三相不平衡电容器配置

附录1：外文资料翻译A1.1 不平衡电力系统电容器设置摘要—本文提出一个针对三相不平衡的电力系统采用的电容器设置方法。

这种方法不仅使功率损失和电容器费用降到最小，而且使当前电力系统中谐波引起的畸变降到最小。

提出的方法是在平衡的和不平衡的操作条件下都能实现这个目标。

当不平衡的系统接近于由他们的正向序列单相等值时，本文的一个目标就是讲述在电容器设置研究结果上的一些重大区别。

此外，还讲述了在电容器设置中考虑谐波畸变的作用。

并且提供了配电测试电力系统的数字例子来说明此方法。

关键词：优化，电容器设置，损失最小化，谐波畸变，不平衡操作，配电系统。

1绪言配电系统在各个地点都安装有电容器，为了获得期望的电压波形，合适的功率因素和减少馈线功率损失。

当处理一个包含几条馈线和他们旁路的大规模配电系统时，决定这些电容器的最佳安装地点和安装容量成为一个复杂的优化问题。

除此之外，还有其他问题需要说明，例如电容器大小、电压和馈线负载的运行限值。

针对平衡的配电馈线的有效解决方法已经被开发了[1,2]。

这些解决方法主要运用于公式化问题中的正向序列网络模型和连带的功率流动。

因此，结果不能直接运用在包含缺相馈线的系统中，不对称负载的馈线或者单相或两相馈线的电容器组。

三相不平衡的配电系统将在[3,4]中研究，其中模拟退火算法和遗传算法分别用于解决这个更加复杂的问题。

在[5]中，一种被简化的公式和MINOS优化包裹用于解决同一个问题。

最近，配电系统中存在由非线性负载和控制设备产生的不需要的谐波。

对安装有电容器的配电网，谐波会导致过电压。

在[6]中提出了这个问题，并且介绍了一种使谐波过电压最小化的方法[6]。

一种避免汇合问题和合并电容器的分离属性以及安装电容器组电压畸变的实用方法，在[7]被开发并且被提出。

这种方法针对三相平衡的操作条件并且仅能分析正向序列网络。

在本文，[7]中讲述的内容将延伸到更加普遍的三相不平衡的操作情况下。

几条配电馈线分为几段，混有单相、两相和三相负载。

这样的系统和那些含有三相不平衡负载的系统一样，可以用本文当前的方法研究。

除损失和电容器设施之外的费用，还有就是谐波畸变引起的费用，将在[8-9]中讨论。

因此，问题被公式化，在这种情况下网络损失和谐波还有电容器的设置费用一起减到最小。

本文首先提出问题说明。

然后描述了三相功率流动和线性谐波分析模块的细节，这部分组成了主要算法。

其次是采用开发的程序和测试系统得到的仿真结果。

最后一部分提出了结论和对未来工作的展望。

2问题的描述当前方法的目标是确定最佳的地点和每相电容器组的大小，使得电容器的总成本，网络的总功率损失和网络的谐波畸变减到最小。

这个目标的实现决定于网络三相功率流动值，母线电压和电流强度的极限值，以及谐波指数和安装的电容器组的总数量。

因此，它可以被公式化作为以下优化问题：最小化J(X,U) = J C + J L + J IH条件电压，电流限制电容器数量限制目标函数和限制条件在下面被定义。

2.1 目标函数， J目标函数假设一带有特定负载的系统。

如果能把这个函数延伸为一类函数，那么带有任意负载的系统都可以计算，这个在这里不讨论。

所有真仿结果和关于提出的算法的讨论根据特定的负载假设。

然而这种引伸能不是困难的被合并到被提出的算法里。

组成目标函数的三个术语下述：2.1.1 电容器的费用，J C = C C T⋅UC c: 每条母线上电容器组的费用系数U : 每条母线上电容器组的介质系数2.1.2 损失的费用，J L= C L⋅(P G− P D)TCL: 每单位能量损失的费用P G: 在 (X,U)中，总的发电功率P D: 总负载T : 是损失期间(X,U): 是对应于安装的电容器介质系数U的功率流动解决方法。

根据需要损失的费用也很容易被合并。

2.1.3 畸变的费用，J IH畸变费用的评估由[8，9]的作者首先研究。

这些费用被认为是运行费用和老化费用的总和。

运行费用指的是谐波造成的增加损失的费用，老化费用指的是谐波造成的组成部分过早的老化的增加费用。

细节可以在[8，9]找到。

2.2 限制条件在基频条件下，每种方法应该满足三相功率平衡等式。

因此，一种满载的三相功率流动解决方法将用来核实这个限制。

这种解决方法也用来检测在基频下母线电压和线路电流的极限侵害值。

每个电容器单元对母线电压畸变的影响都必须被检测。

这通过解决一个线性三相谐波分析问题完成，所有非线性负载通过他们的谐波电流来表示。

这计算细节在下一部分讨论。

3解决方法上面被描述的优化问题是通过使用一个简单，并且有效的做法解决的，在[7]中，这种方法被开发并成功地应用于单相问题。

该做法背后的主导思想是根据增加大小分离电容器组的连续设置。

它假设，所有母线的任一相在每个优化步骤上可以设置一台增加分离电容器。

如果电容器装设在母线上，为禁止的唯一母线，则这种情况可容易地强行把那条母线从母线名单中分离出来。

此外，如果不同的母线上有不同大小的单位电容器组，单位电容器组的设置能相应地修改以适合每条母线。

所以，与电容器装置的分离属性和适合不同母线设置的分离单位的同一性相关的物理限制，能够自然地计算，不需要任何复杂的逻辑。

在优化做法的每一步，三相功率流动解决方法和对应于每一条母线上单位电容器组附加的母线电压谐波，都必须计算。

注意，这些计算的目的是为了获得和比较所有可能的增加设施的目标函数的价值和选择减少目标函数最多的那个。

也需要注意到，通过目标函数比较在不同母线上单位电容器设置的效果的目的，近似三相功率流动解决方法可以使用。

一旦某个候选被选择，然后一种准确的解决方法可以为选择的配置获得。

一种快速但近似三相功率流动的解决方法在第一部分被实施。

进行三相线性谐波分析是为了畸变的演算。

这些下面将详细讨论。

3.1 最新的快速三相功率流动每次在系统总线上增加一个单位电容器，就会增加对三相功率流动解决方法的影响。

这个增加的变动可以通过它的第一次命令近似值来表达功率流动解决方法来获得。

考虑三相功率流动值通过：f(X,U) = 0 (1)第一命令泰勒算法是：[F x (X0,U0) ] [ dX ] = - [ F u(X0 , U0) ] dui (2)注意du i 是增加在结点i的单位电容器，母线的一相，并且函数Fx和Fu代表与X和U有关的f (X，U)的梯度。

在优化的每一步，最近工作点由(X0,U0)表示，它取决于优化进行到那点的方式。

得到新的功率流动解决方法如下：X’ = X0 + dX (3)这个做法将根据结点的数量重复许多次，除了那些没有允许设置电容器的地方。

在针对所有情况计算目标函数J (X’， U)时，涉及到的目标函数最少的解决方法将被选择。

得到一种满载的三相交流功率流动的解决方法，并且所有的运行限制将被检测。

万一这种解决方法违犯了任何一个限值，这个做法将被作为第二个最佳解决方法重复一遍。

这个做法将继续，直到获得一种可行的解答。

如果找不到可以满足所有限制的解答，则优化做法将被终止。

否则，它将进入重复的下一步优化。

3.2 畸变的演算与基频功率流动解决方法同步，需要一种针对网络中当前高次谐波的解决方法，以便评估电力系统中谐波对应的影响。

这通过解决指定的网络的线性谐波等式完成：[Yn] [Vn ] = [In ] (4)[Yn]，[Vn]和[I n]是三相网络矩阵，以第n个谐波频率评估的母线电压向量和独立电流源向量。

矩阵随着母线上单位电容器的增加而修改。

并且需要注意到，非线性设备的谐波介入根据IEEE PES工作组的推荐来建模。

需要编辑一个基于Matlab的程序来评估上面描述的针对一般三相不平衡系统的做法。

此时，程序应用在单相带负荷的情况，并且电容器设置仅在电容器类型固定的情况下可以被测试。

然而，可以随季节性或每日母线负载的变化容易地修改。

4结论本文提出一种针对三相不平衡电力系统的电容器设置方法。

提出的方法不仅节省了系统损失和电容器费用，而且减小了母线电压谐波畸变和不平衡运行的影响。

在配电网中非线性负载最近的扩散导致谐波污染成为一个重要的问题，要求我们在配电自动化中要考虑电容器大小和装设地点。

本文提出一种简单并且有效的解决方法。

在平衡的和不平衡的运行条件下提供了展示做法的有效率的数字例子。

未来工作将涉及更多目标函数中介入谐波费用的可行的方法。

译自 14th PSCC , Sevilla , 24-28 June 2002A1.2 Capacitor Placement In Unbalanced Power SystemsAbstract– This paper presents a capacitor placement method for three phase unbalanced power systems. The method aims to minimize not only the power losses and capacitor costs, but also the distortions due toharmonics present in the power system. The proposed method is capable of accomplishing this objective for both balanced and unbalanced operating conditions. One of the objectives of this paper is to demonstrate the significant differences in the results of capacitor placement studies when unbalanced systems are approximated by their positive sequence single phase equivalents. Furthermore, the effects of taking harmonic distortions into account during the capacitor placement procedure are also demonstrated.Numerical examples on a distribution test power system are provided to illustrate the method.Keywords: Optimization, capacitor placement, loss minimization, harmonic distortion, unbalanced operation, distribution systems.1INTRODUCTIONPower distribution systems contain shunt capacitors at various strategic locations in order to maintain a desired voltage profile, correct power factor and reduce power losses along feeders. When dealing with a large scale distribution system containing several feeders and their laterals, deciding on the best locations and sizes of these capacitors becomes a complicated optimization problem. In addition to the scale of the problem, there are other issues such as the discrete nature of capacitor sizes, operational limits on voltages and feeder loadings, that need to be addressed. Effective solution algorithms for balanced distribution feeders have been developed [1,2]. These solutions mainly utilize the positive sequence network model and the associated power flows in formulat ing the problem. Hence, the results do not directly apply for systems containing feeders with missing phases, unevenly loaded feeders or shunt capacitors on single or double phase feeders. Three phase unbalanced distribution systems are later studied in [3,4] where simulated annealing and genetic algorithms are respectively used to solve this more complicated problem. A simplified formulation and the MINOS optimization package are used to solve the same problem in [5]. Recently, distribution systems are populated with nonlinear loads or control devices that generate unwanted harmonics in the systems. Harmonics are known to cause overvoltages under certain network configurations involving shunt capacitors. This issue is raised and a solution is proposed for minimizing harmonic overvoltages in [6]. A practical method that avoids convergence problems and incorporates discrete nature of capacitor banks along with the voltage distortions due to the installed capacitors, is developed and presented in [7]. This method is based on the balanced three phase operation and therefore analyzes the positive sequence network only.In this paper, the work reported in [7] will be extended to the more general case of the three phase unbalanced operation. Several distribution feeders are known to have line sections carrying a mixture of single, double or three phase loads. Such systems as well as those having full three phase but unbalanced loads can be studied using the presented method in this paper.In addition to the cost of losses and capacitor installations, it is possible to associate a cost with the harmonic distortions as discussed in [8-9]. Hence, the problem is formulated in such away that both network losses and harmonics are minimized along with the cost of capacitors placed for this purpose.The paper is organized such that the problem description is presented first. This is followed by the sections describing the details of the three phase power flow and linear harmonic analysis modules, which make up t he main computational engines of the overall algorithm. Simulation results obtained using the developed program and a test system will be presented next.Conclusions and suggestions on future work will be presented in the final section.2DESCRIPTION OF THE PROBLEMThe objective of the presented method is to determine the best locations and sizes of shunt capacitors for each phase, so that the total cost of the capacitors, of the total power losses of the network and the harmonic distortion of the network are minimized. This objective should be met subject to the network three-phase power flow equations as well as the limits on the bus voltage and current magnitudes, harmonic indices (HI) and the total number of capacitor units to be installed. Hence, it can be formulated as the following optimization problem: Minimize J(X,U) = J C + J L + J IHSubject to Three-phase PF Eq.s atfundamental and harmonicsV, I – limitsHI limitsLimits on number of cap.swhere the terms of the objective function and constraint list are defined below.2.1 The Objective function, JThe objective function assumes a given loading level for the system. While it is possible to extend this function to a sum of similar functions, so that a number of loading levels can be accounted for, this will not be done here. All simulation results and the discussion of the proposed algorithm will be based on single loading assumption. Such an extension can however be incorporated into the presented algorithm without much difficulty. The three terms that make up the objective function are described below:2.1.1 Cost of Capacitors, J C = C C T⋅UC c: is the cost vector for capacitor units at each bus.U : is the vector of capacitor units placed at each bus.2.1.2 Cost of Losses, J L= C L⋅(P G− P D)TCL: is the cost of a per unit energy lossP G: is the total generation at (X,U)P D: is the total loadT : is the loss duration(X,U): is the power flow solution corresponding to the installed capacitor vector of U.The cost of demand lost can be also incorporated without difficulty.2.1.3 Cost of Distortions,J IHThe evaluation of the cost of the distortions is studied first by the authors in [8, 9]. These costs are assumed to be the sum of the operating costs and the aging costs.The operating costs refer to the costs of the incremental losses caused by the harmonics and the aging costs refer to the incremental costs due to the premature aging of the components caused by the harmonics. Details can be found in [8, 9].2.2 The ConstraintsEach solution should satisfy the three-phase power balance equations at the fundamental frequency. Thus,a full three-phase power flow solution is to be run to verify this constraint. This solution will also be used to check the limit violations at fundamental on bus voltages and line currents. The effect of each capacitor unit on the distortions of the bus voltages will have to be also checked. This is accomplished by solving a linear three-phase harmonic analysis problem where all nonlinear loads are represented by their harmonic current injections. Details of this computation are discussed in a later section.3SOLUTION METHODThe above described optimization problem is solved by using a simple yet effective procedure, which is developed and successfully employed in solving the single phase problem in [7]. The main idea behind the proposed procedure is based on sequential placement of discrete capacitor units of incremental size. It is assumed that each phase of any bus can be placed an incremental discrete capacitor at each optimization step (three-phase bank in case of three-phase phase bank otherwise). If capacitor installations are bus, single- prohibited for any of the buses, then this condition can easily be enforced by leaving out that bus from the bus list. Furthermore, if different buses have different sizes of unit capacitors, then placement of unit capacitors can be accordingly modified to suit each bus. Therefore, the physical constraints that relate to the discrete nature of capacitor installations as well as the nonhomogeneity of the discrete units available for installations at different buses, are naturally accounted for without any complex logic.At each step of the optimization procedure, the three phase power flow solution as well as the harmonics of the bus voltages corresponding to the incremental addition of unit capacitors at each possible bus, will have to be computed. Note that, the purpose of these computations is to obtain and compare the values of the objective function for all possible incremental installations and choose the one that reduces the objective function the most. It is also noted that for purposes of comparing effects of unit capacitor installations at different buses on theobjective function, approximate three-phase power flow solutions can be used. Once a candidate is selected, thenan accurate solution can be obtained for the selected configuration only. A fast but approximate three phase powerflow solution is implemented for the first part. A three phase linear harmonic analysis is carried out for thecalculation of distortions. These are discussed in more detail below.3.1 Fast Three-phase Power Flow UpdateThe three-phase power flow solution is affected incrementally each time a unit capacitor is added at a systembus. This incremental change can be captured by expressing the power flow solution by its first orderapproximation. Consider the three phase power flow equations given by:f(X,U) = 0 (1)The first order Taylor approximation will be:[F x (X0,U0) ] [ dX ] = - [ F u(X0 , U0) ] dui (2)Note that du i is the unit capacitor added at node i, one phase of a bus and the functions F x and F u represent the gradient of f(X,U) evaluated with respect to X and U respectively. At each optimization step, the most recentoperating point will be denoted by (X0,U0) which will depend upon the way the optimization proceeded up till thatpoint. The updated power flow solution will then be obtained as:X’ = X0 + dX (3)This procedure will be repeated as many times as the number of nodes excluding those where no capacitors are allowed to be placed. Upon computing the objective function J(X’,U) for all cases, the solution which yields thesmallest objective function will be selected. A full three phase AC power flow solution will be obtained and alloperating constraints will be checked for this case. In case the solution violates any one of the limits, the sameprocedure will be repeated for the second best solution. This procedure will continue until a feasible solution is obtained. If no solution can be found satisfying all the constraints, then the optimization procedure will beterminated. Otherwise, it will proceed to the next optimization iteration.3.2 Calculation of distortionsIn parallel with the fundamental frequency power flow solution, a solution for each dominant harmonic presentin the network will have to be obtained so that the corresponding effects of the power system harmonics can be evaluated. This is accomplished by solving the network’s linear harmonic equations given by:[Yn] [Vn ] = [In ] (4)where [Y n], [V n ] and [I n ] are the three phase network admittance matrix, bus voltage vector and independentcurrent source vector evaluated at the n-th harmonic frequency. The admittance matrix is modified each time a unit capacitor is added at a bus. Also note that, the harmonic injections of nonlinear devices are modeled according tothe recommendations of the IEEE PES Working Group on Harmonics Modeling and Simulation [10-11] where itis suggested that the phase angles of the injected harmonic current sources be adjusted according to the phaseangle of the fundamental with respect to reference.A Matlab-based program is developed in order to evaluate the above described procedure for general three phase unbalanced system operation. At this time, the p rogram is implemented for a single loading level and therefore the capacitor placement can be tested only for fixed capacitor types. However, it can be easily modified to account for seasonal or daily load variations of the bus loads.4CONCLUSIONSThis paper presents a capacitor placement procedure for three phase unbalanced power systems. 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