Electrical Power and Energy Systems 42 (2012) 306–313

Contents lists available at SciVerse ScienceDirect

Electrical Power and Energy Systems

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

The localness of electromechanical oscillations in power systems

Sudipta Ghosh ⇑, Nilanjan SenroyDepartment of Electrical Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110 016, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 September 2011Received in revised form 22 March 2012Accepted 9 April 2012

Keywords:Small signal stabilityClusteringLocalness indexCoherency

0142-0615/$ - see front matter Crown Copyright � 2http://dx.doi.org/10.1016/j.ijepes.2012.04.004

⇑ Corresponding author. Tel.: +91 9891 317925.E-mail addresses: sudiptarit@gmail.com (S. G

(N. Senroy).

An innovative index, indicative of the relative localness of electromechanical oscillations in electricpower systems, is introduced in this paper. The Lindex is calculated using the normalized participation fac-tors obtained from a small signal analysis of the system. With the help of simple representative examplesthe efficacy of the index to understand power system dynamic behavior, like coherency identification isestablished.

Crown Copyright � 2012 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Power system response to small disturbances is oscillatory innature. Monotonic instability due to small disturbances is rare, asa result of extensive and widespread use of modern continuousacting regulators. The focus of this paper is on oscillations thatare electromechanical in nature, involving excursions of synchro-nous generator rotor angles and shaft speeds. Such oscillationsare in the range of 0.1–2 Hz, and are a common cause for concernin large weakly meshed power systems. In multi-machine powersystems, various generators participate in different modes of oscil-lations in varying degrees. The participating generators form pat-terns that well characterize the nature of the electromechanicalmodes of oscillations. Sustained, poorly damped and sometimesundamped oscillations introduce additional operating constraints,and severely hamper system reliability. The analysis of such elec-tromechanical oscillations constitutes the first step towardsdesigning controllers to improve their damping characteristics.

Electromechanical oscillations in power systems are adequatelydescribed and studied using small signal stability models employ-ing linearization techniques. Broadly, two types of electromechan-ical oscillations are observed. Low frequency oscillations (less than1 Hz) involving one part of the system against another, are termedas global or inter-area oscillations. The dynamic characteristics ofthe overall power system are best understood by analyzing theseinter-area modes. On the other hand, a local mode of oscillationinvolves a single generator or a group of adjacent generators and

012 Published by Elsevier Ltd. All r

hosh), nsenroy@ee.iitd.ac.in

its frequency is relatively higher – varying from 0.5 to 2 Hz. Thedamping action of the amortisseur windings located on the rotorof synchronous machines, manifests itself during the local modeoscillations [1]. In some cases, the readily available frequency ofoscillation may hint at the nature of the mode, yet it is not a reli-able tool to classify modes into local or inter-area types. In general,the frequency of an electromechanical mode may be negativelycorrelated with the number of machines participating in the mode,however as Fig. 1 shows, this correlation is not clearly defined.

Although an important step in the stability analysis of a powersystem, the classification of electromechanical oscillations into lo-cal/inter-area types is straightforward and performed in a some-what routine manner. Nonetheless, in large systems it has beennoticed that while distinguishing an inter-area mode from a localmode is straightforward, it is not always easy to determine themost global or local mode. Similarly, the relative ‘localness’ of amode is not immediately clear from mere inspection of the partic-ipating generators. Alternatively, the well established link [2] be-tween the energy of a mode and its description in the complexplane (using eigenvalues) suggests that the localness of the modeis closely related to the modal energy. Thus, an index that summa-rizes the modal energy information of various modes can beexploited to develop an efficient localness ranking strategy. Sucha ranking strategy would be particularly useful in system-wide sta-bility studies such as dynamic aggregation of generators and modelorder reduction of large power systems.

As part of a small signal stability analysis, the calculation of par-ticipation factors helps in associating generator states with themodes or eigenvalues of the system [3,4]. If a large number of dis-persed generators are associated with a single mode, it is identifiedas an inter-area mode, while only a few generators are associated

ights reserved.

0 0.5 1 1.5 2 2.5 3 3.50

5

10

15

20

25

30

35

40

mode frequency [Hz]

Num

ber o

f par

ticip

atin

gm

achi

nes

in m

ode

Fig. 1. The number of machines participating in an electromechanical mode versusits frequency, in modified IEEE 50-machine test system.

Table 1

S. Ghosh, N. Senroy / Electrical Power and Energy Systems 42 (2012) 306–313 307

with a local mode. Further, it has also been recognized that in clas-sically modelled power systems, with negligible line resistancesand machine damping, the machine state participation factorsindicate the energy of the corresponding machines in the mode[5,6]. All the participation factors for a mode sum to unity, indicat-ing the relative contribution of individual machines to the total en-ergy of that mode. In a normalized participation factor matrix eachentry compares individual mode-machine associations, with themaximum association for that particular mode to a machine.Therefore, a vector consisting of all the normalized participationfactors for a mode can be used as a pointer to the total modal en-ergy [7–9].

This paper presents an innovative index, Lindex, that is relatedto the individual modal energy and can be used to rank modesaccording to their localness. The index is heuristic in nature, de-signed intuitively from the patterns formed by the normalizedparticipation factors of electromechanical modes. In addition toranking, an important design objective of the index is its abilityto cluster the modes according to their localness. Thus, not onlythe relative localness of the modes is revealed, effective separa-tion between local and inter-area modes are also accomplished.Experiments on a mechanical mass-spring system analogy, alongwith power system examples demonstrate the utility of the Lindex

to understand evolving system dynamic behavior. Finally, as anillustrative application, the index is used to select modes thatform an eigenspace, from which the dynamic coherency betweengenerators is recognized. The groups are compared with the gen-erator groups obtained if the slow coherency algorithm was used[10–13].

The rest of the paper is structured as follows. Section 2 summa-rizes the general state-space framework for small signal stabilityanalysis of power systems. The Lindex is designed and formulatedin Section 3. The correlation of Lindex with simple RLC networkand one mechanical three mass spring system [3] is illustrated inSection 4. Section 5 discusses some important characteristic ofLindex with help of Kundur’s four machine test system [14], whereasSection 6 presents the application of this index for coherencygrouping recognition in a 10 generator 39 bus test system [15].Section 7 is the concluding section.

Different test systems.

Test system Description Number ofelectromechanical modes

Test system A Four machine system [14] FourTest system B Six machine system [15] SixTest system C Ten machine test system [16] Ten

2. Modal analysis of power system

For small deviations around an operating equilibrium point, amulti-machine power system may be linearized in state spaceform as

_xD ¼ AxD ð1Þ

where xD is the vector consisting of the deviations in the systemstates, and A is the state matrix [1]. The dimensions of the A-matrixare determined by the number of states of the system. Although thetotal number of states increases for detailed modelling of the gener-ators as compared to classical models, the number of generatorelectromechanical states remains the same. As the focus is on elec-tromechanical oscillations, the A-matrix considered herecorresponds to the two generator electromechanical states – rotorangle and speed. For a system of m electromechanical states, theeigen-analysis of the A matrix (m �m) will produce m eigenvalues,ki, and corresponding right and left eigenvectors – vi (m column vec-tors), and ui (m row vectors) respectively. The normalized eigenvec-tors associated with ith eigenvalue ki satisfy (2).

Avi ¼ kivi

uiA ¼ kiuið2Þ

The element vki i.e. the kth element of right eigenvector vi mea-sures the activity of the kth state in the ith mode, whereas the kthelement of the left eigenvector ui indicates the importance of thisactivity in defining the mode. The coupling between a particularstate and a mode is revealed by the participation factor [1], pki = u-kivik measuring the participation of the kth state in the ith mode; pki

lies between 0 and 1. The participation factors for a mode are fur-ther normalized on the maximum participation for that mode.After normalization, pki has a maximum value of one indicatingthe maximum participation of a generator in a mode.

3. The localness index

For an N-generator system, the localness index of the ith elec-tromechanical mode is calculated from the normalized participa-tion factors as

Lindex;i ¼XN

k¼1

ð1� pkiÞn ð3Þ

The number of terms in the summation corresponds to the syn-chronous generators in the system. The exponent of each term, n, ischosen to give the property of clustering to the Lindex. The primarypurpose of Lindex is to rank all modes according to their localness.The modes ranked at the bottom of such a list would be inter-areamodes and the modes grouped at the top would be local modes.Additionally, the modes should form natural clusters, when rankedaccording to their Lindex. In experiments with three test systems(Figs. 14–16 in the Appendix and Table 1), different values of theexponent n were considered for their usefulness as a clusteringtool. For all the electromechanical modes of a system, the Lindex

was calculated using different values of the exponent n in (3).The individual mode Lindex values were used to rank and clusterthe modes to see which value of n yielded best mode clusters.The clustering validity was evaluated using the ‘silhouette score’,for every point (mode in this case) [17,18].

The silhouette score for a point in a cluster compares the simi-larity of the point to other members of its cluster, versus itsremoteness from points in other clusters. Accordingly, points in

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

45

50

Mode

Mac

hine

Fig. 3. Visual representation of the participation factor matrix for the modified IEEE50-machine test system. The dots represent non-zero participation of a machine ina mode. The modes are sorted in order of increasing Lindex.

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

45

50

Mode

Mac

hine

Fig. 4. Visual representation of the participation factor matrix for the modified IEEE50-machine test system. The dots represent non-zero participation of a machine ina mode. The modes are sorted in order of increasing frequency.

308 S. Ghosh, N. Senroy / Electrical Power and Energy Systems 42 (2012) 306–313

the middle of a cluster would have a high silhouette score (theoret-ically 1.0), while a wrongly clustered point would have a low sil-houette score (theoretically �1.0). Points lying at the clusterboundaries have relatively lower silhouette scores. The average sil-houette score of all the points in an ensemble is a good indicator ofhow well the ensemble has been clustered. If too many clusters areformed in an effort to reduce miss-clustering, the points at theboundaries of the clusters would reduce the average silhouettescore.

Fig. 2 plots the average silhouettes score across all the modes ineach test system, for optimally defined clusters on the basis ofmodal Lindex calculated using different values for n in (3). A euclid-ean distance based k-means clustering algorithm was used to formthe clusters. Careful inspection of Fig. 4 reveals that the best clus-tering of modes occurs when the exponent, n, in (3) is close to thenumber of machines in the system. Thus, when n is selected to befour for test system A, six for test system B, and ten for test systemC, the clustering is optimum. Therefore, it is proposed that theexponent in (3) correspond to the number of synchronous genera-tors in the system under consideration i.e. n = N.

In further experiments with larger power systems, the Lindex wasused to sort the participation factor matrix in increasing order. Forthe modified IEEE 145 bus, 50-generator test system [19], Fig. 3presents a visual representation of the participation factor matrixsorted in increasing order of Lindex. The first mode from the left isthe most global in nature, with maximum number of generatorsparticipating. From left to right, the localness of the modesdecreases indicating reducing participation of generators. The mostlocal mode is the 50th mode in which only two machines partici-pate. Such an ordering of modes is not possible on the basis of theirfrequency. Fig. 4 shows the same participation factors sorted inincreasing order of mode frequency. It may be observed that themode, thus, ranked at 50 has four machines participating, whilethe mode ranked at 49 has two machines participating – therebyconfirming that the localness of the mode is not revealed by its fre-quency. Fig. 5 reinforces the idea that the number of machines par-ticipating in a mode varies linearly with the Lindex of the mode.

In the next section, two simple representative examplesare presented, where the system dynamic behavior is capturedby Lindex.

0 2 4 6 8 10 12 14 16 18 200.4

0.6

0.8

1

mea

n si

lhou

ette

sco

re

(a)

0 2 4 6 8 10 12 14 16 18 200.85

0.9

0.95

1

mea

n si

lhou

ette

sco

re

(b)

0 2 4 6 8 10 12 14 16 18 20

0.70.80.9

1

mea

n si

lhou

ette

sco

re

(c)

Fig. 2. Mean silhouette scores for modes clustered using their individual Lindex

values calculated using equation (3). The value of the exponent in (3) was variedfrom 1 to 20 to obtain the curves. Different power systems are considered – (a) 4-machine system, (b) 6-machine system, and (c) 10-machine system.

15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

mode Lindex

Num

ber o

f par

ticip

atin

gm

achi

nes

in m

ode

Fig. 5. Scatter plot of number of machines participating in a mode versus its Lindex.

4. Two simple representative examples

Two dynamic systems – an RLC network and a mechanical massspring system are employed as representative examples to demon-

1 2 3 4 5 6 7 8 9 103

3.5

4

4.5

5

Lin

dex

0 70.1

0.3

0.5

L3 (henry)

dam

ping

Lindex

damping

Fig. 7. Effect of L3, on the localness of mode B for system of Fig. 6.

x1 x3

m1

m2

m3Masses : m1, m3=100; m2=1All spring coefficients k1,k2,k3

and k4 are equal to 1.

k1 k2 k3 k4x2

Fig. 8. Three mass-spring system.

Table 3Modal analysis of three mass spring system.

Frequency Participation factor Lindex

x1 x2 x3

Mode 1 0.2256 Hz 0.0025 1 0.0025 1.984Mode 2 0.0225 Hz 1 0 1 1Mode 3 0.0159 Hz 1 0.0101 1 0.97

S. Ghosh, N. Senroy / Electrical Power and Energy Systems 42 (2012) 306–313 309

strate the ability of the Lindex to capture aspects of the systemdynamic behavior due to change in parameters.

4.1. RLC network system

Consider a network, shown in Fig. 6, consisting of resistances –R1, R2, inductances – L1, L2, L3, L4, and capacitances – C1, C2 excitedby voltage sources – e1, e2 and e3. A sixth order model of the net-work exhibits two oscillatory modes – modes A and B as shownin Table 2. Mode A is localized to L1 and C1, and is of a higher fre-quency than mode B, which is relatively more global due to partic-ipation of L3, L4 and C2. With an increase in the inductance of L3, thelocalness of mode B dropped i.e. mode B became global in charac-ter. This is also accompanied by a concomitant reduction in damp-ing of this particular mode (Fig. 7).

4.2. Mechanical mass-spring system

Consider a mechanical system shown in Fig. 8 [3] consisting ofthree masses connected via four springs. This system is chosen be-cause of the similarity of its dynamic behavior to a power systemwith classically modelled generators and negligible line losses.Small transversal perturbations, expressed by linear differentialequations involving the positions and velocities of the threemasses, lead to a sixth order model exhibiting three oscillatorymodes. Selected results from the modal analysis of the systemare presented in Table 3. There are two global modes (modes 2and 3) and one local mode involving m2. It is observed that themost global mode has the lowest Lindex value corresponding tothe lowest frequency mode i.e. mode 3. With the masses fixed attheir values mentioned in Fig. 8, it is not possible for all the modesto participate in all the modes equally, i.e. all modes global in nat-ure. However, by modifying the spring constant it is possible tocreate three modes completely localized to the three masses.Fig. 9 shows that when the spring constant k1 is varied from 0.01to 10, the Lindex for each mode tends to its upper bound, i.e. 2, indi-cating strong localization of the individual modes. A similar behav-ior can be observed in multi-machine power systems, as discussedin the subsequent section.

5. Effect of system operating conditions on Lindex

In a power system, like any dynamical system, evolving operat-ing conditions lead to changes in the modal behavior. One impor-tant aspect of this change is level of participation of generators in

R1=2 L1=2H

e1

i1 i2

v1 C1=1F

L2=4HΩ

Fig. 6. Simple RLC n

Table 2Modal analysis of network system (Fig. 6).

Modes Frequency (Hz) Damping ratio Normalized p

i1 i2

Mode A 0.079 0.707 1 0Mode B 0.052 0.494 0 0Mode C 0 1 0 0Mode D 0 0 0 1

different modes – a feature captured in the Lindex. In this section, arepresentative power system is employed to examine the evolu-tion of the Lindex of the mode in response to changing system con-ditions. The test system (test system A) consists of two areasconnected by weak tie lines containing two 900 MVA synchronousgenerators each. Modal analysis at the base case loading of450 MW on each machine, with zero inter-area transfer revealed

R2=3

e3e2

i3 i4

v2 C2=1F

L3=1H L4=8H

etwork system.

articipation factor Lindex

i3 i4 v1 v2

0 0 1 0 40.125 0.875 0 1 3.451 0.003 0 0.140 4.390 0 0 0 5

0 1 2 3 4 5 6 7 8 9 100.8

1

1.2

1.4

1.6

1.8

2

Spring constant, k1

L inde

x

Mode1Mode 2Mode 3

Fig. 9. Effect of spring constant, k1, on the localness of the three oscillatory modesof system of Fig. 8.

-400 -300 -200 -100 0 100 200 300 4000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Power Flow (Area 1-Area 2) (MW)

L inde

x

Fig. 10. Effect of inter-area power transfer on Lindex,i of the inter-area mode in testsystem A.

0.045

0.05

Reduced Gen 1 output compensatedby reducing load in Area 1

310 S. Ghosh, N. Senroy / Electrical Power and Energy Systems 42 (2012) 306–313

three stable oscillatory modes and one ‘zero’ mode (Table 4). Of theoscillatory modes, two are local while one is an inter-area mode.While computing the eigenvalues, the ‘zero’ mode may not actuallyget computed as zero. This is primarily because of mismatches inthe power flow solution and computational limitations of theeigenvalue calculation algorithm. In some cases, it may also appearto be an unstable mode for the same reasons. The presence of ‘zero’modes in power system small signal stability analysis bears a di-rect relationship with the selection of an appropriate angle refer-ence, as well as considerations of mechanical damping [1].

In Table 4, the comparison of the Lindex for each mode reveals theindividual mode local character. At the base case modes 1 and 2 arelocal to areas 1 and 2. Mode 3 and 4 are inter-area modes, whoseLindex values are very small compared to the other modes’ Lindex.

Maintaining zero inter-area power transfer, when the stress inarea 1 was reduced i.e. generators 1 and 2 outputs reduced to1 MW, the inter-area modes 3 and 4 became local in nature. TheLindex were further observed at different values of machine inertia.In Table 4, cases A–C indicate a growing similarity with themechanical system presented in Section 4.2, with the spring con-stant analogous to the synchronizing power coefficient in a powersystem. Thus, the localness of modes 3 and 4 (formerly inter-areamodes) increase dramatically with reduction in area loading whenthe difference in machine inertias is significant. It is also interest-ing to observe that the zero eigen-mode which was previously aninter area mode localizes in area 1 with reduction in area loadingand with variation in machine inertias. The shaded columns in

Table 4Variation of mode Lindex with changing operating conditions – two-area, fourgenerator system1 (test system A).

1 Inter-area power transfer kept zero for all the cases.2The unshaded modes indicate inter-area modes. The shaded modes are local toeither one of the two areas.

Table 4 represent modes local to either of the two areas. The un-shaded columns indicate inter area modes.

The localness, Lindex, for the natural modes of other test systemsB and C are shown in Table 5.

In the test system A, when the inter-area power transfer wasvaried, the inter-area mode changes its character and becomeslocalized. This is shown in Fig. 10, where the Lindex of the inter-areamode has a minimum value of 0.03 for zero inter-area powertransfer.

In further experiments in test system A, the active power outputof generator 1, located in area 1, was reduced. The load in area 1was also reduced to match the reduced generation, such that theinter-area power transfer remained zero, and all other generatoroutputs remained steady at 450 MW. Fig. 11 shows the variationof inter-area mode (mode 3) Lindex with increasing operating re-serve of Gen 1. It is seen that with increasing operating reserve,the inter-area mode gets more localized. Additionally, more appli-cation of this innovative Lindex, can be found in [20]. Where, it hasshown that it is possible to predict the generator with maximum

0 20 40 60 80 1000.025

0.03

0.035

0.04

Gen 1 operating reserve (%)

L inde

x

Fig. 11. Variation of Lindex of inter-area mode with increasing operating reserve ofgen 1 in test system A.

10

15

20GEN 4GEN 5GEN 6GEN 7GEN 8GEN 10

Angle curves for coherent generators according to slow modes

S. Ghosh, N. Senroy / Electrical Power and Energy Systems 42 (2012) 306–313 311

participation in the inter area mode as a result of incoming doublyfed induction generator (DFIG) technology.

A useful application of the knowledge of the relative localnessof modes is in the context of dynamic aggregation of generatorsparticipating in those modes. In the next section, variations inthe slow coherency based dynamic aggregation techniques areproposed, that rely on the information of individual mode Lindex

values.

0 2 4 6 8 10 12 14 16 18 200

5

Gen

erat

or A

ngle

(deg

)0 2 4 6 8 10 12 14 16 18 20

0

5

10

15

20

Time (sec)

GEN 4GEN 5GEN 6GEN 7GEN 9

Angle curves for coherent generators according to localness modes

Fig. 12. Time-domain simulation of generator swing curves for coherent groupsformed on the basis of slow modes or low Lindex modes. The disturbance is a three-phase bolted fault on bus 14.

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

60

rato

r Ang

le (d

eg)

GEN 4

GEN 5

GEN 6

GEN 7

GEN 8

GEN 10

Angle curves for coherent generators according to slow modes

Angle curves for coherent generators

6. Application of Lindex in coherency identification

Dynamic aggregation of generators in a power system is a use-ful technique to reduce the order of large power systems to enablecomputational efficiency [21–24]. When more than one generatorsbehave similarly i.e. swing together, they are referred to as coher-ent generators. In the time domain, the swing curves of the angulardeviations from steady state for a group of coherent generators areidentical. Hence, such a group of generators may be represented bya single equivalent generator, thus reducing the overall order of thesystem considerably.

The coherency phenomenon in power systems is observed indisturbed generators located far away from the source of the dis-turbance. Hence, their response is linear, oscillatory and amenableto spectral interpretations. In the frequency domain, the coherencybetween generators is established as follows. The eigen-decompo-sition of the system of generators leads to the relevant eigenvaluesand eigenvectors. For any mode, the phase of the correspondingeigenvector indicates direction of the oscillations of all the gener-ators. Hence, comparing the phases of different machines for amode reveals the coherency between the machines for that mode.Such a coherency pattern, when established for the slow modes ofthe system, is referred to as slow coherency [10–13].

Low frequency modes or slow modes in power system do notnecessarily imply inter-area modes, as has been demonstrated inprevious sections. Instead of focusing on the slow modes to calcu-late the coherency between various generators, it is proposed hereto select system modes on the basis of low Lindex values. These mayor may not match the slow modes of the system, however thecoherency calculated with respect to these modes would ensurecoherency with respect to inter-area modes.

In this section, a ten-machine test system C is employed, whosemodal frequencies and Lindex values are provided in Table 5. Thecoherency between the generators was calculated on the basis ofboth the slow modes as well as the modes with low Lindex values.The resultant generator groups are shown in Table 6. The largestgenerator group formed on the basis of the low frequency modes

Table 5Electromechanical modes Lindex values for different test systems.

Test system B (six machine system)Lindex 0.081 1.310 2.702 3.919 3.984 3.992Frequency 0.016 0.609 0.477 0.959 1.277 2.389

Test system C (ten machine system)Lindex 1.703 4.721 4.838 4.773 5.758 6.082Frequency 0.121 0.286 0 0.339 0.322 0.244Lindex 6.236 7.729 7.801 7.815Frequency 0.349 0.392 0.394 0.385

Table 6Variation of grouping according to the slow method and proposed Lindex, method.

TYPE Group1 Group2 Group3 Group4

Slow modes 4,5,6,7,8,10 1,3 2 9Lindex method 4,5,6,7,9 1,8,10 3 2

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

Time (sec)

Gen

e

GEN 4

GEN 5

GEN 6

GEN 7

GEN 9

according to localness modes

Fig. 13. Time-domain simulation of generator swing curves for coherent groupsformed on the basis of slow modes or low Lindex modes. The disturbance is a three-phase bolted fault on bus 8.

consist of generators 4, 5, 6, 7, 8 and 10. When the grouping wasdone on the basis of low Lindex modes, the composition of this groupchanged slightly with generators 8 and 10 replaced by generator 9.

Area 1 Area 2

Fig. 14. Test system A.

Fig. 15. Test system B.

2

1

34 5

7

6

9

810

1

2

3 5

6

7

8

9

10

11

12

13

14

15

16

17

1819 20

2122

23

24

25

26

2728 29

430

31

32

33

34

35

3637

38

39

Fig. 16. Test system C.

312 S. Ghosh, N. Senroy / Electrical Power and Energy Systems 42 (2012) 306–313

The coherency of the generators was verified from time-domainsimulations of the test system C. Figs. 12 and 13 plot the swing

curves for the largest groups of generators of Table 6, for a boltedfault on bus 14 and 10 respectively. Both the figures confirm thatthe groups obtained on the basis of low Lindex modes exhibit bettercoherency than the groups obtained on the basis of slow modes.

7. Conclusions

In this paper an index, Lindex, is introduced to measure the local-ness of an electromechanical mode in a power system. Simple rep-resentative examples have been employed to explore the ability ofthe index to capture evolving power system dynamic behavior dueto changing system conditions. An illustrative application of the in-dex to help recognize dynamic coherency between generators hasalso been presented. Potential applications include dynamic aggre-gation studies, coherency identification, controller placement anddesign as well as model order reduction of power systems. Whileno analytical basis for the development of the Lindex is available,its link with the energy of a mode is a possible direction of futureresearch. Similarly, further research is required to compute appro-priate thresholds on Lindex to enable better discrimination betweenlocal and inter-area modes.

Appendix A

See Figs. 14–16.

S. Ghosh, N. Senroy / Electrical Power and Energy Systems 42 (2012) 306–313 313

References

[1] Kundur P. Power system stability and control. New York: McGraw-Hill; 1994.[2] MacFarlane AGJ. Use of power and energy concepts in the analysis of

multivariable feedback controllers. Proc IEEE 1969;116(8):1449–52.[3] Pérez-Arriaga IJ, Verghese GC, Schweppe FC. Selective modal analysis with

applications to electric power systems. Part I: heuristic introduction. Part II:the dynamic stability problem. IEEE Trans Power Apparatus Syst 1982;PAS-101(9):3117–34.

[4] Hamdan AMA. Coupling measures between modes and state variables inpower-system dynamics. Int J Control 1986;43(3):1029–41.

[5] Chan SM. Small signal control of multiterminal dc/ac power systems, PhDdissertation. Massachusetts Institute of Technology; 1981.

[6] Messina AR, Ochoa M, Barocio E. Use of energy and power concepts in theanalysis of the inter-area mode phenomenon. Electr Power Sys Res2001;59(2):111–9.

[7] Han S, Xu Z, Wu C. Mode shape estimation and mode checking for IAO usingcorrelation analysis technique. Electr Power Sys Res 2011;81(6):1181–7.

[8] Wu C, Zhang J. Determining of low frequency oscillations based on poweroscillation flows in the tie lines. In: Proc IEEE power & energy society generalmeeting, Calgary, AB; 2009. p. 1–5.

[9] Banejad M. Identification of damping contribution from power systemcontroller, Ph.D. dissertation, Environment and Eng research centre,Queensland university of Technology; 2004.

[10] Chow JH. Time scale modeling of dynamic networks with applications topower systems. New York: Springer-Verlag; 1982.

[11] Guangyue Xu, Vittal V. Slow coherency based cut set determination algorithmfor large power systems. IEEE Trans Power Syst 2010;25(2):877–84.

[12] You H, Vittal V, Xiaoming Wang. Slow coherency based islanding. IEEE TransPower Syst 2004;19(1):483–91.

[13] Bo Yang, Vittal V, Heydt GT. Slow coherency controlled islanding #8212; ademonstration approach on the August 14, 2003, black out scenario. IEEETrans Power Syst 2006;21(4):1840–7.

[14] Klein M, Rogers GJ, Kundur P. A fundamental study of inter area oscillations inpower systems. IEEE Trans. Power Syst 1991;6:914–21.

[15] Padiyar KR. Power system dynamics stability and control. 2nd ed. BSPublications; 2008. p. 547–51.

[16] Starrett SK. Data files for EPRI software using 6-gen system; 2008. <http://www.eece.ksu.edu/~starret/881/dfiles6.html>.

[17] Kaufman L, Rousseeuw PJ. Finding groups in data: an introduction to clusteranalysis. Hoboken, NJ: John Wiley & Sons, Inc.; 1990.

[18] Rousseeuw PJ. Silhouettes: a graphical aid to the interpretation and validationof cluster analysis. J Comput Appl Math 1987;20:53–65.

[19] Zárate-Miñano R, Milano F, Conejo AJ. An OPF methodology to ensure small-signal stability. IEEE Trans Power Syst 2011;26:1050–61.

[20] Ghosh S, Senroy N. Effect of generator type on the type of adjacentsynchronous machines in electromechanical oscillations. Proc IEEE PEDES,New Delhi, India; 2010. p. 1–3.

[21] Podmore R. Identification of coherent generators for dynamic equivalents. IEEETrans Power Appl Syst 1978;PAS-97(4):1344–54.

[22] Miah AM. Study of a coherency based simple dynamic equivalent for transientstability assessment. IET Gener Transm Distrib 2011;5(4):405–16.

[23] Chakraborty A, Chow JH, Salazar A. A measurement based framework fordynamic equivalencing of large power systems using wide area phasormeasurements. IEEE Trans Smart Grid 2010;2(1):68–81.

[24] Marinescu B, Mallem B, Rouco L. Large scale power system dynamicequivalents based on standard and border synchrony. IEEE Trans Power Syst2010;25(4):1873–82.