Equivalent swing equation, coherent machines

When several machines in one plant swing together (coherent machines), they can be replaced by one equivalent machine. Its inertia constant is the MVA-weighted sum of the individual inertias, expressed on the system base.

Derivation

The swing equation of machine $i$ on its own rating $S_{i}$ is

$$\frac{2H_i}{\omega_s}\frac{d^2\delta_i}{dt^2} = P_{mi} - P_{ei}\quad \text{(pu on } S_i)$$

Convert it to a common system base $S_{sys}$ by multiplying both sides by $S_i/S_{sys}$:

$$\frac{2H_{i,sys}}{\omega_s}\frac{d^2\delta_i}{dt^2} = P_{mi} - P_{ei}\quad \text{(pu on } S_{sys}), \qquad H_{i,sys} = H_i\frac{S_i}{S_{sys}}$$

The machines are coherent, so they swing together: $\delta_1 = \delta_2 = \dots = \delta$. Adding the $n$ equations gives

$$\begin{aligned} \frac{2}{\omega_s}\left(\sum_{i=1}^{n} H_{i,sys}\right)\frac{d^2\delta}{dt^2} &= \sum P_{mi} - \sum P_{ei} \\ \frac{2H_{eq}}{\omega_s}\frac{d^2\delta}{dt^2} &= P_m - P_e \end{aligned}$$

where

$$H_{eq} = \sum_{i=1}^{n} \frac{H_i\,S_i}{S_{sys}}, \quad P_m = \sum P_{mi}, \quad P_e = \sum P_{ei}$$

This is the equivalent swing equation: a single machine with inertia $H_{eq}$ and the total power of the plant.

Numerical

Given $H = 6$ MJ/MVA for each unit, ratings 50, 100 and 200 MVA, and a base of 100 MVA:

$$\begin{aligned} H_{eq} &= \frac{6\times 50 + 6\times 100 + 6\times 200}{100} \\ &= \frac{300 + 600 + 1200}{100} = \frac{2100}{100} \\ &= 21\ \text{MJ/MVA} \end{aligned}$$

Check: the total stored energy is $2100$ MJ. On a 100 MVA base this gives $2100/100 = 21$ MJ/MVA.

Answer: $H_{eq} = 21$ MJ/MVA on a 100 MVA base.