Speed response without governor and isochronous governor

Transfer function without governor

Without a governor, the turbine gate is fixed, so $\Delta P_m = 0$ and the speed is decided only by rotor inertia and load damping.

Inertia (rotating mass): for small changes, in per unit, the swing equation is

$$2H\frac{d\,\Delta\omega}{dt} = \Delta P_m - \Delta P_e$$

where $H$ = inertia constant (s), $\Delta\omega$ = speed deviation (pu). Taking the Laplace transform,

$$\Delta\Omega(s) = \frac{1}{2Hs}\big[\Delta P_m(s) - \Delta P_e(s)\big]$$

Load damping: a composite load has frequency-insensitive parts (lighting, heating) and frequency-sensitive parts (motors). So

$$\Delta P_e = \Delta P_L + D\,\Delta\omega$$

where $\Delta P_L$ = non-frequency-sensitive load change and $D$ = load damping constant (% change in load per % change in frequency, typically 1–2).

Substituting,

$$\begin{aligned} 2Hs\,\Delta\Omega(s) &= \Delta P_m(s) - \Delta P_L(s) - D\,\Delta\Omega(s) \\ \Delta\Omega(s) &= \frac{\Delta P_m(s) - \Delta P_L(s)}{2Hs + D} \end{aligned}$$

With $\Delta P_m = 0$:

$$\frac{\Delta\Omega(s)}{\Delta P_L(s)} = \frac{-1}{2Hs + D}$$

 dPm=0  +    +--------------+
 ------>(+)->|      1       |----+--> d(omega)
         ^ - |   2Hs + D    |    |
         |   +--------------+    |
       dPL

For a step load $\Delta P_L$:

$$\Delta\omega(t) = -\frac{\Delta P_L}{D}\left(1 - e^{-tD/2H}\right)$$

The speed falls exponentially with time constant $2H/D$ and settles at $-\Delta P_L/D$, which is a large error because $D$ is small.

Incorporating an isochronous governor

An isochronous (constant speed) governor uses integral control. It measures speed, compares with the reference, and integrates the error to move the gate:

$$\Delta Y(s) = -\frac{K}{s}\,\Delta\Omega(s)$$

The gate keeps moving as long as there is any speed error, so speed returns exactly to the set value.

                          dPL
                           | -
 +-------+  dPm            v      +-----------+
 |turbine|--------------->(+)---->| 1/(2Hs+D) |--+--> dw
 | Gt(s) |                 +      +-----------+  |
 +-------+                                       |
     ^ dY                                        |
 +-------+    -                                  |
 |  K/s  |<---(+)<-------------------------------+
 +-------+     ^ +
               | dw_ref = 0

Here $G_t(s)$ is the turbine transfer function (for hydro, $\frac{1 - T_w s}{1 + 0.5T_w s}$).

Working: a load rise makes speed fall; the integrator output grows, gates open, $P_m$ rises above $P_e$, the rotor accelerates and speed comes back to the reference. In steady state $\Delta\omega = 0$ and $\Delta P_m = \Delta P_L$.

The isochronous governor works well for a single unit supplying an isolated load. It cannot be used on several units in parallel, because each would try to fix the frequency at its own set point and they would fight for the load; for that, speed droop is added.