Reverse Kron reduction of Multi-phase Radial Network

TL;DR

The paper addresses the problem of identifying the full three-phase admittance matrix $Y$ of a radial, unbalanced network from measurements at a subset of nodes by inverting the Kron-reduced mapping $\bar{Y} = Y/Y_{22}$. It introduces a reverse Kron reduction framework built on invariant structures that are preserved under Kron reduction, enabling the reconstruction of $Y$ from $\bar{Y}$ via a sequence of maximal-clique decompositions and reverse steps (Algorithms 1–3). The key contributions are (i) a graph-theoretic decomposition showing $G(\bar{Y})$ consists of edge-disjoint maximal cliques connected by hidden-node trees, (ii) a forward Kron-reduction procedure that grows a maximal clique in a relabeled permuted matrix while preserving an invertible structure, and (iii) a reverse Kron-reduction algorithm that identifies sibling nodes and reconstructs the full $Y$ under a uniform-line assumption. Collectively, this work extends prior single-phase Kron-reduction approaches to unbalanced three-phase radial networks, enabling exact topology and line-parameter identification from partial measurements with practical implications for distribution-grid monitoring and control.

Abstract

We consider the problem of identifying the admittance matrix of a three-phase radial network from voltage and current measurements at a subset of nodes. These measurements are used to estimate a virtual network represented by the Kron reduction (Schur complement) of the full admittance matrix. We focus on recovering exactly the full admittance matrix from its Kron reduction, i.e., computing the inverse of Schur complement. The key idea is to decompose Kron reduction into a sequence of iterations that maintains an invariance structure, and exploit this structure to reverse each step of the iterative Kron reduction.

Figures (5)

• Figure 1: Decomposition of admittance matrix $Y$ according to edges. Shaded nodes are measured nodes and unshaded nodes are hidden nodes.
• Figure 2: (a) Original graph $G(Y)$. (b) Kron reduction $G(\bar{Y})$. (c) Graph $G(\bar{Y}')$ after removing internal measured nodes.
• Figure 3: Original graph $G(Y)$ and Kron reduction $G(\bar{Y}')$ after internal measured nodes are removed. The resulting admittance matrix $\bar{Y}'$ takes the form in \ref{["fig:G(Y)andG(barY'); eq:barY'"]} with the 4 maximal cliques ordered left to right.
• Figure 4: Example \ref{['eg:IterativeKronReduction.1']}: Iterative Kron reduction. The graph $G_0$ underlying $A^0$ is a tree with shaded and unshaded nodes. The unshaded nodes correspond to $A_{22}$ and are to be Kron reduced. In each iteration, only entries (corresponding to a maximal clique in $A^{l+1}$) marked by red triangles or red squares are updated; other entries remain the same as their values in $A^0$.
• Figure 5: Reversible iterative forward Kron reduction with three hidden nodes. The focus of iterative Kron reduction and its reverse process will be on the sequence of permuted matrices and their maximal clqiues $(\hat{A}^0, C^l), \dots, (\hat{A}^k, C^k)$.

Theorems & Definitions (22)

• Definition 1: Block symmetry and block row sum
• Lemma 1: Low2022
• Remark 1: Assumption \ref{['Assumption:y_jk']}
• Remark 2: Assumption \ref{['Assumption:HiddenNodes']}
• Remark 3: Identification of $\bar{Y}$
• Theorem 2: Boundary and internal measured nodes
• proof
• Remark 4
• Example 1
• Example 2