*2.1. Simulation Model*

2.1.1. Permanent Magnet Synchronous Generator model

As defined in [31] the PMSG model behaviour is described in Rotor Reference Frame (RRF) as follows:

$$\frac{d}{dt}\dot{i}\_d = \frac{1}{L\_d}\upsilon\_d - \frac{R}{L\_d}\dot{i}\_d + \frac{L\_q}{L\_d}p\omega\omega\_m\dot{i}\_q\tag{3}$$

$$\frac{d}{dt}\dot{t}\_q = \frac{1}{L\_q}v\_q - \frac{R}{L\_q}\dot{i}\_q - \frac{L\_d}{L\_q}p\omega\_m\dot{i}\_d - \frac{\lambda\_l p\omega\_m}{L\_q} \tag{4}$$

$$T\_c = \frac{3}{2} p [\lambda i\_q + (L\_d - L\_q) i\_d i\_q] \tag{5}$$

where *R* is the resistance of the stator windings, *p* is the number of pole pairs, *Te* is electrical torque, *Ld* and *Lq* are the *dq* axis inductances, *ω<sup>m</sup>* is angular velocity of the rotor, *λ* is amplitude of induced flux, *vd* and *vq* are *dq* voltages and *id* and *iq* are *dq* currents. Equations (3) and (4) represent the ouput currents and volatges in *dq* frame and Equation (5) calculates electromagnetic torque.

*Lq* and *Ld* represent the relation between the phase inductance and the rotor position due to the saliency of the rotor. For a round rotor, there is no variation in the phase inductance therefore *Ld* = *Lq* = *Lab* 2 .

## 2.1.2. Wind Turbine Model

The Wind turbine is modelled using the Matlab Simulink wind turbine model with a nominal mechanical output power of 800 kW and a base wind speed of 14 m/s. The output of this block is applied to the generator shaft in per unit of generator ratings. We assume a direct drive system where mechanical efficiency (*η*\_*m*) is 1. This wind turbine characteristic can be seen in Figure 5.

**Figure 5.** Wind Turbine Model Power Characteristic Curve.

A PID blade pitch angle controller is used with a rate of change limitation of eight degrees per second. This is to account for the fact that pitch angle cannot be varied instantaneously. This rate of change limitation value can be found in the NREL 5 MW reference wind turbine report [32]. A full control and simulation diagram can be found for both the pilot generator in Figure 6 and for non-pilot generators in Figure 7. For the pilot generator, the control system utilises a frequency setpoint and feedback loop to maintain the farm bus frequency. This pilot generator is set to a chosen power level and excluded from the farm power control loop. Non-pilot generators use a power reference and feedback loop to vary their active power contribution to the bus. The setpoint for these turbines is determined by a farm power level supervisor which takes the current farm power level and set point and distributes individual power levels to the turbines. For further information on the direct interconnection algorithm, see [13,24,29].

**Figure 6.** Control and Simulation Diagram for the Pilot Generator.

**Figure 7.** Control and Simulation Diagram for the Non-Pilot Generators.
