design of power system stabilizer using power rate reaching law

Design of Power System Stabilizer using Power Rate Reaching Law based SlidingMode Control Technique

Vitthal Bandal, Student Member, IEEE, B. Bandyopadhyay, Member, IEEE, and A. M. Kulkarni,

LIST OF SYMBOLSGp(s) generic power system stabilizer transfer func-

tionKs stabilizer gainωB system base frequency ( 377 rad/sec at 60 Hz)Tw washout time constant

T1, T2, T3, T4 lead/lag time constantGf (s) filtering in stabilizer

E′q voltage proportional to the field flux linkagesof machine

T ′d d-axis transient open circuit time constant

xd d-axis synchronous reactance of machinex′

d d-axis transient reactance of machineEfd generator field voltage

δ machine shaft angular displacement (degree)ω rotor speed ( rad./sec.)

Sm machine slipSm0 nominal slip of the machine

id direct axis armature current (pu)H inertia constant(sec.)D damping coefficient

KE AVR gainTE AVR time constant(sec.)xe line reactance (pu)

Pg0 mechanical power on the shaft of machinePe electrical power output of machinexq q-axis synchronous reactance of machine

Vref reference input voltageVs correction voltageA state (plant) matrix of the systemB control input matrixC output matrixx state vectorsy output vectorst timeu stabilizing signalT transposes switching function

Abstract— The paper presents a new method for designof power system stabilizer (PSS) using discrete time powerrate reaching law based sliding mode control technique. Thecontrol objective is to enhance the stability and to improvethe dynamic response of a single machine infinite bus (SMIB)system, operating in different conditions. The control rules areconstructed using discrete time power rate reaching law basedsliding mode control. We apply this controller to design powersystem stabilizer for demonstrating the efficacy of the proposedapproach.

Vitthal Bandal is a Research Scholar with Systems and Control Engi-neering, Indian Institute of Technology Bombay, Mumbai-400076, INDIA(e-mail:[email protected])

Prof. B. Bandyopadhyay is with Systems and Control Engineering,Indian Institute of Technology Bombay, Mumbai-400076, INDIA (e-mail:[email protected])(corresponding author)

Prof. A. M. Kulkarni is with Electrical Engineering department,Indian Institute of Technology Bombay, Mumbai-400076, INDIA (e-mail:[email protected])

I. INTRODUCTION

Power system stabilizer (PSS) units have long been re-garded as an effective way to enhance the damping of elec-tromechanical oscillations in power system [1]. The action ofPSS is to extend the angular stability limits of a power systemby providing supplemental damping to the oscillation ofsynchronous machine rotors through the generator excitation[2]. This damping is provided by an electric torque appliedto the rotor that is in phase with the speed variation. Oncethe oscillations are damped, the thermal limits of the tie-linesin the system may then be approached. This supplementarysignal is very useful during large power transfers and lineoutages [3].

Over the past four decades, various control methods havebeen proposed for PSS design to improve overall systemperformance. Among these, conventional PSS of the lead-lag compensation type [1], [4], [5] have been adopted bymost utility companies because of their simple structure,flexibility and ease of implementation. However, the per-formance of these stabilizers can be considerably degradedwith the changes in the operating condition during nor-mal operation. Since power systems are highly nonlinear,conventional fixed-parameter PSSs cannot cope with greatchanges in the operating conditions. There are two mainapproaches to stabilizing a power system over a wide rangeof operating conditions, namely adaptive control and ro-bust control [6]. Adaptive control is based on the idea ofcontinuously updating the controller parameters accordingto recent measurements. However, adaptive controllers havegenerally poor performance during the learning phase, unlessthey are properly initialized. Successful operating of adap-tive controllers requires the measurements to satisfy strictpersistent excitation conditions. Otherwise the adjustment ofthe controller’s parameters fails. Robust control provides aneffective approach to dealing with uncertainties introducedby variations of operating conditions.

Among many techniques available in the control literature,H∞ and variable structure have received considerable atten-tion in the design of PSSs. The H∞ approach is appliedby Chen [6] to PSS design for a single machine infinitebus system. The basic idea is to carry out a search overall possible operating points to obtain a frequency bound onthe system transfer function. Then a controller is designedso that the worst-case frequency response of the closedloop system lies within prespecified frequency bounds. Itis noted that the H∞ design requires an exhaustive searchand results in a high order controller. On the other hand the

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variable structure control is designed to drive the system to asliding surface on which the error decays to zero [7]. Perfectperformance is achieved even if parameter uncertainties arepresent. However, such performance is obtained at the costof high control activities (chattering) [8].

In this paper a PSS design for SMIB system using discretetime power rate reaching law based sliding mode controltechnique is proposed. In the sliding mode controller aswitching surface is designed. When the sliding mode occurs,the system dynamic behaves as a robust state feedbackcontrol system. A discrete time power rate reaching lawbased sliding mode controller is investigated, which is usedto minimize the chattering. Simulations results for singlemachine infinite bus (SMIB) system are presented to showthe effectiveness of the proposed control strategy in dampingthe oscillation modes.

The paper is organized as follows. Section II presentsbasics of power system stabilizer and power system analysis.Section III presents the review on multirate output feedback.Section IV presents discrete time power rate reaching lawbased sliding mode controller design; the same is usedfor PSS design of SMIB system as discussed in sectionV. Conclusions are drawn in Section VI. The controller isvalidated using non-linear model simulation.

II. POWER SYSTEM STABILIZER

A. Basic concept

The basic function of a power system stabilizer is to extendstability limits by modulating generator excitation to providedamping to the oscillation of synchronous machine rotorsrelative to one another. The oscillations of concern typicallyoccur in the frequency range of approximately 0.2 to 3.0Hz, and insufficient damping of these oscillations may limitability to transmit power. To provide damping, the stabilizermust produce a component of electrical torque, which is inphase with the speed changes. The implementation detailsdiffer, depending upon the stabilizer input signal employed.However, for any input signal, the transfer function ofthe stabilizer must compensate for the gain and phase ofexcitation system, the generator and the power system,which collectively determines the transfer function from thestabilizer output to the component of electrical torque whichcan be modulated via excitation system [4].

B. Classical Stabilizer implementation procedure

Implementation of a power system stabilizer implies ad-justment of its frequency characteristic and gain to producethe desired damping of the system oscillations in the fre-quency range of 0.2 to 3.0 Hz. The transfer function of ageneric power system stabilizer may be expressed as

Gp(s) = KsTws (1 + sT1) (1 + sT3)

(1 + Tws) (1 + sT2) (1 + sT4)Gf (s)

where Ks represents stabilizer gain and Gf (s) representscombined transfer function of torsional filter (if required) andinput signal transducer. The stabilizer frequency characteris-tic is adjusted by varying the time constant Tw, T1, T2, T3

and T4. A torsional filter may not be necessary with signalslike power or delta-P-omega signal [9].

A power system stabilizer can be most effectively appliedif it is tuned with an understanding of the associated powercharacteristics and the function to be performed by the sta-bilizer. Knowledge of the modes of power system oscillationto which the stabilizer is to provide damping establishes therange of frequencies over which the stabilizer must operate.Simple analytical models, such as that of a single machineinfinite bus (SMIB) systems, can be useful in determining thefrequencies of local mode oscillations during the planningstage of a new plant. It is also desirable to establish theweak power system conditions and associated loading forwhich stable operation is expected, as the adequacy of thepower system stabilizer application will be determined underthese performance conditions. Since the limiting gain of thesome stabilizers, viz., those having input signal from speedor power, occurs with a strong transmission system, it isnecessary to establish the strongest credible system as the“tuning condition” for these stabilizers. Experience suggestthat designing a stabilizer for satisfactory operation with anexternal system reactance ranging from 20% to 80% on theunit rating will ensure robust performance [10].

C. Power System Analysis

Analysis of practical power system involves the simul-taneous solution of equations consisting of synchronousmachines ,associated excitation system , prime movers, in-terconnecting transmission network, static and dynamic (motor ) loads, and other devices such as HVDC converters,static var compensator. The dynamics of the machine rotorcircuits, excitation systems, prime mover and other devicesare represented by differential equations. This results inthe complete system model consisting of large number ofordinary differential and algebraic equations [9].

1) Generator Equations: The machine equations ( forjthmachine ) are

dE′qj

dt=

−1T

′d0j

[E′qj − (xdj − x

′dj)idj − Efdj ], (1)

dδj

dt= ωB(Smj − Smj0), (2)

dSmj

dt=

−12H

[Dj(Smj − Smj0) − Pmj + Pej ]. (3)

Model 1.0 is assumed for synchronous machines by ne-glecting the damper windings. In addition, the followingassumptions are made for simplicity [11].

1. The loads are represented by constant impedances.2. Transients saliency is ignored by considering xq= x

′d.

3. Mechanical power is assumed to be constant.4. Efd is single time constant AVR.

2) State space model of power system (Machine model1.0): The state space model of a SMIB power system, theblock diagram of which is shown in Fig. 1 can be obtained


+

−

+_

+

_

12Hs

B

∆Τm

∆

∆

∆

K

K

K

K

1 + s T

K

1 + s T K

K

K

∆Ε

T

T

∆δ

Σ Σ

Σ sω

6

3d0

5

E

q

2

4

1

e1

e2

∆Ε

v

∆Sm

5

ref

fd

E

Fig. 1. Block diagram of a Single Machine Infinite Bus (SMIB) system

using generator, transformer, network and loadflow data asgiven below [11],

.x= Ax + B (�Vref + �Vs) , (4)

y = Cx, (5)

wherex denotes the states of the machine and are given as

x = [Sm, δ, Efd, E′q]. Similarly, y = Sm denotes the output

equation of the machine and C is the output matrix.( C =[1, 0, 0, 0] ).

Where Sm is machine slip and is given by,

Sm =(ω − ωB)

(ωB),

δ is machine shaft angular displacement in degrees, Efd isgenerator field voltage in pu and E′

q is voltage proportionalto field flux linkages of machine in p.u.

The elements of matrix A are dependent on the operatingcondition.

III. REVIEW ON MULTIRATE OUTPUT FEEDBACK

In the following, multirate output feedback is brieflyreviewed.

Multirate Output Feedback is the concept of sampling thecontrol input and sensor output of a system at different rates.It was found that multirate output feedback can guaranteeclosed loop stability, a feature not assured by static outputfeedback [12] while retaining the structural simplicity ofstatic output feedback. Much research has been performed inthis field [13]–[17]. In multirate output feedback, the controlinput [15], [17] or the sensor output [16] is sampled at afaster rate than the other. In this paper, the term multirateoutput feedback is used to refer the situation wherein thesystem output is sampled at a faster rate as compared to thecontrol input.

It was found that state feedback based control laws ofany structure may be realized by the use of multirate outputfeedback, by representing the system states in terms of thepast control inputs and multirate sampled system output [18],[19].

IV. DISCRETE TIME POWER RATE REACHING LAW

BASED SLIDING MODE CONTROL

Consider a SISO plant described by a continuous timelinear model

x = Ax + Bu, (6)

y = Cx.

Where x ∈ Rn, u ∈ R, y ∈ R and the matrices A, B andC are of appropriate dimensions.

Let ( Φτ ,Γτ , C) be the system given by Eqn.(6) sampledat sampling interval τ seconds and is represented as,

x(k + 1) = Φτx(k) + Γτu(k), (7)

y(k) = Cx(k). (8)

In [20] a power rate reaching law approach for continuoustime systems had been proposed. The discrete power ratereaching law can be directly obtained from the continuouspower rate reaching law as,

s(k + 1) − s(k) = −kτ |s(k)|αsgn(s(k)) (9)

where τ > 0, is the sampling period, 0 < kτ < 1 and0 < α < 1. s(k) is the switching function defined as afunction of system states as,

s(k) = cT x(k) (10)

Hence,

s(k + 1) = cT x(k + 1) (11)

So, from Eqns. (7), (9) and (11)

s(k + 1) − s(k) = cT [Φτ − I]x(k) + cT Γτu(k) (12)

Comparing the Eqns.(9) and (12), the control law isobtained as follows [21],

u(k) = −(cT Γτ )−1[cT Φδx(k) + ρ|s(k)|αsgn(s(k))] (13)

where, Φδ = Φτ − I and ρ = kτ . Thus, a discrete timesliding mode control based on power rate reaching law isobtained. This control law is designed using the states of thesystem. Switching gain c can be obtained using the proceduregiven in [22]

As, in practice all states of the system are not availablefor measurement and therefore control derived with the helpof only output information of the system will be more usefulfrom practical point of view. A generalized expression for theswitching surface and the control using output informationonly has been derived and is given as [21],

x(k) = ΦτC−10 yk + [Γτ − ΦτC−1

0 D0]u(k − 1), (14)

s(k) = cT (ΦτC−10 yk + [Γτ − ΦτC−1

0 D0]u(k − 1)) (15)


u(k) = −(cT Γτ )−1[cT ΦδΦτC−10 yk

−(cT Γτ )−1[cT Φδ(Γτ − ΦτC−10 D0)u(k − 1)

−(cT Γτ )−1ρ|s(k)|αsgn(s(k))] (16)

Thus, it can be seen from the Eqns. (15) and (16) that thestates of the system are needed neither for switching functionevaluation nor for the feedback purpose.

V. CASE STUDY

A single machine infinite bus power system is consideredhere for PSS design using power rate reaching law basedsliding mode control technique.

A. Linearization of power system

The Nonlinear differential equations governing the be-havior power system can be linearized about a particularoperating point to obtain a linear model which representsthe small signal oscillatory response of a power system. ASIMULINK based block diagram including all the nonlinearblocks can also be used to generate the linear state spacemodel of the system.

B. Classical power system stabilizer design for a powersystem

The classical power system stabilizer (PSS) is designed inthe following way.

The eigenvalue analysis of a power system is carried outand the participation ratio of the machine towards instabilityin the network, is estimated. Power system stabilizer usingphase compensation technique is designed according to theparticipation ratio of the machine towards instability, tillsatisfactory closed loop performance of the power systemis achieved. The above design of classical power systemstabilizer (PSS) is iterative in nature and optimal tuning ofparameters is based on the experience. If the power charac-teristics of the system changes, then the whole procedure ofPSS design has to be repeated. So, design of classical powersystem stabilizers cannot be considered robust in nature forall operating points. The proposed power rate reaching lawbased sliding mode control technique used for power systemstabilizer design is robust in nature for all the models andis not iterative .

C. Design of PSS using discrete time power rate reachinglaw based sliding mode control for Single Machine InfiniteBus (SMIB) system

The single machine infinite bus power system data isconsidered for designing PSS using power rate reaching lawbased sliding mode control. The block diagram of the systemis shown in Fig. 1.

The following parameters are used for simulation of thesingle machine infinite bus system model [11]:

H = 5 sec., D = 0, T′do = 6 sec., KE = 100, TE = 0.02

sec., xe = 0.2 p.u.

0 1 2 3 4 5 6 7 8 9 101.12

1.14

1.16

1.18

1.2

1.22

1.24

1.26

1.28

1.3

Time in sec.

Del

ta (

radi

ans)

Single Machine Infinte Bus System

POWER RATE REACHING LAW BASED SMC PSSCLASSICAL PSS

(a) Delta responses

0 1 2 3 4 5 6 7 8 9 10−2

−1

0

1

2

3

4x 10

−3

Time in sec.

Slip



(b) Slip responses

Fig. 2. Delta and slip responses with classical PSS and PSS using powerrate reaching law based sliding mode control technique for Pg0=0.5 pu,Vref=1.0 pu and Xe=0.25 pu

As discussed in the previous section, the SISO linearizedmodel of entire system obtained at nominal operating condi-tion is obtained, which is represented by Eqn. (6). The powerrate reaching law based sliding mode control given by Eqn.(16) is then applied to the actual nonlinear system to carryout simulations.

D. Simulation with Non-linear model

The slip of the machine is taken as output. This outputsignal of the controller and a limiter is added to Vref signal.This is used to damp out the small signal disturbancesvia modulating the generator excitation. The disturbanceconsidered here is a self clearing fault which is cleared after0.1 second. The limits of PSS output are taken as ±0.1.Simulation results for SMIB system for various operatingconditions, with power rate reaching law based sliding modecontroller and classical controller are shown in Fig. 2 to Fig.5.

As shown in plots, the proposed controller is able to dampout the oscillations in 1 to 2 seconds after clearing the fault.Even in some cases where, classical PSS cannot damp outthe oscillations, the proposed controller is able to damp out


0 1 2 3 4 5 6 7 8 9 100.9

1

1.1

1.2

1.3

1.4

1.5

Time in sec.

Del

ta (

radi

ans)



(a) Delta responses

0 1 2 3 4 5 6 7 8 9 10−6

−4

−2

0

2

4

6

8x 10

−3

Time in sec.

Slip



(b) Slip responses


the oscillations.

VI. CONCLUSION

This paper proposes, the design of PSS for single machineinfinite bus (SMIB) system by power rate reaching law basedsliding mode control technique. It is found that designedcontroller provides good damping enhancement for variousoperating points of SMIB power system. The proposedcontroller results in a better response behavior to damp outthe oscillations. The conventional power system stabilizer isdynamic in nature and is required to be tuned according topower characteristics where as, the proposed controller isnon-dynamic in nature and a single controller structure isable to damp out the oscillations for all models.

Simulation results from a nonlinear power system aregiven to demonstrate the applicability and effectiveness ofthe proposed approach.

The proposed approach can be extended to a multimachinesystem by considering multimachine power system as a setof interconnected single machine systems.

0 1 2 3 4 5 6 7 8 9 100.9

1

1.1

1.2

1.3

1.4

1.5

Time in sec.

Del

ta (

radi

ans)



(a) Delta responses

0 1 2 3 4 5 6 7 8 9 10−6

−4

−2

0

2

4

6

8

10x 10

−3

Time in sec.

Slip



(b) Slip responses


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[8] F. Rashidi, M. Rashidi, and H. Amiri, “An adaptive fuzzy slidingmode control for power systems stabilizer,” in Industrial ElectronicsSociety,2003, IECON’03. The 29th Annual Conference of the IEEE,November 2003, pp. 626–630.


0 1 2 3 4 5 6 7 8 9 100.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Time in sec.

Del

ta (

radi

ans)



(a) Delta responses

0 1 2 3 4 5 6 7 8 9 10−8

−6

−4

−2

0

2

4

6

8

10x 10

−3

Time in sec.

Slip



(b) Slip responses


[9] P. Kundur, Power System Stability and Control. McGraw-Hill, Inc.Newyork, 1993.

[10] E.V.Larsen and D.A.Swann, “Applying power system stabilizers part-iii: General concepts,” IEEE Trans. on Power Apparatus and Systems,vol. PAS-100, no. 6, pp. 3034–3046, June 1981.

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LIST OF FIGURES

1 Block diagram of a Single Machine Infinite Bus(SMIB) system . . . . . . . . . . . . . . . . . . 3

2 Delta and slip responses with classical PSSand PSS using power rate reaching law basedsliding mode control technique for Pg0=0.5 pu,Vref=1.0 pu and Xe=0.25 pu . . . . . . . . . . 4

3 Delta and slip responses with classical PSSand PSS using power rate reaching law basedsliding mode control technique for Pg0=1.0 pu,Vref=1.0 pu and Xe=0.25 pu . . . . . . . . . . 5

4 Delta and slip responses with classical PSSand PSS using power rate reaching law basedsliding mode control technique for Pg0=1.25pu, Vref=1.0 pu and Xe=0.25 pu . . . . . . . . 5

5 Delta and slip responses with classical PSSand PSS using power rate reaching law basedsliding mode control technique for Pg0=1.35pu, Vref=1.0 pu and Xe=0.25 pu . . . . . . . . 6


design of power system stabilizer using power rate reaching law

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