Geometry optimization for a class of switched-reluctance motors: A bi-objective approach

Abstract

This study focuses on a class of a 3-phase Switched Reluctance Motor. The aim of the paper is to optimize torque and iron losses as a function of the geometry. To enhance the efficiency of the motor, a procedure of automated optimal design is adopted. Two materials are evaluated and compared: in particular, a Soft Magnetic Composite material might be a good alternative for rotor and stator cores instead of a laminated steel. The analysis model of the motor is based on 2D Finite Element Method (FEM) simulation, while the design optimization is based on evolutionary computing.

Keywords

Finite elements methods multi-objective optimization soft magnetic composite switched reluctance motor

1. Introduction

Following the rapid pace of development, Switched Reluctance Motors (SRM) are increasingly used in a broad range of applications over the past few decades due to many advantages such as simple structure with non-winding construction in rotor side, fail safe because of high tolerances, robustness, low cost with no permanent magnet in the structure, and also possible operation at high temperatures or under strong temperature variations [1]. The basic theory, design and modeling of SRM have been presented in the literature [2, 3].

In domestic appliance applications, switched reluctance motors have been deeply investigated by many electrical machine researchers over the past few decades. Switched reluctance motors and drives with only one or two phase windings have been targeted so that applications for the technology are being created in low cost, high volume markets such as domestic appliances, air conditioning and automotive auxiliaries [4]. Recently, improving the general performance of machines is a constant demand for many industrial applications. This issue gave rise to the automated optimal design of machines under given constraints. In fact, to avoid costly and highly time-consuming of prototyping and testing methods in the industry, simulation is considered to be a significant part of design processes.

Automated optimal design of motors has been applied since 90’s [5] and nowadays it is being used successfully [6]. Multi-objective optimization methodology has been used to solve several optimization problems in different areas of engineering, ranging from power electrical devices to MEMS actuators like e.g. those proposed in [7, 8, 9, 10].

2. Device properties and characteristics

The basic motor specifications of a given 12/8 SRM and the physical sizes are reported in Table 1. The geometry and design parameters of targeted switched reluctance motor are shown in Fig. 1 [11].

Table 1
Size of switched reluctance motor

Parameters	Value	Symbol
Stator Outer Radius	70 mm	$R_{SO}$
Stator Inner Radius	41.75 mm	$R_{SI}$
Shaft Radius	8 mm	$R_{Sh}$
Air Gap Length	0.25 mm	AG
Rotor Outer Radius	41.5 mm	$R_{OR}$
Rotor Pole Width	11 mm	TW
Rotor Inner Radius	26.5 mm	$R_{IR}$
Stator Pole Height	15.25 mm	$H_{s}$
Back Iron Thickness	13 mm	BIT
Stator Tooth Outer Span	15.14 ( ${}^{\circ}$ )	$\alpha$
Stator Tooth Inner Span	12.58 ( ${}^{\circ}$ )	$\beta$
Number of coil Turns	140	N
Motor power	450 W	P
Supply Voltage	230 V	V
Supply frequency	50 Hz	f
Rotational speed	14000 rpm	n
Axial length of the motor	46.6 mm

Figure 1.

Geometry and design variables of SRM with phase a control mode.

Phase configuration consists of four series-connected coils per phase, driven by a unipolar current with a constant value of 1 A. In the simulation, it is assumed that currents in the motor windings simulate 1-phase control mode ( $i_{a}\neq 0$ , $i_{b}=$ 0, $i_{c}=$ 0).

Since in the SRM, different parts of the magnetic circuit are characterized by different flux density waveforms that are non-sinusoidal; therefore, the frequency of magnetic fields in the different parts of the SRM are different. The frequency in the different flux density waveforms is directly related to the base frequency ( $f_{\textit{base}}$ ) of the SRM, which corresponds to the motor phase current.

$f_{\textit{base}}=\frac{nN_{r}}{60}$ (1)

where $f_{\textit{base}}$ is measured in Hz, $n$ is the speed in rpm, and $N_{r}$ is the number of rotor poles. Stator frequency is considered to be $f_{\textit{base}}$ and the rotor’s frequency is $f$ that is derived by:

$f=\frac{f_{\textit{base}}}{N_{r}}$ (2)

Although the flux density in some of SRM zones, such as stator poles and stator yoke, is not identical, for the purpose of calculating core losses, these zones can be treated as a single zone [12]. Also for the rotor, the whole rotor including rotor pole and rotor yoke is considered as one zone. Therefore, in the simulation, two different frequencies for stator and rotor are considered, as follows:

$\displaystyle f_{\textit{stator}}=f_{\textit{base}}=\frac{14000{\kern 1.0pt}8}% {60}=1866\ \text{Hz}\ \ \text{and}\ f_{\textit{rotor}}=f=\frac{1866}{8}=233\ % \text{Hz}$ (3)

2.1 Material selection

In this paper, two motors with a different configuration have been investigated. In the first -commercial one- the stator and rotor are built entirely by non-oriented laminated steel sheet (called M330-50A) with motor physical sizes listed in Table 1.

Table 2
Prototype material

Prototype name	Stator material	Rotor material
M330	M330-50A	M330-50A
SMC	SMC	SMC HR

In turn, in the second one, which has the same physical sizes of the first one, the stator and the rotor are made of Soft Magnetic Composite material (SMC). The second prototype has been manufactured at Tele and Radio Research Institute (Warsaw) [11]. The prototypes compositions are listed in Table 2.

2.2 Soft magnetic composite properties and measurements

Soft magnetic composites, also called dielectromagnetics, are prepared by bonding technology. The principle feature of these materials is that iron particles are insulated by a thin organic or inorganic coating [11]. SMC materials have a lot of advantages, among the others, cheaper than electric steel, good magnetic and mechanical properties, high resistivity, possibility of tailoring properties of elements due to the designer requirement, possibility producing elements with complicated shape and three-dimensional distribution of magnetic flux.

Figure 2.

(a) Grains of SMC powder, (b) The sample of SMC ring.

Figure 3.

Prototype, stator and rotor made of SMC.

The magnetic measurements of soft magnetic composites have been done at Tele and Radio Research Institute laboratories. The samples of composites in ring-shape with dimension: 75 mm external diameter, 55 mm internal diameter and thickness 10 mm were prepared. The power losses P ${}_{tot}$ were gotten from measurements of the AC hysteresis cycle according to the IEC Standards 60404-6. Total power losses were measured at maximum flux density $B_{m}=$ 0.1 T to 1.4 T, over the frequency range from 50 to 2000 Hz. During measurements of the total power losses, the shape factor of the secondary voltage was equal to 1.11 $\pm$ 1.5%. Maximum measurement error of the total energy losses was equal 3%. Measurements of power losses were conducted by recording individual points and the integration area of the hysteresis. The sample of SMC grains and ring, stator and rotor of SRM (prototype) are shown in the Figs 2 and 3 [11].

The measurement results, BH curves and loss curves, for SMC and SMC HR are reported and compared with the Laminated electrical steel material in the Figs 4 and 5.

Figure 4.

Specific B-H curve, 50 Hz.

Figure 5.

Specific loss curves at 200 Hz and 2000 Hz.

The iron losses, divided into hysteresis losses and dynamic eddy current losses, both depend upon magnetic properties of the materials used to construct the core. The loss curve can be modeled by the following model:

$\displaystyle P=\beta f^{2}B_{m}^{2}\varepsilon^{2}+\eta B_{m}^{1.6\div 2}f(W/% m^{2})$ (4)

where $\eta$ is hysteresis constant, $\beta$ is eddy current constant, $\varepsilon$ is a form constant (0.35 $\ll\varepsilon\ll$ 0.5), $B_{m}$ is the peak of magnetic flux density in $T$ and $f$ is the frequency in Hz. The loss curve for M330 has been calculated by the Eq. (4) for two different frequencies.

Figure 6.

Detail of the mesh.

Figure 7.

Magnetic induction field map, $I=$ 1 A.

3. Motor model and field analysis

In order to perform the analysis of the magnetic field, a finite-element model of the device was developed, taking into account the non-linear B-H curve of the iron lamination and SMC components [13]. The simulation was carried out by a commercial software [14] and the 2D finite element model of the motor is implemented for both prototypes. A mesh with maximum element size of 0.5 mm, a detail of which is shown in Fig. 6, is considered.

Moreover, in simulation, the axial stack length has been set equal to one meter. A typical solution for the field analysis of prototype when rotor and stator are aligned is shown in Fig. 7 for current I $=$ 1 A.

To determine the torque and iron losses, the rotation of the rotor along the midline of the air gap has been simulated in discrete steps of 0.4 ${}^{\circ}$ over the 45 ${}^{\circ}$ polar pitch. As a consequence, 114 non-linear field solutions were completed for a given device geometry [13]. The computation of each non-linear field solution lasts about 8 s on a computer equipped with a CPU working at 3.6 GHz and with 16 GB of RAM.

Figure 8.

Motor torque and iron losses vs rotor position for prototype M330, $I=$ 1 A, $f=$ 50 Hz.

Figure 9.

Motor torque and iron losses vs rotor position for prototype SMC, $I=$ 1 A, $f=$ 50 Hz.

The torque and total losses versus rotor position over 45 degrees for both prototypes are shown in Figs 8 and 9.

The following twofold remark can be put forward. First of all, there is a minor changing in the trend of the torque curve and its value among the two prototypes (see Figs 8 and 9). Moreover, the value of total iron losses has decreased noticeably in the prototype SMC. Magnetic induction field along the air-gap midline for two rotor positions is also shown in Figs 10 and 11.

Figure 10.

Magnetic induction along the air-gap midline at rotor initial position (see Fig. 1), $I=$ 1 A, $f=$ 50 Hz.

Figure 11.

Magnetic induction along the air-gap midline with rotor aligned with stator poles, $I=$ 1 A, $f=$ 50 Hz.

Additionally, the flux linkage in SRM is a non-linear function of both phase current and rotor position (i.e. the relative position of the rotor pole with the stator pole). The aligned position means that the pole of the rotor and the pole of the stator are in the same direction, while the unaligned position means that the rotor of the specific phase is out-of-phase with respect to the stator. The flux linkage of the motor phase varies at different positions and different current. In Fig. 12 the flux linkage as a function of current at different rotor positions is shown.

Figure 12.

Flux linkage as a function of current and rotor position.

4. Inverse problems: shape design

4.1 Optimization

As the design of SRM for a particular application is a compromise between various performance criteria, improvement of a performance parameter may result in the degradation of other important features. Consequently, the designer has to search for solutions that are feasible with respect to all performance parameters. In standard optimization, one objective should be considered at a time. As the number of objectives increases, the optimization procedure would be more difficult to design in terms of cost, complexity and numerical analysis. In this study, two performance indicators (objective functions) should be fulfilled. Moreover, neither the drive and control system nor the current or voltage waveforms applied to the machine are considered in optimization.

Figure 13.

Prototype geometry and design variables.

For optimal shape design of SRM, the geometric sizes of the motor are considered as unknown parameters. As more variables are considered as unknowns, the complexity and thus computation time of the optimization process will be increased. For optimization of targeted SRM design, three geometrical parameters, namely rotor radii ( $x_{1}$ and $x_{2}$ ) and the back iron thickness ( $x_{3}$ ), are considered as unknown variables (see Fig. 13).

Then, the following inverse problem is investigated:

find the optimal values of geometric variables by giving the material properties (i.e. B-H magnetization curve, and P-B loss curve) and the power supply (one phase on, equal to 1 A), in order to maximize the static torque and minimize the iron losses subject to the problem constraints.

4.2 Design problem

The SRM geometry is optimized with considering the three design variables (vector X) as shown in Fig. 13. Imposing the geometrical limits, the overall diameter of the device must not exceed 140 mm; moreover, to ensure sufficient tolerance necessary for mass production, the Air-gap Length (AG), which is the distance from the outer rotor surface to the inner stator surface, is fixed to 0.25 mm. The ranges of continuous-valued design variables along with value of the variables for the prototype are reported in Table 3.

Table 3
Boundaries of the design variables

Design Variables	$x_{1}$	$x_{2}$	$x_{3}$
X(mm)	(mm)	(mm)	(mm)
Min	30.5	17	6
Max	48.5	30.97	24
Prototype	41.5	26.50	13

Table 4

Genetic optimization parameters

Parameters	Value/type
Population size	20
Selection function	Tournament
Cross over function	Intermediate
Mutation function	Adaptive feasible
Elite count	1
Crossover fraction	0.8
Pareto fraction	1
Stopping criterion	50 Generations

Figure 14.

Objective space, prototype M330 (triangle), non-dominated solutions (circle) for optimization of prototype M330.

The problem constraints are the following: $x_{1}>x_{2}$ $\cdot$ 1.283, and $x_{1}+x_{3}<$ 54.5, where $x_{1}$ is the rotor outer radius, $x_{2}$ is the rotor inner radius and $x_{3}$ is the back iron thickness. The former constraint allows to obtain a rotor with salient poles, while the latter allows to obtain a machine the external radius of which is smaller than the outer radius of the prototype i.e. 70 mm.

They divide the search space into two feasible and infeasible regions of $\Omega\in R^{3}$ and they are handled by means of the penalty method.

Figure 15.

Flux and field distributions of three optimal designs and prototype M330.

Developing a script in the numerical computing environment [15] is the first stage of the optimization process to allow the machine topology to be remotely compiled in the Finite Element software for each call during the optimization. Furthermore, it enables parameters to be introduced as variables. Then, determining the optimal value for these parameters is formulated to provide compromise solutions between power losses in the iron core ( $f_{1}$ ) and torque density ( $f_{2}$ ). The design criteria are defined by:

$\displaystyle\textit{Min}\ f_{1}(x)=\int_{S(x)}P\left[{B(x)}\right]^{2}ds,\ \ % x\in\Omega\ \text{(iron losses)}$ (5) $\displaystyle\textit{Max}\ f_{2}(x)=\int r_{0}\times f\ dV,f=\int_{\Omega}% \overline{\nabla}\cdot{\overline{\overline{T}}}d\Omega=\int_{\Gamma}{\overline% {\overline{T}}}\cdot\overline{n}\ d\Gamma\ \ \text{(torque)}$ (6)

where $P$ is the specific power loss in the ferromagnetic material subject to induction $B$ , $S$ is iron volume, $r_{0}$ is the vector going from the origin to an element of volume in the body, $V$ is the volume of the body, $f$ is the force on body calculated by $\overline{\overline{T}}$ Maxwell’s stress Tensor, $\Gamma$ is a closed surface enclosing the body and $\bar{n}$ is the outward normal versor respectively. Due to this fact that the induction, iron volume and force values are dependent on $x$ (the design variables), thus the inverse problem should be investigated for finding optimal values of geometric parameters for a given material to satisfy problem criteria.

Table 5

Comparison of solutions obtained for objective functions

Performance	$R_{OR}$	$R_{IR}$	BIT	Torque	Loss	weight
Parameters	[mm]	[mm]	[mm]	[Nm]	[W/Kg]	[kg/m]
Solution A	30.50	18.90	23.75	5.80	78.16	31.15
Solution B	48.12	17.40	6.31	8.99	222.42	20.98
Solution C	40.23	17.72	6.66	7.38	162.59	23.79
Prototype M330	41.50	26.50	13	7.92	139.36	26.64
Solution D	30.50	17	24	5.07	66.25	30.40
Solution E	48.19	17.57	6.21	7.42	125.18	20.21
Solution F	42.42	20.07	6.80	6.58	110.27	22.80
Prototype SMC	41.50	26.50	13	6.85	102.57	25.76

Figure 16.

Objective space, prototype SMC (triangle), non-dominated solutions (circle) for optimization of prototype SMC.

The iron losses and the torque are functions of the rotor position, which means that, in order to calculate $f_{1}$ and $f_{2}$ (iron losses and torque as maximum value over the rotor angle), many FEA have to be run: if one analysis every 0.5 ${}^{\circ}$ is done, considering that losses and torque are symmetric every 45 ${}^{\circ}$ (as shown in Figs 8 and 9), 90 analysis problems has to be solved for each tentative geometry. It is worth to notice that, from a numerical viewpoint, their calculation could be done with a reduced computational cost: because of geometrical symmetries, the peak of losses always occurs around 15 ${}^{\circ}$ and the peak of torque around 29 ${}^{\circ}$ , so only few analyses around those points could be done. In our calculations, only 4 FEAs has been performed for calculating $f_{1}$ and $f_{2}$ of each tentative geometry i.e. 32 s.

Figure 17.

Flux and field distributions of three optimal designs and prototype SMC.

In this study, a multi-objective genetic algorithm optimization method based on Pareto-optimal solutions is implemented for solving the problem by coupling a numerical computing environment [15] with a commercial Finite Element software [14].

4.3 Optimization process and problem formulation

In the last decades, different optimization techniques inspired by nature were investigated. Genetic algorithms (GA) are a very useful and widely studied optimization technique which is used for searching very large spaces that represent potential solutions [17]. Due to desirable GA technique advantageous such that variables and objectives do not need to be weighted, they are commonly chosen for electrical machine multi-objective optimization. In GA optimization, a population, which is a set of individuals, is evaluated at each iteration (generation) in terms of fitness function and scored. To expand the current population, highly-fit individuals are combined by a crossover operator to produce offspring. Meanwhile, to increase variation in the search space, a mutation operator is performed at a certain probability level. This process continues until a suitable required solution has been generated.

Figure 18.

Pareto front solutions comparison as a function of design variables.

Multi-objective optimization takes into account simultaneously two or more objective functions by exploiting the concept of non-dominated ranking of solutions in the objective space [16]. It involves minimizing or maximizing multiple objective functions simultaneously subject to a set of constraints that are often contradictory, as the minimization of an objective leads to an increase of another goal, so the solution we seek is always a compromise between these objectives.

When implementing a GA process, there are several specific parameters which influence the individuals selected for the next generation and the children these parents produce. The speed of convergence and performance of optimization results are significantly dependent on the values assigned to these variables. The multi-objective genetic optimization parameters are reported in Table 4 [15].

5. Results

The following pictures show the objective function space of investigated problem computed by means of a multi-objective genetic algorithm for both prototypes. The set of solutions on the Pareto front (marked by red circles) are examples of best compromise solution between conflicting design criteria, i.e. torque and losses [13]. To illustrate the specific improvements several solutions are selected from both optimization results and compare with initial design (prototype). In particular, Figs 14 and 16 show the optimization results for the M330 prototype and SMC prototype. Figures 15 and 17 depict the flux and field distributions of three optimal designs and compare for each prototype.

In terms of physical size, results reveal that by moving along Pareto front while torque and total iron losses are increased, the rotor outer radius is increased and back iron thickness is decreased. the rotor inner radius makes not dominant contribution to quality characteristics as depicted in Fig. 18.

Therefore, the outer dimension of rotor and back iron thickness have a larger impact on the optimization objectives. Regarding the objective functions, a SRM made entirely with SMC material will have the torque almost similar to SRM with laminated material but it will decrease the iron losses especially at high frequencies of magnetic flux in the motor magnetic circuit.

Moreover, each solution from the Pareto optimal set would have its own drawbacks and advantageous such as being noisy, not stable or even heavier in terms of weight or more efficient in power. Therefore, the designer would keep the priority of each one based on their importance.

The comparison of selected solutions as a function of design variables, torque, total iron losses and the motor weight is reported in Table 5.

It should be considered that a posteriori analysis of the Pareto front associated with a given design problem facilitates the task of the decision maker and possibly helps to identify innovative solutions [13].

6. Conclusion

The results have proven that the optimal shape design problem is well posed and it is possible to identify solutions which improved both torque and total iron losses. Moreover, motor built by SMC material would have a major impact on decreasing total iron losses. The simulation results have shown that this kind of optimization problem is also suitable for industrial design because it would be affordable in terms of computational cost and time.

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Geometry optimization for a class of switched-reluctance motors: A bi-objective approach

Abstract

Keywords

1. Introduction

2. Device properties and characteristics

Table 1 Size of switched reluctance motor

Table 2 Prototype material

4.1 Optimization

Table 3 Boundaries of the design variables

6. Conclusion

References

Table 1
Size of switched reluctance motor

Table 2
Prototype material

Table 3
Boundaries of the design variables