Original articles
Optimal current waveforms for torque control of permanent magnet synchronous machines with any number of phases in open circuit

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Abstract

Polyphase permanent magnet synchronous motors are well suited for electromechanical actuation systems demanding a high level of reliability. They are indeed able to run on a reduced number of phases and therefore to make the actuation system fault tolerant. The paper introduces a Lagrangian formulation for determining the optimal waveforms of the phase currents allowing the motor to develop the needed torque with one or more than one phase lost in open circuit. The method is then adapted to find sub-optimal sinusoidal currents in fault tolerant mode for machines with sinusoidal EMFs. Eventually the paper compares for this type of machine the oversizing imposed for maintaining the performance unchanged in fault tolerant operation mode, with or without imposing to the currents to remain sinusoidal.

Introduction

In aerospace applications electromechanical actuators trend to replace hydraulic ones in the frame of more electrical and power optimized aircraft concepts [8], [16]. The most common way to reach the reliability level needed in these applications relies on a passive redundancy using two actuators in parallel, one active the other one in standby. In case of a failure of the active actuator, a clutch system allows to disconnect it and to switch to the other one. A less costly and less bulky solution consists in designing actuators able to withstand the most probable types of fault without loosing their ability to perform their mission [3], [10], [14], [18], [23], [24], [25], [27].

Fig. 1 gives a schematic view of the type of the electromechanical actuation system under study. A permanent magnet synchronous machine ensures the high dynamics needed for actuating the load via a mechanical transmission system. The motor is fed by a PWM voltage sourced inverter taking its power from a DC bus or a battery. An inner control loop regulates the torque developed by the motor by imposing the waveforms of the currents flowing in its phases on the basis of a torque reference and of the rotor position. The torque reference comes from an outer loop integrating a position and/or a speed control depending on mission profile.

Even if, as far as we know, no precise statistics are available in the literature, it appears that with a proper design of the mechanical parts (bearings, transmission system) [2], of the motor [9], [12], [22], and of the control electronics [13], the main hazard of failure lies inside the power electronics and may be limited to the loss of feeding of some phases by using a proper architecture for the power converter [1], [21].

The reliability of the actuation system can then be significantly increased as it is possible to palliate such a fault. Indeed the performance of the actuation system can be kept unchanged by a proper reconfiguration of the control of the currents flowing in the remaining phases. This reconfiguration, leading to a fault tolerant operation, mainly relies on a redefinition of the link between the motor phase currents and the torque after fault occurrence.

A general expression of the optimal currents in normal operation mode is given in Refs. [20], [26] on the basis of a Lagrangian formulation. Currents in fault tolerant mode are computed in Refs. [7], [11], [15], [17] but only for machines having a given number of phases (ranging from 4 to 8 or 9) by using some tricks or approximations which are only valid for the number of phases considered. It is therefore worthy to extend the Lagrangian formulation to the determination of the optimal currents in fault tolerant operation mode. This extension must be carried out by considering that the sum of the currents must be zero-valued when the motor phases are star-connected with an isolated neutral point.

In particular for machines with sinusoidal EMFs, the Lagrangian formulation of the optimal currents yields sinusoidal currents for normal operation mode, but not for any fault tolerant mode. Hence, for that type of machines, in order to make the current control easier [5], [6], it can be interesting to reformulate the problem in order to obtain sinusoidal currents both in normal and fault tolerant operation mode, even if this implies that the solution found is suboptimal.

Eventually it is useful to determine the increase of Joule's losses and of motor phase peak current resulting from the reconfiguration of the currents in fault tolerant operation mode. Hence one could evaluate the oversizing of the motor and of the power electronics that results from those increases for any of the possible solutions.

The above considerations explain the content of this paper, which is an extended and enhanced version of [4]. In Section 2 we recall how to obtain with a Lagrangian formulation the optimal currents in normal mode, what may be the number of phases or the shape of the EMFs, with or without the constraint of a zero-valued sum of the currents.

In Section 3 we extend the formulation to fault tolerant operation mode for any number of lost phases.

In Section 4, for machines with sinusoidal EMFs, by limiting the analysis to the loss of one phase, we reformulate the problem under the constraint of keeping the currents sinusoidal both in normal and fault tolerant operation modes.

In Section 5, for the case of machines with sinusoidal EMFs, we determine, by considering the most usual case of a single fault, the increase of the Joule's losses in the motor and of the phase current peak value in order to be able to evaluate the price to pay for making the actuator fault tolerant.

Section snippets

General expression of optimal currents

Surface mounted permanent magnet segment synchronous motors with small slot openings, as those considered in this paper [19] (Fig. 2) have very small reluctant and cogging torques. These torque components may therefore be neglected. Hence the torque expression can be reduced to its electrodynamic component, i.e. the component coming from the interaction of the stator currents with the fluxes produced by the magnets, yielding for a n-phase motor:

T=k=1nikfkwith ik the current in phase k and fk

General expression of the optimal currents

When we loose the feeding of some phases, the currents in these phases go to zero. So the terms corresponding to these currents disappear in Eq. (1). Therefore as the optimal currents are proportional to the fk, a simple trick for finding the optimal currents in fault tolerant mode consists in setting to zero the fk's corresponding to the lost phases. This yields:

ik=Treffkκ=1nfκ2withfk=0forlostphases.

This yields:

  • Current references in the faulted phases equal to 0. This is in

Sinusoidal current expressions in fault tolerant mode for machines with sinusoidal EMFs

As shown in Fig. 4(a)–(d), for machines with sinusoidal EMFs, the optimal currents in fault tolerant mode lose the sinusoidal shape they have in normal operation mode. This section determines for this type of machine the expressions of suboptimal currents which still aim to develop the needed torque while minimizing Joule's losses, but with the constraints of remaining sinusoidal in fault tolerant mode. The analysis is restricted to a single fault operation.

Sizing issues for machines with sinusoidal EMFs

This section evaluates for machines with sinusoidal EMFs two factors allowing to perform an estimation of the oversizing needed for reaching the same performance in fault tolerant mode as in normal mode: the increase of the Joule's losses and the increase of the peak value of the currents. A more precise determination of the oversizing should include supplementary aspects among those:

  • the influence of the increase of the peak values of the currents on the iron core saturation and the resulting

Conclusion

The reliability of electromechanical actuation systems based on polyphase PM synchronous motors can be substantially increased if the control of the torque developed by the motor includes the possibility to face the loss of feeding of one phase or more than one phase. Indeed, as it has been stated in the introduction, with a proper design of the motor, of the mechanical parts and of the control and power electronics, this is the most probable risk of failure.

In PM synchronous motors, the

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