A Mixed Integer Quadratic Programming Model for Topology Identification in Distribution Network
配电网拓扑识别的混合整数二次规划模型

Index Terms-Distribution networks, mixed integer quadratic programming, topology identification.
关键词-配电网络，混合整数二次规划，拓扑识别。

TOPOLOGY identification in distribution networks is a key function in state estimation. Traditional topology identification methods require redundant real-time measurements, which is difficult to achieve in distribution networks. There are few studies on topology identification in distribution network [1], [2]. Based on the works in [3] and [4], an MIQP based topology identification method is proposed, which works well with limited real-time measurements and pseudo-measurements data.
配电网络中的拓扑识别是状态估计的关键功能。传统的拓扑识别方法需要冗余的实时测量数据，这在配电网络中难以实现。关于配电网络拓扑识别的研究较少[1],[2]。基于文献[3]和[4]的工作，提出了一种基于 MIQP 的拓扑识别方法，该方法在有限的实时测量和伪测量数据下表现良好。

Let denote the state vector comprising bus voltage magnitudes, and real and reactive power flows in the branches. Also, let be the measurement vector; let binary variables represent the status of branch (either closed or open); and let be the weight of the th measurement. The objective of topology identification model is to minimize the weighted square of measurement residues:
设 表示包含母线电压幅值及支路有功和无功潮流的状态向量；同时，设 为量测向量；二元变量 表示支路 的状态（闭合或断开）；并设 为第 个量测的权重。拓扑辨识模型的目标是最小化量测残差的加权平方和：

Manuscript received June 09, 2014; revised August 28, 2014 and November 24, 2014; accepted December 29, 2014. Date of publication January 30, 2015; date of current version December 18, 2015. This work was supported in part by the National Key Basic Research Program of China (2013CB228203), the National Science Foundation of China (51477083). Paper no. PESL-00080-2014.
稿件收到日期：2014 年 6 月 9 日；修订日期：2014 年 8 月 28 日及 11 月 24 日；接受日期：2014 年 12 月 29 日。发表日期：2015 年 1 月 30 日；当前版本日期：2015 年 12 月 18 日。本工作部分受国家重点基础研究发展计划（2013CB228203）、国家自然科学基金（51477083）资助。论文编号：PESL-00080-2014。

The authors are with the Department of Electrical Engineering, Tsinghua University, Beijing 100084, China (e-mail: wuwench@tsinghua.edu.cn).
作者来自中国北京清华大学电子工程系，邮编 100084（电子邮箱：wuwench@tsinghua.edu.cn）。

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
本文中一个或多个图表的彩色版本可在线获取，网址为 http://ieeexplore.ieee.org。

Digital Object Identifier 10.1109/TPWRS.2015.2394454 where represents measurement functions; is the set of all real-time and pseudo measurements; and represent real and reactive power on line (power loss is omitted here); and represent the impedance on line represents the voltage magnitude of bus ; and represents the set of all nodes that are connected to bus .
数字对象标识符 10.1109/TPWRS.2015.2394454 中， 表示测量函数； 是所有实时与伪测量值的集合； 和 分别代表线路 上的有功和无功功率（此处忽略功率损耗）； 和 表示线路 的阻抗； 表示母线 的电压幅值；而 则代表与母线 相连的所有节点的集合。

Equation (3) is necessary to ensure the radial operation of the network, where is the total is number of buses and denotes the number of substations. Together with power balance constraints implied in the measurement functions, the network must be radial.
公式(3)是确保网络径向运行的必要条件，其中 表示总母线数， 代表变电站数量。结合测量函数中隐含的功率平衡约束，网络必须保持径向结构。

Equations (4) and (5) are the power balance constraints for zero injection bus [4]. Here, is the set of zero injection buses. represents a small value of load to ensure zero injection buses are not isolated.
式(4)和(5)为零注入母线的功率平衡约束条件[4]。其中， 表示零注入母线的集合。 代表一个小的负荷值，以确保零注入母线不会被孤立。

Equation (6) is the voltage and power constraint derived from Distflow equations [5]. The quadratic terms in (6) can be dropped and can be rearranged by introducing a large positive number to avoid conflict [3]:
式(6)是从 Distflow 方程[5]推导出的电压和功率约束条件。式(6)中的二次项可以省略，并通过引入一个较大的正数 重新排列以避免冲突[3]：

where . 其中 。

The equations for that relate measurements and states are:
与测量和状态相关的 方程为：

where and represent the real and reactive power measurements of load on bus , which are usually pseudo measurements; and represent the real and reactive power measurements on branch represents the square of the voltage magnitude measurement of bus ; and represents the measurement error vector.
其中， 和 分别表示母线 上负荷的有功和无功功率测量值，这些通常为伪测量； 和 表示支路 上的有功和无功功率测量值； 表示母线 电压幅值测量值的平方；而 表示测量误差向量。

To linearize the constraints, disjunctive constraints are used as follows:
为了线性化约束条件，采用了如下形式的析取约束：

Fig. 1. IEEE 33-bus test system.
图 1. IEEE 33 节点测试系统。

TABLE I 表一

MEASUREMENT PLACEMENT 测量位置

By introducing (14) and (15), and in (4), (5), and (9)-(12) can be placed by and .
通过引入(14)和(15)，(4)、(5)以及(9)-(12)中的 和 可替换为 和 。

Finally, the model in (1) (6) can be revised to an MIQP model with linear constraints:
最后，模型（1） （6）可被修正为具有线性约束的 MIQP 模型：

This model can be solved using an available commercially optimization software such as CPLEX.
该模型可利用如 CPLEX 等现有的商用优化软件进行求解。

As shown in Fig. 1, the IEEE 33-bus network is a system including a substation and 37 branches [5]. The normally closed branches are shown in solid lines and normally opened branches are displayed by dashed lines.
如图 1 所示，IEEE 33 节点网络是一个包含一个变电站和 37 条支路的系统[5]。通常闭合的支路以实线表示，而通常打开的支路则用虚线显示。

The measurements were assumed normally distributed as follows:
假设测量值呈如下正态分布：

where denotes measurement vector; is obtained using the distribution system load flow; ; and is the variance of the th measurement. For a given maximum measurement error (denoted as ) in percentage about the mean , the standard deviation of the error can be computed as follows [6]:
其中， 表示测量向量； 是通过配电系统潮流计算得到的； ；而 是第 次测量的方差。对于给定的最大测量误差（以平均值 的百分比表示为 ），误差的标准偏差可按如下方式计算[6]：

Define the accuracy rate for result of th identification as
将第 次识别结果的准确率定义为

where is the total potential independent loops, for the 33-bus system, there are 5 looping branches, so denotes the number of branches between wrong opened branch and the real branch in loop denotes the number of branches in loop .
其中， 表示总的无潜在独立回路，对于 33 节点系统，存在 5 个回路分支，因此 表示错误断开分支与实际回路分支之间的分支数， 表示回路中的分支数， 表示回路中的分支数。

The measurement placement is shown in Table I. The measurement weight is set to .
测量位置如表 I 所示。测量重量 设定为 。

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