# Third-Monthly Hydropower Scheduling of Cascaded Reservoirs Using Successive Quadratic Programming in Trust Corridor

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

_{1}and W

_{2}are the weights with ${W}_{1}\gg {W}_{2}$ to prioritize the firm power output (F) over the energy production; i and t are subscripts for reservoir and time-step, respectively; P

_{it}is the power output in MW in time-step t.

- (1)
- The water balance,

_{it}= storage in hm

^{3}at the beginning of time-step t of reservoir i; Q

_{it}= outflow in m

^{3}/s in time-step t from reservoir i; I

_{it}= local inflow in m

^{3}/s in time-step t into reservoir i; q

_{it}= generating discharge in m

^{3}/s in time-step t of hydroplant i; splt

_{it}= spillage in m

^{3}/s in time-step t from reservoir i; I

_{it}= local inflow in m

^{3}/s in time-step t into reservoir i; I

_{it}= local inflow in m

^{3}/s in time-step t into reservoir i; $\Omega (i)$ = set of reservoirs immediately upstream of reservoir i; $\Delta t$ = the number of days in time-step t; V

_{i}

^{ini}and V

_{i}

^{end}= initial and target storages in hm

^{3}at the beginning and end of the planning horizon, respectively.

- (2)
- Upper and lower bounds on storage or release,

^{3}of reservoir i; ${V}_{it}^{\mathrm{max}}$ = upper bound on the storage at the beginning of t of reservoir i, equal to the flood control limited storage during flooding seasons and the normal storage during dry seasons; Q

_{i}

^{min}and Q

_{i}

^{max}= lower and upper bounds on the release from reservoir i in time-step t.

- (3)
- Firm hydropower output,

- (4)
- The hydropower output determined by,

_{i}= power generating efficiency in MW.s/m

^{4}; h

_{it}= water head in time-step t of hydroplant i; G

_{i}

^{max}(.) = capacity of hydropower output of i, a function of water head; ${c}_{i}^{(0)},{c}_{i}^{(1)},{d}_{i}^{(0)}$ and ${d}_{i}^{(1)}$ = coefficients to be estimated to fit the power output capacity with piecewise linearization; Z

_{i}

^{u}(.) and Z

_{i}

^{d}(.) = forebay and tailwater elevations, dependent on the water storage and release, respectively, of reservoir i; ${a}_{i}$, ${\widehat{V}}_{i}^{(0)}$ and ${\widehat{Z}}_{i}^{\mathrm{u}}$ = coefficients/parameters to be estimated for the relationship curve between storage and forebay water level; ${\beta}_{i}$, ${\widehat{Q}}_{i}^{(0)}$ and ${\widehat{Z}}_{i}^{\mathrm{d}}$ = coefficients/parameters to be estimated for the relationship curve between outflow and tailwater level.

## 3. Solution Techniques

**x**

_{1}), which, if better than the base solution (

**x**

_{0}) on the original objective, will serve as the base solution, with the trust level restored to its initial one. If the solution has not been improved, then either terminate the procedure if the convergence has been achieved or shrink the trust level by a percentage, for instance, 80%.

#### 3.1. Formulation of a QP Problem

**x**representing the decision variables including: ${\overline{V}}_{it}$, ${Q}_{it}$, ${q}_{it}$, etc.

**x**

_{0}) that represents the values of decision variables: ${\overline{V}}_{it}^{(0)},{Q}_{it}^{(0)},\text{}{q}_{it}^{(0)}\text{}\mathrm{and}\text{}{V}_{it}^{(0)}$, and updated with

**x**

_{1}to represent the decision variables: ${\overline{V}}_{it}^{(1)},{Q}_{it}^{(1)}\text{}\mathrm{and}\text{}{q}_{it}^{(1)}$, which will determine the original objective with,

#### 3.2. Finding a Feasible Solution

## 4. Case Studies

#### 4.1. Engineering Background

^{3}/s on average, abundant and stable over the years. It is rich in hydropower resources, with a total drop of 3300 m, dropping more than 1 m every kilometer. It flows downstream and contributes to a total exploitable hydropower resource of 100 TW. The lower reaches of the Jinsha River are the most important transportation and regional economic centers in southwest China. Two major storage reservoirs, Baihetan and Xiluodu, have annual operability, making it very important to coordinate the cascaded reservoirs to maximize the benefits of the third-monthly hydropower scheduling.

#### 4.2. Comparison with Dynamic Programming (DP)

#### 4.3. Solution Efficiency

#### 4.4. Convergence of the Method

#### 4.5. Results in Detail

## 5. Conclusions

- (1)
- Successive quadratic programming (SQP) can derive results consistent with dynamic programming (DP), which is well known and capable of securing the global optimal solution to a small-scale problem.
- (2)
- The present procedure has the computational time increasing linearly as the number of reservoirs increases, taking about 1 min to solve the problem involving all four cascaded hydropower reservoirs.
- (3)
- The convergence of the SQP can be achieved at the fifth iteration, with objective functional value monotonically increasing and improving the most significantly in the second iteration by about 20%, while in the following iterations by only about 1%.
- (4)
- The cascaded hydropower reservoirs coordinate very well to maximize the firm hydropower output at the top priority by regulating their storage capacities to give a constant power yield during the dry seasons, and then making full use of their installed capacities to convert the coming inflows into hydropower energy as much as possible during the flood seasons.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Number | Name | Installed Capacity (MW) | Storage Capacity (GL) | Dam Height (m) | Water Level (m) | Operability | ||
---|---|---|---|---|---|---|---|---|

Flood | Normal | Dead | ||||||

1 | Wudongde | 10,200 | 7408 | 270 | 952 | 975 | 950 | Seasonal |

2 | Baihetan | 16,000 | 20,600 | 289 | 785 | 825 | 760 | Annual |

3 | Xiluodu | 12,600 | 12,670 | 285.5 | 560 | 600 | 540 | Annual |

4 | Xiangjiaba | 6400 | 5163 | 380 | 370 | 380 | 370 | Seasonal |

Number of Reservoirs | Computing Time(s) | Number of Variables | Number of Constraints | ||
---|---|---|---|---|---|

Model 1 | Model 2 | Model 1 | Model 2 | ||

4 | 61.784 | 725 | 725 | 1348 | 1644 |

3 | 46.478 | 544 | 544 | 1020 | 1242 |

2 | 17.981 | 363 | 363 | 1242 | 840 |

1 | 7.444 | 182 | 182 | 364 | 438 |

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**MDPI and ACS Style**

Liu, S.; Luo, J.; Chen, H.; Wang, Y.; Li, X.; Zhang, J.; Wang, J.
Third-Monthly Hydropower Scheduling of Cascaded Reservoirs Using Successive Quadratic Programming in Trust Corridor. *Water* **2023**, *15*, 716.
https://doi.org/10.3390/w15040716

**AMA Style**

Liu S, Luo J, Chen H, Wang Y, Li X, Zhang J, Wang J.
Third-Monthly Hydropower Scheduling of Cascaded Reservoirs Using Successive Quadratic Programming in Trust Corridor. *Water*. 2023; 15(4):716.
https://doi.org/10.3390/w15040716

**Chicago/Turabian Style**

Liu, Shuangquan, Jingzhen Luo, Hui Chen, Youxiang Wang, Xiangyong Li, Jie Zhang, and Jinwen Wang.
2023. "Third-Monthly Hydropower Scheduling of Cascaded Reservoirs Using Successive Quadratic Programming in Trust Corridor" *Water* 15, no. 4: 716.
https://doi.org/10.3390/w15040716