Source Code for Module pypower.dSbus_dV

 1  # Copyright (C) 1996-2011 Power System Engineering Research Center (PSERC) 
 2  # Copyright (C) 2011 Richard Lincoln 
 3  # 
 4  # PYPOWER is free software: you can redistribute it and/or modify 
 5  # it under the terms of the GNU General Public License as published 
 6  # by the Free Software Foundation, either version 3 of the License, 
 7  # or (at your option) any later version. 
 8  # 
 9  # PYPOWER is distributed in the hope that it will be useful, 
10  # but WITHOUT ANY WARRANTY; without even the implied warranty of 
11  # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 
12  # GNU General Public License for more details. 
13  # 
14  # You should have received a copy of the GNU General Public License 
15  # along with PYPOWER. If not, see <http://www.gnu.org/licenses/>. 
16   
17  """Computes partial derivatives of power injection w.r.t. voltage. 
18  """ 
19   
20  from numpy import conj, diag, asmatrix, asarray 
21  from scipy.sparse import issparse, csr_matrix as sparse 
22   
23   
24 -def dSbus_dV(Ybus, V): 
25      """Computes partial derivatives of power injection w.r.t. voltage. 
26   
27      Returns two matrices containing partial derivatives of the complex bus 
28      power injections w.r.t voltage magnitude and voltage angle respectively 
29      (for all buses). If C{Ybus} is a sparse matrix, the return values will be 
30      also. The following explains the expressions used to form the matrices:: 
31   
32          S = diag(V) * conj(Ibus) = diag(conj(Ibus)) * V 
33   
34      Partials of V & Ibus w.r.t. voltage magnitudes:: 
35          dV/dVm = diag(V / abs(V)) 
36          dI/dVm = Ybus * dV/dVm = Ybus * diag(V / abs(V)) 
37   
38      Partials of V & Ibus w.r.t. voltage angles:: 
39          dV/dVa = j * diag(V) 
40          dI/dVa = Ybus * dV/dVa = Ybus * j * diag(V) 
41   
42      Partials of S w.r.t. voltage magnitudes:: 
43          dS/dVm = diag(V) * conj(dI/dVm) + diag(conj(Ibus)) * dV/dVm 
44                 = diag(V) * conj(Ybus * diag(V / abs(V))) 
45                                          + conj(diag(Ibus)) * diag(V / abs(V)) 
46   
47      Partials of S w.r.t. voltage angles:: 
48          dS/dVa = diag(V) * conj(dI/dVa) + diag(conj(Ibus)) * dV/dVa 
49                 = diag(V) * conj(Ybus * j * diag(V)) 
50                                          + conj(diag(Ibus)) * j * diag(V) 
51                 = -j * diag(V) * conj(Ybus * diag(V)) 
52                                          + conj(diag(Ibus)) * j * diag(V) 
53                 = j * diag(V) * conj(diag(Ibus) - Ybus * diag(V)) 
54   
55      For more details on the derivations behind the derivative code used 
56      in PYPOWER information, see: 
57   
58      [TN2]  R. D. Zimmerman, "AC Power Flows, Generalized OPF Costs and 
59      their Derivatives using Complex Matrix Notation", MATPOWER 
60      Technical Note 2, February 2010. 
61      U{http://www.pserc.cornell.edu/matpower/TN2-OPF-Derivatives.pdf} 
62   
63      @author: Ray Zimmerman (PSERC Cornell) 
64      @author: Richard Lincoln 
65      """ 
66      ib = range(len(V)) 
67   
68      if issparse(Ybus): 
69          Ibus = Ybus * V 
70   
71          diagV = sparse((V, (ib, ib))) 
72          diagIbus = sparse((Ibus, (ib, ib))) 
73          diagVnorm = sparse((V / abs(V), (ib, ib))) 
74      else: 
75          Ibus = Ybus * asmatrix(V).T 
76   
77          diagV = asmatrix(diag(V)) 
78          diagIbus = asmatrix(diag( asarray(Ibus).flatten() )) 
79          diagVnorm = asmatrix(diag(V / abs(V))) 
80   
81      dS_dVm = diagV * conj(Ybus * diagVnorm) + conj(diagIbus) * diagVnorm 
82      dS_dVa = 1j * diagV * conj(diagIbus - Ybus * diagV) 
83   
84      return dS_dVm, dS_dVa 
85