# Dynamic Control of Yearly Drawdown Level of Overyear Regulation Reservoir in Cascade System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Joint Optimal Scheduling Model

#### 2.2. Multidimensional DP for Solving the Model

#### 2.3. Dynamic Control Bound Construction of Yearly Drawdown Level

## 3. Case Study

#### 3.1. Study Area

#### 3.2. Results and Analysis

_{i}

^{y}, which is obtained by its own frequency arrangement. If we want to obtain the optimal inflow process of the whole basin under a certain frequency P

_{s}, it is equivalent to calculate the y corresponding to the smallest e

_{y}, which is shown in the following Formula (9):

_{s}(a year). Conversely, a similar method can be used to derive the overall inflow frequency of each year for the river basin, such as the actual inflow frequency of the whole river basin in the year y.

## 4. Conclusions

- (1)
- The inflow series of the station in the middle of the basin and the calculated overall inflow series can both well represent the inflow situation of the whole basin. In these two cases, the relationship between the inflow frequency and the scatter points of the yearly drawdown level is relatively centralized and stable. The results of the inflow series of stations that are located in the upper and lower parts of the basin are relatively poor.
- (2)
- Under the control mode of the fixed yearly drawdown level, the maximum annual average power generation of the cascade system is 100.580 billion kWh, and the corresponding optimal yearly drawdown level is 2785 m, which is the dead water level. That is to say, if the yearly drawdown level is to be fixed, the dead water level is the best for the Lianghekou reservoir.
- (3)
- For the dynamic control bounds constructed by the two selected inflow series, the results calculated from their upper boundaries, lower boundaries and mean values are not significant, and their maximum differences are 0.110 billion kWh and 0.107 billion kWh, respectively. This shows that the results of the dynamic control bounds constructed by the two selected inflow series both have little fluctuation, which can well cope with the impact of inflow uncertainty on scheduling results.
- (4)
- The dynamic control bound constructed based on the overall inflow of the river basin is slightly better than that based on the inflow of the Jinxi station. In all the three cases where the upper boundary, lower boundary and mean value are used as the control rules, the power generation of the former is 0.01% higher than that of the latter.
- (5)
- By constructing a dynamic control mode of the yearly drawdown level, 63.4%~76.3% of the benefits of the lifting space of yearly drawdown level optimization can be realized by the dynamic control bound proposed in this paper, which has a very remarkable effect.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Db: | the index of beginning-state variables of downstream reservoirs. |

De: | the index of end-state variables of downstream reservoirs. |

E: | power generation over the whole scheduling period (kWh). |

Ep_{t}^{i}: | the evaporation capacity of the ith reservoir in the tth stage (m^{3}/s). |

f _{t}^{*}(V): | the sum of optimal outputs from the present stage t to the last stage T. |

H_{t}^{i}: | the average water head of the ith hydropower station in the tth stage (m). |

I_{t}^{i}: | the average interval inflow of the ith reservoir in the tth stage (m^{3}/s). |

K^{i}: | the output coefficient of the ith hydropower station. |

N_{t}^{i}: | the output of the ith hydropower station in the tth stage (kW). |

N^{i}_{t,min}: | the lower limit of N^{i}_{t}. |

N^{i}_{t,max}: | the upper limit of N^{i}_{t}. |

N_{t}(V_{t−1,}Q_{t}): | the total output of the tth stage. |

q_{t}^{i}: | the outflow through the turbines of the ith reservoir in the tth stage (m^{3}/s). |

Q^{i}_{t}: | the average discharge of the ith reservoir in the tth stage (m^{3}/s). |

Q_{t}: | the discharge flow determined by V_{t−1} and V_{t}. |

Q_{t} = (Q_{t}^{1}, Q_{t}^{2}, …, Q_{t} ^{n})’: | the decision variable vector. |

Q^{i}_{t,min}: | the lower limit of Q^{i}_{t}. |

Q^{i}_{t,max}: | the upper limit of Q^{i}_{t}. |

T: | the total number of stages over the whole scheduling period. |

D_{t:} | a set of feasible decisions that satisfy the constraints of the reservoir. |

Δt: | the duration of a stage (h). |

Ub: | the index of beginning-state variables of upstream reservoirs. |

Ue: | the index of end-state variables of upstream reservoirs. |

V^{i}_{t}: | the storage volume of the ith reservoir in the tth stage (m^{3}). |

V_{t}: | the storage state at the beginning of the stage t. |

V^{i}_{t,min}: | the lower limit of V^{i}_{t}. |

V^{i}_{t,max}: | the upper limit of V^{i}_{t}. |

V_{0}^{i}: | the storage volume of the ith reservoir at the beginning of the first stage. |

V_{b}^{i}: | the storage volume of the ith reservoir at the beginning of the whole scheduling period. |

V_{T}^{i}: | the storage volume of the ith reservoir at the end of the Tth stage. |

V_{e}^{i}: | the storage volume of the ith reservoir at the end of the whole scheduling period. |

V_{t−1} = (V_{t−1}^{1}, V_{t−1}^{2}, …, V_{t−1}^{n})’: | the state variable vector. V_{t}^{1}, V_{t}^{2}, and V_{t}^{3} are discretized, i.e., (V_{t}^{1,1}, V_{t}^{1,2}, …, V_{t}^{1,M}), (V_{t}^{2,1}, V_{t}^{2,2}, …, V_{t}^{2,M}) and (V_{t}^{3,1}, V_{t}^{3,2}, …, V_{t}^{3,M}). |

${F}_{\mathrm{t}}^{*}({V}_{t-1})$: | |

${F}_{T+1}{}^{*}({V}_{T})$: | the optimal cumulative output of a storage volume combination V_{T} at the end of the Tth stage (kW). |

W_{t}^{i}: | the average discharge of abandoned water of the ith reservoir in the tth stage (m^{3}/s). |

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**Figure 1.**Flowchart of multidimensional DP in solving joint optimal scheduling of cascade reservoirs.

**Figure 2.**Reverse recursion procedure of multidimensional DP for joint optimal scheduling of cascade reservoirs.

**Figure 3.**Optimal control scheme of yearly drawdown level of overyear regulation reservoir: dynamic control bound model.

**Figure 5.**Overyear optimal water level process of Lianghekou reservoir based on multidimensional DP.

**Figure 7.**Scatter charts of inflow frequency and yearly drawdown level of Lianghekou reservoir with different incoming flow frequencies as references.

**Figure 8.**Dynamic control bound of yearly drawdown level constructed by inflow series of Jinxi station.

**Figure 9.**Dynamic control bound of yearly drawdown level constructed by the overall inflow series of the whole basin.

**Figure 10.**Overyear average water level processes of Lianghekou reservoir under different control rules.

Item | Unit | Lianghekou | Yangfanggou | Jinxi | Jindong | Guandi | Ertan | Tongzilin |
---|---|---|---|---|---|---|---|---|

Normal level | m | 2865 | 2088 | 1880 | 1646 | 1330 | 1200 | 1015 |

Dead level | m | 2785 | 2094 | 1800 | 1640 | 1321 | 1155 | 1010 |

Annual average runoff | m^{3}/s | 664 | 896 | 1200 | 1220 | 1430 | 1670 | 1928 |

Mean annual precipitation | mm | 897 | None | None | 932 | 1077 | 1038 | 1040 |

Temperature | °C | 10.9 | 13.7 | 13.7 | 13.7 | 18.6 | 19.8 | 19.7 |

Flood control level | m | 2845.9 | None | 1859 | None | None | 1190 | None |

Regulation performance | --- | Overyear | Daily | Yearly | Daily | Daily | Seasonal | Daily |

Range of operating water level in dry season | m | [2845.9, 2865] | 2092 | [1859, 1880] | 1644 | 1328 | [1190, 1200] | 1013.5 |

Range of operating water level in flood season | m | [2785, 2865] | 2092 | [1800, 1880] | 1644 | 1328 | [1155, 1200] | 1013.5 |

**Table 2.**Annual power generation of cascade system under different yearly drawdown levels of Lianghekou reservoir.

Yearly Drawdown Level/m | Total Power Generation/Billion kWh |
---|---|

2785 | 100.58 |

2790 | 100.55 |

2795 | 100.49 |

2800 | 100.43 |

2805 | 100.30 |

2810 | 100.16 |

2815 | 99.95 |

2820 | 99.72 |

2825 | 99.43 |

2830 | 99.08 |

2835 | 98.64 |

2840 | 98.10 |

2845 | 97.37 |

**Table 3.**Power generation results under the three control modes based on the obtained dynamic control bound.

Computation Method | Total Power Generation/Billion kWh | Increment Compared to Fixed Water Level Mode |
---|---|---|

Upper boundary of the dynamic control bound constructed based on Jinxi inflow series | 101.255 | 0.67% |

Lower boundary of the dynamic control bound constructed based on Jinxi inflow series | 101.173 | 0.59% |

Mean value of the dynamic control bound constructed based on Jinxi inflow series | 101.283 | 0.70% |

Upper boundary of the dynamic control bound constructed based on the overall inflow series | 101.267 | 0.68% |

Lower boundary of the dynamic control bound constructed based on the overall inflow series | 101.183 | 0.60% |

Mean value of the dynamic control bound constructed based on the overall inflow series | 101.290 | 0.71% |

Fixed yearly drawdown level (determined by discretized water levels) | 100.580 | 0.00% |

Optimal calculation results based on multidimensional DP | 101.520 | 0.93% |

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## Share and Cite

**MDPI and ACS Style**

Chang, Z.; Jiang, Z.; Yuan, X.
Dynamic Control of Yearly Drawdown Level of Overyear Regulation Reservoir in Cascade System. *Water* **2023**, *15*, 665.
https://doi.org/10.3390/w15040665

**AMA Style**

Chang Z, Jiang Z, Yuan X.
Dynamic Control of Yearly Drawdown Level of Overyear Regulation Reservoir in Cascade System. *Water*. 2023; 15(4):665.
https://doi.org/10.3390/w15040665

**Chicago/Turabian Style**

Chang, Zongye, Zhiqiang Jiang, and Xiaohui Yuan.
2023. "Dynamic Control of Yearly Drawdown Level of Overyear Regulation Reservoir in Cascade System" *Water* 15, no. 4: 665.
https://doi.org/10.3390/w15040665