Source Code for Module pypower.d2Sbr_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, 
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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 complex power flow w.r.t. voltage. 
18  """ 
19   
20  from numpy import ones, conj 
21  from scipy.sparse import csr_matrix 
22   
23   
24 -def d2Sbr_dV2(Cbr, Ybr, V, lam): 
25      """Computes 2nd derivatives of complex power flow 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 branch power flows. Takes sparse connection 
30      matrix C{Cbr}, sparse branch admittance matrix C{Ybr}, voltage vector C{V} 
31      and C{nl x 1} vector of 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      nl = len(lam) 
46      ib = range(nb) 
47      il = range(nl) 
48   
49      diaglam = csr_matrix((lam, (il, il))) 
50      diagV = csr_matrix((V, (ib, ib))) 
51   
52      A = Ybr.H * diaglam * Cbr 
53      B = conj(diagV) * A * diagV 
54      D = csr_matrix( ((A * V) * conj(V), (ib, ib)) ) 
55      E = csr_matrix( ((A.T * conj(V) * V), (ib, ib)) ) 
56      F = B + B.T 
57      G = csr_matrix((ones(nb) / abs(V), (ib, ib))) 
58   
59      Haa = F - D - E 
60      Hva = 1j * G * (B - B.T - D + E) 
61      Hav = Hva.T 
62      Hvv = G * F * G 
63   
64      return Haa, Hav, Hva, Hvv 
65