[PSA] Chapter 8 - Generator Modeling

8.0 Introduction

• Exciter : complete voltage control system, including an error detector, feedback loop, automatic voltage regulator(AVR)

8.1 Exciter System Block Diagram

Measurement block

• Gain of measurement block is unity in steady-state

Amplifier block

• Amplifier gain $K_A$ is typically in the range 25 to 400
• $T_A$ in range 0.02 to 0.4 sec
• Amplifier voltage has limitations (exciter field winding can supply limited voltage)

Exciter block

• Voltage of dc generator is proprotional to product of speed times air-gap flux per pole
• Due to saturation effect of magnetic circuit, the flux is a nonlinear function of dc generator field current

• Consider inverse-relationship between $v_F $ and $i$

\[i = i_0 + f(v_F) = {1\over k} v_F + f(v_F) \tag{8.1}\] \[v_R = Ri + {d\lambda \over dt}\]

• Assume that dc generator output voltage $v_F$ is proportional to dc generator air-gap flux,

\[v_F = \beta\lambda\]

• We can get following equation by converting $v_F$ into $E_{fd}$

\[v_R = E_{fd} + E_{fd}S_E(E_{fd})+T_E{dE_{fd} \over dt}\]

Where $T_E \triangleq k/R\beta, S(v_F) \triangleq kf(v_F)/v_F$, Saturation function

8.2 Generator Models

Case I : Open Circuit

• $I_a = 0$, we have $E’_a = E_a = V_a$
• Recall relationship between $E’_a $ and $E_{fd}$

\[T'_{do} {d\vert E'_a \vert \over dt} + \vert E_a \vert = E_{fd}\]

We can substitute $E’_a $ with $E_a$

Can get transfer function

\[G_g(s) = {\vert \hat V_a\vert \over \hat E_{fd}}={1\over 1+sT'_{do}}\]

Case II : Impedance Load

• Assume that isolated generator supplying an impedance load $Z$.
• By considering relationships,

\[\vert E_a \vert = {1\over \sigma} \vert E_a' \vert \tag {8.13}\]

For inductive loads, $\sigma<1$

Can get desired transfer function by defining some auxiliary variables

\[G_g(s) = {k_v\sigma \over 1+s\sigma T'_{do}}\]

Where $k_v \triangleq \vert Z \vert \vert K \vert$, $K \triangleq K_q+jK_d$, $I_d = K_d\vert E’_a \vert, I_q = K_q\vert E’_a \vert$

8.3 Stability of Excitation System

• higher loop gain → smaller voltage error
• but consequently excitation system can be unstable
• Solution : add rate feedback to exciter