Motor Airgap Torque Harmonics Due to Cascaded H-Bridge Inverter Operating with Failed Cells

Abstract

This paper proposes the expressions for the motor airgap torque harmonics induced by a cascaded H-bridge inverter operating with failed cells. These variable frequency drive systems (VFDs), are widely used in oil and gas applications, where a torsional vibration evaluation is a critical challenge for field engineers. This paper proposes mathematical expressions that are crucial for an accurate torsional analysis during the design stage of VFDs, as required by international standards such as API 617, API 672, etc. By accurately reconstructing the electromagnetic torque from the stator voltages and currents in the (αβ0) reference frame, the obtained expressions enable the precise prediction of the exact locations of torque harmonics induced by the inverter under various real-world operating conditions, without the need for installed torque sensors. The neutral-shifted and peak-reduction fault-tolerant control techniques are commonly adopted under faulty operation of these VFDs. However, their effects on the pulsating torques harmonics in machine air-gap remain uncovered. This paper fulfils this gap by conducting a detailed evaluation of spectral characteristics of these fault-tolerant methods. The theoretical analyses are supported by MATLAB/Simulink 2024 based offline simulation and Typhoon based virtual real-time simulation results performed on a (4.16 kV and 7 MW) vector-controlled induction motor fed by a 7-level cascaded H-bridge inverter. According to the theoretical analyses- and simulation results, the Neutral-shifted and Peak-reduction approaches rebalance the motor input line-to-line voltages in the event of an inverter’s failed cells but, in contrast to the normal mode the carrier, all the triplen harmonics are no longer suppressed in the differential voltage and current spectra due to inequal magnitudes in the phase voltages. These additional current harmonics induce extra airgap torque components that can excite the lowly damped eigenmodes of the mechanical shaft found in the oil and gas applications and shut down the power conversion system due torsional vibrations.

1. Introduction

1.1. Investigated System and Background

In the oil and gas (O&G) industries, where the system’s reliability and availability are critical requirements, the cascaded H-bridge (CHB) inverter emerged as a valuable VFD due to its high modularity and fault-tolerant capability [1,2,3,4].

As commercialized in [2], the investigated system shown in Figure 1 is based on a CHB inverter operating with fault-tolerant control techniques (FTCTs) [5]. The phase disposition (PD) PWM is adopted as it provides the best voltage harmonic performances among the multi carrier PWMs [6,7,8]. Also, field-oriented control (FOC) is adopted due to its fast dynamic response and acceptance in the industry [2].

During the normal mode, each H-bridge cell generates a voltage $v_{i,x}$ = $\{-V_{DC},0,V_{DC}\}$, $\forall x\in L=\{\mathrm{1,2}, 3,\dots ,k\}$ and the inverter output LN voltage $v_i\left(t\right)$ is the sum of the individual voltages produced by the series-connected cells $\forall i\in 𝒫=\{a, b, c\}$.

During the failed mode, the related cells are bypassed using a contactor leading to $v_{i,x}=0$. If the physical and/or control part of the CHB inverter is not readjusted, it will produce unbalanced line-to-neutral (LN) and line-to-line (LL) voltages and currents at the motor terminal due to the inequal number of per phase operative cells.

To improve the reliability of the CHB inverter, several hardware- and software-based FTCTs have been previously developed [6,7,8,9,10]. However, the manufacturers of these VFDs often adopt cheaper software-based solutions such as the Neutral-shifted and Peak-reduction solutions [2,9,10,11].

These FTCTs modify the magnitudes and/or phase-shift angles of the PWM reference voltages to rebalance the CHB inverter output LL voltages while the LN voltages remain unbalanced [9,10]. In contrast to the normal mode [11], the zero-sequence harmonics appear in the LL voltage and current spectra during the abnormal (failed and corrected) modes. Moreover, the extra components of the torque created by these unsuppressed harmonics could excite the eigenmodes of the mechanical shafts found in O&G industries and shut down the system due to torsional vibrations.

1.2. Motivations and Importance of This Work

The torsional vibration issue is a critical challenge in the O&G industries because the torsional natural frequencies (TNFs) of their large mechanical shafts could be excited by the motor airgap torque harmonics induced by the abnormal modes of the CHB inverter and cause catastrophic failures. To ensure the reliability of these power conversion systems, international standards such as the API 617 recommend a torsional analysis at the design stage of such systems. It is beneficial to perform an accurate torsional analysis with a precise localization of the VFD-induced torque harmonics during the system’s normal and abnormal modes.

Additionally, several vendors propose VFDs based on the CHB inverter operating with a Neutral-shifted approach [2,9,10,11]. However, only its time domain performances are usually addressed. The effects of failed cells on the spectral characteristics of the CHB inverter as well as on the motor torque spectrum remain largely uncovered. These analyses are useful for the efficient design of such power conversion systems.

For the R&D engineers, this paper is a design tool helping them to develop reliable VFDs based on the CHB inverter operating with failed cells. The proposed expressions of the motor airgap torque harmonics are useful for torsional analysis purposes, as required by API 617. Moreover, the performance comparisons of the Neutral-shifted and Peak-reduction approaches provide them with the most suitable solution for these VFDs. Also, they can assess the effects of the drive parameters (number of failed cells, carrier and operating frequencies, etc.) on the system’s performance, leading to optimized and reliable VFDs.

For the field engineers, this paper is a root cause failure analysis tool. When a fault occurs in the power conversion system, the proposed expressions assist them in assessing whether the failure results from the CHB inverter. For instance, the torque harmonic localized at two times the operating frequency is useful to identify the machine’s failures caused by the unbalanced voltages produced during the abnormal operation modes of the CHB inverter.

1.3. State of the Art of Existing Fault-Tolerant Methods

In [12,13], the Neutral-shifted approach was introduced to enhance the reliability of the CHB inverter. Based on the number of operative cells per phase, this method adjusts the phase-shift angles of the PWM reference voltages to rebalance the inverter’s output LL voltages while keeping the LN voltages unbalanced. The experimental results indicate that the magnitudes of the current harmonics increase as the number of failed cells rises. But the impacts of the Neutral-shifted approach on the spectral characteristics of the CHB inverter and the motor torque spectrum have not been covered.

A generalized Neutral-shifted approach has been proposed in [14] to address simultaneously the effects of unequal DC voltages and failed cells on the CHB inverter load. The simulation and experiment results performed on an RL load validate its effectiveness in rebalancing the LL voltage and currents during the abnormal operation modes of the CHB inverter. However, the spectral characteristics of this approach, as well as its suitability for motor drive applications, have not been addressed.

Moreover, in [15], the Neutral-shifted approach proposed for the grid connected photovoltaic systems modifies both the phase-shift angles and magnitudes (by injecting a min-max homopolar component) of the PWM reference voltages to rebalance and maximize the CHB inverter output LL voltages. This approach extends the Neutral-shifted approach proposed in [12,13] since it considers the fractional DC link voltages generated by the batteries and photovoltaic panels. However, its applicability has not yet been addressed for motor drives. When using the Neutral-shifted approach, the unsuppressed zero-sequence harmonics in the LL voltage and current spectra induce extra torque components that could excite the TNFs of mechanical shafts found in the O&G industries.

The FOC based on the Peak-reduction method is proposed in [16] for the CHB inverter. Depending on the number of operative cells per phase and the motor setpoints (speed and torque), a dynamic homopolar component is injected in the PWM reference voltages to rebalance the motor input LL voltages. Since the Peak-reduction method affects the magnitudes of the LN voltages, the zero-sequence harmonics remain unsuppressed in the LL voltage spectra. Nevertheless, the work has been limited to minimize the motor torque ripple and ensure an equal power distribution between the healthy cells.

Additionally, the Peak-reduction method proposed in [17] computes the necessary homopolar component to rebalance the LL voltages based on the motor (active and reactive) power. This approach ensures equal power sharing between the healthy cells while keeping each reference voltage within its linear modulation region. However, the effects of the Peak-reduction method on the CHB inverter output voltage and the motor torque harmonics have not been addressed.

In [18], the neutral-voltage modulation (NVM) technique has been proposed to address both failed cells and unequal DC link voltages in the CHB inverter. Based on the number of healthy cells and the magnitude of each DC link voltage, a dedicated min-max homopolar component is synthesized and injected into the PWM reference voltages to rebalance and maximize the CHB inverter output LL while keeping each phase within the linear modulation region. The effectiveness of this approach has been validated using experimental results obtained from an RL load. Moreover, the performances of the NVM with both the PD-PWM and phase-shifted PWM have been compared in [19]. The simulation results obtained for an RL load indicate that the NVM with PD-PWM yields the best LL voltage and total harmonic distortion of current. Furthermore, an enhanced NVM approach has been introduced in [20]. Compared to the traditional NMV [18], the necessary common mode voltage is significantly reduced while keeping each phase within its linear modulation region. The simulation and experiment results performed on an RL load confirm the effectiveness of this approach.

In this paper, NVM techniques are not considered, as they are merely a variant of the Peak-reduction method proposed in [21].

The effects of the stator’s unbalanced voltages on the motor performances have also been addressed in the literature. The limitations of the most relevant contributions are outlined here. In [22,23], the expressions of the torque harmonics due to unbalanced voltages with only basebands have been proposed. These expressions predict the frequencies of the torque harmonics due to unbalanced voltages. But the case of unbalanced voltages with baseband, carrier, and sideband harmonics, such as those generated by the CHB inverter operating with failed cells, remains uncovered.

Moreover, in [24,25], the impacts of failed cells on the CHB inverter output voltages and the motor torque spectra have been addressed based on the fast Fourier transform of the simulated waveforms. According to simulation and experimental results, the zero-sequence harmonics are unsuppressed in the LL voltage and current spectra during the CHB inverter’s abnormal modes. Consequently, the torque components induced by these harmonics could excite the TNF of the mechanical shafts found in the O&G industries and shut down the system due to torsional vibrations. However, the expressions of the motor torque harmonics induced by the CHB inverter’s abnormal modes, as required by international standards, remain uncovered. Most of the VFD OEMs (Original Equipment Manufacturers) do not provide such expressions as required by the API 617 standard. They provide only the torque spectrum at selected motor operating points for torsional analysis purposes

To the best of the authors’ knowledge, the analytical effects of unbalanced voltages due to the CHB inverter operating with failed cells on the motor’s torque spectrum have not yet been addressed. Additionally, the effects of the Neutral-shifted and Peak-reduction methods on the CHB inverter and motor torque’s spectral characteristics remain unexplored. This paper addresses the aforementioned scientific gaps.

1.4. Major Contributions

The major contributions of this paper are given as follows:

(1)

The proposed expressions of the motor airgap torque harmonics induced by the CHB inverter’s abnormal modes are ready to use for torsional analysis purposes as required by international standards such as API 617. To the best of authors’ knowledge, such expressions have never been reported in any published paper. They are useful for R&D engineers to build reliable VFDs based on the CHB inverter operating with failed cells, such as those commercialized in [2,11,26,27].

(2)

The proposed correlation between the CHB inverter output voltage harmonics and the torque’s components during abnormal modes can be used to implement selective harmonic elimination algorithms.

The paper is organized as follows: Section 2 discusses the impacts of failed cells on the CHB inverter output voltage harmonics. The expressions of the motor torque harmonics due to the CHB inverter’s abnormal modes are established in Section 3. Section 4 outlines the relevant features of the Neutral-shifted and peak-reduction FTCTs. Section 5 displays the simulation results, and the spectral characteristics of these FTCTs and their impacts on the torque spectrum are also discussed. Section 6 concludes this paper.

2. Voltage Harmonics Generated by the CHB Inverter Operating with Failed Cells

2.1. Voltage Harmonics Generated by the CHB Inverter Operating in Normal Mode

In normal mode, the expressions of the LN voltages generated by the CHB inverter using the PD-PWM are provided in [8] for both the linear and overmodulation regions. The magnitude of each voltage harmonic depends on the number of series-connected cells and modulation index. These expressions have been simplified and are presented in Equation (1):

$$v_i\left(t\right)= V_{\mathrm{0,1}}^i\mathit{cos}\left({\omega }_{0}t+{\theta }_{\mathrm{0,1}}^v\right)$$

(1a)

$$+\sum _{n_v=2}^{\infty }{V}_{\mathrm{0,2}{n}_v-1}^i\mathit{cos}\left(\left(2n_v-1\right){\omega }_{0}t+{\theta }_{\mathrm{0,2}{n}_v-1}^v\right)$$

(1b)

$$+\sum _{m_v=1}^{\infty }{V}_{2m_v-\mathrm{1,0}}^i\mathit{cos}\left(\left(2m_v-1\right){\omega }_{c}t\right. \left.+{\theta }_{2m_v-\mathrm{1,0}}^v\right)$$

(1c)

$$+\sum _{m_v=1}^{\infty }\sum _{n_v=-\infty }^{\infty }{V}_{2m_v,2n_v+1}^i\mathit{cos}\left(2m_v{\omega }_{c}t+\left(2n_v+1 \right){\omega }_{0}t\left.+{\theta }_{2m_v,2n_v+1}^v\right)\right.$$

(1d)

$$+\sum _{m_v=1}^{\infty }\sum _{\begin{array}{c}{n}_v=-\infty \\ n_v\ne 0\end{array}}^{\infty }{V}_{2m_v-\mathrm{1,2}{n}_v}^i\mathit{cos}\left(\left(2m_v-1\right){\omega }_{c}t\right.+2n_v{\omega }_{0}t\left.+{\theta }_{2m_v-\mathrm{1,2}{n}_v}^v\right)$$

(1e)

where $V_{m_v,n_v}^i$ is the magnitude of the voltage harmonic on phase $i\in 𝒫=\{a, b, c\}$, and its detailed expressions are given in [27]; ($m_v$, $n_v$) are arbitrary integers related to the carrier- (${\omega }_c$), and fundamental (${\omega }_0$) frequencies; and the angular frequency (${\omega }_{m_v,n_v}$) and the phase-shift angles (${\theta }_{m_v,n_v}^v$) are given by Equations (2a) and (2b), respectively.

$${\omega }_{m_v,n_v}=m_v{\omega }_c+n_v{\omega }_0$$

(2a)

$${\theta }_{m_v,n_v}^v=m_v{\theta }_c+n_v{\theta }_0^i$$

(2b)

where ${\theta }_c$ is the phase-shift angle of the carrier signals and ${\theta }_{\mathrm{0,1}}^v={\theta }_0^i\in \{0,-2\pi /3, 2\pi /3 \}$ is the phase-shift angle of the fundamental voltage in phase $i\in 𝒫=\{a, b, c\}$.

During normal mode, the baseband harmonics given in Equation (1b) appear only in the overmodulation region [8]. Also, the carrier harmonics are given in Equation (1c) and the triplen ($n_v$) is a multiple of 3. In Equations (1b, 1d–1e), harmonics are suppressed in all the LL voltage and current spectra, as they are common-mode components [8].

2.2. Voltage Harmonics Generated by the CHB Inverter Operating in Failed Mode

The frequencies and phase-shift angles of the voltage harmonics, as indicated by Equation (2), are independent of the number of operational cells per phase. As a result, the voltage harmonics observed in the spectra of both healthy and failed phases are localized exactly at the same frequencies, but their magnitudes differ. Additionally, the carrier and triplen harmonics are suppressed only in the LL voltage spectra of the healthy phases, as they are in-phase and have equal magnitudes in the corresponding LN voltage spectra.

Furthermore, Figure 2 summarizes the impact of failed cells on the CHB inverter’s output voltage spectra, showing baseband harmonics in Figure 2a,c when the failed phase is in the overmodulation region.

Based on the values of ($m_v$, $n_v$), the following types of harmonics appear in the LN and LL voltages generated by the CHB inverter during the abnormal modes:

• Fundamental, $m_v=0$; $n_v=1$, it is the useful component.
• Baseband harmonics, $m_v=0; n_v>1$, for the PD-PWM, they are localized at odd multiples of the fundamental frequency as shown in Equation (1b). Their ranks are given in Equation (3):

  $$n_v=2ϵ+1, \forall ϵ=\mathrm{0,1},\mathrm{2,3},\dots$$

  (3)

They appear only the LN voltage spectrum of the failed phases. In the LL voltage and current spectra, they are in the form of positive ($n_{pos}$), negative ($n_{neg}$), and zero-sequence ($n_z$) harmonics, as given in Equations (4a)–(4c), respectively:

$$n_{pos}=6l+1$$

(4a)

$$n_{neg}=6l-1$$

(4b)

$$n_z=3\left(2l+1\right), \forall l=\mathrm{0,1},\mathrm{2,3},\dots$$

(4c)

The triplen harmonics given in Equation (4c) appear only in the LL voltage spectra of the phases operating with an unequal number of healthy cells.

iii.

Carrier harmonics, $m_v>0$ and $n_v=0$, as shown in Equation (1c) they appear at odd multiples of the carrier frequency. Their ranks are given in Equation (5):

$$m_v=2\gamma +1, \forall \gamma =\mathrm{0,1},\mathrm{2,3},\dots$$

(5)

They only appear in the LL voltage spectra involving the healthy and failed phases.

iv.

Sideband harmonics, $m_v\ne 0$ and $n_v\ne 0$, as shown in Equations (1d)–(1e), they are located around even and odd multiples of the carrier frequency, respectively. Their ranks are given in Equations (6) and (7), respectively.

a.

Sideband harmonics around even multiples of carrier frequency

$$\begin{array}{l}{m}_v=2\gamma , \forall \gamma =1, \mathrm{2,3},\dots \\ n_v=2ϵ+1,\forall ϵ=0, \pm 1, \pm 2, \pm 3, \dots \end{array}$$

(6)

In the LL voltage and current spectra, they appear as given in Equation (4).

b.

Sideband harmonics around odd multiples of carrier frequency

$$\begin{array}{l}{m}_v=2\gamma -1, \forall \gamma =1, \mathrm{2,3},\dots \\ n_v=2ϵ,\forall ϵ=0, \pm 1, \pm 2, \pm 3, \dots \end{array}$$

(7)

In the LL voltage and current spectra, they appear as given in Equation (8).

$$n_{pos}=6l-2$$

(8a)

$$n_{neg}=6l+2$$

(8b)

$$n_z=3\left(2l\right), \forall l=\mathrm{0,1},\mathrm{2,3},\dots$$

(8c)

The triplen harmonics given in Equations (4c) and (8c) appear only in the differential modes of the phases operating with an unequal number of healthy cells.

3. Motor Airgap Torque Harmonics Due to the CHB Inverter Operating with Failed Cells

This section develops the expressions for the motor torque due to the CHB inverter operating with failed cells, based on [28,29,30]. The block diagram of the adopted methodology is shown in Figure 3, where P represents the number of motor poles. In this paper, the voltage drop due to the stator’s impedance is not considered because it is negligible for multi-megawatt motors operating around their rated speed and power [28,29,30]. As a result, the airgap flux is the time-domain integral of the motor input voltages only.

In large motors, such as those found in certain compressor drivetrains, the stator voltage drop is neglected before the flux is estimated to compute the motor airgap torque. The rationale is that most variable-speed drives operate between 65% and 105% of their nominal speed and power. This assumption has no impact on the location of the torque harmonic components in the frequency domain, even for low-power motors. Only their magnitudes are slightly less accurate. For torsional analysis purposes, these results can still be used to predict operating points that may threaten shaft survivability due to increased shear stress resulting from the variable speed motor airgap. All offline and real-time simulations discussed in this paper have been performed considering the stator impedance. The results show that the calculated torque components are consistent with the simulations.

Additionally, the analysis is performed in the steady state due to the larger mechanical time constants compared to the electrical constraints.

3.1. Generic Expressions of the Airgap Torque Harmonics Due to the Stator’s Unbalanced Voltages

The expressions for the stator’s unbalanced voltages and currents generated by the CHB inverter during abnormal modes are given in Equations (9a) and (9b).

$$v_j\left(t\right)=V_{m_v,n_v}^j\mathit{cos}\left({\omega }_{m_v,n_v}t+{\theta }_{m_v,n_v}^v\right)$$

(9a)

$$i_j\left(t\right)=I_{m_i,n_i}^j\mathit{cos}\left({\omega }_{m_i,n_i}t+{\theta }_{m_i,n_i}^i\right)$$

(9b)

where $V_{m_v,n_v}^j$ and $I_{m_i,n_i}^j$ are the magnitudes of the voltage and current harmonics on phase $j\in 𝒫=\left\{a,b,c\right\}$, such as $V_{m_v,n_v}^a\ne V_{m_v,n_v}^b\ne V_{m_v,n_v}^c$ and $I_{m_i,n_i}^a\ne I_{m_i,n_i}^b\ne I_{m_i,n_i}^c$;

($m_i,n_i$) are constant integers related to the current harmonics; ${\omega }_{m_v,n_v}=m_v{\omega }_c+n_v{\omega }_0$ and ${\omega }_{m_i,n_i}=m_i{\omega }_c+n_i{\omega }_0$ are their respective angular frequencies; ${\theta }_{m_v,n_v}^v=m_v{\theta }_c+n_v{\theta }_0^j$ and ${\theta }_{m_i,n_i}^i=m_i{\theta }_c+n_i\left({\theta }_0^j+{\delta }_0\right)$ are their phase-shift angles, such as ${\theta }_0^j\in \{0,-2\pi /3, 2\pi /3\}$; and ${\delta }_0$ is the phase-shift angle between the fundamental voltage and current.

In the (αβ) stationary reference frame, the airgap torque is expressed in Equation (10):

$$t_e\left(t\right)={\displaystyle \frac{3}{2}}{\displaystyle \frac{P}{2}}\left({\phi }_{\alpha }\left(t\right)i_{\beta }\left(t\right)-{\phi }_{\beta }\left(t\right)i_{\alpha }\left(t\right)\right)$$

(10)

where (${\phi }_{\alpha }\left(\mathrm{t}\right),{\phi }_{\beta }\left(\mathrm{t}\right)$) and ($i_{\alpha }\left(t\right)$, $i_{\beta }\left(t\right)$) are the components of the stator flux and current.

The (αβ) components of any three-phase quantity ($x$ can be voltage, current, or flux) are obtained using the magnitude-invariant Clarke transformation given in Equation (11):

$$\left[\begin{array}{c}{x}_{\alpha }\\ x_{\beta }\\ x_0\end{array}\right]={\displaystyle \frac{2}{3}}\left[\begin{array}{ccc}1& -1/2& -1/2\\ 0& \sqrt{3}/2& -\sqrt{3}/2\\ 1/2& 1/2& 1/2\end{array}\right]\left[\begin{array}{c}{x}_a\\ x_b\\ x_c\end{array}\right]$$

(11)

As a result, the ($\alpha \beta$) components of the stator’s flux induced by the three phase unbalanced voltages, such as those given in Equation (9a), can be expressed as given in Equation (12):

$${\displaystyle \genfrac{}{}{0pt}{}{{\phi }_{\alpha }\left(t\right)=\frac{V_{11,m_v,n_v}}{m_v{\omega }_c+n_v{\omega }_0}\mathit{sin}\left({\omega }_{m_v,n_v}t\right)-\frac{V_{12,m_v,n_v}}{m_v{\omega }_c+n_v{\omega }_0}\mathit{cos}\left({\omega }_{m_v,n_v}t\right)}{{\phi }_{\beta }\left(t\right)=\frac{V_{21,m_v,n_v}}{m_v{\omega }_c+n_v{\omega }_0}\mathit{sin}\left({\omega }_{m_v,n_v}t\right)-\frac{V_{22,m_v,n_v}}{m_v{\omega }_c+n_v{\omega }_0}\mathit{cos}\left({\omega }_{m_v,n_v}t\right) }}$$

(12)

Also, the ($\alpha \beta$) components of the stator’s unbalanced current can be written as given in Equation (13):

$${\displaystyle \genfrac{}{}{0pt}{}{i_{\alpha }\left(t\right)=I_{11,m_i,n_i}\mathit{cos}\left({\omega }_{m_i,n_i}t+{\delta }_{ni}\right)+I_{12,m_i,n_i}\mathit{sin}\left({\omega }_{m_i,n_i}t+{\delta }_{ni}\right)}{i_{\beta }\left(t\right)=I_{21,m_i,n_i}\mathit{cos}\left({\omega }_{m_i,n_i}t+{\delta }_{ni}\right)+I_{22,m_i,n_i}\mathit{sin}\left({\omega }_{m_i,n_i}t+{\delta }_{ni}\right)}}$$

(13)

where $V_{\mu \nu ,m_v,n_v}$ and $I_{\mu \nu ,m_i,n_i}$ are parameters depending, respectively, on the magnitudes of the voltages and currents given in Equations (9a)–(9b). Their detailed expressions are given in the Appendix A, with $\mu ,\nu \in \left\{\mathrm{1,2}\right\}$. Also, ${\delta }_{ni}=n_i{\delta }_0$.

By considering Equations (12) and (13) in Equation (10), the expressions of the airgap torque induced by the stator’s unbalanced voltages and currents are given in Equation (14):

$$t_e\left(t\right)={\displaystyle \frac{T_{1,m,n}}{m_v{\omega }_c+n_v{\omega }_0}}\mathit{cos}\left({\omega }_{1}t+{\theta }_{1,m,n}\right)$$

(14a)

$$+{\displaystyle \frac{T_{2,m,n}}{m_v{\omega }_c+n_v{\omega }_0}}\mathit{cos}\left({\omega }_{2}t+{\theta }_{2,m,n}\right)$$

(14b)

$${\omega }_1=\left(m_v+m_i\right){\omega }_c+\left(n_v+n_i\right){\omega }_0$$

(14c)

$${\omega }_2=\left(m_v-m_i\right){\omega }_c+\left(n_v-n_i\right){\omega }_0$$

(14d)

where $(T_{1,m,n},T_{2,m,n})$ are the magnitudes of the torque harmonics. The expressions of their phase-shift angles $\left({\theta }_{1,m,n},{\theta }_{2,m,n}\right)$ are given in the Appendix; $(m,n)$ are arbitrary constant integers identifying the torque harmonics.

The interaction between the stator’s flux and current during the abnormal modes produces a motor torque having two components localized at the frequencies ${\omega }_1$ and ${\omega }_2$ given, respectively, in Equations (14c) and (14d):

• The natural torque harmonics are defined by Equations (14a) and (14c) [28];
• While those defined by Equations (14b) and (14d) appear only during the abnormal modes.

3.2. Relevant Torque Harmonics Due to the CHB Inverter Operating with Failed Cells

Since the term “$m_v{\omega }_c+n_v{\omega }_0$” is in the denominator of Equation (14) and for any PWM inverter, the relation ${\omega }_c\gg {\omega }_0$ is verified, and the torque harmonics have larger magnitudes when $m_v=0$ [30].

3.2.1. Effects of Voltage and Current Harmonics Localized at the Same Frequency

During normal mode [30], the DC component of the torque is created by the interaction between the voltage and current harmonics localized at the same frequencies. To assess the effects of abnormal modes on the motor torque spectrum, let us set $m_v=m_i$ and $n_v=n_i$; as a result, Equation (14) becomes Equation (15):

$$t_e\left(t\right)={\displaystyle \frac{T_{\mathrm{1,2}{m}_v,2n_v}}{m_v{\omega }_c+n_v{\omega }_0}}\mathit{cos}\left(2\left(m_v{\omega }_c+n_v{\omega }_0\right)t+{\theta }_{1,m_v,n_v}\right)$$

(15a)

$$+{\displaystyle \frac{T_{\mathrm{2,0},0}}{m_v{\omega }_c+n_v{\omega }_0}}\mathit{cos}\left({\theta }_{2,m_v,n_v}\right)$$

(15b)

During abnormal modes, this interaction creates two groups of torque harmonics:

• In Equation (15a), torque components at two times the frequency of the considered voltage and current harmonics are produced. However, they have low magnitudes when $m_v>0$ due to “$m_v{\omega }_c+n_v{\omega }_0$” in their denominator.
• In Equation (15b), the DC component is also created. Its magnitude is proportional to the phase-shift angle between the voltage and current harmonics. Figure 4a shows this interaction for the specific case of $m_v=m_i=0$ and $n_v=n_i=0$.

3.2.2. Baseband Harmonics of the Motor Torque

• Interaction between the fundamental current and baseband harmonics of voltage

By setting ($m_i=0$, $n_i=1$) and ($m_v=0$, $n_v=2ϵ+1$), Equation (14) becomes Equation (16):

$$t_e\left(t\right)=T_{\mathrm{1,0},n_v+1}\mathit{cos}\left(2\left(ϵ+1\right){\omega }_{0}t+{\theta }_{\mathrm{1,0},n_v}\right)+T_{\mathrm{2,0},n_v-1}\mathit{cos}\left(2ϵ{\omega }_{0}t+{\theta }_{\mathrm{2,0},n_v}\right)$$

(16)

Each voltage harmonic of rank $n_v$ create two torque harmonics of rank $n_v-1$ and $n_v+1$. Thus, a positive sequence harmonic of voltage and its related negative sequence, as given in Equations (4a)–(4b), create the torque components of orders ($n_{t1}$, $n_{t2}$, $n_{t3}$) given in Equation (17):

$$n_{t1}=n_{neg}-1$$

(17a)

$$n_{t2}={\displaystyle \frac{n_{neg}+n_{pos}}{2}}$$

(17b)

$$n_{t3}=n_{pos}+1$$

(17c)

Furthermore, the zero-sequence harmonics given in Equations (4c) and (8c) create the torque components of orders given in Equation (18):

$$n_{t1,z}=n_z-1$$

(18a)

$$n_{t2,z}=n_z+1$$

(18b)

These interactions are shown in Figure 4b,c. In Equations (17) and (18), the torque components localized at two times the frequency of the voltage harmonics are not shown since they have lower magnitudes compared to those localized at $\left(n_v-1\right){\omega }_0$ and $\left( n_v+1\right){\omega }_0$.

2.

Interaction between the fundamental voltage and baseband harmonics of current

In Equation (14), by setting ($m_v=0$, $n_v=1$) and ($m_i=0$, $n_i=2ϵ+1$), the airgap torque harmonics are expressed by Equation (19):

$$t_e\left(t\right)=T_{\mathrm{1,0},n_i+1}\mathit{cos}\left(2\left(ϵ+1\right){\omega }_{0}t+{\theta }_{\mathrm{1,0},n_i}\right)+T_{\mathrm{2,0},{1-n}_i}\mathit{cos}\left(2ϵ{\omega }_{0}t+{\theta }_{\mathrm{2,0},n_i}\right)$$

(19)

These harmonics appear at the same frequency as those given in Equation (16). However, they have larger magnitudes because the denominators in their expressions depend only on the fundamental frequency “${\omega }_0$”.

The baseband harmonics of the torque result from the superposition of the components given in Equations (15a), (16) and (19). They are localized, as given in Equation (20):

$$t_e\left(t\right)=T_{0,n}\mathit{cos}\left(2n{\omega }_{0}t+{\theta }_{0,n}\right),\forall n=\mathrm{1,2},3,\dots$$

(20)

3.2.3. Effects of the Carrier Harmonics of the Voltage and Current

During abnormal modes, the carrier harmonics are also converted into torque harmonics since they are uncancelled in the LL voltage spectra.

• Interaction between the fundamental current and carrier harmonics of voltages

By setting the parameters in Equation (14) to ($m_i=0$, $n_i=1$) and ($n_v=0$, $m_v\ne 0$), the corresponding harmonics of the torque are given by Equation (21). They are negligible due to the term “$m_v{\omega }_c$” in the denominators of their expressions.

$$t_e\left(t\right)={\displaystyle \frac{T_{1,m_v,1}}{m_v{\omega }_c}}\mathit{cos}\left(m_v{\omega }_{c}t+{\omega }_{0}t+{\theta }_{1,m_v,1}\right)$$

(21a)

$$+{\displaystyle \frac{T_{2,m_v,1}}{m_v{\omega }_c}}\mathit{cos}\left(m_v{\omega }_{c}t-{\omega }_{0}t+{\theta }_{2,m_v,1}\right)$$

(21b)

2.

Interaction between the fundamental voltage and carrier harmonics of current

By setting ($m_v=0$, $n_v=1$) and ($m_i\ne 0$, $n_i=0$), Equation (14) becomes Equation (22):

$$t_e\left(t\right)={\displaystyle \frac{T_{1,m_i,1}}{{\omega }_0}}\mathit{cos}\left(m_i{\omega }_{c}t+{\omega }_{0}t+{\theta }_{1,m_i,1}\right)$$

(22a)

$$+{\displaystyle \frac{T_{2,m_i,1}}{{\omega }_0}}\mathit{cos}\left(m_i{\omega }_{c}t-{\omega }_{0}t+{\theta }_{2,m_i,1}\right)$$

(22b)

These torque harmonics are located at the same frequency as those given in Equation (21). But they have larger magnitudes because the denominators of their expressions depend only on (${\omega }_0$).

With $m_i=2\gamma +1$, they are the first sideband harmonics around odd multiples of the carrier frequency. This interaction is shown in Figure 5 when $\gamma =0$.

3.2.4. Sideband Harmonics of the Motor Torque

• Interaction between the fundamental current and sideband harmonics of the voltage

With ($m_i=0$, $n_i=1$) and ($m_v\ne 0$ and $n_v\ne 1$), the related torque harmonics have very low magnitudes due to the presence of “$m_v{\omega }_c+n_v{\omega }_0$” in the denominators of their expressions. In this paper, they are neglected.

2.

Interaction between the fundamental voltage and sideband harmonics of the current

By setting ($m_v=0$, $n_v=1$) and ($m_i\ne 0$, $n_i\ne 0$), Equation (14) becomes Equation (23):

$$t_e\left(t\right)={\displaystyle \frac{T_{1,m_i,n_i}}{{\omega }_0}}\mathit{cos}\left(m_i{\omega }_{c}t+\left(n_i+1\right){\omega }_{0}t+{{\theta }_{1,m_i,n}}_i\right)$$

(23a)

$$+{\displaystyle \frac{K_t}{{\omega }_0}}{T}_{2,m_i,n_i}\mathit{cos}\left(m_i{\omega }_{c}t+\left(n_i-1\right){\omega }_{0}t+{\theta }_{2,m_i,n_i}\right)$$

(23b)

This interaction produces the significant sideband harmonics of torque because the denominators of Equation (23) depend only on the fundamental frequency (${\omega }_0$). These sidebands harmonics are located as follows:

■

By considering Equation (6) in Equation (23), they appear where even multiples of (${\omega }_c$) are paired with even multiples of (${\omega }_0$), as given in Equation (24a).

■

The carrier harmonics of the torque are included in (Equation (24a), when $ϵ=0$). They appear at even multiples of the carrier frequency (${\omega }_c$).

■

By considering Equation (7) in Equation (23), the related harmonics appear where odd multiples of (${\omega }_c$) are paired with odd multiples of (${\omega }_0$), as given in Equation (24b):

$$t_e\left(t\right)=T_{m_i,n_i}\mathit{cos}\left(2\gamma {\omega }_{c}t\pm 2ϵ{\omega }_{0}t+{\theta }_{m_i,n_i}\right),\forall \gamma =1,2,\dots ; \forall ϵ=0,1,2,\dots$$

(24a)

$$t_e\left(t\right)=T_{m_i,n_i}\mathit{cos}\left(\left(2\gamma -1\right){\omega }_{c}t\pm \left(2ϵ+1\right){\omega }_{0}t+{{\theta }_{m_i,n}}_i\right)$$

(24b)

This interaction is illustrated in Figure 6 for the specific cases of $\gamma =1$.

The expressions of the torque harmonics induced by the CHB inverter’s abnormal modes are resumed in Equation (25). Their phase-shift angles are omitted for the sake of simplicity.

$$t_e\left(t\right)=T_{DC}+\sum _{n=1}^{\infty }{T}_{0,2n}\mathit{cos}\left(2n{\omega }_{0}t\right)$$

(25a)

$$+\sum _{m=1}^{\infty }\sum _{n=0}^{\infty }{T}_{2m,\pm 2n}\mathit{cos}\left(2m{\omega }_{c}t\pm 2n{\omega }_{0}t\right.)$$

(25b)

$$+\sum _{m=1}^{\infty }\sum _{n=0}^{\infty }{T}_{\left(2m-1\right),\pm \left(2n+1\right)}\mathit{cos}\left({\left(2m-1\right)\omega }_{c}t\pm \left(2n+1\right){\omega }_{0}t\right)$$

(25c)

4. Effects of Neutral-Shift and Peak-Reduction Methods on the Motor Torque Spectrum

4.1. Operating Principles of the Neutral-Shifted and Peak-Reduction Methods

Figure 7 summarizes the fault-tolerant control techniques proposed in the literature for the CHB inverter [5]. Among the multi-carrier PWM fault-tolerant control methods, the Neutral-shifted and Generalized Neutral-shifted methods [12,13,14] adjust the phase-shift angles of the PWM reference voltages. While the Peak-reduction [16,17,18,19,20,21] and geometric methods [31,32,33] modify their magnitudes by injecting them with a homopolar component. As shown in Figure 8, this paper focuses on the Neutral-shifted (called Generalized Neutral-shifted in [5]) and Peak-reduction methods.

The block diagram of the Neutral-shifted method is shown in Figure 8a. The necessary phase-shift angles (${\theta }_{ab}$, ${\theta }_{ca}$) to rebalance the CHB inverter output LL voltages are computed from the number of healthy cells ($c_{i,x}$) per phase using a set of nonlinear equations [12]. To maximize the LL voltages while also extending the linear modulation region, the optimum values of (${\theta }_{ab}$, and ${\theta }_{ca}$) are calculated and a min-max homopolar component is injected into the modified reference voltages.

The Neutral-shifted method suffers from high computational effort due to the resolution of nonlinear equations. Additionally, it is not applicable to all possible fault conditions in the CHB inverter due to the triangle inequality theorem [9].

The block diagram of the Peak-reduction method is shown in Figure 8b. To rebalance the CHB inverter output LL voltages, the Peak-reduction method adjusts only the magnitudes of the reference voltages by injecting them with a homopolar component. The modification coefficients $n_i$ are calculated as the ratio of the total number of series-connected cells $k$ and to the number of operative cells $k_i$ per phase.

Peak reduction is easily implemented for closed-loop control. However, the high-magnitude homopolar component injected into the reference voltages induces unequal power sharing among the healthy cells [16,17]. Also, the Peak-reduction method is not applicable when all the cells in a phase are bypassed because $k_i=0,$ leading to $n_i\to \infty$.

4.2. Effects of the Neutral-Shifted and Peak-Reduction Methods on the CHB Inverter Output Voltage Harmonics and Motor Airgap Torque Harmonics

When the PD-PWM is implemented, the number of operative cells per phase of the CHB inverter affects only the magnitudes of the generated voltage harmonics. In contrast to the phase-shifted PWM, the frequencies of these voltage harmonics remain unchanged. Consequently, during the failed and corrected modes, the CHB inverter output voltage harmonics are localized exactly at the same frequencies, as given in Equations (3)–(8). However, the behaviors of the carrier harmonics, triplen baseband, and triplen sideband harmonics are modified according to the selected fault-tolerant method.

The Neutral-shifted method, shown in Figure 8a, modifies the magnitudes and phase-shift angles of the PWM reference voltages. In contrast to the failed mode, the triplen baseband and triplen sideband harmonics given, respectively, in Equations (4c) and (8c) are no longer zero-sequence harmonics due to the alteration of the $2\pi /3$ phase-shift angle of a three-phase system. Consequently, they are uncancelled in all the LL voltage spectra. Also, the carrier harmonics given in Equation (3c) remain zero-sequence harmonics, but they are no longer common-mode components due to unequal magnitudes in the individual phases. As a result, they are cancelled only in the LL voltage spectra of healthy phases.

As shown in Figure 8b, the Peak-reduction method modifies only the magnitudes of the reference voltages. Unlike the Neutral-shifted method, the carrier and triplen harmonics remain zero-sequence components, as in the normal and failed modes. But they are no longer common-mode components due to unequal magnitudes in the individual phases. As a result, they are suppressed only in the LL voltage spectra of the healthy phases.

Regardless of the selected fault-tolerant control methods, the expressions of the CHB inverter output voltage harmonics are given in Equations (3)–(8). Because the motor airgap torque components are produced by the interaction between the voltage (producing the flux) and current harmonics, during the corrected mode, the airgap torque harmonics are also localized at the same frequencies and in the same form as those found during the failed mode and given in Equation (25).

The correlations between the CHB inverter output voltage/current harmonics and the airgap torque components during the failed and corrected modes are given in

Table 1. Correlations between voltage/current harmonics and torque components during the failed and corrected operation modes.

$({\mathit{m}}_{\mathit{v}},{\mathit{n}}_{\mathit{v}})$$/({\mathit{m}}_{\mathit{i}},{\mathit{n}}_{\mathit{i}})$	Voltage/Current Harmonics	Torque Harmonics
(0, 1)	${\omega }_0$	$0,2{\omega }_0$
baseband harmonics
(0, 5)	${5\omega }_0$	${4\omega }_0$$, {6\omega }_0$$, {8\omega }_0$
(0, 7)	${7\omega }_0$
(0, 3)	${3\omega }_0$	${2\omega }_0$$, {4\omega }_0$
Sideband harmonics around odd multiples of the carrier frequency
(1, 0)	${\omega }_c$	${\omega }_c\pm {\omega }_0$
(1, −2)	${\omega }_c-2{\omega }_0$	${\omega }_c\pm 3{\omega }_0$$, {\omega }_c\pm {\omega }_0$
(1, 2)	${\omega }_c+2{\omega }_0$
(1, −4)	${\omega }_c-4{\omega }_0$	${\omega }_c\pm 3{\omega }_0$$, {\omega }_c\pm 5{\omega }_0$
(1, 4)	${\omega }_c+4{\omega }_0$
Sideband harmonics around even multiples of the carrier frequency
(2, −1)	$2{\omega }_c-{\omega }_0$	$2{\omega }_c\pm 2{\omega }_0$$, 2{\omega }_c$
(2, 1)	$2{\omega }_c+{\omega }_0$
(2, −3)	$2{\omega }_c-3{\omega }_0$	$2{\omega }_c\pm 2{\omega }_0$$, 2{\omega }_c\pm 4{\omega }_0$
(2, 3)	$2{\omega }_c+3{\omega }_0$

. The torque harmonic localized at ($2{\omega }_0$) is shown only for the fundamental (${\omega }_0$) and the triplen ($3{\omega }_0$) harmonic of voltage/current because they have the largest contributions to its magnitude.

5. Simulation Results and Discussions

The studied vector-controlled induction motor is fed by a seven-level CHB inverter operating with fault-tolerant control methods. Its implementation block diagram is shown in Figure 9. The parameters given in

Table 2. Simulation parameters.

Parameters	Values
$\mathrm{Motor} \mathrm{rated} \mathrm{power}, P_n$	$7 \mathrm{M}\mathrm{W}$
$\mathrm{Motor} \mathrm{rated} \mathrm{LL} \mathrm{voltage}, U_n$	4.16 kV
Motor poles, p	4
$\mathrm{Stator}/\mathrm{rotor} \mathrm{resistance}, R_s/R_r$	$0.3 \mathrm{\Omega }/0.22 \mathrm{\Omega }$
$\mathrm{Stator}/\mathrm{rotor} \mathrm{leakage} \mathrm{inductance}, L_{ls}=L_{lr}$	$0.4 \mathrm{m}\mathrm{H}$
$\mathrm{Magnetizing} \mathrm{inductance}, L_m$	$10.66 \mathrm{m}\mathrm{H}$
$\mathrm{Inertia} \mathrm{coefficient}, J$	$130 \mathrm{k}\mathrm{g}.{\mathrm{m}}^2$
$\mathrm{DC} \mathrm{link} \mathrm{voltage}, V_{dc}$	$1.05 \mathrm{k}\mathrm{V}$
$\mathrm{Carrier} \mathrm{frequency}, f_{cr}$	$1 \mathrm{k}\mathrm{H}\mathrm{z}$
$\mathrm{Sampling} \mathrm{time}, T_s$	$5 \mathrm{\mu }\mathrm{s}$

have been used for the offline (in MATLAB) and virtual hardware in the loop (VHIL) simulations (Typhoon HIL control center). Moreover, the PWM reference voltages do not include any homopolar component in either the normal or failed modes. During the abnormal modes ($k_a$-$k_b$-$k_c$), there are $k_i$ operative cells in phase $i$, $\forall i\in \{a,b,c\}$.

5.1. Time Domain Simulation Results for the Normal Condition, Failed Modes, and Corrected Modes (3-3-2)

Figure 10 shows the offline simulation results for the normal and abnormal modes (3-3-2). During the normal mode (blue arrows), the CHB inverter output (LN and LL) voltages and currents are balanced. The motor torque is a quasi-straight line with a torque ripple of about $t_r~7.64\%$. This high torque ripple results from the low-frequency torque harmonics induced by the sideband harmonics of currents generated by the CHB inverter operating with PD-PWM. But it can be reduced by injecting a homopolar component into the reference voltages.

While during the failed mode (purple arrows), the motor input (LN and LL) voltages and currents become unbalanced due to one failed cell in phase C. Also, the currents are distorted due to high oscillations in the (q-axis) component of the current since the maximum available voltage from the CHB inverter is reduced [11]. Consequently, the magnitudes of the torque oscillations increase greatly, as shown in Figure 10d (purple arrows), and the torque ripple is 6.5 times higher than during the normal mode.

When the Neutral-shifted (NS) method is used, the CHB inverter output LL voltages are balanced while the LN voltages are unbalanced, as displayed in Figure 10 (dark purple arrows). But the motor input currents are slightly unbalanced since the first triplen baseband harmonic (due to the homopolar component injected into the reference voltages) is unsuppressed in the current spectra. The magnitudes of the torque oscillations have been reduced compared to the failed mode and the torque ripple is 0.8 times lower.

Additionally, when the Peak-reduction (PR) method is employed, the motor input LL voltages are also balanced while the LN voltages are unbalanced, as shown in Figure 11a,b (red arrows). However, the motor input currents are significantly distorted and unbalanced because the homopolar component injected into the PWM reference voltages induces a high-magnitude triplen baseband harmonic in the current spectra. Compared to the failed mode, the magnitudes of the torque oscillations have been slightly reduced, as shown in Figure 11d (red arrow), and the torque ripple is 0.88 times lower.

The high oscillations in the motor torque during the abnormal modes result from the extra torque components created by the unsuppressed carrier and triplen harmonics in the LL voltage and current spectra. Thus, addressing the spectral characteristics of the Neutral-shifted and Peak-reduction methods is necessary to build reliable VFDs based on the CHB inverter operating with failed cells.

For further validation purposes, the VHIL simulations have been performed in the Typhoon HIL control center. The simulation model and SCADA panel are shown in Figure 12 and Figure 13, respectively. The results obtained from Typhoon have been exported into MATLAB to improve the resolution of the plots shown in Figure 14.

As shown in Figure 14(a1–d1), the CHB inverter output (LN and LL) voltages and currents are unbalanced during the failed mode 3-3-2. Also, the motor torque oscillates with high magnitudes, and the torque ripple is about 52%. Compared to the offline results, this torque ripple has been increased by 13% due to high oscillations in the (q-axis) component of the current since the maximum available voltage from the CHB inverter is reduced [11].

When using the Neutral-shifted method, the CHB inverter output LL voltages and currents are balanced while the LN voltages are unbalanced, as shown in Figure 14(a2–d2). Also, the oscillations in the motor torque have been greatly reduced. The torque ripple is 0.52 times lower compared to failed mode. Compared to the offline results shown in Figure 10, the deviation in the torque ripple is 3%.

Moreover, when the Peak-reduction method is used, the CHB inverter output LL voltages are balanced while the LN voltages are unbalanced, as shown in Figure 14(a3–d3). However, the motor input currents are distorted, leading to high oscillations in the motor torque, and a torque ripple of about 51% is obtained. Compared to the offline simulation results, the torque ripple has been significantly increased by 26%.

These VHIL simulations validate the effectiveness of the Neutral-shifted and Peak-reduction methods. But the torque ripple is greatly increased for the failed mode and when using the Peak-reduction method.

5.2. Frequency Domain Simulation Results for the Failed and Corrected Modes (3-3-2)

Figure 15 shows the CHB inverter output voltage and current spectra during the failed mode (3-3-2). The relevant LN voltage harmonics shown in Figure 15(a1–c1) appear as follows:

■

Fundamental, $f_0$ = 60 Hz;

■

Baseband harmonics, $5f_0$ = 300 Hz, $7f_0$ = 420 Hz;

■

Carrier harmonic, $f_c$ = 1000 Hz;

■

Sideband around odd multiples of carrier frequency,

$\left(1\times f_c\right)$:$|f_c-4f_0|$ = 760 Hz; $|f_c+4f_0|$ = 1240 Hz; $|f_c-6f_0|$ = 640 Hz;

■

Sideband around even multiples of carrier frequency,

$\left(2\times f_c\right)$: $|2f_c-15f_0|$ = 1100 Hz; $|2f_c-9f_0|$ = 1460 Hz; $|2f_c-7f_0|$ = 1580 Hz.

As expected, the baseband harmonics appear only in the voltage spectrum (${\mathrm{V}}_{\mathrm{c}}$) of the failed phase while the voltage spectra (${\mathrm{V}}_{\mathrm{a}}$ and ${\mathrm{V}}_{\mathrm{b}}$) of the healthy phases are unaffected. These results are consistent with the theoretical analyses provided in Section 2.

During the CHB inverter’s failed modes, the zero-sequence harmonics are no longer common-mode components. Consequently, in Figure 15(a2–c2), the carrier ($f_c$), and the triplen (red arrows), such as $|f_c\pm 6f_0|$, appear only in the LL voltage spectra (${\mathrm{V}}_{\mathrm{b}\mathrm{c}}$ and ${\mathrm{V}}_{\mathrm{c}\mathrm{a}}$) of the phases that have an unequal number of healthy cells. They are suppressed in the differential voltage ${\mathrm{V}}_{\mathrm{a}\mathrm{b}}$ of the healthy phases. As a result, the THD of ${\mathrm{V}}_{\mathrm{a}\mathrm{b}}$ is lower than those of ${\mathrm{V}}_{bc}$ and ${\mathrm{V}}_{\mathrm{c}\mathrm{a}}$.

Moreover, in Figure 15(a3–c3), the carrier $f_c$ and $|f_c\pm 6f_0|$ appear with high magnitudes only in the current spectrum ${\mathrm{I}}_{\mathrm{c}}$ of the failed phase because they are uncancelled in ${\mathrm{V}}_{\mathrm{c}\mathrm{a}}$. Thus, the THD of ${\mathrm{I}}_{\mathrm{c}}$ is higher than those of ${\mathrm{I}}_{\mathrm{a}}$ and ${\mathrm{I}}_{\mathrm{b}}$. These zero-sequence harmonics will be converted into the motor torque components.

Figure 15d shows the torque spectra during the normal (blue dashed lines) and failed (3-3-2, purple solid lines) modes. The extra torque components are created by the zero-sequence harmonics found in the LL voltage and current spectra during the failed mode. Also, the relevant torque’s harmonics appear as follows:

■

DC component, 0 Hz.

■

Baseband harmonics, $2f_0$ = 120 Hz; $4f_0$ = 240 Hz, $6f_0$ = 360 Hz. As predicted by Equation (25a), they appear at even multiples of the fundamental frequency.

■

Carrier harmonic, $2f_c$ = 2000 Hz. It appears at even multiple of the carrier frequency as predicted by Equation (25b), when $n=0$.

■

Sideband around odd multiples of the carrier frequency,

($1\times f_c$): $|f_c-17f_0|$ = 20 Hz; $|f_c-15f_0|$ = 100 Hz; $|f_c-f_0|$ = 940 Hz. They appear when $m$ and $n$ are odd numbers, as predicted in Equation (25c).

These results are consistent with the theoretical analyses provided in Section 3.

To validate the results provided in Table 2, some of the voltage/current harmonics and the torque components have been labeled with the same-colored arrows. For instance, the conversion of the voltage/current harmonics into the torque components in Figure 15 is given as follows:

■

The fundamental of the LL voltages and currents $f_0$ = 60 Hz creates the DC (0 Hz) and $2f_0$ = 120 Hz components of torque (green arrows).

■

The baseband harmonic of the LL voltages and currents $7f_0$ = 420 Hz creates the torque baseband $6f_0$ = 360 Hz and $8f_0$ = 480 Hz (turquoise solid lines).

■

The triplen sideband harmonic of the LL voltages and currents $|f_c-6f_0|$ (red dashed arrow) produce the torque harmonics $|f_c-7f_0|$ = 580 Hz and $|f_c-5f_0|$ = 700 Hz.

Figure 16 shows the spectral characteristics of the CHB inverter and the motor torque during the corrected mode (3-3-2) using the Neutral-shifted method. Since the frequencies of the voltage harmonics are unaffected by the Neutral-shifted method, the LN voltage harmonics of the healthy and failed phases shown in Figure 16(a1–a3) appear at the same frequency but with unequal magnitudes. In contrast to the failed mode (Figure 15), the baseband harmonics, such as $3f_0$ and $5f_0$, also appear in the voltage spectra (${\mathrm{V}}_{\mathrm{a}}$ and ${\mathrm{V}}_{\mathrm{b}}$) of the healthy phases due to the homopolar component injected into the reference voltages.

As expected, in Figure 16(a2–c2), the triplen $\left|2f_c\pm 3f_0\right|$ are unsuppressed in all the LL voltage spectra. Because, when using the Neutral-shifted method, they are neither zero-sequence nor common mode components. Also, the triplen $3f_0$ appearing in ${\mathrm{V}}_{\mathrm{a}}$ and ${\mathrm{V}}_{\mathrm{b}}$ is suppressed in their differential voltage spectrum (${\mathrm{V}}_{\mathrm{a}\mathrm{b}}$) because for this operation mode (3-3-2), it has the same magnitudes (due to an equal number of healthy cells) in these phases. Consequently, the THD of ${\mathrm{V}}_{\mathrm{a}\mathrm{b}}(5.5\%)$ is lower than that of ${\mathrm{V}}_{\mathrm{b}\mathrm{c}}(8.9\%)$ and ${\mathrm{V}}_{\mathrm{c}\mathrm{a}}(8.83\%)$.

In Figure 16(a3–c3), the relevant current harmonics are low-frequency components such as the fundamental ($f_0$), and the baseband ($3f_0$, $7f_0$) components. They have higher magnitudes in the spectrum (${\mathrm{I}}_{\mathrm{c}}$) of the failed phase (c). The triplen $\left|2f_c\pm 3f_0\right|$ appear in all the current spectra since they are uncancelled in the LL voltages. However, the triplen ($3f_0$ and $9f_0$) that are suppressed in ${\mathrm{V}}_{\mathrm{a}\mathrm{b}}$ appear in ${\mathrm{I}}_{\mathrm{a}}$ because the currents shown in Figure 10 are slightly distorted due to the homopolar component injected into the reference voltages. As expected, the THD of ${\mathrm{I}}_{\mathrm{c}}(16.7\%)$ is higher than those of ${\mathrm{I}}_{\mathrm{a}}(7.9\%)$ and ${\mathrm{I}}_{\mathrm{b}}(6.6\%)$.

Figure 16d shows the motor torque spectra during the failed (3-3-2, pink lines) and corrected (3-3-2, green lines) modes using the Neutral-shifted method. During the corrected mode, the torque harmonics appear at the same frequencies as those found during the failed mode and given in Equation (25) because the frequencies of the voltage harmonics are unaltered by the Neutral-shifted method.

Also, the relevant torque harmonics are the baseband ($2f_0$, $4f_0$, etc.) and sideband harmonics (|$f_c-17f_0$|, |$f_c-15f_0$|, |$2f_c\pm 4f_0$|, etc.) around the first carrier. Compared to the failed mode, the magnitudes of the torque harmonics, such as $\left|2f_c\pm 2f_0\right|$, and $\left|2f_c\pm 4f_0\right|$, are increased due to the triplen $\left|2f_c\pm 3f_0\right|$ found in the LL voltage and current spectra. Also, the magnitude of the torque component $4f_0$ is increased because the triplen $3f_0$ appears in all current spectra with high magnitude. While the magnitude of |$f_c-17f_0$|, |$f_c-15f_0$| and |$f_c\pm 3f_0$| are damped because the sideband harmonics around $f_c$ (excepted $f_c\pm 2f_0$) are suppressed in the LL voltage and current spectra. Consequently, the torque ripple is 0.8 times lower than during the failed mode.

Figure 17 shows the spectral characteristics of the Peak-reduction method during the corrected mode (3-3-2). In Figure 17(a1–c1), the baseband harmonics, such as $3f_0$, $5f_0$, etc., appear in all the LN voltage spectra due the homopolar component injected into the PWM reference voltages.

The carrier and triplen harmonics are no longer common mode components, $3f_0$, $f_c$, $\left|2f_c\pm 3f_0\right|$, etc., are unsuppressed in the LL voltage spectra (${\mathrm{V}}_{\mathrm{a}\mathrm{b}}$ and ${\mathrm{V}}_{\mathrm{c}\mathrm{a}}$) shown in Figure 17(a2–c2). As expected, the carrier $f_c$ is cancelled in the differential voltage spectrum of the healthy phase (${\mathrm{V}}_{\mathrm{a}\mathrm{b}}$). Due to the homopolar component injected in the reference voltages, the triplen $\left|2f_c\pm 3f_0\right|$ that should be suppressed in ${\mathrm{V}}_{\mathrm{a}\mathrm{b}}$ has been increased in magnitudes compared to those found in ${\mathrm{V}}_{\mathrm{a}}$ and ${\mathrm{V}}_{\mathrm{b}}$. But the THD of the ${\mathrm{V}}_{\mathrm{a}\mathrm{b}} (5.1\%)$ remains lower than that of ${\mathrm{V}}_{\mathrm{a}\mathrm{b}} (8.3\%)$ and ${\mathrm{V}}_{\mathrm{c}\mathrm{a}} (8.26\%)$ due to the high-magnitude ($3f_0$) harmonic in these LL voltages.

In Figure 17(a3–c3), the significant harmonics in the current spectra are low-frequency components around the fundamental ($f_0$), such as $3f_0$, $7f_0$ and |$2f_c-29f_0$|. They have a higher magnitude in the spectrum (${\mathrm{I}}_{\mathrm{c}}$) of the failed phase. The triplen $3f_0$ appears in all the current spectra because it is uncanceled in all the LL voltages. Also, the THD of ${\mathrm{I}}_{\mathrm{c}} (15.5\%)$ is higher than those of ${\mathrm{I}}_{\mathrm{a}} (8\%)$ and ${\mathrm{I}}_{\mathrm{b}} (9.7\%)$.

Figure 17d shows the torque spectra during the failed and corrected (3-3-2) modes using the Peak-reduction method. Since the frequencies of the voltage harmonics are unaffected by the Peak-reduction method, these harmonics appear exactly as those found during the failed mode and given by Equation (25).

Compared to the failed mode, the magnitudes of the torque harmonics localized around the 2nd carrier, such as $\left|2f_c\pm 2f_0\right|$ and $\left|2f_c\pm 4f_0\right|$, are increased in magnitude due to the triplen $\left|2f_c\pm 3f_0\right|$ found in the LL voltage and current spectra. Also, the magnitude of the torque harmonic $4f_0$ is increased due to the unsuppressed triplen $3f_0$ in the LL voltage and current spectra. Because the sidebands localized around the first carrier are damped in the LL voltage and current spectra, the magnitudes of the torque harmonics, such as $\left|f_c-17f_0\right|$, $\left|f_c\pm 15f_0\right|$, and $\left|f_c\pm 3f_0\right|$, are damped. As a result, the torque ripple is damped by 0.8 times compared to the corrected mode.

The frequency domain results for the VHIL simulations during the failed and corrected modes using the Neutral-shifted and Peak-reduction methods are shown in Figure 18, Figure 19 and Figure 20, respectively.

During the failed mode (3-3-2), the baseband harmonics such as ($5f_0$) appear only the spectrum of the failed phase ($V_c$), as shown in Figure 18(a1–c1). Moreover, in Figure 18(a2–c2), the carrier ($f_c$) and triplen ($f_c\pm 12f_0,f_c\pm 6f_0$, etc.) harmonics are unsuppressed in the LL voltage spectra of the healthy and failed phases ($V_{bc}$ and $V_{ca}$). Also, in the current spectrum ($I_c$) of the failed phase shown in Figure 18(a3–c3), the carrier ($f_c$) and the triplen ($f_c-12f_0$) harmonics are uncancelled. These results are consistent with the theoretical analyses provided in Section 2 and confirm the accuracy of the expressions of voltage harmonics given in Equations (3)–(8).

Additionally, in Figure 18d, the torque baseband harmonics ($2f_0$ and $6f_0$) are localized at even multiples of the fundamental frequency, as predicted by Equation (25a). Also, the torque sideband harmonics, such as ($2f_c-4f_0$ and ${2f}_c-2f_0$), appear when even values of $m$ and $n$ are paired, as given in Equation (25b). Meanwhile, ($f_c-13f_0$, $f_c-11f_0$, etc.) appear when odd values of $m$ and $n$ are paired, as given in Equation (25c). These results validate the accuracy of the expressions of the motor torque harmonics given in Equation (25).

As expected, in Figure 19(a1–c1), the LN voltage harmonics induced by the Neutral-shifted method in spectra of the healthy and failed phases are located at the same frequencies. But they have a higher magnitude in the voltage spectrum of the failed phase ($V_c$). Also, the baseband harmonic ($3f_0$) appears in all the LN voltage spectra. Moreover, in Figure 19(a2–c2), triplen sidebands ($2f_c\pm 3f_0$) are uncancelled in all the LL voltage spectra. Additionally, in the current spectra shown in Figure 19(a3–c3), the triplen harmonics, such ($3f_0$ and $2f_c\pm 3f_0$), are uncancelled. The $3f_0$ harmonic appears in the current spectra because the motor input currents shown in Figure 14(c2) are distorted.

As expected, in Figure 19d, the torque baseband harmonics ($2f_0$ and $4f_0$) are localized at even multiples of the fundamental frequency. Also, the torque sideband harmonics, such as ($f_c-11f_0$, $f_c\pm 3f_0$, etc.), appear when odd values of $m$ and $n$ are paired. Additionally, the sideband harmonics, such as ($2f_c\pm 4f_0$ and ${2f}_c\pm 2f_0$), appear when even values of $m$ and $n$ are paired. These virtual HIL results align with the offline simulations provided in Figure 16.

The frequency domain VHIL simulations during the corrected mode (3-3-2) when using the Peak-reduction method are shown in Figure 20. As expected, the LN voltage harmonics induced by the Peak-reduction method in the spectra of the healthy ($V_a$ and $V_b$) and failed ($V_c$) phases appear at the same frequencies. Also, the baseband harmonics ($3f_0$, $5f_0$, etc.) appear in all the LN voltage spectra due to the injected homopolar component in the reference voltages.

Moreover, in Figure 20(a2–c2), the triplen harmonics ($3f_0$ and $2f_c\pm 3f_0$) are uncancelled in all the LL voltage spectra since they are no longer common-mode harmonics. Also, the carrier ($f_c$) is suppressed only in the LL voltage spectrum ($V_{ab}$) of the healthy phases. Additionally, in Figure 20(a3–c3), the triplen harmonics (red arrows) are uncancelled in all the current spectra.

Furthermore, in Figure 20d, the baseband harmonics of the motor torque appear effectively at even multiples of the fundamental frequency. Meanwhile, the sideband harmonics of the motor torque appear when odd values of $m$ and $n$ are paired (purple arrows) as well as when even values of $m$ and $n$ are paired (blue arrows). These results confirm the accuracy of the offline simulations shown in Figure 17.

5.3. Performance Comparisons of the Fault-Tolerant Control Techniques for the CHB Inverter

Figure 21 compares the performances of the NS and PR methods for several corrected modes with the carrier frequency ($f_c$) set to $1 \mathrm{k}\mathrm{H}\mathrm{z}$ and the fundamental frequency ($f_0$) ranging from 35 to 60 Hz. The investigated power quality (PQ) parameters include the LL voltage (${\mathrm{V}}_{\mathrm{a}\mathrm{b}}$) THD, the current (${\mathrm{I}}_{\mathrm{a}}$) THD, and the motor torque ripple.

During the corrected mode 3-3-2 shown in Figure 21(a1–c1), both approaches produce a voltage with almost the same THD. This result is consistent with Figure 16 and Figure 17. But the NS produces the lowest current THD around the rated frequency (60–50 Hz) because the ($3f_0$) harmonic in the current spectrum has a lower magnitude compared to that generated by the PR. However, the PR produces the lowest current THD at lower operating frequencies (35–45 Hz) because the magnitude of the harmonic ($3f_0$, due to the homopolar component) decreases within 35–45 Hz. As a result, the PR induces also the lowest torque ripple within 35–50 Hz because the relevant torque component ($2f_0$) has a lower magnitude when using the PR compared to the NS for this specific abnormal mode (3-3-2).

During the corrected mode (3-2-2) shown in Figure 21(a2–c2), these approaches produce the same voltage THD. But in Figure 19(c2), the NS produces the highest current THD due to the harmonics ($f_c$ and $3f_0$) with higher magnitudes than those found in the current spectrum produced by the PR. Moreover, in Figure 17(c2), the NS produces the highest torque ripple at 60 Hz because the induced torque component ($2f_0$) has a higher magnitude than that of the PR. But around 55–60 Hz, the NS produces the lowest torque ripple because the induced torque component ($2f_0$) has a lower magnitude than that of the PR method.

The mean values of these PQ parameters have been calculated for a more accurate comparison. As shown in Figure 21d, the lowest mean values of the voltage and current THDs are obtained using the PR. But with failed cells in multiple phases (3-2-2 and 2-3-3) of the CHB inverter, the NS provides a lower torque ripple.

Consequently, for motor drive applications where torsional vibrations are a critical concern, the Neutral-shifted method is the most suitable solution. However, for applications such as grid-connected converters that require low voltage and current THDs, the Peak-reduction method is the most appropriate.

6. Conclusions

This paper analyzed the effects of failed cells on the CHB inverter output voltage harmonics, and their expressions have been derived. The impacts of these voltage harmonics on the induced motor airgap torque spectrum have also been analyzed. Unlike in the normal mode, the carrier and triplen harmonics are unsuppressed in the line-to-line voltage spectra because they have unequal magnitudes in the line-to-neutral voltages. Consequently, they are also converted into torque harmonics. In this paper, the expressions of the motor airgap torque induced by the CHB inverter’s abnormal modes have also been established using the interaction between the stator’s unbalanced voltages (producing the flux) and currents in the (αβ0) reference frame. The theoretical developments and (offline and VHIL) simulation results carried out on a seven-level CHB inverter feeding a 7 MW vector-controlled induction motor show that (i) the interaction between the stator’s sinusoidal unbalanced voltages and currents creates DC (0 Hz) and pulsating (localized at two times the operating frequency) torque components. (ii) The interaction between the fundamental voltage (or current) and current (or voltage) harmonic of rank $n_i$ induces two torque components localized at $n_i-1$ and $n_i+1$. (iii) During the failed modes, the relevant torque harmonics, mastered by $2f_0$, are low-frequency components located around the DC (0 Hz). Thus, in the event of faults in the power conversion system, the field engineers can use this $2f_0$ torque component to identify the failures induced by the inverter’s abnormal modes.

Additionally, the spectral performances of the Neutral-shifted and Peak-reduction methods have also been addressed. Because the conventional fault-tolerant control methods do not alter the frequencies of the voltage harmonics generated by the CHB inverter, the torque components appear at the same frequencies during the failed and corrected modes.

Future investigations will propose an enhanced fault-tolerant control method capable of eliminating certain torque harmonics that traditional Neutral-shift and Peak-reduction methods cannot cancel when operating a CHB inverter with faulty cells. This method aims to reduce pulsating torques and improve the longevity of electrical drive systems.