Abstract
Small, but fast fluctuating consumptions in a pipe network can lead to severe transient flows. Such fluctuations are stochastic in nature and, cannot be explicitly identified and analyzed for the systems in operation. This study introduces a mathematical model for taking into account the fluctuating consumptions in analysis and design of pipe networks. For this purpose, the fluctuations are simulated by two successive triangular pulses with adjustable geometry. A manyobjective optimization problem is developed to find the worst maximum and minimum pressure heads. A new scheme of genetic algorithms is developed to find the extreme values of all pressure heads in only one single simulation run. The proposed model is applied to two water distribution networks. It is found that, the transient flow caused by fast fluctuating consumptions could seriously affect the system’s performance. For instance, analyzing a real pipe network reveals that, when the fluctuations are critically combined, the nodal pressure heads averagely change by 29 % (13 to 88 %) and −43 % (−16 to −177 %) with respect to the initial steady state.
Introduction
Surge analysis of a piping system is the hydraulic simulation of transient pipe flows and nodal pressures often generated by fast operation of valves and pumps in the system. Of many challenges facing water utilities, one critical but toooftenforgotten issue is protecting the system from excessive transient pressures. Surge analysis is essential to estimate the worstcase scenarios in the water distribution systems (Boulos et al. 2005). Sudden changes to a steadystate flow cause transientstate flows in piping systems. The main transient exciters in water supply networks are the operation of valves and pumps (predictable exciters) as well as fast fluctuations in the system’s water demands (unpredictable exciters). For designing a new piping system or assessing an aged one, analyzing critical transient scenarios is highly required. A transient flow analysis is intended to determine the reliability of water supply and the integrity of hydraulic devices when waterhammer waves occur (Duan et al. 2010). For this purpose, first, main transient excitations e.g., pump failures or fast maneuver of valves are identified in the system. Then, by predicting the worstcase scenarios, the highest and lowest node pressures are analyzed. On this basis, the system performance is evaluated and, the transient protection methods and devices are proposed and designed. Hence, one needs to know exactly how and from where the transient effects are initiated and propagated over the network. This is while, the origin of transient flows caused by unpredictable events might be quite unknown. In practice, the fluctuating consumptions could be resulted from rapid maneuver of taps as well as startup and shutdown of household pumps. In general, these fluctuations are considered negligible however, they are inherently stochastic and out of any control. The fluctuations suddenly excite the system from several different locations in the network and can generate severe transient conditions. A main difficulty for simulating this kind of transient flows is that, the critical combinations of fluctuating consumptions could not be explicitly determined. Finding the critical combinations of fluctuations and their issued dynamic effects needs to solve a complicated manyobjective optimization problem. This study deals with this problem and introduces a methodology for that. In what follows, first, a brief review of the relevant literature is given. Then, the governing equations and the analysis method are presented. For simulating the stochastic fluctuating consumptions, a conceptual model is proposed. Then, to analyze the worst transient effects, a manyobjective optimization problem is developed. To solve the problem, a new scheme of genetic algorithms is introduced. Afterwards, two examples are analyzed using the proposed approach.
Hydraulic transient analysis is important in the design and exploitation of pressurized water pipe systems to guarantee their security, reliability and good performance in different normal operational conditions (Ramos et al. 2009). There are many investigations and several standard references (Chaudhry 1987; Almeida and Koelle 1992; Wylie and Streeter 1993) that comprehensively address different aspects of transient flows in pipelines. Most previous investigations have studied deterministic priorknown transient excitations. There are few studies about unusual stochastic transient excitations. In this context, Foster (2003) dealt with this issue that the operation of some processes in water treatment plants can cause serious transient pressures in the associated pipelines and distribution systems. That work especially focused on the operation of microfiltration systems and the transients caused by them. It was found that beside the standard wellknown transient generators in a piping system, rapid maneuver of flow control valves under normal, backwash, and emergency operations in microfiltration systems would generate potentially damaging high and low transient pressures in the network. Martino et al. (2008) discussed that a pressurized water system may be subjected to high pressure surges because of the expulsion of a trapped air pocket through an orifice at the downstream end of pipes. This issue can also be referred to as unusual transient events in piping systems having leakage and worn out devices. Jung et al. (2009) studied the effects of pressuresensitive demands on surge analysis of pipe systems. It was discussed that when a transient state is initiated, nodal demands could be affected by pressure variations in the system. In fact, that study dealt with this issue that, there is an interaction between the transient pressures and the system withdraws. They introduced an implicit relationship between the pressure and node consumption. The results of two demand models including the pressuresensitive and lumped, were obtained quite different even within the first cycle of a surge wave which is normally used to estimate the critical pressures. Duan et al. (2010) considered that some analysis parameters such as the wave speed and friction factor of pipes are subjected to uncertainties. They investigated the effects of uncertainties on the network transient responses by incorporating the Monte Carlo simulation method into the standard waterhammer equations. Ebacher et al. (2011) studied a pipe network under transient conditions caused by lowpressure events at a water treatment plant. It was found that these events result in negative pressures at several locations in the system. After validating the model with the field data, it was concluded that the positive pressures match well with the measured pressures however, as the pressures reach negative values, the simulated pressures become larger than the recorded values. Haghighi and Keramat (2012) developed a model for considering uncertainties in transient analysis of pipe networks. They used the fuzzy stets theory and the method of simulated annealing. The interaction between the uncertain parameters (the pipe friction factors, wave speeds and nodal demands) and the transient pressures was studied. It was found that even small uncertainties in input variables can result in large uncertainties in the transient responses.
Governing Equations
The transient flow in a pressurized pipe is described by the conservative laws of mass and momentum. The governing partial differential equations and their mathematical proofs are found in details in Chaudhry (1987) and Wylie and Streeter (1993). For most engineering applications the continuity and momentum equations are respectively adopted as the following,
in which x = distance along pipe, t = time, g = gravitational acceleration, A = crosssectional area of pipe, D = pipe diameter, a = wave speed and f = DarcyWeisbach friction factor and the transient responses Q = instantaneous discharge and H = instantaneous piezometric head.
The above equations are traditionally solved using the Method of Characteristics (MOC) introducing the initial and boundary conditions. In the present study, the transient flow is generated and amplified by fast fluctuating consumptions initially introduced with steady values to the network nodes. Herein, a standard transient simulation model is utilized as a function of the fluctuating consumptions to evaluate the time histories of transient pressures. It is worth mentioning that, this simulation model has been already developed and used in the previous studies by Haghighi and Shamloo (2011), Haghighi and Ramos (2012) and Haghighi and Keramat (2012).
For the hydraulic analysis of water distribution networks for both purposes of design and condition assessment, water demands are estimated and applied to the network nodes. For sizing the pipes, pumps and reservoirs, the peak demands at the end of the design period are considered. For both steady and unsteady analyses, a rational assumption is that the water demands change hourly. As a consequence, the water supply and demands can be considered constant in the modeling if the simulation time period is less than one hour. Accordingly, fast fluctuating consumptions are ignored in the simulations. In reality, every consumer can independently influence the flow pattern in the network even though, this influence is trivial and fast. In large pipe networks having too many service connections, this issue results in a stochastic pattern of flow distribution. The severity of a fluctuating consumption is defined as the maximum deviation of its instantaneous quantity from its steady state. If the time duration of fluctuations is considered to be in minutes (macro fluctuations), the generated transient flow would not be very significant. However, if the fluctuations occur in few seconds (minor fluctuations), the exited transient flow could be considerable. Since, the random fluctuations may occur in several different locations simultaneously, they could result in severe transient conditions. Different nodal consumptions have different fluctuating patterns. The combination of these patterns governs the intensity of the generated transient flow in the system.
In this study, the minor fluctuations are simplified in the simulation by a conceptual model shown in Fig. 1. Through this model for node j, the steady consumption C _{ j } is affected by two successive triangular pulses to simulate both positive and negative transient waves. The flow simulation is started from a steadystate condition. The maximum time duration of fluctuations is defined T _{ f }. During this time period, the consumption C _{ j } remains steady until time A from where, it linearly changes to C _{ j } ± ΔC _{ j } ^{1} at time B while, ΔC _{ j } ^{1} is the fluctuations severity in the first pulse. Then, the consumption becomes C _{ j } again at time C where, the first pulse ends. Similarly, the second pulse initiates from time D from where, the consumption linearly changes to C _{ j } ± ΔC _{ j } ^{2} at time E while, ΔC _{ j } ^{2} is the fluctuations severity in the second pulse. Finally, the consumption value becomes C _{ j } again at time F. Based on the described pattern, for each consumption node, there are eight variables governing the induced transient flow. These variables are consisting of the timepoints t _{ A }, t _{ B }, t _{ C }, t _{ D }, t _{ E }, t _{ F } and the severity heights ΔC _{ j } ^{1} and ΔC _{ j } ^{2} . The maximum severity of the fluctuations might be limited to ± (5 − 10) % of the corresponding consumption and, the maximum lag between two successive timepoints might be limited to (2–5) seconds. It is worth noting that, the fluctuations pattern is a userdefined factor that can be modified based on the engineering judgments.
Transient Analysis Model
In general, the transient flow analysis is carried out to compute the extreme values of pressure heads in a pipe network. The node pressure heads must be in a safe range to protect the system from damages associated with the waterhammer waves. Evaluating the worst extreme values of nodal pressure heads caused by fluctuating consumptions leads to develop a complex manyobjective optimization problem. For this purpose, the following optimization problem is introduced to each node \( j \) according to the described fluctuations pattern.Minimize:
Subject to:
where, i = 1 to N, j = 1 to M and, N and M are respectively the number of consumption nodes and the locations in which the transient pressures are evaluated, function ϕ is “min(H _{ j })” to determine the worst minimum pressure head and is “− max(H _{ j })” to determine the worst maximum pressure head in node j. The multiplier α indicates the maximum severity of the fluctuations in node i and, β is the least lag time between two successive timepoints in the fluctuations pattern.
To solve the above problem for each node j, it is required to apply a single nonlinear optimization solver twice. In context of optimization, each node has two objective functions and each function has 8N decision variables consisting of six fluctuations timepoints (t _{ A }, t _{ B }, t _{ C }, t _{ D }, t _{ E }, t _{ F }) and two fluctuations severity heights (ΔC _{ j } ^{1} , ΔC _{ j } ^{2} ) where i = 1 to N. For analyzing the worst maximum and minimum pressure heads in M nodes, a manyobjective optimization problem with 2M objective functions and 8N decision variables is formed. This problem is neither single nor traditional multiobjective since, the tradeoff between the objectives is not important here. In this problem, all functions are independent but intended to be optimized simultaneously. Application of common singleobjective solvers to this problem is computationally burdensome and inefficient. The current problem has some special features that help solve it more efficiently if, they are appropriately utilized in the modelling. First, all functions have the same decision variables and dimension and decision space. Second, the purpose of the optimization is only to find the extreme values of M nodal pressure heads while, the tradeoff between them is not of interest. Third, all functions are evaluated simultaneously by the same transient simulation model once a combination of fluctuating consumptions is introduced. In other words, each single transient simulation run contains all function values in a certain set of decision variables. Consideration of these features in the applied optimization model makes the whole process of transient analysis more efficient and systematic. On this basis, a new scheme of manyobjective genetic algorithm is developed and coupled to the transient simulation model as described in the following section.
Optimization Model
A simple continuous GA from Goldberg (1989) is adopted here as the main core of the algorithm. Then, it is upgraded with some special operators for the selection, pairing and elitism to find the extreme values of all objective functions simultaneously. The transient flow analysis problem introduced in the previous sections can be defined as a general mathematical programming model as follows.
While, x _{ j } ^{min} ≤ x _{ ij } ≤ x _{ j } ^{max} and y _{ j } ^{min} ≤ y _{ kj } ≤ y _{ j } ^{max} ; i = 1 to U, j = 1 to 6N and k = 1 to 2N
where, U is the number of objective functions, at most 2M in the this problem, G is the vector of objective functions and, x and y are the problem decision variables respectively associated with the timepoints and the severity heights of N fluctuating consumptions in the network. Also, x _{ j } ^{min} , y _{ j } ^{min} and x _{ j } ^{max} , y _{ j } ^{max} are respectively the lower and upper bounds of the decision variables. Row \( i \) in the above matrix serves the objective function g _{ i } in G. To solve the problem, a manyobjective genetic algorithm (MOGA) is developed here as follows.

1
Similar to every GA, the MOGA starts with an initial population of chromosomes. The initial population with NP chromosomes is randomly produced in the range [0, 1]. Mathematically, a GA population is a matrix with NP rows and 8N columns containing normal values of decision variables r _{ i } encoded in [0, 1]. Every row in the population matrix is a solution alternative that needs to be decoded first to be introduced to the simulation model as the following.
$$ \begin{array}{cc}\hfill {x}_j={x}_j^{\min }+{r}_i\times \left({x}_j^{\max }{x}_j^{\min}\right)\hfill & \hfill i\kern0.5em \mathrm{and}\kern0.5em j=1\kern0.5em \mathrm{t}\mathrm{o}\kern0.5em 6N\hfill \end{array} $$(11)$$ \begin{array}{cc}\hfill {y}_j={y}_j^{\min }+{r}_i\times \left({y}_j^{\max }{y}_j^{\min}\right)\hfill & \hfill i=6N+1\kern0.5em \mathrm{t}\mathrm{o}\kern0.5em 8N,\kern0.5em j=1\kern0.5em \mathrm{t}\mathrm{o}\kern0.5em 2N\hfill \end{array} $$(12) 
2
The transient simulation model is run against every chromosome in the population. For every chromosome, the maximum and minimum nodal pressure heads are evaluated. This results in a fitness matrix with NP × U members as the following,
$$ \mathrm{Fitness}\kern0.5em \mathrm{matrix}=\left[\begin{array}{ccc}\hfill {g}_{1,1}\hfill & \hfill \dots \hfill & \hfill {g}_{1,U}\hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ {}\hfill {g}_{NP,1}\hfill & \hfill \dots \hfill & \hfill {g}_{NP,U}\hfill \end{array}\right] $$(13)In this matrix, each column is corresponding to a certain function g _{ i } and, each row is corresponding to a certain chromosome in the population.

3
With respect to each objective function i, each chromosome k is given a rank number R _{ i,k }. For each objective function, the best chromosome rank number is one and the worst rank number is NP. On this basis, a ranking matrix is formed. Also, for each chromosome an averaged rank number \( {\overline{R}}_k \) is calculated by averaging the row members of the ranking matrix. \( {\overline{R}}_k \) averagely represents the fitness of chromosome k with respect to all objective functions.
$$ \mathrm{Ranking}\kern0.5em \mathrm{matrix}=\left[\begin{array}{ccc}\hfill {R}_{1,1}\hfill & \hfill \dots \hfill & \hfill {R}_{1,U}\hfill \\ {}\hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ {}\hfill {R}_{NP,1}\hfill & \hfill \dots \hfill & \hfill {R}_{NP,U}\hfill \end{array}\right]\to \overline{R}=\left[\begin{array}{c}\hfill {\overline{R}}_1\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {\overline{R}}_{NP}\hfill \end{array}\right] $$(14) 
4
In this step, some good chromosomes are participated in producing the new generation. To mark the good and bad chromosomes in the ranked population, the following strategy is used:

a)
The best chromosome in each column with R = 1 is known as an elite chromosome in the population and is transferred to the mating pool. With this assumption, the number of elites is at least equal to the number of objective functions. However, in some cases, one may decide to save a few more elite chromosomes for some functions. In fact, the number of elites denoted by N _{ e }, for each objective function is a weighting factor for handling more intractable functions.

b)
Some chromosomes are not elite but are relatively good in all objective functions. To identify good chromosomes, a userdefined threshold rank number termed R _{ good } between 1 and NP is exploited. The chromosomes with the averaged rank number \( \overline{R}\ge {R}_{good} \) are marked “good” and transferred to the mating pool. By this criterion, NR _{ good } chromosomes are selected from the ranked population. Accordingly, there will be \( {\displaystyle \sum_1^U{N}_e}+N{R}_{good} \) parents in the mating pool waiting for paring to generate the next generation. The remaining chromosomes are marked “bad” and discarded from the evolution cycle.

a)

5
In this step, the couples are selected. Using the tournament selection method, the matchmaker selects the parents from the mating pool. For each parent, two chromosomes, x and y, are randomly picked. Between them, x wins the tournament if one of the following conditions is met. Otherwise, y is selected as a parent.

a)
x is an elite chromosome and y is not.

b)
x and y are both elite chromosomes but x has a better averaged rank number (\( {\overline{R}}_x<{\overline{R}}_y \)).

c)
None of x and y are elite but x has a better averaged rank number.

a)

6
The selected parents are mated to generate \( NP\left({\displaystyle \sum_1^U{N}_e}+N{R}_{good}\right) \) offspring in return for the discarded chromosomes. For this study, the blend crossover method (BLXα) proposed by Eshelman and Shaffer (1993) is adopted. Every couple produces two children and the new generation as a combination of children and parents is created.

7
Finally, a few genes in the new population, except the elite chromosomes, are randomly mutated. The mutation ratio μ is a critical parameter to every GA and needs to be initially calibrated.

8
The stopping criteria are checked for all objective functions. If all functions converged, the optimization is terminated otherwise, the algorithm goes back to step 2 where, the previous population is replaced by the new one and the search is continued.
Example 1
The proposed model is applied to a hypothetical pipe network (Fig. 2) originally introduced by Pudar and Liggett (1992). The network was then used as a benchmark in several investigations (Liggett and Chen 1994; Vitkovsky et al. 2000 and 2003; Kapelan et al. 2003; Jung and Karney 2006; Shamloo and Haghighi 2010; Haghighi and Ramos 2012). The network is consisting of 11 pipes, 7 junctions, 5 loops and a reservoir with 30 m piezometric head at node 1 that supplies the system by gravity. All pipes have 254 mm diameter size, 762 m length and 1316 m/s wave speed. Figure 2 demonstrates the network’s layout, the node and pipe names, DarcyWeisbach friction factors and the nodal consumptions. It is assumed that all consumptions encounter stochastic minor fluctuations according to the pattern of Fig. 1. The system is initially under a steadystate flow. The fluctuations time duration T _{ f } is considered to be 120 s and the fluctuations severity is considered to be in − 0.1C _{ j } ≤ ΔC _{ j } ^{1} , ΔC _{ j } ^{2} ≤ 0.1C _{ j } for every nodal consumption. Also, the least lag time between two successive timepoints is assumed 2 s. On this basis, the proposed model is applied to the network to find the worst transient pressures in all nodes except the reservoir one. For every node, there are two objective functions including the highest and lowest pressure heads which, are respectively intended to be maximized and minimized. The transient simulation model is set up as a function of the fluctuations parameters. For every consumption node, there are eight decision variables. As a result, there are totally 48 decision variables for six consumption nodes.
The decision variables are produced by the MOGA and introduced to the transient simulation model to evaluate the objective functions. In this example, there are 12 objective functions to find extreme values of pressure heads in nodes 2 to 7. For managing the problem solution, the MOGA is applied to the problem in two steps. The maximum pressure heads are maximized in one simulation run and, the minimum pressure heads are minimized in another simulation run.
To determine the MOGA parameters, some sensitivity analyzes were carried out. For this purpose, the population size NP, the proportion of good chromosomes NR _{ good }/NP and the mutation ratio μ are taken into consideration. First, a basic value is assigned to each parameter by the engineering judgments. Then, each parameter is changed over a rational bound about its basic value while, the other parameters are kept constant. For each change, the MOGA is run for a limited number of generations, 20 generations for this example, and the optimization performance is evaluated. Also, to mitigate the effects of randomness in the MOGA, the sensitivity analysis is repeated 10 times for each parameter. Upon the average results, the MOGA parameters are decided as follows; NP = 60, NR _{ good }/NP = 0.4 and μ = 0.03. Figures 3 and 4 depict the trend of maximizing and minimizing the extreme values of transient pressure heads in 150 generations. The maximum generation number in the MOGA was initially set to be 1000 however, in both runs the MOGA converged in about 100 generations. It is found that, when the stochastic fluctuations are critically combined, they can result in significant transient pressures in the network. This issue is more clearly concluded from Fig. 5 in which, the time histories of the worst transient pressures in all nodes are presented. Each pressure timehistory is resulted from a unique combination of the fluctuating consumptions. For instance, Fig 6(a) and (b) show the critical combinations of the fluctuations in the whole network that respectively result in the worst extreme pressures in node 3. In the initial steady condition, the pressure head at node 3 is 16.47 m. When the critical fluctuations of Fig. 6(a) occur in the network, a transient state initiates that instantaneously decreases the corresponding pressure head at node 3 to the critical value −6.04 m (see the dash line in Fig. 5 for node 3). This is while, if the fluctuations of Fig. 6(b) happen, the pressure head at node 3 instantaneously increases to 34.25 m. In other words, it is found that the minor fluctuations (± 10 % of the initial consumptions) in this example, can result in about − 136.7 % to 108 % instantaneous variations in the steadystate pressure head at node 3. Similar investigations for the other nodes reveal that the pressure heads in the entire network could averagely experience − 107.7 % to 92.8 % instantaneous variations with respect to the initial steady state.
Also, Table 1 summarizes the analyzed extreme pressure heads in all consumption nodes.
Example 2
In this example, a real case study namely Baghmalek water distribution network, from Haghighi and Keramat (2012), is analyzed. The network shown in Fig. 7 is consisting of 37 steel pipes, 28 nodes and a reservoir at node 1. Table 2 presents the nodal consumptions and topographic elevations. Table 3 shows the length, diameter and Darcy–Weisbach friction factor of pipes. The wave speed for all pipes is 1100 m/s. In this example, the pipe network is operated under a normal steady condition according to the consumptions presented in Table 2. There is neither pump failure nor rapid valve manoeuvre in the system. However, it is assumed that the nodal consumptions have at most ± 5 % minor fluctuations. In this network, the transient pressure heads in all consumption nodes except the reservoir one are intended to be analyzed. Hence, there are totally 54 objective functions to find the worst extreme pressure heads in the network nodes. The fluctuations pattern of Fig. 1 is introduced to all nodes thereby, the timepoints and severity of all fluctuations (216 variables in total) are the decision variables.
To solve the problem, the parameters of MOGA were determined as the following: NP = 400, NR _{ good }/NP = 0.5 and μ = 0.025. Similar to the previous example, this problem is also solved in two steps. In both runs, after almost 500 generations the optimization converged. Figure 8 demonstrates the extreme transient pressure heads in the networks. Upon this analysis, the steady pressure heads averagely experience 29 % (ranging from 12.93 % to 88.38 %) and − 42.62 % (ranging from − 16.04 % to − 176.61 %) instantaneous variations. The minimum transient pressures are more significantly affected by the fluctuations in comparison with the maximum pressures. Nodes 6 and 7 include the largest pressure variations in the network. These nodes might experience the critical negative pressure heads of 7.81 and −8.50 m while, their steady pressure head is about 10 m. This analysis, in addition to the traditional transient flow simulations, provides a better insight into the system configuration. For instance, from this analysis, nodes 6 and 7 are identified as weak points of the system. At these nodes, the network is required to be modified by some modifications in the pipe diameters and the network’s layout. In the previous study by Haghighi and Keramat (2012) it was discussed that, in practice, fast maneuver of the valve located at node 2 is the main transient exciter in the network. This issue was simulated and, the effects of input uncertainties on the network’s transient responses were analyzed to find the worst transient pressures. However, not only nodes 6 and 7 were not identified as weak points in that simulation but also, it was reported that the nodes close to the reservoir (like nodes 6 and 7) are safe.
Conclusion
Fluctuating consumptions as usual events in all pipe networks can cause significant transient conditions. These fluctuations are stochastic in nature and considered as unpredictable transient exciters. To analyze a pressurized pipe network under fluctuating consumptions, a mathematical model was developed in context of a manyobjective optimization problem coupled to a standard transient simulation model. A new scheme of genetic algorithms namely MOGA was introduced to solve the problem more efficiently.
The proposed model was applied to a hypothetical pipe network and a real water supply pipe network from the literature. It was found that, the critical combinations of fluctuating consumptions can generate significant transient flows in the system. For each node there are two critical combinations of fluctuations resulting in the worst maximum and minimum transient pressure heads. Both examples revealed that in some combinations the nodal pressures may instantaneously violate the safe pressure limits. Considering the fluctuating consumptions in the hydraulic analysis, provides a better insight into the network configuration and performance. By the proposed analysis approach, the network’s weak points associated with the critical high and low pressures are identified. As discussed for the Baghmalek pipe network, these weak points may not be easily identified by traditional transient flow analyzes which, are carried out according to priorknown deterministic transient scenarios.
The proposed analysis algorithm is dependent on the fluctuations pattern initially introduced to the nodal steady demands. This pattern is a userdefined factor in the model. In this study, a simple scheme with two successive triangular pulses was utilized in the simulations. In reality, this pattern needs to be more studied and modified based on the real data, statistical analyzes and engineering judgments.
Abbreviations
 A :

Pipe crosssectional area
 a :

Wave speed
 C :

Nodal consumption
 D :

Pipe diameter
 f :

DarcyWeisbach friction factor
 g :

Gravitational acceleration
 G :

Vector of objective functions
 H :

Piezometric head
 M :

Number of objective functions
 N :

Number of network nodes
 L :

Pipe length
 Q :

Pipe discharge
 R :

Rank number
 r :

Random real value
 t :

Time
 NP :

Population size
 N _{ e } :

Number of elite chromosomes
 NR _{ good } :

Number of good chromosomes
 \( \overline{R} \) :

Averaged rank number
 R _{ good } :

Threshold rank number for good chromosomes
 T _{ f } :

Fluctuations simulation time
 t _{ A } , t _{ B } , t _{ C } , t _{ D } , t _{ E } , t _{ F } :

Fluctuations time coordinates
 x ^{min}, x ^{max}, y ^{min}, y ^{max} :

Bounds of decision variables
 ΔC ^{1} and ΔC ^{2} :

Fluctuations severity heights
 μ :

Mutation ratio
 ϕ :

Objective function
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Haghighi, A. Analysis of Transient Flow Caused by Fluctuating Consumptions in Pipe Networks: A ManyObjective Genetic Algorithm Approach. Water Resour Manage 29, 2233–2248 (2015). https://doi.org/10.1007/s1126901509386
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Keywords
 Fluctuating consumption
 Transient pressure head
 Pipe network
 Manyobjective optimization