电力系统机组组合问题的主辅分解计算方法研究
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摘要
机组组合问题是电力系统经济调度领域的一项最基本的工作。该问题可以归纳为一个典型的大规模非线性混合整数规划问题。如果在问题整体求解过程中对不同约束缺乏有针对性的分类处理措施,则容易产生算法设计难度大、计算效率低等问题。本文根据机组组合问题不同约束作用特性的差异,探讨了如何从分解的方式构造新型计算方法,以达到降低算法设计难度、提高计算效率的目的。
由于电力系统机组组合问题存在众多的约束且其特性各有差异。不同的约束对问题求解的影响程度是不同的。本文首先比较全面地分析了机组组合问题各种约束的作用特性,并从线性与非线性、整数与非整数、等式与不等式、简单变量与函数约束等角度对有关约束进行了分类,给出了不同类型约束的可行处理对策,为进一步研究新型分解方法提供了基础。
在约束作用特性分类及处理对策分析的基础上,论文提出了一种求解电力系统机组组合问题的主辅分解方法,并设计了具体的算法流程和计算步骤。分解后的主子问题除包括原问题的目标函数外,还通过拉格朗日乘子的形式将系统有功功率平衡约束和系统旋转备用约束增广进目标函数;分解后的辅助子问题包括机组有功功率上下限约束处理子问题、机组有功功率变化率限值约束处理子问题、机组最小运行和停机时间约束处理子问题、系统运行机组数量约束处理子问题、网络线路潮流约束处理子问题。其中,前两个辅助子问题直接嵌入到主子问题的迭代过程,中间两个辅助子问题置于主子问题迭代过程的外层进行处理,最后一个辅助子问题置于最外层作为结果校验。这种内外分层结构的迭代方式降低了算法设计的难度,减少了不必要的重复计算,提高了计算效率。
本文最后以一个包含17个机组的算例系统为例进行了仿真分析。算例结果表明,本文提出的主辅分解计算方法,由于实施了约束的分类处理,显著降低了主子问题迭代计算的难度,简化了有关辅助子问题处理的复杂性,因此可以改善机组组合问题计算的性能,有助于提高算法的实用性。
Unit commitment is one of the most basic works in the field of power system economic dispatch. It can be summed up as a typical large-scale nonlinear mixed-integer programming problem. If lacks the target-oriented classified processing measure to the different constraint in the whole solution process, then it’s easy to cause the algorithm to be difficult to design and low counting yield . According to the different constraint function characteristic in unit commitment, this article discussed how to structure a new computational method through the decomposition, and then achieves the goal of reducing the algorithm design difficulty and enhancing counting yield.
There has numerous constraints with different characteristic in electric power system generation unit commitment; their influence to the solution is also different. This article first comprehensively analyze each kind of the constraint’s characteristic in unit commitment, and then carry on the classification to the related constraint with different view as linear and non-linear, integer and non-integer, equality and inequality, simple variable and function constraint, gives out feasible processing countermeasure to different constraint, provides the foundation to further study to the new decomposition method.
On the foundation of constraint characteristic classification and countermeasure analysis, this article proposes a kind of main-auxiliary decomposition method to solve the unit commitment problem, and design the concrete algorithm flow and computation steps. After the decomposition,the main-auxiliary problem does not only includes the original objective function, but also augments the system load balance constraints and the spinning reserve constraints into the objective function through the Lagrange multiplier's form; the auxiliary sub-questions after decomposition includes: unit available generation capacity constraints sub-questions, unit ramp-up/ramp-down limits constraints sub-questions, unit minimum-up/minimum-down time constraints sub-questions, unit availability constraints sub-questions, network line power flow constraints sub-questions; the first two sub-questions are directly inserted into the main iterative process, the middle two sub-questions are put into the outer layer of main iterative process to carry on processing, the last one is put into the most outer layer as check of the solution. This inside and outside iterative way reduces the algorithm design difficulty, and nonessential double counting, raises the counting yield.
Finally, this article use an example system contained 17 units to carry on the simulation analysis. The result indicates that: the main-auxiliary decomposition method this article proposed, because it has implemented constraints classified process, the main iterative computation difficulty is obviously reduced, the related sub-questions complexity is simplified, so it can improve the performance of the unit commitment computation, and is helpful to improve the practicality of algorithm.
由于电力系统机组组合问题存在众多的约束且其特性各有差异。不同的约束对问题求解的影响程度是不同的。本文首先比较全面地分析了机组组合问题各种约束的作用特性,并从线性与非线性、整数与非整数、等式与不等式、简单变量与函数约束等角度对有关约束进行了分类,给出了不同类型约束的可行处理对策,为进一步研究新型分解方法提供了基础。
在约束作用特性分类及处理对策分析的基础上,论文提出了一种求解电力系统机组组合问题的主辅分解方法,并设计了具体的算法流程和计算步骤。分解后的主子问题除包括原问题的目标函数外,还通过拉格朗日乘子的形式将系统有功功率平衡约束和系统旋转备用约束增广进目标函数;分解后的辅助子问题包括机组有功功率上下限约束处理子问题、机组有功功率变化率限值约束处理子问题、机组最小运行和停机时间约束处理子问题、系统运行机组数量约束处理子问题、网络线路潮流约束处理子问题。其中,前两个辅助子问题直接嵌入到主子问题的迭代过程,中间两个辅助子问题置于主子问题迭代过程的外层进行处理,最后一个辅助子问题置于最外层作为结果校验。这种内外分层结构的迭代方式降低了算法设计的难度,减少了不必要的重复计算,提高了计算效率。
本文最后以一个包含17个机组的算例系统为例进行了仿真分析。算例结果表明,本文提出的主辅分解计算方法,由于实施了约束的分类处理,显著降低了主子问题迭代计算的难度,简化了有关辅助子问题处理的复杂性,因此可以改善机组组合问题计算的性能,有助于提高算法的实用性。
Unit commitment is one of the most basic works in the field of power system economic dispatch. It can be summed up as a typical large-scale nonlinear mixed-integer programming problem. If lacks the target-oriented classified processing measure to the different constraint in the whole solution process, then it’s easy to cause the algorithm to be difficult to design and low counting yield . According to the different constraint function characteristic in unit commitment, this article discussed how to structure a new computational method through the decomposition, and then achieves the goal of reducing the algorithm design difficulty and enhancing counting yield.
There has numerous constraints with different characteristic in electric power system generation unit commitment; their influence to the solution is also different. This article first comprehensively analyze each kind of the constraint’s characteristic in unit commitment, and then carry on the classification to the related constraint with different view as linear and non-linear, integer and non-integer, equality and inequality, simple variable and function constraint, gives out feasible processing countermeasure to different constraint, provides the foundation to further study to the new decomposition method.
On the foundation of constraint characteristic classification and countermeasure analysis, this article proposes a kind of main-auxiliary decomposition method to solve the unit commitment problem, and design the concrete algorithm flow and computation steps. After the decomposition,the main-auxiliary problem does not only includes the original objective function, but also augments the system load balance constraints and the spinning reserve constraints into the objective function through the Lagrange multiplier's form; the auxiliary sub-questions after decomposition includes: unit available generation capacity constraints sub-questions, unit ramp-up/ramp-down limits constraints sub-questions, unit minimum-up/minimum-down time constraints sub-questions, unit availability constraints sub-questions, network line power flow constraints sub-questions; the first two sub-questions are directly inserted into the main iterative process, the middle two sub-questions are put into the outer layer of main iterative process to carry on processing, the last one is put into the most outer layer as check of the solution. This inside and outside iterative way reduces the algorithm design difficulty, and nonessential double counting, raises the counting yield.
Finally, this article use an example system contained 17 units to carry on the simulation analysis. The result indicates that: the main-auxiliary decomposition method this article proposed, because it has implemented constraints classified process, the main iterative computation difficulty is obviously reduced, the related sub-questions complexity is simplified, so it can improve the performance of the unit commitment computation, and is helpful to improve the practicality of algorithm.
引文
1 Narayana Prasad Padhy. Unit Commitment-A Bibliographical Survey. IEEE Trans. Power Syst. , May. 2004,19(2):1196-1205.
2 Gerald B. Sheble, George N. Fahd. Unit Commitment Literature Synopsis. IEEE Trans. Power Syst. Feb. 1994, 9(1): 128-135.
3 Arthur I. Cohen, Vladimir Brandwain, Show-Kang Chang. Security Constrained Unit Commitment for Open Market. Power Industry Computer Applications, July. 1999. PICA '99. Proceedings of the 21st 1999 IEEE International Conference:39-44.
4 Narayana Prasad Padhy. Unit Commitment Problem under Deregulated environment-A Review. Power Engineering Society General Meeting, 2003, IEEE:1088-1094.
5娄素华等电力系统机组启停优化问题的改进DPSO算法.中国电机工程学报, 2005, 25(8):32-35.
6 R. H. Kerr, J. L. Scheidt, A. J. Fontana, and J. K. Wiley. Unit Commitment. IEEE Trans. on Power App. Syst., May 1966, 85:417-421.
7 K. Hara, M. Kimura, and N. Honda.A method for planning economic unit commitment and maintenance of thermal power systems. IEEE Trans. on Power App. Syst., May 1966, 85:427-736.
8陈皓勇,王锡凡.机组组合问题的优化方法综述.电力系统自动化,1999,23(4):51-56.
9于尔铿等.发电竞价算法(一)——排队法.电力系统自动化,2001,2:16-19
10 R. R. Shoults, S. K. Chang, S. Helmick and W. M. Grady. A practical approach to unit commitment , economic dispatch and savings allocation for multiple-area pool operation with import/export constraints. IEEE Trans. on Power App. Syst, May./Apr.1980, 99:625-635.
11 F. N. Lee. Multi-area unit commitment. IEEE Trans. on Power Syst., Aug.1989, l7:1208-1218.
12 F. N. Lee. The application of commitment utilization factor to thermal unit commitment. IEEE Trans. on Power Syst., May 1991, 6: 691-698.
13 F . N. Lee. Short-term thermal unit commitment-a new method. IEEE Trans on PWRS, 1988, 3(2):421-428.
14于尔铿等发电竞价算法(三)——动态规划法.电力系统自动化, 2001,3:19-22.
15 Lowery, P. G. Generating unit commitment by dynamic programming. IEEE Trans. on PAS, May. 1966, 85(5): 422-426.
16 G. W. Chang, Y. D. Tsai, C. Y. Lai, and J. S. Chung. A Practical Mixed Integer Linear Programming Based Approach for Unit Commitment. Power Engineering Society General Meeting, 2004. IEEE, Vol.1:221-225.
17 Dan Streiffert, Russ Philbrick, and Andrew Ott. A Mixed Integer Programming Solution for Market Clearing and Reliability Analysis. 2005 IEEE Power Engineering Society General Meeting, vol.3: 2724-2731.
18 T. S. Dillon, K. W. Edwin, H. D. Kochs, and R. J. Taud. Integer programming approach to the problem of optimal unit commitment with probabilistic reserve determination. IEEE Trans. on Power App. Syst., Nov./Dec. 1978, 97:2154-2166.
19 A. I. Cohen and S. H. Wan. A method for solving the fuel constrained unit commitment problem. IEEE Trans. Power Syst., Aug. 1987, 2:608-614.
20 R. Nieva, A. Inda, and I. Guillen. Lagrangiaan reduction of search-range for large-scale unit commitment. IEEE Trans. Power Syst., May. 1987, 2: 465-473.
21 S. K. Tong and S. M. Shahidehpour. Combination of Lagranian-Relaxation and linear programming approaches for fuel-constrained unit commitment problems. Proc, Inst.Elect. Eng. Gen. Transm. Dist.,May. 1989,136: 162-174.
22 A. Merlin and P. Sandrin. A new method for unit commitment at Electricite De France. IEEE Trans. Power App. Syst. , Aug. 1983,102: 1218-1225.
23 K. Aoki, M. Itoh, T. Satoh, K. Nara, and M. Kanezashi. Optimal long-term unit commitment in large scale systems including fuel constrained thermal and pumped-storage hydro. IEEE Trans. Power Syst., Aug. 1989, 4: 1065-1073.
24 K. Aoki, T. Satoh, and M. Itoh. Unit commitment in large scale power systems including fuel constrained thermal and pumped strorage hydro. IEEE Trans. Power Syst., May. 1987, 2:1077-1084.
25 F. Zhuang and F. D. Galiana. Toward a more rigorous and practical unit commitment by Lagrangian Relaxation. IEEE Trans. Power Syst., May.1988,3:763-773.
26 C. Wang and S. M. Shahidepour. Ramp-rate limits in unit commitment and economic dispatch incorporating rotor fatigue effect. IEEE Trans. Power Syst., Aug. 1994,9: 1539-1545.
27 H. Ma and S. M. Shahhhidepour. Unit commitment with transmission securith and voltage constrains. IEEE Trans. Power Syst., May. 1999,14:757-764.
28 S. Takriti and J. R. Birge. Using interger programming to refine Lagrangian-based unit commitment solutions. IEEE Trans. Power Syst., Feb. 2000, 15: 151-156.
29 C. P. Cheng, C.W. Liu, and C. C. Liu. Unit commitment by Lagrangian Relaxation and genetic algorithm. IEEE Trans. Power Syst., May. 2000,15:707-714.
30夏清等.考虑电网安全约束条件的机组组合新方法.清华大学学报(自然科学版), 1999, 39(9):14-17.
31 A. H. Mantawy, Y. L. Abdel-Magid, and S. Z. Selim. A simulated annealing algorithm for unit commitment. IEEE Trans. on Power Syst., Feb. 1998, 13:197-204.
32郭泽军等.遗传_模拟退火算法在电厂机组负荷分配中的应用.职大学报,2006,2:69-70.
33蔡兴国,初壮.用遗传算法解算机组组合的研究.电网技术, 2003, 27(7):36-39.
34 S. Mukhtari, J. Singh, and B. Wollenberg. A unit commitment expert system. IEEE Trans Power Syst., Feb. 1988, 3:272-277.
35 Z. Ouyang and S. M. Shahidehpour. Short-term unit commitment expert system. Elect. Power Syst., Dec. 1990, Res: 1-13.
36 S. K. Tong and S. M. Shahidehpour. Hydro thermal unit commitment with probabilistic constraints using segmentation method. IEEE Trans. Power Syst., Feb. 1990, 5:276-282.
37 H. Sasaki, M. Watanabe,and R. Yokoyama. A solution method of unit commitment by artificial neural networks. IEEE Trans Power Syst., Aug. 1992, 7: 974-981.
38 C. Wang and S. M. Shahidepour. Effect of ramp-rate limits on unit commitment and economic dispatch. IEEE Trans. Power Syst., Aug. 1993,8: 1341-1350.
39 M. P. Walsh and M. J. O. Malley. Augmented Hopfield network for unit commitment and economic dispatch. IEEE Trans. Power Syst., Nov.1997,12: 1765-1774.
40高炜欣等. Hopfield神经网络在机组组合问题中的应用.计算机应用. 2009, 29(4):1028-1031.
41 M. P. Walsh and M. J. O. Malley. Augmented Hopfield network for unit commitment and economic dispatch. IEEE Trans. Power Syst., Nov.1997, 12: 1765-1774.
42 H. Mori and O. Matsuzaki. Embedding the priority into Tabu search for unit commitment. Proc. IEEE Winter Meeting,2000:344-349.
43 H. Mori and T. Usami. Unit commitment using tabu search with restricted neighborhood. Proc. Intell. Syst. Applicat. Power Syst. ,1996: 422-427.
44 A. Rajan, C. C. Mohan, and M. R. Manivannan. Neural based tabu search method for solving unit commitment problem. Proc.5th Int. Conf. Power Syst. Manage. Contr ,2002:180-185.
45 W. M. Lin and F. S. Cheng, and M. T. Tsay. An improved tabu search for economic dispatch with multiple minima. IEEE Trans. Power Syst., Feb. 2002, 17: 108-112.
46 A. H. Mantawy, S. A.Soliman, and M. E. El-Hawary. A new tabu search algorithm for the long-term hybro scheduling problem. June. 2002, in Proc. Large Eng. Syst. Conf. Power Eng: 29-34.
47王锡凡.现代电力系统分析.科学出版社,2003.3
48 C. C. Su and Y. Y. Hsu. Fuzzy dynamic programming: an application to unit commitment. IEEE Trans. Power Syst. Aug. 1991,6:1231-1237.
49 X. Hindsberger and H. F. M Ravn. Multiresolution modeling of hydro-thermal systems. in Power Eng. Soc. Int. Conf. Power Comput. Applicat., 2001:5-10.
50 S. J. Huang and C. L. Huang. Application of genetic-based neural network to thermal unit commitment. IEEE Trans. Power Syst. ,May. 1997,12:654-660.
51 A. H. Mantawy, Y. L. Abdel-Magid, and S. Z. Selim. Intergrating genetic algoritms, Tabu search and simulated annealing for the unit commitment problem. IEEE Trans. Power Syst. ,Aug. 1999,14:829-836.
52 N. P. Padhy. Unit commitment using hybrid models: a comparative study for dynamic programming, expert system, fuzzy system and genetic algorithms.Elect. Power Energy Syst. ,2000 23:827-836.
53李晓磊等.基于动态搜索线性混合整数法的机组组合新算法.电力系统自动化, 2008, 32(21):18-21.
54韩学山,柳焯.考虑机组爬坡速度和网络安全约束的经济调度解耦算法.电力系统自动化,2002, 26(13):32-37.
55王民量,张伯明,夏清.考虑机组爬坡速率和网络安全的经济调度新算法.电力系统自动化, 2000,10:14-20.
56汤奕,于继来,周苏荃.电力网络源流路径电气剖分算法.电力系统自动化, 2006, 30(22):28-33
57汤奕.电力网络源流路径电气剖分方法研究.哈尔滨:哈尔滨工业大学工学博士学位论文. 2006.
58于继来,汤奕.交流支路和节点的联合电气剖分.中国电机工程学报, 2007, 27(16):37-42.
2 Gerald B. Sheble, George N. Fahd. Unit Commitment Literature Synopsis. IEEE Trans. Power Syst. Feb. 1994, 9(1): 128-135.
3 Arthur I. Cohen, Vladimir Brandwain, Show-Kang Chang. Security Constrained Unit Commitment for Open Market. Power Industry Computer Applications, July. 1999. PICA '99. Proceedings of the 21st 1999 IEEE International Conference:39-44.
4 Narayana Prasad Padhy. Unit Commitment Problem under Deregulated environment-A Review. Power Engineering Society General Meeting, 2003, IEEE:1088-1094.
5娄素华等电力系统机组启停优化问题的改进DPSO算法.中国电机工程学报, 2005, 25(8):32-35.
6 R. H. Kerr, J. L. Scheidt, A. J. Fontana, and J. K. Wiley. Unit Commitment. IEEE Trans. on Power App. Syst., May 1966, 85:417-421.
7 K. Hara, M. Kimura, and N. Honda.A method for planning economic unit commitment and maintenance of thermal power systems. IEEE Trans. on Power App. Syst., May 1966, 85:427-736.
8陈皓勇,王锡凡.机组组合问题的优化方法综述.电力系统自动化,1999,23(4):51-56.
9于尔铿等.发电竞价算法(一)——排队法.电力系统自动化,2001,2:16-19
10 R. R. Shoults, S. K. Chang, S. Helmick and W. M. Grady. A practical approach to unit commitment , economic dispatch and savings allocation for multiple-area pool operation with import/export constraints. IEEE Trans. on Power App. Syst, May./Apr.1980, 99:625-635.
11 F. N. Lee. Multi-area unit commitment. IEEE Trans. on Power Syst., Aug.1989, l7:1208-1218.
12 F. N. Lee. The application of commitment utilization factor to thermal unit commitment. IEEE Trans. on Power Syst., May 1991, 6: 691-698.
13 F . N. Lee. Short-term thermal unit commitment-a new method. IEEE Trans on PWRS, 1988, 3(2):421-428.
14于尔铿等发电竞价算法(三)——动态规划法.电力系统自动化, 2001,3:19-22.
15 Lowery, P. G. Generating unit commitment by dynamic programming. IEEE Trans. on PAS, May. 1966, 85(5): 422-426.
16 G. W. Chang, Y. D. Tsai, C. Y. Lai, and J. S. Chung. A Practical Mixed Integer Linear Programming Based Approach for Unit Commitment. Power Engineering Society General Meeting, 2004. IEEE, Vol.1:221-225.
17 Dan Streiffert, Russ Philbrick, and Andrew Ott. A Mixed Integer Programming Solution for Market Clearing and Reliability Analysis. 2005 IEEE Power Engineering Society General Meeting, vol.3: 2724-2731.
18 T. S. Dillon, K. W. Edwin, H. D. Kochs, and R. J. Taud. Integer programming approach to the problem of optimal unit commitment with probabilistic reserve determination. IEEE Trans. on Power App. Syst., Nov./Dec. 1978, 97:2154-2166.
19 A. I. Cohen and S. H. Wan. A method for solving the fuel constrained unit commitment problem. IEEE Trans. Power Syst., Aug. 1987, 2:608-614.
20 R. Nieva, A. Inda, and I. Guillen. Lagrangiaan reduction of search-range for large-scale unit commitment. IEEE Trans. Power Syst., May. 1987, 2: 465-473.
21 S. K. Tong and S. M. Shahidehpour. Combination of Lagranian-Relaxation and linear programming approaches for fuel-constrained unit commitment problems. Proc, Inst.Elect. Eng. Gen. Transm. Dist.,May. 1989,136: 162-174.
22 A. Merlin and P. Sandrin. A new method for unit commitment at Electricite De France. IEEE Trans. Power App. Syst. , Aug. 1983,102: 1218-1225.
23 K. Aoki, M. Itoh, T. Satoh, K. Nara, and M. Kanezashi. Optimal long-term unit commitment in large scale systems including fuel constrained thermal and pumped-storage hydro. IEEE Trans. Power Syst., Aug. 1989, 4: 1065-1073.
24 K. Aoki, T. Satoh, and M. Itoh. Unit commitment in large scale power systems including fuel constrained thermal and pumped strorage hydro. IEEE Trans. Power Syst., May. 1987, 2:1077-1084.
25 F. Zhuang and F. D. Galiana. Toward a more rigorous and practical unit commitment by Lagrangian Relaxation. IEEE Trans. Power Syst., May.1988,3:763-773.
26 C. Wang and S. M. Shahidepour. Ramp-rate limits in unit commitment and economic dispatch incorporating rotor fatigue effect. IEEE Trans. Power Syst., Aug. 1994,9: 1539-1545.
27 H. Ma and S. M. Shahhhidepour. Unit commitment with transmission securith and voltage constrains. IEEE Trans. Power Syst., May. 1999,14:757-764.
28 S. Takriti and J. R. Birge. Using interger programming to refine Lagrangian-based unit commitment solutions. IEEE Trans. Power Syst., Feb. 2000, 15: 151-156.
29 C. P. Cheng, C.W. Liu, and C. C. Liu. Unit commitment by Lagrangian Relaxation and genetic algorithm. IEEE Trans. Power Syst., May. 2000,15:707-714.
30夏清等.考虑电网安全约束条件的机组组合新方法.清华大学学报(自然科学版), 1999, 39(9):14-17.
31 A. H. Mantawy, Y. L. Abdel-Magid, and S. Z. Selim. A simulated annealing algorithm for unit commitment. IEEE Trans. on Power Syst., Feb. 1998, 13:197-204.
32郭泽军等.遗传_模拟退火算法在电厂机组负荷分配中的应用.职大学报,2006,2:69-70.
33蔡兴国,初壮.用遗传算法解算机组组合的研究.电网技术, 2003, 27(7):36-39.
34 S. Mukhtari, J. Singh, and B. Wollenberg. A unit commitment expert system. IEEE Trans Power Syst., Feb. 1988, 3:272-277.
35 Z. Ouyang and S. M. Shahidehpour. Short-term unit commitment expert system. Elect. Power Syst., Dec. 1990, Res: 1-13.
36 S. K. Tong and S. M. Shahidehpour. Hydro thermal unit commitment with probabilistic constraints using segmentation method. IEEE Trans. Power Syst., Feb. 1990, 5:276-282.
37 H. Sasaki, M. Watanabe,and R. Yokoyama. A solution method of unit commitment by artificial neural networks. IEEE Trans Power Syst., Aug. 1992, 7: 974-981.
38 C. Wang and S. M. Shahidepour. Effect of ramp-rate limits on unit commitment and economic dispatch. IEEE Trans. Power Syst., Aug. 1993,8: 1341-1350.
39 M. P. Walsh and M. J. O. Malley. Augmented Hopfield network for unit commitment and economic dispatch. IEEE Trans. Power Syst., Nov.1997,12: 1765-1774.
40高炜欣等. Hopfield神经网络在机组组合问题中的应用.计算机应用. 2009, 29(4):1028-1031.
41 M. P. Walsh and M. J. O. Malley. Augmented Hopfield network for unit commitment and economic dispatch. IEEE Trans. Power Syst., Nov.1997, 12: 1765-1774.
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