## 1 Introduction  
With the development of renewable energy generation and energy storage technologies, isolated AC-DC converters become more and more attractive. Isolated AC-DC converters generally include two-stage AC-DC-DC structures [1; 2; 3; 4; 5; 6] and single-stage ones. The two-stage structures possesses good performance, however, because of the large DC link capacitor, the power density and lifetime are difficult to increase [7; 8; 9], which is the main disadvantage of this structure. Over the past few years, single-phase isolated quasi-single-stage (QIS) AC-DC converters without DC link electrolytic capacitors have been put forward [10; 11; 12; 13]. Their front-end rectifier operates at the grid frequency, for which the switching loss can be ignored [14]. Its back dual-active-bridge (DAB) converter can adopt an optimized phase shift modulation. In [15], a triple-phase-shift (TPS) modulation strategy is proposed to extend the range of the soft-switching operation. A closed-form analytical solution for the modulation scheme of full-range zero-voltage-switching (ZVS) and near-minimum circulating currents is presented in [16], which improves the efficiency and lifetime of the converter. Thanks to the efforts of researchers, this type of converter provides an effective solution to enhance the power density and efficiency and lengthen the lifetime of grid-connected power-conversion systems. Since this type of converter needs to be generally connected to power grid, its quality of the grid current waveform ought to be ensured, otherwise the corresponding grid-connected standard cannot be reached. Based on its working principle, the voltage and current of the back-end DAB converter are processed in sinusoidal form, which illustrates that port voltage and average transfer current in a switching cycle are different from those in the contiguous cycle. Instantaneous transformer currents in any two contiguous switching cycles should be uncoupled and independently controlled so as to achieve the goal of average transfer current without steady-state error. In [17], an improved triple-phase-shift (ITPS) modulation strategy is put forward, in order to completely eliminate circulating currents and coupling of transformer currents in the contiguous switching cycles, thus realizing the accurate control of AC side transfer currents. As a result, AC side currents of this converter are sine waves. Under ideal conditions, namely, if the grid voltage waveform is sinusoidal without any harmonics, the grid current waveform will also be sinusoidal because the ITPS ensures a sinusoidal waveform of the AC side average current of the converter. However, an LC filter is set up between the converter and grid, which can transform converter's AC side discrete currents to be continuous sine waveforms of grid currents. The LC filter introduces an impedance link between the grid and the converter, and as a result, the grid current is determined not only by AC side currents together but also by grid voltages. Since low-order harmonics are generally included in actual grid voltages, grid currents are difficult to ensure good sine waves, even though AC side currents of this converter are perfect sine waves. For the grid-connected type converters, the general solution to eliminate the influence of grid harmonics on grid currents is to construct the closed-loop control configuration of grid currents. In [18], a ripple controller with improved voltage feed-forward scheme and quasi-proportional-resonant compensator was designed to realize good voltage ripple suppression for a Q1S AC-DC converter with an audio susceptibility model. A Q1S current-fed resonant AC-DC converter is proposed in [19]. Its closed-loop control adopts a dual-loop structure, whose inner loop is used to control the grid current and the outer loop regulates the output voltage. A PI controller is used in both inner loop and outer loop.