We report three-dimensional modelling of plasmonic solar cells in which electromagnetic simulation is directly linked to carrier transport calculations. To date, descriptions of plasmonic solar cells have only involved electromagnetic modelling without realistic assumptions about carrier transport, and we found that this leads to considerable discrepancies in behaviour particularly for devices based on materials with low carrier mobility. Enhanced light absorption and improved electronic response arising from plasmonic nanoparticle arrays on the solar cell surface are observed, in good agreement with previous experiments. The complete three-dimensional modelling provides a means to design plasmonic solar cells accurately with a thorough understanding of the plasmonic interaction with a photovoltaic device.