= Electric power balance during a demand step
<Electric power balance during a demand step> tracks where the additional <energy> comes from when an <electrical load> suddenly requests more <power>. At an explicitly chosen system boundary, <conservation of energy> gives
$$
\frac{dE_s}{dt}=P_{\rm in}-P_{\rm load}-P_{\rm loss},
$$
where $E_s$ includes the <energy storage> inside that boundary. A negative right-hand side means that stored <energy> is decreasing. An external <electric power system>, an <electric battery> and a rotating <electrical generator> supply different physical buffers; they should not all be counted as independent incoming supplies if some are already included in $E_s$.
For an original lumped rotating-machine example, take <moment of inertia> $J=10\,\mathrm{kg\,m^2}$ and initial <angular velocity> $\omega_0=100\,\mathrm{rad\,s^{-1}}$. Suppose mechanical input initially remains unchanged while electrical output increases by $2\,\mathrm{kW}$, with losses already accounted for. The <rotational kinetic energy> then obeys
$$
\tfrac12J\omega(t)^2=\tfrac12J\omega_0^2-\Delta P\,t,
\qquad \omega(t)=\sqrt{\omega_0^2-\frac{2\Delta P}{J}t}.
$$
After $0.5\,\mathrm s$, $\omega\approx98.995\,\mathrm{rad\,s^{-1}}$: the extra $1\,\mathrm{kJ}$ came from rotation. This stipulated model assumes the extra delivered <power> can be maintained during that interval; it does not predict the load's own response to falling <electric potential difference> or <frequency>.
For a rotating <electrical generator> with $p$ magnetic pole pairs, the generated electrical <frequency> is $f=p\omega/(2\pi)$. A speed change therefore links the <energy> balance to a measurable waveform change. The DOE derivation relates generated <frequency> to pole count and rotational speed.<DOE Fundamentals Handbook Electrical Science>[Volume 3, Module 10, AC Generator Theory, printed p.6, equation 10-1]{ref} This rotating-machine argument does not describe every electronically controlled supply.
A proposed demand-step test records the imposed <electrical load>, source and storage <power>, terminal <electric potential difference>, <frequency> and timing of control action. The important distinction is between an initial buffer and a sustainable new operating point. If available continuous <power> remains too small, increasing stored <energy> alone does not remove that limit; <electrical load shedding> can instead reduce the requested output.
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