SimuVeda

Engineering Ā· Simulator 084

Grid Frequency

Every power grid runs at one frequency, 50 or 60 Hz, as long as generation matches demand. Suddenly lose a power station or switch on a big load, and the frequency starts to fall. Change the grid's inertia, the generators' droop response and automatic control, and see how deep the dip goes and where the frequency settles.

How the model works

All the spinning generators act like one big flywheel. When load exceeds generation, the missing power comes out of their kinetic energy and they slow down. In per-unit (p.u.) terms the swing equation is

2H dĪ”f/dt = Ī”P_m āˆ’ Ī”P_L āˆ’ D Ī”f

where H is the inertia constant (seconds of rated energy stored in rotation), Ī”P_L the load step, Ī”P_m the extra mechanical power from generators, and D the load damping (motors draw less power at lower frequency). Just after the step the rate of change of frequency is RoCoF = āˆ’Ī”P_L fā‚€ / 2H.

Primary control (droop): each governor raises output in proportion to the frequency error, Ī”P_m ā†’ āˆ’Ī”f / R, through a lag of time constant T_g. With a 5% droop, a 5% frequency drop would call for 100% more power. It stops the fall but leaves a steady error:

Ī”f_ss = āˆ’Ī”P_L / (1/R + D)

Secondary control (AGC): an integral controller slowly shifts the governors' set-points until the frequency is back at nominal.

A single-area, linearized model with one equivalent generator. No tie lines, no under-frequency load shedding, no inverter-based fast response, and no limits on generator ramp rates or reserves.

Guided experiments

  1. Lose 5% of generation on a 50 Hz grid with H = 5 s, 5% droop and a 2 s governor. Prediction: frequency starts falling at 0.25 Hz/s, reaches a nadir of about 49.74 Hz after roughly 2 s, then settles at 49.88 Hz, the droop steady state 50 Ɨ (1 āˆ’ 0.05/21).
  2. Low inertia. The same trip on a grid with H = 2 s (many inverter-based sources). Prediction: the initial fall is 2.5 times faster (0.62 Hz/s) and the nadir is deeper, about 49.63 Hz, but the steady state is unchanged.
  3. Slow governors. Governor time constant 6 s. Prediction: the nadir drops to about 49.60 Hz and arrives later (about 4 s), since generation responds more slowly.
  4. No primary control. Governors switched off. Prediction: only load damping resists the fall, so frequency sinks for a minute or more toward 47.5 Hz. A real grid would shed load long before that.
  5. Automatic generation control. Droop plus AGC. Prediction: the dip is the same, but the frequency is pulled back to 50.00 Hz within about a minute.

Challenge

A 60 Hz grid with 4% droop and D = 1 suddenly gains 3% more load. Where does the frequency settle with primary control only? What RoCoF follows the step if H = 4 s?

Reveal the explanation

Ī”f_ss = āˆ’0.03 / (1/0.04 + 1) = āˆ’0.03 / 26 ā‰ˆ āˆ’0.00115 p.u., so the frequency settles at 60 Ɨ (1 āˆ’ 0.00115) ā‰ˆ 59.93 Hz. RoCoF = āˆ’0.03 Ɨ 60 / (2 Ɨ 4) = āˆ’0.225 Hz/s.

References

  • Kundur, P., Power System Stability and Control (1994), ch. 11.
  • ENTSO-E (2020), Frequency Stability Evaluation Criteria for the Synchronous Zone of Continental Europe.
  • Machowski, J., Lubosny, Z., Bialek, J. W. and Bumby, J. R., Power System Dynamics, 3rd ed. (2020), ch. 9.