# The Impact of the Speed and Temperature Variation on the Electric Vehicles Reliability

Authors: Amir Ghaedi; Mehrdad Mahmoudian; Reza Sedaghati
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

The transportation system contains many fossil fuel-based automobiles equipped with the internal combustion engine that results in the pollution of the environment and greenhouse gas emissions. In recent years, to replace these automobiles with clean choices, electric vehicles are developed. So far, three kinds of electric vehicles including hybrid, plug-in, and full-electric vehicles are introduced. In the hybrid and plug-in electric vehicles, both the internal combustion engine and electric motor are used to move the vehicle. However, in the full-electric vehicle, the movement of the vehicle is done only by the electric motor. Due to the development of the electric vehicles in the transportation system, different aspects of these vehicles such as reliability must be studied. The reliability indices of the electric vehicles are affected by the failure rate of the composed components. Thus, to exactly determine the reliability performance of the electric vehicles, the failure rate of the main composed components affected by different parameters such as speed of the vehicle and temperature is taken into account. In the present paper, to accurately study the reliability of all-electric vehicles, the impact of variation in the temperature and vehicle speed on the failure rate of the composed components including battery, inverter, electric motor, and other static and rotation parts of the full-electric vehicle and consequently the failure rate of the vehicle is investigated. To determine the impact of operating temperature on the failure rate of composed components, the Arrhenius law is proposed. Based on the variation in the vehicle failure rate in terms of the vehicle speed and temperature, the reliability of the electric vehicle at different conditions is determined. It is concluded from numerical results performed in the paper that the failure rate of the understudied full-electric vehicle varies between 3.5 and 6 failures per year when the temperature varies between 0 and 50°C and the vehicle speed varies between 0 and 200 km/h.


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## Body

## 1. Introduction

Due to the pollution of the environment, greenhouse gas emissions, global warming, and change in the climate arisen from the spread use of fossil fuels, different countries in the world limit the use of fossil fuels. Among different use of fossil fuels, the transportation system has a significant share in the pollution of the environment. Thus, to reduce the air pollution arisen from fossil fuels, the clean choices can be used in the transportation system for the replacement of fuel-based automobiles. In the conventional automobiles, fossil fuels are entered into the internal combustion engine to create the power required for the movement of the vehicle. Electric vehicles as clean choices can be used in the transportation system to reduce the pollution of the air. Due to the development in the technologies of the electric vehicles and the increasing growth of the use of these vehicles, different aspects of them such as reliability must be taken into account. For this purpose, many researchers have studied the reliability of the electric vehicles. Paper [1] investigates the reliability of the electric vehicles based on the probabilistic graphical model. In the proposed framework of this paper, exponential and Weibull distributions are used to model the failure rates of the composed components of the electric vehicles. Paper [2] studies the grid-to-vehicle and vehicle-to-grid strategies for charging and discharging the batteries used in the electric vehicles. In this paper, the impact of these strategies on the reliability and durability performance of the Li-ion cells of the vehicle batteries is investigated. Paper [3] evaluates the reliability of the power train system of battery electric vehicles and its composed components including the motor, motor controller, power distribution unit, and battery system. In this article, a fault tree model associated with the power train system of the electric vehicle and its composed components is developed. Based on the proposed method of the paper, the trends in the reliability indices of the power train system against the service life of the battery-electric vehicle are determined. Paper [4] compares the reliability between an electric vehicle and a mechanical one. For this purpose, two vehicles including Renault Zoe as the electric vehicle and Renault Kadjar as the mechanical vehicle are taken into account. In addition to the reliability, the price, energy, autonomy, torque, battery and engine lifespan, power, and pollution of the two vehicles are compared in the paper. Paper [5] investigates the reliability of the plug-in hybrid electric vehicle chargers based on the two-phase interleaved unidirectional charger topology. For reliability evaluation of the charger, the MIL-HDBK-217 is used to calculate the component level reliability. Besides, the Markov model is suggested to investigate the reliability of the fault-tolerant two-phase interleaved charger structure considering the impact of the components repair. Paper [6] proposes the eight switches bridge converter to be used in the electric vehicles to improve their reliability and performance. This converter can be used in the electric vehicles to drive the brushless DC motor. For verifying the performance of this topology, the simulation results in MATLAB software and experimental results are given in the paper. The reliability of the electric converters as the key component of the electric vehicles is evaluated in the paper [7]. In this paper, the failure rate of the composed components of the converter in the wear-out period is considered to be variable. For this purpose, the Weibull distribution function is used for modeling the hazard rate of the composed components of the electric converters that can be used in the electric vehicles. The electric converters are used in different renewable energy-based power plants such as wind and tidal turbines, photovoltaic systems, and tidal barrages. In [8–12], reliability studies of the renewable generation units considering the variable failure rate of components are performed. In [8], a reliability evaluation of the current type tidal turbines equipped with the doubly fed induction generator (DFIG) is performed. In this paper, the impact of the variation in the speed of tidal streams on the failure rate of the back-to-back converter connected to DFIG is studied. Two converters named the rotor side converter and grid side converter are taken into consideration and their failure rate of them versus the speed of the tidal streams is determined. In [9], reliability studies of the wind and tidal turbines connected to the grid are performed. In this paper, the failure rate of the electric converters used in the wind turbines is considered to be variable with variation in the wind speed, and the failure rate of the electric converter used in the tidal turbines is considered to be variable with variation in the speed of tidal streams. Thus, the reliability of these renewable energy-based power plants considering the variable failure rate of the composed components is investigated in the paper. Paper [10] studies the reliability of the reservoir-based tidal power plant considering the variation in the height of the tidal. In this paper, the impact of variation in the tidal level on the failure rate of composed components of the barrage type tidal power plant including turbine, permanent magnet synchronous generator, back-to-back converter, transformer, and cable is taken into account. Paper [11] evaluates the reliability of wind turbines equipped with the permanent magnet synchronous generator using the Monte Carlo simulation approach. In this paper, the failure rate of the composed components of the wind unit is considered to vary with variation in the wind speed. In [12], adequacy assessment of the barrage-type tidal generation unit is performed using the analytical approach based on the capacity outage probability table. In this paper, the impact of the variation in the tidal level on the failure rate of the turbine, generator, transformer, and cable is studied and an average value for the failure rate of the composed components is used in the reliability calculation of the generation unit.In [13], a reliability evaluation of a distribution network containing electric vehicles considering quasidynamic traffic flow and energy transfer from the vehicle to the grid is performed. To investigate the reliability performance of distribution networks including the electric vehicles, the traveling chain theorem is suggested for describing the travel of electric vehicles. In this paper, the impact of penetration level of electric vehicles, discharging threshold, and storage capacity of batteries on the reliability indices of the distribution network and electric vehicles is studied. Paper [14] studies the impact of electric vehicles on the reliability indices of the power system containing wind turbines. In this paper, the mathematical model of the electric vehicles in random charging mode is determined and so, the variation in the reliability indices of the power system considering the variation in the penetration level of the electric vehicles in the power system is analyzed. In [15], the impact of the electric vehicle charging stations on the reliability performance of distribution networks is investigated. For this purpose, the radial 33-bus IEEE reliability test system is taken into consideration and the reliability indices of this distribution system considering the electric vehicle charging station are calculated. In [16], the impact of loading associated with the charging and discharging of large-scale electric vehicles on the distribution network is studied. In this paper, the charging and discharging modes of the electric vehicles in the charging stations are analyzed, and different aspects of distribution networks affected by electric vehicles including reliability, economy, environmental impacts, and compatibility with renewable resources are investigated. Paper [17] studies the impact of electric vehicles equipped with energy transfer from the grid to the vehicle and also from the vehicle to the grid on the reliability and performance of the distribution network. In this paper, different reliability indices including the expected duration of curtailment and expected energy not supplied are calculated to investigate the impact of different penetration levels of electric vehicles on the reliability and availability of the distribution network. In [18], the impact of the technology associated with the connection of electric vehicles to the power system on the reliability and performance of the distribution network is investigated. The reliability indices of the power system before and after electric vehicle addition to the power system are calculated during a year. To analyze the reliability performance of the distribution network containing electric vehicles, a suitable model is developed to estimate the energy of electric vehicles during 24 hours. The excess power associated with emergencies such as failures and outages of units can be transferred to the grid using the proposed model. In [19], the reliability performance of DC and AC networks containing electric vehicles is evaluated. In this paper, a modular structure containing multiple sources is developed to use the fast charge, connection, and transfer technology of electric vehicles. Based on the reliability evaluation of the system, an optimal structure is determined using the Markov process, block diagram, and source priority methods. Paper [20] studies the reliability performance of composite power systems containing hybrid and all-electric vehicles. In this paper, a stochastic model is proposed for two types of electric vehicles considering the characteristics of these vehicles including battery storage capacity, traveling distance until the vehicle completely discharged, and discharge rate. Besides, the impact of different driver patterns including recharge time, arrival and departure times, driving time during the day, and also, different charging strategies on the reliability indices of the composite power system is studied. In [21], to evaluate the reliability, availability, and service continuity of grid-connected electric vehicles, a suitable method based on the Markov structure is proposed. In this paper, a Markov-based reliability model of the main components of the electric vehicles including battery, electric motor, motor driver, controller, charging unit, and energy management unit is developed. In [22], MATLAB-Simulink software is used to analyze the reliability performance of electric vehicles. In this paper, lead-acid battery, electric motor, and motor controller as the main components of the electric vehicle are taken into consideration and the impact of driving cycle variation, thermal stresses, and failure analysis on the reliability performance of the electric vehicle is studied. Paper [23] proposes that electric vehicles as energy storage systems to improve the reliability of the distribution system. In this paper, the electric vehicles batteries are used to continue the service to the consumers during the peak load and load curtailment times. The model associated with the storage capacity of batteries is developed considering two-way energy transfer including battery to grid and also grid to battery. Besides, the optimal location of charging stations, optimal times of charging and discharging of batteries, and optimal rate of charging and discharging of them are determined to improve the reliability, decrease the losses and increase the voltage regulation of the distribution network. In the proposed optimization problem, the output energy of the batteries of the electric vehicle is maximized, the expected energy not supplied to the consumers is minimized and the losses of the distribution network are minimized. In [24], a reliability assessment of power electronic components used in hybrid and all-electric vehicles is performed. In this paper, the reliability-based design method of two well-known power electronic devices including battery chargers and inverters for driving electric motors are introduced. in [25], reliability models of electric vehicles based on the Markov theorem are developed and used for the adequacy assessment of microgrids containing electric vehicles. In this paper, electric vehicle batteries as energy storage systems are used to improve the reliability of the distribution network. The batteries of the electric vehicles are modeled as smart energy storage devices that can be used as controllable loads for transferring the electric power from the grid to vehicle and also as controllable energy sources for transferring the electric power from the vehicle to the grid. In [26], the reliability of an electric motor used in the electric van is evaluated using the fault tree analysis approach. In this paper, the electric motor and its controller are considered as a system, and the impact of the failure of this system on the overall failure of a vehicle is evaluated. In [27], an approach is proposed to investigate the reliability and availability of a solar electric vehicle equipped with a standby plug-in facility. In this paper, a composite reliability model based on the Markov theorem is developed to consider the stochastic failure and repair characteristics of the main components of the electric vehicles including the power supply system, energy storage system, motor-drive system, controllers, and energy management system. In [28], reliability modelling of different types of electric vehicles including series, parallel and compound hybrid electric vehicles, series, parallel, and compound plug-in hybrid electric vehicles, and the all-electric vehicle is performed. To develop the reliability model of each type of electric vehicle, the failure of composed components and its impact on the overall failure of the vehicle are considered. In this paper, the availability of all types of the electric vehicles is calculated and from a reliability point of view, a comparison among them is performed. A summary of the literature review is given in Table 1.Table 1
A summary of the literature review.
ReferencesReviewed topicsDisadvantages[1]Reliability evaluation of electric vehicles considering exponential and Weibull distribution for failure rateNot considering variable failure rate and failure of composed components[2, 3, 5–7, 21, 22, 24, 26, 27]Reliability evaluation of electric vehicles considering constant failure rate of composed components including the battery, motor, motor controller, two-phase interleaved unidirectional charger, eight switches bridge converter, motor driver, charging unit, energy management unit, lead-acid battery, power electronic devices, and energy storage system.Not considering the variable failure rate[4]Reliability comparison between electric and conventional vehiclesNot considering variable failure rate and failure of composed components[8–12]Determining variable failure rate of composed components of wind turbine, current type tidal turbines, and barrage type tidal units arisen from variation in the wind speed, tidal current speed, and tidal heightNot considering the reliability of electric vehicles[13–20, 23, 25]Reliability evaluation of distribution network, generation power system, composite power system, DC and AC power system containing electric vehiclesThe electric vehicles are modeled as energy storage systems and do not consider the failure of composed components and also variable failure rate.[28]Reliability evaluation of different types of electric vehiclesNot considering the variable failure rateIt is concluded from reviewed papers published in recent years, so far, to evaluate the reliability performance of the electric vehicles, a constant value is considered for the failure rate of the vehicle. The failure rate of the electric vehicle is dependent on the failure rate of composed components. Due to the variation in the vehicle speed and temperature, the failure rate of composed components of the electric vehicles including battery, inverter, electric motor, and other components varies. Thus, to accurately investigate the reliability performance of the electric vehicles, the dependency between the failure rate of these effective components on the vehicle speed and temperature must be considered. In the current paper, the impact of variation in temperature and vehicle speed on the failure rate of composed components of electric vehicles is studied. For this purpose, both electrical and mechanical components of the electric vehicles are taken into account. The variable failure rate resulted from this study can be used in the reliability evaluation of electric vehicles to exactly calculate the reliability indices of these vehicles. Thus, the contributions of this paper would be as follows:(i)
Developing a reliability model for full-electric vehicles based on the reliability modelling of composed components including battery charger, battery, inverter, electric motor, and other main components(ii)
Deriving the equations associated with the dependency of the failure rate of composed components of the full-electric vehicle including battery, inverter, electric motor, and other static and rotation parts on the vehicle speed and temperature(iii)
Investigating the impact of variation in the vehicle speed and temperature on the failure rate and availability of full-electric vehiclesAccording to the aims of this paper, it is organized as follows: The second section introduces different types of electric vehicles and their composed components. In the third section, the impact of variation in temperature and vehicle speed on the failure rate of composed components of full-electric vehicles is studied. The reliability model of full-electric vehicles is developed in the fourth section. Numerical results associated with the reliability evaluation of full-electric vehicles are given in the fifth section. The last section is devoted to the conclusion of the paper.


## 2. Different Types of Electric Vehicles

So far, three types of electric vehicles named hybrid, plug-in, and full-electric vehicles are used in the transportation system. In the hybrid electric vehicle, the movement of the vehicle can be done by both an internal combustion engine and an electric motor. Based on this technology, the battery is charged only by power produced by the internal combustion engine. Thus, the battery is not connected to the external power grid. The second type of electric vehicle is a plug-in hybrid electric vehicle. To move the plug-in electric vehicle, an internal combustion engine and electric motor can be implemented. In the plug-in electric vehicle, in addition to the power generated by the internal combustion engine, the battery can be charged by the external power grid through the charging stations, while, in the hybrid electric vehicle, the battery is charged only by the power generated by the internal combustion engine. To further reduce the pollution produced by fossil fuels, the third type of electric vehicles named the full-electric vehicle is introduced. In the full-electric vehicle, the movement of the vehicle is done only by the electric motor supplied from the batteries [29]. In modern transportation systems, to charge the batteries used in this technology of electric vehicles, charging stations are placed at suitable distances. Figure 1 presents the structure and main components of a full-electric vehicle.Figure 1
The structure of a full-electric vehicle.In full-electric vehicles, the battery is charged by the electricity of charging stations through a battery charger. The charging stations are connected to the bulk power system. Thus, the electric vehicles can be used as flexible loads. At peak hours, when the load of the power grid is high, the electric vehicles can discharge their batteries and transfer their power to the grid. At low loads, the electric vehicles can charge their batteries to supply the power required for movement. An electric converter connected to the battery is used to supply the electric motor for driving the vehicle. For movement of the vehicle, the mechanical power produced by the electric motor is transferred to the wheels through the transmission system composed of the gearbox, clutch, differential, and axle. The other main components of a full-electric vehicle are the chassis, body, and control system. In the electric vehicles, different electric motors including DC motors equipped with brushes, brushless DC motors, switched reluctance motors, axial flux iron less permanent magnet motors, and permanent magnet synchronous motors can be used [30]. In this paper, a three-phase permanent magnet synchronous motor is assumed to be used in the understudied electric vehicle. High efficiency, high power density, and high reliability are the advantages of the permanent magnet synchronous motors which results in the wide application of these motors in the electric vehicle industries. The control of permanent magnet synchronous motors is easy, and their performance in terms of maximum generated torque per current of them is excellent which results in the optimal extended speed operation.Thus, the electric converter of this technology for electric vehicles would be a three-phase inverter. To change the speed of the electric vehicle equipped with the permanent magnet synchronous motor, the output frequency of the inverter should be changed. The stator windings of the permanent synchronous generator are sinusoidally distributed, and the relation between the mechanical rotation speed of the rotor and the frequency of the power supply would be as [31](1)n=120fP,where n, f, and P are the speed of the motor in rotation per minute (rpm), the output frequency of the inverter (Hz), and the number of poles of the stator winding of the motor. The torque required for driving the vehicle is dependent on the conditions of the road (uphill, downhill, or other ramps), and the inertia of the vehicle is determined based on the structure and mass of the vehicle and also the mass of the occupants of the vehicle. In this study, the required torque for driving the vehicle is assumed to be constant and equal to τ. Thus, the mechanical power required to drive the vehicle would be [31](2)Pmn=τω=2τπn60,where Pm and ω are the mechanical power generated by the motor and the angular speed of the motor in radian per second and τ is the required torque for driving the load connected to the motor. The input electric power of the permanent magnet synchronous motor supplied by the inverter can be calculated as [31](3)Pe=3VIcosθ,where Pe, V, I, and cosθ are the input power of the motor stator, the line-to-line voltage of the stator windings, the current of the stator windings, and the power factor of the motor. In this paper, to change the speed of the permanent magnet synchronous motor, the simultaneous variation of the voltage and frequency is performed to keep the voltage to frequency ratio (V/f) that is proportional to the magnetic flux of the motor core. This method prevents from the saturation of the stator and rotor cores. The induced voltage of the stator windings in the synchronous motor is determined as [31](4)V=kϕω,where k and φ are a constant and the magnetic flux of the motor. The synchronous motor used in the understudied electric vehicle of this paper is based on the permanent magnet technology, and thus, the flux of the motor is constant. Thus, the voltage of the motor is proportional to the speed of the rotor. For changing the speed of the electric vehicle, the generated frequency and voltage of the inverter would be based on (2) and (4). According to the torque and speed of the vehicle, and consequently, the power required for driving the electric vehicle, the output voltage, frequency, and current of the inverter are determined. Then, the discharge rate of the battery for obtained values of voltage and current is calculated. Thus, the following steps for calculating the quantities of devices are proposed:(i)
The required torque of the vehicle is considered to be constant. Thus, based on the speed of the vehicle, the required mechanical and consequently electric power of the permanent magnet synchronous motor is determined(ii)
According to the rotation speed and electric power of the permanent magnet synchronous motor, the current and voltage of the motor, and consequently the output frequency and voltage of the inverter are determined(iii)
According to the current and voltage of the inverter, the voltage and current of the battery are determined


## 3. The Impact of Speed and Temperature on Failure Rate of Electric Vehicles Components

In this paper, a two-state Markov model is used to present the reliability performance of a device applied in the electric vehicles. In this model, a device can be in perfect or failed states. This model is presented in Figure2. The failure and repair rates are transition rates between perfect to failed states and failed to perfect states, respectively [32].Figure 2
Markov model of each device.In this stage, the impact of variation in the speed and temperature on the failure rate of composed components of the full-electric vehicle is studied. The battery charger is a rectifier that transfers the AC power of the charging stations to DC power required for charging the battery used in the electric vehicle. When an electric vehicle is connected to the charging station, the vehicle is stopping. Besides, the charging stations are equipped with canopies that can maintain the station’s temperature at a suitable value. Thus, the failure rate of the battery charger is not affected by the vehicle speed and temperature. In [33], the impact of different discharge rates on voltage, temperature, and useful life cycle of lithium-ion batteries is studied. To determine the impact of vehicle speed on the failure rate of batteries, the following steps are proposed:(i)
The discharge rates of the battery at different vehicle speeds are determined.(ii)
The useful life cycle of the battery or the mean time to failure of the battery at different discharge rates can be determined based on results obtained in [33].(iii)
The failure rate of the battery versus different discharge rates is determined as [32](5)fr=8760MTTF,wherefr and MTTF are failure rate in number of failure occurrences per year and mean time to failure in hours, respectively. To determine the impact of temperature variation on the failure rate of batteries used in the electric vehicles, the Arrhenius law is proposed. According to the Arrhenius law, the failure rate of an understudied device such as a battery is dependent on its operating temperature of it as [34](6)frT=frT0e−Ea/k1/T−1/T0,wherefr (T), fr (T0), Ea, k, T, and T0 are the failure rate of the battery at temperature T, the failure rate of battery at temperature T0, activation energy, Boltzman constant, ambient temperature, and initial test temperature both in Kelvin, respectively. Another approach that can be used to determine the impact of discharge rate on the failure rate of the battery is as follows:(i)
Determine the temperature of the battery at different discharge rates based on results obtained in [33](ii)
Determine the failure rate of the battery at different discharge rates based on the Arrhenius law.Due to the lack of complete data required for the second approach, to determine the failure rate of the batteries in terms of different discharge rates, the first approach is used, in this paper. In this stage, the failure rate of electric converters used in the electric vehicles versus the vehicle speed and temperature is determined. The electric converters are composed of semiconductor devices such as diodes, thyristors, IGBTs, or other transistors. The failure rate of semiconductor devices is significantly affected by the variation in the temperature of the junction. To determine the failure rate of these devices, the power loss arisen from the current passing them is calculated. As can be seen in Figure3, in this paper, the understudied inverter is considered to contain 6 switches based on IGBT technology that are placed in parallel with 6 reverse diodes.Figure 3
The understudied inverter.With the failure of every semiconductor (IGBT or diode), the operation of the inverter is failed. Thus, in the reliability modeling of the inverter, the IGBTs and diodes would be series. To determine the failure rate of the inverter, the failure rate of the semiconductors must be summed. For obtaining the failure rate of the semiconductor devices, the current passing through them must be determined to calculate generated power loss and consequently, the temperature rise occurred in their junction. Then, according to the Arrhenius law, the failure rate of the semiconductor devices versus obtained temperature can be calculated. The temperature rise of different devices is dependent on the power loss created in them resulting in heat generation. According to the thermal modeling of the semiconductor devices, their temperature can be calculated as [35](7)Tscd=Ta+Plrth+Pscdrch,where Tscd, Ta, Pl, rth, Pscd, and rch are the operating temperature of each semiconductor device, the ambient temperature, the inverter power loss that creates heat generation, the thermal resistance between the heat sink and semiconductor medium, the thermal loss associated to every semiconductor device and the thermal resistance between the semiconductor medium and junction. The understudied inverter is composed of 6 IGBTs and 6 diodes. Thus, for determining the inverter power loss, the power loss of the semiconductor devices including 6 IGBTs and 6 diodes must be summed as [8](8)Pl=6×Pscd−IGBT+Pscd−diode.To determine the thermal loss of every semiconductor device such as IGBT, diode, or thyristor, the following equation can be used [8]:(9)Pscd=Pscd−cond+Pscd−rec=12U0Imπ+rIm24±mcosφU0Im8+rIm23π+UdcUnomPsw.The power loss of the semiconductor device is calculated by summing the power loss of them associated with the conduction (Pscd-cond) and recovery (Pscd-rec) modes of their operation. In (9), U0, Unom, and Udc are the voltage drop on the semiconductor devices, the reference, and the dc-link voltage. Besides, Psw is the power loss associated with the switching of semiconductor devices, r is the resistance of the semiconductor device, m is modulation index, ϕ is the angle between voltage and current, and Im is the peak value of the current associated with every phase of the inverter. The equations associated to determine the power loss of IGBTs and diodes are different. For IGBTs and diodes, the sign of the second term of equation (9) would be positive and negative, respectively. Besides, the values of parameters such as U0, r, Unom, and Udc used in (9), are different for IGBT and diode. In (9), the peak value of the inverter current must be placed which is calculated as [8](10)Im=2Pe3V,where Pe and V are the electric power and the RMS voltage of the inverter, respectively. The failure rate of the semiconductor devices is dependent on the operating temperature, junction electrical stress, and operation modes. In [35], the equation for calculating the failure rate of the semiconductor devices such as diodes and IGBTs is given as(11)frscd=fr1a1g1+fr2a2g2+fr3g3+fr4.In (11), the failure rate is obtained in times of occurrence per 1000000 hours. The parameters fr1, fr2, fr3, and fr4 are, respectively, the operation-mode base failure rate, nonoperation-mode base failure rate, the temperature cycles failure rate, and the failure rate of junction electrical stress. The parameters a1 and a2 are, respectively, the operation-mode and nonoperation-mode acceleration factors that are calculated as [8](12)a1ora2=e−Ea/k1/Tscd−1/T0.In (12), the values of activation energy related to the operation and nonoperation states are different. Besides, the operating and test temperature must be placed in the equation in Kelvin. The parameters g1, g2, and g3 are operation-mode duty cycle factor, nonoperation-mode duty cycle factor, and thermal cycle acceleration factor. These factors can be calculated as [8](13)g1=1−Pnondcop,(14)g2=Pnondcnonop,(15)g3=Tscd−TadT2,where dcop, dcnonop, and dT are determined based on the semiconductor devices and would be constant parameters. Besides, Pnon is the occurrence probability of the nonoperation mode of the battery that supplies the input power of the inverter. With the failure of every semiconductor device, the inverter is failed, and so based on the equivalent failure rate of a system composed of series components, the failure rate of the understudied inverter can be obtained as [8](16)frinv=6×frIGBT+frdiode,where frinv, frIGBT, and frdiode are failure rates of the inverter, IGBT, and diode, respectively. According to the current passing through the phases of the inverter and the ambient temperature, the failure rate of the semiconductor used in the inverter and consequently the failure rate of the inverter can be calculated using equations (7) to (16). In this stage, the dependency of the failure rate of the permanent magnet synchronous motor on the vehicle speed and temperature is studied. A permanent magnet synchronous motor is composed of electrical and mechanical parts. Thus, the failure rate of a permanent magnet synchronous motor can be calculated by summing the failure rates of the mechanical and electrical parts as [32](17)frsm=frsm−e+frsm−m,where frsm, frsm-e, and frsm-m are the failure rate of the permanent magnet synchronous motor, the failure rate of the electrical part of the synchronous motor such as windings, and the failure rate of the mechanical part of the permanent magnet synchronous motor such as rotor and bearings, respectively. To determine the failure rate of the electrical parts of the synchronous motor, the operating temperature must be calculated, for this purpose, the power loss created in the motor is determined as [31](18)Psm−loss=3rmim2,where Psm-loss, rm, and im are the power loss of the permanent magnet synchronous motor, the electrical resistance of each stator winding of the motor, and the RMS value of the current passing through stator windings of the synchronous motor. According to the thermal modeling of the permanent magnet synchronous motor, the equivalent thermal resistance is determined and so, the operating temperature of the motor can be determined as [11](19)Tsm=Ta+rsm−thPsm−loss,where Tsm and rsm-th are the operating temperatures of the permanent magnet synchronous motor in Kelvin and the equivalent thermal resistance of the motor. According to the obtained operating temperature of the synchronous motor and the use of the Arrhenius law, the failure rate of the electrical parts of the permanent magnet synchronous motor is calculated. To study the impact of the variation in the temperature and vehicle speed on the failure rate of the permanent magnet synchronous motor, both the ambient temperature and the power loss of the motor are affected. For the mechanical parts of the permanent magnet synchronous motor such as the rotor and other components used in the full-electric vehicles that are mainly the mechanical parts, the dependency between the failure rate of these components and the affective variables, i.e., the vehicle speed and temperature, must be determined. In [36–40], it is concluded that the failures of the mechanical parts at small and high speeds are frequently arisen from the fatigue limit of the components. Based on this result, the useful lifetime or the mean time to failure of the mechanical components is proportional to the speed. Besides, it is deduced in these papers that when the temperature of the mechanical components made of carbon or alloy steels is 70 to 1000 F, their failure rate of them can be determined as [36–40](20)frT=tmfTtmfT0frT0,where fr (T), fr (T0), tmf (T), and tmf (T0) are the failure rate of the device at temperature T, the failure rate of the device at temperature T0, temperature modification factor at temperature T, and temperature modification factor at temperature T0, respectively. The temperature modification factor at each temperature in Fahrenheit is calculated as an empirical formula (21) [36–40]. To determine the effect of operating temperature on the tensile strength of steel, 145 tests are performed on 21 different carbon and alloy steels. The results are presented in Table 2 [36]. Then, a fourth-order polynomial curve fits the obtained data as presented in (21) [36].Table 2
The results of 145 tests [36].
Temperature ( F)tmf701.0001001.0082001.0203001.0244001.0185000.9956000.9637000.9278000.8729000.79710000.698(21)tmfT=0.975+0.000432T−1.15×10−6T2+1.04×10−9T3−5.95×10−13T4.


## 4. Reliability Model of Full-Electric Vehicles

With the failure of each main component of the full-electric vehicle including battery charger, battery, inverter, permanent magnet synchronous motor, transmission system, wheels, chassis, body, and control system, the operation of the electric vehicle is failed. Thus, from a reliability point of view, these components are series in the reliability model of the full-electric vehicle. The equivalent failure rate of series components is calculated by summing the failure rates of composed components. Besides, the probability of healthy state of series components can be determined by multiplying the probability of a healthy state of composed components. To determine the failure rate and repair rate of the full-electric vehicle, equations (22) and (23) are used [41].(22)frfev=∑k=1nfrk,(23)rrfev=frfev∑k=1nfrk/rrk,where frfev, rrfev, frk, rrk, and n are the equivalent failure rate of the full-electric vehicle, the equivalent repair rate of the full-electric vehicle, the failure rate of the composed components, the repair rate of the composed components, and the number of composed components. The availability of the system can be calculated as [41](24)A=rrrr+fr,where A, fr, and rr are the availability, failure rate, and repair rate of the system. The flowchart associated with the reliability evaluation of the full-electric vehicle considering the variation in the failure rate of composed components arisen from the variation in the vehicle speed and ambient temperature is depicted in Figure 4. According to this flowchart, the equivalent availability of the full-electric vehicle considering the variation in the vehicle speed and temperature is calculated.Figure 4
The flowchart of reliability evaluation of full-electric vehicle considering variation in the speed of vehicle and temperature.


## 5. Numerical Results

In this part, the dependency of failure rates of the composed components of the full-electric vehicle on the temperature and vehicle speed is studied. Besides, the equivalent failure rate of the full-electric vehicle versus the variation in the temperature and vehicle speed is investigated. For this purpose, a full-electric vehicle based on the permanent magnet synchronous motor technology is considered. The failure and repair rates of the composed components of the understudied full-electric vehicle at an ambient temperature of 25°C and a rated speed of 100 km/h are illustrated in Table3 [28]. The characteristics of the permanent magnet synchronous motor used in the understudied full-electric vehicle are presented in Table 4 [31]. Besides, the characteristics of the semiconductor devices used in the full-electric vehicles are presented in Table 5 [8].Table 3
Failure rate and repair rate of the composed components of the understudied full-electric vehicles [28].
ComponentsFailure rate (occ./yr)Repair rate (occ./yr)Battery charger0.00421000Battery0.0135876Inverter0.0413800Permanent magnet synchronous motor0.09852500Transmission system0.21250Control system0.00012000Other main rotation components2.15600Other main static components2800Table 4
Characteristics of the permanent magnet synchronous motor [31].
ParametersValuesPower200 kWVoltage400 VEfficiency95%Power factor0.6Rated speed6000 rpmMaximum speed12000 rpmRated torque300 NmTable 5
Characteristics of the semiconductor devices [8].
k0.00008617 eV/Krm0.05 ohmfr10.0001306fr20.001589fr30.00053fr40.00295dcop0.23dcnonop0.77dT80Ea-op0.3Ea-non-op0.4rch-IGBT15 K/kwrch-diode23 K/kwU0-IGBT1.5 kVU0-diode0.84 kVrdiode0.09 ohmrIGBT0.06 ohmm0.5Udc600 VUnom750 VPsw3 kHzrth0.55 K/kwThe battery used in the understudied full-electric vehicle is considered to be based on lithium-ion batteries. The characteristics of these batteries are given in [33]. In [33], for a lithium-ion battery cell, the variation in number of useful life cycles versus different discharge rates is presented in Figure 5. In MATLAB software, a 4th-order polynomial function is fitted on the curve that results in the following equation:(25)Nc=10004.1dr4+6.6dr3−400dr2+1100dr−1000,where Nc is the number of useful life cycles and dr is the discharge rate of the batteries in Ah.Figure 5
The variation of number of useful life cycles versus discharge rate for lithium-ion batteries.According to the approach proposed in this paper, the failure rate of the batteries used in the understudied full-electric vehicle considering the variation in the vehicle speed and temperature is determined and presented in Figure6. As can be seen in the figure, the failure rate of the batteries used in the full-electric vehicle increases with an increase in temperature and vehicle speed. However, the rate of increase in the failure rate of the batteries is higher when the temperature and vehicle speed of the full-electric vehicle is higher. Thus, the reliability of the batteries used in the electric vehicles will be lower at the higher temperature and speed of the vehicles. Figure 7 presents the failure rate of the inverter used in the full-electric vehicles versus temperature and vehicle speed. It is deduced from this figure that the failure rate of the inverter is independent of the vehicle speed. However, an increase in temperature results in an increase in the failure rate of the inverter. By changing the speed of the vehicle, the inverter current remains constant and this leads to a constant inverter failure rate. Figure 8 presents the failure rate of the permanent magnet synchronous motor used in full-electric vehicles.Figure 6
Battery failure rate versus vehicle speed and temperature.Figure 7
Inverter failure rate versus vehicle speed and temperature.Figure 8
Permanent magnet synchronous motor failure rate versus vehicle speed and temperature.As can be seen in the figure, the failure rate of the permanent magnet synchronous motor increases with an increase in the temperature and vehicle speed. The motor is composed of electrical and mechanical components. The current of the motor remains constant with changing the speed of the vehicle, and so the failure rate of the electrical parts of the motor remains constant. However, the failure rate of the rotation components is proportional to the speed of rotation, and so, by increasing the speed of the vehicle, the failure rate of the mechanical parts of the motor increases. Besides, when the temperature increases, the failure rate of both electrical and mechanical components increases, too. To determine the variation in the failure rate of the electrical and mechanical components arisen from the variation in the temperature, the Arrhenius law, and the temperature modification factor are used, respectively. Figures9 and 10 present the failure rate of the other rotation and static parts of the full-electric vehicle. It is deduced from these figures that the failure rate of the static parts of the full-electric vehicle is independent of the vehicle speed. And is dependent only on the temperature. When the ambient temperature increases, the failure rate of the static parts increases, too. However, the failure rate of the rotation parts of the vehicle is dependent on both the temperature and speed of the vehicle. By increasing the temperature and vehicle speed, the failure rate of the rotation parts of the vehicle increases, too.Figure 9
Static parts failure rate versus vehicle speed and temperature.Figure 10
Rotation parts failure rate versus vehicle speed and temperature.In the reliability model of the full-electric vehicle, all mentioned components are series. Thus, the equivalent failure rate of the vehicle is obtained by summing the failure rate of each component. Using (22) and (23) the equivalent failure rate and repair rate of the full-electric vehicle are calculated and illustrated in Figures 11 and 12, respectively. The availability of the full-electric vehicle is calculated by dividing the repair rate of the vehicle by the summation of the failure and repair rates. The availability of the full-electric vehicle considering the variation in the temperature and vehicle speed is calculated and presented in Figure 13.Figure 11
Equivalent failure rate versus vehicle speed and temperature.Figure 12
Equivalent repair rate versus vehicle speed and temperature.Figure 13
Availability of full-electric vehicle versus vehicle speed and temperature.


## 6. Conclusion

In this paper, the availability of the full-electric vehicle considering the variation in the ambient temperature and vehicle speed is determined. For this purpose, the main composed components of a typical full-electric vehicle that affects the reliability of the vehicle are taken into account. The effective components of a full-electric vehicle in the reliability model include battery charger, battery, inverter, motor, transmission system, wheels, chassis, body, and control system. In the reliability model of a full-electric vehicle, the mentioned components are series and so, the failure of each component results in the total failure of the vehicle. To determine the impact of the variation in the temperature on the failure rate of the mechanical components, the time modification factor is utilized, and so the failure rate of the mechanical components considering the variation in the temperature is calculated. For determination of the impact of the variation in the speed of the vehicle on the failure rate of the electrical components, the Arrhenius law is used. For this purpose, the failure rate of the electrical parts of the vehicle including the battery, inverter, and the permanent magnet synchronous motor is obtained by calculating the temperature rise of these components due to the power loss created at different vehicle speeds. In addition to the variation in the temperature rise created by different speeds, the impact of the variation in the ambient temperature is considered in the Arrhenius law, too. For the rotation parts of the vehicle, the impact of variation in the vehicle speed on the component’s failure rate is determined based on the fatigue stress. According to the associated equations driven in this paper, the variation in the component’s failure rate considering the variation in temperature and vehicle speed is determined. It is concluded from the obtained figures that the variation in the temperature and speed vehicle results in the variation in the failure rate of the composed components. However, the failure rate of the static mechanical components is not dependent on the variation in the vehicle speed. The full-electric vehicle studied in this paper is considered equipped with a permanent synchronous motor. In future works, the reliability performance of full-electric vehicles equipped with other technologies of electrical motors would be investigated.



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*Source: 4876218-2022-07-25.xml*