Voltage tracking design for electric pow

In construct to the state feedback design at the bifurcation point via the tuning of the Static Var Compensator as proposed in Saad, et al, 2005, a sliding mode control SMC scheme using

Voltage Tracking Design for Electric Power

Systems via SMC Approach

Der-Cherng Liaw, Shih-Tse Chang and Yun-Hua Huang

Abstract— This paper presents a output tracking design for

regulating the load voltage of the electric power system Based

on a model of electric power system proposed by Dobson

and Chiang (1988), the saddle-node bifurcation and Hopf

bifurcations were observed (Wang, et al, 1994) by treating the

reactive power as system parameter Those bifurcations were

found to lead to the appearance of the dynamic or the static

voltage collapses of the power systems In construct to the

state feedback design at the bifurcation point via the tuning

of the Static Var Compensator as proposed in (Saad, et al,

2005), a sliding mode control (SMC) scheme using SVC was

employed in this study to construct a load voltage tracking

design control law for regulating the load voltage and hence

providing the stability of electric power systems The numerical

simulations demonstrated that the proposed control scheme not

only could provide the regulation of the load voltage but also

prevent and/or delay the appearance of bifurcation phenomena

and chaotic behavior.

I INTRODUCTION

In the recent years, the study of voltage collapse

phe-nomena in the electric power systems has attracted lots

of attention (e.g., [1]-[8]) It is due to the fact of facing

the growing load demands in power systems but with little

addition of the power generation and transmission facilities

That leads the power systems to be operated near the stability

limits As the load demands become too heavy to be offered,

the magnitude of load voltage falls sharply to a very low

level, which is referred as the so-called “voltage collapse.”

A practical power system is a large electric network

con-taining components such as generators, loads, transmission

lines and voltage controllers In [2] and [3], Dobson and

Chiang introduced a simple dynamical model for electric

power systems, which consist of a generator, a nonlinear

load and an infinite bus Based on that model, several results

have been published regarding the nonlinear phenomena of

electric power systems (e.g., [2]-[4], [6], [7], [9]) In addition,

the occurrence of voltage collapse had been believed to

be attributed to the existence of saddle-node bifurcation

of electric power systems [4]-[5] However, it has been

shown that voltage collapse may arise from the existence

of Hopf bifurcation, which is prior to the appearance of

This work was supported in part by the Nation Science Coucil, Taiwan,

R.O.C under Grants NSC95-2221-E-009-339

Der-Cherng Liaw is with the Department of Electrical and Control

Engineering, National Chiao Tung University, Hsinchu, 300, Taiwan, R.O.C.

dcliaw@cc.nctu.edu.tw

Shih-Tse Chang is with the Department of Electrical and Control

En-gineering, National Chiao Tung University, Hsinchu, 300, Taiwan, R.O.C.

stchang.ece94g@nctu.edu.tw

Yun-Hua Huang is with the Department of Electrical and Control

Engi-neering, National Chiao Tung University, Hsinchu, 300, Taiwan, R.O.C.

saddle-node bifurcation [6]-[8] It is known that the voltage regulation issue is usually solved by the setting of either the tap changing ratio (e.g., [9]-[11]) or the extra adding capacitive load in practical electric power systems However, both schemes are only available for discrete tuning The Static Var Compensator (SVC) has recently been considered

as a control actuator for improving system stability (e.g., [12]-[15]) For instance, direct state feedback linearization was proposed to simplify the design of controller as in [14] The study of the impact on the voltage collapse with state feedback control via the tuning of SVC was also given

in [12]-[14] A washout filter-aided feedback design was proposed in [15] to delay the occurrence of the system instability and/or voltage collapse

It is known that the sliding mode control (SMC) scheme possesses the advantages of fast response and less sensitivity

to system uncertainties and/or disturbances than those by other methods To compensate system uncertainties and/or disturbances, several types of SMC schemes have been proposed (e.g., [16]-[18]) For instance, second-order sliding mode control scheme [16] and reliable control via sliding mode approach [18] have been proposed to enhance the performance of sliding mode control designs Due to those advantages, the sliding mode control scheme has been widely employed to design the control laws for a variety of ap-plications (e.g., [19]-[23]) Instead of directly controlling the system behavior at the bifurcation point as proposed

in [15], in this paper we consider a different approach by the load voltage regulation design of the electric power systems via the tuning of the SVC Such a design will be achieved by using sliding mode control scheme, which might also eliminate and/or delay the occurrence of bifurcation phenomena and system instabilities

The organization of this paper is as follows A output tracking control scheme is proposed in Section II It is followed by the application to the load voltage control design in electric power systems in Section III Numerical simulations are given in Section IV to demonstrate the effectiveness of the proposed design Finally, Section V gives the conclusions

II OUTPUTTRACKINGDESIGN

Consider a class of nonlinear systems as given by

˙η = f2(x, η) + d(x, η) + g2· u, (2) where x ∈ Rn, η ∈ R, and the system input u ∈ R

In addition, d denote the system uncertainty The output

function of system is given as y = h(x) ∈ R In the

following, we consider to construct a control law for the

inputu such that the system output will approach a desired

trajectoryyd(t) That is, y → yd as time t increases to ∞

Lete = y − yd We then have

˙e = ˙y − ˙yd,

= ∇h(x) · {f1(x) + g1(x)η} − ˙yd, (3)

where∇h(x) denotes the gradient of h(x)

To follow the design procedure for sliding mode control

(see e.g., [17]), we first treat the state variableη as a virtual

input and find a control lawη = η(x, yd) such that the

sub-dynamics of system as given in (1) will make the output

error functione of (3) to be asymptotically stable at e = 0

Choose V1(e) = 1

2e2 as a Lyapunov function candidate for the error dynamics as in Eq (3) We then have

˙

V1(e) = e · ˙e,

= e · {∇h(x) · [f1(x) + g1(x)η] − ˙yd},

withη = φ(x, yd), where

φ(x, yd) = [∇h(x)g1(x)]−1{−∇h(x) · f1(x)

+ ˙yd− ke}, for k > 0 (5)

By applying the Lyapunov stability criteria, we then have

the next stability result

Lemma 1: The origin e = 0 of the error dynamics (3)

will be asymptotically stabilizable by the virtual input η if

∇h(x) · g1(x) 6= 0 for all x ∈ D ⊂ Rn, where the subsetD

contains the neighborhood ofe = 0

Next, we construct control laws for the system input u

such that the state variableη will approach the desired input

φ(x, yd) as defined in (5) with the appearance of system

uncertainties

Choose the sliding surfaces = η − φ(x, yd) We then have

˙s = ˙η − ˙φ(x, yd),

= f2(x, η) + d(x, η) + g2· u − ˙φ(x, yd),

= f2(x, η) − f2(x, φ(x, yd)) + d(x, η) + g2· ure

− ˙φ(x, yd) + ˙φ(x, yd)|η=φ(x,y d ), (6)

where we choose u = ueq + ure with the assumption of

g26= 0 and

ueq = 1

g2

· {−f2(x, φ(x, yd)) + ˙φ(x, yd)|η=φ(x,yd)} (7) For the case of d(x, η) = 0 and ure = 0, it is clear from

the condition of ˙s = 0 that all the states lying on the sliding

surface s = 0 will always be kept staying on the manifold

s = 0 by the choice of ueq as in (7) This provides that

the sliding surface s = 0 to be an invariant manifold when

d(x, η) = 0 and ure = 0 Moreover, from Lemma 1, the

error dynamics as in (3) will be asymptotically stable ate =

0 as long as the system state lying on the sliding surface

s = 0 (i.e., η = φ(x, yd)) with the control input u = u Following the design of sliding mode control, we next focus

on the dynamical behavior on the boundary layer That is,

to construct the extra control input ure to drive the system state on the manifold of s 6= 0 to enter the sliding surface

s = 0

Suppose the system uncertaintyd(x, η) satisfy the follow-ing inequality:

where || · || denotes the norm function and ρ(x, η) is a nonnegative continuous function In order to compensate the uncertainties, we choose the extra controlure as

ure = 1

g2

{−β(x, η) · sgn(s) − f2(x, η) +f2(x, φ(x, yd)) + ˙φ(x, yd)

− ˙φ(x, yd)|η=φ(x,y d )}, (9) whereβ(x, η) ≥ ρ(x, η) + γ with γ > 0 and sgn(·) denotes the sign function Here, we also assume thatg26= 0 TakingV2(s) = 1

2s2as a Lyapunov function candidate for (6) with the control input u = ueq+ ure as defined in (7) and (9), we then have

˙

V2(s) = s · {−β(x, η) · sgn(s) + d(x, η)}

where | · | denotes the absolute value From the above discussions, we can conclude that 12d|s|dt2 ≤ −1

2γ · |s| This gives that|s(t)| ≤ |s(0)| − γt, ∀ t ≥ 0, which implies that the nonzero value of s will reach the sliding mode s = 0

in a finite time We then have the next result for the output tracking design

Theorem 1: Supposeg26= 0 and ∇h(x)·g1(x) 6= 0 for all

x ∈ D ⊂ Rn, where the subsetD contains the neighborhood

ofe = 0 Then the output function y = h(x) for system (1)-(2) will approach the desired outputyd(t) via sliding mode control Moreover, one of the choices for the control input

u = ueq+ ure whereueq and ure are as given in (7) and (9), respectively

Note that, it is known that the sign function used in the sliding mode control design might cause chattering problem

In practical application, the sign function used in (9) can be replaced by a saturation function for relaxing the chattering issue An example is given in Section IV for numerical demonstrations of the proposed design

III APPLICATION TOELECTRICPOWERSYSTEMS

In this section, we will apply the design as proposed in Section II to the load voltage tracking design of electric power systems First, we recall the mathematical model proposed by Dobson and Chiang ([2], [3]) for electric power systems Typical dynamical behavior of the power system will also be discussed It is followed by the load voltage tracking design for the power system via the tuning of the SVC controller

A Dynamics of Electric Power Systems

In the following, we recall the mathematical model

pro-posed by Dobson and Chiang ([2], [3]) for electric power

systems as given by

M ˙ωm = Pm− dmωm+ Em2Ymsin θm

+EmYmV sin(δ − δm− θm), (12)

kqω˙δ = −kqv 2V2− kqvV + Q(δm, δ, V )

T kqωkpvV˙ = kpωkqv2V2+ (kpωkqv− kqωkpv)V

+kqω(P (δm, δ, V ) − P0− P1)

−kpω(Q(δm, δ, V ) − Q0− Q1), (14) where δm, ωm, δ and V denote the generator phase angle,

generator angular speed, load voltage phase angle and load

voltage, respectively Here, the nonlinear PQ load are given

as

P (δm, δ, V ) = (Y0′sin θ′0+ Ymsin θm)V2

−EmYmV sin(δ − δm+ θm)

−E′0Y0′V sin(δ + θ0′), (15) Q(δm, δ, V ) = −(Y0′cos θ0′ + Ymcos θm)V2

+EmYmV cos(δ − δm+ θm) +E′0Y0′V cos(δ + θ′0), (16) with

E′

0 = E0[1 + C2Y0−2− 2CY0−1cos θ0]−1/2, (17)

Y0′ = Y0[1 + C2Y0−2− 2CY0−1cos θ0]1/2, (18)

θ′

0 = θ0+ tan−1{ CY

−1

0 sin θ0

1 − CY0−1cos θ0

Detailed definitions of each system parameter and derivations

of the model equations above can be referred to (e.g., [2],

[3])

The typical dynamical behaviors of system (11)-(14) with

respect to the variation of the extra demanded reactive load

Q1 with P1 = 0 are obtained by using the codes AUTO

[24] and Matlab as depicted in Figures 2-4 It is shown in

Figures 2-4 that the electric power systems might exhibit

Hopf bifurcation and saddle-node bifurcation, and even the

chaotic behavior as the value of Q1 varies Those results

agree with the previous findings (e.g., [2]-[5], [10]-[11]) and

might cause the undesired dynamical behavior and/or the

drastic change of the load voltage In the next subsection,

we will seek a control law for possibly eliminating the

appearance of Hopf bifurcation and providing the regulation

of the load voltage

B Load Voltage Tracking Design

In the recent years, the SVC’s have been considered as a

control scheme for voltage regulation in the electric power

systems The configuration of the SVC’s is considered as

a fixed capacitor connected in parallel with a thyristor

con-trolled reactor (e.g., [15]), which might provide an additional

reactive power when it is connected in parallel with the PQ load That is, the overall effective reactive load Q1 will become the summation of the original demanded reactive loadQoand the added reactive loadQadded

1 from the SVC’s The mathemetical model for the SVC’s was proposed as (e.g., [15], [22]):

˙

TSV C

(KSV C· u − B), (20)

withBmin ≤ B ≤ Bmax Here,B denotes the susceptance

of the SVC,KSV C is the gain for the SVC, TSV C denotes the time constant andu denotes the control input In addition, the added reactive load by the SVC’s is given as (see, e.g., [15])

Qadded1 = BV2 (21)

Let x1 = δm, x2 = ωm, x3 = δ, x4 = V and x5 = B Then we can rewrite system (11)-(14) with SVC control as follows:

˙x2 = 1

M{Pm− dmx2+ E

2

mYmsin θm

+EmYmx4sin(x3− x1− θm)}, (23)

˙x3 = 1

kqω

{−kqv2x2− kqvx4+ Q(x1, x3, x4)

T kqωkpv

{kpωkqv 2x24+ (kpωkqv− kqωkpv)x4

+kqω(P (x1, x3, x4) − P0− P1)

−kpω(Q(x1, x3, x4) − Q0− Q1− x5x24)} (25)

TSV C

where the extra demanded reactive load has become Q1+

x5x2 Here,Q1denotes the original demanded reactive load while x5x2 is for the added reactive load created by the SVC’s

A washout filter type controller has been proposed in [15]

to delay the appearance of bifurcation phenomena via the tuning of the SVC’s Instead of directly controlling the sys-tem stability at the bifurcated operating point, in this paper,

we propose a different approach for the load voltage tracking design by using sliding mode control scheme Details are given as follows

Let the output of system (22)-(26) be the load voltagex4 DenoteVd(t) the desired load voltage for system (22)-(26) Comparing the mathematical model of the electric power systems as given in (22)-(26) with the form presented in (1)-(2), we havex = (x , x , x , x )T,η = x ,h(x) = x

f1(x) =







f11(x)

f12(x)

f13(x)

f14(x)





 ,

g1(x) =







0 0

−1

k qω · x2 4

k pω

T k qω k pv · x2





,

f2(x, η) = −1

TSV C

g2 = KSV C

TSV C

(28) withf11(x) = x2 and

f12(x) = 1

M{Pm− dmx2+ E

2

mYmsin θm

+EmYmx4sin(x3− x1− θm)}, (29)

f13(x) = 1

kqω{−kqv 2x24− kqvx4+ Q(x1, x3, x4)

T kqωkpv

{kpωkqv 2x24 +(kpωkqv− kqωkpv)x4

+kqω(P (x1, x3, x4) − P0− P1)

−kpω(Q(x1, x3, x4) − Q0− Q1)} (31)

To apply Theorem 1 to load voltage tracking design for

system (22)-(26), we have the two conditions: g2 = KSV C

T SV C

and∇h(x) · g1(x) = (kpω/T kqωkpv) · x2

The next result for the load voltage tracking of system

(22)-(26) follows readily from Theorem 1

Theorem 2: The load voltageV of system (22)-(26) will

approach any desired voltageVd(t) via sliding mode control

if KSV C

T SV C 6= 0 and (kpω/T kqωkpv) · x2 6= 0 Moreover, one

of the choices for the control input u = ueq+ ure where

ueq and ure are as given in (7) and (9), respectively, with

yd= Vd(t),

Remark 1: Note that in Theorem 2 above, the parameters

kpω, T , TSV C, kqω, and kpv, are generally nonzero In

addition, in practical electrical power system the value of the

load voltagex4is non-negative The two conditions given in

Theorem 2 in practical application will then be reduced as

KSV C6= 0, while the attraction region for the exhibition of

sliding motion will bex4> 0

IV NUMERICALSIMULATIONS

An example model is considered in this section to

demon-strate the effectiveness of the proposed control design as

given in Section III Here, we adopt the value of system

parameter for system (11)-(14) from [15] as listed in Table I

In addition, we chooseP1= 0 and treat the extra demanded

reactive powerQ as system bifurcation parameter

A Open-Loop Dynamics

First, we present the numerical results for the uncontrolled system (11)-(14) Bifurcation diagram for the example model

is obtained by using code AUTO [24] as depicted in Fig

1 The figure shows that the system will exhibit both Hopf bifurcation and saddle-node bifurcation, respectively, as de-noted by “HB” and “SNB.” Here, the solid-line denotes the stable system equilibria while the dashed-dot-line is for the unstable ones The corresponding data for the two bifurcations is given in Table II In fact, we do have observed several nonlinear dynamical behavior by using code Matlab

as depicted in Fig 2 and 3 In those figures, we observe the period-1 oscillation as depicted in Fig 2(a) and 3(a), respectively, for the phase portraits and timing responses In addition, chaotic-like behaviors are also observed in Figs 2(b)-2(c) and Figs 3(b)-3(c) for the corresponding phase diagrams and time responses A drastic voltage change is found in Figs 2(d) and 3(d), which is very close to the saddle-node bifurcation point

TABLE I

D ATA FOR SYSTEM PARAMETERS

K pω = 0.4 p.u K pv = 0.3 p.u K qω = −0.03 p.u.

K qv = −2.8 p.u K qv2 = 2.1 p.u T = 8.5 p.u.

P 0 = 0.6 p.u Q 0 = 1.3 p.u M = 0.01464

Y 0 = 3.33 p.u θ 0 = 0 deg E 0 = 1.0 p.u.

C = 3.5 p.u Y m = 5.0 p.u θ m = 0 deg.

E m = 1.05 p.u P m = 1.0 p.u d m = 0.05 p.u.

TABLE II

L OCATION FOR BIFURCATION POINTS

HB 2.98021 0.267426 0.0475028 0.873124 SNB 3.02578 0.30292 0.0610163 0.795136

B Closed-Loop Dynamics

In the following, we consider to apply Theorem 2 to the electrical power systems with the sliding mode control effort via the tuning of the SVC Here, we adopt the values from [15] as TSV C = 0.01 second and KSV C = 1 In addition, for the control design, we choose k = 1 and β(x, η) = 5

To relax the undesired chattering behavior, here, we use a saturation function to replace the sign function for the sliding mode control in the numerical simulations as defined by (see, e.g., [17])

sat(z) =







−1, for z < −1

z, for |z| ≤ 1

1, for z > 1

Two cases are considered below to demonstrate the ef-fectiveness of the proposed design First, we consider the case of the desired load voltage beingVd(t) = 1 − 0.05e−t

switched at time t = 30 second with the control gain β(x, η) = 5 As shown in Fig 3, we do find that the system instabilities prior to the control action are diminished

and the load voltage approaches the desired load voltage

Vd = 1.0 after t ≥ 30 second Note that, as shown in Fig

3(d), the voltage collapse appears at around t = 19 second

for Q1 = 2.9899 In order to effectively recover from the

instability, we set the control law to switch att = 18 second

for the case ofQ1= 2.9899 For the other case, we select the

same desired load voltage asVd(t) = 1 − 0.05e−t with the

system uncertainty d(x, η) = 0.01 In order to compensate

the system uncertainty, the control gainβ(x, η) is increased

to 10 instead of 5 The corresponding time responses of the

load voltage phase anglex3and load voltagex4are obtained

as depicted in Figs 4 and 5, respectively As depicted in the

numerical simulations above, the proposed control effort not

only successfully provide the tracking of the load voltage

but also eliminate the appearance of the undesired system

behavior such as bifurcation type oscillation or the chaotic

behavior

In this paper, we focus on the load voltage tracking

design for a power system by using sliding mode control

scheme It is achieved via the tuning of the SVC’s The

simulations demonstrate the effectiveness of the proposed

design for the power systems with or without the appearance

of system uncertainty Comparing with other existing control

schemes by using SVC’s (e.g., [15]), no exact knowledge of

the operating condition is required in the proposed design

This might relax the computation efforts for solving the

operating point for the controller design The problem of the

implementation issue of real time control is not considered

in this study, however, it can be a research topic for the

further study In this paper, we only consider the case of

whichP1 = 0 and Q1 is varied The same technique might

be applicable to the case ofP16= 0 for the further study

The authors are very grateful to reviewers’ comments

[1] S Abe, Y Fukunaga, A Isono and B Kondo, “Power system voltage

stability,” IEEE Transactions on Power Apparatus and Systems, vol.

PAS-101, no 10, pp 3830-3840, 1982.

[2] I Dobson, H.-D Chiang, J S Throp and L Fekih-Ahmed, “A model

of voltage collapse in electric power systems,” Proc 27th IEEE

Conference on Decision and Control, Austin, Texas, pp 2104-2109,

Dec 1988.

[3] H.-D Chiang, I Dobson, R J Thomas, J S Throp and L

Fekih-Ahmed, “On voltage collapse in electric power systems,” IEEE

Trans-actions on Power Systems, vol 5, no 2, pp 601-611, 1990.

[4] I Dobson and H.-D Chiang, “Towards a theory of voltage collapse

in electric power systems,” Systems & Control Letters, vol 13, pp.

253-262, 1989.

[5] H G Kwatny, A K Pasrija, and L Y Bahar, “Static bifurcation

in electric power networks : loss of stability and voltage collapse,”

IEEE Transactions on Circuits and Systems, vol CAS-33, pp

981-991, 1986.

[6] H O Wang, E H Abed, R A Adomaitis and A M A Hamdan,

“Control of nonlinear phenomena at the inception of voltage collapse,”

American Control Conference, San Francisco, California, pp

2071-2075, June 1993.

[7] H O Wang, E H Abed, and A M A Hamdan, “Bifurcations,

chaos, and crises in voltage collapse of a model power system,” IEEE

Transactions on Circuits and Systems–I: Fundamental Theory and

Applications, vol 41, no 3, pp 294-302, 1994.

[8] M Yue and R Schlueter, “Bifurcation Subsystem and Its Application

in Power System Analysis,” IEEE Transactions on Power Systems, vol.

19, no 4, pp 1885-1893, 2004.

[9] D.-C Liaw, K.-H Fang and C.-C Song, ”Bifurcation Analysis of

Power Systems with Tap Changer,” Proc 2005 IEEE International

Conference on Networking, Sensing and Control (ICNSC’05), Tucson,

Arizona, U.S.A., March 19-22, 2005.

[10] C.-C Liu and K T Vu, “Analysis of tap-changer dynamics and

con-struction of voltage stability regions,” IEEE Transactions on Circuits

and Systems, vol 36, no 4, pp 575-589, 1989.

[11] H Ohtsuki, A Yokoyama and Y Sekine, “Reverse action of on-load

tap changer in association with voltage collapse,” IEEE Transactions

on Power Systems, Vol 6, No 1, pp 300-306, 1991.

[12] R G Kaasusky, C R Fuerte-Esquivel, and D Torres-Lucio, “Assess-ment of the svc’s effect on nonlinear instabilities and voltage collapse

in electric power systems,” IEEE Power Eng Society Winter Meeting,

vol 4, pp 2659-2666, 2003.

[13] K N Srivastava and S C Srivastava, “Elimination of dynamic

bifurcation and chaos in power systems using FACTS devices,” IEEE

Trans on Circuit and Systems-I, vol 45, no 1, pp 72-78, 1998.

[14] Y Wang, H Chen, R Zhou, and D J Hill, “Studies of voltage stability

via a nonlinear svc control,” IEEE Power Eng Society Winter Meeting,

vol 2, pp 1348-1353, 2000.

[15] M S Saad, M A Hassouneh, E H Abed, and A Edris, “Delaying instability and voltage collapse in power systems using svc’s with

washout filter-aided feedback,” Proc 2005 American Control Conf.,

pp 4357-4362, Portland, June 8-10, 2005.

[16] G Bartolini, A Ferrara, E Usai, and V I Utkin, “On multi-input

chattering-free second-order sliding mode control,” IEEE Trans

In-dustrial Electronics, vol 45, no 9, pp 1711-1717, 2000.

[17] H K Khalil, Nonlinear Systems, 3rd ed., Prentice-Hall, 2002.

[18] Y.-W Liang and S.-D Xu, “Relilable control of nonlinear systems via

variable structure scheme,” IEEE Trans Automatic Control, vol 51,

no 10, pp 1721-1725, 2006.

[19] D.-C Liaw, Y.-W Liang, and C.-C Cheng, “Nonlinear control for

missile terminal guidance,” ASME J Dyn Sys., Meas., Control, vol.

122, no 4, pp 663-668, 2000.

[20] D.-C Liaw, and J.-T Huang, “Robust stabilization of axial flow

compressor dynamics via sliding mode design,” ASME J Dyn Sys.,

Meas., Control, vol 123, no 3, pp 488-495, 2001.

[21] D.-C Liaw, C.-C Cheng and Y.-W Liang, “Three-dimensional

guid-ance law for landing on a celestial object,” AIAA J Guidguid-ance Control

and Dynamics, vol 23, no 5, pp 890-892, 2000.

[22] K.-H Cheng, C.-F Hsu, C.-M Lin, T.-T Lee, and C Li, “Fuzzy-neural sliding-mode control for DC-DC converters using asymptotic

Gausian membership functions,” IEEE Trans Industrial Electronics,

vol 54, no 3, pp 1528-1536, 2007.

[23] F.-J Lin, L.-T Teng, and P H Shieh, “Intelligent sliding-mode control

using RBFN for magnetic levitation system,” IEEE Trans Industrial

Electronics, vol 54, no 3, pp 1752-1762, 2007.

[24] E Doedel, AUTO 86 User Manual, Computer Science Dept.,

Concor-dia Univ., Jan 1986.

2.7 2.75 2.8 2.85 2.9 2.95 3 3.05 3.1

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

x4

HB

SNB

Fig 1 Bifurcation diagram with respect to Q 1 and x 4

0.25 0.26 0.27 0.28 0.29

0.868

0.87

0.872

0.874

0.876

x1 (a)

x4

0.1 0.2 0.3 0.4 0.8

0.85 0.9

x1 (b)

0 0.1 0.2 0.3 0.4

0.75

0.8

0.85

0.9

x1 (c)

x4

0 0.1 0.2 0.3 0.4 0.75

0.8 0.85 0.9

x1 (d)

Fig 2 Dynamical behavior in phase-diagram at the varied value of Q 1 :

(a) Q 1 = 2.98201, (b) Q 1 = 2.9891, (c)Q 1 = 2.98975, and (d)Q 1 =

2.9899.

0.865

0.87

0.875

0.88

Time (sec)

(a)

x4

0.75 0.8 0.85 0.9 0.95

Time (sec) (b)

x4

0.75

0.8

0.85

0.9

0.95

Time (sec)

(c)

x4

0.4 0.6 0.8 1

Time (sec) (d)

x4

Fig 3 Time response of power systems at the varied value of Q 1 : (a)

Q 1 = 2.98201, (b) Q 1 = 2.9891, (c)Q 1 = 2.98975, and (d)Q 1 =

2.9899.

0 10 20 30 40 50

−0.15

−0.1

−0.05 0 0.05 0.1

Time (sec) (a)

x3

0 10 20 30 40 50

−0.2

−0.1 0 0.1 0.2

Time (sec) (b)

x3

0 10 20 30 40 50

−0.2

−0.1 0 0.1 0.2

Time (sec) (c)

x3

0 10 20 30 40 50

−0.2

−0.1 0 0.1 0.2

Time (sec) (d)

x3

Fig 4 Time response of state x 3 with SVC-control for V d (t) = 1.0 − 0.05e −t : (a) Q 1 = 2.98201, (b) Q 1 = 2.9891, (c)Q 1 = 2.98975, and (d)Q 1 = 2.9899.

0 10 20 30 40 50 0.85

0.9 0.95 1 1.05

Time (sec) (a)

x4

0 10 20 30 40 50 0.7

0.8 0.9 1 1.1

Time (sec) (b)

0 10 20 30 40 50 0.7

0.8 0.9 1 1.1

Time (sec) (c)

x4

0 10 20 30 40 50 0.7

0.8 0.9 1 1.1

Time (sec) (d)

Fig 5 Time response of state x 4 with SVC-control for V d (t) = 1.0 − 0.05e −t with system uncertainty: (a) Q 1 = 2.98201, (b) Q 1 = 2.9891, (c)Q 1 = 2.98975, and (d)Q 1 = 2.9899.