Optimal Design of Dual Rotor Single Stator PMSM Drive for Automobiles Lucian Nicolae Tutelea, Ion Boldea
Sorin Ioan Deaconu
Electrical Engineering Department Politechnica University of Timisoara Timisoara, Romania
[email protected] ,
[email protected]
Electrical Engineering and Industrial Informatics Department Politechnica University of Timisoara Timisoara, Romania
[email protected]
Abstract—This paper describes a proposed dual rotor single stator brushless PMSM motor/generator system for the parallel HEV or the series HEV, capable to deliver independently torque at the two rotors, fed from a single PWM converter with dual frequency output by adequate vector control. The mathematical model of machine is an analytical model based on the equivalent magnetic circuit considering magnetic saturation. The objective function is a global cost function that could be balanced, between initial active material cost or their total weight and machine efficiency, by choosing adequate weighting factors. Design with MATLAB, quasi 3D FEM analysis and the optimal design via Hooke Jeeves method of this drive system is the core of the paper.
tube and their key FEM validation can reduce the computation effort [17],[18]. The optimization methods are looking generally for the global optimum but from manufacturing reasons the found optimum should be also robust in terms of small variations of optimization variables [19]. In this paper the results of Hooke Jeeves algorithm [20], [21] are considered for an optimal design of PMSM for automotive applications.
Keywords-optimal design; axial airgap machine; dual PM rotor; single stator; single PWM inverter; quasi 3D FEM analysis
I.
INTRODUCTION
Electric propulsion system on HEVs is part of ICE vehicle electrification, to reduce fuel consumption, CO2 pollution, increase human safety and comfort. Fuel economy for city driving is strong, up to 33% (average) or more, and CO2 pollution reduction percentage is similar or better [1][6]. Notable electric machines design optimization effort in the last decade is due to manufacturers’ competition and due to market needs. The large number of optimization variables as well the multi-objective of the optimization design and their nonlinearities and discontinuities make this task difficult [7]. Some approaches are considering stochastically methods to model the real problem [8],[9] through analytical functions. Others are using stochastically methods to find good and poor correlations between some variables and same objectives [10] and then select only the variables that have sufficient effect on machine performance. Another similar approach is to use artificial intelligence principles for robust identification of the complex machine model [11]. A large category of optimization methods are based on biological metaphor as a polymorphism of the life evolutionary principle as they occur in the ant colony optimization algorithm [12], particle swarm optimization, [13], territorial dispute between groups [14] and different kinds of genetic algorithms [15],[16]. Analytical models based on the field
II.
THE DUAL ROTOR SINGLE STATOR BRUSHLESS PMSM PROPOSED SYSTEM
In existent full hybrid electric cases (Toyota Prius 4) there are two electric machines, required to maintain the ICE around its “sweet” point (torque/speed for maximum efficiency or minimum emission) in most operation conditions. This paper aims at discussing the replacement at the two electric machines with a single stator and two rotors, to reduce overall size and cost. In fig. 1 we propose an axial synchronous machine with single stator and PM dual rotors to rotate independently, each shaft having at the side towards the stator a disk of solid steel on which the permanent magnet poles are placed in circular and symmetric manner [22], [23]. The other end of the shaft is connected to the thermal engine (ICE), respectively, to the gears towards the drive wheels. Ring Planet
ICE
2p1 PM pole rotor
NS slots with Gramme-ring AC coils
To drive line
Sun
2p2 PM pole rotor
Clutch
Single inverter with two Frequency control
Battery Power filter
a)
Transmision 4:1
ICE
Differential
Wheel
RmPM1
Wheel
Rmry2
RmPM1
RmPM2
ePM1
ePM2
Rmag1
Rmag2
Power filter AC-DC Bidirectional inverter with two frequency control
Rmry1
Rmsz1
RmPM2
Rmsz2
Battery
b) Figure 1. Proposed PMSM single stator and dual rotors with single inverter a) for planetary geared parallel HEV; b) for series HEV
Rms1
Rms2 es2
es1
Rmsy
Figure 3. The equivalent magnetic circuit for a pole taking into accounts the armature reaction and the dispersion
Figure 2. Three-dimensional exploded view of PMSM single stator and dual rotors
The brushless electric drive system with independent rotors is a single synchronous machine having two surface PM rotors with different number of poles, resulting in twoaxial air-gaps, and a stator with two independent groups of three phase windings with coils disposed around the tooth, one placed on one side and another side of the slotted magnetic core, the two groups can be connected in parallel or in series, and supplied from a single inverter. A 3D drawing of the machine is shown in fig. 2. III.
THE OPTIMIZATION PROBLEM
To reduce the very large 3D FEM computation effort an alternative method is introduced: the quasi 3D model. In the proposed quasi-3D modeling the axial-flux PM machine may be considered to be composed of several linear machines with different length according to the circumferential length. The overall performance of an axialflux machine is obtained by summing the performance of individual machines and by neglecting the possible flux flow in the depth direction of the machine. The number of computation planes needed for computation depends on the purpose of the computation. A finite element program (Opera 13.0) enabling the modeling of linear movement is recommended [23], [24].
The equivalent magnetic circuit for a pole (the machines model) is presented in fig. 3 [25]. By Rmry1,2 was noted the magnetic reluctance of the portion from the rotor disc that corresponds to one pole, by RmPM1,2 the magnetic reluctance of a permanent magnet, by ePM1,2 the mmf of a permanent magnet, by Rmag1,2 – the magnetic reluctance of the air-gap, by Rmsz1,2 the magnetic reluctance of one stator teeth, by Rmsy the magnetic reluctance of the portion from the stator yoke that corresponds to one pole, by es1,2 the one pole armature mmf, by Rmss1,2 the magnetic leakage reluctance of the slot and by RmPM1,2 the leakage reluctance of the permanent magnet. The magnetic circuit of machine 1 and machine 2 are coupled only through stator yoke. The optimal design of the PMSM motor is subject to multi-objective criteria and constructive constrains such as: reducing the initial cost, reducing the motor size and weight, improving the motor efficiency, reducing the torque pulsations and limit the components temperature to a feasible level and avoid PM demagnetization. The multiobjective criteria should be aggregated somewhere in a single objective function if the design objective is to obtain a unique solution. This could be a total cost function including penalty for unmet constrains. The objective function, Ct becomes [18]:
Ct Ci CE Ca C p ,
(1)
where Ci is the initial cost, CE the cost of the lost energy, Ca additional cost to consider the impact of the machine size on the mechanical structure and finally Cp is the penalty cost. The energy efficiency optimization is introduced by
considering the energy loss during the expected run time of the machine. From many numerical examples, it could be observed that, the energy loss cost is much larger than the initial cost. The optimization process is dominated by efficiency improvement rather than by initial cost guaranteed reduction in these cases. The manufacturers and also many consumers are interested in initial cost reduction with a minimum efficiency, given by governmental regulations or through free market. In this example the penalty cost CP has the following structure (2):
C P CTemp Cdemag CT C In , where
(2)
CTemp is the over-temperature penalty cost; Cdemag is the demagnetization penalty cost;
CT is the lack of torque at maximum speed nmax penalty cost;
hPM 1,2 – height of PM for machine 1, 2; HC – coercitive field of PM; Ksdemag – safety factor; Kcdemag – the proportional constant. The penalty cost for PM demagnetization is usually introduced considering that the ‘PM’ is demagnetized when the total flux through the permanent magnet becomes less than 10% due to stator demagnetization current component. The demagnetization penalty cost is computed for both machines and the maximum value is considered. In automotive applications a large constant power speed range is required (fig. 4). The lack of torque at maximum speed penalty cost, CT depends in direct ratio with lack of torque and initial cost (5), (6):
CT CT1 CT2
CIn is the over current at maximum speed nmax penalty cost.
(5)
500
450
400
350 T2 Torque [Nm]
CTemp is computed considering that the surpassing of the maximum allowable winding (and core and PM) temperature, leads to immature aging of the equipment. To avoid this , within the optimization algorithm we introduce a penalty cost for over-temperature Ctemp in the objective function. The penalty cost CTemp may vary either linearly or nonlinearly but monotonously with the over-temperature. In this case it is computed as in (3) [18].
I t1,2 – peak of mmf at maximum load for machine 1, 2;
where
300 Treq2 250
200
k T Tmax Ci CTemp T 0
if if
T Tmax , T Tmax
T1
150
(3)
100 Treq1
where
50
T is the windings temperature;
0
2000
4000
6000 Speed [rpm]
8000
10000
12000
10000
12000
a) Torque/speed
Tmax is the windings allowed temperature; 250
KT is the proportional constant for over-temperature.
Iq2
The over-temperature is computed for each winding and the maximum value is considered in over-temperature penalty cost function (3).
200
150 Current [A]
The stator mmf could demagnetize the PMs and then the demagnetization penalty cost was used to reinforce the PMs:
Id2
Iq1 100
Cdemag1,2
I t k s demag 1,2 0.9 hc demag Ci , hPM 1,2 H C I t1,2 k s demag 0 .9 , if h H PM C 1 ,2 I k t s demag 0 0.9 if 1,2 hPM 1,2 H C
50
Id1
(4)
0
0
2000
4000
6000 Speed [rpm]
8000
b) Required current/speed Figure 4. Mechanical characteristics
CT1,2
where
Tmax req1,2 Tmax1,2 Ci , Tmax req1,2 if Tmax req1,2 Tmax1,2 0 if Tmax req1,2 Tmax1,2
START Optimization vector initialization
, (6)
Geometry error
Tmax req1,2 is required torque at rated constant
power at maximum speed for machines 1, 2; Tmax1,2 is the deliverable maximum torque at nmax. The lack of torque penalty cost is computed for both machines and their sum is considered in total penalty cost (2). In order to produce the required torque, the current should be less than 110% rated current. The over current at maximum speed penalty cost (7), (8), CIn, was introduced to secure this requirement:
C In C In1 C In2
C In1,2
where
Geometry dimensions computation
I s2max ki I s 2 1,2 1 ,2 Ci , 2 Is if I s max1,2 ki I s1,2 0 , if I s max1,2 ki I s1,2
, (7)
Yes Error message
No Performance computation Cost function evaluation
STOP
Grid Search movement (Optimization vector variation) Geometry dimensions computation
Yes
Geometry error No Performance computation Cost function evaluation
, (8)
Cost = 2 Initial cost
No
Are there better solution? Yes Gradient computation
I s max1,2 is the stator current (rms value) at
maximum required torque for machines 1, 2; I s1,2 is the rated current for machines 1, 2; Ki is the overloading current factor at nmax. We may add penalty costs for other technical constraints. For example, we may add a penalty for lower power factor, which is related to the converter kVA rating costs, also for the outer diameter or for machine axial length, etc. The larger the specific costs of these constraints, the higher the probability that the optimization design will observe them. For the optimization design of PMSM we introduce the following optimization variables: Tangential force per surface unit fs (N/m2), slot width of machine 1, wst1 (mm), slot width of machine 2, wst2 (mm), total yoke height hsy (mm), PM height of rotor 1 hPM1 (mm), PM height of rotor 2 hPM2 (mm), height of rotor 1 yoke hry1(mm), height of rotor 2 yoke hry2(mm), stator current density Js(A/mm2), ratio between inner and outer radius kr, ratio between PM1 width and pole pitch alpm1, ratio between PM2 width and pole pitch alpm2, Skewed PM angle beta (rad).
Gradient movement (Optimization vector variation algorithm gradient) Geometry dimensions computation
Yes
Geometry error No Performance computation Cost function evaluation
In this the better solution?
Yes
Cost = 2 Initial cost
No
Is it the smallest step?
Yes
No
Step decrease
Store the best solution END
Figure 5. Hooke Jeeves optimization – block diagram
Geometric Dimensions Evolution
These 13 variables are grouped in a vector Var0. The variable vector Var is not directly initialized by the designer, but he (she) will attribute minimum and maximum values Varmin, Varmax together with corresponding resolution ΔVar. The machine performance is computed considering the magnetic circuit nonlinearities due to magnetic core saturation.
The grid search movement starts with a large step and, when the algorithm fails to bring better motors, the search step is reduced gradually and the algorithm is restarted until the search step becomes smaller than a given value. It was noticed that a large initial search step (around 20% of the variable range) gives a better convergence to the global optimum, despite of the rough approximation of the numerical derivative for large step. Small values of initial search step often lead to a local minimum (fig. 5).The geometry dimensions and performance computation blocks are parts of general design and they are evaluated for every optimization vector produced by the optimization algorithm. If an error occurs during geometry computation or performance computation, the cost will be set two times larger than the initial cost and this way the unfeasible points will be automatically discarded by the algorithm. Finally, the features of the best motor and the evolution from initial vector to the best motor are presented. IV.
OPTIMIZATION RESULTS
The rated dates used in optimization of the axial dual PM rotor single stator and the results of optimization are given in Table I.
wst1 wst2 hsy hry1 hry2
45 40 35 (mm)
The algorithm was implemented in a MATLAB code using input and output file and graphics user interface. By setting a flag, in the input file, the user can decide to provide the initial values of optimized variables or automatically starts from the middle point of the given range.
50
30 25 20 15 10
0
10
20
30 step number
40
50
60
Figure 6. Stator and rotor geometric dimensions evolution wst1 , wst2stator slot width of machine1 respectively of machine2 , hsy – stator yoke height, hry1, hry2 – rotor1 respectively rotor2 yoke height
The probability to reach the global optimum using Hooke Jeeves (HJ) algorithm could be increased by starting the algorithm several times from different points of the optimization variable space. The few sample results for the HJ optimization evolution to the best design are shown in fig. 6, 7, 8, 9 and 10. The stator and rotor geometric dimensions are presented in Fig. 6. The total cost (objective function) and partial cost evolutions are presented in fig. 7 and losses evolution in fig. 8. The efficiency is increasing from around 0.962 and 0.966 respectively to over 0.968 (fig. 9) and mass evolution is presented in fig. 10. Objective Function Evolution 9000 8000
PARAMETERS AND MACHINES DIMENSIONS
Parameters Power [kW] Base speed [rot/min] Maximum speed [rot/min] Rated voltage [V] Pole pairs Number of phase Inner diameter [mm] Outer diameter [mm] PM height [mm] Air-gaps [mm] Total stator yoke height [mm] Slot height [mm] Slot width [mm] Total axial length [mm] Total machines weight [kg] PM type
Machine M1 55 4200 5500
Machine M2 110 2200 12000 220
7
5 3 220 394
3 2 50
5000
total initial energy temp demag
4000
2000 1000
65 32.7
188.1 133 Vacodym 677AP
Br 1.13T, H C 860
6000
3000
6.3
29.5 30.2
7000
Cost (USD)
TABLE I.
kA at 200 C m
0
0
10
20
30 step number
40
Figure 7. Objective function evolution
50
60
Losses Evolution
II). Due to high saturation in rotor disk (yoke) and stator tooth corner, and reduced order of analytical model it could be observed a notable difference. The mesh has 68 regions, 12 symmetry pairs and 76430 elements.
2500 Pcu1 Pcu2 Pfe1 Pfe2
2000
TABLE II.
1500
PARAMETERS COMPARISON
(W)
Parameters
Torque of machine 1 [Nm] Torque of machine 2 [Nm] Inductance of machine 1 [mH] Inductance of machine 2 [mH] Linkage PM flux of machine 1 [mWb] Linkage PM flux of machine 2 [mWb]
1000
500
0
0
10
20
30 step number
40
50
60
Figure 8. Losses evolution Electric Efficiency Evolution 0.972
In quasi 3D analysis the rotating of rotor 1, 2 is converted in translation between rotor 1, rotor 2 and stator. The real scale between the machine’s radial and axial dimensions is presented in fig. 11 (with magnetic field lines produced by the rated current). The total distribution of the flux density for one intermediate position of rotor 1 and rotor 2 is presented in fig. 12, and the air-gaps flux density for rotor 1 and rotor 2 are presented in fig. 13, fig.14 and fig. 15.
m1 m2
0.971
FEM MATLAB quasi 3D optimization 138.8 125 438.1 477.5 0.5 0.567 0.35 0.398 73.8 63.7 191.5 206.4
0.97 0.969
The linkage flux, machine torque, and inductances were computed using Simpson formula on inner, medium and outer layers finite elements results.
0.968 0.967 0.966
Y [mm]
0.965 0.964 0.963 0.962
X [mm] 0
10
20
30 step number
40
50
60
Figure 11. Real scale of the machine. Magnetic field lines with rotor 1 and rotor 2 in intermediate position at rated current
Figure 9. Efficiency evolution m1 – machine1, m2 – machine2 Mass Evolution 180 PM Total
160 140 120
(Kg)
100 80 60 40 20 0
0
10
20
30 step number
40
50
60
Figure 10. Mass evolution
The results obtained with Matlab optimization was compared with results from quasi 3D FEM validation (Table
Figure 12. The total distribution of the flux density at full load in intermediate position
used with good results in order to control proposed machines. In fig. 17 are presented d-q inductances for machines 1 and 2. The d axis inductances become larger than q axis inductance for deep flux weakening, that is increasing the speed range. Linkage PM flux 0.2 c2
b2
a2 0.15
0.1 a1
c1
b1
PM flux (Wb)
0.05
Figure 13. Axial component of the air-gap flux density for rotor 1 at I = 0 (----) and at rated current ()
0
-0.05
-0.1
-0.15
-0.2
0
50
100
150
200 Position (deg)
250
300
350
400
Figure 16. Linkage PM flux for machines 1 and 2 and three phase d-q inductances 0.55 Ld2
Lq2
0.5
0.45
Figure 14. Axial component of the air-gap flux density for rotor 2 at I = 0 (----) and at rated current ()
Inductances (mH)
0.4 Ld1
Lq1
0.35
0.3
0.25
0.2
0.15 -800
-600
-400
-200 0 200 Current magnitude (A)
400
600
800
Flux density [T]
Figure 17. D-q inductances for machines 1 and 2 versus current FEM Torques 500 T2 400
300 200 T1 Torque (Nm)
100 T1cg 0
Figure 15. Axial air-gap PM flux density for three layer (inner, medium, outer)
-100
The linkage PM flux are quite sinusoidal (fig.16) despite of fractionary tooth wound windings. The largest harmonics is the third harmonic and it represents only 2.21% of fundamental for M1, respectively 0.7% of fundamental for M2. Classical vector control with PWM inverter could be
-300
T2cg
-200
-400 -500
0
50
100
150
200 Position (deg)
250
300
350
Figure 18. Interaction and cogging torques computed with FEM
400
The interaction torque T1, T2, (fig. 18) was computed as a difference between total torque and cogging torque from FEM. The cogging torque is acceptable for automotive power drive application: 1.8% for generator (M1) and 2.9% for traction machine (M2). The torque and PM flux computed in FEM ( Table II) are larger than values from analytical model for M1 (about 10% for torque) and they are smaller than values from analytical model for M2 (about 8% for torque). These results are in concordance with the saturation level on M1 respectively M2 rotor disk noticed in fig. 12. V.
[7]
[8]
[9]
[10]
CONCLUSIONS
In this paper, a drive system to produce dual, independent, electromechanical torque output using an axial-air-gap machine with a single stator and two different PM rotors has been introduced in terms of topologies and optimal design. Due to its special configuration and principle of operation, this machine can offer the advantages of high torque density and large speed range. The proposed analytical model was validated by FEM with an accurate of 8-10% for torque, respectively 13% for inductances. This is acceptable but further it could be reduced by a complex analytical model. The maximum computed torque versus speed shown that both machines are able to produce the request torque on the entire speed range. This paper show that it is possible to design an assembly of two machines (generator 55 kW/motor 110 kW) for hybrid vehicle with a mass of only 133kg, an outer diameter 394 mm and an axial length of 188 mm. When the optimization variable vector is large (13 in our case) Hooke Jevees algorithm can not find firmly the global optimum but however they find a notably better than initial solution. Large values of the initial variation step could improve the HJ chance to find solutions close to global optimum. The further studies in relation to prototyping are needed to fully prove the eventual practicality of the proposed system.
[11]
[12]
[13]
[14]
[15] [16]
[17]
[18]
[19]
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