2. Electric Vehicle System and Dynamic Modeling
In this section, first, the power-train modeling of the EV is detailed. Then, the longitudinal and lateral dynamic model is proposed. The architecture of the power train is shown in
Figure 1. The motor’s command torque is dynamically coupled through simple gearbox and transmitted to front-wheels via a conventional differential drive.
The power train modeling parameters and lookup tables were obtained through its technical specifications and laboratory experimentation in a commercial EV detailed in [
24].
It is essential to mention that for EV modeling, the altitude of 2550 m above sea level was considered. This information was factored into the calculation of the air density. Given that vehicle driving at higher altitudes, the air density would have lower values, and so the air resistance as well. According to [
25], the air density of 0.96 kg/m
3 was considered.
On the other hand, the rolling resistance coefficient was calculated in relation to different types of roads and variable weather conditions. Taking into account the real road scenarios, the value for this parameter is 0.017.
The Aerodynamic Drag Coefficient (
) is a rather complex parameter, and in practice, wind tunnels and coast down tests are often used to obtain it. For this document, the Kia Soul EV manufacturer provided the 0.35 value for
. Finally, the value of a vehicle’s frontal area can be estimated as the multiplication of width and height. However, as the shape differs between model vehicles, this value is perhaps not applicable. Nevertheless, various estimations can be found in the literature [
4,
26]. Based on the EV’s model, the frontal area’s estimate is found according to the weight and the
of the vehicle [
26].
The parameters are shown in
Table 2.
2.1. Dynamic Modeling
A system model is required for designing the vehicle motion control, with motor torque as input and EV speed as output. Tire forces influence the vehicle’s longitudinal dynamics model, the aerodynamic drag force of the vehicle, rolling resistance forces, and the gravitational force related to the inclination of the vehicle as shown
Figure 2 [
4,
27]. The external longitudinal forces acting on the vehicle are described in (
1) as follows.
The torque
applied to the front wheels through the differential drive causes the vehicle to move. The aerodynamic drag force is defined as shown in Equation (
2):
EV is driven by a transmission system between motor and wheel to improve vehicle performance. The primary function is to transfer power from the electric motor to the wheels, allowing the torque and motor speed to fulfill performance requirements [
28]. The torque generated by the electric motor is distributed in the front wheels via differential bevel gear. Considering the effort to overcome these forces, the differential drive allows the drive wheels to turn at different speeds when turning a corner or maneuvers while driving the vehicle. Another essential feature is distributing equal torques on each of the wheels, even when rotating at different speeds.
2.2. Battery Model
This model describes the battery state of charge (SOC) and the batteries’ output voltage using experimental data gathered during the driving of the EV on pre-established routes. In addition, the temperature effect was considered [
29].
The state of charge is formulated as:
where,
is the instantaneous current of the battery, considered positive for discharge and negative for discharge. The factor
Q is the nominal capacity measured in ampere-hour (
), the factor 0.997 is defined as the product of parameters that depend on the performance of the battery (charge-discharge) and the number of cycles of the battery [
30,
31]. Finally,
is an expression that depends on the battery temperature expressed as follows:
where,
a,
b, and
c are coefficients of the second-order polynomial found. Considering the equivalent circuit or the Thevenin model, which consists of an array of a Resistor–Capacitor (RC) network in series with a voltage source [
29,
32,
33,
34], the proposal is replacing the RC circuit with a transfer function as shown in
Figure 3.
Where R refers to the internal resistance of the battery. This model considers negative current for the discharge battery process; in contrast, for the charging process, the current is positive. To determine open-circuit voltage () several experiments of charge-discharge under low constant current were performed.
According to the data obtained during experiments on the EV battery during a previous research detailed in [
29], the fitted curve for the relationship between
and
is shown in
Figure 4 using 5th order regression,
is represented as shown in Equation (
5):
The model of the battery presented in this subsection determines the impact of temperature on the performance of the cell. The model shows better performance at low temperatures (25 °C), with a lower discharge rate. For temperatures higher than 35 °C, the discharge rate increases; this can be observed in the SOC decreasing rapidly for high temperatures, as shown in Equation (
3).
2.3. Inverter and Electric Motor
The inverter is a key component of the EV, similar to the Engine Management System (EMS) of combustion vehicles, determining driving behavior. The inverters design and different topologies aim to transfer the battery pack’s energy in direct current to the motor, modifying the voltage and frequency according to its needs. The inverter is also responsible for transforming the energy obtained by the regenerative brake to power the batteries. As a result, the performance of the EV is directly related to the inverter efficiency [
35,
36,
37].
To determine inverter performance there should to measure of battery power, motor performance, and power directly at the EV’s front wheels. The measurement experiment tests were made using the LPS 3000 dynamometer bench manufactured by MAHA Maschinenbau Haldenwang GmbH and Co. KG in Haldenwang, Germany.
In addition, the information obtained through the OBD II port directly from the ECU of EV is used by the authors in [
38] in order to generate an analysis of losses and efficiency curves in the vehicle subsystems as shown in
Figure 5. Before starting with the experiments, MAHA LPS 3000 recommends establishing the following conditions: tires pressure must be 30 PSI, the tire tread temperature must reach 30 °C, secure the vehicle with tension straps, and follow the measurement protocol that governs the dynamometer bank [
38,
39].
After analyzing data from experiments, the inverter efficiency curve as a function of the motor’s rotational speed is shown in
Figure 6. The inverter in this study shows minimum efficiency of 94% at high speed and maximum efficiency of 99% at low speed.
The electric motor of the test EV is a Permanent Magnet Synchronous Motor (PMSM) and has advantages of high efficiency and torque current ratio, high power density, and wide speed range. These features are suitable for automotive applications, especially for HEV and EV [
24,
38,
40,
41,
42]. Technical specifications declare the nominal parameters of the PMSM as 81.4 kW of maximum power, 400-V voltage, and 285 Nm maximum torque. The non-linear effects generated by the PMSM model, mechanical elements and the internal losses are considered within the lookup tables that were generated through experimentation in the dynamometric bank.
According to data of experiments, the transmission and torque efficiency curves as a function of the motor’s rotational speed are shown in
Figure 7, where the mechanical transmission torque efficiencies improve when increasing motor speed. It is essential to mention that
Figure 7 includes all losses between PMSM and gearbox, such as inertial losses, losses in couplings, and lubricant losses.
The PMSM mechanical power was calculated as follows
The efficiency factor
shown in
Figure 6 is obtained through mechanical power tests in the dynamometric bank and electrical power obtained from OBD data. Rewriting the Equation (
6) in terms of torque, voltage, and current, it can be expressed as:
The equation for motor torque can be expressed as follows
where
is the motor rotational speed in (rpm). The mechanical power output of the transmission is calculated by Equation (
10).
where
represents the efficiency factor shown in
Figure 6. Substituting from Equation (
7) into Equation (
10), the mechanical power output and torque of the transmission is obtained as
The EV model and its subsystems were implemented in MATLAB/Simulink. Experiments and power tests have previously validated the model feasibility and efficacy on a dynamometer bank. The measured experimental data also was used for the parameters adjustment of the mathematical model. The presented EV model leads to the calculation of torque, PMSM rotational speed, battery pack power, and the resulting energy consumption in each subsystem.
3. Energy Efficiency Optimizer
In order to propose an optimization algorithm for the driving patterns to achieve maximum efficiency energy, it is important to understand the traction management and the energy consumption flow, from the battery to the EV wheels. In addition, the power conditions and limitations of battery discharge EV during its operation were considered as studies showed in [
43,
44].
Figure 8 shows an overall diagram where each subsystem is interconnected. The proposal includes the electric vehicle’s dynamic model described in Equation (
1), the battery model presented in
Figure 3 and
Figure 4, the PMSM, and the inverter model as lookup tables obtained from
Figure 6 and
Figure 7a. The proposed optimizer requires as input the torque, battery power from OBD data, and dynamometer lookup tables and the output of the optimizer is the reference signal
.
For establishing a strategy for the optimizer, the objective function is obtained from the value analysis of the inverter and mechanical transmission efficiency lookup tables shown in
Figure 6 and
Figure 7. The total energy-efficiency function between power supplied for battery and power wheel of the system is given as:
It is essential to mention that the
represents the efficiency function between the battery and the transmission drive of EV. The decision variable of the optimization problem in (
14) is the rotational velocity generated by PMSM (Ns) expressed in rpm. The speed correction generated by the optimizer on the driving pattern can be assigned in real-time, improving the EV’s overall efficiency during its operation. The energy efficiency objective function was formulated according to the Equations (
6)–(
10), and inequality constraints are given as follows
where
j is the value obtained during the DC with a sampling period of 0.5 s and
is the maximum variation between real and optimal velocity reference that depends on driving patterns. The optimal vehicle velocity depends on the value of
found by the optimizer, radio of tire, and transmission ratio, as shown in Equation (
15).
To evaluate the optimizer’s performance, real data collected on specific routes is used where the speed profile is determined by limits speed conditions and patterns driving.
According to the established speed limits by Ecuador’s traffic law, three test routes were established.
Figure 9 shows data for a route on highway roads where the speed limit is 90 km/h, this route is considered as DC 1 in the analysis.
The DC 2 is considered the route generated by driving the EV in urban areas, where the maximum limit speed established is 50 km/h. The data for this route is shown in
Figure 10 describe a typical driving pattern inside the city.
Figure 11 shows a combination of highways and urban areas, where variations in driving patterns can be seen. This route is considered as DC 3.
The objective is to find a suitable optimization algorithm, among strategies used in the literature such as GA, SA and PSO.
Table 3 presents the fitness values and computational time requirements for each algorithm. PSO obtains the best results on maximizing energy efficiency and minor computational time.
In this study, the PSO, introduced for Eberhart and Kennedy [
45,
46,
47,
48] for swarm behavior and social cooperation, is applied to solve the discontinuous and highly nonlinear objective function and inequality constraints.
Each iteration adjusts the particle position according to its own experience and neighboring, where it is established in the best position encountered by the swarm. The direction that the population takes is defined by particles neighboring the main particle and the swarm history experience. The velocity
and position
of particles are updated by the following equations:
where,
k is the iteration number,
is the best value
in iteration k, pbest is the best position of the best particle,
and
are random numbers in the range of
and the particles number is defined since
. The PSO convergence depends on the learning factors
and
, the inertial weight w, the maximum generation k of a PSO stage, and the number of particles
N.
The objective function solution in Equation (
14), is given in two-dimensional lookup tables with the desired motor rotational speed in a specific range of the desired speed. The proposed solving process for optimal driving employs the process described in the flowchart illustrated in
Figure 12.
The variation of the rotation speed of the PMSM and the EV operating points are simulated under different driving pattern conditions, where the initial positions and the convergence of the particles during the execution of the algorithm are shown in
Figure 13. The performance of the algorithm shows that all particles converge towards the same point in an average of 6 iterations for each scenario while the EV is driven. As a result, the swarm’s collective behavior converges to the same state, suggesting that a global minimum has been found.
Step 1 Parameter settings: the maximum number of iterations N, particle size X, the inertial weight factor , acceleration coefficients and , random numbers for and , and constraint conditions (, and );
Step 2 Fitness calculations and evaluation: compute the best value
and position
of the particle that maximizes the objective function in Equation (
14) determined for
driven pattern sample;
Step 3 Compare value
and previous
: If
is greater than
then update new velocity
and position
of particles using Equations (
16) and (
17), otherwise keep the previous values.
Step 4 If the maximum iteration is met, terminate the algorithm. Otherwise, go to step 2.
Step 5 Repeat the process for sample .
4. Results and Analysis
In order to evaluate the energy efficiency algorithm presented in this paper, the dynamic modeling for a commercial EV and experimental tests presented in
Section 2, and real DCs were used. The simulation model was built with Matlab/Simulink is shown in
Figure 8. All parameters used in the simulation are described in
Table 2 and the solution for optimization problem (
14) where using PSO flowchart shown in
Figure 12 for solving the optimization problem.
The algorithm proposal aims to make small changes (
) in speed reference without affecting the driver’s behavior. In other words, the algorithm intends to make small changes in the speed, improving the efficiency of the vehicle, without removing control of EV of the driver,
Figure 14.
Variations of the delta factor are considered, ranging
factor between 0% and 10% with increases of 1% to determine the optimizer’s performance. The results obtained from the simulations are presented in
Table 4.
Table 4 shows that the SOC value of the battery, according to the model described in
Figure 3 and
Figure 4, presents modifications below
without applying the optimizing algorithm. Equation (
5) shown the VOC in the function of SOC where the battery voltage keeps the value without variations compared to the original value.
The optimizer algorithm proposed to maximize the EV’s energy efficiency. The (EEOptimizer) is designed to make speed adjustments in the vehicle depending on the data of mechanical torque, battery power, and a maximum variation of . These adjustments are made by following the trajectory of a real-world driving cycle.
Given the variety and complexity of vehicle driving patterns, it is essential to consider several of them to evaluate optimizer performance. These driving patterns are created from various speed characteristics described as driving styles: conservative, moderate, and aggressive [
5,
20].
The information is obtained by evaluating the EV on specific routes and real driving conditions described in
Figure 9,
Figure 10 and
Figure 11 that include different driving styles. During the simulation, the following information from ECU is required: power in the wheel, battery power, energy efficiency, torque and SOC for each speed variation of
defined in the optimization problem.
According to the performance of the PSO optimization algorithm presented in
Figure 13, the best locations of efficiency during the entire simulation time are located as a function of speed. In each DC, a specific driving pattern is presented, where results of the efficiency improvement for each DC are shown in
Figure 14.
The value of 0% for
refers to the fact that the VE operates without the optimization algorithm through DC. Before starting with the simulations, it is necessary to take into account that the initial value of the SOC is 1 and a simulation time is 3000 s for each DC. According to the results shown in
Table 4 and
Figure 14, it is possible to determine that the highest energy efficiency value is reached when
is 10%. However, it is important to note that the evolution in the increase in efficiency is greater when the variation
is between 5% and 6%.
Figure 14a shows that for a 5% variation in speed the efficiency increases to 63% of the efficiency when
is 10% and 74% when delta is 6%. For DC 2,
Figure 14b shows the efficiency of 68% and 72% when
is 5% and 6%, respectively. Finally,
Figure 14c presents 66% and 89% of the maximum efficiency value.
It is important to note that the power saving of the DC 3 is better than the others; This result corresponds to the test path that has moderate and aggressive style components in its driving pattern, therefore, the algorithm has several search spaces in the look-up tables.
On the other hand, when
= 5%, it is considered the best solution given that the SOC value remains fixed while the energy efficiency increases for all DC simulations. In this scenario, it is verified that there is an improvement in the EV’s energy efficiency without causing additional consumption in the battery pack.
Figure 15b shows that the proposed algorithm does not generate significant changes in the speed adjustment in EV during its operation, ensuring the following of the trajectory at the reference speed.
Figure 15a presents the result of the variations carried out by the optimizer. The efficiency of the EV during operation shows an increase during the entire simulation process. This result verifies that the proposed optimizer adapts to any driving style, improving efficiency throughout the EV travel; besides, it can be applied for long driving times. Another advantage of this approach is the computational time required to execute the proposed algorithm. The formulation was carried out according to (
14), the generated search tables and the metaheuristic algorithm used, present an average execution time of 0.55 milliseconds.
Figure 16 shows the calculation time required by the algorithm to generate a solution for each sample j of the DC. The sampling time of the vehicle measurements from OBD is 0.5 s, which implies the proposal can be implemented.
To examine the effectiveness of the optimizer, the operation of the algorithm is compared with strategies reviewed in the literature that show the use of DP and iDP as an alternative to find a solution to this problem. However, the computational cost increases depending on the amount of data it must process. The proposed optimizer keeps its computational cost in low levels, specifically 0.55 milliseconds average for any DC. Another aspect that can be emphasized is that during the formulation of the optimization problem, a factor is proposed such that guarantees that the algorithm works within a specific band according to the EV’s speed. Finally, only the PSO algorithm is considered to generate the search for the best solution, because no prior training or identification of additional DC is necessary, avoiding hybrid algorithms. These features show the advantages of the proposed method over the strategies presented in the literature.