used to broaden the stator tooth and claw respectively. The angleθ3maxis the maximum allowable deflection angle of stator tooth, for which the two adjacent stator teeth contact each other. The angleθ4maxcorresponds to the angle, for which the two edges of the stator claw are parallel with each other.
θ2
θ1
(a) Basic modelling
θ3max
θ3
(b) Modification of stator tooth
θ3
Claw
θ4 θ4max
(c) Modification of claw
Figure 6.3. Front view of modelling procedure of PMTFM-CP
Based on the above basic geometric variables, totally 10 free parameters are defined and listed in Table 6.1. It is noted that most of them are defined as ratio to achieve a general valid conclusion.
6.1.2 Design objectives
Due to the time-consuming 3D FEM, the PMTFM-CP is optimized only for one working point with help of the stationary FEM, namely the PMTFM-CP is fed with the maximum phase current along the q-axis. Considering the requirements of the application in the electric vehicle, five objectives are defined in this work.
With help of the 3D FEM, the flux linkageΨuof the phase u, the inner torque T for different electrical rotor positionsθcan be determined. On this basis, the torque and flux linkage of the other two phasesΨvandΨwcan be obtained by translatingΨufor the electrical angle 120° and 240° respectively. After then, the flux linkageΨdalong the direct axis andΨqalong the q-axis can be calculated:
6.1 Free design parameters and design objectives
Para-Description Definition
meter
Ltooth axial length of the stator tooth g
Lclaw axial length control of the stator claw d/(a + d) Lshort axial length adjustment of bevel at claw tip c/(b + c) Rshort axial length adjustment of bevel at claw tip e/(e + f ) Wbasic basic angle control of the stator tooth and claw θ2/θ1
Wtooth angle adjustment of the stator tooth θ3/θ3max
Wclaw angle adjustment of the stator claw θ4/θ4max
LPMR ratio of the axial length of PM to one phase LPM/Lphase
RRYR ratio of the radial length of SMC concentrator to PM Ryoke/RPM
p the pole-pair number of PMTFM-CP p
Table 6.1. Free design parameters of PMTFM-CP
Ψd=2
3 ( Ψu cos(θ) + Ψvcos(θ− 120°) + Ψw cos(θ− 240°) ) (6.1) Ψq=2
3 ( Ψu cos(θ+ 90°) + Ψv cos(θ− 30°) + Ψw cos(θ− 150°) ) (6.2) Subsequently, the stationary voltages along the d- and q-axis udand uqcan be calculated, whereωelis the electrical rotational speed of PMTFM-CP and RSis the resistance of one phase.
ud= RS · id − ωel · Ψq (6.3) uq= RS · iq + ωel · Ψd (6.4) As the first optimization objective, the characteristic speedωcharais defined as the mechanical speed, at which the voltage limit is achieved. It can be deter-mined through solving the (6.5), where umax is the amplitude of the maximum allowable phase voltage.
u2d + u2d = u2max (6.5)
Correspondingly, theωcharacan be calculated using (6.6).
ωchara= ωel
p · 2 · π (6.6)
The second objective is the copper losses, which is one of the dominant losses when the PMTFM-CP is fed with the maximum phase current. It can be calcu-lated with (6.7), where imaxis the maximum effective phase current.
Pcu= i2max · RS (6.7)
The corresponding output torque at the characteristic speedωcharais the third objective and can be calculated using (6.8), where pfeis equal to the iron losses of all SMC parts at the speedωchara.
Tmax= T − pfe
ωchara · 2 · π (6.8)
The fourth objective is the efficiencyηchara of PMTFM-CP at the characteris-tic speedηcharacalculated with (6.9). Hereby, the friction losses and the eddy current losses in PMs are not taken into account.
ηchara= Tmax · ωchara
Tmax ·ωchara + pfe + pcu
(6.9) The last objective is the torque ripple ratioαTdetermined using (6.10), which is defined as the ratio between the torque ripple to the average output torque Tmax.
αT=T(θ)max − T(θ)min
Tmax (6.10)
All of the above mentioned five design objectives are summarized and listed in Table 6.2. Besides the predefined geometric constraints, the properties of the inverters and the required PM mass listed in Table 1.2, there are some other con-ditions applied in the design process. First of all, the electrical machine is only fed with the current component iqalong the q-axis considering the small differ-ence between the inductances along the d- and q-axis. Besides, the number of coils per phase is determined as 5 to achieve an appropriate compromise among
6.1 Free design parameters and design objectives
the objectives. At last, the ratio between the angle of PM to the concentrator remains constant equal to 1/2.
Objective Unit Description
ωchara min−1 characteristic speed of PMTFM-CP
Pcu W copper losses
Tmax N m maximum torque atωchara
ηchara % efficiency of PMTFM-CP atωchara
αT % torque ripple ratio
Table 6.2. Design objectives of PMTFM-CP
6.1.3 Design procedure of PMTFM-CP
The design procedure of PMTFM-CP in this work is illustrated in Fig. 6.4, which is based on a combination of many standard methods. Because the 3D FEM is ex-tremely time-consuming, it is impossible to optimize all the 10 free parameters of PMTFM-CP listed in Table 6.1. Therefore, a design of experiment method (DoE), the Taguchi method, is firstly applied to evaluate the importance of each design parameter in consideration of the defined five objectives. Subsequently, the selected critical parameters are further optimized with the nondominated sorting genetic algorithm 2 (NSGA2) and the multi-objective particle swarm optimization (MOPSO) [O8, O9, S11].
Determination of boundaries objectives free variables
DoE with Taguchi method
Determination of the most important
parameters
Important parameters
Sampling based on 3D FEM
Development of the mathematical model with RBF
Verifying the accuracy of the model with FEM
Optimization with MOPSO and NSGA2
Model
Pareto solutions
Selection based on Fuzzy logic
The optimal design of PMTFM-CP
Figure 6.4. Design process of PMTFM-CP
The genetic algorithms (GA) are based on an analogy to the evolution process in nature and often adopted to optimize electrical machines. Among many differ-ent GA, the newly developed NSGA2 is distinguished by its effectiveness and reliability, which is suitable for the multi-objective optimization. In comparison, the PSO is a heuristic search technique and simulates the social behaviour of a flock of birds searching food. The MOPSO is an extension of PSO, which aims to solve the multi-objective optimization problem. Both methods have been widely used in many different cases.
Because for both algorithms a large number of PMTFM-CP designs should be analysed, a mathematical model based on the radial basis function (RBF) is de-veloped and verified with FEM in this work, so that the five objectives depending on the design parameters can be analytically fast calculated. Since the five ob-jectives are normally in conflict with each other, there is no explicit design with the best values for all objectives. Therefore, the results of both optimization methods are a lot of outstanding “trade-off” designs, which are called nondomi-nated solutions or Pareto front representing the best possible comprises among the objectives. Afterwards, the optimal design is selected from the Pareto front with the help of fuzzy logic (FL). Finally, the optimal design is validated with the 3D FEM.