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A supply deficit pushes the frequency of an operating synchronous grid downward. Complete loss of voltage in a blackout is not the same as a slowly declining frequency. This educational model assumes continued energization and represents one area by a single frequency.

Separate inertia from corrective response

Inertia affects the rate of change; primary response changes power to counter the deficit. Inertia alone does not remove it. Background is available in the grid-inertia introduction and the 2020 Inertia and the Power Grid overview. All numbers below come from our stated assumptions, not real grid-event measurements.

Equations and units

Let f₀=50 Hz and x=(f−f₀)/f₀. Additional generation p and demand increase/supply loss d are per-unit values on the same power base:

2H\frac{dx}{dt}=p-d-Dx,\qquad T_g\frac{dp}{dt}=-p-\frac{x}{R},\qquad f=f_0(1+x)

H is inertia constant [s], D frequency-sensitive load response, R droop and Tg governor-response time constant [s]. Use D=1, R=0.05 and Tg=0.5 s. R=0.05 means a steady −0.05 pu frequency deviation requests +1 pu generation. Headroom, saturation and deadband are omitted.

Without primary response, replace −x/R in the second equation by zero. D remains, so this is not a system without all response. Start with x=0 and p=0; step d to 0.05 at 2 s.

Reproduce the calculation

Save the script and run python3 engineering_labs.py. Execution was checked with Python 3.12.3, NumPy 1.26.4 and Matplotlib 3.6.3. Fourth-order Runge–Kutta integrates 0–30 s with a 0.01 s step. Halving the step to 0.005 s changes results by at most about 1.8×10⁻⁹ Hz. That checks discretization, not physical model validity.

The CSV includes time and four frequency curves. The summary records minimum, endpoint and initial rate.

Read the results

H [s] Primary response Initial RoCoF [Hz/s] Minimum over 0–30 s [Hz] Frequency at 30 s [Hz]
2 No −0.625 47.502 47.502
5 No −0.250 47.652 47.652
2 Yes −0.625 49.786 49.881
5 Yes −0.250 49.846 49.881
Diagram 1 · Use the button to switch views
Synthetic grid frequency following a five-percent deficit; seconds and hertz.

Immediately after the disturbance, p and x are still zero, giving initial RoCoF −f₀d/(2H). Greater H slows the initial drop. Primary response recovers from the nadir but does not restore exactly 50 Hz.

Verify the steady offset analytically

Set derivatives to zero with primary response:

x_{\infty}=-\frac{d}{D+1/R}=-\frac{0.05}{21},\qquad f_{\infty}\simeq49.88095\ \mathrm{Hz}

H does not appear in this steady-state equation. Distinguish changing inertia from secondary control restoring nominal frequency. Without primary response, the model tends to 47.5 Hz; the H=5 case is still approaching that value at 30 s.

Limits and next experiment

Voltage, spatial network differences, loss of synchronism, protection, load shedding and storage limits are omitted. Curves reaching the 47 Hz range are formal linear-model outputs, not a prediction that real grids continue operating there without protection.

First fix H and change only R or Tg. Then cap generation response to distinguish faster response from available headroom. The power learning guide connects this model to transmission and storage foundations.

What to read next

Study real-system inertia and response beyond this simplified model.Grid Inertia and Frequency Regulation Primer — When the Spinning Mass Disappears, Who Protects the Frequency?Continue the seriesSmart Grid Primer — Distribution Networks That Support Two-Way Power Flow With an Information LayerExplore another aspect of this fieldPower Battery Primer — From Cell Chemistry to Grid-Scale Storage and the BMS
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