**1. Introduction**

With the research boom of renewable DC power sources, research on DC microgrids is gradually expanding. Because of the obvious advantages of DC microgrid technology, its application in the field of vehicle power system is becoming more and more extensive. In the DC microgrid of electric vehicles, converters are often used to realize voltage conversion between the DC bus and load. For DC power systems, maintaining the stability of the DC bus voltage is the foundation for ensuring the stable operation of vehicles. The application of a large number of power conversion devices subject to strict closed-loop control leads to an increase in the proportion of constant power loads in the system, which greatly reduces the stability of the system when the power of such loads fluctuates [1]. The research shows that the constant power load always exhibits negative-impedance characteristics and brings an instability effect to the system. The negative-impedance characteristics of a constant power load can cause significant voltage oscillations in the system when there are significant changes in the CPL (constant power load), thereby reducing power quality and posing safety hazards [2,3].

Therefore, how to keep the DC bus voltage of electric vehicles quickly adjusted and stable is the key problem with the DC microgrid. The DC microgrid of electric

**Citation:** Jia, Y.; Wang, D.; Sun, G.; Ni, Y.; Song, K.; Li, Y. High-Order Sliding-Mode Control Strategy for Improving Robustness of Three-Phase Interleaved Bidirectional Converter. *Sustainability* **2023**, *15*, 9720. https://doi.org/10.3390/ su15129720

Academic Editors: Prince Winston David and Praveen Kumar B

Received: 27 May 2023 Revised: 16 June 2023 Accepted: 17 June 2023 Published: 18 June 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

vehicles is a system composed of many parts, its instability phenomena are various, and the mechanism of the system instability is also complicated. Since most distributed power generations, energy storage devices, and loads in the DC microgrid need to be connected with the DC bus through converters, and these power electronic converter devices have nonlinear characteristics, the power electronic system composed by them also has nonlinear characteristics [4].

In practical applications, the system model will be affected by various disturbances, such as the uncertainty of inductance and magnetic characteristics, the instability of input voltage, the disturbance of load, etc. At present, many advanced nonlinear control methods have been applied to the converter, such as active disturbance rejection control, adaptive control, sliding-mode control, etc. The sliding-mode control has the advantages of simple operation, high precision, good stability, and robustness in practical applications. Slidingmode control technology and a DC-DC converter work well together because they are both based on a variable-switching strategy [5]. For the SMC (slide-mode control) method, Reference [6] ensured large signal stability and a fast dynamic response. In order to further improve the transient dynamics of the system, a simple finite-time convergence SMC method is adopted in the converter system [7,8]. However, it is difficult to maintain high accuracy in the event of external disturbances or changes in internal components. Therefore, modern advanced control methods are studied, such as sliding-mode control, adaptive control, optimal control, predictive control, etc. The above control method basically solves the problem of output-voltage instability. However, these methods cannot quickly track and suppress interference. For closed-loop systems, it is difficult to achieve a good voltage output performance under interference.

Considering the perturbations and uncertainties existing in practical applications, it is difficult to measure them with actual sensors, but the designed observer can achieve accurate estimation and compensation of perturbations. In Reference [9], an expanded state observer was designed to realize the estimation of load changes, and a sliding-mode controller was designed to improve the anti-interference performance of the system. Reference [10] designed the unknown input observer, which has low sensitivity to noise and only needs to adjust one parameter, which is easy to implement in the actual system. Reference [11] proposed a sliding-mode control method based on the disturbance observer, which can converge to the neighborhood near the reference voltage in a finite time. However, the above observer can only accurately estimate the slow time-varying perturbations [12]. The perturbations in the actual system are more complex, and there may be higher-order polynomial perturbations. Reference [13] proposed a passive controller based on interconnection and damping allocation, which is robust and easy to implement. However, it can lead to a slow transient response.

In Reference [14], the GPI (generalized proportional integral) observer was designed to achieve an accurate estimation of slow and fast time-varying disturbances, and it was combined with the backstepping method to deal with the unmatched load disturbance. The basic idea of the backstepping method is to decompose a complex system into multiple subsystems, which are recursive from backward to forward through the design of the virtual control law. Interference factors are designed into each subsystem, but in the design process, there may be a high-order derivative of the virtual control function in the controller, which is more complicated to calculate.

In these controllers, in order to maintain the stability of the output voltage, the switch gain is required to be greater than the upper limit of the disturbance. However, in some low-order sliding-mode control laws, an excessive switching gain can lead to significant voltage fluctuations, resulting in unstable output voltage in the practical-implementation literature [15,16]. The interference estimation and compensation technology provide a feasible method to alleviate the chattering phenomenon in Reference [17]. In Reference [18], the nonlinear disturbance observer was used to estimate the uncertain power change, which provides a new way of thinking for dealing with CPL problem. It is difficult for the observer to obtain satisfactory estimation accuracy when dealing with a fast time-varying CPL. There

is also no consideration of supply voltage fluctuations. Performance degradation is caused by these factors. Reference [19] proposes a distributed current-sharing control method. The outer loop is the voltage droop control with the purpose of embedding virtual impedance, while the inner loop is the PI control, which can improve the dynamic and steady-state performance of the system. References [20,21] compensated for the virtual impedance coefficient by actively detecting line impedance to achieve current equalization, and they improved the voltage drop through voltage observer feedforward compensation control. Reference [22] proposed an algorithm for compensating for a current imbalance caused by resistance mismatch. By perturbing the duty cycle of one phase and measuring the deviation of other phase duty cycles, the degree of parameter mismatch is estimated, and current balance is achieved through appropriate compensation coefficients.

There are many effective error estimation methods, including the unknown input observer (UIO) [23], disturbance observer (DOB) [24], and extended state observer (ESO) [25]. The disturbance-observer-based control (DOBC) has been proven to effectively reduce unknown external disturbances and system uncertainty. Due to the fact that these observer techniques are model based, a large amount of information needs to be considered when establishing interference observers. However, both DOB and ESO can only estimate constant and stage constant perturbations and cannot estimate polynomial perturbations.

Due to the uncertainty of converter parameters and the influence of concentrated disturbances, there is an increasing amount of research on the precise estimation of disturbances, using the estimated values as feedforward compensation to improve the antiinterference performance of control. Disturbance-observer-based control (DOBC) considers the parameter changes, load changes, and input voltage fluctuations of the filter as external disturbances to the system. The disturbance observer is used to nominal the controlled object and observe these disturbances through the disturbance observer. The observed values are then fed forward to the output of the voltage control loop to counteract the impact of disturbances on the system. Another disturbance estimation technique is the extended state observer (ESO), which treats both internal uncertainties and external disturbances as total disturbances, treats the total disturbance as a new system state, estimates the system state and disturbances through internal calculations, and then designs the controller by combining the estimated values with the improved sliding-mode control method, so that the output voltage of the converter can track the reference signal quickly [26].

In order to suppress load resistance interference and input voltage changes, an SMC method based on GPI observer for three-phase interleaved parallel DC/DC converter is proposed. Estimate the disturbance and state of the system by designing a GPI observer [27]. Then, based on the estimated values obtained from the GPI observer, a composite controller is constructed using SMC technology, which enables the output voltage to asymptotically track the reference voltage [28]. The simulation results show that, compared with the sliding-mode control method based on NDO, this control method can track the reference value faster and improve the steady-state performance of the system. Meanwhile, this method reduces costs in practical systems.

The GPI observer sliding-mode control method based on the second-order slidingmode algorithm proposed in this article can achieve small switching gain without sacrificing interference suppression by combining interference estimation, ensuring the stability of the output voltage. Compared with other observer control methods, it has strong robustness against disturbances.

The main work content of this paper is divided into three parts:


• Through the simulation study of MATLAB/Simulink in the interference of input voltage and CPL power and the evaluation of the proposed composite controller, the correctness of the proposed controller is proved.
