# A Monthly Hydropower Scheduling Model of Cascaded Reservoirs with the Zoutendijk Method

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

_{1}and W

_{2}are the weights with ${\mathrm{w}}_{1}\gg {\mathrm{w}}_{2}$ to prioritize the firm power output (F) in MW over the total power output of all the hydropower plants; i and t are the subscripts for the reservoir and time-step, respectively; P

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

- 2
- The water balance,

_{it}= the storage in hm

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

_{it}= the outflow in m

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

_{it}= the local inflow in m

^{3}/s in time-step t into reservoir i; $\Omega (i)$ = the 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}= the initial and target storages in hm

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

_{it}= the spillage in m

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

_{it}= the generating discharge in m

^{3}/s in time-step t from hydropower plant i.

- 2
- Upper and lower bounds on storage and release, respectively,

^{3}of reservoir i; ${V}_{it}^{\mathrm{max}}$ = the 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}= the lower and upper bounds on the release from reservoir i in time-step t.

- 3
- Firm hydropower output,

- 4
- Hydropower output and the capacity of generating discharge,

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

^{4}; h

_{it}= the water head in time-step t of hydropower plant i in m; G

_{i}

^{max}(.) = the capacity of generating discharge of i in m

^{3}/s, a function of water head; Z

_{i}

^{u}(.) and Z

_{i}

^{d}(.) = the forebay and tailwater elevations in meters, dependent of the water storage and release, respectively, of reservoir i.

## 3. Solution Techniques

^{th}cycle of updating water heads; ${V}_{it}^{[n]}$ and ${Q}_{it}^{[n]}$ = the storage and release at the solution derived at the beginning of the n

^{th}cycle of updating water heads.

#### 3.1. Reformulation into Lagrange Dual Problem

_{i}(…)” in objective (16) gives a nonlinear programming problem with linear constraints, which allows its solution with methods of feasible directions.

#### 3.2. Initial Feasible Solutions

_{it}) for all the possible combinations (i, t), and ${q}_{it}^{[0]}$, ${V}_{it}^{[0]}$,${Q}_{it}^{[0]}$, and $sp{l}_{it}^{[0]}$ that determine the water head (${\widehat{h}}_{it}^{[0]}$),

#### 3.3. Application of the Zoutendijk Method

^{th}cycle of updating water heads, and initiate a solution for the Zoutendijk method to be applied,

^{th}iteration,

**d**) by

**d**

^{(k)}.

## 4. Case Studies

#### 4.1. Engineering Background

#### 4.2. Results of Curve Fitting

_{i}

^{u}(.) and Z

_{i}

^{d}(.) were fitted into two exponential functions to let them be partially derivable so that the Zoutendijk method could be applied to the hydropower scheduling model. Table 3 and Table 4 present the parameters calibrated and the fitting errors, measured by the mean squared error (MSE). The results show that the fitting accuracy was mostly above 99.5%, suggesting that Equations (12) and (13) are scientifically credible in representing the forebay and tailwater elevations, respectively.

#### 4.3. Scheduling Results of the Cascaded Hydropower Plants

^{th}iteration. In general, the algorithm had an excellent performance in solving the scheduling problems of cascaded hydropower reservoirs.

## 5. Conclusions

- (1)
- The Zoutendijk method is well applicable to the monthly hydropower scheduling of cascaded reservoirs, deriving results that are reasonable and reliable;
- (2)
- The exponential functions used to fit the forebay and tailwater curves showed a very high fitting accuracy at more than 99.5%, showing a great prospect to make segmented curves derivable when formulating a hydropower scheduling problem.
- (3)
- The Zoutendijk method can significantly increase the total hydropower production while ensuring the highest firm power output of the whole cascade.
- (4)
- This solution procedure is very fast in securing the optimum to the problem.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Abbreviations | Corresponding Meaning |
---|---|

LP | Linear programming |

MILP | Mixed integer linear programming |

HOF | Hydropower output function |

NLP | Nonlinear programming |

DP | Dynamic programming |

HP | Heuristic programming |

PSO | Particle swarm optimization |

EA | Evolutionary algorithms |

GSA | Gravity search algorithm |

GA | Genetic algorithm |

MGCL-PSO | Multi group cooperation operation particle swarm optimization |

Number | Name | Annual Inflow (m ^{3}/s) | Installed Capacity (MW) | Storage Capacity (10 ^{8} m^{3}) | Dam Height (m) | Operability |
---|---|---|---|---|---|---|

1 | Wunonglong | 743 | 990 | 2.65 | 137.5 | Daily |

2 | Huangdeng | 902 | 1900 | 1.418 | 203 | Seasonal |

3 | Xiaowan | 1210 | 4200 | 145.57 | 1245 | Over-year |

4 | Manwan | 1230 | 1670 | 3.716 | 1002 | Seasonal |

5 | Dachaoshan | 1340 | 1350 | 7.42 | 906 | Seasonal |

6 | Nuozhadu | 1740 | 5850 | 217.776 | 821.5 | Over-year |

**Table 3.**Parameters in ${Z}_{i}^{\mathrm{u}}(V)={\alpha}_{i}\cdot {(V-{\widehat{V}}_{0})}^{{\beta}_{i}}+{\widehat{Z}}_{\mathrm{u}}$.

$\mathit{i}$ | Name | ${\mathit{\alpha}}_{\mathit{i}}$ | ${\widehat{\mathit{V}}}_{0}$ | ${\mathit{\beta}}_{\mathit{i}}$ | ${\widehat{\mathit{Z}}}_{\mathbf{u}}$ | Average MSE |
---|---|---|---|---|---|---|

1 | Wunonglong | 0.13 | −5.43 | 1.01 | 1868.89 | 0.00% |

2 | Huangdeng | 0.08 | 648.49 | 0.91 | 1582.38 | 0.02% |

3 | Xiaowan | 0.04 | 4662 | 0.82 | 1165.98 | 0.05% |

4 | Manwan | 0.11 | 136.59 | 0.89 | 980.42 | 0.05% |

5 | Dachaoshan | 0.11 | 425.35 | 0.84 | 884.54 | 0.00% |

6 | Nuozhadu | 0.02 | 9554 | 0.82 | 760 | 0.06% |

**Table 4.**Parameters in ${Z}_{i}^{\mathrm{d}}(Q)={\lambda}_{i}\cdot {(Q-{\widehat{Q}}_{0})}^{{\theta}_{i}}+{\widehat{Z}}_{\mathrm{d}}$.

$\mathit{i}$ | Name | ${\mathit{\lambda}}_{\mathit{i}}$ | ${\widehat{\mathit{Q}}}_{0}$ | ${\mathit{\theta}}_{\mathit{i}}$ | ${\widehat{\mathit{Z}}}_{\mathbf{d}}$ | Average MSE |
---|---|---|---|---|---|---|

1 | Wunonglong | 0.002 | 199.96 | 0.97 | 1818.23 | 0.02% |

2 | Huangdeng | 0.005 | 149.99 | 0.89 | 1473.8 | 0.03% |

3 | Xiaowan | 0.002 | −0.01 | 1.00 | 990.83 | 0.00% |

4 | Manwan | 0.014 | 197 | 0.76 | 896.36 | 0.02% |

5 | Dachaoshan | 0.009 | 300 | 0.9 | 811.63 | 0.05% |

6 | Nuozhadu | 0.38 | −31.98 | 0.45 | 591.49 | 0.00% |

Firm Output (MW) | Total Output (MW) | CPU Time (Second) | Number of Optimizations | Is Optimal |
---|---|---|---|---|

146,476.11 | 2,529,067 | 7.539 | 86 | Yes |

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**MDPI and ACS Style**

Zhou, B.; Feng, S.; Xu, Z.; Jiang, Y.; Wang, Y.; Chen, K.; Wang, J. A Monthly Hydropower Scheduling Model of Cascaded Reservoirs with the Zoutendijk Method. *Water* **2022**, *14*, 3978.
https://doi.org/10.3390/w14233978

**AMA Style**

Zhou B, Feng S, Xu Z, Jiang Y, Wang Y, Chen K, Wang J. A Monthly Hydropower Scheduling Model of Cascaded Reservoirs with the Zoutendijk Method. *Water*. 2022; 14(23):3978.
https://doi.org/10.3390/w14233978

**Chicago/Turabian Style**

Zhou, Binbin, Suzhen Feng, Zifan Xu, Yan Jiang, Youxiang Wang, Kai Chen, and Jinwen Wang. 2022. "A Monthly Hydropower Scheduling Model of Cascaded Reservoirs with the Zoutendijk Method" *Water* 14, no. 23: 3978.
https://doi.org/10.3390/w14233978