Optimization Design of Surface-mounted Permanent Magnet Synchronous Motors Using Genetic Algorithms

The permanent magnet synchronous motor (PMSM) has gained widespread popularity in various industrial applications due to its simple structure, reliable performance, compact size, high efficiency, and adaptability to different shapes and sizes. Its exceptional characteristics have made it a focal point in industrial settings. The PMSM can be categorized into two primary types based on the arrangement of the permanent magnets (PM): interior permanent magnet (IPM) and surface-mounted permanent magnet (SPM). In the IPM, the magnets are embedded into the rotor, while in SPM, they are mounted on the rotor's surface. The utilization of PMs eliminates the need for excitation currents due to their high flux density and significant coercive force. This absence of excitation losses contributes to a notable increase in efficiency. In this study, a multi-objective optimal design approach is introduced for a surface mounted PMSM, aiming to achieve maximum efficiency while minimizing material costs. The optimization task is accomplished using a genetic algorithm. Furthermore, the motor designs are simulated using the finite element method (FEM) to assess and compare designs before and after the optimization process.


Introduction
The permanent magent synchronous motor (PMSM) is widely employed in diverse industrial applications due to its simplicity, reliability, compact size, high efficiency, and its ability to be adapted to various shapes and sizes [1].Particularly, in the low speed, the PMSMs have extensive usage in fields such as ship propulsion, lifting, mining, and oil field exploitation [2], [3].The PMSMs can be classified into two main types based on the arrangement of the permanent magnet: interior permanent magnet (IPM), where the magnets are embedded into the rotor, and surface-mounted permanent magnet (SPM), where the magnets are mounted on the rotor's surface.The PMSMs are excited by permanent magnets (PMs) rather than excitation currents due to their high flux density and significant coercive force.This characteristic eliminates excitation losses, resulting in a substantial increase in efficiency [4].The three primary magnet materials used in the PMSMs are ferrite, samarium-cobalt (SmCo), and neodymium-iron-boron (NdFeB).
For high-torque applications, the fractional-slot concentrated winding is commonly employed.The term fractional-slot winding refers to a situation where the number of slots per pole per phase is a non-integer value and less than one.When each coil of the winding is wound around a single tooth of the stator, it is referred to as a concentrated winding.Motors with this type of winding are particularly advantageous in low-speed direct drive transmission systems due to their high efficiency, reduced end turn length, high slot fill factor, and low copper loss.However, fractional-slot configurations have drawbacks, including slightly lower winding factor and higher harmonic content in the magnetomotive force (MMF), distribution compared to integral slot machines [5], [6].Therefore, the motor must be designed with a suitable combination of the number of pole pairs and slots based on its intended applications.
The genetic algorithm (GA) is an heuristic search frequently employed for generating effective solutions to optimization and search problems.Inspired by natural evolution, the GA utilizes techniques such as inheritance, mutation, selection, and crossover to generate solutions for optimization problems.Its versatility makes it a powerful tool for global optimization and analysis of large datasets [7].Numerous researchers have utilized the genetic algorithm to explore optimal designs for the PMSMs [8]- [10].However, the discrete variation of variables and their lack of complete correlation with each other and other parameters can lead to changes in output power after the optimization process.
In this study, a surface-mounted permanent magnet synchronous motor (SPMSM) employing NdFeB N35 magnets is designed.Subsequently, the genetic algorithm is applied to determine the optimal dimensions that achieve maximum efficiency while minimizing material costs (such as electrical steel, copper, and magnets) with power conservation.

Analytical model of SPMSM
The rotor volume, expressed as the product of D 2 and L, plays a crucial role in determining the motor's dimensions, where D represents the inner diameter of the stator and L denotes the active length of the machine.Several methods have been proposed to calculate this parameter [11]- [13].However, to determine the rotor volume, a coefficient concept that depends on the cooling systems, as presented in [14], is employed.
where   is the motor torque.Typically, for motors with an output power of up to 10 hp and air cooling, the specific volume (V 0 ) ranges from 5 to 7 in 3 /(ft•lb).On the other hand, for motors with a power greater than 10 hp and water or liquid cooling, the specific volume is typically between 2 and 5 in 3 /(ft•lb).
The magnet is a critical component of the PMSM, as it generates the magnetic field in the air gap.In this paper, NdFeB N35 magnets are utilized.The fundamental flux density in the air gap can be determined using the following equation: where   represents the electrical angle of the PM, and   is the flux density of the magnet.The thickness of magnet can be calculated as: where   is the permeability of the magnet and and  is the length of air gap.The model for separating iron losses can be categorized into three components: hysteresis loss, eddy current loss, and additional loss.
In this design, the additional loss, which accounts for a small portion of the motor iron loss [15], is disregarded.The motor design in this paper utilizes the M530A-50A (European standard EN 10106-1996) non-oriented electrical steel.For the non-oriented model, the parameter α ranges from 1.6 to 1.8, while the values of  ℎ and   for the M530A-50A steel can be found in [16].
The iron loss encompasses the total loss occurring in the stator, rotor, and teeth of the motor.It can be calculated using the loss separation method, as follows: =     +     +  ℎ  ℎ .( 5) As previously mentioned, the PMSM is excited by permanent magnets instead of excitation currents, resulting in the absence of copper loss in the rotor.Therefore, the copper loss in the PMSM is solely attributed to the stator winding.The calculation for copper loss is as follows: where   is the number of slots,  is the phase current and  is the resistance of the winding in each tooth.The total loss is the sum of the iron loss and the copper loss, which is given as below: =   +   (7)

Optimization method
The diagram of the optimization process using genetic algorithm is presented as: The optimization of Permanent Magnet Synchronous Motors (PMSMs) involves a multi-objective problem with numerous variables and constraints.The selection of variables is critical as it directly impacts the output results.
In this study, a careful consideration led to the inclusion of seven variables in the optimization process.These variables include the inner stator diameter (y 1 ), air gap length (y 2 ), height of the stator yoke (y 3 ), height of the rotor yoke (y 4 ), width of the tooth (y 5 ), current density (j), and the number of turns per tooth (n).The variables y1 , y 3 , y 4 , and y 5 are associated with iron loss and steel volume, whereas y 2 , j, and n pertain to magnet dimensions and copper loss.All seven of these variables have the potential to influence both the motor's dimensions and its losses, ultimately leading to variations in the objective functions.The proposed constraints must be precisely formulated in order to ensure that the optimization results meet all the operational requirements of the Permanent Magnet Synchronous Motor (PMSM).This research paper focuses on three parameters, namely the stator volume, rotor volume, and magnet volume, each accompanied by an equality constraint.The winding's slot fill factor is maintained within the range of 0.45 to 0.55.Additionally, references [17]- [20] provide a typical range for the flux density values of the stator yoke, rotor yoke, and tooth of the PMSM, resulting in four inequality constraints.
The objective functions must be established, specifically the minimization of total loss (Ploss) and material cost (C).
The material cost encompasses the expenses associated with electrical steel, copper, and magnets, which can be described as follows: where   ,   ,   are the cost per kilogram of electrical steel, copper, and magnet respectively.The mass of the different parts of the motor in ( 8) is calculated as: Where the factors  1 ,  The other papermeters   ,   ,   ,   ,   , ℎ  , ℎ  ,   ,   ,   and   represent the volume of stator, rotor, magnet, the operating remanence of the magnets at working temperature, the flux density above the magnets, the wedge height, the tooth tip depth, the opening width of the semi-closed slot; the mass density of electrical steel, copper, and magnet respectively.The cost function becomes:  =  1 () =  1 ( 1 ,  2 ,  3 ,  4 ,  5 ,  6 ,  7 ) (14) The loss component parts of each part of the core is calculated as: 16) where:  5), ( 6), ( 7) together, the total loss function becomes:

Numerical and optimal results
The test problem is a practical SPMSM given in Table 2.

Rated speed 600 rpm
The distribution of flux density (B) on the stator and rotor before optimization (top) and after optimization (bottom) is pointed out in Figure 2. It can be seen that before optimization, the value of B is observed to be 2.185 (T), which subsequently increases to 2.208 (T) after the optimization process.This means that there is a slight increase in the flux density on the tooth, but the total loss decreased as a result.This change will not have any adverse effects on the motor's operational parameters.The distribution of cogging torque for two different cases (before and after optimizations) is shown in Figure 3.The average cogging torque demonstrated a favorable decrease of 1.34 Nm, declining from 3.1 Nm to 1.76 Nm.This outcome aligns with the desired result of reducing cogging torque.Figure 4 presents the distribution of the back EMF.Upon optimization, the root mean square (rms) value of the back electromotive force (EMF) decreased by 2.8V, going from 317.7V to 314.9V.Nevertheless, both values, both before and after optimization, remained smaller than the terminal voltage of 380V.Hence, the operational performance of the motor will not be impacted by this change.The torque characteristic is given in Figure 5.It shows that the optimization process, the motor's torque output exhibited smoother performance, resulting in a reduction in torque ripple as previously mentioned.The average torque output experienced a slight decline from 35.28 Nm before optimization to 35.02 Nm after optimization.However, it remained slightly higher than the desired value.By controlling the advanced phase angle, we were able to increase the torque output back to its preoptimized level.
The pareto front plot of the objective functions is also presented in Figure 6.It is evident that the two objective functions exhibited inverse changes, indicating that a smaller total loss corresponded to a larger material cost, and vice versa.By examining the scatter plot of the Pareto front, we selected the minimum value for the sum of the two objective functions as the basis for calculating the motor parameters.The optimization results on main parameters of the SPMSM is given in Table 3.The torque ripple percentage decreased by 2.82%, going from 9.26% to 6.44%.Additionally, the motor's efficiency improved from 94.298% to 94.835%.The power factor remained relatively stable, while the total material cost decreased from 358.02$ to 342.8$.Furthermore, the torque output becomes smoother after optimization, indicating a decrease in torque ripple.Although there is a slight increase in the flux density on the tooth, it does not adversely affect any motor parameters during operation.These positive outcomes highlight the favorable comparison between the optimized design and the original configuration.The model of thermal equivalent circuit after optimization is presented in Figure 7. Figures 8 and 9 show the radial and axial view results of the thermal model after optimization already gien in Figure 7.The Temperature distributions in the stator slot and in in the rotor and PM after optimization are pointed out in Figures 10 and 11.
Based on the obtained results from Figure 7 to Figure 11, the maximum temperature of the motor is reduced from 78. 4⁰C to 73.2⁰C due to a reduction of the total losses.Although these two values of the maximum temperature are both still in the safe threshold for the operation of the motor, but they once again demonstrate the correctness and effectiveness of the motor optimization model.

Conclusion
This research paper focuses on the design of a SPMSM using a multi-objective optimization approach that incorporates a genetic algorithm.The dimensions of a 2.2 kW SPMSM with a fractional-slot concentrated winding were computed.The main objective of the optimization process was to minimize two crucial factors: total loss and material cost.To validate the effectiveness of the selected variables and constraints in the optimization method, the motor's efficiency was evaluated using the FEM.The results obtained indicate that the optimized design significantly reduces material costs compared to the original design while maintaining the desired efficiency level, but it also brings about desirable changes in other parameters such as cogging torque and torque ripple.Future work could involve the development and optimization of additional goal functions, such as torque output, cogging torque, and torque ripple.Other optimization methods such as particle swarm optimization, cultural algorithm, bee algorithm, and more, could be attempted to tackle the optimization problems and achieve even better results.

Figure 1 .
Figure 1.Main steps of a multi-objective genetic algorithm.

Figure 2 .
Figure 2. Flux density distribution on the stator and rotor before optimization (top) and after optimization (bottom).

Figure 3 .
Figure 3. Distribution of cogging torque for two different cases: before and after optimizations

Figure 4 .
Figure 4. Distribution of the back EMF two different cases: before and after optimizations.

Figure 5 .
Figure 5. Distribution of the torque two different cases: before and after optimizations.

Figure 8 .
Figure 8. Radial view result of the thermal model after optimization.

Figure 9 .
Figure 9. Axial view result of the thermal model after optimization.

Figure 10 .Figure 11 .
Figure 10.Temperature distribution in the stator slot after optimization.

Table 1 .
Upper and lower boundaries of optimization variables.

Table 2 .
Main parameters of the SPMSM.

Table 3 .
Optimization results on main parameters of the SPMSM.