Formulations and Models
This section provides an overview of the formulations and optimization models used in the package.
Nomenclature
Indexes
\(n, m\) |
Bus indices |
\(g\) |
Generator index |
\(l\) |
Transmission line index |
Parameters
\(S_{\text{base}}\) |
Base power [MVA] |
\(P^D_n\) |
Active power demand at bus \(n\) [MW] |
\(\theta_{\text{max}}, \theta_{\text{min}}\) |
Voltage angle limits [radians] |
\(\overline{P}_g, \underline{P}_g\) |
Generation limits for \(g\) [MW] |
\(c_{2,g}, c_{1,g}, c_{0,g}\) |
Cost coefficients for \(g\) |
\(x_l\) |
Reactance of line \(l\) [p.u.] |
\(\overline{P}_l\) |
Line capacity for \(l\) [MW] |
\(C^F_{l,n}, C^T_{l,n}\) |
Incidence matrices for “from” and “to” buses of line \(l\) |
\(C^G_{g,n}\) |
Incidence matrix for generator \(g\) at bus \(n\) |
Variables
\(P_g\) |
Active power generation of \(g\) [MW] |
\(\theta_n\) |
Voltage angle at bus \(n\) [radians] |
\(P^F_l, P^T_l\) |
Power flow on line \(l\) (from/to) [MW] |
DC Optimal Power Flow (DC-OPF)
The DC-OPF formulation is a simplified version of the AC-OPF that assumes:
Voltage magnitudes are fixed at 1.0 p.u.
Reactive power flows are ignored.
Small angle differences between buses.
Objective Function
The objective is to minimize the total generation cost:
Constraints
Active Power Balance:
\[\sum_{g} C^G_{g,n} \cdot P_g - P^D_n = \sum_{l} \left( C^F_{l,n} \cdot P^F_l + C^T_{l,n} \cdot P^T_l \right), \quad \forall n\]Line Flow Equations:
\[P^F_l = \frac{1}{x_l} \cdot \sum_{n} \left( C^F_{l,n} - C^T_{l,n} \right) \cdot \theta_n, \quad \forall l\]\[P^T_l = \frac{1}{x_l} \cdot \sum_{n} \left( C^T_{l,n} - C^F_{l,n} \right) \cdot \theta_n, \quad \forall l\]Line Flow Limits:
\[-\overline{P}_l \leq P^F_l, P^T_l \leq \overline{P}_l, \quad \forall l\]Generator Limits:
\[\underline{P}_g \leq P_g \leq \overline{P}_g, \quad \forall g\]Voltage Angle Reference:
\[\theta_{\text{slack}} = 0\]