Reliability-based optimization of water distribution networks Open Access

As humankind is evolving, there is always a demand for a much better solution. The solution that may fit only one objective is not seen as a satisfactory option as multiple goals need to be fulfilled. Meeting multiple goals from a single solution in technical terms is known as multi-objective optimization. A priori and posteriori are two approaches that can be used to solve any multi-objective optimization problem. In the a priori approach, the decision-maker must decide the importance of all objectives i.e. weightage or priority before the start of the analysis, and corresponding solutions are obtained then. A different set of solutions can be obtained by varying the weights. However, a non-dominated set of Pareto Front is directly obtained in the a posteriori approach and a decision-maker is allowed to choose a solution from this set. The a priori approach uses the scalarization method to create the objective function which further utilizes an evolutionary algorithm (EA) to reach the optimal points. A multi-objective evolutionary algorithm (MOEA) is used to find the optimal results for the a posteriori approach. MOEA's working principle is similar to an EA except these are based on the concept of non-dominance and crowding distance to find the optimal solution, which is considered complex when compared with the scalarization method (Jahanpour et al. 2018).

A water distribution system (WDS) comprises various parts such as reservoirs, pumps, pipes, valves, and so on, that are required to supply the water at the consumer end with adequate pressure, of sufficient quality and quantity. Since WDS plays an important role for any developed society, it is obvious to state that the optimal design of a water distribution network is one of the most researched fields. The main objective in the design of any distribution network is to reduce network costs. Hence, optimization of WDN is often seen as a single objective problem with the main aim of reducing the network cost. However, the optimal solution (optimal solution corresponds to the minimum network cost while satisfying all constraints of the problem) is not practical if implemented since it comprises diameters that are designed to just satisfy the minimum pressure head at the demand node. A slight change in this pressure may occur due to pipe breakout or sudden increase in demand node that may lead to network failure. Since WDN are a costly infrastructure, it is not possible to redesign them. In addition to this, the expansion of the network is not possible in all places due to the constraint of the space. Thus, at the design stage, if only the efficiency of WDN is considered, it may save a lot of effort and cost at the later stage. Hence, minimizing the network cost should not be the only objective to satisfy a longer-running network. Problems that are associated with satisfying more than one objective from a solution are known as multi-objective optimization problems. It was not until the authors Gessler & Walski (1985) that the first multi-objective optimization model was suggested for the distribution system. In most cases, authors chose reduction of network cost as the first objective. The second objective can be a reduction in emission of greenhouse gases from any WDN, reduction of pressure deficit, improving the reliability of the network, and so on (Jetmarova et al. 2018). Gheisi et al. (2016) presented detailed literature on different types of reliability and various measures to calculate them. Liu et al. (2017) conducted a comparative analysis of four different reliability indices in addition to introducing two new reliability indices, namely API (available power index), PHRI (pipe hydraulic resilience index). In the present work, scalarization is adopted to optimize the network using different weights for a two-objective optimization problem. The scalarization method combines all the objectives of the problem into one by assigning different weights to different objectives (Gunantara 2018), thus converting a multi-objective problem into a single objective problem and hence easy to analyze. The a priori method of optimization has proved its efficiency for generating a set of non-dominated solutions that may be used as a pivot point for different techniques (Torres et al. 2012).

Suribabu (2017) developed a normalized function combining the cost and reliability of the network as two objectives into one. The author found the function to be efficient using Differential Evolution (DE) as an optimization technique and under different weights of cost and reliability. The present work validates this normalized function to obtain a set of non-dominated solutions for three different benchmark problems using the Jaya optimization technique as the algorithm emerged to be an efficient performer when tested on various optimization problems. Dede (2018) incorporated the single objective optimization of steel structures using Jaya and found it to be a better performer. Hao et al. (2021) adopted the Jaya technique for the calcination process of cement. The detailed reviews on the various applications of Jaya as the optimization technique can be found in Alsajri et al. (2018).

There is not a well-defined definition for the term “reliability of distribution network” (Xu & Goulter 1999). However, WDS is designed considering that total energy supplied by the pumps or elevated reservoir must be able to deliver the water at sufficient pressure despite all the losses occurring in the pipe. However, due to sudden breakout in the network, which may occur due to a sudden increase in the demand capacity such as fire demand or pipe failure, losses occurring in the pipes may increase dramatically and may affect the users downstream of the failure junction. Nonetheless, if the network can absorb this head loss and continue its supply, the failure may not lead to any drastic effects and may be fixed during odd hours of the day. These networks that absorb such failure losses are said to be more resilient (Goulter & Morgan 1984) and thus have immense hydraulic power. Todini (2000) took advantage of this surplus power in any distribution network and introduce a technique to determine the reliability of a WDN and therefore becomes the first to introduce the surrogate measure of calculating the reliability of the WDN. Thereafter many surrogate measures are developed and have been implemented for the optimal design of WDN. Prasad & Park (2004) introduce the network resilience index with the addition of diameter uniformity in Todini's index of reliability. Raad et al. (2010) carried out the study of comparing various reliability indices and found Todini's RI to be more reliable.

The present work incorporates the RI for the optimal design of the three benchmark networks.

The Scalarization method converts a multi-objective problem into a single objective by assigning weights to each objective and combining them. Selecting a higher weight for any objective prioritizes it over the other objectives. Tchebycheff scalarization using interactive weights was proposed by Steuer & Choo (1983). Thereafter Tchebycheff has been used extensively for scalarization. Many new scalarization approaches that are used to generate Pareto solutions for non-convex problems can be found in Xia et al. (2021). The author also developed a combined scalarization technique. Kasimbeyli et al. (2019) compared six different scalarization techniques, namely – weighted sum, epsilon constraint, Chebyshev method, Pascoletti-Serafini, conic and Benson method. The present work uses weighted sum by assigning higher weights to the cost and then reducing it subsequently. The decrease in weight of cost is equal to the increase in weight of reliability, such that the summation of all weights is always equal to one.

C represents the cost of the network obtained, C_min is optimal network cost determined by substituting the optimal diameters that are obtained by considering minimizing the network cost as the only objective. C_max is the maximum network cost, obtained by substituting the maximum available commercial diameter of the network to all the pipes and thus calculating its cost, RM_max is a resilience measure of the network that corresponds to C_max, RM_min is the resilience measure of the network that corresponds to C_min. RM is the resilience measure of the network at cost C. w₁ and w₂ are weights assigned to the cost and resilience measure of the network, respectively. The summation of w₁ and w₂ is always equal to 1. Since EA starts with random numbers, the probability that the RM value of a solution is less than RM_min is very high. For such situations, RM – RM_min will become negative, which reduces the value of Z_min, and such a solution will be selected as the best solution. To avoid this during any time in the optimal design, if RM<RM_min is found, RM is substituted as RM_min+0.000001 so that RM – RM_min will always be equal to 0.000001. A similar concept may help in dealing with solutions that have zero resilience value. C_min, C_max, R_min, R_max values for the Two Loop, Hanoi, and Go-Yang networks are given in Table 1.

r₁ and r₂ in Equation (10) are random numbers that are generated between 0 and 1. The best and worst solutions are represented by indexes b and w respectively. Index i represents iteration number, j represents the position of the variable in any solution, k is the index of the position of solution in a population set. As the algorithm advances in the iterations, the best and worst values are improvised and an optimal solution is obtained, thus the algorithm becomes victorious as the termination criterion is reached and hence named Jaya (A Sanskrit word that means – Victory over the evil). The pseudo-code for the methodology is given below.

Initialize trials, iterations, population size, number of variables, w₁ and w₂

Trial t=1

while t is less than t-maximum

  Generate initial population matrix at random

  Iteration i=1

  while i is less than i-maximum

    Initialize EPANET and determine cost matrix along with Resilience index (using Equations(3) and (1))

    Determine objective function using Equations (2) and (8)

    Identify best, worst and their corresponding indices

    Update the population using Equation (10)

    Compare updated solution with previous solution

    if updated solution is better than previous solution

      update the population and move to next iteration

    else

      retain the population and move to next iteration

    i++

  end while

  n++

end while

• 1.

  The problem is initialized by the number of variables, number of iterations, weighing factor w₁ and w₂, lower and upper limit of the decision variable.

• 2.
  Initial solutions are generated randomly using Equation (11). The solutions are generated randomly to avoid any biasedness.

  

  (11)

x_ij is generated random value of the decision variable, is the lower and are the upper limits of the decision variables. is a random number generated between 0 and 1 and distributed uniformly.

• 3.
  WDN belongs to the category of non-combinatorial discrete optimization, but the solutions that are generated using Equation (11) are continuous and are thus converted to discrete values using Equation (12).

  

  (12)

•  4.

  EPANET 2 (Rossman 2000) is called and simulated pressure heads, demand are determined at each node for every solution in the population set.

•  5.
  If solutions are found to violate the minimum pressure requirement constraint, it is penalized. The penalty for such a solution is determined using Equation (13).

  

  (13)

  where P_min is the minimum pressure value among all the nodes for a solution; that is, simulated using EPANET 2, P_req is the required minimum pressure for any node, λ is a penalty constant.
• 6.
  Total Cost; that is, the fitness function, is calculated as a sum of penalty value and the cost of the network.

  

  (14)
• 7.

  Reliability is calculated using the resilience index defined in Equation (1).

• 8.

  Cost by Reliability is further calculated, which becomes the new fitness value for the Jaya algorithm. For normalized objective function Z_min is calculated.

•  9.

  The solution giving the minimum and maximum cost by reliability or Z_min value (normalized function) is taken as the best and worst solutions respectively.

• 10.

  Once the best and the worst solutions are identified, a new population is generated using Equation (10).

• 11.

  Steps 4–8 are followed for the new set of solutions. The new population is compared with the old population and the better-performing (with minimum fitness value) solutions are selected and passed to the next generation.

• 12.

  The process for a pair of weights is continued until a stopping criterion is reached.

Two loop network (Alperovits & Shamir 1977) is a hypothetical network with a reservoir fixed at an elevation of 210 m. The network has 14 commercially available pipe diameters and 8 pipes each 1,000 m long with a Hazen-William coefficient of 130, arranged in two loops, which leads to a total search space of 1.48 * 10⁹. A minimum pressure head of 30 m is required for the network at each node. The other hydraulic data of the network such as demand and node elevation are given in Figure 1(a). The optimal cost of the network is 419,000 (Cunha & Sousa 1999) with the optimal diameters as given in Table 2. Different solutions obtained by the Jaya technique by varying the weights are given in Table 3. As can be seen from Table 3 for w₁ = 0.95 & w₂ = 0.05, the optimized cost increase by 32.21% over the optimal cost, while reliability increases to 91.17%, which is very appealing. It is also to be noted that for w₁ = 0.80, 0.75, and 0.70 similar cost and reliability are achieved. A similar trend is seen for w₁ = 0.45, 0.40, 0.35. No improvement in diameters is obtained for such solutions by varying the weights. Thus, considering the designer's preference for the importance of the two objectives, any solution can be picked. Reliability as high as 0.8998 is obtained when w₂ becomes 0.95, which is very near to the R_max; however, the cost at this reliability is 25,52,000, which is less when compared with the C_max; that is, 44,00,000. Thus, maximum reliability can be attained with much less cost. Optimal diameters obtained for a certain set of weights are given in Table 4. No comparison is available in the literature for this normalized function and hence it has not been compared here. For every pair of weights, 10 trial runs are performed with a population size of 50 and 200 iterations as the termination criteria. Convergence is obtained in just 30–50 iterations for every trial run, which highlights the searching capability of Jaya for a non-linear discrete optimization problem. The Pareto Front obtained for the network is shown in Figure 1(b).

The Hanoi network in the present work is taken from Fujiwara & Khang (1990). The network consists of 34 pipes and a reservoir at an elevation of 100 m (Figure 2(a)). There are six commercial sizes available for the Hanoi network, as shown in Table 5, which leads to a search space of 2.865 * 10²⁶, which is large and computationally expensive and hence is a challenging problem for any computational technique. The hydraulic data of the network is given in the supplementary material. The optimal cost of the network is obtained as 6.081 M$ (Zheng et al. 2013) when only minimizing the network cost is considered. The diameters corresponding to this cost are shown in Table 2. The optimal diameters obtained when Equation (2) is optimized using Jaya are also shown in Table 2. The cost and resilience index for different weights to obtain a Pareto Front is shown in Table 6. As the weightage of the resilience index is increased, higher-cost solutions are obtained, since larger size diameters are needed to meet the resilience of the network. There is an increase of 83.85% in reliability for resilience as 0.3530 and cost of $9798084.93 i.e 61.21% more than optimized cost using cost and resilience index weights as 0.05 and 0.95 respectively. The designer can choose among any of the solutions given in Table 6 based on the availability of cost and desired level of reliability to be achieved. The results obtained by Jaya are similar to those obtained by Suribabu (2017). The slight change in diameter values from Suribabu (2017) is marked in bold in Table 7. DE applied by Suribabu (2017) took as many as 2,000 iterations to reach these results; Jaya converges to the same results in less than 100 iterations for all the successful trial runs, which again highlights the computational efficiency of Jaya. The comparison of diameters obtained by Suribabu (2017) to Jaya is shown in Table 7. For every pair of weights, 10 trial runs are performed with a population size of 75 and 200 iterations as the termination criteria. The Pareto Front obtained for the network is shown in Figure 2(b).

The Go-Yang network (Figure 3(a)) consists of 30 pipes and 22 nodes arrange in 9 loops with a ground reservoir fixed at an elevation of 71 m. A pump of 4.52 kW capacity is provided to supply the water to the network. A 15 m minimum pressure head is required above the ground elevation at all the nodes in the network. The Hazen-Williams coefficient for all the pipes in the network is taken as 100. The hydraulic data of the network such as ground elevation, nodal demand, and so on, are given in the supplementary material. The optimal cost of the network (when only minimizing the network cost is considered) is reported as $177,010 (Zheng et al. 2012) with the optimal diameters as given in Table 2. Equation (2) is optimized using Jaya for the Go-Yang network and with just 5.85% increase in cost from the optimal value, 94.05% increase in resilience index is obtained, with the diameters given in Table 2, which is captivating and thus highlights the importance of considering reliability in the optimization model and to not rely on just minimizing the network cost. Table 8 shows the value of the cost and resilience index obtained for different pairs of weights. The maximum cost of the network is $329725.64 with an RI of 0.9941; however, for the weights w₁ = 0.95 and w₂ = 0.05 a cost as high as $211604.5 is reported with an RI of 0.9922, which highlights that by giving just 5% weightage to the reliability, maximum reliability can be achieved with a much lesser cost. With nearly the same resilience index, $118,121.14 is saved, which is huge and is more than the optimal network cost. The diameters obtained for a certain set of weights are shown in Table 9. For each set of weights, 10 trial runs are performed with a population size of 75 and 200 iterations as the termination criteria. For all the trials, the results are obtained in less than 100 iterations. The Pareto Front for the obtained results is shown in Figure 3(b) for the Go-Yang network.

In this paper, a new methodology is proposed that eases the process of generating a set of non-dominated solutions for a multi-objective optimization problem. The computational methodology presented is known to converge to the results previously reported in the literature in many fewer iterations. A normalized function is applied for multi-objective optimization in three different benchmark problems using the Jaya algorithm, a newly developed parameter-less technique. The proposed methodology can generate a set of non-dominated solutions by deciding the priority of the objectives between them. The technique is found to be more convenient when compared with the a posteriori approach. No knowledge of the non-dominance is required, and different sets of Pareto Fronts are obtained that may be used as a pivot solution for further analysis. In addition to this, the involvement of more objectives in a problem may make it very hard to solve using the concept of non-dominance. However, for the method described in the present work, the designer just needs to decide the weights of every objective to obtain the desired results. It is worth mentioning that the a posteriori approach can lead to the non-dominance set in a single run while to generate the same Pareto Front in the a priori approach, different runs are made by varying the weights of the objective that makes the a priori approach more convenient and simpler. The present work indicates that by including the resilience index in the optimization model, better solutions are obtained that are less expensive but highly reliable. Jaya belongs to a parameterless algorithm and hence involves no constants, which further eases the analysis as no efforts are spent on tuning the algorithm. Similar results are obtained in 40–50 iterations using Jaya when compared with the 2,000 iterations taken by Differential Evolution for the Hanoi network; this further reaffirms the trust in Jaya as an evolutionary algorithm capable of solving multi-objective optimization.

All relevant data are included in the paper or its Supplementary Information.