directmethod-basedtransientstabilityanalysisforpower ... · in the aspect of transient stability...

Research ArticleDirect Method-Based Transient Stability Analysis for PowerElectronics-Dominated Power Systems

Yanbo Che ,1 Zihan Lv,1 Jianmei Xu ,1 Jingjing Jia,1 and Ming Li2

1Key Laboratory of Smart Grid of Ministry of Education, Tianjin University, Tianjin 300072, China2Qinghai Nationalities University, Xining 810000, China

Correspondence should be addressed to Yanbo Che; [email protected]

Received 5 July 2019; Revised 2 October 2019; Accepted 8 October 2019; Published 4 January 2020

Academic Editor: Zhiwei Gao

Copyright © 2020 Yanbo Che et al. +is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

With the extensive application of power electronics interfaced nonsynchronous energy sources (NESs) in modern power systems,the system stability especially the transient stability is prominently deteriorated, and it is crucial to find a comprehensive andreasonably simple solution. +is paper proposes a direct method-based transient stability analysis (DMTSA) method whichconcludes the key steps as follows: (1) the system modeling of Lyapunov functions using mixed potential function theory and (2)the stability evaluation of critical energy estimation. A voltage source converter- (VSC-) based HVDC transmission system issimulated in a weak power grid to validate the proposed DMTSAmethod under various disturbances.+e simulation results verifythat the proposed method can effectively estimate the transient stability with significant simplicity and generality, which ispractically useful to secure the operation and control for power electronics-dominated power systems.

1. Introduction

At present, NESs such as renewable power generation,flexible AC transmission systems, HVDC transmissionsystems, energy storage devices, and DC microgrids areincreasingly widely employed in power systems [1–5]. Underthe considerable NES penetration, the system dynamiccharacteristics can potentially deviate from those of thetraditional AC power system. To contain system transientinstability issues such as short-term frequency variations andvoltage fluctuations is a major challenge for system operators[6, 7]. In addition, in Europe and America, with the de-velopment of power electronics-dominated power system, itinevitably brings some problems to the system, such as thecomplexity of design in some devices, difficulty in setting theparameter of relay protection [8], difficulty in system sta-bility and limit analysis [9], harmonic suppression and re-active power compensation [10], and so on. By virtue of fast-switched power electronics, NES possesses an enhancedcontrollability of the system and can potentially improve thetransmission performance of the system. Compared with thetraditional AC power system, the topology of the power

electronics power system changes with the switching actionof the power and electronic devices, and the whole systemhas multiple time-scale control interactions. In addition,strong nonlinearity and dynamic characteristics exist inpower electronic converters, and modulation and limitingphenomena occur when using pulse width modulation(PWM) which has saturation nonlinearity. +ese charac-teristics will seriously affect the stability of the powerelectronics-dominated power system.

Stability issues associated with NES are classified intothree levels based on their scales: device level which relates tothe control and operation of the device itself; subsystem levelwhich deals with small-scale NES-dominated systems suchas microgrid, ship, and aircraft electrical systems; and thepower grid level which includes generation, transmission,and distribution [11]. A lot of research studies have beendone for small signal stability analysis of power electronics-dominated power system (PEDPS). In [2], a detailedmathematical model of multiterminal transmission systembased on VSC (voltage source converter) is established, andthe small signal stability of the hybrid network is studied byusing eigenvalues and the associated participation factor

HindawiMathematical Problems in EngineeringVolume 2020, Article ID 4295094, 9 pageshttps://doi.org/10.1155/2020/4295094

mailto:[email protected]://orcid.org/0000-0003-3601-3244https://orcid.org/0000-0003-2630-1664https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/4295094

matrix. Zeng et al. and Zhang et al. [12, 13] introducedexisting small signal stability analysis methods specific formicrogrid applications, and the key affective parameterswere determined. However, these methods can only judgethe stability of the system near the steady working point andcannot evaluate the boundary of the system stable region andthe stability margin around the equilibrium point [11]. It isdifficult to comprehensively analyze the stability of thesystem, so it is necessary to study the transient stability of thepower electronics-dominated power system.

In the aspect of transient stability analysis of powerelectronics-dominated power systems (PEDPS), threemethods are mainly used—time domain simulation method[14–16], artificial intelligence method [17–19], and directmethod [20–22]. Baharizadeh et al. [15] analyzed the pro-posed control strategy for hybrid microgrids based on thetime domain simulation method and studied the large signalstability of the system. In [16], three kinds of transientstability simulation models for multiterminal VSC-MTDCtransmission system are studied and tested by the timedomain simulation method, and the accuracy and speed ofthe model are compared. In [23, 24], the large disturbancestability problem of more electric aircraft power systemsunder generator failures and load sudden changes is studiedby time domain simulation. As the most mature and reliablemethod, time domain simulation is often used as a standardto check other methods but is difficult to be directly used foronline stability analysis. As the order of power system statevariables increases, the amount of computation in timedomain simulation will be greatly increased. In addition,time domain simulation method can only qualitatively judgewhether the system is stable but cannot get quantitativeinformation such as stability margin of the system. It lacksguiding significance for the engineering design.

Olulope et al. [18] applied artificial neural network to thetransient stability analysis of hybrid distributed generationsystem and predicted the critical clearing time of the system.Balu et al. [25] presented a power system transient stabilityassessment method based on composite artificial neuralnetwork composed of Kohonen network and some radialbasis function (RBF) networks, in which the training timewas short and the classification was high. In the transientstability analysis of the power system, the artificial neuralnetwork method mainly takes the functions of data pre-processing and postprocessing and has the characteristics offast and intuitive computation [17–19, 25]. However, whenthe actual data are not consistent with the presupposed dataduring the analysis process, the application of artificialneural network can cause difference between the analysisresult and the actual stability index.

Griffo and Wang [21] used the direct method to analyzestability of an electric aircraft power system under largedisturbances, and the effectiveness of unstable boundaryprediction is verified by experiment. Hu [22] proposed toanalyze transient stability of power electronic converters bythe direct method and designed the parameters of the device.When the direct method is applied to the transient stabilityanalysis of power electronics-dominated power system, itcan not only judge the stability of the system qualitatively but

also get the stability margin of the system quantitatively. +eadvantages of direct method are: (1) considering nonline-arity and adapting to larger system with fast calculation (2)the stability can be directly judged by energy criterion, (3)giving the stability margin of the system. However, there arealso limitations such as the construction of energy functionis difficult and the judgment result is conservative in thedetailed model of the system [22, 26].

To summarize, the system’s differential algebraic equa-tions are solved by the time domain simulation method toobtain the trajectory of the system state quantity changeswith the number of generators over time. +e main ad-vantages are that the mathematical model is exhaustive, is ofhigh reliability, and can adapt to large-scale power systems.However, the amount of calculation is large, the real-timestable control cannot be achieved, and information on thedegree of system stability is not accessible. In the artificialneural network, the mathematical model is not necessary,and solving nonlinear equations is not required but fastrecall speed is achieved. It can simulate arbitrarily complexnonlinear relations, but the instability mechanism of thesystem cannot be explored by the method. Once the systemchanges, all the data need to be reset, which is difficult toachieve in engineering practice. +e direct method analyzesthe stability of the system from the perspective of energy.Solving the Lyapunov equation to accurately judge thestability of the system is of great necessity but simulating thesystem’s differential equations is not. Its prospect is the mostpromising due to the state stability analysis method, fastcalculation speed, and the quantitative stability index pro-vided. +e direct method can quickly and effectively judgethe stability of the system, and the insufficiency of the energyfunction construction can be solved by rational modelingand introducing the mixed potential function theory.+erefore, for a power electronics-dominated power system,the Lyapunov direct method is used to analyze its transientstability as well as to calculate the stability criterion andverify the simulation results in time domain.

In this paper, based on the direct method, an analyticalmethod suitable for transient stability analysis of PEDPS isdeveloped. Firstly, a Lyapunov function of the system isestablished by applying the mixed potential function theory;secondly, a detailed analysis procedure is presented; finally,the transient stability of a power systemmodel incorporatinga VSC-HVDC system is analyzed to verify the feasibility ofthis method. +e method is proved to be a simplified an-alytical and a practical criterion for specific engineeringdesign.

2. Transient Stability Analysis Process Based onDirect Method

As discussed in Section 1, PEDPS is a complex and fasttime-varying nonlinear system with multiple time-scalecontroller interactions. Transient stability analysis usingthe direct method is to identify the mathematical dynamicmodel of a system, to derive its Lyapunov function, and toevaluate the stability criteria directly from the criticalenergy.

2 Mathematical Problems in Engineering

At first, the model for the power electronics-dominatedpower system analyzed must be determined under an ap-propriate approach [27]. +e high penetration of differenttypes of power electronics in one grid brings out variousnonlinear properties such as controller delays and switchingharmonics. +e process of transient stability analysis basedon a detailed system model is deemed to be very compli-cated. In addition, the proposed transient stability analysis ofthe direct method is not applicable for all nonlinear systems.+erefore, it is necessary to simplify the system model bytheoretical analysis.

Secondly, the transient energy function, key to evaluatethe stability of a dynamic system in the direct method, mustbe established. For a given system, Lyapunov defined apositive definite scalar function P(x) as a fictitious gener-alized energy function to evaluate system stability by the signof _P(x) � dP(x)/dt. For a given system, if a positive definitescalar function P(x) can be found and _P(x) is negative, thenthe system is asymptotically stable. P(x) is called as Lya-punov function. +is criterion is only a sufficient conditionfor evaluating the stability of system; i.e., if a Lyapunovfunction that satisfies the criteria is found, the system sta-bility is assured; however, the system stability could not bedenied if such function P(x) is not found.

For a PEDPS with complex structure, its Lyapunovfunction usually cannot be obtained directly, so the systemstability analysis is usually based on some approximation. Atpresent, there are many ways to obtain suitable globalLyapunov function for various system elements, and ap-proximations have to be applied onto system elementsdepending on the pursued detail levels. However, mostmethods cannot be well applied to some systems of specialtypes, and oversimplified Lyapunov function may lead toconservative results [22]. Relying on the topological de-scription of general nonlinear circuits, Jeltsema andScherpen [28] proposed a mixed potential function methodto deduce scalar functions. Moreover, these scalar functionsin turn can be used to construct Lyapunov function to studythe stability of those nonlinear circuits. +e mixed potentialfunction method is suitable for transient stability analysis ofpower electronics-dominated power systems. +e basic ideais to establish the mixed potential function of P according tothe system model at first, and then the Lyapunov function isestablished based on the stability theory of mixed potentialfunction [29, 30].

Finally, the critical energy of the system is solved. +estability criterion can be obtained according to the Lyapunovfunction, while the critical energy is determined by stabilitycriterion. In addition, for the sake of lower conservatism, theestimate value of attraction domain must be expanded as faras possible when solving the critical energy of the system.Moreover, in order to determine the accuracy of theoreticalcalculation value, after the theoretical stability criterion ofthe system is obtained by the direct method, this paper willcombine time domain simulation experiment to test thestability domain results solved. Such a combination use ofthe direct method and time domain simulation method cangreatly enhance the credibility of the results by taking ad-vantages of both.

2.1. Identifying System Model. Different system models areselected for different analysis levels, and the resultant modelmust be reasonably and appropriately simplified. Fortransient stability analysis of PEDPS at the device level,detailed model of power electronic converter is oftenadopted, and power grid is simplified as an ideal powersupply with impedance. For analyses at subsystem level orglobal power system level, it is necessary to establish areasonable equivalent model for any converter [31].

2.2. Constructing a Model of Lyapunov Function

2.2.1. Introducing Mixed Potential Function. Firstly, establisha mixed potential function P [28], according to componentsand topological relations in a system, with a general form of

P(i, v)) � − A(i) + B(v) + D(i, v), (1)

where i are v are inductor currents and capacitor voltages inthe circuit, respectively, A(i) and B(v) are potential func-tions related to current and voltage, respectively, andD(i, v) � iT · c · v is the energy of capacitor and part of anonenergy storage element in the circuit in which c is aconstant coefficient matrix related to the circuit topology.Stability theorems of mixed potential function need to beused to obtain the stability criterion of the system underlarge disturbance if A(i) is only related to some variables ini1, . . . , ir, and B(v) is only related to some variables inUr+1, . . . , Ur+n.

Secondly, verify the correctness of the function P by

Ldiρdt

�zP

ziρ, ρ � 1, 2, ..., r,

− Cdvσdt

�zP

zvσ, σ � r + 1, r + 2, . . . , r + s.

(2)

Establishing Lyapunov Function, let

Pi �zP(i, v)

zi,

Pv �zP(i, v)

zv,

Aii �z2A(i)

zi2,

Bvv �z2B(v)

zv2.

(3)

+en, a Lyapunov function of the system can beestablished as

P∗(i, v) �

u1 − u2

2· P(i, v) +

12P

Ti · L

− 1· Pi +

12P

Tv · C

− 1· Pv,

(4)

where L is the diagonal matrix of inductors, C is the diagonalmatrix of capacitive elements, u1 is the minimum eigenvalueof matrix L− 1/2 · Aii · L− 1/2, and u2 is the minimum eigen-value of matrix C− 1/2 · Bvv · C− 1/2.

Mathematical Problems in Engineering 3

+e differential form of this Lyapunov function is

_P∗(i, v) �

dP∗(i, v)dt

. (5)

If P∗(i, v) is positive definite and _P∗(i, v) is negativedefinite, then the system is asymptotically stable. However,this condition is only a sufficient condition for judging thestability of the system. Even if P∗(i, v) is not positive and_P∗(i, v) is negative, the system cannot be considered un-

stable. A Lyapunov function satisfying this condition can beestablished by further designed rational parameters.

2.3. Estimating Critical Energy

2.3.1. Solving Stability Criterion. +e stability criterion ofthe whole system can be obtained by the precondition ofsystem working in steady-state equilibrium operating pointsand the condition of working in transient stability,respectively.

+e precondition of system working in steady-stateequilibrium operating points can be solved by the equivalentcircuit structure. +e condition of working in transientstability, according to (4), is if all i, v in a circuit belong to acertain area, they satisfy

u1 + u2 > 0, (6)

and if |i| + |v|⟶∞, thenP∗(i, v)⟶∞. (7)

As t⟶∞, all the solutions of the studied system willtend to steady equilibrium work point, and the system willeventually operate stably.

2.4. Determining Critical Energy. When u1 + u2 � 0, thesystem is critically stable, and the critical voltage vmin can becalculated. Taking vmin into the Lyapunov function (4), thecritical energy of the studied system can be estimated as

P∗(i, v) � minP∗ i, vmin( . (8)

+e concrete process of transient stability analysis basedon the direct method is shown in Figure 1.

3. Model Application of Direct Method onto aVSC-HVDC System

High-voltage direct current transmission system con-structed by voltage source converter (VSC-HVDC) is a newtype of DC transmission technology developed on the basisof full controlled power devices such as voltage sourceconverter (VSC) technology, GTO, and IGBT. +e VSC-HVDC system contains a large number of power electronicsconversion devices, which can realize specific functions inthe links of transmission, distribution, transformation, andutilization, and is a typical power electronics-dominatedpower system. A two-terminal VSC-HVDC system is shownin Figure 2, and we will analyze its transient stabilityaccording to the process proposed in the above section.

3.1. EquivalentModel. It is supposed that the rectifier on theleft part controls the DC voltage constant and the inverter onthe right part controls the DC power constant. If the in-fluence of the converter’s switching characteristics isneglected, the controller’s bandwidth is considered to beinfinitely high, and the inverter has sufficient response speed.+en, the structure of this VSC-HVDC system can besimplified as an ideal circuit shown in Figure 3 [32].

In Figure 3, the rectifier is considered as an ideal voltagesource, and veq is a DC output voltage; R and L represent theresistance and inductance of the DC line; and C represents thecharging capacitance in DC side of the inverter. +e con-trolled current source represents the inverter with a constantload of Pcpl. In the simplified circuit, although the RLC pa-rameters are constant values, the circuit is still nonlinear dueto the existence of the controlled current source.

3.2. Lyapunov Function. +e state equation for the equiv-alent circuit shown in Figure 3 is

LdiLdt

� Veq − RiL − v,

Cdvdt

� iL −Pcpl

v.

(9)

Identifying system model

Establishing Lyapunov functionP∗(i, v)

Estimating critical energy

Power electronics-dominatedpower system

Yes

No

Parameterreconfiguration

P∗(i, v) is positive,P∗(i, v) is negative

Figure 1: Concrete process of DMTSA based on the direct method.

VSC VSCDC transmission line

Reactor Reactor

Filter Filter

Figure 2: VSC-HVDC system.


+e current potential function of the power supply voltageveq for nonenergy storage elements and resistance R is

A iL( � iL

0− RiL( diL +

iL

0VeqdiL. (10)

+e voltage potential function of constant load Pcpl fornonenergy storage elements is

B(v) � i

0

− Pcpl

idi � −

Pcpl

i × i +

v

0

Pcpl

vdv. (11)

+e energy of C is

D(i, v) � −iL − Pcpl

vv. (12)

+erefore, the mixed potential function of the wholecircuit is

P iL, v( � −12

R · i2L +

v

0

Pcpl

vdv + iL · Veq − v . (13)

According to the model parameters shown in (13) andthe circuit shown in Figure 3, we have

zP iL, v(

ziL� Veq − RiL − v � L

diLdt

,

zP iL, v(

zv�

Pcpl

v− iL � − C

dvdt

.

(14)

Equation (14) satisfies the constraint conditions in (2), sothe expression of the mixed potential function model iscorrect. Combined with the stability definition of the abovemixed potential function, Aii and Bvv are expressed as

Aii �z2A iL(

zi2L�

z2 1/2R · i2L − VeqiL zi2L

� R,

Bvv �z2B(v)

zv2�

z2 − Pcpl + v

0Pcpl/vdv

zv2� −

Pcpl

v2.

(15)

In the equivalent circuit, the equivalent inductance andcapacitance are represented by only one inductance symbolL and one capacitance symbol C. +erefore, both L− 1/2 · Aii ·L− 1/2 and C− 1/2 · Bvv · C− 1/2 are first-order matrices, andtheir minimum eigenvalues are u1 and u2, respectively,

u1 � min λ L− 1/2

· Aii · L− 1/2

�R

L,

u2 � min λ C− 1/2

· Bvv · C− 1/2

� −Pcpl

Cv2.

(16)

According to (3), (13), and (16), the global Lyapunovtype function of the system can be deduced as

P∗

iL, v( �12

R

L+

Pcpl

Cv2min · P iL, v( +

12L

Veq − Ri − v 2

+12C

Pcpl

v− i

2

,

(17)

and its differential equation isdP∗ iL, v(

dt�12

R

L+

Pcpl

Cv2min ·

dP iL, v( dt

+1L

· Veq − Ri − v

· −dvdt

− Rdidt

+1C

Pcpl

v− i · −

Pcpl

v2−didt

.

(18)

+e 3D graph of this Lyapunov function is shown inFigure 4, and the contour lines of its differential values areshown in Figure 5.

As can be seen from Figures 4 and 5, the Lyapunovfunction of the system is positive within a large range nearthe equilibrium point, and the derivative dP∗(iL, v)/dt< 0 isnegative. +erefore, the system is asymptotically stableaccording to Lyapunov stability theorem.

3.3. Stability Criterion and Critical Energy. According toFigure 3, the steady-state equation is

i �Pcpl

v,

v � Veq − RiL.

(19)

As shown in Figure 6, curves 1 and 2 are the charac-teristic curves of power supply and load in the system, re-spectively. Curve 2 moves to curve 3 or curve 4 as the voltagerises or decreases slightly. +e equilibrium working point ofthe system can be obtained when the power supply currentequals to the load current, i.e., the intersection points of Aand B [33]. An equilibrium working point is said to be stableif the system at that point can restore steady state undersmall disturbance. Obviously, in Figure 4, point A is not asteady-state equilibrium working point but point B is.

From equation (19), the voltage and current of thesystem at point B are

Veq

R LiLPcpl/v

CV

i

Figure 3: +e simplified circuit of two-terminal VSC-HVDC.


v �Veq

2±

��

Veq

2

2

− RPcpl

,

iL �Pcpl

v.

(20)

For the two-terminal VSC-HVDC system, if(Veq/2)

2 − RPcpl < 0, the voltage v is plural and the system isunstable. +e system is in a state of critical stable state when(19) has only one real number solution. +erefore, for thetwo-terminal VSC-HVDC system, one of the necessaryconditions for the system to work in steady state is(Veq/2)

2 − RPcpl > 0 or

Rmax <V2eq

4Pcpl. (21)

Formula (21) gives the criterion of the steady-stateequilibrium working point, which describes the relationshipbetween the equivalent supply voltage, the equivalent re-sistance, and the constant load power, and is also the pre-condition for the stability of that system.

In addition, based on the stability theorem of the abovemixed potential function, formulas (6) and (16), the suffi-cient condition for the system to be in transient stable state is

R>��L

C

� Rmin. (22)

Based on equilibrium point determination and mixedpotential function theory, the stability criterion of the systemshown in Figure 3 is

��L

C

According to (20) and (25), a steady-state equilibriumpoint of the two-terminal VSC-HVDC system is (98.4V,10.2A), and vmin is 79.1V. It is assumed that the DC controlvoltage of the rectifier is reduced from 100V to 80V after0.04 seconds. At this time, with these parameters, thecondition of (21) is satisfied, but (22) is not, so the system isunstable. +e state space trajectory computed by using themixed potential function theory is shown in Figure 7. It canbe seen that the trajectory moves from steady equilibrium

point to divergent state after a period of time, and the systemis unable to keep stable at this time.

+e time domain simulation results of the system sta-bility after changing DC voltage are shown in Figure 8.When system parameters are not satisfied (19), the stabilityresult is as shown in Figure 8(a), and the system becomesunstable after 0.04 seconds; when system parameters meetthe requires (22), the stability results are as shown inFigure 8(b), and the system resumed stability after a period

Table 1: Equivalent circuit parameter setting.

Parameter Value UnitsVeq 100 VF 400 HzL 3 mHC 3000 μFR 0.16 Ω.Pcpl 1000 W

Divergent state

130

120

110

100

90

80

70

60–20 –10 0 10 20 30 40

Figure 7: State space trajectory.

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.260

80

100

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

–200

2040

60

80

100

Vol

tage

(V)

Vol

tage

(V)

Curr

ent (

A)

Time (s)

Time (s)

Time (s)

(a)

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Vol

tage

(V)

Vol

tage

(V)

Curr

ent (

A)

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2Time (s)

Time (s)

Time (s)

60

80

100

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2–20

02040

60

80

100

(b)

Figure 8: Time domain simulation results.


of oscillation. +us, the feasibility and accuracy of transientstability analysis of PEDPS by the direct method are alsoverified.

+e numerical method is used to “explore” simulationswhen estimating the critical energy. +e stability region ofthe system is solved by judging whether the initial points canconverge to the equilibrium point, and the results arecompared with the stable domain results calculated by themixed potential function theory. +e result is shown inFigure 9, in which the calculation results are moved to theorigin for the convenience of observing the change range ofthe system variables. It can be seen that when the transientstability of the PEDPS is analyzed by the direct method, theestimated critical energy is conservative to some extent, butit can already cover the larger range near the equilibriumpoint. +e result is very close to the large signal stabilityboundary of the system. In addition, compared with digitalsimulation, the computation time using the direct method isgreatly reduced.

4. Conclusion

Compared with traditional power system, PEDPS is time-varying or non-autonomous, and its topology changes withswitching actions of power electronic devices. In addition,each power electronic converter has the characteristics ofstructural complexity and saturation nonlinearity, whichbrings great difficulties to transient stability analysis of thewhole system.

In this paper, based on the direct method and by in-troducing the mixed potential function theory, a completeset of transient stability analysis processes including systemmodeling, Lyapunov function construction, and criticalenergy estimation is proposed for the power electronics-dominated power system. In addition, a VSC-HVDC systemis taken as an example for detailed transient stability analysiswhen a large disturbance occurs, and the stability criterion ofthe system is obtained. +e analysis and simulation resultsshow that the proposed process can be effectively applied to

the transient stability analysis when large disturbance ap-pears in the system. Furthermore, the introduction of mixedpotential function theory also simplified the construction ofthe global Lyapunov function.

For the stability analysis of PEDPS with very complexstructure, what we proposed in this paper is providing a setof simple and practical methods that could realize fast andaccurate stability determination while avoiding complicatedsolving steps. Characterizing with simplicity and univer-sality, such method provides not only reference value for thetheoretical study of PEDPS transient stability analysis butalso a practical stability criterion for specific engineeringdesign.

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request.

Conflicts of Interest

+e authors declare that they have no conflicts of interest.

Acknowledgments

+is work was sponsored by the Basic Research Program ofQinghai Province (no. 2018-ZJ-764).

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–60 –40 –20 0 20 40 60–300

–200

–100

0

100

200

300

The unstable region

The stableregion

The calculated value ofstable domain boundry

Current (A)

Vol

tage

(V)

Figure 9: Critical energy estimation results.


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