# Robust Optimal Operation of Water Distribution Systems

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## Abstract

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## 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

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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 |

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**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