Perspective Chapter: Optimizing μ-PMU Placement for Estimating Asymmetrical Distribution Network States – Introducing a Novel Stochastic Two-Stage Approach

Abstract

Distribution system state estimation plays a crucial role in supplying essential data for system monitoring and control. However, the presence of uncertain parameters such as the variable output of distributed generation (DG), random meter errors, and inaccurate network parameters poses a significant challenge to achieving situational awareness (SA) in distribution systems. To address these challenges, this study introduces an innovative two-stage stochastic programming model. In the first stage, the model focuses on the optimal placement of μ-PMUs with the objective of minimizing installation costs, maximizing measurement redundancy, and enhancing system observability. This is particularly important in the context of partially zero injection nodes (PZIN). In the second stage, the model performs state estimation for three-phase asymmetric DG-integrated distribution systems, aiming to improve SA. The application of the proposed model resulted in the identification of optimal μ-PMU locations in the presence of PZINs and various contingencies. The distribution system state estimation achieved high accuracy and a low error percentage. The feasibility and effectiveness of this methodology were validated using the modified IEEE 85-bus distribution system. In addition, we incorporated extended Kalman filter (EKF) state estimation to compare with the weighted least square method.

1. Introduction

1.1 Concept and incitement

Monitoring in real-time is crucial for implementing control and protection functions in an electric distribution network. Efficiently leveraging current measurements is essential for producing the most precise estimation of the system state that corresponds to the available data. Distribution System State Estimation (DSSE) is acknowledged as a critical component of the distribution management system. Despite significant progress in TSSE over time, it is essential to recognize the fundamental differences between transmission and distribution networks. Consequently, the functional needs of TSSE cannot be directly translated to DSSE [1]. Distinctive features of distribution systems encompass load imbalances, elevated R/X ratios, and reduced observability stemming from inadequate real-time monitoring. On the other hand, transmission systems possess a surplus of measurements, ensuring ongoing system observability through redundancy [2]. Observability in distribution systems faces constraints due to the restricted availability of real-time measurements. The incorporation of μ-PMUs into distribution systems improves the accuracy of DSSE by furnishing precise details on voltage magnitude and phase angle. In today’s distribution networks, the performance requirements for DSSE are becoming more rigorous, primarily driven by challenges stemming from the integration of DG and the adoption of sophisticated technology. The main impediments and difficulties linked to DSSE are concisely delineated in [3].

In transmission networks, the inherent assumption is the operation’s symmetry, necessitating all calculations to be performed on single-phase positive sequences [4]. Distribution systems possess unique attributes, such as a radial structure, unbalanced loads, and a deficiency in real-time measurements, distinguishing them from transmission systems. In situations like these, the application of state estimation methods originally tailored for transmission systems is impractical for distribution systems, necessitating the adoption of three-phase state estimation. The scarcity of real-time measurements is a distinctive characteristic of DSSE. To ensure system observability, the incorporation of pseudo measures becomes necessary, representing the active and reactive powers of both loads and distributed generation (DG) [5].

1.2 Literature review

The authors in Refs. [6, 7, 8, 9, 10, 11] propose several approaches to modeling the mathematical problem, application of μ-PMUs, and locating the metering instruments utilized in Distribution System State Estimation (DSSE). In [12], a pioneering hybrid approach was introduced for state estimation in power systems. This method combines weighted least squares with the integration of phasor measurement units and supervisory control and data acquisition systems. In Ref. [13], a novel algorithm is presented for intricate linear state estimation. This algorithm considers the noise parameter and incorporates data from distribution phasor measurement units to improve the SA of distribution systems. Utilizing high-quality data from these units, this method facilitates the observability of unbalanced and dynamic distribution systems. Regularized state estimation technique was developed in Ref. [14] for robust monitoring of the distribution network. The primary objective was to achieve precise tracking of the system state on a faster time scale, thereby enhancing reliability according to the needs of the new operating environment. Ref. [15] introduces a method designed to enhance the reliability of the state estimation process and equipment performance within power system substations. This involves strategically situating channels and sources of phasor measurement units. The method consists of two stages. In the first stage, the application of a genetic algorithm optimizes the placement of phasor measurement units. Following this, the second stage employs an innovative approach to optimize measurements and allocate channels effectively.

Authors in paper [16] introduce an innovative approach for evaluating the condition of unbalanced three-phase distribution systems. This methodology utilizes branch currents as state variables, calculated in rectangular coordinates. Employing Phasor Measurement Units (PMUs) located at monitored busses, the approach provides branch currents and nodal voltages. Inequality constraints in DSSE involve non-monitored load busses, constrained by daily load variations from the preceding estimation period. The approach proposed by the authors in [17] innovatively combines the weighted least squares (WLS) and Levenberg-Marquardt methods. This fusion relies on incorporating data sourced from smart meters, significantly contributing to the implementation of real-time voltage control strategies. A novel state estimation framework has been introduced in [18], incorporating equality constraints related to voltage-dependent loads and zero injection. This framework aims to improve both the accuracy of state estimation and the effectiveness of bad data detection. Reference [19] introduces a unique method for static harmonic state estimation in distribution networks. This approach utilizes optimization techniques and assumes the strategic positioning of a small number of phasor measurement devices along the feeders. In [20], the author advocates a data-driven strategy to correct measurement errors inherent in the traditional weighted least squares methodology.

Authors in paper [21] present an innovative decentralized state estimation approach employing a multi-agent system to address the intricacies of distribution systems. The key idea involves dividing the system into smaller subsystems for more efficient and rapid estimation. Subsequently, a metaheuristic algorithm, specifically the artificial bee colony algorithm, is applied to solve the state estimation problem in a multi-agent context. Reference [22] introduces an original framework for the multi-objective optimization of Phasor Measurement Unit (PMU) placement. The goal is to minimize the total PMU installation cost, optimize the selection of current channels, and diminish errors in state estimation. To address this mixed-integer nonlinear programming problem, the non-dominated sorting genetic algorithm II is employed as the solution approach. Reference [23] introduces a comprehensive method for estimating and monitoring the overall state of low-voltage distribution networks, specifically those with significant integration of photovoltaic systems. The effectiveness of the proposed technique is showcased in scenarios with high imbalance. Validation of the efficiency of the suggested low-voltage linear state estimation method is carried out through testing on a representative imbalanced residential network. A comprehensive modeling strategy, as suggested in [24], focuses on integrating the effects of Zero-Injection Nodes (ZIN) to optimize the positioning of Phasor Measurement Units (PMUs) and guarantee full observability within power transmission systems. Achieving globally optimal solutions involves the utilization of an intelligent BTS algorithm in the presentation of optimal PMU placement strategies. In Ref. [25], an optimal placement model for Phasor Measurement Units (PMUs) in transmission networks is presented. This model, utilizing multi-objective mixed-integer linear programming, considers variables including the installation cost of PMUs, observability constraints, and the capability to identify gross errors. A framework for investigating gross errors is introduced in [26], which employs μ-PMUs to separate measurement errors. This approach involves analyzing gross errors in the measurements acquired from both smart meters and SCADA devices as a post-processing step within the non-linear DSSE framework.

The measurements of both active and reactive power injection demonstrate non-linear features, without a readily available direct solution. The determination of the measurement function requires an iterative approach, as highlighted in Ref. [27]. Asymmetric distribution systems and the inherent non-linear nature of measurement data prompt the formulation of models in this research field as non-linear programming problems. Diverse solvers are employed to tackle and resolve challenges posed by these problems. A variety of potent metaheuristic algorithms are employed to attain globally optimal solutions, including a combination of particle swarm optimization and chaotic gravitational search algorithms [5], genetic algorithm [15], artificial bee colony algorithm [21], non-dominated sorting genetic algorithm-II [22], and a hybrid multi-objective particle swarm optimization krill herd algorithm [28]. Established solvers such as Gurobi and CPLEX [29] are applied to address challenges in DSSE and obtain accurate solutions for the system state. This study introduces a DSSE model aimed at improving observability and reducing computational complexity and calculation time in DG-based distribution networks. Employing the Taylor series approach, this model approximates and linearizes the non-linear functions related to active and reactive power, leading to a more thorough state estimation compared to existing DSSE techniques. The final step involves computing the estimated state vector using the weighted least square method.

1.3 Novelties and contribution

Despite notable advancements in DSSE, there remain several challenges and deficiencies that require careful consideration. In brief, the inadequacies identified in previous references can be summarized as follows:

1. In distribution networks, there is a multitude of distributed smart meters; however, only a limited subset of these devices can simultaneously sample voltage and power values. Additionally, the data collection process for these devices, reliant on wireless signals for data transfer to an access point, is susceptible to data loss. Given these practical challenges, the quantity of measurements in distribution networks falls significantly short of ensuring system observability.

2. Distribution networks with asymmetrical structures exhibit distinct characteristics, including the presence of single-phase and two-phase loads at specific busses. Current research employs Fully Zero Injection Nodes to minimize the need for μ-PMUs. In the context of FZIN, a bus in an asymmetrical structure catering exclusively to a single-phase load is designated as a non-zero injection bus. It’s important to note that two phases of this bus exhibit zero injection characteristics, potentially enhancing network observability with a reduced number of μ-PMUs. However, relying solely on FZIN may not fully demonstrate the effectiveness of zero injection properties.

To address the shortcomings identified in prior literature, this chapter presents an innovative two-stage programming model. In the first phase, the emphasis is on resolving the optimal μ-PMU placement problem. The objective is to minimize the installation cost of μ-PMUs while ensuring extensive observability in the presence of PZIN under various contingencies. The second phase involves conducting a three-phase asymmetrical DSSE to enhance SA. The principal novel contributions of this chapter are delineated as follows:

1. To overcome the first limitation (a), this paper presents a DSSE model that employs the Taylor series approach for approximations. The objective is to linearize the non-linear functions linked to active and reactive power, ensuring an exhaustive state estimation of the network that exceeds the capabilities of existing DSSE methods.

2. To address the second limitation (b), the proposed optimal μ-PMU placement problem incorporates the concept of PZINs. This concept pertains to busses with solely one or two zero injection phases, and it is employed to improve the overall observability of three-phase asymmetrical distribution networks.

Besides the key advancements mentioned previously, this paper introduces several contributions outlined as follows:

1. Utilizing both single-phase and three-phase μPMUs in radial distribution networks with asymmetrical structures to minimize the required number of installed μ-PMUs. To implement this concept, a conventional model for the optimal placement of PMUs in symmetrical transmission networks is adapted, and a distinct model is derived. In this unique model, the objective function and constraints align with the asymmetrical characteristics of radial distribution networks.

2. Considering various contingency situations in distribution networks, including the failure of a single μPMU and an individual line outage, within the model designed for the asymmetrical and radial structure of distribution networks.

3. The proposed approach is formulated as a MILP problem, and the CPLEX solver is utilized to minimize and achieve the best global solution.

4. Incorporating the optimal placement of μ-PMUs and DSSE issues into a two-stage programming problem, where the decision variables determining the optimal μ-PMU placement (first stage) serve as input data for the DSSE (second stage).

1.4 Chapter organization

A comparative analysis of Kalman filters and WLS for state estimation in distribution systems is presented in Section 2. In Section 3, we outline the problem formulation, consisting of two stages. The first stage focuses on the placement of μ-PMUs, while the second stage is dedicated to addressing state estimation. Section 4 provides a summary of the solver utilized in the context of two-stage stochastic programming. The outcomes of simulations and associated discussions are detailed in Section 5. Ultimately, Section 6 encapsulates the concluding remarks.

2. Comparative analysis of EKF and WLS for state estimation

Among the numerous state estimation techniques available, Kalman filters and WLS are widely used in distribution systems for their robustness and accuracy. This discussion aims to compare these two techniques, examining their principles, applications, and evaluating their performance in different scenarios.

2.1 Kalman filter

Kalman filters are recursive algorithms designed for state estimation in dynamic systems. They operate on a prediction-correction cycle, combining a system model with measurements to update the state estimate. In distribution systems, Kalman filters are commonly employed for real-time tracking and estimation of variables such as voltage, current, and power flow. The applications of Kalman filters in distribution systems are listed as follow:

1. Power system state estimation: Kalman filters excel in tracking the state of dynamic power systems, handling uncertainties and noise in measurements effectively.

2. Sensor fusion: Kalman filters are employed for integrating data from various sensors in distribution systems, enhancing the accuracy of state estimation.

3. Fault detection: Kalman filters can be used to identify faults in distribution systems by analyzing deviations between predicted and measured values.

2.2 WLS method

WLS is a statistical method used for estimating unknown parameters by minimizing the sum of squared weighted residuals. In the context of distribution systems, WLS is often applied to linear regression problems, making it suitable for static state estimation. The application of WLS in distribution systems are listed as follow:

1. Load flow analysis: WLS is utilized for estimating the state variables in load flow studies, providing a reliable representation of the system’s current operating conditions.

2. Bad data detection: WLS helps identify and mitigate the impact of erroneous measurements in distribution systems, improving the overall accuracy of state estimation.

3. Parameter estimation: Weighted Least Squares is used for estimating parameters in distribution system models, aiding in system planning and optimization.

2.3 Comparative analysis

1. Computational complexity: Kalman filters typically involve more complex computations due to their recursive nature and reliance on dynamic models. WLS, being a static technique, generally has lower computational requirements.

2. Robustness to Non-linearity: Kalman filters are well-suited for nonlinear systems, while WLS is more effective in linear scenarios. The choice between the two depends on the nature of the distribution system and the extent of non-linearity.

3. Handling of measurement noise: Kalman filters are designed to handle measurement noise more effectively than WLS, making them preferable in applications where accurate tracking of dynamic variables is crucial.

3. Problem formulation

3.1 Extended Kalman filter formulation

The EKF consists of two primary stages: the a priori phase, which involves prediction or processing based on the previous state estimate computed in the prior iteration, indicated by superscript “−”, and the a posteriori phase, involving measurement update or estimation. These estimates are computed before any system measurements are taken.

The EKF operates under specific assumptions:

• Gaussian and uncorrelated noise characterize both the measurement and process.

• The system is observable, as defined mathematically below:

$${\mathrm{x}}_{\mathrm{k}}=\mathrm{g}\left({\mathrm{u}}_{\mathrm{k}},{\mathrm{x}}_{\mathrm{k}-1}\right)+{\mathrm{\epsilon }}_{\mathrm{k}}$$E1

$${\mathrm{z}}_{\mathrm{k}}=\mathrm{h}\left({\mathrm{x}}_{\mathrm{k}}\right)+{\mathrm{\delta }}_{\mathrm{k}}$$E2

Here, ${\in }_{\mathrm{k}}$ represents the process noise and ${\mathrm{\delta }}_{\mathrm{k}}$ denotes the measurement noise, both having a zero mean. The functions g and h necessitate linearization through the computation of their Jacobian matrices.

In every time step (k), the EKF algorithm defines the estimation $x_k$ through its mean ${\mathrm{\mu }}_{\mathrm{k}}$ and covariance ${\sum }_k$. EKF follows a recursive process with two sequential steps, continuing until an accurate estimation is reached. The determination of both the mean and covariance occurs through separate steps:

1. Prediction

   $${\overline{\mathrm{\mu }}}_{\mathrm{k}}=\mathrm{g}\left({\mathrm{u}}_{\mathrm{k}},{\mathrm{\mu }}_{\mathrm{k}-1}\right)$$E3
   $$\sum _{\mathrm{k}}^{-}=\mathrm{G}\left({\mathrm{u}}_{\mathrm{k}},{\mathrm{\mu }}_{\mathrm{k}-1}\right)\sum _{\mathrm{k}-1}{\mathrm{G}}^{\mathrm{T}}\left({\mathrm{u}}_{\mathrm{k}},{\mathrm{\mu }}_{\mathrm{k}-1}\right)+\mathrm{Q}$$E4

2. Correction

$${\mathrm{K}}_{\mathrm{k}}=\sum _{\mathrm{k}}^{-}{\mathrm{H}}^{\mathrm{T}}\left({\overline{\mathrm{\mu }}}_{\mathrm{k}}\right){\left[\mathrm{H}\left({\overline{\mathrm{\mu }}}_{\mathrm{k}}\right)\sum _{\mathrm{k}}^{-}{\mathrm{H}}^{\mathrm{T}}\left({\overline{\mathrm{\mu }}}_{\mathrm{k}}\right)+\mathrm{R}\right]}^{-1}$$E5

$${\mathrm{\mu }}_{\mathrm{k}}={\overline{\mathrm{\mu }}}_{\mathrm{k}}+{\mathrm{K}}_{\mathrm{k}}\left[{\mathrm{z}}_{\mathrm{k}}-\mathrm{h}\left({\overline{\mathrm{\mu }}}_{\mathrm{k}}\right)\right]$$E6

$$\sum _{\mathrm{k}}=\left[\mathrm{I}-{\mathrm{K}}_{\mathrm{k}}\mathrm{H}\left({\overline{\mathrm{\mu }}}_{\mathrm{k}}\right)\right]\sum _{\mathrm{k}}^{-}$$E7

where $\mathit{g}\left({\mathit{u}}_{\mathit{k}},{\mathit{\mu }}_{\mathit{k}-1}\right)$ is linearized around the mean ${\mu }_{k-1}$, ${\left.\mathrm{G}\left({\mathrm{u}}_{\mathrm{k}},{\mathrm{\mu }}_{\mathrm{k}-1}\right)=\frac{\left({\mathit{u}}_{\mathit{k}},{\mathit{\mu }}_{\mathit{k}-1}\right)}{{\mathit{\delta x}}_{\mathit{k}-1}}\right|}_{{\mathit{x}}_{\mathit{k}-1}={\mathit{\mu }}_{\mathit{k}‐1}}$ and Q is the covariance of the noise matrix. The matrices H, h and R are the same as in the WLS method. The measurement $z_k$ is only incorporated in the algorithm during the correction step. $K_k$ represents the Kalman gain, determining how much the new state estimate is influenced by measurements. Thus, each iteration integrates a fresh set of measurements for correction. However, it’s reasonable to assume that the average remains steady, given the power system’s stability across consecutive time steps. Therefore, (3) becomes:

$${\overline{\mathrm{\mu }}}_{\mathrm{k}}={\mathrm{\mu }}_{\mathrm{k}-1}$$E8

Thus, it follows that g is independent from ${\mathit{u}}_{\mathit{k}}$ and ${\left.\mathrm{G}\left({\mathrm{u}}_{\mathrm{k}},{\mathrm{\mu }}_{\mathrm{k}-1}\right)=\frac{\left({\mathit{u}}_{\mathit{k}},{\mathit{\mu }}_{\mathit{k}-1}\right)}{{\mathit{\delta x}}_{\mathit{k}-1}}\right|}_{{\mathit{x}}_{\mathit{k}-1}={\mathit{\mu }}_{\mathit{k}‐1}}$ becomes an identity matrix. The updated EKF formulation is:

1. Prediction

   $${\overline{\mathrm{x}}}_{\mathrm{k}}={\mathrm{x}}_{\mathrm{k}-1}$$E9
   $$\sum _{\mathrm{k}}^{-}={\sum }_{\mathrm{k}-1}+\mathrm{Q}$$E10

2. Correction

$${\mathrm{K}}_{\mathrm{k}}=\sum _{\mathrm{k}}^{-}{\mathrm{H}}^{\mathrm{T}}\left({\overline{\mathrm{x}}}_{\mathrm{k}}\right){\left[\mathrm{H}\left({\overline{\mathrm{x}}}_{\mathrm{k}}\right)\sum _{\mathrm{k}}^{-}{\mathrm{H}}^{\mathrm{T}}\left({\overline{\mathrm{x}}}_{\mathrm{k}}\right)+\mathrm{R}\right]}^{-1}$$E11

$${\mathrm{x}}_{\mathrm{k}}={\overline{\mathrm{x}}}_{\mathrm{k}}+{\mathrm{K}}_{\mathrm{k}}\left[{\mathrm{z}}_{\mathrm{k}}-\mathrm{h}\left({\overline{\mathrm{x}}}_{\mathrm{k}}\right)\right]$$E12

$$\sum _{\mathrm{k}}=\left[\mathrm{I}-{\mathrm{K}}_{\mathrm{k}}\mathrm{H}\left({\overline{\mathrm{x}}}_{\mathrm{k}}\right)\right]\sum _{\mathrm{k}}^{-}$$E13

3.2 μ-PMU placement problem

For the purpose of SA and monitoring, PMUs can supply precise, high-resolution, and directly synchronized phasor measurements [30, 31]. The development of μ-PMUs, utilizing the same synchrophasor technology, is aimed at improving the intelligent operation of power distribution systems, thereby enabling their observability. When compared to siting PMUs in high-voltage transmission networks, μ-PMUs prove to be significantly more cost-effective for widespread deployment in distribution systems. This may potentially address the challenge of limited observability. The subsequent formulation is presented to ascertain the optimal allocation of μ-PMUs in asymmetrical systems.

3.2.1 Objective function

In the first stage, the objective function pertaining to the allocation of μ-PMUs is defined by Eq. (14). This specified objective consists of two elements, where WX_j and VP_j are associated with the three-phase and single-phase aspects, respectively. The objective is to identify the most favorable network locations for the installation of μ-PMUs, with the overarching aim of minimizing the total investment cost while maximizing the observability index of the distribution system. In subsequent sections, various contingencies, including PZIN, are integrated into this objective function.

$$Min\sum _{j\in {\mathrm{\Omega }}_B}\sum _{n\in \left\{a,b,c\right\}}{\mathit{WX}}_j+{\mathit{VP}}_j^n$$E14

$$3X_j+2\times \left[P_j^a+P_j^b+P_j^c\right]\le 4j=1,\dots ,\mathrm{N}$$E15

Eq. (15) guarantees that each bus is limited to having either a three-phase μ-PMU or, at most, two single-phase μ-PMUs installed.

3.2.2 Observability constraints under the normal operating conditions

The constraint on topological observability, influenced by both the location of μ-PMUs and zero injection properties, is formulated as Eqs. (16)–(18) [32, 33].

$$f_i=\sum _{j\in {\mathrm{\Omega }}_B}\sum _{n\in \left\{a,b,c\right\}}{a}_{\mathit{ij}}^n{\mathit{XX}}_j^n+\sum _{j\in {\mathrm{\Omega }}_B}\sum _{n\in \left\{a,b,c\right\}}{a}_{\mathit{ij}}^n{Y}_{\mathit{ij}}^n{Z}_j^n\ge 3-{\mathit{off}}_i^n$$E16

$${\mathit{XX}}_j^n=X_j+P_j^{n}i,j=1,\dots ,\mathrm{N},\mathrm{n}\in \left\{a,b,c\right\}$$E17

$$\sum _{j\in {\mathrm{\Omega }}_B}\sum _{n\in \left\{a,b,c\right\}}{a}_{\mathit{ij}}^n{Y}_{\mathit{ij}}^n=Z_j^n,i=1,\dots ,\mathrm{N}$$E18

The observability function, fi, assigned to bus i, serves as an indicator of its observability, taking into consideration the impact of PZINs. The PZINs must exceed a $3-{\mathit{off}}_i^n$ to guarantee the observability of bus i. It is noteworthy that if there is no connection associated with the phase of a bus, observing that specific phase becomes unnecessary. This principle is articulated in Eq. (19).

$${\mathit{off}}_i^n\times P_i^n=0,i=1,\dots ,\mathrm{N},n\in \left\{a,b,c\right\}$$E19

In addition to constraints (16)–(19), two additional constraints are formulated regarding the placement of PZINs in the distribution system to optimize the proposed objective function. This is dictated by the following rule:

1. If a PZIN be connected to a single-end bus, the installation of μ-PMUs is disallowed for all of its zero-injection phases. Conversely, their zero-injection characteristics remain unconstrained. This specification is formally denoted as (20).

   $${\mathit{ZB}}_j^n{\mathit{XX}}_j^n=0,j=1,\dots ,\mathrm{N},\mathrm{n}\in \left\{a,b,c\right\}$$E20

2. If an end node considered as a PZIN, the placement of μ-PMUs is limited for each of its zero-injection phases. This restriction enhances observability by leveraging the PZIN characteristic. These conditions are formally expressed in Eqs. (21) and (22).

   $${\mathit{ZBR}}_j^n{\mathit{XX}}_j^n=0,j=1,\dots ,\mathrm{N},\mathrm{n}\in \left\{a,b,c\right\}$$E21
   $$\mathit{\text{if}}{\mathit{ZBR}}_j^n=1,Y_{\mathit{jj}}^n=1$$E22

3.2.3 Observability constraints under single μ-PMU outage contingency

In order to depict the occurrence of a single μ-PMU outage, constraint (16) undergoes a modification aimed at improving observability through at least two distinct approaches, detailed in Eq. (23).

$$f_i=\sum _{j\in {\mathrm{\Omega }}_B}\sum _{n\in \left\{a,b,c\right\}}\left(\begin{array}{l}{a}_{\mathit{ij}}^n{\mathit{XX}}_j^n+\\ 2\times \left(\sum _{j\in {\mathrm{\Omega }}_B}\sum _{n\in \left\{a,b,c\right\}}{a}_{\mathit{ij}}^n{Y}_{\mathit{ij}}^n{Z}_j^n\right)\end{array}\right)\ge 2\times \left(3-{\mathit{off}}_i^n\right),i=1,\dots ,\mathrm{N}$$E23

The coefficient of 2 in the left part of inequality constraint (23) signifies that if a bus’s observability is improved by the PZIN, it maintains observability even in the case of a single μ-PMU outage. This resilience is due to each bus in the system being directly or indirectly observed through a minimum of two distinct pathways. It’s crucial to recognize that the radial structure of distribution networks results in some busses being linked to the network via a single line. Although it is theoretically feasible to independently place a μ-PMU at each end bus for multiple observation paths, this is not deemed a cost-effective solution. Consequently, constraint (23) is substituted with (24), where end busses, excluding critical ones, are observed from at least one pathway, while the remaining busses are observed from at least two pathways.

$$f_i=\sum _{j\in {\mathrm{\Omega }}_B}\sum _{n\in \left\{a,b,c\right\}}\left(\begin{array}{l}{a}_{\mathit{ij}}^n{\mathit{XX}}_j^n+\\ 2\times \left(\sum _{j\in {\mathrm{\Omega }}_B}\sum _{n\in \left\{a,b,c\right\}}{a}_{\mathit{ij}}^n{Y}_{\mathit{ij}}^n{Z}_j^n\right)\end{array}\right)\ge \left[2\times \left(3-{\mathit{off}}_i^n\right)\right]-R_i^n,i=1,\dots ,\mathrm{N}$$E24

3.3 The DSSE problem

The objective of state estimation is to determine accurate values for state variables using real-time measurements obtained from μ-PMUs within the energy management system. The Weighted Least Squares State Estimation is framed as the minimization of the objective function (J(x)), as elaborated in [4].

$$J\left(x\right)=\sum _{u=1}^m{w}_u{e}_u^2$$E25

Subjected to:

$$z=h\left(x\right)+e$$E26

3.3.1 Measurement function linearization process

With the exception of voltage magnitudes, most h measurements demonstrate nonlinearity. In this section, Taylor series approximations of sin and cos terms are employed to linearize these nonlinear functions.

$$I_{\mathit{ij}}^m={\overline{Y}}^m.\mathrm{\Delta }{V}_{\mathit{ij}}^m$$E27

Eq. (15) represents the complex power flow ${\mathrm{S}}_{\mathrm{ij}}^{\mathrm{m}}$ by phase-m.

$$S_{\mathit{ij}}^m=V_i^m\times {\left(I_{\mathit{ij}}^m\right)}^{\ast }=V_i^m\times {\left(\sum _{m\in \left\{a,b,c\right\}}{\overline{Y}}^m.\mathrm{\Delta }{V}_{\mathit{ij}}^m\right)}^{\ast }$$E28

Accordingly, (15) was extended as Eq. (16),

$$S_{\mathit{ij}}^m=V_i^m\measuredangle {\delta }_i^m.{\left(\sum _{m\in \left\{a,b,c\right\}}\left[\left({\overline{G}}^m+j{\overline{B}}^m\right).\left(\left|V_i^m\right|\measuredangle {\delta }_i^m-\left|V_j^m\right|\measuredangle {\delta }_j^m\right)\right]\right)}^{\ast }$$E29

Let us denote:

$$\Delta {\delta }_{\mathit{ij}}^m={\delta }_{\mathit{i}}^m-{\delta }_{\mathit{j}}^m\mathrm{for}\mathrm{m}\in \left\{a,b,c\right\}$$E30

$${cos}_{\mathit{ij}}^m=cos\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^m\right)\text{and}{sin}_{\mathit{ij}}^m=sin\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^m\right)$$E31

3.3.2 Derivation of non-linear active and reactive power flow equations

$$S_{\mathit{ij}}^m=\left|V_i^m\right|\measuredangle {\delta }_i^m\sum _{n\in \left\{a,b,c\right\}}\sum _{i,j\in {\Omega }_B}\left[\begin{array}{l}{\overline{G}}^{\mathit{mn}}\left|V_i^n\right|\measuredangle -{\delta }_i^n-{\overline{G}}^{\mathit{mn}}\left|V_j^n\right|\measuredangle -{\delta }_j^n\\ -j{\overline{B}}^{\mathit{mn}}\left|V_i^n\right|\measuredangle -{\delta }_i^n+j{\overline{B}}^{\mathit{mn}}\left|V_j^n\right|\measuredangle -{\delta }_j^n\end{array}\right]$$E32

The Eq. (32) was simplified as follows:

$$S_{\mathit{ij}}^m=\left|V_i^m\right|\sum _{n\in \left\{a,b,c\right\}}\sum _{i,j\in {\mathrm{\Omega }}_B}\left[\begin{array}{l}{\overline{G}}^{\mathit{mn}}\left|V_i^n\right|\measuredangle \mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn}}-{\overline{G}}^{\mathit{mn}}\left|V_j^n\right|\measuredangle \mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn}}\\ -j{\overline{B}}^{\mathit{mn}}\left|V_i^n\right|\measuredangle -{\delta }_{\mathit{ii}}^{\mathit{mn}}+j{\overline{B}}^{\mathit{mn}}\left|V_j^n\right|\measuredangle \mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn}}\end{array}\right]$$E33

Eq. (33) is separated into its components, distinguishing between its real and imaginary parts, expressed as (34).

$$\begin{array}{ll}{S}_{\mathit{ij}}^m& =\left|V_i^m\right|\sum _{n\in \left\{a,b,c\right\}}\sum _{i,j\in {\mathrm{\Omega }}_B}\left\{\left|V_i^n\right|\left[{\overline{G}}^{\mathit{mn}}{.cos}_{\mathit{ii}}^{\mathit{mn}}+{\overline{B}}^{\mathit{mn}}{.sin}_{\mathit{ii}}^{\mathit{mn}}\right]-\left|V_j^n\right|\left[{\overline{G}}^{\mathit{mn}}{.cos}_{\mathit{ij}}^{\mathit{mn}}+{\overline{B}}^{\mathit{mn}}{.sin}_{\mathit{ij}}^{\mathit{mn}}\right]\right\}\\ & +j\left|V_i^m\right|\sum _{n\in \left\{a,b,c\right\}}\sum _{i,j\in {\mathrm{\Omega }}_B}\left\{\left|V_i^n\right|\left[{\overline{G}}^{\mathit{mn}}{.sin}_{\mathit{ii}}^{\mathit{mn}}-{\overline{B}}^{\mathit{mn}}{.cos}_{\mathit{ii}}^{\mathit{mn}}\right]-\left|V_j^n\right|\left[{\overline{G}}^{\mathit{mn}}{.sin}_{\mathit{ij}}^{\mathit{mn}}+{\overline{B}}^{\mathit{mn}}{.cos}_{\mathit{ij}}^{\mathit{mn}}\right]\right\}\end{array}$$E34

The actual segment of Eq. (34) outlines the equation for active power flow concerning phase m and line ij, as follows:

$$P_{\mathit{ij}}^m=\left|V_i^m\right|\sum _{n\in \left\{a,b,c\right\}}\sum _{i,j\in {\mathrm{\Omega }}_B}\left\{\begin{array}{l}\left|V_i^n\right|\left[{\overline{G}}^{\mathit{mn}}{.cos}_{\mathit{ii}}^{\mathit{mn}}+{\overline{B}}^{\mathit{mn}}{.sin}_{\mathit{ii}}^{\mathit{mn}}\right]-\\ \left|V_j^n\right|\left[{\overline{G}}^{\mathit{mn}}{.cos}_{\mathit{ij}}^{\mathit{mn}}+{\overline{B}}^{\mathit{mn}}{.sin}_{\mathit{ij}}^{\mathit{mn}}\right]\end{array}\right\}$$E35

The imaginary component of Eq. (34) delineates the equation for reactive power flow related to phase m and line ij, as follows:

$$Q_{\mathit{ij}}^m=\left|V_i^m\right|\sum _{n\in \left\{a,b,c\right\}}\sum _{i,j\in {\mathrm{\Omega }}_B}\left\{\begin{array}{l}\left|V_i^n\right|\left[{\overline{G}}^{\mathit{mn}}{.sin}_{\mathit{ii}}^{\mathit{mn}}-{\overline{B}}^{\mathit{mn}}{.cos}_{\mathit{ii}}^{\mathit{mn}}\right]\\ -\left|V_j^n\right|\left[{\overline{G}}^{\mathit{mn}}{.sin}_{\mathit{ij}}^{\mathit{mn}}+{\overline{B}}^{\mathit{mn}}{.cos}_{\mathit{ij}}^{\mathit{mn}}\right]\end{array}\right\}$$E36

Eqs. (37)–(40) represent the outcomes of the Taylor series approximations for sin and cos functions

$${sin}_{\mathit{ii}}^{\mathit{mn}}=sin\left(\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn}}\right)\approx sin\left(\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn},r}\right)+cos\left(\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn},r}\right).\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn}}$$E37

$${cos}_{\mathit{ii}}^{\mathit{mn}}=cos\left(\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn}}\right)\approx cos\left(\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn},r}\right)-sin\left(\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn},r}\right).\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn}}$$E38

$${sin}_{\mathit{ij}}^{\mathit{mn}}=sin\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn}}\right)\approx sin\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)+cos\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right).\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn}}$$E39

$${cos}_{\mathit{ij}}^{\mathit{mn}}=cos\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn}}\right)\approx cos\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)-sin\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right).\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn}}$$E40

The operational points for voltage magnitudes were determined regionally for each line section by utilizing the combined data from μ-PMUs, aiming to improve the accuracy of state estimation. In the context of a specific line section denoted as ij, with q representing the count of μ-PMU voltage measurements, the local operating point of the voltage magnitude, denoted as |V_r|, is calculated using Eq. (41).

$$\left|V_r\right|\approx \frac{1}{q}\sum _{i=1}^q\left|V_i\right|$$E41

3.3.3 Derivation of linearized active and reactive power flow equations

In Eq. (35), the nonlinear active power Eq. (42) for phase m is depicted. To linearize the nonlinear sin(.) and cos(.) equations, they have been replaced with Taylor series approximations (37)–(40), as detailed below:

$${\overline{P}}_{\mathit{ij}}^m=\left|V_i^m\right|\sum _{n\in \left\{a.b,c\right\}}\sum _{i,j\in {\mathrm{\Omega }}_B}\left\{\begin{array}{l}\left|V_i^n\right|\left[\begin{array}{l}{\overline{G}}^{\mathit{mn}}.\left(cos\left(\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn},r}\right)-sin\left(\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn},r}\right).\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn}}\right)\\ +{\overline{B}}^{\mathit{mn}}.\left(sin\left(\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn},r}\right)+cos\left(\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn},r}\right).\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn}}\right)\end{array}\right]\\ -\left|V_j^n\right|\left[\begin{array}{l}{\overline{G}}^{\mathit{mn}}.\left(cos\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)-sin\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right).\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)\\ +{\overline{B}}^{\mathit{mn}}.\left(sin\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)+cos\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right).\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn}}\right)\end{array}\right]\end{array}\right\}$$E42

The assumption $\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn},r}\approx \mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}$ is made that the distance between two adjacent busses in the distribution network is minimal. Based on this assumption, Eq. (42) is revised as (43):

$${\overline{P}}_{\mathit{ij}}^m=\left|V_i^m\right|\sum _{n\in \left\{a.b,c\right\}}\sum _{i,j\in {\mathrm{\Omega }}_B}\left\{\begin{array}{l}\left|V_i^n\right|\left[\begin{array}{l}{\overline{G}}^{\mathit{mn}}.\left(cos\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)-sin\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right).\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn}}\right)\\ +{\overline{B}}^{\mathit{mn}}.\left(sin\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)+cos\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right).\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn}}\right)\end{array}\right]\\ -\left|V_j^n\right|\left[\begin{array}{l}{\overline{G}}^{\mathit{mn}}.\left(cos\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)-sin\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right).\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)\\ +{\overline{B}}^{\mathit{mn}}.\left(sin\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)+cos\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right).\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn}}\right)\end{array}\right]\end{array}\right\}$$E43

Eq. (43) was rearranged as:

$${\overline{P}}_{\mathit{ij}}^m=\left|V_i^m\right|\sum _{n\in \left\{a,b,c\right\}}\sum _{i,j\in {\mathrm{\Omega }}_B}\left\{\begin{array}{l}\left({\overline{G}}^{\mathit{mn}}.cos\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)+{\overline{B}}^{\mathit{mn}}.sin\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)\right).\left(\left|V_i^n\right|-\left|V_j^n\right|\right)\\ +\left(-{\overline{G}}^{\mathit{mn}}.sin\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)+{\overline{B}}^{\mathit{mn}}.cos\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)\right).\left(\left|V_i^n\right|.\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn}}-\left|V_j^n\right|.\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn}}\right)\end{array}\right\}$$E44

The simplification of the regional operating points for voltage magnitudes is expressed by Eq. (44).

$${\overline{P}}_{\mathit{ij}}^m=\sum _{n\in \left\{a,b,c\right\}}\sum _{i,j\in {\mathrm{\Omega }}_B}\left\{\begin{array}{l}\left|V_r\right|\left({\overline{G}}^{\mathit{mn}}.cos\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)+{\overline{B}}^{\mathit{mn}}.sin\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)\right).\left(\left|V_i^n\right|-\left|V_j^n\right|\right)\\ +\left|V_r^2\right|\left(-{\overline{G}}^{\mathit{mn}}.sin\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)+{\overline{B}}^{\mathit{mn}}.cos\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)\right).\left(\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn}}-\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn}}\right)\end{array}\right\}$$E45

As earlier described in (30) and (31), it is clear that $\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn}}={\delta }_i^m-{\delta }_i^n$ and $\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn}}={\delta }_i^m-{\delta }_j^n$. Therefore, $\mathrm{\Delta }{\delta }_{\mathit{ii}}^{\mathit{mn}}-\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn}}=-\left({\delta }_i^n-{\delta }_j^n\right)$. The linearized active power function for phase-m${\overline{P}}_{\mathit{ij}}^m$ was constructed by replacing this for (46), as follows:

$${\overline{P}}_{\mathit{ij}}^m=\sum _{n\in \left\{a,b,c\right\}}\sum _{i,j\in {\mathrm{\Omega }}_B}\left\{\begin{array}{l}\left|V_r\right|\left({\overline{G}}^{\mathit{mn}}.cos\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)+{\overline{B}}^{\mathit{mn}}.sin\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)\right).\left(\left|V_i^n\right|-\left|V_j^n\right|\right)\\ -\left|V_r^2\right|\left(-{\overline{G}}^{\mathit{mn}}.sin\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)+{\overline{B}}^{\mathit{mn}}.cos\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)\right).\left({\delta }_i^n-{\delta }_j^n\right)\end{array}\right\}$$E46

Eq. (47) represents the result of linearizing the reactive power function ${\overline{Q}}_{\mathit{ij}}^m$ for phase-m.

$${\overline{Q}}_{\mathit{ij}}^m=\sum _{n\in \left\{a,b,c\right\}}\sum _{i,j\in {\mathrm{\Omega }}_B}\left\{\begin{array}{l}\left|V_r\right|\left({\overline{G}}^{\mathit{mn}}.sin\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)-{\overline{B}}^{\mathit{mn}}.cos\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)\right).\left(\left|V_i^n\right|-\left|V_j^n\right|\right)\\ -\left|V_r^2\right|\left(-{\overline{G}}^{\mathit{mn}}.cos\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)+{\overline{B}}^{\mathit{mn}}.sin\left(\mathrm{\Delta }{\delta }_{\mathit{ij}}^{\mathit{mn},r}\right)\right).\left({\delta }_i^n-{\delta }_j^n\right)\end{array}\right\}$$E47

With reference to the voltage magnitude operating points defined in Eq. (41) and employing Taylor series approximations on (37)–(40), the refined linearized approximations for the active and reactive power equations were revealed as,

$${\overline{P}}_{\mathit{ij}}^m=\sum _{n\in \left\{a,b,c\right\}}\sum _{i,j\in {\mathrm{\Omega }}_B}\left\{\left|V_r\right|.{\mathrm{\Phi }}_P.\left(\left|V_i^n\right|-\left|V_j^n\right|\right)-\left|V_r^2\right|.{\mathrm{\Psi }}_P.\left({\delta }_i^n-{\delta }_j^n\right)\right\}$$E48

$${\overline{Q}}_{\mathit{ij}}^m=\sum _{n\in \left\{a,b,c\right\}}\sum _{i,j\in {\mathrm{\Omega }}_B}\left\{\left|V_r\right|.{\mathrm{\Phi }}_Q.\left(\left|V_i^n\right|-\left|V_j^n\right|\right)-\left|V_r^2\right|.{\mathrm{\Psi }}_Q.\left({\delta }_i^n-{\delta }_j^n\right)\right\}$$E49

Following the linearization procedure, the measurement function presented in (1) was reformulated as a linear function (50).

$$z=H.x+e$$E50

Utilizing the WLS method, the computed estimated state vector was determined according to Eq. (51).

$$\widehat{x}=\left[{\left(H^T{U}^{-1}H\right)}^{-1}{H}^T{U}^{-1}\right].z$$E51

Eq. (52) illustrates the diagonal matrix that includes the respective measurement variances.

$$U=diag\left({\sigma }_1^2,{\sigma }_2^2,{\sigma }_3^2,\dots ,{\sigma }_m^2\right)$$E52

Eq. (53) demonstrates the measurement vector, encompassing voltage magnitude, voltage angle, as well as active and reactive power flows for each phase.

$$Z={\left[\left|V_i^a\right|,\left|V_i^b\right|,\left|V_i^c\right|,{\delta }_i^a,{\delta }_i^b,{\delta }_i^c,P^a,Q^a,P^b,Q^b,P^c,Q^c\right]}^T$$E53

4. Two-stage stochastic programming

Addressing decision-making challenges in an uncertain setting involves the introduction of an innovative stochastic two-stage programming model designed to optimize both μ-PMU placement and DSSE problems. This methodology entails the classification of decision variables into two main categories: “here-and-now” and “wait-and-see,” as detailed in Refs. [34, 35]. In this methodology, the first phase entails establishing the real values of bus voltage and branch current phasors while guaranteeing the observability of the system. The outcomes from the initial stage subsequently act as input for the succeeding stage. The first stage involves dual variables identified as ‘d’ (representing voltage phasors) and variables ‘c’ (representing current phasors), necessitating immediate decisions in the present, before uncertainties are resolved. Simultaneously, operational variables ‘xt’ function as components for a “wait-and-see” approach, and the determination of variable ‘b’ in the second stage can be made once all uncertain parameters have been observed. The formulated stochastic two-stage optimization is presented as follows:

$$\begin{array}{l}\left(\text{First}\_\text{stage}\right):\underset{c,d}{max}{F}_f\left(c,d\right)+E\left[F_f\left(c,d,\xi \right)\right]\\ \mathrm{S}.\mathrm{t}.{\phi }_f\left(c,d\right)=0\\ {\psi }_f\left(c,d\right)\le 0\\ c\in {\Re }^1\end{array}$$E54

$$\begin{array}{l}\left(\text{Second}\_\text{stage}\right):F_s\left(c,d,\xi \right)=\underset{x_t}{min}{F}_s\left(b,c,d,x_t,\xi \right)\\ \mathrm{S}.\mathrm{t}.{\phi }_s\left(b,c,d,x_t,\xi \right)=0\\ {\psi }_s\left(b,c,d,x_t,\xi \right)\le 0\\ b\in {\left\{0,1\right\}}^m,x_t\in {\Re }^n\end{array}$$E55

Where, the objective function is separated into two parts: a deterministic element denoted as F_f, which reflects decisions, and the expected value of a stochastic element, F_s. The stochastic component is influenced by the actualization of uncertain parameters ‘n’ during the operational stage.

5. Simulation results and discussion

This chapter introduces a two-stage stochastic programming model designed for asymmetrical network state estimation, aiming to enhance SA through the incorporation of μ-PMU data. Figure 1 visually outlines the proposed model. In the initial stage, a dynamic and efficient mathematical programming challenge, as outlined in Ref. [36], harnesses the capabilities of the CPLEX optimizer to address the optimal placement of μ-PMUs amidst the presence of PZINs and various contingencies. The use of this optimizer streamlines decision optimization, leading to enhanced efficiency, cost reduction, and increased profitability. Its proficiency in handling large-scale, real-world problems aligns well with the swift processing requirements of contemporary interactive decision optimization applications. Moreover, in the second phase, the linear approximation of active and reactive power functions is employed to represent the linear DSSE. Finally, the Weighted Least Squares algorithm is utilized to address the DSSE problem integrated with three-phase DG. The resolution of the proposed two-stage programming problem is conducted within the MATLAB environment, as elaborated in Ref. [37].

Figure 1.

The flowchart of the proposed two-stage programming problem.

This chapter investigates four scenarios for the operation and analysis of the distribution system under different operational conditions, allowing for a comparative assessment of the outcomes.

• Scenario I: Normal operational circumstances

• Scenario II: Normal operational circumstances with consideration of PZINs

• Scenario III: Contingency operational conditions

• Scenario IV: Contingency operational conditions with consideration of PZINs

5.1 Case study

To offer a thorough understanding of asymmetrical distribution networks, we have opted for the modified IEEE 85-bus test feeder, as presented in Ref. [38]. The single-line diagram of this system is illustrated in Figure 2, comprising 85 busses and 84 branches, operating at 11 kV with a 100 MVA base. The phases are indicated by red, blue, and green lines corresponding to a, b, and c phases, respectively. Furthermore, the system incorporates nine Distributed Generation sources, including wind turbines, solar panels, and combined heat and power systems, connected to various busses. Table 1 provides technical details for these sources. It is important to note that all μ-PMUs are expected to be of the same type, ensuring uniform costs. The cost estimation for each μ-PMU device, including installation, is $1000 across all test systems, as detailed in the reference [40].

Figure 2.

A modified asymmetric three-phase 85-bus distribution test system.

DG type	Location (bus)	Lower bound (kW)	Upper bound (kW)	Cost coefficient ($/kWh)
WT1	4	200	450	0.02
WT2	31	200	450	0.02
WT3	60	100	360	0.02
PV1	20	150	350	0.05
PV2	45	150	350	0.05
PV3	85	150	350	0.05
CHP1	28	300	700	0.01
CHP2	49	350	750	0.01
CHP3	68	300	700	0.01

Table 1.

Technical data of utilized DG units [39].

5.1.1 Scenario I

Determine optimal locations and quantities of single-phase and three-phase μ-PMUs, along with their associated redundancy indices and installation costs, to achieve complete observability in asymmetrical distribution networks during normal operational conditions. This evaluation, excluding considerations for partially zero injection nodes, is concisely summarized in the initial row of Table 2.

Network name	Contingency statue	PZINs locations	Optimal places of μ-PMUs	No. of μ-PMUs	Measurement Redundancy	Obj. Fun. value	Exe. time (s)
Modified asymmetric three-phase 85 bus distribution test system	Under normal operating condition	—	2, 3, 5, 7, 13, 14, 19, 27, 29, 32, 34, 35, 41, 49, 50, 52, 53, 58, 60, 67, 70, 73, 81, 83: All three phases; 21: Phases a & b; 26, 46: Phases a & c; 10, 61, 65: Phase a	Three-phase units: 27; Single-phase units: 3	114	27.0637	1.1075
	Under normal operating condition	2, 3, 5, 7, 9, 10, 12, 13, 27, 29, 32, 34, 35, 41, 48, 49, 52, 58, 64, 65, 67, 68, 70, 73, 81	2, 4, 7, 12, 14, 19, 26, 29, 33, 41, 50, 53, 57, 64, 65, 67, 70, 73, 83: All three phases; 21: Phases a & b; 46: Phases a & c; 61: Phase a	Three-phase units: 21; Single-phase units: 1	79	19.2640	1.5553
	Single line outage or single μ-PMU loss	—	2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 17, 19, 23, 24, 26, 27, 29, 31, 32, 34, 35, 36, 38, 39, 40, 41, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 64, 65, 67, 68, 69, 70, 73, 81, 82, 83, 84: All three phase; 21, 22: Phases a & b; 37, 45, 46, 47: Phases a & c; 61, 62, 78, 85: Phase a	Three-phase units: 56; Single-phase units: 4	189	58.1235	1.3018
	Single line outage or single μ-PMU loss	2, 3, 5, 7, 9, 10, 12, 13, 27, 29, 32, 34, 35, 41, 48, 49, 52, 58, 64, 65, 67, 68, 70, 73, 81	1, 2, 3, 5, 7, 8, 9, 11, 14, 15, 19, 23, 26, 28, 29, 32, 34, 40, 41, 44, 48, 50, 51, 53, 54, 58, 64, 65, 66, 67, 68, 69, 70, 73, 75, 79, 81, 83, 84; 21, 22: Phases a & b; 37, 46, 47: Phases a & c; 61, 62: Phase a	Three-phase units: 44; Single-phase units: 2	147	42.8835	1.7802

Table 2.

Results of optimal allocation in a combination of single and three-phase μ-PMUs with and without considering PZINs under contingencies.

5.1.2 Scenario II

In the prior analysis, the emphasis was on determining the most suitable allocation of μ-PMUs in typical operational scenarios, excluding considerations of PZINs. The results of investigating the optimal μ-PMU allocation under standard operational conditions, now accounting for PZINs, are outlined in the second row of Table 2. This underscores the effectiveness of the proposed methodology. Noteworthy is the observation that integrating PZINs into the model leads to a reduction in the required number of μ-PMUs and a lowering of the objective function value. The findings underscore that a decrease in the number of μ-PMUs corresponds to a reduction in the measurement redundancy index, ultimately enhancing measurement efficiency.

5.1.3 Scenario III

The third row of Table 2 displays the optimal locations and quantities of μ-PMUs, installation costs, and the resulting measurement redundancy index, ensuring full observability of the system under various contingencies, without considering PZINs. When a μ-PMU is placed at a specific node, it not only enhances visibility at that bus but also indirectly contributes to the visibility of neighboring busses. Consequently, in the event of a single line outage or μ-PMU failure, certain busses may lose observability. To address this issue, an optimal μ-PMU allocation approach is employed to identify the best locations for maintaining complete system observability after any contingency. It is evident that, with the occurrence of one or more events, the number of installed μ-PMUs in the system increases, leading to a significant rise in the objective function value. Consequently, there is no economic justification for scenario III.

5.1.4 Scenario IV

Table 2 provides a detailed breakdown of the optimal setup involving μ-PMUs, installation costs, and the measurement redundancy index for the proposed distribution network. This comprehensive information is specifically located in the fourth row of the table and is examined across various contingencies, ensuring the network’s complete observability following the incorporation of PZINs. As anticipated, additional μ-PMUs are deemed necessary to ensure observability under diverse contingencies. The viability of scenario III hinges on the essential integration of PZINs to minimize the objective function, leading to a simultaneous reduction in both the required number of μ-PMUs and the overall system objective function value. The incorporation of PZINs into the allocation problem proves to be a pivotal factor, resulting in a noteworthy decrease in the essential μ-PMUs and the corresponding system objective function value.

In this segment, we present the outcomes of the three-phase state estimation, where the estimated values are compared with the actual values derived from the initial power flow computations for each specific scenario. The data input for the state estimator comprises a dataset featuring active and reactive power flow, voltage magnitude and angle, in addition to DGs input. This dataset is constructed using a series of voltage and current phasors generated by μ-PMUs. The provided passage delves into the outcomes of the proposed methodology aimed at estimating active power flow within a three-phase 85-bus distribution network. In Figure 3, a visual representation showcases the comparison between estimated and actual values across phases (a, b, and c). The maximum variances in active power flow estimation stand at 0.0716, 0.0918, and 0.1799 per unit for lines 17, 17, and 7, respectively. Similarly, Figure 4 exhibits the actual and estimated values for three-phase reactive powers at designated load points. The highest variances in reactive power flow estimation in phases a, b, and c are 0.1477, 0.0697, and 0.0387 per unit, associated with lines 5, 7, and 7, respectively. Figures 2 and 3 underscore the remarkable proximity between estimated and actual values, providing compelling evidence for the efficacy of the two-stage programming approach for DSSE.

Figure 3.

Estimated and actual values of the active power flow.

Figure 4.

Estimated and actual values of the reactive power flow.

In Figure 5, the voltage magnitudes for the three phases in the 85-bus model are presented, comparing the estimated values to the actual measurements. In cases where phases are not present, a placeholder value of one has been employed. Visual inspection indicates a substantial agreement between the estimated and actual voltage magnitudes. Similarly, Figure 6 displays the estimated and actual voltage angles, assuming a value of zero for the phases that are not present. The proposed methodology consistently achieves heightened accuracy in estimating these parameters. As demonstrated in the presentation, the state estimation model outlined in this study successfully converges and accurately determines the state variables in the adapted asymmetric three-phase 85-bus distribution test system, which incorporates DGs. The simulation results affirm the effectiveness of both the suggested state estimation model and the extended per-unit system. Figure 7 illustrates the estimated and actual values of DG production for all DG types. The most notable disparity in DG production estimation is observed at 0.0265 per unit, specifically associated with PV_3.

Figure 5.

Estimated and actual values of voltage magnitudes.

Figure 6.

Estimated and actual values of voltage angles.

Figure 7.

Estimated and actual values of DG production amount.

In the proposed DSSE approach, the variances for active and reactive power flows are set at 5 × 10⁻⁴ and 2 × 10⁻⁴, respectively. Similarly, the variances governing voltage magnitude and voltage angle are adjusted to 2 × 10⁻⁴ and 6 × 10⁻⁴, respectively. Over the course of this investigation, the solution variables undergo a total of 17 updates. During each update, the freshly revised solutions are systematically juxtaposed with their predecessors, and only those demonstrating superior accuracy are chosen as the definitive solutions. Figures 8 and 9 illustrate the forecasted errors in estimating active and reactive power flows. Significantly, the proposed method demonstrates improved accuracy, with differences between estimated and actual values leading to errors in voltage magnitude and phase angle, benchmarked against load flow results. In Figure 10, the errors in voltage magnitude for all nodes are presented, obtained through the proposed state estimation after addressing the μ-PMU placement problem in the initial stage. More precisely, our observations reveal that the proposed technique achieves markedly heightened accuracy when addressing this specific challenge. In Figure 11, we depict the expected errors in estimation, concentrating specifically on the estimated voltage phase angle at each node within the 85-bus distribution system. This illustration highlights variations across three separate phases. Recognizing the crucial significance of phase angle measurement in power system state estimation—where even slight angle fluctuations can trigger system instability—the presented results underscore the efficacy of the suggested technique. Following the strategic placement of optimal μ-PMUs at designated busses, the state estimation reveals minimal deviation in angle errors. In order to highlight the effectiveness and merits of the proposed model in comparison to previous studies, diverse analyses have been undertaken.

Figure 8.

Active power flow error.

Figure 9.

Reactive power flow error.

Figure 10.

Voltage magnitude error.

Figure 11.

Voltage angle error.

The proposed method for distribution system state estimation utilizes the EKF approach in addition to the weighted least square (WLS) method. The results of the state estimation using the EKF approach are presented in Table 3. The actual values are obtained from the backward-forward power flow method, while the estimated values are obtained from the EKF approach. Comparing the actual and estimated values, it can be observed that the estimated values closely approximate the actual values. This demonstrates that the proposed state estimation method is highly efficient and robust, with minimal errors. Numerical analysis reveals that the EKF approach is slightly more accurate than WLS. However, it should be noted that the EKF utilizes both past and present measurements, whereas WLS only considers measurements from the current time-step. Therefore, intuitively, the EKF should outperform WLS if its underlying process model hypothesis is correct.

Voltage magnitude		Voltage angle		Active power
Actual value	Estimated value	Actual value	Estimated value	Actual value	Estimated value
1.0117	1.0112	0	0	1.4343	1.4328
1.0106	1.0101	0.0394	−0.162	1.3975	1.3909
1.009	1.0085	0.099	0.2828	1.2832	1.2815
1.0071	1.0066	0.1818	0.1882	1.3748	1.3696
1.0059	1.0054	0.2052	0.1993	1.0979	1.1028
1.0023	1.0018	0.3155	0.394	1.0589	1.0599
1.0002	0.9997	0.386	−0.0981	1.0214	0.9834
0.9915	0.991	0.7059	0.6558	0.3817	0.4087
0.9913	0.9908	0.7235	0.9318	0.0051	0.0025
0.9916	0.9911	0.7512	0.9277	0.051	0.0321
0.9922	0.9917	0.7805	0.8379	0.1429	0.2705
0.9913	0.9908	0.7693	0.7934	0.019	0.0583
0.9914	0.9909	0.77	0.798	0.0706	0.0789
0.9912	0.9907	0.7674	0.7984	0.0353	0.4900
0.991	0.9905	0.7658	0.7966	0.0353	0.0819
1.0103	1.0133	0.0333	0.0681	0.1121	0.1070
1.0085	1.0115	0.0869	0.0949	0.2754	0.1960
1.0037	1.0067	0.1512	0.1412	0.2185	0.1650
1.0023	1.0053	0.1178	0.1047	0.1061	0.0763
1.0019	1.0049	0.1062	0.1082	0.0707	0.0401
1.0012	1.0042	0.0922	0.0893	0.0353	0.034
1.0007	1.0037	0.079	0.0817	0.056	0.0432
1.0022	1.0052	0.1153	0.1045	0.0353	0.0458
0.9999	0.9997	0.3782	0.3943	0.5958	0.6099
0.9895	0.9893	0.7269	0.7336	0.5592	0.5892
0.9879	0.9877	0.7468	0.7607	0.4464	0.3546
0.9861	0.9859	0.792	0.7752	0.3896	0.3329
0.9855	0.9853	0.8185	0.7587	0.3333	0.2399
0.9844	0.9842	0.8791	0.9708	0.2767	0.191
0.9837	0.9835	0.9476	0.9529	0.241	0.1799
0.9833	0.9853	0.9843	0.9691	0.3055	0.259
0.9828	0.9848	0.9785	0.9542	0.1992	0.1734
0.9826	0.9846	0.9776	0.9731	0.1852	0.0642
0.9813	0.9833	0.9767	0.9769	0.0647	0.0547
0.9811	0.9831	0.9952	0.9612	0.0353	0.0228
0.981	0.983	0.9936	0.9689	0.056	0.597
0.9877	0.9897	0.7418	0.7508	0.056	0.0349
0.9856	0.9876	0.778	0.7337	0.056	0.0441
0.9841	0.9861	0.8715	0.851	0.1061	0.0949
0.9824	0.9844	0.9664	0.9679	0.0707	0.0426
0.9816	0.9836	0.9487	0.9452	0.0353	0.0449
0.9815	0.9835	0.9462	0.9484	0.0353	0.0351
0.9815	0.9835	0.9446	0.9622	0.1202	0.0851
0.9801	0.9821	0.9464	0.9526	0.0847	0.0682
0.9792	0.9812	0.9269	0.9287	0.0493	0.0350
0.9788	0.9808	0.9156	0.9152	0.014	0.0102
0.9787	0.9807	0.9136	0.9145	0.0293	0.0241
0.9811	0.9831	1.0191	1.0171	0.1185	0.0865
0.9814	0.9834	1.0328	1.0218	0.0924	0.1484
0.981	0.983	1.0244	1.0933	0.056	0.0560
0.9807	0.9827	1.018	1.008	0.1478	0.1176
0.979	0.981	0.9684	0.9649	0.0914	0.803
0.9786	0.9806	0.9579	0.9589	0.056	0.0515
0.9782	0.9782	0.9501	0.9458	0.056	0.0573
0.9787	0.9787	0.9606	0.9581	0.014	0.0222
0.9813	0.9813	1.0309	0.998	0.3765	0.0383
0.9906	0.9906	0.744	0.8056	0.3202	0.3461
0.9889	0.9889	0.8173	0.848	0.056	0.0573
0.9888	0.9888	0.8147	0.8397	0.2635	0.2590
0.9882	0.9882	0.8737	0.8609	0.1122	0.1160
0.9873	0.9873	0.8534	0.8427	0.056	0.0610
0.9868	0.9868	0.8395	0.8329	0.215	0.2248
0.9878	0.9878	0.8733	0.8455	0.2009	0.2309
0.9866	0.9866	0.8747	0.8649	0.07	0.0710
0.9865	0.9865	0.8715	0.8563	0.056	0.0562
0.9864	0.9864	0.8689	0.8547	0.1307	0.1316
0.9861	0.9861	0.8835	0.8567	0.0393	0.0312
0.9861	0.9861	0.9217	0.8926	0.1477	0.1431
0.9846	0.9846	0.8816	0.8657	0.0914	0.0814
0.9841	0.9841	0.8712	0.8594	0.0353	0.0351
0.9839	0.9839	0.8664	0.8562	0.056	0.036
0.9861	0.9861	0.8809	0.8593	0.0915	0.0851
0.985	0.985	0.8947	0.878	0.056	0.0554
0.985	0.9846	0.8909	0.8755	0.0353	0.0345
0.9847	0.9843	0.8859	0.8735	0.056	0.0555
0.9838	0.9834	0.8635	0.8538	0.014	0.0141
0.9865	0.9861	0.8712	0.8565	0.056	0.0561
0.9912	0.9908	0.7424	0.8023	0.0353	0.0350
0.986	0.9856	0.8786	0.8584	0.1617	0.2101
0.9901	0.9897	0.7403	0.7621	0.1054	0.1317
0.9897	0.9893	0.7308	0.7509	0.056	0.059
0.9896	0.9892	0.7295	0.7494	0.0493	0.0632
0.9892	0.9888	0.7174	0.7341	0.014	0.0150
0.989	0.9886	0.714	0.7286	0.0896	0.0410
0.9921	0.9917	0.787	0.7943

Table 3.

Results of distribution system states obtained from EKF method.

A comprehensive comparative analysis is presented in Table 4. The results from the table unmistakably indicate that the proposed method substantially improves the accuracy of DSSE when contrasted with conventional approaches. Particularly noteworthy is the observation that the proposed method yields significantly lower maximum errors in bus voltage magnitude and voltage angles compared to alternative methodologies, emphasizing its superior performance.

Ref.	Proposed system	Method to solve	Model	No. of μ-PMU	Max error in bus voltage magnitude (%)	Max error in bus voltage angle (%)
[1]	Three-phase 13-bus distribution system	Modified Krawczyk operator algorithm	Mixed integer non-linear optimization model	4	2.510	3.281
[2]	Three-phase IEEE 33-bus distribution system	Safety barrier interior point optimization method	Non-linear optimization model	11	2.137	3.485
[3]	Three-phase IEEE 34-bus distribution system	Weighted least square method	Linear optimization model	12	2.568	—
[4]	Three-phase IEEE 37-bus distribution system	Hybrid PSO with chaotic gravitational search algorithm	Non-linear optimization model	13	1.508	1.602
[5]	Three-phase IEEE 69-bus distribution system	Safety barrier interior point optimization method	Non-linear convex optimization model	24	3.381	3.972
This ref.	Three-phase 85-bus distribution system	Weighted least square method	Nonlinear model with Taylor series approximations	30	1.980	2.750

Table 4.

Comparison of optimal solutions in different test systems.

6. Conclusion

Conclusion: This chapter presents a unique two-stage programming model. In the first stage, a strategy is formulated for the optimal placement of μ-PMUs, aiming to minimize installation costs while maximizing measurement redundancy and system observability, especially in the presence of PZINs. Following this, the second stage introduces an innovative DSSE model. This model employs the Taylor series approach for approximation, enabling the linearization of non-linear functions related to active and reactive power. The innovative nature of this approach ensures a comprehensive state estimation of the network, setting it apart from existing DSSE techniques. Following that, the linear DSSE problem is tackled using the WLS method to ascertain the states of the proposed system. Furthermore, the proposed method for distribution system state estimation, utilizing the EKF approach in addition to the WLS method, demonstrates high efficiency and robustness with minimal errors, as evidenced by the closely approximated estimated values to the actual values obtained from the backward-forward power flow method. Numerical analysis reveals the slight superiority of the EKF approach over WLS, attributed to its utilization of both past and present measurements, thus enhancing performance, particularly in scenarios where its underlying process model hypothesis holds. In essence, the suggested state estimation technique is put to the test in a practical case study conducted within the Matlab simulation environment. The results obtained from simulating an 85-bus distribution network highlight the approach’s suitability for large-scale systems, underscoring its effectiveness and precision. Summarily, the conclusions derived from the simulation findings can be succinctly outlined as follows:

1. The proposed model is designed with the primary objective of minimizing the required μ-PMUs, leading to cost-effective installations, increased redundancy in measurements, and enhanced observability, particularly in the presence of PZINs and diverse contingencies.

2. The results of the investigation demonstrate that the proposed model provides a dependable, resilient, sufficiently accurate, and stable state estimation in distribution systems incorporating DGs, all accomplished with a minimal quantity of μ-PMUs.

3. The applicability of the proposed model extends seamlessly to any distribution system, regardless of its size, structure, load level, or the nature of distributed energy resources involved.

4. This initiative is expected to elevate SA and ensure the dependable operation.

5. The strategic deployment of μ-PMUs has been demonstrated to enhance CPU timings, thereby optimizing the computational efficiency of the system.

6. The results of the numerical experiments underscore the significant impact of the asymmetrical structure of distribution networks on real-world operations. This highlights the importance of taking into account techniques for single-phase state estimation.

Conflict of interest

The authors declare no conflict of interest.

Nomenclature

$\mathit{i},\mathit{j}$	Bus index
$\mathit{n}$	Phase index
$\mathit{m}$	Measurement index
$\mathit{W},\mathit{V}$	The price of both three and single-phase μ-PMU
${\mathit{X}}_{\mathit{j}}$	The amount of binary variable is 1, if a three-phase μ-PMU exist at node j
${\mathit{P}}_{\mathit{j}}^{\mathit{n}}$	The amount of binary variable is 1, if a single-phase μ-PMU exist at node j and phase n
${\mathit{a}}_{\mathrm{ij}}^{\mathit{n}}$	The (i,j) member of the incident matrix associated with phase n
${\mathit{Y}}_{\mathrm{ij}}^{\mathit{n}}$	An auxiliary binary variable
${\mathit{Z}}_{\mathit{j}}^{\mathit{n}}$	The amount of binary parameter is 1, if a ZIN exists at node j and phase n
${\mathit{off}}_{\mathit{i}}^{\mathit{n}}$	The amount of binary parameter is 1, if node i of phase n is off
${\mathit{ZB}}_{\mathit{j}}^{\mathit{n}}$	The amount of binary parameter is 1, if a ZIN exist at the end node j and phase n
${\mathit{ZBR}}_{\mathit{j}}^{\mathit{n}}$	Binary parameter, it is 1, for phase n of PZIN j that is also an end bus
${\mathit{R}}_{\mathit{i}}^{\mathit{n}}$	Binary parameter, it is 1 for phase n of non-critical end bus i
${\mathit{w}}_u$	Weight coefficient
$\mathit{e}$	Vector of measurement errors in m dimensions
$\mathit{z}$	Vector of measurements in m dimensions
$\mathit{x}$	Vector of states in n dimensions
$\mathit{h}$	Function for measurements
${\mathit{I}}_{\mathrm{ij}}^{\mathit{m}}$	Currents in different phases
${\mathit{\Delta V}}_{\mathrm{ij}}^{\mathit{m}}$	Voltage drop between nodes i and j for phase m
$\mathit{Y}$	Matrix of admittances
${\mathit{S}}_{\mathrm{ij}}^{\mathit{m}}$	Flow of complex power
${\mathit{\delta }}_{\mathit{i}}^{\mathit{m}}$	Voltage angle at the ith bus and mth phase
${\mathit{V}}_{\mathit{i}}^{\mathit{m}}$	Voltage magnitude at the ith bus and mth phase
${\overline{\mathit{G}}}^{\mathit{m}},{\overline{\mathit{B}}}^{\mathit{m}}$	Conductance and susceptance in the admittance matrix
${\mathit{P}}_{\mathrm{ij}}^{\mathit{m}}$	Active power flow in phase m
${\mathit{Q}}_{\mathrm{ij}}^{\mathit{m}}$	Reactive power flow in phase m
$\phi$	Constraints of equality
$\psi$	Constraints of inequality
$F_f$	Objective function for the first stage
$F_s$	Objective function for the second stage