3.2 Finite Element Modeling
3.2.2 Torque calculation
As previously described by Eqn. 3.6, the air-gap average electromagnetic torque is ex- pressed as a function of the stator winding currents and the ux linkages. This approxi- mation provides a fairly accurate result in static analysis. Alternatively, there are other torque calculation methods implemented in either transient or multiple-step static analy- sis that enable torque ripple or cogging torque calculation estimations. This is because in these cases, the rotation of one of the machine components tends to distort the elements in air-gap region. Hence the torque calculation techniques need to be able to account for the torque at each time-instant of component rotation. Two methods are available in SEMFEM that facilitates the movement of the nite element meshes applied in static time-stepped simulations [157]:
Air-gap element (AGE) technique Moving-band (MB) solver
In the AGE method, Fourier series expansion is used to express the magnetic vector potential in Eqn. 3.37, from which the air-gap eld is analytically solved within some boundary conditions. Maxwell stress tensor method is then employed to calculate the torque from the series representation as follows:
T = L
µ0
Z θ2
θ1
r2BrBθdθ (3.43)
where L is the machine stack length.
The moving-band solver implements the Coulomb's virtual work method to calculate both the forces and torques on a moving body. It is a highly accurate method that utilizes the principle of virtual works by integrating the elds over virtually deformed elements situated between the stationary and moving parts. The integration over virtually distorted domain is usually asserted in terms of localized coordinates instead of global ones. For each static step, the torque expression used in this method is given by:
T = L µ0 Nmb X e=1 Z Ωe −BTG-1∂G ∂θ + 1 2 B2 |G| ∂G ∂θ dΩ (3.44)
where B = (Bx,By), B = kBk, G is the Jacobian matrix of the global nodal coordinates
with respect to local element coordinates, |G| is the determinant of G, Ωeis an individual
element's area, and Nmb the number of moving band elements. Performance comparison
between these two techniques as implemented in SEMFEM for magnetically geared PM machines was also done by Gerber and Wang [157]. The authors concluded that MB solver is relatively fast and accurate for obtaining the average torque, whereas the AGE is more suitable for cogging and ripple torques but at the expense of high computational cost. On its part, MagNet uses a tunable Maxwell stress method to compute the forces and torques on the components of an analyzed model [158]. It de nes two or more dierent components as a set of connected regions separated, and each completely enclosed by the virtual air material. The torque calculation formulation in this case is developed from the conventional Maxwell stress tensor method as demonstrated by McFee et al [159].
Chapter 4
Design Optimization of PMV Machines
In this chapter, the FEA based design optimization of permanent magnet vernier machine is described. In spite of the fact that they can be more time consuming than analytical methods, the 2D-FEA is used as a machine modeling tool during the design optimiza- tion stage as they are comparatively more accurate, which justi es their choice over the former. As discussed in the previous chapters, the PMV machine has an internal mag- netic gearing eect. Hence, in addition to the normal steps followed in the design of a conventional PMSM, it is imperative to decide on the value of the gearing ratio that will render acceptable torque quality for a PMV machine, which will also be able to achieve high torque density concurrently. Consequently, a number of dierent machine candidates were investigated in order to select suitable pole/slot combinations, which will fairly sat- isfy the application requirements. Furthermore, the optimization procedure and employed algorithms are brie y explained in order to provide a good understanding of how the nal machine dimensions were established. Lastly, the simulated performance of one selected optimum PMV machine is also presented and compared to that of an existing PMSM machine.4.1 Employed Optimization Algorithms
There are dierent optimization techniques available, which in most instances, are grouped as gradient-based and non-gradient based algorithms. The latter group is generally praised of its high reliability to reach a global optima, though it can be time and memory expen- sive for a very large sets of optimization variables. Gradient-based algorithms are time e cient, but are likely to be trapped in local optima due to their direct dependence on the function gradient at each step. To ensure that a global optimum is obtained under these techniques, nal results obtained from multiple optimization iterations with dier- ent starting points are expected to be within a ne margin. Considering that a large number of pole/slot combinations need to be investigated for dierent PMV machines, a gradient-based algorithm, namely, Modi ed Method of Feasible Directions (MMFD) was implemented at the initial stages of the optimizations [63].
Most gradient based optimization methods are usually designed to implement only one objective function at a time. That means if there is a need to simultaneously optimize more than one objectives, they have to be combined into a single function de ned as a weighed sum of individual objectives. But this approach has a drawback in that some ob- jectives may not be completely optimized as they mainly depend on the magnitude of the
CHAPTER 4. DESIGN OPTIMIZATION OF PMV MACHINES 64 weight coe cients assigned to them relative to others. Therefore, in the foregoing design, a single objective MMFD was used to conduct initial screening of candidate machines to be optimized. Once this was achieved, a multi-objective optimization whereby an objec- tive function is de ned as a vector of two or more performance indexes was performed on a few selected machines by utilizing the Non-Dominated Sorting Genetic Algorithm-II (NSGA-II). That is, all elements of vector objective function are simultaneously optimized to nd a set of variables satisfying all the inequality or equality constraints imposed on the search space. This gives a cluster of solutions from which we can select the best opti- mized points called pareto optimals. A pareto front can then be obtained by plotting a set of pareto optimals in an objective space of two elements of vector objective function. It gives a boundary between the feasible and infeasible design options. For instance, a PMV machine optimized for two performance indexes, being total active mass and e ciency, under certain constraints not mentioned in this example, was found to have a solution space plotted in Fig. 4.1.
(a)
Figure 4.1: Pareto front example
All the points in this graph represent feasible designs, whereas the blue colored points form the pareto front. That is, there can be no other designs that will dominate those points that lie along the pareto front line. In addition, the same pareto front provides a clear picture of the design trade-os between the two chosen performance indexes, revealing that improvement of one element comes at the deterioration of the other. Thus one good design point can be picked on the pareto front line according to the speci c design requirements and further evaluations done on it.