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We construct mathematical models to represent the relationship between the thermal stress and the deterioration rate for electrical insulation. The Arrhenius-log-normal model has been used generally for such a deterioration model due to the thermal stress. The Arrhenius law is based on the chemical reaction theory between the absolute temperature and the activity of materials. On the other hand, as for the log-normal distribution, we have been only followed the traditional statistical treatment such that the deterioration could be represented by the normal distribution when logarithmic time is used. The Arrhenius-log-normal model is a combination of these two models. However, in the International Electrotechnical Commission 60216-1, deterioration due to the thermal stress is represented by the mechanical strength, and the time showing 50% mechanical strength to the initial strength is defined as the failure time. We assume here the generalized Pareto distribution model, the generalized logistic distribution model, or the normal distribution model for such a model. Thus, in this paper, we construct new mathematical models combined by the Arrhenius law with the generalized Pareto distribution model, the generalized logistic distribution model, or the normal distribution model.

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Assuming that the Arrhenius law holds between the thermal stress and the lifetime, and that the logarithmic lifetime follows some consistent probability distributions at a constant stress. Under such life models, it is crucial to show the optimum test design from an efficiency viewpoint. It would also be useful to know the semi-optimum test plan in which the efficiency is close to that in the optimum one and the test condition is simple. The optimization target is to find the optimum number of test specimens at each test stress level, and we consider the case that the number of stress level is three. The criterion for optimality is measured by the root mean squared error for the lifetime in use condition. To take into account the reality, we used the parameter values in a real experimental case. Comparing the optimum results with those using the conventional test method where test specimens are equally allocated to each test stress level, we have found that there is only a small difference between the optimum test result and the conventional test result if linearity of the Arrhenius plot is required. We may regard the conventional test plan as one of the semi-optimum test plans. We have checked the consistency between the theoretical results and the simulation results.Index Terms -Optimum test plan, semi-optimum test plan, thermal deterioration, Arrhenius law, normal distribution, generalized Pareto distribution, generalized logistic distribution, method of least squares, maximum likelihood estimation method

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