# A Single-Loop Repetitive Voltage Controller with an Active Damping Control Technique

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## Abstract

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## 1. Introduction

## 2. System Description

_{dc}, the full-bridge inverter, which consists of four switching devices, and the LC filter.

_{o}are represented as:

_{inv}, and i

_{o}represent the filter inductance, the filter capacitance, the inverter output voltage, and the load current, respectively. From (1), the open-loop output impedance Z

_{o}(s) is defined as follows:

_{inv}and v

_{o}except considering i

_{o}is written as seen below.

_{o}(s) and G

_{vo}(s) show that they inherit the characteristics of band-pass and low-pass filters without damping coefficients, respectively.

_{o}(s) and G

_{vo}(s) in the frequency domain with the given parameters, L = 2.9 mH and C = 120 μF. In order to reduce the output voltage distortion caused by the load current, |Z

_{o}(s)| should be as low as possible. However, |Z

_{o}(s)| is getting enlarged before the resonant frequency, 269.8 Hz. This means that high order harmonic components affect the shape of the output voltage. For the high frequency range over the resonant frequency, the magnitude of the output impedance is reduced. However, such a high frequency region where the high frequency current is already attenuated by the LC filter is not of interest in practice. From Figure 2, the phase margin of G

_{vo}(s) can be evaluated as 0°. This means the output voltage may see an extremely severe oscillation during a transient. The main reason is that the phase changes rapidly around the resonant frequency. In fact, it is well-known that the resonance should be dampened to stabilize a voltage control loop. One popular approach is employing a voltage controller cascaded with an inner inductor current controller. However, the inner current loop may introduce an iteration delay, and it could deteriorate the transient performance.

## 3. Proposed Single-Loop Voltage Control Strategy with the Active Damping Control

_{L}is measured, and the active damping gain K

_{d}is multiplied. After that, it is subtracted from the output of the feedback controller v

_{fb}. Hence, the following equation can be established:

_{o}and v

_{fb}is rewritten as seen in Equation (5).

_{d}and C as in Equation (5). Since C is a fixed value in an inverter system, the damping coefficient can be adjusted by selecting the value of K

_{d}. Figure 4 illustrates the frequency responses of Equation (5) with different K

_{d}. Equation (5) and Figure 4 clearly indicates that the damping factor of the transfer function can be affected by K

_{d}. With K

_{d}= 0, no damping coefficient is realized, so that high resonant peak persists. As the value of K

_{d}increases, the resonant peak is attenuated, and no resonance can be achieved even in theory. It should be mentioned that the phase margin is secured with increasing K

_{d}. This means the voltage loop with the active damping control can be more stable compared to that without K

_{d}.

_{p}, K

_{r}, ω

_{c}, ω

_{o}, and T

_{s}are the proportional gain, the resonant gain, the damping coefficient, the resonant frequency, and the sampling period, respectively, and

_{rp}, N, and α represent the repetitive control gain, the entire number of samples in one electrical cycle, and the number of samples for the angle advance to improve the stability of the control loop [17]. Here, the zero-phase delay low pass filter q(z) in Equation (7) can be written as seen below to prevent a high frequency noise [18].

_{ff}is directly added, and it compensates for the admittance effect in the entire control loop [19]. With this configuration, the voltage error e

_{o}between the voltage reference ${v}_{o}^{*}$ and v

_{o}is characterized as follows:

_{d}. Consequently, the resonant peak caused by the LC filter can be easily mitigated by increasing K

_{d}as described in Figure 4, and the control loop can be easily stabilized. One simple approach to determine K

_{d}is using the fact that there is no resonant peak in the frequency response in a generalized second order transfer function when the damping ratio of the transfer function is larger than the unity value. By applying this rule to Equation (5), the criterion to select K

_{d}is derived as follows:

_{d}without the resonant peak is calculated as 9.83. However, this does not guarantee a stable performance under the proposed control structure, because the entire stability highly relies on the co-operation with the other controllers, especially the repetitive controller.

_{d}, it may be reasonable to establish the assumption that the active damping component, K

_{d}$\times $ i

_{L}, should be less than the maximum synthesizable voltage physically with the given power stage. From this, the constraint as below can be written:

_{max}and I

_{Lpk}are the maximum achievable modulation index and the peak value of the inductor current under full load condition. Since the maximum output power of the power stage in this paper is 1.5 kW, I

_{Lpk}is simply calculated as 9.64 A. By choosing V

_{dc}and M

_{max}as 400 V and 0.9, the maximum allowable K

_{d}is obtained as 37.34. In sum, the range of K

_{d}considering the experimental parameters is represented as:

_{p}= 10, K

_{r}= 25, ω

_{c}= 62.8 rad/s, ω

_{o}= 377 rad/s, T

_{s}= 50 μs, K

_{d}= 35, K

_{rp}= 2.5, N = 333, and α = 2. In the figure, Z

_{org}(z), Z

_{opr}(z), and Z

_{orep}(z) represent the output impedances without any controller, with the active damping and the PR controller, and with all controllers including the repetitive control. Usually, the output impedances of a UPS should be as low as possible in both the fundamental and the harmonic frequencies to minimize the voltage distortion in wide frequency ranges. For Z

_{org}(z), the magnitude at the fundamental frequency 60 Hz is acceptable, but the high resonant peak is observed near the 5th order harmonic range. Although no severe resonant peak exists with Z

_{opr}(z), there is no magnitude attenuation at each harmonic frequency. The best performance is expected with Z

_{orp}(z) in Figure 6, because multiple notches which have extremely low magnitudes exist at the fundamental and the harmonic frequencies.

_{vo}(z) is the z-domain expression of the open-loop control-to-output voltage transfer function G

_{vo}(s) as expressed in Equation (5).

_{d}are examined. Note that the repetitive control gain K

_{rp}is fixed to 2.5. As shown in the figure, the trajectory stays inside the unit circle with K

_{d}= 35, and it is expected that the control loop is apparently stable. However, when K

_{d}is zero, the trajectory violates the border of the unit circle, so that a stable operation cannot be guaranteed.

_{rp}and K

_{d}are changed in the given range. Here, K

_{rp}is evaluated from 1 to 4, and K

_{d}is adjusted from 0 to 40. For convenience of the view, the magnitude of |H(z)| is only expressed up to 1.0. If the maximum |H(z)| is equal or higher than 1.0, it means that the root trajectory passes the border of the unit circle. Hence, it is a necessary condition that the maximum value of |H(z)| should be less than the unity for stable operation.

_{d}less than roughly 20 can easily become unstable regardless of K

_{rp}. Apparently, higher K

_{d}brings lower the maximum value of |H(z)| with a given K

_{rp}. It is also interesting that higher K

_{rp}induces lower amplitude of |H(z)|. However, this does not mean achieving better stability with higher K

_{rp}, because the stability margin of the entire control may be reduced. As can be seen in Figure 8, a stable operation of the entire control loop is expected from the selected control gains, K

_{d}= 35 and K

_{rp}= 2.5. In sum, the step-by-step flowchart for selecting the active damping and the repetitive control gains are shown in Figure 9.

## 4. Simulation Results

_{s}, have been considered.

_{d}. The conduction angle of the thyristor bridge is selected as 30° in the simulation. In order to clearly show the load current, the scaling factor 20 was multiplied to the original value. In Figure 12a, K

_{d}was selected as 14. In the figure, the output voltage is eventually diverged. Obviously, the voltage controller cannot be stabilized. However, the output voltage in Figure 12b where K

_{d}was chosen as 35 is very well regulated as we have analyzed before.

_{d}= 35 is much better than another one. In order to analyze these results, the root trajectories of H(z) with different K

_{d}are compared in Figure 15. In the figure, the trace of root moving with K

_{d}= 14 contacts the border of the unit circle. This means the entire control loop cannot be stabilized under this condition, because the small gain theorem expressed in Equation (13) is violated.

## 5. Experimental Results

_{dc}is adjusted to be 400 V.

_{o}. In Figure 17, the oscillatory voltage response is generated by the low control bandwidth, low damping, and high LC resonant peak. Both the output voltage and the current contain severe harmonic components.

_{d}= 35 is shown in Figure 18. Now, the control gains and parameters are recovered as analyzed in the previous sections. In contrast with the case in Figure 17, the output voltage v

_{o}is very well regulated and stable, and the voltage error e

_{o}is also well restrained in Figure 18. There is no severe voltage dip on v

_{o}even in the rapid current transient because of the stable operation of the active damping and the repetitive control.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 12.**Simulation results using the proposed method with different K

_{d}. (

**a**) K

_{d}= 14; (

**b**) K

_{d}= 35.

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**MDPI and ACS Style**

Cho, Y.; Byen, B.-J.; Lee, H.-S.; Cho, K.-Y. A Single-Loop Repetitive Voltage Controller with an Active Damping Control Technique. *Energies* **2017**, *10*, 673.
https://doi.org/10.3390/en10050673

**AMA Style**

Cho Y, Byen B-J, Lee H-S, Cho K-Y. A Single-Loop Repetitive Voltage Controller with an Active Damping Control Technique. *Energies*. 2017; 10(5):673.
https://doi.org/10.3390/en10050673

**Chicago/Turabian Style**

Cho, Younghoon, Byeng-Joo Byen, Han-Sol Lee, and Kwan-Yuhl Cho. 2017. "A Single-Loop Repetitive Voltage Controller with an Active Damping Control Technique" *Energies* 10, no. 5: 673.
https://doi.org/10.3390/en10050673