d2AIbr_dV2

D2AIBR_DV2   Computes 2nd derivatives of |complex current|^2 w.r.t. V.

   -----  DEPRECATED - Please use D2ABR_DV2 instead    -----
   -----  See wrapper code in D2AIBR_DV2 for example.  -----

   The derivatives can be take with respect to polar or cartesian coordinates
   of voltage, depending on the 8th argument.

   [HAA, HAV, HVA, HVV] = D2AIBR_DV2(DIBR_DVA, DIBR_DVM, IBR, YBR, V, MU)
   [HAA, HAV, HVA, HVV] = D2AIBR_DV2(DIBR_DVA, DIBR_DVM, IBR, YBR, V, MU, 0)

   Returns 4 matrices containing the partial derivatives w.r.t. voltage
   angle and magnitude of the product of a vector MU with the 1st partial
   derivatives of the square of the magnitude the branch currents.

   [HRR, HRI, HIR, HII] = D2AIBR_DV2(DIBR_DVA, DIBR_DVM, IBR, YBR, V, MU, 1)

   Returns 4 matrices containing the partial derivatives w.r.t. real and
   imaginary part of complex voltage of the product of a vector MU with the
   1st partial derivatives of the square of the magnitude the branch currents.

   Takes as inputs sparse first derivative matrices of complex current,
   complex current vector, sparse branch admittance matrix YBR, voltage
   vector V and nl x 1 vector of multipliers MU. Output matrices are sparse.

   Example:
       f = branch(:, F_BUS);
       [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
       [dIf_dV1, dIf_dV2, dIt_dV1, dIt_dV2, If, It] = ...
               dIbr_dV(branch, Yf, Yt, V, vcart);
       Ybr = Yf;
       dIbr_dV1 = dIf_dV1;
       dIbr_dV2 = dIf_dV2;
       Ibr = If;
       [H11, H12, H21, H22] = ...
             d2AIbr_dV2(dIbr_dV1, dIbr_dV2, Ibr, Ybr, V, mu, vcart);

   Here the output matrices correspond to:
     H11 = d/dV1 (dAIbr_dV1.' * mu)
     H12 = d/dV2 (dAIbr_dV1.' * mu)
     H21 = d/dV1 (dAIbr_dV2.' * mu)
     H22 = d/dV2 (dAIbr_dV2.' * mu)

   See also DIBR_DV, DABR_DV.

   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

0001 function [H11, H12, H21, H22] = ...
0002     d2AIbr_dV2(dIbr_dV1, dIbr_dV2, Ibr, Ybr, V, mu, vcart)
0003 %D2AIBR_DV2   Computes 2nd derivatives of |complex current|^2 w.r.t. V.
0004 %
0005 %   -----  DEPRECATED - Please use D2ABR_DV2 instead    -----
0006 %   -----  See wrapper code in D2AIBR_DV2 for example.  -----
0007 %
0008 %   The derivatives can be take with respect to polar or cartesian coordinates
0009 %   of voltage, depending on the 8th argument.
0010 %
0011 %   [HAA, HAV, HVA, HVV] = D2AIBR_DV2(DIBR_DVA, DIBR_DVM, IBR, YBR, V, MU)
0012 %   [HAA, HAV, HVA, HVV] = D2AIBR_DV2(DIBR_DVA, DIBR_DVM, IBR, YBR, V, MU, 0)
0013 %
0014 %   Returns 4 matrices containing the partial derivatives w.r.t. voltage
0015 %   angle and magnitude of the product of a vector MU with the 1st partial
0016 %   derivatives of the square of the magnitude the branch currents.
0017 %
0018 %   [HRR, HRI, HIR, HII] = D2AIBR_DV2(DIBR_DVA, DIBR_DVM, IBR, YBR, V, MU, 1)
0019 %
0020 %   Returns 4 matrices containing the partial derivatives w.r.t. real and
0021 %   imaginary part of complex voltage of the product of a vector MU with the
0022 %   1st partial derivatives of the square of the magnitude the branch currents.
0023 %
0024 %   Takes as inputs sparse first derivative matrices of complex current,
0025 %   complex current vector, sparse branch admittance matrix YBR, voltage
0026 %   vector V and nl x 1 vector of multipliers MU. Output matrices are sparse.
0027 %
0028 %   Example:
0029 %       f = branch(:, F_BUS);
0030 %       [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
0031 %       [dIf_dV1, dIf_dV2, dIt_dV1, dIt_dV2, If, It] = ...
0032 %               dIbr_dV(branch, Yf, Yt, V, vcart);
0033 %       Ybr = Yf;
0034 %       dIbr_dV1 = dIf_dV1;
0035 %       dIbr_dV2 = dIf_dV2;
0036 %       Ibr = If;
0037 %       [H11, H12, H21, H22] = ...
0038 %             d2AIbr_dV2(dIbr_dV1, dIbr_dV2, Ibr, Ybr, V, mu, vcart);
0039 %
0040 %   Here the output matrices correspond to:
0041 %     H11 = d/dV1 (dAIbr_dV1.' * mu)
0042 %     H12 = d/dV2 (dAIbr_dV1.' * mu)
0043 %     H21 = d/dV1 (dAIbr_dV2.' * mu)
0044 %     H22 = d/dV2 (dAIbr_dV2.' * mu)
0045 %
0046 %   See also DIBR_DV, DABR_DV.
0047 %
0048 %   For more details on the derivations behind the derivative code used
0049 %   in MATPOWER information, see:
0050 %
0051 %   [TN2]  R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and
0052 %          their Derivatives using Complex Matrix Notation", MATPOWER
0053 %          Technical Note 2, February 2010. [Online]. Available:
0054 %          https://matpower.org/docs/TN2-OPF-Derivatives.pdf
0055 %          doi: 10.5281/zenodo.3237866
0056 
0057 %   MATPOWER
0058 %   Copyright (c) 2008-2019, Power Systems Engineering Research Center (PSERC)
0059 %   by Ray Zimmerman, PSERC Cornell
0060 %
0061 %   This file is part of MATPOWER.
0062 %   Covered by the 3-clause BSD License (see LICENSE file for details).
0063 %   See https://matpower.org for more info.
0064 
0065 %% default input args
0066 if nargin < 7
0067     vcart = 0;      %% default to polar coordinates
0068 end
0069 
0070 d2F_dV2 = @(V, mu)d2Ibr_dV2(Ybr, V, mu, vcart);
0071 [H11, H12, H21, H22] = d2Abr_dV2(d2F_dV2, dIbr_dV1, dIbr_dV2, Ibr, V, mu);