## 1. Introduction

In recent years, multiphase induction machines (IMs) have been considered to be such a viable alternative in comparison to three-phase machines due to their fault tolerance capabilities with no extra hardware, lower torque ripple and better power splitter per phase which result very attractive to the research community for various industrial applications where a high-performance control strategy, as well as, reliability are required [

1]. Presently, some applications of multiphase IMs that are being investigated include wind energy generation system [

2], hybrid electric vehicles (EV) [

3] and ship propulsion. In the applications mentioned above, multiphase IMs can be used under different conditions, such as healthy and post-fault operations [

4,

5]. From the point of view of control, the most common control strategy to regulate multiphase IMs is the field-oriented control (FOC), which is constituted by an inner current control loop, to obtain the references voltages, and an outer speed control loop for speed regulation [

6]. However, several new control approaches have been carried out for the inner current control loop in multiphase IMs, some of them are: sliding mode control [

7], resonant control [

8] and model predictive control (MPC) [

9]. Although there are other controllers such as the well-known proportional-integral (PI) controllers [

10], the preferred choice is the MPC due to the fact that it shows a good transient behavior and facilitates the inclusion of nonlinearities in the system as described in [

11,

12], and in [

13] where a comparative study between MPC and PI-PWM control has been addressed. In this context, the MPC strategy produces the reference voltage through the instantaneous discrete states of the power converter according to the minimization of a predefined cost function. However, the classic MPC strategy presents some limitations regarding to the application of only one vector in the whole sampling period. This results in current ripples as well as large voltages at low sampling frequency. Besides, the variable switching frequency develops a spread spectrum, decreasing the performance of the system in terms of useful power [

14].

To overcome this subject, a predictive-fixed switching current control strategy, named (PFSCCS) from now on, applied to a two-level six-phase voltage source inverter (VSI) is presented in this paper. The strategy is based on a modulation concept employed with the MPC scheme, which has been studied for different power converters such as the mentioned two-level six-phase VSI described in [

15,

16] and also other topologies presented in [

17,

18]. In the proposed current strategy, three vectors have been considered at every sampling period, composed by two active vectors (taking only into account the largest vectors) and null vector, where their corresponding duty cycles are achieved according to the switching states and a switching pattern has also been used before being applied to VSI in order to generate a fixed switching frequency. Whereas, for the speed control loop, a PI controller has been developed by a technique shown in [

19].

The main focus of this work is the implementation of the PFSCCS so as to reduce the (x-y) currents compared to the classic MPC strategy using a six-phase IM supplied through a two-level six-phase VSI. In that context, both simulation and experimental validations have been included to demonstrate the capability of the proposed technique. In addition, the effectiveness of the PFSCCS is tested under steady-state and transient requirements, respectively, incorporating the mean square error (MSE) and the total harmonic distortion (THD) analysis.

The paper is organized as follows: the model of the six-phase IM and VSI are presented in

Section 2. In

Section 3 are described the speed controller, classic MPC and the proposed current controller based on modulated model predictive control.

Section 4 shows the performance of the proposed control through simulation and experimental results in steady-state and transient conditions. Finally,

Section 5 summarizes the conclusion.

## 2. Six-Phase IM Drive Model

The six-phase IM, supplied by a two-level six-phase VSI with a DC-Link voltage source (

${V}_{dc}$), is taken into account in this work. The simplified topology is presented in

Figure 1. The six-phase IM is a dependant of time system, for this reason it is possible to represent it through a group of equations in order to define a model of the real system.

In that sense, vector space decomposition (VSD) strategy [

20] has been used to translate the actual six dimensional plane, formed through the six phases of the six-phase IM, into three two dimensional rectangular sub-spaces in the stationary reference frame, named as (

$\alpha $-

$\beta $), (

x-

y) and (

${z}_{1}$-

${z}_{2}$) frame, by applying the amplitude invariant decoupling Clarke conversion matrix

T [

21]. The (

$\alpha $-

$\beta $) frame contains the variables that provide the torque and flux regulation, unlike the (

x-

y) frame which is linked with the energy losses. The zero elements mapped in the (

${z}_{1}$-

${z}_{2}$) frame are not examined due to the adopted topology (isolated neutral points).

Moreover, the model of the VSI must be included in the system. Thus, due to the discrete nature of the VSI, it is necessary to define an amount of

${2}^{6}$ different switching states which represent every state of each VSI leg specified as

${S}_{m}=({S}_{a},\dots ,{S}_{f})$, where

${S}_{m}$ is considered as binary number, i.e.,

${S}_{m}=$0 or

${S}_{m}=$1. Therefore, the stator phase voltages can be projected into (

$\alpha $-

$\beta $)-(

x-

y) frame by considering the vector

${S}_{m}$ and the

${V}_{dc}$ voltage employing the VSD strategy. In

Figure 2, the 64 control alternatives (48 active and one null vectors) are depicted in the (

$\alpha $-

$\beta $)-(

x-

y) frame.

By considering the mentioned analysis, the six-phase IM can be performed by employing the state-space representation as follows:

being

$x\left(t\right)={({x}_{1},\dots ,{x}_{6})}^{T}$ the state vector constituted by stator-rotor currents of the six-phase IM, shown in Equation (

3),

$u\left(t\right)={({u}_{1},\dots ,{u}_{4})}^{T}$ is the input vector constituted by the stator voltages, presented in Equation (

4). While

${M}_{1}\left(t\right)$ and

${M}_{2}\left(t\right)$ are matrices obtained by the electrical parameters of the six-phase IM. The process noise is defined as

$v\left(t\right)$ and

${K}_{n}$ represents the noise weight matrix.

Consequently, by taking into account the state-space representation in Equation (

2) and the state vectors, it is feasible to establish the following equations:

where the electrical variables of the six-phase IM are represented by

${R}_{s}$,

${R}_{r}$,

${L}_{m}$,

${L}_{lr}$ and

${L}_{ls}$,

${\omega}_{r}$ represents the rotor electrical speed and the coefficients (

${r}_{1},\dots ,{r}_{5}$) are defined as:

Besides, in order to produce the stator phase voltages, which are dependant of the

${V}_{dc}$ voltage and the vector

${S}_{m}$, an ideal six-phase VSI has been used [

21] as it is defined in Equation (

7).

In turn, the stator phase voltages can be mapped into (

$\alpha $-

$\beta $)-(

x-

y) frames defined as follows:

where Equation (

9) is considered the output vector, denoted by

$y\left(t\right)$, and

$n\left(t\right)$ is the measurement noise. Finally, the mechanical equations of the six-phase IM are specified as:

where

${J}_{i}$ defines the inertia coefficient,

${B}_{i}$ is the friction coefficient,

${T}_{e}$ represents the generated torque,

${T}_{L}$ is the load torque,

${\omega}_{m}$ is the rotor mechanical speed, which is related to the rotor electrical speed as

${\omega}_{r}=P{\omega}_{m}$,

${\psi}_{\alpha s}$ and

${\psi}_{\beta s}$ are the stator fluxes, and

P is the number of pole pairs.

## 4. Simulation and Experimental Results

First, simulations have been performed in a MATLAB/Simulink R2014a environment so as to verify the feasibility of the PFSCCS using a six-phase IM shown in

Figure 1. Numerical integration using first order Euler’s algorithm has been applied to calculate the progress of the studied system. The simulation parameters of the six-phase IM are listed in

Table 1.

The effectiveness of the presented control technique for the six-phase IM has been evaluated under a load condition

$({T}_{L}=2$ Nm), the sampling frequency is 8 kHz,

${V}_{dc}$ is 400 V and the

d-axis current reference (

${i}_{ds}^{*}$) has been set in 1 A, while for the gains of the two degree PI controller with a saturation, can be found in [

19]. Moreover, for the proposed control,

${\lambda}_{xy}=0.1$, defined in (

15), has been used in order to give more emphasis to the (

$\alpha $-

$\beta $) stator current tracking.

The performance of the proposed technique is compared in transient and steady-state conditions. Both proofs, simulation and experimental results, are analyzed in terms of mean squared error (MSE) and total harmonic distortion (THD) obtained between the reference and the measured stator currents in the (

$\alpha $-

$\beta $) and (

x-

y) sub-spaces for MSE test and the THD is obtained in the (

$\alpha $-

$\beta $) sub-space. The MSE is computed as follows:

where the stator current reference is represented through the superscript *, the measured stator current is defined by

${i}_{\sigma s}$ taking into account that

$\sigma $ includes the (

$\alpha $-

$\beta $)-(

x-

y) frame and

N is the number of studied samples. While, the THD is obtained as follows:

where

${i}_{s1}$ corresponds to the fundamental stator current whereas

${i}_{sk}$ is the harmonic stator current (multiple of the fundamental stator current).

In

Figure 5 the performance of the stator currents in the (

$\alpha $-

$\beta $)-(

x-

y) frame can be seen in steady-state condition. According to the simulations results, shown in

Table 2, the proposed technique has a good behavior considering the MSE and THD analysis of the stator currents at different rotor mechanical speeds. In addition, it can be noticed that at lower speeds, the stator currents ripple in the (

$\alpha $-

$\beta $) frame is slightly smaller than at higher mechanical rotor speeds, in the same way that occurs for the (

x-

y) currents.

For the experimental proofs the PFSCCS, previously described, is examined in the test rig shown in

Figure 6 in order to prove its effectiveness, employing a six-phase IM supplied through two tradictional three-phase VSI, being analogous to a six-phase VSI and the

${V}_{dc}$ voltage is obtained by means of a DC power source. A dSPACE MABXII DS1401 real-time rapid prototyping bench including Simulink version 8.2 has been used to manage the two-level six VSI. Once the results are acquired, these have been analyzed through MATLAB/Simulink R2014a code. Employing stand-still with VSI proofs and AC time domain strategies, the electrical parameters have been acquired [

26,

27].

Table 1 summarizes these results. Current sensors LA 55-P s (frequency bandwidth since DC up to 200 kHz) have been used for the experimental measurements. The current measurements have been then turned to digital format by means of a 16-bit A/D converter. The six-phase IM angle has been measured with a 1024-pulses per revolution (ppr) incremental encoder in order to estimate the rotor speed and also a 5 HP eddy current brake has been used to insert a fixed mechanical load on the system.

Taking this into account, the experimental results have been analyzed with the same tests that simulations results as figures of merit. The stator currents reference in the (

x-

y) frame have been established to zero, i.e.,

${i}_{xs}^{*}={i}_{ys}^{*}=0$ A so as to decrease the losses in the copper. The amounts for the process noise (

${\widehat{Q}}_{w}$ =

$0.0022$) and the measurement noise (

${\widehat{R}}_{v}$ =

$0.0022$) is estimated by means of the strategy proposed in [

23]. The dynamic behavior of the proposed technique has been evaluated with two different values of

${\lambda}_{xy}$, defined in (

15), giving more weight to (

$\alpha $-

$\beta $) stator currents tracking. In the developed tests, the sampling frequencies have been fixed in 8 kHz for PFSCCS and 8 kHz and 16 kHz for classic MPC, respectively, due to the fact that the PFSCCS uses two active vectors and null vector twice in a sampling period and this procedure doubles the switching frequency compared to the sampling frequency. In that sense, tests have been included in order to expose a fair comparison between the classic MPC and PFSCCS at the mentioned sampling frequencies and also to show the performance of both techniques. For the rotor mechanical speeds, two operation points have been considered, 500 rpm and 1000 rpm, respectively, in steady-state condition. Furthermore, for a transient response, a reversal rotor mechanical speed test from 500 rpm to −500 rpm has been considered for PFSCCS and from 1500 rpm to 200 rpm for classic MPC and PFSCCS. The obtained results between classic MPC and PFSCCS are reported in

Table 3, where the proposed current control technique has demonstrated a good tracking of the current references considering the MSE and THD in the (

$\alpha $-

$\beta $)-(

x-

y) frame.

Figure 7 presents the trajectories of the stator currents in the (

$\alpha $-

$\beta $)-(

x-

y) frame of the PFSCCS applied to the six-phase IM. In this test two different values of the tuning parameter (

${\lambda}_{xy}$) have been considered, in

Figure 7a,

${\lambda}_{xy}$ = 0.05 has been considered and

${\lambda}_{xy}$ = 0.1 in

Figure 7b. The rotor mechanical speed has been set to 500 rpm at 8 kHz. The figure shows that (

x-

y) currents decrease when

${\lambda}_{xy}$ increases, which imply that the selection of this parameter has a strong influence on the behavior of the system. Further, the (

$\alpha $-

$\beta $) current tracking has a slightly better performance considering

${\lambda}_{xy}$ = 0.1.

In addition,

Figure 8a shows the harmonic content of the measured stator current (

${i}_{\alpha s}$) through THD analysis and also, in

Figure 8b has been included the switching voltage in the six-phase VSI showing the pattern of the proposed modulation strategy.

On the other hand,

Figure 9 exposes the transient response of the proposed control for a step response in

q axis. The transient response has been included through a reversal test from rotor mechanical speed (500 rpm to −500 rpm) at 8 kHz. Both cases report fast responses considering the overshoot and settling time, which were of 6.14% and 6 ms, respectively, for

Figure 9a and 4.85% and 6.12 ms, respectively for

Figure 9b. The criterion of the 5% has been selected. Finally, a experimental transient response from a step change of 1500 rpm to 200 rpm between classic MPC and PFSCCS has been depicted in

Figure 10 in order to show the performance of the proposed strategy, which it has demonstrated that it can be used in industrial applications (e.g., regenerating braking).