dAbr_dV

dAbr_dV(dFf_dV1, dFf_dV2, dFt_dV1, dFt_dV2, Ff, Ft)
dAbr_dV() - Partial derivatives of squared flow magnitudes w.r.t voltage.
[DAF_DV1, DAF_DV2, DAT_DV1, DAT_DV2] = ...
            DABR_DV(DFF_DV1, DFF_DV2, DFT_DV1, DFT_DV2, FF, FT)
returns four matrices containing partial derivatives of the square of
the branch flow magnitudes at "from" & "to" ends of each branch w.r.t
voltage components (either angle and magnitude, respectively, if polar,
or real and imaginary, respectively, if cartesian) for all buses, given
the flows and flow sensitivities. Flows could be complex current or
complex or real power. Notation below is based on complex power. The
following explains the expressions used to form the matrices:

Let Af refer to the square of the apparent power at the "from" end of
each branch,

    Af = abs(Sf).^2
       = Sf .* conj(Sf)
       = Pf.^2 + Qf.^2

then ...

Partial w.r.t real power,
    dAf/dPf = 2 * diag(Pf)

Partial w.r.t reactive power,
    dAf/dQf = 2 * diag(Qf)

Partial w.r.t V1 & V2 (e.g. Va and Vm, or Vr and Vi)
    dAf/dV1 = dAf/dPf * dPf/dV1 + dAf/dQf * dQf/dV1
    dAf/dV2 = dAf/dPf * dPf/dV2 + dAf/dQf * dQf/dV2

Derivations for "to" bus are similar.

Examples:
    %% squared current magnitude
    [dFf_dV1, dFf_dV2, dFt_dV1, dFt_dV2, Ff, Ft] = ...
            dIbr_dV(branch(il,:), Yf, Yt, V);
    [dAf_dV1, dAf_dV2, dAt_dV1, dAt_dV2] = ...
            dAbr_dV(dFf_dV1, dFf_dV2, dFt_dV1, dFt_dV2, Ff, Ft);

    %% squared apparent power flow
    [dFf_dV1, dFf_dV2, dFt_dV1, dFt_dV2, Ff, Ft] = ...
            dSbr_dV(branch(il,:), Yf, Yt, V);
    [dAf_dV1, dAf_dV2, dAt_dV1, dAt_dV2] = ...
            dAbr_dV(dFf_dV1, dFf_dV2, dFt_dV1, dFt_dV2, Ff, Ft);

    %% squared real power flow
    [dFf_dV1, dFf_dV2, dFt_dV1, dFt_dV2, Ff, Ft] = ...
            dSbr_dV(branch(il,:), Yf, Yt, V);
    dFf_dV1 = real(dFf_dV1);
    dFf_dV2 = real(dFf_dV2);
    dFt_dV1 = real(dFt_dV1);
    dFt_dV2 = real(dFt_dV2);
    [dAf_dV1, dAf_dV2, dAt_dV1, dAt_dV2] = ...
            dAbr_dV(dFf_dV1, dFf_dV2, dFt_dV1, dFt_dV2, Ff, Ft);