Joint State Estimation and Sparse Topology Error Detection for Nonlinear Power Systems

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1 Joint State Estimation and Sparse Topology Error Detection for Nonlinear Power Systems SangWoo Park, Reza Mohammadi-Ghazi, and Javad Lavaei Abstract This paper proposes a new technique for robust state estimation in the presence of a small number of topological errors for power systems modeled by AC power flow equations. The developed method leverages the availability of a large volume of SCADA measurements and minimizes the l 1 norm of nonconvex residuals augmented by a nonlinear, but convex, regularizer. We first study the properties of the solution obtained by the designed estimator and argue that, under mild conditions, this solution identifies a small subgraph of the network that can be used to find topological errors in the modeling of the state estimation problem. Then, we propose a method that can efficiently detect the topological errors by searching over this small subgraph. Furthermore, we develop a theoretical upper bound on the state estimation error to guarantee the accuracy of the underlying state estimation technique. The efficacy of the developed framework is demonstrated through numerical simulations on IEEE test systems. I. INTRODUCTION Supplying economical energy is the lifeblood of modern civilization, and is highly related to the reliability and efficiency of energy infrastructures, such as power systems that provide clean and convenient energy for industrial and individual uses. An important challenge in operating these infrastructures is to protect them against progressive failures of stressed components that may lead to blackouts [1], [2]. To prevent such events, the power system condition should be continuously monitored so that, if needed, required actions can be taken. This condition monitoring is performed through real-time state estimation that aims to recover the underlying system voltage phasors, given supervisory control and data acquisition (SCADA) measurements and a model that encompasses the system topology and specifications [3], [4]. Noting that the measurement noise and errors in modeling the system topology can significantly affect the accuracy of state estimation, dealing with bad data and identifying topological errors have received considerable attentions in the past few years. Topological errors are often initiated by the misconception of the system operator about the ON/OFF switching status of a few lines in the network due to faults or unreported network reconfigurations. A. Previous studies on topological error detection The existing topological error detection methods follow either statistical or geometric approaches. Bayesian hypothesis This work was supported by the ONR grant N , DARPA grant D16AP2, AFOSR grant FA , NSF grant and ARO grant W911NF The authors are with the Department of Industrial Engineering and Operation Research, University of California Berkeley, CA s: {spark111, mohammadi, lavaei}@berkeley.edu testing [5], collinearity tests [6], fuzzy pattern machine [7] are examples of statistical techniques for topology error detection. These methods usually need prior information about the state of the system and/or a decently-sized dataset from previous measurements. Geometric techniques, on the other hand, aim to design state estimation techniques that are robust against topological errors and measurement noise to some extent. One such technique is a heuristic method using normalized Lagrange multipliers of the least-squares state estimation problem [8]. Recent studies, such as the one proposed in [9], improve this technique; however, they may fail to detect certain scenarios that are called critical parameter and measurement pairs. Another important approach in this category is based on the synchronized point-on-wave measurements provided by phasor measurement units (PMU). Using these measurements, the state estimation optimization problem becomes linear and hence, a least absolute value (LAV) estimator can be used to eliminate bad measurements, detect topological errors, and provide a robust state estimator [1], [11], [12]. The caveat of these methods is their absolute reliance on PMU measurements, which makes this framework inapplicable to the highly nonconvex state estimation problem based on nonlineaer power flow measurements. Only few studies have investigated this fully nonlinear, non-convex problem with power measurements, e.g., [13] where a semidefinite programming (SDP) relaxation is proposed to convexify the nonlinear LAV state estimator without topological errors; however, no theoretical guarantees have been proposed to ensure the recovery of a high-quality solution. Moreover, the computational demand of solving the surrogate SDP problem may restrict the application of this method to small-sized problems in practice. These issues motivate further research on developing robust state estimation techniques with the capability of handling nonconvexities associated with various types of measurements. In this study, we propose a local search algorithm to find the global solution of the nonlinear LAV state estimator. The proposed method provides a robust approach both for estimating the system s states in presence of a modest number of topological errors and for pinpointing such errors. To this end, the main contributions of this work can be summarized as: (1) proposing an algorithm for detecting sparse topological errors and finding the state of the power system using a nonlinear LAV (NLAV) state estimator with local search algorithms, (2) formulating a regularized NLAV state estimator to handle problems with noisy measurements, (3) providing error bounds and investigating the properties of the proposed mathematical framework. As explained later in this paper, local

2 search algorithms would efficiently find global solutions of the underlying NLAV and regularized NLAV estimators since spurious local solutions of the problem are expected to disappear whenever the number of measurements is relatively larger than the degree of the freedom of the power flow problem. However, the results of this paper are all valid even if one uses an SDP relaxation of the proposed nonconvex estimators instead of a local search algorithm. The remainder of this paper is organized as follows. Preliminaries are presented in Section II, followed by main results in Section III. A detailed case study on the IEEE 57-bus system is presented in Section IV. Concluding remarks are drawn in Section V. The proofs are provided in the appendix. B. Notations Throughout this paper, boldface lower (resp. upper) case letters represent column vectors (resp. matrices) and calligraphic letters stand for sets and graphs. The symbols R and C denote the sets of real and complex numbers, respectively. R N and C N denote the spaces of N-dimensional real and complex vectors, respectively. S N and H N stand for the sets of N N complex symmetric and Hermitian matrices, respectively. The symbols ( ) T and ( ) denote the transpose and conjugate transpose of a vector/matrix. Re( ), Im( ), rank( ), Tr( ) and null( ) denote the real part, imaginary part, rank, trace, and null space of a given scalar or matrix. The notations x 1, x 2 and X F denote the l 1 -norm and l 2 -norm of vector x respectively, and the Frobenius norm of matrix X. The relation X means that the matrix X is Hermitian positive semidefinite. The (i, j) entry of X is denoted by X i,j. The notation X[S 1, S 2 ] denotes the submatrix of X whose rows and columns are chosen from the index sets S 1, and S 2, respectively. I N shows the N N identity matrix. The symbol diag(x) denotes a diagonal matrix whose diagonal entries are given by the vector x, whereas diag(x) forms a column vector by extracting the diagonal entries of the matrix X. The imaginary unit is denoted by j. λ i (X) denotes the i-th smallest eigenvalue of the matrix X. Given a graph G, the notation G(V, E) implies that V and E are the vertex set and the edge set of this graph, respectively. The operator vec( ) vectorizes its matrix argument. II. PRELIMINARIES Consider an electric power network represented by a graph G(V, E), where V := {1,..., N} and E := {1,..., L} denote the sets of buses and branches, respectively. Let v k C denote the nodal complex voltage at bus k V, whose magnitude and phase are given as v k and v k. The net injected complex power at bus k is denoted as s k = p k + q k j. Define s l,f = p l,f + q l,f j (resp. i l,f ) and s l,t = p l,t + q l,t j (resp. i l,t ) as the complex power injections (resp. currents) entering the line l E through the from and the end of the branch. Note that the currents i l,f and i l,t may not add up to zero due to the existence of transformers and shunt capacitors. Denote the admittance of each branch (s, r) of the network as y s,r. Let v and i be the vectors of nodal complex voltages and net current injections, respectively. The Ohm s law dictates that i = Y v, i f = Y f v, and i t = Y t v, (1) where Y = G + jb S N is the admittance matrix of the power network, whose real and imaginary parts are the conductance matrix G and susceptance matrix B, respectively. Furthermore, Y f C L N and Y t C L N represent the from and to branch admittance matrices. The injected complex power can thus be expressed as p + qj = diag(vv Y ). Let {e 1,..., e N } denote the canonical vectors in R N. Define E k := e k e T k, Y k,p := 1 2 (Y E k + E k Y ), Y k,q := j 2 (E ky Y E k ). Moreover, let {d 1,..., d L } be the canonical vectors in R L. Define Y l,pf, Y l,pt, Y l,qf and Y l,qt associated with the l-th branch from node i to node j as Y l,pf Y l,qf := 1 2 (Y f d l e T i + e i d T l Y f ), Y l,pt := 1 2 (Y t d l e T j + e j d T l Y t ) := j 2 (e jd T l Y f Yf d l e T i ), Y l,qt := j 2 (e jd T l Y t Yt d l e T i ) (3) The traditional measurable quantities can be expressed as (2) v k 2 = Tr(E k vv ) (4a) p k = Tr(Y k,p vv ), q k = Tr(Y k,q vv ) (4b) p l,f = Tr(Y l,pf vv ), p l,t = Tr(Y l,pt vv ) (4c) q l,f = Tr(Y l,qf vv ), q l,t = Tr(Y l,qt vv ) (4d) These equations show that the nodal and line measurements can be expressed as simple quadratic functions of the complex voltage vector v. In this paper, we only focus on traditional voltage and power measurements. However, if we have linear PMU measurements (e.g., certain entries of v and i), they can be regarded as quadratic equations with a zero leading term and the results of this paper are all valid in this scenario as well. To proceed with the paper, we make two definitions: Definition 1. Given a power system model Ω characterized by the tuple (Y, Y f, Y t ) and an index set of measurements M = {1,..., m} specifying m measurements of the form (4), the mapping from the measurement index set to the set of measurement matrices is defined as C Ω (M) {M j (Ω)} j M (5) where each M j (Ω) corresponds to one of the matrices E k, Y k,p, Y k,q, Y l,pf, Y l,pt, Y l,qf, Y l,qt defined in (2) and (3), depending on the type of measurement j. Definition 2. Given a system model Ω and a set of measurement matrices C Ω (M) = {M j (Ω)} j M associated with the model, define the function h Ω M ( v) : R2n 1 R m as the mapping from the real-valued state of the power system to the vector of noiseless measurement values: h Ω M( v) [v T M 1 (Ω)v,... v T M m (Ω)v] T (6)

3 where v [Re{v[Q] T } Im{v[O] T }] T R 2n 1 is the realvalued state vector of the power system s operating point with Q denoting the set of all buses and O indicating the set of all buses except for the slack bus. The Jacobian of the function h Ω M at a point v R2n 1 is defined as J Ω M( v) = [ M1 (Ω) v M m (Ω) v ] (7) where Mj (Ω) is the real-valued symmetrization of M j (Ω), j M. To further clarify this notation, note that a general n n Hermitian matrices can be mapped into a (2n 1) (2n 1) real-valued symmetric matrix as follows: [ ] Re{X[Q, Q]} Im{X[Q, O]} X = (8) Im{X[O, Q]} Re{X[O, O]} III. MAIN RESULTS Let { M j ( Ω)} j M be the set of measurement matrices corresponding to the hypothetical model Ω that the power system operators has, and { M j (Ω)} j M be the set of measurement matrices corresponding to the true model Ω. Assume that Ω and Ω are sparsely different in the sense that there is a small subset of lines in the system for which the operator misunderstands their ON/OFF statuses. In this work, we only focus on sparse errors for two reasons. If Ω and Ω are very different, then the state is not observable from a static set of measurements and dynamic time-stamped data is required. Second, topological errors often occur due to low probability events and it is unlikely that the operator s model would be significantly different from the true model. In this section, we first state the most widely used state estimation formulation and discuss its properties. Then, we develop a novel state estimation technique based on l 1 -norm optimization and analyze its features. A. Nonlinear least-squares state estimation Nonlinear least-squares (NLS) is the most common state estimation technique, which was first proposed by Schweppe [14], [15]. Given an n-bus power system that is characterized by a set of parameters Ω as defined in Definition 1, the NLS state estimator aims to recover the state of the system by solving the optimization problem min v v T Mj ( Ω) v b j 2 2 (9) where b i is the i th element of the measurement vector b R m that is b = h Ω M( z) + η (1) where z is the true underlying state of the system and η is the noise vector. We make the observability assumption that J M ( z) has full row rank. Recent studies have shown that local search algorithms, such as Gauss-Newton, are able to find a global solution of this nonconvex problem in the case where the number of measurements is relatively higher than the degree of the freedom of the system and η is zero [4], [16]. Similar to other estimators, this method requires that the matrices M j (Ω) s be known. However, the model that power system operators use referred to as hypothetical model throughout this paper may be different from the true system due to the presence of a small number of topological errors in modeling the power system associated with faults or recent changes in the switching status of some lines. This small percentage of modeling error leads to outliers in the measurements and could impact the solution of the state estimation problem over a large portion of the network. This is due to the incapability of the l 2 -norm in dealing with outliers. Thus, the algorithm should be revised in order to make a robust state estimation when sparse topological errors are involved. B. Proposed NLAV formulation To design a joint state estimator and sparse topological error detector, we propose the following optimization problem: min f( v) = vt M v v R 2n 1 + ρ v T Mj ( Ω) v b j (11) where M is a regularization matrix and ρ is a regularization coefficient. As will be discussed later, these two parameters assist with the convexification of the problem for finding a global solution using local search algorithms. Let v denote an optimal solution of (11). This expanded real-valued vector can be mapped back to a complex voltage vector v by setting the phase at the slack bus to be zero. Consider the residual r = {r 1,..., r m } R m, where r j = v T M j ( Ω) v b j, j M (12) We assume that the number of measurements is high enough so that the corresponding overdetermined power flow problem has the unique solution z. Having access to redundant measurements is the key for a real-world state estimation problem, and it also enables finding z in the noise-free case using a local search algorithm. This property has been proven for the NLS in matrix completion [17] and power system state estimation problems [4], [16], and a similar argument holds for the NLAV. To compare the true state z of the system with the solution v of (11), a few definitions will be introduced below. Definition 3. Define N M as the indices of the measurements corresponding to the lines that the operator misrepresented their presence in the network. Due to the assumption made earlier about the sparsity of topological errors, the cardinality of N is O(1). Definition 4. Given a regularization matrix M, a system model Ω, and a set of measurement matrices {M j } Ω j M, define Hµ Ω (M ) M + m i=1 µ jm j (Ω). A vector µ R m is called a dual certificate for the voltage vector v C N of the system model Ω if it satisfies the following three conditions: H Ω µ (M ), H Ω µ (M )v =, rank(h Ω µ (M )) = N 1 (13)

4 (a) (b) (c) (d) Fig. 1: (a) A hypothetical power system, (b) the state estimation error graph G, (c) the extended error graph Ḡ, and (d) the residual graph R. Unsolvable nodes are marked as cross, the blue line is the only erroneous line, and dotted lines correspond to the edges added to G to obtain Ḡ. Definition 5. A node i V is called unsolvable if the i th entry of z v is nonzero. Definition 6. Define the following four subgraphs of G: 1) The state estimation error graph G(S, E S ) is an induced subgraph of G such that S is the set of unsolvable nodes of G and E S is the set of all branches whose from and to nodes are both in S. 2) The extended error graph Ḡ(S, D S) is a subgraph of G that includes the state estimation error graph and the edges of G that are directly connected to the nodes in S. 3) The residual graph R(U, F) is a subgraph of G where F is the set of all lines whose associated line measurement residual errors in the vector r are nonzero, and U is the set of nodes that either have nonzero nodal measurement residual errors or include an endpoint of any line in F. 4) The extended residual graph R(Ũ, F) is a graph obtained from the residual graph in the same way the extended error graph was constructed from the state estimation error graph. Definition 6 is illustrated for a small system in Figure 1. Each unsolvable node is marked by a cross. The graphs introduced above will be used next to design an estimator. C. Estimation error In this section, we provide a theoretical upper bound on the state estimation error obtained by the NLAV problem (11) and uncover certain properties of the vector of residual errors. Theorem 1. Suppose that the power system operator has a hypothetical network model Ω and a set of measurement indices M. Assume that there exists a dual certificate µ for the vector z for the model Ω. Consider an arbitrary coefficient ρ satisfying the inequality ρ max j M µ j. Then, there exists a scalar β > such that v v T β z z T F 4g( z, η, ρ) v 2 λ 2 (H Ω µ ( M (14) )) where g( z, η, ρ) is equal to ρ z T ( M j (Ω) M j ( Ω)) z + η j (15) j N Proof. The proof is provided in Appendix A. Another inequality can be derived from the proof of the above theorem to show that β is close to 1. Moreover, the obtained bound is similar to the one in [18] although the contexts of these two results are completely different. Consider the noiseless scenario η = for now. Recall that z and v are the true and recovered states of the system, respectively. The error v v T β z z T F is based on possible mismatches in N 2 elements, and therefore v v T β z z T F should be O(N 2 ) in general. On the other hand, v 2 is O(N) because voltage magnitudes are close to 1 per unit for real-world systems. Moreover, if the sparsity graph of H Ω µ has a high algebraic connectivity (as expected), then its second smallest eigenvalue grows with N. Roughly speaking, this implies that the vectors z and v do not match in at most O(1) entries, after thresholding out those scalar mismatches that are very small (e.g., on the order of 1 N ). On the other hand, since the proposed estimator is based on the l 1 -norm (rather than l 2 -norm), diminishing residues do not exist if N is sufficiently large. Although this statement is not formalized here due to space restrictions, the vector z v is considered to be sparse in the rest of this paper. This makes the state estimation error graph G have O(1) vertices. Under the practical assumption that each bus in the network has a bounded degree, the extended error graph Ḡ has O(1) vertices as well. An alternative proof will be provided in the next subsection that connects the O(1) mismatch to the uniqueness of the solution of the overdetermined power flow problem. Definition 7. A line is called erroneous if it is modeled incorrectly by the system operator. Theorem 2. Suppose that the noise vector η is zero. Under the setting of Theorem 1, the following statements hold: 1) Assume that N = 1. Then, the extended error graph Ḡ is connected and includes that line. Moreover, the residual graph R is a subset of Ḡ. 2) In a general case where N 1, each connected component of the extended error graph Ḡ contains at least one unsolvable node. Proof. The proof is provided in Appendix A.

5 Theorem 2 enables us to develop a technique for detecting topological errors. Note that in the case N > 1 if the erroneous lines are sufficiently far from each other, Part 1 of this theorem applies to each connected component of the extended error graph. However, since the operator does not know the true solution, forming the extended error graph may not be possible. We have observed in many simulations that the extended residual graph is a surrogate for the extended error graph. If there is a pathological case in which the extended residual graph does not include an erroneous line, it means that the l 1 estimator finds spurious voltages at the endpoints of the erroneous line in such a way that the measurement equation for the correct model is satisfied. This leads to some type of undetectability. D. Algorithm Based on the above results, we propose Algorithm 1 for sparse topological error detection. First, let us define D L as the set of lines that are detected as erroneous. This algorithm evaluates the effect of each line in R on the accuracy of the solution by removing that line from the hypothetical model, re-solving the NLAV problem with the updated model, and add that line to D L if the norm of residual errors is reduced. Algorithm 1 Subgraph search algorithm Initialize: Ω t = Ω, C Ω t(m), C Ω (M), D L =, ɛ >, δ >, µ > 1- Solve the NLAV problem (11) with Ω t, C Ω t(m), C Ω (M). 2- Find the extended residual graph R = (Ũ, F). while r 2 > δ do for l F do Update Ω t by changing the ON/OFF status of line l. Re-solve (11) with Ω t, C Ω t(m), C Ω (M) to obtain the and r update if r update 2 < r 2 then Add l to D L and set Ω = Ω t. end if end for end while outputs v update So far, we have assumed that the estimator has a quadratic penalty term defined by the matrix M satisfying three conditions. Algorithm 1 may still be used if M =. In this case however, the estimation error bound provided in Theorem 1 can no longer be used. To obtain a new bound, define t as where t =arg min K S n AT vec(k) A = s.t. rank(k) = 2, K F = 1 (16) [ vec( M 1 ( Ω)) vec( M m ( Ω)) ] (17) It is straightforward to verify that t > if and only if there does not exist any set of noise-free measurements for the model Ω that results in two different solutions with no errors (residues). So, the number t quantifies how good the types of the measurements are as far as the uniqueness of the solution of the over-determined power flow problem is concerned. Following the proof of Theorem 1, it follows from equation (21) that A T vec( v v T z z T ) 1 = A T vec(k) 1 2 z T ( M j ( Ω) M j (Ω)) z + 2 η j (18) j N Now, one can obtain a lower bound on (18) as t K 1 2g( z, η, 1) (19) which is equivalent to v v T z z T 1 2 t g( z, η, 1). IV. NUMERICAL EXAMPLES In this section, we offer a case study on the IEEE 57-bus system to evaluate the efficacy of the proposed algorithm in estimating the state and detecting the topological errors. To be able to visualize the results, we only consider the scenario N = 2. We have tested the results on the IEEE 118-bus system with a modest number of erroneous lines, but they are reported very briefly due to space restrictions and the similarity of the outcome to the case study for the IEEE 57-bus system. It is worth mentioning that the fmincon function in MATLAB is used in the simulations of this section as a local search algorithm. A. Simulation setup We randomly selected 2 lines, out of 8 lines, of the 57- bus system and switched them off by changing their resistance and reactance to infinity. To satisfy the observability condition for the system, the two lines were chosen in such a way that they did not share a common bus. We consider this system as the true system, with the associated measurement matrices { M j (Ω)} j M. For the hypothetical model that is accessible to the power system operator, we used the measurement matrices corresponding to the original benchmark system with no change to its line configurations. The two randomly chosen lines in this study are branches #8 and #67. To generate a legitimate state, we assume that the voltage magnitudes are close to unity and the angles are small. The measurement set is constructed by: (1) using all nodal measurements but only 7% of line measurements at random; (2) taking no measurements from the erroneous lines. The reason for considering 7% line measurements is that this conforms with the current practice for power systems and it is high enough to enable finding the state (global optimum of the estimation problem in the noise-free case) using a local search algorithm. This property has been proven for the NLS in matrix completion [17] and power system state estimation problems [4], [16], and a similar argument holds for the NLAV. For numerical reasons, we normalize each measurement matrix M j (Ω) with respect to the largest absolute value of its entries in all of our analyses before using it in the proposed NLAV estimator.

6 B. Performance of nonlinear least square state estimator Before delving into the performance analysis of the proposed NLAV method, it is useful to study the traditional NLS method. Figures 2(a) and 2(d) show the state estimation and residual errors of NLS when topological errors exist. It follows from these plots that the sate estimation errors are considerably high, which means the NLS lacks robustness in presence of topological errors. Moreover, there is no sparsity pattern in these plots, and the high peaks are not associated with the end points of the erroneous lines. This implies that one needs to search over all possible combinations of lines to find the erroneous lines, which is numerically intractable for large systems. C. Performance of the proposed NLAV estimator In this section, we present the result of the proposed algorithm for estimating the state of the 57-bus system with the topological errors that were described in Section IV-A. The state estimation and the residual errors after the first run of the algorithm are shown in Figures 2(b) and 2(e). The two peaks in Figure 2(b) are associated with the state estimation errors at the ending nodes of the erroneous lines, and the error is zero at other buses. Comparing this result with that of NLS, one can observe the superiority of the NLAV estimator in finding the true state in the presence of topological errors. Another important observation is the sparse pattern of the state estimation error vector, as shown in Figure 2(e). Furthermore, the highest peaks of the residual vector in this plot correspond to the nodes/lines that are directly connected to the erroneous lines. This implies that the erroneous lines can be found by searching over only the lines that are associated with the highest peaks of the residual vector. By doing so, as stated in Algorithm 1, both erroneous lines are correctly detected without any false positive detection. Finally, to verify the efficacy of the proposed method for larger systems, we also performed simulations on the 118-bus system by considering 2, 3 and 5 erroneous lines. Using Algorithm 1, we were able to correctly detect all topological errors in the network in the absence of measurement noise. D. Noisy measurements Dealing with noisy measurements is an important challenge when it comes to implementing the state estimator in the realworld. In this section, we study the effect of noise on the outcome of the algorithm and demonstrate the efficacy of the proposed regularization technique in ameliorating the effect of noise. We considered 1% measurement noise and repeated the previous analysis for the scenario with two missing lines without any regularization by setting M =. Figures 2(c) and 2(f) show the state estimation errors and residual errors. It can be observed that some spurious peaks appear in the state estimation error. This increases the size of the associated graph R and, hence, requires a larger set of lines to search through in order to find the erronous lines. Doing the same analysis by considering the effect of a nonzero regularization M, we obtain the state estimation errors depicted in Figure 3. Comparing this plot with Figure 2(c) clearly shows that the spurious peaks are suppressed and the correct peaks are magnified as the result of regularization. Note that M was chosen as the product of the admittance of the system and its transpose. V. CONCLUSION This paper proposes a new technique to solve the state estimation problem for power systems in the presence of sparse topological errors and to detect such modeling errors. The developed method aims to minimize the l 1 -norm of nonconvex residuals in addition to a convex quadratic penalty term. We proved that, under mild conditions, the proposed method can efficiently detect the topological errors by searching over the lines of a small graph inferred by the solution of the estimator. Two upper bounds are derived on the estimation errors, and the results are demonstrated on benchmark systems. A. Proof of Theorem 1: APPENDIX Consider the NLAV problem (11). One can create lower and upper bounds on the optimal objective value as follows: v T M v + ρ v T M j ( Ω) v z T Mj (Ω) z ρ η j (a) v T (b) z T (c) z T M v + ρ M z + ρ v T M j ( Ω) v z T Mj (Ω) z η j z T Mj ( Ω) z z T Mj (Ω) z η j M z + ρ j N z T Mj ( Ω) z z T Mj (Ω) z + ρ η j where (a) is due to the triangle inequality and (b) is due to the optimality of v. The equality (c) follows from M j ( Ω) = M j (Ω) whenever j / N. Combining the above lower and upper bounds leads to v T M v z T M z + ρ ρ j N v T M j ( Ω) v z T Mj (Ω) z z T Mj ( Ω) z z T Mj (Ω) z + 2ρ η j (2) By adding and subtracting z T Mj ( Ω) z in the absolute value of the left-hand side, one can write: v T M v z T M z + ρ v T M j ( Ω) v z T Mj ( Ω) z j N 2ρ z T ( M j ( Ω) M j (Ω)) z + η j = 2g( z, η, ρ) (21)

7 State estimation error State estimation error Node #8, the "from" bus of line #8 Node #52, the "to" bus of line #8 State estimation error Node #8, the "from" bus of line #8 Spurious peaks due to noise Node #52, the "To" bus of line # Node number (a) Node number (b) Node number (c) For line #8.6.5 For line #8 Residual error Residual error.4.2 For line #67 Residual error For line # Measurement tag (d) Measurement tag (e) Measurement tag (f) Fig. 2: State estimation error for: (a) NLS with noiseless measurements, (b) NLAV with noiseless measurement, (c) NLAV with noisy measurement and without regularization. Residual error for: (d) NLS with noiseless measurements, (e) NLAV with noiseless measurement, (f) NLAV with noisy measurement and without regularization. In (d), (e) and (f), the x-axis shows the measurement tag, which does not have the same numbering as lines or nodes due to a random generation of line measurements. State estimation error Node #8, the "from" bus of line #8 Node #52, the "to" bus of line # Node number Fig. 3: State estimation errors based on the proposed method with regularization in the noisy case. Now, consider the optimization problem v T M v z T M z + ρ v T Mj ( Ω) v z T Mj ( Ω) z min v,t where v R 2n 1 and t R m. An equivalent formulation is v T M v z T M z + ρ min v,t s.t. v T Mj ( Ω) v z T Mj ( Ω) z t j, j M v T Mj ( Ω) v + z T Mj ( Ω) z t j, j M t j (22) Let p + j s and p j s be the nonnegative Lagrange multipliers for the first and second sets of constraints. The Lagrangian can be written as L( v, t, p +, p ) = v T M v z T M z + (ρ p + j p j )t j + {(p + j p j )( vt Mj ( Ω) v z T Mj ( Ω) z)} (23) By define d( v, p +, p ) = min t L( v, t, p), since p + j + p j = ρ for every j M, it yields that ( ) d( v, p +, p ) = v T M + (p + j p j ) M j ( Ω) v z T ( M + ) (p + j p j ) M j ( Ω) z (24) Note that d( v, p) gives a lower bound to the optimization problem (22). We select p + and p in such a way that µ = p + p, where µ is a dual certificate that exists by assumption. Then, equation (24) becomes equal to Tr{H Ω µ ( M ) v v T } = Tr{H Ω µ ( M )X} (25) where X = v v T. Now, consider an eigen-decomposition of H Ω µ ( M ) = UΛU T, where Λ = diag(λ N,..., λ 1 ) such that

8 λ N λ 1. Define [ ] X x X := x T = U T XU (26) α where X is the (N 1) th -order leading principle submatrix of X. It holds that Tr ( H Ω µ ( M )X ) = Tr ( Λ X ) λ 2 (H Ω µ ( M ))Tr ( X) Combining this with (21) leads to (27) Tr ( X) 2 g( z, η, ρ)/λ2 (H Ω µ ( M )) (28) By defining z = z/ z 2 and ṽ = v / v 2, the matrix X can be decomposed as X = U XU ] T = [Ũ [ ] [Ũ ] X x T z x T α z T = Ũ XŨ T + Ũ x zt + z x T Ũ T + α z z T (29) The matrix X can also be expressed as [ṽ V ] diag( v 2 2,,..., ) [ ṽ V ] T (3) Hence, plugging (3) into (26) gives α = v 2 2( z T ṽ) 2 Since X, through Schur complement, we have X α 1 x x T. Also, using α = Tr(X) Tr( X) and X F Tr( X), we obtain X α z z T 2 F = Ũ XŨ T + Ũ x zt + z x T Ũ T 2 F = Ũ XŨ T 2 F + 2 z x T Ũ T 2 F = X 2 F + 2 x 2 2 X 2 F 2Tr 2 ( X) + 2Tr(X)Tr( X) 2Tr(X)Tr( X) 4g( z, η, ρ) λ 2 (H Ω µ ( M Tr(X) (31) )) Plugging back in X = v v T gives X α z z T 2 F = v v T α z 2 z z T 2 F 2 4g( z, η, ρ) λ 2 (H Ω µ ( M )) Tr( v v T ) (32) By defining β = α/ z 2 2 = ( z T ṽ) 2 v 2 2/ z 2 2, this reduces to v v T β z z T 2 4g( z, η, ρ) F λ 2 (H Ω µ ( M )) Tr( v v T ) (33) B. Proof of Theorem 2: A sketch of the proof will be provided here. By contradiction, assume that there is an unsolvable node i that does not belong to G. This implies that the j-th entry of v and z are the same for all neighbors j of node i. As a result, if the i-th entry of v is replaced by the correct voltage z i in the vector v, the objective value in (11) is reduced, which contradicts the optimality of v. Also, if there is a single erroneous line which is not in the extended error graph, the condition f( v ) f( z) is violated. REFERENCES [1] K. Clements and P. Davis, Detection and identification of topology errors in electric power systems, IEEE Transactions on Power Systems, vol. 3, [2] Y. Lin and A. Abur, Probabilistic load-dependent cascading failure with limited component interactions, in Proceedings of the International Symposium on Circuits and Systems, 24. [3] A. MONTICELLI, Electric power system state estimation, Proceedings of the IEEE, vol. 88, 2. [4] R. Zhang, J. Lavaei, and R. Baldick, Spurious critical points in power system state estimation, in Hawaii International Conference on System Sciences, 218. [5] E. M. Lourenco, A. J. A. S. Costa, and K. A. Clements, Bayesian-based hypothesis testing for topology error identification in generalized state estimation, IEEE Transactions on power systems, vol. 19, 24. [6] E. M. Lourenco, A. J. A. S. Costa, K. A. Clements, and R. A. Cernev, A topology error identification method directly based on collinearity tests, IEEE Transactions on power systems, vol. 21, 26. [7] D. Singh, J. P. Pandey, and D. S. Chauhan, Topology identification, bad data processing, and state estimation using fuzzy pattern matching, IEEE Transactions on power systems, vol. 2, 25. [8] K. A. Clements and A. S. Costa, Topology error identification using normalized lagrange multipliers, IEEE Transactions on power systems, vol. 13, [9] Y. Lin and A. Abur, A computationally efficient method for identifying network parameter errors, in Innovative Smart Grid Technologies Conference (ISGT). IEEE, 216. [1] M. Korkali and A. Abur, Robust fault location using least-absolutevalue estimator, IEEE Transactions on power systems, vol. 28, pp , 213. [11] B. Donmez and A. Abur, Sparse estimation based external system line outage detection, in Power Systems Computation Conference (PSCC), 216. [12] Y. Lin and A. Abur, Robust state estimation against measurement and network parameter errors, IEEE Transactions on power systems, 218. [13] Y. Weng, M. D. Ilic, Q. Li, and R. Negi, Convexification of bad data and topology error detection and identification problems in ac electric power systems, IET Generation, Transmission & Distribution, vol. 9, pp , 215. [14] F. C. Schweppe and J. Wildes, Power system static-state estimationn, part i: Exact model, IEEE T. Power. Ap. Syst., pp , 197. [15] F. C. Schweppe, Power system static-state estimationn, part iii: Implementation, IEEE T. Power. Ap. Syst., pp , 197. [16] R. Zhang, J. Lavaei, and R. Baldick, Spurious local minima in power system state estimation, preprint, 218. [17] R. Ge, J. D. Lee, and T. Ma, Matrix completion has no spurious local minimum, in Advances in Neural Information Processing Systems (NIPS), 216. [18] Y. Zhang, R. Madani, and J. Lavaei, Conic relaxations for power system state estimation with line measurements, IEEE Transactions on Control of Network Systems, 217.