# Qualifying Coordination Mechanism for Cascade-Reservoir Operation with a New Game-Theoretical Methodology

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study Area and Data

#### 2.1. Study Area

#### 2.2. Date Collection

## 3. Methodology

#### 3.1. Centralized Model (Model I)

#### 3.2. Non-Cooperative Model (Model II)

#### 3.3. Integrated Game-Theoretical Model

#### 3.3.1. Coordination Model (Model III(a))

#### 3.3.2. Benefits Compensation Model (Model III(b))

#### 3.4. Evaluation Criterion

#### 3.5. Solving Method

## 4. Results and Discussion

#### 4.1. Nash Equilibrium Strategies of Multi-Reservoir Operation

#### 4.2. Assessment of Game-Theoretical Model

#### 4.2.1. Coordination Assessment

**,**the amount of water flowing into the downstream reservoirs is increased [35]. It is indicated that through coordination, the hydropower generation of the downstream reservoir is improved by scarifying the priority of the upstream reservoir in generating hydropower.

#### 4.2.2. Benefits Compensation Assessment

#### 4.3. Factors Affecting the Efficiency of Coordination

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Flowchart of the solving process of the game-theoretical model: (

**a**) coordination model; (

**b**) benefits compensation model.

**Figure 4.**Solutions of the cascade-reservoir operation through the coordination model: (

**a1**) strategies of two coalitions under the high flow scenario; (

**b1**) strategies of two coalitions under the medium flow scenario; (

**c1**) strategies of two coalitions under the low flow scenario; (

**a2**) best response curves of coalitions under the high flow scenario; (

**b2**) best response curves of coalitions under the medium flow scenario; and (

**c2**) best response curves of coalitions under the low flow scenario.

**Figure 5.**Total hydropower generation of cascade-reservoir system based on Nash equilibrium strategies: (

**a**) high flow scenario; (

**b**) medium flow scenario; (

**c**) low flow scenario.

**Figure 6.**Hydropower generation deviations of four reservoirs obtained based on Model III(a) compared to Model II.

**Figure 7.**Storage of cascade reservoirs based on Models II and III(a) under the three flow scenarios.

**Figure 8.**Percentage changes of the hydropower generation of two coalitions obtained by Model III(a) under the three flow scenarios, compared to Model II.

Reservoir | XLD | XJB | TGD | GZB |
---|---|---|---|---|

Normal water level (m) | 600 | 380 | 175 | 66 |

Flood limit water level (m) | 560 | 370 | 145 | 64.5 |

Dead water level (m) | 540 | 370 | 145 | 62 |

Firm power (MW) | 3790 | 2009 | 4990 | 1130.5 |

Install capacity (MW) | 12,600 | 6000 | 22,500 | 2940 |

Output coefficient | 8.7 | 8.7 | 8.8 | 8.5 |

Statistics | Hydrological Station | |
---|---|---|

Pinshan Station | Yichang Station | |

maximum daily flow (m^{3}/s) | 16,964.52 | 50,312.90 |

coefficient of variation | 0.15 | 0.10 |

z test | 0.85 $\uparrow $ | −2.35 $\downarrow $ |

Parameters | Value |
---|---|

Iteration | 500 |

Population | 100 |

Crossover rate | 0.8 |

Mutation rate | 0.1 |

**Table 4.**The individual and overall hydropower generation of cascade-reservoir system based on the three models under the three flow scenarios.

Reservoirs | $\mathbf{High}\text{}\mathbf{Flow}\text{}({\mathbf{10}}^{\mathbf{8}}\text{}\mathbf{kW}\cdot \mathbf{h})$ | $\mathbf{Medium}\text{}\mathbf{Flow}\text{}({\mathbf{10}}^{\mathbf{8}}\text{}\mathbf{kW}\cdot \mathbf{h})$ | $\mathbf{Low}\text{}\mathbf{Flow}\text{}({\mathbf{10}}^{\mathbf{8}}\text{}\mathbf{kW}\cdot \mathbf{h})$ | ||||||
---|---|---|---|---|---|---|---|---|---|

Model I | Model II | Model III(a) | Model I | Model II | Model III(a) | Model I | Model II | Model III(a) | |

XLD | 702.96 | 705.19 | 703.33 | 581.93 | 582.74 | 580.87 | 512.01 | 513.77 | 508.00 |

XJB | 346.23 | 344.99 | 345.98 | 292.50 | 292.42 | 291.00 | 258.38 | 258.28 | 255.97 |

TGD | 1003.23 | 997.96 | 1002.26 | 981.93 | 973.39 | 982.81 | 810.17 | 795.36 | 813.28 |

GZB | 184.71 | 185.68 | 184.92 | 179.04 | 179.37 | 178.61 | 164.61 | 165.45 | 164.22 |

Total | 2237.13 | 2233.82 | 2236.49 | 2035.40 | 2027.92 | 2033.29 | 1745.17 | 1732.86 | 1741.47 |

**Table 5.**Hydropower generation of each reservoir based on Nash-Harsanyi bargaining theory under the three flow scenarios.

Reservoirs | $\mathbf{High}\text{}\mathbf{Flow}\text{}({\mathbf{10}}^{\mathbf{8}}\text{}\mathbf{kW}\cdot \mathbf{h})$ | $\mathbf{Medium}\text{}\mathbf{Flow}\text{}({\mathbf{10}}^{\mathbf{8}}\text{}\mathbf{kW}\cdot \mathbf{h})$ | $\mathbf{Low}\text{}\mathbf{Flow}\text{}({\mathbf{10}}^{\mathbf{8}}\text{}\mathbf{kW}\cdot \mathbf{h})$ | |||
---|---|---|---|---|---|---|

Model III(a) | Model III(b) | Model III(a) | Model III(b) | Model III(a) | Model III(b) | |

XLD | 703.33 | 705.86 (703.33 + 2.53) | 580.87 | 584.08 (580.87 + 3.21) | 508.00 | 515.92 (508.00 + 7.92) |

XJB | 345.98 | 345.65 (345.98 − 0.33) | 291.00 | 293.76 (291.00 + 2.76) | 255.97 | 260.43 (255.97 + 4.46) |

TGD | 1002.26 | 998.63 (1002.26 − 3.63) | 982.81 | 974.73 (982.81 − 8.08) | 813.28 | 797.52 (813.28 − 15.76) |

GZB | 184.92 | 186.35 (184.92 + 1.43) | 178.61 | 180.72 (178.61 + 2.11) | 164.22 | 167.60 (164.22 + 3.38) |

**Table 6.**Hydropower generation of the participating reservoirs based on Models II and III under the three flow scenarios.

Patterns | Reservoir | High Flow (10^{8} kW·h) | Medium Flow (10^{8} kW·h) | Low Flow (10^{8} kW·h) | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Model II | Model III(a) | Model III(b) | Model II | Model III(a) | Model III(b) | Model II | Model III(a) | Model III(b) | ||

Pattern I | XLD | 705.19 | 696.98 | - | 582.74 | 573.42 | - | 513.77 | 504.14 | - |

XJB | 344.99 | 347.92 | - | 292.42 | 293.56 | - | 258.28 | 259.51 | - | |

Total | 1050.19 | 1044.90 | - | 875.16 | 866.98 | - | 772.05 | 763.65 | - | |

Pattern II | XJB | 344.99 | 343.98 | 346.11 | 292.42 | 288.33 | 294.60 | 258.28 | 251.04 | 261.81 |

Coalition 2 | 1183.64 | 1186.89 | 1184.76 | 1152.77 | 1161.23 | 1154.96 | 960.82 | 975.12 | 964.35 | |

Total | 1528.63 | 1530.87 | 1530.87 | 1445.19 | 1449.56 | 1449.56 | 1219.10 | 1226.16 | 1226.16 |

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**MDPI and ACS Style**

Xu, Y.; Fu, X.; Qin, J.
Qualifying Coordination Mechanism for Cascade-Reservoir Operation with a New Game-Theoretical Methodology. *Water* **2018**, *10*, 1857.
https://doi.org/10.3390/w10121857

**AMA Style**

Xu Y, Fu X, Qin J.
Qualifying Coordination Mechanism for Cascade-Reservoir Operation with a New Game-Theoretical Methodology. *Water*. 2018; 10(12):1857.
https://doi.org/10.3390/w10121857

**Chicago/Turabian Style**

Xu, Yuni, Xiang Fu, and Jianan Qin.
2018. "Qualifying Coordination Mechanism for Cascade-Reservoir Operation with a New Game-Theoretical Methodology" *Water* 10, no. 12: 1857.
https://doi.org/10.3390/w10121857