The energy sector is going through a large transformation due to public demands of renewable energy sources. However, a major issue is that these energy sources are intermittent. If designed correctly, district heating systems can naturally contain energy storing units, for example by storing heat in the isolated pipes that make up the heat grid. Additionally, this makes it easier to reuse and transport already generated heat to other users. This paper proposes a mathematical model of such a grid, where excess energy can be retracted from one user and distributed to other users using a network of heat pumps. In some cases, one can balance residual heat production with the heat consumption, temporarily eliminating the need for a centralized heating plant. Existence conditions for stable steady states of such a network with general topology are given. Finally, energy optimal stable steady states are obtained through convex optimization.