Rotor design optimization of a 4000 rpm permanent magnet synchronous generator using moth flame optimization algorithm

ABSTRACT


Introduction
Thin (0.20 and 0.35 mm) lamination is often used in high efficiency alternator designs.However, this alternator design significantly increases manufacturing costs.In addition, the availability of this special lamination in the market is limited.It is very difficult to obtain from the market in small quantities, and it is possible to obtain it in case of high purchases in the sector.However, the supply of raw materials for a small amount of production in the sector is a difficult task.The production of high speed alternators with standard materials (eg M530-50A etc. lamination) is the preferred production method, especially in mass production.However, this causes core loss.In this case, alternator design optimization becomes important [1,2].There are four different structures in high speed generators: (i) salient-pole generator, (ii) asynchronous generator (SG), (iii) permanent magnet synchronous generator (PMSG), and (iv) switched reluctance generator.PMSG is the easiest and most common generator type to manufacture.There are many studies on PMSG design optimization in the literature.The related remarkable studies are as follows: Fang et al. [3] investigated the optimization of SG design.They used double layer interior PMSG (IPMSG).The break angles and length of the PM segments were determined as the factors, while rotor saliency ratio, motor efficiency performance, and back electromotive force (back EMF) are measured as responses.They used finite element analyis (FEA) and the equivalent circuit approach.During the optimization phase, response surface methodology (RSM) is applied.Li and Liu [4] investigated the optimization of the PMSG shape and assessed magnetic field intensity and magnetic flux density (MFD).Kurt et al. [5] studied on optimizing axial flux PMSG (AFPMSG) design using the Taguchi method.
The factors were internal radius, pole number, magnet thickness, and pole utilization factor; and the response was air-gap MFD (AGMFD).They performed optimization using the simplex algorithm, and validated the results using FEA.Hasanien and Muyeen [6] employed genetic algorithm (GA) and RSM to offer an optimum design technique for the controller used in the frequency converter of a variable speed wind turbine (VSWT) driven PMSG.
The purpose was to find the best parameters for the PI controllers.Settling time, maximum percentage overshoot (MPOS), maximum percentage undershoot (MPUS), and steady-state error of the voltage profile are the measured responses.The performance of the specified parameters acquired with GA-RSM are then compared to those obtained with a generalized reduced gradient (GRG) approach that takes into account both unsymmetrical and symmetrical errors.Oh et al. [7] conducted research on the design of an IPMSM with concentrated windings.Authors minimized the PM eddy-current loss by moving the magnets at the rotor and keep the MFD in the air gap at a desired range.They used rotor shape and rpm as factors.Several rotor design parameters (bridge size and magnet height) were taken into account in Neubauer et al.'s [8] study.They focused on the IPMSG's rotor design optimization in order to determine their impact on the MFD and machine's performance.A direct-driven surface-mounted PMSG (D-SPMSG)'s magnetic flux linkage optimization was researched by Xie et al. [9].They used FEA to do simulations and determined winding arrangements, PM specifications and dimensions, as the factors those will be optimized.Demir and Akuner [10] investigated the critical rotor pole data of a line-start PMS motor (LSPMSM).They employed Taguchi for determining factor values of LSPMSM critical rotor pole data in order to optimize efficiency and power factor.They considered the magnet duct dimensions, width of the rib, magnet width, magnet thickness as the factors.Sabioni et al. [11] optimized the design of a 10 kW axial-flux PMSG for direct-coupled wind turbines using non-dominated sorting GA-III (NSGA-III).Material cost, efficiency, outer diameter, and weight are chosen as responses.The authors considered the electromagnetic factors and the dimensions during the design procedure.In order to enhance the design of SPMSG, Dai et al. [12] combined Taguchi and GA.They made an effort to increase efficiency while reducing expense and THD of surface-mounted permanent magnet synchronous machines (SPMSMs).FEA is used to verify the results.They selected the design parameters namely inner radius of rotor, height of air gap, height of magnet, height of stator yoke, width of stator tooth, height of slot, pole-arc to pole-pitch ratio, core length, pole shift angle, magnet eccentricity, magnet segmentation as the factors those have to be optimized.Gul et al. [13] investigated the DSSR PMSG (dual-stator single-rotor PMSG) optimization.They employed GA to efficiently reduce the cost of the DSSR PMSG and weight.The authors additionally investigated into the MFD distribution.They used power distribution factor, thickness of magnet, axial length, thickness of rotor yoke, thickness of stator yokes, ratio of magnet pole arc to pole pitch as the factors.Semon et al. [14] used RSM for rotor design optimization of a V-type IPMSM to reduce THD while maintaining the desired range of airgap MFD.They used geometrical dimensions of rotor pole shape as the factors.To optimize the THD and voltage regulation rate, Jun et al. [15] explored the design optimization of PMSG for wind power generators using the Taguchi.They tried to determine the optimum fator levels of polar arc coefficient, air gap length, PM thickness, number of turns per phase coil.Karimpour et al. [16] optimized the design of IPMSG using FEA and Taguchi in order to improve efficiency, THD, and the amplitude of induction EMF.They used stator tooth width, stator slot depth, magnet thickness, magnet width, magnet inset, magnet spread angle as the factors.In the following year, Karimpour et al.'s [17] research focused on improving the IPMSG design using FEA and the Taguchi Method.They took into consideration the effects of the magnet inset, magnet thickness, stator tooth width, slot depth, and magnet breadth.The efficiency of the generator, output power, and torque ripple were the measured responses to be optimized.Agrebi et al. [18] employed GA to optimize the design parameters (ratio of the bore radius to the active length of the machine, ratio of the slot depth to the bore radius of the machine, pole pairs number, current surface density, rated power, rated angular rotation speed, induction in the stator yoke, slot number per pole and per phase) of a direct drive PMSG (DDPMSG) for a smart wind turbine, and the results were validated by FEA.Alemi-Rostami et al. [19] investigated the construction of a step-bystep strategy for the design of a PMSG to achieve improved efficiency and power factor while requiring less voltage regulation and PM volume.The second motivation is to demonstrate to the readers the utility of MFO with regard to these design problems.As demonstrated by the "No Free Lunch", none of the offered methods in the literature can handle all difficulties with optimization alone [20,22].Therefore, there is always a need to investigate the performance of newly presented algorithms on the presented problems in the literature.

Mathematical modeling
In this paper, it is aimed to determine the optimum level of the factors (embrace, offset, outer diameter, and magnet thickness) to hold the MFD distribution within a desired range while maximizing the efficiency.To do this, the mathematical relation between these responses and factors have to be calculated in the first stage (then MFO will be run through these models in order to do the optimization).
Regression modeling is used to do this.The terms in regression models might be linear, quadratic, or interaction.The models those have these three terms together are referred to as a full quadratic model.The general representation for this type of regression model is given in Eq. ( 1).This model is used to fit mathematical models to the Maxwell simulations presented in Section 4 [23][24][25][26].
is the response for uth run.Responses of this study are the MFD distributions (stator teeth flux density, stator yoke flux density, MFD) and efficiency, which means that in Section 4 four separate regression equations will be calculated.X terms are the factors ( 1 : embrace,  2 : offset,  3 : outer diameter, and  4 : magnet thickness).    terms represents the interaction terms (in this study maximum 3 interaction terms can be used such as  1  2  1  3 ,  2  3 ).  is the prediction error (PE) for the uth experimental run [23][24][25][26].The regressin models are calulated by the aid of Minitab Statistical Package in this study.The  2 (coefficient of determination) calculations and "Analysis of Variance (ANOVA)" are also performed by the aid of Minitab. 2 is used to assess if the factors included in the mathematical model are adequate to account for the change in response and it is expected to R 2 be nearer to 1 (which means 100 percent).Finally, the significance for each model have to be determined before the performing optimization with MFO algorithm.To do this "Analysis of Variance (ANOVA)" is used.ANOVA is a statistical hypothesis test that use the F-test to measure the significance of a model.ANOVA includes two hypotheses (H0 and H1).H0 indicates that the regression model is insignificant, whereas H1 indicates that it is significant.So, H1 must be true in order to employ these models during the optimization phase.If the "p-value" (in this study Minitab is used to calculate the P-Value) is lower than the type-I error (α) then this means the model is significant (H1 is true) [23][24][25][26].In this study, the confidence level is choosen as 95% (which means α=5%).After completing the modeling, MFO algorithm was utilized for optimization by running through these models [23][24][25][26].

Moth flame optimization (MFO) algorithm
Population based optimization algorithms are widely classified into three main categories: physic-based (such as gravitational search algorithm (GSA), and etc.), evolutionary-based (such as biogeography-based optimization (BBO) algorithm, human evolutionary model, and etc.), and swarm-based (such as artificial bee colony (ABC) algorithm, grey wolf optimizer (GWO) algorithm, and etc.) algorithms.
MFO is a nature-inspired swarm-based meta-heuristic optimization algorithm which is inspired from the moths' navigation mechanism.This mechanism is called transverse orientation.Moths use a very effective method for covering great distances in a straight line at night by maintaining a stable angle with respect to the moon.Because the light source (such as the moon) is far away from the moth, and this ensures that it will fly in a straight line.Regardless of the effectiveness of transverse orientation, moths fly in a spiral around artificial light sources (such as the street lamps and etc.).This is the inefficiency of the transverse orientation and this result in a useless or lethal spiral fly path for moths [20,21].
In MFO algorithm, the behaviors of the moth-flames are mathematically modeled.In this mathematical model, the candidate solutions are represented by the moths and the factors (input variables) are represented by the positions of the moths (moths can fly in hyper dimensional space).Because of being MFO a population-based algorithm, the set of moths must be represented in a matrix form [20,21]: In Eq. ( 3), n represents the number of moths and d represents the number of factors (dimensions).It is also assumed that for each moth, there is an array OM with nx1 dimensions for storing the corresponding fitness values (those are the return value of the fitness (objective) function for each moth).Each moth's position vector (for example, the first row in the matrix M) is passed to the fitness function, and the output of the fitness function is assigned to the corresponding moth as its fitness value in the OM matrix (for example the components of OM matrix are: OM1 for the first row in the matrix M, OM2 for the second row in the matrix M, and etc.).Flames are another important component of the proposed algorithm.Flames are also represented by a matrix which has an equal dimension (nxd) with M matrix [20,21]: As been at M matrix, it is also assumed that for each flames, there is an array OF with nx1 dimensions for storing the corresponding fitness values for the flames where n is the moths' number.Moths and flames are both solutions, it should be mentioned.The difference is in how we handle and update them at each iteration.Moths are genuine search agents that wander across the search space, whereas flames represent the best moth position obtained thus far.In other terms, flames can be compared to flags or pins dropped by moths when hunting for food.As a result, each moth looks for a flag (flame) and changes it when a better solution is found.With this technique, a moth's optimal answer is never lost [20,21].MFO algorithm approximates the global optimal of the optimization problems by using the I, P, and T functions.The I function generates a random population of moths (M matrix) and their fitness values (OM array).The main function, the P function, moves the moths around the search space.This function receives the matrix M and eventually returns its updated version.If the termination criterion is satisfied, the T function returns true; otherwise, it returns false (which is represented by M=P(M) while T(M) is equal to false).In the I function, any random distribution may be used [20,21]: (, ) = (() − ()) * () + () (5) where i=1:n, j=1:d, ub and lb are the upper and lower bounds respectively.Following the I function's initialization, the P function (main function) is executed iteratively until the T function returns true.The P function moves the moths around the search space inspiring from the transverse orientation.Each moth's position in proximity to a flame is updated using the equation below [20,21]: where   and   are the i-th moth and j-th flame.S is the spiral function which is the moth's primary updating mechanism.The initial point of the spiral must be start from the moth and final point must be the flame's position.The fluctuation should not exceed the search space [20,21]: (  ,   ) =   .  .cos(2) +   (7) In this logarithmic spiral function the spiral flying path of moths is simulated where b is a constant term that defines the shape of S,   is the absolute distance of the i-th moth for the j-th flame (|  −   |), and t is a random number between [-1, 1].The t in the equation defines how close the next position of the moth should be to the flame (where t=1 is the furthest and t=-1 is the closest).By changing t, a moth can converge on any point in the vicinity of the flame.The moth's next position would be within a hyper ellipse, which may be assumed in all directions around the flame as a result.Because it governs how moths update their positions around flames, the spiral movement is the most important component of the proposed method.Because of the spiral equation, a moth can fly "around" a flame rather than in the space between them.As a result, the exploration and exploitation of the search space can be ensured.The spiral equation enables a moth to fly ''around" a flame rather than in the space between them.As a result, the exploration and exploitation of the search space can be ensured.The position updating guarantees the exploitation around the flames.To increase the likelihood of finding better solutions, the best solutions obtained thus far are referred to as flames.
As a result, the matrix F always contains the n most recent best solutions obtained thus far.During optimization, the moths must update their positions in relation to this matrix.To emphasize exploitation, it is assumed that t is a random number in the range [r; 1], with r decreasing linearly from -1 to -2 over the course of the iteration.It is worth noting that r is referred to as the convergence constant.Moths using this method tend to exploit their corresponding flames more precisely proportional to the number of iterations.Another issue to consider is that the position updating of moths in relation to n different locations in the search space may impair the exploitation of the most promising solutions.An adaptive system for the number of flames is suggested to address this issue [20,21]: where l is the current iteration count, N denotes the maximum number of flames, and T denotes the total number of iterations.The pseudocode for P function is as follows [20,21]:

Results and discussions
In this paper, we studied on a 16-poled 4000 rpm 3 kVA PMSG.This PMSG is designed in the Maxwell environment.Some important design parameters of this PMSG are presented in Table 2.The PMSG has a rated power factor of 0.8.All winding material in the Maxwell design is ordinary copper.For lamination, M530-50A Si-Fe is employed.Finally, an H-Class insulating material is chosen.The goal of the first step is to use regression modeling to discover the mathematical relationship between the parameters (embrace ( 1 ), offset ( 2 ), outer diameter ( 3 ), magnet thickness ( 4 )) and the responses (stator teeth flux density ( 1 ), stator yoke flux density ( 2 ), MFD ( 3 ), and efficiency ( 4 )).An experiment is designed to carry out this phase.

Rotor
The flowchart for the mathematical modeling and optimization phase is presented in Figure 2. Table 3 shows the factor levels for this experimental design.Regression models will be generated for both factor levels with coded and uncoded versions.During the optimization phase, we actually require the coded model.However, in order to demonstrate the true mathematical relationship to the readers, the original models with uncoded factor values are also generated.As a result, uncoded and coded factor levels are presented jointly in Table 4.The experiment has been planned.Eq. ( 9) is used to code the data.
Maxwell simulations are used to execute 25 experimental runs, and the results are in Table 4.The disadvantage of manufacturing real PMSG prototypes -which is not appropriate due to expensesis avoided.Minitab is used for fitting the regression models and performing the significance tests.Eq. ( 10) and Table 5 shows the general representation of the fitted regression model and the coefficients of the uncoded (original) models, respectively.Programming MFO and optimization are done in Matlab.In order to use these equations in the Matlab environment for the MFO approach, the models must be constructed for coded factor values that range from -1 to 1.In this manner, the models are made independent of the units, making multi-objective optimization simple.Table 6 provide the regression models for the levels of coded factors.The regression models for the responses ( 1 ,  2 ,  3 , and  4 ) have  2 statistics of 99.29%, 98.88%, 93.93%, and 99.28%, respectively.Table 7 displays the regression models' prediction capabilities.The  ̂ values in this table represent the projected outcomes from using the coefficients presented in Table 5.For each response, the prediction error percentage (PE(%)) is also provided.ANOVA is used for determining the significance of the model and summarized in Table 8 (confidence level: 95%) and the confirmation test results are given in Table 7.When there are replicates (multiple observations with identical x-values) in the data, Minitab displays the lack-of-fit test.Replicates are considered "pure error" because only random variation can cause differences in observed response values.In this study, the experimental design is conducted without replicates, so lack-of-fit is not observed in the Minitab reports.Instead of lack of fit test, the p-value test results of ANOVA for the mathematical models are presented in Table 8.The results of the ANOVA indicates that P-values for each mathematial model is less than 0.05which means that the models are significant.Also the confirmation tests are performed and presented in Table 9.The PE(%) results presented in Table 7 indicates that the prediction performance of the mathematical models for the observations those are not used in the modeling phase is good.As conclusion, results indicate that, the regression models provided in Table 5 and Table 6 are significant, according to the findings.
After conducting numerous preliminary tests, it is determined to use 30 search agents in the algorithm.
There may be up to 100 iterations.Through a series of early experiments, the number of iterations and the number of search agents were determined.In these preliminary studies, several combinations were tested by gradually increasing the maximum number of iterations from 50 to 5000 and the number of search agents from 10 to 100.A constrained continuous optimization problem is used to model the issue.For this reason the models with coded factor levels presented in Table 6 are used and the factors in this model are then optimized using the MFO technique while adhering to the specified restriction.The goal function and the constraint for the factors are given in Eqs.(11) and (12).According to these functions; While the target for stator teeth flux density ( Keep in mind that the signs provided in the Z equation must be reversed in the Matlab code (see [20,21,25,26] for further information).The CPU time is 5 seconds (on a PC which has 4GB RAM and Intel i5 2.4 GHz processor).In the previous studies published in the literature, the statistical results of the algorithms on multimodal test function are presented by Mirjalili [20] to determine the performance of MFO.Since the multi-modal functions have an exponential number of local solutions, there results show that the MFO algorithm is able to explore the search space extensively and find promising regions of the search space.Results of [20] indicated that the MFO algorithm highly outperforms other well-known algorithms (GA, PSO, and etc.).Performance index figure that shows the reduction values of the objective function (fitness) during each iteration is presented in Figure 3.As shown in Figure 6, the slot surface between the rotor and the stator (on the surface of the lamination) is still in the green region from top to bottom.In addition, in the range of 1.4 -1.6 Tesla, which is between normal and forced zone.Orange zones can be calledfoced zones.Other colour from orange to red can be called over loaded but in this design we avoided from red zones and stay in max orange zone and do not affect the efficiency.In general, no negative magnetic flux effect -that will decrease the efficiency of the optimized PMSGis observed.Because the red parts are rather minor and the green areas are predominate.The standard lamination (M530-50A) used in this study is normally used in alternators where the magnet structure is thin.In this study, it has been concluded that in the optimized high-speed alternator design, thick magnets should be used in order to achieve the lamination M530-50A saturation and the desired tesla value.In order to use thinner magnets, it is necessary to use more efficient lamination material.The results show that it would be appropriate to use thick magnets in order to obtain the desired Tesla values in highspeed alternators using M530-50A lamination material (the magnet thickness of the unoptimized generator is not specified for commercial confidentiality reasons).

Conclusion
In this work, 16-poled, 4000 rpm, 3 kVA PMSG rotor design optimization is carried out.Goal is to optimize the factor levels of embrace, offset, outer diameter (mm), and magnet thickness (mm) for maximizing efficiency and keeping the magnetic distributions in a desired range.The calculated second order regression equations are fitted to the Maxwell simulation data, and MFO, a successful and recently developed optimization method inspired by nature, is utilized to run through these models.In this study the goal is to show the readers how the MFO algorithm can be used to obtain the desired response values by using the fewest possible experimental runs.Efficiency of the PMSG is maximized to 96.43% and the magnetic distributions are determined as 1.66, 0.24, and 0.95 Tesla for stator teeth flux density, stator yoke flux density, and MFD; respectively.The optimum factor levels for embrace, offset, outer diameter, and magnet thickness are calculated as 0.68, 30, 161.56, and 8.92 respectively.For the optimized factor levels, simulations for confirmation are performed using Maxwell.According to the Maxwell confirmation results: the efficiency is calculated as 94.85%, and magnetic distributions are calculated as 1.64, 0.26, and 0.93 Tesla for stator teeth flux density, stator yoke flux density, and MFD; respectively.Results proved that MFO and regression modeling is effectively used for these type of problems.MFO's striking advantage over previously employed nature-inspired algorithms (such as PSO, GA, and others) is its ability to execute optimization with a relatively low number of iterations (100 for this study).Therefore, we can draw the conclusion that MFO, like the previously discussed nature-inspired algorithms, can be employed successfully for optimization in this field.There is no limitation of MFO and it has only one controllable parameter (that is the number of search agents).So optimization results can not be further improved significantly.Additionally, in the lamination used in the non-optimized stator currently produced by the manufacturer, the thickness of the magnet has been reduced without reducing the useful flux value induced by the magnet in the rotor.This reduced value of 1.08 mm for each magnet corresponds to a total of approximately 17.28 mm for 16 poles considering the total pole in the rotor, which corresponds to two magnets.Compared to the non-optimized PMSG, there was a slight decrease in weight while the size remained the same.This work can be expanded for higher power groups in the future researches.Also the performance of MFO can be compared with the other well-known optimization algorithms in the future researchs.

Figure 1 .
Figure 1.Pseudo code for MFO Until the T function returns true, the P function is executed.The best moth is returned as the best attainable approximation of the optimum when the P

Figure 2 .
Figure 2. Flowchart for the mathematical modeling and optimization phase.

Figure 3 .
Figure 3. Best fitness values in each iteration.MFO is calculated the optimized factor levels as  1 =0.68 (coded value: 0.23),  2 =30 (coded value: 1),  3 =161.56(coded value: 0.77), and  4 =8.92 (coded value: 0.73).For this optimized factor level combination; the stator teeth flux density is calculated as 1.66 Tesla, stator yoke flux density is calculated as 0.24 Tesla, MFD distribution is calculated as 0.95 Tesla, and efficiency is calculates as 96.43% by MFO algorithm.Maxwell simulations are used for the confirmations and responses are calculated as: stator teeth flux density=1.64Tesla, stator yoke flux density=0.26Tesla, MFD=0.93 Tesla, and efficiency= 94.85%.Structure of the optimized PMSG, voltage graph of optimized PMSG, and MFD ditributon for the optimized PMSG are displayed in Figures4-6respectively.The THD is calculated as 0.21 for the PMSG.The outcomes show that the maximum efficiency has been attained and that the MFD distribution is within acceptable bounds (Figure6's green zone).

Figure 4 .
Figure 4. Structure of the optimized PMSG.

Figure 5 .
Figure 5. Voltage graph of the optimized PMSG.

Table 2 .
Basic design specifications for a 3 kVA PMSG.

Table 5 .
Regression coefficients for the uncoded factor levels.

Table 6 .
Regression coefficients for the coded factor levels.

Table 7 .
The accuracy of the models' predictions.

Table 8 .
Summary for ANOVA results.