Research Article  Open Access
Mathematical Modeling and Fault Tolerance Control for a ThreePhase SoftSwitching Mode Rectifier
Abstract
This study primarily focuses on the design of an intelligent threephase softswitching mode rectifier (SSMR). Firstly, the smallsignal dynamic model of a singlephase SSMR is derived together with the design of its controller. Then, the developed singlephase SSMR is connected to form an intelligent threephase 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 threephase SSMR design.
1. 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 switchingmode rectifier (SMR) with power factor regulation [1] is used to make the rectifierinduced 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 hardswitching mode SMR and the modified switching control signals of the pulsewidth modulation (PWM) to complete the softswitching mode rectifier (SSMR) operation [2]. In general, large electrical equipments are fed with a threephase 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 [3–5] 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 singlestage threephase 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 switchbased threephase boost AC/DC converters have a very simple architecture but contain a large amount of low older harmonics. Six active switchbased threephase boost AC/DC converters can obtain a better power factor and harmonic control characteristics but involve a more complex control strategy.
Some threephase SMRs constructed using three or two separate singlephase SMR modules were presented in [10–13]. Though the three connected singlephase modules in a threephase connected SMR [12] can directly apply the power factor control and softswitching technology of a singlephase module, when one of the threephase modules fails, 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 Tconnected threephase SMR, which is constructed using two singlephase SMRs and one centertapped autotransformer. The threephase line drawn currents are made balanced by applying unbalanced twophase voltages to power the twosingle SMRs. However, as any module is randomly disabled, this threephase SMR cannot online detect the fault occurrence and continuously perform the threephase SMR operation through automatic switch connection arrangement. To overcome these problems, this study proposes that singlephase 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 threephase SSMR can continue to maintain the threephase 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.
2. SinglePhase SSMR
Figure 1 shows the power circuit of the singlephase boost SSMR adopted in this study. The proposed zerovoltage transition (ZVT) SSMR system design adopts the current switch control method that uses rampcomparison pulsewidth modulation under the continuous current conduction mode (CCM).
2.1. 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 offstates 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.
2.2. 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 proportionalintegrated (PI) controller, then the general rule of the crossover frequency of current control loop gain should be less than the switching frequency (i.e., ) and should be applied for the design of the current controller [10], which obtains
2.3. Deriving the Converter Model
The ZVT SSMR circuit configuration in Figure 1 was divided into a slowvariable subsystem and a fastvariable 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 smallsignal 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 dashedline 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 , and is the proportional constant of the rectifier.
At the DC operating point of , , , , and , after experiencing a smallsignal disturbance and ignoring highorder terms of AC, the system can obtain the following.(a) The AC component of the input current is where , and .(b) AC component of output current is 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 to when can be derived from Figure 6 as follows:
3. Proposed Intelligent ThreePhase SSMR
The threephase 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 singlephase 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 threephase SSMR system can continue to maintain the threephase balanced power supply under a reduced load capacity while still maintaining excellent power quality characteristics.
3.1. Proposed Circuit Configuration of a SinglePhase SSMR Module Connected to Form a ThreePhase SSMR
The proposed circuit configuration and control structure of the intelligent threephase SSMR has two more centertapped autotransformers and three more toggle switches than a traditional threephase connected SSMR system, as shown in Figures 7(a) and 7(b) [13, 16]. The control structure of the intelligent threephase SSMR in Figure 7(b) has added current distributing factors , fault diagnosis, and troubleshooting logic components. The fault diagnosis and troubleshooting logic unit controlled the switching of the proposed intelligent threephase SSMR toggle switches SW1~SW3 and the selection of current distribution factors . SW1~SW3 in each module switche to the ⓐ point shown in Figure 7(a) under normal operation, with the selection of current distribution factors as 1. When each module of the proposed intelligent threephase SSMR is operating under normal conditions, its circuit configuration is complete the same as the threephase connected SSMR. Therefore, the total capacity and the phasor diagram of voltage and current of the intelligent threephase SSMR are similar to those of a traditional threephase connected SSMR.
(a)
(b)
3.2. Design of a Voltage Control Loop Controller
Under good current tracking characteristics, the proposed voltage control loop of a threephase 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 , and is the transfer function of the converter. Based on the ease of implementation consideration, the voltage controller uses a PI controller.
The proposed smallsignal equivalent circuit model for the threephase SSMR converter is plotted in Figure 9 based on the singlephase SSMR determined in Section 2.3. The transfer function of to during that can be obtained from Figure 9 is
When the module load in each phase reaches W, the voltage controller will first use a simple proportional controller (Pcontroller) and set to 8 V (the actual command value is ). In cases where the input inductor current is able to completely track the command current, the current amplitude command signal 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 listed in Table 2 into (9), the converter transfer function can be obtained as

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)steadystate error = 0;(ii)overshoot = 0;(iii)output voltage dip caused by a unitstep load change W is set to ;(iv)voltage dip restore time: sec.
The voltage controller parameters that can be obtained using voltage controller quantitative design steps [15] are and .
3.3. Power Supply Situation during Any Phase Fault in the Proposed Intelligent ThreePhase SSMR
The circuit configuration and the phasor diagram of the associated voltage and current of the proposed intelligent threephase 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 ⓑ, 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, and the obtained and are where is the proportionality constant of the rectifier, and and are current distribution factors of SSMR and SSMR , respectively. In (11), the purported condition must be met by satisfying At the moment when , , , and , the total capacity of the proposed modified connected SSMR during a fault in SSMR is
When a fault occurs in the phase SSMR module, the output power values of SSMR and SSMR are and .
The circuit configuration and the phasor diagram of the related voltages and currents are plotted in Figure 11, and the obtained and are where is the proportionality constant of the rectifier, and and are current distribution factors of SSMR and SSMR , respectively. In (14), the purported condition must be met by satisfying At the moment when , , , and , the total capacity of the proposed modified connected SSMR during a fault in SSMR is
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 and are current distribution factors of SSMR and SSMR , respectively. In (17), the purported condition must be met by satisfying At the moment when , , , and , the total capacity of the proposed modified connected SSMR during a fault in SSMR is
Based on the previous analysis, when a fault occurs in any phase module of the proposed intelligent threephase SSMR, the total capacity is reduced to that of normal operation. When a fault occurs to any phase module and during normal threephase module operation, the switch state of toggle switches SW1~SW3 and the current distribution factors used in each module are as shown in Tables 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 ⓐ point, the logic level is set to high. When it is at the ⓑ 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 truthvalue table in Table 5. The truthvalue 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 can be used to control the position of the toggle switches SW1~SW3 and the selection of a current distribution factor in Figures 10 to 12. When a fault occurs in any phase module of the proposed threephase modified connected SSMR system, the detected faulty module can again perform a nofault module current distribution factor selection and can automatically toggle the position of switches SW1~SW3 to change the circuit configuration. The threephase 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.

 
Note: × indicates not taking into consideration. 

4. Simulation Results of the Proposed Intelligent ThreePhase 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 W, then the system reduces the load to continue providing power. This ensures that nofault modules would not be burnt down by a power overload. Figures 14, 15, and 16 are the PSIM software generated waveforms representing relevant restored voltages and currents after the troubleshooting of faults from a sudden failure of any of the SSMR , SSMR , and SSMR modules under a reduced power load when the proposed SSMR supply load is 1800 W. As is evident from the intelligent threephase SSMR shown in the figures, when each module is operating normally, the threephase input line voltage and current are nearly in phase to produce a threephase balance power supply. When one module fails, the input line voltage and current of the threephase can maintain in phase and supply the threephase balance power, while the amplitude of the input voltage and current and phasor diagram of each nofault module is also consistent with positions that are in phase. Therefore, it has a good power factor and low harmonic characteristics. The output DC voltage during the dynamic response to a module fault and the post fault troubleshooting power restoration can also use the designed voltage controller to retain a high regulation performance.
(a)
(b)
(a)
(b)
(a)
(b)
The comparisons of circuit structure, switching characteristics, power quality, fault tolerance, and reliability of the power supply between the proposed threephase SSMR and some of the existing ones are made in Table 6. It shows that the proposed threephase SSMR possesses the advantages of flexibility, reliability, superior power quality, and online fault detection tolerance.
5. Conclusions
For the application of a threephase power feeding in largecapacity power electronic equipment, this study adopted singlephase single module SSMR to assemble an intelligent threephase SSMR system power supply. When any module in this threephase SSMR module experiences a fault, the proposed intelligent threephase SSMR system can use intelligent online fault diagnosis and a fault troubleshooting strategy to immediately reduce the load capacity online to continue maintaining the threephase balanced power supply, keeping the threephase SSMR system from shutting down for rerouting. This greatly enhances the reliability of the power supply, while the threephase input voltage and current are still almost in phase; so, the power supply side can still possess the characteristics of good power factor and low harmonics.
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Copyright
Copyright © 2013 KueiHsiang Chao and ChinTsang Hsieh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.