Capacitor voltage-balancing systems are usually applied to power electronic circuits. The main issue in these systems is equalising the voltage of a large number of capacitors connected to a voltage source in serial or parallel arrangements. A particular application of the capacitor voltage-balancing systems is found in modular multilevel converters (MMC). The voltage balancing of the floating capacitors in the submodules of the MMC during its pre-charging operating stage is a key issue since it is a critical task for the correct operating of these converters. The stability analysis of an active voltage balancing strategy for pre-charging MMCs is addressed in this paper. The adopted voltage-balancing strategy consists in adding a resistance to each submodule of the MMC by means of a controlled switch. These switches are being controlled by a sliding mode control algorithm with multiple boundaries (discontinuity surfaces of high co-dimension). These systems are essentially discontinuous piecewise smooth dynamical systems (Filippov systems) commanded by electronic switches. In this paper, the local stability of the voltage balanced system is analytically proven for an arbitrary number of submodules. In addition, a detailed analysis of the global dynamics of this system with two submodules and two switching boundaries sliding mode control is presented. Simulation results obtained on a MMC with 10 submodules are shown to validate the theoretical analysis.

电容器电压平衡系统通常应用于电力电子电路。这些系统中的主要问题是均衡连接到串联或并联布置的电压源的大量电容器的电压。在模块化多电平转换器（MMC）中可以找到电容器电压平衡系统的特殊应用。在MMC的预充电操作阶段，浮动电容在子模块中的电压平衡是一个关键问题，因为这对于这些转换器的正确操作是至关重要的任务。本文讨论了一种用于MMC预充电的有源电压平衡策略的稳定性分析。所采用的电压平衡策略包括通过受控开关向MMC的每个子模块添加电阻。这些开关由具有多个边界（高维不连续表面）的滑模控制算法控制。这些系统本质上是由电子开关控制的不连续的分段光滑动力系统（Filippov系统）。在本文中，通过分析证明了任意数量的子模块的电压平衡系统的局部稳定性。此外，还详细分析了具有两个子模块和两个开关边界滑模控制的系统的整体动力学。显示了在具有10个子模块的MMC上获得的仿真结果，以验证理论分析。这些系统本质上是由电子开关控制的不连续的分段光滑动力系统（Filippov系统）。在本文中，通过分析证明了任意数量的子模块的电压平衡系统的局部稳定性。此外，还详细分析了具有两个子模块和两个开关边界滑模控制的系统的整体动力学。显示了在具有10个子模块的MMC上获得的仿真结果，以验证理论分析。这些系统本质上是由电子开关控制的不连续的分段光滑动力系统（Filippov系统）。在本文中，通过分析证明了任意数量的子模块的电压平衡系统的局部稳定性。此外，还详细分析了具有两个子模块和两个开关边界滑模控制的系统的整体动力学。显示了在具有10个子模块的MMC上获得的仿真结果，以验证理论分析。