new load flow methods and techniques by ahmed mustafa

NEW LOAD FLOW METHODS

AND TECHNIQUES

BY

AHMED MUSTAFA MOHAMMED AL-JAALI

INDEX NO.074008

SUPERVISOR

DR. ABDUL-RAHMAN ALI KARRAR

REPORT SUBMITTED TO

University of Khartoum

In partial fulfillment of the requirement of the degree of

B.Sc. (HONS) Electrical and Electronic Engineering

(POWER SYSTEMS ENGINEERNG)

Faculty of Engineering

Department of Electrical and Electronic Engineering

September 2012

ii

DECLARATION OF ORIGINALITY

I declare that this report entitled “New Load Flow Methods and Techniques” is my

own work except as cited in the references. The report has not been accepted for any degree and

is not being submitted concurrently in candidature for any degree or other award.

Signature: _________________________

Name: ____________________________

Date: _____________________________

iii

Dedication

To the late Dr. Fayiz M. Al-Sadig, a teacher, scientist, and a true father…

iv

Acknowledgment

All praise is due to Allah Lord of all the worlds, the One God to whom praise is due

forever.

I would like to express my deep gratitude to our supervisor Dr. Abdul-Rahman Karrar,

without him, after Allah, this work wouldn‟t see the light. He was a real mentor, teacher and

close advisor. He never spared neither time nor effort to help us in our work by all means

possible.

Special thanks is for my project partner, Oday Hassan who was a real mate throughout

the project work and helped me and exhibited much of patience and cooperation.

Finally, I would like to thank the University of Khartoum for giving me the

vi

Abstract

Load flow Analysis is probably one of the most important studies of the performed on a

power system since it evaluates steady state operating conditions of the system. Principal

information obtained from a load flow is the voltage angle and magnitude at each bus hence the

flow of active and reactive power in each line. Load flow analysis is required by other power

system studies such as transient stability, contingency analysis and voltage stability. Numerical

analysis is the main tool used to solve the load flow problem to obtain the desired results.

A problem that faces the numerical methods is that they do not converge near the voltage

stability limit, this lead to the emergence of what‟s known as Continuation Power Flow, which is

a method that performs load flow beyond the voltage stability limit. A numerical technique is

employed to achieve this task. Continuation Power Flow employs arrives at the full PV curve

which is vital to the voltage stability analysis.

Project scope is to discover and implement new methods and techniques of load flow

that overcome the problem of Jacobian singularity and arrive at the full continuation curve.

New methods were discovered; their mathematical expressions were derived and

programmed by MATLABTM

. For the sake verification, the programs were tested on IEEE -14-

bus test system and results were compared with other results obtained by PSAT, Power System

Analysis Toolbox used on MATLABTM

and they were satisfactory.

vii

المستخلص

حث أنها تقم ربائة، الكه د كون واحدا من أهم الدراسات الت تجرى على نظام القدرةتحلل سران الحمولة ق

عند كل جهد المعلومات األساسة المستقاة من تحلل سران الحمولة ه زاوة و مقدار ال. التشغل للحالة المستقرة ظروف

: مطلوب من قبل دراسات نظم قدرة أخرى مثلتحلل سران الحمولة . التفاعلةالقدرة و ، و سران القدرة النشطة عقدة

التحلل العددي هو األداة األساسة المستخدمة لحل معادالت سران . تحلل الطوارئ و استقرار الجهداالستقرار العابر،

.الحمولة للحصول على النتائج المرغوبة

ة و ه أنها ال تتقارب بالقرب من حد استقرار الجهد ، هذا أدى إلى ظهور طرقة تحلل هنالك مشكلة تواجه الطرق العدد

تستخدم . جددة و ه تحلل استمرارة سران القدرة، و ه طرقة تؤدي تحلل سران حمولة ما بعد نقطة استقرار الجهد

الجهد الكامل و هو منحنى -صل إلى منحنى القدرةتحلل استمرارة سران القدرة . تقنة تحلل عددي لتحقق هذه المهمة

.حوي ف تحلل استقرار الجهد

نطاق المشروع هو اكتشاف و تطبق طرق و أسالب جددة لتحلل سران الحمولة تتجاوز مشكلة تفرد مصفوفة

.الجاكوبان و الوصول إلى منحنى االستمرارة الكامل

MATLABابرها الراضة و تمت برمجتها باستخدام تم اكتشاف طرق جددة، و تم اشتقاق تع TM

لغرض .

PSATو تمت مقارنة النتائج مع نتائج متحصل علها من IEEE -14-busالتأكد، تم اختبار البرامج على نظام االختبار TM

. و كانت النتائج مرضة

viii

1 Table of Contents

DECLARATION OF ORIGINALITY ....................................................................................... ii

Dedication iii

Acknowledgment ...................................................................................................................... iv

Abstract vi

vii المستخلص

1 Table of Contents .................................................................................................. viii

Chapter 1 1

1.1 Overview ......................................................................................................................1

1.2 Problem Statement ........................................................................................................1

1.3 Motivation ....................................................................................................................2

1.4 Project Objectives .........................................................................................................2

1.5 Report Layout ...............................................................................................................2

2 Chapter 2 ...................................................................................................................4

2.1 Overview ......................................................................................................................4

2.2 Load-Flow Analysis ......................................................................................................4

2.3 Newton-Raphson Method for Solution of System of Non-Linear Equations: .................4

2.4 Load-Flow Problem(Glover, Sarma, & Overbye, 2008): ...............................................6

2.5 Load-Flow Solution by Newton-Raphson:.....................................................................9

2.6 Continuation Power-Flow (Zakaria) ........................................................................... 13

2.6.1 General principle of the continuation power flow ................................................. 13

2.6.2 Formulation of the Power Flow Equations ........................................................... 14

2.6.3 Application of Continuation Algorithm ................................................................ 16

2.6.4 Predicting the Next Solution ................................................................................ 17

2.6.5 Parameterization and the Corrector Step ............................................................... 19

2.6.6 Choosing the Continuation Parameters ................................................................. 20

2.6.7 Sensing the Critical Point ..................................................................................... 20

3 Chapter 3 ................................................................................................................. 22

3.1 Introduction ................................................................................................................ 22

ix

3.2 Sin/Cos method ........................................................................................................... 22

3.3 The loading parameter (Lambda) method .................................................................... 27

3.3.1 Lambda Change Algorithm: ................................................................................. 27

3.3.2 Voltage Change Algorithm .................................................................................. 31

3.4 Impedance Load Flow ................................................................................................. 36

3.4.1 Single Injection Network Model: ......................................................................... 36

3.4.2 Starting values for admittances............................................................................. 36

3.4.3 Power and Admittance Mismatch Equations ........................................................ 37

3.4.4 Generator Reactive Power Limit Check ............................................................... 40

4 Chapter 4 ................................................................................................................. 42

4.1 Introduction ................................................................................................................ 42

4.2 The Sin Cos method results: ........................................................................................ 42

4.3 Loading Parameter method results: ............................................................................. 43

4.4 Impedance method results: .......................................................................................... 50

4.5 Results from PSAT ..................................................................................................... 51

4.6 Discussion of the results: ............................................................................................. 52

5 Chapter 5 ................................................................................................................. 53

5.1 Project Review ............................................................................................................ 53

5.2 Future Work ................................................................................................................ 54

6 Bibliography............................................................................................................ 55

8 APPENDIX A ...................................................................................................... A-1

9 APPENDIX B ....................................................................................................... B-1

10 APPENDIX C ....................................................................................................... C-1

11 Appendix D .......................................................................................................... D-1

x

List of Figure

Table of Figures

Figure 1 :Transmission Line Pi Model .........................................................................................7

Figure 2: Illustration of the Predictor-Corrector Scheme Used in the Continuation Power Flow 14

Figure 3 : Sin Cos Method algorithm ......................................................................................... 26

Figure 4: Loading Parameter method ......................................................................................... 35

Figure 5: Single Injection Network Model ................................................................................. 36

file:///C:/Users/oday/Desktop/FINAL/NEW%20LOAD%20FLOW%20METHODS%20FINAL%20J.docx%23_Toc334536813





Chapter 1 Introduction

1

Chapter 1

INTRODUCTION

1.1 Overview

There is no doubt that load flow analysis is one of the most important and basic

study among other power system studies such as voltage stability and contingency

analysis. The load flow analysis problem have been formulated and clearly defined. It

gives the values of state variables updates(voltage magnitude and angle) at each bus

bar. Accordingly, it‟s important in determining the steady state Solution of the power

flow problem would have been impossible in the absence of computer system because

of the tremendous difficulty of solving this problem by hand, not only this, but as the

system grows larger and larger, solution by hand tend to be more impossible. The rise

of computer systems and rapid improvements in their technology participated primarily

in the load flow analysis by its easiness, improving the precision, and speed of the

analysis.

In the literature, many methods have been developed to define the load flow

problem and many numerical methods have been developed, each of which has its

advantages and disadvantages. Newton-Raphson method is one the most used methods

out there because it converges quickly and arrives at the solution after a few iterations.

1.2 Problem Statement

In the conventional load flow methods, there‟s a drawback, which is if we increase the

load at some bus bars, we can find the corresponding voltage. As we increase the load, we arrive

at some point where the Newton-Raphson solution does not converge at all because the Jacobian

matrix cannot be inverted. This case is called: Jacobian matrix singularity. It was found that the

Jacobian singularity occurs at the practical maximum loading point. This singularity is one of the

biggest problems encountered in the Newton-Raphson solution or any other method. This


2

problem inhibits the load flow program to continue after reaching the maximum loading point is

reached.

Jacobian singularity received great attention and was studied and methods have been

developed to continue the load flow, even beyond the maximum loading point. The most popular

one is the predictor-corrector continuation load flow, it adapts an algorithm that predicts the

solution of the next loading value using a predictor step, then a corrector step gives the exact

value.

1.3 Motivation

Load flow analysis is such an interesting challenging field of study. Computer systems

technology made it possible to study new methods and program them.

1.4 Project Objectives

Objective is to discover new methods that overcome the Jacobian singularity, develop

programs that perform the new methods, and finally test these programs on some well known

testing systems such as IEEE-14-Bus testing system.

1.5 Report Layout

The rest of this thesis is organized as follows:

Chapter 2- Literature Review: it provides the reader with the very basic knowledge

considering the project and all necessary equations and concepts that will be used along

the following chapters. It starts with the definition of load flow analysis. Solution by

numerical analysis technique, Newton-Raphson, namely is presented to serve as a tool for

the solution of the load flow problem. Next, Newton-Raphson technique is applied to the

load flow problem as steps. Finally, concept of continuation load flow is presented and

so the algorithm of solving this problem numerically.

Chapter 3- Methodology: this chapter introduces the new methods and techniques

discovered. Every method is introduced as a concept, next, an algorithm and a flow chart

is provided for each method.


3

Chapter 4- Results: Results are presented here in forms of tables and PV curves, and they

are compared with results obtained from other methods and PSAT. Those results are

obtained after implementing the methods to IEEE-14-Bus test system.

Chapter 5- Conclusion and future work: In this chapter, conclusion of the work done and

future work and recommendation are presented.

Chapter 2 Literature Review

4

2 Chapter 2

Literature Review

2.1 Overview

In this chapter, a review of load-flow analysis is presented, as well as continuation

load-flow. The chapter begins with a section about the conventional load- flow analysis, its

definition, problem formulation and numerical solution, Newton-Raphson method of solution

was chosen among different methods because it serves the problem to be solved by the project.

The next section is an overview of the continuation load-flow analysis, which is overcomes the

problem of singularity faced in the conventional load-flow and leads to plotting the full P-V

curve.

2.2 Load-Flow Analysis

The load-flow analysis is the computation of voltage magnitude and phase angle at each

bus in a power system under balanced three-phase steady-state conditions. As a by-product of

this calculation, real and reactive power flows in equipment such as transmission lines and

transformers, as well as equipment losses can be computed.

The starting point for a load-flow problem is a single line diagram of the power system,

from which the input data for computer solutions can be obtained. Input data consist of bus data,

transmission line data, and transformer data.(Glover, Sarma, & Overbye, 2008)

Load flow studies are necessary for planning, economic operation, scheduling and

exchange of power between utilities. It‟s also required for many other analysis such as transient

stability, voltage stability, contingency analysis and state estimation.(Das, 2006)

2.3 Newton-Raphson Method for Solution of System of Non-Linear

Equations:

A set of nonlinear algebraic equations in matrix form is given by


5

(2.1)

Where y and x are N vectors and f(x) is an N vector of functions. Given y and f(x),

We want to solve for x. Rewriting equation (2.1)

(2.2)

Adding Dx to both sides of equation (2.2), where D is an invertible matrix,

Dx = Dx + y - f(x)

(2.3)

Multiplying by

x = x + [y – f(x)] (2.4)

The old values x(i) are used on the right side of equation (2.4) above to generate

the new values x(i+1) on the left side. That is,

x(i+1) = x(i) + [y – f(x)]

(2.5)

For non linear equations, the matrix D must be specified. For Newton-Raphson method,

specifying D is based on Taylor‟s series expansion of f(x) about an operating point

(2.6)

Neglecting the higher order terms in equation (2.6), and solving for x


6

(2.7)

The Newton-Raphson method replaces by the old value x(i) and x by the new value

x(i+1) in equation (2.7). Thus,

(2.8)

Where

(2.9)

The NXN matrix whose elements are partial derivatives shown in Equation (2.9) are

called the Jacobian matrix

2.4 Load-Flow Problem(Glover, Sarma, & Overbye, 2008):

Bus admittance matrix is used to solve the load-flow problem. The starting point in

obtaining the data that may be fed into the load-flow program is the single line diagram of the

system. Transmission lines are represented by their per-phase nominal π equivalent model as in

the figure. For each line, numerical values for the series impedance Z, the total line charging

admittance Y, the two buses to which the line is connected, and maximum MVA ratings are

necessary so that the computer can determine all the elements of the N X N bus admittance

matrix of which the typical element is

(2.10)


7

The bus admittance matrix can be constructed from the line and transformer input

data. To form the matrix we use the following rule:

Diagonal elements: = Sum of admittances connected to bus i.

Off-diagonal elements: = -(Sum of admittances connected between buses i and

n.

Other essential information includes transformer ratings and impedance, shunt capacitor

ratings, and transformer tap settings. In advance of each load-flow study certain bus voltage and

power injections must be given known values, as discussed below.

The voltage at a typical bus i of the system is given in polar coordinates by

(2.11)

And the voltage at another bus j is similarly written by changing the subscript from i to j.

the net current injected into the network at bus i in terms of elements of is given by the

summation

(2.12)

Let and denote the net real and reactive power entering the network at the bus i.

then the complex conjugate of the power injected at bus i is

Figure 1 :Transmission Line Pi Model


8

(2.13)

In which we substitute from (2.13) to obtain

(2.14)

Expanding this equation and equating the real and reactive parts, we obtain

(2.15)

(2.16)

Equations above constitute the polar form of the power flow equations; they provide the

calculated values for the net real power and reactive power entering the network at the

typical bus i. let denote the scheduled power being generated at bus “i” and “ denote the

scheduled power demand of the load at that bus. Then,

(2.17)

(2.18)

Each bus i is categorized into one of the following bus types:


9

1- Swing bus (or slack bus): there‟s only one swing bus in the system, which for

convenience is numbered bus 1. The swing bus is reference bus for which ,

per unit., is input data. Load-flow program computes and .

2- Load bus: and are input data. The load-flow program computes and . Most

buses are load buses.

3- Generator bus (Voltage controlled bus), and are input data. The power flow

program computes and . Examples are buses to which generators, switched shunt

capacitors, or static VAR systems are connected. Maximum and minimum VAR limits

that this equipment can supply are also input data.

2.5 Load-Flow Solution by Newton-Raphson:

Equations (2.15) and (2.16) are analogous to the nonlinear equation y = f(x). We define x,

y and f vectors for the power flow problem as:

;

(2.19)

(2.20)


10

Where all V, P and Q terms are in per-unit and terms are in radians. The swing bus

variables and are omitted since they are already known. Equations (2.15) and (2.16) have

the following form:

(2.21)

(2.22)

i=2,3,…..,N

The Jacobian matrix has the following form:

(2.23)

Which can be divided into four quadrants J(1), J(2), J(3) and J(4) where

When


11

(2.25)

(2.27)

When

(2.28)

(2.29)

(2.30)

(2.31)

(2.26)


12

Now, we apply the Newton-Raphson method to the load-flow problem in the following

steps:

Use (2.21) and (2.22) to compute

Use the equations above to calculate the Jacobian matrix.

Solve this equation for and

Compute

Starting with initial value V(0), the procedure continues until convergence is obtained or

until the number of iterations exceeds a specified maximum. Convergence criteria are often

based on ( ), (called power mismatches) rather than , (phase angle and voltage

magnitude mismatches).

For each voltage-controlled (generator) bus, the magnitude is already known, and the

function is not needed. Therefore, we couldn‟t omit from the x vector and from the

y vector. We could also omit from the Jacobian matrix the column corresponding to partial

derivatives with respect to and the row corresponding to partial derivatives with respect to

(x).

At the end of each iteration we compute from (2.22) and for

each generator bus. If the computed value of exceeds the limit, then the bus type is changed

to load bus with set to its limit value. The load-flow program also computes the new values

of .


13

2.6 Continuation Power-Flow (Zakaria)

The Jacobian matrix becomes singular at the voltage stability limit; consequently,

conventional load-flow algorithms are prone to convergence problems at the operating

conditions near the stability limit. The continuation power-flow analysis overcomes this problem

by reformulating the load-flow equations so that they remain well-conditioned at all possible

loading conditions. This allows the solution of the load-flow problem for stable, as well as

unstable equilibrium points (that is, for both upper and lower portions of the P-V

curve)(Kundur). The purpose of continuation power flow is to find a continuum of load flow

solutions for a given load/generation change scenario (i.e. computation direction). It‟s capable of

solving the whole PV curve. The continuation power flow finds the solution path of a set of load-

flow equations that are reformulated to include a continuation parameter.

The continuation power-flow method described here uses locally parameterized

continuation method that belong to a general class of methods for solving nonlinear algebraic

equations known as path-following methods.

2.6.1 General principle of the continuation power flow

The general principle behind the continuation power flow is rather simple. It employs a

predictor-corrector scheme to find a solution path of a set of power flow equations that have been

reformulated to include a load parameter. As shown in Figure 2. Initial solution (A), a tangent

predictor is used to estimate the solution (B) for a specified pattern of load increase. The

corrector step then determines the exact solution (C) using conventional load-flow analysis with

the system load assumed to be fixed.

The voltages for a further increase in load are then predicted based on a new tangent

predictor. If the new estimated load (D) is now beyond the maximum load on the exact solution,

a corrector step with loads fixed would not converge; therefore, a corrector step with a fixed

voltage at the monitored bus is applied to find the exact solution (E). As the voltage stability

limit is reached, to determine the exact maximum load the size of load increase has to be reduced

gradually during the successive predictor steps.


14

2.6.2 Formulation of the Power Flow Equations

In order to apply a locally parameterized continuation technique to the power flow

problem, a load parameter must be inserted into the equations. While there are many ways this

could be done, only a simple example using constant power load model will be considered here.

First, let λ represent the load parameter such that

Where corresponds to the critical load. We desire to incorporate into

(2.32)

(2.33)

Where

Figure 2: Illustration of the Predictor-Corrector Scheme Used in the

Continuation Power Flow


15

(2.34)

(2.35)

The subscript L, G and T respectively denote bus load, generation and injection. The

voltage at bus i is and is the element of the system admittance

matrix .

To simulate different load change scenarios, the and can be modified as

(2.36)

(2.37)

Where

Let , then (2.36) and (2.37) can be rewritten as

(2.38)


16

(2.39)

Where

, = Original load at bus i, active and reactive respectively;

= Multiplier designating the rate of load change at bus i, as changes;

= Power factor angle of load change at bus i.;

= Apparent power which is chosen to provide appropriate scaling of .

The active power generation can be modified to

(2.40)

Where

= Active generation at bus i in the bus case;

= Constant specifying the rate of change in generation as varies.

If these new expressions are substituted into the power flow equations, the result is

(2.41)

(2.42)

Notice that values of , , can be uniquely specified for every bus in the system.

This allows for a very specific variation of load and generation as changes.

2.6.3 Application of Continuation Algorithm

The power flow equations for particular bus i will be reformulated to contain a load

parameter . The next step is to apply a continuation algorithm to the system of reformulated


17

power flow equations. If (F) is used to denote the whole set of equations, the problem can be

expressed as

(2.43)

Where represents the vector of bus voltage angles, V represents the vector of bus

voltage magnitudes. As mentioned, the base case solution is known via a

conventional power flow and the solution path is being sought over a range of A. In general, the

dimension of E will be ( ), where 1and 2are the number of P-Q and P-V buses

respectively.

2.6.4 Predicting the Next Solution

Once a base solution has been found ( =0), a prediction of the next solution can be made

by taking an appropriately sized step in a direction tangent to the solution path. Thus, the first

task in the predictor process is to calculate the tangent vector. This tangent calculation is derived

by first taking the derivative of both sides of the power flow equations

Factorizing:

On the left side of this equation is a matrix of partial derivatives multiplied by a vector of

differentials. The former is the conventional load flow Jacobian augmented by one column ( ),

while the latter is the tangent vector being sought. There is, however, an important barrier to

overcome before a unique solution can be found for the tangent vector.


18

The problem arises from the fact that one additional unknown was added when A was

inserted into the power flow equations, but the number of equations remained unchanged. Thus,

one more equation is needed.

This problem can be solved by choosing a non-zero magnitude (say one) for one of the

components of the tangent vector. In other words, if is used to denote the tangent vector;

This results in

(2.45)

Where is an appropriately dimensioned row vector with all elements equal to zero

except the , which equals one. If the index is chosen correctly, letting imposes a

non-zero norm on the tangent vector and guarantees that the augmented Jacobian will be

nonsingular at the critical point. Whether +1 or -1 is used depends on how the state variable

is changing as the solution path is being traced. If it is increasing a +1 should be used and if it

is decreasing a -1should be used.

Once the tangent vector has been found by solving (2.45) the prediction can be made as

follows:

(2.46)

Where (*) denotes the predicted solution for a subsequent value of (loading) and is a

scalar that designates the step size. The step size should be chosen so that the predicted solution

is within the radius of convergence of the corrector. While a constant magnitude of can be used

throughout the continuation process.


19

2.6.5 Parameterization and the Corrector Step

Now that a prediction has been made, a method of correcting the approximate solution is

needed. The best way to present this corrector is to expand on parameterization, which is vital to

the process.

Every continuation technique has a particular parameterization scheme. The

parameterization provides a method of identifying each solution along the path being traced. The

scheme was used here known as local parameterization. In local parameterization the original set

of equations is augmented by one equation that specifies the value of one of the state variables.

In the case of the reformulated power flow equations, this means specifying either a bus voltage

magnitude, a bus voltage angle, or the load parameter .

In equation form this can be expressed as follow:

(2.47)

Where is an appropriate value for the element of .

Then the new set of equations would be

(2.48)

Now, once a suitable index and value of are chosen, a slightly modified Newton-

Raphson power flow method (altered only in that one additional equation and one additional

state variable are involved) can be used to solve the set of equations. This provides the

corrector needed to modify the predicted solution found in the previous section.

Actually, the index used in the corrector is the same as that used in the predictor and

will be equal to , the predicted value of . Thus, the state variable is called the

continuation parameter. In the predictor it is made to have a non-zero differential change

(d ) and in the corrector its value is specified so that the values of other state


20

variables updatescan be found. How then does one know which state variable should be used as

the continuation parameter.

2.6.6 Choosing the Continuation Parameters

There are several ways of explaining the proper choice of continuation parameter

Mathematically, it should correspond to the state variable that has the largest tangent vector

component. More simply put, this would correspond to the state variable that has the greatest rate

of change near the given solution. In the case of a power system, the load parameter is

probably the best choice when starting from the base solution. This is especially true if the base

case is characterized by normal or light loading. Under such conditions, the voltage magnitudes

and angles remain fairly constant under load change. On the other hand, once the load has been

increased by a number of continuation steps and the solution path approaches the critical point,

voltage magnitudes and angles will likely experience significant change. At this point would be

a poor choice of continuation parameter since it may change only a small amount in comparison

to the other state variables. For this reason, the choice of continuation parameter should be re-

evaluated at each step. Once the choice has been made for the first step, a good way to handle

successive steps is to use

Where t is the tangent vector with a corresponding dimension, m = (2 + +1) and

index corresponds to the component of the tangent vector that is maximal. When the

continuation parameter is chosen, the sign of its corresponding tangent component should be

noted so that the proper value of +1 or -1 can be assigned to , in the subsequent tangent vector

calculation.

2.6.7 Sensing the Critical Point

The only thing left to do amid the predictor-corrector process is to check to see if the

critical point has been passed. This is easily done if one keeps in mind that the critical point is

where the loading (and therefore ) reaches a maximum and starts to decrease. Because of this,

the tangent component corresponding to X (i.e., d ) is zero at the critical point and is negative

beyond the critical point. Thus, once the tangent vector has been calculated in the predictor step,


21

a test of the sign of the d component will reveal whether or not the critical point has been

passed.

Chapter 3 Methodology

22

3 Chapter 3

Methodology and Design

3.1 Introduction

In this chapter, we will discover, formulate and develop algorithms for new methods and

techniques of load flow analysis. The first method is just a new representation of the load flow

problem and will be solved by Newton-Raphson method. The second one is a method that

emp;oys loading parameter and is dedicated for the continuation load flow because it overcomes

the Jacobian singularity problem. Also, Newton-Raphson is used here to find the solution.

Finally, a method that converts the load into impedance representation is discovered.

For all the methods here, system data is required. For the sake of convenience, system

data was saved in two matrices: BusData and LineData. A small piece of code calculates the

matrix of the system in the form of angle and magnitude ( ) were “i” and “j” are

two buses between which the line is connected.

3.2 Sin/Cos method

Here, a different representation was given to the load flow equations. This was done by

accommodating the two state variables updates(voltage angle ( ) and voltage magnitude (Vm))

into one term. The load flow problem was then described by the following equation:

(3.1)

dY=J*dX

Where

dY: Active and reactive power mismatch matrix

and : Active and reactive power calculation mismatch respectively.


23

: Jacobian matrix.

: Voltage magnitude of bus bar “i”.

: Voltage angle of bus bar “i”.

dX: state variables updatesmatrix.

: .

: .

Using this new representation of the state variables; Jacobian matrix elements were

derived, and now the problem became well defined and could be solved numerically using

Newton-Raphson method for the solution of non-linear equations.

The following steps were applied to implement the method:

1. From BusData and LineData system‟s data was brought, and starting with a flat voltage

profile (i.e. , ) initial active and reactive power flows at each bus were

calculated using the following equations:

(3.2)

(3.3)

2. Power mismatch matrix (dY) was constructed as follows:


24

(3.4)

(3.5)

3. Full Jacobian matrix was constructed. Expressions of the elements were derived starting

from the expression of the load flow problem in (3.1). For the sake of convenience,

Jacobian matrix was divided into four quadrants; each quadrant has a certain expression

for its elements.

Jacobian quadrants J1 through J4 expressions:

a. Diagonal elements:

I.

II.

III.

IV.

b. Off-diagonal elements:

I.

II.

III.

IV.

4. Next, Jacobian matrix (J), power mismatch matrix (dY) and state variables updates

matrix (dX) were compacted into matrices (MatA), (MatY) and (MatX) respectively.


25

Those new matrices do not have zero rows and columns corresponding to slack P and Q

and generator Q equations. The equation dY=J*dX was changed into:

MatY=MatA*MatX.

5. Equation in (4) was solved for “ ” as follows: .

6. Voltage magnitude and angle values were updated.

7. Generator Reactive Rower Limit Check algorithm was applied.

8. New iteration of the procedure above was started.

9. The algorithm exits if the predetermined tolerance or the predetermined number of

iterations is exceeded without reaching the desired accuracy.

10. Finally, from the results of voltage angle and magnitude, active and reactive powers were

calculated for each bus.

11. Total results were tabulated for all buses. This included: voltage angle and magnitude and

active and reactive power


26

Figure 3 : Sin Cos Method algorithm


27

3.3 The loading parameter (Lambda) method

This method performs load analysis flow for the system; it also overcomes the maximum

loading point problem encountered before due to Jacobian singularity, hence, it gives complete

PV curves for the system‟s buses.

A loading parameter, Lambda “ ”, was defined which increased the load at each load bus

in constant steps, and then power flow was performed for each step using Newton-Raphson

method. As Lambda increased, load flow was performed and voltage at the bus decreased, until a

point where the load flow solution doesn‟t converge to a solution was reached, at this point, the

loading parameter, Lambda (which was known) and the voltage (which was unknown) were

interchanged. Here, the unknown is the Lambda and the voltage is known. Next, voltage was

increased in steps, too. At each step load flow was performed to obtain the corresponding

Lambda.

To get the continuation curve, the algorithm was divided into two parts: Lambda change

part and Voltage change part.

3.3.1 Lambda Change Algorithm:

An initial value=1 was assigned to , the algorithm was next performed, then, was

increased in by a fixed step, and the algorithm is applied again.

1. From BusData and LineData system‟s data was brought, and starting with a flat voltage

profile (i.e. , ) initial active and reactive power flows at each bus were

calculated using the following equations:

(3.6)

(3.7)


28

2. Before divergence (maximum loading point), the load flow problem had the following

form

[dY]=[J]*[dX]

(3.8)

Where:

dY= power mismatch matrix.

J=Jacobian matrix.

dX= State variables matrix.

In the above equation, of one load bus was replaced with , and this row was

denoted by “x”, and so were the corresponding rows in matrices J and dX.

3. The power mismatch matrix (dY) was constructed as follows:

a. If the row is not “x”:

= Active power mismatch =

= Reactive power mismatch =

b. If the row is “x”:

(3.9)

Where

(3.10)


29

(3.11)

(3.12)

(3.13)

4. Full Jacobian matrix was constructed. Expressions of the elements were derived starting

from the expression of the load flow problem in equation (3.8). For the sake of

convenience, Jacobian matrix was divided into four quadrants; each quadrant had a

certain expression for its elements.

a. For the diagonal elements:

I.

II.

III.

i- for the elements corresponding to row “x”

ii- for elements corresponding to rows other than “x”

IV.




30

b. For the off-diagonal elements:

I.

II.

III.



IV.



5. Jacobian matrix (J), power mismatch matrix (dY) and state variables matrix were

compacted into matrices (MatA), (MatY) and (MatX) respectively. Those new matrices

do not have zero rows and columns corresponding to slack P and Q and generator Q

equations. The equation dY=J*dX has a new form: MatY=MatA*MatX.

6. Equation in “5” was solved for “ ” as follows: . Next, bus bars

voltage angles and magnitudes were updated, and maximum error was calculated.

7. Next, generator‟s reactive power limit check algorithm was applied.

8. If the solution converges (accuracy achieved within the predetermined number of

iteration), value of will be increased by the predetermined increase step, and the

algorithm is applied again, otherwise, control will be passed to the Voltage Increase

Algorithm, with the last values of and Vm taken as initial values for it.


31

9. Finally slack bus and generator‟s bus active and reactive powers were computed.

10. Results were plotted in the form of Lambda V.S. Vm for each bus bar.

3.3.2 Voltage Change Algorithm

The problem was put in the following form:

(3.14)

1- Power mismatch matrix dY forming was started by first computing the active and

reactive power at each bus bar using the following equations:

(3.15)

(3.16)

Next, the element of the dY matrix was evaluated by this expression

Where

(3.17)


32

The rest of the dY elements had the following equations

2- Next, the Jacobian matrix (full Jacobian matrix) was constructed quadrant by

quadrant as follows:

a- For the diagonal elements:

I.

II.

i- element corresponding row “x”

ii- elements corresponding rows other than “x”

III.

i- for the element corresponding to row “x”


IV.


ii- for the elements corresponding to column “x”

iii- for elements corresponding to rows or columns other than “x”


33

b- For the off-diagonal elements (row “i”, column “k”)

I.

II.

III. i- for the element corresponding to row “x”

ii- the elements corresponding to rows other than “x”

IV.

i- for the element corresponding to row “x”

Jik

ii-

3- Jacobian matrix (J), power mismatch matrix(dY) and state variables matrix (dX),

were compacted and the resulting compact matrices are: MatA, MatY and MatX

respectively. The new matrices do not have zero rows and columns corresponding to

slack P and Q and generator Q equations. And, instead of : dY=J*dX, the new

equation to be solved was: MatY=MatA*MatX.

4- Equation MatY=MatA*MatX was solveed for MatX by inverting the compact

Jacobian matrix as follows: .


34

5- Bus bars voltage angles and magnitudes were updated, maximum error was calculated

and was compared with the desired accuracy to decide whether the solution is close to

the right, or another iteration is required.

6- Generator reactive power limits were checked for generator bus bars using the Check

Limit algorithm.

7- If the current iteration number was larger than the maximum iteration number and yet

the maximum error is still larger than the accuracy, the algorithm would exit the

iterations, otherwise, another iteration would start from step 1 above.

8- Whether the desired accuracy was reached or the maximum number of iteration was

reached without the solution divergence, slack bus and generator bus active and

reactive powers were calculated as follows

For the slack bus:

(3.18)

(3.19)

For the generator bus:

*

(3.20)

(3.21)

9- All results were tabulated for each bus, this include: bus voltage angle and magnitude,

generated active and reactive power and demand active and reactive power.

10- Complete PV curve was plotted for each bus in the system.


35

Figure 4: Loading Parameter method


36

3.4 Impedance Load Flow

3.4.1 Single Injection Network Model:

In this model, the network is represented by a single current injection at the slack bus,

while all other nodes are effectively open-circuited. Loads at load bus bars are modeled as

admittances to ground and absorbed into the networks‟ positive sequence Z-bus matrix (after

conversion into impedances). Generation at generator bus bars are modeled as negative

admittances to ground and likewise absorbed into the Z-bus matrix. Figure 7 shows the basic

concept used here.

3.4.2 Starting values for admittances

A starting set of admittances for the load bus bars are selected to give the correct power

demand at nominal voltage.

Figure 5: Single Injection Network Model


37

2

k

dk

kV

PG ,

2

k

dkk

V

QB

(3.22)

And, since voltage is nominal;

dkk PG , dkk QB (3.23)

For the generator bus bars, the starting value for conductance will be established to

deliver the required power demand at the fixed generator nominal voltage. This conductance will

remain unchanged unless the generator voltage is regulated to limit the reactive power. The

initial susceptance will be set to give the reactive demand at the bus, if exists; otherwise it shall

be zero.

2

gk

dkgk

kV

PPG

, 2

gk

dkk

V

QB

(3.24)

The starting admittances for all load and generator bus bars are added to the networks‟

bus admittance matrix Y. The bus impedance matrix Z is obtained by inverting Y. This will be

the only time Z is obtained through inversion since all subsequent modifications are operated on

Z directly.

3.4.3 Power and Admittance Mismatch Equations

Now that we have an initial network model, we need to check for compliance with

network constraints, ie specified powers and voltages. The bus impedance equations with single

injection at the slack bus are expressed as:

0

s

kkks

skss

k

s I

ZZ

ZZ

V

V

(3.25)


38

Note that the current injections are everywhere zero except at slack bus. The slack bus

current is immediately determined asss

ss

Z

VI . The voltage at any other bus is then:

sksk IZV ss

sks

Z

VZ

(3.26)

We shall call this voltage the open circuit (i.e Thevenin‟s) voltage, THkV . The Thevenin‟s

impedance, looking into bus k, is obtained by setting the slack bus voltage to zero and

eliminating its equation from the Z-bus formation. This will result in:

ss

skkskkTHkTHkTHk

Z

ZZZjxrZ

(3.27)

Let node k be a load bus bar. Now let us add an additional admittance element,

kk BjG between node k and ground, with the objective of correcting the calculated power

at the node to match the specified demand. The equivalent impedance of the added element is

kk jxr , which is typically a large value since the admittance element is typically small.

Incorporating this impedance in the Thevenin‟s model and calculating the power consumed in it

results in:

P

xxrr

rV

kTHkkTHk

kTHk

22

2

Q

xxrr

xV

kTHkkTHk

kTHk

22

2

(3.28)

Where the power mismatches P and Q are found as


39

kTHkdk GVPP 2

kTHkdk BVQQ 2

(3.29)

Equations (6) is solved for kr by eliminating kx , which results in a quadratic equation

with solutiona

acbbrk

2

42 , where

P

QPa

2

222 THkTHkTHk VQxPrb

)( 22

THkTHk xrPc

(3.30)

The question arises whether to use the plus or minus sign in the solution for kr . We

require that the admittance corrections kG , kB vanish together with the power mismatches

P and Q upon arrival at a solution. This should take kr to infinity and can only be achieved if

we use the plus sign in the quadratic equation solution.

kx is then trivially found as krP

Q

and the admittance corrections are finally

22

kk

kk

xr

rG

22

kk

kk

xr

xB

(3.31)

Now let node k be a generator bus bar. We shall only add an additional susceptance

element, kBj between node k and ground, with the objective of correcting the reactive power

at the node to match the specified voltage. The conductance shall not be modified unless there is

a reactive power violation, forcing a voltage regulation. Adding the susceptance is equivalent to

adding a reactance kjx in series with Thevenin‟s impedance THkTHk jxr . The criteria for the

added reactance is that the voltage magnitude at node k should be equal to the specified voltage

gkV . This can be expressed as follows:


40

gk

kTHkTHk

kTHk V

xxr

xV

22

(3.32)

Or

2

22

22

gk

kTHkTHk

kTHk V

xxr

xV

(3.33)

Solving for kx

a

acbbxk

2

42 , where

22

THkgk VVa

22 gkTHk Vxb

)( 222

THkTHkgk xrVc

(3.34)

And the conditions for kx as gkTHk VV are only achieved by taking the minus

sign in the quadratic equation solution form. kB is found as before in equation (9).

3.4.4 Generator Reactive Power Limit Check

The reactive power produced in the generator is expressed as:

dkgkgk QBVQ 2

(3.35)

This should be checked against max

gkQ and min

gkQ at every iteration for limit violations. If

a limit is transgressed, then the reactive power is set at the limit, and both kG and kB are

updated using a slight modification to equation (3.29), i.e


41

kTHkdkgk GVPPP 2

kTHkdkgk BVQQQ 2max (for a maximum limit violation)

(3.36)

kG and kB are then found exactly as in equations (3.31) and (3.34) and the admittances

updated as

22

kk

kk

xr

rG

22

kk

kk

xr

xB

(3.37)

Chapter 4 Results and Discussion

42

4 Chapter 4

Implementation and Results

4.1 Introduction

All the methods were programmed using MATLABTM

. An M File was developed for

each method alone. Other auxiliary M. files were used for many purposes such as to provide the

data of the system under study (bus and line data), form the Ybus matrix and produce the output in

tabulated form and for some cases in graphic form.

All the methods were tested on IEEE 14-Bus test system. To verify the results, PSAT

toolbox of MATLABTM

was used as a reference.

4.2 The Sin Cos method results:

Table 1: Sin Cos Method Results as Applied to IEEE 14-Bus


43

4.3 Loading Parameter method results:

Table 2: Results of Loading Parameter Method as applied to IEEE 14-Bus system (Before the Maximum loading point)


44

0.5 1 1.5 2 2.5 3 3.5 4

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

loading parameter

voltages

data1

data2

data3

data4

data5

data6

data7

data8

data9

data10

data11

data12

data13

data14

Continuation PV Curves Using the Loading Parameter method:

1- For a step =0.5

a- As applied by the Loading Parameter method.


45

b- As implemented by PSAT


46

0.5 1 1.5 2 2.5 3 3.5 4

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

loading parameter

voltages

data1

data2

data3

data4

data5

data6

data7

data8

data9

data10

data11

data12

data13

data14

2- For a step =0.1

a- As applied by the Loading Parameter method


47

b-As implemented by PSAT


48

0.5 1 1.5 2 2.5 3 3.5 4

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

loading parameter

voltages

data1

data2

data3

data4

data5

data6

data7

data8

data9

data10

data11

data12

data13

data14

3- Step=0.01

a- As applied to the Loading Parameter method


49

b-As implemented using PSAT


50

4.4 Impedance method results:


51

4.5 Results from PSAT


52

4.6 Discussion of the results:

1- Results of all the methods were compared with the results obtained from PSAT and they

were almost typical.

2- Loading parameter method has the smallest number of iterations, next the Sin Cos

method, and finally comes the Impedance method. But, concerning the maximum error

(the maximum absolute difference between the actual power and the calculated power)

impedance method gave the smallest error, then came the Loading parameter method,

finally, Sin Cos method. This makes the Impedance method the most accurate one.

3- The Loading Parameter method gave results that were quite with the results obtained by

PSAT. The full continuation curve was also obtained for all the 14 buses.

4- Also, looking at the curves of various steps, tell us that any step can be used.

Chapter 5 Conclusion and Future Work

53

5 Chapter 5

Conclusion

5.1 Project Review

Project scope was to discover new methods and techniques for load flow analysis that

overcome the well known problem of Jacobian matrix singularity encountered in many other

conventional techniques. This problem inhibits the load flow program to function under certain

conditions, primarily voltage stability limit of the PV curve and the points beyond it.

MatLab software was used as a tool to implement the new discovered techniques to

produce the desired program that performs the algorithm quickly, easily, and efficiently as far as

time and precision is concerned.

Three methods were formulated and programmed. The first one is just another

representation of the conventional Newton-Raphson problem. But it has the same problem of

divergence near the voltage stability limit. The second method used a loading parameter that is

used to increase the load at each load bus bas, and this parameter is also accommodated in the

power mismatch matrix in the place of one of the reactive power mismatch of some load bus.

The loading parameter value started from the base case, which is „1‟. As this parameter is

increased in fixed steps, load flow was performed and results (voltage magnitude and angle) are

saved at each increase. This is repeated until the voltage stability limit is reached, then we switch

the roles between the loading parameter and its corresponding voltage magnitude, i.e., we apply

decreases to the voltage magnitude of that same bus bar, starting from its last value before

divergence and obtain the resulting voltage angle and loading parameter. Finally, we end up

having the complete continuation PV curve for each bus bar.

The validity of the methods was tested on IEEE-14-Busand the expected results were

found and compared with results from PSAT for the same system.

Chapter 5 Conclusion and Future Work

54

5.2 Future Work

For further verification of the methods, they should be tested on other testing systems

such as IEEE 30-Bus, IEEE 57-Bus and IEEE 118-Bus and compare the results with reliable

software packages such as PSAT and Etap.

And, for real utilization of the methods, after verification, they should be used in other

power system studies, such as voltage stability to evaluate the stability of the system,

contingency analysis.

55

6 Bibliography

7

Das, D. (2006). Electrical Power Systems. New Age International Publishers.

Glover, J. D., Sarma, M. S., & Overbye, T. J. (2008). Power System Analysis (Fourth Edition).

Thomson.

Kundur, P. Power System Stability and Control. McGraw-Hill, Inc.

Zakaria, A. Voltage Stability Sudy: Sudan National Grid as a Study Case, PHD.

APPENDIXA MATLAB CODE FOR SIN COS

A-1

8 APPENDIX A

MATLABTM Code for Sin Cos Method

% This program should be called by pfdata14bus (power flow 14

bus data program)

%Vm=0; delta=0;

converge=1;

busType=[];%buscode

Vm=[]; delta=[]; Pd=[]; Qd=[]; Pg=[]; Qg=[]; Qmin=[]; Qmax=[];

% Added (6-8-00)

P=[]; Q=[];; % Added (6-8-00)

J=[];dY=[];MatY=[];MatX=[];MatA=[];genBus=[];genVolts=[];

C=[]; S=[];T1=[]; T2=[];

busType=busdata(:,2); Vm=busdata(:,3); delta=busdata(:,4);

Pd=busdata(:,5)/basemva; Qd=busdata(:,6)/basemva;

Pg=busdata(:,7)/basemva; Qg=busdata(:,8)/basemva;

Qmin=busdata(:,9)/basemva; Qmax=busdata(:,10)/basemva;

Qsh=busdata(:,11)/basemva;

S=zeros(size(Vm));%Actually C(i)=Vm(i)*sin(delta(i)) but all

initial values of delta = zeros

C=Vm;%Actually C(i)=Vm(i)*cos(delta(i)) but all initial values

of delta = zeros

for i=1 : nBusbars

%busType(i+nBusbars) = busType(i);

if busType(i) == 2 %generator bus

%busType(i)=0; %in first and second quadrant of

jacobian gen bus is treated as load bus

genBus=[genBus;i];

genVolts=[genVolts;Vm(i)];

end

end

busType;

nGens=length(genBus);

%========================================================

% Start of Newton-Raphson Loadflow iterations

%========================================================

maxerror = 1; converge=1;iter = 0;

while maxerror >= accuracy & iter <= maxiter % Test for max.

power mismatch


A-2

%===========Compute power at each busbar=======

for i=1:nBusbars

T1(i) = 0;

T2(i) = 0;

for j=1 : nBusbars

T1(i) = T1(i)+C(j)*g(i,j)-S(j)*b(i,j);

T2(i) = T2(i)+S(j)*g(i,j)+C(j)*b(i,j);

end

P(i)=C(i)*T1(i)+S(i)*T2(i);

Q(i)=S(i)*T1(i)-C(i)*T2(i);

if (busType(i) ~= 1)

dY(i) = -P(i) + (Pg(i)-Pd(i)); %mismatch equation for

active power (load and generator busbars)

end

if (busType(i) ~= 1) & (busType(i) ~= 2)

dY(i+nBusbars) =-Q(i) + (Qg(i)-(Qd(i)-

Qsh(i)));%mismatch equation for reactive power (only load

busbars)

end

end

%===============================================================

==

% Construct Jacobian Matrix

%===============================================================

==

N = nBusbars;

J =zeros(N,N);

%First Quadrant

for i=1 : nBusbars

if busType(i) ~= 1

for k=1 : nBusbars

if (i ~= k)

J(i, k) =S(i)*g(i,k)-C(i)*b(i,k);

else

J(i, i) =S(i)*g(i,i)-C(i)*b(i,i)+T2(i);

end

end

end

end

%Second quadrant

for i=1 : nBusbars

if busType(i) ~= 1

for k=1 : nBusbars

if (i ~= k)


A-3

J(i, k + N) = C(i)*g(i,k)+S(i)*b(i,k);

else

J(i, i + N) = C(i)*g(i,i)+S(i)*b(i,i)+T1(i);

end

end

end

end

%Third Quadrant

for i = 1 : nBusbars

if busType(i) ~= 1 & busType(i) ~= 2

for k=1 : nBusbars

if (i ~= k)

J(i + N, k) = -S(i)*b(i,k)-C(i)*g(i,k);

else

J(i + N, i) =-S(i)*b(i,i)-C(i)*g(i,i)+T1(i);

end

end

end

end

%Fourth Quadrant



for k=1 : nBusbars

if (i ~= k)

J(i + N, k + N) = S(i)*g(i,k)-C(i)*b(i,k);

else

J(i + N, i + N) =-C(i)*b(i,i)+S(i)*g(i,i)-T2(i);

end

end

end

end

%===============================================================

==

% End of Jacobian Matrix construction

%===============================================================

==

% Prepare Jacobian Matrix for inversion; first copy it into a

compact


A-4

% matrix that does not have zero rows and columns (corresponding

to slack

% P & Q and generator Q equations). The original equations have

the form

% dY = J * dX; The compact form is MatY= MatA*MatX

N = nBusbars;

num1=0;

for i=1 : nBusbars

if busType(i) ~= 1

num1=num1+1;

MatY(num1,1) = dY(i);

num2=0;

for j=1 : nBusbars

if busType(j) ~= 1

num2=num2+1;

MatA(num1,num2)=J(i,j);

end

end

end

end

num1=0;

for i=1 : nBusbars

if busType(i) ~= 1

num1=num1+1;

num2=nBusbars-1;

for j=1 : nBusbars

if busType(j)~= 1 & busType(j)~= 2

num2=num2+1;

MatA(num1,num2)=J(i,j+N);

end

end

end

end

num1=nBusbars-1;

for i=1 : nBusbars

if busType(i) ~= 1 & busType(i)~= 2

num1=num1+1;

MatY(num1, 1) = dY(i+nBusbars);

num2=0;

for j=1 : nBusbars

if busType(j)~= 1

num2=num2+1;

MatA(num1,num2)=J(i+N,j);

end

end


A-5

end

end

num1=nBusbars-1;

for i=1 : nBusbars


num1=num1+1;

num2=nBusbars-1;

for j=1 : nBusbars


num2=num2+1;

MatA(num1,num2)=J(i+N,j+N);

end

end

end

end

MatX=MatA\MatY; % Solve for MatX using inverse(MatA) *

MatY

%Note: Matrices dY, dX and m contain zero rows and columns,

%corresponding to slack and generator busbars. In order to

invert m,

%it must be copied into matrix MatA, which contains no zero

rows or

%columns. Likewise dY is copied into MatY. The result of

inversion,

%MatX is expanded into matrix dX.

%==========Update C, S and calculate max error ===========

num1 = 0;

for i=1 : nBusbars

if busType(i) ~= 1

num1 = num1 + 1;

%dX(i+nBusbars) = MatX(num1, 1);

S(i) = S(i) + MatX(num1, 1);

end

end

num1=nBusbars-1;

for i=1 : nBusbars

if busType(i) == 0

num1 = num1 + 1;

% dX(i) = MatX(num1, 1);

C(i) = C(i) + MatX(num1, 1);

end

if busType(i) ==2


A-6

C(i) =Vm(i)*sqrt(1-(S(i)/Vm(i))^2);

%C(i) =Vm(i)*sin(asin(S(i)/Vm(i)));

end

end

maxerror=max(abs(MatY));

for k = 1 : nGens

i=genBus(k);

T1(i) = 0;

T2(i) = 0;

for j=1 : nBusbars

T1(i) = T1(i)+C(j)*g(i,j)-S(j)*b(i,j);

T2(i) = T2(i)+S(j)*g(i,j)+C(j)*b(i,j);

end

Vm(i) =sqrt((C(i))^2+(S(i))^2);

Q(i)=genVolts(k)/Vm(i)*(S(i)*T1(i)-C(i)*T2(i));

fprintf('%g\n',(Q(i) + Qd(i))*100),

end

%============================= Generator reactive power limits

check =====

if maxerror > accuracy & genlimits == 1 & iter > 0

for k = 1 : nGens

i=genBus(k);

T1(i) = 0;

T2(i) = 0;

for j=1 : nBusbars

T1(i) = T1(i)+C(j)*g(i,j)-S(j)*b(i,j);

T2(i) = T2(i)+S(j)*g(i,j)+C(j)*b(i,j);

end

Vm(i) =sqrt((C(i))^2+(S(i))^2);

Q(i)=genVolts(k)/Vm(i)*(S(i)*T1(i)-C(i)*T2(i));

fprintf('%g ',Vm(i)),fprintf('%g

',genVolts(k)),fprintf('%g ',(Q(i) + Qd(i))*100),

if busType(i) == 2

if Q(i) + Qd(i) > Qmax(i) + 0.01

fprintf('1 '),

Qg(i) = Qmax(i);

busType(i) = 0; % convert gen bus to load bus if

reactive power exceeds max

%Vm(i) = 1;

elseif Q(i) + Qd(i) < Qmin(i) - 0.01

fprintf('2 '),

Qg(i) = Qmin(i);


reactive power exceeds min

%Vm(i) = 1;

end


A-7

else

if Q(i) + Qd(i) < Qmax(i) - 0.01 & Vm(i) <

genVolts(k)

fprintf('3 '),

%Vm(i) = genVolts(k);

busType(i ) = 2; % return to gen bus if within

limits

elseif Q(i) + Qd(i) > Qmin(i) + 0.01 & Vm(i) >

genVolts(k)

fprintf('4 '),


busType(i) = 2; % return to gen bus if within

limits

end

end

fprintf('%g\n',busType(i )),

end

end

%===============================================================

=========

fprintf('\n'),

iter=iter+1;

if iter > maxiter & maxerror > accuracy

fprintf('\nWARNING: Iterative solution did not converge

after ')

fprintf('%g', iter), fprintf(' iterations.\n\n')

fprintf('Press Enter to terminate the iterations and print

the results \n')

converge = 0; pause,

end

if maxerror <= accuracy

end

%===============================

end % end of Load Flow iteration

% Calculation of Vm & delta

for i=1 : nBusbars

if busType(i) == 0

Vm(i) =sqrt((C(i))^2+(S(i))^2);

end


A-8

if busType(i) ~= 1

delta(i) = asin(S(i)/Vm(i));

end

end

%calculate slack bus and generator powers

for i=1 : nBusbars

if busType(i) == 1

Pg(i)=P(i) + Pd(i);

Qg(i)=Q(i) + Qd(i);

elseif busType(i) == 2

Qg(i)=Q(i) + Qd(i);

end

end

%==========================================

APPENDIX B MATLAB CODES FOR LAMBDA CHANGE

B-1

9 APPENDIX B

MATLABTM Code for Lambda Change


bus data program)

Vm=0; delta=0;

converge=1;

busType=[];%buscode


% Added (6-8-00)

P=[]; Q=[]; sum1=[]; sum2=[]; A=[]; % Added (6-8-00)


sum1=[];sum2=[];

busType=busdata(:,2); Vm=busdata1(:,3);

delta=busdata1(:,4);





for i=1 : nBusbars





genBus=[genBus;i];


end

end

busType;


A = b(row, row)*(Pg(row)-Pd(row))+g(row, row)*(Qg(row)-Qd(row));

sig=0;

%========================================================


%=======================================================


while maxerror > accuracy & iter <= maxiter % Test for max.

power mismatch


sum1 = zeros(size(Vm));



B-2

for i=1:nBusbars

for j=1 : nBusbars

%sum1 = sum1 + Vm(j)*Y(i, j)*cos(delta(i)-delta(j)-

theta(i,j));

%sum2 = sum2 + Vm(j)*Y(i, j)*sin(delta(i)-delta(j)-

theta(i,j));

sum1(i) = sum1(i) + Vm(j)*(g(i, j)*cos(delta(i)-

delta(j)) + b(i, j)*sin(delta(i)-delta(j)));

sum2(i) = sum2(i) + Vm(j)*(g(i, j)*sin(delta(i)-

delta(j)) - b(i, j)*cos(delta(i)-delta(j)));

end

P(i) = Vm(i) * sum1(i);

Q(i) = Vm(i) * sum2(i);


if i == row

sum1(i) = sum1(i) - Vm(i)*g(i, i);

sum2(i) = sum2(i) + Vm(i)*b(i, i);

lambdac=Vm(i)*(b(i, i)*sum1(i)+g(i, i)*sum2(i))/A;

dY(nBusbars+row) = lambda - lambdac;

else

dY(i+nBusbars) = -Q(i) +lambda*(Qg(i)-(Qd(i)-


busbars)

end

end


dY(i) = -P(i) +lambda*(Pg(i)-Pd(i)); %mismatch equation

for active power (load and generator busbars)

end

end

%===============================================================

==


%===============================================================

==

N = nBusbars;

J =zeros(N,N);

%First Quadrant

for i=1 : nBusbars

if busType(i) ~= 1

for k=1 : nBusbars

if (i ~= k)


B-3

%J(i, k) = Vm(i)*Vm(k)*Y(i, k)*sin(delta(i)-

delta(k)-theta(i,k));

J(i, k) = Vm(i)*Vm(k)*(g(i, k)*sin(delta(i)-

delta(k)) - b(i, k)*cos(delta(i)-delta(k)));

else

%J(i, i) = -Q(i) - Vm(i)^2*Y(i,i)*sin(theta(i,

i));

J(i, i) = -Q(i) - Vm(i)^2*b(i, i);

end

end

end

end

%Second quadrant

for i=1 : nBusbars

if busType(i) ~= 1

for k=1 : nBusbars

if (i ~= k)

%J(i, k + N) = Vm(i)*Y(i,k)*cos(delta(i)-


J(i, k + N) = Vm(i)*(g(i, k)*cos(delta(i)-

delta(k)) + b(i, k)*sin(delta(i)-delta(k)));

else

%J(i, i + N) = P(i)/Vm(i) + Vm(i)*Y(i,

i)*cos(theta(i,i));

J(i, i + N) = P(i)/Vm(i) + Vm(i)*g(i, i);

end

end

end

end

%Third Quadrant



for k=1 : nBusbars

if (i ~= k)

if i == row

J(i + N, k)= (b(i, i)*Vm(k)*(g(i,

k)*sin(delta(i)-delta(k)) - b(i, k)*cos(delta(i)-delta(k)))-g(i,

i)*Vm(k)*(g(i,k)*cos(delta(i)-delta(k)) + b(i, k)*sin(delta(i)-

delta(k))))*Vm(i)/A;

else


B-4

%J(i + N, k) = -Vm(i)*Vm(k)*Y(i,

k)*cos(delta(i)-delta(k)-theta(i,k));

J(i + N, k) = -Vm(i)*Vm(k)*(g(i,

k)*cos(delta(i)-delta(k)) + b(i, k)*sin(delta(i)-delta(k)));

end

else

if i == row

J(i + N, i) = Vm(i)*(-b(i, i)*sum2(i)+g(i,

i)*sum1(i))/A;

else

%J(i + N, i) =P(i)- Vm(i)^2*Y(i,

i)*cos(theta(i,i));

J(i + N, i) =P(i)- Vm(i)^2*g(i, i);

end

end

end

end

end

%Fourth Quadrant



for k=1 : nBusbars

if (i ~= k)

if i == row

J(i + N, k + N) =(b(i, i)*(g(i,

k)*cos(delta(i)-delta(k)) + b(i, k)*sin(delta(i)-delta(k)))+g(i,

i)*(g(i, k)*sin(delta(i)-delta(k)) - b(i, k)*cos(delta(i)-

delta(k))))*Vm(i)/A;

else

%J(i + N, k + N) = Vm(i)*Y(i, k)*sin(delta(i)-


J(i + N, k + N) = Vm(i)*(g(i, k)*sin(delta(i)-

delta(k)) - b(i, k)*cos(delta(i) - delta(k)));

end

else

if i == row

J(i + N, i + N) =(b(i, i)*sum1(i)+g(i,

i)*sum2(i))/A;

else

%J(i + N, i + N) =Q(i)/Vm(i) - Vm(i)*Y(i,

i)*sin(theta(i,i));

J(i + N, i + N) =Q(i)/Vm(i) - Vm(i)*b(i, i);

end

end

end


B-5

end

end

%===============================================================

==


%===============================================================

==


compact


to slack


the form


N = nBusbars;

num1=0;

for i=1 : nBusbars

if busType(i) ~= 1

num1=num1+1;


num2=0;

for j=1 : nBusbars

if busType(j) ~= 1

num2=num2+1;


end

end

end

end

num1=0;

for i=1 : nBusbars

if busType(i) ~= 1

num1=num1+1;

num2=nBusbars-1;

for j=1 : nBusbars


num2=num2+1;


end

end

end

end

num1=nBusbars-1;

for i=1 : nBusbars


B-6


num1=num1+1;


if i== row

deleted= num1;

end

num2=0;

for j=1 : nBusbars

if busType(j)~= 1

num2=num2+1;


end

end

end

end

num1=nBusbars-1;

for i=1 : nBusbars


num1=num1+1;

num2=nBusbars-1;

for j=1 : nBusbars


num2=num2+1;


end

end

end

end


MatY



invert m,


rows or


inversion,


%==========Update angles, voltages and calculate max error

===========

num1 = 0;

for i=1 : nBusbars


B-7

if busType(i) ~= 1

num1 = num1 + 1;

% dX(i) = MatX(num1,1);

delta(i) = delta(i) + MatX(num1,1);

end

end

num1=nBusbars-1;

for i=1 : nBusbars


num1 = num1 + 1;

% dX(i+nBusbars) = MatX(num1, 1);

Vm(i) = Vm(i) + MatX(num1, 1);

end

end



check =====

if maxerror > accuracy & genlimits == 1

for k = 1 : nGens

i=genBus(k);

sum2 = 0;

for j=1 : nBusbars


theta(i,j));

sum2 = sum2 + Vm(j)*(g(i, j)*sin(delta(i)-delta(j))

- b(i, j)*cos(delta(i)-delta(j)));

end

Q(i) = genVolts(k)*sum2;

%fprintf('%g ',Vm(i)),fprintf('%g


if busType(i) == 2

if Q(i) + lambda*Qd(i) > Qmax(i)+0.01

% fprintf('1'),

Qg(i) = Qmax(i);



%Vm(i) = 1;

elseif Q(i) + lambda*Qd(i) < Qmin(i)-0.01

%fprintf('2'),

Qg(i) = Qmin(i);



%Vm(i) = 1;

end


B-8

else

if Q(i) + lambda*Qd(i) < Qmax(i)-0.01 & Vm(i) <

genVolts(k)

%fprintf('3'),

% Vm(i) = genVolts(k);


limits

elseif Q(i) + lambda*Qd(i) > Qmin(i)+ 0.01 & Vm(i)

> genVolts(k)

% fprintf('4'),



limits

end

end

% fprintf('%g\n',busType(i)),

end

end

%===============================================================

=========

%fprintf('\n'),

iter=iter+1;



after ')



the results \n')

converge = 0;% pause,

end


end

%===============================



Pd=Pd*lambda;

Qd=Qd*lambda;

for i=1 : nBusbars

if busType(i) == 1


B-9

Pg(i)=P(i) + Pd(i);

Qg(i)=Q(i) + Qd(i);


Pg(i)=Pg(i)*lambda;

Qg(i)=Q(i) + Qd(i);

end

end

%==========================================

APPENDIX C MATLAB CODE FOR VOLTAGE CHANGE

C-1

10 APPENDIX C

MATLABTM Code for Voltage Change


bus data program)

Vm=0; delta=0;

converge=1;

busType=[];%buscode


% Added (6-8-00)

P=[]; Q=[]; sum1=[]; sum2=[]; A=[]; % Added (6-8-00)


sum1=[];sum2=[];

busType=busdata(:,2); Vm=busdata1(:,3);

delta=busdata1(:,4);





for i=1 : nBusbars





genBus=[genBus;i];


end

end

busType;


A = b(row, row)*(Pg(row)-Pd(row))+g(row, row)*(Qg(row)-Qd(row));

Vm(row)= v;

sig=0;

%========================================================


%=======================================================


while maxerror >= accuracy & iter <= maxiter % Test for max.

power mismatch




C-2


for i=1:nBusbars

for j=1 : nBusbars

%sum1 = sum1 + Vm(j)*Y(i, j)*cos(delta(i)-delta(j)-

theta(i,j));


theta(i,j));

sum1(i) = sum1(i) + Vm(j)*(g(i, j)*cos(delta(i)-

delta(j)) + b(i, j)*sin(delta(i)-delta(j)));

sum2(i) = sum2(i) + Vm(j)*(g(i, j)*sin(delta(i)-

delta(j)) - b(i, j)*cos(delta(i)-delta(j)));

end

P(i) = Vm(i) * sum1(i);

Q(i) = Vm(i) * sum2(i);


if i == row

sum1(i) = sum1(i) - v*g(i, i);

sum2(i) = sum2(i) + v*b(i, i);

vc=lambdac*A/(b(i, i)*sum1(i)+g(i, i)*sum2(i));

dY(nBusbars+row) = v-vc;

else

dY(i+nBusbars) = -Q(i) +lambdac*(Qg(i)-(Qd(i)-


busbars)

end

end


dY(i) = -P(i) +lambdac*(Pg(i)-Pd(i)); %mismatch equation

for active power (load and generator busbars)

end

end

%===============================================================

==


%===============================================================

==

N = nBusbars;

J =zeros(N,N);

%First Quadrant


C-3

for i=1 : nBusbars

if busType(i) ~= 1

for k=1 : nBusbars

if (i ~= k)

%J(i, k) = Vm(i)*Vm(k)*Y(i, k)*sin(delta(i)-


J(i, k) = Vm(i)*Vm(k)*(g(i, k)*sin(delta(i)-

delta(k)) - b(i, k)*cos(delta(i)-delta(k)));

else

%J(i, i) = -Q(i) - Vm(i)^2*Y(i,i)*sin(theta(i,

i));

J(i, i) = -Q(i) - Vm(i)^2*b(i, i);

end

end

end

end

%Second quadrant

for i=1 : nBusbars

if busType(i) ~= 1

for k=1 : nBusbars

if (i ~= k)

%J(i, k + N) = Vm(i)*Y(i,k)*cos(delta(i)-


J(i, k + N) = Vm(i)*(g(i, k)*cos(delta(i)-

delta(k)) + b(i, k)*sin(delta(i)-delta(k)));

else

%J(i, i + N) = P(i)/Vm(i) + Vm(i)*Y(i,

i)*cos(theta(i,i));

J(i, i + N) = P(i)/Vm(i) + Vm(i)*g(i, i);

end

if k == row

J(i , k + N) = -(Pg(i)-Pd(i));

end

end

end

end

r=-A*lambdac/(b(row, row)*sum1(row)+g(row,row)*sum2(row))^2;

%Third Quadrant



for k=1 : nBusbars

if (i ~= k)

if i == row

J(i + N, k)= r*(b(i, i)*Vm(k)*(g(i,

k)*sin(delta(i)-delta(k)) - b(i, k)*cos(delta(i)-delta(k)))-g(i,


C-4

i)*Vm(k)*(g(i,k)*cos(delta(i)-delta(k)) + b(i, k)*sin(delta(i)-

delta(k))));

else

%J(i + N, k) = -Vm(i)*Vm(k)*Y(i,

k)*cos(delta(i)-delta(k)-theta(i,k));

J(i + N, k) = -Vm(i)*Vm(k)*(g(i,

k)*cos(delta(i)-delta(k)) + b(i, k)*sin(delta(i)-delta(k)));

end

else

if i == row

J(i + N, i) =r*(-b(i, i)*sum2(i)+g(i,

i)*sum1(i));

else

%J(i + N, i) =P(i)- Vm(i)^2*Y(i,

i)*cos(theta(i,i));

J(i + N, i) =P(i)- Vm(i)^2*g(i, i);

end

end

end

end

end

%Fourth Quadrant



for k=1 : nBusbars

if (i ~= k)

if i == row

J(i + N, k + N) =r*(b(i, i)*(g(i,

k)*cos(delta(i)-delta(k)) + b(i, k)*sin(delta(i)-delta(k)))+g(i,

i)*(g(i, k)*sin(delta(i)-delta(k)) - b(i, k)*cos(delta(i)-

delta(k))));

else

%J(i + N, k + N) = Vm(i)*Y(i, k)*sin(delta(i)-


J(i + N, k + N) = Vm(i)*(g(i, k)*sin(delta(i)-

delta(k)) - b(i, k)*cos(delta(i) - delta(k)));

end

if k==row

J(i + N, k + N) = -(Qg(i)-(Qd(i)- Qsh(i)));


C-5

end

else

if i == row

J(i + N, i + N) =A/(b(i, i)*sum1(i)+g(i,

i)*sum2(i));

else

%J(i + N, i + N) =Q(i)/Vm(i) - Vm(i)*Y(i,

i)*sin(theta(i,i));

J(i + N, i + N) =Q(i)/Vm(i) - Vm(i)*b(i, i);

end

end

end

end

end

%===============================================================

==


%===============================================================

==


compact


to slack


the form


N = nBusbars;

num1=0;

for i=1 : nBusbars

if busType(i) ~= 1

num1=num1+1;


num2=0;

for j=1 : nBusbars

if busType(j) ~= 1

num2=num2+1;


end

end

end

end

num1=0;

for i=1 : nBusbars

if busType(i) ~= 1

num1=num1+1;


C-6

num2=nBusbars-1;

for j=1 : nBusbars


num2=num2+1;


end

end

end

end

num1=nBusbars-1;

for i=1 : nBusbars


num1=num1+1;


num2=0;

for j=1 : nBusbars

if busType(j)~= 1

num2=num2+1;


end

end

end

end

num1=nBusbars-1;

for i=1 : nBusbars


num1=num1+1;

num2=nBusbars-1;

for j=1 : nBusbars


num2=num2+1;


end

end

end

end


MatY



invert m,


rows or


C-7


inversion,


%==========Update angles, voltages and calculate max error

===========

num1 = 0;

for i=1 : nBusbars

if busType(i) ~= 1

num1 = num1 + 1;

% dX(i) = MatX(num1,1);

delta(i) = delta(i) + MatX(num1,1);

end

end

num1=nBusbars-1;

for i=1 : nBusbars

if i== row

num1 = num1 + 1;


lambdac = lambdac + MatX(num1, 1);

elseif busType(i) ~= 1 & busType(i) ~= 2

num1 = num1 + 1;


Vm(i) = Vm(i)+ MatX(num1, 1);

end

end



check =====

if maxerror > accuracy & genlimits == 1

for k = 1 : nGens

i=genBus(k);

sum2 = 0;

for j=1 : nBusbars


theta(i,j));

sum2 = sum2 + Vm(j)*(g(i, j)*sin(delta(i)-delta(j))

- b(i, j)*cos(delta(i)-delta(j)));

end

Q(i) = genVolts(k)*sum2;

%fprintf('%g ',Vm(i)),fprintf('%g


if busType(i) == 2

if Q(i) + lambdac*Qd(i) > Qmax(i)+0.01

% fprintf('1'),


C-8

Qg(i) = Qmax(i);



%Vm(i) = 1;

elseif Q(i) + lambdac*Qd(i) < Qmin(i)-0.01

%fprintf('2'),

Qg(i) = Qmin(i);



%Vm(i) = 1;

end

else

if Q(i) + lambdac*Qd(i) < Qmax(i)-0.01 & Vm(i) <

genVolts(k)

%fprintf('3'),

% Vm(i) = genVolts(k);


limits

elseif Q(i) + lambdac*Qd(i) > Qmin(i)+ 0.01 & Vm(i)

> genVolts(k)

% fprintf('4'),



limits

end

end

% fprintf('%g\n',busType(i)),

end

end

%===============================================================

=========

%fprintf('\n'),

iter=iter+1;



after ')



the results \n')

converge = 0;% pause,

end


C-9

%===============================



Pd=Pd*lambdac;

Qd=Qd*lambdac;

for i=1 : nBusbars

if busType(i) == 1

Pg(i)=P(i) + Pd(i);

Qg(i)=Q(i) + Qd(i);


Pg(i)=Pg(i)*lambdac;

Qg(i)=Q(i) + Qd(i);

end

end

%==========================================

APPENDIX D MATLAB CODE FOR IMPEDANCE METHOD

D-1

11 Appendix D

MATLABTM code for Impedance method


bus data program)

t=cputime;

dbG=0

dgG=0

Vm=0; delta=0;

converge=1;

busType=[];%buscode

Vm=[]; delta=[]; Pd=[]; Qd=[]; Pg=[]; Qg=[]; Qmin=[];

Qmax=[];zGen=[];vGen=[];vLoad=[]; % Added (6-8-00)

P=[]; Q=[];; % Added (6-8-00)

J=[];dY=[];dX=[];MatY=[];MatX=[];MatA=[];genBus=[];loadBus=[];zeroloadBus=

[];

xPlot=0;

yPlot=0;

busType=busdata(:,2); Vm=busdata(:,3); delta=[];





for i=1 : nBusbars

if busType(i) == 1 %slack bus

slack=i;

end

if busType(i) == 0

Vm(i)=1;

delta(i)=0;

end

end

for i=1 : nBusbars

if busType(i) == 2 %gen bus

genBus=[genBus;i];

end

end


for i=1 : nBusbars

if busType(i) == 0 && (Pd(i) ~= 0 || Qd(i) ~= 0 || Qsh(i) ~=0) %load

bus

loadBus=[loadBus;i];

else


D-2

zeroloadBus=[zeroloadBus;i];

end

end

nLoads=length(loadBus);

nZero=length(zeroloadBus);

yy=y;

%===========add initial admittances to Y matrix=======

for i=1 : nLoads

k=loadBus(i);

gL(i)=Pd(k);

bL(i)=Qsh(k)-Qd(k);

dgL(i)=0;dbL(i)=0;

yy(k,k)=yy(k,k)+gL(i)+bL(i)*1i;

end

for i=1 : nGens

k=genBus(i);

gG(i)=-Pg(k)/Vm(k)^2+Pd(k)/Vm(k)^2;

bG(i)=(Qsh(k)-Qd(k))/Vm(k)^2;

dgG(i)=0;dbG(i)=0;

yy(k,k)=yy(k,k)+gG(i)+bG(i)*1i;

end

Z=inv(yy);

n=nBusbars;

%========================================================

% Start of Loadflow iterations

%========================================================


while maxerror >= accuracy & iter <= maxiter % Test for max. power

mismatch

%disp('startgen')

for i=1:nGens

k=genBus(i);

j=slack;

vGen(i)=Z(k,j)*Vm(j)/Z(j,j);

delta(k)=angle(vGen(i));

%=======================

Vth=abs(vGen(i));

Zth=Z(k,k)-Z(k,j)*Z(j,k)/Z(j,j);

rGen=real(Zth);xGen=imag(Zth);

vDiff=Vth^2-Vm(k)^2;

if vDiff==0

dgG(i)=0;

dbG(i)=0;

else

disc=xGen^2+vDiff*(rGen^2+xGen^2)/Vm(k)^2;

if disc >0

dgG(i)=0;

dbG(i)=-

acc*vDiff/Vm(k)^2/(xGen+sqrt(xGen^2+vDiff*(rGen^2+xGen^2)/Vm(k)^2));

%check


D-3

%Vth*abs(1/dbG(i))/sqrt(rGen^2+(xGen-1/dbG(i))^2)

else

Vm0=sqrt(Vth^2*(rGen^2+xGen^2)/rGen^2);

%Vm0

%Vm(i)

vDiff=Vth^2-Vm0^2;

dgG(i)=0;

dbG(i)=-acc*vDiff/Vm0^2/xGen;

disp('Hi')

k

%Vm(k)=sqrt(Vth^2*(rGen^2+xGen^2)/rGen^2);

%pause

end

bG0=bG(i)+dbG(i);

end

%===========================

%reactive power limits check

if genlimits && iter>2

Qg(k)=Vm(k)^2*bG0+Qd(k)-Qsh(k);

if Qg(k)>Qmax(k)

%disp('wak')

% k

% iter

% Qg(k)

% Qmax(k)

dP=Pg(k)-Pd(k)+Vth^2*gG(i);

dQ=Qmax(k)-Qd(k)+Qsh(k)-Vth^2*bG(i);

rG=real(Zth);xG=imag(Zth);

% dP

if dP~=0

a=dP+dQ^2/dP;b=(2*rG*dP+2*xG*dQ-Vth^2);c=dP*(rG^2+xG^2);

dr=(-b+sqrt(b^2-4*a*c))/(2*a);

dx=dQ/dP*dr;

else

a=dQ;b=2*xG*dQ-Vth^2;c=dQ*(rG^2+xG^2);

dr=0;

dx=(-b+sqrt(b^2-4*a*c))/(2*a);

end

dgG(i)=-acc*dr/(dr^2+dx^2);

dbG(i)=+acc*dx/(dr^2+dx^2);

end

end

gG(i)=gG(i)+dgG(i);

bG(i)=bG(i)+dbG(i);

if dgG(i)~=0 || dbG(i) ~=0

zG=(dgG(i)-dbG(i)*1i)/(dgG(i)^2+dbG(i)^2);

piv=Z(k, k)+ zG;

Z=Z-1/piv*Z(:, k)*Z(k, :);

end

end


D-4

%==========Update load impedances and calculate max error ===========

% disp('startloads')

for i=1:nLoads

k=loadBus(i);

j=slack;

vLoad(i)=Z(k,j)*Vm(j)/Z(j,j);

Vm(k)=abs(vLoad(i));

delta(k)=angle(vLoad(i));

%=======================

Vth=abs(vLoad(i));

dP=Pd(k)-Vth^2*gL(i);

dQ=Qd(k)-Qsh(k)+Vth^2*bL(i);

Zth=Z(k,k)-Z(k,j)*Z(j,k)/Z(j,j);

rL=real(Zth);xL=imag(Zth);

if dP~=0 || dQ~=0

if dP~=0

a=dP+dQ^2/dP;b=(2*rL*dP+2*xL*dQ-Vth^2);c=dP*(rL^2+xL^2);

dr=(-b+sqrt(b^2-4*a*c))/(2*a);

dx=dQ/dP*dr;

else

a=dQ;b=2*xL*dQ-Vth^2;c=dQ*(rL^2+xL^2);

dr=0;

dx=(-b+sqrt(b^2-4*a*c))/(2*a);

end

dgL(i)=acc*dr/(dr^2+dx^2);

dbL(i)=-acc*dx/(dr^2+dx^2);

%===================================

%dgL(i)=Pd(k)/abs(vLoad(i))^2-gL(i);

%dbL(i)=-Qd(k)/abs(vLoad(i))^2-bL(i);

%===================================

gL(i)=gL(i)+dgL(i);

bL(i)=bL(i)+dbL(i);

if dgL(i)~=0 || dbL(i) ~=0

zL=(dgL(i)-dbL(i)*1i)/(dgL(i)^2+dbL(i)^2);

piv=Z(k, k)+ zL;

Z=Z-1/piv*Z(:, k)*Z(k, :);

end

end

end

maxerror=max(abs([dgL dbL dgG dbG]));

if iter==0

xPlot=0;

%yPlot=[dgL dbL dgG dbG];

yPlot=[dbG];

else

xPlot=[xPlot;iter];

%yPlot=[yPlot;[dgL dbL dgG dbG]];


D-5

yPlot=[yPlot;[dbG]];

end

if iter==2

%acc=max(abs(yPlot(1,:)))/(4/3*max(abs(yPlot(1,:))))

end

iter=iter+1;


fprintf('\nWARNING: Iterative solution did not converge after ')


fprintf('Press Enter to terminate the iterations and print the results

\n')

converge = 0; pause,

end


end

%===============================


Islack=Vm(slack)/Z(slack,slack);


Pg(slack)=real(Vm(slack)*conj(Islack))+Pd(slack);

Qg(slack)=imag(Vm(slack)*conj(Islack))+Qd(slack);

for i=1:nGens

k=genBus(i);

Vm(k)=abs(vGen(i));

Qg(k)=Vm(k)^2*bG(i)+Qd(k)-Qsh(k);

end

for i=1:nZero

k=zeroloadBus(i);

Vm(k)=abs(Islack*Z(k,slack));

delta(k)=angle(Islack*Z(k,slack));

end

%==========================================

cpuTime=cputime-t

APPENDIX E IEEE 14-BUS DATA

E-1

new load flow methods and techniques by ahmed mustafa

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