D2SBR_DV2 Computes 2nd derivatives of complex brch power flow w.r.t. voltage.
The derivatives can be take with respect to polar or cartesian coordinates
of voltage, depending on the 5th argument.
[HAA, HAV, HVA, HVV] = D2SBR_DV2(CBR, YBR, V, MU)
[HAA, HAV, HVA, HVV] = D2SBR_DV2(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 complex branch power flows.
[HRR, HRI, HIR, HII] = d2Sbr_dV2(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 complex branch power flows.
Takes sparse connection matrix CBR, sparse branch admittance matrix YBR,
voltage vector V and nl x 1 vector of multipliers MU. Output matrices are
sparse.
Examples:
f = branch(:, F_BUS);
Cf = sparse(1:nl, f, ones(nl, 1), nl, nb);
[Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
Cbr = Cf;
Ybr = Yf;
[Haa, Hav, Hva, Hvv] = d2Sbr_dV2(Cbr, Ybr, V, mu);
Here the output matrices correspond to:
Haa = d/dVa (dSbr_dVa.' * mu)
Hav = d/dVm (dSbr_dVa.' * mu)
Hva = d/dVa (dSbr_dVm.' * mu)
Hvv = d/dVm (dSbr_dVm.' * mu)
[Hrr, Hri, Hir, Hii] = d2Sbr_dV2(Cbr, Ybr, V, mu, 1);
Here the output matrices correspond to:
Hrr = d/dVr (dSbr_dVr.' * mu)
Hri = d/dVi (dSbr_dVr.' * mu)
Hir = d/dVr (dSbr_dVi.' * mu)
Hii = d/dVi (dSbr_dVi.' * 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
[TN4] B. Sereeter and R. D. Zimmerman, "AC Power Flows and their
Derivatives using Complex Matrix Notation and Cartesian
Coordinate Voltages," MATPOWER Technical Note 4, April 2018.
[Online]. Available: https://matpower.org/docs/TN4-OPF-Derivatives-Cartesian.pdf
doi: 10.5281/zenodo.32379090001 function [H11, H12, H21, H22] = d2Sbr_dV2(Cbr, Ybr, V, mu, vcart) 0002 %D2SBR_DV2 Computes 2nd derivatives of complex brch power flow w.r.t. voltage. 0003 % 0004 % The derivatives can be take with respect to polar or cartesian coordinates 0005 % of voltage, depending on the 5th argument. 0006 % 0007 % [HAA, HAV, HVA, HVV] = D2SBR_DV2(CBR, YBR, V, MU) 0008 % [HAA, HAV, HVA, HVV] = D2SBR_DV2(CBR, YBR, V, MU, 0) 0009 % 0010 % Returns 4 matrices containing the partial derivatives w.r.t. voltage angle 0011 % and magnitude of the product of a vector MU with the 1st partial 0012 % derivatives of the complex branch power flows. 0013 % 0014 % [HRR, HRI, HIR, HII] = d2Sbr_dV2(CBR, YBR, V, MU, 1) 0015 % 0016 % Returns 4 matrices containing the partial derivatives w.r.t. real and 0017 % imaginary part of complex voltage of the product of a vector MU with the 0018 % 1st partial derivatives of the complex branch power flows. 0019 % 0020 % Takes sparse connection matrix CBR, sparse branch admittance matrix YBR, 0021 % voltage vector V and nl x 1 vector of multipliers MU. Output matrices are 0022 % sparse. 0023 % 0024 % Examples: 0025 % f = branch(:, F_BUS); 0026 % Cf = sparse(1:nl, f, ones(nl, 1), nl, nb); 0027 % [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch); 0028 % Cbr = Cf; 0029 % Ybr = Yf; 0030 % [Haa, Hav, Hva, Hvv] = d2Sbr_dV2(Cbr, Ybr, V, mu); 0031 % 0032 % Here the output matrices correspond to: 0033 % Haa = d/dVa (dSbr_dVa.' * mu) 0034 % Hav = d/dVm (dSbr_dVa.' * mu) 0035 % Hva = d/dVa (dSbr_dVm.' * mu) 0036 % Hvv = d/dVm (dSbr_dVm.' * mu) 0037 % 0038 % [Hrr, Hri, Hir, Hii] = d2Sbr_dV2(Cbr, Ybr, V, mu, 1); 0039 % 0040 % Here the output matrices correspond to: 0041 % Hrr = d/dVr (dSbr_dVr.' * mu) 0042 % Hri = d/dVi (dSbr_dVr.' * mu) 0043 % Hir = d/dVr (dSbr_dVi.' * mu) 0044 % Hii = d/dVi (dSbr_dVi.' * mu) 0045 % 0046 % For more details on the derivations behind the derivative code used 0047 % in MATPOWER information, see: 0048 % 0049 % [TN2] R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and 0050 % their Derivatives using Complex Matrix Notation", MATPOWER 0051 % Technical Note 2, February 2010. [Online]. Available: 0052 % https://matpower.org/docs/TN2-OPF-Derivatives.pdf 0053 % doi: 10.5281/zenodo.3237866 0054 % [TN4] B. Sereeter and R. D. Zimmerman, "AC Power Flows and their 0055 % Derivatives using Complex Matrix Notation and Cartesian 0056 % Coordinate Voltages," MATPOWER Technical Note 4, April 2018. 0057 % [Online]. Available: https://matpower.org/docs/TN4-OPF-Derivatives-Cartesian.pdf 0058 % doi: 10.5281/zenodo.3237909 0059 0060 % MATPOWER 0061 % Copyright (c) 2008-2019, Power Systems Engineering Research Center (PSERC) 0062 % by Ray Zimmerman, PSERC Cornell 0063 % and Baljinnyam Sereeter, Delft University of Technology 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 < 5 0071 vcart = 0; %% default to polar coordinates 0072 end 0073 0074 nl = length(mu); 0075 nb = length(V); 0076 0077 A = Ybr' * sparse(1:nl, 1:nl, mu, nl, nl) * Cbr; 0078 if vcart 0079 H11 = A + A.'; 0080 H12 = 1j * (A - A.'); 0081 H21 = -H12; 0082 H22 = H11; 0083 else 0084 diagV = sparse(1:nb, 1:nb, V, nb, nb); 0085 0086 B = conj(diagV) * A * diagV; 0087 D = sparse(1:nb, 1:nb, (A*V) .* conj(V), nb, nb); 0088 E = sparse(1:nb, 1:nb, (A.'*conj(V)) .* V, nb, nb); 0089 F = B + B.'; 0090 G = sparse(1:nb, 1:nb, ones(nb, 1)./abs(V), nb, nb); 0091 0092 H11 = F - D - E; 0093 H21 = 1j * G * (B - B.' - D + E); 0094 H12 = H21.'; 0095 H22 = G * F * G; 0096 end