UNIVERSITY OF NAIROBI FACULTY OF ENGINEERING DEPARTMENT OF ELECTRICAL AND INFORMATION ENGINEERING ECONOMIC DISPATCH FOR A HYBRID HVDC AND HVAC SYSTEM PROJECT INDEX: PRJ 057

BY WASWA LEWIS SAKWA F17/1441/2011 SUPERVISOR: PROF. N. O. ABUNGU

EXAMINER: MR. GEVIRA OMONDI PROJECT REPORT SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE AWARD OF THE DEGREE OF BACHELOR OF SCIENCE IN ELECTRICAL AND ELECTRONIC ENGINEERING OF THE UNIVERSITY OF NAIROBI 2016 Submitted on: 18TH MAY 2016

Dedication To my parents- from whom I gather my daily inspiration, and to people and places far yonder, who believed and encouraged me; for in that conviction you gave me a chance to experience much and taught me the one true philosophy- From those to whom much has been given, much will be required.

Acknowledgement I am grateful to God, for His unending Grace and the serenity He accorded me through my studies and during the period within which this project was done. For I am nothing without Your grace. This project would not have come to completion without the support, motivation and guidance of my supervisor, Prof. Nicodemus Odero Abungu. Your nurture and preparation for the project is definitely beyond measure and I cannot quantify my gratitude. You remain a special person and mentor in my academic journey. Sincerest appreciation to Mr. Peter Musau for your patience with me, for your unrelenting effort and for the numerous proof reads and checks of this work. Yours is support unforgotten and true. I forever remain indebted to you. Special gratitude to Dr. Cyrus Wekesa for his series of lectures which were instrumental in the timely solution of the Project problem. And to my classmates, Edwin Owuor Oloo, Bernard wuod Owinga, John Kipkoech, Daniel Bundi, Ken Ngala and the others in whose absence the project completion wouldn’t have been possible. Your indelible advice and sharing is forever cherished. Special mention is Kassim Makata for his encouragement throughout the project. Finally, the possibilities of this fete would not have been realized were it not for Dr. Otmar and Carina Schimana. Your glorious actions have brought me this far and forever I am grateful. And for you I thank God.

Declaration and Certification

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

2) 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.

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

4) 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.

5) 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: ……………………………………………………………………………………… This Project is submitted to the Department of Electrical and Information Engineering, University of Nairobi with my approval as the Student Supervisor: Prof. Nicodemus Odero Abungu Date: …………………………..

NAME: WASWA LEWIS SAKWA

ADMISSION NUMBER: F17/1441/2011

COLLEGE OF ARCHITECTURE AND ENGINEERING

SCHOOL OF ENGINEERING

DEPARTMENT OF ELECTRICAL AND INFORMATION ENGINEERING

BSC. ELECTRICAL AND ELECTRONIC ENGINEERING

A DYNAMIC ECONOMIC DISPATCH FOR A COMBINED HVDC AND

HVAC TRANSMISSION USING PARTICLE SWARM OPTIMIZATION

ALGORITHM

Table of Contents Dedication ........................................................................................................................................... 1

Acknowledgement ............................................................................................................................... 2

Declaration and Certification ............................................................................................................. 3

List of Abbreviations .......................................................................................................................... 6

Abstract ............................................................................................................................................... 7

Chapter 1 .............................................................................................................................................. 8

1.1.1 Economic Load Dispatch ...................................................................................................... 8

1.1.2 Hybrid System .................................................................................................................... 8

1.1.3 HVAC and HVDC Hybrid System ........................................................................................... 8

1.1.4 Economic Load Dispatch for a Hybrid HVAC and HVDC system............................................ 8

1.2 Problem Statement ..................................................................................................................... 9

1.2.1 Project Objectives .............................................................................................................. 9

1.2.2 Project Questions ............................................................................................................... 9

1.2.3 Scope of the Work .............................................................................................................. 9

1.3 Organization of Report ............................................................................................................... 10

Chapter 2 ........................................................................................................................................... 11

2.1 Economic Load Dispatch Literature Review ................................................................................ 11

2.1.1 Significance of Economic Dispatch .................................................................................. 11

2.1.2 Stages of System Control.................................................................................................. 12

2.2 HVDC Literature Review .......................................................................................................... 12

2.2.1 Introduction to HVDC ..................................................................................................... 12

2.2.2 The Converter Station ...................................................................................................... 12

2.2.3 Types of HVDC links and configurations ........................................................................ 13

2.2.4 Need for HVDC ................................................................................................................ 13

2.2.5 Factors to Consider in HVDC transmission line Design ................................................. 13

2.2.6 Comparison between HVAC and HVDC ........................................................................ 14

2.2.7 HVDC losses ..................................................................................................................... 15

2.2.8 Combination of HVAC and HVDC in the project .......................................................... 16

2.3 Principle of equal incremental rates .......................................................................................... 16

2.4 Economic Load Dispatch Neglecting Losses .............................................................................. 17

2.5 Economic Load Dispatch including Transmission Losses ......................................................... 18

2.6 Economic Load Dispatch Problem Statement and Formulation ............................................... 19

2.6.1 Problem objectives ........................................................................................................... 19

2.7 Problem constraints .................................................................................................................... 20

2.7.1 HVDC constraints ............................................................................................................ 20

2.7.2 HVAC constraints ............................................................................................................ 21

2.7.3 HVAC/HVDC hybrid constraints .................................................................................... 22

Chapter 3 ........................................................................................................................................... 23

3.1Methodology ................................................................................................................................. 23

3.1 Deterministic methods ................................................................................................................ 23

3.1.1 Newton’s Method ............................................................................................................. 23

3.1.2 Quadratic programming .................................................................................................. 23

3.1.3 Nonlinear Programming .................................................................................................. 23

3.2 Heuristic Methods....................................................................................................................... 24

3.2.1 Genetic algorithm (GA) ...................................................................................................... 24

3.2.2 Particle Swarm method .................................................................................................... 24

3.2.3 Simulated annealing (SA) ................................................................................................... 25

3.2.4 Tabu search ...................................................................................................................... 25

3.3 Summary of methodologies ...................................................................................................... 25

3.4 Review of Particle Swarm Optimization technique ................................................................... 26

3.4.1 Implementation ................................................................................................................ 27

3.4.2 Representation of a swarm............................................................................................... 27

3.4.3 Swarm initialization ......................................................................................................... 28

3.5 Objective function evaluation ..................................................................................................... 28

3.6 Initialization of best positions ..................................................................................................... 28

3.7 Movement of the particles .......................................................................................................... 28

3.8 Updating the Best and Worst Positions ...................................................................................... 28

3.9 Stopping Criterion ...................................................................................................................... 28

3.10.0 The Pseudo code ................................................................................................................... 29

Chapter 4 ........................................................................................................................................... 31

Results and Analysis ......................................................................................................................... 31

Chapter 5 ........................................................................................................................................... 39

Conclusion and Recommendations ................................................................................................... 39

Recommendation .............................................................................................................................. 39

Appendix 1 ........................................................................................................................................ 40

Appendix 2 ........................................................................................................................................ 41

References ......................................................................................................................................... 63

List of Abbreviations

ED: Economic Dispatch

ELD: Economic Load Dispatch

DED: Dynamic Economic Dispatch

CED: Classical Economic Dispatch

HVAC: High Voltage alternating Current

HVDC: High Voltage Direct Current

FACTS: Flexible Alternating Current Transmission System

MTDC: Multi- Terminal Direct current

GA: Genetic Algorithm

PSO: Particle Swarm Optimization

SA: Simulated annealing

TS: Tabu Search

Abstract Scarcity of Energy resources, increasing power generation cost and ever-growing demand of electrical energy necessitates optimal economic dispatch in today’s power systems. And with large interconnections of electric networks and the energy crisis around the world coupled with continuous rise in prices, it is essential to reduce the running charges of the electric energy by reduction of the fuel consumption for meeting a particular load demand. In Economic Load Dispatch (ELD), the generations are not fixed but are allowed to take values again with certain limits to meet particular load demands with minimum fuel consumption. This thus implies that ELD is really the solution of large number of load flow problems and choosing one which is optimal in the sense that it meets minimum cost of generation should be a priority for power engineers. The project is geared towards the formulation and implementation of an Economic Load Dispatch for a hybrid system composed of two forms of transmission- The High voltage alternating current (HVAC) as well as High Voltage Direct Current (HVDC) The proposed test system consists of the analytical computation and simulation of a six unit thirty bus test system has been performed using the MATLAB environment and with the aid of The Particle Swarm Optimization methodology. The results are analyzed to determine the best mode of transmission in form of reduced losses and reduced cost.

Chapter 1

1 INTRODUCTION

1.1 Economic Dispatch of a Hybrid HVAC and HVDC system using Particle Swarm

Optimization Technique.

1.1.1 Economic Load Dispatch Economic load dispatch is defined as the process of calculating the generation of the generating units so that the system load is supplied entirely and most economically subject to the satisfaction of the constraints. The objective of the economic dispatch is to minimize the total system fuel cost by adjusting the power output of each of the generators connected to the grid. It involves the determination of power output of each plant, and each unit in the plant such that the overall cost of the fuel needed to supply the system load is minimized. [5] 1.1.2 Hybrid System A hybrid system is a system that comprises of two or more systems of operation. It exhibits characteristics of both systems. In Power systems, hybrid systems can be in terms of generation or transmission where there is the combination of the conventional power generation utilities with the unconventional ones. Or for the case of transmission where there is an adoption of both the transmission as direct and alternating currents. 1.1.3 HVAC and HVDC Hybrid System This is a system that employs the conventional HVAC system of generation but a hybrid technique of transmission. This can be employed through two forms which include: a) Parallel operation of a DC link with an AC network. b) Superimposing the DC current on the AC wave on the same transmission line

In the case of parallel DC/AC operation, there is an increase in the transmission capacity of the AC line in as the angle of operation is increased significantly from about 30° to about 80°. And guided by the power transmission equation P = ����

� sin� this represents close to 95% increase. [1] [2] [3]

In the case of Superposition of the DC on the AC waves, there is a general cost effective use of the existing infrastructure. [2] 1.1.4 Economic Load Dispatch for a Hybrid HVAC and HVDC system This refers to the process of calculating and allocating the power generation to power generating units in such a way that the total system demand is supplied most economically. [4]. In this power system, the generation is in the AC mode and the transmission is in both the AC and DC modes. In this system, the ED for the HVDC system involves allocation of generating units in a power system with HVDC lines considering the operational limitations of the generator and HVDC transmission [5]. The ED for the HVAC considers the transmission losses for HVAC lines as well as the limitations of the generating facilities. The ED for the hybrid system considers the combined transmission losses for the two systems and the limitation of the generating facilities. Thus Economic load dispatch for the a hybrid HVAC and HVDC transmission system considers the optimal selection of the available and operational generating sources to be able to meet the system

demand in an environment constrained by both HVAC and HVDC transmission losses as well as generation limits. 1.2 Problem Statement The project seeks to develop an economic dispatch solution for a power system that incorporates AC generators but in two different scenarios. The solution is to be developed separately considering the generator constraints as well as the transmission losses for a combined HVAC/HVDC transmission system. The solution therefore requires a HVDC ED to be developed separately and later in combination with the HVAC system. This project is to be carried out in a MATLAB environment aided by the use of Particle Swarm Optimization (PSO) algorithm. The algorithm has been therefore studied in detail. 1.2.1 Project Objectives The major objectives of this project are outlined below:

1. To develop flexible and extensible computational framework as general environment for implementing the various algorithms for economic dispatch solution.

2. To formulate the economic dispatch problem for HVAC generation with and without

losses putting into consideration the generator constraints

3. To formulate the economic dispatch problem for HVDC transmission with losses and without losses

4. To formulate the economic dispatch problem for a combined HVAC and HVDC transmission both with and without losses where there is superposition of DC on the AC wave using a double AC transmission line

5. To analyze the Performance of the combined HVAC/HVDC hybrid system as compared to either a HVDC transmission system or a solitary HVAC transmission system.

1.2.2 Project Questions

ü How to carry out optimal allocation of generation units to the available generating units so

as to meet the load demand factoring in transmission losses for a hybrid HVAC/HVDC transmission system

ü Is the proposed technique effective in solving the Economic Dispatch problem above? 1.2.3 Scope of the Work This Project shall focus on the formulation of Economic dispatch for -:

a) The High Voltage Alternating Current (HVAC) Generation and its Generational losses b) The High Voltage Direct Current (HVDC) Transmission coupling in the Transmission

losses for the HVDC transmission c) The Combined HVAC and HVDC transmission taking into account the losses along the

transmission lines as well. The project seeks to carry out comparison between the solitary transmissions (HVAC and HVDC) and the Hybrid transmission.

1.3 Organization of Report This project report is subdivided into five sections namely-: Ø Chapter 1- Introduction. Introduction to the concepts of High voltage Direct Current, its

need in the growing energy markets, its technologies. In this chapter there is brief introduction to economic load dispatch and its importance in the power system operation. Both the scope and the objectives of this project are handled in the introduction section.

Ø Chapter 2- Literature Review. There is an in depth analysis of Concepts of HVDC, its comparison with HVAC both technical and economical. There is also an analysis of the combined AC/DC systems. In this section under the Economic Load Dispatch review, problem statement and formulation is handled.

Ø Chapter 3- Methodology. This section contains the implementation procedures for the ED based on Particle Swarm Optimization (PSO) technique

Ø Chapter 4 – Results and Analysis

Ø Chapter 5 - Conclusion

Chapter 2 Literature Review

2.1 Economic Load Dispatch Literature Review Economic dispatch, by definition is an on-line function, carried out after every 15- 30 minutes or on request in Power Control Centers. It is defined as the process of calculating the power generation of the generating units in the system in such a way that the total system demand is supplied most economically. Thus the actual economic dispatch problem is non-convex in nature requiring the accurate, robust and fast solution methodology. [6] [4]

ELD problem involves the solution of two different problems

- Unit commitment or Pre dispatch problem where in it is required to select optimally out of the available generating sources to operate to meet the expected load and provide a specified margin of operating reserve over a specified period of time.

- On- line ED where it is required to distribute the load among the generating units actually paralleled with the system in such a manner as to minimize the total cost of supplying the minute to minute requirements of the system. [7] [4]

Economic dispatch is generation allocation problem and defined as the process of calculating the generation of the generating units so that the system load is supplied entirely and most economically subject to the satisfaction of the constraints To determine the economic distribution of load between the various units, it is important that we express the variable operating costs in terms of output power. Given that most of our electric energy will continue to come from fossils and nuclear fuels for many years, it is imperative that the fuel consumption and the quantities must be taken into consideration. Therefore generational economics are related to the fuel cost economics realizing that the other costs that are a function of power can be included in the cost of fuel

2.1.1 Significance of Economic Dispatch

The operational planning of the power system involves the best utilization of the available energy resources subjected to various constraints to transfer electrical energy from generating stations to the consumers with maximum safety of personal/equipment without interruption of supply at minimum cost. In modern complex and highly interconnected power systems, the operational planning involves steps such as load forecasting, unit commitment, economic dispatch, maintenance of system frequency and declared voltage levels as well as interchanges among the interconnected systems in power pools etc. [8] [7] [9] Dynamic ED determines the optimal generation schedule of on-line generators so as to meet the predicted load demand over a time horizon satisfying the constraints that are dealt with in the problem formulation. It takes into consideration the generating ramp rate limits, valve point effect as well other non-convex problems. [10], [11] [8]

2.1.2 Stages of System Control There are three stages in system control, namely generator scheduling or unit commitment, security analysis and economic dispatch. [4]

• Generator scheduling involves the hour-by-hour ordering of generator units on/off in the system to match the anticipated load and to allow a safety margin.

• With a given power system topology and number of generators on the bars, security analysis assesses the system response to a set of contingencies and provides a set of constraints that should not be violated if the system is to remain in secure state.

• Economic dispatch orders the minute-to-minute loading of the connected generating plant so that the cost of generation is a minimum with due respect to the satisfaction of the security and other engineering constraints.

2.2 HVDC Literature Review

2.2.1 Introduction to HVDC

The use of DC for day to day application is much older than that of AC. The first central electric station was installed by Edison in New York in 1882 and it operated at 110V DC. The use of transformers for transmitting power over long distances and at a higher voltage justified the use of AC especially where the electric energy to be harnessed was from Hydro Electric sources available from far flung load centres. To couple up this, there was the use of Polyphase induction motors which up to now serve the majority of industrial and residential purposes which were simpler and rugged in construction as compared to the DC motors of the same rating.

The electric power grid is experiencing increased needs for enhanced bulk power transmission capability, reliable integration of large-scale renewable energy sources, flexible power flow controllability and interconnections between asynchronous AC networks around the world. However, it has become a challenge to increase power delivery capability and flexibility with conventional AC expansion options in meshed, heavily loaded HVAC networks. As such, upgrading electric power grids with advanced transmission technologies such as HVDC systems becomes more attractive in many cases so as to achieve the needed capacity improvement while satisfying strict environmental and technical requirements.

Alternating Current pervasive use could be attributed to the earlier problems encountered in the DC Distribution and generation. Commutators in DC machines imposed limitations on voltage, speed and size due to commutation problem. Operating a machine at a high voltage requires a relatively larger diameter commutator which then restricts the speed of the machine due to centrifugal force. A low speed machine is costlier and heavier than a high speed machine of equal rating. For all these reasons, there is a necessity to generate, transmit and distribute power in Alternating current form. There are however technical reasons for the use of DC [1] [12] [13]

2.2.2 The Converter Station

The three main elements of a HVDC system are the converter station at the transmission and receiving ends, the transmission medium and the electrodes. The converter stations at each end are replicas of each other and therefore consist of all the needed equipment for going from AC to DC or vice versa.

The main components of converter stations are converter transformers, thyristor valves, VSC Valves, DC filters, AC filters and capacitor banks .Conversion of electrical current from AC to DC using a rectifier at the transmitting end, and from DC to AC using an inverter at the receiving end is the fundamental process that occurs in a HVDC system. Two basic converter technologies used in modern HVDC transmission systems are traditional/classical line commutated current source converters (CSCs) and self-commutated voltage source converters (VSCs). [17] [1] 2.2.3 Types of HVDC links and configurations HVDC configurations can be used for both VSC and CSC converter topologies and are classified into monopolar, bipolar and homopolar .The monopolar link has only one conductor and the ground serves as the return path. The link normally operates at negative polarity as there is reduced radio interference and less corona loss. This configuration is usually preferred in the case of cable transmissions with submarine connections. The bipolar links have two conductors, one operating at positive polarity and the other operating at negative polarity. The advantage of this configuration is the fact that one of the poles can continue to transmit power as a monopolar link with a ground return path if the other link is out of service. Theoretically, the ground current is zero in the case of this configuration since both poles operate with equal current. This is the most common configuration for modern HVDC transmission lines. The Homopolar links have two or more conductors having the same polarity (usually negative) and always operate with ground or metallic path as return. This configuration is not being used recently. [18] [1] [12] [13] [17] 2.2.4 Need for HVDC In spite of the widespread use of alternating form of current in generation, transmission and distribution of power the High Voltage Direct current is finding application in areas which use of AC is unlikely. These include. [19] [6] [12]

i) Due to Large charging currents, use of HVAC for underground transmission was prohibited for long distances

ii) Parallel operation of AC with DC in which case there is an increased stability limits for the system or interconnection of 2 large AC systems by DC transmission tie line. The DC line is an asynchronous link between 2 rigid systems where otherwise slight difference in frequency of the two large systems would produce serious problems of Power Transfer Control in the small capacity link.*

iii) DC transmission is important where there is need for addition of Power in feed without significantly increasing the short circuit level of the receiving AC System.

iv) Transmission of bulk power where AC would be uneconomical, impractical or subject to environmental restrictions

v) Improvement of AC system performance by fast and accurate control of HVDC power

It is indisputable that generation by AC, transmission by DC and distribution by AC are the most economical aspects of economic power dispatch that cannot be ignored. Thus, AC and DC are complementary for optimal design of a HVDC transmission system, many factors need to be considered which include power capacity to be transmitted, type of transmission medium, distance of transmission, voltage levels etc. [1]

2.2.5 Factors to Consider in HVDC transmission line Design For optimal design of a HVDC transmission system, many factors need to be considered which include-: [3] [12] [13] Ø Power capacity to be transmitted

Ø Type of transmission medium, Ø Distance of transmission, Ø Voltage levels, Ø Temporary and continuous overload, Ø Status of the network on the receiving end, Ø Environmental conditions and other safety and regulatory requirements.

However it’s still difficult to attach a price tag on the cost of a HVDC system. The choice of DC transmission voltage level has a direct impact on the total installation cost. At the design stage, an optimization is done finding out the optimum DC voltage from investment and losses point of view. In the evaluation of losses, the energy cost and the time horizon for utilization of the transmission have to be taken into account. Finally the depreciation period and desired rate of return (or discount rate) should be considered. Therefore, to estimate the costs of an HVDC system, it is recommended that life cycle cost analysis is undertaken. One great challenge in economic transmission planning is to quantify the benefits of a transmission upgrade project and to accurately evaluate the economic impacts. To evaluate the economic impact of a transmission project, it is necessary to quantify if the proposed project leads to more efficient ED and brings economic benefits to the consumers in the markets. [1] 2.2.6 Comparison between HVAC and HVDC HVDC transmission can be compared with the HVAC transmission basically from two points of view, the technical and the economic points of view respectively. These are discussed briefly in the next subsections. Economic point of view is thus weightier as far as this project is concerned. 2.2.6.1 Technical Comparisons As aforementioned, in the need to have HVDC connections in transmission, it is worth noting that HVDC transmission overcomes some of the technical problems which are usually associated with the HVAC transmission. These include but are not limited to-:

i) Interconnection of asynchronous networks ii) Congestion management iii) Submarine and underground power transmission iv) Reduction of skin effect v) Power flow control vi) Stability analysis and connecting a remote generating plant to the distribution grid

2.2.6.2 Economic Comparisons In terms of economics, HVDC is suitable for bulk power delivery over long distances at reduced cost due to reduced number of conductors and insulators, bears little environmental impact, and allows reserve sharing at increased efficiency. Also, HVDC uses light and cheap towers and there is less phase-to-phase and phase-to-ground clearances. In the Bipolar links there is reduced power loss as there are two conductors. The HVDC have bundled conductors leading to reduced corona losses. The total capital cost of a transmission system must be equal to the sum of the capital cost of the substations plus the capital cost of the lines. The variation of total costs for HVAC and HVDC as a function of line length is as shown in the figures that follow. As illustrated, there is a break-even distance beyond which the total costs of the DC option will be lower than the AC transmission option. The break-even distance depends on several factors such as

transmission medium cable or Overhead line and different local aspects such as permits, cost of local labour etc. For overhead lines it’s in the range of 500 to 900 km, for submarine cables between 25 and 50 km and twice as far for underground cables.

figure2.2: Cost/Distance curve 1

Transmission cost as a function of line length for both an AC and DC system. [12] [1] The investment costs for HVDC converter stations are higher than for HVAC substations. However, the costs of transmission medium overhead lines and cables, land acquisition/right-of-way costs operation and maintenance costs are lower in the HVDC case. Moreover, Initial loss levels are higher in the HVDC system, but they do not vary with distance, In contrast with HVDC system where loss levels increase with distance. Based on the latest research and developments, HVDC systems offer many promising technical and economic benefits thus is a promising approach to best utilizing transmission grid for maximized social welfare. However, major technology breakthroughs are required to overcome problems inherent in HVDC systems such as; increased cost of converter stations, problem of circuit breaking, HVDC system control, effective cooling voltage transformation challenge and reactive power requirement and the generation of harmonics. [16] [19] [19] 2.2.7 HVDC losses A HVDC system incurs losses during the rectification and inversion processes- which accounts for most of the losses in the system. It however incurs lesser transmission line losses as compared to a HVAC system. Typical overall loses in HVDC transmission are 30% to 50% less than HVAC transmission. [6] [12] Corona losses in power HVDC projects that span up to around 1800MW, the corona loss is given as 0.13% and the copper losses are witnessed too. The converter transformer used in HVDC experience normal operating loss due to the flow of sinusoidal current and has additional losses due to harmonics that flow through the device. [2] Active power loss of the converts can be expressed as a quadratic function of the phase current of the converters valves. It is approximated with a quadratic equation below. P conv losses =� +�Ic + �I2 c and Ic is the converter current given by the quotient of the converter apparent power and the voltage. [2] [3] [13]

2.2.8 Combination of HVAC and HVDC in the project This project used the theoretical relation that exists between the losses in the HVAC system and the HVDC system losses. As asserted above, the HVAC losses are determined by the use of Power flow analysis or by the direct use of B coefficients. The losses henceforth found in such a system are deemed to be 2.5 times the losses that are found in the HVDC system. [12]. Thus while combining the two, it was also imperative that the HVDC lines take a portion of the system and the HVAC system assume a fraction of the transmission lines in the implemented system. In the project, generator 2 was determined to supply an approximate of 115.9MW of power and this portion was allocated to the HVDC lines while the HVAC lines supplied the rest of the fraction. This was a weighting factor of about 0.4 of the total network capacity (283.4MW). 0.6 of the generation was therefore transmitted by the HVAC lines. This has been dealt with in detail in the results and analysis section of the report. 2.3 Principle of equal incremental rates

Classic ED problem is effectively solved when the incremental cost of production cost of one plant is equal to the incremental rate of the next plant on the bus bar. That is to say that

������

= ������

= ������

= λ

The principle of incremental rates is also applied on the determination of the cost incurred in supplying for an extra increment in the load. It is important to note that for there is a definite increase in the transmission losses whenever there is an increment in the load.

If ∆PD is the change in the load and if the ∆PD is supplied by the plant alone and in doing so, the generation required at a plan is∆Pn which includes transmission losses in addition to increase in demand, the cost of power will thus be

������

∆Pn = cost in $/hr

The cost of generation therefore takes into consideration the losses and the demand. In other words, total Demand is given by the total generation less the transmission losses. To mitigate in this losses, there is need to express the relation between the load and the demand in a fraction for the true fuel cost to be accounted for. [6, 7, 8, 15]

������

∆��∆��

= λ

Also, ∆PD = ∆Pn - ∆PL

We can thus write out our equation as λ =������

. ∆��∆���∆��

= ������

. �

�� ∆��∆��

Thus we can conclude that if we factor in the losses the incremental cost change as above. It can be rewritten as

������

. �

�� ������

= λ

This could be described as λ = ������

.ai where ai = �

�� ������

thus we can write the coordination

equation of the economic power operation as λ = ������

.ai i= 1, 2, 3…n

In summary for a thermal system economic power dispatch, the solution is worked out as follows:

ü Pick a set of starting values PG0i that sum up to the demand ü Calculate the incremental fuel costs ü Calculate the incremental losses as well as the total losses ü Calculate the value of λ and PGi according to the coordination equation and the power

balance constraint ü Compare PGi from above with starting point. If there is no significant change, terminate the

procedure, otherwise go to point 2 [6] [5] [16] [7]

2.4 Economic Load Dispatch Neglecting Losses

The economic load dispatch problem is defined as

Min FT = ∑ ������ subject to PD= ∑ ���

��� where Ft and PD are the fuel input to the nth unit and the total load demand. Pn is the generation of the nth unit as well.

We make use of the Lagrangian Multiplier, we obtain the auxiliary function as

F= FT + �(PD -∑ ����� n) where λ is the Lagrangian multiplier. The constraint is that the demand

under which the objective function is must be minimized using the Lagrangian.

This is such that we need to get the extreme values of the Lagrange function.

������

= ������

− λ =0

This can be written as �������

= λ.

Considering incremental fuel rates then �������

= �������

=λ

�������

Is the incremental production cost of plants I Kshs per MWHR. The incremental production cost of a given plant over a limited range is represented as

�������

= FnnPn + fn where Fnn is the slope of incremental cost curve, fn is the intercept of incremental production cost curve.

If the constraint of Power output is put in place as well, the problem can be redefined as

MinF = F1 (PG1) + F2 (PG2) + F3 (PG3) +…FN (PGn) =∑ FiPGi

∑Pgi = PD

PGmin ≤ PGi ≤ PGmax

It is important that the equal incremental principle be maintained in the calculation process after which the output limits for each unit is determined. If the power output of the unit is out of the limit, then the corresponding new limit is set as below.

If Pgk≥ Pgk, PGK = PGK max

If PGK ≤ PGKmin, PGK = PGKmin

We then handle the violated units as negative loads to be able to re-establish new demands of the system.

P’DK = -PGK where K is the numbers of the violated units- K= 1,2,…nK

We then re-compute the power balance such that

∑Pgi = PD + ∑P’DK and finally we recursively go back to the start to ensure that the power balance constrain holds.

2.5 Economic Load Dispatch including Transmission Losses

The optimal dispatch problem that includes losses can be defined as-:

Min FT = ∑ Fi subject to PD + PL - ∑Pgi = 0 and PL is the total system loss. By making use of the Lagrangian

F = FT + �(PD +PL- ∑ ����� �i)

To be able to determine incremental transmission loss, then �����

= ������

+ �(������

− 1) = 0

We can therefore write ������

+ �(������

) = � and ������

is the incremental transmission loss and it is expressed in $/MWhr

The equation ������

+ �(������

) = � is formed in sets such that we have n equations with n+1 unknowns. These are known as coordination equations as they coordinate the incremental transmission losses with incremental cost of production.

To solve these equations, the loss formulae equation is expressed in terms of generations and approximately expressed as

PL = ∑∑PmBmnPn

Where Pm and Pn are the source loadings and Bmn are the transmission coefficients

The solution of the coordination equation requires that the calculation of ������

. This can be

obtained as ������

= 2 ∑ Bmn Pm and ������

= FnnPn +fn

The coordination equations can be written as

FnnPn + fn + λ ∑2BmnPm = λ

Collecting the Pn coefficients, Pn (Fnn + λ2Bmn) = - λ (∑2BmnPm) – fn + λ

Hence Pn = ����

� ∑ ��� ������

� �����

To solve for optimal load dispatching it is important that the following steps be made-: [6]

- Assume the very suitable value of λ0. It should be made the largest intercept of incremental production.

- Calculate generations based on equal incremental production costs.

- Calculate the generation at all buses using the equation Pn = ����

� ∑ ��� ������

� �����

- Powers to be substituted on the right hand side during the zeroth iteration correspond to the values obtained during the second

- Check if the differences in power at all generator buses between two consecutive iteration is less than a pre-specified level. If not go to step three.

- Calculate losses by the relation PL = ∑∑PmBmnPn ; ∆P = | ∑ PG – PL- PD |

- If the ∆P is less than � stop the calculation and calculate the cost of generation with these values of power.

[6] [7] [4] [16] [11] 2.6 Economic Load Dispatch Problem Statement and Formulation The objective of solving economic dispatch problem is to minimize the fuel cost of electric power system, while satisfying a set of constraints. This can be formulated as follows: 2.6.1 Problem objectives

• The objective of DED is to determine the generation levels for committed units which

minimize the total operating cost over a dispatch period while satisfying a set of constraints. DED problem is given by : Min C (�) = ∑∑ ��(��)�

��� subject to Load generation balance ∑ ��

��� it = Dt + Losst where D is the demand and the Loss is reference to transmission loss. The fuel cost equation is given as below Ci(Pi) = ai + biPi + ciP2i + |ei sin(fi(Pimin - Pi))| [ [12], [4], [10] ]

Where, Fi (Pi) is the fuel cost (Kshs/hr), Pi is the power generated (MW) and ai, bi, ci is the fuel cost coefficients of ith unit. This is done under the following limits-: [12] [11] [13] Ramp rate limits -DRi.T≤ Pt+ 1 – Pti≤ URi.T t= 1, 2…N-1

Line flow limits -Fmax L ≤ Fti ≤ Fimax L = 1, 2… L The transmission losses of a system under DED consideration is given by the following equation. It is evident that we express network loss as a function of power injection at each node where power loss is expressed via B coefficients. This is also called the Kron’s formula for loss calculation [14] [15] PL = ∑∑PiBijPj + ∑PiB0i + B00 where [6] [11] [1] [16] I= Number of generators J= Number of buses on the system

Figure 2.1: Power Cost/output curve 1

2.7 Problem constraints

2.7.1 HVDC constraints In HVDC inequality constraints are usually the operational or physical limits for example, Thermal limits such as-:

- Transmission line constraints - Bus voltage limits should be within the insulation limits - Generating units have a PG min and PG max production limits

The constraints restrict the Economic Dispatch of generators to a range between a maximum and minimum values. These are inclusive of-:

- Power Generation capacity constraints in which PGi Min ≤ PGi ≤ PGi max

- Tap Ratio converter should be restricted such that T min ≤ T ≤ T max

- Ignition angle of the converter should be such that �min ≤ � ≤ � max

- Extinction angle of the converter should be such that � min ≤ � ≤ � max

The DC currents are also constrained such that I dc min ≤ I dc ≤ I dc max The DC voltage is also under the constraint of -: V min ≤ V ≤ V max

Generation capacity and Converter constraint: For stable operation, real power output of each generator is restricted by lower and upper limits as follows. The generator constraints are also limited by the converter limits. Pi min ≤ Pi ≤ P max, i=1, 2… n Power balance constraint: The total power generation must cover the total demand PD and the real power loss in transmission lines PL. Hence, ∑ Pi�

��� = PD + PL MWHR. This equation takes into consideration the losses in a system. These are expounded upon as below-:

The total cost of generation is a function of the individual generation of the sources which can take values within constraints. The cost of generation will depend upon the system constraints for a particular load demand. This means that the cost of generation is not fixed for a particular load demand but depends upon the operating constraints of the sources.

2.7.2 HVAC constraints

a) Generator Constraints:

The KVA loading on the generator is given by √ (P2 p+Q2 p) this should not exceed a pre-specified value Cp because of the temperature rise conditions. [1] [7]

The maximum active power generation of a source is limited by the thermal consideration and also the minimum power generation is limited by the flame instability of a boiler in the case of a thermal source. [1] [4]

If the power output of a generator for optimum system operation is less than the pre-specified value Pmin the unit is not put on the bus bar because it is not possible to generate that low value of power from that unit. Hence the generator powers Pp cannot be outside the range state by the inequality.

Ppmin ≤ Pp ≤ Ppmax

Similarly the maximum and the minimum reactive power generation of a source are limited. The maximum reactive power is limited because of the overheating of the rotor and minimum is limited because of the stability limits of the machine. Hence the generator reactive power Qp cannot be outside the range stated by the inequality [6] [1]

Q p min ≤ Qp ≤ Q p max

b) Voltage Constraints:

It is essential that the voltage magnitudes and phase angles at various nodes should vary within certain limits. The voltage magnitudes should vary within certain limits because otherwise most of the equipment connected to the system will not operate satisfactorily or additional use of voltage regulating devices will make the system uneconomical.

| V p min | ≤ | V p | ≤ | V p max |

�p min ≤ � p ≤ � p max

Where V p and � represent the voltage magnitude and the phase angle. Under normal operation the operational angle of a transmission line is between 30 to 45 degrees.

c) Transmission Line Constraints

The flow of active and reactive power through a transmission line circuit is limited by the thermal capabilities of the circuit and is expressed as

C p ≤ C p max where C p max is the maximum loading capacity of the line in reference.

d) Running Spare Capacity Constraint

This constraint meets the forced outages of one or more alternators on the system and also the unexpected load on the system. It is described such that in addition to meeting the load demand and also losses a minimum spare capacity should be available such that

G ≥ Pp + P so where G is the total generation and P so is some pre-specified power. A well planned system bears P so at its minimum.

2.7.3 HVAC/HVDC hybrid constraints

For a mixed AC/DC system, the power constraint is mainly on the power losses for the combined system since the only point of relation is on the transmission. The constraints may be as follows-: [14, 13]

2.7.3.1 Converter Capacity constraint

Operation of the converter is usually constrained by the maximum current that go through its valves and the maximum allowable dc voltage as elaborated in the DC constraints in this section. The allowable current determines the maximum converter apparent power.

i) Maximum apparent Power- √(P2c + Q2

c) ≤ | VcI valve max | ii) Maximum Reactive Power constraint. The maximum reactive power is limited by the

maximum DC bus voltage Qc max = B r max (V2 max -VcVf�(�����)) which setting��− �� = 0 yields Qc max = B r max (V2 max -VcVf)

iii) Minimum reactive Power Constraint- which is determined as a specific requirement for a particular project.

Generally for a mixed AC/DC system, total active power loss includes

a) Active Power loss from the AC Transmission lines b) Active DC power loss on a cable

c) Active power loss of the converter which was earlier defined.

Chapter 3

3.1 Methodology

Optimization methods can be grouped mainly under two categories- These two categories are [16] [20]

1) Deterministic Methodologies 2) Heuristic Methodologies

3.1 Deterministic methods

These refer to algorithms that follow a rigorous mathematical approach. Strictly speaking, it refers to mathematical programming. These methods include but are not limited to:

3.1.1 Newton’s Method

Newton’s method requires the computation of the second order partial derivatives of the power flow equation and other constraints (the hessian) and is therefore called a second – order method. The necessary conditions of optimality commonly are the Kuhn –Tucker conditions. Newton’s method is favoured for its quadratic convergence properties. 3.1.2 Quadratic programming

Quadratic programming (QP) is a special form of nonlinear programming. The objective function of QP optimization method is quadratic and the constraints are in linear form. Quadratic programming has higher accuracy than linear programming based approaches. Especially the most often used objective function in power system optimization is the generator cost function which generally is a quadratic. Thus there is no simplification for such objective function for a power system optimization problem solved by QP. [21]

3.1.3 Nonlinear Programming

Power system operation problems are nonlinear. Thus nonlinear programming (NLP) based techniques can easily handle power system operation problems with nonlinear programming problem, the first step in this method is to choose a search direction in the iterative procedure which is determined by first partial derivatives of the equations. Therefore these methods are referred to as the first order methods, such as generalized reduced gradient method. NLP based methods have higher accuracy than linear programming based approaches and also have global convergence, which means that the convergence can be guaranteed independent of the starting point, but a slow convergent rate may occur because of zigzagging in the search direction

Other methods under this category include-:

3.1.4 Linear Programming 3.1.5 Network flow Programming 3.1.6 Mixed integer Programming 3.1.7 Interior Point method

3.2 Heuristic Methods

These refer to techniques designed for solving a problem more quickly than the classical methods. It is used to find an approximate solution when the classical methods fail to find any exact solution. These include-:

3.2.1 Genetic algorithm (GA) GA used by John Holland in early seventies basically emulates the optimization philosophy adopted in nature. It is a multi-objective search technique based on principles inspired from the genetic and evolution mechanisms observed in natural systems and populations. Its basic principle is the maintenance of a population of solutions to a problem in the form of encoded information individuals that evolve in time. It combines survival of the strongest among string structured yet random information exchange. In every generation a new set of artificially developed strings is produced using elements of the strongest of the old. An occasional new element is experimented with for enhancement. The algorithm identifies the individuals with the optimizing fitness values and those with lower fitness will naturally get discarded from the population. The features of the working philosophy of the Genetic algorithm on the basis of which it carries out optimization may be listed as follows:

i) Based on multiple searching points, i.e. population based search. ii) Using operators inspired by biological evolution, such as crossover and mutation. iii) Based on probabilistic transition rules, iv) Fast convergence to near global optimum, v) Superior global searching capability in a complex searching surface using little

information of searching space, such as derivative, continuity. The key disadvantage of the GA is that its convergence speed near the global optimum becomes slow. To use GA effectively it is necessary to overcome this deficiency. Also the genetic algorithm cannot assure constant optimization response times. These unfortunate genetic algorithms properties limit GA use in optimization problems. [22] [12]

3.2.2 Particle Swarm method

Particle swarm optimization (PSO) is a population based stochastic optimization technique inspired by the social behaviour of flocks of birds or schools of fish. In PSO the potential solutions called particles, fly through the problem space by following the current optimum particles. The particles change their positions by flying around in a multidimensional search space until a relatively unchanged position has been exceeded. A particle bases its search not only on its personal experiences but also by the information given by its neighbours in the swarm. Each particle keeps track of its coordinates in the problem space, which is associated with the best solution fitness it has achieved so far. The fitness value is also stored. This value is called P best. Another best value that is tracked by the particle swarm optimizer is the 1 best value obtained thus far by any particles in the neighbours of the particle. This location is called 1 best. When a particle takes the whole population as its topological neighbours, the best value is a global best and is called g best. The main advantages of PSO are: easy implementation, single concept, robustness to control the parameters and less computational time compared to other optimization techniques. [18] [21]

3.2.3 Simulated annealing (SA) Simulated annealing is the physical process of heating up a solid until it melts followed by cooling it down until it crystallizes into a state with perfect lattice. During this process, the free energy of the solid is minimized. If the temperature is reduced at a very fast rate, the crystalline state transforms to an amorphous structure, a metastable state that corresponds to a local minimum of energy. Therefore the main point of the process is slow cooling, that leads to crystallized solid state which is a stable state, corresponding to a minimum energy. This is the technical definition of annealing and it is essential for ensuring that low energy state will be achieved SA algorithm is a probabilistic meta-heuristic method for global optimization problems emulating the process of annealing. Starting from an initial point the algorithm takes a step and the function is evaluated. Since the algorithm makes very few assumptions regarding the function to be optimized it is quite robust with respect to non-quadratic surfaces. In economic dispatch problem it’s used for determination of the global or near global optimum dispatch solution quadratic surfaces. In economic dispatch problem it’s used for determination of the global or near global optimum dispatch solution. The disadvantage of SA is its repeated annealing with a schedule is very slow especially if the cost function is expensive to compute. [20] [4] 3.2.4 Tabu search

The Tabu Search (TS) algorithm is a general heuristic optimization approach designed for finding optimal solution to optimization problems. It has a flexible memory to keep the information about the history search and employs it to create and explore the new solutions in the search space. The two main components of TS are the tabu list and the aspiration criterion. The tabu list stores all tabu moves that are not permitted to be applied to the current solution. The tabu list records the move direction, frequency and regency. Aspiration criterion is employed to determine which move should be free in such a case, that is, if a certain move criterion is satisfied, it is then set to be allowable. In general terms, TS is an iterative improvement procedure that starts from some initial feasible solution and attempts to determine a better solution in the manner of a great decent algorithm. TS permits backtracking to previous solutions which may ultimately lead via a different direction, to better solutions. However, TS is characterized by an inability to escape local optima, which usually causes simply descent algorithms to terminate, by using a short term memory of simple solutions. [20] [21] [4] Other methods found under the Heuristic approach include-

3.2.5 Evolutionary Programming 3.2.6 Optimization Neural Network 3.2.7 Ant colony search algorithm 3.2.8 Differential Evolution

3.3 Summary of methodologies

The algorithm that was chosen for the Economic dispatch computation was Particle Swarm Optimization as it provides a solution to the problem by working with a population of individuals (swarm) representing a possible solution

3.4 Review of Particle Swarm Optimization technique

Particle Swarm Optimization Technique was first introduced by Kennedy and Eberhart in 1995, motivated by social behaviour of organisms such as a school f fish or a flock of birds that are in motion. It is upon these social dynamics that the algorithm is based. It exploits a population of individuals to probe promising regions of search space. In this context, the population is called swarm and individuals called particles. [16], [23]

The particles in PSO move in a multidimensional search space during which each particle adjusts its position according to its own experience and the experience of the neighbouring particles. It makes use of the best position encountered by itself and its neighbours.

The direction of the swarm is determined by the set of particles neighbouring the particle and its history experience.

We represent P as the particle position and let U denote the corresponding velocity in search space. The jth particle is represented as

Pi = [Pi1 Pi2 Pi3 Pi4 …PiNG].

This is where there is an NP space search.

The best previous best position of each particle is recorded and represented as

Pbi= [Pbi1, Pbi2 …PbiNG]

The index of the best particle in the group is represented as [G1 G2 G3 …, GNG]. The velocity rates for particles in the group is represented as Vi = [Vi1 Vi2 Vi3…, ViNP]

The modified velocity and position of each particle can be calculated using the current velocity and the distance from Pbij to Gj as shown in the following formulas: [24] [6]

Vijr+1 =w × Vr

ij + C1 × R1 × (Pbrij -Pr

ij) +C2 × R2 × (Grj - Pr

ij) (i=1,2,3,…NP; j= 1,2,3;…NG)

Pr+1ij = Prij +Vr+1 ij

NP is the number of particles in a group

NG is the number of members in a particle

R is the pointer of iterations (generations)

W is the inertia weight factor

C1 and C2 are the acceleration constants

R1 and R2 are uniform random values in the range [0,1]

Vrij is the velocity of the jth member of ith particle at the rth iteration

Prij is the current position of the jth member of the ith particle at the rth iteration

C1 and C2 are stochastic acceleration weighting terms representation. These pull the particles towards the optimum position. Low values of the C1 and C2 allow the particles to roam far from the target regions before being tucked back. High values means abrupt movement towards and past the target regions and may miss out on the optimum positions. The generally used value is 2.

The inertia weight factor should be suitably selected to provide a balance between the global and local explorations thus requiring less iterations on average to find a sufficiently optimal solution.

W= wmax - � � ���� � ����� ��

× IT [16] [6] [24]

IT is the current number of iteration

ITmax is the maximum number of iterations/ generations [6]

3.4.1 Implementation

The Particle Swarm algorithm is implemented by searching the generation of power plants Pi within generator constraint. For every Pi there is a randomization of possible solutions that can fall under the maximum and minimum generation possibilities.

PARAMETER REPRESENTATION

SWARM All randomly generated solutions composed of power generations

PARTICLE An individual power solution, a single member of the swarm

VELOCITY The rate of change of and individual power solution in an iteration

DIMENSION Number of units against the random generations per unit. Makes up an array

Table 3.1: Parameter Representation

3.4.2 Representation of a swarm

Decision variables of the ED solution are real power generations which are used to form a swarm. The set of real power outputs P of all generators would represent the position of particles in the swarm. For a system with NG generators, the particle position is represented by a vector of length NG. The particle in this case are prospective solution of the randomly assigned power generation for a particular bracket of limits.

For an NP particles in the swarm, the complete swarm is represented as the matrix below [6]

������11�12�13 … �1��

.

.

.���1���2���3 … ������

����

The Swarm matrix is therefore an NP × NG where the NG represents the prospective solutions that are found within the bracket of Minimum and Maximum power generation for a Pith unit and NP representing the number of units that are available. The swarm therefore is a representation from which optimum generation for minimal fuel cost is picked by the function of the algorithm. [6]

3.4.3 Swarm initialization

Elements of the swarm matrix are initialized randomly within the real power operating limits.

Velocities of the particles are initialized randomly in accordance to the following inequality

Vjmin ≤ Vij ≤ Vj

max [25] [24]

Through the velocity initialization scheme, new particles satisfying real power operating limit constraints, the maximum velocity is computed as

VjMax = ��� ������ ��

� where � is the chosen number of interval in the jth dimension.

3.5 Objective function evaluation

In order to satisfy the power balance constraint, a generator is arbitrarily selected to be the dependent generator d which is also known as the slack generator. If its output violates its operating limits it is fixed by the maximum and minimum power constraining. A penalty term is introduced in the objective function to penalize its fitness value.

3.6 Initialization of best positions

In PSO ED evaluation, the positions with minimum objective function value is the particles best position. The best position out of all the Pbest ij is taken as Gj

best . [25] [6] [24]

3.7 Movement of the particles

The particles in a swarm are accelerated to new positions by adding new velocities to their present positions. The new velocities are calculated as indicated earlier above.

Pnew ij =Pij +Vijnew and the newly formed powers are still constrained by the boundary values of the generators.

3.8 Updating the Best and Worst Positions

There is an updating of particles in their new positions by the objective function values and the Pij best of the particle j is updated. The best position out of all the new Pij best is taken as Gjbest. An objective value at Gjbest is saved as fbest. [6] [23] [25]

3.9 Stopping Criterion

Various criteria are available to terminate the optimization which may include tolerance, the number of function evaluations and the maximum number of iterations. Normally in PSO, the maximum number of iteration is chosen as the stopping criterion. If the stopping criterion is not the best the procedure is repeated with incremented t value.

3.10.0 The Pseudo code

Step 1: Specify the upper and lower bound generation power of each unit, and calculate Pmax and Pmin. Initialize randomly the individuals of the population according to the limit of each unit including individual dimensions, searching points, and velocities. These initial individuals must be feasible candidate solution that satisfies the practical operation constraints. Step 2: To each individual Pg. of the population, employ the B-coefficient loss formula to calculate the transmission loss PL. Step 3: Calculate the evaluation value of each individual Pgi in the population using the evaluation function f given by (4). Step 4: Compare each individual’s evaluation value with its Pbest. The best evaluation value among the Pbest is denoted as Gbest. Step 5: Modify the member velocity v of each individual Pgi according to (5) Vij (t+1) = ω. Vij (t) +C1*rand ()*(Pbest - PGi (t)) + C2*Rand ()*(Gbest - PGi (t)) i = 1, 2. . . n; d=1, 2 . . . m Where n is the population size, m is the number of units, and the ω value is set by (3). Step 6: If Vij (t+1) > VjMax, then Vij (t+1) = Vij max. If Vij (t+1) < Vjmin, then Vij (t+1) = Vij min. Step 7: Modify the member position of each individual Pgi according to (6) PGi (t+1) = PGi (t) + Vij (t+1) → (6) PGi (t+1) must satisfy the constraints, namely the prohibited operating Zones and ramp rate limits. If PGi (t+1) violates the constraints the PGi (t+1) must be modified toward the near margin of the feasible solution. Step 8: If the evaluation value of each individuals is better than the previous Pbest, the current value is set to be Pbest. If the best Pbest is better than Gbest, the value is set to be best. Step 9: If the number of iterations reaches the maximum, then go to step 10. Otherwise, go to step 2. Step 10: The individual that generates the latest Gbest is the optimal generation power of each unit with the minimum total generation cost.

The Flow Chart: figure: 3.2

NO

YES

START

Define PSO parameters, Generator limits, demand, cost functions, constants C1 and C2,

particles P and the dimensions

Initialize particle with random position P and velocity vector V

Evaluate fitness function for each population and check system constraints

Update particle velocity and position

Update the population local best

Update best local best as Gbest

Are the iterations

MAX

END

Chapter 4

Results and Analysis

The Particle Swarm optimization algorithm was implemented on an IEEE 30 bus system and a dynamic economic dispatch done considering losses in the system. It was estimated that the HVDC losses were generally around 40% to 50% of the HVAC losses in a transmission system. Comparison was carried out between the two systems to determine total losses of each individual system and a hybrid system determined with weighting factors for some buses allotted HVDC transmission and separate ED carried out for the particular system depending on the load bus distribution data.

Analysis of the hybrid system was done on the basis of a demand factor assumed to be done based on the table below.

Table 4.1: Approximate Load allocations of each unit

Generator Unit Rating according to Bus load data

Generator 2 115.9 MW

Generator 8 61MW

Generator 1 24.1MW

Generator 5 36.3MW

Generator 11 19.9MW

Generator 13 47.2MW

It was considered that the total supported demand capacity of the 30 bus system is 283.4MW. Two generator units were chosen to cater for the HVDC system calculated with the weighting factors in consideration while keeping the HVAC assumed generators at zero. The same procedure was carried for the HVAC while keeping the HVDC at zero and the EDs of both of them was carried.

The losses for the hybrid system were carried out and compared to both solitary simulations of HVAC and HVDC losses and costs. The conclusions were worked out based on the same results that are herein indicated.

The hybrid AC/DC system was modelled by the assumption that generator 2 was to supply the DC lines. The total power that can be transmitted by the lines supported by the bus 2 is 115.9MW out of a possible 283.4 MW capacity of the network. The weighting factor for HVDC was considered to be 0.406 and the one for HVAC considered to be 0.594 of the demand capacity of the lines. From the results it is deduced that the Hybrid DC/AC system has lesser losses as separately considered in the tables herein.

It can also be seen that as the weighting factor is increased in favour of the DC transmission system, there is a reduction in the cost of fuel per hour. Increasing the weighting factor towards 1 (implying a purely DC transmission system) reduces the cost significantly and the losses as well. Transmission losses in the hybrid system are lesser compared to the one in the mainstream HVAC transmission system. They are however slightly higher compared to the HVDC transmission system. The hybrid system can use the existing HVAC infrastructure to lighten up the costs.

Table 4.2: Demand =200MW

HVDC HVAC Hybrid AC/DC

Gen. cost (HVDC)

Gen. cost (HVAC)

Hybrid Cost

Pg1 93.75 93.75 93.75 374.375 374.375 374.375

Pg2 43 43 43 122.78 122.78 122.78

Pg3 20 20 20 74.9 74.9 74.9

Pg4 15 15 15 50.625 50.625 50.625

Pg5 15.6941 17.6332 16.8403 53.24 60.673 57.6109

Pg6 14 14 14 46.9 46.9 46.9

Total PG 201.44 203.383 202.59

Total Loss 1.44785 3.37746 2.59384

Total Cost 722.82 730.198 727.191

Table 4.3: Demand =250MW HVDC HVAC Hybrid

AC/DC Gen. cost (HVDC)

Gen. cost (HVAC)

Hybrid Cost

Pg1 93.75 93.75 93.75 374.375 374.375 374.375

Pg2 43 43 43 122.78 122.78 122.78

Pg3 21.9955 24.3222 23.3782 81.5944 90.1033 86.5598

Pg4 30 30 30 115.788 115.788 115.788

Pg5 28 28 28 103.6 103.6 103.6

Pg6 35 35 35 135.625 135.625 135.625

Total Gen.

251.746 254.072 253.128

Total Loss 1.75449 4.07421 3.13452

Total Cost 933.762 942.41 938.727

Table 4.4: Demand =283.4MW (NETWORK CAPACITY) HVDC HVAC Hybrid

AC/DC Gen. cost (HVDC)

Gen. cost (HVAC)

Hybrid Cost

Pg1 93.75 93.75 93.75 374.375 374.375 374.375

Pg2 49.5955 52.29 51.1969 138.952 146.059 143.141

Pg3 49 49 49 226.992 226.992 226.992

Pg4 30 30 30 115.788 115.788 115.788

Pg5 28 28 28 103.6 103.6 103.6

Pg6 35 35 35 135.625 135.625 135.625

Total Gen.

285.345 288.04 286.947

Total Loss 1.95125 4.64411 3.53791

Total Cost 1095.26 1102.37 1099.45

Table 4.5: Demand= 300MW HVDC HVAC Hybrid

AC/DC Gen. cost (HVDC)

Gen. cost (HVAC)

Hybrid Cost

Pg1 97.2768 100.47 99.1514 385.167 394.957 390.903

Pg2 63 63 63 177.18 177.18 177.18

Pg3 49 49 49 226.992 226.992 226.992

Pg4 30 30 30 115.788 115.788 115.788

Pg5 28 28 28 103.6 103.6 103.6

Pg6 35 35 35 135.625 135.625 135.625

Total Gen.

302.277 305.47 304.151

T. Loss 2.27615 5.47979 4.15434

T. Cost 1144.3 1154.05 1150.02

Table 4.6: Demand =330MW HVDC HVAC Hybrid

AC/DC Gen. cost (HVDC)

Gen. cost (HVAC)

Hybrid Cost

Pg1 128.127 132.595 130.74 479.569 493.241 487.565

Pg2 63 63 63 177.18 177.18 177.18

Pg3 49 49 49 226.992 226.992 226.992

Pg4 30 30 30 115.788 115.788 115.788

Pg5 28 28 28 103.6 103.6 103.6

Pg6 35 35 35 135.625 135.625 135.625

Total Gen.

333.127 337.595 335.74

Total Loss 3.13046 7.60227 5.7427

Total Cost 1238.68 1252.36 1246.68 Table 4.7: Line Losses

Demand in MW

HVAC Losses (MW)

HVDC Losses(MW)

Hybrid Losses(MW)

200 3.37746 1.44785 2.59384

250 4.07421 1.75449 3.13452

283.4 4.64411 1.95125 3.53791

300 5.47979 2.27615 4.15434

330 7.60227 3.13046 5.7427

Table 4.7: Demand/Cost Analysis

Demand in MW HVAC cost($/HR) HVDC cost($/HR) Hybrid ($/HR)

200 730.198 722.82 727.191

250 942.41 933.762 938.727

283.4 1102.37 1095.26 1099.45

300 1154.05 1144.3 1150.02

330 1252.36 1238.68 1246.68

Below are the graphical analyses of the demand and cost and the loss analysis of the three systems. It is an obvious conclusion that using HVDC transmission offers a cheaper and much cost effective way of handling the thermal generators. It is also observed that the cost per hour per MW between different demands reduces as the demand increases. This indicates that there is higher cost benefits whenever higher power levels are being generated. The losses increase with an increase in demand across all types of transmission with the highest being recorded in the AC system. The increase in the losses in the AC system is attributed to the rising I2R losses. With increased demand, there comes a necessity to increase the current levels of transmission which subsequently increases the losses of the system. The losses in the HVDC transmission which are mainly the Converter losses are quadratically related to the converter currents. For high rating loads on the system, the IC is high which then increases the level of the lost power in the system. For a system that employs both the AC and the DC transmission models, the above are factors that would contribute to it power loss. In equal measure, the currents are subsequently required to be high enough for higher loads making it have increasing losses. It can be understood that from the demand cost curve, the base costs of the three transmission systems are equal. That is, below the minimum thermal generation, while the losses may vary, the cost of transmission is approximately equal as the power being generated has an equal level of minimum generation. Similarly, for power demand beyond the capacity of operation, the cost of operation stagnates as that is the most amount of power that can be generated and thence a maximum cost.

However, as reflected in the demand loss curve, the HVAC losses are far greater compared to the HVDC losses. Hybrid system losses have about the average of the two depending on the weighting factor imposed on the AC links. This means that a certain number of AC links ought to be turned into DC links and assumed to be served by a converter at either end of their supply. The two systems combined therefore form a hybrid AC/DC system with attributes from both of them. It therefore combines the demerits and the merits of either. As it can be seen, the losses of a hybrid system are reduced but cannot be lesser than those of the DC system.

Figure 4.2: Demand Loss Analysis

Figure 4.3: Cost –Demand analysis of the systems

The results indicate that it is possible to use the existing AC infrastructure to transfer power and avoid heavy losses that are characteristic of the AC system. With the inculcation of the DC power into an AC system, frequency flexibility can be achieved thereby ensuring that in the same power system interconnection, it is allowable to interconnect generations of different frequencies without a problem.

It is also important to note that while the system considered herein replaces specific AC lines linked to certain generators (in this case units 2 and 8), possibilities exist that can employ techniques that allow simultaneous transfer of DC and AC on the same line.

Chapter 5

Conclusion and Recommendations From the results herein obtained, it can be seen that the combination of both the DC and AC system in the power transfer process is an exceptional way of reducing transmission losses and effectively meeting the demand.

The costs of power production can be subsequently scaled down by the adoption of HVDC/HVAC hybrid. While HVDC mode of transmission could be preferred, it is important to note that the initial capital costs of constructing a converter station is much more expensive at the initial stages especially for an existent power system. It is therefore possible to utilize the existing power lines by introducing in the converter stations in an already existing AC system. It reduces the infrastructural costs and improves the efficiency of the power system at large.

In comparison to a pure AC system, the losses that are encountered in the Hybrid system are lesser which therefore reduces the transmission costs per the level of demand.

This property is an inherent characteristic of the DC system in which conceptually most of the conductor is used unlike the AC system which suffers from skin effect enabling them to only utilize a portion of the conductor.

Recommendation It is observed that this project sough to use MATLAB mathematical kit to optimize power in a Hybrid system. The process involved the use of B coefficients in the calculation of losses and further the approximation of DC losses as to being close to 40to 50% of the losses found in the AC system. In this project, the HVDC losses were assumed to be 43% of the HVAC losses.

It is possible however that the project could be done by considering the current levels in the converter, determining the �, � and � constants that are essential in determining the exact DC losses which was not captured in this project.

It is also possible to transfer both HVDC and HVAC power simultaneously through a single line, which is important because it save the cost of infrastructure and utilizes to the maximum the loading capacity of transmission lines. This consideration was not delved into as far as the project is concerned. This sort of hybrid transmission could be more cost effective than the herein considered operation.

It is also possible to carry out an optimization for a hybrid system by the combination of two methodologies which would then improve the results to enable us study more clearly the feasibility of the project from the results that shall be obtained.

Appendix 1 Table 6.1: Generator data for IEEE 30-bus system

Unit PGimax

PGimin

Uri DRi ai

bi

c Ei Fi

1 49 20 12 15 0.070 0.095 45 40.00 0.08 2 157.5 93.75 65 85 0.00 3.060 87.5 0.00 0.00 5 30 15 0

16 0.09 0.025 30 30.00 0.09

8 63 43 1

22 0.020 0.600 60 0.00 0.00 11 28 13 0

09 0.025 3.000 0.000 0.00 0.00

13 35 14 0

16 0.0250 3.000 0.000 0.00 0.00

Table 6.2: Load data for 30-bus System

Bus no. PD

(MW)

QD

(MVAR)

Bus No. PD

(MW)

QD

(MVAR) 1 0.0 0.0 16 3.5 1.6 2 21.7 12.7 17 9.0 5.8 3 2.4 1.2 18 3.2 0.9 4 7.6 1.6 19 9.5 3.4 5 94.2 19.0 20 2.2 0.7 6 0.0 0.0 21 17.5 11.2 7 22.8 10.9 22 0.0 0.0 8 30.0 30.0 23 3.2 1.6 9 0.0 0.0 24 8.7 6.7

10 5.8 2.0 25 0.0 0.0 11 0.0 0.0 26 3.5 2.3 12 11.2 7.5 27 0.0 0.0 13 0.0 0.0 28 0.0 0.0 14 6.2 1.6 29 2.4 0.9 15 8.2 2.5 30 10.6 1.9

Appendix 2 The Code: HVDC Dispatch

% Particle swarm optimization clear all; clc; opf=fopen('pso_eco1.doc','w+'); %GENERATOR VAIABLES no_units=5; %Number of unit %POWER DEMAND Pd=input ('Enter the value of load demand in MW = '); %Pd=[200;250;283.4;300;330;] %GENERATION OF RANDOM NUMBERS RR= [43 63]; % range for random numbers in R RD= [20 49]; % random range for D RE= [15 30]; %random number range for E RH= [13 28]; %random number range for H RM = [14 35]; %random number range for B %COST COEFFICIENTS %a,b,c,e,f constants of fuel cost a1=0.000; %constant for slack bus b1=3.060; %constant for slack bus c1=87.5; %constant for the slack bus a=[0.020 0.0700 0.0900 0.02500 0.02500]; b=[0.600 0.095 0.025 3.00000 3.00000]; c=[60 45 30 0.0000 0.00000]; e=[0 0 40 30 0 0]; f=[0 0 0.008 0.009 0 0]; %GENERATOR LIMITS pmin=[43 20 15 13 14]; % Minimum generation pmax=[63 49 30 28 35]; %Maximum generation %LOSS COEFFICIENTS B= 0.43*[0.000103 0.000181 0.000004 -0.000015 0.000002 0.000030 0.000009 0.000004 0.000417 -0.000131 -0.000153 -0.000107 -0.000010 -0.000015 -0.000131 0.000221 0.000094 0.000050 0.000002 0.000002 -0.000153 0.000094 0.000243 0.00000 0.000027 0.000030 -0.000107 0.000050 -0.00000 0.000358]; %RAMP RATE CONSTRAINTS DRi= [22 15 16 9 16]; URi= [12 12 8 6 8]; %TRANSMISSION LINE CONSTRAINTS Plg= [135 65 35 25 20 30]; %CONVERTER TAP RATIO Tmin = 0.9581; Tmax = 1.0155; %DC VOLTAGE Vdcmin = 0.9;

Vdcmax = 1.1; %CONVERTER ANGLE minimum = -45; maximum = 45; %POPULATION VARIABLES no_part=80; %Population size itermax=1500;%Maximum number of iterations. %INITIAL POPULATION igen= 0; %initializing generation counter %generates random numbers of size population R= randi(RR, no_part, 1); D= randi(RD, no_part, 1); E= randi(RE, no_part, 1); H= randi(RH, no_part, 1); M=randi(RM, no_part, 1); Pop= [1 2 3 4 5]; %forming initial population %INCREMENTAL COST alpha=b; beta=2*c; for i=1:no_units Lambda_min(i)=alpha(i)+beta(i)*pmin(i); Lambda_max(i)=alpha(i)+beta(i)*pmax(i); end lambda_min=min(Lambda_min); lambda_max=max(Lambda_max); lambda_min=lambda_min'; lambda_max=lambda_max'; for i=1:no_part part(i)= unifrnd(lambda_min,lambda_max); end %INITIALIZING MPSO PARAMETERS Pbest=zeros(1,no_part); vel_max=(lambda_max-lambda_min)/10; for i=1:no_part vel(i)= unifrnd(-vel_max,vel_max); end c1=2; c2=2; psi=c1+c2; K=2/abs(2-psi-sqrt(psi*psi-4*psi)); Gbest=0.0; P=zeros(no_part,no_units); tic; for iter=1:itermax for i=1:no_part for k=1:no_units temp=0; for j=1:no_units

if j~=k temp=temp+B(k,j)*P(i,j); end end end end %INEQUALITY CONSTRAINTS temp=2*temp; for j=1:no_units Nr(j)=1-(alpha(j)/part(i))-temp; Dr(j)=(beta(j)/part(i))+(2*B(j,j)); if P(i,j)>pmax(j) P(i,j)=pmax(j); end if P(i,j)<pmin(j) P(i,j)=pmin(j); end end %LOSS CALCULATIONS P_loss=0; for k=1:no_units for j=1:no_units P_loss=P_loss+(P(i,k)*B(k,j)*P(i,j)); end end %GENERATION CALCULATIONS Pgen(i)=0.0; for j=1:no_units Pgen(i)=Pgen(i)+P(i,j); end %ERROR CALCULATIONS error(i)=Pgen(i)-Pd-P_loss; fit(i)= 1.0/(100.0+abs(error(i))/Pd); if Pbest(i)<fit(i) Pbest(i)=fit(i); Pbest_part(i)=part(i); end %PBEST AND GBEST COMPARISON if Gbest<Pbest(i) Gbest=Pbest(i); Gbest_part=Pbest_part(i); end %WEIGHTING FACTOR CALCULATION Wmin=0.4; Wmax=0.9; W=Wmax-((Wmax-Wmin)*iter/itermax); vel(i)=K*(W*vel(i)+c1*rand()*(Pbest_part(i)-part(i))+c2*rand()*(Gbest_part-part(i)));

%VELOCITY UPDATING if abs(vel(i))>vel_max if vel(i)<0.0 vel(i)=-vel_max; end if vel(i)>0.0 vel(i)=vel_max; end end %POSITION UPDATING tpart=part(i)+vel(i); for k=1:no_units ttemp=0; for j=1:no_units if j~=k ttemp=ttemp+B(k,j)*P(i,j); end end end %CONTINUATION OF OPTIMIZATION LOOPING ttemp=2*ttemp; for j=1:no_units Nr(j)=1-(alpha(j)/tpart)-ttemp; Dr(j)=(beta(j)/tpart)+2*B(j,j); tp(j)=Nr(j)/Dr(j); if tp(j)>pmax(j) tp(j)=pmax(j); end if tp(j)<pmin(j) tp(j)=pmin(j); end end tP_loss=0; for k=1:no_units for j=1:no_units tP_loss=tP_loss+(tp(k)*B(k,j)*tp(j)); end end tpgen=0.0; for j=1:no_units, tpgen=tpgen+tp(j); end terror=tpgen-Pd-tP_loss; Error(iter)=terror; tfit= 1.0/(1.0+abs(terror)/Pd); if tfit>fit(i) part(i)=tpart; Pbest(i)=tfit; Pbest_part(i)=part(i); end if Gbest<Pbest(i) Gbest=Pbest(i); Gbest_part=Pbest_part(i);

end end %TERMINATION CRITERION if abs(terror)<0.01 break; end runtime=toc; fprintf(opf,'\n ECONOMIC DISPATCH FOR HVDC USING MODIFIED PSO \n'); fprintf(opf,'\n Problem converged in %d iterations\n',iter); fprintf(opf,'\n Optimal Lambda= %g\n',Gbest_part); for j=1:no_units fprintf(opf,'\n Pgen(%d)= %g MW',j,tp(j)); end fprintf(opf,'\n Total Power Generation = %g MW\n',sum(tp)); fprintf(opf,'\n Total Power Demand = %g MW',Pd); fprintf(opf,'\n Total Power Loss = %g MW\n',tP_loss); fprintf(opf,'\n Error= %g\n',terror); %FUEL COST total_cost=0.0; for j=1 Thermal_Fuel_cost(1)=c(j)+b(j)*tp(j)+a(j)*tp(j)*tp(j)+abs(e(j)*sin(f(j)*(pmin(j)-tp(j)))); end for j=1 fprintf(opf,'\n Thermal Fuel cost of Gen.(%d)= %g $/Hr',j,Thermal_Fuel_cost(1)); end for j=2 Thermal_Fuel_cost(2)=c(j)+b(j)*tp(j)+a(j)*tp(j)*tp(j)+abs(e(j)*sin(f(j)*(pmin(j)-tp(j)))); end for j=2 fprintf(opf,'\n Thermal Fuel cost of Gen.(%d)= %g $/Hr',j,Thermal_Fuel_cost(2)); end for j=3 Thermal_Fuel_cost(3)=c(j)+b(j)*tp(j)+a(j)*tp(j)*tp(j)+abs(e(j)*sin(f(j)*(pmin(j)-tp(j)))); end for j=3 fprintf(opf,'\n Thermal Fuel cost of Gen.(%d)= %g $/Hr',j,Thermal_Fuel_cost(3)); end for j=4 Thermal_Fuel_cost(4)=c(j)+b(j)*tp(j)+a(j)*tp(j)*tp(j)+abs(e(j)*sin(f(j)*(pmin(j)-tp(j)))); end for j=4 fprintf(opf,'\n Thermal Fuel cost of Gen.(%d)= %g $/Hr',j,Thermal_Fuel_cost(4)); end

for j=5 Thermal_Fuel_cost(5)=c(j)+b(j)*tp(j)+a(j)*tp(j)*tp(j)+abs(e(j)*sin(f(j)*(pmin(j)-tp(j)))); end for j=5 fprintf(opf,'\n Thermal Fuel cost of Gen.(%d)= %g $/Hr',j,Thermal_Fuel_cost(5)); end Total_Thermal_Fuel_Cost=Thermal_Fuel_cost(1)+Thermal_Fuel_cost(2)+Thermal_Fuel_cost(3)+Thermal_Fuel_cost(4)+Thermal_Fuel_cost(5); total_cost=total_cost+Total_Thermal_Fuel_Cost; fprintf(opf,'\n Total fuel cost= %g $/Hr\n',total_cost); fprintf(opf,'\n cpu time = %g sec.',runtime); fclose('all');

HVAC DISPATCH % Particle swarm optimization clear all; clc; opf=fopen('pso_eco1.doc','w+'); %GENERATOR VAIABLES no_units=6; %Number of unit %POWER DEMAND Pd=input ('Enter the value of load demand in MW = '); %Pd=[200;250;283.4;300;330] %GENERATION OF RANDOM NUMBERS RK=[93.75 157.5];%range for random numbers in A RR= [43 63]; % range for random numbers in R RD= [20 49]; % random range for D RE= [15 30]; %random number range for E RH= [13 28]; %random number range for H RM =[14 35]; %random number range for B %COST COEFFICIENTS %a,b,c,e,f constants of fuel cost a1=0.000; %constant for slack bus b1=3.060; %constant for slack bus c1=87.5; %constant for the slack bus a=[0.000 0.020 0.0700 0.0900 0.02500 0.02500]; b=[3.060 0.600 0.095 0.025 3.00000 3.00000]; c=[87.5 60 45 30 0.0000 0.00000]; e=[0 0 40 30 0 0]; f=[0 0 0.008 0.009 0 0];

%GENERATOR LIMITS pmin=[93.75 43 20 15 13 14]; % Minimum generation pmax=[157.5 63 49 30 28 35]; %Maximum generation %LOSS COEFFICIENTS B= [ 0.000218 0.000103 0.000009 -0.000010 0.000002 0.000027 0.000103 0.000181 0.000004 -0.000015 0.000002 0.000030 0.000009 0.000004 0.000417 -0.000131 -0.000153 -0.000107 -0.000010 -0.000015 -0.000131 0.000221 0.000094 0.000050 0.000002 0.000002 -0.000153 0.000094 0.000243 0.00000 0.000027 0.000030 -0.000107 0.000050 -0.00000 0.000358]; %RAMP RATE CONSTRAINTS DRi= [85 22 15 16 9 16]; URi= [65 12 12 8 6 8]; %TRANSMISSION LINE CONSTRAINTS Plg= [135 65 35 25 20 30]; %CONVERTER TAP RATIO Tmin = 0.9581; Tmax = 1.0155; %DC VOLTAGE Vdcmin = 0.9; Vdcmax = 1.1; %CONVERTER ANGLE minimum = -45; maximum = 45;

%POPULATION VARIABLES no_part=60; %Population size itermax=1000;%Maximum number of iterations. %INITIAL POPULATION igen= 0; %initializing generation counter %generates random numbers of size population R= randi(RR, no_part, 1); D= randi(RD, no_part, 1); E= randi(RE, no_part, 1); H= randi(RH, no_part, 1); M=randi(RM, no_part, 1); Pop= [1 2 3 4 5 ]; %forming initial population %INCREMENTAL COST alpha=b; beta=2*c; for i=1:no_units Lambda_min(i)=alpha(i)+beta(i)*pmin(i); Lambda_max(i)=alpha(i)+beta(i)*pmax(i); end lambda_min=min(Lambda_min); lambda_max=max(Lambda_max); lambda_min=lambda_min'; lambda_max=lambda_max'; for i=1:no_part part(i)= unifrnd(lambda_min,lambda_max); end

%INITIALIZING MPSO PARAMETERS Pbest=zeros(1,no_part); vel_max=(lambda_max-lambda_min)/10; for i=1:no_part vel(i)= unifrnd(-vel_max,vel_max); end c1=2; c2=2; psi=c1+c2; K=2/abs(2-psi-sqrt(psi*psi-4*psi)); Gbest=0.0; P=zeros(no_part,no_units); tic; for iter=1:itermax for i=1:no_part for k=1:no_units temp=0; for j=1:no_units if j~=k temp=temp+B(k,j)*P(i,j); end end end %INEQUALITY CONSTRAINTS temp=2*temp; for j=1:no_units Nr(j)=1-(alpha(j)/part(i))-temp; Dr(j)=(beta(j)/part(i))+(2*B(j,j)); if P(i,j)>pmax(j)

P(i,j)=pmax(j); end if P(i,j)<pmin(j) P(i,j)=pmin(j); end end %LOSS CALCULATIONS P_loss=0; for k=1:no_units for j=1:no_units P_loss=P_loss+(P(i,k)*B(k,j)*P(i,j)); end end %GENERATION CALCULATIONS Pgen(i)=0.0; for j=1:no_units Pgen(i)=Pgen(i)+P(i,j); end %ERROR CALCULATIONS error(i)=Pgen(i)-Pd-P_loss; fit(i)= 1.0/(100.0+abs(error(i))/Pd); if Pbest(i)<fit(i) Pbest(i)=fit(i); Pbest_part(i)=part(i); end %PBEST AND GBEST COMPARISON

if Gbest<Pbest(i) Gbest=Pbest(i); Gbest_part=Pbest_part(i); end %WEIGHTING FACTOR CALCULATION Wmin=0.4; Wmax=0.9; W=Wmax-((Wmax-Wmin)*iter/itermax); vel(i)=K*(W*vel(i)+c1*rand()*(Pbest_part(i)-part(i))+c2*rand()*(Gbest_part-part(i))); %VELOCITY UPDATING if abs(vel(i))>vel_max if vel(i)<0.0 vel(i)=-vel_max; end if vel(i)>0.0 vel(i)=vel_max; end end %POSITION UPDATING tpart=part(i)+vel(i); for k=1:no_units ttemp=0; for j=1:no_units if j~=k ttemp=ttemp+B(k,j)*P(i,j); end

end end %CONTINUATION OF OPTIMIZATION LOOPING ttemp=2*ttemp; for j=1:no_units Nr(j)=1-(alpha(j)/tpart)-ttemp; Dr(j)=(beta(j)/tpart)+2*B(j,j); tp(j)=Nr(j)/Dr(j); if tp(j)>pmax(j) tp(j)=pmax(j); end if tp(j)<pmin(j) tp(j)=pmin(j); end end tP_loss=0; for k=1:no_units for j=1:no_units tP_loss=tP_loss+(tp(k)*B(k,j)*tp(j)); end end tpgen=0.0; for j=1:no_units, tpgen=tpgen+tp(j); end terror=tpgen-Pd-tP_loss; Error(iter)=terror; tfit= 1.0/(1.0+abs(terror)/Pd); if tfit>fit(i) part(i)=tpart;

Pbest(i)=tfit; Pbest_part(i)=part(i); end if Gbest<Pbest(i) Gbest=Pbest(i); Gbest_part=Pbest_part(i); end end %TERMINATION CRITERION if abs(terror)<0.01 break; end end runtime=toc; fprintf(opf,'\n ECONOMIC DISPATCH FOR HVAC USING MODIFIED PSO \n'); fprintf(opf,'\n Problem converged in %d iterations\n',iter); fprintf(opf,'\n Optimal Lambda= %g\n',Gbest_part); for j=1:no_units fprintf(opf,'\n Pgen(%d)= %g MW',j,tp(j)); end fprintf(opf,'\n Total Power Generation = %g MW\n',sum(tp)); fprintf(opf,'\n Total Power Demand = %g MW',Pd); fprintf(opf,'\n Total Power Loss = %g MW\n',tP_loss); fprintf(opf,'\n Error= %g\n',terror); %FUEL COST total_cost=0.0; for j=1

Thermal_Fuel_cost(1)=c(j)+b(j)*tp(j)+a(j)*tp(j)*tp(j)+abs(e(j)*sin(f(j)*(pmin(j)-tp(j)))); end for j=1 fprintf(opf,'\n Thermal Fuel cost of Gen.(%d)= %g $/Hr',j,Thermal_Fuel_cost(1)); end for j=2 Thermal_Fuel_cost(2)=c(j)+b(j)*tp(j)+a(j)*tp(j)*tp(j)+abs(e(j)*sin(f(j)*(pmin(j)-tp(j)))); end for j=2 fprintf(opf,'\n Thermal Fuel cost of Gen.(%d)= %g $/Hr',j,Thermal_Fuel_cost(2)); end for j=3 Thermal_Fuel_cost(3)=c(j)+b(j)*tp(j)+a(j)*tp(j)*tp(j)+abs(e(j)*sin(f(j)*(pmin(j)-tp(j)))); end for j=3 fprintf(opf,'\n Thermal Fuel cost of Gen.(%d)= %g $/Hr',j,Thermal_Fuel_cost(3)); end for j=4 Thermal_Fuel_cost(4)=c(j)+b(j)*tp(j)+a(j)*tp(j)*tp(j)+abs(e(j)*sin(f(j)*(pmin(j)-tp(j)))); end for j=4 fprintf(opf,'\n Thermal Fuel cost of Gen.(%d)= %g $/Hr',j,Thermal_Fuel_cost(4)); end for j=5 Thermal_Fuel_cost(5)=c(j)+b(j)*tp(j)+a(j)*tp(j)*tp(j)+abs(e(j)*sin(f(j)*(pmin(j)-tp(j))));

end for j=5 fprintf(opf,'\n Thermal Fuel cost of Gen.(%d)= %g $/Hr',j,Thermal_Fuel_cost(5)); end for j=6 Thermal_Fuel_cost(6)=c(j)+b(j)*tp(j)+a(j)*tp(j)*tp(j)+abs(e(j)*sin(f(j)*(pmin(j)-tp(j)))); end for j=6 fprintf(opf,'\n Thermal Fuel cost of Gen.(%d)= %g $/Hr',j,Thermal_Fuel_cost(6)); end Total_Thermal_Fuel_Cost=Thermal_Fuel_cost(1)+Thermal_Fuel_cost(2)+Thermal_Fuel_cost(3)+Thermal_Fuel_cost(4)+Thermal_Fuel_cost(5)+Thermal_Fuel_cost(6); total_cost=total_cost+Total_Thermal_Fuel_Cost; fprintf(opf,'\n Total fuel cost= %g $/Hr\n',total_cost); fprintf(opf,'\n cpu time = %g sec.',runtime); fclose('all'); HYBRID (COMBINED HVAC/HVDC) CODE

% Particle swarm optimization clear all; clc; opf=fopen('pso_eco1.doc','w+'); %GENERATOR VAIABLES no_units=6; %Number of unit %POWER DEMAND Pd=input ('Enter the value of load demand in MW = '); %Pd=[150;210;275;330;385;425] %GENERATION OF RANDOM NUMBERS RK=[93.75 157.5];%range for random numbers in A RR= [43 63]; % range for random numbers in R RD= [20 49]; % random range for D RE= [15 30]; %random number range for E RH= [13 28]; %random number range for H RM =[14 35]; %random number range for B %COST COEFFICIENTS %a,b,c,e,f constants of fuel cost a1=0.000; %constant for slack bus b1=3.060; %constant for slack bus c1=87.5; %constant for the slack bus a=[0.000 0.020 0.0700 0.0900 0.02500 0.02500]; b=[3.060 0.600 0.095 0.025 3.00000 3.00000]; c=[87.5 60 45 30 0.0000 0.00000]; e=[0 0 40 30 0 0]; f=[0 0 0.008 0.009 0 0]; %GENERATOR LIMITS pmin=[93.75 43 20 15 13 14]; % Minimum generation pmax=[157.5 63 49 30 28 35]; %Maximum generation %LOSS COEFFICIENTS B= 0.594*[ 0.000218 0.000103 0.000009 -0.000010 0.000002 0.000027 0.000103 0.000181 0.000004 -0.000015 0.000002 0.000030 0.000009 0.000004 0.000417 -0.000131 -0.000153 -0.000107 -0.000010 -0.000015 -0.000131 0.000221 0.000094 0.000050 0.000002 0.000002 -0.000153 0.000094 0.000243 0.00000 0.000027 0.000030 -0.000107 0.000050 -0.00000 0.000358]; Q= 0.175*[ 0.000218 0.000103 0.000009 -0.000010 0.000002 0.000027 0.000103 0.000181 0.000004 -0.000015 0.000002 0.000030 0.000009 0.000004 0.000417 -0.000131 -0.000153 -0.000107 -0.000010 -0.000015 -0.000131 0.000221 0.000094 0.000050 0.000002 0.000002 -0.000153 0.000094 0.000243 0.00000 0.000027 0.000030 -0.000107 0.000050 -0.00000 0.000358]; %RAMP RATE CONSTRAINTS DRi= [85 22 15 16 9 16]; URi= [65 12 12 8 6 8]; %TRANSMISSION LINE CONSTRAINTS Plg= [135 65 35 25 20 30];

%CONVERTER TAP RATIO Tmin = 0.9581; Tmax = 1.0155; %DC VOLTAGE Vdcmin = 0.9; Vdcmax = 1.1; %CONVERTER ANGLE minimum = -45; maximum = 45; %POPULATION VARIABLES no_part=60; %Population size itermax=1000;%Maximum number of iterations. %INITIAL POPULATION igen= 0; %initializing generation counter %generates random numbers of size population R= randi(RR, no_part, 1); D= randi(RD, no_part, 1); E= randi(RE, no_part, 1); H= randi(RH, no_part, 1); M=randi(RM, no_part, 1); Pop= [1 2 3 4 5 ]; %forming initial population %INCREMENTAL COST alpha=b; beta=2*c; for i=1:no_units Lambda_min(i)=alpha(i)+beta(i)*pmin(i); Lambda_max(i)=alpha(i)+beta(i)*pmax(i); end lambda_min=min(Lambda_min); lambda_max=max(Lambda_max); lambda_min=lambda_min'; lambda_max=lambda_max'; for i=1:no_part part(i)= unifrnd(lambda_min,lambda_max); end %INITIALIZING MPSO PARAMETERS Pbest=zeros(1,no_part); vel_max=(lambda_max-lambda_min)/10; for i=1:no_part vel(i)= unifrnd(-vel_max,vel_max); end c1=2; c2=2; psi=c1+c2; K=2/abs(2-psi-sqrt(psi*psi-4*psi)); Gbest=0.0;

P=zeros(no_part,no_units); tic; for iter=1:itermax for i=1:no_part for k=1:no_units temp=0; for j=1:no_units if j~=k temp=temp+B(k,j)*P(i,j)+Q(k,j)*P(i,j); end end end %INEQUALITY CONSTRAINTS temp=2*temp; for j=1:no_units Nr(j)=1-(alpha(j)/part(i))-temp; Dr(j)=(beta(j)/part(i))+(2*B(j,j)+(2*Q(j,j))); if P(i,j)>pmax(j) P(i,j)=pmax(j); end if P(i,j)<pmin(j) P(i,j)=pmin(j); end end %LOSS CALCULATIONS P_loss=0; for k=1:no_units for j=1:no_units P_loss=P_loss+(P(i,k)*B(k,j)*P(i,j))+(P(i,k)*Q(k,j)*P(i,j)); end end %GENERATION CALCULATIONS Pgen(i)=0.0; for j=1:no_units Pgen(i)=Pgen(i)+P(i,j); end %ERROR CALCULATIONS error(i)=Pgen(i)-Pd-P_loss; fit(i)= 1.0/(100.0+abs(error(i))/Pd); if Pbest(i)<fit(i) Pbest(i)=fit(i); Pbest_part(i)=part(i); end %PBEST AND GBEST COMPARISON if Gbest<Pbest(i) Gbest=Pbest(i); Gbest_part=Pbest_part(i); end

%WEIGHTING FACTOR CALCULATION Wmin=0.4; Wmax=0.9; W=Wmax-((Wmax-Wmin)*iter/itermax); vel(i)=K*(W*vel(i)+c1*rand()*(Pbest_part(i)-part(i))+c2*rand()*(Gbest_part-part(i))); %VELOCITY UPDATING if abs(vel(i))>vel_max if vel(i)<0.0 vel(i)=-vel_max; end if vel(i)>0.0 vel(i)=vel_max; end end %POSITION UPDATING tpart=part(i)+vel(i); for k=1:no_units ttemp=0; for j=1:no_units if j~=k ttemp=ttemp+B(k,j)*P(i,j); end end end %CONTINUATION OF OPTIMIZATION LOOPING ttemp=2*ttemp; for j=1:no_units Nr(j)=1-(alpha(j)/tpart)-ttemp; Dr(j)=(beta(j)/tpart)+2*B(j,j)+2*Q(j,j); tp(j)=Nr(j)/Dr(j); if tp(j)>pmax(j) tp(j)=pmax(j); end if tp(j)<pmin(j) tp(j)=pmin(j); end end tP_loss=0; for k=1:no_units for j=1:no_units tP_loss=tP_loss+(tp(k)*B(k,j)*tp(j)+tp(k)*Q(k,j)*tp(j)); end end tpgen=0.0; for j=1:no_units, tpgen=tpgen+tp(j); end terror=tpgen-Pd-tP_loss; Error(iter)=terror; tfit= 1.0/(1.0+abs(terror)/Pd); if tfit>fit(i)

part(i)=tpart; Pbest(i)=tfit; Pbest_part(i)=part(i); end if Gbest<Pbest(i) Gbest=Pbest(i); Gbest_part=Pbest_part(i); end end %TERMINATION CRITERION if abs(terror)<0.01 break; end end runtime=toc; fprintf(opf,'\n ECONOMIC DISPATCH FOR HVAC USING MODIFIED PSO \n'); fprintf(opf,'\n Problem converged in %d iterations\n',iter); fprintf(opf,'\n Optimal Lambda= %g\n',Gbest_part); for j=1:no_units fprintf(opf,'\n Pgen(%d)= %g MW',j,tp(j)); end fprintf(opf,'\n Total Power Generation = %g MW\n',sum(tp)); fprintf(opf,'\n Total Power Demand = %g MW',Pd); fprintf(opf,'\n Total Power Loss = %g MW\n',tP_loss); fprintf(opf,'\n Error= %g\n',terror); %FUEL COST total_cost=0.0; for j=1 Thermal_Fuel_cost(1)=c(j)+b(j)*tp(j)+a(j)*tp(j)*tp(j)+abs(e(j)*sin(f(j)*(pmin(j)-tp(j)))); end for j=1 fprintf(opf,'\n Thermal Fuel cost of Gen.(%d)= %g $/Hr',j,Thermal_Fuel_cost(1)); end for j=2 Thermal_Fuel_cost(2)=c(j)+b(j)*tp(j)+a(j)*tp(j)*tp(j)+abs(e(j)*sin(f(j)*(pmin(j)-tp(j)))); end for j=2 fprintf(opf,'\n Thermal Fuel cost of Gen.(%d)= %g $/Hr',j,Thermal_Fuel_cost(2)); end for j=3 Thermal_Fuel_cost(3)=c(j)+b(j)*tp(j)+a(j)*tp(j)*tp(j)+abs(e(j)*sin(f(j)*(pmin(j)-tp(j)))); end for j=3 fprintf(opf,'\n Thermal Fuel cost of Gen.(%d)= %g $/Hr',j,Thermal_Fuel_cost(3)); end

for j=4 Thermal_Fuel_cost(4)=c(j)+b(j)*tp(j)+a(j)*tp(j)*tp(j)+abs(e(j)*sin(f(j)*(pmin(j)-tp(j)))); end for j=4 fprintf(opf,'\n Thermal Fuel cost of Gen.(%d)= %g $/Hr',j,Thermal_Fuel_cost(4)); end for j=5 Thermal_Fuel_cost(5)=c(j)+b(j)*tp(j)+a(j)*tp(j)*tp(j)+abs(e(j)*sin(f(j)*(pmin(j)-tp(j)))); end for j=5 fprintf(opf,'\n Thermal Fuel cost of Gen.(%d)= %g $/Hr',j,Thermal_Fuel_cost(5)); end for j=6 Thermal_Fuel_cost(6)=c(j)+b(j)*tp(j)+a(j)*tp(j)*tp(j)+abs(e(j)*sin(f(j)*(pmin(j)-tp(j)))); end for j=6 fprintf(opf,'\n Thermal Fuel cost of Gen.(%d)= %g $/Hr',j,Thermal_Fuel_cost(6)); end Total_Thermal_Fuel_Cost=Thermal_Fuel_cost(1)+Thermal_Fuel_cost(2)+Thermal_Fuel_cost(3)+Thermal_Fuel_cost(4)+Thermal_Fuel_cost(5)+Thermal_Fuel_cost(6); total_cost=total_cost+Total_Thermal_Fuel_Cost; fprintf(opf,'\n Total fuel cost= %g $/Hr\n',total_cost); fprintf(opf,'\n cpu time = %g sec.',runtime); fclose('all');

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