transient stability assessment of power systems described with detailed models using neural networks

Electrical Power and Energy Systems 45 (2013) 279–292

Contents lists available at SciVerse ScienceDirect

Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Transient stability assessment of power systems described with detailedmodels using neural networks

A. Karami ⇑, S.Z. Esmaili 1

Faculty of Engineering, University of Guilan, P.O. Box 41635-3756, Rasht, Iran

a r t i c l e i n f o

Article history:Received 28 January 2012Received in revised form 16 July 2012Accepted 29 August 2012Available online 7 November 2012

Keywords:Transient stabilityCritical clearing time (CCT)Synchronous machine modelsPower System Analysis Toolbox (PSAT)Neural networks (NNs)

0142-0615/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijepes.2012.08.071

⇑ Corresponding author. Tel.: +98 131 6690274 8; fE-mail addresses: [email protected], karam

[email protected] (S.Z. Esmaili).1 Tel.: +98 131 6690274 8; fax: +98 131 6690271.

a b s t r a c t

This paper investigates the use of multi-layered perceptron (MLP) neural network (NN) for assessing thetransient stability of a power system considering the detailed models for the synchronous machines, andtheir automatic voltage regulators (AVRs). Two MLP NNs are employed here to estimate the critical clear-ing time (CCT) and a transient stability time margin (TM), as indicators for measuring power system tran-sient stability for a particular contingency under different system operating conditions. The training ofMLP NNs is accomplished using some carefully chosen system features as the inputs and the CCT and/or the TM as the desired targets. In this paper, the required training and/or testing patterns for the neuralnetwork are obtained by performing time-domain simulation (TDS) on the New England 10-machine 39-bus test system and the IEEE 16-machine 68-bus test system with fourth-order machines models andtheir AVRs using the software tool PSAT (Power System Analysis Toolbox), whereas the proposed neuralnetwork models are implemented in MATLAB. In addition, a neural network based sensitivity method andprincipal component analysis (PCA) are employed to reduce the dimension of the input data vectors. Thesimulation results obtained prove that the trained neural networks give satisfactory estimations for bothCCT and TM.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In power system stability studies the term transient stabilityusually refers to the ability of the synchronous machines to remainin synchronism during the brief period following large distur-bances, such as severe lightning strikes, loss of heavily loadedtransmission lines, loss of generation stations, or short circuits onbuses [1]. In large disturbances system nonlinearities play a dom-inant role. In order to determine transient stability or instabilityfollowing a large disturbance, or a series of disturbances, time-domain simulation (TDS) method is usually employed to solvethe set of nonlinear equations describing the system’s dynamicbehavior. Conclusion about stability or instability can then bedrawn from an inspection of the solution [2]. Critical clearing time(CCT) is a well-known indicator that can be used for measuringpower system transient stability.

The continues increase of load demand along with economicaland environmental constraints for building new power plantsand transmission lines, have led power systems to operate closerto their limits and transient stability problem has become more

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ax: +98 131 [email protected] (A. Karami),

likely to occur [3,4]. In the actual operation phase of an electricpower system there might be many changes in the system operat-ing conditions and/or configuration; therefore, the transient stabil-ity studies that were done in the planning phase of a power system,would not be reliable for the actual system operation, and it is nec-essary to repeat those studies so that system operators could takeproper preventative control operations if insecure system statesencountered.

Considering the fact that time domain simulation of the powersystem dynamic equations is a very time consuming task even byusing today’s modern computers, other techniques such as thetransient energy function (TEF) method [5,6], and the extendedequal area criterion (EEAC) [7,8] have been applied in power sys-tem transient stability studied in order to reduce the computa-tional burden of solving system’s equations. Although in recentdecades significant improvements have been made in the applica-tion of TEF and EEAC, these methods become increasingly complexwhen detailed models are considered [9–13]. Therefore, time-domain simulation is still the most accurate method for transientstability analysis and it can be applied to any level of power systemmodels, but as mentioned before, the main problem of this methodis that it is very time consuming. As a result, in recent years therehave been several attempts in using computational intelligencebased techniques like neural networks (NNs) for transient stabilityassessment (TSA) of power systems, e.g., see [14–25]. Neural

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280 A. Karami, S.Z. Esmaili / Electrical Power and Energy Systems 45 (2013) 279–292

networks are widely used for function approximation and/orclassification problems, because no rigorous mathematical systemmodeling is required in order to train an NN to form the underlyingmapping. Therefore, the application of neural networks in manyscientific fields such as power systems is an area of growinginterest [26].

Sobajic and Pao [14] used NNs for prediction of the CCT for asmall test power system. Djukanovic et al. [15] used individual en-ergy function normalized by the critical value of global energyfunction evaluated at fault clearing time (FCT) to predict energymargin and stability. Pao and Sobajic [16] proposed a combinedusage of unsupervised and supervised learning for TSA. Hobsonand Allen [17] reported that the NNs have difficulty in returningconsistent accurate answers under varying network conditions.In Ref. [18], Aboytes and Ramirez used NNs to predict stability ofa 53 generators system. In Ref. [19], Bahbah and Girgis used therecurrent radial basis function (RBF) and the multi-layered percep-tron (MLP) NNs for dynamic system modeling, the generators’ an-gles and angular velocities prediction for TSA. In Ref. [20], Sawhneyand Jeyasurya employed neural networks to predict a transient sta-bility index, which was obtained by the extended equal area crite-rion method. In Ref. [21], a procedure has been described forextracting rules from a trained MLP NN for reasoning power sys-tems critical clearing time. A similar method has been proposedin Ref. [22] for transient stability preventive control incorporatingseries compensation.

In Ref. [23], a generalized regression neural networks basedclassification has been proposed for transient stability evaluationin power systems. In Ref. [24,25], two methods for transient stabil-ity analysis using neural networks based on adaptive resonancetheory (ART) architecture have been presented. To reduce thenumber of calculations and the respective cut in computationaltime, the major portion of calculations for power system transientstability assessment with a mathematical model has been replacedby an estimation procedure in Ref. [27].

The above-mentioned methods for transient stability monitor-ing using neural networks have led to acceptable results. However,the main drawback of the previously published works in this areais that most of them are based on the so-called classical model forthe power system. However, if accurate and reliable results areneeded, it is necessary to consider detailed machines models andtake into account their control devices [9–11]. This is particularlytrue in today’s power systems in which many electronic devicesare employed to increase the stability limits of the system due todifficulty of building new transmission and distribution systems[12,13].

In a study reported in Ref. [28], a methodology for estimating anormalized power system transient stability margin of powersystems with the classical models by means of a multi-layered per-ceptron neural network has been presented. In this paper, we ex-tend our previous work [28] and estimate the CCT and atransient stability time margin (TM) by using two MLP NNs consid-ering detailed models for the synchronous machines and theirexcitation systems that have considerable effects on transientstability. The Matlab-based free and open source software toolPSAT (Power System Analysis Toolbox) [29], is used here for per-forming time-domain simulations in order to prepare the requiredtraining data for neural network, and the proposed method is ap-plied to New England 10-machine 39-bus test system and the IEEE16-machine 68-bus test system. In addition, a neural networkbased sensitivity method [20,30] and principal component analysis(PCA) [20] are employed to reduce the dimension of the input datavectors.

This paper is organized as follows: Section 2 describes modelsused to represent the multi-machine power system transient per-formance. Section 3 provides the methodology used for imple-

menting the two test systems in PSAT, and Section 4 presents theproposed MLP neural network based method. Section 5 representsthe simulation results. Finally, conclusion is drawn in Section 6.

2. Multi-machine power system model

In this section, the synchronous machine and the automaticvoltage regulator models are described.

2.1. Synchronous machine model

In general, for transient electromechanical phenomena analysisof a power system, power flow algebraic equations for the trans-mission network as well as for the stator windings of the synchro-nous machines, along with the differential equations for the rotorof the synchronous machines are employed. Therefore, themathematical model of a power system can be represented by aset of differential and algebraic equations (DAEs). In this paper,the two-axis model is used to describe the synchronous machines.The fourth-order differential equations of the ith synchronousmachine of an n-machine system are expressed as follows [3,4]:

ddi;

dt¼ xi �xs; i ¼ 1;2; :::;n ð1Þ

dxi

dt¼ 1

MiPmi � ðE0qi � X 0diIdiÞIqi � ðE0di þ X 0qiIqiÞIdi � Diðxi �xsÞh i

;

i ¼ 1;2; :::;n ð2Þ

dE0qi

dt¼ 1

T 0doi

Efdi � E0qi � ðXdi � X0diÞIdi

h i; i ¼ 1;2; :::;n ð3Þ

dE0di

dt¼ 1

T 0qoi

�E0di þ ðXqi � X0qiÞIqi

h i; i ¼ 1;2; :::;n ð4Þ

where di is the angular position of rotor, xi is the rotor speed, E0di

and E0qi are d-axis and q-axis transient voltages for machine i,respectively. xs is the synchronous speed, and Mi is the inertia con-stant and Di is the damping constant for machine i. The variables Idi

and Iqi are the d-axis and q-axis currents, respectively, and theparameter variables T 0doi and T 0qoi are the d-axis and q-axis open cir-cuit time constants, respectively. Xdi and Xqi respectively representd-axis and q-axis synchronous reactance. X0di and X0qi represent d-axis and q-axis transient reactance, respectively. Pmi is the mechan-ical input power for machine i and it is assumed constant.

The link between machine voltages and the power networkphasors is as follows:

E0qi ¼ Vi cosðdi � hiÞ þ RsiIqi þ X0diIdi; i ¼ 1;2; :::;n ð5Þ

E0di ¼ Vi sinðdi � hiÞ þ RsiIdi � X0qiIqi; i ¼ 1;2; :::;n ð6Þ

where Vi is the terminal voltage of the machine i in per unit and hi isits angle, and Rsi is the armature resistance for the machine i.

2.2. Automatic voltage regulator

The automatic voltage regulator (AVR) defines the primary volt-age regulation of the synchronous machines. Here, we used thesimplified IEEE model 1 which can be defined by AVR Type II inPSAT for the excitation system, as shown in Fig. 1. The mathemat-ical model of the AVR is as follows [3,4,29]:

dVri

dt¼ 1

Tai½�Vri þ KaiðVrefi � Vi � VfiÞ�; i ¼ 1;2; :::;n ð7Þ

refV +

− −V

+

a

a

sT

K

+1

minrV

maxrV

rV +

)( fde EfS =

esT+1

1

−fdE

f

f

sT

sK

+1

fV

Fig. 1. IEEE type 1 excitation system.

2 For interpretation of color in Figs. 2 and 5, the reader is referred to the webversion of this article.

A. Karami, S.Z. Esmaili / Electrical Power and Energy Systems 45 (2013) 279–292 281

dEfdi

dt¼ 1

Tei½Vri � ð1þ SeiðEfdiÞÞEfdi�; i ¼ 1;2; :::;n ð8Þ

dVfi

dt¼ 1

Tfi½�Vfi � ð1þ SeiðEfdiÞÞKfiEfi=Tei þ KfiVri=Tei�;

i ¼ 1;2; :::;n ð9Þ

where Vref is the reference voltage of the AVR. Vr and Vf are the out-puts of the AVR and excitation system stabilizer (feedback), respec-tively. Efd is the voltage applied to the synchronous generator fieldwinding. Ta, Te and Tf are the AVR, exciter and excitation system sta-bilizer time constants, respectively. Furthermore, Ka and Kf are gainsof AVR and excitation system stabilizer, respectively. Vrmin and Vrmax

are the lower and upper limits of Vr (the exciter ceiling voltages),respectively. Se is the exciter saturation function which is definedas:

SeðEfdÞ ¼ AeðeBe jEfd j � 1Þ ð10Þ

where Ae and Be are constants chosen to match the open-circuitmagnetization curve at two points, usually Emax

fd ; and 0:75Emaxfd .

3. Power system implementation in PSAT

In this paper the well-known New England 10-machine 39-bustest power system along with the IEEE 16-machine 68-bus testpower system are employed as examples of two large powersystems, to demonstrate the proposed MLP neural network basedapproach. Both test systems were implemented using the softwaretool Power System Analysis Toolbox (PSAT) [29] via the PSAT-Simulink library as shown below.

3.1. Implementing the New England 39-bus test system in PSAT

In this sub-section, the New England 10-machine 39-bus testsystem implementation in PSAT is completely described. It shouldbe emphasized that the methodology of the proposed approach isapplicable to any other system in a similar manner. Therefore,implementation of the IEEE 16-machine 68-bus test system inPSAT will be described briefly in the next sub-section.

Static data for the New England 39-bus test system along withdynamic data for its generators as well as their exciters and AVRscan be found in Ref. [5]. In PSAT, the synchronous machines are ini-tiated after power flow computations. A PV or a slack generator isrequired to impose the desired voltage and active power at the ma-chine bus. Synchronous machines controls such as AVRs or TurbineGovernors are not included in the machine models. Furthermore,the voltage ratings of all system equipments in kV need to be spec-ified in PSAT. However, the New England 39-bus test system dataprovided in Ref. [5] are all based on per unit values. Thus, referring

to the many references, we chose three voltage levels for the sys-tem equipments. The voltage ratings for all machines were decidedto be 13.8 kV. Then, the voltage ratings of the other equipmentswere chosen to be 345 kV and 115 kV, according to the existingtransformers in the system. But, in the New England 39-bus testsystem mentioned in Ref. [5], the slack generator (i.e., bus 1),which is the biggest generator in the system, is directly connectedto the system without any transformer. Therefore, based on thevoltage ratings chosen for the system equipments, the slack gener-ator should be connected to a 345 kV bus, that indeed there is nogenerator working with this voltage level in practice. To cope withthis problem, bus 1 connected to the system through a 13.8/345 kVstep up transformer with a very small reactance of 0.00001 perunit. Thus, the original 39-bus New England 39-bus test systemwas changed to a 40-bus system, as shown in Fig. 2. In Fig. 2, the13.8 kV, 115 kV and 345 kV buses are indicated with black, greenand blue colors, respectively.2

Bus 1 was taken as the slack generator with a constant excita-tion; i.e., all its excitation parameters were set to zero. The remain-ing generation buses (i.e., buses 2–10) were considered as PVbuses, and their excitation systems parameters were obtained from[5]. Each generator was described by a two-axis fourth-order mod-el with a uniform damping (i.e., k = Di/Mi) equal to 0.01. Besidesone slack bus and 9 PV buses, the test system consists of additional30 PQ buses (i.e., buses 11–40). However, only 19 buses in the sys-tem have nonzero real and reactive loads. All loads were modeledas constant impedances.

As shown in Fig. 2, a sample contingence is applied to the NewEngland 39-bus test system. This contingency is a three-phaseshort-circuit ground fault at bus 31. It is assumed that this faultis eliminated by opening the transmission line connected betweenbus 31 and bus 25 in the post-fault system. We represent this con-tingency as Fault 31�–25. This contingence is randomly chosen toillustrate the proposed methodology. However, the number of con-tingences can be increased according to the needs of the user,without any problem related to the formulation. As seen inFig. 2, to define this fault, both a fault model and a breaker modelfrom PSAT were used. We then carried out the time-domain simu-lations to determine the critical clearing time by visualizing gener-ators relative rotor angles with respect to Center of Inertia (COI)reference frame.

The CCT for the above contingency in the base case loading con-ditions was found to be 0.182 s by using a trial-and-error approach.Fig. 3 shows the relative rotor angles with respect to the COI refer-ence frame for Fault 31�–25 for the marginally (critically) stablecase. Vertical and horizontal axes of this figure show generatorsrotor angles in terms of degree and time in terms of second,

Bus9

Bus8

Bus7

Bus6

Bus5

Bus40

Bus4

Bus39

Bus38

Bus37

Bus36

Bus35

Bus34

Bus33

Bus32

Bus31

Bus30

Bus3

Bus29Bus28

Bus27

Bus26

Bus25

Bus24

Bus23

Bus22Bus21

Bus20

Bus2

Bus19

Bus18

Bus17Bus16

Bus15

Bus14

Bus13

Bus12

Bus11

Bus10

Bus1

Fig. 2. PSAT-Simulink implemented model of the New England 39-bus test system.

0 1 2 3 4 5 6 7 8 9 10-50

0

50

100

150

200

time (s)

Rot

or a

ngle

s un

der C

OI (

degr

ee)

θ1θ 2θ 3θ 4θ 5θ 6θ7θ8θ9θ10

Fig. 3. Swing curves for generators after Fault 31�–25 in the New England 39-bussystem using detailed model (with FCT = 0.182 s).

0 1 2 3 4 5 6 7 8 9 10-60

-40

-20

0

20

40

60

80

100

120

140

time (s)

Rot

or a

ngle

s un

der C

OI (

degr

ee)

θ1θ2θ3θ4θ5θ6θ7θ8θ9θ10

Fig. 4. Swing curves for generators after Fault 31�–25 in the New England 39-bussystem using classical model (with FCT = 0.3 s).


respectively. The fault is set to occur at 0.1 s from the beginning ofthe simulation. The fault is then cleared by opening the line 31–25at 0.282 s (i.e., the FCT is assumed to be 0.182 s here, which is equalto the CCT). As expected, the system is marginally stable in thiscase.

To compare the CCT obtained by using the detailed systemmodel with that of using the classical system model, we also per-formed some simulations by employing the classical model for

the New England 39-bus test system. In the classical model of apower system, the synchronous generators are represented electri-cally as constant voltage sources behind their d-axis transient reac-tance, without taking into account any control system (exciters,governors, etc.) [9,11]. In this model, the generator voltages are as-sumed constant, only their phase angles change. In the classicalmodel, each generator is modeled by a second order differentialequation of two dynamic state variables (d, x). Damping constants

'To Bus61'

'To Bus53'

Bus9

Bus8

Bus7

Bus68

Bus67

Bus66

Bus65

Bus64

Bus63

Bus62

Bus60

Bus6

Bus59

Bus58

Bus57

Bus56

Bus55

Bus54

Bus52

Bus5

Bus4

Bus37

Bus3

Bus29

Bus28

Bus27

Bus26

Bus25

Bus24

Bus23

Bus22

Bus21

Bus20

Bus2

Bus19

Bus1

Fig. 5. PSAT-Simulink implemented model of the IEEE 68-bus test system.


are usually neglected in the classical two-order model for the gen-erators; however, we have used here a uniform damping equal to0.01 (as in the case for detailed system model) in the rotor motionto approximate electrical damping generated by damping wind-ings of the generators.

Using the above-mentioned classical model, the CCT corre-sponding to the Fault 31�–25 in the New England 39-bus test sys-tem in the base case loading conditions was found to be 0.3 s,which is very greater than the corresponding value obtained byusing the detailed New England 39-bus test system model, i.e.,0.182 s. The swing of rotor angles oscillations simulated for themarginally stable case (i.e., for FCT = 0.3 s), considering the simpleclassical machines model without exciter dynamics of the NewEngland 39-bus test system is shown in Fig. 4. As can be seen inFig. 4, the system has a very oscillatory behavior due to usage ofthe classical system model. It should be noted that the classicalmodel is only valid for a short period of study, say 1 s or less. How-

ever, as shown in Fig. 4, we have performed time-domain simula-tions for a long period of 10 s even for the case of classical systemmodel. This is deliberately done to show the inaccuracy of theclassical model in estimating the actual behavior of the systemfollowing a contingency.

The question that now arises is that how we can improve theCCT by using a fast-acting excitation system. As is well known,the excitation system of a generator can contribute to the effectivecontrol of system voltage and enhancement of system stability [9].Many research results show that with a fast-acting AVR and a highexciter ceiling voltage, the first rotor angle swing is significantlyreduced and subsequent swings are well damped by introducingauxiliary stabilizing signals [3,4]. To examine the effect of AVRaction on transient stability, we preformed some time domainsimulations in the detailed model of the New England 39-bus testsystem, in which both the system excitation gain (or AVR gain) andthe exciter ceiling voltages of all system machines were replaced

'From Bus54'

'From Bus27'

'From Bus60'

Bus61

Bus53

Bus51

Bus50

Bus49

Bus48

Bus47

Bus46

Bus45

Bus44

Bus43

Bus42

Bus41

Bus40

Bus39

Bus38

Bus36 Bus35

Bus34

Bus33

Bus32

Bus31

Bus30

Bus18

Bus17

Bus16

Bus15

Bus14

Bus13

Bus12

Bus11

Bus10

Fig. 5. (continued)


by 10 times of their rated values as given in Ref. [5]. The CCTcorresponding to Fault 31�–25 for this new model of the NewEngland 39-bus test system was found to be 0.274 s, which is verygreater than its previous value, i.e., 0.182 s. This clearly demon-strates that fast-acting exciters with high ceiling voltages coupledwith high gain AVRs can improve transient stability. Therefore, asexplained in Section 4, we will use the AVR gain along with theexciter ceiling voltage as part of proposed neural networks inputs,because they are two important parameters that affect powersystem transient stability greatly.

3.2. Implementing the IEEE 68-bus test system in PSAT

The IEEE 16-machine 68-bus test system is also implemented inPSAT using the same procedure as described for the case of theNew England 39-bus test system in the previous sub-section. Thisis a reduced order equivalent of the interconnected New England39-bus test system and New York power system. The numericaldata for different model parameters of this system is provided inRef. [31]. It consists of 16 generators, 68 buses and 86 lines. Allthe generators of this test system (i.e., G1–G16) which are con-

nected to buses 1–16, are represented with fourth-order modelswith their AVRs using the PSAT.

Here, bus 16 was taken as the slack generator under manualexcitation control, which means no AVR was assumed to be con-nected to this bus. The generators G1–G9 were equipped with sim-plified IEEE model 1 excitation system (or AVR Type II in PSAT)with the same parameters as those given in Ref. [31]. The rest ofthe generators, i.e., G10–G15, are under manual excitation controllike the G16 (slack bus). The generators damping constants (val-ues) were taken according to the data given in Ref. [31]. Besidesone slack bus and 15 PV buses (i.e., buses 1–15), this test systemconsists of additional 52 PQ buses. However, only 35 buses in thistest system have nonzero real and reactive loads. All loads weremodeled as constant impedances. Fig. 5 shows the single-line dia-gram of IEEE 68-bus test system implemented in PSAT. As for thecase of the New England 39-bus test system, in Fig. 5, the13.8 kV, 115 kV and 345 kV buses are indicated with black, greenand blue colors, respectively.

As shown in Fig. 5, a sample contingence is applied to the IEEE68-bus test system. This contingency is a three-phase short-circuitground fault at bus 57. It is assumed that this fault is cleared by

0 1 2 3 4 5 6 7 8 9 10-100

-50

0

50

100

150

200

Rot

or a

ngle

s un

der C

OI (

degr

ee)

time (s)

θ1θ2θ3θ4θ5θ6θ7θ8θ9θ10θ11θ12θ13θ14θ15θ16

Fig. 6. Swing curves for generators after Fault 57�–58 in the IEEE 68-bus systemusing detailed model (with FCT = 0.222 s).

0 1 2 3 4 5 6 7 8 9 10-100

-50

0

50

100

150

200

Rot

or a

ngle

s un

der C

OI (

degr

ee)

time (s)

θ1θ2θ3θ4θ5θ6θ7θ8θ9θ10θ11θ12θ13θ14θ15θ16

Fig. 7. Swing curves for generators after Fault 57�–58 in the IEEE 68-bus systemusing classical model (with FCT = 0.296 s).


isolating the transmission line connected between bus 57 and bus58 in the post-fault system. We represent this contingency as Fault57�–58. We then carried out the time-domain simulations to deter-mine the critical clearing time by visualizing generators relativerotor angles with respect to COI reference frame.

The CCT for the above contingency in the base case loading con-ditions was found to be 0.222 s by using a trial-and-error approach.

Table 1Damping constants and excitation system data for the New England 39-bus test system.

Generator no. Damping constant (D) AVR gain (Ka)

1 10 02 0.606 6.23 0.716 54 0.5 55 0.52 406 0.692 57 0.528 408 0.486 59 0.69 40

10 0.84 5

Fig. 6 shows the relative rotor angles with respect to the COI refer-ence frame for Fault 57�–58 for the marginally stable case. Verticaland horizontal axes of this figure show generators rotor angles interms of degree and time in terms of second, respectively. The faultis set to occur at 0.1 s from the beginning of the simulation. Thefault is then cleared by opening the line 57–58 at 0.322 s (i.e.,the FCT is assumed to be 0.222 s here, which is equal to the CCT).As expected, the system is marginally stable in this case.

To compare the CCT obtained by using the detailed systemmodel with that of using the classical system model, we also per-formed some simulations by employing the classical model forthe IEEE 68-bus test system. As mentioned before, in the classicalmodel of a power system, each generator is modeled by a secondorder differential equation of two dynamic state variables (d, x)and its excitation system is ignored. Damping constants are usuallyneglected in the classical two-order model for the generators;however, we have used here the same damping constants as thoseused for the detailed IEEE 68-bus system model, in the rotor mo-tion to approximate the electrical damping generated by dampingwindings of the generators.

Using the above-mentioned classical model, the CCT correspond-ing to the Fault 57�–58 in the IEEE 68-bus test system in the base caseloading conditions was found to be 0.296 s, which is bigger than thecorresponding value obtained by using the detailed IEEE 68-bus testsystem model, i.e., 0.222 s. The swing of rotor angles oscillationssimulated for the marginally stable case (i.e., for FCT = 0.296 s), con-sidering the simple classical machines model without exciterdynamics of the IEEE 68-bus test system is shown in Fig. 7. As previ-ously stated, the classical model is only valid for a short period ofstudy, say 1 s or less. However, as shown in Fig. 7, we have deliber-ately performed the time-domain simulations for a long period of10 s to show the inaccuracy of the classical model in estimatingthe actual behavior of the system following a contingency.

As can be seen in Fig. 7, in contrast to the New England 39-bustest system, the IEEE 68-bus test system shows a good stabilitybehavior even by using its classical model. This can be explainedby looking at the damping constants of these two systems as givenin Tables 1 and 2, respectively. All data here are in per unit (pu)with a base power of 100 MVA. Note that in Table 2 no voltage reg-ulator gain and no exciter ceiling voltages for generators 10–16 aregiven, because no exciter is equipped with these generators in theIEEE 68-bus test system. Comparing Tables 1 and 2, we can con-clude that for the 16 generators of the IEEE 68-bus, large amountof damping constants have been chosen in comparison with thedamping constants used for the 10 generators of the New England39-bus test system. This has resulted in more stable behavior of thegenerators in the case of IEEE 68-bus test system.

Similar to the case of the New England 39-bus test system, toexamine the effect of AVR action on transient stability, we pre-formed some time domain simulations in the detailed model ofthe IEEE 68-bus test system, in which both the gain of AVR andthe exciter ceiling voltages of all the system machines were

Upper ceiling voltage (Vrmax) Lower ceiling voltage (Vrmin)

0 01 �11 �11 �110 �101 �16.5 �6.51 �110.5 �10.51 �1

Table 2Damping constants and excitation system data for the IEEE 68-bus test system.

Generator no. Damping constant (D) AVR gain (Ka) Upper ceiling voltage (Vrmax) Lower ceiling voltage (Vrmin)

1 4 40 10 �102 9.75 40 10 �103 10 40 10 �104 10 40 10 �105 3 40 10 �106 10 40 10 �107 8 40 10 �108 9 40 10 �109 14 40 10 �10

10 5.5611 13.612 13.513 6614 10015 10016 100

Table 3Set of contingences along with their transient stability time margins in the base case loading conditions.

Test system Contingency Fault at bus Line tripped Critical clearing time (CCT) (s) Assumed fault clearing time (FCT) (s) Time margin (TM)

New England 39-bus Fault 31�–25 31 31–25 0.182 0.15 0.1758IEEE 68-bus Fault 57�–58 57 57–58 0.222 0.18 0.1892


replaced by 10 times of their rated values given in Ref. [31]. TheCCT corresponding to Fault 57�–58 for this new model of the IEEE68-bus test system was found to be 0.255 s, which is greater thanits previous value, i.e., 0.222 s. This also demonstrates that fast-acting exciters with high ceiling voltages coupled with high gainAVRs are beneficial in improving transient stability. In Tables 1and 2, the rated AVR gains and the exciter ceiling voltages of thegenerators for both New England 39-bus test system and IEEE68-bus test system are also shown for further analysis. As can beseen in Table 2, large amount of damping constants have beenassumed for the generators in the IEEE 68-bus test; so increasingthese damping constants by a factor of 10 has resulted in only asmall improvement in the value of CCT.

4. The proposed methodology

In this section, we represent our multi-layered perceptron neu-ral network based approach by using the New England 39-bus testsystem and the IEEE 68-bus test shown in Figs. 2 and 5, respec-tively. As mentioned earlier, our aim is to estimate the criticalclearing time (CCT) for the Fault 31�–25 in the New England 39-bus test system using an MLP NN. In addition, we want to estimatethe CCT for the Fault 57�–58 in the IEEE 68-bus test system usinganother MLP NN. Here, the CCT is used as an indicator for measur-ing power system transient stability.

In practice, the fault clearing time (FCT) of the system protec-tion devices (relays) are set to a pre-determined value and remainsfixed, at least for a period of time. Therefore, it is possible to defineanother index for measuring system transient stability by compar-ing the CCT with FCT used for the relays operation. This indexwhich is called here as the transient stability time margin (TM),is calculated differently for stable and unstable cases as:

TM ¼CCT�FCT

CCT ; if system is stableðCCT > FCTÞCCT�FCT

FCT ; if system is unstableðCCT < FCTÞ

(ð11Þ

The sign of TM indicates whether the system is stable or unstable.It can be easily shown that the value of TM lies between�1.0 and 1.0.

If TM > 0, the system is stable, and if TM < 0, the system is unstable.This transient stability time margin represents a quantitative mea-sure of the degree of stability (or instability) of the system.

As defined in (11), to obtain the TM we need to know the valueof FCT for the specified fault scenario. Here, for the two mentionedfaults chosen in the two test systems (i.e., Fault 31�–25 and Fault57�–58), the FCTs are assumed to be 0.15 s and 0.18 s, respectivelywhich are less than their corresponding CCT values in the base caseloading conditions. Considering these values for the FCT of the twofault scenarios, the values of TM for the base case loading condi-tions are found to be 0.1758 and 0.1892, respectively. In Table 3,the two fault scenarios along with their assumed FCTs and theirCCTs and TMs in the base case loading conditions are shown.

To estimate the CCT and/or the TM for a particular fault (contin-gency) in a given power system by using neural networks, we needto know to which parameters and conditions the CCT and/or theTM, depend on? It is obvious that different contingences lead todifferent values for the CCT and/or the TM. Furthermore, whenthe pre-fault system’s operating conditions and/or its configuration(topology) change, the corresponding values of CCT and/or TM fordifferent contingences also change. Therefore, the CCT and/or theTM are complex functions of the pre-fault system’s operating con-ditions (operating point) as well as its configuration, fault type andlocation, and the post-fault system configuration. Assuming thatthe pre-fault system’s configuration is fixed, a particular contin-gency can be described by the type of fault, its location, and thecorresponding post-fault topology. For each particular contin-gency, such as the contingencies mentioned above for the two testsystems, the CCT and/or the TM will change only when the pre-fault system operating conditions change. On the other hand, fora particular contingency, the CCT and/or the TM are indeed func-tions of only the pre-fault system operating point. Thus, pre-faultsystem operating conditions can be used to estimate the CCTand/or the TM using MLP NNs. The next question is that what oper-ating conditions will result in a compact and efficient neural net-work? This question has already been answered in Ref. [28].With the same assumptions as used in Ref. [28], the followingpre-fault system operating conditions can be treated as the MLPNN inputs to estimate the CCT and/or the TM corresponding to aparticular fault scenario in a given power system:

Multi-LayeredPerceptron (MLP) Neural Network

(NN)

Voltage magnitudesof all PV buses

Active and reactive generated powers

of all machines

Critical Clearing Time (CCT) or

Time Margin (TM)

Load active and reactive powers

demands at all buses

Total active power generated in

the system

Total reactive power generated in

the system

AVR gain and exciter ceiling

voltage

Fig. 8. Block diagram of the proposed MLP NN based approach.


– terminal voltage magnitudes of all the PV buses (V’s),– generated active powers of all the PV buses (PG’s),– active power demands of all the loads (PD’s),– reactive power demands of all the loads (QD’s).

The New England 39-bus test system consists of 9 PV bus and19 loads. Therefore, the CCT and/or the TM for each fault scenarioin the New England 39-bus test system will be functions of 56(9 + 9 + 19 + 19) system operating conditions. Similarly, the IEEE68-bus test system consists of 15 PV bus and 35 loads. Therefore,the CCT and/or the TM for each fault scenario in the IEEE 68-bustest system will be functions of 100 (15 + 15 + 35 + 35) systemoperating conditions.

It should be noted that the operating conditions of both test sys-tems are changed here without enforcing any limits on maximumand/or minimum generated reactive powers of the generators (i.e.,QG-limits are not taken into account), leading us to use the above-mentioned independent system operating conditions as the MLPNNs inputs. However, other system variables can also be used asthe MLP NNs inputs. To consider the violations of generated reac-tive power of the system generators, which may take place in prac-tice, and also to include additional information regarding thesystem total loading conditions, we have used here the followingadditional inputs for the MLP NNs:

– generated reactive powers of all the system generation buses(QG’s),

– generated active power of the slack bus (PG1),– system total active power generated by its generators (TPG),– system total reactive power generated by its generators (TQG).

Based on the above explanations, for the New England 39-bustest system, the total number of the MLP NN inputs is thus 69(56 + 10 + 1 + 1 + 1). The output of the NNs will be the CCT and/or the TM values for the selected contingency. Similarly, for theIEEE 68-bus test system, the total number of the MLP NN inputsis thus 119 (100 + 16 + 1 + 1 + 1). The output of the NNs will bethe CCT and/or the TM values for the selected contingency.

As previously mentioned in Section 3, the gain of AVR (Ka) andthe exciters ceiling voltages (Vrmin and Vrmax) are two importantfactors affecting power system transient stability. Therefore, wehave assumed here that the gain of AVR and the exciter ceilingvoltages may change due to various unforeseen factors that areassociated with a power system. To consider the variations of thegain of AVR and the exciter ceiling voltages in the values of CCTand/or TM, we need to include these parameters as parts of neuralnetworks inputs as well. In New England 39-bus test system, 9generators are equipped with fast-acting exciters whose relevantparameters are given in Table 1. As can be seen in Table 1, the low-er and upper limits of Vr (exciter ceiling voltages) for all the excit-ers are equal in magnitude. Therefore, we only need to consider 9AVR gains along with 9 upper limits of Vr for the New England 39-bus test system as the neural networks inputs. Thus, the total num-ber of neural networks inputs will be 87 (69 + 9 + 9) for the NewEngland 39-bus test system.

Similarly, in the IEEE 68-bus test system, only 9 generators areequipped with fast-acting exciters whose relevant parameters aregiven in Table 2. As can be seen in Table 2, the lower and upperlimits of Vr (exciter ceiling voltages) for all the exciters are againequal in magnitude. Therefore, we only need to consider 9 AVRgains along with 9 upper limits of Vr for the IEEE 68-bus test sys-tem as neural networks input too. Thus, the total number of neuralnetworks inputs will be 137 (119 + 9 + 9) for the IEEE 68-bus testsystem. Fig. 8 shows a block diagram of the proposed neural net-work to estimate the CCT and/or TM in a given power system.

It is to be noted that the block diagram shown in Fig. 8 can beused to estimate the CCT and/or the TM for all credible contingen-cies in a given power system. However, it is necessary to develop aset of neural networks for a set of contingences for the system un-der study, because different training pattern sets for CCT and/or TMare needed for each contingency. In this paper, we have assumedjust one fault scenario in both test systems for simplicity. Obvi-ously, the generalization capability of the proposed neural net-works depend on the data used in the training process. It is to benoted that since all the above-mentioned system operating condi-tions are easily available in an energy control center (ECC), the CCT

Table 4Summary of the neural networks training results.

Test system Neural network output Number of NN inputs Number of NN hidden neurons MSE for training patterns RMSE for testing patterns

New England 39-bus CCT 87 10 0.005 0.0342New England 39-bus TM 87 10 0.005 0.1461

IEEE 68-bus CCT 137 10 0.005 0.0177IEEE 68-bus TM 137 10 0.005 0.0603

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35Comparison of actual and estimated CCT

Crit

ical

cle

arin

g tim

e (C

CT)

, sec

.

Test points

Actual CCTEstimated CCT

0 5 10 15 20-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6Comparison of actual and estimated TM

Tim

e m

argi

n (T

M),

pu

Test points

Actual TMEstimated TM

(a) (b)

Fig. 9. Comparison of NN estimated output and the corresponding actual output in the New England 39-bus test system.

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3


Crit

ical

cle

arin

g tim

e (C

CT)

, sec

.

Test points


0 5 10 15 20-0.2

-0.1

0

0.1

0.2

0.3

0.4


Tim

e m

argi

n (T

M),

pu

Test points


(a) (b)

Fig. 10. Comparison of NN estimated output and the corresponding actual output in the IEEE 68-bus test system.


and/or TM can be quickly computed once the training of the neuralnetworks is completed.

5. The simulation results

The proposed MLP neural network based method for onlinetransient stability assessment has been applied to the New England39-bus and IEEE 68-bus test systems. The single-line diagrams ofthese systems are shown in Figs. 2 and 5, respectively. In this sec-tion the numerical results are represented. All the computationswere performed on a personal computer with 2.13 GHz Intel CoreProcessor and 2.73 GB of RAM running MATLAB 7.6.

5.1. Training patterns generation

As mentioned in Section 4, the CCT and/or the TM correspondingto a particular fault scenario in a given power system is a function ofsome easily available system operating conditions. For generatingtraining and/or testing patterns of the MLP NN in both test systems,active and reactive powers of all loads as well as voltage magni-tudes of PV buses are varied randomly within specified ranges oftheir base case values. In this paper, we assume that the range ofvariations of the voltage magnitudes of all the PV buses is boundedto 0.9–1.1 times their corresponding base case values. It is furtherassumed that both real and reactive loads at all buses are variedin the range 0.6–1.1 times their corresponding base values. The

vity

met

hod.

Red

uce

din

puts

toth

eN

NN

um

ber

ofN

Nin

puts

Nu

mbe

rof

NN

hid

den

neu

ron

sM

SEfo

rtr

ain

ing

patt

ern

sR

MSE

for

test

ing

patt

ern

s

V2,

V3,

V4,

V5,

V6,

V7,

V8,

V9,

V10

,PG

8,PG

2,PG

3,PG

9,PG

7,PG

4,PD

8,PG

1,V

rmax

-G8,

TPG

,TQ

G20

150.

010.

024

V2,

V3,

V4,

V5,

V6,

V7,

V8,

V9,

V10

,PG

10,P

G8,

PG2,

PG3,

PG9,

PG6,

PG7,

PG4,

PD1,

PG1,

QG

6,TP

G21

150.

010.

1086

V1,

V2,

V11

,V12

,V13

,V15

,V3,

V4,

V5,

V6,

V7,

V9,

PG2,

PG11

,PG

12,P

G13

,PG

14,P

G15

,PG

4,PG

5,PG

6,PG

7,PG

8,PG

9,TP

G25

150.

005

0.00

83

V1,

V2,

V11

,V12

,V13

,V14

,V15

,V3,

V4,

V5,

V7,

V9,

V10

,PG

2,PG

11,P

G14

,PG

3,PG

4,PG

6,PG

7,PG

8,PG

9,PG

1023

150.

005

0.02

88


change in loads is distributed among all generators in proportiontheir generated active powers in the base case conditions.

In addition, the gain of AVR and the exciter ceiling voltages arevaried randomly within specified ranges of their base case values.In this paper, we assume that the range of variations of the AVRgains is bounded to 0.5–1.1 times their corresponding base casevalues. It is further assumed the exciter ceiling voltages are variedin the range 0.8–1.2 times their corresponding base values. Then,the randomly generated sets of operating conditions using theabove procedure is verified by a conventional power flow programto make sure that each of the cases is a feasible power flow solu-tion. The cases for which the power flow does not meet steadystate operating requirements are removed. The verified operatingpoints are used as inputs to the MLP NN. The corresponding CCTand or TM values are used as target outputs for training and/ortesting the MLP NNs.

It should be noted that entering large number of operating con-ditions in the PSAT-Simulink implemented model of a power sys-tem is a very difficult task. To get rid of this problem, we usedthe PSAT script file (m-file) corresponding to PSAT-Simulink imple-mented models for both test systems, which are built by PSATautomatically. As a matter of fact, we carried out our required sim-ulations by using some custom made routines so that we were ableto obtain a complete training and/or testing pattern automatically.

The performance of the proposed MLP NN based method is eval-uated by calculating the Root Mean-Squared Error (RMSE) betweenthe actual Target (CCT or TM) and estimated Target (CCT or TM) ob-tained by the trained neural network. The definition is given by:

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

NP

XNP

p¼1

ðactual TargetðpÞ � estimated TargetðpÞÞ2vuut ð12Þ

where p represents the pattern number, and NP is the total numberof patterns in the corresponding set (training or testing).

Three cases for training the proposed MLP NN were considered.In the first case, all system operating conditions were used as theNN inputs. In the second case, a neural network based sensitivityanalysis was employed to reduce the number of proposed neuralnetworks inputs. In the third case, all system operating conditionswere first reduced by principal component analysis (PCA), and thenthe reduced and transformed inputs (features) were used for train-ing MLP NNs. Results obtained from all the cases are described be-low. The neural network toolbox of MATLAB, Mathmorks Inc., wasused for training the proposed MLP NNs [32].

Tabl

e5

Sum

mar

yof

the

neur

alne

twor

kstr

aini

ngre

sult

sfo

rth

eN

N-b

ased

sens

iti

Test

syst

emN

eura

ln

etw

ork

outp

ut

Nu

mbe

rof

redu

ced

inpu

ts

New

Engl

and

39-b

us

CC

T20

New

Engl

and

39-b

us

TM21

IEEE

68-b

us

CC

T25

IEEE

68-b

us

TM23

5.2. Neural networks training using all system operating conditions

Using the procedure described in the previous sub-section, forboth test systems, 2000 patterns were generated. Among 2000 pat-terns produced, 1500 patterns (75% of total patterns) were ran-domly selected to form the training set, and the remaining 500patterns (25% of total patterns) were used to compose the testingset for the MLP NNs. For the New England 39-bus test system,two MLP NNs with 87 inputs each and one output each corre-sponding to the CCT and TM were trained. Similarly, for the IEEE68-bus test system, two MLP NNs with 137 inputs each and oneoutput each corresponding to the CCT and TM were trained. Beforetraining, the input and output data patterns were scaled so thatthey fell in the range [�1,1].

To train an MLP NN, we need to choose a proper structure forthe neural network along with a suitable activation function forits neurons. It is well known that for the MLP neural networksthe numbers of hidden layers and hidden layers neurons are ob-tained by using a trial-and-error procedure [33]. Therefore, afterseveral trials, four MLP NNs with one hidden layer each, including


10 neurons, were found to be successful in estimating CCT and TMfor both test systems. Moreover, for all the employed MLP NNs,tangent hyperbolic transfer function and linear transfer functionwere used for the hidden layer neurons and the output layer neu-ron, respectively.

In Table 4, error goal, the Mean-Squared Error (MSE) betweenthe actual Target and the estimated Target in the training phasealong with the obtained RMSE for testing patterns for all thetrained MLP NNs are given. All the proposed MLP NNs were trainedusing the resilient back-propagation method, which is one of thefastest techniques for training large neural networks. It took about60 s with 1500 epochs on average to train each of the proposed

MAE ¼ 1Ntp

XNtp

p¼1

estimated TargetðpÞbefore input variation� estimated TargetðpÞafter input variationestimated TargetðpÞ before input variation

��

!ð13Þ

MLP NN. It should be noted that other fast training algorithm, suchas Levenberg–Marquardt could be employed to train the proposedMLP NNs [34].

As can be seen in Table 4, the proposed MLP NNs could estimatetheir corresponding targets with a very small RMSE value for thetesting patterns, proving the generalization accuracy of the trainedMLP NNs for both test systems. To see this better, the estimatedCCT and the corresponding actual CCT for Fault 31�–25 in theNew England 39-bus are compared in Fig. 9a for the first 20 testingpatterns duo to limited space. In addition, in Fig. 9b the estimatedTM and the corresponding actual TM for the same fault are alsocompared. It can be observed from this figure that the trainedMLP NNs estimate the actual CCT and/or TM with reasonable accu-racy at different system operating conditions.

Similarly, the estimated CCT and the corresponding actual CCTfor Fault 57�–58 in the IEEE 68-bus test system are compared inFig. 10a for the first 20 testing patterns duo to limited space. Inaddition, in Fig. 10b the estimated TM and the corresponding ac-tual TM for the same fault are also compared. It can be observedfrom this figure that the trained MLP NNs estimate the actualCCT and/or TM with reasonable accuracy at different system oper-ating conditions for the IEEE 68-bus test system as well.

5.3. Neural networks training using the NN-based reduced set of inputs

As shown in Section 4, the CCT and/or the TM corresponding toa particular fault scenario in the New England 39-bus test systemand the IEEE 68-bus test system are functions of 87 and 137 systemoperating conditions, respectively. However, there are relevantoperating conditions (inputs) that have important informationregarding the CCT and/or the TM in each test system, whereasirrelevant ones contain little information regarding the CCT and/or the TM. Therefore, we try to find inputs (features) that containas much information about the CCT and/or the TM as possible.Here, the neural networks based sensitivity method proposed inRefs. [20,30], is used to select the relevant inputs to the neuralnetworks.

In this approach the neural networks are first trained with allcorrelated inputs; then, the sensitivities of the neural networksoutput, i.e., the CCT and/or the TM, with respect to each inputare obtained by a simple way. Here, one input of the trained MLPNNs at a time is selected and is perturbed by a fixed percentage(e.g., 10%), and the rest of inputs remains constant. The input datais then fed to the trained MLP NNs and the corresponding neuralnetworks outputs (i.e., the CCT and/or the TM) are obtained. The

mean absolute (relative) error (MAE) between the outputs esti-mated by the trained neural network before and after each inputvariation is calculated. The steps are repeated for all the neural net-works inputs and the mean absolute error is calculated. After all in-puts have been selected, then those inputs whose change in valueleads to a big value for the mean absolute error are selected as thebest candidates for the input features (relevant inputs) of the pro-posed neural networks. Rest of inputs with small mean absoluteerror, are not considered as the possible candidates for neural net-works inputs. As a matter of fact, a big value for the MAE meansthat the perturbed variable influences the output greatly. Themean absolute error is calculated as:

where p represents the pattern number, and Ntp is the total num-ber of testing patterns.

We employed here the MLP NNs trained in Section 5.2 for theNew England 39-bus and the IEEE 68-bus test systems for estimat-ing the MAE corresponding to the CCT and/or the TM using theabove-mentioned neural networks based sensitivity method. Theresults obtained are summarized in Table 5 in which the relevantfeatures for each MLP NN are shown. Then the MLP NNs weretrained using only these relevant (reduced) features as the inputs.The results obtained along with some numerical values chosen fortraining each MLP NN are also shown in Table 5.

Comparing the test results shown in Table 5 with those men-tioned in Table 4, it is conclude that NN trained by using the re-duced number of inputs gives more accurate estimation for boththe CCT and TM than the cases in which all system operating con-ditions were used as the neural networks inputs. Results obtainedshow that a small number of inputs may contain all the requiredinformation for estimating the CCT and/or the TM. In addition,one benefit of reduction of neural network inputs is that it resultedin few tools for measuring the required information online.

As can be seen in Table 5, most of the obtained reduced inputs bythe NN-based sensitivity method consist of the voltage magnitudesand active generated powers of those generators which are in thevicinity of the fault locations in both test systems. This is com-pletely in agreement with our previous findings in Ref. [28] inwhich the voltage magnitudes and the active generated power ofthe most advanced machines were used as the relevant NN inputs.

In Fig. 11a the estimated CCT and the corresponding actual CCTfor Fault 31�–25 in the New England 39-bus are compared for thefirst 20 testing patterns duo to limited space. In addition, inFig. 11b the estimated TM and the corresponding actual TM forthe same fault are also compared.

Similarly, the estimated CCT and the corresponding actual CCTfor Fault 57�–58 in the IEEE 68-bus test system are compared inFig. 12a for the first 20 testing patterns duo to limited space. Inaddition, in Fig. 9b the estimated TM and the corresponding actualTM for the same fault are also compared. It can be observed fromthis figure that the trained MLP NNs estimate the actual CCTand/or TM with reasonable accuracy at different system operatingconditions.

5.4. Neural networks training using extracted features by PCA

As previously mentioned in Section 4, the CCT and/or the TMcorresponding to a particular fault scenario in the New England

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3


Crit

ical

cle

arin

g tim

e (C

CT)

, sec

.

Test points


0 5 10 15 20-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


Tim

e m

argi

n (T

M),

pu

Test points


(a) (b)

Fig. 11. Comparison of NN estimated output and the corresponding actual output in the New England 39-bus test system using the NN-based sensitivity method.

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3


Crit

ical

cle

arin

g tim

e (C

CT)

, sec

.

Test points


0 5 10 15 20-0.2

-0.1

0

0.1

0.2

0.3

0.4


Tim

e m

argi

n (T

M),

pu

Test points


(a) (b)

Fig. 12. Comparison of NN estimated output and the corresponding actual output in the IEEE 68-bus test system using the NN-based sensitivity method.

Table 6Summary of the neural networks training results for the PCA-based extracted features.

Test system Neural networkoutput

Number of extractedfeatures

Number ofNN inputs

Number of NNhidden neurons

MSE fortraining patterns

RMSE fortesting patterns

New England 39-bus CCT 48 48 20 0.01 0.0593New England 39-bus TM 48 48 20 0.01 0.2181

IEEE 68-bus CCT 19 19 25 0.08 0.0233IEEE 68-bus TM 19 19 25 0.08 0.0825


39-bus test system and IEEE 68-bus test system are functions of 87and 137 system operating conditions, respectively. However, weshould notice that the some of those operating conditions are, ingeneral, dependent on the other operating conditions. Therefore,it would be better if we first extract some features from all the sys-tem operating conditions, and then use only those features thathave considerable variations amongst the training patterns as theneural networks inputs. This can be done by the principal compo-nent analysis (PCA) [20,32].

PCA orthogonalizes the components of the input vectors, it or-ders the resulting orthogonal components (principal components)so that those with the largest variation come first, and it eliminatesthose components that contribute the least to the variation in thedata set [20]. PCA was performed on the 87 and 137 selected sys-tem operating conditions in the New England 39-bus test system

and the IEEE 68-bus test system, respectively using MATLAB [32].Here, we used the same training and testing patterns as those usedin Section 5.2. The features were reduced by using principal com-ponent analysis and retained those principal components, whichaccounts for 99.9% variation in the data set. PCA reduced the fea-tures to 48 and 19 in the New England 39-bus test system andthe IEEE 68-bus test system, respectively. Then, these principalcomponents were used as the inputs for the proposed MLP NNs.The simulation results obtained are summarized in Table 6, inwhich the RMSE for testing patterns along with some numericalvalues chosen for training each MLP NN are given.

Comparing the test results shown in Table 6 with those men-tioned in Table 4, it is conclude that the NN trained using the re-duced features extracted by PCA, gives less accurate estimationfor both the CCT and TM than the cases in which all system


operating conditions were used as the neural networks inputs.However, the results shown in Table 6 also confirm that the trainednetworks by features produced by PCA can estimate their corre-sponding targets with a good degree of accuracy. Results obtainedshow that a small number of inputs may contain all the requiredinformation for estimating the CCT and/or the TM. However, itshould be noted that the physical meaning of the neural networksinputs is lost when the features calculated by PCA are used as theneural network inputs.

6. Conclusions

In this paper, a multi-layer perceptron (MLP) neural network(NN) based approach was proposed for the online transient stabil-ity assessment of power systems through estimation of the criticalclearing time (CCT) and a transient stability time margin (TM) con-sidering detailed models for the synchronous machines and theirautomatic voltage regulators (AVRs). The New England 10-machine39-bus test power system and the IEEE 16-machine 68-bus testpower system were used as examples to demonstrate the proposedapproach. The software tool PSAT was employed for generating therequired training and/or testing patterns using time-domain simu-lation (TDS) technique. The resilient back-propagation methodfrom MATLAB neural network toolbox was used for fast trainingthe proposed neural network. The proposed neural networks aretrained off-line; so the heavy computational burden is avoided inon-line applications. In addition, a neural network based sensitiv-ity method and the principal component analysis (PCA) were usedto reduce the dimension of the input data vectors. It was shownthat the proposed MLP NNs could estimate the actual CCT and/orTM corresponding to a particular fault scenario with reasonableaccuracy at different system operating conditions; therefore, theycould be employed for quick assessment of the transient stabilityin a power system control center.

Acknowledgements

The authors gratefully acknowledge the financial support pro-vided by University of Guilan in Iran. The first author is also thank-ful to Mr. Ali-Reza Bahmanyar for implanting the IEEE 68-bus testsystem in PSAT.

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