Application of Black-Winged Differential-Variant Whale Optimization Algorithm in the Optimization Scheduling of Cascade Hydropower Stations
Abstract
:1. Introduction
2. Cascade Reservoir Optimization Scheduling Model
- (1)
- Objective function
- (2)
- Constraints
3. Black-Winged Differential-Variant Whale Optimization Algorithm
3.1. Whale Optimization Algorithm
- (1)
- Individual Initialization
- (2)
- Encircling Predation
- (3)
- Spiral Bubble Attack
- (4)
- Random Search
3.2. Whale Optimization Algorithm Improvement Strategy
- (1)
- Combinatorial Chaos Mapping
- (2)
- Adaptive Weights
- (3)
- Improvements Based on the Migration Behavior of Black-Winged Kites
- (4)
- Improvements Based on Differential Mutation
3.3. Test Functions to Verify Performance
3.4. Model Solving Process
4. Case Study
4.1. Overview of the Study Area
4.2. Experimental Results
4.3. Scheduling Results Analysis
5. Conclusions and Future Work
- (1)
- The BDWOA improved optimization and search efficiency by introducing Logistic-Sine-Cosine chaotic mapping, adaptive weights, the black-winged kite migration mechanism, and the differential mutation strategy. These enhancements help it effectively avoid becoming trapped in local optima. Compared to the other five algorithms, the BDWOA demonstrated superior performance across ten test functions, specifically by requiring the fewest iterations for convergence and achieving the smallest objective function values in each set of test functions;
- (2)
- In the application of cascade hydropower station optimization scheduling, the BDWOA demonstrated excellent solution efficiency and effectively increased the overall power generation of the cascade hydropower stations. In the optimization model with the goal of maximizing power generation, the BDWOA significantly outperformed the comparison algorithms. Although the BDWOA algorithm’s results varied slightly across different typical years, it consistently achieved the shortest solving time, the highest power generation, and the smallest standard deviation. For example, in wet years, the solving time was 52 s, the power generation was 2127.86 × 109 kW·h, and the standard deviation was 21.77, outperforming the other five algorithms. This indicated that BDWOA’s convergence process was less prone to becoming trapped in local optima, while also demonstrating a faster convergence speed and higher stability;
- (3)
- The proposed BDWOA algorithm enhances adaptability to complex water resource scheduling problems and demonstrates strong application potential for sustainable energy development. Under the constraints of water level, flow, and power output, the BDWOA was able to develop appropriate scheduling strategies based on varying inflow conditions, making more efficient use of water resources and maximizing the power generation benefits of cascade hydropower stations, thereby providing strong support for the sustainable development of renewable energy.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
BDWOA | Black-winged Differential-variant Whale Optimization Algorithm |
WOA | Whale optimization algorithm |
BWOA | Whale optimization algorithm under the black-winged kite strategy |
DWOA | Whale optimization algorithm under the differential variational strategy |
PSO | Standard particle swarm |
LPSO | Particle swarm under the Levy flight strategy |
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Functions Name | Test Function Expressions | Feasibility Domain | Optimum Value |
---|---|---|---|
F1 | [−100,100] | 0 | |
F2 | [−100,100] | 0 | |
F3 | [−100,100] | 0 | |
F4 | [−30,30] | 0 | |
F5 | [−1.28,1.28] | 0 | |
F6 | [−100,100] | 0 | |
F7 | [−5.12,5.12] | 0 | |
F8 | [−600,600] | 0 | |
F9 | Among | [−50,50] | 0 |
F10 | [−5.12,5.12] | 0 |
Functions Name | Optimum Value | |||||
---|---|---|---|---|---|---|
PSO | LPSO | WOA | DWOA | BWOA | BDWOA | |
F1 | 13,056.59 | 16,615.95 | 3.38 × 10−29 | 5.12 × 10−19 | 2.52 × 10−32 | 1.18 × 10−64 |
F2 | 34,457.68 | 39,553.32 | 7.79 × 10−6 | 1.04 | 1.29 × 10−5 | 5.27 × 10−30 |
F3 | 4.42 | 2.44 | 1.23 × 10−6 | 1.45 × 10−4 | 4.28 × 10−5 | 6.61 × 10−23 |
F4 | 5.04 × 108 | 1.34 × 109 | 18 | 16.83 | 16.22 | 16.17 |
F5 | 10,536.4 | 39,530.14 | 0.97 | 0.25 | 1.15 | 0 |
F6 | 1.79 × 108 | 2,332,424.2 | 0 | 0 | 0.25 | 0 |
F7 | 13,281.01 | 9869.78 | 0 | 10.49 | 0 | 0 |
F8 | 3.83 | 0.07 | 0 | 0.03 | 0 | 0 |
F9 | 1.76 | 0.67 | 0.54 | 0.54 | 0.3 | 0.27 |
F10 | 0 | 6503.18 | 0 | 0 | 2.84 × 10−14 | 0 |
Power Station Characteristic Parameters | Xiluodu | Xiangjiaba | Three Gorges |
---|---|---|---|
Regulation Performance | Inadequate Annual Regulation | Inadequate Seasonal Regulation | Inadequate Annual Regulation |
Total reservoir capacity (109 m3) | 126.7 | 51.63 | 393 |
Flood control reservoir capacity (109 m3) | 46.5 | 9.03 | 221.5 |
Normal water level (m) | 600 | 380 | 175 |
Dead water level (m) | 540 | 370 | 145 |
Guaranteed output (MW) | 3795 | 2009 | 4990 |
Installed capacity (MW) | 12,600 | 6400 | 22,500 |
Average annual power generation (109 kWh) | 640 | 310 | 884 |
Minimum discharge (m3/s) | 1700 | 1700 | 6000 |
Typical Year Scenario | Objective Value (×109 kW·h) | Computation Time (s) | ||||
---|---|---|---|---|---|---|
Methods | Average Value | Best Value | Worst Value | Standard Deviation | ||
High-flow year | BDWOA | 2127.86 | 2176.34 | 2092.11 | 21.77 | 52.03 |
WOA | 1965.38 | 2002.71 | 1907.17 | 26.31 | 64.91 | |
BWOA | 2008.86 | 2045.06 | 1971.69 | 23.47 | 71.72 | |
DWOA | 1991.95 | 2045.22 | 1950.94 | 25.34 | 66.54 | |
LPSO | 2045.01 | 2083.92 | 1771.59 | 23.79 | 66.23 | |
PSO | 1967.44 | 2026.09 | 1771.59 | 28.65 | 69.61 | |
Normal-flow year | BDWOA | 1951.63 | 1984.57 | 1922.92 | 14.24 | 55.81 |
WOA | 1821.72 | 1867.02 | 1779.29 | 23.89 | 70.21 | |
BWOA | 1843.41 | 1905.41 | 1791.98 | 30.89 | 66.67 | |
DWOA | 1833.05 | 1870.01 | 1786.11 | 22.76 | 66.51 | |
LPSO | 1869.25 | 1902.13 | 1833.03 | 17.29 | 67.24 | |
PSO | 1809.95 | 1859.94 | 1771.59 | 21.30 | 70.98 | |
Low-flow year | BDWOA | 1916.71 | 1947.45 | 1883.17 | 15.17 | 52.52 |
WOA | 1791.86 | 1845.44 | 1756.43 | 27.04 | 66.79 | |
BWOA | 1811.41 | 1875.88 | 1771.39 | 25.53 | 68.45 | |
DWOA | 1796.05 | 1845.33 | 1756.35 | 24.38 | 67.21 | |
LPSO | 1823.61 | 1854.09 | 1796.43 | 15.51 | 68.78 | |
PSO | 1774.79 | 1813.72 | 1741.42 | 16.62 | 64.17 |
Runoff Scenario | F-Value | p-Value |
---|---|---|
High-flow year | 114.9086 | 7.7646 × 10−43 |
Normal-flow year | 101.0339 | 3.1967 × 10−40 |
Low-flow year | 107.4228 | 1.8468 × 10−41 |
Algorithm | Typical Year Scenario | ||
---|---|---|---|
High-Flow Year | Normal-Flow Year | Low-Flow Year | |
BDWOA | 2132.64 | 1950.18 | 1918.49 |
WOA | 1969.55 | 1814.12 | 1782.23 |
BWOA | 2010.72 | 1841.71 | 1819.16 |
DWOA | 1988.19 | 1825.59 | 1791.61 |
PSO | 2042.18 | 1865.99 | 1824.07 |
LPSO | 1972.84 | 1809.06 | 1768.64 |
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Zhang, M.; Liu, Z.; Bao, R.; Zhu, S.; Mo, L.; Yang, Y. Application of Black-Winged Differential-Variant Whale Optimization Algorithm in the Optimization Scheduling of Cascade Hydropower Stations. Sustainability 2025, 17, 1018. https://doi.org/10.3390/su17031018
Zhang M, Liu Z, Bao R, Zhu S, Mo L, Yang Y. Application of Black-Winged Differential-Variant Whale Optimization Algorithm in the Optimization Scheduling of Cascade Hydropower Stations. Sustainability. 2025; 17(3):1018. https://doi.org/10.3390/su17031018
Chicago/Turabian StyleZhang, Mi, Zixuan Liu, Rungang Bao, Shuli Zhu, Li Mo, and Yuqi Yang. 2025. "Application of Black-Winged Differential-Variant Whale Optimization Algorithm in the Optimization Scheduling of Cascade Hydropower Stations" Sustainability 17, no. 3: 1018. https://doi.org/10.3390/su17031018
APA StyleZhang, M., Liu, Z., Bao, R., Zhu, S., Mo, L., & Yang, Y. (2025). Application of Black-Winged Differential-Variant Whale Optimization Algorithm in the Optimization Scheduling of Cascade Hydropower Stations. Sustainability, 17(3), 1018. https://doi.org/10.3390/su17031018