Guaranteed Phase & Topology Identification in Three Phase Distribution Grids

Topology identification—which determines the connectivity of network nodes from an available measurement set—is vital in the electric grid, especially in distribution networks [1], which may be switched between multiple operating configurations. Real-time topology awareness is critical for most control and optimization approaches and is essential for detecting unintentional changes caused by faulty equipment or cyberattacks. In three phase electrical networks, phase identification—determining the phase label of each phase at each bus—is part of topology identification. Correct phase labels are important for several applications, including allocating resources to minimize phase imbalance [2]. Together, topology and phase identification determine the entire network connectivity. The literature demonstrates several approaches to phase identification—many heuristic—on simulated or real data sets. In [3] and [4] power balance constraints on load measurements are used to identify customer phase connectivity; this approach suffers from multiple feasible solutions. Another class of techniques applies clustering to power or voltage measurements to categorize phases, as in [5] and [6]. Correlations are a popular clustering distance metric for voltage magnitudes [7, 8, 9]. In [10], voltage correlations are used with voltage angle differences to match phases at two buses. Though some of these methods achieve good performance, they don’t provide theoretical guarantees, stymieing understanding of why and how they work, and if and when they fail. For example, clustering algorithms such as k-means [11] can suffer from local minima that may result in incorrect solutions [12]. Furthermore, a theoretical justification enables understanding how load and network characteristics, such as the radial structure and line impedances, impact the success of an approach. [13] presents a maximum likelihood estimator for phase identification, but it requires knowledge of the grid topology and is non-convex.

Topology identification has been extensively explored in the literature. Several works use voltage correlations or covariances to recover topology, based on heuristics [14], theoretically justified patterns in covariances [15, 16], or conditional independence tests [17, 18]. Another class of approaches uses linear regression to estimates the entire network model, including topology and line impedances, but require phasor current measurements in addition to voltages [19, 20, 21, 22, 23]. All these approaches assume that the system is balanced and can be approximated by a single phase network. This may lead to errors estimating unbalanced distribution networks. [24, 25] present three-phase topology estimation, possibly with incorrect phase labels, using conditional independence or mutual information tests. However the sample requirements of these methods often exceed those of direct/greedy methods such as [1]. Post-topology recovery, [24] uses a distance-based metric or mixed-integer program to classify unknown phases. This still falls short of a theoretically guaranteed, polynomial-time algorithm for phase recovery.

Contribution: This work presents a greedy algorithm for joint phase and topology identification—termed GPT—surpassing the prior work in that it

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  is provably correct under realistic assumptions and runs in polynomial time.

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  is applicable to real networks, where some buses may have a subset of all three phases.

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  requires only voltage measurements; either phasors or magnitudes (with slight differences in performance).

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  utilizes voltage statistics enabling successful phase identification even from voltage magnitudes alone and in the presence of phase shifting transformers [26].

GPT is based on a linearized, multi-phase power flow model mapping nodal current injections to nodal voltage phasors, and requires radial structure and diagonal dominance of line impedance matrices to guarantee correctness. When phases are known, GPT reduces to provable, greedy multi-phase topology learning generalizing prior work in [1] for the balanced setting. When topology is known apriori, it reduces to a local approach to phase identification. We demonstrate GPT’s performance and improvement over prior work in both phase and topology recovery on multiple IEEE test networks simulated in Open-DSS, an open source distribution system simulator [27].

The paper is organized as follows. Section II presents a linearized model for unbalanced, three-phase networks with missing phases which is the theoretical basis of our work. Section III theoretically justifies a proximity metric for phase identification while Section IV justifies a distance metric for topology identification once phases have been identified. Section V puts the parts together to propose GPT: an algorithm for joint phase and topology recovery. Finally, Section VI presents validation on non-linear voltage data for three IEEE distribution test networks, simulated in OpenDSS. We compare performance with algorithms in prior work, demonstrating that our approach outperforms the prior work, and is robust to non-ideal measurements.

Before introducing our unbalanced three phase network model, we review the single phase balanced power flow model. In the single phase case, each line $ij$ has an associated scalar admittance $\mathbf{y}_{ij}$ and impedance $\mathbf{z}_{ij}=\mathbf{y}^{-1}_{ij}$. The single phase voltage phasors and current injections are related by the $(|\mathcal{N}|\times|\mathcal{N}|)$ system admittance matrix $\mathbb{Y}$ as $\mathbf{I}=\mathbb{Y}\mathbf{V}$ with the form:

$\mathbb{Y}$ can be factored into $|\mathcal{N}|\times|\mathcal{E}|$ incidence matrix $A$ and diagonal line impedance matrix $\mathbf{D}$ [28]. Without loss of generality, we choose all edges to be directed away from the network “root”, generally the point of common coupling (PCC) or substation. If edge $ij$ is oriented from $i$ to $j$, the corresponding elements of $A$ are

where $A_{i,ij}$ is the $i^{th}$ element of the column corresponding to edge $ij$. By definition, as $\mathbb{Y}1=0$, $\mathbb{Y}$ is not invertible. An invertible reduced admittance matrix, $\underline{\mathbb{Y}}$, is constructed by choosing a reference node $r$ and removing the corresponding row and column of $\mathbb{Y}$. Since the system is lossless, its inverse relates voltages and currents as follows:

$\underline{\mathbf{V}}$ contains voltages differences to the reference voltage while $\underline{\mathbf{I}}$ contains current injections at non-reference nodes. Let $\mathcal{E}_{i}$ and $\mathcal{E}_{j}$ denote the edge sets on the unique path in the radial system to $r$ from nodes $i$ and $j$ respectively. The value of $\underline{\mathbb{Z}}_{i,j}$ is given by :

Thus, the elements of $\underline{\mathbb{Z}}$ correspond to the impedances of common paths between node pairs and the reference [19, 1].

The unbalanced three phase model follows from the single phase one. To clarify definitions, we use ToyNet (Fig. 1b), a simple, unbalanced, three phase radial network, as a running example. We begin with the model for a multiphase line $ij$, with phases $\mathcal{M}_{ij}\subseteq\{a,b,c\}$. The voltage across $ij$ is related to the current along each phase of the line by line impedance matrix $\mathbf{Y}_{ij}$:

$\mathbf{Y}_{ij}$ is the inverse of the $(|\mathcal{M}_{ij}|\times|\mathcal{M}_{ij}|)$ line impedance matrix, $\mathbf{Z}_{ij}$. Eq. (2) for line $56$ in ToyNet is: $\begin{bmatrix}\textbf{i}_{56}^{a}\\ \textbf{i}_{56}^{c}\end{bmatrix}=\mathbf{Y}_{56}\begin{bmatrix}\textbf{v}_{5}^{a}-\textbf{v}_{6}^{a}\\ \textbf{v}_{5}^{c}-\textbf{v}_{6}^{c}\end{bmatrix}$. The node $i$ current injection, denoted $\mathbf{I}_{i}$, is a vector of injections on each phase of $i$, and is given by the sum of line flows: $\mathbf{I}_{i}=\sum_{ij\in\mathcal{E}}\mathbf{I}_{ij}$. Building up from the current-voltage relations across individual lines in (2), the multi-phase voltages and currents injections across the network are related by $\mathbf{I}=\hat{\mathbb{Y}}\mathbf{V}$. Note this model can describe a network with a subset of phases at some nodes. $\hat{\mathbb{Y}}$ is the multi-phase system admittance matrix with dimensions $(\sum_{i\in\mathcal{N}}|\mathcal{M}_{i}|)\times(\sum_{i\in\mathcal{N}}|\mathcal{M}_{i}|)$. The $i,j$ block of $\hat{\mathbb{Y}}$ is

Block $\hat{\mathbb{Y}}_{i,j}$ is $(|\mathcal{M}_{i}|\times|\mathcal{M}_{j}|)$, so $\mathbf{Y}_{ij}$ must be appropriately zero-padded or reduced if $i$ and $j$ don’t have all the same phases. For ToyNet, $\hat{\mathbb{Y}}$ has the structure visualized in Fig. 2.

Voltages covariances are informative for phase identification. Under Assumptions (7,8), the covariance of the voltage of phase $\phi$ at node $i$ and phase $\psi$ at node $j$, in the three-phase model (6), is given by:

We are interested in the sum of covariances for a particular phase ordering between nodes $i$ and $j$. Consider the case where the phases at $i$ are a subset of those at $j$ ($\mathcal{M}_{i}\subseteq\mathcal{M}_{j}$). Let $\mathcal{O}$ denote the ordering/permutation of phases at $j$, where $\mathcal{O}(\phi)$ denotes the specific phase at $j$ matched to the phase $\phi$ at $i$. Then, the covariance sum for matching $\mathcal{O}$, denoted by $c_{ij}^{\mathcal{O}}$, is:

Let $\underline{\hat{\mathbb{Z}}}_{i}$ denote the rows of $\underline{\hat{\mathbb{Z}}}$ corresponding to all phases at node $i$, and $\underline{\hat{\mathbb{Z}}}_{j}^{\mathcal{O}}$ denote the rows corresponding to the phases at $j$ ordered according to $\mathcal{O}$. Then $c_{ij}^{\mathcal{O}}$ is:

The last equality follows from (5). The contribution of a node $k$ to $c_{ij}^{\mathcal{O}}$ is the dot product of the common path lengths between $i,k$ and $j,k$. The following result shows how $c_{ij}^{\mathcal{O}}$ enables phase matching.

(12) is a condition on every pair of edges, $st$ and $kl$, in the network. It states that for every row subset of line impedance matrix $\mathbf{Z}_{st}$ (corresponding to each nodal phase set $\mathcal{M}_{1},...,\mathcal{M}_{n}$), the matching rows of $\mathbf{Z}_{kl}$ produce the largest vectorized dot product. This is reasonable as real line impedance matrices are diagonally dominant. Condition (12) depends on the particular network considered. In a network where all nodes have all three phases ($\mathcal{M}=\mathcal{M}_{i}=\{a,b,c\}$) the condition on the vectorized dot product involves the full three phase line impedance matrices. if some nodes have a subset of phases, it will involve sub-matrices of impedance matrices. Note that, in general, $cov(\mathbf{v}_{i}^{\phi},\mathbf{v}_{j}^{\psi})>0$ even if $\phi\neq\psi$. If node $i$ has phases $a,b$ and node $j$ has phases $a,c$ ($\mathcal{M}_{i}\not\subseteq\mathcal{M}_{j}$), minimizing $c_{ij}$ will incorrectly match $b,c$. GPT avoids such scenarios by ordering nodes; discussed later. In summary, Theorem 2 allows us to use $c_{ij}^{\mathcal{O}}$ as a proximity metric for phase matching.

We use voltage difference variances for topology recovery. Define $d_{ij}$ to be the sum of the variance of the voltage differences between correctly matched phases of nodes $i,j$. Assuming $\mathcal{M}_{i}\subseteq\mathcal{M}_{j}$, $d_{ij}$ is:

Lemma 2 establishes trends in $d_{ij}$ along one phase.

Applying Lemma 2 to all matched phases between two nodes gives the following result:

If phases at each node are known, Thm. 3 enables correct topology recovery with a greedy algorithm based on distance $d_{ij}$. Note that Lemma 2 and Thm. 3 hold for all uncorrelated injections even with unequal variances. Thus, Assumption 8 can be relaxed for topology learning. We now have the tools for joint phase and topology recovery, detailed in the next section.

We propose GPT (Alg. 1): a greedy algorithm for joint phase and topology identification based on the nodal voltage properties of Sections III-IV. GPT computes $c_{ij}^{\mathcal{O}}$’s (9) exhaustively (for all matching options), selecting maxima for phase matching (Theorem 2). Based on phase matchings, it computes $d_{ij}$’s (13) exhaustively (for all node pairs), selecting minima for topology recovery (Theorem 3). GPT greedily builds a tree with node set $\mathcal{T}$, starting from node $i_{0}$ and iterating till all nodes have been added. In each iteration, a new node is added to the tree by choosing node $i\not\in\mathcal{T}$, which has the minimum value of $d_{ij}$ for all $j\in\mathcal{T}$, using the getNext algorithm. When node $i$ is added to the tree, it’s phases are determined based on their matching to the phases of node $j$.

GPT adds $3$ phase, then $2$ phase, then $1$ phase nodes to the tree. The initial node must be three phase, making the reference an intuitive choice. By adding nodes in this order, GPT implicitly enforces the crucial fact that number of phases never increases moving from the substation to the network ends (a single phase node is never the parent of a three phase node) and avoids issues that can arise when applying a naive greedy algorithm to a network with a variable number of phases at each node. For example, suppose we are recovering the topology of ToyNet. All nodes have been added to $\mathcal{T}$ except $6$, $7$, and $8$. To recover the correct topology, we should connect node $6$ to $5$ first. Then nodes $7$ and $8$ will get connected to $6$ naturally, as $d_{76}<d_{75}$ and $d_{86}<d_{85}$. However, consider $d_{65}$ and $d_{75}$,

We have no guarantee that $d_{65}<d_{75}$ due to the presence of additional phase variance in $d_{65}$ illustrating how an algorithm that doesn’t order nodes by decreasing number of phases may return an incorrect topology, unlike GPT.

We present simulation results of GPT. We measure average errors in phase and topology recovery, normalized by network size:

Further, we evaluate the algorithm’s sensitivity to the following parameters.

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  Measurement noise: We add white noise $\mathbf{n}$ to original measurement $\mathbf{v}_{i}$: $\tilde{\mathbf{v}}_{i}=\mathbf{v}_{i}+\mathbf{n}$, defining $\textnormal{noise level}(\tilde{\mathbf{v}}_{i})=\frac{var(\mathbf{n})}{var(\mathbf{v}_{i})}$. As GPT uses voltage covariances, it depends on relative precision and not absolute accuracy and is immune to the stable transducer errors that afflict distribution PMU data [30].

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  Number of measurement samples: Assuming $120$ Hz distribution PMU measurements [31], we record performance on $1$ second to $1$ minute of voltage data.

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  Load Correlations: We test GPT’s sensitivity to the assumption of uncorrelated injections, by varying the correlations of the loads while maintaining their variance. This is done by setting load covariance matrix $\Sigma=\sigma^{2}((1-\epsilon)\mathbb{I}+\epsilon 11^{T})$. As $\epsilon\rightarrow 1$, injections become more correlated.

Three IEEE distribution test networks are simulated in OpenDSS: the 13 and 34 bus networks have some one and two phases buses, while the 37 bus network has all three phase buses [32]. We modify the models by adding loads at every bus, and by disabling voltage regulators, which invalidate the assumption of voltages driven by injections. We fluctuate the load injections at each phase at each bus, and simulate the network with power flow to obtain non-linear voltages.

Fig. 4 plots topology recovery accuracy for three noise levels ranging from $0$ (no noise) to $10$, with $1$ second to $1$ minute of voltage magnitude or phasor measurements. PMUs are highly precise; and the noise level would realistically be $\sim 0.001$ [33, 34]. Nevertheless, GPT performs well under more noise as measurement samples increase. For all test networks and measurement durations, GPT achieves perfect topology recovery from voltage magnitudes for $0$ and $0.001$ noise. Insets in Fig. 4 show recovered topologies across trials. Note how errors are localized to a few nodes, and lower for voltage magnitudes. Fig 4 also compares performance on voltage magnitudes to Liao2019 [24], showing that GPT outperforms it across scenarios.

Fig. 5 presents GPT’s phase matching error on the same three networks and noise levels averaged across several sample durations (we found phase matching error to be invariant to sample duration). The error is compared to that of the approach in Olivier2018 [7]. The methods have comparable performance, except on the 13 bus network, where GPT outperforms Olivier2018 across SNRs.

Fig. 6 shows topology recovery sensitivity as injections stray from the uncorrelated assumption (7). Error increases rapidly as loads become more correlated. In reality, over short time durations, it is reasonable to assume that injections will be uncorrelated across nodes or can be de-trended [24]. We use at most one minute of data to recover phase and topology: short enough that the uncorrelated assumption should hold well.

Our polynomial time algorithms are suitable for real time application, taking on the order of seconds to recover phase and topology for the IEEE test cases shared here. On the largest 37 bus test case, the algorithm completes in $15$ seconds.

We presented GPT for joint phase and topology identification from voltage measurements in unbalanced three phase networks where each bus can have single, two, or three phases. GPT is polynomial time and relies on analytical trends in nodal voltage statistics. We proved its theoretical correctness under reasonable assumptions on load statistics and line impedances, demonstrating its efficacy on non-linear voltages from three test feeders simulated under realistic conditions. GPT is robust to measurement non-idealities, and outperforms the prior work in both phase and topology recovery.

Extensions of the work include handling of voltage regulators at some nodes and improving performance under load correlations. The linearized unbalanced three phase model presented here can motivate many future analytics, including extensions to settings with limited measurements, where our derived distance metrics could give a sense of the relative proximity of dispersed measurement points if not the precise topology. It is challenging to directly extend this work to non-radial settings as inter-nodal distances become difficult to define precisely in meshed networks. However, the model could inform graph theoretic monitoring approaches of meshed networks, similar to those in [35].