pypower/d2Sbr_dV2.py

   1 # Copyright (c) 1996-2015 PSERC. All rights reserved.
   2 # Use of this source code is governed by a BSD-style
   3 # license that can be found in the LICENSE file.
   4
   5 """Computes 2nd derivatives of complex power flow w.r.t. voltage.
   6 """
   7
   8 from numpy import ones, conj
   9 from scipy.sparse import csr_matrix
  10
  11
  12 def d2Sbr_dV2(Cbr, Ybr, V, lam):
  13     """Computes 2nd derivatives of complex power flow w.r.t. voltage.
  14
  15     Returns 4 matrices containing the partial derivatives w.r.t. voltage angle
  16     and magnitude of the product of a vector C{lam} with the 1st partial
  17     derivatives of the complex branch power flows. Takes sparse connection
  18     matrix C{Cbr}, sparse branch admittance matrix C{Ybr}, voltage vector C{V}
  19     and C{nl x 1} vector of multipliers C{lam}. Output matrices are sparse.
  20
  21     For more details on the derivations behind the derivative code used
  22     in PYPOWER information, see:
  23
  24     [TN2]  R. D. Zimmerman, I{"AC Power Flows, Generalized OPF Costs and
  25     their Derivatives using Complex Matrix Notation"}, MATPOWER
  26     Technical Note 2, February 2010.
  27     U{http://www.pserc.cornell.edu/matpower/TN2-OPF-Derivatives.pdf}
  28
  29     @author: Ray Zimmerman (PSERC Cornell)
  30     """
  31     nb = len(V)
  32     nl = len(lam)
  33     ib = range(nb)
  34     il = range(nl)
  35
  36     diaglam = csr_matrix((lam, (il, il)))
  37     diagV = csr_matrix((V, (ib, ib)))
  38
  39     A = Ybr.getH() * diaglam * Cbr
  40     B = conj(diagV) * A * diagV
  41     D = csr_matrix( ((A * V) * conj(V), (ib, ib)) )
  42     E = csr_matrix( ((A.T * conj(V) * V), (ib, ib)) )
  43     F = B + B.T
  44     G = csr_matrix((ones(nb) / abs(V), (ib, ib)))
  45
  46     Haa = F - D - E
  47     Hva = 1j * G * (B - B.T - D + E)
  48     Hav = Hva.T
  49     Hvv = G * F * G
  50
  51     return Haa, Hav, Hva, Hvv