Dynamic response of excitation system

The dynamic response of an excitation system is how fast and how smoothly the generator terminal voltage follows a change in reference voltage (or recovers after a load change). It depends on the gains and time constants of the amplifier, exciter and generator field, and on stabilizing feedback.

Mathematical model

Basic AVR loop (sensor time constant neglected, unity feedback):

 Vref +   Ve  +-------+   +-------+   +-------+
 --->(+)----->|  KA   |-->|  KE   |-->|  KG   |--+--> Vt
     ^ -      |1+sTA  |   |1+sTE  |   |1+sTG  |  |
     |        +-------+   +-------+   +-------+  |
     +-------------------------------------------+

Open-loop transfer function:

$$G(s) = \frac{K}{(1 + sT_A)(1 + sT_E)(1 + sT_G)}, \qquad K = K_A K_E K_G$$

Closed loop:

$$\frac{\Delta V_t(s)}{\Delta V_{ref}(s)} = \frac{G(s)}{1 + G(s)}$$

Steady-state response

For a step $\Delta V_{ref}$, using the final value theorem:

$$\begin{aligned} \Delta V_t(\infty) &= \frac{K}{1 + K}\,\Delta V_{ref} \\ e_{ss} &= \frac{1}{1 + K}\,\Delta V_{ref} \end{aligned}$$

To keep the voltage error small (e.g. below 1 %), $K$ must be large (about 100 or more).

Transient response and stability

Characteristic equation:

$$(1 + sT_A)(1 + sT_E)(1 + sT_G) + K = 0$$ $$T_A T_E T_G\, s^3 + (T_A T_E + T_E T_G + T_A T_G)\, s^2 + (T_A + T_E + T_G)\, s + (1 + K) = 0$$

By the Routh criterion, the system is stable only if

$$(T_A T_E + T_E T_G + T_A T_G)(T_A + T_E + T_G) > T_A T_E T_G\,(1 + K)$$

So a large $K$ (good accuracy) makes the response more oscillatory and can make it unstable. This is the basic conflict in AVR design.

 Vt
      high K: fast but oscillatory
    |    .-.
    |   /   '-.__.-------------
    |  /    ___.-------------- low K
    | / _.-'   (slower, larger error)
    |/.'
    +---------------------------> t

Improving the dynamic response

• Stabilizing (rate) feedback from exciter output to amplifier input, $\dfrac{sK_F}{1 + sT_F}$: adds damping only during transients, leaving steady-state gain unchanged. Closed loop becomes

$$\frac{\Delta V_t}{\Delta V_{ref}} = \frac{G_A G_E G_G}{1 + G_A G_E H_F + G_A G_E G_G}$$

• Fast exciters (static thyristor or brushless) with small $T_E$, and high ceiling voltage and response ratio, so the field current can be forced quickly.
• Lead-lag compensation or a power system stabilizer (PSS) for damping rotor oscillations.

A good excitation system gives a fast rise, small overshoot, short settling time and a small steady-state voltage error.