Optimization methods try to find the maximum or minimum of a function, where
there may exist restrictions or constraints on the independent variables. Finding
the maximum or minimum of a function which is also the global maximum or minimum
of the function has considerable difficulty and is the complex part of any
optimization method. In engineering, it is generally considered practical to
search only for local solutions.
In the optimization of electrical machines the objective function and constraints
can be computed using either the classical (circuital) approach or a numerical
field computation approach, such as the FEM. The FEM is more accurate than
the classical approach but requires substantially more sophisticated software
and computational time. In numerical field computation problems the standard
prerequisites of local optimization (convexity, differentiability, accuracy
of the objective function) are usually not guaranteed. Deterministic optimization
tools for solving local optimization, such as steepest-descent, conjugate gradient
and quasi-Newton methods, are not ideally suited to numeric electromagnetic
problems. This is due to the difficulties with numerical calculation of derivatives
because of discretization errors and numerical inaccuracy. Recently, noniterative
optimization schemes have been proposed using artificial neural networks. Stochastic
methods of optimization have, however, become more popular over the last few
years for optimization of electrical machines due to their high probability
of finding global minima and their simplicity. Stochastic methods such as simulated
annealing, genetic algorithm (GA) and evolution strategies have been successfully
used for different aspects of electrical machine design.
The population-based incremental learning (PBIL) method is a stochastic nonlinear
programming method which shows many advantages over the present stochastic
methods. Stochastic optimization has the disadvantage that it is not very efficient.
This problem is compounded by the large computation time of the numeric field
computations. A solution to this problem is the use of response surface methodology,
which has been successfully used for optimization of e.g., PM d.c. commutator
motors.
1. **Mathematical formulation of optimization problem**
The optimization of an electrical machine can be formulated as a general con
strained optimization problem with more than one objective, i.e., minimization
of costs, minimization of the amount of PM material, maximization of the efficiency
and output power, etc. Finding the extremum (Extr) of vector optimization problems
is defined as:
Extr F(x)= Extr [f1(x),f2(x),...,fk(x)] (eqn.1)
... where F : _n ?_k gi,hj : _n
?_ x ?_n x =(x1,x2,...,xn) subject to specified equality and inequality constraints:
gi(x) = 0,i =1, 2,...,m (eqn.2) hj (x)=0,j =1, 2,...,p (eqn.3) and specified
limits for the independent variables:
xmin = x = xmax (eqn.4)
In eqns (1) to (3) F(x) is the vector objective function with the objectives
fi(x) to be minimized, x is the vector of design variables used in the optimization,
gi are the nonlinear inequality constraints, hj are the equality constraints,
and xmin and xmax are vectors of lower and upper bounds for the design variables.
In vector optimization problems there is a conflict between the individual
objective functions fi(x) since there exists no solution vector x for which
all objectives gain their individual minimum. Vector optimization problems
can be transformed from multi-objective optimization into single objective
optimization using the method of objective weighting. Although objective weighting
always leads to a non-inferiority (Pareto-optimal) feasible solution, the estimation
of the weighting factors and the optimization starting point are subjective
choices and their influence can rarely be estimated in advance.
A more practical method of optimization is to minimize only one objective
function while restricting the others with appropriate constraints. Most constraints
will be upper or lower bound inequality constraints, which means constrained
optimization procedures are required. The optimization is thus done for a feasible
region, in which all the constraints are satisfied for the design variables.
2. **Nonlinear programming methods**
In nonlinear programming both the objective and constraint functions may be
nonlinear. There is no general agreement to the best optimization method or
approach. The extremely vast subject of nonlinear programming has been divided
into direct search methods, stochastic methods and gradient methods. A short
review of these three classes is described below.
Most numerical field problems have constraints of one form or another. A summary
of the main constrained optimization techniques is also given below.
2.1 *Direct search methods*
Direct search methods are minimization techniques that do not require the
explicit evaluation of any partial derivatives of the function, but instead
rely solely on values of the objective function, plus information gained from
earlier iterations. Direct search methods can loosely be divided into three
classes: tabulation, sequential and linear methods.
Tabulation methods assume that the minimum lies within a known region.
The methods of locating the optimum are: (i) evaluation of the function at
grid points covering the region given by the inequalities, (ii) random searching
assuming that the minimum would be found within a sufficiently large number
of evaluations, or (iii) a generalized Fibonacci search finding the solution
of the multivariate minimization problem by using a sequence of nested univariate
searches.
Sequential methods investigate the objective function by evaluating the function
at the vertices of some geometric configuration in the space of the independent
variables. This method originated from the evolutionary operation (EVOP). EVOP
is based on factorial designs. The objective function is evaluated at the vertices
of a hypercube in the space of the independent variables.
The vertex with the minimum function value becomes the center point of the
next iteration and a new design is constructed about this point. This is a
mutation type search mechanism to direct the search towards the optimal. Fractional
factorial experimentation assumes systematic and symmetric vertices to reduce
the number of objective function evaluations.
The simplex method evaluates the objective function of n independent variables
at n + 1 mutually equidistant points, forming the vertices of a regular simplex.
The vertex with the highest value is reflected in the centroid of the remaining
n vertices, forming a new simplex. The size of the simplex is reduced if one
vertex remains unchanged for more than M consecutive iterations, thus narrowing
the search to the minimum.
Linear methods use a set of direction vectors to direct the exploration.
There is a large number of linear methods available such as:
(i) The alternating variable search method which considers each independent
variable in turn and alters it until a minimum of the function is located,
while the remaining (n - 1) variables remain fixed.
(ii)The method of Hooke and Jeeves which uses exploratory moves and pattern
moves to direct the search towards the minimum by attempting to align the search
direction with the principal axis of the objective function.
(iii)Rosenbrock's method which uses n mutually ortho-normal direction vectors.
Perturbations along each search direction are done in turn and if the result
is no greater than the current best value, this trial point replaces the current
point. This is repeated until the minimum is obtained.
(iv)Davies, Swann and Campey method uses n mutually ortho-normal direction
vectors and a linear univariate search algorithm on each direction in turn.
After each stage is complete, the direction vectors are redefined.
(v) Quadratic convergent methods minimize functions which are quadratic in
the independent variables making use of conjugate directions.
(vi)Powell's method is based on mutually conjugate directions and ensures
that a direction is replaced only if by doing so a new set of direction vectors,
at least as efficient as the current set, is likely to be obtained.
2.2 *Stochastic methods*
Simulated annealing (SA). This method generates a sequence of states based
on an analogy from thermodynamics where a system is slowly cooled in order
to achieve its lowest energy state. This is done using nature's own minimization
algorithm based on the Boltzmann probability distribution. Thus, the design
configuration is changed from one to two with objective functions f1 and f2
with a probability of P = exp[-(f2 - f1)/kT]. If f2 >f1, the state with
probability P is accepted. If f2 <f1, the probability is greater than one
and the new state is accepted. At a given temperature, the configurations are
arbitrarily changed using a random number generator and the designs are also
changed with a dictated probability greater than one. The temperature is thus
lowered for the next round of searches. This makes uphill excursions less likely
and limits the search space. SA ultimately converges to the global optimum.
Multiple-restart stochastic hill-climbing (MRSH). This method initially generates
a random list of solution vectors of the independent variables, using binary
vectors. The solution vector corresponding to the minimum result of the objective
function is used in an iterative loop. A bit of the solution vector is toggled
and evaluated. The minimum for a sufficiently large number of iterations is
assumed to be the minimum of the objective function.
Genetic algorithm (GA). This search method is rooted in the mechanisms of
evolution and natural genetics. A GA combines the principles of survival of
the fittest with a randomized information exchange. GAs generate a sequence
of populations by using a selection mechanism, and use crossover as the search
mechanism to guide the search towards the optimal solution.
Population-based incremental learning (PBIL). This is a combination of evolutionary
optimization and hill-climbing. PBIL is an abstraction of the GA that maintains
the statistics of the GA, but abstracts away the crossover operation and redefines
the role of the population.
2.3 *Gradient methods*
Gradient methods select the direction si, of the n dimensional direction vector,
using values of the partial derivatives of the objective function F with respect
to the independent variables, as well as values of F itself, together with
in formation gained from earlier iterations. The solution is thus improved,
that is:
F(xi+1) = F(xi) xi+1 = xi + hisi (eqn. 5)
…where hi is the step increment and si is the search direction. Types of gradient
optimization methods are:
Methods of steepest descent use the normalized gradient vector at the current
point to obtain a new point using a specified step length.
Newton's methods use a second order truncated Taylor series expansion of
the objective function F(x). The method requires zero, first and second derivatives
of the function at any point.
Quasi-Newton methods use an approximation of the second derivative of the
function which is updated after each iteration.
2.4 *Constrained optimization techniques*
Constrained optimization problems are generally transformed to unconstrained
ones and are then optimized using one of the nonlinear programming methods
described above. Some of the techniques used on constrained problems are:
Feasible direction method attempts to maintain feasibility by searching from
one feasible point to another along feasible arcs. This method assumes a feasible
point can be found when the procedure starts.
Penalty function transforms the optimization problem to include the constraints
which enable F to be maintained while controlling constraint violations by
penalizing them. Exact penalty function is similar to the classical penalty
function except that the absolute value of the constraints are used.
Sequential unconstrained minimization technique is also similar to the classical
penalty function except the penalty coefficient is increased after each step
of the algorithm.
Augmented Lagrangian function or multiplier penalty function uses a Lagrangian
function to which the penalty term is added. The problem is trans formed to
an augmented Lagrangian function which is minimized.
3. **Population-based incremental learning**
In terms of Darwinian models of natural selection and evolution, life is a
struggle in which only the fittest survive to reproduce. GAs based on natural
selection and genetic recombination were first proposed by Holland. GAs generate
a sequence of populations by using a selection mechanism, crossover and mutation
as search mechanisms. In nature, competition for resources such as food means
that the fittest individuals of a species dominate over the weaker ones. This
natural phenomenon is called "the survival of the fittest." The fittest
individuals thus get a chance to reproduce ensuring, implicitly, the survival
of the fittest genes. The reproduction process combines the genetic material
(chromosome) from the parents into a new gene. This exchange of part of the
genetic material among chromosomes is called crossover.
GAs encode the solution as a population of binary strings. Each solution is
associated with a fitness value determined from the objective function. The
main operation in GA is crossover, although mutation plays the role of regenerating
lost genetic material by causing sporadic and random alteration of the bits
of the binary strings. GAs are generally characterized by their population
size, crossover type, crossover rate and elitist selection. These control parameters
affect how well the algorithm performs. The optimum set of parameters is dependent
on the application being optimized.
PBIL is an abstraction of the GA that explicitly maintains the statistics
contained in a GA's population, but abstracts away the crossover operation.
PBIL is in fact a combination of evolutionary optimization and hill-climbing.
The PBIL algorithm uses a real valued probability vector which, when sampled,
reveals a high evaluation solution vector with high probability.
The PBIL algorithm creates a probability vector from which samples are drawn
to produce the next generation's population. As in the GA, the solution is
encoded into a binary vector of fixed length. Initially the values of the probability
vector are set to 0.5. A number of solution vectors, analogous to the population
in GAs, are generated based upon the probabilities of the probability vector.
The probability vector is pushed towards the generated solution vector with
the highest evaluation (fitness value). This probability vector can thus be
considered a prototype for high evaluation vectors for the function space being
explored. Each bit of the probability vector is updated using:
Pi =[Pi × (1.0 - dl)]+(dl × si) (eqn. 6)
...where Pi is the probability of generating a one in the bit position i,
si is the ith position in the solution vector for which the probability vector
is being changed and dl is the learning rate. The learning rate is the amount
the probability vector is changed after each cycle. A new set of solution vectors
is produced after each update of the probability vector. The entries in the
probability vector start to drift towards either 0.0 or 1.0 as the search progresses
to represent a high evaluation solution vector.
Mutation is used in PBIL for the same reasons as in the GA, to inhibit premature
convergence. Mutations perturb the probability vector with a small probability
in a random direction. PBILs are generally characterized by their number of
samples, learning rate, number of vectors to update from and mutation rate.
Fig. 1 shows a flow chart representation of PBIL.
PBIL has been shown to work as well, or better, than the GA. The main advantage
of the PBIL over the GA is that since PBIL is characterized by fewer parameters
and their values are less problem-related, as little problem specific knowledge
as possible is needed.
Fig. 1. Flow chart of PBIL algorithm.
4. **Response surface methodology**
When using a numeric field computation based on a FEM analysis, the objective
function being optimized will require a lengthy computation time. Owing to
this large computation time a faster approach than PBIL is required. Response
surface methodology (RSM) is a collection of mathematical and statistical techniques
useful for analyzing problems of several independent variables and model exploitation.
By careful design of the FEM experiments, RSM seeks to relate the output variable
to the input variables that affect it. The computer experimental result y as
a function of the input independent variables x1,x2,...,xn is y = f(x1,x2,...,xn)+
d(x1,x2,...,xn)+ d_ (eqn.7)
... where d(x) is the bias error and d_ is a random
error component. If the expected result is denoted by E(y)= S, then the surface
represented by S = f(x1,x2,...,xn) is called the response surface.
The form of the relationship between the result and the independent variables
is unknown. A polynomial expression is used as a suitable approximation for
the true functional relationship between y and the independent variables.
A polynomial expression of degree d can be thought of as a Taylor's series
expansion of the true underlying theoretical function f(x) truncated after
terms of dth order. The assumption that low order polynomial models can be
used to approximate the relationship between the result and the independent
variables within a restricted region of the operability space is essential
to RSM.
Second order polynomial models are used to model the response surface.
f(x)= bo + bijxixj (eqn. 8)
...where i<j and the coefficients bo, bi, bii and bij are found using the
method of least squares. These unknown coefficients can be estimated most effectively
if proper computer experimental designs are used to collect the data. Designs
for fitting response surfaces are called response surface designs.
4.1 *Response surface designs*
An experimental design for fitting a second order model must have at least
three levels of each factor, so that the model parameters can be estimated.
Rotatable designs are the preferred class of second order response surface
designs. The variance of the predicted response at a point x, in a rotatable
design, is a function only of the distance of the point from the design center
and not a function of direction.
The collecting of the sample results is essential since a sufficiently accurate
approximation has to be found with a minimum of experiments. If not all the
factorial combinations are employed, the design is called an incomplete factorial
design. The Box-Behnken three level design has been chosen for investigating
the response surface. This is an incomplete factorial design which is a reasonable
compromise between accuracy of the function and the required number of computations.
The design generates second order rotatable designs or near-rotatable designs,
which also possess a high degree of orthogonality.
4.2 *Estimation of errors in response surface fitting*
The errors in design of experimental methods are generally divided into two
types: (i) systematic or bias errors d(x) which are the difference between
the expected value of the response E(y)= S and the approximate objective function
f(x), and (ii) the random or experimental error _ in sampling.
In numeric computer experiments, replicated experiments result in the same
result, so random errors cannot be defined. Only the bias error from systematic
departure of the fixed polynomial from the real response, due to an insufficient
polynomial order, can be calculated. An estimate of the variance of the bias
error is (eqn.9) where se is the standard error of estimate, m is the number
of observations, p is the number of coefficients in the polynomial, yi is the
observed response and ˆ yi is the predicted response. The normalized error
is:
d = se yo (eqn.10)
where yo =(y1 +y2 +...+yn)/m. The accuracy of the response surface E(y) is
varied by changing the size of the investigated region, since only second order
polynomials are used.
Fig. 2. Flow chart of the control program which produces the motor output
characteristics.
5. **Modern approach to optimization of PM motors**
The aim of the optimization procedure is to minimize the cost of the active
material of the motor, while ensuring a rated power and high efficiency. It
is proposed to optimize a PM motor using the PBIL with RSM. The PBIL does not
optimize the performance characteristics directly, but uses polynomial fits
of the performance characteristics of the motor, created using RSM.
The motor characteristics are calculated using FEM ( Section 3) and classical
machine theory for a number of combinations of input parameters. Fig. 2 shows
the logic sequence followed by the control program. The output characteristics
and input parameters are used to fit polynomial equations of second order to
every output characteristic. These polynomials are used as objective functions
and constraints in the PBIL optimization.
The following simplifications are made for the FEM: (a) the end leakage reactance
is calculated using classical theory (Appendix A, eqn (A.10)) since it would
be very difficult to calculate using a two-dimensional FEM; (b) the induced
EMF and inductive reactance are assumed constant throughout the load range,
and equal to the value obtained at rated current; (c) the model is independent
of rotor position.
5.1 *PM **d.c.** commutator motors*
Consider a PM d.c. commutator motor with segmental magnets. The optimization
problem can be stated as minimize the cost:
C(x)= VMcPM + Vwcw + Vccc (eqn.11)
…where VM, Vw, Vc are the volume of the PM, copper conductors and steel core,
respectively, and cPM, cw, cc are the price per m3 for the PM, copper conductors
and electrotechnical steel, respectively. The optimization is subject to the
constraints:
[...]
5.2 *PM synchronous motors*
A surface PM rotor and a buried PM rotor synchronous motor have been considered.
Only the rotor is being optimized since the stator of an existing induction
motor has been used. The objective function attempts to minimize the volume
of PM material used. The optimization problem can be expressed as: minimize
VM(x) subject to the constraints:
Pelm(x) = Pelm(d) Ja(x) = Jath
?(x) = ?d g = gmin (eqn.14) hM,wM = hMmin Dmax <D1in
where VM is the magnet volume, Pelm(d) is the desired electromagnetic power,
Jath is the current density thermal limit at Pelm(d), ?d is the desired electrical
efficiency at Pelm(d), g and hM are the mechanical minimum sizes for the air
gap and PM respectively, and Dmax is the maximum diameter of the outer edge
of the PMs.
The independent variables used in the surface PM rotor design are the air
gap g, PM thickness hM and the overlap angle ß, and in the case of the buried
PM rotor design the air gap g, PM thickness hM and PM width wM (Fig 3). The
motor performance obtained from the FEM, using flux linkage and magnetizing
reactance.
The objective function is not easily expressed in terms of the independent
variables. RSM is thus not used for modeling the objective function directly,
but rather for modeling the performance characteristics of the PM synchronous
motor, used in the constraints, in terms of the independent variables.
The characteristics for a number of combinations of the independent variables
(factors) are calculated using the FEM. The factors range over the whole problem
space and their combinations are determined by the Box-Behnken three level
design method, with 3 factors and 3 levels the design needs 15 runs. Response
surfaces are created for the output characteristics Ef , Xsd, Xsq, Xad and
Xaq. A second order polynomial is fitted to each of these five characteristics
using the least squares method. These five polynomial equations are then used
to calculate the electromagnetic power, efficiency and stator current in the
optimization procedure. The PBIL optimization is used for the minimization
of the PM volume with the constraints stated in eqn (eqn.14).
The volume of a surface magnet per pole is VM/(2p)=(0.25ßp/3600) ×(D2 2out
- D2 Min)lM, where lM is the magnet axial length, D2out and DMin are magnet
diameters according to Fig. 3a. The volume of a buried magnet per pole is VM/(2p)=2hMwMlM
(Fig. 3b).
Fig. 3. Geometric layout of PM rotors showing design variables for: (a) surface
PM rotor, (b) buried PM rotor.
**Numerical examples**
*Numerical example 1 *
Optimization of a surface PM rotor. The stator of a commercial 380 V, 50-Hz,
four-pole 1.5-kW induction motor has been used to design a synchronous motor
with surface type PM rotor. Table 1 shows the results for different constraints
in electromagnetic output power, efficiency and power factor. The results show
that at low power ratings (1.5 kW) the design needs a substantial increase
in PM material if the power factor is constrained to a minimum of 0.9 at rated
power. At the high power rating of 2.5 kW the power factor well exceeds the
design minimum.
The most feasible power rating for this motor should ensure that the stator
winding current density remains well below the specification limits, but also
maximizes the power output. A power rating of 2.2 kW is thus considered appropriate
due to the excellent power factor at rated power and the maximum use of the
stator windings. The stator used in this motor is from a 1.5 kW induction motor.
The increase in power rating to 2.2 kW, when operating as a synchronous motor,
is due to the higher stator winding current density possible in PM synchronous
machines, and the improved efficiency and power factor. The stator winding
current density has increased from 9.93 A/mm^2, for the induction motor, to
10.3 A/mm^2 for the PM synchronous motor (forced ventilation).
Higher time and space harmonics have been neglected and the effects of cogging
torque have also been neglected throughout this optimization procedure. The
cogging torque can significantly be reduced using an appropriate PM overlap
angle. Skewing of the stator teeth is not possible as an existing stator is
being used. Using a PM ß =(k +0.14)t1 where k is an integer and t1 = 3600/s1
is the stator slot pitch (angle), the minimum cogging torque can be achieved.
The overlap angle is thus decreased from ß =73.80 to 71.40. The air gap and
magnet thickness are again optimized for this fixed overlap angle. Table 1
shows the final optimized rotor details.
The performance of this optimized motor is compared with an initial surface
PM design. Fig. 4 compares the electromagnetic power and Fig. 5 compares the
efficiency for the two motors. The optimized surface PM synchronous motor has
superior efficiency at the desired rated power with a reduction in the PM volume
from 15,000 mm^{3} per pole for the initial design to 6838 mm^{3} per pole for the optimized motor.
Fig. 4. Electromagnetic power versus load angle for surface PM motors.
*Numerical example 1 *
Fig. 5. Efficiency versus load angle for surface PM motors. Numerical example
1.
*Numerical example 2*
Optimization of a buried PM rotor. The stator of a commercial 380-V, 50 Hz,
four-pole 1.5-kW induction motor has been used to design a synchronous motor
with buried-type PM rotor. Table 2 shows the results for different constraints
in electromagnetic output power, efficiency and power factor.
The most feasible power rating is again chosen to be 2.2 kW. The current density
in the stator winding is Ja ˜ 10.4 A/mm^{2} (forced ventilation).
Table 2. Optimization of buried PM rotor using RSM
Fig. 6. Electromagnetic power versus load angle for buried PM motors.
*Numerical example 2*
Fig. 7. Efficiency versus load angle for buried PM motors.
The cogging torque can be minimized in the final design by creating asymmetry
of the rotor magnetic circuit. This has a marginal effect on the optimization
point since it does not change any of the optimization parameters.
The performance of this optimized motor is compared with an initial buried
PM motor design. The optimized buried PM motor uses 6627 mm^{3} of
PM material, while the initial design uses 16, 200 mm^{3}. It has also
superior efficiency at rated output power.
The optimization of the surface and buried PM synchronous motors both showed
improved performance over their initial designs. The volume of PM material
was also reduced in both designs. The buried PM motor design is considered
the superior design since it uses the minimum of PM material and has a high
efficiency over a wide power range (Fig. 7).
The RSM using the PBIL is thus seen as an appropriate method for optimization
of PM synchronous motors with the aid of the FEM. This optimization technique
can easily be extended for the optimum design of the whole synchronous motor. |