Source Code for Module pypower.d2Sbus_dV2

 1  # Copyright (C) 2008-2011 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 2nd derivatives of power injection w.r.t. voltage. 
18  """ 
19   
20  from numpy import ones, conj, arange 
21  from scipy.sparse import csr_matrix as sparse 
22   
23   
24 -def d2Sbus_dV2(Ybus, V, lam): 
25      """Computes 2nd derivatives of power injection w.r.t. voltage. 
26   
27      Returns 4 matrices containing the partial derivatives w.r.t. voltage angle 
28      and magnitude of the product of a vector C{lam} with the 1st partial 
29      derivatives of the complex bus power injections. Takes sparse bus 
30      admittance matrix C{Ybus}, voltage vector C{V} and C{nb x 1} vector of 
31      multipliers C{lam}. Output matrices are sparse. 
32   
33      For more details on the derivations behind the derivative code used 
34      in PYPOWER information, see: 
35   
36      [TN2]  R. D. Zimmerman, I{"AC Power Flows, Generalized OPF Costs and 
37      their Derivatives using Complex Matrix Notation"}, MATPOWER 
38      Technical Note 2, February 2010. 
39      U{http://www.pserc.cornell.edu/matpower/TN2-OPF-Derivatives.pdf} 
40   
41      @author: Ray Zimmerman (PSERC Cornell) 
42      @author: Richard Lincoln 
43      """ 
44      nb = len(V) 
45      ib = arange(nb) 
46      Ibus = Ybus * V 
47      diaglam = sparse((lam, (ib, ib))) 
48      diagV = sparse((V, (ib, ib))) 
49   
50      A = sparse((lam * V, (ib, ib))) 
51      B = Ybus * diagV 
52      C = A * conj(B) 
53      D = Ybus.H * diagV 
54      E = diagV.conj() * (D * diaglam - sparse((D * lam, (ib, ib)))) 
55      F = C - A * sparse((conj(Ibus), (ib, ib))) 
56      G = sparse((ones(nb) / abs(V), (ib, ib))) 
57   
58      Gaa = E + F 
59      Gva = 1j * G * (E - F) 
60      Gav = Gva.T 
61      Gvv = G * (C + C.T) * G 
62   
63      return Gaa, Gav, Gva, Gvv 
64