机组组合问题紧外逼近与内外逼近模型方法及优化问题QP-free算法研究
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摘要
在世界各国积极发展低碳经济的背景下,降低能源消耗是实现低碳经济最直接的重要途径。近年来,中国电力工业发展很快,消耗的电绝大部分来自燃煤电厂。因此,在保证电力系统安全稳定运行和可靠供电的前提下,优化燃煤电厂的发电调度模式具有重要的现实意义。
本文以电力系统机组组合问题为研究对象。一方面研究适合于实际工程应用的大规模非线性含整数与连续变量的机组组合问题的高效求解方法。另一方面,鉴于在机组组合问题的求解过程中涉及到求解其连续松弛问题,该问题属于光滑非线性规划问题,本文对求解光滑非线性优化问题的QP-free算法开展了深入细致的研究工作。
首先,在回顾和总结现有发电调度计划的应用研究现状基础上,给出了所研究的机组组合问题的凸可分模型,提出了基于该模型的紧外逼近方法和内外逼近方法。这两类方法能有效地求解非光滑、非线性、大规模、含混合变量、多时段的机组组合问题,为实施节能发电计划提供实现的方法及手段。其次,本着探索求解光滑非线性规划问题结构简单、计算量少、收敛速度快及数值效果更好的算法以适用于大规模电力系统优化问题,提出了不等式约束优化问题一种新的线性方程组构造方法以及新型的QP-free算法。全文共分为6章,归纳如下：
第1章分析研究了发电调度计划制定的要求,阐明了本课题研究的理论及实践意义。围绕机组组合问题及其连续松弛后的光滑非线性规划问题的求解方法两条主线,回顾总结机组组合问题的求解方法和求解非线性规划问题的QP-free算法的研究现状,提炼出尚需研究解决的关键问题,并基于此,引述本文的主要研究工作。
第2章给出本文使用的数学理论基础。针对特殊的凸可分模型给出了紧外逼近法及其算法步骤,详细阐述了内外逼近法的思想和基本原理,以及设计光滑非线性规划问题算法的一些相关理论基础,为后续章节内容提供理论分析和算法两方面的支撑。
第3章通过研究机组组合问题的特点及数学模型,提出一种机组组合问题的凸可分模型,以及基于该模型而获得机组组合问题次优解的紧外逼近确定性全局优化方法。通过将多变量函数分解成多个单变量函数,利用多个已经分离的单变量函数的次梯度不等式进行多步外逼近获得更紧的混合整数线性规划主问题,通过交替求解更紧的混合整数线性规划主问题和非线性规划子问题来逼近机组组合问题的最优解。
第4章通过进一步减小松弛间隙,提出一种内外逼近方法求解机组组合问题。给出了改进的初始外逼近子问题,并提出了新的内逼近混合整数线性规划子问题。通过交替求解一系列混合整数线性规划外逼近子问题与内逼近子问题,产生更好的下界和更好的上界。充分利用内、外逼近子问题的有效性,既改进了迭代间隙又提高了解的质量。分别对10～300机组24时段8个算例以及10～100机组24～96时段系统进行了数值仿真。
第5章基于对约束函数梯度的扰动,提出光滑非线性约束优化问题一个新的可行QP-free算法。新算法不仅保存了现有算法的优点,还具有其他一些良好的特性：算法每次迭代只需求解三个具有相同系数矩阵的线性方程组,计算量小；求解一个线性方程组即可产生可行下降方向,克服了以往至少要求解两个线性方程组,然后再做适当的凸组合方可获得可行下降方向的困难；迭代点均为可行点,并不要求是严格内点；算法中采用了试探性线搜索,可以进一步减少计算量；算法中参数很少,针对数学测试问题而言,数值试验表明算法具有较好的数值效果和较强的稳定性。
第6章概括总结了本文的主要研究工作和成果,展望了有待进一步深入开展的几个研究工作。
Under the background of developing positively the low-carbon economy all over the world, the reduction of the energy consumption is the most direct and important way to realize low-carbon economy. In recent years, the development of China's power industry is rapid, and the most part of the consumed electricity is from coal-fired power plants. Therefore, it has important realistic significance to study the dispatch of coal-fired power plants on the premise of ensuring the safe and stable operation of the power system and reliable power supply.
This thesis aims to study the unit commitment (UC) problem in power systems. On the one hand, this paper aims to study some efficient methods for solving the large-scale, nonlinear unit commitment problems with many integer and continuous varibles. On the other hand, in the process of solving the unit commitment problems, it needs to solve its continuous relaxation problem which is a smooth nonlinear programming (NLP) problem. The QP-free algorithm for solving nonlinear programming problem has been carried out in-depth and meticulous research at the same time.
Firstly, based on the review and summary of the application of the existing generation dispatching, the paper proposes a new convex separable model for the unit commitment problem, and the tighter outer approximation and outer-inner approximation approaches for this model. These two methods can effectively solve the nonlinear, nonsmooth, large-scale and multi-period unit commitment problem with mixed variables, and provide methods and means for achieving the energy-saving generation dispatching. Secondly, with the aim of searching an algorithm with a simple structure, less amount of calculation, fast convergence rate and better numerical performance for the nonlinear programming problems to be suitable for large-scale power system optimization problems, this paper put forward a new method of structuring system of linear equations and a new type of QP-free algorithm for optimization problem with inequality constraints. This thesis contains6chapters as follows:
Chapter1presents the meaning of theory research and practice through the analysis of the requirements of the generation dispatching. This chaper reviews and sums up the status of solution methods of UC problem and around the QP-free algorithm for solving NLP problem around two main lines of the solution methods of UC problem and the smooth NLP problem, and extracts the key issues that still need to be studied and solved. The main research and expectations are described based on these.
Chapter2discusses the mathematical theory foundations used in this paper. For the special convex separable model, this chapter gives a tighter outer approximation method and its algorithm steps, and elaborates the thought and basic principle of the outer-inner approximation method in detail and some theoretical basis of designing specific algorithm for smooth NLP problem. All of them provide support for subsequent chapters in both the theory analysis and algorithm.
Chapter3presents a new convex separable model and a tighter outer approximation methodology which is a deterministic global optimization method based on it by studying the characteristics and mathematical models of UC problem. By decomposing a multivariate function into several univariate functions, a tighter mixed integer linear programming (MILP) master problem is obtained by several step outer approximation using subgradient inequalites of the separated univariate functions. The optimal solution of UC problem is approached by alternately solving the MILP master problems and the NLP subproblems.
In chapter4, an outer-inner approximation approach is presented to solve UC problem by further reducing the relaxation gaps. This chapter gives an improved initial outer approximation subproblem and presents a new inner approximation MILP subproblem. A sequence of MILP outer approximation subproblem and.inner approximation subproblem are solved alternately to yield better lower bound and upper bound. Both the iteration gap and the solutions'quality have been improved by taking advantages of the effectiveness of inner and outer approximation subproblems. Some numerical simulations are carried out on the systems of up to300units with24h, and systems of up to100units with24～96h.
In chapter5, a new feasible QP-free algorithm for solving the smooth nonlinear constrained optimization problems based on the perturbation for the gradients of the constraint functions is presented. It reserves all the advantages of previous algorithms, and there are some other interesting features of the algorithm. At each iteration, only three systems of linear equations with the same coefficient matrix need to be solved, which decreases largely the amount of computations. A feasible descent direction can be obtained by solving only one system of linear equations, while the previous algorithms need to solve one linear system to get a feasible direction and another one to obtain a descent direction, and an improving direction is obtained by doing a convex combination. The iteration points are all feasible without requiring to be strictly interior points. The exploratory line search is introduced to the algorithm, and the computational cost can be further reduced. The parameters in the proposed algorithm are few, and some numerical results for math test problems illustrate that the proposed algorithm is efficient and stable.
Chapter6summarizes the main research work and achievements of this paper and points out the further research works in the future.
本文以电力系统机组组合问题为研究对象。一方面研究适合于实际工程应用的大规模非线性含整数与连续变量的机组组合问题的高效求解方法。另一方面,鉴于在机组组合问题的求解过程中涉及到求解其连续松弛问题,该问题属于光滑非线性规划问题,本文对求解光滑非线性优化问题的QP-free算法开展了深入细致的研究工作。
首先,在回顾和总结现有发电调度计划的应用研究现状基础上,给出了所研究的机组组合问题的凸可分模型,提出了基于该模型的紧外逼近方法和内外逼近方法。这两类方法能有效地求解非光滑、非线性、大规模、含混合变量、多时段的机组组合问题,为实施节能发电计划提供实现的方法及手段。其次,本着探索求解光滑非线性规划问题结构简单、计算量少、收敛速度快及数值效果更好的算法以适用于大规模电力系统优化问题,提出了不等式约束优化问题一种新的线性方程组构造方法以及新型的QP-free算法。全文共分为6章,归纳如下：
第1章分析研究了发电调度计划制定的要求,阐明了本课题研究的理论及实践意义。围绕机组组合问题及其连续松弛后的光滑非线性规划问题的求解方法两条主线,回顾总结机组组合问题的求解方法和求解非线性规划问题的QP-free算法的研究现状,提炼出尚需研究解决的关键问题,并基于此,引述本文的主要研究工作。
第2章给出本文使用的数学理论基础。针对特殊的凸可分模型给出了紧外逼近法及其算法步骤,详细阐述了内外逼近法的思想和基本原理,以及设计光滑非线性规划问题算法的一些相关理论基础,为后续章节内容提供理论分析和算法两方面的支撑。
第3章通过研究机组组合问题的特点及数学模型,提出一种机组组合问题的凸可分模型,以及基于该模型而获得机组组合问题次优解的紧外逼近确定性全局优化方法。通过将多变量函数分解成多个单变量函数,利用多个已经分离的单变量函数的次梯度不等式进行多步外逼近获得更紧的混合整数线性规划主问题,通过交替求解更紧的混合整数线性规划主问题和非线性规划子问题来逼近机组组合问题的最优解。
第4章通过进一步减小松弛间隙,提出一种内外逼近方法求解机组组合问题。给出了改进的初始外逼近子问题,并提出了新的内逼近混合整数线性规划子问题。通过交替求解一系列混合整数线性规划外逼近子问题与内逼近子问题,产生更好的下界和更好的上界。充分利用内、外逼近子问题的有效性,既改进了迭代间隙又提高了解的质量。分别对10～300机组24时段8个算例以及10～100机组24～96时段系统进行了数值仿真。
第5章基于对约束函数梯度的扰动,提出光滑非线性约束优化问题一个新的可行QP-free算法。新算法不仅保存了现有算法的优点,还具有其他一些良好的特性：算法每次迭代只需求解三个具有相同系数矩阵的线性方程组,计算量小；求解一个线性方程组即可产生可行下降方向,克服了以往至少要求解两个线性方程组,然后再做适当的凸组合方可获得可行下降方向的困难；迭代点均为可行点,并不要求是严格内点；算法中采用了试探性线搜索,可以进一步减少计算量；算法中参数很少,针对数学测试问题而言,数值试验表明算法具有较好的数值效果和较强的稳定性。
第6章概括总结了本文的主要研究工作和成果,展望了有待进一步深入开展的几个研究工作。
Under the background of developing positively the low-carbon economy all over the world, the reduction of the energy consumption is the most direct and important way to realize low-carbon economy. In recent years, the development of China's power industry is rapid, and the most part of the consumed electricity is from coal-fired power plants. Therefore, it has important realistic significance to study the dispatch of coal-fired power plants on the premise of ensuring the safe and stable operation of the power system and reliable power supply.
This thesis aims to study the unit commitment (UC) problem in power systems. On the one hand, this paper aims to study some efficient methods for solving the large-scale, nonlinear unit commitment problems with many integer and continuous varibles. On the other hand, in the process of solving the unit commitment problems, it needs to solve its continuous relaxation problem which is a smooth nonlinear programming (NLP) problem. The QP-free algorithm for solving nonlinear programming problem has been carried out in-depth and meticulous research at the same time.
Firstly, based on the review and summary of the application of the existing generation dispatching, the paper proposes a new convex separable model for the unit commitment problem, and the tighter outer approximation and outer-inner approximation approaches for this model. These two methods can effectively solve the nonlinear, nonsmooth, large-scale and multi-period unit commitment problem with mixed variables, and provide methods and means for achieving the energy-saving generation dispatching. Secondly, with the aim of searching an algorithm with a simple structure, less amount of calculation, fast convergence rate and better numerical performance for the nonlinear programming problems to be suitable for large-scale power system optimization problems, this paper put forward a new method of structuring system of linear equations and a new type of QP-free algorithm for optimization problem with inequality constraints. This thesis contains6chapters as follows:
Chapter1presents the meaning of theory research and practice through the analysis of the requirements of the generation dispatching. This chaper reviews and sums up the status of solution methods of UC problem and around the QP-free algorithm for solving NLP problem around two main lines of the solution methods of UC problem and the smooth NLP problem, and extracts the key issues that still need to be studied and solved. The main research and expectations are described based on these.
Chapter2discusses the mathematical theory foundations used in this paper. For the special convex separable model, this chapter gives a tighter outer approximation method and its algorithm steps, and elaborates the thought and basic principle of the outer-inner approximation method in detail and some theoretical basis of designing specific algorithm for smooth NLP problem. All of them provide support for subsequent chapters in both the theory analysis and algorithm.
Chapter3presents a new convex separable model and a tighter outer approximation methodology which is a deterministic global optimization method based on it by studying the characteristics and mathematical models of UC problem. By decomposing a multivariate function into several univariate functions, a tighter mixed integer linear programming (MILP) master problem is obtained by several step outer approximation using subgradient inequalites of the separated univariate functions. The optimal solution of UC problem is approached by alternately solving the MILP master problems and the NLP subproblems.
In chapter4, an outer-inner approximation approach is presented to solve UC problem by further reducing the relaxation gaps. This chapter gives an improved initial outer approximation subproblem and presents a new inner approximation MILP subproblem. A sequence of MILP outer approximation subproblem and.inner approximation subproblem are solved alternately to yield better lower bound and upper bound. Both the iteration gap and the solutions'quality have been improved by taking advantages of the effectiveness of inner and outer approximation subproblems. Some numerical simulations are carried out on the systems of up to300units with24h, and systems of up to100units with24～96h.
In chapter5, a new feasible QP-free algorithm for solving the smooth nonlinear constrained optimization problems based on the perturbation for the gradients of the constraint functions is presented. It reserves all the advantages of previous algorithms, and there are some other interesting features of the algorithm. At each iteration, only three systems of linear equations with the same coefficient matrix need to be solved, which decreases largely the amount of computations. A feasible descent direction can be obtained by solving only one system of linear equations, while the previous algorithms need to solve one linear system to get a feasible direction and another one to obtain a descent direction, and an improving direction is obtained by doing a convex combination. The iteration points are all feasible without requiring to be strictly interior points. The exploratory line search is introduced to the algorithm, and the computational cost can be further reduced. The parameters in the proposed algorithm are few, and some numerical results for math test problems illustrate that the proposed algorithm is efficient and stable.
Chapter6summarizes the main research work and achievements of this paper and points out the further research works in the future.
引文
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