sensors Article
Current Sensor Fault Reconstruction for PMSM Drives Gang Huang 1,2 , Yi-Ping Luo 1 , Chang-Fan Zhang 2 , Jing He 2, * and Yi-Shan Huang 3 1 2 3
*
School of Traffic and Transportation Engineering, Central South University, Changsha 410075, China; [email protected] (G.H.); [email protected] (Y.-P.L.) College of Electrical and Information Engineering, Hunan University of Technology, Zhuzhou 412007, China; [email protected] Hunan CSR Times Electric Vehicle Co., Ltd, Zhuzhou 412007, China; [email protected] Correspondence: [email protected]; Tel.: +86-731-2818-2072
Academic Editor: Vittorio M. N. Passaro Received: 27 October 2015; Accepted: 26 January 2016; Published: 30 January 2016
Abstract: This paper deals with a current sensor fault reconstruction algorithm for the torque closed-loop drive system of an interior PMSM. First, sensor faults are equated to actuator ones by a new introduced state variable. Then, in αβ coordinates, based on the motor model with active flux linkage, a current observer is constructed with a specific sliding mode equivalent control methodology to eliminate the effects of unknown disturbances, and the phase current sensor faults are reconstructed by means of an adaptive method. Finally, an αβ axis current fault processing module is designed based on the reconstructed value. The feasibility and effectiveness of the proposed method are verified by simulation and experimental tests on the RT-LAB platform. Keywords: permanent magnet synchronous motor (PMSM); active flux; sliding mode observers; sensors fault; reconstruction
1. Introduction Permanent magnet synchronous motors (PMSMs) are attractive for electric vehicle and railway traction drive applications thanks to their small volume, light weight, high efficiency, high power density, rugged construction and faster response [1,2]. However, PMSM traction systems are easily influenced by the circumstances of the application environment such as vibration and shock, low and high temperature, humidity and dust, etc., which lead to faults that could directly result in deterioration of the torque performance, and even seriously affect the safety of electric vehicles and railway trains. Therefore, it’s very significant to develop the real-time fault diagnosis and detection systems for PMSM drive systems. A PMSM drive system is usually embedded with at least two feedback current sensors, and faults in any of these sensors may lead to performance degradation [3], therefore, it is necessary to diagnose sensor faults to guarantee the safe and reliable operation of the drive system. Compared to sensor fault diagnosis of PMSM systems, there has been a lot of research on motor body faults or inverter faults [4–10]. Reference [11] proposed an offline method to diagnose current sensor faults for PMSM drive systems. In [12] a parity space approach based on redundancies in a temporal window to diagnose sensor faults was proposed, but the method only detected sudden faults. In [3] the authors proposed an Extended Kalman Filter method to diagnose current sensor faults for PMSM drive systems, but the performance would deteriorate at low speed, and it is not sensitive to slowly varying faults. A residual generation method based on an adaptive observer for phase current sensors was described in [13], but if the threshold is set larger, it will lead to misjudgments.
Sensors 2016, 16, 178; doi:10.3390/s16020178
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The technology of residual generation by means of a filter or observer for fault detection and isolation cannot directly estimate the failures, and this may cause problems such as omission and misjudgment if the threshold chosen is not suitable, so in practical engineering applications, fault diagnosis systems should have higher sensitivity to small and slow-variation faults, and at the same time, they should be robust to various uncertainties, so as to reduce the rates of false positives and missed fault reports. Therefore, this paper presents a current sensor fault reconstruction algorithm for an interior PMSM torque closed-loop drive system based on a sliding mode observer thanks to its better robustness to inaccurate mathematical models, external disturbances, and parameter perturbation [14,15]. First, current sensor faults are equated to actuator ones by a newly introduced state variable. Then, in αβ coordinates, based on the motor model with active flux linkage, a current sliding mode observer is constructed with specific equivalent control methodology to eliminate the effects of unknown disturbances on the system, and the phase current sensor faults are reconstructed by means of an adaptive method. Finally, an αβ axis current fault processing module is designed based on the reconstructed value. This method can accurately identify and reconstruct intermittent offset faults, slow-variation offset faults and abrupt gain faults in real-time, so the fault processing module can restrain the torque oscillation after current sensor fault occurs. The feasibility and effectiveness of the proposed method are verified by simulation and experimental tests on the RT-LAB platform. 2. IPMSM Mathematical Model The stator voltage equations for the IPMSM in the rotating reference dq frame can be expressed as [16]: « ff « ff « ff « ff ud Rs ` DLd ´ωe Lq id 0 “ ` ωe (1) uq ωe Ld Rs ` DLq iq ψr The flux equations in the rotating reference dq frame are: #
ψd “ Ld id ` ψr ψq “ Lq iq
(2)
The electromagnetic torque equation can be described as: Te “
3 n p rψr ` pLd ´ Lq qid siq 2
(3)
where Rs stands for the stator resistance, Ld and Lq stand for the dq axis stator inductances, ud , uq and id , iq are respectively the stator voltages and currents in the dq axis frame, ψr , and ψd , ψq are respectively the permanent magnet flux linkage and the dq axis stator fluxes, ω e and n p are the electrical rotor speed and the number of pole pairs, respectively. D stands for the differential operator. Through coordinate transformation, Equation (1) is transformed to be: «
uα uβ
ff
« “ Rs
iα iβ
ff
« `D
L1 ` L2 cos2θ L2 sin2θ
L2 sin2θ L1 ´ L2 cos2θ
ff «
iα iβ
ff
« ` ωe ψr
ff ´sinθ cosθ
(4)
where L1 = (Ld + Lq )/2, L2 = (Ld ´ Lq )/2, uα , uβ and iα , iβ are respectively the stator voltages and currents in the αβ axis frame, θ is the electrical position. Items with 2θ in Equation (4) show the salient features of IPMSM, and it is difficult for the estimation of the IPMSM status variable. The voltage Equation (1) in the rotating reference dq frame can be reconstructed as: «
ud uq
ff
« “
Rs ` DLq ωe Lq
´ωe Lq Rs ` DLq
ff «
id iq
ff
« ` ωe
0 ˘ ` ψr ` Ld ´ Lq id
ff
« ` `
ff ˘ Ld ´ Lq Did 0
(5)
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«
ff Rs ` DLq ´ωe Lq where is a symmetric matrix. It has nothing to do with the d axis stator ωe Lq Rs ` DLq inductance and eliminates the salient pole phenomenon of IPMSM. Through coordinate transformation, the Equation (5) in the stationary αβ-reference frame is transformed to be: «
uα uβ
ff
«
iα iβ
“ Rs
ff
«
iα iβ
` DLq
ff
« ` ωe
ψr ` pLd ´ Lq qid 0
ff «
0
ff ´sinθ cosθ
ψr ` pLd ´ Lq qid
(6)
3. Sliding Mode Observers Design and Fault Reconstruction According to Equation (6), one has: « D
iα iβ
ff
Rs “´ Lq
«
ff
iα iβ
1 ` Lq
«
uα uβ
ff
« ωe ` Lq
ψr ` pLd ´ Lq qid 0
ff «
0
ff ´sinθ cosθ
ψr ` pLd ´ Lq qid
(7)
The active flux linkage is defined as [17,18]: ` ˘ ψext “ ψr ` Ld ´ Lq id Let ψext,
αβ
(8)
represents the active flux linkage vector in the stationary αβ-reference frame, we have: «
ψext,αβ “
ff ψext,α ψext,β
« “
ψr ` pLd ´ Lq qid 0
ψr ` pLd ´ Lq qid »
” Let x “
iα
iβ
ıT
” , u “
» « C “
1 0
0 1
— ,F“–
uβ
ωe fi Lq ffi fl 0
0
ff
uα
fi
Rs ıT — ´ Lq , A “ — – 0 «
ωe “ ´ J, J “ Lq
ff «
0
0 1
cosθ sinθ
ff (9) »
1 ffi — Lq ffi — Rs fl , B “ – ´ 0 Lq 0
fi 0 ffi ffi 1 fl , Lq
ff ´1 0
. ωe Lq According to Equation (7), the equation of PMSM model with sensor fault can be expressed as: ´
#
.
xptq “ Axptq ` Buptq ` Fψext,αβ ` Ed yptq “ Cxptq ` G f s
(10)
where fs = [fsα fsβ ]T is the stator current sensor fault vector in the stationary αβ-reference frame, d = [d1 d2 ]T is the unknown disturbance vector, d1 and d2 are assumed to be bounded. “¨ ” denotes the derivative. u and y are respectively the input vector and output vector. « ff x is the « state variable, ff 1 0 1 0 G“ ,E“ . 0 1 0 1 Consider a new state variable z which is a filtered version of y, and satisfies: .
z “ ´az ` by
(11)
where a and b are constant. If we select a “ 0, b “ 1, Equation (11) can be rewritten as: .
z “ y “ Cxptq ` G f s
(12)
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According to Equations (10) and (12), a new system can be described as: $ . ’ “ Axptq ` Buptq ` Fψext,αβ ` Ed & xptq . z “ Cxptq ` G f s ’ % w“z
(13)
where w is output vector of the new system. fs appears as an actuator fault in the new system and so the approach described earlier can be adopted. The sliding mode observer can be designed as: #
.
ˆ “ A xptq ˆ ` Buptq ` Fψext,αβ ` Eν1 xptq . ˆ “ C xptq ˆ ` G fˆs ` ν2 zptq
(14)
where “ˆ” denotes the estimated values. v1 and v2 are the sliding mode correction control signal. $ & L x ´ xˆ 1 k x ´ xˆ k ν1 “ % 0 $ & L z ´ zˆ 2 k z ´ zˆ k ν2 “ % 0
x ´ xˆ ‰ 0
(15)
x ´ xˆ “ 0 z ´ zˆ ‰ 0
(16)
z ´ zˆ “ 0
where L1 ą 0 and L2 ą 0 are all to be designed. ˆ es “ f s ´ fˆs . The observer errors are defined as: ex “ x ´ xˆ , ez “ z ´ z, The direct axis current component id of the active flux linkage quickly converge to the given value i˚d when the bandwidth is high enough, this ensures the active flux linkage as a linear constant value [17,18]. The state estimation error equations can be expressed as: .
.
.
e x “ x ´ xˆ “ Aex ` Epd ´ ν1 q .
(17)
.
.
ez “ z ´ zˆ “ Cex ` Ges ´ ν2
(18)
Consider a Lyapunov function candidate: V “ exT ex ` ezT ez ` esT Qes
(19)
where Q is a positive constant. Differentiating the Lyapunov function and according to Equations (17) and (18), then: .
V
.T
.
.T
.
.T
.
“ e x ex ` exT e x ` ez ez ` ezT ez ` es Qes ` esT Qes . “ rAex ` Epd ´ ν1 qsT ex ` exT rAex ` Epd ´ ν1 qs ` pCex ` Ges ´ ν2 qT ez ` ezT pCex ` Ges ´ ν2 q ` 2esT Qes . T T T T T T T T T T “ ex Aex ` pd ´ ν1 q Eex ` ex Aex ` ex Epd ´ ν1 q ` ex Cez ` es Gez ´ ν2 ez ` ez Cex ` ez Ges ´ ez ν2 ` 2esT Qes . T T T T T T “ 2ex Aex ` 2ex Epd ´ ν1 q ` 2ez Cex ` 2es Gez ´ 2ez ν2 ` 2es Qes
«
0 1
ff
«
1 0
0 1
ď 2k ex kk E kpk d k ´ L1 q ` 2ezT Cex ` 2esT Gez ´ 2ezT ν2 ` 2esT Qes . ď 2k ex kk E kpk d k ´ L1 q ` 2k ez kpk C kk ex k ´ L2 q ` 2esT pGez ` Qes q
.
Since A is a symmetric negative definite matrix, C “
1 0
, and E “
ff
(20)
.
, then V
can be expressed as: .
V
.
ď 2k ex kk E kpk d k ´ L1 q ` 2k ez kpk C kk ex k ´
L2 q ` 2esT rGez
.
` Qp f s ´ fˆs qs
(21)
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When the sensor gain changes very slowly as the time and environment change, or the sensor suddenly detects a break, power down or damage, the first derivative of the fault approximates to . zero, that is f s « 0, then: .
.
V ď 2k ex kk E kpk d k ´ L1 q ` 2k ez kpk C kk ex k ´ L2 q ` 2esT pGez ´ Q fˆs q
(22)
For the arbitrary initial conditions zp0q, if the adaptive law algorithm for fault reconstruction is: .
Then:
fˆs “ Q´1 Gez
(23)
V ď 2k ex kk E kpk d k ´ L1 q ` 2k ez kpk C kk ex k ´ L2 q
(24)
.
Select L1 ą k d k ` σ1 and L2 ą C k ex k ` σ2 , where σ1 ą 0, σ2 ą 0, then: .
V ď ´2σ1 k E kk ex k ´ 2σ2 k ez k ď 0
(25)
From the Lyapunov theorem, the system will reach the sliding mode state in finite time. Since iα and iβ are practically calculated from iabc . In the case of only “a” and “b” phase current sensors, according to [13]: $ & iα “ i a i ´ ic 2i ` i a (26) % i β “ b? “ b? 3 3 The effects of the αβ axis sensor faults fsα and fsβ are related to the errors in “a” and “b” phase sensor faults fa and fb as follows: $ & f sα “ f a 2 f ` fa (27) % f sβ “ b? 3 According to Equations (23) and (27), the adaptive law for the faults reconstruction in “a” and “b” phase sensors can be expressed as: . . fˆ “ fˆ “ Q´1 ez1 (28) a
?
.
fˆb “
.
sα
.
3 fˆsβ ´ fˆsα 2
? “Q
´1
3ez2 ´ ez1 2
(29)
4. Fault Diagnosis and Fault Processing When no phase sensor faults occur, the fault reconstruction values are also zero. When sensor faults occur, the fault reconstruction values deviate from their zero value. The faults detection decision rule are expressed in Table 1, Where “1” stands for “sensor faults occur”, “0” stands for “no sensor faults occur”, “Flag i” represents “the fault flags”. Table 1. Faults detection decision rule. fˆb
fˆa
Flag ib
Flag i a
0 0 1 1
0 1 0 1
0 0 1 1
0 1 0 1
After a sensor fault occurs, the torque oscillates since the original current balance is destroyed. The performance of the drive system will decrease. To achieve a performance similar to the pre-fault
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status, it is necessary to study the fault processing strategy. The αβ axis current fault processing Sensors 16, 178 6 of 16 module is 2016, shown in Figure 1. The fault processing equations can be expressed as: Sensors 2016, 16, 178
#
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iαri “iˆαiˆ ´ ffˆsα¨Flag Flagi i a r s a i βri “ iˆβiˆ ´ ffˆˆsβ¨Flag pFlag i i a ‘ Flag ib q
(30)
(30)
i rr i f ss ( Flag aia Flag ib ) (30) ˆ ˆ ioperation r i f s ( Flag ia Flag ib ) wherewhere Flag iFlag ‘ Flag i is a logic AND equation, i and i are the αβ axis feedback currents ia Flag a ar iar and b ib is a logic AND operation equation, br ibr are the αβ axis feedback
after currents fault processing. When phase sensor faultssensor occur, “Flag i” is zero the estimation afteria fault processing. When nooperation phase faults i ” isαβ zero value, theof αβ Flag Flag ib isno a logic AND equation, iaroccur, and i“br Flag arevalue, the axis feedback where axis current is adopted for the αβ axis feedback current. When sensor faults occur, “Flag i” equals estimation of αβ axisprocessing. current is adopted forphase the αβsensor axis feedback current. When faults occur, currents after fault When no faults occur, “ Flag i ”sensor is zero value, the one, the difference between estimation of αβ and fault reconstruction is “estimation Flag i ”equals the difference between the current estimation ofits αβcorresponding axis current andsensor its corresponding of αβone, axisthe current is adopted foraxis the αβ axis feedback current. When faults occur, adopted for the αβ axis current. fault reconstruction adopted for the αβ axisthe feedback current. “ Flag i ”equals one,isfeedback the difference between estimation of αβ axis current and its corresponding fault reconstruction is adopted for the αβ axis feedback current.
Figure 1. αβ axis current fault processing module. Figure 1. αβ axis current fault processing module. Figure 1. αβ axis current fault processing module.
5. Simulations and Analysis
5. Simulations and Analysis
5. Simulations and Analysis The effectiveness of the fault reconstruction and fault processing method for PMSM drive
The effectiveness the fault reconstruction andand faultfault processing method forof PMSM drive system system inofthe was simulated experimentally. The block diagram the simulation Theproposed effectiveness ofpaper the fault reconstruction processing method for PMSM drive proposed in the paper was simulated experimentally. The block diagram of the simulation setup is setup shown in Figure 2. systemis proposed in the paper was simulated experimentally. The block diagram of the simulation shown in Figure 2. in Figure 2. setup is shown
Figure 2. Block diagram of current sensor fault reconstruction and fault processing system for IPMSM. Figure 2. Block diagram of current sensor fault reconstruction and fault processing system for Figure 2. Block diagram of current sensor fault reconstruction and fault processing system for IPMSM. IPMSM.
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The parameters for the interior PMSM of this study are tabulated in Table 2. The given speed 7 of 16 and torque are 200 rad/s and 500 Nm, respectively. Let system unknown disturbances d1 and d2 be random noise, which value range is [−50 50]. To reduce chattering and eliminate high-frequency The parameters forthe thechattering, interior PMSM of this study are tabulated in Table 2. be Theused given and interference caused by a successive approximation function can to speed substitute torque rad/s and Nm, cases respectively. Let system unknown disturbances d1 and d2 be random for the are sign200 function sgn 500 (·). Two are discussed: noise, which value range is [´50 50]. To reduce chattering and eliminate high-frequency interference Tableapproximation 2. Interior PMSM parameters. caused by the chattering, a successive function can be used to substitute for the sign function sgn (¨ ). Two cases are discussed: Quantity Symbol Value Rs Stator resistance 0.02 Ω Table 2. Interior PMSM parameters.
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Lq
Q axis inductance
0.001500 H
Symbol Ld
DQuantity axis inductance Stator resistance Inertia Q axis inductance Magnetic flux D axis inductance Number of pole pairs Inertia Damping coefficient Magnetic flux Number of pole pairs DC-bus voltage Damping coefficient DC-bus voltage
(A) Intermittent offset fault on “b” phase sensor
Value H 0.003572 Ω2 1000.02 kg·m 0.001500 0.892 WbH 0.003572 H 4 pairs 100 kg¨ m2 0.0010.892 Nm·s/rad Wb 4 pairs 1500 V
Rs J Lq r Ld P J ψr B PVdc B Vdc
0.001 Nm¨ s/rad 1500 V
The “b” phase sensor fault can be expressed as: (A) Intermittent offset fault on “b” phase sensor 0 t 0.1s The “b” phase sensor fault can be expressed as: 30 0.1s t 0.3s $ ’ f b’ 0.3st ăt 0.1s 00 0.5s (31) ’ ’ ’ ´30 0.1s ď t ă 0.3s & 50 0.5s t 0.7 s (31) fb “ 0 0.3s ď t ă 0.5s ’ ’ ’ t s 20 0.7 ’ ´50 0.5s ď t ă 0.7s ’ % ´20 t ě 0.7s The three-phase stator currents of the current sensor output are shown in Figure 3, and the iβ are shown in Figures and 5, respectively. “a” actual and estimatedstator values of iα and The three-phase currents of the current sensor output4are shown in Figure 3,The and phase the actual phase “b”values current faultshown reconstruction shown in The Figures andand 7, and estimated of iαsensors and iβ are in Figures values 4 and 5are respectively. phase6 “a” respectively. The electromagnetic torque is shown in are Figure 8. As the simulation figures, phase “b” current sensors fault reconstruction values shown in shown Figuresin 6 and 7 respectively. The when an intermittent faultin is Figure added,8.the balancefigures, is destroyed, theintermittent amplitudes electromagnetic torqueoffset is shown Asoriginally shown in current the simulation when an of phase and phase “c” currents slightly, the the measured phase “b” sensor offset fault“a” is added, the originally currentincrease balance is destroyed, amplitudes of phase “a” andcurrent phase produces intermittent offset. The phase electromagnetic torque produces produces a corresponding “c” currentsan increase slightly, the measured “b” sensor current an intermittent offset. equiamplitude oscillation. adaptive law for fault reconstruction accurately reconstruct the The electromagnetic torqueThe produces a corresponding equiamplitudecan oscillation. The adaptive law intermittent offset faults. the fault value becomes zero at t offset = 0.3 faults. s–0.5 s,When the current and the for fault reconstruction canWhen accurately reconstruct the intermittent the fault value torque are back becomes zero at to t =normal. 0.3 s–0.5 s, the current and the torque are back to normal. 150 ia ib ic
100
iabc(A)
50
0
-50
-100
-150 0
0.1
0.2
0.3
0.4
0.5 t(s)
0.6
0.7
0.8
Figure 3. The measured value of the three-phase current.
0.9
1
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150 150 iα iα
100 100
iα(A) iα(A) iα(A)
50 50 0 0 -50 -50 -100 -100 -150 -150 0 0
estimation of of iα iα estimation
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5 t(s) t(s)
0.6 0.6
0.7 0.7
0.8 0.8
0.9 0.9
1 1
Figure4. 4. The The actual actualand andestimated estimatedvalue valueof ofiiiααα... Figure Figure 4. The actual and estimated value of 100 100 50 50
iβ(A) iβ(A) iβ(A)
0 0 -50 -50 -100 -100 -150 -150 -200 -200 0 0
iβ iβ estimation of of iβ iβ estimation
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5 t(s) t(s)
0.6 0.6
0.7 0.7
0.8 0.8
0.9 0.9
1 1
Figure5.5. 5.The Theactual actualand andestimated estimatedvalue valueof ofiiiβββ.. Figure Figure The actual and estimated value of 0.05 0.05
phase ‘a’ current sensor fault(A) phase‘a’ ‘a’current currentsensor sensorfault(A) fault(A) phase
0.04 0.04 0.03 0.03 0.02 0.02 0.01 0.01 0 0 -0.01 -0.01 -0.02 -0.02
fault fault estimation of of the the fault fault estimation
-0.03 -0.03 -0.04 -0.04 -0.05 -0.05 0 0
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5 t(s) t(s)
0.6 0.6
0.7 0.7
0.8 0.8
0.9 0.9
Figure 6. 6. The The current current sensor sensor fault fault and and its its reconstruction reconstruction in in phase phase “a”. “a”. Figure Figure 6. The current sensor fault and its reconstruction in phase “a”.
1 1
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10 10
phase ‘b’ ‘b’ current current sensor sensor fault(A) fault(A) phase
00 -10 -10 -20 -20 -30 -30
fault fault
-40 -40 -50 -50 -60 -60 estimation estimation of of the the fault fault
-70 -70 -80 -800 0
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5 t(s) t(s)
0.6 0.6
0.7 0.7
0.8 0.8
0.9 0.9
11
Figure 7. The current sensor fault and its reconstruction reconstruction in in phase phase “b”. “b”. 900 900 800 800 700 700
Te(Nm) Te(Nm)
600 600 500 500 400 400 300 300 200 200 100 100 00 -100 -1000 0
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5 t(s) t(s)
0.6 0.6
0.7 0.7
0.8 0.8
0.9 0.9
11
Figure 8. The electromagnetic torque. Figure 8. The electromagnetic torque.
When the given speed increases to 300 rad/s at t = 0.8 s, the frequency of the current increases, When speed increases to 300 rad/s atof t =the 0.8electromagnetic s, the frequencytorque of the also current increases, accordingly,the thegiven equiamplitude oscillation frequency increases, the accordingly, the equiamplitude oscillation frequency of the electromagnetic torque also increases, the change of the speed does not affect the observation of the current and the fault reconstruction. The change theprocessing speed doesmodule not affect the observation current and the fault current of fault is introduced at t = of 0.9the s, the phase “b” and thereconstruction. β axis currents The still current fault processing module is introduced at t = 0.9 s, the phase “b” and the β axis currents still experience the corresponding offset since the fault still exists, but the electromagnetic torque experience the corresponding since thereaches fault still exists, the electromagnetic torque oscillation oscillation disappears and theoffset torque value the givenbut value. disappears and the torque value reaches the given value. (B) Slow-variation offset fault on phase “a” sensor and abrupt gain fault on phase “b” sensor (B) Slow-variation offset fault on phase “a” sensor and abrupt gain fault on phase “b” sensor The sensors faults can be expressed as: The sensors faults can be expressed as: t 0.3s # 0 f aa 0 (32) t ă 0.3s f a “ 100 tanh(t ) t 0.3s (32) 100tanhptq t ě 0.3s # 0 t 0.1s t ă 0.1s f bb 0 fb “ 0.5ibbb tt ě s 0.5i 0.1 0.1s
(33) (33)
The three-phase stator currents of the current sensor output are shown in Figure 9, and the actual The three-phase stator currents of the current sensor output are shown in Figure 9, and the and the estimated values of iα and iβ are shown in Figures 10 and 11 respectively. The phase “a” and actual and the estimated values of iαα and iββ are shown in Figures 10 and 11, respectively. The phase “a” and phase “b” current sensors faults reconstruction values are shown in Figures 12 and 13,
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phase “b” current sensors faults reconstruction values are shown in Figures 12 and 13 respectively. The respectively. The The electromagnetic electromagnetic torque is is shown in in Figure Figure 14. 14. As As shown shown in in these these figures, figures, when when the the respectively. electromagnetic torque is shown in torque Figure 14.shown As shown in these figures, when the faults are added, the faults are added, the originally current balance is destroyed, the amplitudes of the β axis and phase faults are added, originally currentthe balance is destroyed, amplitudes the β axis increase and phase originally current the balance is destroyed, amplitudes of the βthe axis and phase of “b” currents at “b” currents increase at t = 0.1 s and produce a slowly varying offset at t = 0.3 s, the measured phase “b” currents increase at t = 0.1 svarying and produce a slowly varying offset atphase t = 0.3“a” s, the measured phase t = 0.1 s and produce a slowly offset at t = 0.3 s, the measured sensor current and “a” sensor sensor current current and and the estimation estimation of of αα axis axis current current produce produce aa slowly slowly varying varying offset at at tt == 0.3 0.3 s. s. The “a” the estimation of α axisthe current produce a slowly varying offset at t = 0.3 s. Theoffset gain sensor faultThe in gain sensor fault in phase “b” produces a corresponding equiamplitude torque oscillation at t = 0.1 0.1 gain sensor fault in phase “b” produces a corresponding equiamplitude torque oscillation at t = phase “b” produces a corresponding equiamplitude torque oscillation at t = 0.1 s–0.3 s, The amplitude s–0.3 s, The The amplitude amplitude of of the the electromagnetic electromagnetic torque torque oscillation oscillation gradually increases increases when when aa slowly slowly s–0.3 of the s,electromagnetic torque oscillation gradually increases whengradually a slowly varying offset fault in the varying offset fault in the phase “a” sensor is introduced at t = 0.3 s whose initial value is about 29.17. varying offset fault the phase at “a” is introduced t = 0.3 s whose initial value is about 29.17. phase “a” sensor is in introduced t =sensor 0.3 s whose initial at value is about 29.17. The adaptive law for The adaptive law law for for fault fault reconstruction reconstruction can can accurately reconstruct reconstruct the the slowly slowly varying varying offset offset fault fault The faultadaptive reconstruction can accurately reconstructaccurately the slowly varying offset fault and abrupt gain fault. and abrupt abrupt gain gain fault. fault. When When the the given given torque torque increases to to 1000 Nm Nm at t = 0.5 0.5 s, s, the the frequency frequency of of the the and When the given torque increases to 1000 Nm atincreases t = 0.5 s, the1000 frequencyatoft =the current does not change current does not change and the amplitude of the current increases. The torque change does not current not change and theincreases. amplitudeThe of torque the current increases. torque change does and the does amplitude of the current change does notThe affect the observation of not the affect the observation of the current and the fault reconstruction. The current fault processing affect of the currentThe andcurrent the fault The current fault processing currentthe andobservation the fault reconstruction. faultreconstruction. processing module is introduced at t = 0.7 s, module is is introduced introduced at at tt == 0.7 0.7 s, s, the three-phase three-phase currents and and the the αβ αβ axis axis current current are are still in in the the module the three-phase currents and the αβthe axis current arecurrents still in the corresponding fault statusstill since the corresponding fault status since the fault still exists, but the electromagnetic torque oscillation corresponding status since the fault stilloscillation exists, butdisappears the electromagnetic torque oscillation fault still exists,fault but the electromagnetic torque and the torque value reaches disappears and the torque value reaches 1000 Nm. disappears 1000 Nm. and the torque value reaches 1000 Nm. 400 400 300 300 200 200
ia ib ic ia ib ic
iabc(A) iabc(A)
100 100 00 -100 -100 -200 -200 -300 -300 00
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Figure 9. 9. The The measured measured value value of of three-phase three-phase current. current. Figure 300 300 250 250 200 200 150 150
iα iα
iα(A) iα(A)
100 100 50 50 00 -50 -50 -100 -100 -150 -150 -200 -200 00
estimation of iα estimation of iα
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Figure 10. 10. The actual actual and and estimated estimated value value of of iα. Figure Figure 10. The The actual and estimated value of iiαα..
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-200 estimation estimationofofiβiβ -200 -200 -300 -3000 -300 00
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Figure 11. The actual and estimated values of Figure 11. The actual and estimated values of Figure11. 11.The Theactual actualand andestimated estimatedvalues valuesof ofiiiiββββ... Figure 80 80 80
phase ‘a’ current sensor fault(A) phase current sensor fault(A) phase ‘a’‘a’ current sensor fault(A)
70 70 70 60 60 60 50 50 50 40 40 40
estimation of the fault estimationofofthe thefault fault estimation
30 30 30 20 20 20 10 10 10 0 00
-10 -100 -10 00
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Figure 12. The current sensor fault and its reconstruction in phase “a”. Figure12. 12.The Thecurrent currentsensor sensorfault faultand andits itsreconstruction reconstructionin inphase phase“a”. “a”. Figure Figure 12. The current sensor fault and its reconstruction
phase ‘b’ current sensor fault(A) phase current sensor fault(A) phase ‘b’‘b’ current sensor fault(A)
150 150 150 estimation of the fault estimationofofthe thefault fault estimation
100 100 100 50 50 50
fault fault fault
0 00 -50 -50 -50
-100 -1000 -100 00
0.1 0.1 0.1
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0.5 0.5 0.5 t(s) t(s) t(s)
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0.8 0.8 0.8
0.9 0.9 0.9
Figure 13. The current sensor fault and its reconstruction in phase “b”. Figure13. 13.The Thecurrent currentsensor sensorfault faultand andits itsreconstruction reconstructionin inphase phase“b”. “b”. Figure Figure 13. The current sensor fault and its reconstruction
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1 1
Figure 14. The electromagnetic torque. Figure Figure 14. 14. The The electromagnetic electromagnetic torque. torque. Figure 14. The electromagnetic torque.
. Hardware-in-the-Loop Experiments 6..Hardware-in-the-Loop Experiments Hardware-in-the-Loop Experiments . Hardware-in-the-Loop Experiments
To verify the proposed algorithm, experiments have been carried out on a RT-LAB
To verify algorithm, experiments have been been a RT-LAB To verify theproposed proposed experiments have outout on15on aand RT-LAB To verify the the proposed algorithm, experiments have carried oncarried acarried RT-LAB hardware-in-the-loop hardware-in-the-loop system. Thealgorithm, configuration and been platform areout shown in Figures 16, hardware-in-the-loop system. The configuration and platform are shown in Figures 15 hardware-in-the-loop The configuration platform in Figures 15 andand 16, 16, system. The configurationsystem. and platform are shown in and Figures 15 andare 16 shown respectively. respectively. respectively. respectively.
Figure 15. The configuration of RT-LAB experimental system. Figure 15. The configuration of RT-LAB experimental system. Figure 15. The configuration of RT-LAB experimental system.
Figure 15. The configuration of RT-LAB experimental system.
Figure 16. Photo of the RT-LAB experimental setup. Figure 16. Photo of the RT-LAB experimental setup.
Figure 16. Photo of the RT-LAB experimental setup.
Figure 16. Photo of the RT-LAB experimental setup.
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The controller and PMSM PMSMmodels modelsuse useblocks blocks from The controllerisisa aTMS320F2812 TMS320F2812 DSP, DSP, The The inverter inverter and from thethe RT-Events toolbox of RT-LAB OP5600. In this system, the PWM switching frequency is set at 5 KHz. RT-Events toolbox of RT-LAB OP5600. In this system, the PWM switching frequency is set at 5 KHz. The sampling period resultsare areshown shownininFigures Figures1717and and The sampling periodisischosen chosenasas20 20µs. μs.The The experimental experimental results 18.18.
(a)
(b)
(c) Figure 17. Experimental results when an intermittent fault is applied to the “b” phase current sensor. Figure 17. Experimental results when an intermittent fault is applied to the “b” phase current (a) phase currents and electromagnetic torque responses (iabc: 75 A/div; Te: 500 Nm/div; t: 100 sensor. (a) phase currents and electromagnetic torque responses (iabc: 75 A/div; Te: 500 Nm/div; ms/div); (b) actual and estimation αβ axis stator currents responses (iα: 200 A/div; iβ: 150 A/div; t: t: 100 ms/div); (b) actual and estimation αβ axis stator currents responses (iα: 200 A/div; iβ: 150 A/div; 100 ms/div); (c) phase currents fault and its reconstruction (fa: 1 A/div; fb: 50 A/div; t: 100 ms/div). t: 100 ms/div); (c) phase currents fault and its reconstruction (fa: 1 A/div; fb: 50 A/div; t: 100 ms/div).
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(a)
(b)
(c) Figure 18. Experimental results when a slow-variation fault and a gain fault are respectively applied Figure 18. Experimental results when a slow-variation fault and a gain fault are respectively applied to to the “a” phase current and “b” phase current sensors. (a) phase currents and electromagnetic the “a” phase current and “b” phase current sensors. (a) phase currents and electromagnetic torque torque (iabc: 150 A/div; Te: 1000 Nm/div; t: 100 ms/div); (b) actual and estimation αβ axis stator (iabc: 150 A/div; Te: 1000 Nm/div; t: 100 ms/div); (b) actual and estimation αβ axis stator currents (iα: currents (iα: 500 A/div; iβ: 600 A/div; t: 100 ms/div); (c) phase currents fault and its reconstruction 500 A/div; iβ: 600 A/div; t: 100 ms/div); (c) phase currents fault and its reconstruction (fa: 50 A/div; (fa: 50 A/div; fb: 100 A/div; t: 100 ms/div). fb: 100 A/div; t: 100 ms/div).
Through comparison with the simulation results, the adaptive law of the fault reconstruction comparison with the simulation adaptive lawoffset of thefaults fault and reconstruction canThrough effectively reconstruct intermittent offsetresults, faults, the slowly varying abrupt gaincan faults of reconstruct the current sensor, and the αβ axis current fault processing can abrupt restraingain the torque effectively intermittent offset faults, slowly varying offsetmodule faults and faults of
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the current sensor, and the αβ axis current fault processing module can restrain the torque oscillation after a fault occurs, and the drive system has better dynamic and higher real-time performance. 7. Conclusions This paper presents a phase current sensor fault reconstruction method for an interior PMSM torque closed-loop drive system based on a sliding mode observer. The motor model with active flux linkage is constructed first. Next a sliding mode current observer is designed in αβ coordinates to eliminate the effects of unknown disturbances by using a specific equivalent control methodology. Then, the phase current sensor faults are reconstructed by means of an adaptive method. Finally, an αβ axis current fault processing module is designed based on the reconstructed value. This method can accurately identify and reconstruct intermittent offset faults, slowly varying offset faults and abrupt gain faults in real-time, and the αβ axis current fault processing module can restrain the torque oscillation of the system after a fault occurs. The feasibility and effectiveness of the proposed method are verified by simulation and experimental tests on the RT-LAB platform. Acknowledgments: This work was supported by the Natural Science Foundation of China (Nos. 61273157 and 61473117). Author Contributions: Gang Huang proposed the method and wrote the original paper. Yi-Ping Luo supervised the writing process. Chang-Fan Zhang and Jing He provided revisions. Yi-Shan Huang performed the main experiment. Conflicts of Interest: The authors declare no conflict of interest.
References 1. 2. 3.
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7. 8. 9. 10.
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Li, X.; Zhang, W. An Adaptive Fault-Tolerant Multi Sensor Navigation Strategy for Automated Vehicles. IEEE Trans. Veh. Technol. 2010, 59, 2815–2819. Kondo, K. Control System of Permanent Magnet Synchronous Motor for Railway Vehicle Traction. Q. Rep. RTRI 2000, 41, 94–99. [CrossRef] Foo, G.H.B.; Zhang, X.; Vilathgamuwa, D.M. A Sensor Fault Detection and Isolation Method in Interior Permanent-Magnet Synchronous Motor Drives Based on an Extended Kalman Filter. IEEE Trans. Ind. Electron. 2013, 60, 3485–3495. [CrossRef] Bolognani, S.; Zordan, M.; Zigliotto, M. Experimental Fault-tolerant Control of a PMSM Drive. IEEE Trans. Ind. Electron. 2000, 47, 1134–1141. [CrossRef] Rosero, J.A.; Romeral, L.; Ortega, J.A.; Rosero, E. Short-Circuit Detection by Means of Empirical Mode Decomposition and Wigner-Ville Distribution for PMSM Running Under Dynamic Condition. IEEE Trans. Ind. Electron. 2009, 56, 4534–4547. [CrossRef] Sahraoui, M.; Ghoggal, A.; Zouzou, S.E.; Benbouzid, M.E. Dynamic Eccentricity Cage Induction Motors-Simulation and Analytical Study of its Spectral Signatures on Stator Currents. Simul. Model. Pract. Theory 2008, 16, 1503–1513. [CrossRef] Riheira, R.L.; Jacobina, C.B.; Silva, E.R.C.; Lima, A.M.N. Fault Detection of Open-Switch damage in Voltage-Fed PWM Motor Drive Systems. IEEE Trans. Power Electron. 2003, 18, 587–593. Roux, W.L.; Harley, R.G.; Habetler, T.G. Detecting Rotor Faults in Lower Power Permanent Magnet Synchronous Machines. IEEE Trans. Power Electron. 2007, 22, 322–328. [CrossRef] Masrur, M.A.; Chen, Z.; Murphey, Y. Intelligent Diagnosis of Open and Short Circuit Faults in Electric Drive Inverters for Real-Time Applications. IET Power Electron. 2010, 3, 279–291. [CrossRef] Ebrahimi, B.M.; Roshtkhari, M.J.; Faiz, J.; Khatami, S.V. Advanced Eccentricity Fault Recognition in Permanent Magnet Synchronous Motors Using Stator Current Signature Analysis. IEEE Trans. Ind. Electron. 2014, 61, 2041–2052. [CrossRef] Jeong, Y.-S.; Sul, S.-K.; Schulz, S.; Patel, N. Fault Detection and Fault-Tolerant Control of Interior Permanent-Magnet Motor Drive System for Electric Vehicle. IEEE Trans. Ind. Appl. 2005, 41, 46–51. [CrossRef] Berriri, H.; Naouar, M.W.; Slama-Belkhodja, I. Easy and Fast Sensor Fault Detection and Isolation Algorithm for Electrical Drives. IEEE Trans. Power Electron. 2012, 27, 490–499. [CrossRef]
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Grouz, F.; Sbita, L.; Boussak, M.; Khlaief, A. FDI Based on an Adaptive Observer for Current and Speed Sensors of PMSM Drives. Simul. Model. Pract. Theory 2013, 35, 34–49. [CrossRef] Zhang, C.F.; He, J.; Jia, L.; Xu, C.J.; Xiao, Y.Y. Virtual Line-Shafting Control for Permanent Magnet Synchronous Motor Systems Using Sliding-Mode Observer. IET Control Theory Appl. 2015, 9, 456–464. [CrossRef] Alwi, H.; Edwards, C. Robust Fault Reconstruction for Linear Parameter Varying Systems Using Sliding Mode Observers. Int. J. Robust Nonlinear Control 2013, 24, 1947–1968. [CrossRef] Huang, G.; Luo, Y.P.; Zhang, C.F.; Huang, Y.S.; Zhao, K.H. Current Sensor Fault Diagnosis Based on a Sliding Mode Observer for PMSM Driven Systems. Sensors 2015, 15, 11027–11049. [CrossRef] [PubMed] Liu, J.X.; Li, T.C.; Yang, G.J. Rotor Position and Speed Estimation for PMSM. Electr. Mach. Control 2009, 13, 690–694. Hasegawa, M.; Matsui, K. Position Sensorless Control for Interior Permanent Magnet Synchronous Motor Using Adaptive Flux with Inductance Identification. IET Electr. Power Appl. 2009, 3, 209–217. [CrossRef] © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).
Current Sensor Fault Reconstruction for PMSM Drives Gang Huang 1,2 , Yi-Ping Luo 1 , Chang-Fan Zhang 2 , Jing He 2, * and Yi-Shan Huang 3 1 2 3
*
School of Traffic and Transportation Engineering, Central South University, Changsha 410075, China; [email protected] (G.H.); [email protected] (Y.-P.L.) College of Electrical and Information Engineering, Hunan University of Technology, Zhuzhou 412007, China; [email protected] Hunan CSR Times Electric Vehicle Co., Ltd, Zhuzhou 412007, China; [email protected] Correspondence: [email protected]; Tel.: +86-731-2818-2072
Academic Editor: Vittorio M. N. Passaro Received: 27 October 2015; Accepted: 26 January 2016; Published: 30 January 2016
Abstract: This paper deals with a current sensor fault reconstruction algorithm for the torque closed-loop drive system of an interior PMSM. First, sensor faults are equated to actuator ones by a new introduced state variable. Then, in αβ coordinates, based on the motor model with active flux linkage, a current observer is constructed with a specific sliding mode equivalent control methodology to eliminate the effects of unknown disturbances, and the phase current sensor faults are reconstructed by means of an adaptive method. Finally, an αβ axis current fault processing module is designed based on the reconstructed value. The feasibility and effectiveness of the proposed method are verified by simulation and experimental tests on the RT-LAB platform. Keywords: permanent magnet synchronous motor (PMSM); active flux; sliding mode observers; sensors fault; reconstruction
1. Introduction Permanent magnet synchronous motors (PMSMs) are attractive for electric vehicle and railway traction drive applications thanks to their small volume, light weight, high efficiency, high power density, rugged construction and faster response [1,2]. However, PMSM traction systems are easily influenced by the circumstances of the application environment such as vibration and shock, low and high temperature, humidity and dust, etc., which lead to faults that could directly result in deterioration of the torque performance, and even seriously affect the safety of electric vehicles and railway trains. Therefore, it’s very significant to develop the real-time fault diagnosis and detection systems for PMSM drive systems. A PMSM drive system is usually embedded with at least two feedback current sensors, and faults in any of these sensors may lead to performance degradation [3], therefore, it is necessary to diagnose sensor faults to guarantee the safe and reliable operation of the drive system. Compared to sensor fault diagnosis of PMSM systems, there has been a lot of research on motor body faults or inverter faults [4–10]. Reference [11] proposed an offline method to diagnose current sensor faults for PMSM drive systems. In [12] a parity space approach based on redundancies in a temporal window to diagnose sensor faults was proposed, but the method only detected sudden faults. In [3] the authors proposed an Extended Kalman Filter method to diagnose current sensor faults for PMSM drive systems, but the performance would deteriorate at low speed, and it is not sensitive to slowly varying faults. A residual generation method based on an adaptive observer for phase current sensors was described in [13], but if the threshold is set larger, it will lead to misjudgments.
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The technology of residual generation by means of a filter or observer for fault detection and isolation cannot directly estimate the failures, and this may cause problems such as omission and misjudgment if the threshold chosen is not suitable, so in practical engineering applications, fault diagnosis systems should have higher sensitivity to small and slow-variation faults, and at the same time, they should be robust to various uncertainties, so as to reduce the rates of false positives and missed fault reports. Therefore, this paper presents a current sensor fault reconstruction algorithm for an interior PMSM torque closed-loop drive system based on a sliding mode observer thanks to its better robustness to inaccurate mathematical models, external disturbances, and parameter perturbation [14,15]. First, current sensor faults are equated to actuator ones by a newly introduced state variable. Then, in αβ coordinates, based on the motor model with active flux linkage, a current sliding mode observer is constructed with specific equivalent control methodology to eliminate the effects of unknown disturbances on the system, and the phase current sensor faults are reconstructed by means of an adaptive method. Finally, an αβ axis current fault processing module is designed based on the reconstructed value. This method can accurately identify and reconstruct intermittent offset faults, slow-variation offset faults and abrupt gain faults in real-time, so the fault processing module can restrain the torque oscillation after current sensor fault occurs. The feasibility and effectiveness of the proposed method are verified by simulation and experimental tests on the RT-LAB platform. 2. IPMSM Mathematical Model The stator voltage equations for the IPMSM in the rotating reference dq frame can be expressed as [16]: « ff « ff « ff « ff ud Rs ` DLd ´ωe Lq id 0 “ ` ωe (1) uq ωe Ld Rs ` DLq iq ψr The flux equations in the rotating reference dq frame are: #
ψd “ Ld id ` ψr ψq “ Lq iq
(2)
The electromagnetic torque equation can be described as: Te “
3 n p rψr ` pLd ´ Lq qid siq 2
(3)
where Rs stands for the stator resistance, Ld and Lq stand for the dq axis stator inductances, ud , uq and id , iq are respectively the stator voltages and currents in the dq axis frame, ψr , and ψd , ψq are respectively the permanent magnet flux linkage and the dq axis stator fluxes, ω e and n p are the electrical rotor speed and the number of pole pairs, respectively. D stands for the differential operator. Through coordinate transformation, Equation (1) is transformed to be: «
uα uβ
ff
« “ Rs
iα iβ
ff
« `D
L1 ` L2 cos2θ L2 sin2θ
L2 sin2θ L1 ´ L2 cos2θ
ff «
iα iβ
ff
« ` ωe ψr
ff ´sinθ cosθ
(4)
where L1 = (Ld + Lq )/2, L2 = (Ld ´ Lq )/2, uα , uβ and iα , iβ are respectively the stator voltages and currents in the αβ axis frame, θ is the electrical position. Items with 2θ in Equation (4) show the salient features of IPMSM, and it is difficult for the estimation of the IPMSM status variable. The voltage Equation (1) in the rotating reference dq frame can be reconstructed as: «
ud uq
ff
« “
Rs ` DLq ωe Lq
´ωe Lq Rs ` DLq
ff «
id iq
ff
« ` ωe
0 ˘ ` ψr ` Ld ´ Lq id
ff
« ` `
ff ˘ Ld ´ Lq Did 0
(5)
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«
ff Rs ` DLq ´ωe Lq where is a symmetric matrix. It has nothing to do with the d axis stator ωe Lq Rs ` DLq inductance and eliminates the salient pole phenomenon of IPMSM. Through coordinate transformation, the Equation (5) in the stationary αβ-reference frame is transformed to be: «
uα uβ
ff
«
iα iβ
“ Rs
ff
«
iα iβ
` DLq
ff
« ` ωe
ψr ` pLd ´ Lq qid 0
ff «
0
ff ´sinθ cosθ
ψr ` pLd ´ Lq qid
(6)
3. Sliding Mode Observers Design and Fault Reconstruction According to Equation (6), one has: « D
iα iβ
ff
Rs “´ Lq
«
ff
iα iβ
1 ` Lq
«
uα uβ
ff
« ωe ` Lq
ψr ` pLd ´ Lq qid 0
ff «
0
ff ´sinθ cosθ
ψr ` pLd ´ Lq qid
(7)
The active flux linkage is defined as [17,18]: ` ˘ ψext “ ψr ` Ld ´ Lq id Let ψext,
αβ
(8)
represents the active flux linkage vector in the stationary αβ-reference frame, we have: «
ψext,αβ “
ff ψext,α ψext,β
« “
ψr ` pLd ´ Lq qid 0
ψr ` pLd ´ Lq qid »
” Let x “
iα
iβ
ıT
” , u “
» « C “
1 0
0 1
— ,F“–
uβ
ωe fi Lq ffi fl 0
0
ff
uα
fi
Rs ıT — ´ Lq , A “ — – 0 «
ωe “ ´ J, J “ Lq
ff «
0
0 1
cosθ sinθ
ff (9) »
1 ffi — Lq ffi — Rs fl , B “ – ´ 0 Lq 0
fi 0 ffi ffi 1 fl , Lq
ff ´1 0
. ωe Lq According to Equation (7), the equation of PMSM model with sensor fault can be expressed as: ´
#
.
xptq “ Axptq ` Buptq ` Fψext,αβ ` Ed yptq “ Cxptq ` G f s
(10)
where fs = [fsα fsβ ]T is the stator current sensor fault vector in the stationary αβ-reference frame, d = [d1 d2 ]T is the unknown disturbance vector, d1 and d2 are assumed to be bounded. “¨ ” denotes the derivative. u and y are respectively the input vector and output vector. « ff x is the « state variable, ff 1 0 1 0 G“ ,E“ . 0 1 0 1 Consider a new state variable z which is a filtered version of y, and satisfies: .
z “ ´az ` by
(11)
where a and b are constant. If we select a “ 0, b “ 1, Equation (11) can be rewritten as: .
z “ y “ Cxptq ` G f s
(12)
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According to Equations (10) and (12), a new system can be described as: $ . ’ “ Axptq ` Buptq ` Fψext,αβ ` Ed & xptq . z “ Cxptq ` G f s ’ % w“z
(13)
where w is output vector of the new system. fs appears as an actuator fault in the new system and so the approach described earlier can be adopted. The sliding mode observer can be designed as: #
.
ˆ “ A xptq ˆ ` Buptq ` Fψext,αβ ` Eν1 xptq . ˆ “ C xptq ˆ ` G fˆs ` ν2 zptq
(14)
where “ˆ” denotes the estimated values. v1 and v2 are the sliding mode correction control signal. $ & L x ´ xˆ 1 k x ´ xˆ k ν1 “ % 0 $ & L z ´ zˆ 2 k z ´ zˆ k ν2 “ % 0
x ´ xˆ ‰ 0
(15)
x ´ xˆ “ 0 z ´ zˆ ‰ 0
(16)
z ´ zˆ “ 0
where L1 ą 0 and L2 ą 0 are all to be designed. ˆ es “ f s ´ fˆs . The observer errors are defined as: ex “ x ´ xˆ , ez “ z ´ z, The direct axis current component id of the active flux linkage quickly converge to the given value i˚d when the bandwidth is high enough, this ensures the active flux linkage as a linear constant value [17,18]. The state estimation error equations can be expressed as: .
.
.
e x “ x ´ xˆ “ Aex ` Epd ´ ν1 q .
(17)
.
.
ez “ z ´ zˆ “ Cex ` Ges ´ ν2
(18)
Consider a Lyapunov function candidate: V “ exT ex ` ezT ez ` esT Qes
(19)
where Q is a positive constant. Differentiating the Lyapunov function and according to Equations (17) and (18), then: .
V
.T
.
.T
.
.T
.
“ e x ex ` exT e x ` ez ez ` ezT ez ` es Qes ` esT Qes . “ rAex ` Epd ´ ν1 qsT ex ` exT rAex ` Epd ´ ν1 qs ` pCex ` Ges ´ ν2 qT ez ` ezT pCex ` Ges ´ ν2 q ` 2esT Qes . T T T T T T T T T T “ ex Aex ` pd ´ ν1 q Eex ` ex Aex ` ex Epd ´ ν1 q ` ex Cez ` es Gez ´ ν2 ez ` ez Cex ` ez Ges ´ ez ν2 ` 2esT Qes . T T T T T T “ 2ex Aex ` 2ex Epd ´ ν1 q ` 2ez Cex ` 2es Gez ´ 2ez ν2 ` 2es Qes
«
0 1
ff
«
1 0
0 1
ď 2k ex kk E kpk d k ´ L1 q ` 2ezT Cex ` 2esT Gez ´ 2ezT ν2 ` 2esT Qes . ď 2k ex kk E kpk d k ´ L1 q ` 2k ez kpk C kk ex k ´ L2 q ` 2esT pGez ` Qes q
.
Since A is a symmetric negative definite matrix, C “
1 0
, and E “
ff
(20)
.
, then V
can be expressed as: .
V
.
ď 2k ex kk E kpk d k ´ L1 q ` 2k ez kpk C kk ex k ´
L2 q ` 2esT rGez
.
` Qp f s ´ fˆs qs
(21)
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When the sensor gain changes very slowly as the time and environment change, or the sensor suddenly detects a break, power down or damage, the first derivative of the fault approximates to . zero, that is f s « 0, then: .
.
V ď 2k ex kk E kpk d k ´ L1 q ` 2k ez kpk C kk ex k ´ L2 q ` 2esT pGez ´ Q fˆs q
(22)
For the arbitrary initial conditions zp0q, if the adaptive law algorithm for fault reconstruction is: .
Then:
fˆs “ Q´1 Gez
(23)
V ď 2k ex kk E kpk d k ´ L1 q ` 2k ez kpk C kk ex k ´ L2 q
(24)
.
Select L1 ą k d k ` σ1 and L2 ą C k ex k ` σ2 , where σ1 ą 0, σ2 ą 0, then: .
V ď ´2σ1 k E kk ex k ´ 2σ2 k ez k ď 0
(25)
From the Lyapunov theorem, the system will reach the sliding mode state in finite time. Since iα and iβ are practically calculated from iabc . In the case of only “a” and “b” phase current sensors, according to [13]: $ & iα “ i a i ´ ic 2i ` i a (26) % i β “ b? “ b? 3 3 The effects of the αβ axis sensor faults fsα and fsβ are related to the errors in “a” and “b” phase sensor faults fa and fb as follows: $ & f sα “ f a 2 f ` fa (27) % f sβ “ b? 3 According to Equations (23) and (27), the adaptive law for the faults reconstruction in “a” and “b” phase sensors can be expressed as: . . fˆ “ fˆ “ Q´1 ez1 (28) a
?
.
fˆb “
.
sα
.
3 fˆsβ ´ fˆsα 2
? “Q
´1
3ez2 ´ ez1 2
(29)
4. Fault Diagnosis and Fault Processing When no phase sensor faults occur, the fault reconstruction values are also zero. When sensor faults occur, the fault reconstruction values deviate from their zero value. The faults detection decision rule are expressed in Table 1, Where “1” stands for “sensor faults occur”, “0” stands for “no sensor faults occur”, “Flag i” represents “the fault flags”. Table 1. Faults detection decision rule. fˆb
fˆa
Flag ib
Flag i a
0 0 1 1
0 1 0 1
0 0 1 1
0 1 0 1
After a sensor fault occurs, the torque oscillates since the original current balance is destroyed. The performance of the drive system will decrease. To achieve a performance similar to the pre-fault
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status, it is necessary to study the fault processing strategy. The αβ axis current fault processing Sensors 16, 178 6 of 16 module is 2016, shown in Figure 1. The fault processing equations can be expressed as: Sensors 2016, 16, 178
#
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iαri “iˆαiˆ ´ ffˆsα¨Flag Flagi i a r s a i βri “ iˆβiˆ ´ ffˆˆsβ¨Flag pFlag i i a ‘ Flag ib q
(30)
(30)
i rr i f ss ( Flag aia Flag ib ) (30) ˆ ˆ ioperation r i f s ( Flag ia Flag ib ) wherewhere Flag iFlag ‘ Flag i is a logic AND equation, i and i are the αβ axis feedback currents ia Flag a ar iar and b ib is a logic AND operation equation, br ibr are the αβ axis feedback
after currents fault processing. When phase sensor faultssensor occur, “Flag i” is zero the estimation afteria fault processing. When nooperation phase faults i ” isαβ zero value, theof αβ Flag Flag ib isno a logic AND equation, iaroccur, and i“br Flag arevalue, the axis feedback where axis current is adopted for the αβ axis feedback current. When sensor faults occur, “Flag i” equals estimation of αβ axisprocessing. current is adopted forphase the αβsensor axis feedback current. When faults occur, currents after fault When no faults occur, “ Flag i ”sensor is zero value, the one, the difference between estimation of αβ and fault reconstruction is “estimation Flag i ”equals the difference between the current estimation ofits αβcorresponding axis current andsensor its corresponding of αβone, axisthe current is adopted foraxis the αβ axis feedback current. When faults occur, adopted for the αβ axis current. fault reconstruction adopted for the αβ axisthe feedback current. “ Flag i ”equals one,isfeedback the difference between estimation of αβ axis current and its corresponding fault reconstruction is adopted for the αβ axis feedback current.
Figure 1. αβ axis current fault processing module. Figure 1. αβ axis current fault processing module. Figure 1. αβ axis current fault processing module.
5. Simulations and Analysis
5. Simulations and Analysis
5. Simulations and Analysis The effectiveness of the fault reconstruction and fault processing method for PMSM drive
The effectiveness the fault reconstruction andand faultfault processing method forof PMSM drive system system inofthe was simulated experimentally. The block diagram the simulation Theproposed effectiveness ofpaper the fault reconstruction processing method for PMSM drive proposed in the paper was simulated experimentally. The block diagram of the simulation setup is setup shown in Figure 2. systemis proposed in the paper was simulated experimentally. The block diagram of the simulation shown in Figure 2. in Figure 2. setup is shown
Figure 2. Block diagram of current sensor fault reconstruction and fault processing system for IPMSM. Figure 2. Block diagram of current sensor fault reconstruction and fault processing system for Figure 2. Block diagram of current sensor fault reconstruction and fault processing system for IPMSM. IPMSM.
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The parameters for the interior PMSM of this study are tabulated in Table 2. The given speed 7 of 16 and torque are 200 rad/s and 500 Nm, respectively. Let system unknown disturbances d1 and d2 be random noise, which value range is [−50 50]. To reduce chattering and eliminate high-frequency The parameters forthe thechattering, interior PMSM of this study are tabulated in Table 2. be Theused given and interference caused by a successive approximation function can to speed substitute torque rad/s and Nm, cases respectively. Let system unknown disturbances d1 and d2 be random for the are sign200 function sgn 500 (·). Two are discussed: noise, which value range is [´50 50]. To reduce chattering and eliminate high-frequency interference Tableapproximation 2. Interior PMSM parameters. caused by the chattering, a successive function can be used to substitute for the sign function sgn (¨ ). Two cases are discussed: Quantity Symbol Value Rs Stator resistance 0.02 Ω Table 2. Interior PMSM parameters.
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Lq
Q axis inductance
0.001500 H
Symbol Ld
DQuantity axis inductance Stator resistance Inertia Q axis inductance Magnetic flux D axis inductance Number of pole pairs Inertia Damping coefficient Magnetic flux Number of pole pairs DC-bus voltage Damping coefficient DC-bus voltage
(A) Intermittent offset fault on “b” phase sensor
Value H 0.003572 Ω2 1000.02 kg·m 0.001500 0.892 WbH 0.003572 H 4 pairs 100 kg¨ m2 0.0010.892 Nm·s/rad Wb 4 pairs 1500 V
Rs J Lq r Ld P J ψr B PVdc B Vdc
0.001 Nm¨ s/rad 1500 V
The “b” phase sensor fault can be expressed as: (A) Intermittent offset fault on “b” phase sensor 0 t 0.1s The “b” phase sensor fault can be expressed as: 30 0.1s t 0.3s $ ’ f b’ 0.3st ăt 0.1s 00 0.5s (31) ’ ’ ’ ´30 0.1s ď t ă 0.3s & 50 0.5s t 0.7 s (31) fb “ 0 0.3s ď t ă 0.5s ’ ’ ’ t s 20 0.7 ’ ´50 0.5s ď t ă 0.7s ’ % ´20 t ě 0.7s The three-phase stator currents of the current sensor output are shown in Figure 3, and the iβ are shown in Figures and 5, respectively. “a” actual and estimatedstator values of iα and The three-phase currents of the current sensor output4are shown in Figure 3,The and phase the actual phase “b”values current faultshown reconstruction shown in The Figures andand 7, and estimated of iαsensors and iβ are in Figures values 4 and 5are respectively. phase6 “a” respectively. The electromagnetic torque is shown in are Figure 8. As the simulation figures, phase “b” current sensors fault reconstruction values shown in shown Figuresin 6 and 7 respectively. The when an intermittent faultin is Figure added,8.the balancefigures, is destroyed, theintermittent amplitudes electromagnetic torqueoffset is shown Asoriginally shown in current the simulation when an of phase and phase “c” currents slightly, the the measured phase “b” sensor offset fault“a” is added, the originally currentincrease balance is destroyed, amplitudes of phase “a” andcurrent phase produces intermittent offset. The phase electromagnetic torque produces produces a corresponding “c” currentsan increase slightly, the measured “b” sensor current an intermittent offset. equiamplitude oscillation. adaptive law for fault reconstruction accurately reconstruct the The electromagnetic torqueThe produces a corresponding equiamplitudecan oscillation. The adaptive law intermittent offset faults. the fault value becomes zero at t offset = 0.3 faults. s–0.5 s,When the current and the for fault reconstruction canWhen accurately reconstruct the intermittent the fault value torque are back becomes zero at to t =normal. 0.3 s–0.5 s, the current and the torque are back to normal. 150 ia ib ic
100
iabc(A)
50
0
-50
-100
-150 0
0.1
0.2
0.3
0.4
0.5 t(s)
0.6
0.7
0.8
Figure 3. The measured value of the three-phase current.
0.9
1
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150 150 iα iα
100 100
iα(A) iα(A) iα(A)
50 50 0 0 -50 -50 -100 -100 -150 -150 0 0
estimation of of iα iα estimation
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5 t(s) t(s)
0.6 0.6
0.7 0.7
0.8 0.8
0.9 0.9
1 1
Figure4. 4. The The actual actualand andestimated estimatedvalue valueof ofiiiααα... Figure Figure 4. The actual and estimated value of 100 100 50 50
iβ(A) iβ(A) iβ(A)
0 0 -50 -50 -100 -100 -150 -150 -200 -200 0 0
iβ iβ estimation of of iβ iβ estimation
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5 t(s) t(s)
0.6 0.6
0.7 0.7
0.8 0.8
0.9 0.9
1 1
Figure5.5. 5.The Theactual actualand andestimated estimatedvalue valueof ofiiiβββ.. Figure Figure The actual and estimated value of 0.05 0.05
phase ‘a’ current sensor fault(A) phase‘a’ ‘a’current currentsensor sensorfault(A) fault(A) phase
0.04 0.04 0.03 0.03 0.02 0.02 0.01 0.01 0 0 -0.01 -0.01 -0.02 -0.02
fault fault estimation of of the the fault fault estimation
-0.03 -0.03 -0.04 -0.04 -0.05 -0.05 0 0
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5 t(s) t(s)
0.6 0.6
0.7 0.7
0.8 0.8
0.9 0.9
Figure 6. 6. The The current current sensor sensor fault fault and and its its reconstruction reconstruction in in phase phase “a”. “a”. Figure Figure 6. The current sensor fault and its reconstruction in phase “a”.
1 1
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10 10
phase ‘b’ ‘b’ current current sensor sensor fault(A) fault(A) phase
00 -10 -10 -20 -20 -30 -30
fault fault
-40 -40 -50 -50 -60 -60 estimation estimation of of the the fault fault
-70 -70 -80 -800 0
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5 t(s) t(s)
0.6 0.6
0.7 0.7
0.8 0.8
0.9 0.9
11
Figure 7. The current sensor fault and its reconstruction reconstruction in in phase phase “b”. “b”. 900 900 800 800 700 700
Te(Nm) Te(Nm)
600 600 500 500 400 400 300 300 200 200 100 100 00 -100 -1000 0
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5 t(s) t(s)
0.6 0.6
0.7 0.7
0.8 0.8
0.9 0.9
11
Figure 8. The electromagnetic torque. Figure 8. The electromagnetic torque.
When the given speed increases to 300 rad/s at t = 0.8 s, the frequency of the current increases, When speed increases to 300 rad/s atof t =the 0.8electromagnetic s, the frequencytorque of the also current increases, accordingly,the thegiven equiamplitude oscillation frequency increases, the accordingly, the equiamplitude oscillation frequency of the electromagnetic torque also increases, the change of the speed does not affect the observation of the current and the fault reconstruction. The change theprocessing speed doesmodule not affect the observation current and the fault current of fault is introduced at t = of 0.9the s, the phase “b” and thereconstruction. β axis currents The still current fault processing module is introduced at t = 0.9 s, the phase “b” and the β axis currents still experience the corresponding offset since the fault still exists, but the electromagnetic torque experience the corresponding since thereaches fault still exists, the electromagnetic torque oscillation oscillation disappears and theoffset torque value the givenbut value. disappears and the torque value reaches the given value. (B) Slow-variation offset fault on phase “a” sensor and abrupt gain fault on phase “b” sensor (B) Slow-variation offset fault on phase “a” sensor and abrupt gain fault on phase “b” sensor The sensors faults can be expressed as: The sensors faults can be expressed as: t 0.3s # 0 f aa 0 (32) t ă 0.3s f a “ 100 tanh(t ) t 0.3s (32) 100tanhptq t ě 0.3s # 0 t 0.1s t ă 0.1s f bb 0 fb “ 0.5ibbb tt ě s 0.5i 0.1 0.1s
(33) (33)
The three-phase stator currents of the current sensor output are shown in Figure 9, and the actual The three-phase stator currents of the current sensor output are shown in Figure 9, and the and the estimated values of iα and iβ are shown in Figures 10 and 11 respectively. The phase “a” and actual and the estimated values of iαα and iββ are shown in Figures 10 and 11, respectively. The phase “a” and phase “b” current sensors faults reconstruction values are shown in Figures 12 and 13,
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phase “b” current sensors faults reconstruction values are shown in Figures 12 and 13 respectively. The respectively. The The electromagnetic electromagnetic torque is is shown in in Figure Figure 14. 14. As As shown shown in in these these figures, figures, when when the the respectively. electromagnetic torque is shown in torque Figure 14.shown As shown in these figures, when the faults are added, the faults are added, the originally current balance is destroyed, the amplitudes of the β axis and phase faults are added, originally currentthe balance is destroyed, amplitudes the β axis increase and phase originally current the balance is destroyed, amplitudes of the βthe axis and phase of “b” currents at “b” currents increase at t = 0.1 s and produce a slowly varying offset at t = 0.3 s, the measured phase “b” currents increase at t = 0.1 svarying and produce a slowly varying offset atphase t = 0.3“a” s, the measured phase t = 0.1 s and produce a slowly offset at t = 0.3 s, the measured sensor current and “a” sensor sensor current current and and the estimation estimation of of αα axis axis current current produce produce aa slowly slowly varying varying offset at at tt == 0.3 0.3 s. s. The “a” the estimation of α axisthe current produce a slowly varying offset at t = 0.3 s. Theoffset gain sensor faultThe in gain sensor fault in phase “b” produces a corresponding equiamplitude torque oscillation at t = 0.1 0.1 gain sensor fault in phase “b” produces a corresponding equiamplitude torque oscillation at t = phase “b” produces a corresponding equiamplitude torque oscillation at t = 0.1 s–0.3 s, The amplitude s–0.3 s, The The amplitude amplitude of of the the electromagnetic electromagnetic torque torque oscillation oscillation gradually increases increases when when aa slowly slowly s–0.3 of the s,electromagnetic torque oscillation gradually increases whengradually a slowly varying offset fault in the varying offset fault in the phase “a” sensor is introduced at t = 0.3 s whose initial value is about 29.17. varying offset fault the phase at “a” is introduced t = 0.3 s whose initial value is about 29.17. phase “a” sensor is in introduced t =sensor 0.3 s whose initial at value is about 29.17. The adaptive law for The adaptive law law for for fault fault reconstruction reconstruction can can accurately reconstruct reconstruct the the slowly slowly varying varying offset offset fault fault The faultadaptive reconstruction can accurately reconstructaccurately the slowly varying offset fault and abrupt gain fault. and abrupt abrupt gain gain fault. fault. When When the the given given torque torque increases to to 1000 Nm Nm at t = 0.5 0.5 s, s, the the frequency frequency of of the the and When the given torque increases to 1000 Nm atincreases t = 0.5 s, the1000 frequencyatoft =the current does not change current does not change and the amplitude of the current increases. The torque change does not current not change and theincreases. amplitudeThe of torque the current increases. torque change does and the does amplitude of the current change does notThe affect the observation of not the affect the observation of the current and the fault reconstruction. The current fault processing affect of the currentThe andcurrent the fault The current fault processing currentthe andobservation the fault reconstruction. faultreconstruction. processing module is introduced at t = 0.7 s, module is is introduced introduced at at tt == 0.7 0.7 s, s, the three-phase three-phase currents and and the the αβ αβ axis axis current current are are still in in the the module the three-phase currents and the αβthe axis current arecurrents still in the corresponding fault statusstill since the corresponding fault status since the fault still exists, but the electromagnetic torque oscillation corresponding status since the fault stilloscillation exists, butdisappears the electromagnetic torque oscillation fault still exists,fault but the electromagnetic torque and the torque value reaches disappears and the torque value reaches 1000 Nm. disappears 1000 Nm. and the torque value reaches 1000 Nm. 400 400 300 300 200 200
ia ib ic ia ib ic
iabc(A) iabc(A)
100 100 00 -100 -100 -200 -200 -300 -300 00
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5 t(s) t(s)
0.6 0.6
0.7 0.7
0.8 0.8
0.9 0.9
11
0.9 0.9
11
Figure 9. 9. The The measured measured value value of of three-phase three-phase current. current. Figure 300 300 250 250 200 200 150 150
iα iα
iα(A) iα(A)
100 100 50 50 00 -50 -50 -100 -100 -150 -150 -200 -200 00
estimation of iα estimation of iα
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5 t(s) t(s)
0.6 0.6
0.7 0.7
Figure 10. 10. The actual actual and and estimated estimated value value of of iα. Figure Figure 10. The The actual and estimated value of iiαα..
0.8 0.8
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400 400 400 300 300 300 200 200 200
iβ iβiβ
iβ(A) iβ(A) iβ(A)
100 100 100 0 00
-100 -100 -100 estimation of iβ
-200 estimation estimationofofiβiβ -200 -200 -300 -3000 -300 00
0.1 0.1 0.1
0.2 0.2 0.2
0.3 0.3 0.3
0.4 0.4 0.4
0.5 0.5 0.5 t(s) t(s) t(s)
0.6 0.6 0.6
0.7 0.7 0.7
0.8 0.8 0.8
0.9 0.9 0.9
1 11
Figure 11. The actual and estimated values of Figure 11. The actual and estimated values of Figure11. 11.The Theactual actualand andestimated estimatedvalues valuesof ofiiiiββββ... Figure 80 80 80
phase ‘a’ current sensor fault(A) phase current sensor fault(A) phase ‘a’‘a’ current sensor fault(A)
70 70 70 60 60 60 50 50 50 40 40 40
estimation of the fault estimationofofthe thefault fault estimation
30 30 30 20 20 20 10 10 10 0 00
-10 -100 -10 00
fault fault fault
0.1 0.1 0.1
0.2 0.2 0.2
0.3 0.3 0.3
0.4 0.4 0.4
0.5 0.5 0.5 t(s) t(s) t(s)
0.6 0.6 0.6
0.7 0.7 0.7
0.8 0.8 0.8
0.9 0.9 0.9
1 11
Figure 12. The current sensor fault and its reconstruction in phase “a”. Figure12. 12.The Thecurrent currentsensor sensorfault faultand andits itsreconstruction reconstructionin inphase phase“a”. “a”. Figure Figure 12. The current sensor fault and its reconstruction
phase ‘b’ current sensor fault(A) phase current sensor fault(A) phase ‘b’‘b’ current sensor fault(A)
150 150 150 estimation of the fault estimationofofthe thefault fault estimation
100 100 100 50 50 50
fault fault fault
0 00 -50 -50 -50
-100 -1000 -100 00
0.1 0.1 0.1
0.2 0.2 0.2
0.3 0.3 0.3
0.4 0.4 0.4
0.5 0.5 0.5 t(s) t(s) t(s)
0.6 0.6 0.6
0.7 0.7 0.7
0.8 0.8 0.8
0.9 0.9 0.9
Figure 13. The current sensor fault and its reconstruction in phase “b”. Figure13. 13.The Thecurrent currentsensor sensorfault faultand andits itsreconstruction reconstructionin inphase phase“b”. “b”. Figure Figure 13. The current sensor fault and its reconstruction
1 11
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Te(Nm)
Te(Nm)
Te(Nm)
1400
0.1 0.1 0.1
0.2 0.2 0.2
0.3 0.3 0.3
0.40.4 0.4
0.5 0.5 t(s) t(s) 0.5 t(s)
0.6 0.6 0.6
0.9 1 0.7 0.7 0.8 0.8 0.9 0.7 0.8 0.9
1 1
Figure 14. The electromagnetic torque. Figure Figure 14. 14. The The electromagnetic electromagnetic torque. torque. Figure 14. The electromagnetic torque.
. Hardware-in-the-Loop Experiments 6..Hardware-in-the-Loop Experiments Hardware-in-the-Loop Experiments . Hardware-in-the-Loop Experiments
To verify the proposed algorithm, experiments have been carried out on a RT-LAB
To verify algorithm, experiments have been been a RT-LAB To verify theproposed proposed experiments have outout on15on aand RT-LAB To verify the the proposed algorithm, experiments have carried oncarried acarried RT-LAB hardware-in-the-loop hardware-in-the-loop system. Thealgorithm, configuration and been platform areout shown in Figures 16, hardware-in-the-loop system. The configuration and platform are shown in Figures 15 hardware-in-the-loop The configuration platform in Figures 15 andand 16, 16, system. The configurationsystem. and platform are shown in and Figures 15 andare 16 shown respectively. respectively. respectively. respectively.
Figure 15. The configuration of RT-LAB experimental system. Figure 15. The configuration of RT-LAB experimental system. Figure 15. The configuration of RT-LAB experimental system.
Figure 15. The configuration of RT-LAB experimental system.
Figure 16. Photo of the RT-LAB experimental setup. Figure 16. Photo of the RT-LAB experimental setup.
Figure 16. Photo of the RT-LAB experimental setup.
Figure 16. Photo of the RT-LAB experimental setup.
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The controller and PMSM PMSMmodels modelsuse useblocks blocks from The controllerisisa aTMS320F2812 TMS320F2812 DSP, DSP, The The inverter inverter and from thethe RT-Events toolbox of RT-LAB OP5600. In this system, the PWM switching frequency is set at 5 KHz. RT-Events toolbox of RT-LAB OP5600. In this system, the PWM switching frequency is set at 5 KHz. The sampling period resultsare areshown shownininFigures Figures1717and and The sampling periodisischosen chosenasas20 20µs. μs.The The experimental experimental results 18.18.
(a)
(b)
(c) Figure 17. Experimental results when an intermittent fault is applied to the “b” phase current sensor. Figure 17. Experimental results when an intermittent fault is applied to the “b” phase current (a) phase currents and electromagnetic torque responses (iabc: 75 A/div; Te: 500 Nm/div; t: 100 sensor. (a) phase currents and electromagnetic torque responses (iabc: 75 A/div; Te: 500 Nm/div; ms/div); (b) actual and estimation αβ axis stator currents responses (iα: 200 A/div; iβ: 150 A/div; t: t: 100 ms/div); (b) actual and estimation αβ axis stator currents responses (iα: 200 A/div; iβ: 150 A/div; 100 ms/div); (c) phase currents fault and its reconstruction (fa: 1 A/div; fb: 50 A/div; t: 100 ms/div). t: 100 ms/div); (c) phase currents fault and its reconstruction (fa: 1 A/div; fb: 50 A/div; t: 100 ms/div).
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(a)
(b)
(c) Figure 18. Experimental results when a slow-variation fault and a gain fault are respectively applied Figure 18. Experimental results when a slow-variation fault and a gain fault are respectively applied to to the “a” phase current and “b” phase current sensors. (a) phase currents and electromagnetic the “a” phase current and “b” phase current sensors. (a) phase currents and electromagnetic torque torque (iabc: 150 A/div; Te: 1000 Nm/div; t: 100 ms/div); (b) actual and estimation αβ axis stator (iabc: 150 A/div; Te: 1000 Nm/div; t: 100 ms/div); (b) actual and estimation αβ axis stator currents (iα: currents (iα: 500 A/div; iβ: 600 A/div; t: 100 ms/div); (c) phase currents fault and its reconstruction 500 A/div; iβ: 600 A/div; t: 100 ms/div); (c) phase currents fault and its reconstruction (fa: 50 A/div; (fa: 50 A/div; fb: 100 A/div; t: 100 ms/div). fb: 100 A/div; t: 100 ms/div).
Through comparison with the simulation results, the adaptive law of the fault reconstruction comparison with the simulation adaptive lawoffset of thefaults fault and reconstruction canThrough effectively reconstruct intermittent offsetresults, faults, the slowly varying abrupt gaincan faults of reconstruct the current sensor, and the αβ axis current fault processing can abrupt restraingain the torque effectively intermittent offset faults, slowly varying offsetmodule faults and faults of
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the current sensor, and the αβ axis current fault processing module can restrain the torque oscillation after a fault occurs, and the drive system has better dynamic and higher real-time performance. 7. Conclusions This paper presents a phase current sensor fault reconstruction method for an interior PMSM torque closed-loop drive system based on a sliding mode observer. The motor model with active flux linkage is constructed first. Next a sliding mode current observer is designed in αβ coordinates to eliminate the effects of unknown disturbances by using a specific equivalent control methodology. Then, the phase current sensor faults are reconstructed by means of an adaptive method. Finally, an αβ axis current fault processing module is designed based on the reconstructed value. This method can accurately identify and reconstruct intermittent offset faults, slowly varying offset faults and abrupt gain faults in real-time, and the αβ axis current fault processing module can restrain the torque oscillation of the system after a fault occurs. The feasibility and effectiveness of the proposed method are verified by simulation and experimental tests on the RT-LAB platform. Acknowledgments: This work was supported by the Natural Science Foundation of China (Nos. 61273157 and 61473117). Author Contributions: Gang Huang proposed the method and wrote the original paper. Yi-Ping Luo supervised the writing process. Chang-Fan Zhang and Jing He provided revisions. Yi-Shan Huang performed the main experiment. Conflicts of Interest: The authors declare no conflict of interest.
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