DIMIS_DV Computes partial derivatives of current balance w.r.t. voltage.
The derivatives can be take with respect to polar or cartesian coordinates
of voltage, depending on the 3rd argument.
[DIMIS_DVM, DIMIS_DVA] = DIMIS_DV(SBUS, YBUS, V)
[DIMIS_DVM, DIMIS_DVA] = DIMIS_DV(SBUS, YBUS, V, 0)
Returns two matrices containing partial derivatives of the complex bus
current balance w.r.t voltage magnitude and voltage angle, respectively
(for all buses).
[DIMIS_DVR, DIMIS_DVI] = DIMIS_DV(SBUS, YBUS, V, 1)
Returns two matrices containing partial derivatives of the complex bus
current balance w.r.t the real and imaginary parts of voltage,
respectively (for all buses).
If YBUS is a sparse matrix, the return values will be also. The following
explains the expressions used to form the matrices:
Imis = Ibus + Idg = Ybus * V - conj(Sbus./V)
Polar coordinates:
Partials of V & Ibus w.r.t. voltage angles
dV/dVa = j * diag(V)
dI/dVa = Ybus * dV/dVa = Ybus * j * diag(V)
Partials of V & Ibus w.r.t. voltage magnitudes
dV/dVm = diag(V./abs(V))
dI/dVm = Ybus * dV/dVm = Ybus * diag(V./abs(V))
Partials of Imis w.r.t. voltage angles
dImis/dVa = j * (Ybus* diag(V) - diag(conj(Sbus./V)))
Partials of Imis w.r.t. voltage magnitudes
dImis/dVm = Ybus * diag(V./abs(V)) + diag(conj(Sbus./(V*abs(V))))
Cartesian coordinates:
Partials of V & Ibus w.r.t. real part of complex voltage
dV/dVr = diag(ones(n,1))
dI/dVr = Ybus * dV/dVr = Ybus
Partials of V & Ibus w.r.t. imaginary part of complex voltage
dV/dVi = j * diag(ones(n,1))
dI/dVi = Ybus * dV/dVi = Ybus * j
Partials of Imis w.r.t. real part of complex voltage
dImis/dVr = Ybus + conj(diag(Sbus./(V.^2)))
Partials of S w.r.t. imaginary part of complex voltage
dImis/dVi = j * (Ybus - diag(conj(Sbus./(V.^2))))
Examples:
Sbus = makeSbus(baseMVA, bus, gen);
[Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch);
[dImis_dVm, dImis_dVa] = dImis_dV(Sbus, Ybus, V);
[dImis_dVr, dImis_dVi] = dImis_dV(Sbus, Ybus, V, 1);
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
[TN3] B. Sereeter and R. D. Zimmerman, "Addendum to AC Power Flows and
their Derivatives using Complex Matrix Notation: Nodal Current
Balance," MATPOWER Technical Note 3, April 2018. [Online].
Available: https://matpower.org/docs/TN3-More-OPF-Derivatives.pdf
doi: 10.5281/zenodo.3237900
[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 [dImis_dV1, dImis_dV2] = dImis_dV(Sbus, Ybus, V, vcart) 0002 %DIMIS_DV Computes partial derivatives of current balance w.r.t. voltage. 0003 % 0004 % The derivatives can be take with respect to polar or cartesian coordinates 0005 % of voltage, depending on the 3rd argument. 0006 % 0007 % [DIMIS_DVM, DIMIS_DVA] = DIMIS_DV(SBUS, YBUS, V) 0008 % [DIMIS_DVM, DIMIS_DVA] = DIMIS_DV(SBUS, YBUS, V, 0) 0009 % 0010 % Returns two matrices containing partial derivatives of the complex bus 0011 % current balance w.r.t voltage magnitude and voltage angle, respectively 0012 % (for all buses). 0013 % 0014 % [DIMIS_DVR, DIMIS_DVI] = DIMIS_DV(SBUS, YBUS, V, 1) 0015 % 0016 % Returns two matrices containing partial derivatives of the complex bus 0017 % current balance w.r.t the real and imaginary parts of voltage, 0018 % respectively (for all buses). 0019 % 0020 % If YBUS is a sparse matrix, the return values will be also. The following 0021 % explains the expressions used to form the matrices: 0022 % 0023 % Imis = Ibus + Idg = Ybus * V - conj(Sbus./V) 0024 % 0025 % Polar coordinates: 0026 % Partials of V & Ibus w.r.t. voltage angles 0027 % dV/dVa = j * diag(V) 0028 % dI/dVa = Ybus * dV/dVa = Ybus * j * diag(V) 0029 % 0030 % Partials of V & Ibus w.r.t. voltage magnitudes 0031 % dV/dVm = diag(V./abs(V)) 0032 % dI/dVm = Ybus * dV/dVm = Ybus * diag(V./abs(V)) 0033 % 0034 % Partials of Imis w.r.t. voltage angles 0035 % dImis/dVa = j * (Ybus* diag(V) - diag(conj(Sbus./V))) 0036 % 0037 % Partials of Imis w.r.t. voltage magnitudes 0038 % dImis/dVm = Ybus * diag(V./abs(V)) + diag(conj(Sbus./(V*abs(V)))) 0039 % 0040 % Cartesian coordinates: 0041 % Partials of V & Ibus w.r.t. real part of complex voltage 0042 % dV/dVr = diag(ones(n,1)) 0043 % dI/dVr = Ybus * dV/dVr = Ybus 0044 % 0045 % Partials of V & Ibus w.r.t. imaginary part of complex voltage 0046 % dV/dVi = j * diag(ones(n,1)) 0047 % dI/dVi = Ybus * dV/dVi = Ybus * j 0048 % 0049 % Partials of Imis w.r.t. real part of complex voltage 0050 % dImis/dVr = Ybus + conj(diag(Sbus./(V.^2))) 0051 % 0052 % Partials of S w.r.t. imaginary part of complex voltage 0053 % dImis/dVi = j * (Ybus - diag(conj(Sbus./(V.^2)))) 0054 % 0055 % Examples: 0056 % Sbus = makeSbus(baseMVA, bus, gen); 0057 % [Ybus, Yf, Yt] = makeYbus(baseMVA, bus, branch); 0058 % [dImis_dVm, dImis_dVa] = dImis_dV(Sbus, Ybus, V); 0059 % [dImis_dVr, dImis_dVi] = dImis_dV(Sbus, Ybus, V, 1); 0060 % 0061 % For more details on the derivations behind the derivative code used 0062 % in MATPOWER information, see: 0063 % 0064 % [TN2] R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and 0065 % their Derivatives using Complex Matrix Notation", MATPOWER 0066 % Technical Note 2, February 2010. [Online]. Available: 0067 % https://matpower.org/docs/TN2-OPF-Derivatives.pdf 0068 % doi: 10.5281/zenodo.3237866 0069 % [TN3] B. Sereeter and R. D. Zimmerman, "Addendum to AC Power Flows and 0070 % their Derivatives using Complex Matrix Notation: Nodal Current 0071 % Balance," MATPOWER Technical Note 3, April 2018. [Online]. 0072 % Available: https://matpower.org/docs/TN3-More-OPF-Derivatives.pdf 0073 % doi: 10.5281/zenodo.3237900 0074 % [TN4] B. Sereeter and R. D. Zimmerman, "AC Power Flows and their 0075 % Derivatives using Complex Matrix Notation and Cartesian 0076 % Coordinate Voltages," MATPOWER Technical Note 4, April 2018. 0077 % [Online]. Available: https://matpower.org/docs/TN4-OPF-Derivatives-Cartesian.pdf 0078 % doi: 10.5281/zenodo.3237909 0079 0080 % MATPOWER 0081 % Copyright (c) 2019, Power Systems Engineering Research Center (PSERC) 0082 % by Baljinnyam Sereeter, Delft University of Technology 0083 % and Ray Zimmerman, PSERC Cornell 0084 % 0085 % This file is part of MATPOWER. 0086 % Covered by the 3-clause BSD License (see LICENSE file for details). 0087 % See https://matpower.org for more info. 0088 0089 %% default input args 0090 if nargin < 4 0091 vcart = 0; %% default to polar coordinates 0092 end 0093 0094 n = length(V); 0095 0096 if vcart 0097 if issparse(Ybus) 0098 diagSV2c = sparse(1:n, 1:n, conj(Sbus./(V.^2)), n, n); 0099 else 0100 diagSV2c = diag(conj(Sbus./(V.^2))); 0101 end 0102 0103 dImis_dV1 = Ybus + diagSV2c; %% dImis/dVr 0104 dImis_dV2 = 1j*(Ybus - diagSV2c); %% dImis/dVi 0105 else 0106 Vm = abs(V); 0107 Vnorm = V./Vm; 0108 Ibus = conj(Sbus./V); 0109 if issparse(Ybus) 0110 diagV = sparse(1:n, 1:n, V, n, n); 0111 diagIbus = sparse(1:n, 1:n, Ibus, n, n); 0112 diagIbusVm = sparse(1:n, 1:n, Ibus./Vm, n, n); 0113 diagVnorm = sparse(1:n, 1:n, V./abs(V), n, n); 0114 else 0115 diagV = diag(V); 0116 diagIbus = diag(Ibus); 0117 diagIbusVm = diag(Ibus./Vm); 0118 diagVnorm = diag(V./abs(V)); 0119 end 0120 0121 dImis_dV1 = 1j*(Ybus*diagV - diagIbus); %% dImis/dVa 0122 dImis_dV2 = Ybus*diagVnorm + diagIbusVm; %% dImis/dVm 0123 end