Evaluation of Suitability of Different Transient Stability Indices for Identification of Critical System States
EEvaluation of Suitability of DifferentTransient Stability Indices for Identification ofCritical System States
A. Sajadi,
Member, IEEE , R. Preece,
Member, IEEE , and J. V. Milanovi´c,
Fellow, IEEE
School of Electrical and Electronic EngineeringUniversity of ManchesterOxford Rd, Manchester M13 9PL, UKCorresponding Author: [email protected]
Abstract —Power system stability indices are used as measuresto evaluate and quantify the response of the system to externallarge disturbances. This paper provides a comparative analysisof established transient stability indices. The indices studied inthis paper include rotor-angle difference based transient stabilityindex (TSI), rate of machine acceleration (ROMA), transientkinetic energy (TKE), and transient potential energy (TPE). Theanalysis is performed using the 3-machine, 9-bus standard testsystem under a realistic range of loading levels. The aim of thestudy is to determine their suitability for reliable identificationof critical system conditions considering system uncertainties.
Index Terms —Power System Dynamics, Transient Stability,Stability Index
I. I
NTRODUCTION
Transient stability analysis investigates the dynamic behav-ior of a given system in respect to the time following a largeexternal disturbance. The external disturbance could be in forma short term fault, such as short circuit faults on transmissionlines or generators with a successful clearance, or a long termfault, such as an outage of generation unit(s) or a disconnectionof lines [1]. Following a fault, regardless of its type, theoscillations that are excited by the fault should be damped suchthat ringing decays within the first few cycles to few secondsfollowing the fault. Otherwise, the transient behavior of thesystem may dominate the system response that the systemtrajectory diverges from the stability region associated withthe pre-fault equilibrium point and potentially lead to system-wide failures [2]. Therefore, transient stability analysis are akey stage in planning and operation studies of power systems.Transient stability index is a measure to quantify the dis-tance between any given operating point of the system (pre-fault equilibrium point) and the critical operating point of thesystem (the margin of the stability region). In other words, itis an indication of power system stability limit at any givenoperating point. Thus, the accuracy of this measure is cruciallyimportant for power system industry to ensure a reliable, stableand secure delivery of power to consumers.Future power systems will be associated with a greaterdegree of uncertainty and complexity because of involvementof highly intermittent power generation sources and power-electronics and overall change in the paradigm of power networks. The higher complexity of the system can be alsointerpolated as higher dimensionality of the system. As a re-sult, it is essential to develop new probabilistic analytical toolsto quantify involved risk in future power systems operation andensure stability and security of electrical energy delivery.The goal of this work is to identify and validate robuststability indicator(s) that can be utilized for different facets ofpower system stability risk analysis. There is significant inter-est in identifying critical system conditions and understandingthe corrective action required to reduce their criticality. Aproper understanding of the performance of different transientstability indicators will enable fast and reliable identificationof appropriate actions.To reach this goal, transient stability indices introduced andestablished in literature are compared. The considered indicesinclude rotor-angle difference based transient stability index(TSI) [3], rate of machine acceleration (ROMA) [4], tran-sient kinetic energy (TKE) [5], and transient potential energy(TPE) [6]. To ensure greater applicability, only simulation-based methods have been investigated as they are more easilyimplemented on existing practical network models and willinherently include the effects of discontinuous system elements(such as controller limits or saturation effects). The analyticalmethods, such as the use of energy functions, are not includedin present analysis due to their computational complexity andlimited applicability for large and complex power systems.Numerical results are illustrated on a 3-machine, 9-busstandard test system [7] under a range of feasible loadinglevels. The disturbances considered in this study include powersystem faults at various busbars representing various levelsof event’s severity. Then, corresponding transient stabilityindices are computed for all of the studied loading levelsand considered faults in all dimensions of operational space.Finally, the sensitivity and smoothness of each index ismeasured to evaluate their suitability for further probabilisticstudies. In further work, the most suitable metric will beused as indicator to predict a broader probabilistic surface andregion of stability for system operation. This will be helpfulto identify the risks and costs associated with operation of thesystem. a r X i v : . [ ee ss . S Y ] J a n he contribution of this study is to identify the most prac-tical and efficient transient stability index to conduct furtherstudies regarding operation and control of power systems witha higher degree of complexity and uncertainty. In particular,the aim is to identify a metric that accurately reflects thetransient stability in terms of sensitivity to system parameterchanges in higher dimensions and fault severity. This studyconsiders change of loading level as the only variable change.II. T RANSIENT S TABILITY I NDICES
Transient stability can be defined by the system’s ability tomaintain its operation following a fault [1]. And the longesttime that system’s trajectory remains within the stability regionassociated with the pre-fault equilibrium point before reachingthe critical operating point at which the instability begins,is called critical clearing time (CCT) [1]. Considering thesedefinitions, the transient stability index can be defined bya quantification of system’s strength to sustain its transientstability. Fig. 1 illustrates a visualization of transient stabilityindex.Fig. 1: Conceptual visualization of transient stability indicesin power systemsThe following section briefly describes the time-domainbased transient stability indices considered in this comparativestudy. The full theoretical background on these indices areextensively described in the quoted references.
A. Rotor Angle Difference Based Transient Stability Index(TSI)
This index relies on maximum rotor angle separation be-tween any two given generators and is given by (1) [3].
T SI = 360 − δ max
360 + δ max × (1)In (1), δ max is maximum rotor angle difference between anytwo generators in the system immediately after fault inception.The closer the value of T SI to 100 is, the more stable thepower system is.
B. Rate of Machines Acceleration (ROMA)
The rate of acceleration or deceleration of generators’rotors in a power system is an indication of its iner-tia and, therefore, the rate of the frequency deviation[4]. Thus, the rate of machines’ acceleration, similar to
Rate of Change of F requency ( ROCOF ) [8], can bedefined by (2). ROM A = max (cid:42) da P F T i dt (cid:43) ≈ max (cid:42) ∆ a P F T i ∆ t (cid:43) (2)In (2), ∆ a P F T i and ∆ t are finite differences of rotoracceleration and time of i -th machine immediately after faultoccurrence to approximate the differentials. C. Transient Kinetic Energy (TKE)
The generators’ transient kinetic energy immediately afterthe fault clearance is defined by (3) [5].
T KE = (cid:88) i =1 J i · ∆ ω i (3)In (3), J i and ∆ ω i are angular momentum of the rotor atsynchronous speed and speed deviation of i -th generator. D. Transient Potential Energy (TPE)
The generators’ transient potential energy immediately afterthe fault clearance is defined by (4) [6].
T P E = (cid:90) t clear t fault (cid:104) ∆ P G i − ∆ P G j (cid:105) ∆ f ij · dt (4)In (4), ∆ P G i and ∆ P G j refer to transient active power ofany given pair of generators i and j and ∆ f ij is the frequencydifference between them. t fault and t clear is the time at whichfault occurs and clears, respectively.III. C OMPUTATIONAL I MPLEMENTATION
In this section, the case study used in this paper andconsiderations and data toolkit used for computational imple-mentation are described.
A. Power Systems Simulation
This study used a 3-machine, 9-bus standard test system(also known as P. M. Anderson 9-Bus), shown in Fig. 2, as acase study. This system consists of 3 synchronous machinesand 3 loads [9].The power system simulation was carried out in DigSILENTPowerFactory software package, v15.2.8. Based on previouslycarried out studies and available information in literature [10],[11], [12], five faults were considered, as described in TableI. The applied faults were balanced 3-phase faults with clear-ance time of 10 cycles.The load and generation dispatch datasets were generatedusing optimal power flow (OPF) in MATPOWER package.The load and generation data were developed for three sce-narios:ig. 2: One-line diagram of a 3-machine, 9-bus standard testsystem TABLE I: Description of Faults
Fault No. Faulted Bus Faulted Line1 4 4-62 5 5-73 6 6-94 7 7-85 8 8-9 • By proving 1 degree of freedomin change of loads, corresponding to change of one loadas the only variable while two other loads remain fixed. • By proving 2 degrees of freedomin change of loads, corresponding to change of two loadsas the variables while the other load remains fixed. • By proving 3 degrees of freedomin change of loads, corresponding to change of all threeloads as the variables.To generate load and generation datasets, for each scenario,loads were varied from 30% to 100% of their nominal con-sumption capacity with steps of 2%, resulting in investigationof 46,656 operating conditions for each fault. For each step,an OPF solution with a homogeneous cost function for allgenerators was run. The reason for using 2% steps was tohave a reasonable computational process time with sufficientdata point to capture the continuous possible load variation.Finally, the datasets were used to run the electromagnetictransient (EMT) simulation in DigSILENT PowerFactory.
B. Data Analytics
The overarching aim of this study is to identify the transientstability indices that have the greatest potential for furtheruse when identifying critical system conditions. These indicesmust therefore be sensitive to parameter changes and varyas conditions vary. Moreover, they must also be smooth inthese variations in order to provide confidence that they areuseful as predictive indices. This is of particular importancewhen the study expands to a multi-dimensional search (inmultiple parameters) and a multi-dimensional surface (and notonly a line) is produced. In this way, it is more likely thatthe global, rather than local minima (with respect to stabilityperformance) is identified. To compute the transient stability indices, discussed inprevious section, and measure their features, the results fromthe EMT simulation in DigSILENT PowerFactory simulationtool were imported in MATLAB and aforementioned transientstability indices were computed.Following section describes the toolkits used for data ana-lytics after computation of these indices.
1) Data Standardization:
Since this is a comparative study,the computed transient stability indices are required to bestandardized as they may vary in different ranges. Data nor-malization refers to adjusting data measured on different scalesto a common scale to bring them into a meaningful alignment.The standardization technique sued in this study is given by(5). x (cid:48) = xmax ( x ) (5)In (5), x (cid:48) and x are standardized data and original datapoints, respectively, and max ( x ) is maximum value of thedataset that x belongs to.It should be emphasized that the data standardization tech-nique used in this study is to only cap the datasets at theirmaximum value as a common reference point. A rescale(normalization) between maximum and minimum will maskof the actual sensitivity and smoothness of the datasets.Standardizing at the maximum value avoids this.
2) Sensitivity Analysis:
To measure the sensitivity of eachoutput dataset y to a change of operational variables, load inthis study denoted x , sensitivity index (SNI) defined by (6)was used [13]. SN I = − log (cid:42) (cid:90) dydx dx (cid:43) (6)In (6), dydx is the instantaneous slope of y ( x ) . And integrationis to measure the size of the slope for all data points. Thesmaller the value of SN I is, the more sensitive the dataset isto a change of variables. The logarithmic scale is to avoidappearance of a significant order of decimal digits in theresults. It should be noted that this is a comparative study andthe scale used for comparison of the results does not influencethe outcome of this research.In a n -dimensional search space, the total sensitivity of theoverall surface is given as (7). SN I overall = n (cid:88) i =1 SN I i n ! (7)In (7), SN I i is the SN I of the dataset of n -th dimension.
3) Smoothness Analysis:
To measure the smoothness ofeach output dataset y to a change of operational variables, x (load in this study), a smoothness index (SMI) defined by(8) was used [13]. SM I = − log (cid:42) (cid:90) (cid:20) d ydx (cid:21) dx (cid:43) (8)n (8), d ydx is the curvature of y ( x ) . And integration isto measure the size of the curvature for all data points. Thegreater the value of SM I is, the smoother the dataset is. Thelogarithmic scale, similar to the
SN I , is to avoid appearanceof a significant order of decimal digits in the results.In a n -dimensional search space, the total smoothness ofthe overall surface is given as (9). SM I overall = n (cid:88) i =1 SM I i n ! (9)In (9), SM I i is the SM I of the dataset of n -th dimension.IV. R ESULTS AND D ISCUSSIONS
In this section, the results obtained using different transientstability indices for assessment of transient stability of the3-machine, 9-bus standard test system shown in Fig. 2, arepresented.
A. Data Standardization
In this study, the data from investigated transient stabilityindices were standardize at the 100% loading level of system,for all three loads, as the common reference point. Fig. 3illustrates the standardized values of studied indices in one-dimension with load 1 as the only variable, following fault1.
30 40 50 60 70 80 90 100
Loading Factor [%] I nd i c e s V a l ue TSI TKETPEROMA
Fig. 3: Standardized values of studied transient stability indicesin a single-dimensional analysis: Load 1 changes as the onlyvariable, following fault 1 in the studied test systemThe plots illustrated in Fig. 3 represent studied stabilityindices. It is evident that the indices were standardized witha common reference point of 1 while their range is mappedrelatively. Amongst the indices,
T SI is the only index whosevalue increases as the loading level of the system decreases.Whereas the values of
ROM A , T KE , and
T P E decreasesproportional to decrease of loading level of system.In a physical sense, the lower the loading level of the systemis, the lower amount of transient energy in the network todissipate following the fault clearance is. As a result, transientkinetic and potential energies and machines’ acceleration willbe reduced proportional to lower levels of system loading. Consequently, the rotor angle differences in the system willreduce leading to an increase of
T SI value.Thus, the lighter the loading level of system becomes, theless severe the faults become as the transient stability marginof system increases.The results from analytical investigation and multi-dimensional analysis on these indices are shown in next twosections.
B. Sensitivity Analysis
In this section, results from sensitivity analysis of thestudied stability indices are presented. As a reminder, thesmaller the value of
SN I is, the more sensitive the dataset is toa change of variables. Sensitivity of a transient stability indexis crucially important as it ensures an accurate reflection of thetransient stability in terms of sensitivity to system parameterchanges.
1) One-dimensional Analysis:
The values of computed
SN I for single-dimensional surface corresponding to a changeof load 1 as the only variable are presented in Table II. Itshould be noted that results from a change of load 2 and load3 as the only the variable in the system with other two loadskept constant, are similar.TABLE II: The results from sensitivity analysis of stabilityindices – single dimensional – Change of load 1 as the onlyvariable while loads 2 and 3 remain constant
Index TSI ROMA TKE TPEFault 1 .
59 1 .
94 1 .
63 1 . Fault 2 .
61 1 .
85 1 .
55 1 . Fault 3 .
59 1 .
97 1 .
54 1 . Fault 4 .
49 2 .
11 1 .
59 1 . Fault 5 .
47 2 .
25 1 .
60 1 . From the results shown in Table II, it can be seen that
T KE , consistently, holds the minimum
SN I values amongstthe studied indices for all studied faults, ranging from 1.54to 1.63. Whereas the
T SI indicates the highest
SN I values,consistently, for all faults, ranging from 2.47 to 2.61, outlining60% greater values of
SN I than
T KE does. The
SN I valuesfor
T P E and
ROM A are second and third highest.It can be concluded that the in single-dimensional analysis,
T KE is the most sensitive index and
T P E , ROM A , and
T SI follow.
2) Two-dimensional Analysis:
The values of computed
SN I for two-dimensional surface corresponding to a changeof loads 2 and 3 as the only variables are presented in TableIII. It should be noted that results from a change of loads 1and 3 and loads 1 and 2 as the only the variables in the systemwith other load kept constant, are similar.The
SN I values for
T KE , consistently, are the minimumamongst the studied indices for all studied faults, ranging from1.68 to 1.90. The
SN I values for
T SI are the highest amongstthe studied indices for all faults, ranging from 2.64 to 2.69.The
SN I values for
T P E and
ROM A are second and thirdhighest by ranging within 1.80 and 2.01 and 2.01 and 2.21.ABLE III: The results from sensitivity analysis of stabilityindices – two dimensional – Change of loads 2 and 3 as thevariables while load 1 remains constant
Index TSI ROMA TKE TPEFault 1 .
62 2 .
01 1 .
81 1 . Fault 2 .
69 2 .
01 1 .
71 1 . Fault 3 .
64 2 .
03 1 .
68 1 . Fault 4 .
68 2 .
18 1 .
88 1 . Fault 5 .
65 2 .
21 1 .
90 2 . It can be concluded that the in two-dimensional analysis,
T KE is the most sensitive index and
T P E , ROM A , and
T SI follow.
3) Three-dimensional Analysis:
The values of computed
SN I for sensitivity analysis of a three-dimensional surfacecorresponding to a change of loads 1, 2 and 3 as the variablesare presented in Table IV.TABLE IV: The results from sensitivity analysis of stabilityindices – three dimensional – Change of loads 1, 2 and 3 asthe variables
Index TSI ROMA TKE TPEFault 1 .
70 2 .
03 1 .
86 1 . Fault 2 .
75 2 .
02 1 .
86 1 . Fault 3 .
71 2 .
03 1 .
85 1 . Fault 4 .
69 2 .
15 1 .
91 1 . Fault 5 .
66 2 .
09 1 .
92 1 . The results shown in Table IV indicate a similar informationas was previously indicated in Tables II and III.
T KE and
T SI consistently feature the smallest and largest
SN I valuesfor all studied faults which reflects the highest and lowestsensitivity amongst studied indices, respectively, by rangingwithin 1.85 and 1.92 for
T KE and 2.66 and 2.75 for
T SI . T P E and
ROM A stand second and third in ranking ofsensitivity of indices.
T KE shows the greatest level of sensitivity because it is afunction of the generators’ angular momentum and the squareof the speed deviation. The greater the loading of the systemis, the greater speed deviations are. As a result, this indexincreases with the square of the system loading. Similarly,
T P E is a quadratic function of the system loading as thisindex is computed as the product of frequency deviation andactive power deviation of a pair of generators.
ROM A isthe third sensitive index and is a first order function of thesystem loading. This index is computed using the generatorsacceleration which a function of systems loading. Finally,
T SI is the least sensitive index as is a first order function of theangular difference of generators. The generators’ angles varyproportionally to the system loading with a similar rate and,therefore, the difference between them (and therefore the
T SI )changes at a slower rate compared to the other studied indices.Fig. 4 visualizes the
SN I for studied indices in multipledimensions. The results shown in this figure reveal the con-sistency of used sensitivity measure and the appropriatenessof using this measure for this work as it has been suitablyadapted for multi-dimensional analysis. S N I V a l u e Sensitivity of Studied Transient Stability Indices
Fault 1 Fault 2 Fault 3 Fault 4 Fault 5
Fig. 4: The results from sensitivity analysis of studied transientstability indices
C. Smoothness Analysis
In this section, results from smoothness analysis of thestudied stability indices are presented. As a reminder, thegreater the value of
SM I is, the smoother the dataset is.Smoothness of a transient stability index is very importantas it ensures its suitability for further probabilistic studiesof the system as the system parameters change in multipledimensions.
1) One-dimensional Analysis:
The values of computed
SM I for single-dimensional surface corresponding to achange of load 1 as the only variable are presented in TableV. The results from a change of load 2 and load 3 as the onlythe variable in the system with other two loads kept constant,are similar.TABLE V: The results from smoothness analysis of stabilityindices – single dimensional – Change of load 1 as the onlyvariable while loads 2 and 3 remain constant
Index TSI ROMA TKE TPEFault 1 .
27 6 .
45 7 .
70 8 . Fault 2 .
38 6 .
37 7 .
63 8 . Fault 3 .
33 6 .
24 7 .
56 8 . Fault 4 .
46 6 .
53 8 .
07 9 . Fault 5 .
50 5 .
61 8 .
01 9 . The results shown in Table V reveal that the values of
SM I for
T SI are, consistently, the greatest amongst all studiedindices for all five fault events, ranging from 10.27 to 10.50.Whereas the values of this metric for
ROM A is the smallestamongst the indices for all studied scenarios, ranging within5.61 and 6.45. The values of
SM I for
T P E and
T KE arestand second and third in smoothness ranking. These resultsoutline that for a 1-dimensional surface, the
T SI is the mostsmooth index and
T P E , T KE , and
ROM A follow.
2) Two-dimensional Analysis:
The values of computed
SM I for two-dimensional surface corresponding to a changeof loads 2 and 3 as the variables are presented in Table VI.The results from a change of loads 1 and 3 and loads 1 and2 as the only the variables in the system with other load keptconstant, are similar.ABLE VI: The results from smoothness analysis of stabilityindices – two dimensional – Change of loads 2 and 3 as thevariables while load 1 remains constant
Index TSI ROMA TKE TPEFault 1 .
19 5 .
05 6 .
48 6 . Fault 2 .
30 4 .
78 6 .
45 6 . Fault 3 .
24 4 .
78 6 .
38 7 . Fault 4 .
40 5 .
02 6 .
74 7 . Fault 5 .
44 4 .
21 6 .
91 7 . The results shown in Table V reveal that
T SI and
ROM A indicate the greatest and smallest values for
SM I for allstudied fault events, ranging from 8.19 to 8.44 for
T SI and4.21 to 5.05 for
ROM A . The values of this metric for
T P E and
T KE range within 6.48 and 7.87 and 6.38 and 6.91,respectively. These results conclude that for a 2-dimensionalsurface, the
T SI is the most smooth index and
T P E , T KE ,and
ROM A follow.
3) Three-dimensional Analysis:
The values of computed
SM I for smoothness analysis of a three-dimensional surfacecorresponding to a change of loads 1, 2 and 3 as the variablesare presented in Table VII.TABLE VII: The results from smoothness analysis of stabilityindices – three dimensional – Change of loads 1, 2 and 3 asthe variables
Index TSI ROMA TKE TPEFault 1 .
94 5 .
07 6 .
86 6 . Fault 2 .
02 4 .
77 6 .
82 6 . Fault 3 .
98 4 .
72 6 .
73 6 . Fault 4 .
12 5 .
00 7 .
06 6 . Fault 5 .
16 4 .
34 7 .
23 7 . The results presented in Table VII highlight that the valuesof smoothness metric for
T SI and
ROM A are consistentlythe largest and smallest for all studied faults. This suggeststhat the three-dimensional surfaces created using these twotransient stability indices are the most and least smooth sur-faces amongst the studied surfaces, respectively. The surfacesconstructed by using
T P E and
T KE indices are second andthird in terms of smoothness amongst investigated indices.The reason for the
T SI to show the greatest level ofsmoothness can be justified by its simple and linear relation-ship to the rotor angle difference, as presented by (1).
ROM A is computed using rate of change of acceleration of generators,defined by (2), which varies significantly depending on opera-tional point of each generator prior to fault.
T KE is computedusing summation of quadratic functions with different weightsin which different angular momentum of generators are afactor, given by (3). Thus, its behavior is non-linear and,therefore, its smoothness is weakened.
T P E is computedusing integral of dot product of two dynamic variables ofpair-generators, their frequency deviation and transient activepower, defined by (4). Therefore, it is a non-linear function andits smoothness in response to change of system’s variables isinfluenced.Fig. 5 visualizes the
SM I for studied indices in multiple dimensions. The results shown in this figure reinforce theconsistency of this smoothness measure and how suitably itcaptured the smoothness of indices in an multi-dimensionalanalysis. S M I V a l u e Smoothness of Studied Transient Stability Indices
Fault 1 Fault 2 Fault 3 Fault 4 Fault 5
Fig. 5: The results from smoothness analysis of studiedtransient stability indices
D. Discussion
In two previous sections, the results from sensitivity analysisand smoothness analysis of four commonly used transientstability indices in power industry,
T SI , ROM A , T P E , and
T KE , are shown. The results included single-dimensionalanalysis which reflects variation of a single load in thesystem, two-dimensional analysis which addresses variation oftwo loads in a system, and three-dimensional analysis whichaddresses variation of three and all of the loads in this system.By looking at the consensus among all presented results inthis paper, a clear consistency across multi-dimensional anal-ysis can be seen. This highlights the suitability of suggesteddata analysis methods.The point to note from the results presented in section IV-B,
Sensitivity Analysis , is that, regardless of dimensionality of thesystem,
T KE is the most sensitive index for transient stabilityanalysis in power systems.
T P E , ROM A , and
T SI also aresensitive to a change of variables in the system, however, bylesser degrees.The point to note in this study, from the results presentedin section IV-C,
Smoothness Analysis , is that, regardless ofdimensionality of the system, the surface created by the
T SI stability index is the smoothest for transient stability analysisin power systems. The surfaces created by the
T P E , T KE ,and
ROM A are smooth as well, however, to a lower degree.The main purpose of this study has been to identify themost suitable transient stability index that can be utilized fordifferent facets of power system stability risk analysis andto conduct further studies regarding operation and control ofpower systems with a higher degree of complexity and un-certainty. Ideally, a single index is desired to, simultaneously,offer the highest level of sensitivity and smoothness. However,each of them have their own limitations. Thus, the desiredstability index can be scrutinized and identified by a fair .41.61.822.22.42.62.8
SNI S M I Higher Sensitivity HigherSmoothness
TSIROMA TPE TKE (a) One-dimensional
SNI S M I Higher Sensitivity HigherSmoothness
TSIROMA TKETPE (b) Two-dimensional
SNI S M I Higher Sensitivity
TSI
HigherSmoothness
ROMA TPE TKE (c) Three-dimensional
Fig. 6: Smoothness vs sensitivity for studied indicestrade-off between these two measures. Fig. 6 visualizes thesmoothness vs. sensitivity for each of the studied indices invarious dimensional space for the different faults considered.By considering the presented results in various dimensionsand presented plot in Fig. 6, it can be concluded that
T P E isthe most suitable transient stability index for the purpose offurther studies. This index offers consistently high levels ofboth sensitivity and smoothness.
T SI offers the greatest level of smoothness with respectto variability of system’s operational condition. However,its major weakness is its lower sensitivity which makes itless attractive. Similarly,
T KE shows the greatest level ofsensitivity while its lower smoothness is a disadvantage forthis index. Finally,
ROM A features low sensitivity and lowsmoothness and is the least valuable stability index in thissense. V. C
ONCLUSION
This work attempted to identify the stability indicators thatcan be used for different facets of future power system stabilityrisk analysis with higher dimensionality and complexity. Toevaluate the suitability of the desired index, its sensitivity toa change of variable and operational conditions of system aswell as smoothness of the surface created by the given indexin a multiple-dimensional space were investigated.This research provided a comparison among the transientstability indices established in literature. These indices in-cluded rotor-angle difference based transient stability index(TSI), rate of machine acceleration (ROMA), transient kineticenergy (TKE), and transient potential energy (TPE). A 3-machine, 9-bus standard test system was used as a case study.The results suggest that
T P E is the most suitable transientstability index for the purpose of further studies as it offersconsistently high levels of both sensitivity and smoothness.Further investigation will include developing mathemati-cal framework to efficiently identify operating conditions orsystem contingencies that will lead to instability in a high-dimensioned search-space using the identified suitable indicesin larger power systems such as NETS-NYPS and Great Britain networks. It will focus on development of an efficientmethod to find critically unstable system conditions of thesystem, including low probability high impact events, to ensurethat sufficient samples are used and various facets of systemoperation are included.A
CKNOWLEDGMENT
The authors would like to thank Research Councils UKfor financial support of this research through the HubNetconsortium (grant number: EP/N030028/1).R
EFERENCES[1] J. Machowski, J. Bialek, and J. Bumby,
Power system dynamics: stabilityand control . John Wiley & Sons, 2011.[2] A. Sajadi, R. Kolacinski, and K. Loparo, “Transient voltage stability ofoffshore wind farms following faults on the collector system,” in
Powerand Energy Conference at Illinois . IEEE, 2016, pp. 1–5.[3] L. Shi, S. Dai, Y. Ni, L. Yao, and M. Bazargan, “Transient stabilityof power systems with high penetration of DFIG based wind farms,” in
Power & Energy Society General Meeting, 2009. PES’09. IEEE . IEEE,2009, pp. 1 – 6.[4] E. Telegina, “Impact of rotational inertia changes on power systemstability.”[5] P. Kundur, N. J. Balu, and M. G. Lauby,
Power system stability andcontrol . McGraw-hill New York, 1994, vol. 7.[6] C. S. Saunders, M. M. Alamuti, and G. A. Taylor, “Transient stabilityanalysis using potential energy indices for determining critical generatorsets,” in
PES General Meeting— Conference & Exposition, 2014 IEEE
IEEE Trans. on Circuits and SystemsI: Fundamental Theory and Apps. , vol. 42, no. 5, pp. 252–265, 1995.[11] L. Mariotto, H. Pinheiro, G. Cardoso Jr, A. Morais, and M. Muraro,“Power systems transient stability indices: an algorithm based on equiv-alent clusters of coherent generators,”
IET generation, transmission &distribution , vol. 4, no. 11, pp. 1223–1235, 2010.[12] G. Dhole and M. Khedkar, “Antigen energy function: a new energyfunction for transient stability assessment,”