The word savanna usually means vast territories of grass plains, covered with herbage, isolated islets of undersized trees and bushes. Its distinctive features are a long dry period without rain falling on winter and spring months, and plentiful rains and thunderstorms in summertime. Fluctuations in temperature reach 20 °C and more.
Such climate is considered to be the most favorable for cactus types growing on these plains. The most popular of them are coryphanta and mammillaria. These types of cactus excellently bear dry and cool winter and need a lot of warmth, sun and watering in summer, but they are afraid of water stagnation near their roots. The soil for these cacti should contain less humus, than the soil for forest cactus species, and it should have an admixture of loam.
By the way, the majority of savanna cacti, especially undersized ones like mammillaria, grow well in penumbra. In natural conditions they hide from the sun in thick grass.