a time-domain explicit integration algorithm for fast ...researcharticle a time-domain explicit...

Research ArticleA Time-Domain Explicit Integration Algorithm for FastOvervoltage Computation of High-Voltage Transmission Line

Lei Zhang Zhongyi Xu and Jing Ye

College of Electrical Engineering amp New Energy China ree Gorges University Yichang Hubei 443002 China

Correspondence should be addressed to Jing Ye 1731464640qqcom

Received 21 January 2020 Revised 3 June 2020 Accepted 15 June 2020 Published 12 October 2020

Academic Editor Chaudry M Khalique

Copyright copy 2020 Lei Zhang et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

is paper presents a multiple time-step solver based on a time-domain explicit integration algorithm for improving thecomputational speed of high-voltage transmission line electromagnetic transient (EMT) simulation For weakly rigid EMTmodelsof high-voltage transmission lines the previous precise Runge-Kutta integration method with a small time step is adopted forstrongly rigid nonlinear EMTmodels of high-voltage transmission lines a an improved precise integration method using a largetime step is used for the solver of EMT simulation Practical simulations for overvoltages of high-voltage transmission line showthat multiple time-step EMT solver can be applied to different cases of EMT simulation for transmission line

1 Introduction

As is known to all the EMT simulation program (EMTP)[1 2] is widely used to calculate the overvoltage of powersystems Overvoltage calculation often is implemented bysolving different ordinary differential or partial differentialequations that are used to describe different electrical phe-nomena or processes erefore the numerical algorithms ofthe differential equation become a key part of the EMTsimulation In EMT-type simulation tools including EMTPPSCADEMTDC [1] NETOMAC and RTDS EMTP uses theimplicit trapezoidal integration method with second-orderaccuracy and A-stability to discretize the differential equationof basic electrical components (inductors and capacitors) andthe discrete time-domain solution of state variables is ob-tained by solving the algebraic equation [2 3]

e implicit trapezoidal integration method is limited inintegration accuracy and numerical stability On the one handwhen the power grid topology changes abruptly in EMTsimulation voltage or current simulation waveform willproduce a numerical oscillation phenomenon [2 3] A typicaland simple simulation example is the calculation of overvoltageof overhead transmission line without load being switchedsuddenlyWhen the trapezoidal method is used to calculate theovervoltage at the end of the transmission line serious

numerical oscillation occurs in the simulation result issimple case shows that the trapezoidal method cannot accu-rately simulate the EMTprocess for the change of topology inpower grids On the other hand with the continuous access todistributed clean energy the number of power grid nodes hasincreased dramatically resulting in a very large calculationscale for EMT simulation For such systems the computationeffect of implicit numerical integration algorithms does notperform as fast as explicit integration algorithmsis ismainlybecause themathematical model of distributed circuit elementscontains many power electronic devices limiting links andmultiscale transient processes [4 5]

To solve this problem multirate simulation and parallelcomputing are used to perform EMTs of large power grids[6 7] A Parareal [8] parallel computation method is pro-posed to be applied to EMTsimulation and it is verified thatthe parallel algorithm can obtain an effective accelerationratio but the algorithm cannot suppress the numericaloscillation Graphics processing units are used to accelerateEMTs of power systems and adopt thread-oriented con-version and automatic code generation technology [9]Reference [10] proposed themultirate EMTsimulation usinglatency technology is method uses the extended implicittrapezoidal integration method to realize the synchroniza-tion process between subnets with different simulation steps

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 8437617 11 pageshttpsdoiorg10115520208437617

but its computational amount is large Reference [11]proposed the concept of ldquodouble-layer grid separationrdquo forthe topology characteristics of AC-DC hybrid system ismethod greatly improved the simulation efficiency of AC-DC hybrid power grids

To replace the role of implicit trapezoidal integration inEMT simulation researchers have begun to seek the ap-plication of higher-order and A-stable or L-stable numericalalgorithms in EMT simulation e two-stage diagonallyimplicit Runge-Kutta method (2S-DIRK) [12] is applied toEMT simulation e 2S-DIRK is L-stable as same as theimplicit Euler method so it can effectively avoid numericaloscillation [2] Two-stage three-order Radau IIA method[13] is used for a new parallel method for solving nonlinearEMT state-space equation at multiple time points and thenthe Newton method is used to solve it iteratively e al-gorithm not only has high computational efficiency but alsocan avoid numerical oscillation

Both the precise integration method [14] and the dif-ferential quadrature (DQ) method [15] have high accuracyand strong stability and both of them have been widely usedin the field of computational mechanics [16 17] In order toimprove the applicable scope of the precise integrationmethod the precise Runge-Kutta method is proposed whichis a new algorithm formed by the precise integration methodand the explicit Runge-Kutta method with the fourth order[18] e calculation efficiency and precision of the preciseRunge-Kutta method need to be improved so it cannot bedirectly used in the EMTsimulation of the power systems Inorder to improve the EMT simulation efficiency via an im-proved precise integration method this paper makes full useof the advantages of the DQ and precise integrationmethod toconstruct a new method to solve the problem e improvedprecise integration method has high accuracy and goodstability and it can use a large time step to improve thesimulation efficiency of EMTs Compared with the implicittrapezoidal integration using a small time step the calculationerror of this method meets the needs of practical engineeringcalculation and the calculation efficiency is considerable

Compared with the implicit numerical algorithm theexplicit numerical algorithm has higher execution efficiencyHowever the existing explicit methods such as the explicitEuler method have too low numerical precision to be directlyused in power system transient simulation Fortunately thenew method proposed in this paper is essentially an explicitsingle-step high-precision integration method e idea ofapplying the improved precise integration method to elec-tromagnetic transient simulation proposed in this paper is thebiggest advantage over the existing implicit electromagnetictransient simulation algorithm [19 20] in calculation effi-ciency which will be confirmed later in this paper

e remainder of this paper is organized as follows InSection 2 an improved precise integration method based onDQ with different grids is presented in detail In Section 3numerical simulations for overvoltages of high-voltagetransmission lines with different time scale and complexityare given to verify the accuracy and effectiveness of theimproved precise integration method Finally the conclu-sions derived from the study are presented in Section 4

2 Improved Precise Integration Method

Generally the state-space model of EMTs in power systemscan be described as an initial value problem of the first-orderordinary differential equation

_x(t) f(x(t)) t isin t0 T1113858 1113859

x t0( 1113857 x01113896 (1)

where t represents time variable f(t x(t)) is a function oftime t and state variable x x0 is the value of state variable atinitial time t0

Formula (1) can always be rewritten into ordinary dif-ferential equations consisting of homogeneous and non-homogeneous parts by a certain transformation and

_x Hx + g(x t) (2)

where H isin Rntimesn is the coefficient matrix related to systemparameters To the right of the medium sign in equation (2) thefirst item is called homogeneous part and the second item iscalled nonhomogeneous term e exact solution of the ho-mogeneous equation of equation (2) on the computer is asfollows

x exp(H middot t)x0 (3)

In the single-step integration equation (3) has the fol-lowing discrete recursive calculation format

xk+1 T times xk k 0 1 2 (4)

e precise integration method (PIM) [14] proposed byZhong uses additive theorem and incremental storage tocalculate exponential matrix T exp(h times H)

T exp(H middot h) expH middot h

m1113888 11138891113890 1113891

m

(5)

where m 2N (where N 15) For very small η hm wecan use Taylor series to expand the exponential matrix T in(5) into M terms (where M 4) then there are

exp(H middot η) I + Ta

Ta Hη +(Hη)

2

2+

(Hη)3

3+

(Hη)4

4

(6)

Substituting formula (6) into formula (5) yields

T I + Ta1113858 1113859m

I + Ta1113858 11138592Nminus1

middot I + Ta1113858 11138592Nminus1

(7)

According to the matrix multiplication the followingrecursive operations are performed iteratively

Tai 2lowastTaiminus1 + Taiminus1 lowastTaiminus11113872 1113873 i 1 2 N (8)

where Ta0 Ta At the end of the cycle

T I + TaN (9)

In the single integration step [tk tk+1] the solution of thenonhomogeneous dynamic equation (2) can be expressed asfollows

2 Mathematical Problems in Engineering

xk+1 Txk + 1113946tk+1

tk

eH tk+1minusτ( )g(x τ)dτ (10)

In formula (10) the exponential matrix T exp(h times H)

in the first term on the right side of equation (10) has beenobtained by the precise integrationmethod while the secondintegration is related to the characteristics of the powersystem which is called the Duhamel integration term Be-cause the calculation of the first term can be achieved by PIMon computer the numerical error mainly comes from thenumerical calculation error of Duhamel integration term Inthis paper Duhamel integration terms are calculated bytime-domain differential quadrature scheme based on dif-ferent grids

21 Uniform Grid Taking the time-domain differentialquadrature method [15] as an example the integrationformat of Duhamel integration term in equation (10) isderived For the initial value problem _z w(t z) the s-thformula of the DQ scheme with s-stage s-order can beexpressed as follows

zk+1 zk + h 1113944s

j1bjw tk + cjh 1113957zj1113872 1113873 (11)

where h is the time step cj is the grid point bj are theintegration coefficients related to the grid points of DQM

e DQM uses uniform grid and the formula ofDuhamel integration term xk+1 in equation (10) is as follows

xk+1 h 1113944s

i1bie

H tk+1minus1113957ti( 1113857g 1113957xk+is1113957ti( 1113857 i isin (1 s) (12)

where 1113957ti tk + (i times hs)e explicit numerical method can be used for esti-

mation and then the second equation in formula (11) can bedirectly used for single-step numerical integration Specif-ically when s 4 the four-order explicit Runge-Kuttamethod is used to calculate the estimated value 1113957xk+is(i

1 2 s) as follows

1113957xk+14 xk +h

4 times 6S1 + 2S2 + 2S3 + S4( 1113857 (13)

where S1 Hxk + g(tk xk)

S2 H xk +h

4 times 2S11113888 1113889 + g tk +

h

4 times 2 xk +

h

4 times 2S11113888 1113889

S3 H xk +h

4 times 2S21113888 1113889 + g tk +

h

4 times 2 xk +

h

4 times 2S21113888 1113889

S4 H xk +h

4S31113888 1113889 + g tk +

h

4 xk +

h

4S31113888 1113889

(14)

e other values of 1113957xk+12 1113957xk+34 and 1113957xk+1 are calculatedin turn according to formula (13) and the approximate valueof Duhamel integration term xk+1 in equation (12) is ob-tained as follows

xk+1 h b1K1 + b2K2 + b3K3 + b4K4( 1113857 (15)

where K1 T1g(1113957xk+141113957t1) K2 T2g(1113957xk+121113957t2) K3 T3g

(1113957xk+341113957t3) K4 g(1113957xk+11113957t4) T1 e(Htimes3h)4 T2 eHtimesh2T3 eHtimesh4 there is

T2 T3 times T3

T1 T2 times T3

T T1 times T3

(16)

It can be seen from formula (16) that only the firstexponential matrix T3 can be calculated by the PIM in thecalculation of this method so that the amount of calculationwill not be too large because of the increase in the number ofintegration nodes After finding the approximate value xk+1of the Duhamel integration term the approximate value ofxk+1 can be obtained by substituting it into equation (10)

22 Nonequidistant Grids e commonly used nonequi-distant grids are Legendre grid Chebyshev grid and Che-byshev-Gauss-Lobatto grid [21] Consider the distribution ofChebyshev grid points on regularized interval [0 1] where

ck 12

1 minus cos2k minus 12(s minus 1)

π1113888 11138891113890 1113891 (k isin (1 s minus 1))

c0 0

cs 1

(17)

Similarly when s 4 the four-order explicit Runge-Kutta method is used to calculate the estimated value1113957xk+ci

(i 1 2 s) as follows

1113957xk+ci 1113957xk+ciminus1

+cih

6S1 + 2S2 + 2S3 + S4( 1113857 (18)

In equation (18) 1113957xk+c0 xk

S1 H1113957xk+ciminus1+ g tk + ciminus1h 1113957xk+ciminus1

1113872 1113873

S2 H 1113957xk+ciminus1+

cih

2S11113888 1113889 + g tk + ciminus1h +

cih

2 1113957xk+ciminus1

+cih

2S11113888 1113889


cih

2S21113888 1113889 + 1113957g tk + ciminus1h +

cih


+cih

2S21113888 1113889

S4 H 1113957xk+ciminus1+ cihS31113872 1113873 + g tk + ciminus1h + cih 1113957xk+ciminus1

+ cihS31113872 1113873

(19)

en the approximation xk+1 of the Duhamel integrationterm is as follows

xk+1 h 1113957b11113957K1 + 1113957b2

1113957K2 + 1113957b31113957K3 + 1113957b4

1113957K41113872 1113873 (20)

where 1113957K1 T1g(1113957xk+c11113954t1) 1113957K2 T2g(1113957xk+c2

1113957t2) 1113957K3 T3g

(1113957xk+c3 t3) 1113957K4 g(1113957xk+11113957t4) where T1 eH(1minus c1)h T2

eHtimesh(1minus c2) T3 eHtimesh(1minus c3) T T1 times T3e approximation of xk+1 can be obtained by

substituting xk+1 into equation (10) Obviously using

Mathematical Problems in Engineering 3

nonuniform grids needs to calculate (s minus 1) exponentialmatrices which requires more extra calculation than usinguniform grid

3 Numerical Simulations

31 Accuracy Test for Improved Precise Integration MethodAs shown in Figure 1 this circuit represents the equivalentschematic diagram of a 515 kV bus switching AC filter in asubstation Ls is the equivalent inductance of the AC grid Rs

represents the equivalent resistance of the AC grid es denotesthe equivalent electromotive force of the AC system α is theinitial phase of power supply es e capacitor C represents asimplified equivalent circuit of an AC filter bank

When the circuit breaker K is closed at t 0 s the ACfilter is fully discharged and there is no current in the branchof inductor In Figure 1 there are

LsCd2uc

dt2 + RsC

duc

dtuc + uc es (21)

d0t

iC

uC

1113890 1113891

minusRs

Ls

minus1Ls

1C

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

iC

uC

1113890 1113891 +

es

Ls

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)

In general the system is usually an underdamped orattenuated oscillation system At this time the character-istic equation of differential equation (21) has a pair ofconjugate complex roots with a negative real part Underthis condition the voltage across the AC filter can beobtained as

uC(t) K1 sin ωαt( 1113857 + K2 cos ωαt( 1113857( 1113857eminusαt

+ Ucm sin ωt + φc( 1113857

(23)

where es(t) Um sin(ωt + φs) ω 100π rads

Ucm Um

R2s +[ωL minus 1ωC]

21113969

1ωC

φc ϕs minusπ2

minus arctanωLs minus 1ωC

Rs

K1 minusαUcm sinφc minus ωUcm cosφc

ωα

K2 minusUcm sinφc

(24)

In Figure 1 the decay constant of resistance induc-tance and capacitor series circuit is α Rs(2Ls) reso-nance angular frequency of this circuit is ω0 1

LsC

1113968 and

natural angular frequency of this circuit is ωα

ω20 minus α2

1113969

[22 23]In this simulation the lumped electrical parameters are

Rs 159Ω Ls 3537mH and C 3124 μF e pro-posed methods were used to solve equation (22) andimplemented with the MATLAB scripting language enwe calculated the voltage of the capacitor C by improved

precise integration method (with uniform grid and s 4)precise Runge-Kutta integration method (four-order explicitRunge-Kutta is used with precise integration method) thetwo-stage diagonally implicit Runge-Kutta (2S-DIRK withsecond order in accuracy) and the trapezoidal method (TR)or critical damping adjustment [19] Figures 2ndash4 are thecomputational results of this simulation test calculated byprecise differential integration method (PDIM h 01ms)precise Runge-Kutta integration method [18 24] (PRKMh 01ms) the two-stage diagonally implicit Runge-Kutta(2S-DIRK h 001ms) and critical damping adjustment(CDA h 001ms) or TR (h 001ms) And the curve ofdifference values by these three methods relative to the truesolution is also given in Figures 3 and 4 e simulationstarts from the zero initial state e simulation ends att 01 s

As shown in Figures 2ndash4 PDIM has a better com-putational result compared with TR and 2S-DIRK in thebigger time step It seems that the PRKM is as accurate asPDIM for this case And even with large time step thecomputational accuracy of PDIM is two orders ofmagnitude higher than that of the trapezoidal methodand 2S-DIRK In addition in order to further verify theeffectiveness of the proposed method in suppressingnumerical oscillations a nonlinear reactor switchingcircuit is used for simulation analysis In Figure 5 thelumped electrical parameters are equivalent resistancers 10Ω and equivalent inductor ls 20mH epiecewise linear current-flux curve of lnl is given inFigure 6 e peak voltage of the sinusoidal voltagesource is 445

2

radickV and es(t) 445

2

radicsin(120πt)kVe

PDIM and the trapezoidal method (TR) give completelydifferent results and their differences are quite large eresult obtained by the trapezoidal method is shown inFigure 7 where sustained numerical oscillations areobserved while the results obtained by PDIM are not ecalculated results obtained using a time step of 001 msare shown in Figure 7

32 Single-Phase Transmission Line with Nonlinear Induc-tance Load is case is a high-voltage transmission linewith a nonlinear inductance load as shown in Figure 8 InFigure 8 e(t) is the excitation AC voltage source 1113957Rs and 1113957Ls

are the internal resistance and internal inductance of thevoltage source at the sending end the switch S is suddenlyclosed at time t 0 s e total length of the line isL 100 km and the distributed parameters of the

sin(100t + φs)515 2

3es =

RsLs

C

K t = 0s

AC filter

iCiC

uC

Figure 1e equivalent circuit of a 515 kV bus switching AC filter


transmission line are resistance R0 inductance L0 and ca-pacitance C0 e nonlinear load is composed of load re-sistance RL and the nonlinear inductor LL in parallelconnection where the relationship between the magneticlink ϕL and current iL of the nonlinear inductor isϕL a tanh(b times iL) the electrical parameters in this case areshown in Table 1

e electrical model used to describe the EMTprocess ofthe high-voltage transmission line shown in Figure 8 is thetelegraph equation e telegraph equation is a hyperbolicpartial differential equation and it is necessary to transformit into ordinary differential equations in the form of equation(2) before different numerical methods are used for EMTsimulation We divide the whole transmission line into Msections uniformly as described in Figure 9 e Π-typecascade equivalent circuit model was carried out on thetransmission lines in Figure 8 So the resistance inductanceand capacitance of each section are as follows

PDIMAnalytical solution

u C (t

)(kV

)ndash400

ndash200

0

200

400

600

001 002 003 004 005 006 007 008 009 010Time (s)

Figure 2 Computational result of capacitor voltage by precise differential integration method

CDAPDIM

PRKM

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

Abs

olut

e err

or

0005 001 0015 0020Time (s)

Figure 3 e curve of computational difference values by thesethree methods

PDIM

TRDIRK

0005 001 0015 0020Time (s)

10ndash4

10ndash3

10ndash2

10ndash1

100

Abs

olut

e err

or


es (t)

rs ls

lnl

t = 0sinl

unl

Figure 5 e equivalent circuit diagram of a saturated reactorcharged by an AC grid

i(kA)02

1H10mH

ϕ

Figure 6 A nonlinear current-flux curve of the saturated reactor


r R0L

M

l L0L

M

c C0L

M

(25)

As shown in Figure 9 it is easy to establish the followingfirst-order linear ordinary differential equations (ODEs) for

L = 100km

RL

Rs

LL

Ls

SSingle-phase homogeneous transmission line

t = 0s˜ ˜

e (t) = 220 2sin(100πt + 90deg)

Figure 8 Schematic of equivalent circuit of single-phase transmission line with nonlinear inductance load

u nl (

t)(kV

)

times 10ndash3

Trapezoidal methodPDIM

46 47 48 49 545t (s)

ndash60

ndash40

ndash20

0

20

40

60

80

(a)

u nl (

t)(kV

)

ndash80

ndash60

ndash40

ndash20

0

20

40

60

80

001 002 003 004 005 0060t (s)

(b)

i nl (t

)(kA

)

001 002 003 004 005 0060t (s)

ndash05

0

05

1

15

2

25

3

(c)

Figure 7 Calculated results of the saturated reactor obtained by PDIM and TR (a) Local snapshot results of the saturated reactor voltagecalculated by PDIM and TR (b) Calculated result of the saturated reactor voltage obtained by PDIM (c) Calculated result of the saturatedreactor current obtained by PDIM

Table 1 System parameters of example 32

Electrical parameters Parameter values1113957Rs 2Ω1113957Ls 006HR0 007ΩkmL0 208 times 10minus3HkmC0 12 times 10minus9FkmRL 96Ωa 840 times 102 Vmiddotsb 595 times 10minus 3 Aminus1


the transmission line using Kirchhoffrsquos voltage and currentlaw [22 23]

LS

di0

dt+ RSi0 + u1 e(t)

ldim(t)

dt um(t) minus um+1(t) minus rim(t) m isin (1 M)

cdum(t)

dt imminus1(t) minus im(t) iM+1

uM+1

RL

+ iL m isin (1 M + 1)

LL

diL

dt uM+1

LL a tanh biL

iL

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

Considering the boundary conditions at the head andend of the transmission line in Figure 9 equation (26) isarranged into the following matrix form

_y(t) Ay(t) + μ(t) (27)

where A isin R(2M+3)times(2M+3) is the constant sparse matrix μ(t)

is (2M + 3) sparse column vector which is the excitationsource of EMT simulation for the transmission line

y(t) i0 i1 iM u1 uM+1 iL1113858 1113859T

μ(t) e(t)Ls

0 0 0 0 uM+1LL

minus iL1113876 1113877T

(28)

0 0 0 0 0

000

0

0000

0

000

0

0

0

0

0 0 0

0

0

0 0 1

0

A =

hellip hellip hellip hellip

helliphellip

hellip

hellip

hellip hellip hellip

hellip

hellip

hellip


hellip

hellip

hellip hellip

hellip hellip

hellip

hellip hellip

hellip

hellip

hellip

RsLs

ndash 1Ls

ndash

rlndash

rlndash

rlndash

1lndash1

l

1lndash

1cndash

1c

1c

1c

1c

1c

1c

1c

1cRL

ndash

1l

1lndash1

l

(29)

u1 u2 u3 uM + 1uM

c c c c ci0 i1 i2 iM iM + 1

r l r l r l

RL LL

iL

e (t)

Rs Ls˜ ˜

Figure 9 Equivalent model of Π-type lumped parameter circuits


In Figure 8 the number of interval segments is M 30after the transmission line is discretized in spatial domaine CDA and the PDIM (with uniform grid and s 4) areused to solve equation (27) to obtain the voltage variationcurve of the nonlinear inductance load e switch S isinitially open and is closed at t 0 s e CDA method usessmall time step h 10 times 10minus 6 s and PDIM uses a larger timestep 1113954h 10 h to calculate transient voltage of the trans-mission line And the sending and receiving end voltage areshown in Figures 10 and 11

As shown in Figures 10 and 11 computational results ofterminal voltage waveform by the CDA and PDIM are ingood agreement However CDA is an implicit method andthe Newton-Raphson formula must be used to calculate thevoltage waveform of propagative transmission line Duringthe calculation the Newton-Raphson formula solves thenonlinear algebraic equations with two iterations isprocess is time-consuming when using small step simula-tions e simulation efficiency of the two methods iscompared in Table 2 e simulation platform is MATLABR2012a e tablet PC processor is AMD Ryzen 5 3500Uwith Radeon VegaMobile Gfx 210GHze tablet PC uses a64-bit operating system and the capacity of RAM is 8GBe simulation starts from the zero initial state except for iLwhich is given a small initial value so that the calculation canbe performed e total simulation ends at t 60ms esimulation acceleration ratio of this example is defined as theratio of the small time-step simulation time of the trape-zoidal method (TR) to the large time-step simulation time ofPDIM

As can be observed in Table 2 the PDIM is significantlymore efficient at handling nonlinear EMT simulations thanthe trapezoidal method Obviously as the simulation timestep of PDIM increases the corresponding acceleration ratioalso changes significantly

In Figure 12 the calculation result of PDIM using largetime step is almost consistent with the simulation waveformof trapezoid method with small time step which shows thatPDIM has good numerical stability and high precision fornonlinear EMT models

33 Lightning Overvoltage Calculation of Substation Busis case is a simulation example of lightning tower over-voltage calculation Figure 13 is a simplified equivalentcircuit diagram of lightning overvoltage calculation ofsubstation bus During lightning stroke the lightningchannel is simulated by resistor parallel ideal current sourceand the resistance re is the resistance of lightning channelwhen lightning stroke hits the top of tower the ideal switchK closes Rch 10Ω represents the impact grounding re-sistance of tower the tower is modeled by lossless trans-mission line whose wave impedance and wave velocity are100Ω and 27 times 108ms respectively and the transmissionline length is L1 50m Lightning current is simulated bydouble exponential wave Its wavefront time and half peaktime are 2650 μs and the peak value of lightning currentiL(t) is 100 kA

e expression of lightning current is as follows

iL(t) aIL eminusαt

minus eminusβt

1113872 1113873 (30)

where a 1058 α 15 times 10minus2 μsminus1 and β 186 μsminus1In this case the electrical model of lightning overvoltage

simulation for tower is established by using telegraphequation After spatial interpolation and discretization usingthe fourth- and second-order interpolation formulas[25 26] the following ordinary differential equations areobtained by taking the number of space segments N 30

PDIMCDA

u 1(t)

(kV

)

0

50

100

150

200

250

300

350

05 1 15 2 25 3 35 40Time (ms)

Figure 10 Computational results of sending terminal voltage byimproved PIM with uniform grid

PDIMCDA

u M+1

(t)(

kV)

0

50

100

150

200

05 1 15 2 25 3 35 40Time (ms)

Figure 11 Computational results of receiving terminal voltage byimproved PIM with uniform grid

Table 2 Comparison of calculation time between the twoalgorithms

Algorithm Time step (h) Time consumed (s) Speedup ratioTR h 10 times 10minus6 s 4655 1PDIM h 10 times 10minus5 s 075 62PDIM h 20 times 10minus5 s 036 129PDIM h 30 times 10minus5s 026 178


x

Hx + δ(t) (31)

where constant coefficient matrices H isin R61times61 and δ(t) arethe input excitation sources of overvoltage at the top oflightning tower

Equation (31) is solved by PDIM (with Chebyshev grid ands 4) and the trapezoidal method respectively e simulationstep of the two methods is h 001μs e simulation resultsare shown in Figure 14 As shown in Figure 14 when t 4 μsthe voltage value of u(t) at the end of the line is about 9845 kVand 9516 kV for the head of the line In Figures 15 and 16 withthe time prolonging it can be seen that the voltage and currentof sending end and receiving end at the end of the transientprocess are almost the same And the steady-state currentvalues are almost 9986 and 9843 kA which are the currentvalues at the beginning and end of the transmission line re-spectively which shows the correctness of the simulation re-sults of this case that its real steady-state current value of thelossless transmission line is near 99 kA

In the calculation of lightning overvoltage it can be seenfrom Figure 16 that because of the fast-changing rate of doubleexponential lightning current the two algorithms can onlyaccurately simulate the changing waveform of lightningovervoltage by using smaller simulation steps As can be ob-served in difference value of two methods in Figure 15 thesimulation results of PDIM and the trapezoidal method are

u M+1

(t)(

kV)


ndash50

0

50

100

150

200

05 1 15 2 25 30t (ms)

(a)

u M+1

(t)(

kV)


05 1 15 2 25 30t (ms)

ndash50

0

50

100

150

200

(b)

Figure 12 Computational results of receiving terminal voltage by improved PIM with uniform grid using bigger time steps (a) PDIM withh 20 times 10minus5 s (b) PDIM with h 30 times 10minus5s

t = 0s

r e =

1kΩ

ie (t)Rch

L1 = 50m

Kndash

Figure 13 A simplified model for lightning overvoltagecalculation

e head of linee end of line

u(t)(

kV)

05 1 15 2 25 3 35 40t (μs)

0

1000

2000

3000

4000

5000

Figure 14 Computational results of voltage waveform by PDIMwith Chebyshev nodes

∆u(t)

(kV

)

05 1 15 2 25 3 35 40t (μs)

ndash15

ndash10

ndash5

0

5

10

15

Figure 15 Receiving end voltage difference waveform of PDIMwith Chebyshev nodes and TR


almost similar is example shows that for systems with veryfast change frequency PDIM is also competent

4 Conclusions

Aimed at the simulation efficiency of EMT simulation forovervoltages of the high-voltage transmission line an improvedprecise integration method based on DQM is proposed in thispaper e improved precise integration method inherits thecharacteristics of high precision and strong stability of the PIMand DQM And PDIM improves the approximate calculationmethod of Duhamel integration term in the calculation ofnonhomogeneous differential equations by the traditional PIMCompared with the numerical results of CDA method or thetrapezoidal method with small time step the advantage ofPDIM with larger time step is verified in the simulation effi-ciency of EMT simulation for high-voltage transmission lines

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

e authors gratefully acknowledge the support from theNational Natural Science Foundation of China (NSFC)through its Grant no 52007103 Natural Science Foundationof Hubei Province through its Grant no 2019CFB331 andScience and Technology Project of State Grid Corporationof China through its Grant no 5200-201956111A-0-0-00

References

[1] H-C Seo and G-H Gwon ldquoSystematization of the simula-tion process of transformer inrush current using EMTPrdquoApplied Sciences vol 9 no 12 p 2398 2019

[2] T Noda K Takenaka and T Inoue ldquoNumerical integrationby the 2-stage diagonally implicit Runge-Kutta method forelectromagnetic transient simulationsrdquo IEEE Transactions onPower Delivery vol 24 no 1 pp 390ndash399 2009

[3] T Noda T Kikuma and R Yonezawa ldquoSupplementarytechniques for 2S-DIRK-based EMT simulationsrdquo ElectricPower Systems Research vol 115 pp 87ndash93 2004

[4] P Chirapongsananurak and S Santoso ldquoMulti-time-scalesimulation tool for renewable energy integration analysis indistribution circuitsrdquo Inventions vol 2 no 2 p 7 2017

[5] C Wang X Fu P Li et al ldquoMultiscale simulation of powersystem transients based on the matrix exponential functionrdquoIEEE Transactions on Power Systems vol 32 no 3pp 1913ndash1926 2017

[6] J Han S Miao J Yu and Y Dong ldquoMulti-rate and parallelelectromagnetic transient simulation considering nonlinearcharacteristics of a power systemrdquo Energies vol 11 no 2p 468 2018

[7] A Abusalah J O Saad and L U Gerin-LajoieKaraagacldquoCPU based parallel computation of electromagnetic tran-sients for large power gridsrdquo Electric Power Systems Researchvol 162 pp 57ndash63 2018

[8] G Kocar S A Dimitrovski and M S StarkeSimunovicldquoParareal in time for fast power system dynamic simulationsrdquoIEEE Transactions on Power Systems vol 31 no 3pp 1820ndash1830 2016

[9] Y Song Y Chen S Huang and Y Xu ldquoEfficient GPU-basedelectromagnetic transient simulation for power systems withthread-oriented transformation and automatic code genera-tionrdquo IEEE Access vol 6 pp 25724ndash25736 2018

[10] F A Moreira and J R Marti ldquoLatency techniques for time-domain power system transients simulationrdquo IEEE Trans-actions on Power Systems vol 20 no 1 pp 246ndash253 2005

[11] M Armstrong J R Marti L R Linares and P KundurldquoMultilevel MATE for efficient simultaneous solution ofcontrol systems and nonlinearities in the OVNI simulatorrdquoIEEE Transactions on Power Systems vol 21 no 3pp 1250ndash1259 2006

[12] R Alexander ldquoDiagonally implicit Runge-Kutta methods forstiff ODErsquosrdquo SIAM Journal on Numerical Analysis vol 14no 6 pp 1006ndash1021 1977

[13] J J B De Swart ldquoA simple ODE solver based on 2-stageRadau IIArdquo Journal of Computational and Applied Mathe-matics vol 84 no 2 pp 277ndash280 1997

[14] W X Zhong ldquoOn precise integration methodrdquo Journal ofComputational amp Applied Mathematics vol 163 no 1pp 59ndash78 2004

[15] R Bellman and J Casti ldquoDifferential quadrature and long-term integrationrdquo Journal of Mathematical Analysis andApplications vol 34 no 2 pp 235ndash238 1971

[16] L Li S Zhou X Du J Song and C Gao ldquoNumerical study onthe seismic response of fluid-saturated porousmedia using theprecise time integration methodrdquo Applied Sciences vol 9no 10 p 2037 2019

[17] S A Eftekhari and A A Jafari ldquoA simple and accurate mixedFE-DQ formulation for free vibration of rectangular and skewmindlin plates with general boundary conditionsrdquoMeccanicavol 48 no 5 pp 1139ndash1160 2013

[18] S Zhang W Z Deng and W Li ldquoA precise Runge-Kuttaintegration and its application for solving nonlinear dy-namical systemsrdquo Applied Mathematics and Computationvol 184 no 2 pp 496ndash502 2007


i(t)(

kA)

05 1 15 2 25 3 35 40t (μs)

0

20

40

60

80

100

120

Figure 16 Computational results of current waveform by PDIMwith Chebyshev nodes


[19] J Lin and J R Marti ldquoImplementation of the CDA procedurein the EMTPrdquo IEEE Transactions on Power Systems vol 5no 2 pp 394ndash402 1990

[20] D Shu X S XieZhang and Q Jiang ldquoHybrid method fornumerical oscillation suppression based on rational-fractionapproximations to exponential functionsrdquo IET GenerationTransmission amp Distribution vol 10 no 11 pp 2825ndash28322016

[21] L Brugnano and D Trigiante ldquoBoundary value methods thethird way between linear multistep and Runge-Kuttamethodsrdquo Computers amp Mathematics with Applicationsvol 36 no 10ndash12 pp 269ndash284 1998

[22] M Gunther Electrical Circuits Springer Berlin HeidelbergBerlin Germany 2015

[23] N M Morris Transient Solution of Electrical Circuits Mac-millan Education UK London UK 1993

[24] M Fu and H Liang ldquoAn improved precise Runge-Kuttaintegrationrdquo Acta Scientiarum Naturalium UniversitatisSunyatseni vol 48 no 5 pp 1ndash5 2009 in Chinese

[25] A C Cangellaris S Pasha J L Prince and M Celik ldquoA newdiscrete transmission line model for passive model orderreduction and macromodeling of high-speed interconnec-tionsrdquo IEEE Transactions on Advanced Packaging vol 22no 3 pp 356ndash364 1999

[26] Q Li Y Wang H Rao et al ldquoExtended critical dampingadjustment method for electromagnetic transients simulationin power systemsrdquo Applied Sciences vol 8 no 6 p 883 2018


but its computational amount is large Reference [11]proposed the concept of ldquodouble-layer grid separationrdquo forthe topology characteristics of AC-DC hybrid system ismethod greatly improved the simulation efficiency of AC-DC hybrid power grids

To replace the role of implicit trapezoidal integration inEMT simulation researchers have begun to seek the ap-plication of higher-order and A-stable or L-stable numericalalgorithms in EMT simulation e two-stage diagonallyimplicit Runge-Kutta method (2S-DIRK) [12] is applied toEMT simulation e 2S-DIRK is L-stable as same as theimplicit Euler method so it can effectively avoid numericaloscillation [2] Two-stage three-order Radau IIA method[13] is used for a new parallel method for solving nonlinearEMT state-space equation at multiple time points and thenthe Newton method is used to solve it iteratively e al-gorithm not only has high computational efficiency but alsocan avoid numerical oscillation

Both the precise integration method [14] and the dif-ferential quadrature (DQ) method [15] have high accuracyand strong stability and both of them have been widely usedin the field of computational mechanics [16 17] In order toimprove the applicable scope of the precise integrationmethod the precise Runge-Kutta method is proposed whichis a new algorithm formed by the precise integration methodand the explicit Runge-Kutta method with the fourth order[18] e calculation efficiency and precision of the preciseRunge-Kutta method need to be improved so it cannot bedirectly used in the EMTsimulation of the power systems Inorder to improve the EMT simulation efficiency via an im-proved precise integration method this paper makes full useof the advantages of the DQ and precise integrationmethod toconstruct a new method to solve the problem e improvedprecise integration method has high accuracy and goodstability and it can use a large time step to improve thesimulation efficiency of EMTs Compared with the implicittrapezoidal integration using a small time step the calculationerror of this method meets the needs of practical engineeringcalculation and the calculation efficiency is considerable

Compared with the implicit numerical algorithm theexplicit numerical algorithm has higher execution efficiencyHowever the existing explicit methods such as the explicitEuler method have too low numerical precision to be directlyused in power system transient simulation Fortunately thenew method proposed in this paper is essentially an explicitsingle-step high-precision integration method e idea ofapplying the improved precise integration method to elec-tromagnetic transient simulation proposed in this paper is thebiggest advantage over the existing implicit electromagnetictransient simulation algorithm [19 20] in calculation effi-ciency which will be confirmed later in this paper

e remainder of this paper is organized as follows InSection 2 an improved precise integration method based onDQ with different grids is presented in detail In Section 3numerical simulations for overvoltages of high-voltagetransmission lines with different time scale and complexityare given to verify the accuracy and effectiveness of theimproved precise integration method Finally the conclu-sions derived from the study are presented in Section 4

2 Improved Precise Integration Method

Generally the state-space model of EMTs in power systemscan be described as an initial value problem of the first-orderordinary differential equation

_x(t) f(x(t)) t isin t0 T1113858 1113859

x t0( 1113857 x01113896 (1)

where t represents time variable f(t x(t)) is a function oftime t and state variable x x0 is the value of state variable atinitial time t0

Formula (1) can always be rewritten into ordinary dif-ferential equations consisting of homogeneous and non-homogeneous parts by a certain transformation and

_x Hx + g(x t) (2)

where H isin Rntimesn is the coefficient matrix related to systemparameters To the right of the medium sign in equation (2) thefirst item is called homogeneous part and the second item iscalled nonhomogeneous term e exact solution of the ho-mogeneous equation of equation (2) on the computer is asfollows

x exp(H middot t)x0 (3)

In the single-step integration equation (3) has the fol-lowing discrete recursive calculation format

xk+1 T times xk k 0 1 2 (4)

e precise integration method (PIM) [14] proposed byZhong uses additive theorem and incremental storage tocalculate exponential matrix T exp(h times H)

T exp(H middot h) expH middot h

m1113888 11138891113890 1113891

m

(5)

where m 2N (where N 15) For very small η hm wecan use Taylor series to expand the exponential matrix T in(5) into M terms (where M 4) then there are

exp(H middot η) I + Ta

Ta Hη +(Hη)

2

2+

(Hη)3

3+

(Hη)4

4

(6)

Substituting formula (6) into formula (5) yields

T I + Ta1113858 1113859m

I + Ta1113858 11138592Nminus1

middot I + Ta1113858 11138592Nminus1

(7)

According to the matrix multiplication the followingrecursive operations are performed iteratively

Tai 2lowastTaiminus1 + Taiminus1 lowastTaiminus11113872 1113873 i 1 2 N (8)

where Ta0 Ta At the end of the cycle

T I + TaN (9)

In the single integration step [tk tk+1] the solution of thenonhomogeneous dynamic equation (2) can be expressed asfollows


xk+1 Txk + 1113946tk+1

tk





zk+1 zk + h 1113944s

j1bjw tk + cjh 1113957zj1113872 1113873 (11)



xk+1 h 1113944s

i1bie




1 2 s) as follows

1113957xk+14 xk +h

4 times 6S1 + 2S2 + 2S3 + S4( 1113857 (13)


S2 H xk +h

4 times 2S11113888 1113889 + g tk +

h

4 times 2 xk +

h

4 times 2S11113888 1113889

S3 H xk +h

4 times 2S21113888 1113889 + g tk +

h

4 times 2 xk +

h

4 times 2S21113888 1113889

S4 H xk +h

4S31113888 1113889 + g tk +

h

4 xk +

h

4S31113888 1113889

(14)


xk+1 h b1K1 + b2K2 + b3K3 + b4K4( 1113857 (15)



T2 T3 times T3

T1 T2 times T3

T T1 times T3

(16)



ck 12


π1113888 11138891113890 1113891 (k isin (1 s minus 1))

c0 0

cs 1

(17)




+cih

6S1 + 2S2 + 2S3 + S4( 1113857 (18)



1113872 1113873


cih

2S11113888 1113889 + g tk + ciminus1h +

cih


+cih

2S11113888 1113889


cih

2S21113888 1113889 + 1113957g tk + ciminus1h +

cih


+cih

2S21113888 1113889


+ cihS31113872 1113873

(19)


xk+1 h 1113957b11113957K1 + 1113957b2

1113957K2 + 1113957b31113957K3 + 1113957b4

1113957K41113872 1113873 (20)


1113957t2) 1113957K3 T3g










LsCd2uc

dt2 + RsC

duc

dtuc + uc es (21)

d0t

iC

uC

1113890 1113891

minusRs

Ls

minus1Ls

1C

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

iC

uC

1113890 1113891 +

es

Ls

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)



+ Ucm sin ωt + φc( 1113857

(23)


Ucm Um


21113969

1ωC

φc ϕs minusπ2


Rs


ωα

K2 minusUcm sinφc

(24)


LsC

1113968 and


ω20 minus α2

1113969





2


2

radicsin(120πt)kVe




sin(100t + φs)515 2

3es =

RsLs

C

K t = 0s

AC filter

iCiC

uC






u C (t

)(kV

)ndash400

ndash200

0

200

400

600

001 002 003 004 005 006 007 008 009 010Time (s)


CDAPDIM

PRKM

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

Abs

olut

e err

or

0005 001 0015 0020Time (s)


PDIM

TRDIRK

0005 001 0015 0020Time (s)

10ndash4

10ndash3

10ndash2

10ndash1

100

Abs

olut

e err

or


es (t)

rs ls

lnl

t = 0sinl

unl


i(kA)02

1H10mH

ϕ



r R0L

M

l L0L

M

c C0L

M

(25)


L = 100km

RL

Rs

LL

Ls


t = 0s˜ ˜

e (t) = 220 2sin(100πt + 90deg)


u nl (

t)(kV

)

times 10ndash3


46 47 48 49 545t (s)

ndash60

ndash40

ndash20

0

20

40

60

80

(a)

u nl (

t)(kV

)

ndash80

ndash60

ndash40

ndash20

0

20

40

60

80

001 002 003 004 005 0060t (s)

(b)

i nl (t

)(kA

)

001 002 003 004 005 0060t (s)

ndash05

0

05

1

15

2

25

3

(c)






LS

di0

dt+ RSi0 + u1 e(t)

ldim(t)


cdum(t)


uM+1

RL

+ iL m isin (1 M + 1)

LL

diL

dt uM+1

LL a tanh biL

iL

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)


_y(t) Ay(t) + μ(t) (27)



y(t) i0 i1 iM u1 uM+1 iL1113858 1113859T

μ(t) e(t)Ls

0 0 0 0 uM+1LL

minus iL1113876 1113877T

(28)

0 0 0 0 0

000

0

0000

0

000

0

0

0

0

0 0 0

0

0

0 0 1

0

A =


helliphellip

hellip

hellip


hellip

hellip

hellip


hellip

hellip

hellip hellip

hellip hellip

hellip

hellip hellip

hellip

hellip

hellip

RsLs

ndash 1Ls

ndash

rlndash

rlndash

rlndash

1lndash1

l

1lndash

1cndash

1c

1c

1c

1c

1c

1c

1c

1cRL

ndash

1l

1lndash1

l

(29)

u1 u2 u3 uM + 1uM


r l r l r l

RL LL

iL

e (t)

Rs Ls˜ ˜









iL(t) aIL eminusαt

minus eminusβt

1113872 1113873 (30)



PDIMCDA

u 1(t)

(kV

)

0

50

100

150

200

250

300

350

05 1 15 2 25 3 35 40Time (ms)


PDIMCDA

u M+1

(t)(

kV)

0

50

100

150

200

05 1 15 2 25 3 35 40Time (ms)





x

Hx + δ(t) (31)




u M+1

(t)(

kV)


ndash50

0

50

100

150

200

05 1 15 2 25 30t (ms)

(a)

u M+1

(t)(

kV)


05 1 15 2 25 30t (ms)

ndash50

0

50

100

150

200

(b)


t = 0s

r e =

1kΩ

ie (t)Rch

L1 = 50m

Kndash



u(t)(

kV)

05 1 15 2 25 3 35 40t (μs)

0

1000

2000

3000

4000

5000


∆u(t)

(kV

)

05 1 15 2 25 3 35 40t (μs)

ndash15

ndash10

ndash5

0

5

10

15




4 Conclusions


Data Availability




Acknowledgments


References




















i(t)(

kA)

05 1 15 2 25 3 35 40t (μs)

0

20

40

60

80

100

120












xk+1 Txk + 1113946tk+1

tk





zk+1 zk + h 1113944s

j1bjw tk + cjh 1113957zj1113872 1113873 (11)



xk+1 h 1113944s

i1bie




1 2 s) as follows

1113957xk+14 xk +h

4 times 6S1 + 2S2 + 2S3 + S4( 1113857 (13)


S2 H xk +h

4 times 2S11113888 1113889 + g tk +

h

4 times 2 xk +

h

4 times 2S11113888 1113889

S3 H xk +h

4 times 2S21113888 1113889 + g tk +

h

4 times 2 xk +

h

4 times 2S21113888 1113889

S4 H xk +h

4S31113888 1113889 + g tk +

h

4 xk +

h

4S31113888 1113889

(14)


xk+1 h b1K1 + b2K2 + b3K3 + b4K4( 1113857 (15)



T2 T3 times T3

T1 T2 times T3

T T1 times T3

(16)



ck 12


π1113888 11138891113890 1113891 (k isin (1 s minus 1))

c0 0

cs 1

(17)




+cih

6S1 + 2S2 + 2S3 + S4( 1113857 (18)



1113872 1113873


cih

2S11113888 1113889 + g tk + ciminus1h +

cih


+cih

2S11113888 1113889


cih

2S21113888 1113889 + 1113957g tk + ciminus1h +

cih


+cih

2S21113888 1113889


+ cihS31113872 1113873

(19)


xk+1 h 1113957b11113957K1 + 1113957b2

1113957K2 + 1113957b31113957K3 + 1113957b4

1113957K41113872 1113873 (20)


1113957t2) 1113957K3 T3g










LsCd2uc

dt2 + RsC

duc

dtuc + uc es (21)

d0t

iC

uC

1113890 1113891

minusRs

Ls

minus1Ls

1C

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

iC

uC

1113890 1113891 +

es

Ls

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)



+ Ucm sin ωt + φc( 1113857

(23)


Ucm Um


21113969

1ωC

φc ϕs minusπ2


Rs


ωα

K2 minusUcm sinφc

(24)


LsC

1113968 and


ω20 minus α2

1113969





2


2

radicsin(120πt)kVe




sin(100t + φs)515 2

3es =

RsLs

C

K t = 0s

AC filter

iCiC

uC






u C (t

)(kV

)ndash400

ndash200

0

200

400

600

001 002 003 004 005 006 007 008 009 010Time (s)


CDAPDIM

PRKM

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

Abs

olut

e err

or

0005 001 0015 0020Time (s)


PDIM

TRDIRK

0005 001 0015 0020Time (s)

10ndash4

10ndash3

10ndash2

10ndash1

100

Abs

olut

e err

or


es (t)

rs ls

lnl

t = 0sinl

unl


i(kA)02

1H10mH

ϕ



r R0L

M

l L0L

M

c C0L

M

(25)


L = 100km

RL

Rs

LL

Ls


t = 0s˜ ˜

e (t) = 220 2sin(100πt + 90deg)


u nl (

t)(kV

)

times 10ndash3


46 47 48 49 545t (s)

ndash60

ndash40

ndash20

0

20

40

60

80

(a)

u nl (

t)(kV

)

ndash80

ndash60

ndash40

ndash20

0

20

40

60

80

001 002 003 004 005 0060t (s)

(b)

i nl (t

)(kA

)

001 002 003 004 005 0060t (s)

ndash05

0

05

1

15

2

25

3

(c)






LS

di0

dt+ RSi0 + u1 e(t)

ldim(t)


cdum(t)


uM+1

RL

+ iL m isin (1 M + 1)

LL

diL

dt uM+1

LL a tanh biL

iL

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)


_y(t) Ay(t) + μ(t) (27)



y(t) i0 i1 iM u1 uM+1 iL1113858 1113859T

μ(t) e(t)Ls

0 0 0 0 uM+1LL

minus iL1113876 1113877T

(28)

0 0 0 0 0

000

0

0000

0

000

0

0

0

0

0 0 0

0

0

0 0 1

0

A =


helliphellip

hellip

hellip


hellip

hellip

hellip


hellip

hellip

hellip hellip

hellip hellip

hellip

hellip hellip

hellip

hellip

hellip

RsLs

ndash 1Ls

ndash

rlndash

rlndash

rlndash

1lndash1

l

1lndash

1cndash

1c

1c

1c

1c

1c

1c

1c

1cRL

ndash

1l

1lndash1

l

(29)

u1 u2 u3 uM + 1uM


r l r l r l

RL LL

iL

e (t)

Rs Ls˜ ˜









iL(t) aIL eminusαt

minus eminusβt

1113872 1113873 (30)



PDIMCDA

u 1(t)

(kV

)

0

50

100

150

200

250

300

350

05 1 15 2 25 3 35 40Time (ms)


PDIMCDA

u M+1

(t)(

kV)

0

50

100

150

200

05 1 15 2 25 3 35 40Time (ms)





x

Hx + δ(t) (31)




u M+1

(t)(

kV)


ndash50

0

50

100

150

200

05 1 15 2 25 30t (ms)

(a)

u M+1

(t)(

kV)


05 1 15 2 25 30t (ms)

ndash50

0

50

100

150

200

(b)


t = 0s

r e =

1kΩ

ie (t)Rch

L1 = 50m

Kndash



u(t)(

kV)

05 1 15 2 25 3 35 40t (μs)

0

1000

2000

3000

4000

5000


∆u(t)

(kV

)

05 1 15 2 25 3 35 40t (μs)

ndash15

ndash10

ndash5

0

5

10

15




4 Conclusions


Data Availability




Acknowledgments


References




















i(t)(

kA)

05 1 15 2 25 3 35 40t (μs)

0

20

40

60

80

100

120

















LsCd2uc

dt2 + RsC

duc

dtuc + uc es (21)

d0t

iC

uC

1113890 1113891

minusRs

Ls

minus1Ls

1C

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

iC

uC

1113890 1113891 +

es

Ls

0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (22)



+ Ucm sin ωt + φc( 1113857

(23)


Ucm Um


21113969

1ωC

φc ϕs minusπ2


Rs


ωα

K2 minusUcm sinφc

(24)


LsC

1113968 and


ω20 minus α2

1113969





2


2

radicsin(120πt)kVe




sin(100t + φs)515 2

3es =

RsLs

C

K t = 0s

AC filter

iCiC

uC






u C (t

)(kV

)ndash400

ndash200

0

200

400

600

001 002 003 004 005 006 007 008 009 010Time (s)


CDAPDIM

PRKM

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

Abs

olut

e err

or

0005 001 0015 0020Time (s)


PDIM

TRDIRK

0005 001 0015 0020Time (s)

10ndash4

10ndash3

10ndash2

10ndash1

100

Abs

olut

e err

or


es (t)

rs ls

lnl

t = 0sinl

unl


i(kA)02

1H10mH

ϕ



r R0L

M

l L0L

M

c C0L

M

(25)


L = 100km

RL

Rs

LL

Ls


t = 0s˜ ˜

e (t) = 220 2sin(100πt + 90deg)


u nl (

t)(kV

)

times 10ndash3


46 47 48 49 545t (s)

ndash60

ndash40

ndash20

0

20

40

60

80

(a)

u nl (

t)(kV

)

ndash80

ndash60

ndash40

ndash20

0

20

40

60

80

001 002 003 004 005 0060t (s)

(b)

i nl (t

)(kA

)

001 002 003 004 005 0060t (s)

ndash05

0

05

1

15

2

25

3

(c)






LS

di0

dt+ RSi0 + u1 e(t)

ldim(t)


cdum(t)


uM+1

RL

+ iL m isin (1 M + 1)

LL

diL

dt uM+1

LL a tanh biL

iL

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)


_y(t) Ay(t) + μ(t) (27)



y(t) i0 i1 iM u1 uM+1 iL1113858 1113859T

μ(t) e(t)Ls

0 0 0 0 uM+1LL

minus iL1113876 1113877T

(28)

0 0 0 0 0

000

0

0000

0

000

0

0

0

0

0 0 0

0

0

0 0 1

0

A =


helliphellip

hellip

hellip


hellip

hellip

hellip


hellip

hellip

hellip hellip

hellip hellip

hellip

hellip hellip

hellip

hellip

hellip

RsLs

ndash 1Ls

ndash

rlndash

rlndash

rlndash

1lndash1

l

1lndash

1cndash

1c

1c

1c

1c

1c

1c

1c

1cRL

ndash

1l

1lndash1

l

(29)

u1 u2 u3 uM + 1uM


r l r l r l

RL LL

iL

e (t)

Rs Ls˜ ˜









iL(t) aIL eminusαt

minus eminusβt

1113872 1113873 (30)



PDIMCDA

u 1(t)

(kV

)

0

50

100

150

200

250

300

350

05 1 15 2 25 3 35 40Time (ms)


PDIMCDA

u M+1

(t)(

kV)

0

50

100

150

200

05 1 15 2 25 3 35 40Time (ms)





x

Hx + δ(t) (31)




u M+1

(t)(

kV)


ndash50

0

50

100

150

200

05 1 15 2 25 30t (ms)

(a)

u M+1

(t)(

kV)


05 1 15 2 25 30t (ms)

ndash50

0

50

100

150

200

(b)


t = 0s

r e =

1kΩ

ie (t)Rch

L1 = 50m

Kndash



u(t)(

kV)

05 1 15 2 25 3 35 40t (μs)

0

1000

2000

3000

4000

5000


∆u(t)

(kV

)

05 1 15 2 25 3 35 40t (μs)

ndash15

ndash10

ndash5

0

5

10

15




4 Conclusions


Data Availability




Acknowledgments


References




















i(t)(

kA)

05 1 15 2 25 3 35 40t (μs)

0

20

40

60

80

100

120















u C (t

)(kV

)ndash400

ndash200

0

200

400

600

001 002 003 004 005 006 007 008 009 010Time (s)


CDAPDIM

PRKM

10ndash5

10ndash4

10ndash3

10ndash2

10ndash1

100

Abs

olut

e err

or

0005 001 0015 0020Time (s)


PDIM

TRDIRK

0005 001 0015 0020Time (s)

10ndash4

10ndash3

10ndash2

10ndash1

100

Abs

olut

e err

or


es (t)

rs ls

lnl

t = 0sinl

unl


i(kA)02

1H10mH

ϕ



r R0L

M

l L0L

M

c C0L

M

(25)


L = 100km

RL

Rs

LL

Ls


t = 0s˜ ˜

e (t) = 220 2sin(100πt + 90deg)


u nl (

t)(kV

)

times 10ndash3


46 47 48 49 545t (s)

ndash60

ndash40

ndash20

0

20

40

60

80

(a)

u nl (

t)(kV

)

ndash80

ndash60

ndash40

ndash20

0

20

40

60

80

001 002 003 004 005 0060t (s)

(b)

i nl (t

)(kA

)

001 002 003 004 005 0060t (s)

ndash05

0

05

1

15

2

25

3

(c)






LS

di0

dt+ RSi0 + u1 e(t)

ldim(t)


cdum(t)


uM+1

RL

+ iL m isin (1 M + 1)

LL

diL

dt uM+1

LL a tanh biL

iL

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)


_y(t) Ay(t) + μ(t) (27)



y(t) i0 i1 iM u1 uM+1 iL1113858 1113859T

μ(t) e(t)Ls

0 0 0 0 uM+1LL

minus iL1113876 1113877T

(28)

0 0 0 0 0

000

0

0000

0

000

0

0

0

0

0 0 0

0

0

0 0 1

0

A =


helliphellip

hellip

hellip


hellip

hellip

hellip


hellip

hellip

hellip hellip

hellip hellip

hellip

hellip hellip

hellip

hellip

hellip

RsLs

ndash 1Ls

ndash

rlndash

rlndash

rlndash

1lndash1

l

1lndash

1cndash

1c

1c

1c

1c

1c

1c

1c

1cRL

ndash

1l

1lndash1

l

(29)

u1 u2 u3 uM + 1uM


r l r l r l

RL LL

iL

e (t)

Rs Ls˜ ˜









iL(t) aIL eminusαt

minus eminusβt

1113872 1113873 (30)



PDIMCDA

u 1(t)

(kV

)

0

50

100

150

200

250

300

350

05 1 15 2 25 3 35 40Time (ms)


PDIMCDA

u M+1

(t)(

kV)

0

50

100

150

200

05 1 15 2 25 3 35 40Time (ms)





x

Hx + δ(t) (31)




u M+1

(t)(

kV)


ndash50

0

50

100

150

200

05 1 15 2 25 30t (ms)

(a)

u M+1

(t)(

kV)


05 1 15 2 25 30t (ms)

ndash50

0

50

100

150

200

(b)


t = 0s

r e =

1kΩ

ie (t)Rch

L1 = 50m

Kndash



u(t)(

kV)

05 1 15 2 25 3 35 40t (μs)

0

1000

2000

3000

4000

5000


∆u(t)

(kV

)

05 1 15 2 25 3 35 40t (μs)

ndash15

ndash10

ndash5

0

5

10

15




4 Conclusions


Data Availability




Acknowledgments


References




















i(t)(

kA)

05 1 15 2 25 3 35 40t (μs)

0

20

40

60

80

100

120












r R0L

M

l L0L

M

c C0L

M

(25)


L = 100km

RL

Rs

LL

Ls


t = 0s˜ ˜

e (t) = 220 2sin(100πt + 90deg)


u nl (

t)(kV

)

times 10ndash3


46 47 48 49 545t (s)

ndash60

ndash40

ndash20

0

20

40

60

80

(a)

u nl (

t)(kV

)

ndash80

ndash60

ndash40

ndash20

0

20

40

60

80

001 002 003 004 005 0060t (s)

(b)

i nl (t

)(kA

)

001 002 003 004 005 0060t (s)

ndash05

0

05

1

15

2

25

3

(c)






LS

di0

dt+ RSi0 + u1 e(t)

ldim(t)


cdum(t)


uM+1

RL

+ iL m isin (1 M + 1)

LL

diL

dt uM+1

LL a tanh biL

iL

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)


_y(t) Ay(t) + μ(t) (27)



y(t) i0 i1 iM u1 uM+1 iL1113858 1113859T

μ(t) e(t)Ls

0 0 0 0 uM+1LL

minus iL1113876 1113877T

(28)

0 0 0 0 0

000

0

0000

0

000

0

0

0

0

0 0 0

0

0

0 0 1

0

A =


helliphellip

hellip

hellip


hellip

hellip

hellip


hellip

hellip

hellip hellip

hellip hellip

hellip

hellip hellip

hellip

hellip

hellip

RsLs

ndash 1Ls

ndash

rlndash

rlndash

rlndash

1lndash1

l

1lndash

1cndash

1c

1c

1c

1c

1c

1c

1c

1cRL

ndash

1l

1lndash1

l

(29)

u1 u2 u3 uM + 1uM


r l r l r l

RL LL

iL

e (t)

Rs Ls˜ ˜









iL(t) aIL eminusαt

minus eminusβt

1113872 1113873 (30)



PDIMCDA

u 1(t)

(kV

)

0

50

100

150

200

250

300

350

05 1 15 2 25 3 35 40Time (ms)


PDIMCDA

u M+1

(t)(

kV)

0

50

100

150

200

05 1 15 2 25 3 35 40Time (ms)





x

Hx + δ(t) (31)




u M+1

(t)(

kV)


ndash50

0

50

100

150

200

05 1 15 2 25 30t (ms)

(a)

u M+1

(t)(

kV)


05 1 15 2 25 30t (ms)

ndash50

0

50

100

150

200

(b)


t = 0s

r e =

1kΩ

ie (t)Rch

L1 = 50m

Kndash



u(t)(

kV)

05 1 15 2 25 3 35 40t (μs)

0

1000

2000

3000

4000

5000


∆u(t)

(kV

)

05 1 15 2 25 3 35 40t (μs)

ndash15

ndash10

ndash5

0

5

10

15




4 Conclusions


Data Availability




Acknowledgments


References




















i(t)(

kA)

05 1 15 2 25 3 35 40t (μs)

0

20

40

60

80

100

120













LS

di0

dt+ RSi0 + u1 e(t)

ldim(t)


cdum(t)


uM+1

RL

+ iL m isin (1 M + 1)

LL

diL

dt uM+1

LL a tanh biL

iL

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)


_y(t) Ay(t) + μ(t) (27)



y(t) i0 i1 iM u1 uM+1 iL1113858 1113859T

μ(t) e(t)Ls

0 0 0 0 uM+1LL

minus iL1113876 1113877T

(28)

0 0 0 0 0

000

0

0000

0

000

0

0

0

0

0 0 0

0

0

0 0 1

0

A =


helliphellip

hellip

hellip


hellip

hellip

hellip


hellip

hellip

hellip hellip

hellip hellip

hellip

hellip hellip

hellip

hellip

hellip

RsLs

ndash 1Ls

ndash

rlndash

rlndash

rlndash

1lndash1

l

1lndash

1cndash

1c

1c

1c

1c

1c

1c

1c

1cRL

ndash

1l

1lndash1

l

(29)

u1 u2 u3 uM + 1uM


r l r l r l

RL LL

iL

e (t)

Rs Ls˜ ˜









iL(t) aIL eminusαt

minus eminusβt

1113872 1113873 (30)



PDIMCDA

u 1(t)

(kV

)

0

50

100

150

200

250

300

350

05 1 15 2 25 3 35 40Time (ms)


PDIMCDA

u M+1

(t)(

kV)

0

50

100

150

200

05 1 15 2 25 3 35 40Time (ms)





x

Hx + δ(t) (31)




u M+1

(t)(

kV)


ndash50

0

50

100

150

200

05 1 15 2 25 30t (ms)

(a)

u M+1

(t)(

kV)


05 1 15 2 25 30t (ms)

ndash50

0

50

100

150

200

(b)


t = 0s

r e =

1kΩ

ie (t)Rch

L1 = 50m

Kndash



u(t)(

kV)

05 1 15 2 25 3 35 40t (μs)

0

1000

2000

3000

4000

5000


∆u(t)

(kV

)

05 1 15 2 25 3 35 40t (μs)

ndash15

ndash10

ndash5

0

5

10

15




4 Conclusions


Data Availability




Acknowledgments


References




















i(t)(

kA)

05 1 15 2 25 3 35 40t (μs)

0

20

40

60

80

100

120


















iL(t) aIL eminusαt

minus eminusβt

1113872 1113873 (30)



PDIMCDA

u 1(t)

(kV

)

0

50

100

150

200

250

300

350

05 1 15 2 25 3 35 40Time (ms)


PDIMCDA

u M+1

(t)(

kV)

0

50

100

150

200

05 1 15 2 25 3 35 40Time (ms)





x

Hx + δ(t) (31)




u M+1

(t)(

kV)


ndash50

0

50

100

150

200

05 1 15 2 25 30t (ms)

(a)

u M+1

(t)(

kV)


05 1 15 2 25 30t (ms)

ndash50

0

50

100

150

200

(b)


t = 0s

r e =

1kΩ

ie (t)Rch

L1 = 50m

Kndash



u(t)(

kV)

05 1 15 2 25 3 35 40t (μs)

0

1000

2000

3000

4000

5000


∆u(t)

(kV

)

05 1 15 2 25 3 35 40t (μs)

ndash15

ndash10

ndash5

0

5

10

15




4 Conclusions


Data Availability




Acknowledgments


References




















i(t)(

kA)

05 1 15 2 25 3 35 40t (μs)

0

20

40

60

80

100

120












x

Hx + δ(t) (31)




u M+1

(t)(

kV)


ndash50

0

50

100

150

200

05 1 15 2 25 30t (ms)

(a)

u M+1

(t)(

kV)


05 1 15 2 25 30t (ms)

ndash50

0

50

100

150

200

(b)


t = 0s

r e =

1kΩ

ie (t)Rch

L1 = 50m

Kndash



u(t)(

kV)

05 1 15 2 25 3 35 40t (μs)

0

1000

2000

3000

4000

5000


∆u(t)

(kV

)

05 1 15 2 25 3 35 40t (μs)

ndash15

ndash10

ndash5

0

5

10

15




4 Conclusions


Data Availability




Acknowledgments


References




















i(t)(

kA)

05 1 15 2 25 3 35 40t (μs)

0

20

40

60

80

100

120













4 Conclusions


Data Availability




Acknowledgments


References




















i(t)(

kA)

05 1 15 2 25 3 35 40t (μs)

0

20

40

60

80

100

120












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