Fitting the frequency-dependent parameters in the Bergeron line model

Electric Power Systems Research 117 (2014) 14–20

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Fitting the frequency-dependent parameters in the Bergeron line model Pablo Torrez Caballero, Eduardo C. Marques Costa ∗ , Sérgio Kurokawa UNESP – University Estadual Paulista, Faculdade de Engenharia de Ilha Solteira – FEIS, Departamento de Engenharia Elétrica, Ilha Solteira, SP, Brazil

a r t i c l e

i n f o

Article history: Received 12 May 2014 Received in revised form 15 July 2014 Accepted 18 July 2014 Keywords: Bergeron line model Transmission line modeling – TLM Electromagnetic transients Time-domain analysis Frequency-dependent parameters Power system modeling

a b s t r a c t A new transmission line modeling – TLM is proposed based on the well-established Bergeron method. The conventional Bergeron model is characterized by the line representation through concentrated longitudinal and transversal parameters, i.e., the line electrical parameters are represented by means of electric circuit elements R, L, G and C. The novel approach of this research is the inclusion of the frequency effect in the longitudinal parameters in the Bergeron line representation. This new feature enables to extend the application of the Bergeron method for simulations of transients composed of a wide range of frequencies.

1. Introduction In general terms, the mathematical modeling of dynamic systems is a simplified and practical method to define initial and boundary conditions for the first step on real projects which are extended since electronic devices, for the most variable applications, up to complex power systems. Thus, the continuous improvement and creation of new computational tools to simulate this “first step” are definitely not outdated. Initially, a brief review on transmission line modeling – TLM is described emphasizing the current state of the art and the main problems in the area. There are several transmission line models available in the technical literature to study electromagnetic transients in power transmission systems. Basically, these models may be classified into two general groups: by lumped parameters and by distributed parameters. In the first group, transmission lines are modeled from the representation by lumped elements, i.e., line is modeled by an equivalent representation by means of resistive, inductive and capacitive circuit elements. These models are developed directly in the time domain and are easily integrated to other time-variable power elements, also modeled in the time domain, such as: capacitors, relays, non-linear loads and many other power components. Since

∗ Corresponding author. Tel.: +1 551791080608. E-mail addresses: [email protected], [email protected] (E.C. Marques Costa), [email protected] (S. Kurokawa). http://dx.doi.org/10.1016/j.epsr.2014.07.023 0378-7796/© 2014 Elsevier B.V. All rights reserved.

© 2014 Elsevier B.V. All rights reserved.

the electrical behavior of most power components are well-known in the time domain, this characteristic represents one of the main advantages in TLM by lumped elements [1]. The line modeling by distributed parameters is developed directly from the frequency-dependent parameters based on the line representation by a two-port circuit in the frequency domain. From this approach, the line modeling and simulations are carried out in the frequency domain and time-domain results are obtained using inverse transforms [2]. The frequency-dependent parameters of the line are accurately represented using frequency-domain models; however, these models have restrictions for inclusion of time-variable elements in the simulation process, since most power components are well known and easily modeled in the time domain [3]. Despite line models by lumped elements are developed in the time domain, the frequency effect on the longitudinal parameters can be included in the line modeling using fitting methods [4]. New frequency-dependent models by lumped parameters have been recently published in the technical literature on power system modeling. These models are developed directly in the time domain from the line representation by cascade of ␲ circuit and the frequency effect on the electrical parameters is modeled by fitting the rational functions Rfit (␻) and Lfit (ω) (resistance and inductance) based on the longitudinal impedance of the line, Z(␻), properly calculated taken into account the earth-return impedance (soil effect) and the skin effect on the cables. The frequency-dependent line model described in reference [5] shows to be robust and accurate for the most of transient conditions on a conventional power transmission system. However, depending of the transmission system

P.T. Caballero et al. / Electric Power Systems Research 117 (2014) 14–20

characteristics (source, line and load) and transient conditions, the frequency-dependent model based on cascade of lumped elements shows to be costly in computational terms, depending of the quantity of line equivalent segments used in the cascade and total simulation time. Furthermore, some hard unbalanced conditions lead to discrete inaccuracies, because the multi-phase modeling is based on the intrinsic use of a constant and real transformation matrix to calculate each propagation mode of the line. Thus, the line representation by frequency-dependent cascade of ␲ circuits shows to be efficient for several situations; however, some difficulty in the modeling and inaccuracies are observed for specific cases. From this last statement, the current research proposes a new time-domain line model based on the well-known Bergeron method and using the same fitting procedure applied in the line model referred in [5]. The Bergeron method, also known as method of the characteristics, was firstly proposed to solve hydraulic systems and after applied to electrical problems, more specifically, electromagnetic wave propagation along a lossless line [6]. In this case, the line modeling is carried out considering only the longitudinal inductance L and the shunt capacitance C, which means that the line resistance R and transversal conductance G are neglected. Thereafter, Dommel proposed a nodal solution combining the method of the characteristics for transmission lines and the integration method of the trapezoidal rule for lumped parameters. From this development, the electromagnetic transient program – EMTP was created and a new line model for lossless transmission lines was proposed [7]. In reference [8], an extension of the Bergeron’s method of characteristics was developed including the line losses. In addition, the inclusion of the frequency effect on the longitudinal parameters was evaluated from the application of inverse transforms, which limits significantly the application of the Bergeron line model for several practical operations in transmission systems, most of them including non-linear and time-variable elements in a single line section. The main contribution of Snelson, in reference [8], is the inclusion of the line losses in the Bergeron model using lumped resistances at both line terminals, concentrating the line losses at the sending and the receiving ends of the line section. Based on the contributions firstly proposed by Dommel and after by Snelson, this paper proposes a new line model taken into account the inclusion of the frequency effect on the longitudinal parameters in the Bergeron line model, maintaining its characteristic robustness and simplified modeling. The sum of these contributions results in an accurate line model capable to simulate electromagnetic transients composed of a wide range of frequencies, emphasizing that most of line models prior developed by lumped parameters as well as the classical Bergeron model have restrictions for simulations including a wide range of frequencies. Thus, the inclusion of the frequency effect in the Bergeron line model is the major contribution of this research. This paper is structured into three parts. The first part is an introduction of the classical Bergeron model for lossless lines and for lossy lines using constant parameters. The second part describes the inclusion of the frequency effect in the Bergeron line model using vector fitting. The third part validates the proposed timedomain model comparing with two well-established line models: a frequency-domain model using numerical Laplace transform [2] and a frequency-dependent cascade of ␲ circuits [4,5].

2. The Bergeron line model The Bergeron’s method was firstly applied to lossless transmission lines. This means that only the line inductance per unit of length (p.u.l.) L and the p.u.l. capacitance C were included in the model, whereas the longitudinal p.u.l. resistance R and the p.u.l.

15

Fig. 1. Equivalent impedance circuit of the Bergeron line model.

conductance G are neglected. In fact, the method of characteristics could be applied for lossy transmission lines, but the resulting ordinary differential equations could not be directly integrated. Thus, considering a single-phase line with length l, the current and voltage at a point x along the line are related by: −

∂e L  ∂i = ∂x ∂t

(1)

−

∂i C  ∂e = ∂x ∂t

(2)

Where the first hand of (2) and (3) represents the voltage and the current as a function of space x through the line, i.e., voltage and current wave propagation along the line as a function of the time t. The general solution of (1) and (2) is expressed [7]: i(x, t) = f1 (x − vt) + f2 (x + vt)

(3)

e(x, t) = Z0 f1 (x − vt) + Z0 f2 (x + vt)

(4)

Where f1 and f2 are arbitrary functions of (x ± vt). Function f1 represents the forward wave propagation along the line with velocity v (also known as propagation or phase velocity) whereas f2 represents the wave propagation in a back forward direction. Term Z0 is the surge impedance, also known in the technical literature as characteristic impedance of the line. Terms Z0 and v are expressed as follows [7]:



Z0 =

L ; C

v= √

1

L C 

(5)

Multiplying (3) by the characteristic impedance Z0 and adding in (4), the following formulation is obtained [6]: e(x, t) + Z0 i(x, t) = 2Z0 f1 (x − vt)

(6)

e(x, t) − Z0 i(x, t) = −2Z0 f2 (x + vt)

(7)

Analyzing (6), (e + Z0 i) is constant for (x − vt). The same instance is valid for the voltage (e − Z0 i) in relationship to (x + vt). These constants are intrinsic related to the propagation characteristics and differential equations of a lossless transmission line. Since (x ± vt) is constant, the traveling time of an electromagnetic wave from the sending end to the receiving end of the line is also constant and is expressed as follows: =

1 √   l LC

v

(8)

The equivalent circuit for a lossless line is described in Fig. 1. The forward wave is constant from the node m to the node k at instant t − . The same is observed for a back forward wave from the node k to m. From this analysis, the following time-domain expression is given: em (t − ) + Z0 im,k (t − ) = ek (t) + Z0 (−ik,m (t))

(9)

From (9), it is possible to verify that the forward wave, which takes  seconds to reaches the sending end of the line (node k),

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Fig. 3. Equivalent circuit used for fitting the line longitudinal parameters.

Fig. 2. Equivalent impedance circuit considering the line losses.

is equal in magnitude to the back forward wave. This behavior is expected because the line has no losses in the successive wave reflections between the two terminals (nodes k and m). By association from (9) with the two-port equations, the following formulation is described for the currents at both line terminals: Im,k (t) =

1 e (t) − Ik (t − ) Z0 k

(10)

Im,k (t) =

1 em (t) − Im (t − ) Z0

(11)

Terms Ik and Im are equivalent current sources at the sending and receiving ends of the line, respectively. Sources Ik and Im are known at state time t from the past history at time t − , as expressed in (12) and (13), respectively. Ik (t − ) =

1 em (t − ) − im,k (t − ) Z0

(12)

Im (t − ) =

1 e (t − ) − ik,m (t − ) Z0 k

(13)

Eqs. (10)–(13) are the time-domain formulation for a lossless line represented by the equivalent impedance circuit in Fig. 1. However, the losses can be represented concentrating the losses at each terminal of the line. Lumped resistances are included at the sending and receiving ends of the line in order to approach the line losses, as described in Fig. 2. The total series impedance is concentrated at both terminals of the line represented by the equivalent impedance circuit in Fig. 2. Thus, (12) and (13) can be reformulated as follows [8]: Ik (t − ) = −

1 em (t − ) − im,k (t − ) (Z0 + R/2)

(14)

1 ek (t − ) − ik,m (t − ) (Z0 + R/2)

(15)

Im (t − ) = −

The term R/2 is added to the characteristic impedance Z0 . Term R is the total resistance of the line, where part is concentrated at the node k and the other half at the node m, as described in Fig. 2. Although this approach has been used in most EMTP versions, entitled as Bergeron model, the concentrated resistance R is constant, which means that the Bergeron model for lossy lines has not properly taken into account the frequency effect on the longitudinal parameters. Most transient phenomena in power systems are composed of a wide range of frequencies which means that the conventional Bergeron model has restrictions in simulations from fast and impulsive transients, e.g.: an atmospheric impulse. From this statement, the proposal of this research is to include the frequency effect in the Bergeron line model without inverse transforms, directly in the time domain.

resistance is basically the sum of the earth-return resistance (soil effect) and resistance component due to the skin effect at lower frequencies. The longitudinal impedance is significantly influenced by the soil effect resulted from the return current through the soil. Thus, from the prior statements, the frequency effect in TLM is an important issue to represent properly the various electromagnetic transient which power transmission systems are subject. In this section, the proposed frequency-dependent Bergeron model is described in details. Firstly, a brief review on vector fitting is presented in order to show how the frequency-dependent parameters are synthesized for representation directly in the time domain by means of a single rational function and lumped elements. In sequence, the new Bergeron line model is described step by step applying the proposed fitting method. 3.1. Fitting the longitudinal impedance The fitting procedure basically consists of approximating a rational function Zfit (ω) to the longitudinal impedance of the line Z(ω). The poles, zeros and residues of the fitted impedance Zfit (ω) are associated with a RL circuit, which is the main step to represent the frequency effect in TLM directly in the time domain. The referred equivalent circuit is described in Fig. 3. The rational function that represents the equivalent RL circuit in Fig. 3 is expresses as follows: Z(ω) ≈ Zfit (ω) = R0 + jωL0 +

n  i=1

jωRi

(jω + Ri /Li )

(16)

Term ω is the angular velocity. The p.u.l. resistance R0 and p.u.l. inductance L0 are values for ω = 0. The frequency range is fitted as a function of the quantity of RL blocks in the electric circuit in Fig. 3. The calculated line resistance R(ω) and fitted resistance Rfit (ω), for a range of 1 MHz, are shown in Fig. 4 as follows. From the same way as in Fig. 4, the inductances L(ω) and Lfit (ω) are in Fig. 5. There are several fitting procedures available in the technical literature to obtain an approximated rational function using tabulated values (poles, zeros and residues) from a general function, such as Z(ω). However, the vector fitting algorithm applied in the proposed line model was firstly developed by Gustavsen and Semlyen [9]. This algorithm has shown to be accurate and robust for smooth

3. Including the frequency effect in the Bergeron model As prior described in this paper, the series impedance of a transmission line is variable with the frequency. The longitudinal

Fig. 4. Line p.u.l. resistance R(ω) (curve 1) and fitted p.u.l. resistance Rfit (ω) (curve 2).

P.T. Caballero et al. / Electric Power Systems Research 117 (2014) 14–20

17

Rn Ln dikn (i − ikn ) = 2 k0 2 dt

(19)

In (17)–(19), terms ik0 , ik1 to ikn are the currents in the resistor R0 and inductors L1 to Ln , respectively. Following the Kirchoff’s current law, the current through the resistors R1 and Rn are (ik0 − ik1 ) and (ik0 − ikn ), respectively. Thus, the general expression is developed associating the past-history current source Ik (t − ) to the state current ik0 , which is equivalent to the current ik,m (t) in Fig. 6. Vs − R0 ik0 − R1 (ik0 − ik1 ) − · · · − Rn (ik0 − ikn ) + Z0 Ik (t − ) = Z0 ik0 (20) Fig. 5. Line p.u.l. inductance L(ω) (curve 1) and fitted p.u.l. inductance Lfit (ω) (curve 2).

The first-order system, based on (17)–(20), can be presented as state equations from the following form: [I˙ k ] = [Ak ][Ik ] + [Bk ][S]

and resonant responses with high order and wide frequency bands. Furthermore, the same method has been efficient in the development of prior time-domain models to simulate electromagnetic transients in power systems [1,5,9].

The proposed development consists basically to replace the constant resistance, referred to the line losses in Fig. 2, by the equivalent circuit described in Fig. 3. This procedure requires a new formulation of the Bergeron model which results in a system of differential equations with dimension as long as the number of RL blocks in the equivalent circuit in Fig. 3.

R1 L1

−1 +

R



1

n

⎢ Z0 + R i=1 i ⎢ ⎢  ⎢ ⎢ R1 [Ak ] = ⎢ R2 ⎢ L2 Z0 + n Ri i=1 ⎢ ⎢  ⎣ Rn R1

n Ln

Vector [Ik ] is composed of the currents in inductors L1 to Ln , its transposed form is expressed as: [Ik ]T = [ ik1

···

ik2

ikn ]

(22)

The current derivates of (22) are expressed by the following transposed vector:

3.2. Frequency-dependent Bergeron line model

⎡

(21)

Z0 +

R i=1 i

Substituting the resistance R (Fig. 2) by the equivalent circuit expressed by Zfit (ω), the frequency-dependent Bergeron circuit is restructures as follows in Fig. 6. In Fig. 6, the frequency-dependent impedance is concentrated at the nodes k and m of the equivalent circuit, from the same way described for the Bergeron model with losses. However, the constant resistance R is replaced by the impedance Zfit (ω), in (16), directly in the time domain. The number of RL blocks is defined based on the type of electromagnetic transient to be analyzed. For an input signal composed of low frequencies, e.g. a switching operation, no more than three or four RL circuits are necessary. Otherwise, for fast and impulsive transients, as an atmospheric impulse for example, several RL blocks are required to cover the entire frequency range which a steep-front wave is composed. Considering the frequency-dependent Bergeron circuit in Fig. 6, the relationship of voltage on the resistors and the inductors from the node k is expressed in (17)–(19). R1 L1 dik1 (i − ik1 ) = 2 k0 2 dt

(17)

R2 L2 dik2 (i − ik2 ) = 2 k0 2 dt

(18)

 di di k1 k2

T

[I˙ k ] =

dt

···

dt

dikn dt

 (23)

The state matrix [Ak ] is constant and expressed as a function of R and L values of the circuit in Fig. 3 and also from the characteristic impedance Z0 , calculated based on the line parameters for direct current. Matrix [A] is expressed as: R1 L1 R2 L2 Rn Ln





R

2

n

Z0 +

R i=1 i

−1 +



 ···

R2

Z0 +

n

R i=1 i

..



R

Z0 +

···

2

n

.

···

R i=1 i

R1 L1 R2 L2 Rn Ln

Z0 +





R

n

n

R i=1 i



Rn

Z0 +

n

−1 +

R i=1 i R

Z0 +

n

n

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ (24) ⎥ ⎥ ⎥ ⎦

R i=1 i

Matrix [Bk ] is also developed as a function of Z0 and from the R and L tabulated values of the circuit in Figs. 3 and 6. Matrix [Bk ] has dimension n per 2 and expresses as:

⎡

R1 L1





1

n

⎢ ⎢ Z0 + i=1 Ri  ⎢ ⎢ R2 1 ⎢

⎢ L2 Z0 + n Ri [Bk ] = ⎢ i=1 ⎢. ⎢. ⎢. ⎢  ⎣ Rn 1

n Ln

Z0 +

R i=1 i

R1 L1



⎤

Z

0

n

⎥ ⎥ ⎥ ⎥ R2 Z0 ⎥

n ⎥ L2 Z0 + R i i=1 ⎥ ⎥ .. ⎥ ⎥ . ⎥ ⎦ Rn Z0

n Ln



Z0 +

Z0 +

R i=1 i

(25)

R i=1 i

Equation (26) describes the vector [S], which is composed of the voltage source Vs , that can be represented by a time-variable input signal Vin (t), and the historical current source Ik (t − ).



[S] =

Vin (t) Ik (t − )



(26)

From the state-space formulation in (21), current ik,m (t) and voltage Vk (t) can be analytically expressed as follows:

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P.T. Caballero et al. / Electric Power Systems Research 117 (2014) 14–20

Fig. 6. Frequency-dependent Bergeron circuit.

ik,m (t) =

n 

Z0 +

q=1

+

1

Z0 +

n



Rq ikq

R i=1 i

From the Bergeron model representation by state equations, current ik,m (t) and voltage Vm (t) are analytically expressed based on the same development in (27) and (28):

n

R i=1 i

Vin +

Z

n

Z0 +

n 

Vk (t) = Vin (t) − ik,m (t)

R i=1 i

Ik (t − )

(27)

ik,m (t) =

n 

Z0 + ZL +

q=1





R1

n

−1 +

⎢ Z0 + ZL + R i=1 i ⎢ ⎢  ⎢ ⎢ R1 [Am ] = ⎢ R2

⎢ L2 Z0 + ZL + n Ri i=1 ⎢ ⎢  ⎣ Rn R1

n Ln

Z0 + ZL +

R i=1 i

R1 L1 R2 L2 Rn Ln



Z0 + ZL +

n

T

T

T

[Im ] = [Ik ] = [ ik1 and [I˙ m ] = [I˙ k ] =

ik2

 di di k1 k2 dt

dt

−1 +



 ···

R2 Z0 + ZL +

n

R i=1 i



R2 Z0 + ZL +

n

R i=1 i

R1 L1



···

···

ikn ]

dikn dt

R i=1 i

Im (t − )

(35)

..

.

···

R1 L1 R2 L2 Rn Ln



Z0 + ZL +





Rn

n

R i=1 i



Rn Z0 + ZL +

−1 +

n

R i=1 i Rn

Z0 + ZL +

n

⎤ ⎥ ⎥ ⎥ ⎥ ⎥(29) ⎥ ⎥ ⎥ ⎥ ⎦

R i=1 i

Where the output voltage Vout (t) is expressed as: Vout (t) = −im,k (t)ZL

(36)

4. Validation of the frequency-dependent Bergeron model

(32)

R i=1 i

Vector [S], in (21), is substituted by the past-history current source Im (t − ) from the node m. Thus, the state equation in (21) can be rewritten from the current source in node m of the equivalent circuit in Fig. 6: [I˙ k ] = [Am ][Ik ] + [Bm ]Im (t − )

Ri (1 + iki )

(31)

 ⎤

Z0

Z0 + ZL +

n

(30)



n ⎥ ⎢ Z0 + ZL + R ⎥ ⎢ i=1 i ⎥ ⎢ ⎥ ⎢ R2 Z0 ⎥ ⎢ ⎢ L2 Z0 + ZL + n Ri ⎥ [Bm ] = ⎢ ⎥ i=1 ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎢. ⎥ ⎢ ⎦ ⎣ Rm Z0

n Lm

Z0 Z0 + ZL +

Since the state matrixes are obtained from the fitted parameters (R and L elements of the equivalent circuit in Fig. 3), the first-order system with the state equations can be solved using numerical or analytical integration methods [5].

Thus, vector [Im ] can be expressed only in terms of ik . Terms [B] and [S], in (21), are also modified. Matrix [B] is reformulated as a single vector composed of a single column:

⎡

+

i=1

···

R i=1 i



n 

Vm (t) = Vout (t) − im,k (t)



R2

The load ZL can be modeled as a constant or time-variable impedance, since the proposed line model is developed in the time domain. The vector with the currents through inductors L1 to Ln could be renamed as [Im ]; however: T

R i=1 i

(28)

The formulation for the node m is similar to the development from the node k, since the parameters of the equivalent circuit Zfit (ω) are the same for both sides of the equivalent circuit in Fig. 6. Considering a generic load, represented by an impedance ZL at node m, the state matrix [Am ] is expressed in (29). R1 L1

n

(34) Ri (1 + iki )

i=1

⎡



Rq ikq

(33)

This section aims to evaluate the accuracy of the frequencydependent model based on the Bergeron line representation. In order to validate the proposed model, results are compared to the well-established Universal Line Model – ULM and a frequencydependent cascade of ␲ circuits with 200 sections for a 100 km length transmission line. The ULM is accurate and useful to validate other models considering simple cases without non-linear and time-variable elements in the transmission system modeling. The ULM is developed from the distributed parameters of the line in the frequency domain, where the time-domain results are obtained applying inverse transforms [2]. On the other hand, the cascade of ␲ circuits is directly developed in the time domain by lumped parameters and the resulted differential equations (state equations) are solved using integration methods [5]. Initially, two extreme cases are simulated to highlight voltage and current transients in transmission lines. The first case consists of a switching impulse applied at the sending end of the transmission line with the receiving end open. The second case is the same transmission line with a switching impulse at the sending

P.T. Caballero et al. / Electric Power Systems Research 117 (2014) 14–20

19

Fig. 7. Voltage transient at the open receiving end of the transmission line: frequency-dependent Bergeron model (1), ULM (2) and cascade of ␲ circuits (3).

end, however, with the receiving end in short (approximately 3.7 , same resistance value estimated for an electric arc) [5]. Results obtained from the open-circuit test are described in Fig. 7. Fig. 7a shows the voltage transients at the receiving end of the line obtained using the frequency-dependent Bergeron model (curve 1), ULM (curve 2) and frequency-dependent cascade of ␲ circuits (curve 3) for a 30-ms simulation. In Fig. 7b, a detailed profile of the voltages in Fig. 7a is presented for a time window of 1.7 ms, showing in details the first voltage wave reflection on the open receiving end. The three models show a similar behavior in Fig. 7a. However, Fig. 7b shows some oscillations in the first wave reflection simulated using the frequency-dependent cascade. On the other hand, the proposed Bergeron model and the ULM are free of highfrequency oscillations. Considering a transmission line with the receiving end in short (short-circuit test), the current transients simulated from a switching impulse are described in Fig. 8. Curve 1 is the current simulated using the frequency-dependent Bergeron model, curve 2 is the transient current at the receiving end simulated using the ULM and dotted curve 3 is the current transient obtained from the cascade of lumped elements. The transient behaviors obtained using the three models are similar, but the same oscillations prior verified in Fig. 7b are also observed in simulation using the cascade of ␲ circuits.

Fig. 8. Current transient at the open receiving end of the transmission line: frequency-dependent Bergeron model (1), ULM (2) and cascade of ␲ circuits (3).

electromagnetic phenomena are time variable and have a wellknown modeling in the time domain. On the other hand, the frequency-domain modeling of these same elements is usually a complex procedure. This statement represents the main advantage of time-domain modeling compared to models developed in the frequency domain and inverse transforms [5]. To verify the performance of the proposed model considering time-variable elements, a sinoidal voltage source at 60 Hz and peak of 1 p.u.l. is switched at the sending end of the transmission line with the receiving end connected to a time-variable load with power factor 0.98. After approximately 16 ms the load profile changes to a low impedance circuit. Thus, two time-variable events are simulated using the proposed model: a switching followed by load variation. Fig. 9 shows the voltage profile at the receiving end of the line simulated using the proposed Bergeron model. The first transient is observed on the terminal connected to the load, between 0 and 5 ms, resulted from the switching impulse applied at the sending end of the line. The second transient, from 15 up to 30 ms, is resulted because the load impedance variation.

5. Frequency-dependent Bergeron model and time-variable elements Since the proposed frequency-dependent model was validated by comparison with two other well-established line models, the following step is to evaluate the proposed model for simulations considering time-variable and non-linear elements. The flexibility to include other power devices in transmission line models is an important characteristic to simulate the several operation conditions which power transmission systems are subject. The main issue is that many of these power devices and

Fig. 9. Voltage transients at the receiving end of the line considering time-variable elements.

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P.T. Caballero et al. / Electric Power Systems Research 117 (2014) 14–20

Fig. 10. Current transients at the receiving end of the line considering time-variable elements.

A low value of the load resistance is set which changes the load characteristics. Fig. 10 shows the current transient occurred from the switching and load variation. A careful analysis of the current transient in Fig. 10 shows that the current curve has an inverse signal compared to the voltage transient. This behavior is verified because the current im,k has a negative signal in the second part of the Bergeron circuit in Fig. 6. The shape and magnitude of the current wave in Fig. 10 is exactly the same, except for the signal. This way, the maximum current peak is over of 0.01 p.u.l. The exact current profile could be described as – im,k , however, im,k is presented in Fig. 10 in order to discuss this particularity. 6. Conclusions The inclusion of the frequency effect in the Bergeron line model using fitting techniques is the original contribution of the research. The new Bergeron model was validated based on results simulated using the well-established Universal Line Model – ULM, which is developed from the electrical parameters of the line in the frequency domain. In addition, the proposed Bergeron model was compared to other time-domain model by state equations. All line models presented similar behavior in the simulation results. However, high-frequency oscillations are observed in simulations using the time-domain model by lumped parameters and state equations whereas the proposed time-domain model and the ULM present no oscillations. The main attributes of the frequency-dependent Bergeron model can be resumed as follows: - simple modeling using only a single Bergeron circuit; - development direct in the time domain, without use of inverse transforms;

- inclusion of time-variable and non-linear power elements in the modeling and simulation process; - less high-frequency oscillations compared to other time-domain models based on lumped parameters and state equations; - similar results compared to line models by distributed parameters, showing a good accuracy; - as the proposed model is represented by a single Bergeron circuit, differently of a cascade with several ␲ circuits, the processing performance (processing time) is significantly better than other time-domain models represented by state equations, because there is just a few differential equations for the currents and voltages in the proposed Bergeron model whereas the line model by state equations has more than 1000 differential equations (for a cascade with 200 equivalent circuits and five RL blocks, for example). In short, based on the several attributions listed in the last paragraph, the proposed frequency-dependent models shows to be accurate, has a simple development and a good computational performance due to the simplified system of differential equations. Acknowledgement This research was supported by Fundac¸ão de Amparo à Pesquisa do Estado de São Paulo – FAPESP (Proc. 13/00974-5). References [1] M.S. Mamis, M.E. Meral, State-space modeling and analysis of fault arcs, Electr. Power Syst. Res. 26 (2005) 46–51. [2] P. Gómes, F.A. Uribe, The numerical Laplace transform: an accurate technique for analyzing electromagnetic transients on power system devices, Int. J. Electr. Power Energy Syst. 31 (2–3) (2009) 116–123. [3] M.S. Mamis, Computation of electromagnetic transients on transmission lines with nonlinear components, IEE Proc. Gener. Transm. Distrib. 150 (2) (2003) 200–204. [4] A. Ametani, N. Nagaoka, T. Noda, T. Matsuura, A simple and efficient method for including frequency-dependent effects in transmission line transient analysis, Int. J. Electr. Power Energy Syst. 19 (1997) 255–261. [5] E.C.M. Costa, S. Kurokawa, J. Pissolato, A.J. Prado, Efficient procedure to evaluate electromagnetic transients on three-phase transmission lines, IET Gener. Transm. Distrib. 4 (9) (2010) 1069–1081. [6] H.F. Branin, Computer methods of network analysis, Proc. IEEE 55 (1967) 1787–1801. [7] H.W. Dommel, Electromagnetic Transient Program Reference Manual (EMTP Theory Book), Dept. Electrical Engineering, University of British Columbia, Vancouver, Canada, 1989. [8] J.K. Snelson, Propagation of travelling waves in transmission lines: frequency dependent parameters, IEEE Trans. Power Appl. Syst. PAS-91 (1972) 85–91. [9] B. Gustavsen, A. Semlyen, Combined phase and modal calculation of transmission line transients based on vector fitting, IEEE Trans. Power Deliv. 13 (2) (1998) 596–604.