# Multi-Objective Model Predictive Control for Real-Time Operation of a Multi-Reservoir System

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Internal Model or Reservoir System Model

#### 2.2. Multi-Objective Optimization

#### 2.2.1. Multi-Objective Optimization in the Classical MPC

#### 2.2.2. Multi-Objective Optimization in MOMPC

#### 2.3. Multi-Criteria Decision Making

#### 2.3.1. Weighted-Sum Method

#### 2.3.2. Maximin Method

#### 2.3.3. Maxisum Method

#### 2.4. Performances of the System under Alternative Operating Rules

## 3. Case Study: A Multi-Reservoir System in the Sittaung River Basin

#### 3.1. Control Objectives

- The first objective (${J}_{1}$) is to maintain the target water levels of reservoirs (or desired storage volume) for irrigation and hydropower generation in the dry season. Therefore, the deviations of reservoirs water level from its reference level are minimized by:$$\begin{array}{c}\mathrm{min}{J}_{1}\left(u\right)=\sum _{i=1}^{{N}_{r}}\sum _{k=1}^{N}{\alpha}_{h,i}({h}_{i}^{k}-{h}_{ref,i}^{k}{)}^{2}+\sum _{i=1}^{11}\sum _{k=1}^{N-1}{\alpha}_{\mathsf{\Delta}u,i}(\mathsf{\Delta}{u}_{i}^{k}{)}^{2}\\ \mathsf{\Delta}{u}_{i}^{k}={u}_{i}^{k}-{u}_{i}^{k-1}\end{array}$$
- The second objective (${J}_{2}$) is to reduce the flood risk at the Taungoo city. A soft constraint [29] is implemented to minimize the water level deviations above the safety water level at a flood control station, which is defined as:$$\begin{array}{cc}\hfill \mathrm{min}{J}_{2}\left(u\right)& =\sum _{k=1}^{N}{\alpha}_{h,p}{({h}_{p}^{k}-{u}_{*}^{k})}^{2}+\sum _{k=1}^{N}{\alpha}_{{u}_{*}}({u}_{*}^{k}{)}^{2}\hfill \\ \hfill & {u}_{*}^{k}=\left\{\begin{array}{cc}{h}_{p}^{k}& \mathrm{if}\phantom{\rule{0.277778em}{0ex}}{h}_{p}^{k}\le {h}_{p,sl}\\ {h}_{p,sl}& \mathrm{if}\phantom{\rule{0.277778em}{0ex}}{h}_{p}^{k}>{h}_{p,sl}\end{array}\right.\hfill \end{array}$$
- The last objective (${J}_{3}$) is to maximize electric energy production at the three reservoirs which is defined as [65]:$$\mathrm{max}{J}_{3}\left(u\right)=\sum _{i=1}^{{N}_{r}}\sum _{k=1}^{N}{\eta}_{i}g\gamma {u}_{i}^{k}({h}_{i}^{k}-{h}_{tw,i}^{k})\times {10}^{-6}$$

#### 3.2. Model Description

#### 3.3. Simulation Settings

## 4. Results

#### 4.1. Pareto Fronts and Trade-Offs

#### 4.2. Performance of the System

#### 4.3. Overall Performance

#### 4.4. Comparison of Results

## 5. Discussion

#### 5.1. The Use of GA in MPC Formulation

#### 5.2. Selection of a Decision-Making Method

#### 5.3. Limitations of the Method

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Sittaung river basin (locations of reservoirs are represented with the red diamonds and a flood control point (Taungoo city) is shown with a green dot.).

**Figure 3.**(

**a**–

**c**) Two dimensional Pareto front and selected solutions using different decision criteria, (

**d**) Three dimensional Pareto front and selected solutions using different decision criteria (The objective costs are normalized and the black arrows indicate the direction of increasing preference of each objective.The circle’s size and colour indicate performance of each alternative in terms of all objective).

**Figure 4.**(

**a**–

**c**) Two dimensional Pareto front and selected solutions using different decision criteria, (

**d**) Three dimensional Pareto front and selected solutions using different decision criteria (The objective costs are normalized and the black arrows indicate the direction of increasing preference of each objective. The circle’s size and colour indicate performance of each alternative in terms of all objective).

**Figure 5.**(

**a**–

**d**) Inflows, outflows and water levels of reservoirs 1 (

**a**), 4 (

**b**), 8 (

**c**) and 9 (

**d**) under DR-1.

**Figure 8.**Parallel line plot for the eight decision rules(Each line represents the overall performance of the three objectives with respect to a particular decision rule and the black arrows indicate the direction of good preference.).

NSGA-II Parameters | Setting Value |
---|---|

Population size | 200 |

Maximum number of generations in each run | 1000 |

Crossover rate | 0.80 |

Mutation rate | 0.35 |

Stopping criteria | 1. The average change in the spread of the Pareto front over generation (=100) is less than or equal to function tolerance (${10}^{-4})$. |

2. The maximum number of generations is reached. |

Decision Rule | Method | w |
---|---|---|

DR-1 | weighted-sum | [1,0,0] |

DR-2 | weighted-sum | [0,1,0] |

DR-3 | weighted-sum | [0,0,1] |

DR-4 | weighted-sum | [0.6,0.2,0.2] |

DR-5 | weighted-sum | [0.2,0.6,0.2] |

DR-6 | weighted-sum | [0.2,0.2,0.6] |

DR-7 | maximin | - |

DR-8 | maxisum | - |

Decision Rule (Weight) | Indicator | Overall Performance (Average) | ||
---|---|---|---|---|

Reliability of Meeting Storage Capacity V (%) | Reliability of Meeting Flood Control E (%) | Reliability of Maximizing Hydropower Generation P (%) | ||

DR-1 (1,0,0) | 93 | 54 | 74 | 74 |

DR-2 (0,1,0) | 50 | 100 | 68 | 73 |

DR-3 (0,0,1) | 0 | 0 | 100 | 33 |

DR-4 (0.6.0.2.0.2) | 92 | 50 | 74 | 72 |

DR-5 (0.2,0.6.0.2) | 67 | 60 | 82 | 69 |

DR-6 (0.2,0.2,0.6) | 25 | 9 | 97 | 44 |

DR-7 | 62 | 22 | 85 | 56 |

DR-8 | 62 | 23 | 82 | 56 |

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**MDPI and ACS Style**

Myo Lin, N.; Tian, X.; Rutten, M.; Abraham, E.; Maestre, J.M.; van de Giesen, N. Multi-Objective Model Predictive Control for Real-Time Operation of a Multi-Reservoir System. *Water* **2020**, *12*, 1898.
https://doi.org/10.3390/w12071898

**AMA Style**

Myo Lin N, Tian X, Rutten M, Abraham E, Maestre JM, van de Giesen N. Multi-Objective Model Predictive Control for Real-Time Operation of a Multi-Reservoir System. *Water*. 2020; 12(7):1898.
https://doi.org/10.3390/w12071898

**Chicago/Turabian Style**

Myo Lin, Nay, Xin Tian, Martine Rutten, Edo Abraham, José M. Maestre, and Nick van de Giesen. 2020. "Multi-Objective Model Predictive Control for Real-Time Operation of a Multi-Reservoir System" *Water* 12, no. 7: 1898.
https://doi.org/10.3390/w12071898