## 1 Introduction  
Pulsed power technology permits the electric energy to be stored at relatively low power, and be transformed into the electromagnetic energy by pulse and then released into the load to achieve much higher instantaneous power [1]. Pulsed power technology plays an important role in controlled nuclear fusion [2], high-energy accelerator [2], and electromagnetic weapons [3; 4; 5; 6]. For example, the pulse generator can generate high instantaneous power and low average power, which may benefit the production of plasma [7; 8]. In a relatively high-voltage and high-impedance pulsed power system with the lowest possible voltages in initial energy storage and switching, which can be used to design the electron ring accelerator [2]. The pulsed power system is mainly composed of the primary energy-storage systems, pulse compression systems, and high-power load systems [9; 10; 11]. As the key components in primary energy-storage systems, the liquid mediums are often used for energy storage and insulation because of their excellent characteristics such as high-energy storage density, great self-healing effects, and good thermal conductivity [12; 13]. The commonest liquid mediums include water medium and oil medium [14; 15]. Due to the rapid decreases in water resistivity, the oil medium would be more applicable to a pulse generator. Moreover, the breakdown strength of the oil medium can exceed 300 kV/cm after purification treatment. However, solid impurities are inevitably introduced when pouring the oil medium into the pulse generator, which substantially affects the insulation performance of transformer oil [16; 17; 18]. In practical production, the insulation performance of transformer oil not only affects the operational reliability of pulsed power system, but it also largely determines the cost of oil-insulated electrical equipment [19; 20; 21]. Therefore, it is of great significance to ensure the safe operation of the pulsed power system by improving the insulation performance of transformer oil [22; 23; 24; 25]. In traditional dielectric theory, impurities in transformer oil gather and recombine into a conductive \"bridge\" under the applied electric field, which leads to the reduction in insulation performance. However, [PERSON] et al. discovered that the breakdown strength of transformer oil under positive impulse breakdown voltage can be improved by introducing the Fe\({}_{3}\)O\({}_{4}\) nano-particles, which contradicted the traditional principle that the insulation performance of liquid mediums is proportional to purity [26]. Researchers have since paid attention to nano-modification technology. With the depth of the research work, many researchers have realized that some parameters such as the type, shape, and surface modification of nano-particles will have a great influence on the modification effect [27; 28; 29; 30]. Specifically, the insulation performance of transformer oil can be characterized by the breakdown strength, which is related to the pre-breakdown phenomenon defined as the \"streamer\" in transformer oil [31; 32; 33; 34]. In order to explain the modification effect and streamer morphology in nano-liquid, several breakdown models and mechanisms have been proposed. [PERSON] et al. established an electrodynamic model for the formation of the streamer in nano-liquids and then proposed the corresponding nano-particles electron-trapping theory [24; 25]. [PERSON] et al. proposed a shallow trap model associated with the charge-hopping process in nano-liquids, which demonstrated that the improvement in shallow trap density will promote the formation of a uniform internal electric field and reduce the charge accumulation in the streamer head [35; 36]. [PERSON] et al. proposed a barrier ionization model that analyzed the effect of nano-particles on the electron capture process and insulation performance [37; 38]. [PERSON] et al. proposed a new model to illustrate the streamers in liquid mediums based on the mechanical stress generated by the electric field [31].