Prediction of Remaining Life of Power Transformers

Reproduction of the work by Hong, Y., Meeker, W. Q., & McCalley, J. D. (2009). Prediction of remaining life of power transformers based on left truncated and right censored lifetime data. Annals of Applied Statistics, 3(2), 857-879.

Author: Jiaxiang Cheng, Nanyang Technological University

Data Preparation

You may prepare the data set alighed with master_data.xlsx. The general structure and information included in our data set are introduced here:

• id: the identification of the samples (transformers);
• status: = 0 when the sample is censored (survived), = 1 when the sample failed already;
• trunc: = 0 when the sample is left-truncated, = 1 when the sample is not;

How to know if a sample is left-truncated or not? when preparing the data set, get clear about when the observation of failures started. This is important as there would be bias without consideration of left truncation problem. In our data set, the start of observing failures is set as 2010. So if the sample was installed before 2010, then this sample is regarded as left-truncated, trunc = 0. Otherwise, if the sample was installed after 2010, then it's not left-truncated with trunc = 1.

• time: the lifetime of the sample. For the censored data, it is the last record of survived lifetime. For the failed samples, it is the age of transformer when it failed;
• time_lt: is equal to (2010 - installation year), only for the left-truncated samples;
• manuf: the manufacturer, if you have the information;

clear: this is only for our data set, as we need to clear some data we don't use. Not necessary for common studies.

Loading Data

After you prepared your own data set, you can run loading_data.R to load your data, as follows:

library(readxl)

data_set_raw <- read_excel("master_data.xlsx", sheet = "master")
data_set <- data_set_raw[which(data_set_raw$clear == 0),]

id <- data.matrix(data_set[, "id"])
manuf <- as.factor(data.matrix(data_set[, "manuf"]))
time <- data.matrix(data_set[, "time"])
time_lt <- data.matrix(data_set[, "time_lt"])
status <- data.matrix(data_set[, "status"])
trunc <- data.matrix(data_set[, "trunc"])

ndim <- dim(id)[1]

For normal case, you only need one line to load the data:

data_set <- read_excel("your_file.xlsx", sheet = "your_sheet")

Maximum Likelihood Estimation

After loading your data set or ours, you can do the maximum likelihood estimation with mle.R to obtain the estimated scale and shape parameters for the two-parameter Weibull distribution. If you get left truncation problem into consideration, you only need to run the following lines:

library(survival)
library(ggplot2)
library(survminer)
library(Metrics)
library(maxLik)

logLikFun <- function(param) {
  beta <- param[1]
  ita <- param[2]
  c <- status[1:ndim]
  v <- trunc[1:ndim]
  t <- time[1:ndim]
  tL <- time_lt[1:ndim]
  f = log(beta / ita) + (beta - 1) * log(t / ita) - (t / ita) ^ beta
  fu = (t / ita) ^ beta
  fuL = (tL / ita) ^ beta
  sum(c * v * f + c * (1 - v) * (f + fuL) + (1 - c) * v * (-fu) + (1 - c) *
        (1 - v) * (fuL - fu))
}

mle <- maxLik(logLik = logLikFun, start = c(beta = 2, ita = 100))
summary(mle)

shape_lt <- mle$estimate[1]
scale_lt <- mle$estimate[2]

Thus, you already obtained the shape and scale parameter as shape_lt and scale_lt.

However, if you are interested, you can also run the whole script and you will obtain the comparison plot between the estimation results with and withou consideration of left truncation problems, as follows:

Bootstrap Samples

At this stage, you can already get the naive prediction of remaining life interval for individuals by running pred_naive_indiv.R. But you may obtain quite a large interval as the naive prediction doesn't consider the uncertainties of the estimated parameters.

So, instead, bootstrap method is used to compensate the uncertainty of parameter estimation. By running bootstrap.R, you may obtain the a number of bootstrap samples generated through random weighted bootstrap. Here you can define the number of bootstrap samples you want:

# initialize the matrix for saving the bootstrap samples
B <- matrix(ncol = 2)
numB <- 100 # the number of bootstrap sample

You can change numB. Normally, for a reliable bootstrap application, the number should be 10000 or more. But due to the limited device I have, I only tried at maximum of 1000, but it should have been at least 10000 instead.

Prediction of Remaining Life for Individuals

After obtaining the bootstrap samples, you can start the prediction of remaining life of individual transformer with predict_individual.R. And you can customize which transformers you want to predict, as follows:

# the number of transformer to predict, this can be customized
nsample <- 100

You can change the nsample here, and then the prediction will be conducted for the first nsample transformers. This can be modified according to your own interests. No need to start counting from the beginning.

Then after running the codes, you will have you intervals predicted corresponding to you selected samples in interval_pred, as follows:

# the prediction of remaining life with 90% confidence interval for individuals
interval_pred <- interval[1:nsample + 1,]

The lower and upper bounds will be recorded accordingly in this matrix, like this:

	V1	V2
1	185.1475	216.6756
2	189.3531	221.7536
3	187.1826	219.0999

You can do the visualization with the matrix in your own style. And I prefer to visualize it using Matlab as follows. It can be seen from the following figure that with more bootstrap samples, the prediction intervals are getting converged:

Prediction of Number of failures for Population

Now you can predict the number of failures in the future within the current population by running predict_population.R. Please notice if any new samples added in the later years the population should be updated and the prediction would be updated as well.

# the number of bootstrap samples used for simulation
ndim_bern <- 100 # cannot be greater than numB
tcut <- 2020 # the cutting year, end of current observation
tpred <- 2030 # the end year of prediction
tdelta <-
  1 # the accuracy of the prediction like 2021, 2022 or 2021.2, 2021.3..

In this part, you can still choose your bootstrap samples to use by changing ndim_bern, which have been generated with bootstrap.R. And still, the number of bootstrap samples should be 10000 and more to achieve optimal results. The computational complexity is even more tremendous for this part. And you can also change the current year tcut, the end of observation, according to your own data set. And the prediction horizon and accuracy can be adjusted as well by tpred and tdelta.

Then you will receive the results in K_B, the number of rows will be decided by your prediction horizon and accuracy, and the number of columns will be determined by the number of bootstrap samples you used, the more the better. And then you can visualize the results freely. Normally, you will want to get a mean prediciton like follows:

Acknowledgement Many thanks for the great work by Hong, Y., Meeker, W. Q., & McCalley, J. D. (2009). Prediction of remaining life of power transformers based on left truncated and right censored lifetime data. Annals of Applied Statistics, 3(2), 857-879. For any issues, please feel free to contact me JIAXIANG002@E.NTU.EDU.SG.