# Virtual Flux Control Methods for Grid-Forming Converters: A Four-Method Comparison
## 1 Introduction  
The global energy landscape has undergone a profound transformation due to the increasing integration of renewable energy sources (RES) into power systems to mitigate carbon emissions. Over the past decade, the annual deployment of RES capacity has surpassed that of conventional generation technologies, such as fossil fuels and nuclear power. Recent reports from the International Renewable Energy Agency (IRENA) [1] and the European Network of Transmission System Operators for Electricity (ENTSO-E) [2] highlight that certain electrical networks are approaching scenarios of 100% RES penetration during specific timeframes. This shift has been facilitated by inverter-based resources (IBRs), which play a crucial role in integrating RES into electrical grids [3, 4]. The increasing integration of IBRs has transformed power systems, bringing both benefits and new challenges. Unlike synchronous generators (SGs), IBRs lack inherentinertia, reducing grid strength and making the system more susceptible to disturbances [5]. To enable the integration of these technologies, grid-following (GFL) converters have traditionally been used as the primary control method. These converters operate as a current source with current control loops to regulate power injection and rely on phase-locked loops (PLLs) for synchronization [6]. However, their dependence on the grid voltage for operation makes them less effective in weak grids, where PLL performance deteriorates, potentially leading to instability [7]. Furthermore, GFL converters can neither operate autonomously in islanded mode nor contribute to system restoration during blackouts [8; 9]. To overcome these limitations, GFM converters have emerged as an alternative solution capable of providing voltage and frequency regulation without requiring an external reference. Unlike GFL converters, which behave as controlled current sources, GFM converters emulate voltage sources, adjusting their output dynamically to support grid stability. This capability allows them to provide essential functions such as inertial response, frequency regulation, blackstart restoration, and improved performance in weak grids [10; 11]. As a result, GFM converters are becoming a key technology in power systems with a high share of renewable energy, ensuring a more stable and resilient electrical network. Table 1 shows a summary of the different control techniques adopted for these types of converters. According to the literature, various control methodologies have been proposed for GFM converters [12; 13]. The first category is known as droop control, a technique designed to mimic the behavior of a SG governor [14; 15; 16; 17; 18]. This approach is widely used due to its simplicity and ease of implementation; however, it does not inherently provide inertial response. To address this limitation, a second category, referred to as synchronous machine-based controllers, has been developed. These controllers allow GFM converters to emulate inertial response and provide damping capabilities, enhancing their dynamic performance [19; 20; 21; 22; 23; 24]. Lastly, a third category consists of nonlinear control algorithms, among which virtual oscillation-based control (VOC) stands out as one of the most widely adopted approaches [25; 26; 27; 28; 29]. On the other hand, there are several techniques for controlling the voltage vector of a power converter regardless of the methodology implemented, with Field-Oriented Control (FOC) being the most widely used. This approach, commonly applied to electrical machines, involves transforming the three-phase system variables into a synchronous reference frame, defined by \(dq\) axes rotating at a predetermined speed. This transformation enables the decoupled control of both the converter's voltage and frequency using two PI controllers, which generate the desired reference voltages, obtaining good dynamics and waveforms [30; 31; 32; 33].