# How does electrical component responds to frequency

Applying Newton's equations for rotating masses to a turbine-generator system gives the following expression (Samarakoon, after Kundur).

$J\frac{d^2\theta}{dt^2}=T_m - T_e$

$\theta$: angle (rad) of the rotor with respect to a stationary reference.
$J$: moment of inertia.
$T_m$: mechanical torque from the turbine.
$T_e$: electrical torque on the rotor.

The Inertial response for a generator is characterised by its Inertia Constant, H, with units of seconds, defined as (Samarakoon, p40):

the ratio of kinetic energy stored at synchronous speed $\omega$ to the generator kVA or MVA rating, $S$.

$$H = \frac{0.5J\omega^2}{S}$$

An equivalent Inertia Constant for an entire system can be estimated: (Ekanayake, Jenkins, Strbac)

$$H_{equivalent} = \sum_{gens} H_{gen}/S_{gen}$$

A value for the GB system (in 2008) was estimated at 9s (by Samarakoon), projected to drop as far as 3s in 2020 with a high wind penetration.

When modelling Inertial response (more commonly referred to as frequency response), a power system can be simplified to a transfer function (Ekanayake, Jenkins, Strbac):

$$\frac{1}{2H_{equivalent}s +D}$$

$D$ is known as the Damping Coefficient - the term encapsulates response from frequency responsive demand (Mu,Wu,Ekanayake,Jenkins,Jia).

To maintin system stability, the frequency must be closely controlled. Traditionally this is achieved through droop controllers on steam turbine generators. Increasingly, however, energy storage and demand response are contributing.

See this question also; https://engineering.stackexchange.com/questions/2245/quantifying-inertia-on-the-electricity-grid

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