Effective Energy Management Strategy with Model-Free DC-Bus Voltage Control for Fuel Cell/Battery/Supercapacitor Hybrid Electric Vehicle System

Abstract

This article presents a new design method of energy management strategy with model-free DC-Bus voltage control for the fuel-cell/battery/supercapacitor hybrid electric vehicle (FCHEV) system to enhance the power performance, fuel consumption, and fuel cell lifetime by considering regulation of DC-bus voltage. First, an efficient frequency-separating based-energy management strategy (EMS) is designed using Harr wavelet transform (HWT), adaptive low-pass filter, and interval type–2 fuzzy controller (IT2FC) to determine the appropriate power distribution for different power sources. Second, the ultra-local model (ULM) is introduced to re-formulate the FCHEV system by the knowledge of the input and output signals. Then, a novel adaptive model-free integral terminal sliding mode control (AMFITSMC) based on nonlinear disturbance observer (NDO) is proposed to force the actual values of the DC-link bus voltage and the power source’s currents track their obtained reference trajectories, wherein the NDO is used to approximate the unknown dynamics of the ULM. Moreover, the Lyapunov theorem is used to verify the stability of AMFITSMC via a closed-loop system. Finally, the FCHEV system with the presented method is modeled on a Matlab/Simulink environment, and different driving schedules like WLTP, UDDS, and HWFET driving cycles are utilized for investigation. The corresponding simulation results show that the proposed technique provides better results than the other methods, such as operational mode strategy and fuzzy logic control, in terms of the reduction of fuel consumption and fuel cell power fluctuations.

1. Introduction

Fuel cell (FC) is a brand-new renewable energy technology with significant emission-reduction potential, low operation temperature, and high energy density. It is a promising candidate for automotive application [1,2,3]. However, using standalone FC is currently hampered by unpredictable natural characteristics like a slow dynamic and an inability to absorb regenerative energy and meet rapid load power demands. Since the FCHEV is entirely an electrical power system, the energy storage system (ESS) (battery (BAT) and supercapacitor (SC)) provides or recovers high power peaks during vehicle acceleration/deceleration [4]. In the literature, the combination of FC with BAT and SC is studied and widely applied in various research fields, such as hybrid electric vehicles [5,6,7], hybrid tramway powertrains [4,8,9], construction machinery [10,11], DC microgrids [12,13], etc. The use of this hybrid configuration has the following reasons: the FC is employed as a primary source to supply the average power of load, the BAT is used to assist FC in absorbing regenerative braking energy or provide peak power, and meanwhile, the SC is employed to meet the high-frequency of load power due to the rapid speed of the vehicle. However, the references mentioned above exhibit that the hybridization of FC with BAT and SC can improve the system performance and downsize the FC and BAT [14]. The hybridization configuration with a multi-source system requires an effective EMS that controls the power flow to reduce fuel consumption and improve system efficiency and durability. On the other hand, the EMS functions as a power splitter for energy from both primary and ESS [7].

On these aspects of extending FC lifetime, improving power performance, and fuel economy of FCHEVs, different EMSs have developed in recent years. The developed EMSs can be divided into two main directions [15]: optimization-based and rule-based EMSs. Regarding optimization-based EMS, there are two sub-classes: offline (global) and online optimization strategies (real-time). Offline optimization-based EMSs use advanced control algorithms to solve an optimal control problem and achieve energy management under a known driving cycle (speed profile). Dynamic programming (DP) [16,17], Pontryagin’s maximum principle (PMP) [18], convex optimization [19], and hierarchical approximate global optimization [20] are the most used offline optimization-based EMSs. However, the aforementioned EMSs for FCHEVs result in heavy calculation capability in practical and strongly rely on the pre-knowledge of drive cycles. On this aspect, online optimization-based EMSs such as model-predictive control (MPC) [6], equivalent consumption minimization strategy (ECMS) [7], and internet of vehicle-based [21] are not subject to the specific drive cycle and their computation amount is relatively small, which can achieve real-time power allocation effectively.

Meanwhile, the rule-based-EMSs have been widely developed for FCHEVs to obtain a real-time power distribution, which use basic rules or maps, do not depend on pre-knowledge of drive cycles, and have no enormous amounts of data to calculate. In [22], a distributed energy management system is proposed for the hybrid power system (HPS) based on a rule-based power distribution strategy by using the charge and discharge limitations of power capability and residual capacity. Similar to [23,24], rule-based-EMSs are also developed for FCHEV to determine the required power of electrical sources and improve the fuel economy by regulating the power distribution of BAT and SC through charge and discharge mechanisms. In addition, in [25], Li et al. developed an operational mode control technique based on droop management to improve the efficiency of a hybrid tramway system.

However, the switching modes of the aforementioned rule-based mechanisms, which are frequently dependent on the on/off tool to control the specific working conditions, remained the drawback of not offering a flexible operation and instability for the charge and discharge of ESSs [26]. In this regard, fuzzy control algorithms have received significant attention for application in FCHEV due to their design flexibility and independence of a complete mathematical system model [12]. Real-time fuzzy-control-based EMS is designed in [27] to extend the battery lifetime and protect the battery from overcharging during repetitive braking energy absorption. The rules-building of fuzzy control depends on a lot of engineering experience, which cannot guarantee fuzzy control optimality; therefore, many researchers have switched to combining fuzzy control with other control approaches. In [28], a low pass filter and fuzzy controller are used to adopt EMS for the FCHEV system, decreasing fuel consumption and prolonging the FC’s lifetime. Moreover, in [29], a power management strategy is designed using fuzzy control and Haar wavelet transform (HWT) to provide optimal power allocation among the energy sources, aiming to improve durability and fuel economy. However, the fuzzy controllers are unable to directly dealing with the uncertainties of fuzzy rules. Therefore, the type-2 fuzzy controls have been adopted and used for the fuel cell hybrid vehicle [30], which provide reliable, practical situations to compensate for fuzzy rule uncertainties. Hence, this paper uses a type-2 fuzzy control combined with Haar wavelet transform (HWT) and a low pass filter to construct an effective EMS for the FCHEV system.

Since FC, BAT, or SC cannot be linked directly to the load in the aforementioned FCHEV-based system, DC-to-DC power converters are required between the DC link and energy sources, which in turn adds more degrees of freedom [31]. In order to guarantee the dynamic exchange of energy among the energy sources and load, the control of DC to DC power converter should be utilized in FCHEV to meet the following objectives: regulate the DC-link bus voltage and power source’s currents while ensuring the stability of the closed loop system. Recently, various power converter controls of the FCHEV systems have been proposed, and the most widely utilized is the proportional-integral (PI) controller due to its practicality and simplicity [32]. In [33], a nonlinear control-based method is presented using the dynamic model of the system. In [34], a back-stepping control method is introduced to regulate the FCHEV, and the global stability of the whole system is demonstrated. In order to achieve optimal tracking performance of the power converter control, terminal sliding mode control is proposed using projection operator adaptive law [35]. Moreover, a second-order sliding mode-based-control strategy is proposed for application in the framework of the more electric aircraft (MEA), and a bidirectional converter is used to regulate the power flow between the power generator and a supercapacitor [36]. The authors investigated the stability properties of the open-loop system using the input-to-state stability (ISS) mechanism and the Lyapunov function. Similar to [37], the control of a dual active bridge (DAB) converter is presented for MEA using two levels: a low-level and a high-level control method. At the low level, a generalized super-twisting algorithm is designed to force the actual value of the battery current to track its given reference trajectory. In contrast, a supervisory controller is introduced at a high level using special switching modules. In [38], a model-based nonlinear controller is proposed for FCHEV, and the Lyapunov theorem is used to verify the stability of the controlled system. The simulation results prove that the designed controller provides satisfactory performance.

Fortunately, the model-free control (MPC) based on an ULM algorithm has received great attention from researchers and widely applied in different fields [39,40], which reduces the dependence on model information and relies only on input/output data if compared to the model-based control approach. The robustness of the MFC method relies on the designed observer/estimator to approximate the unmodeled system dynamics or lumped uncertainties like algebraic observer (AO) [41] and time-delay estimation (TDE) [42]. However, the TDE and AO both have inevitable estimating errors because of the time delays and time windows, respectively. Extended state observer (ESO)-based MFC is also developed [39]. The ESO can only realize the asymptotical observation when the derivative of the lumped disturbance reaches zero, meaning that as time tends to infinity the estimation error will converge to zero. Therefore, the zero estimate error for the unknown lumped uncertainties and the finite-time observation are not considered in the above-mentioned methods.

Motivated by the above discussion, how to properly construct an effective energy management strategy and model-free DC-bus voltage control for an FCHEV system subject to energy source constraints, uncertainties, and external disturbance remains to be resolved and warrants further research. Accordingly, this research proposes an efficient frequency-decoupling-based EMS with model-free DC bus voltage control to improve the power performance, prolong the FC lifetime, and reduce the fuel consumption of FCHEV. Therefore, the main contributions of this paper can be summarized as follows:

• A novel frequency-decoupling technique-based EMS is constructed using an adaptive LP filter, HWT, and IT2FC to determine the appropriate power sharing for different energy sources.
• Considering the dynamic characteristics of load power, an adaptive LP filter based on IT2FC is designed to adjust the output power of SC, maintain the SC state of charge within the predefined region, and ensure a fast response of load power during the vehicle acceleration and deceleration.
• In order to further improve the power performance, decrease the fuel consumption, and maintain the BAT state of charge, another IT2FC combined with HWT is established to provide optimal power of FC with low-frequency components.
• In order to avoid the requirement for accurate modeling and reduce the controller design difficulty, the ULM algorithm is employed to re-formulate the DC/DC power converters thanks to MFC theory, wherein the AMFITSMC based on NDO is proposed to regulate the DC bus voltage and currents of energy sources.
• Using the Lyapunov function, the stability analysis of the AMFITSMC method is investigated.

2. Description and Modeling of FCHEV

As shown in Figure 1, the FCHEV is mainly supplied by three power sources: a proton exchange membrane fuel cell (PEMFC) as a main power source and energy storage systems (battery and supercapacitor) as auxiliary power sources to remedy the slow dynamic response of the PEMFC system and to meet the fast power transient of load demand. The PEMFC is connected to the DC bus via a bidirectional DC/DC converter, and the battery is connected to the DC bus via a bidirectional DC/DC converter, while the supercapacitor is connected directly to the DC bus. On the other side, the traction motor is supplied using a bidirectional DC/AC inverter.

This work aims to develop effective EMS and converter controls to deal with the FCHEV system. However, the EMS and converter controls developed in this article should match the design requirements below:

• High efficiency of the EMS to prolong the FC lifetime, decrease the fuel consumption, and improve the power performance;
• The state of charges of the BAT and SC should be maintained within the desired zones;
• Stabilize the DC bus voltage and power source’s currents with optimal control performance in the presence of different operating conditions;
• The stability of the whole controlled FCHEV system should be ensured.

2.1. Modeling of the Vehicle

The vehicle model is formulated to compute the load power that is supplied by the energy sources. The electrical load power $P_{\mathrm{load}}$ is calculated according to the dynamic proprieties of the electric vehicle as follows [15]

$$P_{\mathrm{load}}=\frac{1}{{\eta }_{\mathrm{inv}}}\left(0.5\rho {\nu }_{\mathrm{EV}}^2{S}_{\mathrm{f}}{C}_{\mathrm{x}}+M\mathrm{g}{C}_r+M\frac{d{\nu }_{\mathrm{EV}}}{dt}\right){\nu }_{\mathrm{EV}}$$

(1)

where ${\nu }_{\mathrm{EV}}$, $C_{\mathrm{x}}$, M, $S_{\mathrm{f}}$, $C_{\mathrm{r}}$, $\rho$, g and ${\eta }_{\mathrm{inv}}$ are the vehicle speed, the aerodynamic drag coefficient, the vehicle mass, the vehicle frontal surface, the rolling resistance coefficient, the air density, the gravitational acceleration, and inverter efficiency, respectively.

2.2. Modeling of the Fuel Cell

The PEMFC with the power module 30 kW is used as the primary energy source in this work to provide an average power of the load demand. The FC stack voltage $V_{\mathrm{FC}}$ can be calculated as follows [43]:

$$V_{\mathrm{FC}}=E-({\eta }_{\mathrm{conc}}+{\eta }_{\mathrm{act}}+{\eta }_{\mathrm{ohmic}}).$$

(2)

where E denotes the reversible voltage, ${\eta }_{\mathrm{conc}}$, ${\eta }_{\mathrm{act}}$, and ${\eta }_{\mathrm{ohmic}}$ represent the concentration, activation, and ohmic losses, respectively. The reversible voltage can be defined as follows:

$$E=1.229-{0.85\times 10}^{-3}(T_{\mathrm{st}}-298.15)+{4.3085\times 10}^{-5}{T}_{\mathrm{st}}[\ln \left(P_{{\mathrm{H}}_2}\right)+\frac{1}{2}\ln \left(P_{{\mathrm{O}}_2}\right)].$$

(3)

where $P_{{\mathrm{O}}_2}$ and $P_{{\mathrm{H}}_2}$ are the oxygen and hydrogen partial pressures, respectively. The FC stack power $P_{\mathrm{FC}}$ can be defined as follows:

$$P_{\mathrm{FC}}=n{V}_{\mathrm{FC}}{I}_{\mathrm{FC}}$$

(4)

where n and $I_{\mathrm{FC}}$ denote the number of cells and the FC current, respectively; the reader can refer to Ref. [43] for more details about the FC model. The FC efficiency can be expressed as follows: [44]:

$${\eta }_{\mathrm{FC}}=\frac{P_{\mathrm{FC}}-P_{\mathrm{AUX}}}{{\dot{m}}_{\mathrm{FC}}{H}_{\mathrm{LHV}}}$$

(5)

where $P_{\mathrm{AUX}}$ represents the power consumed by FC auxiliary subsystems, $H_{\mathrm{LHV}}$ denotes the hydrogen lower heating value, and ${\dot{m}}_{\mathrm{FC}}$ represents the hydrogen consumption rate, which can be defined as follows:

$${\dot{m}}_{\mathrm{FC}}=\frac{n{M}_{{\mathrm{H}}_2}{I}_{\mathrm{FC}}}{2\mathrm{F}}$$

(6)

where $M_{{\mathrm{H}}_2}$ represents the hydrogen molar mass.

2.3. Modelling of Battery

The battery’s state of charge (SOC) is an important key variable in EMS, and it should be maintained in a desired region to protect the battery from overcharge and over-discharge. The following equation describes the $SO{C}_{\mathrm{BAT}}$ as [15,45]

$$SO{C}_{\mathrm{BAT}}=SO{C}_0-{\eta }_{\mathrm{BAT}}\int \frac{I_{\mathrm{BAT}}}{Q_{\mathrm{BAT}}}dt$$

(7)

where $I_{\mathrm{BAT}}$, $Q_{\mathrm{BAT}}$, $SO{C}_0$, and ${\eta }_{\mathrm{BAT}}$ represent the battery current, the battery capacity, the initial value of the battery $SO{C}_{\mathrm{BAT}}$, and the charging and discharging efficiency of BAT, respectively. To calculate the total ${\mathrm{H}}_2$ consumption of the FCHEV, the equivalent ${\mathrm{H}}_2$ consumption concerning the BAT $m_{\mathrm{BAT}}$ should be considered and it can be expressed as follows [46]:

$$m_{\mathrm{BAT}}=\frac{{\alpha }_{\mathrm{ele}}}{3600Q_{\mathrm{BAT}}{\eta }_{\mathrm{BAT}}{\mathrm{C}}_{{\mathrm{H}}_2}}\int P_{\mathrm{BAT}}dt$$

(8)

where ${\mathrm{C}}_{{\mathrm{H}}_2}$ is the hydrogen price and ${\alpha }_{\mathrm{ele}}$ is the electricity price.

2.4. Modelling of Supercapacitor

The supercapacitor (SC) is used to absorb the peak load power of the FCHEV during the vehicle acceleration and deceleration and has enough capacity for energy recycling during the vehicle braking. The supercapacitor $SOC$ can be calculated as follows [47]:

$$SO{C}_{\mathrm{SC}}=\frac{{(V_{\mathrm{SC}}-2I_{\mathrm{SC}}{R}_{\mathrm{SC}})}^2}{V_{\mathrm{SC};\max }^2}$$

(9)

where $V_{\mathrm{SC}}$ denotes the SC terminal voltage, $V_{\mathrm{SC};\max }$ represents the maximum of SC voltage, $I_{\mathrm{SC}}$ indicates for the SC current, $R_{\mathrm{SC}}$ denotes the internal resistance. The equivalent hydrogen ${\mathrm{H}}_2$ concerning the SC $m_{\mathrm{SC}}$ is calculated as follows [48]:

$$m_{\mathrm{SC}}=\frac{3600\Delta SO{C}_{\mathrm{SC}}{f}_{\mathrm{SC}}}{{\rho }_{{\mathrm{H}}_2}{\eta }_{\mathrm{FC}}}$$

(10)

where $f_{\mathrm{SC}}$ is an equivalent factor of the SC and ${\rho }_{{\mathrm{H}}_2}$ is the hydrogen chemical energy density.

2.5. Uni-Directional and Bi-Directional DC/DC Converters Modeling

The DC/DC power converter, which is utilized between the FC and DC bus voltage is a unidirectional boost converter, and its mathematical model can be described as follows [49]:

$$\frac{d{I}_{\mathrm{FC}}}{dt}=\frac{V_{\mathrm{FC}}}{L_{\mathrm{FC}}}-\frac{I_{\mathrm{FC}}{R}_{\mathrm{FC}}}{L_{\mathrm{FC}}}-(1-u_1)\frac{V_{\mathrm{bus}}}{L_{\mathrm{FC}}}+{\phi }_1$$

(11)

$$I_{\mathrm{FC}-\mathrm{out}}=u_1{I}_{\mathrm{FC}}$$

(12)

where $V_{\mathrm{bus}}$ is the DC-link bus voltage, $I_{\mathrm{FC}}$ is the FC current passing through inductor $L_{\mathrm{FC}}$, $V_{\mathrm{FC}}$ is the FC terminal voltage, $R_{\mathrm{FC}}$ is the boost converter resistance, $u_1$ is the control input for switch $S_1$, ${\phi }_1$ denotes the model uncertainties or external disturbance, and $I_{\mathrm{FC}-\mathrm{out}}$ represents the output current of the DC-DC boost converter. While the battery converter is a bidirectional type-buck-boost converter when the battery is in discharging mode, the converter operates as a boost converter, and when the battery is in charging mode, the converter works as a buck converter. Therefore, the mathematical expression of the BAT converter model can be given as follows:

$$\frac{d{I}_{\mathrm{BAT}}}{dt}=\frac{V_{\mathrm{BAT}}}{L_{\mathrm{BAT}}}-\frac{I_{\mathrm{BAT}}{R}_{\mathrm{BAT}}}{L_{\mathrm{BAT}}}-u_{23}\frac{V_{\mathrm{bus}}}{L_{\mathrm{BAT}}}+{\phi }_2$$

(13)

$$I_{\mathrm{BAT}-\mathrm{out}}=u_{23}{I}_{\mathrm{BAT}}$$

(14)

$$u_{23}=[H(1-u_2)+(1-H)u_3]$$

(15)

$$H=\left\{\begin{array}{cc}\hfill 1\hfill & \hfill I_{\mathrm{BAT}}>0\left(\mathrm{boost}\mathrm{mode}\right)\hfill \\ \hfill 0\hfill & \hfill I_{\mathrm{BAT}}<0\left(\mathrm{Buck}\mathrm{mode}\right)\hfill \end{array}\right.$$

(16)

where $u_2$ and $u_3$ denote the control inputs for switches $S_2$ and $S_3$, respectively; and ${\phi }_2$ denotes the model uncertainties or external disturbance.

Then, the DC bus mode can be given as follows:

$$\frac{d{V}_{\mathrm{bus}}}{dt}=u_{23}\frac{I_{\mathrm{BAT}}}{C_{\mathrm{bus}}}+(1-u_1)\frac{I_{\mathrm{FC}}}{C_{\mathrm{bus}}}+\frac{I_{\mathrm{SC}}}{C_{\mathrm{bus}}}-\frac{I_{\mathrm{load}}}{C_{\mathrm{bus}}}+{\phi }_3$$

(17)

where $I_{\mathrm{load}}$ is the load current, $I_{\mathrm{SC}}$ is the supercapacitor current, $V_{\mathrm{bus}}$ is the DC bus voltage, $C_{\mathrm{bus}}$ is the capacitor of DC bus, ${\phi }_3$ denotes the model uncertainties or external disturbance. $u_{23}$ represents the virtual control signal for $S_2$ and $S_3$.

Using (11), (13) and (17), the global system model of DC/DC converters can obtained as follows:

$$\left\{\begin{array}{cc}\frac{d{I}_{\mathrm{FC}}}{dt}\hfill & =\frac{V_{\mathrm{FC}}}{L_{\mathrm{FC}}}-\frac{I_{\mathrm{FC}}{R}_{\mathrm{FC}}}{L_{\mathrm{FC}}}-(1-u_1)\frac{V_{\mathrm{bus}}}{L_{\mathrm{FC}}}+{\phi }_1\hfill \\ \frac{d{I}_{\mathrm{BAT}}}{dt}\hfill & =\frac{V_{\mathrm{BAT}}}{L_{\mathrm{BAT}}}-\frac{I_{\mathrm{BAT}}{R}_{\mathrm{BAT}}}{L_{\mathrm{BAT}}}-u_{23}\frac{V_{\mathrm{bus}}}{L_{\mathrm{BAT}}}+{\phi }_2\hfill \\ \frac{d{V}_{\mathrm{bus}}}{dt}\hfill & =u_{23}\frac{I_{BAT}^{\mathrm{ref}}}{C_{\mathrm{bus}}}+(1-u_1)\frac{I_{\mathrm{FC}}}{C_{\mathrm{bus}}}+\frac{I_{\mathrm{SC}}}{C_{\mathrm{bus}}}-\frac{I_{\mathrm{load}}}{C_{\mathrm{bus}}}+{\phi }_3\hfill \end{array}\right.$$

(18)

3. Energy Management Strategy and Converters Control

According to the dynamic characteristics of three different energy sources, EMS and converters control loops are proposed, as shown in Figure 2. The proposed EMS based on frequency-decoupling and mode-free DC bus voltage control is composed of three parts: the first part is an adaptive LP filter based on interval type–2 fuzzy control–1 (IT2FC–1), which is constructed to provide the initial reference power $P_{\mathrm{FC};\mathrm{in}}^{\mathrm{ref}}$ for fuel cells with the lower frequency components, utilizing load power $P_{\mathrm{load}}$ and Supercapacitor $SOC$; the second part is Harr wavelet transform (HWT) with another interval type–2 fuzzy control–2 (IT2FC–2), which is designed to obtain the actual reference power $P_{\mathrm{FC}}^{\mathrm{ref}}$ for the fuel cell, using the initial reference power $P_{\mathrm{FC};\mathrm{in}}^{\mathrm{ref}}$ and battery $SOC$; the third part is mode-free DC bus voltage control, which is designed to calculate the reference current for battery using ultra-local model with nonlinear disturbance observer (NDO) and integral terminal sliding mode control (ITSMC).

3.1. Design of Adaptive LP Filter Based on Interval Type–2 Fuzzy Controller

The LP filter, which has advantages such as simplicity and short calculation time, can effectively decouple the high-frequency component of the load power demand and deliver it to the supercapacitor in real-time for practical application in FCHEV. The transfer function, which is used to describe the LP filter, which is defined as

$$\mathrm{H}\left(s\right)=\frac{f_s}{s+f_{\mathrm{s}}}$$

(19)

where $\mathrm{H}\left(s\right)$ is the transfer function of LP filter and $f_{\mathrm{s}}$ is the regulating frequency.

The designed LP filter is fed with load power demand, and its regulating frequency $f_s$ is adjusted automatically based on load power demand and $SO{C}_{\mathrm{SC}}$ using interval type-2 fuzzy controller (IT2FC–1). The main objective of IT2FC–1 is to maintain the $SO{C}_{\mathrm{SC}}$ of supercapacitor (SC) within the pr-defined region ($SO{C}_{\mathrm{SC};\min }$ ≤ $SO{C}_{\mathrm{SC}}$ ≤ $SO{C}_{\mathrm{SC};\max }$), which makes the SC possible to effectively supply or absorb the peak power of load demand during the vehicle’s accelerations or deceleration. Nevertheless, the SC has advantages of reducing the load stress on both FC and BAT as well as decreasing their power fluctuations at the same time. The factor of load power demand $I_{\mathrm{P}}=P_{\mathrm{load}}/P_{\mathrm{load};\max }$ and $SO{C}_{\mathrm{SC}}$ are assigned as input signals of IT2FC–1, while the regulating frequency $f_{\mathrm{s}}$ is designated as the output signal of IT2FC–1.

In this paper, the fuzzification is comprised of two membership functions (MFs) for input variables. MFs for the first and second input variables consist of five and four triangular-shaped functions, as shown in Figure 3a and Figure 3b, respectively. The output MFs variable for IT2FC–1 is four crisp singletons consequent as shown in Figure 3c. According to the characteristics of inputs and output, the fuzzy inference is built by 20 rule bases, as listed in

Table 1. Rule base of IT2FC–1.

${\mathit{f}}_{\mathbf{s}}$				${\mathit{I}}_{\mathbf{P}}$
		NL	NM	ZO	PM	PL
$SO{C}_{\mathrm{SC}}$	VS	VS	VS	L	M	S
	S	S	S	L	M	VS
	M	M	M	L	S	VS
	L	L	L	L	S	VS

. For more details about interval type–2 fuzzy, the reader can refer to [25].

Remark 1. 

The linguistic variables: NL is the Negative Large, NM is Negative Medium, ZO is Zero, PL is Positive Large, VS is Very Small, S is Small, M is Medium and L is Large.

3.2. Design of Power Sharing Algorithm Using HWT and IT2FC–2

This subsection is responsible for calculating the optimal power of FC with a low-frequency component. Since the initial reference power $P_{\mathrm{FC};\mathrm{in}}^{\mathrm{ref}}$ still has some high-frequency components that can cause fluctuations in the FC power and affect its lifetime, it cannot be sent directly to FC. Therefore, initial reference power $P_{\mathrm{FC};\mathrm{in}}^{\mathrm{ref}}$ is further decoupled into the high-frequency contents and low-frequency contents using the HWT algorithm. The following equation describes the mathematical expression of the HWT algorithm [50,51]:

$$W(u,\epsilon )=\int S\left(n\right)\frac{1}{\sqrt{\epsilon }}\Omega \left(\frac{n-U}{\epsilon }\right)dt,\epsilon =2^j,U=k{2}^j,k\in Z$$

(20)

where $\Omega$, $S\left(n\right)$, $\epsilon$, W, and U represent the mother wavelet, the input signal of the HWT algorithm, the scalar parameter, wavelet coefficient, and the position parameter, respectively. The mother wavelet $\Phi$ is defined as follows:

$$\Omega \left(t\right)=\left\{\begin{array}{cc}1& if\le n\le 0.5\\ -1& if0.5<n\le 1\\ 0& Otherwise\end{array}\right.$$

(21)

Figure 4 shows the decomposition of three-level HWT that is applied to the input signal $S\left(n\right)$ to decompose into two parts: the first part represents the reference signal by a LP filter $L_0\left(z\right)$ and the second part denotes the detail signal by a HP filter $H_0\left(z\right)$. In this manner, the initial reference power $P_{\mathrm{FC};\mathrm{in}}^{\mathrm{ref}}$ can be directly separated into low and high-frequency components. Thus, reference power $P_{\mathrm{FC};\mathrm{a}}^{\mathrm{ref}}$ for FC can be given as follows:

$$P_{FC;a}^{ref}=S_0\left(n\right)$$

(22)

where $S_0\left(n\right)$ denotes the approximation part of the initial reference power $P_{\mathrm{FC};\mathrm{in}}^{\mathrm{ref}}$ obtained after the decomposition process.

Even though the calculated reference power $P_{\mathrm{FC};\mathrm{a}}^{\mathrm{ref}}$ with low frequency can be sent to FC, the battery SOC should be considered in this stage and maintained within the appropriate region ($SO{C}_{\mathrm{BAT};\min }$ ≤ $SO{C}_{\mathrm{BAT}}$ ≤ $SO{C}_{\mathrm{BAT};\max }$). Therefore, to further improve the power performance, extend the FC lifetime, and protect the battery from overcharging/discharging, another IT2FC–2 is developed to obtain optimal FC power. The given reference power $P_{\mathrm{FC},\mathrm{a}}^{\mathrm{ref}}$ and the battery $SO{C}_{\mathrm{BAT}}$ are used as the input signals of IT2FC–2, while the final reference power of FC $P_{FC}^{ref}$ is assigned as the output signal. Figure 5 highlights the relevant MFs of inputs and output signals. The corresponding fuzzy rules are listed in

Table 2. IT2FC–2 rule bases.

${\mathit{P}}_{\mathbf{FC}}^{\mathbf{ref}}$					${\mathit{P}}_{\mathbf{FC},\mathbf{a}}^{\mathbf{ref}}$
		VS	S	RS	M	RL	L	VL
$SO{C}_{BAT}$	VS	RS	M	RL	L	VL	VL	VL
	S	S	RS	M	RL	L	VL	VL
	M	VS	S	RS	M	RL	L	VL
	L	VS	VS	S	RS	M	RL	L

.

Remark 2. 

In the existing fuzzy control/filter, which is utilized in the energy management for hybrid vehicles [15,28], the influence of load fluctuation on the fuel cell system cannot be minimized. Meanwhile, the balance between the fuel cell lifespan and fuel economy is important for improving the whole FCHEV performance. The purpose of the designed adaptive LP filter is not only to separate the high-frequency part of the load power and distribute it to the SC in real-time, but to maintain the SC state of charge $SO{C}_{\mathrm{SC}}$ within the pr-defined region ($SO{C}_{\mathrm{SC};\min }$ ≤ $SO{C}_{\mathrm{SC}}$ ≤ $SO{C}_{\mathrm{SC};\max }$). Hence, the LP filter is designed according to the dynamic characteristics of the load power and operating constraint of SC using IT2FC–1 to adjust the regulating frequency factor of the LP filter. Reducing the frequency component of load power by using an LP filter is not high enough to distribute it directly to the fuel cell. So, a further frequency reduction stage is needed, and HWT is more convenient to ensure the improvement of power performance, fuel cell lifespan, and fuel economy. Figure 6 shows the histogram of power fluctuations for the obtained $P_{\mathrm{FC};\mathrm{in}}^{\mathrm{ref}}$ from the LP filter compared to the obtained reference power $P_{\mathrm{FC};\mathrm{a}}^{\mathrm{ref}}$ from HWT. It is observed that the presented HWT removed the remaining frequency component of the obtained $P_{\mathrm{FC};\mathrm{in}}^{\mathrm{ref}}$ from the LP filter.

3.3. Adaptive Model-Free Integral Terminal Sliding Mode Control Using Nonlinear Disturbance Observer

This subsection presents an AMFITSMC design, whose control architecture is shown in Figure 2. The referred controller strategy consists of nonlinear disturbance observer (NDO)-based ultra local mode and integral terminal sliding mode control (ITSMC). Wherein the NDO is used for estimating uncertain system dynamics, and ITSMC is developed for achieving tracking performance with rapid convergence speed, finite-time stabilization, and less input chattering. First, the design principle of the NDO-based intelligent proportional integral controller (NDO-iPI) is introduced and followed by AMFITSMC design. Moreover, by means of the Lyapunov theorem, the stability of a closed-loop system via the proposed control strategy is investigated.

3.3.1. NDO-iPI Design

This subsection presents a nonlinear disturbance observer-based-intelligent proportional integral controller (NDO-iPI) using an ultra-local model algorithm, and its design process is given in the following subsections.

Ultra-Local Model (ULM) Algorithm

This paper uses the ULM Algorithm to re-formulate the models of DC bus voltage, FC boost converter, and BAT back-boost converter (18) by knowledge of the input and output signals. Since the ULM algorithm has been defined for a single-input single-output (SISO) system according to [52], three models of ULM are used in this work. Thus, the ULM algorithms for the DC bus voltage, FC, and BAT current loops are formulated as follows:

$$\left\{\begin{array}{c}\frac{d{I}_{\mathrm{FC}}}{dt}={\beta }_{\mathrm{FC}}{u}_1+g_{\mathrm{FC}}\hfill \\ \frac{d{I}_{\mathrm{BAT}}}{dt}={\beta }_{\mathrm{BAT}}{u}_{23}+g_{\mathrm{BAT}}\hfill \\ \frac{d{V}_{\mathrm{bus}}}{dt}={\beta }_{\mathrm{bus}}{I}_{\mathrm{BAT}}^{\mathrm{ref}}+g_{\mathrm{bus}}\hfill \end{array}\right.$$

(23)

where ${\beta }_{\mathrm{bus}}>0$, ${\beta }_{\mathrm{FC}}>0$ and ${\beta }_{\mathrm{BAT}}>0$ are the controller’s gains, $u_1$, $u_{23}$ and $I_{\mathrm{BAT}}^{\mathrm{ref}}$ are controller inputs, $g_{\mathrm{bus}}$, $g_{\mathrm{FC}}$ and $g_{\mathrm{BAT}}$ are uncertain system dynamics for the models of DC bus voltage, FC boost converter and BAT back-boost converter, respectively.

Then, (23) can be rewritten as follows:

$${\dot{x}}_i={\beta }_i{u}_i+g_i,i=1,2,3$$

(24)

Thus, the NDO-iPI controller can be constructed as below:

$$u_i=\frac{1}{{\beta }_i}[-{\widehat{g}}_i+{\dot{x}}_i^{\mathrm{ref}}+K_{\mathrm{P}i}{e}_i+K_{\mathrm{I}i}\int e_{i}dt]$$

(25)

where $e_i=x_i^{\mathrm{ref}}-x_i$ is the tracking error, $x_i^{\mathrm{ref}}$ is the desired trajectory, $K_{\mathrm{P}i}$ and $K_{\mathrm{I}i}$ are the proportional and integral coefficients, respectively, and ${\widehat{g}}_i$ is the estimated value of uncertain system dynamics $g_i$ using NDO, and its description will be given in (29) later.

Substituting (25) into (24), the error equation can be defined as

$${\dot{e}}_i+K_{\mathrm{P}i}{e}_i+K_{\mathrm{I}i}\int e_{i}dt-{\tilde{g}}_i=0$$

(26)

where ${\tilde{g}}_i=g_i-{\widehat{g}}_i$ is the difference between the actual and the estimated value.

According to the Hurwitz criterion [39], and by selecting appropriate values of $K_{\mathrm{P}i}$ and $K_{\mathrm{I}i}$, the steady state error via the closed loop system is ensured.

Nonlinear Disturbance Observer (NDO) Design

In this paper, NDO is used to estimate the uncertain dynamics $g_i\left(t\right)$ via the control input and output signals [53].

Assumption 1. 

The lumped uncertainties $g_i\left(t\right)$ and its first time derivative have bounded values as below:

$$|g_i\left(t\right)|\le {\varsigma }_i,{\dot{g}}_i\left(t\right)\le {\gamma }_i$$

(27)

where ${\varsigma }_i>0$ and ${\gamma }_i>0$ are positive constants.

Remark 3. 

We considered this Assumption 1 in order to theoretically ensure the existence of the constants ${\varsigma }_i$ and ${\gamma }_i$. In this work, Assumption 1 is only used to qualitatively describe the relationship between convergence performance and a key design of NDO.

Suppose that the estimated value of the uncertain dynamics of the controlled system $g\left(t\right)$ and its estimation error are defined as ${\widehat{g}}_i\left(t\right)$ and ${\tilde{g}}_i\left(t\right)=g_i\left(t\right)-{\widehat{g}}_i\left(t\right)$, respectively. Let us define a new vertical variable ${\Gamma }_i\left(t\right)=g_i\left(t\right)-{\mu }_i{x}_i\left(t\right)$. Then, by differentiating ${\Gamma }_i\left(t\right)$, one can obtain

$$\begin{array}{cc}{\dot{\Gamma }}_i\left(t\right)& ={\dot{g}}_i\left(t\right)-{\mu }_i{\dot{x}}_i\left(t\right)\\ & ={\dot{g}}_i\left(t\right)-{\mu }_i[{\beta }_i{u}_i\left(t\right)+g_i\left(t\right)]\\ & ={\dot{g}}_i\left(t\right)-{\mu }_i[{\mu }_i{x}_i\left(t\right)+{\beta }_i{u}_i\left(t\right)]-{\mu }_i{\Gamma }_i\left(t\right)\end{array}$$

(28)

Thus, the proposed observer can be formulated as follows [53]:

$$\left\{\begin{array}{c}{\dot{\widehat{\Gamma }}}_i\left(t\right)=-{\mu }_i{\widehat{\Gamma }}_i\left(t\right)-{\mu }_i[{\mu }_i{x}_i\left(t\right)+{\beta }_i{u}_i\left(t\right)]\\ {\widehat{g}}_i\left(t\right)={\widehat{\Gamma }}_i+{\mu }_i{x}_i\left(t\right)\hfill \end{array}\right.$$

(29)

where ${\mu }_i>0$ is positive constant.

Remark 4. 

According to NDO formula in [53], the variable ${\Gamma }_i$ is defined as ${\Gamma }_i\left(t\right)=g_i\left(t\right)-{\mu }_i{x}_i\left(t\right)$ and used to transfer (24) into general form of NDO (28). As noted from introducing the variable ${\Gamma }_i$, we take into consideration the variation rate of $g_i\left(t\right)$, which is considered as bounded function $|{\dot{g}}_i\left(t\right)|\le {\gamma }_i$ in Assumption 1, and then used to proof the Theorem 1.

Theorem 1. 

Considers the proposed NDO (29) for the system dynamics (24), the estimated error $\tilde{g}$ is converged to bounded region as $|{\tilde{g}}_i|\le \frac{{\gamma }_i}{{\mu }_i}$ for ${\mu }_i>0$.

Proof. 

From (28) and (29), we have ${\tilde{\Gamma }}_i=g_i-{\mu }_i{\widehat{\Gamma }}_i$, ${\dot{\tilde{\Gamma }}}_i=\dot{g}-\mu \tilde{\Gamma }$ and ${\tilde{\Gamma }}_i={\tilde{g}}_i$. Then, the Lyapunov function is defined as follows:

$${\Pi }_i=\frac{1}{2}{\tilde{g}}_i^2$$

(30)

Differentiating (30), one can obtain

$$\begin{array}{cc}{\dot{\Pi }}_i={\tilde{g}}_i{\dot{\tilde{g}}}_i& ={\tilde{\Gamma }}_i{\dot{\tilde{\Gamma }}}_i\hfill \\ & ={\tilde{\Gamma }}_i[{\dot{g}}_i-{\mu }_i{\tilde{\Gamma }}_i]\hfill \\ & =-{\mu }_i{\tilde{\Gamma }}_i^2+{\tilde{\Gamma }}_i{\dot{g}}_i\hfill \end{array}$$

(31)

According to the Yong’s inequality, the part (${\tilde{\Gamma }}_i{\dot{g}}_i$) can be simplified as $|{\tilde{\Gamma }}_i{\dot{g}}_i|\le \frac{1}{2}{\mu }_i{\tilde{\Gamma }}_i^2+\frac{1}{2{\mu }_i}{\dot{g}}_i^2$, and (31) can be rewritten as follows:

$$\begin{array}{cc}{\dot{\Pi }}_i& \le -{\mu }_i{\tilde{\Gamma }}_i^2+\frac{1}{2}{\mu }_i{\tilde{\Gamma }}_i^2+\frac{1}{2{\mu }_i}{\dot{g}}_i^2\hfill \\ & \le -\frac{1}{2}{\mu }_i{\tilde{\Gamma }}_i^2+\frac{1}{2{\mu }_i}{\dot{g}}_i^2\hfill \\ & \le -\frac{1}{2}{\mu }_i{\tilde{\Gamma }}_i^2+\frac{1}{2{\mu }_i}{\gamma }_i^2\hfill \\ & \le -{\mu }_i\Pi +\frac{1}{2{\mu }_i}{\gamma }_i^2\hfill \end{array}$$

(32)

Taking the integration for both sides of (32) yields

$$ln\left\{-{\mu }_i{\Pi }_i+\frac{1}{2{\mu }_i}{\gamma }_i^2\right\}\le -{\mu }_{i}t$$

(33)

Then, (33) can be simplified to

$${\Pi }_i\le \frac{e^{-{\mu }_{i}t}-\frac{1}{2{\mu }_i}{\gamma }_i^2}{-{\mu }_i}$$

(34)

By substituting (30) into (34), the inequality (34) becomes

$$|{\tilde{g}}_i|\le \sqrt{\frac{2e^{-{\mu }_{i}t}-\frac{1}{{\mu }_i}{\gamma }_i^2}{-{\mu }_i}}$$

(35)

According to the result in (35) that $e^{-{\mu }_{i}t}⟶0$, one can conclude that the estimation error will converge to a bounded region $|{\tilde{g}}_i|\le \frac{{\gamma }_i}{{\mu }_i}$ for ${\mu }_i>0$. □

3.3.2. AMFITSMC Design

Based on the designed NDO (29), the control law of AMFITSMC can be constructed as follows:

$$u_i\left(t\right)=\frac{-{\widehat{g}}_i+{\dot{x}}_i^{ref}+K_{\mathrm{p}}e+K_{\mathrm{P}i}\int e_{i}dt}{{\beta }_i}+\frac{u_{\mathrm{TEC}i}\left(t\right)}{{\beta }_i},i=1,2,3$$

(36)

where $u_{\mathrm{TEC}i}\left(t\right)$ represents the sub-controller law for tracking error convergence (TEC) to be designed.

Now, we can define new state variables as $L_{1i}\left(t\right)=\int e_i\left(t\right)dt$ and $L_{2i}\left(t\right)=e_i\left(t\right)$; then, substituting by (36) into (24), a new state space equation can be formulated as follows:

$$\left\{\begin{array}{cc}{\dot{L}}_{1i}\left(t\right)=\hfill & L_{2i}\left(t\right)\hfill \\ {\dot{L}}_{2i}\left(t\right)=\hfill & -K_{\mathrm{P}i}{L}_2\left(t\right)-K_{\mathrm{I}i}{L}_1\left(t\right)+{\tilde{g}}_i\left(t\right)-u_{\mathrm{TEC}i}\left(t\right)\hfill \end{array}\right.$$

(37)

Assumption 2. 

From (37), a positive constant ${\lambda }_i$ exists such that

$$|{\tilde{g}}_i|\le {\lambda }_i$$

(38)

where ${\lambda }_i$ denotes the upper boundary of NDO error.

In order to obtain fast convergence and strong robustness, the integral terminal sliding mode (ITSM) surface is proposed and can be defined as follows:

$${\sigma }_i\left(t\right)=L_{2i}\left(t\right)+{\eta }_{1i}\int [L_{2i}\left(t\right)+{\eta }_{2i}|L_{2i}\left(t\right){|}^{{\nu }_i}sgn\left(L_{2i}\left(t\right)\right)]dt$$

(39)

where ${\eta }_{1i}>0$, ${\eta }_{2i}>0$ and $0<{\nu }_i<1$ are designed parameters. By taking the first time-derivative for (39), one can obtain

$${\dot{\sigma }}_i\left(t\right)=-K_{\mathrm{P}i}{L}_{2i}\left(t\right)-K_{\mathrm{I}i}{L}_{1i}\left(t\right)+{\tilde{g}}_i\left(t\right)-u_{\mathrm{TEC}i}\left(t\right)+{\eta }_{1i}{L}_{2i}\left(t\right)+{\eta }_{1i}{\eta }_{2i}{|L_{2i}\left(t\right)|}^{{\nu }_i}sgn\left(L_{2i}\left(t\right)\right)$$

(40)

Then, the sub-controller law $u_{\mathrm{TEC}i}\left(t\right)$ is developed as

$$u_{\mathrm{TEC}i}\left(t\right)=u_{\mathrm{TEC}i}^{\mathrm{eq}}\left(t\right)+u_{\mathrm{TEC}i}^{\mathrm{re}}\left(t\right)$$

(41)

where $u_{\mathrm{TEC}i}^{\mathrm{eq}}\left(t\right)$ represents the equivalent control law to deal with the known components, and its corresponding equation can be given as follows:

$$u_{\mathrm{TEC}i}^{\mathrm{eq}}\left(t\right)=-K_{\mathrm{P}i}{L}_{2i}\left(t\right)-K_{\mathrm{I}i}{L}_{1i}\left(t\right)+{\eta }_{1i}{L}_{2i}\left(t\right)+{\eta }_{1i}{\eta }_{2i}{|L_{2i}\left(t\right)|}^{{\nu }_i}sgn\left(L_{2i}\left(t\right)\right)$$

(42)

While the reaching law $u_{\mathrm{TEC}i}^{\mathrm{re}}\left(t\right)$ is designed based on adaptive law to ensure that the switching surface can be reached by providing less chattering and achieving better tracking performance. Therefore, $u_{\mathrm{TEC}i}^{\mathrm{re}}\left(t\right)$ is constructed as follows:

$$u_{\mathrm{TEC}i}^{\mathrm{re}}\left(t\right)=[{\widehat{\lambda }}_i+{\rho }_i]sgn\left({\sigma }_i\right)$$

(43)

with

$${\dot{\widehat{\lambda }}}_i={\xi }_i|{\sigma }_i|;i_i={\xi }_i\widehat{\int }{\sigma }_{i}dt$$

(44)

where ${\xi }_i>0$ is the designed parameter. Finally, the control law of the proposed AMFITSMC is given as follows:

$$u_i\left(t\right)=\frac{1}{{\beta }_i}\left[-{\widehat{g}}_i+{\dot{x}}_i^{ref}+{\eta }_{1i}{L}_{2i}\left(t\right)+{\eta }_{1i}{\eta }_{2i}{|L_{2i}\left(t\right)|}^{{\nu }_i}sgn\left(L_{2i}\left(t\right)\right)+[{\widehat{\lambda }}_i+{\rho }_i]sgn\left({\sigma }_i\right)\right]$$

(45)

Theorem 2. 

Consider the system (18) re-formulated by ULM algorithm given in (24), under the designed AMFITSMC given in (45) with adaptive law given in (44), there exist appropriate coefficients ${\eta }_{1i}$, ${\eta }_{2i}$, ${\nu }_i$ and ${\rho }_i$ to ensure the stability of the closed-loop system and the tracking error asymptotically will converge to zero in finite-time.

Proof. 

Define a Lyapunov candidate function $V_i$ as

$$V_i=\frac{1}{2}{\sigma }_i^2+\frac{1}{2{\xi }_i}{\tilde{\lambda }}_i^2$$

(46)

where ${\tilde{\lambda }}_i={\widehat{\lambda }}_i-{\lambda }_i$.

By taking the first time-derivative for (46), one obtains

$$\begin{array}{cc}{\dot{V}}_i\hfill & ={\sigma }_i{\dot{\sigma }}_i+\frac{1}{{\xi }_i}{\tilde{\lambda }}_i{\dot{\tilde{\lambda }}}_i\hfill \\ \hfill & =\sigma [-K_{\mathrm{P}i}{L}_{2i}\left(t\right)-K_{\mathrm{I}i}{L}_{1i}\left(t\right)+{\tilde{g}}_i\left(t\right)-u_{TECi}\left(t\right)+{\eta }_{1i}{L}_{2i}\left(t\right)+{\eta }_{1i}{\eta }_i|L_{2i}\left(t\right){|}^{{\nu }_i}sgn\left(L_{2i}\left(t\right)\right)]\hfill \\ \hfill & +\frac{1}{{\xi }_i}{\tilde{\lambda }}_i{\dot{\tilde{\lambda }}}_i\hfill \\ \hfill & ={\sigma }_i[{\tilde{g}}_i\left(t\right)-[{\widehat{\lambda }}_i+{\rho }_i]sgn\left({\sigma }_i\right)]+|{\sigma }_i|[{\widehat{\lambda }}_i-{\lambda }_i]\hfill \end{array}$$

(47)

By substituting (41) and (44) into (47), it yields

$${\dot{V}}_i={\sigma }_i[{\tilde{g}}_i\left(t\right)-[{\widehat{\lambda }}_i+{\rho }_i]sgn\left({\sigma }_i\right)]+|{\sigma }_i|[{\widehat{\lambda }}_i-{\lambda }_i]$$

(48)

Since ${\tilde{g}}_i\le {\lambda }_i$, (47) becomes

$${\dot{V}}_i\le -{\rho }_i|{\sigma }_i|\le 0$$

(49)

According to the result in (49), it can be concluded that the origin ${\sigma }_i=0$ is ensured and the tracking error will asymptotically converge to zero under the proposed control law (45). □

4. Simulation Results

The proposed method is applied to FCHEV using a MATLAB/Simulink environment. The elements, including the converters and the energy sources, are modeled using SimPowerSystem blocks. FCHEV’s parameters used in the simulation are listed in

Table 3. The main FCHEV’s parameters.

Component	Parameter	Values
Vehicle	Aerodynamic drag coefficient $C_{\mathrm{x}}$	0.275
	Air density $\rho$	1.23 kg·m${}^{-3}$
	Vehicle frontal area $S_{\mathrm{f}}$	2.688 m${}^2$
	Vehicle total mass M	1550 kg
	Rolling resistance coefficient $C_{\mathrm{r}}$	0.014
Fuel cell	Maximum current	300 A
	Rated voltage	265 V
	Maximum net power	30 kW
Battery	Rated capacity $Q_{\mathrm{BAT}}$	20 Ah
	$SO{C}_{\mathrm{BAT};\max }$	90%
	$SO{C}_{\mathrm{BAT};\min }$	30%
Supercapacitor	Storage capacity	160 Wh
	$SO{C}_{\mathrm{SC};\max }$	90%
	$SO{C}_{\mathrm{SC};\min }$	40%
DC-link bus	Rated voltage	420 V
Converters	Inductor $L_{\mathrm{FC}}$, $L_{\mathrm{BAT}}$	2 × 10${}^{-3}$ H
	Resistance $R_{\mathrm{FC}}$, $R_{\mathrm{BAT}}$	0.1 $\Omega$
	Capacitor $C_{\mathrm{bus}}$	0.008 F

. The controller parameters are set as PI: $K_{\mathrm{P}}=(5.5,0.2,0.4)$ and $K_{\mathrm{I}}=(0.32,0.9,0.15)$; NDO-iPI: $K_{\mathrm{P}}=(7,3.5,0.8)$, $K_{\mathrm{I}}=(0.4,1.2,0.3)$, $\beta =(8,5,5)$, and $\mu =(10,10,8)$; and AMFITSMC: ${\eta }_1=(7,5,4)$, ${\eta }_2=(0.8,0.8,0.8)$, $\nu =(0.5,0.5,0.5)$, $\rho =(0.3,0.3,0.3)$, $\mu =(8,8,8)$.

In order to evaluate the proposed method, different typical driving cycles such as WLTP, UDDS, and HWFET are considered [54], as shown in Figure 7. Figure 8a, Figure 9a and Figure 10a present the corresponding load powers.

In order to show the effectiveness and superiority of the proposed EMS, fuzzy logic control [27] and operational mode strategy [55] are chosen for comparison. Figure 8b, Figure 9b and Figure 10b illustrate the power-sharing among the FC, BAT, and SC. The comparison of FC power under WLTP, UDDS, and HWFET driving cycles are depicted in Figure 8c, Figure 9c and Figure 10c, respectively. In contrast, the comparison of battery SOC under WLTP, UDDS, and HWFET driving cycles are displayed in Figure 8d, Figure 9d and Figure 10d, respectively.

It is seen that from Figure 8b, Figure 9b and Figure 10b, the FC can provide normal power as the primary energy source without facing fast power transient, while the BAT can assist the FC, providing the required power at steady state to reduce the FC power fluctuations. Moreover, it is visualized that from Figure 8b, Figure 9b and Figure 10b, the SC can supply the peak power during the vehicle accelerations/deceleration, which meets the rapid response of load power demand and decreases the load stress on both FC and BAT at the same time. It can be observed that from Figure 8d, Figure 9d and Figure 10d, all applied EMSs can keep the SOC of battery within the pre-defined region [30%, 90%], which guarantees the high efficiency of battery charging/discharging. In addition, the performance evaluation in terms of $\Delta SO{C}_{\mathrm{BAT}}$ is inserted in

Table 4. Final values of $SO{C}_{\mathrm{BAT}}$, ${\mathrm{H}}_2$ consumption, and the FC power fluctuation.

EMS	FC Power Fluctuation (W/s)			${\mathbf{H}}_2$ Consumption (L)			Final Value of ${\mathbf{SOC}}_{\mathbf{BAT}}$
	WLTP	UDDS	HWFET	WLTP	UDDS	HWFET	WLTP	UDDS	HWFET
Operational mode strategy	±1000	±1000	±900	22.82	14.38	13.15	73.5	74.95	70.6
Fuzzy logic control	±600	±600	±500	22.4	11.85	12.5	73.35	73.5	70.38
Proposed EMS	±300	±300	±250	19.25	10.21	11.85	71.52	72.6	70.05

. It is seen that from Table 4, $\Delta SO{C}_{\mathrm{BAT}}$ (the difference between the final $SO{C}_{\mathrm{BAT};\mathrm{f}}$ and initial $SO{C}_{\mathrm{BAT};\mathrm{in}}$ 70%) of EMSs based on fuzzy logic control and operational mode strategy is larger than that of the proposed method.

Furthermore, to illustrate the advantages of the proposed EMS in terms of reducing the FC power fluctuation, the FC power transient response compared to operational mode strategy and fuzzy logic control are given as depicted in Figure 11. It is visualized from Figure 11 with the operational mode strategy, FC power fluctuates near ±1000 W/s, ±1000 W/s, and ±900 W/s under WLTP, UDDS, and HWFET driving cycles, respectively. Compared to fuzzy logic control, the FC power fluctuates near ±600 W/s, ±600 W/s, and ±500 W/s under WLTP, UDDS, and HWFET driving cycles, respectively. Meanwhile, the suggested EMS provides a minimum number of FC power fluctuation points, which is decreased to ±300 W/s, ±300 W/s, and ±250 W/s using WLTP, UDDS, and HWFET driving schedules, respectively. It means that the proposed EMS has an advantage and superiority for improvement of FC’s lifetime significantly.

To further assess the effectiveness of the proposed EMS in terms of improvement of the fuel economy for the FCHEV, the equivalent ${\mathrm{H}}_2$ consumption is calculated considering all energy source characteristics using Equations (6), (8) and (10) as follows:

$$m_{{\mathrm{H}}_2}=\int \frac{n{M}_{{\mathrm{H}}_2}{I}_{\mathrm{FC}}}{2F}dt+\frac{{\alpha }_{\mathrm{ele}}}{3600Q_{\mathrm{BAT}}{\eta }_{\mathrm{BAT}}{\mathrm{C}}_{{\mathrm{H}}_2}}\int P_{\mathrm{BAT}}dt+\frac{3600\Delta SO{C}_{\mathrm{SC}}{f}_{\mathrm{SC}}}{{\rho }_{{\mathrm{H}}_2}{\eta }_{\mathrm{FC}}}$$

(50)

Figure 12 shows the comparative analysis of ${\mathrm{H}}_2$ consumption under WLTP, UDDS, and HWFET driving cycles. The consumed ${\mathrm{H}}_2$ quantities for each driving cycle are inserted in Table 4. Compared to fuzzy logic control, the suggested EMS can save about 14%, 13.8%, and 5.2% of consumed ${\mathrm{H}}_2$ under WLTP, UDDS, and HWFET driving cycles, respectively, while the suggested EMS can save about 15.7%, 28.9%, and 9.8% of consumed $H_2$ compared to operational mode strategy.

The tracking performance of DC bus voltage response under WLTP, UDDS, and HWFET driving cycles is displayed in Figure 13. It is evident that from Figure 13, the presented AMFITSMC method can efficiently stabilize the DC bus voltage at its desired trajectory $V_{\mathrm{bus}}=420V$, with less fluctuation compared to NDO-iPI and PI controller. Figure 14 and Figure 15, respectively, show the dynamic responses of FC and BAT currents. It is observed from Figure 14 and Figure 15 that the proposed AMFITSMC technique can stabilize the power converters with satisfactory dynamic performance.

Finally, it is concluded that the suggested EMS shows better performance in terms of ${\mathrm{H}}_2$ consumption reduction and decreasing the FC power fluctuation compared to operational mode strategy and fuzzy logic control under three typical driving cycles. Furthermore, the proposed AMFITSMC technique provides better tracking performance in the regulation of the DC bus voltage and energy storage currents, making it possible for the energy storage system to efficiently store or charge the energy during vehicle acceleration and deceleration.

5. Conclusions

In this paper, an effective EMS with model-free DC bus voltage control is developed for the FCHEV system to minimize the ${\mathrm{H}}_2$ consumption, prolong the FC lifespan, and improve the power performance at the same time. To decrease the load pressure on the FC and BAT, a frequency-separating based on HWT and interval type–2 fuzzy controller is used to decouple high-frequency contents of required power and send them to SC. Furthermore, the proposed AMFITSMC method is used to stabilize DC bus voltage and FC and BAT currents. The simulation results reveal that the proposed method provides satisfactory performance, which can effectively improve the fuel economy and can save about 15.7%, 28.9%, and 9.8% of consumed ${\mathrm{H}}_2$ compared to operational mode strategy under WLTP, UDDS, and HWFET driving conditions, respectively. The proposed EMS can save about 14%, 13.8%, and 5.2% of consumed ${\mathrm{H}}_2$ compared to fuzzy logic control under WLTP, UDDS, and HWFET driving cycles, respectively.