The increasing share of renewable energy in the electricity grid and progressing changes in power consumption have led to fluctuating, and weather-dependent power flows. To ensure grid stability, grid operators rely on power forecasts which are crucial for grid calculations and planning. In this paper, a Multi-Task Learning approach is combined with a Graph Neural Network (GNN) to predict vertical power flows at transformers connecting high and extra-high voltage levels. The proposed method accounts for local differences in power flow characteristics by using an Embedding Multi-Task Learning approach. The use of a Bayesian embedding to capture the latent node characteristics allows to share the weights across all transformers in the subsequent node-invariant GNN while still allowing the individual behavioral patterns of the transformers to be distinguished. At the same time, dependencies between transformers are considered by the GNN architecture which can learn relationships between different transformers and thus take into account that power flows in an electricity network are not independent from each other. The effectiveness of the proposed method is demonstrated through evaluation on two real-world data sets provided by two of four German Transmission System Operators, comprising large portions of the operated German transmission grid. The results show that the proposed Multi-Task Graph Neural Network is a suitable representation learner for electricity networks with a clear advantage provided by the preceding embedding layer. It is able to capture interconnections between correlated transformers and indeed improves the performance in power flow prediction compared to standard Neural Networks. A sign test shows that the proposed model reduces the test RMSE on both data sets compared to the benchmark models significantly.