## 1 Introduction  
Lightning impulse overvoltages are one of the most important causes of damage to electrical power transformers. These transient phenomena can generate extremely high voltages, far exceeding the rated values of the equipment, and can lead to insulation damage, winding penetration and, in extreme cases, the complete destruction of the transformer. The occurrence of these surges can be caused by direct atmospheric discharges (direct lightning strikes), switching operations and maneuvers on the network. If the lightning strike does not strike the line directly, the electromagnetic field that is generated by the discharge may induce voltages in the line conductors and, by coupling, in the transformer windings. Switching operations and network maneuvers can also generate transient surges that can affect transformers [1, 2, 3, 4, 5, 6, 7, 8]. Overvoltages occurring in the transformer can have adverse effects on its structure such as the following [9; 10; 11; 12; 13]:  
* Damage to the insulation: If surges exceed the insulation breakdown voltage, they can lead to internal electrical discharges and the gradual degradation of insulating materials. * Winding perforation: In severe situations, surges can cause physical damage to windings and internal short circuits, as well as major transformer damage. * High currents generated by surges can produce significant electrodynamic forces, which can deform or destroy the mechanical components of the transformer. * Repeated overvoltages can accelerate the aging process of insulation. In order to protect transformers against overvoltages occurring following a lightning impulse, several technical solutions and measures have been adopted, including the following [10; 11; 12; 13; 14; 15; 16]:  
* The use of lightning arresters that directly capture lightning discharges and conduct them safely to earth;
* The use of variable-resistance arresters to limit the amplitude of overvoltages by earthing in the event of the occurrence of excessive voltages;
* The correct sizing of transformer insulation to withstand anticipated surges;
* The use of metallic shields to reduce electromagnetic coupling between conductors and limit induced voltages;
* Continuous monitoring of network parameters and automatic tripping of protective devices in case of anomalies. A wide range of studies have been carried out to understand the transient voltages occurring at the impulse at the transformer windings and to analyze the effects that are generated by this transient phenomenon. In order to correctly assess the risk of damage caused by overvoltages, researchers have carried out detailed research involving the following [4; 5; 6; 7; 8; 9; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25]:  
* Mathematical network modeling: simulating the behavior of the network under surge conditions to determine the level and wave shape of voltage surges that can occur at transformer terminals through a mathematical approach [11; 16; 17; 18; 19; 20; 23; 25];
* Field measurements to record actual overvoltages during atmospheric events to validate mathematical models and calibrate protective equipment [9; 14; 21; 24];
* Assessments of the frequency and severity of surge events to determine the level of risk and to properly size protective systems [4; 5; 6; 7; 8; 15];
* The implementation of standards and regulations defining the requirements for the design and operation of electrical systems to ensure surge protection [4; 5; 6; 7; 8; 22];
* The development of insulating materials with improved characteristics to increase transformers ability to withstand surges and achieve an effective grounding system is essential to dissipate the electrical energy of lightning and reduce step potentials [12; 13; 14; 20; 23]. Due to the fact that the amplitude and frequency of the energizing voltage are time-varying, the transient voltages occurring in the transformer windings at lightning impulse cannot be fitted into a model.