Modular Power Sharing Control for Bearingless Multithree Phase Permanent Magnet Synchronous Machine

This article proposes a modular approachto the power sharing control of permanent magnet synchronous bearingless machine. The selected machine topology features a winding layout with phases distributed into nonoverlapping three phase groups, a solution whose twofold aim is to increase the fault tolerance and to allow for the radial force generation. The three phase subwindings are supplied by standard three-phase inverter, leading to a modular system architecture. A throughout explanation of the methodology used to develop the control algorithm is presented considering the torque and force control in combination with the power sharing management of the machine. Special emphasis is placed on validating the modeling hypotheses based on a finite element characterization of the machine electro-mechanical behavior. The proposed control strategy is also extended to cater the possibility of one or more inverters failure, thus validating the intrinsic advantage of the redundancy obtained by the modularity of the system. An extensive experimental test campaign is finally carried out on a prototyped multi three-phase permanent magnet synchronous drive. The obtained results validate the bearingless power sharing operation in healthy and faulty scenarios, both at steady state and under extreme transient condition.


I. INTRODUCTION
M ULTIPHASE electric drives are recognized for featuring enhanced performance and fault tolerant capability when compared with conventional three phase counterparts [1]. They have been historically adopted in high power applications, such as large generators, ships, and trains propulsion [2] mainly thanks to the possibility of splitting the power (both current and/or voltage) among multiple phases, which allows overcoming power electronics' limitations [3]. Finally, this machine topology has also been adopted in automotive powertrains [4].
A myriad of winding layouts are possible with multiple phases with both overlapped and nonoverlapped arrangements differing by the number of phases and their spatial distribution [5]. The layouts can all be conceptually classified according to the electrical angle between the magnetic axes of consequent phases. This angle is often 360/n or 180/n, being n the total number of phases. However, it can vary according to the number of stator slots and rotor poles. A further multiphase winding variant, usually called with different names (asymmetrical, split-phase, multi-m phase, etc.), is obtained by subdividing the n phases into N sets of m phases. This represents an interesting solution widely adopted in commercial products. In fact, when m = 3 conventional three phase converters can be employed to supply the multi three-phase windings [6]. An interesting feature of multi m-phase machines is the possibility of independently managing the power flows among the different three-phase subwindings, hence achieving the so called power sharing operation [7], [8]. The latter provides the possibility that m-phase subwindings can be simultaneously supplied by diverse inverters, due to a drive architecture layout that, for example, can present more independent power sources. Particularly, the power sharing technique is important in applications requiring a high level of reliability [9]. An attempt to further increase the fault tolerant feature of this winding topology is to physically distribute the m-phase subwindings into N nonoverlapped sets [10], resulting in physical and thermal insulation [11], [12]. Along with the higher fault tolerance, this modular multiphase winding layout presents the further advantage of being able to generate controllable radial forces beside the torque [13], [14]. This additional feature makes this winding layout suitable for bearingless or bearing relief operation in application requiring a high level of reliability [15].
Bearingless operation is particularly convenient in high speed applications [16] or in aseptic environments [17]. Two threephase windings featuring different number of poles were initially employed to generate independent torque and suspension force [18] leading to a reduced stator slot utilization hence decreasing the machine power density. More recently, mainly three combined winding solutions have been proposed, where each winding set contributes to both torque and suspension force production, namely the bridge-configured winding [19], the parallel-configured winding [20] and the multiphase winding [21]. A comprehensive comparison is presented in [22] showing that there is no clear winner, i.e., a configuration requiring the least components number, the easiest control implementation, and the highest fault tolerance. This conclusion makes the modular multi-m phase winding a serious contender in fault tolerant bearingless applications.
This particular drive architecture has already been investigated in [23] for a bearingless application with power sharing capability among modules. However, the presented control strategy was based on the vector space decomposition approach. The latter treats the whole winding layout as a unique system thus loosing the modular approach making the fault tolerant control more complex and challenging.
This article presents a modular power sharing control strategy of a multi three-phase surface mounted permanent magnet synchronous (SMPM) machines suitable for bearingless operation. The proposed control is fully based on the modularity of the multi three-phase drive extending the conventional bearingless control to the management of power sharing among the subwinding and to the fault tolerant control acting in case of module open-phase fault. First, a detailed modeling of the machine electro-mechanical behavior is described in Section II aided by a finite-element analysis (FEA) of the considered machine. Section III outlines the implemented control strategies allowing the power sharing among stator modules during bearingless operation in both healthy and faulty conditions. Then, a brief overview of the control architecture is presented in Section IV along with a detailed FEA aimed at assessing the effects of neglecting the mutual interaction among stator modules and the effect of increasing the harmonic content included in the control. Finally, Section V reports an extensive test campaign on a prototyped machine validating the bearingless power sharing operation in healthy and faulty scenarios, both at steady state and transient conditions.

A. Machine Structure
The modular bearingless SMPM machine is illustrated in Fig. 1(a). The machine is divided into three equal portions  occupied by three three-phase windings and highlighted in red, blue, and yellow, respectively. Each three-phase winding is star-connected with a galvanically isolated neutral point. The symbols + and − indicate the directions of the currents, flowing out and into the plane of the paper, respectively, while the subscript i ∈ [1 , 2 , 3] is adopted to define the numerical order of the winding module. For example, +u 1 means the current direction of the first module phase u is flowing out the plane of the paper. Finally, the main parameters of the machine are listed in Table I.

B. Mathematical Model
The definition of the machine model is the first step required to analyze the electromagnetic behavior of an electric motor, whose understanding is essential for the development of a suitable control algorithm. Owing to the modularity of the multi three-phase SMPM machine, the model can be easily defined as function of the d − q axis variables of each three-phase subwinding (with the d-axis aligned with the center of the north pole of the rotor magnets).
The key inputs of the multi three-phase SMPM machine model are the d-q axis currents of each module, whereas the outputs are the mechanical forces and torque acting on the rotor, hereinafter referred as wrench W . The latter can be defined as a vector of the x − y components of the radial force (F x , F y ) and the torque T acting on the rotor where is the transpose operator. The relationships between the d − q currents of the general i th sector ( i i d , i i q ) and the respective contribution to the wrench of the multi three-phase SMPM motor is function of both axis currents and rotor electrical position, as follows: Under linear operating conditions and neglecting the force contributions due to the interactions between stator m.m.f. harmonics, the wrench produced by the entire machine can be considered the sum of the effects of all the N submodules supplied with the respective d − q currents The functions i f can be evaluated by accurate analytical models or, for a better accuracy, through FE simulations. Although analytical approaches are computationally efficient, their results needs to be FE validated [24]. For this reason, in the following, the characterization of the considered machine, whose details are reported in Table I, is carried out via 2-D FEAs (with the commercial suite MagNet©). Fig. 2 reports the wrench components of the first module of the machine considered throughout this study (whose first phase is placed on the x-axis as shown in Fig. 1(a)). In particular, Fig. 2(a) and (b) shows the wrench components as function of the rotor electrical position when supplied by rated d and q axis current, respectively. Both force components feature a relevant second harmonic whereas the torque, as expected, is dominated by the sixth harmonic. Fig. 1(b 1 ) and (c 1 ) reports the flux density distributions when supplying positive d and q axis rated current, respectively, while subfigures (b 2 ) and (c 2 ) show the same distributions without the permanent magnet contributions. From these flux maps, it is also possible to appreciate that the soft magnetic material is mainly operating in its linear range. Fig. 2(c) and (d) depicts the average wrench components as function of the both d and q axis currents, respectively. It can be noticed that torque and force contributions vary almost linearly with the currents up to double the rated value (i.e., 26A pk ). Above this value, the saturation of the flux path has a relevant effect only when supplying with positive d−axis current (i.e., when strengthening the PM flux). Consequentially, neglecting the reluctance torque, the wrench contribution of each module can be assumed linearly dependent on the respective currents, and so the following matrix equation can be written where 1 K dq (ϑ e ) is a 3 × 2 matrix of wrench coefficients only function of the rotor electrical position (ϑ e ). This approximation is strictly valid when working within the liner region of the wrench-current relationship reported in Fig. 2

(c) and (d).
Outside this area, each coefficient of the matrix 1 K dq would mainly depend from the respective current (e.g., Fig. 2(e). The latter reports the average force produced by the first stator module for a given current amplitude and different phase angles. The locus described by the force vector clearly resembles an ellipse, which uniformly changes dimensions (both axes) when operating in the linear region. This implies that the force vector can be described as function of the current components via constant coefficients and so (4) is valid. Outside the linear region, the force locus deviates from being an ellipse due to the unequal effect of the nonlinearities on the force components. Fig. 2(f) shows the locus described by the force vector as function of the rotor position for the rated current condition and different current phase angles. The figure highlights the fact that the force ripple is mainly produced perpendicular to the magnetic axis of the supplied module (i.e., the y-axis for the first module). In principle, the same identification procedure should be carried out for all the other modules of the machine. However, being each subwinding rotated with respect to the adjacent one, the wrench of the generic i th module can be evaluated by simply taking into account the mechanical shift i Δϑ m of the considered stator module, as follows: where i R is a rotational matrix defined as follows: As an example, for the multisectored machine layout depicted in Fig. 1 wrench produced by all the modules results from (3) and (5) as follows: where the K dq is a 3 × 2N matrix built by pending the columns of the i K dq (ϑ e ) matrices obtained by (5), while the total current vector is The assumption underling (3), i.e., neglecting the mutual interactions among the machine submodules, will be further analyzed and validated in Section IV.

III. CONTROL STRATEGIES
Once the wrench-current function is identified, its inversion needs to be carried out in order to control the machine, i.e., find the current set points, which generate a given reference wrench (W * ). Being the wrench-current relationship (7) an underdetermined system, its inverse problem has more than one solution, i.e., there exist ∞ 2N −3 current solutions providing the same wrench (2N − 3 is the number of unknown minus the number of equations). Among all the possible solutions to this kind of control problems, the result that minimizes the stator copper loss is recognized as valuable one within the field of multiphase machines control [8]. This approach allows fully exploiting the various subwindings capability with a closed-form relationship between currents and wrench. In fact, supposing that the solution to the underdetermined system (7) has also to satisfy the objective of minimizing the joule losses (P J,tot ), the problem to solve can be expressed as follows: where R S is the diagonal matrix of the phase resistances. The latter (8) is a classic quadratic optimization problem, which can be solved applying the Langrange multiplier method, leading to the well-known result where K + dq is the Moore-Penrose inverse of the matrix K dq . In general, the Moore-Penrose inverse represents the solution of an undetermined system of equation minimizing the square of the input variables, which in this case is proportional to the stator Joule loss P J,tot .
As previously mentioned, this is not the only way to solve this class of problems. Although it allows a computational efficient implementation leading to the maximization of the operational efficiency, it does not directly consent to control the power flows among the several submodules as desired in application requiring power sharing operation. In the following two sections, this control strategy is extended in order to consider the power sharing constraints and the open fault of an entire stator module.

A. Power Sharing Operation
The power sharing operation can be defined by introducing the vector of the sharing coefficients Z sh = [ 1 z sh , 2 z sh , . . ., N z sh ] determining the reference q axis currents as follows: where K T is the torque constant and T * is the reference torque. The imposition of the q axis currents via the sharing coefficients fully determine the torque produced by the machine. However, these current components also create a radial force contribution (F q ), which can be evaluated via (7). More precisely, only the even columns of the K dq matrix and the first two rows are needed to evaluate this force contribution. By building up this new submatrix K q (ϑ e ), the force contributions F q are expressed as follows: The remaining degrees of freedom of the system, i.e., the d-axis currents, can be exploited for the production of the reference radial forces (F * ). Indeed, the d axis currents i d produce radial force components, which can be evaluated from (7) where K d is the submatrix of K dq built with its odd columns and the first two rows. In order to produce the desired force reference F * equal to the sum of the d and q axis currents contributions (F d and F q ), the reference d axis currents have to produce the force F * − F q , therefore where K + d (ϑ e ) is the Moore-Penrose inverse of the 3 × N K d (ϑ e ) matrix. It is worth to underline that the d-axis currents do not have any effect on the torque, being the reluctance torque null. Fig. 3 summarizes the proposed bearingless power sharing control strategy for the triple three-phase permanent magnet machine considered in this study.

B. Module Open Fault Condition
Due to the modularity of the torque and force generation of this particular machine topology, it is possible to keep on fully controlling force and torque in power sharing operation also when an entire subwinding is opened. In fact, the control problem outlined in the previous subsection can be easily solved by assuming a reduced number of three-phase subsystems. Indeed, while building the matrix of wrench coefficients K dq , the submatrix i K dq of the faulty module must not be considered.
In case of open fault of the first module (but the same procedure can be extended to the other ones), the considered machine would feature two sharing coefficients 2 z sh and 3 z sh , with 2 z sh + 3 z sh = 1, which determine the split of the torque among the modules 2 and 3 (i.e., 2 i * q and 3 i * q ) while the d axis reference currents needed to produce the reference force would be with where all the wrench coefficients are still function of the rotor electrical position ϑ e . It is worth noticing that for this particular case (i.e., three modules) the pseudoinverse of the matrix is not required because the system of equations is no more underdetermined. This control approach can be used to manage open faults of more than one module, aware that to continue the bearingless operation under power sharing control the minimum number of healthy modules must be two.

IV. CONTROL ARCHITECTURE AND IMPLEMENTATION
At first, this section gives an overview of the control system architecture of the bearingless multi three-phase permanent magnet synchronous machine. Afterward, a detailed FE-based analysis is presented in order to 1) validate the assumption underling (3), which neglects the interaction among the several machine submodules; 2) determine how to implement the machine model inversion (i.e., (9) or (11) and (13) if the power sharing among the modules has to be controlled) on a real time control platform.

A. Control Architecture
A schematic of the bearingless control system with the power sharing among the submodules is illustrated in Fig. 4. The machine three-phase subwindings are supplied by three independent three phase inverters. The position controllers, i.e., two independent PID regulators and the speed loop PI controller determine the reference wrench components from the measured radial shaft positions and angular speed errors. The reference currents are then calculated via the power sharing logic detailed in Section III. The latter are then tracked via six conventional PI regulators. The gate signals of the inverter switches are then obtained through pulsewidth modulation from the references voltages. The current PI controllers, the position PID controllers, and the speed PI regulators are designed according to the pole placement approach.

B. Effect of Increasing the Detail of Wrench Harmonic Content
The bearingless control of a multi three-phase permanent magnet synchronous machine requires the implementation of the control strategy summarized in (9) or in (13) when including the power sharing option on a real time control platform. In particular, the wrench coefficients matrices K + dq , K q , and K + d can be calculated offline once the full matrix K dq has been characterized by FEA or experimental tests. Such matrices, function of the rotor position, can be then stored via look up tables (LUT) on the real time hardware in order to perform the bearingless power sharing control. However, this approach would lead to a significant computational burden given the dimension of these LUTs. In the attempt of relieving this implementation cost, the effect of considering a reduced harmonic content for each element of the wrench coefficient matrix K dq is hereafter reported. In fact, the generic element k r,c of the matrix K dq placed in the row r and column c can be expressed as function of the rotor electrical position (ϑ e = pϑ m ): k r,c (ρ) · cos(ρϑ e + ϕ r,c (ρ)).
Indeed, considering only the dc component of each coefficient would lead to a massive size reduction of the LUT because K dq would not depend anymore from the rotor electrical position. However, this implies disregarding all the ripple force and torque shown in Fig. 2(a) and (b). In this simplified case, the reference d and q axis currents providing a given wrench W * (e.g., F * x = 0 N, F * y = 20 N, T * = 5 N · m) are independent from the rotor electrical position as shown in Fig. 5(a), but the consequent produced wrench feature a significant ripple, as depicted in Fig. 5(b). The latter also reports the FE-calculated (marked lines) wrench components, which match quite well the expected values. The small differences between the expected and the FE values are due to the assumption on which the modeling is based, i.e., null mutual effect among modules.
When considering all the harmonic content of the wrench coefficients k r,c , the force ripple drastically decreases as visible in Fig. 5(f). However, this comes at the cost of a higher computational burden being K dq dependant from the rotor position. In addition, including all the harmonics lead to more frequency components in the reference currents, as shown in Fig. 5(e), which could be difficult to track at high rotational speed. It is worth to underline that the small deviations between the expected and FE calculated wrench components still remains also when considering the full harmonic spectrum of all k r,c . From the previous analyses, these differences are mainly attributed to the mutual interaction among the stator modules. In addition, the torque ripple is almost not affected by the introduction of more harmonics due to the fact that it is mainly due to the cogging effect (no load torque).
A compromise solution between quality of the produced wrench and computational burden consists in considering only the dc and the second (with reference to the electric period) harmonic of each k r,c coefficient, as shown in Fig. 5(c) and (d). The only detrimental effect of this approximation is a slight increase of the force ripple compared with the ideal control solution. As a result, each element of the Moore-Penrose matrices K + dq and K + d can be expressed as function of the rotor position in electrical degrees as follows: which can be implemented only storing the values k + r,c (0) and k + r,c (2) for each of the 18 wrench coefficients k + r,c . Fig. 5(a)-(f) refers to the standard bearingless control of a multi three-phase permanent magnet synchronous machine without the possibility to manage the power flows among stator modules. Fig. 5(g) and (h) reports the reference currents and the expected and FE wrench components when considering a nonuniform power sharing scenario (Z sh = [0.4 0.35 0.25] ). Although the references are calculated considering only the main two harmonics in each wrench coefficient k r,c , the d axis currents include more than two harmonics. This is because the Moore-Penrose inverse of the K d matrix (having each element sum of two harmonics) leads to matrix coefficients featuring a different harmonic content. In other words, the Moore-Penrose matrix operator does not guarantee the preservation of the harmonic spectrum of the coefficients of the inverted matrix.
Also, the differences between the expected and FE wrench components show a higher error when a nonuniform power sharing scenario is considered. It is worth to underline that the wrench mismatch between expected and FE/real values, due to the assumption of negligible interaction among stator modules, is compensated by the actions of the position and speed closedloop controllers. Consequentially, the experimental d and q axis currents reference are expected to feature a higher harmonic content with respect to the predicted ones, shown in Fig. 5(c) and (g).

V. EXPERIMENTAL VALIDATION
The proposed power-sharing technique is validated on a 1.5 kW-3000 r/min prototype bearingless multi three-phase permanent magnet synchronous machine, whose parameters are listed in Table I. In this sections, first a general description of the experimental setup is given and then an extensive test campaign is reported to fully validate the proposed control strategy. In particular, three tests are implemented to verify the performance of the proposed bearingless power sharing control technique in different healthy and faulty operating scenarios.

A. Instrumented Test Rig
The stator of the proposed bearingless SMPM machine is displayed in Fig. 6. It is clearly observed that three three-phase windings are galvanically isolated.   7 shows the instrumented test rig components. The bearingless machine along with the load motor, connected through a universal joint, are shown in Fig. 7(a). At the nondrive-end of the bearingless machine, a self-alignment bearing avoids the axial and x-y displacement of the shaft. The two degrees of the freedom radial movement of the shaft is allowed at the drive-end but it is limited by a backup bearing with a clearance of 150 μ m. The rotor radial x − y positions are measured via two 3300 XL NSv proximity transducers, which are also called eddy current sensors, as shown in Fig. 7(c). The main parameters of the proximity transducers are 10 kHz bandwidth, a linear range from 0.25 to 1.75 mm and an Incremental Scale Factor of 7.87 V/mm. A cylinder of AISI 4140 is mounted on the shaft to maximize the measurement performance, being the sensor calibrated in the factory for acting on this material. The rotor x-y axis positions are regulated by two independent conventional PID controllers [25]. A digital low pass filter is installed in the position controllers to limit the effect of the measurement noise, acquired from the proximity transducers, on the differential component of the position PID controllers. The dominant pole of the position controllers is placed in the real axis of the complex plane, 130 Hz. Each sector of the bearingless machine is supplied by an independent three-phase power inverter, as shown in Fig. 7(b). The inverters are SPWM modulated, and the voltage references are obtained from conventional PI current controllers. The bandwidth of the latter is 1000 Hz. The custom-made control platform, where the control strategy is implemented, is based on the off-the-shelf Microzed board. This platform manages the power converters through fibre optic cables, as shown in Fig. 7(d). The switching frequency is set at 10 kHz.

B. Power Sharing in Healthy Machine Condition
In the first test, the speed controller is disabled and the angular shaft speed is controlled at 3000 r/min by the load motor while the shaft radial position is regulated by the bearingless machine. The power sharing coefficient is kept constant during the whole experiment to Z sh = [0.5 0.7 − 0.2] . The experimental results are shown in Fig. 8. At 0.05 s, the torque reference changes from 0 to 2N · m, resulting in q-axis currents of three modules increasing from 0 to 7.8 A, 10.92 and −3.12 A, respectively, as shown in Fig. 8(a). As explained in Section III, the q-axis current of each sector is determined by the power sharing coefficient and the torque reference. In the meantime, the d-axis currents increase to compensate for the radial force contribution generated by the q-axis currents, as shown in Fig. 8(b). The shaft x-y axes position are displayed in Fig. 8(c) showing a stable operation during the torque transient.

C. Power Sharing in Faulty Machine Condition
The second test, whose results are shown in Fig. 9, verifies the performance of the power-sharing technique when an entire stator module is in open fault. The experimental results are recorded during the transient of the fault, and the pre-and postfault operation is also included. The speed is still set at 3000 r/min by the load motor while the shaft x-y position is controlled by the bearingless machine. This test is constituted by four steps as clearly shown in Fig. 9(a). 1) Before 0.2 s, the power-sharing is set to be uniform, with 5.2 A q-axis currents in all the three sectors. 2) Then, the q-axis currents of three sectors separately change to −6. 24, 9.36

D. Power Sharing During Speed and Position Transient
During the third test, both speed and radial positions are controlled by the bearingless drive in order to asses the system behavior in both position and speed transient under simultaneous power sharing and open module fault conditions. In particular, this test can be divided in five periods described in the following with reference to Fig. 10(a)-(d) showing positions, speed, and d − q axis currents, respectively. 1) Before 0.1 s, the drive is off.
2) Then, at 0.1 s, the drive is activated with the power sharing coefficients equal to Z sh = [−0.4 0.6 0.8] . The shaft moves from its rest position to the airgap center, as shown in Fig. 10(a). The radial suspension force for levitating the rotor is totally generated by the d-axis currents being null the speed set point. Thus, three peaks occur in the d-axis currents at 0.1 s, as shown in Fig. 10(c).  coefficients are changed to Z sh = [0 0.6 0.4] being null the contribution of the first subwinding. 5) At 2.2 s, the first module recovers from its faulty condition, and the power sharing coefficients returns back to the previous healthy value, and consequentially also the currents. 6) At 3.8 s, the speed transient ends, and the q axis currents decrease following the torque reduction. The results highlight the robustness of the proposed control strategy for the power sharing operation of the bearingless drive also when a fault happens in an entire stator module during a speed transient.

VI. CONCLUSION
This work introduced a modular power sharing control technique for bearingless multi three-phase permanent magnet synchronous machines and demonstrated its performance also under one module open phase fault. First a comprehensive description of the radial force and torque generation principle was given, aided by a detailed FEA of the considered machine. This analysis was aimed at assessing one of the hypothesis of the control technique, i.e., the linearity of the force-current relationship. Then, the theoretical fundamentals of the proposed control strategy, allowing both bearingless and power sharing operations in healthy and faulty conditions, were outlined. Further FEAs replicating the real control scenario have also been carried out with the aim of assessing the effect of increasing the force harmonic content used within the control in terms of quality of the force production and computational burden. The analysis also demonstrated the validity of the control hypothesis of negligible coupling between stator modules in the generation of the overall wrench. The proposed control strategy was finally experimentally validated for a wide range of operating scenarios including the bearingless and power sharing operation, in both healthy and faulty conditions also during speed transient. These outcomes represented a step forward with respect to the methods presented in literature and introduce novel elements to be applied in fault tolerant drives for bearingless machines.