coordinated design of statcom and excitation system controllers for multi-machine power systems...

Electrical Power and Energy Systems 49 (2013) 269–279

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Coordinated design of STATCOM and excitation system controllersfor multi-machine power systems using zero dynamics method

Amin Khodabakhshian ⇑, Mohammad Javad Morshed, Moein ParastegariUniversity of Isfahan, Electrical Engineering Department, Iran

a r t i c l e i n f o

Article history:Received 10 August 2012Received in revised form 17 December 2012Accepted 24 January 2013Available online 26 February 2013

Keywords:Excitation systemsFeedback linearizationPower system stabilitySTATCOMZero dynamics

0142-0615/$ - see front matter � 2013 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijepes.2013.01.011

⇑ Corresponding author. Tel.: +98 3117934548.E-mail addresses: [email protected] (A. Khoda

mail.com (M.J. Morshed), [email protected] (M.

a b s t r a c t

FACTs devices are being used in transmission networks for increasing the power transfer limit and sta-bility improvement. They also help damp out both local and inter-area low frequency oscillations. How-ever, uncoordinated design of these devices with excitation systems may deteriorate the power systemperformance. Moreover, power system is a large, complex and nonlinear system, and the controllers thatare designed based on linear control theories may have a detrimental effect on the system performance,especially when there are large disturbances occurring in the system. The design method of a nonlinearcontrol technique, named zero dynamics is given in this paper to design the controllers of STATCOM andexcitation systems coordinately for multi-machine power systems. This technique is able to provide thestability of both external and internal dynamic performances of the system. Simulations results clearlyverify that the proposed method improves the power system stability.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Due to the interactions between generators through the weaktie-lines and the existence of high gain automatic voltage regulators(AVRs), low frequency oscillations normally exist in power systems.As a result of this phenomenon there will be limit on the maximumpower to be transmitted on tie-lines. Two types of oscillations areusually known. One is referred to inter-area modes resulted fromswinging one generation area with respect to other areas. The sec-ond one is associated with swinging of generators existed in onearea against each other and is known as local mode [1].

Power system stabilizers (PSSs) have been widely utilized toadd a signal into the AVR to boost the power system stability. Inthis regard, different methods have been developed to designmultiple stabilizers installed in different generators coordinatelyin order to damp out both inter-area and local modes [2]. With re-spect to the variations of system operating points, although manydesign methodologies have been provided to improve the PSS per-formance, there are still many researches to overcome the tuningproblem for damping both kinds of oscillations in a robust way [2].

Moreover, when there are large disturbances taking place in thesystem, the system operating point varies significantly and nonlinear-ities may have a considerable effect on the system performance. Inthis situation, the controllers designed based on linear models maynot retain stability. Therefore, it is necessary to use control design

ll rights reserved.

bakhshian), Mj.Morshed@g-Parastegari).

methodology in which the nonlinearities of the power system modelcan be considered. Besides, although the excitation control is able toimprove the transient stability, it will fail to keep the system stabilityif a large fault happens near to the generator terminal [3].

Recently because of the world economic growth, the need forexpanding modern power system rises. However, due to environ-mental problems and urbanization it is hard to build up new trans-mission lines. Therefore, the use of the new equipment such asFACTS devices is greatly taken into account. These devices canincrease the transfer power limits and also improve thesteady-state and transient stability [4]. However, as shown in [4]the interactions between the excitation systems and FACTs con-trollers may cause the dynamic instability if they are designedbased on uncoordinated control strategies.

As shown in [4–7] the coordinated design of the excitationsystem without PSS and the supplementary controllers of FACTsdevices could be one of the best solution to damp out both inter-area and local modes as well as boosting the transient stability.In [7], a coordinated design of UPFC controller and excitation sys-tem by using the feedback linearization technique is studied on asingle machine infinite bus (SMIB) system. This technique is alsoused for designing TCPS [6,8], SVC [9,10] and STATCOM [4] control-lers with excitation system simultaneously. In [11] back-steppingmethod is also used to design excitation system and TCSC control-lers coordinately. It is shown that the controllers improve the tran-sient stability, the voltage regulation and the system damping.Hamiltonian and adaptive Hamiltonian methods [12,13] have beenused to design the controllers of STATCOM and excitation systemcoordinately. In [14], H1 theory is also applied to design STATCOM

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Nomenclature

Ts the time constant of STATCOMIqs STATCOM currentdi rotor angle of ith generatorxi and x0 rotor speed and rotor synchronous speed of ith gener-

atorHi moment of inertia of ith generatorPmi and Pei mechanical power and electrical power of ith genera-

torDi damping coefficient of ith generatorE0qi q-axis component of transient internal electro-motive

force of ith generatorx0di d-axis component of transient synchronous reactance of

ith generatorEfdi d-axis component of field voltage of ith generatorT 0doi transient time constant of ith generatorGij, Bij real and imaginary parts of the (i, j)th element of the

system admittance matrixGis, Bis conductance and susceptance of the line between ith

generator and STATCOMX(t) state vector of nonlinear systemynl output vector of nonlinear system

ri relative degree of nonlinear system of the ith subsystemQij Lgj

Lrt�1f htðXÞ of the (i, j)th element of the Q matrix

bi Lrtf htðXÞ of the (i, j)th element of the b matrix

z(t) state vector of linear systemv(t) inputs of the linear systemyl(t) the output of the linear systemufeedback input of the nonlinear systemgk state variable of the nonlinear modelynl,d desirable nonlinear outputv virtual inputs of the linear systemZ transform which converts the nonlinear system to a lin-

ear oneEfdi input voltage of the excitation system of ith generatorus ¼ Iref

qs STATCOM reference currentKd, Kx, Kp the coefficients of the desirable feedback control law

of generator which connect to STATCOM directlyKs the coefficient of desirable feedback control law of

STATCOMa and b constant coefficient to normalize the speed deviation

(Dx) and STATCOM bus voltage deviation

270 A. Khodabakhshian et al. / Electrical Power and Energy Systems 49 (2013) 269–279

and excitation system controllers coordinately to improve thetransient stability of the power system.

The promising solution of designing the controllers of FACTS andexcitation systems together and considering the fact that powersystems are large, nonlinear and time variant, the aim in this paperis to use a nonlinear control method to achieve the improvement inboth dynamic and transient stability performances.

In recent years, the research results of different nonlinear con-trol system theories show that using nonlinear state feedbackand suitable coordinate transformation can exactly linearize an af-fine nonlinear system satisfying certain conditions [15]. The statefeedback calculated by the exact linearization method compen-sates nonlinear characteristics of the original system and trans-forms it into a controllable linear system with good dynamicperformance and ensures the system stability [15,16]. However,it is known that the control law designed by this method is quitecomplicated [16]. Therefore, to solve this drawback another meth-od, called zero dynamics, has been proposed to design the control-lers. This method does not need to exactly linearize all system stateequations, but just a part of them. This approach of the feedbacklinearization algebraically transforms nonlinear system dynamicsinto a partly linear one which is a reduced-order linear systemand an autonomous nonlinear system. A controller can be then de-signed for the reduced-order system. The autonomous system isalso known as zero dynamics of the system which should be stable[17].

External and internal dynamics are also two main parts of thedynamic performance of any system. For internal dynamics havingthe stability is the only requirement. However, for the externaldynamics of the system in addition to the stability, a good perfor-mance must be achieved. In this regard, zero dynamics method isalso able to provide both those objectives [17–20]. In [17] thistechnique has been applied to control the grid current and dc-linkvoltage for maximum power point tracking and to improve the dy-namic response of a three-phase grid-connected photovoltaic sys-tem. In [18,19] zero dynamics method has been used to design theexcitation system controller for a single machine infinite bus(SMIB) power system. This method is also employed in [20] to de-sign STATCOM and excitation system controllers coordinately for aSMIB power system. However, the simple methodology given in

[20] cannot be practical and it is impossible to be used for the realmulti-machine power systems. It is evident that the dynamics ofother machines affect the system performance. Therefore, it isnecessary to give a rigorous method applicable for such a system.In this paper, first a complete formulation of zero dynamics isexpressed for a MIMO system. Then, this technique is used to de-sign the controllers of STATCOM and excitation systems coordi-nately in the n-machine power system. For this purpose, in orderto determine the unknown parameters of the zero dynamicscontroller, the design problem is converted to an optimizationproblem, and is solved by using hybrid Bacteria Foraging Nelder–Mead (BF–NM) algorithm [21,22]. The performance of the pro-posed design is investigated on a multi-machine power systemand simulation results confirm the effectiveness of this methodfor power system stability enhancement.

2. Power system and STATCOM model

2.1. Synchronous static compensator (STATCOM)

A static synchronous compensator (STATCOM) is a regulatingdevice used in AC transmission networks. It is based on a powerelectronics voltage-source converter and can act as either a capac-itor or a reactor to deliver or absorb the reactive power. Usually aSTATCOM is installed to support electricity networks that have apoor power factor and often a poor voltage regulation [23]. It alsoimproves the system damping and the transient stability of thefirst swing of rotor angular velocity for severe fault conditions [23].

2.1.1. STATCOM modelingIn this study, STATCOM is modeled by a shunt current source as

shown in Fig. 1. In this model, STATCOM current is in quadratewith its terminal voltage.

A first order differential equation is used to describe the tran-sient performance of the STATCOM, as follows [20]:

dIs

dt¼ 1

TsIref

qs � Iqs

� �ð1Þ

The bus voltage of STATCOM as can be seen from Fig. 1 may bewritten as follows [20,24]:

Fig. 1. Power system with STATCOM.

A. Khodabakhshian et al. / Electrical Power and Energy Systems 49 (2013) 269–279 271

Vs ¼G1

x1 þ x2þ x1x2

x1 þ x2Is ð2Þ

where G1, x1 and x2 are defined as;

G1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx2E0q1Þ

2 þ ðx1E0q2Þ2 þ 2x1x2E0q1E0q2 cos d12

qð3Þ

x1 ¼ x0d1 þ X1 ð4Þ

x2 ¼ x0d2 þ X2 ð5Þ

2.2. Synchronous generator modeling

For the n-machine power system the nonlinear dynamic modelof the ith generator can be written as follows [13]:

_di ¼ 2pf0ðxi �xi0Þ ð6Þ

_xi ¼ �Di

Hiðxi �xi0Þ �

xi0

HiðPei � PmiÞ ð7Þ

_E0qi ¼1

T 0doi

Efdi � E0qi � ðxdi � x0diÞIdi

h ið8Þ

In Eq. (8) Idi is given as;

Idi ¼Xn

j¼1

E0qiðGij sin dij � Bij cos dijÞ ð9Þ

Now it is assumed that the STATCOM is connected to the buseswhere generators 1 � �m are connected. Other generators which areshown by �mþ 1 � n are not directly connected to any STATCOM. Inthis condition, the output electrical power of ith synchronous gen-erator can be written as follows [14]:

Pei ¼ E0qiVs½Gis cos diþBis sin di�þXn

j¼1

E0qiE0qj½Gij cos dijþBij sin dij� i¼1; . . . ; �m ð10Þ

Pei ¼Xn

j¼1

E0qiE0qj½Gij cos dij þ Bij sin dij� i ¼ �mþ 1; . . . ;n ð11Þ

where Gij, Bij are real and imaginary parts of the (i, j)th element ofthe system admittance matrix and Gis, Bis are the conductance andsusceptance of the line between generator and STATCOM. Also, Vs

is the bus voltage of the STATCOM.

3. Feedback linearization

3.1. Overview

Nonlinear control theories usually provide control methods todeal with systems that are nonlinear, time-variant, or both. How-ever, they require more rigorous mathematical analysis to justifytheir conclusions. Therefore, since many well-established analysisand design techniques such as root-locus, Bode plot, Nyquist

criterion, state-feedback, pole placement exist for linear systems,engineers usually convert nonlinear systems to linear ones to con-trol them using linear control theories. In this regard, there are twomajor methods; the conventional linearization and the input–output linearization methods. The former uses the linear approxi-mation of the dynamics of the system for small ranges of motionsaround one equilibrium point and the latter is the determination ofan algebraically transformation to convert a nonlinear system intoa linear one.

Feedback linearization which is a common approach used incontrolling nonlinear systems involves the transformation of thenonlinear system into one equivalent linear system through thechange of variables and using a suitable control input [15,16].

Dynamics of nonlinear systems can be decomposed to twoparts, external dynamics (observable) and internal dynamics(unobservable). External dynamics can be obtained by a relation-ship between the output and the input. The inputs are designedin such a way that the desirable output is obtained. In addition,the internal dynamics must be bounded. If the system does nothave any internal dynamics, the exact feedback linearizationmethod should be used and if the system, however, has internaldynamics, a partly feedback linearization method should be used.When the outputs should be zero, zero dynamics method is used[15,16].

3.2. Feedback linearization for MIMO system

It is assumed that the nonlinear system is given in the form of

_XðtÞ ¼ f ðXðtÞÞ þXm

i¼1

giðXðtÞÞui

YnlðtÞ ¼ hðXðtÞÞð12Þ

where XðtÞ � ½x1ðtÞ x2ðtÞ � � � xnðtÞ�T 2 Rn is the state vector,u � ½u1 u2 � � � um�T 2 RmYnl � ½ynl;1 ynl;2 � � � ynl;m�

T 2 Rm is the out-put vector, f � ½f1 f 2 � � � f n�

T 2 Rn; g � ½g1 g2 � � � gm�T 2 Rn�m and

h � ½h1 h2 � � � hm�T 2 Rn�m are smooth.The relative degree of ith subsystem in Eq. (12) is considered to

be ri. Therefore, the relative degree of all subsystems can be shownby {r1,r2, . . . ,rm} and the relative degree of the whole system will ber = r1 + � � � + rm. More details are given in Appendix A.

In feedback linearization method the linear model of the systemgiven in Eq. (12) is determined first and then the linear system iscontrolled so that the desirable output is attained. The linear modelof Eq. (12) may be written as follows:

_zðtÞ ¼ AzðtÞ þ vðtÞ

YlðtÞ ¼ CzðtÞð13Þ

where z(t), A, v(t), and Yl(t) are the state vector, the state matrix, theinputs, and the output of the linear system, respectively. In order touse the feedback linearization method, the following conditionsshould be satisfied for all X 2 Rn;

(i) For each subsystem the equation LgiLkf hiðXÞ ¼ 0 should be

satisfied for all 1 6 i 6m, 1 6 j 6m, k < ri � 1, where theoperator L is the Lie derivation [16] (see Appendix B).

(ii) The following matrix should be nonsingular [25].

Q �

Lg1Lr1�1

f h1ðXÞ Lg2Lr1�1

f h1ðXÞ � � � � � � LgmLr1�1

f h1ðXÞ

Lg1Lr2�1


f h2ðXÞ � � � � � � LgmLr2�1

f h2ðXÞ

..

. ...

� � � � � � ...

..

. ...

� � � � � � ...

Lg1Lrm�1

f hmðXÞ Lg2Lrm�1

f hmðXÞ � � � � � � LgmLrm�1

f hmðXÞ

26666666664

37777777775ð14Þ


(iii) The desirable output trajectory yid ð1 6 i 6 mÞ, and its first ri

derivatives are all uniformly bounded.

yid; y

ið1Þd ; . . . ; yiðriÞ

d

h i�� 6 Bid; 1 6 i 6 m ð15Þ

In Eq. (15) Bid is a positive constant.

If all of the above-mentioned conditions are satisfied, a trans-form will exist which converts the nonlinear system to a linearone [25]. The Z : Rn ! Rn relation represents the states deter-mined by mapping and are defined as;

Z ¼ z1 � � � zn½ �T ¼ u1 � � � um g1 � � � gn�r½ �T ð16Þ

where

ui ¼ ui1 ui2 � � � uiri

� �T � L0f hiðXÞ L1

f hiðXÞ � � � Lri�1f hiðXÞ

h iT

ð17Þ

zkðXðtÞÞ � gkðtÞ k ¼ r þ 1; r þ 2; . . . ; n ð18Þ

In Eq. (16) gk are the state variables of the nonlinear model whichappear in the equivalent linear model without change. By using thismapping the inputs of the nonlinear system (ufeedback) can be ob-tained by using the linear system inputs in order to obtain the desir-able nonlinear output ynl,d [25] as follows:

ufeedback ¼ Q�1f�bþ vg ð19Þ

where v are the virtual inputs of the linear system and the vector bcan be determined by using the following equation:

bðXÞ � b1 b2 � � � bm½ �T ¼ Lr1f h1ðXÞ Lr2

f h2ðXÞ � � � Lrmf hmðXÞ

h iT

ð20Þ

More details about the controller design problem are availablein Ref. [25].

If the relative degree of the system r is equal to the degree of thesystem n, the exact feedback linearization will be used to designthe controller and, therefore, the system is completely linearized.However, if the relative degree of the system r is less than the de-gree of the system n, zero dynamics method will be applied to de-sign the controller and, therefore, system is linearized partially.More details can be found in [15].

4. Zero dynamics linearization method for a MIMO system

For MIMO systems, the fundamental notion of zero dynamicsdesign is to search the control laws u1(X), u2(X), . . . ,um(X) such that

the outputs of the system ynl1 ðtÞ; ynl2 ðtÞ; . . . ; ynlm ðtÞ� �

remain equal to

zero at any t P 0 time. Obviously, if the zero dynamics is stable, the‘‘external’’ and ‘‘internal’’ parts of the closed loop system are both sta-ble, and so the whole system is stable. Consequently, the control sys-tem could be optimal as viewed from the output responses.

The relative degree of the whole system given in Eq. (12) is rep-resented by the inequality r = r1 + r2 + � � � + rm < n. Then, by consid-ering Eq. (12) and the assumption that the system outputsynl1 ðtÞ; ynl2 ðtÞ; . . . ; ynlm ðtÞ are zero imply that

h1ðXðtÞÞ ¼ h2ðXðtÞÞ ¼ � � � ¼ hmðXðtÞÞ ¼ 0 ð21Þ

hiðXðtÞÞ ¼ Lf hiðXðtÞÞ ¼ � � � ¼ Lri�1f hiðXðtÞÞ ¼ 0 i ¼ 1; . . . ;m ð22Þ

Under this condition, the states Z = [u g]T (see Eq. (16)) existwhich convert the nonlinear system to a linear one [26].

Z ¼

z1

z2

..

.

zr1

zr1þ1

zr1þ2

..

.

zr1þr2

..

.

zr1þr2þ��þrm�1þ1

..

.

zr1þr2þ��þrm�1

zr

zrþ1

..

.

zn

26666666666666666666666666666666666666664

37777777777777777777777777777777777777775

¼

u11

u12

..

.

u1r1

u2;1

u2;2

..

.

u2;r2

..

.

um;1

..

.

um�1;rm

umr

g1

..

.

gn�r

266666666666666666666666666666666666666664

377777777777777777777777777777777777777775

¼

h1ðXÞLf h1ðXÞ

..

.

Lr1�1f h1ðXÞ

h2ðXÞL1

f h2ðXÞ

..

.

Lr2�1f h2ðXÞ

..

.

hmðXÞ...

Lrm�2f hmðXÞ

Lrm�1f hmðXÞg1ðXÞ

..

.

gn�rðXÞ

2666666666666666666666666666666666666666664

3777777777777777777777777777777777777777775

ð23Þ

Based on Eq. (22) it can be seen that

ui1ðXðtÞÞ ¼ ui2ðXðtÞÞ ¼ � � � ¼ uiriðXðtÞÞ ¼ 0 i ¼ 1; . . . ;m ð24Þ

The last (n � r) rows of matrix Z can be obtained so that it sat-isfies the following condition:

Lgwgi ¼ 0 i ¼ 1;2; . . . ; ðn� rÞ; 8w ¼ 1; . . . ;m ð25Þ

The following equation represents the elements of the Jacobianmatrix of the linear system.

_Z¼

_z1

_z2

..

.

_zr1

_zr1þ1

_zr1þ2

..

.

_zr1þr2

..

.

..

.

_zr1þr2þ��þrm�1

_zr

_zrþ1

..

.

_zn

26666666666666666666666666666666666664

37777777777777777777777777777777777775

¼

_u11

_u12

..

.

_u1r1

_u2ðr1þ1Þ

_u2ðr1þ2Þ

..

.

_u2ðr1þr2Þ

..

.

..

.

_umðr1þr2þ��þrm�1Þ

_umr

_g1

..

.

_gn�r

266666666666666666666666666666666666664

377777777777777777777777777777777777775

¼

u12

u13

..

.

b1ðZÞþXm

l¼1

Q 1lðZÞul

u2ðr1þ2Þ

u2ðr1þ3Þ

..

.

b2ðZÞþXm

l¼1

Q 2lðZÞul

..

.

..

.

umr

bmðZÞþXm

l¼1

Q mlðZÞul

q1ðZÞ...

qn�rðZÞ

2666666666666666666666666666666666666666666664

3777777777777777777777777777777777777777777775

ð26Þ

where _g ¼ qðzÞ can be obtained by the following equations:

qðZÞ¼ q1ðZÞ q2ðZÞ � � � qn�rðZÞ½ �T ¼ �qðZÞþ �p1ðZÞ � � � �piðZÞ � � � �pmðZÞ½ �: u1 � � � ui � � � um½ �T ð27Þ

where

�qðZÞ¼ �q1ðZÞ �q2ðZÞ � � � �qn�rðZÞ½ �T ¼ Lf g1ðXÞ Lf g2ðXÞ � � � Lf gn�rðXÞ½ �Tð28Þ

�piðZÞ ¼ �pi1ðZÞ �pi2ðZÞ � � � �piðn�rÞðZÞ� �T

¼ Lgig1ðXÞ Lgi

g2ðXÞ � � � Lgign�rðXÞ

� �T ð29Þ


The last (n � r) rows of the Jacobian matrix of the vectorfunction _Z ¼ ½ _u _g�T represent zero dynamics of the system. In thelast step of the design, the stability of the zero dynamics given inEq. (30) should be studied._g ¼ qð0; . . . ;0;g1;g2; . . . ;gn�rÞ ð30ÞIt should be noted that in Eq. (30) the first m elements, based on Eq.(24) are zero.

5. Coordinated design of STATCOM and excitation controllersusing zero dynamics

In order to design the controllers of the excitation system andSTATCOM coordinately by using zero dynamics method it is assumedthat the power system consists of two groups of generators. The firstgroup consists of generators which STATCOM affects their voltages di-rectly; the second group consists of generators which STATCOM doesnot affect their voltages directly. This is a valid assumption in a realpower system and it helps simplify the design procedure.

5.1. Controller design for �m generators and STATCOM using zerodynamics

Now �m generators belong to the first group is taken into account. Thissystem ( �m generators and a STATCOM) can be considered as a MIMO sys-tem. The inputs of this system are excitation voltages of �m generators anda STATCOM reference current. The outputs are the changes of the electricpower of generators and the STATCOM bus voltage.

As an example, it is assumed that in the n-machine power sys-tem, the STATCOM is installed between generators 1 and 2 ð �m ¼ 2Þ.The inputs of this MIMO system are excitation voltages of genera-tors 1 and 2, and the STATCOM reference current. This system is athree-input three-output nonlinear system and the outputs of thissystem are the deviation of the electric power of generators andSTATCOM bus voltage as follows:

yi ¼ hi ¼ pei � pmi ¼ DPei; i ¼ 1;2y3 ¼ hs ¼ VS � VS0 ¼ DVS

ð31Þ

In this condition, based on Eq. (23) the elements of the matrix canbe determined by using the following equation:

z1

z2

z3

z4

z5

z6

z7

2666666666664

3777777777775¼

u11ðXÞu21ðXÞu31ðXÞg1ðXÞg2ðXÞg3ðXÞg4ðXÞ

2666666666664

3777777777775¼

h1ðXÞh2ðXÞhsðXÞ

x1

x2

x3

x4

2666666666664

3777777777775

ð32Þ

Considering Eqs. (13) and (26) and substituting uij = 0 imply that

z1

z2

z3

z4

z5

z6

z7

2666666666664

3777777777775¼

u11ðXÞu21ðXÞu31ðXÞg1ðXÞg2ðXÞg3ðXÞg4ðXÞ

2666666666664

3777777777775¼


x1

x2

x3

x4

2666666666664

3777777777775!Eq: ð13Þ

_z1

_z2

_z3

_z4

_z5

_z6

_z7

2666666666664

3777777777775¼ d

dt


x1

x2

x3

x4

2666666666664

3777777777775

0BBBBBBBBBBB@

1CCCCCCCCCCCAþv ¼

dðh1 ðXÞÞdt

dðh2 ðXÞÞdt

dðhsðXÞÞdt

dðx1 Þdt

dðx2 Þdt

dðx3 Þdt

dðx4 Þdt

266666666666664

377777777777775þv

)

_z1

_z2

_z3

_z4

_z5

_z6

_z7

2666666666664

3777777777775¼ ��!Eq: ð21Þ:h1 ðXðtÞÞ¼h2 ðXðtÞÞ¼��¼hm ðXðtÞÞ¼0¼

000

Dx1

�D1H1

Dx1�x0H1

DP1

Dx2

�D2H2

Dx2�x0H2

DP2

2666666666664

3777777777775þ

v1

v2

v s

0000

2666666666664

3777777777775

ð33Þ

where _z4 to _z7 represent the zero dynamics of the system. The inter-nal dynamic stability of zero dynamics for machine 1 can be easilyseen as follows:

_Z4 ¼ Dx

_Z5 ¼ �D1

H1Dx1 �

x0

H1DP

ð34Þ

Since DP1 = 0, DP2 = 0 and DVs = 0, Eq. (34) can be rewritten as

_Z4 ¼ Dx1

_Z5 ¼ �D1

H1Dx1

ð35Þ

In addition,

_Z5 ¼ D _x1 ¼ �D1

H1Dx1 ! Dx1 ¼ c1e�Dt=H ) Lim Dx1 ¼

t!1c1e�Dt=H

� �¼ 0

As a result, the internal dynamic is stable. Moreover, anotherinternal dynamic _Z4 of machine 1 is stable in terms of _Z5 as follows:

_Z4 ¼ D _d ¼ Dx1!R

Dd1 ¼ �c1 � ðH=DÞe�Dt=H

) LimðDd1 ¼ �c1 �t!1ðH=DÞe�Dt=HÞ ¼ 0

Finally, the whole internal dynamics of machine 1 are stable. Thesame approach can be carried out for machine 2 to confirm thestability.

Now, v1, v2 and vs can be expressed as given in Eqs. (36) and(37). It should be noted that _z1 ¼ v1; _z2 ¼ v2; _z3 ¼ vs and z1 =D Pe1, z2 = DPe1, z3 = DVe1.

Therefore, v1 ¼ D _Pe1;v2 ¼ D _Pe2;v s ¼ D _Vs and these values canbe obtained from Eqs. (10) and (2).

v i ¼D _Pei ¼ f 1;iE0qiGiiþ f 1;i

Xn

j¼1

E0qjfGij cos dijþBij sin dijg

þ f 1;i

E0q1

G1ðx1þx2ÞfGis cos diþBis sin dig x2

2E0qiþx1x2E0qr cos dir

h i

þ f 1;ifGis cos diþBis sin digVsþ f 2;i

Xn

j¼1

E0qiE0qjf�Gij sin dijþBij cos dijg

þ f 2;iE0qif�Gis sin diþBis cos digVsþ f 2;iE

0qifGis cos di

þBis sin dig1

G1ðx1þx2Þð�x1x2E0qiE

0qr cosdirÞ

þ f 3E0qifGis cos di

þBis sin digx1x2

x1þx2

þ E0qiGiiþ

Xn

j¼1

E0qjfGij cos dijþBij sin dijg(

þE0q1

G1ðx1þx2ÞfGis cos diþBis sin dig x2

2E0qiþx1x2E0qj cos dij

h iþfGis cos diþBis sin digVsggiiuiþE0qifGis cos diþBis sin dig

x1x2

x1þx2

gssus

i; j2f1;2g and i – j ð36Þ

vs ¼ D _VsðtÞ ¼1

G1ðx1 þ x2Þf1;j x2

2E0qj þ x1x2E0qi cos dji

� �hþf1;i x2

1E0qi þ x1x2E0qj cos dji

� �þ f2;j �x1x2E0qjE

0qi sin dji

_dj

� �þf2;i þx1x2E0qjE

0qi sin dji

_di

� �iþ x1x2

x1 þ x2f3

þ x22E0qj þ x1x2E0qi cos dji

� �gjjuj þ x2

1E0qi þ x1x2E0qj cos dji

� �giiui

þ x1x2

x1 þ x2gssus i ¼ 1; j ¼ 2 ð37Þ

In Eqs. (36) and (37) f1;i ¼ D _E0qi � 1T 0d0i

Efdi; f2;i ¼ D _di

ði ¼ 1;2Þ; f3 ¼ D_Is � 1Ts

us; gii ¼ 1T 0d0i

; gss ¼ � 1Ts

. Furthermore, as men-

tioned before the system inputs ui = Efdi i = 1, 2 and us ¼ Irefqs are

the input voltage of the excitation system of ith generator and


STATCOM reference current, respectively. Eqs. (36) and (37) can berewritten as a function of u1, u2 and us as follows:

v1

v2

v s

264

375 ¼ b1ðXÞ

b2ðXÞb3ðXÞ

264

375þ Q 11ðXÞ Q 12ðXÞ Q 13ðXÞ

Q 21ðXÞ Q 22ðXÞ Q 23ðXÞQ 31ðXÞ Q 32ðXÞ Q 33ðXÞ

264

375 u1

u2

us

264

375 ð38Þ

Using Eq. (14) implies that Q12(X) = Q21(X) = 0. Therefore, the systeminputs for each generator depend on its states.

For the linear system presented in Eq. (13), the input values ofthe linear system; v1, v2 and vs, can be obtained so that the desir-able performance of the system is achieved. For this purpose, thesystem inputs are determined by using the state feedback controltheory. Since the linear system states are Ddi, Dxi, DPi, DVs, thedesirable feedback control law can be determined using the follow-ing equations;

v1 ¼ v iji¼1 ¼ Kd1Dd1 þ Kx1Dx1 þ KP1DPe1

v2 ¼ v iji¼2 ¼ Kd2Dd2 þ Kx2Dx2 þ KP2DPe2

v s ¼ KsDVs

ð39Þ

where Ddi, Dxi, DPi, DVs are the state variables of the linearized sys-tem. Therefore, Eq. (38) represents a linear system with three un-known parameters (ui i = 1, 2 and us). By substituting the virtualinputs (v1, v2 and vs) into Eq. (33), the following state matrix A, gi-ven in Eq. (13), is obtained for the closed loop linear system.

A ¼

0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 1 0 0 0 0 00 � D1

H1�x0

H10 0 0 0

0 0 0 0 0 1 00 0 0 0 0 � D2

H2�x0

H2

2666666666664

3777777777775

ð40Þ

By using the state feedback control theory presented in Eq. (39), theclosed loop state matrix of the system can be written as follows:

Acl ¼

Kd1 Kx1 KP1 0 0 0 00 0 0 0 Kd2 Kx2 KP2

0 0 0 Ks 0 0 00 1 0 0 0 0 00 � D1

H1�x0

H10 0 0 0

0 0 0 0 0 1 00 0 0 0 0 � D2

H2�x0

H2

2666666666664

3777777777775

ð41Þ

The coefficients of the linear controllers that are used in Eq. (41)effectively affect the system performance. Therefore, these control-ler coefficients are calculated such that it improves the system per-formance. In doing so, many methods can be used; e.g. poleplacement [14,27,28] and H1method [26]. Since these methods in-crease the complexity of design problem, in this study one of theintelligent methods is used to determine the feedback gains.

Before going to the next section it should be mentioned that forthe implementation of any controller it is essential that the signalsused in control law be measurable. In this regard, the authors in[29] presented the approach for the small signal stability assess-ment of a multi-machine system using synchronized Phasor Mea-surement Units (PMUs) data. The proposed method does not needany information about the generators, network configuration orline impedances. By installing one PMU on each generator busand using classical model for generator, all of the network and gen-erators parameters needed for small signal stability analysis areestimated using the ambient data registered in the PMUs. In addi-tion, in [30] the proposed control strategy is executed by a nonlin-ear multi-loop controller with rotor angles and speed deviations of

synchronous generators are used as the input signals. The inputsignals, obtained from a PMU, are necessary only from a small areaaround the controlled shunt FACTS devices. Thus, all variables inpower system can be easily measured by using PMUs.

5.2. Controller design for generators which are not directly connectedto the STATCOM

In this case, the performance of each generator can be modeledby a SISO system. Each of these generators is locally controlled andtheir inputs are determined by using zero dynamics method.Therefore, the value of the virtual input can be determined as fol-lows based on Eq. (26);

v i ¼ D _Pei ¼ f1;i

Xn

j¼1

E0qjfGij cos dij þ Bij sin dijg

þ f2;i

Xn

j¼1

E0qiE0qjf�Gij sin dij þ Bij cos dijg þ f1;iE

0qiGii

þ E0qiGii þXn

j¼1

E0qjfGij cos dij þ Bij sin dijg( )

giiui i ¼ 3; . . . ;m ð42Þ

It should be noted that _z1;i ¼ v1;i and z1,i = DPei and, therefore,v i ¼ D _Pei. These values can be obtained from Eq. (10). Since the lin-ear system states are Ddi, Dxi and DPi, the input for each generatordepends on its states and is given as follows:

v iji¼3;...;m ¼ KdiDdi þ KxiDxi þ KPiDPei ð43Þ

5.3. Objective function to determine the unknown parameters ofcontrollers

The design problem is now converted to an optimization prob-lem in which the coefficients of Eq. (39) (Kd, Kx, KP, KS) are deter-mined such that the speed deviation of generators and also thevoltage deviation of the STATCOM bus are minimized. For this pur-pose one performance index should be selected. Every integral per-formance index has certain advantages in control system design.The ITAE criterion tries to minimize time multiplied absolute errorof the control system. The time multiplication term penalizes theerror more at the later stages than at the beginning and hence iteffectively reduces the settling time (ts). This cannot be achievedby IAE (a weighted sum of Integral of Absolute Error) or ISE (aweighted summation of Integral of Squared Error normalized bymaximum error, maximum percentage of overshoot and settlingtime normalized by simulation time). Since the absolute error is in-cluded in the ITAE criterion, the maximum percentage of the over-shoot (Mp) is also minimized. Therefore, ITAE performance index isused as an objective function [31,32] and is given in the followingequation:

Min OF ¼Z t2¼tsim

t1¼0t � a

Xn

i¼1

jDxij þ bjDVSj( )

dt

S:t: : Kmindj 6 Kdj 6 Kmax

dj Kminxj 6 Kxj 6 Kmax

xj

Kminpj 6 Kpj 6 Kmax

pj Kmindi 6 Kdi 6 Kmax

di

Kminxi 6 Kxi 6 Kmax

xi Kminpi 6 Kpi 6 Kmax

pi

Kmins 6 Ks 6 Kmax

s

ð44Þ

In Eq. (44) tsim is the time simulation. The coefficients a and b areused to normalize the speed deviation (Dx) and STATCOM bus volt-age deviation (DVS). The ITAE criteria for the speed and STATCOMbus voltage deviations are illustrated as follows:

G3

Loa

d 1

G1

Loa

d 2

G4

G2

Z1 Z2Z3

Z3Z1Z2

Z3

Z31

2 4

11109876

5 3XT

XT XT

XT

STA

TC

OM

Fig. 2. Two-area power system with STATCOM.

0 1 2 3 4 5 60.992

0.994

0.996

0.998

1

1.002

1.004

1.006

1.008

1.01

Time (sec)

Gen

erat

or S

peed

of G

3 (p

u)

Uncoordinated PSS and STATCOMCoordinated PSS and STATCOM Coordinated Excitation system and STATCOM by Zero Dynamics

Fig. 3. Speed response of G3 for a 200 m-second 3-phase fault at Bus 8.

0 1 2 3 4 5 60.994

0.996

0.998

1

1.002

1.004

1.006

1.008

Time (sec)

Gen

erat

or S

peed

of G

4 (p

u)

Uncoordinated PSS and STATCOMCoordinated PSS and STATCOMCoordinated excitation system and STATCOM by Zero Dynamics

Fig. 4. Speed response of G4 for a 200 m-second 3-phase fault at Bus 8.


f1 ¼Z t2

t1

tjDxðtÞj dt ð45Þ

f2 ¼Z t2

t1

tjDVSðtÞj dt ð46Þ

where DVS(t) = VS(t) � VS, ref(t).In this paper, by combining the Nelder–Mead (NM) technique

and bacterial foraging algorithm (BFA), a hybrid method, namedBF–NM is used to solve the optimization problem. The NM tech-

nique is a local search optimization method and is flexible, robustand easy to be programmed. It has a high convergence speed, butwith a weak search capability. On the other hand, BFA techniquewhich is a promising evolutionary method for handling the optimi-zation problems and has a low convergence speed covers a widesearch region. Therefore, these two techniques are combined[21,22] to obtain the search capability of the intelligent and theaccuracy of conventional methods simultaneously to solve theoptimization problems. More details about BF–NM algorithm arepresented in [21].

0 1 2 3 4 5 60.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

Time (sec)

Term

inal

vol

tage

of G

3 (p

u)


Fig. 5. Terminal voltage response of G3 for a 200 m-second 3-phase fault at Bus 8.

0 1 2 3 4 5 60.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

Time (sec)

Term

inal

vol

tage

of G

4 (p

u)


Fig. 6. Terminal voltage response of G4 for a 200 m-second 3-phase fault at Bus 8.

0 1 2 3 4 5 60.999

1

1.001

1.002

1.003

1.004

1.005

1.006

1.007

Time (sec)

Gen

erat

or S

peed

of G

1 (p

u)

0.5 1 1.5 2 2.5 3 3.5 41

1.00011.00011.00011.00011.00011.0002


Fig. 7. Speed of generator G1 for changes in load at t = 1 s and t = 3 s.


0 1 2 3 4 5 60.998

0.999

1

1.001

1.002

1.003

1.004

1.005

1.006

1.007

Time (sec)

Gen

erat

or S

peed

of G

2 (p

u)0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

1.00011.00011.00011.00011.0001


Fig. 8. Speed of generator G2 for changes in load at t = 1 s and t = 3 s.

0 1 2 3 4 5 6 7 8 9 1010

20

30

40

50

60

70

Time (sec)

mec

hani

cal a

ngle

Diff

eren

t of G

4-G

1 (D

eg) Uncoordinated PSS and STATCOM

Coordinated PSS and STATCOMCoordinated excitation system and STATCOM by Zero Dynamics

0 1 2 3 4 5 634.2

34.4

34.6

34.8

35

Fig. 9. Difference rotor angle generators (G4–G1) for changes in load at t = 1 s and t = 3 s.

0 1 2 3 4 5 6

2

4

6

8

10

12

14

16

18

20

Time (sec)

Rot

or a

ngle

Diff

eren

t of G

4-G

2 (D

eg)

0 1 2 3 4 5 610.45

10.5

10.55

10.6

10.65

Uncoordinated PSS and STATCOMCoordinated PSS and STATCOMCoordinated excitation system and STATCOM byZero Dynamics

Fig. 10. Difference rotor angle generators (G4–G2) for changes in load at t = 1 s and t = 3 s.


Table 1Comparison of the ITAE performance index for all methods.

Zerodynamics

Coordinated PSS andSTATCOM

Uncoordinated PSS andSTATCOM

Three-phase short circuitG1 f1 = 0.0078 f1 = 0.0171 f1 = 0.0212G2 f1 = 0.0071 f1 = 0/0160 f1 = 0.0209G3 f1 = 0.0054 f1 = 0.0143 f1 = 0.0189G4 f1 = 0.0056 f1 = 0.0133 f1 = 0.0191STATCOM f2 = 0.1310 f2 = 0.2307 f2 = 0.2416

Change of load conditionG1 f1 = 0.00013 f1 = 0.0147 f1 = 0.0113G2 f1 = 0.00011 f1 = 0.0130 f1 = 0.0099G3 f1 = 0.0001 f1 = 0.0139 f1 = 0.0108G4 f1 = 0.00016 f1 = 0.0151 f1 = 0.0119STATCOM f2 = 0.0023 f2 = 0.0077 f2 = 0.0054


Now the proposed design procedure to determine the control-lers of STATCOM and excitation system coordinately are summa-rized as follows:

Step (1) Determine the dynamic equations, inputs, outputs andstates of the given power system which have been presented inEqs. (1)–(11) and (31).Step (2) Calculate the vector relative degree (r1,r2, . . . ,rm) of thegiven power system and check the assumption r < n.Step (3) Choose the vector Z = u(X) and check that _Z ¼ _uðXÞ isnonsingular at initial point X = X0.Step (4) Check the condition presented in Eqs. (21) and (22).Step (5) Check the condition of zero dynamics method whichwere presented in Eq. (25).Step (6) Investigate the stability of zero dynamics by Eq. (30)Step (7) Determine the coefficients of the virtual input (v) of Eq.(39) by using the BF–NM algorithm.Step (8) Determine the actual input u of Eq. (38), in which, forinstance, ui = Efdi i = 1, 2 and us ¼ Iref

qs .

6. Simulation results

In this section, the proposed design method is implemented ona multi-machine system. The time domain simulations have beenperformed to investigate the damping performance of the designedsupplementary controllers in the system under study. The perfor-mances of the designed controllers are compared with two othercases when PSS and STATCOM controllers are designed coordi-nately using the method given in [22] and also when both are de-signed separately [1].

6.1. Multi-machine power system

Fig. 2 represents one line diagram of the multi-machine systemgiven in [1]. As shown in this figure, the STATCOM is installed atbus 8. The parameters of this two-area system are given in [1].

Based on the explanations presented in Section 5.1, the coeffi-cients of Eq. (39) are obtained by solving the optimization problempresented in Eq. (44) using the BF–NM algorithm as follows:

Kd1 ¼ 73:4; Kx1 ¼ 18:8; KP1 ¼ �93:5; Kd2 ¼ 71:4;

Kx2 ¼ 20:8; KP2 ¼ �92:5; KV ¼ 98:4

The performance of the power system for short circuit and loadchange conditions are shown in Sections 6.1.1 and 6.1.2,respectively.

6.1.1. Three-phase fault testThe three-phase occurs near bus 8 at t = 1 s and is cleared at

1.2 s. The fault and ground resistances are 0.001 and 0.001 ohmrespectively. The speed deviation of generators G1 and G3 areshown in Figs. 3 and 4, respectively. The terminal voltage deviationof generators G1 and G3 are shown in Figs. 5 and 6, respectively.

By comparing the results of different methods presented in Figs.3–6, it can be seen that the coordinated design of STATCOM andexcitation system using zero dynamics method considerably im-proves the performance of the power system more than othermethods. For example, as shown in Fig. 3 the speed response ofG3 has less overshoot, undershoot and rise time.

6.1.2. Load change testThe performance of the power system is also studied for a

50 MW decrease in load during and the results are shown in Figs.7–10. The speed deviation of generators G1 and G2 are shown inFigs. 7 and 8. In addition, the rotor mechanical angles of generatorsG1 and G2 with respect to G4 are shown in Figs. 9 and 10. As shown

in these figures, the proposed method gives a much better perfor-mance than other methods.

6.2. Comparison of the ITAE performance index for all methods

In order to investigate the effectiveness of zero dynamics meth-od for the coordinated design of STATCOM and excitation systemthe ITAE performance index is also used.

By using Eqs. (45) and (46) the ITAE performance index is calcu-lated. These values are illustrated in Table 1 by f1 and f2. As shownin this table, the ITAE criterion for zero dynamics method is smallerthan those of other methods confirming its superior performance.

7. Conclusions

In this paper the zero dynamics method is used to design thecontrollers of STATCOM and excitation system coordinately. Thedesign strategy includes enough flexibility to set the desired levelof the stability and performance, and to consider the practical non-linear properties of the system. For evaluation of the proposedmethod, its performance is compared with the case when it isequipped with STATCOM and PSS controllers. Simulation and ana-lytical results for a multi-machine power system confirmed theeffectiveness and the robustness of the proposed design techniqueto enhance the dynamic characteristics of the power system.

Appendix A

Consider a nonlinear system with m inputs and m outputs

_XðtÞ ¼ f ðXÞ þ g1ðXÞu1 þ g2ðXÞu2 þ � � � þ gmðXÞum

y1 ¼ h1ðXÞ...

ym ¼ hmðXÞ

where X 2 Rn, f(X) and gi(X) i = 1,2, . . . ,m are n-dimensional smoothvector fields; ui; is the ith control variable, yi(t) is the ith output,hi(X) is a scalar function of X. The concept of the relative degree ofthe system will be introduced below.

Firstly, it should be made clear that for each output yi(t) = hi(X)there exists a corresponding relative degree r. Thus, the relativedegrees of an MIMO system form a set is r = {r1,r2, . . . ,rm}.

Secondly, each sub-relative degree satisfies the following condi-tions in a neighborhood of X0, which is expressed as a definition.

Definition 1. For a MIMO system, if the following conditions areheld in a neighborhood of X0 for each sub-relative degree, namely,for ki < ri � 1

LgiLki

f hiðXÞ ¼ 0 i ¼ 1;2; . . . ;m j ¼ 1;2; . . . ;m


and the m �m matrix

Q �

Lg1Lr1�1


f h1ðXÞ � � � � � � LgmLr1�1

f h1ðXÞ

Lg1Lr2�1


f h2ðXÞ � � � � � � LgmLr2�1

f h2ðXÞ

..

. ...

� � � � � � ...

..

. ...

� � � � � � ...

Lg1Lrm�1

f hmðXÞ Lg2Lrm�1

f hmðXÞ � � � � � � LgmLrm�1

f hmðXÞ

26666666664

37777777775

is nonsingular near X0, r = {r1,r2, . . . , rm} will be the relative degreeset of the system, and each sub-relative degree ri; corresponds tooutput yi(t) = h(X).

As an example for the power system example given in this pa-per, the sub-relative degree r1 for machine 1 can be calculated asfollows:

Lg1h1ðXÞ¼ Lg1

L1�1f h1ðXÞ¼

VsfG1s cos d1þB1s sin d1gþXn

j¼1

E0qjfG1j cos d1jþB1j sin d1jgþ

E0q1G11

266664

377775�

1T 0d01

– 0

Lg2h1ðXÞ¼ Lg2

L1�1f h1ðXÞ¼

Xn

j¼1

E0q1fG1j cos d1jþB1j sin d1jg" #

� 1T 0d02

– 0

Lg3h1ðXÞ¼ Lg3

L1�1f h1ðXÞ¼ E0q1fG1s cos d1þB1s sin d1g

x1x2

x1þx2

��1

Ts– 0

which indicates r1 = 1. In addition, the same process can be done tofind relative degree for machine 2 (r2 = 1) and STATCOM (r3 = 1).Thus, r1 + r2 + r3 = 3 which means that (r1 + r2 + r3) < n as n = 7.Therefore, the system is partially linearized for the selected outputfunctions.

Appendix B

Given a smooth scalar function h(x) of the state x, the gradientof h is denoted byrh ¼ @h

@x. Given a scalar function h(x) and a vectorfield f(x), a new scalar function Lfh is defined and called the Liederivative (or simply, the derivative) of h with respect to f.

Definition 2. Let h: Rn ! R be a smooth scalar function, andf : Rn ! Rn be a smooth vector field on Rn, then the Lie derivativeof h with respect to f is a scalar function defined by Lfh =rhf.

Thus, the Lie derivative Lfh is simply the directional derivativeof h along the direction of the vector f.

Repeated Lie derivatives can be defined recursively as

L0f h ¼ h

Lif h ¼ Lf Li�1

f h� �

¼ r Li�1f h

� �f for i ¼ 1;2; . . .

Similarly, if g is another vector field, then the scalar functionLg Lfh(x) is

LgLf hðxÞ ¼ rðLf hÞg

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