UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
INGENIERÍA ELÉCTRICA Y ELECTRÓNICA
INTRODUCCIÓN En el presente trabajo se muestra la formas de solución de una ecuación de oscilación que van más allá de una solución explícita, con la solución de una formula dada y aunque es posible demostrar de forma abstracta que la mayoría de estas ecuaciones posen solución, por lo menos de manera local, en general resulta muy difícil expresar explícitamente de que solución se trata. Principalmente se debe saber que las ecuaciones que se deben analizar no poseen solución de forma cerrada. Estos mtodos son el mtodo de euler que aunque presente poca precisión, permite !allar una solución aproximada, y el mtodo de euler modificado cuya precisión es mayor, es decir está más próxima al valor verdadero, sin embargo tambin cuenta con error. "e implementó un programa #$%&'%(), para analizar la solución en diferentes puntos mostrando su aproximación al valor verdadero.
ESTABILIDAD TRANSITORIA
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
INGENIERÍA ELÉCTRICA Y ELECTRÓNICA
INTEGRACIÓN NUMÉRICA Para el análisis de la estabilidad transitoria de sistemas de potencia, se debe resolver un conjunto de ecuaciones diferenciales no*lineales de + er orden, con condiciones iniciales conocidas, de la forma. x´ = f ( ( x , r , t ) … … … … … … … … . ( 1)
ondex vector de variable de estado r vector de variables algebraicas t tiempo 'as variables algebraicas están relacionadas las restricciones algbricas de la forma 0
=f ( ( x , r , t ) … … … … … … … ( 2)
que deben ser satisfec!as a cada instante de tiempo. En gene genera ral,l, ecua ecuaci cion ones es dife difere renc ncia iale less no*li no*line neal ales es sólo sólo admi admite ten n solu soluci ción ón numrica, numrica, i. e., calculada calculada paso a paso. paso. iversos algoritmo algoritmoss de integración integración numrica son conocidos y la opción del /mejor0 depende grandemente del problema. %sí siendo, algunas características in!erentes a los mtodos de integración deben ser analizadas para llegar a una conclusión. 'os principales mtodos de integración numrica pueden ser interpretados
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
INGENIERÍA ELÉCTRICA Y ELECTRÓNICA
|
2
|
2
3
|
dx ( t ) d x ( t ) ( ∆ t ) d x ( t ) + x ( t 0 + ∆ t )= x ( t 0 ) + ∆ t + 2 3 dt t t =t ! 2 d t t t =t d t 0
0
t = t 0
3
( ∆ t ) 3!
+ … ( 3)
PRESICIÓN: Esta característica dada, principalmente, por dos factores• •
Error de redondeo1 Error de truncamiento.
'os errore erroress de redon redondeo deoss están están relaci relacion onado adoss a la repres represent entaci ación ón de los n2me n2mero ross en un comp comput utad ador or #arit #aritmt mtic ica a finit finita) a) y pued pueden en ser ser mini minimi miza zado doss utilizando doble precisión y3u otros recursos in!erentes al computador siendo utilizado para resolver el problema. 4 los errores de truncamiento son debidos a la aproximación de la solución real del problema utilizada por el mtodo m todo de integración escogido. El error de truncamiento puede ser analizado a partir de la serie de &aylor aylor de la función, mostrada en en la ecuación #5), y será proporcional proporcional a ( ∆ t ) p + … … … … … … … … … … … … … … … .. ( 4 ) 1
onde 6t es el paso de integración escogido y p es el orden de la serie de &aylor aylor utilizada por el mtodo m todo de integración como aproximación de la solución. 'a solución verdadera del problema, en un instante dado de tempo t n, será
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
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ond onde e 4n es la apro aproxi xima maci ción ón calc calcul ulad ada a por por el mto mtodo do de inte integr grac ació ión, n, O ( ∆ t
p + 1
) es la orden grandeza de la precisión del mtodo y ɛn representa los
demás errores que pueden surgir en el proceso. Estabilidados tipos de inestabilidad pueden ocurrir en la solución de ecuaciones diferenciales ordinarias• •
7nestabilidad in!erente1 y 7nestabilidad inducida.
'a instabilidad in!erente surge cuando errores numricos son amplificados, a cada paso de la solución, !asta dominar completamente el cálculo y !acer el mtodo divergencia de la solución real. 4a la inestabilidad inducida está relacionada con el mtodo de integr integraci ación ón
utiliz utilizado ado
o,
de manera manera más precis precisa, a, con la discre discretiz tizaci ación ón
resultante de la aplicación del mtodo, que depende, tambin, del paso de integración escogido. "e puede mostrar que los mtodos de integración corresponden a resolver una ecuación algebraica de la forma ∝
+∝1 y n+ ∝2 y n−1+ … + ∝k y n−k + 1 + β 0 0 y n + 1
dy n+ 1 dt
+ β 1
dy n dt
+ … + β k
dy n− k +1 dt
=0 ………………………………
onde 8i y 9i son constantes y se desea determinar y n:+, conociendo los valores anteriores de y i #i ; n:+). Esta es una ecuación discreta, lineal, y será estable si todas las raíces
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
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que el mtodo o paso de integración es crítico para la solución, en especial los mtod mtodos os de
RIGIDEZ: >n sistema de ecuaciones diferenciales es dic!o rígido #/stiff0) cuando la razón entre la mayor y la menor menor constantes constantes de tiempo del problema sea sea muyo mayor que +. $todos de integración tradicionales tienen su intervalo de integración definido por la menor constante de tiempo y el tiempo final de la simulación es definido por la mayor mayor constante constante de tiempo. tiempo. e esta forma, forma, un gran n2mero n2mero de puntos deberán ser determinados, aumentando sobremanera el tiempo de computación. %demás de eso, si el mtodo de integración no fuera 'a*estable, puede !aber problemas de inestabilidad numrica.
MÉTODOS DE INTEGRACIÓN NUMÉRICA 'os mtodos de integración numrica, como fue visto en la sección anterior, representan una discretización de la solución, de forma que la ecuación difere diferenci ncial al diferencias diferencias
origin original al
#conti #continua nua))
#discreta) #discreta)
y
es aprox aproxima imada da
apenas apenas
algunos algunos
por por
una una
ecuaci ecuación ón
de
valores valores #correspon #correspondient dientes es a
determinados instantes de tiempo) son calculados, que corresponderán a la solución solución aproximada aproximada obtenida. obtenida. 'a ?igura ?igura + ilustra ilustra este efecto, efecto, en en que que una
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
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Fig. 1: Discretizaci! "e #!a F#!ci! C$!ti!#a
Para Para ejem ejempl plifific icar ar la aplic aplicac ació ión n de los los dive divers rsos os mto mtodo doss de inte integr grac ació ión n num numric rica, a, será será util utiliz izad ado o el sigu siguie ient nte e sist sistem ema a line lineal al de las las ecua ecuaci cion ones es diferenciales-
{
x´ 1= x 2 x 1 ( 0 )= x 10 … … … … … … … … . (7 ) x´ 2=−a 1 x 1−a2 x 2 x 2 ( 0 ) = x 20
cuya solución analítica es de la forma
{
− x 20 t x 20−ℷ 1 x 10 t e + e −ℷ −ℷ ℷ ℷ 2 1 2 1 … … … … … … (8 ) − − ℷ ℷ x x x x t t 2 10 20 20 1 10 x 1 ( t ) = ℷ e +ℷ e 1 2 −ℷ −ℷ ℷ ℷ 2 1 2 1 x 1 ( t )=
ℷ 2 x 10
ℷ 1
ℷ 1
ℷ 2
ℷ 2
onde ℷ
−a2 ± √ a22− 4 a 1
(9)
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
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son las raíces del polinomio característico de la ecuación diferencial de @ do orden asociada a la ecuación #A) o, equivalentemente, son los polos de la función de transferencia de este sistema.
MÉTODO DE EU%ER El mtodo de Euler es el más simple de los mtodos de integración numrica numrica y puede puede ser visualizad visualizado o en la ?igura ?igura @. >tilizando >tilizando el valor de la derivada de la función en el instante de tiempo t t B, se puede escribir que x ( t 0 + ∆ t )= x ( t 0 ) +
|
dx ( t ) … … … … … … … … … … … … … . ( 10 ) dt t t =t 0
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
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{
= 1 =−1 β = 0 β =− ∆ t … … … ( 11) α = α = …= α k = 0 β = β = …= β k =0 ∝
2
0
∝
1
0
3
1
2
3
'a aplicación del mtodo de Euler para la solución del sistema dado por la ecuación #A) resulta en
{
( |) ( |)
( x )n + =( x )n + 1
1
1
( x )n + =( x )n + 2
1
2
d x1 dt
… … … … … … … … … … … … … … … … … .. ( 12 )
n
d x2 dt
∆ t ∆ t
n
El mtodo de Euler puede ser implementado de forma explícita o implícita. En su forma explícita, el valor de las derivadas mostradas en la ecuación #+@) son explícitamente explícitamente calculadas calculadas a cada cada paso del algoritmo. 4a en el caso de !aber expresiones analíticas para las derivadas, estas pueden ser sust sustititui uida das, s, resu resultltan ando do un sist sistem ema a de ecua ecuaci cion ones es que que no depe depend nden en explícitamente de las derivadas. Para el ejemplo dado, la ecuación #A) suministra expresiones analíticas para las derivadas, resultando en el siguiente sistema de ecuaciones-
{
x´ 1= x 2 x 1 ( 0 )= x 10 x´ =−a x a x x ( 0 ) = x
[ ][
⇒
( x 1)n+1 = 1 −a1 ∆ t ( x )
][ ]
( x 1)n ∆ t … ( 13 ) 1− a2 ∆ t ( x )
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
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[ ] [ ( )]
X k =
'
X K =
x 1 x 2
δ ( t k ) = k k
[ ] x 1 x 2
'k
= 'k
ω t k
[ ] ∂ ( δ ( t k k ) )
∂ t ∂ ( ω ( t k ) ) ∂ t
[ ][ ω ( t k )
ω ( t k )
F ( X K , t K ) = ∂ ( δ ( t k ) ) = πf ( ( Pe! + Pax sin δ ( t k ) ) 2
2
∂ t
"
"ustituyendo los respectivos trminos-
[ ][ ] δ ( t k +1)
ω ( t k +1 )
=
δ ( t k )
ω ( t k )
[
ω ( t k )
+ ∆ t πf ( ( Pe!+ P ax sin δ ( t k ) ) "
En forma de ecuaciones se tieneδ ( t k + 1) =δ ( t k ) + ∆ t ∗ω ( t k )
∗( ω ( t k +1 ) = ω ( t k ) + ∆ t ∗(
( P e!+ P ax sin δ ( t k ) ) πf ( "
)
]
]
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
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Pe + Pax sin δ k πf (¿ ( ¿ ¿ " )
¿
ω k +1=ω k + ∆ t ∗¿
MÉTODO DE EU%ER RE,ERSO El mtodo de Euler reverso #/bacFard Euler0) modifica el mtodo de
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
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{
= 1 =−1 β =− ∆ t β =0 … … … … … … ( 15 ) α = α = …= α k = 0 β = β = …= β k =0 ∝
2
∝
0
1
0
3
1
2
3
Este mtodo es más estable que el mtodo de Euler convencional #/forFard Euler0), pero su implementación requiere el cálculo del valor de la derivada en un instante de tiempo para el cual a2n no se conoce el estado. 'a implementación explícita, en este caso, requiere una extrapolación inicial para la determinación del valor del estado en el instante t t B: 6t para permitir el cálculo de la derivada. 'a aplicación de este mtodo para el sistema ejemplo de la ecuación #A) resulta r esulta en
{
( |) ( |)
( x )n + =( x )n + 1
1
1
( x )n + =( x )n + 2
2
1
d x1 dt d x2 dt
∆ t
n+1
… … … … … … … … … … … … … … … … … ( 16 ) ∆ t
n +1
Por otro lado, la implementación implícita continua siendo muy simples, debido a la sustitución de las derivadas por por sus expresiones analíticas-
{
( x )n+ =( x )n + ( x x )n + ∆ t ( x )n + =( x )n ± a ( x x )n+ −a ( x ) n+ 1
2
1
1
1
2
1
2
1
1
1
2
2
1
∆ t
⇒
[
][ ] [ ][ ]
−∆ t ( x )n+ = a ∆ t 1 + a ∆ t ( x )n+ 1
1
2
1
1
2
1
1
0
0
1
( x )n ……………………… ( x )n 1 2
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predic predicció ción n del del valor valor del del estad estado o en el el insta instante nte t t B: 6t y este este valo valorr es utilizado para la corrección y obtención de un nuevo valor del estado en ese instante. El mtodo, por lo tanto, puede ser descrito por los siguientes pasosPaso +- Preditor
HHHHHHHHHHHHH#+I) Paso @- Jorrector
HHHHHHHH#+K) "e debe notar que la ecuación #+I) corresponde a un paso del mtodo
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HHHHHHHHH #@+) Este mtodo tambin puede ser implementado de forma implícita, sustituyendo las expresiones de las derivadas derivadas en las ecuaciones ecuaciones #@B) y #@+)-
HHHHHHHH#@@)
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
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HHH..#@M) b) Jálculo Jálculo del primer primer valor del ángulo ángulo y la velocida velocidad d al final del interval intervalo, o, es decir, en el instante t + ∆ t -
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
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H..#@K) f) Jálculo Jálculo del valor valor corregi corregido do del ángulo ángulo al final final del interval intervalo, o, es decir decir,, en el el instante t + ∆ t -
HHH.#5B) El proceso iterativo se contin2a !asta que se presenten nuevos cambios en el sistema o !asta que se concluya el tiempo de estudio.
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
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El generador entrega una potencia real de Pelec B.I p.u, y Selec B.BAL p.u a la barra de potencia infinita cuyo voltaje es +.B p.u. >na falla trifásica ocurre en la mitad de una línea, y la cual es despejada por la puesta fuera de servicio de la línea, mediante la apertura simultanea de ambos extremos de la línea. l ínea. 'a falla es despejada en B.5 segundos. Cbtener la solución numrica de la ecuación de oscilación, empleando el mtodo de Euler, con un paso de tiempo Tt B.BM segundos. Efectu la simulación numrica !asta t B.M.
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
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'a potencia aparente es dada por-
El voltaje interno de la maquina ER es dado por-
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'a reactancia equivalente entre el generador y la barra de potencia infinita resulta ser-
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
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%U&E" de la perturbación resulta-
><%U&E la perturbación resulta-
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
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So!"i#n ana$%i"a &'%o(o () )!)*+
[ ][ ] δ ( t k + 1)
ω ( t k +1 )
=
δ ( t k )
ω ( t k )
[
ω ( t k )
+ ∆ t πf ( ( Pe− Pax sin δ ( t k ) ) "
]
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
INGENIERÍA ELÉCTRICA Y ELECTRÓNICA
[
ω ( 2) δ ( 3 ) = δ ( ( 2 ) + 0.05 π ∗60∗( 0.8− Pax$$ sin δ ( (26.38 )) ω (3) ω (2)
[ ][ ]
5
[
]
1.926∗180 / p% δ ( 3 ) = 29.147 + 0.05 π ∗60∗( 0.8− 0.65sin δ ( ( 29.147 )) 1.926 ω (3)
[ ][ ] [ ][ ] δ ( 3 ) = 34.665 2.837 ω (3)
5
]
.
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
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[
ω ( 5) δ ( 6 ) = δ ( 5 ) + 0.05 π ∗60∗(0.8− P ax$$$ sin δ ( 5 ) ) ω (6) ω ( 5)
[ ][ ]
5
]
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
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ω (0,0.(1 = 0 , (3.2/10.(1 = #(.32/ rad's
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
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ω 20.(,0.(1 = (.326! , (/.(1 0.(1 = +.& rad's
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UNIVERSIDAD NACIONAL DEL CENTRO DEL PERÚ
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∂ ( δ 2 ( 0.2 + 0.1 ) ) = &.!/ 4 +// = #+/2.+&+ ∂ t
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