[Paper Review] Gauss-Newton Unrolled Neural Networks and Data-driven Priors for Regularized PSSE with Robustness

Qiuling Yang, Alireza Sadeghi+3 more|arXiv (Cornell University)|Mar 3, 2020

Probabilistic and Robust Engineering Design39 references4 citations

TL;DR

This paper proposes Gauss-Newton unrolled neural networks (GNU-NN) with data-driven graph neural network (GNN)-based priors for regularized power system state estimation (PSSE), combining Gauss-Newton iterations with proximal regularization to enable fast, robust, and topology-aware estimation. The method achieves superior accuracy and resilience to bad data and adversarial attacks compared to state-of-the-art baselines on the IEEE 118-bus system.

ABSTRACT

Distributed renewable generation, elastic loads, and purposeful manipulation of meter readings challenge the monitoring and control of today's power systems (PS). In this context, to maintain a comprehensive view of the system in real time, fast and robust state estimation (SE) methods are urgently needed. Conventional PSSE solvers typically entail minimizing a nonlinear and nonconvex least-squares by e.g., the workhorse Gauss-Newton method. Those iterative solvers however, are sensitive to initialization and may get stuck in local minima. To overcome these hurdles and inspired by recent image denoising techniques, this paper advocates a learnable regularization term for PSSE that uses a deep neural network (DNN) prior. For the resultant regularized PSSE problem, a "Gauss-Newton-like" alternating minimization solver is first developed. To accommodate real-time monitoring, a novel end-to-end DNN is constructed by unrolling the proposed alternating minimization solver. Interestingly, the power network topology can be easily incorporated into the DNN by designing a graph neural network (GNN) based prior. To further endow the physics-based DNN with robustness against bad data, an adversarial DNN training method is discussed. Numerical tests using real load data on the IEEE $118$-bus benchmark system showcase the improved estimation and robustness performance of the proposed scheme compared with several state-of-the-art alternatives.

Motivation & Objective

• Address the nonconvexity and initialization sensitivity of conventional Gauss-Newton-based PSSE solvers in power systems.
• Integrate data-driven deep priors, specifically GNN-based priors, to encode power network topology and improve solution quality.
• Develop a real-time capable end-to-end deep neural network by unrolling an alternating minimization solver for regularized PSSE.
• Enhance robustness against bad data and adversarial attacks through distributionally robust training.
• Provide a systematic framework that unifies physics-based modeling with deep learning for power system state estimation.

Proposed method

• Develop an alternating minimization solver for regularized PSSE, where each iteration combines a Gauss-Newton step with a proximal update for the deep prior.
• Unroll the alternating minimization solver into an end-to-end deep neural network architecture, termed Gauss-Newton unrolled neural networks (GNU-NN), to enable real-time inference.
• Incorporate a graph neural network (GNN) with two hops and ReLU activations as a learnable, topology-aware regularizer to encode network structure.
• Formulate a distributionally robust optimization framework using an ambiguity set of probability distributions centered at the nominal data distribution to train the model against worst-case perturbations.
• Use adversarial perturbations generated via (23) and an ℓ₂ transportation cost to simulate worst-case data shifts during training.
• Train the GNU-NN model using the Adam optimizer with Huber loss, fixed learning rate of 10⁻³, over 500 epochs with batch size 32.

Experimental results

Research questions

• RQ1 Can a Gauss-Newton unrolled deep network with a GNN-based prior outperform standard DNNs in PSSE accuracy and convergence robustness?
• RQ2 Does incorporating power network topology via a trainable GNN prior improve estimation performance compared to non-topology-aware models?
• RQ3 To what extent does distributionally robust training enhance the resilience of PSSE models to bad data and adversarial input perturbations?
• RQ4 Can the unrolled architecture achieve real-time performance while maintaining high accuracy in large-scale power systems?
• RQ5 How does the proposed method compare to prox-linear networks and standard feedforward DNNs under both clean and corrupted data conditions?

Key findings

• The proposed GNU-GNN model achieved superior voltage estimation accuracy compared to the prox-linear network, 6-layer FNN, and 8-layer FNN on the IEEE 118-bus system, particularly in dynamic and noisy conditions.
• Voltage magnitude and angle estimates from GNU-GNN closely tracked ground truth across buses 50 and 100 from slots 70 to 90, with minimal deviation.
• Under distributional attacks with γ=0.13 and ℓ₂ cost, GNU-GNN maintained stable and accurate estimation, outperforming all baselines in terms of robustness.
• The estimated voltages for the first 20 buses at slot 80 showed that GNU-GNN provided the most accurate and consistent estimates, even under adversarial input shifts.
• The use of a GNN-based prior significantly improved performance over non-topology-aware models, demonstrating the value of structural inductive bias in PSSE.
• Distributionally robust training enabled the model to generalize under worst-case data shifts, confirming its resilience to bad data and adversarial inputs.