## Abstract

Water distribution networks (WDNs) are an important part of water distribution systems and are responsible for water transportation from the reservoirs to the demand nodes at adequate pressure and velocity. In the present paper, the synthesis of WDN is treated as an optimization problem with a mixed integer nonlinear programming formulation. The objective function to be minimized is the total network cost, considering installation and energy costs, with unknown flow directions, which is the novelty in the paper. Disjunctive programming and linearization techniques are used in the model formulation to avoid nonlinear and nonconvex problems. Two case studies are used to test the model's applicability. Results show that operational costs can represent a significant part of the total cost in sustainable networks. In the first case study, the total cost was better than the literature results (US$ 2,272,538.85 vs. US$ 2,272,387.49) and the operational costs represent ¼ of the total WDN costs. In the second case study, the operation cost corresponds to almost 2/3 of the total WDN cost. These results show the importance of considering operational costs in the WDN design. Also, the consideration of unknown flow directions can lead to better results for the network topology.

## HIGHLIGHTS

Water distribution systems design is treated as an optimization problem.

The optimization problem has a mixed integer nonlinear programming formulation.

Installation and energy costs are considered in the objective function.

Flow directions in node demand loops are considered optimization variables.

Operational costs can represent a significant part of the total water distribution cost.

## INTRODUCTION

Water distribution networks (WDNs) are important systems to serve potable water to demand nodes with adequate pressure and velocity. These networks are composed by one or more reservoirs, consumption nodes, and pipes linking the nodes. Normally, the pipes can form loops in the demand nodes and the water movement can be provided by gravity or by a pumping system when the system elevation is similar to the reservoir elevation.

The design of WDN can be formulated as an optimization problem. The majority of the published papers in the literature consider the installation cost, calculated from a discrete set of available commercial diameters as the objective function to be minimized. The tube length and the flow directions are considered known. The problem constraints are algebraic nonlinear equations and inequalities systems, with possible different solutions.

Different approaches were used in the literature to develop and solve the optimization problems for the synthesis of WDN, involving continuous and discrete variables (Mala-Jetmarova *et al.* 2018).

When loops are present in the node's demand, the nonlinear behaviour of the hydraulic equations can be an additional problem. In the majority of the published papers in the literature, additional software is used to solve the complex system of nonlinear hydraulic equations.

In the present paper, a mixed integer nonlinear programming (MINLP) optimization model is proposed using disjunctive programming and linearization techniques, and a deterministic approach is used to solve it. The objective function to be minimized is the total WDN cost, composed by the installation cost, depending on the available commercial diameters, and by the operational cost, depending on the manometric high of the reservoir and the pumping system. The combination of installation and operational costs in the objective function as well as the consideration of unknown flow directions in the loops are the novelty in the paper. No hydraulic simulators are necessary to solve the problem because the pressure and velocity equations are considered constraints in the model, which is solved in GAMS, using the BARON global optimization solver. Two case studies are used to test the model's applicability.

In the last decades, important stochastic and deterministic approaches have been proposed to solve the WDN design optimization problem. Different models involving linear programming (LP), nonlinear programming (NLP), mixed integer linear programming (MILP), and MINLP formulations were proposed. Obviously, the most realistic and representative models are those that have MINLP formulations and the models are normally nonconvex. Because of this, global optima solutions are not ensured.

Using MINLP models, Bragalli *et al.* (2008, 2012) used a nonconvex continuous relaxation to solve the problem in different solvers in AMPL. D'Ambrosio *et al.* (2015) used mathematical programming to solve the MINLP model by analyzing the modelling aspects in each case, for the dynamics of water in tubes and used spatial branches and linear relaxations.

Cassiolato *et al.* (2019) proposed an MINLP model for the synthesis of WDN. Generalized disjunctive programming (GDP) was used to treat the discrete variables using binary variables. A Big-M reformulation was used and the solver SBB in GAMS was used to solve the problem. The same model was used in Cassiolato *et al.* (2020) with a convex hull reformulation, and GAMS and SBB solvers were used.

Caballero & Ravagnani (2019) presented an MINLP model considering unknown flow directions and a convex hull reformulation was used. The authors also presented an important analysis of the Hazen–Williams equation parameters sensibility.

However, the majority of the published papers in this important field use nondeterministic approaches to solve the problem. Genetic algorithms (GA) were used by Bi *et al.* (2015) and Reca *et al.* (2017). Artificial immune systems (AIS) were used by Eryiğit (2015). Geem (2009) used harmony search (HS). Evolutionary algorithms (EA) were used by Avila-Melgar *et al.* (2016) and Palod *et al.* (2020). Shende & Chau (2019) proposed the simple benchmarking algorithm (SBA) for WDN optimization. With the same objective, an article based on whale optimization algorithm (WOA) was published by Ezzeldin & Djebedjian (2020). Ezzeldin *et al.* (2014), Rao *et al.* (2017), and Surco *et al.* (2017, 2021) used particle swarm optimization (PSO) to solve the optimization problems.

Balekelayi & Tesfamariam (2017) and Mala-Jetmarova *et al.* (2018) presented important reviews considering the distinct approaches used to synthesize WDN and a comparison among them. In the majority of the cited papers, the direction flows are considered known in the network and hydraulic simulators are used to solve the pressure and velocity equations. The most used hydraulic simulator is EPANET (Rossman 2000).

It is given for the WDN of one or more reservoirs and water demand nodes with proper elevations, where loops can exist. It is also given a set of available discrete diameters and pipes linking the demand nodes, with distinct lengths. Each diameter is associated with a cost per length and a specific roughness coefficient. There are upper and lower bounds for the water velocity inside the tubes and the pressure in the node's demand must satisfy a minimum limit. The pumping station considers the hydraulic pumps and the reservoir manometric height for the network water supply.

The WDN design can be thought of as an optimization problem formulated with an MINLP formulation, aiming to minimize the network total cost, subject to a set of algebraic nonlinear equality and inequality constraints, given by the mass balance in each node, the pressure difference between two adjacent nodes, considering the existence of loops; the velocities and pressure calculations, given by the Darcy–Weisbach equation or, in some particular cases, with the Hazen–Williams equation.

## METHODOLOGY

Disjunctions are used in the model development to define the flow directions and the choice of the tube diameter, roughness, and cost, by using binary variables. The manometric reservoir height is calculated by energy balances in the demand nodes.

The following model sets, indexes, parameters, and variables are defined:

**Indexes:**Demand nodes

Available diameter

**Sets:**Available commercial diameters ()

The existing pipes between nodes

*i*and*j*()Demand nodes ()

**Parameters:**Hazen–Williams roughness coefficient for pipe (nondimensional)

Cost per tube length for diameter ($/m)

Node

*i*water demand (L/s)Available commercial diameter

*k*(m)Interest rate (%)

Annual rate of increase in the electric energy (%)

Electricity cost ($/kWh)

Cost to pressurize water due to the elevation ($/m)

- and
Pumping energy minimum and maximum values in pipe (m)

Operational cost actualization factor (nondimensional)

- and
Minimum and maximum values for the friction factor in pipe (adm)

Node

*i*elevation (m)Pipe length (m)

Design lifetime (year)

Pumping hours per year (h/year)

Minimum pressure in node

*i*(m)Pump total volumetric flowrate (m

^{3}/s)- and
Minimum and maximum volumetric flowrate in pipe (m

^{3}/s) Roughness coefficient of pipe with diameter

- and
Minimum and maximum velocity limits in pipe (m/s)

Altimetric height (m)

Hazen–Williams conversion factor

- and
Hazen–Williams parameters (nondimensional)

Darcy–Weisbach roughness coefficient in pipe (m)

Pumps system efficiency (%)

Kinematic viscosity (m

^{2}/s)- and
Minimum and maximum pressure loss in pipe (m)

**Variables:**Energy cost ($)

Installation cost ($)

Hydraulic head in pipe (m)

- and
It is equal to if water goes from the node

*i*() to the*j*() Darcy–Weisbach friction factor in pipe (nondimensional)

- and
Refer to , if water flows from node

*i*to node*j*, and from node*j*to node Logarithm of in pipe

Manometric head of the pumping system (m)

Node

*i*pressure (m)Volumetric flowrate in pipe (m

^{3}/s)- and
Refer to if water flows from node

*i*to node*j*and from node*j*to node Logarithm of in pipe

Reynolds number in pipe (nondimensional)

- and
Refer to if water flows from node

*i*to node*j*and from node*j*to node Total WDN cost ($)

Water velocity in pipe (m/s)

- and
Refer to if water flows from node

*i*to node*j*and from node*j*to node Logarithm of natural in pipe

- and
Refer to Boolean variable and binary variable, where equal True and (1) if water flows from node

*i*to node*j*, or false and (0) if flow from node*j*to node - and
Refer to Boolean variable and binary variable, where equal True and (1) if water flows from node

*j*to node*i*, or false and (0) if flow from node*i*to node Pipe diameter (m)

True (1) if diameter is selected in pipe or false (0), on the contrary

Pipe cost ($)

Pipe roughness coefficient

Pressure loss in pipe (m)

- and
Refer to , if water flows from node

*i*to node*j*, and from node*j*to node Logarithm of in pipe

The problem constraints are algebraic equations, from the mass balance in the demand nodes, energy balance in the node loops, hydraulic equations for the pressure loss and velocity calculation, disjunctions to decide the flow directions, disjunctions to decide the pipe diameter, roughness coefficient and cost, and pressure and velocity limits. The problem has an MINLP formulation and is nonconvex.

The volumetric flowrate is if in the pipe water flows from node *i* to node *j* and is if water flows from node *j* to node *i*. The pumping energy of a hydraulic pump in the pipe is if water flows from node *i* to node *j*, or is if water flows from node *j* to node *i*.

The pressure loss in pipe is if water flows from node *i* to node *j*, or is if water flows from node *j* to node *i*. In each node, a minimum pressure exists and in the reservoir the pressure is considered to be zero.

*f*is the friction factor in the pipe and depends on the Reynolds number (Surco

_{i,j}*et al.*2017):

*et al.*2017):

New variables must be defined for all :

Now, assumes the value of if water flows from node *i* to node *j*. On the contrary, assumes the value of if water flows from node *j* to node *i*. For the variables , , , , , , , and , it is necessary to define upper and lower bounds. The lower bounds are defined as , , , and , for all . Analogously, , , , and are the upper limits.

*i*to node

*j*and false, on the contrary, and is the Boolean variable which is true if water flows from node

*j*to node

*i*and false, on the contrary. If the Darcy–Weisbach equation is used, the exclusive disjunction which defines the flow direction in each pipe is given by:where and are equal to and , respectively, if water flows from node

*i*to node

*j*, and and are equal to and , respectively, if water flows from node

*j*to node

*i*. Moreover, and are the inferior limits and and are the superior limits, for all .

*i*to node

*j*and equal to 0, on the contrary, and the binary variable is equal to 1 if water flows from node

*j*to node

*i*and is equal to 0, on the contrary. The reformulated equations are:

*et al.*2021):

Finally, the optimization problem for the synthesis of WDN considering installation and energy costs, with unknown flow directions, is given by:

Minimize Equation (1)

The same optimization problem can be redefined if the Hazen–Williams equation is used. In this case, the Darcy–Weisbach equation must be replaced by the Hazen–Williams equation and proper equations and inequalities.

### Model application

To test the applicability of the developed model, two examples from the literature were used. In both cases, the pipes diameter, the reservoir manometric head, and the flow directions were the optimization variables to minimize the WDN total cost, and the problems were solved using the global solver BARON in GAMS.

#### Case study 1

*et al.*(2009). Figure 1 presents the network topology. There are one reservoir, eight pipes linking six demand nodes, two loops, and a hydraulic pump with a water catchment level of 30 m, responsible for pumping water to the reservoir. Table 1 presents the nodes and pipe elevation, demand, and pipe lengths. Table 2 presents 10 commercial available diameters, costs, and the Hazen–Williams roughness coefficients. The costs were provided by Gomes

*et al.*(2009) in R$ and converted to USD (USD 1 = R$ 2). For comparison effects, we used the same cost values as used in the paper. Lower and upper bounds for the water velocity are 0.2 and 3 m/s and the minimum required pressure in the nodes demand is 25 m. The pump efficiency is considered 75% in the pump stations and the number of operation hours is 7,300 h/year. The energy cost is 0.1 US$/kWh, the annual interest rate is 12%, and the annual increase rate in the energy is 6%, for a lifetime of 20 years. The Hazen–Williams equation is used and two distinct values for the parameter

*α*were used:

Node/Pipe . | Elevation (m) . | Demand (L/s) . | Pipe length (m) . |
---|---|---|---|

1 | 6.0 | 0.00 | 2,540 |

2 | 5.5 | 47.78 | 1,230 |

3 | 5.5 | 80.32 | 1,430 |

4 | 6.0 | 208.60 | 1,300 |

5 | 4.5 | 43.44 | 1,490 |

6 | 4.0 | 40.29 | 1,210 |

7 | 1,460 | ||

8 | 1,190 |

Node/Pipe . | Elevation (m) . | Demand (L/s) . | Pipe length (m) . |
---|---|---|---|

1 | 6.0 | 0.00 | 2,540 |

2 | 5.5 | 47.78 | 1,230 |

3 | 5.5 | 80.32 | 1,430 |

4 | 6.0 | 208.60 | 1,300 |

5 | 4.5 | 43.44 | 1,490 |

6 | 4.0 | 40.29 | 1,210 |

7 | 1,460 | ||

8 | 1,190 |

Diameter (m) . | Cost (US$/m) . | Roughness coefficient . | Diameter (m) . | Cost (US$/m) . | Roughness coefficient . |
---|---|---|---|---|---|

0.1084 | 23.55 | 145 | 0.3662 | 158.93 | 130 |

0.1564 | 31.90 | 145 | 0.4164 | 187.50 | 130 |

0.2042 | 43.81 | 145 | 0.4666 | 218.12 | 130 |

0.2520 | 59.30 | 145 | 0.5180 | 257.80 | 130 |

0.2998 | 76.12 | 145 | 0.6196 | 320.15 | 130 |

Diameter (m) . | Cost (US$/m) . | Roughness coefficient . | Diameter (m) . | Cost (US$/m) . | Roughness coefficient . |
---|---|---|---|---|---|

0.1084 | 23.55 | 145 | 0.3662 | 158.93 | 130 |

0.1564 | 31.90 | 145 | 0.4164 | 187.50 | 130 |

0.2042 | 43.81 | 145 | 0.4666 | 218.12 | 130 |

0.2520 | 59.30 | 145 | 0.5180 | 257.80 | 130 |

0.2998 | 76.12 | 145 | 0.6196 | 320.15 | 130 |

The problem was solved using the global optimizer solver BARON, in GAMS. Table 3 presents the velocities in the pipes, the pressure in the demand nodes, and flow directions. As can be seen, these values are the same when using or However, costs are different, being US$ 2,272,538.85, when , and US$ 2,272,387.49, when . These values correspond to the global optima in the considered situations. In both cases, operation costs correspond to approximately ¼ of the total WDN costs. Table 4 presents a comparison with the literature results. Different approaches were used in the works of Gomes *et al.* (2009), who used an iterative approach and , _{,} and , and Surco *et al.* (2021), who used PSO to solve the problem with the aid of hydraulic simulators and , and the same values for and .

Pipe/Node . | Velocity (m/s) . | Flow direction . | Pressure (m) . |
---|---|---|---|

1 | 1.39 | R-1 | 30.84 |

2 | 1.13 | 1-2 | 27.10 |

3 | 0.64 | 2-3 | 25.00 |

4 | 0.69 | 4-3 | 26.30 |

5 | 1.33 | 1-4 | 26.62 |

6 | 1.20 | 1-5 | 25.40 |

7 | 0.50 | 5-6 | |

8 | 0.73 | 4-6 |

Pipe/Node . | Velocity (m/s) . | Flow direction . | Pressure (m) . |
---|---|---|---|

1 | 1.39 | R-1 | 30.84 |

2 | 1.13 | 1-2 | 27.10 |

3 | 0.64 | 2-3 | 25.00 |

4 | 0.69 | 4-3 | 26.30 |

5 | 1.33 | 1-4 | 26.62 |

6 | 1.20 | 1-5 | 25.40 |

7 | 0.50 | 5-6 | |

8 | 0.73 | 4-6 |

Pipe . | Gomes et al. (2009)
. | Present paper Diameters . | Surco et al. (2021)
. | Present paper . |
---|---|---|---|---|

1 | 0.6196 | 0.6196 | 0.6196 | 0.6196 |

2 | 0.2998 | 0.2998 | 0.2998 | 0.2998 |

3 | 0.2998 | 0.2520 | 0.2520 | 0.2520 |

4 | 0.2042 | 0.2998 | 0.2998 | 0.2998 |

5 | 0.5180 | 0.5180 | 0.5180 | 0.5180 |

6 | 0.2520 | 0.2520 | 0.2520 | 0.2520 |

7 | 0.2042 | 0.2042 | 0.2042 | 0.2042 |

8 | 0.1564 | 0.2042 | 0.2042 | 0.2042 |

(m) | 15.79 | 13.658 | 13.655 | 13.655 |

(US$) | 1,630,405.75 | 1,662,535.10 | 1,662,535.10 | 1,662,535.10 |

(US$) | 705,244.20 | 610,003.75 | 609,796.65 | 609,852.39 |

(US$) | 2,335,649.95 | 2,272,538.85 | 2,272,331.75 | 2,272,387.49 |

Pipe . | Gomes et al. (2009)
. | Present paper Diameters . | Surco et al. (2021)
. | Present paper . |
---|---|---|---|---|

1 | 0.6196 | 0.6196 | 0.6196 | 0.6196 |

2 | 0.2998 | 0.2998 | 0.2998 | 0.2998 |

3 | 0.2998 | 0.2520 | 0.2520 | 0.2520 |

4 | 0.2042 | 0.2998 | 0.2998 | 0.2998 |

5 | 0.5180 | 0.5180 | 0.5180 | 0.5180 |

6 | 0.2520 | 0.2520 | 0.2520 | 0.2520 |

7 | 0.2042 | 0.2042 | 0.2042 | 0.2042 |

8 | 0.1564 | 0.2042 | 0.2042 | 0.2042 |

(m) | 15.79 | 13.658 | 13.655 | 13.655 |

(US$) | 1,630,405.75 | 1,662,535.10 | 1,662,535.10 | 1,662,535.10 |

(US$) | 705,244.20 | 610,003.75 | 609,796.65 | 609,852.39 |

(US$) | 2,335,649.95 | 2,272,538.85 | 2,272,331.75 | 2,272,387.49 |

When , there is a difference in the total cost between the present paper and Surco *et al.* (2021) results, due to the operation cost. Although the solutions are the same, this difference is caused due to the parameters associated with the pumping system. In the present paper, no numerical approximations were used in the pumping calculation. Besides, the calculated installation costs were the same for distinct values of *α*. The little difference in the operation costs caused distinct values for the manometric head of the reservoir.

In this case study, the operation costs correspond to approximately ¼ of the total network cost. Considering the case with is possible to see how the operational costs in percentage of the total cost for the 20 years used as the lifetime for the pumping system in the network. These results are presented in Table 5. It can be noted that in the first year the operation costs correspond to 2.86% of the total costs and in the 20th year, this percentage corresponds to 26.84%.

Year . | (US$) . | Percentage (%) . | Year . | (US$) . | Percentage (%) . |
---|---|---|---|---|---|

1 | 48,942.86 | 2.86 | 11 | 415,034.38 | 19.98 |

2 | 95,263.78 | 5.42 | 12 | 441,743.25 | 20.99 |

3 | 139,103.23 | 7.72 | 13 | 467,021.30 | 21.93 |

4 | 180,594.13 | 9.80 | 14 | 490,945.16 | 22.80 |

5 | 219,862.30 | 11.68 | 15 | 513,587.39 | 23.60 |

6 | 257,026.83 | 13.39 | 16 | 535,016.64 | 24.35 |

7 | 292,200.39 | 14.95 | 17 | 555,297.89 | 25.04 |

8 | 325,489.66 | 16.37 | 18 | 574,492.65 | 25.68 |

9 | 356,995.58 | 17.68 | 19 | 592,659.12 | 26.28 |

10 | 386,813.68 | 18.87 | 20 | 609,852.39 | 26.84 |

Year . | (US$) . | Percentage (%) . | Year . | (US$) . | Percentage (%) . |
---|---|---|---|---|---|

1 | 48,942.86 | 2.86 | 11 | 415,034.38 | 19.98 |

2 | 95,263.78 | 5.42 | 12 | 441,743.25 | 20.99 |

3 | 139,103.23 | 7.72 | 13 | 467,021.30 | 21.93 |

4 | 180,594.13 | 9.80 | 14 | 490,945.16 | 22.80 |

5 | 219,862.30 | 11.68 | 15 | 513,587.39 | 23.60 |

6 | 257,026.83 | 13.39 | 16 | 535,016.64 | 24.35 |

7 | 292,200.39 | 14.95 | 17 | 555,297.89 | 25.04 |

8 | 325,489.66 | 16.37 | 18 | 574,492.65 | 25.68 |

9 | 356,995.58 | 17.68 | 19 | 592,659.12 | 26.28 |

10 | 386,813.68 | 18.87 | 20 | 609,852.39 | 26.84 |

#### Case study 2

^{2}/s. The pump efficiency is 75% and 7,300 h/year is considered as the operation time. The energy cost is 0.134 $/kWh and the annual interest rate is 10%. The lifetime considered is 25 years and the annual increase in the energy is 6%.

Node/Pipe . | Elevation (m) . | Demand (L/s) . | Pipe length (m) . |
---|---|---|---|

1 | 220.5 | 5.05 | 324 |

2 | 215.6 | 1.91 | 124 |

3 | 210.4 | 3.81 | 184 |

4 | 210.5 | 1.40 | 206 |

5 | 209.5 | 4.35 | 103 |

6 | 213.2 | 3.51 | 202 |

7 | 218.5 | 3.44 | 134 |

8 | 230.7 | 2.48 | 227 |

9 | 211.5 | 3.06 | 167 |

10 | 213.5 | 1.85 | 166 |

11 | 205.5 | 2.86 | 152 |

12 | 208.8 | 6.11 | 168 |

13 | 215.5 | 5.09 | 177 |

14 | 212.6 | 4.06 | 225 |

15 | 207.5 | 8.05 | 254 |

16 | 219.4 | 4.26 | 263 |

17 | 220.5 | 1.21 | 133 |

18 | 321 | ||

19 | 105 | ||

20 | 169 |

Node/Pipe . | Elevation (m) . | Demand (L/s) . | Pipe length (m) . |
---|---|---|---|

1 | 220.5 | 5.05 | 324 |

2 | 215.6 | 1.91 | 124 |

3 | 210.4 | 3.81 | 184 |

4 | 210.5 | 1.40 | 206 |

5 | 209.5 | 4.35 | 103 |

6 | 213.2 | 3.51 | 202 |

7 | 218.5 | 3.44 | 134 |

8 | 230.7 | 2.48 | 227 |

9 | 211.5 | 3.06 | 167 |

10 | 213.5 | 1.85 | 166 |

11 | 205.5 | 2.86 | 152 |

12 | 208.8 | 6.11 | 168 |

13 | 215.5 | 5.09 | 177 |

14 | 212.6 | 4.06 | 225 |

15 | 207.5 | 8.05 | 254 |

16 | 219.4 | 4.26 | 263 |

17 | 220.5 | 1.21 | 133 |

18 | 321 | ||

19 | 105 | ||

20 | 169 |

Diameter (m) . | Cost ($/m) . | Diameter (m) . | Cost ($/m) . |
---|---|---|---|

0.0534 | 24.16 | 0.2042 | 87.62 |

0.0756 | 32.12 | 0.2520 | 118.59 |

0.1084 | 47.09 | 0.2998 | 152.24 |

0.1564 | 63.80 |

Diameter (m) . | Cost ($/m) . | Diameter (m) . | Cost ($/m) . |
---|---|---|---|

0.0534 | 24.16 | 0.2042 | 87.62 |

0.0756 | 32.12 | 0.2520 | 118.59 |

0.1084 | 47.09 | 0.2998 | 152.24 |

0.1564 | 63.80 |

The problem was solved using the solver SBB in GAMS and Table 8 presents the calculated hydraulic variables and the flow directions. The optimized total cost for the WDN was $ 487,254.78. In this case, it can be noted that the operation cost corresponds to almost 2/3 of the total WDN cost. Table 9 presents the pipes diameter, manometric head of the pumping system, and the network costs.

Pipe/Node . | Velocity (m/s) . | Flow direction . | Pressure (m) . |
---|---|---|---|

1 | 1.25 | R-1 | 25.41 |

2 | 2.13 | 1-2 | 27.48 |

3 | 2.03 | 2-3 | 28.84 |

4 | 0.75 | 3-4 | 28.05 |

5 | 0.68 | 4-5 | 28.77 |

6 | 0.46 | 5-6 | 24.01 |

7 | 1.10 | 7-6 | 22.06 |

8 | 1.32 | 8-7 | 15.00 |

9 | 0.44 | 1-8 | 33.23 |

10 | 0.89 | 1-9 | 28.54 |

11 | 1.15 | 9-10 | 29.28 |

12 | 1.48 | 10-11 | 25.76 |

13 | 0.21 | 11-12 | 18.24 |

14 | 1.26 | 3-12 | 18.50 |

15 | 1.63 | 3-13 | 20.61 |

16 | 1.08 | 13-14 | 15.05 |

17 | 1.31 | 14-15 | 16.72 |

18 | 0.97 | 16-15 | |

19 | 1.43 | 17-16 | |

20 | 0.83 | 5-17 |

Pipe/Node . | Velocity (m/s) . | Flow direction . | Pressure (m) . |
---|---|---|---|

1 | 1.25 | R-1 | 25.41 |

2 | 2.13 | 1-2 | 27.48 |

3 | 2.03 | 2-3 | 28.84 |

4 | 0.75 | 3-4 | 28.05 |

5 | 0.68 | 4-5 | 28.77 |

6 | 0.46 | 5-6 | 24.01 |

7 | 1.10 | 7-6 | 22.06 |

8 | 1.32 | 8-7 | 15.00 |

9 | 0.44 | 1-8 | 33.23 |

10 | 0.89 | 1-9 | 28.54 |

11 | 1.15 | 9-10 | 29.28 |

12 | 1.48 | 10-11 | 25.76 |

13 | 0.21 | 11-12 | 18.24 |

14 | 1.26 | 3-12 | 18.50 |

15 | 1.63 | 3-13 | 20.61 |

16 | 1.08 | 13-14 | 15.05 |

17 | 1.31 | 14-15 | 16.72 |

18 | 0.97 | 16-15 | |

19 | 1.43 | 17-16 | |

20 | 0.83 | 5-17 |

Pipe . | Diameter (m) . | Pipe . | Diameter (m) . |
---|---|---|---|

1 | 0.2520 | 11 | 0.0756 |

2 | 0.1564 | 12 | 0.0534 |

3 | 0.1564 | 13 | 0.0534 |

4 | 0.1564 | 14 | 0.0756 |

5 | 0.1564 | 15 | 0.1084 |

6 | 0.0534 | 16 | 0.1084 |

7 | 0.0534 | 17 | 0.0756 |

8 | 0.0756 | 18 | 0.0534 |

9 | 0.1564 | 19 | 0.0756 |

10 | 0.1084 | 20 | 0.1084 |

(m) | ($) | ($) | ($) |

25.466 | 179,816.40 | 307,438.38 | 487,254.78 |

Pipe . | Diameter (m) . | Pipe . | Diameter (m) . |
---|---|---|---|

1 | 0.2520 | 11 | 0.0756 |

2 | 0.1564 | 12 | 0.0534 |

3 | 0.1564 | 13 | 0.0534 |

4 | 0.1564 | 14 | 0.0756 |

5 | 0.1564 | 15 | 0.1084 |

6 | 0.0534 | 16 | 0.1084 |

7 | 0.0534 | 17 | 0.0756 |

8 | 0.0756 | 18 | 0.0534 |

9 | 0.1564 | 19 | 0.0756 |

10 | 0.1084 | 20 | 0.1084 |

(m) | ($) | ($) | ($) |

25.466 | 179,816.40 | 307,438.38 | 487,254.78 |

Table 10 presents the contribution of the operation costs to the total WDN cost during the 25 years considered as the lifetime of the pumping system. It can be noted that in the first year this contribution represents 9.33% and in the last year, 63.1%.

Year . | ($) . | Percentage (%) . | Year . | ($) . | Percentage (%) . |
---|---|---|---|---|---|

1 | 18,512.99 | 9.33 | 14 | 206,001.73 | 53.39 |

2 | 36,352.78 | 16.82 | 15 | 217,023.75 | 54.69 |

3 | 53,543.85 | 22.94 | 16 | 227,644.96 | 55.87 |

4 | 70,109.79 | 28.05 | 17 | 237,879.95 | 56.95 |

5 | 86,073.33 | 32.37 | 18 | 247,742.76 | 57.94 |

6 | 101,456.38 | 36.07 | 19 | 257,246.92 | 58.86 |

7 | 116,280.04 | 39.27 | 20 | 266,405.48 | 59.70 |

8 | 130,564.67 | 42.07 | 21 | 275,231.00 | 60.48 |

9 | 144,329.85 | 44.53 | 22 | 283,735.58 | 61.21 |

10 | 157,594.48 | 46.71 | 23 | 291,930.91 | 61.88 |

11 | 170,376.76 | 48.65 | 24 | 299,828.23 | 62.51 |

12 | 182,694.23 | 50.40 | 25 | 307,438.38 | 63.10 |

13 | 194563,79 | 51.97 |

Year . | ($) . | Percentage (%) . | Year . | ($) . | Percentage (%) . |
---|---|---|---|---|---|

1 | 18,512.99 | 9.33 | 14 | 206,001.73 | 53.39 |

2 | 36,352.78 | 16.82 | 15 | 217,023.75 | 54.69 |

3 | 53,543.85 | 22.94 | 16 | 227,644.96 | 55.87 |

4 | 70,109.79 | 28.05 | 17 | 237,879.95 | 56.95 |

5 | 86,073.33 | 32.37 | 18 | 247,742.76 | 57.94 |

6 | 101,456.38 | 36.07 | 19 | 257,246.92 | 58.86 |

7 | 116,280.04 | 39.27 | 20 | 266,405.48 | 59.70 |

8 | 130,564.67 | 42.07 | 21 | 275,231.00 | 60.48 |

9 | 144,329.85 | 44.53 | 22 | 283,735.58 | 61.21 |

10 | 157,594.48 | 46.71 | 23 | 291,930.91 | 61.88 |

11 | 170,376.76 | 48.65 | 24 | 299,828.23 | 62.51 |

12 | 182,694.23 | 50.40 | 25 | 307,438.38 | 63.10 |

13 | 194563,79 | 51.97 |

## CONCLUSION

In the present paper, an MINLP optimization model for the synthesis of WDN was presented. In the model, the flow directions are considered unknown and binary variables were used to define them in the demand node loops. Also, operation costs are considered in the objective function. The model constraints consider the pressure and velocity limits and hydraulic equations to calculate pressures and velocities and no additional software is needed to calculate them. Both, unknown flow directions and operation costs in the objective function are the novelties in the paper. Some linearization strategies were used to avoid nonlinearities in the model and the environment GAMS was used to solve the problems. Two real case studies were used to test the model applicability.

The objective function to be minimized is the total WDN cost, considering operation and installation costs, depending on a set of available commercial diameters. The operation cost depends on the manometric head of the reservoir. In the first case study, the papers used in the results comparison used known flow directions. Hazen–Williams and Darcy–Weisbach equations were used in the case studies. In both cases, the operation costs represent ¼ and 2/3 of the total cost, which means that it represents an important percentage of the total cost in the WDN.

The linearization techniques used decreased the degree of complexity of the model. It is very useful, considering that the binary variables inserted in the model to consider the flow directions unknown increase the complexity degree. Nevertheless, the solution of the problem becomes more realistic, once the operation costs represent a significant part of the WDN total cost.

For future studies, aiming to improve the current optimization model, uncertainties in the water nodes demand and different operation conditions, and multi-periods of operation could be considered.

## ACKNOWLEDGEMENTS

The authors acknowledge the National Council for Scientific and Technological Development – CNPq (Brazil), processes 311807/2018-6, 428650/2018-0, for the financial support.

## DATA AVAILABILITY STATEMENT

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

## CONFLICT OF INTEREST

The authors declare there is no conflict.

Water network design by MINLP. Rep. No. RC24495, IBM Research, Yorktown Heights, pp. 1–17