# Quantifying inertia on the electricity grid

A lot of discussion about the modernisation of electricity systems is about "inertia". This is usually a qualitative discussion about how turbines (in hydro, coal & gas plants) with lots of kinetic energy in the form of angular momentum and fast responsiveness provide voltage and frequency stabilisation at the scale of quarter-cycle (5ms in 50Hz grids) to a small number of seconds.

However, discussions often stall because it's quite rare to see this "inertial response" quantified, and its source identified. As I understand it, the system itself has very low electrical capacitance, so I guess most of the inertial response comes from the rotation of turbines.

How is inertial response quantified for national electricity systems and what are some typical values of system inertia?

• Keep in mind that all synchronous rotating machinery attached to the grid contributes to its "inertia". This includes motors as well as generators. – Dave Tweed Mar 30 '15 at 14:26
• The appropriate keyword is "transient stability". Correct that the majority of inertia comes from rotating machinery. Old coal-fired generators have very high inertia. New aeroderivative gas turbines (i.e. jet engines strapped to alternators) are lightweight and don't have much inertia. – Li-aung Yip Mar 31 '15 at 6:50

This blog post1 identifies the two main sources of inertia within the power grid:

• "Classic" generation, typically steam turbines
• Large industrial motors

Your understanding is correct in that overall system capacity is comparably low and provides a negligible effect to the system inertia.

From a reliability perspective, system inertia is a good thing. The large rotational mass providing system inertia slows the decline in frequency should there be a sudden change in the generation or load of the system. System inertia helps prevent protective load shedding mechanisms from kicking in by providing time for compensating control systems to adjust generation to the changing environment.

Inertia has become a greater subject of interest as newer renewable generation technologies have increased their footprint on electric grids. Newer renewable technologies connect their generation source to the electric grid through power inverters which do not provide any inertia to the rest of the system. Likewise, renewable technologies are enabling the retirement of older generation technologies which results in less system inertia being available. This decline in inertia is compounded by a decrease in large industrial motors.

1Please note, this source is a bit biased as they sell a product related to grid inertia

This presentation goes into some of the details regarding how system inertia is calculated.

Mechanical dynamics are modeled by the second-order differential equation:

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

$\theta$: 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.

From there, you would need to sum the inertia provided by all of the major contributing sources. This is obviously a non-trivial exercise as generation schedules vary as do production schedules for large industries. You also have to take into account the preferred ramp rate of the generators which will vary based upon fuel source.

To provide a negative answer to your question - I think it's these aspects that makes it so hard to discuss system inertia in a quantified manner. There are too many variables and the environment is dynamic. You could perhaps identify the inertia for a small region, but certainly not for the region of a typical balancing authority or at a national scale.

Some concluding thoughts:

The pessimist might argue that system reliability is doomed due to the decrease in overall system inertia and that we'll see more brownouts and blackouts as part of upgrading the overall power grid.

That outlook is likely a bit too bleak though. Balancing authorities can require that more spinning reserves be available, which can provide fast(er) response generation for localized imbalances within the grid. Likewise, national level energy committees can provide compensation in the arbitrage market for fast voltage and frequency providers such as grid scale bulk electric storage systems (BES).

Obviously, those changes won't come for free - it takes fuel to provide spinning reserves, and grid scale BES aren't cheap. But the challenges are surmountable even if decisions have to be made based upon empirical evidence.

• @EnergyNumbers I believe the equation balances. Per Wikipedia on SI units, the right hand side is in Watts which is kg*m^2*s^-3. The left hand side appears to be kg*m^2*s^-2 * s^-1. Moment of Inertia is kg*m^2 and rotational moment of inertia is kg*m^2*s^-2 – GlenH7 Mar 30 '15 at 15:49

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).

An available proxy for the Inertia Constant is the Primary1 Frequency Control characteristic required by each system operator (MW/Hz). These are compared for 8 different systems by Rebours et al; ranging from 20570MW/Hz for UCTE (Union for the Co-ordination of Transmission of Electricity - European synchronous system) to approximately 600MW/Hz for Belgium.

As lower inertia generators (e.g. wind) displace higher intertia generators (i.e. steam), the inertia constant tends to fall. This means that, to maintain overall stability, generators must react more quickly to sudden changes generation or changes in demand. This is often cited as a limiting factor in the connection of wind, especially to smaller "island" networks (e.g. Lalor, Mullane, O'Malley).

1 - Note: primary/secondary/tertiary response/reserve are defined in different ways on different power systems, as noted by Rebours.