Question 6.4: Steady-State Model of Synchronous Generator Derive a steady-......

In the steady state, all the currents and flux linkages are constant. Also, i_{0} = 0 and rotor damper currents are zero. Equations 6.89 through 6.94 reduce to
v_{d }= -ri_{d} – \frac{dθ}{dt}λ_{q} – \frac{dλ_{d}}{dt}      (6.89)
v_{F} = r_{F}i_{F} + \frac{dλ_{F}}{dt}     (6.90)
v_{D} = r_{D}i_{D} +\frac{dλ_{D}}{dt} = 0     (6.91)
v_{q}= -ri_{q} + \frac{d\theta }{dt}\lambda _{d} – \frac{d\lambda _{q}}{dt}     (6.92)
v_{Q} = r_{Q}i_{Q} + \frac{dλ_{Q}}{dt} = 0   (6.93)
λ_{0}= L_{0}i_{0}      (6.94)

v_{d}=- ri_{d}-ω_{0}λ_{q}
v_{q}=- ri_{q}+ω_{0}λ_{d}               (6.103)
v_{F}= r_{F}i_{F}
where
λ_{d}= L_{d}i_{d}+kM_{F}i_{F}
λ_{F}= kM_{F}i_{d}+L_{F}i_{F}               (6.104)
λ_{q}+L_{q}i_{q}
Substitute values of λ_{d} and λ_{q} from Equation 6.104 to 6.103; then, from Example 6.3, and then from Equations 6.100 to 6.101,
V_{a}= (\frac{v_{q}}{\sqrt{3}}+ j \frac{v_{d}}{\sqrt{3}}) e^{jδ}= (V_{q} + jV_{d}) e^{jδ}       (6.100)
V_{q}= \frac{v_{q}}{\sqrt{3}}   and V_{d}= \frac{v_{d}}{\sqrt{3}}           (6.101)
we can write the following equation:
V_{a} = -r(I_{q} + jI_{d})e^{ jδ} + ω_{0}L_{d}I_{d}e^{ jδ} – jω_{0}L_{q}I_{q}e^{ jδ} +\frac{ 1}{\sqrt{2}}ω_{0}M_{F}i_{F}e^{ jδ}    where   i_{d} =\sqrt{3} I_{d}   and   i_{q} =\sqrt{3} I_{q}.
Define
\sqrt{2} E =ω_{0}M_{F}i_{F}e^{ jδ}       (6.105)
This is the no-load voltage or the open-circuit voltage with generator current = 0. We can then
write
E = V_{a} + rI_{a} + jX_{d}I_{d}e^{ jδ} + jX_{q}I_{q}e^{ jδ}      (6.106)
The phasor diagram is shown in Figure 6.12a. The open circuit voltage on no load is a q-axis quantity and is equal to the terminal voltage.
As the components I_{d} and I_{q} are not initially known, the phasor diagram is constructed by first laying out the terminal voltage V_{a} and line current I_{a} at the correct phase angle \phi, then adding the resistance drop and reactance drop IX_{q}. At the end of this vector, the quadrature axis is located. Now the current is resolved into direct axis and quadrature axis components. This allows the construction of vectors I_{q}X_{q} and I_{q}X_{q}. This is shown in Figure 6.12b.