Inaugural IEEE PES 2005 Conference and Exposition in Africa

Durban, South Africa, 11-15 July 2005

An Investigation into the Capabilities of MATLAB Power System Toolbox for Small

Signal Stability Analysis in Power Systems

M. Ntombela, K.K. Kaberere, K.A. Folly and A. I. Petroianu, SMIEEE

Abstract--With the advancing of computing power there has

been a lot of power system simulation software packages

developed. There exists a need for researchers to evaluate the

capabilities of these tools so users can make informed decisions on

which tool to purchase. The paper presents the capabilities of

MATLAB Power System Toolbox (PST) for linear analysis. The

eigenvalues obtained from PST are compared to those obtained in

the book entitled Power System Stability and Control by Prabha

Kundur.

Index Terms--Small-signal stability analysis, power system

dynamics, eigenvalues, power system components modeling.

I. INTRODUCTION

Power systems are capital intensive big complex systems. It is thus of high risk and often difficult to conduct experiments on such complex systems. With the increase in demand of electricity power systems engineers are forced to operate these systems at their limits with very narrow stability margins which often require the installation of special stabilizing controls whose design rely heavily on the analysis of the linearized system.

The advancement of data processing capabilities of computers has led to the development of many power system simulation tools. This makes it difficult for users to decide which package best suites their area of interest or application. Learning the many different tools available would be time consuming and uneconomical. There exists a need that the capabilities of these simulation tools be published so as to assist the users in making an informed decision as to which simulation package to use. Stability studies for power system planning, operation and control rely immensely on computer based power system simulation tools. Simulation tools use mathematical models that predict the dynamic performance of the system. It is crucial that these power system models be modelled accurately to predict the actual performance of the system.

Small signal stability is the ability of the system to maintain synchronism under small disturbances which occur continually on the system due to the small variations in loads and generation or other small disturbances on the system. A disturbance is considered to be small if the equations that give the response of the system may be linearized for the purpose of analysis [1].

Linear analysis is a powerful technique to study whether a power system is stable or not. The eigenvalues obtained from linear analysis give a complete picture of the stability of the system [1]-[3].

This paper presents an investigation into the capabilities of MATLAB power system toolbox for small signal stability analysis in power systems. The small signal stability of the single machine infinite bus system is investigated. The frequency domain results are validated by performing the step response of the system (time domain).

The paper is organised as follows: x Description of the MATLAB Power System

Toolbox software package is discussed in section II. x Power system component models available in Power

System Toolbox are discussed in section III and V. x Description of the single machine infinite bus test

system is discussed in section IV. x Case studies are formulated and discussed in section

VI.x Data representation, input data and output

representation is discussed in section VII. x Conclusions, possible reasons for the differences in

results are discussed in section VIII.

II. DESCRIPTION OF POWER SYSTEM TOOLBOX (PST)

Power System Toolbox is a collection of MATLAB files that can perform loadflow, transient stability analysis and small signal stability analysis. Although a loadflow is important in its own right it is required in power system dynamics studies to initialize the dynamic models. These dynamic models are coded as MATLAB functions in PST [4], [5].

III. POWER SYSTEM COMPONENTS MODELS AVAILABLE IN PST

There are many component models in PST. See Appendix for a list of all the models available in PST. For the purposes of this paper only the component models employed for the single machine infinite bus test system are discussed in detail.

2420-7803-9327-9/05/$20.00 2005 IEEE

A. Generator Models

1) Classical model

The classical generator model is referred to as the electromechanical model in PST. This model models thegenerator as a constant voltage behind a transient reactance.The classical model is characterised by two states as shown intable 1. 2) Fourth order model

The 4th order model referred to as the transient model inPST. This model models a synchronous machine with thevoltage behind the transient reactance. The 4th order modelextends on the classical model by including the effects of the field winding and one damper winding on the d-axis. The 4th

order model is usually used for academic purposes and will not be discussed further. 3) Sixth order model

The 6th order model is referred to as the subtransientmodel in PST. This model models a synchronous machinewith the voltage behind subtransient reactance. The 6th order model extends on the classical model by including the effectsof the field winding, damper windings, one on the d-axis andtwo damper windings on the q-axis. The second damperwinding on the q-axis improves the accuracy in modellingmultiple paths for circulating eddy currents [10]. This modelis characterised by six states as shown in table 1.

In practice it is common that all generators within the zoneof the power system being studied are represented by thesixth order model and generators far from the area of interest(modeled as the external network or infinite bus) are modeledusing the classical model. It has been found in [6] that the 6thorder model for round rotor generators and the 5th ordermodel for salient pole generators are adequate for smallsignal stability studies. This paper only focuses on theclassical and detailed generator models i.e. the 2nd order andthe 6th order models. Table 1 below summarizes the statesfound in each model.

TABLE I. A table of generator states characterizing the different order models in PST. A 1 indicates a state is present and a if not.

B. Excitation System Models

See Appendix for the three excitation system modelsavailable in PST.

C. Power System Stabilizer Model

The PSS model has two lead-lag blocks and an option of either a speed or power input.

D. Transmission Line

The S -model is used to model the transmission lines.

IV. THE SINGLE MACHINE INFINITE BUS TEST SYSTEM (SMIB)

In this paper we examine the small signal stability of thesingle machine infinite bus system in [1] using PST. The small signal stability is investigated using a differentsimulation tool in [1]. The results obtained in PST arecompared to those in [1].

Fig. 1. The SMIB

Figure 1 shows a thermal generating station consisting of four 555 MVA, 24kV, 60 Hz units represented as one 2220 MVA generating unit G1. The network reactances are in perunit on a 24 kV base. The transformer is a step uptransformer operated a 24 kV on the primary and secondary.

The transformer has an impedance . The lines

L1 and L2 have impedances of j0.5 and j0.93 respectively.The objective of the case studies carried out in this paper is toinvestigate the capabilities of PST on the small signalstability analysis of the test system following a loss of L2.

15.0jX t

The post-fault system operating condition is:

9.0 P 3.0 Q q 360.1TV q 0995.0BV .

V. MODELING OF COMPONENTS IN PST SPECIFIC TO THE SMIB

A. Modelling the Synchronous Generator

There are two well-accepted methods of modelling thesynchronous machine commonly used in power systemswhich are the coupled-circuit method and the operational-impedance method [7].

To this day the synchronous machine is represented bytwo coupled equivalent circuits with time invariantparameters thanks to Parks Transformation. Additional short

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circuited rotor windings are included in the model torepresent the damper windings. In the coupled-circuit methodthe machine is represented by an equivalent circuit with twod- and q-axis rotor windings shown in figure 1(a) and figure1(b) respectively. Usually synchronous machine data isspecified in terms of its subtransient and transient reactances

and time constants ( , , , , etc) often

referred to as derived parameters of the machine. The derivedparameters of the machine can be obtained from the manufacturers or field experiments. The derived parametersare then converted to an equivalent set of coupled-circuit

parameters ( , , , , etc)[7]. This method is

used in the recently developed software package called MATNETEIG developed by the same company that developed PST. The authors are currently researching on thisnew software package.

dX

l fdR

'd

X

fdX

'0dT

"0dT

adX X

ra

xl

xad

xkd

rkd

xf rf

id if

ikdVdVf

iad

Fig. 2(a). D-axis equivalent circuit

ra

xl

xaq

rkq2

iq

Vq

iaq xkq1

ikq1

xkq2

ikq2

rkq1

Fig. 2(b). Q-axis equivalent circuit

The operational-impedance method models the machinedirectly in terms of the derived parameters. PST uses theoperational-impedance method. The representation of themachine is shown in figure 3.

Fig. 3. Operational-impedance generator model

Both these generator models are 6th order generatormodels. It has been found in [7] that provided the saturationis neglected the two models should give identical results.

There is no standard or accepted method of representing generator saturation in small-signal stability however certainprecautions need to be taken when doing so [7]. These arediscussed in detail in chapter 6 of [7]. In PST theseprecautions were taken into consideration and the user shouldbe well aware of the limitations superimposed by the methodof modelling the saturation used.

Generally the only saturation data available for a machineis its open circuit characteristic. It is common practice insmall signal stability studies to assume that the d-axissaturation characteristic of the loaded generator is the same asthe open circuit characteristic. In PST the user is required toinput the two saturation parameters S(1.0) and S(1.2). Whenusing the open circuit characteristic when the generator isloaded an operating point must be identified. The voltagebehind the subtransient reactance is used in identifying theoperating point in PST. This method often assumes that forround rotor machines the d-axis and q-axis saturation

characteristics are the same and that . Several

investigations listed in [7] reveal that this is not the case, theq-axis of round rotor machines saturates significantly morethan d-axis leading to errors in the calculation of the initialrotor angle and field excitation. When using PST the user should be aware of this limitation.

""qd XX

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B. Modelling the Excitation System and Power System

Stabilizer

The exciter model used in the single machine infinite bus test system obtained from [1] closely matches the simplifiedexciter model in PST. For the purposes of this research thesimplified exciter model was studied. Figure 3 obtained from[11] shows the block diagram of the thyristor exciter modelwith AVR (referred to as the simplified exciter model in PST) and a PSS.

Fig. 4. Thyristor excitation system with AVR and PSS

Transfer functions of exciter models and power systemstabilizers can be validated by performing frequency domainexperiments complimented by time domain experiments. As mentioned in the introduction the model for the power systemstabilizer in PST is shown in figure 5.

Fig. 5. Power system stabilizer model in PST

The difference between the model shown in figure 4 and the model shown in figure 5 is that: K in fig. 5 is equal toKSTAB*TW in fig 4.

C. Modeling transmission lines

The S -model is used in the modelling of the lines.

D. Modeling the load

The modelling of the load is a significant factor that contributes to the accurate prediction of the systemeigenvalues/damping [1], [7]. Therefore modelling of loadsshould be done with great care. In this paper the test systemunder investigation has no loads. The author is working onthe well known two-area four-generator system with loads.The load in PST can be modelled as constant power active or reactive, constant impedance, constant current or acombination of the three.

E. Modeling the infinite bus

The infinite bus is modelled as a voltage behind a transientreactance in PST i.e. using the classical generator model. The classical generator modelling the infinite bus should bespecified in PST using the ibus_con matrix. The parametersthat have to be specified for the infinite bus are:

x The MVA rating of the infinite bus, this shouldbe high relative to other machines in the system

x The must be around the common values i.e.0.2 ~ 0.3.

'dx

x The value of the inertia constant H does notinfluence the results obtained.

VI. CASE STUDIES

The following case studies were conducted:x Small signal performance of the system with the

generator modelled using the classical model.x Small signal performance of the system with the

generator modelled using the 6th order model.Three cases have been considered:

o System under manual control with theeffects of saturation taken intoconsideration.

o System with AVR in service.o System with AVR and PSS.

A. SMIB with the Classical Generator Model.

The SMIB in [1] was simulated in PST. See Appendix for the generator data. The results obtained very closely matchedthose in [1] as shown in table II.

TABLE II. Eigenvalues of the SMIB, generator modelled with the classical model

B. SMIB with the Sixth Order Generator Model under

Manual Control and Effects of Saturation taken into

consideration.

See Appendix for the generator data. The two saturationparameters were calculated to be:

124.0)0.1( S , 413.0)2.1( S

Table III shows the eigenvalues obtained in PST arecompared against those obtained in [1].

TABLE III. Eigenvalues of the SMIB with generator modelled with 6th order model.

245

C. SMIB with the Sixth Order Generator Model with

Automatic Voltage Regulation and the Effects of Saturation

taken into consideration.

See Appendix for the excitation system parameters. TableIV shows the eigenvalues obtained in PST compared against obtained in [1].

Table IV. Eigenvalues of the SMIB 6th order generator model withAVR in service.

A step response can also be performed in PST tocomplement the frequency domain results. Figure 6 showsthe step response of this system. The poorly dampedoscillatory mode seen in the frequency domain is evident inthe time domain. The oscillations on figure 7 have a period ofapproximately 0.85 giving a frequency of 1.176 whichcorresponds to the frequency of the oscillatory mode.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

time in seconds

terminalvoltageinp.u.

Fig. 7. Step response of the SMIB with AVR

D. SMIB with Generator Modelled with the Sixth Order

Generator Model under Automatic Voltage Control and a

Power System Stabilizer in service with the effects of

Saturation taken into consideration.

See Appendix for the power system stabilizer parameters.The eigenvalues obtained in PST are compared against those documented in [1] on table V.

Table V. Eigenvalues of the SMIB 6th order generator model with AVR and PSS in service.

A step response of this system with AVR and PSS was also performed. Figure 7 shows the response of the system.The frequency domain depicts that all the modes are welldamped. The time domain step response complements this.All the oscillations are well damped.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

time in seconds

terminalvoltageinp.u.

Fig. 8. Step response of the SMIB with AVR and PSS.

E. Analysis and Presentation of Results.

The prediction of system eigenvalues is not exactly the same i.e. the eigenvalues obtained in PST are not exactly the same as those obtained in [1]. Possible reasons for thedifferences in results obtained in PST and those obtained in[1] is discussed in [11]. Furthermore some of the conclusionsdrawn in [7], [8] are that the noticeable difference in the prediction of system eigenvalues is affected to a large extent by the generator models, the load models the different

246

approaches in constructing the state matrix.The A, B, C and D matrices are accessible. The right and

left eigenvectors are accessible. A matrix of participationfactors and a matrix of normalised participation factors are accessible. The damping ratios and frequencies of the systemeigenvalues are accessible. The states and the order of states used in generator modelling are also accessible.

VII. DATA REPRESENTATION IN PST

PST is not a graphical intensive package. The structure or the topology of the power system is entered in matrix format.The bus and line matrices define the structure of the systemin PST.

The different power system component model parametersare contained in matrices. This can be tiresome when dealing with huge systems with a lot of buses and interconnections.The package comes with the advantage that it runs on theMATLAB environment so it is relatively easy to extract andplot all the system variables. Unlike in other tools where onlythe A matrix is accessible the four important matrices i.e. the A, B, C and matrices, in small signal stability are available inPST.

VIII. CONCLUSIONS

MATLAB PST is on par with the other software packagessince it uses the recommended models. However it shouldalso be noted that there are very few standard IEEE power system component models available in PST. The new package MATNETEIG has been improved in this regard. The fact that the software runs on the MATLAB environmentmakes it easy for users to customise power system componentmodels and improve on the modelling detail of a standardmodel to more closely match the real system under beingmodelled. Another advantage of PST superimposed by thecomputational abilities of MATLAB is that a system with upto 200 buses can be simulated.

PST falls short when it comes to the user interface. Computer have become more and more graphically intensivesince the human brain works better with pictures thannumbers, strings and characters there is a lot of room for improvement in this regard.

A lot of work has been done by the CIGRE TASK FORCE 38.01.07 on identifying the factors which areimportant to modelling power systems for the accuratesimulation of power system oscillations and is documented in[7]. Most these models recommended for use in small signalstability studies have been validated with field experiments.In some cases the models validated in [7] were adjusted to include more detail to make them closely match the field

experiment results. This would be done relatively easy in PST since the component models are coded as MATLABfunctions.

IX. APPENDIX

Models available in PST: Generator Models

x mac_em electromechanical (classical) modelx mac_tra model including the transient effect x mac_sub model including the subtransient

effectx mac_ib a generator as infinite bus modelx Excitation system modelsx smpexc simplified exciter modelx exc_dc12 IEEE type DC1 and DC2 modelsx exc_st3 IEEE type ST3 modelx Power system stabilizer model pssx Simplified turbine governor model tgx Induction motor model mac_indx Induction generator model mac_igenx Static VAR compensator model svc x HVDC line model dc_line, dc_contx Non-conforming load model nc_load

Generator model data for the classical model is as follows:

3.0' d

X, MVAsMWH /.5.3

Generator model data for the 6th order model is asfollows:

16.0 l

X

23.0" d

65.0' q

sMW .5.3

, , , ,

, , , ,

, , , ,

003.0 aR

sd

8'0

25.0" qX

MVA

81.1 d

X

s03.00

sq 1'0

3.0' d

X

76.1 qX

sq 07.0"0

X

X

H

Td

T"

T T

/

Power system stabilizer parameters are as follows:

5.9 STABK , , ,sWT 4.1 sT 154.01 sT 033.02

Excitation system parameters are as follows:

200 AK ,T

.sR 02.0

REFERENCES

[1] P. Kundur, Power System Stability and Control, McGraw-Hill, 1994. [2] J. G. Slootweg, J. Persson, A. M. van Voorden, G. C. Paap ,W. L.

Kling, A Study of the Eigenvalue Analysis Capabilities of Power

247

System Dynamics Simulation Software, 14th PSCC, Sevilla, 24th 28th

June, 2002. [3] K. R. Padiyar, Power System Dynamics: Stability and Control, John

Wiley & Sons, 1996. [4] Graham Rogers, Joe Chow/Cherry Tree Scientific Software, Power

System Toolbox version 2.0 Dynamic Tutorials and Functions, 2003 [5] Graham Rogers, Joe Chow/Cherry Tree Scientific Software, Power

System Toolbox version 2.0 Loadflow Tutorial and Functions, 2003 [6] IEEE Guide for Synchronous Generator Modelling Practices in

Stability Analysis, IEEE Std 1110-1991. [7] Task Force 07 of Advisory Group 01 of Study Committee 38, Analysis

and Control of Power System Oscillations, Final report, December 1996, CIGRE

[8] J. Persson, J. G. Slootweg, L. Rouco, L. Sder, and W. L. Kling, A Comparison of Eigenvalues Obtained with Two Dynamic Simulation Software Packages, Accepted for presentation at 2003 IEEE Bologna Power Tech Conference, 23rd 26th June, Bologna, Italy, Paper 0-7803-7967-5/03.

[9] Graham Rogers, Power System Oscillations, Kluwel Academic Publishers, 2000

[10] Emil Johansson, Jonas Persson, Lars Lindkvist, Lennart Soder, Location of Eigenvalues Influenced by Different Models of Synchronous Machine presented at the Sixth IASTED International Conference POWER AND ENERGY SYSTEMS, May 13-15, 2002, Marina del Rey California, USA.

[11] K. K. Kaberere, K. A. Folly, M. Ntombela, A. I. Petroianu, Comparative Analysis and Numerical Validation of Industrial-Grade Power System Simulation Tools: Application to Small-Signal Stability.

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