Source Code for Module pypower.dAbr_dV

 1  # Copyright (C) 1996-2011 Power System Engineering Research Center (PSERC) 
 2  # Copyright (C) 2011 Richard Lincoln 
 3  # 
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16   
17  """Partial derivatives of squared flow magnitudes w.r.t voltage. 
18  """ 
19   
20  from scipy.sparse import csr_matrix 
21   
22   
23 -def dAbr_dV(dSf_dVa, dSf_dVm, dSt_dVa, dSt_dVm, Sf, St): 
24      """Partial derivatives of squared flow magnitudes w.r.t voltage. 
25   
26      Returns four matrices containing partial derivatives of the square of 
27      the branch flow magnitudes at "from" & "to" ends of each branch w.r.t 
28      voltage magnitude and voltage angle respectively (for all buses), given 
29      the flows and flow sensitivities. Flows could be complex current or 
30      complex or real power. Notation below is based on complex power. The 
31      following explains the expressions used to form the matrices: 
32   
33      Let Af refer to the square of the apparent power at the "from" end of 
34      each branch:: 
35   
36          Af = abs(Sf)**2 
37             = Sf .* conj(Sf) 
38             = Pf**2 + Qf**2 
39   
40      then ... 
41   
42      Partial w.r.t real power:: 
43          dAf/dPf = 2 * diag(Pf) 
44   
45      Partial w.r.t reactive power:: 
46          dAf/dQf = 2 * diag(Qf) 
47   
48      Partial w.r.t Vm & Va:: 
49          dAf/dVm = dAf/dPf * dPf/dVm + dAf/dQf * dQf/dVm 
50          dAf/dVa = dAf/dPf * dPf/dVa + dAf/dQf * dQf/dVa 
51   
52      Derivations for "to" bus are similar. 
53   
54      For more details on the derivations behind the derivative code used 
55      in PYPOWER information, see: 
56   
57      [TN2]  R. D. Zimmerman, I{"AC Power Flows, Generalized OPF Costs and 
58      their Derivatives using Complex Matrix Notation"}, MATPOWER 
59      Technical Note 2, February 2010. 
60      U{http://www.pserc.cornell.edu/matpower/TN2-OPF-Derivatives.pdf} 
61   
62      @return: The partial derivatives of the squared flow magnitudes w.r.t 
63               voltage magnitude and voltage angle given the flows and flow 
64               sensitivities. Flows could be complex current or complex or 
65               real power. 
66      @see: L{dIbr_dV}, L{dSbr_dV} 
67   
68      @author: Ray Zimmerman (PSERC Cornell) 
69      @author: Richard Lincoln 
70      """ 
71      il = range(len(Sf)) 
72   
73      dAf_dPf = csr_matrix((2 * Sf.real, (il, il))) 
74      dAf_dQf = csr_matrix((2 * Sf.imag, (il, il))) 
75      dAt_dPt = csr_matrix((2 * St.real, (il, il))) 
76      dAt_dQt = csr_matrix((2 * St.imag, (il, il))) 
77   
78      # Partial derivative of apparent power magnitude w.r.t voltage 
79      # phase angle. 
80      dAf_dVa = dAf_dPf * dSf_dVa.real + dAf_dQf * dSf_dVa.imag 
81      dAt_dVa = dAt_dPt * dSt_dVa.real + dAt_dQt * dSt_dVa.imag 
82      # Partial derivative of apparent power magnitude w.r.t. voltage 
83      # amplitude. 
84      dAf_dVm = dAf_dPf * dSf_dVm.real + dAf_dQf * dSf_dVm.imag 
85      dAt_dVm = dAt_dPt * dSt_dVm.real + dAt_dQt * dSt_dVm.imag 
86   
87      return dAf_dVa, dAf_dVm, dAt_dVa, dAt_dVm 
88