# Robust Optimal Operation of Water Distribution Systems

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. WDS Operation under Uncertainty

#### 1.2. Robust Optimization

_{i,j}is an uncertain parameter, representing the j coefficient in constraint i (entry i,j in the matrix A), then ${a}_{i,j}\left(\xi \right)={a}_{i,j}^{0}+{\u2206}_{i,j}{\xi}_{i,j}$. Where ${a}_{i,j}^{0}$ is the nominal value (mean), ∆

_{i,j}is the uncertainty level, and ${\xi}_{i,j}$ is the random perturbations. To achieve robust feasibility, which means a feasible solution for any realization of the random variables, the left-hand-side (LHS) of the inequality constraint (2) is maximized over ξ. The rationale behind this is that if the constraint holds for the maximized LHS, it will hold for any value of ξ. Or in other words, if the solution is feasible even against the worst-case scenario, it will meet the constraints in any other (better) scenario. The result is a min-max formulation which is completely deterministic after solving the inner optimization problems over ξ. The equivalent deterministic is obtained by solving the max problems in Equations (3) and (4), and it is referred to as robust counterpart (RC) [19].

^{0}is a vector of the nominal values, and $\Vert \xi \Vert $ is the L-2 norm of the random variables vector ξ. To capture the correlations between random variables, here the uncertainty level ∆ is not a scalar but an affine mapping matrix that defines the correlation links between the different terms of ξ. The construction of the mapping matrix ∆ is based on the covariance matrix of the different uncertain parameters [29]. More explanations of the technique used for the mapping matrix construction are detailed later. Previous RO studies in water resources research considered simple 1D correlations. For example, demand correlation between different consumers at the same time [21] or extreme rainfall events between two basins [25].

- Implementing the RO theory on optimal operation problems with a better approximation of the system’s hydraulics.
- Illustrate the robust optimal operation methodology on a large real-life network.
- Address multiple uncertainties in the same problem including both objective and right-hand side (RHS) uncertainties simultaneously.
- Model multi-dimensional correlated uncertainties to capture the temporal–spatial correlations between the elements of the system.

## 2. Methodology

_{t,s}be state variables representing the volume in tank (s) at time step (t); and Q

_{p,c}, P

_{p,c}are the corresponding flow and power consumption of pump station (p) when operated with the unit combination (c). ${s}_{i}$ and ${s}_{o}$ are the sets of pumps that, respectively, pump water in and out of tank s. Elec

_{t}is the electricity tariff at time step (t). δt is the duration of time step t (1 h), P is the set of all pumping stations, S is the set of all storage tanks, and T is the number of time steps. Using these notations, the deterministic minimal energy costs operation problem can be described by the following linear programming (LP) presented in Equations (6)–(10). A complete description of the formulation can be found in [32].

#### 2.1. Uncertainty Sets

#### 2.2. Robust Counterpart

#### 2.3. Case Studies

#### 2.3.1. Case Study 1: Illustrative Network

^{3}volume, where min and max allowed volumes are 500 and 2800 m

^{3}, respectively. A single aggregative consumer represents the demand in the network.

#### 2.3.2. Case Study 2: Sopron Network

_{k}(t) is the power consumed by pump station (k) in time (t), where (k) points to a pump station that is connected to the power station (ps). The full description of the second case study and its data are detailed in [33]. The network topology is presented in Figure 2.

## 3. Results

#### 3.1. Case Study 1: Illustrative Network Results

#### 3.2. Case Study 2: Sopron Network Results

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## 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 Operation. Environ. Model. Softw.
**2017**, 93, 209–254. [Google Scholar] [CrossRef] [Green Version] - Avni, N.; Fishbain, B.; Shamir, U. Water Consumption Patterns as a Basis for Water Demand Modeling. Water Resour. Res.
**2015**, 51, 8165–8181. [Google Scholar] [CrossRef] [Green Version] - Meniconi, S.; Maietta, F.; Alvisi, S.; Capponi, C.; Marsili, V.; Franchini, M.; Brunone, B. A Quick Survey of the Most Vulnerable Areas of a Water Distribution Network Due to Transients Generated in a Service Line: A Lagrangian Model Based on Laboratory Tests. Water
**2022**, 14, 2741. [Google Scholar] [CrossRef] - Marsili, V.; Meniconi, S.; Alvisi, S.; Brunone, B.; Franchini, M. Stochastic Approach for the Analysis of Demand Induced Transients in Real Water Distribution Systems. J. Water Resour. Plan. Manag.
**2022**, 148, 04021093. [Google Scholar] [CrossRef] - Maiolo, M.; Mendicino, G.; Pantusa, D.; Senatore, A. Optimization of Drinking Water Distribution Systems in Relation to the Effects of Climate Change. Water
**2017**, 9, 803. [Google Scholar] [CrossRef] - Aghapoor Khameneh, P.; Miri Lavasani, S.M.; Nabizadeh Nodehi, R.; Arjmandi, R. Water Distribution Network Failure Analysis under Uncertainty. Int. J. Environ. Sci. Technol.
**2020**, 17, 421–432. [Google Scholar] [CrossRef] - Housh, M.; Salomons, E.; Sela, L.; Simpson, A.R. Water Distribution Systems on the Spot: Energy Market Opportunities for Water Utilities. J. Water Resour. Plan. Manag.
**2022**, 148, 02522002. [Google Scholar] [CrossRef] - McPhail, C.; Maier, H.R.; Westra, S.; van der Linden, L.; Kwakkel, J.H. Guidance Framework and Software for Understanding and Achieving System Robustness. Environ. Model. Softw.
**2021**, 142, 105059. [Google Scholar] [CrossRef] - Maier, H.R.; Guillaume, J.H.A.; van Delden, H.; Riddell, G.A.; Haasnoot, M.; Kwakkel, J.H. An Uncertain Future, Deep Uncertainty, Scenarios, Robustness and Adaptation: How Do They Fit Together? Environ. Model. Softw.
**2016**, 81, 154–164. [Google Scholar] [CrossRef] - Walker, W.E.; Lempert, R.J.; Kwakkel, J.H. Deep uncertainty. In Encyclopedia of Operations Research and Management Science, 3rd ed.; Grass, S.I., FU, M.C., Eds.; Springer: New York, NY, USA, 2013; Volume 1, pp. 395–402. [Google Scholar]
- Pallottino, S.; Sechi, G.M.; Zuddas, P. A DSS for Water Resources Management under Uncertainty by Scenario Analysis. Environ. Model. Softw.
**2005**, 20, 1031–1042. [Google Scholar] [CrossRef] - Napolitano, J.; Sechi, G.M.; Zuddas, P. Scenario Optimisation of Pumping Schedules in a Complex Water Supply System Considering a Cost–Risk Balancing Approach. Water Resour. Manag.
**2016**, 30, 5231–5246. [Google Scholar] [CrossRef] - Schwartz, R.; Housh, M.; Ostfeld, A.; Asce, F. Limited Multistage Stochastic Programming for Water Distribution Systems Optimal Operation. J. Water Resour. Plan. Manag.
**2016**, 142, 06016003. [Google Scholar] [CrossRef] - Charnes, A.; Cooper, W.W. Chance-Constrained Programming. Manag. Sci.
**1959**, 6, 73–79. [Google Scholar] [CrossRef] - Ahmed, S.; Shapiro, A. Solving Chance-Constrained Stochastic Programs via Sampling and Integer Programming. INFORMS Tutor. Oper. Res.
**2008**, 261–269. [Google Scholar] [CrossRef] [Green Version] - Khatavkar, P.; Mays, L.W. Model for Optimal Operation of Water Distribution Pumps with Uncertain Demand Patterns. Water Resour. Manag.
**2017**, 31, 3867–3880. [Google Scholar] [CrossRef] - Grosso, J.M.; Ocampo-Martínez, C.; Puig, V.; Joseph, B. Chance-Constrained Model Predictive Control for Drinking Water Networks. J. Process Control
**2014**, 24, 504–516. [Google Scholar] [CrossRef] [Green Version] - Kim, Y.-O.; Eum, H.-I.; Lee, E.-G.; Ko, I.H. Optimizing Operational Policies of a Korean Multireservoir System Using Sampling Stochastic Dynamic Programming with Ensemble Streamflow Prediction. J. Water Resour. Plan. Manag.
**2007**, 133, 4–14. [Google Scholar] [CrossRef] - Ben-Tal, A.; el Ghaoui, L.; Nemirovski, A. Robust Optimization; Princeton University Press: Princeton, NJ, USA, 2009; ISBN 1400831059. [Google Scholar]
- Chung, G.; Lansey, K.; Bayraksan, G. Reliable Water Supply System Design under Uncertainty. Environ. Model. Softw.
**2009**, 24, 449–462. [Google Scholar] [CrossRef] - Perelman, L.; Housh, M.; Ostfeld, A. Robust Optimization for Water Distribution Systems Least Cost Design. Water Resour. Res.
**2013**, 49, 6795–6809. [Google Scholar] [CrossRef] - Boindala, S.P.; Ostfeld, A. Robust Multi-Objective Design Optimization of Water Distribution System under Uncertainty. Water
**2022**, 14, 2199. [Google Scholar] [CrossRef] - Housh, M.; Ostfeld, A.; Shamir, U. Optimal Multiyear Management of a Water Supply System under Uncertainty: Robust Counterpart Approach. Water Resour. Res.
**2011**, 47, 1–15. [Google Scholar] [CrossRef] - Pan, L.; Housh, M.; Liu, P.; Cai, X.; Chen, X. Robust Stochastic Optimization for Reservoir Operation. Water Resour. Res.
**2015**, 51, 409–429. [Google Scholar] [CrossRef] [Green Version] - Housh, M. Non-Probabilistic Robust Optimization Approach for Flood Control System Design. Environ. Model. Softw.
**2017**, 95, 48–60. [Google Scholar] [CrossRef] - Goryashko, A.P.; Nemirovski, A.S. Robust Energy Cost Optimization of Water Distribution System with Uncertain Demand. Autom. Remote Control
**2014**, 75, 1754–1769. [Google Scholar] [CrossRef] [Green Version] - Jowitt, P.W.; Germanopoulos, G. Optimal Pump Scheduling in Water-Supply Networks. J. Water Resour. Plan. Manag.
**1992**, 118, 406–422. [Google Scholar] [CrossRef] - Ben-Tal, A.; Nemirovski, A. Robust Solutions of Uncertain Linear Programs. Oper. Res. Lett.
**1999**, 25, 1–13. [Google Scholar] [CrossRef] [Green Version] - Yuan, Y.; Li, Z.; Huang, B. Robust Optimization under Correlated Uncertainty: Formulations and Computational Study. Comput. Chem. Eng.
**2016**, 85, 58–71. [Google Scholar] [CrossRef] - Gomes, S.C.; Vinga, S.; Henriques, R. Spatiotemporal Correlation Feature Spaces to Support Anomaly Detection in Water Distribution Networks. Water
**2021**, 13, 2551. [Google Scholar] [CrossRef] - Eck, B.; McKenna, S.; Akrhiev, A.; Kishimoto, A.; Palmes, P.; Taheri, N.; van den Heever, S. Pump Scheduling for Uncertain Electricity Prices. In Proceedings of the World Environmental and Water Resources Congress 2014: Water Without Borders, Portland, OR, USA, 1–5 June 2014; pp. 426–434. [Google Scholar] [CrossRef]
- Perelman, G.; Fishbain, B. Critical Elements Analysis of Water Supply Systems to Improve Energy Efficiency in Failure Scenarios. Water Resour. Manag.
**2022**, 36, 3797–3811. [Google Scholar] [CrossRef] - Selek, I.; Bene, J.G.; Hs, C. Optimal (Short-Term) Pump Schedule Detection for Water Distribution Systems by Neutral Evolutionary Search. Appl. Soft. Comput.
**2012**, 12, 2336–2351. [Google Scholar] [CrossRef]

Flow (m ^{3}/h) | Mean Power (kWatt) | STD Power (kWatt) | S. Energy (kWatt-h/m ^{3}) | Unit 1 | Unit 2 |
---|---|---|---|---|---|

250 | 100 | 10 | 0.4 | 1 | 0 |

250 | 95 | 10 | 0.38 | 0 | 1 |

400 | 172 | 10 | 0.43 | 1 | 1 |

Flow (m ^{3}/h) | Mean Power (kWatt) | STD Power (kWatt) | S. Energy (kWatt-h/m ^{3}) | Well |
---|---|---|---|---|

300 | 126 | 5 | 0.42 | 1 |

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**

Perelman, G.; Ostfeld, A.; Fishbain, B.
Robust Optimal Operation of Water Distribution Systems. *Water* **2023**, *15*, 963.
https://doi.org/10.3390/w15050963

**AMA Style**

Perelman G, Ostfeld A, Fishbain B.
Robust Optimal Operation of Water Distribution Systems. *Water*. 2023; 15(5):963.
https://doi.org/10.3390/w15050963

**Chicago/Turabian Style**

Perelman, Gal, Avi Ostfeld, and Barak Fishbain.
2023. "Robust Optimal Operation of Water Distribution Systems" *Water* 15, no. 5: 963.
https://doi.org/10.3390/w15050963