Small-Signal Stability Constrained Optimal Power Flow Model Based on BP Neural Network Algorithm
Abstract
:1. Introduction
- To overcome the shortcomings of the traditional algorithm that the derivation of small-signal stability index is complicated and computationally intensive, the AI algorithm is introduced to solve the SC-OPF issue in this study;
- To determine the minimum damping ratio and the first-order eigenvalue sensitivity from the optimal system, the BP neural network is successfully employed;
- The simulation cases on the WSCC-9 bus and IEEE-39 bus test system validate that the BP-SCOPF can achieve optimal solutions;
- The results are compared to previous power flow calculation and economic scheduling linear programming method, which shows the superior performance of the BP-SCOPF in dealing with small-signal stability constraint.
2. Basic Models
2.1. SC-OPF Model
- Objective function
- Power flow equality constraints
- Inequality constraints
- Small-signal stability constraint
2.2. BP Neural Network Model
3. Construction Scheme of BP-SCOPF MODEL
3.1. BP Neural Network Architecture
3.2. Small-Signal Stability Constraint Handling
3.3. BP-SCOPF Operation Steps
- Read the system operation data from text files;
- Change the generator output under the principle of system power balance to obtain the input and output of the BP model, as Equation (14);
- Train and test the BP model using samples, judging the fitting performance according to the curve, further, Equations (12) and (13) are error analysis evaluation indicators;
- Take the Equation (1) as the objective function of SC-OPF and set the inequality constraints and small-signal stability constraint, as Equations (3) and (6);
- Carry on the BP-SCOPF iterative computation, with calculating eigenvalue sensitivity by BP algorithm, as show in Equation (16). Predict direction and magnitude of operating parameters by the approximate sensitivity, and then optimize variables during the iterative process;
- Check the eigenvalues and the minimum damping ratio. If all prespecified constraints are satisfied, stop the run and output the optimal solution; otherwise, return to step 3.
4. Case Study
4.1. The BP-SCOPF Model of WSCC-9 Bus System
4.2. A Linear Programming Correction Model Compared with BP-SCOPF Model
4.3. The BP-SCOPF Model of IEEE-39 Bus System
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Parameter | Set Value |
---|---|
Maximum iterations | 100 |
Learning rate | 0.01 |
Goal accuracy | 0.001 |
Input layer neurons | 30 |
Hidden layer neurons | 9 |
Output layer neurons | 1 |
ζ | ΣPGi /MW | PG1 /MW | PG2 /MW | PG3 /MW | QG1 /Mvar | QG2 /Mvar | QG3 /Mvar | |
---|---|---|---|---|---|---|---|---|
PF | 0.0057 | 319.64 | 71.64 | 163 | 85 | 27.10 | 6.59 | −10.92 |
OPF | 0.0034 | 317.64 | 156.23 | 88.57 | 72.84 | 15.20 | 3.82 | −10.42 |
BP-SCOPF | 0.0302 | 319.19 | 123.20 | 25 | 170.99 | 27.45 | 2.62 | −2.39 |
PL1/MW | PL2/MW | PL3/MW | |
---|---|---|---|
Case0 | 125 | 90 | 100 |
Case1 | 110 | 110 | 110 |
Case2 | 130 | 100 | 120 |
Case3 | 150 | 110 | 130 |
Case4 | 160 | 115 | 125 |
PF | OPF | BP-SCOPF | ||||
---|---|---|---|---|---|---|
ζPF | ΣPGi/MW | ζOPF | ΣPGi/MW | ζBP | ΣPGi/MW | |
Case0 | 0.0057 | 319.64 | 0.0034 | 317.64 | 0.0302 | 319.19 |
Case1 | 0.0065 | 334.49 | 0.0054 | 332.92 | 0.0306 | 334.23 |
Case2 | 0.0071 | 354. 41 | 0.0056 | 353.17 | 0.0303 | 354. 31 |
Case3 | 0.0083 | 395.66 | 0.0067 | 393.98 | 0.0305 | 394.98 |
Case4 | 0.0105 | 406.06 | 0.0074 | 404.31 | 0.0308 | 405.29 |
ζLPC | ζCheck | ΣPG/MW | PG1/MW | PG2/MW | PG3/MW | |
---|---|---|---|---|---|---|
Case0 | 0.0302 | 0.0104 | 320.34 | 125 | 90 | 100 |
Case1 | 0.0305 | 0.0218 | 335.94 | 110 | 110 | 110 |
Case2 | 0.0301 | 0.0306 | 356.43 | 130 | 100 | 120 |
Case3 | 0.0306 | −1 | 398.09 | 150 | 110 | 130 |
Case4 | 0.0311 | 0.0519 | 410.76 | 160 | 115 | 125 |
ζBP-SCOPF | ζCheck | ΣPGi /MW | PG1 /MW | PG2 /MW | PG3 /MW | QG1 /Mvar | QG2 /Mvar | QG3 /Mvar | |
---|---|---|---|---|---|---|---|---|---|
Case0 | 0.0301 | 0.0302 | 319.19 | 123.20 | 25 | 170.99 | 27.45 | 2.62 | −2.39 |
Case1 | 0.0301 | 0.0306 | 334.23 | 129.77 | 32.67 | 171.79 | 27.49 | 2.15 | −1.30 |
Case2 | 0.0299 | 0.0303 | 354.41 | 136.67 | 47.03 | 170.71 | 29.69 | 3.44 | −0.91 |
Case3 | 0.0307 | 0.0305 | 394.98 | 173.05 | 52.75 | 169.18 | 36.85 | 6.1 | 1.34 |
Case4 | 0.0304 | 0.0308 | 405.29 | 183.22 | 53.86 | 168.21 | 40.15 | 6.63 | 1.6 |
Gen1 | Gen2 | Gen3 | Gen4 | Gen5 | Gen6 | Gen7 | Gen8 | Gen9 | Gen10 | |
---|---|---|---|---|---|---|---|---|---|---|
/MW | 175 | 355.6 | 455 | 355.6 | 442.4 | 355.6 | 392 | 378 | 581 | 700 |
/MW | 402.5 | 747.5 | 920 | 862.5 | 862.5 | 862.5 | 862.5 | 805 | 1035 | 1380 |
/Mvar | −249.4 | −463.2 | −570.1 | −534.5 | −463.2 | −534.5 | −534.5 | −498.8 | −641.4 | −855.2 |
/Mvar | 249.4 | 463.2 | 570.1 | 534.5 | 463.2 | 534.5 | 534.5 | 498.8 | 641.4 | 855.2 |
PF | OPF | BP-SCOPF | ||||
---|---|---|---|---|---|---|
PGi/MW | QGi/Mvar | PGi/MW | QGi/Mvar | PGi/MW | QGi/Mvar | |
Gen1 | 250 | 203.97 | 402.5 | −42.29 | 402.5 | 181.84 |
Gen2 | 522.28 | 238.40 | 747.5 | 398.90 | 355.60 | 208.73 |
Gen3 | 650 | 251.59 | 614.42 | 193.14 | 455 | 218.51 |
Gen4 | 632 | 152.53 | 442.45 | 72.78 | 767.56 | 181.04 |
Gen5 | 508 | 185.53 | 462.15 | 120.30 | 499.17 | 194.50 |
Gen6 | 650 | 266.36 | 770.57 | 191.15 | 455 | 281.60 |
Gen7 | 560 | 131.70 | 392 | 32.10 | 862.5 | 250.2 |
Gen8 | 540 | 45.01 | 421.50 | 5.88 | 378 | −57.26 |
Gen9 | 830 | 140.99 | 582.03 | −53.59 | 581 | 99.73 |
Gen10 | 1000 | 239.82 | 1287.20 | 8.39 | 1380 | 216.01 |
ΣPGi/MW | 6142.28 | 6122.32 | 6136.33 | |||
ζ | 0.0214 | 0.0209 | 0.0301 |
Case0 | Case1 | Case2 | Case3 | Case4 | |
---|---|---|---|---|---|
1 | 0. 9 | 0.95 | 1.05 | 1.10 | |
ζOPF | 0.0209 | 0.0186 | 0.0194 | 0.0220 | 0.0235 |
ζLPC after check | 0.0308 | 0.0246 | 0.0176 | 0.0285 | 0.0261 |
ζBP-SCOPF after check | 0.0301 | 0.0294 | 0.0305 | 0.0296 | 0.0302 |
BP Training/s | Iteration Round | Average Iteration/s | Total BP-SCOPF/s | |
---|---|---|---|---|
9-bus system | 1.36 | 30 | 0.17 | 5.12 |
39-bus system | 2.01 | 58 | 0. 33 | 19.14 |
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Yang, Y.; Luo, Y.; Yang, L. Small-Signal Stability Constrained Optimal Power Flow Model Based on BP Neural Network Algorithm. Sustainability 2022, 14, 13386. https://doi.org/10.3390/su142013386
Yang Y, Luo Y, Yang L. Small-Signal Stability Constrained Optimal Power Flow Model Based on BP Neural Network Algorithm. Sustainability. 2022; 14(20):13386. https://doi.org/10.3390/su142013386
Chicago/Turabian StyleYang, Yude, Yuying Luo, and Lizhen Yang. 2022. "Small-Signal Stability Constrained Optimal Power Flow Model Based on BP Neural Network Algorithm" Sustainability 14, no. 20: 13386. https://doi.org/10.3390/su142013386