Unbalanced Multi-Phase Grid Topology Estimation

• The paper’s main contribution is a robust methodology leveraging Chow–Liu trees and mutual information to estimate grid topology and phase associations in unbalanced systems.
• It employs voltage sequence transformation and covariance analysis to calculate statistical dependencies, achieving high phase identification accuracy even with measurement noise.
• The approach demonstrates practical scalability and robustness on IEEE feeders, offering reliable grid monitoring, state estimation, and DER integration for modern distribution networks.

Unbalanced multi-phase distribution grid topology estimation refers to the identification of the operational connectivity (lines, branches, switches) and phase associations of electric distribution networks, where lines and loads are not perfectly balanced in terms of phase currents or voltages. This is a fundamental step for grid monitoring, control, and situational awareness, particularly as distributed energy resources (DERs) increase grid complexity. Access to accurate topology and phase information is often limited in practice due to incomplete records, switch operations, and phase labeling errors, especially in low-voltage (LV) feeders. Modern approaches integrate voltage and sometimes current measurements from smart meters, and leverage statistical, graphical, and optimization-based methods that accommodate unbalanced operation, partial observability, and measurement noise.

1. Mathematical Modeling of Unbalanced Multi-Phase Distribution Grids

Unbalanced distribution grids are commonly represented as radial (tree-structured) or weakly meshed graphs $\mathcal{G}=(\mathcal{N},\mathcal{E})$, where each node (bus) may carry a subset of three phases $\{a, b, c\}$. Electrical quantities at each node and branch are vector-valued, capturing each present phase.

For a bus $i$, the phase-to-ground voltage vector is

$$V_{abc,i} = [V^a_i, V^b_i, V^c_i]^T \in \mathbb{C}^{3}.$$

The branch or line connecting nodes $i$ and $k$ is characterized by the 3×3 (or reduced) impedance matrix $Z_{ik}$, often including both self and mutual impedances, as described by Carson’s equations:

• Self-impedance: $z^{aa} \approx r^a + j x_1$
• Mutual impedance: $z^{ab} \approx j x_0$, etc.

Unbalanced systems often leverage the symmetrical-component transformation:

$V_{abc,i} = H V_{pnz,i}$

where $V_{pnz,i} = [V_i^+, V_i^-, V_i^0]^T$ (positive, negative, and zero sequences), $H$ is the standard symmetrical-components matrix ($h = e^{j2\pi/3}$), and the inverse is $V_{pnz,i} = (1/3) H^H V_{abc,i}$. This conversion decouples system dynamics under weak mutual coupling, enabling independent analysis of each sequence.

Power-flow relationships, typically nonlinear in the full AC model, are commonly linearized for estimation. For each sequence $\ell\in\{+, -, 0\}$, voltage increments and injection changes relate via

$\Delta V^\ell = Z^\ell \Delta I^\ell,$

where $Z^\ell$ is the sequence nodal impedance (inherited from the tree topology), and the common assumption is that $\Delta I^\ell_i$ are independent, zero-mean random vectors over short time intervals.

2. Statistical and Information-Theoretic Frameworks

Topology estimation is cast as a probabilistic graphical model inference problem. Under the assumptions of independent, approximately Gaussian injections, the joint distribution of voltage increments in a given sequence obeys the Markov property induced by the true radial structure:

$$P(\Delta V^\ell) = \prod_{i=1}^M P(\Delta V^\ell_i \mid \Delta V^\ell_{\text{pa}(i)}).$$

Conditional independence relationships imply that for any non-adjacent $i, k$ in the tree,

$$\Delta V^\ell_i \perp \Delta V^\ell_k \mid \Delta V^\ell_{\text{pa}(i)}.$$

Within this framework, the optimal tree structure (minimizing Kullback-Leibler divergence between the empirical voltage distribution and a tree-structured model) is found by maximizing total mutual information between connected nodes (Chow-Liu algorithm). For Gaussian vectors $X, Y$, mutual information is computed as

$$I(X;Y) = \frac{1}{2} [\log |\Sigma_{XX}| + \log |\Sigma_{YY}| - \log |\Sigma_{XY}|],$$

where $\Sigma_{XX}$, $\Sigma_{YY}$, $\Sigma_{XY}$ are sample covariances.

3. Algorithmic Strategies for Topology and Phase Identification

Chow–Liu-Based Topology Recovery

A robust, computationally efficient strategy for topology estimation proceeds as follows:

1. Sequence Transformation: Convert measured voltages (and optionally currents) to positive, negative, and zero-sequence components via the $H$-transform.
2. Covariance and Mutual Information Calculation: For each node pair $(i,k)$, compute empirical covariances over time and evaluate $I_{ik}$.
3. Maximum-Weight Spanning Tree: Construct a complete weighted graph and extract the maximum-weight spanning tree (MST) via Kruskal’s algorithm, using mutual information as edge weights.
4. Phase Association via Correlation: For each edge $(i,k)$ in the MST, determine phase correspondence by maximizing:

$$(\phi^*, \psi^*) = \operatorname{argmax}_{\phi,\psi\in\{a,b,c\}} \text{corr}(|\Delta V^\phi_i|, |\Delta V^\psi_k|).$$

Carson's equation ensures that the voltage increments on same phases are most strongly correlated. This phase identification is propagated from the substation outwards.

Computational and Data Requirements

• Covariance and mutual information computation: $O(M^2 N)$ for $M$ nodes, $N$ samples.
• MST construction: $O(M^2 \log M)$.
• A few hundred to a few thousand time samples (minutes to days of smart-meter data) suffice for sub-percent error rates.

Magnitude-Only and Clustering Approaches

When phasor angles are not available, a magnitude-only version of the algorithm applies the sequence transform to absolute voltage values. Empirical performance degrades only modestly; coarser sampling intervals may enhance correlation-based phase identification due to smoothing effects.

4. Validation, Empirical Performance, and Robustness

Algorithms are validated on IEEE standard feeders (37, 123, and 8500-bus test systems) simulated with real-world load profiles including DER penetration (PV, etc.). Results reflect strong robustness:

• For feeders up to 123 buses, topology error is 0% (both with angles and magnitudes), and phase-identification exceeds 99% accuracy, even with 20% PV penetration and with 10% random phase-label flips.
• With realistic measurement noise of ±0.5%, topology error increases to 3–4% and is further reducible via prior knowledge (e.g., feeder structure).
• The largest (8500-bus) case yields initial topology error ∼16%, which drops to ∼4% when low-voltage feeder structure is pre-fixed.
• Data requirements: e.g., 20–30 days of hourly data or 6 hours of 1-minute data for high-accuracy recovery.
• Computation time for the largest system is ∼700 seconds serially or ∼180 seconds in parallel.
• The method is robust to strong load unbalancing, DER variability, measurement noise, and incorrect initial phase tagging.

Test System	Error (noise-free)	Error (0.5% noise)	Data Length Required
37/123-bus	0%	≪1%	20–30 days (hourly data)
8500-bus	16%→4%*	–	20–30 days

*After pre-fixing LV feeder structure.

5. Limitations, Assumptions, and Extensions

The main assumptions are:

• The true distribution network is (approximately) radial.
• Voltage phasors are measured at all (or most) buses.
• Load (injection) increments over short horizons are mutually independent and approximately Gaussian.
• Coupling between sequence networks is weak (justifying the sequence-component decoupling).

Limitations:

• If the system is strongly meshed or contains substantial mutual (off-diagonal) couplings, estimation accuracy diminishes.
• Topology error is sensitive to severe measurement corruption, though the drop is gradual.
• Phase labeling relies on having at least one reference point (e.g., the substation) with correct phase assignment.
• Performance degrades for heavily correlated load profiles or extended periods with little load variation.

Significant extensions include:

• Adaptation to partial observability (when not all nodes are metered).
• Incorporation of historical feeder structure as a prior for improved estimation.
• Generalizations to weakly meshed networks via post-processing.

6. Practical Impact and Applications in Distribution System Monitoring

Accurate unbalanced multi-phase topology and phase estimation is foundational for:

• Monitoring and control of DERs in LV grids, where unbalanced operation is prevalent.
• Diagnostic functions such as event localization, fault detection, and state estimation.
• Enabling advanced applications such as real-time reconfiguration, volt/VAR optimization, and demand response.
• Providing reliable phase labeling despite human error or outdated records.

The described information-theoretic and correlation-based approaches offer high sample efficiency, scalability, and robustness to realistic noise and system heterogeneity. Empirical results confirm feasibility for deployment in smart-meter-enabled distribution systems, making these algorithms suitable for ongoing grid modernization efforts and DER integration policies.

7. Connections to Related Approaches and Research Directions

This information-theoretic approach via Chow–Liu trees is complemented by alternative methodologies, including:

• Graphical model approaches leveraging conditional independence in voltage covariances (Deka et al., 2018).
• Optimization-based formulations using MILP or MIQP integrating measurements, pseudo-loads, and device statuses for jointly estimating topology and state (Gandluru et al., 2019, Soltani et al., 2021).
• Recursive grouping algorithms with Cholesky whitening for hidden node and impedance recovery (Li et al., 2019).
• Greedy, covariance- and variance-based topology and phase assignment methods with guaranteed recovery under explicit linear model assumptions (Bariya et al., 2020).

Future directions lie in integrating these methods with advanced state estimation, optimal sensor placement, and robustification for adverse data conditions, as well as extending to feeders with highly meshed substructures or dynamic topology changes. A plausible implication is that as data acquisition rates and sensor coverage improve, topology and phase identification will become increasingly automated and reliable across the heterogeneity of real-world distribution grids.