For the future of sustainable energy, renewable energy will need to significantly penetrate existing utility grids. While various renewable energy sources are networked with high-voltage DC grids, integration between these high-voltage DC grids and the existing AC grids is a significant technical challenge. Among the limited choices available, the modular multi-level converter (MMC) is the most prominent interface converter used between the DC and AC grids. This subject has been widely pursued in recent years. One of the important design challenges when using an MMC is to reduce the capacitor size associated with each module. Currently, a rather large capacitor bank is required to store a certain amount of line-frequency related circulating energy. Several control strategies have been introduced to reduce the capacitor voltage ripples by injecting certain harmonic current. Most of these strategies were developed using trial and error and there is a lack of a systematic means to address this issue. Most recently, Yadong Lyu has proposed to control the modulation index in order to reduce capacitor ripples. The total elimination of the unwanted circulating power associated with both the fundamental line frequency and the second-order harmonic was demonstrated, and this resulted in a dramatic reduction in capacitor size. To gain a better understanding of the intricate operation of the MMC, this thesis proposes a state-space analysis technique in the present paper. Combining the power flow analysis with the state trajectory portrayed on a set of two-dimensional state plans, it clearly delineates the desired power transfer from the unwanted circulating energy, thus leading to an ultimate reduction in the circulation energy and therefore the required capacitor volume.