion and rehabilitation), [130] (pipe replacements)

Table 1, Table 2 and Table 3 provide a generic set of components used for formulating an optimisation problem involving initial design with subsequent operational management of a WDS. Particular circumstances being considered in different case studies may warrant only a portion of those components to be used.

4.3. Solution Methodology

An enormous effort has been dedicated to the application and development of optimisation methods to solve WDS design optimisation problems since the 1970s. Initially, deterministic methods namely LP [14,114,129], NLP [16,110] and mixed-integer NLP (MINLP) [17,115] were used. In the mid 1990s, after the first popular applications of a GA [20,151], there was a swing towards stochastic methods and they dominate the field since (see Figure 4). A great range of those methods has been applied to optimise design of WDSs to date, inclusive of (but not limited to) a GA [42,45,50,85,86,152,153,154], fmGA [88], non-crossover dither creeping mutation-based GA (CMBGA) [149], adaptive locally constrained GA (ALCO-GA) [155], SA [60], shuffled frog leaping algorithm (SFLA) [103], ACO [104,156], shuffled complex evolution (SCE) [157], harmony search (HS) [105,158,159], particle swarm HS (PSHS) [160], parameter setting free HS (PSF HS) [161], combined cuckoo-HS algorithm (CSHS) [162], particle swarm optimisation (PSO) [106,153,154], improved PSO (IPSO) [163], accelerated momentum PSO (AMPSO) [164], integer discrete PSO (IDPSO) [165], newly developed swarm-based optimisation (DSO) algorithm [150], scatter search (SS) [166], CE [61,62], immune algorithm (IA) [167], heuristic-based algorithm (HBA) [168], memetic algorithm (MA) [107], genetic heritage evolution by stochastic transmission (GHEST) [169], honey bee mating optimisation (HBMO) [170], DE [46,153,154,171], combined PSO and DE method (PSO-DE) [172], self-adaptive DE method (SADE) [173], NSGA-II [70], ES [68], NSES [92], cost gradient-based heuristic method [119], improved mine blast algorithm (IMBA) [174], discrete state transition algorithm (STA) [175], evolutionary algorithm (EA) [109], and convergence-trajectory controlled ACO (ACO_CTC) [176]. The vast majority of those studies solely solve a basic single-objective least-cost design problem (i.e., pipe cost minimisation constrained by the nodal pressure requirement) and use a small number of available benchmark networks (e.g., Hanoi network [49], New York City tunnels [81], two-loop network [14]) to test the proposed optimisation method. The usual result obtained was a better or comparable optimal solution reached more efficiently than by algorithms previously used in the literature, without providing an explanation as to why the selected algorithm performed better for a particular test network. It seems, therefore, that research have been trapped, to some extent, in applying new metaheuristic optimisation methods to relatively simple (from an engineering perspective) design problems, without understanding the underlying principles behind algorithm performance. Moreover, study [177] stresses that there has been “little focus on understanding why certain algorithm variants perform better for certain case studies than others”.

Figure 4. Optimisation methods (of papers from Appendix A Table A1) by year.

Over the past decade, an increase in the use of deterministic and hybrid methods (i.e., a combined deterministic and stochastic method) can be observed from Figure 4. These methods, which are computationally more efficient when comparing to stochastic methods, thus more suitable for large real-world applications, include ILP [51,134], MINLP [147], a combined GA and LP method (GA-LP/GALP) [113,117], combined GA and ILP method (GA-ILP) [178], combined binary LP and DE method (BLP-DE) [179], combined NLP and DE method (NLP-DE) [111], hybrid discrete dynamically dimensioned search (HD-DDS) [180], decomposition-based heuristic [52], optimal power use surface (OPUS) method paired with metaheuristic algorithms [47], and modified central force optimisation algorithm (CFOnet) [181]. However, WDS simulations may still be computationally prohibitive even with more efficient deterministic or hybrid optimisation methods, especially as the fidelity of the model and the number of decision variables increase [22].

The choice of the solution methodology depends on the type of problem being considered, the level of expertise of the analyst and the familiarity with the particular method/tool. Nonetheless, there is often no clear justification provided as to why a particular methodology has been selected over another and/or why an alternative methodology has not been tested. Quite often, this choice is based on the analyst’s preference, level of familiarity, and software availability, rather than on a comparison of the tests performed using two or more solution methodologies. This practice makes it difficult to progress towards the development of meaningful guidelines for the application of different optimisation methods [177]. An interesting research question for further studies would be how to characterise and select the best optimisation method for a particular WDS design problem.

However, that being stated, several attempts have been made to compare or evaluate algorithm performance for both single- and multi-objective WDS design problems, but with an absence of a universally adopted method to date. A methodology for comparing the performance of various single-objective algorithms involves assessing the best solution obtained (which is straightforward contrary to multi-objective optimisation), the convergence speed, and the spread and consistency of the solutions using a number of random starting seeds and evaluations [153,154]. A methodology has also been developed to evaluate the performance of a particular algorithm by assessing the effectiveness of its parameters (such as crossover and mutation) applying their different values [182]. In multi-objective optimisation, in general, performance metrics were proposed and are commonly used to compare performance of various algorithms in terms of the quality of the Pareto fronts obtained (see, for example, [183,184,185]). A comparison of solutions is substantially more complex than in single-objective optimisation as there is no single performance metric both compatible and complete [186]. Possibly for that reason, some WDS design studies have limited their analysis to a visual comparison of solutions only (i.e., two-objective Pareto fronts), which was criticised by [187]. Most recent research, progressively, evaluates the performance and search behaviour of multi-objective algorithms in relation to their parameters and/or WDS features [28] (more such studies are listed in Section 4.3.2).

4.3.1. Computational Efficiency

Numerous advancements have been reported in the literature to improve the computational efficiency of both optimisation algorithms and network simulators. These developments include methods for search space reduction [45,63,88,95,99,120,131,188,189], parallel programming techniques [109], hybridisation of the evolutionary search with machine learning techniques to limit the number of function evaluations [67], surrogate models (metamodels) to replace network simulations [27,43,67,87,143], approximation of the objective function by shortening the EPS [119], and enhanced methods for speedy network simulations for large size WDSs [190].

Various techniques for search space reduction have been proposed, which can be broadly classified as algorithm-based and network-based methods. The algorithm-based techniques include the method for more efficient encoding of decision variables [63,99,131], a self-adaptive boundary search strategy for selection of the penalty factor within the optimisation algorithm to guide the search towards constraint boundaries [88], and the application of an artificial inducement mutation (AIM) to acceleratingly direct the search to the feasible region [95]. The network-based techniques analyse either the network as a whole or individual pipes. The former include a network stratification into upper, middle and lower diameter sets using engineering judgment [188], and the critical path method [45,191]. The latter involve the elimination of certain pipes from the optimisation based on their preliminary capacity assessment [120], application of a pipe index vector (PIV), a measure of the relative importance of pipes regarding their hydraulic performance in the network, which assists in exclusion of impractical and infeasible regions from the search space [189], and introduction of upper/lower bounds on pipe diameters based on the initial analysis [30].

In terms of replacing time consuming network simulations with faster means, three types of a surrogate model have been applied to the design optimisation of WDSs to date. These models include a linear transfer function (LTF) [43], Kriging [67] and ANNs [27,87,143], which are used more frequently than two previous ones. The purpose of a surrogate model is to approximate network simulations (hydraulics and/or water quality), hence reduce the calls of the simulation model during the optimisation. Kriging uses solutions visited during the search to model the search landscape [192]. ANNs can be divided into two groups, offline ANNs, which are trained only once at the beginning of the optimisation, and recently proposed online ANNs, which are “retrained periodically during the optimisation in order to improve their approximation to the appropriate portion of the search space” [27]. ANN metamodels are often used in conjunction with water quality simulations [27,87,143].

All of those methods have shown promise on a limited number of test cases or a specific case study. It would be interesting to conduct a thorough comparison of all of those on a selection of benchmark cases of various sizes and complexity.

4.3.2. Recent Developments

More recently, a number of advancements, such as improving and understanding the algorithm performance and others, have been proposed in the literature indicating potential directions for future research. Some of those developments are a consequence of an appeal of [23,177] “to counteract potential repetition and stagnation in this field” to continually produce too many papers using “an ever increasing number of EA variants” and “theoretical or very simplistic case studies”.

Firstly, to improve the algorithm performance regarding the solution quality, an engineered initial population has been suggested [26,30,44,66,108]. Traditionally, a random (or naïve) initial population of solutions (expressed as pipe sizes) is used as a starting point for algorithms. An engineered initial population, in contrast, is created by taking into account the rules and hydraulics principles of water flow in a pipe network, or by performing pre-optimisation runs under various parameter scenarios (e.g., [30]). Another way to achieve better algorithm convergence, particularly for design problems incorporating water quality, is to run the algorithm with hydraulic analysis only for several first generations, and subsequently add water quality analysis [120]. Furthermore, a postoptimisation technique can be used to refine the solutions that are found by an optimisation algorithm to get closer to the global optimum [193]. Secondly, a range of strategies have been introduced to eliminate the tedious and time demanding process of calibrating algorithm parameters (i.e., selecting the best performing combination of parameter values) for a particular test problem. These strategies involve the use of a statistical analysis [158], evolved heuristics (i.e., hyper-heuristic approach) [68,144,194], and convergence trajectories [176]. Thirdly, several studies focused on analysing algorithm performance [195] and search behaviour [28,48,196] in relation to the WDS design problem features [29]. Lastly, methodological improvements using existing methods have been proposed rather than applying/developing new algorithms, with the aim to improve computational efficiency. Those improvements represent multiple-phase optimisation concepts [30], which can be combined with graph decomposition [46,69] or clustering [90] techniques.

4.4. Test Network

An enormous diversity exists among test networks used in optimisation of WDS design. These networks vary in size, complexity, and the types of network components that they contain (i.e., nodes, pipes, pumps, tanks, reservoirs/sources and/or valves). The simplest networks are small gravity WDSs with one source, a few nodes and pipes (see, for example, [14,60]), or simplified pumped WDSs including only one source, one pump, one pipe and one tank (see, for example, [71]). An example of a large network represents EXNET [82], which is a realistic WDS comprising two sources, control valves and almost 2500 pipes. Figure 5 categorises test networks that are used (in the papers listed in Appendix A Table A1) by network size. In order to be consistent with the previous review [22], network size is expressed by the number of nodes within a network. Networks, for which the number of nodes cannot be identified from the reviewed paper or the references provided, are excluded from the analysis. Figure 5 reveals that nearly a half of the networks (49%) is limited in size to 20 nodes and the majority of the test networks (84%) contains up to 100 nodes. This finding is analogous to operational optimisation of WDSs, where networks with up to 100 nodes represent 80% of applications [22].

Figure 5. Test networks (of papers from Appendix A Table A1) by network size.

Figure 5 illustrates that in the large body of literature, various WDS design formulations and optimisation methods have usually been tested using small, computationally cheap networks. This prevalent usage of small networks is in contrast to the requirement to optimise design of real-world systems that contain hundreds of thousand elements (including pumping stations, tanks and valves) causing a single EPS simulation to take minutes or even hours to execute even on powerful desktop computers. Consequently, large networks are not often considered by optimal design studies. This situation can be remedied by using latest developments in methods capable of generating realistic WDS networks by [55,56,57], who have each developed their own automatic network generation software.

Real-world WDS design optimisation problems normally involve large size, complex-topology networks, comprising a number of elements of various types. Such a problem is often solved by combining a sophisticated simulation model (to potentially analyse both hydraulics and water quality) with a non-trivial optimisation method. The approach ought to satisfy the requirements of a water utility and other stakeholders for objectives, constraints, decision variables, as well as model assumptions. Although studies exist that report on successful solutions to such problems [100,127,197,198,199], they are limited possibly due to the complexity associated with mathematically formulating objectives and constraints and/or finding the best solution. Study [200] even speculates that the real-world considerations need to be explicitly quantified, “if it is possible to do so at all”, otherwise the water industry will apply engineering judgment instead of any optimisation method to design WDSs.

Similar to network size, the frequency of use of test networks varies considerably, as some networks have been used only once, while others have been used repeatedly and by multiple authors. In particular, the prevalence of some networks attributes to their use as benchmark problems to test optimisation algorithms. These benchmark networks, all of which have been used (in the papers listed in Appendix A Table A1) 10 or more times, are listed in Table 4 in order of their usage count. They are, except for the Anytown network, gravity-fed WDSs with the common objective to determine the most economical pipe design. The popularity of those benchmark networks contributed to high percentages of the first two categories in Figure 5, because the majority of them are limited in size to 20 and 100 nodes, respectively.

Table 4. Frequently used test networks.

Test Network Name	No. of Nodes	Network Description	Optimisation Problem	Network Modified Versions	Network Usage Count *
Hanoi network ++ [49]	32	Network organised in three loops supplied by gravity from a single source	New system design (pipes)	Double Hanoi network, triple Hanoi network (both [113])	55
New York City tunnels ++ [81]	20	Tunnel system supplied by gravity from a single source, constituting the primary WDS of the New York city	Existing system strengthening (i.e., pipe paralleling) to meet projected demands	Double New York City tunnels [201]	42
Two-loop network ++ [14]	7	Small network with two loops supplied by gravity from a single source	New system design (pipes)	Adapted to system strengthening and expansion over a planning horizon [118]	40
Balerma irrigation network ++ [50]	447	Large looped network supplied by gravity from four sources, adapted from the existing irrigation network in Balerma, Spain	New system design (pipes)	N/A	20
Anytown network [84]	19	Hypothetical looped system supplied by three parallel pumps from a single source	Existing system strengthening, expansion and rehabilitation (pipes, pumps, tanks) to meet projected demands	** With additional source and tank [53], with additional tank [119] proposed by [83]	15

Notes: * The count of the network usage as in the papers listed in Appendix A Table A1 inclusive of networks’ modified versions. ⁺⁺ Detailed overview of results to a least-cost design problem obtained by various authors is presented in [39]. ** Other modified versions of the Anytown network exist.

5. Future Research

Future research challenges for the optimisation of WDS design are illustrated in Figure 6 and divided into the following four groups: (i) model inputs, (ii) algorithm and solution methodology, (iii) search space and computational efficiency, and (iv) solution postprocessing. As far as model inputs are concerned, there is a requirement to explore how to best represent various types of uncertainties, i.e., stochastic vs. deep uncertainty or aleatoric vs. epistemic uncertainty, in the optimisation process. Additional future uncertainties, for example, climate change, population movements and economic development, might affect planning for optimal WDSs, and make flexible design one of the promising research areas over the next few decades. Another research challenge in regards to model inputs is to compare various approaches to pump decision variables, including VSPs and their coding, in order to determine their advantages, disadvantages and suitability for a particular case. Furthermore and overall, a research question remains how to best formulate the optimisation model for a particular case [22,23].

Figure 6. Future research challenges.

Concerning algorithm and solution methodology, a vast research area represents a progression towards better understanding of algorithm performance and its search behaviour. These aspects need to be further linked to the WDS design problem features including system topology (e.g., existence of pumps/tanks, loops) and initial population used. A related challenge is to eliminate a time consuming process of calibrating algorithm parameters to achieve a satisfactory performance, hence there is a question how to select the best performing combination of parameter values. Moreover, it is important to develop understanding related to the suitability of various optimisation methods for particular design problems and the incorporation of engineering judgement in the search. In relation to staged design, methods for ensuring contiguity among decisions, i.e., that the decisions selected in the previous stages are retained in the subsequent ones, are required.

Recently, there has been an observed increased interest in aspects of the search space and computational efficiency. Indeed, the reduction of the search space and an increase in the computational efficiency are significant particularly for real-world WDS optimisation problems as well as dynamic (i.e., staged and flexible) design, so they are expected to remain important and promising research areas into the future. The research community would benefit from a thorough comparison of existing methods for search space reduction and computational efficiency increase, which could use a selection of benchmark cases of various sizes and complexity. In addition to currently available methods for search space reduction, it might be possible to further improve decision variable coding.

Regarding solution postprocessing, an open question is how sensitive the obtained solution(s) is to the optimisation model used [22,23]. When multi-objective optimisation approach is used, a remaining challenge is to select a practical and representative subset of the non-dominated solutions, which could be useful for the decision makers. Accordingly, there is a need for methods to identify a handful of effective solutions, such as those where a small improvement in one objective leads to a large deterioration in at least one other objective. The existing approaches, including maximum convex bulge/distance from hyperplane, hypervolume contribution, and local curvature [80] are all promising and require a thorough analysis on WDS design problems.

6. Summary and Conclusion

A systematic literature review of optimisation of water distribution system (WDS) design since the end of the 1980s to nowadays has been presented. The publications included in this review are relevant to the design of new WDSs, strengthening, expansion and rehabilitation of existing WDSs, and also consider design timing, parameter uncertainty, water quality and operational aspects. The value of this review paper is that it brings together a large number of publications for design optimisation of WDSs, just under three hundred in total, which have been published over the past three decades. Therefore, it may enable researchers to identify one’s articles of interest in a timely manner. The review analyses the current status, identifies trends and limits in the field, describes a general optimisation model, suggests future research directions. Exclusively, this review paper also contains comprehensive information for over one hundred and twenty publications in a tabular form, including optimisation model formulations (i.e., objectives, constraints, decision variables), solution methodologies used and other important details.

This review has identified the following main limits in the field and future research directions. It was demonstrated that there is still no agreement among researchers and practitioners on how to best formulate a WDS design optimisation model, how to include all relevant objectives and constraints, and whether and how to take into account various sources of uncertainty, while still allowing for an efficient search for the best solution to be achieved. Although a plethora of generic and problem-specific optimisation methods have been developed and applied over the years, there is no consensus on what optimisation method is best for a particular design problem, whether a single or multiple-phase optimisation concept is to be used, and how engineering judgement can best be incorporated in the search. Therefore, a concerted effort by the research community is required to develop methods for objective comparison and validation of various optimisation algorithms and concepts on large, real-world problems. In addition, an analysis of available methods for reducing the search space, increasing computational efficiency, as well as selecting effective Pareto non-dominated solutions representing a practical subset for decision makers, is needed using WDS design problems of various sizes and complexity. In spite of the overwhelming amount of literature that has been published over the past three decades, design optimisation of WDSs faces considerable research challenges in the years to come.

7. List of Terms

• Deterministic dynamic design = staged design over a long planning horizon divided into several construction phases, without considering future uncertainties.
• Deterministic static design = traditional design with a single construction phase for an entire expected life cycle of the system, without considering future uncertainties.
• Dynamic design = staged (i.e., real-life) design capturing the system modifications/growth over a long planning horizon divided into several construction phases (adopted from [118]).
• Hydraulic constraints = constraints arising from physical laws of fluid flow in a pipe network, such as conservation of mass of flow, conservation of energy, conservation of mass of constituent.
• Optimisation approach = single-objective approach or multi-objective approach.
• Optimisation method = method, either deterministic or stochastic, used to solve an optimisation problem.
• Optimisation model = mathematical formulation of an optimisation problem inclusive of objective functions, constraints and decision variables.
• Probabilistic dynamic design = flexible design over a long planning horizon divided into several construction phases, with considering future uncertainties.
• Probabilistic static design = traditional design with a single construction phase for an entire expected life cycle of the system, with considering future uncertainties.
• Simulation model = mathematical model or software used to solve hydraulics and water quality network equations.
• Single pipe design = design which uses pipe sizes/diameters as decision variables (either discrete or continuous).
• Solution = result of optimisation, either from feasible or infeasible domain, so we refer to a ‘feasible solution’ or ‘infeasible solution’, respectively. In mathematical terms though an ‘infeasible solution’ is not classified as a solution.
• Split-pipe design = design which uses pipe segment lengths of a constant (known) diameter as decision variables.
• Static design = traditional (i.e., theoretical) design with a single construction phase for an entire expected life cycle of the system (adopted from [118]).
• System constraints = constraints arising from the limitations of a WDS or its operational requirements, such as water level limits at storage tanks, limits for nodal pressures or constituent concentrations, tank volume deficit etc.