d2ASbr_dV2

D2ASBR_DV2   Computes 2nd derivatives of |power flow|^2 w.r.t. V.

   -----  DEPRECATED - Please use D2ABR_DV2 instead    -----
   -----  See wrapper code in D2ASBR_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] = D2ASBR_DV2(DSBR_DV1, DSBR_DV2, SBR, CBR, YBR, V, MU)
   [HAA, HAV, HVA, HVV] = D2ASBR_DV2(DSBR_DV1, DSBR_DV2, SBR, CBR, 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 of branch power flows.

   [HRR, HRI, HIR, HII] = D2ASBR_DV2(DSBR_DV1, DSBR_DV2, SBR, CBR, 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 of branch power
   flows.

   Takes as inputs sparse first derivative matrices of complex flow, complex
   flow vector, sparse connection matrix CBR, 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);
       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, vcart);
       Cbr = Cf;
       Ybr = Yf;
       dSbr_dV1 = dSf_dV1;
       dSbr_dV2 = dSf_dV2;
       Sbr = Sf;
       [H11, H12, H21, H22] = ...
             d2ASbr_dV2(dSbr_dV1, dSbr_dV2, Sbr, Cbr, Ybr, V, mu, vcart);

   Here the output matrices correspond to:
     H11 = d/dV1 (dASbr_dV1.' * mu)
     H12 = d/dV2 (dASbr_dV1.' * mu)
     H21 = d/dV1 (dASbr_dV2.' * mu)
     H22 = d/dV2 (dASbr_dV2.' * mu)

   See also DSBR_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     d2ASbr_dV2(dSbr_dV1, dSbr_dV2, Sbr, Cbr, Ybr, V, mu, vcart)
0003 %D2ASBR_DV2   Computes 2nd derivatives of |power flow|^2 w.r.t. V.
0004 %
0005 %   -----  DEPRECATED - Please use D2ABR_DV2 instead    -----
0006 %   -----  See wrapper code in D2ASBR_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] = D2ASBR_DV2(DSBR_DV1, DSBR_DV2, SBR, CBR, YBR, V, MU)
0012 %   [HAA, HAV, HVA, HVV] = D2ASBR_DV2(DSBR_DV1, DSBR_DV2, SBR, CBR, 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 of branch power flows.
0017 %
0018 %   [HRR, HRI, HIR, HII] = D2ASBR_DV2(DSBR_DV1, DSBR_DV2, SBR, CBR, 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 of branch power
0023 %   flows.
0024 %
0025 %   Takes as inputs sparse first derivative matrices of complex flow, complex
0026 %   flow vector, sparse connection matrix CBR, sparse branch admittance matrix
0027 %   YBR, voltage vector V and nl x 1 vector of multipliers MU. Output matrices
0028 %   are sparse.
0029 %
0030 %   Example:
0031 %       f = branch(:, F_BUS);
0032 %       Cf =  sparse(1:nl, f, ones(nl, 1), nl, nb);
0033 %       [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
0034 %       [dSf_dV1, dSf_dV2, dSt_dV1, dSt_dV2, Sf, St] = ...
0035 %               dSbr_dV(branch, Yf, Yt, V, vcart);
0036 %       Cbr = Cf;
0037 %       Ybr = Yf;
0038 %       dSbr_dV1 = dSf_dV1;
0039 %       dSbr_dV2 = dSf_dV2;
0040 %       Sbr = Sf;
0041 %       [H11, H12, H21, H22] = ...
0042 %             d2ASbr_dV2(dSbr_dV1, dSbr_dV2, Sbr, Cbr, Ybr, V, mu, vcart);
0043 %
0044 %   Here the output matrices correspond to:
0045 %     H11 = d/dV1 (dASbr_dV1.' * mu)
0046 %     H12 = d/dV2 (dASbr_dV1.' * mu)
0047 %     H21 = d/dV1 (dASbr_dV2.' * mu)
0048 %     H22 = d/dV2 (dASbr_dV2.' * mu)
0049 %
0050 %   See also DSBR_DV, DABR_DV.
0051 %
0052 %   For more details on the derivations behind the derivative code used
0053 %   in MATPOWER information, see:
0054 %
0055 %   [TN2]  R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and
0056 %          their Derivatives using Complex Matrix Notation", MATPOWER
0057 %          Technical Note 2, February 2010. [Online]. Available:
0058 %          https://matpower.org/docs/TN2-OPF-Derivatives.pdf
0059 %          doi: 10.5281/zenodo.3237866
0060 
0061 %   MATPOWER
0062 %   Copyright (c) 2008-2019, Power Systems Engineering Research Center (PSERC)
0063 %   by Ray Zimmerman, PSERC Cornell
0064 %
0065 %   This file is part of MATPOWER.
0066 %   Covered by the 3-clause BSD License (see LICENSE file for details).
0067 %   See https://matpower.org for more info.
0068 
0069 %% default input args
0070 if nargin < 8
0071     vcart = 0;      %% default to polar coordinates
0072 end
0073 
0074 d2F_dV2 = @(V, mu)d2Sbr_dV2(Cbr, Ybr, V, mu, vcart);
0075 [H11, H12, H21, H22] = d2Abr_dV2(d2F_dV2, dSbr_dV1, dSbr_dV2, Sbr, V, mu);