# Droop Control

Droop control is a control strategy commonly applied to generators for primary frequency control (and occasionally voltaqe control) to allow parallel generator operation (e.g. load sharing).

## Contents

## Background

### Physical Intuition

TBA

### Generic Formulation

A more generic formulation of the droop control concept stems from the coupling between active power and frequency, and similarly reactive power and voltage. Recall that the active and reactive power transmitted across a lossless line are:

- [math]P = \frac{V_{1} V_{2}}{X} \sin\delta [/math]

- [math]Q = \frac{V_{2}}{X} (V_{2} - V_{1} \cos\delta) [/math]

Since the power angle [math]\delta \,[/math] is typically small, we can simplify this further by using the approximations [math]\sin\delta \approx \delta \,[/math] and [math]\cos\delta \approx 1 \,[/math]:

- [math]\delta \approx \frac{PX}{V_{1} V_{2}} [/math]

- [math](V_{2} - V_{1}) \approx \frac{QX}{V_{2}} [/math]

From the above, we can see that active power has a large influence on the power angle and reactive power has a large influence on the voltage difference. Restated, by controlling active and reactive power, we can also control the power angle and voltage. We also know from the swing equation that frequency in synchronous power systems is related to the power angle, so by controlling active power, we can therefore control frequency.

## Droop Equations

### Per-Unit Droop Equations

The coupling of active power to frequency and reactive power to voltage forms the basis of frequency and voltage droop control where active and reactive power are adjusted according to linear characteristics, based on the following control equations:

- [math]f = f_{0} - r_{p} (P - P_{0}) \, [/math] ... Eq. 1

- [math]V = V_{0} - r_{q} (Q - Q_{0}) \, [/math] ... Eq. 2

where [math]f \, [/math] is the system frequency (in per unit)

- [math]f_{0} \, [/math] is the base frequency (in per unit)
- [math]r_{p} \, [/math] is the frequency droop control setting (in per unit)
- [math]P \, [/math] is the active power of the unit (in per unit)
- [math]P_{0} \, [/math] is the base active power of the unit (in per unit)
- [math]V \, [/math] is the voltage at the measurement location (in per unit)
- [math]V_{0} \, [/math] is the base voltage (in per unit)
- [math]Q \, [/math] is the reactive power of the unit (in per unit)
- [math]Q_{0} \, [/math] is the base reactive power of the unit (in per unit)
- [math]r_{q} \, [/math] is the voltage droop control setting (in per unit)

These two equations are plotted in the characteristics below:

The frequency droop characteristic above can be interpreted as follows: when frequency falls from [math]f_{0}[/math] to [math]f[/math], the power output of the generating unit is allowed to increase from [math]P_{0}[/math] to [math]P[/math]. A falling frequency indicates an increase in loading and a requirement for more active power. Multiple parallel units with the same droop characteristic can respond to the fall in frequency by increasing their active power outputs simultaneously. The increase in active power output will counteract the reduction in frequency and the units will settle at active power outputs and frequency at a steady-state point on the droop characteristic. The droop characteristic therefore allows multiple units to share load without the units fighting each other to control the load (called "hunting").

The same logic above can be applied to the voltage droop characteristic.

### Alternative Droop Equations

The basic per-unit droop equations in Eq. 1 and Eq. 2 above can be expressed in natural quantities and in terms of deviations as follows:

- [math]r_{p} = \frac{\Delta f}{\Delta P} \times \frac{P_{n}}{f_{n}} [/math]

- [math]r_{q} = \frac{\Delta V}{\Delta Q} \times \frac{Q_{n}}{V_{n}} [/math]

where [math]\Delta f \, [/math] is the frequency deviation (in Hz)

- [math]f_{n} \, [/math] is the nominal frequency (in Hz), e.g. 50 or 60 Hz
- [math]\Delta P \, [/math] is the active power deviation (in kW or MW)
- [math]P_{n} \, [/math] is the rated active power of the unit (in kW or MW)
- [math]r_{p} \, [/math] is the frequency droop control setting (in per unit)
- [math]\Delta V \, [/math] is the voltage deviation at the measurement location (in V)
- [math]V_{n} \, [/math] is the nominal voltage (in V)
- [math]\Delta Q \, [/math] is the reactive power deviation (in kVAr or MVAr)
- [math]Q_{n} \, [/math] is the rated reactive power of the unit (in kVAr or MVAr)
- [math]r_{q} \, [/math] is the voltage droop control setting (in per unit)

## Droop Control Setpoints

Droop settings are normally quoted in % droop. The setting indicates the percentage amount the measured quantity must change to cause a 100% change in the controlled quantity. For example, a 5% frequency droop setting means that for a 5% change in frequency, the unit's power output changes by 100%. This means that if the frequency falls by 1%, the unit with a 5% droop setting will increase its power output by 20%.

The short video below shows some examples of frequency (speed) droop:

## Limitations of Droop Control

Frequency droop control is useful for allowing multiple generating units to automatically change their power outputs based on dynamically changing loads. However, consider what happens when there is a significant contingency such as the loss of a large generating unit. If the system remains stable, all the other units would pick up the slack, but the droop characteristic allows the frequency to settle at a steady-state value **below** its nominal value (for example, 49.7Hz or 59.7Hz). Conversely, if a large load is tripped, then the frequency will settle at a steady-state value **above** its nominal value (for example, 50.5Hz or 60.5Hz).

Other controllers are therefore necessary to bring the frequency back to its nominal value (i.e. 50Hz or 60hz), which are called secondary and tertiary frequency controllers.