Chapter 19 — Governors, turbines, and stabilizers

Chapter 18 closed one of the machine's two loops. The exciter controls the field, and through it the terminal voltage. This chapter closes the other one. The governor controls the valve, and through it the mechanical torque on the shaft — and therefore the speed, and therefore the system frequency.

Then it adds a third element that belongs to neither loop and improves both: the power system stabilizer, a small supplementary signal injected into the exciter that damps the rotor oscillations a high-gain AVR would otherwise aggravate.

Learning objectives

By the end of this chapter you should be able to:

• Explain why generators are governed with droop rather than to a fixed frequency, and compute the steady-state frequency and power sharing that droop produces.
• Trace the TGOV1 block diagram and write the difference equation each stage becomes.
• Explain what a load limiter, an acceleration limiter and a low-value select do in a modern governor, and why the PID needs anti-windup.
• Explain why a high-gain AVR can reduce damping torque, and how a stabilizer fixes it without disturbing the voltage set-point.
• Say why every stabilizer starts with a washout, and what phase compensation is compensating for.
• Predict whether a governor will visibly respond to a disturbance, given the stiffness of the system it is connected to.

19.1 Speed control and droop

The swing equation from Chapter 15 says the rotor accelerates whenever the mechanical torque exceeds the electrical torque:

$$2H\frac{d\omega }{dt}=T_m-T_e-D(\omega -1).$$

Electrical torque is set by the load; the only controllable term is $T_m$, and that is the governor's business. If load exceeds generation the rotor decelerates, frequency falls, and the governor must open the valve.

The obvious control law is isochronous: integrate the speed error and drive it to exactly zero, holding frequency at precisely 60 Hz. It works beautifully for one machine alone. Put two isochronous machines on the same grid and it fails immediately — both are trying to set the same frequency, and any tiny mismatch in their measurements sends one to full load and the other to zero as each fights to impose its own set-point. There is no stable division of load between them.

The universal fix is droop: deliberately make the governor accept a frequency error proportional to its output. With a permanent droop $R$,

$$\mathrm{\Delta }P=-\frac{\mathrm{\Delta }\omega }{R},$$

so each machine picks up load in proportion to $1/R$, and all of them settle at a common, slightly-off frequency that acts as the shared signal telling every governor how much to contribute. A droop of $R=0.05$ — 5 %, the near-universal setting — means a 5 % frequency drop moves the unit from no load to full load.

This is why grid frequency droops under load and is later restored by a slower, system-wide process (automatic generation control) that shifts the governors' references rather than fighting them. Droop is the fast, autonomous, proportional layer; frequency restoration is the slow, coordinated, integral layer on top.

19.2 The TGOV1

TGOV1 is the minimal credible steam turbine-governor: a droop, a valve lag, a reheat lead-lag, and an optional damping term. It is defined in IEEE PES-TR1, and it is the model to reach for when you want the effect of governing without committing to a specific plant.

The TGOV1. Speed deviation $\mathrm{\Delta }\omega =\omega -1$ is divided by the droop $R$ and subtracted from the load reference $P_{ref}$; the governor-valve lag $1/(1+s{T}_1)$ has a non-windup position limit $[V_{min},V_{max}]$; the turbine lead-lag $(1+s{T}_2)/(1+s{T}_3)$ represents reheat; a damping term $D_t\mathrm{\Delta }\omega$ is subtracted; and the resulting mechanical power is converted to torque by dividing by speed.

Stage by stage, with the update each becomes:

Droop. $P_{in}=P_{ref}-\mathrm{\Delta }\omega /R$ with $\mathrm{\Delta }\omega =\omega -1$. Pure algebra, no state. Setting $R=0$ disables droop entirely, which makes the governor isochronous.

Governor / valve lag $1/(1+s{T}_1)$ with a non-windup limit.

$$v_n=v_{n-1}+\frac{\mathrm{\Delta }t}{T_1}(P_{in,n}-v_{n-1}),v_n\leftarrow \mathrm{clamp}(v_n,V_{min},V_{max}).$$

$T_1=0.5$ s lumps the speed relay and the servo. The clamp is on the state, so a valve driven to fully open comes off the stop the moment the demand falls back — the same non-windup discipline as the AVR's regulator limit. The default $V_{min}=0$, $V_{max}=1$ are per-unit valve stroke.

Turbine lead-lag $(1+s{T}_2)/(1+s{T}_3)$. This is where the steam physics lives. With $T_3=7$ s and $T_2=2.1$ s, the DC gain is 1 but the immediate response is only $T_2/T_3=0.3$ of the final value. That is a real and important property of a reheat steam turbine: opening the valve gives you the high-pressure stage's contribution at once, and the reheater's contribution seconds later. A governor that looks sluggish on a steam unit is usually sluggish for this reason and not because of the valve.

Damping $D_t\mathrm{\Delta }\omega$, subtracted from the turbine output. A small speed-proportional term representing frictional and windage effects; zero by default.

Torque conversion. The machine's swing equation takes torque, but the turbine produces power, and $P=T\omega$. So the model divides: $T_m=P_{mech}/\omega$. Near synchronous speed the distinction is a fraction of a percent, but at start-up $\omega$ can be near zero and the division would blow up — so on the very first step the model uses $\omega =1$ rather than the not-yet-computed machine speed. That is a small bumpless-start measure of the same family as the AVR's seeding, and it exists because of the cross-domain delay of §17.4: on the first step the governor has not yet received a speed measurement from the machine.

19.3 The GGOV1

TGOV1 is a caricature. GGOV1 is what a modern governor actually looks like, and it is worth studying not because you will hand-tune all thirty of its parameters but because its structure recurs in every real plant controller.

Four ideas beyond TGOV1:

Selectable droop feedback. TGOV1 always droops on speed. GGOV1's Rselect chooses what the droop signal is measured from: electrical power (the default, and the most common in practice), valve stroke, governor output, or nothing at all — the last giving isochronous control, which is exactly what you want for a single machine running an island.

A PID, with anti-windup that back-calculates. The speed error drives $K_{pgov}$, $K_{igov}$ and an optional filtered derivative. Because a governor spends real time against its limits, the integrator needs anti-windup, and GGOV1 uses the stronger form: after integrating, the model recomputes what the controller output would be and, if it exceeds a limit, solves backwards for the integrator value that would place the output exactly on the limit. Compare this with the DC1A's simpler output clamp — both prevent windup, but back-calculation leaves the integrator holding a value consistent with the limited output, so recovery is immediate and exact.

Low-value select. Three controllers compete for the valve: the speed governor, an acceleration limiter (which caps $d\omega /dt$, important during starting and islanding), and a load limiter (which caps fuel flow, the practical stand-in for an exhaust-temperature limit on a gas turbine). The actuator follows whichever demands least:

$$f_{sr}=min(f_{sr,\text{gov}},f_{sr,\text{accel}},f_{sr,\text{load}}).$$

This is the standard way to compose limiters in prime-mover control: whoever is most conservative wins, automatically and without mode-switching logic.

A rate-limited actuator. The valve cannot slam. Ropen and Rclose bound how fast the stroke may change per second, and only then is the position clamped to $[V_{min},V_{max}]$. Rate limits are a genuine nonlinearity — a model with only position limits will predict a faster response than the plant can deliver.

Two smaller features complete the picture: Flag makes fuel flow proportional to speed (right for a mechanically-driven fuel system), and Teng is a transport delay for engine combustion, realized as the integer-sample ring buffer of §17.2.

19.4 Why a stabilizer is needed

Here is the uncomfortable fact that motivates the third controller. A high-gain, fast-acting AVR — the very thing Chapter 18 argued for — reduces the damping of rotor oscillations.

The mechanism, briefly. Following a disturbance the rotor swings against the system, and the machine's electrical torque can be resolved into two components: a synchronizing component in phase with the angle deviation $\mathrm{\Delta }\delta$, which pulls the rotor back, and a damping component in phase with the speed deviation $\mathrm{\Delta }\omega$, which bleeds the oscillation's energy away. Stability needs both.

A fast AVR is superb at supplying synchronizing torque — that is exactly the field forcing you watched in Lab 18.7. But the exciter and field circuit together introduce phase lag, and by the time the AVR's response reaches the air gap it can be far enough behind the rotor's motion to appear as negative damping at the natural oscillation frequency of about 0.2 to 2 Hz. The result is a machine that holds voltage well and hunts, with a slowly growing swing that the AVR itself is feeding.

The remedy is not to detune the AVR — that would give up the synchronizing torque you need. It is to add a supplementary signal to the AVR's summing junction that is deliberately shaped to produce torque in phase with speed. That signal is $V_S$, and the block that produces it is a power system stabilizer.

The stabilizer's place in the loop. It taps the machine's speed, shapes it, and injects $V_S$ into the exciter's summing junction — the same junction as $V_{REF}$ and $V_C$. Because a washout makes $V_S$ zero in steady state, adding a stabilizer never shifts the voltage set-point.

19.5 PSS1A and STAB1

Both stabilizers in the library take the machine's speed and return $V_S$. Their internals are the same three ideas in a different arrangement.

A washout, always first in spirit. The stage $s{T}_5/(1+s{T}_6)$ is a high-pass filter: it passes the oscillation and blocks the steady value. Its steady-state output is exactly zero, and that property is not a nicety — it is what makes the stabilizer safe. A stabilizer that contributed a constant term would bias the voltage regulator, shifting the terminal voltage away from its set-point and making the AVR and the PSS argue about the operating point. With a washout, the PSS is silent until the machine swings. It is also why a stabilizer needs no initialization recipe of its own: seed the washout's state to the current speed and $V_S$ starts at zero, whatever the operating speed happens to be.

The time constant is long — 10 s by default — because the washout must pass oscillations at 0.2 Hz while rejecting the slow frequency drift that governors are handling.

Phase compensation. Two lead-lag stages $(1+s{T}_1)/(1+s{T}_2)$ and $(1+s{T}_3)/(1+s{T}_4)$. Their job is to advance the phase of the stabilizing signal by however much the exciter and field circuit will subsequently retard it, so that the torque finally produced at the air gap is in phase with $\mathrm{\Delta }\omega$. The defaults, with numerators several times their denominators, give substantial lead across the rotor-oscillation band. Tuning a PSS is, in practice, mostly the business of measuring that lag and choosing these four numbers to cancel it.

Gain and limit. $K_S$ sets how much stabilizing torque you get. Too little and the oscillation persists; too much and the PSS destabilizes some other mode, or amplifies noise. The output clamp — $\pm 0.1$ pu for PSS1A, $\pm 0.05$ pu for STAB1 — bounds the authority a stabilizer can ever exert over the voltage regulator, which is a safety limit rather than a tuning parameter.

The two models differ mainly in packaging. PSS1A is the IEEE 421.5 model and adds an optional second-order input filter $1/(1+s{A}_1+s^2{A}_2)$, used to notch out a shaft torsional mode that would otherwise be amplified into the excitation system. STAB1 folds the gain and washout into a single $KsT/(1+sT)$ block and omits the filter — the classic compact form.

Where the gain sits

In NumaSim's PSS1A the stabilizer gain $K_S$ is applied after the washout, before the lead-lag stages. Because every stage between the input and the gain is linear, this is mathematically identical to applying it at the input — a scalar commutes with a linear filter — so $V_S$ is unaffected by the choice. The one visible consequence is the vw monitoring signal, which reports the washout output before the gain is applied. If you are comparing vw against a hand calculation, do not multiply by $K_S$.

19.6 Initializing the governor

The governor's initialization is the same idea as the exciter's in §18.6, and mercifully simpler.

At the operating point the machine runs at synchronous speed, so $\mathrm{\Delta }\omega =0$ and every droop term vanishes. Each block's input equals its output, and the closed-form steady state collapses to:

$$P_{mech,0}=T_m,P_{ref,0}=P_{mech,0}.$$

The load reference is the scheduled mechanical power. The pairing works exactly as the exciter's does — the governor's Tm output is traced through the resolved connection graph to the machine whose Tm input it drives — and the same warnings apply if the trace fails.

GGOV1 needs one extra number because its droop feeds back from electrical power rather than speed: with Rselect = 1 the steady state requires $P_{ref,0}=R{P}_{e,0}$, so the recipe records the operating-point electrical power as well.

Two caveats worth knowing. On a multi-mass shaft, the machine has several torque inputs $T_{m1}\dots T_{m4}$ and the recipe initializes against the total $T_m$, warning rather than guessing how to split it between stages. And if the governor's set-point is wrong at $t=0$, the symptom is distinctive: the machine's power drifts from the load-flow value with no disturbance applied, and the frequency drifts with it. Compare that with a mis-initialized exciter, where the voltage drifts. The two failure modes are easy to tell apart on a plot.

19.7 Lab: a governor that barely moves, and why that is correct

Open the generator controls disturbance in simulator →

(Full description: generator controls disturbance.)

The same circuit as Lab 18.7, now read from the governor's side. The machine carries a TGOV1 with $R=5\mathrm{\%}$ and $T_1=0.5$ s alongside its DC1A, and takes the same 100 ms three-phase fault at $t=4$ s.

Plot Pmech and the valve position through the disturbance. Across the whole event, Pmech moves by 0.002 pu and the valve by 0.006 — which is to say, essentially nothing, while the AVR next to it swung its output by 1.1 pu.

That is not a broken governor. It is the single most important thing this lab teaches, and the reasoning is worth doing carefully:

1. A droop governor responds only to speed error.
2. The machine is connected to a 500 MVA system source, five times its own rating.
3. A source that stiff holds the bus frequency essentially constant, whatever the 100 MVA machine does.
4. So the speed error never really develops.
5. So the governor has nothing to act on.

The disturbance is the AVR's to handle, and it handles it. Voltage collapses and the exciter responds; frequency does not move and the governor does not. Which controller responds to a disturbance is decided by the disturbance, not by the controller — and a fault on a stiff bus is a voltage event, not a frequency event.

Now make the governor matter. Weaken the source — raise its impedance until its short-circuit capacity is comparable with the machine's rating — or island the machine with its load entirely. Re-run and watch Pmech and the speed. Now the machine's own inertia sets the frequency, a load change moves it, and the droop governor picks up load in proportion to $1/R$ exactly as §19.1 predicts. Two further experiments: change R from 0.05 to 0.02 and confirm the steady-state frequency deviation shrinks by the same factor; and shorten T3 toward T2 to watch the reheat lag disappear and the mechanical power arrive all at once.

19.8 Summary

• Generators are governed with droop, not to a fixed frequency, because two isochronous machines on one grid cannot share load stably. Droop makes frequency the shared signal that divides load in proportion to $1/R$.
• TGOV1 is droop, a non-windup valve lag, a reheat lead-lag whose immediate response is only $T_2/T_3$ of its final value, an optional damping term, and a division by $\omega$ to convert power to torque.
• GGOV1 adds selectable droop feedback, a PID with back-calculated anti-windup, a low-value select against acceleration and load limiters, and a rate-limited actuator.
• A high-gain AVR supplies synchronizing torque but can supply negative damping at rotor-oscillation frequencies. A stabilizer fixes this by injecting $V_S$ shaped to produce torque in phase with $\mathrm{\Delta }\omega$.
• Every stabilizer begins with a washout, so $V_S$ is exactly zero in steady state and adding one never shifts the voltage set-point. The lead-lag stages compensate for the phase lag of the exciter and field circuit.
• Governor initialization back-calculates $P_{ref,0}=P_{mech,0}=T_m$ from the load-flow operating point. A mis-initialized governor makes power drift with no disturbance; a mis-initialized exciter makes voltage drift.
• Whether a governor visibly responds depends on the stiffness of the system. On a bus held by a much larger source, a fault is a voltage event and the governor correctly does almost nothing.

19.9 Problems

Problem 19.1. Two generators share an island. Unit A is 200 MVA with $R=5\mathrm{\%}$; unit B is 100 MVA with $R=4\mathrm{\%}$. Both droops are on the machines' own bases. The island load increases by 15 MW. How is the increase shared, and what is the steady-state frequency deviation?

Solution 19.1

Work in MW per per-unit frequency. Each unit's contribution is $\mathrm{\Delta }P=-\mathrm{\Delta }\omega S_{rated}/R$:

• A: $200/0.05=4000$ MW/pu
• B: $100/0.04=2500$ MW/pu

Total stiffness $6500$ MW/pu. For $\mathrm{\Delta }{P}_{total}=15$ MW, $\mathrm{\Delta }\omega =-15/6500=-0.002308$ pu, i.e. about $-0.138$ Hz on a 60 Hz system.

Sharing: A picks up $4000\times 0.002308=9.23$ MW, B picks up $2500\times 0.002308=5.77$ MW. They sum to 15 MW as required.

Note that B, though half A's size, takes more than half of A's share, because its tighter droop makes it stiffer per MVA.

Problem 19.2. A TGOV1 has $T_2=2.1$ s and $T_3=7.0$ s. The valve steps open by 0.1 pu. What is the mechanical power immediately after the step, and what is it after several minutes? Sketch the shape in between and say what physical feature it represents.

Solution 19.2

A lead-lag $(1+s{T}_2)/(1+s{T}_3)$ has an immediate (high-frequency) gain of $T_2/T_3=2.1/7.0=0.3$ and a DC gain of 1.

Immediately after the step: $0.3\times 0.1=0.03$ pu. After settling: $1.0\times 0.1=0.1$ pu. In between, an exponential rise from 0.03 to 0.1 with time constant $T_3=7$ s.

This is the reheat characteristic of a steam turbine. Opening the valve immediately admits more steam to the high-pressure stage, which delivers about 30 % of the unit's output; the rest of the steam must pass through the reheater before it reaches the intermediate- and low-pressure stages, and the reheater's thermal storage is what sets the 7-second lag. It is the reason steam units are poor at fast frequency response compared with hydro or gas.

Problem 19.3. Explain the difference between the DC1A's output clamp and GGOV1's back-calculated anti-windup. Construct a case where the difference is visible.

Solution 19.3

The DC1A clamps the regulator's integrator state to the limit. The state is therefore never outside the limit, and the output leaves the limit as soon as the input reverses. GGOV1 integrates first, then computes the controller output $K_{p}e+\int +D$, and if that exceeds a limit it solves backwards for the integrator value that puts the output exactly on the limit.

The difference shows up when the controller has a proportional path in parallel with the integrator, which GGOV1 does and the DC1A's simple lag does not. Suppose the output is pinned at $V_{max}$ by a large positive error, so the proportional term is contributing a lot. If you clamped only the integrator, the integrator would sit at $V_{max}$ while the proportional term adds on top — and when the error reverses, the proportional term drops instantly but the integrator is still at the ceiling, so the output stays pinned longer than it should. Back-calculation instead sets the integrator to $V_{max}-K_{p}e-D$, which is below the ceiling by exactly the proportional contribution, so the sum comes off the limit the instant the error turns.

With a pure lag and no parallel path, as in the DC1A, the two schemes coincide.

Problem 19.4. A colleague proposes to eliminate the washout from a stabilizer, arguing that its DC gain of zero "throws away information." Explain what would go wrong, both in steady state and at initialization.

Solution 19.4

In steady state, the stabilizer would output $K_S\omega \approx K_S$ — a large constant — into the exciter's summing junction. The AVR would treat that as a permanent offset on the voltage error and settle at a terminal voltage well away from $V_{REF}$. The stabilizer, meant to be a small dynamic correction, would have become the dominant term in voltage regulation.

At initialization it is worse. §18.6's back-calculation computes $V_{REF}$ assuming the auxiliary inputs are zero at the operating point. A non-zero constant $V_S$ breaks that assumption, so the seeded state would no longer be an equilibrium and the run would begin with a startup transient — the exact failure the whole initialization machinery exists to prevent.

The washout is not throwing information away; it is discarding precisely the component of the signal (the DC value) that the stabilizer has no business acting on. The stabilizer's job is to damp oscillations, and oscillations are what remains after the DC value is removed.

Problem 19.5. In Lab 19.7 the governor barely moves, while in a different study of the same machine it responds strongly. What changed? Give two distinct system conditions that would make the governor the dominant responder, and one disturbance type that would engage the governor but not the AVR.

Solution 19.5

What changed is the stiffness of the system the machine is connected to, not anything about the governor. A droop governor acts on speed error, and speed error only develops if the machine's own power imbalance is large enough to move the frequency.

Two conditions that make the governor dominant: (a) islanding the machine with its load, so its own inertia alone sets the frequency; (b) weakening the external source until its short-circuit capacity is comparable with the machine's rating, so the machine participates meaningfully in frequency regulation.

A disturbance that engages the governor but not the AVR: a load step in an island — say, switching in a large motor. Frequency dips, the governor opens the valve, and the terminal voltage barely moves if the load is largely real power near the machine. The complement is the lab's own case, a bolted fault on a stiff bus: pure voltage event, no frequency event.

Problem 19.6. You attach a stabilizer to a machine, run without a power flow, and see the terminal voltage drift for the first two seconds. Is the stabilizer the cause? Explain how you would isolate it.

Solution 19.6

Almost certainly not. A stabilizer's washout is seeded so that $V_S=0$ at the first step regardless of the operating speed, so it needs no load-flow recipe of its own and contributes nothing at $t=0$.

The likely cause is that the exciter was not initialized — that is what running without a power flow does. Its $V_{REF}$ falls back to the default of 1.0 instead of the back-calculated value, and the voltage drifts to wherever that reference puts it (§18.6).

To isolate it: disconnect the stabilizer's $V_S$ wire and re-run. If the drift is unchanged, it was never the stabilizer. Then apply a power flow and re-run with the exciter properly initialized; the trace should be flat until the first event. Reconnect the stabilizer last and confirm the flat start survives — which it will, precisely because of the washout.

19.10 References

• IEEE PES-TR1, Dynamic Models for Turbine-Governors in Power System Studies, IEEE Power & Energy Society — the source of the TGOV1 and GGOV1 models.
• IEEE Std 421.5, IEEE Recommended Practice for Excitation System Models for Power System Stability Studies, Clause 9 — the PSS1A and the stabilizer family.
• P. Kundur, Power System Stability and Control, McGraw-Hill — Chapter 9 for prime movers and speed governing, Chapter 12 for small-signal stability, the decomposition into synchronizing and damping torque, and PSS tuning.
• IEEE Committee Report, "Dynamic models for steam and hydro turbines in power system studies," IEEE Transactions on Power Apparatus and Systems, 1973 — the origin of the reheat turbine representation.
• E. V. Larsen and D. A. Swann, "Applying power system stabilizers, Parts I-III," IEEE Transactions on Power Apparatus and Systems, vol. PAS-100, no. 6, 1981 — the standard treatment of PSS phase compensation and tuning.

---

Previous: Chapter 18 — Excitation systems and AVRs · Next: Module 6 — System studies.