Risk-Based Design Optimization of Contamination Detection Sensors in Water Distribution Systems: Application of an Improved Whale Optimization Algorithm

Abstract

In the present study, the optimal placement contamination warning systems (CWSs) in water distribution systems (WDSs) was investigated. To this end, we developed a novel optimization model called WOA-SCSO, which is based on a hybrid nature-inspired algorithm that combines the whale optimization algorithm (WOA) and sand cat swarm optimization (SCSO). In the proposed hybrid algorithm, the SCSO operators help to find the global optimum solution by preventing the WOA from becoming stuck at a local optimum point. The effectiveness of the WOA-SCSO algorithm was evaluated using the CEC′20 benchmark functions, and the results showed that it outperformed other algorithms, demonstrating its competitiveness. The WOA-SCSO algorithm was finally applied to optimize the locations of CWSs in both a benchmark and a real-world WDS, in order to reduce the risk of contamination. The statistically obtained results of the model implementations on the benchmark WDS showed that the WOA-SCSO had the lowest average and standard deviation of the objective functions in 10 runs, 131,754 m³ and 0, respectively, outperforming the other algorithms. In conclusion, the results of applying the developed optimization model for the optimal placement of CWSs in the Dortmund WDS showed that the worst-case impact risk could be mitigated by 49% with the optimal placement of at least one sensor in the network. These findings suggest that the WOA-SCSO algorithm can serve as an effective optimization tool, particularly for determining the optimal placements of CWSs in WDSs.

1. Introduction

The quality of water, which is the most critical resource for humans, depends wholly on the local environment, how it is used, treated, and recycled according to the needs of people, as well as anthropogenic activities such as industry, household use, agriculture, and mining. The exponential increase in the global population has led to a rapid surge in the demand for clean water, as evidenced by multiple sources [1,2]. As a result, water quality has become a significant environmental concern due to its impact on aquatic life and the overall health of water ecosystems [3,4]. Meanwhile, water distribution systems (WDSs), which are responsible for transferring and distributing water from main sources to consumers with appropriate pressure and quality, are highly susceptible to quality damage as a result of the ease of access to most of their components [5]. Biological attacks are one of the major disruptive threats to WDSs; thus, the inflow of any chemicals accidentally caused by broken pipes or reverse leakage from connections, or intentionally by actions such as terrorist attacks, quickly spreads throughout the network, reducing customer satisfaction and causing damage [6,7].

In order to deal with network contamination, we must first detect the presence of contamination in the network, then try to stop it by finding its source, and finally prepare the contaminated part of the network for reuse by adopting accurate measures. Therefore, it is ideal but impractical to monitor all the points of a network [8]. Thus, optimal placement of contamination warning systems (CWSs) in WDSs is considered an effective approach to reduce the effects of contamination that has infiltrated into WDSs. In fact, contamination detection sensors must be placed in the network in such a way that they cover most points of the network on the one hand, while the impact of contamination on the consumer population and the costs of their exploitation and implementation are minimized on the other hand [9].

The concept of coverage in the placement design of contamination detection sensors was first proposed by [10], and an integer programming model was developed for the optimal placement of sensors. Then, a variety of models were introduced for the optimal placement of sensors in WDSs, considering the maximum coverage of the network [11,12]. Considering the multi-objective nature of the sensor placement problem in WDSs, other objectives have been investigated, such as minimizing the detection time, the volume of contaminated water, and increasing the maximum coverage of the network [13,14,15,16,17]. The optimal placement of CWSs in WDSs is considered to be a type of non-linear and complex optimization problem because of the need to devise contamination scenarios and use hydraulic and quality simulation models in parallel with the optimization models, and also because this process involves non-linear objective functions. Therefore, we need to use effective optimization methods in solving the aforementioned problem. Most of the primary studies conducted in this area have used linear programming (LP), non-linear programming (NLP), and dynamic programming (DP) methods for the optimal placement of sensors in WDSs [10,18,19].

However, the use of meta-heuristic algorithms has become more popular as a result of the increase in computation time and the failure of classical methods to achieve the optimal placement of sensors in large-scale networks. For example, Ref. [20] developed a simulation–optimization model based on the connection of NSGA-II and EPANET for the optimal placement of contamination detection sensors in WDSs by minimizing the contamination detection time and the population affected by it, and maximizing the probability of accurate contamination detection. Using NSGA–II, Ref. [21] optimized the position of CWSs in a benchmark WDS where the minimization of the contamination detection time and the probability of undetected injection events of the network were considered the objective functions of the study. The researchers of [22] developed a multi-objective simulation–optimization model based on ACO and the EPANET model for the optimal placement of CWSs in WDSs, with the aims of minimizing the amount of contaminated water consumed and maximizing the percentage of network coverage by the sensors. Many meta-heuristic algorithms have been developed in recent years to simplify the calculations related to the design and exploitation of complex engineering systems, or were based on combinations of several existing algorithms and improvements to them [23,24,25,26]. A review of the literature revealed that meta-heuristic algorithms have been extensively applied to address the problem of optimally placing CWSs in WDSs, yielding successful results. However, with the introduction of numerous new algorithms and continuous improvements in existing ones every year, there is always a need to explore and evaluate their efficacy in solving this problem [27,28,29,30,31]. The whale optimization algorithm (WOA) is one of the robust meta-heuristic algorithms based on collective intelligence; it was inspired by the hunting behavior of whales in detecting and hunting prey in nature [32]. This algorithm has been used in a wide range of real-world complex optimization problems, and has obtained favorable results. Among these problems, we can include the optimization of water networks [33,34], reliability-based designs of spillways [35], and water resources analyses [36]. Despite the good performance of the WOA in solving many optimization problems, it has been unable to find optimal solutions for some complex engineering problems due to becoming stuck in local optima, its slow convergence, and its failure to create a proper balance between its search and exploration phases.

Therefore, many studies have investigated the performance of this algorithm using different approaches and have come up with good results. For example, Ref. [37] used gathering strategies to improve the WOA, and used it for feature selection in medicine; Ref. [38] used the gray wolf optimizer (GWO) search method to improve the performance of the WOA in the optimization of complex engineering problems, demonstrating significant success with the improved version of the WOA in comparison with other algorithms. The present study was aimed at using the capabilities of sand cat swarm optimization (SCSO) to improve the convergence speed and accuracy of solutions found by the WOA in optimization problems. For this purpose, after the search phase in the WOA, it utilized the mechanism in SCSO during the exploration phase of the WOA to escape from local optima and achieve an optimal solution. Then, it evaluated the effectiveness of the combined WOA-SCSO algorithm to solve the problem of optimally placing CWSs in a real WDS by minimizing the worst-case impact risk caused by the inflow of contamination. Thus, the major contributions of this study are summarized as follows:

• A hybrid algorithm called the WOA-SCSO algorithm was proposed to improve the exploration, exploitation, and convergence rates.
• The hybrid WOA-SCSO performance was validated over ten CEC’20 benchmark problems and a real-world water engineering problem.
• The performance of the hybrid WOA-SCO algorithm was compared with four recent meta-heuristic algorithms.

The rest of the manuscript is organized as follows: Section 2 presents the problem of sensor placement in WDSs. Section 3 introduces the original whale optimization algorithm (WOA), the sand cat optimizer (SCSO), and the proposed hybrid algorithm. Section 4 presents the different benchmarks and results. Section 5 concludes the study and suggests future study directions.

2. Materials and Method

The flowchart of the proposed simulation–optimization model for optimally locating contamination detection sensors in WDSs using the combined EPANET hydraulic model and hybrid WOA is presented in Figure 1. As is clear from Figure 1, the developed model consisted of two main parts: simulation and optimization. First, injection scenarios were created by establishing a dynamic connection between the EPANET simulator and MATLAB software (the pollution injection scenario was generated using the model developed in [39]). Then, the optimization algorithm performed a trial-and-error process to place the sensors at each iteration step. In each stage of the algorithm iteration, the sensors were placed, and the hydraulic and qualitative simulation models were implemented to calculate the concentration values in the nodes. Next, the concentration values were monitored, and when a concentration value in any of the nodes exceeded the allowable limit for the first time, the corresponding node was designated to be the location for a sensor, and the flow in the network was stopped. This procedure was repeated until the optimization algorithm completed one cycle of iterations. The following sections discuss the two main phases of the proposed simulation–optimization model.

2.1. Simulation Model

We used the EPANET model in this study to perform a quality and hydraulic analysis of the network. The quality equations used in EPANET are as follows [7]:

$$\frac{{\partial C}_i}{\partial t}=-v_i\frac{{\partial C}_i}{\partial x}+r(C_i)$$

(1)

$$C_i=\frac{\sum _{iϵI_j}{Q}_i{C}_i+q_{c,j}{C}_{0j}}{\sum _{iϵI_j}{Q}_i+q_{c,j}}$$

(2)

where $C_i$, $v_i$, and $Q_i$ represent the concentration (mass/volume), the velocity of flow (length/time), and the flow rate (volume/time) in pipe i, respectively. Furthermore, $q_{c,j}$ and $C_{0j}$ denote the external source flow and its concentration entering the WDS at node $j$, respectively. $r$ represents the rate of reaction (mass/volume/time), which is expressed as follows:

$$r=K{C}^n$$

(3)

where $K$ and n are the bulk reaction constant and the reaction order, respectively.

The quality analysis of the WDSs included the simultaneous solutions of Equation (1) to Equation (3) along with equations related to mixing in the reservoir. It should be noted that performing a quality analysis involves performing a hydraulic analysis first. EPANET uses equations of continuity (Equation (4)) and the energy (Equation (5)) laws for the hydraulic analysis [40,41].

$$\sum _{i=1}^{N_j}{Q}_{ij}-q_j=0$$

(4)

where $Q_{ij}$ is the flow rate in pipe $ij$ linking junction $i$ to $j$, $q_j$ is a nodal demand at node $j$, and $N_j$ denotes the number of nodes.

$$\sum _{j=1}^{N_p(i)}{h}_{ij}-\sum _{j=1}^{N_p(i)}H{P}_{ij}=0$$

(5)

where $h_{ij}$ indicates the head loss in pipe j from loop i; $H{P}_{ij}$ is the head added via pumping in pipe $j$; and $Np$ is the total number of pipes included in loop $i$. We used the Hazen–Williams equation in this study to establish a relationship between the continuity and energy law equations, and also to calculate the head loss in the pipes.

2.2. Mathematical Formulation

This study considered the problem of optimally placing contamination detection sensors in WDSs as a single-objective optimization problem with an objective function to minimize the worst-case impact risk of the network. Different approaches can be used to consider the quality risk in a network, where the impact risk is considered here as the volume of contaminated water consumed in the demand nodes. Accordingly, with the assumption of contamination flowing from the k-th node in the s-th time step, the risk created in the j-th node in the t-th time step t is equal to [42,43] the following:

$$m_{t,j,s,k}=C_{t,j,s,k}\times q_{t,j}$$

(6)

where $C_{t,j,s,k}$ and $q_{t,j}$ are the concentration of contamination and water demand at the j-th node in the t-th time step, respectively.

It should be noted that for realistic simulations, a concentration value that is below the allowable concentration value was considered zero so that it could have no role in impact risk calculation. Moreover, the consumption of contaminated water continues until it is detected by the sensors, after which the impact risk becomes zero due to the interruption of water flow in the network, and the lack of water consumption. On the other hand, risk matrices were created to determine the locations of sensors before performing the optimization. In fact, the amount of damage was estimated in all possible modes of contamination. For example, if the network had 100 nodes and 24 time steps, the dimensions of each damage matrix would be equal to $100\times 24$, which is a total of 2400 (possible input from 100 nodes and 24 different time steps). The damage matrix is calculated as follows [42]:

$$M_{s,k}={\left[\begin{array}{ccc}{m}_{\mathrm{1,1},s,k}& \cdots & m_{1,nj,s,k}\\ ⋮& \ddots & ⋮\\ m_{nt,1,s,k}& \cdots & m_{nt,nj,s,k}\end{array}\right]}_{nt\times nj}$$

(7)

where $M_{s,k}$ is the risk matrix, with the assumption of the inflow of contamination from the k-th node at time s. Furthermore, $nj$ and $nt$ are the total number of harvesting nodes and simulation steps, respectively. Given that the risk becomes zero after detection of contamination by the sensor, we used a binary coefficient to modify the risk in the network, as follows:

$$\begin{array}{l}{m}_{t,j,s,k}^{\mathrm{\prime }}=a_{t,s,k}\times m_{t,j,s,k} ,\\ a_{t,s,k}=\left\{\begin{array}{c}1 t\le {\tau }_{s,k}\\ 0 t>{\tau }_{s,k}\end{array}\right.\end{array}$$

(8)

where $m\mathrm{\prime }$ is the modified damage based on the performance of the sensors, and ${\tau }_{s,k}$ represents the time step when the first sensor in the network detects the contamination coming from the k-th node at moment t. $a_{t,s,k}$ is a binary parameter whose value depends on the concentration values and arrangement of the sensors. This parameter also causes the risk to be calculated only before the interruption of flow.

As a result, the total risk of the network, equal to the sum of the risks of all nodes in all time steps, is calculated as follows [42]:

$$G_{s,k}=\sum _{t=1}^{nt}\sum _{j=1}^{nj}{m}_{t,j,s,k}^{\mathrm{\prime }}$$

(9)

Thus, the problem of optimizing the positions of CWSs in WDSs can be mathematically formulated as follows:

$$\begin{array}{c}Z=argmin\left\{F\left(x,G_{s,k}\right)\right\} \\ x^{\in {\left\{\mathrm{1,0}\right\}}^{N_s}}\\ \mathrm{Subject} \mathrm{to} |x|\in x\end{array}$$

(10)

where $F_1$ is the estimated worst-case impact risk, and $\chi$ is the set of binary decision variables that are defined as the index of sensing node, as follows:

$$x_j=\left\{\begin{array}{l}1 \mathrm{i}\mathrm{f} \mathrm{a} \mathrm{s}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{o}\mathrm{r} \mathrm{i}\mathrm{s} \mathrm{i}\mathrm{n} \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{d} \mathrm{a}\mathrm{t} \mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e} j\\ 0 \mathrm{i}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} \mathrm{i}\mathrm{s} \mathrm{n}\mathrm{o} \mathrm{s}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{o}\mathrm{r} \mathrm{i}\mathrm{n} \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{d} \mathrm{a}\mathrm{t} \mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e} j\end{array}\right.$$

(11)

In this study, a meta-heuristic technique was used, based on the nonlinearity and complexity of the multi-objective problem. A new hybrid whale optimization algorithm (WOA) with sand cat swarm optimizer (SCSO) was developed to simultaneously optimize the objective functions. The proposed new algorithm is explained in the following section.

3. Optimization

This section describes the mathematical formulation and mechanism of the developed hybrid WOA-SCO algorithm.

3.1. Whale Optimization Algorithm

WOA is a new meta-heuristic algorithm developed by [32] to solve optimization problems; it was inspired by the behavior of humpback whales based on their bubble-net hunting strategy.

First, the WOA begins by randomly generating the initial population of decision variables; then, it updates the position of the search agents at each iteration using prey encircling operators, the bubble network strategy, and prey search. The prey encircling operator is a behavior simulator for detecting the prey’s location and then encircling it. In the WOA, it is initially assumed that the current best solution is the prey; once it is detected, the position of the search agents relative to it are updated using the following equations [32]:

$$\overrightarrow{D}=|\overrightarrow{C}.\overrightarrow{X^{\mathrm{*}}}\left(t\right)-\overrightarrow{X}(t)|$$

(12)

$$\overrightarrow{X}\left(t+1\right)=\overrightarrow{X^{\mathrm{*}}}\left(t\right)-\overrightarrow{A}·\overrightarrow{D}$$

(13)

where t indicates the current iteration of algorithm, $\overrightarrow{X^{\mathrm{*}}}$ is best solution in iteration $t$, $\overrightarrow{X}$ is the search vector agent’s position, and $\overrightarrow{A}$ and $\overrightarrow{C}$ are vector coefficients that are defined as follows:

$$\overrightarrow{A}=2\overrightarrow{a}·\overrightarrow{r}-\overrightarrow{a}$$

(14)

$$\overrightarrow{C}=2·\overrightarrow{r}$$

(15)

where $\overrightarrow{a}$ is linearly decreased from 2 to 0 over the course of iterations, and $\overrightarrow{r}$ is a vector with random variables in [0, 1]. The mathematical modeling of the network-bubble behavior of whales (exploitation phase) is carried out as follows [32]:

• Shrinking the encircling mechanism: This behavior is achieved by decreasing the value of $\overrightarrow{a}$ in Equation (14). It should be noted that the fluctuation range of $\overrightarrow{A}$ is reduced to a.
• Spiral updating position: First, the distance between the position of the whale and the target prey is calculated, and then the spiral equation between them to imitate the spiral-shaped movement of the whale is defined on the basis of Equations (16) and (17).

$$\overrightarrow{X}\left(t+1\right)=\overrightarrow{D^{\mathrm{\prime }}}·e^{bl}·cos\left(2\pi l\right)+\overrightarrow{X^{\mathrm{*}}(}t)$$

(16)

where $\overrightarrow{D^{\mathrm{\prime }}}=|\overrightarrow{X^{\mathrm{*}}}(t)-\overrightarrow{X}(t)|$ represents the distance between the i-th whale and the prey (the best solution achieved so far), b is a constant to define the logarithmic spiral shape, and I is a random number in [1, −1].

To model the bubble network attack mechanism with the aim of updating the position of the whale during the optimization process, one of the two mechanisms of contraction encircling mechanisms and the spiral model is selected by the whale with a probability of 50%:

$$\overrightarrow{X}\left(t+1\right)=\left\{\begin{array}{c}\overrightarrow{X^{\mathrm{*}}(}t)-\overrightarrow{A}·\overrightarrow{D} if p<0.5\\ \overrightarrow{D^{\mathrm{\prime }}}·e^{bl}·cos\left(2\pi l\right)+\overrightarrow{X^{\mathrm{*}}(}t) if p\ge 0.5\end{array}\right.$$

(17)

where $p$ is a random number in [0, 1].

In the WOA search phase, to update the position of the search agents, the random selection of the search agent is used according to Equations (18) and (19).

$$\overrightarrow{D}=|\overrightarrow{C}·\overrightarrow{X_{rand}}-\overrightarrow{X}|$$

(18)

$$\overrightarrow{X}\left(t+1\right)=\overrightarrow{X_{rand}}-\overrightarrow{A}·\overrightarrow{D}$$

(19)

where $\overrightarrow{X_{rand}}$ is the random position vector of a whale selected from the current population.

3.2. Sand Cat Swarm Optimization

The sand cat swarm optimization (SCSO) algorithm is inspired by the behavior of sand cats in finding and hunting for prey in nature [44].

Like other meta-heuristic algorithms, SCSO begins by randomly generating the initial population (solutions). Then, by evaluating the objective function for each solution, the best solution (the best cat) is determined in each iteration, and the other cats try to reduce their distance from the best cat and close in at each stage, because it is assumed that the best cat is the closest cat to the prey [44]. The position of the sand cat in the search phase is updated based on a random position. To prevent the algorithm from becoming trapped in a local optimum, a sensitivity range is assigned to each cat according to Equation (20):

$$\overrightarrow{r}=\overrightarrow{r_G}\times rand(\mathrm{0,1})$$

(20)

where $\overrightarrow{r_G}$ represents the main sensitivity range, which decreases from 2 to 0 from the beginning of the optimization process to the last iteration.

In fact, $\overrightarrow{r}$ is used for exploitation in the search and exploration phases, where $\overrightarrow{r_G}$ is a parameter used to set the control parameter $R$ in the transfer phase.

$$\overrightarrow{r_G}=s_M-\left(\frac{2\times S_M\times {iter}_c}{{iter}_{Max}}\right)$$

(21)

$$\overrightarrow{R}=2\times \overrightarrow{r_G}\times rand\left(\mathrm{0,1}\right)-\overrightarrow{r_G}$$

(22)

Then, each cat updates its position based on the position of the best solution (Posbc), its current position, and its sensitivity range ($\overrightarrow{r}$).

$$\overrightarrow{Pos}\left(t+1\right)=\overrightarrow{r}.\left({\overrightarrow{Pos}}_{bc}\left(t\right)-rand\left(\mathrm{0,1}\right).{\overrightarrow{Pos}}_c\left(t\right)\right)$$

(23)

Therefore, sand cats can find another possible best position for the prey, which creates another opportunity for the algorithm to find a new local optimum in the search space. The new found position is between the current position and the position of the prey. The attack phase in SCSO is simulated based on Equation (24), which is the distance between ${\overrightarrow{Pos}}_b$ and ${\overrightarrow{Pos}}_c$. In addition, the sensitivity range of the search agent is considered circular. Therefore, the cat′s direction of movement is determined on the circle by a random angle using a roulette wheel. Thus, the sand cat closes in on the prey with every iteration [44].

$$\begin{array}{l}\overrightarrow{{Pos}_{rnd}}=\left|rand\left(\mathrm{0,1}\right).{\overrightarrow{Pos}}_b\left(t\right)-{\overrightarrow{Pos}}_c\left(t\right)\right|\\ \overrightarrow{Pos}\left(t+1\right)={\overrightarrow{Pos}}_{bc}\left(t\right)-\overrightarrow{r}.\overrightarrow{{Pos}_{rnd}}.\mathrm{c}\mathrm{o}\mathrm{s}(\theta )\end{array}$$

(24)

${Pos}_{rnd}$ is a random position. The cats close in on the prey after a number of iterations.

Finally, the position of the search agents (sand cat) in the exploitation and exploration phases is updated in Equation (25).

$$\begin{array}{l}\overrightarrow{{Pos}_{rnd}}=\left|rand\left(\mathrm{0,1}\right).{\overrightarrow{Pos}}_b\left(t\right)-{\overrightarrow{Pos}}_c\left(t\right)\right|\\ \overrightarrow{Pos}\left(t+1\right)={\overrightarrow{Pos}}_{bc}\left(t\right)-\overrightarrow{r}.\overrightarrow{{Pos}_{rnd}}.\mathrm{c}\mathrm{o}\mathrm{s}(\theta )\end{array}$$

(25)

Accordingly, if |R| ≤ 1, the sand cats are directed to attack their prey; otherwise, the cats have to find a new feasible solution in the global region (see Figure 2).

$$\overrightarrow{X}\left(t+1\right)=\left\{\begin{array}{l}{\overrightarrow{Pos}}_b\left(t\right)-\overrightarrow{{Pos}_{rnd}}.\cos \left(\theta \right).\overrightarrow{r} \left|\mathrm{R}\right|\le 1;expolitation\\ \overrightarrow{r}.\left({\overrightarrow{Pos}}_{bc}\left(t\right)-rand\left(\mathrm{0,1}\right).{\overrightarrow{Pos}}_c\left(t\right)\right)\left|\mathrm{R}\right|>1;expoloration\end{array}\right.$$

(26)

A proper balance between the search and exploration phases in SCSO brings about fast and accurate convergence of the algorithm to solve complex and multi-objective problems.

3.3. Hybrid WOA-SCSO

Each optimization algorithm has many strengths and weaknesses in the face of various problems. The purpose of creating hybrid algorithms is to solve various optimization problems using the strengths of the combined algorithms. The WOA is one of the most robust meta-heuristic algorithms, with optimal performance to solve many complex optimization problems; however, for some types of complex problems, it can become trapped in local optima, and subsequently has a very low convergence speed. Therefore, this study used the ability of SCSO to improve the performance of the WOA during the exploration phase. To this end, the SCSO was incorporated into the search phase of the WOA, and the condition approach was utilized after updating the position of each search agent. Algorithm 1 shows the details of this modification.

Algorithm 1. The pseudocode of WOA-SCSO

Generate Initial the whale population Xi where (I = 1, 2, 3, ---, n) Compute the fitness of each solution X* represents the best search agent    While (t < Maximum number of Iterations)        for each solution        Update WOA parameters (a, A, c, L, and p)        If 1 (p < 0.5)          If 2 (|A| < I)             Update the position of the current search agent by the Equations (23)–(26)          Else If2 (|A| > 1)             Randomly choose search agent (XRand)             Update the position of the current search agent by the Equations (23)–(26)          End If 2          Else If l (p > 0.5)             Update the position of the current search agent by the Equation (19)         End If l           End for Check the space limits (if any search agent goes beyond the search phase, then amend it) Compute the fitness of each search agent Update X if there is a better solution in the population (t = t + 1) End while Return X*

The search space for optimally locating contamination sensors is binary or, in other words, the decision variables are zero or one. The proposed WOA-SCOS has a continuous search space. As a result, this problem cannot be solved by the WOA-SCOS. Therefore, to update the search agent position for the WOA-SCOS on a binary search domain, the following sigmoidal equation was included:

$$S\left(x\right)=\frac{1}{1+e^{(-10\times (x-0.5))}}$$

(27)

$$X_{binary}\left(t+1\right)=f\left(x\right)=\left\{\begin{array}{l}1, sigmoid{X}^{\mathrm{*}}\ge r\\ 0, otherwise\end{array}\right.$$

(28)

where $X_{binary}$ represents the binary position based on the Equation (28), and r is a random number in the range of 0 and 1 [23].

4. Results and Discussion

4.1. Case study of Benchmark Functions

Before utilizing the proposed WOA-SCSO algorithm in the problem of optimally placing sensors for contamination detection in WDSs, the CEC’20 benchmark [45] was used to evaluate the algorithm’s efficiency. The results obtained with WOA-SCSO were compared with the WOA, SCSO, hybrid whale optimization algorithm–simulated annealing (WOA-SA) [46], dragonfly algorithm (DA) [47], and month flame optimization (MFO) [48].

For a fair comparison of algorithms, each function was simulated 30 times by each algorithm with a population of 100 and a maximum iteration of 1000.

Table 1. The setting parameters of each algorithm.

Algorithm	Parameter	Value
WOA	$\alpha$	[2, 0]
		[2, 0]
	C	2.rand (0, 1)
	L	[−1, 1]
	B	1
SCSO	$r_G$	[2, 0]
	R	[−$2r_G$, ${2r}_G$]
DA	$\beta$	0.01
	$\alpha$	0.99
SA	Crossover	0.4
	Mutation	0.7
MFO	B	1
	R	[−1, −2]
	T	[r, 1]

presents the adjustment parameters used for the algorithms, which were selected based on suggested values in the reference articles.

Table 2. Simulation results of algorithms for CEC’20 benchmark functions.

F	WOA-SCSO		WOA		SCSO
	Average	STD	Average	STD	Average	STD
1	4630.923	3865.641	29,622,854	14,996,437	4451.761	3723.702
2	1354.157	153.8788	3861.19	570.506	1843.399	358.4839
3	747.1492	9.955987	936.4858	65.65286	754.0862	16.86232
4	1904.242	2.295859	1948.024	22.28444	1904.749	2.079524
5	205,961.7	101,493.9	1,786,564	1,423,187	397,673.7	311,695
6	1624.441	6.94 × 1013	1624.441	6.94 × 1013	1629.656	6.94 × 1013
7	67,541.59	48,168.43	1,089,934	999,417.5	185,219	190,155.1
8	2303.807	16.63077	3946.46	1686.991	2827.785	1038.077
9	2823.531	17.51626	2998.875	66.33284	2850.618	28.56
10	2942.926	28.33319	3030.829	39.80924	2946.545	32.49808
Best	5		0		0
F	WOA-SA		DA		MFO
	Average	STD	Average	STD	Average	STD
1	4103.168	2959.42	4,319,673.969	1,107,564.075	19,258,630	13,458,488
2	2509.475	723.7819	3090.541577	461.1967984	3170.614	600.27
3	770.379	16.60193	906.0590205	29.61939612	840.4546	24.32494
4	1903.312	1.427191	1923.549656	5.342737287	1925.495	8.438489
5	74,101.39	49,371.8	426,743.4619	247,660.423	460,340.8	393,982.6
6	1624.441	6.94 × 1013	2005.250739	6.94 × 1013	1919.427	6.94 × 1013
7	23,614.48	20,063.03	326,989.7085	298,768.1618	372,906.1	296,250.6
8	2381.877	444.3796	3500.107678	1598.297395	2317.165	3.214768
9	2927.662	95.72872	3052.300783	79.92431871	2914.178	38.58321
10	2950.737	32.63042	2987.917673	24.86867008	2997.8	33.16372
Best	4		0		0

shows the statistical results, including the averages and standard deviations of 30 executions for all of the aforementioned algorithms on the benchmark functions, with the best solutions highlighted in bold. As shown in the results, WOA-SCSO provided a better solution than the other algorithms in most functions. As a result, WOA-SCSO ranked first among the algorithms for solving the benchmark functions.

Another parameter that can illustrate the performance of algorithms is their convergence rate in achieving an optimal solution. According to Figure 3, which shows the convergence curve of the algorithms for the benchmark functions, WOA-SCSO had a high-order convergence speed in achieving the optimal solution. Its appropriate convergence rate in solving benchmark problems indicates its high efficiency in solving real-world optimization problems that require fast calculations.

4.2. Case Study of Anytown WDS

To evaluate the robustness of the proposed hybrid model, the Anytown benchmark WDS was selected. This network has 3 pumps, 1 reservoir, 2 tanks, 22 nodes, and 34 pipes. Figure 4 shows the schematic of the Anytown network. Reference [39] used this network to evaluate the software developed for the optimal placement of sensors in the network. Their study, which had the same objective function, used four sensors in the nodes [5,7,10,19] and reported the worst-case impact risk value of the network to be 138,808 m³. In the current study, considering 10% uncertainty for network demand, it was assumed that contamination with a concentration of 10 mg/l flowed into the network from one node for 2 h.

After forming a risk matrix based on the explanation provided in Section 2 and Section 3, each algorithm was executed 10 times with a population of 50 and a maximum iteration of 500. Due to the high costs of installing and operating sensors, the maximum number of sensors was considered to be 4.

Table 3. Validation results.

	Algorithms
	WOA-SA	SCSO	WOA-SCSO	WOA	MFO	DA
1	131,754	137,728.5	131,754	347,665.5	1. × 1018	201,211.5
2	131,754	204,361.5	131,754	211,827	1.39 × 1018	234,097.5
3	131,754	211,827	131,754	234,097.5	131754	201,211.5
4	131,754	207,732	131,754	204,361.5	1.39 × 1018	234,097.5
5	131,754	211,827	131,754	488,260.5	1.39 × 1018	207,732
6	204,361.5	201,211.5	131,754	211,827	131754	201,211.5
7	131,754	207,732	131,754	211,827	1.39 × 1018	234,097.5
8	131,754	201,211.5	131,754	211,827	131754	201,211.5
9	131,754	218,263.5	131,754	218,263.5	1.39 × 1018	131,754
10	131,754	211,827	131,754	599,739	1.39 × 1018	211,827
Average	139,014.75	201,372.2	131,754	293,969.55	9.71656× 1017	205,845.2
STD	21,782.25	21,802.81	0	133,589.8387	6.36098× 1017	28,300.72

presents the values of the objective function obtained from the implementation of the algorithms for the installation of 4 sensors. As the results show, the value of the objective function calculated by WOA-SCSO was constant in all executions, and the standard deviation from the standard deviation of 10 executions was zero, and the average value of the objective function (worst-case impact risk) was 131,754 m³. After WOA-SCSO, the WOA-SA algorithm yielded favorable results, such that the value of the objective function was constant in 9 executions, and different in only 1 execution.

Given the binary nature of the problem, the MFO algorithm provided very poor performance; it violated the constraint in 7 executions, and had only 3 successful executions. The third-best ranking went to SCSO, with an average value of 201,372 m³ and a standard deviation of 21,802 m³. DA and WOA were the next-ranked algorithms in their reductions of the worst-case impact risk of the network.

Figure 5 shows the convergence curve of the algorithms in solving the problem of optimally placing CWSs in the Anytown reference network. As shown in the results, SCSO improved the convergence rate in the WOA as well. Figure 6 also shows the optimal positions of the contamination detection sensors in the Anytown network, according to the optimal solutions of the algorithms.

As the results show, the placement of the sensors was very diverse; their locations differed, although the level of risk was similar in some cases. However, some nodes seemed to play an important role in the network, such as node 19, which was determined as the optimal position for one of the sensors in most of the algorithms. Yet, it can be seen that the positions of the sensors were well-distributed on the network, in order to reduce the possible damage.

As mentioned before, reference [39] developed a novel model for the optimal placement of sensors in the Anytown network using the GA. The worst-case impact risk value reported in that study was 138,808 m³ for the placement of sensors in the nodes [5,7,10,19]. A comparison of the results of that study with those of the WOA-SCSO model indicates the superiority of the model developed on the basis of a novel optimization algorithm.

4.3. Case Study of Baghmalek WDS

The second case study of WDSs was selected from the southwest of Iran. Figure 7 shows the schematic of the Baghmalek water network. This network consists of 90 pipes and 72 nodes, whose water is supplied by gravity through a reservoir. All of the network pipes have a Hazen–Williams coefficient of 130, and were constructed of polyethylene [49]. Figure 8 shows the pattern of water consumption in Baghmalek city.

Due to the high salinity of the region′s soil, and the presence of oil reserves and a refinery in the vicinity of the study area, there is a high risk for contamination of the network water from oil and chemical toxins due to negative pressure. Therefore, it is very important to develop an effective model for the optimal placement of CWSs in the studied network, in such a way that it reduces the worst-case impact risks in the network.

In order to model this network, it was assumed that a dangerous contaminant with a concentration of 10 mg/L flowed into the network from different nodes and at different time steps to cause damage. Moreover, as in the previous example, since the demand parameter is one of the most important parameters affecting a network’s hydraulics, a parameter with uncertainty was considered using normal and log-normal distributions. For this purpose, 1000 samples of the demand factor were generated, each with a definite average value and a standard deviation of 10%. To determine the most appropriate distribution, we followed the recommendation from study [50], which suggested using a normal distribution for demand factors that are less than 1.5 and a log-normal distribution for demand factors greater than or equal to 1.5. To calculate the failure probability of the optimized design, we used a Monte Carlo simulation. According to the theory of Monte Carlo simulations, the failure probability is defined as the ratio of failed designs (i.e., designs that violate constraints) to the total number of scenarios [51].

Accordingly, this network was implemented for several scenarios with a duration of 7 days. In the first scenario, the network without sensors was analyzed, while in the other scenarios, the number of sensors was increased, and their optimal positions were determined in such a way that the qualitative risk value of the network reached the minimum value in the uncertainty conditions of the location and time of contamination. The other assumptions considered for the analysis of this network were as follows:

• The time steps of the hydraulic and quality simulations were 1 h and 15 min, respectively.
• All of the demand nodes had the potential for sensor installation.
• Inflow of contamination from all demand nodes was possible at all hours of the first day of simulation.
• Inflow of contamination was possible from only one node, and the duration of contamination was 2 h.

Table 4. Details of the trade-off diagram.

Number of Sensors	Maximum Impact Risk (l)	Sensor Nodes
1	24,912.95	49.
2	12,697.64	61.
3	10,263.17	22, 54, 51.
4	7942.65	36, 51, 27, 67.
5	7129.02	19, 51, 3, 37, 48.
6	6342.94	12,66, 48, 24,51, 37.
7	5913.10	51, 6, 67, 37, 23, 12, 47.
8	5212.26	67, 51, 18, 61, 39, 3, 42, 27.
10	4855.10	53, 31, 33,19, 51, 45,37, 9, 36, 47.
12	4237.19	42,23,51,29,66,4,62,69,46,24,58, 32.
15	3856.85	48, 53, 51, 46, 16, 14, 52, 30, 27, 41, 19, 3, 66, 27, 11.
20	2992.93	20, 5, 61, 51, 24, 5, 53, 63, 42, 3, 56, 68, 28, 44, 33,39, 19, 54, 24, 47, 8, 21, 37, 9, 18.
25	2680.61	61, 12, 62, 60, 7, 40, 65, 48, 39, 22, 42, 54, 5, 49, 33, 51, 50, 32, 29,65, 41, 19, 27, 1, 19, 53, 10, 22, 47, 15.

presents the results of Baghmalek network modeling. Figure 9 shows the optimal numbers of sensors relative to the risk values. As the results indicate, the maximum damage was equal to 49,825.14 L, which came from the scenario when sensors were not available in the network. Adding a sensor to the network decreased the worst-case impact risk by 49% and the damage to 24,912.95 L. However, adding 1 more sensor decreased the worst-case impact risk by 20% compared to the first scenario.

In general, it can be said that optimal placements of more sensors reduced the amount of damage in the network. Obviously, when there were fewer sensors in the network, the interaction graph had a higher slope. On the other hand, a larger number of sensors led to a decrease in the slope of the graph, and less change in the damage rate.

According to Table 4, the locations of the sensors were diverse, and were not necessarily the same in all of the scenarios. However, some nodes played a significant role in the network, such as 51, which was selected as the optimal position in most scenarios. In any case, in order to minimize the worst-case impact risk caused by the inflow of intentional or accidental contamination, the developed model was able to distribute the proposed locations of the sensors throughout the network. According to the results, the greatest damage in the network was related to the inflow of contamination during the peak time of the network, when the contamination distributed faster throughout the network due to the higher level of demand.

5. Conclusions

We developed an optimization–simulation model in this study based on a novel optimization algorithm that used a combination of the WOA and SCSO for the optimal placement of contamination detection sensors in WDSs under demand uncertainty. For this purpose, the weakness of the WOA during the phase of searching and escaping from local optima was improved using the abilities of SCSO. The performance of the developed algorithm was first evaluated on the benchmark problems of the CEC’20 series, with results indicating an improvement in the performance of the WOA in the search and convergence stage, so that the use of WOA-SCSO provided better performance in most benchmark problems tested compared to the other algorithms. Next, the problem of optimizing the placement of contamination sensors in the water network was formulated as a probabilistic problem. In order to obtain realistic results, we considered changes in demand factors to be uncertain, and modeled them as random parameters that followed normal and log-normal distributions. Examining the modeling results on a benchmark network showed that WOA-SCSO was able to provide the lowest standard deviation from the objective function (the worst-case impact risk) in 10 executions of the model compared to the other algorithms. After validating the optimization–simulation model developed on the benchmark network, we applied it to a quality management scenario of the Baghmalek water distribution network in Iran. We examined the results across several scenarios in applying the developed model on the studied network, taking into account the factor of uncertain demand at different times. The results showed that the use of at least one sensor in a network can decrease the amount of contaminated water consumed in it by 49% compared to a network without sensors. It was also found that the greatest damage occurs in the network when contamination flows into the network during peak demand. Therefore, it is recommended to use high-quality sensors in order to reduce the damage caused by inflows of contamination into the network. To this aim, the proposed algorithm was able to provide appropriate locations for placing sensors in the water network. In order to complete the present study, future research can use the proposed algorithm to solve the important problem of placing sensors in a multi-objective manner and applying the scenarios of draining contaminated water from the network by opening and closing valves immediately after the sensors detect contamination.