Graph-Theory Based Optimal PMU Allocation Considering ZIB Effects

Phasor Measurement Unit (PMU) has called forth a revolution in power system monitoring, control and security study after being invented in mid-1980’s by Professor Arun G. Phadke. This measuring device is very efficient and accurate to provide the time synchronized measurements of voltage, current, frequency and rate of change of frequency of a particular location using signals from Global Positioning System (GPS). Wide Area Monitoring System (WAMS) based PMU is used to monitor systems for Supervisory Control and Data Acquisition (SCADA) because of its reliability and accuracy [1, 2].

Modern power grid needs to monitor ceaselessly. The value of electrical quantities should lie between the accepted limits to ensure the secure and normal operation of electric power systems. This is done by state estimator tool on a completely observable power system. For a completely observable system, all the bus voltage phasors and current phasors are assessable. PMU based complete monitoring system provides fast and accurate measurements of the system elements. A single PMU can observe several buses. Hence, to achieve the complete observability the required number of PMUs are less than the number of buses. As the PMU is an expensive device, the wise placement of this measuring as well as monitoring device can help to minimize the overall power system monitoring cost [3].

From the previous decade, a substantial amount of research on Optimal PMU Placement (OPP) problem solution has been done in literature. Multi-objective PMU placement has been carried to monitor the electric transmission grid by Mazhari et al. [4]. An optimization technique named Cellular Learning Automata (CLA) has been used to solve the problem. A modified edition of another optimization technique named Binary Cuckoo Optimization Algorithm (BCOA) has been used by Dalali et al. [5]. A common Integer Linear Programming (ILP) by Xu et al. [6] and Binary Particle Swarm Optimization (BPSO) was used to solve the OPP problem [7]. These optimization techniques provided better solutions than that of other contemporary methods. However, optimization techniques take significant amount memory space and lots of iterations are involved, and consequently, these techniques are time consuming.

A heuristic approach is used by Roy et al. [8] where the connectivity information of the complete system configuration directs the optimal PMU locations. A Multi-Criteria Decision Making (MCDM) approach to solve OPP problem has been used by Sodhi et al. [9]. The combination of graph theory and Analytical Hierarchy Process (AHP) has been used for the first time to solve this problem [10]. Coherent generator groups are selected to monitor the system by Rios et al. [11]. Graph theory has been used here by the authors. However, graph theory which has not been used in this area extensively. Typically, a graphical representation of an electrical network helps to analyze it deeply [12, 13]. In this paper, some important graph-theoretic terms have been used to identify the strategic PMU placement locations.

The complete organization of the paper has been depicted as follows. Section 2 describes the power system observability types and bus wise observability rules. Then the next section deals with the detailed description of the proposed method. Section 4 analyses the results, and finally, the section 5 concludes the paper.

Graph theory is very useful tool to analyze the power system networks. To achieve the complete the complete observability, the initial target of the method is to form a superior spanning tree containing all the buses of the system. Then the strategic places for installing PMUs throughout the system either for normal operating condition or for considering the effects of Zero Injection Buses. The complete proposed method has been described below-

3.1 Spanning tree formation

Spanning tree formation is the essential part of the proposed method. Only the proper and efficient spanning tree leads the identification of strategic locations for PMU placement. Steps which are followed to form the spanning tree are given below.

Step 1: Selection process starts from bus number 1.

Step 2: The next bus will be selected through the branch of having the lowest valued already selected bus.

Step 3: Stop if all the buses are selected, otherwise go step 2.

Now, the modified configuration of a system contains N number of branches, where N is the number of buses of the test system.

3.2 Optimal PMU placement for normal operating condition from spanning tree

The objective of the proposed method is to achieve the complete observability using the minimum number of PMUs. Optimal PMU placement for complete observability is done as follows-

$B_{i}=\sum_{i=1}^{N} S_{i j} \cdot D_{i}$     (1)

where, the objective is-

$\operatorname{minimum} \sum_{i=1}^{N} D_{i}$      (2)

$D_{i}=\left\{\begin{array}{l}

1, \text { if PMU device is installed at bus } i \\

0, \text { otherwise }

\end{array}\right.$     (3)

and $S_{i j}$ is the network connectivity matrix of the spanning tree whose entries are defined as-

$S_{i j}=\left\{\begin{array}{l}

1, \text { if } i^{\text {th }} \text { node is connected to } j^{\text {th }} \text { node } \\

1, \text { if } i=j \\

0, \text { otherwise }

\end{array}\right.$       (4)

For i, j=1, 2 …… N.

For normal operating condition of the given power system network the strategic locations of the PMUs from the previously formed spanning has been conducted through the following steps-

Step 1: Select the radial buses of the modified configuration of the system.

Step 2: Select the buses which connected to the radial bus and include in a set S. Sort the elements in ascending order according to the number of radial bus connectivity.

Step 3: According to rule 2, install PMU at the buses of set S.

Step 4: Declare the final selected buses for PMU installation are the final solution.

3.3 Optimal PMU placement considering ZIB

Considering the ZIB effects helps further to minimize the required number of PMUs.

ZIBs are removed from the set S and included in a new set Z.

Step 1: Install PMU at the bus elements of S and check the extra observability of the system using rule 2.

Step 2: Check complete observability of the system. If the system is completely observable declare the selected buses as OPP set. Otherwise, select the first element of set Z for PMU placement.

3.4 Percentage of PMU equipped buses

n this section, the percentage of PMU equipped buses has been calculated and checked what percentage of PMU buses is required to make the system completely observable.

$\text { PMU equipped buses }=\frac{\text {Required number PMUs}}{\text {Total number of buses}} \times 100 \%$