Application of an Improved STSMC Method to the Bidirectional DC–DC Converter in Photovoltaic DC Microgrid

Abstract

In a photovoltaic DC microgrid, the intermittent power supply of the distributed generation and the fluctuation of the load power will cause the instability of the bus voltage. An improved super-twisting sliding mode control method based on the super-twisting algorithm is proposed to solve this problem. In this paper, a bidirectional half-bridge buck–boost converter was selected as the research topic. The proposed control method replaces the sign function with the saturation function to further mitigate the chattering effect. The stability of the proposed control method was proven to be finite-time convergent using the Lyapunov theory control. Compared with PI control, linear sliding mode control, and terminal sliding mode control, the proposed control method reduces the system overshoot by up to 33% and greatly improves the response speed; compared with the traditional super-twisting sliding mode control method, the system overshoot is reduced by 6.8%, and the response speed is increased by 38%. The experimental results show that the proposed control method can reduce the fluctuation range of the bus voltage, shorten the time of bus voltage stability, effectively stabilize the bus voltage of the photovoltaic DC microgrid, and maintain strong robustness.

1. Introduction

Energy shortage and environmental pollution are key problems of human development. As a clean energy, solar energy is increasingly used in power systems. Photovoltaic DC microgrids have become a rapidly emerging field in the photovoltaic market [1,2]. During the operation of a photovoltaic DC microgrid, numerous factors will cause large-amplitude oscillations of the DC bus voltage, such as an intermittent power supply caused by weather change, an increase or decrease in electrical equipment, and a transient fault of a power grid [3,4]. To achieve stable and efficient operation of a photovoltaic DC microgrid, the introduction of feedback control to suppress the fluctuation of the bus voltage is required.

Compared with the linear control method, which can only adjust the DC bus voltage in a small range, sliding mode control (SMC), as a typical nonlinear control method, is suitable to stabilize the bus voltage of a DC microgrid due to its discontinuous control rules, strong robustness, and fast response to external disturbances and parameter disturbances. In [5], a two-stage control scheme was adopted for a hybrid AC–DC microgrid, and a sliding mode controller was used as the local controller. Experiments show that this control method can effectively improve the stability and robustness of the system. However, in traditional SMC [6], ASMC [7], and TSMC [8,9] control strategies, the sliding surface with respect to the control input has relative degree (RD) one. The control input acts on the first derivative of the sliding surface. It consists of using a discontinuous control that includes the sign function. Consequently, the chattering effect and undesirable oscillations with a finite amplitude and frequency occur due to unmodeled dynamics and the discrete-time implementation [10,11,12]. Unfortunately, these strategies are not usually applicable in practical plants.

As a typical second-order sliding mode control (SOSMC), super-twisting sliding mode control (STSMC) applies discontinuous control variables to the high-order derivatives of the sliding mode surface, which effectively suppresses chattering and improves the global robustness of the system. It also ensures that the system trajectory converges to the origin within a finite time [13]. In [14], a resonant super-twisting sliding mode controller (RST-SMC) was proposed, which has high disturbance rejection ability and a settling time that has been greatly reduced. The authors of [15] described combining super-twisting control with optimal robust nonlinear control. This control method improves the robustness of the system. In [16], an energy management control method using the super-twisting fractional method was proposed, which can reduce the chattering and steady-state error of the bus voltage in the presence of uncertainty.

In this paper, a photovoltaic DC microgrid is taken as the research object, and the simulation model is established. The improved STSMC method is used to realize the feedback control of a bidirectional DC–DC converter. This paper is organized as follows: Section 2 provides a brief overview of the operation mode and mathematical modeling of the bidirectional DC–DC converter, where the structure of a photovoltaic DC microgrid system is presented in 2.1, the operation mode of a bidirectional half-bridge buck–boost converter is presented in 2.2, and the brief mathematical modeling of a bidirectional buck–boost converter is presented in 2.3. In Section 3, an improved STSMC method is proposed. Section 3.1 and 3.2 present the design process of the improved STSMC method, which is combined with the bidirectional buck–boost converter. The Lyapunov function is used to prove the stability of the system in Section 3.3. In Section 4, the proposed control method is compared with the PI control method, linear SMC, and traditional STSMC. The simulation results show that the proposed control method can effectively improve the system response speed and reduce the system overshoot and the system chattering amplitude. The specific simulation data and results are shown in Section 4. Finally, Section 5 summarizes the main outcome of this paper.

2. Operation Mode and Mathematical Modeling of Converter

2.1. Photovoltaic DC Microgrid System Structure

The structure of a photovoltaic DC microgrid is shown in Figure 1, which is mainly composed of the distributed generation, energy storage device, power electronic converter, and DC load. If each unit wants to connect to the DC bus, it needs to change the voltage to the required level through the corresponding power converter.

Distributed photovoltaic power generation is vulnerable to the influence of external environmental factors, which belong to a fluctuating or even intermittent power supply. In order to solve the unreliability of distributed generation, it is necessary to introduce an energy storage device to “cut peaks and fill valleys” of a DC microgrid, that is, when the electric energy generated by the power supply is greater than the total load, the remaining energy will be stored in the energy storage unit and can also be fed to the public grid. By storing and extracting electric energy, the energy storage system ensures the supply and demand balance between the power generation and power consumption load and realizes the dynamic balance of the DC bus power and the stability of the bus voltage.

2.2. Topology Selection and Operation Mode

Basic structures of non-isolated bidirectional DC–DC converters include the bidirectional buck–boost converter, the bidirectional half-bridge buck–boost converter, the bidirectional Cuk circuit, and the bidirectional SEPIC converter [17]. This paper selected the bidirectional half-bridge buck–boost converter (as shown in Figure 2) as the bidirectional DC–DC converter used in the DC microgrid because, compared with the above converters, the current stress and voltage stress of switching elements and diodes are the smallest, and the conduction loss of active components is also the smallest. The topology of the bidirectional half-bridge buck–boost converter mainly includes a DC bus voltage $V_i$ on the DC microgrid side, battery voltage $V_o$ on the energy storage side, switch $S1$, switch $S2$, capacitor $C1$, capacitor $C2$, diode $D1$, diode $D2,$ and inductance $L$.

The operation mode of the bidirectional half-bridge buck–boost converter includes the buck mode and the boost mode. When the power of the DC bus is greater than the energy storage system, the converter works in buck mode. The theoretical waveform of the inductor current $i_L$ is shown in Figure 3.

The equivalent circuit of the converter working in buck mode is shown in Figure 4, and the specific working process is as follows:

(1)

At t₀ − t₁, S1 is turned on and S2 is turned off. C2 is being charged and i_L increases linearly, as shown in Figure 4a.

(2)

At t₁ − t₂, the circuit works in a dead zone, as shown in Figure 4b, S1 and S2 are turned off, D2 is turned on, and i_L begins to decrease.

(3)

At t₂ − t₃, S1 is turned off and S2 applies the driving signal. At this time, D2 is turned on, and i_L continues to decrease, as shown in Figure 4b.

(4)

At t₃ − t₄, the circuit returns to the dead zone, so the equivalent circuit is still as shown in Figure 4b. i_L continues to decrease. When i_L decreases to the minimum, S1 is turned on and starts the next cycle.

When the power of the DC bus is less than the energy storage system, the converter works in boost mode. The theoretical waveform of the inductor current $i_L$ is shown in Figure 5.

The equivalent circuit of the converter working in boost mode is shown in Figure 6, and the specific working process is as follows:

(1)

At t₀ − t₁, S2 is turned off, S1 applies the driving signal, D1 is turned on, and i_L decreases, as shown in Figure 6a.

(2)

At t₁ − t₂, the circuit works in a dead zone, as shown in Figure 6b. S1 and S2 are turned off, D1 is turned on, and i_L continues to decrease.

(3)

At t₂ − t₃, the equivalent circuit is shown in Figure 6b. S1 is turned off, S2 is turned on, and i_L begins to increase.

(4)

At t₃ − t₄, the circuit returns to the dead zone, so the equivalent circuit is still as shown in Figure 6b. i_L begins to decrease and starts to enter the next cycle.

2.3. Modeling of the Bidirectional DC–DC Converter

When the power is flowing from the DC bus to the energy storage system, the bidirectional DC–DC converter is equivalent to a buck circuit. The DC transfer function of the converter is

$$\frac{V_o}{V_i}=D_{Buck}$$

(1)

Among them, $D_{Buck}$ is the duty cycle of switch S1, $V_i$ is the input voltage of the buck converter, and $V_o$ is the output voltage of the buck converter.

When the energy flows from the energy storage system to the DC bus, the bidirectional DC–DC converter is equivalent to a boost circuit. The DC transfer function of the converter is

$$\frac{V_{o1}}{V_{i1}}=\frac{1}{1-D_{Boost}}$$

(2)

$D_{Boost}$ is the duty cycle of switch S2, $V_{i1}$ is the input voltage of the boost converter, and $V_{o1}$ is the output voltage of the boost converter. In this bidirectional buck–boost converter, $V_{i1}$ should correspond to the output voltage $V_o$ in buck mode, and $V_{o1}$ should correspond to the input voltage $V_i$ in buck mode. Therefore, the relationship between the input and the output in boost mode is

$$\frac{V_i}{V_o}=\frac{1}{1-D_{Boost}}$$

(3)

Because the switch control method of the bidirectional buck–boost converter is the complementary conduction method of the upper and lower switches, there is

$$D_{Buck}=1-D_{Boost}$$

(4)

Substituting (4) into (3),

$$\frac{V_i}{V_o}=\frac{1}{D_{Buck}}$$

(5)

which is consistent with the input–output relationship of the equivalent buck converter.

$$\frac{V_o}{V_i}=D_{Buck}$$

(6)

It can be concluded that the two working modes are equivalent when the upper switch and lower switch are complementary conductive, and the unified current controller can be directly used to control the bidirectional flow of the circuit energy. In this paper, the buck mode was chosen for modeling. There are several assumptions and considerations in the modeling process as follows: one is to ignore the on-resistance and the driving dead time of the switch, and the other is to ignore the internal resistance of the electronic components.

In the distribution network, the DC microgrid can be equivalent to a virtual power load. In the simulation case, the DC microgrid can be directly equivalent to a parallel operation unit which is composed of the distributed power and load. The equivalent circuit when S1 is turned on in buck mode (equivalent to the circuit when D1 is turned on in boost mode) is defined as mode 1; the equivalent circuit when D2 is turned on in buck mode (equivalent to the circuit when S2 is turned on in boost mode) is defined as mode 2 [18]. The simulation model of the DC microgrid can be obtained by equating the load part of the DC microgrid to a current source. Among them, the equivalent current source power of the distributed generation is set as positive, indicating the issued power, which is recorded as $P_{DC}$, $I_{g1}=P_{DC}/V_{in}$, is positive, and the equivalent current source power of the load is set as negative, indicating the absorbed power, which is recorded as $P_L$, $I_{g2}=P_L/V_{in}$, is negative. The equivalent simulation model is shown in Figure 7.

The state equation of mode 1 can be obtained from Figure 7a.

$$\{\begin{array}{c}\frac{d{v}_{C1}}{dt}=\frac{I_{g1}+I_{g2}}{C_1}-\frac{i_L}{C_1}-\frac{V_{in}}{C_1{R}_i}\\ \frac{d{v}_{C2}}{dt}=\frac{i_L-i_2}{C_2}\\ \frac{d{i}_L}{dt}=\frac{V_{in}-V_{Out}}{L}\end{array}$$

(7)

The state equation of mode 2 can be obtained from Figure 7b.

$$\{\begin{array}{c}\frac{d{v}_{C1}}{dt}=\frac{I_{g1}+I_{g2}}{C_1}-\frac{V_{in}}{C_1{R}_i}\\ \frac{d{v}_{C2}}{dt}=\frac{i_L-i_2}{C_2}\\ \frac{d{i}_L}{dt}=-\frac{V_{Out}}{L}\end{array}$$

(8)

where$V_{in}=V_{c1}, V_{out}=V_{c2}$, and $R_{eq}$ represents the equivalent resistance of the system; let $I_g=I_{g1}+I_{g2}$, and $I_g-\frac{V_{in}}{R_i}=-\frac{V_{in}}{R_{eq}}$, giving

$$\left[\begin{array}{c}\frac{d{v}_{C1}}{dt}\\ \frac{d{v}_{C2}}{dt}\\ \frac{d{i}_L}{dt}\end{array}\right]=\left[\begin{array}{lll}-\frac{1}{C_1{R}_{eq}}\hfill & 0\hfill & -\frac{1}{C_1}\hfill \\ 0\hfill & 0\hfill & \frac{1}{C_2}\hfill \\ \frac{1}{L}\hfill & -\frac{1}{L}\hfill & 0\hfill \end{array}\right]\left[\begin{array}{c}{v}_{c1}\\ v_{c2}\\ i_L\end{array}\right]+\left[\begin{array}{c}0\\ -\frac{i_2}{C_2}\\ 0\end{array}\right]$$

(9)

$$\left[\begin{array}{c}\frac{d{v}_{C1}}{dt}\\ \frac{d{v}_{C2}}{dt}\\ \frac{d{i}_L}{dt}\end{array}\right]=\left[\begin{array}{lll}-\frac{1}{C_1{R}_{eq}}\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & \frac{1}{C_2}\hfill \\ 0\hfill & -\frac{1}{L}\hfill & 0\hfill \end{array}\right]\left[\begin{array}{c}{v}_{c1}\\ v_{c2}\\ i_L\end{array}\right]+\left[\begin{array}{c}0\\ -\frac{i_2}{C_2}\\ 0\end{array}\right]$$

(10)

By introducing the duty cycle of the switch, the state average equation can be obtained.

$$\left[\begin{array}{c}\frac{d{v}_{C1}}{dt}\\ \frac{d{v}_{C2}}{dt}\\ \frac{d{i}_L}{dt}\end{array}\right]=\left[\begin{array}{lll}-\frac{1}{C_1{R}_{eq}}\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & \frac{1}{C_2}\hfill \\ 0\hfill & -\frac{1}{L}\hfill & 0\hfill \end{array}\right]\left[\begin{array}{c}{v}_{c1}\\ v_{c2}\\ i_L\end{array}\right]+\left[\begin{array}{c}-\frac{i_L}{C_1}\\ 0\\ \frac{v_{c1}}{L}\end{array}\right]d+\left[\begin{array}{c}0\\ -\frac{i_2}{C_2}\\ 0\end{array}\right]$$

(11)

3. Design of STSMC

3.1. STSMC

The design process of STSMC is as follows [19,20,21,22]:

Consider the nonlinear system

$$\{\begin{array}{c}{\dot{x}}_1=x_2\\ x_2=\mu +f\left(x_1,x_2,t\right)\\ y=x_1\end{array}$$

(12)

where $x_1$ and $x_2$ are the system state variables, $y$ is the system output, $\mu$ is the control variable, and $f\left(x_1,x_2,t\right)$ is the system interference term, generally $\left|f\left(x_1,x_2,t\right)\right|\le L>0$, where $L$ is a positive constant.

The output tracking error of the system is $e=y_c\left(t\right)-y\left(t\right)$, and the linear sliding surface is defined as

$$\sigma =\dot{e}+ce,c>0$$

(13)

After derivation of (13) and substituting the system output tracking error into it,

$$\dot{\sigma }={\ddot{y}}_c+c{\dot{y}}_c-f\left(y,\dot{y},t\right)-c\dot{y}-\mu =\phi \left(y,\dot{y},t\right)-\mu$$

(14)

where $\phi \left(y,\dot{y},t\right)={\ddot{y}}_c+c{\dot{y}}_c-f\left(y,\dot{y},t\right)-c\dot{y}$ is the accumulated disturbance term, $\left|\phi \left(y,\dot{y},t\right)\right|\le M$, where $M$ is a positive constant.

In order to make the linear sliding mode surface converge to zero and remain zero in a finite time, it is necessary to design the SMC algorithm $\mu$.

$$\mu =c{\left|\sigma \right|}^{1/2}\mathrm{sgn}\left(\sigma \right),c>0$$

(15)

Suppose $\phi \left(y,\dot{y},t\right)=0$ in (14). Substituting (15) into (14),

$$\dot{\sigma }=-c{\left|\sigma \right|}^{1/2}\mathrm{sgn}\left(\sigma \right),\sigma \left(0\right)={\sigma }_0$$

(16)

Having a definite integral on both sides of the above equation gives

$${\left|\sigma \left(t\right)\right|}^{1/2}-{\left|{\sigma }_0\right|}^{1/2}=-\frac{c}{2}t$$

(17)

The sliding surface of the system can converge to zero at a certain point $t_r$, that is, $\sigma \left(t_r\right)=0$. According to (17),

$$t_r=\frac{2}{c}{\left|{\sigma }_0\right|}^{1/2}$$

(18)

Therefore, (15) can make the system converge to zero in a finite time.

The above derivation process is carried out under the condition of $\phi \left(y,\dot{y},t\right)=0$. If the accumulated disturbance term $\phi \left(y,\dot{y},t\right)\ne 0$ in (14), then (16) should be

$$\dot{\sigma }=\phi \left(y,\dot{y},t\right)-c{\left|\sigma \right|}^{1/2}\mathrm{sgn}\left(\sigma \right),\sigma \left(0\right)={\sigma }_0$$

(19)

From the above equation, it can be concluded that there is a disturbance term in the system, and the control algorithm $\mu$ cannot make the system converge to zero in a finite time. It is necessary to consider adding a term in the control function, so that the disturbance can be fully compensated. Assuming that $\left|\phi \left(y,\dot{y},t\right)\right|\le M$, where $M$ is a positive constant, then the sliding mode control algorithm can be converted into

$$\{\begin{array}{c}\mu =c{\left|\sigma \right|}^{1/2}\mathrm{sgn}\left(\sigma \right)+\omega \\ \dot{\omega }=b\mathrm{sgn}\left(\sigma \right)\end{array}$$

(20)

Substituting the above equation into (14),

$$\{\begin{array}{c}\dot{\sigma }+c{\left|\sigma \right|}^{1/2}\mathrm{sgn}\left(\sigma \right)+\omega =\phi \left(y,\dot{y},t\right)\\ \dot{\omega }=b\mathrm{sgn}\left(\sigma \right)\end{array}$$

(21)

The added term $\omega$ of the sliding mode control algorithm can fully compensate the accumulated disturbance term in a finite time. That is, when $\omega =\phi \left(y,\dot{y},t\right)$, it can be obtained from (21).

$$\dot{\sigma }=-c{\left|\sigma \right|}^{1/2}\mathrm{sgn}\left(\sigma \right),\sigma \left(0\right)={\sigma }_0$$

(22)

The above equation shows that the system can converge to zero in a finite time. Therefore, in the presence of disturbance, the modified SMC algorithm (20) can make the system converge to zero in a finite time and keep the system stable, which measures up to the anticipated assumptions.

The sliding surface is defined as the output voltage tracking error.

$$S=v_{c1}-V_{ref}$$

(23)

Combined with (20), STSMC can be obtained.

$$\{\begin{array}{c}d=-{\mu }_1{\left|S\right|}^{1/2}\mathrm{sgn}S+d_1\\ {\dot{d}}_1=-{\mu }_2\mathrm{sgn}S\end{array}$$

(24)

3.2. Improved STSMC

The $sgn\left(s\right)$ (as shown in Figure 8) in STSMC is discontinuous at $S=0$, and it is a discontinuous function on the set of real numbers; its motion cannot infinitely approach zero at the end. Therefore, it can be considered to select the appropriate continuous function to reduce the chattering of the system.

In this paper, the continuous saturation function $sat\left(s\right)$ (as shown in Figure 9) is used to replace the discontinuous sign function in (24). The control law can be converted into

$$\{\begin{array}{c}d=-{\mu }_1{\left|S\right|}^{1/2}\mathrm{sat}(S)+d_1\\ {\dot{d}}_1=-{\mu }_2\mathrm{sat}(S)\end{array}$$

(25)

According to (11), (23) and (25), we can obtain

$$\begin{array}{ll}\dot{S}& =\frac{d{v}_{c1}}{dt}=-\frac{1}{C_1{R}_{eq}}{v}_{c1}-\frac{i_L}{C_1}d\\ & =-\frac{1}{C_1{R}_{eq}}{v}_{c1}+\mu 1\frac{i_L}{C_1}{\left|S\right|}^{\frac{1}{2}}\mathrm{sat}(S)+\mu 2\frac{i_L}{C_1}{\displaystyle ʃ}\mathrm{sat}(S)dt\end{array}$$

(26)

where $x_1=S$, and $x_2={\mu }_2\frac{i_L}{C_1}{\displaystyle \int }sat\left(s\right)dt-\frac{1}{C_1{R}_{eq}}{V}_{c1}$, so the above equation can be rewritten as

$${\dot{x}}_1=k_1{\left|x_1\right|}^{\frac{1}{2}}\mathrm{sat}(x_1)+x_2$$

(27)

$${\dot{x}}_2=k_2\mathrm{sat}(x_1)+\dot{\phi }\left(t\right)$$

(28)

where $k_1={\mu }_1\frac{i_L}{C_1}$, $k_2={\mu }_2\frac{i_L}{C_1}$, and $\phi \left(t\right)=-\frac{1}{C_1{R}_{eq}}{V}_{c1}$. $\phi \left(t\right)$ represents time-varying disturbances, which satisfy the definition of the Lipschitz constant. Therefore, $\left|\dot{\phi }\left(t\right)\right|\le \delta$, $\forall t\ge 0$, where $\delta$ is a positive constant.

3.3. System Stability Analysis

In this paper, the Lyapunov candidate function is selected to analyze the stability and the finite time convergence of the systems:

$$V\left(x_1,x_2\right)={\xi }^{T}P\xi$$

(29)

where ${\xi }^T=\left[{\xi }_1,{\xi }_2\right]=[{\left|x_1\right|}^{\frac{1}{2}}sat\left(x_1\right),x_2]$, and the symmetric positive definite matrix $P$ is defined as

$$P=\left[\begin{array}{cc}2k_2+k_1^2/2& -k_1/2\\ -k_1/2& 1\end{array}\right]$$

(30)

Substituting (30) into (29),

$$V\left(x\right)=\left(2k_2+\frac{{k_1}^2}{2}\right)\left|x_1\right|-k_1{x}_2{\left|x_1\right|}^{1/2}\mathrm{sat}(x_1)+{x_2}^2$$

(31)

Obviously, except in $\left\{(x_1,x_2):x_1=0\right\}$, $V\left(x_1,x_2\right)$ is continuous and differentiable, and the derivation can be obtained:

$$\dot{\xi }=\left[\begin{array}{c}\frac{1}{2}{\left|x_1\right|}^{-\frac{1}{2}}\left(k_1{\left|x_1\right|}^{\frac{1}{2}}\mathrm{sat}(x_1)+x_2\right)\\ k_2\mathrm{sat}(x_1)+\dot{\phi }\end{array}\right]=\frac{1}{\left|{\xi }_1\right|}\left(A\xi +B\varphi \right)$$

(32)

where

$$A=\left[\begin{array}{cc}\frac{k_1}{2}& \frac{1}{2}\\ k_2& 0\end{array}\right],B=\left[\begin{array}{c}0\\ 1\end{array}\right],\varphi =\left|{\xi }_1\right|\dot{\phi }$$

(33)

The characteristic polynomial of $A$ is $p\left(s\right)=s^2-\frac{k_{1}s}{2}-\frac{k_2}{2}$; if $k_1<0$ and $k_2<0$, that is, ${\mu }_1<0$,${\mu }_2<0$, the matrix $A$ is a Hurwitz matrix, so $\dot{\xi }$ is asymptotically stable.

For the derivation of $V\left(x\right)$,

$$\begin{array}{ll}\dot{V}\left(x\right)& ={\dot{\xi }}^{T}P\xi +{\xi }^{T}P\dot{\xi }=\frac{1}{\left|{\xi }_1\right|}\left[{\left(A\xi \right)}^T+{\left(B\varphi \right)}^T\right]P\xi +{\xi }^{T}P\frac{1}{\left|{\xi }_1\right|}\left(A\xi +B\varphi \right)\\ & =\frac{1}{\left|{\xi }_1\right|}\left({\xi }^T{A}^{T}P\xi +{\varphi }^T{B}^{T}P\xi +{\xi }^T\mathrm{PA}\xi +{\xi }^T\mathrm{PB}\varphi \right)\\ & =\frac{1}{\left|{\xi }_1\right|}{\left[\begin{array}{c}\xi \\ \varphi \end{array}\right]}^T\left[\begin{array}{cc}{A}^{T}P+PA& PB\\ B^{T}P& 0\end{array}\right]\left[\begin{array}{c}\xi \\ \varphi \end{array}\right]\end{array}$$

(34)

It can be inferred from $\dot{|\phi }\left(t\right)|\le \delta$, $\forall t\ge 0$ that

$$\begin{array}{ll}\dot{V}\left(x\right)& \le \frac{1}{\left|{\xi }_1\right|}\left\{{\left[\begin{array}{c}\xi \\ \varphi \end{array}\right]}^T\left[\begin{array}{cc}{A}^{T}P+PA& PB\\ B^{T}P& 0\end{array}\right]\left[\begin{array}{c}\xi \\ \varphi \end{array}\right]+{\delta }^2{{\xi }_1}^2-{\varphi }^2\right\}\\ & \le \frac{1}{\left|{\xi }_1\right|}{\xi }^T\left(A^{T}P+PA+{\delta }^2{C}^{T}C+PB{B}^{T}P\right)\xi \end{array}$$

(35)

where $C=\left[1 0\right]$.

Defining $A^{T}P+PA+{\delta }^2{C}^{T}C+PB{B}^{T}P=-Q$ gives

$$Q=\left[\begin{array}{cc}-k_1{k}_2-\frac{{k_1}^3}{2}-\frac{{k_1}^2}{4}-{\delta }^2& \frac{k_1}{2}-2k_2\\ \frac{k_1}{2}-2k_2& \frac{k_1}{2}-1\end{array}\right]$$

(36)

When $Q$ is a positive definite symmetric matrix, the following conditions are satisfied:

$$\{\begin{array}{l}{k}_1>2\\ \frac{3k_1}{8}-\frac{{k_1}^2}{16}-{\left[\frac{\left(2-k_1\right)\left(15{k_1}^3+18{k_1}^2+32{\delta }^2\right)}{256}\right]}^{\frac{1}{2}}<k_2\\ k_2<\frac{3k_1}{8}-\frac{{k_1}^2}{16}+{\left[\frac{\left(2-k_1\right)\left(15{k_1}^3+18{k_1}^2+32{\delta }^2\right)}{256}\right]}^{1/2}\end{array}$$

(37)

That is,

$$\{\begin{array}{l}{\mu }_1>\frac{2C_1}{i_L}\\ {\mu }_2>\frac{{\mu }_1{v}_{c1}}{4\left({\mu }_1{v}_{c1}-2L\right)}+\frac{{\delta }^2}{{\mu }_1{v}_{c1}}\end{array}$$

(38)

When the above conditions are satisfied,

$$\dot{V}\le -\frac{1}{\left|{\xi }_1\right|}{\xi }^{T}Q\xi <0$$

(39)

According to the Lyapunov stability theorem, when the controller parameters meet the above conditions, the system is stable.

4. Simulation and Results

In order to verify the effectiveness of the proposed control method in the DC microgrid, the simulation model of the system was built in MATLAB/Simulink. The stable voltage of the DC bus was set to 600 V, the rated voltage of the battery was set to 200 V, the rated capacity was set to 40 Ah, and the state of charge (SOC) value of the battery was set to 80%. The circuit parameters of the bidirectional half-bridge buck–boost converter are shown in

Table 1. Circuit parameters.

Description	Value
Voltage of DC bus	600 V
Voltage of energy storage	200 V
Inductance $L$	10 $\mathrm{mH}$
Capacitor $C1$ of DC bus	1 $\mathrm{mF}$
Capacitor $C2$ of energy storage	1 $\mathrm{mF}$
Internal resistance $R1$ of $C1$	0.001 $\Omega$
Internal resistance $R2$ of $L$	0.001 $\Omega$
Internal resistance $R3$ of $C2$	0.001 $\Omega$
Internal resistance $Ri$ of power supply	0.01 $\Omega$

.

The circuit adopted double closed-loop current feedback control, the outer loop adopted the proposed control, and the inner loop adopted PI control. Among them, the coefficients of the proposed control in the outer loop were set as ${\mu }_1=6$ and ${\mu }_2=4000$, and the control parameters of the PI controller in the inner loop were $K_P=1$ and $K_i=50$.

4.1. Simulation of Proposed Control Method

The simulation duration was set to 2 s, the power of the distributed generation was set to [100, 150, 100, 50] kW every 0.5 s, and the power of the load was set to [−50, −150, −150, −50] kW every 0.5 s. Figure 10 and Figure 11 show the response results of the inductance current and the bus voltage under the proposed control method, respectively.

When the distributed generation power is unstable and the load power fluctuates, the proposed control method can effectively suppress the fluctuation of the bus voltage, which is stable at about 600 V.

4.2. Performance Comparison

The simulation duration was set as 2 s, the power of the distributed generation was set as [100, 150, 100, 50] kW every 0.5 s, and the power of the load was set as [−50, −150, −150, −50] kW every 0.5 s. The bus voltage response comparison result of the proposed control method and STSMC is shown in Figure 12.

It can be seen from the above figure that both the STSMC and the proposed control method can maintain the bus voltage of the DC microgrid at about 600 V. However, the instantaneous response amplitude of the DC microgrid bus voltage is significantly reduced under the proposed control method. Under the proposed control method, the bus voltage overshoot of the DC microgrid is about 609 V, and under STSMC, the bus voltage overshoot is about 650 V. The system overshoot is reduced by about 6.8%.

The power of the distributed power supply was set to 100 kW, and the load power was set to −150 kw. Compared with STSMC, the inductance current response comparison result under the two SMC methods can be obtained, as shown in Figure 13, and the bus voltage response comparison result is shown in Figure 14.

Under STSMC, the inductance current needs to be stabilized after 22 ms, and the bus voltage needs to be stabilized after 21 ms. However, under the proposed control method, the inductance current only needs to be stabilized after 7 ms, and the bus voltage only needs to be stabilized after 13 ms. The instantaneous fluctuation range of the inductance current and the bus voltage under the proposed control method is obviously smaller than that of STSMC. Therefore, compared with STSMC, the proposed control method has a smaller instantaneous fluctuation of the inductance current and bus voltage, a faster response speed, and better robustness.

In addition, the dynamic performances of the proposed control method, PI control, linear SMC, TSMC, and STSMC are compared in Figure 15, where the power changes every 0.5 s, the distributed power is set to [100, 150] kW, and the load power is set to [−50, −150] kW. Under PI control, the system start-up overshoot is about 760 V, and the output voltage response speed is about 0.2 s; under linear SMC, the start-up overshoot of the system is about 910 V, the output voltage tends to be stable after oscillation, and the regulation time is about 0.15 s; under TSMC, the system start-up overshoot is about 865 V, and the output voltage response speed is about 0.13 s. Compared with linear SMC, the proposed control method reduces the system overshoot by about 33%. The effect of the proposed control method is obviously better than the first-order SMC, which can significantly reduce the transient response amplitude of the DC microgrid bus voltage and suppress the chattering phenomenon. Compared with STSMC, PI control, linear SMC, and TSMC, the proposed control method has a smaller fluctuation amplitude and a faster response speed of the DC microgrid bus voltage transient response, reduces the bus voltage chattering phenomenon, and has stronger robustness.

5. Conclusions

In order to solve the problem of DC microgrid bus voltage instability caused by the fluctuation of the distributed generation power and the load power, STSMC was introduced to realize the feedback control of the bidirectional DC–DC converter. In this paper, the dynamic model of a DC microgrid system composed of the distributed generation, load, battery, and bidirectional half-bridge buck–boost converter was established, and the operation mode of the converter was introduced in detail. The continuous saturation function $sat\left(s\right)$ is used to replace the discontinuous function $sgn\left(s\right)$ to weaken the chattering phenomenon. The stability conditions of the system were deduced theoretically, and the system simulation model was established. Comparing the results of the five control methods, the improved super-twisting high-order sliding mode control can significantly reduce the fluctuation range of the bus voltage, shorten the time of bus voltage stability, and maintain better robustness. The proposed control method is also suitable for other converters in other photovoltaic power generation systems. However, this paper only studied the DC microgrid when operated in the islanded mode, and the energy storage unit only considers the battery. In the future, the DC microgrid can be complexed and incorporated into a traditional power grid or connected to a supercapacitor.