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Estudio de modelos mat= emáticos de Líneas de transmisión para simulación de transitori= os electromagnéticos en sistemas de potencia

Mathematical models of transmission lines for electromagnetic transient simulations in power systems<= /span>

Jaimis Leon1, Sérgio Kurokawa 1

1 Facul= tad de Ingeniería Eléctrica, Universidad Estadual Paulista "Júlio de Mesquita = Filho", UNESP, Ilha = Solteira-Brasil.

 

Recibi= do (Received): 14/10/2018=           &= nbsp;           Ac= eptado (Accepted): 05/12/2018

RESUME= N

En este trabajo se estudian tres modelos matemáticos de líneas de transmisión, Cascada de Circu= ito PI, Bergeron y Universal Line Model para simular transitorios electromagnéticos en sistemas de potencia. Se exponen sus principales características, ventajas y desventajas. Asimismo, se muestran los algoritmos en forma de flujogramas para las simulaciones de los transitorio= s. Finalmente, se presentan los resultados de simulaciones de sobretensiones d= e maniobra en una línea de transmisión monofásica utilizando el software MATLAB. Se comprueba que el Universal Line Model es el más preciso.

Palabr= as Clave: Modelos de Líneas de transmisión, transitorios electromagnéticos, dominio del tiempo, modelo matemático.

 

ABSTRACT=

 

In this work we study three mathematical models of transmission lines, Cascade of PI Circuit,= Bergeron and Universal Line Model to simulate electromagnetic transients in power systems. its main features, advantages and disadvantages are discussed. Likewise, the algorithms are shown in the form= of flow charts for the transient simulations. Finally, the results of simulati= ons of maneuvering surges in a single-phase transmission line using the MATLAB software are presented. It is verified that the Universal Line Model is the most accurate.

Keywords: Transmission line = models, electromagnetic transient, time domain, mathematical model.

        &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;            &= nbsp;           &nbs= p;

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1   &nb= sp;  INTRODUCCIÓN

&= nbsp;

         En los sistemas eléctricos de potencia, y en general en las líneas de transmisión (LT), se presentan fenómenos transitorios tales c= omo: sobretensiones temporales, sobretensiones por maniobra y sobretensiones por descargas atmosféricas [1]. Para entender el comportamiento de estos fenómenos, necesitamos modelos de LT que representan con exactitud e= stos transitorios electromagnéticos (TEM).

&= nbsp;

Los mo= delos de las LT varían en complejidad y eficiencia, dependiendo de la precisión requerida. Los parámetros que constituyen estos mod= elos pueden ser constantes o dependientes de la frecuencia, también pueden ser clasificados como parámetros de la LT concentrados o distribuido= s.

&= nbsp;

_______________________________<= /p>

* = Correspondencia:
E-mail: jaimis.leon@unesp.br

&= nbsp;

Un mod= elo clásico para representar la LT, se basa en una cascada de circuitos = π [1-3]. Este modelo es considerado a parámetros concentrados y constantes. Sin embargo, al hacer esta cascada se produce la naturaleza distribuida de los parámetros de la LT.

&= nbsp;

Este m= odelo es utilizado en simulaciones de sobretensiones de maniobra y conmutación. Para análisis de TEM, este modelo es usado para = LT de interés secundario, debido a que presentan en sus respuestas oscilac= iones numéricas.

 

Otro m= odelo de LT es el modelo de Bergeron, que se basa en la teoría de ondas viajantes, por lo que resulta un modelo simple y eficiente [1,4]. Este mode= lo está constituido por parámetros constantes y distribuidos. Una desventaja de este modelo es la no dependencia con la frecuencia de los parámetros longitudinales, pero puede ser utilizado para representar= la LT tanto en simulaciones de sobretensiones por maniobra, conmutación= y descargas atmosféricas.

         =

Por último, el modelo capaz de representar con precisión los TEM = de las LT es el Universal Line Model (= ULM) [5]. Este modelo está constituido por parámetros de la LT distribu= idos y dependientes de la frecuencia, y tiene como base la teoría de onda= s viajantes.

      =

El objetivo del trabajo es determinar cuál de los tres modelos de LT, para simular T= EM, es más preciso. En este sentido son mostrados sus modelos matemáticos y algoritmos que tienen como base flujogramas para las simulaciones. También, es mostrado resultados de sobretensiones de maniobra en una LT monofásica. Se formula como hipótesis que = el ULM es el modelo más preciso para simular transitorios en una LT. La clasificación de los modelos se ilustra en la Figura 1.

 

 

 

 

 

 

=

Figura= 1. Clasificación de los diferentes modelos de LT [1].

 

2&nb= sp;     MODELO DE CASCADA DE CIRCUITOS π= (MODELO A PARÁMETROS CONCENTRADOS CONSTANTES)

=  

El mod= elo de LT a parámetros concentrados es empleado para simulaciones transitor= ias de energización y conmutación. Este modelo de LT puede ser re= presentada por una cascada de circuitos π [1,2]. Cada segmento de circuito π= consta de una resistencia y una inductancia en serie. Completando el circuito `= 0;, hay componentes en paralelo: capacitancia y conductancia. En la Figura 2 se muestra un elemento de circuito π [2,3].

 

Figura= 2. Un circuito π.

 

         = La LT monofásica de longitud d,= representada por la cascada de circuitos π, se muestra en la Figura 3 [2].

 

Figura= 3. LT representada por una cascada de circuitos π.

 

         = En la Figura 3, los parámetros = R y L son la resistencia e induct= ancia de cada circuito π respectivamente. Los parámetros G y C son la conductancia y la capacitancia de cada circuito π respectivamen= te. Los parámetros R, L, G y C son escritos como:

&= nbsp;

(1)<= /span>

         =

         = En la ecuación (1) los parámetros longitudinales R’, L’ y parámetros trasversales C’ y G’= son datos por unidad de longitud y n es el número de circuitos π usados en la representación de = la LT. En base a los circuitos π de la Figura 3 es posible determinar las siguientes ecuaciones diferenciales [2]:

&= nbsp;

(2)<= /span>

(3)<= /span>

         = De las ecuaciones (2) y (3), se nota la influencia de la tensión y corriente eléctrica del circuito π anterior y posterior, respectivamente, en el circuito π especifico.

Si def= inimos las corrientes en los inductores y las tensiones en los capacitores como variables de estado, el sistema de ecuaciones diferenciales resultantes par= a la cascada de circuitos π de la Figura 3 puede ser escrito como [2,3]:

 

(4)<= /span>

         = En la ecuación (4), A y B son matrices de estado para la L= T, [x] es el vector de variables de es= tado y [u] es el vector de entrada de = datos. Estas matrices y vectores son definidos a continuación [2]:

 

(5)<= /span>

(6)<= /span>

(7)<= /span>

         = En las ecuaciones (5), (6) y (7), la matriz A es cuadrada de orden 2n, la mat= riz B y matriz [x] tienen orden 2nx1.

         = El sistema de ecuaciones en (4) puede ser solucionado por diversos méto= dos numéricos que resuelven ecuaciones diferenciales. En [2] se evaluaro= n diferentes métodos de integración, siendo la regla de integración trapezoidal el método más robusto, ya que la precisión= de este se mosto menos sensible en función del aumento del paso de cálculo.

         = La ecuación (8) es resultado de usar el método de integraci&oacu= te;n trapezoidal [2,3].

 

(8)<= /span>

 

         = Simplificando la ecuación (8), obtenemos [2,3]:

(9)<= /span>

         = Donde A’, A’’ y B son matrices constantes que son descritas en = las ecuaciones (10), (11) y (12).

&= nbsp;

(10)=

(11)=

(12)=

         = El algoritmo de este modelo utilizado para realizar las simulaciones de los TE= M tiene como base el flujograma que se muestra en la Figura 4.

 =

Figura= 4. Algoritmo del modelo de cascada de circuitos π.

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3   &nb= sp;  MODELO DE BERGERON (MODELO A PARÁMETROS DIS= TRIBUIDOS CONSTANTES)

 

      El modelo de Bergeron [1,4] es un método de frecuencia constante basado en la teo= ría de las ondas viajantes. Inicialmente este modelo era utilizado para líneas de transmisión sin perdidas, pero después en [4] una extensión del modelo fue desarrollado para incluir las pé= rdidas de la LT. La inclusión de las pérdidas fue obtenida agregando elementos de resistencia concentradas en ambos extremos de la LT sin perdid= a. Aunque estas resistencias se pueden insertar a lo largo de la línea dividie= ndo su longitud total en varias secciones, es mínima la diferencia y el = uso de apenas dos secciones en serie con las resistencias concentradas es perfectamente adecuado. En la Figura 5 se muestra este modelo aproximado de= la LT [4].

=

Figura= 5. Modelo de la LT con pérdidas.

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      Este modelo, proporciona respuestas razonables, siempre que el valor de las resistencias= concentradas sea mucho menor al valor de la impedancia característica de la línea [4].  Sin embrago= , para estudios de alta frecuencia (por ejemplo, portadora de línea de pote= ncia), este modelo puede no ser adecuado. El circuito equivalente de la Figura 5 en representación Norton se muestra en la Figura 6 [4].

&= nbsp;

 <= /o:p>

Figura= 6. Circuito equivalente de la LT con pérdidas.

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      De la Figur= a 6, es obtenido un modelo para cada mitad de LT con pérdidas. Sin embarg= o, como solo los terminales k y m son de interés, son elimi= nadas las variables del punto medio colocando en cascada las dos secciones de med= ia LT. En consecuencia, se obtiene el modelo de LT que se muestra en la Figura= 7 [4].

Figura= 7. Modelo de Bergeron para LT.

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      A partir de= un análisis realizado en [4], se obtiene las expresiones para las corrientes, en el dominio de tiempo, para los terminales emisor y receptor.=

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(13)=

(14)=

      De las ecua= ciones (13) y (14), se observa que las corrientes imk e ikm en el instante t se determinan a p= artir de las corrientes históricas en el tiempo (t-τ). Estas corrientes históricas son mostradas en= las siguientes ecuaciones [4].

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=

 

 

(15)

=

 

 

(16)

Donde:=

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(17)=

(18)=

(19)=

(20)=

         = En la Figura 8, es mostrado el algoritmo en forma de flujograma del modelo de Bergeron para realizar simulaciones de los TEM.

 

Figura= 8. Algoritmo del modelo de Bergeron.

&= nbsp;

4   &nb= sp; UNIVER= SAL LINE MODEL (ULM - MODELO A PARÁMETROS DISTRIBUIDOS E DEPENDIENTES DE= LA FRECUENCIA)

 

El ULM [5] se basa en la formulación del ti= po de onda viajante y en las ecuaciones del Telegrafista, que en el dominio de= la frecuencia se expresan de la siguiente manera [5]:

 

(21)=

(22)=

      Donde V e I son las tensiones y corrientes de fase a lo largo del conductor, Z e Y son respectivamente la impedancia en serie y la admitancia shunt, ambos en = por unidad de longitud.

 

Figura= 9. Diagrama de la línea de transmisión monofásica.

 

      Dado un seg= mente de línea de transmisión de longitud d como el representado en la Figura 9, la solución de (2= 1) y (22) en términos de corrientes se expresan de la siguiente manera [5= ]:

&= nbsp;

(23)=

(24)=

      Los sub&iac= ute;ndices k y m indican los valores de V y I de los terminales emisor y receptor, respectivamente. Yc y = H son respectivamente la admitancia característica y función propagación para las ondas de corriente [5]:

&= nbsp;

(25)=

(26)=

      En la Figur= a 10 [5] es mostrada la representación en el dominio de la frecuencia de los terminales de la línea con dos equivalentes Norton. Ambos equivalent= es conforman el modelo completo de la línea de transmisión monofásica. Téngase en cuenta que este modelo no tiene conexión directa entre los dos extremos de la línea. Su relac= ión es solo a través de las fuentes de corriente auxiliar que implican un retraso de tiempo. Por lo tanto, se dice que los dos extremos están desacoplados en el tiempo.

 

Figura= 10. Diagrama de la línea de transmisión monofásica.

         =

         = Las ecuaciones de la línea descritas en (23) y (24) deben resolvers= e en el dominio del tiempo para simular transitorios electromagnéticos. P= or lo tanto, la forma general en el dominio del tiempo da como resultado las siguientes ecuaciones [5]:

(27)=

(28)=

 

         = Donde las variables en minúscula son la representación en el dominio del tiempo de sus contrapartes en mayúsculas y el símbolo * indica una convolución. La solución de las ecuaciones (27) y = (28) se llevan a cabo de manera muy eficiente a través del análisis del espacio de estados. Esto se hace cuando Yc y H se representan como funciones racionales [5].

&= nbsp;

(29)=

(30)=

 

      Los coefici= entes desconocidos c= p, ap<= /i>, aq<= /i> y f se calculan utilizando la técnica llamada Vector Fitting [5] de la representación de Yc y H.

      Una forma compacta de la ecuación (27) es [5]:

&= nbsp;

(31)=

     =     Donde:

&= nbsp;

(32)=

(33)=

(34)=

 

         = La misma idea de la forma compacta de la ecuación (27) se aplica para la ecuación (28).

         = Finalmente, para implementar el modelo ULM en el dominio del tiempo las convoluciones d= e las ecuaciones (27) y (28), se realizan usando métodos de espacio de est= ado discreto [5] que son de la siguiente forma:

&= nbsp;

(35)=

         = En la Figura 11, se muestra el algoritmo en forma de flujograma del ULM para realizar simulaciones de los TEM.

&= nbsp;

5   &nb= sp;  SIMULACIÓN DE TRANSITORIOS Y ANÁLISI= S DE RESULTADOS

= &= nbsp;

         = Las simulaciones fueron realizadas para una LT monofásica de 100 km de longitud, siendo energizada por dos tipos de fuentes: sinusoidal y escalón unitario, en el terminal emisor. Por otro lado, el terminal receptor se encuentra en dos estados: circuito abierto y conectado a una ca= rga. Para el cálculo de las impedancias por unidad de longitud, la LT fue considerada de acuerdo a la Figura 12. En los tres modelos de la LT son calculadas las tensiones en el terminal receptor. La implementación = de algoritmo de los modelos fue desarrollado en el lenguaje de programaci&oacu= te;n Matlab.

         =

Figura= 11. Algoritmo del ULM.

 

Figura= 12. LT aérea.

 

         = Para todas las simulaciones, se tuvo que V0=3D100 kV y ω=3D377 rad/s y los resultados fueron presentados en valores por unidad (p.u.) siendo 100 kV el valo= r base. Los parámetros longitudinales (resistencia e inductancia) fueron calculadas para un ancho de frecuencia de 0.1 Hz a 1 MHz y sus comportamien= tos son mostrados en la Figura 13. En las Figuras 14-17 se muestran los transitorios de las tensiones en el terminal receptor obtenidas con los mod= elos cascada de circuitos π (LPM), Bergeron (MB) y Universal Line Model (ULM).

         = Para el modelo LPM se utilizaron 100 circuitos π y los parámetros constantes fueron calculados a 60 Hz, observándose la presencia de oscilaciones numéricas que no representan el valor real de las simulaciones y que pueden ser mejoradas a través de filtros o usando circuitos π modificados. Para el modelo MB, los parámetros usad= os fueron calculados, también, a 60 Hz. Para el modelo ULM, los parámetros usados fueron calculados para un ancho de frecuencia de 0= .1 Hz a 1 MHz.

         = En las Figuras 14 y 16 se observa que la tensión se duplica en la prime= ra reflexión. En las Figuras 15 y 17 se observa que, al conectar una carga (5 kΩ) en= el terminal receptor, las sobretensiones se atenúan debido a que la resistencia las amortigua.

 

 

Figura= 13.  Parámetros longitudinales de= LT.

 

Figura= 14.  Energización de LT en circui= to abierto y fuente escalón.

 

Figura= 15.  Energización de LT con carga= de 5 y fuente escalón.

 

Figura= 16.  Energización de LT en circui= to abierto y fuente senoidal.

 

Figura= 17.  Energización de LT con carga= de 5 y fuente senoidal= .

&= nbsp;

6   &nb= sp; CONCLU= SIONES

&= nbsp;

         = En este trabajo se describieron tres modelos matemáticos de líne= as de transmisión para el cálculo de transitorios electromagnéticos, demostrándose que el modelo Universal line Model es el más preciso; validándose así la hipótesis del trabajo.

         = Se comprobó la utilidad práctica de los algoritmos en forma de flujograma que se desarrollaron para cada modelo ya que con ellos se realiz= aron las simulaciones de transitorios electromagnéticos en el modelo monofásico de una línea de transmisión energizada.

&= nbsp;

AGRADE= CIMIENTOS

 

      Agradezco a la Facultad de Ingeniería Eléctrica de la UNESP-Brasil por = el soporte académico. También agradecer al Consejo Nacional de Desarrollo Tecnológico y Científico CNPq= -Brasil por el soporte económico dado para poder realizar este trabajo.=

     

REFERE= NCIAS

&= nbsp;

[1]&= nbsp;  Restrepo L., Caicedo, G., Cas= tro. F. (2008). Modelos de línea de transmisión para transitorios electromagnéticos en sistemas de potencia. Revista Energía y Computación, v. 16, p. 21-= 32.

[2]   Araujo A., Silva R. and Kurokaw= a S. (2014). Comparing Lumped and Distributed Parameters Models in Transmission = Lines during Transient Conditions. IEEE P= ES T&D Conference and Exposition.

[3]   Dommel, H. (1969). Digital computer solution of electromagnetic transients in single and multiphase networks. IEEE Transactions on Power Apparatus a= nd Systems, v. 88, p. 388-399.

[4]   Mamis M., Kaygu= suz A. and Koksal M. (2010). State variable distributed-parameter representation of transmission line for transient sim= ulation. Turkish J Elec= Eng & Comp Sci, <= /span>v. 18, p. 31-42.

[5]   Ramos-Lea&= ntilde;os O., Naredo J. and Gutierrez J. (2012). An Advan= ced Transmission Line and Cable Model in Matlab for= the Simulation of Power System Transients in MATLAB. InTech Open - A Fundamental Tool for Scientific Computing and Engineering Applications. v. 1, p. 269-304.=

 

 

 

<= /span>

Los artículos publicados por TECNIA <= span class=3DSpellE>pueden ser compartidos a través de la li= cencia Creative Commons: CC BY 4.0 Perú. Permisos lejos de este alcance pueden ser consultados a través del corr= eo revistas@uni.edu.pe

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

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J. Leon, et al.

DOI: https://doi.org/10.21754/tecnia.v28i2.550&n= bsp;            = ;            &n= bsp; Revista TECNIA Vol. 28 Nº 1 Agosto – Diciembre 2018

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