Distributed robust state estimation with equality constraints and L1 regularization

Abstract

Large-scale state estimation (SE) problems are frequently encountered in the fields of smart grid, data engineering, bioinformatics, and digital signal processing. Distributed SE, where the SE is solved in a distributed manner, is of importance due to limitation of either computational resources or bandwidth in transmitting the distributed data, or both. Moreover, outliers are likely to be encountered in large-scale SE. There are also challenges of missing data, scaling issues, and parameter constraints that have to be satisfied.

To overcome these challenges, a novel augmented Lagrangian method (ALM)-based Distributed Robust State Estimation (DRSE) framework is proposed in this thesis. It addresses the robust estimation problem under outliers using robust M-estimation, which is efficiently solved using a smoothed iteratively reweighted least squares (IRWLS) scheme to improve the stability of the DRSE. The ALM-based framework also simplifies the incorporation of equality constraints and the decoupling of the computation as well as nonsmooth L1 and group sparsity regularizations, which helps reduce the variance and promote sparsity in certain practical applications. To accelerate this novel framework, a LANCELOT-like procedure is proposed to iteratively update the regularization parameter of the ALM. Moreover, a novel covariance normalization approach is proposed to handle automatically the scaling issues in the decoupled consensus constraints in the ALM.

The proposed framework is applied to two key problems in smart grid and bioinformatics, namely distributed Power System State Estimation (PSSE) and Gene Regulatory Network (GRN) identification from time-course gene microarray data, respectively. In the PSSE problem, we proposed to inpaint the missing data through the auto-regressive (AR) model. On the other hand, a redescending M-estimation function is employed for mitigating the adverse effect of outliers and a novel weight smoothing scheme is proposed to avoid over suppression of measurements in the DRSE. Experimental results show that the proposed algorithm outperforms conventional approaches in terms of convergence rate and accuracy under the presence of outliers. Moreover, the zero-current injection constraints are incorporated as equality constraints, which better satisfying the system property. By utilizing prior information with AR models, missing data in SE can be in-painted through recursive monitoring in consecutive SE. Besides, the LANCELOT-like procedures for updating regularization parameters and the proposed covariance normalization method was shown to improve considerably the convergence performance of the algorithm. Finally, the Levenberg-Macquardt algorithm is adopted to address possible numerical stability under unexpected adverse situations such as singularity of the Hessian matrix during immediate iterations.

For GRN identification, the Non-linear Maximum-A-Posteriori time-varying AR model is employed, which utilizes the L1 regularization and piecewise temporal continuity to address the high-dimensionality and low number of sample problem. Moreover, a novel multi-Laplace prior is proposed to identify possible transcription factors under this framework. Simulation results based on synthetic GRN model and real time-course gene microarray data collected from yeast cell, show that the proposed approach offers highly comparable performance with other state-of-the-art methods using additional physical data. The effectiveness of the proposed framework makes it a valuable tool for solving various distributed non-linear state estimation problems.