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We study several generalizations of the Steiner Tree problem that are motivated by the design of power networks. While Steiner Tree asks for a single minimum-cost tree that connects a given set of terminal vertices, a power network typically consists of multiple trees. Each tree connects to a subset of the terminals, to avoid electrical overloads. The cost of installing a power network is therefore determined by two factors: the total length of the cables in the network and the cost of digging underground trenches into which the cables are placed. Since the digging costs can be substantial, to minimize the total cost of the network it might be necessary to place multiple cables into the same trench. These characteristics lead to variations of Steiner Tree in which the goal is to compute a minimum-cost set of Steiner trees, all with a common root, that together connect a given terminal set while balancing the power demand of the terminals in each tree. Two important variations arise depending on whether the network is intended for low-voltage or high-voltage power. In the low-voltage setting, there is substantial power loss across the cables which effectively means that the maximum depth of any tree in the solution has to be bounded. No such depth bound applies to the high-voltage setting. We investigate the parameterized complexity of several power network design problems, using the number of terminals as the parameter. While this parameterization of the standard Steiner Tree problem is fixed-parameter tractable, many of our variants are W[1]-hard. For low-voltage networks (bounded-depth trees), we present an XP-algorithm for planar inputs, which exploits a nontrivial bound on the treewidth of solution subgraphs. We provide an intricate reduction from Grid Tiling to establish that the resulting algorithm is tight under the Exponential Time Hypothesis. The XP-algorithm extends to the high-voltage setting and to general graphs, albeit at a cost in the running time. For high-voltage networks, we prove that the problem remains W[1]-hard on planar graphs. Finally, we explore a variation of the cost model for sharing digging costs in which both problems become fixed-parameter tractable.