A general multi-objective hyper-heuristic for water distribution network design with discolouration risk

The optimisation of water distribution networks (WDNs) by evolutionary algorithms has gained much coverage in the literature since it was first proposed in the early 1990s. Despite being well studied, the problem and objectives continue to evolve as demands on water companies change. Motivated by the increased focus on reducing the risk of discolouration, this study examines a three objective version of the WDN design problem which takes into account cost, head excess and discolouration risk. Using this formulation, this paper presents a method for producing optimised network designs aimed at reducing discolouration risk in the network design phase and thus reducing the associated long-term maintenance and operational burdens of the system. This paper discusses the use of a discolouration risk model and, using this model, the optimisation of network design, specifically pipe diameters, to produce a range of high quality self-cleaning networks. The network designs are optimised using the Markov-chain hyper-heuristic (MCHH), a new multi-objective online selective hyper-heuristic. The MCHH is incorporated in to the known NSGA-II and SPEA2 and supplied with a range of heuristics tailored for use on the WDN design problem. The results demonstrate an improvement in performance obtained over the original algorithms. doi: 10.2166/hydro.2012.022 om https://iwaponline.com/jh/article-pdf/15/3/700/387024/700.pdf 2019 Kent McClymont (corresponding author) Ed Keedwell Dragan Savić College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, UK E-mail: K.McClymont@exeter.ac.uk Mark Randall-Smith Mouchel, Clyst Works, Clyst Road, Topsham, Exeter EX3 0DB, UK


INTRODUCTION
The UK water industry is tightly regulated by Ofwat, the UK regulatory body, with the performance of water companies closely monitored by a range of indicators; from water quality, customer service (e.g., sufficient pressure) to customer satisfaction. Recently, an emphasis has been placed on customer satisfaction in particular, which is partly measured by monitoring customer complaints and contacts. Motivated by these regulatory demands, water companies are now focusing efforts on reducing the frequency of water discolouration events (Cook ) prior to customer contacts occurring. Indeed, discolouration events (the visible discolouration of water at the tap) have been attributed to approximately 30% of all complaints received by water companies in the UK (Cook ).
It should be noted, however, that the drive to reduce the number of discolouration events is not isolated to the UK, but experienced in many countries, worldwide. Importantly, the phenomenon does not appear to display regional variances, outside of temperature and material differences, with Boxall & Prince () demonstrating the validity of UK models abroad. As such, it can be supposed that any advancement relating to discolouration modelling may have world wide application.
However, despite its apparent prominence in industry, discolouration risk is seldom included in studies that optimise the design of water distribution networks (WDNs). In this paper we investigate the use of discolouration risk as an objective in a multi-objective algorithm and propose a et al. ) and SPEA2 (Zitzler et al. ) in addition to MCHH variants of both these algorithms. The problem is modelled using EPANET (Rossman ) and a state-ofthe-art model of stored material using shear stress (Boxall et al. ; Boxall & Saul ). The experiment is conducted on three benchmark datasets (Two Loops, Hanoi and Anytown) and three real-world networks, described in the Results section. Optimisation results are compared with hyper-heuristics from the literature as well as results presented in (Shie-Yui Liong & Atiquzzaman ) who contrast a shuffled complex evolutionary (SCE) approach to a variety of preceding evolutionary approaches on these datasets.
Finally, the Conclusion section discusses the optimisation results which reveal a correlation between cost and discolouration potential in addition to the trade-off with head excess.

BACKGROUND Modelling discolouration
Despite the varying approaches, only a handful of discolouration risk modelling software packages are available to water companies at present. The most notable of which are: the PODDS I-IV models developed by the University of Sheffield (Boxall et al. ; Boxall & Saul ); the DRM software developed by Mouchel (Dewis & Randall-Smith ); and prototype models founded on discolouration risk ranking (Vreeburg et al. ).
The CTM (Boxall et al. ; Boxall & Saul ) provides a method for calculating the volume of accumulated material within a network, which is measured as turbidity (expressed in Nephelometric Turbidity Units, NTU). The model is also used to calculate the volume of material mobilised given specific hydraulic events in the network. In theory, the impact of mobilised material is proportional to and limited by the material stored in the network and thus by reducing the potential material in the network the associated discolouration risk will also decrease. Taking that relationship into consideration, this paper demonstrates the process of evolving networks that have shear stress characteristics which encourage self-cleaning and, therefore, the prevention of discolouration events by reducing the material stored in the network. The notion of self-cleaning thresholds have been investigated a number of times by Boxall & Prince () who consider shear stress thresholds and Buchberger et al. The implementation of the discolouration model used in the formulation of the WDN design problem is based upon the following equations. The most pivotal aspect of the model's design is the shear stress equation: where τ is shear stress, ρ is water density, g is gravitational acceleration, R h is hydraulic radius, and S 0 is hydraulic gradient. The CTM applies each link's maximum daily shear stress values (known as the daily conditioning shear stress; Cook ) in the calculation of the potential material stored in each link. Using the shear stress obtained above and the relationship given in Boxall & Saul (), it is possible to derive the stored material C (measured as turbidity in NTU) for each pipe in a network, where: C max is the maximum possible material stored in the pipe; τ' s is the layer strength; and k and b are calibrating constants. The problem is complex as the overall hydraulic conditions are affected by each pipe and so changes to one pipe will have a different effect on the overall conditions depending on the sizes of all the other pipes in the network.
As such, each pipe cannot be designed in isolation, but rather as a combination of sizes for all pipes in the network.

METHOD
The novel formulation of the multi-objective WDN design problem and a novel online selective hyper-heuristic (MCHH) is presented in this section.

Multi-objective WDN design problem
As outlined earlier, the formulation of the WDN design problem used in this study incorporates the evaluation of a network's propensity to cause discolouration events. This was done by using discolouration risk as an objective which was calculated using the DPM software based on where the EA is to minimise this potential for pipes to store material.
In order to fairly represent the complete effect of changing pipe diameters in the network, the discolouration risk objective was combined with the traditional objective of minimising cost in addition to minimising head excess.
These three objectives are: 1. Cost of network infrastructure: 2. Sum of cumulative potential material after daily conditioning shear stress for all pipes in the network (L): The f turbidity function returns the stored material in a pipe, given the shear stress (τ' s ), hydraulic radius (R h ) and hydraulic gradient (S 0  selecting that heuristic in future generations based on this score.
In Figure 2, the terms μ and λ represent the parent and child populations. p is the performance score of the current heuristic, given below, γ is a performance threshold that controls the adaption of the weights, and α and β are the reward and penalty scores applied to the weights, respectively.

Heuristic performance
The performance measure calculated in step 2.1.1 (Figure 2 The score is calculated, for each solution in the new child population, as the ratio of solutions it dominates in the parent population and then averages these ratios to produce a single score. This is shown in Equation (6). Theoretically, good heuristics (for moving towards the Pareto front) will have a high probability of generating dominating solutions.
The function p(h, μ, λ) shown in Equation (6)  better. The approach is designed to try and learn these transition sequences to further improve the optimisation process. Firstly, the MCHH constructs a fully connected Markov chain with one state for each heuristic, i.e., each state in the chain is connected to every other state and to itself (see Figure 3)   In addition to applying the MCHH, the experiment was used to study the effect of pipe diameter on network selfcleaning characteristics subject to node head constraints and as such so no other characteristics were altered during the optimisation process. The experiment used EPANET to simulate the hydraulic effects of pipe diameter changes over a 24 hour extended period. The DPM was then applied to calculate the potential material stored in each pipe given the revised hydraulic conditions. The hydraulics, potential for discolouration and cost calculation formed the basis of the objective function for the optimising algorithms.

Optimisers
The optimisation of the pipe diameters was completed using a NSGA-II and SPEA2, two modified variants of these algor-  selecting random heuristics at each generation. The algorithm was also used as a control to demonstrate that the heuristics alone did not provide the significant improvement in performance observed in the MCHH variants.

RESULTS
The results for the three benchmark and three real-world networks are given in Table 1  numbers in bold indicate that the value was the best found for that object on that network across all the algorithms, e.
g., NSGA-II and MCCH variant of NSGA-II both found the lowest cost solutions for Anytown.
The results given in Table 1     All the results clearly demonstrate the MCHH variants' ability to utilise the additional heuristics which is shown by a better performance when compared with Simple Random and TSRoulWheel both of which were supplied with the additional heuristics. On the simpler problems, the MCHH performs much better than NSGA-II and SPEA2. However, it is also noticeable that as the problem complexity increases, the improvement in performance is less profound.

Heuristic weights
The improvement in performance of the MCHH over the The MCHH always initialises all weights to be equal at the start of the search, however, as can be seen in Figure 10, the MCHH quickly biases the use of heuristics to those that provide good results early on in the search. Indeed, while it appears as though the weights start at different points in

Two Loops
The Two Loops network was used to examine the possibility of using discolouration propensity, in the form of stored turbidity, during the process of optimising WDN design. A set of solutions that satisfied the head constraints was found by optimising the network. Of these solutions, two were found that matched the known minimal cost of 419,000 and eliminated all discolouration potentiali.e., satisfied the selfcleaning threshold. These were the same as those found in Savic & Walters (). These results prove that, for a simple WDN, a solution may exist that is both self-cleaning and minimal for cost: the ideal solution.
A further set of solutions that satisfied head constraints with self-cleaning properties but incurred additional cost was identified. These improved upon the head excess of the minimal cost solutions and could be viable alternatives if leakage were a concern. Excessive pressure in a WDN leads to water loss through leakage from joints, fixtures and small breaks which, over a large number of networks, can result in significant costs for water companies (Al-Hemairi & Shakir ). Interestingly, the results show a degree of correlation between cost and discolouration (also shown in Hanoi results in Figure 11). It is hypothesised that this is not a perfect correlation as changes in pipe sizes close to sources will increase the possible shear stress downstream, increasing the cost and reducing discolouration.

Hanoi
The Hanoi benchmark was used to examine whether, for larger networks, it was possible to find solutions with decrease. As this study shows, this is indeed the case and it is not easy to find these solutions in networks larger than Two Loops, where a trade-off surface is more likely. can clearly be seen in Figure 11 that the correlation between cost and discolouration is retained, albeit with some noise.
There is also a significant trade-off between the Head Excess and Discolouration with reduced head excess resulting in increased discolouration risk. In contrast, while there is a trade-off between cost and head excess for the more expensive networks (which suggests bottlenecks in the designs), the relationship between the two objectives becomes less strong as the size of the network decreases. This is shown by the curved edge of the front in the 3D plot in Figure 11 and the increase in the density (number of alternatives) of solutions as the cost scales down. At the lowest cost networks, the trade-off in solutions is found only between discolouration and head excess. If only head excess and cost or discolouration and cost were used as two objective problems, then the problem would be degeneratei.e., the trade-off is lost close to the true Pareto optimal solutions.
The cheapest network satisfying the head-deficit constraint was found at US$6.22 m, comparable with the SCE optimal solution found for EPANET2 (Rossman ).
Interestingly, 16.7% of the pipes for this solution stored over 5 NTU of potential material (based upon recorded visible NTU). However a range of alternative solutions were also located that satisfied the constraints and for an increased cost of US$68,422, this discolouration risk can be reduced to just 11.8% of pipes by adopting a slightly more expensive solution. By including the discolouration risk scores into the optimisation process, it is possible to consider alternative solutions that incur a marginally more expensive up-front cost but significantly reduce later maintenance costs by reducing the discolouration risk and allowing for more selective cleaning schemes to be introduced.

CONCLUSIONS
This paper presents a novel approach to pipe diameter design for WDNs and proposes the MCHH, a multi-objective online selective hyper-heuristic, which is applied to the problem. The experiment demonstrated that by incorporating discolouration propensity, calculated using the DPM, and excess head (thus taking some leakage concerns and three real-world networks.
Two solutions with minimal cost and a self-cleaning feature were found for Two Loop. However, results for Hanoi highlighted the difficulties in finding minimal cost, selfcleaning networks for relatively larger networks. This study does show that a modest increase in cost can also attain a self-cleaning threshold in this larger network a result which may be more applicable to much larger, real WDNs.
As anticipated, the results from the application of this approach to the rehabilitation of real-world networks supports the theory that a trade-off curve of cost against discolouration potential becomes increasingly more varied and gradual as the complexity of the problem increases. The likelihood of finding minimal cost, selfcleaning solutions is also improbable for relatively large networks. Nevertheless, by applying this approach during rehabilitation planning, companies can also consider discolouration potential in addition to new demands and thus improve the self-cleaning properties of their networks for a marginal increase in cost.
In addition, this paper applied the MCHH to the same three objective formulation of the WDN design and rehabilitation problems. The MCHH was incorporated in the well-