Stochastic Modeling of an Inhomogeneous Magnetic Reluctivity

XXIV Symposium
Electromagnetic Phenomena in Nonlinear Circuits
June 28 - July 1, 2016 Helsinki, FINLAND
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Stochastic Modeling of an Inhomogeneous Magnetic
Reluctivity Using the Karhunen-Loève Expansion and the
Hierarchical Matrix Technique
Radoslav Jankoski1,2 , Ulrich Römer1,2 and Sebastian Schöps1,2
1
2
Technische Universität Darmstadt, Institut für Theorie Elektromagnetischer Felder, Darmstadt D-64289, Germany
Technische Universität Darmstadt, Graduate School of Computational Engineering, Darmstadt D-64293, Germany
{jankoski, roemer, schoeps}@gsc.tu-darmstadt.de
Abstract—In this paper the magnetic reluctivity of the core of a
single phase transformer has been modeled as a random field by
using the Karhunen-Loève expansion (KLE). The KLE enables
representing random fields with a minimal number of random
variables but on the other hand it requires computing eigenvalues
and eigenvectors of a generalized eigenvalue problem with dense
matrices. To overcome this difficulty the Lanczos algorithm has
been combined with the hierarchical matrix technique in order
to reduce the complexity of performing arithmetical operation
with dense matrices.
and it can be used for uncertainty quantification (UQ). One
can study the impact of the uncertainties introduced in the
magnetic reluctivity on the output parameters of the electrical
machines. One example, which is presented in this paper,
is the uncertainty of the spatial distribution of the magnetic
reluctivity in a single phase transformer where, for instance,
its inductance can be an output parameter of interest.
I. INTRODUCTION
For optimization of electrical machines that contain magnetic materials, as a part of their construction, having an
accurate knowledge of the magnetic behaviour law expressed
via the magnetic reluctivity plays an important role. However,
in practice there is a lack of knowledge of the magnetic
reluctivity because the industrial process of production introduces uncertainties in this material property. To obtain reliable
results from numerical simulations it is necessary to study
the impact of those uncertainties. The uncertainties in the
material properties are typically modeled as discrete random
variables as it is done in [3]. In this paper we propose to
use the Karhunen-Loève expansion (KLE), to discretize the
random field, due to its error minimizing property, i.e., the
truncation error is minimal when compared to any other mterm approximation in the mean square sense [4].
The KLE of a random field requires the solution of a
computationally expensive generalized eigenvalue problem
with a fully populated (dense) matrix on the left hand side
of the matrix equation. In order to reduce the computational
costs for solving the generalized eigenvalue problem, authors
in [5] have combined the iterative Krylov subspace (Lanczos)
method with the hierarchical matrix technique. The Lanczos
eigenvalue solver requires only matrix-vector multiplications
with a dense matrix as an arithmetical operation. By using the
hierarchical format of matrices the complexity of the matrixvector multiplications is reduced.
Once the KLE is computed the magnetic reluctivity can
be represented via minimal number of a random variables
The system of interest is shown in Fig. 1. The problem has
been simplified by analyzing the system in a 2D representation
in a domain Ω with boundary ∂Ω.
II. P ROBLEM D ESCRIPTION
Fig. 1: Single phase transformer
The behaviour of the system is described with Ampere’s
law in the 2D magnetostatic formulation,
−∇ · (ν(·, ω)∇Az (ω)) = Jz
Az (ω) = 0
in Ω,
on ∂Ω,
(1)
where Az is the z component of the magnetic vector potential
and ν is the magnetic reluctivity of the core that depends on the
outcome of a random event ω. The current sources are given
by the current density Jz . To obtain the KLE for the magnetic
reluctivity the eigenvalues and the eigenfunctions need to be
computed efficiently in the domain where the core of the single
phase transformer is located.
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143
III. K ARHUNEN -L O ÈVE E XPANSION AND H IERARCHICAL
M ATRICES
A. Karhunen-Loève Expansion
The correlation length of the random field is denoted as d and
d = 1 for this computations.
The first eigenfunction is shown in Fig. 2 for illustration
purposes.
The KLE is given for an inhomogeneous magnetic reluctivity as follows:
~
ν(~x, ω) ≈ ν(~x, ξ(ω))
= ν(~x) +
m p
X
λi fi (~x)ξi (ω),
(2)
i=1
where ν is the mean value of the random field, ~x is a vector of
spatial coordinates and ξ~ is a vector of mutually uncorrelated
random variables. The random variable ξi is evaluated as
follows:
Z
1
(3)
ξi (ω) = √
(ν(~x, ω) − ν(~x))fi (~x)d~x.
λi Ωc
The eigenfunctions fi and the eigenvalues λi that appear
in the KLE are obtained by solving the Fredholm integral
equation:
Z
Cov(~x, ~y )fi (~y )d~y = λi fi (~x),
Fig. 2: First eigenfunction
Fig. 3 depicts the magnitude of the largest 30 eigenvalues.
(4)
Ωc
where Cov is the covariance function, which is usually deduced from measurements, and Ωc is the domain of the
transformer core. The domain Ωc is triangulated and the
solution for the eigenfunction fi is approximated as piecewise
constant within one triangle. By applying the Galerkin method
the equation (4) results in a generalized eigenvalue problem,
Af = λBf ,
(5)
with matrices A, B ∈ Rn×n and eigenvector f ∈ Rn . The
number of triangles is denoted as n. Matrix A is a dense and
symmetric and matrix B is diagonal.
B. Hierarchical Matrices
The idea of the hierarchical matrix technique is to find
subblocks of the matrix and perform a low-rank approximation
that enables matrix vector multiplication with almost linear
complexity O(n log n). A certain subblock à in the matrix
A is low rank approximated if it satisfies the so called
admissibility condition which is related to the geometry of
the system. If two blocks of the domain of interest are well
separated by a distance their mutual interaction is negligible
and thus the low-rank approximation is allowed. Fundamental
theory of hierarchical matrices can be found in [2].
IV. R ESULTS
For verification of the numerical algorithm the generalized
eigenvalue problem given with the equation (5) has been
solved by using the open source H2Lib library [1] and our
own implemented Lanczos eigenvalue solver. The covariance
function is assumed to be given in advance instead of deducing
it from measurements,
!
||~x − ~y ||l1
Cov(~x, ~y ) = exp −
.
(6)
d
Fig. 3: Eigenvalues
A rather slow decay of the eigenvalues is observed, i.e.,
only one order of magnitude.
V. C ONCLUSION AND O UTLOOK
An efficient approach for computing the KLE based on the
Lanczos algorithm and the hierarchical matrix technique has
been presented in this paper. The KLE has been computed in a
domain where the core of a single phase transformer is located.
The computational benefits will be discussed in the full paper
where the presented approach is compared with the standard
eigs MATLAB function (also based on Lanczos algorithm). In
future work the KLE will be applied for stochastic modeling
of magnetic materials with a nonlinear, inhomogeneous and
anisotropic magnetic reluctivity.
VI. ACKNOWLEDGMENT
The work in this paper has been supported by the Graduate
School of Computational Engineering (GSCE) at the Technische Universität (TU) Darmstadt.
R EFERENCES
[1] H2lib public repository. https://github.com/H2Lib.
[2] M. Bebendorf. Hierarchical Matrices: A Means to Efficiently Solve
Elliptic Boundary Value Problems. Springer, 2008.
[3] R. Ramarotafika, A. Benabou and S. Clénet. Stochastic modeling of soft
magnetic properties od electrical steels: Application to stator of electrical
machines. IEEE Transaction on Magnetics, 48(10):2573–2584, 2012.
[4] R. G. Ghanem and P. D. Spanos. Stochastic Finite Element Method: a
spectral approach. Springer-Verlag New York, Inc., 1991.
[5] B. N. Khoromskij, A. Litvinenko and H. G. Mathies. Application
of hierarchical matrices for computing the Karhunen-Loève expansion.
Computing, 84:49–67, 2009.
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Proceedings of EPNC 2016, June 28 - July 1, 2016 Helsinki, FINLAND