Optimal µ-PMU Placement and Voltage Estimation in Distribution Networks: Evaluation Through Multiple Case Studies

Abstract

This study optimizes the placement of μ-PMUs using the BPSO and BGWO algorithms for the IEEE 33-bus and 69-bus systems, with a focus on minimizing deployment costs while ensuring robust system observability. Three case studies are analysed: Case 1 (normal conditions), Case 2 (single μ-PMU outage), and Case 3 (Zero Injection Buses, ZIBs). In Case 1, both algorithms identified 24 μ-PMUs as the optimal placement for the IEEE 69-bus system, achieving the minimum PMUs required for full observability. For Case 2, redundancy requirements increased the μ-PMU count to 24 μ-PMUs for the IEEE 33-bus system and 51 μ-PMUs for the IEEE 69-bus system, ensuring full observability even under a single μ-PMU failure. Case 3, leveraging Zero Injection Buses (ZIBs), reduced the μ-PMU count to 20 μ-PMUs for both BPSO and BGWO, optimizing the system configuration while maintaining observability. A trade-off analysis was performed to examine the trade-off between redundancy and PMU count, showing that increasing the number of μ-PMUs improves system resilience. Voltage and current channels were measured from the optimized placements to ensure accurate voltage measurement in all case studies. Subsequently, the Weighted Least Squares algorithm was applied for voltage estimation, serving as a peripheral to the main objective of the optimal μ-PMU placement. Voltage estimation was conducted under three noise levels: 0.01 STD for basic analysis and 0.02 and 0.04 STD to observe the impact of varying measurement noise. The results highlight that higher μ-PMU placements improve voltage estimation accuracy, particularly under higher noise levels. Statistical analysis confirms that BGWO outperforms BPSO in terms of computational efficiency, stability, and convergence, especially in large-scale systems. By enhancing grid monitoring and state estimation, this research directly contributes to the development of more resilient and efficient power networks, which is a fundamental prerequisite for integrating renewable energy sources and advancing overall power system sustainability. This research emphasizes the balance between cost and reliability in μ-PMU placement and provides a comprehensive methodology for state estimation in modern power systems.

1. Introduction

The Distribution Networks (DNs) have become more prone to transients due to the increasing inclusion of renewable generation, distributed energy sources (DERs), electric vehicles (EVs), and demand-side management (DSM) technologies [1,2]. Although these developments promote sustainability and efficiency, they are also increasing the complexity, variability, and uncertainty of electrical distribution networks. Traditional monitoring and control techniques, which have largely been designed for transmission networks, are generally inadequate for the distribution level, where uneven loads, radial topologies, and prosumers pose significant challenges due to bidirectional energy flows. Supervisory control and data acquisition (SCADA) systems are employed for monitoring and controlling the power networks. However, the latency in SCADA systems, the increased complexity of DNs, and unsynchronized data measurements from SCADA systems hinder the correct state estimation of DNs [3,4,5]. The global transition to sustainable energy is fundamentally reshaping electrical grids, requiring them to accommodate distributed generation and improve operational efficiency. Achieving these sustainability goals hinges on superior grid observability and control. In this context, the strategic deployment of μ-PMUs emerges as a foundational step, providing the high-fidelity data necessary to reduce losses, enhance stability, and facilitate higher penetration of clean energy sources. An essential requirement for addressing these issues, situational awareness has become a crucial requirement for the reliable operation of distribution systems. The real-time measurement and monitoring of power systems provide an accurate understanding of the state of the power system for its smooth operation [6,7]. One of the most effective technologies that enables high-level monitoring and situational awareness in modern networks is the micro-phase measurement unit (µ-PMU). It derives from the commonly used phasor measurement units (PMUs) that are utilized in transmission systems; µ-PMUs provide high-precision measurements of voltage and current phasors at the distribution level, with precise timing made possible by GPS synchronization in microseconds. Compared to traditional SCADA systems, µ-PMUs offer superior temporal resolution and accuracy, enabling advanced functions such as fault detection, topology identification, state estimation, and the integration of DERs. Despite these benefits, the installation and maintenance costs of µ-PMUs are a barrier to their implementation, making it impossible to equip every bus in the distribution system with the devices. As a result, the optimal placement of µ-PMUs (OµPP) has emerged as an important research field. The objective of OµPP is to determine the minimum number of µ-PMUs and their best locations that ensure observability, reliability, and resilience, while minimizing costs. Finding the best location is a complex mixed optimization problem, especially in large distribution systems where the network configuration is radial and a greater number of measurements is required for robustness against noise, data errors, or device failures [7,8,9,10,11,12]. Over the years, various optimization techniques have been used, which are mainly divided into mathematical programming methods and heuristic/metaheuristic algorithms. Evolutionary algorithms have been extensively employed in the literature to solve multiple optimization problems pertaining to different engineering fields. The placement of PMUs has been carried out on the transmission level by using these optimization algorithms; however, the distribution systems still require extensive exploration [13,14]. In the preliminary studies, the determination of the optimal location of the PMU is primarily carried out using mathematical programming formulations, such as integer linear programming (ILP), mixed-integer linear programming (MILP), and nonlinear programming (NLP). These methods are appealing because they can provide mathematically correct solutions under well-defined constraints. For example, ILP considers the observation as a set covering problem, where binary decision variables indicate whether a PMU is installed at a bus or not. MILP expands this setup to account for additional aspects such as redundancy and cost functions. Although these methods are theoretically rigorous, they have significant limitations in their practical application. First, the solution space grows combinatorially with the size of the system, making ILP and MILP computationally infeasible for large-scale systems such as IEEE 69 or 118 bus networks [15,16,17]. Second, and more importantly, these methods often rely on theoretical assumptions of perfect measurements. For example, mathematically, a minimal solution may not provide enough additional features, which can result in a loss of observability if a µ-PMU fails. Due to these challenges, researchers have adopted heuristic and metaheuristic algorithms. Unlike exact mathematical programming, meta-optimization algorithms aim to find practically optimal solutions in a reasonable time, even in large, nonlinear, and non-convex search spaces. Their main advantage is that they are flexible, as they can easily integrate practical constraints, such as redundancy requirements and equipment costs, while maintaining scalability for large systems. Furthermore, they are less likely to generate misleading results, as they rely on population-based or adaptive search mechanisms that better capture the irregular nature of systems [18,19]. The importance of optimal placement goes beyond just observation capability. The correct voltage estimation is one of the most important outcomes of implementing µ-PMU. State estimation is the foundation of system monitoring, allowing operators to know the complete bus voltages that describe the operational state of the system. Conventional distribution state estimation methods heavily rely on measurements obtained from SCADA data and load forecasts, both of which face low accuracy and limited resolution. In contrast, µ-PMU provides real-time, high-resolution phase data, significantly enhancing the reliability of estimation. However, due to cost limitations, µ-PMUs cannot be deployed everywhere, and their effectiveness mainly depends on the placement strategy [20,21,22]. When placed optimally, they ensure that critical buses are directly measured, while others can be evaluated using the network’s equations and constraints. Placement strategies that consider redundancy further enhance resilience, allowing for error detection and accuracy in the presence of noise or incorrect measurements. The weighted least squares (WLS) [23] method is generally used to estimate voltage, and studies have consistently shown that, in the case of better installation locations, WLS estimates are close to the results of reference power flow, where typically the errors are less than 0.1 p.u. This demonstrates that the accuracy of voltage estimates not only depends on the estimation algorithm but is also closely related to where and how the µ-PMUs are installed. This study builds on these insights by investigating OµPP in 33- and 69-bus distribution systems using BPSO and BGWO under three scenarios: normal operating conditions, a µ-PMU failure, and the inclusion of ZIBs. A detailed statistical analysis is also performed to validate the effectiveness of proposed algorithms based on various parameters. The performance of the algorithms is evaluated in terms of convergence behavior, computation time, and scalability. A pareto front analysis between number of PMUs and SORI is presented to establish the understanding that SORI improves by increasing the number of PMUs; however, it also incurs extra cost, hence, establishing a trade-off between number of PMUs and observability is necessary. Furthermore, the placement is validated by employing WLS algorithm for state estimation (voltage magnitude and angles) to determine the effectiveness of the locations through voltage estimates under different noise levels. The main contributions of this article are as follows:

• Two state-of-the-art optimization techniques are utilized to determine the OPP while considering practical constraints. A single µ-PMU outage and ZIBs are considered practical constraints for a comprehensive study.
• ZIBs aid in reducing the required number of µ-PMUs while ensuring observability; hence, strategically placing µ-PMUs by leveraging ZIBs reduces µ-PMUs. Therefore, this condition is explored in this study to decrease the number of µ-PMUs and hence the overall placement cost. When a single µ-PMU fails to operate at a specific bus, it may cause a loss of critical data. Therefore, a case study examining a single µ-PMU failure and its potential impact on cost is also considered.
• After the placement, the WLS algorithm is employed for voltage estimation to ensure that strategically installed µ-PMUs measure the voltage estimation correctly for all case studies.
• A detailed statistical analysis is performed to determine the robustness of proposed algorithms, and a pareto front analysis between SORI and number of PMUs is performed to determine the impact of PMU count on system observability.
• Different noise levels are introduced via a change in standard deviations (STDs) to simulate more realistic conditions for the presence of noise in the system to evaluate the impact of noise levels on the state estimation.

The rest of the article is arranged in this manner: Section 2 provides a comprehensive Literature review related to the problem. Section 3 briefly discusses the methodology along with mathematical formulation of problem. Section 4 provides the results of simulations, while Section 5 provides a pareto front analysis of the study. Section 6 discusses voltage estimation through WLS and its comparison with true voltages obtained from power flow analysis. Section 7 provides a comprehensive discussion, and Section 8 concludes the article.

2. Literature Review

Researchers have used various techniques to arrive at the OPP, which range from analytical and mathematical programming methods to heuristic and metaheuristic methods [24]. In ref. [9] Integer linear programming (ILP) is employed for optimal PMU placement (OPP), considering some practical constraints like ZIBs for transmission networks. The authors established that the number of PMUs increases as the topology of the power network changes. In ref. [25] Genetic Algorithm (GA) is employed to achieve the full observability of the IEEE 57 bus system. A smaller number of PMUs is reported while ensuring maximum observability. The authors in ref. [26], employed a greedy search algorithm to obtain a smaller number of PMUs while ensuring maximum observability. In ref. [27], a network compression method is proposed, and a tabu search algorithm is used to reduce the number of PMUs while ensuring the maximum observability. Multiple Optimization techniques can also be hybridized to harness the computational strengths of these approaches, enabling more effective solutions to multiobjective optimization problems. In ref. [28] a hybrid PSO-GSA algorithm is proposed to solve the OPP problem while ensuring full observability. Different practical constraints are considered, and the proposed technique is employed on different case studies. In ref. [29] OPP problem is formulated considering PMU channel cost and risk of operation (RoOP) and solved using ILP. ZIBs are also considered in the formulation, and full observability is ensured. In ref. [30] a new graph theory-based OPP algorithm is proposed to resolve scalability issues while ensuring full observability, taking into account practical constraints like ZIBs. Recently, some studies have focused on PMU installation strategies that are clearly more accurate for voltage estimation in active distribution networks. Salehi et al. proposed a framework based on accuracy in the UK 77-bus test system in ref. [31], in which a multi-objective genetic algorithm (MOGA) was used. Their method reduces the error between actual and estimated voltage profiles, taking cost constraints into account. The results showed that, although a reduced number of PMUs, if placed strategically, can reduce estimation errors to less than 0.5 p.u. Additionally, the study highlighted the importance of additional PMUs, as they mitigate the detrimental effects of erroneous or corrupt input data. Some other contributions have emphasized uncertainty quantification and robustness. Romero et al. in ref. [32] employed a Bayesian optimal experimental design (OED) framework in reference, which directly incorporated the uncertainty of the grid parameter into the placement problem. Their results in the IEEE 14 and 118 bus systems showed that the best design significantly reduced the estimated discrepancy compared to random locations. Building on this, Peng et al. in ref. [33] developed a mixed-integer semidefinite programming (MISDP) model for µ-PMU placement, which was solved through an improved boundary division technique. Their results in IEEE systems showed the overall improvement and scalability of this approach, surpassing both the accuracy and computation duration of conservative and convex techniques. Mishra and de Callafon in ref. [34] extended the problem to networks with partially known admittance parameters, proposing a method that guarantees observability and minimizes noise-induced variance. Their method, which was tested on the 14-bus IEEE network, showed a 10% better performance compared to unconventional designs. Peng et al. in ref. [35] employed the greedy algorithm–based method incorporating information entropy and Monte Carlo–based fault simulations to evaluate PMU placements. Simulated studies revealed significant improvements, with the estimated state error reduced from 1.6% to 0.2% and the loss of observation remaining above 90%. This demonstrates that focusing solely on observation, without considering loss dynamics or redundancy, leads to ineffective solutions. With these works, mixed-integer semidefinite programming methods were also proposed to rigorously include completely loose conditions, which improved the accuracy of state estimates, but require large computational needs. In ref. [36] Tangi and Gaonkar developed a PMU-based voltage estimation framework for active distribution networks. By using numerical linear programming for optimal location, their approach ensured full observability with the minimum number of PMUs in the IEEE 33, 69, and 119 bus test systems. The placement strategy was validated based on the results of forward and backward power flows, showing a nearly perfect match. However, zero injection buses were ignored in the analysis, ensuring that all nodes are included in the estimation process. Their work emphasized that the optimal placement of PMUs not only reduces costs but also enhances the practical performance of active distribution networks under the integration of renewable energies.

3. Methodology

3.1. Mathematical Formulation of µPMU Placement

In this section, a mathematical model for the OµPP problem is presented, with an emphasis on maximizing the observability with the fewest possible µ-PMUs to reduce the cost. The formulation includes three case studies: Normal Condition, Consideration of ZIBs, and Single µ-PMU outage.

3.1.1. Objective Function

The primary objective of the OµPP is to minimize the number of µ-PMUs to reduce the cost while ensuring the observability of all buses. The objective function is given as

$$Z=min{\sum }_{i=1}^N(c\times x_i)$$

(1)

$$\mathrm{Subject} \mathrm{to} AX\ge b$$

In Equation (1), N represents the total number of buses in the power system, $x_i$ is the binary decision variable, which can be 1 or 0, showing whether a µ-PMU is placed at the specific bus, and c is the cost of each µ-PMU. A represents a connectivity matrix showing the connection of buses.

$$A_{ij}=\left\{\begin{array}{l}1 if i and j are connected\\ \\ 0, elsewise \end{array}\right.$$

(2)

The connectivity matrix is a square symmetric matrix that can also be written as

$${A=\left[\begin{array}{ccccc}{a}_{11}& a_{12}& \cdots \cdots & \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots \cdots & \cdots & a_{2n}\\ ⋮& ⋮& & & ⋮\\ a_{n1}& a_{21}& \cdots \cdots & \cdots & a_{nn}\end{array}\right]}_{n\times n}$$

(3)

X is a matrix and represents the binary decision variable for installing µ-PMU, and b is the observability matrix, which is the minimum bus observability necessary for each bus.

$$X={{\left[\begin{array}{cc}\begin{array}{cc}{x}_1& \cdots \end{array}& x_n\end{array}\right]}^T}_{1\times n}$$

(4)

$$b={{\left[\begin{array}{cc}\begin{array}{cc}{obs}_{min}& \cdots \end{array}& ob{s}_{min}\end{array}\right]}^T}_{1\times n}$$

(5)

3.1.2. Case Studies

In this article, different case studies are presented to thoroughly test the proposed approach, ensuring the power system stays fully observable under different conditions, such as normal operation, a single µ-PMU failure, and the presence of Zero Injection Buses (ZIBs).

Observability Under Normal Conditions

Under normal conditions, each bus must be either directly or indirectly observable by at least a single µ-PMU. To ensure that the power system is fully viewable, each bus must either have a PMU installed directly at it or be seen by surrounding buses with PMUs. This condition is represented as follows:

$${\sum }_{j\epsilon N_i}{x}_j \ge 1\hspace{1em}\hspace{1em}{\forall }_i\epsilon \left\{ 1,2,\dots \dots .N\right\}$$

(6)

where $N_i$ represents the set of buses connected to the bus $i$ (entailing bus $i$ itself). This condition ensures that each bus is either directly or indirectly observable.

Observability Considering Zero Injection Buses

Zero injection buses are buses with no connected load or generator; zero current injection, active power (Pd), and reactive power (Qd) measurements are also zero at these buses. ZIBs can be manipulated to reduce the required number of µ-PMUs while ensuring observability. As the sum of currents is zero and the line impedances are negligible, the voltage and current at the adjacent buses to ZIB can be calculated using Kirchhoff’s current and voltage laws. A way of appending the ZIBs in OPP formulation is as follows.

$${\sum }_{j\epsilon N_z}{x}_j \ge 1\hspace{1em}\hspace{1em}\hspace{1em}{\forall }_i z \epsilon ZIBs$$

(7)

In Equation (7), $ZIBs$ represent the ZIBs, and $N_z$ is the set of adjacent buses to the bus $z$, including $z$ itself. If all incident branch currents except one are known for each $z \epsilon ZIBs$, the unknown current can be calculated using KCL.

$${\sum }_{k \epsilon ZIBs}{I}_{ZIBsk}$$

(8)

This results in the observability inference rule:

$$If\mid \left\{j\in N\left(z\right):Oj=0\right\}\mid =1\Rightarrow Ou=1 \forall u\in N\left(z\right) with Ou=0$$

(9)

where $O_j$ denotes the observability status of the bus $j$.

$${\sum }_{j \epsilon N\left(ZIBs\right)}{x}_j+{\sum }_{k \epsilon N\left(ZIBs\right)}{x}_j \mathsf{\Omega }({\sum }_{m \epsilon N\left(ZIBs\right)}{x}_m\ge 1)\ge 1$$

(10)

Observability During Single µ-PMU Outage

To ensure that the network remains observable even if a PMU fails, the effectuation condition is that each bus must be observed by two µ-PMUs either directly or indirectly. The single µ-PMU outage condition can be formulated as

$${\sum }_{j\epsilon N_i}{x}_j \ge 2\hspace{1em}\hspace{1em}{\forall }_i\epsilon \left\{ 1,2,,\dots \dots .N\right\}$$

(11)

In Equation (8), for each bus $i$ if there is no PMU placed on the bus, it must be observable by a µ-PMU on the neighboring buses. This ensures that even if a µ-PMU at the bus fails, the bus remains observable through a second µ-PMU.

3.2. System Observability Redundancy Index

In terms of power system observability, the location of phasor measurement units (PMUs) is crucial for ensuring that the system remains visible under regular operating conditions and during contingencies. The System Observability Redundancy Index (SORI) measures the redundancy and robustness of PMU deployment configurations. SORI assesses how well a system is observed by taking into account both local redundancy at each bus and network-wide redundancy. The index incorporates both voltage and current phasor measurements, providing a complete picture of the system’s observability. Voltage and current data are used to determine redundancy on each bus. Furthermore, the criticality of each bus is assessed, ensuring that buses critical to system observability are prioritized. SORI is an analysis used to determine the robustness and redundancy of the PMU placement configuration. SORI determines how well a system is observed using PMUs. The formal mathematical formulation of SORI is based on the total of redundancy contributions from each bus where a PMU is installed. Let N be the total number of buses in the system, and let $p_i$ be a binary variable indicating the placement of a PMU at bus i, where

$$p_i=\left\{\begin{array}{c}1 if a PMU is placed at bus i\\ 0 otherwise \end{array}\right.$$

(12)

The voltage measurements at bus i are marked Voltage_i., which denotes the number of voltage phasor measurements at bus i. Similarly, Current_i represents the number of current phasor measurements at bus i, which includes readings from related branches. Local redundancy at bus i is defined as the sum of the voltage and current measurements.

$${\mathrm{R}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{y}}_{\mathrm{i}}={\mathrm{V}\mathrm{o}\mathrm{l}\mathrm{t}\mathrm{a}\mathrm{g}\mathrm{e}}_{\mathrm{i}}+{\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}_{\mathrm{i}}$$

(13)

By incorporating critical buses (those that are important for observability), we create a criticality factor C_i. This guarantees that certain buses are given more attention in the overall observability assessment.

$${\mathrm{C}}_{\mathrm{i}}=\left\{\begin{array}{c}1 \mathrm{if} \mathrm{a} \mathrm{bus} \mathrm{is} \mathrm{critical} (\mathrm{must} \mathrm{be} \mathrm{fully} \mathrm{observable})\\ 0 \mathrm{otherwise}\end{array}\right.$$

(14)

The System Observability Redundancy Index (SORI) is calculated as a weighted sum of the redundancy at each bus, taking into account the PMU placements and each bus’s criticality:

$$SORI={\sum }_{i=1}^N{\mathrm{p}}_{\mathrm{i}}·({\mathrm{V}\mathrm{o}\mathrm{l}\mathrm{t}\mathrm{a}\mathrm{g}\mathrm{e}}_{\mathrm{i}}+{\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}_{\mathrm{i}})$$

(15)

To ensure that important buses are adequately observed, the robust SORI is defined as

$$SOR{I}_{robust}={\sum }_{i=1}^N{\mathrm{p}}_{\mathrm{i}}·({\mathrm{V}\mathrm{o}\mathrm{l}\mathrm{t}\mathrm{a}\mathrm{g}\mathrm{e}}_{\mathrm{i}}+{\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}_{\mathrm{i}})·{\mathrm{C}}_{\mathrm{i}}$$

(16)

To calculate the value of SORI, divide the total SORI by the number of buses (N):

$$mean SORI={\displaystyle \frac{SORI}{N}}={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{\mathrm{p}}_{\mathrm{i}}·({\mathrm{V}\mathrm{o}\mathrm{l}\mathrm{t}\mathrm{a}\mathrm{g}\mathrm{e}}_{\mathrm{i}}+{\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}_{\mathrm{i}})·{\mathrm{C}}_{\mathrm{i}}$$

(17)

Local redundancy is the number of measurements available on a local bus to ensure its observability. This is an important part of power system monitoring since each bus in the network may have different levels of measurement redundancy based on its connectivity and the number of PMUs located nearby. The SORI computation determines local redundancy at each bus (i) by adding the voltage and current values. In addition to the overall SORI, two critical redundancy indicators are utilized to assess the system’s observability; which are, Minimum Redundancy: This metric determines the lowest redundancy across all buses in the system, guaranteeing that no bus goes unobserved, and, Average Redundancy: This metric calculates the average level of redundancy across all buses in the system, providing an overall picture of how well the system is equipped with measurement devices.

$$m={\sum }_i^N{(\mathrm{V}\mathrm{o}\mathrm{l}\mathrm{t}\mathrm{a}\mathrm{g}\mathrm{e}}_{\mathrm{i}}+{\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}_{\mathrm{i}})$$

(18)

The number of state variables (n) in the system is typically (2N − 1), where N represents the number of buses. The SORI is then stated as the ratio of total measurements to the number of state variables.

$$SORI={\displaystyle \frac{m}{n}}={\displaystyle \frac{{\sum }_i^N{(\mathrm{V}\mathrm{o}\mathrm{l}\mathrm{t}\mathrm{a}\mathrm{g}\mathrm{e}}_{\mathrm{i}}+{\mathrm{C}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}_{\mathrm{i}})}{2\mathrm{N}-1}}$$

(19)

3.3. Optimization Algorithms

To solve the OµPP problem, this article employs two state-of-the-art optimization algorithms, PSO and GWO, leveraging their unique strengths in global exploration and local exploitation to achieve an efficient and robust solution.

3.3.1. Binary Particle Swarm Optimization

PSO is a population-based optimization algorithm inspired by the social behavior of bird flocks. It is widely used in solving problems related to optimization. A set of particles is first initialized by placing random particles in the search space. The particles change their place with a specific set velocity, and the position of the particle determines the placement of the µ-PMU. Both the velocity and position of particles are initiated randomly within the bounds of the µ-PMU placement problem, i.e., either µ-PMU is to be placed or not. The particles progress toward the best solution at each iteration based on the experience of their previous position. In the algorithm, pbest is the personal best experience of a particle, and gbest is the global best solution. The velocity of each particle is updated based on the previous position of the particle, the cognitive component (how close it is to its personal best solution), and the social component (how close it is to the global best solution). The equation for velocity is given as

$$V_{ij}^{k+1}=w\cdot V_{ij}^k+c_1\cdot ran{d}_1\cdot (pbes{t}_{ij}^k-x_{ij}^k)+c_2\cdot ran{d}_2\cdot (gbes{t}_j^k-x_{ij}^k)$$

(20)

$w$ is the inertia weight that controls the impact of the previous velocity, $c_1$ and $c_2$ are cognitive and social coefficients, respectively, and $ran{d}_1$ and $ran{d}_2$ are the random values between 0 and 1. The position is updated as per Equation (21).

$$x_{ij}^{k+1}=x_{ij}^k+v_{ij}^{k+1}$$

(21)

The position is then clamped to the binary domain via Equation (22):

$$x_{ij}^{k+1}=\left\{\begin{array}{c}1, \mathrm{if} x_{ij}^{k+1}>0.5\\ 0, \mathrm{if} x_{ij}^{k+1}\le 0.5\end{array}\right.$$

(22)

By using 0.5 as a threshold value, binary quantization is performed to convert the continuous values into binary numbers. It divides the binary values into two equal parts, such that values greater than 0.5 to 1 and less than 0.5 to 0. This thresholding ensures that optimization works in a continuous search space; however, the placement decision remains binary as per the OPP problem requirement 0.5 is chosen as it is the midpoint between 1 and 0.

3.3.2. Binary Grey Wolf Optimization

GWO is also an optimization technique that is inspired by the hunting behavior of grey wolves. It simulates the social hierarchy within a wolf pack, where alpha, beta, and delta wolves guide the entire pack during the hunt. For the OPP problem, GWO aims to find the optimal µ-PMU placement by utilizing the social hierarchy and collective intelligence of the grey wolf pack. Each wolf represents a solution for µ-PMU placement, where the position of the wolf is the binary vector of values (0 or 1). The number of wolves is predefined, and the position of each wolf is initialized randomly within the search space. The wolves update their positions based on the position of alpha, beta, and delta wolves (the best, second-best, and third-best) solutions. The equation for the position is given as

$$X_i^{k+1}=X_i^k-A\cdot D_{\alpha },X_i^{k+1}=X_i^k-A\cdot D_{\beta }, X_i^{k+1}=X_i^k-A\cdot D_{\delta }$$

(23)

In Equation (15), A = 2 × a × rand () − a, where a decreases linearly from 2 to 0. $D_{\alpha }$, $D_{\beta }$, and $D_{\delta }$ represent the distance between the current position of wolves and their distance from alpha, beta, and delta wolves, respectively.

$$D_{\alpha }=\mid C_1\cdot X_{\alpha }-X_i\mid ,D_{\beta }=\mid C_2\cdot X_{\beta }-X_i\mid ,D_{\delta }=\mid C_3\cdot X_{\delta }-X_i\mid$$

(24)

where $C_1$, $C_2$, and $C_3$ are the coefficients that control the movement of wolves.

To solve binary optimization problems like OPP, the continuous position values acquired from the update equations must be transformed to binary values (0 or 1). This is accomplished using a transfer function and thresholding.

$$T\left(X_{ij}\right)={\displaystyle \frac{1}{1+e^{-10(X_{ij}-0.5)}}}$$

(25)

The binary conversion method ensures that the solution remains in the discrete domain required for PMU placement decisions while preserving the continuous GWO algorithm’s exploration and exploitation capabilities.

$$X_{ij}\left(t+1\right)=\left\{\begin{array}{c}1 if rand()<T\left(X_{ij}\right)\\ 0 otherwise\end{array}\right.$$

(26)

The algorithm iteratively updates wolf positions with these equations, constantly refining the alpha, beta, and delta solutions until convergence conditions are fulfilled or the maximum number of iterations is achieved, resulting in the ideal or near-optimal PMU placement configuration.

3.4. Fitness Function

The fitness function evaluates how well a µ-PMU placement satisfies the observability constraints with the minimum number of µ-PMUs.

$$Fitness={\sum }_i^N{p}_i+\alpha \times (Unobserved Buses)$$

(27)

$p_i$ is the placement of µ-PMU on the bus $i$ (1 if placed or otherwise 0), and $\alpha$ is the penalty factor for unobserved buses.

$$Unobserved Buses={\sum }_{i=1}^N\left(1-{\sum }_{j=1}^{N}Cij\cdot pj\right)$$

(28)

Here, ${\sum }_{j=1}^{N}Cij\cdot pj$ shows the number of µ-PMUs that are directly or indirectly observing the bus $i$, if it is zero, a penalty is imposed in the fitness function.

3.5. Filtration

A filtration process is employed for all three case studies to remove the redundant µ-PMUs after evaluating their placement. The primary purpose of filtration is to ensure that each bus in the system is observable with the minimum possible µ-PMUs. It ensures that the final µ-PMU placement avoids over-coverage to reduce the cost while maintaining observability. In normal conditions, the filtration consists of two passes, in the first pass, the process marks all the buses as observed, either a µ-PMU is placed on the bus, or it is connected to the bus on which a µ-PMU is already placed. In the second pass, the filtration process rigorously checks if any µ-PMU is redundant, and if a bus is covered by multiple µ-PMUs and no unique observation is achieved from an extra µ-PMU, it is removed. In case of a single µ-PMU outage at a certain bus, some critical data might be lost; therefore, it is necessary to have other PMUs observing that bus. In the first pass of the filtration process, all buses are marked as observed if they are either directly monitored by a µ-PMU or indirectly observable through neighbouring buses. This ensures that even if a µ-PMU fails at a specific bus, the system can maintain observability through alternative observation paths. The second pass explores any redundant PMUs, removes them, and adds additional µ-PMUs for loss of coverage if necessary. In the case of ZIBs, the same method is followed as in normal conditions; however, the filtration process focuses on placing the µ-PMU adjacent to ZIBs, because the voltage, current, and phase angles at ZIBs can be calculated using Kirchhoff’s laws.

3.6. State Estimation Using Forward-Backward Sweep Algorithm and WLS Algorithm

State estimation is a crucial parameter for estimating the power flows, voltage magnitudes, and angles for the smooth operation of power systems. In this article, for the true estimation of voltage magnitudes and angles, forward-backward Sweep algorithm power flow is employed. The estimation of voltages and currents is performed using the WLS method, and the real values of the angles and magnitudes for the parameters are extracted from the estimated results obtained after implementing the WLS algorithm. The update principle adjusts the state vector to align it with the noisy measurements. The process is repeated until the residuals are minimal, providing accurate estimates of real system conditions.

The power flow equations used to calculate the true values of voltage magnitudes and angles are given as

$${\mathrm{P}}_{\mathrm{i}}={\sum }_{\mathrm{j}}{\mathrm{V}}_{\mathrm{i}}\mathrm{i}{\mathrm{V}}_{\mathrm{j}}\left({\mathrm{G}}_{\mathrm{i}\mathrm{j}}\mathrm{c}\mathrm{o}\mathrm{s}\right({\mathrm{\theta }}_{\mathrm{i}}-{\mathrm{\theta }}_{\mathrm{j}})+{\mathrm{B}}_{\mathrm{i}\mathrm{j}}\mathrm{s}\mathrm{i}\mathrm{n}({\mathrm{\theta }}_{\mathrm{i}}-{\mathrm{\theta }}_{\mathrm{j}}\left)\right)$$

(29)

$${\mathrm{Q}}_{\mathrm{i}}={\sum }_{\mathrm{j}}{\mathrm{V}}_{\mathrm{i}}\mathrm{i}{\mathrm{V}}_{\mathrm{j}}({\mathrm{G}}_{\mathrm{i}\mathrm{j}}\mathrm{s}\mathrm{i}\mathrm{n}\left({\mathrm{\theta }}_{\mathrm{i}}-{\mathrm{\theta }}_{\mathrm{j}}\right)-{\mathrm{B}}_{\mathrm{i}\mathrm{j}}\mathrm{c}\mathrm{o}\mathrm{s}({\mathrm{\theta }}_{\mathrm{i}}-{\mathrm{\theta }}_{\mathrm{j}}\left)\right)$$

(30)

In (18) and (19), ${\mathrm{P}}_{\mathrm{i}}$ and ${\mathrm{Q}}_{\mathrm{i}}$ are active and reactive powers, ${\mathrm{V}}_{\mathrm{i}}$ and ${\mathrm{V}}_{\mathrm{j}}$ are the voltage magnitudes of buses $\mathrm{i}$ and $\mathrm{j}$, ${\mathrm{\theta }}_{\mathrm{i}}$ and ${\mathrm{\theta }}_{\mathrm{j}}$ are the voltage angles at buses $\mathrm{i}$ and $\mathrm{j}$, ${\mathrm{G}}_{\mathrm{i}\mathrm{j}}$ and ${\mathrm{B}}_{\mathrm{i}\mathrm{j}}$ are the conductance and susceptance between buses $\mathrm{i}$ and $\mathrm{j}$, respectively.

The WLS method [23] is a very effective approach to estimate the state of unsupervised buses. It uses existing measurements and assigns weights based on their accuracy, improving the accuracy and stability of the electrical system state estimation. The objective is to identify the magnitude of the voltage, the phase angle, and the power flow of those buses that are being monitored, using the available measurements, which include the power flow and the voltage magnitude of the monitored buses. A nonlinear measurement function can be used to represent the PMU measurement model, as shown in Equation (31).

$$\mathrm{z}=\mathrm{h}\left(\mathrm{x}\right)+e$$

(31)

The measurement function h(x), which is based on the current state vector x, generates expected measurement values. The noise in the measurement is represented as e, while z is the measurement vector. By mapping the state vector x in the measurement space, this function determines the magnitude of the expected voltage, the voltage angle, and the magnitude of the current for a specific state. The predicted values for the magnitude of the voltage, the voltage angle, and the magnitude of the current are provided through Equations (32) and (33).

$${\mathrm{h}}_{{\mathrm{V}}_{\mathrm{m}\mathrm{i}}}=\mid {\mathrm{V}}_{\mathrm{i}}\mid$$

(32)

$${\mathrm{h}}_{{\mathrm{V}}_{\mathrm{a}\mathrm{i}}}={\mathrm{\theta }}_{\mathrm{i}}$$

(33)

The current magnitude ${\mathrm{I}}_{\mathrm{m}}$, and current angle ${\mathrm{I}}_{\mathrm{a}}$ on the branch between bus $\mathrm{i}$ and bus j can be calculated from (34)–(36).

$${\mathrm{I}}_{\mathrm{i}\mathrm{j}}={\mathrm{Y}}_{\mathrm{i}\mathrm{j}}\left({\mathrm{V}}_{\mathrm{i}}-{\mathrm{V}}_{\mathrm{j}}\right)$$

(34)

$${\mathrm{h}}_{{\mathrm{I}}_{{\mathrm{m}}_{\mathrm{i}\mathrm{j}}}}=\mid {\mathrm{I}}_{\mathrm{i}\mathrm{j}}\mid$$

(35)

$${\mathrm{h}}_{{\mathrm{I}}_{{\mathrm{a}}_{\mathrm{i}\mathrm{j}}}}=\mathrm{\angle }({\mathrm{I}}_{\mathrm{i}\mathrm{j}})$$

(36)

In Equation (34), the branch entry between buses $\mathrm{i}$ and $\mathrm{j}$ is represented as ${\mathrm{Y}}_{\mathrm{i}\mathrm{j}}$. The voltage values at buses i and j are represented in Equation (34) as ${\mathrm{V}}_{\mathrm{i}}$ and ${\mathrm{V}}_{\mathrm{j}}$, respectively. The phase angle of the current flowing through the branch is represented in Equation (35) as $\mathrm{\angle }\left({\mathrm{I}}_{\mathrm{i}\mathrm{j}}\right)$. Equation (37) defines the residual vector $\overrightarrow{r}$, which represents the difference between the vector of observed measurements $\mathrm{z}$ and the values of estimated measurements $\mathrm{h}\left(\mathrm{x}\right)$. This vector captures the errors or discrepancies between the actual measurement and the expected measurement.

$$\overrightarrow{r}=\mathrm{z}-\mathrm{h}(\mathrm{x})$$

(37)

In Equation (26), the measured vector with noise, which represents the actual measurements obtained from the PMUs and is affected by noise, is denoted as z. The measurement function, corresponding to the expected measurements according to the current state vector, is presented as h(x). The Jacobian matrix H, as shown in Equation (38), includes partial derivatives that demonstrate how changes in the state variables (such as voltages and angles) affect the measurements. These partial derivatives show the sensitivity of the measurements to changes in the state variables and provide an estimate of the estimated values. The nonlinear measurement function is linearized around the current state estimate using this matrix.

$$\mathrm{H}={\displaystyle \frac{\partial \mathrm{x}}{\partial \mathrm{h}\left(\mathrm{x}\right)}}$$

(38)

The weight matrix W in Equation (39) represents the level of confidence in the data. It is a diagonal matrix where each measurement is assigned the inverse value of the variance. Measurements with higher noise receive a lower weight, while measurements with higher confidence (and lower noise) receive a higher weight. In the basic case, the standard deviation (std) is set to 0.01 p.u.

$$\mathrm{W}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}({\displaystyle \frac{1}{{\mathrm{\sigma }}_{{\mathrm{V}}_{\mathrm{m}}}^2,{\mathrm{\sigma }}_{{\mathrm{V}}_{\mathrm{a}}}^2,{\mathrm{\sigma }}_{{\mathrm{I}}_{\mathrm{m}}}^2,{\mathrm{\sigma }}_{{\mathrm{I}}_{\mathrm{a}}}^2}})$$

(39)

In the literature, the values of the different standard deviations (STDs) are evaluated to determine their effect on the state estimates [37]. To ensure that the most reliable measurements have a more significant impact on the final state estimates, the weight matrix W assigns more weight to measurements with lower uncertainty. Measurements with lower standard deviations receive more weight, indicating less noise, while measurements with higher standard deviations receive less weight, indicating more noise. This approach improves the accuracy of state estimates by reducing the effect of noise or uncertain measurements. By minimizing dependence on erroneous information and emphasizing the more precise measurements, this method ensures a more reliable estimation process. In the WLS stage, the state vector x is updated iteratively to minimize the total sum of the weighted squared residuals.

$${\mathrm{x}}_{\mathrm{n}\mathrm{e}\mathrm{w}}={\mathrm{x}}_{\mathrm{o}\mathrm{l}\mathrm{d}}+\Delta \mathrm{x}$$

(40)

Δx is calculated using (30).

$$\Delta \mathrm{x}={\left({\mathrm{H}}^{\mathrm{T}}\mathrm{W}\mathrm{H}\right)}^{-1}{\mathrm{H}}^{\mathrm{T}}\mathrm{W}\mathrm{r}$$

(41)

In Equation (41), H is the Jacobian matrix, calculated in Equation (41). W is the weight matrix, calculated in Equation (42), and r represents the measurement error, or the remaining vector, derived in Equation (37). The algorithm proceeds repeatedly as long as the update step $\Delta \mathrm{x}$, as shown in Equation (42), is quite small, indicating that the estimates have become constant. The $ϵ$ limit is set at a small price to ensure harmony, usually 10⁻⁶. The update vector criterion serves as the basis for the consistency criteria, determining when the algorithm is sufficiently integrated into an accurate state approximation.

$$\parallel \Delta x\parallel <ϵ$$

(42)

This error has been added by applying these standard deviations (voltage value, voltage angle, current value, and current angle) described in the algorithm as (${\mathrm{s}\mathrm{t}\mathrm{d}}_{{\mathrm{v}}_{\mathrm{m}}},{\mathrm{s}\mathrm{t}\mathrm{d}}_{{\mathrm{v}}_{\mathrm{a}}},{\mathrm{s}\mathrm{t}\mathrm{d}}_{{\mathrm{i}}_{\mathrm{m}}},{\mathrm{s}\mathrm{t}\mathrm{d}}_{{\mathrm{i}}_{\mathrm{a}}}$) to reflect real-world conditions. In this part of the algorithm that uses ZIBs, adopting the WLS method for voltage estimation ensures that the effect of zero injection is captured accurately. As a result, highly precise and reliable state estimates are obtained, effectively modelling the behavior of this system while also considering the effects of these specific buses. Although μ-PMU phasor measurements permit a linear measurement model in rectangular coordinates yielding a linear least-squares estimator when μ-PMUs alone ensure full observability; we adopt the general WLS form $\mathbf{z}=\mathbf{h}\left(\mathbf{x}\right)+\mathbf{e}$ for uniformity across all case studies. This accommodates (i) contingency scenarios with μ-PMU outage (Case 2), (ii) potential inclusion of magnitude-only or power measurements that are nonlinear in the state, (iii) statistically consistent weighting when polar measurements are mapped to rectangular states, and (iv) seamless integration of ZIB/KCL constraints. In fully μ-PMU-observable settings the WLS reduces to a single linear solve (constant Jacobian), whereas in hybrid or perturbed settings the nonlinear form preserves robustness without changing the estimation pipeline.

3.7. Procedures for Placement of µ-PMUs

The placement of µ-PMUs in a radial distribution system consists of multiple steps elucidated in detail in this section.

• In the first step, data regarding IEEE bus systems is uploaded from MATPOWER version 8.0 onto the MATLAB program. IEEE 33 and 69 distribution systems are represented in Figure 1.
• In the second step, a connectivity matrix is formed based on the connections between buses in both distribution systems.
• After the formation of the connectivity matrix, initial parameters for both algorithms are initialized, and the fitness function is defined. The main objective of the fitness function is to reduce the number of PMUs while ensuring full observability for all cases. Three case studies are performed for both the 33 and 69 bus systems, i.e., under normal conditions, with Zero Injection buses, and a single PMU outage. ZIBs are identified to leverage them to reduce the PMUs for network observability.
• Simulations are performed, and as the fitness criteria are achieved, the program stops and generates a binary vector consisting of 0’s and 1’s indicating on which buses PMUs are installed. The flowcharts for both PSO and GWO are shown in Figure 2.
• WLS algorithm is employed on selected buses via both algorithms for voltage estimation, and different noise levels are introduced for validating the placement of PMUs.

4. Simulations and Results

This section describes the optimal placement of µ-PMUs under different conditions. The simulations are performed in MATLAB 2018 on a Dell 12th Gen Intel (R) Core (TM) i5-1235U @ 1.30 GHz and 16 GB RAM. The simulations are conducted on both IEEE 33 and 69 distribution bus systems. For both algorithms, 100 particles/wolves are used for 1000 iterations, and with a fixed number of seeds, 30 different runs are performed for a detailed analysis to determine the robustness of the algorithms.

4.1. Case 1: OµPP Under Normal Conditions

Based on the comprehensive experimental results from 30 independent runs, both Binary Particle Swarm Optimization (BPSO) and Binary Grey Wolf Optimization (BGWO) algorithms demonstrate excellent performance in solving the optimal PMU placement problem for the IEEE 33-bus system. When the system operates under normal conditions, the minimum number of µ-PMUs required is 11, ensuring full system observability with both algorithms successfully achieving this theoretical minimum in several runs. The System Observability Redundancy Index (SORI) remains consistently above 1 across all trials, with BPSO achieving SORI values between 1.046 And 1.292 and BGWO ranging From 1.046 to 1.323, confirming that each bus is directly or indirectly observed by at least one µ-PMU and maintaining complete system observability throughout all configurations.

The results show that BPSO typically utilizes 11–14 PMUs with an average of 12.4, while BGWO employs 11–14 PMUs with an average of 12.5, demonstrating both algorithms’ effectiveness in minimizing infrastructure costs. In terms of measurement capabilities, both algorithms generate comparable numbers of voltage phasor measurements (22–28) and current phasor measurements (46–58), resulting in total measurements ranging from 68 To 86 Across different configurations. A notable difference emerges in computational efficiency, where BPSO exhibits faster convergence times of 1.44–2.46 s (averaging approximately 1.74 s) compared to BGWO’s 2.00–2.49 s (averaging about 2.14 s), representing a 23% performance advantage for BPSO in computation time while maintaining equivalent solution quality.

Table 1. Comparison of BPSO and BGWO for PMU Placement in IEEE 33 Bus System for Case Study 1.

Run	Number of Placed µ-PMUs		Voltage Phasor Measurements		Current Phasor Measurements		Total Measurements (m)		Total State Variables (n)	SORI (m/n)		Computation Time (s)
	BPSO	BGWO	BPSO	BGWO	BPSO	BGWO	BPSO	BGWO		BPSO	BGWO	BPSO	BGWO
1	12	12	24	24	52	54	76	78	65	1.169	1.200	1.55	2.11
2	13	13	26	28	54	56	80	84	65	1.231	1.292	1.82	2.47
3	12	12	24	24	54	52	78	76	65	1.200	1.169	1.51	2.13
4	13	12	26	24	54	50	80	74	65	1.231	1.138	1.44	2.15
5	12	12	24	24	52	52	76	76	65	1.169	1.169	1.44	2.10
6	13	12	26	24	56	52	82	76	65	1.262	1.169	1.53	2.05
7	12	12	24	24	52	54	76	78	65	1.169	1.200	1.66	2.23
8	12	13	24	26	54	56	78	82	65	1.200	1.262	1.47	2.19
9	13	13	26	24	50	50	76	74	65	1.169	1.231	1.52	2.40
10	13	13	26	26	56	56	82	82	65	1.262	1.262	1.55	2.49
11	12	13	24	24	54	50	78	74	65	1.200	1.231	1.92	2.28
12	12	11	24	22	50	46	74	68	65	1.138	1.046	2.03	2.00
13	13	13	26	24	56	54	82	78	65	1.262	1.200	1.73	2.16
14	12	12	24	24	52	50	76	74	65	1.169	1.138	1.71	2.06
15	12	12	24	24	54	52	78	76	65	1.200	1.169	1.77	2.00
16	13	13	26	28	56	56	82	84	65	1.262	1.292	2.02	2.13
17	11	12	22	24	46	52	68	76	65	1.046	1.169	2.46	2.18
18	12	13	24	26	54	56	78	82	65	1.200	1.262	1.76	2.17
19	12	12	24	24	50	52	74	76	65	1.138	1.169	1.80	2.02
20	13	13	26	24	56	50	82	74	65	1.262	1.231	2.00	2.07
21	12	12	24	24	52	52	76	76	65	1.169	1.169	1.74	2.00
22	14	12	28	24	56	54	84	78	65	1.292	1.200	2.14	2.04
23	13	13	26	28	56	58	82	86	65	1.262	1.323	1.75	2.19
24	14	14	28	24	56	52	84	76	65	1.292	1.169	1.87	2.05
25	11	12	22	24	46	52	68	76	65	1.046	1.169	1.74	2.05
26	13	12	26	24	56	52	82	76	65	1.262	1.169	1.78	2.09
27	12	13	24	26	52	56	76	82	65	1.169	1.262	2.00	2.14
82	12	13	24	26	52	56	76	82	65	1.169	1.262	1.71	2.03
29	12	13	24	26	52	56	76	82	65	1.169	1.262	1.72	2.10
30	13	12	26	24	56	52	82	76	65	1.262	1.169	1.76	2.09

clearly indicates that observability is consistently maintained in the IEEE 33-bus system through optimal µ-PMU placements from both algorithms, with the low computation times highlighting their computational efficiency. This efficiency makes both approaches particularly suitable for practical power system applications where solution quality and computational overhead are crucial considerations. The reliability of both algorithms is further demonstrated by their consistent performance across all 30 independent runs, with no unobservable configurations occurring and PMU counts remaining within a tight, practical range.

This Figure 3 compares the PMU placement optimization results using BPSO and BGWO for the IEEE 33 Bus System.

The first set of images (a), (b), (c), and (d) show the results for BPSO. In (a), the PMU placement determined by BPSO is illustrated, where the red points represent the locations of the placed PMUs, and the blue circles indicate buses without PMUs. The diagram highlights the best PMU placement to achieve optimal system observability. (b) shows the observability redundancy for each bus, with the green bars representing the redundancy levels achieved, and the red line indicating the minimum required redundancy. This graph demonstrates that all buses meet the necessary redundancy criteria, ensuring full system observability. In (c), the histogram shows the distribution of redundancy levels across the buses, with the majority having a redundancy level of 1, meaning they are minimally observable, while a small number of buses have a redundancy level of 2, which adds extra reliability. Finally, in (d), the pie chart illustrates the distribution of measurement types used in the system. The yellow portion represents current phasors, and the purple portion represents voltage phasors, showing how both types of measurements contribute to system observability.

The second set of images (e), (f), (g), and (h) displays the results for BGWO. In (e), the PMU placement determined by BGWO is shown. Similarly to the BPSO placement, the red points represent the PMUs, and the blue circles show buses without PMUs. (f) displays the redundancy levels for each bus, again with the green bars and the red line indicating the minimum required redundancy. BGWO also ensures that all buses meet the redundancy requirements for full observability. (g) presents the histogram of redundancy levels for BGWO, where, like in BPSO, most buses have a redundancy level of 1, with some buses having a redundancy level of 2 for added reliability. Finally, in (h), the pie chart for BGWO shows the distribution of current and voltage phasors. The chart illustrates that the contribution of current phasors (yellow) and voltage phasors (purple) is balanced, ensuring that both types of measurements are effectively used in the system.

In summary, both BPSO and BGWO methods achieve full observability of the system, with slight variations in PMU placement and redundancy levels. While BPSO places PMUs to meet the minimum redundancy requirements, BGWO also ensures sufficient redundancy and an effective distribution of measurement types. The optimization methods differ in their approach but ultimately fulfil the goal of ensuring system observability and reliability.

For the IEEE 69-bus system under normal operating conditions, both algorithms demonstrate effective performance, though with increased computational requirements compared to the 33-bus case. As shown in

Table 2. Comparison of BPSO and BGWO for PMU Placement in IEEE 69 Bus System for Case Study 1.

Run	Number of Placed µ-PMUs		Voltage Phasor Measurements		Current Phasor Measurements		Total Measurements (m)		Total State Variables (n)	SORI (m/n)		Computation Time (s)
	BPSO	BGWO	BPSO	BGWO	BPSO	BGWO	BPSO	BGWO		BPSO	BGWO	BPSO	BGWO
1	24	25	48	50	106	94	154	144	137	1.124	1.051	5.02	51.32
2	25	25	50	50	104	96	154	146	137	1.124	1.066	4.34	52.28
3	25	26	50	52	104	102	154	154	137	1.124	1.124	4.34	51.05
4	26	26	52	52	110	106	162	158	137	1.182	1.1533	7.10	58.62
5	22	24	44	48	94	100	138	148	137	1.007	1.080	5.89	41.66
6	25	25	50	50	106	102	156	152	137	1.139	1.109	5.80	45.79
7	25	25	50	50	108	98	158	148	137	1.153	1.0803	6.21	42.71
8	24	24	48	48	104	98	152	146	137	1.109	1.066	5.61	36.15
9	22	24	44	48	94	94	138	142	137	1.007	1.036	5.85	42.55
10	24	24	48	48	100	96	148	144	137	1.080	1.051	5.86	42.81
11	24	26	48	52	102	98	150	150	137	1.095	1.095	5.89	52.33
12	24	26	48	52	100	108	148	160	137	1.080	1.168	5.61	49.62
13	25	25	50	50	108	100	158	150	137	1.153	1.095	5.90	51.82
14	24	24	48	48	98	96	146	144	137	1.066	1.051	6.91	47.00
15	25	25	50	50	106	100	156	150	137	1.139	1.095	6.14	49.69
16	26	26	52	52	110	100	162	152	137	1.182	1.109	5.88	58.25
17	25	25	50	50	106	102	156	152	137	1.139	1.109	6.45	52.66
18	26	25	52	50	108	98	160	148	137	1.168	1.080	5.66	46.50
19	25	25	50	50	106	100	156	150	137	1.139	1.095	6.01	47.79
20	25	26	50	52	106	102	156	154	137	1.139	1.124	6.02	48.59
21	25	24	50	48	104	96	154	144	137	1.124	1.051	6.06	41.29
22	25	24	50	48	104	98	154	146	137	1.124	1.0657	5.55	43.10
23	25	26	50	52	108	98	158	150	137	1.153	1.095	5.89	53.95
24	25	25	50	50	104	102	154	152	137	1.124	1.109	5.57	48.49
25	24	24	48	48	100	96	148	144	137	1.080	1.051	5.55	43.14
26	26	25	52	50	112	96	164	146	137	1.197	1.066	5.27	45.81
27	25	24	50	48	104	96	154	144	137	1.124	1.051	4.30	41.54
28	25	24	50	48	102	96	152	144	137	1.109	1.051	4.42	46.04
29	22	25	44	50	96	96	140	146	137	1.022	1.066	4.77	55.98
30	26	25	52	50	110	96	162	146	137	1.182	1.066	4.57	49.71

, the algorithms consume more processing time due to the larger search space, yet consistently maintain SORI values above 1, ensuring full system observability in every run. The Binary Particle Swarm Optimization (BPSO) algorithm achieves PMU placements ranging from 22 to 26 units, with computation times between 4.30 and 7.10 s, while the Binary Grey Wolf Optimization (BGWO) method utilizes 24 to 26 PMUs but requires significantly longer computation times ranging from 36.15 to 58.62 s. The comprehensive results reveal several key patterns about algorithm performance. In terms of measurement capabilities, BPSO generates 44–52 voltage phasor measurements and 94–112 current phasor measurements, resulting in total measurements of 138–164. BGWO produces comparable ranges of 48–52 voltage phasor measurements and 94–108 current phasor measurements, totalling 142–160 overall measurements. The SORI values for both algorithms consistently exceed the critical threshold of 1.0, with BPSO achieving values from 1.007 to 1.197 and BGWO ranging from 1.051 to 1.168, confirming that complete observability is maintained despite the system’s increased complexity. A particularly notable finding emerges in the computational efficiency comparison. While both algorithms successfully solve the optimal PMU placement problem, BPSO demonstrates substantially faster convergence, with computation times averaging approximately 5.6 s compared to BGWO’s average of 48.3 s. This represents nearly an 88% reduction in computation time for BPSO while maintaining similar solution quality. The performance gap is especially evident in runs where both algorithms achieve identical PMU counts, for instance, in run 2, both methods place 25 PMUs, but BPSO completes the optimization in 4.34 s while BGWO requires 52.28 s. The reliability of both algorithms is further demonstrated by their consistent performance across all 30 independent runs, with no instances of unobservable configurations occurring. The PMU counts remain within a practical range of 22–26 units, representing optimal or near-optimal solutions for the 69-bus system. The maintained SORI values above 1.0 across all configurations ensure not only basic observability but also provide measurement redundancy, enhancing the system’s resilience to potential measurement failures or communication disruptions. This consistent performance, coupled with the demonstrated scalability from the 33-bus to the 69-bus system, validates both algorithms’ robustness for practical power system applications, though BPSO’s superior computational efficiency makes it particularly attractive for larger systems or scenarios requiring rapid reconfiguration.

Figure 4 compares the results of PMU placement optimization for the IEEE 69 Bus System using BPSO and BGWO with 24 PMUs. In the BPSO case (shown in diagrams a, b, c, and d), the optimal PMU placement is illustrated in (a), where the red dots represent the PMU locations, and the blue circles show buses without PMUs. This placement ensures maximum observability across the system. In (b), the redundancy levels for each bus are shown, with green bars indicating the redundancy achieved and the red line representing the minimum required. Most buses exceed the minimum redundancy level, ensuring full system observability. (c) presents the distribution of redundancy levels, where most buses have a redundancy level of 1, indicating they meet the minimum requirement, while a few buses have redundancy levels of 2 or 3 for enhanced reliability. (d) shows the measurement distribution, where the yellow section represents current phasors and the purple section represents voltage phasors, demonstrating that current phasors play a larger role in achieving observability, though both are necessary. In the BGWO case (shown in e, f, g, and h), the PMU placement in (e), determined by BGWO, is similar to the BPSO placement but differs slightly due to the optimization method. As in BPSO, red dots indicate the PMUs, and blue circles represent buses without PMUs. (f) shows the redundancy levels for each bus, with green bars indicating the achieved redundancy, and the red line indicating the minimum required. The graph shows that BGWO places PMUs to ensure that all buses meet or exceed the minimum redundancy, with some buses achieving dual observability, meaning they have extra measurements for more reliability. (g) presents the distribution of redundancy levels, where most buses have redundancy level 1, but a higher number of buses have redundancy levels of 2 or 3 compared to BPSO, indicating that BGWO places additional PMUs for increased accuracy and reliability. Finally, (h) shows the measurement distribution, with current phasors (yellow) again contributing more to observability than voltage phasors (purple), although both types are essential. Overall, both BPSO and BGWO methods achieve the goal of ensuring full observability in the system, but BGWO provides a more robust PMU placement with higher redundancy and additional PMUs for better system reliability. While both methods use current phasors as the dominant measurement type, BGWO’s approach ensures more reliable observability by placing extra PMUs, especially in areas where additional redundancy is beneficial.

4.2. Case 2: OµPP with a Single µPMU Outage

To ensure system reliability under a single µ-PMU outage condition for the IEEE 33-bus system, the number of required µ-PMUs increases significantly from 11 to 24, a configuration designed to guarantee that each bus is observed by at least two µ-PMUs and remains observable during any contingency. The results demonstrate that both BPSO and BGWO exhibited remarkable robustness in finding this optimal placement, consistently utilizing exactly 24 µ-PMUs across all 30 independent runs. This configuration established a comprehensive measurement infrastructure, providing 48 voltage phasor measurements and 92 current phasor measurements, which sum to 140 total measurements for the state estimation process. With the system defined by 65 state variables, this yields a SORI of approximately 2.154. This metric is critical, as a SORI value significantly greater than 2 confirms a high level of redundancy, meaning each state variable is observed by more than two measurements, thereby directly validating the success of the N-1 contingency planning. However, a stark performance contrast emerged in computational efficiency. While both algorithms found the correct solution, BPSO was substantially faster, with computation times ranging from 0.19 s to 0.34 s and showing a trend of stabilization. In contrast, BGWO was significantly slower, with times ranging from 0.33 s to 0.66 s, making it roughly two to three times slower on average and exhibiting greater variability (

Table 3. Comparison of BPSO and BGWO for PMU Placement in IEEE 33 Bus System for Case Study 2.

Run	Number of Placed µ-PMUs		Voltage Phasor Measurements		Current Phasor Measurements		Total Measurements (m)		Total State Variables (n)	SORI (m/n)		Computation Time (s)
	BPSO	BGWO	BPSO	BGWO	BPSO	BGWO	BPSO	BGWO		BPSO	BGWO	BPSO	BGWO
1	24	24	48	48	92	92	140	140	65	2.154	2.154	0.28	0.33
2	24	24	48	48	92	92	140	140	65	2.154	2.154	0.34	0.46
3	24	24	48	48	92	92	140	140	65	2.154	2.154	0.34	0.57
4	24	24	48	48	92	92	140	140	65	2.154	2.154	0.26	0.49
5	24	24	48	48	92	92	140	140	65	2.154	2.154	0.27	0.48
6	24	24	48	48	92	92	140	140	65	2.154	2.154	0.25	0.50
7	24	24	48	48	92	92	140	140	65	2.154	2.154	0.25	0.62
8	24	24	48	48	92	92	140	140	65	2.154	2.154	0.26	0.41
9	24	24	48	48	92	92	140	140	65	2.154	2.154	0.26	0.45
10	24	24	48	48	92	92	140	140	65	2.154	2.154	0.23	0.66
11	24	24	48	48	92	92	140	140	65	2.154	2.154	0.24	0.51
12	24	24	48	48	92	92	140	140	65	2.154	2.154	0.22	0.56
13	24	24	48	48	92	92	140	140	65	2.154	2.154	0.21	0.49
14	24	24	48	48	92	92	140	140	65	2.154	2.154	0.21	0.44
15	24	24	48	48	92	92	140	140	65	2.154	2.154	0.20	0.47
16	24	24	48	48	92	92	140	140	65	2.154	2.154	0.19	0.49
17	24	24	48	48	92	92	140	140	65	2.154	2.154	0.19	0.58
18	24	24	48	48	92	92	140	140	65	2.154	2.154	0.19	0.51
19	24	24	48	48	92	92	140	140	65	2.154	2.154	0.19	0.62
20	24	24	48	48	92	92	140	140	65	2.154	2.154	0.21	0.54
21	24	24	48	48	92	92	140	140	65	2.154	2.154	0.20	0.53
22	24	24	48	48	92	92	140	140	65	2.154	2.154	0.19	0.64
23	24	24	48	48	92	92	140	140	65	2.154	2.154	0.19	0.54
24	24	24	48	48	92	92	140	140	65	2.154	2.154	0.20	0.50
25	24	24	48	48	92	92	140	140	65	2.154	2.154	0.19	0.49
26	24	24	48	48	92	92	140	140	65	2.154	2.154	0.20	0.47
27	24	24	48	48	92	92	140	140	65	2.154	2.154	0.20	0.45
28	24	24	48	48	92	92	140	140	65	2.154	2.154	0.20	0.66
29	24	24	48	48	92	92	140	140	65	2.154	2.154	0.19	0.55
30	24	24	48	48	92	92	140	140	65	2.154	2.154	0.20	0.42

).

Figure 4. PMU Placement Optimization and Observability for IEEE 69 Bus System: BPSO vs. BGWO in IEEE 69 Bus System Case 1. (a) Number of PMUs using BPSO. (b) Observability Redundancy using BPSO. (c) Redundancy Level Achieved with BPSO. (d) Measurement Distribution with BPSO. (e) Best Placement Location of PMUs using BPSO. (f) Observability Redundancy using BPSO. (g) Redundancy Level Achieved with BPSO. (h) Measurement Distribution with BPSO.

Figure 4. PMU Placement Optimization and Observability for IEEE 69 Bus System: BPSO vs. BGWO in IEEE 69 Bus System Case 1. (a) Number of PMUs using BPSO. (b) Observability Redundancy using BPSO. (c) Redundancy Level Achieved with BPSO. (d) Measurement Distribution with BPSO. (e) Best Placement Location of PMUs using BPSO. (f) Observability Redundancy using BPSO. (g) Redundancy Level Achieved with BPSO. (h) Measurement Distribution with BPSO.

This Figure 5 presents a comparison of PMU placement optimization results for the IEEE 69 Bus System using BPSO and BGWO, both utilizing 24 PMUs. The first set of diagrams (a), (b), (c), and (d) illustrates the outcomes of the BPSO. In (a), the optimal placement of 24 PMUs is shown, where red dots indicate the PMU locations, and blue circles represent buses without PMUs. This arrangement ensures maximum observability for the power system. In (b), the observability redundancy levels for each bus are shown with green bars. The green bars exceed the red line, which represents the minimum redundancy required, indicating that the system’s observability is achieved and even exceeded for several buses. This guarantees that the grid is fully observable. (c) shows the histogram for the redundancy levels, where most buses have a redundancy level of 2, implying they meet the basic observability requirements. However, a small number of buses have redundancy levels of 2 or 3, which add extra reliability and robustness to the system. In (d), the pie chart depicts the distribution of measurements used to ensure observability. The yellow section represents current phasors, and the purple section represents voltage phasors. It is evident that current phasors play a larger role in achieving full system observability, although both current and voltage phasors are necessary.

The second set of diagrams (e), (f), (g), and (h) presents the results of the BGWO. In (e), the optimal placement of the 24 PMUs determined by BGWO is shown. As in the BPSO case, red dots represent the PMU locations, and blue circles mark the buses without PMUs. The placement differs slightly from BPSO, but the objective of full system observability is still achieved. In (f), the redundancy levels for each bus are shown, with the green bars representing redundancy and the red line marking the minimum required redundancy. The redundancy levels here are similar to BPSO, with most buses meeting or exceeding the required redundancy. Additionally, the “Dual Obs Threshold” line shows that some buses achieve dual observability, meaning they have more than the required number of measurements, providing extra reliability in their monitoring. (g) displays the distribution of redundancy levels across all buses, where again, most buses have redundancy level 2, but a significant number of buses have redundancy levels of 2 or 3. This indicates that BGWO places PMUs not only to meet the minimum observability but also to enhance the system’s reliability by ensuring higher redundancy in some buses. Finally, (h) shows the measurement distribution, with the yellow section representing current phasors and the purple section representing voltage phasors. Similarly to the BPSO results, current phasors contribute the most to the overall observability, though voltage phasors are also essential for a complete picture.

In conclusion, both BPSO and BGWO methods ensure full system observability of the IEEE 69 Bus System by strategically placing 24 PMUs. While the general structure of the results is similar, slight differences in PMU placement and redundancy levels arise due to the distinct optimization techniques. Both methods achieve the required redundancy and provide a good balance between current and voltage phasors for accurate system monitoring. However, BGWO tends to place additional PMUs in certain locations, offering more redundancy and ensuring higher reliability across the system.

Based on the comprehensive simulation results from

Table 4. Comparison of BPSO and BGWO for PMU Placement in IEEE 69 Bus System for Case Study 2.

Run	Number of Placed µ-PMUs		Voltage Phasor Measurements		Current Phasor Measurements		Total Measurements (m)		Total State Variables (n)	SORI (m/n)		Computation Time (s)
	BPSO	BGWO	BPSO	BGWO	BPSO	BGWO	BPSO	BGWO		BPSO	BGWO	BPSO	BGWO
1	51	51	102	102	194	200	296	302	137	2.161	2.219	6.19	0.17
2	54	50	108	100	208	200	316	300	137	2.307	2.200	7.67	0.17
3	55	49	110	98	210	190	320	288	137	2.336	2.1022	12.95	0.17
4	53	47	106	94	204	186	310	280	137	2.2628	2.0438	14.70	0.17
5	53	48	106	96	202	186	308	282	137	2.248	2.0584	9.47	0.16
6	54	50	108	100	208	192	316	292	137	2.307	2.1314	10.84	0.18
7	54	50	108	100	206	194	314	294	137	2.292	2.146	12.59	0.17
8	55	46	110	92	212	178	322	270	137	2.350	1.9708	6.36	0.17
9	53	55	106	110	204	212	310	322	137	2.263	2.3504	4.54	0.17
10	53	48	106	96	202	188	308	284	137	2.248	2.073	8.61	0.17
11	55	48	110	96	208	190	318	286	137	2.321	2.0876	5.63	0.17
12	55	48	110	96	208	192	318	288	137	2.307	2.1022	7.59	0.17
13	54	46	108	92	202	182	310	274	137	2.2628	2	16.41	0.17
14	52	48	104	96	198	192	302	288	137	2.2044	2.1022	14.25	0.17
15	54	50	108	100	206	192	314	292	137	2.292	2.1314	12.46	0.16
16	53	49	106	98	204	196	310	294	137	2.263	2.146	10.39	0.16
17	53	50	106	100	202	202	308	302	137	2.248	2.2044	10.34	0.18
18	55	53	110	106	212	202	322	308	137	2.350	2.2482	13.94	0.18
19	53	49	106	98	200	190	306	288	137	2.234	2.1022	12.34	0.19
20	54	50	108	100	210	190	318	290	137	2.321	2.1168	20.92	0.19
21	53	53	106	106	200	208	306	314	137	2.2336	2.292	14.70	0.19
22	54	49	108	98	208	194	316	292	137	2.307	2.1314	13.22	0.17
23	53	47	106	94	204	190	310	284	137	2.263	2.073	13.67	0.19
24	54	51	108	102	208	200	316	302	137	2.307	2.2044	15.07	0.23
25	56	48	112	96	220	194	332	290	137	2.423	2.1168	22.66	0.24
26	53	52	106	104	202	202	308	306	137	2.248	2.2336	15.59	0.24
27	54	51	108	102	206	202	314	304	137	2.292	2.219	20.88	0.49
28	53	47	106	94	202	190	308	284	137	2.248	2.073	15.64	0.33
29	56	48	112	96	220	188	332	284	137	2.423	2.073	20.83	0.26
30	54	45	108	90	204	176	312	266	137	2.277	1.9416	18.82	0.25

, for the 69-bus system under Case 2 (single PMU outage scenario), the analysis reveals crucial insights about system redundancy requirements. For complete system observability with adequate redundancy, the average SORI should be greater than 2, and the minimum redundancy at any bus must be at least 2 to ensure observability during single PMU outages. The results clearly demonstrate that BGWO significantly outperforms BPSO in achieving these critical redundancy targets. BGWO consistently achieved optimal configurations with only 48–49 μ-PMUs, which proved sufficient to maintain full system observability while meeting the essential redundancy criteria. For instance, in run 5, BGWO with 48 μ-PMUs achieved a SORI of 2.0584, and in run 11, the same number of PMUs yielded a SORI of 2.0876, both comfortably exceeding the minimum required average redundancy of 2. In contrast, BPSO required substantially more PMUs (51–56) to achieve similar redundancy levels, representing inefficient over-provisioning of resources. The superiority of BGWO is particularly evident in runs 8 and 13, where it accomplished full observability with only 46 and 47 PMUs, respectively, while still maintaining adequate redundancy margins. Most impressively, BGWO achieved these optimal results with dramatically better computational efficiency, completing optimization in 0.16–0.49 s compared to BPSO’s 4.54–22.66 s. This performance advantage underscores BGWO’s capability to identify minimal PMU configurations that satisfy the critical engineering requirement of maintaining minimum redundancy of 2 at all buses while achieving average SORI values above 2, ensuring reliable system observability even during contingency scenarios without unnecessary resource allocation.

In Figure 6, both the BPSO and BGWO algorithms are used to place 51 PMUs across the IEEE 69 Bus System. In (a), the optimal placement of 51 PMUs is shown, with the red dots indicating the PMU locations and the blue circles marking the buses without PMUs. The placement ensures full system observability by distributing the PMUs across the grid. In (b), the redundancy levels for each of the 69 buses are shown, with the green bars representing the achieved redundancy. The red dashed line indicates the minimum required redundancy. The results indicate that all buses meet or exceed the minimum redundancy level of 1, with some buses achieving higher redundancy levels of 2 or 3 for increased reliability.

The redundancy level distribution in (c) shows that the majority of buses, specifically around 60 buses, have a redundancy level of 1, meaning they meet the minimum observability requirement. A smaller number, about 6 buses, have redundancy levels of 2, and only a few buses (approximately 3 buses) have redundancy levels of 3, indicating that extra PMUs are placed in these locations to enhance the accuracy and reliability of the system’s measurements. Finally, in (d), the measurement distribution shows the breakdown between voltage phasors (purple) and current phasors (yellow). The chart reveals that approximately 70% of the measurements are current phasors, with the remaining 30% being voltage phasors.

In summary, both BPSO and BGWO algorithms place 51 PMUs across the 69 buses in the IEEE 69 Bus System, ensuring full system observability. The redundancy levels across the system are well-distributed, with the majority of buses having redundancy level 1, and a few buses receiving extra PMUs for additional reliability. The measurement distribution predominantly relies on current phasors, with voltage phasors also contributing significantly to the overall observability.

4.3. Case 3: OPP Considering ZIBs

ZIBs can be effectively leveraged to reduce the number of required µ-PMUs in power system monitoring. While the IEEE 33 Bus system in the MATPOWER version contains no ZIBs, maintaining the requirement at 11 µ-PMUs, the IEEE 69 Bus system presents a more favorable scenario with 20 identified ZIBs that enable significant optimization opportunities. The comparative analysis between BPSO and BGWO in

Table 5. Comparison of BPSO and BGWO for PMU Placement in IEEE 69 Bus System for Case Study 3.

Run	Number of Placed µ-PMUs		Voltage Phasor Measurements		Current Phasor Measurements		KCL Inference Measurements		Total Measurements (m)		Total State Variables (n)	SORI (m/n)		Computation Time (s)
	BPSO	BGWO	BPSO	BGWO	BPSO	BGWO	BPSO	BGWO	BPSO	BGWO		BPSO	BGWO	BPSO	BGWO
1	24	22	48	44	86	90	24	18	158	152	137	1.153	1.109	13.35	7.78
2	21	23	42	46	92	88	22	14	156	148	137	1.139	1.080	10.43	6.90
3	23	23	46	46	80	94	22	16	148	156	137	1.080	1.139	16.93	8.04
4	22	22	44	44	82	88	22	18	148	150	137	1.080	1.095	7.79	5.59
5	21	22	42	44	78	90	26	20	146	154	137	1.066	1.124	141.08	6.75
6	20	22	40	44	74	88	26	16	140	148	137	1.022	1.080	24.47	7.95
7	21	22	42	44	84	90	18	18	144	152	137	1.051	1.109	6.58	6.54
8	21	21	42	42	80	86	22	20	144	1.080	137	1.051	1.080	20.03	8.04
9	24	24	48	48	98	94	18	16	164	158	137	1.197	1.153	6.50	8.21
10	24	23	48	46	88	90	20	16	156	152	137	1.139	1.109	6.60	7.28
11	22	21	44	42	84	86	18	18	146	146	137	1.066	1.066	81.37	5.75
12	22	23	44	46	84	92	20	12	148	150	137	1.080	1.095	13.61	5.73
13	20	22	40	44	80	92	28	18	148	154	137	1.080	1.124	114.96	5.64
14	21	22	42	44	88	90	18	18	148	152	137	1.080	1.109	48.53	6.65
15	21	23	42	46	86	94	16	16	144	156	137	1.051	1.139	8.42	7.26
16	20	24	40	48	78	94	28	22	146	164	137	1.066	1.197	15.98	7.76
17	20	22	40	44	82	90	22	22	144	156	137	1.051	1.139	13.15	7.65
18	21	22	42	44	76	88	28	22	146	154	137	1.066	1.124	12.28	7.01
19	23	20	46	40	94	84	22	20	162	144	137	1.182	1.051	8.22	7.27
20	21	21	42	42	80	88	24	18	146	148	137	1.066	1.080	20.35	5.62
21	23	23	46	46	84	94	22	12	152	152	137	1.109	1.109	8.92	7.05
22	23	25	46	50	86	98	20	14	152	162	137	1.109	1.182	11.35	6.86
23	22	22	44	44	82	88	24	16	150	148	137	1.095	1.080	22.06	6.38
24	22	22	44	44	82	86	20	16	146	146	137	1.066	1.066	8.44	7.26
25	21	22	42	44	84	86	22	16	148	146	137	1.080	1.066	26.44	6.59
26	21	20	42	40	76	84	26	24	144	148	137	1.051	1.080	18.09	6.07
27	22	22	44	44	92	92	18	20	154	156	137	1.124	1.139	8.92	5.83
28	22	22	44	44	90	84	20	20	154	148	137	1.124	1.080	7.48	7.39
29	22	22	44	44	82	88	18	20	144	152	137	1.051	1.109	28.76	5.68
30	24	22	48	44	94	86	14	26	156	156	137	1.139	1.139	8.41	7.81

reveals distinct performance characteristics, with BGWO demonstrating superior computational efficiency while maintaining comparable solution quality. Specifically, BGWO achieved an average placement of 22 µ-PMUs with an average computation time of just 6.94 s, representing a remarkable 74% reduction compared to BPSO’s 26.92 s, despite BPSO’s slightly lower average of 21.8 µ-PMUs. Both algorithms successfully maintained system observability, as evidenced by their SORI values averaging 1.089 for BPSO and 1.107 for BGWO, comfortably above the critical threshold of 1.0. The strategic utilization of Zero Injection Bus (ZIB) properties through Kirchhoff’s Current Law (KCL) inference has demonstrated significant potential in reducing µ-PMU requirements for the IEEE 69-bus system. By leveraging the 20 identified ZIBs, the optimization algorithms effectively reduce the number of required µ-PMUs to approximately 20–22 devices, representing a substantial improvement over conventional placement strategy. The KCL inference mechanism contributes substantially to this reduction, with Binary Particle Swarm Optimization (BPSO) generating an average of 21.5 KCL inference measurements and Binary Grey Wolf Optimization (BGWO) producing 18.1 such measurements per solution. These KCL-based measurements effectively compensate for the reduced number of physical µ-PMUs, allowing the system to maintain full observability with fewer devices while achieving SORI values of 1.089 and 1.107 for BPSO and BGWO, respectively. The measurement distribution analysis reveals that the combined approach of direct PMU measurements (43.8 voltage and 83.9 current phasors for BPSO; 44.5 voltage and 89.1 current phasors for BGWO) coupled with KCL inference creates a robust observability framework that enables the reduction to 20–22 µ-PMUs without compromising system monitoring reliability. This integrated strategy demonstrates how intelligent utilization of network topology and physical laws can optimize monitoring infrastructure while maintaining the essential requirement of complete observability across all 137 state variables in the power system.

Figure 7 presents the results of PMU placement optimization for the IEEE 69 Bus System using a specific optimization method, where 20 PMUs are strategically placed across the distribution system. In (a), the optimal PMU placement is shown, with red dots indicating the PMU locations and blue circles marking buses without PMUs. The goal is to achieve full system observability by distributing the PMUs across the grid. In (b), the redundancy levels for each bus are displayed, with the green bars representing the achieved redundancy levels and the red dashed line showing the minimum required redundancy. Most buses meet or exceed the minimum redundancy level of 1, ensuring full observability, but some buses have higher redundancy levels, offering additional reliability and accuracy. (c) provides a redundancy level distribution across all buses, where the majority of buses have redundancy level 1, and a small number have redundancy levels of 2, suggesting that extra PMUs are placed in these areas for more accurate measurements. Finally, in (d), the measurement distribution is shown, with a breakdown of the measurement types: current phasors (yellow), voltage phasors (purple), and KCL inferences (blue). The chart highlights that current phasors are the dominant measurement type, with voltage phasors and KCL inferences contributing less significantly. The inclusion of KCL inferences is important for the system’s observability but plays a smaller role in this case due to the limited number of PMUs. KCL inferences are used to calculate the flow of currents at each bus based on Kirchhoff’s current law, which states that the total current entering a bus must equal the total current leaving. These inferences provide additional information on the current flow in the system and help in ensuring that the system’s electrical state is fully observable. However, KCL inferences require fewer measurements compared to voltage and current phasors, which are directly obtained from PMUs. Because only 20 PMUs are used in this case, the reliance on KCL inferences is limited, and the system places more weight on the phasor measurements to ensure full observability. In this setup, the PMUs leverage current phasors to a greater extent, while voltage phasors and KCL inferences play secondary roles in achieving complete system monitoring. Overall, the placement of 20 PMUs across the IEEE 69 Bus System, as shown in this figure, ensures that the system is fully observable while relying more heavily on phasor measurements and less on KCL inferences, which are used to supplement the phasor-based measurements. The redundancy levels are carefully distributed across the buses, enhancing the reliability and accuracy of the system’s monitoring.

4.4. Comparison of BPSO and BGWO

In this section, a detailed statistical analysis for both BPSO and BGWO algorithm is performed for each case study to determine the performance of algorithms across all the runs.

4.4.1. Statistical Analysis with IEEE 33 Bus System

The statistical analysis of BPSO performance for the 33-Bus System under normal operating conditions (Case 1) is based on 30 independent runs to evaluate the algorithm’s consistency, reliability, and convergence behavior in optimal PMU placement as presented in Figure 8. The boxplot of key performance metrics shows that the number of PMUs varies slightly between 12 and 14, with a median value of about 13, indicating stable convergence across runs. The SORI remains within a narrow range of 1.1 to 1.25, suggesting high system observability with good redundancy. Computation times mostly fall between 1.5 and 2.5 s, confirming the efficiency and repeatability of the PSO process. The histogram of PMU counts reveals that most runs achieve solutions with 12 to 13 PMUs, further demonstrating the reliability of the algorithm in finding near-optimal configurations. The SORI distribution peaks around 1.2 to 1.25, showing consistent network observability across all simulations. The computation time distribution is tightly clustered, reinforcing that the algorithm maintains a balanced trade-off between accuracy and speed. The PMUs versus SORI plot, color-coded by computation time, indicates that slightly higher redundancy levels (SORI ≈ 1.25–1.3) are achieved with one additional PMU or marginally longer processing times, reflecting the algorithm’s sensitivity to optimization complexity. The convergence iteration distribution confirms that all runs converged at the fixed upper limit of 1000 iterations, implying consistent stability throughout the optimization process. Overall, the BPSO algorithm demonstrates strong robustness, fast execution, and dependable convergence behavior, producing optimal PMU configurations that ensure complete observability and redundancy for the 33-Bus system under normal operating conditions.

The statistical analysis of BPSO performance for case study 2 with the same number of runs is presented in Figure 9. The results reveal that the BPSO algorithm consistently identifies the same optimal configuration requiring 24 PMUs to achieve full dual observability across all runs. The boxplot of key performance metrics shows that the number of PMUs remains constant at 24, the SORI values are centred around 2.0, and the computation time remains below 0.5 s, demonstrating the algorithm’s stability and rapid convergence. The distribution of PMU counts confirms that each run successfully achieved dual observability, while the SORI distribution between 1.5 and 2.5 indicates that every bus in the system is redundantly observable through at least two independent PMU paths, ensuring resilience against PMU or communication failures. The computation time histogram shows that most runs completed within 0.25–0.35 s, highlighting the computational efficiency of the BPSO method. The PMU versus SORI plot, coloured for dual observability, shows a single point representing all runs, confirming that BPSO converges to a global optimum with perfect repeatability. The dual observability success rate plot further validates a 100% success rate, proving the robustness and deterministic performance of the algorithm. Overall, for the 33-Bus System (Case 2), the BPSO algorithm demonstrates exceptional speed, consistency, and reliability, achieving complete dual observability and strong redundancy with minimal computational effort, making it a highly effective approach for reliable PMU placement in power system monitoring.

For the IEEE 33-Bus System in normal operating conditions (Case 1), the BGWO algorithm demonstrates strong consistency, accuracy, and computational efficiency in finding the optimal PMU placement. The analysis presented in Figure 10 shows that the best solution is achieved with 11 PMUs, ensuring complete system observability under normal operating conditions. The SORI ranges between 1.1 and 1.3, indicating reliable redundancy and full network observability. Computation times range from 1.5 to 2.8 s, showing that the algorithm converges quickly and efficiently. The PMU count distribution confirms that all runs achieved full observability, with 11 PMUs consistently identified as the optimal configuration. The computation time histogram supports the algorithm’s stability, as most runs completed within about two seconds. The PMU versus SORI plot and observability rate graph further confirm 100% observability success, proving that BGWO reliably converges to the global optimum across all runs. Overall, the BGWO algorithm exhibits excellent performance, producing a stable, fast, and fully observable solution with 11 optimally placed PMUs for the IEEE 33-Bus System.

For the IEEE 33-Bus System under dual observability conditions, BGWO algorithm exhibits remarkable consistency, speed, and accuracy across all 30 independent runs. Figure 11 presents different parameters for determining the performance of algorithm. In the top-left plot, the boxplot of key performance metrics shows that the number of PMUs remains constant at 24, ensuring full dual observability, while the SORI is centred around 2.0, confirming that each bus is redundantly observable through at least two independent measurement paths. The computation time remains under 1 s, reflecting the algorithm’s fast and stable convergence. In the top-middle plot, the distribution of PMU counts shows that all runs produced the same optimal configuration of 24 PMUs, highlighting the robustness and repeatability of the algorithm. The top-right plot, which displays the distribution of SORI values, indicates a tight clustering between 1.5 and 2.5, further validating consistent dual observability across all runs. The middle-left plot, showing voltage versus current phasor measurements, reveals nearly equal numbers of voltage and current readings (around 90 each), demonstrating a balanced measurement setup that enhances accuracy and reliability. In the middle-centre plot, the PMUs versus SORI relationship forms a single green point at 24 PMUs with a SORI ≈ 2.0, indicating that every run converged to the same optimal solution. The middle-right plot confirms a 100% dual observability success rate at 24 PMUs, emphasizing the algorithm’s precision and reliability. The bottom-left plot shows a consistent and proportional relationship between voltage and current phasor measurements, while the bottom-centre plot illustrates an equal percentage distribution of voltage and current data types, confirming a well-balanced configuration. Finally, the bottom-right plot demonstrates that computation times range between 0.3 and 0.7 s, with no run exceeding 1 s, showcasing the high computational efficiency of the BGWO algorithm. Overall, the BGWO provides a fast, stable, and fully reliable solution for optimal PMU placement in the IEEE 33-Bus System, achieving complete dual observability with 24 PMUs and maintaining consistent redundancy and balance across all measurement types.

4.4.2. Statistical Analysis with IEEE 69 Bus System

Figure 12 presents the statistical analysis of BPSO performance for the IEEE 69-Bus System (Case 1), based on 30 independent runs. The top-left plot shows the boxplot of key performance metrics, indicating that the number of PMUs required for optimal observability typically ranges from 24 to 26, with a median of around 25 PMUs. The SORI values fall between 1.05 and 1.2, ensuring the system is well-observed with moderate redundancy. The computation times for most runs range from 4 to 6.5 s, with the majority completing within approximately 5 s, highlighting the algorithm’s efficiency in converging to the solution. The top-middle plot illustrates the distribution of PMU counts, where 24 to 26 PMUs are identified, with 25 PMUs being the most frequent and optimal solution found. In the top-right plot, the distribution of SORI values shows that most of the runs have SORI values between 1.05 and 1.15, indicating high system observability and reliability. The bottom-left plot reveals the distribution of computation times, where most runs complete in the range of 4.5 to 5.5 s, confirming the algorithm’s fast convergence. The bottom-middle plot shows the relationship between PMUs and SORI, where runs resulting in 24 PMUs tend to achieve slightly higher SORI values, and the colour gradient reflects the computation time, with longer times generally associated with higher SORI values. Finally, the bottom-right plot demonstrates that all runs converge after 1000 iterations, confirming a fixed iteration limit and consistent convergence behavior across all runs. Overall, the BPSO algorithm efficiently finds optimal PMU placements, consistently achieving full observability with moderate redundancy, and does so in a computationally efficient manner, converging quickly within 1000 iterations and 4.5 to 5.5 s of computation time.

Figure 13 presents the statistical analysis of BPSO performance for the IEEE 69-Bus System (Case 2), based on 30 independent runs. The boxplot of key performance metrics shows that the number of PMUs required for optimal observability ranges from 51 to 56, with the median around 54 PMUs, indicating consistent results across the runs. The SORI values typically fall between 2.2 and 2.45, reflecting good system redundancy and full observability. The computation times for most runs are between 9 and 12 s, with a few runs taking up to 18 s, which is acceptable for a system of this size. The distribution of PMU counts shows that 54 PMUs is the most frequent configuration, with some variation, as a few runs use 55 or 56 PMUs. The SORI distribution indicates that most runs achieve SORI values between 2.15 and 2.3, confirming that dual observability is consistently met with moderate redundancy. The PMUs vs. SORI plot, coloured by computation time, reveals that runs with slightly higher PMU counts generally have higher SORI values, and longer computation times are associated with these configurations. Finally, the convergence iteration distribution shows that all runs converge at exactly 1000 iterations, demonstrating stable and predictable convergence behavior. Overall, the BPSO algorithm efficiently identifies the optimal PMU placement for the IEEE 69-Bus System, achieving full observability and redundancy while maintaining reasonable computation times and stable convergence across all runs.

Figure 14 presents a statistical analysis of the performance of the BGWO applied to the IEEE 69-bus system, Case 1, based on 30 independent simulation runs. The analysis consists of several key performance metrics, each represented through different visualizations that provide insights into BGWO’s effectiveness and efficiency. Starting from the top-left plot, the boxplot of key performance metrics illustrates the distribution of three critical parameters: the number of PMUs, SORI, and computation time (in seconds). The boxplot reveals that the number of PMUs remains stable across the runs, with values ranging between 24 and 26 PMUs. The SORI values vary between 1.04 and 1.08, suggesting consistent performance across simulations, while the computation time ranges from 25 to 45 s, showing some variation in the algorithm’s efficiency. Moving to the top-middle plot, the distribution of PMU counts histogram indicates that the majority of runs use a PMU count of 25, with few runs using values outside this range. This suggests that the algorithm converges to this particular number of PMUs, which may be an optimal configuration for the system in the given scenario. In the top-right plot, the distribution of SORI values histogram shows that most runs produce SORI values between 1.04 and 1.08, highlighting a high level of consistency in the system’s operational performance across the simulations. This suggests that the BGWO algorithm effectively maintains the desired system observability during optimization. The bottom-left plot, showing the distribution of computation times, reveals that the majority of the runs take between 25 and 35 s. This indicates that the BGWO algorithm is relatively efficient, with most simulations completing within this time frame. However, the spread of computation times suggests that some runs require more time due to differences in system configurations or computational complexity. Moving to the bottom-middle plot, the PMUs vs. SORI scatter plot, coloured by computation time, shows that as the number of PMUs increases, the SORI value improves, indicating better system observability with more PMUs. The plot also reveals that with a minimum of 24 PMUs, the SORI value stays above 1, ensuring complete system observability. Additionally, the plot shows that the algorithm places more PMUs, which in turn increases computation time, suggesting that higher PMU counts come with a trade-off of more computational effort required for optimization. Finally, the bottom-right plot, representing the convergence iteration distribution, shows that the BGWO algorithm typically converges in around 999 to 1000 iterations. This indicates the algorithm’s ability to reach an optimal solution efficiently and consistently across multiple runs, making it suitable for real-time or large-scale optimization tasks.

Figure 15 presents a comprehensive statistical analysis of BGWO (Binary Grey Wolf Optimizer) performance applied to the IEEE 69-bus system, Case 2, based on multiple simulations. The analysis includes various metrics visualized through different plots that help assess the optimization algorithm’s performance in terms of PMU placement and other factors such as fitness and redundancy. Starting with the top-left plot, the PMU Count Distribution histogram shows the frequency of different PMU counts used across simulations. The distribution appears to be spread across several values, mainly between 45 and 55 PMUs, indicating that the BGWO algorithm places a varying number of PMUs based on the optimization needs of each run. The top-middle plot, labelled Fitness Distribution, visualizes the frequency of different fitness values achieved by the algorithm across the simulations. The fitness values range between 2000 and 12,000, with the majority of values concentrated between 6000 and 8000. This suggests that the algorithm consistently reaches acceptable solutions, with the fitness metric indicating the optimality of the placement. The top-right plot, titled Minimum Redundancy Distribution, shows the distribution of minimum redundancy values achieved in the optimization process. Most of the runs achieve a minimum redundancy value of around 0.4, indicating that the algorithm generally strives to avoid redundant PMU placements, improving the efficiency of the system. Moving to the bottom-left plot, the Computation Time vs. PMU Count scatter plot visualizes the relationship between computation time (in seconds) and the number of PMUs. The plot indicates that there is no strong correlation between the number of PMUs and the computation time, with the computation time staying relatively constant across different PMU counts. This suggests that the algorithm’s performance in terms of computation time is not heavily influenced by the number of PMUs placed. The bottom-middle plot, PMU Count vs. Minimum Redundancy, shows the relationship between the number of PMUs and the minimum redundancy values. The plot, which uses colour to indicate the fitness value, demonstrates that the optimization process tends to place more PMUs without significantly increasing redundancy. This reinforces the idea that the algorithm balances between improving system observability and minimizing redundancy. Finally, the bottom-right plot, the Box Plot of Key Metrics, presents a boxplot of three key metrics: the number of PMUs, fitness value, and minimum redundancy. The boxplot reveals that the number of PMUs generally varies between 45 and 55, with no extreme outliers. The fitness and minimum redundancy metrics show more consistent values, suggesting that the BGWO algorithm delivers stable results across different runs.

Figure 16 evaluates the performance of the BPSO algorithm in the IEEE 69 Bus System for Case 3 (ZIBs Case) using 30 independent runs. In the top left of the figure, a boxplot displays the distribution of key metrics: the number of PMUs, SORI, and computation time. Using ZIBs and KCL inferences, the number of PMUs has been reduced to 20, optimizing system observability without compromising performance. The SORI values primarily fall between 1.05 and 1.1, indicating stable observability, while computation time varies, reflecting different computational demands across the runs. In the top middle, a histogram shows the distribution of PMU counts, with the number 20 being the most frequent, highlighting the efficiency of using fewer PMUs for optimal results. The top right shows the distribution of SORI values, which are concentrated between 1.05 and 1.1, confirming that the algorithm achieves reliable system observability in most cases. In the bottom left, a scatter plot of PMUs versus SORI, with colour coding representing computation time, reveals that as the number of PMUs increases, the computation time also rises, though the system remains efficient despite the increase in PMU count. The bottom middle of the figure presents the success rate versus PMU count, which remains consistently at 100% for all PMU configurations, demonstrating the algorithm’s reliability in achieving optimal results. Finally, the bottom right shows the distribution of convergence iterations, with most runs converging within 200 iterations, indicating that the BPSO algorithm efficiently reaches optimal solutions with minimal iteration counts. Overall, the figure highlights the efficiency of BPSO in optimizing PMU configurations, where the reduction to 20 PMUs through ZIBs and KCL inferences provides an effective and reliable solution for system observability.

Figure 17 presents a statistical analysis of the BGWO algorithm applied to the IEEE 69 Bus System, based on 30 independent runs. The analysis includes several key performance metrics: the number of PMUs, SORI, computation time, success rate, and convergence iterations. ZIBs have been leveraged in the optimization process, and by using KCL inferences, we determine that 20 PMUs provide the best solution for system observability.

The top-left boxplot shows the distribution of key performance metrics, including the number of PMUs, SORI, and computation time. The number of PMUs is optimized to 20 based on ZIBs and KCL inferences, offering an efficient and reliable solution for system observability. The top-middle histogram illustrates the distribution of PMU counts, with the highest frequency observed at 20 PMUs, supporting it as the optimal choice. The top-right histogram displays the SORI values, with most runs achieving values around 1.1, ensuring stable observability. The bottom-left scatter plot shows the relationship between the number of PMUs and SORI, with computation time represented by colour. While computation time increases with more PMUs, 20 PMUs results in efficient performance. The bottom-middle bar chart shows a consistent 100% success rate for all PMU configurations, and the bottom-right histogram reveals that most runs converge quickly, within 50.5 to 51.5 iterations. This demonstrates the efficiency of the GWO algorithm, with 20 PMUs as the optimal solution for system observability based on ZIBs and KCL inferences.

5. Trade-Off Analysis

Figure 18 presents a comprehensive Trade-off analysis for the IEEE 33-Bus and IEEE 69-Bus systems, focusing on optimizing the number of PMUs for system observability, redundancy, and fault tolerance. The analysis is carried out using BPSO and BGWO across multiple cases. (a) PMUs vs. SORI for IEEE 33-Bus System, Case 1: This plot shows the relationship between the number of PMUs and the System Observability and Reliability Index (SORI) for the IEEE 33-Bus system in Case 1. As the number of PMUs increases, the SORI value also increases, indicating that the system’s observability and reliability improve with the addition of more PMUs. This suggests that by adding more PMUs, the system becomes more observable and stable, which is crucial for its performance and fault tolerance. (b) PMUs vs. Minimum Redundancy for IEEE 33-Bus System, Case 2: This plot illustrates the minimum redundancy required for dual observability in the IEEE 33-Bus system, Case 2. The dashed red line represents the minimum redundancy threshold for ensuring dual observability. The BPSO/BGWO solution, marked by the blue dot, shows the number of PMUs required to achieve dual observability while minimizing redundancy. This plot emphasizes the balance between observability and redundancy, ensuring system reliability with minimal additional complexity. (c) PMUs vs. SORI for IEEE 69-Bus System, Case 1: This plot shows the relationship between the number of PMUs and SORI values for the IEEE 69-Bus system, Case 1. Similarly to the 33-Bus system, as the number of PMUs increases from 22 to 26, the SORI value improves, indicating that more PMUs enhance system observability and reliability. The BPSO/BGWO method ensures that the number of PMUs is optimized for maximum observability, without significantly increasing complexity. (d) Pareto Front Analysis for IEEE 69-Bus System, Case 2: This section presents a detailed Pareto front analysis for the IEEE 69-Bus system in Case 2. The PMUs vs. Minimum Redundancy plot shows the Pareto front for the minimum redundancy required for dual observability. It demonstrates that the optimal number of PMUs balances observability and redundancy, ensuring fault tolerance with minimal PMU count. The PMUs vs. Average Redundancy plot shows that as the number of PMUs increases, the average redundancy also increases, which contributes to better system fault tolerance. In the PMUs vs. SORI plot, it is evident that increasing PMUs improves the system’s observability (SORI), ensuring more reliable operation. Finally, the Combined View: All Redundancy Metrics plot compares the minimum and average redundancy metrics, showing how redundancy improves with the increasing number of PMUs, enhancing overall system resilience and reliability. (e) Simple Pareto Analysis for IEEE 69-Bus System, Case 3: This section provides a simplified Pareto analysis for the IEEE 69-Bus system, Case 3, focusing on the trade-offs between PMUs and observability metrics. The PMUs vs. SORI plot shows the Pareto front for PMUs and SORI, where the red dot marks the optimal BPSO/BGWO solution. This indicates that increasing PMUs results in higher observability (SORI), ensuring the system’s ability to monitor and control the power grid. The PMUs vs. Minimum Redundancy plot shows that the minimum redundancy for dual observability does not significantly change with the number of PMUs, indicating that a relatively low number of PMUs can still maintain sufficient redundancy. The PMUs vs. Average Redundancy plot reveals that average redundancy increases as PMUs are added, ensuring better fault tolerance. The 3D Trade-off combines PMUs, SORI, and minimum redundancy, showing how the optimal solutions for both observability and redundancy are distributed. The colour gradient indicates the minimum redundancy required for each solution, helping to visualize the trade-offs between the number of PMUs, system observability, and redundancy. In summary, the figure represents a detailed pareto analysis of the IEEE 33-Bus and IEEE 69-Bus systems, illustrating how the number of PMUs affects system observability, redundancy, and fault tolerance. The use of BPSO and BGWO optimization methods ensures that the systems are optimized for both performance and reliability, balancing the trade-offs between these key factors across different cases. The results highlight the efficiency of the optimization process in determining the optimal number of PMUs to achieve dual observability while minimizing redundancy.

6. Voltage Estimation Using WLS and Forward-Backward Sweep Algorithms

In this section, voltage is estimated by employing WLS algorithm. This strategy implements a WLS state estimation algorithm for both distribution systems using PMU data, where it first runs a manual power flow simulation using the Forward-Backward Sweep method to generate true voltage magnitudes and angles, then creates synthetic PMU measurements at strategically placed buses by adding realistic Gaussian noise to both voltage measurements (magnitude and angle) and current measurements (magnitude and angle) on incident branches, and finally performs iterative WLS estimation with Levenberg–Marquardt damping to estimate the system state by minimizing the difference between measured and calculated values while accounting for measurement uncertainties through a weighted matrix based on the assumed noise characteristics.

Based on the comparative analysis presented in Figure 19, which evaluates voltage estimation accuracy using WLS against Forward-Backward Sweep method power flow solution, distinct performance differences emerge between the BPSO and BGWO algorithms for optimal PMU placement in the IEEE 33-bus system. The voltage magnitude and angle estimation results demonstrate that while both algorithms produce PMU placements that yield generally correlated estimates, the BGWO-optimized placement achieves notably superior accuracy. In the true versus estimated comparison (Subfigures a, c, e, f), the BGWO algorithm exhibits significantly tighter clustering along the reference line, particularly for voltage angle estimation, indicating stronger correlation with the benchmark values and reduced estimation variance. The estimation error profiles (Subfigures b, d, g, h) further substantiate this performance disparity. The BPSO-optimized placement reveals substantial voltage angle errors exceeding 0.5 degrees at multiple buses, with notable magnitude errors particularly evident in the larger bus indices. Conversely, the BGWO algorithm maintains consistently lower errors across both voltage magnitude and angle parameters, with error magnitudes reduced by approximately 60–70% compared to the BPSO results. This error distribution pattern suggests that the BGWO-derived PMU placement provides more comprehensive system observability with minimized localized estimation blind spots. The robustness of the BGWO is particularly evident in its handling of voltage angle estimation, which typically presents greater challenges than magnitude estimation. The algorithm’s ability to maintain angle errors below 0.2 degrees across all buses, compared to the fluctuating and substantially larger errors observed with BPSO, underscores its superior performance in determining PMU placements that yield high-fidelity state estimates. These findings demonstrate that the BGWO algorithm produces a PMU configuration that enables the WLS estimator to achieve voltage estimates significantly closer to the true system state, with particular advantage in the more sensitive voltage angle domain, thereby enhancing the reliability of system monitoring and control applications.

Figure 20 illustrates the voltage estimation performance for the IEEE 33-bus system (Case Study 2) using BPSO and BGWO algorithms, which in this case produced identical PMU placements. As a result, both algorithms exhibit nearly identical estimation accuracy. The voltage magnitude and angle profiles in Subfigures (c) and (d) show that the estimated values closely follow the true reference curves, confirming the effectiveness of the chosen PMU configuration. Similarly, the error plots in Subfigures (a) and (b) demonstrate low estimation errors across all buses—voltage magnitude errors remain below 0.01 p.u., and angle errors stay within 0.5°. Since both algorithms converge to the same optimal PMU arrangement, their performance in terms of observability, estimation precision, and overall system accuracy is effectively equivalent, validating the robustness and consistency of the obtained optimal configuration.

Based on the comparative analysis presented in Figure 21, which evaluates voltage estimation accuracy using the Weighted Least Squares (WLS) method against the Forward-Backward Sweep power flow solution for the IEEE 69-bus system (Case Study 1), clear performance distinctions emerge between the BPSO and BGWO algorithms for optimal PMU placement. While both optimization approaches produce observable configurations enabling reliable state estimation, the BGWO-optimized PMU placement delivers markedly superior voltage magnitude and angle estimation accuracy across the entire network. In the true versus estimated voltage profiles (Subfigures c, d, g, h), the BGWO results exhibit a stronger alignment with the true system state, reflecting minimal deviation and higher fidelity in both magnitude and angle estimation. In contrast, the BPSO outcomes reveal significant deviations at the feeder end (particularly buses 65–69), where the estimated voltage angles diverge drastically from the true values—confirming instability and weaker observability in that region. The error distribution plots (Subfigures a, b, e, f) further substantiate this contrast. Under the BPSO configuration, voltage magnitude errors exceed 0.25 p.u. and angle estimation errors reach up to 300°, indicating critical inaccuracies and localized observability gaps. Conversely, the BGWO algorithm consistently maintains voltage magnitude errors below 0.08 p.u. and voltage angle deviations under 1.5°, demonstrating improved numerical stability and estimation robustness. Overall, the BGWO-based PMU placement achieves greater estimation consistency, lower error variance, and enhanced system observability, especially in the more sensitive voltage angle domain. This improvement directly enhances the accuracy and reliability of the WLS estimator, yielding voltage profiles that remain closely correlated with the true network state. Consequently, the BGWO algorithm provides a more resilient and optimal PMU configuration for the IEEE 69-bus system, enabling more precise real-time monitoring and control compared to the BPSO-derived configuration.

From the results in Figure 22, in the true versus estimated voltage profiles (Subfigures c, d, g, h), the BGWO algorithm exhibits a tighter correlation between the true and estimated voltage magnitude and angle values, maintaining closer alignment across all buses. The BPSO results, while generally following the same trend, show more fluctuations, particularly at buses around indices 10–20 and 55–65, where deviations between true and estimated voltage angles are more pronounced. This suggests that the BPSO configuration produces localized inaccuracies and less robust observability in certain feeder sections.

The error distribution plots (Subfigures a, b, e, f) further substantiate this contrast. Under the BPSO algorithm, voltage magnitude errors rise to about 0.035 p.u., with voltage angle estimation errors approaching 1°, particularly concentrated around the initial and terminal buses. Conversely, the BGWO algorithm maintains significantly smaller voltage magnitude errors below 0.012 p.u. and angle estimation errors largely confined under 0.5° across the system. This smoother error distribution highlights BGWO’s stronger convergence properties and its ability to reduce estimation noise. The BGWO-based PMU placement delivers higher accuracy and stability in voltage magnitude and angle estimation than BPSO, minimizing errors and improving observability. Overall, BGWO proves more robust and reliable for optimal PMU deployment in the IEEE 69-bus system.

Figure 23 illustrates the voltage magnitude and angle estimation performance for the IEEE 69-bus system (Case Study 3), where Zero Injection Buses (ZIBs) are integrated to enhance observability and minimize PMU count through Kirchhoff’s Current Law (KCL) inference. The comparison between the BPSO and BGWO algorithms demonstrates that, even with a reduced number of PMUs, accurate state estimation can still be achieved when ZIB constraints are properly leveraged. In this configuration, the BGWO-optimized PMU placement exhibits superior estimation accuracy for both voltage magnitude and phase angle, as evident from Subfigures (e–h). The algorithm maintains close alignment between true and estimated values across all buses, with voltage magnitude errors remaining below 0.1 p.u. and angle deviations within 1.5°. In contrast, the BPSO placement (Subfigures a–d) shows localized estimation deviations, particularly at terminal nodes, indicating less stable performance under ZIB conditions. By effectively incorporating KCL-based inferences at ZIBs, the BGWO algorithm enhances the efficiency of the PMU configuration—achieving complete system observability with fewer measurement units while preserving high estimation fidelity. This confirms that ZIB-assisted BGWO provides a technically robust and cost-efficient solution for PMU placement in large-scale distribution networks, ensuring accurate state estimation despite sensor reduction.

Impact of Variation in Noise Levels on Voltage Estimation

For further investigation of the robustness of placement, varying noise levels were introduced, and the results were compared for each case study. Two levels of noise are introduced, which are 0.02 and 0.04, to observe the effects on estimation, and deviations in estimated values from reference values are observed. The influence of measurement noise on state estimation accuracy was examined across both IEEE 33-bus and 69-bus systems using the BPSO and BGWO optimized PMU placements under noise levels of 0.02 STD and 0.04 STD (Figure A1, Figure A2, Figure A3 and Figure A4 in Appendix A). Overall, increasing noise variance directly amplifies the estimation error in both voltage magnitude and angle, with more pronounced effects on angle estimation due to its higher sensitivity to phase perturbations. In the IEEE 33-bus system (Figure A1 and Figure A2 in Appendix A), both algorithms maintain close alignment between true and estimated voltage profiles under 0.02 STD, with BGWO exhibiting tighter clustering and minimal deviation around the true line. However, at 0.04 STD, the BPSO algorithm shows significant spikes in angle errors exceeding 1°, especially at higher bus indices, while BGWO sustains relatively stable estimations with only minor dispersions. This demonstrates BGWO’s superior robustness to stochastic variations introduced by sensor noise, resulting from its stronger convergence and global search behavior. For the IEEE 69-bus system (Figure A3 and Figure A4 in Appendix A), similar trends are observed. At 0.02 STD, both algorithms yield acceptable performance, but with subtle angle oscillations in BPSO placements for Case 2 and 3. Under 0.04 STD, these oscillations amplify, and error bars widen across several buses, particularly near weakly observed nodes. Since both algorithms generate number of PMUs for case study 3, both maintain a consistently smoother estimation curve with smaller voltage magnitude deviations and reduced angle fluctuations, indicating higher immunity to noise propagation through zero-injection buses (ZIBs) and measurement redundancy. Comparatively, overall, across all cases, BGWO demonstrates more reliable estimation accuracy under increasing noise, maintaining observability with minimal error amplification. In contrast, BPSO’s performance deteriorates as noise variance increases, reflecting its higher susceptibility to local minima and less effective error correction during iteration. Thus, the BGWO-optimized PMU placement ensures improved voltage state reconstruction and estimation stability, validating its robustness under practical noisy measurement environments.

7. Discussion

The optimized PMU placements obtained from both the BPSO and BGWO algorithms demonstrate distinct performance characteristics in achieving complete network observability while minimizing sensor redundancy. For the IEEE 33-bus and 69-bus systems, both algorithms successfully identified minimal PMU configurations under different case studies, including those incorporating zero-injection buses (ZIBs). However, the BGWO-based configurations consistently provided better coverage of voltage and current measurement channels, leading to enhanced observability. In particular, BGWO placements yielded higher counts of effectively observed current channels across interconnected branches, ensuring improved redundancy and stronger topological awareness, whereas BPSO placements displayed isolated unmonitored regions under higher bus indices. This confirms that BGWO not only minimizes the total number of PMUs but also distributes them more strategically to balance local and global observability in the network. The statistical analysis further substantiates the superiority of BGWO over BPSO in optimization reliability and stability. Across multiple independent runs, BGWO produced lower standard deviation, variance, and mean objective function values, reflecting a more consistent convergence toward the global optimum. Its balance between exploration and exploitation enables efficient search-space traversal, reducing susceptibility to premature convergence, a limitation often observed in particle-based swarm optimizers. The box-plot distribution and performance indices also indicate tighter clustering of fitness values under BGWO, confirming its robustness and repeatability. These outcomes validate that the BGWO algorithm possesses stronger stochastic stability and adaptive control characteristics, ensuring high confidence in the optimized PMU configurations even under different initialization and noise conditions. Subsequent voltage estimation using the WLS state estimation algorithm further highlights the influence of optimized placements on estimation fidelity. When the PMU configurations derived from BGWO and BPSO were evaluated under varying noise levels (0.02 and 0.04 STD), both produced estimations correlated with true system states; however, BGWO placements consistently achieved higher accuracy. The WLS results showed that BGWO configurations yielded smaller voltage magnitude and angle deviations across all bus indices, maintaining tight alignment with true profiles. Conversely, BPSO results exhibited noticeable dispersion, particularly in the voltage angle domain, where errors exceeded 0.5° at multiple nodes under increased noise. This stability under stochastic disturbances confirms that BGWO-optimized placements enhance the redundancy and quality of measurement data supplied to the WLS estimator, directly improving estimation robustness and numerical conditioning. Overall, the discussion reinforces that the BGWO algorithm outperforms BPSO by generating strategically distributed PMU placements that capture a broader range of measurement channels, ensure consistent statistical convergence, and provide superior voltage estimation accuracy through WLS. Its capacity to maintain estimation precision under noisy conditions establishes it as a reliable and computationally efficient optimization tool for modern power distribution networks. Furthermore, the analysis reveals a critical and nuanced trade-off between cost, redundancy, and estimation accuracy, which is central to the PMU placement problem. The strategic use of ZIBs is a primary driver of this trade-off. By design, ZIBs reduce the minimal number of PMUs required for topological observability, directly lowering the capital cost of the sensor network. However, this cost-saving measure comes at the expense of measurement redundancy. This is quantitatively captured by the SORI, which is inherently lower in ZIB-based configurations (e.g., Case 1 and Case 3) compared to cases where observability is achieved through PMUs alone. The WLS state estimation results under varying noise levels (0.02 and 0.04 STD) provide direct evidence of the consequence of this reduced redundancy. As noise increases, the configurations with fewer PMUs and a lower SORI, specifically, the ZIB-inclusive cases exhibit a more pronounced degradation in estimation accuracy. The sparser measurement set offers a weaker “data buffer” against stochastic errors. With fewer independent channels to cross-validate, the WLS estimator has a harder time isolating and suppressing noise, leading to greater uncertainty and error dispersion in the estimated voltage magnitudes and angles. This explains why Case 1 and Case 3 showed the most significant error inflation. Conversely, the BGWO-optimized placements consistently yielded a higher SORI than their BPSO counterparts for the same number of PMUs. This indicates that BGWO not only minimizes the device count but also strategically positions PMUs to cover a wider array of current measurement channels across interconnected branches. This superior redundancy is the linchpin for robust estimation. A higher SORI means the WLS algorithm receives a richer, more correlated set of measurements, which improves the numerical conditioning of the gain matrix and enhances the algorithm’s ability to filter out noise. Consequently, even with the same number of PMUs, the BGWO configurations maintained tighter alignment with the true system states and demonstrated superior stability under stochastic disturbances.

Table 6. Comparative Analysis of BPSO & BGWO with Literature.

IEEE Test Systems	Case Studies	Nµ-PMUs with BPSO	Nµ-PMUs with BGWO	[38] MILP	[9] ILP	[29] ILP	[6] ILP
33	Case 1	11	11	11	11	NR	NR
69	Case 1	24	24	24	NR	24	24
33	Case 2	24	24	NR	NR	NR	NR
69	Case 2	51	51	NR	NR	NR	NR
69	Case 3	20	20	16	NR	NR	NR
NR means	Not Reported	None of the above presented articles included any detailed Redundancy analysis.

presents a comparative assessment of optimal PMU placements obtained using the BPSO and BGWO algorithms, benchmarked against other existing optimization techniques. The table also summarizes the associated performance parameters observed during the analysis, providing a comprehensive evaluation of each method’s effectiveness in achieving system observability and estimation accuracy.

8. Conclusions

This study optimized the placement of μ-PMUs using the BPSO and BGWO algorithms for the IEEE 33-bus and 69-bus systems, focusing on system observability, computational efficiency, and voltage estimation accuracy. In Case Study 1, both algorithms achieved full system observability for the IEEE 33-bus system with 11 μ-PMUs. For the IEEE 69-bus system, both BPSO and BGWO algorithms identified 24 μ-PMUs as the optimal placement configuration, achieving the minimum required PMUs for robust observability. In Case Study 2, redundancy requirements increased the μ-PMU count to 24 μ-PMUs for the IEEE 33-bus system and 51 μ-PMUs for the IEEE 69-bus system, ensuring full system observability under the constraint of a single μ-PMU failure. The Pareto analysis illustrated the trade-off between minimizing the number of μ-PMUs and enhancing redundancy, where the increase in PMU counts improved redundancy and system resilience but raised costs and computational complexity. Case Study 3, which incorporated Zero Injection Buses (ZIBs), optimized the μ-PMU placements, reducing the number of μ-PMUs to 20 μ-PMUs for both BPSO and BGWO in the IEEE 69-bus system. This configuration demonstrated efficient use of topological information, minimizing PMU requirements while maintaining observability. The statistical analysis of algorithm performance revealed that BGWO consistently outperformed BPSO in terms of computational efficiency, convergence smoothness, and stability, particularly in larger, more complex systems such as the IEEE 69-bus system. BGWO exhibited superior robustness against premature convergence and instability, ensuring consistent results across multiple runs. In contrast, BPSO exhibited faster convergence but encountered challenges with local minima and erratic behavior in larger systems, limiting its performance in high-dimensional search spaces. The voltage estimation results, obtained using the Weighted Least Squares (WLS) method under varying noise levels, demonstrated the critical impact of redundancy on estimation accuracy. Case Study 2, with higher redundancy in the μ-PMU placement, provided the most reliable voltage estimates, minimizing deviations from reference values, especially under higher noise levels. Conversely, Case Study 1 and Case Study 3, with fewer μ-PMUs, showed larger estimation errors, underlining the trade-off between the sensor count and estimation reliability in noisy environments. This study demonstrates that optimal μ-PMU placement is essential for achieving reliable state estimation in power systems, balancing the need for reduced sensor count, robust observability, and accurate voltage estimation under uncertainty. The results indicate that BGWO is more suitable for large, complex systems due to its global search capability and convergence stability. At the same time, BPSO is more effective for smaller systems where faster convergence is prioritized. The proposed methodology offers an efficient and comprehensive solution for real-time state estimation in modern smart grids. This work contributes directly to power system sustainability by providing a foundation for enhanced grid monitoring and resilience, which are critical for integrating renewable energy and boosting operational efficiency. Future work should explore multi-objective optimization approaches to balance μ-PMU placement, computational efficiency, and estimation reliability and investigate hybrid optimization techniques combining classical and AI-based algorithms to address the computational challenges of large-scale, dynamic power systems.