Fault Diagnosis and Tolerant Control of Three-Level Neutral-Point Clamped Inverters in Motor Drives

: This paper presents an extension theory-based assessment method to perform fault diagnosis for inverters in motor driving systems. First, a three-level neutral-point clamped (NPC) inverter is created using the PSIM software package to simulate faults for any power transistor in the NPC-type inverter. Fast Fourier transformation is used to transform the line current signals in the time domain into a spectrum in the frequency domain for analysis of the corresponding spectrum of features of the inverter for faults with di ﬀ erent power transistors. Then, the relationships between fault types and speciﬁc spectra are established as characteristics for the extension assessment method, which is then used to create a smart fault diagnosis system for inverters. Fault-tolerant control (FTC) is used here when the rated output of a faulty inverter is decreased in order to maintain balanced output in three phases by changing the framework of the transistor connection. This is performed to reinforce the reliability of the inverter. Finally, by the simulation and experimental results, the feasibility of the proposed smart fault diagnosis system is conﬁrmed. The proposed fault diagnosis method is advantageous due to its minimal use of data and lack of a learning process, which thereby reduces the fault diagnosis time and makes the method easily used in practice. The proposed fault-tolerant control strategy allows both online and smooth switching in the wiring structure of the inverter.


Introduction
Compared to two-level inverters, multi-level inverters [1][2][3][4][5] exert less voltage stress on switches, feature smaller change rates in their output voltages (dv/dt), and can be applied in high power scenarios. As multi-level inverters incorporate multiple power transistors that are connected in both series and parallel, thus making the output line-to-ground voltage waves embedded with many step-like waveforms, their waveforms become step-like voltage waveforms which are better approximated in the form of sine waves, thus being suitable for reducing harmonic components.
Multi-level inverters are generally divided as diode-clamped, neutral-point clamped (NPC, I-type), cascaded H-bridge (CHB), and flying capacitor inverters. Of these types, three-level diode-clamped inverters [6] are widely used due to their simple circuit and ease of control. The inverter works by dividing, using capacitors, the voltage on DC side equally into three voltage levels (+Vdc/2, 0, and −Vdc/2), allowing the output voltage to change in three modes, thus, with diodes and switches, clamping the output voltage at a neutral point on the DC side. Yet, as the switches may be damaged when operating in high-current or high-temperature conditions over extended periods of time, components aging, faulty drive circuits, and other faults, thereby resulting in an inability to work normally, multi-level inverters should be designed with a fault detection mechanism and fault-tolerant control in the hope of operation without disruption in the event of a faulty inverter component [7][8][9]. power transistor. Subsequently, a fault-tolerant control strategy is used to maintain the three-phase balanced line voltage output if any power semiconductor switch of the inverter fails. Figure 1 illustrates the overall architecture of the extension theory-based fault-diagnosis and fault-tolerant control system proposed herein for NPC-type inverters.  This paper is organized as follows: First, the characteristics of switch faults for three-level NPCtype inverters are discussed in Section 2. Then, in Section 3, the concept of extension theory is described in detail. The procedure for using the extension theory-based fault diagnosis method for a three-level NPC-type inverter is explained in Section 4. Practical tests are detailed in Section 5 to demonstrate the effectiveness of the proposed fault diagnosis method. Finally, in Section 6, faulttolerant control in the event of power switch failure in the three-level NPC-type inverter is analyzed, and the simulation and experimental results are used to prove the feasibility of the method.

Characteristics of Faults for Three-Level Inverters
Considering research on inverter fault diagnosis, this paper targets a three-level NPC inverter as shown in Figure 2. In general, faults for inverters come in three types: short-circuit faults, opencircuit faults, and trigger signal faults. A short-circuit fault arises when a switch component is blown due to an overly high voltage across both ends of the switch. An open-circuit fault refers to a power transistor without a trigger signal to actuate conductivity through it. A trigger signal fault is the receipt, by a switch component, of an incorrect trigger signal for command. This paper is organized as follows: First, the characteristics of switch faults for three-level NPC-type inverters are discussed in Section 2. Then, in Section 3, the concept of extension theory is described in detail. The procedure for using the extension theory-based fault diagnosis method for a three-level NPC-type inverter is explained in Section 4. Practical tests are detailed in Section 5 to demonstrate the effectiveness of the proposed fault diagnosis method. Finally, in Section 6, fault-tolerant control in the event of power switch failure in the three-level NPC-type inverter is analyzed, and the simulation and experimental results are used to prove the feasibility of the method.

Characteristics of Faults for Three-Level Inverters
Considering research on inverter fault diagnosis, this paper targets a three-level NPC inverter as shown in Figure 2. In general, faults for inverters come in three types: short-circuit faults, open-circuit faults, and trigger signal faults. A short-circuit fault arises when a switch component is blown due to an overly high voltage across both ends of the switch. An open-circuit fault refers to a power transistor without a trigger signal to actuate conductivity through it. A trigger signal fault is the receipt, by a switch component, of an incorrect trigger signal for command. The PSIM software package is the ultimate simulation environment for power electronics and motor control and was developed by PowerSIM. Using the PSIM software package, a simulation environment for three-level NPC inverters was created and used to investigate the diagnosis of faults occurring on any switch at any time. It was observed from the simulations and analysis that the output waveform measured on an inverter in normal working conditions was a balanced three-phase waveform. For example, when the inverter has a working frequency at 60 Hz without a faulty power transistor, the output waveforms of its line currents and their frequency spectra are given as shown in Figures 3 and 4, respectively. Figure 3 reveals that the waveforms for the line currents are of the same magnitude and all are sine waves, where each is different in phase by 120°, which is typical of the balanced three-phase characteristic. Regarding the parameter setting of the motor drive, a 300 V DC inverter with a switching frequency of 18 kHz was connected to an induction motor. The frequency spectra for line currents ia, ib, and ic for an inverter working at 60 Hz without a faulty switch are shown in Figure 4. We can see from Figure 4   The PSIM software package is the ultimate simulation environment for power electronics and motor control and was developed by PowerSIM. Using the PSIM software package, a simulation environment for three-level NPC inverters was created and used to investigate the diagnosis of faults occurring on any switch at any time. It was observed from the simulations and analysis that the output waveform measured on an inverter in normal working conditions was a balanced three-phase waveform. For example, when the inverter has a working frequency at 60 Hz without a faulty power transistor, the output waveforms of its line currents and their frequency spectra are given as shown in Figures 3 and 4, respectively. Figure 3 reveals that the waveforms for the line currents are of the same magnitude and all are sine waves, where each is different in phase by 120 • , which is typical of the balanced three-phase characteristic. Regarding the parameter setting of the motor drive, a 300 V DC inverter with a switching frequency of 18 kHz was connected to an induction motor. The frequency spectra for line currents i a , i b , and i c for an inverter working at 60 Hz without a faulty switch are shown in Figure 4. We can see from Figure 4 that the frequency spectra at m f − 5, m f + 1, and m f + 5, multiplied by the working frequency, are very small. Hence, the values of the frequency spectra at such frequencies were used in this paper as the feature spectra for faults, where m f is defined as the frequency modulation index: where f tri is the frequency of a triangular carrier wave, i.e., the switching frequency of inverter (18 kHz here) and f sin is a sine wave frequency (60 Hz here) and also the working frequency of the inverter. Therefore, its frequency modulation index m f is 300. Based on this, the frequency spectra for line currents i a , i b , and i c for an inverter working at 60 Hz without a faulty switch at m f − 5, m f + 1, and m f + 5, multiplied by the working frequency, are 17.7, 18.06, and 18.3 kHz, respectively.       If any switch in the inverter is faulty, the characteristics of the inverter change. For instance, when a switch of the inverter Sa1 + is faulty, a distorted waveform of its output line current ia can be seen as shown in Figure 5. Figure 6 shows the waveform of the line current ib with the fault inverter switch Sb2 − , where the waveform obviously differs from the one in normal working conditions. Figure  7 shows the waveform of the line current ic with the faulty switch Sc2 + , where the distortion is apparent. If any switch in the inverter is faulty, the characteristics of the inverter change. For instance, when a switch of the inverter S a1 + is faulty, a distorted waveform of its output line current i a can be seen as shown in Figure 5. Figure 6 shows the waveform of the line current i b with the fault inverter switch S b2 − , where the waveform obviously differs from the one in normal working conditions. Figure 7 shows the waveform of the line current i c with the faulty switch S c2 + , where the distortion is apparent. It is clear from the above analysis that anomalies can be observed in current frequency spectra when an inverter is faulty, as in the case of an inverter working at 60 Hz with faulty switch Sc2 + , where the resultant current frequency spectrum is given as shown in Figure 8. When compared with the It is clear from the above analysis that anomalies can be observed in current frequency spectra when an inverter is faulty, as in the case of an inverter working at 60 Hz with faulty switch Sc2 + , where the resultant current frequency spectrum is given as shown in Figure 8. When compared with the It is clear from the above analysis that anomalies can be observed in current frequency spectra when an inverter is faulty, as in the case of an inverter working at 60 Hz with faulty switch Sc2 + , where the resultant current frequency spectrum is given as shown in Figure 8. When compared with the It is clear from the above analysis that anomalies can be observed in current frequency spectra when an inverter is faulty, as in the case of an inverter working at 60 Hz with faulty switch S c2 + , where the resultant current frequency spectrum is given as shown in Figure 8. When compared with the frequency spectra without a fault in Figure 4, this demonstrates that the frequency spectrum of line current i c has a feature spectrum with relatively major changes at the positions of m f + 1, m f + 5, and m f − 5.
Energies 2020, 9, x FOR PEER REVIEW 8 of 25 frequency spectra without a fault in Figure 4, this demonstrates that the frequency spectrum of line current ic has a feature spectrum with relatively major changes at the positions of   Relevant data for power transistor faults can be obtained by simulation and analysis. These data can be used with extension theory to create an inverter fault diagnosis system for detecting faults in power semiconductor switches in the main circuits of three-level NPC inverters.

Extension Theory
Extension theory, which was proposed by Dr. Cai Wen in 1983, is a formalized tool for studying and solving problems with qualitative and quantitative methods [29]. The theory structurally consists of matter-element theory and extension mathematics, in which the extension theory uses logic algorithms with matter elements. The matter-element theory deals with extensibility and the change of matter elements, describes them in formalized language, and details computation and reasoning. Extension mathematics builds adaptive mathematical tools from extension sets and correlation functions. As such, extension theory creates matter-element models and uses the property transformation of matter elements for qualitative and quantitative transformation, then determining qualitative and quantitative effects via correlation functions to clearly express the effects of features.
Extension theory is a science of solving contradictory problems by dealing with extensibility and rules and methods of transformation. It features (1) the idea of changing contradictory problems into compatible ones and (2) creates matter-element theory to provide new ways of solving contradictory problems with matter-element transformation based on the extensibility of matter elements. Extension theory also (3) creates extension set theory, giving quantitative descriptions of quantitative change to qualitative changes in the extension domain and critical elements, thus realizing quantitative extension mathematics in the extension theory based on extension sets [26][27][28][29][30].

Basic Concept of Extension Matter-Element Theory
In dealing with contradictory problems, if concepts, characteristics, and their corresponding data are brought together for consideration, it is possible to deduce problem solving methods. Hence, the concept of "matter elements" was introduced [30], which consists of "name" and "characteristic" of a matter and the "value" that corresponds to the characteristic. Matter elements are the basic elements in extension theory that describe matters, denoted by R, N (name), c (characteristic), and v (value) as an expression as follows: Additionally, via the above definition for a matter element, the correlation between the characteristics of these three basic elements and the corresponding value can be expressed by the Thus, Equation (2) for a matter element is converted into the following form: To understand the relationships between basic matter elements, references can be made to the expression by a space of matter elements, as Figure 9 shows, where the name, characteristic, and value are plotted in the x-, yand z-axes, which also displays the feature of the change of such a combination.
In Equation (5), R is called an m-dimensional matter element, whose components are expressed by Rk = (N, ck, vk) (k = 1, 2, …, m). Thus, Equation (5) Therefore, by the way of a multi-dimensional definition, it enables describing any single thing in the real world.
The characteristic value corresponding to a characteristic can be a single point or a single range. In the latter case, such a range is referred to as a classical domain which is contained in a neighborhood domain. Assume point f is any point in the interval  (6), where C is a characteristic and Vp is the value for C, i.e., its classical domain: A matter element, RF, corresponded by F, can be expressed by Equation (7). Similarly, C is the value of the characteristic for F and Vq is the value for C, i.e., its neighborhood domain: In Equation (5), R is called an m-dimensional matter element, whose components are expressed by R k = (N, c k , v k ) (k = 1, 2, . . . , m). Thus, Equation (5)  The characteristic value corresponding to a characteristic can be a single point or a single range. In the latter case, such a range is referred to as a classical domain which is contained in a neighborhood domain. Assume point f is any point in the interval F =< a q , b q >, and F 0 ∈ F, then, the matter element, R 0 , corresponded by F 0 =< a p , b p > can be expressed by Equation (6), where C is a characteristic and V p is the value for C, i.e., its classical domain: Energies 2020, 13, 6302

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A matter element, R F , corresponded by F, can be expressed by Equation (7). Similarly, C is the value of the characteristic for F and V q is the value for C, i.e., its neighborhood domain:

Correlation Function
If F 0 ∈ F, then the correlation function K(f ) can be defined as follows: If set K(f ) ≤ 1, then: where: Correlation functions can be used to determine the membership grade between f and F 0 . Extension correlation functions are given in Figure 10. The figure indicates that if K(f ) > 0, it means that point f currently lies within domain F 0 . If K (f ) < −1, it means point f will not be within either of the two domains, and if −1 ≤ K(f ) ≤ 0, it means that the point lies not within domain F 0 , but within domain F. In extension domains, it is possible to make use of transformation of conditions such that f can belong to the domain of F 0 .

Correlation Function
If F0 ∈ F, then the correlation function K(f) can be defined as follows: If set K(f) ≤ 1, then: where: Correlation functions can be used to determine the membership grade between f and F0. Extension correlation functions are given in Figure 10.

Extension Theory-Based Fault Diagnosis for Inverters
The inverter fault diagnosis system operates when a switch fails to actuate normally as a result of having worked in over-current or high-temperature conditions for an extended period of time or after aging, among other causes. Hence, the present research considers testing switch faults with the framework for three-level NPC inverters as shown in Figure 2, where the positions of faulty transistors are identified by extension theory.
To begin with, the waveforms of the line currents in three phases on the inverter under simulation were subjected to fast Fourier transformation (FFT) to generate the feature frequency spectra for the line currents at m f + 1, m f + 5, and m f − 5 as input signals in the fault diagnosis system that were built based on extension theory. The faults were divided into 12 types by switches S a1 Additionally, from the 84 records for the feature frequency spectra that were measured with twelve different switch faults with the inverter working at a frequency range of 30-90 Hz, and additionally based on the values for the feature frequency spectra of the twelve fault types, the upper and lower limits for the neighborhood domains in each fault type were identified. The classical domains were further set by the feature frequency spectra of each fault type.
The diagnosis and identification process for inverter faults with extension theory is delineated as follows: Step 1: Create a matter-element model for fault characteristics C 1 , C 2 , and C 3 on the frequency spectra for the line current for each fault type.
Step 2: Input uncategorized fault characteristics C 1 , C 2 , and C 3 with the frequency spectra of the line current to create a matter-element model.
Step 3: Assign weights to the characteristics of each fault type (W 1 , W 2 , and W 3 ) which represent the significance of these characteristics. Set W 1 = W 2 = W 3 = 1/3 here.
Step 4: Calculate the correlation between uncategorized fault characteristics on the frequency spectra of the line current and each existing fault type.
Step 5: The maximum correlation value from the calculation determines the category of the uncategorized fault characteristics C 1 , C 2 , and C 3 on the frequency spectra of the line current. Thus, the fault type is identified.

Fault Condition Type
A faulty S a1 Tables 2 and 3 show the data for the characteristic frequency spectra of the line currents with a faulty switch working at 40 and 80 Hz. These measured data were used as inputs in the fault diagnosis system, generating the results of identification shown in Tables 4 and 5. The two tables indicate that the data for every fault were accurately identified. For example, considering F 5 in the test data in Table 2, it is clear from the identification results in Table 4 that F 5 was determined according to the highest output correlation at 0.78784, while the correlation F 12 was the lowest at −0.33376, which suggests that F 12 was least likely.  Table 4. Results for the identification of different switch faults in an inverter working at 40 Hz.  Table 5. Results for the identification of different switch faults in an inverter working at 80 Hz.

Analysis of Tolerant Control
The switch faults of a three-level NPC inverter are divided into two types: external switch faults and internal switch faults. In the framework of a three-level NPC inverter in Figure 2, if any of the external switches Sa1 + , Sa2 − , Sb1 + , Sb2 − , Sc1 + , and Sc2 − are faulty because of an open circuit, then it is necessary to alter both the switching condition of the inverter and the phase angle of sine reference voltage in the pulse width modulation (PWM) control signals to keep the output voltage of the inverter in a balanced three-phase form. The parameters of the induction motor are listed in Table 6. shown in Figure 11. The voltage phasor diagram corresponding to this situation is illustrated in Figure 12a. with an open-circuit fault as an example, the waveforms of the sine reference voltages in three phases are shown in Figure 13.

Analysis of Tolerant Control
The switch faults of a three-level NPC inverter are divided into two types: external switch faults and internal switch faults. In the framework of a three-level NPC inverter in Figure 2, if any of the external switches Sa1 + , Sa2 − , Sb1 + , Sb2 − , Sc1 + , and Sc2 − are faulty because of an open circuit, then it is necessary to alter both the switching condition of the inverter and the phase angle of sine reference voltage in the pulse width modulation (PWM) control signals to keep the output voltage of the inverter in a balanced three-phase form. The parameters of the induction motor are listed in Table 6. When an external switch, for example Sa1 + or Sa2 − , encounters an open-circuit fault, the a-phase half-bridge switches (Sa1 + and Sa2 − ) must be deactivated to activate the internal (neutral-point) switches (Sa1 − and Sa2 + ). Specifically, point a is connected to the neutral point and b-and c-phase are still switched normally. The fault-tolerant control for the occurrence of an open-circuit fault in Sa1 + is shown in Figure 11. The voltage phasor diagram corresponding to this situation is illustrated in Figure 12a. As illustrated in the figure, the phasor positions of the line voltages Vab and Vca without any switch fault in Figure 11 become those of Vab1 and Vca1, with the voltage magnitude decreasing by 0.577 times relative to the original line voltage. The line voltage Vbc1 remains unchanged; however, because the voltage vao is 0, the phase angle of the b-phase voltage should be simultaneously adjusted to be 150° behind that of the a-phase voltage, and the phase angle of the c-phase voltage should be 150° ahead of that of the a-phase voltage. After the occurrence of a fault, the three-phase voltage can still maintain the operation of the balanced three-phase system and the corresponding voltage phasor diagram of the system is shown in Figure 12b. The phasor positions of the line voltages Vab1 and Vca1 presented in Figure 12a shift to those of Vab2 and Vca2, with the magnitude of the line voltage Vbc2 decreasing by 0.577 times relative to that of the original line voltage Vbc1. Considering switch Sa1 + with an open-circuit fault as an example, the waveforms of the sine reference voltages in three phases are shown in Figure 13.   Analysis of the tolerant control strategy for other external switch failures shows the same results and is not be repeated here for the sake of brevity.

Tolerant Control Strategy for Internal Switch Faults
In the framework of a three-level NPC inverter, in order to exercise the tolerant control of internal (neutral point) switches, it is necessary to connect a serial-connected H-bridge switch in parallel to the neutral-point switches in each arm (Sa2 + , Sa1 − ; Sb2 + , Sb1 − ; and Sc2 + , Sc1 − ) as shown in Figure 14. If any of the internal (neutral point) switches Sa2 + , Sa1 − , Sb2 + , Sb1 − , Sc2 + , and Sc1 − in the inverter encounter an open-circuit fault, in order for the inverter to remain in operation, it is necessary to make every parallel-connected tolerant control switch operate at once while altering the switching condition of the inverter. For example, if switch Sc2 + has an open-circuit fault, where, in this case, the c phase output of the faulty inverter cannot connect to neutral point o, the c phase voltage, vco, becomes erroneous and causes current distortion; hence, it is necessary to activate the tolerant control, i.e., disconnecting H-bridge switches in the c phase (Sc1 − , Sc2 + ), maintaining the tolerant bypass switches (Sc + , Sc − ), and allowing the H-bridge switches (Sa1 + , Sa2 − , Sb1 + , Sb2 − , Sc1 + , and Sc2 − ) to actuate two-level switching of voltage P and voltage N, that is, changing the three-level inverter output voltage into two-level output where three-phase balanced output is still maintained. Figures 15 and 16 show the fault-tolerant control and the PWM control signal waveforms for the case of inverter internal switch Sc2 + with an open-circuit fault, respectively. Analysis of the tolerant control strategy for other external switch failures shows the same results and is not be repeated here for the sake of brevity.

Tolerant Control Strategy for Internal Switch Faults
In the framework of a three-level NPC inverter, in order to exercise the tolerant control of internal (neutral point) switches, it is necessary to connect a serial-connected H-bridge switch in parallel to the neutral-point switches in each arm (Sa2 + , Sa1 − ; Sb2 + , Sb1 − ; and Sc2 + , Sc1 − ) as shown in Figure 14. If any of the internal (neutral point) switches Sa2 + , Sa1 − , Sb2 + , Sb1 − , Sc2 + , and Sc1 − in the inverter encounter an open-circuit fault, in order for the inverter to remain in operation, it is necessary to make every parallel-connected tolerant control switch operate at once while altering the switching condition of the inverter. For example, if switch Sc2 + has an open-circuit fault, where, in this case, the c phase output of the faulty inverter cannot connect to neutral point o, the c phase voltage, v co , becomes erroneous and causes current distortion; hence, it is necessary to activate the tolerant control, i.e., disconnecting H-bridge switches in the c phase (Sc1 − , Sc2 + ), maintaining the tolerant bypass switches (Sc + , Sc − ), and allowing the H-bridge switches (Sa1 + , Sa2 − , Sb1 + , Sb2 − , Sc1 + , and Sc2 − ) to actuate two-level switching of voltage P and voltage N, that is, changing the three-level inverter output voltage into two-level output where three-phase balanced output is still maintained. Figures 15 and 16 show the fault-tolerant control and the PWM control signal waveforms for the case of inverter internal switch Sc2 + with an open-circuit fault, respectively.

Tolerant Control Simulations
When applying tolerant control to an open-circuit fault with external switch Sa1 + , for example, its output line voltage and line current is given as shown in Figure 17. In the case of internal (neutral point) switch Sc2 + with an open-circuit fault, its output line voltage and line current are also shown in Figure 18. From Figures 17 and 18, we can observe that if an open-circuit fault occurs at 0.12 s, which would cause the line current of the corresponding phase and three-phase output line voltage to be distorted, particularly so in the affected phase, and the tolerant control strategy is enacted at 0.18 s, then it may observed in the figures that once the tolerant control strategy is enacted that the three-phase output line voltage is downgraded from three-level to two-level output, but it remains a balanced three-phase system despite the fault. Moreover, although the three-phase line current displays a small amount of lag in the beginning of tolerant control due to inductive loads, it maintains a balanced three-phase system output.

Tolerant Control Simulations
When applying tolerant control to an open-circuit fault with external switch Sa1 + , for example, its output line voltage and line current is given as shown in Figure 17. In the case of internal (neutral point) switch Sc2 + with an open-circuit fault, its output line voltage and line current are also shown in Figure 18. From Figures 17 and 18, we can observe that if an open-circuit fault occurs at 0.12 s, which would cause the line current of the corresponding phase and three-phase output line voltage to be distorted, particularly so in the affected phase, and the tolerant control strategy is enacted at 0.18 s, then it may observed in the figures that once the tolerant control strategy is enacted that the three-phase output line voltage is downgraded from three-level to two-level output, but it remains a balanced three-phase system despite the fault. Moreover, although the three-phase line current displays a small amount of lag in the beginning of tolerant control due to inductive loads, it maintains a balanced three-phase system output.  According to the analysis of the tolerant control in Section 6.1, other switch failures in different phases will also produce the same results. As such, additional results are not presented here for the sake of brevity.  According to the analysis of the tolerant control in Section 6.1, other switch failures in different phases will also produce the same results. As such, additional results are not presented here for the sake of brevity. According to the analysis of the tolerant control in Section 6.1, other switch failures in different phases will also produce the same results. As such, additional results are not presented here for the sake of brevity. Figure 19 shows the three-level neutral-point clamped (NPC) inverter used in this study. To verify the experimental results, this study used the digital signal processor TMS320F28335 as the control core and considered the occurrence of open-circuit faults in the switches to test the fault-tolerant control strategy. Figure 20 shows that the occurrence of an open-circuit fault in the switch S a1 + would distort the three-phase output line current and voltage. The fault is particularly severe in the i a , v ab , and v ca phases, and the fault-tolerant control strategy can be implemented 0.015 s after the occurrence of the fault. In this situation, the a phase half-bridge switches (S a1 + and S a2 − ) are deactivated and the neutral-point switches (S a1 − and S a2 + ) are activated. The band c-phase switches still operate normally.

Tolerant Control Experiments
The phase angle of the b phase voltage is simultaneously adjusted such that it is 150 • behind the phase angle of the a phase reference voltage, and the phase angle of the c phase reference voltage is adjusted such that it is 150 • ahead of that of the a phase voltage. After the implementation of the fault-tolerant control strategy, the three-phase output line voltage is reduced from five to three levels ( Figure 20b); however, after the occurrence of the fault, the output line current and voltage are still maintained via the operation of the balanced three-phase system. Therefore, the motor can still operate normally with a reduced load. Based on the tolerance control analysis detailed in Section 6.1, the same results can be obtained when other external switches fail.
If any of the internal switches Sa2 + , Sa1 − , Sb2 + , Sb1 − , Sc2 + , and Sc1 − in the inverter experience an open-circuit fault, in order for the inverter to remain in operation, it is necessary to make every parallel-connected tolerant control switch (as shows in Figure 14) operate at once. If internal (neutral point) switch Sc2 + experiences an open-circuit fault, its output line current and voltage are same as shown in Figure 21. The output line current and voltage still maintain the operation of the balanced three-phase system. Hence, the motor can also operate normally with a reduced load.
Energies 2020, 13, x FOR PEER REVIEW 21 of 25 Figure 19 shows the three-level neutral-point clamped (NPC) inverter used in this study. To verify the experimental results, this study used the digital signal processor TMS320F28335 as the control core and considered the occurrence of open-circuit faults in the switches to test the fault-tolerant control strategy. Figure 20 shows that the occurrence of an open-circuit fault in the switch Sa1 + would distort the three-phase output line current and voltage. The fault is particularly severe in the ia, vab, and vca phases, and the fault-tolerant control strategy can be implemented 0.015 s after the occurrence of the fault. In this situation, the a phase half-bridge switches (Sa1 + and Sa2 − ) are deactivated and the neutral-point switches (Sa1 − and Sa2 + ) are activated. The b-and c-phase switches still operate normally. The phase angle of the b phase voltage is simultaneously adjusted such that it is 150° behind the phase angle of the a phase reference voltage, and the phase angle of the c phase reference voltage is adjusted such that it is 150° ahead of that of the a phase voltage. After the implementation of the fault-tolerant control strategy, the three-phase output line voltage is reduced from five to three levels ( Figure 20b); however, after the occurrence of the fault, the output line current and voltage are still maintained via the operation of the balanced three-phase system. Therefore, the motor can still operate normally with a reduced load. Figure 19. Experimental hardware circuits for the three-level NPC inverter. Figure 19. Experimental hardware circuits for the three-level NPC inverter. Based on the tolerance control analysis detailed in Section 6.1, the same results can be obtained when other external switches fail.

Tolerant Control Experiments
If any of the internal switches Sa2 + , Sa1 − , Sb2 + , Sb1 − , Sc2 + , and Sc1 − in the inverter experience an open-circuit fault, in order for the inverter to remain in operation, it is necessary to make every parallel-connected tolerant control switch (as shows in Figure 14) operate at once. If internal (neutral point) switch Sc2 + experiences an open-circuit fault, its output line current and voltage are same as shown in Figure 21. The output line current and voltage still maintain the operation of the balanced three-phase system. Hence, the motor can also operate normally with a reduced load. According to the previous analysis in Section 6.1, other internal switch failures will also have the same results. According to the previous analysis in Section 6.1, other internal switch failures will also have the same results.

Conclusions
This paper has presented a fault diagnosis system for inverters based on extension theory. The system can identify the positions of faulty power transistors in a three-level NPC inverter. The extension theory-based method, as applied here, can be implemented without requiring massive data quantities for training, thus being able to reduce the data volume and enable faster identification with higher accuracy. In addition, the tolerant control strategy of the system is able to be implemented as soon as any switch in the inverter becomes faulty, where the inverter is enabled to continue supplying power, enhancing the reliability of power supply by the three-level NPC inverter. Finally, the simulation and experimental results suggest that the system is able to correctly pinpoint the positions of faulty power transistors, and, when any switch becomes faulty, it is able to maintain the output line voltage and line current with three-phase balance by means of the tolerant control strategy, which verifies that the method proposed herein is viable.

Conflicts of Interest:
The authors of the manuscript declare that there are no conflicts of interest with any of the commercial identities mentioned in the manuscript.