In the operation of an electricity grid, such as the Dutch transmission system operated by TenneT, decisions need to be made on how to operate components in the grid. These decisions require a complex weighing of the associated costs and effects on the operating state of the grid. Optimal Power Flow (OPF) is a class of mathematical optimization problems that capture the effects of each decision, and aims to find an optimal way of controlling the electricity grid. OPF is well described in scientific literature, and both open-source and proprietary software exists to perform OPF calculations. However, most implementations are limited in their capabilities. Furthermore, applying existing methods to models of real-world grids is not always straightforward. During the calculations, problems can occur that can either slow down or completely prevent the methods from obtaining a solution. We call these convergence problems. In this thesis, two such types of convergence problems have been identified. The first convergence problem involves the modelling of switches, and the second involves the modelling of parallel transformers. Both problems have been investigated and strategies have been found to resolve these problems. The result is an OPF method that convergences on a model of the Dutch transmission network. The implementation of this model is very flexible, allowing it to be applied to a wide variety of OPF problems. This is an important step towards the application of OPF in operation of the Dutch transmission grid. The method has been tested on two different models, one of which is a model of the Dutch transmission grid. Both the computational performance and the resulting solutions have been compared for various OPF calculations.

Vector spherical harmonics are a set of basis functions for vector fields derived from the spherical harmonic functions. They are commonly used in spectral methods in certain areas of applied mathematics. In most of the existing literature they are defined in a way that is heavily dependent on the used coördinate system. In this thesis we define vector spherical harmonics in a coördinate free manner. We first do this on a sphere, using two tensor operators: the exterior derivative and the Hodge-star. We then extend these functions to vector functions in 3-dimensional Euclidian space and define a last type of vector function to form a complete basis for vector fields. We do this in such a manner that the vector spherical harmonics inherit some of the nice properties of the spherical harmonic functions with respect to rotations and rotational symmetric operators.