# Optimization of Active Distribution Network Operation with SOP Considering Reverse Power Flow
## 1 Introduction  
In recent years, as environmental issues such as global warming become increasingly severe, the distributed generation (DG) of distributed renewable energy sources has rapidly developed [1; 2; 3]. The distribution network, as the main carrier of the DG, has seen a continuous increase in the penetration rate of the DG within it [4]. This has highlighted the issue of bidirectional power flow [5] in distribution networks, which can, in severe cases, lead to over-voltage problems [6]. When confronted with significant reverse power flows, traditional regulatory equipment like on-load tap changer (OLTC) and capacitor bank (CB) exhibit slow regulation speeds, an inability to output continuously, and limited operational ranges [7; 8; 9; 10; 11; 12]. Consequently, the distribution network's capacity to handle reverse power flow is insufficient, leading to voltage limit violations, wind and solar curtailments, and other issues that hinder the further development of the distribution network. In addressing the aforementioned challenges, flexible distribution devices, exemplified by soft open point (SOP) [13], have garnered significant attention in recent years. SOPs, composed of fully controllable power electronic devices, are free from the operation frequency limitations of traditional mechanical switches. They allow rapid and precise control [14; 15] of active and reactive power on connected feeders, making them an excellent choice for addressing supply-demand imbalances in distribution networks, thereby greatly enhancing network operational efficiency [16; 17; 18]. Considerable research has already been conducted on SOP in distribution network optimization. For instance, the literature [19; 20]explores the working principle of SOPs in network optimization and establishes an optimal scheduling model by coordinating various controllable resources, thus achieving optimal power flow distribution. The literature [21] proposes a time-segmented reactive compensation strategy. By joint optimization of an SOP and energy storage systems, targeting minimization of average voltage deviation and network loss, operational efficiency is enhanced. The literature [22] develops a two-stage day-ahead and intra-day optimization model incorporating an SOP and other flexible resources, with distributed generation (DG) penetration as the objective function, improving economic efficiency while boosting DG accommodation. The literature [23] establishes an active-reactive coordination optimization model for the network, resolving voltage limit issues by regulating reactive compensation devices, photovoltaic inverters, SOPs, and other adjustable equipment. Additionally, several studies, including [24], address voltage issues arising from two-way power flow due to increased DG penetration by establishing a bidirectional transformer model to quantify control curves under bidirectional flow and mitigating voltage fluctuations through multi-layer regulation of energy storage devices. However, there is limited research on detailed modeling of reverse power flow transmission capacity in distribution networks and on utilizing SOP flexibility to address challenges of insufficient reverse power flow transmission capability. Based on the aforementioned issues, this paper's contributions are as follows:  
1. It analyzes the impact of OLTC parameters, the OLTC's low-voltage side reactive compensation, and DG installation location on the reverse power flow capacity of distribution networks. A refined model is established, enabling precise calculations of reverse power flow capacity considering these influencing factors. 2. Addressing the issue of insufficient reverse power flow capacity in distribution networks, a multi-distribution network joint operation optimization model is established based on an SOP. This interconnects multiple distribution networks via an SOP, enhancing the reverse power flow capacity of individual networks, thus improving the overall operational efficiency of the distribution network. 3.