Optimizing EV Powertrain Performance and Sustainability through Constraint Prioritization in Nonlinear Model Predictive Control of Semi-Active Bidirectional DC-DC Converter with HESS

Abstract

The global transportation sector is rapidly shifting towards electrification, aiming to create more sustainable environments. As a result, there is a significant focus on optimizing performance and increasing the lifespan of batteries in electric vehicles (EVs). To achieve this, the battery pack must operate with constant current charging and discharging modes of operation. Further, in an EV powertrain, maintaining a constant DC link voltage at the input stage of the inverter is crucial for driving the motor load. To satisfy these two conditions simultaneously during the energy transfer, a hybrid energy storage system (HESS) consisting of a lithium–ion battery and a supercapacitor (SC) connected to the semi-active topology of the bidirectional DC–DC converter (SAT-BDC) in this research work. However, generating the duty cycle for the switches to regulate the operation of SAT-BDC is complex due to the simultaneous interaction of the two mentioned constraints: regulating the DC link voltage by tracking the reference and maintaining the battery current at a constant value. Therefore, this research aims to efficiently resolve the issue by incorporating a highly flexible nonlinear model predictive control (NMPC) to control the switches of SAT-BDC. Furthermore, the converter system design is tested for operational performance using MATLAB 2022B with the battery current and the DC link voltage with different priorities. In the NMPC approach, these constraints are carefully evaluated with varying prioritizations, representing a crucial trade-off in optimizing EV powertrain operation. The results demonstrate that battery current prioritization yields better performance than DC link voltage prioritization, extending the lifespan and efficiency of batteries. Thus, this research work further aligns with the conceptual realization of the sustainability goals by minimizing the environmental impact associated with battery production and disposal.

1. Introduction

The automobile industry is increasingly focused on electric vehicles (EVs). The present day’s unique circumstances create opportunities and challenges concerning the design and development of necessary infrastructure and accessories [1,2]. Among these, the energy-storage system (ESS) design is critical in EV construction. Therefore, researchers are inclined to explore efficient battery technologies to provide better power capacity, energy density, minimum cost, and size requirements based on the type of vehicle under consideration [3,4].

The ideal ESS should possess a combination of high energy density, an extended cycle life, minimal self-discharge, efficient performance, eco-friendliness, a wide range of operating temperatures, fast charging capabilities, and the absence of memory effects [5,6]. Safety and economic factors are also primary concerns that must be addressed [7]. Although fuel cells present a viable choice for ESS, they currently have limitations related to operational safety and incorporation with battery technologies [8,9]. batteries, on the other hand, are widely recognized as the preferred practical alternative due to their superior qualities [10]. However, their efficient operation in EVs is only possible under specific conditions of a limited temperature range and the charging/discharging rate [11]. Maintaining a constant current during the process is also crucial to minimize degradation and avoid thermal runaway, as the main performance parameters of lithium–ion batteries are temperature-sensitive.

Moreover, EV operation conditions depend on the dynamic variation of the load, and the peak performance relies on the power density of ESS to handle the transient power flow in the powertrain based on the practical driving pattern [12]. This causes batteries to release a significant amount of energy in short time intervals, leading to faster deterioration of the battery’s health and premature failure. Supercapacitors, with their high-power density and capability of charging and discharging at high C-rates [11], are a good option for handling the transient power flow of the EV powertrain [4]. However, unlike batteries, they fail to provide the desired high energy required for extended operation periods [13].

Therefore, combining batteries and SC to form a hybrid energy storage system (HESS) is an excellent solution to overcome individual limitations due to their complementary behaviour [14]. In such a composition, the battery is designated to supply the constant continuous power requirement at a constant current value. In contrast, SC is strategized to deal with the transient power demands, recovery of the power supplied from the battery at light load conditions, and the absorption of regenerative braking energy. In addition, the inclusion of SC avoids oversizing the battery pack and the consequent increase in weight, cost, and depth of charge/discharge cycles [15]. The battery and SC complement each other in the HESS to incorporate the high energy density and power density required for EV powertrain operation. Its comparison with various other energy storage technologies [14,16] is presented in the Ragone plot in Figure 1.

Constant DC link voltage is expected to be fed to the inverter to drive the motor load in the EV powertrain. Bidirectional configurations are necessary to achieve improved performance and efficiency in power transfer [17]. Interconnection of the battery and SC is conducted with different converter topologies depending on the dynamics of the load side [18] and is presented in Figure 2. A conventional parallel connected fully active topology for HESS, and its detailed operation is provided in [19], in which controlling the energy transfer is easy to implement. However, efficiency drops due to the presence of two converters. To overcome this drawback, the size of the DC link capacitor must be increased, resulting in sizing issues. On the other hand, the passive HESS connection is simple and straightforward. Nevertheless, the power transfer to the DC bus is not controlled. It is further inferred that the semi-active topology is a good trade-off between the two topologies, as mentioned above, in regulating energy transfer based on the comparative study presented in [20,21]. The SAT-BDC has a simple design at low cost, overcoming the drawbacks of both fully active and passive bidirectional topologies [22], and it is best suited for the research problem in consideration. Further, the primary objective of SAT-BDC operation is to ensure that a constant continuous current is drawn from the battery pack by including SC to handle the transient variations of the energy exchange. Simultaneously, it is crucial to maintain a constant DC link voltage at the output of the topology connected to HESS so that the inverter and drive performance is improved in the EV powertrain.

In general, the input current and the output voltage are regulated in a basic bidirectional DC–DC converter (BDC) by the voltage-current (VI) control method by applying PID compensators [22]. However, the conventional PID controllers operate independently and need individual tuning parameters with negligent reciprocation between multiple control loops. They fail to effectively confront the multifaceted current research problem since multiple variables interact simultaneously. Also, tuning the controllers is tedious when power splitting between battery and SC is involved [13,23]. Therefore, a robust and reliable control approach is necessary.

Various approaches to control the energy flow from the HESS are reported in [21,24]. The artificial neural networking and fuzzy logic control approaches show improvement in minimizing the overshoot compared to classic controllers without using the mathematical model of the converter [25]. However, effective data-driven control methods are too complex to design due to the necessity of large amounts of data, which is not suited for low-cost, low-power EV applications [26]. The sliding mode control is tested for BDC in [27], which is further improved by including the extremum-seeking control strategy in [28] without the actual model, which is not flexible enough to be applied for dynamic EV operating conditions. The integrated optimization problem to develop the energy split between a battery and supercapacitor utilizing the dynamic programming approach using a preset cost function is presented in [29]. Further, approximate dynamic programming is employed to develop optimal and flexible switching of BDC, as explained in [30]. However, dynamic programming is computationally expensive and unsuitable for real-time applications. The different control approaches and MPC are compared and summarized in

Table 1. Comparison of different control approaches and model predictive control.

Control Approach Their Features	Model of BDC	Ease of Implementation	Flexibility	Realtime Applicability	Computational Complexity	Tuning Complexity
Sliding mode control [27,28,30]	Necessary	Medium	Not Flexible	Well suited	Low	Medium
Extremum-Seeking Control [28]	Not Necessary	Medium	Less flexible	No	Medium	Medium
Artificial Neural Network [26]	Not Necessary	Depends on the training of the input data	Flexible	Depends on the accuracy of data	High	High
Fuzzy logic control [31,32]	Not Necessary	Depends on sophistication of the rule base	Flexible	Depends on training procedure	High	High
Dynamic programming [33,34,35]	Not Necessary	Easy	Flexible	No	Low but expensive	Low
Model Predictive Control [36,37,38]	An accurate model is necessary	Easy	Less Flexible	Well suited	It depends on the complexity of the model, but efficient	Low

.

It is evident that MPC is best suited for plants with accurate models with multiple control objectives, such as constant battery current and DC link voltage regulation in SAT-BDC. However, computational cost and complexity increase with the system’s nonlinearity [38]. The interconnected control objectives are optimally achieved with proper management of the control output (duty cycle D in this case) to minimize the difference between the actual system state and the desired reference value with better flexibility and controllability. The constraints with suitable safety limits are imposed to ensure the system limitations with proper weightage [39]. Predicting the future behaviour of the system model is quite significant in EV powertrain applications to allocate the power sharing between the battery and SC, which is unique in this control approach. In [40], a flexible design of MPC is applied to a boost converter to achieve better transient operating conditions with a DC power source and a simple constant load. A hybrid MPC is proposed to optimize the switching of the DC–DC converter in microgrids, considering stability against existing disturbances both in transient and steady states [41]. Still, it fails to address the issue of variable current discharge from the battery.

Furthermore, to address the challenges posed by the long-life expectancy of batteries and improved performance of EV powertrain operation, the use of supercapacitors (SCs) has been proposed in the existing literature. However, developing an effective control strategy to exploit transients and improve efficiency is still under consideration. The classical voltage current control approaches separately address the two variables (i.e., output voltage and the input/source current) in a BDC. Also, conventional VI controllers fail to tackle the issue of interactions between the two variables, i.e., the current drawn from the battery and the DC link voltage. The idea of applying the flexible NMPC approach with future predicting behaviour is extended to the specific application to improve the performance of the lithium–ion battery in EV powertrain in this research work. To realize the novel approach, HESS, including the lithium–ion battery and SC, is considered with SAT-BDC having a variable load sequence. Therefore, this research work focuses on exploring the potential of a semi-active topology of the bidirectional DC–DC converter (SAT-BDC) as represented in Figure 2c with SC directly connected to the DC bus with a reliable and robust nonlinear model predictive control to regulate the battery current under the prescribed value so that the objective of improving the battery performance is achieved along with DC-link voltage tracking. The different prioritizations are tested with the constraints above to optimize the performance of the proposed system further. The significant contributions of the proposed research work are:

• Modelling and design of semi-active topology of the bidirectional DC–DC converter (SAT-BDC) to transfer energy from a HESS incorporating the mathematically modelled lithium–ion battery and supercapacitor (SC) to a variable load model in the open loop to validate the concept.
• Investigation of the performance of the applied nonlinear model predictive control (NMPC) to enhance the operation and life span of the lithium–ion battery pack by simultaneously maintaining constant battery current and regulating DC link voltage at the output by tracking its reference value with different prioritizations.
• Simulation of individual case studies to validate the impact of different prioritizations of the selected variables and the difference in the performance of the system.

The rest of the paper is structured in the remaining five sections. The system components and their mathematical models are discussed in Section 2. The configuration of the proposed system is presented along with the application of the NMPC control architecture described in Section 3. The system parameters considered for the simulation study and the detailed result analysis are presented in Section 4. The conclusion is reported in Section 5.

2. Modelling of the System Components

This section describes the physical components with accurate mathematical modelling. A second-order resistive capacitive (2-RC) cell model of the lithium–ion battery is derived from the literature in Section 2.1. A suitable SC model is also designed in Section 2.2. Further, commercially available suitable lithium–ion battery and SC with the nearest matching parameter values are selected. The variable load current sequence is generated in the MATLAB/Simulink platform to replicate the motor drive and inverter operation, discussed in Section 2.3. Further, the mathematical modelling of SAT-BDC is also conducted using state-space analysis in Section 2.4.

2.1. Battery

The lithium–ion battery with high energy density must be able to provide the desired voltage rating and capacity to drive the EV powertrain system continuously. This is achieved by developing a battery pack consisting of several individual lithium–ion cells in a series–parallel combination [31]. The exact electrochemical process inside the cell is difficult to comprehend. Therefore, the behaviour and performance of the lithium–ion cell are realized based on the electric equivalent circuit model (EECM), which is a more straightforward approach to modelling the dynamic operation using voltage sources, resistors, and capacitors, as presented in Figure 3. The accurate electrical battery runtime 2-RC model circuit is chosen [42] in this analysis since the 2-RC model is considered to be the best trade-off between accuracy and sizing of the battery [43,44].

The parameters, namely terminal voltage (Vt), open-circuit voltage (Vo), the battery current (I_L), capacity (C) in Amp-hour, and the state of charge (SoC) of the battery in %, are estimated using the electrical equivalent circuit modelling of the cell. The relevant equations are referred from [43,44,45,46,47] as shown below.

$$C_{capacity}=3600\times C_{initial}\times f_1\left(N\right)\times f_2\left(T\right)$$

(1)

Here, C_capacity is the usable capacity, C_initial is the initial nominal capacity of the battery in Ah mentioned in the nameplate, f₁(N) is the cycle number correction factor, and f₂(T) is the temperature-dependent correction factor, which is set as 1 since it is not varied much. However, the cycle number correction factor is provided as,

$$f_1\left(N\right)=1-\left(k_1\sqrt{N}\right)$$

(2)

where k₁ = 4.5 × 10⁻³ and N are the cycle number, it is observed that the usable capacity of the battery C_capacity does not depend on current variation. Instead, it is represented by a current-controlled source I_batt to influence the voltage value Vo dynamically. Also, R_{selfdischarge} in Figure 3 represents the losses when the battery is unused. Further, Vo is dependent on SoC, and its average value is a monotonically increasing function. The SoC is the ratio of the usable current capacity to its nominal initial capacity value. The relationship between Vo and SoC can be represented by a third-order polynomial equation, as shown below.

$$V_{O\left(SoC\right)}=a_O+a_{1}SoC+a_{2}So{C}^2+a_{3}So{C}^3$$

(3)

Vo depends on SoC, and its average value is a monotonically increasing function. a₀, a₁, a₂, and a₃ are the constant values based on the manufacturing materials of the battery.

In addition, the SoC is defined as follows for discharging operation:

$$SoC=So{C}_O-\int {\displaystyle \frac{I_{batt}}{C_{capacity}}}dt$$

(4)

where SoCo is the initial SoC value of the battery.

The general equation for the terminal voltage Vt is provided as follows:

$$V_t=V_O-I_L\left[R_O+\sum _{i=1}^n{R}_i\left(1-e^{{\displaystyle \frac{-t}{R_i{C}_i}}}\right)\right]$$

(5)

R_O, the series ohmic resistance, is responsible for the voltage drop when the load is introduced. R_i and C_i are the i-th polarization resistance and i-th polarization capacitance. i = 2 in the present research case since the second-order resistive, while the capacitive (2-RC) cell model is chosen. The RC networks are responsible for the transient operating conditions. Therefore, the Equation (5) is reduced to

$$V_t=V_O-I_L{R}_O-V_1-V_2$$

(6)

where

$$V_1=I_1{R}_1$$

(7)

$$V_2=I_2{R}_2$$

(8)

$$\stackrel{•}{V_1}=-{\displaystyle \frac{1}{R_1{C}_1}}{V}_1+{\displaystyle \frac{1}{C_1}}{I}_L$$

(9)

$$\stackrel{•}{V_2}=-{\displaystyle \frac{1}{R_2{C}_2}}{V}_2+{\displaystyle \frac{1}{C_2}}{I}_L$$

(10)

These equations can be further used for SoC estimation and temperature dependency of the parameters of the battery. R₀, R₁, R₂, C_1, and C₂ values depend on the SoC and the battery current value I_L. The values corresponding to the RC parameters at 60% of SoC are chosen because it is the mean average condition in a battery charge–discharge cycle between 100 and 20%, as shown below in

Table 2. Battery specifications and the parameter values corresponding to 60% SoC.

Parameter	Value	Remarks
Battery voltage Vt	48 V	[18]
Battery output current IL	80 A	Average value consideration
DC-link voltage reference VDC	72 V	Based on motor consideration
a0	3.2416	[48] Values of Panasonic NCR18650PF lithium–ion battery
a1	1.3905
a2	1.3781
a3	0.9206
R0	0.0314 Ω
R1	0.0181 Ω
R2	0.0281 Ω
C0	0.1712 F
CV	0.55257 F

.

2.2. Supercapacitor

SC is the high-power density storage applied to the system to instigate transient power management. The actual structure of the SC is depicted in Figure 4a. The operation with the mathematical modelling of the SC is derived from [44]. In this work, the 2-RC model is considered to study the system’s dynamic behaviour [18]. It includes a leakage resistance R_f, two branches with non-linear capacities, and different voltages. The circuit is shown in Figure 4b.

The first branch is the fast/main cell representing the charging phases and modelled by resistance R₁ and non-linear capacitance value C₁, which is calculated using the following formula:

$$C_1=C_0+C_V{V}_1$$

(11)

where C₀ is a constant capacitance, and C_V is a constant value with unit Capacitance/Volt. The nonlinear capacity C₁ is decided by the voltage between the terminals of the main cell V₁.

The voltage V₁ is well-defined as

$$V_1={\displaystyle \frac{-C_0+\sqrt{{C_0}^2+2C_V{Q}_1}}{C_V}}$$

(12)

Whereas the current i₁ is denoted in terms of instantaneous charge Q₁ and capacity C₁ in the following equations.

$$i_1={\displaystyle \frac{d{Q}_1}{dt}}=C_1{\displaystyle \frac{d{V}_1}{dt}}=\left(C_0+C_V{V}_1\right){\displaystyle \frac{d{V}_1}{dt}}$$

(13)

The instantaneous charge Q₁ is defined by

$$Q_1=C_0{V}_1+{\displaystyle \frac{1}{2}}{C}_V{V_1}^2$$

(14)

The second cell is the slow branch representing the redistribution of the charges, which is modelled by resistance R₂ and capacitance C₂ with larger time constants.

The voltage V₂ is provided by

$$V_2={\displaystyle \frac{1}{C_2}}\int i_{2}dt={\displaystyle \frac{1}{C_2}}\int {\displaystyle \frac{1}{R_2}}\left(V_1-V_2\right)dt$$

(15)

The leakage resistance R_f signifies the self-discharge of energy throughout the process, and the leakage current value through the R_f is negligible in the circuit operation. The following equation defines the voltage across the SC:

$$U_{SC}=N_{S-SC}{V}_{SC}=N_{S-SC}\left(V_1+R_1+{\displaystyle \frac{I_{SC}}{N_{P-SC}}}\right)$$

(16)

where U_SC and I_SC are the voltage and current of the SC, respectively, N_P-SC and N_S-SC provide the numbers of parallel and series connections.

The final capacitance value of the SC is provided by

$$C_{SC}=C_1+C_2$$

(17)

In this research work, the 2-RC, as mentioned above, model is utilized. The specified parameters mentioned in the equations are calculated. BMOD0063 P125 B04/B08 by Maxwell Technologies is the nearest available SC module, best suited for practical conditions and theoretical calculations. The finalized values to be validated by the simulation process are arranged in

Table 3. Specifications and the parameter values of supercapacitor.

Parameter	Value	Remarks
DC link Voltage VDC	72 V	[49]
Maximum output current IR	100 A	[50]
Rated capacitance CSC	63 F	Values are taken from the datasheet of the BMOD0063 P125 B08 model.
Maximum initial capacitance	100 F
Rf	18 mΩ
R1	0.8 Ω
R2	1 Ω
C1	0.1712 F
C2	0.55257 F

.

2.3. DC-Link Load Current Generation

For any constant load, the performance of the BDC is already achieved in [28]. However, in this research work, the variable load must be considered to realize the motor drive load connected to an inverter in an EV powertrain under realistic conditions. The load current sequence is generated by adding Gaussian noise with the average value. The average value of the variable output current sequence is kept at 30A for the simulation purpose and is generated in MATLAB to replicate the variable load model. The generated load current waveform is presented in Figure 5.

2.4. Semi-Active Topology of Bidirectional DC-DC Converter (SAT-BDC)

The structure of the SAT-BDC utilized in this work is simple and has a reduced component count because the filter capacitor has been replaced with SC. This structure has the advantage of limiting the voltage ripples during the bidirectional energy flow. Moreover, connecting the SC directly to the output side (DC bus in EV applications) aids the controller in maintaining the constant DC link voltage. In addition, SC acts as a low-pass filter for the lithium–ion battery without any additional components. The filtering effect depends on the SC capacity, which is sufficient to manage the fluctuations during the transient operating condition. As mentioned in [20], the presence of SC effectively minimizes the variations in the battery current level. The circuit diagram of SAT-BDC is depicted in Figure 6.

When switch S₁ is on and switch S₂ is off, the dynamics of the SAT-BDC are governed by (18)–(20) [18].

$$\dot{x}=A_{c1}x+B_{vc}{V}_s+B_{ic}{i}_R$$

(18)

And when the switch S₁ is off and the switch S₂ is on, the dynamics of the converter are governed by

$$\dot{x}=A_{c2}x+B_{vc}{V}_s+B_{ic}{i}_R$$

(19)

The average dynamics over one switching period are thus provided by

$$\dot{\tilde{x}}=\left(A_{c1}d+A_{c2}\left(d-1\right)\right)\tilde{x}+B_{vc}{V}_s+B_{ic}{i}_R$$

(20)

Here

$$A_{c1}=\left[\begin{array}{cc}{\displaystyle \frac{-1}{R_{L}L}}& 0\\ 0& 0\end{array}\right]\hspace{1em}\hspace{1em}\hspace{1em}{A}_{c2}=\left[\begin{array}{cc}{\displaystyle \frac{-1}{R_{L}L}}& {\displaystyle \frac{-1}{L}}\\ {\displaystyle \frac{1}{C_{SC}}}& 0\end{array}\right]$$

$$B_{vc}=\left[\begin{array}{c}{\displaystyle \frac{1}{L}}\\ 0\end{array}\right]\hspace{1em}\hspace{1em}\hspace{1em}{B}_{ic}=\left[\begin{array}{c}0\\ {\displaystyle \frac{-1}{C_{SC}}}\end{array}\right]$$

The circuit parameters of the SAT-BDC topology inductor, the series resistance of the inductor, and the capacitor values are calculated from the design equations [31,51] developed in the steady state. The selected parameter values are tabulated in

Table 4. Specifications and the parameter values of SAT-BDC.

Parameter	Value	Remarks
Number of switches	2
Switching frequency, f	20,000
Inductor L	490 µH	[18]
Series Resistance of inductor RL	90 mΩ
Capacitor CSC	0.0063 F/ 63 F
Average load current IL	50 A
Output Voltage (DC link) VDC	72 V
Input or source voltage Vt	48 V

, as shown below.

$$V_{DC}{I}_R=V_t{I}_L+I_L^2{R}_L{\displaystyle \frac{V_t\left(1-u_{sim}\right)R}{R_L}}$$

(21)

$$L={\displaystyle \frac{{\left(1-u_{sim}\right)}^2{u}_{sim}R}{2f}}$$

(22)

$$C_{SC}={\displaystyle \frac{u_{sim}}{Lfk}}$$

(23)

$$R={\displaystyle \frac{V_{DC}}{I_R}}$$

(24)

where R is the variable load resistance value, k is the source side voltage ripple factor, u_sim is the duty cycle, f is the switching frequency, and R_L is the series effective resistance of the inductor.

3. Configuration of the Proposed System with Nonlinear Model Predictive Control

The mathematical model of an SAT-BDC [31] with an active variable load (generated variable current sequence to replicate the load model of EV drivetrain), as mentioned in Section 2.3, is developed in MATLAB. It is connected to the mathematical model of the battery system under consideration. The open-loop performance is observed at first to verify the validity of the designed mathematical model. The voltage values are selected based on the standard values obtained from the battery and motor design values. Furthermore, NMPC is employed to regulate the two constraints, namely, the inductor current (I_L) (the current discharged by battery) of the SAT-BDC and the DC link voltage (V_DC) across the variable load under consideration simultaneously. The time horizon is chosen to achieve and realize the steady-state condition. The time step in each prioritization is different depending on the simulation parameters. Moreover, the filter capacitor of the converter is replaced by the mathematical model of SC with a suitable value, as mentioned in Section 2.2. The NMPC is modified accordingly. The complete configuration consisting of a lithium–ion battery with an SC acting as the HESS and the modified SAT-BDC, whose operation is regulated by NMPC, is shown in Figure 7. As mentioned earlier, the central objective is to compute the duty cycle of SAT-BDC, ensuring that the battery current remains within its constraints, maintaining a constant value, and precisely tracking the DC-link voltage reference. Both targets are accomplished by suitably applying non-linear model predictive control in this research work.

Proposed Nonlinear Model Predictive Control Strategy

The modes of operation of the SAT-BDC are based on the switching conditions. The NMPC operation depends on the constraints considered. The main objective of the research is to derive an optimal switching strategy to maintain the DC link voltage at the reference level and keep the inductor current at a constant value with the help of SC.

At first, the NMPC model is discretized using Euler discretization to obtain the following matrices:

$$A=I+{\delta }_t\ast \left(A_{c1}d+A_{c2}\left(d-1\right)\right)$$

(25)

$${B_v=\delta }_t\ast B_{v_c}$$

(26)

$$B_i={\delta }_t\ast B_i$$

(27)

A, B_i, and B_v are the constraints developed based on the state space analysis.

Finally, the cost function for the NMPC is provided by

$$f\left(z\right)=\sum _{\left\{k=1\right\}}^N{\left(v_{{DC}_k}-v_{{DC}_{ref}}\right)}^2+{\left(i_{L_k}-i_{L_{k+1}}\right)}^2+{ϵ}^2$$

(28)

where $z=\left[i_{L_1},v_{c_1},\dots ,i_{L_N},v_{c_N},d_1,\dots ,d_N,{ϵ}_1,\dots ,{ϵ}_N\right]$, k is the iteration number, N is the time horizon, and ϵ is the inequality constant to convert the inequalities of the constraints to the equality. The inequality constant is applied to define the soft constraints since there is a mutual interaction that is allowed in NMPC.

The optimization problem is thus provided by

$$z=\underset{z}{\mathit{min}}f\left(z\right)$$

(29)

$$x_{k+1}=A\ast x_k+B_v\ast V_s+B_i\ast I_R$$

(30)

Such that

$$i_{L_{min}}\le i_{L_k}\le i_{L_{max}}+ϵ \forall k\in \left[0,N\right]\mathrm{and}0\le d_k\le 1\forall k\in \left[0,N\right]$$

In this work, since the NMPC employed for the SAT-BDC is flexible, simulations are conducted based on the different priorities of the constraints, namely the battery current and DC link voltage. There is an interaction between the considered constraints inductor current I_L and the DC link voltage V_DC; equality is required to establish and prioritize the constraints. Since NMPC allows establishing the priority of one constraint over other using inequality constraint ϵ, the user can utilize this function to give prominence to the individual constraint based on its significance in the application. In EV applications, maintaining constant battery current is more significant than regulating DC voltage. Hence, the focus is provided to the same. The actual impact of the different prioritization of these variable parameters is studied and analyzed. The choice of prioritization plays a key role in the optimum performance and health of the battery. The results are presented in different cases with the parameters derived in Section 4.

4. Results and Analysis

The detailed study of the SAT-BDC connected to a variable active load current with a HESS involving a lithium–ion battery and SC is conducted in this work to corroborate the conceptual interpretation and mathematical analysis of the proposed system. The converter is controlled by the NMPC approach with the intent to discharge the battery at a constant current value and regulate the DC link voltage, tracking a reference value of 72~V. Different operating conditions and priorities are investigated and presented in five different sub-sections further.

Initially, the open-loop operation of the converter is conducted with only a lithium–ion battery as the source in Section 4.1. Further, the closed-loop operation of the converter topology is conducted with the NMPC approach in Section 4.2. To improve the performance of the battery current, SC replaces the filter capacitance of the BDC to make it a semi-active BDC topology with HESS. The NMPC algorithm is modified accordingly, as described in Section 4.3. The system is examined further by prioritizing the DC link voltage and the inductive current to observe the performance of the system in Section 4.4 and Section 4.5, respectively.

4.1. Open-Loop Optimal Solution without Supercapacitor

The open-loop optimal solution is tested with MATLAB simulation to validate the conceptual modelling and design with three initial conditions: A, B, and C. The open-loop simulation is conducted without applying the objective function to the constraints in NMPC. However, it is allowed to define constraints, and the control output duty cycle is not controlled. Instead, the control output duty cycle in the open-loop simulation is computed based on the theoretical calculations in coding for the next N timesteps.

This is demonstrated for all three different initial conditions: A, B, and C. In Scenario A, all the parameters are considered to have zero initial values. In Scenario B, random values are taken at the initial condition. Scenario C consists of the actual parameter values calculated from the design equations at the initial condition. The parameters selected for the NMPC tuning are presented in

Table 5. Simulation configuration of the open loop for the NMPC approach.

Parameter	Value
Time horizon, N	20 timestep
One timestep for MPC	0.0005 s
Weightage of voltage, current, and slack variables	1, 1, 1000 respectively

. The inductor current I_L, the DC link voltage V_DC, the duty cycle u_sim, and the slack variable ϵ are represented in Figure 8.

In all three scenarios, it is evident that the system’s current remains stable without oscillation, and the voltage progressively approaches its reference value. It should be noted that the chosen time horizon is sufficient for the open-loop system to remain stable, as indicated in the subsequent sections. However, in Scenario A, the system voltage value does not reach a steady state within the designated time horizon. In contrast, Scenario C has reached a steady state, resulting in the optimal solution to maintain a constant duty cycle. For Scenarios A and B, the steady-state outcome is not attained, and the optimal solution involves initially applying a high-duty cycle and gradually reducing it over time. The feasibility of the problem is evident from the plot depicting the value of the slack variable, which remains within acceptable bounds. Figure 8 above shows the open-loop optimal solution computed by the MPC for three different initial conditions. In Scenario C, where the system has reached a steady state, the optimal solution is to keep the same steady-state value without any modification in the feedback provided, and there are no disturbance signals. However, in the practical application of EV, the disturbance signals are present in the form of braking and variable load dynamics. Therefore, NMPC control must be applied to the mentioned variables/constraints.

4.2. Closed Loop Simulation, without Supercapacitor

This section presents the results of the closed-loop system, where the duty cycle for timestep zero is applied at each subsequent timestep. The NMPC can simultaneously control the voltage to a particular reference value and the steady non-oscillatory inductor current in an effective manner. From the user’s point of view, it is a trade-off whether it is more important to reach the DC-link voltage reference quickly or keep the current as constant as possible. When there is no SC, the system has faster system dynamics, which makes it necessary to use a smaller timestep for the NMPC implementation. Still, it also allows for a shorter time horizon. Nonetheless, the chosen time horizon is sufficient to achieve the desired control objectives, as evident from the performance of the controller. The simulation parameters of the configured system are presented in

Table 6. Simulation configuration of the closed-loop operation of SAT-BDC without SC.

Parameter	Value
Time horizon, N	30 timestep
One timestep for MPC	0.0005 s
Weightage of voltage, current, and slack variables	100, 1, 1000 respectively

. Figure 9 represents the respective waveforms of the inductor current with the output current, DC link voltage, duty cycle, and inequality constraints.

Figure 9a shows that a very high current is drawn from the battery, which is not appreciated for its longer life. The presence of SC is necessary to regulate the battery current under a certain safe limit. The voltage and duty cycle waveforms are presented, showing the effectiveness of NMPC control. Since the battery current performance is significant, the current is prioritized 100 times more than the DC-link voltage constraint in NMPC.

4.3. Closed-Loop Simulation with the Integration of Supercapacitor

Incorporating a supercapacitor leads to a more stable output voltage compared to a system without it. However, achieving the desired reference voltage takes longer due to the time delay introduced by the supercapacitor. It is important to note the timescale difference compared to the previous figures. The high capacitance value allows for longer timesteps for the nonlinear MPC, contributing to improved computational efficiency. The configuration parameters of NMPC are listed in

Table 7. Simulation configuration of closed-loop NMPC with the integration of SC in SAT-BDC.

Parameter	Value
Time horizon, N	3000 timestep
One timestep for MPC	0.0005 s
Weightage of voltage, current, and slack variables	1, 1, 1000 respectively
The capacitance C	63 F

. Both constraints are allotted equal weightage. Figure 10 represents the inductor current, DC link voltage, duty cycle, and equality constraints. In this scenario, the system configuration remains the same as in the previous simulation, but the noise on the inductor current increases. Despite the increased noise, the controller effectively controls the DC link voltage, closely tracking its reference. However, more significant variations in the inductor current are observed, highlighting the impact of increased noise on system dynamics.

4.4. Closed-Loop Simulation of HESS with Voltage Prioritization

In the previous section, equal prioritization is provided for both the constraints, and as a result, noise is observed in the inductor current waveform. Hence, the priority for the constraints is altered, and higher priority is provided to DC link voltage.

Table 8. Simulation configurations of closed-loop under voltage priority condition.

Parameter	Value
Time horizon, N	3000 timestep
One timestep for MPC	0.0005 s
Weightage of voltage, current, and slack variables	100, 1, 1000 respectively
The capacitance C	63 F

denotes the simulation configuration of the system under voltage prioritization. Figure 11 represents the corresponding waveforms under voltage-prioritized conditions. In this simulation, the primary control objective is to prioritize the regulation of the DC link voltage. The figures demonstrate that the DC link voltage closely follows its reference, indicating effective control. However, the inductor current exhibits oscillations due to the controller’s prioritization of voltage control.

4.5. Closed-Loop Simulation of HESS with Current Prioritization

In this simulation, the primary control objective is to prioritize maintaining a constant current from the battery source. As a result, the inductor current exhibits significantly fewer variations than the previous scenarios. However, the improvement in current control comes at the expense of the DC-link voltage control.

Table 9. Simulation configuration of closed loop under current priority condition.

Parameter	Value
Time horizon, N	1000 timesteps
One timestep for MPC	0.005 s
Weightage of voltage, current, and slack variables	1, 100, 1000 respectively
The capacitance C	63 F

presents the system configuration parameters and the priority of the constraints considered in the simulation. Figure 12 shows the respective waveforms, and it is visible that the controller struggles to track the voltage reference accurately in this case. Still, voltage variations are under the considered limits and acceptable for the particular application. It must be observed that the time steps required to obtain the steady state are also reduced, which is an added advantage in the practical scenario. This scenario is best suited for the EV powertrain application since the battery current performance is better compared to previous case studies, and the voltage performance is still under limits. In addition, the duty cycle is also between 0.45 and 0.65, which is almost ideal for the operation of BDC.

5. Discussion

In this simulation, the primary control objective is to prioritize maintaining a constant current from the battery source. As a result, the inductor current exhibits significantly fewer variations than the previous scenarios. However, the improvement in current control comes at the expense of the DC-link voltage control. Table 9 presents the system configuration parameters and the priority of the constraints considered in the simulation. Figure 12 shows the respective waveforms, and it is visible that the controller struggles to track the voltage reference accurately in this case. Still, voltage variations are under the considered limits and acceptable for the particular application. It must be observed that the time steps required to obtain the steady state are also reduced, which is an added advantage in the practical scenario. This scenario is best suited for the EV powertrain application since the battery current performance is better compared to previous case studies, and the voltage performance is still under limits. In addition, the duty cycle is also between 0.45 and 0.65, which is almost ideal for the operation of BDC.

The presence of the SC stabilizes the battery performance, which is detected by comparing the waveforms between Section 4.2 and Section 4.3. It is observed that the inclusion of the supercapacitor lessens the switching stress by reducing the average duty cycle value from 0.75 to 0.6. However, in voltage prioritization, the average duty cycle value is increased to 0.75. In contrast, in the current prioritization, the average duty cycle is observed to be 0.55, which provides better performance of the switches in the converter system. This clearly shows that the current prioritization and the equal prioritization condition are much better than the voltage prioritization condition. However, the current prioritization improves the battery performance and life by reducing the current ripples to a great extent, as depicted in Figure 12a. The current prioritization may help increase the battery performance and health due to the improvements in the current waveform. It is evident that voltage prioritization is not preferred in converter systems at low-power EV applications. However, in high-voltage, high-power vehicles, voltage prioritization may be applied, which needs to be explored. Voltage priority is preferred over the current priority of the battery system in distributed generation systems like microgrid applications.

6. Conclusions

This paper explores the challenges associated with controlling the DC link voltage and constant battery current as constraints in the SAT-BDC system with the battery and supercapacitor as sources of HESS. A novel solution is proposed using NMPC, considering the EV powertrain application. Based on the results, NMPC proves to be a powerful optimization technique, capable of precisely regulating the DC link voltage to a specified reference value while maintaining a constant extraction of current from the battery to enhance its life. The effectiveness of NMPC in delivering improved overall performance is demonstrated in this application by utilizing an accurate nonlinear model representation and careful formulation of the optimization problem.

Incorporation of the interaction between the two constraints is conducted through prioritization of the mentioned constraints as a key characteristic in this research. The results of the simulation validate the efficacy of the NMPC approach in regulating the DC link voltage and maintaining a constant current. The integration of a supercapacitor enhances the stability of the output voltage, while the prioritization of control objectives influences the trade-off between the DC link voltage and inductor current control. These findings contribute valuable insights for the practical implementation of NMPC in real-world control systems, helping to optimize performance and achieve desired control objectives based on the application. This research work exhibits the ability of the NMPC method to be more flexible compared to other methods. Based on the results, the current prioritization is better suited for EV powertrain applications to maintain a constant battery output current to increase the performance and life expectancy of the battery, which is the primary source of energy storage.

With its capability to handle multiple control objectives and enforce safety constraints, NMPC emerges as a valuable control strategy for applications, requiring robust and efficient control of complex and nonlinear systems. However, the reliance on accurate models and the associated computational costs warrants further investigation and consideration when applying NMPC in practical implementations. Moving forward, future research could explore advanced model-identification techniques and improved computational strategies to enhance the applicability and scalability of NMPC for diverse control challenges. Additional control objectives can be added to the optimization problem, such as temperature and SoC control of the battery. Overall, the insights gained from this study contribute to the growing body of knowledge in control theory and provide a valuable framework for optimizing DC-link voltage control and constant battery current in practical EV applications.