-
Notifications
You must be signed in to change notification settings - Fork 478
/
d2Sbus_dV2.py
50 lines (39 loc) · 1.63 KB
/
d2Sbus_dV2.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
# Copyright (c) 1996-2015 PSERC. All rights reserved.
# Use of this source code is governed by a BSD-style
# license that can be found in the LICENSE file.
"""Computes 2nd derivatives of power injection w.r.t. voltage.
"""
from numpy import ones, conj, arange
from scipy.sparse import csr_matrix as sparse
def d2Sbus_dV2(Ybus, V, lam):
"""Computes 2nd derivatives of power injection w.r.t. voltage.
Returns 4 matrices containing the partial derivatives w.r.t. voltage angle
and magnitude of the product of a vector C{lam} with the 1st partial
derivatives of the complex bus power injections. Takes sparse bus
admittance matrix C{Ybus}, voltage vector C{V} and C{nb x 1} vector of
multipliers C{lam}. Output matrices are sparse.
For more details on the derivations behind the derivative code used
in PYPOWER information, see:
[TN2] R. D. Zimmerman, I{"AC Power Flows, Generalized OPF Costs and
their Derivatives using Complex Matrix Notation"}, MATPOWER
Technical Note 2, February 2010.
U{http://www.pserc.cornell.edu/matpower/TN2-OPF-Derivatives.pdf}
@author: Ray Zimmerman (PSERC Cornell)
"""
nb = len(V)
ib = arange(nb)
Ibus = Ybus * V
diaglam = sparse((lam, (ib, ib)))
diagV = sparse((V, (ib, ib)))
A = sparse((lam * V, (ib, ib)))
B = Ybus * diagV
C = A * conj(B)
D = Ybus.H * diagV
E = diagV.conj() * (D * diaglam - sparse((D * lam, (ib, ib))))
F = C - A * sparse((conj(Ibus), (ib, ib)))
G = sparse((ones(nb) / abs(V), (ib, ib)))
Gaa = E + F
Gva = 1j * G * (E - F)
Gav = Gva.T
Gvv = G * (C + C.T) * G
return Gaa, Gav, Gva, Gvv