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January  2020, 16(1): 431-443. doi: 10.3934/jimo.2018161

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

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

* Corresponding author: Stéphane Chrétien

Received  June 2017 Revised  June 2018 Published  November 2018

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.

Citation: Stephane Chretien, Paul Clarkson. A fast algorithm for the semi-definite relaxation of the state estimation problem in power grids. Journal of Industrial & Management Optimization, 2020, 16 (1) : 431-443. doi: 10.3934/jimo.2018161
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##### References:
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.
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.
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.
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.
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.
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.
 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
 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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