# AMPS: An Augmented Matrix Formulation for Principal Submatrix Updates with Application to Power Grids

## Abstract

We present an augmented matrix approach to update the solution to a linear system of equations when the coefficient matrix is modified by a few elements within a principal submatrix. This problem arises in the dynamic security analysis of a power grid, where operators need to perform $N-x$ contingency analysis, i.e., determine the state of the system when up to $x$ links from $N$ fail. Our algorithms augment the coefficient matrix to account for the changes in it, and then compute the solution to the augmented system without refactoring the modified matrix. We provide two algorithms, a direct method, and a hybrid direct-iterative method for solving the augmented system. We also exploit the sparsity of the matrices and vectors to accelerate the overall computation. Our algorithms are compared on three power grids with PARDISO, a parallel direct solver, and CHOLMOD, a direct solver with the ability to modify the Cholesky factors of the coefficient matrix. We show that our augmented algorithms outperform PARDISO (by two orders of magnitude), and CHOLMOD (by a factor of up to 5). Further, our algorithms scale better than CHOLMOD as the number of elements updated increases. The solutions are computed with high accuracy. Our algorithms are capable of computing $N-x$ contingency analysis on a $778K$ bus grid, updating a solution with $x=20$ elements in $$1.6 \times 10^{-2}$$ seconds on an Intel Xeon processor.

- Authors:

- Publication Date:

- Research Org.:
- Pacific Northwest National Lab. (PNNL), Richland, WA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1416692

- Report Number(s):
- PNNL-SA-119762

Journal ID: ISSN 1095-7197; KJ0401000

- DOE Contract Number:
- AC05-76RL01830

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: SIAM Journal on Scientific Computing; Journal Volume: 39; Journal Issue: 5; Related Information: pages S809-S827

- Country of Publication:
- United States

- Language:
- English

### Citation Formats

```
Yeung, Yu-Hong, Pothen, Alex, Halappanavar, Mahantesh, and Huang, Zhenyu.
```*AMPS: An Augmented Matrix Formulation for Principal Submatrix Updates with Application to Power Grids*. United States: N. p., 2017.
Web. doi:10.1137/16M1082755.

```
Yeung, Yu-Hong, Pothen, Alex, Halappanavar, Mahantesh, & Huang, Zhenyu.
```*AMPS: An Augmented Matrix Formulation for Principal Submatrix Updates with Application to Power Grids*. United States. doi:10.1137/16M1082755.

```
Yeung, Yu-Hong, Pothen, Alex, Halappanavar, Mahantesh, and Huang, Zhenyu. Mon .
"AMPS: An Augmented Matrix Formulation for Principal Submatrix Updates with Application to Power Grids". United States.
doi:10.1137/16M1082755.
```

```
@article{osti_1416692,
```

title = {AMPS: An Augmented Matrix Formulation for Principal Submatrix Updates with Application to Power Grids},

author = {Yeung, Yu-Hong and Pothen, Alex and Halappanavar, Mahantesh and Huang, Zhenyu},

abstractNote = {We present an augmented matrix approach to update the solution to a linear system of equations when the coefficient matrix is modified by a few elements within a principal submatrix. This problem arises in the dynamic security analysis of a power grid, where operators need to perform $N-x$ contingency analysis, i.e., determine the state of the system when up to $x$ links from $N$ fail. Our algorithms augment the coefficient matrix to account for the changes in it, and then compute the solution to the augmented system without refactoring the modified matrix. We provide two algorithms, a direct method, and a hybrid direct-iterative method for solving the augmented system. We also exploit the sparsity of the matrices and vectors to accelerate the overall computation. Our algorithms are compared on three power grids with PARDISO, a parallel direct solver, and CHOLMOD, a direct solver with the ability to modify the Cholesky factors of the coefficient matrix. We show that our augmented algorithms outperform PARDISO (by two orders of magnitude), and CHOLMOD (by a factor of up to 5). Further, our algorithms scale better than CHOLMOD as the number of elements updated increases. The solutions are computed with high accuracy. Our algorithms are capable of computing $N-x$ contingency analysis on a $778K$ bus grid, updating a solution with $x=20$ elements in $1.6 \times 10^{-2}$ seconds on an Intel Xeon processor.},

doi = {10.1137/16M1082755},

journal = {SIAM Journal on Scientific Computing},

number = 5,

volume = 39,

place = {United States},

year = {Mon Oct 09 00:00:00 EDT 2017},

month = {Mon Oct 09 00:00:00 EDT 2017}

}