# Optimal Power Flow of Radial Networks and Its Variations: A Sequential Convex Optimization Approach

## Abstract

This paper proposes a sequential convex optimization method to solve broader classes of optimal power flow (OPF) problems over radial networks. The non-convex branch power flow equation is decomposed as a second-order cone inequality and a non-convex constraint involving the difference of two convex functions. Provided with an initial solution offered by an inexact second-order cone programming relaxation model, this approach solves a sequence of convexified penalization problems, where concave terms are approximated by linear functions and updated in each iteration. It could recover a feasible power flow solution, which usually appears to be very close, if not equal, to the global optimal one. Two variations of the OPF problem, in which non-cost related objectives are optimized subject to power flow constraints and the convex relaxation is generally inexact, are elaborated in detail. One is the maximum loadability problem, which is formulated as a special OPF problem that seeks the maximal distance to the boundary of power flow insolvability. The proposed method is shown to outperform commercial nonlinear solvers in terms of robustness and efficiency. The other is the bi-objective OPF problem. A non-parametric scalarization model is suggested, and is further reformulated as an extended OPF problem by convexifying themore »

- Authors:

- Publication Date:

- Research Org.:
- Argonne National Lab. (ANL), Argonne, IL (United States)

- Sponsoring Org.:
- National Natural Science Foundation of China (NNSFC); National Science Foundation (NSF); USDOE Office of Energy Efficiency and Renewable Energy (EERE)

- OSTI Identifier:
- 1463673

- DOE Contract Number:
- AC02-06CH11357

- Resource Type:
- Journal Article

- Journal Name:
- IEEE Transactions on Smart Grid

- Additional Journal Information:
- Journal Volume: 8; Journal Issue: 6; Journal ID: ISSN 1949-3053

- Publisher:
- IEEE

- Country of Publication:
- United States

- Language:
- English

- Subject:
- bi-objective optimization; convex optimization; difference-of-convex programming; maximum loadability; optimal power flow; radial network

### Citation Formats

```
Wei, Wei, Wang, Jianhui, Li, Na, and Mei, Shengwei.
```*Optimal Power Flow of Radial Networks and Its Variations: A Sequential Convex Optimization Approach*. United States: N. p., 2017.
Web. doi:10.1109/tsg.2017.2684183.

```
Wei, Wei, Wang, Jianhui, Li, Na, & Mei, Shengwei.
```*Optimal Power Flow of Radial Networks and Its Variations: A Sequential Convex Optimization Approach*. United States. doi:10.1109/tsg.2017.2684183.

```
Wei, Wei, Wang, Jianhui, Li, Na, and Mei, Shengwei. Wed .
"Optimal Power Flow of Radial Networks and Its Variations: A Sequential Convex Optimization Approach". United States. doi:10.1109/tsg.2017.2684183.
```

```
@article{osti_1463673,
```

title = {Optimal Power Flow of Radial Networks and Its Variations: A Sequential Convex Optimization Approach},

author = {Wei, Wei and Wang, Jianhui and Li, Na and Mei, Shengwei},

abstractNote = {This paper proposes a sequential convex optimization method to solve broader classes of optimal power flow (OPF) problems over radial networks. The non-convex branch power flow equation is decomposed as a second-order cone inequality and a non-convex constraint involving the difference of two convex functions. Provided with an initial solution offered by an inexact second-order cone programming relaxation model, this approach solves a sequence of convexified penalization problems, where concave terms are approximated by linear functions and updated in each iteration. It could recover a feasible power flow solution, which usually appears to be very close, if not equal, to the global optimal one. Two variations of the OPF problem, in which non-cost related objectives are optimized subject to power flow constraints and the convex relaxation is generally inexact, are elaborated in detail. One is the maximum loadability problem, which is formulated as a special OPF problem that seeks the maximal distance to the boundary of power flow insolvability. The proposed method is shown to outperform commercial nonlinear solvers in terms of robustness and efficiency. The other is the bi-objective OPF problem. A non-parametric scalarization model is suggested, and is further reformulated as an extended OPF problem by convexifying the objective function. It provides a single trade-off solution without any subjective preference. The proposed computation framework also helps retrieve the Pareto front of the bi-objective OPF via the e-constraint method or the normal boundary intersection method. This paper also discusses extensions for OPF problems over meshed networks based on the semidefinite programming relaxation method.},

doi = {10.1109/tsg.2017.2684183},

journal = {IEEE Transactions on Smart Grid},

issn = {1949-3053},

number = 6,

volume = 8,

place = {United States},

year = {2017},

month = {11}

}