DC Optimal Power Flow

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.

If the generating cost functions for all generators are linear, the DC-OPF formulation becomes a linear programming (LP) problem. Otherwise, it is a convex quadratic programming (QP) problem due to the quadratic term in the objective function.

Objective Function

The objective is to minimize the total generation cost:

\[\text{Minimize: } \sum_{g} \left( a_g \cdot p_g^2 + b_{g} \cdot p_g + c_{g} \right)\]

Constraints

  1. Active Power Balance:

    \[\sum_{g\in\mathcal{G}_n} p_g - p^d_n = \sum_{l} \left( F_{ln} \cdot p^f_l + T_{ln} \cdot p^t_l \right), \quad \forall n\]
  2. Line Flow Equations:

    \[p^f_l = \frac{1}{x_l} \cdot \left( \theta_n - \theta_m\right), \quad \forall (l,n,m): F_{l,n} = 1, T_{l,m} = 1\]
    \[p^t_l = \frac{1}{x_l} \cdot \left( \theta_m - \theta_n \right), \quad \forall (l,n,m): F_{l,n} = 1, T_{l,m} = 1\]
  3. Line Flow Limits:

    \[-\overline{s}_l \leq p^f_l \leq \overline{s}_l, \quad \forall l\]
    \[-\overline{s}_l \leq p^t_l \leq \overline{s}_l, \quad \forall l\]
  4. Generator Limits:

    \[\underline{p}_g \leq p_g \leq \overline{p}_g, \quad \forall g\]
  5. Voltage Magnitude Limits:

    \[\underline{\theta}_n \leq \theta_n \leq \overline{\theta}_n, \quad \forall n\]
  6. Voltage Angle Reference:

    \[\theta_{0} = 0\]