We probe the stability of Watts-Strogatz DC power grids, in which droop-controlled producers, constant power load consumers and power lines obey Kirchhoff's circuit laws. The concept of survivability is employed to evaluate the system's response to voltage perturbations in dependence on the network topology. Following a fixed point analysis of the power grid model, we extract three main indicators of stability through numerical studies: the share of producers in the network, the node degree and the magnitude of the perturbation. Based on our findings, we investigate the local dynamics of the perturbed system and derive explicit guidelines for the design of resilient DC power grids. Depending on the imposed voltage and current limits, the stability is optimized for low node degrees or a specific share of producers.