Open-Phase Fault-Tolerant Pole Transition Control of an Asynchronous Variable-Pole Machine Using Harmonic Plane Decomposition

Automotive applications are revisiting the use of induction machines (IMs) as magnet-free propulsive solutions due to their intrinsic robustness and reliability. Special multiphase configurations are under investigation to reduce the losses further and fulfill the stringent energy-efficiency and compactness requirements of the automotive industry. One of these configurations is known as variable-pole machines (VPMs), which allows the number of magnetic pole pairs to change on the fly. These machines can stretch the torque-speed operating region, exploit the maximum torque capability, and exhibit competitive efficiency. Although fault tolerance has been widely explored for multiphase machines, the same cannot be said for VPMs, because, until recently, complete models to describe their dynamics under any condition, including magnetic pole changing and fault occurrences, were unavailable. This article presents a post-fault control strategy for VPMs with an open-phase fault (OPF), which can operate during pole changing and address the issue of fault-tolerant operation. The effectiveness of the control system is verified by experimental tests carried out with an 18-phase variable-pole IM prototype.

x s Stator variables.x 123 Variables in the fundamental reference frame.
x αβ0 Variables in the stationary αβ0 reference frame.
x σ Leakage quantity in the inverse-model.x dq0 Variables in the rotating dq0 reference frame.
x h Harmonic plane.

x M
Magnetizing quantities in the inverse-model.

Variables ᾱ
Complex number representing pitch angle between two neighboring minimum windings.δ Pitch angle of the minimum winding.C [.] Core Clarke transformation matrix.

T a→b
Transformation matrix from a to b. x [.] Space vector quantity in

I. INTRODUCTION
V ARIABLE-POLE machines (VPMs), as a subcategory of multiphase electrical machines (MPEMs) [1], [2], [3], [4], are gaining popularity in applications where the torque-speed demand is very wide, such as traction applications [1], [2], [3], [4].VPMs are induction machine (IM)-based systems that can electronically change the number of pole pairs on the fly without hardware reconfiguration, such as an electric gearbox.The different phase-pole configurations (PPCs) allow the fulfillment of the drive cycle demands with the potential to be equally or more efficient than synchronous machine counterparts [5].Fig. 1 qualitatively summarizes the enlarged operation area of VPMs.Their characteristics fit well with two main demands of typical traction applications [5], [6].
1) High torque at low speeds can be provided by configurations with a high number of pole pairs.2) High efficiency at high speeds and low torque can be provided by configurations with a low number of pole pairs.A study using finite-element (FE) simulations shows that a VPM is smaller than a fixed-pole IM, for the same driving cycle demand of a long-haul truck [7].Other FE simulations show that VPMs can achieve high efficiency, especially in cruising operations [8], [9].
There are different VPM designs in the literature, e.g., starter generators that use six-phase IMs to provide high-torque capability for short periods at low speed for cranking [10], [11] or propulsion motors with toroidal single-slot stator windings [12], [13], [14].Regardless of the design, the absence of a modeling approach that could describe the behavior of VPMs during a pole-phase reconfiguration was a major drawback.This issue was addressed in [15] and [16], where a unified model independent of the PPC was presented as a solution for modeling and controlling the VPMs.In this approach, called harmonic plane decomposition (HPD), individual spatial harmonics are modeled and controlled, thus shaping the magneto-motive force (MMF) distribution of different PPCs with a control structure independent of the PPC.
The next step in VPM drive development is to include fault-tolerant features.Heavy-duty electric vehicles require a higher level of reliability as unscheduled downtime and cost of ownership are major concerns.In the event of a fault, it is important to have a "limp-home" operation until proper maintenance can be carried out, which can significantly reduce costs compared with an immediate shutdown.In simpler terms, the vehicle should be able to travel to the nearest workshop instead of stopping immediately.
The inherent multiphase configuration of VPMs offers additional degrees of freedom, which can be exploited to increase the torque density and the energy efficiency of the entire drive [17], [18], [19].These additional degrees of freedom potentially enable true fault tolerance (TFL), which for VPMs means the following.
1) Constant torque must be exerted even in the event of a fault.
2) The drive can retain the pole-changing capability to achieve the enlarged mechanical operation area.This article analyzes the issue of open-phase faults (OPFs), which are more likely to occur in a VPM due to the increased number of individual phases.With more terminals and inverter legs exposed to mechanical failures, internal winding ruptures, welding problems, or converter electrical faults, OPFs are a major concern in power systems [20].
The literature shows different approaches to post-fault controllers (PFCs) in IM-based MPEMs [4].Either a model of the healthy machine or a reduced-order model of the faulty machine may be used.The reduced-order model may require adapting the transformation matrix and the modulation strategy [21].Also, switching from the healthy model to the faulty model creates a discontinuity.
Following the idea of model continuity, the healthy model is more advisable for VPMs [15], [16], but the faulty operation requires injecting a compensating current into the non-excited harmonic planes.However, the assumption of perfect sinusoidal windings does not hold for VPMs.A deeper analysis of the post-fault current references shows three different optimal strategies, i.e., minimum copper loss, minimum peak current, and minimum torque ripple [22], [23].
Many implementations are possible for the controllers regulating the post-fault currents [4]: hysteresis control, fuzzy logic or sliding mode control, model predictive control (MPC), and dual PI or PR control.The last two are most suitable for a PFC with a healthy model, especially due to their simple implementation.The idea of combining them into a proportional-integral-resonant (PIR) controller seems natural, so this solution has already been used for a PFC with minimum torque ripple [23].For completeness, it is worth noting that direct torque control (DTC) may also be applied for a PFC with a two-level [24] or three-level inverter [24].
The focus of this article lies on the pole-changing capability under faulty conditions.To the best of authors' knowledge, no literature investigates this aspect.
If pole changing was not possible during fault conditions, the drive would lose a significant part of the operating range illustrated in Fig. 1.Therefore, this article illustrates the structure of a PFC with current injection for a VPM.Furthermore, the current reference for pole transition ensures the fault-tolerant operation with minimum copper losses.The analysis is based on the HPD, which is briefly described in Section II.Section III describes the fault-tolerant strategy for an OPF.Section IV illustrates its implementation using PIR controllers.Section V presents the experimental validation performed on an 18-phase VPM.Both steady-state performance and dynamic performance are shown for pole reconfiguration to demonstrate the validity of the developed approach.

II. HARMONIC PLANE DECOMPOSITION MODEL
The HPD theory provides the foundation for modeling a VPM [15], [16].In contrast to the state-of-the-art vectorspace decomposition (VSD), where the rank of the model is dependent on the number of phases, the HPD creates a unified model for all possible PPCs of a VPM, thus facilitating the pole transitioning control.

A. Transformation Matrices
Utilizing the 123 fundamental reference frame is advantageous for MPEMs with multiple PPCs, such as VPMs, because it is not influenced by the hardware connections or the PPC [25].The magnetic axes of the practical abc reference frame are mapped into axes with phase angles in the range [0, π ) for the 123 fundamental reference frame.
The model dimension depends on the number n mw of minimum (or elementary) windings, which can be regarded as building blocks.For example, Fig. 2 shows the distribution of the magnetic axes in the 123 fundamental reference frame for an IM with n mw = 18.From now on, n mw is assumed even.
The Clarke transformation matrix for the VPM from the 123 fundamental reference frame to the αβ0 stationary reference frame is shown in (1), as shown at the bottom of the page, [15], [16].
The generalization of the Park transformation, which transforms a space vector x αβ0 from the stationary αβ0 reference frame to the synchronous dq0 reference frame, is represented by the block diagonal matrix (2), such that the main-diagonal blocks are 2 × 2 square matrices and all off-diagonal blocks Fig. 2.
Distribution of the winding magnetic axes in the fundamental 123 reference frame.are zero matrices Since the harmonic planes are independent, the angle θ h of rotation in each harmonic plane depends on the field-oriented control strategy.

B. Model Equations
The equivalent circuits of each harmonic plane, shown in Fig. 3, can be defined by using the Clarke transformation in the HPD theory Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
The torque equation of VPMs in HPD is as follows: where the dot product operator "•" of two vectors is defined as the real part of the product between the former vector and the complex conjugate of the latter.Since the HPD shares the same mathematical roots as the VSD, similar assumptions can be applied.These include magnetic linearity and negligible interplane cross coupling.Therefore, the harmonic planes can be considered independent [26].As the remaining part of this article focuses only on the stator quantities, the subscript "s" will be omitted in all stator currents and voltages for simplicity.

III. OPEN-PHASE FAULT
When an OPF occurs in the k f th winding (k f ∈ {1, 2, . . ., n mw }), the corresponding current becomes zero, and the following constraint must be satisfied: The consequence is that the space vectors of the stator currents īαβ0,h (h = 1, 3, . . ., n mw − 1) are not independent anymore.However, to preserve the motor operation during the OPF, it is necessary to maintain the same MMF distribution in healthy and faulty conditions.Since this depends on the torque-producing harmonic plane p that is currently excited, the current reference, īref αβ0, p , must not be altered.During the fault, (5) cannot be fulfilled if the current references in all other non-excited harmonic planes are zero.In other words, to satisfy (5) and maintain the same operating conditions, at least one current space vector īαβ0,h in a harmonic plane other than plane p must not be zero.Many solutions to (5) that keep the MMF and, thus, īref αβ0, p unaltered are possible.This article considers the common solution that minimizes the stator copper losses in the post-fault operation [22].From (5), it is possible to express the current space vectors of the non-excited harmonic planes (h ̸ = p) in terms of the excited harmonic plane (h = p) as follows: The instantaneous stator copper losses, expressed in terms of space vectors, can be calculated as follows: Therefore, to minimize the stator copper losses, it is necessary to minimize the sum of the squared modules of each stator current space vector.To calculate the minimum value, it is convenient to decompose each reference current space vector īαβ0,h into two components (X and Y ) Substituting ( 8) into (6) leads to the following equation: As can be seen, the Y components disappear in ( 9), so they are set to zero to minimize (7), and it is only necessary to determine the X component of īαβ0,h with h ∈ {1, 3, . . ., n mw − 1}.
It can be verified that the optimal solution is when all the components are equal.Thus, the resulting space vectors of the reference currents in the αβ0 stationary reference frame for the non-excited harmonic planes are as follows: In steady-state conditions, the current space vector īαβ0,p must rotate on a circular trajectory with a constant angular frequency ω s, p even during a fault.As shown in Fig. 4, the current space vectors īαβ0,h (h ̸ = p) move along segments with directions ϑ h,k f equal to where ϑ h,k f is a function of the harmonic plane order h and the index k f of the faulty winding.Finally, it is noted that the same strategy for open-fault detection applies to OPFs and open-switch faults (OSFs) [27].Similarly, the presented fault-tolerant pole-transition strategy can be applied to OSFs.
IV. CONTROL SCHEME Fig. 5 depicts the structure of the proposed post-fault control system in all harmonic planes.Area (a) refers to the torque-producing harmonic plane p, and area (b) refers to the generic non-torque-producing harmonic plane h (h ̸ = p) in post-fault operation.In the torque-producing plane p, the motor speed is adjusted by a PI controller, which generates a torque request.Depending on the rotor flux level and the desired torque, the control system calculates the reference current vector īref dq0, p , which is tracked by PI controllers in the dq0 reference frame synchronous with the rotor flux.The angular frequency ω s, p is calculated by an observer, i.e., the so-called modified current-voltage model (MCVM) [28].
The space vectors of the reference currents īref αβ0,h (h ̸ = p) in the non-excited harmonic planes, which do not contribute to the electromagnetic torque (non-torque-producing planes), Fig. 5.
Full speed and current control structure in post-fault operation for (a) torque-producing harmonic plane p and (b) non-torque-producing harmonic plane h ̸ = p.are calculated, as explained in Section III.As the references īref αβ0,h are pulsating on straight lines, removing the Park transformation and using a PIR controller simplify the implementation compared with the dual PI controllers.
Since the analysis of the fault angles in ( 11) is independent of the excited harmonic plane p, the post-fault analysis holds even during a pole transition [15].During a transition from p 1 to p 2 pole pairs, both harmonic planes p 1 and p 2 are excited simultaneously.Hence, the current references for the nonexcited harmonic planes must contain the pulsating currents for both numbers of pole pairs p 1 and p 2 .Moreover, if the number of non-excited harmonic planes decreases, the fault current must be distributed among fewer planes.As a result, the current references in the non-excited harmonic planes become It is noted that the proposed PFC only affects the non-excited harmonic planes and does not interfere with the pole-changing strategy.This method keeps the computational burden limited as the online calculation of the transformation angles is unnecessary for all non-excited harmonic planes, especially if compared to the control system with two PI controllers and two Park transformations for the positive and negative sequences per harmonic plane [4], [22], [23].Since īref αβ0,h is a combination of sinusoidal signals, a PIR controller with two resonant frequencies must be used to achieve zero error at steady state.Its transfer function is as follows: The resonant frequencies of the PIR controller are dynamically tuned depending on the stator angular frequency, so that ω 0,1 = ω s, p 1 and ω 0,2 = ω s, p 2 .The forward open-loop transfer function can be used to prove the stability of these controllers [29].Furthermore, PIR controllers have shown their suitability for PFC in the case without pole transition [23].
However, the omission of the Park transformation has the limitation that multiple OPFs or other fault-tolerant strategies  lead to reference current space vectors that do not pulsate and, thus, require a different implementation of the control [22], [23].

V. EXPERIMENTAL RESULTS
In this section, experimental tests are carried out on a prototype VPM in steady-state operation and during a pole transition under an OPF in the second winding (k f = 2).Table I lists the electrical parameters of the VPM obtained with an offline parameter estimation method designed explicitly for MPEMs [30].The prototype of the VPM is a modified three-phase two-pole IM, with its original parameters listed in Table II.It is noted that, the machine is a reconfigured three-phase IM, so it is not optimized to operate as a VPM.Nevertheless, although maximum power and efficiency are affected, the prototype can be considered a proof of concept.

A. Experimental Setup
The prototype has 36 independent toroidal slot windings (Q s = 36) and 28 rotor bars (Q r = 28).Each phase of the machine, composed of two opposing toroidal slot windings in series, is fed by a full-bridge converter.The entire inverter consists of 36 two-level legs fed from a common dc voltage.Each leg includes a current measurement as feedback through an AD9249 ADC [32].A Zynq ZC706 Evaluation Kit [33] performs the centralized control at f PWM = 8.00 kHz.Clocking, ADC deserialization, carrier-based PWM, and low-level communication are implemented in the FPGA.Core0 implements the control system, while Core1 runs an Ethernet server for data logging and communication.
The VPM is a two-pole 18-phase IM in the base configuration (n mw = 18) supplied by 18 full bridges.Fig. 6 shows a schematic of the hardware configuration.The OPF is obtained in the faulty winding by opening a breaker in series with it.Fig. 7 shows the test bench with the VPM, its encoder, and the torque transducer (Magtrol TM312 [34] and Magtrol 3411 Torque Display [35]).The data from the torque transducer are recorded using a Tektronix 3 Series MDO [36].
1) Steady-State Operation: The machine is operated in two PPCs, [m = 18, p = 1] and [m = 6, p = 3], as summarized in Table III.Table IV lists the tuning gains of the PIR controllers.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE III OPERATIONAL POINTS FOR THE TWO PRESENTED PPCS TABLE IV
CONTROLLER GAINS OF THE PIR-PFC Gains K p and K i are chosen to achieve the desired rise time, and K r is determined empirically.The cutoff frequency is set to ω c = 0.2π rad/s, while the resonant frequency ω 0,h (h ̸ = p) is ω s, p .
2) Pole Transition: The performance of the drive is analyzed during a pole transition from configuration [m = 18, p = 1] to configuration [m = 6, p = 3].The pole transition strategy is the premagnetization method with an instantaneous q-current switch, as described in [15].The controllers of the VPM drive use the same gains as in the steady-state tests.

1) Steady-State Operation:
The time-domain currents i 123 of windings [1,2,9], τ shaft , and ω r are shown in Fig. 8 for the PPCs with p = 1 and p = 3.The VPM operates in speed control with ω ref r = 1000 r/min, and the load drive applies a torque of τ shaft = 10.0 Nm.First, the VPM is in healthy mode.Then, an OPF in the second winding (k f = 2) occurs, and the drive continues with the control strategy for the healthy mode.Finally, the drive switches to the proposed post-fault control strategy.The time instant in which the OPF occurs and the beginning of the proposed fault-tolerant operation are marked.After the OPF happens, the currents become significantly imbalanced.This behavior is visible in the waveform of i 1 shown in Fig. 8(a), whose amplitude increases by 48.04% (5.14 A) compared with that in healthy conditions (3.47 A).The same situation repeats for p = 3 when the amplitude of i 1 rises from 6.72854 to 9.73 A, corresponding to an increase of 44.63%, as shown in Fig. 8(b).Finally, in controlled post-fault conditions, the developed controller achieves equal current amplitudes in the remaining healthy windings.
Figs. 9 and 10 show the loci of the current space vector īαβ0,h in healthy, faulty, and post-fault controlled operating conditions when p = 1 and p = 3, respectively.In healthy operating conditions, shown in Figs.9(a) and 10(a), the currents appear balanced in both PPCs in the torque-producing plane, while the currents īαβ0,h (h ̸ = p) remain around zero in the non-excited planes.Figs.9(b) and 10(b) show the imbalance in the αβ0 harmonic planes caused by the fault.It can be noted that īαβ0,h has a non-negligible magnitude, which increases depending on the order of the harmonic plane.Moreover, the discrepancy between the blue and orange lines reveals that the direction of īαβ0,h may be far from the one preserving the motor torque.Even the locus of the current space vector īαβ0,p in the torque-producing harmonic plane deviates from its ideal circular trajectory.Then, Figs.9(c) and 10(c) show that, when the fault-tolerant control strategy  is activated, the current space vectors īαβ0,h (h ̸ = p) follow the reference directions despite spurious high-order harmonics.It can also be seen that the magnitude of īαβ0,h is the same in all non-excited harmonic planes.
As explained in Section III, the proposed approach aims to reduce the stator copper losses P cu,s (7).Fig. 11 shows P cu,s calculated from the measured currents in both PPCs.2) Pole Transition: Fig. 12 shows the dq0 currents under an OPF in the second phase (k f = 2) during the transition from a PPC with [m = 18, p = 1] to a PPC with [m = 6, p = 3].To simplify the visualization, the current vector īαβ0,h̸ = p in the non-torque-producing planes is multiplied by e − jθ h,k f , and consequently, the resulting q component is zero.For the same reason, only the d component of īαβ0,5 is shown.This figure stands exemplary for all unexcited harmonic planes.Fig. 12. Waveforms of i d,1 and i q,1 in plane p 1 = 1, i d,3 and i q,3 in plane p 2 = 3, and i d,5 in the non-excited harmonic plane with h = 5 during a pole transition with an OPF (k f = 2).
For t < 29.1 s, i d,1 and i q,1 appear undisturbed by the fault, while the d components of the currents in the other planes oscillate according to the developed post-fault strategy.For 29.1 s < t < 29.9 s, the pole change occurs.The magnetizing current i d,1 drops, while i d,3 increases up to its new reference value.Also, ψ R, p 1 =1 decreases with a first-order transient.As a result, i q,1 increases to preserve the torque.Meanwhile, ψ R, p 2 =3 rises.At t = 29.9s, i q,1 is set to zero, and i q,3 generates the torque.
Fig. 12 reveals that the currents in both harmonic planes ( p 1 = 1 and p 2 = 3) follow their references adequately and do not contain significant harmonics even if simultaneously excited.In contrast, in steady state, the d-axis current in the non-excited harmonic plane (here, h = 5) shows a single-harmonic component at angular frequency ω s, p 1 before and ω s, p 2 after the pole changing.During the transition, a superposition of these two frequencies with varying amplitude is observed.Fig. 13 shows the time-domain currents of windings [1,2,9] during the same pole transition.The top and bottom plots, corresponding to configurations with p 1 and p 3 pole pairs, respectively, show that the currents in the faulty winding are zero in steady-state conditions.Furthermore, the currents in healthy windings 1 and 9 are equal in amplitude and frequency.During the pole transition, a superposition of two harmonic frequencies is observable in the middle plot.Both the dq0 and the 123 reference frames confirm that the PIR controllers with two resonant frequencies achieve the fault tolerance for both PPCs simultaneously.
Finally, Fig. 14 shows the measured mechanical rotor speed ω r and torque τ shaft during the pole transition.In fault conditions, the speed error exceeds the 0.5 % band for τ s,0.5 % ≈ 0.75 s.This transient is longer than the one reported in [15], but the tests cannot be directly compared because of the different machine configurations.Furthermore, a different pole transition was presented in [15], i.e., from one to four pole pairs.Nevertheless, this test demonstrates that pole changing is possible, even if the machine operates under an OPF.

VI. CONCLUSION
This article proposes a PFC for an 18-phase VPM, which is capable of managing an OPF.The proposed solution can meet the high-reliability requirements of long-haul trucks and allows the vehicle to continue operating until maintenance is possible, substantially reducing the cost of the fault.The basis of the controller is the HPD theory, which provides a unified model for control system.The PFC has been tested under and transient conditions.Its modular structure makes it easy to adapt to machines with a number of phases.The experimental results demonstrate that the PFC can deliver constant torque operation with different pole and perform pole transitions even under fault conditions, thereby achieving TFL.

Fig. 3 .
Fig.3.Inverse-circuit in harmonic plane h in the stationary αβ0 reference frame.

Fig. 4 .
Fig. 4. Trajectories of the stator current space vectors during the fault in steady-state conditions.

Fig. 6 .
Fig. 6.Schematic of the VPM with 36 toroidal stator windings connected to form 18 phases and 28 rotor bars.Only one inverter leg is shown for clarity.The OPF is induced by a breaker in the faulty winding.

Fig. 11 .
Fig. 11.Stator copper losses for both PPCs.A reduction while applying the PFC is observed.

Fig. 14 .
Fig. 14.Waveform of the measured rotor speed ω r shaft torque τ shaft during the pole transition under an OPF (k f = 2).
Faulty winding.K p , K i , and K r Controller gains of the PIR controller.

TABLE I INVERSE
-MODEL ELECTRICAL PARAMETERS OF THE VPM1

TABLE V AVERAGE
STATOR COPPER LOSSES FOR BOTH PPCS IN HEALTHY, FAULTY, AND PFC OPERATION IN THEIR RESPECTIVE STEADY STATES