Voltage sags are one of the most frequently occurring power quality problems challenging power systems today. The Unified Power Quality Conditioner (UPQC) is one of the major custom power solutions that are capable of mitigating the effect of supply voltage sags at the load or Point of Common Coupling (PCC). A UPQC-Q employs a control method in which the series compensator injects a voltage that leads the supply current by $90^{circ}C$ so that the series compensator at steady state consumes no active power. However, the UPQC-Q has the disadvantage that its series compensator needs to be overrated. Thus it cannot offer effective compensation. This paper proposes a new control scheme for the UPQC-Q that offers minimum power injection. The proposed minimum power injection method takes into consideration the limits on the rated voltage capacity of the series compensator and its control scheme. The validity of the proposed control scheme is investigated through simulation and experimental results.