---
title: "Frequency dependent load and speed deviation"
subject: "Power Plant Equipment"
chapter: 2
key: "semester_7/power_plant_equipment:frequency-dependent-load-tf"
source: https://www.bctengineer.com/q/semester_7/power_plant_equipment/frequency-dependent-load-tf
site: BCT Engineering
---

# Frequency dependent load and speed deviation

Power Plant Equipment · Chapter 2: Power/Frequency Control in Hydro Generator System

- Asked in: 2073 Chaitra, 8 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/power_plant_equipment/frequency-dependent-load-tf
- Whole chapter: https://www.bctengineer.com/md/semester_7/power_plant_equipment/chapter-2

## Question

What do you understand by frequency dependent and independent load? What are the roles of these load on change in frequency? Deduce the transfer function of speed deviation due to load change, including effect of these type of load and inertia.

## Short answer

Frequency-independent load (e.g., lighting) is constant with frequency; frequency-dependent load (e.g., motors) varies, adding damping D. A load increase causes frequency to fall, and D limits the deviation. The transfer function of speed deviation due to load change is ΔΩ(s) = –1/(2Hs + D), with inertia H and damping D.

## Answer

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

```text
 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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which pair correctly classifies the loads?
   A. Motors are frequency independent; heating is frequency dependent
   B. Heating and lighting are frequency independent; motor loads are frequency dependent
   C. Both types change power equally with frequency
   D. Lighting is frequency dependent; pumps are not
   Answer: B. Resistive loads take the same power at any frequency, while motor power varies with speed. (Key point: Load types)

2. What is the effect of the frequency-dependent load when frequency falls?
   A. It takes more power and worsens the fall
   B. It takes less power, reducing the unbalance and giving self-regulation
   C. It has no effect on frequency
   D. It trips the generator
   Answer: B. Motor loads draw less as frequency falls, opposing the change and helping the system reach a new steady frequency. (Key point: Role of damping)

3. With dPm = 0, the transfer function d(Omega)/dPL is:
   A. 1/(2Hs + D)
   B. -1/(2Hs + D)
   C. -D/(2Hs)
   D. -2Hs/(2Hs + D)
   Answer: B. From 2Hs dOmega = dPm - dPL - D dOmega, we get dOmega/dPL = -1/(2Hs + D). (Key point: Speed transfer function)

4. What do H and D determine for a step load change?
   A. H sets the final deviation; D sets the initial slope
   B. H sets the initial rate of fall and time constant 2H/D; D sets the final deviation -dPL/D
   C. Both only set the final deviation
   D. D sets the initial slope; H sets nothing
   Answer: B. The initial slope is -dPL/2H, the time constant is 2H/D, and the final value is -dPL/D. (Key point: Roles of H and D)

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