Feeder Topology Identification using Consumer Load and Power Flow Measurements

Advanced control of distributed energy resources at the consumer level requires full situational awareness of the distribution system. This is typically performed by a topology identification and state estimation. In a distribution system, the topology is usually unknown since it depends on switch configurations on the network. In the case of residential feeders, switches may be passive components that do not provide feedback to DMS systems but are automatically or manually set. We present a method of determining the topology of the network given measurements typically available to the operator. We formulate the problem as a spanning tree identification problem over a graph given nodal and edge power flow information. In the deterministic case of known nodal power consumption and edge power flow we show that the placement of sensors on the network alone determines whether the set of spanning trees can be correctly identified. In the stochastic case where loads are given by forecasts derived from delayed smart meter data, we present a locally optimal sensor placement algorithm and maximum likelihood detector. I. NOMENCLATURE TABLE G = (V,E) Undirected Graph G; Vertices V ; Edges E. T Spanning tree on G. T(G) Set of all spanning trees that are constructed on G. τ Set of edges in constructing extended island graph. M Sensor placement (M⊂ E). M(G) Set of all sensor placements leading to identifiably in T(G). xn, x̂n True and forecasted load of node vn. σ n,Σ Forecast error variance and covariance matrix. Γ(T ,M) Observation matrix for tree T , sensor placement M. s Set of measured power flow. rij Acceptance region of Ti over Tj . Ri Acceptance region of hypothesis hi over all alternatives. g1,2(M) Shorthand notation for (1) maximum / (2) mean missed detection probability over all hypotheses. c Cycle in graph G. C(G) Cycle space of G or set of all possible cycles. R. Sevlian is with the Department of Electrical Engineering and the Stanford Sustainable Systems Lab, Department of Civil and Environmental Engineering, Stanford University, CA, 94305. Email: [email protected]. R. Rajagopal is with the Stanford Sustainable Systems Lab, Department of Civil and Environmental Engineering, Stanford University, CA, 94305. R. Rajagopal is supported by the Powell Foundation Fellowship. Email: [email protected]. FC A fundamental cycle basis of G. FC(T ) Fundamental Cycle Basis of G constructed by T . FCM Fundamental Cycle Basis constructed by sensor placement M. λk λk is kth cycle in FCM. μ(G) Circuit Rank of Graph. n(G) Number of connected components. ∆E Edge exchange operation to generate new tree: T → T ′. I(FC) Set of all edges at the intersection of two or more cycles in FC. K(c) Cycle-Sensor Map indicating all sensors on cycle c.