Long-Term Scheduling of Large-Scale Cascade Hydropower Stations Using Improved Differential Evolution Algorithm
Abstract
:1. Introduction
2. Optimization Model
2.1. Objective Function
2.2. Constraints
- Water balance constraint.
- Hydraulic connection.
- Water level constraint.
- Power generating constraint.
- Outflow constraint.
- Water head equation.
- Boundary condition.
3. Overview of iLSHADE
3.1. DE
- “rand/2”:
- “best/1”:
- “best/2”:
- “current to best/1”:
3.2. iLSHADE
3.2.1. Mutation Strategy “Current to pbest/2-rand”
3.2.2. The PIRFE Strategy
3.2.3. Control Parameters Assignments
4. Numerical Experiment
- Using “current to pbest/2-rand” mutation strategy,
- The p value for mutation strategy is computed as , where is set such that when is selected, at least 2 individuals are needed, and .
- Initial population size , the control parameter of external archive size .
- Historical memory size H = 6; set a final pair of parameters and , other values are initialized to 0.5 and other are initialized to 0.8.
- PIRFE parameter = 50.
5. Implementation of iLSHADE for LSLCHS
5.1. Solution Structure and Initial Population
5.2. Constraint Handling
6. Case Study
6.1. Description of Case Study
6.2. Results and Analysis
7. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Benchmark Function | Name | Domain | O-V | O-S |
---|---|---|---|---|
Sphere | 0 | |||
Schwefel (2.2) | 0 | |||
Schwefel (1.2) | 0 | |||
Rosenbrock | 0 | |||
Step | 0 | |||
Quartic | 0 | |||
Schwefel (2.26) | −418.9n | * | ||
Rastrigin | 0 | |||
Ackley | 0 | |||
Griewank | 0 |
LSHADE Mean (Std Dev) | JADE Mean (Std Dev) | CoDE Mean (Std Dev) | jDE Mean (Std Dev) | DE Mean (Std Dev) | iLSHADE Mean (Std Dev) | |
---|---|---|---|---|---|---|
0.00 × 100 (0.00 × 100) ≈ | 8.61 × 10−36 (7.18 × 10−36) − | 5.93 × 10−38 (7.34 × 10−38) − | 1.15 × 10−38 (1.18 × 10−38) − | 2.73 × 10−46 (1.42 × 10−45) − | 0.00 × 100 (0.00 × 100) | |
6.49 × 10−49 (4.38 × 10−48) − | 8.75 × 10−20 (4.68 × 10−20) − | 7.88 × 10−22 (6.58 × 10−22) − | 1.38 × 10−22 (1.01 × 10−22) − | 8.03 × 10−25 (1.65 × 10−24) − | 3.36 × 10−64 (2.05 × 10−63) | |
1.10 × 10−91 (5.90 × 10−91) − | 2.81 × 10−35 (2.75 × 10−35) − | 2.43 × 10−39 (4.02 × 10−39) − | 1.24 × 10−40 (1.79 × 10−40) − | 4.58 × 10−45 (2.03 × 10−44) − | 0.00 × 100 (0.00 × 100) | |
0.00 × 100 (0.00 × 100) ≈ | 0.00 × 100 (0.00 × 100) ≈ | 1.47 × 10−05 (1.96 × 10−05) − | 7.39 × 10−03 (9.11 × 10−03) − | 7.82 × 10−02 (5.53 × 10−01) − | 0.00 × 100 (0.00 × 100) | |
0.00 × 100 (0.00 × 100) ≈ | 0.00 × 100 (0.00 × 100) ≈ | 0.00 × 100 (0.00 × 100) ≈ | 0.00 × 100 (0.00 × 100) ≈ | 0.00 × 100 (0.00 × 100)≈ | 0.00 × 100 (0.00 × 100) | |
0.00 × 100 (0.00 × 100) ≈ | 1.40 × 10−71 (3.27 × 10−71) − | 6.47 × 10−71 (1.52 × 10−70) − | 1.40 × 10−72 (3.66 × 10−72) − | 8.39 × 10−88 (4.17 × 10−87) − | 0.00 × 100 (0.00 × 100) | |
−4189.83 (2.73 × 10−12) ≈ | −4189.83 (2.73 × 10−12) ≈ | −4189.83 (2.73 × 10−12) ≈ | −4189.83 (2.73 × 10−12) ≈ | −4189.83 (2.73 × 10−12) ≈ | −4189.83 (2.73 × 10−12) | |
0.00 × 100 (0.00 × 100) ≈ | 0.00 × 100 (0.00 × 100) ≈ | 0.00 × 100 (0.00 × 100) ≈ | 0.00 × 100 (0.00 × 100) ≈ | 3.06 × 100 (2.47 × 100) − | 0.00 × 100 (0.00 × 100) | |
3.72 × 10−15 (9.55 × 10−16) ≈ | 3.86 × 10−15 (6.90 × 10−16) − | 4.00 × 10−15 (2.37 × 10−30) − | 3.93 × 10−15 (4.93 × 10−16) − | 3.93 × 10−15 (4.93 × 10−16) − | 3.72 × 10−15 (9.55 × 10−16) | |
0.00 × 100 (0.00 × 100) ≈ | 2.42 × 10−12 (6.51 × 10−12) − | 0.00 × 100 (0.00 × 100) ≈ | 3.22 × 10−04 (2.28 × 10−03) − | 8.58 × 10−02 (6.17 × 10−02) − | 0.00 × 100 (0.00 × 100) | |
− | 2 | 6 | 6 | 7 | 8 | |
+ | 0 | 0 | 0 | 0 | 0 | |
≈ | 8 | 4 | 4 | 3 | 2 |
LSHADE Mean (Std Dev) | JADE Mean (Std Dev) | CoDE Mean (Std Dev) | jDE Mean (Std Dev) | DE Mean (Std Dev) | iLSHAD EMean (Std Dev) | |
---|---|---|---|---|---|---|
1.12 × 10−90 (6.44 × 10−90) − | 0.00 × 100 (0.00 × 100) ≈ | 9.85 × 10−19 (7.04 × 10−19) − | 4.31 × 10−41 (4.35 × 10−41) − | 9.34 × 10−44 (2.74 × 10−43) − | 0.00 × 100 (0.00 × 100) | |
2.09 × 10−42 (1.03 × 10−41) − | 4.11 × 10−27 (4.89 × 10−27) − | 4.01 × 10−12 (1.34 × 10−12) − | 3.48 × 10−24 (1.96 × 10−24) − | 1.16 × 10−05 (8.17 × 10−05) − | 4.88 × 10−58 (1.55 × 10−57) | |
3.85 × 10−81 (1.74 × 10−80) − | 3.58 × 10−49 (7.53 × 10−49) − | 4.22 × 10−19 (3.41 × 10−19) − | 7.27 × 10−43 (1.05 × 10−42) − | 1.16 × 10−46 (6.85 × 10−46) − | 0.00 × 100 (0.00 × 100) | |
1.40 × 10−25 (9.70 × 10−25) − | 1.85 × 10+01 (1.01 × 10+01) − | 1.82 × 10+01 (3.29 × 100) − | 1.15 × 10+01 (8.23 × 100) − | 2.62 × 100 (2.60 × 100) − | 0.00 × 100 (0.00 × 100) | |
0.00 × 100 (0.00 × 100) ≈ | 1.96 × 10−02 (1.39 × 10−01) − | 0.00 × 100 (0.00 × 100) ≈ | 0.00 × 100 (0.00 × 100) ≈ | 4.22 × 100 (7.61 × 100) − | 0.00 × 100 (0.00 × 100) | |
0.00 × 100 (0.00 × 100) ≈ | 0.00 × 100 (0.00 × 100) ≈ | 2.35 × 10−33 (2.93 × 10−33) − | 1.79 × 10−69 (3.86 × 10−69) − | 1.74 × 10−59 (1.10 × 10−58) − | 0.00 × 100 (0.00 × 100) | |
−12,569.49 (1.82 × 10−12) ≈ | −12,567.16 (1.64 × 10+01) − | −12,569.49 (1.82 × 10−12) ≈ | −12,569.49 (1.82 × 10−12) ≈ | −11552.14 (3.68 × 10+02) − | −12,569.49 (1.88 × 10−05) | |
1.74 × 10−16 (6.35 × 10−16) + | 0.00 × 100 (0.00 × 100) ≈ | 8.38 × 10−12 (9.18 × 10−12) + | 4.83 × 100 (3.86 × 100) − | 3.62 × 10+01 (1.44 × 10+01) − | 3.16 × 10−11 (1.66 × 10−10) | |
4.00 × 10−15 (2.37 × 10−30) ≈ | 4.76 × 10−15 (1.46 × 10−15) − | 2.74 × 10−10 (1.08 × 10−10) − | 5.60 × 10−15 (1.77 × 10−15) − | 2.64 × 10−01 (5.21 × 10−01) − | 4.00 × 10−15 (2.37 × 10−30) | |
0.00 × 100 (0.00 × 100) ≈ | 1.55 × 10−03 (3.81 × 10−03) − | 3.05 × 10−17 (1.08 × 10−16) − | 0.00 × 100 (0.00 × 100) ≈ | 7.99 × 10−03 (1.46 × 10−02) − | 0.00 × 100 (0.00 × 100) | |
− | 4 | 6 | 7 | 7 | 10 | |
+ | 1 | 1 | 1 | 0 | 0 | |
≈ | 5 | 3 | 2 | 3 | 0 |
Parameter | Wudongde | Baihetan | Xiluodu | Xiangjiaba |
---|---|---|---|---|
Adjustment ability | Season | Annual | Annual | Season |
Regulating storage (billion m3) | 2.60 | 10.40 | 6.46 | 0.90 |
Hydro plant discharge range (m3/s) | [49,400, 906] | [49,700, 905] | [43,700, 1500] | [49,800, 1500] |
Upriver water level range (m) | [975, 945] | [825, 765] | [600, 540] | [380, 370] |
Installed capacity (MW) | 12000 | 16000 | 13860 | 6400 |
Normal water level (m) | 975 | 825 | 600 | 380 |
Method | Wet Year (1999) | Normal Year (2008) | Dry Year (1969) | ||||||
---|---|---|---|---|---|---|---|---|---|
Max | Mean | Std | Max | Mean | Std | Max | Mean | Std | |
iLSHADE | 2425.03 | 2425.01 | 0.02 | 2268.13 | 2268.11 | 0.01 | 1814.36 | 1814.35 | 0.01 |
LSHADE | 2423.86 | 2422.99 | 0.48 | 2266.88 | 2265.08 | 0.94 | 1812.64 | 1810.87 | 0.99 |
Diff | 1.17 | 2.02 | 1.25 | 3.03 | 1.72 | 3.48 | |||
JADE | 2424.43 | 2420.97 | 1.32 | 2267.51 | 2260.74 | 2.43 | 1814.00 | 1808.67 | 2.234 |
Diff | 0.6 | 4.04 | 0.62 | 7.37 | 0.36 | 5.68 | |||
CoDE | 2423.39 | 2422.62 | 0.30 | 2265.62 | 2264.35 | 0.64 | 1813.08 | 1812.39 | 0.39 |
Diff | 1.64 | 2.39 | 2.51 | 3.76 | 1.28 | 1.96 |
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Wen, X.; Zhou, J.; He, Z.; Wang, C. Long-Term Scheduling of Large-Scale Cascade Hydropower Stations Using Improved Differential Evolution Algorithm. Water 2018, 10, 383. https://doi.org/10.3390/w10040383
Wen X, Zhou J, He Z, Wang C. Long-Term Scheduling of Large-Scale Cascade Hydropower Stations Using Improved Differential Evolution Algorithm. Water. 2018; 10(4):383. https://doi.org/10.3390/w10040383
Chicago/Turabian StyleWen, Xiaohao, Jianzhong Zhou, Zhongzheng He, and Chao Wang. 2018. "Long-Term Scheduling of Large-Scale Cascade Hydropower Stations Using Improved Differential Evolution Algorithm" Water 10, no. 4: 383. https://doi.org/10.3390/w10040383