# Optimal Power Flow in Multiphase Radial Networks with Delta Connections: Preprint

## Abstract

This paper focuses on multiphase radial distribution networks with mixed 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 generation units/loads in the OPF problem. The first model extends traditional branch flow models - and it is referred to as extended branch flow model (EBFM). The second model leverages a linear relationship between per-phase power injections and delta connections, which 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 SDP relaxations can be solved efficiently by a generic optimization solver. Numerical evidences indicate 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 also shown that the SDP solution under BVA has a small optimality gap, while the BVA model is accurate in the sense that it reflects 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:
- 1410967

- Report Number(s):
- NREL/CP-5D00-67852

- DOE Contract Number:
- AC36-08GO28308

- Resource Type:
- Conference

- Resource Relation:
- Conference: Presented at IREP 2017 Bulk Power Systems Dynamics and Control Symposium, 27 August - 1 September 2017, Espinho, Portugal

- Country of Publication:
- United States

- Language:
- English

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

### Citation Formats

```
Zhao, Changhong, Dall-Anese, Emiliano, and Low, Steven H.
```*Optimal Power Flow in Multiphase Radial Networks with Delta Connections: Preprint*. United States: N. p., 2017.
Web.

```
Zhao, Changhong, Dall-Anese, Emiliano, & Low, Steven H.
```*Optimal Power Flow in Multiphase Radial Networks with Delta Connections: Preprint*. United States.

```
Zhao, Changhong, Dall-Anese, Emiliano, and Low, Steven H. Mon .
"Optimal Power Flow in Multiphase Radial Networks with Delta Connections: Preprint". United States.
doi:. https://www.osti.gov/servlets/purl/1410967.
```

```
@article{osti_1410967,
```

title = {Optimal Power Flow in Multiphase Radial Networks with Delta Connections: Preprint},

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

abstractNote = {This paper focuses on multiphase radial distribution networks with mixed 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 generation units/loads in the OPF problem. The first model extends traditional branch flow models - and it is referred to as extended branch flow model (EBFM). The second model leverages a linear relationship between per-phase power injections and delta connections, which 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 SDP relaxations can be solved efficiently by a generic optimization solver. Numerical evidences indicate 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 also shown that the SDP solution under BVA has a small optimality gap, while the BVA model is accurate in the sense that it reflects actual system voltages.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Mon Nov 27 00:00:00 EST 2017},

month = {Mon Nov 27 00:00:00 EST 2017}

}