The penetration of renewable energy resources, such as wind and solar, is continuously growing in power system grids and their interconnection in low short-circuit (LSC) systems (aka weak grids) is becoming more frequent. Converter interfaced resources have controls and dynamics that differ from traditional synchronous generators and require sufficient modeling for potential instabilities to be analyzed and prevented both in planning and operations environment. In this work, instability scenarios related to converter controls of wind turbine generators (WTGs) connected to LSC systems are analyzed, and the limitations of using positive sequence models in these scenarios are demonstrated. In addition, the influence of various converter control parameters on the WTG instability is investigated. The analysis is performed on a portion of the American Electric Power (AEP) network with several hypothetical renewable energy resources.