# Infeasibility Maps: Application to the Optimization of the Design of Pumping Stations in Water Distribution Networks

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- −
- A new constraint was added to the mathematical optimization model. For each PS, this equation allows us to discard all pump models that, due to their specifications, did not manage to supply at least the maximum flow rate during the analysis period.
- −
- The method used network preprocessing to determine in advance the maximum and minimum flow that each PS could provide. This procedure made it possible to limit the search space for solutions to the problem, thus eliminating areas of total infeasibility. An area of infeasibility is limited by ranges of the decision variable where it is impossible to meet the hydraulic constraints of the model. Our proposed method maps these ranges before starting the optimization process, accelerating the convergence of the algorithm using infeasibility maps (IMs).
- −
- This study combined the use of the SC with the mapping of infeasibility zones to rule out unfeasible solutions during the evaluation process in any period, thus avoiding unnecessary hydraulic simulations when it was detected that part of the solution was not viable. Consequently, the IMs reduce the search space of the optimization algorithm. Reducing the search space to increase computational efficiency is a significant challenge faced when optimizing water networks

#### 2.1. Mathematical Model

#### 2.1.1. Mathematical Notation

- −
- N
_{t}: total number of time steps in the optimization process. - −
- N
_{ps}: total number of PSs in the network. - −
- N
_{b}: total number of pump models available in the data set. - −
- F
_{a}: amortization factor. - −
- r: interest rate.
- −
- N
_{p}: total number of project life periods. - −
- H
_{0,i}, A_{i}: characteristic coefficients of the pump head installed in PS_{i}. - −
- E
_{i,}F_{i}: characteristic coefficients of the performance curve of the pump installed in PS_{i}. - −
- Q
_{i,j,k}: represents the discharge of pump k during time step j in PS_{i}. - −
- p
_{i,j}: energy cost in PS_{i}during the time step j. - −
- ϒ: specific gravity of water.
- −
- Δt
_{j}: discretization interval of the optimization period. - −
- m
_{i,j}: the number of FSPs running in PS_{i}at time step j. - −
- n
_{i,j}: the number of VSPs running in PS_{i}at time step j. These values depend on the selected pump model and the system selected to control the operation point. - −
- N
_{B,i}: total number of total pumps. - −
- H
_{Bmax}: maximum head of the largest pump available in the data set. - −
- H
_{max,i}: maximum head supplied by PS_{i}during time analysis. - −
- C
_{pump, i}: purchase cost of a pump installed in PS_{i}. - −
- n
_{i}: number of frequency inverters in PS_{i}. - −
- C
_{facility,i}: cost of accessories including pipes in PS_{i}. - −
- C
_{control,i}: sum of a pressure transducer, flowmeter, and programmable logic controller cost for PS_{i}.

#### 2.1.2. Decision Variables

- −
- X
_{ij}: percentage of the flow supplied from PS_{i}at each time step j. - −
- m
_{i}: number of fixed speed pumps in PS_{i}. - −
- b
_{i}: ID of the pump model to be installed in PS_{i}in the range [1,N_{b}].

#### 2.1.3. Objective Function

#### 2.1.4. Constraints

#### 2.2. Infeasibility Maps

_{ij}determines the fraction of flow that PS

_{i}contributes during period j. This variable can have a range from 0 to 100 (expressed as a percentage) for which 0 indicates that the PS did not supply water in that period; in contrast, a value of 100 indicated that all the flow was supplied by a single PS in the period. Therefore, a huge number of possible combinations exist, and many of them are infeasible solutions.

- The distribution of flow generates sectors of the network where it is not possible to reach the minimum required pressures.
- Some of the PSs must provide a pressure greater than the maximum head of the largest pump that exists in the available catalog.
- The sum of the flows supplied is greater than the demand.

_{i,j}. Thus, it was possible to avoid the evaluation of infeasible solutions, which could be ruled out using hydraulic criteria before starting the optimization process. Unfortunately, the non-linearity of the relationships between the hydraulic variables did not allow these values to be fixed, but this value could be expressed as a function of the piezometric head of a reference PS (PS

_{ref}). This reference pumping station could be any of the pumping stations in the network. Furthermore, PS

_{ref}supplied all the water that was not provided by the rest of the PSs.

_{i}different from PS

_{ref}, it was possible to build a graph called the “Infeasibility Map” (IM), such as the one presented in Figure 1.

_{i}(QPSi). It could have values between zero and the total flow demanded by the network.

_{ref}(H

_{ref}). For any H

_{ref}, all points to the left of the minima curve (red color) are infeasible. This infeasibility was due to the fact that it was not possible to reach the minimum height required in all the nodes of the network. Similarly, at any point to the right of the maximum curve (green color), the head required by PS

_{i}always exceeded the maximum head of the largest pump available in the catalog (H

_{Bmax}), and therefore, it would also be infeasible.

_{ij}.

_{ij}represents a fraction of the difference between the highest value of the maximum flow curve and the lowest value of the minimum flow curve. It is important to note that this range is always less than the total demand. Therefore, it represents a search space reduction for any network, regardless of the topology.

_{i}different from PS

_{ref}[18]. From the solution, the intersection of the input flow x

_{ij}and the respective SC could be obtained. If the resulting point was within the blue region, the solution could not be discarded. Otherwise, the solution was irrefutably infeasible. Outside this range, it would have been impossible to achieve a technically feasible solution.

_{1}, (2) PS

_{2}, and (3) PS

_{3}.

_{ref}. PS

_{1}was selected as the reference station after which the minimum and maximum curves for PS

_{2}and PS

_{3}were calculated. The limits defined by the curves allowed for generating the BSR for each PS

_{i}. In Figure 3, blue and green areas represent the BSRs for PS

_{3}and PS

_{2}, respectively. Regarding the flow supplied by each PS

_{i}, x

_{3j,}and x

_{2j}represent the total percentage of flow supplied by PS

_{3}and PS

_{2}, respectively. Consequently, PS

_{1}must supply the remaining flow with the head Hps

_{1}. Note that each supply source had its own SC. Consequently, the SC was found for both pumping stations.

#### 2.3. Case Study

_{1}and PS

_{2}. However, due to the growth of the city, the pumping equipment was old and susceptible to replacement. There is the possibility of putting a third water source into operation, located at PS

_{3}. The node with the lowest elevation was 190 m, and the elevation of the largest node was approximately 295 m. The minimum operating pressure was 15 m for all network nodes. Information about the nodes and pipelines can be found in the Supplementary Materials.

_{1}was the flow provided by PS

_{1}, Q

_{2}was the flow provided by PS

_{2}, and Q

_{3}was the flow provided by PS

_{3}. Figure 4b shows the modular design proposed by [7]. This scheme was used later to carry out the CAPEX calculations.

#### 2.4. Optimization Method

^{104}. Consequently, the use of a computational method was required to solve the optimization model. Specifically, this work used a pseudo-genetic algorithm (PGA) developed by the authors of [20] to solve problems of an integer nature. Unlike a traditional genetic algorithm (GA), the PGA is based on an integer coding of the solution, and each decision variable can store different values represented by alphanumeric variables.

## 3. Results

_{1}was selected as PS

_{ref}and the IMs for the 24 periods were calculated for each PS

_{i}. For example, Figure 6 shows the resulting IM for PS

_{2}in the period of greatest water demand (hour = 12). The orange zone represents the BSR and the remaining area represents hydraulically infeasible solutions that were not used by the PGA in the optimization process.

^{5}hydraulic simulations were needed. However, this preprocessing was only executed once for the entire experiment and represented a small percentage of the total simulations. For example, for the case study in which 100 experiments were executed, the generation of IMs represented approximately 2% of the total number of simulations (2.0 × 10

^{8}) and decreased in an inversely proportional manner with the number of experiments executed.

_{i}for the best solution obtained by the PGA with and without IMs, respectively.

_{1}, PS

_{2}, and PS

_{3}would have had to be at least ten, two, and two pumps, respectively. In contrast, when using the IMs, the PGA found many solutions in which only PS

_{1}and PS

_{2}were needed. Both solutions were hydraulically feasible, but the solution found using IMs was found to be more efficient, cheaper, and perfectly fit the network requirements. This feature is important for decision-making because if the search space is not correctly explored, unnecessary energy and building costs can be incurred. Table 2details the total yearly costs for the best solutions achieved by the PGA without and with IMs (Figure 8 and Figure 9, respectively).

_{3}because it requires a high level of investment. Second, the optimized design without IMs required 10 pumps running on PS

_{1}. This feature implies a high cost of purchasing this equipment.

_{1}, ND

_{2}, and ND

_{3}are the nominal diameters of the corresponding pipe p, which is used for defining the diameters of elements such as isolation valves or check valves according to the modular design presented in Figure 4b. Similarly, L

_{1}, L

_{2}, and L

_{3}are the lengths of pipes. Furthermore, Table 3 shows the number of fixed-speed pumps (m

_{i}) and the number of variable-speed pumps (n

_{i}). H

_{0}, A, E, and F represent the characteristic and efficiency curve coefficients. Finally, the last row displays the selected model pump from the database. It is important to note that the final solution only considered variable-speed pumps and ruled out fixed-speed pumps. The higher cost of this equipment could be offset by the reduction in energy consumption during the years of the project’s life.

## 4. Conclusions

- −
- The exhaustive construction of the IMs required a significant number of hydraulic simulations. However, this procedure only needed to be done once, representing only 2% of the total number of simulations.
- −
- When IMs were not used, the search space was too large, and the algorithm took a long time to find feasible regions, which were usually local minima. The use of IMs allowed for accelerating the convergence of the optimization algorithms, rapidly evolving toward better solutions. Specifically, the number of simulations required by the IM-guided algorithm managed to reduce the number of hydraulic simulations necessary to achieve convergence in the case study by 60%.
- −
- The use of IMs in the case study achieved savings of 71% compared with the solutions obtained by the optimization algorithm when considering the complete search space. Additionally, the 100 experiments ran using IMs had better solutions than the best solution obtained using the PGA when no IMs were used. An inadequate exploration of the solution space implies unnecessary cost overruns and non-optimal solutions for a given problem.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Mala-Jetmarova, H.; Sultanova, N.; Savic, D. Lost in optimisation of water distribution systems? a literature review of system design. Water
**2018**, 10, 307. [Google Scholar] [CrossRef][Green Version] - Mala-Jetmarova, H.; Sultanova, N.; Savic, D. Lost in optimisation of water distribution systems? a literature review of system operation. Environ. Model. Softw.
**2017**, 93, 209–254. [Google Scholar] [CrossRef][Green Version] - Gupta, A.; Kulat, K.D. A selective literature review on leak management techniques for water distribution system. Water Resour. Manag.
**2018**, 32, 3247–3269. [Google Scholar] [CrossRef] - Bordea, D.; Pro, O.; Filip, I.; Drăgan, F.; Va, C. Modelling, Simulation and Controlling of a Multi-Pump System with Water Storage Powered by a Fluctuating and Intermittent Power Source. Mathematics
**2022**, 10, 4019. [Google Scholar] [CrossRef] - León-Celi, C.F.; Iglesias-Rey, P.L.; Martínez-Solano, F.J.; Savic, D. Operation of multiple pumped-water sources with no storage. J. Water Resour. Plan Manag.
**2018**, 144, 04018050. [Google Scholar] [CrossRef] - Blinco, L.J.; Simpson, A.R.; Lambert, M.F.; Marchi, A. Comparison of pumping regimes for water distribution systems to minimize cost and greenhouse gases. J. Water Resour. Plan Manag.
**2016**, 142, 04016010. [Google Scholar] [CrossRef][Green Version] - Gutiérrez-Bahamondes, J.H.; Mora-Meliá, D.; Iglesias-Rey, P.L.; Martínez-Solano, F.J.; Salgueiro, Y. Pumping station design in water distribution networks considering the optimal flow distribution between sources and capital and operating costs. Water
**2021**, 13, 3098. [Google Scholar] [CrossRef] - Martin-Candilejo, A.; Santillan, D.; Iglesias, A.; Garrote, L. Optimization of the design of water distribution systems for variable pumping flow rates. Water
**2020**, 12, 359. [Google Scholar] [CrossRef][Green Version] - Oshurbekov, S.; Kazakbaev, V.; Prakht, V.; Dmitrievskii, V. Improving reliability and energy efficiency of three parallel pumps by selecting trade-off operating points. Mathematics
**2021**, 9, 1297. [Google Scholar] [CrossRef] - Nagkoulis, N.; Katsifarakis, K.L. Minimization of Total Pumping Cost from an Aquifer to a Water Tank, Via a Pipe Network. Water Resour. Manag.
**2020**, 34, 4147–4162. [Google Scholar] [CrossRef] - Makaremi, Y.; Haghighi, A.; Ghafouri, H.R. Optimization of Pump Scheduling Program in Water Supply Systems Using a Self-Adaptive NSGA-II; a Review of Theory to Real Application. Water Resour. Manag.
**2017**, 31, 1283–1304. [Google Scholar] [CrossRef] - Gutiérrez-Bahamondes, J.H.; Salgueiro, Y.; Silva-Rubio, S.A.; Alsina, M.A.; Mora-Meliá, D.; Fuertes-Miquel, V.S. jHawanet: An open-source project for the implementation and assessment of multi-objective evolutionary algorithms on water distribution networks. Water
**2019**, 11, 2018. [Google Scholar] [CrossRef][Green Version] - Fecarotta, O.; McNabola, A. Optimal location of pump as turbines (pats) in water distribution networks to recover energy and reduce leakage. Water Resour. Manag.
**2017**, 31, 5043–5059. [Google Scholar] [CrossRef] - Carpitella, S.; Brentan, B.; Montalvo, I.; Izquierdo, J.; Certa, A. Multi-criteria analysis applied to multi-objective optimal pump scheduling in water systems. Water Sci. Technol. Water Supply
**2019**, 19, 2338–2346. [Google Scholar] [CrossRef][Green Version] - Weber, J.B.; Lorenz, U. Optimizing Booster Stations. In Proceedings of the Genetic and Evolutionary Computation Conference Companion, Berlin, Germany, 15–19 July 2017; pp. 1303–1310. [Google Scholar] [CrossRef]
- Predescu, A.; Truică, C.O.; Apostol, E.S.; Mocanu, M.; Lupu, C. An advanced learning-based multiple model control supervisor for pumping stations in a smart water distribution system. Mathematics
**2020**, 8, 887. [Google Scholar] [CrossRef] - Gil, F.A.A.; Iglesias-Rey, P.L.; Martínez-Solano, F.J.; Cortes, J.V.L.; Mora-Meliá, D. Methodology for Projects Of Pumping Stations Directly Connected To The Network Considering The Operation Strategy. In Proceedings of the 22nd International Congress on Project management and Engineering, Madrid, Spain, 11–13 July 2018; 2018; pp. 551–563. Available online: http://dspace.aeipro.com/xmlui/handle/123456789/1728 (accessed on 20 March 2023).
- León-Celi, C.F.; Iglesias-Rey, P.L.; Martínez-Solano, F.J.; Mora-Melia, D. The Setpoint Curve as a Tool for the Energy and Cost Optimization of Pumping Systems in Water Networks. Water
**2022**, 14, 2426. [Google Scholar] [CrossRef] - Briceño-León, C.X.; Iglesias-Rey, P.L.; Martínez-Solano, F.J.; Creaco, E. Integrating Demand Variability and Technical, Environmental, and Economic Criteria in Design of Pumping Stations Serving Closed Distribution Networks. J. Water Resour. Plan Manag.
**2023**, 149, 4023002. [Google Scholar] [CrossRef] - Mora-Melia, D.; Iglesias-Rey, P.L.; Martinez-Solano, F.J.; Fuertes-Miquel, V.S. Design of water distribution networks using a pseudo-genetic algorithm and sensitivity of genetic operators. Water Resour. Manag.
**2013**, 27, 4149–4162. [Google Scholar] [CrossRef] - Rossman, L.A. EPANET 2.0 User’s Manual (EPA/600/R-00/057); National Risk Management Research Laboratory: Cincinnatti, OH, USA, 2000. [Google Scholar]
- Lučin, I.; Lučin, B.; Čarija, Z.; Sikirica, A. Data-driven leak localization in urban water distribution networks using big data for random forest classifier. Mathematics
**2021**, 9, 672. [Google Scholar] [CrossRef] - Negrete, M. Modelación Computacional en EPANET de un Sector de la Red de Abastecimiento de Agua Potable de Curicó; Universidad de Talca, Facultad de Ingeniería: Curicó, Chile, 2021. [Google Scholar]
- Mora-Melia, D.; Iglesias-Rey, P.L.; Martínez-Solano, F.J.; Muñoz-Velasco, P. The efficiency of setting parameters in a modified shuffled frog leaping algorithm applied to optimizing water distribution networks. Water
**2016**, 8, 182. [Google Scholar] [CrossRef][Green Version] - Benítez-Hidalgo, A.; Nebro, A.J.; García-Nieto, J.; Oregi, I.; Del Ser, J. jMetalPy: A Python framework for multi-objective optimization with metaheuristics. Swarm Evol. Comput.
**2019**, 51, 100598. [Google Scholar] [CrossRef][Green Version]

Time (h) | PS1 | PS2 | PS3 |
---|---|---|---|

1–8 | 0.094 | 0.092 | 0.09 |

9–18 | 0.133 | 0.131 | 0.129 |

19–22 | 0.166 | 0.164 | 0.162 |

23–24 | 0.133 | 0.131 | 0.129 |

OPEX | CAPEX | Fa ● CAPEX + OPEX | ||||
---|---|---|---|---|---|---|

PGA without IMs | PGA with IMs | PGA without IMs | PGA with IMs | PGA without IMs | PGA with IMs | |

PS1 | EUR 121 | EUR 80.7 | EUR 190,756 | EUR 94,470 | EUR 54,710 | EUR 34,002 |

PS2 | EUR 34 | EUR 31 | EUR 34,544 | EUR 37,077 | EUR 13,754 | EUR 13,054 |

PS3 | EUR 16 | - | EUR 85,618 | - | EUR 12,107 | - |

Total | EUR 80,571 | EUR 47,056 |

PS_{1} | PS_{2} | ||
---|---|---|---|

ND_{1} | 350 | 250 | |

(mm) | ND_{2} | 125 | 125 |

ND_{3} | 350 | 250 | |

L1 | 1.75 | 1.25 | |

(m) | L2 | 3.75 | 3.75 |

L3 | 3.50 | 2.50 | |

m_{i} | 0 | 0 | |

n_{i} | 6 | 3 | |

H_{0} | 27.2632 | 27.2632 | |

A | −0.01416 | −0.01416 | |

E | 0.06929 | 0.06929 | |

F | 0.00158 | 0.00158 | |

Model ID | GNI 50-13/7.5 | GNI 50-13/7.5 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gutiérrez-Bahamondes, J.H.; Mora-Melia, D.; Valdivia-Muñoz, B.; Silva-Aravena, F.; Iglesias-Rey, P.L.
Infeasibility Maps: Application to the Optimization of the Design of Pumping Stations in Water Distribution Networks. *Mathematics* **2023**, *11*, 1582.
https://doi.org/10.3390/math11071582

**AMA Style**

Gutiérrez-Bahamondes JH, Mora-Melia D, Valdivia-Muñoz B, Silva-Aravena F, Iglesias-Rey PL.
Infeasibility Maps: Application to the Optimization of the Design of Pumping Stations in Water Distribution Networks. *Mathematics*. 2023; 11(7):1582.
https://doi.org/10.3390/math11071582

**Chicago/Turabian Style**

Gutiérrez-Bahamondes, Jimmy H., Daniel Mora-Melia, Bastián Valdivia-Muñoz, Fabián Silva-Aravena, and Pedro L. Iglesias-Rey.
2023. "Infeasibility Maps: Application to the Optimization of the Design of Pumping Stations in Water Distribution Networks" *Mathematics* 11, no. 7: 1582.
https://doi.org/10.3390/math11071582