An electrical arc in open air (EAOA) represents one of the most complex and critical phenomena in power systems, whose occurrence on transmission lines directly affects network reliability and stability while playing a decisive role in insulation design, protection strategies, and single-phase reclosing timing. This paper reviews more than seven decades of research, tracing the evolution of arc modeling from simple voltage–current descriptions to advanced multi-dimensional and multi-physics frameworks. The arc is recognized as a nonlinear and multi-factor phenomenon involving coupled electrical, thermal, mechanical, and stochastic interactions, which necessitates the use of hybrid and data-driven models for accurate analysis. A structured set of fundamental concepts is first introduced to establish a unified foundation for arc studies. The historical development of models is then reviewed within several key categories, including static voltage–current models, dynamic conductivity models, recovery-voltage-based models, and advanced electromechanical coupled frameworks. Each group of models is analytically examined with respect to its ability to reproduce arc extinction, stability, and recovery behavior. Particular emphasis is placed on secondary arc models, which simultaneously account for insulation recovery and arc elongation, thereby enabling accurate evaluation of dead times and recovery probabilities. The advanced versions of these models incorporate the dependence of recovery voltage on arc history and the stored energy in the ionized channel, providing unprecedented accuracy in estimating extinction and reignition processes. Global experience confirms that valid voltage–current data obtained from both field and laboratory tests are indispensable for reliable model calibration. Advances in transient analysis software have made it possible to reconstruct the detailed behavior of arcs in overhead lines and high-voltage equipment from ignition to extinction. To demonstrate this, a representative arc model is implemented in a digital simulation environment, with results including arc voltage, instantaneous resistance and conductivity, and voltage–current cyclograms. These results illustrate how mathematical formulations can reproduce the nonlinear and transient behavior of arcs and how such simulations contribute to the design of protection systems and performance assessment of power networks. By combining a comprehensive review with a practical numerical case study, this paper bridges the gap between theoretical foundations, experimental evidence, and software-based simulation. Nevertheless, significant challenges remain, including strong environmental dependence, continual calibration requirements, and the computational burden of high-fidelity models. The future of arc modeling lies in adaptive, data-driven, and multi-physics approaches capable of merging laboratory-level accuracy with engineering-level efficiency to support the reliable operation of modern power systems.
