AC-OPF
Mathematical Formulation
The ACOPF model considered in PGLearn is presented below.
\[\begin{align} \min_{\PG, \QG, \PF, \QF, \PT, \QT, \VM, \VA} \quad & \sum_{i \in \NODES} \sum_{j \in \GENERATORS_{i}} c_j \PG_j + c_0 \label{model:acopf:obj} \\ \text{s.t.} \quad \quad \quad & \sum_{j\in\GENERATORS_i}\PG_j - \sum_{j\in\LOADS_i}\PD_j - \GS_i \VM_i^2 = \sum_{e \in \mathcal{E}^{+}_{i}} \PF_{e} + \sum_{e \in \mathcal{E}^{-}_{i}} \PT_{e} & \forall i &\in \NODES \label{model:acopf:kirchhoff:active} \\ & \sum_{j\in\GENERATORS_i}\QG_j - \sum_{j\in\LOADS_i}\QD_j + \BS_i \VM_i^2 = \sum_{e \in \mathcal{E}^{+}_{i}} \QF_{e} + \sum_{e \in \mathcal{E}^{-}_{i}} \QT_{e} & \forall i &\in \NODES \label{model:acopf:kirchhoff:reactive} \\ & \PF_{e} = \gff_{e}\VM_i^2 + \gft_{e} \VM_i \VM_j \cos(\VA_i-\VA_j) + \bft_{e} \VM_i \VM_j \sin(\VA_i-\VA_j) & \forall e = (i,j) &\in \EDGES \label{model:acopf:ohm:active:from} \\ & \QF_{e} = -\bff_{e} \VM_i^2 - \bft_{e}\VM_i \VM_j \cos(\VA_i-\VA_j) + \gft_{e} \VM_i \VM_j \sin(\VA_i-\VA_j) & \forall e = (i,j) &\in \EDGES \label{model:acopf:ohm:reactive:from} \\ & \PT_{e} = \gtt_{e}\VM_j^2 + \gtf_{e} \VM_i \VM_j \cos(\VA_i-\VA_j) - \btf_{e} \VM_i \VM_j \sin(\VA_i-\VA_j) & \forall e = (i,j) &\in \EDGES \label{model:acopf:ohm:active:to} \\ & \QT_{e} = -\btt_{e} \VM_j^2 - \btf_{e}\VM_i \VM_j \cos(\VA_i-\VA_j) - \gtf_{e} \VM_i \VM_j \sin(\VA_i-\VA_j) & \forall e = (i,j) &\in \EDGES \label{model:acopf:ohm:reactive:to} \\ & (\PF_{e})^2 + (\QF_{e})^2 \leq \overline{S}_{e}^2 & \forall e &\in \EDGES \label{model:acopf:thrmbound:from} \\ & (\PT_{e})^2 + (\QT_{e})^2 \leq \overline{S}_{e}^2 & \forall e &\in \EDGES \label{model:acopf:thrmbound:to} \\ & \dvamin_{e} \leq \VA_i - \VA_j \leq \dvamax_{e} & \forall e = (i,j) &\in \EDGES \label{model:acopf:angledifference} \\ & \VA_\text{ref} = 0 \label{model:acopf:slackbus} \\ & \pgmin_{i} \leq \PG_i \leq \pgmax_{i} & \forall i &\in \GENERATORS \label{model:acopf:pgbound} \\ & \qgmin_{i} \leq \QG_i \leq \qgmax_{i} & \forall i &\in \GENERATORS \label{model:acopf:qgbound} \\ & \vmmin_{i} \leq \VM_i \leq \vmmax_{i} & \forall i &\in \NODES \label{model:acopf:vmbound} \\ & {-}\overline{S}_{e} \leq \PF_{e} \leq \overline{S}_{e} & \forall e &\in \EDGES \label{model:acopf:pfbound} \\ & {-}\overline{S}_{e} \leq \QF_{e} \leq \overline{S}_{e} & \forall e &\in \EDGES \label{model:acopf:qfbound} \\ & {-}\overline{S}_{e} \leq \PT_{e} \leq \overline{S}_{e} & \forall e &\in \EDGES \label{model:acopf:ptbound} \\ & {-}\overline{S}_{e} \leq \QT_{e} \leq \overline{S}_{e} & \forall e &\in \EDGES \label{model:acopf:qtbound} \end{align}\]
Variables
- $\VM \in \mathbb{R}^{N}$: nodal voltage magnitude
- $\VA \in \mathbb{R}^{N}$: nodal voltage angle
- $\PG \in \mathbb{R}^{G}$: active power dispatch
- $\QG \in \mathbb{R}^{G}$: reactive power dispatch
- $\PF \in \mathbb{R}^{E}$: 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 $\eqref{model:acopf:obj}$ 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
- $\eqref{model:acopf:kirchhoff:active}-\eqref{model:acopf:kirchhoff:reactive}$: active and reactive Kirchhoff's current law.
- $\eqref{model:acopf:ohm:active:from}-\eqref{model:acopf:ohm:reactive:to}$: Ohm's law expressing power flows as a function of nodal voltages.
- $\eqref{model:acopf:thrmbound:from}-\eqref{model:acopf:thrmbound:to}$: Thermal limits
- $\eqref{model:acopf:angledifference}$: Voltage angle deviation constraints.
- $\eqref{model:acopf:slackbus}$: this constraint fixes the voltage angle of the reference (slack) bus to zero.
- $\eqref{model:acopf:pgbound}-\eqref{model:acopf:qgbound}$: Active/reactive generation limits
- $\eqref{model:acopf:vmbound}$: Nodal voltage angle limits
- $\eqref{model:acopf:pfbound}-\eqref{model:acopf:qtbound}$: Power flow bounds, derived from thermal limits
Although power flow variable bounds $\eqref{model:acopf:pfbound}-\eqref{model:acopf:qtbound}$ are redundant with thermal limits $\eqref{model:acopf:thrmbound:from}-\eqref{model:acopf:thrmbound:to}$, their inclusion improves the performance of interior-point solvers like Ipopt.
Data format
Primal solution
| Variable | Data | Size | Description |
|---|---|---|---|
| $\VM$ | vm | $N$ | Nodal voltage magnitude |
| $\VA$ | va | $N$ | Nodal voltage angle |
| $\PG$ | pg | $G$ | Active power generation |
| $\QG$ | pg | $G$ | Reactive power generation |
| $\PF$ | pf | $E$ | Active power flow (from) |
| $\PT$ | pt | $E$ | Active power flow (to) |
| $\QF$ | qf | $E$ | Reactive power flow (from) |
| $\QT$ | qt | $E$ | Reactive power flow (to) |
Dual solution
| Constraint | Data | Size |
|---|---|---|
| $\eqref{model:acopf:kirchhoff:active}$ | kcl_p | $N$ |
| $\eqref{model:acopf:kirchhoff:reactive}$ | kcl_q | $N$ |
| $\eqref{model:acopf:ohm:active:from}$ | ohm_pf | $E$ |
| $\eqref{model:acopf:ohm:reactive:from}$ | ohm_qf | $E$ |
| $\eqref{model:acopf:ohm:active:to}$ | ohm_pt | $E$ |
| $\eqref{model:acopf:ohm:reactive:to}$ | ohm_qt | $E$ |
| $\eqref{model:acopf:thrmbound:from}$ | sm_fr | $E$ |
| $\eqref{model:acopf:thrmbound:to}$ | sm_to | $E$ |
| $\eqref{model:acopf:angledifference}$ | va_diff | $E$ |
| $\eqref{model:acopf:slackbus}$ | slack_bus | $N$ |
| $\eqref{model:acopf:pgbound}$ | pg | $G$ |
| $\eqref{model:acopf:qgbound}$ | qg | $G$ |
| $\eqref{model:acopf:vmbound}$ | vm | $N$ |
| $\eqref{model:acopf:pfbound}$ | pf | $E$ |
| $\eqref{model:acopf:qfbound}$ | qf | $E$ |
| $\eqref{model:acopf:ptbound}$ | pt | $E$ |
| $\eqref{model:acopf:qtbound}$ | qt | $E$ |