Digital Twins are a promising concept to integrate model-based design and operational applications. This study focuses on the control-related performance gap of heating, ventilation, and air-conditioning (HVAC) systems using physics-based models in Building Performance Simulation environments as Digital Twins. We present a comparative framework to contrasts the current practice of replicating HVAC controls in Digital Twins and novel approaches to use Digital Twins for the specification of HVAC controls. The application of this framework to air handling units in a case study building underlines, that the replication of either designed or implemented HVAC controls in Digital Twins is work-intensive, associate with significant uncertainties, and inevitably results in a performance gap. The control-related performance gap can be closed when HVAC controls are identically transferred from a Digital Twin Prototype on building controllers which is demonstrated using the BPS software IDA ICE and the IEC 61131-10 XML exchange format.