The COVID-19 pandemic affects all the aspects of modern society worldwide, especially in the power sector. Measures of flexibility enhancement are regarded as solutions to guarantee reliable and flexible electricity supply in such an emergency. This study aims at investigating the impact of flexibility enhancement measures (electricity storage and flexible demand) in different situations of the preliminary COVID-19 pandemic. Case studies in different regions (Denmark, the Netherlands, and the Sichuan province of China) are conducted and assessed using the hourly simulation tool EnergyPLAN. These regions own different electricity supply mix and level of renewable electricity. It is found that the flexible demand measure within one day or one week can hardly eliminate the electricity imbalance caused by either the pandemic or the increasing renewable electricity. The monthly flexible demand is effective for balancing, but its potential in these regions is not enough. However, electricity storage measure enhances the electricity balance even during the most extreme situation of the pandemic. From the economic perspective, electricity storage measure leads to an increase of up to 15% in total system costs, while flexible demand measure has a negligible effect on costs. This study serves as the first step to understand the performance of flexibility enhancement measures in the power sector under the shock of a pandemic.