Analytic estimation of the MMC sub-module capacitor voltage ripple for balanced and unbalanced AC grid conditions

In this paper, a mathematical expression to define the maximum and minimum voltage ripples of the modular multilevel converter (MMC) sub-module (SM) capacitors is proposed. Using the arm averaged model of the MMC, the instantaneous power for the upper and lower arms of the converter is obtained, giving the basis to describe the instantaneous energy of the arms. To calculate the SM capacitors peak voltage values, it is required to obtain an analytic expression for the maximum and minimum energy levels. Due to the presence of terms with different magnitudes, frequencies and phases, finding it can be quite challenging. To overcome this issue, the instantaneous arm energy expression is modified using mathematical assumptions in order to join the different components into a single term which can analytically describes the maximum energy point. Then, this expression is used to calculate the peak values of the arm capacitor voltages. By employing the same principles from the arm level, the final analytical expression for the SM capacitors maximum and minimum voltages is found. Simulation results are carried out in order to validate the accuracy of the proposed analysis for different power delivery conditions.