a quantitative design approach to pss tuning

Electric Power Systems Research 73 (2005) 249–256

A quantitative design approach to PSS tuning

P.S. Rao, E.S. Boje∗

School of Electrical, Electronic and Computer Engineering, University of KwaZulu-Natal, 4041 Durban, South Africa

Received 3 February 2004; received in revised form 3 March 2004; accepted 20 June 2004Available online 25 November 2004

Abstract

This paper presents the application of quantitative feedback design techniques for tuning stabilizers in multi-machine power systems. Thisapproach facilitates easy handling of multiple plant models thereby yielding robust and reliable stabilizer parameters. Methods of incorporatingclosed-loop stability and damping performance requirements into the design are explained. In the proposed sequential tuning technique, boundson the stabilizer frequency response are computed for stability and performance at each of the given set of operating conditions of the system.A manual controller shaping then yields the desired stabilizer parameters. Application to an illustrative textbook example of an 11-bus,four-generator system is also included.©

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2004 Elsevier B.V. All rights reserved.

eywords:Power system dynamic stability; Robustness; Quantitative feedback theory; Power system stabilizer

. Introduction

The damping of low frequency oscillations by the applica-ion of power system stabilizers (PSS) has been extensivelynvestigated. Systematic tuning procedures for the conven-ional lead compensation PSS have been established[1] anduccessfully applied by the utilities. However, robust con-rol approaches that explicitly incorporate the various systemonditions in the design can be used to further simplify orutomate the design and tuning process.

The use of small-signal, linearized models of the system atarious operating conditions for stability analysis and PSS de-ign generates a problem statement which fits into the frame-ork of linear robust control. Much work has been carriedut in the use of mathematical control design methods in theingle machine case. However, the more important problemf PSS design in multi-machine systems remains relativelynexplored. This can be partly attributed to the inadequacies

n existing robust multivariable control theory.Issues of coordination of all stabilizers and damping

ontrollers in a given system and the sensitivity of the

tuning of all these controllers to the changing operacondition of the power system are of paramount imtance in any practical problem. The availability of olocal measurements for feedback at any site furtherstrains the design problem. Hence, from a control pof view, the difficulties are two-fold—those arising duethe multivariable, decentralized nature of the problemthose due to the large parametric uncertainty in the symodel.

Classical multivariable frequency response methodcontroller design which try to exploit diagonal dominancthe plant transfer matrix do not apply readily to the posystems problem due to the high level of interaction betwthe individual machines at the modal frequencies. Recdeveloped robust control theories dealing with the optimtion of system norms are well-suited for MIMO systemscan be so conservative that they fail to yield design soluwhen applied to power system models with large paramuncertainty[2].

The possible outage of generators in a power systesults in a mixed reliability–robustness problem peculiapower systems. Such an event can drastically change th

∗ Corresponding author. Tel.: +27 31 2602718E-mail address:[email protected] (E.S. Boje).

namic characteristics of the system along with changing thenumber of inputs and outputs. This combination of uncer-

378-7796/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
oi:10.1016/j.epsr.2004.06.012

250 P.S. Rao, E.S. Boje / Electric Power Systems Research 73 (2005) 249–256

tainty and loop failure cannot be easily incorporated into mostconventional robust control techniques.

Quantitative feedback theory (QFT) introduced byHorowitz [3], is a conceptually simple framework based onthe shaping of the system frequency responses on the Nicholschart so as to guarantee the attainment of the specified closed-loop performance for an entire set of plant models. This ap-proach is particularly known for its ability to handle paramet-ric uncertainty with minimal conservativeness.

The conventional multivariable QFT design procedureas proposed by Horowitz[3] is of little use in PSS designdue to the difficulty in generating magnitude specificationson the off-diagonal plant elements that allow the applica-tion of Schauder’s fixed point theorem. However, the basicidea of frequency response shaping to satisfy constraints onthe Nichols chart can still be used in various formulations.Further, as the plant is treated merely as a set of complexnumbers at each frequency, incorporation of the generatoroutage cases does not generate any theoretical difficultiesor involve additional complexity in the controller synthe-sis.

There have been a few previous reported applications ofquantitative design methods in power systems. Jacobson etal. [4] have used QFT-like loop shaping to satisfy frequencyresponse bounds derived using dissipativity theory. Quanti-tative techniques have been shown to be quite effective intQ .h ces[ ueo inp o theS

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logy.A ur-g

2

sta-b ntirer es. As apingb ertainb putedf of ac ed ont uallys ncies

The same procedure is repeated sequentially for each of thestabilizers in the system.

Some methods of computing these frequency responsebounds are discussed below.

2.1. Stability

This subsection discusses the derivation of stabilitybounds for decentralized control systems.

Existing results in multivariable frequency response tech-niques allow the stability of a MIMO system to be deducedfrom the stability of the diagonal loops subject to certain diag-onal dominance conditions on the plant. Due to the high levelof interaction between the individual generator rotor swingsin a power system, such results based on diagonal dominanceare of little use.

The structured singular value[9] has also been found use-ful in obtaining stability bounds for decentralized control sys-tems[10] that could be applied to power systems problems[11].

Consider a MIMO system represented by its transfer ma-trix P(s). P can be written as the sum of diagonal and off-diagonal parts asP=Pd +Po. Let the required decentralizedcontroller beK= diag[kii ].A positive feedback convention isused here.

DefineE = P P−1,H= diag[p k /(1− k P )]. Then, un-d

1 fixed

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ollerK

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w t.A llyc int

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inde-p Yang[ mc (asr nts int odelwv

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he single machine case[5,6]. Boje et al.[5] have appliedFT for PSS design in a SMIB system. Eitelberg et al[7]ave applied it for stabilizing SSR oscillations. Referen

4,8]point out the utility of the simple loop shaping techniqf QFT for the multiple plant model problems occurringower systems. Their exposition is, however, restricted tISO case.The present approach agrees with the general spirit of

n that it relies upon manual frequency response shapinhe Nichols chart. However the computation of the bouequires the use of techniques based on the structured sialue, which have not previously been used in conjuncith QFT.The next section explains the proposed methodo

n illustrative textbook example of PSS design for a foenerator system is then presented.

. Stability and performance objectives

The objective of applying stabilizers is to achieve ale, well-damped small disturbance response for the eange of operating conditions that the system experiencequential tuning technique with frequency response shased on QFT is proposed here. For each stabilizer, counds on the controller frequency response are com

rom stability and performance considerations, at eachhosen set of frequency points. These bounds are plotthe Nichols chart. The PSS transfer function is then manhaped such that it satisfies the bounds at all the freque

r

.

o d ii ii ii ii

er the following assumptions:

. the system does not have any unstable decentralizedmodes;

. P andPd have an equal number of RHP poles;

. all the elements ofH are stable (by design).

e have the following result proved by Grosdidier and Mo10].

The closed-loop system with the decentralized contris stable if

Hii| <1

µ(E)∀i (1)

hereµ(·) is the structured singular value[9] of the argumenMatlab toolbox routine[12] has been used to numerica

ompute the value ofµ(·) in all the examples presentedhis paper.

Eq. (1) gives a magnitude bound for the independenign of the individual loops. 1/µ can be interpreted as a radf perturbation within which the elements ofH must lie.

The above bound has been successfully used in theendent design of stabilizers for a sample system by

11]. However, as pointed out in[11], the example systeonsidered does not exhibit a high level of interactionepresented by the magnitude of the off-diagonal elemehe plant transfer matrix). For a general power system mith poorly damped interarea modes, the bound(1) can beery conservative.

Fig. 1shows the plot ofµ(E) for a standard four-machinystem taken from[13]. The system details are given in S

P.S. Rao, E.S. Boje / Electric Power Systems Research 73 (2005) 249–256 251

Fig. 1. Magnitude of|H11| (dotted) and the bound from(1) (solid) for theexample system.

tion 3. The magnitude|H11| is also plotted with a PSS withparameters taken from[13]. The PSS when applied to all fourmachines stabilizes the system. However, it is seen inFig. 1that the inequality(1) is not satisfied around the interareamode frequency of 3.74 rad/s. This is expected since there isa high level of interaction between all four machines at thisfrequency. The bound is found to be useful at higher (localmode) frequencies where the interaction is much lesser.

However, it is possible, to use a similar formulation to di-rectly compute bounds on the PSS response at any frequencygiven a trial PSS (referred to later as a nominal PSS) that sta-bilizes the system. This nominal PSS need not achieve anydamping performance objectives and is introduced merelyas an initial guess. Letko(s) be the transfer function of thenominal PSS, which when applied to all input/output pairsof the system, results in a stable closed-loop. The nominalcontroller for the MIMO system representing the applica-tion of the nominal PSS on all machines would beko(s)I,whereI is the identity matrix of appropriate dimension. Let∆ = diag[∆ii ] be a diagonal additive perturbation in the con-troller matrix as shown inFig. 2.

The system shown inFig. 2remains stable if,

|∆ii| <1

µ(P(I − koP)−1)∀i (2)

larv

Fig. 3. Bound on the PSS response at the interarea mode frequency(3.74 rad/s) plotted in the complex plane.

This condition generates a circular region centered aroundko in the complex plane at each frequency within which allthe frequency responses of all the stabilizers should lie forstability of the closed-loop system.

For the four-generator example system considered here,the nominal PSS function given in[13], which stabilizes thesystem is

ko(s) = 2010s

1 + 10s

1 + 0.05s

1 + 0.02s

1 + 3s

1 + 5.4s(3)

Fig. 3shows the stability bound on the PSS responses at theinterarea mode frequency for the example system. (Thoughit might seem odd that the region includes positive and nega-tive phase regions, it can indeed be verified that the examplesystem can be stabilized even with lag compensation on eachof the machines with suitable gain, so long as the PSS phaselag is not very high.) The system is guaranteed to be stableif the frequency responses of all the stabilizers fitted on thesystem stay within such stability bounds at all frequencies.Since(2) is only a sufficient condition, the converse is nottrue i.e., the system is not necessarily unstable if one of thesebounds is violated. However, if the frequency response ofone of the stabilizers violates the stability bound at a partic-ular frequency, the closed-loop system may have an unstablemode around that frequency.

ed-l mpu-t ivenp ainf inedf thes

n inF thel

hea ition.T tion

this follows from the definition of the structured singualue[11].

Fig. 2. Schematic of the nominal PSS with perturbation.

Any controller transfer function that gives a stable closoop can be used as the nominal PSS in the above coations. Synthesis of such a stabilizing controller for a glant model is not difficult. For instance, a simple static g

eedback that stabilizes a given plant model can be obtarom inspection of a plot of the characteristic gains ofystem[14] and would suffice.

Further, the circular stability region for the PSS showig. 3 can also be expanded by numerically optimizing

ocation of the center so as to minimize the value ofµ(·) [15].While dealing with multiple operating conditions, t

bove bound can be obtained for each operating condhe bound for robust stability would then be the intersec


Fig. 4. Magnitude plots of the plant (dotted) and model (solid) for requireddamping enhancement.

of the individual bounds. This fact can be rigorously provedusing connectedness arguments[15].

2.2. Performance

The oscillatory behaviour of the power system modelarises from the poorly damped dominant poles, which resultin magnitude peaks in the plant frequency response. Bound-ing the relative magnitude of the closed-loop response at thefrequency of oscillation results in an improved damping fac-tor of the dominant modes provided there are no pole-zerocancellations between the plant and the controller. This tech-nique has previously been used in numerousH∞ optimizationbased loop shaping approaches to PSS design[16]. Specialcare needs to be taken in choosing the weighting functions toavoid poorly damped pole-zero cancellations when using anautomated controller synthesis method. In QFT based meth-ods this is not a problem as the controller response is manuallyshaped.

Specified closed-loop magnitude constraints can easily beconverted into controller frequency response bounds.

Consider a SISO plant transfer functionp(s) with a feed-backk(s). Given a boundb(ω) on the closed-loop transferfunction we have,

|t(jω)| = |p(jω)|(1 − k(jω)p(jω))−1| < b(ω) ∀ω

|i ofro redo .

eci-fi gn-i mi-n ateb omi-n cya

be-t tor 1

in the four-generator example. The response when the poorlydamped interarea and local modes are replaced by poles ofthe same frequency and with a damping factor improvementof 5% are also shown. The magnitude of this function givesan approximate value ofb(ω) at the oscillatory frequenciesneeded to achieve a 5% damping factor improvement for theclosed-loop poles.

An acceptable small signal dynamic performance for apower system is specified in terms of the settling time aftera disturbance, which typically imposes a constraint of 10%on the damping factor of the dominant poles. Hence, for asystem with multiple stabilizers, the net contribution of allstabilizers should guarantee this. In multi-machine systems,the contribution from each proposed stabilizer in terms ofdamping factor improvement of each of the critical modescan be decided a priori based upon the relative participationfactors and transfer function residues[18].

The boundb(jω) would depend on the operating conditionof the system, since it is affected by changes in the frequenciesof oscillations and movement of the zeroes. It is, however,quite straightforward to obtain bounds on the PSS responsefor each operating condition and then take the intersection ofthese regions as the requirement for robust performance. Theplots of this bound for the example system are shown in thenext section.

The satisfaction of the bound guarantees the damping per-f re ist can-c

3

rizedb

s fort ndingl s arei rs toe mentsb PSSl ualc andc ges.T ssedi

ngeo localm rs aret

com-p lantms itionad es of

t−1(jω)| = |p−1(jω) − k(jω)| > b−1(ω) ∀ω (4)

.e., at each frequencyω, k(jω) should lie outside a discadiusb−1(ω) centered at the pointp−1(jω). A similar devel-pment can be found in[17], where these discs are transfern to the Nichols chart resulting in quadratic inequalities

The boundb(ω) can be taken as the magnitude of a sped model. In[16] this reference model is obtained by desing a controller that provides the required amount of doant mode damping for a nominal plant model. Approximounds can also be obtained by simply replacing the dant poles in the plantp(s) with poles of the same frequennd the desired damping factor.

Fig. 4shows the magnitude plot of the transfer functionween the AVR input and the rotor speed output of genera

ormance of the closed-loop system if appropriate caaken that the controller does not simply use pole-zeroellations to achieve the magnitude bounds.

. Design procedure

The proposed stabilizer design procedure is summaelow.

A set of representative extremal operating conditionhe given power system are chosen and the correspoinear models are obtained. The poorly damped modedentified and the damping contributions of the generatoach of the modes are decided. Several heuristic arguased on participation factors and residues studied in

iterature[18] can be used for this. Further, the individontributions could change with the system conditiononfiguration, particularly for cases with generator outahe issue of choice of damping contributions is not addre

n this paper.A set of frequency points is chosen in the frequency ra

f interest, so as to cover the range of interarea andode frequencies observed in the system. The stabilize

hen designed sequentially.First the frequency response bounds on the PSS are

uted. The closed-loop system comprising of the linear podel and the nominal MIMO controller (ko(s)I) is con-

tructed and the stability bounds at each operating condnd frequency are computed as explained in Section2. Theominant modes of the plant model are replaced by pol


Fig. 5. Single line diagram of the example system[1].

the same frequency but with an increase in damping factor bythe amount to be contributed by the stabilizer. The magnitudeof this artificial transfer function is used in Eq.(4) to obtainthe performance bound on the PSS. The stability and perfor-mance bounds at each frequency point are computed for eachof the operating conditions of the system. The intersection ofall the stability and performance bounds at each frequencypoint are plotted on the Nichols chart. The PSS frequencyresponse is then manually shaped so as to satisfy the boundat each frequency point. This procedure is repeated for eachstabilizer in the system.

It should be noted that the method does not guarantee sta-bility or performance for any intermediate operating condi-tions not included in the initial set considered for the design.

4. Example

The proposed technique is applied here for the design ofstabilizers for a four-generator system taken from[13]. Thesingle line diagram of the system is shown inFig. 5. Theoperating condition given in[13] is taken as the base case.Four operating conditions are defined for the system as givenbelow.

Case a:Base case, area 1 exports 400 MW to area 2.

l sec-

ced

hesef

eseo withs eedso stem,r each,w

wn inT

rad/sf f thel A seto is

Table 1Electromechanical modes for pour operating conditions, (withoutstabilizers)

Case Mode Eigenvalue ζ ωn

a Interarea 0.0605± 3.7370j −0.016 3.74Local area 1 −0.2409± 5.9829j 0.040 5.99Local area 2 −0.2613± 6.1784j 0.042 6.18

b Interarea 0.0679± 3.6730j −0.019 3.67Local area 1 −0.2575± 5.9885j 0.043 5.99Local area 2 −0.2374± 6.1677j 0.039 6.17

c Interarea 0.0357± 2.8636j −0.013 2.86Local area 1 −0.1923± 5.9850j 0.032 5.99Local area 2 −0.2283± 6.1524j 0.037 6.16

d Interarea 0.0344± 4.2360j −0.008 4.24Local area 1 −0.2112± 6.0586j 0.035 6.06Local area 2 – – –

chosen for the design. The details of the procedure for thedesign of the stabilizer on generator 1 are given below. Thesame procedure is to be followed for the other generators.

4.1. Stability bounds

The stability bounds are computed for each plant modelas explained in Section2. The PSS given in reference[13] istaken as the nominal stabilizing controller for all four oper-ating conditions.

Fig. 6 shows the stability bounds on all the four stabiliz-ers responses at the frequencies of 3.73 and 6.0 rad/s plottedon the Nichols chart. The PSS responses should lie withinthe closed region at 3.73 rad/s and below the bound shownat 6.0 rad/s. (This bound includes the origin since the localmodes are open-loop stable for all four cases.)

4.2. Performance bounds

Magnitude bounds on the closed-loop transfer functionsare computed at the four frequency points for each of the fouroperating conditions.Table 2shows the damping factor con-tributions of the four stabilizers to the three rotor modes of

Case b:Area 1 imports 400 MW from area 2.Case c:Same as base case but with one of the paralletions of one of the 110 km tie-lines out of circuit.Case d:Generator 4 out of circuit, load in area 2 is reduto 1130 MW, area 1 exports 450 MW to area 2.

Further details of the generator and load powers for tour cases are given inAppendix A.

Linear models are obtained for the plant for each of thperating conditions. All four generators are to be fittedtabilizers. The AVR reference values and the rotor spf the generators are the inputs and outputs of the syespectively. Cases a–c have four inputs and outputshile case d has three inputs and outputs.The modes of the system for the base case are sho

able 1.The unstable interarea mode shifts to 2.85 and 4.23

or cases c and d, respectively, while the frequency oocal modes stay at around 6.0 rad/s for all four cases.f four frequency points,ω = (2.85, 3.73, 4.23, 6.0) rad/s,
Fig. 6. Stability bounds on the Nichols chart.


Table 2Damping contributions of the four stabilizers

Generator no. Damping factor contribution

Interarea (%) Local 1 (%) Local 2 (%)

1 3 10 02 3 10 03 3 0 104 3 0 10

the system. In this example the damping contributions of thestabilizers are assumed to be equal i.e., all machines partic-ipating in a particular electromechanical mode are requiredto contribute equally to the damping enhancement of thatmode. As shown inTable 2all four generators participatein the interarea mode and so contribute 3% each (for a total12% desired improvement). Generators 1 and 2 contribute10% each to local mode 1 (for a total 20% desired improve-ment) and do not contribute to damping of local mode 2, asthey do not participate in it.

This choice provides approximately 9% damping factorof the interarea mode even for the generator outage case. Thecontribution of any generator to the local mode it does notparticipate in is naturally fixed at zero.

The bounds on the stabilizer fitted on generator 1 forthe four operating conditions at the frequency 2.85 rad/s areshown (dotted, a–d) inFig. 7. The bound at a particular fre-quency is taken as the intersection of the bounds computedfor a number of closely spaced points around that frequencyto ensure that the bounds are not very sensitive to small inac-curacies in the frequency. This technique is used to generateeach of the bounds shown as dotted lines inFig. 7. The in-tersection (solid) of the bounds corresponding to the fouroperating cases is the bound taken for the design. The PSSresponse at 2.85 rad/s is constrained to lie above this linefor acceptable performance (from(4), Section2.2). The twomc o thel iont ilityb

Fig. 8. Performance (solid) and stability (dotted) bounds on stabilizer fittedon generator 1.

Fig. 8 shows the performance and stability bounds forstabilizer 1 at all the four frequency points. The stabilizerresponse has to lie in the intersections of the performanceand stability bounds at each of the four chosen frequencies.

A stabilizer transfer function for machine 1 can now besynthesized. A lead compensator in cascade with a standardwash-out stage (Tω = 10 s) is considered for the design. It isfound that a simple single stage lead compensator satisfiesthe bounds shown inFig. 8 at each of the four frequencies.The final stabilizer transfer function for machine 1 is thusobtained as:

PSS1(s) = 24.510s

1 + 10s

1 + 0.13s

1 + 0.01s

Fig. 9shows the frequency response of the above PSS onthe Nichols chart. The four frequency points used for thedesign are marked. It can be verified that the four pointslie within the corresponding bounds shown inFig. 8. (Thebounds are not overplotted to avoid cluttering.) Interactiveshaping tools[19] can be used, if required, for shaping thecontroller transfer function to satisfy bounds in the Nicholschart. Further, if a simple transfer function cannot be shapedto satisfy the bounds, higher order functions need to be used.This was not necessary for the example presented here.

The same procedure is repeated for each of the stabilizersin the system. The details are omitted due to space restric-t ge byr ea ative-n fours m ata

witha for

inima of the performance bound at around +20◦ and−130◦orrespond to the migration of the mode at 2.85 rad/s teft and right in the complex plane, respectively. Migrato the right is destabilizing and is obviated when the stabounds are also considered as shown below.

Fig. 7. Performance bounds at 2.85 rad/s on the Nichols chart.

ions. The stability bounds are recomputed at each staeplacing the nominal PSSko with the stabilizers, which havlready been designed. This would reduce the conservess of the stability bounds. The final parameters of thetabilizers and the closed-loop eigenvalues of the systell four operating conditions are given inAppendix A.

Time responses of the resulting closed-loop systemll four generators fitted with stabilizers were simulated


Fig. 9. Nichols plot of PSS1.

various disturbances and operating conditions.Fig. 10showsthe response of the system with all generators fitted with sta-bilizers and at all four operating conditions to a small impulsedisturbance in the voltage reference of generator 1. The sys-tem oscillations in terms of the rotor speed deviations fromsteady state of all generators are seen to be well-damped forall four cases considered.

For application of this technique to large systems it is nec-essary to identify a-priori the generating units contributingto the modes of interest that are to be fitted with stabilizers.Standard methods are available for this. The entire process of

F e fourg

Table 3Base case load flow data

Bus no. P Q V

1a – – 1.032b 0.7778 – 1.013b 0.7989 – 1.034b 0.7778 – 1.015 0.0 0.0 –6 0.0 0.0 –7 −1.074 0.11 –8 0.0 0.0 –9 −1.963 0.2778 –

10 0.0 0.0 –11 0.0 0.0 –

All powers are on a system base of 900 MVA.a Slack bus.b PV bus.

Table 4Loads and tie-line power

Case Bus 7 Bus 8 Tie line flow

P Q P Q (Area 1–2)

a 1.07 −0.11 1.96 −0.28 0.444b 1.96 −0.28 1.07 −0.11 −0.442c 1.07 −0.11 1.96 −0.28 0.463d 1.07 −0.11 1.26 −0.39 0.511

All powers on the system base of 900 MVA.

Table 5Parameters of the four stabilizers

Generator no. PSS parameters

Gain (K) T1 T2

1 24.5 0.13 0.012 10.0 0.13 0.013 12.2 0.09 0.014 8.0 0.2 0.02

Table 6Closed-loop eigenvalues for four operating conditions, (with stabilizers)

Case Mode Eigenvalue

a Interarea −0.653± 3.49jLocal area 1 −0.290± 4.84jLocal area 2 −0.217± 5.70j

b Interarea −0.649± 3.37jLocal area 1 −3.04± 4.83jLocal area 2 −2.07± 5.70j

c Interarea −0.566± 2.65jLocal area 1 −2.68± 4.97jLocal area 2 −2.02± 5.71j

d Interarea −0.862± 4.03jLocal area 1 −2.93± 4.93jLocal area 2 –

computation of the bounds on the frequency response can beautomated and is not an issue. The manual frequency responseshaping, however, needs to be repeated for each stabilizer inthe system.

ig. 10. Small disturbance response: per unit speed deviations of thenerators at all operating conditions.


5. Conclusions

A quantitative design approach for tuning stabilizers inmulti-machine electric power systems has been presented.The technique easily incorporates multiple system configura-tions and thereby generates robust stabilizer parameters. Thecomputation of the bounds discussed in the paper and alsothe controller synthesis can be automated to a large extentusing available CACSD software tools, thereby reducing thedesign effort. The method could also be used for the designof damping controllers for FACTS devices.

Appendix A

Operating conditions for the example system

Table 3shows the load flow data for the base case. Thegenerator powers and terminal voltages remain the same forall four operating conditions.Table 4shows the variationof the loads. The generator and network data is taken from[13, p. 813]. A simple 1.0 (three winding) model is used forall generators. The exciter and AVR are together representedby a first order block with a gainKA = 200 and time constantTr = 0.01 s.

P

:

p

w thepw

C

fourc

R

arts982)

[2] P.S. Rao, I. Sen, Robust pole placement stabilizer design using lin-ear matrix inequalities, IEEE Trans. Power Syst. 15 (February (1))(2000) 313–319.

[3] I. Horowitz, Survey of quantitative feedback theory, Int. J. Control53 (2) (1991) 255–291.

[4] C.A. Jacobson, A.M. Stankovic, G. Tadmor, M.A. Stevens, To-wards a dissipativity framework for power system stabilizer de-sign, IEEE Trans. Power Syst. 11 (November (4)) (1996) 1963–1968.

[5] E. Boje, O.D.I. Nwokah, G. Jennings, Quantitative design of SMIBpower system stabilizers using decoupling theory, in: Proceedings of1999 IFAC World Congress, vol. O, Beijing, China, July 1999, pp.267–272.

[6] P.S. Rao, I. Sen, Robust tuning of power system stabilizers us-ing QFT, IEEE Trans. Control Syst. Technol. 7 (July (4)) (1999)478–486.

[7] E. Eitelberg, J.C. Balda, E. Boje, R.G. Harley, Stabilising SSR os-cillations with a shunt reactor controller for uncertain levels of se-ries compensation, IEEE Trans. Power Syst. 3 (August (3)) (1988)936–943.

[8] A.M. Stankovic, P.C. Stefanov, G. Tadmor, D.J. Sobajic, Dissipativityas a unifying framework for suppression of low frequency oscilla-tions in power systems, IEEE Trans. Power Syst. 14 (February (1))(1999) 192–199.

[9] J. Doyle, A. Packard, The complex structured singular value, Auto-matica 129 (1) (1993) 71–109.

[10] P. Grosdidier, M. Morari, Interaction measures for systems underdecentralized control, Automatica 22 (3) (1986) 309–319.

[11] T.C. Yang, Applying structured singular value to multi-machinepower system stabilizer design, Electric Power Syst. Res. 43 (2)(1997) 113–123.

[July

[ 94.[ l In-

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[ A,

arameters of the stabilizers

Each stabilizer transfer function can be expressed as

ss(s) = KW(s)T1s + 1

T2s + 1

hereW(s) is the wash-out block. The final values ofarameters of the four stabilizers are given inTable 5. Theash-out stage 10s/(10s + l) is common to all four.

losed-loop eigenvalues

Table 6shows the closed-loop eigenvalues for theases considered.

eferences

[1] E.V. Larsen, D.A. Swann, Applying power system stabilizers, PI–III, IEEE Trans. Power Apparatus Syst. 101 (June (6)) (13017–3046.

12] G.J. Balas, J.C. Doyle, K. Glover, A. Packard, R. Smith,µ Analy-sis and Synthesis Toolbox, The Mathworks Inc., Mass., USA,1993.

13] P. Kundur, Power System Stability and Control, McGraw-Hill, 1914] J.M. Maciejowski, Multivariable Feedback Design, Prentice Hal

ternational, UK, 1989.15] P.S. Rao, E. Boje, A note on the use of the structured sin

value in decentralised control, in: 5th International SymposiumQuantitative Feedback Theory and Robust Frequency Domainods, Pamplona, Spain, August 2001, pp. 107–112, ISBN-84-956-3.

16] G.N. Taranto, J.H. Chow, H.A. Othman, Robust redesign of psystem damping controllers, IEEE Trans. Control Syst. Techn(September (3)) (1995) 290–298.

17] Y. Chait, C. Borghesani, Y. Zheng, Single looop QFT designrobust performance in the presence of non-parametric uncertaTrans. ASME, Jl. of Dynamic Systems, Meas. Control 117 (Sepber) (1995) 420–425.

18] X. Yang, A. Feliachi, Stabilization of interarea oscillation mothrough excitation systems, IEEE Trans, Power Syst. 9 (Feb(1)) (1994) 494–502.

19] Borghesani, et al., QFT Toolbox, The Mathworks Inc., MA, USJuly 1994.

a quantitative design approach to pss tuning

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