A Takagi-Sugeno Fuzzy Model of Synchronous Generator … · Zlatka Teˇcec, Ivan Petrovi´c, Jadranko Matuško A Takagi-Sugeno Fuzzy Model of Synchronous Generator Unit for Power

Zlatka Tecec, Ivan Petrovic, Jadranko Matuško

A Takagi-Sugeno Fuzzy Model of Synchronous Generator Unitfor Power System Stability Application

UDKIFAC

621.313.32.072.85.5.6; 5.5.4 Original scientific paper

This paper presents a Takagi-Sugeno (TS) synchronous generator unit model intended for application in an auto-tuning power system stabilizer. The model takes into account all three process variables that can affect synchronousmachine dynamics regarding stability - beside usually used active and reactive power (P and Q) it includes also linereactance (xm). It is shown that the proposed model gives very good results in spite of simple third order localmodel used as the consequent part of the proposed TS structure. This makes the model appropriate for implementa-tion on simple microprocessor platforms. Because P, Q and xm are included as TS model premises, it is enough toidentify parameters of models in consequent part of TS model off-line. In this way possible numerical instability isavoided, which is common to adaptive PSSs that calculate controller’s parameters directly from on-line identifiedplant parameters.

Key words: Modelling, Electrical machines, Takagi-Sugeno model, Power system stabilizer

Takagi-Sugeno model sinkronog generatora namijenjen primjenama u samopodesivim stabilizatorimaelektroenergetskog sustava. U ovome je radu predstavljen Takagi-Sugeno (TS) model sinkronog generatora nami-jenjen primjenama u samopodesivim stabilizatorima elektroenergetskog sustava. Model uzima u obzir sve tri pro-cesne varijable koje utjecu na stanje sinkronog stroja s obzirom na njegovu stabilnost – uz uobicajeno korišteneradnu i jalovu snagu (P i Q) model tako�er ukljucuje i ekvivalentnu mrežnu reaktanciju (xm). Pokazano je da pred-loženi model daje vrlo dobre rezultate unatoc jednostavnom modelu treceg reda u posljedicnom dijelu TS strukture.To cini predloženi model prikladnim za primjenu na jednostavnim mikroprocesorskim platformama. Buduci da suP, Q i xm uzrocne varijable TS modela, dovoljno je off-line identificirati njegov posljedicni dio. Na taj su nacinizbjegnuti problemi numericke nestabilnosti, karakteristicni za adaptivne strukture PSS-a koji parametre regulatoraproracunavaju na temelju on-line identificiranog modela procesa.

Kljucne rijeci: modeliranje, elektricni stroj, Takagi-Sugeno model, stabilizator elektroenergetskog sustava

1 INTRODUCTION

Power system stabilizers (PSSs) are used as a part ofthe synchronous generator automatic voltage regulators(AVRs) in order to damp out the low frequency local andsystem power oscillations. Today’s AVRs commonly in-clude so called conventional PSSs (CPSS), where the termconventional in this context means nonadaptive. Althoughmodern types of CPSS like IEEE PSS2B and PSS4B [1]can be very effective in damping both local and systempower oscillations, tuning of these two types of PSS canbe time consuming and requires expert knowledge dur-ing commissioning. Additionally, problems with CPSSmay occur when a CPSS properly tuned for one operatingregime, has to operate in another very different regime.

To resolve these problems a number of methods, mainlyin the field of adaptive control, has been proposed. Someof the commonly used methods are gain scheduling, indi-

rect and direct adaptive control using artificial intelligence(mostly neural networks and fuzzy logic) [2]. The largestgroup of the proposed adaptive PSSs consists of indirectselftuning power system stabilizers [2],[3-8]. Those PSSsprovide better dynamic performance over a wide rangeof operating conditions, but they suffer from the signifi-cant drawback of requiring model parameter identification,state observation and feedback gain calculations in real-time. Errors in parameter identification can significantlydegrade performance and eventually cause the instabilityof the overall control algorithm. The additional drawbackof a number of adaptive stabilizers presented in the lasttwo decades is the fact that they require implementationin floating point arithmetic despite the facts that today’sdigital AVRs, within which PSSs typically have to be im-plemented, are mostly implemented in 16 or 32-bit fixedpoint processors. Thus additional hardware modifications

ISSN 0005-1144ATKAFF 51(2), 127–137(2010)

AUTOMATIKA 51(2010) 2, 127–137 127

A Takagi-Sugeno Fuzzy Model of Synchronous Generator Unit for Power System Stability Application Z. Tecec, I. Petrovic, J. Matuško

of the existing AVRs need to be done in order to imple-ment such algorithms. The mentioned drawbacks could bereason why, despite of numerous adaptive PSS proposedin the last two decades, vary few of them have been usedin the practice. The problems related to the conventionaland adaptive PSSs have motivated researchers to recentlypropose simpler solutions [9], [10]. Our work is also moti-vated by this paradigm.

In this paper we propose a relatively simple Takagi-Sugeno (TS) fuzzy model of synchronous generator unit,which is suitable for use in an easy-to-implement autotun-ing power system stabilizer. The motivation of using TSfuzzy model is rooted in its ability to accurately modelnonlinear system behavior with a relatively small num-ber of linear models [11, 12]. Smooth transitions betweenthese linear models are achieved by using fuzzy rules. Theproposed TS model combines active power, reactive powerand line reactance signals as premise variables. Due to thefact that these three signals completely describe the stateof synchronous generator units [13, 14], the proposed PSScan use off-line TS model identification process and pro-vide satisfactory performance over the wide range of oper-ation conditions. Similar fuzzy solutions, that can be foundin literature, mainly use only signals of active and reactivepower as premise variables [6, 15], and therefore they re-quire on-line parameter estimation.

The rest of the paper is structured as follows. Section2 describes TS fuzzy model structure, the mathematicalmodel of synchronous generator unit and developing pro-cess of the proposed TS model of the synchronous genera-tor unit. Section 3 shows simulation results of the proposedTS synchronous generator unit model as well as simulationresults of the newly proposed autotuning PSS utilizing theproposed TS model. Conclusions are given in Section 4.

2 TAKAGI-SUGENO MODEL OFSYNCHRONOUS GENERATOR UNITS

2.1 Takagi-Sugeno Fuzzy Model Structure

A Takagi-Sugeno (TS) structure can be used both as aplant model [6], or as a regulator, [15, 16]. In Takagi-Sugeno model approach [11, 12] a plant is described by aset of simple local linear regression models, each valid forparticular operating area. The plant model output is a com-bination of local models outputs according to fuzzy rules,which are defined with variable determining operating area(premise variables). The fuzzy rules are structured as fol-lows:

Ri: if[x1(k) is F i

1

]& . . . &

[xnx(k) is F i

nx

]then yi(k + 1) = pi

0 + pi1m1(k) + pi

nmmnm(k),i = 0, 1, · · · , nr, (1)

where:

Ri – i-th inference rule,

xj – j-th premise variable,

Fj – fuzzy set defined on the universe of discourse ofthe variable xj , used in i-th inference rule

yi – output of the i-th input/output (I/O) local models

p = [a1a2 . . . anab1b2 . . . bnbc1c2 . . . cnc] (2)

– parameters of i-th I/O local model

mT (k) = [−y(k) . . . − y(k − na + 1) u(k − d) .... . . u(k − d − nb + 1) ζ(k − 1) . . . ζ(k − nc)]

(3)

– regression vector where u represent input of themodel, y output of the model and d is process dead time,

nr – number of fuzzy rules.

For given values of the premise variables xj(k), the finaloutput of the fuzzy plant model y is inferred by taking theweighted average of the local model outputs yi:

y(k + 1) =

nr∑i=1

μi (x(k)) yi(k + 1)

nr∑i=1

μi (x(k))(4)

where:

μi (x(k)) =nx∧

j=1μi

j [xj(k)] (5)

are membership functions of the fuzzy set F ij .

It means that a local model in the consequent part ofeach fuzzy rule describes plant dynamics for the operatingarea depending on the premise part. A value of premisesdefines a rule weigh (5), which indicate the contribution ofthe i-th rule to the model output.

In a process of defining structure of local linear modelsin consequent part of TS model as well as premise vari-ables, mathematical model of the plant has to be taken intoaccount.

2.2 Physical Mathematical Model of the GeneratorUnit

As a starting point for the design of the power systemstabilizer based on fuzzy model, machine-infinite bus sys-tem model will be used (Fig. 1). Such a model is generallydescribed by the 7th order nonlinear model given by the

128 AUTOMATIKA 51(2010) 2, 127–137


Fig. 1. Infinite bus system

following equations [17], [18]:

Vd = −ra · id +1ωs

· dψdΣ

dt− ω · ψqΣ

Vq = −ra · iq +1ωs

· dψqΣ

dt+ ω · ψdΣ

e = xad · if +xad

ωsrf· dψf

dt

0 = rD · iD +1ωs

· dψD

dt

0 = rQ · iQ +1ωs

· dψQ

dtdδ

dt= ωs(ω − 1)

dω

dt=

1Tm

· (Tt − Tel)

(6)

where V stands for stator voltage, i for stator currents,ωs for synchronous rotor speed, r for inductance, x for re-actance, Ψ for magnetic flux, ω for rotor speed, δ for loadangle and T for torques. Subscripts are defined as follows:d and q stands for Parks d and q axes of stator winding andD and Q for Parks d and q axes of damping winding.

The transient effects in stator and the effects of ro-tor amortization windings can be neglected, and thereforethe nonlinear model (6) can be reduced to the third ordermodel given with the following equations:

dEq′

dt=

ωsrfx2ad

ωxdΣ′x2f

Vm cos δ + ωse−

−ωsrfxdΣxad

xdΣ′xfEq′

dω

dt=

1Tm

[Tt − 1

ωxdΣ′Eq′Vm sin δ+

+xdΣ′ − xqΣ

ω2xdΣ′xqΣ

Vm2

2sin 2δ − D(ω − 1)

]dδ

dt= ωs(ω − 1)

(7)

Although non-linear relationships are present in powersystems, electromechanical oscillations don’t have nonlin-ear character [19]. Therefore, the linear, small perturbationmodel of the machine-infinite bus system, originally devel-oped by Heffron and Philips [20], can be used in the pro-cess of developing TS model intended for power stabilityapplications.

Linearization of the model (7) gives well-known Hef-fron and Filips model [18] given by equations (8) andgraphically depicted in Fig. 2.

ΔEq′1 + sK3Td0′

K3= −K4Δδ + Δe + K8ΔVm

Δω(s + D

Tm

)= 1

Tm[ΔTt − K2ΔEq′−

−K1Δδ − K9ΔVm]ΔVg = K5Δδ + K6ΔEq′ + K7ΔVm

(8)

It can be seen that the model relates the variables ofelectrical torque, speed, angle, terminal voltage, field volt-age and flux linkages. Constants K1 – K6 depend on themachine system parameters (inertias, reactances, time con-stants) and the operating condition (i.e. the value of activepower, reactive power and line reactance).

From the block diagram shown in Figure 2, input-outputmodel needed for TS model consequent part can be ob-tained. If we want to use TS model in PSS application thenPSS output signal ΔuPSS has to be used as the model in-put, while any signal that contains active power oscillationscan be used as the model output. In the proposed solution,the signal of integral of accelerating power ΔPacc has beenchosen as the model output. This signal is derived withinsignal processing module from frequency of the terminalvoltage and a high-pass filtered integral of electrical poweras in [21]. This signal, which actually represents estimatedrotor speed signal (Δω), is used due to good establishmentin conventional PSS2B type of PSS.

As can be seen from Fig. 2 synchronous generatorinput-output model with PI type AVR included, is of thethird order. Namely, with PI type AVR included, the closed

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Fig. 2. Small perturbation transfer function block diagram

of the machine-infinite bus system

AUTOMATIKA 51(2010) 2, 127–137 129


loop from ΔuPSS to the signal of electrical power ΔPel

can be described as a second order transfer function. Toobtain signal of rotor speed Δω, which is in the proposedstructure of PSS chosen as the model output signal, signalof ΔPel have to pass through the summation and the inte-grator. This makes transfer function from ΔuPSS to Δωto be of the third order.

2.3 Premise Part of the TS Model of Generator Unit

As aforementioned, K1 – K6 depend on the machinesystem parameters and the operating condition. Takinginto account the fact that machine system parameters don’t(or slightly) vary with time, constants K1 – K6 dependonly on the operating condition defined with three vari-ables: active power (P), reactive power (Q) and line reac-tance, represented with value of equivalent reactance (xm).These three variables are therefore used as premise vari-ables in TS model.

Signals P and Q are available in standard AVR and canbe used directly as inputs to the TS model. On the otherhand equivalent reactance xm is typically not available,and therefore it should be estimated.

The equivalent reactance xm represents the reactancesof the transmission lines between unit terminals and infi-nite bus. An estimation method presented in [22] is usedwithin the TS model. In that method the equivalent re-actance, seen from the unit terminals, is evaluated usingthe local measurements (terminal voltage V, P and Q). Theidentification procedure starts whenever dynamic oscilla-tions in those measurements are detected. After a largedisturbance, which necessarily follows change of xm, thenew reactance is estimated in relatively short time (200-500 ms). The estimation algorithm doesn’t give accuratevalue of the reactance but chooses the closest one to itamong predefined values. The algorithm can be summa-rized in the following steps:

1. Phase angle between terminals voltage Vg and infinitebus voltage Vm is calculated from expression

Δδe =τ

1 + τsΔf ≈ −ω2

n

2π

τ

1 + τsΔT, (9)

where Δf and ΔT are deviation from nominal valueof the terminal voltage frequency and period, respec-tively. Constant τ is the filter time constant whichvalue has to be big enough to keep the output of withinreasonable bounds.

2. If signal Δδe, calculated in step 1, reaches prede-fined value Δδemax algorithm activates signal Tran-

sient state.

3. For n predefined values of xm the value of phase angleδe is calculated from equation:

δe = tan−1

⎛⎝ PU2

g

xm− Q

⎞⎠ = f (xm) (10)

is derived from equations for the power conditionsalong a line between the power plant and the load cen-tre:

P =UgUm

xmsin δe (11)

Q =U2

g

xm− UgUm

xmcos δe (12)

4. Difference between minimum and maximum value ofall Δδen signals and measured signal δe is calculatedfor the next 1/fn seconds, where fn represents thesynchronous generator nominal frequency. The clos-est agreement between the phase-angle variation, cal-culated with an assumed xm value, and the measuredphase-angle variation δe determines the xm value forgenerator. As the result the algorithm gives the num-ber that defines line reactance.

The fact that the reactance xm cannot be identified dur-ing steady state operation is only a minor handicap. A sig-nificant change of xm is usually associated with a distur-bance anyway. The accuracy of the identification dependson the number of predefined values of xm. Normally, theselection of 2 to 4 values covering an appropriate range ismostly adequate.

To define membership functions of selected premisevariables their physical characters have to be considered.Signals of active and reactive power change smoothlythrough the operating area of synchronous machine dur-ing normal operation. Relationship between parametersK1 to K6 and values of P and Q is approximately linearduring those changes. On the other hand, equivalent reac-tance xm changes its value discontinuously (between 2-4different values). Additionally, relationships between pa-rameters K1 to K6 and equivalent reactance xm is gener-ally nonlinear. Considering that, membership functions ofpremises are defined as follows.

Membership functions of P and Q have linear shapeswith operating range divided into two or three areas (i.e.fuzzy sets). By validation and comparison of differentnumbers of areas we have found that two areas per vari-able are sufficient: Phigh, Plow, and Qhigh, Qlow (Fig.3). Equivalent reactance xm need to have discontinuousmembership function. By analyzing values of reactancesmeasured on the real power plants during the annual main-tenance, we have found that also two different values are

130 AUTOMATIKA 51(2010) 2, 127–137


sufficient: low (0) and high (1) as shown in Fig. 3. Estima-tion method of xm will therefore give one of two possiblepredefined xm values. One of them will take the value ofxm measured (i.e. estimated) during commissioning, andthe second one will be 0.2 p.u. greater or smaller than mea-sured (depending on the measured value).

µPlow µPhigh1

P(k)Pmin= 0.5 pu

0Pmax=

1pu

µQlow µQhigh1

Q(k)Qmin= -0.2 pu

0Qmax= 0.3pu

µxlow µxhigh1

xm(k)xm_low ˜ 0.2

0xm_high ˜ 0.45

Fig. 3. Membership P, Q and xm functions

Combination of those premise variables gives thepremise part of the proposed TS model:

Ri: if [P (k) is Pj ] & [Q(k), is Qj ] & [xm(k) is xj ]then yi(k + 1) = pi

0 + pi1m1(k) + pi

nmmnm(k),i = 1, · · · , 8,j = low, high. (13)

2.4 Consequent Part of the TS Model of Generatorunit

As discussed in 2.2, the third order linear model can sat-isfactorily describe the generator unit behavior. Thereforewe use the third order autoregressive models with exoge-nous input (ARX models) as the consequent part of the TSmodel. The used ARX models have the following struc-ture:

A(q−1, k)y(k + d + 1) = B(q−1, k)u(k) + c(k) (14)

where number of parameters in A and B polynomials are:na=3, nb=1 and nc=0. This structure is named as ARX310.This simple structure of the local models in consequentpart of the TS model makes the model computationallysimple and implementable on simple processor platforms.

By inserting ARX310 models as the consequent part in(13), the final structure of the TS fuzzy model of the syn-chronous generator with AVR included is obtained:

R1: if [P (k) is Plow] & [Q(k) is Qlow] & [xm(k) is xlow]then y1(k + 1) = a1

1y(k) + a12y(k − 1)

+a13y(k − 2) + b1

1u(k) + c1

R2 : if [P (k) is Plow] & [Q(k) is Qhigh] & [xm(k) is xlow]then y2(k + 1) = a2

1y(k) + a22y(k − 1)

+a23y(k − 2) + b2

1u(k) + c2

R3 : if [P (k) is Phigh] & [Q(k) is Qlow] & [xm(k) is xlow]then y3(k + 1) = a3

1y(k) + a32y(k − 1)

+a33y(k − 2) + b3

1u(k) + c3

R4 : if [P (k) is Phigh] & [Q(k) is Qhigh] & [xm(k) is xlow]then y4(k + 1) = a4

1y(k) + a42y(k − 1)

+a43y(k − 2) + b4

1u(k) + c4

R5 : if [P (k) is Plow] & [Q(k) is Qlow] & [xm(k) is xhigh]then y5(k + 1) = a5

1y(k) + a52y(k − 1)

+a53y(k − 2) + b5

1u(k) + c5

R6 : if [P (k) is Plow] & [Q(k) is Qhigh] & [xm(k) is xhigh]then y6(k + 1) = a6

1y(k) + a62y(k − 1)

+a63y(k − 2) + b6

1u(k) + c6

R7 : if [P (k) is Phigh] & [Q(k) is Qlow] & [xm(k) is xhigh]then y7(k + 1) = a7

1y(k) + a72y(k − 1)

+a73y(k − 2) + b7

1u(k) + c7

R8 : if [P (k) is Phigh] & [Q(k) is Qhigh] & [xm(k) is xhigh]then y8(k + 1) = a8

1y(k) + a82y(k − 1)

+a83y(k − 2) + b8

1u(k) + c8

(15)

The final output of the TS fuzzy model is inferred bytaking the weighted average of the local models outputs yi

Thus, the output signal of model (13) can be defined as[23]:

y(k + d + 1) =nr∑

i=1

vi(k)yi(k + d + 1) (16)

where:

vi(k) =μi (x(k))

nr∑j=1

μj (x(k)). (17)

Considering only membership functions of P and Q, thepremise structure of the proposed TS model gives a surfacelimited with four flat surfaces (model 1 to 4 in Figure 4)representing four linear models. Involving xm as premisevariable of the TS fuzzy model, one more set of four linear

AUTOMATIKA 51(2010) 2, 127–137 131


1

3

2

4

Fig. 4. P-Q premise surface

models is added. In Fig. 4 this would look like anothersurface placed under the existing one. If we insert linearmodels (15) in equation (16) the following expression isobtained:

y(k + d + 1) =nr∑

i=1

vi(k)

{−

na∑j=1

aijy(k − j + d + 1)+

+nb∑

j=1

biju(k − j + 1) + ci

}(18)

which can also be written as:

y(k + d + 1) = −na∑

j=1

[nr∑

i=1

aijv

i(k)]

y(k − j + d + 1)+

nb∑j=1

[nr∑

i=1

bijv

i(k)]

u(k − j + 1)+nr∑

i=1

civi(k).

(19)The final expression for synchronous generator TS modelcan be defined from (19) as:

A(q−1, k)y(k + d + 1) = B(q−1, k)u(k) + c(k) (20)

where:A(q−1,k) = 1 +

na∑j=1

aj(k)q−j

B(q−1,k) =nb∑

j=1

bj(k)q−j+1.(21)

Expressions for ai and bi are given with the followingequations:

aj(k)=nr∑

i=1

aijv

i(k),

bj(k)=nr∑

i=1

bijv

i(k).(22)

Obtained model (20) is time variant linear model thatcan be used within classical control scheme, like pole

placement. Its parameters are calculated on-line accord-ing to (22) by applying previously identified values of theTS model consequent part parameters ai

j and bij .

2.5 Identification Steps of the Consequent PartModels

Parameters ai1, ai

2, ai3 and bi

1 of each of the eight ARXmodels of the TS consequent part can be identified by us-ing recursive least squares algorithm (RLS), [24]. Identifi-cation is performed by an embedded automatic procedure,which start identification of a local model only when thegenerator is in corresponding operating point defined bythe TS model premise part parameters. Other two condi-tions that have to be fulfilled to start identification are: 1)synchronous machine has to be in steady state, and 2) iden-tification error signal (difference between measured andestimated output model signal) have to be greater then apredefined level (i.e. 4%). Identification procedure endssuccessfully when synchronous machine passes throughall eight operating areas defined by the TS premise part.Parameter estimation process stops whenever one of thefollowing conditions occurs: (i) transient state is detected,i.e. large change of the line reactance xm occurs; (ii)synchronous generator run out of operating point definedby TS model premise parts or (iii) identification error be-comes less than defined. The last condition indicates thesuccessful end of the identification procedure. Graphicallythe identification procedure is shown in Fig. 5.

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Fig. 5. Identification procedure of the TS model consequent

part

Changes of the line reactance xm are usually not pos-sible during the PSS commissioning, and therefore onlyparameters of four consequent ARX models, which corre-spond to the actual value of the line reactance, are identi-fied at that time. Parameters of other four consequent ARXmodels are identified during normal operation of the PSS,after the xm estimation algorithm detects sufficiently largechange of line reactance xm.

132 AUTOMATIKA 51(2010) 2, 127–137


3 SIMULATION RESULTS

The nonlinear model of a real power plant is imple-mented in MATLAB/Simulink simulation environment forthe purpose of testing proposed TS model. It consists ofa single machine equipped with control systems and con-nected to infinite bus [25]. Synchronous machine blockfrom MATLAB/Simulink SimPowerSystems [26] is used torepresent a synchronous generator connected to an infi-nite bus (Three-phase programmable voltage source block)through a three-phase transformer and transmission lines.The ability of changing the line reactance is modelled aswell. Equations used to simulate AVR and conventionalPSS (IEEE type PSS2B) are given in [27].

3.1 TS model validation

MATLAB System Identification Toolbox is used to ver-ify structure of the TS model consequent part. Verifica-tion was done through the identification of the simulatednonlinear plant by several different linear ARX structures.During the identification experiment a Pseudo Random Bi-nary Signal (PRBS) with T = 0.05 s, n = 11 and ampli-tude 0.003 p.u. around the steady-state value was usedas input signal uPSS to the nonlinear plant model. Thesignal of integral of accelerating power was recorded asthe output signal. The results of the identification processare summarized in Table 1. The numbers in the end ofARX model names means number of ARX model parame-ters: ARX[na][nb][nc] where parameters na, nb and nc aregiven in (21).

Table 1. Fits in % of investigated ARX modelsModel ARX310 ARX422 ARX720 ARX210na, nb, nd 3,1,0 4,2,2 7,2,0 2,1,0Fit [%] 99,40 99,62 99,81 98,41

It can be seen that difference between measured out-put signal (real Pacc) and ARX310 structure is only 0.6%.Comparison of Bode diagrams of considered ARX struc-tures is shown in Figure 6. From Bode diagrams it can beseen that matching between ARX310 and ARX720 (whichrepresent best matching to nonlinear plant model) in inter-esting frequency domain (0.5 to 3 Hz) is satisfying.

Figure 7 shows comparisons between the output signalof the proposed TS model Pacc,ident and signal of inte-gral of acceleration power obtained on simulated nonlin-ear power plant model (Pacc). Changes in reactive powerQ was realized by applying step changes in AVR referencevoltage. Line reactance had low value (xlow). It can beseen that identification error in all operating condition isless then 4%. Such level of identification error is accept-able for autotuning power system stabilizer application.

3.2 Validation of the Identification ProcedureThe identification procedure is presented in Fig. 8. The

graph a) shows two input identification signals. Signal u(t)is signal applied to AVR when identification process is ac-tive and signal test_PRBS(t) is applied to AVR when iden-tification process is not active in order to estimate errorlevel. Both signals are of PRBS type with small enoughamplitude and therefore they don’t induce any instabilitiesof the synchronous generator.

100 101

10-

3

10-

2

10-

1

-500

-400

-300

-200

-100

0

arx 310arx 422arx 720arx 210

L [

dB]

? [rad/s]

f

[º]

100 101

Fig. 6. Bode diagrams of investigated ARX models

Pac

c [p.

u.]

20 30 40 50 60 70 80 90-10

-8

-6

-4

-2

0

2

4

6

x10-4

Pacc ident

Pacc

20 30 40 50 60 70 80 90

-0.2

0

0.2

0.4

0.6

0.8

1

P, Q

[p.

u.]

P

Q

20 30 40 50 60 70 80 90-3

-2

-1

0

1

2

t [s]

Iden

t. er

ror

[p.u

.]

x 10-5

Fig. 7. TS model results, low xm value

AUTOMATIKA 51(2010) 2, 127–137 133


The graphs b) and c) show values of active and reac-tive power, respectively, and from them the operating pointcan be restored. It can be seen that synchronous gener-ator passes through the four operating points defined bypremise part of the first four rules of TS model (15). Thegraph d) represents measured and estimated Pacc(t) sig-nals together with Ident_enable(t) signal which representsthe state of the identification procedure. High level ofIdent_enable(t) signal indicates that the identification pro-cess active .

The identification procedure can be shorten if the com-missioning staff make the synchronous generator to oper-ate in operating points defined by TS model premise part.However if it is not possible to pass through all four oper-ating points during the commissioning process, the identi-fication procedure will automatically continue when a novisited operating point becomes active later during normaloperation. When identification of all four ARX models forcurrent line reactance value is done, the process of auto-matic tuning of PSS according to the pole placement con-trol algorithm is performed and PSS is turned on.

If the line reactance estimation algorithm detects a suf-ficiently large line reactance change during the normal op-eration, the identification procedure is restarted in order toidentify the remaining four TS consequent ARX models.The identification is performed during normal plant opera-tion, when the system is in the steady-state and when it op-erates in operating points defined by TS premise parts. Du-ration of this second identification process can be shortenif an autotuning PSS structure embeds statistics of the syn-chronous unit the most frequent operating points. This waythe operating point in TS model premise parts could be au-tomatically changed to points in which synchronous unitoperates the most frequently. Until the identification pro-cedure of remaining four linear models is done, PSS oper-ates with parameters optimally tuned during commission-ing.

3.3 Proposed TS Model within an Autotuning PSSWhen the identification of the TS model consequent part

(15) is done and the parameters of the linear model (20) areobtained, an appropriate PSS can be designed. To calcu-late PSS parameters, the pole placement control strategy isused in this paper [28]. The structure of the proposed pole-placement autotuning PSS is shown in Fig. 9. It should besaid that other control approaches can be used as well.

The pole placement autotuning PSS based on the pro-posed TS model is implemented in the TI TMS320F2812fixed-point DSP [29]. Its behavior has been investigatedin comparison to the PSS2B PSS controller, through volt-age reference step changes. Other validation cases like re-sistance to reactive power modulation when the mechan-ical power changes and behavior when the line reactance

50 100 150 200 250 300 350 400 450

50 100 150 200 250 300 350 400 450

Pacc(t)Paccident(t)Ident enable(t)

u(t)test PRBS(t)

t [s]

50 100 150 200 250 300 350 400 450

100 150 300

-0.01

0

0.01

u(t)

[p.

u.]

-0.2

0

0.2

0.4

Q(t

) [p

.u.]

0.6

0.8

1

P(t

) [p

.u.]

-4

-3

-2

-1

0

1

2

x 10-4

Pac

c(t)

[p.

u.]

250 350 400 450200

a)

b)

c)

d)

Fig. 8. Identification procedure

Vg RLS MODEL PARAMETERS

IDENTIFICATION

CONTROLLER

POLE PLACEMENT

GRID AVRSGVg

+- +

UREF

uPSS(k)

uPSS(k)

P, Q

INPUT SIGNALS PROCESSING &

FUZZYFICATION

Takagi Sugeno MODEL

PRBS

0

PSS off or xm change

Ident. enable

uPSS(k)DEFUZZYFICATION

MEASUREMENTS

1 1( , ) ( 1) ( , ) ( ) ( )q k y k d q k u k c k− −+ + = +A B

Fig. 9. The structure of the proposed autotuning PSS; SG –

synchronous generator, AVR – automatic voltage regulator,

UREF SG voltage reference

changes can be found in [29]. To perform fair comparisonthe conventional PSS2B was optimally tuned for nominaloperating point of synchronous generator (P = 1 p.u., Q =0, xm =xm_low).

Figures 10 and 11 show: a) signals of active power, b)signals of reactive power and c) control signals of PSSs.

134 AUTOMATIKA 51(2010) 2, 127–137


560 565 570 575

0.99

1

1.01

1.02

560 565 570 575-0.2

-0.15

-0.1

-0.05

0

560 565 570 575-0.015

-0.01

-0.005

0

t [s]

P(t)

[p.u

.]Q

(t) [p

.u.]

u(t)

[p.

u.]

P(t) APSSP(t) no PSSP(t) PSS2B

Q(t) APSSQ(t) PSS2BQ(t) no PSS

u(t) APSSu(t) PSS2Bno PSS

a)

b)

c)

Fig. 10. System responses on step changes of the voltage

reference value - nominal operating point

For the sake of comparison responses obtained by bothproposed autotuning PSS and PSS2B are shown. Whileboth PSSs provide similar responses in nominal operat-ing point at which they were tuned (Fig. 10) and whichis characterized by low value of the line reactance xm_low,the proposed autotuning PSS provides better responses inanother operating point characterized by high value of theline reactance xm_high.

4 CONCLUSIONSIn the paper a new Takagi-Sugeno synchronous gener-

ator unit model intended for PSS application is proposed.The model takes into account all three variables that canaffect synchronous machine behavior regarding stability:P, Q and xm. Therefore on-line estimation of model pa-rameters becomes unnecessary. Probably only one or twoidentification processes during life time will be sufficientto obtain model parameters needed for design of an auto-tuning PSS.

Because of using the all three variables, proposed modelis particularly suitable if direct calculation of PSS param-eters (i.e. self tuning schemes) and possible numerical in-stability common to those schemes are to be avoided. Theproposed Takagi-Sugeno model can be also used in otherapplications involving synchronous machine control.

Simulation results show that an autotuning PSS utiliz-ing the proposed TS model obtains very good stabilizing

566 570 574 578

0.1

0.15

0.2Q(t) APSSQ(t) PSS2BQ(t) no PSS

Q(t

) [p

.u.]

566 570 574 578

0.99

1.01

1.02

P(t

) [p

.u.]

566 570 574 578-0.01

0.01

0.02

u(t)

[p.

u.]

t [s]

1

0

P(t) APSSP(t) no PSSP(t) PSS2B

u(t) APSSu(t) PSS2Bno PSS

c)

a)

b)

c)

Fig. 11. System responses on step changes of the voltage

reference value – out of nominal operating point

effects. In comparison to optimally tuned conventionalPSS2B, it shows better behavior in operating points thatdiffer from point for which is PSS2B optimally tuned.

ACKNOWLEDGMENT

This research has been supported by the Ministry of Sci-ence, Education and Sports of the Republic of Croatia un-der grant No. 036-0363078-3018 and KONCAR ElectricalEngineering Institute, Croatia.

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Zlatka Tecec received her B.Sc.EE and PhD (inthe field of power system stabilizer) degree fromthe Faculty of Electrical Engineering and Com-puting (FER) in Zagreb, in 2004 and 2010 re-spectively. Since 2004 she has been employedin company Koncar-Electrical Engineering Insti-tute, Zagreb, in the Department of Power elec-tronic and control. Since then she is working inSection for Embedded Systems dealing with de-velopment, integration and testing of embeddedhardware and software applied in the power en-

gineering and rail vehicles. In the field of electric power engineeringshe has been working on development, implementation, maintenance andcommissioning of digital voltage regulators and power system stabilizers.She published more than 20 scientific papers and practical project studies.She has been a licensed engineer of electrical engineering since 2008. AsFER external associate assistant she participated in several courses.

136 AUTOMATIKA 51(2010) 2, 127–137


Jadranko Matuško received his BSs, MSc andPhD degrees in electrical engineering and com-puting from the University of Zagreb, in 1999,2003 and 2008, respectively. Currently he isan Assistant Professor at Department of Electri-cal Machines, Drives and Automation at Facultyof Electrical Engineering and Computing, Za-greb. His scientific interests include identifica-tion and estimation techniques, intelligent controlin mechatronic systems.

Ivan Petrovic received B.Sc. degree in 1983,M.Sc. degree in 1989 and Ph.D. degree in 1998,all in Electrical Engineering from the Faculty ofElectrical Engineering and Computing (FER Za-greb), University of Zagreb, Croatia. He hadbeen employed as an R&D engineer at the In-stitute of Electrical Engineering of the KoncarCorporation in Zagreb from 1985 to 1994. Since1994 he has been with FER Zagreb, where he iscurrently the head of the Department of Controland Computer Engineering. He teaches a number

of undergraduate and graduate courses in the field of control systems andmobile robotics. His research interests include various advanced controlstrategies and their applications to control of complex systems and mobilerobots navigation. Results of his research effort have been implementedin several industrial products. He is a member of IEEE, IFAC - TC onRobotics and FIRA - Executive committee. He is a collaborating memberof the Croatian Academy of Engineering.

AUTHORS’ ADDRESSESZlatka Tecec, Ph.D.Koncar – Electrical Engineering Institute,Fallerovo šetalište 22, 1000 Zagreb, Croatiaemail: [email protected]. Prof. Jadranko Matuško, Ph.D.Department of Electrical Machines, Drives and Automation,Prof. Ivan Petrovic, Ph.D.Department of Control and Computer Engineering,Faculty of Electrical Engineering and Computing,University of Zagreb,Unska 3, 10000 Zagreb, Croatiaemails: [email protected], [email protected]

Received: 2010-02-22Accepted: 2010-05-05

AUTOMATIKA 51(2010) 2, 127–137 137

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A Takagi-Sugeno Fuzzy Model of Synchronous Generator … · Zlatka Teˇcec, Ivan Petrovi´c, Jadranko Matuško A Takagi-Sugeno Fuzzy Model of Synchronous Generator Unit for Power