electromagnetic transient simulation of large power

Electromagnetic Transient Simulation of Large Power Networks

with Modelica

Alireza Masoom1 Jean Mahseredjian1 Tarek Ould-Bachir2 Adrien Guironnet3 1Department of Electrical Engineering, Polytechnique Montreal, Canada,

{alireza.masoom, jean.mahseredjian}@polymtl.ca 2Department of Computer Engineering, Polytechnique Montreal, Canada,

[email protected] 3Réseau de Transport d'Électricité, France, [email protected]

Abstract This paper presents the simulation of electromagnetic

transients (EMTs) with Modelica. The advantages and

disadvantages are discussed. Simulation performance and

accuracy are analyzed through the IEEE 118-bus

benchmark which includes EMT-detailed models with

nonlinearities. The domain-specific simulator EMTP is

used for validations and comparisons.

Keywords: Modelica, Equation-based Modeling, Acausal

modeling, Electromagnetic transients, EMT, Synchronous

machines, Large scale, Nonlinearity, IEEE 118-bus grid.

1 Introduction

Power system simulations are mainly categorized into

phasor-domain and time-domain. In phasor-domain

analysis, voltages and currents are computed as phasors

varying in time. The electrical network is solved in steady-

state and at the fundamental frequency. Power generation

sources, such as synchronous machines, are solved with

their differential equations in time-domain and interfaced

with network equations using phasor equivalents. The

phasor-domain approach is suitable for electromechanical

transients, load flow, and short-circuit studies in very

large-scale grids and can be applied to any linear system

but becomes less accurate with the presence of power

electronics-based equipment e.g., FACTS and HVDC

since this technique cannot represent the faster transients.

Its main advantage remains computational speed for

studying lower frequency transients, but harmonics and

nonlinear models are ignored.

In the time-domain simulation approach, the network

and all integrated components are solved in time-domain

with detailed differential equations. Harmonics and

nonlinearities are modeled accurately. The solution may

include lower frequency interactions, as well as very high-

frequency transients. The time-domain approach allows to

solve networks for electromagnetic transients and is

tagged as the EMT-type solution.

Various specialized EMT-type simulation tools

(Mahseredjian 2009) are currently available and well

adapted for studying various power system phenomena,

including transformer saturation effects, lightning and

switching transients, and integration of inverter-based

resources. Power electronics-based components can be

simulated very accurately.

The EMTP (Mahseredjian, et al. 2007) software used

in this paper is widely used for the EMT simulations. The

cornerstone of this software is the discretization of

component models using the well-known companion

circuit approach. The A-stable trapezoidal and Backward-

Euler numerical integration methods are employed for the

discretization. The latter is used at discontinuity points to

avoid numerical oscillations (Sana, Mahseredjian, et al.

1995). In EMTP, the companion circuits of components

are interconnected (to respect Kirchhoff's first law)

through a sparse matrix solver using the Modified

Augmented Nodal Analysis (Mahseredjian, Dennetière, et

al. 2007) formulation.

In power system modeling, Modelica has been first

considered for phasor-domain simulations. iTesla Power

Systems Library (iPSL) (Vanfretti et al. 2016) is a

comprehensive Modelica package that was generated

through the iTesla project (Lemaitre 2014) for unified

modeling and for facilitating network model exchanges

amongst transmission system operators. PowerGrids

(Bartolini et al. 2019) and ObjectStab (Larsson 2004) are

other advanced libraries developed for electromechanical

and stability analyses, respectively.

Modelica (Fritzson 2014) is an equation-based

language that relies on the description of a system by

differential-algebraic equations. The EMT behavior of

electrical components can be modeled by their differential

equations. The language emphasizes the acausal approach

based on declarative modeling where the model and

solver are decoupled. It significantly improves model

development process and model readability.

The first attempt to develop a Modelica-based EMT-detailed simulator was reported in (Masoom et al. 2020),

where transmission lines incorporating the constant

parameter and wideband (Kocar Mahseredjian 2016)

models were introduced and validated. An EMT-detailed

library was developed and validated using the IEEE 39-

bus network. A challenging problem with Modelica is computational

speed (including compilation and simulation time). Some

DOI10.3384/ecp21181277

Proceedings of the 14th International Modelica ConferenceSeptember 20-24, 2021, Linköping, Sweden

277

solutions are based on numerical optimizations, e.g.,

Jacobian optimization (Kofman, Fernández, and

Marzorati 2021) and simulation in DAE mode (Braun et

al. 2017; Henningsson 2019). An open-source solution

(called DynaꞶo) is specifically designed for power system

simulations in phasor-domain (Guironnet et al. 2018).

DynaꞶo approach is based on hybrid coding with

Modelica and C++ and demonstrates competitive

performance compared to the specific domain packages

(Guironnet et al. 2018). This approach was used for EMT

simulations in (Masoom et al. 2021) as well. Even though

it delivers better performance compared to the pure

Modelica tools, there is still a significant gap in

comparison with EMTP.

The IEEE 118-bus benchmark contains the following

models: synchronous generators (including magnetic

saturation model) with controls, transformers,

transmission lines, nonlinear inductances, and nonlinear

surge arresters. The basic models, such as resistance,

inductance, and advanced models, that is, various models

of transmission line, loads, saturable transformers,

synchronous machine (without saturation), machine

controls, etc. were already presented in previous papers

(Masoom et al. 2020; Masoom et al. 2021). This paper

focuses on the synchronous machine model with

saturation and the nonlinear arrester and comparison of

simulation efficiency in Modelica.

The paper is organized as follows. Two selected

models, namely the synchronous machine and nonlinear

arrester are developed using Modelica in Section 2. The

numerical results for the IEEE 118-bus system are

provided in Section 3.

2 Modelica Model Implementation

In this Section, two nonlinear models are presented and

discussed in more detail. The models have been developed

based on EMTP mathematical representations.

2.1 Synchronous Machine Model with

Magnetic Saturation

The details required to model Synchronous Machine (SM)

depend on the type of transient study. Saturation effects in

SM are important for EMT analysis. Assuming the SM is

modeled by two damping windings on the q-axis (denoted

by kq1, kq2) and one damper winding (kd) and one field

winding (fd) on the d-axis, (1)-(6) provide the flux-based

equations of SM in state-variable form.

𝐯𝑑𝑞0 = 𝐏(θ)𝐯𝑎𝑏𝑐/V𝑠𝑡𝑎𝑡𝑜𝑟,𝑏𝑎𝑠𝑒 (1) 𝑝𝛙 = ω𝑏(𝐀𝛙+ 𝐮) (2) 𝐀 = −(𝐑𝐋−1 +𝐖) (3)

𝐢 = 𝐋−1𝛙 (4) 𝐢𝑎𝑏𝑐 = 𝐏

−𝟏(θ)𝐢𝑑𝑞0 (5)

𝐢𝑎𝑏𝑐,𝑎𝑐𝑡𝑢𝑎𝑙 = 𝐢𝑎𝑏𝑐 . I𝑠𝑡𝑎𝑡𝑜𝑟,𝑏𝑎𝑠𝑒 (6) where:

𝐮 = [v𝑞 , v𝑑 , v𝑓𝑑 , 0, 0, 0]𝑇

(7)

𝛙 = [ψ𝑞 ,ψ𝑑 , ψ𝑓𝑑 ,ψ𝑘𝑑 ,ψ𝑘𝑞1, ψ𝑘𝑞2]𝑇

(8)

𝐢 = [i𝑞 , i𝑑 , i𝑓𝑑 , i𝑘𝑑 , i𝑘𝑞1, i𝑘𝑞2]𝑇

(9)

𝐢𝑑𝑞0 = [−i𝑞 ,− i𝑑 , 0]𝑇

(10)

𝐑 = diag(R𝑎 , R𝑎, R𝑓𝑑 , R𝑘𝑑 , R𝑘𝑞1, R𝑘𝑞2) (11)

In the above equations, the operator 𝑝 is 𝑑

𝑑𝑡 , the vector

𝐯𝑎𝑏𝑐 is the terminal voltage, 𝐯𝑑𝑞0 is the voltage in dq

frame, ω𝑏 is the base angular velocity, 𝐏(θ) is the Park’s

transformation, vectors 𝐮, 𝐢, and 𝛙 denote the stator and

rotor voltages, currents, and flux linkages in the dq frame

and 𝐢𝑎𝑏𝑐 is the stator current. 𝐖6×6 is the rotor speed-

dependent matrix; all elements are zero except 𝑤[1,2] =𝜔𝑟 and 𝑤[2,1] = −𝜔𝑟, 𝐋6×6 is the symmetrical matrix of

inductances in the rotor reference frame, 𝐑6×6 is the stator

and rotor resistance matrix.

The details of saturation effects modeling in the dq axes

are explained in (Karaagac et al. 2011). In magnetic

saturation modeling, the following assumptions are made:

(1) The leakage flux saturation and cross saturation are

ignored. It means only magnetizing inductances, L𝑚𝑑 and

L𝑚𝑞 are saturable. (2) Saturation is determined by the air-

gap flux linkage. (3) The sinusoidal distribution of the

magnetic field over the face of the pole is unaffected by

saturation.

Since the saturation relationship between the total air-

gap flux, ψ𝑇, and the magnetomotive force under loaded

conditions is assumed to be the same as at no-load

conditions, therefore magnetic saturation of stator and

rotor iron can be modeled by the no-load saturation curve which is characterized by a piecewise linear graph

(Karaagac et al. 2017).

Consequently, the mathematical model of saturation is

introduced by:

ψ𝑇 = 𝑓(ψ𝑇,𝑢𝑠) = 𝑓 (√ψ𝑚𝑑,𝑢𝑠2 +ψ𝑚𝑞,𝑢𝑠

2 ) (12)

ψ𝑚𝑑,𝑢𝑠 = L𝑚𝑑,𝑢𝑠i𝑚𝑑

i𝑚𝑑 = i𝑑 + i𝑓𝑑 + i𝑘𝑑 (13)

ψ𝑚𝑞,𝑢𝑠 = L𝑚𝑞,𝑢𝑠i𝑚𝑞

i𝑚𝑞 = i𝑞 + i𝑘𝑞1 + i𝑘𝑞2 (14)

where ψ𝑇,𝑢𝑠 is the total unsaturated air-gap flux, ψ𝑚𝑑,𝑢𝑠 and ψ𝑚𝑞,𝑢𝑠 are the unsaturated magnetizing flux linkages,

L𝑚𝑑,𝑢𝑠 and L𝑚𝑞,𝑢𝑠 are the unsaturated magnetizing

inductances, and i𝑚𝑑 and i𝑚𝑞 are the magnetizing

currents; each on the dq axis, respectively. Throughout the

paper, the subscript sat and us mean saturated and

unsaturated, respectively.

The value of saturated magnetizing flux linkages on the

dq axis (ψ𝑚𝑑,𝑠𝑎𝑡 and ψ𝑚𝑑,𝑠𝑎𝑡) can be corrected by a ratio

of corresponding unsaturated values as illustrated in

Figure 1.a. In EMTP, the magnetic saturation is

represented by a piecewise linear curve as sketched in

Figure 1.b. For the jth operating segment, ψ𝑇 is given by:

Electromagnetic Transient Simulation of Large Power Networks with Modelica

278 Proceedings of the 14th International Modelica ConferenceSeptember 20-24, 2021, Linköping, Sweden

DOI10.3384/ecp21181277

ψ𝑇 = ψ𝑘𝑗 + b𝑗ψ𝑇𝑢 (15)

= ψ𝑘𝑗 + b𝑗L𝑚𝑑,𝑢𝑠i𝑇

i𝑇 = √i𝑚𝑑2 + (

L𝑚𝑞,𝑢𝑠

L𝑚𝑑,𝑢𝑠)2

i𝑚𝑞2 (16)

b𝑗 =L𝑚𝑑,𝑠𝑎𝑡𝑗

L𝑚𝑑,𝑢𝑠 (17)

where b𝑗 is the saturation factor and ψ𝑘𝑗 is the residual

flux. The saturated values L𝑚𝑑,𝑠𝑎𝑡 and L𝑚𝑞,𝑠𝑎𝑡 are

computed as:

L𝑚𝑑,𝑠𝑎𝑡 = b𝑗L𝑚𝑑,𝑢𝑠

L𝑚𝑞,𝑠𝑎𝑡 = b𝑗L𝑚𝑞,𝑢𝑠 (18)

For a salient pole machine, because of large airgap path

along the q-axis, it is only required to correct the ψ𝑚𝑑;

thus:

L𝑚𝑑,𝑠𝑎𝑡 = b𝑗L𝑚𝑑,𝑢𝑠

L𝑚𝑞,𝑠𝑎𝑡 = L𝑚𝑞,𝑢𝑠 (19)

Figure 2 demonstrates the solution procedure for the

electrical equations of SM. In the case of no saturation,

the relationship between field current (i𝑓𝑑) and terminal

voltage (v𝑡) is linear; therefore, the magnetizing

inductances in (20) are constant (q𝑗 = d𝑗 = 1). If

saturation is selected, it is required to compute the

magnetizing inductances at each time point; thus, 𝐋 is

time-variant (q𝑗 = d𝑗 = b𝑗 for round rotor and q𝑗 = 0,

d𝑗 = b𝑗 for salient pole machine). This method results in

implicit equations requiring an iterative solution.

The model discussed above has been implemented for

the first time in Modelica. The model code is illustrated in

Figure 3. The declaration of variables and the conversion

of operational parameters to the standard ones are hidden

to conserve space and only the equation section is

demonstrated. The terminal voltages of SM are

represented by Pk.pin[1].v, Pk.pin[2].v and

Pk.pin[3].v for the phases a, b and c, respectively.

P(theta) represents a pre-defined function for the Park’s

transformation calculations.

Equation (2) is used as a differential equation for the

implemented model; the state vector Phi represents the

flux linkages and the input vector u the voltages. The

system matrix A is time-variant and computed as per (3).

The matrix of parameters for representation of

saturation, SD, is given by a 2-by-n matrix, where n is the

number of points taken from the no-load saturation curve.

The first row of this matrix contains the values of field

currents (physical value), while the second row contains

values of corresponding terminal voltages (per unit).

LinearInterplate(SD1PU, SD2PU, iT) is a function to

interpolate the iT by the two vectors of field current

(SD1PU) and voltage (SD2PU). These two vectors are

calculated in the non-reciprocal per unit. The function

returns the total flux (PhiT) and Lmdsat which the latter is

used for calculation of coefficient b as per (17). The

stator physical currents are represented by Pk.pin[1].i,

Pk.pin[2].i and Pk.pin[3].i for the phases, a, b and c,

respectively.

Other pieces of code represent the mechanical

equations of SM which are not discussed in this paper.

𝐋 =

(

L𝑙𝑠 + q𝑗L𝑚𝑞,𝑢𝑠 0 0 0 q𝑗L𝑚𝑞,𝑢𝑠 q𝑗L𝑚𝑞,𝑢𝑠0 L𝑙𝑠 + d𝑗L𝑚𝑑,𝑢𝑠 d𝑗L𝑚𝑑,𝑢𝑠 d𝑗L𝑚𝑑,𝑢𝑠 0 0

0 d𝑗L𝑚𝑑,𝑢𝑠 L𝑙𝑓𝑑 + d𝑗L𝑚𝑑,𝑢𝑠 d𝑗L𝑚𝑑,𝑢𝑠 0 0

0 d𝑗L𝑚𝑑,𝑢𝑠 d𝑗L𝑚𝑑,𝑢𝑠 L𝑙𝑘𝑑 + d𝑗L𝑚𝑑,𝑢𝑠 0 0

q𝑗L𝑚𝑞,𝑢𝑠 0 0 0 L𝑙𝑘𝑞1 + q𝑗L𝑚𝑞,𝑢𝑠 q𝑗L𝑚𝑞,𝑢𝑠q𝑗L𝑚𝑞,𝑢𝑠 0 0 0 q𝑗L𝑚𝑞,𝑢𝑠 L𝑙𝑘𝑞2 + q𝑗L𝑚𝑞,𝑢𝑠)

(20)

Eq. 2

vdq

ψ

AW

saturation

L-1

inv(*)

Eq. 4

no

yes

LEq. (20)

start

R

P(θ)vabc

iabc

wit

h

satu

rati

on

wit

hout

satu

rati

on

state

-vari

able

Eq

uati

on

s

qj=1, dj=1

Round

rotor

dj=qj=bj

dj=bj, qj==0

Eq. 16

idq P

-1(θ)

i d,q

,fd,k

d,k

q1,k

q2

?

iT

yes

nobj

id,q,fd,kd,kq1,kq2

Lmd,us

Lmq,us

L-1

RL-1

×

--

Figure 2. Solution procedure of synchronous machine with/without magnetic saturation in Modelica (Only electrical

equations are demonstrated).

iT

ψT

ψk2

ψk3

Lmd,sat2

Lmd,sat3

ψT

ψT,us

ψmd,us

ψm

q,u

s

ψmd,sat

ψm

q,s

at

(a) (b)

Lmd,sat1=Lmd,us

ib1

ψb1

ψb2

ib2

Lmd,us imd

Lm

q,u

s i m

qLmd,sat imd

Figure 1. (a): Saturated and unsaturated magnetizing flux linkages in the dq axes of a synchronous machine. (b): Magnetic

saturation characteristic (piecewise-linear approximation).

Session 4A: Applications (2)

DOI10.3384/ecp21181277


279

model SM "Synchronous Machine 6 order including

saturation"

// Declaration of variables and parameters are hidden

due to space limitations.

equation

// Conversion of terminal voltage to pu

vabc= {Pk.pin[1].v,Pk.pin[2].v,Pk.pin[3].v} /Vsbase;

// Conversion from abc frame to dq0 frame

vdq0 = P(theta)*vabc;

// State space electrical equations

der(Phi) = Wb * (A * Phi + u);

A = -(R * inv(L) + W);

i = inv(L) * Phi;

// Implementation of magnetic saturation

imd = Ip[2] + Ip[3] + Ip[4]; //imd = id + ifD + ikd

imq = Ip[1] + Ip[5] + Ip[6]; //imq = iq + ikq1+ ikq2

iT = sqrt(imd^2 + (Lmqus/Lmdus)^2 * imq^2);

(PhiT,Lmdsat) = LinearInterpolation(SD1pu,SD2pu, iT);

b = Lmdsat / Lmdus;

if Sauration then

if RoundRotor then

q=b;

d=b;

else

q=0;

d=b;

end if;

else

q=1;

d=1;

end if;

Lq = Lls + q * Lmqus;

Ld = Lls + d * Lmdus;

Lffd = Llfd + d * Lmdus;

Lkdkd = Llkd + d * Lmdus;

Lkq1kq1 = Llkq1 + q * Lmqus;

Lkq2kq2 = Llkq2 + q * Lmqus;

L= [ Lq , 0 , 0 , 0 ,q*Lmqus ,q*Lmqus ;

0 , Ld , d*Lmdus, d*Lmdus, 0 , 0 ;

0 , d*Lmdus, Lffd , d*Lmdus, 0 , 0 ;

0 , d*Lmdus, d*Lmdus, Lkdkd , 0 , 0 ;

q*Lmqus , 0 , 0 , 0 ,Lkq1kq1 , q*Lmqus;

q*Lmqus , 0 , 0 , 0 , q*Lmqus, Lkq2kq2];

// Conversion from dq0 to abc frame

iabc = inv(P(theta))* idq0;

// Calculations of actual Terminal current

Pk.pin[1].i = -iabc[1] * Isbase;



// Mechanical equations

Te = Phi[2] * idq0[1] - Phi[1] * idq0[2];

Tnet = Tm - Te - D * dw;

Tm = Pm_pu / Wr;

der(dw) = Tnet * (1 / 2 / H);

Wr = 1 + dw;

der(d_theta)= dw * Wb;

theta = d_theta + Wb * time;

// where

// u = {Vq , Vd , Vfd , Vkd , Vkq1 , Vkq2 }

u = {vdq0[1], vdq0[2], vfd , 0 , 0 , 0 };

// Phi = {Phiq , Phid , Phifd , Phikd , Phikq1,Phikq2 }

Phi = {Phi[1], Phi[2], Phi[3], Phi[4], Phi[5] , Phi[6]};

// i = {iq , id , ifd , ikd , ikq1 , ikq2 }

i = {i[1] , i[2] , i[3] , i[4] , i[5] , i[6] };

// Change of sign due to generating mode

idq0 = {-i[1], -i[2], 0};

W[6, 6] = [ 0 , Wr , 0 , 0 , 0 , 0 ;

-Wr , 0 , 0 , 0 , 0 , 0 ;

0 , 0 , 0 , 0 , 0 , 0 ;

0 , 0 , 0 , 0 , 0 , 0 ;

0 , 0 , 0 , 0 , 0 , 0 ;

0 , 0 , 0 , 0 , 0 , 0 ];

R[6, 6] = diagonal({Rs, Rs, Rfd, Rkd, Rkq1, Rkq2});

end SM;

Terminal voltage

Eq.(20)

Eq.(15)Eq.(17)

Eq.(16)

Figure 3. Implementation of synchronous machine model with

magnetic saturation in Modelica. The saturation formulation is distinguished with the blue dashed frame.

2.2 Nonlinear Arrester Modeling

Surge arresters protect the insulation of equipment, e.g., transformers in electrical systems against overvoltage

transients caused by lightning or switching surges. The

voltage and current characteristic of a gapless metal-oxide

surge arrester as illustrated in Figure 4 is a severely

nonlinear resistor with an infinite slope in the normal

operation region and an almost horizontal slope in the

protection region (temporary and lightning overvoltages).

In EMTP, the nonlinear resistance is represented by the

following power function:

i𝑘𝑚 = 𝑝𝑗 (v𝑘𝑚V𝑟𝑒𝑓

)

𝑞𝑗

(21)

where i𝑘𝑚 and v𝑘𝑚 are arrester current and voltage, 𝑗 is

the segment number starting at the voltage V𝑚𝑖𝑛𝑗,

multiplier 𝑝𝑗 and exponent 𝑞𝑗 are coefficients defined for

each V𝑚𝑖𝑛𝑗 and V𝑟𝑒𝑓 is the arrester reference voltage. A

linear function is used for the first segment.

The technique for modeling a nonlinear resistance

(arrester function) is like the one used for the nonlinear

Vmin, 1

Vmin, 2

Vmin, j

Vkm

ikmSymmetricalextension

Vmin, 3

10-2 102 104103

Temporary OVLightning OV

Maximum Continuous voltage

Protection region

23 4

5

1

Figure 4. Voltage-current characteristic of ZnO surge arrester

and operating regions.

model ZnoArrester ZnO arrester model in Modelica

extends Modelica.Electrical.Analog.Interfaces.OnePort;

parameter Real Vref = 516000 Reference voltage

//Exponential segments before flashover

parameter Real T[:, 3] "multiplier p, Exponent q, Vmin_pu";

protected

final parameter Real[:] p = T[:, 1];

final parameter Real[:] q = T[:, 2];

final parameter Real[:] V_min = T[:, 3]*Vref;

equation

i_km = ExponentialInterpolate(V_min, p, q, Vref, v_km);

end ZnoArrester;

Inheritance of OnePort partial class

Constructive equation of surge arrester

Internal parameters of model

Parameters of model

Zno

vkm

ikm

vk vm

Figure 5. Implementation of the ZnO surge arrester model in

Modelica environment.



DOI10.3384/ecp21181277

inductance. Figure 5 illustrates the code for the

implementation of the surge arrester. The parameters of

𝑝𝑗, 𝑞𝑗 and V𝑚𝑖𝑛𝑗 are defined by n -by-3 matrix, T.

ExponentialInterpolate() is a function defined by the

specific class function, where the operating voltage is

searched for the appropriate segment, j. Then, the value of

i𝑘𝑚 is exponentially interpolated using (21). The

properties of partial class OnePort are inherited to apply

the appropriate equations of one-port devices.

As one can see, the implementation of the model is very

straightforward, there is no limitation for connection of

this model in series to current sources, or inductors.

Solutions converge for very small time steps in the range

of nanoseconds without any numerical errors (Masoom et

al. 2021).

3 Model Verification and Validation

This section presents simulation results of the modified

IEEE 118-bus benchmark (Haddadi, Mahseredjian, et al.

2018) which is used to validate the accuracy of the

proposed models. The same test case is also simulated

with EMTP.

Figure 6.a shows the schematic diagram of the IEEE

118-bus network sketched using the developed EMT

library in Modelica. A user-friendly Graphical User

Interface (GUI) with an illustrative icon is designed for

each component model for entering the parameters and

drawing networks easily. The physical connection of

components is carried out by interconnecting the

terminals of appropriate components.

The IEEE-118 bus circuit consists of 54 generating

units with controls (a few power plants contain more than

one SM; the total number of SMs is 69), 177 transmission

lines (RL coupled), 9 three-winding grid transformers,

145 two-winding transformers (91 Yd1-connected load-

serving transformers+ 54 generator transformers), and 91

three-phase loads. The voltage levels are 345kV

transmission, 138kV sub-transmission, 25kV distribution,

and {20, 15, 10.5} kV generation. The network includes

(a)(b)

+Line_70

_75

CP

Portsmth_138_070

SthPoint_138_075

+BRk

+BRm

+

SW

100ms|300ms|0 P_b

P_c

Load075

12-30

LoadBUS

138/25125MVA

LoadTransfo75

47.94MW11.01MVAR25kVRMSLL

m-end

k-end

-1|200ms|0

450ms|1E15s|0

(c)

Faulted zone

m

m

Power plant

PQ Load

138kV Network

345kV Network

PI Model of TL

Rf

R=1

Measuring Point

+ PI_ 3PH L ine _1 _2

Rive rs de _1 38 _0 01

+

PI_3 PH

Lin e_1 _3

Po kag on _1 38 _0 02

lo ad 00 2

Hick ryC k_1 38 _ 00 3

+

PI_3 PH

Lin e_3 _5

Shu nt_Re acto r g

Oliv e_ 13 8_ 00 5

+ PI_ 3PH L ine _2 _1 2

+ PI_ 3PH L ine _3 _1 2

TwinBr ch _1 38 _0 12

+ PI_ 3PH L ine _1 1_ 12

So uth Bnd _1 38 _0 11

+ PI_ 3PH L ine _5 _1 1

+ PI_ 3PH L ine _4 _1 1

NwC ar lsl_1 38 _ 00 4

+

PI_3 PH

Lin e_4 _5

lo ad 00 3

lo ad 01 1

G n wCa rlsl_ Con d

G twin Brc h_ PP

+ PI_ 3PH L ine _1 2_ 11 7

Co re y_ 13 8_ 11 7

+

PI_3 PH Lin e_7 _12

+

PI_3 PH Lin e_1 2_1 6

Ja cksn Rd _1 38 _0 07

+ PI_ 3PH L ine _6 _7

Ka nka ke e_ 13 8_ 00 6

+ PI_ 3PH L ine _5 _6

Co nc or d_ 13 8_ 01 3

+ PI_ 3PH L ine _1 1_ 13

FtWa yne _1 3 8_ 01 5

+ PI_ 3PH L ine _1 3_ 15

+

PI_3 PH Lin e_1 4_1 5

Go sh en Jt_ 13 8_ 01 4

+ PI_ 3PH L ine _1 2_ 14

+ PI_ 3PH L ine _1 5_ 33

Ha vilan d _1 38 _0 33

+ PI_ 3PH L ine _1 5_ 17

So re nso n_ 13 8_ 01 7

+ PI_ 3PH L ine _1 6_ 17

NE_ 13 8_ 01 6

+ PI_ 3PH L ine _1 7_ 31

+

PI_3 PH Lin e_1 7_1 13

De er Crk 2_ 13 8_ 11 3

+ PI_ 3PH L ine _3 1_ 32

+

PI_3 PH Lin e_3 2_1 13 De lawa re _ 13 8_ 03 2 De er Crk _1 38 _0 31 Gr an t_ 13 8_ 02 9

+ PI_ 3PH L ine _2 9_ 31

+ PI_ 3PH L ine _2 7_ 32

+

PI_3 PH Lin e_2 8_2 9

M ullin _1 38 _0 28

+

PI_3 PH Lin e_2 7_2 8

M ad iso n_ 13 8_ 02 7

+ PI_ 3PH L ine _3 2_ 11 4

WM ed fo rd _1 38 _1 14

+ PI_ 3PH L ine _1 14 _1 15

M ed fo rd _1 38 _1 15

+ PI_ 3PH L ine _2 7_ 11 5

+

PI_3 PH Lin e_0 8_0 9

+ PI_ 3PH L ine _0 8_ 30

1

Olive _T1 2 3

Bre e d_ 34 5_ 01 0

Oliv e_ 34 5_ 00 8

+

PI_3 PH Lin e_9 _10

Be qu ine _3 45 _0 09

+ PI_ 3PH L ine _2 5_ 27

+ PI_ 3PH L ine _3 3_ 37

Ea stL ima _1 38 _0 37

+ PI_ 3PH L ine _3 7_ 39

NwL ibr ty_ 1 38 _0 39

+ PI_ 3PH L ine _3 9_ 40

We stEn d_ 13 8_ 04 0

+

PI_3 PH Lin e_3 7_4 0

+ PI_ 3PH L ine _4 0_ 41

+ PI_ 3PH L ine _4 0_ 42

ST iffin _1 38 _0 41 Ho wa rd _1 38 _0 42

+ PI_ 3PH L ine _4 1_ 42

Wo ost er _1 38 _0 53

+ PI_ 3PH L ine _5 3_ 54

Tor re y_1 3 8_ 05 4

+ PI_ 3PH L ine _5 4_ 56

Su nn ysd e_ 13 8_ 05 6

+ PI_ 3PH L ine _5 5_ 56

Wa ge nh ls_ 13 8_ 05 5

+ PI_ 3PH L ine _5 5_ 59

T idd _1 38 _0 59

+ PI_ 3PH L ine _5 6_ 59 _1

+ PI_ 3PH L ine _5 6_ 59 _2 +

PI_3 PH Lin e_5 6_5 8

WNw Phil2 _1 38 _0 58

+ PI_ 3PH L ine _5 6_ 57

WNw Phil1 _1 38 _0 57

+

PI_3 PH Lin e_5 0_5 7

WCa m br dg _1 38 _0 50

+

PI_3 PH Lin e_5 1_5 8

Ne wcm rs t_1 38 _ 05 1

+

PI_3 PH Lin e_5 2_5 3

SCo sh oct _1 38 _0 52

+ PI_ 3PH L ine _5 1_ 52

+

PI_3 PH Lin e_4 9_5 0

Ph ilo_ 13 8_ 04 9

+

PI_3 PH Lin e_4 9_5 1

+

PI_3 PH Lin e_4 9_5 4_2

+

PI_3 PH Lin e_4 9_5 4_1

+ PI_ 3PH L ine _4 2_ 49 _1

+ PI_ 3PH L ine _4 2_ 49 _2

+ PI_ 3PH L ine _3 4_ 37

Ro ckh ill_1 38 _0 34

+ PI_ 3PH L ine _3 4_ 43

SKe nto n_ 13 8_ 04 3

+ PI_ 3PH L ine _4 3_ 44

WM Ver no n_ 1 38 _0 44

+

PI_3 PH Lin e_4 4_4 5

NNe wa rk_ 13 8 _0 45

WL an cst_ 1 38 _0 46

+ PI_ 3PH L ine _4 5_ 46

+ PI_ 3PH L ine _4 5_ 49

Cr oo ksvl_ 1 38 _0 47

+ PI_ 3PH L ine _4 6_ 47

+ PI_ 3PH L ine _4 7_ 49

+ PI_ 3PH L ine _4 6_ 48

Zan es vll_1 38 _0 48

+ PI_ 3PH L ine _4 8_ 49

+

PI_3 PH Lin e_3 4_3 6

Ste rlin g_ 13 8_ 03 6 We stL ima _1 3 8_ 03 5 L inco ln_ 13 8_ 01 9

+ PI_ 3PH L ine _1 5_ 19

+ PI_ 3PH L ine _1 9_ 34

+ PI_ 3PH L ine _3 5_ 36

+ PI_ 3PH L ine _3 5_ 37

M cKinle y_ 13 8_ 01 8

+

PI_3 PH Lin e_1 8_1 9

+ PI_ 3PH L ine _1 7_ 18

1

Sor enso n_T1

2 3

So re nso n_ 34 5_ 03 0

+

PI_3 PH Lin e_2 6_3 0

Tan nr sCk_ 3 45 _0 26

1 YgYgD

2 3

Tan nr sCk_ 1 38 _0 25

+

PI_3 PH Lin e_1 9_2 0

Ad am s_ 13 8_ 02 0

Ja y_ 13 8_ 02 1

+

PI_3 PH Lin e_2 0_2 1

Ra nd olp h_ 13 8_ 02 2

+

PI_3 PH Lin e_2 1_2 2

Co llCrn r_ 13 8 _0 23

+

PI_3 PH Lin e_2 2_2 3

+ PI_ 3PH L ine _2 3_ 25

+ PI_ 3PH L ine _2 3_ 32

T re nto n _1 38 _0 24

+ PI_ 3PH L ine _2 3_ 24

Hillsb ro _1 3 8_ 07 2

+

PI_3 PH Lin e_2 4_7 2

NPo rts mt _1 38 _0 71

+ PI_ 3PH L ine _7 1_ 72

+

PI_3 PH Lin e_7 1_7 3

Sa rg en ts_ 13 8_ 07 3

Po rtsm th _1 3 8_ 07 0

+

PI_3 PH Lin e_7 0_7 1

+ PI_ 3PH L ine _2 4_ 70

+

PI_3 PH Lin e_7 0_7 4

Be llefn t_ 13 8_ 07 4

+

PI_3 PH Lin e_7 4_7 5

Sth Poin t_ 13 8_ 07 5

+

PI_3 PH Lin e_7 0_7 5

+

PI_3 PH Lin e_4 9_6 9

Sp or n_ 13 8_ 06 9

+ PI_ 3PH L ine _4 7_ 69

+ PI_ 3PH L ine _6 9_ 70

+

PI_3 PH Lin e_6 9_7 5

Sp or n_ 34 5_ 06 8

1 YgYgD1

2 3

Ka na wha _1 38 _0 80

+

PI_3 PH Lin e_6 8_1 16 Ca pit lH l_1 38 _0 79 Kyg er Crk _3 45 _1 16

Tur ne r_ 13 8_ 07 7

+ PI_ 3PH L ine _7 5_ 77

+ PI_ 3PH L ine _7 5_ 11 8

WHu nt ng d_ 13 8_ 11 8 Da rr ah _1 38 _0 76

+ PI_ 3PH L ine _7 6_ 11 8

+

PI_3 PH Lin e_7 6_7 7

+

PI_3 PH Lin e_6 9_7 7

Ch em ica l_1 38 _0 78

+

PI_3 PH Lin e_7 7_7 8

+ PI_ 3PH L ine _7 8_ 79

+ PI_ 3PH L ine _7 9_ 80

+ PI_ 3PH L ine _7 7_ 80 _1

+ PI_ 3PH L ine _7 7_ 80 _2

+

PI_3 PH Lin e_8 0_9 7

Su nd ial_ 13 8_ 09 7

+ PI_ 3PH L ine _9 6_ 97

Ba ileys v_1 38 _0 96

+

PI_3 PH Lin e_8 0_9 6

L og an _1 38 _0 82

+

PI_3 PH Lin e_7 7_8 2

+ PI_ 3PH L ine _8 2_ 96

+

PI_3 PH Lin e_9 5_9 6

Ca rsw ell_ 13 8_ 09 5 Switc hb k_ 13 8_ 09 4

+ PI_ 3PH L ine _9 4_ 95

+ PI_ 3PH L ine _9 4_ 96

+ PI_ 3PH L ine _9 4_ 10 0

Gle n Lyn _1 38 _1 00 Ha nc ock _1 38 _1 04

+ PI_ 3PH L ine _1 00 _1 04

+ PI_ 3PH L ine _1 04 _1 05

Ro an ok e_ 13 8_ 10 5

+ PI_ 3PH L ine _1 05 _1 07

Re us en s_1 38 _1 07

+ PI_ 3PH L ine _9 3_ 94

Taze we ll_1 38 _0 93

Sa ltvlle _1 38 _0 92

+

PI_3 PH Lin e_9 2_9 3

+ PI_ 3PH L ine _9 2_ 94

+ PI_ 3PH L ine _9 2_ 10 0

Bla ine _1 38 _1 08

Fra nklin _1 3 8_ 10 9

+

PI_3 PH Lin e_1 05_ 108

+

PI_3 PH Lin e_1 08_ 109

+ PI_ 3PH L ine _9 2_ 10 2

Sm yth e_ 13 8_ 10 2 Wyt he _1 38 _1 01

+ PI_ 3PH L ine _1 01 _1 02

+ PI_ 3PH L ine _1 00 _1 01

Cla yto r_ 13 8_ 10 3

+ PI_ 3PH L ine _1 00 _1 03

+

PI_3 PH Lin e_1 03_ 104

+ PI_ 3PH L ine _1 03 _1 05

F ield ale _1 38 _1 10

+

PI_3 PH Lin e_1 09_ 110 + PI_ 3PH

L ine _1 03 _1 10

Da nv ille _ 13 8_ 11 2 Da nR iv er _1 38 _1 11

+ PI_ 3PH L ine _1 10 _1 11

+ PI_ 3PH L ine _1 10 _1 12

+

PI_3 PH Lin e_9 1_9 2

Ho lsto nT_ 13 8_ 09 1

+ PI_ 3PH L ine _8 9_ 92 _1

Clin chR v_1 3 8_ 08 9

+ PI_ 3PH L ine _8 9_ 92 _2

F re mo nt _1 38 _0 88 Be ave rC k_1 38 _0 85

+ PI_ 3PH L ine _8 5_ 89

+ PI_ 3PH L ine _8 8_ 89

+ PI_ 3PH L ine _8 5_ 88

Ho lsto n_ 13 8_ 09 0

+

PI_3 PH Lin e_8 9_9 0_1

+

PI_3 PH Lin e_8 9_9 0_2

+ PI_ 3PH L ine _9 0_ 91

Pin evlle _1 38 _0 87

Ha za rd _1 38 _0 86

+

PI_3 PH Lin e_8 5_8 6

+ PI_ 3PH L ine _8 6_ 87

Be tsyL ne _1 38 _0 84

+

PI_3 PH Lin e_8 4_8 5

Sp rig g_ 13 8_ 08 3

+ PI_ 3PH L ine _8 3_ 84

+ PI_ 3PH L ine _8 3_ 85

+ PI_ 3PH L ine _8 2_ 83

Clo ve rd l_ 1 38 _1 06

+ PI_ 3PH L ine _1 00 _1 06

+ PI_ 3PH L ine _1 05 _1 06

+ PI_ 3PH L ine _1 06 _1 07

Bra d ley_ 13 8_ 09 8

+ PI_ 3PH L ine _9 8_ 10 0

+ PI_ 3PH L ine _8 0_ 98

+ PI_ 3PH L ine _9 9_ 10 0

Hin to n_ 13 8_ 09 9

+ PI_ 3PH L ine _8 0_ 99

1

Kan awha _T1

2 3

Ka na wha _3 45 _0 81

+ PI_ 3PH L ine _6 8_ 81

M usk ng um _ 34 5_ 06 5

M usk ng um _ 13 8_ 06 6

+ PI_ 3PH L ine _6 5_ 68

1

M uskng um _T1

2 3

+ PI_ 3PH L ine _4 9_ 66 _2

+ PI_ 3PH L ine _4 9_ 66 _1

Na triu m _1 38 _0 62

+ PI_ 3PH L ine _6 2_ 66

+

PI_3 PH Lin e_6 6_6 7

Su mm er fl_ 13 8_ 06 7

+ PI_ 3PH L ine _6 2_ 67

Ka mm er _1 3 8_ 06 1

+

PI_3 PH Lin e_6 1_6 2

+

PI_3 PH Lin e_6 4_6 5

Ka mm er _3 4 5_ 06 4

1 Ka mm er _T1

2 3

+ PI_ 3PH L ine _5 9_ 61

T idd _3 45 _0 63

+ PI_ 3PH L ine _5 9_ 60

SWKa mm e r_ 13 8_ 06 0

+

PI_3 PH Lin e_6 0_6 1

+

PI_3 PH Lin e_6 0_6 2

1 YgYgD3

2 3

+

PI_3 PH Lin e_6 3_6 4

+ PI_ 3PH L ine _3 8_ 65

1

EastL ima _T1

2 3

Ea stL ima _3 45 _0 38

e ar th

+ PI_ 3PH L ine _3 0_ 38

e ar th1

e ar th2

e ar th3

e ar th4

e ar th5

e ar th6

e ar th7

e ar th8

lo ad 11 7

lo ad 03 3

lo ad 01 3 lo ad 00 7

lo ad 01 6

lo ad 01 7

lo ad 02 9

lo ad 11 5 lo ad 11 4

lo ad 02 8

lo ad 04 1

lo ad 03 9

Shu nt_Re acto r_ 37

G

lo ad 02 0

lo ad 02 1

lo ad 02 2

lo ad 02 3

G ka nk ake e_ Co nd G

ftW ayn e_ Co nd

G lin coln _C on d

G m cKinle y_ Co nd

G ta nn rs Ck1 38 _PP

G m ad iso n_ Con d

G b re ed _PP

G d ee rCr k_ PP

G d ela war e_ Co nd

G o live_ Co nd

G p ine vlle_ PP

G h olst on _Co nd

G h olst on T_ Co nd

G d an Rive r_ PP

G fie lda le_ Co nd

G d an ville_ Co nd

G cla yto r_ PP

G b ea ver Ck_ Co nd

G clin ch Rv_ PP

G sa ltvlle _Co n d

G g len Lyn _PP G

h an coc k_C on d

G ro a no ke_ Co nd

G re u sen s_ Con d

G tu rn er _C on d

G b elle fnt _Co nd

G sa rg en ts_ Co nd

G h ills br _Co n d

G tr en to n_ Con d G

p or tsm th_ Co nd

G kyg e rCr k_PP

G m usk ng um 3 45 _PP

G m usk ng um 1 38 _PP G

n atr ium _ Con d

G ka mm e r_ PP

G wL an cst _PP

G we stEn d_ Co nd

G h owa rd _C on d

G to rr ey_ PP

G su nn ysd e_ Co nd

G wa ge nh ls_ Con d

G tid d_ PP

G h into n_ Co nd

lo ad 07 5

lo ad 08 6

lo ad 08 8

lo ad 10 1 lo ad 10 2

lo ad 10 9

lo ad 10 8 lo ad 08 4 lo ad 09 3

lo ad 09 5

lo ad 09 4

lo ad 09 6

lo ad 09 7

lo ad 09 8

lo ad 10 6

G d ar ra h_ Con d

G ka na wh a_ PP

lo ad 11 8

lo ad 07 8

lo ad 07 9

G sp or n_ PP

G p hilo _PP

lo ad 06 7

lo ad 04 7

lo ad 04 5

lo ad 04 8

lo ad 04 3

lo ad 04 4 lo ad 05 0 lo ad 05 1

lo ad 05 7 lo ad 05 8 lo ad 05 2

lo ad 05 3

lo ad 06 0

lo ad 08 3

lo ad 08 2

lo ad 03 5 G

ste rlin g _3 6_ Con d

G ro ckh ill_C on d

G riv er sde _ Con d

lo ad 01 4

BR

BR

To=0 .3 s

SW

Tc=0 .1 s

k = 3 p lug To Pin

k = 2 p lug To Pin1

R1

R= 0.00 01*{ 1,1, 1}

R

R= 1

ZnO1

GPortsmth_Cond

Figure 6. (a): IEEE 118-bus Network including 177 PI-section models of TL sketched using the Modelica GUI. (b): the faulty

zone; a phase-b-to-phase-c fault at k-end of Line_70_75. The powerplant “Portsmth_Cond” is selected for validation of SM with saturation in Case 2, Surge arrester ZnO1 is inserted in the circuit only for Case 3. (c): the sub-circuit of Load75 including a

saturable transformer model and constant-impedance model of load.


DOI10.3384/ecp21181277


281

519 nonlinear inductances and 1909 RLC elements. All

SMs use a single-mass Wye-grounded model including

the normalized saturation characteristics represented by 7

points. The SM control systems consist of exciter type ST1, steam turbine and governor type IEESGO,

synchronous machine phasor and power system stabilizer

type PSS1A. The model of all three-phase transformers

consists of three single-phase units. The nonlinear

magnetization branch is placed on the high-voltage side.

The model uses a piecewise linear current-flux curve

defined by 8 points (in the positive part of the symmetric

characteristic) to represent saturation. All loads are

represented by a constant impedance model.

The transmission lines (TL) are modeled using pi-

sections. The constant parameter line model with

propagation delay is simulated slowly, owing to the very

high computational cost of the Modelica built-in delay

operator. Simulation of the network starts with zero initial

states.

3.1 Case 1: Phase-to-Phase Fault Analysis

For creating a transient disturbance, (see Figure 6.b), a

temporary phase-to-phase fault with a fault resistance of

1 Ω is applied on the phases ‘b’ and ‘c’ of “Line_70_75”

at t =100 ms followed by the isolation of the line at t= 200

ms (i.e. the breakers BRm and BRk open simultaneously

after 6 cycles). The fault is cleared at t = 300 ms, then the

line is reconnected at t = 450 ms.

Re-energizing the TL introduces high-frequency

transient oscillations and allows to investigate the

accuracy of transformer models in nonlinear regions.

For this purpose, the curve of flux versus current for

LoadTransfo75 which is located near the faulty line is

compared with EMTP as well.

Numerical tests are performed using the variable-step

DASSL solver (Petzold 1982) in ODE mode with the

tolerance of 1e-3 and the maximum integration order of 5

in Dymola 2021x. In EMTP, Trapezoidal/Backward Euler

integrator with the step sizes of 1 µs and 5 µs is employed.

The simulation time is 500 ms. The network model in

Modelica contains 96308 acausal DAEs. The total number

of network nodes and the size of the main system of

equations in EMTP are 2533 and 3773, respectively.

Figure 7.a depicts the voltage waveforms of phases a,

b and c at the k-end of Line_70_75 obtained by the two

simulators with different precisions. An excellent

agreement is observed between the results. Figure 7.b

shows the simulation results for the phases b and c in the

interval of [300, 310] ms, i.e., after the fault is removed.

The results produced by Modelica models are almost

identical to EMTP when step size is 1 µs (black curve),

whilst the high-frequency transient oscillations (f=1820

Hz) are not captured by EMTP when 𝛥𝑡 = 5 μs (blue

curve). Figure 7.c depicts the curves of voltage after the

re-energization of TL. The consistent results between

Modelica and EMTP are observed in this period once

more. The close-up view of the phase a voltage waveform

at the instant of closing the breakers BRm and BRk shows

that Modelica voltage waveform rises precisely at t = 450

ms while in EMTP it goes up in the next time point. The close-up illustrates the discontinuity treatment

discrepancies between the two simulators. This is an

0 100 200 300 400 500-1.5

-1

-0.5

0

0.5

1

1.5

300 302 304 306 308 310-0.1

-0.05

0

0.05

0.1

Vo

ltag

e (p

u)

450 460 470 480 490 500

-1

-0.5

0

0.5

1

445

Time (ms) (c)

EMTP, h:5us{

va vb{ vc}Dymola, Tol 1e-3

va vb vc}

EMTP, h:1us{ va vb vc}

{Dymola, Tol 1e-2 va vb vc}

Line_70_75 energizedFault duration

Line_70_75 disconnected

EMTP, h:1usDymola, Tol:1-3

306 306.5 307 307.5 308-0.01

0

0.01

0.02

0.03

EMTP, h:5us

Dymola, Tol:1-2 EMTP, h:1usDymola, Tol:1-3

EMTP, h:5us

(b)

(a)

phase b

phase c

450.01450 450.005

0

0.5

1

EMTP, h:5us

EMTP, h:1us

f =1820 Hz

450.001

Figure 7. (a): Voltage waveforms of phases a, b and c at the k-end of Line_70_75; (b): comparison of results for the phases b

and c for different solvers’ parameters. (c): voltage waveforms after re-energization of Line_70_75; the close-up at the instant

of closing the breakers BRk and BRm.

Current (A)

0 100 200 300 400 500 600 700 800 900 1000-200

0

200

400

EMTPModelica

0 20 40 60 80 100260

300

340

380

Magnetization data

Mag

neti

zin

g F

lux

(W

b)

Breakpoints

600

Figure 8. Current-Flux curve of magnetization branch in the LoadTransfo75 transformer; close-up of Modelica and EMTP

solutions near to the knee-point.



DOI10.3384/ecp21181277

important issue for the simulation of circuits with high-

frequency switching.

For validating the accuracy of nonlinear components,

the magnetization branch curve of LoadTransfo75 (see

Figure 6.c) is examined in Figure 8. Once again, the

results obtained by the two models show an excellent

agreement and transformer operating points (depicted by

the red dashed line) move on the transformer current-flux

characteristics (distinguished by the solid red line). The

iterative solution allows reproducing the nonlinear

function accurately in both tools.

The number of nonlinear components and control

closed loops has a significant impact on the accuracy and

speed of simulation. For example, simulation of the same

network, that is IEEE 118-bus, jams in Simscape

Electrical Specialized Power Systems (SPS) package

(Simscape Electrical 2020) which is comparable to

Modelica environment in some ways. This package is

based on the state-space representation of the linear

network in a loop with external current sources denoting

the nonlinear components. In Modelica, nonlinear

functions are solved simultaneously through iterative

methods which gives the most accurate results.

Table 1 shows the data and run-times of simulations

carried out in Dymola and EMTP. The CPU times are extracted from the average of 5-times “re-simulations”. In

Dymola, simulation is accomplished with 203034 steps in

371.2 s, which yields 1.83 ms for each time step. EMTP

outperforms Dymola with the ratio of 3.37:1 when the

least error is favorite, i.e., 𝛥𝑡 = 1 μs. Tolerance has a significant impact on the CPU time and

the number of time steps for the DASSL since the local

error is tightly coupled with the logic for selecting the step

size and order of integration. In this experiment, the

simulation is repeated with the tolerance of 1e-2 as well.

It causes a considerable increase in the number of time

steps, Jacobian, and function evaluations. Consequently,

the CPU time increases with the ratio of 4:1, whereas the

accuracy of simulation does not change effectively (see

Figure 7.b). The norm of error between these two

simulations is reported 4.8e-3 for phase b. In both

tolerances, the results are practically identical to EMTP

when 𝛥𝑡 = 1 μs . However, it should be noted that the solution methods

in Modelica and EMTP are fundamentally different, and a

direct comparison of variable step solver with fixed-step

one is not so fair. The time steps selected in Table 1 are

for demonstration/comparison purposes; in reality, it is possible to select even higher time steps without

significant loss of accuracy.

0 100 200 300 400 500

Ter

min

al v

olta

ge (

kV)

0

20

10

-10

-20

Time (ms)

204200 201 202 2030

10

20

295 305 315 325-20-10

0

10

20 with saturationwith saturationwithout saturation

0 100 200 300 400 5000

10

20

30

40

50

Fie

ld C

urre

nt (

A)

50 70 90 100

10

20

30

40

150 170 190 2100

10

20

30

Time (ms)

with saturation

without saturation

without saturation

Three-phase fault

(a) (b)

EMTP Modelica }With saturation{ Modelica }EMTP Without saturation{

Figure 9. (a): Waveform of phase-a stator voltage of SM with and without saturation model; the close-up after load rejection. (b):

field current with and without saturation model; the zoomed views during and after fault.

Term

inal cu

rren

t (k

A)

0 100 200 300 400 500

-10

-5

0

5

10

EMTP with saturation

Modelica with Saturation

EMTP without saturation

Modelica without Saturation

Time (ms)

152 156 160 164 168

-2-10123

420 440 460 480 500-0.1

0

0.1without saturation

with saturation

without saturation

with saturationw/o sat

w. sat.

Figure 10. Phase-a stator current with and without saturation

model; zoomed view after removing the fault and load rejection.

Table 1. Case 1: comparison of simulation performance.

Characteristics Dymola EMTP Solver DASSL Trapezoidal /Backward Euler

Tolerance 1e-3 1e-2

∆𝑡: 1 𝜇𝑠 ∆𝑡: 5 𝜇𝑠 ∆𝑡: 10 𝜇𝑠 ∆𝑡𝑀𝐼𝑁 0.115 𝑓𝑠 0.116 𝑓𝑠

∆𝑡𝑀𝐴𝑋 5.79 𝜇𝑠 0.16 𝜇𝑠

No result points 203035 335261 601757 154367 81 661

No accepted steps 203034 335260 Not applicable

f-evaluations 415437 760052 Not applicable

J-evaluations 7393 337458 Not applicable

CPU time (s) 371.2 1510.6 110.1 44.2 23.5

CPU-time for 1

accepted steps 1.83 𝑚𝑠 4.49 𝑚𝑠 0.18 𝑚𝑠 0.28 𝑚𝑠 0.28 𝑚𝑠

Performance ratio 1 0.24 3.37 8.39 15.79


DOI10.3384/ecp21181277


283

3.2 Case 2: Analysis of Saturation in SM

To verify the validity of the SM model in the saturation

region, a large disturbance including a sudden three-phase

short-circuit fault is applied at t = 50 ms near to the

terminals of SM “Portsmth_Cond” and lasts till t =150

ms. The SM protective relays detect the fault and trip the

generator breaker at t = 200 ms. The parameters of both

solvers are the same as Case 1, e.g., Tol=1e-3 in Dymola

and 𝛥𝑡 = 5 μs in EMTP.

Figure 9.a shows the phase-a stator voltage waveforms

of the SM with and without magnetic saturation. As one

can see, the results obtained from Modelica model are

superimposed on EMTP ones. As time elapses, the

difference between the results obtained by saturated and

unsaturated models is more distinguishable on the voltage

curves.

Figure 9.b depicts the field current curves of the SM

with and without magnetic saturation. It is observed that

the inclusion of saturation has an important impact on the

excitation current needed for the generator operation.

Figure 10 illustrates the phase-a stator current graphs.

As one can see, Modelica model yields precisely the same

results as EMTP for both cases (with and without

saturation). The stator current considering magnetic

saturation is lower than without saturation. It is seen that

the effect of saturation on the current in the sub-transient

state is more than the transient state and the difference

decreases as time elapses.

3.3 Case 3: Lightning

In this case, it is assumed that a lightning strike with the

characteristics of 10kA, 8 /20 µs (see Appendix, equation

(22) for the impulse source model) hits the phase-a of

“Line_70_75” when the network is in steady state at t =

95 ms. The surge arresters (see Appendix, Table 3 for the

parameters) are located on the bus “SthPoint_138_075”,

the nearest place to the high-voltage side of the

transformer “LoadTransfo75” to protect it from transient overvoltages (see Figure 6.b). The simulation is run for

130 ms with the step sizes of 0.1 µs (depicted by black

curve) and 10 µs (depicted by red curve) in EMTP. Other

solvers parameters are like Case 1.

Figure 11 shows the phase-a voltage waveforms of the

arrester ZnO1. As one can see the results obtained from

Modelica arrester model are identical to EMTP when

𝛥𝑡 = 0.1 𝜇𝑠. A high frequency transient (1300 Hz) is

created due to the strike of lightning.

Table 2 compares the performances of simulations in

both tools. In Dymola, simulation is accomplished with

51513 steps in 87.2 s, which yields 1.69 ms for each time

step. In this case, Dymola outperforms EMTP’s best

result, that is when 𝛥𝑡 = 0.1 μs, with the ratio of 5.56:1.

This test case is designed to show the potential

advantages of variable time step solvers over fixed time

step ones (like EMTP). It is designed on the purpose to

illustrate the fact that a very smalltime step used for the

short duration of the very high transient has a penalizing

effect on EMTP, but not on Modelica solver. Modelica

integrator expectedly reverts to a very small time step only

for a short duration. It would have been possible to apply

lightning in EMTP at simulation time t = 0 s, and in which

case the performance results would have been much

better, nevertheless, our demonstration remains valid. A

more practical example is the breaker arc model that also

forces the usage of very small time steps and may be

triggered at any point of time. It will effectively give an

advantage to Modelica since in this case, it is required to

capture longer simulation periods.

Table 2. Case 3: comparison of simulation performance.

Characteristics Dymola EMTP

Solver DASSL Trapezoidal /BE

Tolerance 1e-3

∆𝑡: 0.1 𝜇𝑠 ∆𝑡: 10 𝜇𝑠 ∆𝑡𝑀𝐼𝑁 0.623 𝑝𝑠

∆𝑡𝑀𝐴𝑋 5.56 𝜇𝑠

No result points 51514 1335308 21637

No accepted steps 51513 Not applicable

f-evaluations 105576 Not applicable

J-evaluations 1503 Not applicable

CPU time (s) 87.2 485.6 9.9

CPU-time for 1 accepted steps 1.69 𝑚𝑠 0.36 𝑚𝑠 0.45 𝑚𝑠

Performance ratio 1 0.179 8.8

Conclusion

In this paper, Modelica programming language has been

considered for EMT simulations due to its advantages for

creating models at very high abstraction levels. In this

paper, we have emphasized the modeling of synchronous

machine including magnetic saturation and surge arrester.

These two nonlinear models are validated by comparisons

with EMTP in a large grid (IEEE 118-bus benchmark). It

is shown that high-level modeling in Modelica is very

accurate as compared to EMTP. However, the

performance is not satisfactory, except when variable time

step is used advantageously for very high-frequency

transients of short duration in a long simulation interval.

Nonetheless, in comparison with Simscape Electrical SPS

package, the Modelica package demonstrates an excellent

90 95 100 105 110 115 120 125 130-1

-0.5

0

0.5

1

1.5

Arr

est

er

Vo

ltag

e (

pu

)

95 96 97 98-1

-0.5

0

0.5

1

1.5f=1300 Hz

EMTP{ ModelicaΔt: 0.1µs, Δt: 10 µs}

Time (ms)

9896.4 96.8 97.2 97.6

0.2

0.4

0.6

0.81

Figure 11. Voltage waveform of surge arrester ZnO1 on the bus

SthPoint_138_075, DASLL solver: Tol=1e-3, EMTP solver:

Trapezoidal /BE with ∆t=0.1 μs and 10 μs.



DOI10.3384/ecp21181277

performance in the EMT simulation of large-scale

networks composed of many nonlinearities.

The EMT-type package created by Modelica code is

user-friendly, modular, easily expandable, and

modifiable. It can be used for didactic purposes as well.

Furthermore, the EMT-type models can be used for model

exchange and co-simulation incorporating FMI.

This paper presents useful and practical information on

currently available capabilities with Modelica for EMT

simulation of large-scale grids. Future work will be

oriented toward performance improvements and the

inclusion of new models.

Acknowledgments

We would like to acknowledge the continuous support of

the R&D team of RTE in this research.

Appendix

Lightning is represented by an impulse current source

given by:

i(𝑡) = i𝑚[𝑒𝛼𝑡 − 𝑒𝛽𝑡] (22)

where i𝑚 = 24.9 [𝑘𝐴], 𝛼 = −55k [1/s] and 𝛽 = −175k [1/s].

Table 3. Exponential segments before flashover for ZnO1.

Multiplier p Exponent q 𝐕𝒎𝒊𝒏 (𝒑𝒖) 0.163113059479073E+02 0.240279296219978E+02 0.667857269772541E+00

0.134112947529269E+02 0.266219333383985E+02 0.107838745800672E+01

0.383838137212802E+02 0.200870413085749E+02 0.117458089341624E+01

0.115443146532003E+01 0.352906710089203E+02 0.125919561101624E+01

0.407093229412981E+03 0.111310570543275E+02 0.127478619617635E+01

0.256681175704043E+04 0.536270125014350E+01 0.137605475880033E+01

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