Load Frequency Control and Automatic Generation Control

Chapters 31 and 32 decided which units should run and at what output, minimising fuel cost over a day. Both calculations assumed the demand was known. It never is. Load changes second by second — a rolling mill starts, a city's air conditioners cycle, a cloud crosses a solar farm — and no schedule computed an hour ahead can track it. Something must close the gap continuously, and it must do so without any central computer being in the loop for the first few seconds.

The mechanism is already present in the physics. Electrical energy cannot be stored in the network, so at every instant the mechanical power delivered by all the turbines must equal the electrical power consumed by all the loads plus the losses. If it does not, the difference has only one place to go: the kinetic energy of the rotating masses. An excess of load draws energy out of the rotors, they slow down, and the frequency falls. An excess of generation pushes energy in, and the frequency rises.

This makes frequency an exceptionally convenient measurement. Because all synchronous machines in an interconnected system are locked to a common electrical speed (Chapter 26), frequency is a single global number. Measure it anywhere — at a generator terminal, at a distribution substation, at a domestic socket — and you learn the same thing about the balance of the entire interconnection. No communication network is needed to distribute it; the grid distributes it itself, at the speed of the synchronizing torques.

Reactive power behaves in exactly the opposite way, and it is worth fixing the contrast now because Chapter 34 is built on it. Reactive power does not travel far: the voltage at a bus is set mostly by the reactive balance in its own neighbourhood, so voltage is a local quantity and there are as many voltage problems as there are buses. Frequency is one problem for the whole system. The two control tasks are therefore organised completely differently, and they are almost perfectly decoupled — a fact that lets us treat active power and frequency in this chapter and reactive power and voltage in the next.

The control problem then decomposes naturally by time scale. In the first ten seconds the primary control — the speed governor on each turbine — acts alone, arrests the frequency excursion and settles the system at a new frequency slightly away from nominal. Over the next few minutes the secondary control, which is what "automatic generation control" (AGC) means in practice, adjusts the governor reference settings to return frequency to nominal and to restore the scheduled interchange between neighbouring systems. Over tens of minutes the tertiary layer re-dispatches units economically, which is where the machinery of Chapter 31 re-enters. This chapter builds the first two layers from their component models.

Start from the swing equation of Chapter 27, written for a single machine on its own rating as a base:

For stability studies we followed the rotor angle. For frequency control we want the speed itself. Since \(d\delta/dt\) is the deviation of rotor electrical speed from synchronous, and speed in electrical radians is \(2\pi f\), differentiating once fewer times gives an equation directly in the frequency deviation \(\Delta f = f - f_0\):

Here \(\Delta P_m\) is the change in mechanical power input and \(\Delta P_e\) the change in electrical power output, both in per unit on the chosen base; \(H\) is the inertia constant in seconds on the same base. For a whole control area, \(H\) is the MVA-weighted sum of the individual machine inertia constants referred to a common base, exactly as in Chapter 4 — the area behaves like one large machine as far as the common frequency is concerned.

The electrical load must now be modelled with a little more care than "a fixed number of megawatts". Part of any composite load is frequency-sensitive: induction motors driving pumps and fans slow down when the frequency falls, and the mechanical power they demand falls with them, roughly as the cube of speed for a fan. Resistive load — heating, incandescent lighting — is indifferent to frequency. Linearising the composite characteristic about the operating point,

The coefficient \(D\) is the load damping constant. It is positive: a fall in frequency reduces the load, which helps. Its usual magnitude is such that a \(1\%\) change in frequency produces a \(1\%\) to \(2\%\) change in the motor part of the load. Throughout this chapter \(D\) is quoted in MW/Hz, or in per-unit megawatts per hertz when a per-unit base is stated; the two differ only by the power base.

Substituting the load model into the swing equation and taking Laplace transforms with zero initial conditions gives the generator–load block, sometimes called the power-system block:

The result is a simple first-order lag, and both of its parameters have a plain meaning. The gain \(K_p = 1/D\) says that a permanent \(1\) MW surplus, left entirely to the load's own frequency sensitivity, raises frequency until the load has grown by that same \(1\) MW. The time constant \(T_p = 2H/(f_0 D)\) is the ratio of stored kinetic energy to damping; with \(H \approx 5\) s and a damping of about \(1\%\) of rating per hertz at \(50\) Hz it comes out near \(20\) s. That is a slow block, and it is the reason a governor is needed at all: left to itself the system drifts a long way before the load's own damping arrests it.

The generator–load block responds to \(\Delta P_m\), the mechanical power at the turbine shaft. That power does not change the instant a valve moves. Steam admitted to a turbine must fill the steam chest and the inlet piping before it does work on the blades, and this storage is well represented by a single lag.

where \(\Delta P_v\) is the change in valve or gate position expressed in power units. A reheat unit has a second, much slower storage in the reheater, and a fraction \(K_r\) of the power appearing promptly in the high-pressure cylinder:

A hydro turbine is different in kind. Accelerating the water column in the penstock takes time, and closing the gate momentarily raises the pressure and hence the power before the flow falls — a non-minimum-phase behaviour represented by a right-half-plane zero, \(G_T(s) = (1-sT_w)/(1+0.5\,sT_w)\) with a water starting time \(T_w\) of about a second. Hydro governors must therefore be detuned with transient droop compensation, and hydro plant is slower on primary response than its reputation suggests.

Now the governor. Its job is to sense speed and move the valve. In the classical Watt arrangement, flyweights driven from the turbine shaft rise as speed rises and lower a sleeve; the sleeve works a pilot valve that admits oil to a hydraulic servomotor, and the servomotor moves the steam valve. Modern units replace the flyweights with an electronic speed transducer and the linkage with a controller, but the transfer function is the same. The oil servomotor integrates flow into position, and with its own position feedback it behaves as a lag:

The signal \(\Delta P_g\) that drives it is the commanded power change, and constructing it is the interesting part.

Imagine a governor that simply held speed constant: any deviation, however small, drives the valve until the deviation vanishes. This is called isochronous control, and on a single machine feeding an isolated load it works perfectly. Put two such machines in parallel and it fails immediately. Both are trying to enforce their own idea of exactly \(50.000\) Hz. Whichever setting is a hair higher drives its unit to full load while the other backs off to zero, and the load split is decided by measurement error rather than by intention. Two integrators cannot share one controlled variable.

The remedy is to give each governor a proportional characteristic instead: let the commanded power depend on the frequency error rather than its integral. The unit's steady-state characteristic then becomes a straight line of negative slope on a plot of frequency against output power,

\(\Delta P_{\text{ref}}\) is the setting of the speed changer — the operator's or AGC's command, which slides the whole characteristic up and down without changing its slope. The constant \(R\) is the speed regulation or droop, with units of hertz per megawatt. Its reciprocal \(1/R\) is the megawatts of primary response the unit contributes per hertz of frequency depression.

In practice \(R\) is quoted as a percentage, and the definition must be read carefully. A droop of \(R\%\) means that a change of \(R\%\) in frequency drives the unit from no load to full load. Converting to engineering units,

Droop solves the load-sharing problem completely. Because every unit sees the same \(\Delta f\), and each contributes \(-\Delta f/R_i\), the increment is divided among the units in proportion to \(1/R_i\). If all units are set to the same percentage droop, that proportion is the ratio of their ratings — each machine picks up the same fraction of its own capacity, which is exactly the fair and safe outcome. Choosing unequal percentage droops deliberately biases the sharing, which is sometimes done to keep a particular unit on base load.

Assemble the three blocks. The speed changer setting and the droop feedback form the command \(\Delta P_g\); the governor lag turns that into valve position; the turbine lag turns valve position into mechanical power; the mechanical power meets the disturbance \(\Delta P_L\) at a summing point; and the generator–load block converts the net imbalance into a frequency deviation, which is fed back through \(1/R\).

Reading the diagram gives the closed-loop response to a load disturbance. With the supplementary loop open and \(\Delta P_{\text{ref}} = 0\), the forward path from the summing junction to \(\Delta f\) is \(G_g G_T G_p\), and the feedback path is \(1/R\):

For a step load increase \(\Delta P_L(s) = \Delta P_L/s\), the final-value theorem gives the steady-state frequency deviation. Every lag contributes unity at \(s=0\), so the whole dynamic structure collapses:

The same result follows in two lines from the steady-state power balance, and it is worth seeing it that way because the physics is then transparent. In the new steady state the frequency has settled, so the rotors are neither gaining nor losing kinetic energy and \(\Delta P_m = \Delta P_e\). The governors have supplied \(\Delta P_m = -\Delta f/R\); the load has fallen by its own frequency sensitivity, giving relief \(-D\,\Delta f\); together these must cover the disturbance \(\Delta P_L\).

When an area contains \(n\) units on governor control, the individual regulations combine in parallel because each unit sees the same \(\Delta f\):

A unit that is at its maximum output, or that has been taken off governor control to run at a fixed setpoint, contributes nothing to this sum. This is why system operators track responsive spinning reserve rather than reserve alone: capacity that is spinning but blocked from responding does not appear in \(\beta\), and the frequency dip after a loss of generation is set by \(\beta\), not by the reserve on paper.

The steady-state answer is the one examinations ask for, but the transient shape is what an operator actually sees on the frequency trace, and it is instructive to extract it from the same block diagram. The governor and turbine time constants are of the order of a few tenths of a second, while \(T_p\) is tens of seconds. Setting \(T_g = T_T = 0\) as a first approximation leaves a single lag in the loop:

so that

Two features deserve comment. First, the governor loop has shortened the effective time constant by the factor \(1+K_p/R\), which is typically \(30\) to \(60\): a system that would have drifted for half a minute on load damping alone settles in a second or so. Second, the response in this approximation is a pure exponential with no overshoot, which is not what real traces show. Restoring \(T_g\) and \(T_T\) makes the characteristic polynomial cubic, and the two fast lags introduce enough phase lag for the frequency to undershoot and recover — a dip of perhaps twice the final offset, over five to ten seconds. Reheat units, with their extra \(5\)–\(10\) s lag, are slower still, and their contribution to the initial dip is limited to the high-pressure fraction \(K_r\).

Primary control leaves the system running at \(f_0 - \Delta P_L/\beta\). Left alone, successive load changes through the day would accumulate frequency offsets in whichever direction the load happened to move, and synchronous clocks would drift without bound. The remedy is to move the speed-changer settings — to slide the droop characteristics of Figure 33-1 bodily upward until the intersection with the new load falls back on \(f_0\).

How much should they move? From the steady-state balance with a non-zero reference change,

The reference must be advanced by the whole of the load increase. That is unsurprising: at nominal frequency the droop term contributes nothing and the load relief is zero, so the entire extra demand must come from the reference setting. What is less obvious is how to generate \(\Delta P_{\text{ref}}\) automatically, since \(\Delta P_L\) is never measured directly.

The answer is to let the frequency error itself generate the command, through an integrator. Take

The argument for why this works is short and completely general. Suppose the system reaches a steady state. Then every signal is constant, and in particular \(\Delta P_{\text{ref}}\) is constant. But \(\Delta P_{\text{ref}}\) is the output of an integrator whose input is \(\Delta f\); an integrator with a constant non-zero input produces a ramp, not a constant. The only constant input consistent with a constant output is zero. Hence \(\Delta f = 0\) in any steady state that exists at all. The integrator does not need to know \(\Delta P_L\); it simply keeps pushing until the error that drives it disappears, and at that moment \(\Delta P_{\text{ref}}\) is sitting at exactly \(\Delta P_L\).

The negative sign is essential and easy to get wrong. A load increase makes \(\Delta f\) negative; the integral of a negative quantity is negative; multiplying by \(-K_I\) with \(K_I>0\) makes \(\Delta P_{\text{ref}}\) positive, raising generation. Get the sign backwards and the loop drives the frequency away from nominal without limit.

Adding the integrator raises the order of the loop by one and therefore raises a stability question. With the fast lags again neglected, the characteristic equation of the single area with both loops closed is obtained by substituting the supplementary command back into the block diagram:

The factor \(s\) in the numerator is the analytical statement that a step disturbance produces zero steady-state frequency error. The two parameters show the trade-off: raising \(K_I\) raises \(\omega_n\), which speeds up the restoration, but \(\zeta = (1+K_p/R)/(2T_p\omega_n)\) falls as \(\omega_n\) rises, so too large a gain makes the frequency hunt. In practice \(K_I\) is set well below the critically damped value, because the supplementary loop must be slower than the primary loop it is superimposed on, and slower still than the turbine's permitted rate of loading. Ten to thirty seconds for the bulk of the restoration is a realistic target; AGC cycles typically issue new setpoints every two to four seconds.

Real systems are not isolated. Neighbouring utilities interconnect because it lets them share reserve, exchange energy economically and ride through each other's disturbances. Interconnection also creates a new controlled quantity: the power flowing on the tie lines, which is contracted in advance and must be held to schedule.

Model two areas, each represented as in Section 33-5, joined by a tie line of reactance \(X_{12}\). From Chapter 26, the power transferred from area 1 to area 2 is

Load frequency control deals in small deviations about an operating point, so linearise. Holding the voltage magnitudes constant — legitimate because voltage control is a separate and faster loop, Chapter 34 — and writing \(\delta_i = \delta_i^0 + \Delta\delta_i\),

\(T_{12}\) is the synchronizing power coefficient of the tie, in MW per radian — the same quantity that governed small-signal stability in Chapter 28. A strong tie (small \(X_{12}\)) has a large \(T_{12}\).

The block diagram needs \(\Delta P_{12}\) in terms of the frequency deviations it already carries, and angle is the integral of frequency. Since \(\delta\) is in electrical radians and \(f\) in hertz, \(d\delta/dt = 2\pi\,\Delta f\), so

The tie line is an integrator driven by the difference of the two area frequencies. That single fact explains most of the qualitative behaviour of interconnected systems. As long as the two areas are at different frequencies, the tie-line flow keeps changing; flow settles only when \(\Delta f_1 = \Delta f_2\), which is another way of saying that the interconnection is synchronous and there is only one steady-state frequency.

Consider what happens with primary control only — no supplementary loops — when a load step \(\Delta P_{L1}\) occurs in area 1. In the steady state both areas share one frequency, \(\Delta f_1=\Delta f_2=\Delta f\). Write the power balance of each area, taking \(\Delta P_{12}\) positive for flow out of area 1:

The second equation gives \(\Delta P_{12} = \beta_2\,\Delta f\) directly. Substituting into the first,

Governor support across the tie is exactly what one wants for the first few seconds and exactly what one does not want after a minute. The supplementary loop must therefore be given a target that reflects both objectives: return the frequency to nominal and return the tie flow to schedule, with the correction made by the area that caused the imbalance.

Before leaving the tie line, one dynamic consequence is worth recording. Neglect governors and damping for a moment and let the two areas swing against each other through the tie. Combining the two swing equations with \(\Delta\ddot\delta = 2\pi\,d\Delta f/dt\) and eliminating gives, for \(\theta = \Delta\delta_1-\Delta\delta_2\),

These inter-area modes lie between about \(0.1\) and \(0.8\) Hz — much slower than the local machine modes of Chapter 28, because the effective inertias are those of whole areas. They are lightly damped, and a supplementary controller tuned too aggressively will excite them, which is the practical reason AGC is deliberately slow.

Each area needs one scalar error signal to drive its integrator. Two candidates are available to it locally: its own frequency deviation \(\Delta f_i\), and the deviation of its net tie-line interchange from schedule, \(\Delta P_{\text{tie},i}\). Neither alone is adequate. Frequency alone would make both areas respond to any disturbance anywhere, so the innocent area would keep generating for its neighbour indefinitely. Tie flow alone would leave the frequency permanently offset if both areas happened to be in balance with each other but jointly short of generation.

The classical solution, due to Nathan Cohn, is to use a weighted sum called the area control error:

The equilibrium argument of Section 33-7 applies to each integrator separately. In any steady state, \(\mathrm{ACE}_1 = 0\) and \(\mathrm{ACE}_2 = 0\). Writing these out with \(\Delta P_{\text{tie},1} = \Delta P_{12}\) and \(\Delta P_{\text{tie},2} = -\Delta P_{12}\), and remembering that the interconnection has one frequency,

Both objectives are met for any positive bias settings. This is a strong and slightly surprising result: the final state does not depend on the value of \(B\) at all. What does depend on \(B\) is the path taken to get there, and in particular whether the innocent area's controller is disturbed on the way.

To see this, evaluate the two ACEs immediately after primary control has settled but before the supplementary loops have acted, using the results of Section 33-8 for a disturbance \(\Delta P_{L1}\) in area 1. Choose the biases equal to the area frequency response characteristics, \(B_i = \beta_i\):

If the bias is set too low, the disturbed area's ACE understates the imbalance and the neighbour's ACE becomes non-zero, so the neighbour begins correcting a disturbance it did not cause and the two controllers work against each other for a while. If the bias is set too high, the ACE overstates the imbalance and the area overshoots, drawing frequency past nominal. Neither error is unstable — the final state is still \(\Delta f = 0\), \(\Delta P_{12}=0\) — but both waste control effort and worsen the transient. Real biases are reviewed annually against measured system response, and grid codes commonly require \(B\) to be at least a stated fraction of the measured \(\beta\).

One further practical refinement matters. The ACE as written contains a time-error and inadvertent-interchange component: because \(\Delta f\) is only driven to zero, not its integral, synchronous clocks accumulate a small offset. Systems therefore add a slow correction term that runs the frequency a few millihertz off nominal for a few hours to pay back the accumulated time error, and a similar term to pay back accumulated inadvertent energy interchange on the tie.

Finally, the supplementary command must be split among the units of the area. This is where Chapter 31 returns. The economic dispatch calculation, run every few minutes, produces not only the optimal outputs but their sensitivity to a change in total demand — the participation factors \(\alpha_i\), with \(\sum_i \alpha_i = 1\), given by

Two allocations therefore operate in sequence after every disturbance. In the first seconds the increment is divided by droop, in proportion to \(1/R_i\) — a purely technical split that ignores cost entirely. Over the following minutes AGC re-divides it in proportion to \(\alpha_i\) — an economic split that ignores droop entirely. The final state is the economic one; the transient state is the physical one.

Take \(f_0 = 50\) Hz unless stated otherwise. Convert every droop to MW/Hz before doing anything else — most errors in this topic are unit errors, not conceptual ones.