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 one reference bus per energized island:
\[\theta_r = B_r^{-1} \, p_r\]
where:
- $B_r$ is the susceptance-weighted Laplacian with one reference row and column deleted per energized island, including isolated buses
- $p_r = g_r - d_r$ is the net injection vector with the reference entries removed
- $\theta_{\text{refs}} = 0$ 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 (row $e$, written $a_e^\top$, has $+1$ at branch $e$'s from-bus and $-1$ at its to-bus, so $a_e^\top \theta = (A\theta)_e$ is the angle difference across the branch), and $b$ stores the imaginary part of the inverse impedance ($b_e = \operatorname{Im}(1/z_e) < 0$ for inductive branches, so $-b > 0$).
DCNetwork caches the energized island partition used to choose reference buses. Constructors initialize this cache, and topology dependent readers refresh it if direct b or sw edits change which branches are energized. The cache assumes serialized topology mutation: a shared DCNetwork may be read from multiple threads only while its topology fields are not being mutated.
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 row $a_e^\top$ of branch $e$ restricted to the non-reference buses (denoted $a_{e,r}$):
\[\frac{\partial B_r}{\partial \mathrm{sw}_e} = -b_e \, a_{e,r} \, a_{e,r}^\top\]
These derivatives apply while the energized island partition is fixed. Toggling a bridge (a branch whose energization changes the island partition) adds or removes a reference bus and changes the dimension of the reduced system $B_r$, so sensitivities are nonsmooth at that split or merge boundary.
Flow Sensitivity to Switching
Branch flows are $f = W A \theta$ with $W = \operatorname{diag}(-b \circ \mathrm{sw})$, so branch $e$ carries $f_e = -b_e \, \mathrm{sw}_e \,(a_e^\top \theta)$. Both the weight $W$ and the angles $\theta$ depend on $\mathrm{sw}_e$, so the product rule splits the sensitivity into an indirect part (every branch feels the change in angles) and a direct part (only branch $e$'s own weight changes):
\[\frac{\partial f}{\partial \mathrm{sw}_e} = \underbrace{W A \frac{\partial \theta}{\partial \mathrm{sw}_e}}_{\text{indirect (all branches)}} + \underbrace{\frac{\partial W}{\partial \mathrm{sw}_e} A \theta}_{\text{direct (branch } e \text{ only)}} = W A \frac{\partial \theta}{\partial \mathrm{sw}_e} \;-\; b_e \,(a_e^\top \theta)\, e_e\]
The direct term uses $\partial W / \partial \mathrm{sw}_e = -b_e \, e_e e_e^\top$. Only the diagonal entry of $W$ for branch $e$ depends on $\mathrm{sw}_e$, so it adds $-b_e \,(a_e^\top \theta) = -b_e \,(A\theta)_e$ to the flow on branch $e$ alone, where $e_e \in \mathbb{R}^m$ is the unit vector for branch $e$.
Substituting the angle sensitivity $\partial \theta / \partial \mathrm{sw}_e$ we get
\[\frac{\partial \theta_r}{\partial \mathrm{sw}_e} = -B_r^{-1} \frac{\partial B_r}{\partial \mathrm{sw}_e}\, \theta_r = b_e \,(a_{e,r}^\top \theta_r)\, B_r^{-1} a_{e,r}\]
(embedded into the full bus vector with the reference entries held at zero) which makes the flow sensitivity fully explicit:
\[\frac{\partial f}{\partial \mathrm{sw}_e} = b_e \,(a_e^\top \theta) \Big( \underbrace{W A_r\, B_r^{-1} a_{e,r}}_{\text{indirect (all branches)}} \;-\; \underbrace{e_e}_{\text{direct (branch } e \text{ only)}} \Big)\]
where $A_r$ is the incidence matrix with its reference bus columns removed (so $A \,\partial\theta/\partial \mathrm{sw}_e = A_r \,\partial\theta_r/\partial \mathrm{sw}_e$), and we used $a_{e,r}^\top \theta_r = a_e^\top \theta$ since $\theta_{\text{refs}} = 0$. The shared scalar $b_e \,(a_e^\top \theta) = b_e \,(A\theta)_e$ is exactly the negative of branch $e$'s flow when it is in service ($f_e = -b_e (A\theta)_e$ at $\mathrm{sw}_e = 1$), so the whole flow sensitivity scales with how heavily branch $e$ is loaded.
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 buses). The flow sensitivity follows as:
\[\frac{\partial f}{\partial d} = W A \frac{\partial \theta}{\partial d}\]