DC Power Flow

Reduced System

For non-OPF power flow with fixed generation, the DC approximation linearizes the power flow equations using the susceptance-weighted Laplacian. The voltage angles satisfy the reduced system obtained by eliminating the reference (slack) bus:

\[\theta_r = B_r^{-1} \, p_r\]

where:

• $B_r$ is the susceptance-weighted Laplacian with the reference bus row and column deleted (invertible for a connected network)
• $p_r = g_r - d_r$ is the net injection vector with the reference entry removed
• $\theta_{\text{ref}} = 0$ by convention

The susceptance-weighted Laplacian is:

\[B = A^\top \operatorname{diag}(-b \circ \mathrm{sw}) \, A\]

where $A$ is the $m \times n$ incidence matrix and $b$ stores the imaginary part of the inverse impedance ($b_e = \operatorname{Im}(1/z_e) < 0$ for inductive branches, so $-b > 0$).

Switching Sensitivity

Switching sensitivity follows from matrix perturbation theory. For a branch $e$ with switching state $\mathrm{sw}_e \in [0,1]$:

\[\frac{\partial \theta_r}{\partial \mathrm{sw}_e} = -B_r^{-1} \frac{\partial B_r}{\partial \mathrm{sw}_e} \, \theta_r\]

where the perturbation is a rank-1 update from the incidence column of branch $e$ restricted to non-reference buses:

\[\frac{\partial B_r}{\partial \mathrm{sw}_e} = -b_e \, a_{e,r} \, a_{e,r}^\top\]

Flow Sensitivity to Switching

Branch flows are $f = W A \theta$ where $W = \operatorname{diag}(-b \circ \mathrm{sw})$. The flow sensitivity has both indirect (via angle changes) and direct (via the switching coefficient) contributions:

\[\frac{\partial f}{\partial \mathrm{sw}_e} = W A \frac{\partial \theta}{\partial \mathrm{sw}_e} + \text{direct effect on edge } e\]

Demand Sensitivity

Since $p = g - d$ and generation is fixed, $\partial p / \partial d = -I$. The angle sensitivity to demand is:

\[\frac{\partial \theta}{\partial d} = -B_r^{-1}\]

embedded in the non-reference block (with zero rows/columns for the reference bus). The flow sensitivity follows as:

\[\frac{\partial f}{\partial d} = W A \frac{\partial \theta}{\partial d}\]