## 1 Introduction  
As it is the core equipment for power transmission and crucial for the safe and stable operation of the power system, transformers play a vital role. With the commissioning of multiple ultrahigh-voltage transmission lines in our country, the threat posed by transformer failures is expected to increase [1; 2; 3]. The safe operation of transformers largely depends on the reliability and stability of their insulation. The insulation performance of transformer oil directly determines whether these devices can operate safely and stably [4; 5]. Therefore, researching transformer oils with excellent electrical properties is of great significance in improving the safety level of transformers [6; 7; 8; 9]. Nanoparticles exhibit excellent surface and volume effects, making nanotechnology one of the hot topics in the field of insulating materials and the most promising direction for improving the performance of insulating materials [10]. The so-called nanofluid usually refers to stable suspensions formed by dispersing nanoparticles with a particle size that is less than 100 nm into a base liquid via certain dispersion methods. Compared with traditional transformer oil, nanofluids have advantages such as high thermal conductivity, good stability, and reduced equipment wear. Research on nanofluids in recent years has also shown that the addition of nanomaterials can effectively improve the insulation properties of transformer oil [11; 12; 13; 14]. Some scholars have conducted experimental and simulation analyses on the mechanism of nanomaterials, enhancing the breakdown voltage of transformer oil and proposing a series of theoretical models, including the fast electron capture model, potential well model, and shallow trap model. The fast electron capture model suggests that conductive nanoparticles can adsorb fast electrons due to their own characteristics and convert them into slower negative charges, thereby reducing the rate of streamer head development and slowing down the movement rate of positive and negative charges, while weakening the electric field intensity of the streamer head [15]. The potential well model suggests that, under the action of an applied electric field, charges are generated on both sides of the nanoparticle's surface due to polarization, distorting the original electric field. This distortion forms a \"potential well\" near the particle, which can capture free electrons generated during the streamer development process [16; 17; 18]. The shallow trap model suggests that nanoparticles increase the density of shallow traps, impeding the movement of free electrons in transformer oil, thereby improving the breakdown performance of transformer oil modified with nanoparticles [19; 20; 21]. The shallow trap model effectively avoids the limitation of the relaxation time, influencing electron capture in the fast electron capture model while also compensating for the limitation of only considering individual nanoparticles in the potential well model, thereby allowing for a wider range of applications. In recent years, research on TiO\({}_{2}\) nanoparticles has also revealed that TiO\({}_{2}\) nanoparticles can absorb dissolved water in the oil to form a water film, reducing the probability of water molecules forming small bridges [22]. In order to enhance the insulation and thermal conductivity performance of transformer oil, researchers from both domestic and international spheres have conducted experiments using various types of nanoparticles. These nanoparticles can be classified into three categories: conductive nanoparticles (Fe\({}_{2}\)O\({}_{3}\), Fe\({}_{3}\)O\({}_{4}\), and SiC), semiconductor nanoparticles (TiO\({}_{2}\)), and insulating nanoparticles (Al\({}_{2}\)O\({}_{3}\), SiO\({}_{2}\), BN, and AlN) [23].