Optimal solution of PM synchronous motor obtained by gravitational search algorithm

Abstract

This paper presents a novel approach to the efficiency improvement of permanent magnet synchronous motor using gravitational search algorithm as an optimization tool. The gravitational search algorithm (GSA) is a recently developed meta-heuristic optimization algorithm, which so far has proven to be quite suitable for solving power engineering optimization problems. The aim of this research work is to implement this novel optimization algorithm for the efficiency improvement of permanent magnet synchronous motor, where the objective function in the optimization process is the efficiency of the investigated motor. Comparative analysis of the initial and a number of optimized solutions of the motor model is performed.

Keywords

Permanent magnet synchronous motor efficiency optimization optimal design gravitational search algorithm

1. Introduction

Energy efficiency in general became a top theme in the modern world societies. Today energy efficiency strategies are implemented in power production, transmission and distribution as well as energy consumption in industry, commercial buildings and households. In most of the areas the energy efficiency improvement is related with the energy efficiency of power devices. One of those devices that are intensively investigated are the electric motors as one of the biggest consumer of electrical energy in the world. Having in mind this fact, as well as the increased need for improvement of the energy efficiency stimulated many electrical engineers and researchers to put an effort in this area in order to improve the efficiency of different types of motors. It is well known that in the past two decades or so, different approaches and optimization procedures were implemented to the design procedures for a variety of electrical machines. Novel materials were also implemented. The optimal design of electrical machines in general can be performed using a variety of objective functions such as: efficiency, torque, power factor, output torque, cogging torque, volume, mass, cost or sometimes a combination of some of these objective functions known as multi-objective or many-objective optimization. In this research work the efficiency of the motor has been selected as an objective function for the optimal design of the permanent magnet synchronous motor.

It is well known from literature and persona experience that the optimal design of electrical machines is a constrained maximization or minimization problem with a large number of optimization parameters and variety of constraints that have to be fulfilled during the optimization process [1]. The design of a machine can be described by a vector X of n variables stating dimensions or dimension ratios, current densities, flux densities, etc. The design is subject to a set of m constraints which may include specifications arising from thermal, mechanical, manufacturing or standards limits. This makes it a difficult problem to solve by hand and therefore at the early stages scientists and researchers were trying to implement different types of optimization methods. The first methods that were implemented belong to the group of methods named as deterministic methods [2–5]. In those methods the quality of the search is highly dependent on the selection of the starting point, as well as the determination of the first order derivation in some cases that is usually hard to determine. Latter on John Holland [6,7] in 1973 gave an introduction to the Adaptation in Artificial Systems that traced the road towards the promotion of Evolution strategies, Genetic algorithms and similar stochastic algorithms that are based on natural evolution principles and mechanisms. In comparison with the deterministic methods the stochastic methods start their search based on a randomly generated population. These methods work through the optimization area based on the information of the objective function called fitness and not the values of the optimization variables as in the case of the deterministic methods. These are the reasons why those algorithms after their introduction became very popular and were implemented very successfully in optimal design of electromagnetic devices [8–11]. Today there is a large number of scientific works dedicated to implementation of Genetic algorithms in optimal design of electromagnetic devices. Afterwards many other algorithms emerged based on the principles of animal behavior and nature mechanics such as: Particle Swarm Optimization (PSO) [12], Differential Evolution [13,14], Ant Colony [15], Firefly Algorithm [16], Cuckoo Search [17] and many more. Today there are over 100 nature inspired algorithms [18,19]. Among those methods one method has been drawing attention to the optimization and research community. This method is known as Gravitational Search Algorithm (GSA) [20]. The proposed algorithm so far has been very successfully used for different optimization purposes [21–25]. Based on the findings of the authors of this algorithm, GSA has been confirmed to have a bit higher performance in searching ability in relation to other population based stochastic methods, such as GA and PSO. Based on the findings in literature it can be concluded that most of the population based algorithms perform in a similar way and that no specific method can be defined as a dominant one. Most of them are problem dependent and in different optimization procedures perform differently. The aim of this paper, based on the good results and positive experience in the application of GSA in other areas, is to implement this algorithm in the process of energy efficiency improvement of a permanent magnet synchronous motor (PMSM). In this investigation the efficiency of the motor is selected to be the objective function. Comparative analysis of the optimal motor solution based on the objective function value, as well as the values of other important parameters in relation to the initial model is performed and the results are presented in the text that follows.

2. The law of gravity

In 2009 Rashedi et al. [20] proposed a new meta-heuristic searching algorithm shortly named Gravitational Search Algorithm – GSA. The introduced algorithm is based on Newton’s law of gravity and law of motion. The law of gravity states that every particle attracts every other particle and the gravitational force between two particles is directly proportional to the product of their masses and inversely proportional to the square of the distance between them, as presented in Eq. (1): $\begin{eqnarray}\displaystyle F=G\cdot \frac{M_{1}\cdot M_{2}}{R^{2}} & & \displaystyle\end{eqnarray}$ (1) where F is the magnitude of the gravitational force, G is gravitational constant, M₁ and M₂ are the mass of the first and second particle respectively, and R is the distance between the two particles. Newton’s second law says that when a force, F, is applied to a particle, its acceleration, a, depends only on the force and its mass, M [26]: $\begin{eqnarray}\displaystyle a=\frac{F}{M}. & & \displaystyle\end{eqnarray}$ (2)

Based on (1) and (2), there is an attracting gravity force among all particles of the universe where the effect of bigger and closer particle is higher. An increase in the distance between two particles means decreasing the gravitational force between them as it is illustrated in Fig. 1. In this figure, F_1j and causes the acceleration vector a is the force that acting on M₁ from M_j and F₁ is the overall force that acts on M₁ and causes an acceleration of the vector a₁.

Fig. 1.

Influence of other masses on mass M₁.

Additionally, the change of the gravitational constant as a result of the aging of the universe should be taken into consideration and determine using Eq. (3) in which the gravitational constant G decreases with the age [27]. $\begin{eqnarray}\displaystyle G(t)=G(t_{0})\cdot \left(\frac{t_{0}}{t}\right)^{{\beta}}{\beta}<1 & & \displaystyle\end{eqnarray}$ (3) where G (t) is the value of the gravitational constant at time t. G (t₀) is the value of the gravitational constant at the first cosmic quantum-interval of time t₀.

In theoretical physics there are defined three kinds of masses:

Active gravitational mass (M_a), is a measure of the strength of the gravitational field due to a particular object. Gravitational field of an object with small active gravitational mass is weaker than the object with more active gravitational mass.

Passive gravitational mass (M_p), is a measure of the strength of an object’s interaction with the gravitational field. Within the same gravitational field, an object with a smaller passive gravitational mass experiences a smaller force than an object with a larger passive gravitational mass.

Inertial mass (M_i), is a measure of an object resistance to changing its state of motion when a force is applied. An object with large inertial mass changes its motion more slowly, and an object with small inertial mass changes it rapidly.

Based on the previously mentioned aspects the Newton’s laws can be rewritten. The gravitational force, F_ij, that acts on mass i by mass j, is proportional to the product of the active gravitational of mass j and passive gravitational of mass i, and inversely proportional to the square distance between them. On the other hand the acceleration a_ij is proportional to the force F_ij and inversely proportional to inertia mass of i. The above statement can presented by rewriting Eqs (1) and (2) as follows: $\begin{eqnarray}\displaystyle F_{ij}\text{} & =\text{} & \displaystyle G\cdot \frac{M_{aj}\cdot M_{pi}}{R_{ij}^{2}}\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle a_{ij}\text{} & =\text{} & \displaystyle \frac{F_{ij}}{M_{ii}}\end{eqnarray}$ (5) where M_aj is the active gravitational mass of particle i, M_pi is the passive gravitational mass of particle j, and M_ii represents the inertia mass of particle i.

The presented Newtonian laws were very successfully implemented by Rashedi et al. [20] in a new optimization algorithm called Gravitational Search Algorithm (GSA). In the following text the main principles of this algorithm is presented.

3. Gravitational search algorithm – GSA

According to the proposed algorithm, the objects in the algorithm are defined as agents, the performances of which are measured by means of masses. Hence, all these objects attract each other by a gravity force, and this force causes a global movement of all objects towards the objects with heavier masses. Since the heavier masses have higher fitness values; they describe good optimal solution to the problem and they move more slowly than lighter ones representing worse solutions. In GSA, each mass has four particulars: its position, its inertial mass, its active gravitational mass and passive gravitational mass. The position of the mass equalled to a solution of the problem and its gravitational and inertial masses are specified by using a fitness function. In other words, each mass performs a solution and the algorithm is navigated by appropriately adjusting the gravitational and inertia masses.

The proposed GSA could be considered as an isolated system of masses. It is like a small artificial world of masses obeying the Newtonian laws of gravitation and motion. More precisely, masses obey the following laws:

Law of gravity: each particle attracts every other particle and the gravitational force between two particles is directly proportional to the product of their masses and inversely proportional to the distance between them, R. The authors of the algorithm, Rashedi et al. [20], use here R instead of R², because according to their experimental results, R provides better results than R² in all experimental cases that they have performed.

Law of motion: the current velocity of any mass is equal to the sum of the fraction of its previous velocity and the variation in the velocity. Variation in the velocity, known as acceleration of any mass is equal to the force acting on the system divided by mass of inertia.

The authors of the GSA take into consideration a universe of N masses (agents), where the position of the i-th agent is defined as: $\begin{eqnarray}\displaystyle X_{i}=(x_{i}^{1},\ldots ,x_{i}^{d},\ldots ,x_{i}^{n})\quad \text{for}∼i=1,2,3\ldots N & & \displaystyle\end{eqnarray}$ (6) where $x_{i}^{d}$ presents the position of i-th agent in the d-th dimension.

At a specific time ‘t’, the force acting on mass ‘i’ from mass ‘j’ as following: $\begin{eqnarray}\displaystyle F_{ij}^{d}(t)=G(t)\frac{M_{pi}(t)\times M_{aj}(t)}{R_{ij}(t)+{\varepsilon}}(x_{j}^{d}(t)-x_{i}^{d}(t)) & & \displaystyle\end{eqnarray}$ (7) where M_aj is the active gravitational mass related to agent j, M_pi is the passive gravitational mass related to agent i, G(t) is gravitational constant at time t, ϵ is a small constant, and R_ij(t) is the Euclidian distance between two agents i and j. The stochastic characteristic of the algorithm, is defined as a total force that acts on agent i in a dimension d that is randomly weighted and calculated as a sum of dth components of the forces exerted from other agents, as presented in the following equation: $\begin{eqnarray}\displaystyle F_{i}^{d}(t)=\mathop{\sum }_{j=1,j\neq i}^{N}m_{j}\cdot F_{ij}^{d}(t) & & \displaystyle\end{eqnarray}$ (8) where m_j is a randomly generated number between 0 and 1. Therefore, the acceleration of mass i at time t and dimension d is given by: $\begin{eqnarray}\displaystyle a_{i}^{d}(t)=\frac{F_{i}^{d}(t)}{M_{ii(t)}} & & \displaystyle\end{eqnarray}$ (9) where M_ii is the inertial mass of ith agent. In the definition of the algorithm the velocity and the position are also taken into consideration. More details about the algorithm’s performance could be found in reference [21].

The concept of the gravitational forces acting between the agents and a presentation of their movement in the search area for one iteration is presented in Fig. 2. Based on the presentation in Fig. 2 it can be concluded that the agents with smaller masses or smaller fitness values of the objective function tend to move towards the agents with larger masses. In this movement some agents reach better values and some get not so good values. Also the velocity of their movement is different the agents with greater masses move slowly while the agents with smaller masses tend to move faster and therefore their acceleration is different.

Fig. 2.

The concept of the gravitational forces.

The structural presentation of the Gravitational search algorithm is presented with the block diagram in Fig. 3. The GSA optimization is performed using 50 agents during 1500 iterations.

Fig. 3.

Main steps of the Gravitational Search Algorithm.

4. GSA optimization of permanent magnet synchronous motor

In this paper the GSA is applied as an optimisation tool in the process of efficiency improvement of a permanent magnet synchronous motor (PMSM). Several GSA performnce parameters were adjusted to meet the appliation. The investigated motor is a surface mounted permanent magnet motor with 36 slots on the stator and a rotor with 6 samarium-cobalt permanent magnets that have a remanent flux density of B_r = 0. 95 T and coercitive filed of H_c = −720 kA/m. The efficiency of the motor is selected as an objective function in the optimal search process. In this case the following motor parameters are selected to be variable in the optimization process: outside radius of the rotor iron core R_ro, permanent magnet fraction f_m, permanent magnet radial height h_m, air-gap g, and axial active length of the motor L.

An overview of the motor together with the variable optimization parameters is shown in Fig. 4. A proper mathematical model of the motor is developed as one of the contributions of this paper. The mathematical model of the motor consists of equations that define all design motor parameters. In this mathematical presentation of the motor the objective function, in this case the efficiency of the PMSM, is defined and implemented in the gravitational search algorithm [20] and is expressed by the following equation: $\begin{eqnarray}\displaystyle \text{objective}∼\text{function}=\text{efficiency}(R_{ro},f_{m},h_{m},g,L)=\frac{T\cdot {\omega}_{m}}{T\cdot {\omega}_{m}+P_{\text{Cu}}+P_{\text{Fe}}+P_{wf}} & & \displaystyle\end{eqnarray}$ (10) where: T-rated torque, ω_m-rated speed, P_Cu-ohmic power loss, P_Fe-core loss and P_wf-windage and friction losses.

The power losses presented in Eq. (10) are defined with the following equations: $\begin{eqnarray}\displaystyle P_{\text{Cu}}\text{} & =\text{} & \displaystyle 3R_{ph}\cdot I_{ph}^{2}\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle P_{\text{Fe}}\text{} & =\text{} & \displaystyle P_{Fe_{SY}}+P_{Fe_{St}}\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle P_{Fe_{SY}}\text{} & =\text{} & \displaystyle p_{S_{1.5/50}}\cdot \left(\frac{B_{SY}}{1.5}\right)^{2}\cdot \left(\frac{f}{50}\right)^{{\alpha}}\cdot m_{Fe_{SY}}\end{eqnarray}$ (13) $\begin{eqnarray}\displaystyle P_{Fe_{St}}\text{} & =\text{} & \displaystyle p_{S_{1.5/50}}\cdot \left(\frac{B_{St}}{1.5}\right)^{2}\cdot \left(\frac{f}{50}\right)^{{\alpha}}\cdot m_{Fe_{St}}\end{eqnarray}$ (14) where: R_ph – stator winding phase resistance, I_ph – phase current, P_FeSY – iron power losses in stator yoke and P_FeSt – iron power losses in stator teeth, p_S1.5∕50 – specific iron power losses at flux density of 1.5 T and frequency of 50 Hz, B_SY – stator yoke (back iron) flux density, B_St – stator teeth flux density, f – frequency, 𝛼 – Steinmetz constant (1 < 𝛼 < 2), m_FeSY – total stator yoke iron mass and m_Fest – total stator teeth iron mass. This is only a partial presentation of the system of equations for the mathematical model of the motor that have been defined and implemented in the optimization algorithm.

Fig. 4.

Permanent magnet synchronous motor optimisation parameters.

The efficiency improvement is a maximisation problem of the objective function in which the value of the efficiency of the motor is maximised, where the torque is one of the constraints. A presentation of those constrains is shown in Table 1. In the mathematical model of the motor, implemented in the Gravitational search algorithm, all the parameters of the motor are defined in function of the optimisation parameters that are presented in Fig. 4. Each parameter is varied in its own lower and upper boundary value using the following equation: $\begin{eqnarray}\displaystyle \text{Variable}_{n}=LB_{n}+(UB_{n}-LB_{n})\cdot x_{n}\quad \text{where}∼n=1,2,3,4∼\text{and}∼5 & & \displaystyle\end{eqnarray}$ (15) whrere: LB_n is the lower boundary for each variable, UB_n is the upper boundary for each variable, and x_n is a randomly generated real number, separately for each variable, between 0 and 1. The values of the lower and upper boundary, as well as the values of the parameters for the initial solution are presented in Table 2. The lower and upper boundary values are defined as ±10% of the initial solution value of each variable separately. In the same table the optimal sulution is also presented together with the values of each optimised parameter and the value of the objective function-efficiency of the motor. The best efficiency obtained by implementation of GSA in this work is 0.8802, where in comparison in a previous work [28] the efficiency reached by using Genetic alsgorithm (GA) and the same mathematical model is 0.8804, by using Particle swarm optimisation (PSO) is 0.8676 and by using Cuckoo serach (CS) is 0.8801. The values show that the GSA is quite comeptetive with GA and CS and better than PSO. With further adjustments of the GSA search parameters it is forseen that the optimization seach and results can be improved.

Table 1

Optimization constraints

Description	Parameter	Value
Torque	T (Nm)	10
Rated synchronous speed	n (min⁻¹)	1000
Number of phases	N _ph	3
Number of permanent magnets	N _m	6
Number of stator slots	Z	36
PM residual flux density	B_r (T)	0.95
Slot opening	b_s (m)	0.003
Stator back iron flux density	B_mSY (T)	≤1.8
Stator teeth flux density	B_mSt (T)	≤1.76
Maximum conductor current density	J_max (A/m²)	≤6.5 ∗10⁻⁶
Slot fill factor (conducting packing area)	k_Cu (/)	≤0.4

Table 2

Optimization parameters, boundaries and values

Variables	Lower boundary	Upper boundary	Initial model	GSA solution
R_ro (m)	0.03798	0.4642	0.042	0.037805
g (m)	0.00072	0.00088	0.0008	0.000721
f_m (/)	0.81	0.99	0.90	0.914828
h_m (m)	0.0018	0.0022	0.002	0.002179
L (m)	0.081	0.099	0.09	0.098995
Eff	–	–	0.8489	0.8802

Fig. 5.

Objective function change during the GSA search.

The convergence of the objective function of the motor during the best GSA optimisation search for 1500 iterations is shown in Fig. 5. The presented optimal solution is defined as the best because each mass (agent) is defined with its position, inertial mass, active gravitational mass, and passive gravitational mass, and because those parameters have a stochastic nature each search results with a different optimal solution. After a large number of computations the results start to repeat. The values of the parameters and the efficiency for those selected solutions are presented in Table 3.

Table 3

Optimization parameter comparison of the selected solutions

#	R_ro (m)	g (m)	f_m (/)	h_m (m)	L (m)	Eff
1	0.037801	0.000722	0.910471	0.002178	0.098996	0.8801
2	0.037811	0.000721	0.916994	0.00215	0.099000	0.8799
3	0.037837	0.000725	0.924495	0.002178	0.099000	0.8800
4	0.037821	0.000725	0.920646	0.002154	0.098999	0.8798
5	0.037810	0.000725	0.898022	0.00216	0.098998	0.8798
6	0.037804	0.000724	0.918158	0.002182	0.098959	0.8801
7	0.037800	0.000723	0.914223	0.002167	0.098995	0.8800
8	0.037811	0.000726	0.909639	0.002184	0.098988	0.8800
9	0.037801	0.000722	0.900191	0.002173	0.098992	0.8800
10	0.037807	0.000723	0.893511	0.002174	0.099000	0.8799
11	0.037823	0.000721	0.923784	0.002167	0.098998	0.8800
12	0.037813	0.000728	0.907485	0.002159	0.099000	0.8797
13	0.037821	0.000725	0.91555	0.002169	0.098999	0.8799
14	0.037803	0.000725	0.914098	0.002182	0.098998	0.8801
15	0.037804	0.000723	0.900071	0.002135	0.098999	0.8796
16	0.037815	0.000729	0.874352	0.002138	0.098999	0.8791
17	0.037811	0.000734	0.907703	0.002153	0.098998	0.8794
18	0.037809	0.000726	0.907084	0.002129	0.098986	0.8795
19	0.037821	0.000737	0.907132	0.002152	0.098993	0.8793
20	0.037805	0.000721	0.914828	0.002179	0.098995	0.8802

On the other hand a comparative presentation of the influence of the parameters on the efficiency of the solutions for selected parameters is presented in Fig. 6. In Fig. 6 the distribution of the solutions in the area defined by the height of the permanent magnets and the permanent magnet fraction is presented. The values shown next to each solution are the values of the efficiency of the selected and presented solutions. Based on the presented results in Fig. 6 it can be concluded that the solutions tend to group towards larger values of the permanent magnet fraction, larger values of the PM height, smaller values of the air gap and larger values of the axial active length of the motor. Generally speaking the algorithm tends to increase the mass of the active materials in order to improve the efficiency, which in practice is a standard approach towards high efficiency motors.

Fig. 6.

Presentation of the efficiency in funciton of the parameters h_m and f_m.

Fig. 7.

Presentation of the efficiency in funciton of the parameters g and L.

The same analogy is used for the data shown in Fig. 7 from which the presentation of the dependence of the efficiency of the solutions in function of the parameters g and L is shown.

Sometimes an additional analysis is needed in which additional motor properties, parameters of objectives can be involved in the determination of the optimal solution. Especially this is a good approach if there are a wide range of solutions with close results. Therefore, in this paper an additional extended analysis of the results will be implemented.

5. Additional analysis of GSA solutions

It is well known that the design of electrical machines in general is a multi-variable and multi-objective process. As it has been seen previously the multi-variable optimization can be easily solved using stochastic optimization methods. In a single objective function optimization the optimizer is working only with one function and not taking into account the other objectives. One way to improve the search is to perform the single objective approach, but changing the objective function and then at the end to compare the best solutions from each optimization. Another way is to perform a multi-objective or a many-objective optimal design [29], but then the problem of defining the influence of each partial objective function on the general optimization function is still present and is usually a problem dependent. With a bit of experience in this area this problem can be easily solved. In this work an additional analysis will be realised in order to detect how the optimization of the efficiency of the motor influences other selected parameters and quantities. In Table 4 part of those parameters such as air gap flux density–B_g, phase resistance–R_ph, total iron mass–m_Fe, total copper mass–m_Cu, total mass of permanent magnets–m_pm, and total motor mass–m_tot are presented, where in Table 5 an additional parameters like copper power losses–P_Cu, iron power losses–P_Fe, total power losses–P_tot and phase current–I_ph are also presented for the optimized solutions (1–20) and the initial model (0).

Table 4
Selected parameters comparison of the selected solutions (1–20) and the initial model (0)

# B_g (T) R_ph (Ω) m_Fe(kg) m_Cu(kg) m_pm (kg) m_tot(kg)

0 0.5868 0.1564 6.8308 1.5979 0.3650 8.7937

1 0.6161 0.0915 6.8777 2.3460 0.3982 9.6219

2 0.6169 0.0918 6.8920 2.3380 0.3958 9.6259

3 0.6207 0.093 6.9174 2.3074 0.4046 9.6294

4 0.6176 0.0922 6.9013 2.7638 0.3983 9.6270

5 0.6093 0.0900 6.8404 2.3848 0.3895 9.6148

6 0.6188 0.0922 6.8938 2.3267 0.4021 9.6226

7 0.6166 0.0916 6.8840 2.3421 0.3976 9.6237

8 0.6152 0.0914 6.8735 2.3475 0.3989 9.6200

9 0.6116 0.0904 6.8477 2.3756 0.3927 9.6160

10 0.6086 0.0897 6.8303 2.3924 0.3900 9.6127

11 0.6205 0.0928 6.9136 2.3138 0.4021 9.6294

12 0.6124 0.0909 6.8649 2.3613 0.3934 9.6195

13 0.6167 0.0919 6.8900 2.3354 0.3989 9.6243

14 0.6171 0.0918 6.8850 2.3379 0.4005 9.6235

15 0.6087 0.0899 6.8424 2.3883 0.3856 9.6163

16 0.5966 0.0874 6.7701 2.4560 0.3752 9.6013

17 0.6108 0.0908 6.8608 2.3652 0.3923 9.6183

18 0.6105 0.0905 6.8588 2.3720 0.3875 9.6184

19 0.6098 0.0907 6.8589 2.3662 0.3919 9.6171

20 0.6181 0.0919 6.8893 2.3348 0.4004 9.6244

#	B_g (T)	R_ph (Ω)	m_Fe(kg)	m_Cu(kg)	m_pm (kg)	m_tot(kg)
0	0.5868	0.1564	6.8308	1.5979	0.3650	8.7937
1	0.6161	0.0915	6.8777	2.3460	0.3982	9.6219
2	0.6169	0.0918	6.8920	2.3380	0.3958	9.6259
3	0.6207	0.093	6.9174	2.3074	0.4046	9.6294
4	0.6176	0.0922	6.9013	2.7638	0.3983	9.6270
5	0.6093	0.0900	6.8404	2.3848	0.3895	9.6148
6	0.6188	0.0922	6.8938	2.3267	0.4021	9.6226
7	0.6166	0.0916	6.8840	2.3421	0.3976	9.6237
8	0.6152	0.0914	6.8735	2.3475	0.3989	9.6200
9	0.6116	0.0904	6.8477	2.3756	0.3927	9.6160
10	0.6086	0.0897	6.8303	2.3924	0.3900	9.6127
11	0.6205	0.0928	6.9136	2.3138	0.4021	9.6294
12	0.6124	0.0909	6.8649	2.3613	0.3934	9.6195
13	0.6167	0.0919	6.8900	2.3354	0.3989	9.6243
14	0.6171	0.0918	6.8850	2.3379	0.4005	9.6235
15	0.6087	0.0899	6.8424	2.3883	0.3856	9.6163
16	0.5966	0.0874	6.7701	2.4560	0.3752	9.6013
17	0.6108	0.0908	6.8608	2.3652	0.3923	9.6183
18	0.6105	0.0905	6.8588	2.3720	0.3875	9.6184
19	0.6098	0.0907	6.8589	2.3662	0.3919	9.6171
20	0.6181	0.0919	6.8893	2.3348	0.4004	9.6244

For the extended analysis selected data from Tables 4 and 5 is used to graphically present some of the relationships between certain motor parameters. Therefore, in Fig. 8 the influence of the total iron power losses–P_Fe and the copper power losses–P_Cu on the efficiency of each solution are presented. The best and worst solution regarding the value of the efficiency are highlighted. It is obvious that the copper power losses have great impact on the efficiency of the solutions, due to their high value, and therefore the selected best solution regarding the efficiency is also the best solution regarding the copper power losses, while the worst solution has the highest value for the copper power losses. On the other hand the iron power losses have a significantly lower value and therefore a smaller influence on the efficiency of the solutions. That is why the selected best solution does not have the smallest iron power losses value, while the worst solution has the smallest value for the same losses. Similar analysis can be performed for the presented results in Fig. 9, where the presented solutions are distributed regarding the total power losses and the total mass of the motor. The best solution has the smallest value for the total power losses, while the worst solution has the highest value for the total power losses in comparison to the other solutions that are distributed between the values of those two solutions. Regarding the total mass of the motor the worst solution has the smallest mass among all the solutions, while the best solution regarding the efficiency has a mass that is in the range of the majority of solutions.

Fig. 8.

Presentation of the efficiency in relation to P_Fe and P_Cu.

Fig. 9.

Presentation of the efficiency in relation to total power losses and total motor weight.

Table 5

Power losses and phase current comparison of the selected solutions (1–20) and the initial model (0)

#	P_Cu (W)	P_Fe(W)	P_Fe + P_w(W)	P_tot(W)	I_ph (A)
0	147.0911	17.3571	39.3571	186.4482	17.7048
1	100.3057	20.3185	42.3185	142.6242	19.1163
2	100.5433	20.3858	42.3858	142.9292	19.1068
3	100.3544	20.4901	42.4901	142.8445	18.9633
4	100.6894	20.4236	42.4236	143.1130	19.0771
5	100.9546	20.1329	42.1329	143.0875	19.3347
6	100.3004	20.4076	42.4076	142.7080	19.0412
7	100.4317	20.3565	42.3565	142.7882	19.1130
8	100.4944	20.2976	42.2976	142.7920	19.1411
9	100.5675	20.1783	42.1783	142.7457	19.2611
10	100.8083	20.0859	42.0859	142.8942	19.3508
11	100.2929	20.4832	42.4832	142.7761	18.9842
12	100.9193	20.2505	42.2505	143.1697	19.2362
13	100.5448	20.3690	42.3690	142.9137	19.0956
14	100.3569	20.3597	42.3597	142.7166	19.0884
15	101.1626	20.1463	42.1463	143.3090	19.3689
16	102.2635	19.7801	41.7801	144.0435	19.7453
17	101.3253	20.2314	42.2314	143.5568	19.2907
18	101.2842	20.2243	42.2243	143.5085	19.3161
19	101.5464	20.2133	42.2133	143.7598	19.3160
20	100.1513	20.3793	42.3793	142.5306	19.0567

This brings us to a conclusion that the single objective optimization besides taking care of the value of the objective function to some extent has an influence also on some other parameters of the motor. But unfortunately the single objective optimization does not take care of all the motor parameters such as the total mass in this case. Therefore, one way to tackle this problem is the perform different optimization procedures using different objective functions and then put all those solutions in a Pareto analysis or use a multi-objective or many-objective optimization approach [29]. Additional motor parameters that can be taken into consideration in such an analysis are the values of the power over mass ratio, power over volume ratio or even cogging torque as one of the important performance parameters of a permanent magnet motors. Additionally for a more detailed investigation a traditional finite element method analysis for the initial solution and the best solution is performed and will be presented in the following text. The comparative analysis of the three motor models is going to be extend by using the commercial program Motor-Solve by Infolytica [30] that is based on the finite element method.

6. Finite element analysis of the GSA optimal solution

In the Motor-Solve program [31] a proper modelling of the initial and optimized motor models is performed. All the necessary data regarding the dimensions, materials, supply and load are applied for the two analyzed motor models. The first parameter that has been calculated is magnetic field in the two dimensional domain of the motor taking into account the axial length of the motor. Based on the presented magnetic field distributions, especially for the optimized solution, it can be concluded that the values of the flux density in the segments of the motor of special interest (stator back iron and stator teeth) are smaller than the values prescribed as constraints. The magnetic field distribution for the two motor models is presented in Figs 10 and 11.

Fig. 10.

Magnetic field distribution for the initial motor.

Fig. 11.

Magnetic field distribution for the gravitational search algorithm solution model.

The next motor parameter that is going to be presented and compared is the air gap flux density distribution for the analyzed two motor models. The average values of the air gap flux density calculated analytically from the optimization process are presented in Table 6. From the presented air gap flux density distribution and the analytically calculated average values, shown in Figs 12 and 13, it can be concluded that there is also a good agreement between those values.

Table 6

Air gap flux density comparative data

Method	Air gap flux density	Initial model	GSA solution
Determined from optimisation	B_gGSA (T)	0.630	0.668
Determined from FEM	B_gFEM (T)	0.621	0.659

Fig. 12.

Air gap flux density distribution for the initial motor.

Fig. 13.

Air gap flux density distribution for the gravitational search algorithm solution.

Fig. 14.

Electromagnetic torque distribution for the initial motor.

Fig. 15.

Electromagnetic torque distribution for the gravitational search algorithm model.

The load torque of the motor is parameter that in the optimization process is defined as a constraint and is equal to 10 Nm. Using the finite element analysis program the electromagnetic torque can be obtained. In general the value of this torque is slightly larger that the load torque, as it is in the case of the invastigated motor. The initial motor and the GSA model have the same avarge value of the electromagnetic torque, as shown in Fig. 14 and Fig. 15, respectively. This means that the torque as a constraint during the optimization is satisfied and reached.The good agreement of the values proves the well determined mathematical model of the permanent magnet synchronous motor, as well as the good performance of the optimization method.

7. Conclusion

In this paper the authors are giving a brief presentation of a novel technique for optimal design of electrical machines, named Gravitational Search Algorithm that has been introduced and implemented. The efficiency of the motor has been selected as an objective function in the optimal design. Based on this methodology a set of optimal solutions of the permanent magnet synchronous motor were reached, due to the stochastic nature of the optimization method. In those solutions the value of the motor efficiency varies between 0.8791 and 0.8802. Afterwards an extended analysis has been implemented in order to evaluate all the solutions using other motor quantities and parameters. In addition finite element analysis of the initial and best GSA solution is performed and the magnetic field distribution and air gap flux density distribution is analysed. Based on the presented data and graphs it is evident that the optimized GSA solution is a better solution in all aspects in relation to the initial model. As a future work additional investigations will be performed in which several other solutions from the GSA optimization will be included in the comparative performance analysis using also additional motor parameters.

References

Liu

and Slemon

, An improved method of optimization for electrical machines, IEEE Transactions on Energy Conversion6(3) (1991), 492–496.

Gottvald

Preis

Magele

Biro

and Savini

, Global optimization methods for computational electromagnetics, IEEE Transactions on Magnetics28(2) (1992), 1537–1540.

D.-H.

Park

S.-C.

and Im

J.-W.

, Design of single-sided linear induction motor using the finite element method and SUMT, IEEE Transactions on Magnetics29(2) (1993), 1762–1766.

Brisset

and Brochet

, Optimization of switched reluctance motors using deterministic methods with static and dynamic finite element simulations, IEEE Transactions on Magnetics34(5) (1998), 2853–2856.

Boules

, Design optimization of permanent magnet DC motors, IEEE Transactions on Industry Applications26(4) (1990), 786–792.

Holland

J.H.

, Genetic algorithms and the optimal allocation of trials, SIAM J. Comput.2(2) (1973), 88–105.

Holland

J.H.

, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbour, 1995.

Üler

G.F.

Mohammed

O.A.

and Koh

C.S.

, Design optimisation of electrical machines using genetic algorithms, IEEE Transactions on Magnetics31(3) (1995), 2008–2011.

Üler

G.F.

and Mohammed

O.A.

, Ancillary techniques for the practical implementation of GAs to the optimal design of electromagnetic devices, IEEE Transactions on Magnetics32(3) (1996), 1194–1197.

10.

Sim

D.J.

Jung

H.K.

Hahn

S.Y.

and Won

J.S.

, Application of vector optimization employing modified genetic algorithms to permanent magnet motor design, IEEE Transactions on Magnetics33(2) (1997), 1888–1891.

11.

Wurtz

Richomme

Bigeon

and Sabonnadiere

J.C.

, A few results for using genetic algorithms in the design of electrical machines, IEEE Transactions on Magnetics33(2) (1997), 1892–1895.

12.

Kennedy

Eberhart

R.C.

and Shi

, Swarm Intelligence: The Morgan Kaufmann Series in Evolutionary Computation, Morgan Kaufmann Publishers, San Francisco, 2001.

13.

Storn

and Price

, Differential evolution: A simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization11 (1997), 341–359.

14.

Price

Storn

and Lampinen

, Differential Evolution: A Practical Approach to Global Optimization, Springer, 2005.

15.

Dorigo

Maniezzo

and Colorni

, Ant system: optimization by a colony of cooperating agents, IEEE Transactions on Systems, Man, and Cybernetics – Part B26 (1996), 29–41.

16.

Yang

X.S.

, Firefly algorithms for multimodal optimisation, in: Proceedings fifth Symposium on Stochastic Algorithms, Foundations and Application Watanabe

and Zeugmann

(eds), Lecture Notes in Computer Science, 5792,2009, pp. 169–178.

17.

Yang

X.S.

and Deb

, Engineering optimization by cuckoo search, International Journal of Mathematical Modelling and Numerical Optimization1(4) (2010), 330–343.

18.

Mirjalili

Dong

J.S.

and Lewis

, Nature-Inspired Optimizers–Theories, Literature Reviews and Applications, Springer, 2020.

19.

Yang

X.S.

, Nature-Inspired Optimization Algorithms, Elsevier, 2014.

20.

Rashedi

Nezamabadi-pour

and Saryazdi

, GSA: A gravitational search algorithm, Information Sciencies179 (2009), 2232–2248.

21.

Geleta

D.K.

and Manshahia

M.S.

, Gravitational search algorithm-based optimization of hybrid wind and solar renewable energy system, Computational Intelligence (2020), 1–27.

22.

Radosavljevic

Jevtic

Arsic

and Klimenta

, Optimal power flow for distribution networks using gravitational search algorithm, Electrical Engineering96(4) (2014), 335–345.

23.

Duman

Güvenç

Sönmez

and Yörükeren

, Optimal power flow using gravitational search algorithm, Energy Conversion and Management59 (2012), 86–95.

24.

Duman

Sönmez

Güvenç

and Yörükeren

, Optimal reactive power dispatch using a gravitational search algorithm, IET Generation, Transmission and Distribution6(6) (2012), 563–576.

25.

Beigvand

S.D.

Abdi

and La Scala

, Combined heat and power economic dispatch problem using gravitational search algorithm, Electric Power Systems Research133 (2016), 160–172.

26.

Holliday

Resnick

and Walker

, Fundamentals of Physics, John Wiley and Sons, 2014.

27.

Mansouri

Nasseri

and Khorrami

, Effective time variation of G in a model universe with variable space dimension, Physics Letters A259 (1999), 194–200.

28.

Di Barba

and Wiak

(eds) Optimal Design Exploiting 3D Printing and Metamaterials, Chapter 4: Petkovska, L. and Cvetkovski, G. - Innovative motors and shape optimisation, IET, 2021, (in print).

29.

Di Barba

Mognaschi

M.E.

Rezaei

Lowther

D.A.

and Rahman

, Many-objective shape optimisation of IPM motors for electric vehicle traction, International Journal of Applied Electromagnetics and Mechanics60(S1) (2019), S149–S162.

30.

Infolytica. User’s manual, 2017.

31.

Motor SolveInfolytica Co., User’s manual, 2017.