Joint Entropy Based Multi-Objective Evolutionary Optimization of Water Distribution Networks

2.1 Informational Entropy under Multiple Operating Conditions

Alternative hypotheses for maximizing the entropy when there are multiple operating conditions include: (a) maximizing the largest entropy value across all the operating conditions; (b) maximizing the smallest entropy value across all the operating conditions, to alleviate the worst-case performance; and (c) maximizing the sum of the separate entropies from all the operating conditions.

A basic property of entropy as a measure of uncertainty is that the joint entropy of two or more independent probabilistic schemes is the sum of their separate entropies (Shannon 1948, Tanyimboh 1993: 73-77). Accordingly, for independent operating conditions, the joint entropy is the sum of the separate entropies. The appeal of this interpretation lies in the fact that it accounts for all the operating conditions. Furthermore, it does not require any additional assumptions or criteria. Therefore, viewed as a basic property of the network, the logical conclusion is that the sum of the entropies should be maximized. This was the fundamental hypothesis of the research.

2.2 Network Design Optimization Model

Flow entropy may be included in the design optimization of a water distribution network to minimize the cost without sacrificing resilience completely. In this way, redundancy is safeguarded and deployed to the best possible advantage, to help address any unanticipated flow re-routing and short-term increases in demand. The objectives were thus cost minimization and entropy maximization. Only the initial construction cost was considered in this research. Other costs and additional factors may be incorporated relatively easily (Siew et al. 2014). Though relevant, these wider considerations were not the main focus of the investigation.

The constraints were the constitutive equations (i.e. conservation of mass and energy) and the minimum node pressure constraints. The equations for the conservation of mass and energy were satisfied by embedding the EPANET 2 hydraulic simulation model (Rossman 2000) in the evolutionary algorithm. The minimum node pressure constraints were addressed by introducing an additional objective that considers the feasibility of the solutions.

Michalewicz (1995) classified the approaches for addressing constraints in evolutionary algorithms as: (a) repairing, (b) modifying, (c) rejecting and (d) penalizing strategies. The most common practice is to degrade infeasible solutions by applying penalties, with greater constraint violations incurring higher penalties. Excessively high penalties may confine the search to the feasible region of the solution space. However, searching through the feasible and infeasible regions improves the efficiency and yields better solutions than searching in the feasible regions only (Glover and Greenberg 1989). Designing penalty functions and calibrating the associated parameters is a complex task and requires extensive fine-tuning (Dridi et al. 2008). A penalty-free formulation was developed to obviate the difficulties.

C _i (d _i , L _i ) is the cost of pipe i with diameter d _i and length L _i while np is the number of pipes. The set D comprises the available discrete pipe diameter options. S is the flow entropy. H _n and \( {H}_n^{req} \) are, respectively, the available and required residual heads at node n. The required head corresponds to the pressure above which the demand is satisfied in full. The decision variables are the pipe diameters. The hydraulic simulation model EAPANET 2, that ensures energy and flow conservation, also provides the nodal heads.

3.1 Solution of the Optimization Problem

NSGA II was employed as it is efficient and used widely by many researchers in various disciplines. The evolutionary optimization procedure in NSGA II (Deb et al. 2002) is based on Pareto-dominance and global elitism. It maintains diversity by seeking an even distribution of the nondominated solutions in the objective space using the crowding distance. The crowding distance is a measure of the spatial distribution of the solutions in the objective space and is based on the average distance between a solution and its neighbours.

The source code of NSGA II (in C++) was modified and linked to the hydraulic simulation model EPANET 2. EPANET 2 (Rossman 2000) is public domain software for modelling water distribution networks that performs both steady state and dynamic simulations, and water quality modelling. A procedure that calculates the flow entropy for any given network configuration was developed, tested and incorporated in the optimization algorithm. The multiobjective optimization algorithm thus produced can handle both single and multiple operating conditions.

While investigating the methodology it was observed that the number of infeasible solutions in the Pareto-optimal sets exceeded the number of feasible solutions. This may be because the nodal head deficit function f ₂ in Eq. 5 does not distinguish between solutions with different levels of surplus head, as all feasible solutions have a deficit of zero. Hence, for feasible solutions, the Pareto-dominance is based on the cost and entropy only. By contrast, for infeasible solutions, the Pareto-dominance is based on three objectives. It is known that more solutions become nondominated as the number of objectives increases (Ishibuchi et al. 2015), and this may have contributed to the imbalance between the number of feasible and infeasible solutions in the Pareto-optimal sets achieved.

Therefore, on discarding the infeasible solutions at the end of each optimization run, all the nondominated feasible solutions achieved from the start to the end of the optimization were incorporated in the Pareto-optimal set. Membership of the final Pareto set of feasible solutions was based on Pareto-dominance and the selection was done using additional software developed (in Perl) in the research. Furthermore, the nondominated solutions from all the optimization runs were combined and sorted based on Pareto dominance to obtain a single unified nondominated set.

3.2 Resilience Evaluation Procedure

Two of the criteria used to assess the proposed maximum entropy design approach were the hydraulic capacity reliability and failure tolerance that emphasize different aspects of resilience (Tanyimboh et al. 2001). Pressure-driven analysis (Tsakiris and Spiliotis 2014) based on the logistic pressure-discharge relationship (Tanyimboh and Templeman 2010) was used to simulate the effects of pipe failures.

The definition used for the hydraulic capacity reliability is the network’s ability to fulfil on average the required nodal demands at adequate pressure, under normal and abnormal operating conditions. The pipe failure model used was taken from Cullinane et al. (1992). Also, the pipe failure tolerance provides an estimate of the fraction of the total demand that the network can satisfy on average when one or more components are out of service and its importance was emphasized previously (Gheisi and Naser 2013, 2015).

4.1 Problem Specifications

This example is based on the network shown in Fig. 1 that serves part of the city of Ferrara (Creaco et al. 2010, 2012). It consists of 49 nodes, 76 pipes and 29 loops. The total demand is 367 l per second. Each reservoir has a head of 30 m. Manning’s roughness coefficient is 0.015 for all the pipes the total length of which is about 25.2 km. The elevation and minimum required head at the demand nodes are 0 m and 28 m respectively.

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The available pipe diameter options and their costs per metre (in mm and €/m) are: {(150, 271.94), (200, 299.43), (250, 328.01), (300, 359.54), (350, 399.03), (400, 438.63), (450, 461.34), (500, 502.78)}. There are 8 pipe diameter options, so the solution space comprises 8⁷⁶ or 4.313 × 10⁶⁸ infeasible and feasible solutions. A 3-bit binary substring with 8 (i.e. 2³) substrings was used. The GA parameters were: N _E  = 500,000, N _G  = 1000, N _S  = 500, N _R  = 30, p _c  = 1.0 and p _m  = 1/228 = 0.004 (this is the probability that any single bit would mutate by being reversed from 0 to 1 or vice versa) where 228 is the chromosome length. N _E is the maximum number of function evaluations or hydraulic simulations allowed; N _G is the number of generations; N _S is the population size; N _R is the number of independent runs with random initial populations; and p _c and p _m are the crossover and mutation probabilities. A single-point crossover operator was used to produce two offspring from two parents. The average CPU time for a single optimization run was about 90 min on a PC (Intel Core 2 Duo, 3.5 GHz, 3GB RAM); in other words, 45 h or 1.9 days in total for the 30 optimization runs. The results are summarised in the supplementary data that is available online, in Table C1 in Appendix C.

4.2 Effectiveness of the Optimization Approach

It was thought that the surplus heads at the critical nodes would increase as the entropy increased. However, there was no correlation between the surplus heads and entropy or cost as shown in Fig. 2. The surplus heads were relatively small with an average of 0.027 m. As the pipe diameters were discrete and thus small surpluses were virtually unavoidable given that only eight pipe diameter options were available, the methodology developed can be considered satisfactory with respect to the optimality of the solutions.

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The results of individual optimization runs revealed apparent gaps in some of the Pareto fronts and an imbalance between the feasible and infeasible solutions. It is possible that the number of feasible solutions for this network is relatively small in comparison to the size of the solution space. It may be noted that the maximum headloss allowed between the supply and demand nodes was only 2 m based on the available head of 30 m at the supply nodes and the stipulated minimum residual head of 28 m at the demand nodes. In fact the present results are consistent with Saleh and Tanyimboh (2016). While the cost and entropy values (for the peak demands) were similar, the present formulation achieved 74 nondominated feasible solutions with 15 million function evaluations in total compared to only 27 nondominated feasible solutions in Saleh and Tanyimboh (2016) with 20 million evaluations in total. Also, the average surplus head was 2.7 cm compared to 3.5 cm in Saleh and Tanyimboh (2016).

In Saleh and Tanyimboh (2016) the least expensive feasible solution cost was €8.011 million and the entropy was 4.7968. The cost of the most expensive solution was €10.285 million and the entropy was 7.2989. Herein, the corresponding values were: €8.074 million and 5.125, for the peak demands, for the least expensive solution; and €10.108 million and 7.127, for the peak demands, for the most expensive solution. Thus the solutions from the proposed methodology are very competitive. In any case, Saleh and Tanyimboh (2016) did not consider multiple operation conditions; so the present solutions have more advantages.

The value of the entropy increased rapidly from the start of the optimization, followed by a second phase with relatively steady progress as shown in Fig. 3a. Only the feasible solutions were included in the results shown in Fig. 3a. The reason is that the entropy values for infeasible solutions are unrealistic, as the flow rates derived from demand-driven analysis are misleading in that the corresponding energy loss due to pipe friction exceeds the total energy of the system. Perhaps this apparent anomaly could be avoided in the future by adopting pressure-driven analysis for more realistic simulations of the infeasible solutions.

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Infeasible solutions help drive the optimization, e.g. by sustaining the search near the feasibility boundaries and avoiding a purely interior search that may be suboptimal (Siew et al. 2014). Figure 3b shows the progress of the infeasible solutions. The mean deficit of the solutions increased as the optimization progressed, as the solutions with the smallest deficits (i.e. the solutions located close to the feasible region) improved and eventually became feasible. It was observed that, after decreasing rapidly initially, the number of feasible solutions increased gradually until the end of the optimization (Czajkowska 2016).

4.3 Effectiveness of the Joint Entropy Formulation

Figure 4 shows the trade-off between flow entropy and cost for the non-dominated feasible solutions. There was a positive coefficient of correlation of 0.991 between the flow entropy and the mean diameter. Larger pipe diameters improve the hydraulic reliability by increasing the pipe flow capacities and lowering the pipe failure rates. This is consistent with previous results in the literature. There is evidence also, of strong positive correlation between the flow entropy and uniformity of the pipe diameters that improves flow re-distribution further.

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It is obvious from Fig. 4 that no operating condition had the highest entropy value in all the solutions. For the relatively expensive solutions (€8.5 million and above) Fire Flows 2 and 1 had the highest and lowest entropy values, respectively. On the other hand, Fig. 4 shows that when considering the less expensive solutions (up to €8.5 million) there was no obvious trend. This provides empirical evidence that maximizing the total entropy is the most appropriate approach for multiple operating conditions. Also, the surplus heads for the fire flows in Fig. 5 reveal that none of the fire flows dominated the other in every solution.

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Furthermore, maximizing the largest or smallest entropy value would consider only one operating condition out of three. The operating condition with the largest or smallest entropy could change in successive generations. This could potentially make the algorithm unstable or inefficient through lack of continuity. While the entropy values for different operating conditions were comparable in the present example, there may be significant differences in other situations.

Similar results were achieved on several additional networks in which the total, maximum and minimum entropy maximization approaches were investigated separately. Entropy maximization based on a single loading condition was investigated also. The results showed that multiple operating conditions improved the resilience (i.e. reliability and pipe failure tolerance) and maximizing the sum of the entropies gave the best solutions (Czajkowska 2016).