A frequency response does not start the instant a loss occurs

PHYSICS · Subjective · Fermi University Physics · Mechanics & Dynamical Systems · Earth, Climate & Energy Physics · Rotation, rigid bodies and accelerated frames

Energy from Physics, Ch The grid with variable renewables: power flow, inertia, adequacy and electrification

Problem

A frequency response does not start the instant a loss occurs. After a loss of generation of size

\Delta P

on a system of stored rotational energy

E_k

and nominal frequency

f_0

, suppose nothing responds for a delay

T_d

, after which the response ramps linearly from zero to

\Delta P

over a further time

T_r

. Ignore the damping from demand. (a) Show that the frequency nadir is

\Delta f_{\rm nadir} = f_0\Delta P(2T_d + T_r)/(4E_k)

. (b) Evaluate it for

f_0 = 50\,\mathrm{Hz}

E_k = 150\,\mathrm{GW\,s}

\Delta P = 1.8\,\mathrm{GW}

T_d = 0.5\,\mathrm{s}

and

T_r = 2\,\mathrm{s}

. (c) The stored energy halves to

75\,\mathrm{GW\,s}

and the delay stays

0.5\,\mathrm{s}

. Find the ramp time that keeps the nadir at the value of part (b), and say by what factor the ramp had to be shortened. (d) At

75\,\mathrm{GW\,s}

, find the longest delay for which any ramp, however fast, can still hold the nadir at that value.

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