Numerical Optimization Methodology for the
Design of Power Equipment
Gaurav Tewari*
Student Member, IEEE
My M. Hua
Student Member, IEEE
Alexander V. Mamishev
Member, IEEE
Department of EE, University of Washington, Seattle,WA98195
* Massachusetts Institute of Technology, Cambridge, MA 02139
Abstract: Numerical optimization using recurrent finite-element
simulations is a promising direction in design of power equipment.
Previously, the speed and memory limitations of computers made
such an approach unfeasible in comparison with analytical
techniques. Recently, exponential improvements in the computer
industry induced shifts toward numerical algorithms.
The methodology presented here allows an entry-level engineer
with the basic knowledge of electromagnetism to design electric
power equipment for maximum energy efficiency and several other
optimization goals. The search for an optimal set of key design
criteria is demonstrated for the case when the number of possible
variations of design far exceeds computationrrt capabilities of
modern computers.
Keywords: optimization, numerical analysis, energy-efficiency,
magnetic fields
I. INTRODUCTION
A. Objectives
The primary objective of this ongoing project is to create
the methodology and software capable of solving various
optimization problems in the field of electromagnetic
analysis and design.
The adequacy of the developed tools has been tested with a
representative problem: minimization (of magnetic field at a
given location by changing the spacing between the power
line conductors, cancellation
loops,, and ground wire,
Details of calculations presented in this paper are not an
exhaustive treatment of the field minimization problem, but
rather a numerical example of the optimization approach.
B. Background
Significant efforts are directed by the industry and
academics worldwide towards the dev,eloprnent of new
optimization algorithms [1-3]. The increasing speed and
memory capabilities of modern co~putms make possible
algorithmic approaches that could not be implemented ten
years ago. Typical examples include optimization of shape
of induction heating coil windings, shape of poles in
magnetic circuits, lamination of an induction motor, and the
geometry of a superconducting magnetic energy storage
system [4], The example presented here is directly related to
power industry concerns. Possible negative health effects of
power frequency magnetic fields have remained a subject of
controversy for more than ten years [5]. The desire to
minimize magnetic fields from power lines in specific
locations stem from research work conducted over the last
two decades on possible health effects from overhead power
line conductors [6,7]. Even though no conclusive evidence
has been presented to justify the claims of negative health
impact of magnetic fields from power lines, some of utility
companies have been involved in a redesign of their
overhead transmission lines to avoid potential litigation
complications.
For idealized geometry, the calculation of magnetic fields
generated by power line conductors at each specific location
is a trivial algebraic task. However, the presence of manmade and natural objects, uneven profile of ground surface,
and non-homogeneous
conductivity
of the ground at
different depths affect the resulting distribution of magnetic
fields. For this case, the finite element method makes it
possible to compute the magnetic field distribution with
reasonable accuracy. A sample case study is included.
II. SOFTWARE DESCRIPTION
In most cases, the geometrical and physical properties of a
device can be defined parametrically. In this scenario, the
operation principles of the device itself are known in
advance, and the engineer’s task is to find such set of
parameter that would provide the optimum performance.
Analytical techniques are useful and may provide the best
theoretical solutions. However, these techniques are usually
limited
to classical
simple
shapes
and geometric
arrangements.
For more complicated
cases, numerical
simulation is necessary.
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Existing software modules, augmented by our code, are used
as the building blocks in the proposed project. They include
a Maxwell core, parametric analysis module, and Matlab
optimization toolbox.
A versatile electromagnetic simulation package, Maxwell, by
Ansoft Corp. can mesh space in both two-dimensional and
three-dimensional geometry. The two-dirnensiona~ cases can
be either in axisymmetric or in rectangular coordinates.
Voltage and current excitations in dc and ac modes are
available.
The parameterized variables include linear and angular
dimensions of the physical objects, the dielectric and
magnetic properties of all materials, the distribution of
voltage, charge, and current excitations, and the frequency of
operation. Each of these variables can be either a simple
independent variable or a function dependent on other userdefined variables. A sweep of variables is possible with the
parametric analysis module.
Data manipulation, system calls, visualization of the output
and related tasks have been implemented in Matlab, In
addition, the Matlab optimization
toolbox has been
that a variety of
interfaced with Maxwell in such
optimization methods could be tried for each specific
project. A user-friendly interface has been developed in
Matlab GUIDE (Graphical User Interface Development
Environment). The optimization module makes possible the
adaptive variation of the variables during the execution of
the optimization program. The choice of the new values of
the variables comes from the optimization library.
wi~y
Electromagnetic
FE Simulation I
] Maxwell 2D: ] Maxwell 3D: I
1
Electrostatic
Magnetostatic
~ Eddy Current
I AC Conduction
~ DC Conduction
111,IMPLEMENTATION
Figure 1 shows the relationship of the developed interface to
commercial software packages and to types of problems it
can solve. To achieve maximum flexibility, the optimization
routines and functions are implemented within the Matlab
environment. The changes to the Maxwell input files allow
automatic variation of design parameters
during the
optimization process without entering the actual Maxwell
interface. The field computations are initiated by system
function calls.
The screenshot of the graphical user interface (GUI) is
shown in Figure 2 and its flowchart is shown in Figure 3.
The GUI results from the effort to generalize use of A4atlab’s
optimization routines with Maxwell projects. Any number
of Maxwell parameters can now be chosen as variables to be
optimized, and their initial values can be specified through
the interface. An option of excluding specific variables from
the optimization has also been added to complement this
change.
Furthermore, all of the optimization algorithms
offered by Matlab are available to solve various userspecified cost functions. Finally, the writing of these cost
functions has been simplified to exclude any need to
explicitly call Maxwell,
They are now simple Matlab
functions of vector input and output.
In addition, figure 2 also shows the details of a generic
optimization process itself, as dictated by the Maxwell
interface. The first five steps are executed within Maxwel/
environment, and the last four are executed within Matlab
environment. The list of input variables and cost function
components may be expanded using indirect computation of
variables of interest.
Electrostatic
Magnetostatic
Eddy Current
L—
I
/
1
L..
I
~$
❑
/
i
+
Simulation:
Our Software
-
Commercial Software
~
Applications
Figure 2: Screenshot of GUI module
Figure 1: The module for solving optimization and inverse
problems interfaces with several software packages. Interfacing
with Maxwell is implemented through system calls and input file
mampulation.
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B
A
c
r%
Start Maxwell if
Ansoft directory
name is valid
“Start MaxwelV
button activated
I
-1
I
I
Is the name of
the Optimization
routine valid?
Figure 4: A two-dimensional representation of an idealized threephase power line with elements of landscape.
I
Figure 3: Schematic depiction of GUI organization
The underlying program path structure was generalized so
that our module can be interfaced with different Maxwell
projects simply by entering their path. The generalized
variable names allow the optimization module to be used
with all the applications of Maxwell, including: permittivity,
conductivityy, force, and torque calculations. In addition to
the aforementioned functionality, our module also includes
all of Matlab’s algorithms, a simplified method for writing
the cost functions, the ability to exclude variables from and
the ability to include constraints in the optimization problem.
IV. APPLICATION EXAMPLE:
MINIMIZATION OF MAGNETIC FIELDS
A. Model Setup.
The model includes an idealized three-phase power line
along with landscape elements, which cannot be modeled
analytically. Landscape elements include a house, a tree,
uneven ground terrain, and a metal pipe underneath the
house. The model enables us to target a local field reduction,
for example, near the house. This is a two-dimensional
model, although a three-dimensional model could be used if
4 shows
the two-dimensional
necessary.
Figure
representation.
The tree is located 5 meters to the left of phase A, and has a
height of 10 meters. The house is located 4 meters to the
right of phase C, and has a maximum height of 6 meters
above the ground level. The center of the metal pipe, is
located 2 meters below ground level. The pipe has a
diameter of 1 cm.
B. Materials and Excitations
The tree is defined to have a relative permittivity of 10,
relative permeability
of 1, and conductivity of 10”4
siemens/meter. The house is defined to have a relative
permittivity
of 10, relative permeabilityy of 1, and
conductivity of 10-7 siemens/meter. The ground is defined to
have a relative permittivity of 6, relative permeability of 1,
and conductivity of 10-5 siemens/meter. The metal pipe
under the house is taken to be a perfect conductor. The
background is modeled as air. Three-phase balanced voltage
and current are assigned.
C. Constraints
The constraints involved in this model were targeted at
maintaining reasonable and realistic values for the horizontal
and vertical displacements of the 3 power lines, and
guaranteeing a minimum amount of separation between the
In other words, the design must provide
conductors.
sufficient electrical insulation distances between each phase
and ground as well as between phases themselves and then
limited to a reasonable width of the pole cross arm. The
order of magnitude of values selected for these parameters is
close to industrial standards. However, it does not
necessarily meet specifications of the particular power line
design, since it is unnecessary for a feasibility study. The 3
conductors were forced to lie within an imaginary box of
height 10 meters and width 32 meters. This was achieved by
ensuring that:
xi <16,
20<y,
J% -~,)’
<30,
+(Y, -Y,)2
iel,2,3
iel,2,3
<15> (i, j)= (2,3)
(1)
(2)
(3)
where Xi, XJ and y,, y, are the horizontal and vertical
coordinates of each phase with respect to the origin located
under the middle wire on the ground level, and i k j.
Inequality (1) ensures that the conductors are close enough
to the supporting tower, inequality (2) guarantees that
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ground insulation clearances and the maximum suspension
height requirements are met, and inequalities (3) and (4)
guarantee the required insulation distance between the
phases.
The cost function
f is defined as
f=(a:+a; ),
where
& and
2.Y are
the
horizontal
and
(5)
vertical
components
of the magnetic flux linkage. If one or more constraint
violations are detected, then the cost function is modified
from its above form proportionally to the magnitude of the
constraint violation, in order to provide meaningful feedback
for the optimization routines. The horizontal displacement of
the first conductor is measured to the left of the origin, and,
thus, should be treated as a negative value. For the sake of
simplifying the optimization process, only the absolute
magnitude of this displacement was used during the solution
finding process. The negative sign preceding ~ in the
constraints above has accounted for its negative value.
computations is fairly small for a six-variable optimization.
The starting values were selected judiciously, which helped
to improve the convergence. Figure 5 shows the major result
of this feasibility study by analyzing the resulting magnetic
flux magnitude and density, Shown on the left vertical axis
is the magnetic flux magnitude.
The magnetic flux
magnitude at the specified location has reduced from about
0.5 mWb to about 0.1 mWb, or, by a factor of five, Shown
on the right vertical axis is the magnetic flux density, which
is more commonly used in the studies of this type. From the
magnetic flux magnitude in Webers, the magnetic flux
density in milligauss may be readily computed by
multiplying the flux magnitude by 109. This conversion
assumes 10 cm long and 10 cm high integration path at the
point of interest.
6“E-07~—~
60
D. Optimization Algorithm
A separate future paper will discuss selection of algorithms
and operating conditions that ensure reasonable
fast
convergence. The Nelder-Mead simplex search was the most
successful approach to numerical optimization. This is a
direct search method, which does not require computation of
gradients or any other derivative information, For instance,
if x is a vector of length n, then a simplex in n-dimensional
space is characterized by the n-+-l distinct vectors, which
form the vertices of x. In two-dimensional space, a simplex
is simply a triangle.
The search progresses by generating a new point in or near
the current simplex under consideration. By comparing the
value of the specified cost function at the new point with its
value at the vertices of the simplex, the algorithm determines
whether one of the vertices of the current simplex ought to
be replaced by the new point. Thus a new simplex,
corresponding to a new “setup,” is generated and the process
is repeated until the diameter of the simplex becomes
smaller than the tolerance specified by the user.
The termination tolerances for the vector of input parameters
and the cost function were set to be 0.5 cm and 2e-010
Webers, respectively. Further, the maximum and minimum
step sizes, as deemed to be most appropriate for the
configuration at hand from trial runs, were established to be
0.5 cm and 5.0 cm.
60
80
100 120 140160180200
Setup
Figure 5: Magnetic flux magnitude vs setup number (left vertical
axis) and Magnetic flux density YS setup number (right vertical
axis) as the optimization algorithm searches for the best possible
arrangement
6 differs in shape from
The cost function,
shown
in Figure
the previous two functions only by the presence of spikes
caused by constraint violation penalties
1.E-04
]
1.E-05
1c0
.G
2 1.E-06~
%
0
0 1.E-07-
,0-8.
~~~~
0
E. Results.
10
O.E+OO~TmZT
0
20 40
20
40
60
80 100120140160180200
Setup
In this example, it took about 200 steps to converge to an
optimum value. Such a number of the cost function
Figure 6: Value of cost function vs setup number.
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..
.
. .
.
.
. ..
More lnslghts mto the optlmlzatlon process can be gamed by
analyzing changes in the conductor coordinates.
Figure 7
shows the variation of vertical coordinates, with an obvious
high degree of cross-correlation between phases A (yl) and
C (y3), a behavior that is sometimes characteristic of simplex
search algorithms. Figure 8 shows the variation of the
horizontal components. It is clear that the optimum was
achieved after about 150 steps, but the convergence criteria
were too strict to stop optimization at that point.
=.~—–
—7
19.~—
13.5
14.5
15.5
Horizontal displacement
Figure 9: History of coordinates
16.;
of phase A, xl (m)
of conductor A (x,, y,) vs setup
number.
.
“g 17
.a)
i
> 15~
O
32
I
20
40
60
80 100120140160180200
Setup
F]gure 7: Vertical displacement
setup number.
F
~—
of conductors
A, B and C vs
16.5 T-
] 0.22
E- 28E
8
ru26—$
.;24.—
-e
’22-
s
20 o~,r
0.2
Horizontal displacement, x2 (m)
0.2!
Figure 10: History of coordinates of conductor B (x*, y2) vs setup
number.
‘
12.5~0.10
O 20 40 60 80 100120140160180200
Setup
Figure 8: Horizontal displacement of conductors A and C vs setup
number (left vertical axis) and horizontal displacement
of
conductor B vs setup number (right vertical axis)
The same data can be visualized from the geometrical
position point of view, as shown in Figures 9, 10 and 11.
The overall tendency was to try to move the conductors as
far as possible to the left and arrange them in an
approximately triangular configuration. Notice that although
an ideal equilateral triangle can be formed within the
specified constraints, it does not represent an optimum
solution because of close proximity of the house and other
landscape elements.
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32
VII. REFERENCES
[1]. T. Onuki and Y. Toda, “optimal design of hybrid magnet in maglev
system with a both permanent and electro magnets,” IEEE Trarrxmwns on
Magnetics, vol. 29, pp. 1783-1786, March 1993.
[2]. S. R. H. Hoole and M. K.. Haldar, “Optimization of electromagnetic
devices: Circuit models, neural networks and gradient methods in concert,”
IEEE Transactions on Magnetics, vol. 31, pp. 2016--2019, May 1995.
[3]. J. A. Ramirez and E. M. Freeman, “Shape optimization
electromagnetic devices including eddy currents, ” IEEE Transactions
Magnetics, vol. 31, pp. 1948-1951, May 1995.
[4]. P. Alotto, A. V. Kuntsevitch, C. Magele, G. Molinari, C. Paul, K. Preis,
M. Repetto, and K. R. Richter, “Multiobjective
optimization
in
magnetostatics: a proposal for benchmark problems, ” IEEE Tnmsactions on
Magrrerk’s, vol. 32, pp. 1238--1241, May 1996.
8
5
15
15.5
Horizontal displacement,
Figure 11: History of coordinates
16
x3 (m)
16.5
of conductor C (xj, ys) vs setup number.
V. CONCLUSIONS
[5], K. Fitzgerald, 1. Nair, and G. Morgan, “Electromagnetic
jury’s still out,” IEEE Spectrum, pp.22-35. Aug. 1990.
Fields: The
[6]. W. Kaune and L. Zaffanella, “Analysis of magnetic fields produced far
from electric power lines,” IEEE Tramacrirms of Power Dehvery, vol. 7,
pp. 2082--2091, Oct. 1992.
[7]. R.G. Olsen and C.E. Lynn, “Modeling of extremely low frequency
magnetic field sources using mrrltipole techniques,” IEEE Transactions of
Power Delivery, vol. 11, pp. 1563--1570, July 1996.
A brief summary of the results follows:
1
of
on
A versatile
tool for solving a variety of
optimization
problems through adaptive finite
element analysis has been developed.
2
Applications
include power equipment design,
insulation coordination, management of electric and
magnetic field distribution, MEMS and sensor
optimization, etc.
3
Problems already solved involved up to 6 variables.
4
The computation
weeks.
5
Future work should concentrate on the stability of
tasks
highly
intensive
computational
and
development of tools for treating discrete problems.
time varies between minutes and
The problem of minimization of power line magnetic fields
in the specified location has been solved for a geometrical
arrangement that does not have closed-form analytical
solutions.
A six-variable optimization process has been
demonstrated for the case where geometrical constraints
This
were represented as penalties to the cost function.
study demonstrates an industrial potential of the proposed
approach and a proper operation
of the developed
optimization interface.
VIII. BIOGRAPHIES
Gaurav Tewari is currently a graduate student in the Department of EECS
at MJT. He has worked as a research assistant at the MIT Laboratory for
Electronic and Electromagnetic systems as well as the MIT Media Lab, and
as an intern at fBM Research. He is a recipient of MIT Robert Fano Award
for outstanding student research project, winner of the fEEE Lance Stafford
Outstanding Research Paper Award, and a recepient of a Regional Power
Engineering Scholarship.
My Hua is currently an undergraduate student at the Department of
Electrical Engineering, University of Washmgtom She IS currently UW
Mary Gates Scholar and a recipient of American Public Power Association
Demonstration
of
Energy-Efficient
Developments
Undergraduate
Scholarship. She is conducting research m the area of power-related
numerical computation and signal processing.
Alexander Marnishev received an equivalent of B.S. degree from the Kiev
Polytechnic Institute, Ukraine, in 1992, MS. degree from Texas A&M
University in 1994, and a Ph. D. degree from MIT in 1999, all m electrical
engineering. Currently he is an Assistant Professor and Director of SEAL
(Sensors, Energy, and Automation Laboratory) in the Department of
Electrical Engineering, University of Washington, Seattle. Prof. Mamishev
is the author of about 40 journal and conference papers, and one book
chapter. His research interests include sensor design, electric power and
electrical insulation, electromagnetic,
optimization, and inverse problem
theory. He serves as a reviewer for tEEE Transactions on Power Delivery
and tEEE Transactions on Dielectrics and Electrical Insulation.
VI. ACKNOWLEDGMENTS
The authors gratefully acknowledge the support of the
American Public Power Association Demonstration EnergyEfficient Developments program.
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