Domestic and international regulations stipulate that PVs must have voltage-reactive power support function to mitigate the overvoltage and voltage fluctuation problems caused by large-scale distributed photovoltaic(PV) integration. However, this control is mianly designed for balanced voltage scenarios, and rarely considers unbalanced operation conditions, posing risks of voltage exceeding limits and/or aggravated voltage unbalance. To fill such void, based on the instantaneous power theory, this paper proposes an improved voltage support control strategy for unbalanced grid conditions, which adds an ancillary negative-sequence voltage suppression module to the conventional voltage-reactive power control. First, the mathematical relationship between the positive/negative-sequence voltage and active/ reactive powers is derived, and the voltage support mechanisms under unbalanced voltages conditions are analyzed. Then, with the control target of maintaining the voltage at the point of common coupling(PCC) within its continuous operation range, the desired positive and negative sequence voltage combinations are established. In this way, the corresponding negative sequence voltage reference can be determined according to the positive sequence voltage regulated by the voltage-reactive power control. Furthermore, relying on the relationship between the negative sequence voltage and power, the minimum required negative sequence active and reactive powers can be derived using the Lagrange multiplier method, which further deteremines the negative sequence current reference and guarantees an effective voltage mitigation under unbalanced conditions with the minimum power capacity. Finally, the validity and effectiveness of the above analysis and the proposed strategy are verified under various operating conditions in Matlab/Simulink.