Mathematical Modeling and Fault Tolerance Control for a Three-Phase Soft-Switching Mode Rectifier

This study primarily focuses on the design of an intelligent three-phase soft-switching mode rectifier (SSMR). Firstly, the smallsignal dynamicmodel of a single-phase SSMR is derived together with the design of its controller.Then, the developed single-phase SSMR is connected to form an intelligent three-phase SSMR. When any of the phase modules in the proposed intelligent threephase SSMR experiences a fault, it can continue to supply power automatically under reduced load capacity while still maintaining good power quality characteristics. Finally, some simulation results were used to demonstrate the effectiveness of the proposed intelligent three-phase SSMR design.


Introduction
Traditional rectifiers contain a large amount of harmonic currents, which reduce the power factor of input AC side and greatly deteriorate the power quality.To enhance power quality, a switching-mode rectifier (SMR) with power factor regulation [1] is used to make the rectifier-induced current form a sine wave with the power factor near to 1.A traditional hardswitch mode SMR comes with reduced power conversion efficiency due to a larger switching loss and possesses greater switching stress and electromagnetic interference (EMI).It uses the auxiliary resonant branches connected to the original power circuit on the hard-switching mode SMR and the modified switching control signals of the pulse-width modulation (PWM) to complete the soft-switching mode rectifier (SSMR) operation [2].In general, large electrical equipments are fed with a three-phase power.A considerable variety of threephase circuit configurations have been derived from singlephase switching rectifier circuits.Among them, the circuit architecture of a boost converter [3][4][5] is the simplest in form, easiest to control, and superior in performance; these are the primary reasons for its wide use.To date, a variety of circuit configurations and control technologies for the single-stage three-phase boost AC/DC converter have been proposed.Some of these configurations use a single active switch [6,7], while others use six active switches [8,9].Single active switch-based three-phase boost AC/DC converters have a very simple architecture but contain a large amount of low older harmonics.Six active switch-based three-phase boost AC/DC converters can obtain a better power factor and harmonic control characteristics but involve a more complex control strategy.Some three-phase SMRs constructed using three or two separate single-phase SMR modules were presented in [10][11][12][13].Though the three connected single-phase modules in a three-phase Δ-connected SMR [12] can directly apply the power factor control and soft-switching technology of a single-phase module, when one of the three-phase modules fails, Δ-connected can change to -connected.It can continue to provide power under a reduced load condition, which translates into improved system reliability, but it is done at the expense of power quality [12].In [13], the authors proposed a modified T-connected three-phase SMR, which is constructed using two single-phase SMRs and one center-tapped autotransformer.The three-phase line drawn currents are made balanced by applying unbalanced twophase voltages to power the two-single SMRs.However, as any module is randomly disabled, this three-phase SMR cannot online detect the fault occurrence and continuously perform the three-phase SMR operation through automatic switch connection arrangement.To overcome these problems, this study proposes that single-phase SSMR modules should be connected together to form an intelligent threephase SSMR that not only has a simple connection structure but also possesses automatic online fault detection functions.
In the case of a module experiencing a fault, the intelligent three-phase SSMR can continue to maintain the three-phase balance of high power quality electricity supply without having to shut down for fault module maintenance, thus greatly enhancing the quality and reliability of the power supplied by the system.

Single-Phase SSMR
Figure 1 shows the power circuit of the single-phase boost SSMR adopted in this study.The proposed zero-voltage transition (ZVT) SSMR system design adopts the current switch control method that uses ramp-comparison pulsewidth modulation under the continuous current conduction mode (CCM).

Scheme of Control Loop.
The control block diagram of the proposed SSMR, as shown in Figure 2, contains both inner and outer control loops.The inner loop is the current control loop, and the outer loop is the voltage control loop.The role of the current control loop is to raise the power factor, and the role of the voltage control loop is to provide stability control for the output DC voltage.
According to the on-and off-states of circuit switches and diodes in Figure 1, a switching period can be divided into seven operating modes.Their main waveform variables are as shown in Figure 3.

Design of a Current Control Loop Controller.
The state average method can be used to derive the current loop gain transfer function [10].If the current controller   () chooses to use a proportional-integrated (PI) controller, then the general rule of the crossover frequency of current control loop gain should be less than the switching frequency 1/2 (i.e.,   < 0.5  ) and should be applied for the design of the current controller [10], which obtains

Deriving the Converter Model.
The ZVT SSMR circuit configuration in Figure 1 was divided into a slow-variable subsystem and a fast-variable subsystem [14].The slow subsystem consisted of main storage (filter) components for input and output, while the fast subsystem consisted of resonant components with a state variable filter (slow system variables) as   () and   () and resonant state variables (fast system variables) as   () and   ().From the fast subsystem perspective, the slow state variables in the entire switching period   could be considered as constants.
In contrast, from the slow subsystem perspective, only the average effect of fast resonant state variables could be seen.Therefore, when deriving the mathematical model for the converter, the moving average function of the fast variables  should be calculated first and then substituted into the slow variables to obtain the averaging model for the slow variables.Finally, the average power method was used to derive the small-signal model of the converter [15].From the circuit configuration in Figure 1, the slow filter perspective could be used to depict the average equivalent circuit of the slow system that was shown in Figure 4.The dashed-line portion indicates the average effect of fast variables on the slow variables system.Additionally,   is the serial equivalent resistance of the boost inductor   .
Then, the voltage   of the resonant capacitor   and the moving average function of the sum   =   +   of the currents   and   flowing through both the diode  and the auxiliary diode   were separately obtained.The seven operation modes [10] in Figure 3 were used to obtain   ,   , and   under various mode solutions, listed in Table 1.The results listed in Table 1 could be used to separately obtain the moving average function of   and   .
(a) Obtain moving average function   of   as (b) Obtain moving average function   of   () as Since the system in Figure 1 shows a very small loss to be negligible, the averaged equivalent circuit of the slow system shown in Figure 4 can be plotted like Figure 5.Its average value of   () during the input voltage period   is where  1 = 2 √ 2/, and  is the proportional constant of the rectifier.At the DC operating point of ⟨  ⟩   =   ,  () =  () , î = Î , ⟨  ()⟩   =   , and   =   , after experiencing a small-signal disturbance and ignoring highorder terms of AC, the system can obtain the following.
where   ≜  1  () , and where Equations ( 5) and ( 6) can be used to draw the smallsignal equivalent circuit model of the converter, as shown in Figure 6.The transfer function of Δ î to Δ  when Δ () = 0 can be derived from Figure 6 as follows:

Proposed Intelligent Three-Phase SSMR
The three-phase rectifier circuit system possesses advantages such as higher capacity, higher output voltage ripple frequency, and smaller ripple amplitude.Therefore, to increase the system capacity and the quality of the power supply, the single-phase SSMR in this study is connected to form a threephase SSMR for the power supply.When any of the phase modules experiences a fault, the three-phase SSMR system can continue to maintain the three-phase balanced power supply under a reduced load capacity while still maintaining excellent power quality characteristics.SW1∼SW3 and the selection of current distribution factors  1 ∼  3 .SW1∼SW3 in each module switche to the h a point shown in Figure 7(a) under normal operation, with the selection of current distribution factors  1 ∼  3 as 1.When each module of the proposed intelligent three-phase SSMR is operating under normal conditions, its circuit configuration is complete the same as the three-phase Δ-connected SSMR.Therefore, the total capacity and the phasor diagram of voltage and current of the intelligent three-phase SSMR are similar to those of a traditional three-phase Δ-connected SSMR.

Design of a Voltage Control Loop Controller.
Under good current tracking characteristics, the proposed voltage control loop of a three-phase SSMR can be reasonably expressed as in the block diagram of Figure 8, where   is the conversion coefficient of a load power disturbance on the voltage,   is the expressed voltage sensing conversion factor, which is set as   = 0.02(/), and   () is the transfer function of the converter.Based on the ease of implementation consideration, the voltage controller   () uses a PI controller.
The proposed small-signal equivalent circuit model for the three-phase SSMR converter is plotted in Figure 9 based on the single-phase SSMR determined in Section 2.3.The transfer function of Δ î to Δ  during Δ () = 0 that can be obtained from Figure 9 is When the module load in each phase reaches   = 300.4W, the voltage controller will first use a simple proportional controller (P-controller)   () =   = 5 and set  *  to 8 V (the actual command value is 8 V/0.02 = 400 V).In cases where the input inductor current   () is able to completely track the command current, the current amplitude command signal Î is measured to be 6.4 A, with an actual output voltage of 360 V at the specific moment.The rest of the parameters required for seeking the transfer function of the converter are listed in Table 2.By substituting the parameters If the system is to achieve high performance load regulation characteristics, the step load dynamic response is generally required to possess characteristics such as zero steadystate error, zero overshoot, smallest possible maximum voltage dip, and quickest possible restore time.The output voltage specification requirements on the step response are (i) steady-state error = 0; (ii) overshoot = 0; (iii) output voltage dip caused by a unit-step load change Δ  = 600 W is set to Δ ,max = 8 V; (iv) voltage dip restore time:   = 0.8 sec.
The voltage controller parameters that can be obtained using voltage controller quantitative design steps [15] are   = 1.3 and   = 9.46.

Power Supply Situation during Any Phase Fault in the
Proposed Intelligent Three-Phase SSMR.The circuit configuration and the phasor diagram of the associated voltage and current of the proposed intelligent three-phase SSMR during  phase module failure are shown in Figure 10.Assume that when the  phase SSMR module is experiencing a fault, the position of SW3 will be switched to i b , and at that specific moment, the output power values of SSMR  and SSMR  are     and   .Their circuit configuration and the phasor diagram of related voltages and currents are plotted in Figure 10 where  is the proportionality constant of the rectifier, and  2 and  3 are current distribution factors of SSMR  and SSMR , respectively.In (11), the purported   =   condition must be met by satisfying At the moment when where  is the proportionality constant of the rectifier, and  1 and  3 are current distribution factors of SSMR  and SSMR , respectively.In ( 14), the purported   =   condition must be met by satisfying At the moment when When a fault occurs in a  phase SSMR module, the output power values of SSMR  and SSMR  are   and   .The circuit configuration and the phasor diagram of the correlated voltages and currents are plotted in Figure 12, and the obtained   and   are where  is the proportionality constant of the rectifier, and  1 and  2 are current distribution factors of SSMR  and SSMR , respectively.In (17), the purported   =   condition must be met by satisfying Based on the previous analysis, when a fault occurs in any phase module of the proposed intelligent three-phase SSMR,   3 and 4.During normal operation, current flowing through the module can be detected; therefore, the logic level of that module during normal operation is set to high.In contrast, when a module experiences a fault, the logic level is set to low.In Table 3, when the switch is placed at the h a point, the logic level is set to high.When it is at the i b point, it is set to low.In Table 4, when the current distribution factor is 1, the logic level is set to high; when the current distribution factor is 0.75, the logic level is set to low.The logic levels demonstrated by the previous results are summarized in a logic control circuit truth-value table in Table 5.The truth-value table listed in Table 5 has been simplified using Karnaugh map, and its logic control circuit can be realized using the logic circuit in Figure 13.The logic circuit in Figure 13   change the circuit configuration.The three-phase balanced power supply is continuously maintained under a reduced load capacity, allowing it to still regulate the input voltage and current into almost the same phase.This achieves an excellent supply quality of almost 1.0 in the power factor.

Simulation Results of the Proposed Intelligent Three-Phase SMR
Since each single module of the proposed intelligent threephase SSMR system is rated at 600 W, the total power is 1800 W. Therefore, when a fault occurs in any module in this system, if the load capacity is greater than 1800/ √ 3 W, then the system reduces the load to continue providing power.This ensures that no-fault modules would not be burnt down by a power overload.power restoration can also use the designed voltage controller to retain a high regulation performance.The comparisons of circuit structure, switching characteristics, power quality, fault tolerance, and reliability of the power supply between the proposed three-phase SSMR and some of the existing ones are made in Table 6.It shows that the proposed three-phase SSMR possesses the advantages of flexibility, reliability, superior power quality, and on-line fault detection tolerance.

Conclusions
For the application of a three-phase power feeding in largecapacity power electronic equipment, this study adopted single-phase single module SSMR to assemble an intelligent three-phase SSMR system power supply.When any module in this three-phase SSMR module experiences a fault, the proposed intelligent three-phase SSMR system can use intelligent online fault diagnosis and a fault troubleshooting strategy to immediately reduce the load capacity online to continue maintaining the three-phase balanced power supply, keeping the three-phase SSMR system from shutting down for rerouting.This greatly enhances the reliability of the power supply, while the three-phase input voltage and current

Figure 3 :Figure 4 :
Figure 3: Voltage and current waveforms of the main components of ZVT SSMR.

Figure 5 :Figure 6 :
Figure 5: Slow speed perspective average model equivalent circuit during the period   of input voltage   .

Figure 9 :Figure 10 :Figure 11 :
Figure 9: Small-signal equivalent circuit model of the proposed three-phase SSMR converter.

Figure 12 :
Figure 12: The circuit configuration and the phasor diagram of related voltages and currents of the proposed intelligent three-phase SSMR during  phase fault.

𝑤 1 :
2 = 0.75 : 1. (18) At the moment when |  | = , |  | = ( √ 3/2), |  | = ( √ 3/2), and |  | = , the total capacity of the proposed modified -connected SSMR during a fault in SSMR  is is reduced to 1/ √ 3 that of normal operation.When a fault occurs to any phase module and during normal three-phase module operation, the switch state of toggle switches SW1∼SW3 and the current distribution factors used in each module are as shown in Tables

Figure 13 :Figure 14 :
Figure 13: Logic control circuit of the proposed intelligent three-phase SSMR.

1 VFigure 15 :
Figure 15: Simulated results of a continuous power supply when the load drops from 1800 W to 1800/ √ 3 W during a failure of a SSMR  module in the proposed intelligent three-phase SSMR: (a) simulated waveforms of output DC voltages and input voltages and currents for each phase; (b) simulated waveforms of input voltages and currents of various modules.

1 VFigure 16 :
Figure 16: Simulated results of a continuous power supply when the load drops from 1800 W to 1800/ √ 3 W during a failure of SSMR  module in the proposed intelligent three-phase SSMR: (a) simulated waveforms of output DC voltages and input voltages and currents for each phase; (b) simulated waveforms of input voltages and currents of various modules.

Table 2 :
Parameters required for finding the transfer function of the converter.
, and the obtained   and   are , and |  | = , the total capacity of the proposed modified -connected SSMR during a fault in SSMR  is The circuit configuration and the phasor diagram of the related voltages and currents are plotted in Figure 11, and the obtained   and   are (13)13)When a fault occurs in the  phase SSMR module, the output power values of SSMR  and SSMR  are   and   .
, and |  | = , the total capacity of the proposed modified -connected SSMR during a fault in SSMR  is

Table 3 :
Positions of toggle switches SW1∼SW3 under various operating conditions.

Table 4 :
Current distribution factors used under various operating conditions.
Note: × indicates not taking into consideration.

Table 5 :
Truth-value table of a logic control circuit of the proposed intelligent three-phase SSMR.