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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<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3390/w10030307">doi:10.3390/w10030307</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7g7cx57xpzhpzeg55lw4qeazly">fatcat:7g7cx57xpzhpzeg55lw4qeazly</a> </span>
more »... esign 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. 2 of 103 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)  , nonlinear programming (NLP) [15, 16] , and others  , 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  and natural genetics  . Nonetheless, it was not until the 1990s when these methods became more popular  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  . 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  . Some of the most recent developments include the use of an engineered (as opposed to a random) initial population to improve the algorithm convergence  , application of online artificial neural networks (ANNs) to replace network simulations  , analysis of the algorithm search behaviour  in relation to the WDS design problem features  , and reduction of the search space  to increase computational efficiency. 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,          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. 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). Application Areas 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  . This possible variability results in iterative methods being required to solve pipe flows in looped networks, such as that described in  . 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        . Popular test networks used to solve such a problem are the two-loop network , Hanoi network  and Balerma irrigation network  . 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  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  . The lack of large and complex test networks has recently been addressed by a number of researchers [55-57] 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] . Water 2018, 10, 307 4 of 103 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,    ). 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  . In studies of newly designed WDSs, in addition to the economic measure, the other objectives considered were the pressure deficit [30, 62,     or excess [68,69] at network nodes, the penalty cost for violating the pressure constraint , greenhouse gas (GHG) emissions [71-76] or emission cost , water discolouration risk  and water quality . 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 . Water 2018, 10, x FOR PEER REVIEW 4 of 91 Water 2018, 10, 307 5 of 103 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  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  and EXNET  . 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  . 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 , fast messy GA (fmGA)  and non-dominated sorting genetic algorithm II (NSGA-II)  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  (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)  and cross entropy (CE)  . 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  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  or integer  decision variables to identify the pipes selected for replacement, and continuous  or integer  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  added the expected pipe repair costs to the rehabilitation cost of the Water 2018, 10, 307 6 of 103 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) , 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  , 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  , pressure violations at network nodes [68, 95] , velocity violations in pipes  causing potential sedimentation problems and subsequent water discolouration, water quality (i.e., disinfectant) deficiencies at network nodes  , and potential fire damage expressed as lack of available fire flows  . To generate multi-objective optimal solutions, those studies use mainly metaheuristics or hyperheuristics, such as structured messy GA (SMGA) , NSGA-II , non-dominated sorting evolution strategy (NSES)  , and evolution strategy (ES)/SPEA2 in a hyperheuristic framework with evolved mutation operators  . 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  . 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"  . Although some studies solved the Anytown network problem as initially formulated [84 ], for example, study  by enumeration and  using GA, others included new aspects to the (original or modified) problem. Those aspects represent, for example, water quality  inclusive of the construction and operation costs of treatment facilities , 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 . These studies used SMGA , GA [53,93], and ε-NSGA-II  to solve the problem. Study  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  . This problem was solved by multiple authors within the Water Distribution Systems Analysis (WDSA) conference series  . Another example of large and real-world system expansion is presented in  . Apart from the decision variables for the BWN-II, it also includes selections Water 2018, 10, 307 7 of 103 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 , some eliminated variables were included in separate optimisation problems. Study  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  , 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  that can be solved separately, the involvement of engineering expertise and manual interventions  to reduce the search space, or the use of surrogate and meta-modelling to speed up the simulation process  . 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  . Problem Elements
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