Passing clouds can create unacceptable rapid voltage variations in weak distribution networks with a significant solar PV penetration. This paper investigates the benefit of a PV inverter ramp-rate control strategy using battery energy storage for the mitigation of rapid voltage variations. By controlling the discharging/charging operation of the energy storage appropriately, not only the inverter ramp-rate can be maintained at a desired level, but also the voltage variations can be controlled within acceptable limits. Results, from simulation using a real Australian distribution feeder with real load demand and PV output profile, show that with a high penetration of PV, the voltage variations can increase beyond the allowable limit stipulated by the IEC 61000-2-2 standards. However, with the proposed ramp-rate control, the variation can be maintained well below the limits.