Download Optimized Design of Salient Pole Synchronous Generators

Optimized Design of Salient Pole Synchronous Generators
O.W. Andersen
Steinhaugen 43, N-7049 Trondheim, Norway
[email protected] http://home.c2i.net/owand/
ABSTRACT
A program for optimized design of salient pole synchronous
generators, Synop, was made by the author for mainframe
computers as early as in 1970. It has been improved continuously since that time and now runs on personal computers. It covers all sizes of air cooled machines from small
two pole generators of a few kilowatts up to the largest
hydroelectric generators.
If the penalty function has the proper form and the input data
make it possible, the penalty is reduced to zero as the criterion
function cost+penalty is minimized. In this way the technical
requirements are met, without any need for the program to be
told specifically how this is to be accomplished.
Keywords: Optimized design, synchronous generators, salient
poles.
1 INTRODUCTION
The objective is to design to the lowest cost, including the cost
of losses, at the same time meeting all performance requirements automatically.
A few thousand alternative designs are usually evaluated in less
than 5 seconds on a modern computer. All the designs are
realistic, in the sense that turns per coil are rounded to the
nearest integers and numbers of slots and parallel circuits meet
the requirements of winding balance.
Since in this way the cost as a function of the design variables
is discontinuous, the method of optimization must be able to
handle such functions. It is described in the references and in
more detail at the author’s web site (TRA2 user’s manual). The
description here will be very brief.
2 OPTIMIZING METHOD
The optimizing method is a Monte Carlo method, based on
random numbers. It was first used in a transformer program
made for National Industri in Norway (now ABB) in 1965.
The cost of the generator plus the cost of losses is minimized.
Some of the performance requirements are taken care of during
the design synthesis, for others a penalty term is added to the
cost for designs that do not meet the requirements. A typical
penalty function is shown below.
% penalty
Permissible range
with penalty = 0
Transient reactance
Fig. 1. Penalty function for transient reactance.
Fig. 2. Hydroelectric generator.
The design variables are:
Stator:
Rotor:
1. Inside diameter
8. Flux density in pole base
2. Slot width
9. Thickness of field coil
3. Ratio slot/tooth width
10. W/cm2 surface loading
4. A/cm stator loading
5. W/cm2 surface loading
6-7. Flux densities in teeth and yoke
If the customer is willing to pay extra for each percent the
synchronous reactance is below a certain limit, xd is also a
design variable.
For each design variable, min and max limits are specified in
the input, and five starting points for the optimization are based
on these limits. The optimization proceeds from each starting
point with small changes in the variables between successive
alternative designs. The changes are partly random, but the
method is such that the optimization proceeds systematically in
the direction of better designs as long as possible. The best end
point from the five starting points becomes the starting point
for a final stage of the optimization, where the changes are
smaller between successive designs, and it usually becomes
possible to get even closer to the true optimum.
In addition to the transient reactance, other technical
requirements that are met by means of penalty functions are for
temperature rise in stator and rotor, total and open circuit losses
and clearance between field coils.
3 DESIGN SYNTHESIS
The complete optimizing program consists basically of three
parts, design synthesis, analysis and the optimizing logic.
Design synthesis is the process by which a design is produced,
on the basis of:
A set of values for the design variables.
The machine rating and other technical specifications.
Design choices made by the program user (winding types, etc.).
The manufacturer’s normal design practice.
Since min and max limits are specified for the design variables,
they should be as meaningful as possible as design parameters.
It must be possible for the synthesis to proceed in a straightforward manner from a set of values, which is chosen by the
optimizing logic for each alternative design. Some of the variables listed earlier will now be explained in more detail.
A/cm stator loading is total current in a stator slot divided by
the slot pitch (distance between slot centerlines). A higher
value gives a shorter machine, but with a tendency of higher
copper losses, higher excitation requirement and higher reactances. It is a very important design variable.
Stator flux densities are at open circuit, since they are of
importance for the open circuit core loss and don’t change
much with the load. On the other hand, the flux density in the
pole base is at rated load, since it is only of importance for the
excitation requirement and varies considerably with the load.
W/cm2 surface loading is loss per unit length divided by the
surface area through which the loss is dissipated. It is directly
related to the temperature rise. For the stator winding, loss is
taken as dc loss, but could have included the small addition for
eddy current loss. Surface area is slot circumference below the
wedge times unit length. For the field winding, the surface area
is only for the outer surface, since very little heat is dissipated
inward through the pole insulation and radially through the
veneers.
The next step is to establish the number of slots. The program
should try to come as close as possible to the desired value for
the stator loading, because it is such an important variable. For
this reason, the desired slot pitch may have to be adjusted.
Based on the slot pitch and the inside stator diameter (variable
1), an approximate number of slots can be found. It also has to
be adjusted, due to the requirements of winding balance.
Now the cross section of the slot can be designed, including the
depth, arrangement and dimensions of the strands, and insulation allowances. The insulation is given by the voltage. The
division of the bar into parallel strands is determined largely by
the permissible eddy current loss. The slot depth is found by
trial and error until it gives the desired value of W/cm2 surface
loading (variable 5).
The stator stack length is found on the basis of the desired tooth
density (variable 6). At the same time, the number of ventilating ducts is determined, assuming a standard duct pitch and
duct width. The outside stator diameter is found on the basis of
the desired yoke density (variable 7). This now completes the
synthesis of the stator.
The air gap is determined from the specified synchronous
reactance, and the program then proceeds inward with the pole
tip, pole body, field winding and rotor rim.
The width of the pole tip (pole chord) is determined from a
fixed per unit pole arc, covering perhaps 70% of the pole pitch.
The outer edge must be deep enough for the mechanical
strength at overspeed. The pole tip can be shaped as a circular
arc with a smaller radius than the stator, so that the air gap is
perhaps 50% larger at the outer edge than at the center. This
keeps the harmonic content in the air gap flux wave within
reasonable limits, but manufacturers have different ideas about
this.
The width of the pole body is primarily determined by the
desired flux density in the pole base (variable 8), and the depth
primarily by the thickness (variable 9) and the W/cm2 surface
loading (variable 10) of the field coil. The synthesis is complicated by the fact that the pole depth affects the leakage flux,
and thereby the width required to carry the flux. Vice versa the
width affects the available space for the coil, and thereby the
depth required to get the desired surface loading. Final dimensions can therefore only be found after a number of loops and
corrections.
Some rounding of dimensions is normal design practice. The
synthesis starts with the rounding of inside stator diameter to
the nearest 10 mm and slot width to the nearest 0.5 mm. Such
rounding contributes to making the criterion function discontinuous and the optimization more difficult, but is only of
marginal importance in this respect compared with the adjustments of numbers of turns, slots and parallel circuits.
It is convenient to have the thickness of the field coil expressed
in a per unit system, so that the value one corresponds to the
surface of the coil being flush with the pole tip.
The program now establishes the number of parallel circuits in
the stator winding. An approximate slot pitch is given by the
slot width (variable 2) and the ratio slot/tooth width (variable
3). Assuming that a normal two layer one turn bar winding is
used, the approximate current per circuit is calculated as the
A/cm stator loading (variable 4) times half the slot pitch.
Knowing this, the program can establish an approximate
number of circuits from the kVA and specified min and max
limits for the stator voltage (the voltage can also be fixed). The
number of circuits is adjusted to the nearest number that gives a
balanced winding, depending on the number of poles.
In low speed generators, the optimum diameter is often such
that the rotor inertia and the rim stress are both at their limits
simultaneously. It is interesting to observe how this is achieved
accurately and automatically, even though there is nothing
specifically about this in the program.
For a hydroelectric generator, the rotor synthesis is completed
by finding the depth of the rim from the required inertia and the
permissible stress at overspeed.
It is important to note that the values for the independent
variables had to be adjusted during the design synthesis in
order to achieve a practical design. Therefore a distinction must
be made between the actual values that were obtained, and the
desired values that were aimed for.
4 DESIGN ANALYSIS
Because typically more than a thousand alternative designs are
evaluated, it is not practical to employ the most sophisticated
and time consuming methods of analysis, such as finite element
magnetic field calculations, for every one of them. However,
such analysis has been used to improve conventional formulas,
for example for pole leakage flux and stator leakage reactance.
Every time a new design is made, tests versus calculations
should be checked for similar machines, where this is available.
For such complex machines, calculations are approximate,
based on simplifications, assumptions and experience. Some
correction factors must be specified in the program input.
In the calculation of excitation, the hard part is the addition of
saturation ampereturns. This is partly because they are very
sensitive to small differences in flux density, the shape of the
flux in the air gap changes with saturation and load, and
ampereturns at the junction pole-rim is affected by the
roughness of the surfaces that are joined together at this point.
Joints between segments and local flux density variations near
wedges, dovetails and ventilating ducts also complicate
matters.
loss is of a similar nature as described earlier for the open
circuit condition.
Another significant part of the stray load loss is the loss
occurring in the clamping structure at the stator ends, due to
flux mostly set up by the stator current in the end windings.
What happens here is also of a very complex nature, but it is
possible to come to some conclusions about the dependency
upon diameter, pole pitch and stator loading.
Windage and friction loss is a function of air gap diameter,
stack length and revolutions per minute (RPM). The nature of
the function can be determined from statistical analysis of tests
on similar machines. Insight can be gained from the study of
airflow and friction in the bearings, especially about
possibilities for loss reduction, but a purely theoretical
calculation of the total loss is probably hopeless.
Temperature rises in stator and rotor are calculated based on
equivalent thermal circuits, calculated losses and estimated
airflow and speeds. Again, due to the uncertainties involved,
comparisons with tests on similar machines are essential.
5 FINITE ELEMENT ANALYSIS
Even more difficult is the calculation of open circuit core loss.
Simply adding up loss in the stator teeth and yoke based on
calculated flux densities and Epstein tests produces a loss
which must be multiplied by a factor of at least 2 to get the loss
in the actual machine.
The discrepancy is explained partly by loss in the pole face,
due to variations in flux density here caused by the movement
of the pole across stator teeth and slots. Most of the loss usually
occurs in the pole punchings, which are quite thick and only
partly insulated from each other with naturally occurring
surface layers of oxide. They are partly short-circuited by burrs
and through the uninsulated amortisseur (damper) bars. Part of
the loss is also due to slot frequency currents induced in the
amortisseur bars themselves and is heavily affected by the ratio
of slot pitches in stator and rotor.
Local flux densities in the stator core contain harmonics, which
add considerably to the eddy current losses in the punchings.
Harmonics in air gap flux density due to the shape of the salient
poles are reflected in the teeth. In the yoke, harmonics also
occur because saturation changes the flux density distribution
between inner and outer radius from being non-uniform near
the zero points in the cycle to being nearly uniform near the
peaks. Through the cycle, maximum flux density in the yoke is
flattened near the teeth and peaked near the outer edge.
Stray load loss is calculated and measured at short circuit. Only
the part of it which is due to slot leakage flux inducing eddy
currents in the stator winding can be calculated with some
accuracy.
The MMF from the stator winding at short circuit is of the
same order of magnitude as the MMF from the field winding at
open circuit. Due to the concentration of MMF in stator slots,
slot harmonics are very pronounced. The resulting pole face
After a run is completed, an input file for a companion special
purpose finite element program is produced automatically, and
the magnetic field in the air gap and between the poles can be
calculated for the final design with a simple command.
In this way, so-called pole shape factors can be accurately
determined. Two of them are the ratios total/fundamental air
gap flux and max actual/fundamental air gap flux density.
Another factor relates the sinusoidally distributed stator MMF
to the concentrated MMF from the field poles. Less field MMF
than stator MMF is required to produce the same fundamental
flux.
The harmonic content of the air gap flux is analyzed with
excitation from the field winding and from the stator winding
in the direct and quadrature axes.
The analysis can also be used to set up equivalent circuits in the
two axes with branches for individual amortisseur bars. Among
other things, this enables the designer to calculate more
accurately subtransient reactances and analyze such phenomena
as unbalanced loads and short circuits.
REFERENCES
1 O.W. ANDERSEN, "Optimum Design of Electrical
Machines", IEEE Transactions on Power Apparatus and
Systems, Vol. 86, June 1967, pp. 707-711.
2 O.W. ANDERSEN, "Optimized Design of Electric Power
Equipment", IEEE Computer Applications in Power, pp. 11-15,
Jan. 1991.
3 O.W. ANDERSEN, "Teaching and Demonstration of
Optimized Design", ICEM 96, Vigo, Vol. III, pp. 488-490.