online updating of synchronous generator …heffron phillips model of the smib system, is shown in...

Online Updating of Synchronous GeneratorLinearised Model Parameters and PSS Tuning

Piyush Warhad Pande, S. Chakrabarti, and S. C. SrivastavaDepartment of Electrical Engineering, Indian Institute of Technology Kanpur

Email: [email protected], [email protected], [email protected]

Abstract—A new approach for the online tuning of Power Sys-tem Stabilizer, using synchrophasor measurements, is presentedin this paper. The methodology involves estimation of Theveninequivalent of the system using the synchrophasor data obtainedfrom Phasor Measurement Unit installed at the generator bus.The Thevenin equivalent so obtained is then used to updatethe parameters of the small signal model of the generator andsubsequently the PSS parameters are tuned using residue basedmethod. The scheme is validated on real time digital simulatorfor WSCC 9 bus system.

Index Terms—Power system stabilizer, phasor measurementunit, residues, Thevenin equivalent

I. INTRODUCTION

LOW Frequency Oscillations (LFOs) are frequently en-countered in power systems. The oscillations are gener-

ally classified as local modes, inter-area modes, and controlmodes depending on the frequency of oscillations and the par-ticipating states of the system [1]. These low frequency oscilla-tions are detrimental to the system operation and, therefore, area major challenge from the perspective of system operation.Utilities usually install Power System Stabilizers (PSS) toensure reasonable damping of the LFOs. PSS has emergedas a vital component of the generator excitation system andhelps in maintaining the integrity of the interconnected system.

The conventional strategy adopted by utilities to tune thePSS involves setting of the PSS parameters based on anassumed Single Machine Infinite Bus (SMIB) model of thegenerator [3], [2]. Moreover, the first order model of theAutomatic Voltage Regulator (AVR) is considered for theexcitation system [3], [2]. The combined small signal dynamicmodel is called Heffron Phillips model, which involves sixconstants that govern the position of the poles of the systemin the S-plane. These six constants, however, depend uponthe operating point, i.e, the generator power output and theThevenin equivalent of the line between the generator and theinfinite bus [1], [3], [4]. Based on an assumed generator modeland the Thevenin equivalent of the line, the settings of the PSSare derived. These fixed settings are usually kept constant untilthe PSS is tuned again in the context of a major change in thesystem operating condition.

The fixed setting PSS provides good damping for the regionsin the vicinity of the operating point for which the PSS istuned. For other operating conditions, the system may exhibitpoor damping [5]–[7].

An adaptive PSS, that takes into consideration the changein the operating point and also the change in the Theveninimpedance of the external system can therefore provide betterdamping to the rotor modes.

With the advent of the synchrophasor technology, theThevenin equivalent of the system can be tracked at a bus.Considerable work has been reported for online trackingof Thevenin equivalent. Some of the prominent works arereported in [8]–[11]. The constants, ‘K1−K6’, in the HeffronPhillips model of the generator, can be updated if the Theveninequivalent of the system is known at the generator bus. Theseconstants so obtained can then be used to tune the PSS.The Thevenin impedance can be assumed as the externalimpedance and the Thevenin voltage can be assumed as thevoltage of the equivalent external system.

This paper proposes a methodology to tune the PSS onlineusing the Thevenin impedance of the external system, obtainedusing the least square algorithm [8]. The Thevenin equivalentso obtained is then used to update the values of ‘K1 − K6’in the Heffron Phillips model of the generator. Subsequentlythe residue based algorithm [12] is used to tune the PSSparameters. The methodology is tested on WSCC 9 Bussystem and the closed loop implementation is done on thereal time digital simulator.

II. GENERATOR MODEL

The generator is modeled considering an SMIB approxi-mation of the generator and power system dynamics. Thegenerator is assumed to be connected to an infinite bus.The generator has a static excitation system and an AVRthat has a high gain and fast response. To study the smallsignal stability, the damper windings are neglected as theyusually produce positive damping torques on the rotor. Thegenerator is represented using one axis flux decay modeland the stator resistance is assumed to be negligible. Withthese simplifying assumptions, and using the variables listedin Table I, the generator and exciter dynamics are modeledusing the differential equations shown below

E′q = − 1

T ′d0(E′q + (Xd −X ′d)Id − Efd) (1)

δ = ω − ωs (2)

ω =ωs2H

[TM − (E′qIq + (Xd −X ′d)IdIq +D(ω − ωs))] (3)

978-1-5386-6159-8/18/$31.00 c© 2018 IEEE

Proceedings of the National Power Systems Conference (NPSC) - 2018, December 14-16, NIT Tiruchirappalli, India

TABLE IACRONYMS AND ABBREVATIONS FOR MATHEMATICAL FORMULATIONS

δ Generator rotor angle

ω Generator rotor speed

ωs Synchronous speed of the generator

E′q Quadrature axis induced voltage behind transient reactance

Efd Field Voltage

H Machine p.u inertia constant

TM Generator mechanical power input

D Damping coefficient

T ′d0,T ′

q0 Direct and quadrature axis transient flux decay constant

Xd,X′d Direct axis synchronous, and transient reactance

Id,Iq Direct and quadrature axis stator current

KA,TA Gain, and time constant of the voltage regulator

Vref Desired terminal voltage

Vt Actual terminal voltage

TAEfd = −Efd +KA(Vref − Vt) (4)

The generator and the excitation system dynamics arenonlinear in nature. The system can be converted into lineartime invariant form by linearizing it around a steady stateoperating condition. The state space representation of thesystem dynamics is as shown below

x = Ax + Bu (5)

where , x = [∆E′q ∆δ ∆ω ∆Efd]T ;

A =

−1

K3T ′d0

−K4

T ′d0

0 1T ′d0

0 0 ωs 0

−K2

2H−K1

2H−Dωs

2H 0

−KAK6

TA

−KAK5

TA0 −1

TA

;

B = [0 0 0 KA

TA]T ; and u = [∆Vref ];

Equation (5) represents a fourth order system with perturbedvalues of rotor speed (∆ω), load angle (∆δ), internal voltage(∆E′q) and field voltage (∆Efd) as the state variables. Thedetails of constants ‘K1−K6’ are discussed in [13]. The blockdiagram of the linearized system, more often known as theHeffron Phillips model of the SMIB system, is shown in Fig1. Here the input to the system is the AVR reference voltageand the output is the rotor speed deviation.

The constants, ‘K1 − K6’, in the Heffron Phillips modeldepend on the operating conditions of the system and theexternal Thevenin equivalent as seen at the generator bus.These constants thus dictate the position of the poles of theclosed loop transfer function of the system. Therefore to tunethe PSS for satisfactory damping at the current operatingpoint, the knowledge of Thevenin equivalent of the systemis necessary.

K1K1

ΔPm

1/sM1/sM ωo /sωo /s

DD

K5K5

K6K6

K4K4K2K2

K3/(1+sK3T’do)K3/(1+sK3T’do) Kex /(1+sTex )Kex /(1+sTex )

XX+

__

_

Δω

+

+

_

+

_

+

ΔVt

ΔVr

PSSPSS

+

Δδ

XX

XX

XX

Fig. 1. Heffron Phillips Model

EFD

AVR

Vt

Rth+jXth

Eth

PMU

Rest of the system

Fig. 2. Equivalent system

GainWashout

FilterPhase

compensation Vs_max

Vs_min

VsΔω

Fig. 3. Basic Structure of PSS

III. POWER SYSTEM STABILIZER

The LFOs of a generator are a result of small disturbancesregularly occurring in the power systems. PSS is used toenhance the damping of these oscillation modes by providing asupplementary input to the AVR [14]. To ensure enhancementin damping, PSS needs to supply a torque in phase with thedeviation of the speed of the generator and, therefore, mustprovide a phase lead to the speed signal. This is achievedby providing a lead lag compensator in the feedback pathbetween speed to the AVR summing junction. Figure 3 showsthe schematic representation of a conventional PSS.

IV. ESTIMATION OF THEVENIN EQUIVALENT OF THESYSTEM

Considerable work has been done for online estimationof Thevenin equivalent of the power system using the data


obtained from synchrophasors [8]–[11]. In this paper, a sim-ple least square error based method is used to estimatethe Thevenin equivalent of the system at the generator bus.Consider the system shown in Fig 2. Let Zth = Rth + jXth

and Eth = Ex+jEy be the Thevenin impedance and Theveninvoltage, respectively, of the external system as seen from thegenerator bus. Let Vt = u+jw and It = a+jb be the voltageand current phasors reported by the Phasor Measurement Unit(PMU). Then the following equations hold true.

Vt − ZthIt = Eth (6)

Vt = Eth + ZthIt (7)

Substituting the variables in rectangular coordinates , weget:

u+ jw = (Rth + jXth)(a+ jb) + Ex + jEy (8)

u = Ex +Rtha−Xthb (9)

w = Ey +Xtha+Rthb (10)

uw

=

1 0 a −b

0 1 b a

Ex

Ey

Rth

Xth

(11)

It and Vt are directly available from PMU. As the numberof variables is more than the number of equations, (11) can besolved by taking multiple measurements and using the leastsquare error minimization algorithm. This will provide the esti-mated value of the Thevenin equivalent. Let ‘n’ be the numberof measurements taken to estimate the Thevenin equivalent.The least square estimates of the Thevenin equivalent can thenbe obtained as shown below.

z = (BTB)−1BTd (12)

where, z = [Ex Ey Rth Xth]T ;

B =

1 0 a1 −b10 1 b1 a1...

......

...

1 0 an −bn0 1 bn an

and d = [u1 w1 . . . un wn]T

V. RESIDUE BASED PSS TUNING

Residues are the constants in the numerators of the partialfraction expansion of the system transfer function. For asystem with a set of ‘n’ distinct poles, corresponding ‘n’residues can be obtained. The system transfer function G(s)is described by its partial fraction expansion as shown below.

G(s) =

n∑i=1

ris− pi

(13)

Let the transfer function from the reference voltage of theautomatic voltage regulator and the speed deviation of thegenerator be G(s) = ∆ω/∆Vref . Also, let Q(s) be thetransfer function of the speed input PSS. The output from thePSS goes to the summing junction of the AVR, as shown inFig 4. The stabilizing signal, ∆VS , stabilizes the critical modeof oscillation by pushing the unstable/less damped eigenvaluesdeeper into the left half of the s-plane, as shown in Fig 5.

G(s)G(s)

Q(s)Q(s)

ΔVrefΔω

++

X

ΔVs

Fig. 4. Generator and exciter transfer function G(s) and PSS transferfunction Q(s) with the feedback path disabled

λ1

λ2

Δ λ

S Plane

Fig. 5. Horizontal shift in eigenvalue

Let Q(s) be the transfer function of PSS, given by

Q(s) = KpssH(s) = KpssGc(s)Gw(s) (14)

where, the transfer function, Gw(s), represents the washoutblock; Gc(s) represents the phase compensation block; andKpss represents the PSS gain.

The phase compensation transfer function, Gc(s), providesthe necessary compensation. Let ‘p’ be the number of lead-lag blocks required for the necessary compensation, then thestructure of Gc(s) will be

Gc(s) =

[1 + T1s

1 + T2s

]p(15)

Assuming that the washout block is appropriately designed,the aim of PSS tuning is to ascertain the values of the PSSgain, Kpss, the time constants, T1, T2, and the order ‘p’,


such that a reasonable damping criterion can be satisfied.The transfer function ∆ω/∆Vref , when the feedback loop isclosed, is given by,

W (s) =G(s)

1−G(s)Q(s)=

G(s)

1−KpssG(s)H(s)(16)

The closed-loop poles are obtained by solving the charac-teristic equation,

1−G(s)Q(s) = 0 (17)

The closing of the feedback loop shifts the open-loop eigenval-ues of the system. Let the open-loop transfer function, G(s),when the system is excited by the eigenvalue (λc) be :

G(λc) =∆ω

∆Vref=

rcs− λc

(18)

where, rc represents the residue of the pole of G(s).Once the feedback loop is closed, the associated character-

istic equation becomes,

1− rcs− λc

Q(λc) = 0 (19)

Let, ∆λc, be the amount by which the eigenvalue of theclosed loop system shifts from the open-loop eigenvalue, λc.The new pole, s = λc + ∆λc, should satisfy the characteristicequation (19). Therefore, the following relationships holds truefor the closed-loop pole.

(λc + ∆λc)− λc −KpssrcH(λc + ∆λc) = 0 (20)

For small shifts, the transfer function H(s) can be repre-sented by a first order Taylor series expansion in the neigh-borhood of s = λc.

H(λc + ∆λc) = H(λc) +

(∂H(s)

∂s

∣∣∣∣s=λc

)∆λc (21)

From (21) and (20), the shift in the eigenvalue is given bythe following expression.

∆λ =KpssrcH(λc)

1− rcKpss∂H(λc)∂λc

(22)

If the gain of the PSS is chosen such that,

|rcKpss∂H(λc)

∂λc| << 1 (23)

then the shift in eigenvalue is given by,

∆λ = KpssrcH(λc) (24)

Let the residue of the eigenvalue, λc be |rc|∠θc, where θc =arg(rc). For a mode, λc, to undergo a horizontal left shift inthe s-plane, as shown in Fig 5, the argument of ∆λc shouldbe ±180◦.

arg{rcH(λc)

}= ±180◦ (25)

Therefore, the phase compensation φ that the PSS mustprovide is,

φ = arg{H(λc)

}= ±180− arg(rc) (26)

Equation (26) provides the information to choose T1, T2and ‘p’ such that arg

{rcH(λc)

}= ±180◦.

The upper limit on gain Kmax can be found from (23) bythe following condition.∣∣∣∣rcKmax

∂H(λc)

∂λc

∣∣∣∣ = 1 (27)

Kmax =1∣∣∣∣rc ∂H(λc)∂λc

∣∣∣∣pu (28)

The gain of PSS should be lower than 10% of Kmax to ensurethat the condition in (23) is satisfied.

VI. PROPOSED ALGORITHM

The oscillation characteristics of a generator depends on thecurrent operating point and the external Thevenin equivalentas seen from the generator bus. Since the operating point ofthe system changes owing to different load generation scenarioand changes in network topology, the Thevenin equivalent ofthe system varies. Therefore, fixed settings of PSS may notprovide optimal damping to the oscillatory modes. Adaptabil-ity in the PSS can be incorporated by estimating the Theveninequivalent of the network online and then tuning the PSS usingan SMIB approximation of the generator. In this case, theThevenin equivalent voltage acts as the infinite bus voltageand the Thevenin equivalent impedance acts as the externalimpedance. The methodology to tune the Power System Sta-bilizer using the aforementioned approach is outlined below.• Estimate the Thevenin equivalent of the system using

(12).• Update the values of ‘K1 −K6’ in the Heffron Phillips

model.• Based on the current operating point, generator output

(Po Qo) and the values of ‘K1 − K6’, obtained in theprevious step, calculate the state space formulation of thegenerator dynamics.

• Calculate the rotor modes and the damping using themodel thus obtained.

• Calculate the parameters of the PSS, which include thelead-lag block time constants and PSS gain, using residuebased method.

• Set the new values of T1, T2 and KPSS as the new PSSparameters.

VII. SIMULATION RESULTS

The PSS tuned using the proposed methodology will behenceforth referred as Thevenin PSS (TPSS) in this paper. Theproposed method of online tuning of the PSS is tested on theWSCC 9 bus system in the real time digital simulator. Figure 8shows the WSCC 9 bus system. In this system, the PMUs areplaced at all the three generator buses. The machine and exciter


data of WSCC 9 bus system are given in [15]. The Theveninequivalent of the system is estimated using the voltage andcurrent phasors obtained from the PMUs using the least squareformulation discussed in section IV. The PSS of generator 2 ischosen to be tuned online. In the test cases considered, the realpower is changed from 160 MW to 240 MW. The real power isincreased in steps of 40MW. Corresponding to each real poweroutput, the reactive power is kept at 20 MVAR and 40 MVAR.Table II shows the Thevenin equivalent obtained using the leastsquare method and the subsequent TPSS parameters obtainedusing the residue algorithm. It can be inferred from table IIthat as the operating conditions change, the thevenin equivalentof the system, as seen from the generator bus also changes.This results in different settings of the TPSS for each of theoperating conditions. Two more scenarios are considered totest the performance of the proposed algorithm. The scenariosare named Scenario-1 and Scenario-2. In Scenario-1 the loadat bus 8 is set at 160 MW and 35 MVAR. Gen-2 is loadedat 188 MW (0.98 p.u) that absorbs -15 MVAR (0.074 p.u) ofreactive power from the grid. In Scenario-2, the load at bus8 is increased to 180 MW and 150 MVAR. Gen-2 is loadedat 188 MW (0.98 p.u) and delivers 66 MVAR (0.34 p.u) ofreactive power to the grid. For both of these scenarios, theThevenin equivalent is calculated and the TPSS is tuned. Figs6 and 7 show the TPSS obtained using the proposed algorithm.The response of the generator is studied for disturbances. Twodisturbances are considered namely step change in the realpower load at bus 8 and step change in the AVR referencevoltage of Gen-2. Figures 9 and 10 show the power output ofgenerator 2, with and without TPSS, for a 5% step up of realpower at bus 8. Figures 11 and 12 show the real power outputof the generator, with and without TPSS, for a 5% step downof the reference voltage of the AVR. From figs. 9 to 12 it canbe observed that in each of the scenarios, the TPSS improvesthe damping of LFOs considerably. The improvement in thedamping of the oscillatory modes is evident by the lowersettling time of the the oscillations.

The performance of TPSS is also compared with the perfor-mance of a Conventional PSS (CPSS) designed for normal op-erating condition of the generator (Po = 0.62 p.u, Qo = 0.05p.u). The parameters of the CPSS are determined using theresidue based method [12] and the parameters are kept fixedfor all the operating conditions. The parameters of the CPSSare T1 = 0.18 sec, T2 = 0.08 sec and Kpss = 20 p.u. Twodisturbances are considered to compare the performance ofCPSS with TPSS. The first disturbance considered is a 5%step change in the real power load at bus 5 while the seconddisturbance is a 5% change in the reference voltage of theAVR. The disturbances are given for the loading conditionsof scenario 2. Figures 13 and 14 show the performance ofTPSS with CPSS for each of the disturbances. It is observedthat TPSS provides better damping for the oscillatory modes ascompared to CPSS with fixed settings. This is evident from thelower settling time of the oscillatory modes when TPSS is inservice. This is also a result of the fact that CPSS is designedconsidering the normal operating condition of the generator

and the parameters are kept fixed for every operating condition.However, the TPSS is adaptive and changes the parameters asthe operating conditions change.

Fig. 6. PSS obtained for Scenario-1

Fig. 7. PSS obtained for Scenario-2

Gen 1

Gen 2

2

1

37 8 9

5 6

4

Gen 3Load

Load Load

PMU

Fig. 8. WSCC 9 bus system

Fig. 9. Real power output of Gen-2 for Scenario-1

Fig. 10. Real power output of Gen-2 for Scenario-2


TABLE IITHEVENIN EQUIVALENT AT THE GENERATOR BUS AND THE

CORRESPONDING PSS SETTINGS OBTAINED

P0 Q0 Zth (p.u) Eth (p.u) T1 T2 Kpss

160 20 0.14 + j0.51 0.93− j0.43 0.160 0.064 26

160 40 0.16j + 0.61 0.97− j0.54 0.160 0.059 26

200 20 0.19 + j0.65 0.87− j0.69 0.150 0.048 34

200 40 0.18 + j0.67 0.93− j0.71 0.150 0.039 34

240 20 0.23 + j0.72 0.77− j0.92 0.145 0.031 36

240 40 0.22 + j0.74 0.85− j0.95 0.145 0.025 36

Fig. 11. Real power output of Gen-2 for change in Vref of AVR in scenario-1

Fig. 12. Real power output of Gen-2 for change in Vref of AVR in scenario-2

Fig. 13. Real power output of Gen-2 in scenario-2 for CPSS and TPSS

Fig. 14. Real power output of Gen-2 in scenario-2 for CPSS and TPSS

VIII. CONCLUSION

A new algorithm for online tuning of the PSS utilizingthe synchrophasor measurements is proposed in this paper.In order to tune the PSS of a generator, the external systemis approximated by its Thevenin equivalent, whose parametersare estimated using the real time synchrophasor measurementsand a least square error minimization technique. Subsequentlythe values of ‘K1−K6’ are updated in the small signal modelof the system and the PSS is tuned using residue method.The developed algorithm is tested on the WSCC 9 bus systemsimulated on real time digital simulator. It is observed thatthe proposed methodology is able to estimate the Theveninequivalent of the system whose parameters are effectively usedto update the constants in the linearized SMIB dynamicalmodel of the system which are further used to tune the powersystem stabilizer. The resulting tuned PSS has shown goodperformance in all the test cases considered.

REFERENCES

[1] G. Rogers, Power system oscillations. Springer Science & BusinessMedia, 2012.

[2] E. Larsen and D. Swann, “Applying power system stabilizers: Part i- iii:Practical considerations,” IEEE Transactions on Power Apparatus andsystems, no. 6, pp. 3034–3046, 1981.

[3] F. P. Demello and C. Concordia, “Concepts of synchronous machinestability as affected by excitation control,” IEEE Transactions on powerapparatus and systems, vol. 88, no. 4, pp. 316–329, 1969.

[4] P. Kundur, M. Klein, G. Rogers, and M. S. Zywno, “Application ofpower system stabilizers for enhancement of overall system stability,”IEEE Transactions on Power Systems, vol. 4, no. 2, pp. 614–626, 1989.

[5] A. Ghosh, G. Ledwich, O. Malik, and G. Hope, “Power system stabilizerbased on adaptive control techniques,” IEEE transactions on powerapparatus and systems, no. 8, pp. 1983–1989, 1984.

[6] M. Klein, G. J. Rogers, and P. Kundur, “A fundamental study ofinter-area oscillations in power systems,” IEEE Transactions on powersystems, vol. 6, no. 3, pp. 914–921, 1991.

[7] Y. Liu, Q. Wu, H. Kang, and X. Zhou, “Switching power systemstabilizer and its coordination for enhancement of multi-machine powersystem stability,” CSEE Journal of Power and Energy Systems, vol. 2,no. 2, pp. 98–106, 2016.

[8] K. Vu, M. M. Begovic, D. Novosel, and M. M. Saha, “Use of localmeasurements to estimate voltage-stability margin,” IEEE Transactionson Power Systems, vol. 14, no. 3, pp. 1029–1035, 1999.

[9] I. Smon, G. Verbic, and F. Gubina, “Local voltage-stability index usingtellegen’s theorem,” IEEE Transactions on Power Systems, vol. 21, no. 3,pp. 1267–1275, 2006.

[10] S. Corsi and G. N. Taranto, “A real-time voltage instability identificationalgorithm based on local phasor measurements,” IEEE transactions onpower systems, vol. 23, no. 3, pp. 1271–1279, 2008.

[11] S. M. Abdelkader and D. J. Morrow, “Online tracking of theveninequivalent parameters using pmu measurements,” IEEE Transactions onPower Systems, vol. 27, no. 2, pp. 975–983, 2012.

[12] M. J. Gibbard and D. J. Vowles, “Reconciliation of methods of compen-sation for psss in multimachine systems,” IEEE Transactions on PowerSystems, vol. 19, no. 1, pp. 463–472, 2004.

[13] P. Kundur, N. J. Balu, and M. G. Lauby, Power system stability andcontrol. McGraw-hill New York, 1994, vol. 7.

[14] A. Yaghooti, M. O. Buygi, and M. H. M. Shanechi, “Designingcoordinated power system stabilizers: a reference model based controllerdesign,” IEEE Transactions on Power Systems, vol. 31, no. 4, pp. 2914–2924, 2016.

[15] P. M. Anderson and A. A. Fouad, Power system control and stability.John Wiley & Sons, 2008.


online updating of synchronous generator …heffron phillips model of the smib system, is shown in...

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