Sun Yi Cai Shun1,3 Lin Yingqian Wang Yunchong Shen Jianxin

（1. College of Electrical Engineering Zhejiang University Hangzhou 310027 China 2. Zhejiang Provincial Key Laboratory of Electrical Machine Systems Hangzhou 310027 China 3. Department of Electrical and Electronic Engineering Nanyang Technological University 639798 Singapore）

Abstract The main challenge of the PM-assisted synchronous reluctance machine (PMASynRM) design is to determine numerous parameters for the requirement of multi-objectives. According to the top-level design concept, the optimization for PMASynRMs can be regarded as multi-parameter and multi-objective optimization problems (MOOPs). In this paper, the high-dimensional optimization problem is transformed into two low-dimensional optimization sub-problems. The analytical model algorithm has been established to solve the first sub-problem. Then, the optimization algorithms including the particle swarm optimization (PSO), the standard genetic algorithm (GA) with elitist strategy, and the pattern search (PS) are used for the second sub-problem. It is revealed that the optimization with PS algorithm is superior, in aspects of optimized machine performance and optimization efficiency, compared with that of PSO and GA algorithms. Furthermore, four PMASynRMs have been optimized with the developed process coupled 2D-FEA simulation, and significant performance improvement has been achieved after optimization. Finally, a 7.5kW@3 000r/min prototype machine is manufactured and tested to validate the top-level design pattern.

Keywords: PM-assisted synchronous reluctance machines (PMASynRM), top-level design pattern, multi-objective optimization, analytical model algorithm, optimization methods

0 Introduction

The synchronous reluctance machine (SynRM) has various inherent merits, such as robust rotor structure, low rotor loss, simple manufacture, low cost, and high temperature tolerance[1]. However, it is difficult for SynRMs to satisfy the requirements of high torque density, high power factor, high efficiency and a wide constant power speed range in traction applications such as hybrid electric vehicles (HEVs)[2-3]. Compared with SynRMs, PM-Assisted synchronous reluctance machines (PMASynRMs) have better performances in the aspects of torque density, power factor and efficiency[4-5]. Although PMASynRM with rare-earth PMs possesses enhanced torque density, power factor and efficiency[6], the cost is higher and the unstable supply of rare-earth PMs is another concern[2]. Thus, PMASynRMs with ferrite magnets are perceived as promising candidate and have been widely investigated in decades[7-10]. In Ref.[8-9], the PMASynRM with ferrite magnets exhibits comparable power density and efficiency as conventional rareearth permanent-magnet synchronous machines (PMSMs) for vehicle applications. In Ref.[10], compared with PMSMs, PMASynRMs with the same power can achieve similar efficiency and power density for the application of commercial washer while the material cost is significantly reduced.

The optimization for PMASynRMs includes torque density improvement, torque ripple reduction, efficiency improvement and et al. Various techniques, such as flux-barrier position optimization[11], PM position and shape optimization[12-13], stator chording and rotor skewing[14-15], asymmetric flux-barrier arrangement[16], optimal combination of the number of rotor flux barriers and stator slots[17-18]have been proposed and applied to enhance torque density and reduce torque ripple in PMASynRMs. Moreover, in Ref.[19], the influence of split ratio and PM position on efficiency is investigated. In Ref.[20], 4-pole/36-slot structure with distributed winding is used to achieve high efficiency of PMASynRMs. However, these methods focus on one or several features of PMASynRMs, rather than on the overall design of PMASynRMs.

In this paper, the optimization for PMASynRMs can be regarded as MOOPs, which is called as the toplevel design pattern. There are 7 optimization objectives including average torque, torque ripple, power factor, efficiency, line-line voltage, current density and slot filling factor. There are 25 optimization variables including stator and rotor geometry parameters, winding parameters, and excitation parameters. It is worth noting that it is quite difficult to establish the analytical model of PMASynRMs with high accuracy when the effect of the saturation in the electrical steel material is considered[21-22]. In this paper, all performance of PMASynRMs is calculated by 2D-FEA during the whole process of optimization.

There are two main difficulties for solving this MOOP: A large number of optimization variables; The sensitivity of optimization objectives to different variables varies greatly, and discrete variables coexist with continuous variables.

When some optimization algorithms such as MO-GA, MO-PSO are used to solve this MOOP, not only a lot of 2D-FEA calculations are needed, but also the whole solution might be divergent. In Ref.[23], the response surface methodology is used to reduce the computation. Since the torque ripple of PMASynRMs is sensitive to many variables, the high accuracy response surface is required to predict torque ripple. However, it is quite difficult to get high accuracy response surface.

For the purpose of solving this complex MOOP, the high-dimensional MOOP is transformed into two low dimensional single-objective optimization problems (SOOPs) by the weighted sum method. Although it is difficult to obtain the Pareto optimal solution of this MOOP in this way, the computation is reduced significantly. Since the sensitivity of optimization objectives to variables varies greatly, it is necessary to classify optimization objectives and variables. There are two advantages of reducing dimension of optimization problem, one is to reduce the computation and the other is to accelerate convergence. Subsequently, the analytical model algorithm and the pattern search (PS) are used to solve these two sub-problems. Finally, three 7.5kW@3 000r/min PMASynRMs with different winding types and pole pairs, and one 18.5kW@3 000r/min PMASynRM are optimized successfully.

1 Problem Formulation

For PMASynRMs, the resultant d-axis and q-axis of rotor synchronous reference frame are defined according to the rotor topology as Fig.1.

Fig.1 Stator stationary and rotor synchronous reference frame

1.1 Objectives and parameters

For the PMASynRM optimization, the optimization objectives are included in Tab.1.

Tab.1 Optimization objectives

Subsequently, all parameters for the PMASynRM can be divided into three categories: the stator geometric parameters, the rotor geometric parameters, which are shown in Tab.2, and the winding parameters including winding configuration such as slot/pole combination, coil pitches, and different winding types such as double-layer or single-layer windings, concentrated or distributed windings. Moreover, the rotor geometric parameters are illustrated in Fig.2.

Fig.2 Rotor geometric parameters

The values of the parameters, the winding parameters and the maximal value of the lamination core lengtheflare given by designers. Since the width of the flux bridge is determined by manufacturing and the rotor mechanical strength, the value of the flux-bridge width should also be given by designers. Thus, the values ofloa,lob,locare determined before the optimization starts. The remaining parameters in Tab.2 are determined by the optimization program automatically.

Then, there are eight optimization variables of the stator geometric and excitation parameters, which are. Moreover, all optimization variables in the rotor are shown in

Tab.2 Stator geometric and rotor geometric parameters

In this paper, the number of the flux barriersbnis set to 3. Moreover, the shape of the PM in the second flux barrier is the same as that in the third flux barrier. Therefore, the number of optimization variables in the rotor is 17 and the number of all optimization variables is 25. The physical limitations of the four PMASynRMs to be optimized, viz. 7.5kW-4p18s-DL, 7.5kW-6p36s-DL, 7.5kW-4p36s-SL and 18.5kW-4p36s-SL, are shown in Tab.3.

Tab.3 Physical limitations of four PMASynRMs

1.2 Cost function and flux-barrier geometric constraints

At first, the cost functionC(X)of the PMASynRM is shown in

whereρiis the weight coefficient,Xis all optimization variables,Fis the finite element analysis function, andare target values.

Subsequently, flux-barrier geometric constraints contain flux-barrier shape constrains and the relative position constrains of two adjacent flux barriers.

For the flux-barrier shapes, there are three typical shapes, which are shown in Fig.3.

Fig.3 Three typical flux-barrier shapes

All these three kinds of flux barriers exist in the optimization without flux-barrier shape constrains. At first, for the PMASynRM with the inverted flux barrier in Fig.3b, the torque ripple is much greater than the PMASynRMs with the flux barriers in Fig.3a and Fig.3c. Furthermore, all electromagnetic performances of both PMASynRMs with the flux-barrier shapes depicted in Fig.3a and Fig.3c are almost the same. Since the arc wing constrains are nonlinear, which increase the difficulty of optimization, only the flux barrier with the line wing in Fig.3a is allowed in the optimization. For thejt hflux barrier, the shape constrainsf(Xr)can be expressed by

For the relative positions of two adjacent flux barriers, there are four typical locations, which are shown in Fig.4.

Fig.4 Four typical locations

There are four kinds of the relative positions of two adjacent flux barriers in the optimization without the relative position constrains. Obviously, the location in Fig.4d is not allowed in the optimization. For both locations in Fig.4b and Fig.4c, there are severe saturation in the magnetic circuit. Although the torque ripple of the PMASynRMs with these two locations are satisfied, the average torque and the power factor become worse due to the severe cross-saturation effect in the d and q-axis magnetic circuit. Moreover, the optimization algorithm is more likely to fall into the locally optimal solution when the PMASynRMs have these two locations. Therefore, PMASynRMs with the relative position of the two adjacent flux barriers in Fig.4a have better performances. Then, the relative position constrainsg(Xr)of the (j-1)th andjth flux barriers can be expressed by

where Δθis the minimal allowed angle between two adjacent flux-barrier ends, and ΔPis the minimal allowed distance between two adjacent flux-barrier middle. In the paper,Δθ=and ΔP=0.02Di1.

2 Optimization Algorithms and Results

At first, the optimization target values and PMASynRMs to be optimized are shown in Tab.4.

Tab.4 Optimization target values and PMASynRMs

Subsequently, there are 24 continuous variables and one discrete variableNas mentioned in Section 1. It takes a considerate amount of time to solve this high dimensional optimization problems directly, and the whole solution might be divergent. In this paper, the high-dimensional optimization problem is transformed into two low-dimensional optimization sub-problems by the weighted sum method. Since torque ripple is sensitive to many variables, the optimization for the torque ripple is excluded in the sub-problemⅠ. Then, variables such as flux-barrier span angle and flux-barrier position, which affect torque ripple significantly are also excluded in the sub-problemⅠ. The optimization objectives of the first low-dimensional optimization sub-problem are. Since there are abundant harmonics in the line-line voltage due to salient rotor and field harmonics in PMASynRMs, the objectiveVlineis replaced by the amplitude of the fundamental line-line voltage. The maximal value ofis 515V for this optimization. Moreover, considering that the average torque might be reduced when optimizing the torque ripple in the second sub-problem, the objectiveTemis adjusted bykTTemin the sub-problemⅠ. HerekTis the torque safety factor andkT=1.03in this paper. Thus, the optimization objectives of the subproblemⅠareand the optimization objectives of the sub-problem Ⅱ are.

The sub-problem I can be expressed by

whereC(X1)is the cost function and the weighted coefficientsρiare set as 1.0. It is worth noting that during actual design, these weight coefficients can be adjusted according to the designer’s experience.

The sub-problem Ⅱ can be expressed by

whereC(X2)is the cost function and the weighted coefficientsρiare set as 1.0. It is worth noting that during actual design, these weight coefficients can be adjusted according to the designer’s experience.

2.1 Sub-problemⅠ

At first, the optimization variables should be initialized. For the PMs shape, since increasing PM width is a more effective way to improve the power factor, the values of the PM widthlm1,lm2are fixed by the maximal values of the PMs width max(lm1),max(lm2). The number of coil turnsNcan be initialized by any integer. For the currentI, the value can be initialized by the rated current, whereΩis the mechanical angular speed of the rotor. The initial values of the variables in the sub-problem Ⅰ are shown in Tab.5. Moreover, the four initial PMASynRMs are shown in Fig.5.

Tab.5 Initial values of variables in sub-problemⅠ

Fig.5 Initial PMASynRMs

Subsequently, the solution process for the subproblemⅠcan be divided into two parts.

For the first part, since the values of the coil turnsNand the lamination core lengthlefare not appropriate at the beginning of the optimization, the value of the average torque of PMASynRMs deviates far from the target value. Then, the optimization strategy is that the value of the air gap thickness is fixed in this part. The values of the average torque and the fundamental voltage are proportional to the lamination core length. Since the reluctance torque is dominated in the PMASynRM, the value of the average torque is approximately proportional to the square of the number of coil turns and the square of the amplitude of the current. Moreover, the value of the fundamental voltage is approximately proportional to the amplitude of the current and the square of the number of coil turns. Thus, the iterations for the lamination core lengthlef, the number of coil turnsNand the amplitude of the currentIcan be described in

where max(lef)is the maximal value of the lamination core length.

When only the lamination core lengthlefis adjusted in the 2D-FEA simulation, the values of the average torqueTem, line-line voltageVline, iron lossWiron, copper lossWcopperand efficiencyηof the PMASynRMs will change in direct proportional tolef, while the values of the power factorcosϕ, torque rippleTrip, current densityJand slot filling factorSfremain the same. Let the lamination core length coefficient be, then the average torque, line-line voltage and iron loss can be adjusted toklTem,klVlineandklWiron, respectively. Since the change of the lamination core length will affect the phase resistance, the copper loss needs to be recalculated. The updated copper loss is noted asWc′opperand the efficiency can be adjusted to. Then, the values ofTarare updated to.

If the number of coil turnsNcalculated by Eqn.(7) is changed, the number of coil turnsNwill be adjusted and the currentIcan remain the same. Otherwise, the value of the currentIis adjusted by Eqn.(7).

After adjusting the value ofNorI, the width of the stator yokehyand the stator toothhtneed to be adjusted. If the current densityJ≤J*and the slot filling factorSf≤, the optimization variableshyandhtremain the same. IfJ≤J*andSf＞, the optimization variableshyandhtcan be adjusted by

Subsequently, when the value of the phase angleαis smaller than that of the angle with maximal power factor, the value of the power factor is positively correlated with the value of the phase angle. Thus, the iteration for variableαcan be adjusted as followings at the case of cosϕ＜cosϕ*

where Δαcan be set as 1 or 2.

Moreover, when voltage phasor leads current phasor, the value of the power factor is positively correlated with the thickness of the PMs, the iterations for variablesbm1,bm2can be expressed as followings at the case of cosϕ＜cosϕ*

where max(bmj)is the maximal value of the PM thickness. It is worth noting that the adjustments forαandbmjare implemented alternatively.

For the second part, if the value of cost is satisfied, the whole solution procedure for the optimization subproblemⅠis finished. Otherwise, it is necessary to adjust the air-gap length. There are three different cases. The first and second cases are that the values of the average torqueTemand the power factorcosϕare less than target values, then the adjustment for the value of the air-gap length is shown in wheremin(g)andmax(g)are the minimal and maximal values of the air-gap length respectively.

For the last case, the value of the efficiency is unsatisfied, the air-gap length can be adjusted according to the values of the copper lossWcopperand the iron lossWiron. If the value of the copper loss is greater than that of the iron loss, the air-gap length should decrease. When the air-gap length decreases, the excitationNIor the lamination core length of the PMASynRM decreases, which leads to the reduction of the copper loss. The reduction rate for the copper loss is greater than or equal to linear reduction. In this case, although the iron loss increases due to the flux density increasing in the stator and rotor cores, the increment of the iron loss is less than the reduction of the copper loss. There are two reasons for the slow growth in the iron loss. The first reason is that there is no severe saturation existing in the stator and rotor cores, which means the flux density increasing will not cause dramatically increased iron loss. The other reason is due to the reduction of the excitationNIor the lamination core length. Thus, when the air-gap length decreases in this case, the efficiency increases. Then, if the value of the copper loss is less than that of the iron loss, the air-gap length should increase, which leads to magnetic saturation alleviation and iron loss reduction. Although the copper loss increases, the total loss decreases and the efficiency increases. Then, the adjustment for the value of the air-gap length is shown in

Subsequently, the analytical model algorithm for the sub-problemⅠcan be described in Tab.6. In this paper, the sub-problemⅠof the four PMASynRMs to be optimized can be solved within 10~20 generations. However, in case the optimization is divergent, the condition of the maximal generations is needed and the maximal generations are set as 50, which is sufficiently high and reasonable.

Finally, the optimization results of the 7.5kW-4p36s-SL for the sub-problemⅠincluding the values of the normalized costC(X1)and the optimized objectivesTarare shown in Fig.6. Moreover, the optimized values of the variables in the sub-problemⅠare shown in Tab.7 and the optimized objectivesTarof all PMASynRMs are shown in Tab.8.

Tab.6 Analytical model algorithm

For the 7.5kW-6p36s-DL, the power factor is much less than that of the 7.5kW-4p36s-SL, which is caused by the saliency ratio and the PM amount per pole. For the first reason, the power factor is positively correlated with the saliency ratio, while the saliency ratio is negatively correlated with the pole pair. For the second reason, the maximal value of the PM width of the 7.5kW-6p36s-DL is shorter than that of the 7.5kW-4p36s-SL, which leads to the armature flux linkage from PMs decreasing. However, the power factor is positively correlated with the armature flux linkage from PMs when voltage phasor leads current phasor. In Tab.8, the power factor of the 7.5kW-4p18s-DL is a little low due to the rich harmonics in the winding MMF caused by the fractional winding. It is worth noting that the torque ripple is sensitive to both fluxbarrier parameters and saturation in the rotor core. The 7.5kW-4p36s-SL and 18.5kW-4p36s-SL machines have different flux-barrier parameters and saturation in the rotor cores, therefore, they exhibit different torque ripple, the former being lower, as can be seen from Tab.8.

Fig.6 Optimization results of the 7.5kW-4p36s-SL

Tab.7 Optimized values of variables in sub-problem Ⅰ

Tab.8 Optimized objectives of all PMASynRMs

2.2 Sub-problem Ⅱ

For the sub-problemⅡ, since the cost functionC(X2)depends on FEA numerical solutions, the algorithm for the sub-problem Ⅱ cannot depend on the gradient. In this paper, the particle swarm optimization (PSO), the standard genetic algorithm (GA) with elitist strategy, and pattern search algorithm (PS) are applied in the sub-problem Ⅱ.

The PSO[24]and GA[25-26]are heuristic optimization algorithms based on random search methods inspired by biological evolution. For the PS algorithm, there are two alternative search methods including the axial search and the pattern search. The axial search is used to determine the new base points and the direction that is conducive to the decrease of fitness value. The pattern search is along the direction of two best points in the adjacent generations, which tries to make the fitness value decrease faster[27]. Assume that the number of variables isDand the best point in thejth generation isxj=[p1p2…pD]. At the beginning, the first base pointfor the first generation is set asx0.Then, the axial search can be described by

wheref(⋅)is the fitness function,δis the axial search step,ei=[00 … 1 … 0]is called as the perturbation vector.

After the axial search finished, if, the axial search is successful and the best point in the second generation is. Otherwise, the best point in the second generation isx1=x0, and the axial stepδis reduced toβδ, hereβis the reduction rate,β∈(0,1). Then, the first base point in the second generation is determined by the pattern search, which is shown in

whereλis the acceleration factor.

Finally, the axial search and the pattern search are implemented alternatively until the termination condition of the PS algorithm is satisfied.

Subsequently, the PSO, GA and PS are applied on the sub-problem Ⅱ of the 7.5kW-4p36s-SL. Since lots of 2D-FEA calculations are extremely time-consuming, the number of populations in both PSO and GA are set to 30. The maximal iterations are 300 times. The optimization results for the sub-problem Ⅱ are shown in Tab.9 and Fig.7. It can be seen that both search efficiency and optimization results of the PS are much better than those of the GA and PSO. Compared with the GA and PSO, the calculation times of the 2D-FEA of the PS are reduced by an order of magnitude. Therefore, whether from the optimization results or from search efficiency, the PS algorithm is more suitable for the sub-problemⅡsolution.

Tab.9 Optimization results from GA, PSO and PS

Fig.7 Value of cost C(X2)for the 7.5kW-4p36s-SL

Then, the PS algorithm is used to optimize all remaining PMASynRMs. The initial and optimized values of the variables in the sub-problem Ⅱare shown in Tab.10 and Tab.11, and the optimized objectivesTarof all PMASynRMs are shown in Tab.12. According to the optimization results in Tab.12, the optimized objectives of the 7.5kW-4p36s-SL and 7.5kW-6p36s-DL are satisfied. The optimized objectives of the remaining PMASynRMs are very close to the targets. Finally, the four PMASynRMs after optimizations are shown in Fig.8.

Tab.10 Initial values of variables in sub-problem Ⅱ

Tab.11 Optimized values of variables in sub-problem Ⅱ

Tab.12 Optimized objectives of all PMASynRMs

Fig.8 PMASynRMs after optimizations

3 Experimental Validation

3.1 The prototype machine and experimental setup

The 7.5kW-4p36s-SL machine which has been optimized in Section 2 was prototyped, so as to verify the optimization and analysis, and parameters of the prototype PMASynRM are shown in Tab.13. It is worth noting that the stator, rotor and winding parameters of the prototype PMASynRM are kept the same as the optimized 7.5kW-4p36s-SL. Moreover, the stator and rotor core of the prototype machine are depicted in Fig.9. Then, the experimental setup of the testing rig for loaded experiments is depicted in Fig.10. The prototype machine is connected to a load machine through a torque sensor to provide measurement of the torque on the shaft. The prototype machine is powered by controller kit with vector control algorithm.

Tab.13 Parameters of prototype PMASynRM

Fig.9 Tested machine

Fig.10 Experimental setup

3.2 Experimental results

Fig.11 shows the waveform of the EMF from both FEM simulation and the tested results at the speed of 600r/min. It can be seen that the EMF results from experiments agree well with the FEA simulation results. Tab.14 shows the data of measured torque, power factor, efficiency and rated current under the rated operation with the data of loadTl=24.0N⋅mand 3 000r/min. As shown in Tab.14, the prototype PMASynRM has premium efficiency and its efficiency is much higher than the IE4 index (91.7%). The measured power factor is slightly smaller than FEA results, which is caused by the current harmonics, while the ideally sinusoidal current is applied in the simulation. Furthermore, the prototype machine is tested with different current phase angles under various loading torque while the amplitude of winding current remains the same. The measured torque characteristics at the peak current of 21.2A and 11.3A as shown in Fig.12, have good agreement with the FEA simulation results. It is worth noting that when the currentIis 21.2A and the current phase angleαis 60.8°, the lineline voltage of the prototype PMASynRM reaches the voltage limit of the controller. Moreover, it can be noticed that the tested torque is slightly smaller than FEA, which might be caused by the flux leakage that exists in the edge of the stator and rotor.

Fig.11 EMF@600r/min

Tab.14 Measured results of prototype PMASynRM

Fig.12 Torque characteristics at 3 000r/min

In general, the experiments agree well with the FEA simulation results and prove the top-level design method.

4 Conclusion

The paper proposed a top-level design pattern for PMASynRMs. At first, the high-dimensional MOOP for PMASynRMs can be transformed into two lowdimensional SOOPs. Thanks to the analytical model algorithm and the PS algorithm, two optimization subproblems are solved successfully with good optimization results and modest calculation times of 2D-FEA. When optimizing the torque ripple in the sub-problem Ⅱ, the flux-barrier geometric constraints including fluxbarrier shape constrains and the relative position constrains of two adjacent flux barriers can improve the search efficiency and help the PS algorithm overstep the local optimum. It is revealed that the optimization with PS algorithm is superior in the aspect of optimization performance and calculation efficiency compared with PSO and GA. Based on the above optimization process, four PMASynRMs with different stator/rotor pole number combinations and winding configurations are optimized, and target performance is achieved. Finally, a 7.5kW@3 000r/min prototype machine is manufactured and tested. The experimental results agree well with the 2D-FEA simulation results. Moreover, the top-level design pattern can also be migrated into other types of electric machines.