Frequency dependent load and speed deviation

Loads are of two types according to how their power depends on frequency:

• Frequency independent loads take the same power at any frequency. Examples: resistive loads such as lighting and heating.
• Frequency dependent loads change their power with frequency. Example: motor loads (fans, pumps, compressors), whose power varies with speed. For small changes their effect is written as $D\,\Delta\omega$.

So the total electrical load change is

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

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

Role on frequency change

• A rise in frequency-independent load $\Delta P_L$ creates an unbalance $P_e > P_m$, which causes the frequency to fall.
• The frequency-dependent load opposes the change: as frequency falls, motor loads take less power. This "self-regulation" reduces the unbalance and helps the system reach a new steady frequency even without a governor.
• A larger $D$ gives a smaller frequency deviation and faster settling.

Transfer function of speed deviation

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) and $\Delta\omega$ = 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: as shown above, the composite load (frequency-insensitive part plus motor part) gives

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

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 the turbine power constant ($\Delta P_m = 0$, no governor action):

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

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

For a step load $\Delta P_L$:

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

• Inertia $H$ sets the initial rate of fall ($-\Delta P_L/2H$) and the time constant $2H/D$.
• Damping $D$ sets the final deviation $-\Delta P_L/D$. If all load were frequency independent ($D = 0$), $\Delta\Omega = -\Delta P_L/(2Hs)$: speed would keep falling linearly without limit.