Fault Tolerant Control of Five-Phase Induction Motor Drive

Abstract

The fault-tolerant capability of multiphase induction motor (IM) drives without adding extra hardware has been an interesting research subject in recent times. Regardless of the application and reliability requirements, fault tolerance is obtained by the software. Although different types of faults may occur, the most considered is the open-phase one which leads to a reduction in the number of active phases in the multiphase drive. Predictive current control (PCC) was recently proposed in the literature for managing the post-fault operation of the drives when an open-phase fault is considered. In PCC, the motor torque is controlled indirectly by controlling the motor current. Predictive torque control (PTC) can control the motor torque directly. However, PTC-based fault tolerant control of a five-phase IM (FPIM) drive has not been published in the literature. Hence, this fault-tolerant capability using the PTC method for an FPIM has been discussed in this chapter. Detail about the modeling of an FPIM, fault detection, and controller for both healthy and faulty conditions of IM has been discussed. The same model of the drive under both healthy and faulty conditions is considered. But the applied cost function is changed in a faulty condition.

1. Introduction

Since the late 1990s, Multiphase motors are getting gradual popularity over their counterpart three-phase motors in various application fields, especially where critical safety is required, for example, propulsion applications, electric aircraft, electric vehicles, etc. [1]. Multiphase motors can drive the load smoothly, even if one or more phases of the motors are damaged. Among the multiphase motors, the five-phase induction motors (FPIMs) are widely used [2]. They have become a useful replacement for the three-phase motors because of higher reliability, higher power handling capacity without exceeding the current handling capacity of semiconductor switches and insulations, and better torque performance. Although multiphase drives are generally claimed to be ‘fault-tolerant’, this term is somewhat broad since many different types of faults may appear in an electrical drive, including both inverter and machine faults that may lead to short-circuit (phase, inverter switch, inter-turn) or open-circuit (inverter switch, phase or line) faults. Among the aforementioned possible faults, the probability of open-phase fault (OPF) in a drive system is high [2].

Due to the advancement of faster microprocessors, model predictive control (MPC) has received wide attention over existing control algorithms for industrial drives such as field-orientation control (FOC), direct torque control (DTC), and proportional-resonant (PR) control. MPC uses plant models and digital control platforms and allows system constraints and restrictions in a very intuitive way. In different applications of motor drives, MPC is found superior to DTC, FOC, and PR controllers for its increased flexibility and faster torque response [2, 3]. The MPC for FPIM drives has two variants: one is predictive torque control (PTC) and another one is predictive current control (PCC). Both types of MPC can effectively control the torque, flux, and thus speed of the induction motors. In PCC, the motor torque and flux are controlled indirectly by controlling the motor current. On the other hand, in PTC, the motor torque and flux are controlled directly and thus comparatively faster torque response is achieved [3]. A PCC has been recently proposed for managing the post-fault operation of the drives when an OPF is considered [4]. The faulty situation assumes zero stator current while freewheeling diodes can continue conducting in a non-controlled mode. An analysis is presented on the post-fault operation of the five-phase drive when the freewheeling diodes of the faulty phase are still conducting. Another PCC is presented in [5] for a six-phase IM drive and it is reported that PCC misbehaves in a post-fault situation if there is a significant delay in fault detection. The aforementioned PCC approaches need to be reconfigured when a fault occurs. A universal reconfiguration-less PCC approach is presented in [6] and it is shown that the system is naturally fault-tolerant. However, the fault-tolerant capability of an FPIM using PTC has not been stated yet in the literature. In this chapter, the performance of a PTC-based FPIM drive in both healthy and faulty conditions has been discussed.

2. System modeling

The block diagram of the proposed PTC for the FPIM is shown in Figure 1. The motor is fed by a two-level five-phase (2L–5Ф) voltage source inverter (VSI). The inverter is driven by the control signals which are generated by the PTC controller (a variant of MPC). The control system has four sub-sections: FPIM, 2L–5Ф inverter, PTC controller, and fault identification. The reference torque is generated by an outer speed loop and PI controller. PTC uses the measured stator current and dc-link voltage, and also receives the status of the machine whether it is healthy or faulty from the fault identification sub-section to predict the control objectives such as stator current (α−β), harmonic current (x−y), stator flux and torque. The predicted control objectives are compared with their corresponding references, and their error costs are calculated for all possible voltage vectors of a 2L–5Φ inverter. Based on the calculated error costs, the optimization function selects a switching state which yields minimum error cost and then applies the voltage vector corresponding to the selected switching state to the motor through the inverter. If OPF occurs, the fault identification section identifies the fault and signals the controller to reconfigure it. Because of the reconfigured control structure, the motor can run smoothly in a faulty condition with a de-rated torque capacity.

Figure 1.

Block diagram of the proposed fault-tolerant PTC of a five-phase induction motor drive.

2.1 Five-phase induction motor modeling

The FPIM with a squirrel cage rotor has distributed windings that are symmetrically displaced by $\theta =\frac{2\pi }{5}$ in between phases. The machine is powered by a 2L–5Φ VSI. As a general rule, the number of possible switching states N is,

$$N=A^B$$E1

Here, A is the number of possible states of each leg of the inverter and B is the number of phases of the inverter. So, for this instance, the number of switching states N =$2^5=32$. This means there are 32 possible stator voltage vectors (30 active and 2 zero vectors). If $S_i$ is denoted as the switching state of each inverter leg where i = {a, b, c, d, e}, the switching state is ${\left[S_a{S}_b{S}_c{S}_d{S}_e\right]}^T$ where $S_iϵ\left\{0,1\right\}$ designates that when the lower switch is on and the upper switch is off. The switching state and the dc link voltage $V_{\mathit{dc}}$ may then be used to calculate the voltage for each stator phase using Eq. (2).

$$\left[\begin{array}{c}{V}_{\mathit{as}}\\ V_{\mathit{bs}}\\ V_{\mathit{cs}}\\ V_{\mathit{ds}}\\ V_{\mathit{es}}\end{array}\right]=\frac{V_{\mathit{dc}}}{5}\left[\begin{array}{ccccc}4& -1& -1& -1& -1\\ -1& 4& -1& -1& -1\\ -1& -1& 4& -1& -1\\ -1& -1& -1& 4& -1\\ -1& -1& -1& -1& 4\end{array}\right]\left[\begin{array}{c}{S}_a\\ S_b\\ S_c\\ S_d\\ S_e\end{array}\right]$$E2

Applying the current-invariant decoupling of Clarke’s transformation, Eq. (2) can be mapped into two orthogonal stationary subspaces, α−β and x−y, plus a zero-sequence component, as shown in Eq. (3). Here, the stator voltage vectors have been identified using a decimal number equivalent to the binary number $\left[\begin{array}{ccccc}{S}_a& S_b& S_c& S_d& S_e\end{array}\right]$.

$$\left[\begin{array}{c}{V}_{\mathit{\alpha s}}\\ V_{\mathit{\beta s}}\\ V_{\mathit{xs}}\\ V_{\mathit{ys}}\\ V_{\mathit{zs}}\end{array}\right]=\frac{2}{5}\left[\begin{array}{ccccc}1& cos\theta & cos2\theta & cos3\theta & cos4\theta \\ 0& \mathit{sin}\theta & sin2\theta & sin3\theta & sin4\theta \\ 1& cos2\theta & cos4\theta & cos\theta & cos3\theta \\ 0& sin2\theta & sin4\theta & sin\theta & sin3\theta \\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right]\left[\begin{array}{c}{V}_{\mathit{as}}\\ V_{\mathit{bs}}\\ V_{\mathit{cs}}\\ V_{\mathit{ds}}\\ V_{\mathit{es}}\end{array}\right]$$E3

Considering a sinusoidally distributed magnetomotive force (MMF), uniform air gap, symmetrically distributed windings, minimal magnetic saturation, and core losses, and with the application of a series of voltage equilibrium equations derived from the stator and rotor electromagnetic circuits, the five-phase IM may be represented in a stationary reference frame. Given this information, the functioning of the multiphase machine is defined by Eqs. (4)–(12).

$$V_s=I_s{R}_s+\frac{d{\psi }_s}{\mathit{dt}}$$E4

$$0=I_r{R}_r+\frac{d{\psi }_r}{\mathit{dt}}-{\mathit{j\omega }}_e{\psi }_r$$E5

$${\psi }_s=L_s{I}_s+L_m{I}_r$$E6

$${\psi }_r=L_m{I}_s+L_r{I}_r$$E7

$${\tau }_e=\frac{5}{2}p{I}_m\left\{{\psi }_r^{\ast }.I_s\right\}$$E8

$$j\frac{d{\omega }_m}{\mathit{dt}}={\tau }_e-{\tau }_l$$E9

$${\omega }_e=p{\omega }_m$$E10

$$\frac{d{I}_s}{\mathit{dt}}=\frac{1}{{\tau }_{\sigma }}\left[-I_s-\frac{K_r}{R_{\sigma }}\left(\frac{-{\psi }_r+j{\tau }_r{\omega }_e{\psi }_r}{{\tau }_r}\right)+\frac{1}{R_{\sigma }}{V}_s\right]$$E11

$$\frac{d{\psi }_r}{\mathit{dt}}=\frac{1}{{\tau }_r}\left(L_m{I}_s-{\psi }_r+j{\omega }_e{\tau }_r{\psi }_r\right)$$E12

The symbols or variables in Eqs. (4)–(12) have their usual meanings, as shown in Table 1 given in Appendix. Integrating Eq. (9) with respect to time, the rotor angular speed ${\omega }_m$ can be derived and it is directly related to the rotor angular frequency ${\omega }_e$ by the pole pairs $p$, which is formulated in Eq. (10). Replacing the variable $I_r$ with the variables ${\psi }_r$ and $I_s$ in Eq. (5) forms the modified rotor voltage in Eq. (12). Extracting $I_r$ from Eq. (7) and putting its value in Eq. (6) develops the relationship between stator and rotor flux. Differentiating this stator flux with respect to time and placing it in the stator voltage equation emerges Eq. (11) in a compact form which aids in the prediction of stator current for the next sampling instant. To anticipate the torque, rotor flux, and stator current predictions are necessary. The compact factors used in IM modeling (Eqs. (11)–(12)) are given in Appendix (Table 2).

2.2 FPIM modeling under faulty condition

If a fault occurs in the machine, the controller detects the fault first. Then, the controller reconfigures itself to accommodate the fault and continues to operate the drive under faulty condition. Eq. (13) is verified when the motor drive is in healthy operation. But when an OPF occurs (assume the faulty phase is ‘a’), the summation of healthy phase voltages is no longer equal to zero ($\sum V_{\mathit{ks}}\ne 0,\text{here}k\in \left\{b,c,d,e\right\})$. Then there comes the relation between α component and x component of stator current as in Eq. (15). The five vector space decomposition (VSD) [4] phase currents are no longer independent. Since phase voltage symmetry should remain valid in the post-fault situation, and the voltage of the faulty phase ‘a’ is no longer controllable due to its oscillation caused by the back EMF as in Eq. (14), the phase voltages must be modified accordingly to arrive at Eq. (16).

$$V_{\mathit{as}}+V_{\mathit{bs}}+V_{\mathit{cs}}+V_{\mathit{ds}}+V_{\mathit{es}}=0$$E13

$$V_{\mathit{as}}=I_{\mathit{as}}{R}_s+\frac{d{\lambda }_{\mathit{as}}}{\mathit{dt}}=\frac{d{\lambda }_{\mathit{as}}}{\mathit{dt}}=\mathit{\text{Back}}{\mathit{EMF}}_{\mathrm{a}}\left(\text{with}{I}_{\mathit{as}}=0\right)$$E14

$$I_{\alpha }+I_x=0\Rightarrow \frac{-I_x}{I_{\alpha }}=1$$E15

$$\left[\begin{array}{c}{V}_{\mathit{bs}}\\ V_{\mathit{cs}}\\ V_{\mathit{ds}}\\ V_{\mathit{es}}\end{array}\right]=\frac{V_{\mathit{dc}}}{4}\left[\begin{array}{cccc}3& -1& -1& -1\\ -1& 3& -1& -1\\ -1& -1& 3& -1\\ -1& -1& -1& 3\end{array}\right]\left[\begin{array}{c}{S}_b\\ S_c\\ S_d\\ S_e\end{array}\right]-\frac{L_m\frac{d{I}_{\mathit{\alpha s}}}{\mathit{dt}}+L_m\frac{d{I}_{\mathit{\alpha r}}}{\mathit{dt}}}{4}.\left[I_{\mathit{4}}\right]$$E16

Where, [I₄] is the identity matrix of order 4. The second term on the right-hand side of Eq. (14) is the counter EMF in terms of VSD variables. An asymmetrical system of equations depicting a two-phase machine with asymmetrical d−q windings results from the VSD performed by the model proposed in [7] based on an orthogonal decoupling matrix. Since, there are four healthy phases, the number of available switching states is reduced from 2⁵ = 32 to 2⁴ = 16 in the faulty condition. So, to reduce the complexity, a non-orthogonal reduced-order transformation matrix is deduced from Eq. (3) which is shown in Eq. (17). The healthy motor model still can be used. The steady-state α−β current references are circular, not ellipsoidal in both pre-fault and post-fault operations.

$$\left[\begin{array}{c}{V}_{\mathit{\alpha s}}\\ V_{\mathit{\beta s}}\\ V_{\mathit{ys}}\\ V_{\mathit{zs}}\end{array}\right]=\frac{2}{5}\left[\begin{array}{cccc}cos\theta -1& cos2\theta -1& cos3\theta -1& cos4\theta -1\\ sin\theta & sin2\theta & sin3\theta & sin4\theta \\ sin2\theta & sin4\theta & sin6\theta & sin8\theta \\ 1& 1& 1& 1\end{array}\right]\left[\begin{array}{c}{V}_{\mathit{bs}}\\ V_{\mathit{cs}}\\ V_{\mathit{ds}}\\ V_{\mathit{es}}\end{array}\right]$$E17

The new reduced-order Clarke transformation matrix permits the same set of α−β and x−y equations in post-fault conditions as in healthy operating conditions. It should be noted that the controller is not informed of the inactive phase of the machine during the occurrence of the OPF. So, PTC still will regulate the machine with an incorrect model. The delay in fault identification has a more significant impact on PTC than FOC. That is why there should be an integrated fault detection technique in PTC as fast as possible.

Several pieces of literature have been published, focusing on different fault-tolerant techniques based on VSD variables (named VSDFD) [4, 8] and observing the phase currents [9]. Reference [8] analyzes the fault occurrence using fault indices that are directly dependent on x−y current components. Fault identification using x−y current components is simpler as their values are equal to zero while the machine is in healthy condition and these current components have no contribution to the production of flux or torque. The procedure for generating fault indices involves using the transformation matrix of Eq. (3) to produce the phase current equations and setting them equal to zero ($I_{\mathit{ph}}=0,$the OPF condition). This results in the following set of equations:

$$R_a=-\frac{I_x}{I_{\alpha }}$$E18

$$R_b=\frac{I_x}{0.38I_{\alpha }+1.17I_{\beta }-0.73I_y}$$E19

$$R_c=\frac{I_x}{2.62I_{\alpha }-1.90I_{\beta }+3.08I_y}$$E20

$$R_d=\frac{I_x}{2.62I_{\alpha }+1.90I_{\beta }-3.08I_y}$$E21

$$R_e=\frac{I_x}{0.38I_{\alpha }-1.17I_{\beta }-0.7266I_y}$$E22

Here, $R_k$(k ϵ {a, b, c, d, e}) is denoted as the fault index of different phases. In normal operating conditions, $R_k=0,$ whereas and in faulty condition, $R_k=1$. That is why this method is particularly advantageous to detect and localize the fault. Since, phase ‘a’ is considered as the faulty phase here, $R_a=-\frac{I_x}{I_{\alpha }}=1$. But the controller must ensure that if there is an open phase fault apart from phase ‘a’, it should consider the faulty phase as ‘a’ and reorganize the other phases in sequential order. This action will allow the post-fault transformation matrix to ensure a smooth transition from pre-fault mode to post-fault mode and proper control performance. For instance, if we now consider phase ‘b’ as the faulty phase, then $R_b$ of Eq. (19) will be equal to 1 and this phase will now be marked as phase ‘a’ and the other phases will be reorganized sequentially.

The ratios $R_k$ can be integrated using the moving average method. Then a hysteresis band (2ε), as shown in Figure 1, is applied to the ratios of $R_k$. This hysteresis band ensures lower ripples (tends to zero) to the fault indices of healthy phases and generates filtered values of fault index ratio. These filtered values create new fault indices ($e_k)$, as shown in Figure 1, during the period of the moving average. Finally, the fault indices are then compared to a threshold and determine the fault flag symbolized as $F_k\in \left\{0,1\right\}$.

When any OPF occurs, the subspace orthogonal voltage components, α−β and x−y, must be calculated according to the modified transformation matrix as shown in Eq. (17). So, the following functions (Eqs. (23)–(27)) can be used to calculate the orthogonal subspace voltage components.

$$v_{\mathit{\alpha s}}=\frac{2}{5}\left(\left(cos\theta -1\right)v_{\mathit{bs}}+\left(cos2\theta -1\right)v_{\mathit{cs}}+\left(cos3\theta -1\right)v_{\mathit{ds}}+\left(cos4\theta -1\right)v_{\mathit{es}}\right)$$E23

$$v_{\mathit{\beta s}}=\frac{2}{5}\left(sin\theta v_{\mathit{bs}}+sin2\theta v_{\mathit{cs}}+sin3\theta v_{\mathit{ds}}+sin4\theta v_{\mathit{es}}\right)$$E24

$$v_{\mathit{xs}}=-v_{\mathit{\alpha s}}$$E25

$$v_{\mathit{ys}}=\frac{2}{5}\left(sin2\theta v_{\mathit{bs}}+sin4\theta v_{\mathit{cs}}+sin6\theta v_{\mathit{ds}}+sin8\theta v_{\mathit{es}}\right)$$E26

2.3 Two-level five-phase inverter

The 2L–5Ф VSI has ten switches (two switches per leg), as shown in Figure 2. The number of total states is 2⁵ = 32 with which two zero vectors and thirty active vectors. The voltage vector expressions in α−β and x−y planes can be written as,

Figure 2.

FPIM drive schematic diagram.

$$v_{\mathit{\alpha \beta }}=v_{\alpha }+j{v}_{\beta }=2/5\left(v_a+a{v}_b+a^2{v}_c+a^3{v}_d++a^4{v}_e\right)$$E27

$$v_{\mathit{xy}}=v_x+j{v}_y=2/5\left(v_a+a^2{v}_b+a^4{v}_c+a^6{v}_d++a^8{v}_e\right)$$E28

where, $a=e^{j2\pi /5}$and$v_a\dots v_e$ are the phase voltages.

The relation between phase voltage$v_x$ of phase x and switching states is,

$$v_x=S_x\ast V_{\mathit{dc}}$$E29

where the switching states of phase x are

$$S_x=\left\{\begin{array}{c}1,\text{if the upper switch is}\mathrm{on},\text{the lower switch is}\mathrm{off};\\ 0,\text{if the upper switch is}\mathrm{off},\text{the lower switch is}\mathrm{on}.\end{array}\right.$$

The subspace voltage components produced by a 2L–5Ф VSI are plotted in α−β and x−y planes, as shown in Figure 3.

Figure 3.

Voltage vectors (a) v_β vs. v_α (b) v_y vs. v_x..

The controller generates the optimum switching signal for the inverter as per the minimization of an objective function. The selected switching signal is then applied to the inverter. The inverter then supplies the required voltage to the induction motor.

2.4 PTC algorithm

The PTC works in three steps: the generation of the available voltage vectors for the inverter, the prediction of control objectives, and the selection of an optimal voltage vector by minimizing a predefined cost function. The number of available voltage vectors for 2L–5Ф inverter is 32 which are 16 if an OPF occurs in any one of the phases. The controller selects an optimum voltage vector by using the discrete mathematical models of the FPIM and inverter. The control objectives such as stator currents (α−β, x−y), stator flux, and torque are predicted by the discrete mathematical models of IM (Eqs. (4)–(12)) and inverter (Eqs. (27)–(29)). The two-steps ahead prediction strategy proposed in [10] is used for predicting the control objective. Hence, the control objectives are predicted for the time instant k + 2. The predicted control objectives are then evaluated by a predefined cost function. The cost function is designed based on the torque and flux errors, harmonics components i.e. x−y component, and high current protection constant. The mathematical formulation of the cost function with the two-steps ahead prediction is given below.

$$g=\left|{\tau }_e^{\ast }\left(k+2\right)-{\tau }_e^p\left(k+2\right)\right|+{\lambda }_f\left|\left|{\psi }_s^{\ast }\left(k+2\right)\right|-\left|{\psi }_s^p\left(k+2\right)\right|\right|+K+\left|\left|i_{\mathit{sxy}}^{\ast }\left(k+2\right)\right|-\left|i_{\mathit{sxy}}^p\left(k+2\right)\right|\right|$$E30

where K is a high current protection constant and${\lambda }_f$ is the weighting factor. The harmonic reference current $i_{\mathit{sxy}}^{\ast }$ is zero in healthy condition. Note that the cost function (30) must require slight modification in faulty operation. The new $i_{\mathit{sxy}}^{\ast }$ are generated by the fault identification sub-section, as shown in Figure 1, for the cost function in a faulty condition. In Eq. (30), the reference current $i_{\mathit{sx}}^{\ast }$ is set as $-i_{\mathit{s\alpha }}^{\ast }$ and the reference current $i_{\mathit{sy}}^{\ast }$ is set as zero for minimum copper loss in the faulty situation. The cost function (30) is modified as follows in a faulty condition.

$$g=\begin{array}{l}\left|{\tau }_e^{\ast }\left(k+2\right)-{\tau }_e^p\left(k+2\right)\right|+{\lambda }_f\left|\left|{\psi }_s^{\ast }\left(k+2\right)\right|-\left|{\psi }_s^p\left(k+2\right)\right|\right|\\ +K+\left|-i_{\mathit{s\alpha }}^{\ast }\left(k+2\right)-i_{\mathit{sx}}^p\left(k+2\right)\right|+{\lambda }_h\left|0-i_{\mathit{sy}}^p\left(k+2\right)\right|\end{array}$$E31

Where, ${\lambda }_h$ is a weighting factor for i_sy current. The switching state which yields minimum $g$ is applied to the inverter; this is done by the optimization function, as shown in Figure 1. For reliable/safe operation during the fault, the number of motor control variables of the faulty phase is reduced. Hence, the motor intends to run in de-rated capacity by control reconfiguration, and thus the fault-tolerant capability of the drive system is achieved.

3. Mode of operation

An FPIM fed by a 2L–5Φ is controlled by the PTC algorithm, and the motor is operated under both healthy and faulty conditions. The parameters of the motor and controller are given in Appendix (Table 3). The performance of the PTC for both healthy and faulty conditions of the motor has been illustrated in the following sub-section.

3.1 Healthy mode

3.1.1 No-load operation

The no-load speed, torque, stator current, stator flux, and fault indices responses under healthy condition are shown in Figure 4. The motor is driven at 500 rpm with no load. The stator flux vector is maintained constant at 0.55 wb. It can be seen that the motor behavior is good. The machine takes only 0.4 sec to track the reference speed, which is very fast. The estimated developed torque accurately tracks the reference torque and the ripple in the torque response is satisfactory. The stator current and estimated stator flux are presented in the α−β plane. The harmonic component of stator current (i.e. x−y component) is represented in the x−y plane which is nearly zero. It is noticed that stator currents and stator flux construct a circular path at the steady-state condition with no load. Fault indices also indicate that each phase is in healthy condition as the fault index of each phase remains below the threshold value of 0.15.

Figure 4.

(a) No-load speed, (b) torque, (c) stator current, (d) stator flux, and (e) fault indices response at no-load under healthy conditions.

3.1.2 Loaded operation

The loading behavior of the drive system is shown in Figure 5. A load of 56% of the nominal load (4.7 N-m) is suddenly applied to the motor at t = 2 sec. There is a tiny speed drop in the motor due to this load disturbance. However, the controller recovers the speed within a short time as shown in Figure 5(a). Hence, the system is robust against load disturbance. The torque ripple is low, as can be seen in Figure 5(b). The stator current and stator flux contain less harmonics and can be seen in Figures 5(c) and 5(d) respectively. As per the healthy condition, fault indices of all phases remain close to zero as shown in Figure 5(e).

Figure 5.

(a) Speed, (b) torque, (c) stator current, (d) stator flux, and (e) fault indices response at 56% of nominal load under healthy conditions.

3.1.3 Speed reversal operation

The speed transient behavior of the motor drive is shown in Figure 6. The motor is driven at 500 rpm, and a reverse speed of −500 rpm is commanded at t = 6 sec with 56% of nominal torque. The motor follows the command speed quickly without a negligible over/undershoot. Hence, the speed transient behavior of the PTC-based drive is good under healthy condition. From Figure 6(a), it can be seen that the machine takes only 0.5 sec to follow the reverse speed command. Figure 6(c) describes that while reversing the direction of rotation, stator currents pass through the condition where for fraction of a second these become DC, and the current in phase ‘a’ tends to be zero for a short time interval. So, the controller is bound to detect that there is an open phase fault in phase ‘a’ and the fault index of phase ‘a’ tries to surpass the threshold value of 0.15 in Figure 6(d) but this phenomenon does not stretch out for a long time. So, the fault index again is forced to be maintained at close to zero. Setting a high threshold value of fault indices at speed transient can solve this problem, but it will introduce fault detection delay which may lead the system unstable at faulty situation. Hence, a trade-off is required between the threshold value of fault indices and the fault detection delay.

Figure 6.

Speed reversal test (a) speed, (b) torque, (c) current, (d) fault indices response at 56% of nominal load under healthy condition.

3.2 Faulty mode

3.2.1 Fault detection

The behavior of the PTC-based FPIM drive is tested in a faulty condition. Only OPF behavior is considered, and the fault is detected using the VSDFD technique. The machine is driven at 500 rpm with 56% of the nominal load, and the VSDFD algorithm is executed from the beginning when the motor was running in healthy condition. When the machine reaches the steady state, phase ‘a’ is disconnected from the inverter at t = 4 sec, as shown in Figure 7. Figure 7(a) justifies the relation which takes place between the x component and α component of stator current (Eq. (15)) when an OPF occurs in phase ‘a’. From Figure 7(b), it can also be seen that the fault indices are almost null in pre-fault or healthy condition, and the fault index ‘e_a’ increases in the post-fault situation. Notice, all other indices of remaining healthy phases, except ‘e_a’, in a post-fault situation have similar values (i.e. almost null) as pre-fault situation. A threshold value of indices is set as 0.15 to detect the fault and avoid false alarms as well, and it can be seen in Figure 8. Once the value of an index becomes greater than or equal to 0.15, the controller detects a fault in the corresponding phase. In Figure 8, it can be seen that the controller takes only 17 ms to detect the fault in phase ‘a’, which is 51% of the fundamental current cycle.

Figure 7.

Open switch fault detection (a) relation between I_α and I_x and (b) fault indices.

Figure 8.

Zoom-in figure of the fault indices of the marked portion in Figure 7(b).

3.2.2 Loaded operation

There is a smooth transition from the pre-fault to the post-fault situation of the motor drive, as shown in Figure 9. As the fault is injected in phase ‘a’ at time t = 4 sec, the phase current ‘i_a’ is zero in a post-fault condition. The controller tracks the reference speed of 500 rpm quickly in the post-fault situation and also yields similar torque responses in pre-fault and post-fault situations. However, the phase currents become unbalanced in the faulty situation. The phase ‘b’ and phase ‘e’ currents are the same but higher than the phase ‘c’ and phase ‘d’ currents. In order to avoid over-current flow through the stator winding the machine is operated in a de-rated torque condition which is 56% of nominal torque. Figure 9(e) shows that the α−β current components trace a circle.

Figure 9.

(a) Speed, (b) torque, (c) current, (d) fault indices response at 56% of nominal load torque in OPF condition.

However, the x−y current components trace a straight line along the x-axis because the y component is maintained at zero (for minimum copper loss) in the post-fault situation. Figure 9(f) shows that the α−β flux traces a circle which is expected. Similar speed, torque, current, and flux responses are also achieved with 28% of nominal load torque as shown in Figure 10. Figure 11 shows the speed, torque, current, flux, and fault indices responses of the drive system at 70% of the nominal load torque. It can be seen that the controller cannot track the reference torque properly. It is because there is a stator current rating limit at i_s = 2.3A to protect the stator winding from over-current. However, the controller detects the OPF properly and maintains a smooth transition from pre-fault to post-fault.

Figure 10.

(a) Speed, (b) torque, (c) current, (d) fault indices response at 28% of nominal load torque in OPF condition.

Figure 11.

(a) Speed, (b) torque, (c) current, (d) fault indices response at 70% of nominal load torque in OPF condition.

3.2.3 Speed reversal operation

Figure 12 demonstrates the rated-speed transient behavior of the machine under OPF situation while the machine carries 56% of the nominal load. Figure 12(b) shows that electrical torque is satisfactorily following the reference torque with less ripple. It can also be seen that the stator ‘a’ phase current is zero because of OPF. The stator current and the stator flux behavior are also satisfactory as can be seen in Figure 12(d) and (e) respectively.

Figure 12.

(a) Speed, (b) torque, (c) stator current and (d) fault indices response at faulty condition under speed reversal with 56% of nominal load.

4. Conclusions

Predictive torque control (PTC) of a five-phase induction motor (FPIM) with fault-tolerant capability is discussed in this chapter. Both healthy and faulty conditions (i.e. OPF) of the motor drive have been analyzed. The vector space decomposition fault detection (VSDFD) technique is used in PTC to detect the fault. Based on the detected fault, PTC reconfigures its control structure so that it can control the motor in a faulty situation. Basically, the controller sets a new harmonic current reference in the cost function for handling the post-fault situation. The VSDFD algorithm is executed in both healthy and faulty conditions. If there is no fault occurs, the VSDFD algorithm generates an output of zero for each phase, otherwise one. Based on this zero or one output, the controller determines whether it needs to reconfigure itself or not. There are two challenges of a fault detection algorithm that is applicable for multiphase drive: one is fault detection speed and another one is a smooth transition from pre-fault to the post-fault situation. It is shown that the controller can detect the fault within 17 ms which is fast, and this detection time is a fraction of the fundamental current cycle. The proposed PTC can control the motor in both healthy and faulty conditions effectively. Moreover, a smooth transition is maintained from pre-fault to post-fault situations. The controller yields similar speed, torque, stator flux, and stator current responses at any load torque within the rating of the machine in both healthy and faulty conditions. Hence, the proposed PTC for the FPIM drive is fault-tolerant and robust against load disturbance.

Acknowledgments

This research is funded by University Grant Commission (UGC), Bangladesh. I also thank Khulna University of Engineering & Technology (KUET), Khulna, Bangladesh for allowing me to conduct this research.

Appendix

Symbols	Variables
$V_s$	Stator voltage
$I_s$	Stator current
$I_r$	Rotor current
${\psi }_s$	Stator flux
${\psi }_r$	Rotor flux
${\tau }_e$	Electromagnetic torque
${\tau }_l$	Load torque
${\omega }_m$	Rotor angular speed
${\omega }_e$	Rotor angular frequency
$p$	Pole pairs

Table 1.

Machine variables.

Factor	Symbols	Expression
Total leakage factor	σ	$1-\frac{L_m^2}{L_s{L}_m}$
Rotor coupling factor	${\tau }_e$	$\frac{L_m}{L_r}$
Equivalent resistance ref. to the stator	${\tau }_e$	$R_s+K_r^2{R}_r$
Transient stator time constant	${\tau }_e$	$\frac{L_{\sigma }}{R_{\sigma }}$
Leakage inductance	${\tau }_e$	$\sigma L_s$
Rotor time constant	${\tau }_e$	$\frac{L_r}{R_r}$

Table 2.

Compact factors used in machine modeling.

Parameter	Value	Parameter	Value	Parameter	Value
Ts	50 μs	p	3	Vdc	300 V
Kp, Ki	1.09, 38.48	Lm	681.7 mH	J	0.02 kg m^2
${\lambda }_f$	15	Ls = Lr	761.63 mH	${\varphi }_{\mathit{nom}}$	0.55 Wb
${\lambda }_h$	0.7	Rs	12.85 Ω	Tnom	4.7 N-m

Table 3.

Controller and machine parameters used in the simulation.

Nomenclature

θ

phase displacement angle

${\mathit{S}}_{\mathit{i}}$, i = {a, b, c, d, e}

switching state of each inverter leg

${\mathit{V}}_{\mathit{dc}}$

DC link voltage

${\mathit{V}}_{\mathit{\text{is}}}$, i = {a, b, c, d, e}

phase voltages

${\mathit{I}}_{\mathit{s}}$, ${\mathit{I}}_{\mathit{r}}$

stator and rotor currents

${\mathit{\psi }}_{\mathit{s}}$, ${\mathit{\psi }}_{\mathit{r}}$

stator and rotor flux

${\mathit{\tau }}_{\mathit{e}}$

electromagnetic torque

σ

total leakage factor

${\mathit{K}}_{\mathit{r}}$

rotor coupling factor

${\mathit{R}}_{\mathit{\sigma }}$

equivalent resistance ref. to the stator

${\mathit{\tau }}_{\mathit{\sigma }}$

transient stator time constant

${\mathit{L}}_{\mathit{\sigma }}$

leakage inductance

${\mathit{\tau }}_{\mathit{r}}$

rotor time constant

${\mathit{\omega }}_{\mathit{m}}$

rotor angular speed

${\mathit{\omega }}_{\mathit{e}}$

rotor angular frequency

R_i

fault indices