D2SBUS_DV2 Computes 2nd derivatives of power injection w.r.t. voltage.
The derivatives can be take with respect to polar or cartesian coordinates
of voltage, depending on the 4th argument.
[GAA, GAV, GVA, GVV] = D2SBUS_DV2(YBUS, V, LAM)
[GAA, GAV, GVA, GVV] = D2SBUS_DV2(YBUS, V, LAM, 0)
Returns 4 matrices containing the partial derivatives w.r.t. voltage angle
and magnitude of the product of a vector LAM with the 1st partial
derivatives of the complex bus power injections.
[GRR, GIR, GIR, GII] = D2SBUS_DV2(YBUS, V, LAM, 1)
Returns 4 matrices containing the partial derivatives w.r.t. real and
imaginary parts of voltage of the product of a vector LAM with the 1st
partial derivatives of the complex bus power injections.
Takes sparse bus admittance matrix YBUS, voltage vector V and nb x 1 vector
of multipliers LAM. Output matrices are sparse.
Examples:
[Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
[Gaa, Gav, Gva, Gvv] = d2Sbus_dV2(Ybus, V, lam);
Here the output matrices correspond to:
Gaa = d/dVa (dSbus_dVa.' * lam)
Gav = d/dVm (dSbus_dVa.' * lam)
Gva = d/dVa (dSbus_dVm.' * lam)
Gvv = d/dVm (dSbus_dVm.' * lam)
[Grr, Gri, Gir, Gii] = d2Sbus_dV2(Ybus, V, lam, 1);
Here the output matrices correspond to:
Grr = d/dVr (dSbus_dVr.' * lam)
Gri = d/dVi (dSbus_dVr.' * lam)
Gir = d/dVr (dSbus_dVi.' * lam)
Gii = d/dVi (dSbus_dVi.' * lam)
For more details on the derivations behind the derivative code used
in MATPOWER, 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 [G11, G12, G21, G22] = d2Sbus_dV2(Ybus, V, lam, vcart) 0002 %D2SBUS_DV2 Computes 2nd derivatives of power injection w.r.t. voltage. 0003 % 0004 % The derivatives can be take with respect to polar or cartesian coordinates 0005 % of voltage, depending on the 4th argument. 0006 % 0007 % [GAA, GAV, GVA, GVV] = D2SBUS_DV2(YBUS, V, LAM) 0008 % [GAA, GAV, GVA, GVV] = D2SBUS_DV2(YBUS, V, LAM, 0) 0009 % 0010 % Returns 4 matrices containing the partial derivatives w.r.t. voltage angle 0011 % and magnitude of the product of a vector LAM with the 1st partial 0012 % derivatives of the complex bus power injections. 0013 % 0014 % [GRR, GIR, GIR, GII] = D2SBUS_DV2(YBUS, V, LAM, 1) 0015 % 0016 % Returns 4 matrices containing the partial derivatives w.r.t. real and 0017 % imaginary parts of voltage of the product of a vector LAM with the 1st 0018 % partial derivatives of the complex bus power injections. 0019 % 0020 % Takes sparse bus admittance matrix YBUS, voltage vector V and nb x 1 vector 0021 % of multipliers LAM. Output matrices are sparse. 0022 % 0023 % Examples: 0024 % [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch); 0025 % [Gaa, Gav, Gva, Gvv] = d2Sbus_dV2(Ybus, V, lam); 0026 % 0027 % Here the output matrices correspond to: 0028 % Gaa = d/dVa (dSbus_dVa.' * lam) 0029 % Gav = d/dVm (dSbus_dVa.' * lam) 0030 % Gva = d/dVa (dSbus_dVm.' * lam) 0031 % Gvv = d/dVm (dSbus_dVm.' * lam) 0032 % 0033 % [Grr, Gri, Gir, Gii] = d2Sbus_dV2(Ybus, V, lam, 1); 0034 % 0035 % Here the output matrices correspond to: 0036 % Grr = d/dVr (dSbus_dVr.' * lam) 0037 % Gri = d/dVi (dSbus_dVr.' * lam) 0038 % Gir = d/dVr (dSbus_dVi.' * lam) 0039 % Gii = d/dVi (dSbus_dVi.' * lam) 0040 % 0041 % For more details on the derivations behind the derivative code used 0042 % in MATPOWER, see: 0043 % 0044 % [TN2] R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and 0045 % their Derivatives using Complex Matrix Notation", MATPOWER 0046 % Technical Note 2, February 2010. [Online]. Available: 0047 % https://matpower.org/docs/TN2-OPF-Derivatives.pdf 0048 % doi: 10.5281/zenodo.3237866 0049 % [TN4] B. Sereeter and R. D. Zimmerman, "AC Power Flows and their 0050 % Derivatives using Complex Matrix Notation and Cartesian 0051 % Coordinate Voltages," MATPOWER Technical Note 4, April 2018. 0052 % [Online]. Available: https://matpower.org/docs/TN4-OPF-Derivatives-Cartesian.pdf 0053 % doi: 10.5281/zenodo.3237909 0054 0055 % MATPOWER 0056 % Copyright (c) 2008-2019, Power Systems Engineering Research Center (PSERC) 0057 % by Ray Zimmerman, PSERC Cornell 0058 % and Baljinnyam Sereeter, Delft University of Technology 0059 % 0060 % This file is part of MATPOWER. 0061 % Covered by the 3-clause BSD License (see LICENSE file for details). 0062 % See https://matpower.org for more info. 0063 0064 %% default input args 0065 if nargin < 4 0066 vcart = 0; %% default to polar coordinates 0067 end 0068 0069 n = length(V); 0070 0071 diaglam = sparse(1:n, 1:n, lam, n, n); 0072 if vcart 0073 E = diaglam * conj(Ybus); 0074 F = E + E.'; 0075 G = 1j * (E - E.'); 0076 0077 G11 = F; %% Grr 0078 G21 = G; %% Gir 0079 G12 = G21.'; %% Gri 0080 G22 = G11; %% Gii 0081 else 0082 Ibus = Ybus * V; 0083 diagV = sparse(1:n, 1:n, V, n, n); 0084 0085 A = sparse(1:n, 1:n, lam .* V, n, n); 0086 B = Ybus * diagV; 0087 C = A * conj(B); 0088 D = Ybus' * diagV; 0089 E = conj(diagV) * (D * diaglam - sparse(1:n, 1:n, D*lam, n, n)); 0090 F = C - A * sparse(1:n, 1:n, conj(Ibus), n, n); 0091 G = sparse(1:n, 1:n, ones(n, 1)./abs(V), n, n); 0092 0093 G11 = E + F; 0094 G21 = 1j * G * (E - F); 0095 G12 = G21.'; 0096 G22 = G * (C + C.') * G; 0097 end