Jinqun XU ,Hupeng LIN ,Hong GUO,*

a School of Automation Science and Electrical Engineering,Beihang University,Beijing 100083,China

b Ningbo Institute of Technology,Beihang University,Ningbo 315800,China

KEYWORDS Dynamic Neighborhood Genetic Learning Particle Swarm Optimization(DNGL-PSO);Permanent magnet synchronous motor;Power density;Efficiency of motor;Electric propulsion motor

Abstract To maximize the power density of the electric propulsion motor in aerospace application,this paper proposes a novel Dynamic Neighborhood Genetic Learning Particle Swarm Optimization (DNGL-PSO) for the motor design,which can deal with the insufficient population diversity and non-global optimal solution issues.The DNGL-PSO framework is composed of the dynamic neighborhood module and the particle update module.To improve the population diversity,the dynamic neighborhood strategy is first proposed,which combines the local neighborhood exemplar generation mechanism and the shuffling mechanism.The local neighborhood exemplar generation mechanism enlarges the search range of the algorithm in the solution space,thus obtaining highquality exemplars.Meanwhile,when the global optimal solution cannot update its fitness value,the shuffling mechanism module is triggered to dynamically change the local neighborhood members.The roulette wheel selection operator is introduced into the shuffling mechanism to ensure that particles with larger fitness value are selected with a higher probability and remain in the local neighborhood.Then,the global learning based particle update approach is proposed,which can achieve a good balance between the expansion of the search range in the early stage and the acceleration of local convergence in the later stage.Finally,the optimization design of the electric propulsion motor is conducted to verify the effectiveness of the proposed DNGL-PSO.The simulation results show that the proposed DNGL-PSO has excellent adaptability,optimization efficiency and global optimization capability,while the optimized electric propulsion motor has a high power density of 5.207 kW/kg with the efficiency of 96.12%.

1.Introduction

Due to high efficiency,high power density,and high reliability,the Permanent Magnet Synchronous Motor(PMSM)has been widely used in electric propulsion aircraft,1which has an important influence on flight performance.To improve the load capacity and flight range of aircraft,the PMSM is required to be with both high power density and high efficiency.However,these two design indexes are contradictory.2,3With the increase of power density,the efficiency of the PMSM will decrease.Therefore,the optimization design of the PMSM with high power density and high efficiency is extremely challenging.4In recent years,the PMSM optimization algorithms have received more and more attention,such as Particle Swarm Optimization(PSO),5,6Genetic Algorithm(GA),7Artificial Bee Colony (ABC) algorithm,8Pigeon-Inspired Optimization (PIO),9etc.

With the advantages of simple operation and fast convergence,the PSO has been widely used in the motor optimization field.However,when dealing with complex optimization problems,the basic PSO tends to fall into local optimum,which limits its performance in multi-variable and multiconstrained motor optimization.10,11To further improve the optimization performance,several learning strategies based PSO was proposed.Zhan et al.proposed the orthogonal learning strategy for PSO to obtain an optimal solution by using an orthogonal experimental design,which can guide particles in better directions by generating efficient exemplars.12Cheng and Jin proposed the Social Learning PSO algorithm (SLPSO),in which each particle learns from particles with better fitness value in the current swarm to expand the population diversity.13Chen et al.proposed a new biogeography based learning strategy,which uses the migration method based on biogeography to obtain high-quality exemplars,and improves the particle updating performance of PSO.14Yang et al.proposed a level-based learning strategy,which divides particles into different levels for evolutionary computation,and expands the search scope while maintaining rapid convergence.15

Gong et al.proposed the Genetic Learning Particle Swarm Optimization (GL-PSO) to improve the global search ability and search efficiency of PSO by hybridizing PSO with GA in a highly cohesive way.16Compared with the abovementioned learning PSO variants,the GL-PSO has become an outstanding representative of learning PSO with its highly scalable framework and excellent performance on complex optimization problems.The GL-PSO consists of two layers:the exemplar generation and the particle update component.The framework of exemplar generation is derived from the genetic algorithm,including crossover,mutation,and selection,which has the advantage of global and fast search capability.17In the particle update component,particles learn from the exemplars,which provides a good direction for searching the optimal solution and accelerates convergence.However,the GL-PSO generates exemplars using a global topology with a fixed particle neighborhood.In a global topology,all particles are connected without duplication,which leads to the lack of population diversity and falling into the local optimal solution.18

To improve the performance of GL-PSO,extensive research on various types of topologies has been carried out.Lin et al.introduced a ring topology into exemplar generation to enhance population diversity.19The ring topology is a topology where neighboring particles are connected.Chen et al.proposed a new crossover strategy for cultivating highquality exemplars.20Two different crossover operations based on global topology are adopted in parallel,which overcomes the problem of global topology easily falling into a local optimal solution to a certain extent.Xu et al.proposed an improved PSO variant (LSERPSO) which adopts a new scaling mutation strategy.21In the scaling mutation strategy,the combination of a ring topology and a global topology makes a good balance of exploration and exploitation in the process of optimization.The existing studies mostly improve the performance of the GL-PSO by adjusting the original framework of the exemplar generation.Essentially,these algorithms still use various types of static topology.Compared with static topology,dynamic topology has stronger robustness to overcome the interference of local optimal solution with the final optimization result.22However,there is a lack of research on the application of dynamic topology in GL-PSO.

Motivated by this,this paper proposes a novel Dynamic Neighborhood Genetic Learning Particle Swarm Optimization(DNGL-PSO) to maximize the power density of the electric propulsion motor with high efficiency.The main contributions of this paper are fourfold.First,a dynamic neighborhood genetic learning strategy is proposed,which is based on the general framework of combining GA and particle swarm.The proposed dynamic neighborhood method can improve the population diversity and realize the global optimization with a larger fitness value,which consists of the exemplar generation mechanism and the shuffling mechanism.Second,an exemplar generation mechanism in local neighborhoods is proposed for the exemplar generation to obtain a better evolution direction and global best solution with larger fitness value,which includes the crossover operation,mutation operation,selection operation,and the exemplar tournament selection operation.When the global optimal solution cannot update its fitness value,the shuffling mechanism is adopted to exchange local neighborhood members.The exemplar generation and the shuffling mechanism work together in a highly cohesive way to ensure that the members in local neighborhoods change dynamically,which helps to improve the population diversity and avoids premature convergence.Third,the shuffling mechanism based on a Roulette Wheel Selection(RWS) operator is proposed,which can effectively select the neighborhood member and ensure that particles with larger fitness values are selected with a higher probability.Fourth,a global learning based particle update approach is proposed,which improves the adaptability and computational efficiency of the DNGL-PSO,and achieves a good balance between the expansion of the early search range and the acceleration of the later local convergence.Finally,the optimization design of PMSM is conducted to verify the effectiveness of the proposed DNGL-PSO.

The rest of this paper is organized as follows.The mechanisms of the basic GL-PSO and DNGL-PSO with details are presented in Section 2 and Section 3,respectively.Section 4 presents the optimization problem of the PMSM,followed by the simulation results in Section 5.The experimental validation is presented in Section 6.Conclusions are drawn in Section 7.

2.Basic GL-PSO

In canonical PSO,the particles are initialized randomly,and the solution space is searched by tracking two extremums.One is the personal best solution called pbest,and the other is the global best solution called gbest.23The particle update component is expressed as

wherexiandvidenote the position and velocity of PSO,respectively;ω denotes the inertia weight;c1andc2are accelerate coefficients;r1andr2are random numbers in the range of [0,1];piandgddenote pbest and gbest,respectively.

According to Eq.(1),each particle learns from its pbest and gbest to update its velocityviand positionxi.According to Eq.(2),each particle also has a velocityvithat determines its positionxi.

After a certain number of iterations,if pbest and the current gbest are highly coincidental,xicannot be further updated and stagnated,and it becomes trapped in a local optimum and cannot obtain the best solution.

To solve this issue,a genetic learning strategy is introduced in the PSO,which contains crossover,mutation,selection operation,and exemplar tournament selection operation.The algorithm framework of GL-PSO is shown in Algorithm 1,and its velocity update is shown by

whereeidenotes the exemplar of GL-PSO,ω denotes the inertia weight,cis the accelerate coefficient,andris a random number in the range of [0,1].This means thatxilearns fromeiinstead of its pbest or gbest.

(1) Crossover operation

The expression of the crossover operation is given as

whererdis a random number within the range of [0,1].The number of particles and dimensions areMandD,respectively.kis a random integer within the range of [1,M],oiis the offspring generated by the crossover operation,andfdenotes the fitness function.

Incrossoveroperation,theparticleiwithalargerfitnessvalue survives,and its pbest is combined with the current gbest to form an offspringoi.Otherwise,particlekwill replace particlei.

(2) Mutation operation

wherelbdenotes the lower boundary of the optimization variables,andubdenotes the upper boundary of the optimization variables.The offspring mutates with a certain probability and generates new offspringoi.The mutation probability is equal to the valuepm.

(3) Selection operation

where the variableeidenotes the exemplar,which remains unchanged if it has a larger fitness value.

(4) Exemplar tournament selection operation

When the exemplar of particle ceases improving for sg generations,the exemplar tournament selection operation triggers.20%ofMexemplars are randomly selected to join the tournament,and then the winner with a larger fitness value will replace the current exemplar of the particle.

3.DNGL-PSo

3.1.Dynamic neighborhood genetic learning strategy

To improve the population diversity and avoid falling into the local optimum,the DNGL-PSO is proposed.

In the Dynamic Neighborhood Genetic Learning (DNGL)strategy,first,we randomly divide the whole population into non-overlapping local neighborhoods.Second,in the local neighborhoods,the high-qualified exemplars are generated through the local neighborhood exemplar generation mechanism.Finally,the particleilearns from exemplar and gbest.This makes the particleiconverge to the global optimal solution continuously.When its gbest cannot be updated,the dynamic neighborhood is triggered by the shuffling mechanism to increase the population diversity and obtain the gbest with a larger fitness value.As the number of iterations increases,the gbest converges to the global optimum,and the dynamic neighborhood genetic learning strategy is implemented,as shown in Fig.1.In this way,the dynamic neighborhood strategy and genetic learning strategy are combined cohesively through the local grouping and shuffling mechanism.

In a local neighborhood,the generation mechanism of high-qualified exemplars plays an important role in the operation of the algorithm.The exemplar generation mechanism includes operations such as crossover operation,mutation operation,selection operation,and exemplar tournament selection operation,as shown in Fig.1.In the crossover operation,a local neighborhood optimal solution called localgbest and particle’s pbest is used to guide the evolution of the exemplar,as shown in Eq.(7)

Fig.1 Framework of DNGL-PSO.

X=[x1,x2,...,xN],i=1,2,...,N,where X represents the population of the local neighborhood,andNrepresents the size of the local neighborhood.In Eq.(7),rdis a random number with the range of [0,1],glrepresents the localgbest.

Exemplars are generated from the particle’s pbest and localgbest in its local neighborhood,which is the core operation of producing high-quality offspring.The mutation operation is to induce mutation with a certain probabilitypmand avoid falling into local optimum,as shown in Eq.(5).The selection operation is to inherit the particles with a larger fitness value,so that offspring maintains excellent performance,as shown in Eq.(6).The exemplar tournament selection is a screening operation,which involves 20% of the population to solve the problem of stagnant exemplar updates.

Compared with GL-PSO,the advantages of DNGL-PSO are as follows:

(1) In GL-PSO,exemplars come from gbest and pbest.In DNGL-PSO,exemplars come from localgbest and pbest.In the search process,the exemplar guides the evolution direction of the particlei,and the particle comes from pbest and localgbest in its neighborhood,thus avoiding the problem that the exemplar directly approaches the current gbest and falls into the local optimum.Therefore,in this way,the search range of the solution space is expanded,and the quality of obtained exemplar is improved.

(2) In DNGL-PSO,when gbest cannot be updated undergmiterations,a shuffling mechanism is introduced to dynamically adjust the neighborhood members,which is reflected in Section 3.2.In this way,the framework of the algorithm changes from static topology to dynamic topology,that is,the members of the local group are no longer fixed,which increases the diversity of exemplar and is beneficial to the evolution of particlesito gbest with larger fitness value.At the same time,the complexity of the algorithm topology is also reduced,as shown in Fig.2.

Fig.2 Global topology and dynamic neighborhood topology.

(3) The addition of the global learning component learns from exemplar and gbest of particlei,as shown in Eq.(7).Exemplar is generated from localgbest and pbest in local neighborhoods.The addition of the global learning component helps exemplar to guide the evolution direction,avoiding the slow convergence rate of local neighborhood indulging in exploitation.Exploration is the ability to explore the solution space on a large scale,whereas exploitation is the ability to search for the global optimal solution in a small range.Therefore,the combination of local and global search is introduced to promote the balance between exploration and exploitation,which is achieved by adjusting the coefficient c1and c2in Section 3.4.

3.2.Shuffling mechanism

To avoid the gbest getting stuck in a local optimal solution,a novel shuffling mechanism is proposed.If gbest cannot be updated after several iterations,the shuffling mechanism is triggered.The details of implementation are shown as follows:

(1) We randomly select dimensions of the particlekwith the largest fitness value in the neighborhood,

wheredrrepresents thed-dimensional information from particleiand its subscriptris in the range of[1,D];rdis a random number with the range of [0,1];rmis the probability value.Therefore,the information of high-quality particles is integrated into particlei.

(2) Implementation of the screening operation: By operating the RWS operator in Section 3.3,half of the number of particles is preserved.This is equivalent to screening out half of the particles with smaller fitness from the current local neighborhoods,as shown in Fig.3.

Fig.3 Screening operation.

(3) Implementation of the assignment operation:Half of the particles are randomly arranged and distributed to each local neighborhood,as shown in Fig.4.

Fig.4 Assignment operation.

Note that the shuffling mechanism is not a simple mechanism of randomly exchanging particles in each local neighborhood,but makes particles with larger fitness value more likely to stay in the current local neighborhood by the RWS operator.This ensures that the particles in the local neighborhood move in the evolution direction of the optimization goal.The shuffling mechanism fundamentally increases the diversity of exemplar,thus finding a new gbest with larger fitness.Therefore,the shuffling mechanism allows particles to search the solution space more thoroughly in the optimization process,thereby obtaining higher population diversity and improving the performance of the optimization algorithm.

3.3.Roulette wheel selection operator

The essence of the RWS operator is that the selection probability of each individual is proportional to its fitness value.The greater the fitness,the greater the selection probability.The process of using the RWS operator to select particles is shown as follows:

(1) Weight coefficient calculation

Each particle has its fitness value,and a weight coefficient is set for each particle so that the particle with higher fitness values has a higher probability of being selected.The weighting coefficient is expressed as

wherefmaxandfminrepresent the maximum and minimum fitness values of particles in the local neighborhood,respectively;f（xi）is the fitness value of particlei.

(2) Probability calculation

The probability of particleibeing selected is calculated according to its weight coefficient,as shown in

wherep1,iis the probability of selecting particlei,andNrepresents the total number of particles.The probabilityp1,iis equivalent to the span of the roulette.As shown in Fig.5,the larger the span is,the easier it is to be selected.

Fig.5 Graphical illustration of RWS operator.

(3) Probability accumulation

The probabilities of all particles are accumulated in sequence for subsequent calculations.

(4) Particle selection

Randomly selectrin the range of [0,1],and ifsi＞r,then choose the particlei.Particleiis the particle selected by the RWS operator.

Compared to the random recombination of local neighborhoods,the application of the RWS operator prevents particles with a small fitness value from interfering with the evolution direction of the optimization target,because their weight coefficient is small and the probability of being selected is small.In this way,the selected particles can guide the evolution direction more effectively without destroying the diversity of the population.This selection mechanism expands the search space,enriches the population diversity,and eventually obtains a better global optimum.

3.4.Global learning component

By implementing the local neighborhood exemplar generation mechanism and the shuffling mechanism,high-quality exemplars are generated,and then each particle learns from the exemplars to complete the particle update.However,GLPSO has no global learning component,which leads to the risk of deviating from the optimization objective in the evolutionary direction.Moreover,it is difficult to dynamically adjust the balance between exploration and exploitation.The particlexilearns from exemplar and gbest to update its velocity by

In DNGL-PSO,there are two learning components in the update of particle’s velocity.One is the learning component based on the local neighborhood exemplar generation mechanism,and the other is the global learning component,as shown in Eq.(12).The learning component based on the local neighborhood exemplar generation mechanism focuses on the local search,while the global learning component focuses on the global search.In the process of optimization,the acceleration coefficientci,1andci,2are adjusted to achieve a good balance between local search and global search by

wherec1,max,c2,maxandc1,min,c2,mindenote the upper and lower limits of the value range ofci,1andci,2respectively;iter andNiterrepresent the current iteration number and the total iteration number respectively.In this way,as the number of iterations increases,ci,1becomes smaller andci,2becomes larger,as shown in Eq.(13).

Correspondingly,the weight of local search is getting smaller,and the weight of global search is getting larger.The above parameter setting corresponds to the evolutionary behavior of particles.On one hand,in the early stage of algorithm calculation,particles are encouraged to search the whole search space,instead of wandering around the local optimal solution.On the other hand,it is very important to effectively enhance the convergence of the algorithm to the global optimum in the later stage,which improves computational efficiency.The pseudo-code of DNGL-PSO is shown in Algorithm 2 and its flowchart of DNGL-PSO is shown in Fig.6.

Fig.6 Flowchart of DNGL-PSO.

3.5.Complexity analysis of DNGL-PSO

The time complexity of the traditional GL-PSO algorithm includes initialization operation (Tini),crossover operation(Tcro),mutation operation (TMut),selection operation (TSel),evaluation operation (Teva),and the position and velocity update(Tupd)of each particle.We suppose thatDis the dimension of the optimization particle,Mis the population size,Niteris the number of iterations of the optimization algorithm,andFis the computational load of the fitness function.The time complexity of GL-PSO can be estimated as follows:

Therefore,the time complexity of GL-PSO isO（（D+F）MNiter）,which is proportional to（D+F）M.

Similar to the standard GL-PSO,the computational cost of the DNGL-PSO includes the calculation cost of the crossover operation in the local neighborhood(TLN),shuffling operation(TShu),mutation operation (TMut),and selection operation(TSel).

Therefore,the time complexity of DNGL-PSO can be calculated as

which is proportional to（D+F）M.

In summary,the computational complexity of DNGL-PSO and GL-PSO are at the same order of magnitude.In other words,DNGL-PSO has better performance than GL-PSO without increasing computational complexity.

4.Problem formulation

The optimization design of the PMSM in the field of electric propulsion is an optimization problem with a single objective,multiple variables,and multiple constraints.In this paper,DNGL-PSO is used to optimize the electromagnetic parameters of the PMSM,in which the optimization goal is to maximize the power density of the PMSM with the required high efficiency as a constraint.

The design specification of the PMSM is shown in Table 1.The output power of the PMSM is 20 kW,the rotating speed is 3000 r/min,the DC bus voltage is 270 V,and the slot-pole combination of the PMSM is 22 poles and 24 slots.The efficiency of the PMSM shall be no less than 95%.

Table 1 Basic parameters of PMSM.

The basic assumptions about PMSM optimization are shown as follows:

(1) The two-dimensional equivalent magnetic circuit method is adopted to calculate the magnetic field of the PMSM.The default magnetic field does not change along thezaxis direction.

(2) The permanent magnet is uniformly magnetized.

(3) The end effect of the PMSM is negligible.

(4) The armature response of the PMSM is negligible.

To ensure the structural strength of the PMSM,several constraints including electromagnetic and thermal constraints are presented,as shown in Table 1.

(1) To ensure the basic requirements of power density greater than 4.0 kW/kg,the weight of the PMSM cannot exceed 5 kg.In this paper,the weight of the PMSM is the electromagnetic weight of the PMSM,which only includes the weight of the stator core,rotor core,stator winding,and permanent magnet.

(2) Efficiency is larger than 95%.

(3) The magnetic circuit of the PMSM should not be saturated.

(4) The slot fill factor of the PMSM cannot exceed 60%.

(5) The maximum operating temperature of the PMSM should not exceed the safe use range.Since the thermal load AJ is positively correlated with the operating temperature of the PMSM,the thermal load AJ is selected as the constraint condition.According to the practical engineering experience,the limited thermal load is 1000 A2/mm3.

The variables that have a great influence on the power density of the PMSM are selected as the optimization variables,including the stator inner diameterDi,the axial length of the PMSMLm,the number of turnsNS,the air gap length δ,the pole arc coefficient α,the magnetic pole thicknesshM,and the stator tooth widthWt.The range of these parameters is shown in Table 1.In addition,for the structural strength consideration in practical engineering,the stator teeth and yoke take the different magnetic flux density,2.3 T for the stator teeth and 2.0 T for the stator yoke.

The power density optimization of the PMSM can be expressed as

wherefis the power density of the PMSM,which is the function of the optimization variables,such as the stator inner diameterDi,the axial length of the PMSMLm,the number of turnsNs,the air gap length δ,the pole arc coefficient α,the magnetic pole thicknesshM,and the stator tooth widthWt.

To simplify the calculation of the electromagnetic field of the motor,the magnetic field in the motor is converted to an equivalent multi-segment magnetic circuit,and the magnetic flux is approximately considered to be uniformly distributed in each magnetic circuit.The calculation of the magnetic field is converted to the calculation of the magnetic circuit,and the magnetic reluctance of each section of the magnetic circuit multiplied by the magnetic flux in the magnetic circuit is equal to the magnetomotive force.In this way,the calculation efficiency of the electromagnetic field of the motor is improved by the 2-D Equivalent Magnetic Circuit (EMC) model,as shown in Fig.7.

Fig.7 Two-dimensional equivalent magnetic circuit model.

Furthermore,the electric propulsion motor usually adopts the SmCo permanent magnet,which has high magnetic property and low temperature coefficient.Therefore,the remanence change of the permanent magnet caused by the temperature can be neglected.

The 2-D EMC model is calculated as

whereRtrepresents the magnetic reluctance of the stator tooth,Ryrepresents the magnetic reluctance of the stator yoke,Rrrepresents the magnetic reluctance of the rotor,Rδrepresents the magnetic reluctance of the air gap,Fmrepresents the magnetomotive force provided by the permanent magnet to the external magnetic circuit,Fcrepresents the permanent magnet magnetomotive force source,Farepresents the armature magnetomotive force,and φmrepresents the magnetic flux.

In addition,Fig.7 shows the magnetic performance of the motor under the load condition.When the parameterFais set to zero,the magnetic performance of the motor under the noload condition is calculated.

Note that the proposed DNGL-PSO algorithm is used for the power density optimization with the corresponding efficiency constraint of the electric propulsion motor in this paper,which can be considered as a single-objective optimization problem.Actually,the proposed DNGL-PSO algorithm can also be used for the multi-objective optimization design of the electric propulsion motor,in which the optimization objective function can be formulized as the superposition of multiple optimization objectives with the corresponding weighted coefficients.

5.Simulation results

5.1.Comparisons with other algorithms

To verify the effectiveness of the proposed DNGL-PSO,the power density optimization of the PMSM is conducted.The parameter settings of DNGL-PSO are shown in Table 2.All these algorithms use the same algorithm parameters in the optimization process.

To evaluate the performance of these algorithms,multiple runs have been carried out to obtain the motor optimization results of the DNGL-PSO,GGLPSOD,GL-PSO,PSO,and PIO,and the statistical results of 50 runs are provided in Table 3.Note that the optimized motor with the proposed DNGL-PSO algorithm has the maximum power density with a relatively short time cost.For these optimization algorithms,the optimization result with maximum power density is selected as the final motor solution,as shown in Fig.8.

Table 2 Parameters of DNGL-PSO.

Table 3 Statistical results of various algorithms.

As can be seen from Fig.8,the power density of the PMSM with the DNGL-PSO is significantly better than that of PIO,PSO,GL-PSO and GGLPSOD.Since the PSO algorithm converges prematurely to a suboptimal result,the power density of the PMSM with the PSO only reaches 4.363 kW/kg.The PIO algorithm is an improved variant algorithm of PSO,and its optimization ability is stronger than PSO.The power density of the PMSM with the PIO goes to 4.136 kW/kg at the begin-ning,increases to 4.536 kW/kg after 50 iterations,and finally converges to 4.712 kW/kg after 100 iterations.GL-PSO has a strong ability to develop population diversity continuously.The power density of the PMSM with the GL-PSO reaches 4.525 kW/kg at the beginning of the iteration.Due to the gbest and its fitness value update continuously,the power density of the PMSM with the GL-PSO approaches to 5.000 kW/kg after the 70th iteration,and finally converges to 5.041 kW/kg.It can be seen that GL-PSO has a better performance than the PSO and PIO.Starting from 4.458 kW/kg,the DNGL-PSO algorithm has a relatively optimal solution 5.044 kW/kg after 45 iterations and finally converges to 5.207 kW/kg for the power density of the PMSM.Compared with GL-PSO,DNGL-PSO has a strong search ability,which can find high-quality solutions quickly,and update the global optimal solution constantly.

Fig.8 Comparison of DNGL-PSO,GGLPSOD,GL-PSO,PIO,and PSO.

The optimization results with the maximum power density of the motor by various algorithms are selected as the final motor solution,as shown in Table 4.

Table 4 Parameters of DNGL-PSO,GGLPSOD,GL-PSO,PSO,and PIO.

Table 5 Key contact thermal resistances and effective interface gaps between components of motor.

The curve about the relationship between motor efficiency and power density is shown in Fig.9.The final optimized PMSM with the DNGL-PSO is shown in Fig.10.

Fig.9 Relationship between motor efficiency and power density.

Fig.10 Final optimized PMSM with DNGL-PSO.

Note that the DNGL-PSO algorithm can expand the global search range,increase the population diversity,and finally improve the performance of the PMSM.In summary,in the high power density optimization of the PMSM,the convergence speed,search ability,and stability of DNGL-PSO are better than GGLPSOD,GL-PSO,PIO,and PSO.

5.2.Further verification of DNGL-PSO

To further verify the accuracy and correctness of the proposed DNGL-PSO,the electromagnetic field calculation of the motor based on the finite element method is completed by Maxwell software,and the temperature field calculation of the motor is completed by MotorCAD software.

Fig.11 shows the magnetic flux density corresponding to DNGL-PSO motor solution.The maximum magnetic density of the stator teeth reaches 2.3 T,and most of the magnetic density of the stator yoke is below 2.0 T.The magnetic circuit of the motor is not saturated,which meets the design requirements of the motor.

Fig.11 Magnetic flux density of DNGL-PSO solution.

Fig.12 shows that the motor output torque increases linearly with the increase of the stator current.The torque is proportional to the stator current.The rated torque of the motor is 63.7 N·m.The output torque value of the DNGL-PSO solution is consistent with the result calculated by the finite element method.

Fig.12 Relationship between the output torque and the phase current of DNGL-PSO solution.

The problem of motor heat dissipation cannot be ignored.Excellent heat dissipation capacity is an important guarantee for the safe and stable operation of the motor.The cooling method is the forced air cooling,and its velocity is 25 m/s.The load of the thermal model is the rated load condition with the torque of 63.7 N·m and the rotational speed of 3000 r/min.The accuracy of the temperature inside the motor is dependent on the accuracy of the key contact thermal resistances in the direction of heat transfer inside the motor.The key contact thermal resistances include the contact thermal resistance of the stator core and the housing,the contact thermal resistance between stator core and slot-liner,the contact thermal resistance between permanent magnet and rotor core,and the contact thermal resistance between rotor core and shaft.The calculation formula of the contact thermal resistance is given as

where δ is the length of the effective interface gap between components,λ is the thermal conductivity of the interface gap,andAis the contact area during thermal conduction.

In this paper,the key contact thermal resistances and effective interface gaps between components are shown in Table 5.

Fig.13 shows that under the air-cooled condition of 25 m/s,the maximum steady-state temperature of the motor reaches 175.8 °C at the end of the winding,and the average temperature of the winding is 158.8°C,which does not exceed its limit temperature 220 °C.Therefore,the thermal design of the motor meets the requirements of heat dissipation inside the motor.24

Fig.13 Steady-state motor temperature of DNGL-PSO solution.

Through the electromagnetic calculation based on the finite element method and the calculation of temperature field inside the motor,it is proved that the DNGL-PSO solution has a good performance in output torque and heat dissipation.

6.Experimental validation

Through the 2-D magnetic circuit calculation method adopted in this paper,the motor solution is obtained and the prototype is manufactured.The output power of the motor is 20 kW,the rotating speed is 3000 r/min,and the rated torque is 63.7 N m.The motor performance is tested through experiments to verify the accuracy and effectiveness of the 2-D magnetic circuit calculation method used in the calculation of motor performance.On this basis,different algorithms are used to complete the motor optimization design,and the results are accurate and effective.

The prototype motor is shown in Fig.14.The experimental platform is established to test the performance of the PMSM as shown in Fig.15.

Fig.14 Prototype motor.

Fig.15 Experimental platform.

The whole motor test platform consists of a dynamometer,a power analyzer,a motor drive controller,a host computer,etc.The dynamometer can drive the motor at a given speed and provide various load torques.Meanwhile,the torque sensor(blue test power machine controller)is used to measure torque and rotating speed.

The results of motor torque test are shown in Fig.16.When the motor outputs the rated torque,the phase current of the finite element method is 123 A,the phase current calculated by the 2D EMC model is 124.5 A,and the phase current measured by experimental test is 127.7 A.The result shows that there is little performance deviation among the experiment test,the finite element method and the 2D EMC model,which means that the 2D EMC model has a high calculation accuracy.

Fig.16 Relationship between motor torque and phase current.

The motor efficiency performance comparison of the experiment,the finite element method and the 2D EMC model is conducted,which is shown in Fig.17.The result shows that there is little performance deviation among the experiment test,the finite element method andthe 2D EMC model,which meansthat the 2D EMC model has a high calculation accuracy.

Fig.17 Measured motor efficiency at 1500 r/min.

In addition,the motor efficiency under the rated condition is measured by the power analyzer,and its value is 95.5%.The power density of the motor is obtained according to the total motor weight of 9.35 kg and the output power of 20 kW.The power density of the total motor is 2.14 kW/kg.

7.Conclusions

In this paper,a novel DNGL-PSO for the electric propulsion motor is proposed to maximize the power density,which can deal with the insufficient population diversity and non-global optimal solution issues.The DNGL-PSO framework is proposed,which consists of the dynamic neighborhood module and the particle update module.The dynamic neighborhood strategy is proposed to improve the population diversity by combining the exemplar generation mechanism in local neighborhoods with the shuffling mechanism.The exemplar generation mechanism in local neighborhoods is proposed to obtain high-quality exemplars by performing the crossover operation,mutation operation,selection operation,and exemplar tournament selection operation.The shuffling mechanism module is proposed to dynamically change the members in the local neighborhood to avoid falling into the local optimal solution.The roulette wheel selection operator applied in the shuffling mechanism can ensure that the particles with larger fitness value are selected with a higher probability.The global learning based particle update approach is proposed to achieve a good balance between exploration and exploitation.The simulation results show that the proposed DNGL-PSO has excellent adaptability,optimization efficiency and global optimization capability in dealing with the optimization design of the electric propulsion motor,while the optimized electric propulsion motor has a high power density of 5.207 kW/kg with the efficiency of 96.12%.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationship that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No.: 52177028),Aeronautical Science Foundation of China (No.201907051002),the Fundamental Research Funds for the Central Universities,China (No.YWF21BJJ522),and the Major Program of the National Natural Science Foundation of China (No.51890882).