Minimum-Output-Current-Ripple Control of Current-Fed Three-Level Phase-Shift Full-Bridge Converter

Abstract

Electrified ports using medium-voltage DC (MVDC) renewable energy microgrids require current-fed dc/dc converters in application scenarios such as battery or ultracapacitor charging units and hydrogen production systems. This paper designs a three-level phase-shift full-bridge (TL-PSFB) converter that interfaces with the MVDC microgrid. Its operation in the current source mode requires a wide output voltage range and small output current ripple. Firstly, the dual-output TL-PSFB topology is introduced, and the principle of phase-shift pulse width modulation (PS-PWM) is presented. Secondly, the principle of the traditional constant-conduction-duty-cycle (CCDC) strategy is analyzed. Then, a minimum-output-current-ripple (MOCR) strategy is proposed by analyzing the relationship between output current ripple, conducting-duty cycle, and phase-shift duty cycle, and a constant current control combined with the MOCR strategy is designed. The output current ripple of the MOCR strategy is smaller than that of the CCDC strategy in a full range of operating conditions. Under the same output current ripple design index, the value and loss of the filter inductor can be reduced with the MOCR strategy. In addition, the MOCR strategy can widen the output voltage regulation range and increase the bus voltage utilization without causing significant changes to the total harmonic distortion (THD) of primary voltage. Finally, experimental results verify the correctness of the theoretical analysis.

1. Introduction

Inland and offshore shipping is facing an energy transition due to the economic and environmental concerns of fossil fuels. Currently, there are some new energy vessels [1] and electrified port projects applied [2]. Compared with the port microgrid directly supplied by the power grid, the addition of renewable energy can significantly reduce the utilization rate of fossil fuel. Port microgrids using renewable energy generation have immediate environmental benefits as well as long-term economic benefits. Due to the large randomness of power supply and load of renewable energy microgrid, the AC microgrid is prone to frequency fluctuation and grid oscillation problems. As a new form of network formation, a medium-voltage DC (MVDC) microgrid can avoid the above problems and has the advantages of high reliability of power supply, fast response, and scalability. The typical structure of an MVDC renewable energy microgrid is shown in Figure 1.

The wind power generation, photovoltaic power generation, and hydroelectric power generation on the source side of the renewable energy microgrid, and the charging device on the load side all have strong intermittensity. The connection of the source side to the public network can solve the short-term high power demand of the load, while the hydrogen production system on the load side can effectively solve the local consumption of renewable energy generation and bring new energy types. Ships with different working scenarios can choose energy sources such as hydrogen fuel, battery, and ultracapacitor storage.

MVDC input dc/dc converters for load-side charging devices or hydrogen production power sources are important components of medium-voltage DC renewable energy microgrids. The common topologies include the non-isolated Buck converter and Boost converter, isolated series resonant converter, phase-shift full-bridge converter, and bidirectional full bridge converter, etc. In MVDC applications, series or multi-level connections are usually used on the basis of the above topologies to increase output voltage level. Switching device withstand voltage is half of the bus voltage in the three-level phase-shift full-bridge (TL-PSFB) topology which can meet the requirements of MVDC input [3,4]. The TL-PSFB topology adopts an intermediate frequency transformer to achieve isolation and voltage transformation, and can flexibly configure the output side voltage according to the power of the electrolyzer or the voltage of the energy storage device. Magnetic coupling can effectively isolate source and load faults.

Some scholars have performed much research on the TL-PSFB converter controlled in voltage mode. The TL-PSFB converter and the basic topological derivation are described by Prof. Xinbo Ruan in [4]. In [5], a three-phase TL-PSFB converter topology was designed with voltage mode control. Literature [6] proposes a first-order sliding mode controller, which can eliminate the output voltage error by adding an integrator and achieve a better control effect. The principle analysis, modeling, and control strategy of the voltage mode controlled TL-PSFB converter are studied in the literature mentioned above.

Higher power density and efficiency has always been the goal of the power electronic converter. Increasing the switching frequency of the power device is the most direct method, however, it will cause a sharp increase in the loss caused by hard switching of switching devices. Therefore, many scholars have been studying soft switching strategies, and the main methods can be divided into two categories: proposing a new modulation strategy and improving the topology structure. The new modulation strategies reported in the literature mainly include an asymmetric duty cycle modulation strategy [7], double phase-shift modulation strategy [8], and triple phase-shift modulation strategy [9], etc. These strategies can extend the soft-switching range of the converter to a certain extent without changing the topology or adding additional energy storage elements, but they cannot achieve soft-switching in the full operating range. In [10,11,12,13,14,15], LLC resonant TL-PSFB converter was studied to achieve a wider range of soft switching with the help of resonant element energy storage. Dr. Bor-Ren Lin designed a new hybrid topology, including a three-level half-bridge converter and unregulated voltage half-bridge converter structure, which has the advantages of small circulation, wide soft-switching range, and small output current ripple [16,17,18]. In addition, soft switching of the primary-side switching device can be achieved by adding a clamp diode [19] or fly-across capacitor to the primary side [20,21,22]. Prof. Yan Li’s focus is on the intermediate AC side. She proposed a hybrid TL-PSFB converter to accommodate a wide range of output voltages by adding a clamp inductor on the AC side, and the converter has good soft switching performance [23]. Soft switching of the secondary diode has also been noticed in the literature [24,25,26]. In addition, some scholars have done some improving research on the topology in some other aspects. Literature [27,28] proposed an improved modulation strategy to equalize the thermal stress of power devices or supporting capacitors. A modulation strategy to reduce the common-mode voltage was proposed in the literature [29]. In order to solve the reverse recovery problem of the secondary rectifying diode, the active clamping strategy was proposed in [30,31].

Only literature [19] (hundred kW class) and [5] (MW class) have publicly reported the application of TL-PSFB converter in large-capacity DC conversion applications. Large-capacity TL-PSFB converters have high losses and complex heat dissipation design. Traditional topology with improved modulation or control strategy do not fundamentally solve the problem of narrow soft-switching range and are prone to soft-switching failure. Resonant topology or other improved topology increases power loop components, has complex topology and low reliability, and the added devices also bring losses. Therefore, the soft switching strategy is not the focus of the design of a large capacity TL-PSFB converter.

Previous research has focused on the voltage-mode controlled TL-PSFB converter, while hydrogen power or energy storage converters require a current source, the major difference between them is that current mode control has no static operating point. In addition, the output current ripple of the converter directly affects the efficiency of the electrolyzer or energy storage system, which requires the converter to have a smaller output current ripple.

This paper is organized as follows. In Section 2, a dual-output TL-PSFB topology is proposed. The principle of phase-shift pulse width modulation (PS-PWM) is introduced, and the design method of the traditional CCDC strategy is analyzed. In Section 3, the relationship between output current ripple and conducting-duty cycle and phase-shift duty cycle is analytically calculated, and a constant current control with the MOCR strategy is proposed. Section 4 compares and analyzes the performance of the two strategies in terms of output current ripple, filter inductance parameters, DC voltage utilization, and the total harmonic distortion (THD) of primary voltage. In Section 5, the proposed modulation method’s effectiveness is verified by experiments. Finally, Section 6 summarizes the main work of this paper.

2. Principle of Dual-Output TL-PSFB Converter

2.1. Topology and Principle of PS-PWM

In order to improve the power density of the converter, a dual-output TL-PSFB converter topology is designed in this paper, as shown in Figure 2. The dual-output TL-PSFB converter mainly includes support capacitors, diode-clamped three-level H-bridge, three-winding medium-frequency transformer, uncontrolled rectifier bridge, output filter inductor, and input and output diodes. The diode-clamped three-level H-bridge inverter unit converts the input DC voltage into a bipolar five-level square wave with adjustable pulse width, which is divided into two output channels after the step-down by the three-winding medium-frequency transformer. Then it is connected to two independent sets of uncontrolled rectifier bridges. Only inductive filtering is used at the rectifier output, reducing unnecessary passive components compared to previous LC or LCL filters.

PS-PWM is adopted in the dual-output TL-PSFB converter, and its principle is shown in Figure 3. In the figure, d_θ is the conducting-duty cycle, d_α is the phase-shift duty cycle, and Ts is the switching period. The diode-clamped three-level H-bridge in Figure 2 uses the same d_θ for the odd-numbered switching devices and a complementary conducting-duty cycle 1 − d_θ for the even-numbered switching devices. The driving pulse of the right bridge leg lags behind that of the left bridge leg, and the phase difference of the driving pulse of the left and right bridge legs’ switches is changed by controlling the carrier delay time of the right bridge leg d_αT_s. PS-PWM consists of two degrees of freedom: d_θ and d_α, both in the range of [0, 1/2].

By analyzing the corresponding relationship between d_θ and d_α, the output voltage of the diode-clamped three-level H-bridge can be obtained, including the three cases as shown in Figure 4. In order to distinguish different conditions of output voltages more clearly, the clamping duty cycle d_γ is introduced, where d_γ = 1/2 − d_θ.

2.2. CCDC Strategy

The design principle of the CCDC strategy is to minimize THD of the primary voltage u_ab. The root mean square (RMS) of the nth harmonic component of u_ab is noted as U_nrms (n = 1, 2, 3…). THD of u_ab is defined as,

$$\mathrm{THD}=\frac{1}{U_{1\mathrm{rms}}}{({\displaystyle \sum _{n=2}^{\infty }{U}_{n\mathrm{rms}}^2})}^{1/2}$$

(1)

where the relationship between the rms value U_rms of u_ab and the rms value of each harmonic component is,

$$U_{\mathrm{rms}}^2={\displaystyle \sum _{n=1}^{\infty }{U}_{n\mathrm{rms}}^2}$$

(2)

as known from the calculation,

$$\mathrm{THD}=\frac{1}{U_{1\mathrm{rms}}}{(U_{\mathrm{rms}}^2-U_{1\mathrm{rms}}^2)}^{1/2}$$

(3)

According to the waveform of u_ab in Figure 4, the Fourier series expression of u_ab can be obtained by analytical calculation shown in (4).

$$\begin{array}{l}{u}_{ab}(t)={\displaystyle \sum _{n=1}^{\infty }(b_n\sin nt)},\\ b_n=\frac{8U_{\mathrm{d}}}{n{T}_{\mathrm{s}}}\sin n{d}_{\mathsf{\theta }}\pi \sin n{d}_{\mathsf{\alpha }}\pi (n=1,3,5,7\cdots )\end{array}$$

(4)

In addition, the relationship between the RMS value of u_ab, d_θ and d_α is shown in (5).

$$U_{\mathrm{rms}}=\{\begin{array}{c}{U}_d\sqrt{d_{\mathsf{\alpha }}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0<d_{\mathsf{\alpha }}\le d_{\mathsf{\gamma }}\\ U_d\sqrt{2d_{\mathsf{\alpha }}+d_{\mathsf{\theta }}-\frac{1}{2}}\hspace{1em}\hspace{1em}\hspace{1em}{d}_{\mathsf{\gamma }}<d_{\mathsf{\alpha }}\le d_{\mathsf{\theta }}\\ U_d\sqrt{d_{\mathsf{\alpha }}+2d_{\mathsf{\theta }}-\frac{1}{2}}\hspace{1em}\hspace{1em}\hspace{1em}{d}_{\mathsf{\theta }}<d_{\mathsf{\alpha }}\le \frac{1}{2}\end{array}$$

(5)

Combining (4) and (5) with (2) and (3), THD of u_ab can be obtained as shown in (6).

$$\mathrm{THD}=\{\begin{array}{c}\sqrt{\frac{2{\pi }^2{d}_{\mathsf{\alpha }}}{16{\sin }^2\pi d_{\mathsf{\theta }}{\sin }^2\pi d_{\mathsf{\alpha }}}-1}\hspace{1em}\hspace{1em}0<d_{\mathsf{\alpha }}\le d_{\mathsf{\gamma }}\\ \sqrt{\frac{\pi (4\pi d_{\mathsf{\alpha }}-1/2+2\pi d_{\mathsf{\theta }})}{16{\sin }^2\pi d_{\mathsf{\theta }}{\sin }^2\pi d_{\mathsf{\alpha }}}-1}\hspace{1em}\hspace{1em}{d}_{\mathsf{\gamma }}<d_{\mathsf{\alpha }}\le d_{\mathsf{\theta }}\\ \sqrt{\frac{\pi (2\pi d_{\mathsf{\alpha }}-1/2+4\pi d_{\mathsf{\theta }})}{16{\sin }^2\pi d_{\mathsf{\theta }}{\sin }^2\pi d_{\mathsf{\alpha }}}-1}\hspace{1em}\hspace{1em}{d}_{\mathsf{\theta }}<d_{\mathsf{\alpha }}\le \frac{1}{2}\end{array}$$

(6)

Partial differential solution of (6) is performed; that is, let ∂THD/∂d_α = 0 and ∂THD/∂d_θ = 0, it can be obtained,

$$\{\begin{array}{l}4\pi d_{\mathsf{\alpha }}-\frac{1}{2}+2\pi d_{\mathsf{\theta }}=2\tan \pi d_{\mathsf{\alpha }}\hspace{1em}\hspace{1em}{d}_{\mathsf{\gamma }}<d_{\mathsf{\alpha }}\le d_{\mathsf{\theta }}\\ 2\pi d_{\mathsf{\alpha }}-\frac{1}{2}+4\pi d_{\mathsf{\theta }}=\tan \pi d_{\mathsf{\alpha }}\hspace{1em}\hspace{1em}{d}_{\mathsf{\theta }}<d_{\mathsf{\alpha }}\le \frac{1}{2}\end{array}$$

(7)

$$\{\begin{array}{l}4\pi d_{\mathsf{\alpha }}-\frac{1}{2}+2\pi d_{\mathsf{\theta }}=\tan \pi d_{\mathsf{\theta }}\hspace{1em}\hspace{1em}{d}_{\mathsf{\gamma }}<d_{\mathsf{\alpha }}\le d_{\mathsf{\theta }}\\ 2\pi d_{\mathsf{\alpha }}-\frac{1}{2}+4\pi d_{\mathsf{\theta }}=2\tan \pi d_{\mathsf{\theta }}\hspace{1em}\hspace{1em}{d}_{\mathsf{\theta }}<d_{\mathsf{\alpha }}\le \frac{1}{2}\end{array}$$

(8)

Two sets of solutions can be obtained by combining (7) and (8) as follows: the first extremum point is: d_θ1 = 0.42, d_α1 = 0.35; the second one is: d_θ2 = 0.35 and d_α2 = 0.42, which are the two extreme points shown in Figure 5.

The primary voltage is input to the uncontrolled rectifier bridge via the medium-frequency transformer. Similarly, the Fourier decomposition of the rectifier bridge output voltage u_r is performed according to the relative relationship between d_θ and d_α.

$$u_{\mathrm{r}}=\{\begin{array}{l}\frac{2U_{\mathrm{d}}{d}_{\mathsf{\alpha }}}{N_{\mathrm{T}}}+{\displaystyle \sum _{n=1}^{\infty }\frac{4U_{\mathrm{d}}}{n{T}_{\mathrm{s}}{N}_{\mathrm{T}}}\cos 2\pi n{d}_{\mathsf{\theta }}\sin 2\pi n{d}_{\mathsf{\alpha }}\cos n\omega t}\hspace{1em}\hspace{1em}0<d_{\mathsf{\alpha }}\le d_{\mathsf{\gamma }}\\ \frac{2U_{\mathrm{d}}{d}_{\mathsf{\alpha }}}{N_{\mathrm{T}}}+{\displaystyle \sum _{n=1}^{\infty }\frac{4U_{\mathrm{d}}}{n{T}_{\mathrm{s}}{N}_{\mathrm{T}}}\cos 2\pi n{d}_{\mathsf{\theta }}\sin 2\pi n{d}_{\mathsf{\alpha }}\cos n\omega t}\hspace{1em}\hspace{1em}0<d_{\mathsf{\alpha }}\le d_{\mathsf{\gamma }}\\ \frac{2U_{\mathrm{d}}{d}_{\mathsf{\theta }}}{N_{\mathrm{T}}}+{\displaystyle \sum _{n=1}^{\infty }\frac{4U_{\mathrm{d}}}{n{T}_{\mathrm{s}}{N}_{\mathrm{T}}}\sin 2\pi n{d}_{\mathsf{\theta }}\cos 2\pi n{d}_{\mathsf{\alpha }}\cos n\omega t}\hspace{1em}\hspace{1em}{d}_{\mathsf{\theta }}<d_{\mathsf{\alpha }}\le \frac{1}{2}\end{array}$$

(9)

According to (9), the maximum DC component of the output voltage of the rectifier bridge is determined by d_θ, and the output voltage can be regulated within the range of the maximum DC component by adjusting d_α under the condition that d_θ remains unchanged.

The d_θ of the CCDC strategy is fixed at one of the two minimum points in Figure 5, and the upper limit of the DC component of the output voltage of the rectifier bridge is determined by d_θ according to (9). In order to improve the utilization of the DC voltage at the primary side and obtain a wider regulation range, the CCDC strategy is usually set d_θ = 0.42 and d_α = 0.35.

3. MOCR Strategy

In this section, the output current ripple of the converter is analytically calculated, and the variation trend of the output current ripple with d_θ and d_α is analyzed. Then, a MOCR strategy is proposed, and a constant current control with the MOCR strategy is designed.

3.1. Analysis of Output Voltage and Current Ripple of the Rectifier Bridge

According to (9), the output voltage of the rectifier bridge is a DC component superimposed several times of AC harmonic components, so the DC component of the output voltage can be ignored when analyzing the harmonics of the filter inductor, and the load side can be equivalent to a series of resistor and capacitor. Different loads (such as ultracapacitors, batteries, electrolyzers, etc.) affect the RC parameters. The equivalent circuit at the output end of the rectifier bridge is shown in Figure 6.

Transform the AC harmonic component of the output voltage of the rectifier bridge obtained by (9) into a complex domain expression,

$${\stackrel{·}{U}}_{\mathrm{r}n}=\{\begin{array}{cc}{\displaystyle \sum _{n=1}^{\infty }\frac{2\sqrt{2}{U}_{\mathrm{d}}}{n{T}_{\mathrm{s}}{N}_{\mathrm{T}}}\cos 2\pi n{d}_{\mathsf{\theta }}\sin 2\pi n{d}_{\mathsf{\alpha }}\angle 0}& 0<d_{\mathsf{\alpha }}\le d_{\mathsf{\gamma }}\\ {\displaystyle \sum _{n=1}^{\infty }\frac{2\sqrt{2}{U}_{\mathrm{d}}}{n{T}_{\mathrm{s}}{N}_{\mathrm{T}}}\cos 2\pi n{d}_{\mathsf{\theta }}\sin 2\pi n{d}_{\mathsf{\alpha }}\angle 0}& d_{\mathsf{\gamma }}<d_{\mathsf{\alpha }}\le d_{\mathsf{\theta }}\\ {\displaystyle \sum _{n=1}^{\infty }\frac{2\sqrt{2}{U}_{\mathrm{d}}}{n{T}_{\mathrm{s}}{N}_{\mathrm{T}}}\sin 2\pi n{d}_{\mathsf{\theta }}\cos 2\pi n{d}_{\mathsf{\alpha }}\angle 0}& d_{\mathsf{\theta }}<d_{\mathsf{\alpha }}\le \frac{1}{2}\end{array}$$

(10)

The harmonic component of the output current is,

$$\begin{array}{cc}\hfill \stackrel{·}{I_{\mathrm{o}n}}& =\frac{{\stackrel{·}{U}}_{\mathrm{r}n}}{R_{\mathrm{C}} + R_{\mathrm{Lf}} + jn\omega L_{\mathrm{f}} + \frac{1}{jn\omega C}}\hfill \\ & =\frac{{\stackrel{·}{U}}_{\mathrm{r}n}}{R_{\mathrm{C}} + R_{\mathrm{Lf}} + j\frac{n^2{\omega }^2{L}_{\mathrm{f}}C - 1}{n\omega C}}\hfill \end{array}$$

(11)

Substitute (10) into (11) to obtain,

$${\stackrel{·}{I}}_{\mathrm{o}n}=\{\begin{array}{l}{\displaystyle \sum _{n=1}^{\infty }\frac{\frac{2\sqrt{2}{U}_{\mathrm{d}}}{n{T}_{\mathrm{s}}{N}_{\mathrm{T}}}\cos 2\pi n{d}_{\mathsf{\theta }}\sin 2\pi n{d}_{\mathsf{\alpha }}}{\mathrm{Re}(\mathrm{Z})}\angle -\mathsf{\varphi }\hspace{1em}\hspace{1em}0<d_{\mathsf{\alpha }}\le d_{\mathsf{\gamma }}}\\ {\displaystyle \sum _{n=1}^{\infty }\frac{\frac{2\sqrt{2}{U}_{\mathrm{d}}}{n{T}_{\mathrm{s}}{N}_{\mathrm{T}}}\cos 2\pi n{d}_{\mathsf{\theta }}\sin 2\pi n{d}_{\mathsf{\alpha }}}{\mathrm{Re}(\mathrm{Z})}\angle -\mathsf{\varphi }\hspace{1em}\hspace{1em}{d}_{\mathsf{\gamma }}<d_{\mathsf{\alpha }}\le d_{\mathsf{\theta }}}\\ {\displaystyle \sum _{n=1}^{\infty }\frac{\frac{2\sqrt{2}{U}_{\mathrm{d}}}{n{T}_{\mathrm{s}}{N}_{\mathrm{T}}}\sin 2\pi n{d}_{\mathsf{\theta }}\cos 2\pi n{d}_{\mathsf{\alpha }}}{\mathrm{Re}(\mathrm{Z})}\angle -\mathsf{\varphi }\hspace{1em}\hspace{1em}{d}_{\mathsf{\theta }}<d_{\mathsf{\alpha }}\le \frac{1}{2}}\end{array}$$

(12)

where,

$$\begin{array}{c}\mathrm{Re}(Z)=\sqrt{{\left(R_{\mathrm{C}}+R_{\mathrm{L}}\right)}^2+{\left(\frac{n^2{\omega }^2{L}_{\mathrm{f}}C - 1}{n\omega C}\right)}^2}\\ \mathsf{\varphi }=\arctan \left[\frac{n^2{\omega }^2{L}_{\mathrm{f}}C - 1}{n\omega C\left(R_{\mathrm{C}} + R_{\mathrm{L}}\right)}\right]\end{array}$$

(13)

Express the output current harmonics into the form in the real number field as

$$i_{\mathrm{o}n}=\{\begin{array}{l}{\displaystyle \sum _{n=1}^{\infty }\frac{\frac{2\sqrt{2}{U}_{\mathrm{d}}}{n{T}_{\mathrm{s}}{N}_{\mathrm{T}}}\cos 2\pi n{d}_{\mathsf{\theta }}\sin 2\pi n{d}_{\mathsf{\alpha }}}{\mathrm{Re}(\mathrm{Z})}\cos (n\omega t-\mathsf{\varphi })}\hspace{1em}\hspace{1em}0<d_{\mathsf{\alpha }}\le d_{\mathsf{\gamma }}\\ {\displaystyle \sum _{n=1}^{\infty }\frac{\frac{2\sqrt{2}{U}_{\mathrm{d}}}{n{T}_{\mathrm{s}}{N}_{\mathrm{T}}}\cos 2\pi n{d}_{\mathsf{\theta }}\sin 2\pi n{d}_{\mathsf{\alpha }}}{\mathrm{Re}(\mathrm{Z})}\cos (n\omega t-\mathsf{\varphi })}\hspace{1em}\hspace{1em}{d}_{\mathsf{\gamma }}<d_{\mathsf{\alpha }}\le d_{\mathsf{\theta }}\\ {\displaystyle \sum _{n=1}^{\infty }\frac{\frac{2\sqrt{2}{U}_{\mathrm{d}}}{n{T}_{\mathrm{s}}{N}_{\mathrm{T}}}\sin 2\pi n{d}_{\mathsf{\theta }}\cos 2\pi n{d}_{\mathsf{\alpha }}}{\mathrm{Re}(\mathrm{Z})}\cos (n\omega t-\mathsf{\varphi })\hspace{1em}\hspace{1em}{d}_{\mathsf{\theta }}<d_{\mathsf{\alpha }}\le \frac{1}{2}}\end{array}$$

(14)

Figure 7 shows the comparison between the calculated waveforms and the simulated waveforms for the output currents at different d_θ and d_α, the trend of the calculated results is generally consistent with that of the simulation results, which proves the correctness of the calculated results. There is a certain error between the calculated and simulated results, which is mainly due to the fact that the calculated results do not take into account non-ideal factors such as duty cycle loss.

The output current ripple is equal to the maximum value of the filter inductor current minus the minimum value in one cycle.

$$△i_{\mathrm{o}}=\max {(i_{\mathrm{o}})}_{T_s}-\min {(i_{\mathrm{o}})}_{T_s}$$

(15)

Combining (14) and (15), the variation of output current ripple under different combinations of d_θ and d_α can be calculated, as shown in Figure 8 and Figure 9.

3.2. Constant Current Control Combined with the MOCR Strategy

As can be seen from Figure 9, as d_θ and d_α increase, there is a changing path that minimizes the output current ripple. That is,

$$\{\begin{array}{cc}{d}_{\mathsf{\alpha }},d_{\mathsf{\theta }}\le \frac{1}{4}& \{\begin{array}{c}{d}_{\mathsf{\alpha }}=\frac{1}{4};d_{\mathsf{\theta }}=0\to \frac{1}{4}:\mathrm{RouteI}\\ d_{\mathsf{\theta }}=\frac{1}{4};d_{\mathsf{\alpha }}=0\to \frac{1}{4}:\mathrm{RouteII}\end{array}\\ \frac{1}{4}<d_{\mathsf{\alpha }},d_{\mathsf{\theta }}\le \frac{1}{2}& d_{\mathsf{\alpha }}=d_{\mathsf{\theta }}=\frac{1}{4}\to \frac{1}{2}:\mathrm{RouteIII}\end{array}$$

(16)

Equation (16) is the mathematical description of the minimum output current ripple control rate. Since d_θ determines the upper limit of the output voltage, transition path Route II→Route III is selected to ensure that the control variable is unique. That is, when d_θ is less than 1/4, maintain d_θ = 1/4 and adjust d_α to control the output voltage. As the output voltage increases, control d_θ is equal to d_α.

Combined with the above analysis, this paper designs a constant current control combined with the MOCR strategy, as shown in Figure 10. The controller collects the output current for feedback, and d_α is calculated by the PI controller, and d_θ is calculated by the MOCR strategy. Since the converter adopts a secondary dual output structure, a two-output maximum current feedback strategy is used to prevent the output from being overloaded during actual operation. In Figure 10, d_δ is limited to zero level, which is set to prevent the same level jump as the bus voltage. In engineering, the value is set according to the dv/dt limit of switching devices and transformers.

4. Comparison of CCDC and MOCR

4.1. Output Current Ripple and Filter Inductor Characterization

Since d_α can only be adjusted within the range less than d_θ in the CCDC strategy, the output ripple of the two control strategies is compared below in the range [0, 0.42). Figure 11 shows the comparison of the normalized results of output current ripple under the same filtering inductance condition. It can be seen from the figure that the maximum ripple current modulated by the MOCR strategy is 28% smaller than that controlled by the traditional CCDC strategy, and the ripple current modulated by the MOCR strategy is all less than the CCDC strategy in the whole range of working conditions.

In Figure 11, d_α corresponding to the maximum current ripple under the two strategies is obtained by the analytical calculation method, thus, the minimum filter inductance value satisfying the maximum ripple index can be calculated.

Table 1. Different output current ripple indexes requiring the value of minimum filter inductance corresponding to two strategies.

∆ io	CCDC: Lf/mH	MOCR: Lf/mH
5% Io	29.39	22.52
10% Io	14.70	11.26
15% Io	9.80	7.51
20% Io	7.35	5.63
25% Io	5.88	4.50

compares the values of minimum filter inductance corresponding to the two modulation methods under different output current ripple indexes. According to the data in the table, under different output current ripple indexes, the MOCR strategy can reduce the value of the minimum output filter inductance by about 23%.

According to the inductor multi-objective optimization design method studied in [32] and the parameters in Table 1, the output filter inductor is designed. Figure 12 shows the comparison of the volume and weight of the minimum inductance required by the two strategies corresponding to different output current ripple indexes. It can be seen from the figure that the volume and weight of the output filter inductor can be optimized to a certain extent by using the MOCR strategy studied in this paper under different requirements of ripple indexes, which will improve the power density of the converter.

The loss of filter inductor mainly includes core loss and winding loss. The winding losses can be decomposed into the sum of ac and dc losses,

$$L_{\mathrm{Cu}}=I_{\mathrm{o}}^2{R}_{\mathrm{dc}}+{\displaystyle \sum _{n=1}^{\infty }{i}_{\mathrm{o}n}^2}{R}_{\mathrm{ac}n}$$

(17)

where R_dc is the dc resistance of the filter inductor, and this parameter can be calculated from the average turn length and resistivity of the inductor winding. R_dc-n is the winding ac resistance corresponding to the ripple current of different frequencies. The relation between ac resistance and ac resistance is shown in (18). F_R is the ac resistivity, which can be obtained by the Dowell model [33].

$$R_{\mathrm{ac}n}=F_{\mathrm{R}}{R}_{\mathrm{dc}}$$

(18)

Generally, core loss is calculated by loss separation theory and mainly includes hysteresis loss, eddy current loss, and other losses. According to the calculation formula of magnetic core loss derived in [32], combined with (17), the total loss of inductance can be calculated. Figure 13 shows a comparison of the output filter inductance loss with different parameters. It can be seen from the figure that, under the same output current ripple index requirements, the MOCR strategy can reduce the loss of output filtering inductor by up to 29.31%, which improves the efficiency of the converter to a certain extent.

4.2. Utilization of DC Voltage and THD of Primary Voltage Characterization

The utilization of DC voltage is the ratio of the fundamental wave amplitude of u_ab to U_d. Combining with (4). The variation of DC voltage utilization with d_θ and d_α can be obtained by combining (4).

$$\frac{U_{\mathrm{ab}}}{U_{\mathrm{d}}}=\frac{8}{T_{\mathrm{s}}}\sin d_{\mathsf{\theta }}\pi \sin d_{\mathsf{\alpha }}\pi$$

(19)

The theoretical regulation range of both d_θ and d_α is [0, 0.5). A fixed d_θ of 0.42 for the CCDC strategy leads to lower utilization of the primary-side DC voltage. This leads to a larger transformer ratio required to achieve the designed output voltage, which will increase the primary side current and, in turn, leads to an increased transformer and primary side switching device losses. Besides, the circuit parameters of the converter are designed according to the rated working condition in general. However, d_α in the light load working condition is far away from 0.35, which will result in the increase of the THD of u_ab, as shown in Figure 14, which violates the initial design principle.

Figure 14 compares the THD of u_ab with the CCDC and MOCR strategies, respectively, over the full range of operating conditions, and it can be seen from the figure that the difference in THD performance between the two strategies under the corresponding operating conditions is not significant. However, it should be noted that the CCDC strategy is optimized for the THD of u_ab, but the optimization range is limited to the operating conditions around d_θ of 0.42 and d_α of 0.35. The CCDC strategy does not bring optimal performance for a wide range of output requirements.

5. Experimental Verification

In this paper, a 1:1 test prototype is designed as shown in Figure 15, and the parameters of the main circuit are shown in

Table 2. The parameters of the main circuit of the test prototype.

Parameters	Value	Parameters	Value
Ud/kV	4	NT	4:3:3
Uo/kV	0–2	Lr/μH	40
Io/A	180	Lf/mH	8
f/kHz	1	C1, C2/mF	4

. The output current ripple index is designed to be less than 20% I_o. The prototype experiment uses the ultracapacitor as the load and adopts the constant current charging mode.

Figure 16 shows the experimental waveform of the output current of the ultracapacitor charged from 0 V to the rated voltage at the constant current 180 A. The maximum current ripple is 34.41 A with the CCDC strategy and 25.37 A with the MOCR strategy, which is 26.27% less. The output current ripple in the whole charging process is smaller than that of the CCDC strategy, and the trend of ripple current during charging is consistent with the theoretical analysis shown in Figure 11, which verifies the correctness of the above theoretical analysis.

In the initial stage, the converter prototype is designed according to the theoretical analysis results of the CCDC strategy. In order to meet the output current ripple index, the output filter inductor is taken to be larger, and its weight and size are also larger. The physical picture is shown in Figure 17a, with a weight of 142 kg and a three-dimensional size of 211 mm (height) × 329 mm (width) × 298 mm (depth).

According to the theoretical analysis results and experimental results of the MOCR strategy studied in this paper, the output filter inductance value of the prototype can be reduced to 6 mH; the three-dimensional figure is shown in Figure 17b. The weight is 120 kg, and the three-dimensional size is 203 mm (height) × 309 mm (width) × 277 mm (depth). Compared with the filter inductor before optimization, weight, volume and loss are respectively reduced by 13.98%, 14.94%, and 11.54%.

6. Conclusions

For the TL-PSFB converter operating in a current-fed mode in DC microgrid, this paper analyzes and points out the problems such as poor current ripple performance and non-compliance with the static operating point design principle of the traditional CCDC strategy. A MOCR strategy is proposed and the performance of the two strategies is quantitatively compared in four aspects. Both experimental and theoretical analyses prove that the proposed MOCR strategy has better performance than the traditional CCDC strategy. In fact, the MOCR strategy adds a degree of control freedom to the CCDC strategy. Adopting the idea of software-based hardware functions avoids the use of high-order filters when a small current ripple index is required. To a certain extent, the power density and efficiency of the converter are improved and the engineering cost is reduced.