The energy sector is the largest contributor to global greenhouse gas emissions. Therefore, to effectively combat climate change, conventional fossil-fueled power generation is being replaced by power generation from renewable energy sources, such as solar panels or wind farms. This energy transition in combination with an increased electrification of the transportation and heating sectors puts significant stress on the electric power transmission grid. In this thesis, we investigate four different aspects of the impact on the transmission grid. First, the capabilities of the grid to support the connection of large solar farms in the Alps is investigated. Using a data-driven approach, a set of generation dispatches over a full year is created to subsequently investigate the stability of the Swiss transmission grid with an additional large alpine solar farm connected to it. Our findings show that the current grid is not able to support such a solar farm. However, after completion of the currently planned grid extensions, a connection is possible. Second, oscillations of the voltage frequency between large areas of the grid are studied. We develop a description of these oscillations based on matrix perturbation theory. We show that given the right aggregation of our system, the perturbation theory results remain valid even in the strongly coupled case. Furthermore, we provide convergence criteria on a per-mode basis and are able to validate the shape predictions for each mode. We demonstrate the validity of our method on synthetic networks and models of real large-scale power grids. Third, we use the flexibility of renewable production to introduce a novel control strategy. Conventional generators by design provide rotational inertia to the power grid. This rotational inertia can provide or absorb energy over short time scales, thereby improving the grid stability. While renewable energy sources cannot provide physical inertia, they can emulate inertia when connected to an external energy storage like a battery. We suggest changing the amount of virtual inertia provided to the grid based on the rate of change of frequency. The performance of the control is evaluated using four performance measures, quantifying both short- and long-term behavior. The performance is greatly improved both for the current grid as well as for a future scenario with a significantly increased penetration of renewable energy sources. Additionally, we show how the impact of the adaptive inertia scheme can be maximized by strategically placing the control either homogeneously over the grid or in the remote parts of it. Fourth, we build a continuous model of the power grid. Initially, we distribute the physical parameters over the grid using a diffusion algorithm. While this already leads to satisfactory results, we improve on these first results by using physics-informed machine learning. The line susceptances can be obtained from a set of steady-state solutions. Following this, the damping and inertia of the system are obtained from dynamical simulations. We find that this leads to good approximations of the discrete dynamics while using only a quarter of the degrees of freedom. These results lead the way to future applications of online learning, in which the physical parameters of the system can be learned from noise measurements. Additionally, they provide an accurate reduced model to facilitate stability calculations.