With the growing of using photovoltaic (PV) units in power distribution systems, the role of high-performance power electronic converters is increasing. In this paper, modelling and control of Modular Multilevel Converter (MMC) are addressed for grid integration of PV units. Designing a proper controller for MMC is crucial during faulty conditions to make the converter stable and provide proper dynamic performance. To achieve this goal, a dynamic model of MMC is presented which it includes symmetrical components of voltage and current. Then, adaptive robust current controllers are developed based on sliding mode and fuzzy controllers for MMC and then the robustness and stability of the controllers are proved by the Lyapunov theory. To implement the proposed controllers under unbalanced grid voltage fault, positive and negative sequences current controllers are implemented to compensate the effect of grid voltage fault and load power variation. Finally, numerical results are shown to evaluate the performance of MMC. In the end, the experimental results are given to prove the controller performance. The outcome indicates that the proposed current controllers are more effective under voltage disturbance conditions and could satisfy the stability of MMC.