04608712
TRANSCRIPT
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Abstract - Unified Power Flow Controller (UPFC) is a
multifunction Flexible AC Transmission System (FACTS) device
with capability of performing such several functions as active
and reactive power flow, voltage and stability control in a power
system. It is well-known that strong interactions exist between
active and reactive power flow control functions of a UPFC.
From another point of view, there are also interactions within
internal parts of UPFC and also between UPFC and power
system. In this paper, considering power system and UPFC’s
model as a unit including series and shunt converters, common
DC link and thevenin equivalent circuits of power system oneither sides of UPFC, in addition to interaction analysis, a
Singular Value Decomposition (SVD) controller will be designed
for active and reactive power flow control. In order to evaluate
the performance of the proposed controller, computer simulation
using MATLAB/SIMULINK software has been provided
comparing performance of the proposed controller with
Decoupling Matrix (DM) and Proportional-Integral (PI)
controllers. Simulation results have clearly confirmed the
competence of SVD over DM and PI.
I. I NTRODUCTION
Fast growing of power electronics provided facilities tomake use of FACTS devices. The capabilities of FACTS have
been clarified in improving such areas as active and reactive
power flow, voltage, stability, oscillation damping . . .[1,2,3].
These devices in the most perfect and applicable form, until
now, have introduced themselves as UPFC. UPFC is a flexible
device capable to perform several functions of which, active
and reactive power flow control can be mentioned as the most
important one. Fig. 1 shows the simplified schematic diagram
of a UPFC.
As shown in this figure, UPFC is composed of two fully-
controlled series and shunt converters connected to each other
through a common DC link in one hand, and to power system
through the corresponding series and shunt transformers, on
the other. In the context of power flow control, these
Fig. 1, Configuration of a UPFC system,
converters are able to exchange active and reactive power
through the DC link. In this respect, shunt converter is just
responsible for providing the series converter with active
power, while both can send and receive reactive power from
the power network independently. Controlling the active
power by the series converter involves in both the voltage
amplitude and phase to be changed which, in turn, leads to
some changes in reactive power [4,5]. Interactions between
the internal parts of UPFC in one hand and between UPFC
and power system on the other, along with interacting activeand reactive power flow control, therefore, deteriorates the
UPFC's performance. Form the control's point of view, it isdesired to have different parts as independent as possible so
that more effective control to be achievable. Aiming at
improvement of active and reactive power flow control
while considering the reduction of interactions, a method
based on d-q axis have been developed in [6] for the first
time. Addressed in [4], authors suggested a SVD based
controller in which, they just focused on interactions
between functions ignoring other ones including those
between UPFC's internal parts and UPFC and power
system. Another method called DM with the purpose of
decoupling the whole UPFC's Multi-Input Multi-Output(MIMO) system into some Single-Input Single-Output
(SISO) ones, which is based on the inverse transfer function
of the system have been proposed in [8], but, totally, it
cannot be generalize to all the systems. In this paper,
considering power system and UPFC’s model as a whole
including series and shunt converters, common DC link and
thevenin equivalent circuits of power system on either sides
of UPFC, beside interaction analysis, a SVD controller will
be designed for active and reactive power flow control. In
order to evaluate the performance of the proposed
controller, computer simulation using
MATLAB/SIMULINK software has been provided.
Simulation results have clearly proved the competence of
SVD controller over DM and PI ones.
II. POWER SYSTEM AND UPFC'S MODEL
The one-line diagram of the power system and UPFC is
shown in fig. 2. In this figure, v se and v sh denote series and
shunt converters, respectively. R sh and L sh are resistance and
leakage inductance of shunt transformer. Power system of
either sides of UPFC is shown by its thevenin equivalent
circuit. v s, R s, L s, vr , Rr , Lr are thevenin equivalent voltage
source and impedance of left and right side of UPFC,
respectively. Considering this one-line diagram of the three-
Dynamic SVD Controller Design of UPFC for
Power Flow Control Considering InteractionsM. Ghanbari, S. M. Hosseini
PhD Student of Islamic Azad University Science and Research Branch, Tehran, Iran
Faculty Member of Islamic Azad University Ali Abad Katool Branch, Iran.mmm gh 53 yahoo.com, Mhoseini346 gmail.com
978-1-4244-1706-3/08/$25.00 ©2008 IEEE.
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Fig. 2, Single- phase equivalent circuit of a three-phase UPFC system
phase UPFC, the state space presentation of the whole system
will be as follows:
⎥⎦
⎤⎢⎣
⎡
−−
+−+⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡
+
+
+⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
rv s svr
rv s sh sv sh
shv
sev
r s s
s s sh
shv
sev
shv
sev
v Lv L
v L Lv L
v
v
L L L
L L L
i
i
d c
ba
g i
i
dt
d
)(
1
(1)
Where,
sr shr sh sh L R L R L Ra −−−=
r s sr L R L Rb +−=
s sh sh s L R L Rc −=
r shr s s sh L R L R L Rd −−−=
sr shr sh s L L L L L L g ++=
The three phase differential equation (with v=a, b, c) in (1)
can be transformed into an equivalent two-phase (d, q) system
equations using Park’s transformation [6]. The transformed
equations in the d–q reference frame can be written as follows:
)2()(
)(
00
00
0000
0
0
0
0
1
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−−
−−
+−
+−
+
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+
+
+
+
+
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
rq s sqr
rd s sd r
rq sh s sq sh
rd sh s sd sh
shq
shd
seq
sed
r s s
r s s
s s sh
s s sh
shq
shd
seq
sed
b
b
b
b
shq
shd
seq
sed
v Lv L
v Lv L
v L Lv L
v L Lv L
v
v
v
v
L L L
L L L
L L L L L L
i
i
i
i
d g wb
g wd b
ca g w
c g wa
g
i
i
i
i
dt
d
where, wb=2π f b is the fundamental frequency of the supply
voltage. Other variables in (2) can be defined as follows:
)( sq sd sjvvv += : Sending end voltage
)( rqrd r jvvv += : Receiving end voltage
)( seq sed se jiii += : Series converter current
)( shq shd sh jiii += : Shunt converter current
)( seq sed se jvvv += : Series converter voltage
hsq shd sh jvvv += ( : Shunt converter voltage
A new control strategy (based on d-q rotating frame),
representation for a UPFC system has been presented by the
authors [9]. The principle of this new control strategy is toconvert the measured three-phase currents and voltages in to
d-q values and the current references are calculated from
desired active and reactive power references and measured
voltage by using (3) [10],
22
22
)(
3
2
)(32
qd
d ref qref
qref
qd
qref d ref
dref
vv
vQv P i
vvvQv P i
+
+=
+−=
(3)
The power flow control is then realized by using properly designed controllers to force the line currents to
flow their references. It is desired that the UPFC control
system has a fast response with minimal interaction between
the real and reactive power flow.
A simple way to design a controller for a complex system isto obtain the state space equations of the system. Based on
(1) - (2), the state space model of the system including
UPFC is described by (4).
Cx y
Bu Ax x
=
+=&(4)
where C is a unitary matrix, ],,,[ shq shd seq sed iiii x =& is the
states vector and ],,,[ shq shd seq sed vvvvu = is the input
vector and ],,,[ shq shd seq sed iiii y = is the output vector.
Using the state space presentation, the transfer function
matrix G can be derived as:
B ASI C sG 1)()( −−= (5)
Note that d and q subscripts point to direct and quadratic
axes components. As mentioned before, series and shuntconverters are coupled through a common DC link. The Vdc
is influenced by active power balance between two
converters. If P se> P sh, it decrease while for P se< P sh, it
increases, where P se and P sh are active powers of series and
shunt converters, respectively.
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Fig. 3, dc equivalent circuit
Approximate equivalent circuit, which is shown in Fig. 3,can be used for dc-link modeling,. Where, resistor R simulates
the losses of a switching in both series and shunt converters.
Referring to this figure, the dynamic behavior of the capacitor
can be shown as:
RC
V ii
C dt
dV
iiii
dc sed shd
dc
R sed shd c
−+=
−+=
)(1
(6)
The equation (6) is a linear differential equation from which
Vdc can be obtained.
III. I NTERACTION ANALYSIS OF UNCONTROLLED SYSTEM
As mentioned above, within UPFC, active and reactive
powers are desired to be controlled independently. In park's
frame and UPFC system, the real and reactive power in series
(shunt) inverter depends on id and iq of series (shunt) inverter's
current respectively. So the control of real and reactive power
can be reduced to the control of d and q axes currentsrespectively. In the following, the interactions between d and
q axis currents of both converters, i.e., the state space of
model's outputs are analyzed.
The eigen values of uncontrolled system (iλ , i=1,2,3,4)
are: -667.9+314.2i, -667.9+314.2i, -1000.24+314.2i,
-1000.24+314.2i. Since all the real parts are negative, the
whole system will be stable [11]. As shown in fig. 5 (a), thestep response of the system implies existence of strong
interactions between unpaired inputs and outputs which
sounds as an obstacle to independent control of P and Q.
An accurate method for stating both static and dynamic
interactions in MIMO systems is Relative Gain Array (RGA)
which is defined as [12]:
T
G GG s )()( 1−×=Λ (7)
Where,×
and T are for element by element multiplicationand transpose of matrix. Fig. 5(b)-5(c) show the off-diagonal
and diagonal elements of )( sGΛ versus frequency. It can be
seen that the diagonal RGAs in s=0 are more positive and
greater than off-diagonal ones showing suitable pairing
between inputs and outputs for decentralized control. Bearing
in mind that the ideal RGA are equal to unit
matrix, I sG =Λ )( , fig. 5(b)-5(c) clearly show the exact
amount of interactions for different frequencies.One measure to assess systems regarding interactions is to
check whether a system is diagonally dominance or not. This
concept can be easily determined using Gershgorin circles
[11]. These circles are plotted for the current model of
power system and UPFC in fig. 5(d). Since circles exclude -
1, system is stable, while including the origin means that
system is not diagonally dominance emphasizing existence
of interactions between inputs and outputs.
IV. SVD CONTROLLER DESIGN
The SVD of a matrix G is defined as follows [11,12,13]:
T V U G Σ= (8)
Where, Σ is a scaling diagonal matrix with elementsiσ
(singular values of G) in descending order. V and U are
rotation matrix of inputs and outputs, respectively. The
closer iσ (for i=1, 2, 3, 4) to each other, the better, as this
leads to control independent of input-output directions.The SVD can be used to obtain decoupled equations
between linear combinations of sensors and linear
combinations of actuators, given by the columns of U and V ,
respectively. If sensors are multiplied by U T and control
actions are multiplied by V , as in fig. 4, then the loop, in the
transformed variables, is decoupled, so a diagonal controller
D K (such as a set of PIs) can be used. Usually, the sensor
and actuator transformations are obtained using the DC
gain, or a real approximation of G(jω), where angular
frequency ω is around the desired closed-loop bandwidth.
From fig. 5, controller K can be written as:
V K U K DT
= (9)
Applying this controller provide us with the new
diagonally dominant system GK Gnew = . While evaluation
of SVD as a function of s gives dynamic decoupled
controller, here, s=0 has been chosen for controller as this
provides good decoupling even at other frequencies. Using
system data given in Table I, 0G is obtained as:
⎥⎥⎥
⎥
⎦
⎤
⎢⎢⎢
⎢
⎣
⎡
−−
−−= −
38.1076.358.550.2
76.338.1050.258.506.694.278.961.4
94.206.661.478.9
10 30G (10)
Then SVD of 0G is as follows:
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⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
×
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
×
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=Σ=
−
0.02050.6903-00.7232
0.69030.02050.7232-0
0.72290.02140.69060
0.02140.7229-00.6906-
0.0452000
00.045200
000.17350
0000.1735
)10(
0.22120.67560.64990.2690
0.6756-0.22120.26900.6499-
0.2613-0.6530-0.63800.3134
0.65300.2613-0.31340.6380-
5
0000
T V U G
(11)
TABLE IParameters of the UPFC control system
R S 0.5p.u
Rr 0.4p.u
R sh 0.3p.uw 2π 50
wLS 0.15p.u
wLr 0.18p.u
wL sh 0.1p.u
1/wC 0.04p.u
Rloss 40p.u
vS = 1,S
δ =5, vr = 1,r δ =-5, v sd =0.9962, v sq=-
0.0872, vrd =0.9962, vrq=0.0872
Vectors0U
and0V
will be used in controller K
directly, while
D K should be derived as:
1
0)(−Σ= sl K D (12)
Where1
0
−Σ is the inverse matrix of 0Σ and )( sl is a PI
controller which can be stated as:
s
k k sl i
p +=)( (13)
Fig. 6 displays the block diagram of control system for
UPFC's power flow control considered based on SVD
controller.
Fig. 4, Block diagram of SVD controller
V. I NTERACTION ANALYSIS OF CLOSED LOOP
CONTROLLED SYSTEM
Block Diagram of controlled system is shown in fig. 6.
The eigen values of the system including transfer function
G(s) followed by the SVD controller have been brought inTABLE II, from which, the stability of the new combination
can be inferred. Fig. 7(a) shows the step response of thesystem. It is obvious that SVD controller has cleared out
unwanted interactions between unpaired inputs and outputs.
Off-diagonal and diagonal RGA are shown in Fig. 7(b)-7(c),
respectively. As these figures shows, the SVD designed
with S =0 not only decreases the interactions in this
frequency, but also improves interactions of other
frequencies. Shown in fig. 7(d), the Gershgorin circles do
not include the origin anymore implying the controlled
system is diagonally dominant.
VI. SIMULATION RESULTS
Through applying the previously designed SVD
controller to the original UPFC system, simulation using
MATLAB/SIMULINK software has been done. Fig. 8
shows the simulation results for a change in active power
from 1p.u to 1.3p.u. In order to compare the SVD
controller's performance, PI and DM controllers
characterized by values in Tables III-VI have been
simulated and resulted i sed , i seq , i shd , i shq , V dc, P and Q were
compared with those of SVD. It can be observed that the
SVD controller has competence over DM and PI with
regard to rise time, settling time and overshoot. Fig. 9 shows
the outputs obtained as a result of changes in P and Q from
1 to 1.3p.u and -0.2 to 0.5p.u, respectively. These results
undoubtedly prove the capabilities of SVD for realization of
independent active and reactive power flow control.
Great deals of problems related to MIMO systems such
as UPFC are caused by unwanted interactions betweenunpaired inputs-outputs. In this paper, considering power
system and UPFC’s model as a unit including series and
shunt converters, common DC link and thevenin equivalent
circuits of power system on either sides of UPFC, beside
interaction analysis, a SVD controller was designed for
active and reactive power flow control. In order to evaluate
the performance of the SVD controller, computer simulation
using MATLAB/SIMULINK software has been provided,
comparing SVD with DM and PI.
TABLE II
Eigen values of controlled system
i )( ivalues Eigen λ
1 -1.0014 + 0.3142i
2 -1.0014 - 0.3142i
3 -0.6669 + 0.3142i
4 -0.6669 - 0.3142i
5 -0.0010 + 0.0000i
6 -0.0010 - 0.0000i
7 -0.0010 + 0.0000i
8 -0.0010 - 0.0000i
i =9 to18 0
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Simulation results have clearly proved the competence of
SVD controller over two other ones.
AKNOWLEDGEMENT
This study is a part of a research project entitled "Studyingand Analysis of Static and Dynamic FACTS Devices Modelsand MIMO Controller Simulation and Design for One of Such
Devices in a Typical Power System" supported by Islamic
Azad university branch of Ali Abad Katool.
R EFERENCES
[1] N. H. Hingorani, " Flexible AC transmission system ", IEEE Spectrum,April 1993, pp. 40-45.
[2] L. Gyugyi, “A unified power flow control concept for flexible ac
transmission systems,” ZEE Proceedings-C, vol. 139, no. 4, July 1992, pp. 323-331.
[3] N. G. Hingorani and L. Gyugyi, "Understanding FACTS: Concept and
Technology of flexible AC transmission systems", IEEE Press, 1999.[4] Q. Yu, S. D. Round, L. E. Norum, T. M. Undeland, "Dynamic Control of
a Unified Power Flow Controller", IEEE 1996, pp. 508 - 514.[5] C.M.Yam and M.H.Haque, "Dynamic Decoupled Compensator for UPFC
Control", Proc. 2002 IEEE Power System Technology, pp.1482-1487.
[6] C. Schauder and H. Mehta, “Vector analysis and control of advanced static
var compensators", IEE Proc.-C (140) (No. 4) (1993) 299–306.
[7] C.M. Yam, M.H. Haque, "A SVD based controller of UPFC for power
flow control", ELSEVIER, B. V., PP. 76-84, July 2006.[8] E. M. Farahani, S. Afsharnia, "DM for UPFC's Active & Reactive Power
Decoupled Control", IEEE ISIE, pp. 1916-1921, July 2006.
[9] Y. H. Song and A. T. Johns, "Flexible AC transmission systems(FACTS)", IEE Power and Energy Series 30, 1999.
[10] M. T. Bina, "Nonactive and Harmonics Power Control", Khajehnasir, 1th
, 2003.
[11] P. Albertos, A. Sala, " Multivariable Control Systems: An Engineering
Approach", Springer-Verlag London, 2004.
[12] S. Skogestad, I. Postlethwaite , "MULTIVARIABLE FEEDBACK CONTROL: Analysis and design", JOHN WILEY & SONS, 2 th, 2001.
[13] J. M. Maciejowski, "Multivariable Feedback Design", Addison Wesley,1th, 1989.
Fig. 5, Block diagram of UPFC system with controller
TABLE III
Proportional and integral gain of
the conventional PI controller
k p 7
k i 15
TABLE IVProportional and integral gain of
the PI controller for DC voltage
k pdc 1
k idc 6
TABLE V
Proportional and integral gain of
Z matrix parameter for DM
gain 500
t S 40
TABLE VI
Proportional and integral gain of l(s) for SVD
k p 0.5
k i 91
Fig. 6, Interaction results of uncontrolled UPFC system(a) Step response (b) off-diagonal RGA
(c) Diagonal RGA (d) Gershgorin circles
Fig. 7, Interaction results of controlled UPFC system(a) Step response (b) off-diagonal RGA
(c) Diagonal RGA (d) Gershgorin circles
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Fig. 8, Outputs for a change in P from 1p.u to 1.3p.u(a) V dc (b) i sed (c) i seq (d) i shd (e) i shq (f) P (g) Q
Fig. 9, Outputs for simultaneous changes in P from 1p.u to 1.3p.u and Qfrom -0.2p.u to 0.5p.u (a) V dc (b) i sed (c) i seq (d) i shd (e) i shq (f) P (g)
Q