### Abstract

Obtaining the convex hull of the alternating current (AC) power flow equations is largely beneficial in the convexification of the optimization problems in power systems where the AC power flow constraints are taken into account. In this paper, the AC power flow formulation in rectangular coordinates which is a non-convex quadratic equality is equivalently decomposed into four quadratic inequalities whose coefficient matrices are positive semidefinite. Two of the quadratic inequalities are convex while the other two are not. The proposed decomposition facilitates the convexification of the power flow equations much. A convex hull formulation is proposed for the resulting non-convex quadratic inequalities. This formulation is applied to study the convex hull of the AC power flow.

Original language | English (US) |
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Title of host publication | 2016 IEEE Power and Energy Society General Meeting, PESGM 2016 |

Publisher | IEEE Computer Society |

Volume | 2016-November |

ISBN (Electronic) | 9781509041688 |

DOIs | |

State | Published - Nov 10 2016 |

Event | 2016 IEEE Power and Energy Society General Meeting, PESGM 2016 - Boston, United States Duration: Jul 17 2016 → Jul 21 2016 |

### Other

Other | 2016 IEEE Power and Energy Society General Meeting, PESGM 2016 |
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Country | United States |

City | Boston |

Period | 7/17/16 → 7/21/16 |

### Fingerprint

### Keywords

- AC power flow
- Convex hull
- Convex relaxation
- Non-convex quadratic equality
- Rectangular coordinates

### ASJC Scopus subject areas

- Energy Engineering and Power Technology
- Nuclear Energy and Engineering
- Renewable Energy, Sustainability and the Environment
- Electrical and Electronic Engineering

### Cite this

*2016 IEEE Power and Energy Society General Meeting, PESGM 2016*(Vol. 2016-November). [7741777] IEEE Computer Society. https://doi.org/10.1109/PESGM.2016.7741777

**The convex hull of the AC power flow equations in rectangular coordinates.** / Li, Qifeng; Vittal, Vijay.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*2016 IEEE Power and Energy Society General Meeting, PESGM 2016.*vol. 2016-November, 7741777, IEEE Computer Society, 2016 IEEE Power and Energy Society General Meeting, PESGM 2016, Boston, United States, 7/17/16. https://doi.org/10.1109/PESGM.2016.7741777

}

TY - GEN

T1 - The convex hull of the AC power flow equations in rectangular coordinates

AU - Li, Qifeng

AU - Vittal, Vijay

PY - 2016/11/10

Y1 - 2016/11/10

N2 - Obtaining the convex hull of the alternating current (AC) power flow equations is largely beneficial in the convexification of the optimization problems in power systems where the AC power flow constraints are taken into account. In this paper, the AC power flow formulation in rectangular coordinates which is a non-convex quadratic equality is equivalently decomposed into four quadratic inequalities whose coefficient matrices are positive semidefinite. Two of the quadratic inequalities are convex while the other two are not. The proposed decomposition facilitates the convexification of the power flow equations much. A convex hull formulation is proposed for the resulting non-convex quadratic inequalities. This formulation is applied to study the convex hull of the AC power flow.

AB - Obtaining the convex hull of the alternating current (AC) power flow equations is largely beneficial in the convexification of the optimization problems in power systems where the AC power flow constraints are taken into account. In this paper, the AC power flow formulation in rectangular coordinates which is a non-convex quadratic equality is equivalently decomposed into four quadratic inequalities whose coefficient matrices are positive semidefinite. Two of the quadratic inequalities are convex while the other two are not. The proposed decomposition facilitates the convexification of the power flow equations much. A convex hull formulation is proposed for the resulting non-convex quadratic inequalities. This formulation is applied to study the convex hull of the AC power flow.

KW - AC power flow

KW - Convex hull

KW - Convex relaxation

KW - Non-convex quadratic equality

KW - Rectangular coordinates

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U2 - 10.1109/PESGM.2016.7741777

DO - 10.1109/PESGM.2016.7741777

M3 - Conference contribution

VL - 2016-November

BT - 2016 IEEE Power and Energy Society General Meeting, PESGM 2016

PB - IEEE Computer Society

ER -