Data Collection and Component Probability Estimation

2.3.2.1 Engine and Other Main Components

The reliability of the engine and the HV battery are based on the number of miles the vehicle travels. Since the reliability of other components are determined on the number of years, we need to translate failure in number of miles to failure in number of years. We calculate the average number of miles traveled per year in the United States. The U.S. Federal Highway Administration records the average annual miler per driver by age

group (OHPI, 2016) (see Table 2.2). We weight the average number of miles driven by

each age group by the proportion of the population in the United States according to age (Joyce A. Martin et al., 2015). Based on these two data sources, a car averages 12,826 miles per year.

The engine in the 2004 Toyota Hybrid is the Toyota 1NZ-FE/FXE engine. According to WikiMotors, the official life span of this engine is 120,000 miles (WikiMotors, 2016). Assuming 12,826 miles in a year, the M T T F of the engine is 120000/12826 = 9.4 years.

Table 2.2: Average Annual Miles per Driver

Age Male Female Total Percentage

15-19 8,206 6,873 7,624 7.10% 20-34 17,976 12,004 15,098 20.30% 35-54 18,858 11,464 15,291 27.90% 55-64 15,859 7,780 11,972 11.80% 65+ 10,304 4,785 7,646 13.10% Total=80.2%

Average 16,550 10,142 13,476 Weighted average=12,826

The M T T F s for PCU, reduction gear, planetary gear, MG1, and MG2 are calculated based on the M T T F of engine and data from Ping et al. (2010). The authors present the experimental data of mean time between failures (M T BF ) of main components in the hybrid electric transit bus. We assume that proportional relation of M T BF between main components in the hybrid electric transit bus is the same with the proportional relation of M T T F of main components in the hybrid system we analyzed in this paper.

For example, we can calculate the M T T F of PCU by using equation (2.16):

MTBF of Engine

MTBF of PCU (in hybrid electric transit bus)

= MTTF of Engine

MTTF of PCU (in hybrid system of 2004 Toyota Prius)

(2.16)

From above method and assumption, we calculate the M T T F for the mechanical system, the electrical equipment, and the PCU in the 2004 Toyota Prius.

The mechanical system in this paper includes the wheels, planetary gear, and reduc- tion gear; the electrical equipment includes MG1 and MG2. But Hu et al. only list the M T BF of the mechanical system and electrical equipment. We divide the M T T F of the mechanical system and electrical equipment to obtain the M T T F for each component. Except for the HV battery, the main components in this paper follow the exponential distribution. We assume a failure in one component in the mechanical system results in failure of the entire mechanical system. According to the property of exponential distri-

bution, the M T T F of wheel, the M T T F of planetary gear, and the M T T F of reduction

gear can be obtained from the equation (2.17).

1 MTTF of wheel + 1 MTTF of planetary gear + 1 MTTF of reduction gear = 1 MTTF of mechanical system (2.17)

We assume the M T T F of each of these three components are identical. In a similar way, the M T T F of MG1 and the M T T F of MG2 can be derived.

2.3.2.2 HV Battery

The 2004 Toyota Priuss battery is a nickel metal hybrid battery. To our knowledge, no official data on the lifespan of this HV battery exists. The Panasonic EV Energy Ni-MH handbook (PanasonicCorporation, 2016) claims this kind of battery can be recharged over 500 times, but translating this recharging information to the lifetime of the HV battery requires additional data, which is not available.

Given this lack of reliable data on the lifetime of the HV battery, we estimate the probability that the HV battery fails each year based on a survey of the number on the number of miles the HV battery lasts. An online poll conducted in PRIUSchat asked users how many miles their HV battery lasted (PriusChat, 2013). Although this survey is not scientific and there is no way to verify if the users are truthful, the survey provides

some information that can be used to estimate the reliability of a HV battery. Table2.3

shows the result of the survey of hybrid battery, and we use 12,826 miles per year to estimate the failure in terms of number of years.

We use the Bayesian approach described in Subsection 2.3 to estimate the reliability of the HV battery and assume a Weibull distribution for failure. The likelihood functions

come from equations (2.6) and (2.7). If the battery fails before 100,000 miles (or 7.8

Table 2.3: Survey of Battery Performance

How is your Gen 2 Prius (2004-2009) Hybrid Battery Doing?

Failed below 100,000 miles (7.8 years) 6 vote(s)

Failed between 100,000 and 150,000 miles (7.8 years-11.7 years) 8 vote(s)

Failed between 150,000 and 200,000 miles (11.7 years-15.6 years) 5 vote(s)

Failed at over 200,000 miles (15.6 years) 1 vote(s)

Has not failed below 100,000 miles (7.8 years) 42 vote(s)

Has not failed between 100,000 and 150,000 miles (7.8 years-11.7 years) 37 vote(s)

Has not failed between 150,000 and 200,000 miles (11.7 years-15.6 years) 19 vote(s)

Has not failed at over 200,000 miles (15.6 years) 8 vote(s)

100,000 and 150,000 miles (7.8 and 11.7 years), we use the likelihood function in equation (2.6) where t2 = 11.7 and t1 = 7.8. The formula is similar for when the battery fails between 150,000 and 200,000 miles (11.7 and 15.6 years). If the battery fails over 200,000 miles (15.6 years), we use the likelihood function in equation (2.7) where t3 = 15.6. If

the battery has not failed below 100,000 miles (7.8 years), we use equation (2.7) where

t3 = 7.8. If the battery has not failed between 100,000 and 150,000 miles (7.8 and 11.7

years), that means the battery has not failed before 11.7 years. Thus, we use equation (2.7) where t3 = 11.7. Equation (2.7) is similarly used when the battery has not failed between 150,000 and 200,000 miles (11.7 and 15.6 years).

The prior distributions for β and λ are gamma distributions where the parameters for the gamma distributions are chosen so that the gamma distribution resembles a uniform distribution. We use WinBUGS to run the Gibbs sampler and run 3 separate chains each with a burn-in phase of 1500 samples. The burn-in phase means that first 1500 samples are discarded. After the burn-in phase, we record 1000 samples for each chain.

Figure 2.13 depicts the 3 chains for (a) β and (b) λ. With 3 chains, we have a total of

3000 samples. The different lines represent different chains or runs in the simulation. The mean value for β is 2.808 with a standard deviation of 0.3487. The mean value for λ is 4.284×10−4with a standard deviation of 3.333×10−4. We use the 3000 samples for β and λ with the Weibull distribution to simulate failure times for the battery. Figure2.14

(a) Samples of β

(b) Samples of λ

Figure 2.13: Gibbs sampler results for β and λ

depicts the histogram of this simulation.

The M T T F for this simulation is 16.2731 years. As Figure 2.14 shows, the HV

battery has a 0.2647 probability of lasting more than 20 years and a 0.0247 probability of lasting more than 30 years. These results seem to overestimate the lifetime of a battery. Although a battery could last 20 years, it seems very unlikely that a battery will last 30 years or more. Thus, we desire a method that will use the survey and limit the results to lifetimes that appear more reasonable.

We set an upper limit during the Bayesian optimization by assuming that the battery never lasts longer than a predefined number of years. Most vehicles fail before 300,000 miles, and we set 300,000 miles, or 23.39 years, as the upper bound for the four likelihood equations without an upper limit. Running 3000 samples in the Bayesian optimization reveals that the mean value of β is 3.845 with a standard deviation of 0.3571and the

simulated values of β and λ with the Weibull distribution reveals the follow histogram for the lifetime of the battery in Figure 2.15. The M T T F of this distribution is 13.70 years with only a 0.0637 probability the battery lasts longer than 20 years.

Figure 2.14: Histogram of Failure Time

We repeat the Bayesian optimization simulation with an upper limit of 250,000 miles

(19.49 years) and 200,000 miles (15.59 years). Figure 2.16 depicts the time to failure

for the battery using a limit of 19.49 years. Running 3000 samples in the Bayesian optimization reveals that the mean value of β is 4.575 with a standard deviation of 0.4607 and the mean value for λ is 1.028 × 10−5 with a standard deviation of 1.226 × 10−5. Using the simulated values of β and λ with the Weibull distribution predicts the distribution of the battery lifetime in Figure2.16. The MTTF of this distribution is 13.11 years with only a 0.0150 probability the battery lasts longer than 20 years.

Figure 2.17 depicts the time to failure with a limit of 15.59 years. Running 3000

samples in the Bayesian optimization reveals that the mean value of β is 5.671 with a

standard deviation of 0.7739 and the mean value for λ is 2.059 × 10−6 with a standard

deviation of4.442 × 10−6. Using the simulated values of β and λ with the Weibull dis-

Figure 2.15: Histogram of Failure Times With Upper Limit of 300,000 Miles

M T T F of this distribution is 12.06 years with a 0 probability the battery lasts longer than 20 years.

Table 2.4 displays the results on the reliability of the HV battery for different upper limits. We need to determine which probability distribution is the most reasonable. The average age of vehicles in the United States is 11.5 years (Gardner, 2015), which suggests that perhaps we should choose a smaller upper limit so that there is close to a 0.5 probably the battery fails before 11.5 years. However, the average age of the vehicle does not really explain how long a battery will last because many factors influence why a person disposes of a vehicle and purchases a new one.

Table 2.4: Probability that HV Battery Fail Before a Given Time Period

Upper Bound P(1 year) P(5) P(10) P(15) P(20)

No upper bound 0.0003 0.0207 0.1793 0.4473 0.7353

300,000 miles 0.0003 0.0167 0.1877 0.6150 0.9363

250,000 miles 0 0.0100 0.1780 0.7007 0.9850

Figure 2.16: Histogram of Failure Times With Upper Limit of 250,000 Miles

Selecting 250,000 miles, or 19.5 years, as the upper limit seems most reasonable to us. The M T T F is 13.1143 years and there is only a 0.0150 probability the battery will last longer than 20 years. (Even though the limit is 19.5 years, a battery can last longer than 19.5 years. The upper limit assumes that a battery has not lasted longer than 19.5 years, but it is still possible a battery could last longer than 19.5 years.)