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A new algorithm for analysis of SVC’s impact on bifurcations, chaos and voltage collapse in power systems D. Padma Subramanian a,, R.P. Kumudini Devi b , R. Saravanaselvan b a Department of Electrical and Electronics Engineering, SRM Valliammai Engineering College, Kattankulathur-603203 b Department of Electrical and Electronics Engineering, College of Engineering, Guindy, Anna University, Chennai-25 a r t i c l e i n f o  Article history: Received 2 June 2006 Received in revised form 10 January 2011 Accepted 28 January 2011 Available online 27 March 2011 Keywords: Continuation algorithm Bifurcation approach Voltage collapse SVC a b s t r a c t In this paper, a numerical algorithm, based on initial value problem, using local parameterisation contin- uation technique is proposed for tracing stable and unstable steady state periodic solution branches of power systems. Bifurcation diagrams of steady state solutions are constructed by the application of the proposed algorithm. From the bifurcation diagrams, the existence of various bifurcation points such as, unstable Hopf bifurcation (UHB), stable Hopf bifurcation (SHB), cyclic fold bifurcation (CFB), saddle node bifurcation (SNB) and period doubling bifurcation (PDB) are identied. With the use of tools of nonlinear dynamics, voltage collapse points, and chaotic solutions due to period doublings are unearthed. Simula- tions have been carried out to analyse the sensitivity of the system with respect to load reactive power and compensating capacitor. The impact of SVC on Hopf bifurcations and occurrence of SNB are investi- ga ted . The alg orithm is validated by ap ply ing it to a sta nd ar d pow er sys temrepo rte d in lit er atu re an d the results obtained are presented. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Evolution of periodic solutions of power systems may lead to abnormal solutions, viz, chaotic solutions, subharmonic solutions and dynamic voltage collapse points. Hence a numerical method that computes both stable and unstabl e period ic solutio ns plays an important role in nonlinear dynamic analysis [1–7]. The pack- ages BIFPACK [5] , AUTO [6] and MATCONT [7] have been used suc- cessively on a wide range of problems. However, considering the characte ristics of periodic solution s of autonomo us systems, an alternate, initial value problem based approach is proposed in this paper. In addition, the effect of SVC on Hopf bifurcations, SNB and voltage stability are analysed. Ap ril le and Tr ick [1] , propo sed a New ton algo rith m for det ermi- nation of a peri odic solution and per iod of autonomous syst em. Holodniok and Kubíc ˇ ek [2] described an algorithm for the contin- uation of periodic solutions of the ordinary differential equation based on the shooting method and on the arc-length continuation algorithm DERPAR. Aluko and Chang [3] applied a Euclidean arc- leng th con tin uat ion pro cedure wit h adap tive step sizi ng to the boundary value problem solved by Newton–Fox method that xes its phase by an automatically determined xed-component index device. Deodel et al. [4] proposed an integral phase condition that minimizes the changes in the prole of a periodic solution. In the references cited above, [1] describes an approach to get one peri- odic steady state solu tio n and its per iod, where as app roac h to get continuum of periodic solutions is not given. Refs. [2–7] treat the problem as boundary value problem to obtain range of stable and unstable periodic solutions. The bifurcations and hence chaos exhibited by power systems with the variation in system parame- ters (e.g., load) have been reported widely in the literature [8,9] . Howeve r, in these refe ren ces, simula tion resu lts are obta ine d using AUTO, subject to initial conditi on, boundary condit ion and integral constraints. This paper presents a simple and easy to implement algorith m based on initial value pro blem for the bifurcatio n analy- sis of bot h stab le and unsta ble steady state per iodi c solu tion branches of autonomous systems. By the computation of Floquet multipliers, stability of converged solutions is analysed. Most of the power systems today are operating under stressed con diti on s and are thr eatened by the poss ibil ity of volt age instabil - ity/collapse. Extensive literature exists on this subject from static as wel l as dynamic con siderations. Stat ic ana lysis is ina dequ ate when the system dynamics is taken into account. There has been con siderable int erest in ana lysis of dyn amic volt age inst abil ity based on bifurcation approach in recent years. It has been shown [10] that voltage collapse may arise from the existence of Hopf bifurcation, which is prior to the appearance of SNB. With increas- ing popularit y of Flexible AC Transmissio n System (FACTS ) devices and SVC be ing the we ll- un de rs to od and wi de ly accepte d FAC TS de - vice, it is worth exploring the effect of SVC on elimination of dy- namic bifurcation s, chaos and impro vemen t of voltage stability. In [11–17], the effects of FACTS controllers on eliminating Hopf bifurcation and chaos have been studied. The effect of AVR gain 0142-0615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2011.01.033 Corresponding author. E-mail address: [email protected](D. Padma Subramanian). Electrical Power and Energy Systems 33 (2011) 1194–1202 Contents lists available at ScienceDirect Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
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A new algorithm for analysis of SVC’s impact on bifurcations, chaos and voltagecollapse in power systems

D. Padma Subramanian a ,⇑ , R.P. Kumudini Devi b , R. Saravanaselvan b

a Department of Electrical and Electronics Engineering, SRM Valliammai Engineering College, Kattankulathur-603203b Department of Electrical and Electronics Engineering, College of Engineering, Guindy, Anna University, Chennai-25

a r t i c l e i n f o

Article history:Received 2 June 2006Received in revised form 10 January 2011Accepted 28 January 2011Available online 27 March 2011

Keywords:Continuation algorithmBifurcation approachVoltage collapseSVC

a b s t r a c t

In this paper, a numerical algorithm, based on initial value problem, using local parameterisation contin-uation technique is proposed for tracing stable and unstable steady state periodic solution branches of power systems. Bifurcation diagrams of steady state solutions are constructed by the application of theproposed algorithm. From the bifurcation diagrams, the existence of various bifurcation points such as,unstable Hopf bifurcation (UHB), stable Hopf bifurcation (SHB), cyclic fold bifurcation (CFB), saddle nodebifurcation (SNB) and period doubling bifurcation (PDB) are identied. With the use of tools of nonlineardynamics, voltage collapse points, and chaotic solutions due to period doublings are unearthed. Simula-tions have been carried out to analyse the sensitivity of the system with respect to load reactive powerand compensating capacitor. The impact of SVC on Hopf bifurcations and occurrence of SNB are investi-gated. The algorithm is validated by applying it to a standard power systemreported in literature and theresults obtained are presented.

Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Evolution of periodic solutions of power systems may lead toabnormal solutions, viz, chaotic solutions, subharmonic solutionsand dynamic voltage collapse points. Hence a numerical methodthat computes both stable and unstable periodic solutions playsan important role in nonlinear dynamic analysis [1–7] . The pack-ages BIFPACK [5] , AUTO [6] and MATCONT [7] have been used suc-cessively on a wide range of problems. However, considering thecharacteristics of periodic solutions of autonomous systems, analternate, initial value problem based approach is proposed in thispaper. In addition, the effect of SVC on Hopf bifurcations, SNB andvoltage stability are analysed.

Aprille and Trick [1] , proposed a Newton algorithm for determi-nation of a periodic solution and period of autonomous system.Holodniok and Kubíc ˇ ek [2] described an algorithm for the contin-uation of periodic solutions of the ordinary differential equationbased on the shooting method and on the arc-length continuationalgorithm DERPAR. Aluko and Chang [3] applied a Euclidean arc-length continuation procedure with adaptive step sizing to theboundary value problem solved by Newton–Fox method that xesits phase by an automatically determined xed-component indexdevice. Deodel et al. [4] proposed an integral phase condition thatminimizes the changes in the prole of a periodic solution. In thereferences cited above, [1] describes an approach to get one peri-

odic steady state solution and its period, whereas approach toget continuum of periodic solutions is not given. Refs. [2–7] treatthe problem as boundary value problem to obtain range of stableand unstable periodic solutions. The bifurcations and hence chaosexhibited by power systems with the variation in system parame-ters (e.g., load) have been reported widely in the literature [8,9] .However, in these references, simulationresults are obtained usingAUTO, subject to initial condition, boundary condition and integralconstraints. This paper presents a simple and easy to implementalgorithm based on initial value problem for the bifurcation analy-sis of both stable and unstable steady state periodic solutionbranches of autonomous systems. By the computation of Floquetmultipliers, stability of converged solutions is analysed.

Most of the power systems today are operating under stressedconditions and are threatened by the possibility of voltage instabil-ity/collapse. Extensive literature exists on this subject from staticas well as dynamic considerations. Static analysis is inadequatewhen the system dynamics is taken into account. There has beenconsiderable interest in analysis of dynamic voltage instabilitybased on bifurcation approach in recent years. It has been shown[10] that voltage collapse may arise from the existence of Hopf bifurcation, which is prior to the appearance of SNB. With increas-ing popularity of Flexible AC Transmission System (FACTS) devicesand SVC being the well-understood and widely accepted FACTS de-vice, it is worth exploring the effect of SVC on elimination of dy-namic bifurcations, chaos and improvement of voltage stability.In [11–17] , the effects of FACTS controllers on eliminating Hopf bifurcation and chaos have been studied. The effect of AVR gain

0142-0615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved.doi: 10.1016/j.ijepes.2011.01.033

⇑ Corresponding author.E-mail address: [email protected] (D. Padma Subramanian).

Electrical Power and Energy Systems 33 (2011) 1194–1202

Contents lists available at ScienceDirect

Electrical Power and Energy Systems

j ou rna l h omep age : www. e l s ev i e r. com/ loca t e / i j epe s

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on the bifurcation of subsynchronous resonance in power systemsis explored in [18] . Voltage stability assessment and improvementof steady state stability margins in power systems are analysed in[19,20] .

The main theme of this paper is to present and validate the pro-posed algorithm for tracing stable and unstable periodic branchesof power system. Further, the effectiveness of SVC in eliminatingHopf bifurcations, delaying the voltage collapse and improvingthe voltage stability of the system is also analysed.

2. Algorithm

The various steps in the proposed algorithm are explainedbelow.

2.1. Calculation of steady state periodic solution and its period [1,22]

Consider an autonomous system of ordinary differentialequations

_ x ¼ f ð xÞ ð1 Þ

wherex

andf

aren

-vectors andf

is continuous inx

and has a con-tinuous rst partial derivative with respect to x for all x. Let x(t ) bethe solution to Eq. (1) with initial state x(0) = x0 and let p(t ) be anonconstant periodic solution with period T . Then p(t + t 0) is a solu-tion for any constant t 0 . Furthermore, _ p (t ) is a solution of the equa-tion of rst variation on the orbit p(t ). Conventional numericalintegration methods are computationally inefcient for tracingthe stable and unstable periodic branches of autonomous systems,for two reasons:

(i) The period of oscillation T is not known.(ii) Fromthe theory of ordinary differential equations, the sensi-

tivity matrix always has an eigenvalue k = 1 whenever x0 lieson a periodic orbit. This implies that the Jacobian is singulareven on a stable orbit p( t ) of Eq. (1) .

Integrating bothsides of Eq. (1) fromtime 0 to time t , we obtain

xðt Þ ¼Z t

0 f ð xðt ÞÞdt þ x0 , xðt ; x0 Þ ð2 Þ

Right-hand side of Eq. (2) is denoted by x(t , x0) to emphasise thatthe solution x(t ) at any time t depends on the initial state x0 .

F ðT ; x0 Þ, xðT ; x0 Þ ¼Z T

0 f ð xðt ÞÞdt þ x0 ð3 Þ

The period T is now added to the arguments of the function F (T , x0 )because it is an unknown variable that must be determined alongwith the unknown initial state x0 = p0 . It follows from Eq. (3) that,

the initial state x0 = p0 which give rise to a periodic solution p (t )with no transient component must satisfy the equation

x0 ¼ F ðT ; x0 Þ ð4 Þ

In Eq. (4) , we now have n + 1 variables f x01 ; x02 ; . . . :; x0n ; T g to bedetermined. Recall that if x = p(t ) is a periodic solution of autono-mous system, then so is x = p (t + t 0) where t 0 is any instant, because f ( x) does not contain t as an argument; _ pðt Þ ¼ f ð pðt ÞÞ implies

_ pðt þ t 0 Þ ¼ f ð pðt þ t 0 ÞÞ. In other words Eq. (1) admits an innitenumber of periodic solutions each one differing from the othersby a translation in time. Hence a value for one of the n initial statesf x01 ; x02 ; . . . ; x0kÀ1 ; x0kþ 1 ; . . . ; xn g, say, x0k ¼ pk0

can be assumed andremaining initial states which fall on the same instant t = t 0 as thatof x0k ¼ pk0

can be determined. Hence theprocedure is equivalent to

that of choosing a new time origin of periodic solutions. The compo-nent k must be chosen such that

iÞ f k x T ð jÞ; xð jÞ0 ! f m x T ð jÞ; xð jÞ

0 ;

m ¼ 1 ; 2 ; . . . ; k À 1 ; k þ 1 ; . . . ; n ð5 Þ

iiÞ xð jÞ0 k

¼ xð jÞ0 k

T ð jÞ; xð jÞ0 , pð jÞ

k0ð6 Þ

Only precaution here is that the assumed value for pk0 , must fallwithin the range of values taken on by x

k= p

k(t ) over one period,

i.e., pkmin < pk 0< pkmax where pkmin and pkmax denote, respectively,the minimum and maximum values of pk(t) over one period. Oncethe value of initial state is xed by assumption, we dene thenew n  1 unknown vector

y, ½ x0 1 x0 2. . . x0 kÀ1 Tx0 kþ 1 xn

t ð7 Þ

Eq. (4) can be recasted into the equivalent form

H ð y Þ, x0 À F ðT ; x0 Þ ¼ 0 ð8 Þ

The systemof n algebraic equations in n unknowns can be solved byNewton–Raphson iteration

y ð jþ 1 Þ ¼ y ð jÞ À I ðÀkÞ ÀF 0ðT ð jÞ; xð jÞ0

h iÀ1

xð jÞ0 À x T ð jÞ; xð jÞ

0

h ið9 Þ

where I(Àk) denotes a diagonal matrix with unit diagonal elementsexcept for the kth element, which is zero because the Jacobian ma-trix H 0ð y Þ, @ H ð y Þ=@ y in question here requires taking partial deriv-atives with respect to the variable in y , and the kth element of y is T , not x0k . The matrix F 0ðT ; x0 Þ, @ F ðT ; x0 Þ=@ y is given by

F 0ðT ; x0 Þ ¼

@ x1 ðT ; x0 Þ@ x0 1

Á Á Á@ x1 ðT ; x0 Þ@ x0 kÀ1

@ x1 ðT ; x0 Þ@ T

@ x1 ðT ; x0 Þ@ x0 kþ 1

Á Á Á@ x1 ðT ; x0 Þ@ x0 n

@ x2 ðT ; x0 Þ@ x0 1

Á Á Á@ x2 ðT ; x0 Þ@ x0 kÀ1

@ x2 ðT ; x0 Þ@ T

@ x2 ðT ; x0 Þ@ x0 kþ 1

Á Á Á@ x2 ðT ; x0 Þ@ x0 n

Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á

@ xkðT ; x0 Þ@ x0 1

Á Á Á@ xk ðT ; x0 Þ@ x0 kÀ1

@ xkðT ; x0 Þ@ T

@ xkðT ; x0 Þ@ x0 kþ 1

Á Á Á@ xkðT ; x0 Þ@ x0 n

Á Á Á Á Á Á Á Á Á Á

Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á Á

@ xn ðT ; x0 Þ@ x0 1

Á Á Á@ xn ðT ; x0 Þ@ x0 kÀ1

@ xn ðT ; x0 Þ@ T

@ xn ðT ; x0 Þ@ x0 kþ 1

Á Á Á@ xn ðT ; x0 Þ@ x0 n

266666666666666666664

377777777777777777775ð10 Þ

Except for column k, the im th element of F 0ðT ; x0 Þ can be computedas follows

@ xiðT ; x0 Þ@ x0 m x0 ¼ xð jÞ

0

¼ xi T ; xð jÞ

0 þ D x0 À xi T ; xð jÞ0 D x0 m

ð11 Þ

where D x0 ¼ D x01D x02

. . . D x0n Ãt is a small perturbation vector,

x T ; xð jÞ0 and x T ; xð jÞ

0 þ D x0 are corresponding solutions of Eq. (1)

evaluated at the end of one period t = T. The kth column is evaluatedas

@ x jðT ; x0 Þ@ T ¼ f jð xðT ; x0 ÞÞ; j ¼ 1 ; 2 ; . . . ; n ð12 Þ

The Newton–Raphson iteration given by Eq. (9) will convergequickly whenever the initial guess is sufciently close to the correctsolution.

2.2. Prediction of next solution [21]

Having known the converged solution ( x0) and its period( T ), Eq.(3) can be written as

F ð x0 Þ ¼Z T

0 f ð xðt ÞÞdt þ x0 ð13 Þ

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To trace thepossible solution path as thebifurcation parameter var-ies, the latter has to be considered as an additional variable

Dene

F ð x0 ; l Þ ¼Z T

0 f ð xðt Þ; l Þdt þ x0 ð14 Þ

F ( x0 , l ) is a function of x0 and l . The initial state vector x0 which

give rise to steady state solution x(t ) with no transient componentmust satisfy the following equation

b F ð x0 ; l Þ ¼ x0 À F ð x0 ; l Þ ¼ 0 ð15 Þ

The tangent vector is evaluated by solving the augmented set of equations with the continuation parameter. The predicted solutionis given by

ð16 Þ

where ‘‘ ⁄ ’’ stands for the predicted solution, and r is the step size.

2.3. Correction process based on a local parameterisation

By using the predicted values of the variables, Eq. (15) is solvedsimultaneously with Eq. (17) by applying Newton–Raphsonmethod.

xi ¼ g ð17 Þ

Eq. (17) represents the local parameterisation process, which iden-ties each solution along the path being traced. The Jacobian matrixfor the correction process is evaluated by numerical differentiationas in the prediction stage.

2.4. Calculation of the stability of the converged solution

The stability of the converged solution is evaluated by means of Floquet theory, based on monodromy matrix [23] . The nxnmonodromy matrix M of the periodic solution x( x0) with periodT and initial state vector x0 is dened by

M ¼ @ xð x0 ; T Þ=@ x0 ð18 Þ

For a value of the parameter l , the monodromy matrix M(l ) is cal-culated by numerically differentiating the n -vector of solution x( x0 , T ) with respect to the n-vector x0 evaluated at the convergedsolution. The monodromy matrix M(l ) has n-eigenvalues, whichare called Floquet multipliers or characteristic multipliers k1 (l ),

k2(l ), . . . , kn(l ). For autonomoussystem, oneof themultipliers is al-ways equal to unity. These multipliers determine the local stabilityof the solution by the following rules:

x(t ) is stable if | k j|<1 for all j = 1, . . . , n x(t ) is unstable if | k j|>1 for some j. On the stable periodic orbit,

the n-multipliers are always inside the unit circle. The multipliersare function of the parameter under consideration l . When theparameter is allowed to vary, some of the multipliers may crossthe unit circle. The multiplier crossing the unit circle is called thecritical multiplier k(l c ).

Different types of bifurcations occur depending on where a crit-ical multiplier or pair of complex conjugate multipliers cross theunit circle. Three associated types of bifurcations are shown inFig. 1 , where it shows the path of the critical multiplier only, i.e.,the eigenvalue with | k(l c )| = 1. It can be seen from Fig. 1 a, thatthe critical multiplier goes outside the unit circle along the positivereal axis, with k(l c ) = 1. In Fig. 1 b, the multiplier goes outside theunit circle along the negative real axis with k(l c ) = À1. In Fig. 1 c,a pair of complex conjugate multipliers crosses the unit circle witha non-zero imaginary part.

All three sketches refer to a loss of stability when l passesthrough l c . Onthe other hand, changing the arrows point to the re-versedirectionillustrates a gain of stability, i.e., a critical multiplierenters the unit circle. In the case of Fig. 1 a, typically, turning pointof the periodic orbit occurs (cyclic fold bifurcation) with a gain orloss of stability. In the case of Fig. 1 b, system oscillates with period2 (Flip Bifurcation). In the case of Fig. 1 c, the phenomenon of bifur-cation into a torus occurs (Torus Bifurcation), which is also calledsecondary Hopf bifurcation.

These steps lead us to the following algorithm.

(i) Compute the transient response xðt ; xð jÞ0 Þof the network from

t = 0 to t = T ( j) with initial state x0 ¼ xð jÞ0 .

(ii) Choose k such that Eq. (5) is satised, and choose xð jÞ0k

as givenby Eq. (6) .

(iii) Compute the sensitivity matrix

F 0 T ð jÞ; xð jÞ0 , @ F ðT ; x0 Þ

@ x0j x0 ¼ xð jÞ

0

by using thenumerical differentiation formula given by Eq. (11) .Thekth column of this matrix is computed using Eq. (12) :

(iv) Compute y ( j+1) and hence xð jþ 1Þ0 and T ( j+1) from Eq. (9) .

(v) Return to step i) unless xð jþ 1Þ0 À xð jÞ

0 < e and x T ð jÞ; xð jÞ0 À

xð jÞ0 k < d where e and d are user specied small positive

numbers. Steps (i)–(v) gives one steady state periodic solu-tion and its period. To get a continuum of solutions, the fol-lowing steps along with steps (ii)–(v) are repeated.

Fig. 1. Thelocus of the eigenvalue k as the parameter l is varied: (a)Cyclic fold bifurcation CFB; (b)Flip bifurcation FB; (c)Torusbifurcation TB or secondary Hopfbifurcation.

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(vi) Predict the next solution along the tangent direction to thesolution path.

(vii) Correct the predicted solution by using Newton–Raphsonalgorithm.

(viii) Calculate the stabilityof the converged steady state solution.

3. System descriptions and modelling

The description of various power system models used in thisstudy are as follows.

3.1. Power system model

The power system considered for investigation is shown inFig. 2 . It consists of two generator buses and a load bus. The SVCis placed at the load bus. One of the generator bus is treated as aslack bus and the other generator is described by the swingequation:

M _x m ¼ Àdm x m þ P m þ V m VY m sin ðd À dm À H mÞ þ V 2mY m sin H m

ð19

Þwhere M , dm , and P m are generator inertia, damping, and mechanicalpower respectively. The load is modelled by a simplied inductionmotor in parallel with constant P –Q load and constant impedanceas described in [24] . The induction motor model species the realand reactive power demands in terms of load voltage andfrequency.

The load model is described by

P ¼ P 0 þ P 1 þ K pw_d þ K pv ðV þ T _V Þ ð20 Þ

Q ¼ Q 0 þ Q 1 þ K qw_d þ K qv V þ K qv 2 V 2 ð21 Þ

where P 0 , Q 0 are the constant real and reactive powers of the induc-tion motor and P 1 , Q 1 are the constant P –Q load and K pw , K pv, K qw , K qv

and K qv2 are constants associated with the dynamic load. The realand reactive powers supplied to the load by the network are

P ¼ ÀV 00 VY 00 sin d þ H 00À ÁÀ V mVY m sin ðd À dm þ H m Þ

þ Y 00 sin H 00 þ Y m sin H mÀ ÁV 2 ð22 Þ

Q ¼ V 00 VY 00 cos d þ H 00À Áþ V mVY m cos ðd À dm þ H mÞ

À Y 00 cos H 00 þ Y m cos H mÀ ÁV 2 ð23 Þ

The load bus includes a capacitor as part of its constant impedancerepresentation in order to maintainthe voltage magnitude at a nom-inal and reasonable value. Instead of including the capacitor in the

circuit, it is convenient to account for the capacitor by adjusting V 0and Y 0 to give Thevenin equivalent of the circuit with the capacitor.Primes are used to indicate Thevenin equivalent parameters [24] .

V 00 ¼V 0

1 þ C 2 Y À20 À 2 CY À1

0 cos H 0 1 =2 ð24 Þ

Y 00 ¼ Y 0 1 þ C 2 Y À20 À 2 CY À1

0 cos H 0 1 =2

ð25 Þ

H 00 ¼ H 0 þ tan À1 CY À1

0 sin H 0

1 À CY À10 cos H 0( ) ð26 Þ

3.2. Static VAR compensator

An SVC is a shunt connected static VAR generator/absorberwhose output is adjusted to exchange capacitive or inductive cur-rent so as to maintain or control specic variables, typically SVCbus voltage. The SVC can be assumed in its basic form as a contin-uous, variable susceptance, which is adjusted in order to achieve aspecied nodal voltage magnitude. In this paper, a simplied rstorder model of SVC has been considered to represent the dynamicsof its control action utilised for damping the dynamic bifurcations.This is shown in Fig. 3 .

From Fig. 3 , the equations describing the dynamics of SVC arewritten as:

_

B ¼K SVC

T SVC ðV ref À V Þ ÀB

T SVC ð27 Þ

Bmin B Bmax ð28 Þ

where B is the susceptance, K SVC is the SVC gain, T SVC is the timeconstant and V ref is the reference voltage. In order to model the

Fig. 2. A simple power system. dm = generator angle, x = generator freq, d = load angle, v = load voltage magnitude.

Fig. 3. Block diagram representation of static VAR compensator.

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SVC hard limiters, a tanh function is employed, such that for a sus-ceptance B with limits ±B LIMIT, the output of the limiter is given by,

BSVC ¼ BLIMIT tanhB

BLIMIT ð29 Þ

The net effect of the variable susceptance introduced to the systemby the SVC is modelled as an additional reactive power source,

Q SVC = BÁV 2

.

3.3. Power system model with SVC

The equations describing the dynamics of the power systemmodel with SVC are:

_dm ¼ x m ð30 Þ

M _x m ¼ ÀDx m þ P m þ V mVY m sin ðd À dm À H m Þ

þ V 2m Y msinH m ð31 Þ

K qw_d ¼ ÀK qv V À K qv 2 V 2 þ Q þ Q SVC À Q 0 À Q 1 ð32 Þ

TK qw K pv _V ¼ K pwK qv 2 V 2 þ ðK pwK qv À K qw K pv ÞV þ K pwðQ 0

þ Q 1 À Q À Q SVC Þ ÀK qw ðP 0 þ P 1 À P Þ ð33 Þ

Eqs. (30)–(33) along with (27)–(29) describe the dynamics of thesystem shown in Fig. 2 . The Network, generator and load parame-ters except that of SVC are taken from [24] .

4. Simulation results and discussion

The algorithm is tested by conducting bifurcation analysis on athree-node power system equipped with SVC, Fig. 2 . Simulationshave been carried out to analyse the sensitivity of the system withrespect to load reactive power and capacitor. The bifurcation dia-grams have been constructed using the proposed algorithm. Toolsof nonlinear dynamics, viz, time response plots and phase plots areused to unravel the chaotic behaviour and voltage collapse points

of the system. The reactive power demand Q 1 in per unit of theconstant P –Q load is taken as bifurcation parameter in all the casesreported in this paper.

4.1. 3-Bus system without SVC

4.1.1. Bifurcation analysisTypical values of various system parameters considered for sim-

ulation are given in Appendix A . Fig. 4 shows the bifurcation dia-gram of the periodic solution branch superimposed on thestationary branch. Three critical points, viz, S1, S2 and S3 aredetected.

At points S1 and S2 ( Fig. 4 ), a pair of complex conjugate eigen-values cross the imaginary axis. At these two points the systembehaviour is dominated by generator angle and angular velocity.At point S3, one real eigenvalue becomes zero, which is the SNBpoint. Movement of a complex conjugate eigenvalue at and aroundthe bifurcation points (S1 and S2) is shown in Fig. 5 .

Six different bifurcation points are identied from Fig. 4 and aresummarised in Table 1 .

At Q 1 = 10.9446, i.e., at point S1, there is a UHB with the emer-gence of an unstable limit cycle. At this value of Q 1 , the complexeigenvalues are ±j 3.7456 and unstable oscillation has frequency f = 0.59756 Hz and period T = 1.67346 s. At Q 1 = 11.4066, i.e., atpoint S2, there is a SHB. The point S3, at Q 1 = 11.41136, is theSNB point where one real eigenvalue becomes zero. This reactive

power demand corresponds to the system steady state operatinglimit, and the system has no operative solution for Q 1 > 11.41136.

4.1.2. Period doubling routes to chaosThe periodic solution that emanates from UHB undergoes a per-

iod doubling bifurcation at Q 1 = 10.87327 (PDB1 in Fig. 4 ). The per-iod of the period 2 solution is 3.4996 s, roughly twice thefundamental period. At Q 1 = 10.88094, period 2 solution bifurcatesto period 4 solution. Further period doubling bifurcations occur asQ 1 is increased and result in chaos at Q 1 = 10.89. The chaos disap-pears at Q 1 = 10.894. When Q 1 is decreased from 11.389 (PDB2),the second period doubling route to chaos occur. This is demon-strated using the time domain and phase plots as shown in Fig. 6 .

4.1.3. Voltage collapseTime domain plots are used to unearth the voltage collapse

point from the bifurcation diagram( Fig. 4 ). For values of Q 1 rangingbetween 10.894 (at which chaos disappears) and 10.9446 (atwhich UHB appears), the voltage at the load bus exhibits oscilla-tions and leads to a sudden voltage drop – a voltage collapse , Fig. 7 .

4.2. 3-Bus system with SVC

The SVC’s impact on Hopf bifurcations and voltage collapse of the simple power system shown in Fig. 2 . is unraveled in this sec-tion. The SVC is placed at the load bus. The SVC parameters are gi-ven in Appendix A . The error signal is computed as the differencebetween reference value V ref and measured value V . The control-

ler’s output is SVC susceptance, which is allowed to vary betweenBmin and Bmax by using a wind-up limiter.

PDB1

ssb SS1

UHB

U

usbusb

ssb

PDB2

S2SHB

S3SNB

Q1

v CFB

Fig. 4. Bifurcation diagram of load voltage without SVC for C = 12: S – stableperiodic branch, U – unstable periodic branch, ssb – stable stationary branch, usb –unstable stationary branch.

S1

S2Real

I m a g i n a r y

Fig. 5. Movement of a complex conjugate eigenvalue as Q 1 varies.

Table 1

Summary of bifurcation points in Fig. 4.

Type of bifurcation Q 1 Description

CFB 10.818 Cyclic fold bifurcationPDB1 10.87327 Period doubling bifurcation 1UHB 10.9446 Unstable Hopf bifurcationPDB2 11.389 Period doubling bifurcation 2SHB 11.4066 Stable or supercritical Hopf bifurcation

SNB 11.41136 Saddle node bifurcation

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4.2.1. A. Bifurcation analysisFig. 8 shows the bifurcation diagram of the periodic solution

branch superimposed on the stationary branch in the presence of SVC. Three critical points R1, R2, and R3 are observed. At Q 1 = 11.32109, i.e., at point R1, there is a UHB with the emergenceof an unstable limit cycle.

-0.25

0

0.25

(i)

(ii)

(iii)

(iv)

(a)

(a)

(b)

(a) (b)

(b)

(a) (b)

Time (s)

ω

Time (s)

ω

Time (s)

ω

Time (s)

ω

δ m

ω

δ m

δ m

ω

ω

ω

δ m

150 170 190

Fig. 6. Time domain and phase plots of PDB2: i. (a) Time plot of Period 1 solution at Q 1 = 11.389, i. (b) Phase plot of Fig. 6. i. (a). ii. (a). Time plot of Period 2 solution atQ 1 = 11.386, ii. (b). Phase plotof ii. (a). iii. (a).Timeplot of Period4 solutionat Q 1 = 11.383, iii.(b). Phase plot of iii. (a). iv.(a). Time plot of chaotic solution at Q 1 = 11.379, iv.(b).Phase plot of Fig. 6. iv. (a).

Fig. 7. Voltage collapse at Q 1 = 10.94.

R1(UHB )

(SNB)

ssb

usb

usb

R2(SHB)

PDB

U

Q 1

v

R3

Fig. 8. Bifurcation diagram of load voltage with SVC for C = 12.

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It can be observed from Figs. 4 and 8 that the UHB occurs at ahigher value of Q 1 in the presence of SVC. The loci of complex con- jugate eigenvalues shown in Fig. 9 conrm the occurrence of SHBat Q 1 = 11.5579. The frequency and period of stable oscillation atSHB point are 0.6035 Hz and 1.6567 s respectively. It is also to benoted that SHB and SNB occur at lower part of Q 1–V curve.

4.2.2. Period doubling route to chaosThe period doubling route to chaos in the presence of SVC (point

PDB in Fig. 8 ) is demonstrated using three-dimensional plots,Fig. 10 .

Extensive analysis have been performed for various values of SVC gains. The results reveal that the Hopf bifurcation is eliminated

Real I m a g

i n a r y

Fig. 9. Movement of complex conjugate eigenvalues as Q 1 varies.

Fig. 10. Cascade of perioddoubling with SVC: (a)P1 oscillation at Q 1 = 11.524958. (b)P2 oscillation at Q 1 = 11.523342. (c)P4 oscillation at Q 1 = 11.522001.(d) P8 oscillation atQ 1 = 11.520969. (e) Chaos at Q 1 = 11.517934.

Real

I m a g

i n a r y

Fig. 11. Movement of complex conjugate eigenvalues as Q 1 varies.

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for K svc = 25. For the entire range of bifurcation parameter(Q 1 = 10.0 to 11.6) considered, for K svc = 25, the loci of complex pairof eigenvalues are shown in Fig. 11 . The Figure conrms the elim-ination of Hopf bifurcation since Floquet multipliers are not cross-ing the imaginary axis.

4.2.3. Voltage collapseFig. 12 shows that the voltage collapse point occur at

Q 1 = 11.3521. Comparison of Figs. 7 and 12 reveals that the voltagecollapse point has been put off in the Q 1–V curve in the presence of SVC. It can also be observed from gures that the amplitude of oscillations preceding the voltage collapse is less with SVC.

4.3. Effect of capacitance

The effect of capacitance on the dynamic behaviour of the testsystem is brought out in this section. Simulations as in Sections4.1 and 4.2 have also been carried out for C = 3. The bifurcation dia-grams of load voltage are shown in Fig. 13 . In the absence of SVC,Figs. 4 and 13 a, the system exhibits almost similar sequence of solutions and bifurcations for both capacitor values. In the pres-ence of SVC, Figs. 8 and 13 b, it canbe observed that for lower valueof capacitance, there are no period doublings and hence chaos.Lower value of capacitance shifts the occurrence of bifurcationsto a lesser value of Q 1 irrespective of the presence of SVC. Table 2summarizes the various Q 1 values at which bifurcations occur inFig. 13 .

5. Conclusion

In this paper, a numerical algorithm, based on initial valueproblem, using local parameterisation continuation technique isproposed for tracing stable and unstable steady state periodic solu-tion branches. The proposed numerical algorithm is validated byconducting bifurcation analysis on a standard power system re-ported in literature. The results show that the algorithm is effective

as a tool for construction of bifurcation diagrams of steady statesolutions of autonomous systems.

Existence of Hopf bifurcations points to the possibility of perioddoublings, chaos and voltage collapse and hence reduces the dy-namic stability of the nominal operating point. For representativevalues of SVC gain, delaying and nally elimination of Hopf bifur-cation is demonstrated. Elimination of Hopf bifurcation is con-rmed through eigenvalue plot. Simulation results also show thatthe voltage collapse point and SNB have been put off in the Q 1–Vcurve by the SVC. Voltage stability margin of the system is there-fore improved signicantly with the inclusion of SVC.

For lower value of capacitance, the system exhibits almost sim-ilar sequence of solutions except that the various bifurcationpointsoccur at a lesser value of Q 1. In the presence of SVC, it is observedthat there are no period doublings and hence chaos for lower valueof capacitance.

Appendix A

The parameters considered for simulation, except those for SVCare taken from [24] and are given below for easy reference. All theparameter values are in p.u. except angles which are in electricaldegrees.

Network parameters:

Y 0 ¼ 20 :0 H 0 ¼ À5 :0 ; V 0 ¼ 1 :0 ; Y m ¼ 5 :0 ; H m ¼ À0 :08726 :

Generator parameters:

V m ¼ 1 :0 ; P m ¼ 1 :0 ; M ¼ 0 :3 ; D ¼ 0 :05

Load parameters:

K pw ¼ 0 :4 ; K pv ¼ 0 :3 ; K qw ¼ À0 :03 ; K qv ¼ À2 :8 ; K qv 2 ¼ 2 :1 ;T ¼ 8 :5 ; P 0 ¼ 0 :6 ; Q 0 ¼ 1 :3 ; P 1 ¼ 0

SVC parameters:

K SVC ¼ 2 :0 ; T SVC ¼ 0 :01 ; BLIMIT ¼ 1 :0

Time (s)

v

Fig. 12. Voltage collapse at Q 1 = 11.3521.

Q1

UHBSNB

U

PDB2

PDB1

SHBssb

CFBv

usb

(a)

ssbUHB

U

SHBSNB

v

Q 1

(b)

Fig. 13. Bifurcation diagram of load voltage for C = 3: (a) without SVC (b) with SVC.

Table 2

Summary of bifurcation points in Fig. 13 .

Type of bifurcation Without SVC With SVCQ 1 Q 1

CFB 6.18181 –PDB1 6.21801 –UHB 6.23981 6.41338PDB2 6.53861 –

SHB 6.543068 6.702317 a

SNB 6.543116 6.701049 a

a Points that at SHB and SNB, Q 1 values corresponds to lower part of Q 1–V curve.

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