Battery State of Charge Management for an Electric Vehicle Traction System

Abstract

This chapter introduces a battery state of charge (SOC) management technique designed for an electric vehicle traction system that incorporates an indirect field-oriented induction motor drive. The primary goal of this technique is to restrict the change in battery SOC from exceeding a maximum limit, by compensating for the motor speed tracking performance. It employs a fuzzy-tuned model predictive controller (FMPC), where a fuzzy logic controller (FLC) adjusts the input weight in the objective function to ensure that the change in battery SOC does not exceed the maximum permitted value while regulating the motor speed. The various components of the EV traction system are thoroughly modeled, and simulations are conducted using MATLAB/Simulink 2018b. The simulation results, carried out using the New European Drive Cycle (NEDC), verify that the technique limits the change in SOC while controlling the motor speed. This approach offers the advantage of maintaining precise control over the battery bank SOC, which distinguishes it from conventional speed regulators.

1. Introduction

One of the primary concerns associated with electric vehicles (EVs) pertains to their limited operational range. Additionally, the shortage of charging infrastructure and the extended charging duration remain significant challenges. In addressing these challenges, this chapter aims to shed light on a range of battery energy management (BEM) strategies outlined in existing literature while also introducing an innovative technique that holds promise for the EV market.

Regarding EVs, the BEM strategies can be segregated into two categories [1]. The first category involves the development of rules before initiating the system. Those rules dictate the behavior of the system during operation. Strategies falling within the second category are distinguished by their cost function and require an optimization technique to achieve the system’s objective. In the literature, the first category of strategies involves the use of a fuzzy logic controller (FLC) for managing multiple power sources such as combustion engines, ultra-capacitors, and batteries [2]. An FLC allocates power demand among these sources to maximize each source’s efficiency. In an alternative approach, an FLC is designed to consider battery SOC, input reference speed, and commanded vehicle acceleration to determine the battery’s power output, albeit with a trade-off of sacrificing a certain degree of motor performance to achieve battery energy conservation [3]. In the works [4], an advanced energy management system was developed. This system oversees both the torque signal and the SOC of the battery, subsequently generating the electric throttle signal to control the motor’s speed. Additionally, in Suhail et al.’s study [5], a neural FLC was introduced for efficient management of regenerative braking in a hybrid EV. This controller continuously monitors the engine’s speed and power, while accurately calculating the necessary torque for the given situation [5]. When the delivered power surpasses the required amount, the regenerative braking system initiates the process of charging the battery bank using the surplus power generated by the engine [5].

Among the strategies in the second category, dynamic programming (DP) is the most frequently employed optimization technique due to its ability to settle on the optimal solution [6]. In order to reduce the computational complexity, alternative approaches, such as coupling convex programming with a model predictive controller (MPC), can achieve a sub-optimal solution. Furthermore, the equivalent consumption and minimization strategy (ECMS) also obtained a sub-optimal solution [6].

The MPC-based BEM techniques primarily focus on solving receding horizon algorithms, predicting velocity profiles, and generating SOC reference trajectories. An adaptive ECMS (A-ECMS), and a fuzzy adaptive ECMS (Fuzzy A-ECMS) were compared [6]. They improve upon the original ECMS by dynamically estimating the optimal equivalent factor online, in contrast to the static value set by the user in ECMS. They continuously evaluate the current battery SOC against the desired SOC and adjust the optimal equivalent factor accordingly to minimize errors. The Fuzzy A-ECMS technique showed more robustness to various driving conditions as compared to the A-ECMS technique. In Ref. [7], an FLC monitors changes in the battery’s first and second derivative of SOC and generates an input weight $R$ for the MPC cost function. When sudden high acceleration occurs, an increase in $R$ prompts the MPC to restrict the EV’s acceleration to a safe level, minimizing battery energy consumption. In Ref. [8] a synthesized velocity profile prediction method is utilized to obtain driving velocity profiles. DP was then used to calculate optimal battery SOC trajectory and constraints at various set points [9]. These set points are then integrated into an MPC, which controls the maximum battery power output to track the optimal battery SOC at each set point. In Ref. [10], the road gradient was used in conjunction with an MPC, to generate a velocity profile for the vehicle. The MPC accelerated the vehicle when traveling up the road slope and decelerated the vehicle when traveling down the road slope. This was done prior to the occurrence of the road slope. Consequently, the power requirement from the battery was reduced. Furthermore, Zhao et al. [11] combined the wavelet neural network with the MPC to generate the reference SOC trajectory over a prediction horizon. This technique utilized particle swarm optimization to aid the wavelet neural network in generating the global SOC trajectory, which was used as a reference in the MPC. Furthermore, Chen et al. [12] adapted a long short-term memory velocity predictor. It gauged the vehicle’s speed and power demand of the vehicle. Subsequently, an MPC strategically allocates load power between an ultra-capacitor and a battery through a DC-DC converter. This was carefully structured to guarantee that they operated at their highest efficiency and to minimize the overall power dissipation.

Inspired by the techniques discussed in Refs. [6, 7], this chapter introduces a novel SOC tracking method capable of restricting the maximum change in SOC at the cost of speed-tracking performance degradation. The chapter’s scope focuses on designing and testing this technique through simulation and excludes the method for obtaining the SOC reference trajectory. The test results suggest that the proposed SOC tracking method successfully regulates the SOC degradation, and maintains it at the desired SOC reference. The testing was performed on the New European Drive Cycle (NEDC), and the average of the magnitude of the deviation from the SOC reference was found to be 0.00095 for the proposed SOC tracking technique compared to the 0.0037 obtained by the A-ECMS and the 0.0019 obtained by the Fuzzy A-ECMS techniques [6].

This chapter comprises five sections, with the introduction as the first section. Section 2 introduces the SOC tracking technique. Section 3 describes the EV traction system components and controllers. Section 4 presents the simulation methodology and results, and finally, Section 5 concludes the chapter.

2. Description of the state of charge tracking technique

Among the BEM strategies explored in Ref. [7], the fuzzy-tuned model predictive controller (FMPC) technique stands out as having broader potential applications within the EV traction system. It offers the possibility of fine-tuning the approach to achieve a similar outcome to the Fuzzy A-ECMS strategy detailed in Ref. [6], particularly in terms of SOC tracking. However, before delving into these adjustments, it is essential to grasp the fundamental workings of the FMPC technique and gain a comprehensive understanding of the overall system.

2.1 Fuzzy-tuned model predictive controller

Figure 1 illustrates a flowchart detailing the FMPC BEM technique, as discussed in Ref. [7]. The primary aim of this approach is to mitigate variations in the speed regulating current signal, denoted as $i_{\mathit{sq}}^{\ast }$, to address abrupt accelerations. These rapid speed increases are reflected in sudden surges in battery bank current, leading to a rapid decline in battery bank SOC over a short period. This not only reduces the battery’s runtime but also contributes to a shorter battery lifespan [7].

Figure 1.

Flowchart for the FMPC BEM technique [7].

To counteract these issues, the technique monitors the battery bank current and estimates the SOC. The rate of change of SOC, denoted by the first derivative of SOC, is obtained by taking the difference between the current sample of SOC and the preceding SOC sample. Furthermore, the second derivative in SOC is obtained by taking the difference between the current and preceding sample for the change in SOC. Those variables are processed by the FLC and $G_{\mathit{MPC}}$ gain. The final result is the parameter $R$ that impacts the MPC objective function. This parameter is used to penalize variations in $i_{\mathit{sq}}^{\ast }$. Additionally, the technique incorporates motor speed and drive cycle information into the MPC block. The chosen drive cycles are the NEDC drive cycle, representing a smooth driving behavior, and the US06 drive cycle, representing an aggressive driving behavior [7]. These drive cycles provide a comprehensive assessment of the FMPC BEM technique’s ability to regulate speed across a range of driving habits. The MPC, equipped with the estimated model of the EV traction system, solves the cost function, which has been adjusted with the input weight $R$, using a receding horizon algorithm. The resulting $i_{\mathit{sq}}^{\ast }$ signal effectively regulates motor speed while suppressing abrupt acceleration patterns. This design assists in preventing abrupt surges in battery current during acceleration and enables a smoother transition to the steady-state value of the battery bank’s discharge current.

2.2 Proposed modification for the state of charge tracking

For the SOC tracking to be effective, it is crucial for the input weight $R$ to adapt based on the error between the battery bank SOC and the reference SOC. Figure 2 presents the flowchart outlining the SOC tracking approach. This method utilizes an MPC for speed regulation, while an FLC assesses the difference between the SOC reference trajectory and the actual battery SOC. Subsequently, it generates an input weight $R$ that constrains the MPC’s speed regulating signal, $i_{\mathit{sq}}^{\ast }$. This approach possesses the capability to tightly constrain the motor tracking performance to an extensive degree by closely adhering to the SOC reference trajectory. The effectiveness of this scheme was evaluated through simulation, with testing conducted using the NEDC drive cycle. Subsequent sections will detail the design of the EV traction system components and the Simulink model employed in this study.

Figure 2.

Flowchart for the proposed SOC tracking technique.

3. Component modeling for an electric vehicle traction system

3.1 Lithium-ion battery bank model

3.1.1 Chen and Mora’s model

The Chen and Mora (CM) circuit model is a comprehensive representation that captures the dynamic attributes of the battery’s terminal voltage, and variations in battery parameters concerning SOC, and has undergone extensive experimentation over the last decade [13]. Figure 3 depicts the CM equivalent circuit battery model utilized in this study. The left section of the circuit represents the variation in battery SOC due to the fluctuations in battery current. On the other hand, the variations in battery terminal voltage in response to the battery current are shown on the right. In this model, the state variable $x_1$ represents the battery’s SOC, while $x_2$ corresponds to the voltage across $R_{\mathit{ts}}\Vert C_{\mathit{ts}}$, and $x_3$ corresponds to voltage across $R_{\mathit{tl}}\Vert C_{\mathit{tl}}$. The parallel combination $R_{\mathit{ts}}\Vert C_{\mathit{ts}}$ characterizes the short-term terminal voltage dynamics in response to fluctuations in discharge current, while the parallel combination $R_{\mathit{tl}}\Vert C_{\mathit{tl}}$ characterizes the long-term terminal voltage dynamics in response to variations in discharge current [13]. Eqs. (1)–(4) describe the CM equivalent circuit model [13].

Figure 3.

CM lithium-ion equivalent battery circuit model.

$${\dot{x}}_1\left(t\right)=-\frac{1}{C_C}i\left(t\right),C_C=3600{\mathit{Cf}}_1{f}_2{f}_3$$E1

$${\dot{x}}_2\left(t\right)=-\frac{x_2\left(t\right)}{R_{\mathit{ts}}\left(x_1\right)C_{\mathit{ts}}\left(x_1\right)}+\frac{i\left(t\right)}{C_{\mathit{ts}}\left(x_1\right)}$$E2

$${\dot{x}}_3\left(t\right)=-\frac{x_3\left(t\right)}{R_{\mathit{tl}}\left(x_1\right)C_{\mathit{tl}}\left(x_1\right)}+\frac{i\left(t\right)}{C_{\mathit{tl}}\left(x_1\right)}$$E3

$$y=E_o\left(x_1\right)-x_2\left(t\right)-x_3\left(t\right)-i\left(t\right)R_s\left(x_1\right)$$E4

In this model, the SOC, denoted as $x_1$, varies within the range of $\left[0,1\right]$. The states $x_2$ and $x_3$ are positive while the battery is discharging, and their sign depends on whether the battery is charging or discharging. Eq. (1) contains the parameter $C$ which represents the capacity in ampere-hours (A.h). Furthermore, the impact of temperature on battery performance, the number of charging and discharging cycles, and the effect of self-discharging are taken into account through the factors $f_1$, $f_2$, and $f_3$. They are set to 1 in this work. Eq. (4) depicts the states $x_2$ and $x_3$ and their impact on the terminal voltage $y$. In addition, the impact of the battery’s series resistance is taken care of by multiplying $i\left(t\right)$ and $R_s$ then subtracting the product from the open-circuit voltage $E_o$. Eqs. (5)–(10) define the variables $R_s$, $C_{\mathit{tl}}$, $R_{\mathit{tl}}$, $C_{\mathit{ts}}$, $R_{\mathit{ts}}$, and $E_o$ [13].

$$E_o\left(x_1\right)=-a_1{e}^{-a_2{x}_1}+a_3+a_4{x}_1-a_5{x}_1^2+a_6{x}_1^3$$E5

$$R_{\mathit{ts}}\left(x_1\right)=a_7{e}^{-a_8{x}_1}+a_9$$E6

$$R_{\mathit{tl}}\left(x_1\right)=a_{10}{e}^{-a_{11}{x}_1}+a_{12}$$E7

$$C_{\mathit{ts}}\left(x_1\right)=-a_{13}{e}^{-a_{14}{x}_1}+a_{15}$$E8

$$C_{\mathit{tl}}\left(x_1\right)=-a_{16}{e}^{-a_{17}{x}_1}+a_{18}$$E9

$$R_s\left(x_1\right)=a_{19}{e}^{-a_{20}{x}_1}+a_{21}$$E10

The lithium battery’s $E_o$ curve was recorded through the technique described in Ref. [13]. Furthermore, MATLAB was employed to determine the variables $a_1$ through $a_6$ in Eq. (5). The remaining parameters of the lithium-ion battery model, as specified in Eqs. (6)–(9), are acquired using the APE technique [13]. The $R_s$ constants $a_{19}$ through $a_{21}$ can be obtained by fitting Eq. (10) with curve $R_s\left(x_1\left(t\right)\right)$ versus $x_1\left(t\right)$ [13]. The values for parameter $a_1$ through $a_{21}$ are provided in Table 4 in Ref. [7].

3.1.2 SOC estimation by coulomb counting

The SOC illustrates the available capacity as a percentage of the rated capacity [13]. The mathematical formulation for the Coulomb counting (CC) method is shown in Eq. (11).

$$\mathit{SOC}\left(t\right)=\mathit{SOC}\left(t_o\right)-\frac{1}{C_c}{\int }_0^{t}i\left(t\right)\mathit{dt}$$E11

The starting SOC is denoted by $\mathit{SOC}\left(t_o\right)$, while the parameter $C_c$ stands for the capacity of the battery, calculated according to Eq. (1). The variable $i\left(t\right)$ signifies the discharge current, with positive values indicating discharging and negative values signifying charging.

3.2 Description of the controllers

3.2.1 Model predictive controller

The model predictive control (MPC) technique utilizes the system’s model to calculate the desired control signal required which will lead the system toward the required value. Before formulating the equations for the MPC, we need to select the prediction horizon, denoted as $N_p$, which signifies how many samples into the future the controller can forecast. The second variable is the control horizon, represented as $N_c$. It indicates the number of control variables that can be manipulated, with the condition that $N_p$ must be greater than or equal to $N_c$.

Consider a system in which the output at a particular sampling instant, denoted as $k$, is not directly influenced by the control signal. This system can be described using the discrete-time state-space model as shown in Eqs. (12) and (13).

$$x_m\left(k+1\right)=A_m{x}_m\left(k\right)+B_{m}u\left(k\right)$$E12

$$y\left(k\right)=C_m{x}_m\left(k\right)$$E13

Eqs. (14) and (15) define the difference between the current and previous values of the control signal, denoted as $\mathrm{\Delta }u$, and the state variable $\mathrm{\Delta }{x}_m$.

$$\mathrm{\Delta }u\left(k\right)=u\left(k\right)-u\left(k-1\right)$$E14

$$\mathrm{\Delta }{x}_m\left(k\right)=x_m\left(k\right)-x_m\left(k-1\right)$$E15

We can formulate Eqs. (16) and (17) by merging Eqs. (12)–(15)

$$\mathrm{\Delta }{x}_m\left(k+1\right)=A_m\mathrm{\Delta }{x}_m\left(k\right)+B_m\mathrm{\Delta }u\left(k\right)$$E16

$$\begin{array}{l}y\left(k+1\right)=y\left(k\right)+C_m\mathrm{\Delta }{x}_m\left(k+1\right)\\ =y\left(k\right)+C_m{A}_m\mathrm{\Delta }{x}_m\left(k\right)+C_m{B}_m\mathrm{\Delta }u\left(k\right)\end{array}$$E17

Eqs. (16) and (17) can be employed to construct the augmented state-space model of the system as depicted in Eqs. (18) and (19).

$$x\left(k+1\right)=\mathit{Ax}\left(k\right)+B\mathrm{\Delta }u\left(k\right)$$E18

$$y\left(k\right)=\mathit{Cx}\left(k\right)$$E19

where $x\left(k\right)=\left[\begin{array}{c}\mathrm{\Delta }{x}_m\left(k\right)\\ y\left(k\right)\end{array}\right],A=\left[\begin{array}{cc}{A}_m& 0_m^T\\ C_m{A}_m& 1\end{array}\right],B=\left[\begin{array}{c}{B}_m\\ C_m{B}_m\end{array}\right],C=\left[0_{m}1\right].$

and the empty spaces in the matrix are filled by the zero matrix $0_m=\left[0,0,\cdots 0\right]$.

At instant $k$, Eq. (20) describes the system states for future samples. This is obtained by expanding $x\left(k+1\right)$ in Eq. (18).

$$\begin{array}{ll}x\left(k+1|k\right)& =\mathit{Ax}\left(k\right)+B\mathrm{\Delta }u\left(k\right)\\ & ⋮\\ x\left(k+N_p|k\right)& =A^{N_p}x\left(k\right)+A^{N_p-1}B\mathrm{\Delta }u\left(k\right)+\cdots +\\ & A^{N_p-N_c}B\mathrm{\Delta }u\left(k+N_c-1\right)\\ & \end{array}$$E20

Likewise, Eq. (21) can be derived by expanding $y\left(k+1\right)$ in Eq. (19).

$$\begin{array}{ll}y\left(k+1|k\right)& =\mathit{CAx}\left(k\right)+\mathit{CB}\mathrm{\Delta }u\left(k\right)\\ & ⋮\\ y\left(k+N_p|k\right)& ={\mathit{CA}}^{N_p}x\left(k\right)+{\mathit{CA}}^{N_p-1}B\mathrm{\Delta }u\left(k\right)+\cdots +\\ & {\mathit{CA}}^{N_p-N_c}B\mathrm{\Delta }u\left(k+N_c-1\right)\\ & \end{array}$$E21

Eqs. (22) and (23) describe the matrix $\mathrm{\Delta }U$, with length $N_c$, containing the changes in the control signal starting with instant $k$. Meanwhile, the matrix $Y$, with length $N_p$, describes the predicted output for the system.

$$\mathrm{\Delta }U={\left[\mathrm{\Delta }u\left(k\right)\mathrm{\Delta }u\left(k+1\right)\cdots \mathrm{\Delta }u\left(k+N_c-1\right)\right]}^T$$E22

$$Y={\left[y\left(k+1|k\right)y\left(k+2|k\right)\dots y\left(k+N_p|k\right)\right]}^T$$E23

Merging Eqs. (20)–(23) yields Eq. (24).

$$Y=\mathit{Fx}\left(k\right)+\mathrm{\Phi \Delta }U\left(k\right)$$E24

where $F=\left[\begin{array}{c}\mathit{CA}\\ {\mathit{CA}}^2\\ ⋮\\ {\mathit{CA}}^{N_p}\end{array}\right]$ and $\mathrm{\Phi }=\left[\begin{array}{ccccccc}\mathit{CB}& & 0& & \cdots & & 0\\ \mathit{CAB}& & \mathit{CB}& & \cdots & & 0\\ ⋮& & ⋮& & \ddots & & ⋮\\ {\mathit{CA}}^{N_p-1}B& & {\mathit{CA}}^{N_p-2}B& & \cdots & & {\mathit{CA}}^{N_p-N_c}B\\ & & & & & & \end{array}\right]$. The vector containing the reference signal of the system has a length of $N_p$ and is defined by Eq. (25).

$$R_s^T=\left[11\cdots 11\right]r\left(k\right)$$E25

Eq. (26) contains the cost function $J$.

$$J={\left(R_s-Y\right)}^T\overline{Q}\left(R_s-Y\right)+\mathrm{\Delta }{U}^T\overline{R}\mathrm{\Delta }U$$E26

where $\overline{R}$ is an $N_c\times N_c$ input weight matrix, and $\overline{Q}$ is an $N_c\times N_c$ output weight matrix. The specific values of these weight matrices can be adjusted depending on the system’s operational requirements. The ratio of the input weight $R$ to the output weight $Q$ serves to penalize variations in the control signal during the system’s operation. In this work, $Q$ was kept at 1, while the modification was performed on $R$ while the system was running. Eq. (27) can be obtained by expanding Eq. (26).

$$J={\left(R_s-\mathit{Fx}\left(k\right)\right)}^T\left(R_s-\mathit{Fx}\left(k\right)\right)$$E27

The partial derivative with respect to $\mathrm{\Delta }U$ is taken and equated to zero yielding Eqs. (28) and (29).

$$\frac{\partial J}{\mathrm{\partial \Delta }U}=-2{\mathrm{\Phi }}^T\left(R_s-\mathit{Fx}\left(k\right)\right)+2\left({\mathrm{\Phi }}^T\mathrm{\Phi }+\overline{R}\right)\mathrm{\Delta }U$$E28

$$\frac{\partial J}{\mathrm{\partial \Delta }U}=0\to \mathrm{\Delta }U={\left({\mathrm{\Phi }}^T\mathrm{\Phi }+\overline{R}\right)}^{-1}{\mathrm{\Phi }}^T\left(R_s-\mathit{Fx}\left(k\right)\right)$$E29

The element $\mathrm{\Delta }u\left(k\right)$ is obtained from matrix $\mathrm{\Delta }U$ and is used to update $u\left(k-1\right)$. This results in the updated control signal $u\left(k\right)$.

3.2.2 Fuzzy logic controller

The architecture of the fuzzy logic controller (FLC) is illustrated in Figure 4. This FLC generates the change in the input weight at time instant $k$, denoted by $\mathrm{\Delta }R\left(k\right)$. It monitors the error $e\left(k\right)$ and changes in error $\mathrm{\Delta }e\left(k\right)$ of the battery SOC, denoted by ${\mathit{SOC}}_{\mathit{\text{battery}}}\left(k\right)$, and the SOC reference, denoted by ${\mathit{SOC}}_{\mathit{\text{reference}}}\left(k\right)$. Furthermore, Figure 4 depicts the three stages that the signal passes through before the controller issues a command. The fuzzifier generates linguistic variables from the given signal. Next, the inference mechanism correlates the linguistic variables with the rule base and then produces a linguistic output. Finally, the defuzzifier creates the control signal from the linguistic outputs. The change in the weight $\mathrm{\Delta }R\left(k\right)$ is produced by the FLC and is added to the current value $R\left(k-1\right)$ to form $R\left(k\right)$.

Figure 4.

Fuzzy logic controller block diagram.

The surface representing the fuzzy inference system is depicted in Figure 5. When the values of $e\left(k\right)$ and $\mathrm{\Delta }e\left(k\right)$ lie within [0.5, 1], the resultant $\mathrm{\Delta }R\left(k\right)$ is positive, indicating a large increase in the input weight. Conversely, when $e\left(k\right)$ and $\mathrm{\Delta }e\left(k\right)$ are between [−1, −0.5], there is a significant drop in $\mathrm{\Delta }R\left(k\right)$. When $e\left(k\right)$ and $\mathrm{\Delta }e\left(k\right)$ fall between [−0.5, 0.5], the magnitude $\mathrm{\Delta }R\left(k\right)$ depends on their point of intersection with the surface. A minor increase or decrease in $\mathrm{\Delta }R\left(k\right)$ is applied until $e\left(k\right)$ and $\mathrm{\Delta }e\left(k\right)$ approach zero.

Figure 5.

Fuzzy logic controller surface.

3.3 Induction motor drive

Eqs. (30)–(35) describe the induction motor (IM) model in the synchronously rotating $\mathit{dq}$-coordinate system [7].

$$V_{\mathit{sd}}=R_s{i}_{\mathit{sd}}+\frac{d{\lambda }_{\mathit{sd}}}{\mathit{dt}}-{\omega }_e{\lambda }_{\mathit{sq}}$$E30

$$V_{\mathit{sq}}=R_s{i}_{\mathit{sq}}+\frac{d{\lambda }_{\mathit{sq}}}{\mathit{dt}}-{\omega }_e{\lambda }_{\mathit{sd}}$$E31

$$V_{\mathit{rd}}=R_r{i}_{\mathit{rd}}+\frac{d{\lambda }_{\mathit{rd}}}{\mathit{dt}}-{\omega }_{\mathit{sl}}{\lambda }_{\mathit{rq}}$$E32

$$V_{\mathit{rq}}=R_r{i}_{\mathit{rq}}+\frac{d{\lambda }_{\mathit{rq}}}{\mathit{dt}}-{\omega }_{\mathit{sl}}{\lambda }_{\mathit{rd}}$$E33

$$T_{\mathit{em}}=\frac{3p}{2}\frac{L_m}{L_r}\left({\lambda }_{\mathit{rq}}{i}_{\mathit{rd}}-{\lambda }_{\mathit{rd}}{i}_{\mathit{rq}}\right)$$E34

$$\frac{d{\omega }_m}{\mathit{dt}}=\frac{1}{J}\left(T_{\mathit{em}}-T_L-B{\omega }_m\right)$$E35

The variables $V$, $i$, and $\lambda$ correspond to the voltages, currents, and fluxes, where the $\mathit{dq}$ coordinate system components for the rotor are denoted by subscript $r$, and the stator is denoted by subscript $s$. The variables $T_L$, $T_{\mathit{em}}$, $B$, and $J$ are the load torque, motor torque, coefficient of friction, and motor inertia. While $R_s$ is the the stator resistance and $R_r$ is the rotor resistance. Additionally, ${\omega }_m$, ${\omega }_e$, and ${\omega }_{\mathit{sl}}$ denote the rotor speed, the speed at which the $d$-axis is rotating, and the speed at which the rotor axis is rotating, respectively. Furthermore, ${\omega }_e$ was intentionally set equated to ${\omega }_{\mathit{\text{sync}}}=2\mathit{\pi f}$ which represents the synchronous speed.

The electrical coupling equations are represented by Eqs. (30)–(34), and Eq. (35) represents the mechanical coupling equation of the induction machine. Eq. (36) illustrates the relationship between the $\mathit{dq}$-fluxes and the $\mathit{dq}$-currents in matrix form.

$$\left[\begin{array}{c}{\lambda }_{\mathit{sd}}\\ {\lambda }_{\mathit{sq}}\\ {\lambda }_{\mathit{rd}}\\ {\lambda }_{\mathit{rq}}\end{array}\right]=\left[\begin{array}{cccc}{L}_s& 0& L_m& 0\\ 0& L_s& 0& L_m\\ L_m& 0& L_r& 0\\ 0& L_m& 0& L_r\end{array}\right]\left[\begin{array}{c}{i}_{\mathit{sd}}\\ i_{\mathit{sq}}\\ i_{\mathit{rd}}\\ i_{\mathit{rq}}\\ \end{array}\right]$$E36

Eqs. (37) and (38) represent the conversion matrices used in the system. The variables are converted to the $\mathit{dq}$ coordinate system components, manipulated, then transformed back to $\mathit{abc}$ components.

$$\left[\begin{array}{c}{i}_{\mathit{sd}}\\ i_{\mathit{sq}}\end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}cos\left({\theta }_e\right)& cos\left({\theta }_e-\frac{2\pi }{3}\right)& cos\left({\theta }_e-\frac{4\pi }{3}\right)\\ -sin\left({\theta }_e\right)& -sin\left({\theta }_e-\frac{2\pi }{3}\right)& -sin\left({\theta }_e-\frac{4\pi }{3}\right)\end{array}\right]\left[\begin{array}{c}{i}_a\\ i_b\\ i_c\end{array}\right]$$E37

$$\left[\begin{array}{c}{i}_a\\ i_b\\ i_c\end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{cc}cos\left({\theta }_e\right)& -sin\left({\theta }_e\right)\\ cos\left({\theta }_e+\frac{4\pi }{3}\right)& -sin\left({\theta }_e+\frac{4\pi }{3}\right)\\ cos\left({\theta }_e+\frac{2\pi }{3}\right)& -sin\left({\theta }_e+\frac{2\pi }{3}\right)\end{array}\right]\left[\begin{array}{c}{i}_{\mathit{sd}}\\ i_{\mathit{sq}}\end{array}\right]$$E38

where ${\theta }_e$ represents the angle that the $d$-axis makes with the stationary axis [7].

3.3.1 Indirect field-oriented control (IFO)

To perform indirect field-orientation (IFO) we fix ${\lambda }_{\mathit{rq}}$ to zero. Therefore, we can obtain Eq. (39) by combining Eqs. (32) and (36). This rotor flux is calculated using Eq. (39). In addition, Eq. (33) merged with Eq. (36), will result in Eq. (40) [7]. Eq. (40) serves the purpose of calculating the rotor slip.

$${\lambda }_{\mathit{rd}}=L_m{i}_{\mathit{sd}}$$E39

$${\omega }_{\mathit{sl}}=\frac{L_m}{{\tau }_r}\frac{i_{\mathit{sq}}}{{\lambda }_{\mathit{rd}}}$$E40

where ${\tau }_r$ is the rotor time constant.

The state-space representation of the IFO IM drive can be derived by combining Eqs. (34) and (36), resulting in

$$T_{\mathit{em}}=\frac{3p}{2}\frac{L_m}{L_r}\left({\lambda }_{\mathit{rd}}{i}_{\mathit{sq}}\right)$$E41

A relationship governing the speed ${\omega }_m$ and the current $i_{\mathit{sq}}$ can be obtained by combining Eqs. (35) and (41).

$$\frac{d{\omega }_m}{\mathit{dt}}=\frac{1}{J}\left(\frac{3p}{2}\frac{L_m}{L_r}\left({\lambda }_{\mathit{rd}}{i}_{\mathit{sq}}\right)-T_L-B{\omega }_m\right)$$E42

Under a specific load $T_L$, differentiating Eq. (42) results in

$$\frac{d^2{\omega }_m}{{\mathit{dt}}^2}=-\frac{B}{J}\frac{d{\omega }_m}{\mathit{dt}}+\frac{3p}{2}\frac{L_m}{L_r}\frac{{\lambda }_{\mathit{rd}}}{J}\frac{{\mathit{di}}_{\mathit{sq}}}{\mathit{dt}}$$E43

The state-space representation of the IFO IM drive is provided in Eqs. (44) and (45) [7].

$$\left[\begin{array}{c}{\ddot{\omega }}_m\\ {\dot{\omega }}_m\end{array}\right]=\left[\begin{array}{cc}-\frac{B}{J}& 0\\ 1& 0\end{array}\right]\left[\begin{array}{c}{\dot{\omega }}_m\\ {\omega }_m\end{array}\right]+\left[\begin{array}{c}\frac{3p}{2}\frac{L_m}{L_r}\frac{{\lambda }_{\mathit{rd}}}{J}\\ 0\end{array}\right]\frac{{\mathit{di}}_{\mathit{sq}}}{\mathit{dt}}$$E44

$${\omega }_m=\left[\begin{array}{cc}0& 1\end{array}\right]\left[\begin{array}{c}{\dot{\omega }}_m\\ {\omega }_m\end{array}\right]$$E45

The comprehensive EV traction system, encompassing the IFO IM drive, the battery bank, and the SOC tracking scheme is illustrated in Figure 6. The battery bank current is used to estimate the battery SOC, and an FLC compares the battery bank SOC and reference SOC, then generates an input weight $R$ for the MPC objective function. The outer controller loop comprising the input weight $R$, the MPC, and the motor speed ${\omega }_m$ is responsible for generating the reference $i_{\mathit{sq}}^{\ast }$ current. The $d$-axis current $i_{\mathit{sd}}^{\ast }$ and $q$-axis current $i_{\mathit{sq}}^{\ast }$ are responsible for regulating the flux and torque of the induction motor, respectively. The “Slip Calc.” block carries out rotor flux estimation using Eq. (39) and subsequently calculates the slip using Eq. (40). Moreover, PI controllers for the inner current loops ensure that the stator $q$ and $d$ component currents are tracking the reference $q$ and $d$ component currents, respectively. The $q$ and $d$ component voltages are generated from the inner PI controllers and converted to the reference $\mathit{abc}$ component sinusoidal voltages used to produce the inverter gating signals.

Figure 6.

IFO IM drive with SOC tracking.

4. Simulation results

4.1 Testing methodology

4.1.1 Drive cycle description

A driving cycle consists of data points that represent the velocity of a real-life vehicle measured over time. The New European Drive Cycle (NEDC) consists of both urban and extra-urban driving stages. It is a combination of straight acceleration and constant speed periods, and it is depicted in Figure 7. The urban driving cycle lasts for 800 seconds, starting at second 11 and ending at second 785. It consists of three constant speed peaks, and they are repeated four times as shown in the rectangle. The constant speed peaks are arranged in ascending order with the peaks occurring at 187, 400, 625, and 437 RPM, respectively. The extra-urban driving stage spans 370 seconds, beginning at second 800 and concluding at second 1170. The speed peaks occur at 625, 875, 1250, and 1500 RPM.

Figure 7.

NEDC drive cycle.

4.1.2 Simulink model description

Figure 8 shows the Simulink model for the EV traction system used in this work. A 580 $V$ lithium-ion battery bank with a capacity of 4000 $\mathit{mAh}$ provides the required voltage and power for the EV traction system. The inverter produces the required voltage and frequency to control the speed of the IM. Furthermore, the IM ratings were 0.5 kW, 415 V, and 50 Hz. The references in the system are the red block representing the reference motor $d$-axis current $i_{\mathit{sd}}^{\ast }$, the drive cycle reference block, and the SOC reference block. The slip frequency estimation block estimates the rotor flux and slip using Eqs. (39) and (40), respectively. The battery bank SOC estimation is performed using Eq. (11), and the SOC is sent to the FLC. The FLC compares the battery SOC with the reference SOC and generates the $R$ for the MPC cost function. The MPC generates the speed-regulating signal $i_{\mathit{sq}}^{\ast }$ while giving consideration to the reference SOC. The motor $\mathit{dq}$ currents are regulated by the inner PI controllers. While the $\mathit{dq}$ voltages are converted to the reference $\mathit{abc}$ sinusoidal voltages $V_a^{\ast }$, $V_b^{\ast }$, and $V_c^{\ast }$ that are used by the inverter to generate the required voltages for the motor. The simulation sampling time is 10 kHz.

Figure 8.

MATLAB/Simulink model for the EV traction system with SOC tracking.

4.1.3 State of charge reference

The scope of this work focused on devising a technique to regulate the SOC of the EV traction system rather than producing an SOC reference signal. Therefore, the SOC reference signal used in this work is just a limitation to the maximum change in SOC. Figure 9a shows the SOC deterioration over the NEDC drive cycle. The rate of change of SOC, denoted by $\mathrm{\Delta }\mathit{SOC}$, is obtained by taking the difference between the current and previous sample for the SOC signal. Figure 9b illustrates the inverted version of the $\mathrm{\Delta }\mathit{SOC}$ over-the-NEDC drive cycle, and is used as the primary evaluation metric for the SOC tracking technique. The inverted $\mathrm{\Delta }\mathit{SOC}$ will be referred to as $\mathrm{\Delta }\mathit{SOC}$ for simplicity.

Figure 9.

Battery bank (a) SOC decay (b) inverted $\mathrm{\Delta }\mathit{SOC}$ with the SOC tracking technique.

The $\mathrm{\Delta }\mathit{SOC}$ graph is divided into five regions as shown in Figure 10. The regions are classified as follows:

• Region 1: Represents the urban stage of the NEDC drive cycle, and is characterized by steady acceleration, deceleration, and constant speed. No limitations are set on the $\mathrm{\Delta }\mathit{SOC}$. This region represents the impact of the motor current on the $\mathrm{\Delta }\mathit{SOC}$ when the motor performance is unrestricted. It also acts as the reference point for generating the maximum permitted $\mathrm{\Delta }\mathit{SOC}$ value for regions 2 through 4. The peak values for the $\mathrm{\Delta }\mathit{SOC}$ occur at 0.15, 0.19, and 0.255 Coulombs.

• Region 2: Represents the urban stage of the NEDC drive cycle, and is characterized by steady acceleration, deceleration, and constant speed. The $\mathrm{\Delta }\mathit{SOC}$ is restricted to 90% of its original value. The maximum permitted $\mathrm{\Delta }\mathit{SOC}$ value is obtained by multiplying the $\mathrm{\Delta }\mathit{SOC}$ of region 1 by 90%. The maximum permitted $\mathrm{\Delta }\mathit{SOC}$ value is shown by the 3 red bars in region 2.

• Region 3: Represents the urban stage of the NEDC drive cycle, and is characterized by steady acceleration, deceleration, and constant speed. The $\mathrm{\Delta }\mathit{SOC}$ is restricted to 80% of its original value. The maximum permitted $\mathrm{\Delta }\mathit{SOC}$ value is obtained by multiplying the $\mathrm{\Delta }\mathit{SOC}$ of region 1 by 80%. The maximum permitted $\mathrm{\Delta }\mathit{SOC}$ value is shown by the 3 red bars in region 3.

• Region 4: Represents the urban stage of the NEDC drive cycle, and is characterized by steady acceleration, deceleration, and constant speed. The $\mathrm{\Delta }\mathit{SOC}$ is restricted to 70% of its original value. The maximum permitted $\mathrm{\Delta }\mathit{SOC}$ value is obtained by multiplying the $\mathrm{\Delta }\mathit{SOC}$ of region 1 by 70%. The maximum permitted $\mathrm{\Delta }\mathit{SOC}$ value is shown by the 3 red bars in region 4.

• Region 5: Represents the extra-urban stage of the NEDC drive cycle, and is distinguished by its high speed. The $\mathrm{\Delta }\mathit{SOC}$ is restricted to 80% of the $\mathrm{\Delta }\mathit{SOC}$ when the motor is running at 1500 $\mathit{RPM}$ (0.55 Coulombs). The maximum permitted $\mathrm{\Delta }\mathit{SOC}$ value is shown by the red bar in region 5.

Figure 10.

Battery bank maximum permitted $\mathrm{\Delta }\mathit{SOC}$ values on regions 2 through 5.

4.2 Test results

Figure 11 overlaps the $\mathrm{\Delta }\mathit{SOC}$ readings with the predefined maximum permitted values. Region 1 serves as an example of what the $\mathrm{\Delta }\mathit{SOC}$ values in regions 2 through 4 would have been if the SOC tracking scheme had not enforced maximum permitted $\mathrm{\Delta }\mathit{SOC}$ values. Regions 2 through 4 clearly demonstrate that the $\mathrm{\Delta }\mathit{SOC}$ was constrained and prevented from exceeding the established maximum permitted value, confirming the effectiveness of the SOC tracking method. Additionally, in region 5, we observe that when the $\mathrm{\Delta }\mathit{SOC}$ remained below the maximum permitted value, it was allowed to adapt freely. However, once the $\mathrm{\Delta }\mathit{SOC}$ began to rise, beyond the 1000-second mark, it was capped at the maximum permitted value. Moreover, during these capping periods, the magnitude of the error $\mathrm{\Delta }\mathit{SOC}$ was recorded and the average was obtained. Figure 12 displays the average of the absolute value of the error during the capping periods. The absolute average error for the capping periods varies between $\left[0.00052,0.0013\right]$ and the average for the complete NEDC drive cycle is 0.00095. This result is comparable to the ECMS, A-ECMS, and Fuzzy A-ECMS techniques which yielded 0.0003, 0.0037, and 0.0019 deviations off the reference SOC while testing on the NEDC drive cycle.

Figure 11.

Battery bank $\mathrm{\Delta }\mathit{SOC}$ and maximum permitted $\mathrm{\Delta }\mathit{SOC}$ values with the SOC tracking technique.

Figure 12.

$\mathrm{\Delta }\mathit{SOC}$ absolute average error at the maximum permitted $\mathrm{\Delta }\mathit{SOC}$ values with the SOC tracking technique.

Figure 13 illustrates the response of the motor’s speed during the NEDC drive cycle. In region 1, it is evident that the unrestricted SOC tracker technique adeptly regulates the motor’s speed. However, as we move into regions 2 through 4 (where the maximum permitted $\mathrm{\Delta }\mathit{SOC}$ values range from 90 to 70%), we observe a noticeable decline in the motor’s speed tracking performance. The more stringent the $\mathrm{\Delta }\mathit{SOC}$ constraint, the larger the drop in the motor’s speed-tracking performance. Consequently, in region 4, we witness a substantial deterioration in motor speed tracking performance compared to regions 2 and 3. Moving to region 5, we note that before the 1000-second mark, the $\mathrm{\Delta }\mathit{SOC}$ remained below its maximum permitted value, allowing for effective regulation of the motor speed. Conversely, after the 1000-second mark, the $\mathrm{\Delta }\mathit{SOC}$ reached its maximum permitted value and was held at that boundary, resulting in a decline in speed tracking performance.

Figure 13.

Motor speed response with the SOC tracking technique.

Figure 14 presents the fluctuation in the input weight parameter, denoted as $R$ throughout the NEDC drive cycle. It is important to note that the input weight $R$ plays a direct role in determining the generation of the motor speed-regulating current $i_{\mathit{sq}}^{\ast }$, which was discussed in the previous section. Specifically, as the value of $R$ increases, it imposes a more significant constraint on $i_{\mathit{sq}}^{\ast }$. Since $i_{\mathit{sq}}^{\ast }$ directly influences the motor current, a lower $i_{\mathit{sq}}^{\ast }$ results in drawing less current from the battery bank, eventually leading to a reduced $\mathrm{\Delta }\mathit{SOC}$.

Figure 14.

Input weight R with the SOC tracking technique.

From the previous statement, it is apparent that enforcing a stricter limit on $\mathrm{\Delta }\mathit{SOC}$ necessitates a higher value for $R$. This becomes evident when comparing the values of $R$ in regions 2 through 4, which have the same speed reference. As $\mathrm{\Delta }\mathit{SOC}$ constraints become more stringent (from 90 to 70% maximum permitted $\mathrm{\Delta }\mathit{SOC}$), the value of $R$ increases correspondingly.

Furthermore, an increase in the reference speed demands a higher value for $i_{\mathit{sq}}^{\ast }$, consequently increasing the $\mathrm{\Delta }\mathit{SOC}$ value. If there is a maximum permitted $\mathrm{\Delta }\mathit{SOC}$ value, then the input weight $R$ must rise in tandem with the increasing reference speed to restrict $i_{\mathit{sq}}^{\ast }$ further, and thereby maintain the $\mathrm{\Delta }\mathit{SOC}$ at the maximum permitted value. This phenomenon is observable when examining region 5. Before the 1000-second mark, the $\mathrm{\Delta }\mathit{SOC}$ value below the maximum permitted value, $R$ remained at the minimum level. However, after the 1000-second mark, as the $\mathrm{\Delta }\mathit{SOC}$ value reached the maximum permitted value, the input weight $R$ increased in line with the reference speed signal to uphold the $\mathrm{\Delta }\mathit{SOC}$ at its maximum permitted value.

5. Conclusions

This chapter introduces a method for tracking the state of charge (SOC) using a fuzzy-tuned model predictive controller. The mathematical models for the components of an electric vehicle traction system were developed and tested on Simulink. The simulation was conducted using the New European Drive Cycle, during which the motor speed and battery SOC were continuously monitored. The outcomes of the simulation demonstrate the effectiveness of the SOC tracking technique in regulating motor speed when there are no SOC restrictions. Furthermore, it successfully maintains the battery SOC within the defined maximum permitted value, albeit with some trade-offs in motor speed tracking performance. The absolute average deviation from the reference for the SOC tracking technique was lower than the Fuzzy A-ECMS, and A-ECMS techniques which yielded 0.00095, 0.0019, and 0.0037, respectively. However, the ECMS technique with a fixed optimal equivalent factor had the lowest deviation of 0.0003. In other words, there is still room for improvement in the SOC tracking technique. Furthermore, the robustness of the technique on different driving behaviors is yet to be tested. In summary, the simulation results provide substantial evidence supporting the effectiveness of the SOC tracking technique.

Future research directions include:

• Testing the robustness of the technique on an aggressive drive cycle such as the US06 drive cycle.

• Testing the effectiveness of the technique on a reference SOC instead of a fixed rate of change of SOC.

• Incorporating the $i_{\mathit{sd}}$ current in the MPC objective function, and monitoring the impact on the absolute average deviation from the reference SOC.

Abbreviations

EV	electric vehicle
BEM	battery energy management
SOC	state of charge
DP	dynamic programming
MPC	model predictive controller
FLC	fuzzy logic controller
FMPC	fuzzy model predictive controller
ECMS	equivalent consumption minimization strategy
A-ECMS	adaptive equivalent consumption minimization strategy
Fuzzy A-ECMS	fuzzy adaptive equivalent consumption minimization strategy
CM	Chen and Mora
CC	Coulomb counting
IM	induction motor
IFO	indirect field orientation
PI	Proportional-integral controller
NEDC	New European drive cycle