Combined heat and power (CHP) economic dispatch solved using Lagrangian relaxation with surrogate subgradient multiplier updates

This paper presents a flexible algorithm to solve the combined heat and power (CHP) economic dispatch problem. The CHP economic dispatch is solved in two levels known as the lower level and higher level. The higher level is the optimization of the surrogate dual function for the relaxed global constraints in which the surrogate subgradient is used to update the Lagrangian multipliers. Coherently, the lower levels are the optimization of the subproblems taking in count each of its local constraints. Flexibility for the choice of algorithm is given at the lower levels optimization techniques with the condition that the algorithm is able to improve its search at each iteration. It is also seen that simple step size rules such as the ‘square summable but not summable’ and ‘constant step size’ could be used easily and leads the method to convergence. In addition this paper illustrates the ear clipping method used to modify the common nonconvex feasible region of CHP benchmark problems to a convex region which subsequently enhances the search for an optimal solution. The algorithm is then justified through a numerical test on three benchmark CHP problem with a nonconvex feasible region. Results prove that the algorithm is reliable and could be easily implemented even on a much complex and nonconvex problems.     

Optimization-based scheduling of hydrothermal power systems withpumped-storage units

This paper presents an optimization-based method for scheduling hydrothermal systems based on the Lagrangian relaxation technique. After system-wide constraints are relaxed by Lagrange multipliers, the problem is converted into the scheduling of individual units. This paper concentrates on the solution methodology for pumped-storage units. There are, many constraints limiting the operation of a pumped-storage unit, such as pond level dynamics and constraints, and discontinuous generation and pumping regions. The most challenging issue in solving pumped-storage subproblems within the Lagrangian relaxation framework is the integrated consideration of these constraints. The basic idea of the method is to relax the pond level dynamics and constraints by using another set of multipliers. The subproblem is then converted into the optimization of generation or pumping; levels for each operating state at individual hours, and the optimization of operating states across hours. The optimal generation or pumping level for a particular operating state at each hour can be obtained by optimizing a single variable function without discretizing pond levels. Dynamic programming is then used to optimize operating states across hours with only a few number of states and transitions. A subgradient algorithm is used to update the pond level Lagrangian multipliers. This method provides an efficient way to solve a class of subproblems involving continuous dynamics and constraints, discontinuous operating regions, and discrete operating states     

系统分解协调算法的抽水蓄能电站优化调度

建立了包含抽水蓄能电站的电网统一调度优化模型,即以调度周期内火电燃料成本为最小目标函数,满足系统及各机组约束条件。利用系统分解协调思想,开发了一个结合拉格朗日松弛方法和粒子群优化算法的混合算法,将原优化问题分解为两层优化问题。上层拉格朗日算子优化利用次梯度算法求解,下层各子问题利用粒子群优化算法求解,经过迭代寻优得到最优对偶解后,利用一个启发式算法求得满足系统约束及各机组运行约束的原问题的可行解。最后通过算例验证了模型的合理性及算法的有效性。     

基于原对偶内点法的电压无功功率优化

文章以电压无功优化的二次规划模型为基础,对原对偶内点法进行了扩展,使之能处理电压无功优化控制中大量的不等式的约束。文章提出了一种壁垒参数的选取方法和控制计算步长的策略,并采用了一种有效的预测校正方法来提高算法的收敛性。     

Scheduling of hydrothermal power systems

The authors present a method for scheduling hydrothermal power systems based on the Lagrangian relaxation technique. By using Lagrange multipliers to relax system-wide demand and reserve requirements, the problem is decomposed and converted into a two-level optimization problem. Given the sets of Lagrange multipliers, a hydro unit subproblem is solved by a merit order allocation method, and a thermal unit subproblem is solved by using dynamic programming without discretizing generation levels. A subgradient algorithm is used to update the Lagrange multipliers. Numerical results based on Northeast Utilities data show that this algorithm is efficient, and near-optimal solutions are obtained. Compared with previous work where thermal units were scheduled by using the Lagrangian relaxation technique and hydro units by heuristics, the new coordinated hydro and thermal scheduling generates lower total costs and requires less computation time     

具有相同机组水火电调度问题的新算法

对Lagrangian松弛法求解水火电调度问题时由机同机级引起解震荡现象进行了研究。通过一个例子分析了震荡产生的根本原因。对此，在松弛函数中引入了惩罚项并采用了伪次梯度法来修正乘子。新算法在求解低层子问题时并不同时求解，使震荡现象在很大程度上得以克服，同时可大幅度地降低偶解对约束的违反程度。通过简单的例子和对一个包含两组机同机组的短期发电调度问题的计算表明，对偶解的约束违反程度明显地降低，解震荡明显地减弱且最后可行解的质量有显著的改善。     

基于拉格朗日函数鞍距分配的广义内点法

不等式约束的处理是电力系统优化分析中比较困难的问题。文中根据拉格朗日函数的鞍点理论,将优化问题的等式约束进行松弛,形成计及等式约束的原始问题以及相应的对偶问题。通过定义原始和对偶问题之间的鞍距,并将鞍距在不等式约束之间进行分配,从而形成不同的针对不等式约束拉格朗日乘子的修正方程,进一步形成不同的优化算法。推导表明,内点罚函数法只是拉格朗日鞍点理论应用的一个特例。所提出的基于拉格朗日函数鞍距分配的广义内点法可以在电力系统优化分析中进行应用,将其应用于大规模间歇式电源接入情况下的电力系统最大传输能力问题中时,IEEE 30节点系统的计算结果及IEEE 14节点系统中不同算法的比较结果表明,此算法能够有效处理潮流问题不等式约束。     

电力市场发电机组检修计划的快速算法

电力市场下发电机组检修计划的安排不当是造成近几年大停电的原因之一。电力市场机组检修计划需要兼顾可靠性与经济性两个方面。采用Benders分解法可以将这个高维度、非线性、混合整数随机规划问题分解为主问题和子问题求解：主问题是一个多目标整数规划问题,而子问题则是一个非线性随机问题。针对Benders分解法求解效率不高的问题,求解主问题时利用了机组检修连续性的特点,对Balas隐枚举法中的前向搜索部分和回溯部分进行了改进;求解子问题时,利用了半不变量法的偏导数解析表达式来求对偶乘子。对IEEE-RBTS和IEEE-RTS系统中所有发电机组安排了全年的检修计划,测试表明所提出的算法快速、有效。     

Improvements on the duality based method used in solving optimalpower flow problems

To improve the previously developed dual-type (DT) method used in solving optimal power flow (OPF) problems with large number of thermal-limit constraints, the authors propose two new techniques in this paper. The first one is a graph-method based decomposition technique which can decompose the large-dimension projection problem, caused by the large number of thermal-limit constraints, into several independent medium-dimension projection subproblems at the expense of slight increment of the dual problem's dimension. The second technique is an active-set strategy based DT method which can solve the medium-dimension projection subproblems efficiently. They have used the DT method embedded with these two new techniques in solving numerous OPFs with large number of thermal-limit constraints. The test results show that the proposed techniques are very efficient and effectively improve the DT method for handling large number of thermal-limit constraints     

考虑限幅约束的多区域互联电网全分布式潮流算法

多区域互联电网中有大量限幅非光滑约束,增加了潮流模型的非凸性。针对传统基于启发式规则处理限幅的分布式潮流算法在计算过程中容易出现收敛性问题,提出了一种能鲁棒性处理非光滑约束的互联电网全分布式潮流计算方法。首先根据分层分区调度模式对电网进行分区,并将模型中非光滑约束进行光滑处理,然后基于具有二阶收敛速度的双层交替方向非精确牛顿法,将潮流问题转化为求解最优步长增量的问题。基于零空间法(Null-Space)对状态变量的系数矩阵进行降维处理,采用共轭梯度(Conjugate Gradient, CG)算法更新每个分区的对偶乘子,乘子更新过程中利用二阶信息提高了算法的收敛性。多区域间进行分布式计算时无需协调层参与,仅需通信少量边界信息,因此通信负担轻。最后,以30节点和182节点系统为测试算例,验证了所提方法在设置恶劣初值和处理非光滑约束时具有较高的精确性和较好的收敛性。     

多区域输电阻塞管理的拉格朗日松驰分解算法

提出一种新的基于增广拉格朗日松驰的区域分解最优潮流算法,将一个大的最优潮流问题分解成多个区域子问题,并用此算法求解多区域电力市场输电阻塞管理问题,与现有的其他方法相比,该算法的主要优点在于无需在原始网络模型的基础上增加任何虚拟发电机或负荷,通过将该算法与电力市场实时平衡机制相结合,多区域有力阻塞管理问题可以分解为单区域二次规划子问题。这些子问题可以顺序求解也可以并行求解。采用这一方法,所有的区域市场独立调度员在得到不到其他区域网络信息的情况下仍然可以相互协作消除网络阻塞,在这一过程中,惟一需要进行区域间交换的信息是与区域间“耦合”约束相对应的拉格朗日乘子,最后,通过分析3区域的IEEE RTS－96标准测试系统说明了该方法的有效性。     

A Method for Solving the Fuel Constrained Unit Commitment Problem

The unit commitment problem involves finding the hourly commitment schedule for the thermal units of an electrical system, and their associated generation, over a period of up to a week. For some utilities, contractual or other factors limit the amount of fuel available to certain of the units or plants. This paper describes a new method which solves the unit commitment problem in the presence of fuel constraints. The method uses a Lagrangian decomposition and successive approximation technique for solving the unit commitment problem where the generation, reserve and fuel constraints are adjoined onto the cost function using Lagrange multipliers. All important operating constraints have been incorporated including minimum up and down times, standby operation, ramping limits, time-dependent start-up cost, spinning and supplemental reserve. The method is being applied to a production-grade program suitable for Energy Management Systems applications.     

一种基于Karmarkar内点法的最优潮流算法

以原－对偶内点算法（Ｋａｒｍａｒｋａｒ内点法的一种变形）为基本算法解算最优潮流问题，综合考虑非线性目标函数和约束条件，结合牛顿法最优潮流先进的稀疏矩阵技术，并且提出了一种新的原－对偶内点算法迭代步长选取原则和障碍参数修正策略。算例表明本算法有较好的数值稳定性，优化结果精确，对不等式约束有较强的处理能力，显示了内点算法应用于大规模电力系统优化问题的良好前景。     

具有爬升约束机组组合的充分必要条件

在Lagrangian松弛框架下，很难确定机组组合问题的一个可行解是否可通过调整对偶机组组合而获得。对于具有爬升约束的机组组合调度问题来说，由于机组出力在连续的2个开机区间的耦合性，求解可行解就更困难。在Lagrangian松弛框架下，开发1个机组组合新方法的核心是如何获得1个可行的机组组合。文中采用Benders分解可行性条件严格证明了在给定时段，机组组合可行的充分必要条件：即在该时段一个相应于系统负载平衡约束和旋转各用约束的不等式组成立。该条件不需要求解经济分配问题，就可以判定机组组合的可行性。有了此条件，可在发电功率经济分配前知道机组组合是否可行，若不可行，则可通过调整机组组合状态而获得可行的组合。该条件对于构造一个求解机组组合问题的系统方法是重要且有效的。数值测试表明该条件是判定机组组合可行性的有效方法。     

考虑网络安全约束的机组组合新算法

市场机制驱使电网运行于安全极限的边缘,考虑网络安全约束的机组组合问题变得尤为重要,基于对偶原理的拉格朗日松弛法是解决这一问题的有效途径。文章提出了一种解决网络安全约束下的机组组合问题的新算法,在拉格朗日对偶分解的基础上结合变量复制技术,通过引入附加人工约束将网络约束嵌入单机子问题中,实现在机组组合中考虑网络安全约束。该算法摆脱了现有各种处理手段在解决网络安全约束的机组组合问题时将网络安全约束与机组启停相分离的不足,揭示了安全经济调度和安全约束下的机组组合在概念上的区别和联系。     

一种考虑时间关联约束的安全经济调度解法

针对有时间关联约束的一类安全经济调度问题,以机组输出功率速率约束为例,建立Dantzig-Wolfe分解的主从优化问题及其迭代机制来求解。主问题是仅计及时间关联约束的优化问题,从问题是按研究期间所划分时段数构成若干静态子优化问题。主问题在由从问题确定的解空间内寻优;从问题依据主问题解所对应的拉格朗日乘子来修正其目标,以间接松弛时间关联约束。在给出主从问题交替求解收敛条件及其论证基础上,提出了详细的计算方法和特殊问题的处理手段。5机系统、IEEE 24节点系统和IEEE 118节点系统的计算和分析表明,文中方法能有效解决带有时间关联约束的一类安全经济调度问题,具有对大规模系统实际应用的前景。     

Optimal reactive dispatch through primal-dual method

This paper presents an algorithm for the optimal reactive dispatch problem. The algorithm is based on Newton's method which it works with an augmented Lagrangian function associated with the original problem. The function aggregates all the equality and inequality constraints. The first order necessary conditions for optimality are reached by Newton's method, and by updating the dual variables and the penalty terms associated with the inequality constraints. The proposed approach does not have to identify the set of binding constraints and can be utilized for an infeasible starting point. The sparsity of the Hessian matrix of the augmented Lagrangian is completely exploited in the computational implementation. Tests results are presented to show the good performance of this algorithm     

Topology error identification using normalized Lagrange multipliers

This paper introduces a method for topology error identification based on the use of normalized Lagrange multipliers. The proposed methodology models circuit breakers as network switching branches whose statuses are treated as operational constraints in the state estimation problem. The corresponding Lagrange multipliers are then normalized and used as a tool for topology error identification, in the same fashion as measurement normalized residuals are conventionally employed for analog bad data processing. Results of tests performed with the proposed algorithm for different types of topology errors are reported     

求解离散无功优化问题的精确连续化方法

提出了一种求解离散无功优化问题的新算法。该方法对离散变量进行二进制编码,从而将离散变量转化为若干0-1变量的线性组合,并将二进制变量约束转化为等价的互补约束,再利用非线性互补函数将互补约束转化为等价的非光滑方程,对其进行光滑化处理后,将原问题转化成可微的非线性规划问题,并采用非线性原对偶内点算法求解。此外,还对离散变量编码的原理以及应用原对偶内点法中遇到的求导难点进行了介绍,并通过几个不同规模试验系统的计算分析,验证该方法能有效处理离散变量,而且具有良好的收敛性和精确性。     

卫星通信系统中适用于16QAM信号的一种预失真方法

提出了一种用于卫星通信系统中由记忆功率放大器引起的16QAM信号非线性失真的高效数字自适应补偿方法。该方法使用通用记忆多项式模型和间接学习结构构造预失真系统,并使用QR分解递归最小二乘算法动态更新预失真参数。针对QR分解递归最小二乘算法不能直接更新系数矢量的问题,提出了一种称为＂权值冲洗＂的高效权值更新方法。计算机仿真结果表明该方法具有收敛速度快、系统性能稳定、补偿效果好等特点,并适合于大规模电路硬件实现。