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Distributed Robust Power System State Estimation

Vassilis Kekatos, Georgios B. Giannakis

arXiv:1204.0991v2stat.MLmath.OC

TL;DR

System-wide PSSE is difficult to centralize because of interconnection scale, communication, privacy, and reliability constraints. The paper develops an ADMM-based decentralized framework that extends to robust estimation, and simulations show accurate estimates within few exchanges while jointly identifying corrupted measurements.

  • Problem

    Interconnection-scale PSSE is needed, but centralized estimation faces computational, communication, privacy, and reliability constraints.

  • Method

    The paper develops an ADMM-based decentralized PSSE framework using local solvers and extends it with sparse bad-data modeling for joint state estimation and corruption identification.

  • Results

    10^-3 accuracy is reached in approximately 10 iterations or 6.2 msec for decentralized LSE, and in less than 20 ADMM iterations or 15.2 msec for D-RPSSE with 10% bad measurements.

  • Takeaways & Limitations

    The framework provides accurate decentralized estimates, converges to centralized counterparts, avoids local-observability requirements, and jointly identifies corrupted measurements.

Abstract

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Deregulation of energy markets, penetration of renewables, advanced metering capabilities, and the urge for situational awareness, all call for system-wide power system state estimation (PSSE). Implementing a centralized estimator though is practically infeasible due to the complexity scale of an interconnection, the communication bottleneck in real-time monitoring, regional disclosure policies, and reliability issues. In this context, distributed PSSE methods are treated here under a unified and systematic framework. A novel algorithm is developed based on the alternating direction method of multipliers. It leverages existing PSSE solvers, respects privacy policies, exhibits low communication load, and its convergence to the centralized estimates is guaranteed even in the absence of local observability. Beyond the conventional least-squares based PSSE, the decentralized framework accommodates a robust state estimator. By exploiting interesting links to the compressive sampling advances, the latter jointly estimates the state and identifies corrupted measurements. The novel algorithms are numerically evaluated using the IEEE 14-, 118-bus, and a 4,200-bus benchmarks. Simulations demonstrate that the attainable accuracy can be reached within a few inter-area exchanges, while largest residual tests are outperformed.

I. INTRODUCTION

Modern power-system monitoring requires interconnection-level estimation, but centralized and independently local approaches face practical, privacy, observability, and reliability constraints. The paper frames a decentralized alternative that coordinates local estimation while supporting conventional and robust formulations.

  • Motivation: Interconnection-scale PSSE is needed because long-distance tie-line transfers and renewable intermittency require accurate, more frequent system-wide state acquisition.
  • Motivation: Centralized PSSE is difficult to scale because interconnections may contain thousands of buses, require real-time processing, and need resilience to corrupted data.
  • Proposed direction: The proposed framework targets a middle ground by coordinating local control centers with low exchange overhead, privacy support, and compatibility with existing PSSE solvers.
  • Proposed direction: The framework also supports robust estimation by jointly estimating states and identifying corrupted measurements rather than relying only on conventional post-estimation bad-data tests.
  • Limitations of existing approaches: Independent local estimation can be suboptimal, locally unobservable, and inconsistent on shared boundary-bus estimates over critical tie lines.
  • Limitations of existing approaches: Centralized estimation improves accuracy but incurs impractical computational and communication load, increased vulnerability, and disclosure of internal system structure.

IV. DECENTRALIZED PSSE

The decentralized PSSE formulation converts shared-state consistency into an optimization problem solved by ADMM. Its updates separate local computation from neighbor exchanges and converge to centralized least-squares solutions without requiring local observability.

  • Problem reformulation: Shared boundary states are tied through auxiliary variables x_kl, with constraints enforcing agreement between each area’s local copy and the shared variable.
  • ADMM procedure: ADMM introduces one multiplier for each shared-state constraint and alternates local-state minimization, auxiliary-variable updates, and multiplier ascent.
  • ADMM procedure: The local minimization decouples across control areas, enabling existing PSSE solvers to perform the principal computations.
  • Communication and implementation: Only shared boundary-bus states are exchanged between interested neighboring areas; measurements and regression matrices remain undisclosed.
  • Convergence: Under mild conditions, ADMM converges to the centralized LSE when the system is globally observable and to one of multiple LSEs when it is not.
  • Convergence: The method does not require local observability, as verified by removing a measurement that leaves one area locally unobservable in the IEEE 14-bus test.

V. DECENTRALIZED BAD DATA ANALYSIS

Grossly corrupted SCADA measurements motivate a decentralized robust PSSE framework that identifies bad data while estimating system states.

  • SCADA failures, calibration issues, wiring errors, parameter uncertainty, and malicious injections can produce grossly corrupted measurements.

A. Interconnection-Wide Bad Data Identification

The paper models bad data as sparse measurement corruption and uses compressed-sensing ideas to jointly recover the system state and faulty measurements through a convex relaxation.

  • The model z = Hx + o + w represents measurements as state responses, noise, and an unknown sparse corruption vector.A nonzero o(i) indicates that measurement z(i) is bad.
  • Recovering x and o jointly reveals the state while identifying faulty measurements, despite the corrupted model being unobservable even when H has full column rank.
  • The ℓ0 formulation is NP-hard, motivating a convex ℓ1 relaxation for practical robust estimation.
  • The convex quadratic program jointly estimates the state and bad data while suppressing corrupted measurements’ effects on the state estimate.

B. Interesting Links

The paper connects robust PSSE with residual-based tests and existing robust estimators, clarifying equivalences for single outliers and limitations for multiple bad data.

  • Residual-based χ2 and largest normalized residual tests rely on residuals from the unconstrained LSE and are applied after PSSE.
  • The ℓ0 optimization with one expected bad datum is equivalent to the largest normalized residual test in identifying the bad measurement.
  • For multiple bad data, the connection between sparse optimization and the largest normalized residual test is unclear.
  • As λ approaches infinity, the convex estimator becomes LSE; as λ approaches zero, it becomes the least-absolute-value estimator.
  • For finite λ > 0, the estimator corresponds to Huber’s M-estimator, with λ = 1.34 giving 95% asymptotic efficiency under Gaussian noise without bad data.
  • The residual sensitivity matrix P couples the outlier minimization across areas, complicating decentralized implementation.

C. Distributed Robust Algorithm

D-RPSSE extends decentralized ADMM-based PSSE to jointly estimate states and bad data while preserving local computation, limited exchange, and existing solver compatibility.

  • The robust decentralized formulation partitions corruption variables by area, so bad-data variables remain local and need not be shared.
  • ADMM offers two update choices for the additional corruption variables; optimizing them with shared-state variables is selected for computational efficiency.
  • Neighboring areas exchange shared state variables while each area updates its local state, auxiliary, dual, and corruption variables.
  • The local state update is an LSE modified by replacing measurements with bad-data-compensated values, while consensus updates remain unchanged.
  • The corruption update thresholds residuals: small tentative absolute residuals yield zero corruption, whereas larger residuals receive a shrunk nonzero correction.
  • D-RPSSE preserves software compatibility by adding pseudo-measurements and offsets to local measurements, jointly acquiring robust estimates and bad-data identification without repeated PSSE.

VI. SIMULATED TESTS

The simulations evaluate decentralized state estimation on IEEE 14-, 118-, and 4,200-bus benchmarks using PMU-based measurements and area partitions. Interconnection-wide estimation improves accuracy substantially over internal and local alternatives.

  • Test systems: The evaluation uses IEEE 14- and 118-bus systems plus a 4,200-bus system generated from IEEE benchmarks.The admittance matrices and underlying states are obtained using MATPOWER.
  • Measurement setup: Measurements comprise PMU bus-voltage and line-current recordings represented in rectangular coordinates.The IEEE 14-bus case uses 46 measurements, while the IEEE 118-bus case uses 77 voltage and 205 current meters.
  • Area partitioning: The IEEE 14-bus grid is divided into four areas, and the IEEE 118-bus grid is divided into three areas.The partitions support comparisons among internal, local, and global estimators.
  • Comparative results: Across 100 Monte Carlo runs, the global estimator roughly doubles accuracy, while extending internal estimates to boundary buses yields only marginal improvement.Accuracy is measured by the average standard deviation per state.

A. Testing the Decentralized LSE

The decentralized LSE converges to the centralized solution and reaches measurement-limited accuracy in few iterations, including when an area is locally unobservable. Compared with the method of [29], it requires far fewer iterations and therefore less inter-area communication.

  • Error metrics: The per-area error curves compare convergence to the centralized solution with accuracy relative to the underlying state across areas.The figures report these two error types for the IEEE 14- and 118-bus tests.
  • Convergence: The ADMM iterations converge to the centralized LSE when the system is globally observable.If the system is unobservable, they converge to one of the multiple centralized LSEs.
  • IEEE 14-bus results: The algorithm of [29] requires around 10,000 iterations to attain the same accuracy.Its per-iteration complexity is smaller, but the method does not fully exploit local PSSE solvers.
  • Without local observability: Removing the line (6, 11) current measurement makes area 3 locally unobservable, yet the developed method still converges.The IEEE 14-bus experiment therefore tests the method without local observability.
  • IEEE 118-bus results: The IEEE 118-bus decentralized solution attains the desired statistical accuracy within 5-10 iterations.This result is reported for the per-area error curves in Fig. 4.

B. Testing the Decentralized Robust Estimator

The decentralized robust estimator is evaluated on IEEE 14-bus, 118-bus, and 4,200-bus systems under corrupted-measurement scenarios. It approaches centralized accuracy with modest iteration counts and favorable timing as network size increases.

  • Centralized robust estimation: The IEEE 14-bus tests evaluate centralized bad-data methods across four scenarios, with corruption introduced by multiplying affected measurement components by 1.2.Performance is measured by the ℓ2-norm between the true state and the estimate, averaged over 1,000 Monte Carlo runs.
  • Centralized robust estimation: Huber’s estimator slightly outperforms LNRT when multiple measurements are corrupted, while ordinary LSE performance deteriorates.Across scenarios, Huber’s estimator converges in 6–12 iterations and within 1.3 msec; on the IEEE 118-bus grid, its average time is 3.2 msec versus 81 msec for LNRT.
  • Decentralized robust estimation: On the IEEE 14-bus benchmark under (S3), D-RPSSE reaches 10^-3 accuracy in 30 iterations and 12.1 msec, versus 5 msec for LSE.Figure 5 reports per-area error curves for the robust decentralized algorithm.
  • Large-network evaluation: On the 4,200-bus grid, decentralized LSE reaches 10^-3 accuracy in approximately 10 iterations or 6.2 msec, while centralized LSE takes 93.4 msec.With 10% bad measurements, D-RPSSE reaches 10^-3 in fewer than 20 ADMM iterations or 15.2 msec, compared with 193.5 msec for the centralized robust estimator.

VII. CONCLUSIONS

The paper presents distributed and robust state estimation in a systematic framework that avoids local observability requirements and supports accurate local estimates with limited neighboring-area exchanges.

  • Conclusions: The proposed algorithms waive local observability requirements while maintaining backward compatibility with existing PSSE formulations.The framework also accommodates constraints such as zero-injection buses and operational limits, along with different maximum-likelihood estimators.
  • Conclusions: With a few minimal exchanges between neighboring areas, local control centers can estimate their assigned interconnection states accurately and identify unintentionally or intentionally corrupted data.The conclusion also identifies generalized state estimation, non-convex problems, and re-weighted estimators as future research directions.

APPENDIX

The appendix derives simplified ADMM updates for distributed PSSE by eliminating auxiliary variables and exploiting averaged shared states and multipliers.

  • Lemma and proof: For adjacent areas, the updated Lagrange multipliers satisfy v^t_l,k = 0 for every iteration r > 0.This lemma follows from the decoupled auxiliary-variable update and the paired multiplier updates.
  • Update simplification: The area-wise optimization in step (9a) is separable across areas, and its squared-norm terms can be further decomposed over vector entries.Entries shared among multiple neighboring areas require collecting the corresponding copies and multipliers.
  • Update simplification: The auxiliary variable x_kl equals the average of the shared state-variable copies across areas k and l at each iteration.Eliminating these auxiliary variables allows step (9b) to be dropped from the updates.
  • Compact iterations: Averaged shared variables and weighted Lagrange multipliers are introduced to recursively rewrite the ADMM updates in the compact iteration form (10).The recursive multiplier updates and collected definitions yield the stated iterates.
  • Compact iterations: The appendix concludes that the simplified definitions and recursive updates directly recover iteration set (10).This establishes the equivalence between the expanded ADMM updates and the compact implementation.
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