selected as state variable (i.e., independent variable). Hence, the voltage-current relation of machine in matrix form is expressed as follows:
where
[v] = [vq1, vd1, vq2, vd2, vKq, vfr, vKd]T
[i] = [iq1, id1, iq2, id2, iKq, ifr, iKd]T
[z] is the impedance matrix defined in the Appendix.
Using the concept of Taylor series expansion, the linearization of machine equations (1.40), (1.41), (1.42), and (1.43) results in a set of equations, which are expressed in matrix form:
where matrix elements are explained in the Appendix.
Grid voltage with constant magnitude and frequency is presented in synchronous reference frame. Hence, it is advantageous to relate the variables in synchronously rotating reference frame (i.e., and
whereas β shows the phase difference between the terminal voltage of phases a and x. Numerically, the numerical value of both α and β is 30° electrical.
The linearized version of above of nonlinear differential equation (1.45) with suitable approximation (cosΔδk = 1 and sinΔδk = Δδk) results in
Inverse transformation of above equation yields
where Fr and Fe represent the d-q performances indices under steady state.
(1.48)
(1.49)
Substituting Equations (1.46) and (1.47) into Equation (1.44) results in
(1.50)
and is rearranged as
(1.51)
Simplified version of above equation can be written as
In above expressions, the additional subscript “0”
where
In above linearized model of machine, the effect of mutual coupling between stator winding sets abc and xyz is considered (by using mutual leakage reactance, xlm and xldq). Results are presented in consideration of the asymmetrical six-phase synchronous machine (α = 30° electrical) in comparison with its three-phase equivalent.
1.4 Dynamic Performance Results
Dynamic analysis of any electrical system is of prime importance to understand its operating characteristic under different conditions changing suddenly. Therefore, the mathematical model developed in previous section has been effectively used to analyze the behavior of grid connected six-phase synchronous generator under sudden change in active load. For this purpose, a set of differential equations that describe the synchronous machine operation were simulated in MATLAB/Simulink environment. Simulation has been carried out for a machine of 3.2 kW, 6 poles, whose parameters are mentioned in the Appendix.
Computer traces in Figures 1.2 and 1.3 are showing the dynamic behavior of six-phase synchronous generator due to a step change in output active power from 0% to 50% of rated/base value at time t = 5 s and further increase in output power by 50% (i.e., at full load) at time t = 15 s. It is assumed that the terminal voltage and frequency is constant irrespective of the change in load torque. Input phase voltage was maintained constant at 240 V, 50 Hz, operating at 0.85 power factor (lagging). Initially, generator is operating at no load condition at synchronous speed. At time t = 5 s, a step increase in output power, and hence, increase in prime mover applied torque Tl is considered. This resulted in the increase in rotor speed immediately, following the step increase in prime mover torque as shown in Figure 1.2a, where the load angle δ increases in Figure 1.2c. The rotor speed continues to increase till the accelerating torque on the rotor vanishes. It can be noted from Figure