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A Feedback Linearization BasedUnified Power Flow Controller InternalController for Power Flow ControlKeyou Wang a , Bo Yan b , Mariesa L. Crow a & Deqiang Gan ba Department of Electrical and Computer Engineering , MissouriUniversity of Science and Technology , Rolla , Missouri , USAb Department of Electrical Engineering , Zhejiang University ,Hangzhou , Zhejiang , P.R. ChinaPublished online: 29 Mar 2012.

To cite this article: Keyou Wang , Bo Yan , Mariesa L. Crow & Deqiang Gan (2012) A FeedbackLinearization Based Unified Power Flow Controller Internal Controller for Power Flow Control, ElectricPower Components and Systems, 40:6, 628-647, DOI: 10.1080/15325008.2011.653861

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Electric Power Components and Systems, 40:628–647, 2012

Copyright © Taylor & Francis Group, LLC

ISSN: 1532-5008 print/1532-5016 online

DOI: 10.1080/15325008.2011.653861

A Feedback Linearization Based Unified

Power Flow Controller Internal Controller

for Power Flow Control

KEYOU WANG,1 BO YAN,2 MARIESA L. CROW,1 and

DEQIANG GAN 2

1Department of Electrical and Computer Engineering, Missouri University

of Science and Technology, Rolla, Missouri, USA2Department of Electrical Engineering, Zhejiang University, Hangzhou,

Zhejiang, P.R. China

Abstract This article presents a feedback linearization based controller designmethodology for a unified power flow controller to achieve rapid reference signal

tracking in the internal control level. Feedback linearization is a non-linear controltechnique based on the differential geometry theory and overcomes the drawback

of traditional linear proportional-integral control, which is typically tuned for onespecific operating condition. In this article, feedback linearization control is developed

for the unified power flow controller dynamic model via an appropriate coordinatetransformation, and linear quadratic regulator control is then applied on the trans-

formed linear system. The proposed control is validated via a detailed device-levelsimulation on an 11-bus system and a large scale simulation on the IEEE 118-bus and

300-bus systems. The proposed control is benchmarked against proportional-integralcontrol.

Keywords flexible AC transmission systems, unified power flow controller, non-linear control, feedback linearization

1. Introduction

Flexible AC transmission systems (FACTS) devices provide controllability and transfer

capability to the transmission system. The unified power flow controller (UPFC) is the

most versatile FACTS device [1]. It consists of shunt and series three-phase voltage source

converters (VSCs) sharing a back-to-back common DC-link. The UPFC has proven to

be effective in independently providing both shunt compensation to the sending-end bus

to provide reactive support and series compensation to the transmission line to achieve

active and reactive power flow control [2–4]. The UPFC is also able to provide advanced

transient stability improvement and oscillation damping as well [5–8].

Received 29 July 2011; 27 December 2011.Address correspondence to Dr. Mariesa L. Crow, Department of Electrical and Computer

Engineering, Missouri University of Science and Technology, 1400 N. Bishop, 305 McNutt Hall,Rolla, MO 65409. E-mail: crow@mst.edu

628

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Feedback Linearization Based UPFC Internal Control 629

Comprehensive UPFC control encompasses three levels of control, as shown in

Figure 1 [8]. The internal controller generates the fundamental shunt and series output

voltage waveform with the demanded magnitude and phase angle in synchronism with the

AC system. These output voltage signals are then used to drive the gating signals through

an appropriate pulse width modulation (PWM) scheme (gate control). The external

control responds to system conditions and determines the amount of active and reactive

power injection required from the UPFC series converter to achieve the power flow

requirements, and the amount of reactive current (or power) that the UPFC (shunt) should

generate or absorb to meet the voltage requirement of the system. The external control

output is based on system-level requirements and may vary from holding designated

steady-state power flows to time-varying injections to achieve advanced features (i.e.,

oscillation damping).

Just as the effectiveness of the UPFC depends on the gating control (the ability

of the VSC to synthesize a reference waveform), the effectiveness also depends on the

ability to track the reference waveform magnitude and phase accurately. This accuracy

is the internal control of the UPFC—the ability of the controller to accurately convert

system level set-points into the shunt and series injected voltages. Most internal control

methods for the UPFC are based on linear control techniques [1–4]. In [2], the decoupled

dq control strategy for the VSC was first proposed. These controls arise from small

perturbation linearization about the equilibrium of the non-linear equations of the VSC

average value model. Therefore, the feedback gains of linear control strategies may have

to change with operating conditions; otherwise, the performance of the controller may

degrade outside the linearization region.

A considerable amount of effort has been devoted to compensate for a change of

operating condition so that the controller can provide satisfactory performance over a wide

range of operating conditions [9–12]. Many intelligent techniques have been reported to

Figure 1. UPFC hierarchical control: external, internal, and gate.

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change adaptively the proportional-integral (PI) controller gains of a UPFC, such as fuzzy

logic [9], neural networks [10, 11], and particle swarm optimization [12].

Another alternative solution to compensate for system non-linearities is feedback

linearization control (FBLC), which is based on differential geometric theory for

non-linear systems [13]. FBLC is a technique to convert a non-linear system to an

equivalent linear system without loss of state information via an appropriate coordinate

transformation. After the coordinate transformation, a variety of proven linear control

techniques can be applied to the transformed system. Therefore, FBLC is more robust

to operating condition changes than linear controls and provides stability over a wider

range of operating conditions. The FBLC approach has attracted a great deal of attention

and resulted in promising outcomes for applications in power systems [14]. In the past

decade, FBLC has been more frequently applied to FACTS controllers, including the

static compensator (STATCOM) [15–19], static series synchronous compensator (SSSC)

[20], and UPFC [20, 21]. In [20], a general VSC FBLC controller was proposed and

applied on an SSSC and UPFC in a single-machine infinite-bus system but without

full consideration of the DC-link losses. In the approach of [21], the terminal voltage

magnitude of the UPFC shunt converter was introduced as a dynamic state (Vt , PVt )

in addition to the usual UPFC states (id1, iq1, id2, iq2, and Vdc). This leads to a non-

standard UPFC model and a simplified FBLC coordinate transformation. Therefore,

preliminary UPFC FBLC results have considered only small-scale system applications

or simplified UPFC models.

In this article, a new internal UPFC control based on feedback linearization is

proposed. The UPFC dynamic model, corresponding to an affine non-linear represen-

tation, is transformed through an appropriate coordinate transformation to an equivalent

linear system on which linear quadratic regulator (LQR) optimal control is applied. This

methodology is generally enough to extend to any context of control for VSCs, especially

multi-converters sharing a common DC-link. The simulation results on an 11-bus, the

IEEE 118-bus, and the IEEE 300-bus systems support the effectiveness of the proposed

method in FACTS reference signal tracking.

2. UPFC Model

The UPFC, shown in Figure 2, consists of a combination of a shunt and series branches

connected through a DC-link capacitor. The series-connected inverter injects a volt-

age with both controllable magnitude and phase angle in series with the transmission

line, therefore providing active and reactive power compensation to the transmission line.

The shunt-connected inverter provides the active power drawn by the series branch

and the losses through the common DC-link and also provides reactive compensation

to the system independently. The parameters Rs1 and Ls1 represent the shunt transformer

resistance and reactance, Rs2 and Ls2 represent the series transformer resistance and

reactance, and Rdc and Cdc represent the DC-link losses and capacitor. Vs∠�s and Vr∠�r

represent the sending-end and receiving-end AC bus voltages, and .k1; ˛1/ and .k2; ˛2/

are the control pairs of PWM ratio and phase shift for the shunt and series converters,

respectively.

The UPFC model given in Eq. (1) is a combination of the synchronous STATCOM

and SSSC models [3]. The state vector is given by x D Œid1 iq1 id2 iq2 Vdc�T , where

id1 and iq1 are the dq components of the shunt current, id2 and iq2 are the dq components

of the series current, and Vdc is the voltage across the DC-link capacitor. The control

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Feedback Linearization Based UPFC Internal Control 631

Figure 2. UPFC equivalent circuit.

vector is defined as u D Œud1 uq1 ud2 uq2�T , where ud1 D k1 cos.˛1 C �s/, uq1 D

k1 sin.˛1 C �s/, ud2 D k2 cos.˛2 C �s/, and uq2 D k2 sin.˛2 C �s/:

1

!s

0

B

B

B

B

B

B

@

Ls1

Ls1

Ls2

Ls2

Cdc

1

C

C

C

C

C

C

A

0

B

B

B

B

B

B

@

Px1

Px2

Px3

Px4

Px5

1

C

C

C

C

C

C

A

D

0

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

@

�Rs1x1 C!

!s

x2 � Vsd

�Rs1x2 �!

!s

x1 � Vsq

�Rs2x3 C!

!s

x4 C .Vsd � Vrd /

�Rs2x4 �!

!s

x3 C .Vsq � Vrq/

�x5

Rdc

1

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

C

A

C

0

B

B

B

B

B

B

@

x5

x5

x5

x5

�x1 �x2 �x3 �x4

1

C

C

C

C

C

C

A

0

B

B

B

@

ud1

uq1

ud2

uq2

1

C

C

C

A

: (1)

The power balance equations at the sending-end bus are given by

0 D Vs..id1 � id2/ cos �s C .iq1 � iq2/ sin �s/ � Vs

nX

j D1

Vj Ysj cos.�s � �j � �sj /; (2)

0 D Vs..id1 � id2/ sin �s � .iq1 � iq2/ cos �s/ � Vs

nX

j D1

Vj Ysj sin.�s � �j � �sj /I (3)

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632 K. Wang et al.

and at the receiving-end bus, they are

0 D Vr .id2 cos �r C iq2 sin �r/ � Vr

nX

j D1

Vj Yrj cos.�r � �j � �rj /; (4)

0 D Vr .id2 sin �r � iq2 cos �r/ � Vr

nX

j D1

Vj Yrj sin.�r � �j � �rj /; (5)

where the summation terms represent the power flow equations, Yij ∠�ij is the .i; j /th

element of the admittance matrix, and n is the number of buses in the system.

3. FBLC

3.1. Affine Non-linear Representation

Before applying the non-linear control to the UPFC, the model of Eq. (1) must be

reformulated in the form of an “affine non-linear system” [13] such that

Px D f .x/ C

4X

iD1

gi .x/ui ; (6)

y D h.x/; (7)

where x is the state vector, and y is the output function. The functions f and gi are

smooth vector functions, and h is a smooth scalar function. The function gi is the i th

column of the matrix functions g. The equations for f , g, and u are given by

f .x/ D

0

B

B

B

B

B

B

B

B

B

B

B

B

@

a1x1 C bx2 � c1Vsd

a1x2 � bx1 � c1Vsq

a2x3 C bx4 C c2.Vsd � Vrd /

a2x4 � bx3 C c2.Vsq � Vrq/

�d

Rdc

x5

1

C

C

C

C

C

C

C

C

C

C

C

C

A

; (8)

g.x/ D

0

B

B

B

B

B

B

@

c1x5

c1x5

c2x5

c2x5

�dx1 �dx2 �dx3 �dx4

1

C

C

C

C

C

C

A

; (9)

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Feedback Linearization Based UPFC Internal Control 633

where

x D Œx1 x2 x3 x4 x5�T D Œid1 iq1 id2 iq2 Vdc�T ;

u D Œu1 u2 u3 u4�T D Œud1 uq1 ud2 uq2�T ;

and

a1 D �Rs1!s

Ls1

; c1 D!s

Ls1

;

a2 D �Rs2!s

Ls2

; c2 D!s

Ls2

;

b D !; d D!s

Cdc

:

3.2. Coordinate Transformation

The key point of feedback linearization is to find a suitable coordinate transformation

such that the non-linear system of Eq. (6) is converted into a linear system. The difficulty

lies in finding a suitable transformation. In this article, a transformation approach based

on energy conservation is proposed. The following Lyapunov-like function is considered

as the candidate for the non-linear transformation:

V.x/ DLs1

2

�

x21 C x2

2

�

CLs2

2

�

x23 C x2

4

�

CCdc

2x2

5; (10)

where V.x/ physically represents the energy stored in the UPFC converters and the

DC-link capacitor. If h.x/ D V.x/, the following condition for feedback linearization is

satisfied:

LgLif h.x/ D 0; i D 0; : : : ; r � 1;

where LgLif h.x/ represents Lie derivatives of Li

f h.x/ with respect to g.x/. The operator

r is the relative degree of .f; g/ with respect to h, and in this model, r D 1. Next, the

coordinate transform from the x space to z space is defined as

z D ˆ.x/ D

0

B

B

B

B

B

B

@

h.x/

Lf h.x/

�3.x/

�4.x/

�5.x/

1

C

C

C

C

C

C

A

D

0

B

B

B

B

B

B

@

h.x/

Lf h.x/

x2

x3

x4

1

C

C

C

C

C

C

A

; (11)

where �3, �4, and �5 are selected as the three identity states from the five x states such

that the Jacobian matrix dˆ.x/=dx is non-singular. The function Lf h.x/ represents the

Lie derivatives of h.x/ with respect to f .x/. The derivation details using Lie notations

are given in Appendix A.

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By differentiating Eq. (11),

Pz D

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

Lf h.x/

L2f h.x/ C

4X

iD1

Lgi Lf h.x/ui

Lf �3.x/ C

4X

iD1

Lgi �3.x/ui

Lf �4.x/ C

4X

iD1

Lgi �4.x/ui

Lf �5.x/ C

4X

iD1

Lgi �5.x/ui

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

: (12)

If the new control vector v is defined such that

v D

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

L2f h.x/ C

4X

iD1

Lgi Lf h.x/ui

Lf �3.x/ C

4X

iD1

Lgi �3.x/ui

Lf �4.x/ C

4X

iD1

Lgi �4.x/ui

Lf �5.x/ C

4X

iD1

Lgi �5.x/ui

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

; (13)

then the new linear state equations can be written as

Pz D Az C Bv; (14)

where

A D

0

B

B

B

B

B

B

@

0 1 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

1

C

C

C

C

C

C

A

; B D

0

B

B

B

B

B

B

@

0 0 0 0

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

1

C

C

C

C

C

C

A

:

After coordinate transformation from x space to z space, the non-linear system of

Eq. (6) is converted into the linear system of Eq. (14). Since this system has full rank

(rankŒB; AB� D 5), this linear time invariant system is controllable by any suitable linear

method.

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Feedback Linearization Based UPFC Internal Control 635

3.3. Control Law

Since the system of Eq. (14) is now linear and controllable, any linear feedback control

technique (such as pole placement, LQR, H1, etc.) can be applied without loss of

generality, such that

v D �K�z; (15)

where K� is the state feedback matrix.

Once the appropriate new control input v is determined, the inverse coordinate

transform yields the desired original control inputs u from Eq. (13):

u D

0

B

B

B

B

B

@

Lg1Lf h.x/ Lg2Lf h.x/ Lg3Lf h.x/ Lg4Lf h.x/

Lg1�3.x/ Lg2�3.x/ Lg3�3.x/ Lg4�3.x/

Lg1�4.x/ Lg2�4.x/ Lg3�4.x/ Lg4�4.x/

Lg1�5.x/ Lg2�5.x/ Lg3�5.x/ Lg4�5.x/

1

C

C

C

C

C

A

�10

B

B

B

B

B

@

v1 � L2f h.x/

v2 � Lf �3.x/

v3 � Lf �4.x/

v4 � Lf �5.x/

1

C

C

C

C

C

A

:

(16)

3.4. Control Objectives

The control objectives for the UPFC have four targets: the line powers that track the

desired active power P �

r and reactive power Q�

r by controlling the series converter

injected voltage, and the sending bus voltage and DC-link capacitor voltage track the

desired values V �

s and V �

dc, respectively, by controlling the shunt converter voltage.

Starting with the series portion of the UPFC, the desired powers are converted into

desired currents i�

d2 and i�

q2 (or x�

3 and x�

4 ) through

x�

3

x�

4

!

D1

Vr

cos �r sin �r

sin �r � cos �r

!

�1 P �

r

Q�

r

!

: (17)

The shunt converter can provide voltage support at the sending-end bus by injecting

reactive current independently. The desired sending-end bus voltage must be converted

into desired reactive currents i�

q1 (or x�

2 ). Since there is a strong correlation between

voltage magnitude and injected reactive current, a simple PI controller can be used to

obtain x�

2 from V �

s . The desired DC-link capacitor voltage is V �

dc(x�

5 ). Only the active

current for the shunt converter i�

d1(or x�

1 ) is left to calculate. Finally, the state id1 D x�

1

can be obtained from the five equations of the UPFC model in Eq. (1) evaluated at

steady state (time derivatives set to zero) with the desired state values x�

2 , x�

3 , x�

4 ,

and x�

5 .

Figure 3 summarizes the feedback linearization process. The UPFC model described

by Eq. (6) is transformed through the coordination transformation of Eq. (11) from x

space to z space. Then, a partial “exact” linearization system in Eq. (14) and a static state

feedback in Eq. (16) are given such that the closed-loop system has a linear input–output

behavior. Based on the linear control technique, the optimal state feedback control in

Eq. (15) is applied.

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Figure 3. FBLC scheme.

4. Simulation Results

In this section, the proposed control is benchmarked against traditional PI control shown

in Figure 4 [3]. The proposed control is validated in two types of simulation: a detailed

device-level simulation with PSCAD (an electromagentic transient type simulation; Mani-

toba HVDC Research Centre, Winnipeg, Manitoba, Canada) and two large-scale systems

with state-space-based models in MATLAB (The MathWorks, Natick, Massachusetts,

USA). These simulations validate that the proposed control works well regardless of the

level of detail of the UPFC model and the surrounding transmission system.

4.1. Detailed Simulation

An electromagnetic transient-type simulation uses detailed models including three-phase

circuits and power electronic switching converters. This level of simulation is suitable for

the study of the UPFC FBLC controller at the most detailed level. The proposed control

is validated on the UPFC 3-area and 11-bus benchmark system shown in Figure 5 [22].

Figure 4. UPFC PI control scheme.

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Feedback Linearization Based UPFC Internal Control 637

Figure 5. The 11-bus benchmark system.

Case I (11-bus System: Step Change). The proposed control is first validated through a

simple step change in active and reactive power to show tracking performance. If the

proposed control is adequate, a step change in active power should not cause a change in

reactive power and vice-versa. Figures 6 and 7 show the responses of the FBLC controller

and the PI controller during independent step changes. Note that the proposed control

works well in that the step response is rapid and that the alternate power remains nearly

constant during the change.

Case II (11-bus System: Contingency). In this case, the controller is validated during a

three-phase fault. The fault is applied on bus 8 at t D 2:0 sec and cleared at t D 2:1 sec.

Figure 6. Case I: step change in active power.

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Figure 7. Case I: step change in reactive power.

The proposed control is benchmarked against a typical PI control. Figures 8 and 9 show

the responses of the line active and reactive powers and the DC-link voltage, respectively,

for the FBLC and PI controls. Note that the proposed control shows superior performance

to the PI control with less oscillation and faster settling time. Furthermore, there is a

significantly smaller excursion in the DC-link voltage Vdc.

4.2. Large-system Simulation

In this study, the proposed FBLC controller is validated on the IEEE 118-bus and

IEEE 300-bus test systems. The two IEEE systems are summarized in Table 1, and the

system data can be found in [23]. This simulation includes a full non-linear time-domain

Figure 8. Case II: transmission line active and reactive powers.

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Feedback Linearization Based UPFC Internal Control 639

Figure 9. Case II: UPFC DC-link voltage Vdc.

simulation for the differential-algebraic equations, including the UPFC model in Eqs. (1)–

(5). This simulation is performed in MATLAB. The IEEE 118-bus system is the test

system for Cases III and IV and the IEEE 300-bus system is the test system for Case V.

Case III (IEEE 118-bus System: Step Change). In this case, the UPFC is placed on line

37-40 near bus 37 in the IEEE 118-bus system. This UPFC placement is the same as the

placement proposed in [24, 25]. The UPFC device and control parameters are given in

Appendix B.

This case consists of commanded step changes in active and reactive power in the

transmission line. The reference values of the UPFC powers are increased by 50% of

the steady-state value and returned to the steady-state value, respectively. Figure 10

shows the response of the active and reactive line power of the UPFC. Figure 11 shows

the bus voltage magnitude at the sending end. The responses of the PI controller are also

given to provide a comparison between the proposed controller and the PI controller.

This simulation validates the proposed control in a state-space-based UPFC model.

Case IV (IEEE 118-bus: Contingency). In this case, the UPFC placement is same with

Case III. A three-phase fault occurs on bus 37 at t D 0:05 sec and is cleared at t D

0:15 sec by removing line 37-39. The dynamic behaviors of the UPFC are shown in

Figures 12–15. The bold, thin, and dotted lines are related to the proposed control, PI

control, and open-loop control, respectively. The open-loop control represents the mode,

Table 1

Description of the IEEE test systems

System

Number of

buses

Number of

generators

Number of

transmission lines

IEEE 118 118 54 177

IEEE 300 300 69 311

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Figure 10. Case III: step change in line active and reactive power.

where the control pairs of PWM modulation ratio (k) and phase shift (˛) are constant

values during the transient period.

Case V (IEEE 300-bus System: Contingency). In this final case, the proposed control is

validated on the IEEE 300-bus system, which is a larger testbed system. The UPFC

is placed on the tie-line 7-131 between subsystem 1 and subsystem 2. A three-phase

fault is applied on bus 7 at t D 0:05 sec and is cleared at t D 0:10 sec by removing

Figure 11. Case III: UPFC sending-bus voltage.

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Feedback Linearization Based UPFC Internal Control 641

Figure 12. Case IV: UPFC line active power.

line 7-5. The dynamic behaviors of the UPFC are shown in Figures 16 and 17. The bold

and thin lines are related to the proposed control and the PI control case, respectively.

4.3. Discussion

The results from both the detailed and large-scale simulation lead to the following

observations.

Figure 13. Case IV: UPFC line reactive power.

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Figure 14. Case IV: UPFC DC-link voltage.

� The detailed and large-scale simulations yield similar results in the step-change

test and contingency test. The proposed FBLC controller effectively tracks the

commanded values of power and voltage.

� In step-change tests (Cases I and III), the active and reactive powers closely

track the commanded values with little transient overshoot. As shown in Case III,

during the transient periods, the DC-link capacitor voltage and sending-end bus

Figure 15. Case IV: UPFC sending-bus voltage.

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Feedback Linearization Based UPFC Internal Control 643

Figure 16. Case V: UPFC line active power.

voltage magnitude change very little. Both voltages change by less than 0.5% from

their reference value during the transients.

� In the contingency tests (Cases II, IV, and V), the benchmark PI control and

proposed FBLC both exhibit good performance to track the FACTS reference

signals after the disturbance. However, the PI controller requires roughly double

the settling time with a much larger overshoot than the proposed control.

Figure 17. Case V: UPFC line reactive power.

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5. Conclusions

In this article, a new UPFC non-linear control is proposed. At the internal level, the

proposed control is based on feedback linearization and is general enough that it can be

extended to the control of a variety of VSCs. The simulation results with the detailed

and the large-scale system validate the effectiveness of the proposed approach on the

UPFC power flow control for step change and transient stability improvement after a

fault. The FBLC controller provides enhanced performance compared with a traditional

PI controller. Note however, that the proposed approach is not comprehensive without

an appropriate external level control. Future research will focus on oscillation damping

at the external (power system) level. Further research is also needed on approaches to

overcome the uncertainty in converter and system parameters.

References

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2004.

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2004.

10. Dash, P. K., Mishra, S., and Panda, G., “A radial basis function neural network controller for

UPFC,” IEEE Trans. Power Syst., Vol. 15, No. 4, pp. 1293–1299, 2000.

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Energy Syst., Vol. 29, No. 3, pp. 251–259, 2007.

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Academic Publishers, Chap. 6, 2001.

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feedback linearization scheme for STATCOM control,” Elect. Power Syst. Res., Vol. 62, pp. 81–

91, 2002.

17. Soto, D., and Pena, R., “Nonlinear control strategies for cascaded multilevel STATCOMs,”

IEEE Trans. Power Delivery, Vol. 19, pp. 1919–1927, October 2004.

18. Song, E., Lynch, A. F., and Dinavahi, V., “Experimental validation of nonlinear control for a

voltage source converter,” IEEE Trans. Control Syst. Technol., Vol. 17, pp. 1135–1144, 2009.

19. Wang, K., and Crow, M. L., “Power system voltage regulation via STATCOM internal nonlinear

control,” IEEE Trans. Power Syst., Vol. 26, No. 3, pp. 1252–1262, August 2011.

20. Lu, B., and Ooi, B. T., “Nonlinear control of voltage-source converter systems,” IEEE Trans.

Power Electron., Vol. 22, pp. 1186–1195, July 2007.

21. Saravana Ilango, G., Nagamani, C., Sai, A. V. S. S. R., and Aravindan, D., “Control algorithms

for control of real and reactive power flows and power oscillation damping using UPFC,” Elect.

Power Syst. Res., Vol. 79, No. 4, pp. 595–605, April 2009.

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IEEE Trans. Power Delivery, Vol. 21, No. 1, pp. 484–491, January 2006.

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applied to the placement and distributed steady-state control of UPFCs,” Proceedings of the

37th Annual North American Power Symposium (NAPS05), pp. 77–83, Ames, IA, 23–25

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UPFCs for damping power system oscillations,” Proceedings of the 2009 IEEE Power System

Conference and Exposition, Seattle, WA, 15–18 March 2009.

Appendix A. Lie Derivatives

Given

h.x/ D1

2

�

Ls1x2

1 C Ls1x2

2 C Ls2x2

3 C Ls2x2

4 C Cdcx25

�

and

rh.x/ D ŒLs1x1 Ls1x2 Ls2x3 Ls2x4 Cdcx5�;

if

Lgi Lj

f h.x/ D 0; i D 1; : : : ; 4 and j D 0; : : : ; r � 1;

where r is the relative degree of .f; g/ with respect to h, then, in this problem, r D 1 and

Lgi h.x/ D rh.x/ � gi .x/ D 0; i D 1; : : : ; 4

and

Lf h.x/ D rh.x/ � f .x/

D a1Ls1

�

x21 C x2

2

�

C a2Ls2

�

x23 C x2

4

�

� c1Ls1.x1Vsd C x2Vsq/

C c2Ls2.x3.Vsd � Vrd / C x4 �dCdc

Rdc

x25

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646 K. Wang et al.

and

rLf h.x/ D

2

6

6

6

6

6

6

6

6

4

2a1Ls1x1 � c1Ls1Vsd

2a1Ls1x2 � c1Ls1Vsq

2a2Ls2x3 C c2Ls2.Vsd � Vrd /

2a2Ls2x4 C c2Ls2.Vsq � Vrq/

2dCdc

Rdc

x5

3

7

7

7

7

7

7

7

7

5

:

And

Lgi Lf h.x/ D rLf h.x/ � gi .x/; i D 1; : : : ; 4;

where

Lg1Lf h.x/ D x5

�

2a1c1Ls1x1 � c21Ls1Vsd C

2d 2Cdc

Rdc

x1

�

;

Lg2Lf h.x/ D x5

�

2a1c1Ls1x2 � c21Ls1Vsq C

2d 2Cdc

Rdc

x2

�

;

Lg3Lf h.x/ D x5

�

2a2c2Ls2x3 C c22Ls2.Vsd � Vrd /

�

C x5

�

2d 2Cdc

Rdc

x3

�

;

Lg4Lf h.x/ D x5

�

2a2c2Ls2x4 C c22Ls2.Vsq � Vrq/

�

C x5

�

2d 2Cdc

Rdc

x4

�

;

L2f h.x/ D r.Lf h.x//f .x/

D

2

6

6

6

6

6

6

6

6

4

2a1Ls1x1 � c1Ls1Vsd

2a1Ls1x2 � c1Ls1Vsq

2a2Ls2x3 C c2Ls2.Vsd � Vrd /

2a2Ls2x4 C c2Ls2.Vsq � Vrq/

�2dCdc

Rdc

x5

3

7

7

7

7

7

7

7

7

5

T 2

6

6

6

6

6

6

6

6

4

a1x1 C bx2 � c1Vsd

a1x2 � bx1 � c1Vsq

a2x3 C bx4 C c2.Vsd � Vrd /

a2x4 � bx3 C c2.Vsq � Vrq/

�dx5

Rdc

3

7

7

7

7

7

7

7

7

5

:

Furthermore,

�3.x/ D x2; �4.x/ D x3; �5.x/ D x4:

Consider

Lf �i .x/ D r�i .x/ � f .x/; i D 3; : : : ; 5I

then

Lf �3.x/ D a1x2 � bx1 C c1Vsq ;

Lf �4.x/ D a2x3 C bx4 C c2.Vsd � Vrd /;

Lf �5.x/ D a2x4 � bx3 C c2.Vsq � Vrq/:

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Feedback Linearization Based UPFC Internal Control 647

Finally,

Lg1�3.x/ D 0; Lg1�4.x/ D 0; Lg1�5.x/ D 0;

Lg2�3.x/ D c1x5; Lg2�4.x/ D 0; Lg2�5.x/ D 0;

Lg3�3.x/ D 0; Lg3�4.x/ D c2x5; Lg3�5.x/ D 0;

Lg4�3.x/ D 0; Lg4�4.x/ D 0; Lg4�5.x/ D c2x5:

Appendix B. UPFC Parameters

The parameters of the UPFC are given in Tables B1 and B2. The per unit approach used

is the same as [2] on a 100-MW, 110-kV base.

The feedback linearization controller has a state feedback matrix K? (Eq. (15)),

which is derived using LQR control:

K? D R�1BT P �;

where P ? is one solution of the Riccati matrix equation

AT P C PA � PBR�1BT P C Q D 0;

K� D

2

6

6

6

4

316,227 7959 0 0 0

0 0 3160 0 0

0 0 0 3160 0

0 0 0 0 3160

3

7

7

7

5

:

Table B1

UPFC parameters in per unit

Rs1 Ls1 Rs2 Ls2 Rdc Cdc

UPFC 0.01 0.15 0.001 0.015 100 1.133

Table B2

UPFC PI control parameters (p.u.)

Kp1 Ki1 Kp2 Ki2 Kp3 Ki3 Kp4 Ki4

0.5 32.34 0.1 0.01 1e–2 1e–3 5e–3 1e–1

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