PaperlayerView exact revision on arXiv

Source-linked AI summary

Data-Driven Learning-Based Optimization for Distribution System State Estimation

Ahmed S. Zamzam, Xiao Fu, Nicholas D. Sidiropoulos

arXiv:1807.01671v3eess.SP

TL;DR

DSSE is difficult because sparse measurements create a non-convex, initialization-sensitive estimation problem. The paper trains a shallow neural network on measurement-state pairs to initialize Gauss-Newton, and experiments show improved accuracy, convergence reliability, and computation time, with training-cost design further improving performance.

  • Problem

    Sparse measurements make DSSE non-convex and initialization-sensitive, while accurately learning the complete measurement-to-state mapping is difficult.

  • Method

    A shallow neural network learns to map measurements to a point near the network state, which initializes Gauss-Newton for refinement.

  • Results

    The proposed approach converged in all 1000 tested cases, whereas Gauss-Newton failed in 28, and took almost four times less time.

  • Takeaways & Limitations

    The hybrid architecture offers fast and reliable DSSE convergence with superior estimate accuracy and computation time in the tested IEEE-37 scenarios.

Abstract

from arXiv · show

Distribution system state estimation (DSSE) is a core task for monitoring and control of distribution networks. Widely used algorithms such as Gauss-Netwon perform poorly with the limited number of measurements typically available for DSSE, often require many iterations to obtain reasonable results, and sometimes fail to converge. DSSE is a non-convex problem, and working with a limited number of measurements further aggravate the situation, as indeterminacy induces multiple global (in addition to local) minima. Gauss-Newton is also known to be sensitive to initialization. Hence, the situation is far from ideal. It is therefore natural to ask if there is a smart way of initializing Gauss-Newton that will avoid these DSSE-specific pitfalls. This paper proposes using historical or simulation-derived data to train a shallow neural network to `learn to initialize' -- that is, map the available measurements to a point in the neighborhood of the true latent states (network voltages), which is used to initialize Gauss-Newton. It is shown that this hybrid machine learning / optimization approach yields superior performance in terms of stability, accuracy, and runtime efficiency, compared to conventional optimization-only approaches. It is also shown that judicious design of the neural network training cost function helps to improve the overall DSSE performance.

I. INTRODUCTION

Distribution-system state estimation is difficult because measurements are scarce, the optimization is non-convex, and Gauss-Newton is initialization-sensitive. The paper addresses this by learning an initialization from historical or simulated data before optimization.

  • State estimation supports real-time monitoring and control tasks by inferring bus voltage magnitudes and angles from measurements and physical laws.These tasks include Volt/VAr optimization, feeder reconfiguration, and restoration.
  • Distribution systems rely on pseudo-measurements because they have far fewer real-time measurements than transmission networks.Pseudo-measurements are derived from short-term load and renewable-energy forecasts.
  • WLS DSSE is non-convex, while limited measurements can introduce multiple local minima and make Gauss-Newton highly dependent on initialization.Poor initialization may require many iterations or lead to non-convergence.
  • The proposed approach trains a shallow neural network to map measurements to a state near the true network state, then uses that output to initialize Gauss-Newton.Historical data can be reused to simulate network operations and create measurement-state training pairs.
  • Shallow learning is chosen instead of end-to-end deep estimation because exact state mapping is difficult, deep networks are costly to train, and sequential inference can slow estimation.The hybrid design retains Gauss-Newton refinement after neural-network initialization.

A. Network Representation

The distribution network is represented as a multi-phase graph whose buses and lines carry phase-specific voltage and network variables.

  • The network comprises N + 1 multi-phase buses and L edges, with bus 0 representing the substation connection to the transmission grid.Buses and lines have associated phase sets, and each bus voltage is collected into a phase-indexed phasor vector.

B. Problem Formulation

DSSE estimates network states from noisy measurements and pseudo-measurements using a weighted least-squares formulation. Nonlinear measurement mappings make the resulting optimization non-convex.

  • DSSE recovers bus voltage phasors from real-time measurements and pseudo-measurements, which contain measurement and forecast errors.Measurement noise and forecast error are modeled as zero-mean quantities with specified variances.
  • Pseudo-measurement forecast-error variance can be estimated from historical forecast data, whereas measurement-noise variance depends on measuring-equipment accuracy.Load and generation forecasting supplies the pseudo-measurements used in the formulation.
  • The measurement vector combines L_m real measurements and L_s pseudo-measurements, linked to the state by h(v).Weights are inversely proportional to the corresponding noise variances.
  • The WLS objective is non-convex because nonlinear measurement mappings appear inside squared residual terms.The mapping nonlinearity is the stated source of non-convexity.

C. Available Measurements for DSSE

DSSE uses heterogeneous phasor and real-valued measurements, supplemented by pseudo-measurements when real-time data do not make the network observable. Their nonlinear relationships produce a difficult fourth-order objective.

  • Phasor measurements of nodal voltages and current flows are modeled as linear functions of the state vector.Complex quantities are represented through separate real and imaginary measurements.
  • Voltage magnitudes, current magnitudes, and active and reactive power flows are real-valued measurements modeled nonlinearly, generally as quadratic functions of the state.These measurements can come from SCADA, distribution automation, intelligent electronic devices, and PMUs.
  • Real-time measurements are often insufficient to determine the network state, so pseudo-measurements help achieve observability.They are derived from load and generation forecasts using historical and location-based information.
  • Because measurement functions are quadratic and appear in the WLS objective, the objective becomes fourth order in the state variables.The resulting objective is described as very challenging to optimize.
  • Gauss-Newton iteratively linearizes first-order optimality conditions and performs well near the true state, but lacks a theoretical convergence guarantee here.Its practical behavior depends on the chosen state-variable representation and initialization.

III. PROPOSED APPROACH: LEARNING-AIDED DSSE OPTIMIZATION

The paper trains a shallow neural network to map measurements to a neighborhood of the true state, using that output to initialize Gauss-Newton. A relaxed training objective reduces network and data complexity while retaining a useful initialization, with practical benefits for real-time DSSE and modest topology changes.

  • Learning-aided DSSE: The neural network approximates a mapping from measurements z to estimated network states using a shallow architecture with one hidden layer.The network combines transformed input measurements through hidden neurons to produce the state estimate.
  • Learning-aided DSSE: The proposed “learn to initialize” strategy maps available measurements to a point near the true latent state, then uses it to initialize Gauss-Newton.This avoids attempting to learn the exact end-to-end DSSE mapping with a large or deep network.
  • Relaxed training objective: The relaxed cost function accepts outputs within a radius ϵ of the target state, rather than requiring exact state reproduction.The objective seeks ∥v_j − g_T(z_j)∥_2^2 ≤ ϵ for as many training samples as possible.
  • Relaxed training objective: Controlling ϵ can reduce the required neuron count and training-sample complexity while still producing an initialization that supports rapid Gauss-Newton convergence.The paper links this reduction to the lower precision required for initialization compared with exact DSSE solution prediction.
  • Practical deployment: Keeping the network shallow and its hidden-layer size moderate makes online evaluation simple enough for real-time operation.The approach combines offline learning with optimization-based DSSE during online estimation.
  • Practical deployment: The relaxed objective is reported to preserve initialization effectiveness under modest network reconfiguration, even when training uses the original topology.The paper attributes this robustness to learning a neighborhood rather than an exact solution mapping.

IV. EXPERIMENTAL RESULTS

Experiments on the IEEE-37 feeder evaluate the shallow neural network’s initialization quality and its effect on Gauss–Newton state estimation. Relaxing the training cost improves generalization, accuracy, convergence, and computational efficiency.

  • Experimental setup: The IEEE-37 feeder is a highly unbalanced benchmark with single-, two-, and three-phase connections and six buses hosting distributed energy resources.The experiments use PMU, current-magnitude, and pseudo-measurements across the feeder.
  • Experimental setup: 100,000 loading and generation scenarios train a shallow neural network with 103 inputs, 2,048 hidden nodes, and 210 outputs.Historical load and generation data are processed through a linearized power-flow solver to generate training measurements and voltage profiles.
  • Training-cost effect: With ϵ = 1, the shallow NN output-distance histogram is less spread than with the conventional cost function, ϵ = 0.The comparison uses 1,000 loading and generation scenarios.
  • Training-cost effect: Positive ϵ values reduce computations by up to 25% and can improve estimation accuracy by almost 5 times.Relaxing the approximation requirement improves generalization across loading and generation profiles.
  • Estimator comparison: The proposed architecture achieves almost 10 times better average accuracy than the Gauss–Newton estimator and greatly reduces fitting error.The comparison evaluates 1,000 scenarios using PMU-informed Gauss–Newton initialization.
  • Estimator comparison: Across 1,000 trials, Gauss–Newton diverges in 28 scenarios, whereas the proposed architecture converges in all cases and takes almost four times less time.The runtime reduction follows from requiring only a few Gauss–Newton iterations after neural-network initialization.

V. SYSTEM RECONFIGURATION

The study tests the learning-based estimator under feeder reconfiguration, including faults, energized tie lines, and altered switch states. It retains performance gains under modest topology changes, while severe changes favor a different initialization or retraining.

  • Topology changes: Topology changes are important because inaccurate network configuration makes accurate state estimation untenable.The estimator would fit measurements to an incorrect model when the generating function differs from the assumed model.
  • Test scenarios: The evaluation considers three reconfiguration scenarios involving faults, energized tie lines, and changed switch states.The neural network is trained using data from the original network topology.
  • Results: Under significant reconfiguration, projecting the flat voltage profile onto the PMU-defined linear space performs better than neural-network initialization.For severe changes expected to persist, the shallow network can be retrained in a few minutes to match the underlying physical model.

VI. CONCLUSION

The paper presents a data-driven DSSE architecture that trains a neural network to estimate the network state, then refines that estimate with Gauss-Newton. Experiments on the IEEE-37 feeder suggest fast, reliable convergence and superior accuracy and computation time.

  • VI. CONCLUSION: The proposed architecture trains a neural network using historical load and energy-generation data to approximate the network state.The network accommodates several measurement and pseudo-measurement types.
  • VI. CONCLUSION: The neural-network estimate is fed to Gauss-Newton for refinement.
  • VI. CONCLUSION: The IEEE-37 test feeder evaluates the approach in scenarios including distributed energy sources.
  • VI. CONCLUSION: The combined approach offers fast and reliable convergence to the optimal solution in realistic experiments.
  • VI. CONCLUSION: The learning approach shows superior performance in estimate accuracy and computation time.

VII. APPENDIX A PROOF OF PROPOSITION 1

The appendix proves that shallow neural networks can represent the relevant vector-valued mapping by approximating each scalar output separately and concatenating the resulting networks. It also establishes optimality for the stated approximation cost problem.

  • VII. APPENDIX A PROOF OF PROPOSITION 1: The proof relies on a universal approximation result for continuous functions using continuous sigmoidal functions.
  • VII. APPENDIX A PROOF OF PROPOSITION 1: A vector-valued function is represented as separate scalar-valued functions, one for each output element.
  • VII. APPENDIX A PROOF OF PROPOSITION 1: Finite training samples are normalized into an L-dimensional unit cube before applying the scalar approximation result.
  • VII. APPENDIX A PROOF OF PROPOSITION 1: The scalar approximators are combined by parallel concatenation to obtain a shallow neural network with K outputs.
  • VII. APPENDIX A PROOF OF PROPOSITION 1: The construction achieves the target approximation condition for the vector-valued function by controlling the scalar approximation error.
  • VII. APPENDIX A PROOF OF PROPOSITION 1: The constructed parameters achieve zero cost and are therefore optimal for the stated problem.

APPENDIX B AN EXAMPLE NETWORK

The appendix examines continuity of the state-estimation mapping in a 3-bus balanced lossless network. Under a positive lower bound on the relevant susceptance-voltage products, the mapping from selected measurements to phase differences is continuous.

  • APPENDIX B AN EXAMPLE NETWORK: The example uses a 3-bus balanced lossless network with voltage magnitudes and line power flows measured.
  • APPENDIX B AN EXAMPLE NETWORK: The power-flow equations relate measured quantities to voltage magnitudes, line susceptances, and phase differences.
  • APPENDIX B AN EXAMPLE NETWORK: The state-estimation mapping takes P12, P13, |v1|, and |v2| as measurements and returns θ12 and θ13 as network states.
  • APPENDIX B AN EXAMPLE NETWORK: The mapping is continuous when the relevant susceptance-voltage product is bounded below by any ϵ > 0.
  • APPENDIX B AN EXAMPLE NETWORK: The continuity proof composes the measurement-to-angle function with sin−1(·), which is continuous on the stated domain.
Loading 1807.01671v3…