Steady-State and Transient Stability

Chapter 27 finished with a single nonlinear differential equation and two ways of attacking it: linearise about the operating point, or integrate step by step. The choice is not a matter of taste. It is dictated by how far the rotor angle moves, and that in turn is dictated by the size of the disturbance.

A power system is disturbed continuously. Loads switch on and off, tap changers step, wind output fluctuates, and the aggregate of all this is a permanent low-level jitter in every rotor angle on the network. None of it moves \(\delta\) more than a degree or two. For disturbances of that size the sine in \(P_e = P_{max}\sin\delta\) may be replaced by its tangent at the operating point without appreciable error, the equation becomes linear, and the whole apparatus of linear system theory applies. The question it answers is: does this operating point hold? That is steady-state stability.

A power system is also occasionally struck hard. A three-phase fault removes a transmission path entirely for a hundred milliseconds or more; the rotor angle swings through sixty or eighty degrees. No linearisation survives that, because the very feature that decides the outcome — the fact that \(P_e\) turns over and starts falling past \(90^\circ\) — is the part a linear model throws away. The question now is: does the machine survive this particular event? That is transient stability.

Between the two sits a third classification with a longer history than clear boundaries. Dynamic stability, now more often called small-signal stability, is also a small-disturbance problem, but it retains the machine's control equipment — exciter, voltage regulator, governor — in the model rather than treating the field flux as frozen. The distinction matters because those controllers act on the same one-to-two-second time scale as the electromechanical swing, and their effect on damping can be either sign. Section 28-5 pursues this.

Take the swing equation with damping from Chapter 27 and perturb the operating point. Write \(\delta = \delta_0 + \Delta\delta\), where \(\delta_0\) satisfies \(P_m = P_{max}\sin\delta_0\), and expand the sine:

The constant term cancels against \(P_m\), leaving only the increment, and the swing equation becomes an ordinary linear second-order equation in \(\Delta\delta\):

Put it in standard form by dividing through by the coefficient of the second derivative:

The characteristic equation \(s^2 + 2\zeta\omega_n s + \omega_n^2 = 0\) has roots \(s = -\zeta\omega_n \pm j\omega_n\sqrt{1-\zeta^2}\), and the whole of steady-state stability is contained in where those roots sit. Four cases exhaust the possibilities, and each is a physically distinct machine behaviour.

Imagine loading a generator gradually. The governor admits more steam, \(P_m\) rises in small increments, and after each increment the rotor settles at a slightly larger angle where \(P_{max}\sin\delta\) again equals \(P_m\). Provided the field current is held fixed, \(|E|\) does not change and the machine simply climbs its own power-angle curve.

The climb cannot continue indefinitely. At \(\delta = 90^\circ\) the curve reaches its crest, \(P_s = P_{max}\cos 90^\circ = 0\), and the spring holding the rotor to the network has gone slack. Any further increment of mechanical power cannot be met electrically, the rotor accelerates, \(\delta\) advances past \(90^\circ\), \(P_e\) falls, and the machine runs away. The crest of the power-angle curve is therefore not merely the largest power the machine can transmit; it is the boundary of stable operation.

Two refinements matter in practice. First, no system is operated at its steady-state limit. A margin is specified, usually as

with \(P_0\) the scheduled output. A margin of \(30\%\) to \(50\%\) is typical, and it exists precisely because the limit above assumes an infinitesimal disturbance, whereas real disturbances are finite.

Second, for a salient-pole machine the reluctance term of Chapter 26 shifts the crest. With \(P_e = \frac{|E||V|}{X_d}\sin\delta + \frac{|V|^2}{2}\left(\frac{1}{X_q}-\frac{1}{X_d}\right)\sin2\delta\), the synchronising coefficient is

Setting \(P_s=0\) gives a critical angle below \(90^\circ\) — typically \(70^\circ\) to \(80^\circ\) — but the maximum power itself is higher than the cylindrical-rotor value, because the reluctance term adds to the fundamental over most of the range. Saliency helps the limit and hurts the angle at which it occurs.

The formula \(P_{SSSL}=|E||V|/X\) is unambiguous only once \(E\) and \(X\) are pinned down, and the correct pair depends on what is assumed to remain constant while the load is increased. This is the single point at which steady-state stability calculations go wrong most often.

If the machine has no voltage regulator, the field current is fixed by the operator and the flux is set by it. The voltage that stays constant is the one behind the synchronous reactance, \(E_q\) behind \(X_d\), and the calculation proceeds in two steps: find \(E_q\) from the present operating condition, then compute the limit with the same \(E_q\).

If the machine has a fast automatic voltage regulator, the field current is not fixed at all. The regulator raises it as the load angle grows, holding the terminal voltage — or, in the idealised limit, the voltage behind the transient reactance. Since \(X_d'\) is three or four times smaller than \(X_d\), the effective transfer reactance collapses and the limit rises dramatically. This is why the classical model of Chapter 26 uses \(E'\) behind \(X_d'\).

The idealised extreme is a regulator so fast and so powerful that it holds the terminal voltage \(V_t\) rigidly. The machine then behaves as a constant-voltage source at its own terminals, the transfer reactance is only that of the external network, and the limit rises further still.

The idealisation of a regulator that instantaneously holds a voltage constant is convenient for computing a limit and misleading about everything else. A real excitation system is a feedback loop with gain, time constants and its own dynamics, and it interacts with the electromechanical oscillation of Section 28-2 in a way that took the industry two decades to understand.

The mechanism is this. When the rotor swings forward, the terminal voltage dips, because more current is drawn through the machine and line reactance. The regulator sees the dip and boosts the field. The boost raises \(|E'|\) and hence \(P_e\), which opposes the forward swing — so far so good, and this is the synchronising action that raises the steady-state limit. But the field circuit has a time constant \(T_{d0}'\) of several seconds, so the boost arrives late. By the time the extra flux appears, the rotor may already be swinging back, and the extra power then assists the backward swing instead of opposing it. A torque in phase with the angle is synchronising; a torque in phase with the speed is damping; and a delayed synchronising action delivers a component that is in antiphase with the speed. That component is negative damping.

The remedy is not to slow the regulator down, because its speed is what buys transient stability. The remedy is to add a second input to the exciter that is deliberately proportional to speed deviation, so that its contribution to torque is in phase with speed and therefore purely damping. That auxiliary controller is the power system stabiliser, and it is developed in Chapter 33 alongside the rest of the excitation and frequency control apparatus. Its signal is usually derived from shaft speed, from electrical power, or from a combination that rejects the slow components a governor should handle.

The lesson generalises beyond excitation. Any controller acting on the swing time scale — a governor, an HVDC converter control, a static var compensator — modifies \(D\) as well as \(P_s\), and the sign of its contribution depends on its phase lag at the oscillation frequency. Designing for damping is therefore an exercise in phase compensation at around one hertz, which is exactly the problem the frequency-response methods of a control course are built for.

Now abandon the linearisation entirely. A three-phase fault appears on a line; the breakers clear it in a few cycles; the question is whether the machine holds. Chapter 26 supplied the three power-angle curves and Chapter 27 supplied the equation of motion. Assembling them gives the sequence of events, and the sequence is always the same.

Before the fault, the machine sits at \(\delta_0 = \sin^{-1}(P_m/P_{max,1})\) on the tall pre-fault curve, in equilibrium.

At fault inception, the transfer reactance jumps and \(P_{max}\) collapses to \(P_{max,2}\). The rotor cannot move instantaneously, so \(\delta\) is unchanged and the operating point drops vertically onto the low curve. Mechanical input now exceeds electrical output by a large margin, and the rotor accelerates.

During the fault, \(\delta\) increases along the low curve. The accelerating power stays positive throughout, because \(P_{max,2}\) is generally smaller than \(P_m\) — for a fault at the sending bus it is zero and the acceleration is uniform, the parabolic case of Chapter 27.

At clearing, the faulted circuit is isolated and the operating point jumps vertically onto the post-fault curve at whatever angle \(\delta_c\) the rotor has reached. If it lands above the \(P_m\) line, the accelerating power is now negative and the rotor begins to decelerate — but it is still turning faster than synchronous speed, so \(\delta\) continues to increase while the excess kinetic energy is given back.

After clearing, one of two things happens. Either the rotor's excess speed reaches zero at some angle \(\delta_{max}\) short of the post-fault unstable equilibrium \(\delta_u = 180^\circ - \sin^{-1}(P_m/P_{max,3})\), in which case the machine swings back and is stable; or \(\delta\) passes \(\delta_u\) with speed still to spare, \(P_e\) falls below \(P_m\) again, and the machine pulls out of step.

The kinetic-energy language used above is not decoration. Multiply the swing equation by \(d\delta/dt\) and integrate once, and the result is an energy statement: the accelerating power integrated over angle is the kinetic energy stored in the rotor's excess speed. The rotor gains energy while \(P_m>P_e\) and gives it back while \(P_e>P_m\), and stability is the question of whether there is enough angle left over which to give it back. That observation is the equal-area criterion in embryo, and it is the reason damping must be excluded from the model — a dissipative term would break the energy bookkeeping the criterion depends on.

Six quantities decide the outcome, and each acts on a different part of the picture. Knowing which lever moves which term is what separates an engineer from a formula.

Clearing time. The dominant factor by a wide margin. A longer fault means more angle accumulated at constant acceleration and more kinetic energy stored, and both grow as \(t_c^2\). Halving the clearing time roughly quarters the energy that must subsequently be dissipated.

Fault type. Chapter 24 ordered the faults by severity of current; stability orders them by severity of power loss, and the order is the same. A three-phase fault shorts the positive-sequence network directly and kills the transfer completely at the fault point. An unsymmetrical fault appears in the positive-sequence network as a shunt impedance — \(Z_2\) for a line-to-line fault, \(Z_2+Z_0\) for a single line-to-ground, \(Z_2\parallel Z_0\) for a double line-to-ground — so some power continues to flow throughout. The single line-to-ground fault is the mildest and by far the commonest; the three-phase fault is the harshest and rarest, and it is the one systems are designed against.

Fault location. A fault at the sending bus gives \(P_{max,2}=0\) and is the worst case. The further along the line the fault, the larger the residual transfer, and the less severe the disturbance.

Inertia. \(H\) appears in the denominator of the acceleration. More inertia means a slower swing and more time for the breakers, but the benefit goes as \(\sqrt H\) in clearing time, not linearly. Inverter-based generation with no rotating mass, discussed in Chapter 39, removes inertia from the system entirely, and that is one of the central problems of the modern grid.

Transfer reactance. It sets all three values of \(P_{max}\). Anything that reduces \(X\) — a second circuit, a higher voltage, series compensation — raises the post-fault curve and increases the angle over which the rotor can decelerate.

Pre-fault loading. A heavily loaded machine starts at a large \(\delta_0\), which leaves less angular room before \(\delta_u\), and has a small \(P_s\), which makes it a weak spring. Both effects push in the same direction, and this is the quantitative reason that transmission limits computed for stability fall as loading rises.

Every practical measure for improving stability can be located in the swing equation, and organising them that way makes a long list memorable. There are only three places to intervene: reduce the accelerating power during the fault, increase the decelerating power after it, or increase the inertia that resists both.

Reduce the duration and severity of the disturbance. Fast circuit breakers — two-cycle rather than eight — and fast protective relaying are the most cost-effective stability investment available, and they are the subject of Chapters 35 and 36. Single-pole switching, in which only the faulted phase is opened for a single line-to-ground fault, keeps two phases transmitting throughout. High-speed auto-reclosing restores the healthy circuit after the arc has deionised, so that the post-fault curve becomes the pre-fault curve; since about \(80\%\) of overhead-line faults are transient, this succeeds most of the time.

Raise the post-fault power-angle curve. Anything that lowers the transfer reactance does this. Double-circuit lines and bundled conductors (Chapter 8) reduce \(X\) directly. Series capacitors cancel part of the line reactance and are the most direct method of all, with the caveat of subsynchronous resonance noted in Chapter 38. Higher transmission voltage reduces per-unit reactance for a given power. Machines with lower \(X_d'\) cost more but transmit more.

Act on the machine during the swing. A fast, high-ceiling excitation system boosts \(|E'|\) within a few hundred milliseconds of the fault, lifting the post-fault curve when it is most needed — the transient benefit that Section 28-5 warned must be paid for with a stabiliser. Fast valving momentarily closes the turbine intercept valves to cut \(P_m\), attacking the accelerating power at its source. A braking resistor switched in at the generator bus creates artificial electrical load, doing the same job from the other side. Both act within one swing.

Control the network electronically. Static var compensators hold mid-line voltage during and after the fault, keeping the effective \(P_{max}\) high. HVDC links and FACTS controllers can modulate their power in response to measured speed deviation, injecting damping directly. Chapter 38 develops these.

The threefold division into steady-state, dynamic and transient stability is the classical one and it is what examinations ask for. It is organised by the size of the disturbance, and it takes for granted that the thing at risk is synchronism. Both assumptions have been overtaken by events.

Since the 1990s, following a series of collapses that were plainly not losses of synchronism, the industry has classified stability by which variable fails to hold rather than by how hard the system was pushed. Three families are recognised.

Rotor angle stability is everything this Part has discussed: the ability of synchronous machines to remain in step. It subdivides into small-disturbance (the steady-state and dynamic problems of this chapter) and large-disturbance or transient stability.

Voltage stability is the ability of the system to hold acceptable voltages at all buses after a disturbance. Its failure mode is a progressive, irreversible voltage decline driven by reactive-power shortage and by load restoration through tap changers and thermostats — a mechanism with no rotor angle in it at all. It is the subject of Chapter 34.

Frequency stability is the ability of the system to hold frequency after a severe imbalance between generation and load, typically following islanding. It is governed by the aggregate inertia and by the response of governors and load shedding, and it belongs to Chapter 33.

The classical names map onto the new scheme cleanly enough: steady-state and dynamic stability are both small-disturbance rotor angle stability, and transient stability keeps its name. What the new scheme adds is the recognition that a system can be perfectly synchronised and still collapse.

Take \(f = 50\) Hz and work in per unit on a common base. State clearly, in every steady-state stability problem, which voltage you are holding constant and why.