In this project we will consider the control of a synchronous generator supplying electricity to the grid. We will focus on the problem of frequency stability. The frequency at which the generator rotates is directly proportional to the frequency of the currents and voltages in the power grid. Hence, keeping the rotational frequency constant is of fundamental importance to the correct operation of the grid. A simplied model to study the stability of a single generator connected to the grid is the so called swing equation given by:

In this model !s is the synchronous rotational velocity 2 50 rad/s. The constant D = 5 represents the damping induced by mechanical and electrical losses and the constant H has the value 4. The term Pm = 1=2 represents the mechanical power that is transformed by the generator into electrical power and the term U sin  represents the electrical power and describes the eect the grid has on the generator. We will take U as the input since we can change its value by the voltage across the excitation coil in the generator's rotor.

A synchronous generator, when operating in steady state, maintains a constant angular velocity achieved by matching the supplied mechanical power Pm with the power lost through dissipation H (is) and the electrical power U sin . When a fault occurs and a transmission line is tripped (opened) there is a sudden change in the electrical power and some generators will accelerate while other generators will decelerate. Once the fault is cleared and the transmission line is re-closed, dierent generators will be rotating at dierent angular velocities and voltages and currents in the power grid will no longer be sinusoids with a frequency of 50 Hz.

The objective of this project is to design a controller that will resume the steady state operation of the generator at 50 Hz. Please justify all your answers including relevant plots if necessary.