This paper presents an impedance-based method to systematically investigate the interaction between multiple trains and traction network, focusing on evaluating the harmonic instability problem. Firstly, the interaction mechanism of multi-train and traction network is represented as a feedback interconnection of the two subsystems, i.e., an equivalent output impedance of the traction network and an equivalent input admittance of the multi-train. Then the harmonic instability is evaluated through a series of pole-zero diagrams drawn from the closed loop transfer matrix of the multi-train and network system (MTNS). The interaction system is unstable and the harmonic instability will happen if there are some high frequency poles of the closed loop system locating in the right half plane. This method is used for analyzing the harmonic instability phenomena, the characteristics, influential factors and potential mitigation schemes. The theoretical results are further validated by the simulations and experiments. Index Terms-electrified railways, harmonic instability, impedance-based analysis, multi-train and network interaction. I. INTRODUCTION A. Motivations Y the end of January 2017, the total mileage of the China high-speed railway (HSR) has exceeded 22 thousand kilometers, accounting for more than 60% in the world. It is vital to ensure the security and reliability of such a large scale transportation system. However, the current, voltage and phase-locked loop controllers of the high-speed trains (HSTs) may interact with each other and these controllers can further interact with the traction network, resulting in gradual amplification and instability in harmonic voltage and current of the traction network. This phenomenon has been named harmonic instability in [1-4]. These harmonic instability problems have been reported in China HSR lines: 1) The amplification phenomenon of the 47 th to 55 th harmonic current components occurred on the China HSR line in 2011. It led the control system of the HSTs to be unstable, the lightning arrestors of the HST to be burned, circuit breaker to