## 1 Introduction  
Power quality phenomena in alternating current (AC) electrified railway have drawn more attention in the last decades, especially, after the evolution in the power electronics field [1]. Railway operators have an absolute interest to run the electrified trains with the lowest possible costs. In this context, various solutions were proposed to overcome the power quality deterioration in the electrified railway, e.g., the flexible AC transmission systems (FACTS) presented in [1; 2; 3; 4; 5]. On the other side, the modular multilevel converter (MMC) is an attractive solution for medium-voltage applications due to the lower harmonics, lower switching losses, MMC flexibility, and scalability [6; 7] since other multilevel converters, such as the neutral point clamp and the flying capacitor multilevel converters, are quite hard to extend to higher levels [8]. Therefore, MMC has been enhanced to be combined with the FACTS family. As example, MMC has been dedicated to operating as a rail power conditioner (RPC) with different topologies in [6; 9], and an MMC based static synchronous compensator for railway applications is proposed in [10]. Half-bridge converters have been used in high power applications due to their high efficiency, low cost and simple control compared to the full-bridge converters [11]. In this context, the half-bridgeAC/DC/AC MMC based RPC using a V/V power transformer, denominated as a simplified rail power conditioner (SRPC), is used in this study. The main disadvantage of the half-bridge MMC is the fact that it requires the double of the voltage with respect to the output neutral, and therefore, two capacitors with a mid-neutral point are required. In any case, the SRPC total power rating is divided between the MMC submodules (SMs), then, a higher number of MMC SMs signifies lower power ratings required for each SM, and a better output performance to synthesize a multilevel output waveform. A SM consists of a storage capacitor (with voltage \(V_{SM}\)) and a half-bridge converter [12]. The circulating current control is important to suppress the harmonic contents produced in the MMC arm currents due to the SM capacitor voltage ripples or the inner DC-voltage differences between the MMC SMs. The magnitude of the circulating current has an influence on the MMC arm current. It increases the total root mean square (RMS) values of the MMC arm currents, which leads to extra power losses, and second-order harmonics will result in further generation of other higher-order harmonics, such as the 4 th, 6 th, 8 th, etc., [13]. The circulating currents do not contribute to the MMC output currents synthesized by the power compensator, since the circulating currents flow only in the MMC arms [14]. As a conventional method to suppress circulating currents, inner passive filters are used, but they do not totally solve the circulating currents problem [15]. Furthermore, the large passive component values may lead to bulky implementation of the MMC [16]. The circulating current can be reduced by increasing the inductors between the MMC arms or by increasing the SM capacitor. Nevertheless, the size of the inductors and the cost of SM capacitors have to be taken into account [17]. On the other hand, as addressed in [18], the existence of a small circulating current component in the MMC arm current can be useful to have a control freedom, thus optimizing the MMC performance. In some applications, circulating currents can be used in a useful way, e.g., during the phase-to-ground fault, in which the MMC works in an asymmetrical operation whereby the power will flow through the upper and the lower MMC arms, leading to an imbalance of arm capacitor energy. In that case, improved control strategies were developed based on circulating current injection [19; 20]. Concerning the circulating current elimination, a method based on repetitive controller and proportional-resonant (PR) controllers using a phase-shifted pulse-width modulation (PWM) is presented in [8].