standard method to evaluate the machine parameters [25, 26].
Detailed mathematical simulation of six-phase synchronous machine is based on determination of voltage and torque equations in integral form with flux linkage per second and speed as state variable, winding currents as output, connected grid voltage and prime mover torque as input variables. The voltage equations (1.1) to (1.7) together with the flux linkages equations (1.8) to (1.14) are firstly solved for the currents, which are then substituted in the voltage equations. Integral forms of mathematical equations are as follows:
Figure 1.1 Equivalent circuit representation of a six-phase synchronous machine.
(1.19)
(1.20)
(1.21)
(1.22)
(1.23)
(1.24)
(1.25)
Current in terms of flux is written as follows:
(1.27)
(1.28)
(1.29)
(1.30)
(1.31)
ψmq and ψmd are defined using state variables as follows:
(1.33)
(1.34)
where
(1.35)
(1.36)
(1.37)
Developed electromagnetic torque (Te) and rotor dynamic equations for the machine can be expressed as follows:
(1.38)
where
(1.39)
the developed torque associated with the first winding set abc, and
the developed torque associated with the second winding set xyz.
where Te is overall developed electromagnetic torque and Tl is the prime mover torque. In the mathematical modeling, both motoring and generating mode is possible. In this chapter, generating mode is considered, where mechanical input torque Tl is fed from prime mover to generate the electrical output power, resulting the flow of current from machine to connected utility grid.
Input voltages vabcs and vxyzs in stationary frame coordinate are transformed directly to
1.3 Linearization of Machine Equations for Stability Analysis
In above section, current and flux linkage per second are both related to each other [Equation (1.26) to (1.32)], one variable vector (current or flux linkage per second) can be taken as state variable. The choice of state variable is generally determined by the application [15]. Here, current is selected as state variable (i.e., independent variable). Hence, the voltage-current relation of machine in matrix form is expressed as follows:
