## 1 Introduction  
In recent years, wind turbine generators have been significantly increasing in size, in terms of hub height, rotor diameter, and generator capacity, in order to convert more power from wind energy with higher efficiency and lower investment and operating costs. In 2021, the power capacity of wind turbine generators had reached 15 MW [1], and in 2035, the capacity of wind turbine generators is predicted to reach up to 17 MW [2]. Along with the increasing capacity, the connection of the wind turbines to grid power systems by traditional two-level or three-level converters requires multiple electronic devices such as IGBTs in series or parallel connections to achieve very high power capacity and operating voltage. The Modular Multilevel Converter (MMC) is a promising solution for high-power-capacity wind power generation. Compared to other conventional converter technologies, the MMC is a novel converter concept that has many advantages, such as having a simple structure and a flexible design that enables one to expand the number of levels and to replace the submodules easily, so that maintenance becomes easier [3,4]. In addition, connecting the MMC to a grid system without a transformer is possible [5]. The implementation of an MMC in a wind power generator, however, has not been extensively reported. The majority of reports have investigated the stability of grid-connected wind farms using MMC-based High Voltage Direct Current (HVDC) transmission systems [6, 7, 8, 9, 10], but only a few papers have discussed the topology concept of the implementation of MMC-controlled wind turbine generators. The performance and topology of MMCs for 2 MW 0.69 kV and 10 MW 10 kV has been discussed in [11]; however, the study only discusses the grid side converter and concentrates mainly on loss distribution between submodules. In [12], the application of an MMC-controlled multi-phase PMSG was proposed. The fault tolerant control strategy of the MMC is validated via simulation analysis with detailed model representation. However, a comprehensive discussion regarding the modelling of MMC-controlled wind turbine generators for a grid connectivity study has not been widely discussed. A feasibility study on the stability of wind turbine generators connected to a grid system is particularly important in a wind farm design. The dynamic behavior under steady-state and transient conditions of a wind farm will affect the voltage and frequency stability of the grid system. A large variation in the energy production of a wind farm or lack of synchronism due to a short circuit fault can have a great impact on power system stability and power quality [13; 14]. Therefore, modeling and simulation analyses of the grid connectivity of a wind farm in the early stages of development of the facility is essential. In this paper, detailed and average models of grid-connected MMC-controlled Permanent Magnet Synchronous Generator (PMSG)-based direct drive wind turbines are proposed. The models can be used as representations of the wind turbine generator for the study of grid connectivity. PMSG was chosen on the basis that most manufacturers use this type of generator for their large-capacity wind turbine generators [15; 16; 17] due to its high efficiency and attractive features suitable for the wind turbine concept. In the detailed model, the MMC includes a detailed representation of power electronic IGBT converters, and the MMC circuit is configured by a sub-module that has a two-level half-bridge configuration composed of two IGBTs with anti-parallel diodes and a capacitor. As a switching phenomenon with multi-carrier modulation technique is considered in the modeling, the model should be discretized at a relatively small simulation time step (10 ms). The detailed model is suitable for analyzing the dynamic performance of control systems and harmonics over a short duration. In the average model, the MMC circuit with power electronic IGBT modules is represented by equivalent voltage sources generating AC voltages.