There are a number of well-established and widely used control strategies for AFE rectifiers. Figure 23 summarizes the existing control techniques [ 8 42 ]. Some of the well-established and widely used control strategies for AFE rectifiers are highlighted in blue in Figure 23 and will be further discussed in this section:
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Some of these control techniques are better suited for maintaining the DC bus voltage, while others allow decoupled control of active and reactive power flow. They also vary in the number and types of required sensors, and the complexity of calculations. In this section, first, grid synchronization is discussed, then VOC, DPC, MPC, and HCC are described and compared. Moreover, the performance of the aforementioned control systems is compared in a simulation using a high-fidelity electro-thermal model of the two-level six-switch boost-type AFE rectifiers in terms of disturbance rejection, THD, efficiency, thermal behavior, etc.
The performance and accuracy of PLL heavily depend on the grid conditions. Therefore, novel methods for robust PLL that can work in unstable or weak grid conditions are being developed [ 44 46 ].
Phase Locked Loop (PLL) has been used since for radio synchronization, and its applications have expanded since [ 45 ]. Figure 24 shows the basic synchronous reference frame (SRF) PLL that consists of several stages. First, the voltages in three-phase are converted to thedomain using Clarke&#;s transformation. Then, fromtousing an estimated anglefor Park&#;s transformation. This is the Phase Detector (PD) stage of the PLL [ 45 ]. The q-axis component of the voltage is then passed through the PI controller, acting as a Loop Filter (LF) to obtain the estimated frequency. The next stage, Voltage-Controlled Oscillator (VCO), uses this frequency to obtain the estimation of phase angle
Some of the control systems described in the following subsections are implemented in the synchronous reference frame, meaning that they require a series of Clarke&#;Park and inverse Clarke&#;Park transformations. The Park transformations rely on the angle of the three-phase voltage vector [ 43 ]. This angle is essential to ensure that the voltage of the grid-connected converter is synchronized with the grid voltages [ 44 45 ].
The VOC requires two AC voltage sensors, three AC current sensors, and one DC voltage sensor. Moreover, the grid voltage angle is used in Clarke&#;Parke transformations; therefore, a PLL is required.
To obtain the desired inverter voltage,and, which can then be transformed into three-phase values and passed on to the modulator, a feed-forward compensation to decouple direct and quadrature axes and the grid voltages,and, are added, as shown in Equation ( 2 ).
The corresponding outputs of thecurrent PI controllers are denoted asand. They represent the voltage that needs to be applied over the boost inductor to achieve the desired current,or, as shown in ( 1 ).
The two inner control loops control the direct and quadrature axis currents. In the standard implementation, a PI controller is used for each of these control loops as well.
The VOC of an AFE rectifier uses cascaded control loops. As shown in Figure 25 , the outer control loop controls the DC link voltage using a proportional-integral (PI) controller. The PI&#;s output represents the DC current required to keep the DC link voltage at the desired level. This DC current can be translated into the respective d-axis current referenceusing a power balance equation.
The DPC requires two AC voltage sensors, three AC current sensors, and one DC voltage sensor. No PLL is required. There are implementations of DPC where the AC voltages are estimated using other measurements, and, therefore, they can omit AC voltage sensors [ 49 ].
If the value oforis equal to one, it means that the active or reactive power needs to be increased, while zero means it needs to be decreased. The control system uses the values of, and the grid voltage angleto decide the next switching state using a switching table. For the purposes of this paper, a switching table and corresponding voltage vectors from [ 49 ] have been used. The grid voltage angle sectoris selected according to Equation ( 3 ) below. Therefore, the first sector is between angles &#;30and 0
As shown in Figure 26 , the outer PI control loop to control the DC link voltage generates a DC current reference, which is further used to calculate the active power reference. The reactive power reference is set to zero for unity power factor operation [ 48 ]. The instantaneous active and reactive power is estimated from the grid side measurements. The error between these estimations and the reference values is then passed to the hysteresis comparator, and the results are further used to calculateand
In Direct Power Control (DPC), the control action is selected from a table of the converter switching states based on the instantaneous difference between the reference and estimated values of active and reactive power [ 47 ].
There is no formal way to evaluate the stability of MPC controllers [ 57 ]. There are many variations in MPC; therefore, this study is focused on OSV-MPC. OSV-MPC requires two AC voltage sensors, three AC current sensors, and one DC voltage sensor. PLL is optional.
The FCS-MPC applied to AFE rectifiers chooses from a finite set of control actions to minimize the cost function. The two common types of FCS-MPC are the Optimal Switching Vector MPC (OSV-MPC) and the Optimal Switching Sequence MPC (OSS-MPC). The OSV-MPC is the most popular type of MPC used for power electronics applications [ 57 ]. For OSV-MPC, the available control actions are the eight possible output voltage vectors that correspond to the eight valid switching states (000, 001, 010, 011, 100, 101, 110, 111) of the AFE rectifier. The OSV-MPC results in a variable switching frequency, which might not be desirable in some cases. OSS-MPC takes care of this problem by using a limited number of switching sequences as a control set, which results in a constant switching frequency. Therefore, both types of FCS-MPC generate gate signals directly and do not require a modulator.
CCS-MPC generates a continuous control signal, which is then converted to gate signals using modulators [ 58 ]. Therefore, in the case of an AFE rectifier, the expected output of the MPC would be the rectifier voltage references in, ordomains. The advantage of the CCS-MPC is that the converter operates at a constant switching frequency, and a long prediction horizon is possible without special search algorithms due to lower computational cost. However, the formulation of the CCS-MPC strategies is complex [ 57 ].
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If the MPC is implemented in a natural reference frame (), as in Figure 27 or a stationary reference frame (), it does not require PLL. However If the MPC is implemented in a synchronous () reference frame, as shown in Figure 28 , it requires PLL [ 52 56 ].
Normally, the MPC is used in the &#;inner&#; control loop in combination with an &#;outer&#; voltage PI. The MPC controls the active and reactive power or AC currents [ 50 52 ]. While it is possible to include the control of DC link voltage in the MPC formulation, it would not be accurate unless the load current is compensated for.
Model Predictive Control (MPC) uses the mathematical model of the system to predict the outcome of a control action; it then chooses the action that minimizes the cost function. The cost function represents the difference between the reference and predicted system states.
The hysteresis band is the main tunable parameter of the HCC. By decreasing the magnitude of the hysteresis band, ripple can be reduced, while the switching frequency will increase. In the standard implementation, the hysteresis band is constant. However, it can also be sinusoidal or adaptive [ 59 ]. The width and type of the hysteresis band define the switching frequency, ripples, and overall efficiency of the system. In the case of constant and sinusoidal hysteresis band HCC, the switching frequency is variable. However, constant switching frequency can be achieved using the adaptive HCC. The variable switching frequency is normally not desired due to the difficulty of filter design and unpredictable noise issues [ 42 ].
The current level higher than the reference current by a magnitude of the hysteresis band is called the &#;upper band&#;, and the current level lower than the reference current by a magnitude of the hysteresis band is called the &#;lower band&#;. If the measured current is above the upper band ( i p h > i p h * + H B ), then the upper switch is on. If the measured current is below the lower band ( i p h < i p h * &#; H B ), then the lower switch is on.
The Hysteresis Current Control (HCC) is used to control the current flowing through the AFE rectifier. The simplest implementation of HCC is shown in Figure 29 . The HCC receives a reference for the DC current from the outer control loop. This reference is multiplied by normalized phase voltages to obtain the three-phase sinusoidal reference currents for the current controller.
In this method, the sinusoidal modulation signals generated by the controller are compared to the carrier signal. The frequency of this carrier signal defines the converter switching frequency. The carrier signals are usually triangular. The use of carrier-based PWM allows the interleaving of carrier signals between different converters or phases, which might be useful for decreasing ripples in the converter.
The VOC, continuous control set MPC uses Pulse Width Modulation (PWM). While DPC, HCC, and finite control set MPC to generate gate signals directly without the use of a modulator. There are several types of PWM: carrier-based PWM, Space vector PWM, pre-programmed PWM, and closed-loop PWM [ 58 ]. Some sources also mention lookup table-based modulation [ 11 ]. Carrier-based modulation and space vector modulation are the most commonly used methods.
In order to further compare the performance of various control systems, they have been implemented in a MATLAB Simulink environment. The simulated AFE system consists of six C3MK SiC-MOSFETs, three inductors on the AC side, one capacitor on the DC side, three AC voltage sensors, three current sensors, and one DC voltage sensor. The parameters of the rectifier are given in Table 4 . The AC-side filter inductors are designed to keep the grid current THD under 5% at full load and the switching frequency at 20 kHz. A forced air cooling system is used to keep the junction temperature at a reasonable level.
μ s sampling time, and the simulation step time is 0.1 μ s. As mentioned earlier, the converter was designed for an AFE with a 20 kHz switching frequency; therefore, the switching frequency for the modulation carrier waveform for VOC has been set to 20 kHz. The operating frequency of HCC is adjusted by changing the hysteresis window. For DPC and OSV-MPC, the sampling time of the inner control loop that selects the gate signals is adjusted to bring the average switching frequency as close to 20 kHz as possible.The system is simulated at full load condition, and then a 20% drop in the load is applied. The simulation results are summarized in Table 5 . All the control systems are running at 1s sampling time, and the simulation step time is 0.1s. As mentioned earlier, the converter was designed for an AFE with a 20 kHz switching frequency; therefore, the switching frequency for the modulation carrier waveform for VOC has been set to 20 kHz. The operating frequency of HCC is adjusted by changing the hysteresis window. For DPC and OSV-MPC, the sampling time of the inner control loop that selects the gate signals is adjusted to bring the average switching frequency as close to 20 kHz as possible.
The THD of grid currents, efficiency, SiC MOSFET mean junction temperature, and swing is given for full load conditions. Then, the overall time to simulate the full AFE with each control system for one second is given. Next, the disturbance rejection behavior of each control approach is demonstrated in terms of the voltage overshoot and the time it takes to return to the setpoint.
When the control system is engaged in a pre-charged rectifier, it takes up to 25 ms to reach the setpoint. As shown in Figure 30 , the DPC is the fastest to reach the setpoint, while the other three perform similarly.
In steady state full load operation, the OSV-MPC demonstrates the highest voltage ripple of 11 V. Generally speaking, the voltage ripple is inversely proportional to their switching frequencies, as shown in Figure 31 Figure 31 b shows the harmonic content of the DC link voltage with different control systems. VOC has a fixed switching frequency, so the harmonics around the switching frequency are the most prominent. The DPC, MPC, and HCC have higher harmonic content on average, but it is distributed throughout the range of frequencies.
When the load drop of 20% occurs, all control systems see approximately a 30 V increase in voltage before settling back to the setpoint within 20 ms, as shown in Figure 32 . Again, DPC is the fastest to react.
While it was attempted to obtain a 20 kHz switching frequency in all converters, it is not easily tunable for the control systems that do not use a modulator. Therefore, DPC and MPC are switching at lower frequencies, as shown in Figure 33 . The switching frequency of VOC stays at 20 kHz. The switching frequency of HCC is 20 kHz on average. However, it varies between 18 and 22 kHz. For DPC and MPC, the switching frequencies average around 17.5 and 18.7 kHz, respectively, with a spread of around 2 kHz from the average.
The thermal behavior of the converter is evaluated using an electro-thermal model from [ 64 ]. Even though the converter and the cooling system are the same for all cases, the choice of control system affects even the MOSFET junction temperature, as shown in Figure 34 . The swing of the junction temperature is the main stressor of the switches [ 65 ]. The VOC and HCC have roughly the same switching frequency. However, the VOC demonstrates higher average temperatures and higher temperature swings.
To conclude, all the control systems compared in this section use a similar number of sensors. Most of them have sensorless implementations that skip either current or voltage sensors in the literature. VOC has a fixed switching frequency, which makes it easier to design the filters, estimate losses, and design the cooling system. Moreover, using a carrier-based PWM makes it a suitable option for parallel converter systems with carrier interleaving. DPC, MPC, and HCC do not require modulators, which reduces the computational burden, but they have variable switching frequencies, which is a disadvantage. Moreover, as demonstrated in the above simulation, it is not easy to tune the average switching frequency for DPC and MPC. While for HCC, the switching frequency may change based on the operating conditions [ 42 ].
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