Frequency fluctuations
in power systems
Advisor:
László P. Csernai
Student:
Szabolcs Horvát
University of Bergen
June, 2007
Abstract
A basic difficulty with using and producing electric energy is that—
apart from short periods of time and small quantities—electric energy cannot be stored. It must be produced at the same time as it
is used. To maintain the balance between power production and
consumption, the power plants that produce the energy must be
continuously regulated.
Power system control is based on the phenomenon that in case of
an imbalance, the AC frequency of the network changes. During
normal operation of the electric grid, the system frequency kept
near a set value, but it is allowed to fluctuate within certain bound.
In this work we survey the causes of frequency fluctuations, and
explore the possibility of gathering information about the state of the
system (and perhaps increasing control performance) by measuring
these fluctuations.
We reach the conclusion that frequency fluctuations are mainly
caused by random demand changes in the system, but their nature
is also influenced by the state of the generators that are participating
in power control.
1
Acknowledgements
I am indebted to my supervisor, Professor László P. Csernai, for his
professional guidance, but also for his encouragement and support
throughout the past year. Without his help I would not have been
able to complete this work.
I would like to thank Hermod Nitter, my companion in this project,
for useful discussions on the topic of power system control.
The Quota Scholarship provided by the Norwegian Government
made it possible for me to study at the University of Bergen.
But my greatest gratitude goes to my mother and father for their understanding, and the emotional support they provided, even while I
was far from home.
June, 2007
Szabolcs Horvát
2
Contents
1 Notations and basics
1.1 Alternating current (AC) . . . . . .
1.1.1 Reactance . . . . . . . . . .
1.1.2 The phasor representation
1.1.3 Impedance . . . . . . . . . .
1.1.4 AC power . . . . . . . . . . .
1.2 Generators . . . . . . . . . . . . . .
1.2.1 A simple generator . . . . .
1.2.2 The synchronous machine
1.2.3 Coupling of generators . . .
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4
4
5
5
6
7
8
8
9
12
2 Power system control
13
2.1 Practical governor models . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.1 The flyball governor . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.2 Speed droop governor . . . . . . . . . . . . . . . . . . . . . . 19
3 Data analysis and simulations
3.1 Data sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Description of the model . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
21
22
29
4 Summary
37
Bibliography
38
3
B
θ
A
Figure 1: Rotating wire loop in magnetic field.
1 Notations and basics
1.1 Alternating current (AC)
The simplest possible electric generator is a rectangular wire loop that is rotating
in a homogeneous magnetic field. Since the sides of the loop are moving, the
free charges inside experience a Lorentz force, and an electric current is induced.
Let us calculate the induced electromotive force! To make the calculations
simpler, we shall analyse this system from the rotating reference frame attached
to the wire loop.
According to Faraday’s law of induction, the integral of the electric field E
around a loop C is proportional to the rate of change of magnetic flux through
the surface S enclosed by the loop:
d
E · dl = −
dt
C
I
Z
S
B · d S.
If the area enclosed by the loop is A, and the angle between the surface normal
of the wire loop and the magneticRfield is denoted by ϑ, the electromotive force
induced in the loop will be V = C E · d l = − ddt AB cos ϑ = AB ϑ̇ sin ϑ. The dot
indicates the time derivative. If the loop is rotating with a steady angular velocity
4
ω, the induced voltage will vary harmonically:
V = AB ω sin ωt .
1.1.1 Reactance
Let us look at the relation between voltage and current in the case of simple
circuit elements!
The voltage measured between the poles of a resistor is related to the current
by Ohm’s law: V = R · I .
The voltage on a capacitor is proportional to the charge Q carried by it,
V = Q/C , hence the current is proportional to the time derivative of the voltage:
I = Q̇ = C V̇ . In the case of sinusoidally alternating voltage, the current varies
sinusoidally too, but it lags behind by a quarter cycle:
I =C
d
V0
V0 cos ωt = −C ωV0 sin ωt =
sin ωt .
dt
XC
The quantity X C = −1/ωC is called capacitive reactance.
When the current in an inductor (coil) changes, an electromotive force that
is proportional to the rate of this change is induced. The voltage measured
between the poles of the inductor is V = L I˙. For a harmonically varying current
I = I 0 sin ωt , the voltage is
V =L
d
I 0 sin ωt = LωI 0 cos ωt = I 0 X L cos ωt ,
dt
so the phase difference between V and I is the opposite as in the case of a
capacitor. The quantity X L = ωL is called inductive reactance.
Reactance is similar to resistance in that it is the ratio of the peak values of
alternating voltage and current (it is measured in ohms), but it is used when
there is a phase difference of ±π/2 between the two. Conventionally X is positive
when the current is lagging behind the voltage.
1.1.2 The phasor representation
It is advantageous to treat harmonically varying quantities as the projection of a
rotating two dimensional vector. This way the peak amplitude is equal to the
length of the vector, while the phase difference between two quantities may be
visualised as the angle between two vectors (fig. 2).
When this vector is represented as a complex number, it is referred to as a
phasor. To distinguish a complex phasor from the instantaneous value of the real
quantity it represents, we shall write a tilde above it: Ṽ = V0 e i ϕ . The projection
is usually taken as the real part of phasors: V = ℜ(Ṽ ) = V0 cos ωt .
5
R
Ž
IC
Ž
V
Ž
IR
L
C
Ž
IL
Figure 2: The relative phases of currents passing through a resistor, an inductor
and a capacitor connected in parallel.
Since projection is a linear operation, phasors may be added or multiplied
by scalars just like any real-valued quantity, but care should be taken when
multiplying them because ℜ( Ã B̃ ) 6= AB .
1.1.3 Impedance
An advantage of the phasor notation is that the relation between alternating
voltage and current may be expressed
to Ohm’s law. In the
¡
¢in a form analogous
d
i ωt
i ωt
˜
= i ωCV0C e , so ṼC / I˜C = i X C . For an
case of a capacitor IC = C d t V0C e
inductor, ṼL / I˜L = i X L .
The ratio of complex voltage and current is called impedance and is usually
denoted by Z = Ṽ / I˜. It can be easily proven that the resulting impedance of more
complicated circuits (e.g. series or parallel connection) can be calculated in
exactly the same way as resistances. The real part of impedance is the resistance,
while the imaginary part is the reactance: Z = R + i X .
The phase difference between the voltage measured on an element of impedance
Z and the
/R). The ratio of the peak values is
p current is ϕ = arg Z = arctan(X
p
∗
∗
∗
2
˜
˜
V0 /I 0 = (Ṽ Ṽ )/( I I ) = Z Z = |Z | = R + X 2 . (Z ∗ denotes the complex conjugate of Z .)
It should be noted that this simple mathematical notation is only usable for
systems in a steady state (i.e. for purely harmonically varying quantities). Even
though the angular frequency, ω, is not explicitly written out in this description,
the impedance of a non purely resistive element does depend on ω.
6
1.1.4 AC power
The power consumed by a circuit element is at any instant equal to the product
of voltage and current: p = V · I .
In the resistive case P = I 2 R = I 02 R sin2 ωt = 12 I 02 R(1 − cos 2ωt ). In practical
applications one usually deals with time scales much longer than 1/ω so let us
calculate the time average of dissipated power:
1
P avg =
T2 − T1
Z
T2
T1
2
R I 2 d t = R I rms
,
where T2 − T1 À 1/ω and
s
I rms =
1
T2 − T1
Z
T2
I2 dt
T1
is called the root mean square (RMS) value
The RMS value of a
p of the current.
1 2
sinusoidally varying current is I rms = I 0 / 2, and P avg = 2 I 0 R. For AC voltage
and current, usually the RMS values are specified instead of the peak values V0
and I 0 .
In the purely inductive or capacitive case P = V · I = I 02 X sin ωt · cos ωt =
1 2
2 I 0 X sin 2ωt . Note the instantaneous value of the power is oscillating between
positive and negative values, but the average is P avg = 0. This means that the
circuit element in question consumes and releases energy periodically, but no
power is dissipated. Energy is temporarily stored as the energy of an inductor’s
magnetic field or capacitor’s electric field.
In the general case of Z = R + i X , the phase difference between voltage and
current is ϕ = arctan(X /R) and
P = V · I = V0 I 0 sin(ωt + ϕ) · sin ωt
V0 I 0
V0 I 0
=
(1 − cos 2ωt ) · cos ϕ +
sin 2ωt · sin ϕ .
|2
{z
} |2
{z
}
p
q
Both the amplitude and mean value of the first term, p, is P 0 = Vrms I rms cos ϕ,
while the second term, q, has amplitude Q 0 = Vrms I rms sin ϕ but mean 0. p
behaves like the power dissipated in a pure resistor. It represents a non-zero net
energy flow, so it is called real power or active power. q is similar to the power
on a reactance: it represents an oscillating energy flow with mean 0. It is called
reactive power.
∗
It is easy to check that the complex quantity S = Ṽrms · I˜rms
is S = P 0 + iQ 0
(fig. 3). S is called the apparent power. Thus P 0 = |S| cos ϕ, where we call cos ϕ
the power factor.Power factors are usually described as “lagging” (when the
7
φ
Reactive power, Q₀
p
Ap
S
er,
w
o
nt p
are
Real power, P₀
Figure 3: The power triangle
current lags behind the voltage) or “leading” (in the opposite case) to indicate
the sign of ϕ.
Conventionally, a circuit element with Q 0 > 0 is said to consume reactive
power while one with Q 0 < is said to produce it. But it is important to note that
no net power is produced or consumed in this case. There is an asymmetry in
the interpretation of the magnitude of real and reactive power, P 0 and Q 0 . While
P 0 is usually used as the mean power, i.e. the net energy flow, Q 0 is just the
amplitude of oscillation. In spite of the terminology, the energy associated with
reactive power is never consumed while the system is in a steady state, but it is
conserved.
To distinguish between them, different units are used for real power (watt),
reactive power (VAR, volt-ampère reactive) and apparent power (VA).
1.2 Generators
1.2.1 A simple generator
Now let us take another look at the simple generator of figure 1, and try to get an
intuitive understanding of its behaviour when a load is attached to it.
When a resistor is connected to the wire, a current I starts to circulate that
creates the wire loop’s own magnetic field. The wire loop gains a magnetic
moment µ = I A, which interacts with the external magnetic field B. The torque
exerted on the wire loop is
T=
d
d
B·µ =
B AI cos ϑ = −B AI sin ϑ.
dϑ
dϑ
8
A
C
N
B‘
B
S
C
A‘
Figure 4: Arrangement of the armature windings of a synchronous generator.
The three electrically separated sections (labelled A, B and C) are positioned
120◦ apart, so that the voltages induced in them will also have phase differences
of a third of a cycle.
If we neglect the inductance of the loop, then I = V /R = AB ω sin ωt and
T = −ω(AB sin ωt )2
1
R
This torque has opposite sign than ω (it tries to slow the rotation down), so
an external force is needed to keep the rotation speed steady. The external
energy source that rotates the wire loop, and ultimately provides the energy
that is dissipated in the resistor, is called the prime mover. In the case of a real
generator this may be for example a steam or water turbine. The rotating wire
loop corresponds to the armature of a real generator and its own magnetic field
is called the armature reaction.
The mechanical power drawn by the rotating wire is −T ω = R1 (AB ω sin ωt )2 .
As we expect, this is exactly equal to V 2 /R, the power dissipated in the resistor.
The more power is dissipated, the greater the external torque must be to keep the
wire rotating. If the torque is insufficient, the rotation speed decreases because
the power will be drawn from the kinetic energy of the rotating part.
1.2.2 The synchronous machine
The most common type of generator used in utility power systems is the synchronous generator. Let us examine the structure of this generator-type!
9
Figure 5: The contribution of the three phases to the armature reaction sums up
into a field of constant magnitude.
In most generator designs, the rotating part, the rotor, contains the magnet
that provides the excitation field, while the armature winding is part of the
stationary part, the stator. Except in the case of very small generators it is not
practical to use constant magnets, as their magnetic field is relatively weak
(compared to their size), so electromagnets are used. These have the additional
advantage that the intensity of their magnetic field is adjustable. The rotor
magnet usually draws its current from an external source, the exciter, which may
be a smaller DC generator mounted on the same shaft with the main machine.
Alternatively, the excitation current may also be taken directly from the AC grid
(and rectified), but a generator employing this approach is unable to start up
without an already functioning AC source.
The armature of a synchronous machine is made up of three separate conductors. These are arranged in such a way that the voltages induced in them
change sinusoidally and have phase differences of 120◦ . This is how three phase
electric power is produced. The three phases are conventionally labelled A, B
and C (see figure 4).
The armature reaction (and therefore the reaction torque) of the simple
single phase generator discussed in the previous section pulsates as the machine
is rotating. One advantage of a three phase system is that the magnitude of the
armature reaction is constant.
It can be demonstrated that n two-dimensional vector quantities, spaced
360◦ /n angles apart, sum up to a steadily rotating vector if their magnitudes
oscillate sinusoidally and are shifted 360◦ /n relative to each other. The specific
case of n = 3 is illustrated on figure 5.
Let us describe the vectors in a reference frame that is rotating with the same
angular frequency ω as their magnitudes. With a suitable choice of phase, the
10
vectors’ components are
µ
¶
³
³
³
k´
k´
k´
vn = sin ωt + 2π · cos ωt + 2π , sin ωt + 2π
,
n
n
n
k = 1..n.
Summing the vectors up we get
n
X
¶
³
³
n µ1
X
k´ 1 1
k´
sin 2ωt + 4π , − cos 2ωt + 4π
vn =
n 2 2
n
k=1
k=1 2
¶
´
³
³
³ n´ 1 X
n µ
k´
k
+
sin 2ωt + 4π , − cos 2ωt + 4π
.
= 0,
2
2 k=1
n
n
|
{z
}
0
The second term is just a set of equally spaced vectors of the same lengths, whose
sum is zero. The sum does not depend on ω, hence in the static reference frame
it is a rotating vector of constant magnitude.
This result can be applied to the armature reaction of the synchronous generator, which is called the stator field. This type of generator is called synchronous
for the very reason that the excitation field and stator field are rotating in accordance, in other words: they maintain a fixed position relative to each other. As a
result, the torque is also constant.
In situations when it is impractical to rotate the generator with the desired
network frequency (usually 50 Hz or 60 Hz), a rotor with four or more magnetic
poles is used.
The induced voltage follows the magnetic flux with a phase lag of 90◦ . Correspondingly, when the load is purely resistive, the stator field is perpendicular to
the rotor field. Otherwise it deviates from this position by an angle that is equal
B.
rotor field
C.
A.
stator field
Figure 6: Relative position of the stator field and rotor field. A: Power factor of
unity. B: Lagging power factor. C: Leading power factor.
11
to the phase lag ϕ of the current (fig. 6). We should note that a purely resistive
load is not a realistic situation inasmuch as the generator itself has a non-zero
inductance. And since most loads are inductive too, a lagging power factor is the
most common.
The torque acting on a magnetic dipole is the cross product of the dipole’s
magnetic moment and the external field. Therefore the magnitude of the torque
on the rotor is determined by the perpendicular component of the stator field.
The stator field is a function of the armature current, which in turn depends on
the load connected to the generator. As appliances are turned on and off, the load
changes, so the torque must be adjusted continually to maintain equilibrium
and keep the rotation speed constant. This is usually done by an automatic
control system, which is called the governor (fig. 8).
The induced electromotive force is determined by the rotation speed (which
is kept constant) and the magnitude of the rotor field. Assuming that the prime
mover can exert a great enough torque, while the rotor field and stator field are
perpendicular, the voltage is constant, no matter how large the load is. But when
the power factor differs from 1, the stator field has a component that is parallel
to the rotor field, effectively changing the magnitude of the rotor field and the
induced voltage. A lagging power factor causes a reduction in voltage, a leading
power factor causes an increase. This can be compensated for by changing the
excitation current.
1.2.3 Coupling of generators
When there are multiple generators in an AC system, all of them rotate at the
same angular frequency, and their output must have approximately the same
phase at any time. It can be shown that if one of the generators speeds up, and
its phase begins to get ahead of the others, the torque on its rotor will increase,
preventing any further increase in the phase difference. Thus the generators
tend to be “locked” together.
When two generators are operating in parallel and there is a slight phase
difference, ϕ, between their output voltages, an electromotive force ∆Ṽ exists in
the “inner” circuit comprised by the two generators. Suppose that generator 2 is
lagging behind generator 1. As it is illustrated on figure 7, ∆Ṽ is ahead of the first
generator’s output voltage by approximately 90◦ . It is a very good approximation
to neglect the generators’ resistance compared to their inductive reactance, so
the current in the inner circuit, I˜ (which is called the circulating current), has the
same phase as the output voltage of generator 1. Thus the circulating current
will cause an increase in the perpendicular component of the stator field in the
first generator, and a decrease in the second generator.
The slight phase angle that a generator has compared to the rest of the
12
~
ΔV
~
V₁
φ
~
I
~
V₁
~
I
~
V₂
~
V₂
Figure 7: Two generators operating in parallel. When there is a phase difference
between the electromotive forces of the two generators, the one which is ahead
experiences a greater torque due to the increased armature reaction.
network is called the power angle or voltage angle. It is directly related to the
generator’s share of power production.
2 Power system control
Electrical generators are usually operated as part of a large interconnected network, the power grid. If one generator fails, in a large enough network the
remaining generators can quickly fill newly created gap between power consumption and power production, and prevent an outage.
A group of generators which have AC connections between them form a
synchronous area. As explained in the previous section, within such a network,
every generator must rotate with the same frequency, and thus the network
frequency is the same everywhere. Today, synchronous areas can be very large,
and contain the power network of several countries. Examples of such large
networks in Europe are the UCTE (Union for the Co-ordination of Transmission
of Energy), which encompasses most of mainland Europe, and NORDEL (Nordic
Electricity System), whose members are the Scandinavian countries.
Synchronous areas can be subdivided into control areas, which are controlled
by a single company, the transmission system operator (TSO). The responsibility
of transmission system operators is to operate and maintain the electric grid
in their area, and ensure that the system frequency does not deviate from a
13
Feedback
ω−ω₀
Turbine
Valve
Generator
steam flow
50 Hz
Boiler
Figure 8: The torque is continually adjusted to keep the system frequency at
a constant value, usually 50 Hz or 60 Hz. For example, in the case of a steam
turbine, the steam flow (which determines the torque exerted by the turbine) is
adjusted using a valve.
prescribed value, which is 50.0 Hz in most of Europe.
A difficulty with electric energy production is that electric energy cannot
be stored, except for very small quantities and short times. Therefore it must
be produced at the same time at which it is consumed. The production must
match the consumption at every instant. This is no easy task—the power plants
need to be continuously regulated to ensure that their power output matches
the demand of the customers.
The control methods are based on the fact that when the balance between
power consumption and power production is upset, the system frequency starts
to change. As it was shown in the previous section, when the load on a generator
increases, the reaction torque becomes larger. If this change is not compensated
for by increasing the driving torque, the turbine will start to slow down.
Because the total power consumption in a network cannot be measured
directly (and efficiently), the balance is maintained by keeping the system frequency constant.
In the UCTE synchronous area, four levels of control are used (fig. 8), each
one working at a longer time-scale than the previous ones.
The primary control works at the level of individual generators. We shall
study it in greater detail in section 2.1. Primary control restores the balance
of production and consumption in case of a disturbance. But it does not readjust the system frequency to the prescribed value. It merely ensures that the
14
restore mean to match with UT
System
frequency
activate
restore normal operation
limit deviation
Primary
control
take over
free reserves
Secodary
control
take over
free reserves
Tertiary
control
Time
control
activate on long term
Figure 9: Frequency control of a synchronous area.
frequency does not keep drifting away from the standard value when the balance is upset. Figure 13A illustrates the behaviour of the system under primary
control—when there is a jump in power consumption, the frequency stabilises
at a new point, and stays constant afterwards. The difference between the new
and old stable values is called the quasi-steady-state deviation.
It is called “quasi-steady-state” because on the characteristic time scale of
primary control this deviation can be considered constant. The value of the
system frequency is brought back to the prescribed value by the secondary
controller. Secondary control is also automatic, but it is centralised: it works at
the level of control areas. Its function is to restore the frequency to the prescribed
value of 50 Hz, and to restore power exchanges with the neighbouring control
areas to the scheduled values.
While every generator needs to have some kind of speed regulator (which is
in fact the primary controller), not all of them participate in secondary control.
Different generators can react to changes with different speeds, and have different ranges of power output where they can operate. Therefore some of them are
15
operated as base-load plants, i.e. they operate at a constant power output.
Tertiary control functions at an even longer time scale than secondary control. Its role is to restore the set working point (power output) of generators
participating in secondary control by redistributing the load between them or by
starting up additional generators. This may be done automatically or manually.
The final level of control, having the least impact on short time scales, is the
time control. The fictional time corresponding to system frequency is called the
synchronous time. According to UCTE regulations, synchronous time may not
deviate from the Coordinated Universal Time (UCT) by more than 30 seconds.
This is ensured by measuring the time deviation each day (at 8 a.m. CET), and
if it is greater than 20 seconds, setting the prescribed system frequency for the
following day 10 mHz higher or lower than 50 Hz. A deviation of 10 mHz from
the prescribed 50 Hz causes a shift of 17.28 seconds in synchronous time during
a single day.
2.1 Practical governor models
In this section we shall examine some practical mechanisms used for speed
governing, and derive simple mathematical models for them.
The principle behind any such governing mechanism is to compare the
rotation speed to a pre-determined value, and if there is a difference, make
adjustments to the torque driving the machine. There are several possibilities of
how exactly these adjustments should be made in order to maintain the stability
of the system. A few simple approaches are considered below.
2.1.1 The flyball governor
One of the simplest methods for regulating the rotation speed of machines is
to exploit the centrifugal force. The centrifugal governor, also called the flyball
governor, has already been in use as early as the 18th century. Its first use to
regulate steam engines is attributed to James Watt himself.
The centrifugal governor consists of two heavy weights, called flyballs, that
are rotating together with the shaft (fig. 10). The flyballs are held together by a
spring, but as they rotate, the centrifugal force pushes them apart. The faster the
rotation, the greater their distance from the axis becomes, and thus the angular
velocity is “converted” to a displacement parameter. This displacement can be
mechanically transferred to a valve (for example, through a system of levers),
and used to adjust the steam flow.
Let us make a simple, linearised model for this governor system! The inertia
of the flyballs is vanishingly small compared to the inertia of the turbine they
are coupled to. Therefore on the time scales where the rotation speed changes it
16
displacement
flyball
steam valve
Figure 10: The centrifugal governor. This simple device can be used to regulate the steam flow with a valve (as shown on figure 8), and keep the network
frequency constant.
is a good approximation to consider the governor system to be in equilibrium,
i.e. the spring force and the centrifugal force are balanced at any time.
We shall denote the distance of the flyballs from the axis with r , and the
elongation of the spring by x. Values at the standard operating point (when the
angular velocity is the prescribed ω0 ) will be noted with a subscript 0, and the
deviations from them will be noted with a prime:
r = r0 + r 0
x = x0 + x 0
When the displacements are not very large, i.e. |r 0 | ¿ r 0 etc., the relation between
r 0 and x 0 can be approximated as r 0 = αx 0 . It follows from the principle of energy
conservation that when the centrifugal force is balanced by the spring, the two
0
forces are related as F spring = xr 0 F c.f. = αF c.f. . (The infinitesimal work, F d x, done
at one end of a lever system must be equal to the work at the other end.)
The balance of forces can be written as
F c.f. = m(ω0 + ω0 )2 (r 0 + r 0 ) =
1
1
k(x 0 + x 0 ) = F spring .
α
α
m is the mass of the flyballs, and k is the spring constant. Let us expand these
expressions and ignore those terms that are higher than first order in any of the
primed quantities:
mω20 r 0 + mω20 r 0 + 2mω0 r 0 ω0 =
17
1
1
kx 0 + kx 0 .
α
α
Substituting r 0 /α for x 0 and using the fact that the equation must hold for ω0 = 0,
r 0 = x 0 = 0, we get
¶
µ
k
2
mω0 − 2 r 0 = −2mω0 r 0 ω0 .
α
It is a requirement that for a constant angular velocity the system must be
stable, i.e. when r 0 is changed by a small amount, the resulting force imbalance
should act to return it to its initial value. For ω0 = 0, this constraint leads to
k/α2 > mω20 . In other words, the spring must have a minimal stiffness for the
system to be stable.
We have established that for relatively small deviations, ω0 is proportional to
0
r and x 0 : ω0 = C · r 0 = αC · x 0 , where C is a positive constant.
The flyball mechanism can convert variations in rotation speed to displacement, but it cannot exert a force great enough to open or close a large valve.
Therefore some force amplifier is must be used in practice, such as the hydraulic
device illustrated on figure 11. This particular device does not simply enlarge a
displacement by a multiple, because for as long as the pilot valve is open, the
piston keeps moving. As a linear approximation we can regard the position
of the pilot valve, x, as directly determining the rate of change in the position
y of the piston: −(const.) · x = ẏ. In other words this valve–piston system is a
mechanical integrator.
In order to be able to study this governor system in more detail, let us look at
how the turbine behaves. If its moment of inertia is Θ, and the torque acting on
x
piston
pilot valve
y
pressure
flow control valve
Figure 11: Hydraulic mechanism (actuator) used to operate the flow control
valve. The more the pilot valve is opened, the greater the force on the piston,
therefore the input displacement (x) determines not the magnitude, but the rate
of change of the output: ẏ = −(const.) · x.
18
it is T , then its motion is described by the equation
ω̇Θ = T.
This is called the swing equation. The torque is the difference of the driving
torque, Td and the electrical reaction torque, Tr . The latter is depends on the
electrical load connected to the generator, and cannot be controlled. The driving
torque must be constantly adjusted to balance the reaction torque and to correct
any deviations from the prescribed system frequency. There is also a small
damping torque, arising from various electrical and mechanical effects, which
increases with the rotation speed. We shall use Tdamping = −Dω as a linear
approximation.
Now that we have all the pieces, let us put the equations together. The total
torque is T = Td − Tr − Dω. For simplicity, let us assume that the electric load
does not change in time, so Tr is constant. The driving torque is determined
by the position of the steam-flow valve, so T = −Dω + C 1 y + (const.), where C 1
denotes a constant. Taking the derivative of this equation, we get Ṫ = −D ω̇ +
C 1 ẏ = −D ddt (ω0 + ω0 ) − C 2 x. But x, the input to the hydraulic force-amplifier,
is proportional to the deviation (ω0 ) from the preferred rotation velocity, so
Ṫ = −D ω̇0 −C 3 ω. Using the swing equation, we get
ω̈0 Θ = −D ω̇0 −C 3 ω0 ,
which (for C 3 > 0) is the differential equation of a damped harmonic oscillator. A
small disturbance of the rotation velocity dies off after a number of oscillations,
no matter what the value of Tr is. But notice that the damping term in this
equation represents the natural damping of the rotor itself, and the governor
mechanism itself has no “built-in” damping.
The type of control described here is called integral control because the
feedback is through the time-integral of the regulated variable’s error.
2.1.2 Speed droop governor
The governor discussed in the previous section does not have very good stability
properties as it relies on the natural damping of the rotor. Better results can be
achieved if the hydraulic integrator is converted to a displacement amplifier by
means of a summing beam (a lever) that provides feedback to the pilot valve
from the piston’s position (fig. 12). In this device the position of the pilot valve
is the sum of the input displacement x and a constant multiple of the output
y. Its behaviour is described by the differential equation ẏ = −(const.) · (βy + x).
The steady state solution of this equation is y = −x/β, so we see that this device
works as a displacement amplifier. By making the feedback proportional to the
frequency-deviation, we got a proportional controller.
19
If the driving torque corresponding to y = 0 is Td0 , and we ignore the damping term, then the swing equation is
Θ
d
(ω0 + ω0 ) = Td0 + (const.) · y − Tr .
dt
The steady state solution is Tr − Td0 = (const.) · y = −(const.) · ω0 . Hence this type
of governor does not keep the rotation speed constant. But it does limit the
deviation from the nominal value as the reaction torque changes.
The power output of a rotating machine is ω · Tr . Since the relative change
in rotation speed cannot be very large under normal operation, we can say
that the reaction torque is simply proportional to the power output. Thus the
relationship between the power load and rotation speed of a generator is linear,
as illustrated in figure 13A.
The slope of this curve on fig 13A is characteristic for every generator. It is
usually expressed as the ratio of relative changes in rotation speed and power
output,
ω0 /ω0
(in %),
s =− 0
P /P 0
and is called the droop of the generator. In this formula P 0 is the rated real power
output of the generator, while P 0 is the deviation from this value.
When there are several interconnected generators, the proportion in which a
load increase is shared between them is determined by their droop values.
x
y
pressure
flow control valve
Figure 12: Hydraulic displacement-amplifier (servomechanism). A lever (the
“summing beam”) functions as a feedback mechanism that limits the displacement of the piston. Unlike the hydraulic device of 11, this system has a steady
state. In the steady state, the piston position y is proportional to the input
displacement x.
20
A
B
Ω
Ω'
P
time
5
10
15
20
Figure 13: A: The relationship between the power load of the generator and the
rotation speed when the speed droop governor is operating. B: Response of
the speed droop governor to a sudden drop in power consumption: following
the power drop at time 5, the rotation speed stabilises at a new, higher value.
(Because this example is meant to illustrate the behaviour of this governor type
and because machines of different sizes have different reaction times, exact time
units are not specified.)
3 Data analysis and simulations
In this section we shall analyse frequency and demand measurement data from
two synchronous areas. A simple model of a control area will be constructed,
and studied using computer simulation. The results will be compared with the
measured data.
3.1 Data sources
Frequency measurement data was obtained from two sources: the UCTE, and
the National Grid Company (United Kingdom).
Real-time frequency data for the UCTE synchronous area is available on
the UCTE website [6]. A computer program was used to continuously monitor
the site and retrieve the data. The data has a time resolution of 2 seconds and
frequency resolution of 1 mHz.
The National Grid Company of the United Kingdom provides both frequency
and power consumption measurements [7]. The frequency data has a resolution
of 10 mHz and is updated every 15 seconds. The demand is given in megawatts
and is updated every minute. However, National Grid’s data source—especially
their frequency measurements—proved to be much less reliable than UCTE’s.
Data was usually unavailable for several short periods during a day, and measure21
ment times were often mismatched, thus further reducing the time resolution.
Because of this we shall mainly work with the UCTE frequency measurements.
Let us plot the frequency and demand data for intervals of different lengths.
Frequency variations for a weekday are plotted on fig. 18 for the UCTE network
and fig. 17 for the National Grid network. Note how UCTE operates its grid
within stricter frequency bounds than National Grid. There are additional plots
on which the data of a shorter period is magnified (figures 16, 19, and 20).
It is interesting to note that almost all big frequency jumps happen at sharp
hours. This is especially visible on fig. 18. We can verify this observation quantitatively by calculating the standard deviation of the frequency values, and
looking at how it changes during a day. The results of this calculation are plotted
on fig. 22. It is not surprising that most frequency jumps are at sharp hours,
because our society functions by the 24-hour clock. Consumers tend to start
and finish activities (and therefore turn on and turn off electric machines) at
integer hours; work hours start and end at sharp hours. The scheduling of power
exchanges is also done for hourly or half-hourly periods. But it is important
to note that these effects should be taken into consideration in the regulation
of the electric network. At times when large frequency jumps are likely, more
regulating capacity is needed.
On shorter time scales there is no discernible order in the frequency fluctuations (figures 16, 19, and 20).
The distribution of the frequency values in the UCTE network is plotted
on fig. 21. It can be seen that (in accordance with the regulations), the system
frequency tends to stay within 100 mHz of the prescribed value of 50 Hz. The
histogram was made using data from a number of time periods, approx. 25 days
in total.
The demand data from National Grid is plotted on figures 14, 15 and ??. On
these plots, the most apparent feature is of course the daily variations, but if we
zoom in to shorter time scales, comparable with 1 hour, the random fluctuations
become visible. Unfortunately the time resolution of the data is not good enough
to go to an even shorter time scale and make a comparison with the frequency
fluctuations.
3.2 Description of the model
Let us construct a simple model that can be used for numerical simulation of
frequency variations in a synchronous area.
As discussed in section 1.2.3, under normal operating conditions all machines in an interconnected AC grid are rotating with the same frequency. Therefore we shall use a single angular frequency, ω, for the complete control area.
Notations will be used as follows: ω0 = ω−ω0 – the deviation from the prescribed
22
Demand HGWL
45
40
35
30
25
5
6
7
8
9
Day of month HApril, 2007L
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
Demand HGWL
Figure 14: Power demand variation during April, 2007 in the electric network
of the National Grid Company (United Kingdom). During weekends the top
demand is less than on weekdays. The period from the 6th to the 9th was a
holiday (Good Friday to Easter Monday).
40
35
30
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23
0
hour of day
Demand HGWL
Figure 15: Demand variation during one day (11th of April, 2007, Monday) in the
National Grid network. At this time scale the curve is dominated by the daily
variations—power consumption is greater during the day than during the night.
Random variations are only visible at shorter time scales. Frequency data for the
same day is plotted on fig. 17.
43.4
43.3
43.2
43.1
43.0
42.9
42.8
9
10
11
12
hour of day
Figure 16: A section of fig. 15, magnified. At this time scale the random changes
are visible.
23
11th of April, 2007, Monday
50 200
frequency HmHzL
50 100
50 000
49 900
49 800
49 700
0
1
2
3
4
5
6
7
8
9
time Hhour of dayL
10 11 12 13 14 15 16 17 18 19 20 21 22 23
0
Figure 17: Frequency variations during the 11th of April, 2007, in the National
Grid network. The frequency scale of this graph is the same as that of figure 18
(UCTE frequency data), therefore they are directly comparable. Note how frequency control is less tight than in the UCTE grid. Spacing of horizontal lines is
25 mHz.
24
frequency HmHzL
15th of March, 2007, Thursday
50 100
50 050
50 000
49 950
49 900
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23
0
frequency HmHzL
16th of March, 2007, Friday
50 100
50 050
50 000
49 950
49 900
0
1
2
3
4
5
6
7
8
9
time Hhour of dayL
10 11 12 13 14 15 16 17 18 19 20 21 22 23
0
frequency HmHzL
Figure 18: Frequency variations during two consecutive weekdays in the UCTE
synchronous area. Notice that most abrupt frequency changes happen at sharp
hours. This is partly because of the behaviour of consumers, and partly because
the scheduling for power exchanges between adjacent control areas is made for
hourly or half-hourly intervals. See also figure 22 on how standard deviation of
frequency fluctuations is changing during a day.
50 100
50 050
50 000
49 950
49 900
15
15.25
time Hhour of dayL
15.5
15.75
16
Figure 19: Frequency variations during a one hour period (15th of March, 2007).
25
frequency HmHzL
50 100
50 050
50 000
49 950
49 900
54 000
54 060
time Hseconds elapsed since the start of the dayL
54 120
54 180
54 240
54 300
54 360
54 420
54 480
54 540
54 600
Figure 20: Frequency variations during a ten minute period (15th of March, 2007).
Time is in seconds elapsed since the start of the day, i.e. this is approximately
from 15:00 to 15:10.
49 950
50 000
50 050
50 100
f HmHzL
Figure 21: Distribution of frequency values in the UCTE area. The histogram
was made using data gathered during a 25-day period. It can be seen that the
frequency does not usually deviate from 50 Hz by more than 50 mHz.
26
40
D f HmHzL
35
30
25
20
15
10
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
hour of day
Figure 22: This plot was made by dividing days into 15-minute intervals, and
calculating the standard deviation of frequency values (∆ f ) for each interval.
Each point on the graph represents one interval. The intervals are centred
around 0, 15, 30 and 45 minutes; e.g. an interval around 12 o’clock lasts from
11:52:30 to 12:07:30. The data was averaged for 4 days. The vertical lines are
drawn at sharp hours.
Note how all large values of ∆ f occur at sharp hours (compare with fig. 18).
Other than the peaks at sharp hours, there is no discernible daily variation in
∆f .
This graph is based on the frequency data from the UCTE network.
27
angular frequency; p i – the power output of the i th machine; d – the total power
demand in the system; Θi – the inertial momentum of the i th machine.
Machines usually respond to control with some delay. This delay may be
caused by several factors, for example, in the case of a steam turbine, re-heating
might be necessary before the power output can be increased; or the servomechanism that operates the steam-flow valve may respond with a noticeable delay.
We shall take into consideration all these delay-effects by introducing a “control
variable”, v i , for each generator. The control mechanism adjusts the variable v i ,
but the actual power output follows v i with some delay. As a first order linear
approximation, we can use the equation ṗ i = (v i − p i )/Ti , where Ti is a time
constant. Obviously, the steady state solution is v i = p i .
For the control equation, we can use proportional-integral (PI) control (as
described in section 2.1). The proportional term represents primary control,
while the integral term, which is responsible for bringing the frequency deviation
ω0 back to 0, represents secondary¡ control.
¢control variable is set to the
R The
1
0
0
sum of these two terms: v i = −G i · ω + τ ω d t . τi is the characteristic time
i
constant of secondary control, while G i is a gain factor representing the “stiffness”
of control.
Adding the only missing piece, the swing equation, we get the complete
system of differential equations describing our model:
M
z
0
ω̇ (t ) ω0 ·
µ}|
X
{¶
Θi =
µ
X
i
¶
p i (t ) − d (t )
i
ω0 (t )
v̇ i = −G i · ω̇ (t ) +
τi
v i (t ) − p i (t )
ṗ i (t ) =
Ti
µ
0
¶
Note that in the swing equation ω0 was substituted for ω. This is a good
approximation since the relative frequency deviations are very small: ω0 /ω0 ¿ 1.
Because there is a single angular frequency for the complete network, and
because the moments of inertia appear only in the swing equation, the network can be considered to have a single parameter that characterises its inertia.
P
ω0 i Θi can be replaced by a parameter M that scales linearly with the number
of generators.
A computer program written in the C++ language was used to numerically
solve these equations for a power consumption that changes arbitrarily in time.
Because the demand d (t ) may not vary smoothly, it was not possible to employ
28
higher order numerical integrators, so the straightforward Euler method was
used.
3.3 Simulation results
We are primarily interested in how the model behaves, and how it responds to
changes of parameters and different demand functions (d (t )). For this, it is not
necessary to use real physical units; to keep things simple, all parameters will be
given as numbers.
Changing parameter values is mathematically equivalent to choosing different units, so it is useful to fix the value of some of the parameters to eliminate
any redundancy. In reality, the number of relevant parameters for this system is
only two. For the inertial parameter M a value of 1/generator will be used (it will
be numerically set to the number of generators). For the controller gains G i , we
shall use values around 1.0. Only the two time constants will be varied.
Also note that the swing equation contains the difference of the total producP
tion ( i p i ) and the demand d . The magnitude (as opposed to the derivative) of
these quantities does not appear at any other place in the equations (except the
equation which relates v i to p i ). Therefore, provided that the initial values of
v i are the same as p i , the behaviour of the system does not change if the same
P
constant is added to both d and i p i . In other words, if we have a solution of
this system of equations for a particular d (t ) function, this solution can be transformed into a solution for another system with the demand function d (t ) + d 0
P
by simply adding constants to p i in such a way that i p i is also increased by d 0 .
For reasons of simplicity (and to avoid redundancy), the initial conditions will
always be set to p i = d = 0 for all i . This should not be interpreted as 0 demand
and production, but as 0 deviation from a positive demand and production.
First, let us look at the how the model behaves when the demand is suddenly
increased. The results of the simulation are plotted on figures 23 and 24. As
expected, after the increase, the power output stabilises at the new value of d .
The higher the delay factor the more time is needed for the oscillations to die off.
Now let us see what happens under the same load conditions as before when
there is more than one generator. The power outputs of two generators are
plotted on fig. 25. The generator with the quicker response (i.e. the one with
the smaller delay factor) takes the higher share of the final load. Otherwise their
behaviour is similar in the sense that both power outputs oscillate at the same
frequency before stabilisation. In our model this is true for any number of generators because there is a single angular frequency, and a single swing equation
for the complete system. No matter how many generators are there, and what
are their parameters, because of the frequency locking, all of them respond to
a disturbance with oscillations of the same frequency, but perhaps different
29
amplitudes and damping factors. This system is not capable of producing random frequency fluctuations, even if all generators have different parameters.
Therefore the source of the frequency fluctuations visible in measurement data
must lie in the short time-scale demand fluctuations.
We saw that in practice the power consumption changes randomly on small
time scales. The simplest model for a randomly changing but continuous curve
is the random-walk. We can use a random-walk curve to model the changes in
demand, and see what frequency fluctuations arise in the system.
These simulations were run with approximately 100 generators. Generator
and control parameters were chosen randomly (with a uniform distribution)
from intervals. The exact parameters for each run of the model are documented
in the figure captions. Since the secondary control is centralised, it is reasonable
to assume that generators respond similarly to it, regardless of their type, so τi
were chosen from a narrow interval. The gains G i were drawn from the interval
0.5 .. 2.0 for each run. The delay constants Ti were allowed to change over an
order of magnitude to reflect the differences between types.
As we expect, the production follows the demand, but it is smoothed out
because of the finite response time of the control mechanism (fig. 26). A smaller
delay constant allows the system to follow the demand more precisely (fig. 28).
Though this is not very apparent on the figures where the secondary control is
strong enough, on figure 29, where the effect of secondary control was reduced
by giving τi large values, it can be seen that the frequency curve is very similar
to the negative of the demand curve, but large variations are reduced on a long
time scale. This is in accordance with the linear relationship that exists between
power and frequency variations when proportional control is used.
Furthermore it can be observed that a larger number of generators can exert
a tighter control over frequency. As we compare figures 26 and 27, we see that
a tenfold reduction in the number of generators available for control increases
the frequency variations by ten times.
Unfortunately the demand data at our disposal does not have a high enough
resolution to make it possible to compare frequency and demand fluctuations,
and see a direct correlation.
But our second observation, namely that a larger number of controllers can
exert a tighter control, could be used to assess the state of the power grid from the
frequency measurements. Figure 22 shows that—apart from the hourly peaks—
the magnitude of frequency fluctuations is usually constant throughout the day.
This means the there are sufficient control reserves at any time to maintain
the frequency within set bounds. However, we can expect that if the number
of machines participating in control is reduced, for example because some of
the machines reach the bounds of their control range, then the magnitude of
frequency fluctuations will increase.
30
â pi , d
i
1.2
1
0.8
0.6
0.4
0.2
t
50
100
150
50
100
150
200
250
300
Ω'
t
-0.2
200
250
300
-0.4
-0.6
-0.8
-1
-1.2
Figure 23: Response of a single generator to a sudden increase in demand at
time 100. On the first figure, the tick line is the demand, while the thin line is
the total production. The second figure is a plot of the frequency deviation. As
expected, the frequency changes in the inverse direction than the demand, i.e. a
demand increase is accompanied by a frequency drop. Parameter values: G = 1,
τ = 20, T = 1. Here, and in all subsequent plots of simulation results, values of
the power demand and power production must be interpreted as an offset from
a positive value.
31
â pi , d
i
1.5
1.25
1
0.75
0.5
0.25
t
50
100
50
100
150
200
250
300
Ω'
0.5
t
-0.5
150
200
250
300
-1
-1.5
-2
Figure 24: This is the same system as the one on fig. 23, but parameter values
are different: G = 1, τ = 20, T = 5. A system with a higher delay-constant is
more unstable, i.e. it oscillates for a longer time before stabilisation, and the
amplitude of oscillations is higher.
p1 , p2
0.8
0.6
0.4
0.2
t
50
100
150
200
250
300
Figure 25: Power output of two coupled generators under the same conditions
that are described in the caption of fig. 23. Parameter values: G = 1 and τ = 10.
One of the generators has a delay factor of T1 = 1, the other T2 = 3. The generator
with the higher delay factor can responds a bit less quickly than the other one,
therefore it takes the smaller share from the final load increase. Because the
system has a single frequency, the two generators oscillate synchronously before
their power output is stabilised. This is true for any number of generators.
32
â pi , d
i
t
100
200
300
400
500
-100
-200
-300
Ω'
0.3
0.2
0.1
t
100
200
300
400
500
-0.1
-0.2
Figure 26: Behaviour of a 300-generator system. The demand function is generated as a random-walk curve. Generator parameters are uniformly distributed
in the following intervals: G = 0.5 .. 2.0, τ = 30 .. 40, T = 0.5 .. 20.
On the first plot, the thin black curve is the power demand, while the thick grey
curve is the total power production. On the second plot we see the frequency
variation.
The production follows the demand curve closely, but with a slight delay.
33
â pi , d
i
25
t
100
200
300
400
500
-25
-50
-75
-100
-125
Ω'
2
1
t
100
200
300
400
500
-1
-2
Figure 27: This simulation was run with the same settings as the one whose
results are plotted on fig. 26, but with only 30 generators. G = 0.5 .. 2.0, τ =
30 .. 40, T = 0.5 .. 20. Now the frequency deviations are 10 times greater. This
is because fewer generators are only capable of proportionally smaller total
increase/decrease in power output during a fixed amount of time.
34
â pi , d
i
200
150
100
50
t
100
200
300
400
500
Ω'
0.4
0.2
t
100
200
300
400
500
-0.2
-0.4
-0.6
Figure 28: Behaviour of a 100-generator system with the following parameters:
G = 0.5 .. 2.0, τ = 30 .. 40, T = 0.5 .. 5. Due to the smaller delay factors, the demand
curve is now more closely followed by the production curve.
35
â pi , d
i
300
250
200
150
100
50
t
100
200
300
400
500
Ω'
t
-0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
100
200
300
400
500
Figure 29: Behaviour of a 100-generator system with the following parameters:
G = 0.5 .. 2.0, t au = 200 .. 250, T = 0.5 .. 5. Using large values for τ means having
a very slow (and “weak”) integral control (secondary control), so the dominant
behaviour is that of the proportional (primary) control.
The most striking difference between this figure and fig. 28 is that the frequency varies more loosely. In fact, the frequency curve mirrors the demand
curve. This is in accordance with the linear relationship between the frequency
and power deviations that we have derived for proportional control in sec. 2.1.2
36
4 Summary
The purpose of power system control is to maintain the delicate balance of
power production and power consumption. This is necessary because large
quantities of electric energy cannot be stored. Therefore electric power must be
transmitted to the consumers as soon as it is produced.
It is not possible to measure the difference between power production and
power consumption directly, so power generation control is achieved through
measurement of another related parameter, the frequency of the alternating
current. When the demand exceeds the production in a network, the frequency
starts to fall. The deviation of the frequency from a set value can be used as
feedback to adjust the power production and ensure that balance is maintained.
In fact, power system control means ensuring that the system frequency does
not deviate from a prescribed value.
Under normal operation the frequency is allowed to fluctuate only between
certain bounds, which are set in the regulations of the system operator. This
work investigates the cause of these fluctuation, and explores the possibilities to
use frequency measurements for a better system-level control.
The behaviour of a regulated network of interconnected generators was
studied with computer simulations, and the results were compared with high
resolution frequency measurements from two different power networks.
It has been found that the control system of generators does not produce
random fluctuations by itself. Instead, the frequency variations reflect the short
time-scale fluctuations of power demand. However, the properties of frequency
fluctuations do depend on the state of generators participating in frequency
control, therefore it is possible to assess the state of the network from frequency
measurements.
37
Bibliography
[1] P. M. Anderson and A. A. Fouad, Power System Control and Stability (IEEE
Press, Wiley Interscience, 2003)
[2] Alexandra von Meier, Electric Power Systems: A Conceptual Introduction
(IEEE Press, Wiley Interscience, 2006)
[3] UCTE Operation Handbook, final version 2.5/24.06.2004, available from
http://www.ucte.org/
[4] A villamosenergia-rendszer szabályozása, available from the Hungarian
Transmission System Operator Company (MAVIR),
http://www.mavir.hu/
[5] Remus Fetea, Reactive power: A strange concept?, Physics Teaching in Engineering Education 2000
[6] Real-time frequency data for the UCTE grid is available on the UCTE main
page: http://www.ucte.org/
[7] National Grid real-time operational data is available from the website
http://www.nationalgrid.com/uk/Electricity/Data/
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