# Graphical models for optimal power flow

## Abstract

Optimal power flow (OPF) is the central optimization problem in electric power grids. Although solved routinely in the course of power grid operations, it is known to be strongly NP-hard in general, and weakly NP-hard over tree networks. In this paper, we formulate the optimal power flow problem over tree networks as an inference problem over a tree-structured graphical model where the nodal variables are low-dimensional vectors. We adapt the standard dynamic programming algorithm for inference over a tree-structured graphical model to the OPF problem. Combining this with an interval discretization of the nodal variables, we develop an approximation algorithm for the OPF problem. Further, we use techniques from constraint programming (CP) to perform interval computations and adaptive bound propagation to obtain practically efficient algorithms. Compared to previous algorithms that solve OPF with optimality guarantees using convex relaxations, our approach is able to work for arbitrary tree-structured distribution networks and handle mixed-integer optimization problems. Further, it can be implemented in a distributed message-passing fashion that is scalable and is suitable for “smart grid” applications like control of distributed energy resources. In conclusion, numerical evaluations on several benchmark networks show that practical OPF problems can be solved effectively using this approach.

- Authors:

- California Inst. of Technology (CalTech), Pasadena, CA (United States)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Univ. of Michigan, Ann Arbor, MI (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE Office of Electricity Delivery and Energy Reliability (OE); USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1345156

- Report Number(s):
- LA-UR-16-24896

Journal ID: ISSN 1383-7133

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Constraints

- Additional Journal Information:
- Journal Volume: 22; Journal Issue: 1; Journal ID: ISSN 1383-7133

- Publisher:
- Springer

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 24 POWER TRANSMISSION AND DISTRIBUTION; 97 MATHEMATICS AND COMPUTING; Computer Science; Energy Sciences; Information Science

### Citation Formats

```
Dvijotham, Krishnamurthy, Chertkov, Michael, Van Hentenryck, Pascal, Vuffray, Marc, and Misra, Sidhant.
```*Graphical models for optimal power flow*. United States: N. p., 2016.
Web. doi:10.1007/s10601-016-9253-y.

```
Dvijotham, Krishnamurthy, Chertkov, Michael, Van Hentenryck, Pascal, Vuffray, Marc, & Misra, Sidhant.
```*Graphical models for optimal power flow*. United States. doi:10.1007/s10601-016-9253-y.

```
Dvijotham, Krishnamurthy, Chertkov, Michael, Van Hentenryck, Pascal, Vuffray, Marc, and Misra, Sidhant. Tue .
"Graphical models for optimal power flow". United States.
doi:10.1007/s10601-016-9253-y. https://www.osti.gov/servlets/purl/1345156.
```

```
@article{osti_1345156,
```

title = {Graphical models for optimal power flow},

author = {Dvijotham, Krishnamurthy and Chertkov, Michael and Van Hentenryck, Pascal and Vuffray, Marc and Misra, Sidhant},

abstractNote = {Optimal power flow (OPF) is the central optimization problem in electric power grids. Although solved routinely in the course of power grid operations, it is known to be strongly NP-hard in general, and weakly NP-hard over tree networks. In this paper, we formulate the optimal power flow problem over tree networks as an inference problem over a tree-structured graphical model where the nodal variables are low-dimensional vectors. We adapt the standard dynamic programming algorithm for inference over a tree-structured graphical model to the OPF problem. Combining this with an interval discretization of the nodal variables, we develop an approximation algorithm for the OPF problem. Further, we use techniques from constraint programming (CP) to perform interval computations and adaptive bound propagation to obtain practically efficient algorithms. Compared to previous algorithms that solve OPF with optimality guarantees using convex relaxations, our approach is able to work for arbitrary tree-structured distribution networks and handle mixed-integer optimization problems. Further, it can be implemented in a distributed message-passing fashion that is scalable and is suitable for “smart grid” applications like control of distributed energy resources. In conclusion, numerical evaluations on several benchmark networks show that practical OPF problems can be solved effectively using this approach.},

doi = {10.1007/s10601-016-9253-y},

journal = {Constraints},

number = 1,

volume = 22,

place = {United States},

year = {Tue Sep 13 00:00:00 EDT 2016},

month = {Tue Sep 13 00:00:00 EDT 2016}

}

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