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This paper investigates the operation of space-vector-based hysteresis current control in the synchronous reference frame (SRF) for grid-connected voltage source converters. The space-vector-based approach allows for the systematic application of zero-voltage vectors and prevents high switching frequencies caused by phase interaction. The average switching frequency can be minimized by always selecting the zero-voltage vector that is closest to the previous switching state. Implementation in the SRF ensures a simple structure based on two three-level hysteresis comparators and a switching table that fits well into the traditional structure of vector-oriented current control. For operation in a grid-connected converter with an LCL filter, the current references to the hysteresis controllers are given by a dc-link voltage controller and an active damping mechanism for suppressing filter oscillations. Results from simulations and laboratory experiments are presented to illustrate the features of the investigated approach compared to traditional phase-current hysteresis controllers.