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d2ASbr_dV2.py
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d2ASbr_dV2.py
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# 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 |complex power flow|**2 w.r.t. V.
"""
from scipy.sparse import csr_matrix
from pandapower.pypower.d2Sbr_dV2 import d2Sbr_dV2
def d2ASbr_dV2(dSbr_dVa, dSbr_dVm, Sbr, Cbr, Ybr, V, lam):
"""Computes 2nd derivatives of |complex power flow|**2 w.r.t. V.
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 square of the magnitude of branch complex power flows.
Takes sparse first derivative matrices of complex flow, complex flow
vector, sparse connection matrix C{Cbr}, sparse branch admittance matrix
C{Ybr}, voltage vector C{V} and C{nl 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}
@see: L{dSbr_dV}
@author: Ray Zimmerman (PSERC Cornell)
"""
il = range(len(lam))
diaglam = csr_matrix((lam, (il, il)))
diagSbr_conj = csr_matrix((Sbr.conj(), (il, il)))
Saa, Sav, Sva, Svv = d2Sbr_dV2(Cbr, Ybr, V, diagSbr_conj * lam)
Haa = 2 * ( Saa + dSbr_dVa.T * diaglam * dSbr_dVa.conj() ).real
Hva = 2 * ( Sva + dSbr_dVm.T * diaglam * dSbr_dVa.conj() ).real
Hav = 2 * ( Sav + dSbr_dVa.T * diaglam * dSbr_dVm.conj() ).real
Hvv = 2 * ( Svv + dSbr_dVm.T * diaglam * dSbr_dVm.conj() ).real
return Haa, Hav, Hva, Hvv