## 1 Introduction  
Owing to the development of distributed generations (DGs) and the requirement of environment protection, more DGs are integrated into distribution networks [1, 2]. Despite holding the advantages of being flexible, economic, and environmentally friendly, the integration of DGs bring significant challenges to the stable operation of distribution networks, resulting in the problems of bi-directional power flow, voltage out-of-limit, and line overload [3, 4]. Traditional methods for voltage regulation includes on-load tap changer (OLTC) and capacitor bank (CB) [5, 6]. However, utilizing OLTC requires abundant reactive power in a system. CB may cause harmonic amplification and significant impact current. Thus, new methods have to be explored for distribution networks to improve power quality. A flexible interconnection device (FID) is a power electronic device. It can flexibly control power flow through voltage source converter (VSC) technology. Thus, it is used for feeder connection in distribution networks [7; 8; 9]. FID usually replaces the tie switches for the connection of two or more feeders [10]. FID can control active and reactive power flow separately. It shows the advantages of continuous adjustment and rapid response [11; 12], which are applicable for operation optimization. [PERSON] et al. [13] constructed a stochastic scenario-based optimization model to simultaneously optimize the siting and sizing of soft open points (SOPs) and tie switches. The optimal operation strategy of these devices is obtained during the network reconfiguration process, thereby significantly improving the operational efficiency and system stability of the distribution network. [PERSON] et al. [14] proposed a multi-objective robust optimization model based on various controllable devices as a flexible distribution switch and energy storage. The proposed method can obviously decrease network losses, increase the profit, and improve voltage profiles. [PERSON] et al. [15] introduced a novel mathematical optimization algorithm for maximizing hosting capacity (HC) through sequential network reconfiguration and SOP placement. A new SOP allocation index is proposed, which effectively improves distribution network performance and optimizes DG penetration. [PERSON] et al. [16] designed a hybrid multi-terminal soft open point (MT-SOP) to efficiently improve distribution system interconnection capacity. A case study showed that the hybrid MT-SOP increased the utilization of converters and reduced system loss. [PERSON] et al. [17] presented a cooperative control strategy of a flexible interconnection device and energy storage. The proposed control strategy can achieve power mutual aid between feeders, leading to the stable operation of the power grid. A photovoltaic (PV) system, as a typical DG, can selectively provide active power and reactive power based on inverter control [18]. The IEEE 1547-2018 standard indicates that inverter-based distributed energy resources can actively participate in voltage regulation by modulating the active and reactive power output [19]. Several studies have concentrated on the optimization of a PV inverter. [PERSON] et al. [20] proposed a harmonic suppression and reactive power compensation strategy for a distribution network based on a photovoltaic multifunctional grid connected inverter. Except for the active power output of PV, the remaining capacity of the inverter was utilized to offer reactive power compensation for better power quality. [PERSON] et al. [21] propounded an advanced nonlinear control scheme based on adaptive integral back stepping for the optimization of a three-phase grid-connected photovoltaic inverter. The scheme aims to maximize the power output of the photovoltaic system, precisely control active and reactive power, and reduce harmonic injection. [PERSON] et al.