# Long-, Medium-, and Short-Term Nested Optimized-Scheduling Model for Cascade Hydropower Plants: Development and Practical Application

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## Abstract

**:**

## 1. Introduction

## 2. Forecast-Based Reservoir Operation: A Review

## 3. Project and Its Operation

#### 3.1. Project Overview

#### 3.2. Reservoir Operating Rule Curves

- (1)
- The falling stage lasts from early November to 10 June of the following year, during which time the water level falls gradually from 175 m. During a year of normal incoming flow, the reservoir’s water level at the end of April will not be lower than 155 m, the falling low water level in dry season. In May, the reservoir can be operated for increased output, gradually reducing the water level. In general, the water level will fall to 155 m at the end of May and to 145 m by 10 June.
- (2)
- The flood season lasts from 11 June to 10 September, during which time the water level fluctuates between 144.9 and 146.5 m.
- (3)
- The impounding stage lasts from 11 September to the end of October; the level starts from the flood control level and recovers to at least 158 m by the end of September and to 175 m (normal water level) by the end of October, the end of the impounding stage.

## 4. Methodology

#### 4.1. Nesting Method for Multiple-Time-Scale Optimized-Scheduling Model

#### 4.1.1. Division of Periods for Scheduling Cascade Reservoir

- First layer: The scheduling period is the year. For June and for September, the scheduling interval is 10 days, and for the rest of the year, the scheduling interval is the month.
- Second layer: The scheduling periods are the falling stage, the flood season, and the impounding stage. From January to April of the falling stage, the scheduling interval is the month; from 1 May to 10 June of the falling stage, the scheduling interval is 10 days. For the flood season, the scheduling interval is the day. For the impounding stage (11 September to the end of October), the scheduling interval is 10 days. For November and December of the falling stage, the scheduling interval is the month.
- Third layer: The scheduling period is the month, and the scheduling interval is 10 days.
- Fourth layer: The scheduling period is 10 days, and the scheduling interval is the day.
- Fifth layer: The scheduling period is the day, and the scheduling interval is the hour.

#### 4.1.2. Interactions of Multiple-Time-Scale Nested Optimized-Scheduling Model

#### 4.1.3. Technical Process for Real-Time Generation of Reservoir Scheduling Scheme

**Step 1**: Taking into account the accuracy of the cascade reservoir group runoff forecast, as well as the time lag effect of the upper and lower cascade reservoir group flow, divide the scheduling period into five time scales: the year, the annual cycle (falling stage, flood season, and impounding stage), month, 10-day period, and day. Establish the five-layer nested structure of scheduling periods with time scales corresponding to the scheduling periods.**Step 2**: Specific to the cascade reservoirs, establish the scheduling model with the goal being to take the fullest advantage of the storage capacity of these reservoirs. Constraints include water balance, hydraulic connection, generating unit output, reservoir storage capacity, reservoir outflow, power plant output, power load, and water level constraints.**Step 3**: To balance computational efficiency and calculation accuracy, use both CIMAR-IDP and the IGA method to solve the scheduling model at the same time.**Step 4**: Use the scheduling schemes generated earliest by the two methods to guide the actual scheduling. Generally speaking, IGA is fast, and the scheduling scheme from this algorithm is produced earlier than that produced by CIMAR-IDP.**Step 5**: If a scheduling scheme is obtained by CIMAR-IDP, use it as the benchmark scheme to revise or replace the scheme obtained via IGA as the initial value for the next layer.

#### 4.2. Optimized Scheduling and Solution Method

#### 4.2.1. Formation of Optimal Reservoir Operation Problem

**Objective function**

**Constraints**

**(1) Water balance**

**(2) Hydraulic connection**

**(3) Output function**

**(4) Reservoir storage capacity**

**(5) Reservoir outflow**

**(6) Reservoir outflow**

**(7) System load**

**(8) Water level:**

- Reservoir upper/lower water level constraint$$\underline{{Z}_{j,t}}\le {Z}_{j,t}\le \overline{{Z}_{j,t}}$$
- Reservoir water level change amplitude constraint$$\left|\begin{array}{c}{Z}_{j,t+1}-{Z}_{j,t}\end{array}\right|\le \u25b3{Z}_{j}$$
- Water level control at the end of scheduling period$${Z}_{je}={Z}_{je}^{\ast}$$

#### 4.2.2. Method for Solving Scheduling Model

**Dynamic programming and its adaption**: the variables, equations, and penalty for DP are detailed as follows.

- (1)
- Stage and stage variable

- (2)
- State and state variable

- (3)
- Decision variable

- (4)
- State transition equation

- (5)
- Recursive equation

- (6)
- Penalty function

#### IDP Algorithm

**Step 1**: Propose an initially feasible scheduling line (that is, a feasible track) ${Z}_{t}^{0}(t=1,2,\dots ,T)$ in line with the constraints (the initial and ending conditions). The feasible scheduling line should be a water level change curve such that the corresponding reservoir operations to regulate the inflow will be within the allowable range of variation for the reservoir. It is generally not difficult to draw up the initially feasible scheduling line, especially for the condition that water is not allowed to be abandoned from the reservoir; in this case, the highest water level should be chosen since that would allow the hydropower plant to generate the maximum power output. An initially feasible scheduling line estimated in this way might be very close to the optimal scheduling one. For this case, see Figure 5.

**Step 2**: Taking the initially feasible scheduling line as the center, select several water levels at increments (steps) of $\u25b3Z$ above and below the line, forming a strategy “corridor” of several discrete values. At points t = 1 and t = T, $\u25b3Z$ = 0.

**Step 3**: Within the scope of the strategy corridor thereby formed, use the dynamic programming method to iterate the optimal scheduling line ${Z}_{t}^{\ast}$ within the scope of this strategy corridor chronologically.

**Step 4**: If $\left|\begin{array}{c}{Z}_{t}^{\ast}-{Z}_{t}^{0}\end{array}\right|\ge \epsilon $, set ${Z}_{t}^{0}={Z}_{t}^{\ast}$ (t = 1, 2, …, T), and recalculate according to Steps 2–3. If $\left|\begin{array}{c}{Z}_{t}^{\ast}-{Z}_{t}^{0}\end{array}\right|<\epsilon $, it means that the selected step length cannot be optimized. In this case, use the scheduling line obtained as the initial scheduling line, and continue decreasing the step length $\u25b3Z$ to perform the optimization calculation until a step length is reached at which the accuracy requirement is satisfied. At that point, the optimal scheduling line ${Z}_{t}^{\ast}$ is the solution.

#### CIMAR-IDP

**Step 1**: Assign one initial scheduling line ${Z}_{i,t}^{0}$ (i = 1, 2, …, n; t = 1, 2, …, T) to each reservoir.

**Step 2**: Fix ${Z}_{i,t}^{0}$ (i = 2, 3, …, n; t = 1, 2, …, T), and perform the proposed optimized scheduling calculation against the first reservoir to obtain the optimal scheduling line ${Z}_{1,t}^{\ast}$. When calculating, pay attention to the hydraulic connection among the reservoirs. Assign the sum of the output values of the whole cascade as the output value.

**Step 3**: Fixing ${Z}_{1,t}^{\ast}$ and ${Z}_{i,t}^{0}$ (i = 3, 4, …, n; t = 1, 2, …, T), perform the optimization calculation against the next reservoir to obtain the optimal scheduling line ${Z}_{2,t}^{\ast}$.

**Step 4**: Continue in this manner to obtain the optimal scheduling lines ${Z}_{1,t}^{\ast}$, ${Z}_{2,t}^{\ast}$, …, ${Z}_{n,t}^{\ast}$ for all reservoirs.

**Step 5**: If $\left|\begin{array}{c}{Z}_{i,t}^{\ast}-{Z}_{i,t}^{0}\end{array}\right|<\epsilon $, the optimal scheduling line at this point is the optimal solution. If $\left|\begin{array}{c}{Z}_{i,t}^{\ast}-{Z}_{i,t}^{0}\end{array}\right|\ge \epsilon $, set ${Z}_{i,t}^{0}$=${Z}_{i,t}^{\ast}$ and return to Step 2.

#### GA and Its Adaptation

^{”}population generated from the mutation.

#### Improved GA

**(1) Initial population generation of uniform design**

**(2) Improvements to operators**

**(a) Improvement to crossover operator**

**Step 1**: As shown in Figure 6a, the water level at time is constrained by two time frames, the one before and the one after. The water level of feasibility region1 ( $\left[\begin{array}{c}\underline{Z{F}_{pos,j}},\overline{Z{F}_{pos,j}}\end{array}\right]$) at time $pos$ can be estimated directly based on the water balance and the upper/lower limits of the output.

**Step 2**: Similarly, as shown in Figure 6b,${V}_{pos-1,j}=Z\_{V}_{j}\left({p}_{{i}_{2},j,pos-1}\right)$, ${V}_{pos+1,j}=$$Z\_{V}_{j}\left({p}_{{i}_{1},j,pos+1}\right)$, and another individual ${p}_{k+1,t}^{{}^{\prime}}$ is generated according to the above cross- over operation.

**(b) Improvement to mutation operator**

## 5. Results and Discussion

#### 5.1. Test of IGA Performance

#### 5.1.1. Test Parameter Settings

**Convergence**: A unified consideration of local convergence and global convergence is conducted, and if within the designated number of iterations the optimal solution has been maintained for $Snum$ generations when the algorithm stops, then it is deemed a convergence.**Convergence rate**: the ratio of the number of convergences to the total number of experiment runs.**Average calculation time, average power generation, standard deviation**: values calculated from multiple repeated tests for statistical breakdown.

#### 5.1.2. Analysis of Test Results

- 1.
- For average conditions, the optimal solution produced by IGA, with better global convergence, is closer to the global optimal solution than that of GA.
- 2.
- A feasible solution may become an infeasible one because of damage by GA; the maximum damage rate is 21.8%. The average electric energy under IGA may be higher, mainly because the improvement due to operator inspection reduces the proportion of individuals that are damaged; thus, the algorithm will be able to find the optimal solution in a more stable and effective manner.
- 3.
- IGA has a high convergence rate and a small standard deviation in electric energy, meaning that the convergence is more stable. The difference between GA and IGA is more obvious when the population size is small, mainly because the initial population of uniform design has better representativeness, but a randomly generated population has high distribution randomness in solution space, and the difference between the two will decrease as the population size increases. Only for higher population sizes can the genetic diversity of GA be guaranteed. Additionally, only by this method can the probability of GA being locally concentrated in solution space be reduced and calculation accuracy be improved.
- 4.
- The main advantage of IGA is that it uses a small population size to rapidly obtain high-accuracy convergence, but as the population size gradually increases, the advantage of IGA is no longer so obvious. This is because the IGA is added with threshold estimation; as the population size increases, the increase in calculation time is much greater than that with GA.

#### 5.2. Real-World Implementation

#### 5.2.1. Reservoir Scheduling and Its Data Management

#### 5.2.2. Display of Calculation Results and Analysis of Scheduling Effect

**Process 1**: The yearly optimal scheduling model is solved to obtain the water levels at the end of the falling stage, flood season, and impounding stage, which are 155.0 m, 146.5 m, and 175.0 m, respectively. See Table 4 for the results of the scheduling scheme for this example year.

**Process 2**:

- 1.
- The falling stage model uses the falling stage water level obtained from the yearly model as the terminal water level to prepare the reservoir falling scheme for the falling stage. See Figure 10a for the scheduling scheme for the falling stage of the TG reservoir, and see Table 5 for the scheduling results.
- 2.
- The flood season model uses the water level of the flood season obtained from the yearly model as the terminal water level to prepare the reservoir falling scheme for the flood season. See Figure 10b for the scheduling scheme for the TG reservoir for the flood season, and see Table 6 for the scheduling results.
- 3.
- The impounding stage model uses the water level of the impounding stage obtained from the yearly model as the terminal water level to prepare the reservoir falling scheme for the impounding stage. See Figure 11a for the scheduling scheme for the TG reservoir for the impounding stage, and see Table 7 for the scheduling results.

**Process 3**: For the example case of January (in the falling stage), the monthly model uses the water level at the end of January calculated by (a) in Process 2 as the terminal water level for the monthly model to prepare the January scheduling scheme for the TG reservoir. See Figure 11b for the January scheduling scheme for the TG reservoir, and see Table 8 for the scheduling results.

**Process 4**: For the example case of January (in the falling stage), the water level at the end of the first 10 days of January calculated in Process 3 is used as the terminal water level for the 10-day model to prepare the scheduling scheme for the first 10 days of January. See Figure 12a for the TG reservoir operation plan for the first 10 days of January, and see Table 9 for the scheduling results.

**Process 5**: For the example case of January (in the falling stage), the water level at the end of 1 January calculated in Process 4 is used as the terminal water level to prepare the hourly reservoir operation plan for 1 January. See Figure 12b for the TG reservoir operation plan for 1 January, and see Table 10 for the scheduling results.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Three Gorges Project | Gezhouba Project | ||
---|---|---|---|

Design flood level (m) | 175 | Design flood level (m) | 66 |

Check flood level (m) | 180.5 | Check flood level (m) | 67 |

Normal pool level (m) | 175 | Water level during normal operation (m) | 66 |

Drawdown level in dry season (m) | 155 | Minimum operating level (m) | 63 |

Top level of flood control (m) | 175 | Maximum operating level (m) | 66.5 |

Flood control level (m) | 145 | Minimum level in flood season (m) | 63 |

Regulating storage (10${}^{8}$ m${}^{3}$) | 165 | Regulating storage (10${}^{8}$ m${}^{3}$) | 0.85 |

Flood control capacity (10${}^{8}$ m${}^{3}$) | 221.5 | Flood control capacity (10${}^{8}$ m${}^{3}$) | / |

Installed capacity (10${}^{4}$ KW) | 2240 | Installed capacity (10${}^{4}$ KW) | 271.5 |

Guaranteed output (10${}^{4}$ KW) | 499 | Guaranteed output (10${}^{4}$ KW) | 104 |

Time | Reservoir Inflow (m${}^{3}$/s) | Upper Limit of Water Level (m) | Lower Limit of Water Level (m) | Lower Limit of Output (${10}^{4}$ kW) | Upper Limit of Output (${10}^{4}$ kW) |
---|---|---|---|---|---|

January | 4290 | 175 | 155 | 499 | 1820 |

February | 3840 | 175 | 155 | 499 | 1820 |

March | 4370 | 175 | 155 | 499 | 1820 |

April | 6780 | 175 | 155 | 499 | 1820 |

May | 12,100 | 175 | 155 | 499 | 1820 |

June | 24,100 | 146 | 144.9 | 499 | 1820 |

July | 25,000 | 146 | 144.9 | 499 | 1820 |

August | 26,000 | 146 | 144.9 | 499 | 1820 |

September | 23,500 | 146 | 144.9 | 499 | 1820 |

October | 18,200 | 175 | 155 | 499 | 1820 |

November | 10,000 | 175 | 155 | 499 | 1820 |

December | 5800 | 175 | 155 | 499 | 1820 |

**Table 3.**Statistics for solutions to optimal scheduling of power generation produced by different algorithms.

Algorithm | Population Size | Convergence Rate (%) | Average Calculation Time (s) | Average Power Generation (10${}^{8}$ kWh) | Standard Deviation of Power Generation (10${}^{8}$ kWh) |
---|---|---|---|---|---|

GA | 32 | 65 | 8.457 | 970.02 | 3.42 |

GA | 60 | 81 | 12.861 | 972.64 | 2.09 |

GA | 150 | 98 | 23.354 | 974.88 | 1.55 |

GA | 200 | 99 | 30.862 | 975.33 | 1.45 |

IGA | 32 | 100 | 4.946 | 976.2 | 0.72 |

IGA | 60 | 100 | 11.013 | 976.25 | 0.63 |

IGA | 150 | 100 | 26.251 | 976.58 | 0.56 |

IGA | 200 | 100 | 38.092 | 976.73 | 0.46 |

CIMAR-IDP (Ref) | 1305 | 977.2 |

Time | Reservoir Inflow (m${}^{3}$/s) | Power Generation Discharge (m${}^{3}$/s) | Abandoned Water (m${}^{3}$/s) | Power Generation Water Head (m) | Average Output (10${}^{4}$ kWh) | Power Generation (10${}^{8}$ kWh) |
---|---|---|---|---|---|---|

1 January | 4502 | 5600 | 0 | 107.45 | 541 | 40.25 |

1 February | 4542 | 5603 | 0 | 105.65 | 532.2 | 35.77 |

1 March | 5682 | 5600 | 0 | 105.18 | 529.9 | 39.42 |

1 April | 8715 | 6674 | 0 | 109.04 | 655 | 47.16 |

1 May | 10,137 | 16,299 | 0 | 98.01 | 1459.7 | 108.6 |

1 June | 12,240 | 17,918 | 0 | 83.56 | 1353.6 | 32.49 |

11 June | 16,210 | 16,210 | 0 | 79.52 | 1123.8 | 26.97 |

21 June | 17,040 | 17,040 | 0 | 79.42 | 1178.9 | 28.29 |

1 July | 22,539 | 22,539 | 0 | 78.68 | 1536.3 | 114.3 |

1 August | 27,787 | 25,906 | 1881 | 77.91 | 1740.6 | 129.5 |

1 September | 28,830 | 25,906 | 2924 | 77.75 | 1736 | 41.67 |

11 September | 26,670 | 17,059 | 0 | 86.17 | 1345.7 | 32.3 |

21 September | 24,010 | 8840 | 0 | 101.2 | 813.1 | 19.51 |

1 October | 15,355 | 15,355 | 0 | 108.13 | 1495.6 | 111.27 |

1 November | 15,825 | 15,825 | 0 | 108.06 | 1540.5 | 110.92 |

1-December | 5866 | 5866 | 0 | 108.9 | 574.9 | 42.78 |

**Table 5.**Implementation effect statistics of falling stage scheduling scheme for TG reservoir (varying scheduling intervals).

Time | Reservoir Inflow (m${}^{3}$/s) | Power Generation Discharge (m${}^{3}$/s) | Abandoned Water (m${}^{3}$/s) | Power Generation Water Head (m) | Average Output (10${}^{4}$ kWh) | Power Generation (10${}^{8}$ kWh) |
---|---|---|---|---|---|---|

1 January | 4502 | 5601 | 0 | 107.45 | 541 | 40.25 |

11 January | 4542 | 5600 | 0 | 105.5 | 531.3 | 36.98 |

1 February | 5682 | 5601 | 0 | 105.2 | 530.1 | 39.44 |

1 March | 8715 | 6640 | 0 | 108.99 | 651.5 | 46.91 |

1 April | 9845 | 9845 | 0 | 108.64 | 962.9 | 23.11 |

1 May | 9024 | 17,157 | 0 | 104.24 | 1610.4 | 38.65 |

11 May | 11,415 | 21,388 | 0 | 93.69 | 1819.9 | 48.05 |

21 May | 12,240 | 17,918 | - | 83.56 | 1353.6 | 32.49 |

Time | Water Level (m) | Time | Water Level (m) | Time | Water Level (m) |
---|---|---|---|---|---|

11 June | 146.5 | 12 July | 146.5 | 12 August | 146.5 |

12 June | 146.5 | 13 July | 146.5 | 13 August | 146.5 |

13 June | 146.5 | 14 July | 146.5 | 14 August | 146.5 |

14 June | 146.5 | 15 July | 146.5 | 15 August | 146.5 |

15 June | 146.5 | 16 July | 146.5 | 16 August | 146.5 |

16 June | 146.5 | 17 July | 146.5 | 17 August | 146.5 |

17 June | 146.5 | 18 July | 146.5 | 18 August | 146.5 |

18 June | 146.5 | 19 July | 146.5 | 19 August | 146.5 |

19 June | 146.5 | 20 July | 146.5 | 20 August | 146.5 |

20 June | 146.5 | 21 July | 146.06 | 21 August | 146.5 |

21 June | 146.5 | 22 July | 145 | 22 August | 146.46 |

22 June | 146.5 | 23 July | 145.34 | 23 August | 146.43 |

23 June | 146.5 | 24 July | 146.5 | 24 August | 146.5 |

24 June | 146.5 | 25 July | 146.5 | 25 August | 146.5 |

25 June | 146.5 | 26 July | 146.5 | 26 August | 146.47 |

26 June | 146.5 | 27 July | 146.5 | 27 August | 145.94 |

27 June | 146.5 | 28 July | 146.5 | 28 August | 145.38 |

28 June | 146.5 | 29 July | 146.5 | 29 August | 145 |

29 June | 146.5 | 30 July | 146.5 | 30 August | 146.32 |

30 June | 146.5 | 31 July | 146.5 | 31 August | 146.5 |

1 July | 146.5 | 1 August | 146.5 | 1 September | 146.5 |

2 July | 146.5 | 2 August | 146.5 | 2 September | 146.5 |

3 July | 145.87 | 3 August | 146.5 | 3 September | 146.5 |

4 July | 145.26 | 4 August | 146.5 | 4 September | 146.5 |

5 July | 145.21 | 5 August | 146.5 | 5 September | 146.5 |

6 July | 146.12 | 6 August | 146.5 | 6 September | 146.5 |

7 July | 145.98 | 7 August | 146.5 | 7 September | 146.5 |

8 July | 146.22 | 8 August | 145.85 | 8 September | 146.5 |

9 July | 146.5 | 9 August | 145 | 9 September | 146.5 |

10 July | 146.5 | 10 August | 145.52 | 10 September | 146.5 |

11 July | 146.5 | 11 August | 146.5 | 11 September | 146.5 |

**Table 7.**Implementation effect statistics of impounding stage scheduling scheme for TG reservoir (varying scheduling intervals).

Time | Reservoir Inflow (m${}^{3}$/s) | Power Generation Discharge (m${}^{3}$/s) | Abandoned Water (m${}^{3}$/s) | Power Generation Water Head (m) | Average Output (10${}^{4}$ kWh) | Power Generation (10${}^{8}$ kWh) |
---|---|---|---|---|---|---|

11 September | 26,670 | 17,059 | 2924 | 86.17 | 1345.7 | 32.3 |

21 September | 24,010 | 8840 | 0 | 101.2 | 813.1 | 19.51 |

1-Oct | 18,370 | 18,370 | 0 | 107.76 | 1781.5 | 42.76 |

11-Oct | 13,670 | 13,670 | 0 | 108.29 | 1333.2 | 32 |

21-Oct | 14,145 | 14,145 | 0 | 108.25 | 1379.1 | 36.41 |

1-Nov | 15,825 | 15,825 | 0 | 108.06 | 1540.5 | 110.92 |

1-Dec | 5866 | 5866 | 0 | 108.9 | 574.9 | 42.78 |

**Table 8.**Implementation effect statistics of January operation plan for TG reservoir (10-day period).

Time | Reservoir Inflow (m${}^{3}$/s) | Power Generation Discharge (m${}^{3}$/s) | Abandoned Water (m${}^{3}$/s) | Power Generation Water Head (m) | Average Output (10${}^{4}$ kWh) | Power Generation (10${}^{8}$ kWh) |
---|---|---|---|---|---|---|

1 January | 4863 | 5603 | 0 | 108.61 | 547.8 | 13.15 |

11 January | 4411 | 5601 | 0 | 108.19 | 545.8 | 13.1 |

21 January | 4256 | 5650 | 0 | 107.49 | 545.9 | 14.41 |

**Table 9.**Implementation effect statistics of operation plan for TG reservoir for first 10 days of January (daily period).

Time | Reservoir Inflow (m${}^{3}$/s) | Power Generation Discharge (m${}^{3}$/s) | Abandoned Water (m${}^{3}$/s) | Power Generation Water Head (m) | Average Output (10${}^{4}$ kWh) | Power Generation (10${}^{8}$ kWh) |
---|---|---|---|---|---|---|

1 January | 4810 | 5514 | 0 | 108.9 | 540.5 | 1.3 |

2 January | 4730 | 5552 | 0 | 108.97 | 544.5 | 1.31 |

3 January | 4580 | 5519 | 0 | 108.93 | 541.1 | 1.3 |

4 January | 4740 | 5562 | 0 | 108.98 | 545.5 | 1.31 |

5 January | 4730 | 5552 | 0 | 108.93 | 544.3 | 1.31 |

6 January | 4670 | 5609 | 0 | 108.89 | 549.7 | 1.32 |

7 January | 4910 | 5614 | 0 | 108.71 | 549.5 | 1.32 |

8 January | 5190 | 5542 | 0 | 108.55 | 541.7 | 1.3 |

9 January | 5230 | 5582 | 0 | 108.58 | 545.7 | 1.31 |

10 January | 5040 | 5627 | 0 | 108.5 | 549.7 | 1.32 |

**Table 10.**Implementation effect statistics of 1 January operation plan for TG reservoir (hourly period).

Time | Reservoir Inflow (m${}^{3}$/s) | Power Generation Discharge (m${}^{3}$/s) | Abandoned Water (m${}^{3}$/s) | Power Generation Water Head (m) | Average Output (10${}^{4}$ kWh) | Power Generation (10${}^{8}$ kWh) |
---|---|---|---|---|---|---|

0:00 | 4810 | 5515 | 0 | 111.83 | 551 | 551 |

1:00 | 4810 | 5515 | 0 | 111.83 | 551 | 551 |

2:00 | 4810 | 5515 | 0 | 111.82 | 551 | 551 |

3:00 | 4810 | 5511 | 0 | 111.82 | 550.6 | 551 |

4:00 | 4810 | 5515 | 0 | 111.82 | 551 | 551 |

5:00 | 4810 | 5515 | 0 | 111.82 | 551 | 551 |

6:00 | 4810 | 5514 | 0 | 111.81 | 550.9 | 551 |

7:00 | 4810 | 5515 | 0 | 111.81 | 551 | 551 |

8:00 | 4810 | 5511 | 0 | 111.81 | 550.5 | 551 |

9:00 | 4810 | 5515 | 0 | 111.81 | 550.9 | 551 |

10:00 | 4810 | 5515 | 0 | 111.8 | 550.9 | 551 |

11:00 | 4810 | 5515 | 0 | 111.8 | 550.9 | 551 |

12:00 | 4810 | 5515 | 0 | 111.8 | 550.9 | 551 |

13:00 | 4810 | 5515 | 0 | 111.8 | 550.9 | 551 |

14:00 | 4810 | 5515 | 0 | 111.79 | 550.9 | 551 |

15:00 | 4810 | 5511 | 0 | 111.79 | 550.5 | 550 |

16:00 | 4810 | 5515 | 0 | 111.79 | 550.9 | 551 |

17:00 | 4810 | 5515 | 0 | 111.79 | 550.9 | 551 |

18:00 | 4810 | 5514 | 0 | 111.78 | 550.8 | 551 |

19:00 | 4810 | 5515 | 0 | 111.78 | 550.9 | 551 |

20:00 | 4810 | 5511 | 0 | 111.78 | 550.4 | 550 |

21:00 | 4810 | 5515 | 0 | 111.78 | 550.8 | 551 |

22:00 | 4810 | 5515 | 0 | 111.77 | 550.8 | 551 |

23:00 | 4810 | 5515 | 0 | 111.77 | 550.8 | 551 |

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**MDPI and ACS Style**

Shang, L.; Li, X.; Shi, H.; Kong, F.; Wang, Y.; Shang, Y.
Long-, Medium-, and Short-Term Nested Optimized-Scheduling Model for Cascade Hydropower Plants: Development and Practical Application. *Water* **2022**, *14*, 1586.
https://doi.org/10.3390/w14101586

**AMA Style**

Shang L, Li X, Shi H, Kong F, Wang Y, Shang Y.
Long-, Medium-, and Short-Term Nested Optimized-Scheduling Model for Cascade Hydropower Plants: Development and Practical Application. *Water*. 2022; 14(10):1586.
https://doi.org/10.3390/w14101586

**Chicago/Turabian Style**

Shang, Ling, Xiaofei Li, Haifeng Shi, Feng Kong, Ying Wang, and Yizi Shang.
2022. "Long-, Medium-, and Short-Term Nested Optimized-Scheduling Model for Cascade Hydropower Plants: Development and Practical Application" *Water* 14, no. 10: 1586.
https://doi.org/10.3390/w14101586