The installed capacity of photovoltaic (PV) generation in residential distribution system is growing rapidly around the world. While integrating PV into the distribution systems can provide a range of benefits, a severe issue caused by high-level PV penetration is the reverse power flow along the distribution feeder. Due to the high R/X ratio in the distribution systems, real power exporting can lead to a comparable impact on grid voltage to reactive power, which raises the feeder voltage significantly. On the other hand, the stochastic PV output variation can result in fast and uncertain voltage deviation. Thus, it is necessary to regulate the voltage and improve voltage stability against the stochastic PV generation in the distribution system. In this paper, a multi-objective control strategy for volt-var control (VVC) of distribution feeders with a high PV penetration is proposed. This control corresponds to an optimal reactive power flow problem with uncertain variables. We develop a multi-objective stochastic optimization model considering power loss, voltage deviation and voltage collapse proximity indicator (VCPI)-based voltage stability index. The model is solved by a multi-objective genetic algorithm. Numerical test results show that the obtained VVC solution is robust against stochastic PV output variation.