The design of WDN can be formulated as an optimization problem. The WDN installation cost is the most used objective function. Normally, the problem is formulated as the minimization of the installation cost, related to the tube diameter, subjected to a set of constraints involving mass balances in the demand nodes, energy balances if network loops are present, pressure and velocity limits. It is considered a set of commercial tubes with proper costs and rugosity coefficients to be chosen, aiming to minimize the total WDN cost. It is usual to solve the hydraulic equations using additional software. EPANET (Rossman, 2000) is the most used hydraulic simulator. Distinct formulations have been used, as Linear Programming (LP), Nonlinear Programming (NLP), Mixed Integer Linear Programming (MILP) and Mixed Integer Nonlinear Programming (MINLP). Optimization models with MINLP formulations are more representative for real problems and, recently, the use of this kind of optimization models by research groups have been increased. In general, global optimality cannot be ensured due to the nonlinear and nonconvex behavior of the model.
In a recent paper, Mala-Jetmarova et al. (2018) presented a detailed review of types of WDN optimization problems and methods used to solve the problem. Designs of new WDNs, expansion and rehabilitation of existing water distribution systems, strengthening, design timing, parameters uncertainty, water quality and operational considerations were reviewed. As pointed by the authors, different deterministic and stochastic approaches have been used to solve the optimization problem. Stochastic approaches are used in large scale problems, where deterministic approaches normally fail. Some of important methods used are Particle Swarm Optimization (PSO), in Ezzeldin et al. (2014), Surco et al. (2017) and Surco et al. (2021), Genetic Algorithms (GA), in Savic and Walters (1997) and Kadu et al. (2008), Harmony Search (HS), in Geem (2006), Ant Colony Optimization (ACO), in Zecchin et al. (2006), Simulated Annealing (SA), in Cunha and Sousa (1999) and Honey Bee Mating Optimization (HBMO), in Mohan and Babu (2009).
There are less papers focusing on deterministic approaches to solve the WDN optimization problem. It is because the intrinsic limitations of deterministic solvers in getting trapp in local optima solutions in nonlinear problems and in the difficulties in using global optimality methods in large scale problems. However, important advances have been published in this important research field. Bragalli et al (2012) proposed, to the optimization of WDN with fixed topologies, a nonconvex continuous Non-Linear Programming (NLP) relaxation and an MINLP search approach. Raghunathan (2013) used linearization techniques and global optimization considering tailored cuts in MINLP formulation problems. D’Ambrosio et al (2015) presented a complete review of Mathematical Programming approaches in the optimization of WDN considering the notion of the network design and the network operation. Caballero and Ravagnani (2019) proposed an MINLP model considering unknown flow directions in the network loops and used global optimization techniques to solve the problem. Cassiolato et al. (2019) used Generalized Disjunctive Programming to reformulate an MINLP model, developed by Surco et al (2017), using a Big M approach. In Cassiolato et al. (2020), a hull reformulation was proposed in the problem and the model was solved, reducing the relaxation gap and improving the overall numerical performance. As an extension, Cassiolato et al. (2023) a Mixed Integer Non-Linear Programming (MINLP) model was developed to the synthesis of WDN considering the minimization of the WDN total cost, given by the sum of installation and operational costs. Cassiolato et al. (2022a) considered in the model unknown flow directions and SBB and BARON solvers were used to achieve the problem solution. Cassiolato et al. (2022b) considered installation and energy costs, with unknown flow directions.
Balekelayi and Tesfamariam (2017) reviewed three approaches to the WDN synthesis, the use of deterministic and non-gradient methods and real time optimization and compared some population-based algorithms to solve the problem for a case study.
As mentioned before, the majority of the published papers use non-deterministic approaches and consider fixed and known flow directions and a hydraulic simulator to solve the velocities and pressures calculation. In real WDN, variations in nominal values can occur and these variations can influence in the optimum network operation conditions, causing an unappropriated behavior. So, the evaluation of the uncertainties in distinct operation periods is a recent and important research field. The uncertainties in the demand nodes or in the tubes rugosity due to the use in long times are problems that need to be considered in the final stage of the WDN design. Branisavljevimc et al. (2009) used a Genetic Algorithm to find optimal solutions considering uncertainties in the water nodal demand by a Monte Carlo simulation. Sivakumar et al. (2015) studied the uncertainties in the tube rugosity and evaluated the tube flowrate and the different pressures between two adjacent nodes. Dongre and Gupta (2017) considered uncertainties in the water demand and in the tubes rugosity using fuzzy logic. Geranmehr et al. (2019) also used a fuzzy model to evaluate uncertainties in the nodes demand in the reservoir and in the rugosity coefficient. Calvo et al. (2018) considered non-correlated functions of log-normal probability distributions to the management of valves. Salcedo-Díaz et al. (2020) modeled the uncertainties in the nodes demand by a set of correlated scenarios generated by a Monte Carlo simulation, assuming a log-normal probability distribution.
In the present paper, the existence of uncertainties in the nodes demand in the synthesis of WDN is considered. The optimization model has an MINLP formulation and disjunctive programming is used to deal with integer variables relating the tubes diameter, cost and rugosity coefficient. The objective function is the total WDN installation cost and the constraint are mass balances in the demand nodes, energy balances in the network loops and velocity and pressure limits. No additional software is required for the hydraulic calculations and the model was coded in GAMS and a deterministic approach is used to solve the problem. The uncertainties in the nodes demand are thought as a set of correlated scenarios generated by a Monte Carlo simulation, assuming a log-normal probability distribution. Two case studies were used to test the applicability of the developed model.
Optimization model
The system is modeled as a set of reservoirs and node demands, described by their elevation and expected demand values, and a set of pipes with initial and final nodes, pipe length and chosen from a set of commercial diameters. To each diameter is associated a cost per length and a specific rugosity coefficient. Between the demand nodes it can exist closed loops. For each demand node there is a minimum pressure limit and the velocity in the tubes is between an upper and a lower bounded.
The design of the WDN is thought as an optimization problem with MINLP formulation, in which the objective function to be minimized is the network installation cost, subject to a set of algebraic constraints composed by a mass balance in each node, pressure difference between two adjacent nodes, considering the existence of loops, the equation for the volumetric flow rate in each pipe and Hazen-Williams equation for the pressure loss calculation, forming a nonlinear equations system. Complete the constraints set the inequalities for the velocity inside the tubes and pressure in the demand nodes limits. Disjunctive programming is used to determine the optimal WDN topology, with the attribution of binary variables and linear equations.
The uncertainties in the water demand nodes are modeled using a finite set of scenarios sampling from a probability distribution. The problem must be solved in three stages. In the first stage the problem is solved without considering the existence of uncertainties. The decisions taken in the first stage before considering the uncertainties are given for the design variables. In the second stage, the decisions taken after considering the uncertainties allow to calculate operational variables. At last, in the third stage, the decisions are given by the design and operational variables.
The indexes, sets, variables and parameters are described as:
Indexes | |
i, j | Demand node |
K | Available diameter |
S | Scenario |
Sets | |
\(\mathcal{D}\) | Available commercial diameters (k) |
\({\mathcal{E}}_{i,j}\) | There exist a pipe between node i and node j (i-j) |
\(\mathcal{N}\) | Demand nodes (i, j) |
\(\mathcal{S}\) | Scenarios (s) |
Parameters | |
\(CostD\left({D}_{k}\right)\) | Cost per length of pipe with diameter \({D}_{k}\) [$/m] |
\({d}_{i,s}\) | Water demand for node i in the scenario s [volume/time] |
\({D}_{k}\) | Available commercial diameter k [m] |
\({e}_{1}\) | Annual interest ratio [%] |
\({Ep}_{i,j}^{min}\) and \({Ep}_{i,j}^{max}\) | Minimum and maximum values for the pump energy in pipe i-j [m] |
\(FAI\) | Annualization factor for the installation cost [year-1] |
\({h}_{i}\) | Node i elevation [m] |
\({L}_{i,j}\) | Pipe i-j length [m] |
\({n}_{a}\) | Design life time [year] |
\({P}_{i}^{min}\) | Minimum pressure in node i [m] |
\(pro{b}_{s}\) | Probability of occurrence of scenario s [%] |
\({q}_{i,j}^{min}\) and \({q}_{i,j}^{max}\) | Minimum and maximum values for the volumetric flowrate in pipe i-j [m3/s] |
\({R}_{k}\) | Rugosity coefficient in pipe with diameter \({D}_{k}\) [non-dimensional] |
\({v}_{i,j}^{min}\) and \({v}_{i,j}^{max}\) | Minimum and maximum values for the velocity in pipe i-j [m/s] |
\(\alpha\) | Hazen-Williams numerical conversion factor [depends on the system being used] |
\(\beta\) e \(\gamma\) | Hazen-Williams equation coefficients [non-dimensional] |
\({\Delta }{P}_{i,j}^{min}\) and \({\Delta }{P}_{i,j}^{max}\) | Minimum and maximum values for the pressure loss in the pipe i-j [m] |
Boolean variables | |
\({W}_{i,j}^{1}\) | True, if water flows from node i to node j or false, on the contrary |
\({W}_{i,j}^{2}\) | True, if water flows from node j to node i or false, on the contrary |
\({Y}_{i,j,k}\) | True, if in the pipe i-j diameter \({D}_{k}\) is selected or false, on the contrary |
Binary variables | |
\({w}_{i,j}^{1}\) | 1, if water flows from node i to node j or 0, on the contrary |
\({w}_{i,j}^{2}\) | 1, if water flows from node j to node i or 0, on the contrary |
\({y}_{i,j,k}\) | 1, if in the pipe i-j diameter \({D}_{k}\) is selected or false, on the contrary |
Variables | |
\(C{p}_{i,j,s}\) | Pump in the pipe i-j annualized operational cost in scenario s [$/year] |
\({Cost}_{i,j}\) | Pipe i-j cost [$] |
\({Diam}_{i,j}\) | Pipe i-j diameter [m] |
\({E}_{i,j,s}^{pow}\) | Pump energy in pipe i-j in scenario s [kW] |
\(E{p}_{i,j,s}\) | Pipe i-j pump in scenario s [m] |
\({Ep}_{i,j,s}^{1}\) \(\left({Ep}_{i,j,s}^{2}\right)\) | Equal to \(E{p}_{i,j,s}\) if water flows from node i (j) to node j (i) in scenario s |
\(expTAC\) | Expected total annual cost [$/year] |
\({P}_{i,s}\) | Pressure in node i in scenario s [m] |
\({q}_{i,j,s}\) | Volumetric flowrate in pipe i-j in scenario s [m3/s] |
\({q}_{i,j,s}^{1}\) \(\left({q}_{i,j,s}^{2}\right)\) | Equal to \({q}_{i,j,s}\) if water flows from node i (j) to node j (i) in scenario s |
\({\stackrel{-}{q}}_{i,j,s}\) | Logarithm of \({q}_{i,j,s}\) in pipe i-j in scenario s |
\({Rug}_{i,j}\) | Rugosity coefficient in pipe i-j [nondimensional] |
\(TA{C}_{s}\) | Total annual cost in scenario s [$/year] |
\({v}_{i,j,s}\) | Water velocity in pipe i-j [m/s] |
\({v}_{i,j,s}^{1}\) \(\left({v}_{i,j,s}^{2}\right)\) | Equal to \({v}_{i,j,s}\) if water flows from node i (j) to node j (i) in scenario s |
\({\stackrel{-}{v}}_{i,j,s}\) | Logarithm of\({v}_{i,j,s}\) in pipe i-j in scenario s |
\({{\Delta }P}_{i,j,s}\) | Pressure loss in pipe i-j in scenario s [m] |
\({\Delta }{P}_{i,j,s}^{1}\) \(\left({\Delta }{P}_{i,j,s}^{2}\right)\) | Equal to \({{\Delta }P}_{i,j,s}\) if water flows from node i (j) to node j (i) in scenario s |
\({\Delta }{\stackrel{-}{P}}_{i,j,s}\) | Logarithm of\({{\Delta }P}_{i,j,s}\) in pipe i-j in scenario s |
The WDN is evaluated by its total annual cost (TAC), given by the annual installation cost plus the annual pump energy cost. For each scenario \(s\in \mathcal{S}\), a value for \(TA{C}_{s}\) is calculated and, to evaluate the performance of the WDN under uncertainties in a unique metric, the expected value for TAC is minimized, given by:
$$E\left[TAC\right]=\sum _{s\in \mathcal{S}}pro{b}_{s}\cdot TA{C}_{s}$$
1
In this equation \(pro{b}_{s}\) is the inverse of the number of generated scenarios, being used the same probability of occurrence for all scenarios.
The model constraints are the algebraic equations and inequalities that must be solved in each scenario, which constitute a nonlinear system.
Mass balance in each demand node:
$$\sum _{j\in {\mathcal{E}}_{j,i}}\left({q}_{j,i,s}^{1}-{q}_{j,i,s}^{2}\right)-\sum _{j\in {\mathcal{E}}_{i,j}}\left({q}_{i,j,s}^{1}-{q}_{i,j,s}^{2}\right)={d}_{i,s}, \forall i\in \mathcal{N} \text{a}\text{nd} s\in \mathcal{S}$$
2
Energy balance in the network loops:
$${P}_{i,s}+{h}_{i}+E{p}_{i,j,s}^{1}-E{p}_{i,j,s}^{2}={P}_{j,s}+{h}_{j}+{\Delta }{P}_{i,j,s}^{1}-{\Delta }{P}_{i,j,s}^{2}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}$$
3
Being \({P}_{i}^{min}\le {P}_{i,s}\), for all \(i\in \mathcal{N}\) and \(s\in \mathcal{S}\).
The choice of the pipe diameter depends on its cost, rugosity coefficient, volumetric flowrate and pressure loss, given by the Hazzen-Williams equation. Logarithms can be used to linearize the last two equations. The exclusive disjunction can be thought:
$$\begin{array}{c}\underset{\_}{\vee }\\ k\in \mathcal{D}\end{array} \left[\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}{Y}_{i,j,k}\\ {Diam}_{i,j}={D}_{k}\end{array}\end{array}\\ {Cost}_{i,j}={L}_{i,j} CostD\left({D}_{k}\right)\end{array}\\ {Rug}_{i,j}={R}_{k}\\ {\stackrel{-}{v}}_{i,j,s}={\stackrel{-}{q}}_{i,j,s}-\text{ln}\left(\frac{\pi }{4}{D}_{k}^{2}\right)\\ \varDelta {\stackrel{-}{P}}_{i,j,s}=\text{ln}\left(\alpha {L}_{i,j}\right)+\beta {\stackrel{-}{q}}_{i,j,s}-\text{ln}\left({R}_{k}^{\beta } {D}_{k}^{\gamma }\right)\end{array}\right], \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}$$
4
This disjunction can be written using a convex hull reformulation (Grossmann and Lee, 2003). The binary variable \({y}_{i,j,k}\) associated to the pipe i-j with diameter \({D}_{k}\), for all \(k\in \mathcal{D}\), is equal to 1 if in the pipe i-j the diameter \({D}_{k}\) is selected and 0, on the contrary. In this way:
\({Diam}_{i,j}=\sum _{k\in \mathcal{D}}{D}_{k} {y}_{i,j,k}, \forall i,j\in {\mathcal{E}}_{i,j}\) | (5) |
\({Cost}_{i,j}=\sum _{k\in \mathcal{D}}{L}_{i,j} CostD\left({D}_{k}\right) {y}_{i,j,k}, \forall i,j\in {\mathcal{E}}_{i,j}\) | (6) |
\({Rug}_{i,j}=\sum _{k\in \mathcal{D}}{R}_{k} {y}_{i,j,k}, \forall i,j\in {\mathcal{E}}_{i,j}\) | (7) |
\({\stackrel{-}{v}}_{i,j,s}={\stackrel{-}{q}}_{i,j,s}-\sum _{k\in \mathcal{D}}\text{ln}\left(\frac{\pi }{4}{D}_{k}^{2}\right) {y}_{i,j,k}, \forall i,j\in {\mathcal{E}}_{i,j} \text{a}\text{nd} s\in \mathcal{S}\) | (8) |
\({\Delta }{\stackrel{-}{P}}_{i,j,s}=\text{ln}\left(\alpha {L}_{i,j}\right)+\beta {\stackrel{-}{q}}_{i,j,s}-\sum _{k\in \mathcal{D}}\text{ln}\left({R}_{k}^{\beta } {D}_{k}^{\gamma }\right) {y}_{i,j,k}, \forall i,j\in {\mathcal{E}}_{i,j} \text{a}\text{nd} s\in \mathcal{S}\) | (9) |
\(\sum _{k\in \mathcal{D}}{y}_{i,j,k}=1, \forall i,j\in {\mathcal{E}}_{i,j}\) | (10) |
The original variables can be found by exponentiation:
\({e}^{{\stackrel{-}{v}}_{i,j,s}}={v}_{i,j,s}, \forall i,j\in {\mathcal{E}}_{i,j} \text{e} s\in \mathcal{S}\) | (11) |
\({e}^{{\stackrel{-}{q}}_{i,j,s}}={q}_{i,j,s}, \forall i,j\in {\mathcal{E}}_{i,j} \text{e} s\in \mathcal{S}\) | (12) |
\({e}^{{\Delta }{\stackrel{-}{P}}_{i,j,s}}={{\Delta }P}_{i,j,s}, \forall i,j\in {\mathcal{E}}_{i,j} \text{e} s\in \mathcal{S}\) | (13) |
The flow direction in each pipe is given by the exclusive disjunction:
$$\left[\begin{array}{c}\begin{array}{c}{W}_{i,j}^{1}\\ {v}_{i,j,s}={v}_{i,j,s}^{1}\\ {q}_{i,j,s}={q}_{i,j,s}^{1}\end{array}\\ {{\Delta }P}_{i,j,s}={{\Delta }P}_{i,j,s}^{1}\\ {Ep}_{i,j,s}={Ep}_{i,j,s}^{1}\\ {v}_{i,j}^{min}\le {v}_{i,j,s}\le {v}_{i,j}^{max}\\ {q}_{i,j}^{min}\le {q}_{i,j,s}\le {q}_{i,j}^{max}\\ {{\Delta }P}_{i,j}^{min}\le {{\Delta }P}_{i,j,s}\le {{\Delta }P}_{i,j}^{max}\\ {Ep}_{i,j}^{min}\le {Ep}_{i,j,s}\le {Ep}_{i,j}^{max}\end{array}\right]\underset{\_}{\vee }\left[\begin{array}{c}\begin{array}{c}{W}_{i,j}^{2}\\ {v}_{i,j,s}={v}_{i,j,s}^{2}\\ {q}_{i,j,s}={q}_{i,j,s}^{2}\end{array}\\ {{\Delta }P}_{i,j,s}={{\Delta }P}_{i,j,s}^{2}\\ {Ep}_{i,j,s}={Ep}_{i,j,s}^{2}\\ {v}_{i,j}^{min}\le {v}_{i,j,s}\le {v}_{i,j}^{max}\\ {q}_{i,j}^{min}\le {q}_{i,j,s}\le {q}_{i,j}^{max}\\ {{\Delta }P}_{i,j}^{min}\le {{\Delta }P}_{i,j,s}\le {{\Delta }P}_{i,j}^{max}\\ {Ep}_{i,j}^{min}\le {Ep}_{i,j,s}\le {Ep}_{i,j}^{max}\end{array}\right], \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}$$
14
This disjunction can be written using a convex hull reformulation. The binary variable \({w}_{i,j}^{1}\) is equal to 1 if water flows from node i to node j and 0, on the contrary, and the binary variable \({w}_{i,j}^{2}\) is equal to 1 if water flows from node j to node i e 0, on the contrary. In this way:
\({v}_{i,j,s}={v}_{i,j,s}^{1}+{v}_{i,j,s}^{2}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) | (15) |
\({v}_{i,j}^{min} {w}_{i,j}^{1}\le {v}_{i,j,s}^{1}\le {v}_{i,j}^{max} {w}_{i,j}^{1}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) | (16) |
\({v}_{i,j}^{min} {w}_{i,j}^{2}\le {v}_{i,j,s}^{2}\le {v}_{i,j}^{max} {w}_{i,j}^{2}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) | (17) |
\({q}_{i,j,s}={q}_{i,j,s}^{1}+{q}_{i,j,s}^{2}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) | (18) |
\({q}_{i,j}^{min} {w}_{i,j}^{1}\le {q}_{i,j,s}^{1}\le {q}_{i,j}^{max} {w}_{i,j}^{1}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) | (19) |
\({q}_{i,j}^{min} {w}_{i,j}^{2}\le {q}_{i,j,s}^{2}\le {q}_{i,j}^{max} {w}_{i,j}^{2}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) | (20) |
\({{\Delta }P}_{i,j,s}={{\Delta }P}_{i,j,s}^{1}+{{\Delta }P}_{i,j,s}^{2}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) | (21) |
\({{\Delta }P}_{i,j}^{min} {w}_{i,j}^{1}\le {{\Delta }P}_{i,j,s}^{1}\le {{\Delta }P}_{i,j}^{max} {w}_{i,j}^{1}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) | (22) |
\({{\Delta }P}_{i,j}^{min} {w}_{i,j}^{2}\le {{\Delta }P}_{i,j,s}^{2}\le {{\Delta }P}_{i,j}^{max} {w}_{i,j}^{2}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) | (23) |
\({Ep}_{i,j,s}={Ep}_{i,j,s}^{1}+{Ep}_{i,j,s}^{2}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) | (24) |
\({Ep}_{i,j}^{min} {w}_{i,j}^{1}\le {Ep}_{i,j,s}^{1}\le {Ep}_{i,j}^{max} {w}_{i,j}^{1}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) | (25) |
\({Ep}_{i,j}^{min} {w}_{i,j}^{2}\le {Ep}_{i,j,s}^{2}\le {Ep}_{i,j}^{max} {w}_{i,j}^{2}, \forall i,j\in {\mathcal{E}}_{i,j} \text{e}\text{and} s\in \mathcal{S}\) | (26) |
\({w}_{i,j}^{1}+{w}_{i,j}^{2}=1, \forall i,j\in {\mathcal{E}}_{i,j}\) | (27) |
The uncertainties in the demand nodes imply in the use of possible pumps which could be necessary to satisfy the water demands in the network, depending on the scenario being evaluated. The pump energy located in the pipe i-j in scenario s and the pump annualized operation cost are given by:
$${E}_{i,j,s}^{pow}=\frac{9.81}{0.82}{q}_{i,j,s} E{p}_{i,j,s}, \forall i,j\in {\mathcal{E}}_{i,j} \text{a}\text{nd} s\in \mathcal{S}$$
28
$$C{p}_{i,j,s}=0.24\cdot \text{8,000} {E}_{i,j,s}^{pow}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}$$
29
The pump efficiency is 0.82, the energy cost per kWh is 0.24 and the number of total hours of pumping is 8,000.
The total annual cost in each scenario is composed by the pipe installation annual cost and by the pumps annual operation cost, given by:
$$TA{C}_{s}=\sum _{i,j\in {\mathcal{E}}_{i,j}}\left(FAI\cdot Cos{t}_{i,j}+C{p}_{i,j,s}\right), \forall s\in \mathcal{S}$$
30
The annualization factor of the installation cost for the extension of the WDN design in \({n}_{a}\) years, subject the annual interest rate \({e}_{1}\), is:
$$FAI=\frac{{e}_{1}{\left(1+{e}_{1}\right)}^{{n}_{a}}}{{\left(1+{e}_{1}\right)}^{{n}_{a}}-1}$$
31
As a result, the complete optimization model for the WDN synthesis considering uncertainties in the demand nodes and unknown flow directions is:
min | \(\sum _{s\in \mathcal{S}}pro{b}_{s}\cdot TA{C}_{s}\) |
s. a | \(\sum _{j\in {\mathcal{E}}_{j,i}}\left({q}_{j,i,s}^{1}-{q}_{j,i,s}^{2}\right)-\sum _{j\in {\mathcal{E}}_{i,j}}\left({q}_{i,j,s}^{1}-{q}_{i,j,s}^{2}\right)={d}_{i,s}, \forall i\in \mathcal{N} \text{and} s\in \mathcal{S}\) |
| \({P}_{i,s}+{h}_{i}+E{p}_{i,j,s}^{1}-E{p}_{i,j,s}^{2}={P}_{j,s}+{h}_{j}+{\Delta }{P}_{i,j,s}^{1}-{\Delta }{P}_{i,j,s}^{2}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \({x}_{i,j}=\sum _{k\in \mathcal{D}}{D}_{k} {y}_{i,j,k}, \forall i,j\in {\mathcal{E}}_{i,j}\) |
| \({Diam}_{i,j}=\sum _{k\in \mathcal{D}}{D}_{k} {y}_{i,j,k}, \forall i,j\in {\mathcal{E}}_{i,j}\) |
| \({Cost}_{i,j}=\sum _{k\in \mathcal{D}}{L}_{i,j} CostD\left({D}_{k}\right) {y}_{i,j,k}, \forall i,j\in {\mathcal{E}}_{i,j}\) |
| \({Rug}_{i,j}=\sum _{k\in \mathcal{D}}{R}_{k} {y}_{i,j,k}, \forall i,j\in {\mathcal{E}}_{i,j}\) |
| \({\stackrel{-}{v}}_{i,j,s}={\stackrel{-}{q}}_{i,j,s}-\sum _{k\in \mathcal{D}}\text{ln}\left(\frac{\pi }{4}{D}_{k}^{2}\right) {y}_{i,j,k}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \({\Delta }{\stackrel{-}{P}}_{i,j,s}=\text{ln}\left(\alpha {L}_{i,j}\right)+\beta {\stackrel{-}{q}}_{i,j,s}-\sum _{k\in \mathcal{D}}\text{ln}\left({R}_{k}^{\beta } {D}_{k}^{\gamma }\right) {y}_{i,j,k}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \(\sum _{k\in \mathcal{D}}{y}_{i,j,k}=1, \forall i,j\in {\mathcal{E}}_{i,j}\) |
| \({e}^{{\stackrel{-}{v}}_{i,j,s}}={v}_{i,j,s}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \({e}^{{\stackrel{-}{q}}_{i,j,s}}={q}_{i,j,s}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \({e}^{{\Delta }{\stackrel{-}{P}}_{i,j,s}}={{\Delta }P}_{i,j,s}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \({P}_{i}^{min}\le {P}_{i,s}, \forall i\in \mathcal{N} \text{and} s\in \mathcal{S}\) |
| \({E}_{i,j,s}^{pow}=\frac{9.81}{0.82}{q}_{i,j,s} E{p}_{i,j,s}, \forall i,j\in {\mathcal{E}}_{i,j} \text{an}\text{d} s\in \mathcal{S}\) |
| \(C{p}_{i,j,s}=0.24\cdot \text{8,000} {E}_{i,j,s}^{pow}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \(TA{C}_{s}=\sum _{i,j\in {\mathcal{E}}_{i,j}}\left(FAI\cdot Cos{t}_{i,j}+C{p}_{i,j,s}\right), \forall s\in \mathcal{S}\) |
| \({v}_{i,j,s}={v}_{i,j,s}^{1}+{v}_{i,j,s}^{2}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \({v}_{i,j}^{min} {w}_{i,j}^{1}\le {v}_{i,j,s}^{1}\le {v}_{i,j}^{max} {w}_{i,j}^{1}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \({v}_{i,j}^{min} {w}_{i,j}^{2}\le {v}_{i,j,s}^{2}\le {v}_{i,j}^{max} {w}_{i,j}^{2}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \({q}_{i,j,s}={q}_{i,j,s}^{1}+{q}_{i,j,s}^{2}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \({q}_{i,j}^{min} {w}_{i,j}^{1}\le {q}_{i,j,s}^{1}\le {q}_{i,j}^{max} {w}_{i,j}^{1}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \({q}_{i,j}^{min} {w}_{i,j}^{2}\le {q}_{i,j,s}^{2}\le {q}_{i,j}^{max} {w}_{i,j}^{2}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \({{\Delta }P}_{i,j,s}={{\Delta }P}_{i,j,s}^{1}+{{\Delta }P}_{i,j,s}^{2}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \({{\Delta }P}_{i,j}^{min} {w}_{i,j}^{1}\le {{\Delta }P}_{i,j,s}^{1}\le {{\Delta }P}_{i,j}^{max} {w}_{i,j}^{1}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \({{\Delta }P}_{i,j}^{min} {w}_{i,j}^{2}\le {{\Delta }P}_{i,j,s}^{2}\le {{\Delta }P}_{i,j}^{max} {w}_{i,j}^{2}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \({Ep}_{i,j,s}={Ep}_{i,j,s}^{1}+{Ep}_{i,j,s}^{2}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \({Ep}_{i,j}^{min} {w}_{i,j}^{1}\le {Ep}_{i,j,s}^{1}\le {Ep}_{i,j}^{max} {w}_{i,j}^{1}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \({Ep}_{i,j}^{min} {w}_{i,j}^{2}\le {Ep}_{i,j,s}^{2}\le {Ep}_{i,j}^{max} {w}_{i,j}^{2}, \forall i,j\in {\mathcal{E}}_{i,j} \text{and} s\in \mathcal{S}\) |
| \({w}_{i,j}^{1}+{w}_{i,j}^{2}=1, \forall i,j\in {\mathcal{E}}_{i,j}\) |
The three-stages procedure for considering uncertainties in WDN nodes demand can be described as:
i) In the first step the expected TAC is minimized with a probability of occurrence of 100% for the original scenario. In this step the optimum design is achieved considering a unique nominal value for each uncertain parameter. In this case, the annual pumping cost is zero.
ii) In the second step the expected TAC with probability of occurrence in all scenarios is minimized but with fixed values for the discrete variables, given by the solution obtained in the first step. In this case, the annual installation cost is fixed. The solution found in this stage is called deterministic solution.
iii) In the third stage the TAC is minimized with probability of occurrence in all scenarios with no fixed values for the variables. In this way, the optimal design considering uncertainties in the nodes demand is achieved. The solution found in this stage is called stochastic solution.