A Passivity-Based Control Integrated with Virtual DC Motor Strategy for Boost Converters Feeding Constant Power Loads

Abstract

This article proposes a nonlinear control strategy to address the voltage instability issue caused by the boost converter with an uncertain constant power load (CPL). This strategy combines a passivity-based controller (PBC) with a virtual DC motor controller (VDCM). Initially, a PBC is designed for the boost converter, which enhances the robustness of the converter with CPL perturbations in the DC bus voltage. To overcome the limitations of PBC, including steady-state errors resulting from variations in load or input voltage, the VDCM is incorporated, simulating the characteristics of a DC motor. This addition improves the system’s inertia and damping, making it more stable and significantly enhancing its dynamic performance. The efficacy and stability analysis of the proposed control strategy is validated through both simulation and experimentation.

1. Introduction

The continuous deterioration of the natural environment, the increasing scarcity of traditional fossil fuels, and the advancement of power electronics have led to widespread attention and research on DC/DC conversion as a major form of power conversion [1]. Boost converters, in particular, are widely employed in applications that require stepping up low voltages to higher levels, such as renewable energy systems, electric vehicles, and portable electronic devices [2]. However, constant power loads (CPLs), prevalent in DC systems, exhibit complex impedance characteristics that can negatively impact power quality, leading to voltage oscillations and reduced damping [3]. This phenomenon is a major contributor to system instability.

To address this issue, many linear control strategies have been proposed and can be broadly divided into two categories: passive damping methods and active damping methods [4]. Passive damping methods primarily eliminate the impact of CPLs by incorporating hardware devices such as capacitors, resistors, and inductors [5], but this method increases the cost and size of the system, making it less efficient. Active damping methods stabilize the system by adding virtual resistance [6,7], virtual impedance control [8], and so on. However, these linear methods are limited by their reliance on small-signal models near the operating point and may fail when large perturbations occur, leading to instability. Therefore, advanced nonlinear control techniques need to be introduced. Popular nonlinear approaches include model predictive control (MPC), backstepping control, sliding mode control (SMC), and passivity-based control (PBC). A pseudo-extended Kalman filter was introduced in [9] to stabilize microgrid with CPL using stochastic nonlinear MPC, but the computation is complex and dependent on model accuracy. A backstepping controller based on droop control was used in [10] to mitigate the CPL problem. However, the design process is cumbersome and relies on a nonlinear disturbance observer, which is sensitive to system parameters. Additionally, two fast sliding mode-based controllers were proposed in [11] for buck converters with CPLs, based on an output voltage regulation approach and output power regulation scheme, respectively. Nevertheless, these require a high switching frequency of the converter. In contrast to other nonlinear control strategy methods, PBC stands out for its simplicity, highly efficient, easy realization, and thus has become one of the most effective applied nonlinear techniques [12]. PBC achieves a global asymptotically stable equilibrium by injecting a virtual resistance matrix in a specific way and reshaping the dissipated energy of the system, which is a crucial operation for system control. Thereby, it ensures the stability in the closed-loop control system and has the advantage of enabling the flexible plug-and-play operation of distributed power supplies [13,14,15,16,17,18,19,20,21]. A composite current-constrained controller was proposed in [13], which incorporates perturbation estimation and a nonlinear penalty term into the passivity-based control law. In [15], an alternating component passivity-based controller was designed by damping and interconnection injection.

While PBC is effective, the limitation is that it cannot eliminate the output voltage steady state errors caused by load or power supply variations [16]. To solve this problem, a nonlinear disturbance observer (NDO) has been proposed to compensate for the steady-state error generated by PBC [17,18,22], but this requires high computational resources and real-time performance. Additionally, the adaptive extended Kalman filter (AEKF) has been designed in [19], which relies heavily on priori information. Both approaches utilize load information to address this issue, thereby increasing algorithmic complexity. In comparison, control methods that eliminate steady-state error without requiring load information typically combine PBC with other control algorithms. For example, a complementary proportional-integral (PI) controller has been proposed in conjunction with an adaptive interconnection and damping assignment passivity-based controller (IDA-PBC) [20], but the control process is lengthy and prone to overshoot. Literature [21] combines integral sliding mode control (ISMC) with PBC. However, it fails to address the inherent drawbacks of sliding mode control. Furthermore, the high-speed response of the front-end power electronic converter can threaten the stable operation of the DC system when disturbances occur due to the frequent switching of distributed power sources or loads. Inspired by virtual synchro-nous motor control (VSG) in AC microgrids [23], virtual DC motor (VDCM) control has been proposed to provide inertia and damping [24]. Ref. [25] compares the control model of a DC motor with the conventional double closed-loop control to highlight the performance of the VDCM. The VDCM contains the DC motor armature equations and the mechanical rotation equations. It is known through the mechanical rotation equations that the control contains an integral part, which usually serves to eliminate the steady state error of the system.

In summary, the traditional PBC can effectively suppress the oscillation caused by the CPL, but the issue of steady-state error in its control performance remains insufficiently studied. To address this issue, this paper proposes a robust control strategy for boost converter against CPL perturbations, which combines PBC and VDCM. The Euler–Lagrange (EL) model of the boost converter is employed to design PBC, ensuring system stability and passivity. This approach enables flexible plugging and unplugging of distributed power supplies in microgrids, with better adaptability to actual circuit topologies and a more implementable structure. The VDCM serves as a patch for PBC, compensating for steady-state errors caused by various perturbations and increasing the system’s inertia and damping to improve its robustness. The efficacy of the proposed method has been validated through simulations using MATLAB/Simulink R2022a software and experiments on the dSPACE1104 experimental platform. The results demonstrate that the proposed method not only possesses robustness similar to that of PBC but also eliminates steady-state errors through VDCM, addressing the limitations of PBC.

The remainder of this paper is structured in the following manner: Section 2 lays out the system model, and Section 3 delves into the details of the closed-loop control design for the PBC with VDCM. The simulation and experimental sessions are then described in Section 4. Lastly, Section 5 presents a summary of the work.

2. System Configuration and Modeling

The topology employed in this paper, where a CPL is loaded through a boost converter, is shown in Figure 1. Typical examples of CPL include a DC/DC converter connected to resistors and a DC/AC inverter driving a motor [26], where the system may oscillate and become unstable due to frequent switching of the switching devices. Assuming that the converter operates in a continuous conduction mode (CCM), the average state equation can be derived as follows:

$$\left\{\begin{array}{l}L{\displaystyle \frac{d{i}_L}{dt}}=u_{in}-\left(1-\mu \right)u_C\\ C{\displaystyle \frac{d{u}_C}{dt}}=\left(1-\mu \right)i_L-{\displaystyle \frac{P_{CPL}}{u_C}}\end{array}\right.,$$

(1)

where $u_{in}$ represents the input voltage, $L$ is the inductance, $C$ is the high-voltage side capacitor, $i_L$ denotes the inductance current of the converter, $u_C$ is the capacitor voltage on the high-voltage side of the converter, $P_{CPL}$ is the power of CPL, and $\mu$ is the duty cycle.

Define

$$\begin{gathered} X=\left[\begin{array}{l}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{l}{i}_L\\ u_C\end{array}\right], A=\left[\begin{array}{cc}L& 0\\ 0& C\end{array}\right], B=\left[\begin{array}{cc}0& \left(1-\mu \right)\\ -\left(1-\mu \right)& 0\end{array}\right], \\ R=\left[\begin{array}{cc}0& 0\\ 0& {\displaystyle \frac{P_{CPL}}{u_C^2}}\end{array}\right],U=\left[\begin{array}{l}{u}_{in}\\ 0\end{array}\right]. \end{gathered}$$

Equation (1) can be rewritten in the EL equation as follows, thus establishing the basis for the controller design detailed in the subsequent chapter:

$$A\dot{X}+\left(B+R\right)X=U.$$

(2)

The proposed controller is designed to regulate the output voltage of the boost converter feeding CPL and better address the impacts of CPL and input voltage disturbances, thus achieving the following objectives:

$$\underset{t\to \infty }{\lim }\left[X-X_d\right]=0,$$

(3)

where $X_d=\left[\begin{array}{l}{x}_{1d}\\ x_{2d}\end{array}\right]=\left[\begin{array}{l}{I}_L\\ U_C\end{array}\right]$ is the reference point value.

3. Design of Virtual DC Motor Compensated Passivity-Based Control

To address the instability caused by CPLs, a passive-based controller has been designed, and its stability has been analyzed. Furthermore, VDCM has been incorporated to increase the system’s inertia and eliminate steady-state errors. The overall framework of the proposed method is shown in Figure 2.

3.1. Design of Passivity-Based Control

To mitigate the extreme cycle oscillation caused by CPL, the PBC applied to the boost converter was implemented by injecting virtual damping resistance, as shown in Figure 2a. This approach adds two virtual resistances to the placement of the converter. In this sense, the system is entirely passive and stable. Additionally, the CPL was configured as a Buck converter under PI control with resistive loads. The duty cycle output by the controller is compared with a triangular wave to generate a PWM signal, which controls the switching devices of the converter. The design of passivity-based controllers undergoes two necessary stages [17,18]:

• Energy shaping stage:

Let $X=X_d+\tilde{X}$, where $\tilde{X}$ is the systematic error deviating from the reference point when the system oscillates. According to Equation (2), the equation can be obtained as follows:

$$A\left({\dot{X}}_d+\dot{\tilde{X}}\right)+\left(B+R\right)\left(X_d+\tilde{X}\right)=U.$$

(4)

Equation (4) is expressed as follows:

$$A\dot{\tilde{X}}+\left(B+R\right)\tilde{X}=U-\left[A{\dot{X}}_d+\left(B+R\right)X_d\right].$$

(5)

II.

Damping injection stage:

By injecting the virtual damping matrix $R_d$ into Equation (5), it can be concluded that

$$A\dot{\tilde{X}}+\left(B+R_Z\right)\tilde{X}=U-\left[A{\dot{X}}_d+\left(B+R\right)X_d-R_d\tilde{X}\right],$$

(6)

where

$$\begin{gathered} R_Z=\left[\begin{array}{cc}{R}_{1d}& 0\\ 0& {\displaystyle \frac{1}{R_{2d}}}+{\displaystyle \frac{P_{CPL}}{u_C^2}}\end{array}\right], \\ {R}_d=R_Z-R=\left[\begin{array}{cc}{R}_{1d}& 0\\ 0& {\displaystyle \frac{1}{R_{2d}}}\end{array}\right]. \end{gathered}$$

By adding virtual resistors, the transient energy dissipation and Lyapunov stability can be ensured. The series resistance of the inductor circuit ($R_{1d}$) is sufficient to ensure energy dissipation in the inductor and effectively suppress inductor current ripple. The virtual resistance in the parallel capacitor circuit ($R_{2d}$) is sufficiently small to minimize energy dissipation in the capacitor and suppress output voltage ripple. It should be noted that the case of ${\mathrm{u}}_C=0$ generally occurs only during equipment startup, when the load side has no voltage and the CPL does not activate, thus exerting no impact on the controller. In practical operation, protective actions will be triggered to shut down the converter once the load voltage drops below a specific threshold [27]. Therefore, $\tilde{X}=0$, and Equation (6) can be rewritten as

$$U-\left[A{\dot{X}}_d+\left(B+R\right)X_d-R_d\tilde{X}\right]=0.$$

(7)

Rearrange Equation (7) and get the following equations:

$$u_{in}-L{\dot{x}}_{1d}-\left(1-\mu \right)x_{2d}+R_{1d}\left(x_1-x_{1d}\right)=0,$$

(8)

$$-C{\dot{x}}_{2d}+\left(1-\mu \right)x_{1d}-{\displaystyle \frac{P_{CPL}}{x_{2d}}}+{\displaystyle \frac{1}{R_{2d}}}\left(x_2-x_{2d}\right)=0.$$

(9)

The PBC formula in Equation (8), enables accurate tracking of voltage reference value during steady-state operation. However, due to the randomness and fluctuation of CPL, steady-state errors are introduced, which further expand the output voltage error. To mitigate this issue, the VDCM approach, discussed in the subsequent section, is employed. Therefore, $I_L$ needs to be modified as $I_{L\_ref}$ through VDCM. It should be noted that $I_L$ represents the reference voltage generated by the PBC voltage control loop, while $I_{L\_\mathit{ref}}$ corresponds to the adjusted current reference value obtained by processing $I_L$ through the VDCM compensation module. In addition, the boost converter of direct control via inner and outer loops is ineffective in regulating the DC bus voltage ($x_2$) to a stable equilibrium point due to the non-minimum phase characteristics [18]. Consequently, auxiliary loop control is necessary [17,28]. The duty cycle can be expressed as $1-\mu =u_{in}/U_C$, and according to Equations (8) and (9), the duty cycle $\mu$ and the reference value of inductance current of PBC $I_L$ are defined as follows

$$I_L={\displaystyle \frac{P_{CPL}}{u_{in}}}+{\displaystyle \frac{U_C}{R_{2d}{u}_{in}}}\left(U_C-u_C\right),$$

(10)

$$\mu =1-{\displaystyle \frac{1}{x_{2d}}}\left[u_{in}-R_{1d}\left(I_{L\_ref}-i_L\right)\right].$$

(11)

According to Equation (9), the auxiliary controller can be obtained as

$${\dot{x}}_{2d}={\displaystyle \frac{1}{C}}\left[-{\displaystyle \frac{P_{CPL}}{x_{2d}}}+{\displaystyle \frac{1}{R_{2d}}}\left(u_C-x_{2d}\right)+\left(1-\mu \right)I_{L\_ref}\right].$$

(12)

3.2. Stability Analysis of PBC

Lyapunov stability analysis is commonly employed to analyze the stability of PBC. Based on Equations (6) and (7) and the positive definite matrix A, the equation and a Lyapunov function $V\left(x\right)$ can be derived:

$$A\dot{\tilde{X}}+\left[B+R_Z\right]\tilde{X}=0,$$

(13)

$$V(x)={\displaystyle \frac{1}{2}}{\tilde{X}}^{T}A\tilde{X}>0\left(\forall \tilde{x}\ne 0\right).$$

(14)

The derivative of $V\left(x\right)$ related to time can be written as

$$\dot{V}\left(x\right)={\tilde{X}}^{T}A\dot{\tilde{X}}.$$

(15)

Based on Equation (13), $\dot{\tilde{X}}$ can be expressed as

$$\dot{\tilde{X}}=-A^{-1}\left[B+R_Z\right]\tilde{X}.$$

(16)

The combination of Equations (15) and (16) yields

$$\dot{V}\left(x\right)=-\left[{\tilde{X}}^T{R}_Z\tilde{X}+{\tilde{X}}^{T}B\tilde{X}\right].$$

(17)

Since $B$ is an antisymmetric matrix, ${\tilde{X}}^{T}B\tilde{X}=0$. If the matrix $R_Z$ is positive, then

$$\dot{V}\left(x\right)=-{\tilde{X}}^T{R}_Z\tilde{X}<0.$$

(18)

Therefore, the system satisfies Lyapunov stability and asymptotic stability.

3.3. Virtual DC Machine Control Strategy

As mentioned above, the inherent limitation of PBC is that its propensity to generate steady-state errors in response to disturbances in the load or power supply. To address this shortcoming, VDCM control is incorporated. This control strategy features an integrator that tracks the desired current value in real time, thereby eliminating the steady-state error. Furthermore, VDCM enhances the inertia and damping of the DC system by simulating the external characteristics of the DC motor, as shown in Figure 3. The armature current $I_{\mathrm{a}}$ is served as the reference value $I_{\mathit{ref}}$ on output side of the converter. The output voltage of the DC machine $U_{\mathrm{o}}$ simulates the output voltage $U_C$ of the boost converter.

The VCDM can be mathematically modeled using the armature equation and the mechanical rotation equation of the DC motor [24], in which the mechanical rotation equation is expressed as

$$J{\displaystyle \frac{d\omega }{dt}}=T_m-T_e-D\left(\omega -{\omega }_N\right),$$

(19)

$$T_e={\displaystyle \frac{P_e}{\omega }},$$

(20)

where $J$ represents the moment of inertia, $D$ denotes the damping coefficient, $T_m$ is the mechanical torque, $T_e$ is the electromagnetic torque, $\omega$ signifies angular velocity, ${\omega }_N$ is the rated angular velocity, and $P_e$ represents the electromagnetic power.

The armature equation of the DC machine is written as follows:

$$U_{\mathrm{o}}=E-R_a{I}_a,$$

(21)

$$E=C_T\varphi \omega ,$$

(22)

where $E$ is the armature induced electromotive force, $R_a$ denotes the armature equivalent resistance, $C_T$ is the torque coefficient, and $\varphi$ is flux per pole.

The VDCM control block diagram is illustrated in Figure 2b, which is consistent with the aforementioned VDCM control formula. The input and output of the VDCM control are illustrated in Figure 2a. In conventional VDCM control, a voltage PI controller generates the deviation power $\Delta P$, which is then added $P_{\mathit{ref}}$ to obtain the mechanical power $P_m$. After passing through the VDCM block, the adjusted reference current is output, thereby enhancing the system’s damping and inertia. Similarly, the reference current $I_L$, obtained from the voltage error via the PBC voltage loop, should undergo feedback adjustment through the VDCM. The resulting control signal $I_{L\_\mathit{ref}}$ is fed into the subsequent PBC current loop to eliminate steady-state voltage errors and improve system stability. After being controlled by VDCM, the reference current on output side of the converter $I_{\mathit{ref}}$ is converted into the reference inductor current $I_{L\_ref}$ through the ratio $U_C/U_{in}$.

4. Results

4.1. Simulation Results

To validate the stability of the proposed method and its effectiveness in suppressing steady-state errors compared to traditional PBC under load variations, we apply MATLAB/Simulink R2022a software testing under CPL, reference voltage and input voltage variations. The simulation model is shown in Figure 4, employing the ode1 (Euler) solver. The reference voltage is set to 27 V. The parameters of boost converter and CPL, as well as the control parameters, are listed in

Table 1. Main parameters of simulation and experiment.

Boost Converter Parameters	Value
Input voltage $u_{in}$/$\mathrm{V}$	12
Input filter inductor $L$/$\mathrm{m}\mathrm{H}$	2
Capacitance $C/\mathsf{\mu }\mathrm{F}$	1000
Switching frequency $f/\mathrm{k}\mathrm{H}\mathrm{z}$	10
Control Parameters of Boost Converter
PBC gains $R_{1d}/\mathsf{\Omega }$	${10}^6$
PBC gains $R_{2d}/\mathsf{\Omega }$	$0.05$
Moment of inertia $J/\mathrm{k}\mathrm{g}·{\mathrm{m}}^2$	$0.5$
Damping coefficient $D$	$0.3$
Rated angular velocity ${\omega }_N/\mathrm{r}\mathrm{a}\mathrm{d}·{\mathrm{s}}^{-1}$	$2\pi ·50$
Torque coefficient $C_T$	$18.48$
Flux per pole $\varphi /\mathrm{W}\mathrm{b}$	$0.0698$
Armature equivalent resistance $R_a/\mathsf{\Omega }$	$1$
Voltage loop proportional gain $K_{vp}$	0.6
Voltage loop integral gain $K_{vi}$	20
Current loop proportional gain $K_{ip}$	1
Current loop integral gain $K_{ii}$	25
CPL Parameters
Load voltage $v_{ou{t}_{buck}}/\mathrm{V}$	$15$
Load power $P_{const}/\mathrm{W}$	$4.5/27$
Filter inductor $L_f/\mathrm{m}\mathrm{H}$	$2$
Output capacitance $C_f/\mathsf{\mu }\mathrm{F}$	$100$
Control Parameters of CPL
proportional gain $K_p$	3
Integral gain $K_i$	20

. The subsequent text presents and analyzes the simulation results.

4.1.1. Constant Power Load Variations Test

The simulation results of the CPL variations test are presented in Figure 5. In this test, the output voltage is regulated at 27 V, with the initial CPL of 4.5 W. At 1 s, the CPL suddenly increases to 27 W and then returns to 4.5 W at 2 s. The comparative analysis with dual-loop PI control reveals that the proposed method reaches the reference value approximately 90 ms faster, accompanied by a smaller bus voltage overshoot of around 3.1 V. Moreover, the proposed method can eliminate steady-state errors of capacitor voltage caused by load changes as shown in Figure 5a, which is a limitation of PBC. Figure 5b shows the waveform of the inductor current. It can be observed that the current ripple under the traditional PI control is larger than that under the other two control methods. Therefore, it is evident that the proposed method has stronger control stability and faster return to the reference value. To investigate the response of various parameters to the capacitor voltage, simulation results are presented with a single parameter modified while the others are held constant in Figure 6. The simulation reveals that $R_{2d}$ in Figure 6a has a significant impact on the tracking of the reference point, and a smaller value is selected to achieve a better control effect. The effect of different values of $R_{1d}$ in Figure 6b has lower sensitivity of the system, but closer inspection reveals that larger values result in smoother performance. Additionally, $J$ in Figure 6c has a substantial influence on the dynamic response of the capacitor voltage, and choosing the appropriate $D$ in Figure 6d is crucial for controlling the voltage to converge to the reference value.

4.1.2. Reference Voltage Variation Test

The simulation results of the reference voltage variation test are shown in Figure 7. At 1 s, the reference voltage is stepped down from 27 V to 22 V. A comparative analysis reveals that the output voltage of both PBC and PBC + VDCM exhibits no overshoot, whereas classical PI control exhibits an overshoot of approximately 0.5 V, as shown in Figure 7a. Furthermore, the settling time of the proposed method is 53 ms less than that of PI control and 12 ms less than that of PBC. The inductor current variation amplitude of PBC + VDCM is smaller than that of the other two control methods, and its ripple is also smaller than that of classical PI control as shown in Figure 7b. In summary, the proposed method demonstrates excellent dynamic response and superior performance in terms of voltage and current regulation.

4.1.3. Input Voltage Variation Test

The simulation results of the input voltage variation experiment are shown in Figure 8. In this test, the system carries a CPL of 4.5 W, and the input voltage is ramped up from 12 V to 18 V in 1 s. The simulation results demonstrate that the PBC + VDCM strategy exhibits superior performance. Similarly to the PI controller, it eliminates steady-state error, while its overshoot is approximately 0.5 V smaller and its settling time is roughly 109 ms shorter, as shown in Figure 8a. In contrast, the PBC fails to eliminate steady-state error. Furthermore, the inductor current ripple is minimized, and the current reaches its stabilization point quickly as illustrated in Figure 8b, highlighting the excellent dynamic response of the proposed method.

4.1.4. Phase Plot

Given that the proposed control method is nonlinear and the boost converter is a second-order system, a global view of the proposed controller’s behavior can be obtained by phase plot. Figure 9 shows the phase plot of the Boost converter with a 27 W CPL under the proposed control method. The state variables, namely the inductor current and capacitor voltage, are plotted as time-varying curves. Different colors are used to represent different initial states of the state variables. It can be seen that the phase trajectories fitted by the two state variables under different initial conditions remain within the region of attraction and converge to the equilibrium point, thereby proving the stability of the system.

4.2. Experimental Results

To further validate the effectiveness of the proposed method, a boost converter with CPL experimental platform is constructed, based on the dSPACE1104, as shown in Figure 10. Multiple DC power supplies are employed to drive the circuit board, supply power to the converter’s input side, and facilitate experiments on power supply disturbances. The current and voltage of the converter are sampled into dSPACE via data lines. dSPACE runs the algorithm and outputs the control variables to the PWM generator, which generates PWM signals to control the converter’s switching devices. A computer is utilized to initialize dSPACE and adjust the control parameters. Voltage and current waveforms are monitored using an oscilloscope. The output voltage of the boost converter is controlled at 27 V. The converter experiment utilized the same parameter set as the simulation, as listed in Table 1.

4.2.1. Experimental Test of CPL Variations

Figure 11 illustrates the comparative experimental results of the boost converter under different control strategies for the constant power load, which transitions from 4.5 W to 27 W and back to 4.5 W. As depicted in Figure 11a, under classical PI control, the output voltage waveform ($u_C$) exhibits a noticeable drop and a relatively long transient process following a sudden load change before returning to the reference value. Figure 11b demonstrates that under PBC, the voltage drops considerably after the load increase, and the steady-state error remains uncompensated. In contrast, Figure 11c shows that the voltage fluctuation is essentially negligible under the proposed control strategy. The dynamic and steady-state performance of the boost converter is significantly enhanced by the proposed control, attributable to its PBC and VDCM characteristics.

4.2.2. Experimental Test of Reference Voltage Variation

Figure 12 presents a comparison of experimental results illustrating the variation in reference voltage from 27 V to 22 V under different control strategies. As depicted in Figure 12a, under classical PI control, the voltage waveforms exhibit overshooting and require an extended period for stabilization. Figure 12b,c display the experimental results for the PBC and the proposed control method, respectively. The longer control time observed in the proposed control method, compared to the PBC, can be attributed to the VDCM increasing the system’s damping and inertia, which enhances stability at the constant reference voltage value.

4.2.3. Experimental Test of Input Voltage Variation

Figure 13 presents the experimental comparison results of the input voltage abruptly changing from 12 V to 18 V under different control strategies. The data clearly indicates that the proposed method, as depicted in Figure 13c, not only exhibits a superior response time compared to the classical PI control shown in Figure 13a, but also effectively mitigates the steady-state error associated with the PBC illustrated in Figure 13b following the input voltage change.

5. Conclusions

This paper proposes a control method that combines PBC and VDCM to maintain the stability of the DC bus voltage under disturbances, such as large fluctuations in CPL. The PBC can show the good robustness and low overshoot under disturbance condition. The VDCM is applied to improve system’s inertia and damping, while compensating for the steady-state error caused by PBC during load or input voltage changes. Both the simulation and experimental results demonstrate that the PBC + VDCM control method exhibits strong robustness and dynamic characteristics, characterized by small overshoots in bus voltage and inductor current, and rapid convergence to the reference value. The proposed control method has application potential in components of DC microgrids such as photovoltaics, electric vehicle charging piles, and energy storage systems.

However, the research in this paper still has certain limitations, and future research work can focus on the following two aspects:

(1)

Verifying the applicability of the proposed control method in other types of DC/DC converters, such as topological structures like buck converters and bidirectional buck/boost converters;

(2)

DC microgrids incorporating DC devices such as photovoltaics, energy storage systems, and electric vehicle charging piles are typical multi-converter systems. Therefore, it is necessary to further explore the adaptability and scalability of the proposed method in such multi-converter systems, encompassing aspects such as coordinated control of multiple converters and their interactive influences.