Low-Observability Distribution System State Estimation by Graph Computing with Enhanced Numerical Stability

Abstract

In distribution systems, limited measurement configurations and communication constraints often result in a low success rate of data acquisition, posing challenges to both system observability and the real-time performance required by state estimation (SE). Consequently, the distribution system SE (DSSE) relies heavily on pseudo-measurements to supplement real-time data. However, existing pseudo-measurement models generally fail to adequately account for topology changes and may lead to numerical instability issues. To resolve these challenges, this paper presents a graph computing-based DSSE method with enhanced numerical stability for low-observability distribution systems. Specifically, a graph neural network (GNN) is employed to dynamically learn the coupling relationships between bus and branch electrical quantities to improve the credibility of pseudo-measurements. Additionally, a loop belief propagation (LBP) algorithm based on factor graphs is designed to capture the statistical discrepancies between real-time and pseudo-measurements, thus avoiding the impact of pseudo-measurement modeling on numerical stability. Numerical results on IEEE 33-bus and 95-bus test systems demonstrate that the proposed method effectively adapts to topology variations and significantly improves the accuracy of both pseudo-measurement modeling and DSSE.

1. Introduction

With the continued integration of distributed energy resources (DERs) [1,2], flexible loads, and energy storage systems, the fluctuation and randomness of power in the distribution system are significantly enhanced. This trend imposes higher requirements on the real-time perception capability of system states [3]. However, in most current distribution systems, the redundancy of measurement configurations is relatively low. Moreover, due to issues such as untimely maintenance of measurement equipment and insufficient data transmission and storage resources, the success rate of the real-time data acquisition is relatively low, which fails to meet the requirements of SE for both real-time performance and observability simultaneously [4]. In this context, the DSSE heavily relies on supplementing real-time measurements with pseudo-measurements [5]. However, existing pseudo-measurement modeling methods generally rely on the assumption of a fixed topology and fail to adequately consider the impact of uncertainty differences between pseudo-measurements and real-time measurements on the numerical stability of SE, which results in the low credibility of the estimation results. Therefore, it is necessary to comprehensively consider topology variations and pseudo-measurement uncertainties to develop a highly credible DSSE method, thus ensuring the secure and reliable operation of distribution systems [6,7].

Currently, pseudo-measurement modeling is primarily performed by learning historical system state information of distribution systems to supplement missing measurements [8]. In this regard, some existing works employ time series analysis for pseudo-measurement modeling [9]. However, these approaches exhibit limited adaptability to nonlinear loads and rely heavily on historical data, making it difficult to rapidly adapt to abrupt power fluctuations. Recent studies have increasingly adopted artificial intelligence (AI) methods for pseudo-measurement modeling. These methods demonstrate superior capability in capturing the nonlinear characteristics of power loads through training with high-dimensional data, thus precisely characterizing complex power fluctuations of sources and loads [10,11,12]. For instance, some studies have utilized artificial neural networks (ANNs) [13] to generate high-accuracy pseudo-measurements, which were further combined with Gaussian mixture models (GMMs) to obtain the distribution of pseudo-measurement errors [13]. The authors of [14] proposed an attention-enhanced recurrent neural network (A-RNN) method, in which pseudo-measurement modeling accuracy was improved by analyzing features in both time and frequency domains. The method presented in [15] employed convolutional neural networks (CNNs) to extract high-dimensional features from load data. In [16], a game-theoretic relevance vector machine (RVM) approach was developed to enhance the robustness of pseudo-measurement generation. The authors of [17] combined one-dimensional convolutional neural networks (1D-CNN) with transfer learning to generate accurate pseudo-measurement data while maintaining high accuracy even under topology variations. Moreover, a spiking neural network (SNN)-based pseudo-measurement model was proposed in [18] to enhance modeling accuracy in scenarios with load fluctuations. Ref. [19] introduced an extreme learning machine (ELM) approach utilizing historical operational data to resolve the challenges of insufficient accuracy and efficiency in traditional pseudo-measurement models. To capture the temporal characteristics of distribution system operation states, some studies adopted a CNN–long short-term memory (CNN–LSTM) model [20] to enhance the temporal continuity of pseudo-measurements. However, the above pseudo-measurement modeling methods critically depend on the assumption of fixed distribution system topology, which is inconsistent with practical systems. Although transfer learning-based methods [17] are capable of alleviating the challenges caused by topology changes, the requirement for repeated model retraining under frequent topology switching scenarios significantly compromises the real-time performance of state estimation. Therefore, further investigation is required to enable effective pseudo-measurement modeling while ensuring its accuracy and reliability under dynamic topology conditions.

Based on pseudo-measurement modeling, distribution system operating states are typically estimated through static SE [16] or forecasting-aided SE [17]. Among static methods, the weighted least squares (WLS) method [21,22,23] is one of the most widely used techniques in practical applications. Under the assumption that measurement noise follows a zero-mean Gaussian distribution, WLS can provide minimum-variance unbiased estimates [24]. To enhance the robustness of SE, the weighted least absolute value (WLAV) estimator was proposed in [25] to mitigate the impact of bad data on estimation results. Moreover, with the capability of utilizing data from multiple time instances, forecasting-aided SE has received increasing attention in recent studies. In [18], pseudo-measurements were incorporated into the prediction step of unscented particle filtering to track the dynamic fluctuations of distribution system states. The authors of [19] applied an ELM-based pseudo-measurement model to predict the real and imaginary parts of bus voltages, thereby avoiding the iterative process of traditional nonlinear methods and improving both computational efficiency and accuracy. The method proposed in [26] employed a model that integrates a Transformer-based architecture enhanced with generalized robust principal component analysis (GRPCA) in conjunction with power flow calculations to generate bus-level pseudo-measurements and further adopted the adaptive interpolation strong tracking extended Kalman filter (AISTEKF) to improve prediction accuracy and estimation stability under the scenario of non-Gaussian noise and missing data. However, since the errors of pseudo-measurements are typically much larger than those of real-time field measurements, both static and forecasting-aided SE methods may suffer from numerical stability issues arising from excessive disparities between covariance matrix elements when processing these two types of data simultaneously, which can severely degrade estimation accuracy or even lead to divergence.

Recently, GNNs, as an emerging branch of deep learning in recent years, have demonstrated strong representation capabilities for graph-structured data by simultaneously modeling node features and network topology [27]. Ref. [27] provided a comprehensive review of GNNs and highlighted their significant advantages in modeling graph-structured data, such as power distribution networks. Ref. [28] further explained the topological generalization ability of GNNs within the framework of geometric deep learning. Some recent studies have explored the use of GNNs in DSSE. Ref. [29] proposed a distributed graph convolutional estimator capable of tracking node status changes under decentralized settings. Ref. [30] incorporated physical power models into the design of the EleGNN framework, which significantly improved estimation accuracy and robustness under missing measurements. Furthermore, the belief propagation (BP) theory has attracted considerable attention due to its high numerical stability and asynchronous computation capability. The authors of [31] introduced the BP algorithm into the field of the power system SE for the first time. Further, in [32], the SE problem under the extended direct current (DC) model was formulated as a factor graph, and the application of BP to SE problems was elaborated in detail. Ref. [33] proposed a fast real-time state estimator based on BP, which allows for more effective utilization of prior information and takes into account the evolving characteristics of power systems. Moreover, a large-scale multi-area SE method based on factor graphs was introduced in [34], which utilizes the sparsity of power systems and enables the distributed calculation of SE in non-overlapping areas without a centralized coordinator. In [35], a distributed SE approach was developed by integrating the WLS method for individual subsystems with the Gaussian BP (GBP) algorithm to obtain local solutions. References [36,37] investigate the convergence of the LBP algorithm in loopy graph structures and its relationship with Gibbs measures, respectively, providing theoretical support for its feasibility in power system state estimation. Existing studies have demonstrated that BP algorithms are free from numerical stability issues by eliminating the requirements of matrix calculations [33]. Moreover, the underlying probabilistic graphical structure of BP facilitates the incorporation of prior information about measurements and states, thus alleviating the impact of missing data [38], while complex-domain Gaussian belief propagation [39], although providing a distributed SE solution in the complex domain, does not explicitly incorporate pseudo-measurement modeling, which is crucial under low-observability conditions.. However, existing BP-based approaches rely heavily on a large number of real-time measurements to construct factor graphs; thus, they are only applicable to transmission systems with high measurement redundancy, while their feasibility and effectiveness can hardly be guaranteed in poorly observable distribution systems.

To address the challenges of low numerical stability and poor adaptability to frequent topology changes in SE for low-observability distribution systems, this paper proposes a graph computing-based SE method with enhanced numerical stability. First, a GNN is employed for pseudo-measurement modeling to obtain the values and variances of injected power. This approach enables the model to capture temporal correlations in state variations while accurately learning spatial dependencies among different measurement points, thereby improving adaptability to topology changes in distribution systems. Second, a hybrid factor graph is constructed to integrate both pseudo-measurements and real-time measurements, allowing for a precise representation of the relationships between measurements and system states. Then, an LBP algorithm is designed for the hybrid factor graph to perform fully distributed SE with improved numerical stability. This algorithm takes into account the temporal evolution of system states and prior information while avoiding global computation and high-dimensional matrix inversion. Finally, the effectiveness of the proposed method is validated through comprehensive case studies.

2. Basic Principles of SE

Conventional SE algorithms are based on physical models and aim to obtain the optimal estimate of the current operating state of the distribution system by processing field measurement data. The measurement equations of distribution systems can be expressed as follows [40,41]:

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

(1)

where $\mathbf{z}$ denotes the real-time measurement vector, including power flow, power injection, and voltage amplitude measurements; $\mathbf{x}$ is the state variable vector, including voltage magnitudes and phase angles; $\mathbf{e}$ is the measurement error vector, which is assumed to follow a zero-mean Gaussian distribution; and $h\left(\mathbf{x}\right)$ denotes the measurement functions.

The measurement functions $h\left(\mathbf{x}\right)$ can be expressed as:

$$\left\{\begin{array}{cc}\hfill P_i& =\sum _{j=1}^N{V}_i{V}_j\left(G_{ij}cos({\theta }_i-{\theta }_j)+B_{ij}sin({\theta }_i-{\theta }_j)\right)\hfill \\ \hfill Q_i& =\sum _{j=1}^N{V}_i{V}_j\left(G_{ij}sin({\theta }_i-{\theta }_j)+B_{ij}cos({\theta }_i-{\theta }_j)\right)\hfill \\ \hfill P_{ij}& =-V_i^2{G}_{ij}+V_j^2\left(G_{ij}sin({\theta }_i-{\theta }_j)-B_{ij}cos({\theta }_i-{\theta }_j)\right)\hfill \\ \hfill Q_{ij}& =-V_j^2{B}_{ij}+V_i{V}_j\left(G_{ij}sin({\theta }_i-{\theta }_j)-B_{ij}cos({\theta }_i-{\theta }_j)\right)\hfill \end{array}\right.$$

(2)

where $P_i$ and $Q_i$ represent the active and reactive power injections at bus i, respectively; $P_{ij}$ and $Q_{ij}$ represent the active and reactive power flows on branch $ij$, respectively; $G_{ij}$ and $B_{ij}$ denote the conductance and susceptance of branch $ij$, respectively; $V_i$ and $V_j$ are the voltage magnitudes at buses i and j, respectively; and ${\theta }_i$ and ${\theta }_j$ denote the voltage-phase angles at buses i and j, respectively.

The WLS method aims to minimize the weighted sum of squared measurement residuals. The objective function can be formulated as follows:

$$J\left(\mathbf{x}\right)={(\mathbf{z}-h\left(\mathbf{x}\right))}^T{\mathbf{R}}^{-1}(\mathbf{z}-h\left(\mathbf{x}\right))$$

(3)

where $J\left(\mathbf{x}\right)$ is the objective function to be minimized, and $\mathbf{R}$ is the covariance matrix of the measurement errors, where its diagonal elements represent the variances of the corresponding measurement noise.

The optimization problem is typically solved using the Gauss–Newton iterative method as follows:

$$\Delta \mathbf{x}={\left(\mathbf{G}\left(x_k\right)\right)}^{-1}\mathbf{H}{\left(x_k\right)}^T{\mathbf{R}}^{-1}(\mathbf{z}-h\left({\mathbf{x}}_k\right))$$

(4)

$${\mathbf{x}}_{k+1}={\mathbf{x}}_k+\Delta \mathbf{x}$$

(5)

where k and $k+1$ denote the iterations k-th and $(k+1)$-th, respectively; $\mathbf{H}\left(x_k\right)$ and $\mathbf{G}\left(x_k\right)$ represent the Jacobian matrix and the gain matrix in the iteration k-th, respectively, as shown in (6) and (7). The iteration stops when the state update $\Delta \mathbf{x}$ is less than a predefined convergence threshold.

$$\mathbf{H}\left(x_k\right)=\left[{\displaystyle \frac{\partial h\left(\mathbf{x}\right)}{\partial \mathbf{x}}}\right]{|}_{\mathbf{x}={\mathbf{x}}_k}$$

(6)

$$\mathbf{G}\left(x_k\right)=\mathbf{H}{\left(x_k\right)}^T{\mathbf{R}}^{-1}\mathbf{H}\left(x_k\right)$$

(7)

However, due to insufficient measurements and the low success rates of real-time data acquisition in distribution systems, such methods may suffer from inadequate estimation accuracy or even fail to converge [6].

3. Fully Distributed SE with GNN-Based Pseudo-Measurements

For distribution systems, the connectivity among buses plays a critical role in state estimation. GNNs are capable of effectively capturing the topological dependencies within power systems, enabling accurate representation of spatial relationships among different measurement points and thereby mitigating the impact of topology changes. In this paper, a GNN-based approach is presented for pseudo-measurement modeling, and a factor graph framework is utilized to perform SE based on the generated pseudo-measurement data.

The overall framework of the proposed method is illustrated in Figure 1. First, a graph-based dataset is constructed using historical measurement data and topological information from the distribution system. This dataset is then used to train the GNN, which outputs pseudo-measurement data ${\mathbf{z}}_p$ for buses with missing or unavailable measurements. Then, ${\mathbf{z}}_p$ are combined with real-time measurements ${\mathbf{z}}_{rt}$ and zero-injection measurements ${\mathbf{z}}_z$ to form a hybrid measurement set. Finally, the LBP algorithm is applied to perform fully distributed state estimation for the distribution system, obtaining the estimated value of state variables.

3.1. GNN-Based Pseudo-Measurement Modeling

Conventional modeling approaches are unable to account for topological changes, whereas GNNs not only capture temporal correlations and variations, but also accurately represent spatial dependencies among different measurement points. To overcome the limitations of conventional models in handling topological relationships, this study leverages the capability of GNNs to effectively model complex topological structures.

The pseudo-measurement modeling process consists of two main stages: offline training and online testing. In the offline training phase, a graph-based dataset is constructed using historical measurement data and topological information, which is leveraged to train the GNN to obtain a well-performing model. During the online testing phase, real-time measurement data and current topology information are converted into a graph-based dataset. The trained GNN model is applied to predict pseudo-measurements, and a Gaussian Mixture Model (GMM) is subsequently adopted to model the pseudo-measurement errors, yielding both the error distribution and variance.

In pseudo-measurement modeling, the distribution system is represented as a graph, where $\mathcal{V}$ denotes the set of nodes (buses) and $\mathcal{E}$ denotes the set of edges (branches). The node features are represented by ${\mathbf{X}}_V$, where $N_V$ is the number of nodes and $F_V$ is the number of node features. The edge features are represented by ${\mathbf{X}}_E$, where $N_E$ is the number of edges and $F_E$ is the number of edge features. In the context of DSSE, the measurement data $\mathbf{Z}$ are provided by the SCADA, and include the active and reactive power injections $P_i,Q_i$ at measured buses, as well as the active and reactive power flows $P_{ij},Q_{ij}$ on measured branches.

The forward propagation process of the proposed GNN consists of two main phases: message passing and readout [42]. During the message-passing phase, multiple rounds of information exchange are performed. For a specific node $vs.$, the process can be expressed as:

$$m_v^{t+1}=\sum _{w\in \mathcal{N}\left(v\right)}{M}_v^t\left(h_v^t,h_w^t,e_{vw}\right)$$

(8)

where $m_v^{t+1}$ indicates the updated information of node $vs.$ at iteration $t+1$; $\mathcal{N}\left(v\right)$ denotes neighboring nodes of node $vs.$; $h_v^t$ represents the state of node $vs.$ at time t; $e_{vw}$ signifies the edge feature linking node $vs.$; and w; $M_v^t(·)$ indicates the message function at time t. Edge modeling follows a similar approach to node modeling, defined by Equations (9):

$$m_{e_{vw}}^{t+1}=M_e^t\left(h_v^t,h_w^t,h_{e_{vw}}^t\right)$$

(9)

where $M_e^t(·)$ represents the edge message-passing function at time t. The node and edge message-passing functions used in this study are defined as follows:

$$m_v^{t+1}=W_n^t{h}_v^t+\sum _{w\in \mathcal{N}\left(v\right)}{h}_w^t{f}^t\left(h_{e_{vw}}^t\right)$$

(10)

$$m_{e_{vw}}^{t+1}=W_e^t{h}_{e_{vw}}^t+f^t\left(h_v^t,h_w^t\right)$$

(11)

where $W_n$ and $W_e$ denote the learnable parameter matrix, respectively. $f(·)$ is the activation function of the fully connected layer.

A certain level of discrepancy exists between the injected power predicted by the GNN and the actual value in the distribution system. The probability density function (PDF) of this prediction error cannot be accurately described using a simple Gaussian distribution, as the variance of measurement errors directly affects the weighting of the data, and consequently impacts the accuracy of the state estimation results. To ensure the precision of DSSE, this paper employs the Gaussian Mixture Model (GMM) to approximate the distribution of pseudo-measurement errors produced by the GNN. Specifically, the GMM consists of a weighted sum of multiple Gaussian components, allowing it to flexibly approximate probability distributions of arbitrary shapes in a smooth and accurate manner [43].

$$f\left(z_p\right|\gamma )=\sum _{i=1}^M{w}_{i}f\left(z_p\right|{\mu }_i,{\sigma }_i^2)$$

(12)

where $z_p$ represents the pseudo-measurement; M is the number of components in the model; $w_i$ is the weight of the i-th Gaussian component, satisfying $w_i>0,{\sum }_{i=1}^M{w}_i=1$; and ${\sigma }_i^2$ denote the mean and variance of the i-th Gaussian component, respectively. $\gamma$ represents the set of parameters in the GMM, which are typically estimated using the expectation maximization (EM) algorithm [44].

In this study, the pseudo-measurement error $e_p$ output of the GNN is modeled using M Gaussian components. The correlation between the error and the j-th Gaussian component is described by the marginal density function $f\left(j\right|e_{pi},\gamma )$, defined as the following:

$$f\left(j\right|e_{pi},\gamma )={\displaystyle \frac{w_{j}N(u_j,{\sigma }_j^2)}{{\sum }_{k=1}^M{w}_{k}N(u_k,{\sigma }_k^2)}}$$

(13)

where $N(u_j,{\sigma }_j^2)$ denotes a Gaussian distribution with mean $u_j$ and variance ${\sigma }_j^2$.

3.2. Factor Graph-Based Fully Distributed SE for Distribution Systems

In DSSE, the uncertainty associated with pseudo-measurements is typically much higher than that of real-time measurements. As a result, the covariance matrix often exhibits ill-conditioned characteristics due to the mismatch in the magnitude of its diagonal elements, leading to numerical instability in the estimation process. To address this issue, this paper proposes a distribution system state estimation method based on a hybrid factor graph. Specifically, a factor graph model is constructed using a hybrid measurement set consisting of pseudo-measurements $z_p$, real-time measurements $z_{rt}$, and zero-injection measurements $z_z$. The LBP algorithm is then applied in the hybrid factor graph to avoid matrix operations and, therefore, eliminate numerical stability issues.

According to the characteristics of different types of measurements, this paper constructs three types of factor nodes: the pseudo-measurement factor node $f_p$, the real-time measurement factor node $f_{rt}$, and the zero-injection measurement factor node $f_z$, corresponding to pseudo-measurements $z_p$, real-time measurements $z_{rt}$, and zero-injection measurements $z_z$, respectively. In addition, variable nodes and their edges connecting to the respective factor nodes are established based on the dependency relationships between each type of measurement and the state variables, thereby forming the hybrid measurement factor graph. The hybrid factor graph not only serves a computational function but also retains data, which allows for the effective storage of prior information throughout the estimation process. This facilitates consideration of the temporal evolution of system states and enhances the numerical stability of the LBP algorithm.

Given that the relationship between measurements and state variables in distribution systems is generally nonlinear, this paper first establishes the mapping between each measurement and the corresponding state variable increments through the hybrid measurement factor graph.

To clearly illustrate the structure of the proposed hybrid measurement factor graph, consider the three-bus system shown in Figure 2a, with the corresponding hybrid measurement factor graph depicted in Figure 2b. It should be noted that, in addition to the aforementioned factor nodes $f_p$, $f_{rt}$, and $f_z$, each variable node is also connected to an initialization factor node. For variable nodes with either measurement data or pseudo-measurements, the corresponding initialization factor node is assigned the mean and variance based on the difference between the measured (or pseudo-measured) value and the estimated value, reflecting the associated measurement error. For variable nodes without any available measurement or pseudo-measurement data, the initialization factor node is assigned a mean close to zero and an infinite variance to indicate the absence of prior information [39]. For brevity, the initialization factor nodes are omitted in Figure 2. In the system depicted in Figure 2, the set of variable nodes is presented by V. Voltage magnitude is constructed as a real-time measurement factor node $f_{rt}$. Zero-injection power is constructed as a zero-injection measurement factor node. Injection power is constructed as a pseudo-measurement factor node.

In the proposed hybrid measurement factor graph, an LBP algorithm is designed to perform system-wide state estimation through messages passing between adjacent nodes. Specifically, the message exchange is categorized into two types: messages sent from variable nodes to factor nodes, and messages sent from factor nodes to variable nodes. Once convergence is determined based on the message updates, marginal inference is performed at each variable node to obtain the corresponding state variable estimates.

(1) Messages from variable nodes to factor nodes

Consider a local factor graph that centers on the variable node $\Delta x_i$ in Figure 3a. The message transmitted from variable node $\Delta x_i$ to its adjacent factor node $f_s$ is explicated as follows. In Figure 3a, the sets $F_p\backslash f_s$, $F_z\backslash f_s$, and $F_{rt}\backslash f_s$, respectively, denote the pseudo-measurement factor node set, zero-injection measurement factor node set, and real-time measurement factor node set connected to variable node $\Delta x_i$, excluding the factor node $f_s$. When the outgoing message $u_{\Delta x_i\to f_s}$ from the variable node to the factor node follows a Gaussian distribution, this message can be characterized by its mean $c_{\Delta x_i\to f_s}$ and variance $s_{\Delta x_i\to f_s}$ [45]. The message propagated from variable node $\Delta x_i$ to factor node $f_s$ is formulated as the product of all incoming messages from factor nodes connected to $\Delta x_i$ except $f_s$.

$$c_{\Delta x_i\to f_s}=\left(\sum _{f_a\in F_s\backslash f_s}{\displaystyle \frac{c_{f_a\to \Delta x_i}}{s_{f_a\to \Delta x_i}}}\right)s_{\Delta x_i\to f_s}$$

(14)

$$s_{\Delta x_i\to f_s}=1/\left(\sum _{f_a\in F_s\backslash f_s}{\displaystyle \frac{1}{s_{f_a\to \Delta x_i}}}\right)$$

(15)

where $F_s\backslash f_s$ denotes the set of zero-injection measurement factor nodes $f_z$, real-time measurement factor nodes $f_{rt}$, and pseudo-measurement factor nodes $f_p$ neighboring the variable node $\Delta x_i$, excluding $f_s$ (i.e., $F_s\backslash f_s$ encompasses the subsets $F_z\backslash f_s$, $F_{rt}\backslash f_s$, and $F_p\backslash f_s$); $c_{f_a\to \Delta x_i}$ and $s_{f_a\to \Delta x_i}$ denote the mean and variance of messages transmitted from zero-injection measurement factor node $f_z$, real-time measurement factor node $f_{rt}$, or pseudo-measurement factor node $f_p$ in the set $F_s\backslash f_s$ to the variable node $\Delta x_i$, respectively. After obtaining all incoming messages from connected factor nodes in $F_s\backslash f_s$, the variable node computes the outgoing message $u_{\Delta x_i\to f_s}$ and propagates it to the factor node $f_s$.

(2) Messages from factor nodes to variable nodes

Consider a local factor graph that centers on the factor node $f_s$ in Figure 3b. The message transmitted from factor node $f_s$ to its adjacent variable node $\Delta x_i$ is explicated as follows. The factor node $f_s$ can be a zero-injection measurement factor node $f_z$, real-time measurement factor node $f_{rt}$, or pseudo-measurement factor node $f_p$. When the outgoing message $u_{f_s\to \Delta x_i}$ from the factor node to the variable node follows a Gaussian distribution, this message can be characterized by its mean $c_{f_s\to \Delta x_i}$ and variance $s_{f_s\to \Delta x_i}$.

$$c_{f_s\to \Delta x_i}=\left(c_s-\sum _{\Delta x_b\in B\backslash \Delta x_i}{H}_{\Delta x_b}·c_{\Delta x_b\to f_s}\right)/H_{\Delta x_i}$$

(16)

$$s_{f_s\to \Delta x_i}=\left(s_s+\sum _{\Delta x_b\in B\backslash \Delta x_i}{H}_{\Delta x_b}^2·s_{\Delta x_b\to f_s}\right)/H_{\Delta x_i}^2$$

(17)

where $B\backslash \Delta x_i$ denotes the set of variable nodes neighboring the factor node $f_s$, excluding $\Delta x_i$; $c_s=z_s-h_s\left(x\right)$ denotes the residual of the factor node $f_s$, where $z_s$ is a measurement value of zero-injection measurement, real-time measurement, or pseudo-measurement; and $h_s\left(x\right)$ is the calculated value of zero-injection measurement, real-time measurement, or pseudo-measurement. $s_s$ denotes the variance of the factor node $f_s$. The coefficient $H_{\Delta x_q}$ denotes Jacobian matrix elements of the measurement function associated with the factor node $f_s$.

$$H_{\Delta x_q}={\displaystyle \frac{\partial h_s(x_i,x_1,\dots ,x_K)}{\partial x_q}},q=i,1,\dots ,K$$

(18)

After obtaining all incoming messages from connected variable nodes in $B\backslash \Delta x_i$, the factor node $f_s$ computes the outgoing message $u_{f_s\to \Delta x_i}$ and propagates it to the variable node $\Delta x_i$.

(3) Marginal inference

After convergence, the marginal of the variable node $\Delta x_i$ is obtained as the product of all incoming messages into the variable node $\Delta x_i$. The mean and variance of $\Delta x_i$ are then derived using the following expressions:

$$\widehat{\Delta x_i}=\sum _{f_c\in F_s}\left({\displaystyle \frac{c_{f_c\to \Delta x_i}}{s_{f_c\to \Delta x_i}}}\right)s_{\Delta x_i}$$

(19)

$$s_{\Delta x_i}=1/\left(\sum _{f_c\in F_s}{\displaystyle \frac{1}{s_{f_c\to \Delta x_i}}}\right)$$

(20)

where $F_s$ denotes the set of factor nodes neighboring the variable node $\Delta x_i$; $s_{\Delta x_i}$ denotes the variance of the i-th variable node; $\Delta {\widehat{x}}_i$ denotes the estimated increment of the state variables. By iteratively updating the state variables using Equation (5) until convergence is achieved, the estimated state variables of the distribution system are obtained.

4. Case Study

4.1. Accuracy Evaluation of Pseudo-Measurement Modeling

The proposed pseudo-measurement modeling method is evaluated on the IEEE 33-bus and 95-bus test systems, whose system topologies are shown in Figure A1 and Figure A2, respectively. The measurement configuration is as follows: active power injection $P_i$, reactive power injection $Q_i$, and voltage magnitude $V_i$ at all load buses are obtained through the SCADA system, along with the active and reactive power flows $P_{ij}$, $Q_{ij}$ measurements on selected branches. The true values for the case studies are derived from power flow calculations, upon which zero-mean Gaussian noise is added to simulate measurement data. The training data are derived from the one-day dataset in [24], with a temporal resolution of 15 min. To simulate diverse operating conditions, we generated a 100-day dataset by applying up to 30% zero-mean random perturbations to the active and reactive loads. Four distinct topological configurations were simulated to represent realistic network reconfigurations, such as line switching and load transfers. Specifically, topology 1 is used from day 1 to day 20, topology 2 from day 21 to day 80, and topology 3 from day 81 to day 100. Topology 4, which is not included in the training data, is used exclusively for testing to evaluate the model’s generalization capability under untrained network configurations. The total number of data points is 9600, and the first 90 days are used for training while the last 10 days form the test set. SCADA measurements of voltage magnitudes, power injections, and line flows are assumed to be available at selected buses and branches. The detailed topology configurations are listed in

Table 1. Modified topology cases in the IEEE 33-bus and 95-bus test systems.

Test System	Case Index	Closed Switches	Open Switches
IEEE 33-bus	1	21–8, 9–15, 25–29	12–22, 18–33
	2	21–8, 9–15	12–22, 18–33, 25–29
	3	12–22, 18–33, 25–29	9–15, 21–8
	4	All switches	—
IEEE 95-bus	1	39–60, 41–65, 65–82	19–26
	2	39–60, 30–33	65–82
	3	19–26, 30–33	39–60
	4	65–82, 19–26, 30–33	41–65

. Furthermore, the standard deviations of measurement errors and the corresponding measurement placement schemes used in the simulations are summarized in

Table 2. Measurement configuration and uncertainty.

System	Measurement Type	Measurement Location (Bus/Branch)	Standard Deviation
IEEE 33-bus	Voltage magnitude and bus injection power	1–33	Voltage magnitude: 0.5%, Power: 1%
	Missing data	15, 16, 17, 18, 24, 26, 27, 28 (8)	—
	Power flow	1–2, 3–4, 14–15, 6–26	1%
	Missing data	3–14, 14–15 (2)	—
IEEE 95-bus	Voltage magnitude and bus injection power	1, 3, 5, 6, 7, 9, 10, 11, 13, 15, 17, 18, 21, 23, 25, 27, 29, 31, 32, 33, 35, 36, 37, 38, 41, 42, 44, 46, 48, 50, 51, 54, 56, 58, 59, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 80, 81, 86, 90, 92, 95 (53)	Voltage magnitude: 0.5%, Power: 1%
	Missing data	15, 16, 17, 18, 24, 26, 27, 28 (8)	—
	Power flow	1–93, 1–94, 20–19, 34–35, 34–33, 34–38, 83–82, 83–83, 83–84 (8)	1%
	Missing data	20–19, 83–82, 83–82, 83–84 (3)	—

.

The GNN model is implemented in PyTorch 2.0.1. For the IEEE 33-bus system, the GNN is configured with three layers and 64 neurons per layer, trained over 30 epochs with a batch size of four. For the IEEE 95-bus system, the model is also configured with three layers, but with 96 neurons per layer, trained over 30 epochs and a batch size of eight. The number of components in the GMM used to model pseudo-measurement errors is set to four. To validate the effectiveness of the proposed method, a deep neural network (DNN) is selected as a benchmark for comparison. The accuracy of pseudo-measurement modeling is evaluated using the mean absolute percentage error (MAPE), defined as follows:

$$\mathrm{MAPE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|{\displaystyle \frac{S^{\mathrm{pseudo}}-S^{\mathrm{real}}}{S^{\mathrm{real}}}}\right|\times 100\%$$

(21)

where n is the number of buses, $S^{\mathrm{pseudo}}$ represents the pseudo-measurement values, and $S^{\mathrm{real}}$ represents the true values.

A total of 100 consecutive time slices are selected for statistical analysis of the pseudo-measurement modeling results. The error distribution between the GNN-generated pseudo-measurements (normalized values) and the corresponding true active power injection values (also normalized) at buses with missing measurements is shown in Figure 4. As observed, in both test systems, the absolute error between the true values and the pseudo-measurements remains within 0.04 for the majority of data points, indicating that the proposed method achieves high accuracy in pseudo-measurement modeling.

Table 3. Comparison of MAPE results for pseudo-measurements under Topology 4.

Systems	IEEE 33-Bus		IEEE 95-Bus
	Proposed	DNN	Proposed	DNN
P Pseudo	12.451%	17.213%	14.074%	18.249%
Q Pseudo	15.238%	18.135%	16.752%	20.365%

presents a detailed comparison of the MAPE for bus injection pseudo-measurements under Topology 4, which was excluded from the training data to evaluate model generalization. The results are shown for both the IEEE 33-bus and 95-bus systems, comparing the performance of the proposed GNN-based approach against a conventional DNN baseline. As observed in the table, the proposed method consistently achieves lower MAPE values across both P and Q pseudo-measurements in both test systems. Specifically, in the IEEE 33-bus system, the proposed model reduces the MAPE of active power from 17.213% to 12.451%, and reactive power from 18.135% to 15.238%. In the larger and more complex IEEE 95-bus system, similar improvements are observed, with MAPE reductions from 18.249% to 14.074% for active power, and from 20.365% to 16.752% for reactive power. These results demonstrate that the GNN-based pseudo-measurement model achieves higher accuracy than the DNN.

4.2. Accuracy Evaluation of LBP with Pseudo-Measurements

To evaluate the effectiveness of the proposed algorithm under different measurement uncertainty scenarios, both the proposed LBP method based on the hybrid factor graph and the conventional WLS method are tested on the aforementioned systems. In this section, two measurement noise scenarios are considered:

1.

Normal noise scenario: the standard deviation of voltage magnitude, injection power, and branch power are set to 0.5%, 1%, and 1%, respectively.

2.

High noise scenario: the standard deviation of voltage magnitude is increased to 1%, while the standard deviations of injection power and branch power are both increased to 10%.

$$MAE={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|V_i^{\mathrm{esti}}-V_i^{\mathrm{true}}\right|$$

(22)

where $V_i^{\mathrm{esti}}$ denotes the estimated voltage magnitude and $V_i^{\mathrm{true}}$ denotes the true voltage magnitude.

The estimation accuracy of the proposed LBP algorithm is compared with that of the conventional WLS method on the IEEE 33-bus and 95-bus systems. The results are averaged over 100 Monte Carlo simulations. The MAE values under both normal and high measurement noise scenarios for the two methods are summarized in

Table 4. Average MAE values for different methods in the normal measurement noise scenario.

Algorithm	IEEE 33-Bus		IEEE 95-Bus
	${\mathit{U}}_{\mathit{av}}/\mathit{pu}$	${\mathit{\theta }}_{\mathit{av}}/\mathit{rad}$	${\mathit{U}}_{\mathit{av}}/\mathit{pu}$	${\mathit{\theta }}_{\mathit{av}}/\mathit{rad}$
LBP	$1.00\times {10}^{-3}$	$1.98\times {10}^{-4}$	$2.56\times {10}^{-3}$	$2.72\times {10}^{-3}$
WLS	$2.76\times {10}^{-3}$	$8.04\times {10}^{-4}$	$4.21\times {10}^{-3}$	$4.91\times {10}^{-3}$

and

Table 5. Average MAE values for different methods in the large measurement noise scenario.

Algorithm	IEEE 33-Bus		IEEE 95-Bus
	${\mathit{U}}_{\mathit{av}}/\mathit{pu}$	${\mathit{\theta }}_{\mathit{av}}/\mathit{rad}$	${\mathit{U}}_{\mathit{av}}/\mathit{pu}$	${\mathit{\theta }}_{\mathit{av}}/\mathit{rad}$
LBP	$1.89\times {10}^{-3}$	$1.23\times {10}^{-3}$	$6.75\times {10}^{-3}$	$6.34\times {10}^{-3}$
WLS	$3.03\times {10}^{-3}$	$2.45\times {10}^{-3}$	$9.84\times {10}^{-3}$	$2.67\times {10}^{-2}$

, respectively. It can be observed that the proposed method consistently outperforms the WLS estimator in terms of estimation accuracy. This improvement is primarily attributed to the superior numerical stability of the LBP algorithm, which is particularly advantageous in distribution systems with a high number of zero-injection buses and pseudo-measurements. The proposed method can accommodate extreme variances in measurement errors, whereas the WLS estimator tends to exhibit reduced accuracy or even convergence failure under such conditions. Moreover, the WLS method does not exploit prior information of the distribution system, while the proposed method, through its factor graph structure, retains prior knowledge and effectively incorporates the temporal evolution of system states, thus mitigating the impact of pseudo-measurement modeling errors on the estimation results. For the IEEE 33-bus system, the bus-level estimation errors under the normal measurement noise scenario are illustrated in Figure 5. It can be seen that while both algorithms meet the accuracy requirements, the proposed method achieves significantly lower overall estimation errors. Even for buses where WLS shows large errors, the proposed method maintains high estimation accuracy. Figure 6 shows the estimation errors under the high measurement noise scenario for the IEEE 33-bus system. Compared with the normal noise scenario, both methods experience an increase in estimation error; however, only the proposed method maintains error levels at the order of 10–3. These results collectively demonstrate that the proposed method offers more accurate SE for low-observability distribution systems with limited measurement configurations and numerous zero-injection buses.

5. Conclusions

This paper proposes a graph computing-based state estimation method with enhanced numerical stability for low-observability distribution systems. By employing a G to model the topological dependencies within the distribution system, the proposed method enables the construction of a pseudo-measurement model with high reliability and accuracy. Subsequently, a GMM is introduced to capture the uncertainty in pseudo-measurement errors. Moreover, a fully distributed state estimation algorithm suitable for low-observability scenarios is implemented through factor graph computation, effectively mitigating potential numerical instability issues caused by zero-injection measurements and pseudo-measurements. Comprehensive case studies under various operating conditions on the IEEE 33-bus and 95-bus distribution systems verified the effectiveness of the proposed method.