# Multi-Objective-Based Tuning of Economic Model Predictive Control of Drinking Water Transport Networks

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## Abstract

**:**

## 1. Introduction

- (i)
- by adding a steady-state target optimization layer between the RTO and the MPC [6],
- (ii)
- by considering the dynamics of the system in the real-time optimization stage by replacing the RTO by a Dynamic RTO (DRTO) [13], or
- (iii)

## 2. Problem Formulation

#### 2.1. EMPC Applied to DWTN

#### 2.2. Multi-Objective MPC of DWTN

- To provide a reliable water supply in the most economic way, minimizing water production and transport costs, written as$${J}_{1}\left(k\right)=({\alpha}_{1}+{\alpha}_{2}\left(k\right))u\left(k\right)\Delta t,$$
- To guarantee the availability of enough water in each reservoir to satisfy its underlying demand, keeping a safety stock in order to face uncertainties and avoid stock-outs. This objective is reached by minimizing$${J}_{2}\left(k\right)=\epsilon {\left(k\right)}^{\top}\epsilon \left(k\right),$$
- To operate the DWTN under smooth control actions. This is reached by minimizing$${J}_{3}\left(k\right)=\Delta u{\left(k\right)}^{\top}\Delta u\left(k\right),$$

## 3. Pareto Front Calculation of Multi-Objective Optimization Problems

**Definition**

**1.**

#### 3.1. Normalization

#### 3.2. (Normalized) Weighted Sum (WS)

#### 3.3. (Enhanced) Normalized Normal Constraint ((E)NNC)

## 4. Tuning Strategies for Multi-Objective EMPC

#### 4.1. Decision-Making Strategy for Multi-Objective Optimization

#### 4.1.1. DM Based on a Management Point

#### 4.1.2. DM Procedure and Prioritization

**Remark**

**1.**

**Remark**

**2.**

#### 4.2. Tuning Strategy Proposals

#### 4.2.1. Histogram-Based Weights Selection

**Step 1**. Calculate the number of water-demand combinations for the Pareto front for the specified objective functions.**Step 2.**Select, for all Pareto fronts, the preferred solution according to the decision-making procedure described above.**Step 3.**Make a histogram of the occurrence of the different sets of selected weights for the Normalized Weighted Sum.**Step 4.**Select, in the histogram, the weights with the highest number of occurrences and use the weights for implementation in the MPC.**Step 5.**Evaluate the controller in an online setting (without computing the entire Pareto set in each iteration).

#### 4.2.2. Model-Based Weights Selection

**Steps 1**to**3**are identical to the previous approach.**Step 4.**Calculate a regression model of the preferred set of weights as a function of the average of water demands.**Step 5.**Evaluate the controller in an online setting, i.e., in each MPC, use the regression model for the calculation of weights based on the water demand.

**Remark**

**3.**

## 5. Application Example

#### 5.1. Aggregate Model of the Barcelona DWTN

#### 5.2. Pareto Front Generation for the DWTN Problem

`TOMLAB/CPLEX`solver, and, for the calculation front points (the case of ENNCP), the

`TOMLAB/SNOPT`solver has been deployed. As has been exposed previously, the NWS optimization problems are convex, hence, in order to calculate Pareto front points with this method, only the

`TOMLAB/CPLEX`solver has been used.

#### 5.3. Solver Errors

- Infeasibility problem errors;
- Resource limit errors, related to the maximum number of iterations; and
- Numerical errors, related to ill-conditioning issues.

**Remark**

**4.**

#### 5.4. Key Performance Indicators

**Economic KPI:**This performance indicator is related to the water production and transport costs (4), and is defined as

**Safety KPI:**This performance indicator is related to the volume-regulation strategy of the tanks. It has been defined as

**Smoothness KPI:**This performance indicator is related to the smoothness of the control movements, and is defined as

**Remark**

**5.**

#### 5.5. DM Strategy Simulations

#### 5.5.1. DM Exploiting ENNCP

#### 5.5.2. DM Exploiting NWS

`CPLEX`because they are convex optimization problems. The rest of the points (63 of them) are solved with

`SNOPT`. In the case of simulations with the NWS method, all the Pareto front points have been calculated with

`CPLEX`because all the optimization problems are convex. Again, similar results are observed.

#### 5.6. Weight Variations and Measured Disturbances

#### 5.7. Tuning Strategy

#### 5.8. Results Discussion

## 6. Conclusions and Further Work

## Author Contributions

## Funding

## Conflicts of Interest

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Type of Component | Quantity |
---|---|

Water storing tanks | 17 |

Pumping stations | 26 |

Valves | 35 |

Nodes | 11 |

Sectors of consume | 25 |

Priority Percentages | Economic KPI | Safety KPI | Smoothness KPI | |||
---|---|---|---|---|---|---|

Day 2 | Day 3 | Day 2 | Day 3 | Day 2 | Day 3 | |

[100 100 100] | 34.3553 | 33.7995 | 3873.5 | 3888.2 | 0.0040 | 0.0042 |

[100 75 50] | 34.0348 | 33.5804 | 3900.5 | 3878.1 | 0.1699 | 0.1549 |

[50 100 75] | 39.6096 | 38.9024 | 4195.9 | 3931.9 | 0.0495 | 0.0538 |

[75 50 100] | 38.1425 | 36.4246 | 3716.1 | 3678.3 | 0.0019 | 0.0006 |

**Table 3.**KPIs for the DM Strategy (with condition (22)) using the ENNCP Method.

Priority Percentages | Economic KPI | Safety KPI | Smoothness KPI | |||
---|---|---|---|---|---|---|

Day 2 | Day 3 | Day 2 | Day 3 | Day 2 | Day 3 | |

[100 100 100] | 34.3553 | 33.7995 | 3873.5 | 3888.2 | 0.0040 | 0.0042 |

[50 30 20] | 34.4205 | 33.7557 | 4853.6 | 4541.7 | 0.2184 | 0.2632 |

[20 50 30] | 49.6496 | 48.7163 | 3360.7 | 3359.5 | 0.0186 | 0.0034 |

[30 20 50] | 46.9891 | 43.6090 | 3658.0 | 2537.8 | 0.0003 | 0.0002 |

Priority Percentages | Economic KPI | Safety KPI | Smoothness KPI | |||
---|---|---|---|---|---|---|

Day 2 | Day 3 | Day 2 | Day 3 | Day 2 | Day 3 | |

[100 100 100] | 34.3305 | 33.6452 | 3809.5 | 3822.2 | 0.0039 | 0.0035 |

[100 75 50] | 34.0022 | 33.4155 | 3501.8 | 3371.0 | 0.0086 | 0.0092 |

[50 100 75] | 42.7737 | 42.1820 | 4068.1 | 4000.4 | 0.0024 | 0.0018 |

[75 50 100] | 35.0110 | 34.2817 | 3578.0 | 3827.3 | 0.0028 | 0.0026 |

**Table 5.**KPIs for the DM Strategy (with condition (22)) using the NWS Method.

Priority Percentages | Economic KPI | Safety KPI | Smoothness KPI | |||
---|---|---|---|---|---|---|

Day 2 | Day 3 | Day 2 | Day 3 | Day 2 | Day 3 | |

[100 100 100] | 34.3305 | 33.6452 | 3809.5 | 3822.2 | 0.0039 | 0.0035 |

[50 30 20] | 33.8902 | 33.1499 | 4704.4 | 4886.1 | 0.2069 | 0.2217 |

[20 50 30] | 50.0738 | 48.7135 | 3309.9 | 3353.7 | 0.0032 | 0.0034 |

[30 20 50] | 48.3035 | 49.6586 | 3402.5 | 2178.9 | 0.0002 | 0.0001 |

Tuning Strategy | Economic KPI | Safety KPI | Smoothness KPI | |||
---|---|---|---|---|---|---|

Day 2 | Day 3 | Day 2 | Day 3 | Day 2 | Day 3 | |

Original MPC | 34.4477 | 34.5007 | 3921.7 | 3912.3 | 0.0105 | 0.0103 |

Normalised MPC | 34.5643 | 34.6338 | 3837.6 | 3838.3 | 0.0026 | 0.0025 |

Histogram-Based Weighting | 34.1424 | 34.2004 | 3324.7 | 3337.2 | 0.0017 | 0.0017 |

Adaptive Weighting | 33.4410 | 33.0017 | 3135.9 | 3023.0 | 0.0007 | 0.0006 |

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**MDPI and ACS Style**

Ocampo-Martinez, C.; Toro, R.; Puig, V.; Van Impe, J.; Logist, F.
Multi-Objective-Based Tuning of Economic Model Predictive Control of Drinking Water Transport Networks. *Water* **2022**, *14*, 1222.
https://doi.org/10.3390/w14081222

**AMA Style**

Ocampo-Martinez C, Toro R, Puig V, Van Impe J, Logist F.
Multi-Objective-Based Tuning of Economic Model Predictive Control of Drinking Water Transport Networks. *Water*. 2022; 14(8):1222.
https://doi.org/10.3390/w14081222

**Chicago/Turabian Style**

Ocampo-Martinez, Carlos, Rodrigo Toro, Vicenç Puig, Jan Van Impe, and Filip Logist.
2022. "Multi-Objective-Based Tuning of Economic Model Predictive Control of Drinking Water Transport Networks" *Water* 14, no. 8: 1222.
https://doi.org/10.3390/w14081222