凸规划技术在水火联合调度问题中的应用
|
摘要
当代电力行业的发展有着非常明显的时代特征,清洁能源和智能电网是其两大主题词。在水电资源或其它清洁能源丰富的地区,大力开发清洁能源与火电互补调度是非常必要的。我国水电资源丰富,在倡导开发水电的同时,如何充分利用已开发的部分水电能源也很重要。水火联合调度问题不论从建模的角度还是从求解算法的角度来讲都是非常困难的。建模的难点在于水力发电本身就是个复杂的物理过程,加之开发同一水域的梯级水电能源会令决策变量时空耦合。算法方面,即便对某些因素进行简化,多数现存的优化算法也很难对这个大规模、非凸的模型进行高效、精准地求解。因此,有必要探寻新的水火联合调度问题求解思路。
本论文在研究凸规划特别是近年来线性锥规划的快速发展基础上,结合几种处理非凸等式约束方法,从模型和算法两个方面研究了大规模水火联合调度问题。首先,在详细分析了水轮机发电过程及不同调度时期各类水电站出力特点后,总结出水火联合调度问题的两个瓶颈问题为高次非凸等式约束和整数变量。基于现代内点理论的锥规划恰巧可以同时处理这两个问题。半定规划求解一个非凸标准水火联合调度算例表明,半定规划在解的全局最优性和快速性方面均优于其它算法。然而,半定规划求解非凸问题的本质为松弛方法。为了克服半定规划在直接求解含有高次非凸等式约束问题时会出现解的秩不为1的情况,本论文首次提出了先用高精度的凸化方法将水电非凸等式约束转化为0-1混合整数约束,再用改进的半定有效不等式方法对问题进行求解的思想。全文共6章,各章节主要内容可归纳如下：
第1章—绪论：从模型和算法两个角度回顾国内外水火联合调度研究现状。
第2章—线性锥规划基本原理：介绍本工作在求解水火联合调度问题中所要应用的数学手段—锥规划的相关性质。着眼于介绍半定规划、二阶锥规划模型及与其他模型之间的转化关系；这些模型的凸性、对偶性及相关的内点算法；它们在几类问题中的应用及适用的求解器。
第3章—线性半定规划求解简单水火电力调度系统：以一个简单的水火电力调度算例为例阐述使用线性半定规划求解该类问题的基本原理和过程方法。通过将原问题从一般向量空间提升到半定锥空间,新的半定规划模型将二次非凸等式约束松弛为半定锥上的线性函数。用集成了线性锥规划内点算法的软件包求解转化后的模型。实验结果指出原模型中缺少弃水变量上限导致解不合理。而其他文献所用的算法因为无法保证解的全局性而忽略了这个问题。在给出了一个上界之后,半定规划的解依然优于其他算法的解,进一步证明了线性半定规划的有效性。
第4章—水火电力网联合调度问题的凸性分析：在前几章的基础上提出了水电调度问题的分段二次凸化方法。为了达到利用最新凸优化技术的目的,该方法首先对水火联合调度问题的难点—水电子系统中的非凸等式约束进行详细分析,通过逐步精炼水火联合调度模型中难以处理的和只给问题增加计算量而不增加计算难度的约束和变量,并通过将各调度时期的水量、水头和水库特性结合起来分析,提出了可以直接应用凸规划技术的水火联合调度模型。借助成熟的线性分段技术思想,提出了逼近效果更好的分段二次凸化方法,针对水电子系统中两个具体的约束：水电出力函数和弃水函数等式约束,给出相应的具体的数学表达式。本章得出结论,水火联合问题可以通过逼近、引进变量等手段转化为一个可利用成熟锥规划技术求解的混合整数规划问题。
第5章—分段二次凸化半定有效不等式方法求解水火联合调度系统的仿真与分析：在前几章基础上,用几组数值试验和算例仿真进一步验证本文提出的处理非凸等式约束的基于改进半定有效不等式方法的效果。一方面分析半定有效不等式方法对一个整数线性规划例子生成的松弛模型紧性方面存在的问题,再分析改进的半定有效不等式方法对该例子的更紧的松弛。另一方面用改进的半定有效不等式方法求解大规模的水火联合调度问题。数值试验KNAPSACK问题,和仿真算例32-HTC、153-HTC两个大规模的水火仿真系统可以说明本文所提出的凸化配合改进的半定有效不等式算法可以可靠、快速地求解非凸、混合整数水火联合调度问题。
第6章—总结与展望：总结了本文的主要工作和创新点,并指出了可进一步开拓的研究方向。
Electric power industry is now advancing with distinguish modern marks. Clean energy and smart grid are two key words of it. It is necessary to utilize clean energy to the best and have it participated in the power generation coordination when there are huge storage of hy-dropower and alike around the area. China is rich in hydropower resources. How to utilize the exiting hydropower is as important as advocating hydropower development. Hydrothermal coordination problems are of difficulty not only because of the mathematical modeling pro-cess but also because of the corresponding optimization algorithms. The struggling of mathe-matical modeling is caused by the complex process of hydropower generation. The fact that developing cascaded hydropower plants within the same watershed can bring a coupling ef-fect on both spatial aspect and time aspect. Though the problems can be simplified by getting rid of certain factors, not all the algorithms can guarantee a fast and accurate optimal solution.
This dissertation tried to solve the problems from angels of both mathematical modeling and optimization algorithm, with the help with several methods of handling nonconvex equal-ity constraints based on some recently developed convex optimization skills, especially the ones regarding to conic optimization. At first, the whole process of generating hydropower was examined, followed by the conclusion that the bottleneck problems are nonconvex equal-ity constraints and integer variables. Luckily, conic programming based on modern interior point method has the potential of dealing with both of these two issues at the same time. Through solving a standard test data set of hydrothermal coordination problem, semidefinite programming was proved to be better than other existing algorithms with respect to computa-tional time and the quality of the solution. However, semidefinite programming is born to be a relaxation method. To overcome the problem that the rank of the solution matrix might not equal to one when the problem contains nonconvex equality constraints, this dissertation pro-posed a solution that transfers nonconvex equality constraints into0-1mixed integer con-straints by convexification with high accuracy first, then solves the problem by improved semidefinite valid inequality method. This dissertation is divided into6chapters, which can be summarized as follows:
Chapter1on introduction:Literature review of both domestic and abroad on hydrother-mal coordination problem from angels of mathematical modeling and mathematical optimiza-tion methods.
Chapter2on basic knowledge of conic programming:Introducing principal concepts of conic programming, which are the key mathematically tools in solving hydrothermal coordi- nation problems, focusing on aspects of introduction of semidefinite programming and second order cone programming, the relationship between these optimization methods, the convexity of these methods, their duality and related interior point algorithms, the applications of these methods, and the solvers that can be passed along to solve these mathematical optimization problems.
Chapter3on semidefinite programming on a simple hydrothermal coordination problem: This chapter elaborated how to solve a simple mathematical model of hydrothermal coordina-tion with basic semidefinite programming. By lifting the normal vector space produced by the original problem into a higher dimensional semidefinite cone, the new semidefinite program-ming problem was a linear one on semidefinite cone relaxed from the original model with nonconvex equality constraints. The resulting model can be solved by efficient solvers inte-grated with conic interior point methods. The results showed that the original data set had an issue of lacking upper bounds for spillage water variables. While the exiting literatures worked on the same data set failed to report this issue because the methods they used were not truly global ones. After fixing upper bounds for the data set, the results by semidefinite pro-gramming were still the best among all the methods, further validating the fact that semidefi-nite programming method is very effective.
Chapter4on convexity of hydrothermal coordination problems:Based on the previous chapters, this chapter presented a piece-wise quadratic convexification method for hydro-thermal coordination problem. To take the advantages of newly developed convex program-ming techniques, the proposed method focused on analysis of the most difficult part of the problem, i.e., the nonconvex equality constraints of the hydrothermal coordination problem first, and then provided a convex programming-friendly model by analyzing storage, water net head, and reservoir characteristics all together over different spans of scheduling horizon. Based on well-developed piecewise linearization techniques, the dissertation presented a more accurate piecewise quadratic convexification method. And it also provided a detailed mathe-matical expression designed for hydropower generation equality constraints and spillage equality constraints. It concluded that the hydrothermal coordination problem can be trans-ferred into a mixed integer programming problem by means of approximation and introduc-tion of variables.
Chapter5on simulation and analysis of hydrothermal coordination problem based on quadratic convexification semidefinite valid inequality method:Several testing problems and simulations were applied to validate the effectiveness of the proposed method. On one hand, this chapter presented the tightness problems caused by solving a mixed integer programming problem with semidefinite valid inequality relaxation method, followed by the analysis of im-proved semidefinite valid inequality method for a tighter relaxation. On the other hand, the improved semidefinite valid inequality method was used for large-scale hydrothermal coordi-nation problem. A testing KNAPSACK problem,32-HTC simulation problem, and153-HTC simulation problem were used to testify that the proposed improved semidefinite valid ine-quality method can solve the mixed integer hydrothermal coordination problem reliably and efficiently.
Chapter6on conclusions and future works:The framework of the dissertation along with the creativities was summarized. Several further follow-up studies were pointed out as well.
本论文在研究凸规划特别是近年来线性锥规划的快速发展基础上,结合几种处理非凸等式约束方法,从模型和算法两个方面研究了大规模水火联合调度问题。首先,在详细分析了水轮机发电过程及不同调度时期各类水电站出力特点后,总结出水火联合调度问题的两个瓶颈问题为高次非凸等式约束和整数变量。基于现代内点理论的锥规划恰巧可以同时处理这两个问题。半定规划求解一个非凸标准水火联合调度算例表明,半定规划在解的全局最优性和快速性方面均优于其它算法。然而,半定规划求解非凸问题的本质为松弛方法。为了克服半定规划在直接求解含有高次非凸等式约束问题时会出现解的秩不为1的情况,本论文首次提出了先用高精度的凸化方法将水电非凸等式约束转化为0-1混合整数约束,再用改进的半定有效不等式方法对问题进行求解的思想。全文共6章,各章节主要内容可归纳如下：
第1章—绪论：从模型和算法两个角度回顾国内外水火联合调度研究现状。
第2章—线性锥规划基本原理：介绍本工作在求解水火联合调度问题中所要应用的数学手段—锥规划的相关性质。着眼于介绍半定规划、二阶锥规划模型及与其他模型之间的转化关系；这些模型的凸性、对偶性及相关的内点算法；它们在几类问题中的应用及适用的求解器。
第3章—线性半定规划求解简单水火电力调度系统：以一个简单的水火电力调度算例为例阐述使用线性半定规划求解该类问题的基本原理和过程方法。通过将原问题从一般向量空间提升到半定锥空间,新的半定规划模型将二次非凸等式约束松弛为半定锥上的线性函数。用集成了线性锥规划内点算法的软件包求解转化后的模型。实验结果指出原模型中缺少弃水变量上限导致解不合理。而其他文献所用的算法因为无法保证解的全局性而忽略了这个问题。在给出了一个上界之后,半定规划的解依然优于其他算法的解,进一步证明了线性半定规划的有效性。
第4章—水火电力网联合调度问题的凸性分析：在前几章的基础上提出了水电调度问题的分段二次凸化方法。为了达到利用最新凸优化技术的目的,该方法首先对水火联合调度问题的难点—水电子系统中的非凸等式约束进行详细分析,通过逐步精炼水火联合调度模型中难以处理的和只给问题增加计算量而不增加计算难度的约束和变量,并通过将各调度时期的水量、水头和水库特性结合起来分析,提出了可以直接应用凸规划技术的水火联合调度模型。借助成熟的线性分段技术思想,提出了逼近效果更好的分段二次凸化方法,针对水电子系统中两个具体的约束：水电出力函数和弃水函数等式约束,给出相应的具体的数学表达式。本章得出结论,水火联合问题可以通过逼近、引进变量等手段转化为一个可利用成熟锥规划技术求解的混合整数规划问题。
第5章—分段二次凸化半定有效不等式方法求解水火联合调度系统的仿真与分析：在前几章基础上,用几组数值试验和算例仿真进一步验证本文提出的处理非凸等式约束的基于改进半定有效不等式方法的效果。一方面分析半定有效不等式方法对一个整数线性规划例子生成的松弛模型紧性方面存在的问题,再分析改进的半定有效不等式方法对该例子的更紧的松弛。另一方面用改进的半定有效不等式方法求解大规模的水火联合调度问题。数值试验KNAPSACK问题,和仿真算例32-HTC、153-HTC两个大规模的水火仿真系统可以说明本文所提出的凸化配合改进的半定有效不等式算法可以可靠、快速地求解非凸、混合整数水火联合调度问题。
第6章—总结与展望：总结了本文的主要工作和创新点,并指出了可进一步开拓的研究方向。
Electric power industry is now advancing with distinguish modern marks. Clean energy and smart grid are two key words of it. It is necessary to utilize clean energy to the best and have it participated in the power generation coordination when there are huge storage of hy-dropower and alike around the area. China is rich in hydropower resources. How to utilize the exiting hydropower is as important as advocating hydropower development. Hydrothermal coordination problems are of difficulty not only because of the mathematical modeling pro-cess but also because of the corresponding optimization algorithms. The struggling of mathe-matical modeling is caused by the complex process of hydropower generation. The fact that developing cascaded hydropower plants within the same watershed can bring a coupling ef-fect on both spatial aspect and time aspect. Though the problems can be simplified by getting rid of certain factors, not all the algorithms can guarantee a fast and accurate optimal solution.
This dissertation tried to solve the problems from angels of both mathematical modeling and optimization algorithm, with the help with several methods of handling nonconvex equal-ity constraints based on some recently developed convex optimization skills, especially the ones regarding to conic optimization. At first, the whole process of generating hydropower was examined, followed by the conclusion that the bottleneck problems are nonconvex equal-ity constraints and integer variables. Luckily, conic programming based on modern interior point method has the potential of dealing with both of these two issues at the same time. Through solving a standard test data set of hydrothermal coordination problem, semidefinite programming was proved to be better than other existing algorithms with respect to computa-tional time and the quality of the solution. However, semidefinite programming is born to be a relaxation method. To overcome the problem that the rank of the solution matrix might not equal to one when the problem contains nonconvex equality constraints, this dissertation pro-posed a solution that transfers nonconvex equality constraints into0-1mixed integer con-straints by convexification with high accuracy first, then solves the problem by improved semidefinite valid inequality method. This dissertation is divided into6chapters, which can be summarized as follows:
Chapter1on introduction:Literature review of both domestic and abroad on hydrother-mal coordination problem from angels of mathematical modeling and mathematical optimiza-tion methods.
Chapter2on basic knowledge of conic programming:Introducing principal concepts of conic programming, which are the key mathematically tools in solving hydrothermal coordi- nation problems, focusing on aspects of introduction of semidefinite programming and second order cone programming, the relationship between these optimization methods, the convexity of these methods, their duality and related interior point algorithms, the applications of these methods, and the solvers that can be passed along to solve these mathematical optimization problems.
Chapter3on semidefinite programming on a simple hydrothermal coordination problem: This chapter elaborated how to solve a simple mathematical model of hydrothermal coordina-tion with basic semidefinite programming. By lifting the normal vector space produced by the original problem into a higher dimensional semidefinite cone, the new semidefinite program-ming problem was a linear one on semidefinite cone relaxed from the original model with nonconvex equality constraints. The resulting model can be solved by efficient solvers inte-grated with conic interior point methods. The results showed that the original data set had an issue of lacking upper bounds for spillage water variables. While the exiting literatures worked on the same data set failed to report this issue because the methods they used were not truly global ones. After fixing upper bounds for the data set, the results by semidefinite pro-gramming were still the best among all the methods, further validating the fact that semidefi-nite programming method is very effective.
Chapter4on convexity of hydrothermal coordination problems:Based on the previous chapters, this chapter presented a piece-wise quadratic convexification method for hydro-thermal coordination problem. To take the advantages of newly developed convex program-ming techniques, the proposed method focused on analysis of the most difficult part of the problem, i.e., the nonconvex equality constraints of the hydrothermal coordination problem first, and then provided a convex programming-friendly model by analyzing storage, water net head, and reservoir characteristics all together over different spans of scheduling horizon. Based on well-developed piecewise linearization techniques, the dissertation presented a more accurate piecewise quadratic convexification method. And it also provided a detailed mathe-matical expression designed for hydropower generation equality constraints and spillage equality constraints. It concluded that the hydrothermal coordination problem can be trans-ferred into a mixed integer programming problem by means of approximation and introduc-tion of variables.
Chapter5on simulation and analysis of hydrothermal coordination problem based on quadratic convexification semidefinite valid inequality method:Several testing problems and simulations were applied to validate the effectiveness of the proposed method. On one hand, this chapter presented the tightness problems caused by solving a mixed integer programming problem with semidefinite valid inequality relaxation method, followed by the analysis of im-proved semidefinite valid inequality method for a tighter relaxation. On the other hand, the improved semidefinite valid inequality method was used for large-scale hydrothermal coordi-nation problem. A testing KNAPSACK problem,32-HTC simulation problem, and153-HTC simulation problem were used to testify that the proposed improved semidefinite valid ine-quality method can solve the mixed integer hydrothermal coordination problem reliably and efficiently.
Chapter6on conclusions and future works:The framework of the dissertation along with the creativities was summarized. Several further follow-up studies were pointed out as well.
引文
[1]Wikipedia. Electric power industry. http://en.wikipedia.org/wiki/Electric power_indus- try,2012.
[2]USEI Administration. What is Energy:Energy Basics.2011-06-16.
[3]H Campbell. Global warming is not the world's top problem, energy is. http://www.science20.com/science 20/blog/global warming not worlds top problem energ y-117173,2013.
[4]D Sun. Towards efficient system operation:Generation perspective 2012[C]. IEEE PES general meeting panel session:Smart grid drivers for improving generator performance, in Power and Energy Society General Meeting,2012 IEEE,2012, pp.1.
[5]R R Tan, et al.. Pinch analysis approach to carbon-constrained planning for sustainable power generation. Journal of Cleaner Production,2009,17(10):940-944.
[6]Wikipedia. Smart grid. http://en.wikipedia.org/wiki/Smart grid,2013.
[7]E M Lightner and S E Widergren. An Orderly Transition to a Transformed Electricity System[J]. IEEE Transactions on Smart Grid,2010,1(1):3-10.
[8]U.S. Congress. Energy independence and security act of 2007.2007, H.R.6.
[9]S Randalls. History of the 2° C climate target[M]. Wiley Interdisciplinary Reviews:Climate Change,2010.
[10]钟志亮.雾霾围城彰显我国能源结构问题[J].广西电业,2013(1)：1.
[11]张超,陈武.关于我国2050年水电能源发展战略的思考[J].北京理工大学学报(社会科学版),2002,4(1)：4.
[12]Z K Shawwash, et al.. The B.C. Hydro short term hydro scheduling optimization model[J]. IEEE Transactions on Power Systems,2000,15(3):1125-1131.
[13]K U Rao. Computer Techniques and Models in Power Systems. New Delhi:I. K. International Pvt Ltd,2009.
[14]X A Sun. Advances in Electric Power Systems:Robustness, Adaptability, and Fairness. Doctor of Philosophy, Sloan School of Management, Massachusetts Institute of Technology,2011.
[15]A J Wood and B F Wollenberg. Power Generation Operation and Control (2nd Edition) [M]. New York:John Wiley & Sons,1992:250-252.
[16]K Nolde, et al.. Medium term scheduling of a hydro-thermal system using stochastic model predictive control[J]. Automatica,2008,44(6):1585-1594.
[17]A Frangioni, et al.. Tighter Approximated MILP Formulations for Unit Commitment Problems[J]. IEEE Transactions on Power Systems,2009,24(1):105-113.
[18]A Frangioni, et al.. Solving unit commitment problems with general ramp constraints [J]. International Journal of Electrical Power & Energy Systems,2008,30(5):316-326.
[19]R Fuentes-Loyola, et al.. A performance comparison of a primal-dual interior point method vs. Lagrangian relaxation to solve the medium term hydro-thermal coordination problem[C] Power Engineering Society Summer Meeting,2000. IEEE,2000(4):2255-2260.
[20]R Fuentes-Loyola and V H Quintana. Medium-term hydrothermal coordination by semidefinite programming[J]. IEEE Transactions on Power Systems,2003,18(4):1515-1522.
[21]H Wei, et al.. Large scale hydrothermal optimal power flow problems based on interior point nonlinear programming[J]. IEEE Transactions on Power Systems,2000,15(1):396-403.
[22]韦化,李滨,杭乃善等.大规模水火电力系统最优潮流的现代内点理论分析[J].中国电机工 程学报,2003,23(4)：4-8.
[23]I Farhat and M El-Hawary. Interior point methods application in optimum operational scheduling of electric power systems[J]. IET Generation, Transmission & Distribution,2009,3(11): 1020-1029.
[24]M E El-Hawary and M Kumar. Optimal Parameter Estimation for Hydro-Plant Performance Models in Economic Operation Studies[J]. IEEE Transactions on Power Systems,1986,1(4): 126-131.
[25]M E El-Hawary and D H Tsang. The Hydrothermal Optimal Load Flow, A Practical Formulation and Solution Techniques Using Newton's Approach[J]. IEEE Transactions on Power Systems, 1986,1(3):157-166.
[26]S O Orero and M R Irving. A genetic algorithm modelling framework and solution technique for short term optimal hydrothermal scheduling[J]. IEEE Transactions on Power Systems,1998,13(2): 501-518.
[27]M Basu. Economic environmental dispatch of hydrothermal power system[J]. International Journal of Electrical Power & Energy Systems,2010,32(6):711-720.
[28]S Kumar and R Naresh. Efficient real coded genetic algorithm to solve the non-convex hydrothermal scheduling problem[J]. International Journal of Electrical Power & Energy Systems, 2007,29(10):738-747.
[29]N Sinha, et al.. Fast evolutionary programming techniques for short-term hydrothermal scheduling[J]. IEEE Transactions on Power Systems,2003,18(1):214-220.
[30]L Lakshminarasimman and S Subramanian. Short-term scheduling of hydrothermal power system with cascaded reservoirs by using modified differential evolution[J]. IEE Proceedings on Generation, Transmission and Distribution,2006,153(6):693-700.
[31]B Yu. et al.. Short-term hydro-thermal scheduling using particle swarm optimization method[J]. Energy Conversion and Management,2007,48(7):1902-1908.
[32]Z Jingrui, et al.. Small Population-Based Particle Swarm Optimization for Short-Term Hydrothermal Scheduling[J]. IEEE Transactions on Power Systems,2012,27(1):142-152.
[33]P K Hota, et al.. An improved PSO technique for short-term optimal hydrothermal scheduling[J]. Electric Power Systems Research,2009,79(7):1047-1053.
[34]A H A Rashid and K M Nor. An algorithm for the optimal scheduling of variable head hydro and thermal plants[J]. IEEE Transactions on Power Systems,1993,8(3):1242-1249.
[35]J P S Catalao, et al.. Scheduling of Head-Sensitive Cascaded Hydro Systems:A Nonlinear Approach[J]. IEEE Transactions on Power Systems,2009,24(1):337-346.
[36]D Vo Ngoc, et al.. A newly improved particle swarm optimization for economic dispatch with valve point loading effects[C]. Power and Energy Society General Meeting,2011 IEEE,2011: 1-8.
[37]D C Walters and G B Sheble. Genetic algorithm solution of economic dispatch with valve point loading[J]. IEEE Transactions on Power Systems,1993,8(3):1325-1332.
[38]L dos Santos Coelho and V C Mariani. Combining of chaotic differential evolution and quadratic programming for economic dispatch optimization with valve-point effect[J]. IEEE Transactions on Power Systems,2006,21(2):989-996.
[39]J Wu, et al.. A Hybrid Method for Optimal Scheduling of Short-Term Electric Power Generation of Cascaded Hydroelectric Plants Based on Particle Swarm Optimization and Chance-Constrained Programming[J]. IEEE Transactions on Power Systems,2008,23(4):1570-1579.
[40]C J Rehn, et al.. Voltage optimization using augmented Lagrangian functions and quasi-Newton techniques[J]. IEEE Transactions on Power Systems,1989,4(4):1470-1483.
[41]C Li, et al.. Stochastic optimization of interconnected multireservoir power systems[C]. Power Industry Computer Application Conference,1989, PICA'89,1989:203-210.
[42]D B Das and C Patvardhan. New multi-objective stochastic search technique for economic load dispatch[J]. IEE Proceedings on Generation, Transmission and Distribution,1998,145(6):747-752.
[43]W Lingfeng and C Singh. Stochastic combined heat and power dispatch based on multi-objective particle swarm optimization. Power Engineering Society General Meeting,2006. IEEE,2006: 8.
[44]X Lu, et al.. Global potential for wind-generated electricity. Proceedings of the National Academy of Sciences, June 22,2009.
[45]W Qianfan, et al.. A Chance-Constrained Two-Stage Stochastic Program for Unit Commitment With Uncertain Wind Power Output[J]. IEEE Transactions on Power Systems,2012,27(1): 206-215.
[4.6]V S Pappala, et al.. A Stochastic Model for the Optimal Operation of a Wind-Thermal Power System[J]. IEEE Transactions on Power Systems, vol.24:940-950,2009.
[47]R A Jabr. Robust self-scheduling under price uncertainty using conditional value-at-risk[J]. IEEE Transactions on Power Systems,2005,20(4):7-10.
[48]Z Yu, et al.. On convexity issues of short-term hydrothermal scheduling[J]. International Journal of Electrical Power & Energy Systems,2000,22(6):451-457.
[49]R A Jabr. Modeling network losses using quadratic cones[J]. IEEE Transactions on Power Systems,2005,20(1):505-506.
[50]L Lovasz and A Schrijver. Cones of matrices and set-functions and 0-1 optimization[J]. SIAM Journal on Optimization,1991,1:166-190.
[51]S H M Hashimoto, et al.. Efficient linear programming algorithm for the transmission network expansion planning problem[J]. IEE Proceedings on Generation, Transmission and Distribution, 2003,150(5):536-542.
[52]H M Khodr, et al.. A linear programming methodology for the optimization of electric power-generation schemes[J]. IEEE Transactions on Power Systems,2002,17(3):864-869.
[53]C B Somuah and N Khunaizi. Application of linear programming redispatch technique to dynamic generation allocation[J]. IEEE Transactions on Power Systems,1990,5(1):20-26.
[54]N Karmarkar. A new polynomial-time algorithm for linear programming[J]. Combinatoria1984, 1984,4(4):373-395.
[55]R A Jabr, et al.. A homogeneous linear programming algorithm for the security constrained economic dispatch problem[J]. IEEE Transactions on Power Systems,2000,15(3):930-936.
[56]D Vlaisavljevic, et al.. Fuzzy linear programming based optimal power system rescheduling including preventive redispatch[J]. IEEE Transactions on Power Systems,1999,14(2):525-531.
[57]A Farag, et al.. Economic load dispatch multiobjective optimization procedures using linear programming techniques[J]. IEEE Transactions on Power Systems,1995,10(2):731-738.
[58]J L Martinez Ramos, et al.. Short-term hydro-thermal coordination based on interior point nonlinear programming and genetic algorithms[C]. Power Tech Proceedings,2001 IEEE Porto, 2001:3-6.
[59]H Wei, et al.. A decoupled solution of hydro-thermal optimal power flow problem by means of interior point method and network programming[J]. IEEE Transactions on Power Systems,1998, 13(2):286-293.
[60]V H Quintana, et al.. Interior-point methods and their applications to power systems:a classification of publications and software codes[J]. IEEE Transactions on Power Systems,2000, 15(1):170-176.
[61]M Christoforidis, et al.. Long-term/mid-term resource optimization of a hydro-dominant power system using interior point method[C]. Power Industry Computer Application Conference,1995. Conference Proceedings.,1995 IEEE,1995:164-171.
[62]J Medina, et al.. A clipping-off interior-point technique for medium-term hydro-thermal coordination[J]. IEEE Transactions on Power Systems,1999,14(1):266-273.
[63]J Piantadosi, et al.. Stochastic dynamic programming (SDP) with a conditional value-at-risk (C-VaR) criterion for management of storm-water[J]. Journal of Hydrology,2008,348(3-4):320-329.
[64]刘子龙.水库确定性优化调度动态规划法调度模型及应用[J].人民长江,1999,30(10)：46-48.
[65]冯平,李少飞,安伟哲.三门峡水库电站运行调度的优化研究[J].天津大学学报,2004,37(4)：294-296.
[66]刘攀,郭生练等.求解水库优化调度问题的动态规划-遗传算法[J].武汉大学学报(工学版),2007,40(5)：1-6.
[67]A Castelletti, et al.. Neuro-dynamic programming for designing water reservoir network management policies[J]. Control Engineering Practice,2007,15(8):1031-1038.
[68]H Brannlund, et al.. Optimal Short Term Operation Planning of a Large Hydrothermal Power System Based on a Nonlinear Network Flow Concept [J]. IEEE Transactions on Power Systems, 1986,1(4):75-81.
[69]R E Rosenthal. A nonlinear network flow algorithm for maximization of benefits in a hydroelectric power system[J]. Operations Research,1981:763-786.
[70]C A Li, et al.. Implementation of network flow programming to the hydrothermal coordination in an energy management system[J]. IEEE Transactions on Power Systems,1993,8(3):1045-1053.
[71]G Xiaohong, et al.. Nonlinear approximation method in Lagrangian relaxation-based algorithms for hydrothermal scheduling[J]. IEEE Transactions on Power Systems,1995,10(2):772-778.
[72]N J Redondo and A J Conejo. Short-term hydre-thermal coordination by Lagrangian relaxation: solution of the dual problem[J]. IEEE Transactions on Power Systems,1999,14(1):89-95.
[73]W S Sifuentes and A Vargas. Hydrothermal Scheduling Using Benders Decomposition: Accelerating Techniques[J]. IEEE Transactions on Power Systems,2007,22(3):1351-1359.
[74]G S Lauer, et al.. Solution of Large-Scale Optimal Unit Commitment Problems[J]. IEEE Transactions on Power Apparatus and Systems,1982, PAS-101(1):79-86.
[75]S Salam, et al.. Comprehensive algorithm for hydrothermal coordination[J]. IEE Proceedings Generation, Transmission and Distribution,1997,144(5):482-488.
[76]M Shahidehopour and F Yong. Benders decomposition:applying Benders decomposition to power systems[J]. Power and Energy Magazine, IEEE,2005,3(2):20-21.
[77]W Sifuentes and A Vargas. Short-term hydrothermal coordination considering an AC network modeling[J]. International Journal of Electrical Power & Energy Systems,2007,29(6):488-496.
[78]H M Khodr, et al.. Distribution Systems Reconfiguration Based on OPF Using Benders Decomposition [J]. IEEE Transactions on Power Delivery,2009,24(4):2166-2176.
[79]J H Holland. Adaptation in Natural and Artificial Systems. University of Michigan Press,1975.
[80]R C Bansal. Optimization methods for electric power systems:an overview[J]. International Journal of Emerging Electric Power Systems,2005,2(1):1.
[81]H Zhang, et al.. Daily hydrothermal scheduling with economic emission using simulated annealing technique based multi-objective cultural differential evolution approach[J]. Energy,2013,50(0): 24-37.
[82]K P Wong and Y W Wong. Short-term hydrothermal scheduling part. I. Simulated annealing approach[J]. IEE Proceedings on Generation, Transmission and Distribution,1994,141(5): 497-501.
[83]K P Wong and Y W Wong. Short-term hydrothermal scheduling. II. Parallel simulated annealing approach[J]. IEE Proceedings on Generation, Transmission and Distribution,1994,141(5): 502-506.
[84]K K Mandal, et al.. Particle swarm optimization technique based short-term hydrothermal scheduling[J]. Applied Soft Computing,2008,8(4):1392-1399.
[85]C Samudi, et al.. Hydro thermal scheduling using particle swarm optimization[C]. Transmission and Distribution Conference and Exposition,2008. T&D. IEEE/PES,2008:1-5.
[86]Y Peng, et al.. Optimal operation of cascade reservoirs based on generalized ant colony optimization method[C].2010 Sixth International Conference on Natural Computation (ICNC), 2010:2647-2651.
[87]H Shyh-Jier. Enhancement of hydroelectric generation scheduling using ant colony system based optimization approaches[J]. IEEE Transactions on Energy Conversion,2001,16(3):296-301.
[88]R Moeini and M H Afshar. Extension of the constrained ant colony optimization algorithms for the optimal operation of multi-reservoir systems[J]. Journal of Hydroinformatics,2012, in press.
[89]R Naresh and J Sharma. Two-phase neural network based solution technique for short term hydrothermal scheduling[J]. IEE Proceedings on Generation, Transmission and Distribution,1999, 146(6):657-663.
[90]R.-H Liang and Y.-Y Hsu. Short-term hydro-scheduling using Hopfield neural network. IEE Proceedings on Generation, Transmission and Distribution,1996,143(3):269-275.
[91]I J Ramirez-Rosado and J A Dominguez-Navarro. New multiobjective tabu search algorithm for fuzzy optimal planning of power distribution systems [J]. IEEE Transactions on Power Systems, 2006,21(1):224-233.
[92]T A A Victoire and A E Jeyakumar. Unit commitment by a tabu-search-based hybrid-optimisation technique[J]. IEE Proceedings on Generation, Transmission and Distribution,2005,152(4): 563-574.
[93]X Yuan and Y Yuan. Application of cultural algorithm to generation scheduling of hydrothermal systems[J]. Energy Conversion and Management,2006,47(15-16):2192-2201.
[94]Y Lu, et al.. A hybrid multi-objective cultural algorithm for short-term environmental/economic hydrothermal scheduling[J]. Energy Conversion and Management,2011,52(5):2121-2134.
[95]G Zhuangzhi, et al.. Multi-objective optimization scheduling for hydrothermal power systems based on electromagnetism-like mechanism and data envelopment analysis[J]. Proceeding of the CSEE,2013,33(4):53-61.
[96]A H Mantawy, et al.. Integrating genetic algorithms, tabu search, and simulated annealing for the unit commitment problem[J]. IEEE Transactions on Power Systems,1999,14(3):829-836.
[97]A Y Saber, et al.. Fuzzy unit commitment scheduling using absolutely stochastic simulated annealing[J]. IEEE Transactions on Power Systems,2006,21(2):955-964.
[98]C C Asir Rajan. Hydro-thermal unit commitment problem using simulated annealing embedded evolutionary programming approach[J]. International Journal of Electrical Power & Energy Systems,2011,33(4):939-946.
[99]L Ruey-Hsun. A noise annealing neural network for hydroelectric generation scheduling with pumped-storage units[C]. Power Industry Computer Applications,1999. PICA'99. Proceedings of the 21st 1999 IEEE International Conference,1999:171-176.
[100]F E1-Turky and E E Perry. BLADES:an artificial intelligence approach to analog circuit design[J]. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems,1989, 8(6):680-692.
[101]O Gusikhin, et al.. Intelligent systems in the automotive industry:applications and trends[J]. Knowl.Inf. Syst.,2007,12(2):147-168.
[102]V C Ramesh and X Li. A fuzzy multiobjective approach to contingency constrained OPF[J]. IEEE Transactions on Power Systems,1997,12(3):1348-1354.
[103]胡国强,贺仁睦.梯级水电站多目标模糊优化调度模型极其求解方法[J].电工技术学报,2007,22(1)： 154-159.
[104]刘彦奎,王曙明.模糊随机优化理论[M].北京：中国农业大学出版社,2006.
[105]D H Wolpert and W G Macready. No free lunch theorems for optimization[J]. IEEE Transactions on Evolutionary Computation,1997,1(1):67-82.
[106]R Horst and P M Pardalos. Handbook of Global Optimization[M]. Kluwer,1995.
[107]J D Pinter. Global Optimization in Action[M]. Kluwer,1996.
[108]H Tuy. Convex Analysis and Global Optimization[M]. Kluwer,1998.
[109]A Ben-Tal and A S Nemirovskii. Lectures on Modern Convex Optimization:Analysis, Algorithms, and Engineering Applications[M]. Society for Industrial and Applied Mathematics,2001.
[110]Y Nesterov, et al.. Semidefinite Programming Relaxations of Nonconvex Quadratic Optimization[M]. Handbook of Semidefinite Programming. US:Springer US,2000:361-419.
[111]M X Goemans and D P Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming[J]. J. ACM,1995,42(6):1115-1145.
[112]Y Nesterov. Squared Functional Systems and Optimization Problems[M]. High Performance Optimization. US:Springer,2000:405-440.
[113]Y Nesterov. Semidefinite relaxation and nonconvex quadratic optimization[J]. Optimization methods and software,1998,9(1-3):141-160.
[114]Y Ye. Approximating quadratic programming with bound and quadratic constraints[J]. Mathematical programming,1999,84(2):219-226.
[115]P A Parrilo. Semidefinite programming relaxations for semialgebraic problems[J]. Mathematical programming,2003,96(2):293-320.
[116]R T Rockafellar. Convex Analysis[M]. Princeton University Press.1970.
[117]J-B Hiriart-Urruty and C Lemar'echal. Convex Analysis and Minimization Algorithms[M]. Springer,1993.
[118]J-B Hiriart-Urruty and C Lemar'echal. Fundamentals of Convex Analysis [M]. Spinger,2001.
[119]J M Borwein and A S Lewis. Convex Analysis and Nonlinear Optimization:Theory and Examples[M]. Springer,2006.
[120]Y Nesterov and A Nemirovskii. Interior Point Polynomial Algorithms in Convex Programming[M]. Society for Industrial and Applied Mathematics,1994.
[121]R Rockafellar. Lagrange Multipliers and Optimality[J]. SIAM Review,1993,35(2):183-238.
[122]X Bai and H Wei. Semi-definite programming-based method for security-constrained unit commitment with operational and optimal power flow constraints[J]. IEE Proceedings on Generation, Transmission and Distribution,2009,3(2):182-197.
[123]X Bai and H Wei. A semidefinite programming method with graph partitioning technique for optimal power flow problems[J]. International Journal of Electrical Power & Energy Systems, 2011,33(7):1309-1314.
[124]R A Jabr. Rank-constrained semidefinite program for unit commitment[J]. International Journal of Electrical Power & Energy Systems,2013,47(0):13-20.
[125]R Quan, et al.. Tighter relaxation method for unit commitment based on second-order cone programming and valid inequalities[J]. International Journal of Electrical Power & Energy Systems,2014,55(0):82-90.
[126]E Parrilla and J Garcia-Gonzalez. Improving the B & B search for large-scale hydrothermal weekly scheduling problems[J]. International Journal of Electrical Power & Energy Systems,2006,28(5): 339-348.
[127]A Borghetti, et al.. Lagrangian heuristics based on disaggregated Bundle methods for hydrothermal unit commitment[J]. IEEE Transactions on Power Systems,2003,18(1):313-323.
[128]C Jaeseok, et al.. A fuzzy branch and bound-based transmission system expansion planning for the highest satisfaction level of the decision maker[J]. IEEE Transactions on Power Systems,2005, 20(1):476-484.
[129]A I Cohen and M Yoshimura. A Branch-and-Bound Algorithm for Unit Commitment[J]. IEEE Transactions on Power Apparatus and Systems, PAS-102(2):444-451,1983.
[130]C Chern-Lin and S C Wang. Branch-and-bound scheduling for thermal generating units[J]. IEEE Transactions on Energy Conversion,1993,8(2):184-189.
[131]M Madrigal and V H Quintana. An interior-point/cutting-plane method to solve unit commitment problems[C]. Power Industry Computer Applications,1999. PICA '99. Proceedings of the 21st 1999 IEEE International Conference,1999:203-209.
[132]M Murai and M Kato. Cutting plane methods for Lagrangian relaxation-based unit commitment algorithm[J]. Electrical Engineering in Japan,2002,141(3):18-32.
[133]A L Ott. Evolution of computing requirements in the PJM market:Past and future [C]. Power and Energy Society General Meeting,2010 IEEE,2010:1-4.
[134]D Streiffert, et al.. A mixed integer programming solution for market clearing and reliability analysis. Power Engineering Society General Meeting,2005. IEEE,2005(3):2724-2731.
[135]J J Shaw. A direct method for security-constrained unit commitment[J]. IEEE Transactions on Power Systems,1995,10(3):1329-1342.
[136]P Interconnection. PJM'S new generation-scheduling software to save customers estimated ﹩56 million unit commitment system uses first of its kind technieque. PJM News Release,2004.
[137]R E Bixby, et al.. MIP:Theory And Practice-Closing The Gap. System modelling and optimization:methods, theory, and applications. Kluwer Academic Publishers,2000.
[138]J Ostrowski, et al.. Tight Mixed Integer Linear Programming Formulations for the Unit Commitment Problem[J]. IEEE Transactions on Power Systems,2012,27(1):39-46.
[139]A Frangioni, et al.. Sequential Lagrangian-MILP approaches for Unit Commitment problems[J]. International Journal of Electrical Power & Energy Systems,2011,33(3):585-593.
[140]D Han, et al.. Outer Approximation and Outer-Inner Approximation Approaches for Unit Commitment Problem[J]. IEEE Transactions on Power Systems,2013, in press.
[141]A Frangioni and C Gentile. A computational comparison of reformulations of the perspective relaxation:SOCP vs. cutting planes[J]. Operations Research Letters,2009,37(3):206-210.
[142]S Boyd and L Vandenberghe. Convex Optimization[M]. Cambridge University Press,2004.
[143]F Alizadeh and D Goldfarb. Second-order cone programming[J]. Mathematical programming, 2003,95(1):3-51.
[144]G Anupam and O D Ryan. (2011). Linear and Semidefinite Programming. Available: http://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15859-f11/www/notes/lecture12.pdf.
[145]Y Nesterov and A S Nemirovsii. Interior point polynomial algorithms in convex programming vol. 13[M]. Philadelphia, USA:SIAM,1994.
[146]Y Nesterov and A S Nemirovsii. Conic formulation of a convex programming problem and duality[J]. Optimization methods and software,1992,1(2):95-115.
[147]F Alizadeh-Dehkharghani. Combinatorial optimization with interior point methods and semi-definite matrices. Ph.D., University of Minnesota, Minneapolis, MN, USA,1991.
[148]L Vandenberghe and S Boyd. Semidefinite Programming[J]. SIAM Review,1994,38(0):49-95.
[149]E De Klerk. Interior Point Methods for Semidefinite Programming. M.Sc., Technische Universiteit Delft, Delft,1997.
[150]E De Klerk, et al.. On primal-dual path-following algorithm in semidefinite programming. New trends in mathematical programming. The Netherlands:Kluwer Academic Publishers,1998: 137-157.
[151]K M Anstreicher and M Fampa. A long-step path following algorithm for semidefinite programming problems. Department of management sciences, University of Iowa, Iowa City, 1996.
[152]L Faybusovich. Semidefinite Programming:A Path-Following Algorithm for a Linear-Quadratic Functional[J]. SIAM Journal on Optimization,1996,6(4):1007-1024.
[153]F Alizadeh-Dehkharghani, et al.. Primal-dual interior point methods for semidefinite programming: convergence rates, stability and numerical results[J]. SIAM Journal on Optimization,1998,8(3): 746-768.
[154]Y Nesterov and M J Todd. Self-scaled barriers and interior-point methods for convex programming[J]. Mathematics of Operations Research,1997,22(1):1-42.
[155]R D C Monteiro. Primal-dual path-following algorithms for semidefinite programing[J]. SIAM Journal on Optimization,1997,7(3):663-688.
[156]M Kojima, et al.. Interior point methods for the monotone semidefinite linear complementarity problem in symmetric matrices[J]. SIAM Journal on Optimization,1997,7(1):88-124.
[157]C Helmberg, et al.. An interior-point method for semidefinite programming[J]. SIAM Journal on Optimization,1996,6(0):342-361.
[158]C Helmberg. Semidefinite Programming for Combinatorial Optimization [M]. Konrad-Zuse-Zentrum fur Informationstechnik,2000.
[159]M X Goemans and F Rendl. Semidefinite programming in combinatorial optimization[J]. Mathematical programming,1999,79(1997):143-161.
[160]F Juhasz. The asymptotic behaviour of lovasz' 9 function for random graphs[J]. Combinatorica, 1982,2(2):153-155.
[161]F Rendl. Semidefinite Relaxations for Integer Programming.50 Years of Integer Programming 1958-2008[M]. Springer Berlin Heidelberg,2010:687-726.
[162]D Jibetean and M Laurent. Semidefinite Approximations for Global Unconstrained Polynomial Optimization[J]. SIAM J. on Optimization,2005,16(2):490-514.
[163]J B Lasserre. Global optimization with polynomials and the problem of moments[J]. SIAM Journal on Optimization,2001,11(3):796-978.
[164]N Z Shor. An approach to obtaining global extremums in polynomial mathematical programming problems[J]. Cybernetics and Systems Analysis,1987,23(5):695-700.
[165]S Liu, et al.. A robust optimization approach to wind farm diversification[J]. International Journal of Electrical Power & Energy Systems,2013,53(0):409-415.
[166]Conic Programming Solvers. Available:http://www.gamsworld.org/cone/solvers.htm.
[167]S Soares, et al.. Optimal Generation Scheduling of Hydrothermal Power Systems[J]. IEEE Transactions on Power Apparatus and Systems,1980, PAS-99(3):1107-1118.
[168]K K Mandal and N Chakraborty. Differential evolution technique-based short-term economic generation scheduling of hydrothermal systems[J]. Electric Power Systems Research,2008,78(11): 1972-1979.
[169]G S C M E El-Hawary. Optimal economic operation of electric power systems, vol. Mathematics in science and engineering:Academic Press,1979.
[170]A Saxena, et al.. Convex relaxations of non-convex mixed integer quadratically constrained problems:Extended formulations[J]. IBM Corp. Tech. Rep. RC24621,2008.
[171]K M Anstreicher. Comparing Convex Relaxations for Quadratically Constrained Quadratic Programming.2010.
[172]J Lavaei and S H Low. Zero Duality Gap in Optimal Power Flow Problem[J]. IEEE Transactions on Power Systems,2012,27(1):92-107.
[173]卢强.电力改革要以建立安全经济运营的电力系统为目标.2013,Available: http://www.sgcc.com.cn/xwzx/gsyw/2013/04/290331.shtml.
[174]M A Abido. Environmental/economic power dispatch using multiobjective evolutionary algorithms[J]. IEEE Transactions on Power Systems,2003,18(4):1529-1537.
[175]CO2 emissions from fuel combustion highlights. I E Agency,2012.
[176]张晓花,赵晋泉,陈星莺.节能减排多目标机组组合问.题的模糊建模及优化[J].中国电机工程学报,2010,30(22)：71-77.
[177]P D Brown, et al.. Optimization of Pumped Storage Capacity in an Isolated Power System With Large Renewable Penetration[J]. IEEE Transactions on Power Systems,2008,23(2):523-531.
[178]Wikipedia. Water turbine. Available:http://en.wikipedia.org/wiki/Water_turbine.
[179]聂启蓉.平班水电站水轮机模型验收试验[J].红水河，2007,vo1.26(1)：4.
[180]A J Wood and B F Wollenberg. Power Generation Operation and Control:John Wiley & Sons, 1984.
[181]M Piekutowski, et al.. Optimal short-term scheduling for a large-scale cascaded hydro system[J]. IEEE Transactions on Power Systems,1994,9(2):805-811.
[182]S P Bradley, et al.. Applied mathematical programming:Addison-Wesley Pub. Co.,1977.
[183]C E Lin and G L Viviani. Hierarchical Economic Dispatch for Piecewise Quadratic Cost Functions[J]. IEEE Transactions on Power Apparatus and Systems,1984, PAS-103(6):1170-1175.
[184]A Astorino, et al.. Piecewise-quadratic Approximations in Convex Numerical Optimization[J]. SIAM Journal on Optimization,2011,21(4):1418-1438.
[185]J H Park, et al.. Economic load dispatch for piecewise quadratic cost function using Hopfield neural network[J]. IEEE Transactions on Power Systems,1993,8(3):1030-1038.
[186]M Farivar and S H Low. Branch Flow Model:Relaxations and Convexification—Part I[J]. IEEE Transactions on Power Systems,2013,28(3):2554-2564.
[187]A L Diniz and M E P Maceira. A Four-Dimensional Model of Hydro Generation for the Short-Term Hydrothermal Dispatch Problem Considering Head and Spillage Effects [J]. IEEE Transactions on Power Systems.2008,23(3):1298-1308.
[188]E Barron, et al.. Functions Which Are Quasiconvex under Linear Perturbations[J]. SIAM Journal on Optimization,2012,22(3):1089-1108.
[189]T Pock, et al.. Global Solutions of Variational Models with Convex Regularization[J]. SIAM Journal on Imaging Sciences,2010,3(4):1122-1145.
[190]A L Diniz. Test cases for unit commitment and hydrothermal scheduling problems[C]. Power and Energy Society General Meeting,2010 IEEE,2010:1-8.
[191]J Briet, et al. Grothendieck inequalities for semidefinite programs with rank constraint. Available: http://arxiv.org/abs/1011.1754.
[2]USEI Administration. What is Energy:Energy Basics.2011-06-16.
[3]H Campbell. Global warming is not the world's top problem, energy is. http://www.science20.com/science 20/blog/global warming not worlds top problem energ y-117173,2013.
[4]D Sun. Towards efficient system operation:Generation perspective 2012[C]. IEEE PES general meeting panel session:Smart grid drivers for improving generator performance, in Power and Energy Society General Meeting,2012 IEEE,2012, pp.1.
[5]R R Tan, et al.. Pinch analysis approach to carbon-constrained planning for sustainable power generation. Journal of Cleaner Production,2009,17(10):940-944.
[6]Wikipedia. Smart grid. http://en.wikipedia.org/wiki/Smart grid,2013.
[7]E M Lightner and S E Widergren. An Orderly Transition to a Transformed Electricity System[J]. IEEE Transactions on Smart Grid,2010,1(1):3-10.
[8]U.S. Congress. Energy independence and security act of 2007.2007, H.R.6.
[9]S Randalls. History of the 2° C climate target[M]. Wiley Interdisciplinary Reviews:Climate Change,2010.
[10]钟志亮.雾霾围城彰显我国能源结构问题[J].广西电业,2013(1)：1.
[11]张超,陈武.关于我国2050年水电能源发展战略的思考[J].北京理工大学学报(社会科学版),2002,4(1)：4.
[12]Z K Shawwash, et al.. The B.C. Hydro short term hydro scheduling optimization model[J]. IEEE Transactions on Power Systems,2000,15(3):1125-1131.
[13]K U Rao. Computer Techniques and Models in Power Systems. New Delhi:I. K. International Pvt Ltd,2009.
[14]X A Sun. Advances in Electric Power Systems:Robustness, Adaptability, and Fairness. Doctor of Philosophy, Sloan School of Management, Massachusetts Institute of Technology,2011.
[15]A J Wood and B F Wollenberg. Power Generation Operation and Control (2nd Edition) [M]. New York:John Wiley & Sons,1992:250-252.
[16]K Nolde, et al.. Medium term scheduling of a hydro-thermal system using stochastic model predictive control[J]. Automatica,2008,44(6):1585-1594.
[17]A Frangioni, et al.. Tighter Approximated MILP Formulations for Unit Commitment Problems[J]. IEEE Transactions on Power Systems,2009,24(1):105-113.
[18]A Frangioni, et al.. Solving unit commitment problems with general ramp constraints [J]. International Journal of Electrical Power & Energy Systems,2008,30(5):316-326.
[19]R Fuentes-Loyola, et al.. A performance comparison of a primal-dual interior point method vs. Lagrangian relaxation to solve the medium term hydro-thermal coordination problem[C] Power Engineering Society Summer Meeting,2000. IEEE,2000(4):2255-2260.
[20]R Fuentes-Loyola and V H Quintana. Medium-term hydrothermal coordination by semidefinite programming[J]. IEEE Transactions on Power Systems,2003,18(4):1515-1522.
[21]H Wei, et al.. Large scale hydrothermal optimal power flow problems based on interior point nonlinear programming[J]. IEEE Transactions on Power Systems,2000,15(1):396-403.
[22]韦化,李滨,杭乃善等.大规模水火电力系统最优潮流的现代内点理论分析[J].中国电机工 程学报,2003,23(4)：4-8.
[23]I Farhat and M El-Hawary. Interior point methods application in optimum operational scheduling of electric power systems[J]. IET Generation, Transmission & Distribution,2009,3(11): 1020-1029.
[24]M E El-Hawary and M Kumar. Optimal Parameter Estimation for Hydro-Plant Performance Models in Economic Operation Studies[J]. IEEE Transactions on Power Systems,1986,1(4): 126-131.
[25]M E El-Hawary and D H Tsang. The Hydrothermal Optimal Load Flow, A Practical Formulation and Solution Techniques Using Newton's Approach[J]. IEEE Transactions on Power Systems, 1986,1(3):157-166.
[26]S O Orero and M R Irving. A genetic algorithm modelling framework and solution technique for short term optimal hydrothermal scheduling[J]. IEEE Transactions on Power Systems,1998,13(2): 501-518.
[27]M Basu. Economic environmental dispatch of hydrothermal power system[J]. International Journal of Electrical Power & Energy Systems,2010,32(6):711-720.
[28]S Kumar and R Naresh. Efficient real coded genetic algorithm to solve the non-convex hydrothermal scheduling problem[J]. International Journal of Electrical Power & Energy Systems, 2007,29(10):738-747.
[29]N Sinha, et al.. Fast evolutionary programming techniques for short-term hydrothermal scheduling[J]. IEEE Transactions on Power Systems,2003,18(1):214-220.
[30]L Lakshminarasimman and S Subramanian. Short-term scheduling of hydrothermal power system with cascaded reservoirs by using modified differential evolution[J]. IEE Proceedings on Generation, Transmission and Distribution,2006,153(6):693-700.
[31]B Yu. et al.. Short-term hydro-thermal scheduling using particle swarm optimization method[J]. Energy Conversion and Management,2007,48(7):1902-1908.
[32]Z Jingrui, et al.. Small Population-Based Particle Swarm Optimization for Short-Term Hydrothermal Scheduling[J]. IEEE Transactions on Power Systems,2012,27(1):142-152.
[33]P K Hota, et al.. An improved PSO technique for short-term optimal hydrothermal scheduling[J]. Electric Power Systems Research,2009,79(7):1047-1053.
[34]A H A Rashid and K M Nor. An algorithm for the optimal scheduling of variable head hydro and thermal plants[J]. IEEE Transactions on Power Systems,1993,8(3):1242-1249.
[35]J P S Catalao, et al.. Scheduling of Head-Sensitive Cascaded Hydro Systems:A Nonlinear Approach[J]. IEEE Transactions on Power Systems,2009,24(1):337-346.
[36]D Vo Ngoc, et al.. A newly improved particle swarm optimization for economic dispatch with valve point loading effects[C]. Power and Energy Society General Meeting,2011 IEEE,2011: 1-8.
[37]D C Walters and G B Sheble. Genetic algorithm solution of economic dispatch with valve point loading[J]. IEEE Transactions on Power Systems,1993,8(3):1325-1332.
[38]L dos Santos Coelho and V C Mariani. Combining of chaotic differential evolution and quadratic programming for economic dispatch optimization with valve-point effect[J]. IEEE Transactions on Power Systems,2006,21(2):989-996.
[39]J Wu, et al.. A Hybrid Method for Optimal Scheduling of Short-Term Electric Power Generation of Cascaded Hydroelectric Plants Based on Particle Swarm Optimization and Chance-Constrained Programming[J]. IEEE Transactions on Power Systems,2008,23(4):1570-1579.
[40]C J Rehn, et al.. Voltage optimization using augmented Lagrangian functions and quasi-Newton techniques[J]. IEEE Transactions on Power Systems,1989,4(4):1470-1483.
[41]C Li, et al.. Stochastic optimization of interconnected multireservoir power systems[C]. Power Industry Computer Application Conference,1989, PICA'89,1989:203-210.
[42]D B Das and C Patvardhan. New multi-objective stochastic search technique for economic load dispatch[J]. IEE Proceedings on Generation, Transmission and Distribution,1998,145(6):747-752.
[43]W Lingfeng and C Singh. Stochastic combined heat and power dispatch based on multi-objective particle swarm optimization. Power Engineering Society General Meeting,2006. IEEE,2006: 8.
[44]X Lu, et al.. Global potential for wind-generated electricity. Proceedings of the National Academy of Sciences, June 22,2009.
[45]W Qianfan, et al.. A Chance-Constrained Two-Stage Stochastic Program for Unit Commitment With Uncertain Wind Power Output[J]. IEEE Transactions on Power Systems,2012,27(1): 206-215.
[4.6]V S Pappala, et al.. A Stochastic Model for the Optimal Operation of a Wind-Thermal Power System[J]. IEEE Transactions on Power Systems, vol.24:940-950,2009.
[47]R A Jabr. Robust self-scheduling under price uncertainty using conditional value-at-risk[J]. IEEE Transactions on Power Systems,2005,20(4):7-10.
[48]Z Yu, et al.. On convexity issues of short-term hydrothermal scheduling[J]. International Journal of Electrical Power & Energy Systems,2000,22(6):451-457.
[49]R A Jabr. Modeling network losses using quadratic cones[J]. IEEE Transactions on Power Systems,2005,20(1):505-506.
[50]L Lovasz and A Schrijver. Cones of matrices and set-functions and 0-1 optimization[J]. SIAM Journal on Optimization,1991,1:166-190.
[51]S H M Hashimoto, et al.. Efficient linear programming algorithm for the transmission network expansion planning problem[J]. IEE Proceedings on Generation, Transmission and Distribution, 2003,150(5):536-542.
[52]H M Khodr, et al.. A linear programming methodology for the optimization of electric power-generation schemes[J]. IEEE Transactions on Power Systems,2002,17(3):864-869.
[53]C B Somuah and N Khunaizi. Application of linear programming redispatch technique to dynamic generation allocation[J]. IEEE Transactions on Power Systems,1990,5(1):20-26.
[54]N Karmarkar. A new polynomial-time algorithm for linear programming[J]. Combinatoria1984, 1984,4(4):373-395.
[55]R A Jabr, et al.. A homogeneous linear programming algorithm for the security constrained economic dispatch problem[J]. IEEE Transactions on Power Systems,2000,15(3):930-936.
[56]D Vlaisavljevic, et al.. Fuzzy linear programming based optimal power system rescheduling including preventive redispatch[J]. IEEE Transactions on Power Systems,1999,14(2):525-531.
[57]A Farag, et al.. Economic load dispatch multiobjective optimization procedures using linear programming techniques[J]. IEEE Transactions on Power Systems,1995,10(2):731-738.
[58]J L Martinez Ramos, et al.. Short-term hydro-thermal coordination based on interior point nonlinear programming and genetic algorithms[C]. Power Tech Proceedings,2001 IEEE Porto, 2001:3-6.
[59]H Wei, et al.. A decoupled solution of hydro-thermal optimal power flow problem by means of interior point method and network programming[J]. IEEE Transactions on Power Systems,1998, 13(2):286-293.
[60]V H Quintana, et al.. Interior-point methods and their applications to power systems:a classification of publications and software codes[J]. IEEE Transactions on Power Systems,2000, 15(1):170-176.
[61]M Christoforidis, et al.. Long-term/mid-term resource optimization of a hydro-dominant power system using interior point method[C]. Power Industry Computer Application Conference,1995. Conference Proceedings.,1995 IEEE,1995:164-171.
[62]J Medina, et al.. A clipping-off interior-point technique for medium-term hydro-thermal coordination[J]. IEEE Transactions on Power Systems,1999,14(1):266-273.
[63]J Piantadosi, et al.. Stochastic dynamic programming (SDP) with a conditional value-at-risk (C-VaR) criterion for management of storm-water[J]. Journal of Hydrology,2008,348(3-4):320-329.
[64]刘子龙.水库确定性优化调度动态规划法调度模型及应用[J].人民长江,1999,30(10)：46-48.
[65]冯平,李少飞,安伟哲.三门峡水库电站运行调度的优化研究[J].天津大学学报,2004,37(4)：294-296.
[66]刘攀,郭生练等.求解水库优化调度问题的动态规划-遗传算法[J].武汉大学学报(工学版),2007,40(5)：1-6.
[67]A Castelletti, et al.. Neuro-dynamic programming for designing water reservoir network management policies[J]. Control Engineering Practice,2007,15(8):1031-1038.
[68]H Brannlund, et al.. Optimal Short Term Operation Planning of a Large Hydrothermal Power System Based on a Nonlinear Network Flow Concept [J]. IEEE Transactions on Power Systems, 1986,1(4):75-81.
[69]R E Rosenthal. A nonlinear network flow algorithm for maximization of benefits in a hydroelectric power system[J]. Operations Research,1981:763-786.
[70]C A Li, et al.. Implementation of network flow programming to the hydrothermal coordination in an energy management system[J]. IEEE Transactions on Power Systems,1993,8(3):1045-1053.
[71]G Xiaohong, et al.. Nonlinear approximation method in Lagrangian relaxation-based algorithms for hydrothermal scheduling[J]. IEEE Transactions on Power Systems,1995,10(2):772-778.
[72]N J Redondo and A J Conejo. Short-term hydre-thermal coordination by Lagrangian relaxation: solution of the dual problem[J]. IEEE Transactions on Power Systems,1999,14(1):89-95.
[73]W S Sifuentes and A Vargas. Hydrothermal Scheduling Using Benders Decomposition: Accelerating Techniques[J]. IEEE Transactions on Power Systems,2007,22(3):1351-1359.
[74]G S Lauer, et al.. Solution of Large-Scale Optimal Unit Commitment Problems[J]. IEEE Transactions on Power Apparatus and Systems,1982, PAS-101(1):79-86.
[75]S Salam, et al.. Comprehensive algorithm for hydrothermal coordination[J]. IEE Proceedings Generation, Transmission and Distribution,1997,144(5):482-488.
[76]M Shahidehopour and F Yong. Benders decomposition:applying Benders decomposition to power systems[J]. Power and Energy Magazine, IEEE,2005,3(2):20-21.
[77]W Sifuentes and A Vargas. Short-term hydrothermal coordination considering an AC network modeling[J]. International Journal of Electrical Power & Energy Systems,2007,29(6):488-496.
[78]H M Khodr, et al.. Distribution Systems Reconfiguration Based on OPF Using Benders Decomposition [J]. IEEE Transactions on Power Delivery,2009,24(4):2166-2176.
[79]J H Holland. Adaptation in Natural and Artificial Systems. University of Michigan Press,1975.
[80]R C Bansal. Optimization methods for electric power systems:an overview[J]. International Journal of Emerging Electric Power Systems,2005,2(1):1.
[81]H Zhang, et al.. Daily hydrothermal scheduling with economic emission using simulated annealing technique based multi-objective cultural differential evolution approach[J]. Energy,2013,50(0): 24-37.
[82]K P Wong and Y W Wong. Short-term hydrothermal scheduling part. I. Simulated annealing approach[J]. IEE Proceedings on Generation, Transmission and Distribution,1994,141(5): 497-501.
[83]K P Wong and Y W Wong. Short-term hydrothermal scheduling. II. Parallel simulated annealing approach[J]. IEE Proceedings on Generation, Transmission and Distribution,1994,141(5): 502-506.
[84]K K Mandal, et al.. Particle swarm optimization technique based short-term hydrothermal scheduling[J]. Applied Soft Computing,2008,8(4):1392-1399.
[85]C Samudi, et al.. Hydro thermal scheduling using particle swarm optimization[C]. Transmission and Distribution Conference and Exposition,2008. T&D. IEEE/PES,2008:1-5.
[86]Y Peng, et al.. Optimal operation of cascade reservoirs based on generalized ant colony optimization method[C].2010 Sixth International Conference on Natural Computation (ICNC), 2010:2647-2651.
[87]H Shyh-Jier. Enhancement of hydroelectric generation scheduling using ant colony system based optimization approaches[J]. IEEE Transactions on Energy Conversion,2001,16(3):296-301.
[88]R Moeini and M H Afshar. Extension of the constrained ant colony optimization algorithms for the optimal operation of multi-reservoir systems[J]. Journal of Hydroinformatics,2012, in press.
[89]R Naresh and J Sharma. Two-phase neural network based solution technique for short term hydrothermal scheduling[J]. IEE Proceedings on Generation, Transmission and Distribution,1999, 146(6):657-663.
[90]R.-H Liang and Y.-Y Hsu. Short-term hydro-scheduling using Hopfield neural network. IEE Proceedings on Generation, Transmission and Distribution,1996,143(3):269-275.
[91]I J Ramirez-Rosado and J A Dominguez-Navarro. New multiobjective tabu search algorithm for fuzzy optimal planning of power distribution systems [J]. IEEE Transactions on Power Systems, 2006,21(1):224-233.
[92]T A A Victoire and A E Jeyakumar. Unit commitment by a tabu-search-based hybrid-optimisation technique[J]. IEE Proceedings on Generation, Transmission and Distribution,2005,152(4): 563-574.
[93]X Yuan and Y Yuan. Application of cultural algorithm to generation scheduling of hydrothermal systems[J]. Energy Conversion and Management,2006,47(15-16):2192-2201.
[94]Y Lu, et al.. A hybrid multi-objective cultural algorithm for short-term environmental/economic hydrothermal scheduling[J]. Energy Conversion and Management,2011,52(5):2121-2134.
[95]G Zhuangzhi, et al.. Multi-objective optimization scheduling for hydrothermal power systems based on electromagnetism-like mechanism and data envelopment analysis[J]. Proceeding of the CSEE,2013,33(4):53-61.
[96]A H Mantawy, et al.. Integrating genetic algorithms, tabu search, and simulated annealing for the unit commitment problem[J]. IEEE Transactions on Power Systems,1999,14(3):829-836.
[97]A Y Saber, et al.. Fuzzy unit commitment scheduling using absolutely stochastic simulated annealing[J]. IEEE Transactions on Power Systems,2006,21(2):955-964.
[98]C C Asir Rajan. Hydro-thermal unit commitment problem using simulated annealing embedded evolutionary programming approach[J]. International Journal of Electrical Power & Energy Systems,2011,33(4):939-946.
[99]L Ruey-Hsun. A noise annealing neural network for hydroelectric generation scheduling with pumped-storage units[C]. Power Industry Computer Applications,1999. PICA'99. Proceedings of the 21st 1999 IEEE International Conference,1999:171-176.
[100]F E1-Turky and E E Perry. BLADES:an artificial intelligence approach to analog circuit design[J]. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems,1989, 8(6):680-692.
[101]O Gusikhin, et al.. Intelligent systems in the automotive industry:applications and trends[J]. Knowl.Inf. Syst.,2007,12(2):147-168.
[102]V C Ramesh and X Li. A fuzzy multiobjective approach to contingency constrained OPF[J]. IEEE Transactions on Power Systems,1997,12(3):1348-1354.
[103]胡国强,贺仁睦.梯级水电站多目标模糊优化调度模型极其求解方法[J].电工技术学报,2007,22(1)： 154-159.
[104]刘彦奎,王曙明.模糊随机优化理论[M].北京：中国农业大学出版社,2006.
[105]D H Wolpert and W G Macready. No free lunch theorems for optimization[J]. IEEE Transactions on Evolutionary Computation,1997,1(1):67-82.
[106]R Horst and P M Pardalos. Handbook of Global Optimization[M]. Kluwer,1995.
[107]J D Pinter. Global Optimization in Action[M]. Kluwer,1996.
[108]H Tuy. Convex Analysis and Global Optimization[M]. Kluwer,1998.
[109]A Ben-Tal and A S Nemirovskii. Lectures on Modern Convex Optimization:Analysis, Algorithms, and Engineering Applications[M]. Society for Industrial and Applied Mathematics,2001.
[110]Y Nesterov, et al.. Semidefinite Programming Relaxations of Nonconvex Quadratic Optimization[M]. Handbook of Semidefinite Programming. US:Springer US,2000:361-419.
[111]M X Goemans and D P Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming[J]. J. ACM,1995,42(6):1115-1145.
[112]Y Nesterov. Squared Functional Systems and Optimization Problems[M]. High Performance Optimization. US:Springer,2000:405-440.
[113]Y Nesterov. Semidefinite relaxation and nonconvex quadratic optimization[J]. Optimization methods and software,1998,9(1-3):141-160.
[114]Y Ye. Approximating quadratic programming with bound and quadratic constraints[J]. Mathematical programming,1999,84(2):219-226.
[115]P A Parrilo. Semidefinite programming relaxations for semialgebraic problems[J]. Mathematical programming,2003,96(2):293-320.
[116]R T Rockafellar. Convex Analysis[M]. Princeton University Press.1970.
[117]J-B Hiriart-Urruty and C Lemar'echal. Convex Analysis and Minimization Algorithms[M]. Springer,1993.
[118]J-B Hiriart-Urruty and C Lemar'echal. Fundamentals of Convex Analysis [M]. Spinger,2001.
[119]J M Borwein and A S Lewis. Convex Analysis and Nonlinear Optimization:Theory and Examples[M]. Springer,2006.
[120]Y Nesterov and A Nemirovskii. Interior Point Polynomial Algorithms in Convex Programming[M]. Society for Industrial and Applied Mathematics,1994.
[121]R Rockafellar. Lagrange Multipliers and Optimality[J]. SIAM Review,1993,35(2):183-238.
[122]X Bai and H Wei. Semi-definite programming-based method for security-constrained unit commitment with operational and optimal power flow constraints[J]. IEE Proceedings on Generation, Transmission and Distribution,2009,3(2):182-197.
[123]X Bai and H Wei. A semidefinite programming method with graph partitioning technique for optimal power flow problems[J]. International Journal of Electrical Power & Energy Systems, 2011,33(7):1309-1314.
[124]R A Jabr. Rank-constrained semidefinite program for unit commitment[J]. International Journal of Electrical Power & Energy Systems,2013,47(0):13-20.
[125]R Quan, et al.. Tighter relaxation method for unit commitment based on second-order cone programming and valid inequalities[J]. International Journal of Electrical Power & Energy Systems,2014,55(0):82-90.
[126]E Parrilla and J Garcia-Gonzalez. Improving the B & B search for large-scale hydrothermal weekly scheduling problems[J]. International Journal of Electrical Power & Energy Systems,2006,28(5): 339-348.
[127]A Borghetti, et al.. Lagrangian heuristics based on disaggregated Bundle methods for hydrothermal unit commitment[J]. IEEE Transactions on Power Systems,2003,18(1):313-323.
[128]C Jaeseok, et al.. A fuzzy branch and bound-based transmission system expansion planning for the highest satisfaction level of the decision maker[J]. IEEE Transactions on Power Systems,2005, 20(1):476-484.
[129]A I Cohen and M Yoshimura. A Branch-and-Bound Algorithm for Unit Commitment[J]. IEEE Transactions on Power Apparatus and Systems, PAS-102(2):444-451,1983.
[130]C Chern-Lin and S C Wang. Branch-and-bound scheduling for thermal generating units[J]. IEEE Transactions on Energy Conversion,1993,8(2):184-189.
[131]M Madrigal and V H Quintana. An interior-point/cutting-plane method to solve unit commitment problems[C]. Power Industry Computer Applications,1999. PICA '99. Proceedings of the 21st 1999 IEEE International Conference,1999:203-209.
[132]M Murai and M Kato. Cutting plane methods for Lagrangian relaxation-based unit commitment algorithm[J]. Electrical Engineering in Japan,2002,141(3):18-32.
[133]A L Ott. Evolution of computing requirements in the PJM market:Past and future [C]. Power and Energy Society General Meeting,2010 IEEE,2010:1-4.
[134]D Streiffert, et al.. A mixed integer programming solution for market clearing and reliability analysis. Power Engineering Society General Meeting,2005. IEEE,2005(3):2724-2731.
[135]J J Shaw. A direct method for security-constrained unit commitment[J]. IEEE Transactions on Power Systems,1995,10(3):1329-1342.
[136]P Interconnection. PJM'S new generation-scheduling software to save customers estimated ﹩56 million unit commitment system uses first of its kind technieque. PJM News Release,2004.
[137]R E Bixby, et al.. MIP:Theory And Practice-Closing The Gap. System modelling and optimization:methods, theory, and applications. Kluwer Academic Publishers,2000.
[138]J Ostrowski, et al.. Tight Mixed Integer Linear Programming Formulations for the Unit Commitment Problem[J]. IEEE Transactions on Power Systems,2012,27(1):39-46.
[139]A Frangioni, et al.. Sequential Lagrangian-MILP approaches for Unit Commitment problems[J]. International Journal of Electrical Power & Energy Systems,2011,33(3):585-593.
[140]D Han, et al.. Outer Approximation and Outer-Inner Approximation Approaches for Unit Commitment Problem[J]. IEEE Transactions on Power Systems,2013, in press.
[141]A Frangioni and C Gentile. A computational comparison of reformulations of the perspective relaxation:SOCP vs. cutting planes[J]. Operations Research Letters,2009,37(3):206-210.
[142]S Boyd and L Vandenberghe. Convex Optimization[M]. Cambridge University Press,2004.
[143]F Alizadeh and D Goldfarb. Second-order cone programming[J]. Mathematical programming, 2003,95(1):3-51.
[144]G Anupam and O D Ryan. (2011). Linear and Semidefinite Programming. Available: http://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15859-f11/www/notes/lecture12.pdf.
[145]Y Nesterov and A S Nemirovsii. Interior point polynomial algorithms in convex programming vol. 13[M]. Philadelphia, USA:SIAM,1994.
[146]Y Nesterov and A S Nemirovsii. Conic formulation of a convex programming problem and duality[J]. Optimization methods and software,1992,1(2):95-115.
[147]F Alizadeh-Dehkharghani. Combinatorial optimization with interior point methods and semi-definite matrices. Ph.D., University of Minnesota, Minneapolis, MN, USA,1991.
[148]L Vandenberghe and S Boyd. Semidefinite Programming[J]. SIAM Review,1994,38(0):49-95.
[149]E De Klerk. Interior Point Methods for Semidefinite Programming. M.Sc., Technische Universiteit Delft, Delft,1997.
[150]E De Klerk, et al.. On primal-dual path-following algorithm in semidefinite programming. New trends in mathematical programming. The Netherlands:Kluwer Academic Publishers,1998: 137-157.
[151]K M Anstreicher and M Fampa. A long-step path following algorithm for semidefinite programming problems. Department of management sciences, University of Iowa, Iowa City, 1996.
[152]L Faybusovich. Semidefinite Programming:A Path-Following Algorithm for a Linear-Quadratic Functional[J]. SIAM Journal on Optimization,1996,6(4):1007-1024.
[153]F Alizadeh-Dehkharghani, et al.. Primal-dual interior point methods for semidefinite programming: convergence rates, stability and numerical results[J]. SIAM Journal on Optimization,1998,8(3): 746-768.
[154]Y Nesterov and M J Todd. Self-scaled barriers and interior-point methods for convex programming[J]. Mathematics of Operations Research,1997,22(1):1-42.
[155]R D C Monteiro. Primal-dual path-following algorithms for semidefinite programing[J]. SIAM Journal on Optimization,1997,7(3):663-688.
[156]M Kojima, et al.. Interior point methods for the monotone semidefinite linear complementarity problem in symmetric matrices[J]. SIAM Journal on Optimization,1997,7(1):88-124.
[157]C Helmberg, et al.. An interior-point method for semidefinite programming[J]. SIAM Journal on Optimization,1996,6(0):342-361.
[158]C Helmberg. Semidefinite Programming for Combinatorial Optimization [M]. Konrad-Zuse-Zentrum fur Informationstechnik,2000.
[159]M X Goemans and F Rendl. Semidefinite programming in combinatorial optimization[J]. Mathematical programming,1999,79(1997):143-161.
[160]F Juhasz. The asymptotic behaviour of lovasz' 9 function for random graphs[J]. Combinatorica, 1982,2(2):153-155.
[161]F Rendl. Semidefinite Relaxations for Integer Programming.50 Years of Integer Programming 1958-2008[M]. Springer Berlin Heidelberg,2010:687-726.
[162]D Jibetean and M Laurent. Semidefinite Approximations for Global Unconstrained Polynomial Optimization[J]. SIAM J. on Optimization,2005,16(2):490-514.
[163]J B Lasserre. Global optimization with polynomials and the problem of moments[J]. SIAM Journal on Optimization,2001,11(3):796-978.
[164]N Z Shor. An approach to obtaining global extremums in polynomial mathematical programming problems[J]. Cybernetics and Systems Analysis,1987,23(5):695-700.
[165]S Liu, et al.. A robust optimization approach to wind farm diversification[J]. International Journal of Electrical Power & Energy Systems,2013,53(0):409-415.
[166]Conic Programming Solvers. Available:http://www.gamsworld.org/cone/solvers.htm.
[167]S Soares, et al.. Optimal Generation Scheduling of Hydrothermal Power Systems[J]. IEEE Transactions on Power Apparatus and Systems,1980, PAS-99(3):1107-1118.
[168]K K Mandal and N Chakraborty. Differential evolution technique-based short-term economic generation scheduling of hydrothermal systems[J]. Electric Power Systems Research,2008,78(11): 1972-1979.
[169]G S C M E El-Hawary. Optimal economic operation of electric power systems, vol. Mathematics in science and engineering:Academic Press,1979.
[170]A Saxena, et al.. Convex relaxations of non-convex mixed integer quadratically constrained problems:Extended formulations[J]. IBM Corp. Tech. Rep. RC24621,2008.
[171]K M Anstreicher. Comparing Convex Relaxations for Quadratically Constrained Quadratic Programming.2010.
[172]J Lavaei and S H Low. Zero Duality Gap in Optimal Power Flow Problem[J]. IEEE Transactions on Power Systems,2012,27(1):92-107.
[173]卢强.电力改革要以建立安全经济运营的电力系统为目标.2013,Available: http://www.sgcc.com.cn/xwzx/gsyw/2013/04/290331.shtml.
[174]M A Abido. Environmental/economic power dispatch using multiobjective evolutionary algorithms[J]. IEEE Transactions on Power Systems,2003,18(4):1529-1537.
[175]CO2 emissions from fuel combustion highlights. I E Agency,2012.
[176]张晓花,赵晋泉,陈星莺.节能减排多目标机组组合问.题的模糊建模及优化[J].中国电机工程学报,2010,30(22)：71-77.
[177]P D Brown, et al.. Optimization of Pumped Storage Capacity in an Isolated Power System With Large Renewable Penetration[J]. IEEE Transactions on Power Systems,2008,23(2):523-531.
[178]Wikipedia. Water turbine. Available:http://en.wikipedia.org/wiki/Water_turbine.
[179]聂启蓉.平班水电站水轮机模型验收试验[J].红水河，2007,vo1.26(1)：4.
[180]A J Wood and B F Wollenberg. Power Generation Operation and Control:John Wiley & Sons, 1984.
[181]M Piekutowski, et al.. Optimal short-term scheduling for a large-scale cascaded hydro system[J]. IEEE Transactions on Power Systems,1994,9(2):805-811.
[182]S P Bradley, et al.. Applied mathematical programming:Addison-Wesley Pub. Co.,1977.
[183]C E Lin and G L Viviani. Hierarchical Economic Dispatch for Piecewise Quadratic Cost Functions[J]. IEEE Transactions on Power Apparatus and Systems,1984, PAS-103(6):1170-1175.
[184]A Astorino, et al.. Piecewise-quadratic Approximations in Convex Numerical Optimization[J]. SIAM Journal on Optimization,2011,21(4):1418-1438.
[185]J H Park, et al.. Economic load dispatch for piecewise quadratic cost function using Hopfield neural network[J]. IEEE Transactions on Power Systems,1993,8(3):1030-1038.
[186]M Farivar and S H Low. Branch Flow Model:Relaxations and Convexification—Part I[J]. IEEE Transactions on Power Systems,2013,28(3):2554-2564.
[187]A L Diniz and M E P Maceira. A Four-Dimensional Model of Hydro Generation for the Short-Term Hydrothermal Dispatch Problem Considering Head and Spillage Effects [J]. IEEE Transactions on Power Systems.2008,23(3):1298-1308.
[188]E Barron, et al.. Functions Which Are Quasiconvex under Linear Perturbations[J]. SIAM Journal on Optimization,2012,22(3):1089-1108.
[189]T Pock, et al.. Global Solutions of Variational Models with Convex Regularization[J]. SIAM Journal on Imaging Sciences,2010,3(4):1122-1145.
[190]A L Diniz. Test cases for unit commitment and hydrothermal scheduling problems[C]. Power and Energy Society General Meeting,2010 IEEE,2010:1-8.
[191]J Briet, et al. Grothendieck inequalities for semidefinite programs with rank constraint. Available: http://arxiv.org/abs/1011.1754.