Title : A multi-stage linear state estimation method for three-phase unbalanced distribution grids
Abstract:
This paper addresses the challenge of state estimation in distribution grids with limited observability and complex three-phase network behavior. Unlike transmission systems, distribution grids are characterized by sparse measurement infrastructure, unbalanced operation, and nonlinear three-phase power flow equations, which make accurate and computationally efficient state estimation difficult. To address these challenges, this paper proposes a novel iterative linear method for three-phase state estimation in distribution networks. The proposed approach performs state estimation using real-time measurements from a limited number of meters, installed at less than 10% of the network nodes, together with historical aggregated energy consumption data available at each node.
To enhance observability, a forecasting framework is first employed to estimate nodal active and reactive power consumption. The historical measurements are preprocessed through timestamp alignment, duplicate aggregation, and missing-data treatment using interpolation and forward/backward filling techniques. A multi-level ensemble learning model based on a voting regressor framework is then applied to generate 24-hour-ahead forecasts using weighted combinations of decision trees, random forests, and gradient-boosted models (XGBoost). The final forecasted time step is subsequently used for phase disaggregation and state estimation.
The proposed state estimation framework captures the complexity of the distribution grid through the development of sensitivity matrices that linearize the relationship between nodal power injections and voltage profiles. These matrices are incorporated into a linear optimization problem to estimate the system states efficiently. Since the sensitivity matrices depend on the network operating point, a multi-stage iterative framework is introduced to update the matrices at each iteration and progressively improve the estimation accuracy.
The proposed method is validated on a real distribution test system using real-time measurements collected from a limited set of nodes. The framework estimates voltage magnitudes at unmeasured nodes, and its performance is evaluated using actual smart meter data. The results demonstrate that the proposed approach achieves high estimation accuracy despite the limited measurement availability and the nonlinear characteristics of three-phase distribution systems. In particular, the proposed estimator achieves a mean absolute voltage estimation error of less than 0.75 V across the evaluated scenarios.
The findings indicate that the proposed iterative linear state estimation framework provides an effective and computationally efficient solution for monitoring modern distribution grids with sparse measurements. The proposed approach offers a practical alternative to conventional nonlinear state estimation techniques while maintaining high accuracy, making it suitable for real-time applications in active distribution networks with high penetration of distributed energy resources.
