IEEE Transactions on Power Apparatus and Systems, Vol. PAS-104, No. 12, December 1985
3409
DYNAMIC BEHAVIOUR OF AGC SYSTEMS
INCLUDING THE EFFECTS OF NONLINEARITIES
A. Norched,
Member IEEE
-Ontario Hydro
Toronto, Canada
1.B. Shahrodi,
Member IEEE
University of Toronto
Toronto, Canada
Abstract - A general mathematical model capable of
representing multi-area, multi-unit systems has been
used to study the dynamic behaviour of AGC systems.
The eigenvalues of the system are obtained and
classified. The modes which play the most Important
role in the system dynamics are identified and
related to the system control loops. The effects of
the nonlinearities are assessed by their position in
the control loops.
It is shown that the adverse
effects of the nonlinearities are pronounced when the
linear model at operating point, has lightly damped
critical modes.
The effects of the nonlinearities
can be minimized by increasing the damping of these
modes using methods outlined in the paper.
2. SYSTEM EIGENVALUES
A functional block diagram of a one unit control area
in an interconnected power pool is shown in
Fig. 1 [1].
In the multiple loop control system of Fig. 1, where
control actions take place at different speeds,
dominant modes can be related to separate control
loops (7]. The parameter changes within a particular
loop, affect mainly the corresponding dominant mode.
1. INTRODUCTION
Linear models have been widely used to study AGC
The studies done were either
systems [1,2,3].
low
limited
to
order
or
systems
were
too
to
mathematically
oriented
offer
a
physical
understanding of the system dynamic behaviour. As
for the effects of the system nonlinearities, the
number of published studies is relatively small. The
lack of a general method for the study of large scale
nonlinear systems has prompted a series of works
using simulation techniques (4,5].
For this study,
detailed multi-area, multi-unit AGC
model, with the option of including the system
nonlinearities, has been developed [61.
A large
number of AGC systems, with various configurations
and
control
are
strategies,
examined.
The
eigenvalues of the system are categorized and related
to the structure of the AGC model. The oscillatory
dominant modes are identified.
These modes are
referred to as 'critical modes'. Methods to improve
the stability of the system through increasing the
damping of the critical modes are presented.
Simulation results are used to compare the dynamic
behaviour of the nonlinear system with that of the
corresponding linear model. The time response of the
system is obtained for step load changes representing
emergency conditions as well as for morning pick-up
conditions. The latter is simulated by low-amplitude
random signal superimposed on ramp loads.
The
conditions under which the nonlinear effects become
dominant are identified and methods to alleviate
these effects are presented.
Figure 1:
Control loops in
an
interconnected
area.
a
the
of
phenomena
results
The
study
explain
encountered in AGC systems such as the commonly
observed tie-line oscillations.
85 WM 073-2
A paper recommended and approved
by the IEEE Power System Engineering Committee of
the IEEE Power Engineering Society for presentation
at the IEEE/PES 1985 Winter Meeting, New York, New
York, February 3
8, 1985. Manuscript submitted
August 10, 1984; made available for printing
November 28, 1984.
-
The natural modes of
an
AGC system
can
be classified
as:
(a)
(b)
(c)
(d)
a.
Critical Modes,
Slow Non-Oscillatory Modes,
Fast Non-Oscillatory Modes, and
Well-Damped Oscillatory Modes.
Critical Modes
The critical modes are lightly damped oscillatory
modes which are subject to instability as the AGC
controller gains are varied. They are pronounced in
most of the system variables and play a significant
role in their time responses. These modes are:
(i)
Af-mode,
(ii)
APtie-mode, and
(iii)
AGC-APtie-mode.
Some of the frequently observed tie-line oscillations
can be attributed to the poor damping of one or more
of these modes.
Af-mode
Af-mode is an oscillatory mode mainly associated
with the governor loop. The frequency of this mode,
is in the range of 0.01-0.3 Hz. This wide range of
frequency is due to the difference in the type of
units.
The faster the response of the unit, the
higher the frequency of the Af-mode and the more
damped this mode is.
0018-9510/85/1200-3409$01.00©1985
IEEE
3410
AP
tie-mode
This oscillatory mode is mainly associated with the
'natural loop'. The frequency of this mode is the
highest among the frequencies of the critical modes
and is in the range of 0.2-0.6 Hz.
APtie-mode
is typically poorly damped. This mode represents the
The
electromechanical oscillations of the areas.
faster the response of the units, the more damped the
APtie-mode is.
Table 1
AGC System Parameters
fo
Nominal frequency
Turbine time constant
Governor time constant
ACE Smoother time constant
Governor droop
Transient regulation
Proportion of power developed
before reheater
Reheater time constant
Nominal load in an area
Synchronizing coefficient
in the MW base of area 2
Water starting time
Transient droop time constant
Speedchanger servomotor time
constant
for a non-reheat unit
for the first reheat unit
for the second reheat unit
for a hydraulic unit
Frequency -bias gain
Proportional gain
Integral gain
Unit inertia constant
for a non-reheat unit
for the first reheat unit
for the second reheat unit
for a hydraulic unit
TT
TG
TSM
R
r
C
TR
AGC-APti e-mode
PB
T12
The frequency of this oscillatory mode is the lowest
When there is no
among the critical modes.
integrator in the AGC controller, this mode loses its
The AGC-APtie-mode is almost
oscillatory form.
independent of the governor and AGC-Af-loops.
Unlike the other two critical modes, this mode is not
subject to instability as the AGC integrator gain
The frequency of this mode is in the
increases.
The damping and the
range of 0.001-0.02 Hz.
frequency of this mode are highly dependent on the
AGC controller gains and the speed of unit response.
TW
Tr
b.
Slow Non-Oscillatory Modes
The presence of these modes is the result of the
slow-response components of the system such as a
hydraulic unit with a transient droop governor. The
hydraulic slow mode is dominant in the time response
It is almost
of the hydraulic unit output.
insensitive to the variations of system parameters.
c.
B
kp
kj
H
Table 2:
8.0
sec.
0.5 pu.
0.1 pu.
2.0
8.0
sec.
sec.
0.5 sec.
0.8 sec.
0.6 sec.
0.3 sec.
0.3
0.13
0.13
5.0
8.0
6.0
3.0
sec.
sec.
sec.
sec.
Eigenvalues of a Two-Area System With
and Without Linearized Boiler Models
Fast Non-Oscillatory Modes
These modes appear due to the fast-responding
components of the system such as the turbine, the
permanent droop governors, and the servomotors. The
fast non-oscillatory modes are almost insensitive to
the variations of the system loop gains.
Some of
them are almost equal to the open loop poles of the
fast-responding components.
d.
TSR
60.0 Hz
0.3 sec.
0.08 sec.
0.3 sec.
3.0 Hz/pu.
24.0 Hz/pu.
0.3
Well-Damped Oscillatory Modes
Similar to fast non-oscillatory modes, these modes do
not play a significant role in the response of the
system due to their high damping. In this group of
modes, only the AGC-Af-mode is worth mentioning.
This mode is associated with AGC-Af loop and
appears in oscillatory form only when the total area
inertia is low (less than 10.0 s in. this study). The
frequency of this mode is in the order of 0.02 Hz and
its damping ratio is in the range of 0.6-0.9. This
mode is pronounced in the response of the area
frequency and the unit output, especially when the
Af-mode is highly damped, i.e., when the unit is
fast-responding.
SOURCE OF EIGEN
VALUES OR MODES
NO BOILER
EFFECT INCLUDED
LINEARIZED
BOILER MODEL
hf-mode
-0.0652±jO .1685
-0.1139±jl1.1838
.079±j0.0775
-0.0648+j0.1653
-0.0190,-0.0236
-0.0189,-0 .0217
-0.125,-0.1586
-0.2993,-0.3883
-0.9374,-0.9643
-1.250,-1.249,
-1.667,-2.005,
-3.333,-3.333,
-3.333,-1.999
-O.125,-0.1586
-0.3043,-0.3874
-0.9380,-0.9644
-1.250,-1249,
-1.667,-2.005,
-3.333,-3.333,
-3.333,-1.999
-3.333,-3.333,
-3.333,-3.092
APtie-mode
AGC-APtie-mode
Hydraulic slow
mode
Reheater mode
ACE smoother
Hydraulic turbine
Servomotor
Steam turbine
Governor
Boiler
Sample Results
The eigenvalues of a two area system are calculated.
Area 1 consists of one non-reheat unit, two reheat
units and one hydraulic unit and Area 2 consists of
one
non-reheat unit, one reheat unit and one
hydraulic unit. The system parameters used are given
in Table 1 and the calculated eigenvalues are
classified in Table 2. The eigenvalues shown are for
the two cases: a) when boiler dynamics are ignored,
and b) when the linearized boiler model given in the
appendix is included.
Figure 2 shows the time response of the incremental
frequency of each area to a 0.1 pu. load change in
-O
-3. 333,-3.333,
-3.333,-3.084,
-2.998
-12.5,-12.5
-12.5,-12.5,
-12.5,-12.596,
-12.652
-0.1104+j 1.1818
-0.0:777±jO.0718
-3.007
-12.5,-12.5,
-12.5,-12.5,
-12.5,-12.594
-12.694
-O.0078±jO.0110,
-O.0079±jO.0109,
-O.0118±jO.0107,
-O.0118±jO.0113,
-O.OlOl+jO.0112,
-O.0973±jO.0623,
-0.0869+jO.0788,
-0.0900+jO.0818,
-0.0912+JO.0718,
-0.0964+j 0.0630,
-O.3012,-0.2875,
-.2954.
-0.2954±jO.0003
The high-frequency oscillations of the
and the low frequency oscillations of
Af-mode can clearly be seen in these figures.
area
two.
APtie-mode
_ tS_.
j,
90%
:: : . r
FREOUENCY OF AREA * 1
Figure 3:
..._
.
...
....
~~~~~~~~~~~~.
2't ' -_,,.
1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~....
...
a
.
7i.
..
li.s0
0
9I.W
2
Figure 2: Time response of
0.1 pu, step load
3.
a
two-area
change in
system
area
-Valvre po3itin_
Variation of the coeCficient 'k4'
as a function of unit loading.
of the position feedback of the
causes
servomotors
instability of
to
In
order
AGC-Af-modes.
and
AGC-APtie
restabilize the system, the following measures can be
taken:
The opening
speedchanger
to
2.
fast-responding units are dominant in the
area, the integrator should be removed from the
AGC controller. In case it is desired to damp
oscillations
electromechanical
the
out
(APtie-mode) as well, the unit MW-feedback
(a) If
DYNAMIC STABILITY OF THE AGC SYSTEM
The results of the study show that the dynamic
stability of the system is governed by the amount of
damping of the system's critical modes. The effects
of structural and parameteric changes on the damping
of these modes are sumnarized below:
I.
-
feedback.
TlIM IN SECONDS
FREQUENCY OF AREA
:: t80~-v --- |: -)
Throughout the study, it is assumed that all the
speedchanger servomotors are equipped with position
....
.-r- -4----*----... ----_-_,,,,
- 4-
b
.*..f--t
Minor Feedback Loops
III.
1!{
: : ::::
3411
Loading of the Units
Higher loading of the units results in more damping
of the critical modes. In other words, a system with
fewer units loaded to near their maximum continuous
rating is more stable than a system with a larger
number of lightly loaded units.
II. Boiler Dynamics
The nonlinear model of a drum-type boiler along with
its linearized form are described in the Appendix.
The linearized model of each boiler introduces five
of
Most
the
boiler
additional
eigenvalues.
For the sample
eigenvalues are lightly damped.
system of section 2, the system eigenvalues with the
inclusion of the boiler model are given in Table 2.
It is clear from the table that the presence of the
boiler has a negligible effect on the system's
(dotted line in Fig. 1) is implemented.
(b) If the integral controller remains in the AGC
loop, the unit MW plus frequency feedback should
be implemented.
IV. The Addition of a New Generating Unit
The addition of a slow responding generating unit to
the AGC system results in the reduction of the
damping of the critical modes. However, the addition
of a fast-responding unit has a negligible effect on
these modes.
V.
Interconnection of Areas
The interconnection of two areas with lightly damped
Af-modes results in a poorly damped or unstable
other
In
words,
slow-responding
APtie-mode.
units should not be dominant in interconnected
a
of
interconnection
the
areas.
However,
fast-responding area to a slow-responding area may
result in a stable APtie-mode.
eigenvalues.
Interconnected areas of equal sizes have a poorer
dynamic performance than those with different sizes.
An increase in the size of one of the interconnected
The simulation results indicate that the boiler modes
APtie-modes to increase.
only in the time response of the variables
inside the boiler control loops. The boiler effect
the variables outside the boiler model is
on
negligible and can be accounted for by considering
(see
only the gain K4 relating Ams to AV
Appendix).
This gain accounts for the di?ference
between drum and throttle pressures. Figure 3 shows
the variation of k4 as a function of the valve
opening (or the unit loading). Only when the boiler
is loaded close to its nominal output is its effect
pronounced. It is noted that the examined boilers
are of drum-type with medium size drums.
appear
areas
causes
the
damping
of
the
Af-
and
VI. The Number of Units on AGC
The opening of the AGC loops for some of the system
units increases the damping of the Af-mode and the
This
is
especially
APtie-mode significantly.
true when the units without AGC are the larger ones.
Therefore, it is advantageous to use smaller units
for AGC leaving the larger ones without AGC. This is
consistent with the industry practice where very
large units are not normally subjected to continuous
and high-frequency control signals.
3412
Three-Area System Tie-Line Topology
VII.
The topology of tie-line connectilons is dictated by
factors other than AGC. However, it is worthwhile to
examine it.s effects on the AGC system stability.
Figure 4 shows three power pools each consisting of
three control areas interconnected by two tie-lines
The connection of
as shown by the solid lines.
areas 2 and 3 (by the dotted line) has little effect
on the Af-mode due to the low sensitivity of this
The cases (b)
mode to the APtie control action.
and (c) of Fig. 4 are more prone to stability
problems as they include the hydraulic unit with
The
transient droop which has a slow response.
connection of the areas 2 and 3 causes stability
problems in these two cases.
1
NR
1
RH,RH
1
NR
2
Effects of the Boiler
As shown in Se ction 3, the linear drum-type boiler
model does not play a major role 'in the dynamilcs of
the thermal unit.
The boiler control loops act
almost indlependently of the AGC control, loops and
have their own dynamics and eigenvalues.
The
nonlinear boiler m'odel is found to have si'milarly
negligible effects on the AGC system [6].
Figure 4: Three-area pools examined.
NRH = Non-reheat Unit
RH = Reheat Unit
HYD = Hydraul'ic Unit
4.
Limit cycles result in inicreasing the RNS vallue~of
the control error as well as the we'ar and tear in
the system hardware. The abrupt changes in the unit
output can be sufficientl'y damped, if the line'ar part
of the system, follow ing the nonlinear element, acts
as a low-pass filter.
Consequently, abrupt changes
due to the governor backlash a're more attenuated in
the
of
units
than
output
slow-responding
fast-responding ones. On the other hand, the low
frequency oscillations are not attenuaLted, silnce the
linear part of the system, as a low-pass filter,
cannot affect such silgnals.
A large number of AGC systems with various system
structures aknd operating conditilons are examined.
The conclusions reached are listed below.
I.
33
3NR
(H)
(A)
(2) undesirable system response like abrupt changes
in unit'outputs, and
(3) low frequency oscill'ations leading to a sl'ow-down
in the operation of the control loops.
Figure 6 shows the response of an isolated non-reheat
unit to a step of 0.4 pu. load cha'nge. Despite this
abnormally large load change, the nonlilnear boiler
model has a neglilgible effect on the dynamics of the
THE EFFECTS OF NONLINEARITIES
The dynamic behaviour of the AGC system under the
effect of nonlinearities is related to the dynamics
of its model linearized around the operating point.
Approxim'ating a nonlinear element by a variable gailn,
ki,: a necessary but not sufficient condition for
the syste'm instability is that ki would produce
instability in the equivalent linear system [8].
in sections 2 and 3, the dynamics of the AGC model
was shown to be dominated by the system critical
modes. These modes may become un'stable as a result
of control loop gain variations. A nonlin'ear element
can cause instability, if 'it provides a condition
under which one of the critical modes becomes
The condition for an unstable critical
unstable.
mode depends on the locatilon of the nonlinear elemen-t
in the system and the course of possible equivalent
gain (ki) variation. Filgure 5 shows the locations
of some of the AGC nonlinear elements (9,10].
state variables outside the boiler model. However,
its effect is pronounced in the state 'variables
associated with the boiler (dr'um pressure in the
figure). The figure also shows that the linearized
............
..l
..
0.0
10.10
30.00
10.00
FREOUENCY OF AREA
0.10
5080
TIME IN SECONDS
U.0so
7.080
U.W
00.00
70.00
lIMO00.0
00.00
1
zU~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-
'0..
.....
08.10
08.0
Figure 5:
In
addition
nonlinear
form
(1)
AGC system nonlinearities.
to
stability
the
element
can
problem
generate
limit
DRUM PRESSURE DEVIATION OF
stated
above,
cycles
in
a
the
of:
steady
state
undamped
conditions,
osc-illations
30.00
around
the
steady
0.00
50.o
TIME IN SECONDS
UNITo1
Figure 6: The response of an isolated reheat unit to
0.4 pu. step load change.
(x) nonlinear boiler model.
(*) linearized boiler model.
(+) coefficient k4 only.
(A) without any boiler effects.
3413
thea
and
model
approximate model (with only
coefficient k4 considered) have virtually no effect
the system dyn'amics, verifying the results
on
obtained in section 3.
a
k.
II. Effects-of the Control Signal Deadband
I A''A
'i
...
TINE IN SECIONDS
OF
FREQUENCY
AREA
..1.
'
1~~
ci
.. ..
ILU
~~~~....
.. ....
..
(b) with
E.;.
w
!W.
.f\
1
---
ci
ua
A
-NJ
V;v
V.
V,
V"
=a
.........................n.....................V
...v
.....VV.\I..
...v
cc,;.
9-
I.M
HAD
to.60
FREQUENCY
Figure
-j
OF
8:
Figure
The
7:
of
response
morning pick-up
The
~Study
in
deadband
of
width
0.02
deadband
of
width
0.008
of
the
Governor
the
of
'and
APtie-modes
AGC
linear
Af-
the
gain
model
lightly
are
ratio
of
backlash
may
8a,_
changes
pic'k-up
governor
the
the
of
of
limit
the
output
shows
where
the
of
width
0.008%
pu.
pu.
backlash.
no
03
W
0.
33
3.3..
.
0.0
3.00
....
or
may
linear
figure.
the
parametric
or
leading
condition.
backlAsh
.0
..
[N...
NON
GENERATION OF ....T.UN
....EC
.T
.
W.
S...
.N.......
Figure ...10 .The response........ of.n.soatd.aeaofon
one reheat.. unit..... to... a
an
non ehea
increases.
the
of
one
on
governor
(Fig.
cycles
to
response
the
the
power
abrupt
backlash
9)
generating
changes,
interesting
output
and/or
of
the
for
effect
units
low-amplitude load changes
can
clearly be seen in
output changes in a stepw'ise manner.
This
an
.....T.I....
of
-leading
stable
The
to
system
MW-loop
APtie-mode
.......
An
of
the
.W.
The
backlash
effects
model
unit
of
two-area
a
change.
this
For
The response of an isolated area of one
reheat and one hydraulic units to a
0.1 pu step load.change.
(x) w-ith backlash.
(6) no backlash.
the
of
by
1
FREQUENCY OF AREA
Figure 9:
Therefore,
the
the
system.
such
high-frequency
traceked.
of
backlash
8b.
effect
in
in
seen
the
of
'in Fig.
Figure 10.
backlash
with
a
lowest
the
under
under
damping.
of
restabilization
Another
has
modes.
load
under
implementation
the
.appearance
step
the
of
its
damping
shown
with
1..
the
Atf-
backlash
response
stabilized
be
in
improvement
be
mode
changes
structural
the
pu.
clearly
can
critical
is
loop.
when
if
the
instability
to
APtie-mode
of
unstable
the
the
to
reduced
critical
shows
8a
0.1
a
instabil'ity
Fig.
high
a
governor
Thus,
i-mode
Figure
to
effect
showed
the
damped,
the
among
~~~~~~~~~~~ ~~~~~~~~~~~~~prone
system
(x)
to
2.
pu.
APtie-modes
and
variations
two-area
effect.
system
two-area
change
load
pu.
unstable.
them
a
a
'in area
wildth 0.02%
step
Backlash
these' modes are
of
damping
decreases.
loop
gain
In
of
response
pu.
a
1..
governor
damping
-
TliN [N SECODS
sensitivity
make
to
deadband.
no
of
parameter
.The
70.80
"-m
2.
area
+)with
Effects
The
system
two-area
a
condition
(6) 'With
(x)
III.
..
IN SECONDS
0.1
(&)
. ..
AREA
OF
FREQUENCY
IN SECONDS
TINE
The
Wu. YU
GU. uu
19. w
10.00
10.3.
AREA
TINE
....
..
the MW-feedback.
U-M
M-M
Wan
As nn
21
0V
(a)..........
..W
W
.W 3 .W
2
0.3
11
x
A
3~~~~~~~~~~~~~~~~~~~.
CCj
-
t
...I
h.PI1/'.~~~~~~V Vv v v v rt
The control silgn'al deadband approximated as a
variable gain, reduces the, AGC loop gain. Therefore,
the stability of the system is not jeopardized by the
The critical modes
presence of this deadband:
to
namely
instability,
subject
APtie- and
hf-modes, do not bee-ome unstable as a result of AGC
control loop gain reduction.
As for the response of the system to morning pick-up
conditions, the results show that the system load
tracking deteriorates and the RMS of the control
error incr'eases apprecilably (Fig. 7) under the effect
of the deadband.
I
...:.......
effect
is
the
abrupt
plants.
mornilng
the
of
is
that
are
Fig.
not
10
rat liitsoaeareahighcopaed
or gThe rseroneo
tothe rate ~~~~ofceheang oflod unernormaeopeatnt ting
oremergency pic-upcondiios. Thre oren herefeto
Thegvale
3414
*,
the system response can be ignored. This is verified
for step load changes (Fig. 11) as well as morning
pick-up conditions.
. N
tL
IL
JE
-- *-
;-'''*,'*-'';-''*''--''';..'..''..'..'..''..
'...
*.
'-..
;'.'*...
-.
",..
-..'*.....'*.;
6
U._: :
.
1@LM
HADZ
Figure 11:
U.0
w.
i__
....
.....
N'
......O _
.
ION Of
on
5.
Ks-_-__,.
..:..
**
REHEAT ; ........
UNI T
C~~~~~~~~ENERAT ................
...............
o
.....
an
aTIN
U
T iME IN SECOMD
iani.
n
N
U.E "N
As an
The response of an isolated area of one
reheat and one hydraulic units to a 0.1
step load change.
(x) with valve rate limits.
(+) with speedchanger rate limits.
(A)
no
nonlinearity.
the other hand, the speedchanger rate limits play
major role in the unit response, especially for
thermal units where the limits correspond to +1-4% of
their maximum continuous rating (MCR) per minute.
For hydraulic units the limits can reach +100% of the
unit MCR per minute.
The effects of speed limiters are summarized below:
On
a
(1) The speedchanger
rate limiter has almost no
effect during the normal operating conditions.
(2) The stability of the system is not threatened by
this limiter since the Af- and APtie-modes
do not become unstable when the AGC loop gain is
reduced.
(3) The
speedchanger
rate
limiter
causes
an
appreciable slow-down in the AGC control action.
This effect is pronounced when the rate limits
are low or the rate of change of the load is high
(Fig. 11). Low frequency limit cycles, referred
to here as AGC-limiter oscillations emerge in
such cases.
V.
The Simultaneous Effects of the Speedchanger
Rate Limiter and the Governor Backlash
__.
.
-,
.................-A
.......O.
M
*
14
64
P ~
i
-.
..
-.
.._
_
t- .
. F :.
FI1- .:
..........UNIT
, ......N
....GENERATION OF NON-RHT UNIT IN
.......
AREAwl
',
.............
TIME IN SECONDS
Figure 12 The response of an isolated area of one
non-reheat and one hydraulic units to a
0.1 pu step load change
(A) with speedchanger rate limits and
the backlash.
(+) with speedchanger rate limits only.
(x)
no
Simulation studies were carried out for step load
changes representing emergency conditions and ramp
loads with random components representing morning
pick-up conditions in a wide range of system
configurations and operating conditions. The results
show that: the presence of nonlinearities can result
in destabilizing the lightly damped critical modes of
the system. The system regains stability when the
damping of these modes is increased using suggested
measures.
Nonlinearities also result in undesirable
abrupt changes in unit outputs and/or cause an
increase in control errors.
ACKNOWLEDGEMENTS
The authors wish to express their thanks to Prof.
A. Semlyen
of
the
Department
of
Electrical
Engineering, University of Toronto for his support
throughout this study.
Thanks are also due to
Mr. P.L. Dandeno of Ontario Hydro for his suggestions
during the preparation of this paper and to the NRC
for financing the computer expenses.
REFERENCES
[1]
121
[4]
[5]
----
-4
nonlinearity.
CONCLUSIONS
An AGC model for multi-area, multi-unit systems was
developed to study system dynamics with and without
the effects of nonlinearities.
Based
on
the
eigenvalue analysis, critical modes were identified
and related to the system control loops. The effects
of the system configuration and the loop parameters
on the damping of these modes were examined.
[3]
The step responses of the examined systems show that
the simultaneous effects of the rate limiter and
governor backlash is mostly a combination of their
individual effects.
Generally, when the rate of
change of the load is high, AGC limiter oscillations
dominate.
Figure 12
shows
the
high-frequency
backlash limit cycles superimposed on the low
frequency AGC limiter oscillations in response to a
4*.
step load change. On the other hand, the backlash
limit cycles are pronounced when the rate of change
of the load is low. Thus, for a simulated morning
pick-up, the system response is only under the
backlash effect.
O.I. Elgerd, Electric Energy Systems Theory,
McGraw-Hill Book Co., 1971, Chapter 9.
G.
Quazza,
control
"Non-interacting
of
interconnected power systems", IEEE Trans. Power
App. Syst., vol. PAS-85, pp. 727-741.
G. Quazza, "Criteria for equitable participation
of areas in tie-line power and frequency control
of an interconnected power systems", Automazione
e Strumentazione, March 1966.
F.P. deMello, R.J. Mills and W.F. B'Rolls,
"Automatic generation control, part I: process
modelling", IEEE Trans. Power App. Syst., vol.
PAS-92, pp. 710-715, March-April 1973.
R.A.
Schlueter, J. Perminger, et al., "A
nonlinear
model
dynamic
of
the
electric
coordinated systems", IEEE Winter Meeting, Paper
No. C75088-0, New York,
N.Y., January 26-31,
1975.
E.B. Shahrodi,
"Modelling and analysis of AGC
N.A.Sc.
systems",
Thesis,
of
Department
Electrical Engineering, University of Toronto,
January 1981.
[7] John W. Brewer, Control Systems Design, Analysis
and Simulation,
Prentice Hall, Inc., 1974,
Chapter 16.
and
[81 R.E.
"Physical
mathematical
Kalman,
mechanisms of instability in nonlinear automatic
control
Trans.
Vol.
systems",
ASME,
79,
pp. 553-563, 1957.
[9] IEEE Committee, "Dynamic models for steam and
hydro turbines in power systems studies", IEEE
Trans., vol. PAS-92, pp. 1904-1915, Nov./Dec.
1973.
(10] C. Concordia, L.K.
Kirchmayer, E.A. Szymanski,
"Effect of speed-governor deadband on tie-line
power and frequency control performance", AIEE
[6]
3415
Trans. on Power Apparatus and Systems, vol. 75,
pp. 429-435, August 1957.
APPENDIX
and the quantities at the operating point are denoted
by a subscript 'o' (PDo, PTo, etc).
The values chosen for the storage constant, cb, are
between 90 and 300 seconds [41. The definition of
the boiler parameters and their typical values are
given in Table Al.
Boiler Model
The drum-type boiler model used in the study is shown
in Figure A.l.
Table A.1
Data for Drum tyRe Fossil Fuelled Boiler
Tw
Tf
TD
Tr
Waterwall time constant
Fuel system time constant
Fuel firing system delay time
Lead-lag compensator time
Boiler integrator gain
Proportional-integral ratio
of gains
Superheater friction drop
coefficient
Throttle pressure setpoint
Valve position setpoint
Steam flow setpoint
Boiler storage constant
KI
TI
K
PT0
PVO
msO
Cb
'igure A.1:
The nonlinear boiler
model
a drum-type boiler, the throttle pressure PT is
ferent from the drum pressure PD by the
rheater drop PDT. This drop is given as
su
PT
PD
where
mS
=
Vp
=
is
= k
(1)
2s
(2)
VPPT
valve position,
steam flow, and
the superheater friction drop coefficient.
=
k
PDT
=
=
Equations (1) and (2) are included in the simulation
using the boiler model shown in Fig. A.1.
process
In this figure,
fl(PD,Vp)
=
f2(PDV)
=
+4k Vp
Vp f1(P
1)
/(2kVp
VP)
,
The model of Fig. A.1 is linearized
pT
s
where
=l
2
as
shown below:
APD + k2 AVp
k3 APD
+
k4 pV
V2
bf 1 /bpD=
(1+4k
k1=
1
D
k
)
~f /bVp
1
=
k4 = bf2/bVp
(1+4kc
kI
k3 =6f2JbPD
=
Po
VPO
2
PDo
-1/2
p
-1/2
PDo)
Po
pTo+ k2
PO
g
2~/V
20 2Apo2
/VVP
4.0 sec
10. 0 sec
6.0 sec.
69.0 sec.
0.03
26.0
0.06
1.0 pu.
0.5 pu.
0.5 pu.
90.0 300.0 sec.