# How does load affect frequency on the power grid?

This story about the use of battery/freewheel based Frequency Regulators confused me about how the 60hz frequency of the North American power grid was set--saying that it was kept at that frequency by balancing load and supply. I used to think that it was only voltage which was affected by this balance, and that the frequency was determined by the speed of the rotors in the generator. EG see Wikipedia's page on Alternator Synchronization.

Can someone help me understand the physics behind this?

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The physics is actually much easier than it seems at first glance.

Power generators are engines just like the everyday ones we see all around in our cars, lawnmowers, snowblowers, etc.

Except for new power sources like some wind and solar systems with electronic inverters, the vast majority of power is supplied by large rotating AC generators turning in synch with the frequency of the grid. The frequency of all these generators will be identical and is tied directly to the RPM of the generators themselves, generally 3600 RPM for gas turbines and 1800 RPM for nuclear plants. If there is sufficient power in the generators then the frequency can be maintained at the desired rate (i.e. 50Hz or 60Hz depending on the locale).

The power from the individual generators will lead the grid in phase slightly by an amount roughly corresponding to the power they deliver to the grid.

An increase in the power load is accompanied by a concurrent increase in the power supplied to the generators, generally by the governors automatically opening a steam or gas inlet valve to supply more power to the turbine. However, if there is not sufficient power, even for a brief period of time, then generator RPM and the frequency drops.

This is much like what happens to a car on cruise control if you start going up a hill, if the hill is not too steep you can maintain speed, once you reach the limits of the torque supplied by the engine, the car and engine slow down. If the combined output of all the generators cannot supply enough power then the frequency will drop for the entire grid. All the generators slow down just like your car engine on a hill.

For large grids the presence of many generators and a large distributed load makes frequency management easier because any given load is a much smaller percentage of the combined capacity. For smaller grids, there will be a much larger fluctuation in capacity as delays in matching power supplied are harder to manage when the loads represent a relatively larger percentage of the generated power.

So a battery systems like the one in the article is really designed to keep short-term fluctuations in power requirements from dropping the frequency because of lags in the governors and generators which require a finite time to adjust to the new power requirements. These "frequency regulator" power stations can supply very high power for short bursts to keep the power requirements even so that the other generators don't see too much load faster than they can respond due to mechanical limitations.

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I think I understand. It's like how turning the crank of a hand generator gets harder when the resistance across the contacts is increased, so the increase of load on the power grid literally makes it harder to turn the alternator's axle, which--unless countered by an increase in power--leads to the rotor slowing down, and therefore the frequency. Am I rite? – C. Lawrence Wenham Jan 9 '11 at 22:01
Yes, that's it exactly. The load makes it harder to turn the all the generator shafts which then requires more power to compensate and if the total power requirements exceed the total power available on the grid then all the generators slow down since they are connected together. – inflector Jan 9 '11 at 22:06
Would it even be logical to assume that all those automatic governors of underlying energy source vary their settings based on the frequency? That seems to be the easiest way to tell whether the grid i over- or under-powered? – Cray Dec 1 '11 at 8:45
It's also interesting to note that distributors sometimes intentionally lower the frequency (albeit by an amount that will rarely exceed 1%) for extended periods in consideration of stresses on the system. For the reasons mentioned in the article, varying frequency is harder to get away with than varying voltage (which can easily change by 10% from the standard). Of course, distributors also have higher reliability subsystems for special industrial customers than for residential consumers. – Chris White Oct 22 '12 at 8:21
Fun fact: The generators at the Glen Canyon Dam in Page, Arizona spin at 150 RPM. 48 poles on the rotor! – james large Feb 25 at 23:35

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.

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Higher load in the grid -> lower overall frequency and vice versa