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Forecasts of regional electricity net-demand, consumption minus embedded generation, are an essential input for reliable and economic power system operation, and energy trading. While such forecasts are typically performed region by region, operations such as managing power flows require spatially coherent joint forecasts, which account for cross-regional dependencies. Here, we forecast the joint distribution of net-demand across the 14 regions constituting Great Britain's electricity network. Joint modelling is complicated by the fact that the net-demand variability within each region, and the dependencies between regions, vary with temporal, socio-economical and weather-related factors. We accommodate for these characteristics by proposing a multivariate Gaussian model based on a modified Cholesky parametrisation, which allows us to model each unconstrained parameter via an additive model. Given that the number of model parameters and covariates is large, we adopt a semi-automated approach to model selection, based on gradient boosting. In addition to comparing the forecasting performance of several versions of the proposed model with that of two non-Gaussian copula-based models, we visually explore the model output to interpret how the covariates affect net-demand variability and dependencies. The code for reproducing the results in this paper is available at https://doi.org/10.5281/zenodo.7315105, while methods for building and fitting multivariate Gaussian additive models are provided by the SCM R package, available at https://github.com/VinGioia90/SCM.