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Distributed Estimation via Iterative Projections with Application to Power Network Monitoring

Fabio Pasqualetti, Ruggero Carli, Francesco Bullo

arXiv:1103.0579v2math.OCcs.SYeess.SY

TL;DR

Large, distributed power networks need state estimates from noisy measurements, while centralized computation can be impractical. The paper proposes cooperating control centers’ distributed estimation and false-data detection methods. The methods converge in finite time, and a finite-memory approximation and its accuracy are characterized.

  • Problem

    Large, spatially distributed systems motivate decentralized estimation, while power-system operators need an estimate of the entire network state.

    Source: §1 · p. 1

  • Method

    The paper develops incremental and diffusive distributed estimators using local information, then uses the estimation methods for distributed false-data detection.

    Source: §6 · p. 16

  • Results

    Both estimation methods converge in finite time, and the diffusive method has an asynchronous, scalable implementation whose efficiency is analyzed and illustrated with a practical example.

    Source: §6 · p. 16

  • Takeaways & Limitations

    The procedures are presented as applicable beyond power networks, including state estimation and false-data detection in sensor networks.

    Source: §2 · p. 5

    False data w in Im(H) can evade detection, so the setup assumes that the corruption is detectable from the measurements.

    Source: §2 · p. 5

Abstract

from arXiv · show

This work presents a distributed method for control centers to monitor the operating condition of a power network, i.e., to estimate the network state, and to ultimately determine the occurrence of threatening situations. State estimation has been recognized to be a fundamental task for network control centers to ensure correct and safe functionalities of power grids. We consider (static) state estimation problems, in which the state vector consists of the voltage magnitude and angle at all network buses. We consider the state to be linearly related to network measurements, which include power flows, current injections, and voltages phasors at some buses. We admit the presence of several cooperating control centers, and we design two distributed methods for them to compute the minimum variance estimate of the state given the network measurements. The two distributed methods rely on different modes of cooperation among control centers: in the first method an incremental mode of cooperation is used, whereas, in the second method, a diffusive interaction is implemented. Our procedures, which require each control center to know only the measurements and structure of a subpart of the whole network, are computationally efficient and scalable with respect to the network dimension, provided that the number of control centers also increases with the network cardinality. Additionally, a finite-memory approximation of our diffusive algorithm is proposed, and its accuracy is characterized. Finally, our estimation methods are exploited to develop a distributed algorithm to detect corrupted data among the network measurements.

1 Introduction

Large, geographically distributed systems make centralized estimation difficult, yet power-grid operators need a network-wide state estimate to support safe operation. The paper develops distributed estimation and false-data detection methods for cooperating control centers, alongside a finite-memory approximation.

1 Introduction

  • Wide system distribution and high dimensionality motivate decentralized methods, while operators still need estimates of the entire network state.The paper targets state estimation and false-data detection by control centers using local information and cooperation.Source: §1 · p. 1, §1.1 · p. 2

1.1 Control centers, state estimation and cyber security in power networks

  • Measurements travel from remote terminal units through SCADA to control centers, where noise can impair estimation and downstream control and optimization.The paper also focuses on false-data attacks that corrupt measurements sent to control centers.Source: §1.1 · pp. 2–3

1.2 Related work on distributed estimation and projection methods

  • Prior distributed estimation methods may converge only asymptotically, require global model knowledge, or lack final-error guarantees, while row-projection methods motivate the paper’s finite-time algorithms.The proposed methods are Kaczmarz (row-projection) methods and can compute weighted least-squares solutions.Source: §1.2 · p. 3

1.3 Our contributions

  • The paper proposes incremental and diffusive algorithms for minimum-variance estimation that converge in finite time using local information, and a distributed false-data detector.The detection method inherits the estimation method’s convergence properties and characterizes corrupted vectors guaranteed to be detected.Source: §1.3 · p. 3
  • A finite-memory approximation of the diffusive method offers each control center a good estimate of its subnetwork under stated assumptions, with error bounds independent of network dimension.The paper also illustrates its procedures on the IEEE 118-bus system.Source: §1.3 · p. 4

2 Problem setup and preliminary notions

The setup models a power-network state through noisy, approximately linear measurements and defines distributed estimation and false-data detection problems for cooperating monitors. It also states a detectability assumption and notes that monitor-count selection depends on the scenario.

State and measurement model

  • The network state comprises bus voltage angles and magnitudes, estimated from measurements such as voltages, currents, and power flows.The adopted measurement model linearizes a nonlinear relation around the origin; noise has zero mean and positive-definite covariance.Source: §2 · p. 4

State estimation problem

  • The minimum-variance estimate is a weighted least-squares solution, whose centralized computation requires complete knowledge of H and Σ and inversion of H^TΣ^-1H.The weighted residual is measured using Σ^-1.Source: §2 · p. 4

Distributed estimation problem

  • Monitors know only their local measurement and model blocks, so the stated estimation problem is to compute the network’s minimum-variance estimate through cooperation.The setup permits neighboring monitors to exchange information.Source: §2 · p. 5

False-data detection problem

  • False data is attacker-chosen measurement corruption, and the detection problem is distributed; vectors in Im(H) can evade detection, so the setup assumes the attack is detectable.For w in Im(H), x + H†w is a valid network state.Source: §2 · p. 5

Complexity and scope

  • Method complexity depends on state dimension and monitor count, while the paper leaves scenario-specific monitor-count selection open.The authors suggest fewer monitors absent severe computation or communication constraints, and more monitors otherwise.Source: §2 · p. 5

Generality

  • The methods are presented for power systems but are also applicable to state estimation and false-data detection in other settings, including sensor networks.The cited sensor-network example concerns dynamical systems.Source: §2 · p. 5

3 Optimal state estimation and false data detection via distributed computation

The section develops incremental and diffusive distributed procedures for minimum-variance state estimation, using local measurement information to reduce computation while retaining centralized estimation accuracy. It also applies the distributed estimate to false-data detection.

3 Optimal state estimation and false data detection via distributed computation

  • The procedures use only subsets of measurements, reduce computational burden, and retain the estimation accuracy of a centralized method.The incremental and diffusive approaches use different cooperation patterns; the detection procedure can use either estimate.Source: §3 · p. 6

3.1 Incremental solution to a set of linear equations

  • The incremental procedure returns the minimum-norm solution satisfying z = Hx̂, which is orthogonal to Ker(H).The final monitor returns the solution after the measurements are partitioned into row blocks.Source: §3.1 · p. 6
  • For k≤n, each monitor’s complexity is O(k^2n), versus O(mn^2) for the centralized computation; smaller blocks require more communication rounds.With k=m, no communication is needed; with k=1, m−1 rounds are required.Source: §3.1 · p. 7

3.2 Incremental state estimation via distributed computation

  • The incremental weighted-least-squares estimate equals the minimum-variance estimate only as ε→0+, and ε can be chosen to meet a desired accuracy.When ε is fixed, the estimates differ.Source: §3.2 · p. 8

3.3 Diffusive state estimation via distributed computation

  • On a connected monitor graph, every monitor computes x̂(ε) in d steps; as ε→0+, this approaches the minimum-variance estimate.Here d is the graph diameter.Source: §3.3 · pp. 9–10
  • With each monitor transmitting within every interval T, the asynchronous diffusive method computes x̂(ε) at each monitor within dT.The communication graph must be connected.Source: §3.3 · p. 10

3.4 Detection of false data via distributed computation

  • False data w is detectable when it changes the estimation residual beyond threshold Γ; lowering Γ increases failure sensitivity but also false alarms.The threshold depends on measurement-noise magnitude.Source: §3.4 · p. 11

4 A finite-memory estimation technique

Algorithm 2 lets monitors estimate only their local subnetworks using local-sized vectors. With a sufficiently sparse measurement matrix, local estimation error decays exponentially with iterations and becomes negligible before termination.

4 A finite-memory estimation technique

  • Algorithm 2 is designed to let each monitor use local, small-dimension vectors to estimate only a specified subnetwork.The paper notes that knowing only a subnetwork may suffice for distributed estimation and control strategies.Source: §4 · p. 11
  • The monitor-interconnection graph reflects the sparsity of the measurement matrix, with edges where either cross-region block is nonzero.Competence regions cover the network and may overlap.Source: §4 · p. 12
  • For sufficiently sparse H, each monitor’s local estimation error decays exponentially with iterations and becomes negligible before Algorithm 2 terminates.The stated bound applies for h < diam(Gh).Source: §4 · pp. 12, 14
  • As ε decreases, the estimate θbus(ε) converges to the minimum variance estimate θbus,wls.Figure 2(a) plots the normalized error between these estimates against ε.Source: Figure 2 · p. 14

5 An illustrative example

The examples examine distributed estimation and false-data detection on the IEEE 118-bus system, then illustrate finite-memory estimation on a 400-bus grid. They report increased estimation accuracy with additional measurements and negligible local error before Algorithm 2 terminates.

5 An illustrative example

  • The examples demonstrate the methods developed in the preceding sections, including the IEEE 118-bus case and the finite-memory scalability example.The section introduces these as illustrative examples of method effectiveness.Source: §5 · p. 14, §5.1 · p. 14, §5.2 · p. 15

5.1 Estimation and detection for the IEEE 118 bus system

  • For fixed ε, estimation error decreases with increasing measurements; the distributed algorithm handles up to 5 ¯ N measurements at the same or lower computational complexity than centralized estimation with ¯ N measurements.Source: Figure 3 · p. 15
  • The corrupted first-area injection is detected because its residual exceeds the threshold; residuals below threshold in areas 2–5 suggest regional identification.The injected error is uniformly distributed from 0 to wmax, approximately 10% of the nominal real injection value; identifying corrupted measurements is left for future research.Source: §5.1 · p. 15

5.2 Scalability property of our finite-memory estimation technique

  • On the 400-bus, 16-control-center grid, local estimation error becomes negligible before Algorithm 2 terminates.The example estimates regional voltage angles using real-power injections measured only within each region.Source: §5.2 · p. 16, Figure 4 · p. 16

6 Conclusion

The paper proposes two distributed methods for minimum-variance network-state estimation using local measurements and model knowledge. It also presents an asynchronous scalable implementation of the diffusive method and a distributed cyber-attack detection algorithm.

6 Conclusion

  • Two distributed algorithms compute the minimum variance estimate of network state from noisy measurements using only local measurements and model knowledge.The methods use incremental and diffusive cooperation, respectively, and both converge in finite time.Source: §6 · p. 16
  • The diffusive estimation method has an asynchronous, scalable implementation whose efficiency is supported by rigorous analysis and a practical example.Source: §6 · p. 16
  • The paper develops a distributed algorithm for detecting cyber-attacks against network measurements and characterizes its detection performance.Source: §6 · p. 16

APPENDIX

The appendix proves the incremental procedure’s stated invariant by induction and supplies supporting linear-algebra arguments for Theorem 3.2. Its displayed excerpts also state a kernel identity and outline how it is established.

6.1 Proof of Theorem 3.1

  • The proof of Theorem 3.1 proceeds by induction, maintaining that the estimate is orthogonal to the measurement matrix’s kernel and reproduces the measurements.The induction step uses consistency of the linear equations to establish the measurement relation for the next estimate.Source: §6.1 · pp. 16–17

6.2 Proof of Theorem 3.2

  • Lemma 6.1 states that Ker((H†)T) equals Ker(H), and its proof establishes both set inclusions using pseudoinverse identities.Source: §6.2 · p. 17
  • Theorem 3.2’s proof verifies intermediate equations using pseudoinverse manipulations and a sufficient condition, then concludes that the required equation holds.Source: §6.2 · pp. 17–18

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