Effects of Zero-Sequence Transformations and Min-Max Injection on Fault-Tolerant Symmetrical Six-Phase Drives with Single Isolated Neutral

Abstract

Recently, there has been increased interest in the study of multiphase machines due to their higher fault-tolerant capability when compared to their conventional three-phase counterparts. For six-phase machines, stator windings configured with a single isolated neutral (1N) provide significantly more post-fault torque/power than two isolated neutrals (2N). Hence, this configuration is preferred in applications where post-fault performance is critical. It is well known that min-max injection has been commonly used for three-phase and multiphase machines in healthy condition to maximize the modulation limit. However, there is a lack of discussion on min-max injection for post-fault condition. Furthermore, the effects in terms of the common-mode voltage (CMV) in modulating signals has not been discussed. This paper investigates the effect of min-max injection in post fault-tolerant control on the voltage and speed limit of a symmetrical six-phase induction machine with single isolated neutral. It is shown that the min-max injection can minimize the amplitude of reference voltage, which maximizes the modulation index and post-fault speed of the machine. This in turn results in a higher post-fault power.

I. INTRODUCTION

The recent increase in interest on multiphase machines is, to a large extend, motivated by the rising demand for fault-tolerant drives in applications such as electric vehicles, off-shore wind farms, more-electric aircraft, etc., where the reliability of the drives are of great concern. Studies on the fault-tolerant capabilities, topologies and control of multiphase machines have flourished over the past two decades [1].

A recent generalized study on six-phase machines, which included symmetrical six-phase (S6), asymmetrical six-phase (A6) and dual three-phase (D3) machines, demonstrated that the post-fault capabilities of six-phase machines depend heavily on the winding configuration [2]. Furthermore, the study suggested that for a six-phase machine supplied with a six-leg inverter, the single isolated neutral (1N) configuration is superior to the two isolated neutrals (2N) configuration in terms of post-fault power/torque. For the case of a S6 machine, the 1N configuration can yield a maximum of 54.2% more post-fault torque than 2N. Hence, the 1N configuration should be selected in applications where fault-tolerance is prioritized. Despite the clear post-fault advantage, it is known configuring a six-phase machine with 1N allows additional zero-sequence current to flow in the machine and demands more control effort than the 2N configuration. Nevertheless, there has been a lack of discussions on the effect of post-fault control on the voltage and speed limit of S6 induction machines with the 1N configuration using min-max injection.

In this paper, a study on min-max injection for the post-fault control of a S6 induction machine with the 1N configuration is presented. This study is focused on the fault-tolerant control performance of the S6-1N using different common-mode voltages (CMVs) injected to the modulating signals. It is demonstrated that there are multiple phase voltage references (i.e. inverter leg voltages) that can give rise to the same machine phase voltages. This means that for the same post-fault currents, the voltage references generated by the controller may be different depending on the choice of the current regulator structure. With the carrier-based PWM method, a higher voltage references may result in premature over-modulation, which restricts the maximum operating speed (which in turn effects the maximum post-fault power) of the machine. Subsequently, it is demonstrated here that, regardless of the CMV, the use of zero-sequence (min-max) signal injection can minimize the amplitude of the reference voltage, which maximizes the modulation index.

Over the years, several fault-tolerant control strategies have been proposed for multiphase machines [3]-[6], including six-phase machines [7]-[12]. Despite their differences, the majority of these methods are based on the field oriented control (FOC) method [13], where the machine phase variables are transformed into stationary or rotating reference frame variables using a suitable transformation matrix before being controlled using controllers such as the PI-controller, PR-controller, predictive controller, etc. The control strategies rely on the choice of decoupling transformation (or extended Clarke transformation) matrix [T] which can be broadly classified into two categories: control using reduced-order transformation (ROT) [8], [10], [14]-[17] and control using full-order transformation (FOT) [9], [18]-[21]. Between the two, the FOT approach provides a simpler control that uses the same transformation matrices and the same machine parameters, with minimal changes to the control structure when the machine transit from healthy into post-fault operation. In terms of the current controller, a dual synchronous reference frame PI controller (DSRF-PI) or PR controller is recommended for the control of non-torque/flux producing currents since these currents are generally AC quantities [3], [12], [21], [22].

For the FOT fault-tolerant control of a six-phase machine using the vector space decomposition (VSD) approach, a 6×6 decoupling transformation matrix [T] is used to transform the phase variables into α, β, x, y, 0₊ and 0_- decoupled components

\(\begin{array}{cccccccccc} {[\alpha} & \beta & x & y & 0_{+} & 0_{-}]^{T} &= &[T] & & \\ & & & \cdot {[a_{1}} & b_{1} & c_{1} & a_{2} & b_{2} & c_{2}]^{T} \end{array}\)       (1)

For machines with distributed windings, it is generally accepted that α-β components contribute to flux and torque production, while x-y and 0₊0_- components only lead to losses (i.e. non torque/flux producing current). However, in post-fault operation, these loss-producing x-y and 0₊0_- currents can be manipulated to drive a six-phase machine according to different operation requirements such as Minimum Loss (ML), Maximum Torque (MT) [2] or Full- Range Minimum Loss (FRML) [24].

In the literature, there are two six-phase decoupling transformation matrices that are commonly considered for a S6 machine. They are given as follows:

\(\begin{array}{;} \left[T_{i n d}\right]=\frac{1}{\sqrt{3}}, \\ \left[\begin{array}{cccccc} 1 & \cos (\theta) & \cos (2 \theta) & \cos (\gamma) & \cos (\theta+\gamma) & \cos (2 \theta+\gamma) \\ 0 & \sin (\theta) & \sin (2 \theta) & \sin (\gamma) & \sin (\theta+\gamma) & \sin (2 \theta+\gamma) \\ 1 & \cos (2 \theta) & \cos (\theta) & -\cos (\gamma) & -\cos (\theta+\gamma) & -\cos (2 \theta+\gamma) \\ 0 & \sin (2 \theta) & \sin (\theta) & \sin (\gamma) & \sin (\theta+\gamma) & \sin (2 \theta+\gamma) \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 \end{array}\right] \end{array}\)       (2)

\(\begin{array}{;} \left[T_{comb}\right]=\frac{1}{\sqrt{3}}, \\ \left[\begin{array}{cccccc} 1 & \cos (\theta) & \cos (2 \theta) & \cos (\gamma) & \cos (\theta+\gamma) & \cos (2 \theta+\gamma) \\ 0 & \sin (\theta) & \sin (2 \theta) & \sin (\gamma) & \sin (\theta+\gamma) & \sin (2 \theta+\gamma) \\ 1 & \cos (2 \theta) & \cos (\theta) & -\cos (\gamma) & -\cos (\theta+\gamma) & -\cos (2 \theta+\gamma) \\ 0 & \sin (2 \theta) & \sin (\theta) & \sin (\gamma) & \sin (\theta+\gamma) & \sin (2 \theta+\gamma) \\ 1/\sqrt{2} & 1/\sqrt{2} & /\sqrt{2} & 1/\sqrt{2} & 1/\sqrt{2} & 1/\sqrt{2} \\ 1/\sqrt{2} & 1/\sqrt{2} & 1/\sqrt{2} & -1/\sqrt{2} & -1/\sqrt{2} & -1/\sqrt{2} \end{array}\right] \end{array}\)       (3)

where θ = 2π/3 and γ = π/3.

The main difference between the two decoupling matrices [T_comb] and [T_ind] lies in the transformations for the 0₊ and 0_- components, which are the last two rows of the matrices.

If the six-phase windings are treated as two separate sets of three-phase windings, i.e. a₁b₁c₁ and a₂b₂c₂, the zero-sequence currents for each respective winding set can be defined as:

\(\begin{array}{l} i_{01}=i_{a 1}+i_{b 1}+i_{c 1} \\ i_{02}=i_{a 2}+i_{b 2}+i_{c 2} \end{array}\)       (4)

It is easy to see that the zero-sequence currents in [T_ind] represent the zero-sequence currents of individual three-phase windings, i.e.:

\(\left[T_{ind}\right]: \quad i_{0+}=\frac{1}{\sqrt{3}} i_{01} \quad i_{0-}=\frac{1}{\sqrt{3}} i_{02}\)       (5)

On the other hand, [T_comb] considers the zero-sequence currents as the sum and difference of two three-phase zero-sequence currents, as follows:

\(\left[T_{c o m b}\right]: i_{0+}=\frac{1}{\sqrt{6}}\left(i_{01}+i_{02}\right) \quad i_{0-}=\frac{1}{\sqrt{6}}\left(i_{01}-i_{02}\right)\)       (6)

For the S6-2N, the zero-sequence current cannot flow. Therefore, the 0₊ and 0_- currents can be neglected regardless of the choice of a transformation matrix. However, for the S6-1N, the zero-sequence currents can flow, depending on the choice of transformation matrix:

• For [T_comb], both the 0₊ and 0_- currents can flow. As a matter of fact, the two currents are not independent. However, they have equal magnitudes and opposite signs. Therefore:

\(i_{0+}+i_{o-}=0.\)       (7)

• For [T_comb], the 0₊ current is zero, while the 0- current can flow.

From the way the 0₊ and 0_- currents are defined, it seems as though [T_ind] is more suitable for the 2N configuration, while [T_comb] is more suitable for the 1N configuration. In fact, for the S6-1N, it is obvious that choosing [T_comb] over [T_ind] reduces the number of required current controllers from 6 to 5, since i₀₊ does not need to be controlled. To date, this is the only apparent advantage of [T_comb] over [T_ind] found in the literature.

However, since i₀₊ = -i_o-, as pointed out in (7), [T_ind] permits the use of a single controller for one of the variables, while the other variable does not need to be specifically controlled. The number of independent controllers depends on the degrees of freedom, which should be constant and independent of the employed axis transformation.

It can be said that both transformations have been used interchangeably in the past, and the advantage of one over the other was not clearly discussed. This was the motivation for this work, which demonstrates that the two transformations give rise to different controller output voltages, which affects the maximum modulation limit. This in turn, affects the maximum motor speed during post-fault. Furthermore, the authors demonstrate that min-max injection, which is commonly regarded as useless for S6-1N machines, can help extend the maximum modulation limits in post-fault operations.

This paper is structured as follows. Section II describes the fault-tolerant control of a S6-1N using full-order transformation based on a dual synchronous reference frame PI current controller considering different common-mode voltage injections. Section III briefly demonstrates the implementation of min- max injection. Experimental results and discussions are given in Section IV. Finally, some conclusions are presented in Section V.

II. REFERENCE VOLTAGE ISSUE IN FOT FAULT-TOLERANT CONTROL

The overall control structure of an FOT fault-tolerant drive for a S6-1N is shown in Fig. 1. For i = a₁, b₁, c₁, a₂, b₂, c₂, the phase voltage for phase-i of the machine v_in is defined as the voltage of terminal-i with respect to the neutral point n. The inverter leg voltage v_i0, on the other hand, is defined as the voltage of the machine terminal with respect to the DC-link midpoint 0.

Fig. 1. Fault-tolerant control scheme with a single isolated neutral connection.

The rotor flux oriented control (RFOC) method based on SRF-PI and DSRF-PI controllers [21], as shown in Fig. 2, is used to make the machine stator currents follow the designated post-fault current references [2]. The controller generates reference voltages in the stationary reference frame (αβxy0+0-), which are then transformed into the phase reference voltages v_a1^*, v_b1^*, v_c1^*, v_a2^*, v_b2^*, v_c2^* using an inverse decoupling transformation [T]^-1. Depending on the choice of [T], slight modifications of the controllers become necessary. If [T_ind] is used, one DSRF-PI controller is used to control the 0_- current. Then the resulting control effort of the 0_- current is multiplied by -1 to obtain the 0₊ current. On the other hand, if [T_comb] is used, only one DSRF-PI controller is need for 0_- current control, and \(v_{0+}^{*}\) can be set to zero to further simplified the controller. Using the carrier based PWM method, these phase voltage references are converted into modulation signals, which are then compared with triangular carrier signals to generate switching signals for the VSI. It is important to recognized that in the linear modulation region, with the control delay ignored, the PWM block and the inverter can be considered as a unity gain. Thus, the reference voltages, \(v_{i}^{*}\) become equal to the leg voltages v_i0.

Fig. 2. Fault-tolerant control scheme with DSRF (dual PI) controllers in the x-y current loop.

In the fault-tolerant control of an n-phase machine using the FOT approach, n-voltage references are generated even when one or more of the phases of the machine is open-circuited. Obviously, the reference voltages for the faulted phase(s) are not applied onto the machine. However, the overall control still provides all six reference voltages. As a result, there are multiple sets of possible voltage references that can be applied by the controller, which gives the same currents (hence operating points) for the machine.

To illustrate this scenario, experimental results from a S6-1N rig controlled using the FOT approach and shown in Fig. 1 and Fig. 2, are used for discussion here. Information on the experimental rig is given in the Appendix. The S6-1N machine was subjected to a single open circuit fault (1 OCF) so that phase-a1 is open-circuited. Then it was controlled to run at 800 rpm, with i_d = 1.3A and i_q = 2.41A. Fig. 3 shows results obtained with the FOT control using [T_comb] (right plot) and [T_ind] (left plot).

Fig. 3. Test 1: 1 OCF at 800 rpm based on [T_ind] (left plots) and [T_comb] (right plots). From top to bottom. (a) Modulating signals. (b) Phase currents. (c) d-q currents.

As can be seen from these results, the reference voltages (which are analogous to the modulating signals) in Fig. 3(a) for the two control methods are significantly different. However, the post-fault currents are the same. It should be recalled that the reference voltage directly controls the inverter phase voltage and not the machine phase voltage. For the same phase currents, the phase voltages of the machine have to be the same for both cases. Hence, it can be deduced that there is a common-mode voltage that is present in the inverter leg voltages when one of the transformations is used.

To understand the concept of this common-mode voltage V_cm, each phase of the S6-1N machine can be represented as a Thevenin equivalent circuit, i.e. the EMF behind an RL-branch as shown in Fig. 1. For the purpose of discussion, it is assumed that 1 OCF occurs at phase-a1.

Using Kirchoff’s voltage law, it is easy to see that (not including the faulted phase) the inverter leg voltage and the machine phase voltage of phase-i can be related by the following expression:

\(v_{i n}=v_{i 0}-v_{n 0}\)       (8)

where v_n0 is the voltage of the neutral point with respect to the DC-link midpoint.

Hence, V_cm can be calculated using the following expression:

\(V_{c m}=\left(V_{a 1} *+V_{b 1} *+V_{c 1} *+V_{a 2} *+V_{b 2} *+V_{c 2} *\right) / 6\)       (9)

Using (9) and the reference voltage signals in Fig. 3(a), the V_cm generated when using [T_ind] and [T_comb] can be determined as shown in Fig. 4(a). As can be seen from this figure, an AC common-mode voltage is visible in the [T_ind] approach. Meahwhile the [T_comb] approach has almost zero common-mode voltage.

Based on (8), the actual machine phase voltages can be obtained by subtracting V_cm from the inverter leg voltages (reference voltages) as shown in Fig. 4(b). The corresponding line-to-line voltages can then be determined, as shown in Fig. 4(c). As expected, the two approaches give the same machine phase and line voltages. Hence, the same currents and operating point.

Fig. 4. 1 OCF at 800 rpm based on [T_ind] (left plots) and [T_comb] (right plots). From top to bottom. (a) Common-mode signal, V_cm. (b) Phase voltages. (c) Line voltages.

Even though V_cm eventually gets cancelled out and the machine’s operating points are the same for the two approaches, the V_cm causes the [T_ind] approach to have a higher modulating signal and it tends to go into premature over-modulation, which restricts the maximum operating region [25].

Therefore, some conclusions can be extracted from the analysis with regard to the common-mode voltage signal.

i. Control using different transformation matrices gives different common-mode voltage signals that result in different modulating signals.

ii. The common-mode voltage signal is cancelled out in the phase voltages, so that there is no difference in the amplitude of the line voltage applied to the machine, provided that the modulating signals are within their linear limits.

These test and analysis results can be extended to OCF with up to three phases, and arrive at the same conclusions.

III. MIN-MAX INJECTION IN FAULT-TOLERANT CONTROL

Based on the discussion in the previous section, the following deductions can be made:

i. There is a need to minimize the voltage references (hence the modulating signals) so that the linear modulation region can be maximized.

ii. An additional common-mode signal, or zero-sequence signal can be freely added to all of the phase voltage references to modify the amplitude of the modulating signals without affecting the control outcome.

Hence, this paper proposes the use of min-max injection for S6-1N to maximize the linear modulation region. It should be noted that min-max injection is commonly used for 3-phase and even multiphase under healthy conditions to extend the linear modulation region. However, whether it is applicable to faulty conditions has never been discussed.

The min-max injection method essentially adds a common signal (i.e. zero-sequence signal) to all of the reference voltages (hence modulating signals) [25]. For PWM using the bipolar carrier, the zero-sequence signal to be added is given as:

\(v_{Z S I}=0.5 *\left(\max \left(v_{i}^{*}\right)+\min \left(v_{i}^{*}\right)\right)\)       (10)

with i = a₁, b₁, c₁, a₂, b₂, c₂

From (9), (10) can be written in the general form as:

\(\begin{aligned} v_{Z S I} &=0.5 *\left(\left(\max \left(V_{n}+V_{c m}\right)+\min \left(V_{n}+V_{c m}\right)\right)\right.\\ &=V_{c m}+0.5 *\left(\max \left(V_{n}\right)+\min \left(V_{n}\right)\right) \end{aligned}\)       (11)

By subtracting this zero sequence signal from the reference voltages, V_cm is automatically removed as shown in equation (12) for phase V_a1n.

\(\begin{array}{l} V_{i}=V_{i}^{*}+v_{Z S I} \\ V_{a 1 n}=\frac{5}{6}\left(V_{a 1}\right)-\frac{1}{6}\left(V_{b 1}+V_{c 1}+V_{a 2}+V_{b 2}+V_{c 2}\right) \\ V_{a 1 n}=\frac{5}{6}\left(V_{a 1}^{*}+v_{Z S I}\right)-\frac{1}{6}\left(V_{b 1}^{*}+v_{Z S I}+V_{c 1}^{*}+v_{Z S I}+V_{a 2}^{*}\right. \\ \quad \quad \quad \quad \quad \quad \quad \quad \left.+v_{Z S I}+V_{b 2}^{*}+v_{Z S I}+V_{c 2}^{*}+v_{Z S I}\right) \\ V_{a 1 n}=\frac{5}{6}\left(V_{a 1}^{*}\right)-\frac{1}{6}\left(V_{b 1}^{*}+V_{c 1}^{*}+V_{a 2}^{*}+V_{b 2}^{*}+V_{c 2}^{*}\right) \end{array}\)       (12)

Furthermore, the modulating voltage signals are shifted to be symmetrical along the horizontal-axis, which results in a maximized modulation index. It is must be noted that min-max injection reduces the magnitude and yields the same modulating signals since it essentially reverts to the basic voltages that are required by the machine (phase voltages).

IV. RESULTS AND DISCUSSIONS

In order to verify the effectiveness of min-max injection in extending the maximum modulation limit for a post-fault S6-1N, experimental results for fault scenarios up to 3 OCFs were conducted. In the post-fault control, the flux current i_d is maintained at its rated value of 1.3A. Meanwhile, the torque current is reduced so that the maximum post-fault phase current is equal to the rated phase current of a 3.55A peak.

The motor speed was gradually increased until the maximum modulation limit was achieved (i.e. when the highest modulating signal(s) is equal to 1). Table I summarizes the post-fault torque and maximum achievable speed before and after min-max injection considering different fault scenarios. It should be highlighted that even though flux current i_d is maintained at its rated value, the post-fault torque current i_q (hence post-fault torque T’) differs depending on the derating factor. Furthermore, since machine voltage under the faulted mode is dominated by the q-axis voltage, which depends on the flux-current (i_d) and not the torque current (i_q) [26], the evaluation of the maximum speed is not significantly affected by different values of i_q.

TABLE I INDEPENDENT FAULT SCENARIOS FOR A SYMMETRICAL MACHINE CONSIDERING MIN-MAX INJECTION

As expected, the two approaches yielded different maximum speeds, with [T_ind] having the lower speed limit due to the presence of V_cm. Nevertheless, with the min-max injection, the maximum attainable speeds for both methods became the same.

To further illustrate the impact of min-max injection, Fig. 5 shows a comparison of the maximum speed before and after injection. As can be seen from the figure, the use of min-max injection effectively reduces the amplitude of the modulating signals and extends the maximum modulation limit. Consequently, the machine can achieve a higher operable speed and yield a higher power in the post-fault mode, without violation of the voltage limit. It is worth noting that applying min-max injection to either of the transformation matrixes essentially results in the same modulating signals and maximum speeds.

Fig. 5. Comparison of the maximum achievable speed before and after min-max injection between [T_ind] and [T_comb] for: (a) 1 OCF. (b) 2 OCFs. (c) 3 OCFs.

Fig. 6, Fig. 7 and Fig. 8 illustrate the modulating voltages, phase currents and common-mode voltages before and after min-max injection for 1 OCF, 2 OCFs and 3 OCFs, respectively. Based on Table I, since the power is the product of the torque and the speed, the maximum post-fault power of the machine is affected. For all of the cases, the use of min-max injection increases the maximum modulation index and the maximum post-fault operating speed.

Fig. 6. Test 1: 1 OCF with min-max injection at the rated speed based on [T_ind] (left plots) and [T_comb] (right plots). From top to bottom. (a) Modulating signals. (b) Phase currents. (c) Common-mode signal, V_cm. (d) Phase voltages. (e) Line voltages.

Fig. 7. Test 1: 2 OCFs (scenario 2a) with min-max injection at the rated speed based on [T_ind] (left plots) and [T_comb] (right plots). From top to bottom. (a) Modulating signals. (b) Phase currents. (c) Common-mode signal, V_cm. (d) Phase voltages. (e) Line voltages.

Fig. 8. Test 1: 3 OCFs (scenario 3a) with min-max injection at the rated speed based on [T_ind] (left plots) and [T_comb] (right plots). From top to bottom. (a) Modulating signals. (b) Phase currents. (c) Common-mode signal, V_cm. (d) Phase voltages. (e) Line voltages.

As shown in [27], the ability to run above the base speed can be advantageous for post-fault operation since the additional increase in speed can compensate the low post-fault torque and increase the overall post-fault power. Furthermore, as predicted, once min-max injection is used, the maximum postfault speeds of the machine using either of the transformations are the same.

The modulating index should be decided with respect to the line voltages. This means that if the line-to-line voltage is less than ±Vdc, there is no overmodulation issue. Fig. 9 shows the normalized line-to-line voltage at the maximum speed before and after min-max injection for 1 OCF with 1N based on the MT and ML modes, respectively. Referring to Fig. 9 and Table II, it can be seen that min-max injection can give a higher speed when compared to the MT and ML modes without using min-max injection. Furthermore, the value of the line-to-line voltage after min-max injection at the maximum speed is the same as that with the dc link voltage.

Fig. 9. Normalized line voltage amplitude at the maximum speed with and without min-max injection (mm) for 1 OCF based on Maximum Torque (MT) or Minimum Loss (ML) connected with a single isolated neutral (1N).

TABLE II MAXIMUM LINE VOLTAGE BASED ON DIFFERENT OPERATING MODES AND NEUTRAL CONFIGURATIONS

V. CONCLUSIONS

This paper investigated the effect of a post-fault control method for a S6-1N machine in terms of voltage and speed limit using min-max injection. It was demonstrated that the voltage limit has a direct impact on the speed limit. The line voltage amplitude, on the other hand, depends on the zero sequence components. Although both transformations can effectively control the machine in the post-fault mode, they yield different modulating signals. As a result, they have different maximum speed limits. When comparing the two transformations, [T_comb] is found to be better in terms of the number of current controllers required and the maximum modulation limit. It was further demonstrated that min-max injection can be utilized to improve the post-fault performance of a S6-1N machine. The use of min-max injection maximizes the modulation limit and post-fault speed of the machine, giving overall better post-fault power.