Research on Automotive Bidirectional CLLC Resonant Converters Based on High-Order Sliding Mode Control

Abstract

A CLLC resonant converter’s gain is easily influenced by the operating frequency, and when the operating frequency is adjusted over a wide range, the efficiency of the converter is greatly reduced. Traditional closed-loop control strategies also have disadvantages such as slow dynamic response and vulnerability to load. In this paper, a high-order sliding mode control (SMC) design method is proposed based on the current problems and the characteristics of automotive CLLC resonant converters. A sliding mode surface based on the output voltage characteristics of the CLLC converter includes higher-order differential terms for voltage and current and an error integral term for the output voltage, which reduces the operating frequency range of the converter and improves its dynamic responsiveness, thus increasing its efficiency. In order to verify the accuracy of the algorithm, a simulation model is built in MATLAB to verify the stability of the controller by varying the input voltage and the magnitude of the load and to verify the dynamics by abruptly varying parameters such as load and voltage. Comparing high-order SMC with PID control also shows that high-order SMC is more suitable for automotive converters.

1. Introduction

With the further intensification of the global energy crisis, research into new energy technologies and energy storage technologies is expanding in all countries. Due to the limitations of new energy sources, a unidirectional or bidirectional DC–DC converter is generally required to maintain the stability of energy transfer, such as in electric vehicles (EVs) which need a DC–DC converter that can achieve bidirectional flow of energy to achieve stable operation of the EVs; the topology of automotive DC–DC converters and their control technology have therefore developed rapidly in recent years. Among the most common DC–DC converter topologies are the dual active bridge (DAB) converter [1] and the resonant converter [2]; CLLC resonant converters are attracting a lot of attention for their excellent performance and are increasingly being used in new energy vehicles. CLLC resonant converters have a symmetrical structure compared with conventional converters. Furthermore, the resonant technology enables soft-switching of the primary and secondary semiconductors, which helps reduce converter losses. The operating modes, parameter designs, and associated characteristics of CLLC resonant converters have been discussed in [3,4,5,6].

To achieve smooth operation of the vehicle, the vehicle bidirectional CLLC converter needs to have a faster response and be immune to interference [7], so the closed-loop control algorithm used in the converter is particularly important. At present, the research and application of closed-loop control strategies for resonant converters is still mainly based on PID frequency control. In [8], MPC and PID closed-loop controllers were implemented for CLLC resonant converters. By varying the input voltage and load magnitudes, the feasibility and characteristics of the two control strategies were evaluated. A composite optimal trajectory control that combines PID control and optimal trajectory control is proposed in [9], which switches between the two control modes according to the size of the load, thus improving the efficiency of the converter at light loads. A hybrid control method is proposed in [10], which incorporates both PID control and direct digital control (DDC). The DDC controller calculates the operating frequency in real time for this condition, and its output is superimposed on the PID controller output. Shortening the transition process is achieved by abruptly changing the switching frequency to the steady-state operating frequency, thus reducing the excessive voltage and current stress in the resonant tune and maintaining a stable output voltage. Two closed-loop control methods for improving converter efficiency are proposed above but essentially add extra modes of operation on the basis of PI, thus making them more difficult to apply in practice. The introduction of PFM in the PID control strategy to regulate the converter has the following problems: (i) To meet the gain requirements, the frequency of the resonant converter usually varies over a wide range, making it challenging to set the parameters for the PID. (ii) When the parameters such as the load on the converter vary considerably, it is difficult for the PID controller to ensure the stability of the system. (iii) When disturbances are serious, the PID controller takes longer to regulate, increasing the risk of soft-switching loss.

In order to solve these problems, the application of SMC in the converter is proposed in [11,12]. The SMC is a nonlinear control method based on a large-signal model, with features such as wide stability range, fast dynamic response, robustness, and simplicity. Furthermore, SMC is naturally suitable for switching converters. Conventional SMC suffers from chattering phenomenon, which is detrimental to the steady-state operation of the converter; so, it is generally necessary to introduce differential terms to eliminate the effects of chattering phenomenon. In [13], the characteristics of DC–DC converters and SMC are discussed, as well as the difficulties and solutions associated with applying SMC to DC–DC converters. SMC has been successfully applied to quasi-resonant converters in [14,15], based on the input–output linearization method, high-order SMC can be applied to resonant converters [16]. A single-order, sliding mode controller was first proposed in [17] for the CLLC resonant converter, where the linear combination of the output voltage and its integral term are set to the sliding surface according to the large-signal model of the CLLC converter, thus ensuring that the gain requirements and system stability can be met under a wide range of frequency variations, rapidly changing loads, etc. The stability and speed of the closed-loop system is also verified by a small-signal model. Due to the absence of differential terms, the output voltage has a greater ripple and chattering phenomenon.

In this paper, a new design method for a high-order sliding surface is proposed to improve the SMC algorithm. In order to overcome the shortcomings of current SMC algorithms, as well as to adapt to the characteristics of new energy vehicles, some voltages and their integral terms, etc., are set as sliding surface according to the mathematical model of the CLLC resonant converter and the input–output linearization theory. At the same time, additional higher-order differential terms were added to the sliding surface to weaken the chattering phenomenon and improve the control effect. The accuracy was verified by building the sliding mode controller in MATLAB and varying the input voltage and output load. In addition, it is further demonstrated that the application of SMC to resonant converters is beneficial in improving the dynamic responsiveness, robustness and efficiency of the converters over a wide range of operating conditions, while maintaining gain.

This paper is organized as follows: Section 2 describes the characteristics of the on-board CLLC resonant converter and develops a mathematical model based on the equivalent circuit using the EDF method; Section 3 designs the sliding surface and performance parameters based on the developed mathematical model of the CLLC resonant converter and its characteristics; Section 4 builds a simulation model in MATLAB to verify the rationality of the designed sliding mode controller and provides a comparative analysis; and Section 5 puts forward conclusions with relevant discussions.

2. Analysis of Automotive Bidirectional Full-Bridge CLLC Resonant Converters

2.1. Analysis of the Topology of Power Main Circuits

The energy flow of EVs is shown in Figure 1, and this paper focuses on the bidirectional DC–DC converter in the electric drive subsystem [18]. The bidirectional DC–DC converter varies the output voltage of the power battery to a range suitable for efficient motor operation and also feeds the power generated by the motor back into the power battery, thus extending the driving range. These are some characteristics of bidirectional DC–DC converters for automotive applications: (i) In order to reduce the size and mass of the vehicle, the bidirectional DC–DC converter should be as light and small as possible while still providing adequate power. (ii) To extend EV range, DC–DC converters should provide the highest possible transfer efficiency. (iii) In EVs, there is a high rate of starts and stops and a wide range of battery voltage conversions, so they require a DC–DC converter with good dynamic response and immunity to interference.

To meet the above requirements, the CLLC converter topology was selected. Compared with the LLC structure, a resonant capacitor and a resonant inductor are added to the secondary side to form a symmetrical structure, which makes the forward and reverse operating states consistent and solves the problem that the gain can be greater than 1 when working in the reverse direction; the circuit topology is shown in Figure 2.

The bidirectional full-bridge CLLC resonant converter consists of two full-bridge circuits and a resonant network. Switches $S_1$–$S_4$ and $S_5$–$S_8$ form the two full-bridge circuits, respectively, and $L_{r1}$, $L_{r2}$, $C_{r1}$, $C_{r2}$, and $L_m$ form the two resonant networks in different states; high-frequency transformer T acts as an electrical isolator. The converter operates in the forward direction by applying complementary drive signals to $S_1$, $S_4$ and $S_2$, $S_3$ to achieve inversion when the body diodes of $S_5$–$S_8$ are used to achieve rectification. In reverse operation, the primary and secondary switches swap operating states for reverse power transfer. In this topology, the primary and secondary switches can achieve soft-switching, effectively improving the conversion efficiency of the converter.

2.2. Modeling Analysis Based on the EDF Method

2.2.1. Nonlinear Equation of State for CLLC Resonant Converters

The CLLC resonant converter is controlled by pulse frequency modulation (PFM), as PFM does not satisfy the “small ripple assumption” that is a prerequisite for the use of state space averaging, and this method is no longer applicable. There are two main modeling methods for resonant converters: equivalent circuit models and the EDF method. As the equivalent circuit model method is more complex, the EDF method is used in this paper.

Replacing the full-bridge inverter circuit with a square-wave voltage $V_{ab}$ of amplitude $V_{in}$, and equating the rectifier circuit to a combination of a controlled voltage source and a controlled current source.Therefore, the equivalent circuit of the bidirectional full-bridge CLLC resonant converter shown in Figure 3.

From the above diagram, we can find the nonlinear equation of state for the equivalent circuit:

$$\left\{\begin{array}{c}{L}_1\frac{d{i}_1}{dt}+v_{c1}+L_m\frac{d{i}_m}{dt}=v_{AB}\hfill \\ C_1\frac{d{v}_{c1}}{dt}=i_1\hfill \\ L_m\frac{d{i}_m}{dt}=n\left[L_2\frac{d{i}_m}{dt}+v_{c2}+v_0\mathrm{sgn}\left(i_1-i_m\right)\right]\hfill \\ C_2\frac{d{v}_{c2}}{dt}=n\left(i_1-i_m\right)\hfill \\ \frac{v_0}{R_L}+C_0\frac{d{v}_{c0}}{dt}=n\left|i_1-i_m\right|\hfill \\ v_0=v_{c0}\hfill \end{array}\right.$$

(1)

$i_1$, $i_2$, $i_m$ are the currents flowing through inductors $L_1$, $L_2$, $L_m$, respectively; $v_{C0}$, $v_{C1}$, $v_{C2}$ are the voltages of capacitors $C_0$, $C_1$, $C_2$, respectively; $R_L$ is the output load with a voltage of $v_0$; $\mathrm{sgn}\left(·\right)$ is the signum function whose positive or negative value represents the direction of the current flowing through the primary side of the transformer.

2.2.2. Harmonic Approximation

When the converter operating frequency is near the resonant frequency, the current waveform flowing through the resonant tank can be approximated as a sine wave, assuming only the fundamental component is considered. Assuming an operating frequency of ${\omega }_s$, the resonant voltages $v_{C1}$, $v_{C2}$ and resonant currents $i_1$, $i_2$, $i_m$ can be expressed in the form of an extended descriptor function as follows:

$$\left.\begin{array}{c}{i}_1\left(t\right)=I_{1s}sin\left({\omega }_{s}t\right)+I_{1c}cos\left({\omega }_{s}t\right)=i_{1r}sin\left({\omega }_{s}t+\alpha \right)\hfill \\ i_m\left(t\right)=I_{ms}sin\left({\omega }_{s}t\right)+I_{mc}cos\left({\omega }_{s}t\right)=i_{mr}sin\left({\omega }_{s}t+\beta \right)\hfill \\ i_2\left(t\right)=n\left[\left(I_{1s}-I_{ms}\right)sin\left({\omega }_{s}t\right)+\left(I_{1c}-I_{mc}\right)cos\left({\omega }_{s}t\right)\right]=i_{2r}sin\left({\omega }_{s}t+\gamma \right)\hfill \\ v_{C1}\left(t\right)=V_{C1s}sin\left({\omega }_{s}t\right)+V_{C1c}cos\left({\omega }_{s}t\right)=V_{1c}sin\left({\omega }_{s}t+\theta \right)\hfill \\ v_{C2}\left(t\right)=V_{C2s}sin\left({\omega }_{s}t\right)+V_{C2c}cos\left({\omega }_{s}t\right)=V_{2c}sin\left({\omega }_{s}t+\phi \right)\hfill \end{array}\right\}$$

(2)

where: $i_{1r}=\sqrt{{I_{1s}}^2+{I_{1c}}^2},\alpha =arctan(I_{1s}/I_{1c})$, $i_{mr}=\sqrt{{I_{ms}}^2+{I_{mc}}^2},\beta =arctan(I_{ms}/I_{mc})$, $V_{1c}=\sqrt{{V_{C1s}}^2+{V_{c1s}}^2},\theta =arctan(V_{C1s}/V_{c1s})$, $V_{2c}=\sqrt{{V_{C2s}}^2+{V_{c2s}}^2},\phi =arctan(V_{C2s}/V_{c2s})$, $i_{2r}=\sqrt{{(I_{1s}-I_{ms})}^2+{(I_{1c}-I_{mc})}^2},\gamma =arctan((I_{1s}-I_{ms})/(I_{1c}-I_{mc}))$.

$V_{C1s}$, $V_{C1c}$, $V_{C2s}$, $V_{C2c}$, $I_{1s}$, $I_{1c}$, $I_{ms}$, and $I_{mc}$ are the sine component amplitudes and cosine component amplitudes of $v_{C1}$, $v_{C2}$, $i_1$, $i_m$, respectively. Thus, the derivative of Equation (2) can be obtained as below:

$$\left.\begin{array}{c}\frac{d{i}_1}{dt}=i_{1r}{\omega }_{s}cos\left({\omega }_{s}t+\alpha \right)\hfill \\ \frac{d{i}_m}{dt}=i_{mr}{\omega }_{s}cos\left({\omega }_{s}t+\beta \right)\hfill \\ \frac{d{v}_{C1}}{dt}=V_{1c}{\omega }_{s}cos\left({\omega }_{s}t+\theta \right)\hfill \\ \frac{d{v}_{C2}}{dt}=V_{2c}{\omega }_{s}cos\left({\omega }_{s}t+\phi \right)\hfill \end{array}\right\}$$

(3)

2.2.3. Edf Equations

The Fourier transform of the nonlinear components $v_{AB}$, $sgn\left(i_1-i_m\right)$ and $\left|i_1-i_m\right|$ of Equation (1) is approximated by their DC and fundamental components. The nonlinear equation of state in Equation (1) can then be written as an approximate linear equation of state:

$$\left.\begin{array}{c}{v}_{AB}\approx f_1\left(d,v_{in}\right)sin\left({\omega }_{s}t\right)\hfill \\ \mathrm{sgn}\left(i_1-i_m\right)\approx f_2\left(I_{1s}-I_{ms},I_{1c}-I_{mc}\right)sin\left({\omega }_{s}t\right)+f_3\left(I_{1s}-I_{ms},I_{1c}-I_{mc}\right)cos\left({\omega }_{s}t\right)\hfill \\ \left|i_1-i_m\right|\approx f_4\left(I_{1s}-I_{ms},I_{1c}-I_{mc}\right)\hfill \end{array}\right\}$$

(4)

d is the duty cycle; $f_1\left(·\right)$, $f_2\left(·\right)$, $f_3\left(·\right)$, and $f_4\left(·\right)$ are extended descriptive functions for the harmonic coefficients of the corresponding state variables of the converter under specific operating conditions, and the corresponding Fourier series expansion gives the following specific expressions [19]:

$$\left.\begin{array}{c}{v}_{AB}=\frac{4v_{ab}}{\pi }sin\left({\omega }_{s}t\right)\hfill \\ \mathrm{sgn}\left(i_1-i_m\right)=\frac{4}{\pi }sin\left({\omega }_{s}t+\beta \right)\hfill \\ \left|i_1-i_m\right|=\frac{2}{\pi }{i}_{mr}\hfill \end{array}\right\}$$

(5)

$v_{ab}$ is the sinusoidal component of the output voltage of the full-bridge inverter circuit.

2.2.4. Large-Signal Model for CLLC Resonant Converters

According to the principle of harmonic balance, substitute Equations (2)–(5) into Equation (1), then make the corresponding coefficients on both sides of each equation of (1) equal, and the large-signal model of the bidirectional full-bridge CLLC resonant converter is obtained as follows:

$$\left.\begin{array}{c}\frac{d{i}_{1r}}{dt}=\frac{1}{L_{e3}}\left[\frac{4}{\pi }{v}_{ab}cos\alpha -V_{1c}cos\left(\theta -\alpha \right)\right]-\frac{1}{L_{e1}}\left[\frac{4}{\pi }n{V}_{0}cos\left(\beta -\alpha \right)+V_{2c}cos\left(\phi -\alpha \right)\right]\hfill \\ \frac{d{i}_{mr}}{dt}=\frac{1}{L_{e1}}\left[\frac{4}{\pi }{v}_{ab}cos\beta -V_{1c}cos\left(\theta -\beta \right)\right]-\frac{1}{L_{e2}}\left[\frac{4}{\pi }n{V}_0+V_{2c}cos\left(\phi -\beta \right)\right]\hfill \\ \frac{d{v}_{1c}}{dt}=\frac{i_{1r}}{C_1}cos\left(\alpha -\theta \right)\hfill \\ \frac{d{v}_{2c}}{dt}=\frac{n{i}_{rm}}{C_2}cos\left(\beta -\phi \right)\hfill \\ \frac{d{v}_{co}}{dt}=\frac{2n}{\pi C_0}{i}_{rm}-\frac{v_0}{R_L{C}_0}\hfill \\ v_0=v_{co}\hfill \end{array}\right\}$$

(6)

where: $L_{e1}=\frac{L_m{L}_1+L_m{L}_2+L_1{L}_2}{L_m},L_{e2}=\frac{L_m{L}_1+L_m{L}_2+L_1{L}_2}{L_m+L_1},L_{e3}=\frac{L_m{L}_1+L_m{L}_2+L_1{L}_2}{L_m+L_2}$.

3. SMC of CLLC Resonant Converters

A vehicle’s output load changes rapidly due to its frequent start/stop characteristics, but DC bus voltage must be maintained near the rated voltage even when the load changes. SMC has a high degree of robustness and is insensitive to external disturbances. The SMC is easy to implement and suitable for DC–DC converter power regulation.

3.1. Basic Principles of Sliding Mode Control

The SMC directs the state trajectory of the controlled system onto a pre-designed state space (sliding surface), which slides along the surface and eventually reaches a stable point after reaching the sliding surface.

The basic problems with SMC are described below, nonlinear time-varying switching systems defined:

$$\dot{x}=f(x,u,t)$$

(7)

where x is a n-dimensional state vector; $\dot{x}$ is the derivative of x with respect to time t; and u is the control function as shown in Equation (8).

$$u\left(t\right)=\left\{\begin{array}{c}{U}^{+},S\left(x\right)>0\hfill \\ U^{-},S\left(x\right)<0\hfill \end{array}\right.$$

(8)

when all three of the following conditions can be met, the above form of control method is then referred to as SMC.

• Conditions of arrival: regardless of where the trajectory starts, the arrival condition directs the state trajectory to the sliding surface, as shown in stages A–B in Figure 4. Depending on the conditions of the termination point, it follows that:

  $$\underset{\mathrm{s}\to 0}{lim}S·\dot{S}<0$$

  (9)
• Conditions of existence: This condition ensures that once the system trajectory has reached the sliding surface, the trajectory can continue to start the sliding motion on the sliding surface, as shown in stages B–C in Figure 4. The existence condition can be determined simply by testing the conditions of arrival, i.e., by satisfying Equation (9).
• Conditions of stability: The conditions of stability ensure that the sliding surface is always able to drive the trajectory towards a stable equilibrium point. If the stability conditions are not met, this will result in an unstable sliding system. At this point, the system is satisfying the following mathematical model:

  $$\left\{\begin{array}{c}\dot{x}=f(x,u,t)\hfill \\ S\left(x\right)=0\hfill \end{array}\right.$$

  (10)

3.2. Sliding Mode Controller for CLLC Resonant Converters

According to the above modeling analysis, CLLC resonant converters are a single-input, single-output system. To improve the control effect, a high-order sliding surface can be formulated by adding differential and integral terms to reduce the chattering phenomenon [20].

According to the input-output feedback linearisation method, the output voltage is assumed to have the following response characteristics:

$$b_{{}^5}\frac{d^5{v}_0}{d{t}^5}+b_{{}^4}\frac{d^4{v}_0}{d{t}^4}+b_{{}^3}\frac{d^3{v}_0}{d{t}^3}+b_2\frac{d^2{v}_0}{d{t}^2}+b_1\frac{d(v_0-v_{ref})}{dt}+b_0(v_0-v_{ref})=0$$

(11)

$b_0$, $b_1$, $b_2$, $b_3$, $b_4$, and $b_5$ are parameters that control the system’s performance, and $v_{ref}$ is the desired value of the output voltage.

From Equation (6), it can be deduced that:

$$C_0\frac{d^5{v}_0}{d{t}^5}+\frac{1}{R}\frac{d^4{v}_0}{d{t}^4}=n·\frac{2}{\pi }\frac{d^3{i}_{pm}}{d{t}^3}$$

(12)

Equation (12) can be described as below:

$$a_5\frac{d^5{v}_0}{d{t}^5}+a_4\frac{d^4{v}_0}{d{t}^4}+a_3\frac{d^3{v}_0}{d{t}^3}+a_2\frac{d^2{v}_0}{d{t}^2}+a_1\frac{d{v}_0}{dt}+a_0\frac{d^3{i}_{pm}}{d{t}^3}=0$$

(13)

where: $a_5=L_{eq}{C}_0$, $a_4=\frac{L_{eq}}{R}$, $a_3=a_2=a_1=0$, $a_0=-\mathrm{n}·\frac{2}{\pi }{L}_{eq}$, $a_0=-\mathrm{n}·\frac{2}{\pi }{L}_{eq}$. Combining like terms of Equations (11) and (13), a new equation can be obtained as below:

$$(b_{{}^5}-a_{{}^5})\frac{d^5{v}_0}{d{t}^5}+(b_{{}^4}-a_{{}^4})\frac{d^4{v}_0}{d{t}^4}+(b_{{}^3}-a_{{}^3})\frac{d^3{v}_0}{d{t}^3}+(b_2-a_2)\frac{d^2{v}_0}{d{t}^2}+b_1\frac{d(v_0-v_{ref})}{dt}+b_0(V_0-v_{ref})-a_1\frac{d{v}_0}{dt}-a_0\frac{d^3{i}_{pm}}{d{t}^3}=0$$

(14)

Ideally, the invariance conditions for the sliding mode region of SMC is $S=0$ and $\dot{S}=0$.

In order to guarantee the designed output voltage response characteristics and the invariance condition $\dot{S}=0$ in the sliding mode region, another equation is established as below:

$$\begin{array}{c}\dot{S}=(b_{{}^5}-a_{{}^5})\frac{d^5{v}_0}{d{t}^5}+(b_{{}^4}-a_{{}^4})\frac{d^4{v}_0}{d{t}^4}+b_{{}^3}\frac{d^3{v}_0}{d{t}^3}+b_2\frac{d^2{v}_0}{d{t}^2}+b_1\frac{d(v_0-v_{ref})}{dt}+b_0(v_0-v_{ref})-a_0\frac{d^3{i}_{pm}}{d{t}^3}=0\hfill \end{array}$$

(15)

Therefore, the sliding surface equation is:

$$\begin{array}{c}S=(b_{{}^5}-a_{{}^5})\frac{d^4{v}_0}{d{t}^4}+(b_{{}^4}-a_{{}^4})\frac{d^3{v}_0}{d{t}^3}+b_{{}^3}\frac{d^2{v}_0}{d{t}^2}+b_2\frac{d{v}_0}{dt}+b_1(v_0-v_{ref})+b_0\int (v_0-v_{ref})dt-a_0\frac{d^2{i}_{pm}}{d{t}^2}\hfill \end{array}$$

(16)

If $b_5=a_{{}^5}$; $b_4=a_4+k_{d4}$; $b_3=k_{d3}$; $b_2=k_{d2}$; $b_1=k_p$; $b_0=k_i$; $a_0=k_c$; the treatment turned Equation (16) into (17):

$$\begin{array}{c}S=k_{{}^{d4}}\frac{d^3{v}_0}{d{t}^3}+k_{{}^{d3}}\frac{d^2{v}_0}{d{t}^2}+k_{d2}\frac{d{v}_0}{dt}+k_p(v_0-v_{ref})+k_i\int (v_0-v_{ref})dt+k_c\frac{d^2{i}_{pm}}{d{t}^2}\hfill \end{array}$$

(17)

By designing parameters $k_{d4}$, $k_{d3}$, $k_{d2}$, $k_p$, $k_i$, and $k_c$, the performance of the system can be adjusted. In addition, the output response parameters are assumed without sacrificing design freedom, the voltage characteristics determined by Equation (14) can be adjusted by multiplying both sides of the equation by a factor so that the output response characteristics are not affected. By introducing the Lyapunov function $V=\frac{1}{2}\left(S^{\prime }\right)$ to verify that $S·\dot{S}\le 0$, the designed sliding surface is proved to exist [21].

4. Analysis of Simulation and Experimental Results

Table 1. Design parameters of full-bridge CLLC resonant converter.

Name of the Parameter	Values
Nominal input voltage $V_{in}$ (V)	350
Nominal output voltage $V_{out}$ (V)	300
Resonant frequency $f_s$ (kHz)	125
Power output P (kW)	3

shows the main design parameters and component parameters that were used to verify the results of the theoretical analysis.

As shown in Figure 5, the power simulation circuit and sliding mode controller are built in simulink based on the parameters in

Table 2. Main component parameters.

Name of Components	Symbols	Values
Switches	$S_1-S_8$	SCT3060AL
Transformer (ratio)	$T_r$	26:23
Primary resonant inductor ($\mathsf{\mu }$H)	$L_{r1}$	16.5
Secondary resonant inductor ($\mathsf{\mu }$H)	$L_{r2}$	12.9
Primary resonant capacitor (nF)	$C_{r1}$	99
Secondary resonant capacitor (nF)	$C_{r2}$	126
Primary filter capacitor (nF)	$C_1$	19.4
Secondary filter capacitor (nF)	$C_2$	22.2

and the sliding mode surface designed in Equation (17).

The results of forward and reverse runs are similar, the only difference between them is the parameters of the controller. Therefore, this paper takes the simulation results of forward running as an example to verify the correctness of SMC. The parameters of the components in Figure 5 of the simulation model are set according to Table 1 and Table 2, where the parameters of the Mosfet are set according to those of the SCT3060AL-E SiC Mosfet from Rohm, with an onstate resistance of 0.6 $\Omega$, a parasitic diode drop of 3.2 V, etc.

The switch current waveform as shown in Figure 6, the negative current shows the antiparallel body diode conduction across the switch, causing voltage across the switch $V_{ds}=0$, and the gate signal $V_{gs}$ is applied, resulting in ZVS (zero voltage switching). It is clear from Figure 7 that gate voltage $V_{gs}$ falls to zero and, thereafter, the switch voltage $V_{ds}$ starts rising. The current naturally reduces to zero, and the negative current shows antiparallel body diode conduction causing ZCS (zero current switching). The above experimental results show that when the converter is operated near the resonance point, the SMC facilitates improved system performance of the system without destroying the soft switching. Figure 8 shows the magnetic current and resonant current waveforms, at which point the resonant current varies in a sinusoidal form. The magnetic current increases linearly as the magnetic inductance is clamped by the output voltage. Figure 9 shows that when the input voltage is 350 V, the output voltage is stable at 300 V after a regulation time of about 2 ms, and the overshoot and voltage ripple are relatively small.

Figure 10 and Figure 11 show the output voltage response curves when the input voltage is adjusted to 270 V and 400 V, respectively. At 400 V in particular, the output voltage stabilizes at 300 V after only about 1 ms, but the overshoot is relatively large at this point. Using the SMC, this converter is able to regulate a wide range of input voltages from 270–400 V. Figure 10 and Figure 11 also show that the ripple in the output voltage is very small and the response is fast.

In order to compare the control effect of SMC and PID, the sliding mode controller and PID controller are built in simulink, respectively, and experiments on voltage regulation and variable load are carried out at the same time. The experimental results show the superiority of SMC.

Figure 12 shows the output voltage response curves of SMC and PID control when the input voltage changes abruptly from 350 V to 300 V at 0.1 s. It can be seen that the SMC achieves static stability after about 2 ms of regulation, while the PID takes about 7 ms to stabilize, making it clear that the SMC has superior regulation performance. At the same time, the overshoot of the SMC during regulation is much smaller than that of the PID, and the output voltage ripple after stabilization is within 1.5%, which helps to improve the quality of power transmission.

Figure 13 shows the corresponding output voltage curves for SMC and PID control when the output load changes abruptly from full load to half load at 0.1 s. It can be seen that the SMC returns to a steady state again after about 1 ms of regulation, whereas the PID control takes about 3 ms to stabilize, so the SMC is more capable of dynamic regulation. Furthermore, the voltage dip of the SMC is much smaller than the voltage dip of the PID control during regulation, as well as the ripple of the output voltage, which helps to improve the converter’s stability and efficiency.

In [22], the authors propose that the dynamic characteristics of CLLC resonant converters are susceptible to load, so different controllers are designed according to the bode diagram corresponding to different equivalent loads when designing the controller. When the load changes, it is switched to the controller corresponding to this load. In the paper, voltage switching experiments were carried out in the most extreme case, when the voltage jumped from 250 V to 325 V, the regulation time took about 20 ms. Of course, if the magnitude of the voltage switching is reduced, the regulation time should be reduced. However, the SMC proposed in this paper only takes about 2 ms when the voltage jumps from 350 V to 270 V. In [23], an optimized, extended phase shift technique is proposed to improve the efficiency of the CLLC resonant converter at light loads. The technique introduces phase shifts between the primary bridge legs and between the primary and secondary bridge circuits respectively, and then finds the most efficient phase shift combination based on the relationship between the two phase shift angles and the efficiency, thus increasing the efficiency of the converter. The paper states that the maximum efficiency of the converter is 90.5% at half load. Nevertheless, the sliding mode control used in this paper has a regulation time of around 2 ms when the converter is switched from full load to load, at which point the efficiency of the converter is around 92%, which is significantly higher than the efficiency of the above method.

Through the above comparison experiments, it is concluded that SMC has the advantages of fast response, small overshoot, and strong anti-interference compared with PID. CLLC resonant converters are typical and complex nonlinear power electronic systems whose closed-loop control effect based on PID hardly suffices for complex electric vehicle operating conditions. The topology of the CLLC resonant converter is complex and contains many parameters that make it difficult to model accurately, so control algorithms that rely on accurate modeling are difficult to apply to the CLLC. Because the sliding surface of the SMC is independent of the parameters and perturbations of the controlled object, it responds rapidly and does not require online identification. It makes up for the fact that CLLC converters cannot be modeled accurately. This discontinuous, variable structure control is in line with the switching back and forth (variable structure) nature of the converter, which is able to constrain the system’s operating state to follow a predetermined trajectory in real time, depending on the control requirements of each phase, and therefore has significant advantages for regulating nonlinear systems. Since electric vehicles have complex driving conditions and are subject to many external disturbances, the SMC’s insensitivity to disturbances is also better suited to their actual application scenario.

5. Conclusions

This paper proposes a high-order sliding mode control algorithm based on the characteristics of CLLC resonant converters. The creation of high-order sliding mode surfaces based on output voltage and current enables closed-loop stabilization of the converter output and improves converter performance.

Firstly, the rationality of the CLLC resonant topology is analysed according to the requirements of the on-board bi-directional converter, as well as its equivalent circuit and mathematical model. Then, a high-order sliding mode controller was designed based on the mathematical model and characteristics of the CLLC. Finally, the simulation shows that the controller has good robustness and dynamicity and is able to reduce the chattering phenomenon and ripple of the output voltage, thereby improving power transmission quality and converter efficiency.