transient energy of an individual machine part i

Transient Energy of an Individual Machine PART I: StabilityCharacterization

Wang, S., Yu, J., Foley, A. M., & Zhang, W. (2021). Transient Energy of an Individual Machine PART I: StabilityCharacterization. IEEE Access, 9, 44797-44812. https://doi.org/10.1109/ACCESS.2021.3064969

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https://doi.org/10.1109/ACCESS.2021.3064969

https://pure.qub.ac.uk/en/publications/c8c1e1dd-2316-46c8-8b5f-802dc2932f45

Received February 26, 2021, accepted March 2, 2021, date of publication March 9, 2021, date of current version March 26, 2021.

Digital Object Identifier 10.1109/ACCESS.2021.3064969

Transient Energy of an Individual MachinePART I: Stability CharacterizationSONGYAN WANG 1, JILAI YU 1, AOIFE M. FOLEY 2, (Senior Member, IEEE),AND WEI ZHANG 1, (Senior Member, IEEE)1Department of Electrical Engineering, Harbin Institute of Technology, Harbin 150001, China2School of Mechanical and Aerospace Engineering, Queen’s University Belfast, Belfast BT7 1NN, U.K.

Corresponding author: Songyan Wang ([email protected])

ABSTRACT The individual-machine-equal-area-criterion method already shows potential in the transientstability assessment of multimachine systems. This two-paper series systematically analyzes the transientcharacteristics of the system trajectory from a genuine individual-machine transient energy perspective.In the first paper, the stability property that characterizes a critical machine is investigated by definingthe residual kinetic energy, which only occurs at the maximum individual-machine potential energy point.We indicate that the transient energy conversion inside a critical machine can delineate the trajectory stabilityof the machine. In this way, the mapping between individual-machine kinetic energy and individual-machinetrajectory can hence be established. Simulation results show that the individual-machine transient energydescribes the system trajectory more precisely than the superimposed global transient energy in terms oftransient stability analysis, and we also demonstrate that the conjecture in early individual-machine studiesmight fail in special simulation cases.

INDEX TERMS Transient stability, transient energy, equal area criterion, individual machine.

NOMENCLATUREKE Kinetic energyPE Potential energyCOI Center of inertiaCCT Critical fault clearing timeDLP Dynamic liberation pointDSP Dynamic stationary pointEAC Equal area criterionGTE Global total transient energyGKE Global KEGPE Global PEMDM Most severely disturbed machineMPP Maximum potential energy pointTSA Transient security assessment3DKC 3-dimensional Kimbark curve3DKE 3-dimensional kinetic energy curveGMPP Global MPPIMKE Individual machine kinetic energyIMPE Individual machine potential energyIMPP Individual machine MPP

The associate editor coordinating the review of this manuscript and

approving it for publication was Xiaorong Xie .

IMTE Individual-machine transient energyIMTR Individual-machine trajectoryIVCS Individual machine-virtual COI-SYS machine

systemIMEAC Individual machine EAC

I. INTRODUCTIONA. LITERATURE REVIEWTransient energy depicts the capability of the postfault powernetwork to absorb the accumulated energy during a distur-bance [1], [2]. Once a large disturbance occurs in the powersystem, the complicated motions of the system trajectoriescan be well explained using the conversion between thekinetic energy and potential energy of the system. Motivatedby this transient energy perspective, Athay et al. [3] statedthat the Lyapunovmethodmay yield severe conservativeness,and the real critical transient energy of the system would benear the potential energy at the unstable equilibrium point.Kakimoto et al. [4], [5] replaced the actual time-domainsimulated system trajectory with the sustained fault trajectoryto approximate the critical potential energy. Notably, the twotransient energy methods have played key roles in revealingthe mechanisms of transient stability, and they are milestonesin the history of transient stability analysis.

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S. Wang et al.: Transient Energy of an Individual Machine PART I: Stability Characterization

The methods developed by Athay and Kakimoto are pro-posed using the transient energy of all machines in the system,and these methods are called ‘‘global’’ transient energy meth-ods. Interestingly, another distinctive idea of using individualmachines for transient stability analysis was formed at almostthe same time. Fouad and Stanton [1], [2] believed that theinstability is determined by the motion of some unstablecritical machines if more than one machine tends to losesynchronism. Later, Vittal and Fouad stated that ‘‘systemseparation does not depend on the total system energy, butrather on the transient energy of individual machines orgroups of machines tending to separate from the rest’’, and anindividual-machine-energy-function was first proposed [6],[7]. Based on these previous works, Stanton performed adetailed machine-by-machine analysis of the energy of amultimachine instability [8]. Stanton also defined a partial-energy function to quantify the energy conversion of theindividual machine, and the method was used to measure theenergy of a local transient control action in a real industrialenvironment [9], [10]. In addition, Rastgoufard et al. [11] andHaque [12] used the equal-area-criterion (EAC) characteris-tics of the individual machine to compute the critical clearingtime. Moreover, Ando and Iwamoto [13] presented the poten-tial energy ridge to evaluate the individual-machine stability.These individual-machine thinkings essentially depart fromstate-of-the-art of global methods.

Modern power systems are constantly changing. Facingthe increasing complexity of the power system, model-freehybrid methods that are fully based on real-time measuredtrajectories become challenging. Specifically, the energyflow method was developed to locate the oscillation sourceof the power system [14]. Trajectory sensitivity techniqueshave been used in the computation of stability bound-aries [15], [16]. Pattern recognition techniques that rely onphasor measurement units are used to predict the transientstability of the system [17]. A precise frequency-dependentmodel is established for transient stability analysis of a powersystem with frequency threats [18]. The hardware-softwarereference model has already been developed to validatethe simplified mathematical models in the time-domainsimulations [19]. In addition, well-known trajectory-basedgroup separation methods have been applied in transientstability assessment (TSA) [20], [21]. With the integra-tion of inverter-based distributed generators, the single-machine-equivalent method is applied to analyze the powersystem transient stability considering momentary cessa-tion [22]. Transient stability is also enhanced through theemergency coordinated modulating strategy in the equivalenttwo-machine system considering ultrahigh voltage directcurrent transmissions [23]. Generally, these modern hybridmethods are still based on global transient information.Recently, inspired by individual-machine thinking, a novelhybrid individual-machine EAC method, i.e., the IMEACmethod, was proposed [24], [25]. Later, the authorsreplaced the virtual COI machine with a real machinein the IMEAC, and a novel nonglobal method, i.e., the

reference-machine-based IMEAC method, was estab-lished [26]. The IMEAC method already shows potential inTSA of industrial large-scale power systems.

Compared with those of well-developed hybrid methods,the studies of transient energy are currently fading, andthey have been at a standstill in modern TSA. However,an ineluctable fact is that transient energy may provide deepinsight into the physical nature of transient stability from theangle of energy conversion. Compared with global energyconcepts, studies of individual-machine transient energy(IMTE) are rare. In particular, key concepts, such as thepotential energy surface and the transient energy conversioninside an individual machine, have remained at the stage ofuncompleted discussions and hypotheses [6]–[10]. Againstthis background, the exploration of the IMTE becomes valu-able because it may validate the hypothesis of early individ-ual machine studies. Most importantly, it may also providea theoretical foundation for the recently proposed IMEACmethod [24]–[26].

B. SCOPE AND CONTRIBUTION OF THE PAPERThe two-paper series can be considered the companion papersof the three-paper series [24]–[26]. The two paper seriesexplore the IMEAC method from the perspective of transientenergy. The first paper focuses on the transient characteris-tic of the IMTE, and the companion paper establishes thepotential energy surface of an individual machine. In thefirst paper, based on the definition of the IMTE, it is proventhat the IMTE becomes conserved once a fault is cleared,and the individual-machine potential energy (IMPE) mayreach amaximum at the individual-machinemaximum poten-tial energy point (IMPP), i.e., the dynamic-stationary-point(DSP) or dynamic-liberation-point (DLP), depending on thestability state of the critical machine. Second, followingthe conversion between individual-machine-kinetic-energy(IMKE) and individual-machine-potential-energy (IMPE)inside a critical machine, it is proven that the residualindividual-machine-kinetic-energy (IMKE) at the IMPP canbe used to characterize the stability of the critical machine,and the unity principle [24] is redefined in light of resid-ual IMKE. At the end of this paper, the mapping betweenIMTE and individual machine trajectory (IMTR) of a criticalmachine is established through the 3-dimensional IMKE ofthe machine. The equivalence between IMKE and IMEAC isalso analyzed.

The contributions of this paper are summarized as follows:(i) Conversion between IMKE and IMPE inside a critical

machine is analyzed. This analysis reveals that the transientenergy conversion inside a critical machine has a dominanteffect on the stability of the system.

(ii) The instability of the system is proven to be determinedby the residual IMKE that occurs at the DLP of an unsta-ble critical machine. This feature distinguishes the IMEACmethod from the classic global transient energy theories [3].

(iii) The IMPE is found to vary substantially with theduration of the disturbance in special simulation cases.

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This finding clarifies the misunderstanding in previousindividual-machine works [7].

The remaining paper is organized as follows. In Section II,the background of the IMEAC method is revisited.In Section III, the IMTE is defined. In Section IV, the sta-bility characterization of a critical machine is investigatedusing residual IMKE. In Section V, the physical natureof the IMKE is explored through the equivalence betweenIMKE and IMEAC. In Section VI, simulation cases areprovided to demonstrate the individual-machine transientenergy conversion. Conclusions and discussions are stated inSection VII.

In this two-paper series, all the models and the transientenergy analysis are based on the COI-SYS reference, as inRef. [24]. The fault types in this paper are defined thesame as those in Ref. [24]. The test systems, i.e., TS-1,TS-2 and TS-3, can be found in Refs. [24], [25]. TS-4 isa three-machine system proposed by Anderson and Fouad[27]. TS-5 is a modified three-machine system proposed byPavella et al. [28]. All faults are three-phase short-circuitfaults, which occurred at 0 s. The fault is cleared with-out line switching. The identification of critical machinesfollows the strategy proposed in Refs. [1], [2]. Note thatsome cases in this paper are completely identical to those ofRefs. [24]–[26]. However, they are reanalyzed in a transientenergy form. Importantly, all the key concepts in this paperare analyzed from the individual-machine angle, with theglobal transient energy-based analysis being provided forcomparison.

II. BACKGROUND OF IMEAC METHODA. EQUATION OF MOTION OF AN INDIVIDUAL MACHINEIn the IMEAC method [24]–[26], an ‘‘individual machine’’should be precisely expressed as an ‘‘individual machine inthe COI reference’’.

For an n-machine system with rotor angle δi and inertiaconstantMi, the motion of the ith machine in the synchronousreference is governed by the differential equations{ •

δi = ωi

Mi•ωi = Pmi − Pei

(1)

where

Pei = E2i Gii +

∑n

j=1,j6=i(Cij sin δij + Dij cos δij)

Using the common terminology

Cij = EiEjBijDij = EiEjGij

Pmi mechanical power of Machine i (constant)Ei voltage behind transient reactance of Machine iBij transfer susceptance in the reduced bus admittance

matrixGij transfer conductance in the reduced bus admittance

matrix

FIGURE 1. Motion of the virtual COI-SYS machine in the synchronousreference [TS-1, bus-34, 0.202 s].

The position of the COI of the system is defined byδCOI =

1MT

∑n

i=1Miδi

ωCOI =1MT

∑n

i=1Miωi

PCOI =∑n

i=1(Pmi − Pei)

(2)

whereMT =∑n

i=1MiIn Eq. (2), the COI-SYS can be considered a virtual

machine with its own equation of motion. Notably, motionrepresents the equivalent motion of all machines in thesystem. { •

δCOI = ωCOI

MT•

ωCOI = PCOI(3)

The motion of the virtual COI-SYS machine in thesynchronous reference is shown in Fig. 1.

Because Machine i and the virtual COI-SYS machine aretwo ‘‘individual’’ machines with interactions, a two-machinesystem named the individual machine-virtual-COI-SYSmachine system (IVCS) can be formed by these machines,as shown in Fig. 2.

The relative motion between Machine i and the virtualCOI-SYS machine can be depicted as

•

θi = ω̃i

Mi•

ω̃i = fi(4)

where

fi = Pmi − Pei −Mi

MTPCOI

θi = δi − δCOI

ω̃i = ωi − ωCOI

Eq. (4) depicts the separation of an individual machinewith respect to the virtual COI-SYS machine in the IVCS,which is identical to the motion of an individual machinein the COI-SYS reference. However, for historical reasons,‘‘individual machine’’ or ‘‘critical machine’’ is still used as

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FIGURE 2. Two-machine system formed by Machine i and the virtual COI–SYS machine.

FIGURE 3. Relative trajectory between a critical machine and the virtualCOI-SYS machine [TS-1, bus-34, 0.202 s]. (a) IVCS5 in the synchronousreference. (b) IMTR5 in the COI-SYS reference.

the default term in this paper for simplicity to represent thepair of machines.

The individual-machine trajectory (IMTR) of a criticalmachine in the COI-SYS reference is shown in Fig. 3.

B. MECHANISM OF THE IMEAC METHODReference [24] proved that the EAC strictly holds for a crit-ical machine because the IVCS is a two-machine system,as shown in Fig. 2. In the IMEAC method, the 3-dimensionalKimbark curve (3DKC) in t-θi-fi space is established todemonstrate the mapping between the Kimbark curve of themachine in θi-fi space and the IMTR of the machine in t-θispace, as shown in Fig. 4. This figure shows that by using the3DKC of the machine, the IMTR of the critical machine int-θi space can be successfully mapped to the Kimbark curveof the machine in θi-fi space. In this way, the stability of theIMTR of a critical machine can be evaluated through the EACof the machine [24].

For a multimachine system, the IMTR of each criticalmachine corresponds to its unique 3DKC. In the 3DKC ofeach critical machine, the IMTR of the machine in t-θi space

FIGURE 4. 3DKC of a stable Machine 5 in a multimachine system [TS-1,bus-34, 0.180 s]. (P1: prefault point; P2: fault-clearing point; P3: DSP5).

FIGURE 5. Mappings between the system trajectory and 3DKCs of allindividual machines in the system [TS-1, bus-34, 0.202 s].

is mapped into θi-fi space, and thus, the trajectory stabilityof the IMTR of the machine can be evaluated using the EACof the machine. The mappings between the system trajectoryand the 3DKC of each individual machine in the system areshown in Fig. 5 [24].

After this mapping process, the system operator may onlyobserve critical machines rather than all machines in thesystem to evaluate the stability of the system because onlycritical machines are severely disturbed, and theymight causethe system to become unstable. Therefore, the unity principle[24] is stated as follows:

(i) The system can be considered stable if all criticalmachines are stable.(ii) The system can be considered unstable as long as any

unstable critical machine is found to become unstable.The unity principle explicitly describes the relationship

between the individual-machine stability and the systemstability.When using the IMEACmethod in the real industrialenvironment in the TSA, the critical machines are identified

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FIGURE 6. Use of the IMEAC method in TSA [TS-1, bus-2, 0.430 s].

first once the fault is cleared. After that, the system operatormonitors the IMTR of each critical machine in parallel. Thestability of each critical machine can be evaluated throughits Kimbark curve. Furthermore, the stability of the entiresystem is obtained following the unity principle. The systemcan be evaluated as becoming unstable if any unstable criticalmachine is found to become unstable. The application ofthe IMEAC method in TSA is shown in Fig. 6 for tutorial.In Fig. 6, Machine 9 becomes unstable first amongall machines in the system, and the instability of thismachine further incurs the instability of the entiresystem.

This section is a brief retrospective of the IMEAC method[24], [25]. In the following sections, all these key conceptsare expressed in a transient energy form.

III. DEFINITION OF IMTEA. DEFINITIONS AND THEOREMSThe IMTE of Machine i in a multimachine system is definedas

Vi = VKEi + VPEi (5)

where

VKEi =12Miω̃

2i

VPEi =∫ θi

θ si

[−f (PF)i

]dθi

The first term and second term on the right handside of Eq. (5) represent the individual-machine-kinetic-energy (IMKE) and the individual-machine-potential-energy(IMPE) of the machine, respectively. f (PF)i corresponds to thepostfault system. The prefault point θ si is set as the energyreference point of the machine.

From Eq. (5), the definition of Vi appears similar to that ofglobal transient energy [1]–[5]. However, all the componentsin Vi (Mi, ω̃i, fi and θi) are the parameters of an individualmachine in the COI-SYS reference. Therefore, Vi in Eq. (5)is the transient energy defined in a real individual-machineform.

Two theorems are given first before analyzing IMTE.Theorem 1: IMTE remains conserved during the postfault

period.

Proof: Differentiating Vi in Eq. (5) with t , one obtains•

Vi = Miω̃i•

ω̃i−fi•

θi (6)

Using Eq. (4), we have•

Vi = 0 t > tc (7)

Eq. (7) proves that Vi remains constant after fault clearing.Thus, Theorem 1 holds. Notice that Eq. (7) holds for eachindividual machine in the system regardless of whether it is acritical machine. The complicated interactions among multi-plemachines in the system only cause the conversion betweenthe IMKE and IMPE of an individual machine because theIMTE of the machine remains conserved. In other words,all machines in the system contribute to the transient energyconversion inside the individual machine.Theorem 2: The IMPE of a critical machine reaches a

maximum at the DLP or DSP of the machine.Proof:The point at which the IMPE of a critical machine

reaches a maximum after fault clearing can be denoted as•

VPEi = 0⇒dVPEidθi

•

θi = 0 (8)

Based on Eq. (4), Eq. (8) can be further expressed as

f (PF)i ω̃i = 0 (9)

Following the analysis in Ref. [24], the point where IMPEreaches a maximum is either a DLP (fi = 0) if a criticalmachine becomes unstable or a DSP (ω̃i = 0) if a criticalmachine remains stable. Thus, Theorem 2 holds.

The two theorems indicate the following:(i) Conversion between IMKE and IMPE only occurs

inside an individual machine.(ii) Following (i), no transient energy exchange occurs

between any two machines in the system.(iii) The IMPP of the machine along the time horizon can

be a DSP or DLP.(iv) The minimum IMKE and the maximum IMPE occur

simultaneously at the IMPP because the IMTE remains con-served after fault clearing.

We reanalyze the trajectory stability of the system fromthe IMTE angle. When a multimachine power system is sub-jected to a large disturbance, the equilibrium of each machineis destroyed. During this ensuing transient, each individualmachine may or may not lose synchronism with the system,depending on whether the IMPE absorbing capacity, i.e., themaximum IMPE at the IMPP, is adequate for converting theIMKE of the machine at the end of the disturbance intothe IMPE of the machine.

A tutorial example is given below. The system trajectory isshown in Fig. 7 (a). The variances of the IMPE and IMKEof the two critical machines after fault clearing are shownin Figs. 7 (c) and (d), respectively. In this case, Machine5 becomes unstable, while Machine 4 remains stable.

From Figs. 7 (b-d), the IMKE converts to IMPE insideeach critical machine after fault clearing. IMPE5 and IMPE4

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FIGURE 7. Transient energy conversion inside a critical machine [TS-1,bus-34, 0.219 s]. (a) System trajectory. (b) IMTE. (c) IMPE. (d) IMKE.

reachmaxima at 0.499 s and 0.420 s, respectively. In addition,IMKE5 and IMKE4 reach a minimum at the correspondingIMPP because the IMTE of each machine remains conservedafter fault clearing.

B. THE PROBLEM OF THE STABILITY CHARACTERIZATIONUSING IMPEIn fact, if the system operator observes only the variationin the IMPE along the time horizon, one problem is that hecannot confirmwhether the IMPP in the IMPE curve is a DSPor DLP. In particular, for the case in Fig. 7 (c), IMPE5 mayreach a maximum at IMPP5, and then the machine becomesunstable. However, IMPE4 may also reach a maximum atIMPP4 while the machine remains stable. This result indi-cates that the stability of the machine cannot be evaluated

FIGURE 8. Kimbark curve of Machine 5 [TS-1, bus-34, 0.219 s].

only through the variation in IMPE. In contrast, a DSP orDLP can be clearly identified from the Kimbark curve of themachine [24].

From the analysis above, the key feature for identifyingthe stability of a critical machine should be the occurrenceof the minimum IMKE rather than the maximum IMPE atthe IMPP, as shown in Fig. 7 (d). In particular, the minimumIMKE of Machine 4, i.e., IMKE4 at IMPP4, is strictly zero,which indicates that IMKE4 at the fault clearing point is fullyabsorbed inside Machine 4, and thus, the machine maintainsstable in the first swing. Comparatively, the minimum IMKEof Machine 5, i.e., IMKE5 at IMPP5, is positive, whichreveals that IMKE5 at the fault clearing point cannot be fullyabsorbed inside Machine 5. Furthermore, this slight IMKE5at IMPP5 finally causesMachine 5 to become unstable, whichfurther causes the system to become unstable.

From the analysis above, the minimum IMKE of a criticalmachine that occurs at the IMPP shows a dominant effect onthe stability of a critical machine. In this paper, we define theminimum IMKE of a critical machine that occurs at the IMPPas the ‘‘residual IMKE’’. Residual IMKE can be considered akey factor for identifying the stability of the critical machinein the transient energy form.

In the following sections, the concept of IMTE will beanalyzed with the assistance of IMEAC. This analysis mayhelp readers essentially understand the mechanism of thetransient energy conversion of an individual machine usingprevious IMEAC works [24], [25].

IV. STABILITY CHARACTERIZATION OF A CRITICALMACHINEA. STABILITY CHARACTERIZATION OF AN UNSTABLECRITICAL MACHINE1) STABILITY CHARACTERIZATION IN THE KIMBARK CURVEFORMBased on numerous simulations, the representative Kimbarkcurve of a critical machine becoming unstable is shownin Fig. 8. The corresponding system trajectory is alreadyshown in Fig. 7 (a). Machine 5 becomes unstable inthis case.

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FIGURE 9. Transient energy conversion inside Machine 5 [TS-1, bus-34,0.219 s].

From Fig. 8, for the unstable critical machine case, thecritical machine first accelerates from P1 to P2 during thefault-on period and then decelerates fromP2 to P3 (DLP) afterthe fault is cleared. After that, the critical machine acceleratesagain at the DLP and then becomes unstable. Therefore,the unstable case of the critical machine can be characterizedby the occurrence of the DLP with fi of the machine beingzero:

ω̃i 6= 0, fi = 0 (10)

Following IMEAC, the stability characterization of anunstable critical machine can be expressed as

AACCi > ADECi (11)

2) STABILITY CHARACTERIZATION IN THE TRANSIENTENERGY FORMWe illustrate the above procedure from the transient energyangle. The conversion between IMKE and IMTE of an unsta-ble critical machine is shown in Fig. 9.

From Fig. 9, once a fault occurs, the machine is disturbedby an ‘‘external force’’, and thus, the IMTE starts increasing.At the fault clearing point, the IMTE can be expressed as

V ci = V c

KEi + VcPEi =

12Miω̃

c2i +

∫ θci

θ si

[−f (PF)i

]dθi (12)

In Eq. (12), the first term (V cKEi) can be considered the accu-

mulated disturbed energy during the entire fault-on period.Once the fault is cleared, the IMTE becomes conserved. TheIMKE continues to decrease until it reaches a minimum atthe IMPP of the machine. In addition, the IMPE increases tothe maximum at the IMPP. The IMTE at the IMPP can bedepicted as

VDLPi =VDLP

KEi + VDLPPEi =

12Miω̃

DLP2i +

∫ θDLPi

θ si

[−f (PF)i

]dθi

(13)

During the ensuing transient of the postfault period,the increase in the IMPE from the fault clearing point toIMPP5 is

1VPEi =∫ θDLPi

θ si

[−f (PF)i

]dθi −

∫ θci

θ si

[−f (PF)i

]dθi

=

∫ θDLPi

θci

[−f (PF)i

]dθi (14)

FIGURE 10. Simulations of the fault [TS-1, bus-34, 0.180 s]. (a) Kimbarkcurve of Machine 5. (b) System trajectory.

In Eq. (14), 1VPEi reflects the amount of the IMKE thatconverts to IMPE from the fault clearing point to the IMPPof the machine.

Because V ci is equal to VDLP

i following Theorem 1, usingEqs. (12-14), the residual IMKE at the IMPP of an unstablecritical machine can be depicted as

VDLPKEi = V c

KEi −1VPEi (15)

From Eq. (15), if the amount of IMKE at the fault clear-ing point cannot fully convert to IMPE, the residual IMKE(VDLP

KEi ) will occur at the IMPP of the machine. VDLPKEi can

be considered the key factor that forces a critical machine tobecome unstable.

B. STABILITY CHARACTERIZATION OF A STABLE CRITICALMACHINE1) STABILITY CHARACTERIZATION IN THE KIMBARK CURVEFORMThe representative Kimbark curve of a critical machine main-taining stable is shown in Fig. 10 (a). The correspondingsystem trajectory is shown in Fig. 10 (b).Machine 5maintainsstable in this case.

For the stable critical machine case, a critical machinefirst accelerates from P1 to P2 during the fault-on period andthen decelerates. In contrast to the unstable critical machinecase, the velocity of the machine reaches zero at P3 (DSP).

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FIGURE 11. Transient energy conversion inside Machine 5 [TS-1, bus-34,0.180 s].

In this way, the machine will not separate from the system andwill remain stable. Therefore, the stable case of the criticalmachine can be characterized by the occurrence of the DSPwith the velocity of the machine being equal to zero:

ω̃i = 0, fi 6= 0 (16)

Following the IMEAC, the stability characterization of astable or a critically stable critical machine can be expressedas

AACCi = ADECi (17)

2) STABILITY CHARACTERIZATION IN THE TRANSIENTENERGY FORMHere, we illustrate the stable case above from the transientenergy angle. The conversion between IMKE and IMTE ofan unstable critical machine is shown in Fig. 11.

Once the fault is cleared, the IMTE of the machinebecomes conserved, as shown in Fig. 11. The IMKE contin-ues to decrease until it reaches a minimum at the IMPP of themachine. In addition, the IMPE increases to the maximum atthe IMPP.

From the analysis above, the conversion between IMKEand IMPE of a stable critical machine is quite similar to thatof an unstable critical machine. However, the key differenceis that the residual IMKE of a stable critical machine (VDSP

KEi )is strictly zero. The analysis is given below.

The IMTE at the IMPP with zero IMKE is depicted as

VDSPi = VDSP

PEi =

∫ θDSPi

θ si

[−f (PF)i

]dθi (18)

The increase in the IMPE from the fault clearing point tothe IMPP can be described as

1VPEi =∫ θDSPi

θ si

[−f (PF)i

]dθi −

∫ θci

θ si

[−f (PF)i

]dθi

=

∫ θDSPi

θci

[−f (PF)i

]dθi (19)

Following Theorem 1, the residual IMKE at the IMPP canbe depicted as

VDSPKEi = V c

KEi −1VPEi = 0 (20)

From Eq. (20), the IMKE fully converts to the IMPE at theIMPP because the residual IMKE of a stable critical machine

is strictly zero (including the critically stable case), as shownin Fig. 11. Under this circumstance, a zero residual IMKE(VDSP

KEi ) can be considered the key factor that maintains criticalmachine stable.

C. STABILITY CHARACTERIZATION OF A CRITICALMACHINE USING RESIDUAL IMKEFollowing the analysis in Sections A and B, the residualIMKE can be given in a more general form:

V REKEi = V c

KEi −1VPEi (21)

where

V cKEi =

12Miω̃

c2i

1VPEi =∫ θ IMPPi

θ si

[−f (PF)i

]dθi −

∫ θci

θ si

[−f (PF)i

]dθi

=

∫ θ IMPPi

θci

[−f (PF)i

]dθi

In Eq. (21), V REKEi is the residual IMKE at the IMPP of the

machine. The critical machine becomes unstable if V REKEi is

positive at the IMPP (the IMPP would be the DLP). The criti-cal machine remains stable if V RE

KEi is strictly zero at the IMPP(the IMPP would be the DSP). Therefore, the occurrence ofthe residual IMKE at the IMPP can essentially be used as thestability characterization of the critical machine.

From the analysis above, the stability characterization of acritical machine in the transient energy form can be describedas follows:

(i)A critical machine becomes unstable if its residual IMKEoccurs at the IMPP.(ii) A critical machine remains stable if its residual IMKE

is strictly zero at the IMPP.Statements (i) and (ii) indicate that the stability character-

ization of the critical machine does not require any KE orPE corrections, which are normally required in conventionalglobal methods [1], [2]. This difference is because the tran-sient energy conversion of a critical machine strictly followsthe EAC characteristics. A detailed analysis is given in thecase study.

The fundamental definitions and the characteristics of theIMTE have been analyzed in the above sections. In thefollowing section, the correlations between the IMTE andIMTR, and the equivalence between the IMTE and IMEACwill be analyzed. This analysis may essentially clarify thephysical nature of the IMKE and further provide redefinitionsof the unity principle in a transient energy manner.

V. PHYSICAL NATURE OF IMTE1) MAPPING BETWEEN IMTE AND IMTRIn general, the transient stability of the system should beexplicitly expressed as the transient stability of the ‘‘systemtrajectory’’. In the COI-SYS reference, the IMTR of a crit-ical machine is used to measure the stability of the systemtrajectory. The system becomes unstable once one or more

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FIGURE 12. Kimbark curve and IMTR of unstable Machine 5 [TS-1, bus-34,0.219 s] (a) Kimbark curve of Machine 5. (b) IMTR5.

IMTRs of critical machines infinitize with time, as analyzedin Ref. [24].

If we retrospectively consider the IMEAC method [24],its key idea is to evaluate the stability of the IMTR of thecritical machine through the Kimbark curve of the machine.During the postfault period, once the Kimbark curve of acritical machine goes through the DLP, the machine acceler-ates again, as shown in Fig. 12 (a). Under this circumstance,the velocity of the machine becomes ‘‘infinite’’, and it can bedenoted as ∣∣ω̃i,t ∣∣ = +∞, t = +∞ (22)

The infinity of ω̃i may directly cause the infinity of θi

∣∣θi,t ∣∣ = ∣∣∣∣∫ t

t0ω̃idt

∣∣∣∣ = +∞ t = +∞ (23)

We name the Kimbark curve beneath the DLP the ‘‘insta-bility abyss of the critical machine’’. Falling into the insta-bility abyss is a unique characteristic of a critical machinebecoming unstable because this result never occurs for stablecritical machines or a noncritical machine.

If we describe the phenomenon above from the transientenergy angle, the infinity of ω̃i,t along the time horizonindicates the following:

VKEi,t = +∞, VPEi,t = −∞, t = +∞ (24)

FIGURE 13. Infinity of the IMKE5 [TS-1, bus-34, 0.219 s].

FIGURE 14. 3DKE of a stable critical machine in a multimachine system[TS-1, bus-34, 0.180 s]. (P1: prefault point; P2: fault-clearing point; P3:DSP5).

In Eq. (24), the negative infinity of the IMPE is caused bythe conservation of the IMTE after fault clearing. The IMKEbecoming infinite along the time horizon is shown in Fig. 13.

From Figs. 12 and 13, the infinity of the IMTR stronglycorrelates to the positive infinity of the IMKE (and also thenegative infinity of the IMPE). Briefly, one can conclude thefollowing.For the unstable critical machine, the IMTR of the machine

becomes infinite, and the IMKE of the machine becomespositive infinite along the time horizon.For the stable critical machine, the IMTR of the machine is

bounded, and the IMKE of the machine is bounded along thetime horizon.

Extended from the 3DKC as given in Section II,a 3-dimensional IMKE (3DKE) in t-θi-VKEi space is estab-lished to describe the mapping between the IMKE and IMTR.Using the parameters from the actual simulatedmultimachinesystem trajectory, the 3DKE of a stable critical machine isshown in Fig. 14.

In Fig. 14, the IMTR curve and the IMKE curve can beconsidered the mapping of the 3DKE of the machine in t-θispace and t-VKEi space, respectively. Therefore, the stabilityanalysis of the IMTR of the machine can be transferred fromt-θi space to t-VKEi space using the occurrence of the residualIMKE. In particular, if the residual IMKE occurs in t-VKEispace, the critical machine accelerates, and thus, the IMTRof the machine in t-θi space becomes unstable. If the residual

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IMKE is strictly zero in t-VKEi space, the critical machineremains stable.

From the analysis above, the IMKE in t-VKEi space and theIMEAC in θi-fi space depict the same problem, i.e., the tra-jectory stability of the IMTR of the critical machine in t-θispace.

In the following paper, the relationship between IMTRand IMPE will be re-analyzed by using a novel concept,i.e., the fictional ‘‘ruler IMTR’’ and also the ‘‘constant-θiangle surface’’ in the angle space.

2) EQUIVALENCE BETWEEN IMTE AND IMEACFrom the analysis in Section IV, one finds that the transientenergy conversion inside a critical machine is quite similarto the IMEAC characteristic of the machine. In this section,we use the case of an unstable critical machine to reveal theequivalence between IMTE and the IMEAC.

For an unstable critical machine, the acceleration area fromthe prefault point to the fault clearing point can be depictedas [24]

AACCi =∫ θci

θ si

f (F)i dθi =∫ ω̃ci

ω̃si

Miω̃idω̃i =12Miω̃

c2i (25)

The deceleration area can be expressed as

ADECi =∫ θDLPi

θci

[−f (PF)i

]dθi

=

∫ θDLPi

θ si

[−f (PF)i

]dθi −

∫ θci

θ si

[−f (PF)i

]dθi (26)

Based on Eqs. (12-14) in Section IV, we have{AACCi = V c

KEi

ADECi = 1VPEi(27)

Therefore, the residual IMKE in Eq. (21) can also beexpressed in an EAC form:

V REKEi = AACCi − ADECi (28)

The relationship between IMTE and the IMEAC of a crit-ical machine is shown in Fig. 15. The Kimbark curve ofMachine 5 is already shown in Fig. 8.

From the analysis above, observing the transient energyconversion of the machine can be considered analyzing theEAC of the machine along the time horizon. In addition,although V RE

KEi was originally defined in the transient energyform, the residual IMKE is fully identical to the differencebetween the ‘‘acceleration area’’ and the ‘‘deceleration area’’of the machine. In fact, the strict EAC characteristic of acritical machine fully indicates that the transient energy anal-ysis of a critical machine does not rely on any KE or PEcorrections, as in Refs. [1], [2].

3) REDEFINITIONS OF THE UNITY PRINCIPLE FROM THETRANSIENT ENERGY ANGLEThe equivalence between IMTE and the IMEAC reveals thatIMTE can also be used to measure the trajectory stability of

FIGURE 15. Equivalence between IMTE and the IMEAC of Machine 5[TS-1, bus-34, 0.219 s].

the IMTR of the machine. Similar to the use of the IMEACin TSA, the physical nature of IMTE can be given asIndividual-machine transient energy conversion reflects

the stability of the individual-machine trajectory.Based on this statement, the trajectory-based unity prin-

ciple [24]–[26], as given in Section II, can be redescribedin a transient energy manner. The redefinitions of the unityprinciple are given as follows:

(i) The system can be considered remaining stable if theresidual IMKE of each critical machine is strictly zero at itsIMPP.(ii) The system can be considered becoming unstable as

long as the residual IMKE of any critical machine occurs atits IMPP.

From the redefinitions above, it is the energy conversioninside an unstable critical machine that finally determinesthe instability of the system. Specifically, once the faultis cleared, IMKE and IMPE convert inside each criticalmachine. If the residual IMKE occurs in a critical machine,the machine becomes unstable and further causes the systemto become unstable. Comparatively, if the residual IMKEof every critical machine is strictly zero, then each criticalmachine is stable, ensuring that the system remains stable.This condition essentially provides a departure from thesuperimposed global transient energy view [3].

VI. CASE STUDYA. A TUTORIAL EXAMPLEThe case [TS-1, bus-2, 0.430 s] in Ref. [24] is first consideredin the present case study. In this case, Machines 8, 9 and 1 arecritical machines. Machines 8 and 9 become unstable, whileMachine 1 remains stable. The simulated system trajectory isshown in Fig. 16. The profiles of transient energy of the threecritical machines are shown in Figs. 17 (a-c). The Kimbarkcurves of these critical machines are already shown in Fig. 6.

As shown in Fig. 17, transient energy conversion occursinside each critical machine after fault clearing becausethe IMTE of each critical machine remains conserved.The IMTEs of critical machines are much higher thanthose of non-critical machines because critical machines areseverely disturbed by faults. The residual IMKE9 and residual

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FIGURE 16. Stability characterizations of the multimachine system in transient energy form [TS-1,bus-34, 0.430 s].

FIGURE 17. Stability characterizations of the multimachine system intransient energy form [TS-1, bus-34, 0.430 s]. (a) IMTE. (b) IMKE. (c) IMPE.

IMKE8 are 0.793 p.u. and 0.003 p.u., respectively. In thecase of Machine 9, IMKE9 at the fault clearing point can-not be fully absorbed, which causes the machine to becomeunstable. Furthermore, the instability of Machine 9 causesthe instability of the system following the unity principle.This view is entirely different from the global transient energyangle. In the case of Machine 8, the extremely slight amountof the IMKE8 (0.003 p.u.) could cause themachine to becomeunstable.

TABLE 1. Residual IMKE of each critical machine.

Compared with the unstable critical machine case,the residual IMKE1 is strictly zero at DSP1. This resultindicates that all the IMKE1 at fault clearing is absorbed bythe machine at DSP1, and this full absorption ensures that themachinemaintains stability. In addition, IMTE1 is higher thanthat of Machines 8 and 9, although Machine 1 is stable. Thereason is that Machine 1 has a much larger inertia than theother two unstable critical machines. However, becausethe machine remains stable, IMKE1 and IMPE1 are boundedand never become infinite with time.

The simulations above fully demonstrate that the residualIMKE is the key factor in characterizing the stability of thecritical machine. In addition, energy conversion does notoccur among any machines in the system during the entirepostfault period because the IMTE of each machine remainsconserved throughout this period.

B. VARIATION IN THE IMPP WITH THE DURATION OF THEDISTURBANCEA simulation case is provided to demonstrate the variationin the IMPP with the change in fault duration. The fault isset as [TS-1, bus-34]. The critical-fault-clearing time (CCT)is 0.201 s. tc ranges from 0.050 s to 0.300 s. The simulationstep is set as 0.001 s. Machine 5 is the critical machine in eachcase.

The variation in the IMPP under different tc conditions isshown in Fig. 18. The residual IMKE5 with different tc isshown in Table 2.

From Table 2, with the increase in fault clearing time,the state of Machine 5 changes from maintaining stable tobecoming unstable. The IMPP5 also changes from the DSP to

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FIGURE 18. Variation in IMPP5 with the duration of the disturbance.(a) IMTR5. (b) IMPE5. (c) IMKE5.

TABLE 2. Residual IMKE with different fault clearing times.

the DLP. V cKE5 increases with tc. Comparatively, 1VPE5 first

increases and then decreases with increasing tc. The resid-ual IMKE5 first remains zero when the machine maintainsstable or critically stable, and then increases with tc whenthe machine becomes unstable. The increase in the residualIMKE5 also indicates that the machine is disturbed moreseverely with increasing tc.

C. CLARIFICATION OF THE CONJECTURE IN AN EARLYINDIVIDUAL-MACHINE STUDYIn an early individual-machine study [7], it was conjecturedthat ‘‘this maximum value of the potential energy (maximumIMPE) along the post-fault trajectory of a given machine isessentially independent of the duration of the disturbance’’.From numerous simulations, we state that the maximumIMPE of a critical machine indeed varies slightly with theduration of the disturbance in most cases. However, it mayvary significantly in a few cases. That is, the early individual-machine method, which uses the maximum IMPE along thesustained fault trajectory to replace the real maximum IMPEalong the actual simulated system trajectory, might fail incertain simulation cases.

Case [TS-1, bus-15] is used to demonstrate a special situ-ation. tc is set as 0.495 s, 0.515 s, 0.535 s, 0.550 s, 0.600 sand 0.650 s in series. The CCT is 0.495 s. Machines 7 and5 become unstable in each tc. The variance of IMPE7 andthat of IMPE5 along the time horizon are shown in Figs. 19(a) and (b), respectively.

From Fig. 19 (a), the maximum IMPE7 when tc is 0.650 sincreases by only 5.2% over that of the critically stablecase (tcr = 0.495 s). Therefore, the maximum IMPE7varies slightly with increasing tc. However, the maximumIMPE5 when tc is 0.650 s decreases by approximately93.2% compared with that of the critically stable case.These results reveal that the maximum IMPE of a criticalmachine might also vary ‘‘significantly’’ with increasing tc.Therefore, to ensure the precision of the stability evalua-tion, the computation of the maximum IMPE of a criticalmachine should be based on the actual simulated system tra-jectory rather than the sustained fault trajectory, as proposedin Ref. [7].

D. CRITICAL TRANSIENT ENERGY OF THE SYSTEMFollowing the analysis in Ref. [25], from an individual-machine angle, the critical stability of the most severely dis-turbedmachine (MDM) determines the critical stability of thesystem. Therefore, the transient energy conversion inside theMDM plays the key role in maintaining the critical stabilityof the system.

A critical-stable simulation case is given below. Machine5 is also the MDM in this case [25]. The CCT is 0.215 s. Thecritically stable and critically unstable system trajectories areshown in Figs. 20 (a) and (b), respectively.

From Fig. 20, the comparison between the critically stablecase and the critically unstable case, shows that the infinity ofIMTR5 finally causes the separation of the system. NeitherIMTR4 nor IMTR1 can cause the critical instability of thesystem because they are bounded by time. Following the unityprinciple, the critical stability of the system is completelydecided by the critical stability of Machine 5, i.e., the MDMin this case [25].

The variation in the IMTR above can be preciselydescribed using transient energy conversion insideMachine 5,i.e., the MDM in this case. The energy conversion inside

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FIGURE 19. Variation in the maximum IMPE with the duration of thedisturbance. (a) IMPE7. (b) IMPE5.

critical-stableMachine 5 is shown in Fig. 21. The correspond-ing transient energy of Machine 5 from critical stability tocritical instability is shown in Table 3.

Fig. 21 and Table 3 clearly show that the residual IMKE5can be used as the stability characterization of the criti-cal stability of the machine. For the critically stable case,IMKE5 at the fault clearing point completely converts toIMPE5 at CDSP5 with zero-residual IMKE5. Under thiscircumstance, the critical stability of Machine 5 maintainsthe critical stability of the system. Comparatively, throughthe very slight increase in the duration of the disturbance(from 0.215 s to 0.216 s), IMKE5 at the fault clearingpoint slightly increases, while IMPE5 at CDLP5 slightlydecreases. Under this circumstance, a very low residualIMKE5 (0.0006 p.u.) occurs at CDLP5 and causes Machine5 to become critically unstable. Furthermore, the criticalinstability of Machine 5 causes the critical instability of thesystem.

From the analysis above, the critical stability of the systemis fully decided by the energy conversion inside Machine 5,

FIGURE 20. System trajectory. (a) Critically stable case [TS-1, bus-19,0.215 s]. (b) Critically unstable case [TS-1, bus-19, 0.216 s].

FIGURE 21. Transient energy conversion inside critical-stable Machine 5[TS-1, bus-19, 0.215 s].

TABLE 3. IMPE5 and IMKE5 at IMPP5.

i.e., the MDM in this case. Therefore, from the transientenergy angle, the critical transient energy of the systemshould be defined as the critical transient energy of theMDM.The critical energy point of the entire system should also bedefined as the CDSP of MDM.

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FIGURE 22. Global residual KE at GMPP. (a) System trajectory [TS-1,bus-34, 0.202 s]. (b) GPE and IMPE.

E. ANALYZING RESIDUAL GLOBAL KE FROM ANINDIVIDUAL MACHINE ANGLEA comparison between conventional superimposed globaltransient energy and IMTE is provided. In this case,Machines 4, 5 and 1 are critical machines, and Machine5 becomes unstable. The system trajectory is shownin Fig. 22 (a). The IMPP of each critical machine and globalMPP (GMPP) are shown in Fig. 22 (b). In this case, the globaltransient energy (GTE) is defined as the ‘‘superimposition’’of IMTEs of all machines in the system [1]. GTE is computedusing the actual simulated system trajectory rather than thefictional linear system trajectory.

1) INDIVIDUAL-MACHINE ANGLEFrom an individual-machine angle, Machine 5 reaches DLP5with residual IMKE5 (0.0154 p.u.), and this slight amountof residual IMKE5 causes the machine to become unstable.The instability of Machine 5 further causes the instability ofthe system. Comparatively, Machines 4 and 1 remain stableat DSP4 and DSP1 with zero residual IMKE, respectively.Note that DLP5, DSP4 andDSP1 are quite clear for describingthe stability of the IMTR of each critical machine, as shownin Fig. 22 (a). In this case, it is the conversion betweenIMKE5 and IMPE5 that finally causes the system to becomeunstable.

2) SUPERIMPOSED GTE ANGLECompared with the individual-machine view, global analystsbelieve that the critical transient energy of the system shouldbe defined as the maximum global PE (GPE) that occurs atGMPP when the system becomes unstable. From Fig. 22 (b),one can find that GMPP occurs among IMPPs of criticalmachines in the system. In fact, this phenomenon is not acoincidence because the IMPE of a critical machine has adominant effect on the amount of GPE. Moreover, the resid-ual global KE (GKE) at GMPP is always positive regardlessof whether the system remains stable. The reason is that thevelocities of all the machines cannot reach zero simultane-ously at GMPP. Therefore, the residual GKE cannot be usedas the stability characterization of the system. In addition, onecan find that the system trajectory cannot show distinctivefeatures at GMPP because none of the IMTRs of the criticalmachine reaches its inflection point at GMPP, as in Fig. 22 (a).Specifically, when GMPP occurs, Machine 5 is still deceler-ating in its first swing, while Machines 4 and 1 have alreadyfallen in the second swing for a while.

F. BRIEF DISCUSSION OF THE KIMBARK CURVE ANDTRANSIENT ENERGY IN TSACompared with the EAC, transient energy is rarely usedin TSA. This opinion is also widely accepted in transientstability analysis. The reasons are given below.

(i) The Kimbark curve is depicted in a visible figuremanner using areas.

(ii) The stability of the machine can be evaluated throughthe occurrence of the DLP even though the EAC cannot holdin TSA.

For (i), obtaining the stability margin by using visible areasin the Kimbark curve is more flexible than computing theIMKE and IMPE of the machine (too many variables canbe found in the individual-machine transient energy con-version). For (ii), under the simulation environment with acomplicated, high-order generator model with excitations,the transient energy might fail to remain conserved afterfault clearing, and thus, the residual IMKE cannot be used toidentify the stability of the machine. Comparatively, althoughthe EAC might also fail because of the complexity of thegenerator model (the acceleration area is even smaller thanthe deceleration area), the stability of the machine can still beclearly evaluated through the occurrence of the DLP underthis circumstance.

VII. CONCLUSIONThe following conclusions can be drawn through the analysisin this paper:

(i) The stability of a critical machine can be characterizedthrough the occurrence of the residual IMKE at the IMPP ofthe machine.

(ii) The occurrence of residual IMKE of a critical machinein t-VKEi space is identical to the IMTRof the critical machinebecoming unstable in t-θi space.

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(iii) The computation of the maximum IMPE of a criticalmachine should be based on the actual simulated system tra-jectory because the maximum IMPE may vary significantlywith the duration of the disturbance in some simulation cases.

(iv) The critical transient energy of the system should bedefined as the critical transient energy of the MDM.

(v) The IMTE approach describes the key characteristicsof the system trajectory more effectively than GTE.

In fact, transient energy can be seen as the reflectionof the Newtonian mechanics in the power system transientstability. Using transient energy, the transient characteristicof the multi-machine system can be visually depicted as aball rolling on the energy basin. In the companion paper,the well-known potential energy surface will be completelyredefined in an individual-machine manner. This work mayfurther validate the feasibility of the IMEAC method in TSA.

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[24] S. Wang, J. Yu, and W. Zhang, ‘‘Transient stability assessment usingindividual machine equal area criterion Part I: Unity principle,’’ IEEEAccess, vol. 6, pp. 77065–77076, 2018.

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[27] P. M. Anderson and A. A. Fouad, Power System Control and Stability.Ames, IA, USA: Iowa State University Press, 1977.

[28] M. R-Pavella, J.-L. Horward, B. Lemal, and T. Ph Nguyen, ‘‘Transientstability analysis by scalar liapunov function: Recent improvements andpractical results,’’ Extrait de la Collection des Publications de la Fac-ulte des Sciences Appliquees de l’Universite de Liege, Liège, Belgium,Tech. Rep., 1977.

SONGYAN WANG received the B.S., M.S., andPh.D. degrees from the School of Electrical Engi-neering and Automation, Harbin Institute of Tech-nology (HIT), in 2007, 2009, and 2012, respec-tively. He was a Visiting Scholar with VirginiaTech, Blacksburg, VA, USA, in 2010. From2013 to 2014, he was a Research Fellow withQueen’s University Belfast, U.K. He is currentlyan Assistant Professor with HIT. His researchinterests include power system operation and con-

trol. He is also the Associate Editor of Renewable and Sustainable EnergyReviews.

JILAI YU joined the School of Electrical Engineer-ing and Automation, Harbin Institute of Technol-ogy, in 1992. From 1994 to 1998, he was an Asso-ciate Professor with the School of Electrical Engi-neering and Automation, where he is currently aProfessor and the Director of the Electric PowerResearch Institute. His current research interestsinclude power system analysis and control, opti-mal dispatch of power systems, green power, andsmart grids.

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AOIFE M. FOLEY (Senior Member, IEEE)received the B.E. degree in civil engineering fromthe Irish Power System, University College Cork,Ireland, in 1996, the M.Sc. degree in transporta-tion engineering from the Trinity College Dublin,The University of Dublin, Ireland, in 1999, andthe Ph.D. degree in unit commitment modelingof wind and energy storage from the Irish PowerSystem, University College Cork, in 2011. She iscurrently a Reader with the School of Mechanical

and Aerospace Engineering, Queen’s University Belfast, U.K. She worked inindustry for 12 years for ESB International, Siemens, SWS Energy, and PMGroup, mostly in the planning, design, and project management of energy,telecoms, waste, and pharma projects. Her research interests include energysystem modeling focused on electricity systems, markets and services, windpower, electric vehicles, and smart technologies. She was a Founding Mem-ber of the IEEE VTS, U.K., and the Ireland Chapter in 2011. She was aChartered Engineer in 2001, a Fellow of the Engineers Ireland in 2012, anda member of the IEEE PES and IEEE VTS. She is also the Editor-in-Chiefof Renewable and Sustainable Energy Reviews.

WEI ZHANG (Senior Member, IEEE) receivedthe B.S. and M.S. degrees in power system engi-neering from the Harbin Institute of Technology,Harbin, Heilongjiang, China, in 2007 and 2009,respectively, and the Ph.D. degree in electricalengineering from New Mexico State University,Las Cruces, NM, USA, in 2013.

He is currently an Associate Professor withthe School of Electrical Engineering and Automa-tion, Harbin Institute of Technology. His research

interests include distributed control and optimization of power systems,renewable energy and power system state estimation, and stability analysis.

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transient energy of an individual machine part i

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