Stability formulas

$\delta$ in electrical radians, $t$ in seconds, $f$ the system frequency in Hz, $P_m$ and $P_e$ in per unit on the same base as $H$; $P_a$ is the accelerating power. Equivalent form: $\frac{2H}{{\omega }_s}\ddot{\delta }=P_m-P_e$ with ${\omega }_s=2\pi f$.

Same equation with $\delta$ in electrical degrees (since $\pi$ rad $=180^{\circ }$). Do not mix the two forms.

Unit: seconds (MJ per MVA). In the second form $J$ is in kg m${}^2$, ${\omega }_{sm}$ the synchronous mechanical speed in rad/s and $S_{\text{machine}}$ in VA. Stored energy is $H{S}_{\text{machine}}$ megajoules when $S$ is in MVA.

$p$ is the number of poles. The swing equation above uses the electrical angle.

The stored energy in MJ is fixed; only the MVA it is divided by changes.

Applies to machines that swing coherently (same $\delta$ and speed), each $H_i$ on its own rating $S_i$. The stored energies in MJ add.

$E$ and $V$ are the voltage magnitudes at the two ends, $X$ the total series reactance between them; per unit values give $P_e$ in per unit.

Of the two angles with this sine, the stable operating point is the one below $90^{\circ }$.

Positive for ${\delta }_0<90^{\circ }$ (stable), zero at $90^{\circ }$, negative beyond. Units: per unit power per electrical radian.

Undamped natural frequency, from linearising the swing equation about ${\delta }_0$ with $\Delta \delta$ in radians: $\frac{H}{\pi f}\Delta \ddot{\delta }=-P_s\Delta \delta$.

Reached when load is increased slowly: beyond $\delta =90^{\circ }$ the coefficient $P_s$ becomes negative and the machine falls out of step.

${\delta }_m$ is the maximum swing angle. Valid for the classical model (constant $P_m$, no damping).

$P_m$ jumps from $P_{m0}$ to $P_{m1}$ with the network unchanged, ${\delta }_0={\sin }^{-1}(P_{m0}/P_{\max })$; angles in radians where they multiply $P_{m1}$. Stable only if ${\delta }_m\le \pi -{\delta }_1$ with ${\delta }_1={\sin }^{-1}(P_{m1}/P_{\max })$.

With the same $E$ and $V$, each curve's peak is $EV/X$, so the during-fault and post-fault peaks are $r_1{P}_{\max }$ and $r_2{P}_{\max }$, where $X$ is the transfer reactance between the machine EMF and the infinite bus in each state.

$r_1=0$, $r_2=1$, so ${\delta }_{\max }=\pi -{\delta }_0$; ${\delta }_0$ in radians. Typical case: three-phase fault at the machine terminals cleared without a line being lost.

Valid only when $P_e=0$ during the fault ($r_1=0$) and the rotor starts at synchronous speed, so acceleration is constant; angles in radians, $H$ in seconds on the same base as $P_m$. For $r_1\ne 0$ the swing equation has to be integrated step by step.