Optimizing Low Pass Filter Cut-off Frequency for Energy Management in Electric Vehicles with Hybrid Energy Storage Systems

2.1 System modeling

2.1.1 Electric vehicle model

The electric vehicle model in the form of city car is adopted from [17] the configuration is shown in Figure 1. Its parameters are listed in Table 1, it was adopted from the same source with some modifications. From Figure 1, there are two power sources used which are battery and supercapacitor (SC).

Table 1. EV model parameters

The HESS of battery and supercapacitor is divided into four topologies: passive, semi-active supercapacitor, semi-active battery, and active [18, 19]. In this study, a semi-active supercapacitor with one DC-DC converter on the SC side is used. This configuration has a trade-off between cost and power-sharing capability. Compared to the passive configuration, the semi-active has better energy sharing since there is one DC-DC converter that can control the energy flow. On the other side, compared to an active configuration which uses a DC-DC converter in each energy storage, the semi-active configuration is simpler and cheap.

The load for the HESS in this system is the power for the traction motor (propulsion load) and low voltage power consumption or called auxiliary load. The propulsion load is varies based on the driver's comment through the gas pedal, whereas the auxiliary power treated as constant power.

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Figure 1. HESS configuration in EV [17]

The electric vehicle is modelled using Newton’s motion law which is based on the force acting on the vehicle as illustrated in Figure 2. The Eq. (1) is the equation of motion for a vehicle. v_s is the longitudinal velocity, F_trac is traction force, F_roll is wheel friction force, F_aero is force air friction, F_grade is the force of gravity. Formulas for calculating wheel friction, air friction, and gravitational force are respectively shown in Eqs. (2)-(4). Where M_veh is the vehicle mass, ρ_air is the air density (1.25 kg/m³ under normal conditions), g is the acceleration due to gravity, and δ is the angle of inclination of the road.

$\frac{d v_s}{d t}=\left(F_{\text {trac }}-F_{\text {roll }}-F_{\text {aero }}-F_{\text {grade }}\right) \div M_{\text {veh }}$          (1)

$F_{\text {roll }}=c_{\text {roll }} M_{\text {veh }} g \cos (\delta)$          (2)

$F_{\text {aero }}=\frac{1}{2} \rho_{\text {air }} A_f C_d v_s^2$          (3)

$F_{\text {grade }}=M_{\text {veh }} g \sin (\delta)$          (4)

Figure 3 depicts the block diagram for the simulation design. The black line represents a control signal, whereas the blue line represents an electrical connection. In this instance, we utilize the Urban Dynamometer Driving Schedule (UDDS) as the speed reference, which is derived from a regular drive cycle. This signal is conveyed to the driver block, which uses the speed references to decide whether to power on or brake. This block is made up of a PID control that acts as a driver replacement and processes the difference in speed between reference speeds and observed speeds (v_s). The brake block calculates the portion of braking (%Br) which is assigned for regenerative braking (FB_Reg) and friction braking (FB_Fric) based on Eq. (5). The Motor block then received the signal for the percentage of powering (%Pwr) and regenerative braking which was then multiplied by the maximum Powering/ Braking torque to get the motor torque (T_m) as in Eq. (6). The motor block calculates the motor power using Eq. (7). In the driveline block, the processes inside it calculate the F_tracbased on Eq. (8) utilizing the torque signal from the motor where T_spinloss in losses in the motor spin. The Glider block uses Eq. (1) to determine the vehicle's acceleration and speed. The power required by the system is load power (P_Load) which is motor power (P_m) plus the auxiliary load (P_aux). This power is sent to the EMS block for power allocation to the SC. Whereas, the battery takes the rest of the power required by the system since it is uncontrolled.

$F B_{F r i c}=F B_{\text {max }}-F B_{\text {Reg }}$          (5)

$T_m=\left(\% P w r * T_{m \_\max }\right)+\left(F B_{R e q} * r_w / G\right)$          (6)

$P_m=\left(T_m\left(v_s G / r_w\right)\right) / \mu$          (7)

$F_{\text {trac }}=\left(T_m-T_{\text {spinloss }}\right) * G \div r_w-F B_{\text {Fric }}$          (8)

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Figure 2. Force acting on a moving vehicle [20]

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Figure 3. Simulation design block diagram

2.1.2 Energy storage system (ESS) model

In the EV model used, there are two ESS used. Battery as the main supply and SC to support and as an energy buffer for the battery. The battery used is a lithium-ion (Li-ion) type, while the model used uses the battery model in the MATLAB software which refers to the Generic Battery Model (GBM). GBM itself refers to the model proposed by Tremblay and Dessaint [21] which has been validated experimentally with good accuracy (±5% error) for the 20-100% SoC range on Li-ion batteries.

Furthermore, the battery is modelled with a controlled voltage source and internal resistance. The value of the controlled voltage source is set based on Eq. (9) for discharging conditions and Eq. (10) for charging conditions. All variables used are described in the Nomenclature. Furthermore, Figure 4 shows a block diagram illustration of the battery model used.

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Figure 4. Battery model [21]

$\begin{aligned} V_{b a t}=E_0-R_{b a t} & \cdot i-K \frac{Q}{Q-i t}(i *+i t)+A \cdot \exp (-B \cdot i t)\end{aligned}$          (9)

$\begin{aligned} V_{b a t}=E_0-R_{b a t} \cdot i-K \frac{Q}{i t+0.1 Q} i *-K \frac{Q}{Q-i t} i t +A \cdot \exp (-B \cdot i t)\end{aligned}$          (10)

The supercapacitor model used refers to the non-linear Stern-Tofel model [22]. In the Stern-Tofel model, the total capacitance (C_T) of the supercapacitor is a combination of Helmholtz's capacitance (C_H) and Gouy-Chapman's capacitance (C_GC), the formulation of the model is shown in Eqs. (11)-(14). Furthermore, the self-discharging current equation and supercapacitor voltage are shown in Eq. (15) and Eq. (16), respectively.

Figure 5 shows the block diagram of the supercapacitor model consisting of a voltage source and the supercapacitor's internal resistance. The voltage source used is a controlled voltage source where the voltage value is controlled as shown in Figure 5.

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Figure 5. Supercapacitor model [22]

The battery and SC specification used is listed in Table 1 for its capacity and rated voltage. Whereas the rest of the variable value is using MATLAB default value. For more detail about the battery and SC models see studies [21, 22], respectively.

$C_T=\frac{N_p}{N_S}\left(\frac{1}{C_H}+\frac{1}{C_{G C}}\right)^{-1}$          (11)

$C_H=\frac{N_e \cdot \varepsilon \cdot \varepsilon_0 \cdot A_i}{d}$          (12)

$C_{G C}=\frac{F \cdot Q_T}{2 \cdot N_e \cdot R \cdot T} \sinh \left(\frac{Q_T}{N_e^2 \cdot A_i \sqrt{8 R T \varepsilon \varepsilon_0 C}}\right)$          (13)

$Q_T=\int i_{S C} d t$          (14)

$i_{s d}=N_e \cdot I_f \cdot \varepsilon^{\left(\frac{\alpha F c\left(\frac{V_{\text {init }}}{N_s}-\frac{V_{\max }}{N_s}-\Delta V\right)}{R T}\right)}$          (15)

$\begin{aligned} & V_{S C} =\frac{N_e Q_T d}{N_p N_e \varepsilon \varepsilon_0 A_i} +\frac{2 N_e N_S R T}{F} \sinh ^{-1}\left(\frac{Q_T}{N_p N_e^2 A_i \sqrt{8 R T \varepsilon \varepsilon_0 c}}\right)-R_{S C} i_{S C}\end{aligned}$          (16)

2.2 LPF as EMS

The EMS has an important role for HESS. It manages the power distribution for each ESS. Filter-based energy management is part of the rule-based EMS class. In this study, the type of filter chosen is a low-pass filter (LPF). The EMS design using LPF is illustrated in Figure 6. This process is inside the EMS block of Figure 3. The power demand (P_Load) is sent to LPF. The output of LPF is a low frequency for the battery. Since no DC-DC converter in the battery, the signal output from the LPF is used to get the reference power for the SC by subtracting the power demand signal with it. After that, the SoC limiter needs to be added because the LPF method cannot accommodate the constrain except frequency. To avoid SC being over-charged and over-discharged, the SoC_SC is limited to 50-100%. According to the study [23], the charge of SC is not allowed to go down below 50% of the maximum voltage. The final output is a power reference for SC, P_SC*, which then send to the DC-DC converter. The challenging task of LPF as EMS is finding the best cut-off frequency. Therefore, in this study, the RPFT is proposed.

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Figure 6. EMS design using LPF

2.2.1 Ragone plot

The Ragone plot is a plot that shows the relation between power density and energy density of an energy storage system (ESS). Figure 7 shows a Ragone plot of some ESS. It is seen that the battery has high energy density with low power density. On the other hand, the supercapacitor has a low energy density with a high-power density. From this plot, the cut-off frequency (fc) is calculated using Eq. (17) [24-27]. This method was used by the study [24] to determine the cut-off frequency of LPF EMS before then they add an adaptation algorithm to improve the performance. Based on the Ragone plot in Figure 7 and Eq. (17), the cut-off frequency is chosen at point (10², 10⁴) of the Ragone plot which is 0.01Hz.

The fc is then substituted in Eq. (18) which is the first order LPF transfer function. This LPF will attenuate the signal with a frequency higher than the determined cut-off frequency. The rest with the high frequency is sent to the SC. However, the Ragone plot gives the cut-off frequency for the general battery and drive cycle. Each battery has a different and specific working frequency. Therefore, the specific cut-off frequency is important to optimize the battery performance.

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Figure 7. Ragone plot [28]

$f_c=\frac{\rho_{\text {power }}[\mathrm{W} / \mathrm{kg}]}{\rho_{\text {enerqi }}[\mathrm{J} / \mathrm{kg}]}$          (17)

$L P F=\frac{\omega}{s+\omega}=\frac{1}{\left(\frac{1}{2 \pi f_c}\right) s+1}$          (18)

2.2.2 FFT

The spectrum analysis method is used to know the frequency of the power demand of the electric vehicle. Since every drive cycle has different power demand, this method is not track-independent, which will give a different result if the drive cycle is not the same. There is two spectrum analysis method that can be used which is FFT and Power Spectral Density (PSD). FFT gives the result of the magnitude of each frequency, whereas PSD gives the power density in each frequency. In this study, the FFT method is used.

The step used for finding cut-off frequency using the FFT method is as follows:

a) Apply the power demand signal to FFT.

b) Calculate the magnitude spectrum.

c) Find the cut-off frequency for which the magnitude is below the magnitude spectrum threshold.

In this study, the maximum magnitude of the spectrum is divided by 10 to set the threshold for selecting the cutoff frequency. This threshold is used to exclude any frequencies in the spectrum that have a magnitude below 1/10th of the maximum magnitude. This is done to avoid selecting a cutoff frequency that is affected by noise or other low-level signals in the data. By excluding these low-level signals, we can select a more reliable cutoff frequency for the LPF. The value of 1/10 used in this code is arbitrary and can be adjusted depending on the specific application and the noise level in the data. If the noise level is low, a higher threshold value can be used, while if the noise level is high, a lower threshold value may be more appropriate.

The FFT analysis is done using MATLAB and the result is shown in Figure 8. The bar graph shows the percentage magnitude compared to the maximum magnitude spectrum. This method is already used by studies [28, 29]. Based on Figure 8, the cut-off frequency is 0.009Hz which is close to that of the Ragone plot method.

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Figure 8. FFT of UDDS drive cycle

2.2.3 Proposed method

As the Ragone plot only gives the general approximation for battery cut-off frequency. Hence, another method is required to get the optimal value of the cut-off frequency to ensure the battery in HESS works optimally. The proposed method is RPFT. Fine-tuning is referred to as the Proportional Integral Derivative (PID) control which finds the best PID’s gain result from the initial value. The RPFT step is illustrated in Figure 9. The initial value of the cut-off frequency is from Ragone Plot which is 0.01Hz. Then, searching the best result from the frequency above and below this initial value by adding or subtracting it with the determined step. The forward step (F_step) and backward step (B_step) used in this study is 0.01 and 0.002, respectively. The search is stopped when the new cut-off frequency value has a worse performance compared with the previous one.

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Figure 9. Fine tuning step

The performance of the proposed method is compared to the Ragone plot and FFT based on four criteria which are delta-SoC, distance per kWh, maximum battery current (I bat max), and Battery Current Root Means Square (BCRMS). In terms of distance per kWh, the higher value is better. On the other hand, in terms of delta-SoC, maximum battery current, and BCRMS, the lower value is better. The main objective of this study is to prolong the battery lifetime; therefore, battery peak current and BCRMS are the main criteria. Whereas the others are used to know the effect of the proposed system and EMS on the traveling distance. The battery aging parameter based on the BCRMS is calculated using Eq. (19) [30]. Where T_f  is the simulation time.

BCRMS $=\sqrt{\frac{1}{T_f} \sum_{i=1}^{T f} I_{b a t}^2}$          (19)

The simulation results reveal that the RPFT method, employing an optimized cut-off frequency, surpasses the performance of both the Ragone plot only and FFT methods. This superiority is evident in terms of delta-SoC, distance per kWh, and peak battery current. A heightened cut-off frequency substantially diminishes the power portion allocated to the supercapacitor (SC), thus nudging the system's behavior towards that of a battery-only EV. Conversely, a reduced cut-off frequency augments the power portion for the SC, thereby supplying more power beyond peak power demands. However, it is imperative that the optimal cut-off frequency is precisely calibrated through fine-tuning to achieve the desired results. A decrease in peak currents and BCRMS by as much as 29.80% and 9.99%, respectively, is achieved by the RPFT method, thereby effectively extending the battery's lifespan.

These findings hold substantial implications for HESS EV energy management, offering practical advantages such as improved battery performance, extended lifespan, and decreased maintenance costs. The RPFT method emerges as a promising solution for optimizing energy use in HESS EVs, thereby enhancing the overall efficiency and reliability of these vehicles.

In future studies, the development of an adaptive algorithm, such as one based on fuzzy logic, could be explored. This would allow for the recognition of different drive cycles and dynamic adjustment of the cut-off frequency. Such an advancement would enable the control system to adapt to different driving scenarios, ensuring optimal energy management across a range of conditions.