three phase load flow analysis

First Edition 2008 © MOHAMMAD YUSRI HASSAN 2008

Hak cipta terpelihara. Tiada dibenarkan mengeluar ulang mana-mana bahagian artikel, ilustrasi, dan isi kandungan buku ini dalam apa juga bentuk dan cara apa jua sama ada dengan cara elektronik, fotokopi, mekanik, atau cara lain sebelum mendapat izin bertulis daripada Timbalan Naib Canselor (Penyelidikan dan Inovasi), Universiti Teknologi Malaysia, 81310 Skudai, Johor Darul Ta’zim, Malaysia. Perundingan tertakluk kepada perkiraan royalti atau honorarium. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical including photocopy, recording, or any information storage and retrieval system, without permission in writing from Universiti Teknologi Malaysia, 81310 Skudai, Johor Darul Ta’zim, Malaysia.

Perpustakaan Negara Malaysia Cataloguing-in-Publication Data

Recent developments in three phase load flow analysis / edited by: Mohammad Yusri Hassan. Includes index ISBN 978-983-52-0680-1 I. Electric power system--Load dispatching. I. Mohammad Yusri Hassan. 621.317

Editor: Mohammad Yusri Hassan Pereka Kulit: Mohd Nazir Md. Basri & Mohd Asmawidin Bidin

Diatur huruf oleh / Typeset by Fakulti Kejuruteraan Elektrik

Diterbitkan di Malaysia oleh / Published in Malaysia by PENERBIT

UNIVERSITI TEKNOLOGI MALAYSIA 34 – 38, Jln. Kebudayaan 1, Taman Universiti,

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Contents

v

CONTENTS

Preface ix

Chapter 1 Review of Load Flow Analysis Khalid Mohamed Nor

1

Chapter 2 Single-Phase Load Flow Analysis Khalid Mohamed Nor Hazli Mokhlis Taufiq A Gani

23

Chapter 3 Three-Phase Power-Flow Methods using Sequence Components Mamdouh Abdel-Akher Khalid Mohamed Nor

49

Chapter 4 Unbalanced Distribution Power Flow Analysis using Sequence and Phase Components Mamdouh Abdel-Akher Khalid Mohamed Nor

81

Contents vi

Chapter 5 Representing Single-Phase and Two-Phase Lines with Dummy Lines and Dummy Nodes Mamdouh Abdel-Akher Khalid Mohamed Nor

107

Chapter 6 Unbalanced Three Phase Power Flow with Dummy Lines and Nodes Mamdouh Abdel-Akher Khalid Mohamed Nor

113

Chapter 7 Three Phase Load Flow Analysis with Distributed Generation Syafii Ghazali Khalid M Nor M. Abdel-Akher

137

Chapter 8 Future Potentials and Works Syafii Ghazali and Khalid Mohamed Nor

153

Index

161

Preface

ix

PREFACE

Load Flow is the most used analysis in power system. It is required in virtually all aspect of engineering and technical activity in electrical power engineering. Due to its importance and significance, it was amongst the early area that power system researchers worked on especially with the advent of digital computers. Since its long history many can be forgiven for perceiving that all its well and not much need to be studied anymore about load flow.

In its early stage load flow was formulated as a balanced problem. It does simplify the problem in the sense the size of the variables to be solved is drastically reduced. The balanced problem was solved with many mathematical methods that exploit the intrinsic nature of the load flow problem. Over the years as the load flow algorithms increased in complexities, in terms of control variables and development of HVDC and FACTS, the original problem has to be modified and consequently, the programming code need to updated constantly from time to time. From another aspect, computer and computing technology has developed so fast and so much, that most of the original programs that implemented the load flow algorithm have been ported, from the main-frame, to minicomputer to UNIX workstation and now to the ubiquitous Microsoft windows based personal computer or personal workstation. How do we developed the algorithm and what computing technique that we need to employ that minimized the disruption in program maintenance as well ensure minimum computing resources is expended to achieve this objective?

Preface x

While transmission system may still remain a well balanced three phase networks, the distribution system is not necessarily so. This is because transposition used in transmission lines may not be available at all parts of a distribution system. Furthermore the advent of switching mode supplies that power the dominant portion of consumer loads in single phase, means the load are no longer balanced while the networks is lightly unbalanced. From a restricted point of view, but nonetheless quite significant, many networks in North America and some other parts of the world, use single phase and two phase networks to minimize the cost of transmission lines to supply remote loads over long distances. The load flow problem for this kind of network requires special considerations.

Many developments in load flow analysis is being reported in this monograph. Most materials have been published in refereed journals. They have been updated and edited for additional clarity and of course continuity of presentation. This monograph was conceived earlier as the number of work that was done students working under my supervision increased. The works developed towards a better understanding of the load flow analysis. It is not so much the formulation but rather the modeling and the application of previous works to solve a wider type of problems. This publication is hoped to add values to an already voluminous amount of information about load flow in the literature.

Mohammad Yusri Hassan Facuty of Electrical Engineering Universiti Teknologi Malaysia 2008

Literature Review of Load-Flow Analysis

1

1 LITERATURE REVIEW OF LOAD-

FLOW ANALYSIS

Khalid Mohamed Nor

1.1. Introduction

Power-flow studies is an essential tool in planning and designing the future expansion of power systems as well as determining the best operation of existing power systems. The main objective of an unbalanced three-phase power-flow study is to obtain the individual phase voltages at each bus corresponding to the network specified conditions. The balanced three-phase power-flow studies assume that the system unbalances may be neglected. However, there are many cases in which the unbalances of loads, untransposed transmission lines etc. cannot be ignored. Besides, in distribution systems, there are several examples in which the balanced conditions hypothesis cannot be applied. A typical case would be mixed single-phase, two-phase, and three-phase systems in which it is obviously impossible to use a balanced model hypothesis. These cases must be handled with a complete three-phase power system model, as has been proposed by several authors in the literature.

This chapter presents a survey of the basic formulation and the

various unbalanced power-flow algorithms used for solving and analyzing unbalanced electrical networks. The unbalanced three-

Recent Trends in Power System Operation

2

phase power-flow methods can be categorized into three main groups. Firstly, methods are developed for solving general network structure; therefore, these methods can be applied for solving meshed transmission as well as radial distribution power systems. The second group is particularly intended for radial distribution systems since they consider primarily the radial structure of the network. Most of the methods, in both first and second categories, are developed using power system models established in phase coordinates frame of reference.

Finally, the third category includes few methods that are

established based on symmetrical components frame of reference. These methods cannot solve power systems that contain single-phase and two-phase line segments which is the case in distribution power systems. Therefore, the application of these methods is limited to pure three-phase systems and they are normally used for power quality studies since voltage sag analysis is usually studied based on the concept of symmetrical components.

1.2. Discussion of the frame of reference

Power systems have been traditionally analyzed using symmetrical components matrix transformations [1]. This technique has been widely used in unbalanced electric power systems in steady-state analysis. Symmetrical components transformation decouples symmetrical three-phase power systems into positive-, negative-, and zero-sequence networks. Symmetrical power systems include components such as transposed transmission lines, transformers, and generators which can be decoupled using the symmetrical components transformation. The advantage of the sequence networks is that the problem size is greatly reduced and the formulation of the problem is simplified. In addition, many power system analyses can be integrated such as balanced and unbalanced power-flow.


3

Unfortunately, if a power system includes components such as salient-pole synchronous machines, unbalanced loads, and untransposed lines, the network cannot be uncoupled by the mean of symmetrical components. Besides, the phase shifts introduced by special transformer connections are difficult to be represented and distribution power systems have mixed single-phase, two-phase, and three-phase circuits.

Consequently, the trend was to use the untransformed phase

coordinates for unbalanced power-flow solution and a few methods are developed based on the symmetrical components. However, once the phase coordinates are used, the symmetric power systems components such as transposed transmission lines, transformers, generators etc. cannot longer be used. Hence, a complete three-phase power flow problem should be solved using three-phase power system models in phase coordinates which will be computationally expensive.

1.3 Basic formulation of unbalanced power-flow The network power-flow equations can be formulated

analytically in variety of forms such as node voltage, branch voltages, or based on the network structure in case of distribution systems. In this section, the most common and general formulation based on the node voltages will be introduced since it is the suitable form for many power system analyses [2, 3] The formulation of the network equations in the nodal admittance form results in complex linear simultaneous algebraic equations in terms of node currents. The node voltages can be calculated by the direct solution of the liner equation if the node currents are known. Nevertheless, in a power system, values of powers are known (specified) rather than currents. Thus, the resulting equations in terms of powers quantities, known as the power-flow equations, become nonlinear and must be solved by iterative techniques.


4

The objective of the unbalanced three-phase power-flow is to obtain the individual phase voltages at all buses corresponding to the network specified conditions. The introduction of three-phase voltage magnitudes and angles implies that at each bus, there are six independent constraints required to solve for six unknowns. The six constraints are dependent on the type of buses. There are two main types for buses; they are load (PQ) and generator buses or PV buses. One of the generators has to be selected as slack or swing generator. The three-phase busbars of an unbalanced power system are as follows [3, 4]:

i. Load bus (PQ): at the load bus, both the generated active

power abcgeniP _ and the generated reactive power abc

geniQ _ shown

in Figure 1.1 are zero. The active power abcloadiP _ and the

reactive power abcloadiQ _ drawn from the system by the load

are known from historical record, load forecast, or measurement. The unknowns at the load bus are the three-phase voltages magnitudes and angles.

ii. Voltage controlled bus (PV): any bus of the network at which the voltage is maintained constant is said to be voltage controlled or PV bus. The total three-phase active power abc

geniP _ output is specified together with one voltage magnitude. The latter may correspond to one of the three-phase voltages or any combination of the three-phase voltages. The positive sequence voltage magnitude is usually specified. The unknowns at PV bus are the total reactive power abc

geniQ _ and the voltage angle of the specified voltage magnitude.

iii. Slack bus: The slack bus is one of the generator buses selected as a reference bus. Therefore, both the magnitude and angle of the positive sequence voltage are specified. In this case, the total active and reactive power are not specified, but obtained at the end of the study. The reason is


5

that the active power loss in the network is initially unknown.

pgeniP _

pschiP _

ploadiP _

pcalciP _

Bus i

Bus k

Bus N

(a) Active power-flow

pgeniQ _

pschiQ _

ploadiQ _

Bus i

Bus k

Bus N

pcalciQ _

(b) Reactive power-flow

Figure 1.1 Notation for active and reactive power-flows of phase p

at a general bus i Figure 1.1 shows the active and reactive power-flows of a

phase p at a general bus i in the network. From Figure 1.1, the basic equations of the power-flow problem, which are normally


6

called mismatch equations, can be written for a phase p at a general bus i as follows:

( ) p

calcip

loadip

genip

calcip

schip

i PPPPPP _____ −−=−=Δ (1.1)

( ) pcalci

ploadi

pgeni

pcalci

pschi

pi QQQQQQ _____ −−=−=Δ (1.2)

The active and reactive power mismatches in (1.1) and (1.2)

are calculated according to the bus specification. The active and reactive power are specified for PQ busbars, therefore the active and reactive power mismatches can be computed.

The treatment of mismatch equations of generator buses in

unbalanced three-phase power-flow analysis varies according to the method and the generator model utilized. This is because both the total specified power and the specified positive sequence voltage at the generator terminal busbar are subjected to the condition that the internal voltages, the voltages behind the generator reactance, are balanced. The active power mismatch equation of PV bus can be written as follows [5]:

( ) ( )

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+−−=++

c

b

a

gencba

abccalci

abcloadi

abcgenicbaqq

VVV

VVV

jQPPVaVaVyE

Y***

___**2*

(1.3)

where oa 1201∠= qy the positive sequence reactance of a generator qE the voltage behind the positive sequence

generator reactance genY the generator admittance matrix


7

The calculated reactive power in (1.3) is computed during the iterative solution process form the current injected at the generator terminals from the network side as follows:

( )***_ Im ccbbaa

abccalci IVIVIVQ ++= (1.4)

The solution of (1.3) and (1.4) guarantees a balanced internal

voltage of a generator by calculating Eq during the solution process. The specified active power will have some mismatch after voltage regulator action is introduced, but this mismatch tends to be corrected at successive stages of the solution process [5].

In contrast to the traditional balanced three-phase power-flow,

the slack generator cannot be excluded from the solution process since only one voltage magnitude and angle are specified. Therefore, the constraint of balanced internal bus voltage must be maintained as in the case of the other PV generators. The complete process and flowchart for solving generator terminal voltages can be found in detail in the hybrid power-flow method proposed by [5].

1.4. General network structure methods

The general network structure methods can be used for solving both transmission and distribution systems. These methods are established based on phase coordinates models without any simplifications. The following subsection will give a review for different methods reported in the literature for the general network structure methods.

1.4.1. Early techniques, Newton Raphson, Fast decoupled,

and Gauss-Seidel Due to robust convergence, Newton Raphson three-phase

power-flow was proposed by Wasley and Shlash and later by Birt et. al.[6]. The fast decoupled three-phase Newton Raphson was introduced by Arrillaga and Arnold [7]. The Y-Bus Gauss-Seidel method was introduced by Laughton et al. [8]. These elementary


8

methods made some assumptions about generators either to simplify the problem or to improve the convergence characteristics [5]. Examples of such assumptions are that the real power specified is the total power at the internal busbars, in practice, the specified quantity is the power leaving the terminal busbar, or that the voltage magnitude is maintained at the internal busbar instead of the terminal busbar, or that a small value of the machine positive sequence impedance is specified in order to avoid numerical difficulty [7]. 1.4.2. Z-Bus Gauss method

The assumptions made in the early developments are recovered in the hybrid power-flow method reported by [5]. The hybrid method is an iterative technique which utilizes the implicit Z-BUS Gauss for load busbars combining with Y-BUS Gauss method for generator busbars. The application of the Z-Bus Gauss method for a distribution system power-flow analysis was introduced by Sun et al. [9] and subsequently by Chen et al. [10]. The Z-Bus method was chosen by Chen et al. [10] for distribution system power-flow for the following reasons:

i. Newton Raphson approach is known for its excellent convergence characteristics, but its major shortcoming is the requirements of the Jacobian matrix, with a rank four times that of the Y-Bus be recalculated for each iteration.

ii. The convergence character of the Z-Bus method is highly

dependent on the number of specified voltage buses (PV nodes) in the system. If the only voltage specified bus in the system is the swing bus, the rate of convergence is comparable to the Newton Raphson method, which is the case in distribution systems at this era (1991). At this time, the introduction of PV nodes was not popular in distribution systems and all generation units, usually called


9

as cogenerators, are modeled as specified power injection buses (PQ nodes).

1.4.3. Improved fast decoupled

The model by Arrillaga and Arnold [7] is based on the same hypothesis made by Stott and Alsac [11]. The fast decoupled version by Stott and Alsac [11] was improved by Amerongen [12], Monticelli et al. [13], and Haque [14]. Consequently, fast decoupled three-phase power-flow by Arrillaga and Arnold [7] was improved by Garica et al. (1996). The improved model by Garica et al. [15] is based on the decoupling theory presented by Amerongen [12] and Monticelli et al. [13]. The improved fast decoupled by Garica et al. [15] differs in the way that the decoupled submatrices are built and evaluated. 1.4.4. Reformulations of the Newton Raphson method

The deficiency of the Gauss method [10] is that the introduction of PV nodes affects the convergence behavior whereas the fast decoupled methods [7], [15] are sensitive for high R/X ratio of transmission lines. Therefore, alternatives or reformulations were developed based on the Newton Raphson method which its standard version is computationally expensive for large systems.

1.4.4.1. Newton Raphson in complex form

The Newton Raphson method was introduced in complex form by Nguyen [16]. In this method, the Jacobian matrix is expressed in complex form and hence, the whole solution process is carried out in phasor format. The complex format offers reduced memory requirements in comparison with the standard formulation. However the method introduced some simplifications by neglecting mismatch components arising from voltage changes.

1.4.4.2. Current injection formulation

A new development of the Newton Raphson method based on current mismatch or current injection equations was presented by


10

Costa et al. [17] for balanced three-phase power-flow analysis. In this development, the current injection equations are written in rectangular coordinates and an order of 2n bus admittance matrix is composed of 22× blocks. A new dependent variable ΔQ is introduced for each PV node, together with the additional equation imposing the condition of zero deviation in the bus voltage. Except of PV nodes, the Jacobian matrix has the elements of the off-diagonal blocks equal to those of the admittance matrix. The elements of the diagonal blocks are updated every iteration, according to the load model being considered for a certain PQ bus. The current mismatch formulation was tested on large scale systems and has achieved an average 30% speedup, when test benched against a state of the art production grade balanced three-phase Newton Raphson power-flow.

The performance results of balanced three-phased power-flow

Newton Raphson based on current mismatch formulation encouraged Garcia et al. [18] to develop Newton Raphson unbalanced power-flow using current injection method. The three-phase current injection equations are written in rectangular coordinates in similar manner to the case of the balanced power-flow. The resultant equations are of an order 6n system of equations. The Newton Raphson Jacobian matrix is composed of

66× block matrices and retains the same structure as the three-phase nodal voltage admittance matrix. The three-phase current injection method has the advantage that the recalculated elements of the Jacobian matrix during the solution process are very small. In some situations, for a radial distribution system with no PV nodes, the Jacobian matrix is constant.

Practical experience of the three-phase current injection reports

some convergence problems that were identified when the three-phase current injection method was applied to calculate power-flows in heavily loaded systems [19]. This is because the PV bus representation in the current injection formulation required a change of the corresponding column of the Jacobian matrix.


11

Consequently a new representation for PV nodes is reported by Garcia et al. [19]. The improvement of PV representation made the convergence characteristics of the three-phase current injection method typical to the standard unbalanced three-phase Newton Raphson method.

1.5. Radial network structure methods

Distribution networks are characterized with a radial structure with high R/X ratio of transmission lines. A distribution feeder begins at a substation where the electric supply is transformed from the high voltage transmission system to a lower voltage distribution network to deliver the electric power to customers. Due to the radial structure, a variety of power-flow methods are specially intended for solving distribution networks. These methods consider primarily the radial structure of the distribution network in the formulation of the power-flow algorithm.

1.5.1. Forward/backward sweep analysis

A distribution feeder is nonlinear because most loads are assumed to be constant complex power loads. The approach of the linear system can be modified to take into account the nonlinear characteristics of the distribution feeder (Kersting and Mendive 1974). In this approach, an iterative technique involves mainly two basic steps based on Kirchhoff’s voltage and current laws. The two steps are named as backward sweep and forward sweep and they are repeated until convergence is achieved. This iterative technique is known as the forward/backward sweep analysis method.

The backward sweep is primarily a current or a power-flow

summation with possible voltage updates. The forward sweep is primarily a voltage drop calculation with possible current or power-flow updates. This algorithm is based on the fact that the current at the end of the sub-lateral is zero whereas the voltage at the source node is specified. Therefore, by the application of both the forward/backward sweeps in iterative scheme a radial distribution feeder can be solved.


12

1.5.2. Compensation-based methods The basic forward/backward sweep method presented by

Kersting and Mendive [20] was extended for weakly meshed distribution networks. Firstly, a compensation-based balanced power-flow was proposed by Shirmohammdi et al. [21]. In this method, the authors break the interconnected loops at a number of breakpoints in order to convert the network into a radial structure. The radial network is solved efficiently by the forward/backward sweep method. The flows at different breakpoints are considered by injecting currents at their two end nodes. The solution of the radial network with the additional breakpoints current compensation completes the solution of the weakly meshed network.

Later, an improved compensation-based version was presented

by Luo and Semlyen [22] for large and weakly meshed distribution networks. In this method, branch powers are used instead of branch currents in the method presented by Shirmohammdi et al. [21]. The weakness of this improved version is that: firstly the effect of load and shunt admittance are ignored in the loop impedance matrix, secondly, the shunt admittances are ignored in the main power-flow algorithm, and finally, one single source is considered. The first and second weaknesses were recovered by Haque [23] whereas the voltage dependent load models were incorporated in algorithm and reported in [24]. Later, Haque [25] generalized the method for solving distribution networks having more than one feeding source.

The basic compensation-based method proposed by

Shirmohammdi et al. [21] for balanced power-flow was extended for unbalanced power-flow analysis by Cheng and Shirmohammadi [26]. In this development, different component models are incorporated such as multi-phase operation, dispread generation (PV nodes), distributed loads, voltage regulators, and shunt capacitors with automatic tap controls.


13

1.5.3. Fast decoupled for radial networks At the same time that Cheng and Shirmohammadi [26]

proposed the unbalanced power-flow compensation-based method, Zimmerman and Chiang [27] introduced the fast decupled power-flow for solving unbalanced radial distribution systems. The basic concept of the method is based on the fact that the voltage and current injected into each sub-lateral are given, and hence, it is possible to compute all voltages and currents in the rest of the feeder. This is because the current at the end of radial feeder is zero and the voltage is unknown whereas the voltage at the source is specified.

The fast decoupled method by Zimmerman and Chiang [27]

depends on the numerical characteristics of a distribution line. Therefore, it is possible to make decoupling approximation to the Jacobian matrix required for updating the end voltages. Due to the reduced load flow equations, the fast decoupled method is very fast when it was compared against the former Newton Raphson, implicit Zbus Gauss method, and the forward/backward sweep method. The fast decoupled method uses lateral variables instead of node variables, this makes the method efficient for a given system structure, but it may add some overhead if the system structure is changed regularly, which is common in distribution systems due to switching operations.

The last fast decoupled power-flow method for solving distribution networks was introduced by Lin et al. [28]. The method uses the traditional Newton Raphson algorithm in a rectangular coordinate system. The Jacobian matrix can be decoupled both on phases and on real and imaginary parts based on assumptions such as neglecting the off-diagonal blocks (mutual coupling). Therefore, the memory requirements of the traditional fast decoupled load flow can be reduced to only one-sixth. This fast decoupled method can be executed with performance of 10 to 100 times faster than other methods for test systems of 45 to 270 buses. Although the method has excellent performance


14

characteristics, the assumptions made in this fast decoupled version reduces the accuracy of the method.

1.5.4. Newton Raphson for radial networks

The application of Newton Raphson method for solving balanced radial feeders was presented by Zhang and Cheng [29]. This method is based on the formulation of the Jacobian matrix as UDUT form, where U is a constant upper triangular matrix depending solely on system topology and D is a block diagonal matrix resulting from the radial structure of distribution systems. This formulation allows the conventional Newton Raphson algorithm with forming of the Jacobian matrix, LU factorization and forward/backward substitution be replaced by forward/backward sweeps on radial feeders with equivalent impedances.

Reliable load-flow utilizes Newton Raphson method was

presented by Gómez and Ramos [30]. The purpose of the algorithm is to enhance convergence rate. This is achieved by writing the load-flow equation in alternative variables consisting of 3N equations (2N linear plus N quadratic) to make the load-flow equations more linear. The algorithm is validated with many ill-condition networks reported in the literature and then compared with some of the existing algorithms in terms of convergence characteristics and computation time.

The last application of Newton Raphson for unbalanced radial

distribution systems was proposed by Teng and Chang [31]. In this method, the authors used branch voltages as state variables and employed the Newton Raphson algorithm to solve the load flow problem. By utilizing branch voltages as state variables, a constant Jacobian matrix can be obtained, and a building algorithm for the Jacobian matrix can be developed from the observation of the constant Jacobian matrix. In addition, lower and upper triangular factorization is developed to avoid the time-consuming in computation. The branch voltage Newton Raphson method gave


15

fast execution time when it was compared with the Gauss Z-Bus method [10]. 1.5.5. Direct solution of radial networks

A direct approach was developed by Teng [32], in this approach, special topological characteristics of distribution system were fully utilized to make direct solution possible. The power-flow solution is obtained by developing two matrices and a simple matrix multiplication. The two matrices are the bus-injection to branch-current matrix and the branch-current to bus voltage matrix. Due to the distinctive solution techniques of this method, the consumed time for LU decomposition of the Jacobian matrix or the admittance matrices, or the consumed time for the forward/backward substitution required in the traditional power-flow methods is no longer necessary. 1.5.6. Phase decoupling methods

The fast execution time and low memory requirements can be obtained by phase decoupling methods. Firstly, the phase decoupled methodology for solving radial distribution networks was presented by Lin and Teng [33]. In this method, the Newton Raphson algorithm is used to assure rapid convergence. The Jacobian matrix is formulated based on branch currents. Therefore, the specified powers are represented by equivalent current injections for load buses. The resultant Jacobian matrix has constant elements corresponding to self or mutual impedances. The Jacobian matrix is decoupled based on the fact that the line self-impedance is significantly greater than the mutual impedance, and hence, the off-diagonal blocks are neglected in the Jacobian matrix resulting in the phase decoupled method.

Secondly, the quasi-coupled unbalanced radial power-flow

was presented by Ramos et al. [34]. In this method, a distribution network is decoupled into a three independent single-phase networks. Consequently, three separate balanced power-flow programs can be run. However, the results provided by this process


16

may not be accurate enough in certain circumstances. Therefore, the mutual coupling is considered by an equivalent branch voltage sources or bus current injections for accurate results. In this method, the bus current injections are more appropriate when Newton Raphson is employed whereas branch voltage sources are suitable when forward/backward sweeps are adopted. The quasi-decoupled method has a 30-40 % execution time saving when it is compared with the compensation based method by Cheng and Shirmohammadi [26]. The disadvantage of the quasi-coupled method is that it has accuracy limitations.

1.6. Sequence component methods

The first algorithm used in unbalanced power-flow solution for analyzing three-phase network was presented El-Abiad et al. [35]. This method utilized Gauss Seidel iterative scheme based on the bus impedance matrix established in sequence components. Twenty six years later, Lo and Zhang [36] introduced the decomposed three-phase power-flow method based on symmetrical networks. In this method, the power-flow problem is decomposed into three sub-problems. The sub-problems involve the solution of the positive-, negative-, and zero-sequence networks. The positive sequence network sub-problem is solved using Newton Raphson method. The other two sub-problems are formulated into a set of linear simultaneous equations. The mutuality is included by putting the three-sub-problems into an iterative process.

Due to the mutual coupling of transmission lines, Lo and

Zhang [36] power-flow method requires the construction of NN 33 × admittance matrix in sequence components.

Consequently, Zhang and Chen (1993) introduced the decoupled line model that allows the formulation of the admittance matrices corresponding to each network independently rather than the construction of NN 33 × admittance matrix. The proposed method by Zhang and Chen [37] allows the treatment of the positive sequence network independent on the negative and zero sequence networks. The positive sequence network is solved using the bus


17

admittance matrix, and later Zhang [38] solved the positive sequence using the conventional Newton Raphson and fast decoupled balanced power-flow algorithms whereas parallel processing solution was reported by Zhang et al. [39].

The differences between the decoupled unbalanced power-

flow methods by Zhang [38] and the method by Lo and Zhang [36] are retained to transformer and transmission line models. In Lo and Zhang method [36], the Y-Δ transformer phase shift is incorporated in the NN 33 × admittance matrix. On the other hand, Zhang [38] used the classical transformer model [40] for constructing the decoupled NN 33 × admittance matrix. In this transformer model, the Y-Δ transformer phase shift cannot be suitably incorporated in the solution process [40].

The latest application of symmetrical components for

unbalanced power-flow analysis was introduced by Smith and Arrillaga [41]. In this development, the sequence components mismatches are utilized at generator busbars rather than phase components mismatches. In this method, it is reported that the unbalanced power-flow will find one of two solutions at each load busbar. One of the two solutions corresponds to abnormal levels of zero sequence voltage. The abnormal solution can be avoided by ensuring a path for the zero-sequence current from the load.

1.7. Conclusions

The chapter has presented a survey for the unbalanced power-flow methods. These methods were classified to three main categories. Firstly, methods have the ability to solve general network structure. Secondly, methods are intended specially for solving radial and weakly meshed distribution networks. Finally, methods are developed based on symmetrical components theory. Among of these methods, the phase decoupling or sequence decoupled power-flow methods have the best performance with regards to execution time and memory requirements. However, these methods have inaccurate results or have modeling


18

complexity that prevents them to be used for a general network structure.

The advantage of the sequence decoupled methods is that it is

possible to use only one balanced power-flow for solving the positive sequence network rather than three in case of phase decoupling methods. Besides, the coupling in the sequence components model is lower than its counterpart in phase coordinates model. Also, there are many symmetric elements which are naturally uncoupled in sequence model whereas they are coupled in phase coordinates. Therefore, the decoupling of the unbalanced network into positive-, negative-, and zero-sequence is more suitable than phase decoupling. However, to decouple sequence networks, and at the same time retain the phase coordinates modeling features, the power system sequence component models need to be rewritten and improved to fulfill the unbalanced power systems modeling requirements. This will be described in the next chapter.

References [1] Fortescue C. L. (1918). Method of the symmetrical

coordinates applied to the solution of polyphase networks. AIEE Trans., vol. 37, pp. 1027–1140

[2] Sadat H. (1999). Power system analysis. McGraw-Hill Book Co, pp. 189-190

[3] Grainger J. J. and Stevenson W. D. (1994). Power system analysis. McGraw-Hill Book Co., pp. 332-333

[4] Wasley R. G. and Shlash M. A. (1974). Newton-Raphson algorithm for 3-phase load flow. proc. IEE, 121, July, pp. 630-638

[5] Chen B. K., Chen M. S., Shoults R. R., and Liang C. C. (1990). Hybrid three phase load flow. IEE Proc. Generation,


19

Transmission and Distribution, vol. 137, no. 3 , May, 1990, pp. 177-185

[6] Birt K. A., Graff J. J., McDonaled J. D. and El-Abiad E. H. (1976). Three Phase Load Flow Program. IEEE Trans. On PAS, vol.95, no. 1, pp.59-65

[7] Arrillaga J. and Arnold C. P. (1978). Fast-decoupled three phase load flow. Proc. IEE, vol. 125, no. 8, pp. 734-740

[8] M.A. Laughton and A.O.M. Saleh, "Unified Phase-Coordinate Load Flow and Fault Analysis of Polyphase Networks", International Journal of Electrical Power and Energy Systems, v01.2, no.4, 1980, pp. 18 1-1 92.

[9] Sun D. I. H., Abe S., Shoults R. R., Chen M. S., Eichenberger P., and Farris D. (1980). Calculation of energy losses in a distribution system. IEEE Trans. on PAS, vol. 99, no. 4, July-August, pp. 1347-1356

[10] Chen T. H., Chen M. S., Hwang K. J., Kotas P., and Chebli E. A. (1991). Distribution system power flow analysis- a rigid approach. IEEE Trans. Power Delivery, vol. 6, no. 3, July, pp. 1146-1152

[11] Stott B. and Alsac O. (1974). Fast Decoupled Load Flow”, IEEE trans. on power apparatus, PAS-93, pp. 859-869

[12] Van Amerongen R. (1989). A general Purpose version of the fast decoupled load flow. IEEE Trans. Power systems, vol. 4, no. 2, May, pp. 760-770

[13] Monticelli A., Garica A., Saavedra. (1990). Fast decoupled load flow: hypothesis, derivations, and testing. IEEE Transactions power system, vol. 5, no. 4, November, pp 1425-1431

[14] Haque M. H. (1993). Novel decoupled load flow method. IEE proc. generation, transmission, and distribution, vol. 140, no. 3, May pp. 199-205

[15] Garcia A. V. and Zago M. G. (1996). Three-phase fast decoupled power flow for distribution networks. IEE Proc. Generation, Transmission, and Distribution, vol. 143, no. 2, March, pp. 188-192


20

[16] Nguyen H. L. (1997). Newton–Raphson Method in Complex Form-power system load flow analysis. IEEE Trans. Power Systems, vol. 12, no. 3, August, pp. 1355–1359

[17] Costa V. M. Da, Martins N., Pereira J. L. R. (1999). Developments in the Newton Raphson Power Flow Formulation Based on Current Injection. IEEE Trans. On Power Systems, vol. 14, no. 4, November, pp. 1320-1326

[18] Garcia P. A. N., Pereria J. L. R., Carneiro S. J. R, Da Costa V. M., and Martins N. (1999). Three-phase power flow calculation using the current injection method. IEEE TRANS. on Power Systems, vol. 15, no. 2, May, pp. 508-514

[19] Garcia P. A. N., Pereria J. L. R., Carneiro S. J. R, Vinagre M. P., and Gomes F. V. (2004). Improvements in the representation of PV buses on three-phase distribution power flow. IEEE Trans. on Power Delivery, vol. 19, no. 2, April, pp. 895-898

[20] Kersting W. H. and Mendive D. L. (1976). An application of ladder network theory to the solution of three phase radial load flow problems. IEEE PAS Winter Meeting, New York, IEEE paper no. A76 044-8

[21] Shirmohammadi D, Hong H. W., Semlyen A., and Luo G. X. (1988). A compensation-based power flow method for weekly meshed distribution and transmission networks. IEEE Trans. on power systems, vol. 3, no. 2, may, pp. 753-762

[22] Luo G. X. and Semlyen A.(1990). Efficient load flow for large weakly meshed networks. IEEE trans. on power systems, vol. 5, no. 4, November, pp. 1309-1316

[23] Haque M. H. (1996-a). Efficient load flow method for distribution systems with radial or mesh configuration. IEE proc. generation, transmission, and distribution, vol. 143, no. 1, January, pp. 33-38


21

[24] Haque M. H. (1996-b). Load flow solution of distribution systems with voltage dependent load models. Electric power systems research journal, vol. 36, pp. 151-156

[25] Haque M. H. (2000). A general load flow method for distribution systems. Electric power systems research journal, vol. 54, pp. 47-54

[26] Cheng C. S. and Shirmohammadi D. (1995). A three-phase power flow method for real time distribution system analysis. IEEE Trans. on power systems, vol. 10, no. 2, May, pp. 671-679

[27] Zimmerman R. D. and Chiang H. D. (1995). Fast decoupled power flow for unbalanced radial distribution systems. IEEE Trans. on power systems, vol. 10, no. 4, November, pp. 2045-2052

[28] Lin W. M., Su Y. S., Chin H. C., and Teng J. H. (1999). Three-phase unbalanced distribution power flow solutions with minimum data preparation. IEEE Trans. on power systems, vol. 14, no. 3, August, pp. 1178-1183

[29] Zhang F. and Cheng C. S. (1997). A modified Newton method for radial distribution system power flow analysis. IEEE Trans. on power system, vol. 12, no. 1, February, pp. 389-397

[30] Gómez A. and Ramos E. R. (1999). Reliable load flow technique for radial distribution networks. IEEE trans. on power systems, vol. 14, no. 3, August, pp. 1063-1068

[31] Teng J. H. and Chang C. Y. (2002). A novel and fast three-phase load flow for unbalanced radial distribution systems. IEEE Trans. on power system, vol. 17, no. 4, November, pp. 1238-1244

[32] Teng J. H. (2003). A direct approach for distribution system load flow solutions. IEEE Trans. on power system, vol. 18, no. 3, July, pp. 882-887

[33] Lin W. M. and Teng, J. H. (1996). Phase-decoupled load flow method for radial and weakly-meshed distribution networks. IEE Proc. Generation, Transmission and Distribution, vol. 143 no. 1, January, pp. 39-42


22

[34] Ramos E. R., Expósito A. G., and Cordero G. Ă. (2004). Quasi-coupled three-phase radial load flow. IEEE Trans. on power systems, vol. 19, no. 2, May, pp. 776-781

[35] El-Abiad A. H. and Tarsi D. C (1967). Load flow study of untransposed EHV networks. IEEE PICA, Pittsburgh, pp.337-384, 1967

[36] Lo K. L. and Zhang C. (1993). Decomposed three-phase power flow solution using the sequence component frame. Proc. IEE, Generation, Transmission, and Distribution, vol. 140, no. 3, May, pp 181-188

[37] Zhang X. P. and Chen H. (1994). Asymmetrical three-phase load-flow study based on symmetrical component theory. IEE Proc. Generation, Transmission and Distribution, vol. 141, no. 3, May, pp. 248-252

[38] Zhang X. P. (1996). Fast three phase load flow methods. IEEE Trans. on Power Systems, vol. 11, no. 3, August, pp 1547-1553

[39] Zhang X. P., Chu W. J., and Chen H. (1996). Decoupled asymmetrical three-phase load flow study by parallel processing. IEE Proc. Generation, Transmission and Distribution, vol. 143 no. 1 January, pp. 1996

[40] Arrillaga J. and Watson N. R. (2001). Computer modeling of electrical power systems. Book, 2nd Edition, John Wiley & Sons LTD, pp 11-13, pp 48-51, and pp 111-115

[41] Smith B. C. and Arrillaga J. (1998). Improved three-phase load flow using phase and sequence components. IEE Proc. Generation, Transmission and Distribution, vol. 145, no. 3, May, pp. 245-250

Single-Phase Load Flow Analysis

23

2 SINGLE-PHASE LOAD FLOW

ANALYSIS

Khalid Mohamed Nor Hazli Mokhlis Taufiq A Gani

2.1 Introduction

Load flow analysis is used in planning and designing the future expansion of power systems, as well as in system operation. Load flow also provides steady state condition for other analysis such as stability studies, short-circuit and outage security assessment.

From the basic load flow formulation, such as given in

references [3], more sophisticated algorithm have been developed that handle complex devices such the Unified Power Flow Controller (UPFC) and High Voltage Direct Current (HVDC). Practical load flow analyses also require good graphical user interface and efficient data management system. As power engineering continues to develop many new requirements will continue to drive the need to modify load flow analysis software implementation.

The factors that drive the need for modifications are: a) Data structure changes – addition of new data type or

additional control features such as the UPFC. b) Modification of analysis algorithm – new solution method. c) Change of platform Windows, Linux and web-based. (File

format, input/output, graphical user interface) d) Change of application such as real-time applications.


24

Factors (a) and (b) are caused by the development of new power electronic devices such as Flexible AC Transmission System (FACTS) devices such as the UPFC. Since the load flow software is not designed to simulate these devices directly, the analysis and data structure of the software must be adapted and extended. Minor changes in system functionality and data may propagate through the whole program, which lead to massive reconstruction of the code.

These problems arose in the traditional function-oriented software development methodology approach adopted of implementing load flow algorithm. In the function-oriented approach, a strong dependency exists between the pieces of the code and a strong coupling exists between data structure and procedure [1]. Little discrepancies in the data structures may lead to programming errors that are very hard to detect and thus making modification very difficult.

Change from one platform to other will cause the existing

codes application unable to work in the new platform. The codes will need to be modified to suit particular requirement of the new platform. This problem is usually solved by using library files rather than by using code. However, a library file could not be extended for new requirements. Any extension can only be carried out at source code level. Another problem is that not all type of platforms supports the same type of library files. In the case of real time application, modification is generally only on the data input/output. However data input/output that are coupled tightly with the analysis will cause difficulty in modification.

The problems discussed above makes the tasks of software

maintenance (including upgrading) time consuming and costly. In order to minimize investment and time to maintain and upgrade software application we must try to 1) Retain as much as possible existing codes. 2) Modify by reusing as much as possible elements of the

previous codes.


25

3) Uncouple tight linkage between modules so that a change in any module does not propagate or escalate into other modules.

The above strategies are implemented by taking into account all possibilities of adding new requirement in the development of the algorithm, code implementation and software application architecture.

One of the popular approaches that were used in solving reuse problems for software upgrading is by using Object Oriented Programming (OOP). There are many reports on the application of OOP for power system application [1, 2, 7]. The main focus of reuse was on reusing existing classes through inheritance feature of OOP. Other features by OOP such as aggregation and polymorphism are not intensively explored.

Another approach in software reuse is to use Component-

based Development (CBD). In CBD, the essential building block is called component. A small number of reports have been appeared on the application of CBD in developing power system analysis. Such example is in [7], where OOP and CBD were applied in developing power system analysis software. In this report, different components, which are load flow, fault, mathematical solver and graphical user interface (GUI), were integrated into load flow analysis software. These components can be reused for other application since each of them is independent between each other. Changing on any component will not affect the rests. A part from software application reusability, another issue that is not reported in literature was on the code and algorithm reuse.

2.2 Code Reusability Code reusability uses as much as possible earlier codes whenever the algorithm is modified (changed) to incorporate (due to) new control devices. We have considered the case of full Newton Raphson solver as well the Fast Decoupled solver. In Newton Raphson technique we used matrix partitioning to achieve reusability, whereas in Fast Decoupled we modified the matrix


26

equations while preserving sparsity structure and add a third set of decoupled equation.

2.2.1 Matrix Partitioning in Full Newton-Raphson Implementation In the Newton-Raphson (NR) solution of load flow, devices

control equations may be combined together with nodal mismatch equations [3]. These additional variables and equations will alter the structure of NR matrix equations due to nodal mismatch equations only. As a result, algorithm and codes need to be modified to accommodate the changes whenever new and different control devices are introduced. In order to avoid such drawback, matrix partitioning approach is applied in solving NR method.

Equation 2.1 shows the NR matrix equation that has been coupled with additional equations correspond to the control devices. The matrix equation is partitioned into sub-matrices corresponding to the nodal mismatch equations, which are shown by sub-matrix J, and sub-matrices shaded in gray involve equations of the control devices.

(2.1)

The sub-matrices are representing particular matrices as follows:

J is the Jacobian matrix with dimensions (Nbusx2)x (Nbusx2) due to nodal mismatch equations.

A12 is a matrix of devices equations with dimensions (Nbusx 2) x (Nequ).

A21 is a matrix of devices equations with (Nequ) x (2 x Nbus). A22 is a matrix of devices equations with (Nequ x (Nequ) B1 is power mismatch of active and reactive with

1x (Nbus x 2).

J A12

A21

X1

X2

B1

B2A22


27

B2 is mismatch of control devices equations with 1 x (Nequ). X1 is a vector of voltage magnitude and phase angle ( θΔΔ ,V )

with (Nbus x 2) x 1 X2 is a vector of device variables that need to be calculated

with (Nequ x 1) Nequ – Number of set of equations for a particular device, which every other device would have its own set of equations. Nbus – Number of buses in the power system network. Equation 2.1 can be expanded into:

12121 BXAJX =+ (2.2) 2222121 BXAXA =+ (2.3)

Rearranging equation (2.2):

[ ]21211

1 XABJX −= − (2.4) Substituting equation (2.4) into equation (2.3) and rearranging:

[ ] [ ]11

2122121

2122 BJABXAJAA −− −=− [ ] [ ]1

1212

112

121222 BJABAJAAX −−− −−= (2.5)

Equation (2.5) is solved first and X2 is then substituted into equation (2.4) and solved to get X1. If equation (2.4) is expanded as in equation (2.6), the right hand side of equation (2.6) composes of two terms involving matrix product of J-1 which have been calculated in equation (2.5).

2121

11

1 XAJBJX −− −= (2.6) As an example of load flow application using the above

equations, let us consider a two-bus system shown in Figure 2.1, where a UPFC is connected between bus m and bus k [3].


28

S h u n t C o n v e r te r

S e r ie s C o n v e r te r

kV mV

S ta tV

s s s cV

S ta tV S ta tθ s s s cV s s s cθ

d cV

Fig. 2.1. UPFC schematic diagram

Variables related to UPFC that need to be find along with the usual load flow parameters are SSSCV , SSSCθ , StatV and Statθ . The following are related equations of UPFC variables, which are based on equation 2.1.

[ ] [ ]Tbbmkmk PQPB ΔΔΔ=2 (2.7)

[ ]TStatSSSCSSSCSSSC VVX θθ ΔΔΔ= /][ 2 (2.8)

[ ]

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

=

m

bbm

Stat

bbStat

m

bb

k

bb

m

mkm

m

mk

k

mk

m

mkm

m

mk

k

mk

VP

VV

PV

PPVQ

VQQ

VP

VPP

A

θθ

θθ

θθ

0

0

21(2.9)


29

[ ]

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

=

0

0

12

sssc

msssc

sssc

m

Stat

k

sssc

ksssc

sssc

k

sssc

msssc

sssc

m

Stat

k

sssc

ksssc

sssc

k

VQ

VQ

QV

QV

QV

PV

P

PV

PV

P

A

θ

θθ

θ

θθ

(2.10)

[ ]

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

∂Δ∂

=

Stat

bbStat

sssc

bbsssc

sssc

bb

sssc

mksssc

sssc

mk

sssc

mksssc

sssc

mk

VP

VVP

VP

VQ

VQ

VP

VP

A

θ

θ

θ

0

0

22 (2.11)

Where, mkPΔ and mkQΔ are the power mismatches of power flow from bus m to k and bbPΔ is the active power mismatch of UPFC source.

The main modification of basic load flow codes is in the power mismatches equations, where new subroutines are added to include the effects of power injection at particular buses. Such an example is shown in Figure 1, where the power mismatches at bus k needs to include the power injection of the shunt converter (VStat). The new equations of power mismatches when considering new devices will be as follows:

injected

iold

inew

i PPP −Δ=Δ (2.12) injectedi

oldi

newi QQQ −Δ=Δ (2.13)

where, newi

newi QP ΔΔ , : the new power mismatches

oldi

oldi QP ΔΔ , : the original power mismatches


30

injectedi

injectedi QP , : the injected power of particular devices

The codes to implement the above equations are added in a special routine (such as a class or function) for UPFC and not into the original power calculation, which has been developed. Therefore, the original power calculation codes are preserved.

The above example shows that the NR matrix solution can be separated into two different equations. The first one is to solve the usual variables (V andθ ), and the second one is to solve control devices variables. By doing so, sub-matrix J of the NR matrix equation remains unchanged. New control equations can therefore be added into the codes without modifying the original codes since they are solved separately. Although we need to add new functions for control devices, the same original codes are still used without any change. This approach makes the original NR codes and algorithm reusable.

2.2.2 Modification for Fast-Decoupled Algorithm In Fast Decoupled (FD) load flow, sub-matrices A12 and

A21 in equation 2.1 are approximated to zero. As usual, the two matrices in FD to be solved iteratively are:

[ ] [ ][ ]θΔ=Δ '/ BVP (2.14)

]]["[]/[ VBVQ Δ=Δ (2.15)

where,

QP ΔΔ , : active and reactive power mismatch vectors θΔΔ ,V : voltage magnitude and angle correction vectors

( )221 ikikikikikik XRXBandXB +== // "' (2.16)


31

21 // ""'ik

kiikii

kiikii SBBandXB +−=−= ∑∑

∈∈

(2.17)

The [k,k] element of matrix B” is included with the impedance of UPFC on the shunt converter as follow:

( )22StatStatStatkkkk XRXBB +−= /"'" (2.18)

where,

ikik xR , : resistance and reactance of branch i-k iS : susceptance of branch i-k

The decoupling of the NR equation of 1 leaves the control

devices equations involving sub-matrix A22 to be as in equation 19, where the variables are as defined in the earlier equations.

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΔΔΔ

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

Δ

ΔΔ

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

Δ∂Δ∂

Δ∂Δ∂

Δ∂Δ∂

Δ∂Δ∂

Δ∂Δ∂

Δ∂Δ∂

Δ∂Δ∂

bb

mk

mk

Stat

sssc

sssc

sssc

Stat

bb

sssc

bbsssc

sssc

bb

sssc

mksssc

sssc

mk

sssc

mksssc

sssc

mk

PQP

VV

PVPVP

VQVQVPVP

θ

θ

θθ

θ

θ

0

0

(2.19)

The above equations are solved sequentially i.e. by solving

equation 2.14 first, then 2.15 and 2.19 or simply stated as [P, Q, UPFC] sequence. This process is simplified in the following flow chart by taking FACTS as an example.


32

[ ]PΔ

[ ]QΔ

θ

Figure 2.2 Fast Decoupled Load Flow Algorithm

The above flow chart shows a FD algorithm that has been

modified to include additional analysis. In this flow chart, the FACTS routine is added into the algorithm without changing or modifying any of the previously developed algorithm and codes. In Figure 2.2, the previous algorithm is represented by unshaded boxes and the new added routines are represented by shaded boxes. The routine consists of various tasks (in the form of function) to include FACTS into the load flow analysis. In the case of UPFC for example, the routine of calculate new active mismatch will calculate the injected active power due to the shunt converter at a


33

certain bus that connected to UPFC. This power will then be subtracted from the power mismatch that has been calculated in the previous algorithm to get new power mismatch at certain buses (equations 2.12). By developing this routine, the original codes for calculating the active power mismatches still applicable without any modification. The same explanation also applied in recalculating new reactive power mismatches due to the injected reactive power.

This example showed how code reusability can be obtained

in FD method. The algorithm can be extended to include other FACTS devices such as SSSC and StatCom without affecting previously developed algorithm and codes.

2.3 Object Oriented Reusability

2.3.1 Reusability in Power System Model Object oriented power system model represents each

electrical device as an object. The object has properties of data, methods and relationship with other devices. Data are variables related to the object, while a method is a task to do certain job related to it. Data and method can be set to be publicly accessible or restricted for it own usage. This property makes the internal data and method that are not necessary for other class hidden and therefore prevent their misuse.

Various power system models have been proposed such as

in [1, 2]. However, there is no dominantly accepted model. Every software developer has their own model that suits their own requirements.

In OOP, a new class can inherit from an existing class, or

created as an aggregation of more than one class. Inheritance allows new class to inherit data and method from the parent class. To make the object versatile we can use polymorphism to accept multiple input and output.


34

In order to fully exploit OOP, i.e. to use inheritance and

aggregation, the hierarchy of the device classes must be modeled at the right granularity. Therefore, our approach is to design model based on the primitive objects of linear circuit elements i.e. node or bus, branch and source. These objects are defined as the basic (parents) objects. The reason is because all electrical devices can be modeled as equivalent circuit comprising of these primitive elements. By doing so, the model is flexible to be extended with new device objects.

Using the class diagram notation of Unified Modeling

Language (UML) [ ?] where all classes are used to model power system devices and their relationships are drawn as shown in Figure 4. A base class, Power System Devices is designed to share all common data and method for its derived classes. Basic properties such as the number of devices, the id number and the name of the device are defined in this class. These properties are common and required to all type of devices.

As examples of inheritance are the transformer and

transmission line classes that derived from the Branch class. In Branch class, there are resistance, reactance and susceptance data that are common for transformer and transmission line devices. Derived classes will share these common data and methods. By using inheritance, a derived class will only contain data and methods that are different from the parent’s class.

Under the base class, the Node class represents a node,

while Branch class represents a branch connected between two nodes. Any devices that are connected to one node are derived from the Node class. Classes for devices that are connected between two nodes are derived from the Branch class and for more than two nodes the devices are derived from MultiNodesDevice class. The Node class contains the data and method related for a


35

node such as voltage and phase angle, and calculating nodal power mismatches.

The derived classes from the Node class are Load, AC

generator and Static Shunt Compensator (StatCom) classes. The Transformer class is a two-node device class. The Line class, for transmission line, is a composite class of two nodes and a Branch classes.

The circuit models of the Unified Power Flow Controller (UPFC), High Voltage Direct Current (HVDC), Multi Winding Transformer and Static Series Synchronous Compensator (SSSC) contain internal nodes and shunts. Therefore, these devices are derived from the MultiNodeDevice class, which is a composition of Bus, Node, and Branch class.

UPFC is in effect a combination of SSSC and StatCom

devices. For this reason, we can combine the UPFC class data and method from SSSC and StatCom classes. Under SSSC class there is Synchronous Voltage Source (SVS) class as the derived class.

Fig. 2.3. Power System Model Hierarchy

TPowerSystemDevice

TNode

TStatcomTLoad

TImpedanceLoad TConstantPower

TACGenerator

TBranch

TTransformer TLine

TTwoWinding

TUPFC TTransMultiWindingTHVDC

TMultiNodeDevice

TStatcom TSSSC

TSSSC

TSVS


36

class TUPFC : public MultiNodeDevice private: void Calculate_Pmk(); void Calculate_Qmk(); void Calculate_Pbb(); protected: public: TSSSC *SSSC; TStatcom *STAT; void Develop_Matrice_B(); void Develop_Matrice_C(); ;

This is possible because SVS model is a simplified form of SSSC model by neglecting the coupling transformers impedance. A fragment of C++ code related to the UPFC class is presented as follows.

The UPFC class inherits TMultiNodeDevice class, which is shown in the class declaration. Inheritance relationship enables UPFC class to access and use all the public property of TMultiNodeDevice class.

The aggregation relationship of UPFC with StatCom and SSSC is defined under public type. Both SSSC and StatCom classes are defined here to enable UPFC’s methods to access and use their properties. However, only the public properties of both classes can be used.

Under public type, there are also other methods i.e to develop matrices of B, C and D. These matrices are related to UPFC equations and correspond to the matrices A12, A21 and A22 in equation 1 respectively.

The private methods of UPFC class can only be accessed

and used by UPFC’s methods. Other methods outside this class cannot access them. The private methods in UPFC class are to


37

calculate active and reactive power flow of UPFC source and the power mismatches. These methods are internal usage and therefore, they are defined as private type to encapsulate it from other classes.

The above example codes shows how inheritance and

aggregation are defined. Any new devices that have similar characteristic can inherit an existing class. Thus, with the proposed power system classes, any new devices can be incorporated into the power system model without having to modify the original developed classes.

2.3.2 Load Flow Analysis Classes Similar with other previously discussed classes, all related

load flow data and methods would be encapsulated into a class. Data and method that can be shared are defined as public members and the non-common members are defined as private members of the class. Base class is identified to share the common data and methods. The designed class of load flow analysis is shown in Figure 5.

The LFBase class is a base class contains common data and

method for load flow analysis such as calculating power mismatches. It also contains data preparation functions, which provides the interface between load flow solver and user. The derived classes from the base class are NRaphson and FDecoupled. These classes correspond to NR and FD solution methods respectively.


38

TFDecoupled

TLFBase

TNRaphson

Fig. 2.5. Load Flow analysis class hierarchy

By inheriting the base class, adding or changing other solving techniques such as Z-matrix and Gauss-Seidel can be done without having to rewrite the common methods such as calculating active or reactive power mismatch that have been defined in the base class. Instead of single-phase load flow, three-phase load flow and dc load flow also can be incorporated under LFBase class.

Figure 6 shows the flow chart process in using load flow solver. The Data preparation function in the analysis part is responsible in changing the supply data to a required variable name and structure type used in the load flow analysis solver.

Data Input

Load Flow Solver

Analysis part

User define Datapreparation

Result

Data preparationfunctions

Fig. 2.6. Load Flow analysis process

The raw data is prepared by the Data Input, which is also a component type. The data will be processed first in the Data preparation functions before used in load flow solver. In our case,


39

class PACKAGE TLBase : public TComponent private : protected: int Nline; int NTrans; void Create_Branch(); void Create_Admittance(); public: //Data Preparation Functions void AddBus ( . . .); void AddLine (float* R,float* X,float* S…); //Getting Data Functions void ReadLine (float* R,float* X,float* S…); void ReadTrans(. . .);

the data structures used are based on the developed power system model. A fragment of C++ codes on data preparation functions is shown below:

By having these functions, load flow analysis class is independent from the structure of the data supplied to it. Thus, any data that has different data structure from the load flow analysis can be supplied by using database, reading text file or other approaches as long as it is the right data. These functions can be considered as an interface layer for other application to communicate with load flow solver. Adding new properties or altering the properties of the load flow class can be done without affecting the data input application.

The data preparation functions are designed by identifying

all required data and its relationship. For instance, since resistance (R), reactance (X), susceptance (S) and MVA ratings belong to a transmission line, they are put under the line data preparation i.e. AddLine() function. The process of identifying will help to produce systematic data preparation functions. The functions are also given


40

suitable names according to the task they are assigned, which will help users to recognize each function’s job and it data parameters easily and clearly.

In contrast to ‘Add’ functions, the ‘Read’ functions are to

read and retrieve data from the load flow analysis. The name given to the functions reflect its role in reading particular data values. For instance, ReadBus() function is to read and get values related to a bus such as voltage and phase angle. By having these functions, users do not need to know the variable’s name or power system model that being used in the load flow solver in order to get particular values. What they need to know is only on how to use the functions to get the required variables from load flow solver.

2.3.3 Classes for Sparse Linear Solver The analytical analysis of load flow analysis requires a sparse linear solver to solve sparse linear equation in the form of [ ] [ ] [ ]BXA = . In our software development, a public domain library SuperLU [4] is used as the solver for load flow analysis. It is used since it uses many latest techniques, such as graph reduction technique in matrix factorisation. Furthermore, very unsymmetrical matrices can also be solved using this library. The library comes with four packages containing real and complex solvers, in both single and double precision versions. The source codes in the packages are grouped into particular classes. These classes are necessary in order to produce a reusable sparse linear solver component, which can be extended. Using class diagram representation, all classes are drawn as shown in Figure 2.7.


41

Fig. 2.7. Classes of sparse linear solver diagrams

Four classes, SingleReal, DoubleReal, SingleComplex, and DoubleComplex are derived from the base class, which is MathemathicalSolver class, for real and complex variables with single or double precisions. This base class contains common attributes of data and methods for the derived classes.

The derived classes are defined with polymorphism relationship with the base class. By doing so, user can access similar method with different operation depending on the defined object, whether single or double precision.

This is another form of software reuse, which is the reuse of existing or legacy software of proven capability. The linear sparse solver can be replaced with other solver, which may be a better, or an improved solver or even a proprietary code solver.

2.3.4 Classes for Data Input In order to prepare data for load flow analysis, two simple

standalone classes are developed. They are the ReadIEEEData class and ReadControlDevices class. The ReadIEEEData class is a group of tasks to read network data files based on IEEE common data format. Whereas, the ReadControlDevice class is a group of tasks to read controllable device data file such as for UPFC. Both classes are developed to read files in the text format (.txt). In our case, these classes use conventional variable name, which is

T M a th e m a tic a lS o lv e r

T S in g le R e a l T D o u b le R e a l T S in g le C o m p le x T D o u b le C o m p le x


42

different from the load flow solver that uses the developed power system model. These data input can also be replaced with any other ways of getting data such as by using database. This is the main advantage of separating data input from the load flow solver.

2.3.5 Graphical User Interface The classes for graphical user interface are developed by

reusing a commercial computer-aided drawing (CAD) component DbCAD. DbCAD is a library which has many services for Computer Aided Drawing (CAD) application [5]. DbCAD can manage the graphic entity as single vectors, which are selectable, editable, and displayable in the graphic window with specified properties (color, layer, line type, etc). The selectable and editable vectors are described in one or more standard database tables called graphic databases.

The GUI class is developed into a component and can be

used to draw one-line diagram and capturing database, which will be used to perform analysis such as load flow, symmetrical and unsymmetrical faults.

2.4 Component Development The development of load flow analysis software involves

various activities such as development of GUI, mathematical solver, database, and load flow analysis. These activities require knowledge in various disciplines such as mathematic, computer science and mainly engineering. Therefore, experts in those fields are required to produce the software. Technological changes due to rapid development in computing technology drive the demand to modify features available in these applications. Such changes are likes in the upgrading of computer processor, new input output devices, memory storage and graphical devices.


43

Another problem is if the load flow analysis application is coupled with other application such as GUI in the software, problem of modification will arise since the GUI part will also be affected. Therefore, the load flow analysis application needs to be developed independently from other parts. All of these problems made load flow analysis software development costly, time-consuming and complex tasks.

In order to solve such problems, CBD methodology is used

in developing such software. Each of the requirements can be developed into different components, which are independent between each other. The components then can be integrated into comprehensive power system analysis software.

A component is a part of a larger structure, an element contributing to the composition of the whole, with stable, defined interfaces. In computer science, a software component is defined as a piece of pre-built software with well-defined interfaces and behaviors, accessible only via its interface [6]. However, although all of software artifacts can be components, not all of them will meet their essential characteristics.

Software developers have been already familiar with a kind of

software components called software libraries for years. Typically, one of the popular software libraries is a .lib file. However, some constraints are often found such as: a) Working with libraries needs greater understandings how to fit

them into the application project. Unfortunately, most of libraries are not equipped with well-defined interfaces so that it is very difficult to plug the libraries into the software applications.

b) Moreover, apart of interfaces, many software developers use an approach that does not emphasize a good architecture definition. This makes reconstructions of software applications inevitable whenever a library is replaced with a better or newer version. (different vendor)


44

c) Most of the libraries only work well to a certain type of programming languages, which are used in their development. Problems in incompatible of data types are often found when a library is used in a different compiler or environment in which it was developed.

Technologies in software components evolve. Technologies like ActiveX, COM, DCOM from, Microsoft, VCL from Borland, and Java Bean are introduced and come to the edge. These component technologies are strongly supported by some Integrated Development Environment (IDE) tools. There are many IDEs that support components are available in the market today. Borland C++ Builder and Microsoft Visual C++ are the leading IDE and the best choices to develop both of software components and of software applications.

Borland C++ Builder is chosen as the IDE tool because it offers wide range of component platforms, such as Visual Component Library (VCL, which is produced by Borland), Component Object Model (COM, which is developed Microsoft), and Common Object Request Broker Architecture (CORBA developed by Object Management Group OMG).

The previously discussed classes are developed into

components and then integrated together as load flow analysis software. The interactions between these components in the application are shown in Figure 8.


45

DataBase

GUIData Input

Other Component

Linear Solver

Data PreparationFunctions

Output Functions

NR Method FD Method

Component

Functions

Legends

Fig. 2.8. Components in load flow application

Any of the components such as GUI and linear sparse solver could be replaced by other components. Adding and changing components are possible without affecting other components in the application. Therefore, at any time a better component could be chosen to replace any component.

In the application that has been developed there are two

way of providing data. The data can be prepared using a database component or from a component that read files in the format of text. Therefore, user has the choice to provide data whether by reading text files in the format of IEEE or from database. The data also can be placed into the database via the GUI. Other components such as to read data in other formats or to provide data for other analysis such as fault analysis and stability analysis can also be added into this application. Such addition will not affect the existing components inside this application.

2.5 Effects on Execution Time The impact of proposed implementation of reusability has been investigated by solving load flow with many test cases. Effects of matrix partitioning and component based development


46

were studied. The analysis was done over PC with the specification of Pentium IV, 2.4 GHz, 526 MB RAM.

2.5.1 Effect of Matrix Partitioning Approach Tests were carried out using IEEE 300 bus system, which is

modified to contain UPFCs. The execution times of the NR solution using partitioning and non-partitioning matrix approach are presented in Table 2.1.

Table 2.1 Comparison of execution time between matrix

partitioning and non- partitioning matrix for one complete iteration

CPU time (in second) Number of UPFC in the system Non-Matrix

Partitioning Matrix

Partitioning

% difference

1 0.109375 0.109375 0 2 0.113281 0.128906 13.8% 3 0.113281 0.128906 13.8% 4 0.121094 0.132812 14.1%

From the above results, it can be observed that the time taken in solving the load flow for one iteration in both approaches is quite similar. Although the matrix partitioning requires slightly more execution time compared to the non-partitioning approach, it is not so significant.

2.5.2 Component and Non-Component Execution Time Test This test is conducted to reveal the difference on execution

time between component software and non-component software. In order to do this, a component based software and non-component software with the same algorithm and object-oriented codes were developed. The component-based application is prepared by reusing the load flow and mathematical solver component, which were delivered in the library files. On the other hand, these files are


47

included into the non-component load flow application. In this test, the execution time for solving load flow is taken for both NR and FD methods.

The 118-bus system and 300-bus system were used as the

test system (No FACTS devices were included in the system). The results of complete iteration after convergence are presented in Table 2.2 for NR method and Table 2.3 for FD method.

Table 2.2 Newton Raphson load flow execution time

CPU time (in second)

Data test system Non-component Component

IEEE-118 0.027344 0.039062 IEEE-300 0.289062 0.312500

Table 2.3 Fast Decoupled load flow execution time

CPU time (in second)

Data test system Non-component Component

IEEE-118 <0.001 <0.001 IEEE-300 <0.001 0.01

From the results, it is clear that the component software application consume a slightly more time compared to the non-component. The difference is indeed small, which is around 0.01 to 0.05 seconds. The component application consumed more time as compared with non-component because the component application needs to read library files, which are separated from the application files when they are executed. Therefore, time is consumed in the process of transferring parameters message to and from the functions of the library. Whereas, in the non-component application, calling


48

function takes place in the same memory block of the application, which made the CPU time less than in the component application.

REFERENCES [1] A.F. Neyer, F.F. Wu and K. Imhof, “Object Oriented

Programming For Flexible Software: Example of A Load Flow”, IEEE Transaction on Power Systems, Vol. 5, No. 3, August 1990.

[2] Jun Zhu and L.Lubkeman, “Object-Oriented Development of Software Systems for Power System Simulation”, IEEE Trans. (Power App. Sys), Vol. 12, No. 2, May 1997.

[3] C.R. Fuerter-Esquivel, E. Acha and H. Ambriz-Perez, “A Comprehensive Newton-Raphson UPFC Model for the Quadratic Power Flow Solution of Practical Power Networks” IEEE Trans. on Power System, Vol. 15, No 1, Feb. 2000.

[4] James W.Demmel, John R.Gilbert and Xiaoye S.Li, “SuperLU User Guide”, which is available in http://www.nersc.gov/~xiaoye/SuperLU

[5] ABACO s.r.l. “DbCADev : User and Reference Guide”, ABACO s.r.l . 1999.

[6] Lars-Ola G.Osterlund (Jan 2000), “Component Technology”, IEEE Magazine (Computer Applications in Power, Vol. 13, pp 17-25.

[7] Khalid M. Nor, Taufiq A. Gani, Hazlie Mokhlis, “The Application of Component Based Methodology in Developing Visual Power System Analysis Tool”, Proceeding of the 22nd conference on IEEE PES PICA, Sydney, 2001.

Three-Phase Power-Flow Methods using Sequence Components

49

3 THREE-PHASE POWER-FLOW

METHODS USING SEQUENCE COMPONENTS

Mamdouh Abdel-Akher Khalid Mohamed Nor

1.3. Introduction

Power flow is an important tool in power system planning and operational studies. The single-phase power-flow algorithms assume a balanced power system operation and a balanced power system model. There are many cases where the system unbalance cannot be ignored due to unbalanced loads, untransposed transmission lines and a combination of balanced with unbalanced networks in distribution systems. Therefore a three-phase power-flow program that deals with unbalanced power system will be a useful analytical tool.

A variety of three-phase power-flow algorithms have been

studied for solving unbalanced power systems. These algorithms can be categorized into two groups. The first group solves a general network structure such as Newton Raphson method [1]-[5], Fast Decoupled method [3], [6], the hybrid method [7], and the bus admittance method [8]. The second group considers primarily the radial structure of distribution networks such as the compensation based method [9].


50

Unbalanced power systems can be modeled using phase

components without simplifications [10]. However the advantage of the application of sequence components is that the size of the problem is effectively reduced in comparison to phase components approach. In Newton Raphson power-flow, the size of the problem is reduced from a (6Nx6N) Jacobian matrix to a (2Nx2N) Jacobian matrix for positive sequence power-flow and two (NxN) admittance matrices for negative sequence and zero sequence nodal voltage equations. Zhang [3] showed that the factorization time of an admittance matrix of order (3Nx3N) is 70% more than the total factorization time of three admittance matrices of order (NxN). In addition, the sequence networks, positive sequence, negative sequence, and zero sequence can be solved using parallel computations [3].

Based on sequence components, the three-phase power-flow

problem in [2] is decomposed into three separate sub-problems. The line mutual coupling is included by putting the three sub-problems into an iterative scheme. The transformer phase shifts are included in the solution process by transforming the (6x6) transformer admittance matrix in phase components to its counterparts in sequence components. The (6x6) transformer admittance matrix in sequence components is included in the overall (3Nx3N) admittance matrix.

The decoupled sequence components transmission line model

was introduced in [3]. This line model allows the overall (3Nx3N) sequence component admittance matrix to be decoupled into positive sequence, negative sequence, and zero sequence admittance matrices. The decoupling compensation power-flow methods [3], [8] use the conventional sequence transformer model [10] where the transformer phase shifts are difficult to be incorporated.


51

In comparison to phase components, implementation of sequence components approach so far faced two problems to analyze unbalanced power system. The first is the coupling in the untransposed transmission line in sequence components model still exists. Secondly, the phase shifts introduced by special transformer connections are difficult to be represented [10]. On the other hand, in phase components, the coupling between lines and the phase shifts are included in the phase components model.

All previous implementations of unbalanced three-phase power-flow based on sequence components used a single-phase power-flow which is modified from the traditional balanced power-flow algorithms. Zhang however suggested the possibility of sequence components power-flow to use the traditional balanced single-phase power-flow for solving the positive sequence network [3].

In this work, the sequence components transformer model [2]

and the decoupled sequence components transmission line model [3] are used together for developing an improved sequence power-flow solution. As a result, the untransposed transmission lines as well as the phase shifts introduced by special transformer connections are considered in the sequence components power-flow solution process. In addition, the injected powers and currents due to loads and untransposed transmission lines have been formulated such that the single-phase power-flow programs can be called as routines for solving the positive sequence power-flow.

The proposed sequence power-flow methods are tested with

different case studies and compared with the sequence-decoupled power-flow methods [3], [8] and the hybrid method [7]. The performance of the proposed methods is further examined by comparison with a phase-coordinates Newton Raphson program.


52

3.1. Sequence Component Power System Model 3.1.1. Sequence Generator Model

Figure 3.1 shows the generator model for power-flow programs. The model is represented with three uncoupled sequence circuits [3], [4], [11]. The positive sequence reactance and EMF behind it are not introduced; this actually presents the case in single-phase power-flow model.

VSpecified

V1

y2

V2

y0

V0

Figure 3.1 Sequence component model of generator

If there is unbalance in a power system, current will flow in both

the negative sequence and zero sequence components of the generator model, resulting in three-phase unbalanced voltages at the generator busbar.


53

Table 3.1 Sequence Component Transformer Model Connection Equivalent Circuit/Admittance Matrix Bus P

Bus S Positive Negative Zero

ys.c.P S

ys.c.P S ys.c.P S

ys.c.P S

ys.c.P S ys.c.P S

⎥⎦

⎤⎢⎣

⎡−∠−

∠−

sm

mp

yyyy

3030

⎥⎦

⎤⎢⎣

⎡∠−

−∠−

sm

mp

yyyy

3030

ys.c.P S

ys.c.P S

ys.c.P S ys.c.P S

⎥⎦

⎤⎢⎣

⎡−∠−

∠−

sm

mp

yyyy

3030

⎥⎦

⎤⎢⎣

⎡∠−

−∠−

sm

mp

yyyy

3030

ys.c.P S

ys.c.P S

ys.c.P S ys.c.P S

Where: P primary winding, S secondary winding, star connection, grounded star connection,

delta connection, and ..csmsp yyyy === , where ..csy is the short circuit admittance of a transformer.

3.1.2. Sequence Transformer Model The transformer model is established by transforming the overall

transformer admittance matrix in phase components to its counterparts in sequence components as follows [2]:

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=

−

TT

YT

TY abc

1

012 (3.1)

Where:

02

2

1201and11

111

∠=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

aaaaaT

The resulting (6x6) sequence admittance matrix is used for

constructing the sequence component transformer model that is used for building the decoupled sequence admittance matrices. The


54

models for different transformer connections are summarized in Table 3.1. The positive sequence and negative sequence networks of the Y-∆ transformers, in Table 3.1, are expressed by admittance matrices as the phase shift cannot be suitably incorporated in an equivalent circuit. In Table I, if both mp yy − and ms yy − are not ignored, the transformer equivalent circuits should be modified to the typical symmetrical component π-equivalent model [10].

3.1.3. Sequence Transmission Line Model When a transmission line is balanced or transposed, the

admittance matrix in phase variables will be full and symmetrical. Hence, the transmission line can be represented by three uncoupled sequence circuits as given in (3.2).

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=s

s

s

s

z

z

z

yy

y

yy

y

2

1

0

2

1

0

2012Z012 YY (3.2)

On the other hand, if any transmission line is unbalanced or

untransposed, the phase admittance matrix will also be full but symmetrical in one diagonal axis. Therefore, the sequence admittance matrix will be full and unsymmetrical as follows:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=sss

sss

sss

s

zzz

zzz

zzz

yyyyyyyyy

yyyyyyyyy

222120

121110

020100

222120

121110

020100

2012Z012 YY (3.3)


55

⎯⎯ →⎯ 012_I i ⎯⎯⎯← 012_Ij

Bus i Bus j

012_Vi 012_Vj

Z012Y

2

S012Y

2

S012Y

Fig. 3.2. Sequence coupled line model

The sequence coupled line model is characterized by weak

mutual coupling, so the coupled line model in Fig. 3.2 can be decoupled into three independent sequence circuits [3]. This can be achieved by eliminating the off-diagonal elements in (3.3) by replacing them with certain compensation current injections at both ends of the unbalanced line.

The injected currents for an unbalanced line (see Fig. 3.2)

connected between busbar i and busbar j due to the off-diagonal elements in both the shunt and series admittance matrices can be calculated as follows:- i. Off-diagonal current injection in the series admittance matrix:

⎪⎭

⎪⎬

⎫

−+−=Δ

−+−=Δ

−+−=Δ

)()(

)()(

)()(

112100202

221200101

220211010

jizjizij

jizjizij

jizjizij

VVyVVyI

VVyVVyI

VVyVVyI (3.4)

ii. Off-diagonal current injection in the shunt admittance matrix at any busbar, say bus i, is as follows:

( )( )( )⎪⎭

⎪⎬

⎫

+=Δ

+=Δ

+=Δ

isisii

isisii

isisii

VyVyI

VyVyI

VyVyI

2121102

2121101

2021010 (3.5)


56

The final sequence current injections at the line busbars are:

2or 1or 0_Line_

_Line_

=Δ−Δ=Δ

Δ+Δ=Δ

nijn

jjnnj

ijn

iinni

IIIjbusat

IIIibusat (3.6)

In the traditional balanced power-flow studies, the positive sequence current injection is transformed to positive sequence power injection as follows:

( )( ) ⎪⎭

⎪⎬⎫

Δ=Δ

Δ=Δ*

Line_111Line

*1Line_11Line

jj_j_

ij_i_

IVS

IVS (3.7)

Now (3.3) can be rewritten similar to (3.2) for transposed lines

as follows: -

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=s

s

s

z

z

z

yy

y

yy

y

22

11

00

22

11

00

2YY S012

Z012

(3.8)

Zy11

iBus jBus1_iV 1_jV

Sy11 Sy11

1Line __iSΔ 1Line __jSΔ

a) Positive sequence model


57

Zy22

2_iV 2_jV

Sy22Sy22

2Line __iIΔ 2Line__jIΔ

c) Negative sequence model

Zy00

0_iV 0_jV

Sy00Sy00

0Line __iIΔ 0Line__jIΔ

a) Zero sequence model

Fig. 3.3. Sequence decoupled line model

Fig. 3.3 shows the three decoupled sequence networks that represent the coupled line model in Fig. 2. The mutual coupling is included by the current and power injections at the line busbars. Equations (3.2) and (3.8) are used for constructing the decoupled sequence admittance matrices whereas the coupling effect is considered by the current and power injections in (3.6) and (3.7) respectively.

3.2. Sequence power-flow methods 3.2.1. Busbars Specifications

The sequence power-flow methods use single-phase power-flow specifications which include three types:


58

Slack Busbar: At the slack busbar both the positive sequence voltage magnitude and the angle are specified:

Specified1

Specified1

BusbarSlack 1 θ∠=VV (3.9)

Voltage Controlled Busbars- PV: The PV busbar is a generator busbar in which both the positive sequence voltage magnitude and the total generated power are specified as shown in (3.10):

⎪⎭

⎪⎬⎫

=

=

3Generation TotalBusbar PV1

Specified1

Busbar PV1

PPVV

(3.10)

Load Busbars: For balanced loads, the specified power for each phase is calculated from the total power demand at the load busbar:

( ) cbamjQPSm or or 3TotalTotalSpecified =+= (3.11)

On the other hand, the specified powers for unbalanced loads are specified individually for each phase as:

cbamjQPS mmm or or SpecifiedSpecifiedSpecified =+= (3.12)

The specified three-phase powers in (3.11) and (3.12) are used for calculating the phase components injected currents. The injected current of phase m (m=a, b, or c) is given as follows:

( )*Specifiedmmm VSI = (3.13)

The injected currents in the sequence networks due to loads are calculated by transforming the phase components injected currents in (3.13) to their counterparts in sequence components as shown in (3.14):

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

c

b

a

III

aaaa

III

2

2

2

1

0

11

111

31

(3.14)

Then, the load sequence specifications for the three sequence networks are calculated from the sequence current injections in (3.14). In the traditional balanced power-flow studies, the current injection of the positive sequence network is modified to power


59

injection as given in (3.15):

( )⎪⎪⎭

⎪⎪⎬

⎫

=

=

=

2SpecifiedLoad_2

*11

SpecifiedLoad_1

0SpecifiedLoad_0

II

IVS

II

(3.15)

The total sequence specifications can be evaluated from (3.6), (3.7), and (3.15). Equations (3.6) and (3.7) present the coupling effect in the untransposed transmission lines whereas (3.15) presents the injection due to the actual load in the network. Hence, the final sequence specifications at a busbar i are calculated as follows:

⎪⎪⎭

⎪⎪⎬

⎫

Δ+=

Δ+=

∑

∑

∈=

∈=

M

ikk

kiii

M

ikk

kiii

QQQ

PPP

,1_Line_1

Specified_Load_1

Specified1_

,1_Line_1

Specified_Load_1

Specified_1

(3.16)

⎪⎪⎭

⎪⎪⎬

⎫

Δ−−=

Δ−−=

∑

∑

∈=

∈=

M

ikk

kii_i

M

ikk

ki_i_i

III

III

,1_Line_0

SpecifiedLoad_0

Specified_0

,1Line_2

SpecifiedLoad_2

Specified2_

(3.17)

Where M refers to the total number of untransposed transmission lines connected to busbar i and the summation in (3.16) and (3.17) gives the total injected currents and powers at busbar i due to the untransposed transmission lines.

3.2.2. Positive Sequence Power-flow

Equations (3.9), (3.10), and (3.16) present the positive sequence power-flow specifications that are the same to those used by any single-phase power-flow algorithm. Both (3.9) and (3.10) present the generator specified values that are constant and need not to be updated during the sequence power-flow solution process. The


60

specified values in (3.16) do not mean only the loads in the actual network, but also include the coupling effect. This means that there may be, at certain busbar, no actual load but there is a specified load in the positive sequence network. Equation (3.16) needs to be updated and supplied to the single-phase power-flow routine for each iteration in the overall solution process. The calculated positive sequence power for N busbars power system is:

( )

( )⎪⎪⎭

⎪⎪⎬

⎫

−=

+=

∑

∑

=

=

N

kikikikikkii

N

kikikikikkii

BGVVQ

BGVVP

11_1_1_1_1_1_

Calculated1_

11_1_1_1_1_1_

Calculated1_

cossin

sincos

θθ

θθ (3.18)

The positive sequence mismatch at busbar i is given as:

⎪⎭

⎪⎬⎫

−=Δ

−=ΔCalculated

1_Specified

1_1_

Calculated1_

Specified1_1_

iii

iii

QQQ

PPP (3.19)

For PV busbars, only the active positive sequence power mismatch need to be calculated in (3.19) using the specified generation in (3.10).

The traditional Newton Raphson and Fast Decoupled single-phase power-flow methods were chosen for solving the positive sequence power-flow due to their well established usage in the industry. Other methods, such as current injection method [12] may also be used with appropriate adjustment. The basic Newton Raphson is described by (3.20) while Fast Decoupled is given by (3.21).

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡

1

1

1

1

11

11

ΔQΔP

ΔVΔθ

LJNH

(3.20)

⎪⎭

⎪⎬⎫

=

=

11''

1

11'1

ΔQΔVBΔPθΔB

(3.21)


61

3.2.3. Negative Sequence and Zero Sequence Nodal Voltage Equations

The negative sequence and zero sequence voltages are calculated by solving (3.22) and (3.23) respectively using the specified current injections in (3.17).

Specified222 IVY = (3.22) Specfied000 IVY = (3.23)

3.2.4. Sparse Linear Solver In the proposed implementation, the power system sparsity is

exploited by using SuperLU library routines [13]. SuperLU is a general purpose library for the direct solution of large, sparse, and nonsymmetrical systems of linear equations.

3.2.5. Sequence Power-Flow Algorithm

Fig. 3.4 shows the sequence power-flow solution algorithm. The dashed block shows Newton Raphson single-phase power-flow routine which is completely decoupled from the overall solution process. The Newton Raphson algorithm can be replaced by a Fast Decoupled algorithm as done in this study or by any other single-phase power-flow algorithm. The sequence power-flow solution process starts with: -

1. Constructing the sequence admittance matrices according to the sequence power system models in section 3.2.

2.Factorizing the negative sequence and zero sequence admittance matrices using SuperLU library routines.

3.Calculating the specified generation for positive sequence power-flow which is fixed and need not to be updated during the solution process.

4.Calculating the injected phase components currents due to the specified loads based on an initial set of three-phase voltages using (3.13). (The initial three-phase voltages are used in the first iteration only, in the following iterations, the updated three-phase voltages are used).

5.Transforming the injected phase components currents at each


62

busbar to their counterparts in sequence components using (3.14).

6.Combining the injected sequence components powers and currents of the specified loads and untransposed lines together for calculating the final sequence specified values using (3.16) and (3.17).

7.Solving (3.22) and (3.23) by using the specified negative sequence and zero sequence currents to compute the negative sequence and zero sequence voltages. The positive sequence specified powers in (3.16) are supplied to the single-phase power-flow routine for calculating the positive sequence voltages.

8.The new set of three-phase voltages is calculated. 9.Go to step 4 The process is repeated until certain preset permissible tolerance

is reached. In Fig. 3.4, the convergence is measured using positive sequence power mismatch criterion. Other convergence criteria can be applied such as positive sequence voltage mismatch or phase voltage mismatch.


63

Calculate

aI bI cIc

b

a

SSS

Calculate

1I 2I 0I

Calculate

1S

Calculate

1SΔ

ConvergenceYes

Cal

cula

te U

ntra

nspo

sed

Lin

es I

njec

ted

Cur

rent

Specified Loads

Construct

Factorization ofand

Data PreparationData Structure/Initialization

Solve

Update

Update

111 SVJ Δ=Δ1J

1VNo

Solve

222 IVY =Solve

000 IVY =

Data Structure & Factorization of

1J

Cou

plin

g E

ffec

t

Update

aVbV cV

1Y 2Y 0Y

Dashed Block : Single Phase Power Flow Routine

Posi

tive

Seq

uenc

e Vo

ltage

s

2Y 0Y

Spec

ified

Pow

er

Stop

Fig. 3.4. Sequence Newton Raphson method

3.3. Results and Discussion

The sequence power-flow methods, sequence Newton Raphson and sequence Fast Decoupled, are developed according to the algorithm shown in Fig. 3.4. These methods are coded using two single-phase power-flow programs, Newton Raphson and Fast Decoupled [14]. These routines are used for solving positive


64

sequence network without affecting their capability of solving balanced power systems.

The accuracy of the sequence power-flow methods are tested

and compared with the decoupling compensation power-flow methods [3], [8], and the hybrid method [7] whereas the performance are examined by comparison with a phase-coordinates Newton Raphson program. A 345 kV test system [7] is used for studying the following cases:-

Case A: Configuration ABC-CBA well transposed lines Case B: Configuration ABC-CBA untransposed lines Case C: Configuration ABC-CBA untransposed lines and

unbalanced load at busbar ‘14’, Sa=230+j37, Sb=240+j48, and Sc=280+j65.

Case D: Configuration ABC-ABC- well transposed lines Case E: Configuration ABC-ABC- untransposed lines

All case studies were run on a Pentium 4, 2.66 GHz CPU with a

512 kB cache, 512 MB RAM and a MS Windows 2000 operating system. Case C was used for comparison between performance of the sequence power-flow and the phase coordinates Newton Raphson power-flow program.

The sequence power-flow methods utilize SuperLU linear solver [13] and component technology as a programming methodology [14] whereas the phase-coordinates Newton Raphson program is a commercial grade program based on object-oriented methods using a different sparse linear solver. These differences may have some slight contribution in the performance of the programs. However the large difference in performance between the sequence components methods and the phase-coordinates Newton Raphson program allows reasonable conclusions to be drawn on the performance of the proposed methods.

In addition to the above test cases, the modified IEEE 14 node network [15] is used to test the proposed methods under heavy loading conditions.


65

3.3.1. Balanced Power System and Balanced Load

Case A and case D present the balanced system case study. The results are identical with those obtained from single-phase power-flow programs. The results differ only in the expected 30 degree phase shift accompanied to Y-Δ transformer connections.

3.3.2. Unbalanced Power System and Balanced Load

Case B and case E present the unbalanced power system and balanced load case study. The three-phase busbar voltages, total generation, and total system losses are given in both Table A3.1 and Table A3.2 in the Appendix for case B and case E respectively. The results are identical with those given in Table 5 and Table 6 in [7]. The slight difference in the three-phase busbar voltages at certain busbars, such as phase ‘a’ at busbar ‘32’, is due to the generator model [7]. The positive sequence reactance and the EMF behind it are introduced in [7] while in this work the generator model in Fig. 3.1 is used.

3.3.3. Unbalanced Power System and Unbalanced Load

Case C presents the case of untransposed transmission system and unbalanced load at busbar ‘14’. The results in Table 3.2 are different from those in Table 3.3 where the sequence decoupling compensation methods are used [3], [8]. This is because the sequence power-flow methods are less sensitive to errors in the injected currents of the untransposed transmission lines, i.e. the program will converge but with wrong results. These errors are corrected in this work by the derivation of the current injections due to untransposed transmission lines. Then, these current injections are combined with the sequence current injections due to the actual loads in one set of sequence specifications as was described by (3.16) and (3.17).


66

Table 3.2 Results of Case C-Proposed Methods

5 iterations, positive sequence voltage mismatch 10-4 pu Bus Type Phase A Phase B Phase C 14 0 1.006 20.1 0.999 -100.0 0.980 138.0 32 0 0.986 12.0 0.996 -107.9 0.989 130.8 41 0 0.954 6.6 0.975 -113.1 0.970 126.0 45 0 0.971 5.6 0.989 -114.0 0.984 125.1 70 0 1.015 22.6 1.011 -97.4 0.996 141.3 71 0 0.994 14.7 1.004 -105.3 0.997 133.5 77 0 0.979 9.1 1.000 -110.7 0.996 128.4 97 0 1.004 20.7 1.007 -99.7 0.994 139.7 98 0 1.014 23.9 1.017 -96.5 1.004 143.0 99 2 1.026 -1.7 1.034 -121.7 1.030 117.9 100 0 1.019 25.1 1.022 -95.4 1.010 144.4 101 2 1.026 -0.3 1.034 -120.3 1.030 119.3 102 0 1.015 24.2 1.019 -96.3 1.007 143.4 103 0 1.004 20.4 1.006 -99.8 0.996 139.1 108 2 1.026 -0.3 1.034 -120.3 1.030 119.3 113 3 1.026 0.1 1.034 -119.9 1.030 119.7 114 0 1.018 25.2 1.022 -95.3 1.009 144.4 120 2 1.026 0.7 1.034 -119.4 1.030 120.2 134 0 0.988 9.7 1.006 -110.1 1.001 129.0 181 2 1.026 -1.7 1.034 -121.7 1.030 117.9 190 0 0.984 15.3 0.986 -104.8 0.975 134.0 193 2 1.045 -13.7 1.049 -133.2 1.055 106.4 196 0 0.995 10.4 1.011 -109.4 1.005 129.7 312 2 1.026 -1.7 1.034 -121.7 1.030 117.9

Load 3620.000 650.000 Generation 3652.107 858.231

Tota

l M

VA

Losses 32.080 207.732


67

Sample of the power-flow results for case C is given in Table

3.4, the results include two busbars that have specified values, the first is the load busbar ‘14’ and the second is the generator busbar ‘312’. The load constraint at busbar ‘14’ for phases a, b, and c is equal to Sa=230+j37, Sb=240+j48, and Sc=280+j65 MVA respectively whereas at the generator busbar ‘312’; the total specified power constraint is equal to 250 MW. Table 3.4 shows that the power delivered to the load at each phase satisfies the load constraint at busbar ‘14’. The total power leaves the generator busbar ‘312’ is equal to 81.06+81.72+87.22 = 250 MW which also satisfies the total generation constraint.

Table 3.3 Results of Case C - Power Flow Methods [3], [8] 3 iterations, positive sequence voltage mismatch 10-4 pu

Bus Type Phase A Phase B Phase C 14 0 1.018 19.5 0.998 -100.6 0.967 139.1 32 0 0.998 12.0 0.999 -108.6 0.974 131.2 41 0 0.964 7.1 0.978 -113.8 0.957 126.1 45 0 0.980 6.1 0.993 -114.7 0.971 125.2 70 0 1.024 22.2 1.011 -97.8 0.987 142.0 71 0 1.006 14.8 1.006 -105.9 0.983 133.9 77 0 0.990 9.5 1.003 -111.4 0.982 128.5 97 0 1.012 20.4 1.003 -100.0 0.989 140.2 98 0 1.022 23.6 1.013 -96.8 0.999 143.2 99 2 1.030 -1.6 1.034 -122.0 1.026 118.0 100 0 1.025 24.9 1.017 -95.6 1.008 144.8 101 2 1.030 -0.2 1.035 -120.6 1.025 119.4 102 0 1.023 24.0 1.015 -96.5 1.003 143.9 103 0 1.014 20.1 1.006 -100.3 0.984 139.7 108 2 1.030 -0.2 1.035 -120.6 1.025 119.4 113 3 1.030 0.3 1.034 -120.1 1.026 119.9 114 0 1.025 24.9 1.017 -95.5 1.008 144.9 120 2 1.030 0.8 1.034 -119.6 1.026 120.4


68

3 iterations, positive sequence voltage mismatch 10-4 pu Bus Type Phase A Phase B Phase C 134 0 0.997 10.1 1.008 -110.8 0.988 129.2 181 2 1.030 -1.6 1.034 -122.0 1.026 118.0 190 0 0.995 15.1 0.987 -105.4 0.962 134.6 193 2 1.045 -13.4 1.056 -133.4 1.050 106.1 196 0 1.003 10.7 1.013 -110.1 0.994 129.8 312 2 1.030 -1.6 1.034 -122.0 1.026 118.0 3.3.4. Convergence Characteristics

Table 5 shows the convergence characteristic of the sequence power-flow methods for the studied cases A, B, C, D, and E. The sequence specifications and the three-phase voltages are updated after calculating each sequence voltage (positive-negative-zero). If the sequence specifications are updated only once every iteration step, as shown in Fig. 3.4, the number of iterations required for convergence will increase depending on the degree of the unbalance in the network.

The table shows, in all studied cases, that the sequence Newton

Raphson and sequence Fast Decoupled methods have comparable number of iterations. The impact of the negative sequence and zero sequence networks seemed to have made the convergence characteristics of the sequence Fast Decoupled similar to the sequence Newton Raphson unlike the case of balanced power-flow programs.


69

Table 3.4 Sample of Power-Flow Results Of Case C Bus i Bus j Phase From i To j From j To i

70 103 A 115.95 4.23 -115.33 -6.92 B 115.83 1.37 -115.45 -3.92 C 111.95 2.00 -111.47 -4.85 A 114.16 2.62 -113.60 -5.33 b 115.91 3.17 -115.51 -5.71 c 113.46 0.11 -113.01 -2.98

70 98 a -100.52 -10.99 99.76 6.43 b -102.70 -12.34 103.21 7.11 c -111.09 -19.20 111.94 16.10 a -99.45 -9.65 98.64 5.13 b -102.56 -13.40 103.10 8.17 c -112.97 -17.79 113.79 14.67

70 100 a -131.53 -17.60 130.62 12.93 b -133.03 -18.24 134.03 12.36 c -139.56 -25.01 140.97 22.75 a -130.04 -15.90 129.06 11.24 b -132.86 -19.61 133.88 13.73 c -141.95 -23.31 143.34 21.06

70 14 a 230.26 47.62 -230.00 -37.00 b 240.30 59.89 -240.00 -48.00 c 280.42 82.02 -280.00 -65.00

312 100 a 81.06 22.92 -79.35 -11.94 b 81.72 16.49 -82.74 -12.79 c 87.22 19.16 -87.75 -18.20


70

Table 3.5 Convergence Characteristic of The Proposed Methods

Method Sequence Newton Raphson

Sequence fast decoupled

Phase Voltage

Mismatch 10-3 10-4 10-5 10-3 10-4 10-5

A 5 6 8 4 5 7 B 5 6 8 4 5 7 C 5 6 7 4 5 7 D 5 7 9 4 6 8 E 5 7 9 4 6 8

Table 3.6 Performance In Terms Of Execution Time and CPU Memory

Method CPU Time (p.u)*

CPU Memory (p.u)*

Phase coordinate Newton Raphson 1.00 1.00

Sequence Newton Raphson 0.15 0.24 Sequence Fast Decoupled 0.11 0.20 *Time and memory are normalized based on phase coordinate Newton Raphson


71

0

10

20

30

40

50

2 3 4 5 6 7 8 9 10

R/X Ratio

Num

ber o

f Ite

ratio

ns

Phase Coordinate Newton Raphson

Sequence Newton Raphson

Sequence Fast Decoupled

Fig. 3.5 Performance of sequence power flow methods for high R/X ratio

3.3.5. Time and Memory Requirements In the comparison performance between the proposed sequence

power-flow and the phase-coordinates Newton Raphson method, Case C was run for 100 times. The average CPU time and memory usage are reported in Table 3.6.

The fast execution time and the low memory requirements of the sequence power-flow methods are expected since the execution time and memory requirements mainly depend on the size of the problem to be solved. The Jacobian matrix, negative sequence and zero sequence admittance matrices of the sequence Newton Raphson for N busbars and M branches power system is a (6N+2M) nonzero elements instead of (36N+2M) nonzero elements for the phase-coordinates Jacobian matrix. This is a saving in CPU memory of about 83%. This saving is about the same saving shown in Table VI. The difference can be attributed to other memory requirements arising from the differences in the code implementation and the linear solver.


72

The algorithm execution time totally depends on the solution process. In the sequence Newton Raphson, apart from the negative sequence and zero sequence admittance matrices which are factorized only once in the beginning of iteration process, a (2Nx2N) Jacobian matrix is re-factorized and updated for each iteration. In phase-coordinates Newton Raphson, a (6Nx6N) Jacobian matrix is re-factorized and updated for every iteration step. This leads to a large saving in execution time as shown in Table VI. The result in Table VI is consistent with the experience reported in [3] which has been discussed earlier in section I.

The sequence Fast Decoupled method when compared with the sequence Newton Raphson, is between 20% to 30% faster. This is less saving compared with single-phase power-flow because in three-phase power-flow, the sequence Fast Decoupled and the sequence Newton Raphson have to solve the same negative and zero sequence nodal voltage equations.

3.3.6. Effect of High R/X Ratio

The effect of increasing the line R/X ratio is studied for both the proposed methods and the phase-coordinates Newton Raphson method using the test case C. The R/X ratio is increased till the divergence occurred as shown in Fig. 3.5. The figure shows that the sequence Newton Raphson has similar characteristics to the phase-coordinates Newton Raphson method.

Although sequence Fast Decoupled takes less number of iterations than sequence Newton Raphson for normal R/X ratio as given in Table 3.5, the number of iterations increases when the R/X ratio is increased as shown in Fig. 5. Similar observations were reported in [3], that the sequence Fast Decoupled is less sensitive for unbalanced loads but sensitive for R/X ratio. Also it is noted with some investigations that the characteristic of the proposed sequence Fast Decoupled method is similar to the balanced Fast Decoupled algorithm for high R/X ratio.


73

Table 3.7: Comparative convergence characteristics of the IEEE 14 bus network

Proposed Ref. [A] Positive sequence power mismatch

0.0001 p.u. Methods: A

Balanced power-flow

Methods: B Sequence power-

flow methods

Methods C Loading Factor

NR FD SNR SFD NR Phase NR

0.5 3 4 3 4 3 3 0.7 3 4 3 4 3 3 1.3 3 5 4 5 3 3 1.4 3 5 4 5 3 3 1.5 3 5 4 5 4 4 2.0 3 6 4 6 4 4 2.1 3 6 4 6 4 4 2.4 3 7 4 7 4 4 2.5 4 8 4 8 4 4 2.7 4 8 4 8 4 4 2.8 4 9 4 9 4 4 3.0 4 10 5 10 5 5 3.6 4 16 5 16 5 5 3.9 5 32 6 32 6 6 4.0* 6 38 6 38 7 7 4.1 diverge diverge diverge diverge diverge

* The positive sequence power mismatch at load factor equals 4 is taken to be 0.01 p.u All the loads are assumed to PQ Methods A: the conventional balanced NR and FD power-flow methods; these methods are also used in the sequence power-flow algorithm for solving the positive sequence network, Methods B: sequence decoupled power-flow methods, Methods C: the conventional balanced power-flow (NR) and the phase coordinates unbalanced power-flow (PNR) reported in [A].


74

3.4. Effect of High Loading conditions In this test, the proposed power-flow is compared with the phase

coordinates current injection method (CIM) with heavy loading conditions [15]. This test is carried out using the modified three-phase IEEE 14 node network [15]. In this test, the conventional balanced three-phase power-flow methods in Table 3.7 are the same as those used to solve the positive sequence power-flow problem. The zero-sequence impedances were taken to be three times the corresponding positive sequence impedance [15]. The results of IEEE 14 bus network for the sequence power flow methods as well as methods reported in [15] are summarized in Table 3.7.

Table 3.7 shows that all the methods start to diverge when the loading factor is higher than four. The sequence power-flow method has similar convergence characteristics to the corresponding conventional balanced power-flow methods. In particular, the conventional balanced power-flow method [15] requires 7 iterations at a loading factor equal to 4. In this test, it was found that the proposed conventional balanced power-flow methods, and in consequence, the sequence decoupled power-flow methods cannot converge at a positive sequence power mismatch of a 0.0001 p.u. However, when the positive sequence power mismatch is reduced to a 0.01 p.u, both the conventional balanced and sequence decoupled power-flow methods converge as given in Table 3.7. This shows that the sequence power-flow methods inherit the characteristics of the positive sequence power-flow solver.

3.5. Conclusions

The chapter has presented the formulation and the solution of the three-phase power-flow problem using sequence components. The results of busbar voltages for different studied cases are the same as those obtained from a three-phase power-flow programs developed in phase components.

The improved sequence component transformer model and the decoupled sequence line model overcome the disadvantage of


75

using sequence components and at the same time retaining the advantage of dealing with matrices of smaller size. This advantage leads to a large saving in time and memory when the proposed methods are compared with a phase-coordinates Newton Raphson method. In addition, the proposed sequence Newton Raphson method has similar convergence characteristics of the phase-coordinate Newton Raphson method.

The power and current injections of the sequence networks have

been formulated such that single-phase power-flow programs can be used in the three-phase power-flow algorithm. This allows balanced and unbalanced power-flow algorithms to be integrated in a single application.

REFERENCES [1] K. A. Birt, J. J. Graff, J. D. McDonald, and A. H. El-Abiad,

“Three phase load flow program”, IEEE Trans. on Power Apparatus and Systems, vol. 95, no. 1, pp 59-65, Jan/Feb, 1976.

[2] K. L. Lo and C. Zhang, “Decomposed three-phase power flow solution using the sequence component frame”, Proc. IEE, Generation, Transmission, and Distribution, vol. 140, no. 3, pp 181-188, May 1993.

[3] X. P. Zhang, “Fast three phase load flow methods”, IEEE Trans. on Power Systems, vol. 11, no. 3, pp 1547-1553, Aug. 1996.

[4] B. C. Smith and J. Arrillaga, “Improved three-phase load flow using phase and sequence components”, Proc. IEE Generation, Transmission, and Distribution, vol. 145, no. 3, pp 245-250, May 1998.

[5] P. A. N. Garcia, J. L. R. Pereria, S. Cameiro J. R, V. M. Da Costa, and N. Martins, “Three-phase power flow calculation using the current injection method”, IEEE


76

Trans. on Power Systems, vol. 15, no. 2, pp 508-514, May 2000.

[6] J. Arrillaga and C. P. Arnold, “ Fast decoupled three phase load flow”, Proc. IEE, vol. 125, no. 8, pp 734-740, 1978.

[7] B. K. Chen, M. S. Chen, R. R. Shoults, and C. C. Liang, “ Hybrid three phase load flow”, Proc. IEE Generation, Transmission, and Distribution, vol. 137, no. 3, pp177-185, May 1990.

[8] X. P. Zhang, and H. Chen, “Asymmetrical three phase load flow based on symmetrical component theory”, Proc. IEE Generation, Transmission, and Distribution, vol. 137, no. 3, pp 248-252, May 1994.

[9] C. S. Cheng and D. Shirmohammadi, “A three-phase power flow method for real-time distribution system analysis” IEEE Trans. on Power Systems, vol. 10, pp. 671–679, May 1995.

[10] J. Arrillaga and N. R. Watson, “Computer modeling of electrical power systems”, Book, 2nd Edition, John Wiley & Sons LTD, pp 11-13, pp 48-51, and pp 111-115, 2001.

[11] N-Q Dinh and J. Arrillaga, “A salient-pole generator model for harmonic analysis”, IEEE Trans. on Power Systems, vol. 16, no. 4, pp 609-615, Nov. 2001.

[12] V. M. Da Costa, N. Martins, and J. L. Pereira, ”Developments in the Newton Raphson power flow formulation based on current injections”, IEEE Trans. on Power Systems, vol. 14, no. 4, pp 1320-1326, Nov. 1999.

[13] J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, and J. W. H. Liu, “A supernodal approach to sparse partial pivoting”, SIAM Journal on Matrix Analysis and Applications, vol. 20, no. 3, pp 720 -755, 1999.

[14] K. M. Nor, H. Mokhlis, and T. A. Gani, “Reusability techniques in load-flow analysis computer program” IEEE Trans. on Power System, vol. 19, no. 4, pp. 1754-1762, Nov. 2004.

[15] P. A. N. Garcia, J. L. R. Pereira, S. Jr. Carneiro, M. P. Vinagre, F. V. Gomes “Improvements in the representation


77

of PV buses on three-phase distribution power flow” IEEE Transactions on Power Delivery, No. 2. Vol. 19, pp. 894- 896, April 2004.

[16] Kang Xiaoning, Suonan Jiale, Song Guobing, and Fu Wei Zhiqian Bo, “A Novel Three Phase Load Flow Algorithm Based on Symmetrical Components for Distribution Systems”, universities power engineering conference, UPEC 2008

Appendix

Table A1 Results of Case B

5 Iterations, Positive sequence voltage mismatch 10-4 p.u. Bus Type Phase A Phase B Phase C 14 0 0.991 19.5 0.996 -100.7 0.998 139.3 32 0 0.978 11.7 0.992 -108.4 1.002 131.5 41 0 0.951 6.4 0.969 -113.2 0.979 126.5 45 0 0.969 5.4 0.984 -114.1 0.992 125.5 70 0 1.003 22.3 1.008 -97.9 1.011 142.1 71 0 0.986 14.5 1.000 -105.7 1.010 134.2 77 0 0.977 8.9 0.994 -110.8 1.004 128.8 97 0 1.000 20.3 1.003 -99.8 1.002 140.2 98 0 1.010 23.6 1.013 -96.6 1.012 143.4 99 2 1.029 -1.8 1.031 -121.8 1.030 118.1 100 0 1.017 24.8 1.018 -95.3 1.016 144.7 101 2 1.029 -0.4 1.031 -120.4 1.030 119.5 102 0 1.013 23.9 1.015 -96.2 1.013 143.7 103 0 0.995 20.1 1.003 -100.2 1.008 139.8 108 2 1.029 -0.4 1.031 -120.4 1.030 119.5 113 3 1.029 0.0 1.031 -120.0 1.030 119.9 114 0 1.017 24.9 1.018 -95.3 1.015 144.7 120 2 1.029 0.5 1.031 -119.5 1.030 120.5 134 0 0.986 9.5 1.000 -110.2 1.009 129.4


78

5 Iterations, Positive sequence voltage mismatch 10-4 p.u. Bus Type Phase A Phase B Phase C 181 2 1.029 -1.8 1.031 -121.8 1.030 118.1 190 0 0.974 14.9 0.983 -105.2 0.988 134.8 193 2 1.049 -13.8 1.046 -133.3 1.055 106.6 196 0 0.994 10.1 1.006 -109.4 1.011 130.0 312 2 1.029 -1.8 1.031 -121.8 1.030 118.1

Load 3620.00 650.00 Generation 3651.838 856.595

Tota

l M

VA

Losses 31.834 206.551

Table A2 Results of Case E 5 iterations, positive sequence voltage mismatch 10-4 p.u.

Bus Type Phase A Phase B Phase C 14 0 0.978 19.0 0.999 -100.8 0.998 138.8 32 0 0.943 9.6 0.999 -108.8 1.010 130.4 41 0 0.911 3.6 0.976 -114.0 0.992 124.8 45 0 0.934 2.6 0.989 -114.8 1.005 123.8 70 0 0.991 21.9 1.012 -98.0 1.010 141.6 71 0 0.952 12.6 1.007 -106.2 1.018 133.0 77 0 0.938 6.3 1.001 -111.6 1.017 127.1 97 0 0.993 20.1 1.005 -99.8 1.000 139.8 98 0 1.003 23.4 1.015 -96.6 1.010 143.1 99 2 1.028 -1.8 1.032 -121.8 1.031 118.0 100 0 1.014 24.9 1.019 -95.3 1.012 144.5 101 2 1.027 -0.6 1.031 -120.4 1.032 119.3 102 0 1.007 23.8 1.017 -96.3 1.010 143.4 103 0 0.974 19.3 1.008 -100.4 1.010 139.1 108 2 1.027 -0.6 1.031 -120.4 1.032 119.3 113 3 1.028 0.0 1.032 -119.9 1.030 119.9 114 0 1.014 24.9 1.019 -95.3 1.012 144.6 120 2 1.028 0.6 1.032 -119.4 1.030 120.4 134 0 0.951 7.0 1.005 -110.9 1.021 127.6


79

5 iterations, positive sequence voltage mismatch 10-4 p.u. Bus Type Phase A Phase B Phase C 181 2 1.028 -1.8 1.032 -121.8 1.031 118.0 190 0 0.953 13.9 0.988 -105.4 0.990 134.1 193 2 1.047 -15.9 1.037 -134.4 1.066 105.3 196 0 0.966 7.8 1.008 -110.0 1.022 128.2 312 2 1.028 -1.8 1.032 -121.8 1.031 118.0

Load 3620.00 650.00 Generation 3652.915 943.603

Tota

l M

VA

Losses 32.909 292.776


80

Unbalanced Distribution Power Flow Analysis using Sequence and Phase Components

81

4 UNBALANCED DISTRIBUTION POWER

FLOW ANALYSIS USING SEQUENCE AND PHASE COMPONENTS


4.1. Introduction

Three-phase distribution networks are traditionally modeled in phase coordinates frame of reference. This is because the mutual inductances between different phases of an asymmetrical transmission lines are not equal to each other. Besides the sequence networks are coupled together and cannot be broken into independent circuits and distribution systems contain multi-phase unbalanced laterals. A variety of three-phase power-flow algorithms have been developed based on phase components for solving unbalanced power systems. Some of these algorithms solve a general network structure such as standard Newton Raphson method or its variants [1-5]. However, the three-phase Newton-Raphson method is computationally expensive for large systems due to the size of the Jacobian matrix and its fast decoupled version is sensitive for high line R/X ratios. The admittance or impedance methods [6-9] have convergence characteristics that are highly dependent on the number of the PV nodes in electrical network [8]. There are also unbalanced power-flow methods which


82

consider primarily the radial structure of distribution networks. Therefore, these power-flow methods can solve only radial or weekly meshed systems such as methods given in [10-11].

The advantage of the network decomposition is that the size of

the unbalanced power-flow problem is effectively reduced [12-16], and hence, large computational saving is achieved. There were two studies reported to decompose an electrical network in sequence or phase components. The development using sequence components [12-14] is limited to three-phase networks comprising of three-phase components. On the other hand, the development in phase components is limited to radial networks and cannot solve interconnected unbalanced transmission or distribution networks.

Many improvements have been presented to model the

unbalanced power system elements in sequence components [12-14]. The sequence power-flow method [14] has been used to analyze unbalanced three-phase transmission networks with accurate solution and superior performance characteristics. This method was able to handle the transformer phase shifts, the line coupling, and any number of PV nodes efficiently. The introduction of PV nodes is currently very important aspect whenever distributed generation is considered. However, these methods [12-14] are limited to solve three-phase networks and cannot be applied to solve distribution networks with two-phase and single-phase line segments.

In this chapter, the sequence power-flow method [14] is

extended to solve unbalanced three-phase systems. An unbalanced distribution network is decomposed into a three-phase network with three-phase line segments and unbalanced laterals with two-phase and single-phase line segments. The unbalanced laterals that consist of single-phase and two-phase line segments are replaced by an equivalent current injection at their upstream nodes (i.e. root nodes). Hence, the main three-phase network is solved based on symmetrical components as given in [14].


83

Exact calculation of the equivalent current injections at the

upstream nodes requires the solution of unbalanced laterals without any approximation. This is achieved by utilizing a simplified backward/forward method in phase components to solve the unbalanced laterals. Then, the solution of the whole network is obtained by putting the decomposed networks in a hybrid iterative scheme in both sequence and phase components. The proposed decomposed distribution power-flow utilizes the advantages of the sequence decoupled power-flow such as fast execution time and the ability to handle PV nodes [14].

The proposed power-flow method is validated by solving the

IEEE radial test feeders [19] and the solution of the radial feeders is compared with the results obtained from a public domain radial distribution analysis program (RDAP) which utilizes the backward/forward method. The execution time is measured and compared with a commercial grade three-phase power-flow program utilize Gauss method. The obtained results show that the proposed method is accurate and has fast execution time. The chapter is arranged as follows: the decomposition of distribution networks is presented in section 2. The power system models are given in section 4.3. The forward/backward method is given in section 4.4 whereas the solution process of the unbalanced laterals is demonstrated in section 4.5. The decomposed distribution power-flow is presented in section 4.6. The results are given in section 4.6 and the conclusions are drawn in section 4.8.


84

Figure 4.1 Typical radial distribution network, IEEE 13 bus test

feeder

4.2. Distribution networks 4.2.1. Characteristics

A distribution network is characterized by its radial structure that is different from traditional interconnected transmission networks. The radial structure of distribution network makes the flow of the power in one direction from the source substation towards the customers. An example of a radial distribution network is the IEEE 13 bus test feeder shown in Figure 4.1. This feeder contains the most common features in a distribution network such as:

a. single-phase, two-phase, and three-phase power system elements for lines, loads, and capacitor banks


85

b. voltage regulators or load tap changing transformer (LTC) and distribution transformers

4.2.2. Distribution power-flow methods

Due to characteristics of distribution networks, the trend was to solve distribution network using phase coordinates power-flow methods since the single-phase and two-phase elements cannot be represented by sequence networks. Bringing to mind the classification of the unbalanced power-flow algorithms presented in chapter 2, there are three categories for the solution of unbalanced networks described as follows:

The first category includes the general network structure methods. This category contains methods such as phase-coordinate Newton Raphson three-phase power-flow formulated based on power or current mismatches. These methods can solve meshed transmission network or radial distribution networks.

The second category includes methods that were intended especially for distribution systems since they take the advantage of the radial structure of distribution networks in the problem formulation. One of the most famous techniques for solving radial distribution networks is the forward/backward sweep method.

The third category includes the methods that were formulated using sequence components. In the previous chapter, the sequence decoupled power-flow algorithm is reformulated such that the standard single-phase power-flow methods can be used for solving the positive sequence network. Nevertheless, the sequence decoupled power-flow proposed is limited only for solving three-phase networks and cannot solve distribution networks such as the network given in Figure 5.1 due to the awkwardness of positive sequence problem.

4.2.3. Awkwardness of positive sequence problem

The single-phase and two-phase voltages and currents at the different nodes of the unbalanced laterals will produce an ill condition positive sequence problem. The following simple numerical example demonstrates this fact. Firstly, two-phase


86

network is considered such as the unbalanced lateral connected between buses 645 and 646 in Figure 5.1. The three-phase voltage vector at these buses, 645 or 646, is transformed to its counterparts in sequence components as follows:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

c

b

VV

aaaa

VVV 0

11

111

31

2

2

2

1

0

(4.1)

By assuming that obV 1200.1 −∠= and o

cV 1200.1 ∠= . This is reasonable assumption since the voltages of both phase ‘b’ and ‘c’ are almost around these values. Substituting in (4.1), the sequence voltages can be calculated as follows, and remembering that

oa 1200.1 ∠= :

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∠−∠

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∠∠∠∠=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

0.10.20.1

31

1200.11200.1

0

1200.12400.112400.11200.11111

31

2

1

0

o

o

oo

oo

VVV

(.2)

Therefore, for a two-phase unbalanced lateral comprises of phases ‘b’ and ‘c’, the positive sequence voltage will have a magnitude around 0.6667 p.u. This produces ill condition in the positive sequence network, and hence, single-phase power-flow algorithm cannot handle or solve power system with buses having these values for positive sequence voltage. Similar calculation can be carried for an unbalanced lateral comprises of one phase only such as phase ‘a’ in the unbalanced lateral that is connected between buses 684 and 652 as shown in Figure 4.1. In this case, the positive sequence voltage becomes worse since its magnitude is equal to 0.3333 p.u. 4.3. Decomposition of distribution networks

The difficulty of the application of sequence networks for solving distribution systems can be avoided by decomposition of distribution networks into a main three-phase network and unbalanced laterals. The decomposition of the IEEE 13 node test feeder is shown in Fig. 4.1. The unbalanced laterals are decoupled


87

from the main three-phase circuit and replaced by equivalent current injections.

4.3.1. Main three-phase network

The main three-phase network includes all power system elements having all phases (a-b-c), i.e. complete three-phase network without any single-phase and two-phase circuits. Therefore, the sequence decoupled networks, positive-, negative-, and zero-sequence networks, can be established [14]. Hence, the sequence decoupled power-flow algorithm can be used for solving the main three-phase network. However, the solution requires calculating the injected current at the buses having unbalanced laterals such as bus 632 and 671 of the IEEE 13 bus feeder. The injected currents at an unbalanced lateral plugged at a certain bus represent the total loads, capacitor banks, and the lines losses of that unbalanced lateral.

4.3.2. Unbalanced laterals

A distribution network contains many unbalanced laterals. For example, the IEEE 13 node feeder contains two unbalanced laterals plugged at bus 632 and at bus 671. A decomposed unbalanced lateral involves only two-phase levels. The first level is a two-phase circuit while the second is a single-phase circuit such as the unbalanced lateral plugged at bus 671. Consequently, an unbalanced lateral comprises of a few numbers of buses and branches. Due to this fact, unbalanced laterals represent small, simple, and independent radial distribution networks.


88

Fig. 4.2 Decomposition of the IEEE 13 node test feeder into main three-phase network and unbalanced laterals

Each unbalanced lateral has a source bus, such as bus 632 and

bus 671, in which voltage should be specified to solve the unbalanced lateral having specified values for loads and capacitor banks. Therefore, the unbalanced laterals, that represent small or simple radial networks, can be solved properly and efficiently by a simplified backward/forward sweep algorithm.

4.4. Power System Models 4.4.1. Line Segments and transformers:

For the transformer and line segments, the models are reported in [13]. These models are used to solve the main three-phase network. In the unbalanced laterals solution, the phase component models are used [17].


89

4.4.2. Load Model: The load can be modeled as constant power, constant current,

or constant load model, or any combination of these models. The load may be connected as star or delta with any number of phases. The loads are represented by equivalent current injections. The calculation of the load current injections is summarized in Table 1.In the case of the main three-phase network, the phase current injections are transformed to their counterparts in sequence components. Distributed loads are modeled similar to reference [10].

4.4.3. Voltage controlled nodes model

One of the advantages of the proposed method is the simplicity of the PV node representation. The balanced three-phase program bus specifications are used similar to reference [13-14]. This is very important feature when distribution systems with cogeneration are to be analyzed. Table 4.1: Calculation of Loads Equivalent Current Injections

Load Injected current

PQ ( )*kmm

km VSI =

I ( )*_ Nmmm VSI =

( ) ( )mk

mkm SV argarg −=θ , k

mmkm II θ∠= St

ar

Z 2

_ Nmmm VSy = , kmm

km VyI =

PQ ( )*kmnmn

kmn VSI =

I ( )*_ Nmnmnmn VSI =

kmnmn

kmn II θ∠= , ( ) ( )m

kmn

kmn SV argarg −=θ D

elta

Z 2

_ Nmnmnmn VSy = , kmnmn

kmn VyI =


90

Where n and m refer to phases a, b, or c, where n ≠ m, k refer to the iteration serial, Vm_N the nominal voltages, phase to ground in case of star loads, and Vnm_N the nominal voltages, line to line in case of delta loads

Figure 4.3 Branch numbering scheme, example is for IEEE 13 bus feeder

Fig. 4.4 Forward/backward sweep solution for the IEEE 13 bus

feeder

4.5. Backward/Forward Method


91

In contrast to traditional power-flows algorithms, the forward/backward method requires a certain numbering to the network elements. Example of a numbering scheme is the branch numbering given in [10] and is shown in Fig. 4.3. The backward/forward method requires a certain numbering to the network elements. Examples of numbering schemes are branch numbering in [10]. The IEEE 13 node feeder is arranged according to the branch numbering scheme as shown in Fig. 4.4. The backward/forward sweep method [10] contains mainly three steps to solve an unbalanced distribution radial network. The three steps are: 1. Nodal current calculation: the first step is to calculate all the

current injection at different buses in the feeder due to loads and capacitor banks, and any other shunt elements based on initial voltages. In the next iterations, the nodal currents are calculated using the updated voltages.

2. Backward sweep: At iteration k, starting from branches at the last layer and moving towards the branches connected to the root node, all branch currents are calculated by applying current summation. For a branch L, the branch current is calculated as follows:

∑∈ ⎥

⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Mm

k

cm

bm

am

k

cj

bj

aj

k

cL

bL

aL

JJJ

III

JJJ

_

_

_

_

_

_

_

_

_

(4.1)

Where J is the current that flows on a line section, L and M refers to the set of the line sections connected to node j , I is the current injected due to shunt elements at bus j, and k is the iteration serial. 3. Forward sweep: Nodal voltages are updated in a forward sweep

starting from branches in the first layer toward those in the last layer. The updated voltage are corrected starting from the root node as follows:


92

k

cL

bL

aL

k

ci

bi

ai

k

cj

bj

aj

JJJ

VVV

VVV

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

_

_

_

branch

_

_

_

_

_

_

Z (4.2)

Where Z is branch series impedance, j is the node at which the voltage is updated, i is the node at which the voltage is updated from the former layer.

The backward/forward sweep method solution of the IEEE 13

bus feeder is shown in Fig. 4.3. The figure shows the three main steps for solving radial network. The three steps are repeated until the convergence is recognized.

4.6. Solution of unbalanced laterals

The backward/forward sweep algorithm described in the previous section is usually used for the solution of the whole radial distribution network. In this case, the feeder comprises of three-phase levels. The first level is the main three-phase network, the second is a two-phase circuit, and finally, the third level is for the single-phase circuit. Therefore, the application of the backward/forward algorithm for the solution of the whole network becomes tedious due to complicated data structure required for the solution process. However, if the backward/forward sweep algorithm is applied only for solving the decomposed unbalanced laterals, the algorithm is greatly simplified due to the simplicity of the decomposed unbalanced lateral structure.


93

Fig. 4.5 General network structure of a decomposed unbalanced lateral

Figure 4.5 shows a general unbalanced lateral that may be

plugged at any node in the main three-phase network. The unbalanced lateral mainly comprises of two levels for phases, the single-phase and two-phase levels. It also includes little number of layers, and each layer has few numbers of branches. Consequently, a simple method can be developed based on the backward/forward method to handle the simplified unbalanced lateral radial network.

Usually a distribution network contains many unbalanced

laterals. These unbalanced laterals are decoupled from each other, i.e. each unbalanced lateral is a standalone radial network and having its upstream root node. The application of backward/forward sweep algorithm requires the knowledge of the voltage and the angle at the root node, i.e. each unbalanced lateral should have its own slack bus. However, this condition is not satisfied because these voltages are dependent on the solution of the main three-phase network. This leads to an iterative solution between the different unbalanced laterals and the main three-phase network.

The first two steps in the backward/forward are performed to

calculate the current injection at the root nodes for different unbalanced laterals. The third step is postponed till the main three-


94

phase network is solved. Hence, the voltage at the root nodes become known, the forward sweep step can be performed for updating the node voltages in each unbalanced lateral. The solution in this way results in a hybrid iterative process between the main three-phase network and the unbalanced laterals in sequence and phase components respectively.

Table 4.2: Modified ordering scheme for the hybrid distribution

power-flow

Bus i Bus j Branch ID

Root node ID Layer ID Network ID

650 650R 1 0 0 650R* 632 2 0 0

632 645 3 632 1 632 633 4 0 0 632 671 5 0 0 645 646 6 632 2 633 634 7 0 0 671 692 8 0 0 671 684 9 671 1 671 680 10 0 0 684 611 11 671 2 684 652 12 671 2 692 675 13 0 0

Distribution N

etwork

*Regulated node


95

Table 4.3: Ordering of Table 2 according to root node and layer ID

Bus i Bus j Branch ID

Root node ID Layer ID Network ID

650 650R 1 650R* 632 2 632 633 4 632 671 5 633 634 7 671 692 8 671 680 10 692 675 13

0 0 Three-phase

main network

632 645 3 1 645 646 6 632 2

Unbalanced lateral 1

671 684 9 1 684 611 11 2 684 652 12

671 2

Unbalanced lateral 2

4.7. Decomposed distribution power-flow analysis

As discussed in the previous section, the voltages at the root nodes of the unbalanced laterals depend on the solution of the main three-phase network. Therefore, the forward sweep is performed after the main three-phase network solution. This results in an iterative solution between the main three-phase circuit and the unbalanced laterals solution using the backward/forward sweep method. Hence, the solution of the whole distribution network becomes hybrid solution in both sequence and phase components.


96

Fig. 4.6 Hybrid iterative power-flow solution algorithm

4.7.1. Branch numbering scheme

The branch numbering shown in Fig. 4.3 is for the solution of the whole distribution network using backward/forward sweep method. However, the proposed hybrid distribution power-flow solves the network partially using the backward/forward sweep method. The branch numbering is modified for the excluding the main three-phase network.

The modified branch numbering scheme include, in addition

to the branch and layer identification numbers, the root node identification number. This is because the unbalanced laterals can be specified easily by their root nodes, i.e. upstream nodes. As an example for branch numbering, the data structure for branches of


97

the IEEE 13 bus feeder is arranged in Table 3. In this numbering scheme, the three-phase branches are assigned zero entry for both the layer ID and the root node ID to refer to the main three-phase network. The other branches that represent two-phase circuits and single-phase circuits have entries for both the layer ID and the root node ID.

The structure of the decomposed networks can be obtained by

ordering Table 4.2, firstly by root nodes ID, and then by layer ID. Table 4.3 gives the decomposed network structure. The table gives the decomposed networks of the IEEE 13 node feeder that involves three-phase main network and two unbalanced laterals plugged at nodes 632 and 671.

4.7.2. Solution process Figure 4.4 shows the proposed hybrid iterative distribution

power-flow. The algorithm starts by performing the first and the second steps of the backward/forward method. Then, the current injections due to unbalanced laterals is then passed to the sequence decoupled power-flow algorithm. These currents are transformed with other current injection due to loads, capacitor banks, or current injections due to line coupling. Therefore, the final sequence current injection for buses that have unbalanced laterals is given by:

2or 1or 0,1

Line_nn__LateralUnbalanced

n_Capted_Load_ni_Distribun_Loadn

=∑∈=

Δ++

++=

nM

ikk

ki_

ki_

ki_

kki_

ki_

II

IIII (4.3)

Where M refers to the set of lines connected to node i

The sequence decoupled networks are solved according the formulation in our previous chapter [14]. The standard Newton-Raphson and fast decoupled methods are used for solving the


98

positive-sequence network whereas the negative- and zero-sequence networks are represented by two nodal voltage equations. The negative- and zero-sequence networks are represented by two nodal voltage equations [14]. After solving the sequence networks, the root voltages of the unbalanced laterals are known, and hence, the third step of the backward/forward method can be performed for updating the voltages at the rest of buses in different unbalanced laterals. The process is repeated until convergence is reached by using phase voltages mismatch criterion since the positive sequence voltage or the positive sequence power mismatch criteria will not be accurate.

Radial or meshed main three-phase network

Radial Unbalanced Lateral 1

Radial Unbalanced Lateral i

Radial Unbalanced Lateral N Figure 4.7: Radial or meshed main three-phase network having

radial unbalanced laterals 4.7.3. Meshed distribution networks

The hybrid distribution power-flow solves a distribution network in both sequence and phase components. The main three-phase network is solved using sequence components based on voltage node formulation. The proposed formulation can solve networks with meshed or radial structure. The decomposed unbalanced laterals are solved using backward/forward sweep method which is intended specially for solving radial networks.


99

The proposed hybrid distribution power-flow can solve any distribution network with radial or meshed main three-phase network having any number of unbalanced radial laterals such as the network structure shown in Fig. 4.7. The decomposition of Fig. 4.6 agrees with practical distribution networks since the meshed network usually exists in the main three-phase network and seldom to find meshed two-phase or single-phase networks.

4.8. Results The IEEE 13 bus and the IEEE 123 bus radial distribution

networks are used as the test feeders, the regulators are removed form the original data of the IEEE feeders. The proposed power-flow methods are coded using two balanced three-phase power-flow routines. They are Newton-Raphson and Fast Decoupled methods [20]. These routines are used for solving positive sequence network without affecting their capability of solving balanced power systems. All case studies were run on an Intel, Core 2 Duo, 1.83 GHz CPU with a 1024 MB RAM and a MS Windows XP operating system.

The results obtained from executing the proposed methods are compared with two power-flow methods are widely used for analyzing unbalanced three-phase distribution networks:

a) The Backward/forward method: it is used in the accuracy test of the proposed methods

b) The bus admittance method, it is used to measure the performance of the proposed methods

It is important to mention herein that the sequence power-flow

methods utilize SuperLU library [21] as the sparse linear solver and component technology as a programming methodology [20] whereas the methods ‘a’ and ‘b’ are commercial grade programs developed based on structural programming. Also, method ‘b’ utilizes a different sparse linear solver. These differences may have


100

some slight contribution in the performance of the programs, i.e. the execution time of proposed methods. Usually, component technology and object oriented programming consumes more time when it is compared with structural programming for identical algorithms.

4.8.1. Accuracy

The solution of the IEEE 13 feeder is given in Table 4.4 for the voltage magnitude and the angle for the three-phases. The solution shows that the solution of the proposed unbalanced power flow methods based on sequence networks agree well with the backward/forward solution.

The IEEE 123 bus is the most comprehensive test feeder since it contains variety of unbalanced laterals. Therefore, the IEEE 123 bus feeder presents a good test for the proposed unbalanced power flow based on sequence networks. The results of the three-phase voltages are compared with those calculated using Backward/forward method program. The voltage profile for phase ‘a’ is given in Fig. 4.8. The figure shows that both solutions have the same voltage profile for the three-phases of the IEEE 123 bus radial feeder.


101

Table 4.4: Numerical three-phase voltages of the IEEE 13 node feeder


102

Volatge of Phase 'a'

0.8

0.84

0.88

0.92

0.96

1

1.04

150 152 610 25 35 49 112 67 79 95

Bus ID

Vol

atge

Va,

p.u

.

Hybrid Method

Forward/Backward

Volatge of Phase 'b'

0.8

0.84

0.88

0.92

0.96

1

1.04

150 53 610 18 40 300 100 81

Bus ID

Vol

atge

Vb,

p.u

.

Volatge of Phase 'c'

0.8

0.84

0.88

0.92

0.96

1

1.04

150 152 57 66 31 48 197 87 79

Bus ID

Vol

atge

Vc,

p.u

.

Fig. 4.8 Voltage profile of the IEEE 123 node feeder


103

Table 4.5: Execution Time in Seconds

Proposed decomposed

three-phase power-flow algorithm

Balanced power-flow

solver

Newton Raphson

Fast decoupled

Three-phase power-flow solver

’bus admittance method’

IEEE 13 node

<<0.01 <<0.01 <<0.01

IEEE 123 node

0.05 0.02 0.03

4.8.2. Performance

The performance of the proposed methods in terms of execution time is given in Table 4.5 for both the decomposed three-phase power-flow method and for the three-phase power-flow method based on bus admittance formulation. For the IEEE 13 node feeder it is found the execution time is very small for all methods, and in many cases, it can not be measured. In the case of the IEEE 123 node feeder, when the fast decoupled power-flow algorithm is used as a balanced solver, the hybrid power-flow method requires less time than the thee-phase power-flow solver. On the other hand, when the Newton-Raphson power-flow algorithm is used, the hybrid power-flow algorithm consumes more time when it is compared with the three-phase power-flow solver. However, the advantage of the decomposed Newton-Raphson solver is the introduction of any number of PV nodes as well as the easy implementation and integration with existing balanced power-flow and short circuit calculations packages.


104

4.9. Conclusion The chapter has presented a new formulation for unbalanced

distribution power-flow based on sequence and phase components. An unbalanced distribution network is decomposed into a main three-phase circuit and unbalanced laterals. The main three-phase network is solved using the sequence decoupled power-flow algorithm. The unbalanced laterals are solved using the backward/forward sweep method. This results in a hybrid solution in both sequence and phase components. The advantage of the proposed formulation is that a complicated distribution network is decomposed to many sub-problems. One of these sub-problems is basically a standard balanced three-phase power-flow. In addition, the proposed decomposition is able to The decoupling features of the proposed formulation eventually reduce the size of the distribution power-flow problem which leads to improvements in both execution time and memory requirements.

REFERENCES [1] R. G. Wasley and M. A. Shlash, “Newton-Raphson algorithm

for 3-phase load flow”, proc. IEE, vol. 121, pp. 630-638, July 1974

[2] H. L. Nguyen, “Newton–Raphson method in complex form-power system load flow analysis”, IEEE Trans. Power Systems, vol. 12, no. 3, pp. 1355–1359, August 1997

[3] P. A. N. Garcia, J. L. R. Pereria, S. Cameiro J. R, V. M. Da Costa, and N. Martins, “Three-phase power flow calculation using the current injection method”, IEEE Trans. on Power Systems, vol. 15, no. 2, pp 508-514, May 2000.


[5] A. V. Garcia and M. G. Zago, “Three-phase fast decoupled power flow for distribution networks”. Proc IEE. Generation, Transmission, and Distribution, vol. 143, no. 2, pp. 188-192, March 1996


105

[6] K. A. Birt, J. J. Graff, J. D. McDonald, and A. H. El-Abiad, “Three phase load flow program”, IEEE Trans. on Power Apparatus and Systems, vol. 95, no. 1, pp 59-65, Jan/Feb, 1976.

[7] D. I. H. Sun, S. Abe, R. R. Shoults, M. S. Chen, P. Eichenberger, and D. Farris, “Calculation of energy losses in a distribution system”, IEEE Trans. on PAS, vol. 99, no. 4, pp. 1347-1356, July-August 1980


[9] Chen T. H., Chen M. S., Hwang K. J., Kotas P., and Chebli E. A. (1991). Distribution system power flow analysis- a rigid approach. IEEE Trans. Power Delivery, vol. 6, no. 3, July, pp. 1146-1152

[10] C. S. Cheng and D. Shirmohammadi, “A three-phase power flow method for real-time distribution system analysis” IEEE Trans. on Power Systems, vol. 10, pp. 671–679, May 1995.

[11] J. H. Teng, “A direct approach for distribution system load flow solutions”, IEEE Trans. on power system, vol. 18, no. 3, pp. 882-887, July 2003

[12] K. L. Lo and C. Zhang, “Decomposed three-phase power flow solution using the sequence component frame”, Proc. IEE, Generation, Transmission, and Distribution, vol. 140, no. 3, pp 181-188, May 1993


[14] M. Abdel-Akher, K. M. Nor, and A. H. Abdul-Rashid, “Improved three-phase power-flow methods using sequence components”, IEEE Trans. on power systems, vol. 20, no. 3, 1389-1397, Aug. 2005

[15] W. M. Lin and J. H. Teng, “Phase-decoupled load flow method for radial and weakly-meshed distribution networks”, IEE Proc. Generation, Transmission and Distribution, vol. 143 no. 1, pp. 39-42, January 1996


106

[16] E. R. Ramos, A. G. Expósito, and G. Ă Cordero, “Quasi-coupled three-phase radial load flow”, IEEE Trans. on power systems, vol. 19, no. 2, pp. 776-781, May 2004

[17] J. Arrillaga and N. R. Watson, “Computer modeling of electrical power systems”, Book, 2nd Edition, John Wiley & Sons LTD, pp 11-13, pp 48-51, and pp 111-115, 2001.

[18] W. H. Kersting, " The electric power engineering handbook, Chapter 6: Distribution Systems”, Editor: L.L Grigsby, CRC Press LLC, 2001

[19] W. H. Kersting, “Radial distribution test feeders”, PES winter meeting, vol. 2, no. 28, pp. 908-912, Jan. 28/Feb. 1, 2001: data can be downloaded from internet at: http://ewh.ieee.org/soc/pes/dsacom/test feeders.html


[21] J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, and J. W. H. Liu, “A supernodal approach to sparse partial pivoting”, SIAM Journal on Matrix Analysis and Applications, vol. 20, no. 3, pp 720 -755, 1999.

Representing Single-Phase and Two-Phase Lines with Dummy Lines and Dummy Nodes

107

5 REPRESENTING SINGLE-PHASE AND

TWO-PHASE LINES WITH DUMMY LINES AND DUMMY NODES


5.1 Three-phase line

Sequence symmetrical components transform unbalanced network to three balanced three phase network. This is possible when dealing with a three phase line, but not directly possible with one phase or two phase missing. The missing phases are due to the usage of one line or only two lines from a three phase connections. The missing lines are not connected or used. Usually this is done to save cost of long transmission line to a remote or rural area where the load is small and need not be carried by a full three phase lines.

The three-phase transmission lines are usually modeled by a lumped π network [1]. Fig. 5.1 shows an example of a three-phase line connected between nodes ‘i’ and ‘j’. The series resistances and inductances between nodes are lumped in the middle. The shunt capacitances of the transmission lines are divided into two halves and lumped at nodes connected to the line terminals. The line series impedance and shunt admittance matrices are given by:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=ccij

cbij

caij

bcij

bbij

baij

acac

abij

aaij

abcij

zzzzzzzzz

Z ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=ccij

cbij

caij

bcij

bbij

baij

acij

abij

aaij

abcij

yyyyyyyyy

Y (5.1)


108

Bus i Bus j

Three-phase line Two-phase line Single-phase line

a

b

c

Bus ka

b

c

b

c cbcijy

caijy

bcijz

abijz ca

ijz

aaijz

bbijz

ccijz

bcijy ab

ijy

aaijybb

ijyccijy

caijy

Bus l

bcjkz

bbjkycc

jky

bbjkz

ccjkz

ccjkybb

jky

bciky bc

iky

cckly cc

kly

ddklz

ccijybb

ijyaaijy

abijy

Fig.5.1 Multi-phase system of a radial feeder

The voltage and the current relationship of the three-phase line

shown in Fig. 5.1, between node ‘i' and ‘j’, is given by:

( ) ( )

( ) ( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−

−+=

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−

−−

abcj

abci

abcij

abcij

abcij

abcij

abcij

abcij

abcj

abci

ZZZZ

VV

YY

II

11

11

(5.2)

5.2 Two-phase lines

The two-phase lines are modeled as a three-phase line by using a dummy line and dummy node as shown in Fig. 5.2. The real line connected between node ‘j’ and ‘k’ consists of phase ‘b’ and ‘c’. The upstream node of the dummy line at node ‘j’ is a real node which exists in the actual network. The other node of the dummy line at node ‘k’ is a dummy node. This dummy node does not exist in the real network and does not have a load.

The current that flows from phase ‘a’ at node ‘j’ toward phase

‘a’ at node ‘k’ is zero and therefore, the voltage drop across the dummy line is zero. This is true since there is no coupling between the dummy line and the other real phases of the line, and the injection at the dummy node is zero. Therefore the series impedance of the dummy line can be set to any arbitrary


109

impedance zdd, which will satisfy the ohm’s law relationship for the dummy line.

The series and shunt admittances of the two-phase line shown in

Fig. 1 are given by:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=ccjk

cbjk

bcjk

bbjk

ddjk

abcjk

zzzz

z

00

00Z

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=ccjk

cbjk

bcjk

bbjk

abcjk

yyyy

00

000Y (5.3)

The two-phase line can be represented as a complete three-

phase circuit similar to (5.2). However, the voltage at the dummy node will be identical to that of the upstream node ‘j’. The voltage of the dummy node at node ‘k’ is given by:

ajk

aj

ak VVV Δ+= (5.4-a)

ddjk

ddjk

ajk

ajk zzIV ×==Δ 0 (5.4-b)

5.3 Single-phase lines

The single-phase line which is shown in Fig. 5.1 consists of phase ‘c’ only. The line is connected between node ‘k’ and node ‘l’. Two dummy lines are required to represent the missing phases ‘a’ and ‘b’. The upstream node of the dummy lines, node ‘k’ has dummy node ‘a’ and real node ‘b’. The other terminal of the line at node ‘l’ has two dummy nodes for phase ‘a’ and ‘b’ as shown in Fig. 5.2.


110

Bus ka a

b

a

b

ddjkzBus j

Three-phase line Two-phase line Single-phase line

a

b

c

Bus k

b

c

b

c c

aaijy bb

ijy ccijy

abijy bc

ijy

caijy

bcijz

abijz ca

ijz

aaijz

bbijz

ccijz

bcijy ab

ijy

aaijybb

ijyccijy

caijy bc

jkz

bbjkycc

jky cckly cc

kly

bbjkz

ccklz

ddklz Bus l

ccjkz

ccjkybb

jky

ddklz

bcikybc

iky

Bus i

Fig.5.2 Representation of multi-phase system given in Fig. 5.1 using dummy lines and nodes

Bus i Bus jijmmz

imV j

mV

ijmmy ij

mmyimIΔ j

mIΔ

Fig. 5.3. Sequence decoupled line model

Dummy nodes do not have any load since they do not exist in reality. The currents that flow in the dummy lines are zero as the dummy lines ‘a’ and ‘b’ are not coupled with the real line ‘c’. The series impedances of the dummy lines can be assigned any arbitrary impedance ‘zdd’ similar to the two-phase line model. Similar to the two-phase line, the single-phase line can also be represented by a complete three-phase circuit similar to (5.2). The


111

single-line series impedance and shunt admittance matrices are expressed as follows:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=cckl

ddkl

ddkl

abckl

zz

z

000000

Z ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

cckl

abckl

y00000000

Y (5.5)

The voltage at the dummy node is identical to that of the

upstream node. The voltage of the dummy node at node ‘l’ is given by:

a

klajk

aj

akl

ak

al VVVVVV Δ+Δ+=Δ+= (5.6-a)

ddkl

ddkl

akl

akl zzIV ×==Δ 0 (5.6-b)

bkl

bk

bl VVV Δ+= (5.6-c)

ddkl

bkl

bkl

bkl zzIV ×==Δ 0 (5.6-d)

5.4 Assigning the Series Impedance Value of Dummy Lines The dummy series impedance of both single-phase and two-

phase lines can be assigned to any arbitrary value as given by (5.4-b), (5.6-b), and (5.6-d). However, for easy implementation of numerical calculations, our approach is to assign the series impedance value based on the data of the real phases. For the two-phase line described and the single-phase line given by (5.3) and (5.5), the series impedances are given by (5.7) and (5.8) respectively.

2

ccjk

bbjkdd

jk

zzz

+= (5.7)

cckl

ddkl zz = (5.8)


112

5.5 The application of dummy nodes and dummy lines By using dummy lines and nodes to complete missing phases, the technique to use sequence component load flow can now be applied to any line configuration. This will be discussed in the next chapter. Although this is developed for load flow application, the modeling can be used for unbalanced fault analysis in system where single phase and two phase lines exists. In fact dummy lines and nodes can also be used in harmonic load flows. These are potential that need to be developed further. Discussion on them however has to be limited to this far only as it is outside the scope of this book.

REFERENCE [1] J. Arrillaga and N. R. Watson, “Computer modeling of

electrical power systems”, Book, 2nd Edition, John Wiley & Sons LTD, pp 11-13, pp 48-51, and pp 111-115, 2001.

[2] Anderson, P . ‘ Analysis of Faulted power system,’ Iowa State University Press, 1973

Unbalanced Three Phase Power Flow with Dummy Lines and Nodes

113

6 UNBALANCED THREE PHASE POWER

FLOW WITH DUMMY LINES AND NODES


6.1. Introduction

Three-phase power-flows are used to analyze three-phase

unbalanced networks. The unbalanced condition exists in both power system components as well as operation of power systems. This condition is due to many causes such as unbalanced loads, long untransposed high voltage lines in transmission networks, presence of single-phase lines or two-phase lines, or both type of lines, in a three-phase distribution networks. Therefore, three-phase power-flow methods are traditionally implemented based on complete phase component modeling [1-10]. For fast execution time, some of these methods consider primarily the radial structure of distribution networks [8-10]. They can only solve radial or weekly meshed distribution systems, which mean that they cannot solve unbalanced transmission systems.

The Newton-Raphson method [1] or its variants [2-5] and the

bus admittance or impedance matrix method [6-7] can solve a general network structure. The Newton Raphson method is well known by its robust convergence characteristics, but its major shortcoming is the size of the JACOBIAN matrix [7]. In balanced systems, the fast decoupled method has superior performance


114

characteristics over the Newton Raphson method. However, fast decoupled method is sensitive for high line R/X [5]. The admittance or impedance matrix methods have convergence characteristics that are highly dependent on the number of the PV nodes in an electrical network [7].

Network decomposition effectively reduces the size of the

unbalanced power-flow problem and hence has the potential of achieving large computational savings. The decomposition of a network can be achieved using sequence components [11-13] or based on phase components [14]. In [15], a new method has been proposed which utilizes the positive sequence power-flow solver as the main engine. In this method, each three-phase node has been replaced by three independent nodes. This method takes long time for the reading process because each line is represented by a large number of artificial lines [15].

The application of sequence components decomposition is

limited to a complete three-phase network and cannot solve multi-phase systems. The phase components decomposition can only solve radial and weekly meshed distribution networks and cannot solve a general network structure [14]. The advantage of symmetrical component is that one balanced power-flow is executed rather than three balanced power-flows in the case of phase components decomposition [11-13]. In addition, many of symmetrical component models, such as for generators and transformers, are uncoupled in sequence components but are coupled in the phase components [1]. The decomposed sequence networks, positive sequence, negative sequence, and zero sequence can also be solved by using parallel processing [12], [16].

The focus of this work is to extend the sequence power-flow

methods [11-13] to solve a multi-phase networks with single-phase and two-phase line segments based on symmetrical components. The missed phases of single-phase and two-phase lines are replaced by uncoupled lines and unloaded nodes. Hence, the


115

sequence decoupled asymmetrical line model [12] can be used for single-phase and two-phase lines. In addition, the spot and distributed loads in the form of the ZIP model are considered, capacitor banks and distributed loads with any connection and number of phases are also included. The proposed line model is incorporated in the sequence power-flow methods [12-13]. The solution of the IEEE radial feeders shows that the proposed line model is accurate when it is compared with commercial grade power-flow software program.

This chapter is organized into a few sections. Section 5.2 gives

a quick review for the strength and weakness of the current injection method which is considered one of the latest development adopted for the solution of distribution systems. In section 5.3, the representation of multi-phase line model is presented. In section 5.4, the symmetrical components models for lines, loads, capacitor banks, are presented. Section 5.5 presents the formulation of the power-flow problem based on the symmetrical components. Numerical results for the IEEE 13 node and the IEEE 123 node radial feeders are given in section 5.6. The results show that the sequence power-flow formulation is robust and the proposed line model is accurate. Conclusions are drawn in section 5.7.

6.2. Power flow solution based on phase components

Current injection method (CIM) [5] is one of the recent approaches utilized for the analysis of distribution networks, as has been mentioned earlier in chapter 1. This method is superior to the classical Newton Raphson method due to the reduction in the calculated elements of the Jacobean matrix. The CIM utilizes the classical phase components models [1], consequently a three-phase Jacbian matrix must be constructed.

The CIM has a constant Jacobian matrix only if all loads in the system are assumed to be of constant impedance and all nodes are PQ nodes. However, if PQ load model is considered, or PV nodes exist in the system, the Jacobian matrix is updated every iteration [5]. The advantage of the CIM over the standard NR method is


116

generally in the reduction of the number of the calculated elements in the Jacobian matrix. An improvement for the PV node representation has been proposed for the original CIM method such that its convergence characteristics become similar to the conventional NR method [18]. This has been discussed in chapter 3 by the solution of the modified IEEE 14 node network and the results has been reported in Table 3.7.

6.3. Symmetrical Component Modeling The approach of using dummy nodes and lines converts a

multi-phase system to be a complete three-phase system as shown in Fig. 5.2. Based on this approach, unbalanced three phase load flow in terms of the sequence networks can be conceptually solved [11-13]. The solution of the three-phase power-flow using sequence components requires the construction of the three-phase power system models in terms of their sequence components. The following sections summarize the three-phase models in sequence components.

6.3.1. Sequence Decoupled Asymmetrical Line Model

The series impedance and shunt admittance matrices of the three-phase line given by (5.3) are transformed to their counterparts in sequence components. The resulting series impedance and shunt admittance matrices are given by:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=222120

121110

020100

012

222120

121110

020100

012

ijijij

ijijij

ijijij

ij

ijijij

ijijij

ijijij

ij

yyyyyyyyy

zzzzzzzzz

YZ (6.1)

If a three-phase line is fully transposed, its impedance and

admittance matrices in (5.1) will be symmetrical. Its sequence components will be diagonal matrices. However, if the three-phase line is untransposed, the phase components admittance matrices in


117

(5.1) are full and symmetrical but not phase-wise balanced i.e. acij

abij zz ≠ . Hence, the sequence admittance matrix will be full and

unsymmetrical. The sequence coupled line model can be decomposed into

three independent sequence circuits [12]. This can be achieved by replacing the coupling, i.e. the off-diagonal elements in (9), by an equivalent current compensation as follows:

mj

nmij

li

nlij

mj

minm

ij

lj

linl

ij

ni VyVyVV

zVV

zI ++−+−=Δ )(1)(1 (6.2)

Fig. 5.3 shows the decomposed line model in sequence

components. The coupling among the sequence networks are included by the current compensation calculated using (6.2).

6.3.2. Loads and Capacitor Banks The loads can be spot loads or distributed along the line length.

They can be connected as delta or star and can be modeled as constant power (PQ), constant current (I), or constant impedance (Z) or any combination of these types. The loads are modeled by current injection in phase components. Then, the current injection in phase coordinates is converted to its counterparts in sequence components.

The distributed loads are modeled using the lumped load

model [10]. The model divides the distributed load between the line ends using a specific ratio η. This ratio is calculated based on the voltage magnitudes at the line terminal nodes. It is possible to consider the ratio η to be 0.5 [9]. In this work, the ratio η is calculated per iteration during the power-flow solution process. The representation of the distributed load in the multi-phase three-phase model is shown in Fig. 6.4.


118

Capacitor banks are connected at certain node to compensate reactive power for improving the voltage profile in an electrical network or for reducing the network losses. Capacitor banks may be connected as star or delta. The capacitors are usually specified by their reactive powers at a nominal operating voltage. Therefore, they are modeled similar to the constant PQ loads.

6.4. The Sequence Power-Flow Algorithm 6.4.1. Problem Formulation

The solution based on symmetrical components without any approximation can be demonstrated as follows. The equivalent symmetrical formulation of the power-flow problem based on current injection is given by:

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

012

0122

0121

012

0122

0121

0120122

0121

0122

01222

01221

0121

01212

01211

NNNNNNN

N

N

I

II

V

VV

YYY

YYYYYY

(6.3)

Extracting from (6.3) for m sequence network:

UnbalanceSystem

111⎟⎠

⎞⎜⎝

⎛+−= ∑∑∑

===

lk

N

k

mlik

nk

N

k

mnik

mi

mk

N

k

mmik VYVYIVY (6.4)

Rewriting (6.4): m

SUimi

mk

N

k

mmik IIVY _

1−=∑

=

(6.5)

The second term on the right hand side of (6.5) is the

unbalance terms in the network equations. The system unbalance in the symmetrical component modeling arises mainly from the line coupling. This is because generators and transformers are


119

balanced three phase devices whose symmetrical components are naturally uncoupled.

a

b

c

a

b

c

Bus i Bus j

a) Single-phase distributed load

a

b

c

a

b

c

b) Single-phase distributed load Lumped model Fig. 6.1. Multi-phase distributed load representation

Line 1

Line M

Bus j

Bus k

Bus i

Bus j

Bus k

Bus i

mijLinei __IΔ

Coupled Sequence Network Decomposed Sequence NetworksNetwork m, m=0, 1, 2

mikLinei __IΔ

M Lines are connected to bus i

M Lines are connected to bus i

Line s

Fig 6.2 Current compensation due to system-unbalance of a coupled sequence networks

Consequently, the equivalent system of (6.5) can be

decomposed into three sequence networks as follows:


120

Fig. 6.3. Power-flow algorithm based on symmetrical components

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

mSUN

mSU

mSU

mN

m

m

mN

m

m

mNN

mN

mN

mN

mm

mN

mm

I

II

I

II

V

VV

YYY

YYYYYY

_

_2

_1

2

1

2

1

21

22221

11211

(6.6)

In equation (6.6), the positive, negative, and zero-sequence

admittance matrices have been rewritten as a symmetrical system on the left hand side of the equation while the line coupling or the system unbalance are separated as current injections on the right hand side.


121

The system unbalance described by (6.4) is derived from the (3N×3N) admittance matrix in sequence components. This approach is tedious since it requires the construction of the (3N×3N) admittance matrix in sequence components [12]. This (3N×3N) admittance matrix then needs to be ordered to separate the system unbalance from the coupled sequence networks. This is computationally expensive.

In this work, the unbalance is included by current injection

compensation as shown in Fig. 3. The equivalent sequence current injection due to an individual line is calculated using (105.). However, for a general network, several lines are connected to a certain bus as shown in Fig. 5. Therefore the total current compensation due to the system unbalance can be determined by using (6.2) as follows:

( )∑∈=

Δ=M

isss

mi

mSUi II

,1_ (6.7)

The right hand side of (6.6) uses the current-iteration-voltage

values and therefore the unbalance terms are constant currents per iteration. The decomposed sequence networks given by (6.6) can be modified by combining the unbalance in the system with the current injection at each bus. The final decomposed sequence networks, at iteration r, have the following form:

( ) ( )rmrmm IVY =

+1 (6.8) The total current injection of (6.8) is calculated by combining

the current injection due to loads, distributed loads, capacitor banks, and system unbalance as follows: ( ) ( ) ( )rm

SUirm

i_SErm

i III _−= (6.9)


122

6.4.2. Specified Values of the Decomposed Sequence Networks

The current injections of (6.9) are used to calculate the voltages according to (6.8). The positive sequence problem is solved using balanced three-phase power-flow method. The negative and zero-sequence networks are solved directly since there are no specified values for these networks. In the standard balanced three-phase power-flow programs the specified values are expressed as constant power loads (PQ). Hence, the positive sequence current injection is converted to a complex power injection [12]. The final positive sequence power injections are expressed as follows [12]:

( ) ( ) ( )( )*11Specified

1 ri

ri

ri IVS = (6.10-a)

( ) ( )rir

i II 2Specified

2 = (6.10-b)

( ) ( )rir

i II 0Specified

0 = (6.10-c) In addition to (6.10), there is another additional equation

which represents the specified values of the PV nodes. It is expressed as follows [12]:

Specified

1Terminal_Specified

1ii VV = (6.11-a)

( )Specified

Terminal_Specified1

31

⎟⎠⎞

⎜⎝⎛= ii PP (6.11-b)

6.4.3. Solution of the Decomposed Sequence Networks The positive-sequence network is solved either by using the

standard Newton-Raphson method or the fast decoupled method [17]. Other methods, such as current injection method [18] may also be used with appropriate adjustment. It should be mentioned that the Newton-Raphson and fast decoupled are used without any


123

modifications in their formulation since the final specified values have the same form of the standard balanced three-phase power-flow methods. The negative sequence and zero sequence networks are solved as ordinary nodal voltage equations using the current injection calculated in (6.10-b) and (6.10-c). Finally, the three decomposed sub-problems are solved in an iterative scheme.

6.4.4. Iterative Solution Process

Figure 5.6 shows the proposed sequence power-flow method. It comprises three independent sub-problems corresponding to positive, negative, and zero-sequence networks [11-13]. The sequence power-flow requires the solution of the sub-problems in an iterative scheme. This is because the specified values of the sequence networks are expressed in terms of current injections which have to be updated when the bus voltages are updated. The independent sub-problems offer an opportunity for the proposed method to be implemented in parallel programming which would result in even faster execution [5.16].

Table 6.1: The solution of the IEEE 13 node radial test feeder


124

646645 632 633

634

650

692675611

684

652

671

680

500 ft, 603300 ft, 603 500 ft, 602

2000

ft, 6

01300 ft, 604300 ft, 605

800

ft, 6

07

1000

ft, 6

01100 ft, 606

2000

ft, 6

01

The length and configuration of the lines are marked in bold letters

500 ft, 606

Yg-Yg498 MVA, 4.16/0.48 kV

Z=1.1+j2.0 %

Fig. 6.4 The IEEE 13 node test feeder.

The specified values given by (6.10) and (6.11) account for different types of load models and capacitor banks as well as the system unbalance. Therefore, the decomposed algorithm is able to solve three-phase radial or meshed unbalanced networks with variety of component models. The convergence criteria of the method can be based on positive-sequence power mismatches or positive sequence voltages, or based on phase voltages.

6.5. Results and Discussions The IEEE 13 node and the IEEE 123 node feeders [19] are

used to validate the proposed approach for analyzing multi-phase systems as well as to test different load models. The solution of the IEEE 13 node and the IEEE 123 node is compared with the solution calculated using the radial distribution analysis program (RDAP) [20].


125

0.8

0.84

0.88

0.92

0.96

1

150 152 610 25 35 49 112 67 79 95Bus ID

Vol

atge

Va,

p.u

.

Sequence Newton RaphsonRDAP

0.8

0.84

0.88

0.92

0.96

1

150 53 610 18 40 300 100 81Bus ID

Vol

atge

Vb,

p.u

.


0.8

0.84

0.88

0.92

0.96

1

150 152 57 66 31 48 197 87 79Bus ID

Vol

atge

Vc,

p.u

.


Fig. 6.5. The solution of the IEEE 123 feeder


126

The IEEE 13 node feeder is shown in Fig. 5.7. The loads are the same as the original data except that the distributed load is removed. This is for the sake of comparison between proposed method and the RDAP as the distributed load is modeled differently in the two methods.

6.5.1. Multi-phase system test

In this test, the IEEE 13 node feeder and the IEEE 123 node feeder are solved using the proposed methods and the RDAP solver. The solution of the IEEE 13 node feeder is given in Table II. The voltage profile of the IEEE 123 feeder is shown in Fig. 6.5.

6.5.1.1 Solution of the IEEE 13 node feeder

Table I shows the results of the sequence Newton-Raphson method and the RDAP software. The table shows that the results calculated by the sequence power-flow methods are identical with RDAP, up to four decimal places. The dummy nodes have voltages equal to the upstream nodes voltage since the current flow through the dummy lines is zero.

If the distributed load along the line connected between nodes

632 and 671 is included in the input data of the feeder, the numerical results of the three-phase voltages are slightly different. The differences are only in the fourth figures. This is because the distributed load model adopted in the proposed methods is the lumped load model [9], whereas RDAP software may have used a different model, such as the spot load model at the middle of the line.

6.5.1.2. Solution of the IEEE 123 node feeder

The voltage profile of the IEEE 123 node radial test feeder is shown in Fig. 6.5 for both the sequence Newton-Raphson method and the RDAP software. The Newton Raphson method converges only after 3 iterations using a voltage tolerance of 0.001 p.u.. The


127

RDAP software tolerance setting is adjusted to 0.00001+j0.00001 p.u voltage. The figure shows that the calculated voltage profile of the methods is very close to each other.

Table 6.2: The convergence characteristics of phase ‘A’ in the

IEEE 13 node feeder

Table 6.3: Convergence characteristics of the proposed methods Test System Sequence

Newton Raphson Sequence Fast

Decoupled RDAP Software

Voltage Mismatch

(p.u.)* 10-4 10-5 10-6 10-4 10-5 10-6 10-4 10-5 10-6

IEEE 13 node 3 4 5 3 4 6 3 4 5 IEEE 123

node 3 4 5 3 4 6 3 4 5 *The phase voltage mismatch of a 0.0001 is equivalent to an entry of (0.0001 + j 0.0001) × 0.707 p.u. in the case of the RDAP software

6.5.2. Accuracy and Convergence characteristics

To examine the accuracy and the convergence characteristics of the proposed methods against the RDAP solver, the absolute voltage mismatch of phase ‘a’ of the IEEE 13 node feeder is summarized in Table 6.2 for the first four iterations. The table shows that the sequence Newton-Raphson solution is the most accurate for the same number of iterations. For example, at iteration 4, the sequence Newton-Raphson has the lowest phase


128

voltage mismatch in comparison with both the sequence fast-decoupled and the RDAP solver. The reported numerical results in Table 6.2 shows that the proposed line model and the sequence power-flow methods are accurate when they are compared with the commercial grade radial distribution analysis program.

Table 6.3I shows the convergence characteristics for both the proposed methods, the results shows that both sequence Newton-Raphson and the RDAP solver have identical number of iterations. The sequence fast decoupled has comparable convergence characteristics to both the sequence Newton-Raphson and the RDAP solver. Table III also shows that the sequence power-flow methods inherit the characteristics of the balanced power flow solver. The standard fast-decoupled method usually requires more iterations to converge for the same solution calculated using the Newton-Raphson method.

The results given in Table 6.2 and Table 6.3 show that the dummy lines and dummy nodes are accurately modeled the multi-phase systems in terms of sequence components. The proposed sequence power-flow methods generally exhibit similar or slightly better performance to the RDAP solver as demonstrated in Table II. However, in addition to the radial feeders, the proposed methods have the ability to analyze meshed networks with any number of PV nodes efficiently [12-13]. The proposed line model is also being tested to solve unbalanced fault calculations in multi-phase systems and will be reported in near future.

6.5.3. Influence of the load model

The IEEE 13 node and the IEEE 123 node feeders are used to examine the influence of the load model on the convergence characteristics. The initial load values are gradually increased until the divergence occurred. The results for both the sequence power-flow methods as well as the RDAP solver are shown in Fig. 6.6, Fig. 6.7 and Fig. 6.8 for constant power load model, constant current load model, and constant impedance load model respectively.


129

0

20

40

60

80

1 2 3 4 5 6Factor of Loads

Num

ber o

f Ite

ratio

ns

IEEE

13

Nod

e N

umbe

r of I

tera

tions

IE

EE 1

23 N

ode

0

20

40

60

80

Sequence FD Sequence NR

RDAP


RDAP

Fig. 6.6. The influence of the constant complex power load model on the convergence characteristics

6.5.3.1 Constant power load model

In this test, the three-phase power-flow methods can not converge beyond a load factor of 2.0 and the convergence characteristics of the three methods are similar as shown in Fig. 6.6. However, in the case of the IEEE 123 node feeder a large number of iterations are required for convergence at a load factor of 2.


130

0

20

40

60

80

0

20

40

60

80



RDAP


RDAP

Num

ber o

f Ite

ratio

ns

IEEE

13

Nod

e N

umbe

r of I

tera

tions

IE

EE 1

23 N

ode

Fig. 6.7. The influence of the constant current load model on the

convergence characteristics

6.5.3.2. Constant current load model When constant current load model is considered, the three

methods have similar convergence profile up to a load factor of 3.5 as shown in Fig 6.7. However, beyond this load factor, the sequence Newton Raphson requires large number of iterations for the IEEE 13 node test. On the other hand, the three methods almost have similar convergence characteristics in the case of IEEE 123 node feeder.


131

6.5.3.3. Constant impedance load model Figure 6.7 shows that the fast decoupled method has the better

convergence characteristics than RDAP solver specially in the case of the IEEE 13 node feeder. However, the three-methods have similar convergence characteristics in the IEEE 123 node case study.

The above results in Fig. 6.6, Fig. 6.7, and Fig. 6.8 shows that the sequence fast decoupled has good convergence characteristics for high load factors. This is consistent with the results reported in the literature that the sequence fast decoupled is less sensitive to the load unbalance whereas it is more sensitive to the line R/X ratio [12-13]. The results also shows that the sequence Newton-Raphson is not sensitive to the size of the system since the method has similar convergence characteristics for both IEEE 13 node and the IEEE 123 node feeders for different load models.


132

0

20

40

60

80

0

20

40

60

80



RDAP


RDAP

Num

ber o

f Ite

ratio

ns

IEEE

13

Nod

e N

umbe

r of I

tera

tions

IE

EE 1

23 N

ode

Fig. 6.8. The influence of the constant impedance load model on

the convergence characteristics

6.6. Conclusions

This chapter has presented an approach which can be used to analyze unbalanced networks based on symmetrical components. Missing phases have been represented by unloaded dummy nodes and dummy lines which are not coupled to the real phases. The modified network will have a three-phase structure which can be solved using sequence components. The decomposed positive-, negative-, and zero-sequence networks is solved by using an iterative scheme. The positive sequence network is solved using both the standard Newton-Raphson and fast-decoupled power-flow


133

methods whereas the negative and zero-sequence networks are represented by two nodal voltage equations. The solution of meshed and radial systems shows that the three-phase networks with multi-phases can be solved based on sequence components with accurate results and robust convergence characteristics. In addition, the approach can be generalized to integrate balanced power-flow, unbalanced power-flow, and fault calculations in a single tool.

REFERENCES [1] J. Arrillaga and N. R. Watson, “Computer modeling of

electrical power systems”, Book, 2nd Edition, John Wiley & Sons LTD, pp 11-13, pp 48-51, and pp 111-115, 2001.


[3] A. V. Garcia and M. G. Zago, “Three-phase fast decoupled power flow for distribution networks”. Proc IEE. Generation, Transmission, and Distribution, vol. 143, no. 2, pp. 188-192, March 1996

[4] H. L. Nguyen, “Newton–Raphson method in complex form-power system load flow analysis”, IEEE Trans. Power Systems, vol. 12, no. 3, pp. 1355–1359, August 1997

[5] P. A. N. Garcia, J. L. R. Pereira, S. Carneiro J. R, V. M. Da Costa, and N. Martins, “Three-phase power flow calculation using the current injection method”, IEEE Trans. Power Systems, vol. 15, no. 2, pp 508-514, May 2000.


[7] T. H. Chen, M. S. Chen, K. J. Hwang, P. Kotas, and E. A. Chebli, “Distribution system power flow analysis- a rigid


134

approach”, IEEE Trans. Power Delivery, vol. 6, no. 3, pp. 1146-1152, July 1991.

[8] W. H. Kersting and D. L. Mendive, “An application of ladder network theory to the solution of three phase radial load flow problem“, IEEE PES winter meeting, New York, Jan. 1976.

[9] C. S. Cheng and D. Shirmohammadi, “A three-phase power flow method for real-time distribution system analysis” IEEE Trans. Power Systems, vol. 10, pp. 671–679, May 1995.

[10] J. H. Teng, “A direct approach for distribution system load flow solutions”, IEEE Trans. Power System, vol. 18, no. 3, pp. 882-887, July 2003

[11] K. L. Lo and C. Zhang, “Decomposed three-phase power flow solution using the sequence component frame”, Proc. IEE, Generation, Transmission, and Distribution, vol. 140, no. 3, pp 181-188, May 1993


[13] M. Abdel-Akher, K. M. Nor, and A. H. Abdul-Rashid, “Improved three-phase power-flow methods using sequence components”, IEEE Trans. Power Systems, vol. 20, no. 3, 1389-1397, Aug. 2005

[14] E. R. Ramos, A. G. Expósito, and G. A Cordero, “Quasi-coupled three-phase radial load flow”, IEEE Trans. Power Systems, vol. 19, no. 2, pp. 776-781, May 2004

[15] J. A. Peralta, F. de León, and J. Mahseredjian, “Unbalanced multiphase load-flow using a positive-sequence load-flow program”, IEEE Trans. Power Systems, vol. 23, no. 2, pp. 469-476, May 2008

[16] X. P. Zhang, W. –J. Chu, H. Chen "Decoupled asymmetrical three-phase load flow study by parallel processing", IEE Proc., Generation, Transmission, and Distribution, vol. 143, no. 1, pp. 61 – 65, January 1996


135


[18] V. M. Da Costa, N. Martins, and J. L. Pereira, ”Developments in the Newton Raphson power flow formulation based on current injections”, IEEE Trans. on Power Systems, vol. 14, no. 4, pp 1320-1326, Nov. 1999.

[19] W. H. Kersting, “Radial distribution test feeders”, PES winter meeting, vol. 2, no. 28, pp. 908-912, Jan. 28/Feb. 1, 2001: data can be downloaded from internet at: http://ewh.ieee.org/soc/pes/dsacom/test feeders.html

[20] Radial Distribution Analysis Program (RDAP), which can be downloaded from http://www.zianet.com/whpower/


136

Three Phase Load Flow Analysis with Distributed Generations

137

7 THREE PHASE LOAD FLOW ANALYSIS

WITH DISTRIBUTED GENERATION

Syafii Gahazali Khalid Mohamed Nor

Mamdouh Abdel-Akher

7.1 Distributed Generation

A distributed generation (DG) is an electric power source connected into utilities networks at distribution level with typically ranging from 10 kW up to tens of MW capacity. A study by the Electric Power Research Institute (EPRI) have predicted that DG may account for up to 25% of all new generation going online by the year 2010. The integration of DG will be increase and give the benefit contribution to existing grid power system operation and has been an active area of research today [9-12].

7.1.1 Three Phase DG model The generators can be adequately modeled by their sequence networks since they are designed for maximum symmetry of the phase windings [13]. Fig. 7.1 shows a synchronous generator grounded through a reactor. The generator model given below is


138

for cylindrical synchronous machines. The voltage equation at the terminal bus can be expressed as follows:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

n

n

n

n

c

b

a

cccbca

bcbbba

acabaa

cn

bn

an

cn

bn

an

III

ZIII

ZZZZZZZZZ

EEE

VVV

(7.1)

The standard sequence network generator model can be calculated by transforming (7.1) to sequence components as follows:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

n

n

n

n

a

a

a

cccbca

bcbbba

acabaa

cn

bn

an

an

an

an

III

ZIII

ZZZZZZZZZ

EEE

VVV

2

1

0

2

1

0

AA (7.2)

The matrix A is defined as the symmetrical components transformation matrix :

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

2

2

11

111

aaaaA (7.3)


139

Fig 7.1. Three-phase distributed generation circuit diagram

Where a =1.0 ∠120° Multiplying (7.2) by the inverse of the transformation matrix (7.3), since the generator internal voltages are balanced, the internal voltages of phase ‘b’ and phase ‘c’ can be replaced by the magnitude of phase ‘a’ and the corresponding phase angle resulting in:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ +−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

2

1

0

2

0

2

1

0

0000003

0

0

III

ZZ

ZZE

VVV

q

n

q (7.4)

A synchronous generator is usually expressed by its sequence data, rather than its phase data. Equation (7.4) is used to establish the uncoupled sequence networks shown in Fig. 7.3.


140

7.1.2 PV and PQ Node Model The sequence component model allows using the balanced three-phase power-flow specifications for generators [8], [14]. Normally at a terminal generator bus, both the positive sequence voltage and the total power leaving the terminal of the actual three-phase bus are specified. In Fig. 7.2, the positive sequence voltage magnitude at the terminal bus is the same as the positive sequence network of the generator, and hence, the positive sequence voltage magnitude of the positive sequence network of the generator is specified. The total power specified at the terminal bus is mainly due to the positive sequence network of the generator since there is no induced EMF in both the negative- and zero-sequence networks. Consequently, the specified power of the positive sequence network of is known.

By knowing of injected power and voltage magnitude at generator bus, the DG can modeled as voltage controlled device or PV model. If the Q limit is used, the PV model will be automatically converted to a PQ model. The DG will operate as a PQ injection source rather than voltage controlled device. Figure 3 shows the sequence decoupled power-flow, it comprises of three independent sub-problems corresponding to positive-, negative-, and zero-sequence networks [8]. The power-flow requires the

Figure. 7.2 Distributed Generation sequence components model


141

solution of the sub-problems in iterative scheme. This is because the specified values of sequence networks are expressed in terms of current injections and should be updated per iteration.

The total sequence current is used as the specified values for the sequence networks as shown in Fig. 3. In standard balanced power-flow programs the specified values are expressed as constant power loads (PQ), therefore, the positive sequence current injection is transformed to an equivalent power injection. The final sequence power and current injections are expressed as follows:

ki_

ki_ II 0Specified_0 = (7.5)

( )*11Specified_1ki_

ki_

ki_ IVS = (7.6)

ki_

ki_ II 2Specified_2 = (7.7)

In addition to (6.5-6.6), there are more one equation represents the specified values of the DGs which is expressed as follows:

busterminalatSpecified_1_1_ ii VV = (7.8)

busterminalatSpecified_Total_1_ 3

1ii PP = (7.9)

The positive sequence network has been solved using the standard Newton-Raphson. It should be mention here in that the Newton-Raphson and fast decoupled [8] are used in this paper without any modifications in their formulation since the final specified values of the sequence networks are still having the same form as shown in Fig. 7.3 and given by (7.8) and (7.9). The negative and zero sequence networks are solved using nodal voltage equations. The specified values of the negative- and zero sequence networks are expressed as current injections. Therefore, the solution of both the negative- and zero-sequence networks can be expressed by ordinary nodal voltage equations as follows, using matrix notation:


142

d2_Specifie22 IVY = (7.10) d0_Specifie00 IVY = (7.11)

After solving sequence network, the phase voltage for base can be calculated. The process is repeated until convergence criterion reached. In the program phase voltages mismatch, positive sequence voltage mismatch and the positive sequence power mismatch can be used as convergence criterion.


143

7.2 Description of the Test System The practical distribution data 37-node feeder is an actual feeder [2]. The original data of the feeder has been modified to account for various unbalance factors. The complete data are given in the appendix at the end of the paper. This data is characterized with the following:

1- The loads may be single-phase, two-phase, or three-phase. They are connected as star or delta. They are includes the three main types of the load models, i.e. constant power, constant current, or constant impedance.

Figure. 7.3 Power-flow algorithm with DG based on symmetrical components


144

2- The line matrix is fully coupled and unsymmetrical. One capacitor bank already add at node 35 with size 50 kVar, 40 kVar, and 90 kVar for phase a, phase b and phase c respectively.

7.3 Results and Discussions The proposed model tested and analyzed using practical distribution data 37 node feeder without and with DG. Firstly, the program is validated using radial distribution analysis package (RDAP) for the original feeder as shown in Fig. 7.4. The analysis

Figure. 7.4 Test system 37 node original feeder

Figure. 7.5 Test system 37 node feeder with DG connected


145

is illustrated through three-phase power-flow simulation with various DG size and location as shown in Fig. 7.5. The following different cases are studied:

Case 1: Without DG (Base case) Case 2: 1 unit DG 100 kW place at node 11 Case 3: 1 unit DG 100 kW place at node 23 Case 4: 1 unit DG 100 kW place at node 35 Case 5: 1 unit DG 200 kW place at node 11 Case 6: 1 unit DG 200 kW place at node 23 Case 7: 1 unit DG 200 kW place at node 35


146

Table 7.1 Sample Results For The Phase Voltages Of Case 1: Base Case Of

The 37-Node Feeder


147

The results of the base case are almost identical to the obtained using the RDAP software as shows in table 7.1. This shows that the power system models developed in [8] are accurate for distribution power-flow analysis. The effect of proposed DG model on voltage profile is given in

figure 7.6 for the three-phases. The result shows that, the voltage is decreased by increase of node number without DG installation.

Phase A0.9

0.94

0.98

1.02

1 5 9 13 17 21 25 29 33 37

Vol

tage

(pu)

W ithout DG

DG 100kW at 11

DG 100kW at 23

DG 100kW at 35

DG 200kW at 11

DG 200kW at 23

DG 200kW at 35

Phase B0.9

0.94

0.98

1.02

1 5 9 13 17 21 25 29 33 37

Vol

tage

(pu

)

Phase C0.9

0.94

0.98

1.02

1 5 9 13 17 21 25 29 33 37Bus ID

Vol

tage

(pu)

Figure. 7.6 Effect of DGs on voltage profile


148

However, after DG installed in the network, the voltage profile of

the three-phases has been improved as shown in figure 7.6. For DG located near substation for node 11 of cases 2 and 5, the voltage near to fix for node 1 to 11, but continuously decrease until the end of node. If DG installed at the end of node for cases 4 and 7, the voltage increase at the end of node, but the voltage drop for the middle node. For cases 3 and 6, DG installed at the middle of the node or node 23, the voltage almost fix for all nodes and just small drop for the end of the nodes. Therefore, the best result is when DG located at the middle 23 node among three test cases. Another variation is DG size. The effect of DG size not significant to voltage improvement as shown in the curve, since this the improvement of the voltages is dependent on the amount of the reactive power injected at the PV node. However, if the Q limit is used, in this case the PV node will be automatically converted to a PQ node. In this case the DG will operate as a PQ injection source rather than voltage controlled device. The effect of proposed DG model on line losses is given in Fig.7.7 for all cases. The result shows that the losses are reduced when the size of DG increased from 100 kW to 200kW and location of DG increased from node 11 to node 35. However, the incretion of DG location does not always guarantee the line loss reduction such as

Fig. 7.7 Effect of DGs on network losses


149

for same DG rating 100 kW cases 2, 3 and 4 with increased DG location. Therefore, the DG size give more effect in network loss reduction compared to DG location but the setting of rating and location of the DG are important factors for network loss reduction.

7.4 Conclusion The Chapter has presented three phase DGs model in unbalanced three-phase distribution power-flow and analyzed their effect when they are connected in distribution networks. In this paper, the DG was modeled as PV node with an option to convert to a PQ node when it achieved Q limit. The model was tested and analyzed using a practical 37 node distribution feeder with various size and location of DG. The simulation results show that DG size and location are important factors to improve voltage profile and line losses reduction. The DG location give more effect in voltage profile improvement compared to network loss reduction for DG modeled as PV node. In contrast the DG sizes give more effect in network loss reduction compared to voltage profile improvement.

REFERENCES [1] W. H. Kersting and 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, IEEE paper no. A76 044-8, 1976

[2] P. A. N. Garcia, J. L. R. Pereira, and S. Carneiro Jr, “Voltage control devices models for distribution power flow analysis”, IEEE Trans. on power systems, vol. 16, no. 4 pp. 586-594, November 2001

[3] D. Shirmohammadi, H. W. Hong, A. Semlyen, and G. X. Luo, “A compensation-based power flow method for weekly meshed distribution and transmission networks. IEEE Transaction on Power System, vol. 3, no. 2, pp. 753-762, May 1988.


150

[4] C. S. Cheng and D. Shirmohammadi,” A three-phase power flow method for real time distribution system analysis. IEEE Transaction on Power System, vol. 10, no. 2, pp. 671-679, May 1995

[5] R. D. Zimmerman and H. D. Chiang, “Fast decoupled power flow for unbalanced radial distribution systems,” IEEE Transaction on Power System, vol. 10, no. 4, pp. 2045-2052, November 1995.

[6] F. Zhang and C. S. Cheng,”A modified Newton method for radial distribution system power flow analysis,” IEEE Transaction on Power System, vol. 12, no. 1, pp. 389-397 February 1997.

[7] B. C. Smith and J. Arrillaga, “Improved three-phase load flow using phase and sequence components,” IEE Proc. Generation, Transmission and Distribution, vol. 145, no. 3, , pp. 245-250, May 1998.

[8] M. Abdel-Akher, K. M. Nor, and A. H. Abdul Rashid, “Improved Three-Phase Power-Flow Methods Using Sequence Components”, IEEE Transaction on Power System Vol.20, no 3, pp. 1389-1397, Aug 2005

[9] S. Khushalani, J. M. Solanki and N. N.Schulz,” Development of Three-Phase Unbalanced Power Flow Using PV and PQ Models for Distributed Generation and Study of the Impact of DG Models”, IEEE Transactions on Power Systems, Vol. 22, NO. 3, AUGUST 2007

[10] G. Pepermansa, J. Driesenb, D. Haeseldonckxc, R. Belmansc, and W. D’haeseleer, “Distributed generation: definition, benefits and issues”, Energy Policy, Volume 33, pp.787-798, 2005.

[11] J.A. Pecas Lopes, N. Hatziargyriou, J. Mutale , P. Djapic, and N. Jenkins, “Integrating distributed generation into electric power systems: A review of drivers, challenges and opportunities, “Electric Power Systems Research Volume 77, Issue 9, July 2007 pp. 1189–1203, July 2007.

[12] S. Khushalani, J. M. Solanki, and N. N. Schulz,“Development of Three-Phase Unbalanced Power Flow


151

Using PV and PQ Models for Distributed Generation and Study of the Impact of DG Models,” IEEE Transaction on Power System Vol.22, no 3, pp. 1019-1025, Aug 2007.

[13] J. Arrillaga and N. R. Watson, Computer modeling of electrical power systems. (2nd Edition, John Wiley & Sons LTD, 2001)

[14] M. Abdel-Akher, K. M. Nor, and A. H. Abdul Rashid”, “Revised sequence components power system models for unbalanced power system studies”, Proc. of the Asian power and energy systems conference, AsiaPES, Phuket, Thailand, April 4-7, 2007.

Appendix

The 37 node data Load data

Node Load Ph-1 P-1 Ph-2 P-2 Ph-3 P-3 Model kW kVar kW kVar kW kVar 2 Y-PQ 4 2 5 3 6 3 3 Y-PQ 5 1 4 2 4 2 4 D-PQ 0 0 0 0 4 2 5 Y-Z 5 3 5 3 5 3 6 Y-PQ 7 4 8 5 10 5 7 Y-Z 6 4 8 5 8 5 8 D-Z 5 3 5 3 5 3 9 Y-I 11 6 11 7 13 7 10 Y-I 14 6 11 7 11 7 11 Y-I 5 3 5 3 5 3 12 Y-I 0 0 15 9 20 5 13 D-Z 15 9 15 9 15 5 14 Y-Z 5 3 5 3 5 3 15 Y-Z 15 12 19 12 30 20 16 Y-PQ 23 12 0 0 0 0 17 Y-PQ 5 3 0 0 0 0 18 D-PQ 40 10 23 14 40 25 19 Y-PQ 23 17 23 14 15 15 20 Y-PQ 5 3 5 3 5 3 21 Y-PQ 20 15 26 16 22 16 22 Y-PQ 20 15 26 16 22 16


152

23 Y-PQ 5 3 5 3 5 3 24 D-PQ 25 18 30 19 25 18 25 Y-Z 25 18 30 19 25 18 26 Y-Z 5 3 5 3 5 3 27 Y-Z 33 25 34 21 30 20 28 Y-Z 30 25 34 21 30 20 29 Y-I 5 3 5 3 5 3 30 D-I 45 25 38 23 38 23 31 Y-I 38 25 0 0 0 0 32 Y-I 5 3 5 3 5 3 33 Y-I 40 25 42 26 35 25 34 Y-I 35 20 42 26 35 23 35 Y-PQ 0 0 5 3 5 3 36 D-PQ 40 3 30 3 45 15 37 Y-PQ 40 3 30 3 45 15

Line configuration in Ω/mile :

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+++++++++

06511461504236015800384915350423601580004781466605017015600

38491535050170156000178145760

.....

...........

jjjjjjjjj

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−−

397151645169820164519809583191698208319167655

.........

Ω-1/ mile

Capacitor data: Node 35: Qa=50 kVar, Qb= 40 kVar, and Qc=90 kVar

Future Potentials and Works

153

8 FUTURE POTENTIALS AND WORKS

Syafii Gahazali

Khalid Mohamed Nor

Load flow analysis has come along way from the beginning of its solution on the digital computers. Most challenges in load flow analysis algorithm has been met successfully. However as power system engineering and technology develops, new issues cropped that need new approach and direction. In the coming years there will be greater growth in distributed generation and what has been described as smart grid. Power system consumers will use more single phase loads to drive their electronics and computer based equipment. The scenario will be that, there will be some of form of voltage control to accommodate the generators and greater unbalance in the distribution network. The need for three phase load flow analysis would therefore be greater than in the past. Power flow analysis involves computer hardware. Over the last few decades, computing technology continues to advance rapidly. Naturally power flow algorithm needs to change in order to take advantage of new developments in computing technology. Among the exciting computing technology that has appear promising is the web based computing and parallel programming. The Three-phase load flow algorithm and analysis can be enhanced in many ways and there will always be new developments and new areas for researchers to explore. The


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following will briefly present further enhancements of three-phase load flow algorithm and analysis. 8.1. Parallel Programming From about 1986 to 2002, the performance of microprocessors improved at the rate of 52% per year [1]. Today, its performance is improved by the addition of processors in the same machine. The so called multi-core systems are now quite common. Multi-processor machines are now becoming standard while the speed of single processors have almost stabilized or is increasing slowly compared to its rapid in the past. Therefore in the present trend of computing technology, performance improvements can now be achieved more on the ability to run a program on multiple processors in parallel. In other words, the multicore approach improves performance only when software can perform multiple activities at the same period of time. Unfortunately, it is still very challenging to write algorithms that really take advantage of multiple processors. Most applications presently use a single core processor. They see no speed improvements when run on a multi-core machine, since it is executed serially which in fact means that it is really running as if it is in a single core machine mode. One of the latest compilers that supports parallel processing is MS Visual C++ 2008 with OpenMP standard or Intel® C++ Compiler. The Intel C++ compiler can be integrated into Visual Studio® 2008 for a full development environment. These software can take advantage of systems that have multiple processors, or multiple-core processors. A separate executable building process is created for each available processor. As an example, if the system has four processors, then four executables processes are created. The Compilers can process these executables simultaneously, and therefore overall executable building time is reduced.


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The OpenMP is primarily designed for shared memory multiprocessor using directive based approach [2]. C++ OpenMP provide a standard include file, called omp.h, that provide the OpenMP type definition and library function prototype. Using this library, user can conveniently express potential parallelism in existing sequential code, where the exposed parallel tasks will be run concurrently on all available processors. Usually this results in significant increase in execution speed. Other compiler is the Intel® C++ Compiler that support multi-threaded code development through C or C++. The parallel processing makes a program run faster and efficient because there are more engines (CPUs) running together. By using these techniques the program will be more efficient and faster. The availability of parallel processing hardware and software presents an opportunity and a challenge to apply this new computation technology to solve power system load flow analysis. In three-phase load flow based on sequence component, the positive-, negative- and zero- sequence bus admittance matrices can be created simultaneously using parallel processing techniques as well as pre computation of DG model for co-generator, wind turbine, fuel cell and photovoltaic. In the load flow iterative scheme, the three sequence decoupled networks can be solved simultaneously by using multi core processors. At the end of each iteration, data sets of bus voltage are combined and processed. The computation will be done in parallel and serially. It needs to be pointed out that parallel computing works exceptionally well for large scale computation. Parallel programming can be used in a multiple internal processors (PC based parallel system) arrangement and multiple interconnected computers (multiple computers) environment.


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8.2. Implementation of load flow in parallel processing The applications of parallel technique for three-phase power flow have been reported in [3], and implemented in [4] under a PC 486/33 host computer and a T805-20 transputer board with four interconnected processors. However these parallel system are costly to use. Other parallel programming of load flow has used a cluster of GNU/Linux dual-processor workstations[4]. The workstations are connected via100Mbit/s Ethernet, i.e., the parallel machine consists of hardware readily found in any engineering department. The results showed that the new algorithm can improve computation speed. The message passing computation time can calculated using:

T = tcomm + tcomp Where : tcomm = the communication time

tcomp = the computation time The time of computation also depend of speed of communication media. Other alternative to reduce communication time is by using multi core processors in single computer. PC based parallel system with multi core processors are the latest generation of Intel® multi-core technology i.e quad-core processors. It provides four complete execution cores within a single processor. This parallel system have low cost with predicted high-performance compare to parallel system used by Zang [4] and Feng Tu [5]. There is hardly any report of implementation of load flow analysis or power system analysis on Intel multicore platform. This is not surprising given that the previous works do not seem to be successful in terms of speed up and furthermore the hardware are expensive. The cost of the hardware make the advantage of faster solution not worthwhile. Under the presence scenario, the hardware cost is no longer an issue, in fact the cost of the hardware is the same whether a program is run as a serially or in parallel.


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An important aspect in load is the matrix solver. Recently it was reported that [12] a public domain software the Superlu, a sparse matrix solver, has successfully been shown to have a significant execution speedup when executed in parallel mode using Intel multicore platform. This is a good foundation upon which a competitive parallel load flow analysis can be developed. 8.3. Web-based Programming

The operational and commercial needs of the power industry require information systems to not only perform traditional functions but also support many of the new functions, specially to meet the needs of competition with deregulation. The rapid development of the Internet and distributed computing have opened the door for feasible and cost-effective solutions [8]. Remote monitoring and control is one of the most promising applications of the internet as an excellent medium for providing remote access. Many application programs have been transferred to the Web-based platform and load flow analysis to be one of them.

In ref [6] describe a Web-based, platform-independent load flow analysis package. Three tiers architecture containing client tier, middle tier, and data tier is adopted in this system. It supplies client users with flexible and portable independent user interfaces, and this design can share this package with computers on the Internet. The design concept have tested under six bus power system and shown a good result. The algorithm and software still need to improve in order to speedup of computation time and user friendly application.

8.4. Three-phase Load Flow with Specific DG

Due the increase in the load demand, many generation units are currently installed in distribution networks. Distribution


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feeders are traditionally solved with a single voltage source i.e. substation. With increase of generator installed in distribution networks as known as Distributed Generation (DG) there is a need to add generator model in unbalanced three phase distribution power flow.

The most common of DG technologies installed today are cogeneration, wind power, photovoltaic and fuel cell. These technologies have different electrical characteristics and thus have different impact/result in power system analysis. The wind turbine generator can be used induction type or synchronous type. Distributed generation of induction type driven by wind power have different characteristics, because power output is not fixed by turbine governor setting, but depend on wind speed which is a variable quantity and they usually draw almost fixed reactive power from the associated network [9]. Another source of distributed generator is the photovoltaic (PV) systems, which are commonly known as solar panels. Solar panels are made up of discrete PV cells connected together to be PV modules and PV arrays that convert light radiation into electricity. The PV cells produce direct-current (DC) electricity, which must then be inverted for use in an AC system. The systems can be used as single phase source or three phase source and thus can have the unbalancing impact on the grid connected system. The output power and voltage varies according to sun radiation and modules temperature. Both parameters have to be input for next load flow algorithm with PV connected.

According to literature searching, there is a cogenerator

model as PV and PQ bus already developed in phase component based load flow [10] and in sequence component base load flow [11], but the code need to improve using newer approach of software development such as object components and just a few researcher works focus on wind power and photovoltaic model in distribution power flow. Because this technology is still new and in development stage. Therefore there are still many open questions


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need more research to further improve of specific DG models in three phase distribution power flow and analyze their impact in distribution network planning, operation and control .

REFERENCES [1] David Callahan, “Design Considerations For Parallel

Programming”, MDSN Magazine, 2008 [2] R Chandra, L Dagum, D Kohr, D Maydan, J McDonald, and

R Menon, Parallel Programming in Open MP, Morgan Kaufmann Publishers, Oct 2000.

[3] X. P. Zhang, “Fast three phase load flow methods”, IEEE Trans. on Power Systems, vol. 11, no. 3, pp 1547-1553, Aug. 1996

[4] Zhang X. P., Chu W. J., and Chen H. “Decoupled asymmetrical three-phase load flow study by parallel processing”. IEE Proc. Generation, Transmission and Distribution, vol. 143 no. 1 January, pp. 1996

[5] Feng Tu, Flueck, A.J, “A message-passing distributed-memory parallel power flow algorithm”. Power Engineering Society Winter Meeting, 2002. IEEE,Volume 1, 27-31, Jan. 2002, Page(s):211 - 216

[6] Rong-Ceng Leou; Zwe-Lee Gaing,A Web-based load flow simulation of power systems, Power Engineering Society Summer Meeting, 2002 IEEE Volume 3, July 2002 Page(s):1587 - 1591

[7] Narain G. Hingorani and Laszlo Gyugyi, “ Understanding FACTS Concept and Technology of Flexible AC Transmission System”, 2000, IEEE, Press.

[8] S. Chen, and F. Y. Lu, “Web-Based Simulations of Power Systems, IEEE Computer Applications in Power Vol. 15, Issue: 1 , Jan 2002, pp. 35-40

[9] M. R. Patel,”Wind and Solar Power System”, CRC Press, Boca Raton, Florida, 1999

[10] S. Khushalani, J. M. Solanki and N. N.Schulz,” Development of Three-Phase Unbalanced Power Flow


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Using PV and PQ Models for Distributed Generation and Study of the Impact of DG Models”, IEEE Transactions on Power Systems, Vol. 22, No. 3, August 2007

[11] Syafii, K M Nor, M Abdel-Akher, “Analysis of Three Phase Distribution Networks with Distributed Generation”, 2nd IEEE International Power & Energy Conference (PECon 2008), Johor Bahru, Dec 2008

[12] X.S. Li, “Evaluation of SuperLU on Multi-core Architectures", Journal of Physics: Con-ference Series (to appear). Proc. of SciDAC 2007, July 13-17, 2008, Seattle.

Index

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INDEX

Backward/Forward, 119 Branch numbering, 119, 126 C++ Builder, 64, 65 Capacitor banks, 151 CBD, 41, 63 Constant current load, 167 Constant power load, 166 Current compensation, 154 Current injection, 20, 148 Distributed generation, 187,

197 Distribution networks, 22,

111 Dummy node, 142 Fast Decoupled, 34, 42, 47,

49, 68, 72, 84, 86, 88, 94, 96, 98, 99, 130, 163

Gauss method, 18, 19, 25, 110

GUI, 41, 62, 66 HVDC, 3, 38, 53 IEEE radial feeders, 147 Intel C++ compiler, 192 Jacobian matrix, 18, 20, 21,

25, 26, 27, 28, 29, 43, 72, 97, 98, 107, 146, 148

Line, 53, 77, 117, 149, 189 Losses, 92, 105, 106 LU decomposition, 28

MS Visual C++, 192 Object oriented, 50 OOP, 41, 51 OpenMP, 192 Phase components, 2 Power flow analysis, 190 PQ node, 19, 149, 184, 185 PV node, 18, 19, 20, 21, 22,

24, 108, 110, 117, 134, 146, 149, 157, 165, 184, 185

Reusability, 42, 50, 104, 137, 172

Slack bus, 13 software application, 40, 41,

64, 65, 68 sparse matrix, 195 SuperLU, 59, 69, 85, 86, 90,

131, 199 Three-phase, 34, 35, 36, 103,

107, 124, 134, 136, 138, 145, 170, 191, 196

Transformer, 53, 76 Unbalanced laterals, 115 UPFC, 38, 39, 44, 45, 46, 48,

50, 53, 54, 55, 61, 67, 69 Voltage profile, 133 Zbus, 25

three phase load flow analysis

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