0.2 −0.1 0 τ J J normalized frequency frequency response FJ{ ψ J } FJ{ ψ J }FDI 1 Fig. 10. Test case 4: Frequency response of F J { ψ J } with and without undetectable FDI attack. For the attacked case, F J { ψ J } FDI , the threshold τ J J is exceeded.

In this paper, we consider the problem of blind estimation of states and topology (BEST) in power systems. We use the linearized DC model of real power measurements with unknown voltage phases (i.e. states) and an unknown admittance matrix (i.e. topology) and show that the BEST problem can be formulated as a blind source separation (BSS) problem with a weighted Laplacian mixing matrix. We develop the constrained maximum likelihood (ML) estimator of the Laplacian matrix for this graph BSS (GBSS) problem with Gaussian-distributed states. The ML-BEST is shown to be only a function of the states' second-order statistics. Since the topology recovery stage of the ML-BEST approach results in a high-complexity optimization problem, we propose two low-complexity methods to implement it: (1) Two-phase topology recovery, which is based on solving the relaxed convex optimization and then finding the closest Laplacian matrix, and (2) Augmented Lagrangian topology recovery. We derive a closed-form expression for the associated Cramér-Rao bound (CRB) on the topology matrix estimation. The performance of the proposed methods is evaluated for the IEEE-14 bus test-case system and for a random network. It is shown that, asymptotically, the state estimation performance of the proposed ML-BEST methods coincides with the oracles minimum mean-squared-error (MSE) state estimator, and the MSE of the topology estimation achieves the proposed CRB.

The constrained Cramér-Rao bound (CCRB) is a lower bound on the mean-squared-error (MSE) of estimators that satisfy some unbiasedness conditions. Although the CCRB unbiasedness conditions are satisfied asymptotically by the constrained maximum likelihood (CML) estimator, in the nonasymptotic region these conditions are usually too strict and the commonly-used estimators, such as the CML estimator, do not satisfy them. Therefore, the CCRB may not be a lower bound on the MSE matrix of such estimators. In this paper, we propose a new definition for unbiasedness under constraints, denoted by C-unbiasedness, which is based on using Lehmann-unbiasedness with a weighted MSE (WMSE) risk and taking into account the parametric constraints. In addition to C-unbiasedness, a Cramér-Rao-type bound on the WMSE of C-unbiased estimators, denoted as Lehmann-unbiased CCRB (LU-CCRB), is derived. This bound is a scalar bound that depends on the chosen weighted combination of estimation errors. It is shown that C-unbiasedness is less restrictive than the CCRB unbiasedness conditions. Thus, the set of estimators that satisfy the CCRB unbiasedness conditions is a subset of the set of C-unbiased estimators and the proposed LU-CCRB may be an informative lower bound in cases where the corresponding CCRB is not. In the simulations, we examine linear and nonlinear estimation problems under nonlinear constraints in which the CML estimator is shown to be C-unbiased and the LU-CCRB is an informative lower bound on the WMSE, while the corresponding CCRB on the WMSE is not a lower bound and is not informative in the non-asymptotic region.

Abstract-In many practical parameter estimation problems, prescreening and parameter selection are performed prior to estimation. In this paper, we consider the problem of estimating a preselected unknown deterministic parameter chosen from a parameter set based on observations according to a predetermined selection rule, Ψ. The data-based parameter selection process may impact the subsequent estimation by introducing a selection bias and creating coupling between decoupled parameters. This paper introduces a post-selection mean squared error (PSMSE) criterion as a performance measure. A corresponding Cramér-Rao-type bound on the PSMSE of any Ψ-unbiased estimator is derived, where the Ψ-unbiasedness is in the Lehmann-unbiasedness sense. The post-selection maximumlikelihood (PSML) estimator is presented. It is proved that if there exists an Ψ-unbiased estimator that achieves the Ψ-Cramér-Rao bound (CRB), i.e., an Ψ-efficient estimator, then it is produced by the PSML estimator. In addition, iterative methods are developed for the practical implementation of the PSML estimator. Finally, the proposed Ψ-CRB and PSML estimator are examined in estimation after parameter selection with different distributions.Index Terms-Non-Bayesian parameter estimation, Ψ-Cramér-Rao bound (Ψ-CRB), estimation after parameter selection, postselection maximum-likelihood (PSML) estimator, Lehmann unbiasedness.

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