## 1 Introduction  
For the past decades, the series-connected battery string (SCBS) has been widely used in several applications of energy storage systems for electric vehicles (EVs), hybrid electric vehicles (HEVs), subway and electric railroad [1; 2; 3; 4; 5; 6] that require high voltage and capacity. The balance of the SCBS plays an important role in chemical and electrical characteristics of a battery string. In fact, the battery lifetime and the energy storage capacity are decreased due to an unbalance of the SCBS generated through the repeated charge and discharge operation. Particularly, this imbalance can lead to fire and/or explosion in the worst case [7]. Thus, the series-connected battery string in EVs needs to be balanced with the voltage [8]. In practice, internal and external sources are known as two main reasons that cause the unbalance of the battery cells [8; 9]. For internal sources, the deviation in production results in variations of storage capacity value, total internal impedance, and rated self-discharge. In the case of the external sources, multi-rank rack protection Integrated circuits (ICs) primarily lead the unequal charge from the various series ranks in the pack [8]. Recently, the cell balancing methods have been introduced [10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20] and reviewed [7; 8; 21; 22; 23], as represented in Figure 1. There are two main ways to equalize a battery cell, which are passive and active methods [8; 22]. Resistors are used in the passive cell balancing methods, which are connected in parallel with the battery cells to avoid overcharging through passive components so that one can equalize the low cells. Due to the power consumption of resistors, this method has low efficiency as reported in the literature [8]. The active cell balancing methods are based on inductors, capacitors and switched controllers or converters, the energy of the cells is transmitted from a high to a low voltage level [8, 11, 14]. Therefore, the active balancing method has higher efficiency than the passive balancing method as reported in the literature [7, 8]. The balancing method based on a multi-winding transformer (MWT) [18, 19, 20, 24, 25] has advantages such as fast balancing speed, simple control technique, and repudiation of voltage sensors. Nevertheless, this circuit has the complexity of fabricating MWT, a limited number of windings due to the parameter matching for the turns ratio, and leakage inductance, especially for a large number of cells. The MWT-based balancing circuit was improved by applying forward-flyback conversion [23] to balance the cells. The buck-boost converter-based cell balancing, which reduces switching loss to obtain high efficiency, is applied to EVs and HEVs [12, 23, 26]. However, this balancing technique requires an intelligent control and a large number of switches, resulting in relative complexity and high cost. In addition, a balancing method based on a switched capacitor circuit (SCC) [11, 15, 26, 27] has been widely utilized in energy storage applications thanks to advantages such as simple design and control and elimination of a voltage sensor for each cell. Importantly, the SCC can be employed in the modular method as well. However, the balancing time is long, especially for a large SCBS since the energy is transferred between adjacent battery cells (i.e., cell to cell balancer). It is known that a higher number of cells and/or larger voltage difference between cells result in longer balancing time. There is a limited number of studies on battery balancing topology such as the requirements of voltage sensors, intelligent control, more importantly, the limited number of cells in the SCBS. Therefore, a novel modular cell balancing circuit (MCBC) is proposed to solve these limitations. In this approach, the SCBS, divided into several modules, is a balanced proposed MCBC, which consists of an intra-module and outer-module balancers.