constrained economic dispatchfor hvdc using particle swarm...

UNIVERSITY OF NAIROBI

FACULTY OF ENGINEEING

DEPARTMENT OF ELECTRICAL AND INFORMATION ENGINEERING

SECURITY CONSTRAINED ECONOMIC DISPATCH FOR HVDC USING

PARTICLE SWARM OPTIMIZATION

PROJECT INDEX: 058

SUBMITTED BY CYPRIAN OCHIENG’, F17/1430/2011

SUPERVISOR: PROF. N.O. ABUNGU

EXAMINER:

PROJECT REPORT SUBMITTED IN PARTIAL FULFILLMENT

OF THE REQUIREMENT FOR THE AWARD OFTHE DEGREEOF

BACHELOR OF SCIENCE IN ELECTRICAL AND ELECTRONICENGINEERING

OF THE UNIVERSITY OF NAIROBI 2016

SUBMITTED ON:

DECLARATION OF ORIGINALITY

NAME OF STUDENT: Cyprian Ochieng’

REGISTRATION NUMBER: F17/1430/2011

COLLEGE: Architecture and Engineering

FACULTY/ SCHOOL/ INSTITUTE: Engineering

DEPARTMENT: Electrical and Information Engineering

COURSE NAME: Bachelor of Science in Electrical &

Electronic Engineering

TITLE OF WORK: Security Constrained Economic Dispatch for

HVDC Using Particle Swarm Optimization

(PSO)

I understand what plagiarism is and I am aware of the university policy in this regard.

I declare that this final year project report is my original work and has not been submitted

elsewhere for examination, award of a degree or publication. Where other people’s work

or my own work has been used, this has properly been acknowledged and

referenced in accordance with the University of Nairobi’s requirements.

I have not sought or used the services of any professional agencies to produce this work.

I have not allowed, and shall not allow anyone to copy my work with the intention of

passing it off as his/her own work.

I understand that any false claim in respect of this work shall result in disciplinary action,

in accordance with University anti-plagiarism policy.

Signature: …………………………………………………………………..

Date: ………………………………………………………………………..

CERTIFICATION

This report has been submitted to the Department of Electrical and Information

Engineering University of Nairobi with my approval as supervisor:

Prof. Nicodemus Abungu Odero

Date: .....................................................................................................................

DEDICATION

I dedicate this project to the Almighty God and to my family, lecturers and fellow

students for their support and encouragement.

ACKNOWLEDGEMENTS

I would like to thank God for this life and for His Presence and Guidance throughout my

life and studies.

I would also like to thank my supervisor, Prof. Nicodemus Abungu for his insight,

motivation, support and guidance.

I extend my appreciation to Mr. Musau for his valuable insights into my project, criticism

and encouragement.

I appreciate all my lectures and all staff at the Department of Electrical and Electronic

Engineering, and the entire community of the University of Nairobi for their contribution

towards my degree.

I am thankful to my classmates and friends for their support and availability throughout

my studies and most importantly during the period over which I was working on the

project.

In conclusion I would like to sincerely thank my family for their presence and undying

support throughout my studies.

Contents

DECLARATION OF ORIGINALITY ........................................................................................................ 1

CERTIFICATION ........................................................................................................................................ 2

DEDICATION ............................................................................................................................................... 3

LIST OF TABLES ........................................................................................................................................ 9

LIST OF ABBREVIATIONS ..................................................................................................................... 10

1.1. Definition of terms .......................................................................................................................... 12

1.2. A brief introduction to HVDC system .......................................................................................... 14

1.2.1. Components of an HVDC transmission system ....................................................................... 15

1.2.2. HVDC systems control............................................................................................................... 16

1.2.3. HVDC configurations ................................................................................................................ 17

1.2.4. Reasons for HVDC ..................................................................................................................... 18

1.2.5. Comparison between DC and AC systems ............................................................................... 18

1.3. Problem statement ......................................................................................................................... 20

1.4. Objectives ....................................................................................................................................... 20

1.5. Research Questions ........................................................................................................................ 21

CHAPTER 2 ................................................................................................................................................ 22

2.1. Economic dispatch problem neglecting transmission losses ....................................................... 23

2.1.1. Fuel cost characteristics ............................................................................................................ 24

2.1.2. Problem formulation .............................................................................................................. 25

2.2. Economic dispatch problem considering network losses ...................................................... 27

2.2.1. Transmission line loss equation .............................................................................................29

2.2.2. Losses in HVDC systems ........................................................................................................30

2.2.3. HVDC inequality constraints ................................................................................................. 31

2.3. Review of solution methods .................................................................................................... 32

2.3.1. Particle swarm optimization ................................................................................................... 33

2.3.2. Genetic algorithm ................................................................................................................... 34

2.3.3. Evolutionary Programming .................................................................................................... 34

2.3.4. Linear Programming .............................................................................................................. 35

2.3.5. Lambda iteration .....................................................................................................................36

CHAPTER 3 ................................................................................................................................................ 37

SOLUTION OF SECURITY CONSTRAINED ECONOMIC DISPATCH USING PSO .................... 37

3.1. PSO Algorithm ............................................................................................................................... 37

3.2. Parameter representation ............................................................................................................. 42

3.3. PSO Algorithm for SCED.........................................................................................................42

CHAPTER 4 ........................................................................................ Error! Bookmark not defined.45

RESULTS AND ANALYSIS………………………………………………………………………………………………………………45

4.1. Case Study: IEEE 30-bus System………………………………………………………………………………….45

4.2. Results………………………………………………………………………………………………………………………….46

4.3. Analysis and Discussion………………………………………………………………………………………………….49

CHAPTER 5 ................................................................................................................................................ 51

CONCLUSION AND RECOMMENDATION………………………………………………………………………………..51

5.1 Conclusion……………………………………………………………………………………………………………………….51

5.2 Recommendations…………………………………………………………………………………………………………..51

REFERENCES…………………………………………………………………………………………………………………………………..52

APPENDIX…………………………………………………………………………………………………………………………………….53

LIST OF FIGURES

Fig 1.1: Total cost/distance…………………………………………………….............................……..18

Fig 1.2: Typical transmission line structures for approximately 1000MW…………………....…….19

Fig 2.1: N-thermal units connected to a bus to serve a load.........................................................….....23

Fig2.2: A typical fuel cost characteristics…………..………………………………………………….24

Fig 2.3: N thermal units serving LoadP through a transmission network with losses.…....................26

Fig 3.1: PSO Algorithm..................................…………………………………………….….………....36

Fig 4.1: One line diagram of IEEE 30-bus system [1]............................…............................................43

Fig 4.2: Variation of Real Power Losses with Power Demand for SCED and CED…………....…...47

Fig 4.3: Variation of Optimal Cost with Power Demand for SCED and CED…………...........….…48

LIST OF TABLES

Table 4.1: Optimal generation for SCED and CED using PSO for a total demand of 283.4MW........44

Table 4.2: Optimal generation for SCED and CED using PSO for a total demand of 374.3MW........45

Table 4.3: Optimal generation for SCED and CED using PSO for a total demand of 540MW...........45

Table 4.4: Comparison of Economic Dispatch using PSO and GA for a total demand of 283.4MW..46

LIST OF ABBREVIATIONS

CED Classical Economic Dispatch

ED Economic Dispatch

GA Genetic Algorithm

HVAC High Voltage Alternating Current

HVDC High Voltage Direct Current

IEEE Institute of Electrical and Electronic Engineering

LP Linear Programming

MATLAB Matrix Laboratory

MW Megawatts

OPF Optimal Power Flow

PSO Particle Swarm Optimization

QP Quadratic Programming

SCED Security Constrained Economic Dispatch

HPSO Hybrid Particle Swarm Optimization

ABSTRACT

With increasing number of appliances, at house hold, office and institution level, as well

as facilities such as transport, communication among others, and machinery that require

electrical power, there is need for a reliable and efficient power supply that is at a

reasonable cost. With this in mind there is need for generation facilities, while harnessing

the abundant renewable energy sources provided by nature, to generate the energy that

requires fossil fuel at reduced cost to ensure affordability of the power on the side of the

consumer. This is to be done while ensuring that the whole system of generation and

supply if operating within their defined safe limits.

Considering this growing demand of power, there is need for interconnection of power

grids between nations within a region as well as integration of offshore generated energy

into the inland grid. With these long distances over which transmission is to be done to

achieve this, as well as to harmonize power systems operating at different frequencies,

HVDC technology is necessary.

Security Constrained Economic Dispatch is as an optimization procedure that attempts to

obtain an optimal balance of two conflicting objectives; cost efficiency that aims at

serving the demand with minimum cost, and security that requires electricity to be

delivered to customers without interruption even when a component of the system fails.

SCED problem has been solved with conventional methods as well as intelligent search

methods. In this project, Particle Swarm Optimization method is used to solve the

problem. The PSO algorithm has been implemented on the IEEE 30-bus network with a

load demand of 283.4MW. Six generating units are used to supply the power to meet this

demand.

CHAPTER 1

INTRODUCTION

1.1.Definition of terms

1.1.1. Economic Dispatch

Describes how the real power output of each controlled generating unit in an area is

selected to meet a given load and to minimize the total operating cost in the area. It is the

allocation of generation levels to generating units comprising a power system to

economically serve the load in entirety while remaining within the operational limits of

the generation facilities.

1.1.2. System Security

These are measures put in place to keep the system operating when components fail. A

measure of this security is the ability of the power system to withstand the effects of

contingencies such as generator, transformer or line outage, the effects of which are

monitored with specified security limits. An operationally secure system is one with low

possibility of system collapse or equipment damage.

The following are the three major functions carried out in the energy control center under

system security;

i. System monitoring; gives up-to-date information on the state of the power system

on real time basis with regard to the load and generation change.

ii. Contingency analysis; foresees possible system outages before they actually

occur. They alert the operators to any potential overloads or serious voltage

violation.

iii. Corrective action analysis; allows the operator to alter the operation of the power

system in such a direction as to avert the occurrence of a serious problem due to a

given outage.

1.1.3. Security Constrains

These are limits put in a power system to prevent outage of equipment due to

overstretching of the performance capabilities of the facilities and therefore ensure

continued supply of power to consumers with minimum interruptions.

May be applied as;

A temporary constraint to deal with an outage situation when some assets are not

available.

A permanent constraint when the normal integrated power system capability and

expected generation offers and demand may not result in secure operation.

1.1.4. Security Constrained Economic Dispatch

This is the operation of generating facilities to produce energy at the lowest cost to

reliably serve consumers, recognizing any operational limits of generation and

transmission facilities. The varying nature of demand of energy and the variations in

costs of different types of generating units, together with the known and unknown

conditions on the transmission network determine which generating units to be used to

serve the load most reliably.[13]

1.1.5. HVDC system

This involves the transmission of power at high voltages with the aim of improving

system efficiency and reducing the overall energy cost.

1.2.A brief introduction to HVDC system

Power transmission was formerly done in early 1880s using Direct Current (DC). With

the introduction of transformers, development of induction motors, availability of

synchronous generators and facilities that could convert alternating current to direct

current whenever required, alternating current gradually replaced direct current as a

method of power transmission. However in 1928 the availability of devices like mercury

vapor rectifiers that have the ability to rectify and invert current created possibilities for

high voltage direct current transmission.

With fast development of converters (rectifiers and inverters) that can work at higher

voltages and large currents, dc transmission became a major factor in the planning of

power transmission.

In the early stages all HVDC schemes used mercury arc valves, invariably single phase in

construction in contrast to the low voltage polyphase units used for industrial application.

Around 1960, control electrodes were added to silicon diodes, giving silicon-controlled-

rectifiers (SCRs). Among the early schemes were;[14]

The Gotland Scheme in Sweden, commissioned in 1954, capable of transmitting

20MW of power at -100KV and consisting of a single 96km cable with sea

return.

The Cross Channel link between England and France put in operation in 1961,

two single conductor submarine cables 64km at ±100KV with two bridges each

rated at 80MW.

Sakuma Frequency Changer in 1965 to connect the 50Hz and 60Hz systems of

Japan, capable of transmitting 300MW in either direction at 250KV.

With the growing application of HVDC transmission, need arises to formulate Economic

Dispatch for it. This entails the allocation of generation levels to generating units in a

power system employing HVDC lines to economically serve the load in entirety while

remaining within the operational limits of the generating and transmitting facilities.

The fundamental processes in an HVDC system is the conversion of electrical current

from AC to DC (rectification) at the transmitting end, and from DC to AC at the

receiving end. There are three ways of achieving conversion;[12]

I. Use of Natural Commutated Converters; are most used in HVDC systems today.

A thyristor capable of handling 4000A currents and blocking up to 10KV is used.

Series connection of these thyristors to form a thyristor valve enables them to

block hundreds of KV. The thyristor is operated at net frequency (50Hz or 60Hz)

and change of DC voltage level is achieved by means of adjusting the control

angle of the thyristor.

II. Use of Capacitor Commutated Converters (CCC); this is an improvement of the

thyristor-based-commutation. Commutation capacitors are inserted in series

between the converter transformers and the thyristor valves. These capacitors

improve the commutation failure performance of the converters when connected

to weak networks.

III. Use of forced commutated converters; the valves of these converters are made

with semiconductors with the ability to turn-on and also turn-off. They are known

as Voltage Source Converters (VSCs). Semiconductors normally used to are

Gate-Turn-Off Thyristors (GTOs) and Insulated Gate Bipolar Transistors

(IGBTs). The VSC commutates with high frequency and its operation is achieved

by Pulse Width Modulation (PWM). With PWM it is possible to create any phase

angle and/or amplitude by changing the PWM pattern, which can be done almost

instantaneously. Thus PWM offers possibility to control both active and reactive

power independently, making the PWM VSC a close to ideal component in the

transmission network.

1.2.1. Components of an HVDC transmission system

The three main elements in an HVDC system are;

i. Converter Station

Converter stations at each ends are replicas of each other and thus consists of all

equipment needed to convert from AC to DC and vice versa. The main

components here are;

Thyristor valves – most common way of arranging thyristor valves is in a

twelve-pulse group with three quadruple valves. All communication

between the control equipment at earth potential and each thyristor at high

potential is done with fiber optics.

VSC valves – consists of two level or multilevel converter, phase-reactors

and AC filters. VSC valves, control equipment and cooling equipment are

in enclosures which make transport and installation easy.

Transformers – adapt the AC voltage level to DC voltage level and

contribute to commutation reactance.

AC filters and Capacitor banks – filters are installed in order to limit the

amount of harmonics to the level required by the network. In the

conversion process the converter consumes reactive power which is

compensated for in part by the filter banks and the rest by capacitor banks.

DC filters – reduce the disturbances in telecommunication systems created

by harmonics due to HVDC converters.

ii. Transmission medium

Most frequently overhead line, normally bipolar (that is, two conductors with

different polarity) when over land. HVDC cables are used for submarine

transmission.

iii. Electrodes

Are conductors that provide connection to the earth for neutral. They have large

surface to minimize current densities and surface voltage gradients.

1.2.2. HVDC systems control

Control is done for efficiency and stability of the system, and also for maximum

flexibility of power control without compromising on safety of equipment.

The parameters mostly controlled are;

Direct current from rectifier to inverter cos cosdr di

d

cr L ci

V VI

R R R

rectifier end voltage

inverter end voltage

rectifier resistance

inverter resistance

transmission line resistance

dr

di

cr

ci

L

V

V

R

R

R

Power at the rectifier terminal dr dr dP V I

Power at the inverter terminal 2

di di d dr L dP V I P R I

The means of control is by control of the internal voltages that can be used to control the

voltage at any point along the transmission line and the current flow or power.

This is done by controlling the firing and extinction angles of the rectifiers and inverters

(fast action) or by changing taps on the transformers on the AC side (slow response).

Power reversal is done by reversing the polarity of the DC voltages at both ends, current

flow remains unchanged (since valves can only conduct in on direction)

A control action may be chosen with aim of prevention of large fluctuations in DC

voltage/current due to variations in the AC side voltage, maintenance of direct current

voltage near the rated value or to keep power factor at the receiving and transmitting ends

as high as possible.

Tap changer control is used to keep the converter firing angle ( & ) within the desired

range. They are sized to allow for minimum and maximum steady state voltage variation.

For current limits, the maximum short circuit current is limited to 1.2 to 1.3 times the

normal full load current to avoid thermal damage of equipment. Minimum current limit is

set to avoid ripple in the current that may cause it to be discontinuous or intermittent.

Minimum firing angle limit is set to prevent reversal of power flow as a result of the

inverter station switching to rectification mode in case of a DC fault.

For power control, the current order required to transmit a scheduled power is given by;

, is thescheduled power, thecontrolled voltageoord o d

d

PI P V

V

1.2.3. HVDC configurations

There are three HVDC configurations; monopolar, bipolar and homopolar systems.

Monopolar HVDC systems have either ground return or metallic return.

A monopolar HVDC system with ground return consists of one or more six-pulse

converter units in series or parallel at each end. It can be a cost-effective solution

for an HVDC cable transmission and/or the first stage of a bipolar scheme. At

each end of the line it requires an electrode line and a ground or sea electrode

built for continuous operation.

A monopolar HVDC system with metallic return usually consists of one high

voltage and one medium voltage conductor. A monopolar configuration is used

either in the first stage of a bipolar scheme, avoiding ground currents, or when

construction of ground electrode lines and ground electrodes result in an

uneconomical solution due to a short distance or high value of earth resistivity.

Bipolar HVDC systems consist of two poles, each of which includes one or more

twelve-pulse converter units in series or parallel. There are two conductors, one

with positive and the other with negative polarity. For power flow in the other

direction, the two conductors reverse their polarities. A bipolar system is a

combination of two monopolar schemes with ground return.

Back-to-Back HVDC links are special cases of monopolar HVDC

interconnections where there is no DC transmission line and both converters are

located at the same site. For economic reasons each converter is usually twelve-

pulse converter unit, and the valves for both converters may be located in one

valve hall. The control system, cooling equipment and auxiliary system may be

integrated into configurations common to the two converters.

Generally for a Back-to-Back HVDC link, the DC voltage rating is low and the

thyristor valve current rating is higher in comparison with HVDC

interconnections via overhead lines or cables.

1.2.4. Reasons for HVDC

Some short comings of AC transmission as well as the need to incorporate the upcoming

renewable energy from sources such as solar and wind compel a change and application

of DC technology. Some gaps in high voltage AC transmission are;[7]

Inductive and capacitive elements of overhead lines and cables put limits to the

transmission capacity and the transmission distance for AC transmission links.

Depending on the required transmission capacity, the system frequency and the

loss evaluation, the achievable transmission distance for an AC cable is in the

range of 40 to 100km, mainly limited by charging current.

Direct connection between two AC systems with different frequencies is not

possible.

Direct connection between two AC systems with the same frequency or a new

connection within a meshed grid may be impossible because of system instability,

too high short-circuit levels or undesirable power flow scenarios

1.2.5. Comparison between DC and AC systems

Comparison can be done under technical merits, economic considerations and

environmental issues.

Technical Merits of HVDC

The advantages of a DC link over an AC link are;

A DC link allows for power transmission between AC networks with different

frequencies or networks which cannot be synchronized for some reasons.

Transmission capacity or the maximum length of a DC line or cable is not limited

by inductive and capacitive parameters. The conductor cross-section is also fully

utilized because there is no skin effect.

A digital control system provides accurate and fast control of the active power

flow.

Fast modulation of DC transmission power can be used to damp power

oscillations in an AC grid and thus improve the system stability.

Economic considerations

For a given transmission task, feasibility studies are conducted before the final decision

on implementation of an HVAC or HVDC system can be made. The figure below shows

a typical comparison curve between AC and DC transmission considering AC vs DC

station terminal costs, AC vs DC line costs, and AC vs DC capitalized value of losses.[7]

Fig 1.1 Total cost/distance[7]

The DC curve is not as steep as the AC curve because of considerably lower line costs

per kilometer. For long AC lines the cost of intermediate reactive power compensation

has to be taken into account. The break even distance is in the range of 500 to 800km

depending on factors like country-specific cost elements, interest rates for projects

financing, loss evaluation, and cost of right of way, among others.

Environmental issues

An HVDC transmission system is basically environment friendly because improved

energy transmission possibilities contribute to a more efficient utilization of existing

power plants. The land coverage and the associated right of way cost for an HVDC

overhead transmission line is not as high as that of an AC line. This reduces the visual

impact and saves land compensation for new projects. It is also possible to increase the

power transmission capacity for existing right of way. [7]

Fig 1.2 Typical transmission line structures for approximately 1000MW[7]

There are however some environmental issues which must be considered for the

converter stations. These include audible noise, visual impact, electromagnetic

compatibility and use of ground or sea return path in monopolar operation, among others.

In general it can be said that an HVDC system is highly compatible with any

environment and can be integrated into it without the need to compromise on any

environmentally important issues of today.

1.3.Problem statement

The aim of this paper is to introduce security constraints to the economic dispatch

through the analysis of factors affecting the generation of energy at the generating units

and the transmission of that energy to the demand centers using High Voltage Direct

Current.

To solve this problem, adequate knowledge of economic dispatch, system security and

HVDC as well as its security aspects and transmission is a fundamental requirement.

1.4.Objectives

To obtain an optimal solution to the Security Constrained Economic Dispatch, the

following objectives are to be achieved;

To formulate the ED problem taking into consideration the constrains in

generation of power and its transmission using HVDC

To study the methods of solution of the Security Constrained Economic Dispatch

problem, and

To come up with an optimal solution for the problem

1.5.Research Questions

The process and outcome of the project will attempt to address the following questions;

What are the constraint to be considered while undertaking a Security Constrained

Economic Dispatch for HVDC?

What is the most effective technique to apply in solving this particular SCED

problem?

CHAPTER 2

LITERATURE REVIEW

Economic Dispatch (ED) entails optimal allocation of the outputs of a large number of

participating generators.

Security Constrained Economic Dispatch seeks to optimize the process, taking into

account all the relevant factors pertaining to selection of the generating units to dispatch

so as to deliver a reliable supply of power at the lowest cost possible.

The choice as to whether a generating unit should participate in sharing the load at a

given interval of time is a problem of unit commitment. The unit commitment problem

having been solved, optimal allocation of the available generation units to meet the

forecasted load demand for the time interval in question is done.

ED process has two stages, also referred to as time periods;

a. Unit commitment stage – the operators decide which units to be committed to be

online for each hour, usually for the next 24hrs period, based on load forecast

taking into account the unit’s maximum and minimum output levels, the

minimum time a generator in the unit must run once started, the generating costs

and the costs of environmental compliance and how quickly the output of the unit

can be changed. Also of importance to consider under this stage are the forecasted

conditions that can affect transmission grid, that is, the “security constrained”

aspect of commitment analysis, as well as the generation and transmission

facilities outages, line capabilities (limits and directions) and weather. If security

analysis indicate that optimal ED cannot be done reliably, relatively expensive

generators may have to be opted for.

b. Unit dispatch stage – operators decide in real time the level at which each

available resource (as determined from stage (a)) should be operated, given the

actual load and grid conditions, such that reliability is maintained and overall

production costs are minimized.

Optimization techniques are used to determine not only the optimal outputs of the

participating generators, but also the optimal settings of various control devices such as

the tap settings of Load Tap Changers (LTCs), outputs of VAR compensating devices,

desired settings of phase shifters, among others.

The desired objectives for optimization problems include minimization of the cost of

generation, minimization of the total power loss in the system, minimization of voltage

deviations, and maximization of the reliability of the power supplied to customers. While

formulating the optimization strategy, one or more of these objectives can be taken into

consideration. Determination of the real power output of the generators so that the total

cost of generation in the system is minimized is known as Economic Dispatch.

The majority of generating systems are of three types; nuclear, hydro and thermal (using

fossil fuels such as coal, oil and gas), but due to developments in the technology of

renewable energy, and with rising environmental concerns together with rising demand

for power, these sources are currently complemented by other sources such as wind, solar

and tidal energy. Nuclear plants tend to be operated at constant output power levels.

Operating cost of hydro units do not change much with the output. The operating cost of

thermal plants however change significantly with the output power level, and therefore

are considered in this paper for discussing the ED problem.

2.1.Economic dispatch problem neglecting transmission losses

First the ED problem is considered with the transmission losses neglected (like in the

case of Back-to-Back HVDC systems where systems operating at different frequencies

are joined together and at one location and the transmission distance is essentially zero or

where a group of generators are connected to a particular bus-bar like in the case of

individual generating units in a power plant, or when they are physically located very

close to each other, and thus the transmission losses can be ignored due to the short

distance involved).[10]

Consider the Fig2.1 below showing N-thermal units connected to a bus to serve a load

LoadP .

Fig 2.1 N-thermal units connected to a bus to serve a load LoadP [9]

Input to each unit is expressed in terms of cost rate, say $/h. The total cost rate is the sum

of cost rates of each of the individual units. The essential operating constraint is that the

sum of the power outputs must be equal to the load (neglecting power losses).

2.1.1. Fuel cost characteristics

ED problem is the determination of generation levels with the aim of minimizing the total

cost of generation for a defined level of load. For thermal generating units, the cost of

fuel per unit power output varies significantly with the power output of the unit. In

solving the ED problem, the fuel cost characteristics of the generators are considered

while finding their optimal real power outputs. A typical fuel cost characteristics is as

shown in Fig2.2 below.

Fig2.2 A typical fuel cost characteristics[9]

The labor cost, supply and maintenance are generally fixed. MinP is the output level below

which it is uneconomical or technically infeasible to operate the units. MaxP on the other

hand is the maximum output power limit. In formulating the dispatch problem, fuel costs

are usually represented as a quadratic function of output power as shown by the equation

below.

2( )F P aP bP c (2.1)

2.1.2. Problem formulation

Total fuel cost for operating N generators is given by;

1 1 2 2( ) ( ) ... ( )T N NF F P F P F P (2.2)

1

( )N

i i

i

F P

(2.3)

With transmission losses neglected, total generation should meet the total load.

Therefore, the equality constraint is;

1

N

i Load

i

P P

(2.4)

Based on minimum and maximum power limits of the generators, the following

inequality constraint is imposed;

( ) ( )i Min i i MaxP P P ; 1,2,...i N (2.5)

This is a constrained optimization problem that can be solved by Lagrange multiplier

method.

The Lagrange method is formulated as;

TL F (2.6)

Where 1

N

Load

i

P Pi

accounts for the equality constraint (2.4), is the Lagrange

Multiplier. The necessary condition for TF to be minimum is that the derivative of

Lagrange function with respect to each independent variable is zero. Thus the necessary

conditions for the optimization problem are;

Load

1 1

{ ( ) (P )N N

i i i

i ii i

LF P P

P P

(2.7)

0i

i

F

P

; 1,2,...i N (2.8) and

0i

L

P

(2.9)

Rearranging (2.8),

i

i

F

P

; 1,2,...i N (2.10)

Equation (2.10) states that to minimize the fuel cost, the necessary condition is to have all

the incremental fuel costs equal. Equation (2.10), along with the equality constraint (2.4)

and the inequality constraint (2.5) are the Coordination Equations for Economic Dispatch

with network losses neglected.

Using equation (2.1), fuel cost characteristics of all the generators are expressed as;

2

i i i i i iF a P b P c ; 1,2,...i N (2.11)

Using (2.10), the necessary conditions for the optimal solutions are given by;

2ii i i

i

Fa P b

P

; 1,2,...i N (2.12)

Or

2

ii

i

bP

a

; 1,2,...i N (2.13)

Substituting iP from (2.13) into (2.4), we have;

1 2

Ni

Load

i i

bP

a

(2.14)

Or

1

1

( )2

[ ]1

( )2

Ni

Load

i i

N

i i

bP

a

a

(2.15)

With this, can be calculated from (2.12) and iP be determined for 1,2,...i N from

(2.11).

2.2.Economic dispatch problem considering network losses

This involves economically distributing the load among different plants of a power

system. In this case transmission losses are considered as shown in the schematic below

depicting such a system.

Fig 2.3 N thermal units serving LoadP through a transmission network with losses

For a unit with low incremental cost, operating cost may be higher if the transmission line

losses are very high (for example where there is a large distance between the unit and the

load), thus it becomes necessary to take into consideration the transmission line losses

when determining Economic Dispatch of units in a power system.

The total fuel cost rate is still as expressed by equation (2.2)

1 1 2 2( ) ( ) ... ( )T N NF F P F P F P

The power balance (equality) equation now includes the transmission losses.

0

0N

Loss Load i

i

P P P

(2.16)

LossP is the total transmission loss in the system.

The problem is to find iP s that minimize TF subject to the constraint (2.16).

Using the method of Lagrange multipliers,

TL F , with given as expressed by equation (2.16).

The necessary conditions to minimize TF are as follows;

0i

L

P

; 1,2,...i N

Or

1 1

{ ( ) ( )} 0N N

i i Loss Load i

i ii

F P P P PP

(2.17)

Or

( 1) 0i Loss

i i i

F PL

P P P

; 1,2,...i N (2.18)

This (equation 2.18) is the condition for optimal dispatch.

Rearranging (2.18),

1

i

i

Loss

i

F

P

P

P

(2.19)

The equation above is often expressed as

ifi

i

FP

P

(2.20)

With fiP being the Penalty Factor of the plant, and is given by;

1

1fi

Loss

i

PP

P

(2.21)

Here, Loss

i

P

P

is the incremental loss for bus i .

From Equation (2.20), the minimum cost operation is achieved when the incremental cost

(IC) for each unit multiplied by its penalty factor is same for all generating units in the

system. Relating to the case of units in the same plant, or generators connected to the

same bus, (2.20) implies;

1 21 2

1 2

... Nf f fN

N

FF FP P P

P P P

When units are connected to the same bus, incremental change with transmission loss

with change in generation is the same for all the units, thus;

1 2...f f fNP P P , and therefore;

1 2

1 2

... N

N

FF F

P P P

(2.22)

Which is the same as in the case of units connected to a bus.

Equation (2.19) and (2.16) are collectively known as coordination equations for

Economic Dispatch considering transmission losses.

2.2.1. Transmission line loss equation

Transmission line loss equation, known as Kron’s loss formula is expressed as;

T T

Loss O OOP P BP B P B (2.23)

Where P is the vector of all generator bus net outputs; B is a square matrix; OB is a

vector of same length as P ; OOB is a constant.

The B-terms are called Loss-Coefficients or B-Coefficients, and the N by N symmetrical

matrix B is simply known as the B-matrix.

Equation (2.23) can be written as;

1 1 1

N N N

Loss i ij j iO i OO

i j i

P PB P B P B

(2.24)

Referring to the coordination equation, the equality constraint now becomes;

1 1 1 1

[ ]N N N N

Load i ij j iO i OO i

i j i i

P PB P B P B P

(2.25)

The derivative of Lagrange function now becomes;

1

[1 2 ]N

ij j iO

ji i

L FB P B

P P

(2.26)

The coordination equations are now coupled.

Solution of Economic Dispatch problem in this case is a little complex compared to the

case with network losses neglected.

2.2.2. Losses in HVDC systems

Typically, overall losses in HVDC transmission are 30% to 50% less than HVAC

transmission. Although HVDC incurs losses during the conversion process from AC to

DC, the line losses in HVDC cable are smaller than HVAC cables, and when used over

long distances, lower cable losses compensate for higher conversion losses of HVDC

transmission.

The power losses produced, per converter station, in VSC HVDC technology are more

than the power losses produced per converter station in LCC HVDC technologies. The

power losses in VSC per converter station are 4% to 6% of the total power being

delivered while that per converter stations in LCC HVDC are 2% to 3%. However VSCs

are preferred due to their low levels of harmonics generated hence reduced need to install

filters in offshore substations.

For a VSC HVDC system, losses can be studied under the following stages;

VSC Converter losses; divided into conduction losses and switching losses

Transmission losses, including DC cable losses, coupling transformer losses,

smoothing reactor losses and losses in AC filters.

2.2.3. HVDC inequality constraints

In HVDC the inequality constraints are usually the operation or physical limits. For

instance, a transmission line capacity is constrained by its thermal limit, the bus voltages

are within their insulation limits and generating units have lower and upper output limits.

Such constraints restrict the ED of the generators to range between the maximum and

minimum values, and include;

The power generator capacity constraint

GiMin Gi GiMaxP P P

The tap ratio of the converter

Min MaxT T T

Ignition angle of the converter

Min Max

Extinction angle of the converter

Min Max

DC current

dcMin dc dcMaxI I I

DC voltage

dcMin dc dcMaxV V V

The aim of this project is to minimize the total operating cost of the power system while

meeting the total load plus the transmission losses while operating within the generator

limits and transmission line limits. The transmission losses were taken to be 40% of the

losses obtained by calculation using B coefficients for an HVAC line of equivalent

length.

Mathematically, the aim is to minimize;

2

1 1

( )N N

i i i i i i i

i i

F P a P b P c

,

Subject to the following constraints;

The energy balance equation

0

N

i Loss Load

i

P P P

The generator limits

GiMin Gi GiMaxP P P

The tap ratio of the converter

Min MaxT T T

Ignition angle of the converter

Min Max

Extinction angle of the converter

Min Max

DC current

dcMin dc dcMaxI I I

DC voltage

dcMin dc dcMaxV V V

2.3.Review of solution methods

Two basic approaches are used in the solution;

1. The case of network loss formula, and

2. The case of optimization tools incorporating power flow equations and

constraints.

2.3.1. Particle swarm optimization

Particle Swarm Optimization was proposed by James Kennedy and Russell C. Eberhart in

1995.[15] It is a technique used to explore the search space of a given problem to find the

settings or parameters required to maximize a particular objective. It originates from two

concepts; the idea of swarm intelligence based on the observation of swarming habits of

certain kinds of animals, and the field of evolutionary computation.

Optimization is the mechanism by which the maximum or minimum value of a function

or process is obtained. A search space is defined, with elements called candidate

solutions of the search space. The number of parameters involved in the optimization

problem is the dimensions of the search space. An objective function maps the search

space to the function space. For a known function, calculus may be used to easily provide

the minima or maxima as desired. In real life however, the objective function is not

directly known. Instead, the objective function is a “black box” to which we apply

parameters and receive an output value. The result of this evaluation of a candidate

solution becomes the solution’s fitness. The final goal of an optimization task is to find

the parameters in the search space to maximize or minimize the fitness.[17]

The elements of a candidate solution may be subject to certain constraints, in which case

the task becomes a constrained optimization task.

Each particle is searching for an optimum and is moving, hence has a velocity. Each

particle remembers the position at which it had its best so far (its personal best). To

improve the effectiveness, particles in the swarm co-operate by exchanging information

about what they have discovered in the places they had been to. Thus a particle has

neighbors associated with it, it knows the fitness of those in its neighborhood, and uses

the position of the one with best fitness to adjust its own velocity.

In each time step a particle has to move to a new position by adjusting its velocity. The

adjustment is the sum of its current velocity, a weighted random portion in the direction

of its personal best, and a weighted random portion in the direction of the neighborhood

best.

Particles’ velocity on each dimension are clamped to a maximum velocity MaxV . If the sum

of accelerations would cause the velocity on that dimension to exceed MaxV , a parameter

specified by the user. Then the velocity in that direction is limited to MaxV .

Particle Swarm Optimization is a preferred method of solving ED problems for among

others the following reasons;

With a population of candidate solutions, a PSO algorithm can maintain useful

information about the characteristics of the environment.

PSO, as characterized by its fast convergence behavior, has an inbuilt ability to

adapt to a changing environment.

Some early works on PSO have shown that PSO is effective for locating and

tracking options in both static and dynamic environment.

As compared to other optimization techniques, PSO is a simple concept that is

easy to implement, cheaper, impervious to failure regardless of user input or

unexpected conditions and takes less time to converge.

2.3.2. Genetic algorithm

First proposed by Frazer and later by Bremermann and Raed, Genetic Algorithm was

popularized by the work of Holland. GA models genetic evolution. Features of

individuals are expressed using genotypes. The main driving operators of a GA is

selection which models survival of the fittest and recombination through application of a

crossover operator that models reproduction.[17]

A population of individuals (phenotypes) to an optimization problem is evolved towards

better solutions. Each candidate solutions’ features are mutated. The process starts from a

population of individuals and is an iterative process resulting in successive generations.

For each generation the fitness of each individual is evaluated, the fitness being the value

of the objective function in the optimization problem being solved. The more fit

individuals are stochastically selected from the current population and each individual’s

genome modified to form the next generation. This generation solutions are then used in

the next iteration of the algorithm. The algorithm terminates when a satisfactory fitness

level has been reached for the population.

Although Genetic Algorithm always converges, it does not give assurance that a global

optimum will be obtained. It also lacks a constant optimization response time.

2.3.3. Evolutionary Programming

Evolutionary Programming (EP) was first used by Lawrence J. Fogel in the US in 1960 to

use simulated evaluation as a learning process to generate artificial intelligence. While

EP shares the objectives of imitating natural evolutionary processes, with Genetic

Algorithm and Genetic Programming, it differs substantially in that EP emphasizes the

development of behavioral models and not genetic models. EP considers phenotypic

evolution, it iteratively applies two evolutionary operators; variation through application

of mutation operators and selection.[17]

The evolutionary process, first developed to evolve finite state machines, consists of

finding a set of optimal behaviors from a space of observable behaviors. The fitness

function measures the “behavioral error” of an individual with respect to the environment

of that individual.

EP utilizes four main components of Evolutionary Algorithm (EA).

Initialization; a population of individuals is initialized to uniformly cover the

domain of the optimization problem.

Mutation; the mutation operator introduces variation in the population to produce

new candidate solutions. Each parent produces one or more offspring through

application of the mutation operator.

Evaluation; a fitness function is used to quantify the “behavior error” of

individuals. While the fitness function provides an absolute fitness measures to

indicate how well the individual solves the problem being optimized, survival in

EP is usually based on a relative fitness measure. A score is computed to quantify

how well an individual compares with a randomly selected group of competing

individuals. Individuals that survive to the next generation are selected based on

this relative fitness.

Selection; the selection operator selects the individuals that survive to the next

generation.

The setback with EP is its slow convergence in solving some of the multimodal

optimization problems.

2.3.4. Linear Programming

Linear programming is the most commonly applied form of constrained optimization.

The main elements of any constrained optimization problem are;

Variables (decision variables); values are unknown at start, usually represent

things that can be adjusted or controlled. The goal is to find values of the

variables that provide the best value of the objective function.

Objective function; is a mathematical expression that combines the variables to

express the goal. The requirement is to either maximize or minimize the objective

function.

Constraints; are mathematical expressions that combine the variables to express

limits on the possible solutions.

In linear programming all the mathematical expressions for the objective function and the

constraints are linear, thus it has an inaccurate evaluation of system losses and a limited

ability to find accurate solutions due to its linear approximation of non-linear control

parameters as compared to an exact non-linear model of a power system.[17]

2.3.5. Lambda iteration

When the minimization is constrained with an equality constraint it can be solved using

the method of Lagrange Multiplier. The key idea is to represent a constrained

minimization problem as an unconstrained problem.[11]

Lambda iteration method requires a unique mapping of from a value of lambda

(incremental cost) to each generator’s output. For any choice of lambda the generators

collectively produce an output. The methods starts with values of lambda below and

above the optimal value (corresponding to too little and too much output), and then

iteratively brackets the optimum value. Inclusion of losses impact the necessary

conditions for an optimal economic dispatch. The analytic calculation of the penalty

factor is involving, the problem is that a small change in the generation impacts the flow

and hence the losses throughout the entire system. However using a power flow, the

function can be approximated by making a small change to the output of individual

generators and seeing how the losses change.

CHAPTER 3

SOLUTION OF SECURITY CONSTRAINED ECONOMIC DISPATCH USING

PSO

3.1.PSO Algorithm

The PSO algorithm works by simultaneously maintaining several candidate solutions in

the search space. Initially, the algorithm chooses candidate solutions randomly within the

search space composed of all possible solutions. The algorithm uses the objective

function to evaluate its candidate solutions and operates upon the resultant fitness values.

Fig 3.1. PSO Algorithm

Each particle maintains its position, composed of the candidate solution and its evaluated

fitness, and its velocity. Additionally, it remembers the best fitness value it has achieved

thus far during the operation of the algorithm, referred to as the individual best fitness,

and the candidate solution that achieved this fitness, referred to as the individual best

position or individual best candidate solution. Finally, the PSO algorithm maintains the

best fitness value achieved among all particles in the swarm, called the global best

fitness, and the candidate solution that achieved this fitness, called the global best

position or global best candidate solution.

The PSO algorithm consists of just three steps, which are repeated until some stopping

condition is met:

1. Evaluate the fitness of each particle

2. Update individual and global best fitness and positions

3. Update velocity and position of each particle

The first two steps are fairly trivial. Fitness evaluation is conducted by supplying the

candidate solution to the objective function. Individual and global best fitness and

positions are updated by comparing the newly evaluated fitness against the previous

individual and global best fitness, and replacing the best fitness and positions as

necessary.

The velocity and position update step is responsible for the optimization ability of the

PSO algorithm. The velocity of each particle in the swarm is updated using the following

equation:

1

1 1 2 2

2

]

0 1and

1 , 1 1 1 1

,

0 1

,

,

j j j pbest j gbest j

max max

V t w t V t c r X t X t c r X t X

x

r r

t

x

The index of the particle is represented by j. Thus, ( )jv t is the velocity of particle i at

time t and ( )jx t is the position of particle j at time t. The parameters w , 1c and 2c

( 10 1.2,0 2,w c and 20 2c ) are user-supplied coefficients. The values 1r and

2r ( 1 20 1and 0 1r r ) are random values regenerated for each velocity update. The

value ,j pbestX is the individual best candidate solution for particle i at time t, and gbestX is

the swarm’s global best candidate solution at time t.

Each of the three terms of the velocity update equation have different roles in the PSO

algorithm. The first term 1jw t V t is the inertia component, responsible for keeping

the particle moving in the same direction it was originally heading. The value of the

inertial coefficient w is typically between 0.8 and 1.2, which can either dampen the

particle’s inertia or accelerate the particle in its original direction. Generally, lower values

of the inertial coefficient speed up the convergence of the swarm to optima, and higher

values of the inertial coefficient encourage exploration of the entire search space.

The second term 1 1 , 1 1 j pbest jc r X t X t , called the cognitive component, acts

as the particle’s memory, causing it to tend to return to the regions of the search space in

which it has experienced high individual fitness. The cognitive coefficient 1c is usually

close to 2, and affects the size of the step the particle takes toward its individual best

candidate solution gbestX .

The third term 2 2 1 1gbest jc r X t X t , called the social component, causes the

particle to move to the best region the swarm has found so far. The social coefficient 2c is

typically close to 2, and represents the size of the step the particle takes toward the global

best candidate solution gbestX the swarm has found up until that point.

The random values 1r in the cognitive component and 2r in the social component cause

these components to have a stochastic influence on the velocity update. This stochastic

nature causes each particle to move in a semi-random manner heavily influenced in the

directions of the individual best solution of the particle and global best solution of the

swarm.

In order to keep the particles from moving too far beyond the search space, we use a

technique called velocity clamping to limit the maximum velocity of each particle. For a

search space bounded by the range , ,max maxx x velocity clamping limits the velocity to

the range , ,max maxv v where *max maxv k x .

The value k represents a user-supplied velocity clamping factor, 0.1 1.0.k In

many optimization tasks, such as the ones discussed in the paper, the search space is not

centered on 0 and thus the range , ,max maxx x is not an adequate definition of the search

space. In such a case where the search space is bounded by , ,min maxx x we define

*( ) / 2max max minv k x x .

Once the velocity for each particle is calculated, each particle’s position is updated by

applying the new velocity to the particle’s previous position:

1 j j jX t X t V t

This process is repeated until some stopping condition is met. Some common stopping

conditions include: a preset number of iterations of the PSO algorithm, a number of

iterations since the last update of the global best candidate solution, or a predefined target

fitness value.

For this project, the constrained optimization problem is converted as an unconstrained

optimization using penalty function method.

In fitness penalization of a solution, the fitness function is the sum of the objective

function value and the sum of constraint violation.[5]

i.e. minimize

1 1

( ) 1000*n n

i i i l

i i

F P P D P

Power loss is obtained by DC power flow with the following assumptions made;

1. All voltage magnitudes are equal to 1.0 p.u.

2. The resistances of the branches are ignored; i.e., the susceptance of the branch is

1ij

ij

Bx

3. The angle difference on the two ends of the branch is very small, such that

sin

cos 1

ij i j

ij

4. All ground branches are ignored; i.e.,

0 0 0i jB B

And therefore the DC power flow model is

1 1

2 2

1 1n n

P

PB

P

The DC power flow is a purely linear equation, so only one iteration calculation is needed

to obtain the power flow solution. It is used in calculation of real power flow on

transmission lines and transformers. The power flowing on each line using DC power

flow is;

( )i j

ij ij i j

ij

P Bx

3.2.Parameter representation

The aim is to minimize the operating cost. The optimization is done by PSO technique.

The population of particles P, representing the generators where iP is the ith unit in the

power system, is initialized together with other variables. Each particle is generated

randomly within the allowable range specified by the generation limits of the particular ith

particle.

min maxi i iP P P

The size of the population, representing the number of generators is initialized along with

the initial and final inertia weight, random velocity of the particle, acceleration constant,

maximum generation, the number of iterations and Lagrange’s multiplier.

The fitness of each individual in the population is calculated using the fitness function,

which includes the cost function and the penalty function for violation of the equality and

inequality constraints.

1 1

( ) Equalityconstraints InequalityconstraintsT n

T i i

i i

C C P

Each unit’s position and velocity is updated along with the multipliers i for equality and

i for inequality constraints whose value can be 1 or 0 depending on whether the

particular constraint has been violated or has not been violated respectively.

If the number of iterations reaches the maximum, the individual that generates the latest

value is the optimal generation power of each unit with the minimum total generation

cost.

3.3.PSO Algorithm for SCED

PSO is a population based stochastic optimization technique. Each particle in the

population represents a candidate solution to the problem. All particles start at randomly

initialized positions and fly throughout the search space to find the best possible solution,

while communicating with each other and sharing the best local solutions each of them

has achieved. Based on the local and global information obtained, each particle updates

its position towards a desired global optimum.[4]

The elements of the PSO algorithm are as described below:

Particle, jX t – each particle is a candidate solution vector containing n

optimization variables. jX t is the thj particle at time t described as;

1 2 , , , ,j j j j nX t x t x t x t

The particle vector describes the particle’s position within the search space.

Population, Pop t – is a set of m particles at time t ,

Particle velocity, jV t – is the velocity of the thj particle at the time t in the n-

dimensional search space represented as;

1 2 , , , ,T

j j j j nV t v t v t v t

The velocity of the particle indicates the relative change of the particle within the

solution space with respect to its current position vector. For each time increment

a particle’s velocity demonstrates the time rate of change to the particle’s solution

vector.

Individual best ,j pbestX t – is the best position achieved by the thj particle so far

at time t . Each particle memorizes its best position throughout the entire searching

process.

Global best, gbestX t – is the best solution that has been achieved so far among

all the particles. The information of global best is known to all the particles

through communication among the particles.

The PSO algorithm is implemented in the order below

Initialization – at the start t = 0, all particles are initialized with a randomly

assigned position and velocity value. The thi dimensional position ,j ix of the thj

particle is initialized with a uniform random value between lower and upper

bounds. The thi dimensional velocity of the particle is initialized with a uniform

random value between – imaxv and imaxv , with ( ) ( )– i u i l

imax

im

x xv

N where ( )i ux and

( )i lx are the upper and lower bounds of the of the particles position respectively in

the thi dimension and imN is the minimum number of steps that change a particles

position from its lower bound to its upper bound of the thi dimension, a value

chosen by the user.

Velocity updating – during each iteration cycle the particle velocity is updated

according to the following formula.

1 1 2 2 1 , 1 1 1 1 ]j j pbest gbestV t w t V t c r Xj t Xj t c r X t Xj t

Where w t is the inertia weighting factor, 1c and 2c are two positive constants,

and 1r and 2r are uniform random numbers in [0, 1].

Position updates – with the velocity updated, each particle changes its position

according to the formula;

1 j j jX t X t V t

Process termination – the process stops when a specified stopping criterion is met,

for instance when the number of iterations reach a pre-specified maximum.

CHAPTER 4

RESULTS AND ANALYSIS

The proposed Particle Swarm Optimization algorithm was tested on IEEE – 30 bus

systems and results compared with those obtained from Classical Economic Dispatch

neglecting security constraints, as well as with results obtained from Genetic Algorithm.

The Network topology, load data, line limits and generator cost data for the systems are

taken from [1].

4.1.Case Study: IEEE 30-Bus System

Fig 4.1: One line diagram of IEEE 30-bus system [1]

4.2.Results

The optimal total generation, generation for the individual six generating units, the

optimal generation costs for each unit, the total generation cost and the system power

losses by PSO are given in tables 4.1 and 4.2 for SCED as well as CED for system

demand of 540MW and 840MW. Table 4.3 shows the comparison between the results

from the proposed PSO method with those from GA for a total demand of 840MW.

Table 4.1: Optimal generation for SCED and CED using PSO for a total demand of

283.4MW

Generation No. SCED CED

PG1 166.003 168.289

PG2 43.8662 50.1362

PG5 20.7474 26.4431

PG8 26.9202 23.1219

PG11 17.5307 10.8661

PG13 16.8103 13.2421

Total generation (MW) 291.878 292.098

Total cost ($/hr) 801.712 803.898

Total loss (MW) 8.47783 8.6982


374.3MW


PG1 175.305 193.811

PG2 61.6824 52.011

PG5 30.8379 34.0868

PG8 44.3274 45.164

PG11 40.6352 26.8235

PG13 31.2773 33.2283


Total cost ($/hr) 1172.5 1163.01

Total loss (MW) 9.76504 10.8245


540MW


PG1 109.883 149.128

PG2 80.7397 83.7191

PG5 68.1473 50

PG8 130.108 90.8636

PG11 81.3105 101.135

PG13 82.3566 80


Total cost ($/hr) 2268.03 2180.01

Total loss (MW) 12.545 14.8457

Table 4.4: Comparison of Economic Dispatch using PSO and GA for a total demand

of 283.4MW.

Generation No. PSO GA

PG1 166.003 175.7899

PG2 43.8662 48.2548

PG5 20.7474 22.0974

PG8 26.9202 22.3942

PG11 17.5307 12.3715

PG13 16.8103 11.3669


Total cost ($/hr) 801.712 802.3516

Total loss (MW) 8.47783 9.395

4.3.Analysis and Discussions

Fig 4.2: Variation of Real Power Losses with Power Demand for SCED and CED

0

2

4

6

8

10

12

14

16

0 100 200 300 400 500 600

Rea

l P

ow

er L

oss

es (

MW

)

Demand (MW)

SCED CED

Fig 4.3: Variation of Optimal Cost with Power Demand for SCED and CED

Fig 4.2 is a graph showing the variation of Real Power Losses with Demand for both

SCED and CED. The Real Power Losses increase with increasing Total Demand. In

comparison, the Real Power Losses are lower for the SCED case than or the CED. A

possible explanation for this could be due to the fact that SCED aims at operating within

the power flow limits and as a consequence reduces the power loss in the buses.

Fig 4.3 shows variation of Optimal Cost with power demand for SCED and CED. The

cost of generation is observed to increase with increasing demand, a fact that can be

attributed to higher fuel requirement for higher power generation. It can also be observed

that the cost of generation under SCED is higher than that under CED. The margin

between the two is more pronounced at higher demand. This can be explained from the

fact that higher costs occur when the transmission line constraints are violated. At low

power demand, the power flow is more likely within the bus limits, or if it goes above the

limits it is by a small magnitude. Increasing demand increases the strain on the system

resources in an attempt to meet this demand, and with that, increases the costs.

0

500

1000

1500

2000

2500

0 100 200 300 400 500 600

Op

tim

al C

ost

($/

hr)

Demand (MW)

SCED CED

CHAPTER 5

CONCLUSION AND RECOMMENDATIONS

5.1.Conclusion

The project utilizes the PSO algorithm to solve the security constrained economic

dispatch problem for HVDC. The procedure was tested on the IEEE 30-bus network with

six generators. The results of the various parameters from the SCED for HVDC using

PSO were compared with results of similar published works obtained using Genetic

Algorithm to verify the effectiveness of the proposed PSO algorithm. In this project, the

security aspects considered were the generator active power limits and the real power

flow limits of the buses.

From the comparisons done, the PSO algorithm exhibited the advantages of lower

optimal cost, lower total losses and higher probability of convergence to the global

optimum. The method is therefore appropriate for network flow analysis.

5.2.Recommendations

More research and study is necessary on HVDC systems with regards to security

constraints, and to economic dispatch as a whole.

Power flow in PSO code was found to be rather slow in execution. The computational

time could be reduced by lowering the number of iterations, which on the negative side

could increase the chances of settling at a local minimum thus inhibiting achieving of the

optimal solution, the global minimum.

Use of a Hybrid Particle Swarm Optimization algorithm could enhance the chances of

obtaining best results with a lower computational time. Such a method can be considered

in future projects.

REFERENCES

[1] Jizhong Zhu, Optimization of Power System Operation, New Jersey, Wiley-IEEE

Press, 2009.

[2] Xi-Fan Wang, Yonghua Song, Malcolm Irving, Modern Power Systems Analysis,

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APPENDIX

Appendix Table1: Generator data for IEEE 30-bus system [1]

Generators No.1 No.2 No.5 No.8 No.11 No.13

Pgimax(pu) 2.00 0.80 0.50 0.35 0.30 0.40

Pgimin(pu) 0.50 0.20 0.15 0.10 0.10 0.12

Qgimax(pu) 2.50 1.00 0.80 0.60 0.50 0.60

Qgimin(pu) -0.20 -0.20 -0.15 -0.15 -0.10 -0.15

Cost Function

ai 0.00375 0.0175 0.0625 0.0083 0.0250 0.0250

bi 2.00000 1.7500 1.0000 3.2500 3.0000 3.0000

ci 0.00000 0.0000 0.0000 0.0000 0.0000 0.0000

Appendix Table 2: Load data for IEEE 30-bus system [1]

Bus no. PD(p.u) QD(p.u) Bus no. PD(p.u) QD(p.u)

1 0.000 0.000 16 0.035 0.016

2 0.217 0.127 17 0.090 0.058

3 0.024 0.012 18 0.032 0.009

4 0.076 0.016 19 0.095 0.034

5 0.942 0.190 20 0.022 0.007

6 0.000 0.000 21 0.175 0.112

7 0.228 0.109 22 0.000 0.000

8 0.300 0.300 23 0.032 0.016

9 0.000 0.000 24 0.087 0.067

10 0.058 0.020 25 0.000 0.000

11 0.000 0.000 26 0.035 0.023

12 0.112 0.075 27 0.000 0.000

13 0.000 0.000 28 0.000 0.000

14 0.062 0.016 29 0.024 0.009

15 0.082 0.025 30 0.106 0.019

Appendix Table 3: Line flow limits data for IEEE 30-bus system [1]

constrained economic dispatchfor hvdc using particle swarm...

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