1

Prediction and Enhancement of Power SystemTransient Stability Using Taylor Series

Amirreza Sahami, Student Member, IEEE, and Sukumar Kamalasadan, Senior Member, IEEE

Abstract—Timely information about behavior of a powersystem is important for monitoring and controlling the system.Accurate and prompt transient stability prediction is an effectiveway to reduce the risk of a power system failure and possibleblackouts. In this paper, a method for predicting the generatorsbehavior using Taylor Series has been derived that can be usedto predict the angular changes during transient oscillations andthus the related critical clearing time. The paper also discusses theapplication of this approach for preventive control actions. Theproposed technique is applied on IEEE 39 bus test system andthe advantages, efficiency and error comparisons are presented.

Index Terms—Braking Resistor, Prediction, Transient StabilityEnhancement, Transient Oscillations, Preventive Control.

I. INTRODUCTION

One of the main objectives of power systems is to deliverstable, reliable, and high-quality power to customers. Thequality of delivered electrical power and safety of electricalfacilities are related to the nominal system frequency [1], [2].System reliability is tested with respect to three criteria, the:(N-1) feasibility, voltage stability, and transient stability [3].Transient Stability of a power system refers to the study ofa power system behavior after the system undergoes a largedisturbance. A disturbance creates substantial power imbalancebetween generated power and network demand. Consequently,oscillations happen in the system, making generators angles toswing, and the system to lose its normal condition. In severecases these oscillations can lead to a local or global blackout.

Stability of a system depends on the initial operating con-dition, the nature of the physical disturbance, and the durationof the disturbance even though the study mainly focus onpost-disturbance scenarios of the system. It should be notedthat post-disturbance stable state may be different from pre-disturbance operating point, depending on the sequence ofthe disturbances, and the controllers actions [3], [4]. So, thetransient stability problem is the study of the stability of thepost-disturbance system. Therefore, predicting the behavior ofthe system helps designing controllers to have better actionsthat can prevent a system from collapse and possible blackouts.Predicting and controlling the behavior of modern intercon-nected power systems has a major impact on the economyand national security [5], [6].

Different approaches and studies are reported in the liter-ature for predicting and enhancing transient stability of largepower systems. In [7], it is mentioned that various methods,such as hybrid neural network with optimization, waveletneural network, echo state network are used for predicting theoutput of a wind generator. A new transient stability predictionmethod, combining trajectory fitting (TF) and extreme learningmachine (ELM) based on two-stage process, is proposed in [8].

In [9], a data-based method for transient stability prediction byusing data pre-processing is presented. Ref. [10] uses TaylorSeries expansion to find the state space model of the linearizedmodel of a voltage control voltage source inverter (VCVSI).

For enhancing transient stability, different methods, suchas fast valving of steam stream in turbines, tripping gener-ators, using braking resistors, and controlled opening of tielines are mentioned in [11], [12]. In [13], using dynamicprogramming in a discrete supplementary control for transientstability enhancement in a multi-machine power system isdiscussed. Ref. [14] proposes an optimal controller for StaticVar Compensators (SVCs) to improve transient stability of apower system. In [15], direct feedback linearization (DFL)technique is employed to control excitation system and fastvalving actuators to improve transient stability. In [16], anapproach based on Hybrid neural network-optimization to takepreventive control actions for enhancing transient stability isreported. In [17] a hybrid direct and intelligent method ofreal-time coordinated wide-area controller for improved powersystem transient stability has been presented. Ref. [18] usesgeneration rescheduling to enhance system stability. Ref. [1]presents a new approach for improving transient stability, usingthe concept of the potential energy terms of energy function.In [19], a hybrid method based on offline analysis methodof generator tripping for transient stability enhancement ispresented. In [20], the application of a close-loop wide-areadecentralized power system stabilizer for transient stabilityenhancement is investigated.

In this paper, using piecewise linearization and TaylorSeries, the behavior of system generators is predicted, whichhelps finding critical clearing time and angles. Finding themmakes it possible to take necessary control actions in order toprevent the system collapse. In this paper, after predicting thecritical clearing time, a braking resistor is used to prevent loseof synchronism in the system. The method has been testedon IEEE 9 bus and IEEE 39 bus test systems, and results areprovied and compared with numerical methods. The rest ofthe paper is organized as follows: In section II, the proposedmethod is discussed. Section III shows an illustrative exampleof the proposed architecture. Section IV discusses a generalprediction methodology for a multiple machine system, andSection V discuss the prediction of generators angle and speedfor IEEE 39 bus system. Section VI illustrates and applicationand section VII concludes the paper.

II. PROPOSED METHOD FOR PREDICTING GENERATORS’BEHAVIOR

Phasor measurement units (PMUs) are devices that providereal-time phasor measurements at those locations of a power

2

system network where they are placed. Due to advancementsin the field of relay technology, digital relays can now actas PMUs, which has significantly reduced the cost of PMUs[21], [22]. In what follows, it is assumed that there are PMUsor digital relays at all generator buses, which is a realisticassumption.

Let the dynamics of generators is modeled using (1) and(2).

2H

ωs

dω

dt+Dω = Pm − Pe (1)

dδ

dt= ω − ωs (2)

where ωs is the synchronous speed, which is equal to 1 p.u.Let 2H

ωs= M . So M = 2H . Then (1) can be presented as

Mdω

dt+Dω = Pm − Pe (3)

Assume that the behavior of the system between any twoconsequent time steps is linear. This is a valid assumption sincethe waveform of any stable power system variables are analyticfunctions, except at switching moments. Hence, Taylor seriescan be used to linearize the system dynamics, and δ and ωcan be expanded as

δ(t) = δ(0) + δ′(0)t+ δ

′′(0)

t2

2!+ ...+ δ(n)(0)

tn

n!+ ... (4)

ω(t) = ω(0) + ω′(0)t+ ω

′′(0)

t2

2!+ ...+ ω(n)(0)

tn

n!+ ... (5)

Neglecting terms with order higher than two, and consideringt0 as the initial point,

δ(t0 + ∆t) = δ(t0) + δ′(t0)∆t+ δ

′′(t0)

∆t2

2!+O(∆t3) (6)

δ(t0 + ∆t) = δ(t0) + ω(t0)∆t+ ω′(t0)

∆t2

2!+O(∆t3) (7)

ω(t0 + ∆t) = ω(t0) + ω′(t0)∆t+ ω

′′(t0)

∆t2

2!+O(∆t3) (8)

where O(∆t3) represents neglected terms. From the swingequation we know

Mdω

dt= Pm − Pe −Dω = M ∗ a(t) (9)

Assuming a linear behaviour for the system between twoconsequent moments. Then

dt = ∆t = One T ime Step (10)

So:M

∆ω

∆t= Pm − Pe −Dω (11)

M∆ω = (Pm − Pe)∆t−Dω∆t (12)

dδ

dt= ω − ωs (13)

dδ

dt=

∆δ

∆t= ω ⇒ ∆δ = ω∆t (14)

⇒M∆ω = (Pm − Pe)∆t−D∆δ (15)

∆ω = (Pm − Pe

M)∆t− D

M∆δ (16)

ω(t0 + ∆t) = ω(t0) + (Pm − Pe

M)∆t− D

M∆δ (17)

δ(t0 + ∆t) = δ(t0) + [ω(t0)∆t+ (Pm − Pe

M− D

Mω(t0))

∆t2

2!] ∗ 2πf (18)

Using (17) and (18) behaviours of the generators of thesystem can be predicted. It is worth noting that because afunction that shows the variables behaviour is not an analyticfunction at switching moments, n sample of data is neededto be known to approximate a function with Taylor series oforder n.

III. AN ILLUSTRATIVE EXAMPLE

Consider the network shown in Fig. 1. It is a Single-MachineInfinit-Bus 50 Hz system. A three-phase symmetrical faulthappens at Bus 3 at t=0.1s. According to simulation, CriticallyStable Clearing Time (CSCT) is 0.150s and Critically UnstableClearing Time (CUCT) is 0.151s. The goal is to predict thesystem behavior. For the machine, H = 3.5 and M = 7.

Fig. 1. SMIB Network from Kundur [12]

During the fault the voltage of Bus3 (V B3) is zero. So, noactive power is transferred from the generator to the grid (Pe =0). Assume that the post-fault configuration of the system issame as the pre-fault, or it is known in general. At steady state(until t = 0.1s) the system state is as followsδs.s. = 0.729020rad = 41.77◦, ω = 0.

To predict δ and ω at t = 0.11, (17) and (18) can be used. So7 ∗∆ω = 0.9 ∗ (0.11 − 0.1) = 0.009 and thus ∆ω = 0.0013and ω(t = 0.11) = 0.0013.

From this ω(t = 0.11)simulated = 0.0013 and δ(t =

0.11) = 2πf{

0.9−07

0.012

2! + 0 ∗ 0.01}

+ 0.72902 = 0.7310

and then δ(t = 0.11)simulated = 0.7311Considering this the prediction of desired paramters at

t=0.2s is as follows.7 ∗∆ω = 0.9 ∗ (0.2− 0.1) = 0.09∆ω = 0.0129ω(t = 0.2) = 0.0129ω(t = 0.2)simulated = 0.0129

3

TABLE IPrediction for the network shown in Fig. 1

Time ω δPredicted Simulated Predicted Simulated

0.11 0.0013 0.0013 0.7310 0.73110.20 0.0129 0.0129 0.9310 0.9311

δ(t = 0.2) = 2∗π∗50∗{

0.9−07

0.12

2! + 0 ∗ 0.1}

+0.72902 =

0.9310δ(t = 0.2)simulated=0.9311As it can be seen, using pre-fault condition and via knowl-

edge about Pm and Pe at the very moment after fault, thepredictions are accurate. Table I, and figs. 2, 3 shows a com-parison between the simulated results and predicted results.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

0.5

1

1.5

2

2.5

An

gle

(ra

dia

n)

Angle Machine1

Predicted

Simulated

Fig. 2. Generator Angle.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

-5

0

5

10

15

20

Ro

tor

Sp

eed

(P

.U.)

10-3 Speed M1

Simulated

Predicted

Fig. 3. Generator speed.

IV. GENERAL PREDICTION OF THE BEHAVIOUR OF THESYSTEM IN A MULTI-MACHINE SYSTEM

As could be seen in aforementioned discussions, there isa term Pe in prediction formulas. Pe is the electrical outputof the generator that its behaviour is under study. The mostaccurate prediction happens when the actual output electricalpower of generators (Pe) is known. This way, the acceleratingpower can be found accurately. However, it is not practicallypossible since the swing equation should be numerically solvedto find (Pe). Also, in real-time studies, the actual outputof generators cannot be known beforehand to be used forprediction. Therefore, the output of generators for predictingtheir speed and angle, should be found in another way. Threedifferent approaches can be considered for approximating Pe

during the fault:• Assuming Pe of generators equal to zero.• Assuming Pe as a constant number. This amount is the

amount of Pe one moment after the fault.

• Predicting Pe of generators. Because the behavior of thesystem is predicted for next time step, Taylor Series canbe used. In next session this method is elaborated.

A. Predicting Pe via Taylor series

In order to predict Pe, the behaviour of Pe is consideredlinear between every two consecutive moments, except atswitching times. Hence, Taylor series of Pe can be employed.The expansion of Pe is:

P (t) = P (0) + P ′(0)t+ P ′′(0)t2

2!+ · · · (19)

Mdω

dt= Pm − Pe −Dω (20)

Md2ω

dt2= 0− dPe

dt−Ddω

dt(21)

dω

dt= a(t) (22)

dPe

dt= 0−M d2ω

dt2−Ddω

dt= −M da(t)

dt−Da(t) (23)

Assuming the above equations for one time step and substi-tuting dPe and dt with ∆Pe and ∆t respectively leads to:

∆Pe

∆t= −M∆a

∆t−Da(t) (24)

∆Pe = Pe(0)−M∆a(0)−Da(0)∆t (25)

So, the first order prediction for Pe will be.

Pe(t0 + ∆t) = Pe(t0)−M∆a(t0)−Da(t0)∆t (26)

This equation has been used for predicting electrical powerduring the fault. To increase the accuracy, we may have toadd a higher order term to the prediction equation.

Md3ω

dt3= 0− d2Pe

dt2−Dd

2ω

dt2(27)

Substituting the second term in (21) will result in

Md2a

dt2= −d

2Pe

dt2−Dda

dt(28)

Assuming the above equations for one time step and substi-tuting dPe and dt with ∆Pe and ∆t respectively leads to

M∆2a

(∆t)2= −∆2Pe

(∆t)2−D∆a

∆t(29)

∆2Pe

(∆t)2= −M ∆2a

(∆t)2−D∆a

∆t(30)

P (t) = P (0) + P ′(0)t+ P ′′(0)t2

2!+ · · · (31)

Pe(t0 + ∆t) = Pe(t0)−M∆a(t0)−Da(t0)∆t+1

2(∆t)2

∆2Pe

(∆t2)(32)

Pe(t0 + ∆t) = Pe(t0)−M∆a(t0)−Da(t0)∆t

+1

2(−M∆2a(t0)−D∆a(t0)∆t− D2

Ma(t0)∆t2 +

D2

M2a(t0)∆t2) (33)

4

Pe(t0 + ∆t) = Pe(t0)−M∆a(t0)−Da(t0)∆t

−D2

∆a(t0)∆t+1

2∆t2(

D2

M2a(t0)− D2

Ma(t0))− M

2(∆2a(t0)) (34)

Considering ∆t = TS as a constant time step, we have

∆a(t0) = a(t0)− a(t0 −∆t) = a(t0)− a(t0 − TS) (35)

∆2a(t0) = ∆a(t0)−∆a(t0 −∆t) = a(t0)− 2 ∗ a(t0 −∆t) + a(t0 − 2∆t)

(36)Hence, (34) can be written in discrete form as follows

Pe(i+ 1) = Pe(i)−M∆a(i)−Da(i)∆t− D

2∆a(i)∆t

+1

2∆t2(

D2

M2a(i)− D2

Ma(i))− M

2(∆2a(i)) (37)

Substituting (35) and (36) in (37) leads to (38).

Pe(i+ 1) = Pe(i)−M(a(i)− a(i− 1))−Da(i) ∗ TS − D

2(a(i)− a(i− 1)) ∗ TS

+1

2TS2(

D2

M2a(i)− D2

Ma(i))− M

2(a(i)− 2a(i− 1) + a(i− 2)) (38)

Based on (38), we can predict the output electrical power.Using (17), (18), and (38), angles, speeds, and output electricalpower of generators can be predicted. It is worth remindingthat because 2nd order Taylor series is used, the data for thefirst two moments after fault or after fault removal is requiredfor predicting the system’s variables during the fault and afterthe fault removal, respectively.

PMUs can be used to improve the accuracy of the predictionfor post-fault system. It means that, we may update theinitial point of the prediction using PMU data when the post-fault system is being predicted. It should be mentioned thatthe scope of this work is to predict the behavior of thesystem during the fault so that using direct methods becomespossible without numerically solving the swing equation forduring-the-fault system studies. The prediction also helps toapply predictive controllers and have a more stable system.In addition, considering a sustained fault in a system andpredicting the system behaviour can be used for finding theUEP of a system. Finally, with defining an appropriate criteria,prediction can be used for finding the critical clearing time,and for finding the critical machines, which refer to machinesthat loose synchronism first.

In what follows, the prediction has been used to predictgenerator behavior in a dynamic IEEE 39 bus test system. Theprediction error for desired variable (X), has been calculatedand provided using (39) and (40).

Error(Xti)(%) =Xactual

ti −Xpredictedti

Xactualti

∗ 100 (39)

Mean Error(x) =

∑ni=1 |Error(Xti)|

n(40)

where n is the number of moments that the variables arepredicted. This can be represetned as in (41).

n =t(faultremoval)− t(faultstart)

TimeStep(41)

V. PREDICTION IN IEEE 39 BUS TEST SYSTEM

To test the proposed method, prediction of generator anglesand speed is performed on IEEE 39 bus test system. One-linediagram and the features of the test system are presented infigure 4 and table II. To create system dynamics a symmetricalthree phase fault is applied at bus 16 at t = 0.1 sec. Fault isremoved at t = 0.285 sec. The system is critically stable inthis scenario, meaning that the critical fault duration is 0.158seconds. Machine 2 is the reference (δ2 = 0).

The prediction for system behaviour during the fault isonly based on the PMU data for two time steps after fault.However, prediction for post-fault system (after t = 0.258 sec)is corrected by updating the initial point in the related formulasevery 8 time steps (every 0.08 sec.). The results for machines 4,as the first machines that lose synchrony, are provided in figs.6 to 11. It can be seen that angle and speed prediction error iswithin 1%. For machine power prediction, expect during thesudden change in the power at fault time, the error is withina threshold limit of 1%.

Fig. 4. IEEE 39 Bus Test System

TABLE IIIEEE 39 Bus Features

Buses &Generators

39 Buses10 Generators

Lines &Loads

46 Lines19 Loads

Total Active PowerGeneration (MW) 6147.92 Total Active

Load (MW) 6097.100

Total Reactive PowerGeneration (MVAR) 2487.332 Total Reactive

Load (MVAR) 1409.100

VI. APPLICATION

To test the efficiency of the proposed method in predictingthe system behavior and preventing lose of synchronism, CCTfor fault on bus 16 in IEEE 39 bus test system has beenpredicted. For prediction, a PC has been used with a Core i7-3770-3.4GHz CPU and 8GB of RAM. PASHA and MatLab areused for simulation [5]. Elapsed time for predicting system for2 seconds was 0.078385 seconds. After prediction, to preventlose of synchronism, dynamic braking has been used. dynamic

5

Detect Fault

Happening

Get the PMUs

DATA

Detect Fault

Removal

CONTROLLER

Sen

d C

on

tro

l S

ign

als

to R

elat

ed

Act

uat

ors

(D

yn

amic

Res

isto

r)

Read Generators

Power, Angle,

Speed from Data

Base

Use Equations 17, 18, 38 to

predict Angles, Speed, and

Electrical Power

Find Critical Machines and

related Critical Clearing

Time Based on Desired

Criteria

1

2

3

4

5DATA

BASE

Fig. 5. Schematic of the Prediction Application

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (s)

0.2

0.4

0.6

0.8

1

1.2

Ang

le (

rad

ian

)

Generator 4 Angle

Simulated

Predicted

Fig. 6. Machine 4 Rotor Angle

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (s)

-3

-2

-1

0

1

Ang

le M

4 E

rror

(%

)

Generator 4 Angle Prediction Error

Fig. 7. Error of Machine 4 Rotor Angle Prediction

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (s)

0

0.005

0.01

0.015

0.02

Rot

or S

peed

(P

.U.)

Generator 4 Rotor Speed

Simulated

Predicted

Fig. 8. Machine 4 Rotor Speed

braking uses the concept of applying an artificial electrical loadduring a transient disturbance to increase the electrical poweroutput of generators and thereby reduce rotor acceleration. Oneform of dynamic braking involves the switching in of shuntresistors for about 0.5 second following a fault to reduce theaccelerating power of nearby generators and remove the kineticenergy gained during the fault [12].

Table III shows the machines that tend to lose synchronyfirst and their related critical clearing time and angle. Since

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (s)

-3

-2

-1

0

1

Err

or

W4

(%

)

W4 Prediction Error

Fig. 9. Error of Machine 4 Speed Prediction

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (s)

0

200

400

600

800

1000

Ele

ctri

cal

Po

wer

(M

W)

Generator 4 Output Power

Simulated

Predicted

Fig. 10. Machine 4 Output Power

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (s)

-10

0

10

20

30

40

50

Mac

hin

e4 P

ow

er E

rro

r (%

)

Generator 4 Output Power Prediction Error

Fig. 11. Error of Machine 4 Output Power Prediction

the first machines that tend to lose synchrony with each otherare machine 1 and 4, the braking resistor has been switchedin at bus 33 which is connected to machine 4. Fault happensat t = 0.1 sec. and prediction results are available at t = 0.18sec. The braking resistor is switched in at t = 0.2 sec andis switched out at t = 0.7 sec. The fault is not cleared untilt = 0.315sec. Hence, it shows that although the CCT for theoriginal system is 0.258 seconds, being able to predict thesystem behaviour and taking a simple preventive action, helpsave the system even when that fault is not removed for alonger period than CCT.

0 1 2 3 4 5 6

Time (s)

0

5

10

An

gle

(ra

dia

n)

Relative Angles Between Machine 1& 4

t= 0.7, Resistor is switched out

t= 0.286, Fault is Removed

t= 0.2, Resistor is switched in

t= 0.17, Prediction is done

t= 0.1, Fault Starts

Without Dynamic Resistor

Improved with Prediction

and using Dynamic Resistor

Fig. 12. Relative Angle between generators 4 and 1

6

0 1 2 3 4 5 6 7 8

Time (s)

60

60.2

60.4

60.6

60.8

61

Fre

qu

ency

(H

z)Frquency of Generator 1

Without Dynamic Resistor

Improved with Prediction

and using Dynamic Resistor

Fig. 13. Frequency of Generator 1

0 1 2 3 4 5 6 7 8

Time (s)

59.5

60

60.5

61

61.5

62

62.5

63

Fre

quen

cy (

Hz)

Frequency of Generator 4

Without Dynamic Resistor

Improved with Prediction

and using Dynamic Resistor

Fig. 14. Frequency of Generator 4

TABLE IIICritical Clearing Time

Machines Prediction Machines SimulationCriticalClearing

Angle

CriticalClearingAngle

CriticalClearing

Angle

CriticalClearingAngle

1,4 0.2800 -89.1800 1,4 0.2800 -89.89121,5 0.2900 -89.5500 1,5 0.2800 -87.12741,7 0.3000 -87.1100 1,7 0.2852 -88.14421,6 0.3200 -87.6100 1,6 0.3000 -88.50714,10 0.3300 87.6600 4,10 0.3100 87.65841,9 0.3400 -89.6500 1,9 0.3100 -89.05135,10 0.3600 89.2200 5,10 0.3500 89.56247,9 0.3700 89.4200 7,9 0.3500 89.21156,10 0.4000 89.0600 6,10 0.3900 88.50141,3 0.4300 -88.2800 1,3 0.3900 -88.37954,8 0.4500 89.5700 4,8 0.4100 85.05872,4 0.4500 -88.6400 2,4 0.4200 -86.5300

VII. CONCLUSIONS

In this paper a new approach for transient stability predictionand improvement is proposed. The method is based on usingTaylor Series for predicting critical clearing time. After predic-tion, a dynamic resistor is used to prevent lose of synchronismin the system. The technique was applied on IEEE 39 bussystem and results showed the accuracy and efficiency of themethod.

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