# Enhanced Efficient EMT-Type Model of the MMCs Based on Arm Equivalence
## 1 Introduction  
The modular multilevel converter based high-voltage direct current (MMC-HVDC) has been widely applied in large-scale renewable energy integration and asynchronous power grid interconnection [1; 2; 3]. Compared with the conventional line-commutated converter (LCC) and two- or three-level voltage source converters (VSCs), the MMC has become the most promising and competitive alternative due to its high modularity, no commutation failure, and excellent output waveforms [4; 5; 6; 7]. In order to research the dynamic behaviors of MMCs, significant research efforts have been devoted to develop MMC models for different application purposes on electromagnetic transient (EMT) simulation programs [8; 9]. Most EMT simulation programs use [PERSON]'s algorithm [8]. With the application of trapezoidal integration, all dynamic elements are converted into a Norton equivalent current source in parallel with a conductance. Then, the admittance matrix equation of the full circuit is formulated to solve the circuit, which is the nodal analysis method. The detailed switching modes (DSMs) (e.g., detailed nonlinear insulated-gate bipolar transistor (IGBT)-based model, and simplified IGBT switchable resistance-based model) could accurately present the IGBT switching characteristics and dynamic behaviors of the MMCs [10]. Whereas, with a large count in switching elements, the MMC contributes a significantly large number of switchable nodes to the admittance matrix (\(\boldsymbol{\gamma}\)) of the full circuit. Due to the high switching frequency (typically in the kHz range) [11], the large-sized \(\boldsymbol{Y}\) must be frequently updated and re-inversed at every switchingaction, which is computationally inefficient [12]. As a result, DSMs bring a tremendous computational burden and excessive computing time requirements, due to the massive nodes and minuscule time-step. Consequently, DSMs are practically impossible for system-level studies, especially for HVDC grids. To overcome this drawback of the DSMs in simulation efficiency, detailed equivalent models (DEMs) were proposed in [12], and improved in [13]. Based on the Thevenin equivalent circuit, DEMs eliminate the internal intermediate nodes, and greatly improve the simulation efficiency, with no loss of accuracy. Nevertheless, all SMs are considered separately and all individual capacitor voltage ripples are recorded, leading to the consumption of excessive computational processing and simulation time for large-scale MMC-HVDC grids. Thus, DEMs are still computationally inefficient for simulation scenarios of large-scale MMC-HVDC grids with large numbers of submodules (SMs). Aiming to further improve the simulation efficiency, average value models (AVMs) were developed in [14] and [15], where, the MMC dynamics were imitated by using AC-side controlled voltage sources, and a DC-side controlled current source with an equivalent capacitor. Due to drastic simplification, the AVMs neglect the switching actions of the power devices and the impact of capacitor voltage ripples, which leads to a lack of representation of characteristics of the arm dynamics (e.g., arm currents and SM capacitor voltages). Even worse, they have difficulty achieving accurate results under blocked mode. Although the simulation efficiency of AVMs is satisfactory, they lose accuracy compared to DEMs. Thereby, AVMs are mainly suitable for system-level controller studies, rather than the detailed responses of the MMC-HVDC link, especially for the dynamics of DC voltage and current [1]. Due to the unsatisfactory defects of DEMs and AVMs, this paper proposes an enhanced computationally efficient model based on arm equivalence (AEM) for the MMC on EMT simulation programs. The proposed AEM, not only has an accuracy consistent with the DEMs, but also has a satisfactory simulation efficiency compared to AVMs.