Control strategy with minimal controller reconfiguration of fault tolerant polyphase PMSM drives under open circuit fault of one phase F. Baudart, B. Dehez, F. Labrique, E. Matagne, D. Telteu, P. Alexandre
Abstract—Segment polyphase motors gain interest in applications requiring high performance and high reliability due to their high fault tolerance. This paper sums up the generalization of the vector control strategy of such motor and presents a method to adapt with a minimal reconfiguration the control strategy in the case of an open phase failure. Simulations results validate the method. Index Terms—segment polyphase motor, vector control, open phase failure
I. I NTRODUCTION
M
ORE and more electromechanical actuation systems are used in a range of high performance and high reliability applications as in aeronautics. An actuation system consisting in a permanent magnet synchronous motor (PMSM) driving a load via a mechanical transmission and fed by a power electronic converter reaches the high performance criterion because of its high power to mass ratio. But it not necessarily reaches the high reliability one. The weakest point of an electromechanical actuation system is the power electronic. A fault in the power electronic commonly yields to the loss of control of one phase of the motor. Therefore a way to improve the reliability of the system is to make it able to pursue its mission after a fault occurs. That means isolates the fault, prevent it to perturb the remaining healthy parts and to adapt the system to compensate the loss. This is known as the fault tolerance ability. In applications we are considering this implies the reconfiguration of the control strategy of the remaining healthy phases for compensating the faulty one. In 3-phase classic motors the loss of one phase is critical because one third of the system is lost. Improving the reliability of 3-phase motors by a fault tolerance reconfiguration highly decreases the power to mass ratio. In order to reach the same performances with 2 phases the motor and the power electronic have to be oversized by two. Therefore polyphase motors (i.e. motors with more than 3 phases) are more and more considered: the split up of the power minimizes the impact of the loss of one phase and reduces the perturbations on the system. In literature polyphase motors considered for operating in fault mode F. Baudart, B. Dehez, F. Labrique, E. Matagne are all with the Center of research in mechatronics in Universit´e de Louvain, 1348 Belgium (mail :
[email protected]) P. Alexandre and D. Telteu-Nedelcu are with the Soci´et´e Anonyme Belge de Construction A´eronautique (S.A.B.C.A.), Brussels, 1470 Belgium
usually are motors with four[1], five[2], six[7] and seven[8] phases. Motors with higher number of independant phases are barely studied because systems become more complex and more subject to fault occurence. Segment polyphase motors[1] are a type of motor with fault tolerant features. While improving the reliability of the system this type of polyphase motor reaches performances that can be higher than performances reached by similar classic 3-phase motors[3]. Section II explains the main features of that type of motor while Section III overflies the generalization of the vector control for such machines in normal (healthy) mode. When a fault occurs in one phase an adaptation of the current patterns in the remaining healthy phases is needed in order to still develop the same torque as in normal mode. By making the assumption that the fault leads to an open circuit failure of one phase Section IV gives the sinusoidal current patterns in the remaining healthy phases minimizing the Joule’s losses for a given torque reference. The method for obtaining those currents is explained in [10]. Then it is shown how to adapt with minimal controller reconfiguration the vector control scheme to allow the fault mode operation after a fault occurs in any phase. Section V validates the control scheme by simulating an actuation system with a 6-phase segment motor. II. T HE SEGMENT MOTOR The main characteristic of segment motors is the arrangement of the windings. They are wound around one tooth on two. Instead of the classic motor with overlapping windings the segment motor has phases physically isolated from one to another, each located in a segment of the stator core. This improves the reliability of the system. A fault cannot appear between two phases or propagate from one to another. Fig.1(a) shows the basic structure of a 6-phase motor with 5 pole pairs. The number of pole pairs cannot be equal to the number of phases in order to induce in them a balanced set of EMF but the stator pitch and the rotor pitch have to be as close as possible. This yields the basic rule for selecting the number of pole pairs on the rotor side: this number must be equal to the number of phases plus or minus one. The isolation of the windings also almost eliminates the magnetic coupling between phases. [6] gives a ratio mutual on self inductance of −3.5% for a 4-phase machine and [11] a ratio of −4.4% for a 5-phase one. Therefore the n stator
(a) Segment 6-phase motor
(b) Equivalent circuit of one phase Fig. 1.
Fig. 3. Vector representation of currents references for a 6-phase segment motor in healthy mode
phases of a segment motor can be seen as n decoupled circuits, each made of a resistance R, an inductance L in series with a back EMF e induced by the magnets, as shown in Fig.1(b). The electrical equation of phase k is given by: uk = Rik + L
d ik + ek dt
(1)
with k ∈ [1, . . . , n]. With magnets properly shaped we can make the assumption that the back EMF’s vary sinusoidally with the rotor position: i d h k−1 ψ0 · cos θ − 2π (2) ek = θ˙ n dθ where ψ0 is the maximum value of the flux induced by the magnets and θ the rotor electrical position.
The extended transform of Concordia is defined such that it changes the n reference currents into two currents iα and iβ of a fictitious 2-phase motor and n−2 homopolar currents equals to 0. Those n − 2 currents are defined as homopolar because they do not contribute to the torque production for sinusoidal EMF’s machines. Several forms of extended transform of Concordia can be found in literature, as in [4],[5] and [9]. The extended Park transform given by
cos (θe ) − sin (θe ) 0 P = .. .
sin (θe ) cos (θe ) 0 .. .
0 0 1 .. .
... ... ... .. .
0
0
...
0
III. T ORQUE CONTROL IN HEALTHY MODE A. Equations in the Park frame As shown in Fig.2 the vector control needs a torque reference, currents measurement, position and speed measurements in order to align the currents on their reference patterns. The control applies proper voltages to the motor via the power converter feeding it. In healthy mode, with sinusoidal EMF’s and some usual assumptions, reference currents which ensure a constant torque with minimal Joule’s losses are also sinusoidal and isomorphic to their respective EMF’s, as shown in Fig.3 for the 6-phase case. The vector control needs a change of variables in order to obtain reference values independent of the electrical position. This is done by using an extended form of the Concordia and Park transformations.
Fig. 2.
System
0 0 0 .. . 1
(3)
is then applied to transform the two θ-depending currents iα and iβ into id and iq . The n − 2 homopolar currents are kept unchanged. The vector representations of the two consecutives transformations are shown in Fig.4 in the case of the 6-phase machine. The current references become in the Park frame: id,ref = 0 iq,ref = Tref
2 nψ0
(4)
i0j ,ref = 0 with j ∈ [1, n − 2]. Applying those two transformations on eq.(1) and eq.(2) the electrical equations of the polyphase motor are found to be in
Fig. 4. Vector representation of currents changes using the extended Concordia and Park transformations
the following sinusoidal currents reconfiguration if phase 1 is lost: i′1,ref = 0 Fig. 5.
i′2,ref = i2,ref + c2 i1,ref .. .
Vector control in healthy mode
(9)
i′n,ref = in,ref + cn i1,ref Park reference frame: did ud = Rid + L − ωLiq dt r n ˙ diq uq = Riq + L + ωLid + ψ0 θ dt 2 di0j u0j = Ri0j + L dt
(5)
Eventually the equations in the Park frame are the same as for a 3-phase machine with n−2 homopolar equations instead of one. B. Vector control The control scheme built on the system of equations (5) in the Park frame is given in Fig.5. The Concordia and Park matrix respectively named C and P are applied to the measured phase currents to give the following vector: T idq = id iq i01 . . . i0(n−2) (6)
which is then compared to the currents references, given by system of equations (4): 2 idq,ref = 0 Tref nψ 0
0
...
0
T
(7)
The errors between reference and measured currents are managed by a regulator block (detailled in appendix A). That regulator block and a feed-forward compensation estimator of the EMF’s given by: pn ˙ eˆdq = 0 2 ψ0 θ
0 ...
T 0
(8)
define the voltage references in the Park frame. As in the 3phase case the phase voltage references applied to the motor via a power electornic are then computed using inverse Park and Concordia transforms. IV. T ORQUE CONTROL INCLUDING AN OPEN PHASE FAULT MODE
A. Reconfiguration of reference currents and voltages in fault mode Authors of [10] show that the loss of one phase can be compensated by adding in the remaining phases a component of current isomorphic to the current of the lost phase. The loss of phase 1 for example can be compensated by adding current components isomorphics to i1 to the currents which should flow in normal operation in the still active phases. This yields
With the constraint of having the sum of currents equal to zero (mandatory in a case of a star connection of the windings) ck coefficients minimising Joule’s losses are given by: 1 k−1 1 + 2 cos 2π (10) ck = n n−3 with k ∈ [2, . . . , n] and n the number of phases. An n-leg inverter with an additionnal neutral leg or n single phase inverters do not require that constraint of having a zero sum of currents. In that case ck coefficients are given by: 2 k−1 2π (11) cos ck = n n−2 The constraint of the zero sum of currents yields a more important reconfiguration than without, increasing Joule’s losses and the VA-rating. Joule’s losses could still be reduced by releasing the constraint of keeping the currents sinusoidal. However this decreases is not substantial and the controllers need then a higher bandwith. A more detailled comparison of the three solutions is made in [10]. In order to get the reference currents with any faulted phase we define for each phase a state factor αj equal to 0 when the phase is healthy and equal to 1 when phase j goes in open circuit. Using that state factor, one can define the link between the reference currents in normal and fault operation mode in one expression: ′ i1,ref
i1,ref
−α1
i′2,ref i2,ref α1 c2 . = . + . .. .. .. i′n,ref
in,ref
α1 c n
α2 c n −α2 . .. α2 cn−1
... ... .. . ...
i1,ref αn c 2 αn c3 i2,ref . . .. .. in,ref −αn
(12)
which we rewrite in a compact form as: i′ref = iref + Ck iref = (I + Ck ) iref
(13)
with I the n × n identity matrix. If all state factors are equal to 0 the expression gives the currents in healthy mode. If the state factor of one phase is equal to 1, the reference current of that phase is set to 0 and the components needed to operate in fault mode are added to the other current references. Adapting eq.(1) reference voltages in fault mode are given by: u′ref = Ri′ref + L
d ′ i + eref dt ref
(14)
Fig. 7. Currents adaptations of a 6-phase motor with phase 1 in open circuit in steady state Fig. 6. Voltages adaptations of a 6-phase motor with phase 1 in open circuit in steady state
as shown in Fig.7. Using eq.(13) yields:
C. Adaptation of the vector control d (I + Ck ) iref + eref dt d + eref + Ck Riref + L iref dt (15)
u′ref = R (I + Ck ) iref + L = Riref + L
d iref dt
and eventually: u′ref = uref + Ck (uref − eref )
(16)
This expression allows for computing reference voltages in any faulted mode using reference voltages in normal mode. B. Illustration with a 6-phase machine reconfiguration Assuming phase 1 is lost and the motor is fed by an nleg inverter, we need to compensate the loss of phase 1 using eq.(16) with ck coefficients given by eq.(10), in order to get a zero sum of currents. This yields: u′1,ref = 0 2 u′2,ref = u2,ref + (u1,ref − e1,ref ) 3 u′2,ref = u3,ref (17) 1 u′2,ref = u4,ref − (u1,ref − e1,ref ) 3 u′2,ref = u5,ref 2 u′2,ref = u6,ref + (u1,ref − e1,ref ) 3 as shown in Fig.6. These reconfiguration of voltages applied to the machine should implie the currents flowing in the machine to follow those reference currents: i′1,ref = 0 2 i′2,ref = i2,ref + i1,ref 3 i′2,ref = i3,ref (18) 1 i′2,ref = i4,ref − i1,ref 3 i′2,ref = i5,ref 2 i′2,ref = i6,ref + i1,ref 3
The adaptation consists in keeping the same controller as in normal mode by acting in fault mode: • on the measured currents in order to deliver to the controller currents measurements similar with those obtained in normal operation mode; • on the voltages generated by the controller which are corresponding to thoses needed in normal operation mode to adapt them to the fault mode conditions. Inversing eq.(13) allows using a feed forward action to build from the measured currents i′ and the reference ones, currents ”measurements” i similar to those which should be obtained in normal operation mode: i = i′ − Ck C −1 P −1 idq,ref
(19)
Adaptation of the measured currents is shown in Fig.8 outside left the shaded box which is the vector control in normal mode as shown in Fig.5. For the 6-phase machine with phase 1 lost, measured currents at steady state should tend to as shown in the right diagram of Fig.7. The compensation terms on currents rebuild them to be as in normal mode, as shown in the left diagram of Fig.7. The vector control with a fault operation mode is given in Fig.8. Similarly in order to obtain the reference voltages in fault mode two feed forward terms are added to the voltage references generated by the controller. According to eq.(16) they are based on the reference voltages generated by the controller and the estimated values of the back EMF’s: u′ = u + Ck u − Ck C −1 P −1 edq,ref
Fig. 8.
Vector control with adaptation for running in fault mode
(20)
25
20
20 Torque [N.m]
Torque [N.m]
25
15
10
5
0 0
10
5
10
20
30
40 time [ms]
50
60
70
0 0
80
Fig. 9. Torque response to a torque reference step with non-adapted vector control
Fig. 11. control 20
15
15
10
10
5
5
0 −5 −10
Fig. 10.
20
30
40 time [ms]
50
60
70
80
0 −5 −10
−15 −20 0
10
Torque response to a torque reference step with adapted vector
20
Currents [A]
Currents [A]
15
−15 10
20
30
40 time [ms]
50
60
70
80
Current responses with non-adapted vector control
Adaptation of the reference voltages in fault mode is shown in Fig.8 outside right the box corresponding to the controller. In the former case of the 6-phase machine, voltages on the left diagram of Fig.6 are the ones given by the vector control at steady state. If phase 1 goes in open circuit fault the voltages are adapted by the compensations terms and are then given by the right diagram of Fig.6. V. VALIDATION BY SIMULATION A. Simulation data In order to validate the adapted vector control a simulation model of a 6-phase segment motor with 5 pole pairs has been builded in a Matlab Simulink environment. Motor parameters are given in appendix B. In this model the motor is fed by a 6-leg inverter and the mechanical load consists of an inertial torque J ω, ˙ a viscous torque f ω and a friction torque Cr, ω being the mechanical speed. The inverter bridge generates PWM at a sampling frequency of 12.5 kHz. The same sampling is used in the discrete adapted vector controller. The digital implementation of the torque controller is taken into account by a delay of one sampling (0.08 ms) in the measurements. B. Simulation results Figure 9 shows the torque response of the 6-phase motor to a step of 17 Nm on the torque reference value, the motor being initially at standstill. A small torque ripple at the PWM frequency is due to the use of an inverter for feeding the motor. The torque control is the one represented in Fig.5, i.e. without adaptation. After 30 ms phase 3 is opened. As the torque control is not adapted it is only able to compensate the mean error by increasing the amplitudes of currents in the still active phases as shown in Fig.10. The pulsatory torque component is not compensated.
−20 0
10
Fig. 12.
20
30
40 time [ms]
50
60
70
80
Current responses with adapted vector control
Figure 11 shows the same simulation with the adapted vector control. The fault is supposed to be detected after a delay of 10 ms. State factor α3 of phase 3 is then set to 1 in order to adapt voltages applied to the motor. The adaptation yields a torque response without the pulsatory component. As shown in Fig.12 the compensation feed-forward terms added in fault mode adapt the reference currents such that their vector representation is the one represented in Fig.7 shifted such that phase 3 is the faulted one. VI. C ONCLUSION Segment polyphase motors undeniably have high potentiality in high performance and high reliability applications. The extensions of the Concordia and Park transformations give an easily implementable solution for controlling the system in the Park frame with equations very similar to the well-known 3-phase ones. Another interesting feature of the polyphase segment motor is the possibility of running with a phase lost in open circuit by reconfiguration of the currents in the remaining phases without dramatically deteriorating the performances. Using sinusoidal currents in fault mode allows to keep the vector control by compensating the added components in fault mode with feed-forward terms which are directly determined by reference values. The main interest is keeping the vector control identical either a phase is failed or not, and thus minimizing the reconfiguration of the controller. Further works will validate the results with experimental setup. A PPENDIX A. Regulation scheme and regulation parameters Figure 13(a) shows the current controller scheme in the Park frame for the 6-phase machine. It uses a PI controller and feedforward terms for decoupling the equations in d and q. With
a star connection only five currents are independent so one homopolar controller can be removed. In fault mode another current is lost so another controller can be shunted but this is not mandatory. Figure 13(b) shows the discrete PI controller. For the simulations the proportional gain Kp is 1.08, the integral gain Ki 210 and the time constant T s 8e − 5 sec. B. Motor parameters Phase resistance R = 0.268 mΩ Self inductance L = 1372 µH Mutual inductance M = 55 µH Amplitude of induced flux φ0 = 0.0496 Wb Rotor inertia J = 33 kg·cm2 /rad Viscous friction f = 15 N·cm/(rad/s) Resistive torque Cr = 1 Nm ACKNOWLEDGEMENT Authors affilied to the universit´e catholique de Louvain want to thank S.A.B.C.A. for having funded this research. R EFERENCES [1] B. Mecrow, A. Jack, D. Atkinson, S. Green, G. Atkinson, A. King, et B. Green, ”Design and testing of a four-phase fault-tolerant permanentmagnet machine for an engine fuel pump,” Energy conversion, ieee transactions on, vol. 19, 2004, p. 671-678. [2] L. Parsa et H. Toliyat, ”Five-phase permanent-magnet motor drives,” Industry Applications, IEEE Transactions on, vol. 41, 2005, p. 30-37. [3] M. Rezaei, ”A comprehensive comparison between non-segment and segment polyphase permanent magnet excited synchronous machine”, Proc. ICEM 2008 [4] M.A. Abbas, R. Christen, et T.M. Jahns, ”Six-Phase Voltage Source Inverter Driven Induction Motor”’, Industry Applications, IEEE Transactions on, vol. IA-20, 1984, pp. 1251-1259.
[5] A. Tessarolo, ”On the modeling of poly-phase electric machines through Vector-Space Decomposition: Numeric application cases”, Power Engineering, Energy and Electrical Drives, 2009. POWERENG ’09. International Conference on, 2009, pp. 524-528. [6] C.H. Sneessens, T. Labb´e, F. Baudart, F. Labrique et E. Matagne, ”Model and torque control of a five phase PMSM synchronous motor using tooth concentrated windings technology”, 8th International Symposium on Advanced Electromechanical Motion Systems, 2009 [7] B. McCrow, A. Jack, D. Atkinson, J. Haylock, ”Fault tolerant drives for safety critical applications,” New Topologies for Permanent Magnet Machines (Digest No: 1997/090), IEE Colloquium on, 1997, p. 5/1-5/7. [8] D. Casadei, M. Mengoni, G. Serra, A. Tani, L. Zarri, ”Comparison of different fault-tolerant control strategies for seven-phase induction motor drives”, Power Electronics and Applications, 2009. EPE ’09. 13th European Conference on, 2009, p. 1-9 [9] F. Baudart, F. Labrique, E. Matagne, D. Telteu, P. Alexandre, ”Calcul d’une matrice de Concordia e´ tendue pour des machines synchrones polyphas´ees segment´ees en marche normale et en marche d´egrad´ee”, Sixi`eme e´ dition de la conf´erence Electrotechnique du Futur, EF’09, 2009, Compi`egne [10] F. Baudart, B. Dehez, F. Labrique, E. Matagne, ”Optimal sinusoidal currents for avoiding torque pulsation after the loss of one phase in polyphase SMPM synchronous motor”, 20th International Symposium on Power Electronics, Electrical Drives, Automation and Motion, SPEEDAM’10, 2010, Pisa. [11] B. Mecrow, A. Jack, J. Haylock, J. Coles, ”Fault-tolerant permanent magnet machine drives”, Electric Power Applications, IEE Proceedings Volume 143, Issue 6, 1996 Page(s):437 - 442 Franois Baudart was born in Belgium, on June 25, 1985. He graduated in electromechanical engineering from the Ecole Polytechnique de Louvain (EPL), Louvain-la-Neuve, in 2008. He is currently researcher in the center for research in mechatronics at the same university working toward a PH’D. Bruno Dehez was born in Uccle, Belgium, in 1975. He received the degree in electromechanical engineering and the Ph.D. degree from the Universit´e catholique de Louvain (UCL), Louvain-la-Neuve, Belgium, in 1998 and 2004, respectively. He is currently Senior Researcher and professor in the Centre for Research in Mechatronics (CEREM) at the Ecole polytechnique de Louvain (EPL). His research interests are in the field of dedicated actuator design. Paul Alexandre was born in Ixelles, Belgium, in 1964. He received the degrees ”Ing´enieur civil e´ lectricien” (1987), ”Ing´enieur civil en Automatique” (1989) and ”Docteur en Sciences appliqu´ees” (1997) from the Universit´e Libre de Bruxelles (ULB). He is currently Technical Group Manager at the Mechatronic Department at ”Soci´et´e Anonyme Belge de Constructions A´eronautique” (SABCA) company. Dan Telteu-Nedelcu was born in Craiova, Romania, in 1973. He received the degrees ”Inginer” in electromechanical field (1998), from University of Craiova and ”Docteur en Sciences appliqu´ees” (2004) from the Universit´e catholique de Louvain (UCL). He is currently with the Mechatronic Department at ”Soci´et´e Anonyme Belge de Constructions A´eronautique” (SABCA) company. Francis Labrique was born in Maurage (Belgium) in 1946. He received the degrees ”Ing´enieur civil e´ lectricien” (1970) and ”Docteur en Sciences appliqu´ees” (1983) from the Universit´e catholique de Louvain (UCL). He is currently professor in the Centre for Research in Mechatronics (CEREM) at the Ecole polytechnique de Louvain (EPL). His research activities are in the pole of power electronics and digital control of electrical drives.
(a) ’reg’ block Fig. 13.
(b) ’PI’ block Blocks used in vector control
Ernest Matagne was born in Assesse, Belgium, in 1947; he received the ”Ing´enieur Civil e´ lectricien” and ”Docteur en Sciences Appliqu´ees” Degrees from the Universit´e catholique de Louvain (UCL), Louvain-la-Neuve, Belgium, in 1971 and 1991 respectively. He is currently professor in the Centre for Research in Mechatronics (CEREM) at the Ecole polytechnique de Louvain (EPL). His current research applications are about machine and other magnetic devices design and control.
