# Convex Relaxation of OPF in Multiphase Radial Networks with Wye and Delta Connections

## Abstract

This panel presentation focuses on multiphase radial distribution networks with wye and delta connections, and proposes a semidefinite relaxation of the AC optimal power flow (OPF) problem. Two multiphase power flow models are developed to facilitate the integration of delta-connected loads or generation resources in the OPF problem. The first model is referred to as the extended branch flow model (EBFM). The second model leverages a linear relationship between phase-to-ground power injections and delta connections that holds under a balanced voltage approximation (BVA). Based on these models, pertinent OPF problems are formulated and relaxed to semidefinite programs (SDPs). Numerical studies on IEEE test feeders show that the proposed SDP relaxations can be solved efficiently by a generic optimization solver. Numerical evidence also indicates that solving the resultant SDP under BVA is faster than under EBFM. Moreover, both SDP solutions are numerically exact with respect to voltages and branch flows. It is further shown that the SDP solution under BVA has a small optimality gap, and the BVA model is accurate in the sense that it reproduces actual system voltages.

- Authors:

- National Renewable Energy Laboratory (NREL), Golden, CO (United States)
- California Institute of Technology

- Publication Date:

- Research Org.:
- National Renewable Energy Lab. (NREL), Golden, CO (United States)

- Sponsoring Org.:
- USDOE Advanced Research Projects Agency-Energy (ARPA-E)

- OSTI Identifier:
- 1374123

- Report Number(s):
- NREL/PR-5D00-68859

- DOE Contract Number:
- AC36-08GO28308

- Resource Type:
- Conference

- Resource Relation:
- Conference: Presented at the 2017 IEEE Power & Energy Society (PES) General Meeting, 16-20 July 2017, Chicago, Illinois

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 24 POWER TRANSMISSION AND DISTRIBUTION; optimal power flow; multiphase distribution network; convex relaxation

### Citation Formats

```
Zhao, Changhong, Dall-Anese, Emiliano, and Low, Steven.
```*Convex Relaxation of OPF in Multiphase Radial Networks with Wye and Delta Connections*. United States: N. p., 2017.
Web.

```
Zhao, Changhong, Dall-Anese, Emiliano, & Low, Steven.
```*Convex Relaxation of OPF in Multiphase Radial Networks with Wye and Delta Connections*. United States.

```
Zhao, Changhong, Dall-Anese, Emiliano, and Low, Steven. Tue .
"Convex Relaxation of OPF in Multiphase Radial Networks with Wye and Delta Connections". United States.
doi:. https://www.osti.gov/servlets/purl/1374123.
```

```
@article{osti_1374123,
```

title = {Convex Relaxation of OPF in Multiphase Radial Networks with Wye and Delta Connections},

author = {Zhao, Changhong and Dall-Anese, Emiliano and Low, Steven},

abstractNote = {This panel presentation focuses on multiphase radial distribution networks with wye and delta connections, and proposes a semidefinite relaxation of the AC optimal power flow (OPF) problem. Two multiphase power flow models are developed to facilitate the integration of delta-connected loads or generation resources in the OPF problem. The first model is referred to as the extended branch flow model (EBFM). The second model leverages a linear relationship between phase-to-ground power injections and delta connections that holds under a balanced voltage approximation (BVA). Based on these models, pertinent OPF problems are formulated and relaxed to semidefinite programs (SDPs). Numerical studies on IEEE test feeders show that the proposed SDP relaxations can be solved efficiently by a generic optimization solver. Numerical evidence also indicates that solving the resultant SDP under BVA is faster than under EBFM. Moreover, both SDP solutions are numerically exact with respect to voltages and branch flows. It is further shown that the SDP solution under BVA has a small optimality gap, and the BVA model is accurate in the sense that it reproduces actual system voltages.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Tue Aug 01 00:00:00 EDT 2017},

month = {Tue Aug 01 00:00:00 EDT 2017}

}