## 1 Introduction  
High-voltage large-capacity flexible DC transmission, with its advantages of controllable trend, easy to absorb new energy, and ability to provide reactive power support, has become an important pillar of the future DC grid [1, 2, 3, 4]. Among the various topologies, modular multi-level (MMC) converter is the preferred topology due to the advantages of high scalability and low harmonic [5]. The short-circuit fault is one of the critical contingencies for an AC/DC hybrid system. One of the shortcomings for the half-bridge sub-module structure is that it does not have the ability to clear DC faults, and it needs to be equipped with DC circuit breakers to isolate the fault line [6, 7, 8]. The fault current generated during this period is harmful and critical for the secure and safe operation of DC grid. For properly design of fault current limiting and fault blocking devices, an accurate calculation method of fault current and voltage is very necessary [9, 10]. The traditional short-circuit transient characteristic calculation method for DC grid is electromagnetic transient (EMT) simulation [11, 12, 13, 14]. However, this method suffers from low efficiency and high computation burden. In [15], a generic calculation method for pole-to-pole short-circuit fault currents in DC grids is proposed. Based on the simplified RLCequivalent of MMC, it presents a simple numerical analysis. Literature on MMC modeling shows that the minimum required complexity for dynamic MMC modeling has to include the outer controllers and partial dynamics [16, 17]. In [18], relying on the modified averaged value models, DC fault current simulation is performed for protection design in radial multi-terminal DC systems. In [19], the dynamic analysis is carried out for voltage source converter based multi-terminal DC grids (VSC-MTDC) in order to analytically design a generalized feedback for DC voltage droop control. Although, converter outages and communication loss are considered, the article does not analyze line outages or DC faults. To improve the efficiency of the calculation method, MMCs are equivalent to RLC series circuits and the phase-mode conversion method is adopted to simplify the model of the whole DC grid in [20]. However, when the DC grid is radial or mesh, there are some difficulties in performing this method. In [21], the effect of the AC side on the DC short-circuit current is considered, and it provides accurate estimation for short-circuit currents. However, since the influence of MMC control is not considered, there are still some errors in estimating the fault currents and fault voltages. In [22, 23], the accuracy for all the node voltages and branch currents could be improved by the inclusion of MMC controllers. However, the equivalent current sources for converters in the active power control are treated as constant. However, the active and reactive power take changes very sharply especially several milliseconds after the pole-to-pole fault. Therefore, the assumption that the active power keeps constant during the fault leads to some bias especially in estimating the node voltages. Therefore, to guarantee enough accuracy, the dynamics of the AC active power should be involved. This paper makes in-depth analyses of the AC active power on the transient characteristics. From the perspective of the power balance, it is proved that the deviation of AC power results in a relatively large error in calculating the node voltages. Fortunately, the real-time transmitted AC active power and the dynamics of the MMC controllers can be reflected through improving the model of MMC. Based on this, a MMC is equivalent to a RLC series circuit in parallel with a variable controlled current source. Next, the whole DC grid is discretized by the discrete adjoint model. By this method, the dynamics of the controllers can be easily included. At last, the formulas and solutions for the matrices describing the relationship between the branch currents and the node voltages are given.