Research Journal of Applied Sciences, Engineering and Technology 3(11): 1239-1245, 2011
ISSN: 2040-7467
© Maxwell Scientific Organization, 2011
Submitted: July 26, 2011
Accepted: September 18, 2011
Published: November 25, 2011
A New Approach to Dynamic Control of Synchronous Generator in a Bulk
Electric Power System by Direct Feedback Linearization
S. Sajedi, F. Khalifeh, T. Karimi and Z. Khalifeh
Department of Electrical Engineering, Fasa Branch, Islamic Azad University, Fasa, Fars, Iran
Abstract: Feedback linearization is a common approach used in controlling nonlinear systems. The approach
involves coming up with a transformation of the nonlinear system into an equivalent linear system through a
change of variables and a suitable control input. In this paper a new control structure based on the application
of the method of exact linearization by Direct Feedback Linearization for a synchronous generator is presented.
The power system model consists of a single axis flat rotor machine, connected to an infinite power bus. This
way the sub-transient phenomena are ignored. The global system consisting of a turbine and a generator that
produces the electrical output power is coupled and it presents multiplicative and harmonic non-linearities. The
application of a DFL on the output voltage equation expressed as a function of the electrical system parameters
referred to the generator practically turns the non linear system into two uncoupled subsystems with the
possibility of independently controlling the output voltage and the frequency.
Key words: Control non-linearity, fault-tolerant systems, linearization, linear optimal regulators, power
systems control
INTRODUCTION
The design of most field excitation controllers in
energy generators is based on methods that use
differential geometry to solve the problems related to the
non linear characteristic of the power systems (Isidori,
1989). Recently a different approach to this problem (Gao
et al., 1990, 1992; Wang et al., 1993; Junco et al., 1996)
which uses the method called “Direct Feedback
Linearization” (DFL), has allowed the design of power
systems regulators using techniques well-known in the
study of linear systems. That is, starting with a high-order
differential-equation model, arriving at a linear control
through state variables feedback. Contrary to what is
proposed in previous works (Gao et al., 1990, 1992;
Wang et al., 1993; Junco et al., 1996; Matas et al., 2008),
where a state feedback that exactly linearizes the
relationship field voltage-electrical output power is
carried out, in this paper a DFL-based design strategy
applied on the output voltage equation from the generator
connected to an infinite power bus, allows the
implementation of a voltage linear control practically
independent of the frequency modifications caused by
mechanical power variations.
The results obtained by means of simulation of this
control type show a good response in relation to the
degree of prospective uncoupling, to the dynamic
evolution of the variables and the global stability of the
system, when there are modifications in the work points
in a wide generator operating range. On the other hand
tests carried out taking into account ground faults in the
transmission lines, denote a good behavior of the system
post-fault in relation to its transient stability. Two types of
DFL were kept in mind, one fixed, and the other adaptive
which takes into account the variations of its parameters
due to line reactance modifications during and after
having produced the fault.
The power system model consists of a single axis flat
rotor machine, connected to an infinite power bus. This
way the sub-transient phenomena are ignored. The global
system consisting of a turbine and a generator that
produces the electrical output power is coupled and it
presents multiplicative and harmonic non-linearities. The
application of a DFL on the output voltage equation
expressed as a function of the electrical system parameters
referred to the generator practically turns the non linear
system into two uncoupled subsystems with the
possibility of independently controlling the output voltage
and the frequency.
MATHEMATICAL MODEL
The power system is modeled as a single turbinegenerator machine connected to an infinite power bus
through two parallel transmission line groups (Fig. 1). The
mathematical equations that represent the generator
dynamics of a flat rotor machine, single axis model, are
written following the standard notation. The dynamics of
the turbine and the steam control valve are described by:
Corresponding Author: S. Sajedi, Department of Electrical Engineering, Fasa Branch, Islamic Azad University, Fasa, Fars, Iran
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Res. J. Appl. Sci. Eng. Technol., 3(11): 1239-1245, 2011
where,
xds = xT + xL +xd
x’ds= xT + xL +x’d
xs = xT + xL
Fig. 1: Power system consisting of a turbo generator group
connected to an infinite power bus
P&m (t ) = −
1
TT
Pm (t ) +
KT
TT
X E (t )
(1)
X& E (t ) = −
1
TG
X E (t ) +
[ Pc (t ) −
KG
TG
1
Rω 0 ω ( t )] (
2 )
The mechanical equations of the generator are:
δ&(t ) = ω (t )
ω& (t ) =
D
2H
Exact linearization of the output voltage Vt: The design
requirements for a good robust non- linear voltage
stabiliser, lead to maintaining the value of nominal
voltage Vt within a band of specified tolerance and in a
wide generator operation range. Therefore the voltage Vt
must be included as a feedback signal. According to the
approach of the linearized plant (Gao et al., 1990; Wang
et al., 1993; Junco et al., 1996) as a function of the
variables w, d and Pe, the output voltage Vt is a strongly
non linear function of these variables, and it is not
possible to apply design procedures for linear systems.
To solve this problem, in this paper the application of
the exact feedback linearization technique on the Eq. (11)
is proposed. Differentiating this expression, yields:
(3)
ω (t ) −
ω0
2 H [ Pm ( t ) − Pe ( t )]
V&t2 = 2
(4)
x s2
2
xds
Eq E& q + 2 xsxd
xds
(12)
Vs( Eq cosδ − ωEq senδ )
The generator electrical dynamics is represented by:
E& q′ (t ) = ( E f (t ) − Eq (t )) Td1′ 0
where for the sake of simplicity the time variable has
been suppressed from the notation, and where:
(5)
V&t 2 =
The complete electrical system equations referred to the
generator are represented by:
Eq ( t ) =
Eq, (t ) −
xds
,
xds
( xd − xd, )
,
xds
Vs cosδ (t )
= xad I f (t )
senδ (t ) =
I q (t ) =
Vs
xds
pe (t ) =
Vs Eq ( t )
xds
Pe ( t )
xad I f ( t )
senδ (t )
(6)
d (Vt2 )
dt
(13)
Starting from the generator electrical and dynamic
equations (Eq. 5, 6) with T'd = T’d0 x'ds /xd , the shortcircuit transient-time constant in the direct axis, the
following equation can be written:
(7)
& =
Eq
1
Td,
( k c u f − Eq ) +
( xd − xd, )
,
xds
Vsωsenδ
(14)
(8)
Replacing (14) in (12) and grouping terms, results in:
Q( t ) =
Vs
xds
Eq (t ) cosδ (t ) −
E f ( t ) = kc u f ( t )
Vt 2 (t ) =
+
xs2
2
xds
2 xs xd
2
xds
Eq2 (t ) +
Vs2
xds
(9)
(10)
xd2
2
xds
Vs2
Vs Eq (t ) cosδ (t )
(11)
V&t 2 = −
1
Td,
Vt 2 +
1
Td,
vf
(15)
which represents a first order system in Vt2 and where vf is
expressed in Eq. (16). In accordance with the DFL theory,
the exact cancellation of non linearities takes place if the
excitation voltage uf is obtained starting from vf according
to Eq. (17), written as a function of measurable variables
of the plant:
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Res. J. Appl. Sci. Eng. Technol., 3(11): 1239-1245, 2011
vf =
x s2
2
xds
+ 2Td, [
{2 kcu f ( Eq +
,
( x d − xd )
,
xds
x
−
d
xs
ωVs2 senδ cosδ +
uf =
2
xds
xs2
xd2
x s2
Vs cosδ ) − Eq2
]ωVs Eq senδ + 2Td,
Vs2
x x
2Td′0 dsxs d
( xd −
2 kc xad I f [1 +
(16)
,
xd xds
xs )ωPe
Fig. 2: Block diagram of the linearized plant
V2
( Q+ s )
xds xd
xds
xs . ( x I ) 2
ad f
V2
( Q+ s
xds
,
xd )
( xad I f ) 2
x ds x d
xs
,
xd ( xd − x d )
,
x s xds
}
v f + ( xad I f ) 2 − 2Td′0 ( xd − xd,& 0 −
2 kc xad I f [1 +
−
xd
xs
equations. Pe is a non linear function in Vt, and in the
system under study it is considered as an auxiliary input
with characteristics of “perturbation”. The control scheme
this way defined presents a high uncoupling degree
among the voltage and frequency controls, which can be
used independently to modify the excitation field and the
mechanical power respectively, as it is shown in the
simulations carried out in the following sections.
(17)
xd2 2
V
xs2 s
ωPe −
V2
( Q+ s )
xds
.
( xad I f ) 2
]
Since this procedure leads to a linear equation in Vt2 it is
possible to take it to a linear equation in Vt if one keeps in
mind that:
V&t 2 = −
Vt 2 +
1
,
Td
2VtV&t 2 = −
V&t = −
1
,
2 Td
1
,
Td
1
,
Td
2
vg = K p (Vt 0 − Vt ) + Ki ∫0t (Vt 0 − Vt )dt
vf
Vt +
Vt +
1
,
Td
vf
vf
(18)
Pc = Pm0 + Km[ Pco − ( Pm − Pmo )]
− K XE ( X E − X E 0 ) − Kωω − Kδ (δ − δ0 )
(22)
with Pc0 , Pm0 , XE0 and *0 representing steady state values,
and:
vf
(19)
vt
J = ∫0∞ ( X T QX + U T RU )dt
0
0 ⎤
⎡123 0
⎢ 0 150 0
0 ⎥⎥
Q= ⎢
R=1
⎢ 0
0 160 0 ⎥
⎥
⎢
0
0 120⎦
⎣ 0
deriving vf and applying the DFL technique again, the
system of equations corresponding to the linearized plant
is obtained.
where,
δ&(t ) = ω (t )
ω& (t ) = −
V&t = −
(21)
with Kp = 2 and Ki = 1/ T’d0 and a LQC - type feedback
for the turbine frequency regulator subsystem:
1
, v
2 Td t
making:
vg =
Control law: The following PI-type control for the
voltage regulator subsystem is proposed:
D
2H
1
2 Td,
ω (t ) +
Vt +
ω0
2 H [ pm ( t ) −
1
2 Td,
X T = [δ (t ) ω (t ) Pm (t ) X E (t )]
(20)
⎣
⎦
K = k s Kω km K X E = [1109
. 1317
. 148.731127
. ]
Pe (t )]
vg
The plant block diagram is shown in Fig. 2.
The independence of the Vt voltage control regarding
the other variables is observed in the linearized system of
and considering that the measurements of the angle * are
available. (for example by means of the use of an
observer).
To find the elements of Q (with R = 1) the system
was tested for different operating conditions, measuring
the overshoot and the response time for each state
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Res. J. Appl. Sci. Eng. Technol., 3(11): 1239-1245, 2011
Output voltage [ppu]
variable. Then the results were plotted as a function of Q
and it was observed that the most satisfactory behaviour
of the system takes place for a range of values between
100 and 1000. A later adjustment taking into account the
maximum values of the constants in the feedback loops,
allows defining the main Q finally adopted.
SIMULATION RESULTS
For all the tests carried out, typical values of a plant
were kept in mind (Wang et al., 1993) and furthermore
saturation phenomena were not considered, they were
replaced by constraints in the field control voltage, as it
will be seen in the section 5.2. The values correspond to:
40 =50 r/s, D = 5.0 ppu, H = 4.0 s, T’d0 = 6. 9 s , Tg = 0. 2
s, TT = 2.0 s ,Kg = KT = kc = 1, R = 0.05 ppu , xd = 1.863
S x’d = 0.257 S, xT = 0. 127 S and max ½kc uf½ = 4.
Three types of tests were carried out. The first of
them consisting of a step in the voltage Vt set point, the
second considering a step in the given mechanical power
set point, and the third simulating a symmetrical threephase fault to earth in one of the two transmission line
groups. It was also kept in mind a parameter that
measures the point on the line where the fault takes place
expressed in ppu (0#8<1 ), 200ms after the fault has taken
place, the disconnection of the affected line occurs. The
reactances that compose the pattern of the system are
modified during and after the fault, in accordance with its
location along the transmission lines.
1.1
1.05
1
0
5
10
15
20
Time [sec]
Deviation from synchronous value [Hz]
(a)
2
0
-2
-4
-6
-8
-10
-12
-14
-16
-18
x10
0
-3
5
10
Time [sec]
15
20
(b)
Unconstrained field control voltage: The first test shows
the voltage and the frequency evolution (Fig. 3a, b) for a
voltage step of 10%. It is proven that the response of Vt is
exponential and independent of the evolution of w. The
frequency graph shows a transient owing to the
“perturbation” Pe. The second test allows to check the
uncoupling of the system, (Fig. 4a, b) where a
modification in the frequency for effect of a mechanical
power action, does not affect the output voltage that
maintains a constant value. The third test consists of
simulating a fault in one of the two transmission lines
groups of 200 ms duration previous to the breaking off of
the affected line. In this case, two types of DFL were
used, one fixed taken from the nominal parameters of the
plant, and the other adaptive that contemplates the
variations of the line equivalent reactances during and
after the fault.
The Fig. 5 shows the behaviour of Vt for the two
cases. It can be observed that for the adaptive DFL the
response Vt recaptures the original value linearly, while
for the fixed DFL it presents an offset in the final value
with non linear evolution. It should be remarked that the
adaptive DFL shows the exact linearization achieved on
Vt, but it is not of direct practical implementation. In Fig.
6 the evolution
Fig. 3: Output voltage set point step test (10%). (a) evolution
of the generator output voltage Vt. (b) deviation from
synchronous frequency value 40
of the frequency is shown, where after the transient the
synchronous frequency is reached, while in Fig. 7 the
electrical output power Pe under identical test conditions
for the case of fixed DFL is plotted. There have been
carried out tests using three values for the parameter
(namely 0.001, 0.1 and 0.5).
In Fig. 8 the temporary evolution of the load angle *
is shown for different values of the parameter 8. From
that figure it is possible to observe during and after to the
fault that its evolution remains within the more stable
band of operation of the generator, reaching a new
value as a function of the line conditions after the fault. In
the Fig. 9 and 10 the output voltage of the generator Vt
and the excitation control voltage uf, are shown,
respectively. As it has been mentioned before, the
behaviour of Vt is not too linear but asymptotically stable,
while the excitation control voltage remains, during and
after the fault, within the typical margins (±2.5 to ±3.5)
before the cutting that the limiting circuits produce.
1242
Electrical output power [ppu]
0.7
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
(a)
-0.005
-0.01
-0.015
10
Time [sec]
0.001
0.5
0.1
0.5
0.4
0.3
0.2
0.1
2
20
15
3
4
5
7
6
Time [sec]
Fig. 7: Fault to earth test. Plot of the electrical output power Pe
for different values of 8
2 (b)
1.8
1.4
1.2
1
0.8
0.6
G enerator load angle [deg]
80
0.4
0.2
0
0
4
10
Time[sec]
15
20
Fig. 4: Mechanical power set point step test (10%) (a) evolution
of the frequency referred to the synchronous value w0.
(b) evolution of the generator output voltage Vt
0.15
0.001
75
70
0.1
65
0.5
60
55
50
45
40
2
3
4
5
Time [sec]
7
6
Fig. 8: Fault to earth test. Behavior of the load angle d
for different values of 8
Fixed DFL
1.00
Output voltage (ppu)
0.6
0
5
0
Output voltage [ppu]
Deviation from synchronous value [Hz]
Res. J. Appl. Sci. Eng. Technol., 3(11): 1239-1245, 2011
Adaptive DFL
0.95
0.90
0.85
Output voltage [ppu]
0.80
0.75
0.70
0.65
5
0
10
Time (sec)
15
20
Deviation from synchronous value [Hz]
Fig. 5: Fault to earth test (200 ms duration). Evolution of the
generator output voltage Vt
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
2
0.4
3
4
5
Time [sec]
6
7
0.001
0.3
Fig. 9: Fault to earth test. The evolution of the generator output
voltage Vt is less l dependant
0.2
0.1
0.5
0
0.1
-0.1
-0.2
-0.3
2
3
5
4
Time [sec]
6
7
Fig. 6: Fault to earth test (200 ms duration) with later
desconection of the affected line. Evolution of the
synchronous frequency w in function of 8
Constrained field control voltage: As it can be seen in
Fig. 10, the system so designed does not present
saturation phenomena (the signal uf lies between the
linear excursion band). In the following tests, the
excitation control parameters were modified so as to
reduce the rise time in the output voltage set point step
test (Fig. 3a) by a factor of 10 (Kp = 20, Ki = 10/T’do). The
graphics obtained (Fig. 11a, b) show the evolution of the
output voltage and the practically negligible variation of
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Res. J. Appl. Sci. Eng. Technol., 3(11): 1239-1245, 2011
Excitation control voltage [ppu]
3.5
3
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
3.5
3
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
0.5
2
3
Excitation control voltage [ppu]
0.001
0.1
4
5
Time [sec]
6
7
Output voltage [ppu]
Output voltage [ppu]
1.05
1.0
Deviation from synchronous value [Hz]
10
Time [sec]
15
0.1
20
4
5
Time [sec]
6
7
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
2
0.04
0.02
3
Fig. 12: Fault to earth test, Field control voltage, showing
saturation phenomena for 8 = 0.5, 0.1 and 0.001
1.10
5
0.5
2
Fig. 10: Fault to earth test. Behaviour of the excitation control
voltage uf for three value of 8
0
0.001
3
4
5
Time [sec]
6
7
Fig. 13: Fault to earth test, Output voltage Vt for the values of
8 = 0.5, 0.1 and 0.001
0
-0.02
CONCLUSION
-0.04
-0.06
-0.08
0
5
10
15
20
Time [sec]
Fig. 11: Output voltage step test-10%. Rise time improvement
by a factor of 10, (a) output voltage Vt. (b) output
frequency T
the frequency, as a result of the “perturbation” produced
by the modification in the electrical output power. Then,
the fault tests were repeated for duration of 200 ms. Due
to the new and higher control demand, the field control
voltage exceeds the limits values set at ±3.5, thus
producing the cuttings shown in Fig. 12. It can be
observed that in this case the system remains stable as
shown in the output voltage evolution (Fig. 13). It should
be remarked that the system does not present noticeable
deviations in the evolution of the others variables, i.e.,
frequency, load angle, electrical output power, with
respect to those plotted in Fig. 6, 7 and 8.
The results obtained by means of the simulation
justify the hypothesis outlined in the definition of the
mathematical model, where the implementation of a DFL
on the output voltage equation, allows to linearize and
uncouple the voltage regulator loop and to act on the
frequency loop, considering the electrical output power
variations as an “auxiliary input”. It is observed that in the
face of a change in the given power, the frequency
changes in a controlled form and it does not affect the
voltage regulation, while modifications of the output
voltage alter the frequency during a bounded transient. As
a consequence of the decoupling among the voltage and
frequency variables, it is possible to adjust the control
maintaining system stability, in a wide range of operating
conditions. In tests carried out considering faults in the
transmission lines, the obtained results allow to conclude
that the response post-fault in the case of using the ideal
adaptive DFL is linear and asymptotically stable, while in
the practical case of the fixed DFL is non-linear and
presents an voltage offset product of the variation of the
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Res. J. Appl. Sci. Eng. Technol., 3(11): 1239-1245, 2011
plant parameters with respect to those used in the original
DFL calculation. Also, it can be verified that a more
severe control action that turns the system into non linear
operation, does not produce a loss in the overall plant
stability.
xd
x'd
xL
xad
reactance in the direct axis of the generator
transient reactance on the direct axis of the
generator
reactance of the transmission line
mutual reactance between the excitation winding
and the winding of the stator
NOMENCLATURE
REFERENCES
Pm
XE
Pc
TT
KT
TG
KG
*
4
Pe
D
H
T’d0
E'q
Eq
Ef
If
Iq
Q
Vs
Vt
uf
kc
xT
mechanical power
opening of the steam valve
reference of power
turbine time constant
gain of the turbine
valve time constant
gain of the speed regulator
generator load angle
relative speed to the synchronous 40
active power surrendered by the generator
damping constant
moment of inertia
transient time constant in open circuit
transient emf in the quadrature axis
emf in the quadrature axis
equivalent emf in the excitation coil
current of excitation field
current in the quadrature axis
reactive power
voltage on the infinite bus
output voltage of the generator
control input of the excitation amplifier
gain of the excitation amplifier
reactance of the transformer
Gao, L., L. Chen, Y. Fan and H. Ma, 1990. The DFL
Nonlinear Control with Applications in Power
Systems, Technical Report EE9001, Department of
Automation, Tsinghua Univresty., Beijing, China.
Gao, L., L. Chen, Y. Fan and H. Ma, 1992. A Nonlinear
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Isidori, A., 1989. Nonlinear Control Systems: An
Introduction. 2nd Edn., Springer Verlag, NY, USA.
Junco, S., S.A. Czajkowski and J.C. Nachez, 1996.
Control No lineal de Excitación del Generador
Sincrónico Mediante Inversión Causal, AADECA
’96,7º Congreso latino-americano de Control
Automático, Argentina, pp: 140-145.
Matas, J.C.,M.Guerrero, J.M. Garcia de Vicuna, L. Miret
and J. Feedback, 2008. Linearization of direct-drive
synchronous wind-turbines VIA a sliding mode
approach.IEEE Trans. Power Elec.,23(3):1093-1103.
Wang, Y., D.J. Hill, R.H. Middleton and L. Gao, 1993.
Transient stability enhancement and voltage
regulation of power systems. IEEE Trans. Power
Syst., 8: 627.
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