SOC-OPF

The SOC-OPF model considered in PGLearn is presented below.

Definitions

The second order cone and rotated second-order cone of order $n$ are defined as

\[\begin{align*} \mathcal{Q}^{n} &= \left\{ x \in \mathbb{R}^{n} \ \middle| \ x_{1} \geq \sqrt{x_{2}^{2} + ... + x_{n}^{2}} \right\}\\ \mathcal{Q}_{r}^{n} &= \left\{ x \in \mathbb{R}^{n} \ \middle| \ 2 x_{1} x_{2} \geq x_{3}^{2} + ... + x_{n}^{2}, x_{1}, x_{2} \geq 0 \right\}. \end{align*}\]

Mathematical formulation

The SOC-OPF formulation in PGLearn is the Jabr relaxation of AC-OPF. The formulation is obtained through the change of variable

\[\begin{align*} \wm_{i} &= \VM_{i}^{2} && \forall i \in \NODES \\ \wr_{e} &= \VM_{i} \VM_{j} \cos (\theta_{i} - \theta_{j}) && \forall e = (i, j) \in \EDGES \\ \wi_{e} &= \VM_{i} \VM_{j} \sin (\theta_{i} - \theta_{j}) && \forall e = (i, j) \in \EDGES \end{align*}\]

Note that $\wr_{e}$ and $\wi_{e}$ correspond to the real and imaginary parts of the complex voltage product $V_{i}V_{j}^{*}$, respectively. This transformation implies the non-convex equality

\[(\wr)^{2} + (\wi)^{2} = \VM_{i}^{2} \times \VM_{j}^{2} = \wm_{i} \times \wm_{j}\]

The Jabr relaxation is obtained by relaxing this into the convex constraint

\[(\wr)^{2} + (\wi)^{2} \leq \wm_{i} \times \wm_{j}\]

The resulting SOC-OPF model is presented below.

\[\begin{align} \min \quad & \label{eq:SOCOPF:objective} \sum_{i \in \NODES} \sum_{j \in \GENERATORS_{i}} c_j \PG_j + c_0\\ \text{s.t.} \quad & \label{eq:SOCOPF:kcl_p} \sum_{g \in \GENERATORS_i} \PG_g - \sum_{e \in \EDGES^{+}_{i}} \PF_{e} - \sum_{e \in \EDGES^{-}_{i}} \PT_{e} - \GS_i \wm_{i} = \sum_{j\in\LOADS_i} \PD_j & \forall i & \in \NODES % && [\lambda^{p}] \\ & \label{eq:SOCOPF:kcl_q} \sum_{g \in \GENERATORS_{i}} \QG_{g} - \sum_{e \in \EDGES^{+}_{i}} \QF_{e} - \sum_{e \in \EDGES^{-}_{i}} \QT_{e} + \BS_{i} \wm_{i} = \sum_{j\in\LOADS_i} \QD_j & \forall i & \in \NODES % && [\lambda^{q}] \\ % Ohm's law & \label{eq:SOCOPF:ohm_pf} \gff_{e} \wm_{i} + \gft_{e} \wr_{e} + \bft_{e} \wi_{e} - \PF_{e} = 0 & \forall e = (i,j) & \in \EDGES % && [\lambda^{pf}] \\ & \label{eq:SOCOPF:ohm_qf} -\bff_{e} \wm_{i} - \bft_{e} \wr_{e} + \gft_{e} \wi_{e} - \QF_{e} = 0 & \forall e = (i,j) & \in \EDGES % && [\lambda^{qf}] \\ & \label{eq:SOCOPF:ohm_pt} \gtt_{e} \wm_{j} + \gtf_{e} \wr_{e} - \btf_{e} \wi_{e} - \PT_{e} = 0 & \forall e = (i,j) & \in \EDGES % && [\lambda^{pt}] \\ & \label{eq:SOCOPF:ohm_qt} -\btt_{e} \wm_{j} - \btf_{e} \wr_{e} - \gtf_{e} \wi_{e} - \QT_{e} = 0 & \forall e = (i,j) & \in \EDGES % && [\lambda^{qt}] \\ % Jabr constraints & \label{eq:SOCOPF:jabr} \left( \frac{\wm_{i}}{\sqrt{2}}, \frac{\wm_{j}}{\sqrt{2}}, \wr_{e}, \wi_{e} \right) \in \mathcal{Q}_{r}^{4} & \forall e = (i,j) & \in \EDGES % && [\omega] \\ % Thermal limits & \label{eq:SOCOPF:sm_f} (\overline{S}_{e}, \PF_{e}, \QF_{e}) \in \mathcal{Q}^{3} & \forall e & \in \EDGES % && [\nu^{f}] \\ & \label{eq:SOCOPF:sm_t} (\overline{S}_{e}, \PT_{e}, \QT_{e}) \in \mathcal{Q}^{3} & \forall e & \in \EDGES % && [\nu^{t}] \\ % Voltage angle deviation & \label{eq:SOCOPF:va_diff} \tan(\dvamin_{e}) \wr_{e} \leq \wi_{e} \leq \tan(\dvamax_{e}) \wr_{e} & \forall e & \in \EDGES % && [\mu^{\Delta \theta}] \\ % Variable bounds & \label{eq:SOCOPF:pg_bounds} \pgmin_{i} \leq \PG_{i} \leq \pgmax_{i}, & \forall i & \in \GENERATORS % && [\mu^{pg}] \\ & \label{eq:SOCOPF:qg_bounds} \qgmin_{i} \leq \QG_{i} \leq \qgmax_{i}, & \forall i & \in \GENERATORS % && [\mu^{qg}] \\ & \label{eq:SOCOPF:wm_bounds} \vmmin_{i}^{2} \leq \wm_{i} \leq \vmmax_{i}^{2}, & \forall i & \in \NODES % && [\mu^{w}] \\ & \label{eq:SOCOPF:wr_bounds} \wrmin_{e} \leq \wr_{e} \leq \wrmax_{e} & \forall e & \in \EDGES % && [\mu^{wr}] \\ & \label{eq:SOCOPF:wi_bounds} \wimin_{e} \leq \wi_{e} \leq \wimax_{e} & \forall e & \in \EDGES % && [\mu^{wi}] \\ & \label{eq:SOCOPF:pf_bounds} {-}\overline{S}_{e} \leq \PF_{e} \leq \overline{S}_{e} & \forall e & \in \EDGES % && [\mu^{pf}] \\ & \label{eq:SOCOPF:qf_bounds} {-}\overline{S}_{e} \leq \QF_{e} \leq \overline{S}_{e} & \forall e & \in \EDGES % && [\mu^{qf}] \\ & \label{eq:SOCOPF:pt_bounds} {-}\overline{S}_{e} \leq \PT_{e} \leq \overline{S}_{e} & \forall e & \in \EDGES % % && [\mu^{pt}] \\ & \label{eq:SOCOPF:qt_bounds} {-}\overline{S}_{e} \leq \QT_{e} \leq \overline{S}_{e} & \forall e & \in \EDGES % && [\mu^{qt}] \end{align}\]

Variables

• $\wm \in \mathbb{R}^{N}$: squared nodal voltage magnitude
• $\PG \in \mathbb{R}^{G}$: active power dispatch
• $\QG \in \mathbb{R}^{G}$: reactive power dispatch
• $\wr \in \mathbb{R}^{E}$: real part of voltage product
• $\wi \in \mathbb{R}^{E}$: imaginary part of voltage product
• $\PF \in \mathbb{R}^{E}$: active power flow "from"
• $\QF \in \mathbb{R}^{E}$: reactive power flow "from"
• $\PT \in \mathbb{R}^{E}$: active power flow "to"
• $\QT \in \mathbb{R}^{E}$: reactive power flow "to"

Objective

The objective function $\eqref{eq:SOCOPF:objective}$ minimizes the cost of active power generation.

Constraints

• $\eqref{eq:SOCOPF:kcl_p}-\eqref{eq:SOCOPF:kcl_q}$: Kirchhoff's current law for active and reactive power
• $\eqref{eq:SOCOPF:ohm_pf}-\eqref{eq:SOCOPF:ohm_qt}$: Ohm's law for active/reactive power flows in from and to directions
• $\eqref{eq:SOCOPF:jabr}$: Jabr constraint
• $\eqref{eq:SOCOPF:sm_f}-\eqref{eq:SOCOPF:sm_t}$: thermal limits
• $\eqref{eq:SOCOPF:va_diff}$: voltage angle deviation constraints
• $\eqref{eq:SOCOPF:pg_bounds}-\eqref{eq:SOCOPF:qg_bounds}$: active/reactive generation limits
• $\eqref{eq:SOCOPF:wm_bounds}$: bounds on squared voltage magnitude
• $\eqref{eq:SOCOPF:wr_bounds}-\eqref{eq:SOCOPF:wi_bounds}$: bounds on voltage product variables
• $\eqref{eq:SOCOPF:pf_bounds}-\eqref{eq:SOCOPF:qt_bounds}$: power flow bounds, derived from thermal limits

Data format

Primal solution

Dual solution