Fault Tolerant Control of Five-Phase Permanent Magnet Assisted Synchronous Reluctance Motor based on Dynamic Current Phase Advance AKM Arafat
Seungdeog Choi
Electrical and Computer Engineering University of Akron Akron, USA
[email protected]
Electrical and Computer Engineering University of Akron Akron, USA
[email protected]
Abstract— In this paper, fault tolerant control of a five-phase permanent magnet assisted synchronous reluctance motor (PMaSynRM) has been discussed. Reliable control method under any fault conditions has been predominantly required in critical applications such as hybrid/electric vehicular applications and aerospace industries. The proposed method utilizes the advance vector control of multiphase machine to provide maximum amount of torque under different types of open phase fault conditions. To maximize the amount of torque, the current phase advance in the five-phase reluctance machine has been introduced which varies with saturation effects and load dynamic behavior. Under such condition, the phase advance has been calculated dynamically at different faults and load conditions. Considering that, the optimal set of currents has been injected to provide maximum amount of torque under different open phase fault conditions. Extensive theoretical analysis along with Finite element methods has been carried out to support the proposed method. The experimental results have been provided utilizing the 5hp dynamo system which consists of TI DSP control board and five-phase inverter system.
I. INTRODUCTION In industry applications such as automotive and aerospace, higher reliability is required under multiple failures which alter the smooth operation of electric machines. Hence, improving the fault tolerance capability of the drives is the key feature to support the uninterrupted operation in critical services. There have been numerous efforts accomplished in the past focusing on this fault tolerant control to provide effective solutions under different fault conditions [1-4]. To improve the fault tolerance, multiphase machine has been suggested in many applications [5]. Due to its higher number of degrees of freedom, the controllability becomes even more flexible than compared with conventional three phase drive system. In recent applications, the multiphase machines are equally used in low power cases where higher reliability and greater fault tolerance are the prime concerns. In such applications, even one or more phases lost can be quickly handled to provide significant amount of torque to run the system. In addition,
reduced torque pulsation in multiphase machines made it more suitable to the automotive wheel industry. Among many types of multiphase machines, the five-phase permanent magnet assisted synchronous reluctance motor (PMa-SynRM) has been suggested in vast industrial applications [6]. The lower magnet size and higher utilization of the reluctances made it suitable in designing low-cost higher-torque-density electric motor drives. This type of machines take the advantages of both the synchronous reluctance machine (RSM) and interior permanent magnet machines (IPMSM) [7] which basically improve the overall torque characteristics. In order to preserve the reliability of the drives, various control strategies need to apply under different types of faults. Numerous control methods have been suggested in many literatures to improve the fault tolerance capability of the system [8-12]. In [8], under fault condition redundant no of phases in the inverter side has been suggested along with additional machines in parallel to continue the safe operation. In spite of being easy control, additional hardware requirement made this method cost ineffective. Harmonic injection into the feeding currents has been found one of the popular methods to improve the torque [9-10]. However, this method increases the overall losses which deteriorate the efficiency of the machine. Another widely accepted method has been found which is Back EMF (BEM) based machine modelling [2]. Due to its highly model dependency the application is being restricted in many operations. To improve the performance with the BEM based model, recently iterative learning method has been adopted to reduce the overall torque-ripple [11]. This method is applied as a current control technique for recovering performance in multiphase machines under fault conditions. ILC based control is also applicable where necessity of explicit information of the fault condition can be avoided. However, the learning process and the design of the control algorithm yet leave many conclusions to be made to suit the application. In addition to these, there were more research works which supported higher phase currents in the remaining
This work was supported by the Ohio Third Frontier Technology Validation and Start-Up Fund.
978-1-4673-7151-3/15/$31.00 ©2015 IEEE
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(a) (b) (c) Fig. 1. Cross section of Five-Phase PMa-SynRM under different fault condition: a) Single Phase Fault, b) Two-phase non adjacent fault, and c) Two-phase adjacent fault.
(b) Fig.2. FEA results under different fault conditions: a) Variation of torque, b) Average torque.
Is
δ
In this section, the FEA model of the 5-phase faulty machine has been presented (PMa-SynRM).There are two types of open phase faults can be happened in a five-phase system which are single-phase open and two-phase open fault, which will significantly alter the torque production capability of five-phase system. Three-phase open fault is practically not sustainable which is not considered in this paper. Two-phase fault again can be categorized as adjacent phase fault or nonadjacent phase fault. Fig. 1 shows those examples of FEA fault model of the considered five-phase PMa- SynRM under different fault conditions. The FEA torque results under those faults have been shown in Fig.2.
Iqs
Vs jX qs I qs
EPM
II. FIVE-PHASE MOTOR MODEL UNDER FAULT CONDITION
jX ds I ds Vqs
III. MATHEMATICAL MODEL AND PHASOR DIAGRAM OF THE FIVE-PHASE PMA-SYNRM UNDER FAULT CONDITIONS
rs Is
healthy phases to improve the torque [12]. Literally in a reluctance type of machines, higher than rated current is not possible to push through due to its higher prone to magnetic saturation which eventually reduce the reluctance torque, hence reducing the overall performances. In the proposed 5-phase PMa-SynRM drives, the d axis has been aligned to the stator reference frame. Hence, an angle has been considered between the phase-a current and the stator reference frame which is called as phase advance. The phase advance has been found as an optimum parameter which helps to maximize the torque under different fault conditions. Additionally, it is found that, the phase advance is not a fixed parameter to control. It changes with different fault and loading conditions. In this proposed method, the phase advance has been checked dynamically at different operating condition and calculated optimum value to provide maximum torque under different fault conditions. The finite element analysis has been done to support the theoretical explanation. The experimental results are found utilizing the five-phase inverter controlled by TI DSP (F28335) in a 5hp dynamo system.
γ
Vds
EPM
Ids
Fig.3. Phasor diagram of 5-phase PMa-SynRM.
The mathematical model equations are derived in reference to the d-q rotating frame. The equations are given as follows: Vd = − ω r ( L q ⋅ I q − λ PM ) ; Vq = ω r ( Ld ⋅ I d ) ; Ȝ q = L q ⋅ I q − λ PM ;
(1) Ȝ d = Ld ⋅ I d where V d and I d is the d-axis voltage and current, V q and I q is the q-axis voltage and current, Ld is the d-axis inductance, L q is the q-axis inductance, and λ P M is the permanent magnet
(a)
flux linkage, Ȝ d and Ȝ q is the d-q axis flux linkages. The general electro- magnetic torque of five-phase PMaSynRM can be derived as, 5 p 5 p ªλd I q − λq I d º¼ = ªλPM I d + ( Ld − Lq ) I d I q º¼ (2) Te = 22 ¬ 22¬
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where P is the number of pole pair, I d = I s cos γ , I q = I s sin γ , and γ is the phase advance. The vector diagram with γ is given in Fig. 3. IV. PHASE ADVANCE BASED ON OPTIMAL TORQUE UNDER FAULT CONDITIONS Phase advance ( γ ) is one of the critical parameter of PMaSynRM of which optimization lead to the full utilization of reluctance torque. If the fault current is considered as I F ∠γ F where γ F is the phase advance during the fault. Using the torque equation (2), it can be maximized by differentiating as ∂ Te / ∂ γ F = 0 and Solving for γ F
Te = 5P/ 4 ª¬λPM IF cos γ F + (LdF − LqF )I F cos γ F I F sin γ F º¼
(3)
γ F = sin −1 (−λPM + λPM 2 + 8( LdF − LqF )2 I F 2 / (4( LdF − LqF ) I F )) (4) where LdF and LqF is the d and q axis inductances under fault condition. In (4) it is observed that the optimal phase advance clearly depends on the fault current and the difference of the inductances. Putting the calculated value for flux linkage (FEA), λPM =.05 ( LdF − LqF ) =K1 I F =K2 −1
γ F = sin (−.05 + .052 + 8K12 K 2 2 / (4 K1 K 2 ))
(5)
From the equation (5), to find the optimum γ F , the two variables K1 & K 2 have to be controlled. Here, K1 depends on the motor saturation effects which are the difference of the d-q inductances. To have less effect of this parameter the variation is assumed within the threshold. On the other hand, the K2 is the current under fault condition which can be controlled using proper current feedback.
Jpw = 5P/ 4 ª¬λPM IF cos γ F + (LdF − LqF )I F cos γ F I F sin γ F º¼ − Tl − Bw (7) Having side changes and Considering following assumptions, 1) magnetic saturation is not reached, 2) no frictional torque, 3) variable load torque, and 4) constant fault current magnitude, equation (7) becomes
P3 = P1 cos γ F + P2 cos γ F sin γ F
Where, P1 = λPM IF , P2 = (LdF − LqF )I F and P3 = (Jpw + Tl ) X 4 / 5P Here, the constant value, P2 is attributed to the saturation effects and P3 is attributed to the dynamic effect (load and motor inertia). The trigonometric equation in (8) can be further written as 2 P22 x4 + (P12 − P22 ) x2 − 2PP (9) 1 3 x + P3 = 0 Where x= cos γ F , the solution of the polynomial in equation (9) can be written as γ F = cos−1 ( f ( p3 )) (10) Hence, the load variation also changes the operating point of the phase advance at different fault conditions which can be utilized to get maximum amount of torque.
VI. D-Q INDUCTANCE CALCULATION UNDER DIFFERENT FAULT CONDITIONS To measure the d-q inductances, the self and mutual inductances can be utilized under different open phase fault conditions which use the proposed d-q transformation matrix shown in equation (11). Here J is 2/(A+B+C+D+E) where A, B, C, D, E represent the open phase fault existence. Under normal condition, A=B=C=D=E=1, otherwise each of them are stated as zero under fault condition. The phase advance ( γ ) has been added in the matrix which are calculated under different fault conditions. Laa = Lk + Lg cos 2θ
V. PHASE ADVANCE BASED ON MECHANICAL DYNAMIC BEHAVIOUR The mechanical dynamic equation is given by Jpw = Te − Tl − Bw
(8)
Lbb = Lk + Lg cos 2(θ − 2π 5) Lcc = Lk + Lg cos 2(θ − 4π 5)
(6)
(12)
Ldd = Lk + Lg cos 2(θ − 6π 5) Lee = Lk + Lg cos 2(θ − 8π 5)
Where, p = d dt , Tl is the load torque, B is the coefficient of friction and J is the motor inertia constant. Using the torque equation , the dynamic equation (6) becomes,
ª 2π 4π 6π 2π § · § · § · § ·º « A cos (θ + γ ) Bcos ¨ θ − 5 + γ ¸ C cos ¨ θ − 5 + γ ¸ D cos ¨ θ − 5 + γ ¸ E cos ¨ θ + 5 + γ ¸ » © ¹ © ¹ © ¹ © ¹» « « 2π 4π 6π 2π § · § · § · § ·» + γ ¸ Csin ¨ θ − + γ ¸ D sin ¨ θ − + γ ¸ E sin ¨ θ + + γ ¸» T (θ ) = J « A sin (θ + γ ) Bsin ¨ θ − 5 5 5 5 © ¹ © ¹ © ¹ © ¹» « « » Α Β C D E « » 2 2 2 2 2 ¬ ¼
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(11)
Table. 1. Inductance matrix under different fault conditions.
Single phase fault ª Laa Lab Lac Lad Lae º « Lba Lbb Lbc Lbd Lbe » « » L = « Lca Lcb Lcc Lcd Lce » « » « Lda Ldb Ldc Ldd Lde » « Lea Leb Lec Led Lee » ¬ ¼
Two phase fault (adjacent) ª Laa « Lba « L = « Lca « « Lda « Lea ¬
Two phase fault (non-adjacent)
Lab Lac Lad Lae º Lbb Lbc Lbd Lbe »» Lcb Lcc Lcd Lce » » Ldb Ldc Ldd Lde » Leb Lec Led Lee »¼
4π 6π 8π ) + Lg cos 2(θ − ) + Lg cos 2(θ − ) 5 5 5 2π 6π 8π Lac = Lca = M k + Lg cos 2(θ − ) + Lg cos 2(θ − ) + Lg cos 2(θ − ) 5 5 5 2π 4π 8π Lad = Lda = M k + Lg cos 2(θ − ) + Lg cos 2(θ − ) + Lg cos 2(θ − ) 5 5 5 6π 2π 4π Lae = Lea = M k + Lg cos 2(θ − ) + Lg cos 2(θ − ) + Lg cos 2(θ − ) 5 5 5 6π 8π Lbc = Lcb = M k + Lg cos 2(θ ) + Lg cos 2(θ − ) + Lg cos 2(θ − ) 5 5 4π 8π Lbd = Ldb = M k + Lg cos 2(θ ) + Lg cos 2(θ − ) + Lg cos 2(θ − ) 5 5 4π 6π Lbe = Leb = M k + Lg cos 2(θ ) + Lg cos 2(θ − ) + Lg cos 2(θ − ) 5 5 2π 8π Lcd = Ldc = M k + Lg cos 2(θ ) + Lg cos 2(θ − ) + Lg cos 2(θ − ) 5 5 2π 6π Lce = Lec = M k + Lg cos 2(θ ) + Lg cos 2(θ − ) + Lg cos 2(θ − ) 5 5 2π 4π Lde = Led = M k + Lg cos 2(θ ) + Lg cos 2(θ − ) + Lg cos 2(θ − ) 5 5 Lab = Lba = M k + Lg cos 2(θ −
(13) The self and mutual inductances can be estimated as given in equation (12) and equation (13). Where Lk is the fixed inductance which consisted of leakage and air gap components, Lg is the air gap component of the magnetic field produced by the fundamental space-harmonic component of the stator conductor distribution and M k is the mutual inductances between the phases. Considering those self and mutual inductances modified inductance matrix can be developed as given in Table.1. In the matrix, the diagonal components represent the self-inductances and others components represent the mutual inductance. The matrix takes modified number of rows (R) and columns (C) based on the number of phases fault. For example, for single-phase fault (phase-a), The R-1and C-1 is zero. Therefore, the dimension of the matrix becomes 4x4. For two-phase adjacent fault (phase-a and phase-e) R-1, R-5 and C-1, C-5 becomes zero. Therefore, the dimension of the matrix becomes 3x3. For two-phase nonadjacent fault (phase-b and phase-e) R-2, R-5 and C-2, C-5 becomes zero. By utilizing the inductance matrix and the d-q transformation matrix shown in equation (11) the d-q axis inductances can be calculated theoretically.
From the equation (8) the torque is function of phase advance and saturation factor k where, k is the function of d-q axis inductances. VII. REDUCED TORQUE RIPPLE WITH OPTIMUM PHASE ADVANCE UNDER FAULT CONDITIONS It has been observed that, under fault conditions the torque pulsation increases which creates higher torque ripple in the machine. This phenomenon is also not expected to go beyond a certain threshold. The higher torque ripple is due to the currents harmonics and lowered average torque at open phase fault conditions.
T = f (γ , k) k = f (LdF , LqF )
dT =
∂T ∂T dγ + dk ∂γ ∂k
∂T ∂T dγ + dk dT ∂γ ∂k = Torque ripple = Tavg Tavg
(14)
From the equation (14), it can be seen, utilizing the optimum phase advance and lower magnetic saturation factor the torque ripple can be improved. In conclusion, the torque under fault condition is the function of phase advance and magnetic saturation effects where phase advance itself varies with load. VIII. GRAPHICAL PRESENTATION OF PHASE ADVANCE AND CONTROL SCHEME AT OPEN PHASE FAULT The most advantage of a five-phase machine is that, In case of one or two phase loss fault, the machine can continue the operation as a four phase or three phase machine. In Fig. 4 the three types of phase fault is presented which are the singlephase fault (A=0) shown in Fig. 4(a), two-phase adjacent fault (A, B=0) shown in Fig. 4(b), and two-phase non-adjacent fault (B, E=0) shown in Fig. 4(c). The faulty phases are marked as red line. And the healthy phases are kept as black. In the event of fault, the healthy phases need to be adjusted to maximise the torque output. This phase advance under the different fault condition can significantly improve maximum torque. As the average torque improves the overall torque ripple also decreases. The proposed phase advances ( γ ) under different fault conditions have been shown in equation (15), (16) and
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(17) respectively for single phase open fault, adjacent twophase open fault, and nonadjancent two-phase open fault.
(a)
(b)
The proposed fault tolerant scheme has been shown in Fig. 6. The machine used in the proposed scheme takes the specification as in Table .2. In the block diagram, the d-q axis current has been calculated from the current feedback. Using the equation (9), the phase advance has been calculated. The reference phase advance been compared with the measured phase advance and then added to the rotor position to generate advanced phase current. IX. SIMULATION RESULTS
(c)
The finite element simulation results are shown in this section. The electromagnetic torque has been varied with the proposed phase advanced and calculated the optimal value to provide maximum torque. In Fig.5 the results are shown for normal and single-phase open faults. In 5(a), the maximum torque under normal condition has been calculated at phase advance of 15 degree. Similarly, in Fig.5 (b), under singlephase open fault it has been calculated as 30. Torque under two-phase faults is shown in Fig.7. For two-phase nonadjacent (Fig.7 (a)) and adjacent (Fig.7 (b)) faults the phase advance has been calculated as 110 degree and 50 degree where the torque is maximum. Even in the case of two-phase open fault the toque reduces, it can be maximized by adding the proposed phase advance to the currents.
Fig. 4. Different open phase fault conditions: a) Single-phase open, b) two-phase adjacent open, c) two-phase non adjacent open fault. Table. 2. Specification of PMa-SynRM.
Parameter
Specifications
Number of slot/poles
15/4
Rated current (rms)(A)
15.17
Rated voltage (rms) (V)
67
Power (kW)
3
Rated speed (rpm)
1800
Rated Torque
15
Phases
5
I12 = I1 sin(θ − 720 + γ 1 ) I13 = I1 sin(θ − 1440 + γ 1 )
(15)
I14 = I1 sin(θ − 2160 + γ 1 ) I15 = I1 sin(θ − 2880 + γ 1 )
I 22 = I2 sin(θ − 720 + γ 2 )
(a)
I 23 = I2 sin(θ − 1440 + γ 2 )
(16)
I 24 = I2 sin(θ − 2160 + γ 2 )
I 21 = I2 sin(θ − 00 + γ 3 ) I 23 = I2 sin(θ − 1440 + γ 3 )
(17)
I 24 = I2 sin(θ − 2160 + γ 3 )
Spd ref
PI Id Ref
Sine PWM
(b) Fig.5. FEA simulation results for torque vs phase advance under a) Normal condition, b) Single-phase open fault.
5-phase inverter
PI PI
Phase advance cal
PARK
Encoder
Phase advance ref
Current
Fault current cal Estimate inductances
IPARK
PI
Speed calculation
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(b)
(a)
(c) (b) Fig. 7. FEA simulation results for torque vs phase advance under a) two-phase non adjacent open fault, b) two-phase adjacent open fault.
X. EXPERIMERNTAL RESULTS In this section the experimental results are shown. The 5phase machine has been controlled with complete vector control. The proposed phase advance method has been incorporated in the control strategy. Fig.8 and Fig.9 are showing the torque measurement under different fault conditions at two different loading conditions. Fig. 8 is for 15% rated load and Fig.9 is for 28% rated load condition. In Fig. 8(a) the maximum torque under normal condition has been found as 2.3 Nm at 40 degree phase advance. In the same way, the maximum torque for single-phase open, two-phase non adjacent open and two-phase adjacent open fault have been found at phase advance of 50 degree, 62 degree and 60 degree which are shown in Fig. 8(b), 8(c) and 8(d) respectively.
(a)
(d) Fig. 8. Torque vs Phase advance at 15% rated load, a) Normal condition, b) single-phase open fault, c) two-phase nonadjacent open fault, d) two-phase adjacent open fault.
In Fig. 9, similar results are shown for 28% rated load. Fig. 9(a) is showing the maximum torque has been found at 67 degree phase advance. Similarly, under single-phase open fault, two-phase nonadjacent open and two-phase adjacent open faults the maximum torque have been found at 60 degree, 63 degree and 50 degree which are shown in Fig. 9(b), 9(c) and 9(d) respectively. It is found that, the maximum torque significantly varies with the phase advances at different fault operating conditions which has been shown in finite element analysis as well. Also it is noticeable that, the phase advances also changes with different loading conditions. Hence, to provide maximum torque at different conditions, it is necessary to calculate the phase advance dynamically and generate the advanced currents in the healthy phases. The torque ripple results are not given in this paper which can be assumed to be lower with the proposed phase advance technique as the average torque is increasing.
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been analyzed by using the finite element method. In detail, single-phase open and nonadjacent two-phase open and adjacent two-phase open faults have been taken into account during the calculation. At the open phase faults the generated torque is reduced in the machine. To maximize the reluctance torque in PMa-SynRM the phase advance has been introduced in the control strategy. Using the optimum value of the phase advance the output torque has been maximized. It has been shown that the phase advance varies with the different types of fault and also with different loading conditions. For this reason, it is proposed to adjust the phase advance dynamically in the system to ensure the maximum torque at fault conditions. The experimental results are given using the 5hp dynamo system to validate the proposed theory.
(a)
REFERENCE Suman Dwari and Leila Parsa, “An Optimal Control Technique for Multiphase PM Machines Under Open-Circuit Faults”, IEEE Transactions on industry applications vol. 43, no. 4, July/August, 2007. [2] A. Mohammadpour and Leila Parsa, “A Unified Fault-Tolerant Current Control Approach for Five-Phase PM Motors With Trapezoidal Back EMF Under Different Stator Winding Connections”, IEEE transaction on Power Electronics, vol. 28, no.7, October 2012. [3] A. Mohammadpour, S. Mishra, and Leila Parsa, “Fault-Tolerant Operation of Multiphase Permanent-Magnet Machines Using Iterative Learning Control”, IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 2, no. 2, December 2013. [4] Nicola Bianchi, Silverio Bolognani, and Michele Dai Pré, “Strategies for the Fault-Tolerant Current Control of a Five-Phase Permanent-Magnet Motor”, IEEE Transactions on industry applications, vol. 43, no. 4, July/August, 2007. [5] Leila Parsa, “On Advantage of Multi-phase machines”, IEEE Industrial Electronics Society, IECON, November 2005. [6] S.S.R Bonthu,Jeihoon Baek, and Seungdeog Choi, “Comparison of optimized permanent magnet assisted synchronous reluctance motors with three-phase and five-phase systems”, IEEE Energy Conversion Congress and Exposition (ECCE), 2014. [7] T. J. E. Miller, Alan Hutton, Calum Cossar, and David A. Staton, “Design of a Synchronous Reluctance Motor Drive”, IEEE Transactions on industry applications, vol. 27, no. 4, July/August 1991. [8] N. EaUgrul, W. Soong, G. Dostal, and D. Saxon: “Fault Tolerant Motor Drive System with Redundancy for Critical Applications”, IEEE Power Electronics Specialists Conference,33rd Annual, vol. 3, 2002. [9] Alfio Consoli, Alberto Gaeta, Guiseppe Scarcella, Giacomo Scelba, and Antonio Testa, “HF Injection-Based Sensoreless technique for Fault-Tolerant IPMSM Drives”, IEEE Energy Conversion Congress and Exposition (ECCE), 2010. [10] Suman Dwari and Leila Parsa, “Optimum Fault-Tolerant Control of Multi-phase Permanent Magnet Machines for Open-Circuit and Short-Circuit”, IEEE Applied Power Electronics Conference, APEC, Feb 5-March 1, 2007. [11] Mohammadpour, A., Mishra, S. , Parsa, L., “Fault-Tolerant Operation of Multiphase Permanent-Magnet Machines Using Iterative Learning Control,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 2, no. 2, December 2013. [12] Leila Parsa, A. Toliyat: “Fault-Tolerant Interior-Permanent-Magnet Machines for Hybrid Electric Vehicle Applications”, IEEE Transaction on vehicular technology,vol.56, no.4,July 2007. [1]
(b)
(c)
(d) Fig. 9. Torque vs Phase advance at 28% rated load, a) Normal condition, b) single-phase open fault, c) two-phase nonadjacent open fault, d) two-phase adjacent open fault.
XI. CONCLUSION In this paper, the five-phase PMa-SynRM has been used under fault tolerant control at different faults. Here, two types of open phase faults have been studied theoretically which has
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