Artificial Intelligence for Renewable Energy Systems

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:

(1.26)

(1.27)

(1.28)

(1.29)

(1.30)

(1.31)

(1.32)

ψ mqand ψ mdare 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

(1.40)

the developed torque associated with the second winding set xyz .

(1.41)

(1.42)

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 in rotor reference frame by using the transformation matrix [20, 22] associated with each winding sets abc and xyz , respectively. The voltages applied to the damper windings are not shown, because voltages are zero due to short-circuited windings.

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:

(1.43)

where

[ v ] = [ vq 1, vd 1, vq 2, vd 2, vKq , vfr , vKd ] T

[ i ] = [ iq 1, id 1, iq 2, id 2, 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:

(1.44)

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 to rotor reference frame (i.e., and is given by Equation ( 1.45).