# A Novel Approach to Avoiding Technically Unfeasible Solutions in the Pump Scheduling Problem

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Overview of Pump Scheduling Methodologies

#### 2.1. Objective Function

_{h}is the time-step (hour) of hydraulic simulation, T is the period of hydraulic simulation (hour), γ is the water specific weight (kN/m

^{3}), C

_{t}is the unit energy cost (€, $, £/kWh) at time t, Q

_{p}

_{,t}is the flow (m

^{3}/s) of the pump p, at time t, H

_{p}

_{,t}is the head (m) of pump p, at time t, η

_{p}

_{,t}is the efficiency (dimensional) of the pump p, at time t. Finally, PEN is 0 if all constraints are satisfied, and a very large value (in €, $ or £) if one of the constraints is violated.

#### 2.2. Constraints

_{k}

_{,t}) of the system where Nk tanks are present, to be maintained between the allowable minimum H

_{k}

^{min}and maximum H

_{k}

^{max}at every time instant t. This means that the following two Equations (2) and (3) must be satisfied:

_{k}

_{,0}at t = 0) for each tank k is reached or exceeded in the tank at the end of the calculation period (t = T):

_{n}

_{,t}is the head at node n at time t, Nn the number of nodes and H

_{n}

^{min}the minimum head required at node n.

#### 2.3. Decision Variables

- Triggering-based approach (TBA): the pumps are managed by means of start/end operational times in terms of the length of time of pump operation [43,44], directly in terms of start/end run times [2], or by linking pump control to time constant trigger level [10,45] or time variable trigger level [46].

## 3. Detecting Technically Feasible Solutions: The Problem

#### 3.1. Pattern-Based Approach

_{h}) is assumed equal to 1 h.

#### 3.2. Triggering Based Approach

## 4. An Approach to Identify Technically Feasible Solutions

_{h}) may be set by the user in Epanet2. Once the hydraulic time-step is set, the total number of time-steps in EPS usually corresponds to the ratio between the total duration of simulation and the hydraulic time-step. Nevertheless, time-steps shorter than the hydraulic time-step will occur automatically (thus increasing the number of total time-steps) upon one of the following occurrences (cases): (a) the next output reporting time period arises; (b) a tank becomes empty or full, (c) the next time pattern period arises; and (d) a simple control or rule-based control is activated.

## 5. Overview of the Proposed Pump Scheduling Model

#### 5.1. Decision Variables

_{s}is expressed in minutes. Similarly, the pump OD can be calculated as:

_{2}being a non-negative real number $0\le {x}_{2}\le 1$, with the resulting switch-off time:

#### 5.2. Objective Function and Constraints

#### 5.3. Hydraulic Solver and Pumping Operation Model

#### 5.4. Penalty for Technically Unfeasible Solutions

#### 5.5. Optimization Algorithm

## 6. Case Studies and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Anytown water distribution network model [39].

**Figure 2.**Results of the pump scheduling model by [39] for Anytown network.

**Figure 3.**Van Zyl water distribution network model [48].

**Figure 4.**Results of the pump scheduling methodology by [10] for Van Zyl network, hydraulic time-step 1 h.

**Figure 5.**Results of the pump scheduling methodology by [10] for Van Zyl network, hydraulic time-step 10 s.

**Figure 7.**Results of the proposed pump scheduling methodology for Van Zyl network, hydraulic time-step 1 h.

**Figure 8.**Results of the proposed pump scheduling methodology for Van Zyl network, hydraulic time-step 10 s.

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**MDPI and ACS Style**

Marini, G.; Fontana, N.; Maio, M.; Di Menna, F.; Giugni, M. A Novel Approach to Avoiding Technically Unfeasible Solutions in the Pump Scheduling Problem. *Water* **2023**, *15*, 286.
https://doi.org/10.3390/w15020286

**AMA Style**

Marini G, Fontana N, Maio M, Di Menna F, Giugni M. A Novel Approach to Avoiding Technically Unfeasible Solutions in the Pump Scheduling Problem. *Water*. 2023; 15(2):286.
https://doi.org/10.3390/w15020286

**Chicago/Turabian Style**

Marini, Gustavo, Nicola Fontana, Marco Maio, Francesco Di Menna, and Maurizio Giugni. 2023. "A Novel Approach to Avoiding Technically Unfeasible Solutions in the Pump Scheduling Problem" *Water* 15, no. 2: 286.
https://doi.org/10.3390/w15020286