Implementation of FACTS and Economic … of FACTS and Economic Generation Dispatch in an Interactive...

Martin Geidl

Implementation of FACTS and Economic

Generation Dispatch in an Interactive Power

Flow Simulation Platform

Diploma Thesis PSL0201

Department:

EEH – Power Systems Laboratory, ETH Zurich

Experts:

Prof. Dr. Goran Andersson, ETH Zurich

Univ.-Prof. Dr. Lothar Fickert, TU Graz

Supervisors:

Dipl. El. Ing. ETH Christian Schaffner, ETH Zurich

Ass.-Prof. Dr. Herwig Renner, TU Graz

Zurich, March 2003

Vorwort

Diese Diplomarbeit habe ich im Rahmen meines Erasmus-Studienaufenthaltes an

der ETH Zurich von August 2002 bis Marz 2003 verfasst. In dieser Zeit hatte ich

Gelegenheit mich intensiv mit meinen personlichen Interessensgebieten zu befassen,

die Arbeit hat mir sehr viel Freude bereitet.

Bedanken mochte ich mich bei all jenen, die mich in dieser Zeit begleitet haben:

Meinen Eltern fur die großzugige Unterstutzung und das Vertrauen, meinem Bruder

Stefan fur den privaten IT-Support, Michi furs Coaching, Wolfi und Georg fur die

Zeit in Zurich, den Maxen fur die Wochenenden daheim, Sandra und Thomas fur

die netten Abende.

Bei allen Mitarbeitern des PSL bedanke ich mich fur die freundliche Aufnahme in

der Gruppe und das angenehme Arbeitsklima. Rusejla, Tina und Thilo danke ich

fur die zahlreichen Tipps und die spannenden fachlichen Diskussionen.

Besonderer Dank gilt Prof. Goran Andersson und Prof. Lothar Fickert sowie meinen

Betreuern Dr. Herwig Renner und Christian Schaffner. Sie haben mir die Moglichkeit

gegeben diese Arbeit durchzufuhren und mich dabei bestens betreut.

Vielen Dank!

Zurich, am 7. Marz 2003

i

Abstract

Title: Implementation of FACTS and Economic Generation Dispatch in an Inter-

active Power Flow Simulation Platform

Keywords: SVC, TCSC, Injection Model, KKT, Unlimited Point, Economic Dis-

patch

This diploma thesis reports the implementation of new features in a power flow

simulation platform. The FACTS elements SVC and TCSC are mathematically

described and implemented in the Newton-Raphson power flow computation as well

as in the GUI. The SVC element is equipped with regulation on reactive power,

voltage and firing angle, the TCSC is implemented with regulation on active power

and firing angle. Furthermore an ’Economic Generation Dispatch’ which targets

minimisation of the total generation costs in a network is implemented, whereas the

cost functions of the generators are assumed to be quadratic functions of the active

power output. The inequality constrained optimisation problem is solved with the

so-called ’Unlimited Point’ algorithm.

ii

Kurzfassung

Titel: Implementierung von FACTS und Economic Generation Dispatch in einer

interaktiven Lastfluss-Simulationsumgebung

Stichworter: SVC, TCSC, Injection Model, KKT, Unlimited Point, Economic

Dispatch

In der vorliegenden Arbeit wird eine bestehende Lastfluss-Simulationsumgebung er-

weitert. Die FACTS-Elemente SVC und TCSC werden mathematisch modelliert

und sowohl in der Newton-Raphson Lastflussrechnung als auch im GUI implemen-

tiert. Der SVC wird mit Regelung auf Blindleistung, Spannung und Steuerwinkel

ausgestattet, der TCSC wird mit Regelung auf Wirkleistung und Steuerwinkel imple-

mentiert. Weiters wird ein Modul ’Optimaler Kraftwerkseinsatz’ zur Minimierung

der gesamten Erzeugungskosten in einem Netzwerk implementiert, wobei die Kosten

der Generatoren als quadratische Funktionen der Wirkleistung angenommen werden.

Das ungleichheitsbeschrankte Optimierungsproblem wird mit dem ’Unlimited Point’

Algorithmus gelost.

iii

Contents

List of Acronyms viii

List of Symbols ix

1 Introduction 1

1.1 Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Simulation Platform FlowDemo.net . . . . . . . . . . . . . . . . . . . 1

1.3 Flexible AC Transmission Systems . . . . . . . . . . . . . . . . . . . 3

1.3.1 Thyristor Controlled Reactor . . . . . . . . . . . . . . . . . . 3

1.3.2 Static Var Compensator . . . . . . . . . . . . . . . . . . . . . 4

1.3.3 Thyristor Controlled Series Capacitor . . . . . . . . . . . . . . 5

1.4 Power Flow Computation . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.1 Newton-Raphson . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Economic Generation Dispatch . . . . . . . . . . . . . . . . . . . . . 7

2 FACTS Modelling 8

2.1 FACTS Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Injection Model . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Total Susceptance Model . . . . . . . . . . . . . . . . . . . . . 9

2.1.3 Firing Angle Model . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Implemented Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 TCR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

iv

CONTENTS

2.2.2 Circuit for SVC and TCSC . . . . . . . . . . . . . . . . . . . . 12

2.2.3 SVC Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.4 TCSC Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Implementation of FACTS in the Power Flow Computation 17

3.1 SVC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.1 Reactive Power Regulation . . . . . . . . . . . . . . . . . . . . 18

3.1.2 Voltage Regulation . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.3 Firing Angle Regulation . . . . . . . . . . . . . . . . . . . . . 18

3.2 TCSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Active Power Regulation . . . . . . . . . . . . . . . . . . . . . 19

3.2.2 Firing Angle Regulation . . . . . . . . . . . . . . . . . . . . . 20

3.2.3 Resonance Protection and Angle Limits . . . . . . . . . . . . . 20

3.3 Additional Implementations . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.1 Determination of the Fire Delay Angle . . . . . . . . . . . . . 21

3.3.2 Incremental Changes of SVC and TCSC . . . . . . . . . . . . 22

3.3.3 Switching of SVC and TCSC . . . . . . . . . . . . . . . . . . 22

3.3.4 Power Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3.5 Formatting of the Results . . . . . . . . . . . . . . . . . . . . 24

4 Implementation of FACTS in the Client 25

4.1 SVC in the Client . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1.1 Class Svc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1.2 Class SvcDialog . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 TCSC in the Client . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2.1 Class Tcsc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2.2 Class TcscDialog . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Economic Generation Dispatch 31

v

CONTENTS

5.1 Mathematical Approach . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.1.1 Problem Formulation and Optimality Conditions . . . . . . . 31

5.1.2 Unlimited Point Algorithm . . . . . . . . . . . . . . . . . . . . 32

5.1.3 Optimisation of Total Generation Costs . . . . . . . . . . . . . 33

5.2 Implementation of the Economic Generation Dispatch . . . . . . . . . 37

6 Program Tests 39

6.1 Tests with Single SVC and TCSC Element . . . . . . . . . . . . . . . 40

6.1.1 3-bus Test Network . . . . . . . . . . . . . . . . . . . . . . . . 40

6.1.2 Single SVC Test . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6.1.3 Single SVC Test Results . . . . . . . . . . . . . . . . . . . . . 42

6.1.4 Single TCSC Test . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.1.5 Single TCSC Test Results . . . . . . . . . . . . . . . . . . . . 43

6.2 Combined SVC and TCSC Test . . . . . . . . . . . . . . . . . . . . . 43

6.2.1 6-bus Test Network . . . . . . . . . . . . . . . . . . . . . . . . 44

6.2.2 Test Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . 44

6.2.3 Combined SVC and TCSC Test Results . . . . . . . . . . . . 44

6.3 Test of Economic Generation Dispatch . . . . . . . . . . . . . . . . . 47

6.3.1 Test Arrangement . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.3.2 EGD Test Results . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.4 Known Issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

7 Conclusion 50

7.1 Project Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

A Flowcharts 53

vi

CONTENTS

B Test Data 57

C Variables, Scripts and Functions 64

Bibliography 68

vii

List of Acronyms

Shortcut Full Name

AC Alternating Current

FACTS Flexible Alternating Current Transmission Systems

TCR Thyristor Controlled Reactor

SVC Static Var Compensator

TCSC Thyristor Controlled Series Capacitor

GUI Graphical User Interface

GTO Gate Turn Off

IGBT Isolated Gate Bipolar Transistor

MOV Metal Oxide Varistor

EGD Economic Generation Dispatch

viii

List of Symbols

Symbol Description

iTCR TCR currentt timeα fire delay angleω rated angular frequencyu voltageU voltage, rms-valueL inductanceC capacitanceXL,XC reactance of a reactor, capacitorB susceptanceBL, BC susceptance of a reactor, capacitorBTCR apparent susceptance of a TCRUi, Uj node voltagePij , Pji active power flowQij , Qji reactive power flowδij , δji transmission angleθi, θj phase angleF objective functiong equality constrainth inequality constraintL Lagrangian-functionλ, μ Lagrangian-multipliersz slack variable for inequality constraintsCP cost function of generatorPi active power output of generatora0,i, a1,i, a2,i cost function factorsAP matrix of cost function factorsPgen vector of generator power valuesPgen,max vector of max. generator power valuesJ Jacobian matrixn number of generators, dimension of objective functionm number of equality constraintsp number of inequality constraints

ix

Chapter 1

Introduction

1.1 Assignment

The intention of this work is to implement FACTS (Flexible Alternating Current

Transmission Systems) and Economic Generation Dispatch in the power flow demon-

stration platform FlowDemo.net see [1].

Dealing with the implementation of FACTS, two different tasks have to be per-

formed: At first appropriate mathematical models have to be found, secondly the

devices have to be implemented in the power flow computation as well as in the

Graphical User Interface (GUI).

For the Economic Generation Dispatch an algorithm to solve the inequality con-

strained optimization problem due to Karush-Kuhn-Tucker has to be implemented

in Matlab.

1.2 Simulation Platform FlowDemo.net

FlowDemo.net is an interactive, internet based software to compute and visualise

power flow in an electric transmission system [1]. The program is freely accessible

on the internet on www.FlowDemo.net. The intention of this software is not only

to compute but also to demonstrate and visualise the power flow in an electric

transmission system.

Predefined networks could be loaded and demonstrated in version 1.0, it went online

in 2001. The new version 2.0 provides a GUI with a network editor where the user

1

1.2 Simulation Platform FlowDemo.net

can define individual networks and hold them in a data base. FlowDemo.net 2.0

contains the FACTS elements SVC and TCSC and the new feature Economic Gen-

eration Dispatch.

FlowDemo.net 2.0 consists of several software modules which are implemented in

various programming languages (see Figure 1.1). The modular software design tar-

gets independence of the functional units and extensibility. The four major parts

are the client, the server, the computation engine and the data base.

Figure 1.1: Software assembly of FlowDemo.net 2.0

Client: The client, which includes the Graphical User Interface (GUI) with the

network editor, is implemented in Java. It works either in edit-mode or in run-

mode. In the edit-mode a network editor is available which allows the user to

design, edit, save or load individual networks. As long as the network is edited no

computation and visualisation is done. In the run-mode the power flow is computed

and illustrated.

Server: The server is the central functional unit that manages all software modules

and connections between them. It is also implemented in Java.

Computation Engine: The power flow computation, which is not visible for the

user of FlowDemo.net, is started in the client by clicking on the run-button. At

first network errors, islands or disconnected elements are detected, then the network

data are prepared and finally the power flow computation starts. The matrix-based

procedure uses a Newton-Raphson algorithm which is implemented in Matlab [2],

[3]. When the computation is finished the results are sent back to the client and

the power flow is visualised in the GUI. Matlab is a suitable instrument for this

task, its major advantage over other programming languages is the easily handling

of matrices, especially if they are sparse.

2

1.3 Flexible AC Transmission Systems

Data Base: The user can create individual networks and save them either locally

or in the FlowDemo.net data base. It is also possible to load predefined networks in

’read only’ mode. The network data are kept in tables for busses, generators, loads,

shunts, lines, transformers, capacitors, SVC and TCSC. They contain all input-

data of the network elements but no results of the power flow computation. The

implementation in PostgreSQL is based on the IEEE Power System Applications

Data Dictionary [12].


Flexible Alternating Current Transmission Systems (FACTS) are controllable power-

electronic based devices in electric transmission systems which allow to affect power

flow and/or node voltages dynamically [5], [6], [7].

As mentioned in [5] the first FACTS, based on mechanically switched reactors and

capacitors, were developed in the 1960s. Later, in the 1970s, semiconductor-switches

were applied which offered new opportunities and initialized a rapid development.

Today FACTS are equipped with converter-based technology using high-capacity

thyristors (also GTO-thyristors) or IGB-transistor switches.

FACTS generally work as

• parallel compensation,

• series compensation or

• both, parallel and series compensation.

Typical fields of application are dynamic reactive power compensation, dynamic

power flow control and power flow dispatching, increase of transmission capacity,

improve of steady-state stability (voltage regulation) and enhancing dynamic sta-

bility (oscillation damping).

Some basic FACTS circuits will be shortly described in this chapter, mathematical

descriptions of the devices are given in section 2.

1.3.1 Thyristor Controlled Reactor

A Thyristor Controlled Reactor (TCR) is one of the simplest FACTS elements (see

Figure 1.2). It consists of a linear reactor in series with two bipolar thyristors [7].

3


Changing the current by triggering the thyristors affects on the apparent suscep-

tance, i.e. the apparent susceptance of the TCR is a function of the thyristor firing

angle. Usually a TCR is used in combination with other reactive elements, for

example with a capacitor in parallel.

Figure 1.2: Thyristor Controlled Reactor

1.3.2 Static Var Compensator

The Static Var Compensator (SVC) is a device for dynamic shunt compensation, the

most common type comprises a TCR in parallel with a fixed capacitor1 pictured in

Figure 1.3. The equivalent impedance of this circuit can be affected by triggering the

thyristors, with a certain dimensions of the capacitor and the reactor it is possible to

get an operating area from inductive to capacitive susceptance. In AC transmission

systems usually three of these circuits are connected in delta, with ratings up to

many hundred MVA [7].

Figure 1.3: SVC circuit

1There are several other possible arrangements mentioned in [5].

4

1.4 Power Flow Computation

1.3.3 Thyristor Controlled Series Capacitor

A TCR with a capacitor in parallel can also be used as a series element for series

compensation, the circuit is the same as the mentioned SVC circuit. This arrange-

ment is called Thyristor Controlled Series Capacitor, one of its typical applications

is the compensation of inductive transmission lines: If the TCSC is operated in

capacitive mode, i.e. the apparent susceptance of the circuit is capacitive, it can be

used to compensate a part of the line’s inductivity.

Figure 1.4: TCSC circuit


As described in [3] the power flow computation2 is the task to determine any normal

state of an electric network. This normal state is:

• The network is in a static state, the waveforms of network quantities like

voltages, currents and powers are sinusoidal in time.

• All network elements have linear characteristics.

• The system works under symmetric conditions.3

Given are the values of fixed loads, the impedance of shunt elements, line and trans-

former parameters, FACTS power values, voltage targets or firing angles, generator

voltages and/or power values4 and the topology of the network. The result of the

2also called ’load flow computation’3Otherwise the system has to be transformed in three decoupled systems (M-G-0) and re-

transformed after the computation of each single decoupled system.4That depends on the regulation type of the generator. For the slack generator only the node

voltage is given, for PV-generators the active power and the node voltage is specified and forPQ-generators active and reactive power is fixed.

5


power flow computation includes node voltages and voltage angles, current and

power values.

In [3] the fundamental power flow equation system which is based on linear models

is stated as:

Nr. Equation

1

2

3

A1 c =

⎡⎢⎢⎣

[diag(MNetIi ) 0 ] [diag(MNet

Ui ) 0 ] 0

CT 0 0

0 −E C

⎤⎥⎥⎦

⎡⎢⎢⎣

INport

UNport

UKirch

⎤⎥⎥⎦ =

⎡⎢⎢⎣

0

0

0

⎤⎥⎥⎦

4 PNport + jQNport − diag(UNport)I∗Nport = 0

5 P 0gen′/load − P

gen′/loadNport = 0

6 Q0load − Qload

Nport = 0

7 U0gen − |U gen

Nport| = 0

8 angle0 − � (UKirchSlack) = 0

Equation 1, 2 and 3 include the topological description of the network as well as the

linear models of the network elements. Equation 4 describes the power relationship

of all n-ports. Equation 5 and 6 specify the balance between generated and con-

sumed active and reactive power values. The magnitudes of the node voltages are

added in equation 7, equation 8 defines the slack angle. A more detailed description

is given in [3].

The procedure of power flow computation can be divided in two main parts: Firstly,

the equations based on the linear models have to be formulated, and secondly the

system of linear and nonlinear equations has to be solved.

1.4.1 Newton-Raphson

With the iterative method of Newton-Raphson it is possible to obtain the zeros

of a nonlinear function f(x) = 0 using the Taylor-Series approach. Because it is

an iterative method, the result always contains a so-called mismatch value that

6

1.5 Economic Generation Dispatch

decreases with the number of executed iterations if the computation converges.5 A

detailed description with numeric examples of the Newton-Raphson algorithm is

given in [3].

1.5 Economic Generation Dispatch

The task of Economic Generation Dispatch is to dispatch the total generation power

in a network to the generators due to their economic behavior whereas the total costs

of generation should be minimised. For this purpose every generator is modelled with

a cost function that defines the dependence of the generator costs and the generated

power.6 The inequality constrained optimisation problem is generally formulated as

[15]:

Minimise F (x) Objective function

subject to g(x) = 0 Equality constraints

and h(x) ≤ 0 Inequality constraints

The objective function describes the total generation costs as a function of the gener-

ated active power values, for FlowDemo.net these functions are assumed as quadratic

polynomials. The equality constraint results from the condition that the total gen-

eration power must be the same before and after the dispatch.7 The inequality

constraints represent the power limits of the generators. A detailed discussion of

this problem formulation is done in Chapter 5.

FlowDemo.net solves the inequality constrained optimization problem with the so

called ’Unlimited Point’ method.

5Newton-Raphson does not converge in every case resp. for every function.6In this work only active power is considered.7This equation is not fulfilled after a generation dispatch and re-computation of the power flow:

Due to the new generation pattern the power flow situation in the whole network changes. For thisreason the network losses and therefore the slack power and the total generated power change.

7

Chapter 2

FACTS Modelling

In this Chapter mathematical models of FACTS are discussed. Firstly an overview

of common modelling instruments is given and secondly the applied models are

explained. The priorities for the models implemented in FlowDemo.net are applica-

bility for the computation engine and clearness for the user.

2.1 FACTS Models

There are several ways of modelling FACTS, whereas the suitability of a model

depends on the specific problem. Some basic modelling techniques are mentioned in

the following.

2.1.1 Injection Model

The Injection Model describes the FACTS as a device that injects a certain amount

of active and reactive power to a node, so the FACTS device is represented as a

PQ-element [5], [8], [11]. Figure 2.1 shows the idea of interpreting a FACTS device

as PQ-elements. If the FACTS model does not contain losses, the injected powers

can be written as [8]:

Pij = UiUjBijsin δij (2.1)

Pji = −Pij (2.2)

Qij = U2i Bij − UiUjBijcos δij (2.3)

Qji = U2j Bij − UiUjBijcos δji (2.4)

8

2.1 FACTS Models

Figure 2.1: Injection Model

whereas only the FACTS (no other elements) is connected between bus i and bus j.

Pij and Pji represent active power flow, Qij and Qji are reactive power values. The

transmission angles are

δij = Θi − Θj = � U i − � U j (2.5)

δji = −δij (2.6)

Since this model uses PQ-elements to describe the FACTS, it can be implemented

in a Newton-Raphson computation like PQ-loads. The Injection Model does not

contain internal information about the device, i.e. it is independent from the internal

design of the FACTS.

2.1.2 Total Susceptance Model

This model interprets the FACTS as a shunt (for shunt compensation) or series el-

ement (for series compensation) with varying susceptance B [9]. Due to (2.1)-(2.6)

the power flow through the FACTS depends on B, Pij and Qij = f(B). Figure 2.3

Figure 2.2: Total Susceptance Model

shows a 1-port and a 2-port black box. In network analysis every n-port is repre-

sented by the impedance matrix which can be stated from the T or Π circuit model

of the network element. Inserting the variable B in the 1- and 2-port models for

shunt and series elements leads to the 1-port matrix model for the shunt connected

element: [Ii

]−

[jBij

] [Ui

]=

[0

](2.7)

9

2.2 Implemented Models

Figure 2.3: 1-port (a) and 2-port (b) element

The 2-port model for the series element can be stated as:

⎡⎣ Ii

Ij

⎤⎦ −

⎡⎣ −jBij jBij

jBij −jBij

⎤⎦

⎡⎣ Ui

Uj

⎤⎦ =

⎡⎣ 0

0

⎤⎦ (2.8)

Since it is well known how to implement the 1-port and 2-port models in a power

flow computation, this model is proper for a power flow computation with Newton-

Raphson. Like the Injection Model, the Total Susceptance Model does not describe

the internal design of the FACTS. It does not contain the dependence of B from

any internal value, for example firing angle.

2.1.3 Firing Angle Model

As described in [9] the Firing Angle Model includes the dependence of the FACTS

impedance or power values from the variable firing angles of semiconductor switches.

The firing angle is now considered as a state variable, so that B−1ij = Xij =

f(α,XL, XC) and Pij, Qij = f(α,XL, XC). Such a function f(α,XL, XC) can be

inserted in the Injection Model as well as in the Total Susceptance Model. With

this extended model the user can influence powers by changing firing angles of the

valves.

With the Firing Angle Model we consider the internal circuit as well as values which

affect the power flow through the device, like capacitance, reactance and especially

the firing angle. A major difference between this model and the models mentioned

above is that the Firing Angle Model describes the FACTS internal design.


After having a look on different modelling techniques in Section 2.1 we will now dis-

cuss which and how models are implemented in the computation engine. FlowDemo.net

10


uses more or less all models mentioned in Section 2.1. For each type of regulation

the models are nested in different ways with respect to the firing angle.

Power-Regulation: The injected powers are set by the user, so the injected load(s)

is (are) given. After the Newton-Raphson power flow computation the apparent

susceptance of the FACTS can be determined from the node voltage and the power

value. In the last step the fire delay angle is computed from the apparent susceptance

by adopting the Firing Angle Model (see Section 2.1.3).

Voltage-Regulation: The desired node voltage is given, the injected powers are

results of the power flow computation. After determining the total susceptance of

the device from the voltage and the power value, the fire delay angle is calculated.

Angle-Regulation: The total susceptance can be determined from the user-defined

fire delay angle and the reactance of the capacitor and the reactor. Afterwards the

device is implemented in the power flow computation like a shunt or series element

(SVC or TCSC). The injected loads are result from the computation.

The active power losses of a SVC or a TCSC are neglected in the models because

usually they are below one percent of the reactive power rating of the device [5].

2.2.1 TCR Model

In [6] the current of an elementary single-phase TCR with fire delay angle1 control

is stated as:

iTCR(t) =1

L

∫ ωt

αv(t)dt (2.9)

With a sinusoidal voltage u(t) = U cos ωt we get for the current:

iTCR(t) =U

ωL(sin ωt − sin α) (2.10)

Assuming not only the voltage sinusoidal but also the current, the RMS-value results

in

ITCR(α) =U

ωL

(1 − 2

πα − 1

πsin 2α

)(2.11)

Using ITCR = UBTCR and BL = X−1L = 1

ωLthe apparent susceptance2 of the circuit

can be written as

BTCR(α) = BLπ − 2α − sin 2α

π(2.12)

1The ’fire delay angle’ α, also called ’trigger angle’, is equivalent to the ’prevailing conductionangle’ σ = π − 2α, therefore a TCR can be characterized by either of them. α and σ are definedin [6].

2This is the fundamental frequency equivalent without considering harmonics of the current.

11


Therefore the susceptance of a TCR can be varied continuously from maximum

(α = 0, BTCR = BL) to minimum (α = π2, BTCR = 0).

2.2.2 Circuit for SVC and TCSC

A TCR in parallel with a capacitance is a basic circuit for SVC and TCSC. The

arrangement shown in Figure 2.4 allows to vary the equivalent susceptance continu-

ously. If ωL < 1ωC

the control range covers capacitive and reactive operation mode.

Figure 2.4: Basic circuit for SVC and TCSC

The equations for this circuit can be stated as follows. With a capacitance

BC = X−1C = −ωC (2.13)

connected in parallel to a TCR, the apparent susceptance of the whole circuit results

in

B = BTCR + BC (2.14)

and with (2.12) it can be expressed as

B(α) = BLπ − 2α − sin 2α

π+ BC (2.15)

Figure 2.5 and Figure 2.6 picture the trajectories of B and X = B−1 for a basic

circuit with BL = 1.0 p.u. and BC = −0.2 p.u. Concerning the firing angle three

different operation modes can be defined for this circuit [7]:

Bypassed-Thyristor Mode (α = 0◦): In this mode the thyristors are triggered

to full conductance, the module behaves approximately like a parallel arrangement

of the capacitor and the inductor. If the susceptance of the inductor is higher than

the susceptance of the capacitor, the current through the device is inductive.

Blocked-Thyristor Mode (α = 90◦): The thyristors are blocked, the current

through the reactor gets zero and the arrangement acts just like a fixed capacitor.

12


Partially Conducting Thyristor (0◦ < α < 90◦): In this operation area it

is possible to control the apparent susceptance of the circuit continuously from

inductive to capacitive character.

As generally known, a capacitance in parallel with an inductance has always a

resonance point where BL = −BC . Due to (2.14), the antiresonance-condition for

the basic circuit is

B(αres) = BLπ − 2αres − sin 2αres

π+ BC = 0 (2.16)

This condition can be fulfilled by variation of α, i.e. there is a firing angle αres, where

the circuit oscillates in resonance. In order to prevent the circuit from resonance

state, this firing delay angle αres has to be ’forbidden’, therefore the user should not

be allowed to set α = αres. The determination of the fire delay angle αres where

B(αres) = 0 is explained in 3.3.1. For keeping a ’safety margin’ to resonance, a

protected area for the firing angle is defined as

αres − Δα > α > αres + Δα (2.17)

Antiresonance is not a problem for shunt-connected circuits because the bus voltage

is fixed, so in case of resonance the current gets close to zero what does not endanger

the internal elements of the device. For series connected circuits we can assume the

current or the transmission power as fixed, what leads to a high series voltage drop

if the circuit is in resonance state. This could cause a serious stress for the FACTS

components.

2.2.3 SVC Model

As mentioned in Section 1.3.2, a SVC is a shunt-connected circuit. The equations

of the basic circuit are used here.

Due to the equations of the Injection Model (2.1) and (2.3) and the fact that there

is no bus j for a shunt connected element, what means

Uj = 0 (2.18)

the active and reactive power values of an SVC element result in

Pi = 0 (2.19)

Qi = U2i B (2.20)

13


0 10 20 30 40 50 60 70 80 90−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Fire delay angle a [deg]

Equ

ival

ent s

usce

ptan

ce B

(a)

[p.u

.]

Figure 2.5: Variable susceptance (BL = 1.0 p.u., BC = −0.2 p.u.)

0 10 20 30 40 50 60 70 80 90−15

−10

−5

0

5

10

15

Fire delay angle a [deg]

Equ

ival

ent r

eact

ance

X(a

) [p

.u.]

Inductive operation

Capacitive operation

Figure 2.6: Variable reactance (BL = 1.0 p.u., BC = −0.2 p.u.)

14


Figure 2.7: SVC Model

With (2.15) we get a function for the reactive power that includes the fire delay

angle:

Qi = U2i

(BL

π − 2α − sin 2α

π+ BC

)(2.21)

Considering this equation in matters of resonance where B = 0 we can see that

resonance state would cause a reactive power consumption of Qi = 0.

2.2.4 TCSC Model

The TCSC is a series connected TCR in parallel with a capacitor. For the injected

Figure 2.8: TCSC Model

power values of the TCSC at bus i and bus j the equations (2.1)-(2.6) are used.

Applying (2.15) leads to the transmitted active power

Pij = UiUj

(BL


π+ BC

)sin δij (2.22)

Since the model does not consider any active power losses in the TCSC we get the

same absolute value (but due to the reference system with a different sign) on the

other bus of the TCSC:

Pji = −Pij (2.23)

15


Since losses in lines are usually not neglected, this equation is just valid for the

TCSC element but not for a combination of a TCSC with lines. Because the TCSC

is a reactive element the reactive power values are not equal on both nodes, they

can be stated as

Qij = Ui

(BL


π+ BC

)(Ui − Uj cos δij) (2.24)

Qji = Uj

(BL


π+ BC

)(Uj − Ui cos δji) (2.25)

If the circuit is in resonance state, the expression in the big brackets of (2.24) and

(2.25) gets close to zero. Assuming a certain (desired) power flow Pij + jQij one

can recognise that this would cause very high bus voltages that could endanger the

TCSC.

16

Chapter 3

Implementation of FACTS in the

Power Flow Computation

The FACTS devices have to be implemented in the Matlab computation engine

by integrating SVC and TCSC models. FlowDemo.net 1.0 provides a power flow

computation for networks consisting of the 1-port elements generator, load, shunt,

capacitor and the 2-port elements line and transformer. Version 2.0 includes also

the FACTS devices SVC (1-port) and TCSC (2-port).

The implementation of the FACTS elements is based on the linear models

of the existing elements. There are no new element types in the network

matrices and the Jacobian matrix. Depending on the regulation the

FACTS are included in the element matrices of loads, generators, shunts

and lines.

3.1 SVC

Depending on the regulation type, the SVC can be described like a PV-generator, a

PQ-load or a shunt element with defined susceptance B. The original SVC matrix is

fractionised and each SVC element is embedded in the element matrix of generators,

loads or shunts before the Newton-Raphson computation is started. When the

computation is finished the results relating to the SVC elements are taken out from

the generator, load and shunt result vectors and the power values are saved in the

SVC result vectors. This procedure is shown in Appendix A.

17

3.1 SVC

3.1.1 Reactive Power Regulation

For a Q-regulated SVC the user defines the reactive power Qi the device should

inject. In this case the Injection Model is used, and the SVC is implemented like a

reactive load that consumes a defined amount of reactive power from the connected

bus.

Since Qi is known and Uj is a result of the power flow computation, we can obtain

the equivalent susceptance by converting (2.20) to

B =Qi

U2i

(3.1)

With B, Ui, XC , XL and (2.15) it is possible to determine the fire delay angle

(explained in Section 3.3.1).

3.1.2 Voltage Regulation

If the user wants to control the bus voltage Ui, the SVC has to act like a PV-regulated

generator (with P = 0) according to the Injection Model. A result of the power flow

computation is the power value Qi but not the equivalent susceptance and the firing

angle of the thyristors, they are determined in the same way as described in Section

3.1.1.

3.1.3 Firing Angle Regulation

Other than the mentioned regulations on reactive power and node voltage it is

possible to set the fire delay angle α of the TCR. If this is done, the Total Susceptance

Model is applied after the apparent susceptance B of the SVC is computed from α,

XL and XC . The device is then implemented like a shunt, the computation results

include the power values.

In this regulation mode the reactive power limits are not taken into consideration.

The user can get every reactive power value that results from the specified fire delay

angle. A way to find the firing angles where reactive power values are on the limit

is to run the SVC in reactive power regulation mode, set the desired value on the

limit and take a look at the fire delay angle. To get sure that the minimum reactive

power limit is not violated with firing angle regulation the angle must not be less

than the displayed angle when the SVC worked in reactive power regulation mode

on its lower limit. The maximum reactive power limit is exceed if the angle is greater

18

3.2 TCSC

than the angle which is displayed if the SVC operates in reactive power regulation

on its upper reactive power limit.

3.2 TCSC

For the Thyristor Controlled Series Capacitor two types of regulation are imple-

mented. The first one is the control of the active transmitted power P , the second

is to set the fire delay angle α. Depending on the control mode, the TCSC is imple-

mented as loads on the connected busses or as a line between them. For this reason

the TCSC matrix which includes all TCSC information is split and the TCSC ele-

ments are embedded in the element matrices of loads and lines. The computation

starts with this additional rows in the load and line matrices. After it is finished

the results are assigned to TCSC, lines and loads.

3.2.1 Active Power Regulation

P-regulated TCSC means that the user defines the active power that should be trans-

mitted from bus i to bus j, whereas the reactive powers, the equivalent susceptance

and the firing angle are unknown.

Eliminating B from (2.1)-(2.4) gives the injected reactive power values as functions

of the specified value P and the complex node voltages:

Qij =Pij

sinδij

[Ui

Uj

− cos δij

](3.2)

Qji =Pij

sinδij

[Uj

Ui

− cos δij

](3.3)

With this equations the TCSC can be handled like two loads Pij + jQij at bus i and

−Pij + jQji at bus j.

Using (3.2) and (3.3) to obtain the reactive power values, we have to consider that

the node voltages are not known before the power flow computation and that they

will vary with every iteration of the Newton-Raphson algorithm. Therefore Qij

and Qji have to be recalculated after every iteration step.

When the Newton-Raphson computation is finished, the apparent susceptance of

19

3.2 TCSC

the TCSC is found by transforming (2.22):1

B =Pij

UiUj sin δij

(3.4)

The fire delay angle α can be determined from the equivalent susceptance B = f(α),

XL and XC as explained in 3.3.1.

3.2.2 Firing Angle Regulation

The angle-regulated TCSC is treated like a line with variable reactance X. After

the apparent susceptance B of the TCSC circuit is computed from (2.15), the line

impedance can be set to X = B−1. This TCSC, which is now handled like a line

with Z = jB−1 is added in the line matrix and the computation is started. The

result contains the active and reactive power values.

3.2.3 Resonance Protection and Angle Limits

As discussed in Section 2.2.2, every TCSC circuit has a resonance point where

the total impedance gets infinite high and, assuming a certain current, the voltage

between the TCSC busses gets infinite high too. To protect the TCSC elements from

overvoltage, real circuits include a Metal Oxide Varistor (MOV) in parallel to the

capacitor and the TCR. Another more preventative kind of overvoltage protection

is to avoid a certain band of the firing angle where resonance would put the circuit

at the risk of overvoltage.

For this reason the user is not allowed to set the fire delay angle in an area which

is close to the resonance angle αres, the ’safety margin’ Δα must be kept. Since the

user usually does not know the resonance angle of the circuit the fire delay angle αres

that would cause resonance is computed with (2.16). Afterwards (2.17) is checked

and, if necessary, the angle is corrected automatically:

• If αres − Δα < α ≤ αres it is set to α = αres − Δα

• If αres < α < αres + Δα it is set to α = αres + Δα

1B can also be determined from Qij or Qji. Since the defined value does not include anynumeric miscalculation, we decided for Pij .

20

3.3 Additional Implementations

These are the resonance limits for the fire delay angle. Additional limits result from

the maximum and minimum firing delay, this is

0◦ ≥ α ≥ 90◦

and corrected, if the user defines the firing angle outside this range. In order to

inform the user about any automatically modification of input data, a message is

sent to the GUI if corrections have been done.


3.3.1 Determination of the Fire Delay Angle

For P-, Q- and V-regulated devices, the equivalent susceptance is determined after

the power flow computation.

Due to (2.15) the fire delay angle α is a function of the TCR’s apparent susceptance

B, the susceptance of the reactor BL and the capacitance BC . If these three values

are known it is possible to determine α from a nonlinear implicit equation in the

form of

g(α) = 0 (3.5)

Transforming (2.15) leads to

g(α) = B −(BL


π+ BC

)= 0 (3.6)

This equation can be solved with the method of Newton-Raphson. It uses a Taylor

series approach, therefore it is necessary to find the derivation

∂g(α)

∂α=

∂

∂α

[B −

(BL


π+ BC

)](3.7)

that results in∂g(α)

∂α=

2BL

π(1 + cos 2α) (3.8)

The determination of the fire delay angle is implemented with a Newton-Raphson

algorithm in Matlab. The function is used to determine the resonance firing angle

αres before the power flow computation gets started and to determine the firing angle

after the power flow computation of P-, Q- or V-regulated devices has finished.

21


3.3.2 Incremental Changes of SVC and TCSC

Whenever the user changes only one value of the network, for example a load value,

the program does not start a completely new computation of the whole network.

Only the changed value is sent from the client to the computation engine and the

Newton-Raphson algorithm starts taking the former results as initial values. The

intention behind this procedure is to increase the computation speed, especially

when the client works in run-mode and the user changes a value by clicking on a

puls/minus-button of an element. For SVC and TCSC this incremental changes are

implemented for changes of the target values P , Q, U and α. If the control mode

of a FACTS element is changed, the power flow computation is started from the

beginning.

The implementation allows to do incremental changes in two steps:

1. Several declarations of variables which identify the changing element and the

setting of the new target value have to be done.

2. A script which searches for the changing element and replaces the old value

with the new value before it runs the load flow computation has to be called.

Appendix C Table C.1 gives detailed information about the declarations and the

changing commands for all implemented SVC and TCSC incremental changes. The

content of the declaration variables is listed in Table C.2.

3.3.3 Switching of SVC and TCSC

If an element is switched on or off in the client, the power flow has to be calcu-

lated new. The switching is implemented quite similar to the incremental changes

described in section 3.3.2.

If an element is switched not the whole new network is sent to the computation

engine. Just the information about the switched element is given to the load flow

program. If an element is switched off the current value of the element is defined

with zero and the computation starts with the old results as initial values for the

Newton-Raphson algorithm. The program does not read the new topology of the

network and the incidence matrix is the same as before. Only the given and the

wanted value change.

Since FlowDemo.net 1.0 was designed for only one element of each kind per bus and

22


the FACTS elements are treated like loads, shunts, lines or generators this switching

is not trivial to implement for the FACTS elements. The design does not allow to

switch perhaps the relating ’load’ of a SVC element if there is another ’real’ load on

the same bus.

Another way to implement switching is to delete the element’s line in the element

matrix, rebuild the computation matrices and start Newton-Raphson using a part of

the old results as initial values. After the matrices are prepared for Newton-Raphson,

it is always possible to distinguish between the element matrix rows which come from

a SVC or TCSC and the rows which represent original elements. But it is a problem

that after switching off a FACTS element and clearing the depending element row

in the element matrix, saving this information and recomputing the load flow, it

could happen that other elements are switched before the first element is switched

on again. This could cause a change of the matrix dimension.

It is generally feasible to solve the problem in question, it could be a task for the

future work.

Switching off FACTS elements is implemented in the Matlab code, but in another

way than the switching of the non-FACTS elements. Whenever a SVC or TCSC

element is switched off its relating row in the element matrix is deleted, the matrices

are prepared for the computation and Newton-Raphson starts taking the old voltage

values as start values for the new run. In Table C.3 the commands for FACTS

switching are printed, whereas the declaration variables are defined in Table C.2.

This procedure takes less iterations than a completely new computation would need

to converge. But of course a procedure which sets the current to zero instead of

changing the element matrices would be faster. Whenever a FACTS element is

switched on, the computation starts from the beginning.

3.3.4 Power Limits

Real generators are not able to deliver unlimited power. The mechanical and electri-

cal design of a machine leads to limits for active and reactive power. If a generator

reaches such a power limit, the control unit keeps the value on a constant level (limit)

independent from the power demand in the system. This fact has to be considered

and implemented in the computation software.

In the version 1.0 of FlowDemo.net these borders were checked after the Newton-

Raphson algorithm converged. If limits were violated, the element values were set

to these boundary values and the computation was started again.

23


This procedure is now integrated in the Newton-Raphson algorithm. Whenever the

absolute values of the disturbance vector are below the tolerance value the limits

are checked. If there are any violations the exceeded values are set to their limits

and the computation continues. If no limits were violated the loop ends and the

computation itself is finished.

3.3.5 Formatting of the Results

The results of the Matlab computation program are formatted and packed in a result

vector which includes all voltages, voltage angles and power values in a well defined

order. This vector is sent to the client where the results are displayed.

24

Chapter 4

Implementation of FACTS in the

Client

This Section describes the implementation of the FACTS elements in the Client

of FlowDemo.net. The new elements are represented by Java classes which are

explained in the following.

SVC and TCSC are added in the GUI as new elements. Basically, the implementa-

tion is done by adding classes for both elements to the object oriented Java code.

The new classes are Svc and Tcsc. For configuring SVC and TCSC elements the

user can open a dialog window by double-clicking on an element symbol in the edi-

tor. These dialog boxes are implemented in the classes TcscDialog and SvcDialog.

A toolbar is placed in the upper left corner of the client window and includes but-

tons for all network elements (see Figure 4.1). After clicking a button, the user can

create an element in the edit area.1 Buttons for SVC and TCSC are added in the

class Toolbar.

Figure 4.1: Toolbar with buttons for SVC and TCSC

1In this case an instance of the relating class is generated.

25

4.1 SVC in the Client


Since a Static Var Compensator is a 1-port element, the class for this element

is implemented as a subclass of the class OnePortEl which includes all common

variables and methods for 1-ports.2 The dialog window for the SVC is implemented

in the class SvcDialog.

4.1.1 Class Svc

The class Svc contains new variables which are characteristic for the SVC element,

like the fire delay angle α, the reactance of the capacitor XC and the inductor XL

and the control mode. The inheritance tree starts with the superclass element:

Element → OnePortEl → Svc

The class includes also the graphical design of the SVC element which is shown in

Figure 4.2. The black rectangle between the SVC and the bus is a switch. It can be

operated by clicking on it.

Figure 4.2: SVC symbol in different positions

4.1.2 Class SvcDialog

The class SvcDialog contains the implementation of the SVC dialog window. It is

an inheritance of the superclass Dialog.

Dialog → SvcDialog

Whenever the user clicks on a SVC symbol in the editor, a dialog window opens

where the user can specify the parameters of the device. This window, which is

divided in five major parts, is shown in Figure 4.3.

2That means Svc is a so called ’inheritance’ or ’extension’ of OnePortEl.

26


Figure 4.3: SVC dialog window

Current Values: The current values for reactive power and the fire delay angle are

printed in this field whereas a positive sign for the reactive power means inductive

consumption.

Power Limits: Positive and negative reactive power limits can be specified in this

text boxes. If the power limit is exceeded, it may happen that the desired operating

state cannot be reached. Perhaps the desired voltage can’t be obtained because the

SVC’s power limit is reached. As mentioned in Section 3.1.3, there are no power

limits for the α-regulated SVC.

Regulation: In this part the user can choose the control mode and set the target

value. Of course only the value of the selected control mode is used in the computa-

tion, the other two values can’t be set because they are results of the computation.

Elements: The values of the internal elements XC and XL can be specified in these

boxes. In the ’V’ (voltage) and ’Q’ (reactive power) control mode it might happen

that the desired operating state can not be gained with these elements. In this case

27

4.2 TCSC in the Client

the user is informed by an according warning message.

Step Value +/−: Whenever a ’+’ or ’−’ button is clicked while the client is working

in run-mode, the magnitude of the concerning value changes with this user-defined

step value.


The Thyristor Controlled Series Capacitor is a 2-port element, therefore the class

Tcsc is an extension of the superclass TwoPortEl.

4.2.1 Class Tcsc

This class is implemented with all special features of a TCSC compared with other

2-port elements like fire delay angle α, the reactance of the capacitor XC and the

inductor XL and the control mode. The inheritance tree is:

Element → TwoPortEl → Tcsc

In Figure 4.4 the graphical design of the TCSC element which is also implemented

in this class is shown.

Figure 4.4: TCSC symbol in different positions

4.2.2 Class TcscDialog

The dialog window for the TCSC was implemented in the class TcscDialog which

is an inheritance of the superclass Dialog.

Dialog → TcscDialog

28


This Dialog window consists of two layers which are pictured in Figure 4.5 and

Figure 4.6. The layer can be selected by clicking on a button at the top of the

window.

Figure 4.5: Tcsc dialog window: ’Current Values’

Figure 4.6: Tcsc dialog window: ’Control & Elements’

Current Values: Active, reactive and apparent power flow values are printed in

the upper part followed from the load value in % which is related to the maximal

apparent transmission power (specified in the text box ’Max Apparent Transmission

29


Power’). The voltage drop and the firing angle of the TCSC are printed at the

bottom of the window.

Control & Elements: In the first part of this layer the user can choose a regulation

mode and set the relating target value. Beneath text boxes for XC and XL follow.

Finally the step value for changes with the +/− buttons can be specified (see Section

4.1.2).

30

Chapter 5

Economic Generation Dispatch

The task of Economic Generation Dispatch (EGD) is to minimise the total cost of

active power generation in a network assuming that every generator has quadratic

cost curves related to its own active power generation. The loads are assumed to be

completely inflexible, i.e. of constant value and independent of the market situation.

This kind of Economic Dispatch is also called ’Lambda-Dispatch’, which is done in

electric spot markets. Inputs for the EGD are the total inflexible load and the power

limits of the generators, results are the powers of the generators and the so-called

’Market Clearing Price’ (MCP) or ’System Marginal Price’ (SMP).

The problem of EGD is based on the theory of non-linear inequality constrained

optimisation which is described in Section 5.1.1.

5.1 Mathematical Approach

In this Section the inequality constrained optimisation problem for the EGD is

formulated and an algorithm which is used to solve the problem is described.

5.1.1 Problem Formulation and Optimality Conditions

The inequality constrained optimisation problem is a special case of the equality

constrained optimisation problem. In addition to the equality constraints we get

31


new constraints expressed from inequalities [15]:

Minimise F (x) Objective function

subject to g(x) = 0 m Equality constraints

and h(x) ≤ 0 p Inequality constraints

(5.1)

From optimisation theory we know that this conditions can be fulfilled by minimis-

ing an artificial objective function, the so-called Lagrangian function L. This

function includes the objective function, the equality constraints and the inequality

constraints:

L(x) = F (x) + λTg(x) + μTh(x) (5.2)

The vectors λ and μ include the so called Lagrange multipliers which get additional

unknown variables in this formulation. Every equality constraint gi(x) is multiplied

with an element λi, every inequality constraint hi(x) is multiplied with an element

μi. The objective function and the Lagrangian function have the same argument

vector x, its dimension n is the number of variables which have to be optimised.

dim {x} = n × 1

dim {g(x)} = dim {λ} = m × 1

dim {h(x)} = dim {μ} = p × 1

(5.3)

In the optimum of the Lagrangian function (5.2) the necessary optimality conditions

of the first order, the Karush-Kuhn-Tucker (KKT) optimality conditions, are

fulfilled [15]:

∂F (x)

∂x+

(∂g(x)

∂x

)T

λ +

(∂h(x)

∂x

)T

μ = 0 (5.4)

g(x) = 0 (5.5)

diag {μ}h(x) = 0 (5.6)

h(x) ≤ 0 (5.7)

μ ≥ 0 (5.8)

This system of equality and inequality constraints can be solved with the so-called

’Unlimited Point’ algorithm [16] by transforming it to a set of equality constraints

and solving the new equation system with the Newton-Raphson method.

5.1.2 Unlimited Point Algorithm

The Unlimited Point algorithm is an integrated method to solve the KKT-system by

the use of Newton-Raphson. It is not based on a common optimisation method like

’QP’, ’LP’, ’Simplex’ or ’Interior Point’, it uses just a Newton-Raphson algorithm.

32


To get an equation system which can be solved with Newton-Raphson, the original

KKT-system has to be modified. This transformation is shortly explained in the

following paragraph.

As described in [16] the first inequality constraint (5.7) is transformed into an equal-

ity constraint by adding a positive slack variable which is equal to or greater than

zero. Together with the second inequality constraint (5.8) we can replace both

inequality constraints with one new equality constraint and reformulate the trans-

formed KKT-system:

∂

∂x

(F (x) + λTg(x) + μ2sT

h(x))

= 0 (5.9)

g(x) = 0 (5.10)

h(x) + z2r = 0 (5.11)

diag{μ2s}z2r = 0 (5.12)

This equation system can be solved with a Newton-Raphson algorithm. As men-

tioned in [16] the convergence of this method depends on the variables s and r, a

good convergence can be achieved with r = 2 and s = 1. The notations μ2s and

z2r denotes that every single element of μ and z has an exponent of 2s or 2r. This

exponent is not the exponent of the whole vector!

5.1.3 Optimisation of Total Generation Costs

The theory of KKT and the Unlimited Point algorithm can be used to find an optimal

dispatch of the generated power in a network where the total costs of the active power

generation are at a minimum. In this work the cost functions of the generators are

assumed to be polynomials of second order. Every generator is characterised by the

parameters a0,i, a1,i and a2,i:

CP (Pi) = a0,i + a1,iPi + a2,iPi2 (5.13)

The first parameter a0,i of this polynomial can be interpreted as the fixed costs of a

power generation. The second parameter a1,i represents the costs for primary energy

which are almost linear to the power output. The third parameter a2,i includes the

losses which occur in power stations and other costs which arise quadratically with

the power output. We can write the parameters of the cost functions a0,i, a1,i and

33


a2,i of all generators in a matrix:

AP =

⎡⎢⎢⎢⎢⎢⎢⎣

a0,1 a1,1 a2,1

a0,2 a1,2 a2,2

......

...

a0,n a1,n a2,n

⎤⎥⎥⎥⎥⎥⎥⎦

(5.14)

This matrix keeps all information about the cost characteristics of the generators in

the network. Furthermore we can arrange the active power values of all n generators

in a vector:

Pgen =[

P1 P2 . . . Pi . . . Pn

]T(5.15)

Now we can state the objective function as the total generation costs in the whole

network by summing up the generation costs of the generators.

F (Pgen) =n∑

i=1

AP(1,i) + AP

(2,i)Pgen(i) + AP

(3,i)Pgen(i)2 (5.16)

Since the total inflexible load keeps constant, the total generation before and after

the dispatch must be equal1, therefore we can sum up the generated powers and

state the equality constraint

P1 + P2 + . . . + Pi + . . . + Pn − P′tot = 0

n∑i=1

Pgen(i) − P

′tot = 0 (5.17)

whereas P′tot is the total generation power before the dispatch. After re-dispatching

the generation and recomputing the power flow the slack power will change due to

the changed load flow situation. For this reason the total generation power has to

be recalculated after every dispatch.

The inequality constraints come from the power limits of the generators. The max-

imum active generation power values are written in a vector

Pgen,max =[

P1,max P2,max . . . Pi,max . . . Pn,max

]T(5.18)

The power value of each generator must be lower than or equal to the maximum

value. This leads to the first kind of inequality constraints for the power output of

the generators:

Pi − Pi,max ≤ 0 ∀i = 1 . . . n

Pgen(i) − Pgen,max

(i) ≤ 0 ∀i = 1 . . . n (5.19)

1Taking network losses into account, the total generation changes with the slack power.

34


With this inequality constraint we limit the generators power values with its maxi-

mum power. Another inequality constraint comes from the minimum power output

of the generators which is assumed to be zero.

−Pi ≤ 0 ∀i = 1 . . . n

−Pgen(i) ≤ 0 ∀i = 1 . . . n (5.20)

As described in Section 5.1.2 the inequality constraints are transformed in equality

constraints by adding a positive slack variable:

Pi − Pi,max + z2ri = 0 ∀i = 1 . . . n

−Pi + z2r(n+i) ≤ 0 ∀i = 1 . . . n

Pgen(i) − Pgen,max

(i) + z2r(i) = 0 ∀i = 1 . . . n (5.21)

−Pgen(i) + z2r(n+i) ≤ 0 ∀i = 1 . . . n (5.22)

The Lagrangian-function (5.2) for the optimisation of the total costs of n generators

due to the polynomial cost functions (5.13) is

L(x) = a0,1 + a0,2 + . . . + a0,i + . . . + a0,n

+a1,1P1 + a1,2P2 + . . . + a1,iPi + . . . + a1,nPn

+a2,1P21 + a2,2P

22 + . . . + a2,iP

2i + . . . + a2,nP 2

n

+λ1

(P1 + P2 + . . . + Pi + . . . Pn − P

′tot

)+μ1 (P1 − P1,max) + μ2 (P2 − P2,max) + . . .

+μi (Pi − Pi,max) . . . + μn (Pn − Pn,max)

+μn+1 (−P1) + μn+2 (−P2) + . . .

+μn+i (−Pi) + . . . + μ2n (−Pn)

(5.23)

With (5.16), (5.17) and (5.21) the transformed KKT-system (5.9) can be set up.

The number of equations in this system of n generators is 5n + 1. As recommended

in [16] we use s = 1 and r = 2 for the exponent of the slack variables. Equation

(5.24) represents the transformed KKT-system for n generators.

Preparing the system (5.24) for a matrix-based Newton-Raphson algorithm we have

to arrange the unknown variables of the equation system in a vector x (5.25). After

the iterative computation converged, this vector contains the results.

x =[

P1 . . . Pi . . . Pn λ1 μ1 . . . μi . . . μ2n z1 . . . zi . . . z2n

]T(5.25)

For the implementation of Newton-Raphson we need the Jacobian matrix of the

equation system. It is built by partially differentiating each row of the transformed

35


a1,1 + 2a2,1P1 + λ1 + μ21 + μ2

n+1 = 0

a1,2 + 2a2,2P2 + λ1 + μ22 + μ2

n+2 = 0...

...

a1,i + 2a2,iPi + λ1 + μ2i + μ2

n+i = 0...

...

a1,n + 2a2,nPn + λn + μ2n + μ2

2n = 0

P1 + P2 + . . . + Pi + . . . + Pn − P′tot = 0

P1 − P1,max + z41 = 0

P2 − P2,max + z42 = 0...

...

Pi − Pi,max + z4i = 0...

...

Pn − Pn,max + z4n = 0

−P1 + z4n+1 = 0

−P2 + z4n+2 = 0

......

−Pi + z4n+i = 0

......

−Pn + z42n = 0

μ21z

41 = 0

μ22z

42 = 0...

...

μ2i z

4i = 0...

...

μ22nz

42n = 0

(5.24)

36

5.2 Implementation of the Economic Generation Dispatch

KKT-system (5.24). In (5.26) the Jacobian matrix is stated for a system with n = 2

generators.

J(n=2) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

2a2,1 0 1 2μ1 0 −2μ3 0 0 0 0 0

0 2a2,2 1 0 2μ2 0 −2μ4 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 4z31 0 0 0

0 1 0 0 0 0 0 0 4z32 0 0

−1 0 0 0 0 0 0 0 0 4z33 0

0 −1 0 0 0 0 0 0 0 0 4z34

0 0 0 2μ1z41 0 0 0 4μ2

1z31 0 0 0

0 0 0 0 2μ2z42 0 0 0 4μ2

2z32 0 0

0 0 0 0 0 2μ3z43 0 0 0 4μ2

3z33 0

0 0 0 0 0 0 2μ4z44 0 0 0 4μ2

4z34

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5.26)

Since (5.24) includes 5n+1 equations with 5n+1 unknown variables the dimension

of the Jacobian matrix J and the vector x is:

dim {J} = (5n + 1) × (5n + 1) (5.27)

dim {x} = (5n + 1) × 1 (5.28)

The Jacobian matrix and the right hand side of the equation system (5.24) have to

be computed in each iteration of the Newton-Raphson algorithm.

5.2 Implementation of the Economic Generation

Dispatch

The Economic Generation Dispatch (EGD) is implemented by using the mathemat-

ical approach described in the last Section. The optimisation problem is solved with

the Unlimited Point algorithm as described in Section 5.1.2. It is implemented in

Matlab.

Re-dispatching the generation affects the power flow in the network. The slack

power and the total active generated power change too, therefore re-dispatching and

recomputing have to be repeated several times until the power flow in the network

does not significantly change anymore. The procedure is repeated in a loop that

terminates if the change of the generator power values is below a defined tolerance

value (see flowchart in Appendix A). The iteration steps are:

1. The user specifies the parameters of the polynomial cost function for each

generator in the network.

2. The power flow in the network is computed.

3. The total active power generation is given by the sum of all generator power

values.

37

5.2 Implementation of the Economic Generation Dispatch

4. The total generation power is re-dispatched to the generators in the network

with respect to the power limits and the total generation costs.

5. The power flow in the network is computed again, therefore the slack power

and the total generation change.

6. If the generator power values did not change significantly after the power flow

computation the loop ends, otherwise the algorithm continues with step 3.

Usually Newton-Raphson algorithms use a well defined tolerance value as a criteria

for convergence. If all elements of the right hand side of the equation system, the so-

called disturbance vector, are below the tolerance value, the loop terminates. When

solving (5.24) with Newton-Raphson the disturbance vector is computed in every

iteration from the unknown variables in (5.25). If we consider realistic dimensions

for these variables we can see that they are placed in a wide numerical range. The

generator power values, their power limits and the total generation may vary in

dimensions of many hundred MW. The slack variables μi and zi can be very close

to zero if generators are close to their limits. Therefore the results of the last 2n

equations in (5.24) can be in a very different dimension than the values of all other

equations. If we assume Pi = 100 MW and zi = 10−8 (this is realistic if a generator

is close to a limit) we get a numerical range of 1010. Taking into account that there

is an element z4i in the Jacobian matrix, it is obvious that these circumstances are

likely to cause critical situations in the numeric calculation.

For this reason it is difficult to find a proper tolerance value. If the tolerance value

is too small, the algorithm needs many iterations to converge. If it is to large, the

algorithm converges with inaccurate results.

The solution of this problem is to scale the power values in a lower dimensions

before starting Newton-Raphson and re-scale them after the algorithm converged.

In FlowDemo.net the power values are transformed in the p.u. system.

38

Chapter 6

Program Tests

The features of FlowDemo.net 2.0 which were added in this project are tested by

comparing the power flow results with those of the software package NEPLAN [14].

Firstly, a small 3-bus network with a single SVC and TCSC element is used to test

the different regulation modes. Then a combined arrangement of SVC and TCSC is

tested in a 6-bus network which is a part of the standardised IEEE 14-bus network.

The Economic Generation Dispatch is tested with three generators in the 6-bus

network which is defined in Section 6.2.1. Different cost structures and generator

limits are specified.

With insight to the internal procedure of the computation it is also possible to com-

pare some characteristic quantities like the number of iterations until the Newton-

Raphson algorithm converges or the maximal mismatch value in addition to the

load flow result. Of course for this comparison both programs have to work with the

same parameters for the Newton-Raphson procedure, i.e. the same tolerance value,

maximal number of iterations, step size etc. Therefore in NEPLAN the ’Convergence

mismatch’ has to be set to the same value as in FlowDemo.net and ’Newton Raph-

son’ has to be specified as ’Calculation method’. With these settings NEPLAN does

not completely support the computation of networks including SVC and TCSC. If

the network contains these elements, the ’Calculation method’ should be set to ’Ex-

tended Newton Raphson’. Otherwise the user is informed in the ’Analysis’-Window,

for example ’TCSC not supported in NR and current iteration method. Choose ex-

tended NR’.

Such internal program analysis are not reported in this work, but in general we can

state that:

39

6.1 Tests with Single SVC and TCSC Element

• NEPLAN and FlowDemo.net need about the same number of iterations for simple

networks including SVC and α-regulated TCSC elements.

• NEPLAN needs significantly less iterations to converge if there are P-regulated

TCSC in the network.

The reason for the second point is the different implementation of the TCSC in

NEPLAN and FlowDemo.net. In NEPLAN the classic n-port model is included whereas

in FlowDemo.net an Injection Model is implemented. This model needs an extra

treatment in the Newton-Raphson algorithm, a so-called step size control which

leads to an increase of the iterations until the computation converges (see Section

6.4).

All test results of FlowDemo.net and NEPLAN are printed in Appendix B.


For the single SVC and TCSC tests, where only one SVC or TCSC is part of the net-

work, the 3-bus network described in the following paragraph is used. It is designed

with respect to a typical application of FACTS, the Congestion Management.

6.1.1 3-bus Test Network

The 3-bus network shown in Figure 6.1 represents a situation where FACTS can be

used to manage congestion. The line from bus 1 to bus 2 is overloaded whereas the

path from bus 1 to bus 3 and bus 2 has free transmission capacity. It is possible

to change the power flow situation by using an SVC or a TCSC on bus 3. The

parameters of this network are given in Appendix C Table C.4.

6.1.2 Single SVC Test

The SVC element is tested with all three types of regulations. Since there is only

voltage regulation and no firing angle available in NEPLAN, PQ-loads with P = 0

are used to test reactive power regulated SVC, the fire delay angle is checked by

recalculating the results.

The SVC elements are specified with XC = 1.0 p.u. and XL = 0.1 p.u. for all tests.

The test procedure starts with voltage regulation, all SVC data and some network

40


Figure 6.1: 3-bus network with congestion on line 1-2

Figure 6.2: 3-bus network with single SVC

41


values (bus voltages and angles, generated power) are recorded for different target

values V . Then the results of the SVC’s reactive power are used as desired values

for Q-regulation and the results for the fire delay angle are taken as target values

for the α-regulation. After comparing all results of V -, Q- and α-regulation we can

see if the program works correctly and estimate its numerical accuracy.

6.1.3 Single SVC Test Results

The SVC test results are printed in Appendix B Table B.1. In this Table we can

see the results of the three SVC regulation types in FlowDemo.net and the results

of the voltage regulated SVC in NEPLAN.

V-Regulation: In this regulation mode FlowDemo.net and NEPLAN give the same

results. Concerning two decimals there are no deviations. As we can see in Table

B.1 nr. 1 and nr. 8 the reactive power limits are not exceed.

Q-Regulation: Comparing the results of the reactive power regulation in Table

B.1 nr. 9–18 with the results of the voltage regulation we can see that there are

small deviations of the fire delay angle in the second decimal (nr. 11 and 15). Test

nr. 9 and nr. 19 show that the reactive power limits are not violated.

Angle-Regulation: The results of the angle-regulated SVC differ from those of

NEPLAN only in nr. 24, first decimal of the reactive power. The reason for this is

that the function which describes the FACTS’s susceptance is highly sensible to the

firing angle. A very small deviation of the firing angle can cause a big deviation of

the reactive power value.

6.1.4 Single TCSC Test

The TCSC is tested with regulation of the firing angle α and of the active trans-

mission power P . Since in NEPLAN the so called ’prevailing conduction angle’ and

a different approach for the apparent susceptance is implemented, the firing angle

results cannot be compared with those of FlowDemo.net. Sometimes a certain P is

revised by NEPLAN if it is not allowed due to certain borders and limits. These limits

can be deactivated by triggering the TCSC on the total apparent impedance Xtot.

Because one of the applications of a TCSC is reactive line compensation, the TCSC

is tested in this field of operation. Therefore another bus is inserted in the 3-bus

test network (which becomes a 4-bus network then) and the TCSC is tested with all

42

6.2 Combined SVC and TCSC Test

Figure 6.3: 3-/4-bus network with single TCSC

types of regulation. The network elements are exactly the same as used in Section

6.1.2, the values of the TCSC are XC = 0.3 p.u. and XL = 0.1 p.u.

6.1.5 Single TCSC Test Results

The TCSC test results are printed in Appendix B Table B.2, the results of FlowDemo.net

are compared with those of NEPLAN.

Angle-Regulation: The results differ only for U2 in test nr. 4 and U3 in test nr. 7.

A reason for this inaccuracy could be that the TCSC in NEPLAN is operating with

specified total impedance Xtot. This target value Xtot is calculated from the fire

delay angle, XL and XC .

Power-Regulation: Deviations of the voltage drop du are visible in test nr. 8, 9

and nr. 12, third decimal. All other results are equal with those of NEPLAN.


Beside the tests where only one FACTS element was part of the network, we need

to check the functionality of the program if more than one SVC or TCSC element is

in the network. These tests are done in a more complex 6-bus network that includes

all kinds of network elements in different regulation modes.

The 6-bus network is also used to test the incremental changes and the switching of

FACTS elements. Incremental changes are done for all SVC and TCSC elements by

clicking on the +/− buttons which cause a change of the target value. The compu-

43


tation results are verified by repeatedly setting and re-setting the target value and

checking if the result values are the same as before the change/rechange operation.

This procedure is also used to check the switching of FACTS elements.

6.2.1 6-bus Test Network

For the combined SVC and TCSC tests a 6-bus network was extracted from the

standardised IEEE 14-bus network which is defined in [13]. The 6-bus network in

Figure 6.4 includes bus 1, 2, 5, 6, 12, 13 and all elements which are connected to

them from the original 14-bus network. The network data are printed in Appendix

C Table C.5.

6.2.2 Test Arrangement

Several arrangements of SVC and TCSC are tested and compared with the results of

the same network in NEPLAN. For the network shown in Figure 6.5 the result values

are documented and printed in Appendix B. The new busses 7 and 8 are inserted

to connect the TCSC elements in series with the lines.

In this example three SVC and two TCSC elements are used in different regulation

modes. The first SVC regulates the voltage of bus 3. The second SVC on bus 4

is α-regulated, the third SVC is Q-regulated and connected to bus 6. The TCSC

between bus 7 and bus 2 is regulated on active transmission power P , the TCSC

between bus 8 and bus 5 is regulated on the firing angle α (see Table C.6).

6.2.3 Combined SVC and TCSC Test Results

The results of the combined SVC and TCSC tests are printed in Appendix B Table

B.3 and Table B.4, the target values of the FACTS are colored grey.

In Table B.3 nr. 1 and nr. 2 the power flow results of NEPLAN and FlowDemo.net for

the 6-bus network without any FACTS are compared. Then the network is computed

with the three SVC (nr. 3 and nr. 4) and later with the two TCSC elements (nr. 5

and nr. 6). Finally all FACTS are inserted in the 6-bus network (nr. 7 and nr. 8).

In Table B.4 the target value of each single SVC and TCSC element is changed and

the results of FlowDemo.net are compared with those of NEPLAN.

Since NEPLAN works with values in S.I. units one has to transform the p.u. values used

44


Figure 6.4: 6-bus test network

45


Figure 6.5: 6-/8-bus test network with SVC and TCSC elements

46

6.3 Test of Economic Generation Dispatch

in FlowDemo.net. Although the values are transformed exactly some differences in

the power flow result occur (Table B.3 nr. 1 and nr. 2).

Another possible reason for small deviations of the results is multi-rounding, i.e. some

values may be rounded more than one time and on different digits during the pro-

ceedings in the client, the server and the computation engine.

6.3 Test of Economic Generation Dispatch

The Economic Generation Dispatch is tested in the Matlab environment. The test

results are printed in Appendix B.

6.3.1 Test Arrangement

The Economic Generation Dispatch is tested with three generators in the 6-bus

network pictured in Figure 6.4. An additional generator is connected to bus 4,

it operates in PQ-mode with P = 10 MW and Q = −5 MVar. The program is

tested with different generation limits Pgen,max and different cost function factors

AP resp. CP,i. The numerical performance of the Unlimited Point algorithm is tested

by using cost function factors of very different numerical dimension (see Appendix

B Table B.5 nr. 10–12 and nr. 18–20).

6.3.2 EGD Test Results

The results of the Economic Generation Dispatch tests are printed in Appendix B

Table B.5 and B.6. The result of the ordinary power flow is Pgen = [28.2 22.0 10.0]T

MW, the dispatch starts with the total load of 60.2 MW.

The Unlimited Point algorithm does not converge in the second run of test nr. 9

and nr. 11, it does not converge at all in test nr. 7. The reason for this is the

big difference in the numerical dimension of the cost function factors (see Section

5.2). Nevertheless the dispatch leads to useful results in test nr. 9 and nr. 11. In

test nr. 23 there is no possible solution due to the specified power limits, the total

generation before the dispatch is higher than the sum of the generator power limits.

The column ’∑

CP,i’ in Table B.5 and B.6 gives the total generation costs in the

network before and after the dispatch. When calculating this value it is important to

use the same dimension of Pgen,disp as the dispatch program uses. Since the dispatch

47

6.4 Known Issue

in FlowDemo.net works with power values in p.u. these values have also to be taken

in p.u. for calculating the total costs. This means that the cost factors in the matrix

CP,i are related to power values in p.u. For the tests in Table B.5 and B.6 the base

power value is 1.0 p.u. = 100 MVA.

The column ’total save’ gives the total save of generation costs in the network. They

are computed as the relative difference of the costs before and after the dispatch in

percent:

total save =

∑CP,i before − ∑

CP,i after∑CP,i before

· 100% (6.1)

The total save is always positive except test nr. 21 and nr. 31. The reason for

this is that in these tests the generators are limited lower than they are limited for

the normal power flow computation, so that the more expensive generators have to

deliver more power as they do without any Economic Dispatch.

Of course the total save increases with the difference in the cost factors of the

generators. The more the generators differ in their cost structures the more saving

can be made using an Economic Generation Dispatch.

6.4 Known Issue

One handicap of the TCSC Injection Model implementation is the regulation of the

active transmission power P . In a certain range of the target value P the Newton-

Raphson algorithm converges to a solution which is mathematically correct but

not useful in practice. This solution represents a point on the lower branch of the

TCSC’s nose curve. As a corrective measure a step size control can be used in the

calculation of the new reactive power value after every iteration. This expands the

area where the computation leads to the desired result, but the problem still exists

for high values of P . Several measures are tried to solve the problem in question,

but unfortunately no general solution can be found with the following methods:

1. Initialising the reactive power value of the TCSC with zero and slowly increas-

ing it from iteration to iteration in order to avoid a fast decrease of the end

bus voltage.

2. Initialising and keeping the reactive power value of the TCSC zero for the first

three iterations in order to get stabile voltages and angles before the reactive

power of the TCSC influences them.

48

6.4 Known Issue

3. Re-setting the end bus voltage on a high and the reactive power on a low value

in order to constrain the result in the right area.

4. Changing the initial values of the Newton-Raphson procedure:

- for voltages and phase angles

- for currents and powers

5. Avoiding a change of the sign of the reactive power value, i.e. restrict the

reactive power value whether to be inductive or capacitive.

6. Calculating the new reactive power value only after every second iteration in

order to stabilise the voltage and phase angle values.

7. Including a step size control for the reactive power value, whereas the step size

depends on the change of the TCSC’s end-node voltage from one iteration to

the next. Three different step sizes and step values are used.

8. Using a step size control which does a backward step for all unknown values

(i.e. for the whole vector of unknowns in the Newton-Raphson loop) if the

change of the voltage from one iteration to the next is over a certain limit.

Then the iteration step is repeated with a damping factor c < 1 for the reactive

power value of the TCSC.

When implementing a step size control a lot of questions come up. A criteria which

indicates the need for a change of the step size and a step size value have to be

found. It is also possible to implement more than one step size values whereas

the step size depends on the value of a certain criteria. Another possibility is to

directly determine the step size from an empirical formula which is a function of

state variables, e.g. voltage drop, active and reactive power values.

The success of such a method depends very much on the chosen criteria and step

size value. Only a small change of the step size value can completely change the

result of the computation. Therefore it is not sure that all mentioned methods fail

in every case, it is still possible that a certain criteria in combination with a certain

step size leads to the desired computation result.

49

Chapter 7

Conclusion

This report documents the implementation of two types of FACTS devices and an

Economic Generation Dispatch in the power flow simulation platform FlowDemo.net.

The conclusion will give a brief summary of the work undertaken and then discuss

possible improvements and future work.

7.1 Project Summary

In the first step of this work the existing version of FlowDemo.net has been analysed

and the possibilities of implementing FACTS have been evaluated.

After deciding for the Static Var Compensator (SVC) and the Thyristor Controlled

Series Capacitor (TCSC), these elements were modelled mathematically and a con-

cept for the embedding in the power flow computation was worked out. Firstly, the

SVC was implemented with regulation on reactive power, node voltage and fire delay

angle. Secondly, the TCSC was implemented with regulation on active transmission

power and fire delay angle.

The implementations in Matlab were completed with some additional features like

incremental changes and switching procedures. Problems occurred with the switch-

ing of SVC and TCSC elements and with the computation of P -regulated TCSC.

After the implementations in the power flow computation were finished, the SVC and

TCSC elements were added to the GUI. New classes and dialog boxes for SVC and

TCSC elements were included in the object-orientated Java code and an interface

to the Matlab computation engine was defined.

50

7.2 Discussion

The first part of this work was completed by doing several tests and comparing the

results with those of software package NEPLAN.

The second part of the project was the implementation of an Economic Generation

Dispatch. The target of this procedure is to minimise the total generation costs in the

network. This task is stated as an inequality constrained optimisation problem due

to Karush-Kuhn-Tucker, the problem was solved using an Unlimited Point algorithm

which was implemented in Matlab. The Economic Generation Dispatch was tested

out and its numerical performance was evaluated.

7.2 Discussion

Implemented FACTS Models

As mentioned in Chapter 2 an Injection Model of the SVC and the TCSC is im-

plemented for regulation of reactive and active power. The major advantage of this

model is that the implementation can be easily based on PQ-elements which are

usually available in power flow software. The new FACTS elements do not need to

be included in the Jacobian matrix which therefore stays simple.

The Injection Model seems not to be very convenient for the implementation of a

P-regulated TCSC (see Section 6.4). An implementation of the classic n-port model

(as described in [10]) would probably not cause such difficulties. A step size con-

trol for the Injection Model implementation opens almost endless opportunities of

possible criteria and step size values. Since an implementation of a step size control

improved already the performance of the computation we assume that it is feasible

to solve the problem in this way. The point is to find an efficient combination of a

significant criteria that indicates the need for a change of the step size and a proper

step size value.

Economic Generation Dispatch

The Economic Generation Dispatch, also called ’Lambda Dispatch’, is working cor-

rectly when using realistic values for the cost function factors. If they are placed in

very different numerical ranges it could happen that the Unlimited Point algorithm

does not converge.

51

7.3 Outlook

7.3 Outlook

An element which is missing in FlowDemo.net but available in common power flow

software is the infinite bus (feeder). A simple implementation for power flow studies

could be based on a PV- or slack-generator without reactive power limits. Further-

more one-sided switching of 2-ports is not implemented yet. This feature could be

implemented like the two-sided switching, but only the current values of the opened

end have to be set to zero.

As mentioned in Section 3.3.3 the switching on procedures of SVC and TCSC ele-

ments are not implemented satisfactorily yet. Also the current version of the switch-

ing off procedures, which is not very fast yet, could be improved.

The implementation of the TCSC could be extended with regulation on the TCSC

current. One possibility to realise this is the use of the Injection Model quite similar

to the P -regulation.

FACTS for series and parallel compensation are now implemented, the Unified Power

Flow Controller (UPFC), which represents combined series and parallel compensa-

tion, could be a further development. Since the UPFC is a more complex element

than SVC and TCSC and due to the computation problems that occurred using

the Injection Model for the P -regulated TCSC, we would advice to implement the

classic n-port model of the UPFC.

In terms of optimisation Optimal Power Flow (OPF) would be an interesting new

feature of FlowDemo.net. Therefore elementary and extensive considerations whether

it is possible to implement this with the current software design or not have to be

done.

The Economic Generation Dispatch is now implemented as a so-called Lambda

Dispatch with total inflexible load. The implementation could be enhanced with

flexible loads, i.e. loads which change their values depending on the market situation.

This would introduce aspects of an electricity market in FlowDemo.net.

52

Appendix A

Flowcharts

53

Flowcharts

Qiset by user

Add SVC in load matrix

Start

End

Regulation ?

aiset by user

Compute Xi = f(XL, XC, ai)

Add SVC in line matrix

Newton-Raphson

Extract SVCs fromlines-matrix

Viset by user

Add SVC in gen matrix

Compute susceptanceB = f(Q, Ui)

Extract SVCs fromgen-matrix

Compute firing anglea = f(B, XL, XC)

Extract SVCs fromload-matrix

Compute susceptanceB = f(Q, Ui)


Figure A.1: SVC flow chart

54

Flowcharts

Pijset by user

Add TCSC-loads in loadmatrix

Pij + Qij on bus i-Pij+Qji on bus j

Start

End

Extract TCSC-loadsfrom load-matrix

Regulation ?

aiset by user

Compute Xi = f(ai)

Add TCSC-line in line matrixwith Zline,i = jXi

Newton-Raphson

Extract TCSC-linesfrom lines-matrix


Iteration stepNewton-Raphson

(new Ui, Uj, d)

Compute new reactive powervalues

Qij = f(Pij, Ui, Uj, d)Qji = f(Pij, Ui, Uj, d)

NR convergedOR

max. iterations

Compute susceptanceB = f(Pij, Ui, Uj, d)

Figure A.2: TCSC flow chart

55

Flowcharts

Start

a0, a1, a2 andPgen,max set by

user

Unlimited Point

Solution?

Power flowcomputation

Pgen,old

Power flowcomputation

Pgen

| Pgen - Pgen,old | < ed

ORmax. repeats

End

Pgen,old = Pgen

no

yes

no

yes

Pgen = Pgen,old

Figure A.3: Economic Generation Dispatch flow chart

56

Appendix B

Test Data

57

Test Data

FlowDemo.net

V-RegulationV QSVC a U2 q2 U3 q3 PG QG

p.u. MVar deg p.u. deg p.u. deg MW MVar1 0.90 50.0 62.62 0.96 -5.42 0.95 -2.39 125.6 112.5 *2 0.95 48.1 62.10 0.96 -5.43 0.95 -2.43 125.5 110.13 0.97 34.4 58.99 0.97 -5.46 0.97 -2.67 124.8 93.64 0.99 20.1 56.46 0.98 -5.50 0.99 -2.92 124.3 77.15 1.01 5.1 54.30 0.99 -5.54 1.01 -3.16 123.9 60.66 1.03 -10.4 52.42 1.00 -5.59 1.03 -3.40 123.6 44.17 1.05 -26.7 50.75 1.00 -5.64 1.05 -3.65 123.6 27.68 1.10 -50.0 48.72 1.02 -5.71 1.08 -3.99 123.7 4.9 *

Q-RegulationQ QSVC a U2 q2 U3 q3 PG QG

p.u. MVar deg p.u. deg p.u. deg MW MVar9 60.0 50.0 62.62 0.96 -5.42 0.95 -2.39 125.6 112.5 *

10 50.0 50.0 62.62 0.96 -5.42 0.95 -2.39 125.6 112.511 48.1 48.1 62.11 0.96 -5.43 0.95 -2.43 125.5 110.112 34.4 34.4 58.99 0.97 -5.46 0.97 -2.67 124.8 93.613 20.1 20.1 56.46 0.98 -5.50 0.99 -2.92 124.3 77.114 5.1 5.1 54.30 0.99 -5.54 1.01 -3.16 123.9 60.615 -10.4 -10.4 52.43 1.00 -5.59 1.03 -3.40 123.6 44.116 -26.7 -26.7 50.75 1.00 -5.64 1.05 -3.65 123.6 27.617 -50.0 -50.0 48.72 1.02 -5.71 1.08 -3.99 123.7 4.918 -60.0 -50.0 48.72 1.02 -5.71 1.08 -3.99 123.7 4.9 *

a QSVC U2 q2 U3 q3 PG QG

deg MVar p.u. deg p.u. deg MW MVar19 62.62 50.0 0.96 -5.42 0.95 -2.39 125.6 112.520 62.62 48.1 0.96 -5.43 0.95 -2.43 125.5 110.121 62.11 34.4 0.97 -5.46 0.97 -2.67 124.8 93.622 58.99 20.1 0.98 -5.50 0.99 -2.92 124.3 77.123 56.46 5.1 0.99 -5.54 1.01 -3.16 123.9 60.624 54.30 -10.5 1.00 -5.59 1.03 -3.40 123.6 44.125 52.43 -26.7 1.00 -5.64 1.05 -3.65 123.6 27.626 50.75 -50.0 1.02 -5.71 1.08 -3.99 123.7 4.9

* power limit exceed

NEPLAN

V-RegulationV QSVC a U2 q2 U3 q3 PG QG

p.u. MVar deg p.u. deg p.u. deg MW MVar27 0.90 50.0 62.62 0.96 -5.42 0.95 -2.39 125.6 112.528 0.95 48.1 62.10 0.96 -5.43 0.95 -2.43 125.5 110.229 0.97 34.4 58.99 0.97 -5.46 0.97 -2.67 124.8 93.630 0.99 20.1 56.46 0.98 -5.50 0.99 -2.92 124.3 77.131 1.01 5.1 54.30 0.99 -5.54 1.01 -3.16 123.9 60.632 1.03 -10.4 52.42 1.00 -5.59 1.03 -3.40 123.6 44.133 1.05 -26.7 50.75 1.00 -5.64 1.05 -3.65 123.6 27.634 1.10 -50.0 48.72 1.02 -5.71 1.08 -3.99 123.7 4.9

Nr.

Nr.

Nr.

Nr.

a-Regulation

Table B.1: SVC test data

58

Test Data

FlowDemo.net

a-Regulationa U2 q2 U3 q3 U4 q4 P34 Q34 Q43 du PG QG

deg p.u. deg p.u. deg p.u. deg MW MVar MVar p.u. MW MVar1 0.00 0.99 -5.98 1.02 -2.41 1.01 -4.21 21.5 10.1 -9.3 0.0349 123.9 56.22 15.00 0.98 -6.25 1.03 -1.93 1.00 -4.80 17.6 9.2 -8.1 0.0567 124.0 57.03 30.00 0.97 -7.01 1.05 -0.62 0.98 -6.51 6.1 4.3 -3.5 0.1237 124.6 59.34 45.00 0.82 -8.77 1.31 -1.56 0.66 -8.90 -16.6 -129.2 63.5 0.6585 145.1 744.65 60.00 0.99 -2.08 1.05 -10.46 1.01 5.18 78.5 -21.8 -0.2 0.2822 127.4 47.76 75.00 0.99 -3.11 1.02 -8.33 1.03 2.50 64.3 -4.7 -7.5 0.1935 125.4 49.57 90.00 1.00 -3.22 1.02 -8.10 1.03 2.24 62.7 -3.3 -8.0 0.1846 125.3 49.7

P-RegulationP U2 q2 U3 q3 U4 q4 a Q34 Q43 du PG QG

MW p.u. deg p.u. deg p.u. deg deg MVar MVar p.u. MW MVar8 -10.0 0.95 -8.11 1.09 0.86 0.93 -8.95 37.53 -10.9 7.7 0.2350 126.6 63.39 -5.0 0.96 -7.76 1.07 0.46 0.95 -8.18 35.77 -4.8 3.5 0.1968 125.8 61.9

10 -1.0 0.96 -7.49 1.06 0.10 0.96 -7.57 34.07 -0.9 0.6 0.1690 125.3 60.911 0.0 0.96 -7.42 1.06 0.00 0.96 -7.42 33.59 0.0 0.0 0.1623 125.2 60.712 1.0 0.97 -7.35 1.06 -0.10 0.97 -7.27 33.09 0.8 -0.6 0.1557 125.1 60.413 5.0 0.97 -7.08 1.05 -0.50 0.98 -6.67 30.67 3.7 -2.9 0.1303 124.7 59.514 10.0 0.98 -6.75 1.04 -1.05 0.99 -5.93 26.70 6.5 -5.3 0.1001 124.3 58.5

NEPLAN

X-RegulationX U2 q2 U3 q3 U4 q4 P34 Q34 Q43 du PG QG

Ohm p.u. deg p.u. deg p.u. deg MW MVar MVar p.u. MW MVar15 1210.0 0.99 -5.98 1.02 -2.41 1.01 -4.21 21.5 10.1 -9.3 0.0349 123.9 56.216 1317.3 0.98 -6.25 1.03 -1.93 1.00 -4.80 17.6 9.2 -8.1 0.0567 124.0 57.017 1427.3 0.97 -7.01 1.05 -0.62 0.98 -6.51 6.1 4.3 -3.5 0.1237 124.6 59.318 1521.2 0.82 -8.77 1.30 -1.56 0.66 -8.90 -16.6 -129.2 63.5 0.6585 145.1 744.619 1582.9 0.99 -2.08 1.05 -10.46 1.01 5.18 78.5 -21.8 -0.2 0.2822 127.4 47.720 1609.3 0.99 -3.11 1.02 -8.33 1.03 2.50 64.3 -4.7 -7.5 0.1935 125.4 49.521 1613.3 0.99 -3.22 1.02 -8.10 1.03 2.24 62.7 -3.3 -8.0 0.1846 125.3 49.7

P-RegulationP U2 q2 U3 q3 U4 q4 a Q34 Q43 du PG QG

MW p.u. deg p.u. deg p.u. deg deg MVar MVar p.u. MW MVar22 -10.0 0.95 -8.11 1.09 0.86 0.93 -8.95 -- -10.9 7.7 0.2349 126.6 63.3 *23 -5.0 0.96 -7.76 1.07 0.46 0.95 -8.18 -- -4.8 3.5 0.1969 125.8 61.9 *24 -1.0 0.96 -7.49 1.06 0.10 0.96 -7.57 -- -0.9 0.6 0.1690 125.3 60.9 *25 0.0 0.96 -7.42 1.06 0.00 0.96 -7.42 -- 0.0 0.0 0.1623 125.2 60.7 *26 1.0 0.97 -7.35 1.06 -0.10 0.97 -7.27 -- 0.8 -0.6 0.1558 125.1 60.4 *27 5.0 0.97 -7.08 1.05 -0.50 0.98 -6.67 -- 3.7 -2.9 0.1303 124.7 59.5 *28 10.0 0.98 -6.75 1.04 -1.05 0.99 -5.93 -- 6.5 -5.3 0.1001 124.3 58.5

* not allowed in Neplan

Nr.

Nr.

Nr.

Nr.

Table B.2: TCSC test data

59

Test Data

U1

q 1U

2q 2

U3

q 3U

4q 4

U5

q 5U

6q 6

PG

1Q

G1

PG

2Q

G2

p.u.

deg

p.u.

deg

p.u.

deg

p.u.

deg

p.u.

deg

p.u.

deg

MW

MV

arM

WM

Var

1NEPLAN

1.04

00.

000

1.04

5-0

.570

1.04

7-1

.440

1.07

0-2

.580

1.06

9-3

.000

1.06

0-1

.630

38.7

0-4

0.35

22.0

025

.92

2FlowDemo.net

1.04

00.

000

1.04

5-0

.570

1.04

6-1

.440

1.07

0-2

.580

1.06

9-3

.000

1.06

0-1

.630

38.7

0-4

0.40

22.0

025

.90

wit

h t

hre

e S

VC

3NEPLAN

1.04

00.

000

1.04

5-0

.530

1.05

0-1

.800

1.07

0-2

.930

1.06

9-3

.350

1.06

0-1

.980

39.2

2-4

1.91

22.0

05.

92

4FlowDemo.net

1.04

00.

000

1.04

5-0

.530

1.05

0-1

.800

1.07

0-2

.930

1.06

9-3

.350

1.06

0-1

.980

39.2

0-4

1.90

22.0

05.

90

wit

h t

wo

TC

SC

5NEPLAN

1.04

00.

000

1.04

5-1

.530

1.04

9-2

.230

1.07

0-3

.350

1.06

7-3

.560

1.06

02.

320

38.9

9-4

1.64

22.0

024

.44

6FlowDemo.net

1.04

00.

000

1.04

5-1

.530

1.04

9-2

.230

1.07

0-3

.350

1.06

7-3

.560

1.06

02.

320

39.0

0-4

1.00

22.0

024

.40

wit

h a

ll F

AC

TS

7NEPLAN

1.04

00.

000

1.04

5-1

.510

1.05

0-2

.310

1.07

0-3

.430

1.06

7-3

.640

1.06

0-2

.400

39.0

8-4

2.00

22.0

04.

448

FlowDemo.net

1.04

00.

000

1.04

5-1

.510

1.05

0-2

.310

1.07

0-3

.430

1.06

7-3

.640

1.06

0-2

.400

39.1

0-4

1.90

22.0

04.

40

Qa

Qa

Qa

P72

Q72

Q27

dua

P85

Q85

Q58

dua

MV

arde

gM

Var

deg

MV

arde

gM

Var

MV

arM

Var

p.u.

deg

MV

arM

Var

MV

arp.

u.de

g

wit

h t

hre

e S

VC

3NEPLAN

-17.

56--

39.0

2--

20.0

0--

4FlowDemo.net

-17.

5624

.23

39.0

015

.00

20.0

047

.48

wit

h t

wo

TC

SC

5NEPLAN

20.0

0-0

.01

-0.7

30.

0386

141.

39-6

.60

3.72

-3.7

70.

0076

160.

26

6FlowDemo.net

20.0

00.

00-0

.10

0.03

8736

.93

-6.6

03.

70-3

.80

0.00

7670

.00

wit

h a

ll F

AC

TS

7NEPLAN

-4.2

7--

30.0

2--

20.0

0--

20.0

00.

00-0

.73

0.03

8314

1.49

-6.6

13.

72-3

.77

0.00

7616

0.26

8FlowDemo.net

-4.3

031

.15

30.0

015

.00

20.0

047

.48

20.0

0-0

.10

-0.6

00.

0383

36.9

7-6

.60

3.70

-3.8

00.

0076

70.0

0

SV

C b

us 6

TC

SC

bus

7 -

bus

2T

CS

C b

us 8

- b

us 5

Nr.

SV

C b

us 4

SV

C b

us 3

Nr.

wit

ho

ut

FA

CT

S

Net

wo

rk v

alu

es

FA

CT

S v

alu

es

Table B.3: Combined SVC and TCSC test data

60

Test Data

U1

q 1U

2q 2

U3

q 3U

4q 4

U5

q 5U

6q 6

PG

1Q

G1

PG

2Q

G2

p.u.

deg

p.u.

deg

p.u.

deg

p.u.

deg

p.u.

deg

p.u.

deg

MW

MV

arM

WM

Var

8NEPLAN

1.04

00.

000

1.04

5-1

.720

1.04

0-1

.390

1.07

0-2

.530

1.06

7-2

.730

1.06

0-1

.500

38.7

9-3

7.86

22.0

04.

449

FlowDemo.net

1.04

00.

000

1.04

5-1

.720

1.04

0-1

.390

1.07

0-2

.530

1.06

7-2

.730

1.06

0-1

.500

38.8

0-3

7.70

22.0

04.

40

10NEPLAN

1.04

00.

000

1.04

5-1

.720

1.04

0-1

.390

1.07

0-2

.530

1.06

7-2

.730

1.06

0-1

.500

38.7

9-3

7.86

22.0

04.

4411

FlowDemo.net

1.04

00.

000

1.04

5-1

.720

1.04

0-1

.390

1.07

0-2

.530

1.06

7-2

.730

1.06

0-1

.500

38.8

0-3

7.10

22.0

04.

40

12NEPLAN

1.04

00.

000

1.04

5-1

.720

1.04

0-1

.390

1.07

0-2

.530

1.06

7-2

.730

1.06

0-1

.500

38.7

9-3

7.86

22.0

05.

5613

FlowDemo.net

1.04

00.

000

1.04

5-1

.720

1.04

0-1

.390

1.07

0-2

.530

1.06

7-2

.730

1.06

0-1

.500

38.8

0-3

7.10

22.0

05.

60

14NEPLAN

1.04

00.

000

1.04

5-0

.810

1.04

0-0

.900

1.07

0-2

.040

1.06

7-2

.240

1.06

0-1

.000

38.2

7-3

2.19

22.0

05.

5615

FlowDemo.net

1.04

00.

000

1.04

5-0

.810

1.04

0-0

.900

1.07

0-2

.040

1.06

7-2

.240

1.06

0-1

.000

38.3

0-3

2.10

22.0

05.

60

16NEPLAN

1.04

00.

000

1.04

5-0

.810

1.04

0-0

.900

1.07

0-2

.020

1.06

3-2

.100

1.06

0-0

.980

38.2

0-3

2.18

22.0

08.

7717

FlowDemo.net

1.04

00.

000

1.04

5-0

.810

1.04

0-0

.900

1.07

0-2

.020

1.06

3-2

.100

1.06

0-0

.980

38.2

0-3

2.10

22.0

08.

80

Qa

Qa

Qa

P72

Q72

Q27

dua

P85

Q85

Q58

dua

MV

arde

gM

Var

deg

MV

arde

gM

Var

MV

arM

Var

p.u.

deg

MV

arM

Var

MV

arp.

u.de

g

8NEPLAN

45.5

7--

39.0

2--

20.0

0--

20.0

00.

00-0

.73

0.04

2214

0.33

-6.6

13.

72-3

.77

0.07

5516

0.26

9FlowDemo.net

45.6

090

.00*

39.0

015

.00

20.0

047

.48

20.0

0-0

.10

-0.6

00.

0422

36.6

5-6

.60

3.70

-3.8

00.

0076

70.0

0

10NEPLAN

45.5

7--

-6.6

0--

20.0

0--

20.0

00.

740.

070.

0422

140.

33-6

.61

3.72

-3.7

70.

0755

160.

2611

FlowDemo.net

45.6

090

.00*

-6.6

030

.00

20.0

047

.48

20.0

0-0

.10

0.10

0.04

2336

.64

-6.6

03.

70-3

.80

0.00

7670

.00

12NEPLAN

45.5

7--

-6.6

0--

30.0

0--

20.0

00.

740.

070.

0422

140.

33-6

.61

3.72

-3.7

70.

0755

160.

2613

FlowDemo.net

45.6

090

.00*

-6.6

030

.00

30.0

058

.51

20.0

0-0

.10

0.10

0.04

2336

.64

-6.6

03.

70-3

.80

0.00

7670

.00

14NEPLAN

33.8

8--

-6.6

0--

30.0

0--

-10.

003.

85-3

.94

0.00

8510

8.80

-6.6

13.

72-3

.77

0.00

7616

0.26

15FlowDemo.net

33.9

069

.09

-6.6

030

.00

30.0

058

.51

-10.

003.

90-3

.80

0.00

8526

.30

-6.6

03.

70-3

.80

0.00

7670

.00

16NEPLAN

33.9

8--

-6.6

0--

30.0

0--

-10.

003.

86-3

.95

0.00

8410

8.75

-6.7

06.

93-7

.08

0.01

6114

3.07

17FlowDemo.net

34.0

069

.43

-6.6

030

.00

30.0

058

.51

-10.

003.

90-3

.80

0.00

8526

.28

-6.7

06.

90-7

.10

0.01

6150

.00

* fir

e de

lay

angl

e ou

t of

ran

ge

TC

SC

bus

8 -

bus

5

Net

wo

rk v

alu

es

Nr.

Nr.

FA

CT

S v

alu

esS

VC

bus

3S

VC

bus

4S

VC

bus

6T

CS

C b

us 7

- b

us 2

Table B.4: Combined SVC and TCSC test data

61

Test Data

CP,i Pgen,max Pgen,disp Ptotal total save

a0,i a1,i a2,i MW MW MW before after %1 1 1 100 20.051 1 1 100 20.05 2 10 / 10 60.15 3.74 3.72 0.481 1 1 100 20.051 0 1 100 53.461 1 1 100 3.46 2 12 / 12 60.38 3.46 3.36 2.911 1 1 100 3.461 1 1 100 3.751 0 1 100 53.73 2 12 / 12 61.21 3.52 3.37 4.361 1 1 100 3.731 1 1 100 3.361 1 1 100 3.36 2 12 / 12 60.08 3.64 3.35 7.851 0 1 100 53.361 1 1 100 55.211 2 1 100 5.21 2 11 / 11 60.42 4.16 3.96 4.711 3 1 100 0.001 3 1 100 0.001 2 1 100 5.06 2 11 / 11 60.12 4.52 3.96 12.521 1 1 100 55.061 1 1 100 28.201 10 1 100 22.00 1 not conv. 60.20 15.62 15.62 0.001 100 1 100 10.001 1 1 100 60.561 10 1 100 0.00 2 26 / 27 60.56 7.62 3.97 47.871 20 1 100 0.001 20 1 100 -0.081 10 1 100 0.00 2 26 / not conv. 60.12 11.08 3.95 64.361 1 1 100 60.201 10 1 100 0.001 1 1 100 30.21 2 9 / 9 60.42 6.28 3.79 39.681 1 1 100 30.211 100 1 100 0.221 1 1 100 30.10 2 29 / not conv. 60.42 31.66 4.00 87.351 1 1 100 30.101 1000 1 100 0.001 1 1 100 30.21 2 31 / 31 60.42 285.46 3.79 98.671 1 1 100 30.211 1 0 100 60.511 1 1 100 0.03 2 15 / 15 60.57 3.66 3.61 1.491 1 1 100 0.031 1 1 100 0.031 1 0 100 61.58 3 15 / 15 / 15 61.64 3.69 3.62 2.041 1 1 100 0.031 1 1 100 0.031 1 1 100 0.03 2 15 / 15 60.13 3.73 3.60 3.451 1 0 100 60.071 1 1 100 32.791 1 2 100 16.42 1 10 60.16 3.81 3.80 0.241 1 3 100 10.951 1 3 100 10.931 1 2 100 16.39 2 10 / 10 60.10 3.95 3.80 3.781 1 1 100 32.781 1 1 100 54.421 1 10 100 5.44 2 13 /13 60.40 5.17 3.93 23.871 1 100 100 0.541 1 100 100 0.541 1 10 100 5.42 2 13 / 13 60.11 12.05 3.93 67.411 1 1 100 54.151 1 1000 100 0.061 1 100 100 0.59 2 15 / 15 60.12 87.98 3.96 95.501 1 1 100 59.47

19

20

1

15

16

17

18

11

12

13

14

7

8

9

10

3

4

5

6

SCP,i

2

Nr. repeatsUnlim. Point

iterations

Table B.5: Economic Generation Dispatch test data

62

Test Data

CP,i Pgen,max Pgen,disp Ptotal total save

a0,i a1,i a2,i MW MW MW before after %

1 1 1 100 20.051 1 1 100 20.05 2 10 / 10 60.15 3.74 3.72 0.481 1 1 100 20.051 1 1 100 30.181 1 1 20 20.00 1 11 60.18 3.74 3.74 -0.081 1 1 10 10.001 1 1 10 9.951 1 1 20 20.00 1 11 59.95 3.74 3.74 0.011 1 1 100 30.001 1 1 10 28.201 1 1 10 22.00 1 no solution 60.20 3.74 3.74 0.001 1 1 10 10.00

1 1 1 100 56.981 2 2 100 3.49 2 12 / 12 60.47 4.23 3.97 6.191 3 3 100 0.001 1 3 100 33.691 2 2 100 25.54 2 13 / 13 60.31 4.37 4.35 0.371 3 1 100 1.081 1 2 100 46.181 2 3 100 14.12 2 11 / 11 60.30 4.34 4.23 2.441 3 1 100 0.001 9 2 100 33.781 10 1 100 17.60 1 11 60.18 8.97 8.95 0.121 10 2 100 8.801 9 2 100 35.061 10 1 15 15.00 1 12 60.16 8.97 8.95 0.131 10 2 15 10.101 9 0.2 100 60.561 10 0.1 15 0.00 2 10 / 10 60.56 8.76 8.52 2.711 10 0.2 15 0.001 9 0.2 40 40.001 10 0.1 15 13.47 1 12 60.20 8.76 8.65 1.211 10 0.2 15 6.731 9 0.2 10 10.011 10 0.1 40 35.48 2 13 / 13 60.49 8.76 8.97 -2.371 10 0.2 15 15.00

1 0.700 0.035 1 0.011 0.040 0.002 100 61.63 3 12 / 12 / 12 61.64 3.22 3.03 6.181 0.150 0.070 5 0.001 0.032 0.001 20 20.001 0.029 0.001 30 30.00 2 14 / 14 60.33 3.02 3.02 0.011 0.034 2E-04 40 10.331 3.200 0.100 20 20.001 2.900 0.100 30 30.00 2 11 / 11 60.33 4.89 4.87 0.391 3.400 0.020 40 10.331 3.200 0.100 40 30.381 2.900 0.100 30 30.00 2 11 / 11 60.38 4.89 4.86 0.671 3.400 0.020 20 0.001 3.14 0.12 100 0.011 2.98 0.18 100 36.87 2 13 / 13 60.61 4.87 4.86 0.211 3.07 0.09 100 23.731 3.14 0.12 50 10.341 2.98 0.18 30 30.00 2 12 / 12 60.34 4.87 4.85 0.281 3.07 0.09 20 20.00

32

37

33

34

35

36

28

29

30

31

24

25

26

27

1

21

22

23

SCP,iNr. repeatsUnlim. Point

iterations

Table B.6: Economic Generation Dispatch test data

63

Appendix C

Variables, Scripts and Functions

Element Changing value Declarations Command/Script

TCSC Pij = −Pji first{user} changeTcscP.m

second{user}

num{user}

newP{user}

TCSC α first{user} changeTcscAngle.m

second{user}

num{user}

newAngle{user}

SVC Qi bus{user} changeSvcQ.m

newQ{user}

num{user}

SVC Vi bus{user} changeSvcV.m

newV{user}

num{user}

SVC α bus{user} changeSvcAngle.m

newAngle{user}

num{user}

Table C.1: Declarations and commands for incremental changes

64


Variable Content

first{user} start bus nr. of TCSC

second{user} end bus nr. of TCSC

num{user} linear row index in element matrix

bus{user} connected bus nr. of SVC

newP{user} new desired active power transmission

newQ{user} new desired reactive power value

newV{user} new desired bus voltage

newAngle{user} new desired fire delay angle

Table C.2: Content of declaration variables

Element Switching Declarations Command/Script

TCSC off first{user} tcscOff.m

second{user}

num{user}

TCSC on (all) start.m

SVC off bus{user} svcOff.m

num{user}

SVC on (all) start.m

Table C.3: Declarations and commands for switching

Element Bus Values

Generator 1 slack, Utarget = 1.06 p.u.

Load 3 P = 120 MW, Q = 40 MVar

Line 1-2 X = 0.12 p.u., R = 0.03 p.u., G = 0.00 p.u.

1-3 X = 0.24 p.u., R = 0.06 p.u., G = 0.00 p.u.

2-3 X = 0.16 p.u., R = 0.04 p.u., G = 0.00 p.u.

Table C.4: 3-bus test network data

65


Element Bus Values

SVC (all) XC = 1.0 p.u., XL = 0.1 p.u.

TCSC (all) XC = 0.3 p.u., XL = 0.1 p.u.

Generator 1 slack, Utarget = 1.040 p.u.

2 PV, P = 22.0 MW, Utarget = 1.06 p.u.

Load 2 P = 21.7 MW, Q = 12.7 MVar

3 P = 7.6 MW, Q = 1.6 MVar

4 P = 11.2 MW, Q = 7.5 MVar

5 P = 13.5 MW, Q = 5.8 MVar

Shunt 3 G = 0.00 p.u., B = 0.19 p.u.

5 G = 0.05 p.u., B = 0.19 p.u.

Capacitor 2 PV, Utarget = 1.045 p.u., Qmax/min = 500/-400 MVar

4 PV, Utarget = 1.070 p.u., Qmax/min = 240/-100 MVar

Line 1-2 R = 0.01938 p.u., X = 0.05917 p.u., B = 0.05280 p.u.

1-3 R = 0.05403 p.u., X = 0.22304 p.u., B = 0.04920 p.u.

2-3 R = 0.05695 p.u., X = 0.17388 p.u., B = 0.03460 p.u.

4-5 R = 0.06615 p.u., X = 0.13027 p.u., B = 0.00000 p.u.

4-6 R = 0.06615 p.u., X = 0.13027 p.u., B = 0.00000 p.u.

5-6 R = 0.06615 p.u., X = 0.13027 p.u., B = 0.00000 p.u.

Trafo 3-4 R = 0.00000 p.u., X = 0.25202 p.u., B = 0.00000 p.u.,

turns ratio = 1.00 p.u.

Table C.5: 6-bus test network data

Element Bus Regulation Values

SVC 6 Q XC = 3.00 p.u., XL = 1.00 p.u.

3 V XC = 3.00 p.u., XL = 1.00 p.u.

4 α XC = 3.00 p.u., XL = 1.00 p.u.

TCSC 7-2 P XC = 0.03 p.u., XL = 0.01 p.u.

8-5 α XC = 0.10 p.u., XL = 0.03 p.u.

Table C.6: Values of FACTS in the 6-bus test network

66


Filename Type Call

dispatch.m Script dispatch

kktDispatch.m Function [P nport gen disp,price,kkt msg] =

kktDispatch(Ap,P nport gen,Pmax)

kktRHS.m Function RHS = kktRHS(x,Ap,P nport gen,Pmax)

kktJacobian.m Function J = kktJacobian(x,Ap,P nport gen,Pmax)

Table C.7: Scripts and functions for Economic Generation Dispatch

Variable Content

P nport gen disp vector of dispatched generator power values in p.u.

price Lagrangian-multiplier λ1 of equality constraint g1(x)

kkt msg string message, information about computation

x vector of unknown values (defined in 5.25)

Ap matrix of cost function factors (defined in 5.14)

P nport gen vector of generator power values before dispatch in p.u.

P max vector of maximal generator power values in p.u.

Table C.8: Variables for Economic Generation Dispatch

67

Bibliography

[1] FlowDemo.net: ’Interactive, internet based application to visualize power flow

in an electric transmission system’, EEH – Power Systems Laboratory, ETH

Zurich, 2000.

Available from www.flowdemo.net

[2] S. Baud: ’FlowDemo.NET: Interaktive Webanwendung fur die Demonstration

der Lastflussrechnung und -steuerung’, Diplomarbeit EUS0002, EEH – Power

Systems Laboratory, ETH Zurich, 2001.

[3] R. Bacher: ’Modellierung und Analyse Elektrischer Netze (Power Systems Anal-

ysis)’, Autography, EEH – Power Systems Laboratory, ETH Zurich, 2000.

[4] B. T. Ooi, G. Joos, F. D. Galiana, D. McGills, R. Marceau: ’FACTS Con-

trollers and the Deregulated Electric Utility Environment’, IEEE Electrical and

Computer Engineering, 1998.

[5] Y. H. Song, A. T. Johns: ’Flexible ac transmission systems (FACTS)’, IEEE

Press, U.K., 1999, ISBN 0-85296-771-3.

[6] N. G. Hingorani, L. Gyugyi: ’Understanding FACTS: Concepts and Technology

of AC Transmission Systems’, IEEE Press, N.Y., 2000, ISBN 0-7803-3455-8.

[7] R. M. Mathur, R. K. Varma: ’Thyristor-Based FACTS Controllers for Electrical

Transmission Systems’, IEEE Press, N.Y., 2002, ISBN 0-471-20643-1.

[8] X. P. Zhang, E. J. Handschin: ’Optimal Power Flow Control by Converter

Based FACTS Controllers’, IEEE AC-DC Power Transmission, 2001.

[9] H. Ambriz-Perez, E. Acha, C. R. Fuerte-Esquivel: ’Advanced SVC Models for

Newton-Raphson Load Flow and Newton Optimal Power Flow Studies’, IEEE

Transactions on Power Systems, 2000.

68

BIBLIOGRAPHY

[10] T. Orfanogianni: ’A flexible software environment for steady-state power flow

optimization with series FACTS devices’, Diss. ETH Nr. 13689, EEH – Power

Systems Laboratory, ETH Zurich, 2000.

[11] N. Li, Y. Xu, H. Cheng: ’FACTS-Based Power Flow Control in Interconnected

Power Systems’, IEEE Transactions on Power Systems, 2000.

[12] F. L. Alvarado: ’Solving Power Flow Problems with a Matlab Implementation

of the Power System Applications Data Dictionary’, IEEE Proceedings of the

32nd International Conference on System Sciences, 1999.

[13] Power Systems Test Case Archive: ’IEEE 14 Bus Power Flow Test Case’,

University of Washington, 2003.

Available from www.ee.washington.edu/research/pstca/

[14] BCP Busarello + Cott + Partner Inc.: NEPLAN Version 5.1.0, 2002.

Available from www.neplan.ch

[15] R. Bacher: ’Optimierung liberalisierter elektrischer Energiesysteme’, Autogra-

phy, EEH – Power Systems Laboratory, ETH Zurich, 2002.

[16] G. Tognola: ’Unlimited Point - Ansatz zur Losung des optimalen Lastflussprob-

lems’, Diss. ETH Nr. 12437, EEH – Power Systems Laboratory, ETH Zurich,

1998.

69

Implementation of FACTS and Economic … of FACTS and Economic Generation Dispatch in an Interactive...

Documents

Transcript of Implementation of FACTS and Economic … of FACTS and Economic Generation Dispatch in an Interactive...