Three-phase formulation details

StandardACPForm

Real-valued formulation from:

• Formulation without shunts: Mahdad, B., Bouktir, T., & Srairi, K. (2006). A three-phase power flow modelization: a tool for optimal location and control of FACTS devices in unbalanced power systems. In IEEE Industrial Electronics IECON (pp. 2238–2243).

StandardDCPForm

Applying all of the standard DC linearization tricks to the StandardACPForm

SOCWRForm

Applying the standard BIM voltage cross-product (sine and cosine) substitution tricks to StandardACPForm results immediately in a SOC formulation.

SDPUBFForm

The BFM SDP relaxation as described in:

• Gan, L., & Low, S. H. (2014). Convex relaxations and linear approximation for optimal power flow in multiphase radial networks. In PSSC (pp. 1–9). Wroclaw, Poland. https://doi.org/10.1109/PSCC.2014.7038399

Note that this formulation is complex-valued and additional steps are needed to implement this in JuMP.

SOCNLPUBFForm

The starting point is SDPUBFForm. The SDP constraint can be relaxed to a set of SOC constraints, starting from either the real or complex form of the matrix on which the PSD-ness constraint is applied.

• Kim, S., Kojima, M., & Yamashita, M. (2003). Second order cone programming relaxation of a positive semidefinite constraint. Optimization Methods and Software, 18(5), 535–541. https://doi.org/10.1080/1055678031000148696
• Andersen, M. S., Hansson, A., & Vandenberghe, L. (2014). Reduced-complexity semidefinite relaxations of optimal power flow problems. IEEE Trans. Power Syst., 29(4), 1855–1863.

SOCConicUBFForm

See SOCNLPUBFForm

LPfullUBFForm

Matrix formulation that generalizes simplified DistFlow equations, as introduced in :

• Gan, L., & Low, S. H. (2014). Convex relaxations and linear approximation for optimal power flow in multiphase radial networks. In PSSC (pp. 1–9). Wroclaw, Poland. https://doi.org/10.1109/PSCC.2014.7038399

Note that this formulation is complex-valued and additional steps are needed to implement this in JuMP.

LPdiagUBFForm

This formulation has originally been developed by Sankur et al.

• Sankur, M. D., Dobbe, R., Stewart, E., Callaway, D. S., & Arnold, D. B. (2016). A linearized power flow model for optimization in unbalanced distribution systems. https://arxiv.org/abs/1606.04492v2

This formulation is here cast as only considering the diagonal elements defined in LPfullUBFForm, which furthermore leads to the imaginary part of the lifted node voltage variable W being redundant and substituted out.

LPLinUBFForm

Scalar reformulation of:

• Sankur, M. D., Dobbe, R., Stewart, E., Callaway, D. S., & Arnold, D. B. (2016). A linearized power flow model for optimization in unbalanced distribution systems. https://arxiv.org/abs/1606.04492v2

This formulation was already derived in real variables and parameters.