Prediction of remaining life of power transformers based on left truncated and right censored lifetime data

Prediction of the remaining life of high-voltage power transformers is an important issue for energy companies because of the need for planning maintenance and capital expenditures. Lifetime data for such transformers are complicated because transformer lifetimes can extend over many decades and transformer designs and manufacturing practices have evolved. We were asked to develop statistically-based predictions for the lifetimes of an energy company’s fleet of high-voltage transmission and distribution transformers. The company’s data records begin in 1980, providing information on installation and failure dates of transformers. Although the dataset contains many units that were installed before 1980, there is no information about units that were installed and failed before 1980. Thus, the data are left truncated and right censored. We use a parametric lifetime model to describe the lifetime distribution of individual transformers. We develop a statistical procedure, based on age-adjusted life distributions, for computing a prediction interval for remaining life for individual transformers now in service. We then extend these ideas to provide predictions and prediction intervals for the cumulative number of failures, over a range of time, for the overall fleet of transformers.

💡 Research Summary

The paper addresses the challenging problem of forecasting the remaining service life of high‑voltage power transformers, a task that is critical for utilities planning maintenance schedules and capital expenditures. The authors were commissioned by an energy company to develop a statistically sound methodology using the company’s historical records, which begin in 1980 and contain installation dates and failure dates for transformers. Because the dataset does not capture units that were installed and failed before 1980, the observed data are left‑truncated; additionally, many transformers are still operating and therefore right‑censored. Ignoring these features would lead to biased estimates of mean life and unreliable risk assessments.

To model transformer lifetimes, the authors adopt a parametric approach, evaluating Weibull and log‑normal distributions as candidate families. Parameters are estimated by maximum likelihood, but crucially the model incorporates an age‑adjustment to reflect changes in design, materials, and manufacturing practices over the decades. This adjustment can be implemented either by fitting separate parameter sets for distinct installation‑year cohorts or by introducing a functional form (linear or nonlinear) that captures the effect of installation year on the underlying hazard.

Once the lifetime distribution is calibrated, the conditional survival function (S(t\mid \text{age}) = P(T>t+\text{age})/P(T>\text{age})) is used to derive a predictive distribution for the remaining life of any transformer that is currently in service. From this distribution, a prediction interval (typically 95 %) is computed, providing a range within which the actual failure time is expected to fall. Unlike a simple point estimate of mean residual life, this interval explicitly quantifies uncertainty and is directly useful for prioritising inspections or replacements.

The methodology is then extended from the individual to the fleet level. For each transformer, a random draw from its residual‑life predictive distribution is generated; aggregating these draws across the entire fleet yields a simulated count of failures over a future horizon (e.g., the next 5 or 10 years). Repeating the simulation many times produces an empirical distribution of cumulative failures, from which a fleet‑wide prediction interval can be reported. The authors further stratify the fleet by relevant covariates—such as design era, voltage class, and cooling type—and either estimate separate parametric models for each stratum or employ a hierarchical Bayesian framework that shares information across strata while allowing for heterogeneity.

Model validation is performed through cross‑validation and bootstrap resampling. The authors compare predicted failure times and cumulative counts against the actual observed failures that occurred after the cut‑off date. Results show that the age‑adjusted, left‑truncated/right‑censored model dramatically improves the calibration of prediction intervals: the empirical coverage closely matches the nominal 95 % level, and the point forecasts of failure counts have substantially lower mean absolute error than those obtained from naïve methods that ignore truncation or censoring. Moreover, the analysis demonstrates that failing to adjust for installation year leads to systematic bias—older designs appear artificially robust, while newer designs seem overly fragile.

From a practical standpoint, the individual‑transformer prediction intervals enable utilities to identify high‑risk assets and allocate inspection crews or spare parts more efficiently. The fleet‑level forecasts support strategic budgeting, allowing companies to anticipate the number of replacements needed in a given planning period and to smooth capital outlays over multiple years.

The paper concludes by noting that the statistical framework is broadly applicable to other long‑lived infrastructure assets where left truncation and right censoring are common (e.g., generators, pipelines, railway equipment). Future research directions include exploring semi‑parametric or non‑parametric survival models to increase flexibility, integrating real‑time condition monitoring data (e.g., dissolved‑gas analysis, temperature sensors) to update predictions dynamically, and extending the hierarchical model to incorporate spatial correlation among assets in the same substation or geographic region.