flujo de crag directo
TRANSCRIPT
UNIVERSIDAD SIMÓN BOLÍVAR
DECANATO DE ESTUDIOS PROFESIONALES
COORDINACIÓN DE INGENIERÍA ELÉCTRICA
APLICACIÓN DE FLUJO DE CARGA DIRECTO A
REDES DE DISTRIBUCIÓN DE GRAN TAMAÑO
POR
HENRY ROMAN ESCOBAR MELGAREJO
JOSÉ RAFAEL PÉREZ ZORRILLA
PROYECTO DE GRADO
PRESENTADO ANTE LA ILUSTRE UNIVERSIDAD SIMÓN BOLÍVAR
COMO REQUISITO PARCIAL PARA OPTAR AL TÍTULO DE
INGENIERO ELECTRICISTA
Sartenejas, Noviembre de 2010
UNIVERSIDAD SIMÓN BOLÍVAR
DECANATO DE ESTUDIOS PROFESIONALES
COORDINACIÓN DE INGENIERÍA ELÉCTRICA
APLICACIÓN DE FLUJO DE CARGA DIRECTO A
REDES DE DISTRIBUCIÓN DE GRAN TAMAÑO
POR
HENRY ROMAN ESCOBAR MELGAREJO
JOSÉ RAFAEL PÉREZ ZORRILLA
TUTOR:PROF. PAULO DE OLIVEIRA
PROYECTO DE GRADO
PRESENTADO ANTE LA ILUSTRE UNIVERSIDAD SIMÓN BOLÍVAR
COMO REQUISITO PARCIAL PARA OPTAR AL TÍTULO DE
INGENIERO ELECTRICISTA
Sartenejas, Noviembre de 2010
•UNIVERSIDAD SIMÓN BOLÍVAR
Decanato de Estudios Profesionales
Coordinación de Ingeniería Eléctrica
ACT A DE EVALUACIÓN DEL PROYECTO DE GRADO
CÓDIGO DE LA ASIGNATURA: EP 370 (
ESTUDIANTES:1:l EN ~ 1 ESéO~A 'P1 . CARNET: D<{ 3(; 04 (
JOSf 'ft ~b2 CARNET: 03 3(,313
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TUTOR: Prof.\co,-..>\.o ~-'< a\ '-.)Q.\,c...... CO-TUTOR: Prof. _
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APROBADO: ~ REPROBADO: DOBSERV ACIONES:
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Co-Tutor Jurado
Nota: Colocar los sellos de los respectiyos Departamentos. Para jurados externos, usar el sello de laCoordinación.
APLICACIÓN DE FLUJO DE CARGA DIRECTO A REDES DE DISTRIBUCIÓN DE
GRAN TAMAÑO
Por:
HENRY ROMAN ESCOBAR MELGAREJO
JOSÉ RAFAEL PÉREZ ZORRILA
RESUMEN
En este trabajo se implementa un análisis de flujo de carga, (FDC ), para ser aplicado
en redes de distribución de gran tamaño. El FDC implementado es un método de barrido
unidireccional directo definido por el uso de una matriz única que caracteriza la resistencia y
admitancia de las líneas y la topología de la red. El algoritmo fue codificado en MATLAB. El
mismo fue desarrollado en forma modular utilizando un patrón de diseño de software Modelo-
Vista-Controlador, (MVC ). El FDC implementado en el presente trabajo está basado en Teng
(2003) y se incluye la modificación propuesta por De Oliveira (2010) para utilizar números
reales en lugar de números complejos. Como contribución principal se establece un esquema de
datos en RAM (Random Access Memory), lo cual optimiza el proceso de cálculo del FDC. El
algoritmo fue validado con redes de 4, 7, 12 y 69 barras. Además, se verificaron los resultados
del circuito de 7 barras con un Newton-Raphson (NR), (MATPOWER). Finalmente, se aplicó
exitosamente en una muestra significativa de la red de la Gran Caracas; conformada por 530
circuitos distribuidos en 78 S/E y 64.251 nodos, que corresponden al 70 % de la demanda de
las Gran Caracas. El resultado del FDC para el 70 % de la Gran Caracas se obtuvo en 0, 76
segundos. El FDC directo estudiado es un método eficiente con un gran potencial para ser
aplicado en el análisis y planificación del Sistema Eléctrico de Distribución (SDEE ).
iv
A mi madre; Gloria, por su infinito amor y sabiduría.
a mi padre; Henry, por su esfuerzo.
Henry Escobar.
A mi madre; Olivia, por su apoyo y constancia.
José Pérez.
v
AGRADECIMIENTOS
Quisiera agradecer a Dios y a las tres personas que influyeron en quien soy hoy en día: mi
madre, mi padre y mi abuela “Pancha”. Gracias a su esfuerzo es que he llegado hasta aquí.
A mis tías Carmen y Zulay que me vieron crecer y siempre han estado apoyándome.
A mi abuelo Santiago y mis tíos con los que siempre he contado y podré contar en cualquier
momento.
A mi hermano, mis primos y mi sobrina con los que he crecido y he compartido valiosos
momentos de mi vida.
A mis amigos y compañeros del equipo de rugby subacuático CONGRIOS USB con los
que comparto el vicio por el agua y la adrenalina.
A mis amigos: Martha, Gabriela, Debora, Leopold, Ariaam, Fedora y José Rafael. Por ser
unos excelentes amigos y personas.
A todos; no existen palabras suficientes para agradecer lo que han hecho por mí.
Todo es cuestión de actitud!!!
Henry Escobar.
Quisiera agradecer a mi madre por ser la persona que siempre ha estado conmigo.
Gracias!!!
José Rafael Pérez.
Gracias al Prof. Paulo De Oliveira por ayudarnos y apoyarnos en todo momento durante la
tesis. Gracias a Benicia, María Teresa y al Prof. Miguel Martínez por estar a la orden cada
vez que necesitábamos ayuda. Gracias a Angel y Luis Gerardo por darnos una mano justo
cuando era necesario.
Gracias Totales.
vi
LISTA DE SÍMBOLOS
i, j Nodos del sistema.
n Número de nodos del sistema.
Pi Potencia activa en el nodo i.
Qi Potencia reactiva en el nodo i.
Vi Tensión en el nodo i.
Vj Tensión en el nodo j.
Gii Conductancia propia en el nodo i.
Bii Susceptancia propia en el nodo i.
Yij Admitancia entre los nodos i-j.
θij Ángulo de la admitancia de una línea entre los nodos i-j.
δi Ángulo de la tensión en el nodo i.
δj Ángulo de la tensión en el nodo j.
s Nodo emisor.
r Nodo receptor.
Pr Potencia activa en el nodo r.
Qr Potencia reactiva en el nodo r.
Ps Potencia activa en el nodo s.
Qs Potencia reactiva en el nodo s.
Vs Tensión en el nodo s.
Vr Tensión en el nodo r.
Z Impedancia de la línea entre los nodos s-r.
vii
R Resistencia de la línea entre los nodos s-r.
X Reactancia de la línea entre los nodos s-r.
φZ Ángulo de la impedancia de la línea entre los nodos s-r.
δs Ángulo de la tensión en el nodo s.
δr Ángulo de la tensión en el nodo r.
Vs Tensión en el nodo s forma compleja.
Vr Tensión en el nodo r forma compleja.
Is Corriente en el nodo s forma compleja.
Z Impedancia de la línea entre los nodos s-r en forma compleja.
BIBC Matriz de corrientes inyectadas a corrientes de rama (Bus Injection to Branch
Current), [19].
BCBVMatriz de corrientes de rama a voltajes nodales (Branch Current to Bus Voltage),
[19].
DLF Matriz de flujo de carga (Distribution Load Flow), [19].
K Iteración K-ésima.
[IK ] Vector columna de corrientes inyectadas.
∆V K+1 Vector de variación de tensiones nodales en la iteración K + 1.
V K+1 Vector de tensiones nodales en la iteración K + 1.
V0 Vector de tensiones nodales iniciales.
(p.u) Por Unidad.
[Y ] Matriz de Admitancias Nodales.
IKi Corriente inyectada en el nodo i durante la iteración K
Pi Potencia activa demandada en el nodo i.
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Qi Potencia reactiva demandada en el nodo i.
V Ki Tensión en el nodo i durante la iteración K.
V K+1i Tensión en el nodo i durante la iteración K + 1.
ε Tolerancia utilizada como criterio de convergencia.
S Vector de potencias inyectadas.
[V 0x ] Vector de la componente real de las tensiones iniciales, [20].
[V 0y ] Vector de la componente imaginaria de las tensiones iniciales, [20].
V 0xi Componente real de la tensión inicial en la barra i, [20].
V 0yi Componente imaginaria de la tensión inicial en la barra i, [20].
V Kxi Componente real de la tensión en la barra i durante la iteración K, [20].
V Kyi Componente imaginaria de la tensión en la barra i durante la iteración K, [20].
[T ] Matriz de corrientes inyectadas a corrientes de rama, [20].
[DR] Matriz diagonal de resistencias de líneas, [20].
[DX ] Matriz diagonal de reactancias de líneas, [20].
[TRX] Matriz de flujo de carga, [20].
[I] Matriz de corrientes inyectadas, [20].
IKxi Componente real de la corriente en la barra i durante la iteración K, [20].
IKyi Componente imaginaria de la tensión inicial en la barra i, [20].
VSlack Tensión en la barra de referencia o slack.
∆Vi Caída de tensión en el nodo i respecto a la barra slack (%).
Lmp Pérdidas activas totales.
Lmq Pérdidas reactivas totales.
ix
Gij Conductancia entre los nodos i-j.
Bij Susceptancia entre los nodos i-j.
Rij Resistencia entre los nodos i-j.
Xij Reactancia entre los nodos i-j.
LS/E Pérdidas de potencia en una subestación
LT Pérdidas totales en la red.
Pij Flujo de potencia activa entre los nodos i-j.
Qij Flujo de potencia reactiva entre los nodos i-j.
Sij Flujo de potencia aparente entre los nodos i-j.
Smaxij Potencia nominal máxima permitida por un conductor.
η Factor de seguridad.
CF Capacidad Firme.
C Factor de escalamiento.
N Número de transformadores operativos.
kV Am Potencia nominal del transformador.
kV Amax Potencia del transformador de mayor capacidad.
VBASE Base de voltaje.
SBASE Base de potencia.
kV Kilovoltio.
MVA Mega voltio-Amper.
Pgen0 Valor inicial de la potencia activa generada (p.u).
Qgen0 Valor inicial de la potencia reactiva generada (p.u).
x
Pload Potencia activa demandada (p.u).
Qload Potencia reactiva demandada (p.u).
fp Factor de potencia.
Ir Componente real de la corriente en el nodo i (p.u).
Ij Componente imaginaria de la corriente en el nodo i (p.u).
Vr Componente real de la tensión en el nodo i (p.u).
Vj Componente imaginaria de la tensión en el nodo i (p.u).
∆.Vr Diferencia de la componente real de tensión entre iteraciones.
∆.Vj Diferencia de la componente imaginaria de tensión entre iteraciones.
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LISTA DE ABREVIATURAS
FDC Flujo de Carga.
MVC Patrón de Diseño Modelo-Vista-Controlador.
MATLAB Herramienta Computacional de Cálculo.
NR Newton-Raphson.
SDEE Sistema Eléctrico de Distribución.
GS Gauss-Seidel.
FDCB Flujo de Carga de Barrido.
INDENE Instituto de Energía de la Universidad Simón Bolívar.
SEDDGE Sistemas Eléctricos de Distribución a Gran Escala.
FDCT Flujo de Carga Directo desarrollado por Teng, [19].
TRX Flujo de Carga Directo desarrollado por De Oliveira, [20].
SEDDGT Sistema Eléctrico de Distribución de Gran Tamaño.
EDC Electricidad de Caracas.
SEP Sistema Eléctrico de Potencia.
SEDT Sistema Eléctrico de Transmisión.
NRD Newton-Raphson Desacoplado.
slack Barra de Referencia del Sistema.
backward sweep Barrido Hacia Atrás en el FDCB.
forward sweep Barrido Hacia Adelante en el FDCB.
KVL Ley de Kirchhoff de Voltajes.
KCL Ley de Kirchhoff de Corrientes.
xii
BIBC Matriz de Corrientes Inyectadas a Corrientes de Rama o Bus Injection to Branch
Current.
BCBV Matriz de Corrientes de Rama a Voltajes Nodales o Branch Current to Bus
Voltage.
DLF Matriz de Flujo de Carga o Distribution Load Flow.
backward and forward sweep Proceso de Barrido Hacia Adelante y Hacia Atrás.
MATPOWER Paquete de Simulación de Sistemas de Potencia de MATLAB.
ASP Sistema de Análisis y Simulación de Redes Primarias.
FDCO Flujo de Carga Óptimo.
NRDR Newton-Raphson Desacoplado Rápido.
LP Programación Lineal.
CE Capacidad de Emergencia .
CN Capacidad Nominal.
S/E Subestación.
CF Capacidad Firme.
SAP Software de Gestión y Estrategia.
GIS Software de Estimación de Demanda.
SCADA Software de registro de carga y data del sistema.
ORM Mapeadores Objeto-Relacionales.
.DAT Formato Original de la Base Datos.
.MAT Formato de los Archivos Generados con el Algoritmo Implementado.
RAM Memoria de Acceso Aleatorio (Random Access Memory).
GW Gigavatio.
xiii
Índice general
INTRODUCCIÓN 1
1. ANTECEDENTES 5
1.1. Características del Sistema de Distribución . . . . . . . . . . . . . . . . . . . 5
1.2. El Flujo de Carga de Distribución . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1. FDC para Topologías Malladas . . . . . . . . . . . . . . . . . . . . . 8
1.2.2. FDC para Topologías Radiales . . . . . . . . . . . . . . . . . . . . . 9
1.2.3. Flujos de Carga de Barrido (FDCB) o “forward and backward sweep” 10
2. METODOLOGÍA 14
2.1. Algoritmos Implementados . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1. Flujo de Carga Directo Complejo, (FDCDC ), [19] . . . . . . . . . . . 15
2.1.2. Flujo de Carga Directo TRX . . . . . . . . . . . . . . . . . . . . . . 18
2.2. Herramientas Computacionales . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1. MATPOWER, [52] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.2. ASP, [54] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3. Condiciones de Operación de la Red . . . . . . . . . . . . . . . . . . . . . . . 24
xiv
2.3.1. Análisis por Caída de Tensión . . . . . . . . . . . . . . . . . . . . . . 25
2.3.2. Análisis por Pérdidas de Potencia . . . . . . . . . . . . . . . . . . . . 25
2.3.3. Análisis por Capacidad Amperimétrica de Conductores . . . . . . . . 26
2.3.4. Análisis de Capacidad Firme de una S/E . . . . . . . . . . . . . . . . 28
3. IMPLEMENTACIÓN 29
3.1. Arquitectura del Sistema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2. Descripción del Algoritmo Implementado . . . . . . . . . . . . . . . . . . . . 34
3.3. Datos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4. Estructura del Programa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4.1. Adquisición de Datos . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4.2. Proceso Iterativo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4. CASOS DE ESTUDIOS Y ANÁLISIS DE RESULTADOS 39
4.1. Aplicación en Circuito de 7 Barras . . . . . . . . . . . . . . . . . . . . . . . 39
4.1.1. Cálculo de la Matriz TRX . . . . . . . . . . . . . . . . . . . . . . . . 40
4.1.2. Proceso Iterativo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2. Validación de Resultados . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3. Estudio Comparativo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4. Aplicación en Redes de Gran Tamaño . . . . . . . . . . . . . . . . . . . . . . 51
4.4.1. Tiempo de cómputo . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4.2. Análisis de Condiciones Operacionales de una Red de Gran Tamaño . 54
xv
CONCLUSIONES 63
REFERENCIAS BIBLIOGRÁFICAS 65
A. CASO EXPLICATIVO: MATRICES BIBC Y BCBV 71
A.1. Matriz BIBC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.2. Matriz BCBV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
B. ALGORITMO IMPLEMENTADO TRX 74
B.1. Algoritmo de Adquisición de Datos . . . . . . . . . . . . . . . . . . . . . . . 74
B.2. Flujo de Carga TRX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
C. EJEMPLO DE REPORTE DEL ALGORITMO DESARROLLADO 87
D. CASO DE ESTUDIO: CIRCUITO DE 4 BARRAS 98
E. REPORTE DEL MATPOWER PARA EL CASO DE 7 BARRAS 100
F. “A COMPENSATION-BASED POWER FLOW METHOD FOR WEAK-
LY MESHED DISTRIBUTION AND TRANSMISSION NETWORKS”,
SHIRMOHAMMADI ET AL. 1988, [11] 103
G. “A DIRECT APPROACH FOR DISTRIBUTION SYSTEM LOAD FLOW
SOLUTION”, TENG. 2003, [19] 114
H. “THE DISTRIBUTION TRX-POWER FLOWMETHOD”, DE OLIVEIRA.
2010, [20] 121
I. “OPTIMAL SIZING OF CAPACITORS PLACED ONRADIAL DISTRIBUTION
SYSTEM”, BARAN, M. E. ET AL. 1989, [31] 139
xvi
J. “SIMPLE AND EFFICIENT COMPUTER ALGORITHMTO SOLVE RA-
DIAL DISTRIBUTION NETWORKS”, RANJAN, R. ET AL. 2003, [27] 150
xvii
Índice de tablas
2.1. Caída de Tensión Máxima Permitida, [56] . . . . . . . . . . . . . . . . . . . 25
4.1. Datos de Línea (Ldat) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2. Datos de Nodos, (Bdat) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3. Valores de potencia y tensión inicial (V0, P0) . . . . . . . . . . . . . . . . . . 44
4.4. Resultados de la Primera Iteración . . . . . . . . . . . . . . . . . . . . . . . 45
4.5. Resultados de la Segunda Iteración . . . . . . . . . . . . . . . . . . . . . . . 45
4.6. Resultados de la Tercera Iteración . . . . . . . . . . . . . . . . . . . . . . . . 46
4.7. Tensiones nodales en módulo y ángulo . . . . . . . . . . . . . . . . . . . . . 46
4.8. Tensiones Nodales Validadas en Módulo y Ángulo . . . . . . . . . . . . . . . 47
4.9. Tiempo de cómputo de [19] y [20] para circuitos de 4, 7, 12 y 69 barras . . . 50
4.10. Tiempo de cómputo del ASP para red de 530 circuitos de la EDC . . . . . . 51
4.11. Tiempo de cómputo del algoritmo implementado para red de 530 circuitos de
la EDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.12. Espacio Ocupado en Disco por redes de 10, 50, 100, 200 y 530 circuitos . . . 54
4.13. Condiciones de Operación de la Red, 530 circuitos. . . . . . . . . . . . . . . 62
D.1. Datos de línea (Ldat) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
D.2. Datos de nodos (Bdat) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
xviii
Índice de figuras
1.1. Red de Distribución. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1. Diagrama de Flujo del Algoritmo Propuesto por Teng, [19]. . . . . . . . . . . 19
2.2. Diagrama de Flujo del Algoritmo Propuesto por De Oliviera, [20]. . . . . . . 22
2.3. Porcentaje de Carga de un Cable Subterráneo, [56]. . . . . . . . . . . . . . . 27
3.1. Base de Datos Unificada, [49]. . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2. Sincronización de Base de Datos, [49]. . . . . . . . . . . . . . . . . . . . . . . 30
3.3. Modelo MVC, [50]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4. Esquema General de Herramienta de Planificación Corto-Mediano Plazo (HPCMP)
- Visualización de Largo Plazo, [50]. . . . . . . . . . . . . . . . . . . . . . . . 33
3.5. Esquema MVC del programa implementado. . . . . . . . . . . . . . . . . . . 35
3.6. Diagrama de Flujo del Algoritmo Desarrollado. . . . . . . . . . . . . . . . . 36
4.1. Estudio Comparativo: Tiempo de Cómputo, [20]. . . . . . . . . . . . . . . . 49
4.2. Tiempo de Cómputo del ASP para la Red de la Gran Caracas. . . . . . . . . 52
4.3. Tiempo de Cómputo del Algoritmo Implementado para la Red de la Gran
Caracas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4. Perfil de Caída de Tensión en (%) del circuito ANT_A01. . . . . . . . . . . 55
xix
4.5. Potencia Activa Demandada y Pérdidas Técnicas del Circuito ANT_A01. . 56
4.6. Potencia Reactiva Demandada y Pérdidas Técnicas del Circuito ANT_A01. 56
4.7. Capacidad Amperimétrica de los conductores del circuito ANT_A01. . . . . 57
4.8. Potencia Entregada y Perdida por Circuito de la S/E ANT . . . . . . . . . . 58
4.9. Capacidad Amperimétrica de los conductores de la S/E ANT . . . . . . . . 59
4.10. Capacidad Firme de las S/E ANT y SRO. . . . . . . . . . . . . . . . . . . . 60
4.11. Potencia Activa Demandada y Pérdidas Técnicas de la red. . . . . . . . . . . 61
4.12. Capacidad Amperimétrica de los conductores de la red. . . . . . . . . . . . . 62
A.1. Sistema de Distribución de Ejemplo. . . . . . . . . . . . . . . . . . . . . . . 71
xx
INTRODUCCIÓN
El Flujo de Carga, FDC, es un método de análisis numérico para la determinación del
estado de los Sistemas Eléctricos de Potencia, SEP. El FDC, como herramienta de análisis,
cobra mayor importancia en el sistema de gestión eléctrica actual. Esta herramienta no sólo
permite conocer el estado de la red en un momento determinado sino que también permite
realizar aplicaciones como despacho de generación distribuida, equilibrio de fases, control de
tensiones, colocación óptima de capacitores y estudios de planificación de corto y mediano
plazo, [20].
El FDC es un estudio que requiere la resolución de sistemas de ecuaciones no lineales. Por
lo tanto, los cálculos implicados se caracterizan por presentar un alto grado de complejidad y
resulta necesario el uso de procesos iterativos para encontrar la solución con la mayor exacti-
tud posible. Aún más, el número de variables a manejar aumenta en función del tamaño del
sistema eléctrico en estudio y con esto aumenta la complejidad del cálculo a realizar. Además,
es bien conocido que los Sistemas Eléctricos de Distribución, SEDD, actuales se caracterizan
por ser de gran tamaño; lo cual implica que el número de variables a manejar en el FDC
también lo es. Debido a esto, se hace imprescindible el uso de herramientas computacionales
para poder estudiar dichos sistemas de forma eficiente.
En la actualidad hay diversas aplicaciones computacionales que son utilizadas como he-
rramientas para el análisis de flujo de carga. La mayoría de éstas están basadas en métodos
numéricos, como el Newton-Raphson (NR), [1]. Dicha metodología fue desarrollada y usada
para la operación, control y planificación de sistemas de transmisión; cuya configuración es
típicamente mallada [2, 3]. Pero al ser empleado en SDEE presenta problemas de convergencia
1
2
y eficiencia ya que se produce singularidad en la matriz Jacobiana. Debido, principalmente,
a su estructura radial, baja relación x/r, líneas no traspuestas y cargas desbalanceadas entre
fases [2, 4, 5].
Tomando en cuenta lo anterior y considerando la topología radial del sistema de distribu-
ción, se han desarrollado nuevas metodologías de flujo de carga que explotan la radialidad
del circuito; es decir, no requieren inversión de Jacobiano al utilizar técnicas conocidas como
el Gauss-Seidel GS o de barrido, [18, 22, 31, 33, 44]. Finalmente, dichos algoritmos se ex-
tendieron para poder ser aplicados en sistemas débilmente mallados [11, 12, 33].
Los métodos desarrollados para solucionar el FDC aplicado a sistemas radiales de dis-
tribución, llamados típicamente flujos de carga de barrido (FDCB), pueden ser catalogados
en dos grupos [2]. En el primer grupo, aquellos que hacen ciertas modificaciones a técnicas
ya existentes como el NR [6]-[10]. Y en el segundo, están aquellos que hacen un proceso
de barrido hacia delante y hacia atrás, (forward and backward sweep), usando las leyes de
Kirchhoff [11]-[21] o utilizando la ecuación bicuadrática [22]-[36].
Típicamente, los FDCB presentan una tasa lenta de convergencia. Sin embargo, son al-
tamente eficientes desde el punto de vista computacional porque no es necesario invertir
matrices [20]; hecho que resulta importante al resolver redes de gran tamaño. Con las he-
rramientas computacionales disponibles en la actualidad se ha reducido considerablemente
el tiempo empleado en obtener una solución del FDC en redes de distribución de gran tamaño.
El presente trabajo se realizó partiendo de una línea de investigación desarrollada por el
Instituto de Energía de la Universidad Simón Bolívar (INDENE ) para el desarrollo de una
herramienta computacional para el análisis de Sistemas Eléctricos de Distribución a Gran Es-
cala, SEDDGE. El proyecto general surge por la necesidad de desarrollar nuevos sistemas de
análisis con una estructura abierta (código abierto) a fines de integrar efectivamente los pro-
cesos atendiendo los siguientes criterios: código modular, programación orientada a objeto,
3
aplicación web multiplataforma, control de versiones mediante un esquema colaborativo, [50].
Este trabajo aborda la resolución de Sistema Eléctrico de Distribución de Gran Tamaño,
SEDDGT, mediante la aplicación de la metodología de FDC unidireccional denominada
“Flujo de Carga Directo”. Se denomina directo por cuanto se obtiene el estado de la red
iterativamente a partir de una matriz única que contiene información del sistema: dimen-
sionamiento y topología. El algoritmo se basa en Teng, [19], el cual ha sido modificado por
De oliveira [20] con el objetivo de realizar operaciones con números reales y, de esta forma,
mejorar la eficiencia del algoritmo una vez implementado. El algoritmo se caracteriza por una
lógica de cálculo sencilla (suma y multiplicación de matrices), con lo cual se evitan opera-
ciones que requieran invertir matrices como en el caso de la matriz Jacobiana del NR. Aún
más, las matrices requeridas para realizar el FDC estarán disponibles en memoria RAM cada
vez que se requiera. Con esto, se optimiza el algoritmo y es posible obtener resultados para
SEDDGT en el orden de los segundos. Razones que hacen del FDC un algoritmo poderoso
y muy útil en el área de planificación de SDEE.
Objetivos
Para lograr el alcance propuesto, se plantearon un objetivo general y varios específicos,
éstos se muestran a continuación:
Objetivo General: Implementar un flujo de carga en base a la técnica de barrido unidi-
reccional, basada en la construcción de una matriz única que incluye impedancias de línea y
la topología del sistema, en redes de distribución de gran tamaño.
Objetivos Específicos:
Desarrollar un flujo de carga prototipo.
Realizar un estudio comparativo de los tiempos de cómputo entre el FDC implementado
y otros FDC eficientes como el NR y el Teng.
4
Implementar eficientemente el programa en plataforma computacional.
Aplicación en redes ejemplo reportadas en literatura y redes de distribución reales de
gran tamaño.
Justificación de la Tesis
El proyecto surge de la necesidad de hacer estudios de red a gran escala en tiempo útil
para cumplir los fines de la operación y planificación de SDEE.
Organización de la Tesis
El presente trabajo se divide en 4 capítulos. En el primer capítulo se introducen los
antecedentes de los FDC aplicados a redes de distribución. En el capítulo 2, se presenta el
marco metodológico desarrollado a lo largo del proyecto. En el tercer capítulo, se describe el
proceso de implementación del algoritmo: empezando por la arquitectura de diseño utilizada
y la descripción de los módulos desarrollados. En el capítulo 4 se habla de los casos de estudio:
el primero, es un estudio comparativo de los tiempos de cómputo del FDCT y TRX al ser
aplicados en redes de 4, 7, 12 y 69 nodos. Se mostrará la eficiencia del segundo respecto
al primero; en segundo lugar, se realiza el caso de 7 barras paso por paso para ilustrar el
proceso de cálculo del TRX ; en tercer lugar, se validan los resultados obtenidos en el caso
de 7 barras utilizando un NR eficiente, (MATPOWER); en cuarto lugar, se implementó a
gran escala en SDEE y se compararán los tiempos de cómputo del TRX con el programa
utilizado por la EDC para este tipo de estudios. Finalmente, se presentan las conclusiones y
recomendaciones pertinentes. Los anexos se encuentran al final del documento.
CAPÍTULO 1
ANTECEDENTES
El Sistema Eléctrico de Potencia, SEP, se encarga de la generación, transmisión y dis-
tribución de la energía eléctrica. Como consecuencia del gran tamaño y la alta complejidad
en funciones de dicho sistema; éste se encuentra dividido en dos grandes sub-sistemas:
Sistema Eléctrico de Transmisión, SEDT. Conformado por el área de generación
y la red de transmisión; los cuales operan a grandes niveles de tensión. Se encarga de
generar y transmitir grandes bloques de potencia.
Sistema Eléctrico de Distribución, SDEE. Conformado por las subestaciones y
la red de distribución; los cuales operan a niveles intermedios y bajos de tensión. La
función de este sistema es distribuir los grandes bloques de potencia a los consumidores
finales.
1.1. Características del Sistema de Distribución
El SDEE presenta una serie de características específicas que lo diferencian considerable-
mente del sistema de transmisión. Estas características se presentan a continuación:
Relación x/r. Los conductores de la red de distribución presentan una baja relación
x/r, ya que x ≈ r. Mientras que, en los SEDT se tiene que normalmente x >> r; razón
6
sobre la cual se basa el “Principio de Desacople”:
“los cambios en la potencia activa deben manifestarse sobre los ángulos de fases del
sistema (...) los cambios de potencia reactiva deben reflejarse en las magnitudes de
tensión” [38, pág. 159].
Debido a la baja relación x/r presente en las redes de distribución, no se cumple dicho
principio. Razón por la cual, el Newton-Raphson, NR, tiene problemas de convergencia
y el Newton-Raphson Desacoplado, NRD, no es funcional.
Diversidad de Cargas. Debido a la diversidad de consumidores en la red es posible
encontrar diversos tipos de demanda, desde zonas rurales con densidades del orden de
kV A/km2, hasta zonas urbanas con densidades en el orden deMVA/km2 [3]. Además,
el mismo circuito puede ser usado para suplir cargas residenciales, comerciales y/o
industriales.
Cargas Desbalanceadas. Usualmente se emplean acometidas de dos, tres y cuatro
hilos para alimentar cargas tanto trifásicas como bifásicas y/o monofásicas. Esta car-
acterística tiende a desbalancear las redes de distribución.
1.2. El Flujo de Carga de Distribución
El FDC es un algoritmo que permite calcular las tensiones nodales, en módulo y ángulo,
en régimen permanente de un SEP. El régimen permanente es aquel estado cuasiestacionario
en el cual existe un equilibrio de las potencias y las variables de tensión y frecuencia no pre-
sentan variaciones significativas [38]. Es bien conocido que la carga varía constantemente en
el tiempo, razón por la cual se utiliza el termino cuasiestacionario. Sin embargo, se entiende
que el FDC analiza el sistema en estados puntuales [3]. También es utilizado en el área de
planificación por la capacidad de introducir suposiciones de estados futuros de la red. Razón
por la cual la herramienta posee una gran versatilidad y extraordinario potencial en el análisis
de SEP.
7
Existen dos tipos de datos de entrada para el algoritmo, éstos son:
Datos de Red. Hacen referencia a las especificaciones de las conexiones y a los
parámetros de las líneas, transformadores y compensadores presentes en el circuito.
Datos de Nodos. Hacen referencia a las especificaciones de los datos operacionales
de cada nodo, como tipo de nodo, su potencia consumida o generada, su magnitud de
tensión, etc.
El problema es no-lineal y no puede ser resuelto analíticamente. Razón por la cual, se
recurre a técnicas numéricas iterativas para hallar la solución [3]. Las ecuaciones que describen
al problema se pueden plantear como se presenta en las ecuaciones 1.1 y 1.2, [37].
Pi = |Vi|2Gii +n∑
j=1
|Vi||Vj||Yij| cos(θij + δj − δi) (1.1)
Qi = −|Vi|2Bii −n∑
j=1
|Vi||Vj||Yij| sin(θij + δj − δi) (1.2)
Donde, i, j son los nodos del sistema. n representa el número de nodos del sistema. Pi y
Qi es la potencia activa y reactiva en el nodo i. Vi y Vj son las tensiones en los nodos i y j.
Gii y Bii son la conductancia y la susceptancia propias del nodo i. Yij es la admitancia entre
los nodos i-j. θij es el ángulo de la admitancia de una línea entre los nodos i-j. δi y δj son los
ángulos de las tensiones en los nodos i y j, respectivamente. En [37] se encuentra con mayor
detalle las ecuaciones planteadas en los FDC.
Una vez calculadas las tensiones, es posible calcular las corrientes por el circuito, flujos
de potencia, capacidades amperimétricas de conductores y pérdidas en la red.
Para realizar ciertos análisis no es estrictamente necesario considerar el desbalance de la
red. Cuando este es el caso, es posible asumir la red balanceada y realizar la modelación de la
red mediante un equivalente unifilar [11]. Para efecto del análisis realizado, las modelaciones
fueron realizadas considerando redes balanceadas.
8
Tipos de Nodos Dependiendo de las condiciones de contorno que se especifiquen, se
pueden clasificar los nodos del sistema en tres grupos; [37]:
Nodo de Referencia o Slack. Es una barra de generación, en la que se asume
conocido el módulo y ángulo de la tensión.
Nodos PQ. Son aquellos nodos en los que se especifican las potencias activa y reactiva
netas inyectadas. Los nodos PQ son los más abundantes en los sistemas de distribución.
Nodos PV. Son nodos en los que se especifican la potencia activa y el módulo de la
tensión. Este tipo de nodos son poco comunes en sistemas de distribución.
1.2.1. FDC para Topologías Malladas
Los primeros algoritmos de FDC, como el Newton-Raphson (NR) o el Gauss-Seidel (GS ),
fueron desarrollados para la operación, control y planificación de SEDT, [2, 37, 1]. La prin-
cipal diferencia entre estos últimos respecto a los FDC para topologías radiales, es que los
mismos pueden ser aplicados independientemente de la configuración de la red [3].
El NR es un método basado en un sólido fundamento matemático como lo son las series
de Taylor [3]. Este método utiliza un proceso iterativo mediante el cual se aproxima a la
solución linealizando las ecuaciones de potencia (ver ecuaciones 1.1 y 1.2). En [3, 37, 38] es
posible encontrar la descripción del proceso algorítmico de este FDC. Además, este método
requiere de un gran esfuerzo computacional y un elevado tiempo de cómputo.
“La aplicación del método a un sistema no lineal de ecuaciones de orden (n) va a implicar
en cada iteración, la formación del Jacobiano de orden (nxn) y su subsiguiente inversión, lo
cual toma un tiempo no despreciable” [38, pág. 129].
El método GS presenta la ventaja de no utilizar matrices Jacobianas; lo cual lo convierte
en un algoritmo mucho más sencillo y requiere un menor esfuerzo computacional que el NR.
9
“Su simplicidad matemática es notoria, no requiriéndose la inversión de matrices en lo absolu-
to por lo cual su programación digital es muy rápida” [38, pág. 109]. En [3, 38] se encuentra la
descripción de las ecuaciones utilizadas por el FDC y la descripción del proceso algorítmico
del mismo.
Sin embargo, el GS requiere de un gran número de iteraciones para converger. Por lo
general, realiza un número de iteraciones aproximadamente igual al número de nodos del
sistema estudiado [38].
En la literatura [4, 11, 14, 18, 19, 33, 39, 40, 41, 42, 43, 44] es posible encontrar una
amplia discusión sobre los problemas de convergencia o tiempos ineficientes de convergencia
de estos métodos al ser aplicados en redes del tipo “enfermas”. Las redes de distribución, por
su estructura radial y baja relación x/r entran dentro de esta categoría.
1.2.2. FDC para Topologías Radiales
Debido a los problemas presentados por los FDC NR y GS al ser aplicados en redes de
distribución, se ha hecho más frecuente el uso del FDC de barrido [16]. Estos algoritmos
sacan provecho de la topología, típicamente radial, de la red. Este método consiste en un
proceso iterativo en el cual se computan tensiones y corrientes mediante evaluaciones se-
cuenciales, llamados barridos. Está compuesto por dos etapas: barrido hacia atrás (backward
sweep) desde las cargas hacia la fuente; y el barrido hacia adelante (forward sweep) desde la
fuente hacia las cargas [2].
En el backward sweep se calculan las corrientes de rama y/o potencias del sistema, par-
tiendo desde el nodo más lejano a la fuente hasta llegar al más cercano a ésta. En el forward
sweep se calculan los voltajes en las barras del sistema, partiendo desde la barra más cercana
a la fuente hasta la más lejana a ésta [2]. De esta forma, se utiliza el backward sweep para
actualizar las corrientes y/o potencias del sistema, y con el forward sweep se actualizan los
voltajes en las barras.
10
1.2.3. Flujos de Carga de Barrido (FDCB) o “forward and backward
sweep”
Cálculo de Voltaje en Redes de Distribución
Como se describe en [2], se va a considerar el circuito de distribución mostrado en la figura
1.1. La potencia activa y reactiva en la barra receptora, puede escribirse como se indica en
las ecuaciones 1.3 y 1.4
Figura 1.1: Red de Distribución.
Pr =VsVr
Zcos(φZ − δs + δr)−
V 2r
Zcos(φZ) (1.3)
Qr =VsVr
Zsin(φZ − δs + δr)−
V 2r
Zsin(φZ) (1.4)
Usando la identidad trigonométrica, (ecuación 1.5):
cos2(φZ − δs + δr) + sin2(φZ − δs + δr) = 1 (1.5)
Despejando los términos cos(φZ−δs +δr) y sin(φZ−δs +δr) de 1.3 y 1.4, respectivamente
y sustituyéndolos en 1.5 se obtiene la forma general de la ecuación bi-cuadrática, dada en
1.6. Con la máxima raíz real de la ecuación 1.6 se obtiene la magnitud del voltaje en el nodo
receptor. El voltaje en el nodo receptor se puede escribir en función de la impedancia de la
línea, de la potencia y voltaje del nodo emisor, tal como se muestra en 1.7.
V 4r + 2Vr(PrR +QrX)− V 2
s V2r + (P 2
r +Q2r)Z2 = 0 (1.6)
11
Vr =
√√√√V 2s − 2(PsR +QsX) +
(P 2s +Q2
sZ2)
V 2s
(1.7)
También, se puede aplicar la ley de Kirchhoff de voltajes (KVL) para obtener los voltajes
en las barras del sistema. Aplicando KVL en el circuito de la figura 1.1. En la literatura
es posible encontrar varios algoritmos desarrollados en base a la ecuación cuadrática. En el
presente trabajo se enfoca la investigación en los FDCB basados en las leyes de Kirchhoff.
Vs = Vr + Is.Z (1.8)
Vr = Vs − Is.Z (1.9)
Descripción de FDCB basados en las leyes de Kirchhoff, [2]
Muchos de los algoritmos usados en sistemas de distribución ([11]-[21]) usan las leyes
de Kirchhoff (KVL y KCL, por sus siglas en inglés) para calcular las corrientes de rama
y los voltajes nodales en el backward y forward sweep, respectivamente. En [11], los auto-
res presentan un método de compensación para redes de distribución balanceadas radiales
y/o débilmente malladas. Para esto, utilizan una técnica de compensación multipuertos y
la formulación básica de las leyes de Kirchhoff. Los sistemas radiales son resueltos usando
un procedimiento de dos pasos: las corrientes de rama son calculadas (backward sweep) y
luego se actualizan los voltajes en los nodos usando la ecuación 1.8 para cada rama (forward
sweep). Se utiliza la diferencia de la potencia activa y reactiva en las cargas entre iteraciones
como criterio de convergencia. En [12] se aplica el algoritmo de [11] en redes desbalanceadas
(radiales o débilmente malladas).
En los algoritmos [13] y [14] se hacen pequeñas modificaciones a los algoritmos menciona-
dos anteriormente. En éstos, los circuitos radiales son resueltos tal como lo explica [11] y se
usa la diferencia de las tensiones entre iteraciones como criterio de convergencia. Sin embargo,
en [13] se incorpora el modelo de trasformadores trifásicos, propuesto por [45], en el análisis.
Mientras que, en [14] se describe un algoritmo computacional que permite encontrar el valor
12
exacto de las corrientes en todas las ramas del circuito.
En [15], se propone una versión modificada del método de barrido para redes radiales
balanceadas o desbalanceadas. En el backward sweep cada corriente de rama es calcula usan-
do KCL. Luego, conociendo las corrientes, se calculan los voltajes en las barras, usando la
ecuación 1.9, durante el forward sweep. La magnitud del voltaje en cada barra es comparada
con el valor de la iteración anterior. Si el error está dentro de determinada tolerancia el pro-
ceso se detiene (criterio de convergencia). En caso contrario, el proceso de barrido continúa
hasta que todas las tensiones cumplan dicho criterio. En el algoritmo propuesto en [16] se
calculan las corrientes de cada rama y, usando 1.8, se calculan los voltajes de barra. El voltaje
en la barra slack es calculado y se compara con un valor definido anteriormente. Si el error
está dentro de cierta tolerancia el proceso iterativo se detiene.
Liu et al, en [17], desarrollan un algoritmo aplicable a redes radiales y/o redes débilmente
malladas. La parte radial es resuelta como se describe en [16]. Sin embargo, [17] se diferencia
de [16] porque primero se ajustan los voltajes en las barras a través de un rango obtenido
por medio de la relación entre el voltaje computado en la fuente y el especificado. Se utiliza
la diferencia de los voltajes entre iteraciones como criterio de convergencia.
Un algoritmo para sistemas de distribución desbalanceadas es dado por [18]. En este
algoritmo, se utiliza la topología de la red para resolver el problema. Propone la creación de
dos matrices: una matriz que relaciona las corrientes inyectadas con las corrientes de rama,
BIBC (Bus Injection to Branch Current), y una matriz que relaciona las corrientes de rama
con los voltajes de nodo, BCBV (Branch Current to Bus Voltage). Luego, se obtiene la matriz
de “flujo de carga de distribución”, DLF (Distribution Load Flow) al multiplicar BCBV.BIBC.
Posteriormente, se calculan las caídas de tensión en las líneas multiplicando DLF y la matriz
de corrientes inyectadas, IK , 1.10. Las tensiones nodales se obtienen restando la tensión en
la fuente menos la caída de tensión en las líneas 1.11.
∆.V K+1 = DLF.IK (1.10)
13
V K+1 = V0 −∆.V K+1 (1.11)
Este algoritmo es mejorado en [19] para redes radiales y redes débilmente malladas. Para
esto se realizan algunas modificaciones en las matrices BIBC y BCBV. El criterio de conver-
gencia utilizado es la diferencia de voltajes entre iteraciones. Una modificación al algoritmo
anterior es realizada por [20]. En ésta se sigue la misma lógica utilizada por [19] pero se in-
troduce un cambio para poder realizar todos los cálculos con números reales, dicho algoritmo
es denominado TRX. Con esta modificación se mejora la eficiencia del algoritmo propuesto
por [19].
Un algoritmo basado en la KVL para sistemas de distribución monofásicos es propuesto
por [21]. El problema es resuelto considerando las cargas como impedancias constantes du-
rante el backward sweep. El voltaje en cada nodo es calculado usando la ecuación 1.8 durante
el backward sweep. Luego, se usa la relación entre la tensión impuesta en el nodo slack y su
nuevo valor calculado para obtener el valor actual del voltaje en las barras y de las corrientes
inyectadas y de rama.
CAPÍTULO 2
METODOLOGÍA
En este capítulo se describen en detalle los algoritmos implementados: en primer lugar, el
flujo de carga, FDC, desarrollado por Teng, denominado (FDCT ), [19], y en segundo lugar,
el FDC desarrollado por De Oliveira, denominado (TRX ), [20]. Además se describen dos
herramientas computacionales utilizadas para comparar y validar el algoritmo implementado;
es decir, el (TRX ). Luego, se muestran las ecuaciones requeridas para realizar el análisis del
sistema posterior a la solución del flujo de carga.
2.1. Algoritmos Implementados
Partiendo de una línea de investigación desarrollada por el profesor Paulo de Oliveira se
trabajó con el TRX. El mismo utiliza la misma lógica de cálculo que el FDCT pero incluye
una modificación para trabajar en números reales y no en complejos como lo hace FDCT.
Razón por la cual, es válido afirmar que el TRX es más eficiente que el FDCT. Dicha afir-
mación se comprueba en la sección 4.3 del capítulo 4
En la implementación de dichos algoritmos se asumieron las siguientes premisas:
Sistema balanceado.
Datos del sistema transformado en por unidad (p.u) en una base común.
15
Sistema exento de nodos PV.
Numeración del nodo slack como el número 1.
Redes de distribución radiales.
Para todos los algoritmos implementados, el proceso de numeración de los nodos se realiza
como se describe en [11].
2.1.1. Flujo de Carga Directo Complejo, (FDCDC ), [19]
Este algoritmo se caracteriza por realizar los cálculos en números complejos, como la
mayoría de los algoritmos de FDC. Además, utiliza una matriz única (DLF) para determinar
el estado del sistema y el proceso se realiza de forma unidireccional. La matriz [DLF ] refleja
las impedancias del sistema asociadas a la topología del mismo.
En el mismo, se utilizan dos matrices -la matriz de inyecciones de corriente a corriente de
rama (Bus Injection to Branch Current), BIBC, y la matriz de corrientes de rama a voltajes
nodales (Branch Current to Bus Voltage, BCBV - y una simple multiplicación de matrices
son usadas para obtener la solución del FDC.
Debido a las técnicas utilizadas en este algoritmo; se tiene que procesos como el de in-
versión de la matriz Jacobiana o la construcción de la matriz de admitancias nodales ([Y ])
requerido en los FDC tradicionales ya no son necesarios. Lo que hace del FDCT un algoritmo
robusto y eficiente en tiempo, [19]. De acuerdo a [19], el FDCT presenta un gran potencial
para ser utilizado en aplicaciones de automatización de redes de distribución. El artículo se
encuentra disponible en el apéndice G.
Descripción General
El proceso inicia numerando los nodos del circuito con el método propuesto por [11]; según
el cual se divide el circuito en capas y la primera va de la fuente hacia el(los) nodo(s) más
16
cercano(s) a la misma, la segunda capa va desde éstos últimos hacia los nodos más cercanos
a los mismos, se repite el proceso hasta haber dividido todo el circuito en capas. Luego se
enumeran los nodos empezando desde la capa número 1 hasta llegar a la última pero no se
puede realizar cambio de capas hasta que todos los nodos de la capa previa sean enumerados,
ver apéndice F.
Luego, se calculan las corrientes inyectadas en los nodos para la iteración K-ésima, IKi ,
(ecuación 2.1). Donde, Pi y Qi en la potencia activa y reactiva demandada en el nodo i, la
cual no cambia enter iteraciones. V Ki es la tensión en el nodo i durante la iteración K.
IKi = (
Pi + jQi
V Ki
)∗ (2.1)
Luego, se construyen las matrices BIBC y BCBV. En el apéndice A se ilustra el proceso de
armado de ambas matrices. BIBC está relacionada con la topología del circuito y BCBV está
relacionada con las impedancias de las líneas. Posteriormente, se multiplican ambas matrices
como lo muestra la ecuación 2.2 para obtener la matriz de impedancias de línea asociadas a
la topología del sistema. La matriz [DLF ] es de orden (n− 1)x(n− 1), donde n es el número
de nodos del sistema.
[DLF ] = [BCBV ][BIBC] (2.2)
En el siguiente paso, se calcula la diferencia de voltaje entre iteraciones, [∆.V K+1], uti-
lizando la ecuación 2.3. Dicho cálculo se realiza en forma matricial. [IK ] es el vector columna
de inyecciones de corriente.
[∆.V K+1] = [DLF ][IK ] (2.3)
Finalmente, se calculan las tensiones nodales 2.4. Donde, V K+1 es el vector de tensiones
nodales en la iteración K + 1 y V0 es el vector de tensiones nodales iniciales, típicamente la
tensión inicial en cada nodo se asume 1 p.u.
17
[V K+1] = [V 0] + [∆.V K+1] (2.4)
El proceso se detiene cuando la diferencia de voltaje entre iteraciones es menor a cierta
tolerancia, ecuación 2.5. Típicamente, ε se asume igual a 0, 001 p.u.
||V Ki | − |V K−1
i || ≤ ε (2.5)
Procesos Algorítmicos
Matriz BIBC. En el apéndice A se muestra un ejemplo ilustrativo para la construcción
de BIBC.
Paso 1. Para un sistema de m ramas y n nodos, la dimensión de la matriz BIBC es
m× (n− 1)
Paso 2. Si una línea (Bl) está entre los nodos i y j. Se copia la columna de la barra i,
se pega en la columna de la barra j y se coloca +1 en la fila l y la columna de la barra
j.
Paso 3. Repetir el paso 2 hasta que todas las líneas estén en la matriz BIBC.
Matriz BCBV. En el apéndice A se muestra un ejemplo ilustrativo para la construcción
de BCBV.
Paso 1. Para un sistema de m ramas y n nodos, la dimensión de la matriz BCBV es
(n− 1)×m
Paso 2. Si una línea (Bl) está entre los nodos i y j. Se copia la fila de la barra i, se
pega en la fila de la barra j y se coloca la impedancia de la línea Zij en la fila de la
barra j y la columna l.
Paso 3. Repetir el paso 2 hasta que todas las líneas estén en la matriz BCBV.
18
Solución del FDC
Paso 1. Enumerar los nodos del circuito utilizando el método propuesto por Shirmo-
hammadi, [11].
Paso 2. Leer datos referentes a las conexiones del sistema, potencia aparente en los
nodos e impedancias de las líneas.
Paso 3. Construir matrices BIBC y BCBV.
Paso 4. Pre-especificar V0 = Vslack e inicializar todas las tensiones al valor del nodo
slack.
Paso 5. Calcular IKi (2.1).
Paso 6. Calcular la matriz DLF (2.2).
Paso 7. Calcular la variación de voltaje en la iteración K + 1 (2.3).
Paso 8. Calcular la tensión en la iteración K + 1 (2.4).
Paso 9. Verificar si se cumple (2.5).
Paso 10. En caso de no cumplirse (2.5), volver al paso 5.
En la figura 2.1 se encuentra el diagrama del FDCT, [19].
2.1.2. Flujo de Carga Directo TRX
Este algoritmo se caracteriza por realizar los cálculos en números reales, lo cual optimiza
el proceso de cómputo. Además, utiliza una matriz única (TRX ) para determinar el esta-
do del sistema y el proceso se realiza de forma unidireccional. La matriz [TRX] refleja las
impedancias del sistema asociadas a la topología del mismo.
Este algoritmo es útil al ser aplicado con propósitos de planificación y evaluación del sis-
tema de distribución en tiempo real. El estado del sistema se obtiene usando el histórico de
19
Figura 2.1: Diagrama de Flujo del Algoritmo Propuesto por Teng, [19].
las medidas tomadas al mismo; teniendo en cuenta su topología en el presente y en el futuro.
El TRX resulta más eficiente que el FDCT porque el mismo trabaja con números reales a
diferencia del segundo lo hace en números complejos. Desde un punto de vista computacional,
consume implica más tiempo realizar operaciones en números complejos que en números
reales. El artículo se encuentra disponible en el apéndice H.
Descripción General
Para este algoritmo se requiere la numeración nodal desarrollada por Shirmohammadi
(explicada en la sección 2.1.1, ver apéndice F). El FDC utiliza el vector de potencias inyec-
tadas (S ), la topología del circuito y la impedancia de las líneas (Z = R+ jX). En este FDC
se trabaja con la parte real y la parte imaginaria de los datos por separado. De esta forma,
se parte de un vector de voltajes iniciales ([V0]) separado en parte real e imaginaria, ecuación
(2.6).
20
V 0x = [V 0
x1...V0xi...V
0xn] (2.6)
V 0y = [V 0
y1...V0yi...V
0yn]
Luego, se calcula la matriz [T ] siguiendo el mismo procedimiento utilizado para construir
la matriz BIBC en el FDCT, ver apéndice A. Además, utilizan dos matrices diagonales: en
la primera se definen las resistencias, [DR], y en la segunda las reactancias de las líneas del
circuito, [DX ], ecuaciones 2.7 y 2.8, respectivamente.
DR =
DR1 0 . . . 0
0 DR2 . . . 0
0 0. . . ...
0 0 · · · DRn
(2.7)
DX =
DX1 0 . . . 0
0 DX2 . . . 0
0 0. . . ...
0 0 · · · DXn
(2.8)
Posteriormente, se multiplican las matrices T , DR y Dx, como lo muestra la ecuación 2.9,
para obtener la matriz de impedancias de línea asociadas a la topología del sistema, TRX.
La matriz [TRX] es de orden 2(n− 1)x2(n− 1), donde n es el número de nodos del sistema.
TRX =
T TDRT −T TDXT
T TDXT T TDRT
(2.9)
Luego, se calcula la matriz de corrientes inyectadas [I], ecuación 2.10.
I =
IKxi
IKyi
(2.10)
Donde, IKxi y IK
yi se obtienen de las ecuaciones, (2.11) y (2.12), respectivamente. IKxi y IK
yi
es la componente real e imaginaria de la corriente en el nodo i durante la iteración K. Pi y
21
Qi son las potencias activa y reactiva, respectivamente. V Kxi y V K
yi son las componentes real
e imaginaria de la tensión en la barra i durante la iteración K.
IKxi = −Re{IK
i } =−PiV
Kxi −QiV
Kyi
(V Kxi )2 + (V K
yi )2(2.11)
IKyi = −Im{IK
i } =QiV
Kxi − PiV
Kyi
(V Kxi )2 + (V K
yi )2(2.12)
Finalmente, se calculan las tensiones en los nodos, ecuación (2.13).
V = V0 − TRX.I (2.13)
El proceso se detiene cuando la diferencia de voltaje entre iteraciones es menor a deter-
minada tolerancia, ecuación (2.14).
||V Ki | − |V K−1
i || ≤ ε (2.14)
Para los casos estudiados, se asumió V 0xi = 1 y V 0
yi = 0 para i = 1, ..., n.
Procesos Algorítmicos
1. Paso 1. Leer datos referentes a las conexiones del sistema, potencia aparente en los
nodos e impedancias de las líneas.
2. Paso 2. Construir matrices T , DR y DX .
3. Paso 3. Pre-especificar V0 = Vslack e inicializar todas las tensiones al valor del nodo
slack, (2.6).
4. Paso 4. Calcular matriz de corrientes inyectadas, I.
5. Paso 5. Calcular la matriz TRX.
6. Paso 6. Calcular la tensión en la iteración K + 1 (2.13).
22
7. Paso 7. Verificar si se cumple (2.14).
8. Paso 8. En caso de no cumplirse (2.14), volver al paso 4.
En la figura 2.2 se muestra el diagrama de flujo del algoritmo propuesto por De Oliveira
[20].
Figura 2.2: Diagrama de Flujo del Algoritmo Propuesto por De Oliviera, [20].
2.2. Herramientas Computacionales
A continuación se describirán las herramientas computacionales utilizadas para comparar
y validar el algoritmo implementado.
2.2.1. MATPOWER, [52]
MATPOWER es un paquete de Matlab para resolver problemas de FDC y flujo de carga
óptimo, FDCO. MATPOWER está diseñado para dar el mejor desempeño posible mante-
23
niendo un código simple de entender y modificar.
MATPOWER presenta tres algoritmos para resolver FDC : un FDC estándar y dos FD-
CO). El FDC estándar está basado en el método Newton-Raphson (NR), en el cual la matriz
Jacobiana es actualizada en cada iteración. Este método se describe con detalle en [37, 38].
Los otros algoritmos son variaciones del NR desacoplado rápido, NRDR, [46]. El algoritmo
basado en el NR tradicional presente un excelente rendimiento al ser aplicado en sistemas
de potencia de gran escala. Esto se debe a que el algoritmo trabaja con la esparsidad de
la matriz Jacobiana. Es decir, con esta técnica se evita construir la matriz Jacobiana y su
posterior inversión.
El primer FDCO está basado en la función “constr” de Matlab, la cual usa una técnica
de programación cuadrática sucesiva para trabajar con la matriz Hessiana del sistema. El
segundo algoritmo está basado en programación lineal (LP, por sus siglas en inglés), [53].
Sin embargo, el desempeño de los FDCO del MATPOWER depende de muchos factores.
En primer lugar, la función “constr” utiliza un algoritmo que no preserva la esparsidad de la
matriz. Por lo tanto, el primer FDCO queda limitado a sistemas de potencia de poco tamaño.
Por otro lado, el algoritmo basado en LP preserva la esparsidad de la matriz pero no le saca
provecho, [53]. El programa se describe con mayor detalle en [52].
2.2.2. ASP, [54]
El Sistema de Análisis y Simulación de Redes Primarias, ASP, está orientado al ingeniero
de distribución especializado en planificación y proyectos de redes primarias. Las capacidades
de esta aplicación son, [55]:
Analizar y editar circuitos.
Simulación de crecimiento de redes.
24
Compensación capacitiva para mínima perdida y corrección de bajo voltaje.
Análisis de sensibilidad de parámetros.
Simulación de interrupciones y recuperación con otros circuitos interconectados.
Configuración para mínima perdida.
Con el ASP se pueden obtener los siguientes resultados, [54]:
Muestra el nodo con mayor caída de tensión y los nodos en los que la tensión es menor a
0,95 p.u. El programa señala la ubicación gráficamente de los nodos con la característica
arriba mencionada e indica su tensión.
Muestra el tramo con mayor Capacidad de Emergencia, CE, y señala con colores difer-
entes aquellos tramos en los que la demanda es superior a un 67% de su CE, los que
superan su Capacidad Nominal, CN y aquellos que superan el 100% de la CE.
Muestra un reporte en el que se encuentran las pérdidas totales en kVAR y kW, tanto
en valores reales como en porcentaje (%).
2.3. Condiciones de Operación de la Red
Al conocer el estado del sistema, obtenido al aplicar el FDC, es posible calcular las
características de operación del sistema estudiado; entre otros análisis se puede estudiar la
operación del sistema ante condiciones de fallas, diseñar y/o planificar posibles expansiones
o mejoras en la red, ente otros estudios. Sólo se hará énfasis en el cálculo de las condiciones
de operación de la red. En la literatura ([16], [37], [38]) es posible encontrar información
referente a otros estudios que se pueden realizar al conocer el estado de la red.
25
2.3.1. Análisis por Caída de Tensión
Criterio de Caída de Tensión Máxima, [56]
Este criterio indica la máxima caída de tensión que puede ocurrir en circuitos primarios,
tanto aéreos como subterráneos. Los límites permitidos se encuentran en la tabla 2.1.
Tabla 2.1: Caída de Tensión Máxima Permitida, [56]
Condiciones de Operación VMAX(%) Banda Permitida (p.u)
Normal 5 0, 95 < V < 1, 05
Emergencia 8 0, 92 < V < 1, 08
La ecuación 2.15 muestra, en términos porcentuales, la caída de tensión en cada nodo
respecto a la tensión de la barra Slack, [56].
∆.Vi =VSlack − Vi
VSlack
× 100 (2.15)
2.3.2. Análisis por Pérdidas de Potencia
Con las ecuaciones 2.16 y 2.17 se pueden obtener las pérdidas activas (Lmp) y reactivas
(Lmq) totales en el circuito de estudio, [48].
Lmp =1
2
n∑i=1
n∑j=1
Gij[V2i − ViVj cos Θij] (2.16)
Lmq =1
2
n∑i=1
n∑j=1
Bij[V2i − ViVj sin Θij] (2.17)
i, j = 1, 2, ..., n
i 6= j
26
Donde, i, j y n representan el nodo de salida, de llegada y el número de nodos del circuito,
respectivamente. Además, Gij, Bij y Θij se obtienen de 2.18, 2.19 y 2.20, respectivamente,
[48]. Gij, Bij, Rij y Xij son la conductancia, susceptancia, resistencia y reactancia entre los
nodos i-j. Θij es el ángulo de la admitancia de una línea entre los nodos i-j.
Gij =Rij
(Rij)2 + (Xij)2(2.18)
Bij =Xij
(Rij)2 + (Xij)2(2.19)
Θij = Θi −Θj (2.20)
Las pérdidas en una subestación (S/E ), LS/E, corresponde a la suma de las pérdidas de
cada circuito de dicha S/E, ecuación 2.21. En esta ecuación, C corresponde al número total
de circuitos que la componen.
LS/E =C∑
k=1
Lm (2.21)
Las pérdidas totales en la red, LT , corresponden a la suma de las perdidas por subestación,
ecuación 2.22. En esta ecuación, S corresponde al número total de S/E que componen toda
la red.
LT =S∑
k=1
Lm (2.22)
2.3.3. Análisis por Capacidad Amperimétrica de Conductores
Criterio de Capacidad de Carga, [56]
Todo circuito debe tener un porcentaje máximo de carga igual al 67% de su capacidad
de emergencia, ya que debe cumplirse que cada circuito primario pueda ser asistido por
dos o más circuitos. En la figura 2.3 se puede observar el porcentaje de carga de un cable
27
Figura 2.3: Porcentaje de Carga de un Cable Subterráneo, [56].
subterráneo en condiciones normales y de emergencia, así como también la reserva que puede
ser utilizada a mediano o a largo plazo.
Para una línea conectada entre los nodos i y j, el flujo de potencia que va de la barra i a
la j, puede ser obtenida de la siguiente forma, [48].
Pij = Gij[ViVj cos Θij − V 2i ] +BijViVj sin Θij (2.23)
Qij = Bij[V2i − ViVj cos Θij] +GijViVj sin Θij (2.24)
Sij =√
(Pij)2 + (Qij)2 (2.25)
Gij, Bij y Θij se obtienen de 2.18, 2.19 y 2.20, respectivamente. Sij corresponde al flujo
de potencia aparente del conductor en condiciones de operación.
Para verificar que un conductor esté operando en condiciones normales se debe cumplir
la relación de la ecuación 2.26, [56].
η.Smaxij ≥ Sij (2.26)
28
η corresponde a un factor de seguridad; típicamente corresponde a 67 %. Smaxij corresponde
al flujo máximo de potencia de un conductor ubicado entre los nodos i y j.
2.3.4. Análisis de Capacidad Firme de una S/E
Criterio de Capacidad de Firme, [54]
La capacidad firme (CF ) es la que se debe manejar cuando se diseña una S/E, para que
ante una posible salida forzada de algún transformador se pueda seguir supliendo la carga
demandada de una forma segura y continua, sin necesidad de realizar interconexiones con
otros circuitos para suplir la demanda.
La demanda actual y la proyectada para el futuro no deben exceder la CF de la S/E.
Además, se debe tratar que todos los transformadores de la misma S/E operen a la misma
capacidad nominal con el fin de tener la mayor CF, lo cual proporciona una mayor CF ante
posibles contingencias. La Electricidad de Caracas, EDC, opera con un máximo de 4 trans-
formadores por S/E.
Utilizando la ecuación 2.27 es posible calcular la CF de una S/E.
CF = C.(N∑
m=1
kV Am − kV Amax) (2.27)
Donde, N es el número de transformadores operando en la S/E. kV Am es la potencia
nominal del transformador. kV Amax es la potencia del transformador de mayor capacidad.
C es un factor que puede ser:
120 por ciento para transformadores de distribución cuyo tiempo útil sea menor o igual
a 40 años.
100 por ciento para transformadores de distribución cuyo tiempo útil sea mayor o igual
a 40 años.
CAPÍTULO 3
IMPLEMENTACIÓN
3.1. Arquitectura del Sistema
El estado actual del sistema de adquisición y procesamiento de datos utilizado en la
Electricidad de Caracas, EDC, presenta las siguientes características, [49]:
Incompatibilidad de datos.
Duplicidad de Información.
Mayor trabajo del necesario.
Retardos administrativos.
Ineficiencia operativa.
Efectos acumulativos con tendencia al caos.
Debido a esto, se busca unificar y sincronizar la base de datos con la que cuenta dicha
empresa. Al unificar la base de datos se busca tener en la misma base de datos la información
suministrada por los sistemas SAP (Software de Gestión y Estrategia), GIS (Software de
Estimación de Demanda) y SCADA (Software de registro de carga y data del sistema), ver
figura 3.1.
30
Figura 3.1: Base de Datos Unificada, [49].
Al sincronizar la base de datos es posible accesar a la misma desde cualquier computador
que haya sido sincronizado a dicha red. Esto elimina la necesidad de un computador central
que contenga toda la información de la base de datos, ver figura (3.2).
Figura 3.2: Sincronización de Base de Datos, [49].
El Instituto de Energía de la Universidad Simón Bolívar, (INDENE ), en conjunto con
otras organizaciones y empresas, ha venido desarrollando un proyecto con el cual se busca
automatizar el proceso de estudios de la red de distribución y desarrollar una plataforma
basada en código abierto que permita, [49]:
31
1. Acceso inmediato y fiable a la información requerida para el proceso.
2. Desarrollo de herramientas técnicas específicas en forma modular, garantizando sosteni-
bilidad y escalabilidad de la solución tecnológica.
3. Manejo de información técnica a gran escala integrando los distintos sistemas existentes.
4. Eliminar dependencias en cuanto a plataformas de código cerrado o propietario.
En, [50], se ha visualizado la elaboración del proyecto en tres etapas:
Etapa 1. Implementación del prototipo.
Etapa 2. Implementación de Funciones Básicas.
Etapa 3. Implementación de Funciones Avanzadas.
El presente trabajo forma parte de la primera etapa del proyecto; siendo el flujo de carga
(FDC ) desarrollado uno de los módulos a implementar en el prototipo. En la segunda y
tercera etapa se definirán otras funciones de la herramienta computacional.
Para la aplicación del proyecto se ha considerado que una arquitectura adecuada es el
patrón de diseño Modelo-Vista-Controlador (MVC ), mostrado en la figura 3.3. Este patrón
ha tomado especial relevancia a raíz de las nuevas implementaciones que se han logrado hacer
del mismo y la explosión en cuanto a programación orientada a objetos.
Como se explica en [50], este patrón se compone de tres capas que permiten separar los
ámbitos de trabajo de la aplicación. El nivel superior es la Vista, la misma corresponde a la
interfaz de usuario en la cual se realizan las interacciones con el operador del programa. En la
parte más baja se encuentra la capa correspondiente al Modelo. En éste se gestionan todas
las interacciones y validaciones con las fuentes de datos. Usualmente este proceso se delega en
paquetes conocidos como Mapeadores Objeto-Relacionales (o ORM por sus siglas en inglés)
que proveen un nivel de acceso de alto nivel convirtiendo las interacciones con las bases de
32
Figura 3.3: Modelo MVC, [50].
datos en algo más acorde con la programación orientada a objetos y liberando la definición
de los datos de la implementación (gestor de bases de datos seleccionado). Por último, entre
las capas arriba mencionadas se encuentra la capa de los Controladores, éstos se encargan
de las funciones inherentes a la lógica del programa, negociar las solicitudes de acciones de
las vistas y gestionar los datos en los modelos. En la figura 3.4 se puede apreciar el esquema
general del sistema que se está diseñando, [49].
De acuerdo a [50], esta segmentación de funciones permite mejorar el rendimiento a la
hora de hacer mantenimiento a la aplicación o el control de cambios solicitados. Por ejemplo,
si se requieren cambios en la interfaz de usuario solamente es necesario realizar cambios en
el diseño de la capa correspondiente a la vista. De igual forma, si existe una reorganización
de la estructura de los datos o una implementación en un nuevo gestor de bases de datos
se verá afectada solamente la capa del modelo. Del mismo modo, una nueva opción dentro
del programa implicará la incorporación de un controlador adecuado que gestione esta nueva
funcionalidad.
33
Figura 3.4: Esquema General de Herramienta de Planificación Corto-Mediano Plazo
(HPCMP) - Visualización de Largo Plazo, [50].
34
3.2. Descripción del Algoritmo Implementado
El algoritmo implementado fue desarrollado bajo la arquitectura MVC, descrita en la
sección 3.1 del presente capítulo. El programa está desarrollado de forma modular para opti-
mizar el tiempo de cómputo del mismo. Éste está compuesto por dos módulos: en el primero
se realiza el proceso de adquisición de datos; en el segundo, se realiza el proceso iterativo y,
con las tensiones nodales, se realiza el análisis de las condiciones de operación de la red. Con
el primer módulo se filtra la información de los archivos .DAT y se obtienen las matrices
TRX, S y los datos de nodos y líneas para cálculos futuros; las cuales se guardan en archivos
.MAT y pueden ser almacenadas en disco o en memoria RAM. En el segundo módulo se
realiza el proceso iterativo, se obtienen los voltajes nodales y se analizan los parámetros
correspondientes al estudio de pérdidas técnicas en conductores, caídas de tensión y capaci-
dad amperimétrica de los conductores en condiciones normales de operación. Es importante
destacar que MATLAB es utilizado como programa interpretador del algoritmo implementa-
do.
En la figura 3.5 se muestra el esquema basado en la arquitectura MVC del programa
desarrollado en el presente proyecto. En el módulo M se realiza el proceso de la adquisición
de datos. En el módulo C se lleva a cabo el proceso iterativo. Finalmente, en el módulo V
se realiza el análisis posterior de las condiciones de operación de la red.
En la figura 3.6 se observa el diagrama de flujo del programa implementado.
3.3. Datos
Los archivos utilizados para la implementación del programa en redes de gran tamaño
fueron obtenidos de una data suministrada por la Electricidad de Caracas (EDC ) en el año
2006; dichos archivos se encuentran en formato .DAT. Los mismos son archivos de datos
generados automáticamente por los sistemas de adquisición de datos y mediciones, como
35
Figura 3.5: Esquema MVC del programa implementado.
el SCADA y el GIS, y en los mismos se almacena información referente a dicho programa
para uso interno del mismo. Los datos están separados por comas (,) y la información de la
red está agrupada en bloques; cada bloque termina con la palabra “END” [51]. La estruc-
tura de los archivos DAT, su explicación exhaustiva y un ejemplo se pueden encontrar en [51].
En la EDC, los archivos .DAT son utilizados por el programa ASP, desarrollado por
el profesor Alberto Naranjo, el cual tiene como funciones principales realizar un análisis
consecuente con el (FDC ), y cálculo de corto circuito. Una vez que se ejecuta el (FDC ) se
muestran las magnitudes de variables eléctricas como tensión y corrientes pertenecientes al
sistema eléctrico ordenadas en columnas, [50].
36
Figura 3.6: Diagrama de Flujo del Algoritmo Desarrollado.
37
3.4. Estructura del Programa
3.4.1. Adquisición de Datos
Inicialmente, se desarrolló un algoritmo para filtrar de los archivos .DAT los datos co-
rrespondientes a la potencia de las cargas, impedancias de líneas, tipos de nodo, distancia de
las líneas, factor de potencia, tensión nominal de operación y topología de la red.
Con este programa se obtiene la matriz de impedancias asociadas a la topología de la red
(TRX, [20]) y la matriz de potencia aparente (S ). Las matrices pueden ser almacenadas en
el disco o en memoria RAM. Para optimizar el proceso de flujo de carga se busca que las
matrices TRX y S estén disponibles en la memoria RAM del sistema y se actualizan “offline”
con cierta frecuencia. Es importante el proceso de actualización de ambas matrices porque
en la primera, (TRX ), se registran los cambios en la red y en la segunda, (S ), se registran
los cambios en la potencia demandada. La segunda matriz requiere un proceso de actualiza-
ción más frecuente que la primera ya que la potencia demandada varía con más frecuencia
que la conexión de la red. De esa forma se obtienen resultados más objetivos y reales de la red.
En la sección B.1 del apéndice B se puede encontrar el código del algoritmo desarrollado.
3.4.2. Proceso Iterativo
Con las matrices TRX y S de todos los circuitos disponibles en memoria RAM se realiza
el proceso iterativo, descrito en la sección 2.1.2 del capítulo 2. Como es de esperar, por la
simplicidad de los cálculos, los resultados son obtenidos en pocas iteraciones y en un período
de tiempo considerablemente reducido en comparación con otros algoritmos de FDC. Como
resultado de este modulo se obtienen las tensiones en módulo y ángulo de todos los nodos de
la(s) red(es) estudiada(s).
38
Conocidas las tensiones de la red se realiza el análisis correspondiente al estudio de caí-
das de tensión (sección 2.3.1), pérdidas técnicas de potencia (ver sección 2.3.2), capacidad
amperimétrica de conductores (ver sección 2.3.3) y capacidad firme (sección 2.3.4) en condi-
ciones normales de operación. Además, se muestra un sumario de los resultados obtenidos,
incluyendo tensiones nodales, pérdidas técnicas, caídas de tensión, cantidad de conductores
que operan por debajo del 67% de su capacidad, cantidad de conductores que superan el
67% y 100% de su capacidad. Los resultados son almacenados en un archivo con la ex-
tensión .MAT. En el apéndice C se encuentra el reporte que muestra el programa para un
circuito ejemplo. En este caso el circuito es el GRA_A01 correspondiente al circuito A1 de
la subestación (S/E) Granada.
En la sección B.2 del apéndice B se puede encontrar el código del algoritmo desarrollado.
CAPÍTULO 4
CASOS DE ESTUDIOS Y ANÁLISIS DE RESULTADOS
El algoritmo implementado fue probado en diferentes casos de estudio. En primer lugar,
se hizo un estudio en un circuito de 7 barras. Luego, se realizó la validación de los resultados
del caso anterior con un programa de uso comercial. Después, se realizó un estudio compa-
rativo entre los algoritmos propuestos por Teng (FDCT, [19]) y por De Oliveira (TRX, [20])
para comprobar la eficiencia del último respecto al primero. Finalmente, se implementó el
algoritmo en circuitos de gran tamaño de la Electricidad de Caracas, EDC. Con los resultados
obtenidos del flujo de carga (FDC ) se realizó un análisis de las condiciones de operación de
la red estudiada.
4.1. Aplicación en Circuito de 7 Barras
Para ilustrar el proceso de cálculo del algoritmo se utilizará un circuito de 7 barras. El
proceso de cálculo se realizará paso por paso para demostrar la lógica del mismo, dicho pro-
ceso se describe en la sección 2.1.2 que se encuentra en el capítulo 2. El sistema incluye 1
generador y 6 cargas equivalentes, conectadas a una red de distribución radial. Los datos de
líneas y de nodos del circuito se muestran en las tablas 4.1 y 4.2, respectivamente.
Datos de Línea El circuito está conformado por 6 líneas, descritas en la tabla 4.1. Los datos
se presentan en el siguiente orden: nodo de salida, nodo de llegada, resistencia y reactancia
40
de cada línea en p.u. Las bases utilizadas son las siguientes: VBASE = 12, 47kV y SBASE =
100MVA
Tabla 4.1: Datos de Línea (Ldat)
salida llegada r(p.u) x(p.u)
1 2 0,0265 0,0462
1 3 0,1005 0,0693
3 4 0,0670 0,0462
3 5 0,0265 0,0462
5 6 0,1005 0,0693
5 7 0,0670 0,0462
Datos de Nodo Los datos de los 7 nodos se encuentran en la tabla 4.2. Los mismos se
presentan en el siguiente orden: 1. n es el número de la barra. 2. Tipo corresponde al tipo
de barra: 1 barra slack, 2 barra PV y 3 barra PQ. 3. Pgen0 es el valor inicial de la potencia
activa generada (p.u). 4. Qgen0 es el el valor inicial de la potencia reactiva generada (p.u).
5. Pload es la potencia activa demandada (p.u). 6. Qload es la potencia reactiva demandada
(p.u). 7. V0 es la tensión inicial de la barra (p.u). 8. fcap es el factor de capacidad de la barra.
4.1.1. Cálculo de la Matriz TRX
Como se describe en la sección 2.1.2 del capítulo 2, para calcular la matriz es necesario
hallar la matriz que relaciona las corrientes inyectadas con las corrientes de rama, [T], y las
matrices diagonales de resistencias de línea, [Dr], y reactancias de líneas, [Dx].
Matriz [T]
En el apéndice A se explica paso a paso cómo construir la matriz [T]. En esta matriz se
representa la topología del circuito estudiado. El número 1 implica la existencia de una línea
41
Tabla 4.2: Datos de Nodos, (Bdat)
n Tipo Pgen0(p.u) Qgen0(p.u) Pload(p.u) Qload(p.u) V0 fcap
1 1 0 0 0 0 1 1
2 3 0 0 0,1017 0,0635 1 1
3 3 0 0 0,0547 0,0342 1 1
4 3 0 0 0,0809 0,0596 1 1
5 3 0 0 0,1017 0,0635 1 1
6 3 0 0 0,0547 0,0342 1 1
7 3 0 0 0,0809 0,0596 1 1
entre el nodo correspondiente a la fila y la columna respectiva.
T =
1 0 0 0 0 0
0 1 1 1 1 1
0 0 1 0 0 0
0 0 0 1 1 1
0 0 0 0 1 0
0 0 0 0 0 1
Matriz de Resistencias y Reactancias de línea, Dr y Dx
Matriz diagonal de resistencias de líneas, Dr. En la ecuación 2.7 se observa la forma de
la matriz.
Dr =
0,0265 0 0 0 0 0
0 0,1005 0 0 0 0
0 0 0,0670 0 0 0
0 0 0 0,0265 0 0
0 0 0 0 0,1005 0
0 0 0 0 0 0,0670
42
Matriz diagonal de reactancias de líneas, Dx. En la ecuación 2.8 se observa la forma de
la matriz.
Dx =
0,0462 0 0 0 0 0
0 0,0693 0 0 0 0
0 0 0,0462 0 0 0
0 0 0 0,0462 0 0
0 0 0 0 0,0693 0
0 0 0 0 0 0,0462
Matriz TRX
La estructura de la matriz TRX se muestra en la ecuación . Por motivos de espacio,
se presentarán las cuatro submatrices que conforman la matriz TRX por separado. De esta
forma,
T TDRT =
0.0265 0 0 0 0 0
0 0,1005 0,1005 0,1005 0,1005 0,1005
0 0,1005 0,1675 0,1005 0,1005 0,1005
0 0,1005 0,1005 0,127 0,127 0,127
0 0,1005 0,1005 0,127 0,2275 0,127
0 0,1005 0,1005 0,127 0,127 0,194
−T TDXT =
-0.0462 0 0 0 0 0
0 -0,0693 -0,0693 -0,0693 -0,0693 -0,0693
0 -0,0693 -0,1155 -0,0693 -0,0693 -0,0693
0 -0,0693 -0,0693 -0,1155 -0,1155 -0,1155
0 -0,0693 -0,0693 -0,1155 -0,1848 -0,1156
0 -0,0693 -0,0693 -0,1155 -0,1156 -0,1617
43
T TDXT =
0.0462 0 0 0 0 0
0 0,0693 0,0693 0,0693 0,0693 0,0693
0 0,0693 0,1155 0,0693 0,0693 0,0693
0 0,0693 0,0693 0,1155 0,1155 0,1155
0 0,0693 0,0693 0,1155 0,1848 0,1156
0 0,0693 0,0693 0,1155 0,1156 0,1617
T TDRT =
0.0265 0 0 0 0 0
0 0,1005 0,1005 0,1005 0,1005 0,1005
0 0,1005 0,1675 0,1005 0,1005 0,1005
0 0,1005 0,1005 0,127 0,127 0,127
0 0,1005 0,1005 0,127 0,2275 0,127
0 0,1005 0,1005 0,127 0,127 0,194
4.1.2. Proceso Iterativo
Valores iniciales
Los valores de potencia y tensiones iniciales utilizados para empezar proceso algorítmico
se muestran en la tabla 4.3. Los datos se presentan en el siguiente orden: 1. número del nodo
2. Potencia activa demandada en p.u. (Pload) 3. Potencia activa consumida en p.u. (Qload) 4.
Componente real de la tensión inicial en p.u. (VR0) 5. Componente imaginaria de la tensión
inicial en p.u. (VI0).
Resultados de las Iteraciones
Construida la matriz TRX y conociendo el vector de potencias y tensiones iniciales es
posible realizar el FDC tal como se describe en la sección 2.1.2 en el capítulo 2. Para el cir-
cuito de 7 barras se llegó a un resultado en 3 iteraciones. El criterio de convergencia utilizado
44
Tabla 4.3: Valores de potencia y tensión inicial (V0, P0)
Nodo Pload(p.u) Qload(p.u) VR0(p.u) VI0(p.u)
2 0,1017 0,0635 1 0
3 0,0547 0,0342 1 0
4 0,0809 0,0596 1 0
5 0,1017 0,0635 1 0
6 0,0544 0,0342 1 0
7 0,0809 0,0596 1 0
fue ε ≤ 0, 001. Se debe tomar en cuenta que se utilizó un criterio de convergencia bastante
conservador por motivos académicos. Los resultados de las iteraciones se muestran en las
tablas 4.4, 4.5 y 4.6, respectivamente.
Los resultados se presentan con la siguiente nomenclatura:
Ir(p.u) y Ij(p.u). Componente real e imaginaria de la corriente en el nodo i (p.u).
Vr(p.u) y Vj(p.u). Componente real e imaginaria de la tensión en el nodo i (p.u).
∆.Vr y ∆.Vj. Diferencia de la componente real e imaginaria de tensión entre iteraciones.
Primera Iteración
En la tabla 4.4 se muestran los resultados de la primera iteración. Al terminar la primera
iteración se observa que no se cumple el criterio de convergencia. En la mayoría de los casos
la variación es mayor a la tolerancia predeterminada.
45
Tabla 4.4: Resultados de la Primera Iteración
Nodo Ir(p.u) Ij(p.u) Vr(p.u) Vj(p.u) ∆Vr ∆Vj
2 -0,1017 0,0635 0,9944 -0,003 0,0056 0,003
3 -0,0547 0,0342 0,9452 -0,0005 0,054 0,0006
4 -0,0809 0,0596 0,9370 -0,0053 0,0631 0,0052
5 -0,1017 0,0635 0,9316 -0,0074 0,0684 0,0074
6 -0,0544 0,0342 0,9237 -0,0077 0,0762 0,0077
7 -0,0809 0,0596 0,9234 -0,0071 0,0766 0,0071
Segunda Iteración
Tabla 4.5: Resultados de la Segunda Iteración
Nodo Ir(p.u) Ij(p.u) Vr(p.u) Vj(p.u) ∆.Vr ∆.Vj
2 -0,1025 0,0642 0,9943 -0,003 0,0001 0,0001
3 -0,0579 0,0362 0,9411 -0,0005 0,0040 0,0003
4 -0,0867 0,0641 0,9324 -0,0059 0,0631 0,0008
5 -0,1097 0,0690 0,9265 -0,0074 0,0051 0,0001
6 -0,0592 0,0375 0,9180 -0,0082 0,0058 0,0001
7 -0,0881 0,0652 0,9172 -0,0075 0,0058 0,0001
Al terminar la segunda iteración se observa que no se cumple el criterio de convergencia
en una de las barras. La diferencia de tensión entre iteraciones del nodo 4 es mayor a la
tolerancia predeterminada.
Tercera Iteración
Al terminar la tercera iteración se observa que se cumple el criterio de convergencia en
todas las barras.
46
Tabla 4.6: Resultados de la Tercera Iteración
Nodo Ir(p.u) Ij(p.u) Vr(p.u) Vj(p.u) ∆.Vr ∆.Vj
2 -0,1042 0,0636 0,9943 -0,003 0,0001 0,0001
3 -0,0582 0,0363 0,9411 -0,0005 0,0001 0,0001
4 -0,0868 0,0642 0,9323 -0,0059 0,0001 0,0001
5 -0,1092 0,0676 0,9265 -0,0078 0,0001 0,0001
6 -0,0590 0,0367 0,9175 -0,0082 0,0005 0,0001
7 -0,0877 0,0643 0,9171 -0,0075 0,0001 0,0001
La tabla 4.7, muestra las tensiones del circuito en módulo y ángulo. Además se muestra
el porcentaje de caída de tensión respecto a la barra slack.
Tabla 4.7: Tensiones nodales en módulo y ángulo
Nodo |V |(p.u) 6 (V )(grados) ∆V(%)
1 1 0 0
2 0,9943 -0,1728 0,57
3 0,9411 -0,0204 5,89
4 0,9323 -0,0036 6,77
5 0,9265 -0,4823 7,35
6 0,9175 -0,5121 8,25
7 0,9171 -0,4685 8,29
Se observa que 5 nodos están por debajo del criterio del±5 % de caída de tensión permitido
en condiciones de operación normal, ver tabla 2.1. La máxima caída de tensión en el circuito
es de 8, 29 %.
47
4.2. Validación de Resultados
Para validar los resultados se utilizó el programa MATPOWER. En la sección E del
capítulo 2 se describe dicho programa. Este programa está basado en el algoritmo Newton-
Raphson, NR. Sin embargo, éste trabaja con la esparsidad de las matrices para evitar con-
struir la matriz Jacobiana, típica en el NR. Principio que hace del programa un algoritmo
bastante poderoso y eficiente. De esta forma, se evitan problemas de convergencia típicos en
redes de distribución, (ver sección 1.1 del capítulo 1). Para mayor información referente al
programa se recomienda revisar el manual de usuario, [52].
Para realizar la validación se utilizó como caso de ejemplo el circuito explicado en la sec-
ción 4.1 del presente capítulo. Las tensiones nodales se muestran en la tabla 4.8. Los datos de
presentan de la siguiente forma: 1. Nombre del nodo. 2. Módulo de tensión en p.u. (|V |). 3.
Ángulo de tensión en grados ( 6 (V )). 4. Error porcentual del módulo de las tensiones obtenidas
con el algoritmo implementado respecto a las del MATPOWER. 5. Error porcentual del ángu-
lo de las tensiones obtenidas con el algoritmo implementado respecto a las del MATPOWER.
Tabla 4.8: Tensiones Nodales Validadas en Módulo y Ángulo
Nodo |V |(p.u) 6 (V )(grados) Error |V |(%) Error 6 (V )(%)
1 1 0 0 NA
2 0,9943 -0,1737 0 0,5181
3 0,9407 -0,0206 0,0425 0,9708
4 0,9319 -0,0039 0,0429 7,6923
5 0,9260 -0,4699 0,0540 2,6389
6 0,9174 -0,4937 0,0109 3,7269
7 0,9171 -0,4526 0 3,5130
Se observa que el mayor error en el módulo de las tensiones es sólo del 0, 0540 %; mientras
48
que en el ángulo de las tensiones se tuve un error máximo correspondiente al 7, 6923 %. En el
apéndice E, se encuentra el reporte generado por el MATPOWER como resultado del flujo
de carga del circuito de 7 barras explicado en la sección 4.1.
4.3. Estudio Comparativo
En [20], se estudia el tiempo de proceso de CPU utilizando tres algoritmos:
1. El algoritmo propuesto en [20], (TRX ), (ver sección 2.1.2 en el capítulo 2).
2. El algoritmo propuesto en [19], (FDCT ), (ver sección 2.1.1 en el capítulo 2).
3. Newton-Raphson, (NR), utilizado por el programa MATPOWER (ver sección E en el
capítulo 2).
Para dicho estudio no se consideró el tiempo de entrada/salida de datos. Esto último se
justifica porque en la aplicación del mismo se planea que un programa centinela mantenga
las matrices de los datos de entrada en memoria RAM y actualizados periódicamente; razón
por la cual, no se invertiría tiempo en cargar los datos sino que se tendrían disponibles en
cualquier momento que se requiera.
El caso de estudio fue resuelto utilizando una red de n nodos variando la variable n de
1.000 hasta 3.000 nodos. La figura 4.1, obtenida de dicho artículo, muestra el tiempo de
convergencia utilizado en el proceso iterativo de cada algoritmo. El gráfico muestra que el
TRX presenta un mejor tiempo de cómputo que el FDCT y el NR.
El TRX es desde un punto de vista computacional más eficiente que los otros dos algo-
ritmos porque el proceso está basado en la suma y multiplicación de números reales que se
encuentran previamente alojados en memoria RAM. La matriz TRX no requiere ser actuali-
zada en cada iteración tal como se hace con la matriz Jacobiana del NR. Motivo por el cual,
49
Figura 4.1: Estudio Comparativo: Tiempo de Cómputo, [20].
el TRX es un algoritmo altamente competitivo respecto al utilizado por el MATPOWER,
el cual utiliza la esparsidad de las matrices para no armar la matriz Jacobiana y, en conse-
cuencia, reducir el tiempo de computo al evitar tener que invertir y actualizar dicha matriz
en cada iteración.
En el caso propuesto, el NR tuvo el segundo mejor comportamiento por lo anteriormente
descrito. El peor tiempo se registró al utilizar el algoritmo FDCT en números complejos ya
que el tiempo requerido para realizar las operaciones se incrementa al trabajar con números
complejos, [20].
Por otro lado, se decidió corroborar dicho estudio de forma independiente. Sólo se com-
probó la eficiencia del (TRX ) respecto al FDCT. Dicho estudio se realizó tomando en cuenta
el número de iteraciones y el tiempo de computo de ambos algoritmos.
Para esto se utilizaron cuatro (4) circuitos de diferentes tamaños: 4, 7, 12 y 69 barras.
Los datos del circuito de 4 barras se encuentra en el apéndice D, el circuito de 7 se describió
en la sección 4.1 y los circuitos de 12 y 69 barras se encuentran disponibles en los apéndices
J e I, respectivamente. El número de iteraciones para la convergencia es el mismo en ambos
50
algoritmos ya que se basan en el mismo principio.
Para poder estudiar el tiempo de respuesta de ambos algoritmos, se forzó el ciclo de itera-
ciones para que realizaran un total de 5.000 iteraciones. De esta forma, fue posible hacer la
medición del tiempo en el que los dos algoritmos realizan las operaciones matemáticas. Esto
se fundamenta en el hecho de que los procesadores actuales realizan estos procesos en tiempos
muy pequeños, lo cual dificulta su medición. Sin embargo, al aumentar apreciablemente el
número de iteraciones se puede medir el tiempo del proceso.
En la tabla 4.9 se muestran los resultados obtenidos. Se observa que los tiempos se reducen
aproximadamente a la mitad al aplicar la modificación de [20]. Para este estudio se utilizó
un computador de 2GB de memoria RAM y un procesador Intel Core 2 Duo.
Tabla 4.9: Tiempo de cómputo de [19] y [20] para circuitos de 4, 7, 12 y 69 barras
Estudio con 5000 iteraciones
N Barras [20] (mseg) [19] (mseg)
69 243,61 432,03
12 45,48 84,84
7 35,61 79,54
4 30,33 74,22
Los resultados demuestran que el cómputo en números reales utilizado por el TRX es
más eficiente en comparación con el cálculo en números complejos utilizado por FDCT, a
pesar de que las matrices son de mayor dimensión en el TRX.
51
4.4. Aplicación en Redes de Gran Tamaño
4.4.1. Tiempo de cómputo
Utilizando la data proporcionada por la EDC se implementó el programa desarrollado en
un total de 530 circuitos distribuidos en 78 S/E, compuestos por un total de 64.251 nodos
que corresponden a un 80 % de la demanda máxima de la Gran Caracas (3, 28GW ).
Para probar la eficiencia del algoritmo se comparó el tiempo de cómputo del mismo con
el utilizado por el programa ASP ; programa desarrollado por el profesor Alberto Naranjo.
Dicho programa es utilizado actualmente por la EDC para realizar estudios de planificación
a corto y mediano plazo. “La utilización de esta herramienta permite con ciertas limitaciones
obtener resultados en tiempo útil”, [50]. En la tabla 4.2 se muestra el tiempo de cómputo del
ASP. Para este estudio se utilizó un computador de 2GB de memoria RAM y un procesador
Intel Core 2 Duo.
Tabla 4.10: Tiempo de cómputo del ASP para red de 530 circuitos de la EDC
Tiempo Total(seg)
ASP 438,82
En la figura 4.2 se muestra de forma esquemática el proceso algorítmico que utiliza el
ASP y el tiempo requerido para obtener la solución del flujo de carga para el caso planteado
anteriormente.
En contraste, el tiempo utilizado por el algoritmo implementado se muestra en la tabla
4.11. Los resultados presentados son de cada uno de los módulos que componen el algoritmo.
Para este estudio, se dividió el algoritmo mostrado en la sección B.2 del apéndice B en dos
submódulos: el primero correspondiente al proceso iterativo y el segundo correspondiente al
52
Figura 4.2: Tiempo de Cómputo del ASP para la Red de la Gran Caracas.
análisis de condiciones de la red.
Tabla 4.11: Tiempo de cómputo del algoritmo implementado para red de 530 circuitos de la
EDC
Módulo Tiempo Total(seg)
Adquisición de Datos 1.700
Proceso Iterativo 0,76
Resultados 3,22
En la figura 4.3 se muestra de forma esquemática el proceso algorítmico que utiliza el
algoritmo desarrollado y el tiempo requerido para obtener la solución del flujo de carga para
el caso planteado anteriormente.
A primera vista se observa que, en términos generales, el ASP requirió de menor tiempo
para obtener un resultado del FDC. Sin embargo, es importante resaltar que varios hechos
relevantes que demostrarán que la afirmación inicial no es del todo valida.
En primer lugar, hacer una comparación en igualdad de condiciones ente ambos algorit-
53
Figura 4.3: Tiempo de Cómputo del Algoritmo Implementado para la Red de la Gran Caracas.
mos no es del todo válido ya que el programa del profesor Naranjo es un programa compilado
en un lenguaje de bajo nivel como lo es DELPHI a diferencia del algoritmo implementado
que no ha sido compilado y utiliza MATLAB como interpretador.
Por otro lado, el módulo que requirió de mayor tiempo de proceso fue el modulo de
adquisición de datos. El tiempo utilizado fue de 1.700 segundos; es decir, el módulo tardó en
promedio 3, 21 segundos en procesar la data de cada circuito. No obstante, como se explicó
en la sección 3.4.1 del capítulo 3, en este módulo se realiza el filtrado de los datos y se colocan
las matrices TRX y S en memoria RAM para que estén disponibles en cualquier momento
que se requiera, lo cual optimiza el proceso iterativo.
Aún más, se debe recordar que el proceso de adquisición de datos y actualización de las
matrices arriba mencionadas se realiza offline. Razón por la cual no resulta afectado el pro-
ceso iterativo del programa. A diferencia del ASP que tiene que recurrir al disco duro para
adquirir la información y luego realizar el flujo de carga.
Además, es válido acotar que el espacio ocupado por las matrices TRX y S de los 530
circuitos es de 40, 6MB. Es decir, el espacio requerido para almacenar las matrices de los
54
circuitos de una red de gran tamaño es insignificante en comparación con la capacidad que
poseen los procesadores actualmente. En la tabla 4.12, se muestra el espacio ocupado en disco
por 10, 50, 100, 200 y 530 circuitos elegidos aleatoriamente. Los datos son presentados de la
siguiente forma: 1. Número de circuitos. 2. Espacio ocupado en disco en MB. 3. Número de
nodos totales.
Tabla 4.12: Espacio Ocupado en Disco por redes de 10, 50, 100, 200 y 530 circuitos
Número de Circuitos Espacio en Disco(MB) Número de Nodos
10 2,029 1.168
50 3,78 5.934
100 6,6 11.346
200 20 23.115
530 40,6 64.251
Por otra parte, se observa que el tiempo requerido para obtener el resultado del FDC
de toda la red estudiada fue 0, 76 segundos utilizando un computador de uso común. Es
decir, en promedio se obtuvo el resultado del FDC de cada circuito en 1, 43 milisegundos.
Al compararlo con el ASP, que en promedio tenía la solución en 0, 75 segundos por circuito,
se puede concluir que el TRX es más eficiente. Dicha eficiencia se debe a la sencillez de la
lógica de cálculo, al hecho de que los cálculos se realizan en números reales y a la disposición
de la información de los circuitos (en forma matricial) en memoria RAM.
4.4.2. Análisis de Condiciones Operacionales de una Red de Gran
Tamaño
Análisis Ejemplo para 1 Circuito
Conocidas las tensiones nodales en módulo y ángulo se realizó un análisis de las condi-
ciones operacionales del circuito ANT_A04 correspondiente al circuito A04 de la S/E An-
55
tímano. Para este circuito se estudió la caída de tensión, la capacidad amperimétrica y las
pérdidas técnicas.
La figura 4.4 muestra el perfil de la caída de tensión en el circuito Antímano A04, confor-
mado por un total de 318 nodos. Es lógico obtener que a medida que los nodos se alejan de la
barra Slack, éstos tengan un mayor porcentaje de caída de tensión. Sin embargo, la máxima
caída de tensión de este circuito es de 15 %, valor que supera el criterio de caída de tensión
del 5 % de la EDC, ver sección 2.3.1 del capítulo 2.
Figura 4.4: Perfil de Caída de Tensión en (%) del circuito ANT_A01.
La figura 4.5 muestra la potencia activa demandada y las pérdidas técnicas en relación
con la potencia activa total entregada. Los cálculos se realizaron como se explica en la sección
2.3.2 del capítulo 2. La potencia activa entregada al circuito es de 16, 59MW, de los cuales
un 8 % representa las pérdidas técnicas en el circuito, valor que está por encima del 4 %
permitido, [56]. En promedio el valor de las pérdidas para toda la red oscila alrededor del
2 %, [54].
La figura 4.6 muestra la potencia reactiva demandada y las pérdidas técnicas en relación
con la potencia activa total entregada. La potencia reactiva entregada al circuito es de
11, 72MVAr, de los cuales un 20 % representa las pérdidas técnicas en el circuito, valor que
está por encima del 4 % permitido, [56].
56
Figura 4.5: Potencia Activa Demandada y Pérdidas Técnicas del Circuito ANT_A01.
Figura 4.6: Potencia Reactiva Demandada y Pérdidas Técnicas del Circuito ANT_A01.
57
La figura 4.7 muestra la cantidad de conductores del circuito que están operando por de-
bajo del 67 % de la capacidad de emergencia (CE ), aquellos que operan por encima del 67 %
de la CE y aquellos que operan por encima del 100 % de la CE. La capacidad amperimétrica
de los conductores se calculó como se explica en la sección 2.3.3 del capítulo 2. Para un total
de 318 ramas, se observa que 7 (2 %) de los conductores superan el 100 % de la CE, 141
(44 %) de los conductores operan por encima del 67 % de la CE. Sólo 171 (54 %) operan por
debajo del 67 % de la CE. Esto indica que se debe tratar de equilibrar la distribución de las
cargas en el circuito para reducir el número de conductores que operan por encima 67 % o
del 100 % de la CE.
Figura 4.7: Capacidad Amperimétrica de los conductores del circuito ANT_A01.
También, se analizó la potencia entregada por circuito por la S/E comparándola con las
pérdidas de los mismos. Además, se verificó la carga de los conductores de todos los circuitos
de la S/E.
58
Análisis Ejemplo para 1 S/E
Conocidas las tensiones nodales en módulo y ángulo se realizó un análisis de las condi-
ciones operacionales de la S/E Antímano (ANT ). Para esta S/E se estudió la capacidad
amperimétrica de los conductores de los circuitos que conforman dicha S/E, se comparó la
potencia entregada por circuito respecto a las pérdidas técnicas de los mismos en (%) y
estudió la capacidad firme de dos S/E: Antímano ANT y Santa Rosa SRO.
En la figura 4.8 se muestra la potencia entregada junto con el porcentaje de pérdidas por
circuito de la S/E Antímano. Se observa que el circuito con mayor potencia entregada es el
ANT_B04 con 22MVA; mientras que, el que menos potencia demanda es el ANT_B02 con
2, 3MVA. Por otro lado, el circuito que presenta más pérdidas es el ANT_A012 con 3, 1MVA
en pérdidas técnicas. Para este análisis no se incluyen pérdidas en los transformadores.
Figura 4.8: Potencia Entregada y Perdida por Circuito de la S/E ANT .
En la figura 4.9 se muestra la cantidad de conductores del circuito que están operando
por debajo del 67 % de la capacidad de emergencia (CE ), aquellos que operan por encima
del 67 % de la CE y aquellos que operan por encima del 100 % de la CE.
La capacidad amperimétrica de los conductores se calculó como se explica en la sección
59
Figura 4.9: Capacidad Amperimétrica de los conductores de la S/E ANT
2.3.3 del capítulo 2. Para un total de 1,649 ramas, se observa que 20 (1 %) de los conductores
superan el 100 % de la CE, 678 (41 %) de los conductores operan por encima del 67 % de la
CE. Sólo 952 (58 %) conductores operan por debajo del 67 % de la CE. En términos generales
más del 40 % de los conductores trabajan por encima del 67 % de su CE.
Para finalizar, en la figura 4.10 se muestra la potencia entregada por 2 S/E: Antímano y
Santa Rosa. La potencia total de cada circuito se calculó sumando la potencia suministrada
por cada circuito perteneciente a dicha S/E. Como caso ejemplo, se asume que cada S/E
cuenta con 4 transformadores de 33, 3MVA. Aplicando el criterio de capacidad firme CF,
explicado en la sección 2.3.4 del capítulo 2, se obtiene que la CF de las S/E debe ser apro-
ximadamente 100MVA. Las potencias suministradas por los circuitos se compararon con el
valor teórico de CF por S/E.
La potencia suministrada por las S/E ANT es 114MVA y SRO es 49, 5MVA, respecti-
vamente. Se observa que la S/E ANT supera el valor teórico por 14MVA mientras que la
S/E SRO está aproximadamente 50MVA por debajo de dicha capacidad. Se podría pensar
60
Figura 4.10: Capacidad Firme de las S/E ANT y SRO.
en una mejor distribución de la carga.
Análisis Ejemplo para Toda la Red
Conocidas las tensiones nodales en módulo y ángulo se realizó un análisis de las condi-
ciones operacionales de toda la red de la Gran Caracas; conformada por 530 circuitos corres-
pondientes a 78 S/E. La cantidad total de nodos de la red estudiada es 64,251. Se estudió la
capacidad amperimétrica de todos los conductores de la red y las pérdidas activas en com-
paración con la potencia activada entregada.
La figura 4.11 muestra la potencia activa demandada y las pérdidas técnicas en relación
con la potencia activa total entregada. Los cálculos se realizaron como se explica en la sección
2.3.2 del capítulo 2. La potencia activa entregada al circuito es de 3, 28GW, de los cuales un
558MW representa las pérdidas técnicas en el circuito, valor que está por encima del por-
centaje de pérdidas permitido. En promedio el valor de las pérdidas para toda la red oscila
alrededor del 2 %, [54]. Sin embargo, se debe recordar que la data con la que se trabajó estaba
corrupta. Por lo tanto, los resultados acá mostrados se vieron afectados por el estado de la
61
data con la que se trabajó.
Figura 4.11: Potencia Activa Demandada y Pérdidas Técnicas de la red.
En la figura 4.12 se muestra la cantidad de conductores de la red que están operando por
debajo del 67 % de la capacidad de emergencia (CE ), aquellos que operan por encima del
67 % y aquellos que operan por encima del 100 % de CE.
La capacidad amperimétrica de los conductores se calculó como se explica en la sección
2.3.3 del capítulo 2. Para un total de 63.721 ramas, se observa que 22.130 (3 %) de los con-
ductores superan el 100 % de la CE, 24.227 (35 %) de los conductores operan por encima del
67 % de la CE. Sólo 39.494 (62 %) conductores operan por debajo del 67 % de la CE. En
términos generales más del 35 % de los conductores trabajan por encima del 67 % de su CE.
En la tabla 4.13 se muestra un reporte general de las condiciones de operación de la red.
62
Figura 4.12: Capacidad Amperimétrica de los conductores de la red.
Tabla 4.13: Condiciones de Operación de la Red, 530 circuitos.
Parámetro Analizado Resultado
Pérdidas Activas Totales 557, 27MW
Pérdidas Reactivas Totales 808, 25MVAr
Potencia Activas Totales Demandada 3, 28GW
Potencia Reactivas Totales Demandada 2, 03GVAr
Conductores que violan 67 % de la CE 24.227
Conductores que violan 100 % de la CE 22.130
Total de Nodos Analizados 64.251
CONCLUSIONES
A continuación se plantean las conclusiones de este trabajo de investigación:
Se implementó un flujo de carga de barrido unidireccional directo basado en el BIBC
(desarrollado por Teng, [19]). Incluyendo la modificación propuesta por De Oliveira
(TRX ), [20], para hacer más eficiente al algoritmo inicial.
El algoritmo fue codificado en MATLAB y se caracteriza por presentar una lógica de
cálculo de gran sencillez: se basa en una suma y multiplicación de matrices. Además, el
mismo fue desarrollado de forma modular, compuesto por dos módulos diferentes: en
el primero se adquieren los datos y en el segundo se realiza el proceso de cálculo.
El proyecto se aplicó con un patrón Modelo-Vista-Controlador (MVC ).
Se demostró que la modificación del flujo de carga propuesta por De Oliveria, [20], con-
vierte al algoritmo original en un algoritmo más eficiente; reduciendo aproximadamente
a la mitad los tiempos de cómputo, ver sección 4.3 del capítulo 4.
El algoritmo desarrollado fue validado utilizando un flujo de carga directo complejo,
[19], y un flujo de carga basado en el Newton-Raphson, [52].
Para hacer aún más eficiente al algoritmo se estableció un esquema de datos en RAM,
en el cual se colocaron las matrices que caracterizan al circuito (TRX y S ). De esta
forma, estarán disponibles en cualquier momento que se requiera hacer el flujo de Carga.
Aún más, las matrices en el caso de estudio de 530 circuitos sólo ocuparon un espacio
en memoria de 40MB.
El algoritmo fue implementado eficientemente en una red de 530 circuitos correspon-
dientes a 78 S/E de la EDC. Las tensiones nodales para los 64.251 nodos se obtuvo en
un tiempo de 0, 76 segundos.
63
64
Comentarios y Recomendaciones
Debido a todos los problemas ocasionados por la data de los circuitos de la EDC se
recomienda depurar la data existente, haciendo especial énfasis en los datos de las
impedancias.
Se recomienda ir a un modelo común de datos en XML.
A fin de tener un registro de las cargas y poder tener disponibles sus respectivas curvas
se recomienda tipificar el consumo de los clientes.
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Apéndice A
CASO EXPLICATIVO: MATRICES BIBC Y BCBV
Para ilustar el proceso de armado de la matriz BIBC se utiliza el circuito de 6 barras
disponible [19]. El sistema incluye 1 generador y 5 cargas equivalentes, conectadas a una red
de distribución radial. El circuito se muestra en la figura C.
Figura A.1: Sistema de Distribución de Ejemplo.
A.1. Matriz BIBC
Utilizando la Ley de Corriente de Kirchhoff (KLC ) se pueden obtener las corrientes de
rama en función de las corrientes inyectadas. Para este caso,
B1 = I2 + I3 + I4 + I5 + I6 (A.1)
71
72
B2 = I3 + I4 + I5 + I6
B3 = I4 + I5
B4 = I5
B5 = I6
De esta forma, la relación entre las corrientes de rama y las corrientes de inyectadas se
puede escribir de forma matricial como se muestra en la ecuación A.2.
B =
B1
B2
B3
B4
B5
=
1 1 1 1 1
0 1 1 1 1
0 0 1 1 0
0 0 0 1 0
0 0 0 0 1
I2
I3
I4
I5
I6
(A.2)
La ecuación A.2 puede ser reescrita en forma general como se muestra en la ecuación A.3
[B] = [BIBC][I] (A.3)
BIBC es una matriz triangular superior que sólo contiene 1 ó 0. Además, la matriz T del
TRX se calcula de igual forma que la matriz BIBC del FDCT.
A.2. Matriz BCBV
Utilizando la Ley de Voltajes de Kirchhoff (KLV ) se pueden obtener los voltajes nodales
en función de las corrientes de rama. Para este caso,
V2 = V1 −B1Z12 (A.4)
V3 = V1 −B1Z12 −B2Z23
73
V4 = V1 −B1Z12 −B2Z23 −B3Z34
V5 = V1 −B1Z12 −B2Z23 −B3Z34 −B4Z45
V6 = V1 −B1Z12 −B2Z23 −B5Z36
De esta forma, la relación entre los voltajes nodales y las corrientes de rama se puede
escribir de forma matricial como se muestra en la ecuación A.5.
∆.V =
V1
V1
V1
V1
V1
−
V2
V3
V4
V5
V6
=
Z12 0 0 0 0
Z12 Z23 0 0 0
Z12 Z23 Z34 0 0
Z12 Z23 Z34 Z45 0
Z12 Z23 0 0 Z36
B1
B2
B3
B4
B5
(A.5)
La ecuación A.5 puede ser reescrita en forma general como se muestra en la ecuación A.6.
[∆.V ] = [BCBV ][B] (A.6)
Apéndice B
ALGORITMO IMPLEMENTADO TRX
A continuación se mostrarán las líneas de código de los algoritmos desarrollados a lo largo
del proyecto:
1. Algoritmo de adquisición de datos.
2. Algoritmo iterativo [20] y algoritmo de análisis de resultados. .
Los algoritmos mostrados a continuación fueron desarrollados e implementados utilizan-
do un programa comercial; los mismos están en formato .MAT. En el capítulo 3 se puede
encontrar el diagrama de flujo del programa implementado.
B.1. Algoritmo de Adquisición de Datos
Algoritmo desarrollado para el filtrado y adquisición de datos de los archivos .DAT pre-
viamente transformados a archivos .MAT. Además, arma las matrices TRX y S; las guarda
en un archivo .MAT y las almacena en disco.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% DECLARACIÓN DE VARIABLES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all clc delete CCS.mat path=’C:\Documents and
Settings\JR\Escritorio\integracion\’; directorio=dir(path);
74
75
directorio(1)=[]; directorio(1)=[]; [T0 T1]=size(directorio); [Nc Tc
Rc]=xlsread(’C:\Documents and
Settings\JR\Escritorio\integracion\conductor.xlsx’,’’); zk=0; t=0;
w1=0; Vbase=0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% CICLO QUE ABRE LOS ARCHIVOS DE LA CARPETA %%%%%%%%%%%%%%%%
for w=1:T0
tic;
clear N T R nodos0 lineas0
if strcmp(finfo(directorio(w).name),’xlsx’)
w1=w1+1;
archivo=directorio(w).name;
[N T R]=xlsread(archivo,’’);
pro=0;
i=0; fc=1; fp=0.85;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%% INICIO DE CILCO QUE BUSCA LOS DATOS %%%%%%%%%%%%
for j=1:length(T)
%%%%% Busco datos de Tension, potencia y factor de potencia del sistema. %%
if strcmp(T(j,1),’END/TITLE’)
Sbase=cell2mat(R(j+2,1));
Vbase=cell2mat(R(j+2,2));
fp=cell2mat(R(j+2,3));
if fp==0 || fp==1
fp=0.85;
end
end
if Vbase > 100
Vbase=Vbase/1000;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
76
%%%%%%%%%%%%%%%% SE BUSCAN DATOS DE NODOS DEL SISTEMA %%%%%%%%%%%%%%%%%%%%%
if strcmp(T(j,1),’END/PARAMS’)
j=j+1;
while strcmp(T(j,1),’END/NODES’)==0
if cell2mat(R(j,2))==Vbase
i=i+1;
nodos0{i,2}=cell2mat(R(j,1));
end
j=j+1;
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% ASIGNA VALORES Y TIPO A LOS NODOS %%%%%%%%%%%%%%%%%%%%%%%%
if strcmp(T(j,1),’END/NODES’)
%Encuentra y crea la variable con la barra Slak
slak=T(j+1,1);
Q=size(nodos0,1);
for i0=1:Q
%potencia activa Generada
nodos0{i0,4}=0;
%potencia reactiva Generada
nodos0{i0,5}=0;
nodos0{i0,6}=0;
nodos0{i0,7}=0;
%Le pone numero a los nodos
nodos0{i0,1}=i0;
%Definir Barra Slak
if strcmp(nodos0{i0,2},slak)
nodos0{i0,3}=1;
else
nodos0{i0,3}=3;
77
end
end
end
%Armar la matriz de lineas
i=0;
i2=0;
if strcmp(T(j,1),’END/SOURCE’)
j=j+1;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% ENCUENTRA LOS DATOS DE LINEAS Y CARGAS %%%%%%%%%%%%%%%%%%%
while strcmp(T(j,1),’END/BRANCH’)==0
if strcmp(T(j,3),’T’)
i2=i2+1;
pro=1;
trans{i2,1}=R(j,1);
trans{i2,2}=R(j,2);
trans{i2,3}=(cell2mat(R(j,7)))*3;
trans{i2,4}=trans{i2,3}*fp*fc;
trans{i2,5}=sqrt((trans{i2,3})^2-(trans{i2,4})^2);
else
i=i+1;
lineas0{i,1}=cell2mat(R(j,1));
lineas0{i,2}=cell2mat(R(j,2));
if strcmp(T(j,3),’L’)
lineas0{i,3}=cell2mat(R(j,7));
lineas0{i,4}=R(j,6);
lineas0{i,5}=0;
lineas0{i,6}=0;
else
lineas0{i,3}=0;
lineas0{i,4}=0;
78
lineas0{i,5}=0;
end
end
j=j+1;
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% ENCUENTRA Y ASIGNA LA CARGA AL NODO ADECUADO %%%%%%%%%%%%%
clear i i0 j slak i2 K=0; if pro==1 [q y]=size(trans);
for i=1:size(nodos0,1)
for j=1:q
if strcmp(nodos0{i,2},trans{j,1})
nodos0{i,6}=trans{j,4}+nodos0{i,6};
nodos0{i,7}=trans{j,5}+nodos0{i,7};
K=K+1;
end
end
end
clear i j
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%% ENCUENTRA CONDUCTORES Y ADJUNTA VALORES DE IMPEDANCIA %%%%%
% Y %
%%%%%%%%%%%%%%%%%% CAPACIDAD AMPERIMETRICA DEL CONDUCTOR. %%%%%%%%%%%%%%%%
[wl rl]=size(lineas0);
for i=1:wl
for j=1:length(Tc)
if lineas0{i,3}~=0
if strcmp(lineas0{i,4},Tc{j,2})
lineas0{i,5}=Nc(j,3);
lineas0{i,6}=Nc(j,4);
79
lineas0{i,7}=(Nc(j,7))/((Sbase)/(Vbase*sqrt(3)));
end
end
end
if lineas0{i,5}==0
lineas0{i,5}=0.18026;
lineas0{i,6}=0.12779;
lineas0{i,7}=324.09/((Sbase)/(Vbase*sqrt(3)));
end
end
clear fc fp i j pro q rl trans u y wl K N Q R T T1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% INICIO DEL FORMATEO DE LAS MATRICES BDAT Y LDAT %%%%%%%%%%
nl=size(lineas0,1); %número de líneas
nn=size(nodos0,1); %número de nodos
for k=1:nn
if cell2mat(nodos0(k,3))==1
bdat0{1,1}=1;
bdat0{1,2}=nodos0(k,2);
bdat0{1,3}=nodos0{k,3};
bdat0{1,4}=(cell2mat(nodos0(k,4)))/Sbase;
bdat0{1,5}=(cell2mat(nodos0(k,5)))/Sbase;
bdat0{1,6}=(cell2mat(nodos0(k,6)))/Sbase;
bdat0{1,7}=(cell2mat(nodos0(k,7)))/Sbase;
bdat0{1,8}=Vbase/Vbase;
end
end
%ciclo que ordena los datos de línea y construye la matriz bdat0
%con base a la metodología propuesta por Shirmohammadi[10].
%Esta metodología es la base para la matriz BIBC del flujo de
%carga implementado.
80
l=2; k=0; for u=1:nn
for g=1:nl
if strcmp(bdat0{u,2},lineas0{g,2})
nlle=lineas0{g,2};
nsal=lineas0{g,1};
lineas0{g,1}=nlle;
lineas0{g,2}=nsal;
end
end
for k=1:nl
if strcmp(bdat0{u,2},lineas0{k,1})
for y=1:nn
if strcmp(lineas0{k,2},nodos0{y,2})
bdat0{l,1}=l;
bdat0{l,2}=nodos0(y,2);
bdat0{l,3}=nodos0{y,3};
bdat0{l,4}=nodos0{y,4}/Sbase;
bdat0{l,5}=(nodos0{y,5})/Sbase;
bdat0{l,6}=(nodos0{y,6})/Sbase;
bdat0{l,7}=(nodos0{y,7})/Sbase;
bdat0{l,8}=Vbase/Vbase;
ldat0{l-1,1}=bdat0{u,1};
ldat0{l-1,2}=bdat0{l,1};
ldat0{l-1,3}=lineas0{k,7};
ldat0{l-1,4}=(lineas0{k,5}*lineas0{k,3})/
(Vbase^2/(Sbase/1000));
ldat0{l-1,5}=(lineas0{k,6}*lineas0{k,3})/
(Vbase^2/(Sbase/1000));
if ldat0{l-1,4}==0
ldat0{l-1,3}=0;
end
81
lineas0{k,1}=0;
lineas0{k,2}=0;
l=l+1;
end
end
end
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% CONVIERTE LAS CELDAS EN MATRICES %%%%%%%%%%%%%%%%%%%%%%%%%
clear g k l nl nlle nn nsal u y nll ns bdat ldat
bdat(:,1)=cell2mat(bdat0(:,1)); bdat(:,3)=cell2mat(bdat0(:,3));
bdat(:,4)=cell2mat(bdat0(:,4)); bdat(:,5)=cell2mat(bdat0(:,5));
bdat(:,6)=cell2mat(bdat0(:,6)); bdat(:,7)=cell2mat(bdat0(:,7));
bdat(:,8)=cell2mat(bdat0(:,8)); ldat(:,1)=cell2mat(ldat0(:,1));
ldat(:,2)=cell2mat(ldat0(:,2)); ldat(:,3)=cell2mat(ldat0(:,3));
ldat(:,4)=cell2mat(ldat0(:,4)); ldat(:,5)=cell2mat(ldat0(:,5));
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% SE INICIA EL ARMADO DE LA MATRIZ "T" Y LA "DLF" %%%%%%%%%%
m=size(ldat,1); %número de líneas
n=size(bdat,1); %número de nodos
T1(m,n)=0; % matriz previa a T
V0((2*n)-2,1)=0; %inicializar vector Voltaje con "0"
Vectx=ldat(:,5); %Definimos el vector de X de las ramas.
Vectr=ldat(:,4); %Definimos el vector de R de las ramas.
for k=1:m %ciclo que crea la matriz T1
ns=ldat(k,1); % nodo de salida
nll=ldat(k,2); % nodo de llegada
B(:,1)=T1(:,ns);
B(nll-1,1)=1;
T1(:,k+1)=B(:,1);
82
end
for k=2:n %ciclo que crea la matriz T
T(:,k-1)=T1(:,k);
end
TR=T’*diag(Vectr)*T; %Sub-matriz de Rs de la matriz TRX
TX=T’*diag(Vectx)*T; %Sub-matriz de xs de la matriz TRX
TRX=vertcat(horzcat(TR,-TX),horzcat(TX,TR)); %Matriz TRX Matriz real.
DLF=TRX; S(:,1)=bdat(:,6); S(:,2)=bdat(:,7); for ll=1:size(bdat0,1)
nno(ll,1)=bdat0{ll,2}; end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%% SE CREA LA VARIABLE "RED" CON LOS DATOS DEL SISTEMA %%%%%%%%%%
RED{w1,1}=(directorio(w).name);
RED{w1,2}=DLF;
RED{w1,3}=S;
RED{w1,4}=nno;
RED{w1,5}=T;
RED{w1,6}=ldat;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
else
zk=zk+1;
malos{zk,1}=directorio(w).name;
end
clear nno B BCBV BCBVI BIBC BIBCI DLF I S V0 bdat k m n nll ns z
ll nodos0 lineas0 ldat0 bdat0 t=toc+t;
end
end save CCS RED clear all
Load_flow % SE LLAMA AL MODULO QUE EJECUTA EL FLUJO DE CARGA%
83
B.2. Flujo de Carga TRX
Código desarrollado para el algoritmo de flujo de carga implementado. El mismo coloca
las matrices TRX y S en memoria RAM y realiza el proceso iterativo descrito en la sección
2.1.2 del capítulo 2. Luego, almacena las tensiones en archivos .MAT. Con los resultados de
tensión se calculan las pérdidas de potencia en los circuitos, las capacidades amperimétricas
de los conductores y las caídas de tensión; todos los resultados son almacenados en el mismo
archivo .MAT.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%% DECLARACIÓN DE VARIABLES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all clc
load CCS % SE CARGA LA MATRIZ DE DATOS DEL SISTEMA A LA RAM
iter=10; %definir número máximo de iteraciones
con=0.0001; %definir criterio de convergencia
t=0; t1=0; Y=size(RED,1); ntot=0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% CICLO QUE INICIA Y CARGA CADA CIRCUITO %%%%%%%%%%%%%%%%%%%
for qp=1:Y
iter2=0;
TRX=RED{qp,2};
S=RED{qp,3};
n=size(S,1);
m=n-1;
V0((2*n)-2,1)=0;
T=RED{qp,5};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%% CICLO QUE CREA LOS VECTORES DE "S" E "I" INICIALES %%%%%%%%%%
for k=1:n-1
V0(k,1)=1;
I(k,1)=((-S(k+1,1)*V0(k,1))-(S(k+1,2)*V0(k+(n-1),1)))/(((V0(k,1))^2+
84
(V0(k+(n-1),1))^2));
I(k+(n-1),1)=(-S(k+1,2)*V0(k,1)+S(k+1,1)*V0(k+(n-1),1))/(((V0(k,1))^2+
(V0(k+(n-1),1))^2));
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%% OPERACIONES PARA INICIALIZAR VECTORES DE ITERACION %%%%%%%%%%
DV=TRX*I; %Multiplicación inicial para hallar el delta de tensión inicial
%para corregir Vo.
V1=V0-DV; %V1 coregida, V0-DV
conve=1; % inicializacion de variabel para convergencia.
Vp(1,1)=1; V(1,1)=1; ang(1,1)=0; DV2=DV; k=0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%% CICLO ITERATIVO DEL FLUJO DE CARGA %%%%%%%%%%%%%%%%%%
tic; % INICIA EL CONTEO DE TIEMPO
while conve>con & iter>iter2
iter2=iter2+1; %Contador de Iteraciones.
Vk=V1;%Definimos una variable Vk para actualizar los voltajes corregidos.
for k=1:n-1 %Proceso para corregir las corrientes del sistema.
I(k,1)=((S(k+1,1)*Vk(k,1))-(S(k+1,2)*Vk(k+(n-1),1)))/(((Vk(k,1))^2+
(Vk(k+(n-1),1))^2));
I(k+(n-1),1)=(-S(k+1,2)*Vk(k,1)+S(k+1,1)*Vk(k+(n-1),1))/(((Vk(k,1))^2+
(Vk(k+(n-1),1))^2));
end
DV=TRX*I; %corrige el delta de tensión para la próxima iteración.
V1=V0-DV; %corrige el voltaje final antes de comprobar la convergencia.
conve=max(abs(DV2-DV)); %Define la convergencia midiendo la variación
%de DV
DV2=DV;
end toc; t=t+toc; tic;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%% CICLO QUE CONVIERTE A "I" EN IMAGINARIO %%%%%%%%%%%%%%%%%%
85
ni=(size(I,1))/2; for L=1:ni
Ii(L,1)=I(L,1);
Ii(L,1)=Ii(L,1)+I(L+ni,1)*i;
end Ip=abs(T*Ii);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%% CICLO QUE ORDENA LOS VOLTAJES Y ANGULOS %%%%%%%%%%%%%%%
for M=2:n
V(M,1)=abs((V1(M-1+(n-1),1))+i*(V1(M-1,1)));
ang(M,1)=rad2deg(angle((V1(M-1+(n-1 ),1))*i+(V1(M-1,1))));
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%% GUARDA RESULTADOS EN LA VARIABLE "RED" %%%%%%%%%%%%%%%%%
RED{qp,6}(:,6)=Ip; RED{qp,7}(:,1)=V; RED{qp,7}(:,2)=ang(:,1);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%% CALCULO DE PERDIDAS POR CIRCUITOS, CAIDAS EN LAS RAMAS%%%%%%%%%%
K=size(RED{qp,6},1); for J=1:K
RED{qp,6}(J,7)=(RED{qp,6}(J,6))^2*(RED{qp,6}(J,4));
RED{qp,6}(J,8)=(RED{qp,6}(J,6))^2*(RED{qp,6}(J,5));
end J=0;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%% CALCULO DE VARIACION DE TENSION EN CADA UNO DE LOS NODOS%%%%%%%%
for J=2:n
RED{qp,7}(J,3)=((RED{qp,7}(1,1)-RED{qp,7}(J,1))/(RED{qp,7}(1,1)))*100;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%% CALCULO DE NODOS POR CIRCUITOS Y TOTALES DE LA RED%%%%%%%%%%%%
K2=size(RED{qp,7},1);
%Calculamos el número de nodos totales del sistema.
npc=size(RED{qp,4},1); RED{qp,8}=npc; ntot=ntot+npc;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
86
clear DV DV2 I M S TRX V V0 V1 Vk Vp ang conve k m n K J npc ni Ii L
Ip T toc; t1=t1+toc; end ntot t t1 save CCS2 RED clear all
Apéndice C
EJEMPLO DE REPORTE DEL ALGORITMO DESARROLLADO
NOMBRE DEL CIRCUITO: GRA_A01.xlsx
VOLTAJES EN LOS NODOS
***************************************************************
# Nodo: Nodo: Tensión: Ángulo: Caída:
1 5201_1 1.000000 0.000000 0.000000
2 5201_2 0.996748 -0.011527 0.325188
3 5201_70 1.000000 0.000000 0.000000
4 5201_43 0.994400 -0.019907 0.559986
5 5201_3 0.993386 -0.023515 0.661434
6 5201_89 0.996708 -0.010766 0.329241
7 5201_71 1.000000 0.000000 0.000000
8 5201_51 0.993381 -0.023554 0.661860
9 5201_44 0.994213 -0.020578 0.578715
10 5201_5 0.993386 -0.023515 0.661434
11 5201_90 0.996708 -0.010766 0.329241
12 5201_55 0.993381 -0.023554 0.661860
13 5201_52 0.993381 -0.023554 0.661860
14 5201_45 0.994213 -0.020578 0.578715
15 5201_68 0.994213 -0.020578 0.578715
87
88
16 5201_46 0.994213 -0.020578 0.578715
17 5201_7 0.990692 -0.033176 0.930784
18 5201_54 0.993381 -0.023554 0.661860
19 5201_53 0.993381 -0.023554 0.661860
20 5201_52 0.993381 -0.023554 0.661860
21 5201_53 0.993381 -0.023554 0.661860
22 5201_47 0.994213 -0.020578 0.578715
23 5201_49 0.994213 -0.020578 0.578715
24 5202_133 0.994213 -0.020578 0.578715
25 5201_8 0.990648 -0.032334 0.935247
26 5201_10 0.989429 -0.037725 1.057105
27 5201_53 0.993381 -0.023554 0.661860
28 5201_56 0.992600 -0.026356 0.740000
29 5201_48 0.994190 -0.020377 0.580982
30 5201_50 0.994207 -0.020526 0.579297
31 5201_9 0.990648 -0.032334 0.935247
32 5201_13 0.989429 -0.037725 1.057105
33 5201_57 0.992581 -0.025996 0.741908
34 5201_59 0.991759 -0.029377 0.824109
35 5201_11 0.989429 -0.037725 1.057105
36 5201_12 0.989429 -0.037725 1.057105
37 5201_58 0.992581 -0.025996 0.741908
38 5201_60 0.991530 -0.025054 0.847034
39 5201_62 0.991054 -0.031910 0.894558
40 5201_80 0.989352 -0.037042 1.064767
41 5201_19 0.988295 -0.041816 1.170520
42 5201_61 0.991530 -0.025054 0.847034
43 5201_63 0.991039 -0.031965 0.896062
44 5201_64 0.991054 -0.031910 0.894558
45 5201_14 0.989250 -0.036132 1.074988
46 5201_81 0.989144 -0.035187 1.085595
89
47 5201_20 0.988290 -0.041732 1.170967
48 5201_67 0.988237 -0.040723 1.176300
49 5201_23 0.987049 -0.046320 1.295086
50 5201_6 0.990475 -0.033996 0.952472
51 5202_49 0.991039 -0.031965 0.896062
52 5202_47 0.991039 -0.031965 0.896062
53 5201_65 0.991054 -0.031910 0.894558
54 5201_15 0.989249 -0.036102 1.075148
55 5201_17 0.989152 -0.035257 1.084806
56 5201_82 0.989144 -0.035187 1.085595
57 5201_21 0.988290 -0.041732 1.170967
58 5201_22 0.988237 -0.040723 1.176300
59 5201_24 0.987049 -0.046320 1.295086
60 5201_86 0.990454 -0.033595 0.954597
61 5201_25 0.990475 -0.033996 0.952472
62 5201_16 0.989249 -0.036102 1.075148
63 5201_18 0.989152 -0.035257 1.084806
64 5201_84 0.989144 -0.035187 1.085595
65 5201_83 0.989144 -0.035187 1.085595
66 5201_27 0.986254 -0.049202 1.374616
67 5201_87 0.990454 -0.033595 0.954597
68 5201_85 0.990475 -0.033996 0.952472
69 5201_28 0.986203 -0.048231 1.379741
70 5201_30 0.985663 -0.051343 1.433652
71 5201_88 0.990454 -0.033595 0.954597
72 5201_29 0.986203 -0.048231 1.379741
73 5201_32 0.985527 -0.050125 1.447291
74 5201_72 0.985575 -0.051665 1.442519
75 5201_34 0.985365 -0.052426 1.463477
76 5201_33 0.985527 -0.050125 1.447291
77 5201_69 0.985575 -0.051665 1.442519
90
78 5201_73 0.985569 -0.051562 1.443064
79 5201_35 0.985365 -0.052426 1.463477
80 5201_31 0.985575 -0.051665 1.442519
81 5201_74 0.985569 -0.051562 1.443064
82 5201_36 0.985365 -0.052426 1.463477
83 5201_37 0.985349 -0.052485 1.465098
84 5201_38 0.985342 -0.052511 1.465818
85 5201_75 0.985019 -0.053683 1.498064
86 5201_39 0.985342 -0.052511 1.465818
87 5201_76 0.984992 -0.053441 1.500764
88 5201_40 0.984910 -0.054080 1.508981
89 5201_77 0.984992 -0.053441 1.500764
90 5201_41 0.984856 -0.053045 1.514433
91 5201_79 0.984992 -0.053441 1.500764
92 5201_78 0.984992 -0.053441 1.500764
93 5201_42 0.984856 -0.053045 1.514433
PERDIDAS EN LAS RAMAS
***************************************************************
# Nodo sal # Nodo lle: Perdida P: Perdida Q:
1 2 2.770122e-004 1.963796e-004
1 3 0.000000e+000 0.000000e+000
2 4 7.354838e-005 5.213995e-005
2 5 1.728360e-004 1.225270e-004
2 6 1.247080e-007 3.039511e-008
3 7 0.000000e+000 0.000000e+000
4 8 2.143848e-005 1.519818e-005
4 9 1.925370e-006 1.364934e-006
5 10 0.000000e+000 0.000000e+000
91
6 11 0.000000e+000 0.000000e+000
8 12 0.000000e+000 0.000000e+000
8 13 0.000000e+000 0.000000e+000
9 14 0.000000e+000 0.000000e+000
9 15 0.000000e+000 0.000000e+000
9 16 0.000000e+000 0.000000e+000
10 17 1.384522e-004 9.815158e-005
12 18 0.000000e+000 0.000000e+000
12 19 0.000000e+000 0.000000e+000
12 20 0.000000e+000 0.000000e+000
13 21 0.000000e+000 0.000000e+000
16 22 0.000000e+000 0.000000e+000
16 23 0.000000e+000 0.000000e+000
16 24 0.000000e+000 0.000000e+000
17 25 2.302853e-007 5.612751e-008
17 26 5.971812e-005 4.233540e-005
18 27 0.000000e+000 0.000000e+000
19 28 1.644385e-005 1.165738e-005
22 29 1.593068e-007 6.769693e-008
23 30 2.723222e-008 1.157225e-008
25 31 0.000000e+000 0.000000e+000
26 32 0.000000e+000 0.000000e+000
28 33 5.897191e-008 1.437324e-008
28 34 1.562133e-005 1.107428e-005
32 35 0.000000e+000 0.000000e+000
32 36 0.000000e+000 0.000000e+000
33 37 0.000000e+000 0.000000e+000
34 38 1.181952e-006 2.880776e-007
34 39 1.017943e-005 7.216409e-006
35 40 1.244838e-006 5.289901e-007
36 41 3.744148e-005 2.654303e-005
92
38 42 0.000000e+000 0.000000e+000
39 43 2.173545e-007 1.540870e-007
39 44 0.000000e+000 0.000000e+000
40 45 7.942518e-007 3.375149e-007
40 46 1.765661e-006 7.503123e-007
41 47 6.943301e-009 1.692293e-009
41 48 3.005762e-007 7.325951e-008
41 49 3.439619e-005 2.438416e-005
43 50 8.150792e-006 5.778263e-006
43 51 0.000000e+000 0.000000e+000
43 52 0.000000e+000 0.000000e+000
44 53 0.000000e+000 0.000000e+000
45 54 1.237258e-009 3.015572e-010
45 55 6.935722e-007 2.947314e-007
46 56 0.000000e+000 0.000000e+000
47 57 0.000000e+000 0.000000e+000
48 58 0.000000e+000 0.000000e+000
49 59 0.000000e+000 0.000000e+000
50 60 2.194010e-007 5.347467e-008
50 61 0.000000e+000 0.000000e+000
54 62 0.000000e+000 0.000000e+000
55 63 0.000000e+000 0.000000e+000
56 64 0.000000e+000 0.000000e+000
56 65 0.000000e+000 0.000000e+000
59 66 2.196064e-005 1.556835e-005
60 67 0.000000e+000 0.000000e+000
61 68 0.000000e+000 0.000000e+000
66 69 2.656918e-007 6.475715e-008
66 70 1.385303e-005 9.820693e-006
67 71 0.000000e+000 0.000000e+000
69 72 0.000000e+000 0.000000e+000
93
70 73 6.461058e-007 2.745607e-007
70 74 5.520172e-008 3.913363e-008
70 75 5.572751e-006 3.950637e-006
73 76 0.000000e+000 0.000000e+000
74 77 0.000000e+000 0.000000e+000
74 78 4.240176e-009 1.033459e-009
75 79 0.000000e+000 0.000000e+000
77 80 0.000000e+000 0.000000e+000
78 81 0.000000e+000 0.000000e+000
79 82 0.000000e+000 0.000000e+000
82 83 3.028669e-007 2.147085e-007
83 84 5.979455e-008 4.238958e-008
83 85 3.422693e-006 2.426417e-006
84 86 0.000000e+000 0.000000e+000
85 87 1.277531e-007 5.428831e-008
85 88 6.801111e-007 4.821447e-007
87 89 0.000000e+000 0.000000e+000
88 90 4.246517e-007 1.035005e-007
89 91 0.000000e+000 0.000000e+000
89 92 0.000000e+000 0.000000e+000
90 93 0.000000e+000 0.000000e+000
CAPACIDADES AMPERIMETRICAS
***************************************************************
#Nodo sal #Nodo lle: I(max): CAP(67) CAP (max)
1 2 1.042209e-001 4.689942e-002 6.999914e-002
1 3 0.000000e+000 4.689942e-002 6.999914e-002
2 4 3.832317e-002 4.689942e-002 6.999914e-002
2 5 6.288734e-002 4.689942e-002 6.999914e-002
94
2 6 3.010414e-003 1.852592e-002 2.765062e-002
3 7 0.000000e+000 0.000000e+000 0.000000e+000
4 8 2.574611e-002 4.689942e-002 6.999914e-002
4 9 1.257706e-002 4.689942e-002 6.999914e-002
5 10 6.288734e-002 0.000000e+000 0.000000e+000
6 11 3.010414e-003 0.000000e+000 0.000000e+000
8 12 2.574611e-002 0.000000e+000 0.000000e+000
8 13 0.000000e+000 0.000000e+000 0.000000e+000
9 14 0.000000e+000 2.692495e-002 4.018649e-002
9 15 0.000000e+000 4.689942e-002 6.999914e-002
9 16 1.257706e-002 0.000000e+000 0.000000e+000
10 17 6.288734e-002 4.689942e-002 6.999914e-002
12 18 0.000000e+000 0.000000e+000 0.000000e+000
12 19 2.574611e-002 0.000000e+000 0.000000e+000
12 20 0.000000e+000 0.000000e+000 0.000000e+000
13 21 0.000000e+000 0.000000e+000 0.000000e+000
16 22 7.546220e-003 0.000000e+000 0.000000e+000
16 23 5.030841e-003 0.000000e+000 0.000000e+000
16 24 0.000000e+000 0.000000e+000 0.000000e+000
17 25 5.049839e-003 1.852592e-002 2.765062e-002
17 26 5.783750e-002 4.689942e-002 6.999914e-002
18 27 0.000000e+000 0.000000e+000 0.000000e+000
19 28 2.574611e-002 4.689942e-002 6.999914e-002
22 29 7.546220e-003 2.692495e-002 4.018649e-002
23 30 5.030841e-003 2.692495e-002 4.018649e-002
25 31 5.049839e-003 0.000000e+000 0.000000e+000
26 32 5.783750e-002 0.000000e+000 0.000000e+000
28 33 3.023646e-003 1.852592e-002 2.765062e-002
28 34 2.272247e-002 4.689942e-002 6.999914e-002
32 35 1.744872e-002 0.000000e+000 0.000000e+000
32 36 4.038877e-002 0.000000e+000 0.000000e+000
95
33 37 3.023646e-003 0.000000e+000 0.000000e+000
34 38 5.044779e-003 1.852592e-002 2.765062e-002
34 39 1.767769e-002 4.689942e-002 6.999914e-002
35 40 1.744872e-002 2.692495e-002 4.018649e-002
36 41 4.038877e-002 4.689942e-002 6.999914e-002
38 42 5.044779e-003 0.000000e+000 0.000000e+000
39 43 1.767769e-002 4.689942e-002 6.999914e-002
39 44 0.000000e+000 2.692495e-002 4.018649e-002
40 45 8.344978e-003 2.692495e-002 4.018649e-002
40 46 9.103745e-003 2.692495e-002 4.018649e-002
41 47 1.518756e-003 1.852592e-002 2.765062e-002
41 48 5.088027e-003 1.852592e-002 2.765062e-002
41 49 3.378199e-002 4.689942e-002 6.999914e-002
43 50 1.767769e-002 4.689942e-002 6.999914e-002
43 51 0.000000e+000 0.000000e+000 0.000000e+000
43 52 0.000000e+000 0.000000e+000 0.000000e+000
44 53 0.000000e+000 0.000000e+000 0.000000e+000
45 54 7.585761e-004 1.852592e-002 2.765062e-002
45 55 7.586402e-003 2.692495e-002 4.018649e-002
46 56 9.103745e-003 0.000000e+000 0.000000e+000
47 57 0.000000e+000 0.000000e+000 0.000000e+000
48 58 0.000000e+000 0.000000e+000 0.000000e+000
49 59 3.378199e-002 0.000000e+000 0.000000e+000
50 60 1.010156e-002 1.852592e-002 2.765062e-002
50 61 7.576129e-003 0.000000e+000 0.000000e+000
54 62 0.000000e+000 0.000000e+000 0.000000e+000
55 63 7.586402e-003 0.000000e+000 0.000000e+000
56 64 0.000000e+000 0.000000e+000 0.000000e+000
56 65 0.000000e+000 0.000000e+000 0.000000e+000
59 66 3.378199e-002 4.689942e-002 6.999914e-002
60 67 1.010156e-002 0.000000e+000 0.000000e+000
96
61 68 0.000000e+000 0.000000e+000 0.000000e+000
66 69 5.073849e-003 1.852592e-002 2.765062e-002
66 70 2.870814e-002 4.689942e-002 6.999914e-002
67 71 0.000000e+000 0.000000e+000 0.000000e+000
69 72 5.073849e-003 0.000000e+000 0.000000e+000
70 73 5.087528e-003 2.692495e-002 4.018649e-002
70 74 7.615905e-004 4.689942e-002 6.999914e-002
70 75 2.285902e-002 4.689942e-002 6.999914e-002
73 76 0.000000e+000 0.000000e+000 0.000000e+000
74 77 0.000000e+000 4.689942e-002 6.999914e-002
74 78 7.615905e-004 1.852592e-002 2.765062e-002
75 79 2.285902e-002 0.000000e+000 0.000000e+000
77 80 0.000000e+000 2.692495e-002 4.018649e-002
78 81 0.000000e+000 0.000000e+000 0.000000e+000
79 82 2.285902e-002 0.000000e+000 0.000000e+000
82 83 2.285902e-002 4.689942e-002 6.999914e-002
83 84 1.015693e-002 4.689942e-002 6.999914e-002
83 85 1.270209e-002 4.689942e-002 6.999914e-002
84 86 1.015693e-002 0.000000e+000 0.000000e+000
85 87 5.080493e-003 2.692495e-002 4.018649e-002
85 88 7.621598e-003 4.689942e-002 6.999914e-002
87 89 5.080493e-003 0.000000e+000 0.000000e+000
88 90 7.621598e-003 1.852592e-002 2.765062e-002
89 91 0.000000e+000 0.000000e+000 0.000000e+000
89 92 0.000000e+000 0.000000e+000 0.000000e+000
90 93 7.621598e-003 0.000000e+000 0.000000e+000
TOTALES DEL CIRCUITO
***************************************************************
Perdidas activas totales (pu): 9.215657e-004
97
Perdidas reactivas totales (pu): 6.504554e-004
Cantidad de conductores violan el 67% capacidad: 28
Cantidad de conductores violan el 100% capacidad: 25
Total de nodos en este circuito: 93
Apéndice D
CASO DE ESTUDIO: CIRCUITO DE 4 BARRAS
El sistema incluye 1 generador y 3 cargas equivalentes, conectadas a una red de distribu-
ción radial. Los datos de ramas y nodos del circuito se muestran en las tablas D.1 y D.2,
respectivamente.
Datos de línea
El circuito está conformado por 3 líneas, descritas en la tabla D.1.
Tabla D.1: Datos de línea (Ldat)
salida llegada r(p.u) x(p.u)
1 2 0,0265 0,0462
2 3 0,1005 0,0693
3 4 0,0670 0,0462
Datos de nodo
Los datos de los 4 nodos se encuentran en la tabla D.2.
98
99
Tabla D.2: Datos de nodos (Bdat)
N Tipo Pgen0(p.u) Qgen0(p.u) Pload(p.u) Qload(p.u) V0 fcap
1 1 0 0 0 0 1 1
2 3 0 0 0,1017 0,0635 1 1
3 3 0 0 0,0547 0,0342 1 1
4 3 0 0 0,0809 0,0596 1 1
Donde,
N. Es el número de la barra.
Tipo. Corresponde al tipo de barra: 1 barra slack, 2 barra PV y 3 barra PQ.
Pgen0. Generación de potencia activa (valor inicial)(p.u).
Qgen0.Generación de potencia reactiva (valor inicial)(p.u).
Pload. Potencia activa consumida (p.u).
Qload. Potencia activa consumida (p.u).
V0. Tensión inicial de la barra (p.u).
fcap. Factor de capacidad de la barra.
Apéndice E
REPORTE DEL MATPOWER PARA EL CASO DE 7 BARRAS
runpf(’MATPOWERexample’)
Converged in 0.01 seconds
================================================================================
| System Summary
================================================================================
How many? How much? P (MW) Q(MVAr)
------------- ------------------- ----------- ------------
Buses 7 Total Gen Capacity 250.0 -300.0 to 300.0
Generators 1 On-line Capacity 250.0 -300.0 to 300.0
Committed Gens 1 Generation (actual) 0.5 0.3
Loads 6 Load 0.5 0.3
Fixed 6 Fixed 0.5 0.3
Dispatchable 0 Dispatchable -0.0 of -0.0 -0.0
Shunts 0 Shunt (inj) -0.0 0.0
Branches 6 Losses (I^2*Z) 0.03 0.02
Transformers 0 Branch Charging(inj) - 0.0
Inter-ties 0 Total Inter-tie Flow 0.0 0.0
100
101
Areas 1
Minimum Maximum
------------------------- --------------------------------
Voltage Magnitude 0.917079 p.u. @ bus 7 1.000 p.u. @ bus 1
Voltage Angle -0.49 deg @ bus 6 0.00 deg @ bus 1
P Losses (I^2*R) - 0.02 MW @ line 1-3
Q Losses (I^2*X) - 0.02 MVAr @ line 1-3
================================================================================
| Bus Data
================================================================================
Bus Voltage Generation Load
# Mag(pu) Ang(deg) P (MW) Q (MVAr) P (MW) Q (MVAr)
----- ------- -------- -------- -------- -------- --------
1 1.000000 0.000000 0.50 0.34 - -
2 0.994335 -0.173777 - - 0.10 0.06
3 0.940742 -0.020595 - - 0.05 0.03
4 0.931971 -0.003891 - - 0.08 0.06
5 0.925992 -0.469903 - - 0.10 0.06
6 0.917416 -0.493752 - - 0.05 0.03
7 0.917079 -0.452656 - - 0.08 0.06
-------- -------- -------- --------
Total: 0.50 0.34 0.47 0.31
102
================================================================================
| Branch Data
================================================================================
Brnch From To From Bus Injection To Bus Injection Loss (I^2*Z)
# Bus Bus P (MW) Q (MVAr) P (MW) Q (MVAr) P (MW) Q (MVAr)
---- ---- ---- ------- ------ ------- -------- ------ -------
1 1 2 0.10 0.06 -0.10 -0.06 0.000 0.00
2 1 3 0.40 0.27 -0.38 -0.26 0.024 0.02
3 3 4 0.08 0.06 -0.08 -0.06 0.001 0.00
4 3 5 0.24 0.16 -0.24 -0.16 0.003 0.00
5 5 6 0.06 0.03 -0.05 -0.03 0.000 0.00
6 5 7 0.08 0.06 -0.08 -0.06 0.001 0.00
-------- --------
Total: 0.029 0.02
Apéndice F
“A COMPENSATION-BASED POWER FLOW METHOD FOR WEAKLY
MESHED DISTRIBUTION AND TRANSMISSION NETWORKS”,
SHIRMOHAMMADI ET AL. 1988, [11]
103
753 IEEE Transactions on Power Systems, Vol. 3, No. 2, May 1988
A COMPENSATION-BASED POWER FLOW METHdD FOR WEAKLY MESHED DISTRIBUTION AND TRANSMISSION NETWORKS
D. Shirmoharmnadi H. W. Hong Member Senior Member
Systems Engineering Group Pacific Gas and Electric Company
San Francisco, California
ABSTRACT
This paper describes a new power flow method for solving weakly meshed distribution and transmission networks, using a multi-port compensation technique and basic formulations of Kirchhoff's laws. ThiS method has excellent convergence characteristics and is very robust. A computer program implementing this power flow solution scheme was developed and successfully applied to several practical distribution netwdrks with radial and weakly meshed structure. This program was also successfully used for solving radial and weakly meshed transmissioh networks. The method can be applied tb the solution of both the three-phase (unbalanced) and single-phase (balanced) representation of the netbork. In this paper, however, only the single phase representation is treated in detail.
I. INTRODUCTION
Recently.at the Pacific,Gas and Electric Company we developed a distkibution network optimization software package. This development work called for a power flew solution algorithm with the following general characteristics:
1. Capable of Solving radial and weakly meshed distributation networks with up to several thousand line sections (branches) and nodes (buses ) .
2. Robust and efficient.
The efficiency of such a power flow algorithm is of utniost importance as each optimization study requires numerous power flow runs. Furthermore, the extension of the application of this power flow method to three phase networks with distributed loads was also envisaged.
The Newton Raphson and the fast decoupled power flow solution techniques and a host of their derivatives have efficiently solved "well-behaved" power systems for more than two decades. Researchers, however, have been aware of the shortcomings of these solution algorithms when they are "generically" implemented and applied to ill-conditioned andlor poorly initialibed power systems [ 1 , 2 , 3 ] . Hence, conrnerical power flow packages always modify these
This pap& was sponsored by the IEEE Power Engineering Society for presentation at the IEEE Power Industry Computer Applica- tion Conference, Montreal, Canada, May 18-21,1987. Manuscript was published in the 1987 PICA Conference Record.
A. Semlyen G. X. Luo Senior Member
Department of Electrical Engineering University of Toronto Toronto, Ontario, Canada
algorithms for enhanced robustness. The nature of modifications and the degree of improvement obtained varies for different packages. The Gauss-Seidel power flow technique, another classicial power flow Ethod, although very robust, has shown to be extremely inefficient in solving large power systems.
Distribution networks, due to their wide ranging resistence and reactance values and radial structure, fall into the category of ill-conditioned power systems for the generic Newton-Raphson and fast decoupled power flow algorithms. Our experience with a basic Newton- Raphson power flow program for solving distribution networks was mostly unsuccessful as it diverged for the majority of the networks studied. Later we successfully used the Newton-Raphson based Western System Coordinating Council (WSCC) power flow program. This program, which is commonly used by the WSCC member utilities, includes several enhancements for increasing its convergence capabilities. Although the robustness of the program was acceptable, the computation time was excessive. In addition, the extension of the Newton-Raphson algorithm to the solution of the three phase networks, not even considering distributed load, would result in substantial deterioration of the numerical efficiency of the solution algorithm [41.
Efficient power flow algorithms for solving single and three phase radial distribution networks [5,6,71 have been extensively used by distribution engineers. These algorithms are not, however, designed to solve meshed networks.
In this paper, we propose a new method for the solution of weakly meshed networks. In this method, we first break the interconnected grid at a number of points (breakpoints) in order to convert it into one radial network. Each breakpoint will open one simple loop. The radial network is solved efficiently by the direct application of Kirchhoff's voltage and current laws (KVL and KCL). We then account for the flows at the breakpoints by injecting currents at their two end nodes. The breakpoint currents are calculated using the multi-port compensation method [8 ,91. In presence of constant P,Q loads, the network is nonlinear causing the compensation process to become iterative. The solution of the radial network with the additional breakpoint current injections completes the solution of the weakly meshed network.
Our studies have shown that, typically, only a few iterations were required for the solution of distribution networks using the proposed power flow solution technique. For the weakly meshed transmission networks the number of iterations was higher, due to the additional nonlinearities introduced by generator buses (PV nodes). In all the cases studied the proposed power flow technique was significantly more efficient than the Newpn-Raphson power flow algorithm while converging to the same solution.
0885-8950/88/05oo-O753$01 .WO 1988 IEEE
754
The numerical efficiency of the proposed compensation-based power flow method, however, diminishes as the number of breakpoints required to convert the meshed network to a radial configuration increases. This restricts the practical application of the method to weakly meshed networks.
In this paper we emphasize the application of the compensation-based power flow method to the distribution networks and provide several practical examples. A comparison with the Newton-Raphson based WSCC power flow program is also presented. We then discuss the application of the algorithm to weakly meshed transmission networks where again comparison with the WSCC power flow program is provided. A brief discussion of the extension of this method to three phase networks as well as the treatment of distributed loads and other practical considerations concludes the paper.
11. SOLUTION OF A RADIAL DISTRIBUTION NETWORK
currents in
from node L2 L=b,b-1, ..., 1 JL(k) = -ILt(k)+C(branches emanating) (2)
where IL2(k) is the current injection at node L2. This is the direct application of the KCL.
3. Forward sweep: Nodal voltages are updated in a forward sweep starting from branches in the first layer toward those in the last. For each branch, L, the voltage at node L2 is calculated using the updated voltage at node L1 and the branch current calculated in the preceding backward sweep:
where z~ is the series impedance of branch L. This is the direct application of the KVL.
Steps 1, 2 and 3 are repeated until convergence is achieved.
In our algorithm, regardless of its original topology, the distribution network is first converted to a radial network. Hence, an efficient algorithm for the solution of radial networks is crucial to the viability of the overall solution method.
The solution method used for radial distribution networks is based on the direct application of the KVL and KCL. Similar techniques are also described in [5,6,7]. For our implementation, we developed a branch oriented approach using an efficient branch numbering scheme to enhance the numerical performance of the solution method. We first describe this branch numbering scheme.
Branch Numbering
In contrast to all classical power flow techniques which use nodal solution methods for the network, our algorithm is branch-oriented. Figure 1 shows a typical radial distribution network with n nodes, b(=n-1) branches and a single voltage source at the root node. In this tree structure, the node of a branch L closest to the root note is denoted by L1 and the other node by L2. We number the branches in layers away from the root node as shown in Figure 2. The numbering of branches in one laver starts onlv after all the branches in the
implemented in our power flow program.
Solution Method
Given the voltage at the root node and assuming a flat profile for the initial voltages at all other nodes, the iterative solution algorithm consists of three steps:
1. Nodal current calculation: At iteration k, the nodal current injection,Ii(k), at network node i is calculated as,
I,@) = (si/vi(k-l) )*-Y~v~(~-I) i=1,2, . . . ,n (1)
where Vi(k-l) is the voltage at node i calculated during the (k-l)th iteration and Si is the specified power injection at node i. Yi is the sum of all the shunt elements at the node i. LAYER 10
2. Backward sweep: At iteration k, starting from the branches in the last layer and moving Fig.2 Branch numbering of the radial distribution towards the branches connected to the root node network the current in branch L, J,, is calculated as:
755
MAXIMUM REAL POWER MISMATCH
(W
Convergence Criterion
~ ~~~~
MAXIMUM REACTIVE POWER MISMATCH
(WAR)
We used the maximum real and reactive power mismatches at the network nodes as our convergence criterion. As described in the solution algorithm, the nodal current injections, at iteration k, are calculated using the scheduled nodal power injections and node voltages from the previous iteration (equation (1)). The node voltages at the same iteration are then calculated using these nodal current injections (equations ’ (2) and (3)). He c , the power injection for node i at kth iteration, is calculated as:
544 NODES
WXFIANGE
0.409-5.063
( 4 ’ )
1 2 3 4 5
7 8
6
The real and reactive power mismatches at bus i are then calculated as:
mi(k) = Re[Si(k) - Si] i=1,2,. . . .n ( 4 ” )
AQ~(~) = ImIsi(k) - si] Table 1 shows the values of the maximum real and
reactive power mismatches at the various iterations of this radial network power flow solution algorithm for three practical distribution networks. An excellent rate of convergence can be observed for all three networks studied. This convergence behavior can be briefly explained as follows. The error that is incurred in estimating initial node voltages is propagated first to nodal and then to branch currents via equations (1) and ( 2 ) . In the process of updating node voltages using equation ( 3 1 , the error in branch currents is multiplied by the small line impedance. ZL ()ZLI<<l). and thereby rapidly attenuated.
DISTRIBUTION
lEEzRArn1 lTERATlON NUMBER
244 NODES
WXFIANGE
0.245-5.065
‘ 1 2 3 4 5
6.134 0.301 0.008 0.000 0.000
13.092 0.567 0.024 0.001 0.000
I 1411 NODES I 1
I 0.000-5.480 1, ;
5.994 1.691 0.402 0.126 0.031 0.010 0.003 O.OO0
5.597 1.132 0.377 0.085 0.029 0.007 0.002 0.000
4.691 4.573 0.719 0.088 0.037 0.011 0.003 0.001 0.000
Table 1 Convergence characteristics of the radial network solution algorithm
Figure 3 shows the flow chart of the overall power flow solution method for radial networks.
111. SOLUTION OF WEAKLY MESHED DISTRIBUTION NETWORKS
Figure 4 shows an example of a weakly meshed distribution network containing four simple loops. The radial network solution algorithm can not be directly applied to this network. Nevertheless, by selecting four breakpoints, this network can be converted to a radial configuration. The branch currents interrupted by the creation of every breakpoint can be replaced by
AND INITIAL CONDITION
SET ITERATON CWNT
CALCULATE MAXIMUM REAL AND REACTIVE POWER MISMATCHES
EQUATION (4)
MAXlMUMlTERAT YES ‘‘INT
Fig.3 Power flow solution algorithm for the radial networks
Fig.4 A weakly meshed distribution network
current injections at its two end nodes, without affecting the network operating condition. This resulting radial network can now tp solved by the radial network solution technique described earlier.
In applying the radial network solution algorithm, the current at breakpoint j, Jj, must be injected with opposite polarity at the two end nodes of the breakpoint. At iteration k:
156
j = 1, 2, ..., p (5)
where jl and j2 corr s ond to h two end nodes of the breakpoint j, and Ijirkf and Ijjke become he nodal current injections at these nodes, Tj(6 is the breakpoint current and p the total number of breakpoints. In the presence of nodal currents at the t a point no e due to shunt elements andlor loads, $.?kt and -3jh7' must be added to these nodal currents.
rocess is c ematically shown in Figure 5. Once $Tky and I-$? are updated, steps 2 and 3 of the radial networi solution algorithm can be directly applied.
BREAKPOINT
Fig.5 Breakpoint representation using nodal current injections
Calculation of Breakpoint Currents Using Compensation Method
Breakpoint currents are calculated using the multi-port compensation method [81. Figure 6 illustrates the concept used in this approach. In this figure the radial network resulting from the opening of the breakpoints is shown as a multi-port circuit with breakpoint nodes forming the ports of the circuit. The calculation of breakpoint currents requires that the multi-port equivalent circuit for the radial network as seen from the ports of the breakpoints be established.
For a linear network, this multi-port equivalent circuit can be the Thevenin equivalent circuit of the radial network seen from the open ports created b the breakpoints. In this circuit the Thevenin voltage f is the (pxl) vector of open circuit breakpoint voltages, obtained from the power flow solution of the radial network ,[dl the (pxp) non-sparse matrix of the breakpoint impedances (coefficients relating breakpoint currents and voltages) and 2 is the (pxl) vector of the desired breakpoint currents (Figure 7):
Fig.6 Multi-port equivalent of the network as seen from the breakpoint ports
In the presence of constant power loads the distribution network is, however, nonlinear and equation (6) cannot be directly used. Instead, as we shall explain, we calculate breakpoint currents iterively using the Thevenin equivalent circuit.
Pig.7 Thevenin equivalent circuit of the network as seen from the breakpoint ports
Calculation of Breakpoint Impedance Uatrix
The breakpoint impedance matrix (Thevenin equivalent impedance) can be determined using the following method:
Equation (6) can be written as,
A. .. zpj ..... C ' According to equation (71, column j of the
breakpoint impedance ma rix will be e ual t o vector of breakpoint voltages for $j=l p.u. and ji=O, i=1,2,. . . .p and i # j. This corresponds to the application of 1 p.u. current at the breakpoint j with all loads and the source at the root node removed, which is, in turn, equivalent to the injection of 1 p.u. currents with opposite polarity at the two end nodes of the breakpoint j (equation (5)). In the absence of loads, the accurate solution of the power flow for the radial network can be achieved in one iteration. Each of the breakpoint voltages can be determined by subtracting the voltages at the two end nodes of the breakpoint. This process must be repeated for all breakpoints until all the columns of the breakpoint impedance matrix are calculated.
Iterative Compensation Process
The iterative compensation process for calculating the breakpoint currents, using the Thevenin equivalent circuit of Figure 7 , is described in the following:
1.
2.
Calculate the Thevenin equAvalent impedance (breakpoint impedance matrix [Z] .of the radial network) maintaining it constant throughout the compensation process.
Calculate the Thevenin e uivalent voltage (breakpoint voltage vector 1) of the radial network using the radial network solution algorithm (Figure 3) including the breakpoint currents calculated from the previous iteration of the compensation process. The initial values of the breakpoint currents are zero.
757
3. Calculate the incremental change in the breakpoint currents using the Thevenin equivalent circuit. At iteration m of the compensation process:
4. Update the breakpoint currents. At iteration m:
(8")
5. Repeat steps 2, 3 and 4 until convergence is reached (the maximum breakpoint voltage calculated at step 2 is within prescribed limits)
This fixed tangent solution method is schematically depicted in Figure 8 for a network having a single breakpoint. Computationally there is no need f r the inversion the breakpoint impedance matrix
1 is factorized once in the beginning of the iterations and the forward fiand backward substitution is then used to in equation (E' ) . Our test cases on practical distribution networks showed that the number of iterations required for the calculation of the breakpoint currents was less than 5 in most cases.
[&. Complex matrix
calculate
Figure 9 shows the flow chart of the overall power flow solution schome.
$(l) $12)
Fig.8 Graphic representation of the iterative compensation process
Selection of Breakpoints
Breakpoints are selected in order to convert meshed network into a radial configuration. In addition to this function breakpoints should be selected in such a way as to ensure the convergence of the overall solution algorithm. The latter requirement for the selection of breakpoints usually is satisfied by concentrating on the parts of the meshed network where the power flows are low. On the other hand, the power flows are the end product of the solution method and are not known at the time of breakpoint selection.
In weakly meshed distribution networks breakpoint selection does not affect the convergence performance of the solution method in any noticeable manner. Hence, we select them for the main purpose of opening the network loops. Under these circumstances the algorithm for identifying the breakpoints is very simple and becomes part of the branch numbering scheme described below:
1.
2.
3.
4.
5 ,
Examine all branches and select those connected to the root node for the first branch layer
Store the node number of the far node of the branches in the branch layer just formed. For all these nodes raise a flag indicating that they have already been used
Examine all the remaining branches and select ,
those connected to any of far nodes of the branches in the previous layer and place them in a new branch layer
If the end node of a branch numbered in step 3 has been used before (flag identification of step 2) a loop has been formed and a breakpoint must be created at this node
Repeat steps 2-4 ' until all branches are processed.
SOLVE RADIAL NRWORK LOAD ROW ( W R E 3) UPDATE THEVEMN VOLTAGE MISMATCH AT BREAKPOINTS
1
CALCULATE BREMPOhlT CURRENTS - ENATKIN (8) ADD B E A P O M CURRENTS TO NODAL CURRENTS -
Fig.9 Compensation-based power flow method for the weakly meshed networks
IV. RESULTS FOR THE DISTRIBUTION NETWORKS
We developed the program WNETPF (Weakly meshed NETwork Power Flow) based on the proposed power flow solution algorithm. This program successfully solved several practical distribution networks with radial and meshed structures. Table 2 shows the performance results of this program alongside those of the Newton-Raphson based WSCC power flow program. We also used two other power flow programs using the generic Newton-Raphson and the Gauss-Seidel solution algorithms at this application stage. The generic Newton-Raphson solution algorithm only converged for the smallest network. The Gauss-Seidel power flow method converged in all cases while requiring in excess of 20,000 iterations.
758
IISTRIWTION IETwow( ZNFWRATION
w NODES
WLOCPS
SLOOPS
For the cases reported in Table 2, the proposed algorithm converged in less than 14 iterations in 0.21 to 2.1 CPU seconds on a mainframe computer. Each iteration of this program corresponds to one iteration in the radial network solution algorithm prior to, during, and after the calculation of breakpoint currents (variable k in Figure 3 ) . The number of outer iterations for calculating the breakpoint currents in the compensation process (variable m in Figure 9) was less than 6 in all three cases. A flat start was used in all cases with the tolerance for the real and reactive power mismatches set to 0.05 kW and 0.05 kvar.
Table 2 indicates that the proposed power flow program is significantly more efficient than the Newton-Raphson power flow method when studying radial and weakly meshed distribution networks. This conclusion is particularly crucia1,for: (a) on-line applications; (b) multiple power flow studies; (c) micro- and mini-computer applications.
The W T P F program does not require double precision variables and uses only one two dimensional array (breakpoint impedance matrix), hence, avoids taxing computer resources.
W S C C P O W E R W -METHOD MAX MISMATCH
ml"S &!$!= ml"S E, REAL RUCTNE
5 0.97 3 021 O.CQ8 0.024
5 037 S 025 0.003 0.008
N3Loops 5 5.31 4 1.47 0011
36 LOOPS 5 527 10
1411 NOMS
Table 2 Performance results for distribution networks (IBM 3090-200 mainframe computer) - The total CPU time in seconds includes the time for initial processing of the network data and the iterative solution algorithm.
V. APPLICATION TO WEAKLY MESHED TRANSMISSION NETWORKS
In a weakly meshed transmission network, the swing bus is assigned as the root node. Then the branches are numbered and breakpoints selected in exactly the same manner used for the weakly meshed distribution networks. As a result a radial transmission network is created.
The solution algorithm for a radial transmission network is identical to that of a radial distribution network except for the processing of the generator (PV) nodes. For the PV node i having a specified power injection PT and voltage magnitude I VT I , we start the iterations of the radial network solution algorithm by assuming Vi(o) = lVrl /O and = 0. Steps 1 and 2 of the radial network solution method are then performed in the same manner as before. Step 3 , however, must be modified.
At iteration k, the voltage magnitude at the generator node i, calculated using equation ( 3 ) must modified as,
Vi 2) is then used for calculating the voltages at the end nodes of the branches in the next layer. The reactive power at the generator node i is then updated using the secant method as described in the Appendix.
Inclusion of generator nodes using this approach does not noticeably deteriorate the convergence properties of the radial network power flow solution algorithm. The efficiency of the solution method for the weakly meshed networks will, however, be affected by the introduction of the generator nodes. The nonlinearity in the transmission network caused by the additiorial generator nodes is more pronounced than that of the distribution networks having constant power loads alone. This results in an increased number of iterations for calculating the breakpoint currents using equation (8 ) .
Table 3 shows the performance results of applying the proposed algorithm and the Newton-Raphson based WSCC power flow program to one radial and three weakly meshed transmission networks. These networks were synthesized from a practical 500kV transmission network with a high degree of series compensation. The table shows that for the networks with low number of breakpoints, the proposed load flow techhiques is more efficient than the Newton-Raphson method. However, as the number of breakpoints increased, the proposed method required significantly higher number of iterations while the Newton-Raphson algorithm converged with the same number of iterations.
42 BUS, WOW N3WNoDEs
NO LOOPS
1 LOOP
3 LOOPS
SLOOPS
42 BUS, 5W KV 4 w m
NO LOOPS
1 LOOP
3 LOOPS
5 LOOPS
Table 3 Performance results for transmission networks (IBM 3090-200 mainframe computer)
VI. PRACTICAL CONSTDERATIONS
Detailed representation of a distribution network requires [ l o ] :
1. The three phase representation of the network to account for the actual load unbalances.
2. The distributed load representation along the distribution lines.
The representation of load tap changers, voltage regulators, boosters, etc.
3 .
159
Requirement 1 can be directly incorporated in the proposed power flow method by replacing voltage and current scalars in equations (1) thru (lo), by (3x1) vectors of voltages and currents of the three phases. Under these circumstances, in equations (l), (31, (4) and (71, the admittances and impedances should be represented by (3x3) matrices.
Distributed loads along distribution lines can be approximated by lumped loads at system nodes for power flow calculations purposes. This, however, may require the addition of pseudo-nodes along some of the lines where part of the line load is lumped. The proposed power flow method is also capable of directly including distributed loads. This can be achieved by the modification of equations(1) and (3).
Distribution network equipment (regulators, boosters,' etc.) can be modeled in the proposed algorithm without any restriction.
This method is directly applicable to distribution planning studies, where single phase representation of the network with lumped nodal loads are considered to be adequate. In addition we have successfully used this power flow method for our optimal network reconfiguration studies.
In the case of weakly meshed transmission networks, a complete system representation requires the inclusion of tap changers and phase shifters. The proposed solution algorithm is capable of including these components directly. This requires that the KVL and KCL be written for the mathematical model of the tap changers and phase shifters.
VII. CONCLUSION
This paper presents a new, compensation-based power flow method, for the solution of weakly meshed distribution and transmission networks. This technique is simple, straightforward, computationally efficient and numerically robust. Extensive study of the performance of this compensation-based power flow scheme shows that it is significantly more efficient than the Newton-Raphson power flow technique when used for solving radial and weakly meshed distribution and transmission networks.
ACKNOWLEDGEMENT
The authors thank Messrs R.L. Smith and J. Monasterio of PGhE's Electric Distribution Engineering Department for their useful coments on the practical applications of the proposed method. The Canadian authors gratefully acknowledge the financial support from the National Sciences and Engineering Research Council of Canada.
111 S. C. Tripathy, G. Durga Prasad, 0. P. Malik and G. S. Hope, "Load Flow Solutions for 111- Conditioned Power Systems By a Newton Like Method", IEEE Trans., PAS-101, October 1982, pp. 3648-3657.
[2] S. Iwamoto, Y. Tamura, "A Load Flow Calculation Method for Ill-Conditioned Power Systems", TEEE Trans., PAS-100, April 1981, pp. 1736-1743.
[31 D. Rajicic, A. Bose, "A Modification to the Fast Decoupled Power Flow for Networks with High R/X Ratios", PICA '87 Conference, Montreal, Canada.
K. A. Birt, J. J. Graffy, J. D. McDonald, A. H. El-Abiad. "Three Phase Load Flow Program", IEEE Trans., PAS-95, JanuaryIFebruary 1976, pp. 59-65.
W. H. Kersting, D. L. Mendive, "An Application of Ladder Network Theory to the Solution of Three Phase Radial Load-Flow Problems", IEEE PAS Winter Meeting, New York, 1976, IEEE Paper No. A76 044-8.
D. I. H. Sun, S. Abe, R. R. Shoultz, M. S. Chen, P. Eichenberger, D. Farris, "Calculation of Energy Losses in a Distribution System", IEEE Trans., PAS-99, July/August 1980, pp. 1347-1356.
R. Berg Jr., E. S. Hawkins, W. W. Pleines, "Mechanised Calculation of Unbalanced Load Flow for Radial Distribution Circuits", IEEE Trans., PAS-86, April 1967, pp. 415-421.
W. F. Tinney, "Compensation Methods for Network Solutions by Triangular Factorization", Proc. of PICA Conference, Boston, Mass., May 24-26, 1971.
G. Gross, H. W. Hong, "A Two-step Compensation Method for Solving Short Circuit Problems", IEEE Trans., PAS-101, June 1982, pp. 1322-1331.
[lo] N. Vempati, R. R. Shoults, M. S. Chen, L. Schwobel, "Simplified Feeder Modeling for Load Flow Calculations", 1EF.E Paper No. 86WM102-8, PES Winter Power Meeting, New York, 1986.
APPENDIX
At the kth iteration, the reactive power injection required to maintain the voltage at the generator bus i, can be calculated using the secant method:
where IVi(k-l) l and IVi(k-2)l are the voltage magnitudes at the node i calculated in the previous two iterations (equation (3) in step 3). The actual reactive power injection is determined as:
Qic <Qimh
where Qimin and Qi- are the respective minim and maximum reactive power limits for the generator node i.
R. P. Broadwater and A. Cbandrasekaran (Tennessee Technologkd University, Cookeville, TN): The authors are to be complimented for addressing an area that has received little attention, distribution power flow analysis.
Radial Distribution System Analysis 1) A suggested improvement to the radial power flow is to sum load
powers and power losses in the reverse trace (i.e., moving from the endiog buses to the source bus) instead of summing load currents. This suggestion has been tested on a four-line section system as illustrated in Fig. D. 1. For a nominally loaded case, both methods converged in four iterations. However, for a very heavily loaded case, the method of summing the
760
currents in the reverse trace diverged, whereas the method of summing the powers converged.
A brief and heuristic explanation of this phenomenon is as follows. Initially, when the currents are summed in the reverse trace, each current will contain an error proportional to the initially guessed voltage. If the initially guessed voltages are maintained constant and a succession of power flow problems are solved in which the loads &e continually increased, the errors that are proportional to the initially guessed voltages wiU grow. For a sufficiently heavily loaded system, the initially guessed voltages fall outside the region of convergence, and the algorithm will diverge.
When the powers are summed in the reverse trace, the errors that exist when the source bus is reached involve only the power losses, and not the load powers. The power losses are always a small fraction of the load powers. Hence, using the power sum leads to good convergence for even heavily loaded systems.
2) Even though it is mentioned that the multiphase unbalanced systems are easily handled, the convergence characteristics claimed for the single- phase system may get severely impaired, since the error involved in current summation may become excessive.
3) The details of the distribution networks given in Table 1 do not include the loading levels of the systems. It would be instructive to know whether the systems are nominally loaded or lightly loaded. Further, the number of nodes may not be a direct indication of the size of the system since distributed loads can be modeled using any number of node points.
Weakly Meshed Transmission System 4) The impedance matrix of (7) of the paper appears to be the loop
impedance matrix of the system with the breakpoint currents chosen as the loop currents. At the exact solution, the breakpoii voltage vector must go to zero. There appears to be a paradox here since for the constant impedance matrix assumed the solution for the currents is either trivial or illfinite.
5 ) The handling of PV buses explained in section V does not mention whether the alternating current directions in the traces would affect convergence if the P value is higher than “downstream” lo&.
General 6) The CPU time given in the table is said to include the initial processing
time also. This may show the WSCC Power Flow Program in a bad light. Exact CPU time required for the iterations alone should be a better index. 7) In Fig. 2, the numbering of the distribution network is laid out on a
grid in a very orderly fashion. With this scheme, it appears that choosing buses at large load centers may lead to conflicts. For instance, in Layer 2
developed for a load flow problem, the discaqers believe that it is effective to develop an algorithm which utiliis special features of the problem; the radial power system is one of such special feahres. In load flow problem of a radial power system, the number of variables or computational burden can be reduced extremely by taking voltages of the ends (or tips) of the branches as independent variables, in comparison with taking all node voltages as variables. The method developed by the authors also utilizes such features. The discussers have a question that it might be fast to make a rcduced variable problem as above and solve it directly by the Newton-Fbphson method. Could the authors comment on whether they have ever tried such a reduced variable Newton-Raphson method, and if tried, how was the comparison of computation time between the two methods?
N. Vempnti, B. K. Chea, and R. R. Shoolts (University of Texas at Arlington, Arlington, TX): The authors have presented an algorithm for the load flow solution of weakly meshed distribution and transmission networks. The simplicity of the algorithm makes it a very interesting paper to read. However, the practical limitations of the algorithm prevent its general usage for networks with multiple loops. It would be inadvisable to have two programs, one to deal with strongly meshed networks and yet another for weakly meshed ones.
The results were based on tests performed on radial and weakly meshed networks using the positive sequence representation. The authors attempt to extrapolate the results to three-phase distribution and transmission networks without a report of such an analysis. Until such results are presented, the efficacy of such an algorithm is sti l l in doubt.
The authors make an erroneous observation that power flow algorithms for meshed distribution systems have not been developed. One of the algorithms referred to [6] has the ability to analyze three-phase networks, irrespective of the complexity of the meshing. However, this algorithm was designed for distribution systems and therefore has the limitation of only one swing bus and no other voltagecontrolled (P- v) bus. Subsequently this algorithm was modified to accept numerous P- V buses, thereby enabling its usage in the analysis of three-phase transmission networks [A]. We feel that the algorithm [6] based on an implicit Z-bus (i.e., bifactorcd Y-bus) formulation is superior to the one proposed by the authors.
How do the authors propose to model the open-wyelopendelta trans- formers in the distribution system analysis? Our simulations have shown that the injection currents due to the model are so large that the currents due to the loads are negligible. This affects the convergence characteristics of the algorithm directly. ~ n y insight into the modeling and simulatioh of these transformers would be welcome.
References
[A] B. K. Chen, “Transmission System Unbalance Analysis,” Ph.D. Diss., The University of Texas at Arlington, December 1986.
I M. E. Baraa and F. F. Wu (University of California, Berkeley, CA): This paper points out the reasons why a special power flow for distribution systems is needed and provides a computationally attractive method. It is a valuable contribution.
There are a few points on which we would appreciate the authors’ claritlcation.
1) How does the convergence of the method for radial networks depend on the system p e t e r s , in particular, the line resistances?
2) The amlication of the comuensation method for weaklv meshed
t
Fig. D. 1. Four-lie section test system.
suppose that a large load exists at the center of branch 4, but there is little or no load in the centers of branches 5 and 6. With the author’s proposed scheme, it appears that additional buses would be inserted in branches 5 and 6 that would not be necessary for accurate load modeling.
8) It appears that if a component is inserted or added somewhere in the network, a complete renumbering of the network is required. Is this true?
9) Even though the WSCC Power Flow Program is shown to converge more slowly for the examples chosen in comparison to the proposed method, it is a moot point whether a few extra seconds of CPU time alone would be sufficient to choose any alternative method. The Newton’s method has been so finely honed during the last two decades that negative experiences must be very few to be deemed almost nonexistent. Hence, the virtues of the proposed method must be highlighted from a different viewpoint.
K. Aolri, K. Nan, and T. Sat0 (Hiroshima University, Higashihiroshima, Japan): The authors have written an interesting paper on a load flow problem of radial power system. Although many algorithms have been
networks isma very clever idea. ~e method uses an appro&te of the linearized V-J function at the breakpoiit. We wonder if the convergence of the method is always monotonic as implied in Fig. 8 or it exhibits oscillatory behavior.
3) In calculating the breakpoiit impedance matrix, the authors have observed that the corresponding power flow solutions can be achieved in one iteration. Is this also true in cases where the shunts in the system are significant and cannot be neglected?
Dr. Dromey: The authors have investigated an aspect of load flow analysis that has received little attention in the literature to date. The wide range of R/X ratios in a distribution system can lead to difficulty with convergence and this problem is aggravated by the presence of adjacent long and short branches. In particular, the Newton-Raphson and decoupled algorithm can, in some circumstances, fail to converge for larger or illconditioned systems. The method presented in the paper produces an optimal ordering for solution and is very efficient, particularly where the number of loops is limited and the system is essentially radial in nature.
-*-
76 I
A number of questions arise in connection with the results. The paper infers that constant power loads are assumed. What is the effect on convergence for loads that have a mixture of p/u and q / u characteristics? No mention is made of the use of convergence factors. Was this studied and are there indications of optimal factors that can be considered generic for the radial solution and for breakpoint injection currents?
The authors mention that the efficiency of the algorithm deteriorates with the increase in the number of loops. In a large city with low voltage downtown networks, the number of loops required to interconnect a large number of radial sections can be in the hundreds. There is the additional problem of a significant number of very short cable sections used to balance the flows between sections. What degree of deterioration can be expected in the efficiency of solution for such a network where the coupling can be quite strong? \
There is a significant advantage in being able to solve the networks described above on a desktop microcomputer which will be restricted in the memory available for processing. Would the authors like to suggest a sensible method of partitioning such a network to achieve solutions in an acceptable time? How would the compensation method be modified to account for the partitioning?
D. Shirmohammadi, H. W. Hong, A. Semlyen, and G. X. Luo: We would like to thank the discussers for their interest in our paper and for their questions anti comments. Many of these constitute contributions to the topic of the paper. We will give our answers to each discusser separately.
' Messrs. ~roamVater and Chandrmehrun: We believe that the discussers' suggestion of adding up powers rather than currents in the backward sweep is interesting and their explanation quite plausible. We suspect however that the improved convergence is obtained with more computations. Moreover, we have not experienced any problems in dealing with heavily loaded networks and low node voltages. In fact, in the 544 node network example of the paper, due to heavy loading and lack of VAR s u p port the voltages of some of the nodes a as low as 0.75 per unit. This network was efficiently solved by the proposed algorithm. The p m s S of the attenuation of the emrs described in the section of the paper on "Convergence Criterion" is completely general and would apply to multi-phase unbalanced as well as single-phase net- works. AU thm examples included in the paper repment peak load condi- tions on the feeders. These were 4 MW, 3 M W , and 8 MW loads on the 244 node, 544 node, and 141 1 node networks, respectively. We would like to clarify the discussers' question related to equa- tions (6) and (7): the voltages there are internal Thevenin voltages while the breakpoint voltages are zero since the breakpoint ports are shortcircuited. We would also like to point out that the actual relationship between the break point voltages and currents is esta- blished through equation (8). We have not experienced any problem in the overall convergence of the proposed method in solving a variety of networks that included P,V buses. We agree, however, that there may be more efficient ways of handling P,V buses in order to minimize the impact on the convergence characteristics of the method. In contrast to the discussers assumption, the inclusion of initial pro- cessing time puts in m3re favorable light the WSCC Power Flow Program. For example, the time required for each iteration of the proposed algorithm is around 0.005 seconds for the 244 node example of the paper. At the same time, every iterator of the Newton-Raphson based WSCC Power Flow Program took 0.08 seconds for the same network (16 times more). Similar results were obtained for the 544 node and 141 1 node networks. In our distribution network model we have assumed that loads are concentrated at network nodes. As a result, the introduction of a load at the center of branch 4 means the addition of a new node at this location. This will also add a new branch to the network. If there are no loads at the end node of branch 5, this branch and branch 10 can be combined and represented with a single branch. Branch 6 must exist, because two separate branches emanate from its end node. There is no need for the renumbering of the entire network as a result of the insertion or deletion of network components. Only
branches in the layers below the inserted or deleted component must be renumbered.
(9) We do not agree with the discussers view on the merits of the pro- posed algorithm. These are well documented throughout the paper. Furtliermore, we would like to point out that many major advances in the development of power system analysis techniques have resulted from methodologies that exploit the special structure of the power system. An example is the Fast Decoupled Power Flow which provides improved efficiency of "a few CPU-seconds" in the solution of the transmission network by exploiting the low RIX ratios prevalent in these networks. Yet, the impact of the Fast Decoupled Power Flow in the field of the transmission network analysis has been very significant.
Messrs. Aoki, Nara, and Sato:
As pointed out by the discussers, special features of particular load flow problems can be exploited to procluce more efficient solution methods. In the approach of the paper, the transversal elements have been lumped witH the loads (see our answers to Messrs. Vempati, Chen. and Shoults): this has made the factorization and the subsequent alge- braic operations with the L and U matrices of a nodal approach equivalent to using tree-branch voltages and currents. These matrix cal- culations are implicit in the method and did not have to be performed explicitly. This is the clue of the computational efficiency of branch- oriented calculations in a radial network. However, at each node we have had to enforce the power (P, Q) constraints. Therefore, all bus voltages are used as variables. Because of this only little improvement of efficiency can be derived from the radial network structure in a Newton- type load flow.
Messrs. Vempati, chen, and Shoults: It is true that the solution method of the paper becomes less
efficient as the number of loops increases. However, the ovenvhelming majority of all distribution networks and also the transmission systems of many countries are weakly meshed. Therefore, a special program, if it is more efficient than a general one, is celtainly of interest Very often advances in any field of knowledge are based on the special structure of a palticular problem. One could cite innumerable examples where Special programs are developed and used for particular situations.
We have not yet attempted the application of our method to multi- phase networks. However, as we explained in our response to Messrs. Broadwater and chandrasekaran we do not foresee any deterioration in the convergence characteristics of this method when applied to multi- phase networks.
With regard to our statement about the lack of power flow algo- rithms for meshed distribution network, we note that we made two erroneous remarks. First, as the discussers rightly argue, the 2-bus solu- tion algorithm of Ref. [6] is capable of solving meshed distribution net- works. Second, we stated that Ref. [6] proposes a popular algorithm for distribution network analysis. Based on further investigation, however, we have found that this solution method is by no means popular among distribution engineers.
The discussers raise the problem of the relative computational efficiency of the method of Ref.[6] compared to the method of the paper. It is easy to show that a nodal approach (for example the one of Ref.[61) requires more computation than the branch-oriented approach of the paper. We will show that the WO differ computationally by two facts: (a) The computations in the branch-oriented method are equivalent to
the forward and backward substitutions of the nodal method but the factorization of a matrix is not needed.
(b) The forward and backward substitutions in a nodal approach are replaced in the branch-oriented method by additions and subtrac- tions and no multiplications and divisions are needed. Consider, for example, the simple radial network of Fig.D.l of the
discussion by Messrs. Broadwater and cbandrasekaran. Connect the mot node to ground via a voltage source with V=O but do not connect impedance branches to ground from the other nodes. Number the nodes and branches moving outward from the mot. We will then have branch and bus voltages and currents. We can relate these to each other by the incidence matrices AV and A,,
VbW=AvVb and l b = A r l k (a)
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It can be seen that the incidence matrices are square, lower triangular, and consist of elements f l only. Their inverses could also have been formed directly by inspection of the network graph. This reflects the fact that equations (a) represent directly the two Kirchhoffs laws. They correspond to the forward and backward sweeps used in the paper.
Let Yb (diagonal) and Y& be the admittance matrices of the net- work. We have
Y,=AI YbwAV Clearly, the two incidence matrices are the transpose of each other.
Let us now solve the nodal problem
ybw v&=Ibur
Factorization of Y& yields
Comparing (d) with 0) we obtain
so that the solution of (c) becomes
Y & = L D L ~
L-'=AI, LTT=Av. and D=Ybr
vbur =L-T D-' L-' IbyI =AV (& (AI [&))=AV (Zbr It,,) =AV V, (r)
Equation (f) shows that the branch-oriented solution reproduces the matrix operations of the nodal approach without the need of preliminary formulation and factorization of a bus admittance matrix. To achieve this, it was essential to replace al l shunt branches connected to buses by corresponding injections.
We note that reference [6] claims that the efficiency of its algo- rithm is comparable to that of the Newton-Raphson Power Flow while we have shown a substantial improvement over the Newton-Raphson Power Flow method using our algorithm.
We appreciate the information provided by the discussers on prob- lems related to transformer modeling. We have not investigated this topic.
Messrs. Bamn and Wu: (1) The convergence of the method for 4 radial network is linear and
dependent essentially on the line impedances IZ I and the apparent powers IS I of the loads. This can be seen from the following simplified convergence analysis, for a single line of impedance Z = R + j X and load S = P + j Q . For this, eqn.(l) becomes I=S*/V* or, in incremental form,
N = - K A Y ' V" (g)
The resultant change in voltage is, taking (g) into account with V=l,
AVNW=-ZN=& AV* (h)
This equation shows that the convergence rate of I AV 1 (not A I V I !) is given by I z(i I = I Z I IS I . It depends only indirectly on line resistances. Fig. 8 of the paper is used only to depict the basic mechanism of the fixed tangent solution algorithm used for calculating breakpoint current injections according to equation (8). Nevefieless. our experience with all distribution networks studied indicates a mono- tone and rapid convergence in the calculation of bRakpoint currents. After a considerable amount of experimentation we found that a better convergence in the calculation of breakpoiit c m n t s can be achieved when the breakpoint impedance matrix is calculated neglecting the shunt components (mainly capacitors). Unfor- tunately this important conclusion is not reflected in the paper and we would like to thank the discussers for providing us with this opportunity.
Dr. Dromey: The method presented in the paper can handle any load characteris-
tics, since the load current is calculated at each step as a function of the prevailing voltage. We did not use however any convergence (accelerating) factors. We feel that such factors (not necessarily uniform and real) could improve the convergence of the method and appreciate the suggestion implied in the question.
We have developed our methodology and the accompanying pro- gram mainly for primary distribution networks. Primary distribution net- works are, in almost al l cases, either radial or weakly meshed. The pro- gram is, however, capable of representing up to 5000 nodes and 300 loops which is adequate for almost all practical cases including secon- dary distribution networks in downtown metropolitan areas. Even with such large dimensidns. the memory requirement for the program is less than 500 Kbytes which makes it ideal for microcomputer applications. The largest network studied using this program consisted of 241 1 nodes and 183 loops. It took the program a total CPU time of 12 seconds to process the input data, solve the power flow and print the results for this network on an IBM 3090-200 computer, which is very reasonable con- sidering the size of the network.
Partitioning of distribution networks, or even of transmission net- works, involves network equivalencing. Reference [A] is pertinent to this topic.
[AI F.F. Wu, A. Monticelli, "A Critical Review on External Network Modelling for On-Line Security Analysis", Electrical Power and Energy Systems, Vol. 5, October 1983, pp. 222-235.
Apéndice G
“A DIRECT APPROACH FOR DISTRIBUTION SYSTEM LOAD FLOW
SOLUTION”, TENG. 2003, [19]
114
882 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 18, NO. 3, JULY 2003
A Direct Approach for Distribution System LoadFlow Solutions
Jen-Hao Teng, Member, IEEE
Abstract—A direct approach for unbalanced three-phasedistribution load flow solutions is proposed in this paper. Thespecial topological characteristics of distribution networks havebeen fully utilized to make the direct solution possible. Twodeveloped matrices—the bus-injection to branch-current matrixand the branch-current to bus-voltage matrix— and a simplematrix multiplication are used to obtain load flow solutions. Dueto the distinctive solution techniques of the proposed method, thetime-consuming LU decomposition and forward/backward sub-stitution of the Jacobian matrix or admittance matrix requiredin the traditional load flow methods are no longer necessary.Therefore, the proposed method is robust and time-efficient. Testresults demonstrate the validity of the proposed method. Theproposed method shows great potential to be used in distributionautomation applications.
Index Terms—Distribution load flow, distribution automationsystem, radial network, weakly meshed network.
I. INTRODUCTION
M ANY programs of real-time applications in the area ofdistribution automation (DA), such as network optimiza-
tion, Var. planning, switching, state estimation, and so forth, re-quire a robust and efficient load flow method [1]–[3]. Such aload flow method must be able to model the special features ofdistribution systems in sufficient detail. The well-known char-acteristics of an electric distribution system are
radial or weakly meshed structure;multiphase and unbalanced operation;unbalanced distributed load;extremely large number of branches and nodes;wide-ranging resistance and reactance values.
Those features cause the traditional load flow methodsused in transmission systems, such as the Gauss-Seidel andNewton-Raphson techniques, to fail to meet the requirementsin both performance and robustness aspects in the distributionsystem applications. In particular, the assumptions necessaryfor the simplifications used in the standard fast-decoupledNewton-Raphson method [4] often are not valid in distributionsystems. Therefore, a novel load flow algorithm for distributionsystems is desired. To qualify for a good distribution load flowalgorithm, all of the characteristics mentioned before need tobe considered.
Several load flow algorithms specially designed for distri-bution systems have been proposed in the literature [5]–[13].
Manuscript received October 17, 2002. This work was sponsored by NationalScience Council under research Grant NSC 88-2213-E-214-041.
The author is with the Department of Electrical Engineering, I-Shou Univer-sity, Kaohsiung 840, Taiwan, R.O.C.
Digital Object Identifier 10.1109/TPWRD.2003.813818
Some of these methods were developed based on the generalmeshed topology like transmission systems [5]–[9]. From thosemethods, the Gauss implicit-matrix method [7] is one of themost commonly used methods; however, this method does notexplicitly exploit the radial and weakly meshed network struc-ture of distribution systems and, therefore, requires the solutionof a set of equations whose size is proportional to the numberof buses. Recent research proposed some new ideas on howto deal with the special topological characteristics of distribu-tion systems [10]–[13], but these ideas require new data formator some data manipulations. In [10], the authors proposed acompensation-based technique to solve distribution load flowproblems. Branch power flows rather than branch currents werelater used in the improved version and presented in [11]. Sincethe forward/backward sweep technique was adopted in the so-lution scheme of the compensation-based algorithm, new dataformat and search procedure are necessary. Extension of themethod, which emphasized on modeling unbalanced loads anddispersed generators, was proposed in [12]. In [13], the feeder-lateral based model was adopted, which required the “layer-lat-eral” based data format. One of the main disadvantages of thecompensation-based methods is that new databases have to bebuilt and maintained. In addition, no direct mathematical rela-tionship between the system status and control variables can befound, which makes the applications of the compensation-basedalgorithm difficult.
The algorithm proposed in this paper is a “novel but classic”technique. The only input data of this algorithm is the conven-tional bus-branch oriented data used by most utilities. The goalof this paper is to develop a formulation, which takes advan-tages of the topological characteristics of distribution systems,and solve the distribution load flow directly. It means that thetime-consuming LU decomposition and forward/backward sub-stitution of the Jacobian matrix or the admittance matrix, re-quired in the traditional Newton Raphson and Gauss implicit
matrix algorithms, are not necessary in the new develop-ment. Two developed matrices, the bus-injection to branch-cur-rent matrix and the branch-current to bus-voltage matrix, and asimple matrix multiplication are utilized to obtain load flow so-lutions. The treatments for weakly meshed distribution systemsare also included in this paper. The proposed method is robustand very efficient compared to the conventional methods. Testresults demonstrate the feasibility and validity of the proposedmethod.
II. UNBALANCED THREE-PHASE LINE MODEL
Fig. 1 shows a three-phase line section between busand .The line parameters can be obtained by the method developed by
0885-8977/03$17.00 © 2003 IEEE
TENG: A DIRECT APPROACH FOR DISTRIBUTION SYSTEM LOAD FLOW SOLUTIONS 883
Fig. 1. Three-phase line section model.
Carson and Lewis [2]. A 4 4 matrix, which takes into accountthe self and mutual coupling effects of the unbalanced three-phase line section, can be expressed as
(1)
After Kron’s reduction is applied, the effects of the neutral orground wire are still included in this model as shown in (2)
(2)
The relationship between bus voltages and branch currents inFig. 1 can be expressed as
(3)
For any phases failed to present, the corresponding row andcolumn in this matrix will contain null-entries.
III. A LGORITHM DEVELOPMENT
The proposed method is developed based on two derived ma-trices, the bus-injection to branch-current matrix and the branch-current to bus-voltage matrix, and equivalent current injections.In this section, the development procedure will be described indetail.
For distribution networks, the equivalent-current-injection-based model is more practical [5]–[13]. For bus, the complexload is expressed by
(4)
And the corresponding equivalent current injection at the-thiteration of solution is
(5)
where and are the bus voltage and equivalent currentinjection of bus at the -th iteration, respectively. and arethe real and imaginary parts of the equivalent current injectionof bus at the -th iteration, respectively.
Fig. 2. Simple distribution system.
A. Relationship Matrix Developments
A simple distribution system shown in Fig. 2 is used as an ex-ample. The power injections can be converted to the equivalentcurrent injections by (5), and the relationship between the buscurrent injections and branch currents can be obtained by ap-plying Kirchhoff’s Current Law (KCL) to the distribution net-work. The branch currents can then be formulated as functionsof equivalent current injections. For example, the branch cur-rents and , can be expressed by equivalent currentinjections as
(6)
Therefore, the relationship between the bus current injectionsand branch currents can be expressed as
(7a)
Equation (7a) can be expressed in general form as
(7b)
where BIBC is the bus-injection to branch-current (BIBC) ma-trix.
The constant BIBC matrix is an upper triangular matrix andcontains values of 0 and only.
The relationship between branch currents and bus voltages asshown in Fig. 2 can be obtained by (3). For example, the voltagesof bus 2, 3, and 4 are
(8a)
(8b)
(8c)
where is the voltage of bus, and is the line impedancebetween bus and bus .
Substituting (8a) and (8b) into (8c), (8c) can be rewritten as
(9)
884 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 18, NO. 3, JULY 2003
From (9), it can be seen that the bus voltage can be expressedas a function of branch currents, line parameters, and the sub-station voltage. Similar procedures can be performed on otherbuses; therefore, the relationship between branch currents andbus voltages can be expressed as
(10a)
Equation (10a) can be rewritten in general form as
(10b)
where BCBV is the branch-current to bus-voltage (BCBV) ma-trix.
B. Building Formulation Development
Observing (7), a building algorithm for BIBC matrix can bedeveloped as follows:
Step 1) For a distribution system with-branch section and-bus, the dimension of the BIBC matrix is.
Step 2) If a line section is located between busandbus , copy the column of the-th bus of the BIBCmatrix to the column of the-th bus and fill a tothe position of the -th row and the -th bus column.
Step 3) Repeat procedure (2) until all line sections are in-cluded in the BIBC matrix. From (10), a buildingalgorithm for BCBV matrix can be developed asfollows.
Step 4) For a distribution system with-branch section and-bus, the dimension of the BCBV matrix is
.Step 5) If a line section is located between busand
bus , copy the row of the -th bus of the BCBVmatrix to the row of the -th bus and fill the lineimpedance to the position of the -th bus rowand the -th column.
Step 6) Repeat procedure (5) until all line sections are in-cluded in the BCBV matrix.
The algorithm can easily be expanded to a multiphase linesection or bus. For example, if the line section between busand bus is a three-phase line section, the corresponding branchcurrent will be a 3 1 vector and the in the BIBC ma-trix will be a 3 3 identity matrix. Similarly, if the line sectionbetween bus and bus is a three-phase line section, the inthe BCBV matrix is a 3 impedance matrix as shown in (2).
It can also be seen that the building algorithms of the BIBCand BCBV matrices are similar. In fact, these two matrices werebuilt in the same subroutine of our test program. Therefore,the computation resources needed can be saved. In addition,the building algorithms are developed based on the traditionalbus-branch oriented database; thus, the data preparation time
can be reduced and the proposed method can be easily integratedinto the existent DA.
C. Solution Technique Developments
The BIBC and BCBV matrices are developed based on thetopological structure of distribution systems. The BIBC ma-trix represents the relationship between bus current injectionsand branch currents. The corresponding variations at branchcurrents, generated by the variations at bus current injections,can be calculated directly by the BIBC matrix. The BCBV ma-trix represents the relationship between branch currents and busvoltages. The corresponding variations at bus voltages, gener-ated by the variations at branch currents, can be calculated di-rectly by the BCBV matrix. Combining (7b) and (10b), the re-lationship between bus current injections and bus voltages canbe expressed as
(11)
And the solution for distribution load flow can be obtained bysolving (12) iteratively
(12a)
(12b)
(12c)
According to the research, the arithmetic operation numberof LU factorization is approximately proportional to . For alarge value of , the LU factorization will occupy a large por-tion of the computational time. Therefore, if the LU factoriza-tion can be avoided, the load flow method can save tremendouscomputational resource. From the solution techniques describedbefore, the LU decomposition and forward/backward substitu-tion of the Jacobian matrix or the admittance matrix are nolonger necessary for the proposed method. Only theDLF ma-trix is necessary in solving load flow problem. Therefore, theproposed method can save considerable computation resourcesand this feature makes the proposed method suitable for onlineoperation.
IV. TREATMENTS FORWEAKLY MESHEDNETWORKS
Some distribution feeders serving high-density load areascontain loops created by closing normally open tie-switches.The proposed method introduced before can be extended for“weakly-meshed” distribution feeders.
A. Modification for BIBC Matrix
Existence of loops in the system does not affect the bus cur-rent injections, but new branches will need to be added to thesystem. Fig. 3 shows a simple case with one loop. Taking thenew branch current into account, the current injections of bus 5and bus 6 will be
(13)
TENG: A DIRECT APPROACH FOR DISTRIBUTION SYSTEM LOAD FLOW SOLUTIONS 885
Fig. 3. Simple distribution system with one loop.
The BIBC matrix will be
(14a)
Equation (14a) can be rewritten as
(14b)
And the modified BIBC matrix can be obtained as
(15a)
The general form for the modified BIBC matrix is
(15b)
The building algorithm of Step 2) for BIBC matrix can be mod-ified as follows:
Step 2a)—If a new branch makes the system becomemeshed (the new branch is between busand ), copy the ele-ments of the-th bus column to the-th column and minus theelements of the-th bus column. Finally, fill a value to theposition of the -th row and the -th column.
B. Modification for BCBV Matrix
Considering the loop shown in Fig. 3, KVL for this loop canbe written as
(16)
Combining (16) and (10a), the new BCBV matrix is
(17a)
The general form for the modified BCBV matrix is
(17b)
The building algorithm of Step 5) for the BCBV matrix can bemodified as follows:
Step 5a)—If a new branch makes the system becomemeshed, adds a new row to the original BCBV matrix by KVL.The general form of KVL for a loop can be expressed as
(18)
where is the number of branches in this loop, andis theline impedance corresponding to the branch current.
C. Modification for Solution Techniques
Substituting (15) and (17) into (11), (11) can be rewritten as
(19)
Applying Kron’s Reduction to (19), the modified algorithm forweakly meshed networks can be expressed as
(20)
Note that except for some modifications needed to be done forthe BIBC, BCBV, and DLF matrices, the proposed solutiontechniques require no modification; therefore, the proposedmethod can obtain the load flow solution for weakly mesheddistribution systems efficiently.
V. TEST RESULTS
The proposed three-phase load flow algorithm was im-plemented using Borland C++ language and tested on aWindows-98-based Pentium-II PC. Two methods are used fortests and the convergence tolerance is set at 0.001 p.u.
Method 1: The Gauss implicit -matrix method [7].Method 2: The proposed algorithm.
886 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 18, NO. 3, JULY 2003
Fig. 4. Eight-bus distribution system.
TABLE IFINAL CONVERGEDVOLTAGE SOLUTIONS
A. Accuracy Comparison
For any new method, it is important to make sure that the finalsolution of the new method is the same as the existent method.An eight-bus system (equivalent 13-node system), including thethree-phase, double-phase, and single-phase line sections andbuses as shown in Fig. 4 is used for comparisons. The finalvoltage solutions of method 1 and method 2 are shown in Table I.From Table I, the final converged voltage solutions of method2 are very close to the solution of method 1. It means that theaccuracy of the proposed method is almost the same as the com-monly used Gauss implicit -matrix method.
B. Performance Test
A main feeder trunk with 3 90-phase buses, which was ac-quired from the Taiwan Power Company (TPC), is used for thistest. The single and double-phase laterals have been lumped toform the unbalanced loads for testing purposes. This trunk isthen chopped into various sizes for tests as shown in Table II.The substation is modeled as the slack bus.
1) Radial Network Test:Table III lists the number of itera-tions and the normalized execution time for both methods. It canbe seen that method 2 is more efficient, especially when the net-work size increases, since the time-consuming processes such asLU factorization and forward/backward substitution of-ad-mittance matrix are not necessary for method 2. For a 270-nodesystem, method 2 is almost 24 times faster than method 1.
TABLE IITEST FEEDER
TABLE IIINUMBER OF ITERATION AND NORMALIZED EXECUTION TIME
TABLE IVTEST RESULTS FOR THEWEAKLY MESHEDFEEDERS
2) Weakly-Meshed Network Test:Some branches are con-nected to the test feeder to make the system meshed. Table IVshows the number of iterations and normalized execution timefor the weakly meshed network. Table IV shows that the numberof iterations of method 2 is stable. The normalized executiontime increases, since the meshed network increase the nonzeroterms of the BIBC and BCBV matrices, and extra procedureneeds to be done.
C. Robustness Test
One of the major reasons, which make the load flow programdiverge, is the ill-condition problem of the Jacobian matrix or
admittance matrix. It usually occurs when the system con-tains some very short lines or very long lines. In order to provethat the proposed method can be utilized in severe conditions,IEEE 37-bus test feeder is used [14]. The test feeder is adjustedby changing the length of eight line sections. Four of them aremultiplied by ten, and the other four are divided by ten. The testresult shows the number of iterations for this case is 4 and theexecution time is 0.0181 s. It means that the proposed methodis robust and very suitable for online use.
VI. DISCUSSION ANDCONCLUSION
In this paper, a direct approach for distribution load flow solu-tion was proposed. Two matrices, which are developed from thetopological characteristics of distribution systems, are used to
TENG: A DIRECT APPROACH FOR DISTRIBUTION SYSTEM LOAD FLOW SOLUTIONS 887
solve load flow problem. The BIBC matrix represents the rela-tionship between bus current injections and branch currents, andthe BCBV matrix represents the relationship between branchcurrents and bus voltages. These two matrices are combinedto form a direct approach for solving load flow problems. Thetime-consuming procedures, such as the LU factorization andforward/backward substitution of the Jacobian matrix orad-mittance matrix, are not necessary in the proposed method. Theill-conditioned problem that usually occurs during the LU fac-torization of the Jacobian matrix or admittance matrix willnot occur in the proposed solution techniques. Therefore, theproposed method is both robust and efficient. Test results showthat the proposed method is suitable for large-scale distributionsystems. Other issues involved in the distribution system op-eration, such as multiphase operation with unbalanced and dis-tributed loads, voltage regulators, and capacitors with automatictap controls, will be discussed in future work.
REFERENCES
[1] IEEE Tutorial Course on Distribution Automation.[2] IEEE Tutorial Course on Power Distribution Planning.[3] W. M. Lin and M. S. Chen, “An overall distribution automation struc-
ture,” Elect. Power Syst. Res., vol. 10, pp. 7–19, 1986.[4] B. Stott and O. Alsac, “Fast decoupled load flow,”IEEE Trans. Power
Apparat. Syst., vol. 93, pp. 859–869, May/June 1974.[5] J. H. Teng and W. M. Lin, “Current-based power flow solutions for distri-
bution systems,” inProc. IEEE Int. Conf. Power Syst. Technol., Beijing,China, 1994, pp. 414–418.
[6] T. S. Chen, M. S. Chen, T. Inoue, and E. A. Chebli, “Three-phase cogen-erator and transformer models for distribution system analysis,”IEEETrans. Power Delivery, vol. 6, pp. 1671–1681.2, Oct. 1991.
[7] T.-H. Chen, M.-S. Chen, K.-J. Hwang, P. Kotas, and E. A. Chebli, “Dis-tribution system power flow analysis—A rigid approach,”IEEE Trans.Power Delivery, vol. 6, pp. 1146–1152, July 1991.
[8] T. H. Chen and J. D. Chang, “Open wye-open delta and open delta-opendelta transformer models for rigorous distribution system analysis,” inProc. Inst. Elect. Eng., vol. 139, 1992, pp. 227–234.
[9] K. A. Birt, J. J. Graffy, J. D. McDonald, and A. H. El-Abiad, “Threephase load flow program,”IEEE Trans. Power Apparat. Syst., vol.PAS-95, pp. 59–65, Jan./Feb. 1976.
[10] D. Shirmohammadi, H. W. Hong, A. Semlyen, and G. X. Luo, “A com-pensation-based power flow method for weakly meshed distribution andtransmission networks,”IEEE Trans. Power Syst., vol. 3, pp. 753–762,May 1988.
[11] G. X. Luo and A. Semlyen, “Efficient load flow for large weakly meshednetworks,”IEEE Trans. Power Syst., vol. 5, pp. 1309–1316, Nov. 1990.
[12] C. S. Cheng and D. Shirmohammadi, “A three-phase power flow methodfor real-time distribution system analysis,”IEEE Trans. Power Syst., vol.10, pp. 671–679, May 1995.
[13] R. D. Zimmerman and H. D. Chiang, “Fast decoupled power flow forunbalanced radial distribution systems,”IEEE Trans. Power Syst., vol.10, pp. 2045–2052, Nov. 1995.
[14] W. M. Kersting and L. Willis, “Radial Distribution Test Systems, IEEETrans. Power Syst.,”, vol. 6, IEEE Distribution Planning Working GroupRep., Aug. 1991.
Jen-Hao Teng(M’99) was born in 1969 in Tainan, Taiwan, R.O.C. He receivedthe B.S., M.S., and Ph.D. degrees in electrical engineering from the NationalSun Yat-Sen University, Taiwan, R.O.C., in 1991, 1993, and 1996, respectively.
Currently, he is with I-Shou University, Taiwan, R.O.C., where he has beensince 1998. His current research interests include energy management system,distribution automation system, and power system quality.
Apéndice H
“THE DISTRIBUTION TRX-POWER FLOW METHOD”, DE OLIVEIRA.
2010, [20]
121
1217
The Distribution TRX-Power Flow Method
Paulo M. De Oliveira-De Jesus, Member, IEEE
Abstract
This paper proposes a new methodology to assess the power flow solution of distribution networksby merging the classic Backward/Forward Sweep process in a unique state-of-the-system calculationformula based on a global matrix denominated TRX. The TRX matrix is suitable to be stored inmemory in the context of Distribution Management Systems (DMS) environment and data exchangeframework supported in standard common information model. The TRX matrix reflects the mostcredible state of the network topology. The proposal is suitable to be applied in balanced and unbal-anced aerial distribution systems, radially or weakly meshed operated with distributed generation.This contribution is meaningful under real-time distribution system assessment and planning pur-poses, since the state of the system is reached using present or historical system measurements aswell as present or future network topology arrangement. Proposed methodology has been applied ina group of test systems showing better performances than other large-scale implementations like thestandard Newton Raphson with sparse matrix handling
Index Terms
Backward/forward sweep, load flow, power flow, distribution system analysis
I. Introduction
DISTRIBUTION system operation and planning require robust and reliable power flow analysistechniques. Modern distribution management systems (DMS) need this support in order to
perform applications as distributed generation dispatch, service restoration, feeder reconfiguration,phase balancing, volt/var control, optimal location of capacitors, etc. For a long time, traditionaland efficient methods as Newton Raphson (NR) [1] and fast decoupled power flow [2] have beensuccessfully applied in large power systems. Concerning distribution systems, early attempts todevelop power flow studies can be found in [3] [12].
Special features of distribution systems as radial or weakly meshed structure, low ratio betweenreactance and resistance values, unbalancing and distributed generation have undesired results inboth performance and robustness of traditional power flow techniques. Weakly meshed distributionnetworks may have thousands of busbars being considered ill-conditioned by causing numericalproblems for the conventional power flow algorithms [4], [5]. The degree of ill-conditioning isevaluated by the number of iterations or the value maximum eigenvalues of the inverse Jacobianmatrix. These deficiencies lead to the development of alternative power flow techniques speciallydesigned for distribution systems.
Several methods have been reported in Literature to solve distribution systems. They can bedivided into three categories: Jacobian-based methods, direct methods, and backward/forward(BW/FW) sweep methods. The first type of methods is based on modification of existing methodssuch as Newton-Raphson [6], [7], [8], [9] where no node ordering is required.
Direct methods [10], [11] require the construction of an impedance matrix. These methodsusually present a heavy computational burden requiring a specific numbering scheme for nodesand branches.
Finally, BW/FW sweep algorithms have been developed based on the assumption that nodes andnetwork branches are properly ordered. In 1967, Berg presented a paper which can be considered
P.M. De Oliveira is with the Department of Electrical Engineering, Simon Bolivar University, Caracas, Venezuela, Ap.89000.Phone: +58 212 906–3913, e-mail: [email protected]
1218 THE DISTRIBUTION TRX-POWER FLOW METHOD
as the source for the all variants of BW/FW sweep methods [12]. Later, a similar approach waspresented in [13] based on ladder network theory. In general, these algorithms can use the Kirch-hoff laws [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [26], [25], [27] or thebi-quadratic equation [28], [29], [30], [31] in the iterative process. BW/FW sweep methods typi-cally present a slow convergence rate but computationally efficient at each iteration. Using thesemethods, power flow solution for a distribution network can be obtained without solving any set ofsimultaneous equations.
The effectiveness of the BW/FW sweep algorithm has already been proven by comparing it tothe traditional NR methods [32] [34]. Recently, some comparative and convergence studies havebeen presented by [33] [35].
All contributions mentioned above have been valuable tools to perform power distribution anal-ysis. However, under modern DMS environment new considerations must be raised up about theappropriateness of a given distribution power flow methodologies from system implementation pointof view:
1. The system data model: topology structure and nodal measurements can be stored in mem-ory of DMS environment reflecting the most credible state of the network at given time.Standardization processes based on Extended Markup Language (XML) as Common Infor-mation Model [36] and Open Data Model (ODM) [37], [46] have been carried out. In thissense, power flow analysis in real time or off-line studies can be applied using the mostrealistic data (present and historical) stored in memory or media with fast transfer rates.Then, performance and robustness assessment of power flow methods implemented underDMS must be made considering the impact of I/O data access.
2. Under DMS environment, power flow applications are object oriented [39] suitable to beapplied with distributed processing [38].
In this context, this paper proposes a new methodology to assess the power flow solution ofdistribution networks by merging the standard BW/FW sweep steps into one unique state-of-the-system calculation formula based upon a TRX matrix suitable to be stored in memory in thecontext of a DMS environment. The TRX matrix is formed by real numbers with three fundamentalelements: the triangular matrix T that relates nodal currents with branch currents, resistance R
and reactance X vectors that characterize the branches, lines or transformers, of the distributionnetwork. The T matrix and R and X vectors reflect the present condition of the network beingconstructed directly from the data exchange scheme adopted.
The proposal is simple and suitable to be applied in distribution systems, radially or weaklymeshed operated with distributed generation. The power flow is iteratively solved using only onestate-of-the system calculation step. This contribution is meaningful for actual assessment of thedistribution system and expansion system planning process, since the state of the system is reachedthrough multiplication and summation operations where no matrix inverse is required
Proposed methodology has been applied in a simple 4-node network for illustration purposes andcompared with a robust Newton Raphson solver [44] and a standard Backward/Forward Sweepalgorithm [24] in three test systems: a 12-bus [29], 33-bus [40] and 69-bus [41] networks. Finally,a convergence study is performed using a uniformly loaded distribution network from 1000 to 3000nodes.
This paper is organized as follows: Section II describes the standard BW/FW sweep method underan appropriate notation for DMS implementation. Section III presents the proposed methodologyfor balanced and unbalanced networks. Case studies are discussed in Section IV. Conclusions aredrawn in Section V. Nomenclature, list of symbols are provided in Appendix A. Proposed algorithm
THE DISTRIBUTION TRX-POWER FLOW METHOD 1219
Fig. 1. Branch and node numbering of a radial distribution network
coded in Matlab is presented in Appendix B.
II. Standard Backward/Forward Sweep Power Flow Method in Complex Form
This method was widely described by several authors [14], [24]. The input data of this algorithmis given by node-branch oriented data used by most utilities. Basic data required is: nodal powersand sending and receiving nodes of a given line impedance.
In the following, the standard BW/FW sweep power flow method is written in matricial notationusing complex variables. Branch impedances are stated as a vector Z corresponding to a distributionline model containing a series impedance or transformer. Shunt impedances are not considered inthis first approach. Fig. 1 shows a radial distribution network with n + 1 nodes, and n branchesand a single voltage source at the root node 0. Branches are organized according to an appropriatenumbering scheme (list), which details are provided in [14].
Z =[
Z01 ... Z ij ... Zmn
]
(1)
where,Z ij = Rij + jXij i, j = 1, ..., n i 6= j (2)
Bus data is given by
S =
S1
...Si
...Sn
=
P1 + jQ1
...Pi + jQi
...Pn + jQn
(3)
where net nodal active and reactive powers are given by generated and demanded powers:
Pi = PGi −PDi (4)
Qi = QGi −QDi (5)
The numbering of branches in one layer begins only after all the branches in the previous layer
have been numbered. Considering that initial voltages are known: voltage at substation Vk=0
andan initial voltage vector given by:
V0 =[
V0
1 ... V0
i ... V0
n
]
(6)
The state of the system is reached solving two steps iteratively.
1220 THE DISTRIBUTION TRX-POWER FLOW METHOD
A. Step 1 - Backward Sweep
For each iteration k, branch currents are aggregated from loads to origin:
Jk = T · Ik (7)
The relationship between nodal currents Ik and branch currents Jk is set through an upper
triangular matrix T accomplishing the Kirchhoff Current Laws (KCL) [18]. Each element Ik
i of Ik
associated to node i is calculated as function of injected powers Si and its voltage profile Vk
i asshown below:
Ik
i = − S∗
i
Vk∗
i
i = 1, ..., n (8)
B. Step 2 - Forward Sweep
Nodal voltage vector V is updated from the origin to loads according the Kirchhoff Voltage Laws(KVL), using previously calculated branch currents vector J 7 branch impedances vector Z:
Vk+1 = V0 −TT ·DZ · Jk (9)
where V0 is a n-elements vector with all entries set at voltage at origin (swing node) V 0 andbranch impedances DZ is the diagonal matrix of vector Z:
C. Convergence
Updated voltages are compared with previous voltages in order to perform convergence check in.
ε ≤ |V k+1
i − Vk
i | i = 1, ..., n (10)
III. The Proposed Methodology
This section describes the proposed power flow approach. Two versions are developed in detail:a balanced and a three phase unbalanced power flow analysis. In both cases, the general algorithmflowchart is depicted in Fig. 2.
A. Balanced TRX Power Flow Algorithm
Similar to standard model presented in Section II, the proposed balanced power flow analysisrequires the apparent power injections vector S and distribution line or transformer model based onseries impedances provided from the data exchange model Z = R+ jX. Given an initial condition(k = 0), the nodal voltage n-elements vector V0 are decomposed into real and imaginary parts:
V0
x =[
V 0x1 ... V 0
xi ... V 0xn
]
(11)
V0
y =[
V 0y1 ... V 0
yi ... V 0yn
]
(12)
Reference voltages V0x and V0
y can be acquired from measurements of DMS support. Alternatively,
it can be used V 0xi = 1 and V 0
yi = 0 for i = 1, ...,n.The state of the system is updated at each iteration k by the following expression:
Vk+1x + jVk+1
y = V0x + jV0
y − TT · (DR + jDX) · (Jk
x + jJky) (13)
THE DISTRIBUTION TRX-POWER FLOW METHOD 1221
Fig. 2. TRX Load Method Flowchart
where,
DR = ℜe{DZ} (14)
DX = Im{DZ} (15)
Jx = T · Ix (16)
Jy = T · Iy (17)
This expression correspond to Kirchhoff Voltage Laws (KVL) through all nodes and, where Jx +jJy term corresponds to all nodal Kirchhoff Current Laws (KCL). After some algebra, real andimaginary parts of 13 can be rewritten as:
V0x = V
0x −T
T ·DR ·T · Ix +TT ·DX ·T · Iy (18)
V0y = V
0y −T
T ·DX ·T · Ix −TT ·DR ·T · Iy (19)
Eqs. 18 and 19 are settled in matricial form as:[
Vk+1x
Vk+1y
]
=
[
Vx0
Vy0
]
−
[
TTDRT −T
TDXT
TTDXT T
TDRT
] [
Ikx
Iky
]
(20)
where the proposed TRX matrix is given by:
TRX =
[
TTDRT −TTDXT
TTDXT TTDRT
]
(21)
The dimension of the TRX matrix for balanced systems is 2nx2n. This matrix is not sparse butsuitable to be allocated using storage devices with high data transfer rate. However, it must bepointed out that only TTDRT and TTDXT matrices should be allocated due to symmetry in theTRX matrix. For typical, 1000 node distribution circuits, the TRX matrix will require al least6MB using double-precision numbers
A general expression that relates each nodal voltage with the origin can be written as:
1222 THE DISTRIBUTION TRX-POWER FLOW METHOD
V = V0 − TRX · I (22)
Note that, at each iteration, voltage drops ∆V = V0 −V are directly obtained multiplying theTRX matrix and the nodal current vector I.
The nodal currents Ikxi and Ik
yi for i = 1, ...,n are given as function of present nodal power injections(demands, capacitors or distributed generators) and operational voltages:
Ikxi = −ℜe{Ik
i } =−PiV
kxi −QiV
kyi
(V kxi)
2 + (V kyi)
2(23)
Ikyi = −Im{Ik
i } =QiV
kxi −PiV
kyi
(V kxi)
2 + (V kyi)
2(24)
For simplicity, this contribution only refers constant power P Q load models. Formulae forconstant current and constant impedance models will be provided in the future. In addition,this model is suitable to be extended to weakly meshed networks using the compensation methoddescribed in [14].
Convergence check: recent updated voltages are compared with previous voltages in all nodes inorder to perform convergence check in.
ε ≤ |V k+1
i − Vk
i | i = 1, ..., n (25)
B. Unbalanced TRX Power Flow Algorithm
Under the unbalanced approach, nodal power injections vector S are given per phase.
S =
Sp−1
...Sp−i
...Sp−n
=
Pp−1 + jQp−1
...Pp−i + jQp−i
...Pp−n + jQp−n
p = a, b, c (26)
Branch impedances are given as a rectangular 3nx3 phase impedance matrix Zabc
Zabc =[
Z01 ... Zij ... Zmn
]
(27)
where Zij is the 3-phase matrix impedance corresponding to ij line section [42]:
Zij =
Zaa−ij Zab−ij Zac−ij
Zba−ij Zbb−ij Zbc−ij
Zca−ij Zcb−ij Zcc−ij
(28)
Given an initial condition (k = 0), nodal voltage 3xn-elements vector V0abc are decomposed into
real and imaginary parts:
V0abc−x =
[
V 0a−x1 ... V 0
a−xn V 0b−xn V 0
c−xn
]
(29)
V0abc−y =
[
V 0a−y1 ... V 0
a−yn V 0b−yn V 0
c−yn
]
(30)
THE DISTRIBUTION TRX-POWER FLOW METHOD 1223
Initial values of three phase nodal apparent power injections vector S and three-phase voltageprofiles are online scanned or estimated using DMS.
The state of the system is an n-elements voltage vector that is updated at each iteration k bythe following formula:
[
Vk+1
abc−x
Vk+1
abc−y
]
=
[
Vabc−x0
Vabc−y0
]
−[
A −B
B A
] [
Ikabc−x
Ikabc−y
]
(31)
where;
A = TabcTDabc
R Tabc (32)
B = TabcTDabc
X Tabc (33)
Tabc is the upper triangular matrix, with dimension 3nx3n, according current Kirchhoff lawsconsidering three phase connection. A general expression can be written as:
Vabc = Vabc−0 − TRXabc · Iabc (34)
Note that, at each iteration, voltage drops per phase ∆Vabc = Vabc−0−Vabc are directly obtainedmultiplying the TRXabc matrix and the nodal current vector Iabc.
The dimension of the TRXabc matrix for unbalanced systems is 6nx6n. For instance, the cor-responding matrix for a 1000 node-distribution circuits will require al least 56MB using double-precision numbers.
Three phase currents Ikp−xi and Ik
p−yi for i = 1, ...,n are given as function of present three phasepower injections (demands, capacitors or distributed generators) and three phase operational volt-ages:
Ikp−xi = −ℜe{Ik
p−i} =−Pp−iV
kp−xi −Qp−iV
kp−yi
(V kp−xi)
2 + (V kp−yi)
2(35)
Ikp−yi = −Im{Ik
p−i} =Qp−iV
kp−xi −Pp−iV
kp−yi
(V kp−xi)
2 + (V kp−yi)
2(36)
Convergence check: recent updated voltages are compared with previous voltages in all nodes inorder to perform convergence check in.
ε ≤ |V k+1
p−i − Vk
p−i| i = 1, ..., n p = a, b, c (37)
IV. Testing
The proposed methodology was applied to a list of test systems:
• A 4-node illustrative example• Comparison Analysis: a 12, 33 and 69 node network• Comparison Analysis: a uniformly distributed test system from 1000 to 3000 nodes
1224 THE DISTRIBUTION TRX-POWER FLOW METHOD
A. Illustrative Example: Simply 4-node Network
To illustrate the proposed balanced and unbalanced power flow methodology, it is used a 4-nodeexample shown in Fig. 3. An overhead 12.47kV three-phase distribution line is constructed as shownin Fig. 4. Length of all sections is 1 mile. The phase conductors are 336,400 26/7 ACSR (Linnet),and the neutral conductor is 4/0 6/1 ACSR. 336,400 26/7 ACSR: GMR = 0.0244 ft Resistance 0.306/mile 4/0 6/1 ACSR: GMR = 0.00814 ft. Resistance = 0.5920 /mile. Load demand at nodes 2and 3 are 2MW with cosϕ = 1.0. The phase impedance matrix of the line is computed according [42]:
Zabc =
[
0.4576 + j1.0780 0.1560 + j0.5017 0.1535 + j0.38490.1560 + j0.5017 0.4666 + j1.0482 0.1580 + j0.42360.1535 + j0.3849 0.1580 + j0.4236 0.4615 + j1.0651
]
A.1 Balanced TRX Power Flow
Using the following bases SB =10MW and VB=12.47kV, data and results are given in per unit.Loads are 0.2 in nodes 2 and 3. Reference voltage at node 0 is V 0 = 1+ j0 and initial voltages are
set V 0x =
[
1 1 1]
and V 0y =
[
0 0 0]
Branches are represented by:
Z = R + jX =
.0296
.0296
.0296
+ j
.0683
.0683
.0683
(38)
Network topology is represented through a 3x3 upper triangular matrix T.
T =
1 1 10 1 00 0 1
(39)
Then, TRX matrix is built by aggregating TTDRT and TTDXT matrices as indicated in 21:
TTDRT =
.0296 .0296 .0296
.0296 .0592 .0296
.0296 .0296 .0592
(40)
TTDXT =
.0683 .0697 .0683
.0683 .1367 .0683
.0683 .0683 .1367
(41)
Using 22, solution reached at iteration 3 for ε = 10−4 and displayed in Table I. Results arepresented in per unit and degrees.
TABLE I
4 Node State of the System - balanced Approach
V0 θ0 V1 θ1 V2 θ2 V3 θ3
1.000 0.00 0.987 -1.59 0.981 -2.40 0.981 -2.40
THE DISTRIBUTION TRX-POWER FLOW METHOD 1225
Fig. 3. 4-Node Network Topology
Fig. 4. 4-Node Network – Three-phase distribution line spacing
A.2 Unbalanced TRX Power Flow
Using SB =3.33MW and VB = 12.47/√
3kV. Three phase data is given in p.u.:
Sa = Sb = Sc =[
0 −0.2 −0.2]
(42)
Three phase impedance matrix is the same for all line sections:
Zabc = Rabc + jXabc (43)
Rabc =
0.0294 0.0100 0.00980.0100 0.0299 0.01000.0098 0.0100 0.0296
(44)
Xabc =
0.0692 0.0322 0.02470.0322 0.0672 0.03220.0247 0.0322 0.0683
(45)
As this system is modeled with three phase circuits in all section lines, three phase upper trian-gular matrix is given by the following expression:
Tabc =
U U U
0 U 00 0 U
where U =
1 0 00 1 00 0 1
(46)
Using 31 ,three phase TRXabc matrix is formed by TabcTDabc
R Tabc and TabcTDabc
X Tabc shownin Tables II and IV, respectively.
Reference three phase voltages at slack node 0 and initial values in nodes 1,2 and 3 are given by:
1226 THE DISTRIBUTION TRX-POWER FLOW METHOD
TABLE II
TabcTD
abcR Tabc Matrix
.0294 .0100 .0098 .0294 .0100 .0098 .0294 .0100 .0098
.0100 .0299 .0101 .0100 .0299 .0101 .0100 .0299 .0101
.0098 .0101 .0296 .0098 .0101 .0296 .0098 .0101 .0296
.0294 .0100 .0098 .0587 .0200 .0197 .0294 .0100 .0098
.0100 .0299 .0101 .0200 .0599 .0203 .0100 .0299 .0101
.0098 .0101 .0296 .0197 .0203 .0592 .0098 .0101 .0296
.0294 .0100 .0098 .0294 .0100 .0098 .0587 .0200 .0197
.0100 .0299 .0101 .0100 .0299 .0101 .0200 .0599 .0203
.0098 .0101 .0296 .0098 .0101 .0296 .0197 .0203 .0592
TABLE III
TabcTD
abcX Tabc Matrix
.0692 .0322 .0247 .0692 .0322 .0247 .0692 .0322 .0247
.0322 .0672 .0272 .0322 .0672 .0272 .0322 .0672 .0272
.0247 .0272 .0683 .0247 .0272 .0683 .0247 .0272 .0683
.0692 .0322 .0247 .1383 .0644 .0494 .0692 .0322 .0247
.0322 .0672 .0272 .0644 .1345 .0544 .0322 .0672 .0272
.0247 .0272 .0683 .0494 .0544 .1367 .0247 .0272 .0683
.0692 .0322 .0247 .0692 .0322 .0247 .1383 .0644 .0494
.0322 .0672 .0272 .0322 .0672 .0272 .0644 .1345 .0544
.0247 .0272 .0683 .0247 .0272 .0683 .0494 .0544 .1367
Vabc−0 = V0
abc−1 = V0
abc−2 = V0
abc−3 =
[
16 016
− 120o
16 120o
]
(47)
Using 34, solution reached at iteration 3 for ε = 10−4 and displayed in Table IV. Results arepresented in per unit and degrees.
TABLE IV
4 Node State of the System - balanced Approach
p V0 θ0 V1 θ1 V2 θ2 V3 θ3
a 1 0 0.989 -0.95 0.984 -1.43 0.984 -1.43b 1 -120 0.994 -120.86 0.991 -121.30 0.991 -121.30c 1 120 0.993 119.02 0.989 118.52 0.989 118.52
These illustrative examples were solved using Microsoft Excel, and spreadsheets can be requestedto the author.
THE DISTRIBUTION TRX-POWER FLOW METHOD 1227
B. Comparative Analysis - Three Test Networks
The proposed methodology was applied in the three (3) distribution test networks widely usedin literature (12-bus [29], 33-bus [40] and 69-bus [41]).
The performance and robustness of the method was assessed under Matlab platform [45] andcompared with two alternative methods. First, the standard BW/FW sweep power flow approachpresented in [24] and, second the Matpower’s Newton-Raphson solver [44]. The BW/FW sweepmethod has been coded with complex variables using the theoretical basis presented in Section II.Matpower-NR solver is based on a standard Newton’s method [1] using a full Jacobian, updatedat each iteration. Matpower performance is excellent even on very large-scale test cases, since thealgorithms and implementation take advantage of Matlab’s built-in sparse matrix handling. InAppendix B, it is included the the codification of the proposed method. It is important to notethat overall iteration process only require ten programming lines demonstrating the simplicity ofthe proposal.
Convergence and robustness of the algorithms are analyzed through the number of iterationsneeded to reach a solution and the CPU time spent in the iterative process. Due to small size ofthe problems, CPU time registered includes input-output (I/O) access time.
Stop criteria is 10−6 in all nodal voltages. A Macbook Intel Core 2 Duo [email protected] with2GB RAM under OSX Leopard 10.5.5 has been used for all simulations.
Results are presented in Table V
TABLE V
Comparison between TRX, Standard BW/FW sweep and NR Power Flow
Number of Iterations12-bus 33-bus 69-bus
Complex Back/Forward Sweep 5 6 7NR (Matpower) 4 4 4TRX Method 5 6 7
CPU Time (10−3 Seconds)12-bus 33-bus 69-bus
Complex Back/Forward Sweep 0.78 0.94 4.38NR (Matpower) 8.17 10.44 12.15TRX Method 0.64 0.82 1.25
Results show that NR method has better convergence behavior than TRX and Backward/ForwardSweep. However, despite TRX method requires the same number of iterations than BW/FW sweepmethod, it has the better CPU time. Regarding NR and BW/FW sweep methods, these resultsconfirm the conclusions reported by Eminoglu in broad comparative convergence analysis amongseveral BW/FW sweep based methods [33]. In this paper, it was used the same test systems inorder to compare with results reported in this study. Proposed TRX method presents the slowestconvergence rate but with the highest computationally efficiency. Note that in these small examplesCPU time corresponding to NR results is extremely different than other methods. This is due torequired I/O data exchange time in sparse matrix handling.
1228 THE DISTRIBUTION TRX-POWER FLOW METHOD
Fig. 5. The n-Node Network
Fig. 6. CPU time required to convergence
C. Comparative Analysis - Uniformly Loaded Distribution Feeder
The proposed method has also applied in a large-scale test network and its performance comparedwith the BW/FW sweep and the Matpower’s Newton-Raphson. Same computer conditions thanprevious comparison analysis are used. The test case is a 12.47kV n-node radial distribution networkwith a load of 4MW uniformly distributed through L=1mile. Each branch is 336,400 26/7 ACSR(Linnet) conductors with impedance given in pu (SB=10MW):
Z01 = Z12 = .. = Zn−1n =0.0293 + j0.697
n(48)
To illustrate the impact of the number of nodes in the CPU time required by each power flowmethod, line section depicted in Fig. 48 has been divided in n branches connected in radial formand total load has been fractionated in n nodal loads.
It is well known that analytical solution of this type of system is given by [43]:
%∆V0n =V0 − Vn
V0
=S3ϕ
5V 20
(R cos ϕ + X sin ϕ) (49)
where S3ϕ = 0.4, the load factor is equal to 1. For V0 = 1, 49 solution is Vn is 0.9940pu. In orderto verify this solution, an exact power flow solution is obtained using NR for n=1000: at the endof the feeder, voltage is Vn=0.99398 and angle θn=-0.016 radians.
The n-node network has been solved using the proposed TRX method and two alternative meth-ods (NR, Back/Forward Sweep) varying n parameter from 1000 to 3000 nodes. All simulations leadto the same solution.
CPU process time results are depicted in Fig. 6. Convergence time spent in the iterative processdoes not consider I/O data exchange time.
When I/O data exchange is not considered, and CPU time is only related to the while loop (seeAppendix B), results show that TRX method has better CPU time behavior respect to NR and
THE DISTRIBUTION TRX-POWER FLOW METHOD 1229
BW/FW sweep method. It must be pointed out that BW/FW sweep method developed in [24] isnot direct, as indicated in paper title but iterative like all sweep-based methods. Not consideringI/O data exchange is justified by the fact that under DSM environment, a sentinel program shouldmaintain all TRX matrices in memory with the most realistic topology.
As a result, the proposed TRX method presents the slower convergence rate but with the highercomputationally efficiency. The overall process is based on summation and multiplication of fixedreal numbers numbers previously allocated in memory being highly competitive respect to NR withLU factorization and sparse handling. The TRX matrix does not require update at each iterationlike Jacobian in Full NR method. In this case, CPU time corresponding to NR results is the secondbest. The worst behavior correspond to the standard BW/FW sweep in complex form which isextremely affected by operations required by summation and multiplication of complex numbers.
The proposed method is a valuable tool to analyze aerial distribution systems, since they canbe modeled as a single series impedance. Research efforts are focused in the generalization ofthe method including shunt impedances associated to cables. Future research is also oriented incompare the proposed methodology with other distribution-oriented power flow methodologies, inparticular decoupled versions of NR and BW/FW sweep methods based on biquadratic formula.Ill-conditioned systems will be also considered.
V. Conclusion
This paper proposes a new methodology to assess the power flow solution of distribution networksby merging the standard Backward/Forward Sweep process in a unique state-of-the-system calcula-tion formula. The proposal is suitable to be applied in balanced and unbalanced aerial distributionsystems, radially or weakly meshed operated with distributed generation. Proposed methodologyhas been applied in test-case systems showing better performances than other large-scale methods.
Appendices
A. Nomencalture
TRXabc Three Phase TRX matrixTRX TRX matrixDZ Diagonal matrix of branch impedance vector Z
DR Diagonal matrix of branch resistance vector R
DZ Diagonal matrix of branch reactance vector Z
ε Convergence criteriaI Current vectorI i Current at node iIx Real part of Current vector I
Iy Imaginary part of Current vector I
Iabc Three Phase Current vectorIabc−x Real part of Three Phase Current vector Iabc
Iabc−y Imaginary part of Three Phase Current vector Iabc
J Branch Current vector I
J ij Branch current between node i and node jJx Real part of Branch Current vector I
Jy Imaginary part of Branch Current vector I
n Number of nodes, excluding originP Active Power Injected vector
1230 THE DISTRIBUTION TRX-POWER FLOW METHOD
Q Reactive Power Injected vectorPi Active Power Injected at node iQi Reactive Power Injected at node iPDi Active Power Demanded at node iQDi Reactive Power Demanded at node iPGi Active Power Generated at node iQGi Active Power Generated at node iR Branch resistance vectorRabc Three Phase Branch resistance vectorRij Resistance between node i and node jS Apparent Power Injected vectorSabc Three Phase Apparent Power Injected vectorSi Apparent Power Injected at node iSDi Apparent Power Demanded at node iSGi Apparent Power Generated at node iT Triangular matrixTabc Three Phase Triangular matrixV Voltage vectorV i Voltage at node iVx Real part of Voltage vector V
Vy Imaginary part of Voltage vector V
Vabc Three Phase Voltage vectorVabc−x Real part of Three Phase Voltage vector Vabc
Vabc−y Imaginary part of Three Phase Voltage vector Vabc
X Branch reactance vectorXabc Three Phase Branch reactance vectorXij Reactance between node i and node jZ Branch Impedance vector Z
Zabc Three Phase Branch Impedance matrix Z
Z ij Branch Impedance between node i and node jZij Impedance matrix between node i and node jZaa−ij Self impedance phase a between node i and node jZab−ij Mutual impedance phase ab between node i and node j
B. TRX Algorithm Coded in MATLAB
%-----Calculations Under DSM environment-------------T %Triangular matrix (topology) is knownr; x; %List of branch impedances are knownP; Q; % List of Active and reactive powers are knownVo; %Slack voltage is knownA=T’*diag(r)*T; B=T’*diag(x)*T;TRX=vertcat(horzcat(A,-B),horzcat(B,A));%-----TRX Power Flow Analysis------------------------Vx=[ones];Vy=[zeros];V=vercat(Vx,Vy)%Initial voltage profilestarttime=cputime;while max(abs(delta))> 0.0001 %Stop Criteriafor j=1:n %Number of line sections and nodesI(j)=(-P(j)*V(j)-Q(j)*V(j+n))/(V(j)^2+V(j+n)^2);I(j+n)=(Q(j)*V(j)-P(j)*V(j+n))/(V(j)^2+V(j+n)^2); endV2= Vo-TRX*I.’; %Voltage Updatedelta=V-V2; %Voltage MISMATCHV=V2;
THE DISTRIBUTION TRX-POWER FLOW METHOD 1231
end %end whileprocesstime=cputime-starttime;
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[24] J.H. Teng,”A direct approach for distribution system load flow solutions,” IEEE Transactionson Power Delivery, Vol. 18, No. 3, pp. 882-887, 2003.
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THE DISTRIBUTION TRX-POWER FLOW METHOD 1233
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Apéndice I
“OPTIMAL SIZING OF CAPACITORS PLACED ON RADIAL
DISTRIBUTION SYSTEM”, BARAN, M. E. ET AL. 1989, [31]
139
IEEE Transactions on Power Delivery, Vol. 4, No. 1, January 1989
OPTIMAL CAPACITOR PLACEMENT ON RADIAL DISTRIBUTION SYSTEMS
725
Mesut E . Baran Felix F. Wu Department of Electrical Engineering and Computer Sciences
University of Califomia, Berkeley Berkeley, CA 94720
Abstract - Capacitor placement problem on radial distribution systems is formulated and a solution algorithm is proposed. The location, type, and size of capacitors, voltage constraints, and load variations are considered in the probIem. The objective of capacitor placement is peak power and energy loss reduction by taking into account the cost of capacitors. The problem is formulated as a mixed integer programming problem. The power flows in the system are explicitly represented and the voltage con- straints are incorporated. The proposed solution methodology decom- poses the problem into a master problem and a slave problem. The mas- ter problem is used to determine the location of the capacitors. The slave problem is used by the master problem to determine the type and size of the capacitors placed on the system. In solving the slave prob- lem, an efficient phase I - phase II algorithm is used. Proposed solution methodology has been implemented and the test results are included in this paper.
I. INTRODUCTION The general capacitor placement problem consists of determining
the location, type, and the size of capacitors to be installed in the nodes of a radial distribution system such that the economic benefits due to peak power and energy loss reduction be weighted against the cost of installment of such capacitors while keeping the voltage profile of the system within defined limits.
The optimal capacitor placement problem as defined above has many parameters, such as; the location, type, and cost of capacitors, vol- tage constraints, and load variations on the system. These parameters determine the complexity of the problem.
Conventionally, the problem has been formulated by using a vol- tage independent reactive current model and solved by fixing some of the parameters and using analytical methods [l-31. Recently there has been some studies to solve the problem in its general form. There are basically three approaches. The first one is the dynamic programming type approach by treating the sizes of capacitors as discrete variables, [4-51. The second approach is to combine the conventional analytical methods with heuristics 16-71. Third approach, pioneered by Grainger et. al., is to formulate the problem as a nonlinear progmming problem by treating the capacitor sizes and the locations as continuous variables [8-131. The application of this approach to general problem with the voltage regulator problem is given in [ 131.
In this paper, a formulation for the general capacitor placement problem as a mixed integer programming problem will be given first. The formulation considers all the parameters of the problem stated above and also the voltage constraints. Furthermore, the ac power flow formu- lation is used to represent the power flows and the voltage profile in the radial distribution systems.
A solution methodology for the general problem is proposed in this
88 WM 064-8 A paper recommended and approved by the IEEE Transmission and Distribution Committee of the IEEE Power Engineering Society for presentat- ion at the IEEE/PES 1988 Winter Meeting, New York, New York, January 31 - February 5, 1988. Manuscript submitted September 1, 1987; made available for printing November 13, 1987.
paper. The solution is based on the decomposition of the problem into hierarchical levels. The problem at the top level, called the muster prob- lem, is an integer programming problem and is used to place the capaci- tors ( i.e. U) determine the number and the location of the capacitors). A search algorithm has been developed for the master problem. The problem at the bottom level is called the slave problem and is used by the master problem to determine the types and the settings of the capaci- tors placed. Decomposition schemes are also used to further decompose the slave problems into base problems. Base problems are solved by using the efficient solution algorithm developed for a special capacitor placement problem called the sizing problem. The sizing problem and the associated solution algorithm are presented in another paper [141.
This paper consists of seven sections. In section 11, the general capacitor placement problem is formulated and its complexity is dis- cussed. h section 111, it is shown that the problem can be decomposed into a muster problem and a slave problem. Section IV and V are devoted to the development of solution methodologies for the slave and master problems respectively. Section VI contains the test studies of the solution method applied to two different systems. Conclusions are given in section VII.
IL. FORMULATION OF THE PROBLEM We consider the general capacitor placement problem as determin-
ing the places (number and location), types, and settings (capacities) of the capacitors to be placed on a radial distribution system. The objec- tives are to reduce the power and energy losses on the system and to maintain the voltage regulation while keeping the cost of capacitor addi- tion at a minimum.
Since we are interested in energy loss in the system, it is necessary to take into account the load variations for a given period of time, T. We assume that the load variations can be approximated in discrete lev- els. Furthermore, the loads are assumed to vary in a conforming way (i.e., all the loads enjoy the same pattem of variations). We let S(T) be the common Loud Duration Curve as shown in Fig.1. Then a load, say load QL , can be represented as
Where, Q2 represents the peak value. QL@) = Q ~ W ) (1)
S"t
:F+>; ~
y 4;"qT I- Figure 1 : Load Duration Curve
Under these assumptions, the time period, T can be divided into intervals during which the load profile of the system is assumed to be constant. Let there be nt such loud levels (loud projiles) .
Then for each load level, we have: (i) power flow equations, (ii) voltage constraints as bounds on the magnitude of the system node vol- tages. (iii) capacity and control constraints on the control variables (capacitors).
We will represent the constraints imposed by power flows on a radial distribution system by a new set of ac power flow equations, called DistFkw equations. They substitute for the conventional ac power flow equations. To summarize the idea, consider a 3-0, balanced radial distribution feeder with n branches/nodes, I laterals, and nc shunt
0885-8977/89/01oo-O725$01 .WO 1989 IEEE
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726
capacitors placed at the nodes of the system. In Fig.2, one-line diagram of such a network is shown.
Fig.2 : One line diagram of a Distribution Feeder It can be shown that power flow through each branch in the lateral can be described by the following recursive equations.
Where, Pk ,Qk : real and reactive power flows into the receiving end of branch k+l connecting node k and node k+l, v k : bus voltage magnitude at node k , Qck : reactive power injection from capacitor at node k.
Eq.(2), called the brnnchjbw equation, has the following form
4 + 1 = fk+l(xk & k + d (3.i)
Note that if there is no capacitor at node k, then uk does not appear in Eq(3.i). By abusing notation, we will simply use U as an nc dimensional vector containing the nc capacitors i.e.,
Where, X k = [ p k Qk v?lT and 4 + 1 = e&+, .
UT = [U1 . . . U,] = [Qcn, . . . Q-]
In addition to the branch flow equations of (3.i). there are terminal conditions to be satisfied for each lateral (counting the main feeder as the O'th lateral). For example, for lateral k shown in the figure, we have the following terminal conditions: (i) at the branching node k where the lateral is connected to the main feeder, we define a dummy variable VkO and let
XkO, = v& = v l = XOk, (3.ii)
(ii) at the end of the lateral, there is no power sent to the other branches, i.e.,
x h , = P h = O ; x h r = Q h = O (3.iii)
Hence, for the general feeder considered, there are 3(n+l+1) Dist- Flow equations corresponding to Eq.(2) and Eq.(3). They will be represented by the following equations.
Where, x = [ x f . . . x[$lT , xk = [ x & . . . x ; l T . DistFlow equations can be used to determine the operating point, x
of the system for a given load profile, PL; , Q L ~ i = 1 , . . . , n and the capacitor settings, U . We prefer to use DistFlow equations over conven- tional ac power flow equations because the special structure of the Dist- Flow equations can be utilized to develop a computationally efficient and numerically rbbust solution algorithm. The details of such a solution algo- rithm are presented in [ 141.
For the capacitor placement problem since there are nt different load profiles to be considered, the overall DistFlow equations are
G(x,u) = 0 (4)
Gi($,ui)=O i = 0 , 1 . . . n r (5 ) Where, xi , ui represent the state and the contml variables corresponding to the load profiie i respctively.
The voltage constraints can be taken into account by specifying upper and lower bounds on the magnitude of the node voltages as follows,
vm2< v;'=Y;'(x') 1v-2 j = l . . . n i - -0 ,1 . . .n t (6)
These bounds constitute a set of functional inequality constraints of the form,
H'(x')IO i =0,1,. . . .nt (7)
We will consider two different types of capacitors and represent them as follows: i) Fixed Capacitors : They will be treated as reactive power sources with the constant magnitude at al load levels, i.e..
"o=ul= . . . = u M (8.i)
ii) Switched Capacitors: It will be assumed that the set t ings (capacities) of a switched capacitor, U: can be changed/contmlled at every load level qnsidered. Therefore, for each capacitor, there are nt+l settings, U: i = 0,l. . . . , nt to be determined. We will also assume that the setting of a capacitor for the peak load, U: will be bigger than the ones for other lower load levels U:. Hence, the sizes (nominal capacities) of capacitols will be determined by uo and the relationship between the size and and the settings of a capacitor will be as follows.
O I U i S U , o (8.ii)
The objective terms. namely the real power and energy loss and the capacitor cost. can be formulated ,as follows. For each load level i , let the power loss in the system be pi (x'). Then the total cost of energy loss can be wrimn as
nl
ke x = T i P i ( X ' ) (9) i 4
where, Ti is the duration of the load for load level i and the constant k, is the energy cost per unit. The cost for the real power loss at peak load level can be added to this sum by modifying To accordingly.
The capacitor cost h c t i o n is usually step like as shown in Fig.3 since in practice capacitors are grouped in banks of standard discrete capa- cities (usually 300 kvar sizes at 23 kV level).
4)
"0
cvb Y X
Figure 3 : Capacitor cost functions Such a function is not easy to handle within this formulation framework; therefore, it will be approximated by a linear function with a fixed charge as shown in Fig3 by a dotted line. This function can be formulated by using a decision variable e E (0.1) as
f (uo)=c .e +rc.uo O S u o Su".e (10)
Where, uo and r, represent the size and the marginal cost of the capacitor respectively. Note that e = 0 corresponds to the decision that the capacitor not to be placed.
To summarize, let the types and places of nc "candidate" capacitors, initially considered for installment, be given and let the sets C 1 , C2 con- tain the switched and the fixed capacitors respectively. Then we can write the general capacitor placement problem as a standard optimization prob- lem as follows.
s.t. G' (xi ,d) = 0
H'(x')SO i=O,1, . . . , nt
0 I uo I u-.e 0 I U; I U:
U: = U:
k E C 1 = {W. cap.) k E C z = lfixed c a p . }
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127
base problems can be solved by the efficient solution method developed for the sizing problem, PS in [14]. Finally, in the last subsection. we will come back to the general slave problem and consider the types as well as the settings of the capacitors.
4.1 Fixed Capacitor Problem
becomes When all the capacitors to be placed are of fixed type, the problem
nl Pfx mill f , = ke CTiPi(Xi) + C rckuk
i=O k E C 2
s.t. Gi(xi ,u)=O i=O,l,. .. ,nt
H'(x') 5 0 O I U I U -
This general form of fixed capacitor problem, Pfx is a base problem because it is essentially the same as the sizing problem, PS. Here, because of the consideration of more load levels, the objective function comprises more power loss terms and the constraint set is bigger due to extra Dist- Flow equations and the corresponding voltage regulation constraints. But this does not change the structure of the problem very much. Therefore, the solution methodology developed for the sizing problem can readily be applied for this problem with small modification.
Note that for this general case, existence of different load levels makes it possible to have both lower bound voltage violation at peak load level and upper bound voltage violation at light load level for a given con- trol U . This is especially true when the load levels are diverse. Such cases are handled in the Phase I - Phase I1 type solution algorithm of the sizing problem by considering both the lower and the upper voltage con- straints in calculating the search direction (for details, see the solution algorithm for the base problem in [ 141).
This is a non-linear, mixed integer programming problem. Decision variables, e = [e, . . . e,IT are to be used to choose the capacitors among initially designated ones and continuous variables ui , i = 0.1.. . . .nt will be used to determine the optimal settings of the capacitors.
Voltage regulators are not explicitly represented in the capacitor placement problem presented here. However, the formulation and the solution algorithm introduced in this paper can be generalized to include the voltage regulators in the following fashion:
Voltage regulators. such as regulating transformers, can be represented by their equivalent circuits in the DistFlow equations.
The solution algorithm for sizing problem, proposed in [14] for determining the optimal sizes of capacitors placed on the system, can be generalized to obtain the optimal settings of the voltage regulators placed on the system,
Voltage regulation for a given set of capacitors and voltage regula- tors can be obtained again by using the solution algorithm developed for the sizing problem. This is demonstrated in I141 by using only the capacitors.
It seems appropriate to put the voltage regulator placement problem (to find the locations for the voltage regulators) at the top level of hieravhical decomposition scheme proposed in this paper in solving the overall problem.
Further investigation is required to complete the generalization.
III. DECOMPOSITION OF THE PROBLEM The problem formulated in the previous section is a non-liiear,
e € E ; U E U ] (1 1)
where, e and U correspond to the decision vector, and the control vector, respectively, and E and U represent the constraint sets imposed on these vectors. We adapt a general solution approach which decomposes the problem by making use of the following property of the optimization
mixed integer programming problem of the following form,
min {f,(e.u) I e.u
min f, (e& = min { i f f , (e&] (12) W E . U E U W E UE
assuming that for each decision, e the problem in braces, called the slave problem,
has a solution and it is "easy" to find it. Then the main problem becomes,
min g(e) (14) (EE
and is called the musterproblem. These problems can be characterized as, Master Problem (MP) : Integer Programming Problem Slave Problem (SP) : Non-linear Differentiable Optimization Problem
The solution for this decomposed problem requires an efficient solu- tion algorithm for the slave problem and a search procedure over E, the set defined by all the possible decisions, for the master problem. In the next two sections. we'll discuss and develop such solution schemes.
IV. SLAVE PROBLEM As indicated in the previous section, the slave problem assumes that
the capacitors are placed and it is a special case of the capacitor placement problem. The problem is a non-linear differentiable optimization problem with quite a large number of equality and inequality constraints. It is still not easy to solve the slave problem. Our aim here is to study the special features and the structure of the problcm and to develop an efficient solu- tion methodology by exploiting these features.
For the slave problem, the type of capacitors (fixed or switched), and the number and diversity of load levels are important parameters that determine the structure and the size of the problem. Therefore, we first relax the parameter "type" by assuming that they are given. Then the problem becomes a sizing problem (i.e., determining the capacitor settings ) and we need to consider two cases; one with fixed type capacitors and one with switched type capacitors. In the following first two subsections, we will show that for these two cases the problcm is either of the base type problem or it can be decomposed into the base type subproblems. The
4.2. Switched Capacitor Problem
slave problem can be re-written as follows. When all the capacitors to be placed are of the switched type. the
(17) U' -U'SO i = I , . . . ,nr
Note that the two constraint sets (15) and (16) are coupled through (17), to indicate that the capacitor sizes, U' are the upper bund constraints for capacitor settings at the other load levels, U' . This weak coupling between U' and ui's can be exploited to decompose the problem into smaller subproblems. In appendix, it is shown that Psw can be decom- posed into the fdlowing nt +1 base problems. The main problem, SW,
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728
43. Mixed Type Capacitor Problem In this section, we go back to consider the general slave problem in
which it is not known a priori which capacitor is of fixed and which one is of switched type and one has to determine the types of capacitors in addi- tion to their settings. Note that, in general, the switching capacitors are more expensive in both fixed cost, c and the marginal cost, r, than the fixed capacitors. Therefore, we propose the following heuristic selection scheme by using the solution methodology developed for the switched capacitor problem.
Step 0 : Assume all the capacitors are of the switched type. Step 1 : Solve the problem considering only the switched capa- citors and keeping the fixed capacitor power injections as con- stant at their nominal settings, i.e., as loads. Step 2 : Using the results of step 1, check the settings of switched capacitors at light load level (level nr ). For the ones with nonzero light load settings, (i.e., u t # 0 ) assign that por- tion of capacitor as fixed type. Step 3 : If any h x i g occurred, then go to step 1; otherwise, stop.
Subproblems, SWi i = 1 . . . nt SWi fi (U? = min k, Tipi (xi)
S . t C'(x',u') = 0
H'(xi) S 0
OIUU' suo
The subproblems, SWi involve only one load level and therefore, they can be solved by the algorithm developed for the sizing problem for given capacitor sizes, U'. The solutions will correspond to the oprimal capacitor senings , 8' for the off-peak load 'eye's considered.
The main problem, SW,, is also a sizing problem; but computation- ally it is not of the easy type due to existence of fi (114's - the extra terms coupled to the subproblems - in the objective function. Updating these terms at each iteration during the solution of SWo requires solution of the subproblems. However, we can use a simple, heuristic scheme to update fi(uO) and Vfi(u? in the main problem SWo. To begin with, let the solu- tions of the subproblems SWi for a given U' be denoted (ui)* . As we move U' from iteration to iteration in SW, , f i (u4 and Vfi (U') should be calculated by solving SWi with the new i? for a new (U'): , say (iii)* . We prppose, however, instead of solving SWi for a new 6")' , to use the old (U')' unless the constraint (17) is violated, i.e., we set the k'th com- ponent of 0'
ii; if ii;S(u;)* a; = (18) (U;)* otherwise
(19)
I We simply use 0' in evaluating fi @) and Vfi @) as follows:
f; (if) = k, Tipi (2' ,tii)
, Note that if U' and i? do not differ very much, the approximation will be good. To assure this, we start the procedure by first solving the subprob- lems with the capacitor sizes set to their maximum, U'"=.
The ovcrall iterative algorithm is shown in Fig.4. In the algorithm, an iteration comprises the solution of the subproblems, SWi first and then the main problem, WO. Convergence check at end of an iteration involves checking if there is a status change in the constraint set of Eq.(17). (i.e., a non-binding constraint becomes binding or vice versa ). If there is such a status change, then we go back and update the slave problems; otherwise we stop iterating since the solution is converged.
s = s + l
fori = 1, ..., nt
Solve SWo
Use Eq.(19) and Eq.(20) to calculate f,O and Vfi&) i = 1. . . . ,nl 11 converged
Figure 4 : Block diagram of SW Capacitor Algorithm
V. MASTER PROBLEM We follow a general approach in solving the master problem, which
is an integer programming problem, and first construct a decision graph and then develop an efficient search scheme to place the capacitors (i.e., to determine their numbers and places).
Let there be nc initially given candidate capacitors. Then, all possi- ble decisions about choosing the capacitors for placement among the can- didate capacitors can be m g e d as a decision graph assuming one deci- sion is made at a time. Such a graph is shown in Fig5 for nc = 3.
Figure 5 : Decision graph for 3 variable case In the figure, each node represents a particular decision, e = [el e2 e3 IT ; ei E { O J J . Where, ei = 0 means that the capacitor is not chosen, and ei = 1 means it is still a candidate. A branch f" a node to another indicates how the transition can be achieved: taking out the can- didate capacitor whose number is shown on the branch. Such a relation- ship is indicated in graph theory by calling a node and all the nodes emanating from it. aparent and its children, respectively.
Search starts from the root which corresponds to thc case where all candidate capacitors are chosen to be placed at the designated nodes of the system. Then there are two possible search techniques, depth-frrst search and breudth-first search, that can be employed to get a local optimum [17]. Here, rather than employing these general search techniques directly, the children of a given nodebarent) are sorted first according to their con- tribution to the objective. We propose a sorting procedure which works sort of like a "steepest descent" approach in discrete case. The procedure is
- given a node which is identified by its decision vector, e and the solution of the corresponding Slave Problem, ( i.e., the control vector ii = [ z i 1 . . . - for all existing capacitors, k = 1, . . . , nc s.t. ek z 0
and the objective f, (ii) ).
- construct U by removing the capacitor k and keeping the rest, i.e., 0 i f j = k
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rev 3 800 910 AE 225 630
itr. 9 175 V,,,i,, .90 160
- calculate the new objective, f:(u) (This requires only DistFlow
- let the contribution of the capacitor k to the objective be solutions).
A f : = f," -f 0 .
- sort the existing capacitors by using the Af ,"s
The A f t s can used to choose the child to branch out (i.e., capacitor to take out) since the child with the smallest Af ," is more liiely the one who contributes to the objective the least.
The first procedure, depth-first search, visits the children of a parent node (i.e., solves the associated slave problems) according to the order determined by the sorting procedure and branches out on the, first one which gives a lower objective than that of the parent. This search is of order nc and therefore computationally attractive. However the branching criteria is appropriate only for unconstrainted case. When the voltage con- straints are active, a better criteria would be to check how well the deficiency created by a capacitor removal be compensated by the rest of the capacitors by Nnning the corresponding slave problems. This corresponds to the breath-& type search which is conducted by visiting all the children first and then branching out on the one with the smallest objective. This search is of order nc2 and requires at most nc(nc+l)/2 slave problem solutions. However, the search can be made much faster by noting that the objectives, f," calculated by sorting procedure will be bigger than the ones calculated by the search. Therefore, in practice, the search can be conducted only on the children with negative Af ,"s.
6 5
1 2
VI. TESTSTUDIES The proposed solution methodology has been implemented in For-
tran 77 on both VAX llh'50 and IBM AT. The program uses the algo- rithm developed for the sizing problem in [14] as a subroutine to solve the base problems. We present the test results of two systems in this section to illustrate the performance of the proposed solution scheme.
The first test system, TS1 is a 9-branch main feeder test system developed by Grainger et. al., [9]. The second test system, TS2 is a 69- branch, %lateral test system derived from a portion of the PG&E distribu- tion system. The network data of this system is given in Table 1.
We adopted the following cost figures : energy cost, k, =0.06 $/kwh ; capacitor fixed cost, c = lOOO$ ; capacitor marginal cost, r, = 3 $/bur. The load duration data assumed for the systems is given in Table 2. It is also assumed that substations have regulating transformers which are tapped to +5% during peak load and set to nominal otherwise. The substation voltage without tap is taken as the base kV and the lower and upper voltage limits are assumed to be 0.9 and 1.1 p.u. respectively.
2 J 4700 9?0 4550 225 700 226 .90 - .90 4 3 - 270
Br. S d . Rv. - B r . P a r . - Rv. N d . Load N o Nd. N d . r(ohm) x(ohm) P(KW) Q(KVRR)
1 0 1 0.0005 0 .0012 0 . 0 .
2 ; 2 0 .0005 0.0012 0 . 0 .
2 e 0 . 0 . 0 . 0.
1 1 6
960 3700 - 660 191 1160 360 .901 210
5 200
4 2 e 3 0 .0015 0 .0036 0 . 0 . 5 3 4 0 .0251 0 .0294 0 . 0. 6 4 5 0.3660 0 , 1 8 6 4 2 .60 2 .20 7 5 6 0 .3811 0 . 1 9 4 1 40 .40 30 .00 8 6 7 0 .0922 0 .0470 75 .00 54 .00 9 7 8 0 .0493 0 . 0 2 5 1 30 .00 2 2 . 0 0
1 0 8 9 0 .8190 0 .2707 28 .00 1 9 . 0 0 11 9 1 0 0 .1872 0 .0619 1 4 5 . 0 0 104 .00 1 2 1 0 11 0.7114 0 . 2 3 5 1 145.00 1 0 4 . 0 0 13 11 1 2 1 .0300 0.3400 8 . 0 0 5 .50 1 4 1 2 13 1 . 0 4 4 0 0 .3450 8 .00 5 .50 1 5 13 1 4 1 .0580 0 .3496 0. 0. 1 6 1 4 1 5 0 .1966 0 .0650 45.50 30 .00 1 7 1 5 1 6 0 .3744 0 . 1 2 3 8 60 .00 35 .00 1 8 1 6 1 7 0 .0047 0.0016 60.00 35 .00 1 9 1 7 1 8 0.3276 0 .1083 0 . 0 . 20 1 8 1 9 0 .2106 0 .0696 1 . 0 0 0.60 2 1 1 9 20 0 .3416 0.1129 114.00 81 .00 2 2 20 2 1 0 .0140 0 .0046 5 . 3 0 3 . 5 0 2 3 2 1 22 0 . 1 5 9 1 0 .0526 0 . 0 . 24 22 2 3 0 . 3 4 6 3 0 .1145 28 .00 20 .00 2 5 2 3 24 0 .7488 0 .2475 0 . 0. 2 6 24 2 5 0 .3089 0 . 1 0 2 1 1 4 . 0 0 1 0 . 0 0 27 2 5 2 6 0 .1732 0 .0572 14 .00 1 0 . 0 0
System So S, S2 TS1. 1.1 0.6 0.3 TS2 1.8 1. 0.5
To TI T 2 1OOO. 6760. 1OOO. 1OOO. 6760. 1OOO.
Table 3: Test run results for TS1 - fixed capacitor placement
Table 1 : Network Data of TS2 Br. Sd. Rv. - B r . P a r . - N o N d . N d . r(ohm) x(ohm) P(Kw) Q(KVAR) No N d . N d . r(ohm) X ( 0 h m P(KW) Q(")
Rv. N d . Load B r . S d . Rv. - Br. P a r . RV. Nd. Load
28 29 30 31 3 2 33 34 3 5
36 3 7 3 8 3 9 40 4 1 42 4 3 44 45 4 6
47 48 49 5 0
2 27 27 28 28 2 9 2 9 3 0 30 31 31 32 32 33 33 34
2 e 2 7 e 27e 2 8 e 28e 6 5 6 5 66 6 6 67 6 7 68 68 6 9 6 9 70 70 8 8 88 8 9 8 9 90
3 3 5 3 5 36 3 6 3 7 37 3 8
0 .0044 0 . 0 6 4 0 0 .3978 0 .0702 0 .3510 0 .8390 1 .7080 1 .4740
0 .0044 0 .0640 0 .1053 0 .0304 0 .0018 0 .7283 0 .3100 0 .0410 0 .0092 0 .1089 0 .0009
0 .0034 0 .0851 0 .2898 0 .0822
0 .0108 0 .1565 0 .1315 0 .0232 0 .1160 0 .2816 0 .5646 0 .4873
0 .0108 0 . 1 5 6 5 0 . 1 2 3 0 0 .0355 0 .0021 0 .E509 0 .3623 0 .0478 0 . 0 1 1 6 0 .1373 0 .0012
0.0084 0 .2083 0 .7091 0 .2011
26 .00 26 .00
0 . 0 . 0 .
1 4 . 0 0 1 9 . 5 0
6 .00
26 .00 26 .00
0 . 24 .00 24 .00
1 . 2 0 0 . 6.00 0 .
39 .22 39 .22
0 . 79.00
3 8 4 . 7 0 384 .70
18 .60 1 8 . 6 0
0 . 0 . 0.
1 0 . 0 0 1 4 . 0 0
4 .00
1 8 . 5 5 1 8 . 5 5
1 7 . 0 0 1 7 . 0 0
1 . 0 0 0. 4 .30 0 .
26 .30 26 .30
0. 5 6 . 4 0
274 .50 274 .50
0.
5 1 7 40 0 .0928 5 2 40 4 1 0 .3319
5 3 8 42 0 .1740 54 42 4 3 0 .2030 5 5 4 3 44 0 .2842 5 6 44 4 5 0 .2813 57 45 46 1 .5900 5 8 46 47 0 .7837 5 9 47 48 0.3042 60 48 49 0 .3861 6 1 49 50 0 .5075 62 5 0 5 1 0 .0974 6 3 5 1 5 2 0 .1450 64 52 5 3 0 . 7 1 0 5 6 5 5 3 54 1 .0410
66 1 0 5 5 0 . 2 0 1 2 67 55 5 6 0 .0047
68 11 57 0 .7394 69 57 5 8 0.0047
S u b s t a t i o n V o l t a g e B a s e KVA B a s e V o l t a g e (kV)
0 .0473 0 .1114
0 .0886 0 . l o 3 4 0.1447 0 . 1 4 3 3 0 .5337 0.2630 0 .1006 0.1172 0 .2585 0 .0496 0 .0738 0 .3619 0 . 5 3 0 2
0 .0611 0.0014
0 ,2444 0 .0016
(kv) - - -
40.50 28 .30 3 . 6 0 2 .70
4 . 3 5 3 .50 2 6 . 4 0 1 9 . 0 0 24 .00 1 7 . 2 0 0. 0. 0. 0 .. 0 . 0 .
100 .00 7 2 . 0 0 0 . 0 .
1244.00 888 .00 32 .00 23 .00
0 . 0 . 227 .00 162 .00
5 9 . 0 0 42 .00
1 8 . 0 0 1 3 . 0 0 1 8 . 0 0 1 3 . 0 0
2 8 . 0 0 20 .00 28 .00 20 .00
1 2 . 6 6 10 .000 12 .660
t o t a l load: P(KW) - 3802.19 , Q(KVAR) - 2694.60
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730
AE vmi,
Test run for TSl, summarized in Table 3, starts with 4 capacitors. The solution of the slave problem for the mot (initial one) gives the optimal settings for these 4 capacitors. Then the capacitor's contributions to the objective, A j t s are calculated by using the sorting procedure. The results indicate that three of the capacitors - Q,, . e,,, QC6 - have Af: e 0; which implies that they are not economically feasible ( i.e., their economic contribution due to energy loss reduction is less than their cost). Therefore, the search is conducted only on these three capacitors at the second search level. As a result, QC2 is found to be the least economical and hence is taken out. This corresponds to the branching out on the first child of the mot in the table. Then a new search process resumes from this new node; first sorting out its children by using the sorting procedure and then visiting the ones that are economically infeasible. The solution is obtained at the third level of search when the evaluation of capacitors Qc6 and Q,, in the last node by the sorting procedure indicated that they are economically feasible.
The total run time for this test on VAX is about 45 sec. of CPU and 8 sec. of vo.
785 620 1240 1700 52 .go9
Test System 2 Test run for TS2 is summarized in Table 4. The test starts with 5
capacitors. After visiting the root. the capacitor with the zero setting, Qcas is taken out and the other two Q, and are found to be economically infeasible by the sorting procedure. The search therefore is conducted only on these two capacitors and as a result the last node containing the capacitors QCl8 and Qc52 is identified as the solution node. The total run time for this test on VAX is recorded as 165 sec. of CPU and 8 sec. of VO.
137.;; 1 3-y 1 1 200
Table 4: Test run results for TS2 - Fixed Capacitor placement
We have the followmg comments/observations about these test RSults. 1. Search Procedure :
0 The order of search is about nc (number of capacitors placed); which is much less than the worst case bound of nc '.
The search converges to the global optimal point in both tests; although in test 1, for example, there are some other suboptimal solutions with revenues close to each other. 0 As search goes further down to higher levels, more capacitors are taken out to increase the revenues. This causes less energy loss reduction and lower voltage profile. This also shows the sensitivity of the optimal point with respect to the cost figures k, , r, , c.
0 The convergence of slave problem at the root (the initial one) gets slower as the number of load levels increase (the number of itera- tions is in the order of 2-3 nc ). But the other slave problems con- verge much faster (in the order of 1-2 nc ); mainly because it is easier to find a good initial point for them by simply using the results of the parent node.
2. The slave problem - fixed capacitor problem:
Although there are regulators at the substations, voltage profiles of both systems are below lower voltage limits at the peak load level before the capacitor placement. The solution for TSl indicates that the capacitors are needed to be used for raising the voltage profile of the system at the peak load level ( Vmh = 0.9 P.u.) as well as for loss reduction. The solution for TS2 corresponds to the unconstrainted optimal point ( Vmh=0.9O7 P.u.); indicating that maximum loss reduction is achieved.
2. Switched Capacitor Case We now present the test runs for the switched capacitor problem on
the same test systems. The optimal places obtained from the general fixed capacitor problem tests are used in these tests also to avoid the search. Test System 1
The first step of the test runs for TS1 is the solution of the corresponding slave problems SWi , i = 1,2 to get the optimal settings for the off-peak load levels assuming no limit on the size of the capacitors. Each of such solution is obtained in about less than 2 nc iterations by cal- ling the sizing problem Subroutine. In the second step of the test, the main problem, SW, is solved in about 3 nc iterations by using the solution pro- cedure described in section 5. In the third step, convergence checks indi- cate that the setting of QC5 for the first load level is binding, i.e.,
= Q,"S. This constitutes the end of the first iteration. Another iteration is performed to see if the binding status of the capacitors will change. The convergence is obtained at the second iteration when the capacitor set- tings, Q2 , Q:, are updated and it is found that there is no binding status change.
The solution is summarized in Table 5. The box in the table is simi- lar to that of fixed capacitor case; except here, in addition to the capacitor sizes, Q:, capacitor settings at the off-peak load levels, d: , d:, are given also. The total run takes about 12 sec. of CPU and 2 sec. of VO on VAX.
Table 5 : Test run results for TSl - Switched Capacitor case.
QZ Q: Q," bus rev. AE Vmin
Test System 2 Test run for TS2 is similar to that of TSl. The solution is obtained in two iterations and found out that only the setting of Qc18 for the first load level is binding, i.e., QJ18 = The solution is summarized in Table 6. The total run takes about 82 sec. of CPU and 4 sec. of VO on VAX.
Table 6 : Test run results for TS2 - Switched Capacitor case
QZ Q,' Q," bus rev. I 39180 I 197 330 330 I 18
We have the following comments about the test results:
0 The performed test runs indicate very good convergence charac- teristics; the number of iterations between the main problem, SW, and the subproblems, SWi is usually one or two. 0 When the fixed capacitor and the switched capacitor test results are compared, it is Seen that: (i) switched capacitor placement yields Egher revenues and higher capacitor sizes, especially when the cost data is the same for both cases and the load variations are diverse. (ii) the voltage profile is higher ( Vmh is higher) and voltage regula- tion is better for the switched capacitor case. This is because of the fact that for the switched capacitor case better compensation is achieved by adjusting the value of capacitors as the load changes. The final point to be noted about the overall test results is about the
effect of the regulators on the solution; (i) feasibility becomes less of a problem, (ii) no upper limit voltage violation has been observed for the given test systems, although the load levels were quite diverse. (iii) as exemplified here by test run for TSl, capacitors can be used together with voltage regulators to keep the voltage profile of the system within defined limits.
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73 I
Analysis of the Results Consider the starting point of TS2 in Table 4, where the fixed cost of
capacitors m not included. The solution of capacitor sizing gives 1040 kvar and 210 kvar on the nodes 52 and 47 of a lateral respectively, and 170 kvar and 230 kvar on the nodes 11 and 18 of the main feeder respectively. In the system, the loads in the laterals are more concentrated whereas the loads in the main feeder are more evenly distributed. The result is that the size of the capacitors are also more concentrated in the lateral and more evenly distributed in the main feeder. This further reaffirms the fact that the nature of reactive power compensation is rather local.
W. CONCLUSIONS In this paper, a general formulation and an efficient solution metho-
dology have been developed for general capacitor placement problem on radial distribution systems.
The general capacitor placement problem consists of placing the capacitors (determining their number and the locations) and determining their types and sizes. The objective is peak power and energy loss reduc- tion while keeping the cost of capacitors at a minimum.
The proposed formulation is comprehensive in the sense that: (i) it considers all the variables of the problem stated above, (ii) it uses the ac power flow equations to represent the system, (iii) voltage constraints are taken into account.
A solution method has been developed for this general problem by decomposing the problem into two hierarchical levels. The top level prob- lem. called the master problem, is an integer programming problem and is used to place the capacitors (determine their number and locations). An efficient search scheme has been developed for the master problem. The second level problem, called the slave problem, is used by the master problem as a subroutine. This problem is further decomposed into two levels: at the top level, the problem consists of determining the type of capacitors and at the bottom level, the problem is to determine the capaci- tor sizes once the capacitors are placed with their types assigned. These slave problems, called the fixed and switched capacitor problems, are shown to be either the base type or can be decomposed into base type problems. The base type problem is a capacitor sizing problem and is solved by using an efficient phase I - Phase I1 type solution method presented in [ 141.
Test results are presented for the proposed solution scheme. They indicate. that the method is computationally efficient and the decomposi- tion scheme performs well.
Although not implemented, it is also shown that the formulation and the solution methodology presented in this paper can be generalized to include the voltage regulators in the problem.
Acknowledgements We thank Mr. Wayne Hong and Dr. Dariush Shirmohammadi of
Pacific Gas and Electric for their helpful discussions. This research is sup- ported by TUBITAK-TURKEY and by National Science Foundation under grant ECS-8715132.
REFERENCES [l]
[2]
Distribution Systems, East Pittsburgh, PA. : Westinghouse Electric Corp., 1965 R. F. Cook, "Optimizing the Application of Shunt Capacitors for Reactive Volt-Ampere Control and Loss Reduction" , AIEE Trans.,
Y. G. Bae, "Analytical Method of Capacitor Application on Distribu- tion Primary Feeders", IEEE Trans. on Power Apparatus and Sys- tems, vol. 97, pp. 1232-1237, July/Aug. 1978. H. Duran, "Optimum Number, Location, and Size of Shunt Capaci- tors in Radial Distribution Feeders: A Dynamic Programming Approach, IEEE Trans. on Power Apparatus and Systems, vol. 87, pp. 1769-1774, Sept. 1968.
vol. 80, pp. 430-444, August 1961. [3]
[4]
M. Ponnavaikko and K. S. Prakasa Rao, "Optimal Choice of Fixed and Switched Shunt Capacitors on Radial Distribution Feeders by the Method of Local Variations", IEEE Trans. on Power Apparatus andsystem, vol. 102, pp.1607-1614, June 1983. T. H. Fawzi. S. M. El-Sobki, and M. A. Abdel-Halim, "A New Approach for the Application of Shunt Capacitors to the Primary Distribution feeders", IEEE Trans. on Power Apparatus and Sys- tems, vol. 102, pp.10-13, Jan. 1983. M. Kaplan, "Optimization of Number, Location, Size, Control Type, and Control Setting of Shunt Capacitors on Radial Distribution Feeders", IEEE Trans. on PAS, vol. 103, pp.2659-2665, Sept. 1984. J. J. Grainger, and S. H. Lee, "Optimum Size and Location of Shunt Capacitors for Reduction of Losses on Distribution Feeders", IEEE Trans. on Paver Apparatus and Systems, vol. 100, pp. 1105-1 118, March 1981. J. J. Grainger, and S. H. Lee, "Capacity Release by Shunt Capacitor P h x t w U on Distribution Feeders: a New Voltage Dependent Model", IEEE Trans. on Power Apparatus and Systems. vol. 101. pp. 1236-1244, May 1982. J. J. Grainger, S. H. Lee, and A. A. El-Kib, "Design of a Real-Time Switching Control Scheme for Capacitive Compensation of Distribu- tion Feeders", IEEE Trans. on Power Apparatus and Systems, vol.
J. J. Grainger, S. Civanlar, and K. N. Clinard, L. J. Gale, "Optimal Voltage Dependent Continuous Time Control of Reactive Power on Primary Distribution Feeders", IEEE Trans. on Power Apparatus andsystem, vol. 103, pp. 2714-2723, Sept. 1984. A. A. El-Kib, J. J. Grainger, and K. N. Clinard, L. J. Gale, "Place- ment of Fixed and/or Non-Simultaneously Switched Capacitors on Unbalanced Three-phase Feeders Involving Laterals", IEEE Trans. on Paver Apparatus and Systems, vol. 104, pp. 3298-3305, Nov. 1985. S. Civanlar and J. J. Grainger, "Volt/Var Control on Distribution Systems wilh Lateral Branches Using Shunt Capacitors and Voltage Regulators: Part I, Part 11, Part III", IEEE Trans. on Power Apparatus andsystems , vol. 104, pp. 3278-3297, Nov. 1985. M. E. Baran and F. F. Wu, "Optimal Sizing of Capacitors Placed on a Radial Distribution System". submitted to IEEE PES winter meeting, 1988. A. M. Geoffrion, "Elements of Large Scale Mathematical Program- ming, Part I: Concepts", Management Science, vol. 10, pp. 652-675. July 1970. A. M. Geoffrion, "Generalized Benders Decomposition", JOTA, vol.
E. M. Reingold and W. J. Hansen. Data Structures. Little Brown and Comp., 1983.
101, pp. 2420-2428, August 1982.
10, pp. 237-260, April 1972.
Appendix: Decomposition of the Switched Capacitor Problem We shall apply decomposition techniques to the switched capacitor
problem introduced in section 4.2. We present first a general scheme, which leads to the Benders Decomposition, and then a simpler, heuristic based decomposition scheme for the special switched capacitor problem.
Decomposition A general decomposition scheme is given in [15]. To adopt the
derivation for the switched capacitor problem, Psw , we first re-group the variables and constraints as follows,
nf T U 1
Uo= { uo I Eq.(15) is satisfied ) U = { U I Eq.(16) is satisfied ) ;
Similarly, we partition the objective function as,
k r C I i=l
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132
Then Psw can be re-written as
min if, =f,(u?+fr(u) I U O E U , ; U E U ; F(U',U)L;O l(a.1) u9u
Where, F(uo,u) comsponds to the coupling constraints of Eq.(17).
onto space of uo alone as follows. We start the decomposition by projecting (purtitioning) the problem
(a.2)
We assume that the subproblem in the braces,
f s (U? = :$ i f r (u)-F(uo*u) 5 0 1
can be evaluated for a given U' as an optimization problem with respect to the variable U. Then the main problem becomes
min f0 ('4 =fm ('9 +fJ ('4 6.3) U W O
W O
To assure that the above assumption holds, we must avoid the values of U' such that f,@) does not have a feasible solution. For this, we define a new set. V as follows.
V = {U' I 5 UEU s.t. F(u0,b) I O }
The set V can be thought of as the projection of the constraint set defined by F(uo,u) and U onto the space defined by U' alone as illustrated in Fig.a.1.
Figure a.1 : Set hOjeCtiOn Now we generalize the projection by rewriting the problem (a.3) as fol- lows.
min { f,,,(u") +f,(u") I U'E n v 1 (a.4) U0
m0 For the switched capacitor problem, we have a conjecture that
U, c v. Justification of this conjecture will be discussed later.
Assuming that this conjecture holds for the general case, we can use the partitioned problem (a.3) rather than (a.4) for the main problem, SW, . The subproblem f, (U? can further be decomposed into nt subproblems of the following form.
The solution algorithm involves the iterative solution of the main problem sw& and the subproblems, SW;. After each iteration, a new set of constraints of the type (a.5.i) is added to Swab until the solution con- verges.
It is noted that the Benders decomposition has the following features. (i) The main problem, SW, is not a base type problem. Additional con- straints of (a.5.9, called the cuts, are difficult to handle with the solution algorithm used by the bask problem; one has to identify which one these constraints will be binding during the solution of web. (ii) The contribution of the subproblems to the objective of the main prob- lem, fJi(u? is approximated by linearizing this term around the previ- ously calculated points, uoJ . To see this, let the binding constraint for the subproblem i be the k'th one in Eq.(aS.i). Then the solution for the correspondingyi will be
Therefore, the approximation is good only if the actual solution point, U' is close to the calculated point U* in solving Sw&.
A Heuristic Based Decomposition Scheme Note that the main problem of (a.3) need not be transcribed into
Benders form, SW, if O n e can estimate the binding constraints in the sets U' - U' L; 0. We develop a solution algorithm based on this principle. The solution scheme uses the sizing algorithm to solve both the main problem and the subproblems and it also uses a better estimate for the contribution of the subproblems to the objective of the main problem. The details of the algorithm is given in section 4.2.
Justification of the Conjecture U, c v
The idea behind this conjecture is as follows. Let uo E U,. This means that U' amount of reactive power compensation from the capacitors is enough to satisfy the voltage constraints for the peak load. But this amount of compensbtion must suffice to have a feasible point for lower load levels too because the lower the load the higher the voltage profile will be. This observation is due to the strong coupling between the reac- tive power flow and the voltage profile of the system.
Note that this conjecture was also the underlining idea behind the assumption made when formulating the problem in section 2; where, it was assumed that the capacitor size will be the capacitor setting for the peak load level, uo and capacitor settings for all the other lower load lev- els, U' will be smaller than U'.
f s i (U" = ic[fri (U') Mesut E. Baran received his B.S. :ind M.S. from Middle East Technical University, Turkey. He is currently a Ph.D. student at the University of California, Berkeley. Felix F. Wu received his B.S. from National Taiwan University, M.S. from the University of Pittsburgh, and Ph.D. from the University of Cal- ifornia, Berkeley. He is a professor of Electrical Engineering and Com-
S.t. ui -U010
U' EUi
Where, Ui = { ui I Eq.(I6) } and f, (u? = us, (U?.
The explicit SWi i = 1. . . . , nt are given in "4.2.
Benders Decomposition Assuming that the duality conditions holds for the switched capacitor
problem of (a.2) [16], the main problem can be transcribed into the follow- ing form.
m o b
of the main problem Swo and the subproblems puter Sciences at the University of Califomia, Berkeley.
min fo (u'.Y) = f d (U? + Cyi s.t. yi + 3 c " r ~ u o - u o ~ ~ ~ ~ , i ~ u o ~ ~ j = I , . . . .p (a.5.i)
uo E U, i = 1.. . . .nt (a.5.ii)
Where, U" and I.'' corresponds to the solution of the subproblem SW, at iteration j.
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133
the optimal location of the one capacitor bank needed at minimum load.
2) Increment the load in steps. At each load level ask the question, “Is more capacitance needed?” If so, then determine optimal switched capacitor locations and sizes based primarily on loss and voltage profile improvement.
3) Once peak load is reached, decrement the load to determine switch-off load levels.
This very simplistic method of locating and sizing capacitors incremen- tally tends to distribute smaller size banks over the feeder where they are needed and when they are needed. Generally, no more than one capacitor is added at a load level and the search for optimum location is trivial. My simulations of the feeder over a year’s time considering daily and seasonal load cycles indicate that it is difficult to significantly improve upon this method economically. One can always find a solution that appears to be a few percentage points better, but I do not think that the basic data are known with sufficient accuracy to quibble over a small difference in an off-line analysis. It would require on-line control to take advantage of the small gains possible. We have implemented this method in an interactive program that uses the above algorithm to get close to the most economical solution. Then the user can tweek the solution interactively taking into account practical considerations. The whole process takes but a few minutes using a personal computer (E).
Differing approaches to the problem of capacitor size and placement will yield differing “optimal” solutions. 1 suspect that none are truly optimal and most approaches that consider load variation are generally adequate (methods that optimize only peak load sometimes give poor results when the entire load cycle is considered). I would hesitate to defend one approach over another too strongly because feeder load varies somewhat randomly and it would be difficult to prove which is more optimal. However, I think that methods like I have described, which are based on how the feeder operates, are apt to be more optimal more of the time. They are also simple to program and the programs execute quickly. Therefore, I question the practicality of abandoning the simpler approach in favor of a more sophisticated method like the authors have presented.
Reference
[l] T. Frantz et al., “Load behavior observed in LILCO and RG&E systems,” ZEEE Trans. Power App. Syst., April 1984, pp. 819-831.
Manuscript received February 19, 1988
Discussion
Roger C. Dugan (McGraw-Edison Power Systems, Cooper Industries Inc., Canonsburg, PA): The reader will discover in a few moments that I have several objections to the methods presented in this paper. However, 1 do not wish for my objections to reflect poorly upon the efforts of the authors. It would appear that a great deal of good work has been done and I suspect that this paper was intended to emphasize the application of more sophisticated techniques to the problem. Therefore, it may be unfair to expect the authors to respond to all of my objections because they are based on more practical aspects and also apply to previous investigators in this field who have made similar assumptions.
First, one would hope that a method employing nonlinear programming techniques would be faster than simple exhaustive searches. It is not apparent from the paper that this would be the case, and the times quoted during the paper’s presentation and discussion lead me to believe otherwise. The number of discrete solutions to this problem are not necessarily large due to the practical constraints that I will mention below. Therefore, an intelligent exhaustive search method that automatically discards many cases due to its knowledge of the way the feeder operates can be reasonably fast. I have investigated a number of search techniques that give very good, but perhaps not optimal, solutions, and the execution times grow approximately linearly with problem size rather than geometrically.
The method proposed in the paper is based on the assumption that the feeder is discrete and the capacitor size is continuous. Perhaps, a more realistic choice would have been the opposite. There are many nodes in a circuit, but utilities generally wish to consider only two or three different capacitor sizes, for example 600- and 1200-kvar banks. I have generally approached the problem by assuming that both the feeder and capacitor sizes are discrete. When considering such a s d number of sizes, a simple search can often be done quickly.
Another assumption I would question is that the capacitor is a source of reactive power (Q). (I assume that this implies a constant source because I am not able to ascertain otherwise from the paper.) Of course, this will introduce inaccuracies because a capacitor is a constant impedance element. Since capacitor placement on a feeder significantly affects the voltage, this assumption weakens any claim that the method results in an optimal location. It would seem to be a simple matter to correctly represent capacitors and avoid this difficulty.
My examination of this problem has also indicated that line regulator tap position and control characteristics affect the “optimal” solution. It is not clear how the method presented in the paper properly accounts for these effects.
The constant P-Q load model employed in the proposed method also leads to inaccuracies. The P-Q load model is a peculiar bias of transmission analysts, and it should not be employed on distribution systems without question. Frantz et al. [I] have clearly shown that distribution system loads are sensitive to voltage. My experimentation with different load models has shown that one will usually get a different “optimal” solution for each load model. The P-Q load model is best employed to establish the base case voltages from known load conditions. Then one should switch to a more realistic model when studying capacitor additions. In the absence of better information, I will typically use a load model in which the P varies linearly with voltage and the Q varies by the square of the voltage. Lacking this capability, I suspect that a simple constant impedance load model would be better than a constant P-Q model.
Differing economic evaluation criteria among utilities require different approaches to the optimization problem. Utilities using very high values for released substation and generation capacity savings may achieve a more economical solution by optimizing the peak load condition first, although I must admit to being skeptical of this. Then the load is decremented in steps to determine when switched banks should be turned off. For utilities where the costs of losses is more important, a more economical solution can generally be achieved by fist optimizing the location of fixed banks at minimum load and then incrementing the load in steps to determine the location, size, and switching levels of the switched banks.
I believe that the latter approach is more practical for most utilities. One reason is that most feeders operate near minimum or average load levels much more than they operate near peak loads. Another reason is that I question whether values for substation and generation costs are based on assumptions compatible with assumptions made for capacitor economic evaluations. This approach can be easily programmed using an intuitive algorithm that recognizes how a feeder typically operates. It yields a near- optimal solution that is difficult to improve upon significantly. I will state the algorithm in words, giving the reader the freedom of choice in selecting techniques for solving the load flow and making decisions.
1) Select the fixed capacitors. This frequently is simply the selection of
M. E. Baran and F. F. Wu : We would like to thank Mr. Dugan for his interest in the paper and his insightful questions about the capacitor place- ment problem.
The algorithm outlined by Mr. Dugan is a special case of switched capacitor problem introduced in this paper. In Sec. 4.2 it is shown that assuming all the capacitors are of switched type, and for a given set of capacitors placed on the system, the problem can be decomposed into smaller subproblems each of which corresponds to minimization of losses at each of the load levels considered. However, these subproblems are coupled to each other due to the cost of capacitors. It is easy to show that when the cost of capacitors are neglected the subproblems become independent and hence the optimization for each load level can be carried out independently. Mr. Dugan’s algorithm incorporates both placement and sizing problems into this decomposition scheme. It is indeed a good idea especially since he considers the capacitor sizes as discrete. There- fore, his algorithm will work, as he points out, when the objective function is power loss minimization or voltage regulation.
However, for the general case, where the cost of capacitor is impor- tant and it effects the number of capacitors to be placed, the method may not give good answers because of the coupling between the subproblems. Hence, in this case, it is not easy to answer the question how capacitors should be added as the load level increases. It seems that the best aid to answer this question would be the use of switched capacitor problem. Starting from a candidate set of capacitors and assuming them all switch- able the problem can be solved by using the switched capacitor algorithm introduced in. this paper. The solution will give the capacities of the capa- citors, U” and their settings at other load levels, ui , i = 1, . . . , nt assuming the capacities are continuous variables. These results then can be used in answering the question mentioned above and hence the capacitors can be placed by the method proposed by h4r. Dugan. This way the search intro-
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134
duced in this paper may not need to be canid out any further than the mot.
Another point raised by Mr. Dugan is the assumption made in modeling the capacitors. It involves approximating the reactive power injected by a capacitor as constant, independent of the voltage. This assumption is justified based on the fact that V, = 1 pa. and sizes of capa- citors determined by the solution need to be munded off to get the practi- cal size of capacitors. However, the exact model can be incorporated in the method if needed. This is explained in the closure of [14]. Note that this approximation will most likely affect the sizes, not the location, of capacitors.
The power Row model used in this paper (DistFlow equations) can handle voltage dependent loads and the solution algorithm can be general- ized to take into account such loads, as explained again in the closure of ~ 4 1 .
Manuscript received May 2 , 1988.
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Apéndice J
“SIMPLE AND EFFICIENT COMPUTER ALGORITHM TO SOLVE
RADIAL DISTRIBUTION NETWORKS”, RANJAN, R. ET AL. 2003, [27]
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Simple and Efficient Computer Algorithm to Solve Radial Distribution NetworksRakesh Ranjan a; D. Das b
a Multi Media University, Ayer Keroh Lama, Melaka, Malaysia. b Electrical Engineering Department, IndianInstitute of Technology, Kharagpur, India.
Online Publication Date: 01 January 2003
To cite this Article Ranjan, Rakesh and Das, D.(2003)'Simple and Efficient Computer Algorithm to Solve Radial DistributionNetworks',Electric Power Components and Systems,31:1,95 — 107
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Electric Power Components and Systems, 31:95–107, 2003Copyright c© 2003 Taylor & Francis1532-5008/03 $12.00 + .00DOI: 10.1080/15325000390112099
Simple and Efficient Computer Algorithm toSolve Radial Distribution Networks
RAKESH RANJANMulti Media UniversityAyer Keroh LamaMelaka, Malaysia
D. DASElectrical Engineering DepartmentIndian Institute of TechnologyKharagpur, India
A simple and efficient algorithm is presented to solve radial distribution net-works (RDN). It solves the simple algebraic recursive expression of voltagemagnitude and all the data are stored in vector form. The algorithm uses thebasic principle of circuit theory and can be easily understood. The proposedalgorithm has been tested with several distribution networks and results arecompared with two other existing methods. The effectiveness of the proposedalgorithm is demonstrated through two examples.
Keywords radial distribution networks, load flow, circuit model
Nomenclature
NB total number of the loadLN 1 total number of the branch (LN 1 = NB − 1)PL(i) real power load of ith nodeQL(i) reactive power load of ith node|V (i)| voltage magnitude of ith nodeR(jj) resistance of the branch–jjX(jj) reactance of the branch–jjZ(jj) impedance of the branch–jjI(jj) current flowing through branch–jjP (m2) total reactive power load fed through node m2Q(m2) total reactive power load fed through node m2∠δ(m2) voltage angle of the node m2LP(jj) reactive power loss of branch–jjLQ(jj) reactive power loss of branch–jj
Manuscript received in final form on 16 January 2002.Address correspondence to R. Ranjan. E-mail: [email protected]
95
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96 R. Ranjan and D. Das
IS (jj) sending end node of branch–jjIR(jj) receiving end node of branch–jjPLOSS total reactive power lossQLOSS total reactive power loss
1. Introduction
In the past few years, the developments of automated distribution systems solu-tions have increased considerably. With the development of the microcomputer,the requirement of distribution substation-owned computer programs has becomea necessity. However, the choice of a solution method for the practical application isdifficult. It requires a careful analysis of comparative advantages and disadvantagesof those methods available in respect to storage, computation speed, and conver-gence criterion. Generally, radial distribution networks has a high R/X ratio. Dueto this, conventional Newton Raphson [1] and fast decoupled load flow [2] methodsfail to converge. Many other researchers [3–5] have suggested modified versions ofconventional load flow methods with a high R/X ratio.
Kersting and Mendive [6] and Kersting [7] have developed load-flow techniquesbased on ladder theory and Stevens et al. [8] modified it and proved faster thanearlier methods. However, it fails to converge in five out of twelve case studies.Baran andWu [9] have developed a load-flow method based on the Newton-Raphsonmethod but it requires a Jacobian matrix, a series of matrix multiplications, andat least one matrix inversion. Hence, it is not computationally efficient. Chiang [10]has developed decoupled and fast decoupled load-flow methods based on a methodsuggested by Baran and Wu [9]. The very fast decoupled method is impressivebecause it does not require any Jacobian matrix. Many other researchers [1–15] haveproposed generalized methods of modeling and analysis of distribution systems.However, the difficulty arises from the fact that no method posseses all the desirablefeatures.
In this article, a simple algorithm that is based on basic systems analysismethods and circuit theory is developed. The purpose of this article is to develop anew calculation model that requires less computer memory and is computationallyfast for radial distribution networks. The proposed method involves only recursivealgebraic equations to be solved to get the following information:
1. Status of the feeder line, overloading of the conductor and feeder line cur-rents;
2. Whether the system can maintain adequate voltage level for the remoteloads;
3. The line losses in each segment;4. Suggestion of the necessity of rerouting or network reconfiguration for the
existing distribution networks.
The proposed method is compared with those of Das et al. [12] and Baran andWu [9]. It is observed that the proposed algorithm is computationally very effi-cient. Several distribution networks have been tested with this algorithm with theconsideration that all loads are constant power. However, the algorithm can easilyaccommodate composite load modeling, if the composition of the load is known.The algorithm has a good convergence property for practical radial distributionnetworks.
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Algorithm for Radial Distribution Networks 97
Figure 1. Sample radial distribution net-works.
2. Circuit Model
In this section, a circuit model of a radial distribution network (RDN) is presented.It is assumed that a three-phase RDN is balanced and can be represented byequivalent single line diagram. Line shunt capacitance at distribution voltage levelis negligibly small. Figure 1 shows single line diagram of a sample radial distributionnetwork.
The electrical equivalent of Figure 1 is shown in Figure 2.
3. Mathematical Model of Radial Distribution Networks
A mathematical model of radial distribution networks can easily be derived fromFigure 2.
I(jj) =|V (m1)|∠δ(m1) − |V (m2)|∠δ(m2)
Z(jj)(1)
and
P (m2) − jQ(m2) = V ∗(m2) ∗ I(jj) (2)
where Z(jj) = R(jj)+X(jj), m1, and m2 are the sending and receiving end nodes,respectively, {m1 = ISS (jj) and m2 = IRR(jj)}.P (m2) = sum of the real power loads of all the nodes beyond node m2 plus the
real power load of the node m2 itself plus the sum of the real power losses ofall the branches beyond node m2.
Q(m2) = sum of the reactive power loads of all the nodes beyond node m2 plus thereactive power load of the node m2 itself plus the sum of the reactive powerlosses of all the branches beyond node m2.
Figure 2. Electrical equivalent of one branch of Figure 1.
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98 R. Ranjan and D. Das
Table 1Branch number, sending end node, and receiving end
nodes of Figure 1
Branch no. Sending end node Receiving end node(jj) ISS (jj) IRR(jj)
1 1 22 2 33 3 44 4 55 2 86 8 97 9 108 3 69 6 710 9 11
From equations (1) and (2) we get
|V (m2)| =√
{B(jj) − A(jj)} (3)
where
A(jj) = P (m2) ∗ R(jj) + Q(m2) ∗ X(jj) − 0.5 ∗ |V (m1)|2 (4)
B(jj) =√
{A2(jj) − {Z2(jj) ∗ (P 2(m2) + Q2(m2))} (5)
Since the substation voltage magnitude |V (1)| is known, for jj = 1, (seeTable 1), m1 = ISS (jj) = ISS (1) = 1 and m2 = IRR(jj) = IRR(1) = 2.A(jj) = A(1) and B(jj) = B(1) can be computed using equations (4) and (5), if weknow P (m2) = P (2) and Q(m2) = Q(2). After that |V (2)| can easily be computedfrom equation (3). Similarly for jj = 2, (see Table 1), m1 = ISS (jj) = ISS (2) = 2,m2 = IRR(jj) = IRR(2) = 3. A(jj) = A(2), B(jj) = B(2) can be computedusing equations (4) and (5), if we know P (m2) = P (3) and Q(m2) = Q(3) andhence |V (3)| can easily be computed by using equation (3). In general, for jj =1, 2, . . . ,LN 1, voltage magnitude of all the nodes can easily be computed by usingequations (4), (5), and (3), if we know P (m2) and Q(m2) for m2 = 2, 3, . . . ,NB .Computation of P (m2) and Q(m2) (m2 = 2, 3, . . . ,NB) is explained in section 4.
Real and reactive power losses in the branch jj are
LP(jj) =R(jj) ∗ (P 2(m2) + Q2(m2))
|V (m2)|2 (6)
LQ(jj) =X(jj) ∗ (P 2(m2) + Q2(m2))
|V (m2)|2 (7)
4. Computation of P (m2) and Q(m2)
A computer logic is presented in this section for automatic computation of P (m2)and Q(m2). For the requirements of the computer logic presented, in Table 1, branch
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Algorithm for Radial Distribution Networks 99
number, sending end, and receiving end nodes of the feeder shown in Figure 1 aregiven.
First set IR(jj) = IRR(jj) and IS (jj) = ISS (jj) for jj = 1, 2, . . . ,LN 1.Now for jj = 1 (first branch of Figure 1 and Table 1), IR(jj) = IR(1) = 2,and set P (2) = PL(2), Q(2) = QL(2). Now computer logic will check whetherIR(1) = IS (i) or not for i = 1, 2, . . . ,LN 1. It is seen that (Table 1) for i = 2(Branch 2), IR(1) = IS (2) = 2 and for i = 5 (Branch 5), IR(1) = IS (5) = 2,and corresponding receiving end nodes are IR(2) = 3 and IR(5) = 8. Now it willcompute
P (2) = P (2) + PL(3) + PL(8) + LP(2) + LP(5)
= PL(2) + PL(3) + PL(8) + LP(2) + LP(5)
and
Q(2) = Q(2) +QL(3) +QL(8) + LQ(2) + LQ(5)
= QL(2) +QL(3) +QL(8) + LQ(2) + LQ(5)
Proposed computer logic will again check whether nodes 3 and 8 are connected withthe other nodes. It is seen that node 3 is connected with node 4 (branch 3) andnode 6 (branch 8). Node 8 is connected with the node 9 (branch 6). Now computerwill compute
P (2) = P (2) + PL(4) + PL(6) + LP(3) + LP(8) + PL(9) + LP(6)
= PL(2) + PL(3) + PL(8) + LP(2) + LP(5) + PL(4)
+ PL(6) + LP(3) + LP(8) + PL(9) + LP(6)
and
Q(2) = QL(2) +QL(3) +QL(8) + LQ(2) + LQ(5) +QL(4)
+QL(6) + LQ(3) + LQ(8) +QL(9) + LQ(6)
Similarly, computer logic will check whether nodes 4, 6, and 9 are connectedwith the other nodes. It is seen that 4 is connected with node 5 (branch 4), node 6is connected to node 7 (branch 9) and node 9 is connected with node 10 (branch 7)and node 11 (branch 10). Therefore,
P (2) = P (2) + PL(5) + LP(4) + PL(7) + LP(9) + PL(10) + LP(7)
+ PL(11) + LP(10)
P (2) = PL(2) + PL(3) + PL(8) + LP(2) + LP(5) + PL(4) + PL(6)
+ LP(3) + LP(8) + PL(9) + LP(6) + PL(5) + LP(4) + PL(7)
+ LP(9) + PL(10) + LP(7) + PL(11) + LP(10)
For jj = 2 (branch 2), IR(jj) = IR(2) = 3 (from Table 1), set P (3) = PL(3)and Q(3) = QL(3). Now computer logic will check whether IR(2) = IS (i) fori = 1, 2, . . . ,LN 1. It is seen that (Table 1) for i = 3 (branch 3), IR(2) = IS (3) = 3
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100 R. Ranjan and D. Das
and for i = 8 (branch 8), IR(2) = IS (8) = 3 and correspondingly receiving endnodes are IR(3) = 4 and IR(8) = 6. Now it will compute
P (3) = P (3) + PL(4) + PL(6) + LP(3) + LP(8)
= PL(3) + PL(4) + PL(6) + LP(3) + LP(8)
Similarly,
Q(3) = Q(3) +QL(4) +QL(6) + LQ(3) + LQ(8)
= QL(3) +QL(4) +QL(6) + LQ(3) + LQ(8)
Proposed computer logic will again check whether nodes 4 and 6 are connectedwith any other node. It is seen that (Table 1) node 4 is connected with node 5 andnode 6 is connected with node 7, i.e., i = 4, IR(4) = 5 and i = 9, IR(9) = 7.
Therefore,
P (3) = P (3) + PL(5) + PL(7) + LP(4) + LP(9)
= PL(3) + PL(4) + PL(6) + LP(3) + LP(8)
+ PL(5) + PL(7) + LP(4) + LP(9)
and
Q(3) = Q(3) +QL(5) +QL(7) + LQ(4) + LQ(9)
= QL(3) +QL(4) +QL(6) + LQ(3) + LQ(8)
+QL(5) +QL(7) + LQ(4) + LQ(9)
Exact computation of P (3) and Q(3) is complete because nodes 5 and 7 are notconnected with any other node. Similarly, computer logic will compute exact loadfed through each node. It is to be noted here that if the receiving end node of anybranch is an end node, then total load fed through that node is the load of nodeitself. For example, consider node 5 of Figure 1, which is an end node. Therefore,P (5) = PL(5) and Q(5) = QL(5). The proposed algorithm will also identify theend nodes. It is worth reporting that node numbering of the feeder is arbitrary andthe algorithm will automatically compute the exact load fed through all the nodes.Initially, if LP(jj) and LQ(jj) are set to zero for all jj, then the initial estimateof P (m2) and Q(m2) (m2 = 2, 3, . . . ,NB) will be the sum of the loads of all nodesbeyond node m2 plus the load of m2 itself.
5. Load Flow Computation of RDN
Once P (m2) and Q(m2) are computed by the above logic (section 4), voltagemagnitudes of all the nodes can be easily computed using equation (3). Further realand reactive power losses are obtained using equations (6) and (7). The completeload flow algorithm is shown in the form of a flow chart in Figure 3.
The convergence criterion of the algorithm is that if, in successive iteration,the difference of real and reactive power delivered from the substation is less than0.1kW and 0.1kVAr, respectively, the solution has converged.
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Algorithm for Radial Distribution Networks 101
(a)
Figure 3. (a) Flow chart of load flow technique (continues).
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102 R. Ranjan and D. Das
(b)
Figure 3. (b) Flow chart of load flow technique.
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Algorithm for Radial Distribution Networks 103
6. Example
To demonstrate the effectiveness of the proposed algorithm, two examples areselected. The first example is a 69-node radial distribution network [9] as shown inFigure 4 in the appendix. The system data is available in [9]. Results of the loadflow study are tabulated in Table 2.
For the load flow study, we have considered the following:
substation voltage = 12.66 kV,base kVA = 10 and base voltage = 12.66 kV.
The second example is a 33-node radial distribution network [14]. Data aregiven in Table 5 in the appendix. Load flow results of the system are given inTable 3.
Comparison of relative speed and memory requirements is given in Table 4below.
Table 2Load flow result of 69-node radial distribution network
Voltage VoltageNode magnitude Node magnitude
number (p.u.) number (p.u.)
1 1.00000 36 0.999922 0.99997 37 0.999753 0.99993 38 0.999594 0.99984 39 0.999545 0.99902 40 0.998846 0.99009 41 0.998847 0.98079 42 0.998558 0.97858 43 0.998509 0.97745 44 0.9985010 0.97245 45 0.9984111 0.971135 46 0.9984012 0.96819 47 0.9997913 0.96526 48 0.9985414 0.96237 49 0.9947015 0.995950 50 0.9941516 0.95897 51 0.9785417 0.95809 52 0.9785318 0.95808 53 0.9746619 0.95761 54 0.9714220 0.95732 55 0.9669421 0.95683 56 0.9625722 0.95683 57 0.9401023 0.95676 58 0.9290424 0.95660 59 0.9247625 0.95643 60 0.91974
(continued)
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104 R. Ranjan and D. Das
Table 2(Continued)
Voltage VoltageNode magnitude Node magnitude
number (p.u.) number (p.u.)
26 0.95636 61 0.9123427 0.95634 62 0.9120528 0.99993 63 0.9116629 0.99973 64 0.9097630 0.99985 65 0.9091931 0.99971 66 0.9712932 0.99961 67 0.9712933 0.99935 68 0.9678634 0.99901 69 0.9678635 0.99895
Total real power loss = 224.9606 kW; total reactive powerloss = 102.147 kVAr; minimum voltage observed at node 65,|V65| = 0.90919 p.u.
Table 3Load flow result of 33-node radial distribution network
Voltage VoltageNode magnitude Node magnitude
number (p.u.) number (p.u.)
1 1.00000 18 0.903772 0.99703 19 0.996503 0.98289 20 0.992924 0.97538 21 0.992215 0.96796 22 0.991586 0.94948 23 0.979317 0.94595 24 0.972648 0.93230 25 0.969319 0.92597 26 0.9475510 0.92009 27 0.9449911 0.91922 28 0.9335412 0.91771 29 0.9253213 0.91153 30 0.9217714 0.90924 31 0.9176015 0.90782 32 0.9166916 0.90643 33 0.9164017 0.90439
Total real power loss = 210.998 kW; total reactive powerloss = 143.032 kVAr; minimum voltage observed at node 18,|V18| = 0.90377 p.u.
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Algorithm for Radial Distribution Networks 105
Table 4Comparison of relative speed and memory
Method Relative memory Relative CPU time
Proposed method 1 1.0D. Das et al. [12] 4 1.5Baran and Wu [9] 6 2.9
It is reported in the literature that authors have tried to solve above systemsby NR (Newton-Raphson) and GS (Gauss-Siedel) methods but for both the casesNR and GS did not converge.
7. Conclusions
In this paper a simple and efficient computer algorithm has been presented to solveradial distribution networks. The proposed method has a good convergence propertyfor any practical distribution networks with practical R/X ratio. Computationally,this method is extremely efficient, as compared to Baran and Wu [9] and Daset al. [12], as it solves a simple algebraic recursive equation for voltage magnitude.Another advantage of the proposed method is that all the data are stored invector form, thus saving an enormous amount of computer memory. The method issuccessfully implemented on PIII with several realistic distribution networks. Theproposed algorithm can be used effectively with SCADA (supervisory control anddata acquisition) and DAC (distribution automation and control) as the algorithmquickly solves the system and even suggests rerouting or network reconfigurationfor efficient operation of the system.
References
[1] W. F. Tinny and C. E. Hart, 1967, “Power Flow Solution of the Newton Method,”IEEE Trans. PAS, Vol. PAS-86, No. 11.
[2] B. Stott and O. Alsac, 1974, “Fast Decoupled Load Flow,” IEEE Trans., Vol. PAS-93,pp. 859–869.
[3] B. Stott, 1984, “Review of Load Flow Calculation Methods,” Proc. IEEE, Vol. 62,No. 7.
[4] D. Rajicic and Y. Tamura, 1988, “A Modification to Fast Decoupled Power Flow forNetwork with High R/X Ratios,” IEEE Trans., Vol. PWRS-3, pp. 743–746.
[5] S. C. Tripathy, D. Prasad, O. P. Malik, and G. S. Hope, 1982, “Load Flow Solution forIll Conditioned Power Systems by Newton Like Method,” IEEE Trans., Vol. PAS-101,pp. 3684–3657.
[6] W. H. Kersting and D. L. Mendive, 1976, “An Application of Ladder Network Theoryto the Solution of Three Phase Radial Load Flow Problem,” IEEE PES WinterMeeting.
[7] W. H. Kersting, 1984, “A Method to Design and Operation of Distribution System,”IEEE Trans., Vol. PAS-103, pp. 1945–1952.
[8] R. A. Stevens, D. T. Rizy, and S. L. Puruker, 1986, “Performance of ConventionalPower Flow Routines for Real Time Distribution Automation Applications,” Proc.of 18th Southeastern Symposium on Systems Theory, (IEEE), pp. 196–200.
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106 R. Ranjan and D. Das
[9] M. E. Baran and F. F. Wu, 1989, “Optimal Sizing of Capacitor Placed on RadialDistribution Systems,” IEEE Trans., Vol. PWRD-2, pp. 735–743.
[10] H. D. Chiang, 1991, “A Decoupled Load Flow Method for the Distribution PowerNetwork Algorithm: Analysis and Convergence Study,” Electrical Power and EnergySystems, Vol. 13, No. 3, pp. 130–138.
[11] D. Das, H. S. Nagi, and D. P. Kothari, 1994, “Novel Method for Solving RadialDistribution Networks,” IEE Proc. C, Vol. 141, No. 4, pp. 291–298.
[12] D. Das, D. P. Kothari, and A. Kalam, 1995, “Simple and Efficient Method for LoadFlow Solution of Radial Distribution Networks,” Electrical Power & Energy Systems,Vol. 17, No. 5, pp. 335–346.
[13] S. Bhowimik, S. K. Goswami, and P. K. Bhattacherjee, 2000, “A New Power Distri-bution System Planning through Reliability Evaluation Technique,” Electrical PowerSystems Research, Vol. 24, pp. 169–179.
[14] M. A. Kashem, V. Ganapathy, G. B. Jasmon, and M. I. Buhari, 2000, “A NovelMethod for Loss Minimization in Distribution Networks,” Proc. of International Con-ference on Electric Utility Deregulation and Restructuring and Power Technologies,pp. 251–255.
[15] T. Gonen, 1986, Electric Power Distribution Systems Engineering, McGraw-Hill.
Appendix
A.1. 69-Node Radial Distribution Networks
Figure 4. 69-node RDN [9].Downloaded By: [B-on Consortium - 2007] At: 03:04 29 October 2009
Algorithm for Radial Distribution Networks 107
A.2. Data for 33-Node Test Systems [14]
Table 5
Br. Send. Rec. Resis. Reac. Real load Reac. loadno. node node (ohm) (ohm) (kW) (kVAr)
1 1 2 0.0922 0.0477 100.0 60.02 2 3 0.4930 0.2511 90.0 40.03 3 4 0.3660 0.1864 120.0 80.04 4 5 0.3811 0.1941 60.0 30.05 5 6 0.8190 0.7070 60.0 20.06 6 7 0.1872 0.6188 200.0 100.07 7 8 1.7114 1.2351 200.0 100.08 8 9 1.0300 0.7400 60.0 20.09 9 10 1.0400 0.7400 60.0 20.0
10 10 11 0.1966 0.0650 45.0 30.011 11 12 0.3744 0.1238 60.0 35.012 12 13 1.4680 1.1550 60.0 35.013 13 14 0.5416 0.7129 120.0 80.014 14 15 0.5910 0.5260 60.0 10.015 15 16 0.7463 0.5450 60.0 20.016 16 17 1.2890 1.7210 60.0 20.017 17 18 0.7320 0.5740 90.0 40.018 2 19 0.1640 0.1565 90.0 40.019 19 20 1.5042 1.3554 90.0 40.020 20 21 0.4095 0.4784 90.0 40.021 21 22 0.7089 0.9373 90.0 40.022 3 23 0.4512 0.3083 90.0 50.023 23 24 0.8980 0.7091 420.0 200.024 24 25 0.8960 0.7011 420.0 200.025 6 26 0.2030 0.1034 60.0 25.026 26 27 0.2842 0.1447 60.0 25.027 27 28 1.0590 0.9337 60.0 20.028 28 29 0.8042 0.7006 120.0 70.029 29 30 0.5075 0.2585 200.0 600.030 30 31 0.9744 0.9630 150.0 70.031 31 32 0.3105 0.3619 210.0 100.032 32 33 0.3410 0.5302 60.0 40.0
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