Module dIbr_dV

Computes partial derivatives of branch currents w.r.t. voltage.

Returns four matrices containing partial derivatives of the complex branch currents at "from" and "to" ends of each branch w.r.t voltage magnitude and voltage angle respectively (for all buses). If Yf is a sparse matrix, the partial derivative matrices will be as well. Optionally returns vectors containing the currents themselves. The following explains the expressions used to form the matrices:

   If = Yf * V

Partials of V, Vf & If w.r.t. voltage angles:

   dV/dVa  = j * diag(V)
   dVf/dVa = sparse(range(nl), f, j*V(f)) = j * sparse(range(nl), f, V(f))
   dIf/dVa = Yf * dV/dVa = Yf * j * diag(V)

Partials of V, Vf & If w.r.t. voltage magnitudes:

   dV/dVm  = diag(V / abs(V))
   dVf/dVm = sparse(range(nl), f, V(f) / abs(V(f))
   dIf/dVm = Yf * dV/dVm = Yf * diag(V / abs(V))

Derivations for "to" bus are similar.