# A Special Ordered Set of Type 2 Modeling for a Monthly Hydropower Scheduling of Cascaded Reservoirs with Spillage Controllable

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

_{it}is the power output in MW in time t; ${\eta}_{i}^{QP}$ is the coefficient estimated for hydropower plant i to convert discharge in m

^{3}/s to power in MW.

_{it}demonstrates storage in hm

^{3}of reservoir i at the beginning of time t; $\Omega (i)$ means the set of reservoirs immediately upstream of reservoir i; Q

_{it}is the outflow in m

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

_{it}represents local inflow in m

^{3}/s into reservoir i in time t; $\Delta t$ is the number of days in time t; V

_{i}

^{ini}and V

_{i}

^{end}denote initial and target storages in hm

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

_{it}means the spillage in m

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

_{it}is generating discharge in m

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

^{3}of reservoir i; correspondingly, ${V}_{it}^{\mathrm{max}}$ represents the upper bound on the storage in hm

^{3}at the beginning of t of reservoir I, which equals the flood control’s limited storage during flooding seasons and the normal storage during dry seasons; Q

_{i}

^{min}and Q

_{i}

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

_{i}is the power-generating efficiency in MW.s/m

^{4}; h

_{it}demonstrates water head in time t of hydropower plant i; G

_{i}

^{max}(.) represents the capacity of generating discharge of I, which is a function of water head; Z

_{i}

^{u}(.) and Z

_{i}

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

## 3. Solution Techniques

#### 3.1. Spillage as a Nonlinear Function

_{it}and h

_{it}, only appear in constraints (4) and (7)–(9), where (7) and (8) are all the nonlinear constraints involved in the problem. From (4) and (8), we have

_{it}and h

_{it}) from the problem by replacing (4) and (7) with

#### 3.2. SOS2 Formulation

## 4. Case Studies

#### 4.1. Engineering Background

#### 4.2. Detailed Results of Four Cascaded Hydropower Plants

^{3}/s) and provides detailed results of the optimal scheduling process for the four cascaded hydropower reservoirs on the Lancang River, shown in Figure 3. Zmin means the lowest water level during the scheduling, which generally refers to the dead water level. Likewise, Zmax is the highest water level, which refers to the normal water lever in non-flood seasons and the flood-control water level during the flood seasons, respectively. Z denotes the water level for the current time period in the model. Despite lacking upstream reservoirs for assistance, Xiaowan, renowned for its robust regulating capacity, experiences a limited degree of spillage during the flood season, typically between June and September. Owing to its limited regulation capability, Manwan has spillages that occurred during the flood season. At all hydropower plants, the observed monthly water levels exhibit a consistent pattern characterized by increased water levels during dry seasons and a subsequent decrease during flood seasons. This pattern aims to minimize spillage and maximize hydroelectric generation, ensuring optimal utilization of water resources. During the flood months of July, August, and September, the Dachaoshan reservoir dedicates a portion of its storage capacity to fulfill the essential flood control requirements. This strategic allocation ensures that the reservoir can effectively manage and mitigate flood-related challenges during this specific period.

#### 4.3. Solution Efficiency of SOS2 Modeling

#### 4.4. Comparisons with SQP

#### 4.5. Impacts on Results by Prioritizing the Objectives in Different Ways

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The location of the Lancang River basin in Asia, depicting the hydraulic connections of Xiaowan, Manwan, Dachaoshan, and Nuozhadu.

V | Q | ||||||
---|---|---|---|---|---|---|---|

${\widehat{\mathit{Q}}}^{(0)}:{\mathit{x}}^{(0)}$ | ${\widehat{\mathit{Q}}}^{(1)}:{\mathit{x}}^{(1)}$ | … | ${\widehat{\mathit{Q}}}^{(\mathit{l})}:{\mathit{x}}^{(\mathit{l})}$ | ${\widehat{\mathit{Q}}}^{(\mathit{l}+1)}:{\mathit{x}}^{(\mathit{l}+1)}$ | … | ${\widehat{\mathit{Q}}}^{(\mathit{L})}:{\mathit{x}}^{(\mathit{L})}$ | |

${\widehat{V}}^{(0)}:{y}^{(0)}$ | ${\lambda}^{(0,0)}$ | … | ${\lambda}^{(0,l)}$ | ${\lambda}^{(0,l+1)}$ | … | ${\lambda}^{(0,L)}$ | |

${\widehat{V}}^{(1)}:{y}^{(1)}$ | ${\lambda}^{(1,0)}$ | … | ${\lambda}^{(1,l)}$ | ${\lambda}^{(1,l+1)}$ | … | ${\lambda}^{(1,L)}$ | |

$\vdots $ | $\vdots $ | $\vdots $ | $\ddots $ | $\vdots $ | $\vdots $ | $\ddots $ | $\vdots $ |

${\widehat{V}}^{(k)}:{y}^{(k)}$ | ${\lambda}^{(k,0)}$ | … | ${\lambda}^{(k,l)}$ | ${\lambda}^{(k,l+1)}$ | … | ${\lambda}^{(k,L)}$ | |

${\widehat{V}}^{(k+1)}:{y}^{(k+1)}$ | ${\lambda}^{(k+1,0)}$ | … | ${\lambda}^{(k+1,l)}$ | ${\lambda}^{(0,l+1)}$ | … | ${\lambda}^{(k+1,L)}$ | |

$\vdots $ | $\vdots $ | $\vdots $ | $\ddots $ | $\vdots $ | $\vdots $ | $\ddots $ | $\vdots $ |

${\widehat{V}}^{(K)}:{y}^{(K)}$ | ${\lambda}^{(K,0)}$ | … | ${\lambda}^{(K,l)}$ | ${\lambda}^{(K,l+1)}$ | … | ${\lambda}^{(K,L)}$ |

Name | Installed Capacity (MW) | Storage Capacity (10 ^{8} m^{3}) | Dam Height (m) | Water Level (m) | Operability | ||
---|---|---|---|---|---|---|---|

Flood | Normal | Dead | |||||

Xiaowan | 4200 | 149.14 | 294.5 | 1232 | 1240 | 1166 | Annual |

Manwan | 1670 | 5.02 | 132 | 994 | 994 | 988 | Seasonal |

Dachaoshan | 1350 | 9.40 | 111 | 899 | 899 | 887 | Seasonal |

Nuozhadu | 5850 | 126.70 | 261.5 | 804 | 812 | 765 | Over-year |

Station | Starting Conditions | Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sept | Oct | Nov | Dec | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Xiaowan | Local Inflow (m^{3}/s) | 433 | 394 | 434 | 780 | 1350 | 2098 | 2844 | 3147 | 3781 | 1747 | 1037 | 642 | |

Storage (mcm) | 11,937.09 | 10,380.27 | 7761.05 | 5785.14 | 4642 | 4642 | 5073.75 | 7194.42 | 10,054.05 | 14,557 | 14,557 | 13,841.62 | 11,937.09 | |

Qspl (m ^{3}/s) | 0 | 0 | 0 | 0 | 0 | 36.02 | 76.14 | 83.75 | 89.67 | 0 | 0 | 0 | ||

Qhp (m ^{3}/s) | 1033.63 | 1404.5 | 1196.31 | 1221.02 | 1350 | 1895.41 | 1949.70 | 1960 | 1954.08 | 1747 | 1313.00 | 1376.77 | ||

Power (MW) | 2160.69 | 2742.34 | 2137.16 | 2007.89 | 2133.77 | 3018.06 | 3342.26 | 3739.61 | 4168.22 | 3939.98 | 2950.53 | 2999.469 | ||

Energy (GWh) | 1607.55 | 1908.67 | 1590.05 | 1445.68 | 1587.53 | 2173.01 | 2486.64 | 2782.27 | 3001.12 | 2931.34 | 2124.381 | 2231.61 | ||

Manwan | Local Inflow (m^{3}/s) | 1038.63 | 1408.5 | 1201.31 | 1229.02 | 1364 | 1953.43 | 2055.84 | 2076.75 | 2083.75 | 1765 | 1324.00 | 1383.77 | |

Storage (mcm) | 372 | 372 | 372 | 372 | 372 | 249 | 284.1 | 284.1 | 284.1 | 372 | 372 | 372 | 372 | |

Qspl (m ^{3}/s) | 0 | 0 | 0 | 0 | 0 | 56.40 | 102.61 | 110.94 | 120.02 | 0 | 0 | 0 | ||

Qhp (m ^{3}/s) | 1038.63 | 1408.5 | 1201.31 | 1229.02 | 1411.45 | 1883.48 | 1953.23 | 1965.81 | 1929.82 | 1765 | 1324 | 1383.77 | ||

Power (MW) | 821.59 | 1103.93 | 945.78 | 966.94 | 1073.60 | 1385.82 | 1448.42 | 1457.18 | 1465.22 | 1376.05 | 1039.43 | 1085.05 | ||

Energy (GWh) | 611.27 | 768.34 | 703.66 | 696.19 | 798.76 | 997.79 | 1077.62 | 1084.14 | 1054.96 | 1023.78 | 748.39 | 807.28 | ||

Dachaoshan | Local Inflow (m^{3}/s) | 1200.63 | 1537.5 | 1309.31 | 1338.02 | 1547.45 | 2099.87 | 2591.84 | 2416.75 | 2448.84 | 2060 | 1546 | 1540.77 | |

Storage (mcm) | 740 | 740 | 740 | 740 | 740 | 465 | 637 | 637 | 637 | 740 | 740 | 740 | 740 | |

Qspl (m ^{3}/s) | 0 | 0 | 0 | 0 | 0 | 150.53 | 708.84 | 533.75 | 526.1 | 177 | 0 | 0 | ||

Qhp (m ^{3}/s) | 1200.63 | 1537.5 | 1309.31 | 1338.02 | 1653.55 | 1883 | 1883 | 1883 | 1883 | 1883 | 1546 | 1540.77 | ||

Power (MW) | 859.72 | 1076.78 | 929.52 | 947.95 | 1073.61 | 1169.65 | 1195.14 | 1206.40 | 1241.09 | 1294.89 | 1082.35 | 1078.92 | ||

Energy (GWh) | 639.63 | 749.44 | 691.57 | 682.53 | 798.76 | 842.15 | 889.19 | 897.56 | 893.59 | 963.40 | 779.29 | 802.72 | ||

Nuozhadu | Local Inflow (m^{3}/s) | 1200.63 | 1537.5 | 1323.31 | 1447.02 | 1895.549 | 2461.53 | 2852.84 | 2957.75 | 3069.1 | 2254 | 1614 | 1563.77 | |

Storage (mcm) | 21,749 | 18,776.28 | 18,109.63 | 15,296.7 | 12,203.22 | 10,414 | 12,441.58 | 16,500.31 | 19,337 | 21,749 | 21,749 | 21,749 | 21,749 | |

Qspl (m ^{3}/s) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

Qhp (m ^{3}/s) | 2347.51 | 1794.7 | 2408.55 | 2640.5 | 2585.83 | 1679.28 | 1286.97 | 1863.35 | 2138.54 | 2254 | 1614 | 1563.77 | ||

Power (MW) | 4257.04 | 3175.99 | 4086.57 | 4176.26 | 3818.06 | 2525.50 | 2113.22 | 3262.11 | 3906.89 | 4187.95 | 3026.73 | 2935.59 | ||

Energy (GWh) | 3167.24 | 2210.49 | 3040.41 | 3006.91 | 2840.64 | 1818.36 | 1572.23 | 2427.01 | 2812.96 | 3115.84 | 2179.24 | 2184.08 |

4 × 4 | 8 × 8 | 15 × 15 | 20 × 20 | 25 × 25 | |
---|---|---|---|---|---|

Spillage (m^{3}/s) | 30,731.2 | 13,799.8 | 2431.4 | 2128.3 | 1951.4 |

F (MW) | 5,511,737.2 | 7,067,768.6 | 8,042,369.1 | 8,097,267.3 | 8,093,758.3 |

P (MW) | 72,789,359.6 | 90,731,379.7 | 103,268,755.2 | 103,975,851.6 | 104,052,755.0 |

Obj | 25,146,663.1 | 6,641,342.9 | −5,714,286.2 | −6,072,947.9 | −6,246,391.5 |

Number of variables | 1495 | 4183 | 12,583 | 21,181 | 32,462 |

CPU time (s) | 0.14 | 0.68 | 4.70 | 10.00 | 31.59 |

∑Spillage (m^{3}/s) | F (GW) | ∑P (GW) | Obj (MW) | Number of Variables | CPU Time (s) | |
---|---|---|---|---|---|---|

SOS2 | 2734.58 | 8.09 | 104.05 | −6246.39 | 32,462 | 31.59 |

SQP | 2750.66 | 8.04 | 104.00 | −6074.57 | 725 | 100.19 |

0.58% ↓ | 0.62% ↑ | 0.05 ↑ | 2.83% ↓ | - | 0.68% ↓ |

Experiments | Weights | Spillage | Firm Output | Total Output | ||
---|---|---|---|---|---|---|

W_{1} | W_{2} | W_{3} | (hm^{3}) | (MW) | (GW) | |

1# | 1000 | 1 | 0.001 | 6318.709 | 7914.266 | 104.053 |

2# | 0.001 | 1000 | 1 | 6355.715 | 8138.426 | 104.237 |

3# | 0.001 | 1 | 1000 | 6601.307 | 6917.175 | 107.122 |

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## Share and Cite

**MDPI and ACS Style**

Liu, S.; Qian, G.; Xu, Z.; Wang, H.; Chen, K.; Wang, J.; Feng, S.
A Special Ordered Set of Type 2 Modeling for a Monthly Hydropower Scheduling of Cascaded Reservoirs with Spillage Controllable. *Water* **2023**, *15*, 3128.
https://doi.org/10.3390/w15173128

**AMA Style**

Liu S, Qian G, Xu Z, Wang H, Chen K, Wang J, Feng S.
A Special Ordered Set of Type 2 Modeling for a Monthly Hydropower Scheduling of Cascaded Reservoirs with Spillage Controllable. *Water*. 2023; 15(17):3128.
https://doi.org/10.3390/w15173128

**Chicago/Turabian Style**

Liu, Shuangquan, Guoyuan Qian, Zifan Xu, Hua Wang, Kai Chen, Jinwen Wang, and Suzhen Feng.
2023. "A Special Ordered Set of Type 2 Modeling for a Monthly Hydropower Scheduling of Cascaded Reservoirs with Spillage Controllable" *Water* 15, no. 17: 3128.
https://doi.org/10.3390/w15173128