A 3D-space vector modulation-based sensorless speed control of a fault-tolerant PMSMs fed by 27-level inverter

ABSTRACT This paper exhibits a novel technique to obtain a sensorless control of a permanent magnet synchronous motor post an open-circuit fault condition to achieve a maximum reliability of the whole drive system. The contribution of this work can be expressed in two things. The first contribution is the introducing of a 4-leg 27-level cascaded H-bridge inverter modulated through a 3D-space vector modulation technique to make the whole drive system fault tolerant. Besides that, an algorithm to track the saturation saliency position post an open-circuit fault is used to enhance the reliability of the whole system to meet the safety issues requires in some applications. This algorithm is based on measuring the transient current response of the motor line currents due to the (Insulated Gate Bipolar Transistors (IGBTs) switching actions. Then, according to the operating condition (healthy or post an open-circuit fault), the saliency position signals are constructed from the dynamic current responses. Simulation results are provided to demonstrate the eﬀectiveness of the saliency-tracking technique under healthy and post an open-circuit fault condition. Moreover, the results will verify the maximum reliability for the whole drive system that is achieved in this work through a continuous operation of the drive system post an open-circuit fault condition and under sensorless speed control.


Introduction
Multilevel inverters are a very attractive solution in high-power applications (Ying & Gan, 2012), medium voltage (Marzoughi et al., 2014) including high-speed railways (Qashqai et al., 2015) and renewable energy sources (Yu et al., 2016). This is due to their capability to handle high voltage and at the same time produce an output voltage near sinusoidal (low total harmonic distortion (Bhagwat & Stefanovic, 1983;Lai & Peng, 1996;Okazaki et al., 2016). The multi-level inverter has many topologies such as the cascaded H-bridge, neutral point clamped and flying capacitor inverters (Nabae et al., 1981;P. Hammond, 1997;Meynard et al., 2002). The advantages coming from the elimination of cabling and mechanical sensors in terms of increasing the reliability and reducing the cost has directed the researchers towards sensorless control of two-level and multi-level inverter drive system sensorless techniques. Sensorless control techniques of multi-level inverter drive are similar to those used in two-level inverter drive and can be divided into two main groups. The first group is called model-based group (Adiuku et al., 2015;Kant & Singh, 2018;Kim et al., 2017;Wang & Lu, 2010), and these techniques fail at low and zero speeds. The second group is known as signal injection group dealing with spatial saliencies (Saleh et al., 2009;Saleh & Sumner, 2016a;Scelba et al., 2017). Spatial saliency-tracking methods inject high-frequency current or voltage components in addition to the fundamental one. A suitable demodulation of the stator currents and/or voltages allows to extract useful information about the position of the rotor.
In certain applications like electric vehicles, aerospace systems, military, and hospitals, the shutdown of the electrical systems due to any fault in the inverter may result in a potential risk to human beings and immeasurable economic losses (Haimin et al., 2008;Welchko et al., 2004). Hence, these applications require a reliable inverter that enables the continuous operation of the system post the fault. Therefore, discussion of fault-tolerant inverters and enhanced system reliability attracts much attention among researchers.
Fault-tolerant inverters are reported in many papers in the literature. These faulttolerant inverters make use of the hardware redundancy to achieve post-fault operation (Zhang et al., 2014). Based on the hardware redundancy, the fault-tolerant inverters can be classified into three groups. The first group deals with switch fault to achieve postfault operation using different techniques like inherently redundant witching states (Lezana et al., 2010), DC-bus midpoint connection (Campos-Delgado et al., 2008) and redundant parallel or series switches installation (Rodriguez et al., 2008). The second group deals with leg fault to achieve post-fault operation by adding redundant legs in parallel or series connection to the main legs (Jen-Ren & Lipo, 1993;Mohammadpour et al., 2014). Finally, post-fault operation can be achieved in multi-level inverter using three scenarios including neutral-shift, DC-bus voltage reconfiguration, and redundant module installations are employed (Correa et al., 2006;Lezana & Ortiz, 2009;Yi et al., 2008). Sensorless control of motor drives under open circuit phase faults is proposed (Saleh, & Sumner, 2016b) using a four-leg two-level Permanent Magnet Synchronous Motor (PMSM) drive and in (Saleh & Sumner, 2018) using four-leg multi-level (PMSM) drive. A special fault-tolerant control technique was used in cases of an open-circuit fault to keep the performance of the drive from degradation. This paper is introducing a novel method to achieve sensorless speed control of a permanent magnet synchronous motor post an open-circuit fault condition to achieve a maximum reliability of the whole drive system. The contribution of this paper can be expressed through two achievements. The first contribution is the use of a 4-leg 27-level inverter based on 3-Dimension Space Vector Pulse Width Modulation (3D-SVPWM) modulation technique to maintain continuous system operation post of an open-circuit fault. The second contribution is the introduction of an algorithm to track the saturation saliency of PMSM post the open-circuit fault. These two procedures help to meet the safety procedure for the whole system and increase the reliability of the system. Figure 1 shows the proposed four-leg multi-level inverter drive topology. An overview of each part of this topology is illustrated below.

Three-phase PMSM model
The three-phase PMSM model is given in Equation 1-Equation 5. In this model, the saturation saliency is inherited in the self and leakage inductance equations. Moreover, this model can give trusted result in healthy operating conditions and in the cases of some fault conditions like open-circuit fault: where v a;b;c are the external voltages applied to the motor, r s is the equivalent resistance of the stator winding, i a;b;c are the stator phase currents, ; a;b;c are the total fluxes linking stator windings, L aa , L bb and L cc are the total-inductances of the stator windings and L ab , L ac and L bc are the mutual inductances of the stator windings. φ ma;b;c is the permanent magnet fluxes linking the stator windings. The inductances in the stator windings are functions of rotor angle, defined by:  where L sl ; L x are the stator self and fluctuating inductances per phase respectively. L so is the stator mutual inductance. It is worth to say here that Equation 4 is inherent including the saturation saliency effect which appears in the term (2θÞ. Finally, the permanent magnet fluxes linking the stator winding are: where λ m is the peak flux linkage due to permanent magnet.

27-level inverter
Asymmetric cascaded multi-level inverter is used in this research with switching frequency equals 5 kHz. Each leg of the inverter consists of three H-bridges connected in series and supplied by three independent DC sources a scaled in the power of three as shown in Figure 2. This configuration of the multi-level inverter helps to generate 27-level at the output of the inverter instead of generating seven levels when using symmetric sources. The output of each H-bridge is shown in Figure 2. It can be noted that the switching frequency of the main H-bridge, which manages more than 80% of the total power, is the same frequency of the system, in this case, only 50 Hz. The frequency of the auxiliary H-bridges is also low but increases as the voltage level of the inverter becomes lower in the chain, which means that the H-bridge with DC link voltage (V) has a switching frequency equal to the switching frequency of the multi-level inverter which is 5 kHz. The voltage V that is used is 25 V and the maximum voltage generated from each leg is 325 V.

3D SVPWM
The proposed 3D-SVPWM technique that is adopted in this work is presented (Prats et al., 2003). This technique is very simple and based on the geometrical consideration. Moreover, it is independent of the number of levels of the multi-level inverter. More importantly, it can be used under healthy condition and post an open-circuit fault without modifications. The 3D space vector of the 27-level inverter is shown in Figure 3. The reference voltage will be pointing to sub-cube. This subcube can be identified using the components (a, b, c) which are the integer values of the reference voltage (V_ref) as shown in Figure 4. This cube can be decomposed into six tetrahedrons. These tetrahedrons and the associated PWM waveforms are shown in Figure 5.
The flow diagram of the 3D modulation algorithm for choosing the tetrahedron where the reference voltage is pointing to is shown in Figure 6 Figure 7 shows the indirect rotor field orientation (IRFO) control structure proposed for the 27-level 4-leg PMSM drive. The reliability of this topology has been enhanced by adding the fourth leg to control the zero sequence component of the current in case of unhealthy operating condition in addition to using the 3D-SVPWM technique. The simulation of the PMSM drive system has been carried out using the SABER simulation package in sensored mode, pre, and post an open-circuit fault. The current controllers of the d-q components of the stator currents of the PMSM were designed such that the closed-loop bandwidths were 300 Hz and damping ratios were 0.707, while the speed controller was designed such that the closed-loop bandwidth was 10 Hz and damping ratio is 0.707. These design specifications were chosen for the motor under healthy operating condition as well as under opencircuit fault condition. The PMSM motor parameters are given in Table 1.

Simulation results of a sensored speed control
The implementation of the above vector control structure in hardware is illustrated in the flow chart given in Figure 8 This flow chart shows what is done in each PWM period starting from reading the analog to digital converters of the Digital Signal Processor (DSP) to obtain both the stator currents and the stator dynamic current responses and ending of the generating the PWM signals. The stability analysis of the IRFO control topology that is given in Figure 7 is examined using root locus as shown in Figure 9. The root locus of the closed-loop system shows that the system is stable as the poles of the closed-loop system exist in the left half of the root locus.
The simulation results in Figure 10 show the effectiveness of the PMSM drive system. The motor was running at speed equals to 1300 rpm at full load. In time intervals (2.5 s-3 as well as phase shifting them by 30 degrees away from the faulted phase. These actions are done automatically by the controller in this work without the need to do any extra actions as in Saleh and Sumner (2018). This is related to the using of 3D-SVPWM in which the phase to neutral voltage in each leg can be generated separately unlike the using of 2D SVPWM.

Under healthy condition
In Section 2.1, Equation 3 has shown that the saturation saliency is inherently included in the model of the PMSM and reflected in the term (2θÞ of both the self and mutual inductances. Based on that, the self and mutual inductances will vary according to the position of the rotor (θÞ. This variation will be reflected in the transient response of the three-line currents of the PMSM motor due to the switching actions. So, the saturation saliency can be tracked by measuring these transient current responses during each PWM period. The algorithm to track the saturation saliency under health operating conditions is presented in this section. Figure 11 shows the case 3 switching sequence of the 4-leg 27level inverter under healthy operating condition. The stator circuit when the vectors V0, V1, V2 and V3 are applied is shown in the same Figure. Using the stator dynamic circuits shown in Figure 11, the following equations hold true:    Assuming that the voltage drop across the stator resistances is small and the back emf can be neglected providing the time separation between the vectors is small, the following equations can be obtained using vector V0, V1, V2 and V3: Finally, the saliency position scalars P a , P b and P c can be obtained as follows: By doing the same procedures for other cases, Table 1 can be constructed to track the saliency under healthy operating condition for 4-leg multi-level inverter.

Post an open-circuit fault condition
The algorithm to track the saliency of the 4-leg inverter 27-level inverter under healthy operating condition as given in Table 2 cannot be applied in the cases of an open-circuit fault operating condition. This related to the fact that the dynamic current response of the faulted phase is equal zero and hence the position scalar related to the faulted phase cannot be constructed. To obtain the algorithm to track the saliency under an open-circuit operating condition the new stator dynamic circuits should be considered. Figure 12 shows the case 3 switching sequence of the 4-leg 27-level inverter under phase 'b' opencircuit fault operating conditions. The stator circuit when the vectors V0, V1 and V3 are applied is shown in the same Figure 12. Using the stator dynamic circuits shown in Figure 10, the following equations hold true: By neglecting the voltage drop across stator resistance and the back emf, the following equations can be obtained using vector V0, V1 and V3: Finally, the saliency position scalars P a and P c can be obtained as follows: P b can be deduced from P a and P c as follows:  Table 2. Selection of P a , P b and P c for a star-connected machine driven by 4-leg 27-level inverter under healthy operating condition.
By doing the same procedures for other cases and other phases, Tables 3-5 can be constructed to track the saliency in case of open-circuit fault. Figure 13 shows the results of the simulation to tracking saliency under healthy operating condition as given in Table 2 and under single-phase open-circuit fault as given in Tables 3-5. The motor was running at 500 rpm at full load and under healthy operating condition. Open-circuit faults in phases 'a', 'b' and 'c' were introduced at time intervals (2-2.5), (3-3.5) and (4-4.5), respectively. Figure 13 shows that the new algorithm can track the saliency in the case of an open-circuit fault with the same quality as in healthy conditions.

Fully sensorless speed control of 4-leg multilevel inverter under unhealthy condition
The sensorless speed control for a PMSM motor driven by 4-leg 27-level inverter in healthy mode and post an open-circuit fault has been implemented in simulation in the SABER modelling environment. The estimated position signals Pαβ are used as the input to a mechanical observer (Lorenz & Van Patten, 1991) to obtain the speed ω^ and a cleaned position θ^. Note that the simulation includes a minimum pulse width of 10 µs when di/dt measurements are made, similar to the experimental results of (Saleh et al., 2009). This estimated speed ω^ and position θ^ are used to obtain a fully sensorless speed control as shown in Figure 14. Figure 15 shows the results of sensorless low-speed control of a PMSM motor using the algorithm proposed in this paper. The motor was running in healthy mode at a speed of 30 rpm. At t = 2 s an open-circuit fault is introduced to phase 'a'. Then, at t = 2.5 s a speed step command from 30 rpm to zero was applied to the motor. Between t = 3 s and t = 4 s, the fault in phase 'a' is removed and the motor return to run in healthy mode. At t = 4 s, an open circuit was introduced to phase 'b'. After that, at t = 4.5 s, a speed step command from zero to −30 rpm was applied to the motor. Between time t = 5 s and t = 6 s, the fault is removed from phase 'b' and the motor returned to run in a healthy operating condition. Finally, at time t = 6 s, an open circuit was introduced to phase 'c'. Figure 15 shows that the motor responded to the speed step with a good transient and steady-state response even the motor was running under an open-circuit fault.    Figure 16 shows results similar to the results shown in Figure 15 but with higher speed steps commanded from the system. The figure shows the effectiveness of the system in response to the high-speed steps (from 500 rpm to −500 rpm back to 500 rpm) in the healthy case as well as the case of an open circuit. Figure 17 demonstrates the stability of the fully sensorless system when a load disturbance is applied at low speed (200 rpm) in both healthy modes and under open phase fault conditions. The results show that the system maintains the speed in all the cases.
To validate the effectiveness of the proposed approach in this research, the results obtained in Figures 15-17 are compared with experimental results obtained (Baoming & Peng 2009) for sensorless control of AC drive fed by three-level inverter using modelbased algorithm, those given in Pereda et al. (2009) for DTC control of induction motor fed by a multi-level inverter, and those obtained in Saleh et al. (2009) for sensorless control of induction motor using voltage test pulses. Firstly, the new approach could work at zero speed at full load as shown in Figure 15 similar to the results obtained in all saliencytracking sensorless technique. While the techniques obtained in Baoming and Peng (2009) could not work at zero and low speeds due to the fact that these methods rely on the amplitude of back-EMF, which is proportional to the rotor speed. This quantity will vanish at low and zero speeds. The steady-state error of the speed for the proposed method under healthy condition is ±5 rpm under healthy operating conditions and ±8 rpm under an open-circuit fault condition while it is ±7 rpm in Pereda et al. (2009). Finally, the dynamic response of the new approach is faster than the dynamic response obtained in Pereda et al. (2009) and Baoming and Peng (2009).

Conclusion
This paper has presented a novel technique to track the saturation saliency of a PMSM fed by a 4-leg 27-level inverter in the case of a single-phase open-circuit fault. The new technique is based on measuring the transient current response of the motor line currents due to the 27-level inverter switching actions. The proposed method can track the saliency of the motor under healthy conditions post an open-circuit fault condition. The new technique is applicable to track the rotor slotting saliency in IMs (14*fr). The results have shown the effectiveness of the new technique in increasing the safety measures in critical systems that need a continuous operation. In future, the research will continue in the fields of sensorless control of PMSM under other types of faults such as short-circuit. In addition, other techniques to obtain a fault-tolerant will be investigated to obtain a sensorless control.

Disclosure statement
No potential conflict of interest was reported by the authors.