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Real-time Power System State Estimation and Forecasting via Deep Neural Networks

Liang Zhang, Gang Wang, Georgios B. Giannakis

arXiv:1811.06146v2cs.LGstat.ML

TL;DR

Rapid voltage fluctuations and the scale and nonconvexity of modern grids challenge computationally practical, accurate real-time state monitoring. The paper unrolls a physics-based prox-linear solver into a model-specific DNN for PSSE and uses deep RNNs for nonlinear state forecasting. Tests on IEEE benchmark systems with real load data show improved estimation and forecasting relative to existing alternatives, with nearly order-of-magnitude PSSE improvement on the IEEE 118-bus system.

  • Problem

    Modern grids require real-time monitoring, but existing PSSE methods face computational expense, nonconvexity, missing measurements, and limited ability to represent system dynamics.

  • Method

    The paper develops model-specific prox-linear nets by unrolling a physics-based solver and deep RNNs that forecast states from historical estimated voltages.

  • Results

    Numerical tests on IEEE 57- and 118-bus systems using real load data show improved estimation and forecasting relative to existing alternatives.

  • Takeaways & Limitations

    The proposed prox-linear PSSE is easy to train and computationally inexpensive, while RNN forecasting provides nonlinear voltage prediction and can aid PSSE with missing measurements.

Abstract

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Contemporary power grids are being challenged by rapid voltage fluctuations that are caused by large-scale deployment of renewable generation, electric vehicles, and demand response programs. In this context, monitoring the grid's operating conditions in real time becomes increasingly critical. With the emergent large scale and nonconvexity however, the existing power system state estimation (PSSE) schemes become computationally expensive or yield suboptimal performance. To bypass these hurdles, this paper advocates deep neural networks (DNNs) for real-time power system monitoring. By unrolling an iterative physics-based prox-linear solver, a novel model-specific DNN is developed for real-time PSSE with affordable training and minimal tuning effort. To further enable system awareness even ahead of the time horizon, as well as to endow the DNN-based estimator with resilience, deep recurrent neural networks (RNNs) are also pursued for power system state forecasting. Deep RNNs leverage the long-term nonlinear dependencies present in the historical voltage time series to enable forecasting, and they are easy to implement. Numerical tests showcase improved performance of the proposed DNN-based estimation and forecasting approaches compared with existing alternatives. In real load data experiments on the IEEE 118-bus benchmark system, the novel model-specific DNN-based PSSE scheme outperforms nearly by an order-of-magnitude the competing alternatives, including the widely adopted Gauss-Newton PSSE solver.

I. INTRODUCTION

Growing renewable generation, electric vehicles, and demand response increase load peaks and voltage fluctuations, making real-time monitoring critical. The paper develops model-specific DNNs for PSSE and deep RNNs for forecasting to address computational, nonlinear, observability, and dynamics challenges.

  • Motivation: Renewable generation, electric vehicles, and demand response create unprecedented load peaks and voltage fluctuations, increasing the need for real-time grid monitoring.Monitoring supports instability detection, protection, and energy management.
  • PSSE challenge: Existing PSSE schemes can require many iterations or substantial computation, while increasing grid dynamics and system size motivate real-time large-scale alternatives.
  • Prior approaches: Plain FNN-based PSSE schemes can suffer vanishing or exploding gradients and ignore the underlying power-system model.
  • Contributions: The paper combines data with physical constraints by developing model-specific DNNs for PSSE through unrolling a reduced-complexity prox-linear solver.
  • Contributions: Deep RNNs forecast system states from historical voltage series, capturing nonlinear dependencies while using a fixed parameter count for variable-length inputs.
  • Evaluation: Numerical tests use real load data on IEEE 57- and 118-bus benchmark systems to compare the proposed methods with existing alternatives.

II. LEAST-ABSOLUTE-VALUE ESTIMATION

This section formulates power-system state estimation as recovery of rectangular-coordinate bus voltages from noisy measurements and develops a least-absolute-value solution using a prox-linear solver. It then motivates a reduced-complexity reformulation by contrasting the convex quadratic subproblem with an ADMM implementation’s auxiliary-variable burden.

  • A power network is represented as a graph of buses and transmission lines, with bus voltages and active or reactive power quantities defining the system variables.
  • At each time slot, PSSE recovers the 2N-dimensional rectangular state vector v_t from generally noisy measurements z_t collected at selected buses and lines.The measurement vector includes quantities such as squared voltage magnitudes and nodal or line measurements.
  • The measurement model is z_m = v^⊤H_m v + ϵ_m, where noise and modeling inaccuracies are represented by ϵ_m.
  • Using the outlier-robust LAV criterion, the state estimate is obtained through a prox-linear solver that repeatedly minimizes a regularized, locally linearized cost.The solver is described as having provably fast locally quadratic convergence and as handling the problem’s non-smoothness and nonconvexity.
  • 2M + 2N auxiliary variables and 2M + 4N variable updates per iteration make the ADMM inner loop a motivation for a reduced-complexity Lasso-type reformulation.

III. THE PROX-LINEAR NET

The prox-linear net unrolls a reduced-complexity physics-based solver into a trainable DNN for real-time PSSE. Its architecture incorporates solver structure, skip-connections, and fixed-width layers to support efficient training and inference.

  • Reduced-complexity prox-linear solver: The reduced-complexity solver recasts the inner prox-linear subproblem as a Lasso problem solvable with ISTA iterations.The original ADMM-based inner loop requires many auxiliary-variable updates, motivating the Lasso reformulation.
  • Unrolling the solver: Unrolling the double-loop prox-linear iterations produces a K(I + 1)-layer structured DNN that approximates the physics-based solver.The architecture is obtained by unrolling outer iterations and their inner-loop recursions.
  • Unrolling the solver: The resulting prox-linear net is a hybrid combining plain-vanilla FNN components with the conventional iterative solver.At convergence, the unrolled output is linked to the solver variables through vI+1 = BuI.
  • Training and tuning: The prox-linear net trains its coefficients end-to-end from historical or simulated measurement-voltage pairs and requires tuning only I, K, and η.Soft-thresholding or ReLU activations can be used, with similar performance observed in simulated tests.
  • Network architecture: Skip-connections directly link the input measurements to intermediate and output layers, while each layer uses a fixed number M of hidden neurons.These connections are described as helping avoid vanishing and exploding gradients and enabling efficient DNN training.
  • Real-time PSSE: The proposed architecture combines offline training with real-time inference and is reported to boost performance relative to competing FNN and Gauss-Newton PSSE approaches.The real-time inference and offline training stages are distinguished in the system flow chart.

IV. DEEP RNNS FOR STATE FORECASTING

Deep RNNs model nonlinear power-system state dynamics from historical voltage sequences, and their forecasts can support PSSE when measurements are missing. The monitoring pipeline combines recurrent forecasting with prox-linear-net estimation across successive time slots.

  • Motivation: Missing SCADA measurements can challenge PSSE, motivating state forecasting to provide system awareness ahead of time.The paper identifies meter or communication failures as a source of missing entries in measurement vectors.
  • Forecasting model: State forecasting represents the next voltage state as a nonlinear function of current and lagged states, with r controlling the number of included lags.The measurement function maps states to measurements, while the transition function is approximated using RNN modeling.
  • Deep RNN architecture: RNNs process correlated time-series inputs, while deep RNNs stack recurrent layers to learn hierarchical nonlinear representations.The paper emphasizes scalability to long-memory and variable-length input sequences.
  • Deep RNN architecture: Unfolding the recurrent computational graph produces a deep RNN whose rows represent layers and columns represent time slots.The depicted unfolded network uses single-step delay units to connect recurrent processing across time.
  • Forecasting model: The deep RNN forecasts the next state from historical voltage data, with output-layer and recurrent weights learned end-to-end by backpropagation.This broadens nonlinear prediction beyond single-hidden-layer feed-forward transition models by incorporating memory.
  • Forecasting model: The proposed forecasting approaches focus on one-step prediction but can be generalized to multiple steps ahead with minor modifications.
  • Integrated monitoring: Estimated historical states can feed the RNN, whose forecast becomes a virtual measurement for PSSE and can impute missing measurements at the next time slot.The overall scheme implements prox-linear-net PSSE and deep-RNN forecasting at successive time slots; residuals may also help reveal erroneous data or grid changes.

V. NUMERICAL TESTS

The proposed deep prox-linear-net PSSE and deep RNN forecasting methods were evaluated on IEEE 57- and 118-bus systems using real load data. Training and testing datasets came from the 2012 Global Energy Forecasting Competition load series, with system-specific subsampling.

  • Evaluation setup: IEEE 57- and 118-bus benchmark systems were used to evaluate the deep prox-linear-net PSSE and deep RNN state-forecasting methods.
  • Evaluation setup: Training and testing datasets were generated from real load data in the 2012 Global Energy Forecasting Competition.The load series were subsampled by factors of 5 and 2 for the IEEE 57-bus and 118-bus systems, respectively.
  • Training and reproducibility: All neural networks were trained and tested independently for 20 runs, with results averaged over those runs.The implementation used Adam with a starting learning rate of 10^-3 for 200 epochs, and the 118-bus prox-linear-net code was publicly released.

A. Prox-linear nets for PSSE

The prox-linear net provides a model-specific DNN approach to PSSE, evaluated against Gauss–Newton and plain FNN baselines on 57- and 118-bus systems. It achieves competitive or markedly improved estimation performance while retaining the runtime advantage of neural-network inference.

  • Experimental setup: The experiments compare the prox-linear net with Gauss–Newton and 6- and 8-layer FNNs on 57- and 118-bus systems.The networks use 114 hidden units for the 57-bus system and 236 for the 118-bus system.
  • 57-bus results: 3.49 × 10−4 RMSE: the prox-linear net performs competitively with Gauss–Newton at 3.2×10−4 on the 57-bus system.The 6-layer and 8-layer FNNs obtain 6.35×10−4 and 9.02×10−4, respectively.
  • 57-bus results: 0.0973s runtime: the prox-linear net is nearly as fast as the FNNs and far faster than Gauss–Newton at 14.22s over 1,500 test examples.The corresponding FNN runtimes are 0.0944s and 0.0954s.
  • 118-bus results: 2.97 × 10−4 RMSE: the prox-linear net markedly outperforms Gauss–Newton at 4.71 × 10−2 on the 118-bus system.The 6-layer and 8-layer FNNs obtain 1.645 × 10−3 and 2.366 × 10−3, respectively.
  • 118-bus results: 0.3323s runtime: the prox-linear net is much faster than Gauss–Newton at 183.4s over 3,706 test examples.The 6-layer and 8-layer FNNs require 0.2895s and 0.3315s, respectively.

B. Deep RNNs for state forecating

Deep RNNs forecast voltage magnitudes and angles from voltage time series on 57- and 118-bus systems. Forecasting with estimated voltages performs comparably to forecasting with ground-truth voltages and better than the FNN and VAR(1) baselines.

  • Evaluation: The forecasting evaluation uses normalized RMSE for voltage magnitudes and angles across 57- and 118-bus systems.Figures examine temporal errors for individual buses and spatial errors across multiple buses.
  • 57-bus results: 2.303 × 10−3 RMSE: the RNN using ground-truth voltages slightly outperforms the estimated-voltage RNN at 2.305 × 10−3 on the 57-bus system.The 2-layer FNN and VAR(1) obtain 3.153×10−3 and 6.772×10−3, respectively.
  • 118-bus results: 2.588×10−3 RMSE: the ground-truth-voltage RNN outperforms the estimated-voltage RNN at 2.751×10−3 on the 118-bus system.The 2-layer FNN and VAR(1) obtain 4.249×10−3 and 6.461×10−3, respectively.
  • Cross-system comparison: The estimated-voltage RNN performs better than the FNN and VAR(1) despite those baselines using ground-truth voltage time series.This supports comparable forecasting performance when the RNN receives estimated rather than ground-truth voltages.
  • Illustrative forecasts: The deep RNN approaches perform best in the reported individual-bus and all-bus forecasting examples.The examples cover bus 30 in the 57-bus system and the first 50 buses in the 118-bus system.

VI. CONCLUSIONS

The paper combines model-specific prox-linear DNNs for PSSE with deep RNNs for forecasting to support real-time power-system monitoring. Numerical tests on IEEE 57- and 118-bus systems illustrate the approaches’ merits, while distribution-network specialization remains future work.

  • Conclusions: The proposed monitoring scheme combines physics-based prox-linear DNN estimation with deep RNN forecasting from historical estimated voltages.The prox-linear net is described as easy to train and computationally inexpensive.
  • Future work: The current research agenda includes specializing the DNN-based estimation and forecasting schemes to distribution networks.The paper also identifies on-the-fly RNN algorithms for dynamically changing environments as future work.
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