Accurate global modelling of the magnetic field in the low solar corona (below about 2.5 solar radii) is critical for understanding and predicting the solar wind. The magnetic structure and topology in this region controls how much of the photospheric magnetic flux escapes to form the interplanetary magnetic field (IMF), and how the IMF is structured. In this talk, I will describe ongoing work to advance beyond the simplified potential-field models used in existing space weather forecasts. In particular, we have developed a time-dependent magneto-frictional model that accounts for the build-up of electric currents in the corona. Here, I will use the model to illustrate the likely consequences of non-potentiality for solar wind predictions (both the IMF and proxies for wind speed).