Model and Definitions

LEARNING POWER SYSTEM PARAMETERS FROM LINEAR MEASUREMENTS

6.2 Model and Definitions

show two applications of Theorem 6.4.1 for the uniform sampling of trees and the Erdős-Rényi(𝑛, 𝑝)model in Corollary 6.4.1 and 6.4.2, respectively.

3. (Heuristic) Algorithm: Motivated by the three-stage recovery scheme, a heuris- tic algorithm with polynomial (in𝑛) running-time is reported in Section 6.5, together with simulation results for power system test cases validating its performance in Section 6.6.

Some comments about the above results are as follows:

Outline of This Chapter

The remaining content is organized as follows. In Section 6.2, we specify our models.

In Section 6.3, we present the converse result as fundamental limits for recovery. The achievability is provided in 6.3. We present our main result as the worst-case sample complexity for Gaussian IID measurements in Section 6.4. A heuristic algorithm together with simulation results are reported in Sections 6.5 and 6.6.

Graphical Model

Denote by V = {1, . . . , 𝑛} a set of 𝑛 nodes and consider an undirected graph 𝐺 = (V,E)(with no self-loops) whose edge setE ⊆ V × V contains the desired topology information. The degree of each node 𝑗 is denoted by𝑑_𝑗. The connectivity between the nodes is unknown and our goal is to determine it by learning the associatedgraph matrixusing linear measurements.

Definition 6.2.1(Graph matrix). Provided with an underlying graph𝐺 =(V,E), a symmetricmatrixY(𝐺) ∈S^𝑛×𝑛is called agraph matrixif the following conditions hold:

𝑌_{𝑖, 𝑗}(𝐺) =

















≠0 if𝑖≠ 𝑗 and(𝑖, 𝑗) ∈ E 0 if𝑖≠ 𝑗 and(𝑖, 𝑗) ∉E arbitrary otherwise

.

Remark 9. Our theorems can be generalized to recover a broader class of symmetric matrices, as long as the matrix to be recovered satisfies (1) KnowingY(𝐺) ∈F^𝑛×𝑛 gives the full knowledge of the topology of𝐺; (2) The number of non-zero entries in a column ofY(𝐺)has the same order as the degree of the corresponding node, i.e., |supp(𝑌_𝑗) | =𝑂(𝑑_𝑗). for all 𝑗 ∈ V. To have a clear presentation, we consider specifically the case|supp(𝑌_𝑗) | =𝑑_𝑗.

In this work, we employ a probabilistic model and assume that the graph𝐺 is chosen randomly from acandidacy setC(𝑛)(with𝑛nodes), according to some distribution G𝑛. Both the candidacy setC(𝑛) and distributionG𝑛are not known to the estimator.

For simplicity, we often omit the subscripts ofC(𝑛)andG𝑛.

Example7. We exemplify some possible choices of the candidacy set and distribution:

(a) (Mesh Network) When 𝐺 represents a transmission (mesh) power network and no prior information is available, the corresponding candidacy setG(𝑛) consisting of all graphs with𝑛nodes and𝐺 is selected uniformly at random fromG(𝑛). Moreover, |G(𝑛) |=2(^𝑛₂) in this case.

(b) (Radial Network) When𝐺represents a distribution (radial) power network and no other prior information is available, then the corresponding candidacy set T(𝑛) is a set containing all spanning trees of the complete graph with𝑛buses (nodes) and𝐺 is selected uniformly at random fromT(𝑛); the cardinality is

|T(𝑛) | =𝑛^𝑛−2by Cayley’s formula.

(c) (Radial Network with Prior Information) When 𝐺 = (V,E) represents a distribution (radial) power network, and we further know that some of the buses cannot be connected (which may be inferred from locational/geographical information), then the corresponding candidacy setT𝐻(𝑛)is a set of spanning trees of a sub-graph𝐻 = (V,E𝐻) with𝑛buses. An edge𝑒 ∉E𝐻 if and only if we know𝑒∉E. The size ofT𝐻(𝑛)is given by Kirchhoff’s matrix tree theorem (c.f. [192]).

(d) (Erdős-Rényi (𝑛, 𝑝) model) In a more general setting, 𝐺 can be a random graph chosen from an ensemble of graphs according to a certain distribution.

When a graph𝐺is sampled according to the Erdős-Rényi (𝑛, 𝑝)model, each edge of𝐺 is connected IID with probability𝑝. We denote the corresponding graph distribution for this case byG_ER(𝑛, 𝑝).

The next section is devoted to describing available measurements.

Linear System of Measurements

Suppose the measurements are sampled discretely and indexed by the elements of the set{1, . . . , 𝑚}. As a general framework, the measurements are collected in two matricesAandBand defined as follows.

Definition 6.2.2(Generator and measurement matrices). Let𝑚be an integer with 1≤ 𝑚 ≤ 𝑛. Thegenerator matrixBis an𝑚×𝑛randommatrix and themeasurement matrixAis an𝑚×𝑛matrix with entries selected fromFthat satisfy the linear system (6.1):

A=BY(𝐺) +Z

whereY(𝐺) ∈S^𝑛×𝑛is a graph matrix to be recovered, with an underlying graph𝐺 andZ∈F^𝑚×𝑛denotes the randomadditive noise. We call the recoverynoiselessif Z=0. Our goal is to resolve the matrixY(𝐺)based on given matricesAandB.

In the remaining contexts, we sometime simplify the matrixY(𝐺) asYif there is no confusion.

Applications to Electrical Grids

Various applications fall into the framework in (6.1). Here we present two examples of the graph identification problem in power systems. The measurements are modeled as time series data obtained via nodal sensors at each node, e.g., PMUs, smart switches, or smart meters.

Example1: Nodal Current and Voltage Measurements

We assume data is obtained from a short time interval over which the unknown parameters in the network aretime-invariant. Y∈C^𝑛×𝑛denotes thenodal admittance matrixof the network and is defined

𝑌_{𝑖, 𝑗} :=











−𝑦_{𝑖, 𝑗} if𝑖 ≠ 𝑗 𝑦_𝑖+Í

𝑘≠𝑖𝑦_{𝑖, 𝑘} if𝑖 = 𝑗

(6.5) where 𝑦_{𝑖, 𝑗} ∈Cis the admittance of line(𝑖, 𝑗) ∈ E and 𝑦_𝑖 is the self-admittance of bus𝑖. Note that if two buses are not connected then𝑌_{𝑖, 𝑗} =0.

The corresponding generator and measurement matrices are formed by simultaneously measuring both current (or equivalently, power injection) and voltage at each node and at each time step. For each𝑡 =1, . . . , 𝑚, the nodal current injection is collected in an 𝑛-dimensional random vector𝐼_𝑡 =(𝐼_{𝑡 ,1}, . . . , 𝐼_{𝑡 ,𝑛}). Concatenating the𝐼_𝑡 into a matrix we getI:= [𝐼₁, 𝐼₂, . . . , 𝐼_𝑚]^⊤ ∈C^𝑚×𝑛. The generator matrixV:=[𝑉₁, 𝑉₂, . . . , 𝑉_𝑚]^⊤ ∈ C^𝑚×𝑛is constructed analogously. Each pair of measurement vectors(𝐼_𝑡, 𝑉_𝑡)fromI andVmust satisfy Kirchhoff’s and Ohm’s laws,

𝐼_𝑡 =Y𝑉_𝑡, 𝑡 =1, . . . , 𝑚 . (6.6) In matrix notation, (6.6) is equivalent toI=VY, which is a noiseless version of the linear system defined in (6.1).

Compared with only obtaining one of the current, power injection or voltage measurements (for example, as in [147, 178, 179]), collecting simultaneous current- voltage pairs doubles the amount of data to be acquired and stored. There are benefits however. First, exploiting the physical law relating voltage and current not only enables us to identify the topology of a power network but also recover the parameters of the admittance matrix. Furthermore, dual-type measurements significantly reduce the sample complexity for learning the graph, compared with the results for single-type measurements.

Example2: Nodal Power Injection and Phase Angles

Similar to the previous example, at each time𝑡 =1, . . . , 𝑚, denote by𝑃_{𝑡 , 𝑗} and𝜃_{𝑡 , 𝑗} the active nodal power injection and the phase of voltage at node 𝑗, respectively. The matricesP ∈R^𝑚×𝑛and𝜃𝜃𝜃 ∈R^𝑚×𝑛are constructed in a similar way by concatenating the vectors𝑃_𝑡 = (𝑃_{𝑡 ,1}, . . . , 𝑃_{𝑡 ,𝑛})and𝜃_𝑡 = (𝜃_{𝑡 ,1}, . . . , 𝜃_{𝑡 ,𝑛}). The matrix representation

of the DC power flow model can be expressed as a linear system P = 𝜃𝜃𝜃CSC^⊤, which belongs to the general class represented in (6.1). Here, the diagonal matrix S ∈ R^{|E |×|E |} is the susceptence matrix whose 𝑒-th diagonal entry represents the susceptence on the𝑒-th edge inEandC∈ {−1,0,1}^{𝑛×|E |}is the node-to-link incidence matrix of the graph. The vertex-edge incidence matrix3C∈ {−1,0,1}^{𝑛×|E |}is defined as

𝐶_{𝑗 ,𝑒} :=

















1, if bus 𝑗 is the source of𝑒

−1, if bus 𝑗 is the target of 𝑒 0, otherwise

.

Note thatCSC^⊤specifies both the network topology and the susceptences of power lines.

Probability of Error as the Recovery Metric

We define the error criteria considered in this chapter. We refer to finding the edge set E of𝐺via matricesAandBas thetopology identification problemand recovering the graph matrixYvia matricesAandBas theparameter reconstruction problem.

Definition 6.2.3. Let 𝑓 be a function or algorithm that returns an estimated graph matrixX = 𝑓(A,B) given inputsA andB. The probability of error for topology identification𝜀_T is defined to be the probability that the estimated edge set is not equal to the correct edge set:

𝜀_T :=P ∃𝑖≠ 𝑗

sign(𝑋_{𝑖, 𝑗}) ≠sign 𝑌_{𝑖, 𝑗}(𝐺) (6.7) where the probability is taken over the randomness in𝐺 ,BandZ. Theprobability of error for parameter reconstruction𝜀_P(𝜂) is defined to be the probability that the Frobenius norm of the difference between the estimate Xand the original graph matrixY(𝐺) is larger than𝜂 > 0:

𝜀_P(𝜂) := sup

Y∈Y(𝐺)

P(||X−Y(𝐺) ||_F > 𝜂) (6.8) where || · ||_F denotes the Frobenius norm, 𝜂 > 0 andY(𝐺) is the set of all graph matrices 𝑌(𝐺) that satisfy Definition 6.2.1 for the underlying graph 𝐺, and the probability is taken over the randomness in𝐺,B and Z. Note that for noiseless parameter reconstruction, i.e.,Z = 0, we always consider exact recovery and set 𝜂=0 and abbreviate the probability of error as𝜀_P.

3Although the underlying network is a directed graph, when considering the fundamental limit for topology identification, we still refer to the recovery of an undirected graph𝐺.