## 1 Introduction  
Power-distribution systems are involved in the transport of electricity to consumers using hierarchical voltage levels, i.e., from high to low [1; 2; 3; 4; 5]. Initially, information about power-distribution networks that had been built decades ago was stored in documents and detailed maps [3]. With the dawn of the era of computers, that information is progressively being digitalized [3]. The required amount of information for assets is large, and the criticality of failures makes operation less complex in low-voltage systems than in high- and medium-voltage systems [3]. Hence, efforts have been made to digitalize the high- and medium-voltage systems [3]. Currently, technologies are evolving to digitalize low-voltage systems [3]. However, power-distribution systems are still far from fully digitally transformed systems with real-time operation [3; 6; 7]. In the scientific-research context, mathematical-programming models are being developed for the digital transformation of power-distribution systems [3]. This manuscript aims at developing such a mathematical-programming model, which can be considered a stepping stone towards the digital transformation of power-distribution systems. In addition, some electricity-supply contracts with consumers include clauses that oblige electricity-distribution utilities to pay compensation to consumers in the event of a disruption in the supply of electricity [8]. This depends on the jurisdiction and the specific contract between the electricity-distribution utility and consumers [8]. In general, such clauses, including penalty charges for the electricity-distribution utility, are included in the contract to protect the consumer from such scenarios, and there can be laws or regulations in place in a specific jurisdiction that require or allow for such compensation to be paid in certain circumstances to the electricity consumers [8]. In addition to the suppliers' losses resulting from failure of equipment, those penalty charges and consumer compensations can be a huge burden to the electricity-distribution utilities [8]. Therefore, in order to manage the circumstances, it is essential that they carry out optimal operation and maintenance of equipment with modern technologies. Hence, the reliability of components in electricity-distribution utilities is very important, as failure of those components is associated with very high costs [9; 10]. The hierarchical structure of power-distribution systems is composed of several distinct levels of components, such as distribution transformers (DTs), feeders, switches, fuses, and consumer-side equipment [1]. DTs are crucial equipment in power-distribution utilities, as they constitute high investments and transform primary distribution voltages to lower voltages suitable for consumer equipment [2; 11; 12]. DT failures cause serious problems to distribution-network management, resulting in comparatively huge expenditure for the repair or replacement of other components in the distribution system [2; 9; 10]. The reliability analysis of DTs involves the assessment, avoidance, and governance of failures, along with science, statistics, and risk analysis [9; 10; 13]. Through the use of failure analysis for DTs, the lifespan can be predicted by managing the life cycle and the uncertainties involved with the failure [9; 10; 13]. There is no zero rate of failure for any equipment [14; 15]. Reliability analysis requires time-to-failure data to reveal the patterns of failure occurrences, balancing the reduction in the cost of failures with performance enhancement for digital transformation from the pillars of digitization and digitalization [13; 14; 16]. Predictions of reliability allow the likelihood of the characteristics of failure rates to be evaluated [13; 15]. It is necessary to digitize the wear curves based on the failure data to implement reliability curves of components in power-distribution systems [13].