DC-OPF
Mathematical Formulation
The DCOPF model considered in PGLearn is presented below.
\[\begin{align} \min_{\PG, \PF, \VA} &\quad \sum_{i \in \NODES} \sum_{j \in \GENERATORS_{i}} c_j \PG_j + c_0 \label{model:dcopf:obj} \\ \text{s.t.} \quad & \sum_{j\in\GENERATORS_i}\PG_j - \sum_{e \in \mathcal{E}^{+}_{i}} \PF_{e} + \sum_{e \in \mathcal{E}^{-}_{i}} \PF_{e} = \sum_{j\in\LOADS_i}\PD_j + \GS_i & \forall i &\in \NODES \label{model:dcopf:kirchhoff} \\ & {-}b_{e}(\VA_{i} - \VA_{j}) - \PF_{e} = 0 & \forall e = (i, j) &\in \EDGES \label{model:dcopf:ohm} \\ & \dvamin_{e} \leq \VA_i - \VA_j \leq \dvamax_{e} & \forall e = (i, j) &\in \EDGES \label{model:dcopf:angledifference} \\ & \VA_\text{ref} = 0 \label{model:dcopf:slackbus} \\ & \pgmin_{i} \leq \PG_i \leq \pgmax_{i} & \forall i &\in \GENERATORS \label{model:dcopf:pgbound} \\ & {-}\overline{S}_{e} \leq \PF_{e} \leq \overline{S}_{e} & \forall e &\in \EDGES \label{model:dcopf:thrmbound:from} \end{align}\]
Variables
- $\PG \in \mathbb{R}^{G}$: active power dispatch
- $\PF \in \mathbb{R}^{E}$: active power flow "from"
- $\VA \in \mathbb{R}^{N}$: nodal voltage angle
Objective
The objective function $\eqref{model:dcopf:obj}$ minimizes the cost of active power generation.
PGLearn currently supports only linear cost functions. Support for quadratic functions is planned for a later stage; please open an issue if you would like to request this feature.
Constraints
- $\eqref{model:dcopf:kirchhoff}$: Kirchhoff's current law.
- $\eqref{model:dcopf:ohm}$: Ohm's law expressing power flows as a function of nodal voltages.
- $\eqref{model:dcopf:thrmbound:from}$: Thermal limits.
- $\eqref{model:dcopf:angledifference}$: Voltage angle deviation constraints.
- $\eqref{model:dcopf:slackbus}$: this constraint fixes the voltage angle of the reference (slack) bus to zero.
- $\eqref{model:dcopf:pgbound}$: Active generation limits.
Data Format
Primal variables
| Symbol | Data | Size | Description |
|---|---|---|---|
| $\PG$ | pg | $G$ | Active power dispatch |
| $\VA$ | va | $N$ | Nodal voltage angle |
| $\PF$ | pf | $E$ | Active power flow |
Dual variables
| Associated constraint | Data | Size |
|---|---|---|
| $\eqref{model:dcopf:slackbus}$ | slack_bus | $1$ |
| $\eqref{model:dcopf:kirchhoff}$ | kcl_p | $N$ |
| $\eqref{model:dcopf:ohm}$ | ohm | $E$ |
| $\eqref{model:dcopf:angledifference}$ | va_diff | $E$ |
| $\eqref{model:dcopf:pgbound}$ | pg | $G$ |
| $\eqref{model:dcopf:thrmbound:from}$ | pf | $E$ |