IEEE Transactions on

DAMPINC REPRESENTATION FOR POWER SYSTEM STABILITY STUDIES I 151 Power Systems, Vol. 14, No. 1, February 1999

Power System Damping Ad Hoc Task Force* of the Power System Dynamic Performance Committee

Abstract: The summary of the systems and the in the various power system stablity. care is required in program application study of power required for Recommendations system stability prz are recommended.

Keywords: Power Generator modeling,

paper includes a review and actual damping in electric power

melhods for modeling such damping coriputer programs used to study

It is concluded that great both software development and

to avoid serious errors in the sys:em stability. Judgment is often

reaslmable modeling of damping. are provided to guide the power .ctitioner, and research activities

systems, Stability, Damping, Load modeling.

extend the first swing stability limits. This, coupled with the increased emphasis to utilize every megawatt of transmission capacity, has created transmission capability limitations due to insufficient damping torque. Consequently, multi-million dollar devices are being installed in power systems based upon power system stability studies which indicate insufficient damping. The need to correctly model the true physical damping in the power system has increased in importance. The purpose of this paper is to discuss the factors which affect damping, to suggest damping representation for power system stability studies, and to recommend research activity to improve the modeling of damping for power system stability studies.

1 2.0 Actual Damping

0 Introduction Power system damping consists of several types and

Most power system. by employing compxer by-step time simulalion. are used to determine transfer capability procedures for study is generally study where insufficient synchronizing damping torque.

Until about 1960, study of synchronizng since stability limits insufficient stability aids have be

*B. L. Agrawal C. Concordia, R. G . Fouad, P. Kundur,

stability studies are conducted programs which use a step- The results of such studies

transmission system power and to formulate operating

transmission systems, This form of referred to as transient stability

insa.bility occurs due to either torque or insufficient

enphasis had been placed on the torque (first swing stability) were usually dictated by

synchronizing torque. Numerous en developed and employed to

(Co-Chairman), P. M. Anderson, Farmer (Co-Chairman), A. A.

W. W. Price, C. W. Taylor.

components[l]. Following is a list of the damping components which occur in the power system modeled for transient stability studies:

PE-192-PWRS-0-3-1998 IEEE Power System Dyn2mic Power Engineering Society Power Systems Manusc-ipt available for printing March

1. Damping due to turbine-generator torque vs. speed characteristics (Tm=P,,Jw).

I ( Paper recommended and approved by the Performance Committee of the IEEE

for publication in the IEEE Transactions on submitted December I I. 1997, made

9, 1998

7. Material damping due to stress-strain cycles caused by torsional oscillations.

8. Transient losses associated with the network and gcncrator statom following il transient.

2. Damping due to load-frequency characteristics.

3. Electrical damping due to rotor amortisseur winding currents.

4. Damping (or negative damping) due to excitation system operation including the impact of power system stabilizer(PSS).

5. Damping (or negative damping) due to speed governing.

6. Steam damping for steam turbines.

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3.0 Modeled Damping

Turbine-generator modeling in power system stability program can be broken into two general categories referred to as “classical representation” and “full representation.” Classical representation consists of a constant generator internal voltage in series with the generator’s transient reactance, and constant mechanical power input. There is no inherent damping in this model except for that due to the assumption of constant mechanical power(1tem 1 in the list above). Any additional damping must be represented by a fixed coefficient input by the program user [2,3]. There is no known general method for determining the appropriate damping coefficient for “classical representation.” Limited simulation studies have shown that for a “classical representation” the appropriate damping to be modeled varies significantly depending upon the unit location in the system and the frequency of oscillations (see Section 9 for more details).

Full representation implies that the excitation system is modeled, the generator and its saliency is represented with variable flux linkages for the determination of electrical torque, and the governor and a portion of the energy supply system (steam or hydro) is modeled for the determination of mechanical torque or power. The following material, through Section 8, applies to turbine-generators with full representation,

For turbine-generator modeling in stability programs, it is common to include the amortisseur windings in the generator equations. Therefore, Section 2.0, Item 3 damping is commonly included in the computer model. The excitation system afid governor are usually modeled in detail in stability programs. Therefore, the damping, or negative damping, for Items 4 and 5 will inherently be accounted for. Steam damping is very small relative to the damping of Items 1, 2, and 3 so that Item 6 damping is negligible for transient stability

studies. Material damping occurs in the shafts between turbine-generator masses when the masses have relative torsional motion. For transient stability studies, relative motion of the masses is small and is normally not modeled. Therefore, Item 7 damping is not considered here. The damping associated with network transients decays rapidly and is not significant in the overall damping of low frequency oscillations, This allows the generator equation to be simplified by the elimination of the flux rate of change ( p v terms in the stator voltage equations. Hence, Item 8 damping does not need to be considered here.

From the above discussion of the various components of damping it has been shown that all significant damping is generally accounted for in the power system model, with the possible exception of Speed-Torque damping (Item 1) and Load- Frequency damping (Item 2). Modeling of these components may be inherently incorporated in the program or it may be necessary to provide a specific damping coefficient as an input to the program The methods for modeling damping will vary from one program to the next and are determined by the following factors for the specific program: generator model, type of swing equation, turbine output, and load modeling capability. The following is a brief discussion of the issues associated with each of the factors.

4.0 Generator Model

The rate of change of flux ( p “ j terms for the calculation of armature voltages are generally neglected in stability programs for large power systems. If such terms were included in the generator model, it would be necessary to reduce the time step, and increase the case running time, many fold. For consistency, it would also be necessary to include the differential equations of the network, rather than the usual algebraic representation. Therefore, unless accuracy is severely degraded, it is not practical to include the pY terms in production grade programs where thousands of case studies may be required to

establish the interconnected season. Figure 5 neglecting the significant effect have a short term torque. Without rotor losses that fault are not torque and can be other means[4]. fault damping, damping.)

5.0 Effect of Srbeed Variation and System

operating limits for an power system for one operating

4 of reference 3 shows that pY terms does not have a

on the damping, but could effect on the synchronizing

the pY terms, the short term occur during a close-in 3-phase

moc.eled. This is called braking approximately modeled by

(This is often referred to as even though it is not truly

Fre uenc t- bility programs include

in rotor speed on variation in system uations but some do

neglect the p Y terms. t for armature voltage rotor speed deviations

lecting the p Y/ terms and t of both pY and rotor

ted in the armature spite of this, there is no

ill be affected by terms, and there is no

the effect of speed

offsets the effect

or and network

uctive reactance(X) and ) at nominal frequency. he admittances are not

153

modified during a stability run to account for changes in system frequency.

If the network were completely inductive, then the method described above would be precisely correct, since one would be solving a set of equations involving flux linkage, current, and inductance instead of voltage, current, and inductive reactance. Multiplying the fluxes by speed to obtain voltages would, in fact, be incorrect unless the inductances were also multiplied by frequency to obtain reactances. In the real case with resistive and capacitive elements in the network, some error would be introduced. The resistive elements should use V (speed times flux); the capacitive elements should use V and have their reactances computed by dividing by Erequency. The error due to resistive elements would be small. The error due to capacitive elements may be significant for large frequency swings.

If the above approach is used to interface the generator and the network equations, i.e. neglecting the speed effect on internal voltage, it may still be important to include the speed effect on the terminal voltage value that is fed back to the voltage regulator. Experience has shown that significant error in the excitation system output could result if the speed effect is neglected. For consistency, the speed effect should also be used to compute the electrical power ( P and Q) at the terminal of the generator. This value of P should not be used to replace T, in the swing equation. The electrical torque computed from flux and current should be used. Furthermore, for consistency, voltage and power flow values at other points in the network should be computed by multiplying the “voltage” values from the network solutions (which are really fluxes) by frequency. From the above the following can be concluded:

It is best to include the effect of speed variations in the generator equations and effect of system frequency in network equations. If the effect of speed variations is neglected in the generator equations and the effect of system frequency is neglected in the network equations,

154

it has a canceling effect but some error is unavoidable since all network elements are not similarly dependent on frequency. Under these conditions of neglecting the effect of generator speed variations the electrical torque (T,) is the same as electrical power (P,) and either can be used in the swing equations. However it would be erroneous to replace T, by PJu in the swing equations unless u is 1 .O.

0 Even when the effects of rotor speed deviation are neglected in the generator equations, it is important to include the speed effects in the terminal voltage value that is fed back to the voltage regulator.

6.0 Type of Swing Euuation

Following is the basic swing equation in a correct form which will be used for discussion purposes:

2HAw'+ D A m = T, - T, (1)

Where: H = The stored energy in the rotating

bodies at rated speed divided by the chosen MVA base

dw'= Derivative of speed (= Acceleration)

dw =Deviation from synchronous speed in per unit

D = Damping coefficient input by the user

T,,, = Mechanical torque driving the turbine-generator, in per unit

T,= Retarding torque due to the electrical load on the generator, in per unit

If the stability program utilizes a swing equation in the form of Equation 1 and accurately

calculates T, and T, without neglecting rotor speed variations, D in the swing equation would be set to 0. If rotor speed variations are neglected in armature voltage calculations, and T, is not approximated as P, in the swing equation, an error has been introduced. The error can be offset by setting D equal to an appropriate value.

If the stability program calculates T, in the swing equation by assuming T, = P,, an error has been introduced which can be offset by setting D equal to an appropriate value.

There are other swing equations in forms other than Equation 1, but they must be equivalent to Equation 1 to be valid. For instance, both sides of Equation 1 could be divided by rotor speed to form a valid swing equation in terms of P, and P,.

7.0 Turbine Output

If the govemor is not modeled and the stability program assumes P, equal to some constant value k, then an input D may be required if T, is approximated by P,. If the prime mover is of the type that provides constant torque, D should be 0. It is more likely that a prime mover without govemor response would provide constant power if it is a steam or hydraulic turbine. In this case, D should be set equal to k in per unit if T, is approximated by P,. For other types of prime movers, such as combustion turbines, the appropriate value for D is not apparent. The easiest way to avoid error is to calculate T, from the knowledge of P, and use the true value of P, with D set to zero.

8.0 Electric Load Models

Loads normally produce system damping due to the power-fkequency characteristics of each specific load[5]. Some early power system stability programs did not provide for representation of the power-frequency characteristics in the load model. Therefore, D was adjusted for each turbine- generator modeled to account for load damping. Since there is no direct correspondence between

each generator a~ at best, a crude frequency characi modeled within t of the load and L not account for la

9.0 Classica

The classical genr account for any k amortisseur wind effects correspo neglected. Ho system is not m effects are also machine, the mec constant. If this I

be constant powe prime mover is ac and vice versa.

An eigenvalue i which compares machine model w modeled with fu included investiga

damping of the fu different types investigated.

of D in the Cli

For both systems model had less c intertie mode dai classical machine system and D = have the same loc that the classical one system and These results ind value of D that machine which w approximation foi is important in a the number of

each load this method was, pproximation. The power- istics of each load should be : load model at the location n the swing equation should 1 characteristics.

vlachine Reuresentation

ator representation does not ses in the generator field and gs. Therefore, the damping ling to these losses are :ver, since the excitation eled, any negative damping :glected. For the classical nical input to the machine is :chanical input is assumed to an error is introduced if the ally a constant torque device

iestigation has been made he damping of a classical 1, the damping for a machine representation. The study In of the required magnitude sical model to match the representation model. Two If power systems were

was found that the classical mping. To have the same 3ing it was found that the ust have a D = 0.33 for one 4 in the other system. To mode damping it was found achine must have D > 2 for > 6 for the other system.

ate that there is no single n be applied to a classical provide even a reasonable

lamping. If damping torque 3wer system stability study, whines represented by a

155

classical model must be limited to the absolute minimum.

10.0 Conclusions

An investigation has been made of the various methods currently used to represent damping in power system stability programs. The following can be concluded:

1. Power system stability programs with “full representation” generally model all significant damping with the possible exception of load- frequency damping and prime mover speed- torque damping.

2. The only swing equation that does not introduce damping errors is Equation 1 or an equivalent equation such as

2H Am’ + D AO = (Pm - P,)/o

where, w = I + AW

3. Neglecting the pY terms in armature voltage calculations is not expected to introduce damping errors, but this has not been thoroughly investigated.

4. If rotor speed variations are neglected in armature voltage calculations, no damping error is introduced by assuming T, = P,.

5 . If a machine is not equipped with a governor, or if a governor is not modeled, constant mechanical input is assumed. If this mechanical input is assumed to be constant power, an error is introduced if the prirne mover is actually a constant torque device and vice versa. The program user may select an appropriate value for D in the swing equation to offset the effect of this error.

6. Damping for electrical loads must be modeled at points where the loads are lumped, The load damping value can vary from one load to the next.

156

7.

8.

~

Due to the ever-increasing importance of damping in power systems, there is a need to supply the electric power industry with better software and better information on the application of power system stability programs. Therefore the committee recommends the following:

There is no single value of D that can be applied to a classical machine. The appropriate damping depends on the system configuration, location of the unit in the system, and the oscillatory mode of interest.

Understanding of this unique problem is essential for credible power system stability study. Judgment is often required for reasonable modeling of damping.

1 1 .O Recommendations

1.

2.

3.

4.

5.

Research programs should be initiated to establish the effect on system damping of neglecting pY terms in armature voltage calculations without neglecting the speed variations in armature voltage calculations.

Research programs should be initiated to develop a means of measuring the speed torque damping for the various types of prime movers.

Users should only utilize power system stability programs which have swing equations the same as, or equivalent to Equation 1, if damping is important.

Users should eliminate classical machine models in power system stability studies, except for aggregations of machines distant from the study area.

Damping due to electric load should be modeled at the points where the load is represented by representing the load in as much detail as practical[6].

6. Panel sessions should be held to provide the power system stability practitioner with a basic understanding of the issues and options for the modeling of damping.

1 2.0 Acknowledgments :

The Power System Damping Ad Hoc Task Force was formed in 1995 to investigate and document the recommended methods for the modeling of damping in transient stability studies The Task Force Chairmen would like to thank all of those that contributed their special expertise to assist in the preparation of this report.

13.0 Cited References

1.

2.

3.

4.

5.

6.

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Anderson, P.M. and A.A. Fouad, Power System Control and Stabi~ity, IEEE Press, Piscataway, NJ, 1993.

Kundur, P, Power Syste Stability and Control, McGraw-Hill, 199

Dahl, O.G.C. Electric Power Circuits, Theory and Applications, Volume 11, Stability, McGraw-Hill Book Company, Inc, New York, 1938.

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IEEE Committee Report, IEEE Task Force on Load Representation for Dynamic Performance, “Load Representation for Dynamic Performance

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14.0

a. Concordia, Damping an AIEE Trans p.731-37.

b. Shepherd, 1 Damping Synchronous PAS-80, June

c. deMello, F.P., of Synchron Effected by Trans., v. PA

d. El-Sherbiny, Part I: Imp1 Block Diagri Winding and paper A 75 PES Summer 1975.

e. El-Sherbiny, Part 11: An Contributed b Winding, and with the Effec

Summer Meet 75 528-0, p

f. Kilgore, L.A., “Simplified 1 System A Subsynchrono IEEE Trans. 1977, p. 1840

g. Fouad, A.A., Torsional Os, with series-(:

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1 ther References h.

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[.IC., “Amortisseur Winding ysis of Damping Torque Amortisseur Winding, Field 1. their Interaction, Together of Loading,” IEEE paper A sented at the IEEE PES ig, San Francisco, July 1975.

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m.

nd K.T. Khu, “Damping of llations in Power Systems )mpensated Lines.” IEEE

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Trans., v. PAS-97, n.3, May/June 1978, p. 744- 753.

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Hammons, T.J., “Stressing of Large Turbine- Generators at Shaft Couplings and LP Turbine Final-Stage Blade Roots Following Clearance of Grid System Faults and Fault Synchronization,” IEEE Trans., v. PAS-99, n.4, 1980, p. 1652- 1662.

Ooi, Boon-Teck, “Phase Modulation Theory of Electromechanical Damping in Synchronous Generators,” IEEE Trans.., v. PAS-100, n.5, 1981, p. 2211-2218.

Gosens, J.F., A.J. Calvaer, and L.J. Soenen, “Full Scale Short Circuit and Other Tests on Dynamic Torsional Response of Rodenhuize No.4, 300 MW 3000 rpm Turbogenerator, I: Test Facilities, and 11: Test,” IEEE Trans., v. PAS-100, n.9, 1981, p. 4166-4173 and 4174- 4185.

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