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Position Sensorless Control of Switched ReluctanceMotor Based on Numerical Method
Fei Peng, Jin Ye, and Ali Emadi,McMaster Institute for Automotive Research and Technology (MacAUTO)
Electrical and Computer Engineering DepartmentMcMaster University, Hamilton, Ontario, Canada
Email: [email protected]@[email protected]
Abstract-In this paper, a new position sensorless controlmethod for switched reluctance motor drives is proposed. Rotorposition is initially calculated based on the flux Iinkage-position-phase current characteristics by numerical method. Then, a third-order phase locked loop considering the acceleration variationis designed to undermine the impact of current sampling noiseand numerical residual error on the estimated rotor position.Simulation and experimental results show that the proposed po-sition sensorless control method has achieved sufficient accuracyin terms of position and speed estimation.
Index Terms-Position sensorless control, Numerical method,Phase locked loop, Switched reluctance motor (SRM).
I. INTRODUCTION
Switched reluctance motor (SRM) is an attractive candidateto variable speed drive applications. Compared to the widelyused brushless DC (BLDC) machine and induction machine(1M) drive systems, SRM has features such as simple, robustand low-cost structure, high reliability and satisfactory highspeed performance [1].
A typical SRM drive system is shown in Fig. 1. Nor-mally an SRM is driven by asymmetric half bridges. Currentcontroller is employed to generate switching signals for theasymmetric half bridges according to the current reference androtor position. The current reference is either given by a speedcontroller or a torque distributer. If the current reference comesdirectly from a speed controller, flat top chopping current foreach phase is employed. Due to the strong nonlinearity, insome cases, the flat top chopping current regulation mightnot provide satisfactory performance. Therefore, torque sharingcontrol is used to produce constant torque [2]-[7].
According to Fig. 1, both the current controller and thetorque distributer need the rotor position information. The
Fig. 1. A typical SRM control diagram.
978-1-5090-0737-0/16/$31.00 ©2016 IEEE
rotor position is often obtained through external sensors suchas encoder, resolver, hall sensors or optical couplers. Thesesensors increase the cost of the SRM system and reduce thesystem reliability. In order to compete with the widely usedposition sensorless controlled BLDC and 1M drive system interms of cost and reliability, the sensorless control method forSRM drive has to be developed.
There are two categories of position sensorless controlmethods for SRMs: flux-linkage based methods and inductancebased methods. Since phase inductance is a function of rotorposition, rotor position is detected through injecting currentpulses to the inactive phase in [8], [9]. Commutation happenswhen the injected current peak exceeds a certain threshold.However, current injection methods are limited at higher speed.In [10]-[14] the phase inductance is measured through thecurrent and voltage on the active phase, which relies on theinstant measurement of current derivatives. Therefore, thismethod suffers from the measurement noise. In [15]-[21], therotor position is detected based on flux linkage-current-positioncharacteristics. The current observer or flux observer or hy-brid observer can be designed to estimate the rotor positionindirectly. In [22], [23] instead of comparing the estimatedrotor position with the commutation angles, the estimated fluxis compared with the commutation flux. The flux or currentobserver methods work well at constant speed, but will havesignificant error during acceleration and deceleration. In [24],[25], rotor position is calculated directly based on flux linkage-position-current characteristics with either lookup table ornumerical method, which also suffers from measurement noise.
In this paper, a new position sensorless control methodfor SRM is proposed. The new method is a combination ofdirect calculation method and flux observer method. Rotorposition is firstly calculated by numerical method through therelationship among position, phase current and flux linkage.Then the motion of the SRM including acceleration is modeledand a phase locked loop (PLL) based on this model is designedto estimate the rotor according to the numerically calculatedresults. The dynamic response of this method is improvedthrough the motion model and the impact of measurementnoise and residual error could be reduced through parameterselection of the PLL. Both simulation and experimental resultsare provided to verify the performance of the proposed positionsensorless control method.
Fig. 2. Flux linkage profile of the studied SRM
II. MODEL OF SRM
By neglecting mutual coupling between phases, the phasevoltage equation of SRM is given as:
_ R . d'¢(O, i)U w - w· ~ + dt (1)
where U w is the phase voltage applied on the phase winding;R w is the winding resistance; '¢ is the flux linkage; 0 is therotor position and i is the phase current. Due to its doublesalient structure and saturation, '¢ is a nonlinear function ofboth i and O. Fig. 2 shows the flux linkage profile of the studiedSRM in this paper. i could be measured directly, but 0 is notavailable without position sensor. In order to obtain the rotorposition, '¢ has to be measured by integrating (1):
Fig. 3. 9 for different currents and its simplification.
III. PROPOSED SENSORLESS CONTROL METHOD
A. Position estimation
Since current is directly measured, 6i could be taken as zero.Then, (3) could be written as
With the flux linkage profile shown in Fig. 2, ,¢, and iobtained, 0 could be calculated either directly by lookup tableor by numerical method. In this paper, numerical method isadopted to calculate the rotor position.
Since '¢(0, i) is a function of 0 and i, in a small neighbor-hood of a point in Fig. 2, there is
~(klk-l)=1P(O(klk-l),i(k)) (6)
(7)e(k) = () (klk) - () (klk - 1)
= BO I . (1P (k) - ~ (k Ik - 1))B'¢ i(k)
According to (4), there is
Since the controller is digitally implemented, the estimatedposition for current sampling step comes from the previoussampling step and is donated as 0 (k Ik - 1). Then the corre-sponding estimated flux is obtained as
(3)
(2)
B'¢I . B'¢I6'¢ = -. . 6~ + - .60B~ ()=const BO i=const
where w is the angular speed, a is the angular acceleration.a is determined by factors such as torque production, loadcondition, load inertia, and fraction.
where () (k Ik) is the estimated rotor position of current stepand e(k) is the estimation error.
9 = g~ is the iteration gain of the numerical calculation,whose values at different currents are shown in Fig.3(a). Itis shown that 9 varies a lot with rotor position. Normally,the values are stored in a lookup table for use. In order tosave time and space of the digital controller, only the smallestvalue at each current is taken as shown in Fig.3(b). Thisapproach will not lead to large errors considering relativelyflat curve when the phase is turned on. Smaller gain may leadto slower convergence, which will not affect the stability ofthe numerical iteration.
(4)
(5)w=a
() - 0= ;~ I i=const . (1P - ~)where () is the estimated rotor position near the real positionand "j; is the corresponding estimated flux linkage.
The motion equations of the SRM is:
O=w
B. PLL design TABLE I. CALCULATING THE ESTIMATED POSITIONS AT THE END OFSTART UP STAGE.
In order to reduce the impact of measurement noise andnumerical residual error, a PLL is designed according to (5):
iJ == w+ koe~ == (} + kwe& == kae
(8)
If
Oa > Ob,Oa > Oe,Ob > Oeoa < Ob,Oa > Oe,Ob > oeOa < Ob,Oa < Oe,Ob > OeOa < Ob,Oa < Oe,Ob < Oeoa > Ob,Oa < Oe,Ob < oeOa > Ob,Oa > Oe,Ob < Oe
Then
oa = 360 - 0a, 0e = 360 - 0eOa = 360 - Oa
Oa = 360 - Oa,Ob = 360 - ObOb = 360 - Ob
Ob = 360 - Ob, Oe = 360 - OeOe = 360 - Oe
(10)
Combing (10) and (7), the proposed position estimation isobtained.
According to (8) and (5), the transfer function of the errordynamics of the system becomes
where (), wand (} are the estimated position, angular speedand angular acceleration respectively. e is the estimation error.
(13)()a == ()b + 120()b == ()e + 120
Therefore, a more accurate rotor position could be obtainedby averaging the three calculated positions:
() == ()a + ()b + 120 + ()e + 240 (14)3
IV. DIGITAL IMPLEMENTATION
A. Start Up
Since there is no information about the rotor position beforestart up, a start up sequence is required. On start up, currentpulses are injected to all three phases of the SRM for a shortperiod of time. The position estimators of the three phases startto estimate rotor positions at the same time. Since Fig. 2 issymmetric between (0,180] and (180,360], the three estimatorsshould only estimate the rotor position among (180,360] inthis stage. After the estimated positions converges and becomestable, the rotor position is calculated through the estimatedvalues according to TABLE I.
There is known relationship among the three phase posi-tions, which is:
(9)e 1Q s3 + kos2 + kvs + ka
() (k + 11 k) == () (k Ik - 1) +w(k Ik - 1) . T + ko . e (k) . Tw(k + 11 k) == w(k Ik - 1) + (} (k Ik - 1) . T + kw . e (k) . T&(k + 11 k) == (} (k Ik - 1) + ka . e (k) . T
(9) is a third order system and is stable as long as the gainsof the PLL are positive.
Since the controller is implemented in digital processor, (8)has to be digitalized:
C. Relationship Between Previous Sensorless Control Methods
The proposed position sensorless control method is similarto some previous1Y proposed sensorless control method. Forexample, if the g item is removed from (7), and the PLL isreduced to secon~order, then the position estimator becomes:
¢ (klk -1) = 1jJ (0 (klk -1) ,i (k))e (k) = (1jJ (k) - ¢ (klk - 1))
() (k + 11 k) == () (k Ik - 1) +w(k Ik - 1) . T + ko . e (k) . Tw(k + 11 k) == w(k Ik - 1) + kw . e (k) . T
(11)
which is a typical flux observer based position estimator.
With () obtained, the SRM could be started by keepingturning on the active phase and turning off the inactive phases.
B. Low Speed
Since the flux linkage is calculated by (2) by integrationin a open loop manner, it will lose its accuracy when speedis low and the integration time is long. In this case, currentpulses are injected into the inactive phases to help estimatingthe rotor position and the position estimator of the active phaseis disabled.
C. High Speed
After the SRM gets into higher speed, the calculated fluxlinkage is accurate enough. In this stage, the pulse injection isstopped. Rotor position is estimated by the active phase.
which is a numerical method based direct calculation method.
Therefore, the proposed method could be considered as thecombination of observer based method and direct calculationmethod.
If the third order PLL is removed, then (7) becomes
~ ~ B() I ( ~ )B(k)=B(k-1)+ 81jJ i(k)· 1jJ(k)-1jJ(k-1) (12)
V. SIMULATION RESULTS
Simulation is performed in MATLAB/SIMULINK to verifythe effectiveness of the proposed position sensorless controlmethod. Since speed control and torque control is not themain concern of this paper, only current control is applied. PIcurrent controller is adopted at start up and low speed, whileproportional flux controller with feed-forward is adopted athigh speed. The controller enters high speed mode when thespeed is higher than 410 RPM. Considering the sampling noise
Fig. 4. Diagram of the proposed sensorless control method.
TABLE II. PARAMETERS OF THE EXPERIMENTAL PLANT
Symbol Parameter valueR w SRM winding resistance 0.3 nis switching frequency 20 kHzT Sampling period 0.0001 s
UDC DC bus voltage 300 VBon Tum on angle 216 0
Boff Turn off angle 3300
iref Reference current IDAko PLL gain ko 1000k v PLL gain k v 100000k a PLL gain k a 100000K p Proportional gain of the PI current controller 5Ki Integral gain of the PI current controller 0.01
Kf Proportional gain of the flux controller 0.3
in practice, the control diagram of the system is shown in Fig.4.Speed is controlled by varying the load torque and the speedis control to ramp from zero to 4,500 RPM within 1 second.
The parameters of the system as well as the proposed PLLare shown in TABLE II. The roots of (9) are -887.4438, -111.5460 and -1.0102. They are all located on the negativereal axis which means the response of error does not haveovershoot. The bode diagram of (9) is shown in Fig. 5 withthe parameters shown in TABLE II. It is shown from Fig. 5that the error is significantly attenuated.
A. Start Up and Low Speed
Fig.6 shows the three phase currents and estimated rotorpositions during start up and low speed. The reference currentof 10 A is injected to all three phases. The estimated positionshave well converged after current exceed 5 A in all three
Fig. 5. Bode diagram of (9).
phases. Then the calculation in TABLE I is performed andthe controller enters low speed region. It is shown that thestart up period is very short (about 1ms), which doesn't havemuch impact on the torque production of the SRM.
Since speed is low at this time, the model based fluxcalculation may suffer from integration errors. Fig.7 shows theflux linkage estimation error at low speed. Estimating rotorposition with the active phase may cause significant error.Therefore, current pulses of about 5 A is injected to the inactivephases to estimated the rotor position.
The estimated rotor position of one phase and the real rotor
Fig. 6. Three phase currents and estimated rotor positions.
Fig. 7. Real and calculated flux linkage during start up and low speed.
posItion are shown in Fig.8. It is shown that the proposedestimation method has significant accuracy during start up andlow speed operation.
B. High Speed
The SRM is driven to 4,500 RPM (36,000 RPM in electricspeed) to test its performance at high speed. Fig.9 showsthe estimated and real positions during high speed operation.Fig.l0 shows the real speed and estimated speed from thebeginning. It is indicated that both the position estimation andspeed estimation are accurate.
C. Current Noise
In practice, the current sampling of ADCs may introducenoises to the sampled currents. To simulate this case, whitenoise is introduced to test the performance of the proposedsensorless control method. Fig.ll shows the estimated rotorposition with the sampled current and the real rotor position at4,500 RPM in this case. It is shown that the estimated positionmatches the measured position very well and is merely effectedby the noise. This is due to the noise attenuation ability ofthe third order PLL. Fig.12 shows the position estimation
Fig. 8. Real and calculated rotor position during start up and low speed.
Fig. 9. Three phase currents, estimated and real positions at 4,500 RPM.
Fig. 10. Real and estimated rotor speed from start up.
Fig. 11. The estimated and real positions with current noise at 4,500 RPM.
Fig. 12. The position estimation error with the PLL.
Fig. 13. The position estimation error without the PLL.
error from the beginning. As a comparison, Fig.13 shows theposition estimation error without the third-order PLL. It isshown that, compared to Fig.12 the estimation error increasessignificantly without the PLL.
VI. EXPERIMENTAL RESULTS
Experiments are designed to verify the effectiveness of theproposed position sensorless control method. The experimentsare performed on a 12/8 SRM. It's flux linkage profile isshown in Fig. 2. A BLDC with diode rectifier is used as itsload. Fig.14(a) shows the tested SRM and the load BLDC.A DCIDC converter is connected to the output of the dioderectifier to adjust the load. Fig.14(b) shows the power converterfor the SRM and the DCIDC converter and the load resistor. Aposition sensor is installed on the tested SRM to measure thereal position and speed. But this position and speed are onlyused for comparison purpose, and is not used by the proposedcontroller. The measured and estimated positions and speedsare sent to a PC for record through USB communication at10kHz.
Firstly, the rotor is fixed to test whether the proposedmethod has the ability to start the SRM with heavy load.
Fig. 14. Experimental setup.
Fig. 15. Experimental current and position waveforms starting with standstillrotor.
Fig. 16. The estimated and real positions during start up and low speed.
The currents of phase A and phase B are sampled and shownin Fig. 15(a). Fig. 15(b) shows the estimated and measuredposition obtained in the experiment. It is shown that duringthe start up period, short pules are injected to estimate theinitial position. The estimated initial position is very close tothe the real position. The start up period is very short and thusit doesn't have obvious impact on torque production. Then thecontroller enters low speed operation and pulses are injectedon the inactive phases to estimate the rotor position. In theexperiment, the rotor moves a little bit. At first, phase A isturned on, current pulses are injected to phase B. Then, phase Bis turned on. At this time, the position is estimated by injectingcurrent to phase C. Then phase A is turned off and currentpulses are injected to phase A. The estimated position is veryclose to the real position.
Secondly, the SRM is controlled to run at 1,000 RPMand 4,500 RPM respectively to test the effectiveness of theproposed position sensorless control method. Fig. 16 shows thereal and estimated position at start up and low speed operation.Due to the large cogging torque of the load BLDC, the rotorstarts spinning with vibration. This high dynamic introduceserrors in the beginning. With the increment of the speed, theinertia of the load smooths out the vibration and estimatedposition converges to the real position shortly.
Fig. 17 shows the estimated and real position at 1,000RPM. It is shown that the position estimation is very accurate.Fig. 18 shows the estimated and real position at 4,500 RPM.It is shown that even though there are few sampling point ineach period, the estimated position is still accurate enough.Fig. 19 shows the real and estimated speed, Fig. 20 showsthe corresponding position estimation error. It is shown thatthe estimation response is fast and accurate enough thatthe estimated speed matches the real speed very well. Theestimation error is within ±18° and decreases to be in ±9° asthe speed acceleration decreases.
VII. CONCLUSION
A new position sensorless control method for switchedreluctance motor drives is proposed in this paper. Rotorposition is initially calculated by a numerical method. Then,a third-order phase locked loop considering the accelerationvariation is designed to undermine the impact of currentsampling noise and numerical residual error on the estimatedrotor position. The variation of speed acceleration is consideredin designing the PLL to guarantee tracking accuracy duringacceleration and deceleration. By applying the PLL, the impactof current sampling noise on the estimated rotor position is
Fig. 17. The estimated and real positions at 1,000 RPM.
Fig. 18. The estimated and real positions at 4,500 RPM.
Fig. 19. The estimated and real speed at 4,500 RPM.
Fig. 20. The estimated position error.
well attenuated. Simulation and experimental results showthat the proposed sensorless control method has significantaccuracy on both position and speed estimation.
ACKNOWLEDGMENT
This research was undertaken, in part, thanks to fundingfrom the Canada Excellence Research Chairs Program and theNatural Sciences and Engineering Research Council of CanadaDiscovery Grants Program.
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