[PSA] Chapter 6 - Generator modeling I (Machine Viewpoint)

6.1 Classical Machine description

• Rotor or field : Rotating part
• Stator : Stating part

6.2 Voltage generation

• Basic assumptions : neglect resistance, considering only air-gap flux
  1. Magnetic circuit is linear
  2. Flux density $\mathbf B$, in the air gap, due to $i_F$ alone, is radial and has a spatial distribution as a function of angle given by

\[B = B_{max}\cos{(\alpha - \theta)}\]

• The direction is outward
  1. Each coil consists of $N$ turns concentrated in a single slot.
• $\theta$ : angle the rotor axis (North pole) is looking at
• $\alpha$ : general angle we are looking at

6.3 Open circuit voltage

• with zero current in stator, only the rotor field current $i_F$ produces air-gap flux

• Flux in crosshatched rectangular slot at an angle $\alpha$ is

\[d\phi = \ell rB_{max} \cos (\alpha-\theta)d\alpha\]

• Then, integrate to get flux over all Gaussian surface

\[\begin{aligned}\phi&=\oint \mathbf{B}\cdot d\mathbf{s} \\&=\ell r\int_{-\pi/2}^{\pi/2}B_{\max}\cos(\alpha-\theta)\,d\alpha \\&=2\ell r B_{\max}\cos\theta \\&=\phi_{\max}\cos\theta\end{aligned}\tag{6.2}\]

• Denote $\phi_{max} \triangleq 2 \ell r B_{max}$
• Flux linkage of coil $aa’$

\[\lambda_{aa'}=N\phi_{\max}\cos\theta=\lambda_{\max}\cos\theta\tag{6.3a}\]

• Other phases are similar with only changing its phase by $\pm 120 ^\circ$
• Rotor rotates with given angular frequency, we can assume

\[\lambda_{aa'}=\lambda_{\max}\cos\left(\omega_0 t+\theta_0\right)\tag{6.4}\]

• Also, By Faraday’s law, the EMF induced :

\[\begin{aligned}e_{a'a}&=\frac{d\lambda_{aa'}}{dt} \\&=-\omega_0\lambda_{\max}\sin\left(\omega_0 t+\theta_0\right) \\ &=E_{max}\cos{\big(\omega_0 t + \theta_0 - {\pi \over 2}\big)}\end{aligned}\tag{6.5}\]

• We can say Voltage imposed at the stator $e$ lags $\lambda$ by $90 ^\circ$
• Also, define effective phasor

\[\Lambda_{aa'}=\frac{\lambda_{\max}}{\sqrt{2}}e^{j\theta_0}\tag{6.7}\]

• Define effective Voltage

\[\begin{aligned}E_{aa'}&=\frac{E_{\max}}{\sqrt{2}}e^{j\left(\theta_0-\pi/2\right)} \\&=-j\frac{\omega_0\lambda_{\max}}{\sqrt{2}}e^{j\theta_0} \\&=-j\omega_0\Lambda_{aa'}\end{aligned}\tag{6.8}\]

6.4 Armature Reaction

• Armature (Balanced three-phase stator) current produce own rotating magnetic field
• Armature reaction flux linkage representaged by equivalent inductance

\[\Lambda_{ar} = L_{s1}I_a\]

• • Skip for now *

6.5 Terminal Voltage

• Total air-gap flux is sum of rotor-field flux and armature-reaction flux.

\[\Lambda_{ag}= \Lambda_{aa'}+\Lambda_{ar}\]

• Differentiating every flux to get terminal voltage

\[V_{ag} = E_a-j\omega_0 L_{s1}I_a\]

• Account for windings resistance or leakage inductance ($r, X_l$)

\[V_a = E_a-rI_a-jX_sI_a\]

• Resolve Phasors $I_a, I_b, I_c$ into two groups of components
• Direct axis of the rotor $I_{ad}, I_{bd}, I_{cd}$
• Quadrature axis direction of the rotor $I_{aq}, I_{bq}, I_{cq}$
• Assume that $\phase{I_a} = \theta_0 \pm 90^\circ$, Where stator current (phase a) is perpendicular to Voltage or EMF
• Also can divide $\Lambda_{ar}$ into two terms

\[\Lambda_{ar} = \Lambda_{ad}+\Lambda_{aq}\]

• Define fictitious inductance parameters

\[\begin{aligned} \Lambda_{ad} &= L_{d1}I_{ad} \tag{6.26}\\ \Lambda_{aq} &= L_{q1}I_{aq} \end{aligned}\]

• Multiply by $-j\omega_0$ and substitute (6.26) to remove flux linkage

\[-j\omega_0 \Lambda_{ag} = -j\omega_0 \Lambda_{aa'}-j\omega_0 \Lambda_{ad}-j\omega_0\Lambda_{aq}\] \[V_{ag} = E_a-jX_{d1}I_{ad}-jX_{q1}I_{aq}\]

• Consider $r, X_l$

\[\begin{aligned} V_a &= V_{ag}-rI_a-jX_lI_a \\ &=V_{ag}-rI_a-jX_l(I_{ad}+I_{aq}) \tag{6.29} \end{aligned}\]

• Substitute two equations above

\[\begin{aligned} V_a &= E_a-rI_a-j(X_l+X_{d1})I_{ad}-j(X_L+X_{q1})I_{aq} \\ &=E_a-rI_a-jX_dI_{ad}-jX_qI_{aq} \tag{6.30} \end{aligned}\]

Where $X_d \triangleq X_{d1}+X_l$, $X_q \triangleq X_{q1}+X_l$

• We can show $a’$ and $E_a$ are colinear by getting difference

\[a' = V_a + rI_a +jX_qI_a\] \[E_a = V_A+rI_a+jX_dI_{ad}+jX_qI_{aq}\] \[E_a-a' = j(X_d-X_q)I_{ad}\]

6.6 Power Delivered by Generator

Round rotor

• Neglecting resistance for a round rotor,

\[\begin{aligned} S_G &= V_a I_a^* = V_a\big({E_a-V_a \over Z_G}\big)^* \\ P_G&={\vert E_a \vert \vert V_a \vert \over X_s } \sin \delta_m \\ Q_G&= {\vert V_a \vert (\vert E_a \vert \cos \delta_m - \vert V_a \vert ) \over X_s} \end{aligned}\]

Salient-pole generator

• Also neglecting resistance,

\[S_G = V_aI_a^* = V_a(I_{ad}+I_{aq})^*\]

• We can substitute equations to (6.30)

\[V_a = \vert V_a \vert \phase{-\delta_a} = \vert V_a\vert \cos \delta_m- j \vert V_a \vert \sin\delta_m\]

• Dividing into real and imaginary axis,

\[\vert E_a \vert = \vert V_a \vert \cos \delta_m + j X_d I_{ad}\] \[0 = -\vert V_a \vert \sin \delta_m + X_q I_{aq}\]

• We can express $I_{aq}, I_{ad}$

\[I_{ad} = {\vert E_a \vert - \vert V_a \vert \cos \delta_m \over jX_d}, \ I_{aq} = {\vert V_a \vert \sin\delta_m \over X_q}\]

• Substitute into $S_G$ to get $P_G, Q_G$

\[P_G = {\vert E_a \vert \vert V_a \vert \over X_d} \sin\delta_m + {\vert V_a \vert ^2 \over 2} \bigg({1\over X_q} - {1 \over X_d}\bigg)\sin 2\delta_m\]

• If $X_d = X_q = X_s$, the second term cancels

\[Q_G = {\vert E_a \vert \vert V_a \vert \over X_d} \cos\delta_m + \vert V_a \vert ^2 \bigg({\cos^2 \delta_m\over X_d} - {\sin^2 \delta_m \over X_q}\bigg)\]

6.7 Synchronizing Generator to an infinite bus

• Definition : An infinite bus is an ideal voltage source
  1. Frequency is the same
  2. Phase sequence, Phase is the same
  3. $\vert V_a \vert = \vert E_a \vert = \vert V_\infty \vert$
• If this happens, $I_a = 0, S_G=0$ (Floating)
• When turbine accelerates, $\delta_m = \phase{E_a} - \phase{V_\infty}$ increases, thus $P_G$ increases

• To keep mechanical power constant, $\vert E_a \vert \sin\delta_m=\text{constant}$
• $E_a$ must be located in dotted line, increasing $\delta_m$ decreases $i_F$ (proportional to length of $E_a$)

6.8 Synchronous condencer

• $\delta_m = 0 \Rightarrow P_G=0$ (Unloaded)
• But it delivers reactie power.

\[Q_G = {\vert V_a \vert ( \vert E_a \vert - \vert V_a \vert) \over X_s}\]

• Generator but specifically built to operate in this mode is called synchronous condencer
• Convinent and continuous control of reactive power by adjusting the field current $i_F$