d2Abr_dV2

d2Abr_dV2(d2F_dV2, dF_dV1, dF_dV2, F, V, mu)

d2Abr_dV2() - Computes 2nd derivatives of |branch flow|^2 w.r.t. V.

The derivatives can be take with respect to polar or cartesian coordinates
of voltage, depending on the first 3 arguments. Flows could be complex
current or complex or real power. Notation below is based on complex power.

[H11, H12, H21, H22] = D2ABR_DV2(D2F_DV2, DF_DV1, DF_DV2, F, V, MU)

Returns 4 matrices containing the partial derivatives w.r.t. voltage
components (angle, magnitude or real, imaginary) of the product of a
vector MU with the 1st partial derivatives of the square of the magnitude
of branch flows.

Takes as inputs a handle to a function that evaluates the 2nd derivatives
of the flows (with args V and mu only), sparse first derivative matrices
of flow, flow vector, voltage vector V and nl x 1 vector of multipliers
MU. Output matrices are sparse.

Example:
    f = branch(:, F_BUS);
    Cf =  sparse(1:nl, f, ones(nl, 1), nl, nb);
    [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
    [dSf_dV1, dSf_dV2, dSt_dV1, dSt_dV2, Sf, St] = ...
            dSbr_dV(branch, Yf, Yt, V);
    dF_dV1 = dSf_dV1;
    dF_dV2 = dSf_dV2;
    F = Sf;
    d2F_dV2 = @(V, mu)d2Sbr_dV2(Cf, Yf, V, mu, 0);
    [H11, H12, H21, H22] = ...
          d2Abr_dV2(d2F_dV2, dF_dV1, dF_dV2, F, V, mu);

Here the output matrices correspond to:
  H11 = d/dV1 (dAF_dV1.' * mu)
  H12 = d/dV2 (dAF_dV1.' * mu)
  H21 = d/dV1 (dAF_dV2.' * mu)
  H22 = d/dV2 (dAF_dV2.' * mu)

For more details on the derivations behind the derivative code used in MATPOWER information, see:

[TN2]  R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and
       their Derivatives using Complex Matrix Notation", MATPOWER
       Technical Note 2, February 2010. [Online]. Available:
       https://matpower.org/docs/TN2-OPF-Derivatives.pdf
       doi: 10.5281/zenodo.3237866