# Short-Term Optimal Operation of Cascade Reservoirs Considering Dynamic Water Flow Hysteresis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Influence of Water Flow Hysteresis

^{3}/s; ${\tau}_{i-1}$ is the water flow hysteresis between reservoir $i-1$ and reservoir $i$, unit: h; ${M}_{i}$ is the number of tributaries between reservoir $i-1$ and reservoir $i$; ${\tau}_{j}^{\prime}$ is the water flow hysteresis between outflow of tributary $j$ and inflow of the reservoir $i$, unit: h; $Q{c}_{i-1}(t-{\tau}_{i-1})$ is the outflow of reservoir $i-1$ in period $t-{\tau}_{i-1}$, unit: m

^{3}/s. $Q{c}_{j}^{\prime}(t-{\tau}_{j}^{\prime})$ is the outflow of tributary $j$ in period $t-{\tau}_{j}^{\prime}$, unit: m

^{3}/s.

## 3. Dynamic Water Flow Hysteresis Method Based on the Space Mapping Principle

## 4. Mathematical Model for Short-Term Optimal Operation of Cascade Reservoirs Considering Dynamic Water Flow Hysteresis

#### 4.1. Objective Function

#### 4.2. Constraint Conditions

^{4}m

^{3}; $Q{c}_{i,t}$ is the outflow of the reservoir $i$ in period $t$, unit: m

^{3}/s; $Q{f}_{i,t}$ is the generation flow of reservoir $i$ in period $t$, unit: m

^{3}/s; $Q{q}_{i,t}$ is the abandoned flow of reservoir $i$ in period $t$, unit: m

^{3}/s; $V\_{\mathrm{min}}_{i}$, $V\_{\mathrm{max}}_{i}$ are the lower and upper limits of storage capacity of reservoir $i$, respectively, unit: 10

^{4}m

^{3}; $Q\_{\mathrm{min}}_{i}$ and $Q\_{\mathrm{max}}_{i}$ are the lower and upper limits of the outflow of reservoir $i$, respectively, unit: m

^{3}/s; $N\_{\mathrm{min}}_{i}$ and $N\_{\mathrm{max}}_{i}$ are the lower and upper limits of the output of reservoir $i$, respectively, unit: MW; ${Z}_{i,T+1}$ and ${Z}_{i,\mathrm{end}}$ are the water level of reservoir $i$ in period $T$ and the controlled water level of reservoir $i$, respectively, unit: m; $F(\u2022)$ is the space mapping operator.

## 5. Model Solution

## 6. Case Study

## 7. Conclusions

- Compared with the FWFHM, the space mapping principle is adopted to establish a DWFHM that can fully consider the effect of multi-period outflow on the current period inflow. It can take into account both the change of water flow propagation time caused by the change of the magnitude of discharge and the impact of water flow attenuation caused by factors such as the river channel storage, thus leading to a more accurate calculation result.
- The short-term optimal operation model of cascade reservoirs with dynamic water flow hysteresis is put forward and solved by the IPOA. Compared with the traditional model using the FWFHM, the improved model can effectively increase the generated benefit. The new model can fully consider the impact of flow rate, water head, and the installed capacity of cascade reservoirs. In the non-flood season, when the inflow is small, the priority is given to raising the water head of the Guandi hydropower station, in order to increase the total generated energy. In the flood season, when the inflow is large, the improved model makes full use of the installed capacity of the Jindong hydropower station so as to increase the total generated energy.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Comparison of the calculated results of the FWFHM and the DWFHM; (

**a**) Non-flood season; (

**b**) Flood season.

Items | Unit | Jindong | Guandi |
---|---|---|---|

Normal water level | m | 1646 | 1330 |

Dead water level | m | 1640 | 1321 |

Regulation volume | 10^{8} m^{3} | 0.0905 | 6.06 |

Regulation performance | - | daily regulation | daily regulation |

Installed capacity | MW | 4800 | 2400 |

(MW·h) | Scheme 1 | Scheme 2 | Scheme 3 |
---|---|---|---|

Jindong generated energy | 48,081 | 48,081 | 47,617 |

Guandi generated energy | 19,453 | 17,813 | 21,103 |

Total generated energy | 67,534 | 65,894 | 68,721 |

(MW·h) | Scheme 1 | Scheme 2 | Scheme 3 |
---|---|---|---|

Jindong generated energy | 110,207 | 110,207 | 111,268 |

Guandi generated energy | 46,965 | 45,329 | 45,833 |

Total generated energy | 157,173 | 155,537 | 157,101 |

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**MDPI and ACS Style**

Zhang, Y.; Liu, Y.; Wu, Y.; Ji, C.; Ma, Q.
Short-Term Optimal Operation of Cascade Reservoirs Considering Dynamic Water Flow Hysteresis. *Water* **2019**, *11*, 2098.
https://doi.org/10.3390/w11102098

**AMA Style**

Zhang Y, Liu Y, Wu Y, Ji C, Ma Q.
Short-Term Optimal Operation of Cascade Reservoirs Considering Dynamic Water Flow Hysteresis. *Water*. 2019; 11(10):2098.
https://doi.org/10.3390/w11102098

**Chicago/Turabian Style**

Zhang, Yanke, Yuan Liu, Yueqiu Wu, Changming Ji, and Qiumei Ma.
2019. "Short-Term Optimal Operation of Cascade Reservoirs Considering Dynamic Water Flow Hysteresis" *Water* 11, no. 10: 2098.
https://doi.org/10.3390/w11102098