design of flat torque switched reluctance motor considering asymmetric bridge converter using...
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8/4/2019 Design of Flat Torque Switched Reluctance Motor Considering Asymmetric Bridge Converter Using Response Surfac
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Proceedings of the 2008 International Conference on Electrical Machines Paper ID 1219
978-1-4244-1736-0/08/$25.00 2008 IEEE 1
Design of Flat Torque Switched Reluctance Motorconsidering Asymmetric Bridge Converter using
Response Surface ModelingJae-Hak Choi1, Yon-Do Chun1, Pil-Wan Han1, Dae-Hyun Koo1, Do-Hyun Kang1, Ju Lee2
1 Industry Applications Research Division, Korea Electrotechnology Research Institute, Changwon, Korea2 Department of Electrical Engineering, Hanyang University, Seoul, Korea
E-mail : [email protected]
Abstract-This paper presents an optimum design for obtainingflat torque of switched reluctance motor, which has highfluctuating torque due to its inherent salient structure. In order toreduce the fluctuating torque ripple causing noise and vibration,an optimization design technique has been introduced and
investigated to find geometric and electric variables by means ofcombining finite element analysis considering driving circuits andresponse surface modeling.
I. INTRODUCTION
Switched Reluctance Motor (SRM) has a lot of advantages
such as simple and rugged motor construction, high reliability,
and low cost [1]. However, SRM has some problems that limit
its applications because of its inherent structure. One of the
major problems is the fluctuating torque ripple that causes
undesirable acoustic noise and high vibration. The flat torque
depends essentially on geometric shape parameters and electric
circuit parameters, which have been adopted as two-
dimensional design variables. As shown in Fig. 1(a), thegeometric design variables are relative to the salient pole arc
such as stator pole arc s , and rotor pole arc r . The electric
design variables are relative to turn-on and turn-off angle,
which is decided by switch Qa, Qb and Qc in Fig. 1(b) [2].
(a) Configuration of initial model: switched reluctance motor
(b) Asymmetric bridge converterFig. 1. Cross section of switched reluctance motor and Drive Circuit
Gradient-based nonlinear optimization methods are
inefficient in this application where expensive function
evaluations are required, and in this application where
objective and constraint functions are noisy due to modeling
and cumulative numerical inaccuracy since gradient evaluationresults cannot be reliable. Moreover, it is difficult to be
integrated with analysis software, and they cannot be employed
when only experimental analysis results are available. In this
research an effective optimization method based on a response
surface modeling has been used to overcome aforementioned
difficulties. The optimum design, which minimizes fluctuating
torque ripple could be obtained from this work and has been
verified by experiment and analysis.
II. MODEL AND DESIGN VARIABLE
A. Model Discretion
Fig. 1(a) shows construction of 6/4 SRM with stator pole arc30 and rotor pole arc 30. The stator consists of three phases
and six salient poles with concentrated winding, and the rotor
consists of four salient poles. The drive circuit in Fig. 1(b)
consists of a single-phase diode bridge rectifier that converts
the input AC into DC and an asymmetric bridge converter that
supplies power to the SRM. The specifications of the
manufactured SRM are as follows. The outside diameter,
lamination length, air-gap length are 80.4mm, 80mm, and
0.4mm, respectively
B. Finite Element Analysis Tool
In SRM with no magnetic saturation, the instantaneous
torque is expressed by d/)(dLi/)i,(T =2
21 . Theelectromagnetic torque is proportional to the derivative of the
inductance,L, which is a function of rotor position, , and the
square of winding currents affected by inductance of windings.
Although this mathematical equation is often quoted for SRM,
it is not sufficient for accurate prediction of torque, because the
magnetic saturation effect can not be considered [1]. The finite
element method is essential for the precise calculation of the
nonlinear magnetic saturated torque [3].
Also, the switching conditions and freewheeling diodes of
the motor drive circuit have to be considered in finite element
analysis. To provide continuous torque, the drive circuit must
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be commutated that the switches S1 and S2 of phase-A turn off
and the switches S3 and S4 of phase B turn on. When S1 and S2
of phase-A turn off, the current flows through the freewheeling
diodes D1 and D2. The current is restored to DC link capacitor
or flows through other phases. The current flow of this case is
shown in Fig. 2 as a dotted line. When switches of phase B
turn on, the current flows as the solid line.
Therefore, these kinds of conditions must be considered in
analysis. The time-stepped voltage source Finite Element
Method (FEM) is coded with the circuit equation considered
turn-on, turn-off switches and freewheeling diodes.
C. Significant Design Variables and Formulation
Inductance profile varies with the combination of stator and
rotor pole arcs, and influences the torque characteristics. Each
phase inductance profile shifts 30 in 6/4 SRM. For the flat
torque and maximum average torque, the pole arcs of the stator
and rotor have to be more than 30. If the pole arcs of stator
and rotor are smaller than 30, a large torque ripple will beperiodically generated. It is impossible to obtain flat torque
although the phase current flows ideally as shown in Fig 3(a).
Fig. 3(b) shows inductance profile of A-phase, switching
current of A-phase and torque characteristic of three phase, and
illustrate torque generation principles with pole arc
combination when s =30 and r 30. In order to generateflat torque, the flat current of each phase has to flow in rising-
inductance period (30), and can be possible by adjusting the
on and off. The stator pole arc is set to 30, because widening
the pole arc of rotor is better than widening the pole arc of
stator with respect of high slot fill factor. Torque ripple is
ideally able to be zero while increasing average torque.
Consequently, the pole arcs of rotor and stator, turn-on angle
and turn-off angle among electric and geometric variables are
selected as design variables for optimization. The object
function and design variables are represented by (1)
Object Function:
Minimize torque ripple, Tripp
Subject to:
Average torque, Tave 0.1Nm
Stator & rotor pole arcs s = 30, 30 r < 60,
Turn on angle, (60 s )/2 on + (60 r )/2
Turn off angle, 30off 30 + ( r s ). (1)
Fig. 2. Current flow in the drive circuit
(a)s &r < 30 (b) s =30 andr 30
Fig. 3. Torque generation principles
III. OPTIMIZATION PROCEDURE
The approximate optimization procedure is useful tool to
find optimum value about undefined relative equations
between objective and design variable. This method
approximates objective and constraint functions to quadratic
functions within the reasonable design space and sequentially
optimizes the approximate optimization problems in the
context of the design space adjustment strategy. Approximateoptimization problem is converged by agreement with the
actual function within an acceptable tolerance for error. The
approximate optimization based on a response surface
modeling has been applied to the optimum design of SRM.
Approximate optimization procedure consist of four parts;
Design of Experiments (DOE), Finite Element Method (FEM),
Response Surface modeling (RSM), Optimization. Firstly, The
analysis points through the DOE is well adapted to contain the
combinatorial exploration of numerous finite element
simulations required by the investigations on the effect of all
design variables of a given device [4]. Secondly, the response
values of analysis points are obtained through 2-D FEMcoupled with circuit equations of the converter. Thirdly, a
response surface and equation are estimated from the analyzed
response values by using RSM and regression analysis.
Fourthly, a feasible design region is chosen for an optimum
design. The conjugate gradient method in Microsoft Excel has
been used when deciding the best optimal model with
estimated quadratic regression equations.
A. Response Surface Modeling
The Central Composite Design (CCD) on the various DOE is
well adapted to contain the combinatorial explosion of
numerous finite element simulations required by the
investigations on the effect of all design variables of a given
device [4].RSM is a set of useful mathematical and statistical
technology. RSM statistically approximates the relationship
between the response value from performing FEA and design
variables. To make an approximate function, Least Square
Method and Variable Selection Method are used. To evaluate
the function, Analysis of Variance (ANOVA) is used. Among
many DOEs, CCD that is generally used for polynomial models
is used because maximum information can be obtained with the
number of minimum analysis times for the system.
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Fig. 4 shows the analysis points for CCD when there are two
factors for 2-Levels. The number of the CCD analsysis can be
calculated as follows.
c
k nkn ++= 22 (2)
where kis the number of design variable, 2k is the number of
experiment for the 2k factorial design, 2kis the number of axial
point, and nc is the number of replication for the center point.
RSM statistically approximates the relationship between the
response value, y, from performing FEM analysis and the
design variables with an error, and the equation can be
expressed in (3).
+= ),,,( 21 kxxxfy (3)
where u=f(x1, x2, , xk) is the true response function that has
kdesign variables, and denotes the random error that includes
measurement error on the response and is inherent in theprocess or system.
For most of the response surfaces, the functions for the
approximations are polynomials because of its simplicity,
though the functions are not limited to the polynomials. The
response surface is described as follow.
ji
k
j
k
i
k
ij
ijjjj
k
j
jj xxxxy =
= +==
+++=1
1
1 1
2
1
0 (4)
where represents regression coefficients, x is the design
variable, and kis the number of variables.
B. Approximate Optimization Process
Fig. 5 describes the optimization procedure of the SPSRM indetail. The computational procedure is as follows:
Step 0. Set the initial design and the design space. The
initial design space is assumed 50% ~ 100% of the whole
design space that includes the initial design.
Step 1. Select CCD (2k+2k+nc) sampling points within the
design space. Calculate the sampling points set by Finite
Element Analysis (FEA).
Step 2. Approximate the objective and constraints functions
to quadratic polynomial functions by RSM and ANOVA.
Step 3. Find an approximate optimum using the
approximate objective and constraint functions.
Step 4. Evaluate actual objective and constraints at theapproximate optimum value by real FEA.
Step 5. Check convergence at the approximate optimum
using actual objective and constraints function values. If the
approximate optimization problem is converged, then
terminate the optimization. Otherwise adjust the design
space.
Step 6. Select CCD (2k+2k+nc) design points within the
new design space. The new sampling points set consists of
the previous approximate optimum points and newly
selected design points. Go to Step 2 again.
Fig. 4. Central Composite Design for 2-Levels 2 Factors
Fig. 5. Approximate Optimization procedures
IV. RESULTS COMPARISON AND DISCUSSION
A. Geometric Design Variable Optimization: Rotor Pole Arc
Fig. 6 shows regression analysis results of the rotor pole arcs
on conditions that the stator pole arc is set to 30 as explained
in section II-C, and that the turn-on and turn-off angle is set to
0 and 37.5 to raise phase current as shown in Fig. 3(b). The
quadratic equation of the torque ripple and average torque are
respectively estimated in (5) and (6). Table I show the results
compared with the initial model as shown in Fig. 1.
2)(18.0)(8.158.391 rrripT += (5)
)(0058.04006.0 raveT = (6)
(a) Torque ripple (b) average torqueFig. 6. Regression analysis of torque characteristics with rotor pole arc
s=30
r=44
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B. Electric Design Variable Optimization: Switching Angles
Table II and Table III show the optimum results of the
switching variables on the initial shape model as shown in Fig.
1 and the optimum shape model as shown in Fig. 6,
respectively.
Fig. 7 shows the feasible region of turn-on and turn-off onthe optimum shape. The gray color area indicates the feasible
region and includes optimum value.
Fig. 8 shows the inductance profile and current waveform of
one phase of motor. The optimum turn-on and turn-off of the
initial shape are -6.0 and 45.0 as shown in table II.
TABLE IGEOMETRIC VARIABLES DESIGN RESULTS OF ROTOR POLE ARC
Rotor Pole ArcOptimization
InitialPoint
(FEM)
(1)Approx.Optimum
(RSM)
(2)RealOptimum
(FEM)
ConvergenceError
(1) vs. (2)
Torque ripple (Trip) 73.8% 44.6% 43.5% 2.4%
Torque average (Tave) 0.225Nm 0.144Nm 0.143Nm 0.7%
Rotor pole arc (r) 30.0 43.96 44.0 -at the same switching angle (turn-on angle: 0, turn-off angle: 37.5)
TABLE IIELECTRIC VARIABLES 1STITERATION DESIGN RESULTS OF INITIAL SHAPE
Turn-on and offat initial model
InitialPoint
(FEM)
(1)Approx.Optimum
(RSM)
(2)RealOptimum
(FEM)
ConvergenceError
(1) vs. (2)
Torque ripple (Trip) 73.8% 50.4% 55.9% 9.8%
Torque average (Tave) 0.223Nm 0.327Nm 0.317Nm 3.1%
Turn-on angle (on) 0.0 -6.0 -6.0 -
Turn-on angle (off) 37.5 45.0 45.0 -
at the same initial shape (stator pole arc: 30, rotor pole arc: 30)
TABLE IIIELECTRIC VARIABLES 3RDITERATION DESIGN RESULTS OF OPTIMAL SHAPE
Turn-on and -offat optimum model
ShapeOptimum
(FEM)
(1)Approx.Optimum
(RSM)
(2)RealOptimum
(FEM)
ConvergenceError
(1) vs. (2)
Torque ripple (Trip) 43.5% 37.3% 38.5% 3.1%
Torque average (Tave) 0.143Nm 0.140Nm 0.138Nm 1.4%
Turn-on angle (on) 0.0 -0.1 -0.1 -
Turn-on angle (off) 37.5 35.0 35.0 -
at the same Optimum shape (stator pole arc: 30, rotor pole arc: 44)
Fig. 7. 3rd iteration feasible design region of turn-on and off on optimum shape
The negative torque is generated because the phase current
flowed in the falling inductance period. However, the
switching angles of the optimum shape are -0.1 and 35.0 as
shown in table III. The negative torque is not generated
because the phase current was off before the falling inductance
period.
Fig. 9 and Fig. 10 show the waveform of one phase terminal
voltage and three phase current on initial model and optimum
model, respectively. The currents are measured as ratio of
4.2(A/div) to 3(V/div) in Fig. 9 and as ratio of 2.8(A/div) to
(2V/div) in Fig. 10. Both of the motor voltage and current
waveforms generally regard as important electric parameters
for a prediction of motor performance. The flat-top current
doesnt flow at initial model in Fig. 9, otherwise the flat-top
current flows at optimum model in Fig. 10. Here it can be
known the flat-top current waveform is important to make flat
torque waveform, because the motor torque is proportional to
the square of winding currents. If the rising and falling ofphase current is fast or slow according to the switching angle,
the flat torque can not be obtained.
(a) Initial shape (b) Optimum shapeFig. 8. Inductance profile and current of initial and optimum motor
(a) simulation
(b) experiment (20V/div, 4.2A/div)Fig. 9. The voltage and current of initial model
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Fig. 11 shows the energy conversion loop of initial model
and optimal models. Estimation for the average torque could be
illuminated with areas on the energy conversion loop. It can be
known that the initial model has an advantage with respect of
the high average torque, because the energy conversion loop of
initial shape is much wider than that of optimum model.
Fig. 12 shows the analyzed torque waveform. The torque
ripple is reduced from 73.8% to 43.5% by optimizing the
geometric pole arcs, and is reduced from 43.5% to 38.5% by
optimizing the electric switching angles. The torque ripple is
reduced by about two times than the initial one. The average
torque was satisfied of constraint condition.
(a) simulation
(b) experiment (20V/div, 2.8A/div)Fig. 10. The voltage and current of optimum model
Fig. 11. Energy conversion loop
Fig. 12. Instantaneous torque waveform
V. CONCLUSIONS
Firstly, to minimize torque ripple, the geometric rotor pole
arc has been optimized from initial 30 to optimum 44 by
using the approximate optimization. Secondly, the optimum
combination of electric turn-on and turn-off angle can also be
obtained for each initial and optimum shape. The torque rippleis reduced by about two times than the initial one. It can be
also known that there is the trade-off between a torque ripple
and an average torque. This paper measured the phase currents
and the terminal voltage of both models and then shows that
the analysis method considering the drive circuit is suitable for
SRM optimization. The results prove that the optimization
procedure is efficient in this application where expensive
function evaluations are required and in this application where
objective and constraint functions are noisy due to modeling
and cumulative numerical inaccuracy. The optimization
introduced in this article may be also used effectively for
various electric machines.
REFERENCES
[1] T. J. E. Miller, Switched Reluctance Motors and their control, OxfordUniversity Press, 1993, pp. 53-70
[2] Jae-Hak Choi, Youn-hyun Kim and Ju Lee, "Geometric design of polearcs considering electric parameters in switched reluctance motor,"International Journal of Applied Electromagnetics and Mechanics, vol.19, no.1-4, pp. 275-279, 2004.
[3] Jae-Hak Choi, Tae-Heoung Kim, Yong-Su Kim, Seung-Jun Lee, Youn-Hyun Kim, and Ju Lee, Finite Element Analysis of Switched ReluctanceMotor Considering Asymmetric Bridge Converter and DC Link VoltageRipple,IEEE Transactions on Magnetics, vol.41, no.5, 1640-1643, May2005.
[4] Box, G. E. P. and Willson, K. B., On the Experiment Attainment ofOptimum Conditions,Journal of the Royal Statistical Society, Series B.,13, pp.1~14, 1951.