A Study on the Sensitivity Matrix in Power

System State Estimation by Using Sparse

Principal Component Analysis

Adam Molin1 Henrik Sandberg1 Magnus Johansson2

1KTH Royal Institute of Technology – Department of Automatic Control

2Svenska Kraftnät (Swedish National Grid)

Future Electric Power Systems and the Energy Transition,

Champéry, Feb 5-9, 2017

Motivation

• Grid more frequently at operational limits due to increased

power transfers

• Uncertainties in measurements and network parameters

always present in the state estimator of the Energy

Management System (EMS)

• Better understanding of the data quality inevitable for

driving the power system closer to its limits

Objective

Identify the most relevant data components

(measurements & line parameters) for power

system state estimation

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Traditional approach

• Identification of bad analog devices and parameter estimation

performed separately

• Assumes one category is “better”, and that quality

problems are mainly in the other category

• Accuracy of the two input categories is practically

impossible to quantify

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How can we analyze the impact on the state estimation

quality when uncertainties in both telemetry and

parameters are present?

Background

power system

state estimator

data input estimate

cost

Data input a to power system state estimator:

Network model (power flow equations):

Weighted least squares:

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power system

state estimator

perturbed data perturbed estimate

cost

General approach

Methodology: Sparse PCA on (a variant of) sensitivity matrix

Sensitivity Analysis with Sparsity Constraint

Find the set 𝐷 of data inputs with 𝐷 ≤ 𝑡, whose joint

perturbations impact the state estimate most.

Data input

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Outline

• Computation of sensitivity matrix 𝑺

• Sparse Principal Component Analysis of 𝑺

• Structure of spectral properties of 𝑺

• Numerical illustration of approach

• Conclusion

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Computation of sensitivity matrix

Necessary condition for optimality

Sensitivity equation

Sensitivity matrix:

Method of feasible directions [1]: Compute the

modifications (directions) of the optimal state estimate when

there is a perturbation of the input data, such that the

linearized necessary conditions are still valid.

[1] R. Minguez, A.J. Conejo, “State estimation sensitivity analysis,” Power Systems,

IEEE Trans. on, 2007

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Modified sensitivity matrix 𝒅𝒛 /𝒅𝒂

• Facilitates interpretability ( common in power sys.)

• Normalization with covariance enables the

comparison of entries

state

estimator

perturbed

measurement

estimate perturbed data

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Sparse Principal Component Analysis on 𝑺

Conduct sparse PCA on sensitivity matrix 𝑺 [2]:

Loading vector encodes relevant inputs Non-zero entries in 𝒗 are interpreted as crucial data

inputs for state estimation

Deflate matrix:

Normalized inputs

Rerun after deflation

Constraint on

number of inputs

Maximize impact on output

[2] A. d’Aspremont, et al., “A direct formulation for sparse PCA using semidef.

program.,” SIAM review, 2007

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Two Extreme Choices of Sparsity Parameter 𝒕

Take and replace objective with

returns entry with maximum absolute value.

– Only shows impact of single input/single output

Omit cardinality constraint

returns singular value (SVD)

– upper bound

– non-structured

Or

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Algorithms for sparse PCA

1. SDP-SPCA:

Semi-definite programming relaxation technique [2]

2. Soft thresholding:

ℓ1-regularized PCA inspired by LASSO [3]

[2] A. d’Aspremont, et al., “A direct formulation for sparse PCA using

semidef. program.,” SIAM review, 2007

[3] H. Zou, et al., “Sparse principal component analysis,” Journal of

computational and graphical statistics, 2006.

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SDP-SPCA [2]

Define

SDP relaxation:

Omit rank constraint

ℓ1-relaxation of cardinality constraint

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[2] A. d’Aspremont, et al., “A direct formulation for sparse PCA using semidef.

program.,” SIAM review, 2007

Structure of spectral properties of 𝑺

Calculation of 𝑺 reveals the form:

with

and the Jacobians

(projection matrix)

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Proposition 1:

Let be the SVD with .

Assume that has no singular value at 1.

Then, for any positive singular value, we have .

Structure of spectral properties of 𝑺

From the structure of 𝑺, we have

Same holds for

– Cardinality constrained PCA

– Soft thresholding algorithm

Target set for loading vectors: (either belong to measurement inputs or to line parameters)

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• Consider only measurement perturbations:

(Sensitivity matrix idempotent and symmetric: 𝑆 = Π = ΠTΠ)

• 𝐼 is a critical set of measurements ( 𝐼 = 𝑡 ) if

(Loss of observability)

(Converse statement exists.)

Relation to critical measurements

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Proposition 2: Suppose 𝑣∗ solving (1) satisfies 𝑣∗𝑇Π𝑣∗ = 1.

Then the set of measurements corresponding to the 𝑡 ≤ 𝑡 non-

zero entries in 𝑣∗ is a critical set.

Outline

• Computation of sensitivity matrix 𝑺

• Sparse Principal Component Analysis of 𝑺

• Structure of spectral properties of 𝑺

• Numerical illustration of approach

• Conclusion

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Numerical illustration of approach

• Input data:

20 measurements

2 x 11 branch parameters

(resistance & reactance)

Sensitivity matrix: 20 x 42 – matrix

6-bus system

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Spectral profile of sparse PCA

Adjusted variance (squared singular values, add up to 1)

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PCA component

Numerical illustration of approach

Results from SVD: Sparsity pattern of 𝑉𝑇

Separation into measurement & line parameter inputs

(as stated in Proposition 1)

Otherwise dense structure

singular value > 1

singular value ≤ 1

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Numerical illustration of approach

Results from SDP-SPCA: Sparsity pattern of 𝑉𝑇

Separation into measurement & line parameter inputs

(as stated in Proposition 1)

Sparse structure (𝑡 = 2.5 and 𝑡 = 2)

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Numerical illustration of approach

Results from Soft-Thresholding: Sparsity pattern of 𝑉𝑇

Separation into measurement & line parameter inputs

(as stated in Proposition 1)

Sparse structure (tuned to return 4 entries)

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Interpretation of sparsity pattern

Perturbation of these 3 reactances may lead to 1.44 amplification

of estimation error

If branch parameter variation less than 6.4%, the measurement

data loading vectors dominate the parameter loading vectors

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Comparison with simulated sensitivities

Predicted (from analytical model) vs simulated (Monte Carlo)

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IEEE 24-bus system

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Sensitivity matrix: 81 x 153 – matrix

Singular values

IEEE 24-bus system – Sparse loading vectors (Soft-Thresholding)

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Conclusion

Analysis tool for identifying relevant components for power system state estimation – Valuable for model & sensor calibration

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Approach concatenates sensitivity analysis & sparse PCA

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Reveals information to the Energy Management System (EMS) that is relevant for bad data detection and cyber security studies

Future work:

• Sparsity constraint on principal component to reflect concentration of perturbation impact

• Numerical tractability for large-scale power networks

Reference: “A Study on the Sensitivity Matrix in Power System State Estimation by Using Sparse Principal Component Analysis,” IEEE CDC, 2016

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