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:

\[\text{Minimize: } \sum_{g} \left( c_{2,g} \cdot P_g^2 + c_{1,g} \cdot P_g + c_{0,g} \right)\]

Constraints

  1. Active Power Balance:

    \[\sum_{g} C^G_{g,n} \cdot P_g - P^D_n = \sum_{l} \left( C^F_{l,n} \cdot P^F_l + C^T_{l,n} \cdot P^T_l \right), \quad \forall n\]
  2. Line Flow Equations:

    \[P^F_l = \frac{1}{x_l} \cdot \sum_{n} \left( C^F_{l,n} - C^T_{l,n} \right) \cdot \theta_n, \quad \forall l\]
    \[P^T_l = \frac{1}{x_l} \cdot \sum_{n} \left( C^T_{l,n} - C^F_{l,n} \right) \cdot \theta_n, \quad \forall l\]
  3. Line Flow Limits:

    \[-\overline{P}_l \leq P^F_l, P^T_l \leq \overline{P}_l, \quad \forall l\]
  4. Generator Limits:

    \[\underline{P}_g \leq P_g \leq \overline{P}_g, \quad \forall g\]
  5. Voltage Angle Reference:

    \[\theta_{\text{slack}} = 0\]