Reliability Modeling of Power Transformers with Condition-Based Degradation Warning, Seasonal Load Variation, and Standby Redundancy: A Semi-Markov Approach

Power transformers are critical, capital-intensive assets in electricity distribution networks, and their failure disrupts supply while incurring substantial repair costs. Building on an earlier probabilistic analysis of power transformers with six failure modes and inspection, this paper proposes a twelve-state semi-Markov extension that additionally captures three operational realities not previously modeled: condition-based degradation warnings from dissolved gas or partial discharge monitoring, a regime-switching failure rate driven by seasonal peak load conditions, and a repair structure that separates an immediate in-stock repair from a spare-part-wait state and a standby-covered repair state. Reliability indices—mean time between failures, steady-state availability, and the expected busy period of the repair crew—are derived under the special case of exponential resolution times: MTBF is obtained in closed form via a first-passage-time argument on the up-state cycle, while availability and busy period are obtained from the steady-state solution of the twelve-state generator matrix and evaluated numerically, together with a full sensitivity and relative sensitivity analysis of all thirteen model parameters and, building on that analysis, approximate confidence intervals for MTBF, availability, and busy period obtained by delta method propagation of parameter uncertainty. Because condition monitoring, seasonal exposure, and standby utilization records were not part of the original data collection, illustrative parameter values grounded in transformer loading standards and utility maintenance practice are used to demonstrate the model, and the sensitivity analysis quantifies which of these illustrative values would most change the results once replaced by field estimates. A dedicated ±50% what-if analysis confirms the headline conclusions are robust to substantial mis-specification of the two most sensitive parameters, and a distribution shape analysis further confirms, analytically, that the steady-state indices are invariant to the choice of sojourn time family (exponential, Weibull, or otherwise) once means are fixed, underscoring the genuinely semi-Markov (not merely CTMC) character of the model. The extension is positioned as a direct methodological successor to two prior studies of the same transformer fleet, and a data collection plan is given to convert the illustrative parameters into field-estimated ones.