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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

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