Topology Estimation in Bulk Power Grids: Guarantees on Exact Recovery
Deepjyoti Deka, Saurav Talukdar, Michael Chertkov, Murti Salapaka
TTopology Estimation in Bulk Power Grids:Guarantees on Exact Recovery
Deepjyoti Deka † , Saurav Talukdar ‡ , Michael Chertkov † , and Murti Salapaka ‡ ( † ) Los Alamos National Laboratory, New Mexico, USA, ( ‡ )University of Minnesota Twin Cities, Minneapolis, USA Abstract —The topology of a power grid affects its dynamicoperation and settlement in the electricity market. Real-timetopology identification can enable faster control action followingan emergency scenario like failure of a line. This article discussesa graphical model framework for topology estimation in bulkpower grids (both loopy transmission and radial distribution)using measurements of voltage collected from the grid nodes. Thegraphical model for the probability distribution of nodal voltagesin linear power flow models is shown to include additional edgesalong with the operational edges in the true grid. Our proposedestimation algorithms first learn the graphical model and subse-quently extract the operational edges using either thresholdingor a neighborhood counting scheme. For grid topologies con-taining no three-node cycles (two buses do not share a commonneighbor), we prove that an exact extraction of the operationaltopology is theoretically guaranteed. This includes a majorityof distribution grids that have radial topologies. For grids thatinclude cycles of length three, we provide sufficient conditionsthat ensure existence of algorithms for exact reconstruction. Inparticular, for grids with constant impedance per unit length anduniform injection covariances, this observation leads to conditionson geographical placement of the buses. The performance ofalgorithms is demonstrated in test case simulations.
Index Terms —Concentration matrix, Conditional indepen-dence, Counting, Distribution grids, Graphical lasso, Graphicalmodels, Power flows, Transmission grids.
I. I
NTRODUCTION
The power grid comprises of the set of transmission linesthat transfer power from generators to the end users. Struc-turally, the grid is described as a graph with nodes representingbuses and edges representing the transmission lines. It is worthnoting that high voltage transmission grids are topologicallyloopy (with cycles) while medium voltage distribution gridsare generally radial (no cycles) in structure [1]. The trueoperational topology in either case is determined by the currentbreaker/switch statuses on an underlying set of permissibleedges as shown in Fig. 1. Topology estimation/learning refersto the problem of determining the current operational topologyand is a necessary part for majority of control and optimizationproblems in the dynamic and static regimes of grid operation.Real-time topology estimation can enable timely detectionof line failures and identification of critical lines that affectlocational marginal prices. Such estimation is hampered bythe limited presence of real-time line-based measurementsespecially in medium and low voltage part of the grid. Inrecent years, there has been a surge in installation of nodal/busbased measurement devices like phasor measurement units(PMUs) [2], micro-PMUs [3], FNETs [4], smart sensors that record high-fidelity real-time measurements at nodes/buses(not lines) to enhance observability and then use the infor-mation to control, e.g. smart devices. We analyze the problemof real-time topology estimation using voltage measurementscollected from meters located at the grid nodes through theframework of graphical models.
A. Prior Work
Research in topology estimation in the power grid hasexplored different directions to utilize the available measure-ments under varying operating regimes. Primarily such re-search has focused on distribution grids that are operationallyradial in topology and use measurements from static powerflow models. [5] presents a topology identification algorithmfor radial grids with constant r/x (resistance to reactance)ratio for transmission lines using signs of elements in theinverse covariance matrix of voltage magnitudes. In a similarradial setting, [6] presents the use of conditional independencebased tests to identify the radial topology from voltage mea-surements for general distributions of nodal injections. Greedyschemes to identify radial topologies based on provable trendsin second moments of voltage magnitudes are presented in[7]. This has been extended to cases with missing nodes in[8], [9]. Signature/envelope based identification of topologychanges is proposed in [10], [11]. In [12], a machine learning(ML) topology estimate with approximate schemes is usedin a distribution grid with Gaussian loads. For loopy powergrids, approximate schemes for topology estimation have beendiscussed in [13] but do not have guarantees on exact recovery.For measurements from grid dynamics, [14], [15] discuss theuse of Wiener filters for topology estimation in radial andloopy grids.
B. Contribution
In this work, we present a graphical model [16] basedlearning scheme for topology estimation in general powergrids, both loopy and radial, using linear power flow models.Our approach factorizes the empirical distribution of nodalvoltages collected by the meters and then extracts the truetopology (operational edges) from the estimated graphicalmodel. We show that for general grids, the graphical model ofthe distribution of nodal voltage measurements is a super-setof the original topology. We present algorithms and conditionsunder which the true grid topology can be extracted fromthe estimated graphical model. For radial grids, we show a r X i v : . [ m a t h . O C ] J u l hat a local neighborhood counting scheme or thresholdingin graphical model concentration matrix are capable of re-covering the true topology, irrespective of the line impedancevalues. For grids with cycles, the thresholding scheme isguaranteed to estimate the true topology when minimum cyclelength is greater than three (no triangular cycles), while theneighborhood counting method is able to estimate for caseswith minimum cycle lengths greater than six. Finally forgrids with triangles, we provide sufficient conditions underwhich thresholding based tests are able to recover the truetopology. These conditions depend on the line impedancesand injection fluctuations at the grid nodes. In particular, forgrids with constant impedance per unit length and similarinjection fluctuations, we present sufficiency conditions forexact recovery that depend on geometry of the grid layout.The next section presents nomenclature and power flowrelations in bulk power grids. Section III develops the graph-ical model of power grid voltage measurements under twolinear power flow models and discusses key aspects of itsstructure. Section III includes a discussion of efficient schemesto estimate the graphical model. The first topology learningusing neighborhood counting is presented in Section V alongwith with conditions for exact recovery. Section VI describesthe second learning algorithm that includes thresholding ofvalues in the inverse covariance matrix of nodal voltages.Section VII includes simulations results of our work on IEEEtest cases. Conclusions and future work are included in SectionVIII.II. P OWER G RID : S
TRUCTURE AND P OWER F LOWS
Structure : We represent the topology of the power grid bythe graph G = ( V , E ) , where V is the set of N +1 buses/nodesof the graph and E is the set of undirected lines/edgesthat represent the operational lines. This operational grid isdetermined by closed switches/breakers in an underlying setof permissible lines E full (see Fig. 1). For distribution grids,the operational edge set E represents a tree (radial network)while for transmission grid the operational network is loopyin general. The operational lines are unknown to the observerwho may or may not have access to the permissible edge set E full . As E full may not be known, we consider all node pairsas permissible edges. We denote nodes by alphabets i , j and soon. The undirected edge connecting nodes i and j is denotedby ( ij ) . Let P ji ≡ { ( ik ) , ( k k ) , ... ( k n − j ) } denote a set of n distinct undirected edges in G that connect node i and node j .We call P ji as a path of length n from i to j . If i = j then path P ji is termed a ‘cycle’. By definition, a cycle has length threeor more. Note that for radial networks, there is a unique pathbetween each distinct pair of nodes, while for loopy networksthere can be multiple paths between a node pair. An exampleof a path is shown in Fig. 1. The length of shortest pathbetween two nodes is called the ‘separation’ between them.The set of nodes that share an edge with node i are calledits neighbors, while the set of nodes that are separated by twohops from node i are called its two-hop neighbors. Nodes withdegree one are termed ‘leaves’. The single neighbor of a leaf 𝑎 𝑏 𝑐 𝑑 Fig. 1. Example of a power grid with substations. Substations arerepresented by large red nodes. The operational grid is formed by solid lines(black). Dotted grey lines represent open switches. P ba ≡ { ( ab ) , ( bc ) , ( cd ) } represents a non-unique path from a to d . is called its ‘parent’. Next we discuss the power flow modelsin the grid. Power Flow (PF) Models : Let z ij = r ij + ˆ ix ij denotethe complex impedances of line ( ij ) in the grid ( ˆ i = − )with resistance r ij and reactance x ij . By Kirchhoff’s laws,the complex valued PF equation for the flow of power out ofa node i in grid G is given by, P i = p i + ˆ iq i = (cid:88) j :( ij ) ∈E V i ( V ∗ i − V ∗ j ) /z ∗ ij , (1) = (cid:88) j :( ij ) ∈E v i − v i v j exp(ˆ iθ i − ˆ iθ j ) z ∗ ij , (2)where, the real valued scalars, v i , θ i , p i and q i denote thevoltage magnitude, voltage phase, active and reactive powerinjections respectively at node i . V i (= v i exp(ˆ iθ i )) and P i denote the nodal complex voltage and injection respectively.During normal operation in bulk power systems, one canassume that the grid is lossless and the net power (generationminus demand) in the grid is zero. One bus then is consideredas reference bus, with its power injection equal to negative ofthe sum of injections at all other buses. Further the voltage andphase at all other buses are measured relative to the respectivevalues at the reference bus which has phase and p.u.voltage magnitude. Without a loss of generality, we ignorethe reference node/bus and restrict the power flow analysisto the N non-reference buses in the grid. Under the loss-lessassumption, we use the following well-known relaxations tothe PF equations. DC Power Flow (DC-PF) model [17]: Here all lines in thegrid are considered to be purely susceptive, voltage magnitudesare assumed to be constant at unity and phase differencesbetween neighboring lines are assumed to be small. This leadsto the following linear relation between active power flows andphase angles, ∀ i ∈ V : p i = (cid:88) j :( ij ) ∈E β ij ( θ i − θ j ) , which, in vector form is, p = H β θ. (3)ere, H β is given by, H β ( i, j ) = (cid:80) k :( ik ) ∈E β ik if i = j − β ij if ( ij ) ∈ E otherwise . (4)Thus, H β is the reduced weighted Laplacian matrix for thegrid G with edge weights given by susceptances β ( β ij = x ij x ij + r ij for edge ( ij ) ). The reduction is derived by removingthe row and column corresponding to the reference bus fromthe weighted Laplacian matrix. Linear Coupled Power Flow (LC-PF) model [7], [18]:The a.c. power flow equations in (1) are linearized assumingsmall deviations in both phase difference of neighboring nodes( | θ i − θ j | << for edge ( ij ) ), and voltage magnitudedeviations from the reference bus ( | v − | << ), which leadsto, (cid:20) vθ (cid:21) = (cid:20) H g H β H β − H g (cid:21) − (cid:20) pq (cid:21) , (5)where For radial grids, it can be verified that (cid:20) H g H β H β − H g (cid:21) − = (cid:34) H − /r H − /x H − /x − H − /r (cid:35) where H /r ( H /x ) is the reduced weighted Laplacian with theweight associated with each edge ( ij ) being /r ij ( /x ij ).This makes Eq. (5) equivalent to the LinDistFlow model[19], [20], [21] used in radial networks. Thus the LC-PFequations are a generalization of the LinDistFlow equationsto general loopy grids [7]. Note that both the DC and LCmodels represent an invertible map between nodal injectionsand voltages at the non-substation buses. This property is usedin determining the graphical model of nodal voltages discussednext. III. G RAPHICAL M ODEL OF V OLTAGES
Graphical models represent the structure within a multi-variate probability distribution. In this section, we develop thegraphical model for the distribution of nodal voltages in thelinear power flow models discussed (DC-PF and LC-PF). First,we make the following assumption regarding the injections atthe non-reference nodes in the grid.
Assumption : Injection fluctuations at non-reference nodesin the grid are independent zero-mean Gaussian random vari-ables with non-zero covariances.Note that over short to medium intervals, fluctuations ingrid nodes are due to changes in loads or noise that can beassumed to be independent and uncorrelated. The mean of thefluctuations can be empirically estimated and de-trended andhence ignored without a loss of generality. In fact the Gaussianassumption is not necessary for majority of our analysis butis taken as the most accepted model of disturbance/noise.For the DC-PF, the vector of active injections at all nodes p in grid G is a Gaussian random variable P DC ( p ) ≡N (0 , Σ DCp ) where Σ DCp is the diagonal covariance matrix ofactive injections at the non-reference buses. On the other hand, in the LC-PF model, the injection vector (cid:20) pq (cid:21) comprises ofactive and reactive injections and is modelled by the Gaussianrandom variable P LC ( p, q ) ≡ N (0 , Σ LC ( p,q ) ) . Here Σ LC ( p,q ) = (cid:20) Σ LCpp Σ LCpq Σ LCqp Σ LCqq (cid:21) (6)is the covariance matrix between active and reactive injectionsat the node. Each block in Σ LC ( p,q ) is a diagonal matrix. Notethat as active and reactive injections at each node may becorrelated, Σ LCpq = Σ
LCqp (cid:54) = 0 .We now derive the probability distribution for nodal phaseangles P DC ( θ ) in the DC-PF as, P DC ( θ ) = 1 | J DCP ( θ ) | P DC ( H β θ ) , = 1 | J DCP ( θ ) | (cid:89) i ∈V P DCi (cid:88) j :( ij ) ∈E β ij ( θ i − θ j ) (7)Here J DCP ( θ ) is the constant Jacobian involved in the lineartransformation from phase angles to injections. Note that theprobability distribution for injections is in product form asthe fluctuations are assumed to be independent and holdstrue for non-Gaussian distributions as well. For Gaussianinjections, we can use the linear relation for DC-PF to derivethe distribution for phase angles directly. This follows fromthe result that a linear function of Gaussian random variablesis also Gaussian [22]. The probability distribution of the nodalphase angles is given by: P DC ( θ ) ≡ N (0 , Σ DCθ ) where Σ DCθ = H − β Σ DCp H − β (8)where Eq. (8) follows from Eq. (3). Similarly for the LC-PFmodel, the distribution of voltage magnitudes and phase angles ( v, θ ) is given by a Gaussian random variable with covariancematrix given by: Σ LC ( v,θ ) = (cid:20) H g H β H β − H g (cid:21) − (cid:20) Σ LCpp Σ LCpq Σ LCqp Σ LCqq (cid:21) (cid:20) H g H β H β − H g (cid:21) − (9)In either case, we represent the probability distribution ofthe nodal voltages using a graphical model. We first formallydefine a ‘Graphical Model’. Graphical Model : For a n dimensional random vector X = [ X , X , ..X n ] T , the corresponding undirected graphi-cal model GM [16] has vertex set V GM where each noderepresents one variable. For every node i , its graph neighborsform the smallest set of nodes N ( i ) ⊂ V GM − { i } such thatfor any node j (cid:54)∈ N ( i ) , the distribution at i is conditionallyindependent of j given set N ( i ) , i.e., P ( X i | X N ( i ) , X j ) = P ( X i | X N ( i ) ) . For Gaussian graphical models, it is known thatthe edges in the graphical model for a random vector are givenby the non-zero terms in the inverse covariance matrix (alsocalled ‘concentration’ matrix) [16]. In other words, for node i , node j ∈ N ( i ) if Σ − ( i, j ) (cid:54) = 0 .We first focus on the graphical model for the distributionof phase angles in the DC-PF model. The following theoremgives its graphical model representation. ower grid jk i (a) θ j θ k θ i Graphical Modelin DC-PF (b) v j v k θ i Graphical Model v i θ k θ j in LC-PF (c) v k θ i in LC-PF v i θ k θ j ‘Hybrid’ Graph v j (d)Fig. 2. (a) Power grid graph G (b) Graphical model for phase angles in DC-PF (c) Graphical model for voltage magnitudes and phase angles in LC-PF(d) ‘Hybrid’ graph by combining nodes for same bus in graphical model forLC-PF. Theorem 1.
Consider grid graph G with nodal injection fluc-tuations modelled as independent Gaussian random variables.The graphical model for nodal phase angles in the DC-PFmodel consists of edges between nodes representing phaseangles at all neighbors and two-hop neighbors in G .Proof. To determine the edges in the graphical model, weanalyze the inverse covariance matrix of the phase angles.Using Eqs. (4, 8), it is clear that for i (cid:54) = j Σ DCθ − ( i, j ) = − β ij ( β i Σ DCp ( i,i ) + β j Σ DCp ( j,j ) )+ (cid:88) k (cid:54) = i,j β ik β jk Σ DCp ( k, k ) if edge ( ij ) ∈ E (cid:88) k (cid:54) = i,j β ik β jk Σ DCp ( k, k ) if i, j are two hops away , otherwiseHere β i = (cid:80) j (cid:54) = i β ij . It is thus clear that the inverse covariancematrix, for each node i has non-zero terms at its neighbors andtwo-hop neighbors, which define the edges in the graphicalmodel.An example of a grid and the associated graphical modelfor phase angles in DC-PF are given in Fig. 2(a) and 2(b)respectively. Next we look at the graphical model of voltagemagnitudes and phase angles in the LC-PF model. As eachgrid node is described by voltage magnitude and phase angle,the number of nodes in the graphical model is twice thenumber of grid buses. The following theorem gives the inverseof the covariance matrix Σ LC ( v,θ ) in LC-PF. Theorem 2.
For LC-PF, the inverse covariance matrix Σ LC ( v,θ ) − satisfies Σ LC ( v,θ ) − = (cid:20) J vv J vθ J θv J θθ (cid:21) where J vv = H g D − (Σ LCqq H g − Σ LCpq H β ) − H β D − (Σ LCpq H g − Σ LCpp H β ) J vθ = H β D − (Σ LCqq H β + Σ LCpq H g ) − H β D − (Σ LCpq H β + Σ LCpp H g ) J θv = H β D − (Σ LCqq H g − Σ LCpq H β ) + H g D − (Σ LCpq H g − Σ LCpp H β ) J θθ = H β D − (Σ LCqq H β + Σ LCpq H g ) + H g D − (Σ LCpq H β + Σ LCpp H g ) where matrix D is diagonal with D ( i, i ) = | Σ LCpp ( i, i )Σ LCqq ( i, i ) − Σ LCpq ( i, i ) | Proof.
We derive the inverse of Σ LC ( v,θ ) by inverting each matrixon the right side of Eq. (9). From Assumption 1, it is clear thateach block in Σ LC ( p,q ) (see Eq. (6)) is a diagonal matrix withblocks Σ LCpq = Σ
LCqp . Using Schur’s compliment expansion orby direct multiplication it can be verified that, (cid:20) Σ LCpp Σ LCpq Σ LCpq Σ LCqq (cid:21) − = (cid:20) D − Σ LCqq − D − Σ LCpq − D − Σ LCpq D − Σ LCpp (cid:21) , where, D is a diagonal matrix specified in the theoremstatement. Note that the i th diagonal entry in D is given by thedeterminant of (cid:20) Σ p ( i, i ) Σ pq ( i, i )Σ pq ( i, i ) Σ q ( i, i ) (cid:21) , the injection covariancematrix at node i . Multiplying the individual inverses result inthe expression given in the statement of the theorem.Using the expression for inverse of Σ LC ( v,θ ) , we present thestructure of the graphical model of nodal voltage magnitudesand phase angles in the LC-PF model. Theorem 3.
Consider grid graph G with nodal injection fluc-tuations modelled as independent Gaussian random variables.The graphical model for nodal voltage magnitudes and phaseangles in the LC-PF model consists of edges between voltagemagnitudes and phase angles at the same bus, at all neighbors,and at two-hop neighbors in G .Proof. Consider four terms in the expression for J vθ in Theo-rem 2 where D − , Σ LCqq , Σ LCpq , Σ LCpp are all diagonal matrices.Using analysis presented in Theorem 1, we can show that J vθ ( i, j ) (cid:54) = 0 if i = j or i and j correspond to neighbors ortwo-hop neighbors in G . Thus the node representing voltagemagnitude at i in the graphical model is linked to nodes forphases at i , all neighbors and two-hop neighbors of i . A similarargument for J vv , J θv and J θθ proves the theorem.Theorem 3 thus implies that the graphical model for voltagemagnitudes and phase angles in the LC-PF model does notinclude any edge between voltages corresponding to buses thatare three or more hops away in grid G . Indeed if we combinethe voltage and phase angle at each bus into a ‘hybrid’ nodeand connect such nodes if edges exist between their constituentvoltages and phase angles in the graphical model, we get a‘hybrid’ graph of N nodes that has the same structure asthe graphical model for phase angles in the DC-PF. This isdepicted in Figs. 2(c) and 2(d). emarks: A few points are in order. • For the general case with non-Gaussian but independentinjection fluctuations, the probability distribution for phaseangles in DC-PF is given in Eq. (7). It can be shown thatthe graphical model in this case is also given by the truetopological edges in the grid and edges between the two-hopneighbors (see details for a radial case in [6]). • The graphical model for only voltage magnitude deviations(no phase angles) in the LC-PF model has a similar structureas discussed in Theorem 3, if the resistance to reactance ratioon all lines is the same, i.e., r/x = 1 /α (some constant). Inthis case, the inverse covariance of voltage magnitude fluctu-ations is given by Σ LCv − = H /r Σ LCp + αq − H /r (see [5]).However for non-constant r/x ratio, the graphical modelfor voltage magnitudes alone may include several additionaledges. This is unlike Theorem 3 where the structure is trueeven for non-constant r/x values.In the next section, we discuss methods to estimate theinverse covariance matrix and thereby learn the graphicalmodel of voltages in DC-PF or LC-PF.IV. E STIMATION OF G RAPHICAL M ODEL OF V OLTAGES
We describe the estimation of the inverse covariance matrixof phase angles Σ DCθ − for the DC-PF with Gaussian injectionfluctuations with mean. The inverse covariance matrix forvoltage magnitudes and phase angles for the LC-PF can beestimated using the same techniques. Consider n i.i.d. samplesof phase angle vectors { θ k , ≤ k ≤ n } in the grid. Note thatif sufficient number of samples (much greater than the numberof nodes) are available one can determine Σ DCθ − directly by inverting the sample covariance matrix .On the other hand, if the number of samples are not largeenough (comparable to the number of grid nodes), we considerthe maximum likelihood estimator of a Gaussian graphicalmodel [16] with a constraint for the number of edges in thegrid. Σ DCθ − = arg min S − log det S + (cid:104) S, (cid:88) k θ k θ kT n (cid:105) + λ (cid:107) S (cid:107) (10)Here (cid:80) k θ k θ kT /n represents the empirical covariance matrixof phase angles. Further, | S | is the l -norm of the inversecovariance matrix and is a proxy for the l norm that measuresthe sparsity of the grid edges. The optimization problem (10)is convex and termed Graphical Lasso [23], [24]. Further,one can use a regression based optimization [25] to determineentries in the inverse covariance matrix for each node i in adistributed fashion. Given the estimate of graphical model forthe DC-PF and LC-PF models, we present two methods todistinguish between true topological edges and the ‘spurious’edges between two-hop neighbors in the grid. The first methoduses local counting of nodal neighborhoods in the graphicalmodel to determine the true topological edges and is discussedin the next section. V. T OPOLOGY L EARNING USING N EIGHBORHOOD C OUNTING
Consider power grid G with at least three non leaf nodes(one path of length least exists in G ). Let the graphicalmodel for nodal voltages in DC-PF or LC-PF be known. Theobjective of topology learning is to identify the true edgesbetween the non-reference buses in the grid. We first considerlearning when G is radial. A. Learning Radial Grids
Consider the DC-PF model with nodal phase angles inradial grid G . The following theorem enables us to distinguishbetween true and spurious edges in the graphical model. Theorem 4.
Let GM be the graphical model for nodalphase angles under the DC-PF model with Gaussian injectionfluctuations in radial grid G .(a) For edge ( ij ) in GM , there exists nodes k and l separatedby hops in GM with paths k − i − l and k − j − l iff ( ij ) is a true edge in G between non-leaf nodes i and j (b) Iff ( ki ) is a true edge in G between leaf node k and non-leafnode i , the non-leaf neighbors of k in GM and the non-leafneighbors of i in G are the same.Proof. Note that if k, l are two hops away in GM , they mustbe at least three hops away in G . For the ‘If’ part of statement(a), consider nodes k and l on either side of edge ( ij ) in radialtree G as shown in Fig. 3(a). They are separated by hops in GM and connected by two-hop paths k − i − l and k − j − l . Forthe converse, consider i, j as two-hop neighbors with singlecommon neighbor c in G . Note that existence of path k − i − l and k − j − l in GM requires k and l to be one and/or two-hopneighbors of both i and j in G . As G is radial, this is possibleonly if k and l include c or any of its immediate neighbors.However, this makes the separation between k, l in GM equalto one hop. Thus no k, l exist that are two hops away in GM .The ‘if’ part of statement (b) follows immediately for eachleaf node and its non-leaf parent in G (see Fig. 3(a)). Theconverse statement can be proven through contradiction byconsidering a non-leaf node that is a two-hop neighbor of leafnode k in G . (See [14] for detailed proof of a similar statementas statement (b)).Thus the first statement in the theorem enables the discoveryof non-leaf nodes and topological edges between them ingrid G through counting. The second statement subsequentlyidentifies the edges connected to the leaf nodes leading toexact reconstruction. Note:
We consider the LC-PF model where the graphicalmodel includes nodal voltage magnitudes and phase angles. Asmentioned in the previous section, combining nodes pertainingto voltage magnitude and phase angle at the same bus leadsto a ‘hybrid’ graph with identical structure as the graphicalmodel for phase angles in DC-PF. Thus, Theorem 4 can beused to estimate the true edges in radial grid G under LC − P F model as well. jlu t k (a) ijlu t ks (b)Fig. 3. Graphical model for phase angles in (a) radial grid (b) loopy grid withcycle length . Solid lines denote true topological edges, dashed lines denotespurious edges in the graphical model. Red edges and blue edges representtwo paths between nodes k and l of length in either graphical model. Next, we discuss learning the topology in loopy grids withcycle length greater than . B. Learning Loopy Grids with cycle length greater than The following theorem states that Statement (a) in Theorem4 holds for loopy grid G with minimum cycle length greaterthan and can be used to distinguish between true andspurious edges. Theorem 5.
Let GM be the graphical model for nodalphase angles under the DC-PF model with Gaussian injectionfluctuations in loopy grid G with cycle length greater than .For edge ( ij ) in GM , there exists nodes k and l separated by hops in GM with paths k − i − l and k − j − l iff ( ij ) is atrue edge in G between non-leaf nodes i and j .Proof. If edge ( ij ) exists in G , consider nodes k and l in cycle k − i − j − l − r − r − r − .. − k in G of length or more. Notethat nodes k, l satisfy the condition in the ‘if’ statement forgraphical model GM as shown in Fig. 3(b). For the converse,consider the case where i and j are not neighbors but two-hopneighbors in G . Further, existence of paths k − i − l and k − j − l in GM implies that k, l must be one or two-hop neighbors ofboth i and j in G . From the minimum cycle length constraint, i and j has exactly one common neighbor in G , say node c .Further, k, l cannot both be neighbors of i or j or c in G as thatwould make k, l one hop neighbors in GM . First, consider theconfiguration in G where k is neighbor of i and two hops awayfrom j . This leads to a cycle i − c − j − r − k − i of cycle and is not permissible. Similarly, node l cannot be neighbor of i and two-hop neighbor of j . Next, consider the configurationwhere k = c is the common neighbor of i and j , while l is twohops away from both i and j . This configuration produces acycle i − k − j − r − l − r − i of length for some r (cid:54) = r andviolates the cycle constraint. Finally consider the case where k, l are two hops away from both i and j . This leads to at leastone cycle of type i − c − j − r − ( k or l ) − r − i of length . Thus no configurations are permissible thereby proving thatthe converse of the statement is true for grids with minimumcycle length greater than .Following the argument after Theorem 4, it is clear thatTheorem 5 also holds for the ‘hybrid’ graph for the LC-PFmodel where graphical model nodes for votlage magnitudeand phase angle at the same bus are combined to form a ‘hybrid’ node. The topology reconstruction steps for gridswith minimum cycle length greater than are described inAlgorithm . The tolerance τ is used to determine edgesin the graphical model using entries in the estimated inversecovariance matrix of voltages. Note that Algorithm does notneed additional values on nodal injection covariances or lineimpedances and relies only on nodal voltage samples. In the Algorithm 1
Topology Learning using Neighborhood Count-ing
Input:
Inverse covariance matrix of nodal voltages Σ − V : Σ − V = Σ DCθ − for DC-PF or Σ − V = Σ LC ( v,θ ) − for LC-PF,tolerance τ > Output:
Grid G Construct graphical model GM for voltages in DC or LC-PF with edges ( ij ) for | Σ − V ( ij ) | ≥ τ if data from LC-PF then Construct ‘hybrid’ graph by combining nodes forvoltage magnitude and phase at same bus in graphicalmodel end if Insert edges between non-leaf nodes in G using Theorem5. Draw edges between leaf nodes and parent in G usingStatement (b) in Theorem 4.next section, we discuss another technique to determine thetopological edges from the graphical model.VI. T OPOLOGY I DENTIFICATION USING T HRESHOLDING
Here we analyze the values in the inverse covariance matrixof voltages in the DC or LC-PF models to determine trueedges in grid G from the edges in the graphical model. Firstwe discus the case of radial grids. A. Learning Radial Grids
Consider the DC-PF model for nodal phase angles in radialgrid. The next result distinguishes between one-hop and two-hop neighbors in the grid graph.
Theorem 6.
Let Σ DCθ − be the inverse covariance matrix ofnodal phase angles in radial grid G under the DC-PF modelwith Gaussian nodal injection fluctuations. If Σ DCθ − ( i, j ) < , then ( ij ) is a true edge in G Proof.
Observe the expression for Σ DCθ − ( i, j ) given in The-orem 1. For edge ( ij ) , no k exists with edges ( ik ) , ( jk ) as G is a radial grid. Thus Σ DCθ − ( i, j ) < if ( ij ) is a true edge.Similarly, it can be shown that Σ DCθ − ( i, j ) > if i and j are two-hop neighbors in G .A similar result for voltage magnitudes in the LC-PF modelwith constant r/x ratio is given in [5]. Our next result showsthat the inverse covariance matrix of nodal voltage magnitudesand phase angles in the LC-PF model can in fact be used toestimate the radial topology without the restriction on constant r/x ratio. heorem 7. Let Σ LC ( v,θ ) − = (cid:20) J vv J vθ J θv J θθ (cid:21) be the inversecovariance matrix of nodal voltages in radial grid G underthe LC-PF model with Gaussian nodal injection fluctuations.If J vv ( i, j ) + J θθ ( i, j ) < , then ( ij ) is a true edge in G Proof.
From expressions for Σ LC ( v,θ ) − , D in Theorem 2, J vv + J θθ = H g D − (Σ LCqq + Σ
LCpp ) H g + H β D − (Σ LCqq + Σ
LCpp ) H β (11)As D − (Σ LCqq + Σ
LCpp ) is a diagonal matrix with positive en-tries, a similar analysis as Theorem 6 proves the statement.Theorems 6 and 7 thus provide a simple thresholding based scheme to identify the true edges in radial grid under both DC-PF and LC-PF. Next we show that these results extend to loopygrids without cycles of length . B. Learning Loopy Grids without Triangles
A triangle is a sub-graph with three nodes i, j, k and edges ( ij ) , ( ik ) , ( jk ) . In other words, a triangle represents a cycle oflength . For neighboring buses i, j in loopy grid G withouttriangles, there is no k such that ( ki ) , ( kj ) are edges. Thusthe following corollary holds: Corollary 1.
For loopy grid G with cycle lengths greaterthan with Gaussian nodal injection fluctuations, the result inTheorem 6 and Theorem 7 hold under the DC-PF and LC-PFmodels respectively. The steps in topology learning for grids without trianglesare listed in Algorithm . Algorithm 2
Topology Learning using Thresholding
Input:
Inverse covariance matrix of nodal voltages, Σ DCθ − for DC-PF or Σ − V = (cid:20) J vv J vθ J θv J θθ (cid:21) for LC-PF, tolerance τ < Output:
Grid G for all buses i, j ∈ G do Insert ( ij ) in G if Σ DCθ − ( i, j ) ≤ τ for DC-PF or J vv ( i, j ) + J θθ ( i, j ) ≤ τ for LC-PF. end for The tolerance τ < is selected to account for theestimation error in the inverse covariance matrix due to finitenumber of samples. For exact inverse covariance matrices, thetolerance can be kept at . The estimation steps for the inversecovariance matrix in described in the previous section. Finallywe discuss conditions under which Algorithm is able to learnthe topology for grids with triangles. C. Topology Learning in Loopy Grids with Triangles
Consider a loopy grid G with multiple triangles (cyclesof length ). We are interested in understanding the use ofAlgorithm in identifying the true topology from the graphicalmodel GM . We consider the DC-PF with inverse covariancematrix of phase angles, Σ DCθ − . We first analyze the casewhere an edge in the graph is part of only one triangle (see ji kt u (a) ji k t uk k (b)Fig. 4. Edge ( ij ) in loopy grid with (a) one triangle formed with node k (b)multiple triangles formed with set K = { k , k , k } Fig. 4(a)). The next theorem states a sufficient condition underwhich Algorithm is guaranteed to recover that edge. Theorem 8.
Let edge ( ij ) be part of only one triangu-lar sub-graph on nodes i, j, k with edges ( ij ) , ( jk ) , ( ik ) . Σ DCθ − ( i, j ) < in the DC-PF if the following holds: β ij > − Σ DCp ( j, j ) β ik + Σ DCp ( i, i ) β jk DCp ( i, i ) + Σ DCp ( j, j )) + (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) (cid:32) Σ DCp ( j, j ) β ik + Σ DCp ( i, i ) β jk DCp ( i, i ) + Σ DCp ( j, j )) (cid:33) + Σ DCp ( i, i )Σ DCp ( j, j ) β ik β jk Σ DCp ( k, k )(Σ DCp ( i, i ) + Σ DCp ( j, j )) (12) Proof.
From Theorem 1, Σ DCθ − ( i, j ) < if β ik β jk Σ DCp ( k, k ) < β ij ( β ij + β ik Σ DCp ( i, i ) + β ij + β jk Σ DCp ( j, j ) ) Using conditions for positivity of quadratic functions, thestatement follows.Using similar techniques for edges that are part of multipletriangles (see Fig. 4(b)), we have the following sufficiencyresult.
Theorem 9.
Consider edge ( ij ) in graph G . Let K be the setof nodes that are neighbors of both i and j . Σ DCθ − ( i, j ) < for the DC-PF if the following holds: β ij > − Σ DCp ( j, j )( β i − β ij ) + Σ DCp ( i, i )( β j − β ij )2(Σ DCp ( i, i ) + Σ DCp ( j, j )) + (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) (cid:32) Σ DCp ( j, j )( β i − β ij ) + Σ DCp ( i, i )( β j − β ij )2(Σ DCp ( i, i ) + Σ DCp ( j, j )) (cid:33) + Σ DCp ( i, i )Σ DCp ( j, j )Σ DCp ( i, i ) + Σ DCp ( j, j ) (cid:88) k ∈K β ik β jk Σ DCp ( k, k ) (13) where β i = (cid:88) k :( ik ) ∈E β ik . Under Theorem 9, all edges in a general loopy graph withmultiple triangles can be learned using Algorithm fromthe inverse covariance matrix of phase angles in the DC-PFmodel. Note that this is only a sufficient condition and thegrid topology may be learned even if it is violated. Similarrelations can also be derived for the LC-PF model, but areomitted for space constraints.e now consider two interesting cases for the DC-PF modelthat highlight the rationale behind the previous two theorems.First, we consider the case where covariances of injectionfluctuations are equal at all nodes. This may be reasonable in asmall distribution grid with comparable nodal injections suchthat their fluctuations are similar. In this case the followingcondition ensures correct edge identification using Algorithm . Theorem 10.
Consider edge ( ij ) in graph G where injectioncovariances at all nodes are equal. Let K with cardinality |K| be the set of nodes that are neighbors of both i and j . Σ DCθ − ( i, j ) < if the following holds: β ij > max k ∈K ,r ∈{ i,j } β kr (cid:112) / |K| (14) Proof.
Simplifying the inequality condition in Theorem 9under equal injection covariances leads to β ij ( β i + β j ) > (cid:88) k ∈K β ik β jk ⇒ (2 + |K| ) β ij + β ij ( (cid:88) k (cid:54)∈ K ( ik ) ∈ E β ik + (cid:88) k (cid:54)∈ K ( jk ) ∈ E β jk ) > (cid:88) k ∈K ( β ik − β ij )( β jk − β ij ) This holds true when (2 + |K| ) β ij > |K| max k ∈K ,r ∈{ i,j } ( β kr − β ij ) which leads to the statement in the theorem.We also consider the case where the susceptance per unitlength is the same on all lines. This may be true for meshedurban grids where the lines are built using same material. Inthat setting, we have the following corollary to Theorem 10that ensures correct topology learning by Algorithm . Corollary 2.
Consider graph G with equal injection covari-ances at all nodes and constant susceptance per unit lengthon all lines. Let l ij be the length of edge ( ij ) and K be the setof nodes that are neighbors of both i and j . Σ DCθ − ( i, j ) < for the DC-PF if the following holds: l ij > max k ∈K ,r ∈{ i,j } l kr (cid:112) / |K| (15)This result follows immediately by using β ij = l ij β u where β u is the constant susceptance per unit length on all lines.Note that Corollary 2 is a sufficiency condition based on thegeometry of the grid . For example, for K of cardinality and , the right-side in the inequality becomes max r ∈{ i,j } l ik √ and max k ∈K ,r ∈{ i,j } l ik √ respectively. The lower limit on l ij thusincreases with an increase in the cardinality of K . Further, itsignifies that if line lengths are more equitable, Algorithm can guarantee recovery in the presence of greater number oftriangles in the grid.Finally for completion, we also consider simultaneous pa-rameter and topology learning in grid G when the observer hasaccess to the covariance of nodal injections in the grid nodesalong with the empirical covariance matrix of voltages, created (a) (b) (c)Fig. 5. (a) bus radial system (b) bus loopy grid with minimum cyclelength (c) bus loopy grid with minimum cycle length using measurement samples. The covariance of injections maybe learned from historical data or other off-line methods. Theorem 11.
Consider grid G with covariance matrix ofphase angle Σ DCθ and known diagonal injection covari-ance matrix Σ DCp in the DC-PF model. The edges inthe grid are given by the non-zero off-diagonal terms in Σ DCp / (cid:113) Σ DCp − / Σ DCθ − Σ DCp − / Σ DCp / .Proof. Note that Σ DCp − / Σ DCθ − Σ DCp − / is a posi-tive definite matrix and has a unique square root Σ DCp − / H β Σ DCp − / . Multiplying both side of the squareroot by Σ DCp / gives the reduced weighted Laplacian matrix,where the non-zero off-diagonal terms correspond to gridedges and associated susceptance values.A similar result can be derived for the LC-PF model aswell. It is omitted for brevity. In the next section, we detailnumerical simulations on learning the grid topology using ourgraphical model based framework.VII. N UMERICAL S IMULATIONS
We demonstrate results for Algorithm (neighborhoodcounting) and Algorithm (thresholding) in extracting theoperational edge set E of power grids using nodal voltage mea-surements in DC or LC-PF. We consider a bus radial case[26], [6] and extend it to two loopy grids with minimum cyclelength and (greater than ) as shown in Fig. 5. The nodalinjection fluctuations are modelled by uncorrelated zero-meanGaussian random variables. We use DC-PF and LC-PF modelsto generate i.i.d. samples of nodal voltages and use them toestimate the inverse covariance matrix for the graphical modelthrough Graphical Lasso. The estimated matrix is then input toAlgorithms and to determine the grid topology. Fig. 6 plotsthe average absolute estimation errors (sum of false positivesand false negatives in the estimated topology) in Algorithm using voltage samples generated by LC-PF model for the radialand loopy test systems in Figs. 5(a) and 5(c) respectively. Notethat the minimum cycle lengths in these two cases is greaterthan (sufficient for exact recovery by Algorithm ). Thus,increasing the number of samples leads to a reduction in edgedetection errors. Next, in Fig. 7, we present the performance ofAlgorithm in learning the radial and loopy (minimum cycle
500 1000 1500 2000 250005101520253035404550
Number of voltage magnitude, phase angle samples in LC−PF A b s o l u t e e rr o r s i n e s t i m a t ed t opo l og y
20 bus radial grid20 bus loopy grid (c)
Fig. 6. Accuracy of Algorithm with number of voltage measurementsamples in LC-PF for radial and loopy test systems in Figs. 5(a), 5(c).
100 150 200 250 30001234567
Number of voltage samples A b s o l u t e e rr o r s i n e s t i m a t ed t opo l og y
20 bus radial grid, DC−PF20 bus radial grid, LC−PF20 bus loopy grid (b), DC−PF20 bus loopy grid (b), LC−PF
Fig. 7. Accuracy of Algorithm with number of voltage measurementsamples in DC-PF and LC-PF for radial and loopy test systems in Figs. 5(a),5(b). length ) test systems in Figs. 5(a) and 5(b) using voltagesamples in both DC and LC-PF. The performance of Algorithm outshines that of Algorithm in terms of number of samplesneeded.Finally, we consider the bus loopy IEEE test case[27] that includes node cycles as shown in Fig. 8(a). Theperformance of Algorithm in estimating the true edges usingphase angle measurements from DC-PF is shown in Fig. 8(b).As learning of loopy grid with cycle length is not guaranteedto be exact, the average errors decay to a non-zero value forhigher sample values. Our ongoing work include efforts toselect optimal thresholds in Algorithms and to improvetheir performance. VIII. C ONCLUSION
In this document, we discuss the problem of topologyestimation of a loopy power grid from measurements of nodalvoltages through a graphical model framework. We demon-strate that the estimated graphical model of voltages for twolinear power flow models (DC-PF and LC-PF) includes true (a)
500 1000 1500 2000051015202530
Number of phase angle samples in DC−PF A b s o l u t e e rr o r s i n e s t i m a t ed t opo l og y
14 bus loopy grid (b)Fig. 8. (a) -bus loopy IEEE test case. The nodes part of triangles (3 nodecycles) are colored solid black. (b) Accuracy of Algorithm with number ofphase angle samples in DC-PF for the bus system. edges as well as spurious edges between two-hop neighborsin the power grid. We present two schemes to distinguishbetween true and fictitious edges. The first algorithm is basedon neighborhood counting in the graphical model, while thesecond algorithm uses a thresholding operator on the inversecovariance matrix of nodal voltages. For loopy grids, we showthat the neighborhood counting scheme is able to learn thetopology if minimum cycle lengths are greater than six. On theother hand, the thresholding method recovers the correct topol-ogy as long as cycle lengths are greater than three. Further,for general loopy grids without any cycle length constraint,we provide sufficient conditions based on injections and linesusceptances that enable exact recovery by the thresholdingalgorithm. We demonstrate the performance of our learningalgorithms through numerical simulations. The advantage ofour learning framework lies in the fact that it is entirely datadriven. It requires only samples of nodal voltage magnitudesand/or phase angles as input and does not require additionalinformation on line parameters and injection statistics.R EFERENCES[1] R. Hoffman, “Practical state estimation for electric distribution net-works,” in
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