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Bayesian State Estimation for Unobservable Distribution Systems via Deep Learning

Kursat Rasim Mestav, Jaime Luengo-Rozas, Lang Tong

arXiv:1811.02756v4stat.MLcs.LGstat.AP

TL;DR

The paper tackles state estimation when distribution systems are unobservable and measurements may be unreliable or missing. It combines distribution learning, Monte Carlo-trained deep regression, and Bayesian bad-data filtering; simulations show stronger benchmark performance and much faster online estimation, while the authors note that further evaluation is needed.

  • Problem

    State estimation is needed for distribution systems with limited measurements, but unobservable settings and unreliable or missing data challenge conventional approaches.

  • Method

    The method learns power-injection distributions, uses Monte Carlo samples to train a deep neural network for MMSE estimation, and applies Bayesian bad-data detection and filtering.

  • Results

    The Bayesian deep-learning estimator outperformed pseudo-measurement techniques, achieved one to two orders of magnitude lower ASE, and reduced online computation time by five orders of magnitude versus WLS.

  • Takeaways & Limitations

    Bayesian state estimation via deep learning can support unobservable systems while reducing online computation and handling unreliable, missing, or attacked measurements.

  • Takeaways & Limitations

    The authors state that the Bayesian approach has limitations and requires further research and evaluation beyond the paper’s scope.

Abstract

from arXiv · show

The problem of state estimation for unobservable distribution systems is considered. A deep learning approach to Bayesian state estimation is proposed for real-time applications. The proposed technique consists of distribution learning of stochastic power injection, a Monte Carlo technique for the training of a deep neural network for state estimation, and a Bayesian bad-data detection and filtering algorithm. Structural characteristics of the deep neural networks are investigated. Simulations illustrate the accuracy of Bayesian state estimation for unobservable systems and demonstrate the benefit of employing a deep neural network. Numerical results show the robustness of Bayesian state estimation against modeling and estimation errors and the presence of bad and missing data. Comparing with pseudo-measurement techniques, direct Bayesian state estimation via deep learning neural network outperforms existing benchmarks.

I. INTRODUCTION

The paper addresses state estimation in distribution systems that are often unobservable and proposes Bayesian deep learning to estimate states despite sparse or unreliable measurements. It combines distribution learning, neural-network regression, and Bayesian data cleansing, with simulations reporting advantages over pseudo-measurement methods.

  • Motivation: Distribution systems can be unobservable because sensors are insufficient, poorly placed, faulty, missing, or compromised.Unobservability creates multiple states consistent with the same measurements and removes accuracy guarantees for deterministic estimators such as WLS.
  • Motivation: Bayesian inference uses probabilistic dependencies between measurements and states, so estimation does not require observability and directly minimizes estimation error.Its implementation requires learning an unknown joint distribution and addressing analytically intractable or computationally prohibitive estimation.
  • Proposed approach: The method is designed to combine information from smart meters and faster measurements while mitigating incompatible timescales, delayed measurements, and missing data.Smart-meter data contribute to learning network-state distributions rather than being used directly for state estimation.
  • Proposed approach: The proposed approach learns stochastic power-injection distributions, trains a deep neural network for MMSE state estimation, and embeds power-flow equations in regression learning.A Bayesian bad-data detection and filtering algorithm is also included for unreliable, missing, or attacked measurements.
  • Results: Numerical results report better performance than pseudo-measurement benchmarks, robustness to distribution-learning inaccuracies, and improved Bayesian bad-data filtering.The study also reports that networks with five or more layers and bulging middle sections performed better than flatter or rectangular networks.
  • Scope: The paper identifies unresolved limitations of the proposed Bayesian technique and calls for further research and evaluation beyond the study’s scope.The supplied conclusion passage introduces these limitations but does not enumerate them.

B. Related Work

Prior distribution-system estimators commonly make systems observable through pseudo-measurements before applying WLS, while Bayesian approaches and pre-estimation bad-data filtering remain less explored. The paper instead uses Monte Carlo samples to train a neural network that computes MMSE estimates in real time.

  • Existing approaches: Distribution-system state estimation has long faced limited observability, leading many existing methods to create pseudo-measurements before applying WLS or related techniques.Pseudo-measurements can be interpreted as forecasts of real-time measurements from historical data.
  • Existing approaches: Pseudo-measurement methods use forecasting or machine learning, but the related historical-data approach can be limited by temporal resolution.The cited literature frames these methods as forecasting-aided state estimation.
  • Bayesian estimation: Bayesian state estimation models states probabilistically, but prior work has explored it less extensively and generally requires conditional-statistics computation.The paper positions its approach within this less-explored Bayesian line of research.
  • Bayesian estimation: The paper uses Monte Carlo sampling to train a neural network that computes the MMSE estimate directly from measurements during real-time operation.This differs from approaches that use Monte Carlo sampling to calculate the conditional mean directly.
  • Bad-data detection: Classical bad-data detection is usually post-estimation, requires observability, and may repeatedly re-estimate states after suspected data removal.Such techniques can fail to identify bad data or mistakenly remove good data.
  • Bad-data detection: The proposed bad-data method belongs to the less explored class of pre-estimation detection and filtering techniques.Related alternatives use dynamic-model prediction errors or neural-network autoencoders as detection statistics.
  • System model: The paper’s static power-flow model maps net injections to states and states to measurements, with measurement noise included in the observation relation.The model uses g(·) for the injection-to-state mapping and h(x) for the sensor-dependent measurement function.
  • System model: Bayesian estimation additionally assumes a joint probability model for measurement noise and net injections, with those variables treated as statistically independent.The estimator is defined from the joint distribution of states and measurements under an MMSE objective.

B. Bayesian State Estimation

The paper defines Bayesian state estimation through the MMSE conditional mean and approximates this estimator with an offline-trained neural network for online use.

  • Bayesian estimator: The MMSE estimator minimizes expected squared state-estimation error and equals the conditional mean of the state given measurements.The paper distinguishes this objective from least squares, which minimizes modeling error.
  • Bayesian estimator: The developed framework can also support robust minimum-absolute-error and maximum-a-posteriori Bayesian estimators.
  • Bayesian estimator: Computing the MMSE conditional mean is generally intractable because the joint distribution of states and measurements is unknown or unavailable.
  • Methodology: The methodology separates online neural-network state estimation from offline distribution and regression learning.The paper notes that this partition can become online as data arrive continuously.
  • Methodology: Monte Carlo sampling and power-flow analysis generate state-measurement training samples from estimated net-injection distributions.

B. Deep Neural Network Approximation

The paper converts the intractable nonparametric MMSE regression into a finite-dimensional deep-neural-network learning problem, using physically generated samples when direct training pairs are unavailable.

  • Motivation: The MMSE state estimator is a nonparametric regression on measurements whose unknown joint distribution makes the learning problem infinite-dimensional and intractable.
  • Neural-network approximation: A multilayer neural network provides a finite-dimensional approximation whose dimension is the number of parameters in its weight matrix.The network applies parameterized nonlinear transformations across multiple neuron layers.
  • Training-data challenge: Direct input-output training pairs are unavailable, and the underlying joint state-measurement distribution cannot be learned directly.
  • Training-data construction: Sampling net injections and measurement noise through the physical model produces state and measurement samples for neural-network training.The construction uses x = g(s) and z = h(g(s)) + e.
  • Training algorithm: Empirical-risk minimization with stochastic-gradient methods, particularly Adam, trains the network parameters from the generated samples.
  • Regularization: Structural regularization is introduced because deep networks can overfit when their parameter count is large relative to the training set.

E. Regression Learning: Neural Network Structure

The paper regularizes network structure by clustering statistically similar neuron outputs and replacing each cluster with a representative neuron; experiments indicate that a bulging middle section performs better.

  • Network structure: Network shape affects state-estimation performance and can range from shallow-and-wide to deep-and-narrow, without requiring rectangular layers.
  • Clustering method: The proposed structural regularization clusters strongly correlated neuron outputs within each layer and combines each group into one representative neuron.
  • Clustering method: Hierarchical agglomerative clustering controls the number and sizes of neuron clusters after selecting a similarity measure.
  • Clustering method: A reduced network replaces each neuron cluster with a single neuron, producing a non-rectangular architecture.
  • Results: A network with a bulging middle section seems to perform better in the reported numerical experiments.
  • Distribution learning: The injection distribution is learned from limited historical data using a Gaussian-mixture model estimated by maximum likelihood.The paper presents this as a practical choice while noting that parametric assumptions can be wrong.

G. Computation complexity

The proposed Bayesian estimator has lower online complexity than WLS through neural-network inference, but its offline sampling and training costs can be substantial. The framework also incorporates procedures for missing and bad data, with additional computational trade-offs.

  • Online computation: Online Bayesian state estimation requires roughly O(N^2) computations for an N-dimensional state vector with fixed neural-network depth, versus O(N^3) per WLS iteration.The lower online cost is attributed to the neural-network implementation of the estimator.
  • Offline computation: Offline training cost depends on the training-set size, training algorithm, and iterations needed to reach the target accuracy.Unlike WLS, Bayesian training also incurs nontrivial Monte Carlo sampling cost.
  • Offline computation: Generating training samples can cost roughly O(N^4) because each sample requires power-flow solution with approximately O(N^3) computation and at least O(N) samples are needed.This estimate uses a standard Newton-Raphson power-flow implementation and excludes further training-iteration costs.
  • Missing and bad data: With missing measurements, the exact Bayesian estimate averages state estimates over resampled missing data, while replacing missing values by estimated means is a cheaper heuristic.Resampling is described as costly, motivating mean replacement.
  • Missing and bad data: Bad-data detection uses learned measurement distributions before state estimation, after which detected measurements can be removed and the missing-data procedure applied.The method treats bad data as outliers and contrasts pre-estimation detection with residue-based conventional methods.

V. SIMULATIONS RESULTS AND DISCUSSIONS

Simulations on an 85-bus system evaluate Bayesian state estimation against pseudo-measurement baselines under clean, bad, and missing-data conditions. The deep neural network achieves substantially lower error and much faster online estimation, while filtering mitigates bad-data effects.

  • Simulation setup: The simulations used an 85-bus system with current-magnitude meters on 20% of branches and one slack-bus SCADA meter.Measurement noise was modeled as independent zero-mean Gaussian noise with variance equal to 1% of average net consumption.
  • Distribution learning: Gaussian mixture models with three components were adopted for consumption and solar-generation distributions after distribution-estimation comparisons.Weibull estimates exceeded the 95% confidence interval capturing the true distribution.
  • ASE performance: Across all hours, BSEdnn achieved one to two orders of magnitude lower ASE than both pseudo-measurement-based WLS techniques.The comparison involved conventional pseudo-measurements and neural-network-generated pseudo-measurements.
  • Computational cost: 0.16 millisecond per Monte Carlo run was required after training for BSEdnn, versus 22.515 seconds for WLS with pseudo-measurements.The reported online computation was five orders of magnitude faster for the neural-network estimator.
  • Bad-data robustness: Bad-data detection and filtering drove ASE closer to clean-data performance, while unfiltered BSEdnn still outperformed pseudo-measurement WLS estimators.The detection threshold did not require the mean, standard deviation, or distribution of bad data to be known.

D. Optimizing the Structure of Deep Neural Network

The study examines how network depth and shape affect BSEdnn under a fixed neuron budget. Simulations favor intermediate depth and show that two rounds of correlation-based pruning improve error before a third worsens it.

  • Depth: The depth study tested rectangular networks with fixed totals of 1000, 2000, and 3000 neurons across different depths.The authors note that conclusions from these simulations apply to the studied system and may not be conclusive.
  • Depth: 4 to 6 layers produced the best validation ASE performance among the tested network depths.The result was obtained from validation data sets.
  • Network shape: Correlation-based pruning reduced neurons at the network ends and improved test error after the first pruning round.The pruning procedure began from a rectangular network and combined highly correlated neurons.
  • Network shape: A second pruning round further reduced training, test, and validation errors, producing a wider middle section than the network ends.A third round increased errors, so the locally best procedure ended after the second iteration.

E. Simulation results for a 3120-bus mesh network

A 3120-bus Polish mesh network tests scalability under normal, bad, and missing-data conditions. The deep-neural-network estimator performs well at millisecond-level inference, although training takes more than two hours and must occur offline.

  • Network and setup: The 3120-bus mesh network contained 2664 buses with stochastic consumption and 348 with stochastic generation.The experiment was designed to demonstrate computational scalability and applicability to mesh networks.
  • Network and setup: Only 20% of power-injection measurements were assumed available, and a three-layer network with 1000 neurons per layer was trained.The network was trained using one injection distribution and tested with a different distribution.
  • Results: Millisecond-level state estimation was achieved for the 3120-bus system, while standard nonlinear programming routines failed at this network size.Training took more than two hours, requiring offline preparation several hours before operation.

APPENDIX

The appendix specifies the measurement variables and their relationships to distribution-system states. It includes power injections, branch flows, current magnitudes, and accumulated energy-consumption measurements.

  • Measurement variables: The measurement vector can include phase-specific active and reactive power injections, branch flows, and current magnitudes.Different measurement configurations can be assumed for the state-estimation model.
  • Measurement availability: For distribution systems, only a small subset of the power-flow variable set is typically available.The variables are indexed by buses i and j and phase k.
  • State relationships: Active and reactive power injections and branch power flows are related to state variables through the distribution-system equations.The appendix separately states these relationships for injections and branch flows.
  • Measurement variables: Current-magnitude measurements |I^k_ij| can strengthen state estimation and are related to the system state variables.The appendix identifies current from node i to node j as I^k_ij.
  • Measurement modeling: Energy-consumption measurements are modeled as accumulated values of power injections.The appendix distinguishes active and reactive injections and branch-flow quantities.

B. Input-Output Relation of Deep Neural Networks

The network maps measurements to state estimates through successive affine transformations and nonlinear activations across hidden and output layers.

  • An L-layer network uses L−1 hidden layers followed by an output layer.
  • Each hidden neuron combines outputs from the preceding layer through weighting coefficients and an affine operation.
  • Activation functions transform neuron inputs, with sigmoid and rectifier functions among the common choices.
  • Vector-matrix notation represents each hidden-layer output as an elementwise activation applied after an affine mapping.
  • The complete network is an L-fold iterated nonlinear map from measurements to state estimates.

C. Injection Distribution Learning from Smart Meter Data

The paper learns stochastic power-injection distributions from slower smart-meter data by modeling operating patterns as Gaussian mixtures and recovering fast-timescale parameters with stationary autoregressive models.

  • Smart meters provide cumulative energy measurements more slowly than SCADA, creating the central timescale mismatch for injection-distribution learning.
  • The smart-meter measurement density is modeled as a Gaussian mixture representing K operating patterns with pattern-specific probabilities.
  • The fast-timescale injection variable is assigned the same Gaussian-mixture component count and appearance probabilities as the smart-meter process.
  • The variance calculation uses the fast process's lagged autocovariances under a fixed Gaussian-mixture component.
  • A stationary AR-K model relates each fast-timescale sample to its previous K values plus an IID Gaussian innovation.
  • AR parameters can be estimated from one or more traces using least squares, after which autocovariances follow from a Yule-Walker variation.
  • The learning procedure first obtains Gaussian-mixture coefficients from smart-meter data and then computes component-specific fast-process parameters.
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