Composite Non-Linear Control of Hybrid Energy-Storage System in Electric Vehicle

Abstract

The underlying circuit control is a key problem of the hybrid energy-storage system (HESS) in electric vehicles (EV). In this paper, a composite non-linear control strategy (CNC) is proposed for the accurate tracking current/voltage of the fully-active HESS by combining the exact feedback linearization method and the sliding mode variable structure control technology. Firstly, by analyzing the circuit characteristics of HESS, the affine non-linear model of fully-active HESS is derived. Then, a rule-based energy management strategy (EMS) is designed to generate the reference current value. Finally, the HESS is linearized by the exact feedback linearization method, and the proposed CNC strategy is developed combined with sliding mode variable structure control technology to ensure fast response, high performance, and robustness. At the same time, the stability proof based on the Lyapunov method is given. Moreover, the performance of the CNC strategy is thoroughly investigated and compared with simulation studies with the traditional PI control and a modified sliding mode control, and its effectiveness under different driving conditions is fully verified.

1. Introduction

With the rapid growth of global vehicle ownership, energy shortages and environmental pollution have become increasingly serious, which has a serious negative impact on the sustainable and healthy development of the vehicle industry. As one of the ideal models of new energy vehicles, electric vehicles (EV) have attracted extensive attention and attention because of their zero-emission and low-pollution characteristics. Nevertheless, the battery energy-storage system of an EV has the disadvantages of low power density and short cycle life, which will lead to insufficient power during vehicle driving and increase the maintenance cost of vehicle energy sources [1]. As a new energy-storage element, the supercapacitor has a high power density and long cycle life compared with the battery. It can meet the peak power demand of vehicle driving well, avoid high current discharge of the battery, and make use of its fast charging and discharging speed to efficiently recover vehicle regenerative braking energy and improve energy utilization. The HESS consisting of the battery, the supercapacitor, and the DC/DC converter can meet the driving needs of vehicles well. It is an important development direction of EV energy sources in the future [2].

At present, the research on HESS can be divided into two categories: one is to design top-level EMS to distribute the required power to each energy-storage element reasonably and efficiently to generate reference, such as rule-based [3], fuzzy control [4], dynamic programming [5], and neural networks [6], etc. The other is to design the underlying control strategy by considering the interaction between different components so that the EMS can be implemented as expected. Previously, most researchers were committed to designing EMS to obtain high economy. However, due to the combination of multiple components, the HESS has multi-variable, non-linear, and strong coupling characteristics. How to realize the underlying control under the condition of multi-component coupling and strong non-linearity and ensure the safe and stable operation is also a key technical problem of the HESS. In [7], an advanced topology was proposed for the HESS, which can achieve a smooth transition and maintain stability when the load fluctuates. Nevertheless, its practicability and effectiveness need to be further verified. In [8], a fractional-order proportional-integral-derivative (PID) controller was proposed for the coordinated control of the fuel cell and the supercapacitor in hybrid renewable energy systems. Nevertheless, as a linear controller, PID is very sensitive to parameter changes and is not suitable for the control method of complex non-linear systems. In [9], combined with the port-controlled Hamiltonian (PCH) model and L₂ gain control method, an L₂ gain adaptive robust control (L₂-ARC) strategy was developed to stabilize the output of the HESS. Nevertheless, the control process is more complicated.

Recently, the exact feedback linearization method and the sliding mode variable structure control technology have been widely studied. As a non-linear control method, exact feedback linearization can globally linearize the non-linear system through coordinate transformation and input/output linearization, to further analyze and control [10]. An exact feedback linearization control strategy for voltage source converters (VSCs) in smart distribution systems and microgrids was proposed, which was studied by using the equivalent linear model with non-linear characteristics, and the effective control effect was obtained [11]. Nevertheless, the effectiveness of exact feedback linearization depends on the accurate description of the system’s non-linearity, which is highly dependent on the accuracy of the mathematical model. As a non-linear robust control, sliding mode variable structure control (SMC) technology has been widely applied with satisfactory control performance, such as a robotic system [12], vehicle control [13], DC motor [14], power electronics [15], and power system [16], etc. Based on the fifth-order averaged model, the classical sliding mode current controller and the Lyapunov-based voltage controller were designed to realize the underlying circuit control for the fully-active HESS [17]. For the HESS composed of the fuel cell/battery/supercapacitor, a terminal sliding mode control strategy with adaptive law was presented to realize the stable tracking of the current and maintain voltage stability [18]. A simple multimode HESS was designed for EVs, and an adaptive sliding mode control strategy with hysteresis control was presented combined with the practical application, which effectively improves the system efficiency and ensures the safety of the battery [19]. Nevertheless, the inherent chattering of sliding mode control will harm the control effect.

In this paper, the exact feedback linearization method and the sliding mode variable structure control technology were combined, and a composite non-linear control strategy was proposed for the underlying circuit control of the fully-active HESS. After exact feedback linearization, the HESS establishes the linear switching function, which is helpful to improve the dynamic quality of the sliding mode variable structure. The sliding mode variable structure is insensitive to system uncertainty and disturbance, which eliminates the strong dependence of the exact feedback linearization method on the accuracy of the mathematical model. The main contributions of this work are as follows:

(1)

Through the exact feedback linearization and non-linear coordinate transformation, the original non-linear system is transformed into a linear system, which is convenient for controller design;

(2)

The exponential reaching law and integral sliding mode surface are combined to improve the convergence speed of the system and effectively weaken the chattering of sliding mode variable structure;

(3)

The control performance under different conditions was tested and compared with the traditional PI control and a modified sliding mode control. The results show that the proposed CNC strategy has faster corresponding speed, smaller overshoot, and better steady-state chattering, which proves the effectiveness of the proposed CNC strategy.

The rest of the paper is organized as follows. Section 2 introduces the model of the fully-active HESS. Section 3 gives the proposed CNC control strategy in detail. Section 4 shows the simulation results and analysis on under different strategies. Finally, the conclusion is drawn in Section 5.

2. The Model of Fully-Active Hybrid Energy-Storage System (HESS)

2.1. The Circuit Model

Currently, there are three types of HESS topologies: passive, semi-active and fully active. Considering the control effect, the fully-active HESS is selected as the research object, with its circuit model shown in Figure 1. It consists of a battery pack, a supercapacitor pack, two standard bidirectional DC/DC converters, a DC/AC inverter, and a driving motor. The standard bidirectional DC/DC converter consists of two Insulated Gate Bipolar Transistors (IGBTs), an inductor, and a common capacitor. The switches Q₁, Q₂, Q₃, and Q₄ adopt complementary PWM control method. When Q₁ is on (off), Q₂ is off (on), Q₃ and Q₄ are the same. Compared with the independent control, the complementary control does not need the transition switching of buck and boost circuit, which improves the work efficiency and system response speed. To simplify the research process, a load resistance with variable current i_load is used to substitute for the driving system composed of DC/AC inverter and driving motor, and the influence of series resistance of the battery and the supercapacitor is ignored. Based on Kirchhoff’s law, the state Equations of the HESS circuit model are as follows:

$${\dot{i}}_1=\frac{V_1}{L_1}-i_1\frac{R_{\mathrm{L}1}+R_{\mathrm{on}2}}{L_1}+u_1{i}_1\frac{R_{\mathrm{on}2}-R_{\mathrm{on}1}}{L_1}+(u_1-1)\frac{V_{\mathrm{dc}}}{L_1}$$

(1)

$${\dot{i}}_2=\frac{V_2}{L_2}-i_2\frac{R_{\mathrm{L}2}+R_{\mathrm{on}4}}{L_1}+u_2{i}_2\frac{R_{\mathrm{on}4}-R_{\mathrm{on}3}}{L_2}+(u_2-1)\frac{V_{\mathrm{dc}}}{L_2}$$

(2)

$${\dot{V}}_{\mathrm{dc}}=(1-u_1)\frac{i_1}{C_{\mathrm{dc}}}+(1-u_2)\frac{i_2}{C_{\mathrm{dc}}}-\frac{i_{\mathrm{load}}}{C_{\mathrm{dc}}}$$

(3)

where L₁ and L₂ are the inductors of the battery side and the supercapacitor side, i₁ and i₂ are the currents flowing through them, and R_L1 and R_L2 are the series resistances of L₁ and L₂; C₁ and C₂ are the filter capacitors of the battery side and the supercapacitor side, C_dc is the DC bus capacitor, V₁, V₂, and V_dc are the voltages corresponding to capacitors C₁, C₂, and C_dc; u₁ and u₂ are the duty cycle of switch Q₁ and Q₃; R_on1, R_on2, R_on3, and R_on4 are the on-resistances of the complementary switches Q₁, Q₂, Q₃, and Q₄, respectively. In this study, it can be assumed that they are equal, then the above state equations can be simplified as:

$${\dot{i}}_1=\frac{V_1}{L_1}-i_1\frac{R_{\mathrm{L}1}+R_{\mathrm{on}2}}{L_1}+(u_1-1)\frac{V_{\mathrm{dc}}}{L_1}$$

(4)

$${\dot{i}}_2=\frac{V_2}{L_2}-i_2\frac{R_{\mathrm{L}2}+R_{\mathrm{on}4}}{L_1}+(u_2-1)\frac{V_{\mathrm{dc}}}{L_2}$$

(5)

$${\dot{V}}_{\mathrm{dc}}=(1-u_1)\frac{i_1}{C_{\mathrm{dc}}}+(1-u_2)\frac{i_2}{C_{\mathrm{dc}}}-\frac{i_{\mathrm{load}}}{C_{\mathrm{dc}}}$$

(6)

2.2. The Affine Non-Linear Model

For the multi-input and multi-output (MIMO) affine non-linear systems, the system model is as follows [20]:

$$\{\begin{array}{l}\dot{x}=f(x)+g(x)u\hfill \\ y=h(x)\hfill \end{array}$$

(7)

where x is the n × 1 order state vector, u is the m×1 order control variable, y is the m × 1 order output vector, f(x) is the smooth vector field, g(x) is the n × m order matrix.

The main control objective of the HESS is that the current of the battery and the supercapacitor can accurately track the reference value, and the bus voltage can be stable at the target voltage. Therefore, to realize the current tracking control, it can be assumed that the bus voltage is stable. According to Equations (4)–(6), selecting x = [x₁ x₂]^T = [i₁ i₂]^Tas state variables, y = [y₁ y₂]^T = [h₁(x) h₂(x)]^T = [i₁ i₂]^T as output variables, u = [u₁ u₂]^T as control input variables, the affine non-linear model of the HESS can be obtained as follows:

$$\{\begin{array}{l}\dot{x}=f(x)+g_1(x)u_1+g_2(x)u_2\hfill \\ \begin{array}{l}{y}_1=h_1(x)\\ y_2=h_2(x)\end{array}\hfill \end{array}$$

(8)

where

$$f(x)=\left[\begin{array}{c}{f}_1\\ f_2\end{array}\right]=\left[\begin{array}{c}\frac{V_1}{L_1}-\frac{R_{L1}+R_{\mathrm{on}2}}{L_1}{x}_1-\frac{V_{\mathrm{dc}}}{L_1}\\ \frac{V_2}{L_2}-\frac{R_{L2}+R_{\mathrm{on}4}}{L_2}{x}_2-\frac{V_{\mathrm{dc}}}{L_2}\end{array}\right],g_1(x)=\left[\begin{array}{c}{g}_{11}\\ g_{12}\end{array}\right]=\left[\begin{array}{c}\frac{V_{\mathrm{dc}}}{L_1}\\ 0\end{array}\right],g_2(x)=\left[\begin{array}{c}{g}_{21}\\ g_{22}\end{array}\right]=\left[\begin{array}{c}0\\ \frac{V_{\mathrm{dc}}}{L_2}\end{array}\right].$$

3. The Composite Non-Linear Strategy

This part gives the design scheme of the CNC strategy. Firstly, a rule-based EMS is introduced to generate reference values; secondly, the exact linearization conditions of the HESS are derived; then, the linearized control model of the HESS is obtained by non-linear coordinate transformation; finally, combined with exponential reaching law and integral sliding mode surface, the voltage/current tracking controller is designed, and the stability proof based on Lyapunov theory is given.

3.1. The Rule-Based Energy Management Strategy (EMS)

Before the control strategy design, an energy management strategy (EMS) needs to be designed to distribute the required power, to obtain the power between the battery and the supercapacitor, while generating the current reference value of the battery.

A rule-based EMS is used to manage the energy of HESS in this paper. The specific design is shown in Figure 2a, where P_req is the required power of the load, P_bat is the power of the battery, P_sc is the power of the supercapacitor, P_ch is the charging power of the supercapacitor, P_min is the threshold power, V_sc is the voltage of the supercapacitor, V_scmin is 60% of the rated voltage of the supercapacitor. When P_req < 0, the vehicle is in braking mode, and the supercapacitor is used to receive the power of the load. When P_req > 0, the vehicle is in driving mode, which can be divided into two situations:

(a)

When P_req > P_min: if V_sc > V_scmin, the battery bears the threshold power P_min, and the supercapacitor bears the remained power; if V_sc < V_scmin, the battery bears the required power P_req and to charge the supercapacitor with the charging power P_ch.

(b)

When P_req < P_min: if V_sc > V_scmin, the battery bears the required power P_req; if V_sc < V_scmin, the battery bears the required power P_req and to charge the supercapacitor with the charging power P_ch.

Figure 2b shows the artificial load power and the distributed results of the battery and the supercapacitor. It can be seen that the output power of the supercapacitor fluctuates greatly, while the output power of the battery is smooth, which shows the EMS is helpful to protect the battery.

3.2. The Exact Linearization Conditions

The exact linearization conditions of the affine non-linear model are given below [21]. If the following two conditions are true for the affine non-linear model shown in Equation (8):

(1)

For all x⁰ near x, the rank of matrix [g₁(x) g₂(x) ad_fg₁(x) ad_fg₂(x)] is constant and equal to n.

(2)

The vector field set D = {g₁(x) g₂(x) ad_fg₁(x) ad_fg₂(x)} is involutive at x = x⁰.

Then, there must be output functions [h₁(x) h₂(x)]^T such that the correlation degree r of the system at x = x⁰ is equal to the order n of the system. When the output function satisfies the total relation degree r = n, the linear system expression can be obtained by coordinate transformation directly to realize decoupling. If the output function does not satisfy the total relation degree r = n, it is needed to find another output function ω(x) to meet the condition, and then coordinate transformation is carried out.

Firstly, the exact linearization conditions of the system are verified:

$$\begin{array}{l}a{d}_f{g}_1(x)=\frac{\partial g_1(x)}{\partial x}f(x)-\frac{\partial f(x)}{\partial x}{g}_1(x)=\\ \left[\begin{array}{cc}\frac{\partial g_{11}}{\partial x_1}& \frac{\partial g_{11}}{\partial x_2}\\ \frac{\partial g_{12}}{\partial x_1}& \frac{\partial g_{12}}{\partial x_2}\end{array}\right]\left[\begin{array}{c}{f}_1\\ f_2\end{array}\right]-\left[\begin{array}{cc}\frac{\partial f_1}{\partial x_1}& \frac{\partial f_1}{\partial x_2}\\ \frac{\partial f_2}{\partial x_1}& \frac{\partial f_2}{\partial x_2}\end{array}\right]\left[\begin{array}{c}{g}_{11}\\ g_{12}\end{array}\right]=\left[\begin{array}{c}\frac{V_{\mathrm{dc}}(R_{L1}+R_{\mathrm{on}2})}{L_1^2}\\ 0\end{array}\right]\end{array}$$

(9)

$$\begin{array}{l}a{d}_f{g}_2(x)=\frac{\partial g_2(x)}{\partial x}f(x)-\frac{\partial f(x)}{\partial x}{g}_2(x)=\\ \left[\begin{array}{cc}\frac{\partial g_{21}}{\partial x_1}& \frac{\partial g_{21}}{\partial x_2}\\ \frac{\partial g_{22}}{\partial x_1}& \frac{\partial g_{22}}{\partial x_2}\end{array}\right]\left[\begin{array}{c}{f}_1\\ f_2\end{array}\right]-\left[\begin{array}{cc}\frac{\partial f_1}{\partial x_1}& \frac{\partial f_1}{\partial x_2}\\ \frac{\partial f_2}{\partial x_1}& \frac{\partial f_2}{\partial x_2}\end{array}\right]\left[\begin{array}{c}{g}_{21}\\ g_{22}\end{array}\right]=\left[\begin{array}{c}0\\ \frac{V_{\mathrm{dc}}(R_{L2}+R_{\mathrm{on}4})}{L_2^2}\end{array}\right]\end{array}$$

(10)

It can be seen that the rank of matrix [g₁(x) g₂(x) ad_fg₁(x) ad_fg₂(x)] is 2, which is equal to the system order n. Therefore, condition (1) is satisfied. Obviously, for condition (2), when the system order n = 2, the vector field set D is involution. It can be concluded that there must be a set of output functions such that the correlation degree r of the system is equal to the order n of the system.

Next, verify whether the selected output function [h₁(x) h₂(x)]^T satisfies r = n. The correlation degree of the system can be calculated by the Lie derivative:

$$\{\begin{array}{c}{L}_f{h}_1(x)=\frac{\partial h_1(x)}{\partial x}f(x)=\left[\frac{V_1}{L_1}-\frac{R_1+R_{\mathrm{on}2}}{L_1}{x}_1-\frac{V_{\mathrm{dc}}}{L_1}\right]\\ L_f{h}_2(x)=\frac{\partial h_2(x)}{\partial x}f(x)=\left[\frac{V_2}{L_2}-\frac{R_2+R_{\mathrm{on}4}}{L_2}{x}_2-\frac{V_{\mathrm{dc}}}{L_2}\right]\end{array}$$

(11)

$$\{\begin{array}{cc}{L}_{g_1}{h}_1(x)=\frac{\partial h_1(x)}{\partial x}{g}_1(x)=\left[\frac{V_{\mathrm{dc}}}{L_1}\right]& L_{g_2}{h}_1(x)=\frac{\partial h_1(x)}{\partial x}{g}_2(x)=\left[0\right]\\ L_{g_1}{h}_2(x)=\frac{\partial h_2(x)}{\partial x}{g}_1(x)=\left[0\right]& L_{g_2}{h}_2(x)=\frac{\partial h_2(x)}{\partial x}{g}_2(x)=\left[\frac{V_{\mathrm{dc}}}{L_2}\right]\end{array}$$

(12)

It can be seen from Equations (11) and (12) that under the given output, the total correlation degree r = r₁ + r₂ = 1 + 1 = 2 = n, where r₁ and r₂ are the correlation degree of output functions h₁(x) and h₂(x), respectively. The system is non-linear for the state variable x but linear for the control output variable u. Therefore, the non-linear coordinate transformation can be carried out directly to realize the linearization of the non-linear system.

3.3. Non-Linear Coordinate Transformation

For MIMO (multiple-input multiple-output) systems, one of the input-output feedback linearization methods is to derive the output y of the system until the control input variable u appears. Therefore, the derivation of [y₁ y₂]^T can be obtained as follows:

$$\left[\begin{array}{c}{\dot{y}}_1\\ {\dot{y}}_2\end{array}\right]=A(x)+E(x)\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]$$

(13)

where,

$$A(x)=\left[\begin{array}{c}\frac{V_1}{L_1}-\frac{R_{L1}+R_{\mathrm{on}2}}{L_1}{x}_1-\frac{V_{\mathrm{dc}}}{L_1}\\ \frac{V_2}{L_2}-\frac{R_{L2}+R_{\mathrm{on}4}}{L_2}{x}_2-\frac{V_{\mathrm{dc}}}{L_2}\end{array}\right],E(x)=\left[\begin{array}{cc}\frac{V_{\mathrm{dc}}}{L_1}& 0\\ 0& \frac{V_{\mathrm{dc}}}{L_2}\end{array}\right].$$

Due to E(x) being a non-singular matrix, it can obtain the following results by matrix transformation:

$$\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]=-E^{-1}(x)A(x)+E^{-1}(x)\left[\begin{array}{c}{\dot{y}}_1\\ {\dot{y}}_2\end{array}\right]$$

(14)

3.4. The Voltage/Current Tracking Controller

For the HESS, the control objective is that the current i₁ and the current i₂ track the reference value accurately. However, the feedback linearization requires the high accuracy of the system model. In order to enhance the robustness of the system, sliding mode variable structure control is introduced into the original decoupling system. The HESS after coordinate transformation is a controllable linear system, so the linear switching function can be selected. Let its reference current value be y_ref = [y_1ref y_2ref]^T = [i_1ref i_2ref]^T, the current tracking error is defined as follows:

$$e=\left[\begin{array}{c}{e}_1\\ e_2\end{array}\right]=\left[\begin{array}{c}{y}_1-y_{1\mathrm{ref}}\\ y_2-y_{2\mathrm{ref}}\end{array}\right]=\left[\begin{array}{c}{i}_1-i_{1\mathrm{ref}}\\ i_2-i_{2\mathrm{ref}}\end{array}\right]$$

(15)

In order to eliminate the influence of chattering on the controller, the integral sliding surface is selected as follows:

$$S=\left[\begin{array}{c}{s}_1\\ s_2\end{array}\right]=\left[\begin{array}{c}{c}_{11}{e}_1+c_{12}{\displaystyle \int e_1\mathrm{dt}}\\ c_{21}{e}_2+c_{22}{\displaystyle \int e_2\mathrm{dt}}\end{array}\right]$$

(16)

The derivation of Equation (16) shows that:

$$\dot{S}=\left[\begin{array}{c}{\dot{s}}_1\\ {\dot{s}}_2\end{array}\right]=\left[\begin{array}{c}{c}_{11}{\dot{e}}_1+c_{12}{e}_1\\ c_{21}{\dot{e}}_2+c_{22}{e}_2\end{array}\right]=\left[\begin{array}{c}{c}_{11}{\dot{i}}_1+c_{12}{e}_1\\ c_{21}{\dot{i}}_2+c_{22}{e}_2\end{array}\right]$$

(17)

The exponential reaching law is selected to design [22]:

$$\dot{s}=-ks-\epsilon \mathrm{sgn}(s),k>0,\epsilon >0$$

(18)

where k is the sliding mode gain, ε is the exponential reaching rate. In order to reduce the chattering of the system, the continuous function θ(s) is used instead of the symbolic function sgn(s):

$$\theta (s)=\frac{s}{\left|s\right|+\delta }$$

(19)

where δ is a small positive number. Combined with Equations (17)–(19), the following Equation can be obtained:

$$\left[\begin{array}{c}{\dot{s}}_1\\ {\dot{s}}_2\end{array}\right]=\left[\begin{array}{c}{c}_{11}{\dot{i}}_1+c_{12}{e}_1\\ c_{21}{\dot{i}}_2+c_{22}{e}_2\end{array}\right]=\left[\begin{array}{c}-k_1{s}_1-{\epsilon }_1\theta (s_1)\\ -k_2{s}_2-{\epsilon }_2\theta (s_2)\end{array}\right]$$

(20)

The PWM duty ratio can be solved such that:

$$\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]=\left[\begin{array}{c}\begin{array}{l}-\frac{V_1}{V_{\mathrm{dc}}}+\frac{R_{L1}+R_{\mathrm{on}2}}{V_{\mathrm{dc}}}{x}_1+1-\frac{k_1{L}_1}{c_{11}{V}_{\mathrm{dc}}}(c_{11}{e}_1+c_{12}{\displaystyle \int e_1\mathrm{dt}})-\\ \frac{{\epsilon }_1{L}_1}{c_{11}{V}_{\mathrm{dc}}}\theta (c_{11}{e}_1+c_{12}{\displaystyle \int e_1\mathrm{dt}})+\frac{c_{12}{L}_1}{c_{11}{V}_{\mathrm{dc}}}{e}_1\end{array}\\ \begin{array}{l}-\frac{V_2}{V_{\mathrm{dc}}}+\frac{R_{L2}+R_{\mathrm{on}4}}{V_{\mathrm{dc}}}{x}_2+1-\frac{k_2{L}_2}{c_{21}{V}_{\mathrm{dc}}}(c_{21}{e}_2+c_{22}{\displaystyle \int e_2\mathrm{dt}})-\\ \frac{{\epsilon }_2{L}_2}{c_{21}{V}_{\mathrm{dc}}}\theta (c_{21}{e}_2+c_{22}{\displaystyle \int e_2\mathrm{dt}})+\frac{c_{22}{L}_2}{c_{21}{V}_{\mathrm{dc}}}{e}_2\end{array}\end{array}\right]$$

(21)

The current reference value of battery i_1ref in Equation (15) is obtained by the energy management strategy:

$$i_{1\mathrm{ref}}=P_{\mathrm{bat}}/V_1$$

(22)

The circuit loss is ignored. The current reference value of supercapacitor i_2ref in Equation (15) can be obtained from the power conservation theorem:

$$i_{2\mathrm{ref}}=\frac{V_{\mathrm{dc}}{i}_{\mathrm{load}}-V_1{i}_1}{V_2}$$

(23)

To ensure that the DC bus voltage can be stable at the target voltage, the sliding surface is defined as follows:

$$s_3=V_{\mathrm{dc}}-V_{\mathrm{dcref}}$$

(24)

where V_dcref is the target DC bus voltage. When the system is stable, the DC bus voltage will be stable at the target value, and the tracking error is zero.

Combining the exponential reaching law described in Equations (18) and (19), the sliding mode control law of bus voltage can be obtained:

$$u(\mathrm{t})=-k_3{s}_3-{\epsilon }_3\theta (s_3)$$

(25)

By adding the voltage stability term u(t) to Equation (23), the bus voltage can be controlled to stabilize at the target value. Equation (23) can be rewritten as:

$$i_{2\mathrm{ref}}=\frac{V_{\mathrm{dc}}u(t)+V_{\mathrm{dc}}{i}_{\mathrm{load}}-V_1{i}_1}{V_2}$$

(26)

To ensure the stability of the control system, the Lyapunov function is defined as follows:

$$V=\frac{1}{2}{s}_1^2+\frac{1}{2}{s}_2^2$$

(27)

The differential of Equation (27) can be obtained as follows:

$$\dot{V}=s_1{\dot{s}}_1+s_2{\dot{s}}_2=-k_1{s}_1^2-{\epsilon }_1\frac{s_1^2}{\left|s_1\right|+\delta }-k_2{s}_2^2-{\epsilon }_2\frac{s_2^2}{\left|s_2\right|+\delta }$$

(28)

Due to k_i > 0, ε_i > 0, i = 1,2, and δ is a small positive number. It can be obtained as follows:

$$\dot{V}=s_1{\dot{s}}_1+s_2{\dot{s}}_2=-k_1{s}_1^2-{\epsilon }_1\left|s_1\right|-k_2{s}_2^2-{\epsilon }_2\left|s_2\right|\le 0$$

(29)

Therefore, according to the Lyapunov theory, the stability of the designed control strategy is proved. Figure 3 presents the overall diagram of the proposed EFL-SMC strategy.

4. Simulation and Analysis

In order to evaluate the control performance of the proposed CNC strategy, the simulation model was established in Simulink and compared with the traditional proportional−integral (PI) control and a modified sliding mode control (E-GSMC) [23]. The simulation includes: dynamic response performance (see 4.1), robust tracking performance (see 4.2), and a typical driving cycles test (see 4.3). The selected component parameters of the simulation model are listed in

Table 1. The component parameters of HESS.

Parameter	Value
L1: Battery side inductance (H)	2.6 × 10−3
L2: Supercapacitor side inductance (H)	1.8 × 10−3
R1: Inductor L1 series resistance(Ω)	0.2
R2: Inductor L2 series resistance(Ω)	0.15
Cdc: Load side capacitor (F)	1.5 × 10−3
C1: Battery side capacitor (F)	0.7 × 10−2
C2: Supercapacitor side capacitor (F)	0.5 × 10−2

. In particular, the nominal voltage of the battery pack is 180 V and the initial SOC is set as 90%; the nominal voltage of the supercapacitor pack is 150 V; the target DC bus voltage is 200 V; the frequency of pulse width module is 10 kHz; the P_min and P_ch in the energy management strategy are set to 3000 W and 200 W, respectively. Because of the small fluctuation of bus voltage in the control process, the parameters in the controller are set as follows: c₁₁ = 1, c₁₂ = 2, c₂₁ = 1, c₂₂ = 2, k₁L₁/V_dc = 10, ε₁L₁/V_dc = 3, k₂L₂/V_dc = 20, ε₂L₂/V_dc = 2, k₃ = 2, ε₃ = 0.2, δ = 0.001.

4.1. The Simulation Results of the Dynamic Response Performance

To investigate the dynamic response characteristics, the load required power Preq was set as 0 W→6000 W at times 0 s and 0.05 s. Figure 4, Figure 5 and Figure 6 show the dynamic response curves of the battery current, the supercapacitor current, and the bus voltage under three control strategies (PI, E-GSMC, CNC). It can be seen that the three control strategies have a certain overshoot in the starting process, and the proposed CNC strategy has a shorter steady-state time and less error chattering in the initial stage. When the reference value changes suddenly, it has a faster response speed and smaller overshoot than the PI control strategy and the E-GSMC strategy.

4.2. The Simulation Results of the Robust Tracking Performance

To investigate the robust tracking performance, the load required power Preq was set as 4000 W→7000 W→−2000 W→5000 W→2000 W at times 0 s, 1s, 2 s, 3 s, and 4 s. As shown in Figure 7, there is little difference in the robust tracking performance of battery current under the three control strategies, and the tracking error of the proposed CNC strategy is slightly improved compared with the PI control strategy and the E-GSMC strategy. As shown in Figure 8, the robust tracking performance of supercapacitor current under the three control strategies shows little difference in steady-state tracking, but when the reference current changes, the fluctuation of the PI control strategy is the largest, followed by the E-GSMC strategy, and the proposed CNC strategy is the smallest. As shown in Figure 9, the bus voltage under the three control strategies can be stable at the target voltage, but when the load required power changes, both PI control strategy and E-GSMC strategy fluctuate greatly, while the fluctuation of the proposed EFL-SMC strategy is the smallest, its overshoot decreases significantly, and its steady-state chattering is significantly less than that of the PI control strategy and E-GSMC strategy. The three strategies were simulated three times and the time costs were recorded. As shown in

Table 2. The time costs of three strategies.

Strategies	First Simulation	Second Simulation	Third Simulation
PI	23.85 s	23.73 s	23.65 s
E-GSMC	25.84 s	25.63 s	25.92 s
Proposed CNC strategy	25.29 s	25.35 s	24.94 s

, it can be seen that the time for PI control is shorter because the control process is relatively simple. Compared with the E-GSMC strategy, the time cost of CNC is significantly smaller. These confirm that the proposed CNC strategy has a better robust tracking performance.

4.3. The Simulation Results of the Typical Driving Cycles Test

To investigate the effectiveness of the proposed EFL-SMC strategy, the test was conducted under two typical driving cycles––the Urban Dynameter Driving Schedule (UDDS) and the New European Driving Cycle (NEDC). Figure 10 presents the load required power of the UDDS/NEDC driving cycle and the power split results between the battery and supercapacitor with the rule-based EMS. Here, the load required power was optimally distributed to the battery/supercapacitor. The battery only needs to bear the threshold power, and the peak power was borne by the supercapacitor, which effectively protects the battery. Figure 11, Figure 12 and Figure 13 present the tracking curves of the battery current, the supercapacitor current, and the bus voltage under the UDDS/NEDC driving cycle. It can be seen that both battery current and supercapacitor current can accurately track the target value under two typical driving cycles. The bus voltage can also be stable at the target value with the maximum error less than 3.5 V under the UDDS driving cycle and less than 6.5 V under the NEDC driving cycle. Figure 14 presents the voltages curves of the battery and supercapacitor under the UDDS/NEDC driving cycle. It can be seen that the voltage of the battery is relatively smooth, while the voltage of the supercapacitor fluctuates greatly. These confirm the effectiveness of the proposed CNC strategy under different driving conditions.

5. Conclusions

This paper presents a composite non-linear control strategy (CNC) by combining the exact feedback linearization method and the sliding mode variable structure control technology for the control of the fully-active HESS. Firstly, the affine non-linear model of fully-active HESS was established by deriving the circuit model. Then, a rule-based EMS was designed to distribute the required power. Finally, by combining the exact feedback linearization method with the sliding mode control technology, the CNC strategy was developed to ensure high performance and robustness of the control. According to the simulation results, the effectiveness of the proposed CNC strategy could be confirmed. The developed CNC strategy could reasonably distribute the required power between the battery and the supercapacitor, and realize the robust tracking of the battery/supercapacitor current reference value with maintaining the bus voltage stability. This provides a new idea for the control of the HESS.