A fast algorithm for the semi-definite relaxation of the state estimation problem in power grids

Stephane Chretien; Paul Clarkson

doi:10.3934/jimo.2018161

Article Contents

2020, Volume 16, Issue 1: 431-443. Doi: 10.3934/jimo.2018161

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A fast algorithm for the semi-definite relaxation of the state estimation problem in power grids

Stephane Chretien and
Paul Clarkson

National Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW, UK

* Corresponding author: Stéphane Chrétien

Received: May 31, 2017

Revised: May 31, 2018

Early access: November 2018

Published: October 2019

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Abstract

State estimation in power grids is a crucial step for monitoring and control tasks. It was shown that the state estimation problem can be solved using a convex relaxation based on semi-definite programming. In the present paper, we propose a fast algorithm for solving this relaxation. Our approach uses the Bürer Monteiro factorisation is a special way that solves the problem on the sphere and and estimates the scale in a Gauss-Seidel fashion. Simulations results confirm the promising behavior of the method.

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Mathematics Subject Classification: Primary: 93B, Secondary: 90-08.

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Figure 1. Comparison of Sum of Squared Errors for the IEEE-30 network: New method vs. SDP relaxation (using YALMIP) with noise standard deviation equal to.2 when power is observed at half the number of buses chosen at random.

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Figure 2. Comparison of computation times for the IEEE-30 network: New method vs. SDP relaxation (using YALMPI) with noise standard deviation equal to.2 when power is observed at half the number of buses chosen uniformly at random.

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Figure 3. Example of evolution of the objective function as a function of iteration number for one realisation of a random noise for the IEEE-30 network.

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Figure 4. Example of evolution of the euclidean distance between successive $A$-iterates as a function of iteration number for one realisation of a random noise for the IEEE-30 network.

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Figure 5. Mean Squared Error obtained using the estimator based on the new method with noise standard deviation equal to.2 when power is observed at half the buses. The buses selected for observation were selected uniformly at random.

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Figure 6. Computation times using the new method with noise standard deviation equal to.2 when power is observed at half the buses. The buses selected for observation were selected uniformly at random.

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Result:

$W_{opt}$

Choose $A^{(1,1)} \in \mathbb C^{n\times k}$

$\underline {First\;stage} $

$\begin{array}{l} {\bf{while}}\;s \le S - 1\;{\bf{do}}\\ \;\left| \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\nabla g(A) = 2\;\sum\limits_{l = 1}^L \; ( - {z_l}\;\alpha (H_l^* + {H_l})A\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + 2\;{\alpha ^2}\;{\rm{trace}}({H_l}A{A^*})(H_l^* + {H_l})A).\;\;\;\;\;\;\;\;\;\;\left( 8 \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{{\tilde A}^{(t,s + 1)}} = {A^{(t,s)}} - \eta \nabla g({A^{(t,s)}})\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( 9 \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{A^{(t,s + 1)}} = \frac{1}{{\left\| {{{\tilde A}^{(t,s + 1)}}} \right\|}}\;{{\tilde A}^{(t,s + 1)}}.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {10} \right) \end{array} \right.\\ {\bf{end}} \end{array}$

Set $A^{(t+1,1)}=A^{(t,S)}$.

$\underline {Second\;stage}$

Set

$\begin{align} W_{opt}&= \alpha^{(t+1)} \ A^{(t+1,1)}A^{(t+1,1)^*} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {11} \right) \end{align}$

with
\begin{align} \alpha^{(t+1)} & = \frac{\sum\nolimits_{l=1}^L z_l\ \textrm{trace }(H_l A^{(t+1, 1)}A^{(t+1, 1)^*})}{\sum\nolimits_{l=1}^L \left(\textrm{trace }(H_l A^{(t+1, 1)}A^{(t+1, 1)^*})\right)^2} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \left({12} \right) \end{align}
Algorithm 1: The two stage optimisation procedure

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