Wind turbine installation vessels (WTIVs) are ships that are specifically designed to install offshore wind turbines. These WTIVs have four or six large truss-like legs that are lowered towards the seabed by means of jacking systems. These jacking systems are regulated by control systems to ensure the leg is lowered with a constant velocity regardless of external disturbances. The bottom of these legs are outfitted with spudcans, which have a conical shape to allow for penetration into the seabed. When these spudcans have settled in the seabed, the hull is lifted above the waves. Consequently, wave-exciting forces on the hull are prevented and the motions of the hull are near-zero so that the on-board crane can perform the installation operations with minimised disturbances. The objective of this thesis is to develop and analyse a model that is able to describe and simulate the dynamics of these jacking systems in great detail in response to external loads and the dynamics of the WTIV. The control systems of the jacking systems are included in this model to simulate and evaluate the interaction between the control system and the dynamics of the WTIV. Two conventional control systems are considered: the Volts-per-Hertz (V/Hz) and the direct torque control (DTC) method. In the process of lowering the legs and subsequent platform lifting, a transient phase can be identified during which the spudcans are penetrating the seabed. Due to the periodical motions of the ships, multiple impacts with the seabed are expected. Additionally, the jacking systems and the leg undergo a change of load direction as initially the leg is in tension and the jacking systems are generating power, and afterwards the leg is in compression and the jacking systems are consuming power. This thesis is focused on this seabed penetration phase as this phase introduces complicated dynamics. In literature, no model is available that has the abilities to simulate the WTIVs and its jacking systems with control systems in such level of detail. This research gap is addressed by developing such a simulation model. This model is written in Python and developed using finite element (FE) techniques and solved using numerical time integration. Seabed characteristics are derived using a detailed coupled Eulerian-Lagrangian (CEL) FE models. Multiple control strategies are simulated and evaluated, each differentiating how the velocity and torque setpoints of the jacking systems are calculated. From the simulation model, it is found that in order to achieve load sharing between jacking systems, torque and velocity require to be independently controlled which only the DTC method has the ability to. Furthermore, each of the jacking systems should be provided with its own power supply. Best performance and stability was achieved when each chord of the leg is given a common torque and velocity setpoint, which is equivalent to a common torque and setpoint per leg in reality. Moreover, load sharing can be improved without a control system by increasing the relative stiffness ratio between the chord and the mechanical contact between rack and pinion.