lt
σdl. (2.36)
In the thesis, the detent force is used extensively to describe the total mag-netic tangential forces. The detent force includes both the cogging forces and the end forces associated with the longitudinal ends of a linear electrical ma-chine [105].
2.4.2 Virtual Work
The force, derived through Virtual work, is obtained from the magnetic energy and magnetic co-energy [93, ch.1]:
The force is then calculated through the gradient of the energy [83]. For a system with linear magnetic materials, the gradient of the magnetic energy can be computed directly. For a system with non-linear energy, the magnetic co-energy should instead be used. A couple of things to note is that the co-co-energy can be calculated from the energy and vice versa through Wm = H · B −Wm. Furthermore, the magnetic energy within the soft magnetic material is often very low due to the high permeability and thus a low magnetic field strength H.
The highly concentrated magnetic energy in the air regions between the trans-lator/rotor and stator, gives similar results when using the magnetic energy and co-energy.
2.5 Optimization theory
Optimization methods are used in many aspects of application designs and can ease the workload for a broad search-space. Despite being very different in theory, the implementation of these methods are often quite similar. Two essential parts of the optimization methods are the design parameters and the objective function. The design parameters are often merged into a single vec-tor x(:), where every index of the vecvec-tor represent one design parameter. One
vector x(:)
lb, ub of x(:) settings
tness function f(x(:))
optimal x(:)
Figure 2.5. Illustration of the optimization implementation. Although the algorithms of the optimization methods may differ, they are often similar in the way they are implemented. The optimization variables are defined through a vector x(:) together with the lower and upper boundaries (lb and ub). The fitness function is given as an input function to the algorithm. Further settings, such as the number of iterations must also be stated.
such parameter could be the width of a permanent magnet in an electrical ma-chine. It could also be used to specify different settings such as a winding pattern, meaning that if the variable value with index for the winding pattern (WP) is 1, the winding pattern is set to WP1. If the value is 2, the winding pattern is set to WP2, etc. The objective function, often also called fitness function, is the function that determine how ’good’ a set of parameters are.
The objective function for an electrical machine can be a measure of e.g. out-put power or torque.
In Fig. 2.5, a schematic for the implementation of an optimization method is illustrated. Beyond the design parameters and objective function, the bound-aries of the design parameters can be stated. Relating to the earlier example, it can be the maximum and minimum width of the permanent magnets. It can also be the total number of winding patterns, allowing for e.g winding pat-terns 1-3. The prerequisite for a set of patpat-terns is that the optimization method allows for the usage of only integer values. The possibility for integer-only values is not always available and can restrict the application of the optimiza-tion method in many aspects. One can, of course, round the parameter values to the closest integer but that will reduce the efficiency of the optimization al-gorithm drastically. The possibility for integer-only numbers is often accessed through the settings of the optimization method. Tolerances and the number of iterations before an optimal value is generated can also be specified, depending on the method.
In this thesis, two different optimization methods are evaluated. These are the Genetic Algorithm and Particle Swarm Optimization. These methods are both metaheuristic population based optimization methods, meaning that the design parameters are randomly generated within the search space and further influence the randomization based on the result from the objective function.
Neither of these optimization method depend on the derivative of the solution, as a function of parameter input values.
2.5.1 Genetic Algorithm
The Genetic Algorithm (GA) was developed to mimic the behaviour of a pop-ulation and is based around the performance of the parameter values for one generation, and how good properties are maintained through several genera-tions [106, 107]. The new parameter values, or children, are passed on through their parents in different combinations: they can be passed on as mutations with random changes from one parent; they can be based on a crossover of properties from both parents; or they can be directly passed onto the next generation without any modifications. The configuration for a GA revolves around the Population size and the number of generations. As the name sug-gests, the population size determines the number of different parameter values that will be tested for a generation and the number of generations determine the number of mutations and crossovers. The GA can be accessed through the MATLAB ®2Optimization ToolboxTM 3and has the great benefit of allowing for integer-only values.
2.5.2 Particle Swarm Optimization
The Particle Swarm Optimization (PSO) was initially developed to mimic how birds zeroed in on a target, or food [108]. It was later transitioned into mim-icking a swarm of particles scattered through the parametric space. The way the particles move, or the way that the parametric values are changed, depends on the location of the particle and the location of the particle with the best result. The configuration for the PSO is handled in a similar manner as the Genetic algorithm. Instead of the population size, the PSO is named Swarm Size and instead of the number of generations, the PSO is named simply iter-ations. PSO can also be accessed through MATLAB Optimization Toolbox.
Unlike GA and to the authors knowledge, PSO does not allow for integer-only values of the parameters.
2MATLAB is a registered trademark of The MathWorks, Inc.
3MATLAB Optimization Toolbox is a trademark of The MathWorks, Inc.