## 1 Introduction  
In modern materials science and mechanical engineering, there is a persistent interest in developing effective methods for local hardening of the surfaces of structural steels. The quality and properties of the surface layer are key determinants of wear resistance, corrosion resistance, fatigue resistance, and the overall service life of a product. Electrolytic plasma treatment (EPT) is a promising area in the field of surface engineering, based on the interaction of an electric discharge with the surface of a conductive material immersed in an electrolyte. EPT relies on the creation of a plasma discharge between a metallic workpiece, acting as the cathode, and a liquid electrolyte. Upon exceeding a critical voltage (usually 120-400 V),a localized plasma sheath forms around the cathode, achieving temperatures in the range of 5000-8000 K [1]. This discharge induces a significant thermal impact on the material's surface, causing rapid heating, followed by immediate cooling in the electrolyte, leading to surface hardening and the formation of a hardened microstructure. Of particular interest is the application of EPT to structural cast steel 20 GL, prized for its high processability, strength, and availability. While 20 GL steel is normally weakly susceptible to thermal hardening because of its low carbon content (\(\sim\)0.2%) and the prevalence of a ferrite-pearlite structure, the rapid heating and cooling rates in the EPT process can make it possible to achieve austenitization of the surface layer. This leads to the formation of a hardened structure after quenching, opening avenues for enhancing the wear and corrosion resistance of cast parts without significantly complicating their manufacturing technology. In the work [2], it was demonstrated that increasing the heating rate significantly expands the temperature range in which the phase transformation occurs, specifically the transition from ferrite to austenite. The author also indicated that even with the complete disappearance of ferrite, regions with a high carbon concentration may persist around former carbide particles. These carbon-rich areas can undergo hardening during rapid cooling, as demonstrated on 20 GL steel. Electolytic plasma treatment, in all its variations, employs high-speed heating methodologies. Some works [3; 4] indicate that cathode heating can yield heating rates ranging from 10 to 200 \({}^{\circ}\)C/s under constant voltage conditions. Furthermore, the cooling rate of the component following cathodic heating can be controlled via adjustments to the applied voltage, spanning from 12 to 80 \({}^{\circ}\)C/s within the zone of minimal austenite stability and from 20 to 140 \({}^{\circ}\)C/s in the martensitic transformation range [5]. Continuing the development of research in the field of electrolytic plasma treatment, it is important to note that the thermal processes occurring in the near-surface layer of the metal play a key role in shaping the structure and properties of the processed material [6]. The temperature mode of heating determines the depth and nature of phase transformations, while the cooling rate dictates the possibility of forming hardened structures, such as martensite. Therefore, a quantitative assessment of temperature, heat flow, as well as heating and cooling rates, holds significant practical importance, enabling the prediction of thermal hardening effectiveness across various processing modes. Several studies [7; 8; 9; 10] have explored the thermal and electrophysical phenomena that occur during anodic electrolytic plasma heating, including the modeling of temperature fields, heat flux density distribution, and electric potential within the electrolyte bath. These investigations have determined that the intensity of thermal exposure and the nature of plasma formation are influenced by factors such as the geometry of the treated component, the type of electrolyte used, the electrode material, and the applied voltage regime.