# Inertia

In power systems engineering, "inertia" is a concept that typically refers to rotational inertia or rotational kinetic energy. For synchronous systems that run at some nominal frequency (i.e. 50Hz or 60Hz), inertia is the energy that is stored in the rotating masses of equipment electro-mechanically coupled to the system, e.g. generator rotors, fly wheels, turbine shafts.

## Contents

## Derivation

Below is a basic derivation of power system rotational inertia from first principles, starting from the basics of circle geometry and ending at the definition of moment of inertia (and it's relationship to kinetic energy).

The length of a circle arc is given by:

- [math] L = \theta r [/math]

where [math]L[/math] is the length of the arc (m)

- [math]\theta[/math] is the angle of the arc (radians)
- [math]r[/math] is the radius of the circle (m)

A cylindrical body rotating about the axis of its centre of mass therefore has a rotational velocity of:

- [math] v = \frac{\theta r}{t} [/math]

where [math]v[/math] is the rotational velocity (m/s)

- [math]t[/math] is the time it takes for the mass to rotate L metres (s)

Alternatively, rotational velocity can be expressed as:

- [math] v = \omega r [/math]

where [math]\omega = \frac{\theta}{t} = \frac{2 \pi \times n}{60}[/math] is the angular velocity (rad/s)

- [math]n[/math] is the speed in revolutions per minute (rpm)

The kinetic energy of a circular rotating mass can be derived from the classical Newtonian expression for the kinetic energy of rigid bodies:

- [math] KE = \frac{1}{2} mv^{2} = \frac{1}{2} m(\omega r)^{2}[/math]

where [math]KE[/math] is the rotational kinetic energy (Joules or kg.m^{2}/s^{2} or MW.s, all of which are equivalent)

- [math]m[/math] is the mass of the rotating body (kg)

Alternatively, rotational kinetic energy can be expressed as:

- [math] KE = \frac{1}{2} J\omega^{2} [/math]

where [math]J = mr^{2}[/math] is called the **moment of inertia** (kg.m^{2}).

Notes about the moment of inertia:

- In physics, the moment of inertia [math]J[/math] is normally denoted as [math]I[/math]. In electrical engineering, the convention is for the letter "i" to always be reserved for current, and is therefore often replaced by the letter "j", e.g. the complex number operator i in mathematics is j in electrical engineering.

- Moment of inertia is also referred to as [math]WR^{2}[/math] or [math]WK^{2}[/math], where [math]WK^{2} = \frac{1}{2} WR^{2}[/math]. WR
^{2}literally stands for weight x radius squared.

- Moment of inertia is also referred to as [math]WR^{2}[/math] or [math]WK^{2}[/math], where [math]WK^{2} = \frac{1}{2} WR^{2}[/math]. WR

- WR
^{2}is often used with imperial units of lb.ft^{2}or slug.ft^{2}. Conversions factors:

- 1 lb.ft
^{2}= 0.04214 kg.m^{2} - 1 slug.ft
^{2}= 1.356 kg.m^{2}

- 1 lb.ft

- WR

## Normalised Inertia Constants

The moment of inertia can be expressed as a normalised quantity called the **inertia constant H**, calculated as the ratio of the rotational kinetic energy of the machine at nominal speed to its rated power (VA):

- [math]H = \frac{1}{2} \frac{J \omega_0^{2}}{S_{b}}[/math]

where [math]H[/math] is the inertia constant (s)

- [math]\omega_{0} = 2 \pi \times \frac{n}{60}[/math] is the nominal mechanical angular frequency (rad/s)
- [math]n[/math] is the nominal speed of the machine (revolutions per minute)
- [math]S_{b}[/math] is the rated power of the machine (VA)

## Generator Inertia

The moment of inertia for a generator is dependent on its mass and apparent radius, which in turn is largely driven by its prime mover type.

Based on actual generator data, the normalised inertia constants for different types and sizes of generators are summarised in the table below:

Machine type | Number of samples | MVA Rating | Inertia constant H | ||||
---|---|---|---|---|---|---|---|

Min | Median | Max | Min | Median | Max | ||

Steam turbine | 45 | 28.6 | 389 | 904 | 2.1 | 3.2 | 5.7 |

Gas turbine | 47 | 22.5 | 99.5 | 588 | 1.9 | 5.0 | 8.9 |

Hydro turbine | 22 | 13.3 | 46.8 | 312.5 | 2.4 | 3.7 | 6.8 |

Combustion engine | 26 | 0.3 | 1.25 | 2.5 | 0.6 | 0.95 | 1.6 |

## Relationship between Inertia and Frequency

TBA