Optimal Unit Commitment using Swarm Intelligence
for Secure Operation of
Solar Energy Integrated Smart Grid
Thesis submitted in partial fulfilment
of the requirements for the degree of
Master of Science (by Research)
in
IT in Power Systems
By
Ravikanth Reddy Gaddam
200871002
Advisor: Dr. Amit Jain
Power Systems Research Center
International Institute of Information Technology
Hyderabad – 500032, INDIA
April 2013
Copyright © Ravikanth Reddy Gaddam, 2013
All Rights Reserved
Acknowledgments
This thesis wouldn’t have become a success without the people who have afforded a
constant support throughout. Firstly, I whole heartedly express gratitude to my advisor Dr.
Amit Jain for his valuable guidance over the entire period of my pursuit for MS. His
knowledge and encouragement helped and me a lot in the course of my research. The values
he incorporated will definitely make me a better individual in life.
I would like to extend my appreciation to Dr. Ramamoorty for his valuable inputs and
comments on the work during his visits to the Center as a Distinguished Faculty. Also, the
breadth I have gained in the area of power systems with his lectures is invaluable and will
surely help me in industry.
I feel grateful to be a part of Power Systems Research Center. I extend my thanks to
dear friends at PSRC for their support and joyful times we had during my stay at IIIT
Hyderabad. Especially the company I had with Siva will be most memorable in my life.
Many thanks for his support and assistance without which I wouldn’t have made.
Last but not the least to mention is my family who are always concerned about my
future and well being. I extend my in-depth gratitude to my father, mother and brothers for
their love, patience and support.
Abstract
Majority of the world’s electricity demand at present is met by thermal power
generating stations which purely run on conventional fossil fuels. Utilities are mainly
dependent on these sources spending huge revenue to cater for the ever increasing electricity
demand. This motivates utilities to schedule their generation according to load demand in the
most economical way. Every thermal generating unit is characterized by its distinct
incremental heat rate curve which governs the production cost. An optimal allocation of
generation among the different power generating units can thus save considerable fuel input
and cost. However, a greater amount of fuel cost can potentially be saved by extending this
optimization procedure to decide which of these units would participate in the optimal
allocation. In other words, it is significant to determine whether the unit must be ON/OFF.
Committing thermal generating units among the available ones in this fashion is popular as
Unit Commitment (UC). It is a vital economic optimization problem in power systems with
the main criteria of meeting the load demand while accounting for all the unit operational
constraints at the same time. On the other hand, optimally distributing the generation among
the set of units already scheduled/committed for operation is known as Economic Dispatch
(ED). Present day load demand is increasing at a pace which expansion in power system has
hardly been able to keep up with. The power systems have thus been left stressed. This makes
the system insecure and vulnerable. Hence system security also plays a major role in
generation planning. Therefore, the problem of UC is not merely to commit adequate
generation but distribute it with utmost attention towards system security for achieving smart
grid of future.
In this thesis, the evolutionary technique of Binary Particle Swarm Optimization
(BPSO) is used in solving the UC problem. As for determining the optimal generation
dispatch for the minimal cost, a complete Optimal Power Flow (OPF) is run for each hour’s
commitment over the UC time horizon.
Conventional OPF solves for the optimal dispatch adjusting to all the constraints like
fixed bus voltage limits, line power flows, transformer tap positions etc., resulting in a secure
solution. Apart from these concerns, it would be very useful to compute the loadability and
safety margin of a given system. This is important as it can potentially avoid any system
collapse and keep a measure of system security in true sense.
The work proposed in this thesis makes attempt to find optimal unit commitment
keeping system operation secure by calculating what is known as the global L-index. It is a
mathematical tool which computes how far the current operating condition is from the
collapse point. Inclusion of such a constraint would naturally ensure that the optimal solution
does not violate system voltage security. The major contribution of the proposed work lies in
investigating more number of feasible solutions because the replacement of hard limits on the
bus voltages by the L-index criterion effectively loosens the search space. Thus,
experimentation is carried out by calculating the L-index while relaxing the fixed voltage
limits on all or a few buses at the same time.
A consistent raise in fuel costs and fast depletion of fossil fuels, growing awareness
towards environmental protection has paved the path towards utilization of renewable energy
sources for power production. Hence renewable energy sources are used and deployed in
power systems with more eagerness today. However, they impose a considerable impact on
the system owing to the intermittent nature of availability. The problem of UC becomes more
complex with the integration of renewable sources given the dissimilarities in behavioural
and functional constraints with respect to conventional thermal generation units which need
to be addressed as renewable generation will be integral part of smart grid.
This thesis aims at solving a voltage security constrained UC with the inclusion of
renewable energy sources into the system. In the present work inclusion of one such source
like Solar into the system has been considered. A Solar Thermal Power Plant (STPP) of
suitable capacity is modelled with Concentrated Solar Power (CSP) technology of parabolic
trough type. The standard dimensions of parabolic trough from a commercially installed
power plant are considered for modelling. The optimal generation dispatch function (OPF)
solution, which is an inherent exercise within the master problem of unit commitment, also
accounts for the intermittent output from solar generator present in the same grid.
The proposed method is validated by implementing it on the standard IEEE 14 and 30
bus test systems. One of the thermal generating units in standard test system is replaced by a
solar thermal unit and the UC schedule is determined for the remaining units. The UC results,
illustrated both in the presence and absence of STPP, proves the effectiveness of method.
Results furnished with global L-index criterion with relaxation on hard voltage limits are
found to be convincing besides upholding system security.
Contents
Chapter
Page
1 Introduction ............................................................................................................................. 1
1.1 Unit Commitment and Its Importance .............................................................................. 1
1.2 Economic Dispatch and Optimal Power Flow ................................................................. 2
1.3 UC with Renewable Sources in Power Systems .............................................................. 3
1.4 Voltage Security ............................................................................................................... 4
1.5 Smart Grid and UC Importance in it ................................................................................ 5
1.6 Thesis Contribution .......................................................................................................... 6
1.7 Thesis Organization.......................................................................................................... 8
2 Unit Commitment Techniques: A Review .............................................................................. 9
2.1 Deterministic and Stochastic Methods ............................................................................. 9
2.1.1 Priority List ............................................................................................................ 11
2.1.2 Dynamic Programming ......................................................................................... 11
2.1.3 Integer Programming and Mixed Integer Programming ....................................... 12
2.1.4 Lagrangian Relaxation........................................................................................... 13
2.1.5 Artificial Neural Networks .................................................................................... 13
2.1.6 Genetic Algorithm ................................................................................................. 14
2.1.7 Simulated Annealing ............................................................................................. 15
2.1.8 Tabu Search ........................................................................................................... 16
2.1.9 Ant Colony Optimization ...................................................................................... 17
2.1.10 Particle Swarm Optimization .............................................................................. 18
2.2 Security Constrained Unit Commitment ........................................................................ 20
2.3 UC in Renewable Integrated Power Systems ................................................................. 22
3 Unit Commitment ................................................................................................................. 26
3.1 General ........................................................................................................................... 26
3.2 Generator characteristics ................................................................................................ 27
3.3 Start-up and shut down costs .......................................................................................... 28
3.4 Constraints...................................................................................................................... 29
3.5 Unit Commitment Formulation ...................................................................................... 31
3.6 Economic Dispatch ........................................................................................................ 33
3.7 Optimal Power Flow ...................................................................................................... 37
viii
3.8 Calculation of Bus Injections ......................................................................................... 40
3.9 Calculation of Line Flows .............................................................................................. 41
4 Solar Thermal Power Plant ................................................................................................... 43
4.1 Solar Energy ................................................................................................................... 43
4.2 STPP ............................................................................................................................... 44
4.3 Parabolic Trough Collector ............................................................................................ 46
4.4 Losses in PTC................................................................................................................. 48
4.4.1 Optical Losses ....................................................................................................... 48
4.4.2 Thermal Losses ...................................................................................................... 49
4.5 Thermal Analysis of PTC............................................................................................... 50
4.6 Solar Radiation ............................................................................................................... 51
4.7 Solar Radiation Geometry .............................................................................................. 52
4.8 Example.......................................................................................................................... 55
4.9 Electricity Generation from STPP.................................................................................. 56
4.10 Example........................................................................................................................ 58
5 PSO and Implementation ...................................................................................................... 59
5.1 Particle Swarm Optimization ......................................................................................... 59
5.2 PSO Algorithm ............................................................................................................... 61
5.3 Binary Particle Swarm Optimization ............................................................................. 62
5.4 Representation of UC as a BPSO Problem .................................................................... 63
5.5 Implementation............................................................................................................... 64
5.5.1 Creation of Initial Population ................................................................................ 65
5.5.1.1 Procedure followed in creating Initial Population .................................... 67
5.5.2 BPSO Updation ..................................................................................................... 68
5.5.3 OPF evaluation ...................................................................................................... 69
5.5.4 Details of Algorithm followed for a 24 hour UC schedule ................................... 71
5.5.5 Simulation Results on IEEE Test Systems ............................................................ 74
5.5.5.1 Case 1: UC for standard IEEE 14 bus test system .................................... 74
5.5.5.2. Case 2: UC for standard IEEE 30 bus test system ................................... 76
5.6. UC with STPP integration ............................................................................................. 77
5.6.1 Algorithm for 24 hr UC with STPP Integration .................................................... 79
5.6.2 Simulation Results on IEEE Test Systems ............................................................ 81
5.6.2.1 Case 3: UC for standard IEEE 14 bus test system with STPP integrated . 81
5.6.2.2. Case 4: UC for standard IEEE 30 bus test system with STPP integrated 83
5.7 New Approach to Voltage Security Constrained UC..................................................... 84
5.7.1 Voltage Security Margin: P-V Curve Fundamentals............................................. 85
ix
5.7.2 L-Index Approach ................................................................................................. 87
5.7.3 Proposed UC Algorithm with L-index .................................................................. 89
5.7.3.1 Case 5: UC for IEEE 14 Bus Test System with L-index .......................... 90
5.7.3.2 Case 6: UC for IEEE 30 Bus Test System with L-index .......................... 92
5.7.3.3 Case 7: UC for STPP integrated IEEE 14 Bus Test System with L-index
............................................................................................................................... 93
5.7.3.4 Case 8: UC for STPP integrated IEEE 30 Bus Test System with L-index
............................................................................................................................... 95
6. Summary and Conclusions .................................................................................................. 97
Appendix A: Transmission Loss Coefficients ....................................................................... 100
Appendix B: Standard IEEE 14 and 30 Bus Test Systems Data ........................................... 103
B.1 IEEE 14 Bus Test System Data ................................................................................... 103
B.1.1 Unit Data ............................................................................................................. 103
B.1.2 Line Data............................................................................................................. 103
B.1.3 24 Hour Load Demand ....................................................................................... 104
B.2 IEEE 30 Bus Test System Data ................................................................................... 105
B.2.1 Unit Data ............................................................................................................. 105
B.2.2 Line Data............................................................................................................. 106
B.2.3 24 Hour Load Demand ....................................................................................... 107
B.2.4 Bus Load Factors ................................................................................................ 108
Appendix C: Characteristics of Thermal Generating Units ................................................... 109
C.1 Input – Output Characteristics ............................................................................... 109
C.2 Incremental Heat Rate Characteristics ................................................................... 110
C.3 Unit Heat Rate Characteristics ............................................................................... 110
References .............................................................................................................................. 112
Publications ............................................................................................................................ 119
x
List of Figures
Figure
Page
Figure 3.1 Input-Output Characteristics of a Thermal Generator ............................................ 27
Figure 3.2 Transmission Line π Model .................................................................................... 42
Figure 4.1 Schematic Diagrams of Parabolic Trough and Linear Fresnel CSP Systems ........ 45
Figure 4.2 Schematic Diagrams of Dish type and Power Tower CSP Systems ...................... 45
Figure 4.3 Parabolic Trough Collector .................................................................................... 47
Figure 4.4 Latitude, Hour angle and Declination .................................................................... 53
Figure 4.5 Incidence angle, Surface Azimuth angle and Slope ............................................... 53
Figure 4.6 Solar Thermal Power Plant with Parabolic Trough Collector ................................ 57
Figure 5.1 Star and Ring Topologies ....................................................................................... 60
Figure 5.2 A Simple P-V Curve............................................................................................... 86
Figure 5.3 Single Machine Load System ................................................................................. 88
Figure C.1 Input-Output Characteristics of a Steam Turbine Generator ............................... 109
Figure C.2 Incremental Heat Rate Characteristics of a Thermal Generator .......................... 110
Figure C.3 Unit Heat Rate Characteristics of a Thermal Generator ...................................... 111
xi
List of Tables
Table
Page
Table 4.1 Data of Luz System (LS-3) PTC.............................................................................. 57
Table 5.1 UC for IEEE 14 Bus Test System ............................................................................ 75
Table 5.2 Hourly Min. and Max. Load Bus Voltages for IEEE 14 Bus Test System ............. 75
Table 5.3 UC for IEEE 30 Bus Test System ............................................................................ 76
Table 5.4 Hourly Min. and Max. Load Bus Voltages for IEEE 30 Bus Test System ............. 77
Table 5.5 UC for IEEE 14 Bus Test System with STPP integrated ........................................ 82
Table 5.6 Hourly Min. and Max. Load Bus Voltages for IEEE 14 Bus Test System with STPP
integrated.................................................................................................................................. 82
Table 5.7 UC for IEEE 30 Bus Test System with STPP integrated ........................................ 84
Table 5.8 Hourly Min. and Max. Load Bus Voltages for IEEE 30 Bus Test System with STPP
integrated.................................................................................................................................. 84
Table 5.9 UC for IEEE 14 Bus Test System with L-index ...................................................... 91
Table 5.10 Hourly Min. and Max. Load Bus Voltages for IEEE 14 Bus Test System with Lindex ......................................................................................................................................... 91
Table 5.11 UC for IEEE 30 Bus Test System with L-index .................................................... 92
Table 5.12 Hourly Min. and Max. Load Bus Voltages for IEEE 30 Bus Test System with Lindex ......................................................................................................................................... 93
Table 5.13 UC for STPP integrated IEEE 14 Bus Test System with L-index ......................... 94
Table 5.14 Hourly Min. and Max. Load Bus Voltages for STPP integrated IEEE 14 Bus Test
System with L-index ................................................................................................................ 95
Table 5.15 UC for STPP integrated IEEE 30 Bus Test System with L-index ......................... 96
Table 5.16 Hourly Min. and Max. Load Bus Voltages for STPP integrated IEEE 30 Bus Test
System with L-index ................................................................................................................ 96
Table B.1 IEEE 14 Bus Unit Data ......................................................................................... 103
Table B.2 IEEE 14 Bus Line Data ......................................................................................... 104
Table B.3 IEEE 14 Bus 24 Hour Load Demand .................................................................... 104
B.1.4 Bus Load Factors ...................................................................................................... 105
Table B.4 Bus Load Factors................................................................................................... 105
Table B.5 IEEE 30 Bus Unit Data ......................................................................................... 105
xii
Table B.6 IEEE 30 Bus Line Data ......................................................................................... 107
Table B.7 IEEE 30 Bus 24 Hour Load Demand .................................................................... 108
Table B.8 Bus Load Factors................................................................................................... 108
xiii
List of Abbreviations
ACO
Ant Colony Optimization
ANN
Artificial Neural Network
BB
Branch and Bound
BPSO
Binary Particle Swarm Optimization
CSP
Concentrated Solar Power
DP
Dynamic Programming
ED
Economic Dispatch
EDL
Economic Dispatch including Losses
EP
Evolutionary Programming
GA
Genetic Algorithm
HNN
Hopfield Neural Network
IP
Integer Programming
LR
Lagrangian Relaxation
MIP
Mixed Integer Programming
PL
Priority List
PSO
Particle Swarm Optimization
PTC
Parabolic Trough Collector
PV
Photovoltaic
SA
Simulated Annealing
SCUC
Security Constrained Unit Commitment
STPP
Solar Thermal Power Plant
TS
Tabu Search
UC
Unit Commitment
xiv
Nomenclature
ai, bi, ci
Generator cost coefficients
Ac
Parabolic trough collector/aperature area
Ar
Parabolic trough receiver area
B
Susceptance
C
Collector concentration ratio
Fi(Pgi)
Cost function of ith unit
f(xij)
Fitness value of particle xi during jth iteration
G
Conductance
gbest
Global best value of PSO population
Ii
Current at ith bus
i
Bus or Line index
It
Solar insolation during hour t
L
L-index on ith bus
N
Total number of buses in system
NG
Total number of units in the system
ng
Number of committed units
Nb
Number of load buses
Nl
Number of Transmission lines
Pd
Total active power demand
Presv
Active power reserve
pbest
Personal best value of BPSO particle
Pgi,max
Maximum active power generation limit of ith unit
Pgi,min
Minimum active power generation limit of ith unit
Pgi
Active power generation of ith unit
Ploadi
Active power demand at ith bus
Pi
Active power injection at ith bus
Ploss
Active power loss
Plk
Active power flow on ith transmission line
Plmaxi
Maximum active power flow limit on ith transmission line
Plmini
Minimum active power flow limit on ith transmission line
xv
Pop
BPSO population
PSt
Active power output from Solar Thermal Power plant during hour t
Qgi
Reactive power generation of ith unit
Qloadi
Reactive power demand at ith bus
Qi
Reactive power injection at ith bus
qu
Useful heat energy
ql
Thermal Losses
Rupi
Up ramping rate of ith unit
Rdni
Down ramping rate of ith unit
sig
Sigmoid function
Si
Apparent power at ith bus
tap
Number of buses with tap transformers
tapmax,i
Maximum tap limit of transformer on ith bus
tapmin,i
Minimum tap limit of transformer on ith bus
T
Time horizon in hours for UC scheduling
t
Hour index
Ta
Ambient air temperature
Tiup
Minimum up of ith unit in hours
Tidown
Minimum down of ith unit in hours
Tion
Continuously ON time of ith unit in hours
Tioff
Continuously OFF time of ith unit in hours
Tr
Collector receiver temperature
Uit
ith unit ON/OFF status during tth hour (1--ON, 0--OFF)
Ul
Overall thermal loss coefficient
vi
j
Velocity of particle xi during jth iteration
Vi
Voltage of ith bus
Vi,max
Maximum voltage limit of ith bus
Vi,min
Minimum voltage limit of ith bus
w
Inertia weight
x
BPSO particle
xk j
Status (0 or 1) of kth particle in jth iteration
[xPg]
(ng X 1) state vector of active power generation
xvi
[xV]
(Nb X 1) state vector of load bus voltage
[xδ]
(N-1 X 1) state vector of bus voltage angles
[xt]
(tap X 1) state vector of tap positions of transformers
[xΦ]
(tap X 1) state vector of phase shifter transformers
X
Augmented state vector
Y
Admittance matrix
Z
Impedance
λ
Lagrangian multiplier
δi
Voltage angle of ith bus
δi,max
Maximum voltage angle limit of ith bus
δi,min
Minimum voltage angle limit of ith bus
ηopt
Parabolic trough collector optical efficiency
ηth
Solar field thermal efficiencies
ηR
Rankine cycle efficiency
ηTG
Turbine Generator set efficiency
xvii
Chapter 1
1 Introduction
1.1 Unit Commitment and Its Importance
Over the years, power systems had seen an immense shift from isolated systems to
huge interconnected systems. These interconnected power systems are more reliable and at
the same time have brought up many challenges in the operation from economics and system
security perspective. Power systems can be divided into three main sub-systems called the
Generation, Transmission and the Distribution systems apart from the power consumption at
the end. The behaviour of all sub-systems are interdependent. Each of the sub-systems has its
own behavioural attributes and constraints which govern overall system operation. Power
systems have expanded the reach over a large geography for years to supply and cater to the
ever increasing load demand. With this vast spread due to continuously growing power
requirements, every utility in the world is facing a problem in reliable operation of system.
The need to supply of electricity to consumers with utmost importance towards reliability
inclines utilities to plan at every level. In addition to reliability, an aspect that concerns
utilities in planning is the economics involved in system operation. From the stage of power
generation to the supply at consumer level, there exist many economic considerations. Thus,
the planning steps followed should enable system reliable operation while optimizing the
economics needed.
The power system is subjected to a varying electric load demand with peaks and
valleys at different times in a day completely based on human requirements. A costumer has
to be supplied with power whenever he/she requires. This urges the company to commit (turn
ON) sufficient number of generating units to cater to this varying load at all times. The option
of committing all of its units and keeping them online all the time to counter varying nature
of load is economically detrimental [1] for the utilities. A generating unit requires specific
amount of heat input per Mega Watt to be generated which actually demands a cost in the
form of fuel requirements. Heat demand per MW (heat rate), also known as the input
1
characteristics, varies with each generator, thus varying the fuel cost per MW (incremental
cost) too. This makes the commitment problem significant. So, a prior decision as to which
among the available units should be ON helps the utility earn a considerable saving in
generation costs. The process of this decision making is called Unit Commitment (UC) and is
one of the main optimization problems in power systems which draw focus in its day to day
operation.
Unit Commitment, as the name suggests, is an hourly commitment schedule of units
determined ahead in time, varying from few hours to a week, with the main goal of meeting
load demand. In general, UC schedules are determined a day ahead. The hourly load demand
for UC problem is available from precise load forecasting. The only optimizing criterion in
determining UC schedule is the cost of generation which needs to be minimized over the
planning period while fulfilling all system constraints arising from the generating unit’s
physical capabilities and the transmission system’s network configuration. A generating unit
has various limitations such as minimum up-down time, maximum and minimum generation
limits, and ramp rate limits etc. Similarly transmission network configuration governs the
maximum power flow possible through lines while transformer tap settings and phase shifting
angle limits impose the physical constraints.
UC problem is a combination of two sub-problems. One determines the generating
units to be committed and the other actually focuses on the amount of generation from each
of these committed units. Generating units exhibit different operating efficiencies and
performance characteristics which reflect on the required inputs. Thus the cost of generation
also depends on the amount of output from each committed unit apart from the choice of
generating units. Thus, a UC problem is solved in two stages. The combination of generating
units yielding the least final production cost is thus chosen as the UC schedule for the hour.
1.2 Economic Dispatch and Optimal Power Flow
The problem of computing power output from each committed unit for that hour’s
predetermined schedule for minimum cost is called Economic Dispatch (ED). It is a nonlinear
optimization problem whose control variables are the power outputs of each committed unit
and the solution so found must be bounded between maximum and minimum generation
limits of respective units. The problem of ED is solved based on an equal incremental cost
2
criterion. Since the incremental cost for each unit varies for the same amount of generation,
generation is distributed such that every unit operates with the same incremental cost. This
simply implies that the units bearing heavier incremental cost coefficients tend to share lesser
portion of the load and vice-versa. The problem of ED thus far does not consider the
transmission system effects in the optimization procedure. Losses in the transmission lines
when considered necessitate extra computational burden. However, the equal incremental
cost approach just explained would no longer yield optimal dispatch. This is because the
electrical distance between generation and load comes into picture.
Unit Commitment is the problem of economics with due attention towards system
security. In reliable and smooth functioning of power systems, security plays a major role.
Optimal Power Flow (OPF) solves the same objective of minimizing the cost of generation.
However, OPF results in a more secure solution [1], [2] by considering both the generation
and transmission systems. It can be solved for objectives like minimizing cost of generation,
minimizing transmission losses, minimizing the emissions from generating units, etc. The
work presented in this thesis solves OPF with the objective of minimizing overall generation
cost and ensuring system constraints are not violated at the same time. Control variables
include generator power outputs, transformer tap settings and phase shifting angles restricted
within their minimum and maximum bounds.
1.3 UC with Renewable Sources in Power Systems
The extended dependence on fossil fuel resources have made them fast depleting and
the era of fossil fuels is gradually coming to an end. The large scale usage of fossil fuels over
the years has also created harmful effects on the environment. The outcome of this is the
phenomenon of global warming, a matter of great concern. The employment of alternative
sources of energy is becoming slowly but increasingly prevalent. The alternative sources that
have positive effect on environment are the renewable sources like solar, wind, tidal, biomass
etc. Harnessing these resources pose many technical issues apart from costlier economics.
The main drawback of these sources is their intermittent nature of availability leading to
potential problems in integrating them into the existing power grid for smooth and reliable
operation. The problem of meeting load demand and maintaining system security at the same
time gets compounded with the integration of renewable sources. Given the scenario,
3
formulation of UC schedule in the presence of renewable energy sources becomes tedious.
This thesis addresses the problem of determining a secure UC schedule that minimizes the
production cost of thermal generating units in the presence of a power plant run by solar
energy.
Extracting useful energy from the Sun requires advanced conversion technologies.
Concentrated Solar Power and Solar Pond are a couple of popular technologies to convert
solar thermal energy into useful heat source and finally into electricity. Likewise,
photovoltaic cells can convert the photon energy of sun into direct form of electricity. Today,
establishment of power plants that uses solar energy is one of the most promising solutions
that can cater to the increasing electricity demand. The very recently inaugurated (17th March
2013) world’s largest solar power plant in United Arab Emirates (UAE) named “Shams 1”
forms the best example. The capacity of power plant is 100 MW and sees an area of 2.5 km2
that will power 20,000 homes in UAE. The plant employs Concentrated Solar Power
technology of Parabolic Trough type. A total of 768 parabolic trough collectors are used with
about 2,50,000 mirrors mounted on them. This plant is currently the largest capacity CSP
plant but will not going to hold the status for long. Many CSP plants of larger capacities
around the world are already under construction in the U.S., India and other countries [3].
In designing the solar power plant, there are concerning issues related to technologies
and the quality of output. This thesis presents a considerably detailed study on the existing
Concentrated Solar Power technologies, especially the parabolic trough collectors. The work
here models a solar thermal power plant using parabolic trough collector technology.
1.4 Voltage Security
The issue of voltage security has been a predominant one to deal with in what has
developed into power systems, overloaded beyond capabilities in recent times. Owing to the
economic and resource constraints linked with expansion plans, the overload limits being
crossed is expected now more than ever. There has been very little or in fact no expansion in
transmission capacities to measure against the continuously expanding supply needs. The
infrastructure is thus strained in order to meet the demand. The growing electrical distance
between the generation centres and remote loads is constantly on the rise. As a result of this
stress, the voltage in the system sags considerably and the system is quite near to being
4
insecure. A secure system is one which is capable of remaining in steady operating state post
the occurrence of a disturbance. A system whose voltage is already in the lower range would
obviously be more likely to tip over when a disturbance occurs; it is thus insecure. The
monitoring or evaluation of the system security, especially in terms of voltage levels has thus
become imperative.
The transfer of power through a transmission network is accompanied by voltage
drops between the generation and consumption points. In normal operating conditions, these
drops are in the order of a few per cent of the nominal voltage unit. One of the tasks of power
system planners and operators is to check that under heavy stress conditions and/or
consequent events, all bus voltages remain within acceptable bounds. Only then the system
would be secure in terms of voltage.
This thesis focuses on security more so in the context of voltage profile maintained
for the dispatched generation schedule. The objective of minimizing the production cost
should not be achieved at the expense of foregoing voltage security. Besides being
economically optimal, the determined schedule should be capable to withstand disturbances
and restore voltages. This would not be possible if the voltage profile hangs lower than
specified at the beginning itself. System steady state voltage stability margin is defined as the
distance from the current operating state to the steady state stability limit or the critical point.
Various approaches have claimed the solution towards the same over the years. A global
indicator of voltage stability known as L-index, first introduced by Kessel et al. [4], is by far
the most reasonable measure of the distance to the point of instability. The aim is to ensure
that the voltage profile is not only located away from this point but also has a specified
margin available.
1.5 Smart Grid and UC Importance in it
With the way technologies are developing in modern times, their applications in many
fields are rising at an equal pace for betterment of existing features or to arrive at new
outcomes. Similar to the way that a 'smart' phone these days means a phone with a computer
in it, 'Smart Grid' in electrical terms means computerizing the electric grid that includes many
hardware equipments like transmission lines, transformers, circuit breakers and many more.
Smart Grid aims at bringing in two way communications as well as technologies for
5
automation between central control room and the devices that are monitored in the grid. A
smart grid extends its definition in many ways towards a more efficient, more reliable and
more usage of eco-friendly resources in the grid. It also uses smart metering systems and
smart appliances which can act upon control/automatic settings. It is an intelligent electrical
network that sense and prevents major power outages by isolating the disturbances in the grid
much faster. It also encourages users towards smart appliances that reduce peak demands on
the grid which directly influence/relax the utility’s maximum installed capacity.
The present day increase in energy crisis is a universal concern and increase in use of
environment friendly energy resources, like solar and wind energy, is a way to overcome this
and these concerns can certainly be addressed by developing futuristic smart grids. Thus, the
future smart grids will become more dependent on distributed renewable generations. Most
renewable energy sources are highly intermittent in nature and often uncontrollable. The
usage of more and more eco-friendly energy resources for power generation and smart
appliances into the grid certainly brings up significant fluctuation on the supply side of the
power grid as well as uncertainties on the demand side. These directly throw a challenging
task on the conventional thermal units scheduling. The commitment schedule found under
such uncertainties should elevate system reliability and security. It is always important to
maintain demand and power generation balance to avoid the system entering into unhealthy
state. Thus, accurate and rigorous Unit Commitment procedures are demanding in solving the
scheduling problem for future smart grids [5].
1.6 Thesis Contribution
Taking into consideration the journey of modern day power systems towards
integrating sustainable renewable energy sources, algorithms that analyze system operation
are needed to be more robust. Unit commitment, which is one of the most crucial power
system optimization problems ranks high in this respect. The proven history of harnessing
solar power in bulk through concentrated solar power technology of parabolic trough type
collectors warrants the deployment of solar energized thermal power plants. One of the main
contributions of the thesis is solving the complex combinatorial UC problem in presence of
solar powered thermal plant connected to the grid. To model a solar integrated power system
6
close to a practical scenario, a solar thermal power plant modelled with parabolic trough type
technology is interconnected into the existing power network.
The thesis determines UC schedule adopting stochastic optimization technique of
Binary Particle Swarm Optimization (BPSO). Often in solving the UC problem using BPSO
the effect of transmission losses are neglected in the creation of initial population which may
lead to false computations. The proposed technique follows a unique approach in calculating
approximate losses prone to occur every hour for inclusion in the process of initial population
generation. Further, in optimal allocation of load among the units committed, a full-fledged
optimal power flow is performed for every hour in finding the fitness value of each particle.
This is in contrast with the usual practice of performing a conventional ED every hour.
The other significant and major contribution is determination of UC schedule with
attention towards what is known as system voltage security. The attempt is first of its kind in
UC computation. Given the present trend of ever increasing load demand on power systems,
its elements are operated in an overloaded and stressed environment owing to the
comparatively slow infrastructure developments. As a consequence, bus voltages go below
operating limits endangering normal system operation. This demands a voltage secure UC
schedule for satisfactory system operation. In the thesis, system voltage security is added as
an additional constraint in the OPF evaluation using an indicator called global L-index. It
provides a good measure of the distance of a given system operating state from the collapse
point. Experimentations are carried employing L-index and relaxing the hard voltage limits
on load buses to show the effectiveness. By selecting a desired measure of L-index in feasible
range allows the committed generators and the system to operate far enough from the
collapse point ensuring secure operation.
The work carried out would be very useful for present day power system that is
stepping towards integration of renewable energy sources. The voltage secured UC schedule
will also serve power systems in its reliable and safe operation.
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1.7 Thesis Organization
The thesis is mainly organized into five chapters. The on-going 1st chapter gives an
insight into to the topics that are dealt in the whole thesis. Mainly the problem of Unit
Commitment and its importance in power systems along with a brief note on sub-problems of
Economic Dispatch and Optimal Power Flow is explained. The technique of Particle Swarm
Optimization and its ability in solving UC problem are discussed in short. Utilization of
renewable energy sources, in particularly the solar energy for power production and the
complexity it throws on the problem of UC is put forward, which has great relevance for
futuristic smart grid. Also the problem of how an over loaded system effects system voltage
stability and its normal operation is demonstrated. The concept of global L-index in avoiding
the situation is also discussed. Whereas chapter 2 completely gives an account of what
literature has so far come across in approaching the problem of UC, the security issues
addressed. It also provides a thorough review about UC problem under the impact of
renewable energy sources in the system.
Chapter 3 is completely dedicated towards the problem formulations of Unit
Commitment, Economic Dispatch and Optimal Power Flow. The respective objectives and
constraints involved are discussed. Chapter 4 is devoted for the techniques available in
harnessing solar thermal energy especially the parabolic trough technology. The modelling of
a solar thermal power plant using parabolic trough collector is studied in detail along with an
understanding of solar radiation geometry.
Chapter 5 gives the development of proposed techniques for the UC problem. At first
the concept of PSO and its application towards UC problem is illustrated. Results are
depicted for cases of UC in absence and presence of solar thermal power plant. Later the idea
of L-index is clearly explained and its incorporation into the problem of UC is studied.
Results are also shown for the cases of UC with L-index both in absence and presence of
solar thermal plant. This is followed by a concluding chapter 6 which gives the observations
and conclusions obtained from the thesis.
8
Chapter 2
2 Unit Commitment Techniques: A Review
2.1 Deterministic and Stochastic Methods
UC is a highly non linear optimization problem in power system operation. Given the
highest priority to the reliability of power supply a UC schedule which lays emphasis on the
economic and secure operation is necessary. With the increase in system size, UC problem
becomes more combinatorial and exhaustive. Several methodologies have been proposed to
solve UC problem and increase potential savings in power system operation. In the operation,
system security also plays a major role and thus a great amount of research work is carried
towards the security constrained UC.
Depleting conventional fossil fuels and advancements in the harnessing technologies
of renewable energy sources encourage utilities to go for power production in substantial
amounts from renewable sources. Inclusion of renewable sources in power systems saves a
great deal of generation cost but the intermittent and irregular nature of availability imposes
many challenges over the grid operation. Integration of power plants run by renewable energy
sources into the grid made UC problem more complex. Thus elevating the computational
efforts required in arriving at a secure and economical UC schedule. This chapter gives a
brief review of UC methodologies along with system security issues and renewable energy
source inclusion into the conventional grid. Approaches followed and their effectiveness in
solving the UC problem are also described.
Techniques applied to solve UC problem can be broadly classified into deterministic
and stochastic. Deterministic approaches include methods like Priority List (PL), Integer
Programming (IP), Mixed Integer Programming (MIP), Dynamic Programming (DP),
Lagrangian Relaxation (LR) and the Branch and Bound (BB). These methods are simple but
they suffer from drawbacks like numerical convergence, solution quality and execution time.
On the other hand stochastic search algorithms which are inspired by artificial intelligence
include Artificial Neural Networks (ANN), Genetic Algorithms (GA), Evolutionary
Programming (EP), Simulated Annealing (SA), Particle Swarm Optimization (PSO), Ant
9
Colony Optimization (ACO) and Tabu Search (TS). These are able to overcome the
shortcomings of traditional optimization techniques and can handle complex nonlinear
constraints assuring high quality solutions.
All of the methods under deterministic as well as stochastic have got their own
advantages and disadvantages. Owing to this the other class of strategies called hybrid
methods have emerged. Hybrid techniques are the blend of two or more of the above
mentioned optimization techniques and are not covered in this thesis.
In the year 1966 KERR et al [6] proposed UC program for the GE 412 digital process
control computer. It uses a simple heuristic approach of ‘changing the starting or stopping
time of any unit may result in a change in system fuel cost’. If the change results in a
reduction of system fuel cost, the commitment schedule is adjusted accordingly and another
trial adjustment is attempted. The final commitment schedule is the result of checking all
possible unit configurations which offers minimum system fuel cost.
Reference [7] presents a method to determine UC of thermal generating units
considering some additional constraints like a unit should not be committed more than once
in a day and no more than two units of the same plant should be started up simultaneously.
The problem is solved using branch and bound algorithm and results are depicted for a
network of ten generating units.
A heuristic algorithm based on incremental changes to the unit schedules is proposed
in [8]. The approach sequentially replaces an expensive unit in operation with one or more
less expensive units. It defines the term ‘unit period’ and finds the thermal UC under the
existence of peaking units. The commitment schedule generated is compared with truncated
dynamic programming algorithm and found to offer an approximate two percent (2%)
savings on the daily average cost.
The authors of [9] utilize the advantages of implementing mathematical programming
techniques and acquiring human expert knowledge. A rule based approach is applied to
implement a priority list which obtains a sub-optimal feasible solution. Betterment of this
solution is carried out by an expert system which performs a clever search in conjunction
with the mathematical programming technique.
10
2.1.1 Priority List
A priority list based evolutionary algorithm is proposed in [10] in which an initial
population required for the evolutionary technique is seeded using priority list to improve
efficiency and increase the speed of convergence. The results obtained are compared with
genetic algorithm approach for units ranging from 10 to 100.
An extended priority list technique is presented in [11] where, at first to speed up the
procedure a commitment schedule is obtained using priority list method by disregarding the
operational constraints. Some heuristics depending on the solution necessity relating to must
run units, minimum up/down times are applied later to improve the solution. Effectiveness of
proposed method is compared with GA and EP. Reference [12] presents a similar approach
using PL in the first stage and strategically deciding the unit ON/OFF status based on
minimum up/down times in second step. The best cost computed and time recorded are
compared with [11].
2.1.2 Dynamic Programming
Feasibility of dynamic programming in solving the UC problem is proposed by P.G.
Lowery in [13]. Mainly the difficulties in applying DP to UC are addressed. Results showed
that simple, straight forward constraints are adequate to produce an optimum operating
policy.
In reference [14], Pang et al elaborates on the important operating constraints and
spinning reserve requirements in UC problem formulation. The paper presents truncated
dynamic programming method for commitment of thermal units over the periods of up to 48
hours. It employs a priority list of the available units so as to truncate the search. After
truncating the potential uneconomical schedules at each time step, dynamic programming
method is used in finding the schedule that has least total cost. Walter L. Snyder et al presents
a field proven DP formulation for UC in [15]. To minimize the problem size generating units
are subdivided into ‘classes’ on the basis of similar characteristics. In reference [16], to
minimize the execution time, decomposition and successive approximation techniques are
used to divide the original large scale UC problem. Sub-problems are solved using DP
approach and results are compared with a heuristic method.
11
A new approach for UC using fuzzy dynamic programming is proposed in [17] that
takes into account the errors in forecasted hourly loads using fuzzy set notations. The
approach is superior to conventional dynamic programming method and effectiveness is
demonstrated on Taiwan power system with 6 nuclear units, 48 thermal units, and 44 hydro
units. James A. Momoh et al. presents a neural network based DP called adaptive dynamic
programming in [18]. It is an adaptive process which solves UC by adjusting dynamically to
search for optimal solution when the load changes. In [19] Joon Hyung Partk Et al. brings up
the drawbacks while using priority list, lagrange relaxation, mixed integer linear
programming in conjunction with DP. A modified dynamic programming approach is
illustrated which can be applied to large scale power systems. Reference [20] applied
Sequential DP and Truncated DP apart from conventional DP for the solving UC problem.
2.1.3 Integer Programming and Mixed Integer Programming
T. S. Dillon et al. in [21] proposes UC schedule for hydro-thermal systems using
extensions and modification of Branch and Bound method for integer programming. This
method is computationally practical to realistic systems with associated reserves.
John A. Muckstadt et al. [22] presents a decomposable mixed-integer programming
model for simultaneous economic consideration of unit commitment and short term dispatch
of thermal units. To incorporate the probabilistic nature forecasting, the demand forecast is
characterized as a discrete function in the scheduling model.
A practical mixed integer linear programming technique is proposed in [23] which is
suitable for both traditional and deregulated environments and is tested on actual system data.
The method not only gives the unit schedule and dispatch, it also provides the marginal price
according to system constraints to assist strategic bidding in the power market.
Reference [24] presents a fuzzy linear optimization formulation solved using mixed
integer linear programming routine. Start-up costs are modelled using linear variables. The
fuzzy formulation provides flexibility relaxation in constraint enforcement. On the other hand
[25] proposes a fuzzy mixed integer programming approach for security constrained UC in
the presence of a wind powered unit. More emphasis is given to the forecast uncertainties in
load demand, ancillary services and wind power, which are simulated in fuzzy frame, this
12
study illustrates how the most efficient schedule can be selected based on variation in
forecasted parameters.
2.1.4 Lagrangian Relaxation
A new Lagrangian relaxation algorithm for unit commitment is proposed in [26].
Firstly using Lagrangian dual the UC is solved approximately by ignoring the load or reserve
constraints. A search finds the reserve-feasible dual solution by intelligently adjusting the
Lagrange multipliers. Later for the given reserve-feasible dual solution, by running an
economic dispatch to satisfy the power balance equations, one obtains a primal feasible
solution to the UC problem. The UC problem with time limited reserve constraints is also
treated. On the other hand [27] provides an understanding of practical aspects of the
Lagrangian relaxation methodology in solving thermal UC problem. The method involves
decomposition of the master problem into easy sub-problems and applied to a realistic
system. Over the decades both, UC and Economic Dispatch problems are solved using the
Lagrangian multipliers, each deploying its own set of multipliers. These multipliers differ
from the other owing to the options in solving the respective problems. A. G. Bakirtzis et al.
[28] shows this clear distinction between the Lagrangian multipliers applied in solving the
UC problem and the Economic Dispatch problem. In solving UC by LR a good lower bound
on the primal feasible solution is obtained in [29]. Dual value is a lower bound to the optimal
feasible cost, used to evaluate solution quality. This algorithm uses the different behaviours
of the multipliers when the optimal dual value is overestimated and underestimated and
adjusts the estimate accordingly.
2.1.5 Artificial Neural Networks
H. Sasaki et al. in 1992 explored the possibility of applying the Hopfield Neural
Network (HNN) to combinatorial optimization problems in power systems, in particular to
unit commitment [30]. A large number of inequality constraints included in unit commitment
are handled by dedicated neural networks. Units are dispatched according to the Priority
order of fuel per unit output. A method which solves both UC and its sub problem ED
separately employing neural networks is proposed in [31]. UC problem requiring discrete
variables, is solved using a discrete neuron model where as ED is solved using a continuous
neuron model. Advantage with the approach is decoupling of interdependency of both UC
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and ED which is otherwise solved using single neural network. Each is solved iteratively with
respective Hopfield neural network. Titti Saksornchai et al. employ neural networks in a
different way to reduce errors in UC schedule [32]. The paper describes the procedure to
improve the UC schedule by using hour-ahead and day-ahead short term load forecasting
programs developed using neural networks. The sensitivity analysis of economic impact
shows that the cost of UC scheduling tends to increase as the mean average percentage error
of forecasting data increases.
In [33], the problem of assigning the output power of generators is solved by a single
layer Hopfield neural network. Based on this single layer HNN a multilayer HNN which
solves UC is presented. A hybrid dynamic programming based Hopfield neural network
approach to UC problem is proposed in [34]. The method employs a linear input–output
model for neurons extremely different and is relatively simple, straight forward and efficient
compared to other hybrid methods. The proposed two step process uses a direct computation
Hopfield neural network to generate economic dispatch. Then using dynamic programming
(DP) the generator schedule is produced.
2.1.6 Genetic Algorithm
D. Dasgupta et al. discusses the application of genetic algorithms to determine short
term commitment order of thermal units in [35]. The genetic based UC system evaluates the
priority order of the units dynamically, considering system parameters, operating constraints
at each time period of the scheduling horizon. The method gives feasible best near-optimal
commitment.
A GA in conjunction with constraint handling techniques to solve the thermal UC
problem is presented in [36]. To deal effectively with the constraints of the problem and
reduce the search space of the GA in advance, the minimum up and down-time constraints
are embedded in the binary strings that are coded to represent the on-off states of the
generating units. The other constraints are handled by integrating penalty factors into the cost
function within an enhanced economic dispatch program.
UC problem solved using GA that gives a stable and acceptable solution that is near
optimal based on genetic algorithm with new search operators is developed in [37]. The
advantages achieved by the introduction of genetic operators specific to the problem are
14
mutation that makes the probability of bit change dependent on load demand, production and
start-up costs of units, as well as the transposition searching through local minima. The
method incorporates repair algorithms or penalty factors in the objective function for
infeasible solutions.
A floating point GA is developed in [38]. The floating point chromosome
representation and encoding-decoding schemes are designed based on the load demand. The
evaluation function and its constraints, population size, cross over operation and probability,
mutation operation and probability are characterized in detail. Also the formulated genetic
operators exploit the non-convex solution space and multimodal objective function.
A matrix real coded GA to UC problem is developed in [39] with new repairing
mechanism and window mutation. A real number matrix representation of chromosome is
used that can solve the UC problem through genetic operations. The algorithm avoids the
determination of suboptimal ED and converts the UC problem into single level optimization
which is solved through crossover and mutation.
An improved GA is presented in [40]. This method is different from the traditional
GA, which reserves the best individual in the last population which can speed up the search.
2.1.7 Simulated Annealing
Simulated annealing has been applied to large scale UC problem and tested on up
to 100 units in 1990 [41]. SA generates feasible solutions randomly and moves among these
solutions using a strategy leading to a global minimum with high probabilities. This method
yielded highly near-optimal solutions, and is much faster than dynamic programming. The
method can easily accommodate complicated constraints like crew and short-term
maintenance schedule.
U. D. Annakkage et al. investigates the application of parallel simulated annealing for
unit commitment problems in [42]. Two parallel simulated annealing concepts, speculative
computation and serial subset, are applied. The authors also proposed a combined scheme.
The algorithm is used to reduce the computational burden and can speed up the serial SA for
a particular UC problem by a factor of six.
15
An enhanced SA-approach for solving the UC problem is presented in [43]. Owing to
the large set of constraints intrinsic in the UC problems, a restoration mechanism has been
developed and incorporated into the algorithm to ensure that the candidate solutions produced
are feasible and satisfy all the constraints.
Dimitris N. Simopoulos et al. present an enhanced SA algorithm combined with
dynamic economic dispatch [44]. Three alternative mechanisms for generating feasible trial
solutions in the neighbourhood of the current one, contributing to reduction in CPU time is
presented. On the other hand, [45] and [46] presented incorporation of the unit unavailability
and the uncertainty of the load forecasting in short term UC problem and ramp rate
constrained UC using SA respectively.
C. Christober Asir Rajan et al. employs Evolutionary programming based SA with
cooling and banking constraints in [47]. UC schedule is coded as a string of symbols. The
mutation rate is computed as a function of the ratio of the total cost of schedule to the cost of
best schedule in the current population. On the other hand in [48] the author applies SA to
solve multi-area UC problem. The objective is to determine optimal or near optimal strategy
for units located in multiple areas that are interconnected via tie lines. Tie line transfer limits
are considered in constraints set. The power transfer between areas through tie lines depends
upon the operating cost of generation at each hour and tie line transfer limits.
2.1.8 Tabu Search
An efficient algorithm to solve a long-term Hydro Scheduling Problem (LTHSP) is
proposed in [49]. The algorithm is based on using the short-term memory of the Tabu Search
approach to solve the nonlinear optimization problem in continuous variables of the LTHSP.
The paper introduces new rules for generating feasible solutions with an adaptive step vector
adjustment.
C. Christober Asir Rajan et al. [50] presented an Evolutionary Programming based TS
method to the UC problem in utility system. The essential processes simulated in the
procedure are mutation, competition and selection. Competition and selection are applied to
select from among the parents and the offspring which lead to the best solutions to form the
basis of the subsequent generation. The parents are obtained from a pre-defined set of
16
solution’s i.e. each and every solution is obtained from the TS method. With added
constraints like cooling and banking the work is again reported in [51].
TS based hybrid optimization technique is employed in [52]. A hybrid PSO and
sequential quadratic programming guides the TS. The central idea of TS method is to use the
adaptive memory, which prevents convergence to local optima, by driving the different parts
of search space. The TS method with an improved random perturbation-of-current-solution
scheme is used in solving the UC problem.
An improved TS and PSO algorithm are developed in [53]. Based on the TS method,
since the short memory of computer is used, it is possible to escape from the local minima.
Among the proposed methods TS is faster than PSO, but the PSO method gives better
response.
2.1.9 Ant Colony Optimization
ACO approach to UC is demonstrated in [54] that uses artificial ants (or agents),
which to some extent have memory and are not completely blind, thus will live in an
environment where time is discrete. The state transition, global and local updating rules are
also introduced to ensure the optimality of the solution. The effectiveness of the method is
demonstrated on a 10-unit test system. Enhancement of hydro generation scheduling using
ACO is proposed in [55]. To apply the method to solve this problem, the search space of
multi-stage scheduling is first determined.
A novel ACO algorithm with random perturbation behaviour, based on combination
of general ACO and stochastic mechanism is developed for the solution of optimal UC with
probabilistic spinning reserve determination [56]. Considering the uncertainty in forecasted
load and probability of generating units failure, the probabilistic technique based on security
function method is applied to evaluate the spinning reserve capacity requirements to satisfy a
given security level.
The UC problem with power flow constraints is solved with ant colony search in [57].
To make the system function in a realistic environment, white gaussian noise is added to the
network nodes with proportional loads to the demand forecasted at each hourly stage. Multistage decisions give the ant search competitive edge over other conventional techniques. Not
17
only the minimum cost path is determined, but also based on pheromone deposition other
related features such as pheromone updating, optimal control parameters namely tuning
factor, relative importance of the trail are discussed.
A novel Selective Self Adaptive ACO which improves search performance by
automatically adapting ant populations and their transition probability parameters is reported
in [58]. This assists in reducing search space and recovering a feasible optimality region so
that a high quality solution can be achieved in a very early iterative. The new concept of
Relative Pheromone Updating provides a reasonable evaluation of the pheromone trail
intensity among the agents. A new method which uses Lagrangian multipliers associated with
discrete variables of the thermal UC problem as a source of information for the ant colony
algorithm is proposed in [59].
2.1.10 Particle Swarm Optimization
A clear understanding of how to apply a discrete binary PSO method for solving the
UC problem is given in [60]. The problem is considered as two linked optimization subproblems. The scheduling problem which minimizes the transition cost is solved using binary
PSO and the Economic dispatch problem which minimizes the production cost is treated by
Lambda-iteration method. The feasibility of proposed method is demonstrated on 10 and 26
unit systems.
An improved PSO which adopts orthogonal design in generating the initial population
is proposed in [61]. The generated particles are scattered uniformly over feasible solution
space. The method uses more particles’ information to control the mutation operation and is
similar to the social society in that a group of leads could make better decision. A new
adaptive strategy for choosing parameters which assures the convergence is also proposed.
PSO is used to solve both UC and ED problems in [62]. The discrete version of PSO
is adopted to solve UC problem and real valued PSO is employed in solving ED problem.
Author quoted the blending of real valued PSO with binary valued PSO as ‘Hybrid’ PSO.
Both are run independently and simultaneously adjusting their solutions to achieve an optimal
one.
18
A similar approach which employs fuzzy adaptive PSO to solve UC as well as ED is
quoted in [63]. By analyzing the social model of standard PSO, for UC problem of variable
resource size and changing load demand, a fuzzy adaptive criterion is applied for the PSO
inertia weight, based on the diversity of fitness. The inertia weight is dynamically adjusted
using fuzzy IF/THEN rules to increase the balance between global and local searching
abilities.
A PSO based hydro thermal scheduling is discussed through various possible options
in particle selections in [64]. Hydro generation, thermal generation, water discharge rate and
reservoir volume are among the possible particles for the scheduling problem.
V. S. Pappala et al. developed a new adaptive PSO approach in [65]. The algorithm
provides alternatives for the demerits of PSO such as parameter tuning, selection of optimal
swarm size and problem dependent penalty functions. The method uses an adaptive penalty
function approach and particles do not require any repair strategies for satisfying the
constraints.
References [66], [67], propose similar approach of multi population PSO for UC. The
potential solution schedules are distributed among several clusters based on their
corresponding fitness values. Each cluster contains a cluster best schedule. Each solution then
flies through to its cluster space towards the cluster best, as well as personal best solution
instead of only global best (unlike trivial PSO). Therefore, this algorithm provides a way to
explore larger search space and thus reduces the probability of local trapping. Then global
optimum solution is found from all cluster best particles.
Three versions of PSO algorithm namely Binary PSO, Improved PSO and PSO with
Lagrangian relaxation are demonstrated in [68]. The paper demonstrates binary PSO and
improved PSO for UC which avoids non standard form of equations.
To improve the performance of PSO when applied to UC problem two influential
strategies are derived in [69]. One is asynchronous time varying learning strategy which
alters the constants used in traditional binary PSO based on iteration count and the other is a
new repairing strategy for particles pertaining to minimum up-down constraints.
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2.2 Security Constrained Unit Commitment
The aim of minimizing generator production costs often comes in direct conflict with
the need of ensuring secure operation of a power system. In the viewpoint of supplying load
demand with utmost reliability, system security remains the most important aspect of power
systems engineering. In an attempt to reach a compromise between these two opposing
objectives, security-constrained unit commitment (SCUC) has emerged. The objective of
SCUC is to minimize system operating cost as well as start up cost of the committed units
while satisfying network security constraints such as, transmission line limits and bus voltage
constraints. If the committed set of units does not satisfy these constraints then the analysis
involves coordination of units on/off status and active power outputs while enforcing system
operating constraints which results in a tremendously complex optimization problem.
Several attempts have been made to solve the problem. This section deals with the
methods and constraints adopted in solving SCUC.
In [70] John J. Shaw proposed a new technique to solve the system security issues
while determining the most optimal commitment schedule. The technique mainly includes the
reserve requirements and transmission line flow constraints for the system security. These
security constraints are incorporated into the optimization function at the outset and this is
called the ‘Direct’ method. This is in contrast to the other methods that omit the security
constraints from the optimization stage and consider them only to construct a feasible UC
schedule, referred as ‘Indirect’ methods. Constraints are appended into the objective function
using Lagrange multipliers. The Lagrange multiplier associated with an overloaded line will
discourage generation from the units that contribute to excess flow, and will encourage
generation from units that can relive the overload. The method establishes a mechanism for
distributing generation throughout the system while meeting the load demand and satisfying
security requirements.
In power systems the real power generation is determined by the UC, while the
system voltage profile is mainly controlled by the generation and distribution of reactive
power. In view of this, a UC with transmission security and voltage constraints is given in
[71]. Using Benders decomposition the problem is decomposed into a master problem and
two smaller sub-problems corresponding to transmission and voltage constraints. Lagrangian
20
relaxation with dynamic programming process solves the UC problem without the
constraints. The transmission sub-problem minimizes transmission flows for the worst
contingency case by unit generation and phase shifter adjustments. The reactive sub-problem
examines voltage constraints by reactive power and tap changer adjustments. Benders cut is
generated if any violation is detected on transmission flows or voltage constraints after subproblems are solved. With Benders cuts, UC is solved iteratively to provide a minimum cost
generation schedule while satisfying all constraints. This work does not consider the impact
of real power adjustments on reactive power constraints and vice versa owing to the fact that
both the sub-problems are solved independently. The author extended by incorporating this
feature in [72]. In the initial master problem constraints on reactive power are not taken into
account. The corresponding adjustments of VAR sources are included in the hourly network
security check sub-problems. A full Newton-Raphson AC power flow is conducted and if it
cannot converge or any violations of transmission flows and bus voltages exits, a Benders cut
will be formed and fed back into the next calculation of master UC problem.
SCUC with random disturbances, such as outages of generation units and
transmission lines are modelled as scenario trees using Markov processes, and load
forecasting uncertainties as uniform random variables in [73]. The formulation of stochastic
SCUC problem differs from other deterministic SCUC, since spinning and operating reserve
constraints are relaxed in the latter case. A specific amount of spinning and operating
reserves are considered in the deterministic SCUC problem in order to maintain the security
of power system operation when outages occur or actual hourly demands increase
unexpectedly. In the stochastic SCUC problem, each possible system state is represented by a
scenario in which equipment outages and possible load increases are represented in the
SCUC solution and reserve constraints are relaxed accordingly. That is, a specific
commitment schedule is determined in each scenario for each potential contingency and
reserve constraints are implicitly enforced corresponding to each potential contingency.
Reserve constraint is not enforced explicitly.
Weeraya Poommalee et al. [74] proposed SCUC by Lagrangian Relaxation which
uses GA in updating the Lagrange multipliers. LR solves the UC problem through dual
optimization procedure. In the UC evaluation stage power losses are ignored. MATPOWER
developed OPF algorithm is run on the UC schedule. If the OPF successfully converges
without any violations on system constraints the UC schedule is considered to be optimal
21
otherwise the schedule would be revised. The work also considered the system security under
n-1 contingency for transmission line outage.
An algorithm to solve UC problem with added environmental constraints along with
operational, power flow constraints under contingencies has been developed to plan an
economic and secure generation schedule in [75]. The problem is a bi-objective function with
ED and economic emission dispatch. This is converted to single optimization problem by
introducing a penalty factor. The algorithm proceeds by first finding all the states that satisfy
the load demand. Then for each satisfying state, by removing one line from the system a
contingency analysis is performed by carrying out the optimal power flow using hybrid
Lagrangian multiplier and Newton Raphson power flow algorithm. This contingency analysis
is performed repeatedly for each satisfying state until all the lines are removed once, except
the lines which are connected only either to the load bus or generator bus. Optimal power
flow is carried out for every contingency of that state. By adding minimum transitional costs,
which satisfy unit constraints, to the entire satisfying states an optimum schedule is selected.
Voltage and reactive power constrained UC is solved using Benders decomposition in
[76]. Based upon the practical situation of the weak effect of voltage and reactive power to
the unit on/off states, the primal problem is decomposed into a master problem and a subproblem. UC is solved by the master problem with transmission security constraints based on
a DC power flow. The sub problem is a series of reactive power optimization problems for
given active power schedule. An AC power flow is carried out to clarify whether the voltage
and reactive power are within acceptable limits. If not, Benders cut will be produced to
modify the commitment for master problem. A similar method is also carried out in [77] in
which at the sub-problem level a non-linear optimal power flow is solved.
2.3 UC in Renewable Integrated Power Systems
To sustain the ever increasing electricity demand with the depleting fossil fuels is a
serious problem being faced today. The objective of many nations is to develop ones own
alternative resources to avoid importing the expensive fossil fuels. As a future way to attain
this objective the plan points towards the development of renewable resources and able to
integrate them into the existing grids.
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In 1985 M. Castro et al. presented weekly planning for the large Spanish electric grid
with the integration of solar electric units, based on solar central receiver and photovoltaic
ones [78]. The solar electric generation value is obtained from a solar central simulation
program developed. The planning model used makes the generation planning hour by hour in
periods of a week or multiples (up to one year). The weekly scheduling is done using all units
present in the Spanish power system, including hydro and pump-storage ones and analyzing
the introduction of solar thermal and photovoltaic ones. Overloading in the tie lines between
geographic zones is avoided both in case of normal operation and in the case of failure of a
thermal unit. In this case operation cost associated with the failure unit is considered. The
result obtained in these plannings is the substitution equivalent cost, which is the generation
cost ($/kWh) of the energy replaced when the solar energy units are introduced. This value
allows weekly planning schedule and defines the maximum solar central operation cost that
allows to replace thermal units (mainly fuel, gas and coal ones) by solar ones.
The impact of solar thermal power plants on economy and reliability of utility system
is studied in [79]. An efficient method is developed to evaluate the economical limitations of
integrating the solar thermal power plants to the utility grid as well as the attractiveness of
these plants in terms of the marginal impact on the system economy and reliability. The
economic limitation is formulated by evaluating the annual expected fuel cost saving due to
the inclusion of solar thermal power plant and considering the installation cost incurred in
solar thermal power plants and necessary transmission system.
Short term generation scheduling problem in a small autonomous system with both
conventional and unconventional energy sources and storage battery is worked out in [80].
This is a generation mix of power system with diesel generators, wind turbine generators and
photovoltaic panels. Solution method divides the original optimization problem into two
smaller ones, namely a diesel unit commitment sub-problem and a battery storage policy subproblem. The two sub-problems are coupled through the power balance equation and solved
by Dynamic programming. An optimum battery storage policy is first computed then the
optimal commitment of diesel units is determined.
A new approach is considered in [81] for incorporating Photo Voltaic (PV) and
battery in the thermal unit commitment. It is shown that the peaking generators can be kept
off during peak hours by utilizing PV-battery. The problem is solved in three steps; the first
23
step finds initial feasible solution for thermal units involving minimum battery charging
states at each stage. In the second step a thermal UC problem is solved. In the third step an
optimum PV-thermal dispatch is found based on the thermal commitment.
A fuzzy-optimization approach for solving generation scheduling problem with
consideration of wind and solar energy systems especially PV, is demonstrated in [82]. While
performing the generation scheduling problem by conventional methods the values of hourly
load, available water, wind speed, solar radiation are considered to be forecasted. However,
actually there are always errors in these forecasted values. A characteristic feature of this
approach is that the errors in the forecast hourly load, available water, wind speed and solar
radiation can be taken into account using fuzzy sets. Fuzzy set notations in the hourly load,
available water, wind speed, solar radiation, spinning reserve and total fuel cost are
developed to obtain optimal generation schedule under an uncertain environment. A similar
work is demonstrated in [83] for solving thermal UC problem integrated with PV-battery
system. To avoid the forecasting errors which become severe for large scale power system,
the proposed method applies a fuzzy model where load demand, solar radiation, spinning
reserve and production cost are presented as fuzzy variables. The errors are taken from a
priory statistics due to the absence of actual data. Then a modified differential evolutionary
method is deployed to solve the UC problem.
A short term thermal UC strategy integrated with solar and wind energy systems is
presented in [84] by facilitating GA operated PSO. Load leveling is achieved using renewable
energy sources. Penalty terms are used to discourage the non-potential and erroneous solution
both for equality and in-equality constraints. A battery system is incorporated with solar
energy to supply power in case of peak load shaving and also to facilitate the load leveling by
optimizing battery charge/discharge schedule. This assumes accurate forecasting on wind
speed and solar radiation. In [85] the authors extended the work for thermal UC with wind
energy system considering unexpected load deviation. The method tracks down the load
deviation at a particular hour and using a sophisticated load forecasting technique re-predicts
the load demand for the hours to come. This way a relatively accurate load forecasting is
achieved which will eventually reduce the fuel cost.
An optimal operational planning of smart grid with wind generator, PV facility, diesel
generator, and battery energy storage system is presented in [86]. This method used TS and
24
GA for optimization method and the procedure is divided into two parts. Firstly, actual load
is controlled by controllable load such as electric water heater, heat pump, and electric
vehicles by using TS. Secondly, the schedule of diesel generators commitment problem is
decided for the revised load. Based upon the forecast data, the generated power of Wind
Farms, the generated power of PV, and the storage energy of battery is calculate. The
difference between the load demand and the calculated renewable power gives the deviation
energy. Hence, Diesel Generators’ ON/OFF operation and generator output power are
determined. The amount of charge/discharge for the battery system is decided from the
difference between load demand, renewable power and DG output.
Effects of security constraints on UC with wind generators are examined in [87]. The
work studies the effects of capacity limit of lines connecting wind generators on the optimal
UC schedule. A stochastic characteristic of wind generators is included, which is used to
compute expected energy not served, a measure which quantifies the amount of output that
may not become online from the wind generators at any given hour. It is parameterized and
added to the spinning reserve constraint to counter the risk introduced through the inclusion
of wind generators. This inclusion is analogous to N-1 reliability criteria.
Renewable energy generation such as solar and wind brings new challenges to Unit
Commitment and Economic Dispatch from security perspective due to the uncertainty caused
by its imprecise prediction. It is difficult to predict and control the output of renewable
generation because of their intermittency and the reserve capacity required to deal with
inherent uncertainty. A more literature on security constrained unit commitment
methodologies when renewable energy sources such as single or multiple wind farms of
appreciable capacity are integrated with the conventional thermal units can be found in [8891].
25
Chapter 3
3 Unit Commitment
3.1 General
Economic operation of power system is very important to return profit on the capital
invested and to subside a part of investment itself through proper planning. More
significantly it is important from the perspective of conserving the irreplaceable fossil fuels.
Economic operation results in maximizing the operating efficiencies which in turn minimize
the cost per kilowatt-hour. Total load on power system varies at every instant of time,
generally being higher during the daytime and early evening when industrial loads are high,
lights are on, and so forth, and lower during the late evening and early morning when most of
the population is asleep. In addition, the use of electric power has a weekly cycle, the load
being lower over weekend days than weekdays. Therefore, the option of turning ON enough
units and leave them online, so that the variable load demand is met at all times is not viable
due to the costs involved. This causes some of the units to operate near their minimum
capacity at times, resulting in lower system efficiency and increased economics. Thus, if the
operation of the system is to be optimized, units must be shut down as the load goes down
and must be brought online as it goes up again [1].
Electric utilities have to plan their generation to meet this varying load in advance, as
to which among their available generators are to start-up and when to synchronize them into
the network as well as the sequence in which the operating units must be shut down. The
process of making this decision is well known as ‘Unit Commitment’ [92]. The word
‘commit’ refers to ‘turn ON’ a unit. Thus, the problem of Unit Commitment is to schedule the
ON and OFF times of the generating units with the overall minimum cost while ensuring the
unit’s operational constraints like minimum up/down times, ramp rate limits, maximum and
minimum power generation limits.
Out of the cost incurred in generation, major component is the cost of fuel input per
hour for all the generators, while maintenance cost contributes only to a small extent. This
fuel cost evaluation is more important for thermal and nuclear power stations, which is not
26
the case with hydro stations where the energy is obtained from storing water in dams built for
irrigation purpose and is apparently free. Fuel cost savings can be obtained by proper
allocation of load among the committed units. But the problem of UC minimizes the total
cost which includes both production cost i.e., the fuel cost and costs associated with the startup and shutdown of units. Start-up cost and shutdown cost are categorized by unit type. A
fixed cost is incurred with the shut-down of a unit while the start-up cost is dependent on the
length of time the unit has been down prior to starting. When performing the unit
commitment scheduling a variety of operating constraints and spinning reserve requirements
are observed.
3.2 Generator characteristics
Fundamental constituent in economic operation of a unit is its performance
characteristics, which depicts the relation between input and output. This characteristics
specifies the input energy rate or cost of fuel used per hour as a function of generator power
output. The input-output characteristic of a generating unit is obtained by combining directly
the input-output characteristics of boiler and that of turbine-generator set [93]. A typical
input-output characteristic also called fuel cost curve of a thermal generating unit is convex
as shown in Fig. 3.1. Complete performance characteristics of a thermal generating unit are
detailed in Appendix C.
Figure 3.1 Input-Output Characteristics of a Thermal Generator
27
It can be seen that the characteristics are bounded between minimum and maximum
capacities. The minimum power output limitations are generally caused by boiler’s fuel
combustion stability and design [93] where as maximum limit is determined by the design
capacity of boiler, turbine, generator. These non-linear characteristics are generally
approximated to a quadratic function expressed in terms of unit’s power generation as shown
in Eq. (3.1).
F ( Pg i ) = ai Pg i2 + bi Pgi + ci
(3.1)
Where F(Pgi) represents the cost function, Pgi is the power output and ai, bi, ci are the
coefficients of input-output characteristic of ith unit. These cost coefficients are determined
experimentally. The constant c is equivalent to the fuel consumption or cost incurred in
operating the unit without power output. The slope of this input-output curve is called the
incremental fuel cost of unit.
3.3 Start-up and shut down costs
As mentioned earlier there exists a cost incurred in starting and shutting down a unit,
apart from fuel cost. Certain amount of energy must be expended to bring a unit online
because the temperature and pressure of the thermal unit must build slowly. This energy does
not result in any MW power output and is considered as start-up cost. There are two types of
start up costs called hot start-up cost and cold start-up cost. If the unit’s boiler is allowed to
cool down and then heat back up to operating temperature while turning ON the unit it is
called cooling and the corresponding cost is cold start cost. On the other hand if the boiler is
supplied with sufficient energy to just maintain operating temperature until the unit is brought
online again is called as banking and the cost involved is called hot start cost. This hot start
cost varies directly with the duration of unit being offline. The two costs are as shown, and
are compared while determining the UC schedule and a best approach among them is chosen
[1].
Start up cost for Cold start:
STC = Cc (1 − ε − t /α ) F + C f
(3.2)
Start up cost for Hot start:
STC = Ct t F + C f
(3.3)
28
Where STC is the Start up cost, Cc is the cold start cost in MBtu, F is the fuel cost, Cf is the
fixed cost that includes crew expenses and maintenance expenses, Ct is cost in Mbtu/hour for
maintaining the unit at operating temperature, α is the thermal time constant of the unit and t
the time in hours the unit was allowed to cool.
Shutdown cost is generally taken as a constant value.
3.4 Constraints
The list of constraints is by no means exhaustive and depends on the individual
utility’s rules and reliability measures. Some of the constraints which reduce the freedom in
the choice of starting up and shutting down of units in the system are listed below. These
constraints can be brought in either because of unit technical issues or system operational
requirements.
A thermal unit usually undergoes a gradual temperature changes, and this develops
into a time period of some hours required to bring the unit on-line. When a unit is online its
generation cannot be increased or decreased instantaneously owing to mechanical limitations.
And in general for turning on and turning off a unit in thermal systems requires a crew to
operate. These all issues pose limitations in arriving at optimal UC schedule.
Minimum up time:
Once a unit is committed and running, it should not be turned off immediately. It is an
engineering consideration normally requires that a unit be running for at least a certain
amount of time before it is shutdown.
Minimum down time:
Once the unit is decommitted, there is a minimum time gap before it can be committed and
brought online again.
Crew constraints:
It is due to the limitation of personnel availability in the plant. If a plant consists of two or
more units, both cannot be scheduled at the same time since there is no enough crew to attend
both units while starting up or shutting down.
29
Must run units:
These units include pre-scheduled units which must be on-line. Some units are given a mustrun status during certain times of the year for the reasons of voltage support on the
transmission network i.e. a reliability and/or economic considerations.
Must out units:
Units which are on forced outages and maintenance are unavailable for commitment and are
treated as must-out units.
Units on fixed generation:
These are the units which have been pre-scheduled and have their generation specified for
certain time period. A unit on fixed generation is automatically a must run unit for the
designated time period.
Fuel constraints:
These constraints applies in a system in which some units have limited fuel, or else have
constraints that require them to burn a specified amount of fuel in a given time, presents a
most challenging unit commitment problem.
Constraints which impose indirect influence on UC but should be considered are:
Maximum and Minimum output limits of a unit:
These define the range in which the unit can actually be dispatched, these limits does not
have any direct influence on the starting up and shutting down of the unit.
Ramp rate limits:
These represent the range of change in output over a unit time, used to prevent
undesirable effects on generating units due to rapid changes in loading. When a unit is in the
start-up stage, a pre-warming process must be introduced in order to prevent a brittle failure,
especially when the unit start-up is a long process. Because of the unit physical limitations,
the unit generating capability increases as a ramp function. Similarly, when a unit is in the
shut down process, it will take a while for the turbine to cool down. Before the unit
generating capability decreases to its lower limit, the residual energy is to be used to meet the
30
load demand. Therefore, because of the unit physical limitations, the unit generating
capability increases as a ramp function [46], [94].
Spinning Reserve:
Spinning reserve requirements are necessary in the operation of a power system in
order to achieve minimum load interruptions. Spinning reserve is the term used to describe
the total amount of generation available from all units synchronized (i.e., spinning) on the
system, minus the present load and losses being supplied. Spinning reserve must be carried so
that the loss of one or more units does not cause too far a drop in system frequency. Quite
simply, if one unit is lost, there must be ample reserve on the other units to make up for the
loss in a specified time period [1].
Spinning reserve requirements may be specified in terms of excess megawatt capacity
or some form of reliability measures. Typical rules specify that reserve must be a given
percentage of forecasted peak demand, or that reserve must be capable of making up the loss
of the most heavily loaded unit in a given period of time. The amount of spinning reserve is
an important factor in the assurance of uninterrupted supply to the customers and so is the
distribution of spinning reserve among various generating plants based upon their responding
time and relative distance to the load centres.
3.5 Unit Commitment Formulation
The unit commitment schedule takes many factors into consideration including:
•
Unit operating constraints and costs
•
Generation and reserve constraints
•
Plant start-up and shut-down constraints
•
Network constraints
31
Objective function: Mathematically the objective function of unit commitment problem is
the sum of fuel costs as well as start-up and shut-down cost of all generating units over a time
frame, which needs to be minimized and can be represented as follows:
Total Cost = ∑∑ U it F ( Pg i ) + U it (1 − U it −1 ) STCi + U it −1 (1 − U it ) STDi 
T
NG
(3.4)
t =1 i =1
Subjected to:
ng
∑ Pg
Power balance:
i =1
i
= Pd
ng
Minimum capacity committed:
∑ Pg
i =1
Minimum up/down time:
Unit generation limits:
Unit Ramp rate limits:
i ,max
(3.5)
≥ Pd + Presv
Ti on ≥ Ti up
(3.6)
(3.7)
Ti off ≥ Ti down
Pgi ,min ≤ Pgi ≤ Pgi ,max
Pgit −1 ≤ Pgit ≤ Pgit −1 + Rupi
Pgit −1 ≥ Pgit ≥ Pgit −1 − Rdn i
(3.8)
(3.9)
Where T is time horizon in hours over which UC schedule is desired, t is the hour index, i is
the Generator index, NG is the total number of generators ,Uit is ith unit ON/OFF status (1ON, 0-OFF) during tth hour, F (Pgi) is the Cost function, ng is the number of committed
generators, Pgi is ith unit active power generation, STCi, STDi are ith unit start up cost and shut
down costs respectively, Pgi,max, Pgi,min are maximum and minimum generation limits, Rupi,
Rdni are rate of ramp up and ramp down constants, Pd, Presv are active power demand, reserve
respectively. And Tiup, Tidown are minimum up, down times in hours, Ticold is cold start hour,
Tion, Tioff are continuously on and off timings in hours.
The start up cost of ith unit is decided as follows:
 Hot start cost

STCi = 
Cold start cost
if Ti down ≤ Ti off ≤ (Ti cold + Ti down )
if Ti off > (Ti cold + Ti down
32
)
(3.10)
3.6 Economic Dispatch
Economic dispatch is a sub-problem in determining UC. As mentioned earlier UC
problem minimizes the total cost which includes both generation cost and costs associated
with the start-up and shutdown of units. Precisely it determines which among the available
units should be committed so as to meet the load demand by considering ‘all possible number
of combinations’. On the other hand, ED minimizes only the cost of generation for a given set
of committed generators. Therefore as name suggests ED distributes the given amount of load
on to the committed generators economically while ensuring that all committed units are used
at least at their minimum capacity to fulfil the load requirement. Both the optimization
problems viz., UC and ED are interrelated. Always in finding UC, ED is solved as a subproblem as each of the commitment schedules requires an ED solution. The constraints
involved in solving ED are very limited.
ED constraints:
1. Dispatch should meet the load demand.
2. Unit generation limits
Mathematically ED can be expressed as:
Objective Function: The sum of fuel cost of all committed generators to be minimized.
FT = F1 + F2 + ..... + Fng
(3.11)
ng
= ∑ Fi ( Pgi )
i =1
Subjected to:
ng
Power balance:
∑ Pg
i =1
Unit generation limits:
i
= Pd
(3.12)
Pgi ,min ≤ Pgi ≤ Pgi ,max
(3.13)
Thus, ED is a constrained optimization problem, which is in general solved using
Lagrange multiplier method. It optimizes a Lagrange function which converts a constrained
problem into an unconstrained problem by adding the constraint function to objective
function after multiplying with the Lagrange multiplier as shown in Eq.(3.14).
33
ng


ℓ ( Pgi , λ ) = FT + λ  Pd − ∑ Pgi 
i =1


(3.14)
Where, λ is the Lagrange multiplier. Necessary condition for the objective function to have
minimum is that the partial derivative of Lagrange function with respect to each of its
arguments must be zero. Therefore,
∂ℓ ( Pgi , λ )
∂Pgi
∂ℓ ( Pgi , λ )
∂λ
=
∂FT
−λ = 0
∂Pgi
(3.15)
ng
= Pd − ∑ Pgi = 0
(3.16)
i =1
From equation (3.15)
∂FT
=λ
∂Pgi
i = 1, 2, 3,…ng
(3.17)
From equations (3.11) and (3.17)
ng
∂FT
=
∂Pgi
∂ ∑ Fi ( Pgi )
i =1
∂Pgi
= λ
That is,
dF1 ( Pg1 )
dPg1
=
dF2 ( Pg 2 )
dPg 2
= ..... =
dFng ( Pg ng )
dPg ng
=λ
(3.18)
The above relation shows that the optimal loading of generators is possible at the
point where slopes of the input-output characteristics of all the generators are equal. In other
words the optimal dispatch corresponds to the point of equal incremental cost for all
generators; equation (3.18) is also called coordination equation. To know the values of
Lagrange multiplier or otherwise called incremental cost and individual unit generation,
equation (3.18) is solved as under.
On derivating equation (3.1),
dFi ( Pgi )
dPgi
∴
= 2ai Pgi + bi
(3.19)
2ai Pgi + bi = λ
34
Pgi =
λ − bi
i = 1, 2, 3……ng
2ai
(3.20)
Thus, λ can be calculated using (3.12) and (3.20) as
ng
λ − bi
i =1
2ai
∑
= Pd
ng
∴λ =
Pd + ∑
i =1
bi
2ai
(3.21)
ng
1
∑
2
i =1 ai
Load sharing so obtained from (3.20) for each generator might cross the individual
minimum or maximum generation limits since the formulation so far does not take this into
account. For a practical and technically feasible dispatch the effect of generator limits is to be
considered. This is achieved by a simple principle, if a particular generator loading reaches its
minimum or maximum, its loading is fixed at that value and the balance load is shared
between remaining units again on equal incremental cost basis.
The above formulation of ED neglects the effect of transmission losses in the process
and only holds good if all the committed units are situated at one place, which is not true in
practice. In other word the above formulation of ED is to distribute the generation among the
units in one power plant. Power system is spread over a large geographic area with most of its
generating stations located at remote places and interconnected with transmission lines for
supplying power which results in losses while transporting the generated power. The
transmission losses usually vary from 5 to 15 percent of total system load. Thus, it is
unrealistic to neglect transmission losses and they should be included in the ED formulation.
The following text explains ED problem for the distribution of load among various
generating plants i.e. ED including losses (EDL) [2].
The objective function of ED given in equation (3.11) is now subjected to the
following constraints where Ploss represents active power loss in the system and it is obvious
that the limits on active power generation remain the same.
35
ng
∑ Pg
i =1
i
= Pd + Ploss
(3.22)
Pgi ,min ≤ Pgi ≤ Pgi ,max
In general, transmission losses of the system are approximately expressed as a
function of generator powers using B-coefficients derived in Appendix A. Accordingly the
expression for system losses is
N
N
Ploss = ∑∑ Pi Bij Pj
(3.23)
i =1 j =1
Where N is total number of buses in the system, Bij is the loss coefficient between buses i and
j and Pi, Pj represents active power injections at respective buses.
Therefore, using Lagrange multipliers, the augmented Lagrangian function is
ng


ℓ ( Pgi , λ ) = FT + λ  Pd + Ploss − ∑ Pgi 
i =1


(3.24)
For optimization
∂ℓ ( Pgi , λ )
∂Pgi
=
 ∂P

∂FT
+ λ  loss − 1 = 0
∂Pgi
 ∂Pgi

 ∂P 
∂FT
= λ 1 − loss 
∂Pgi
 ∂Pgi 
∴
∂ℓ ( Pgi , λ )
∂λ
Where
i = 1,2,….ng
(3.25)
ng
= Pd + Ploss − ∑ Pgi = 0
(3.26)
i =1
∂Ploss
∂Pgi
is called the incremental transmission losses and is obtained by partially
derivating Eq. (3.23) as
N
∂Ploss
= ∑ 2 Bij Pg j
∂Pgi j =1
i = 1,2,3….ng
36
(3.27)
From equation (3.25)
∂FT
∂Pgi
λ=
∂P
1 − loss
∂Pgi
 ∂FT
 ∂Pgi
λ =

 Li

Where, Li =
(3.28)
1
th
∂Ploss , is the penalty factor of i plant.
1−
∂Pgi
By using (3.19), (3.25) and (3.27) we obtain Pgi as under
N


2ai Pgi + bi = λ 1 − ∑ 2 Bij Pg j 
j =1


N


2 ( ai + λ Bii ) Pgi = λ 1 − ∑ 2 Bij Pg j  − bi
j =1




2 Bij Pj  − bi
j =1, j ≠ i

2 ( ai + λ Bii )
λ 1 −
Pgi =

N
∑
(3.29)
3.7 Optimal Power Flow
It is very clear from previous section that transmission loss bias the economic
dispatch problem and the coordination equations include the effects of incremental
transmission loss and increased the complexity of problem. Behaviour of network elements
leads many effects on system operation. For instance, when network transmission lines are
considered in formulation, it indicates some of the effects like increase in the total generation
demand due to real power losses, adjustments in the generation schedule in accordance to the
limits on transmission line flows. Thus, it is very important to take into account the effects of
network elements in finding the optimal solution to ensure system security.
Optimal power flow (OPF) is an extension to conventional ED problem; it determines
minimal cost by optimal settings of different control variables in the system. The OPF is a
power flow problem in which certain controllable variables are adjusted to optimize system
37
objectives. Some of the objective functions which are optimized using OPF formulation are
the cost of active power generation, system losses, emission of generating units etc., while
satisfying power flow equations, equipment operation limits and system security. The
controls that an OPF can accommodate are active and reactive power injections, generator
voltages, transformer tap ratios and phase shifter angles [93], [95].
OPF is very different from ordinary power flow. In power flow calculation the
objective is to find bus voltage magnitudes and phase angles at all the buses in the system.
Power flow is a steady state study and gives the snap shot of the whole system operating
state. It is given with scheduled complex loads on all load buses and generated active powers,
voltage magnitudes on all generator buses. The net flow of power from a bus into the system
is termed as injection at that bus. Power flow finds the load bus voltage magnitudes and
phase angles by minimizing the difference between scheduled injection and calculated
injections using techniques like Gauss-Seidal or Newton-Raphson. Scheduled injection at a
bus is the difference between scheduled power generation if any and the complex load at that
particular bus. The power injections at a bus are derived in the next section and calculated
using equations (3.39) and (3.41). Post power flow calculations are carried out by system
operators using the bus voltage magnitudes and corresponding phase angles to find the
current state of the system. These calculations involve line power flows, line losses and
reactive power generation at generator buses. Power system operators have to plan the
adjustments accordingly if these values exceed their corresponding limits to ensure system’s
secure operation.
Optimal power flow is a very large and complex mathematical problem. In general
OPF is posed as minimizing the function F(x,u) while satisfying nonlinear equality
constraints g(x,u) = 0 and nonlinear inequality constraints h(x,u) ≤ 0 on the vectors x and u.
The vector x contains dependent variables including bus voltage magnitudes and phase angles
and the reactive power outputs of generators on voltage controlled buses. The vector u
consists of control variables which are independent and involves active and reactive power
generations, transformer phase shifter angles, transformer tap ratio settings, load shedding,
DC line flow, switched capacitor settings.
OPF problem with the objective function of minimizing the generation cost in thermal
electric power system is discussed here. In the ED solution presented so far, limits on only
38
minimum and maximum active power generations are observed. In OPF many more limits on
power systems equipments can be included like bounds on reactive power generations,
transmission line flows, bus voltage magnitudes. OPF problem finds an optimal profile of
active and reactive power generations along with voltage magnitudes in such a manner as to
minimize the total operating costs.
The objective function is same as the one shown in equation (3.11), whereas the list of
constraints subjected to, varies and are shown below.
OPF constraints:
1. Power Balance in the network.
2. Unit generation limits.
3. Limits on load bus voltage magnitudes.
4. Limits on transmission line flows, transformer tap settings and phase shifter
angles.
Objective function: The sum of fuel cost of all committed generators is to be minimized
FT = F1 + F2 + ..... + Fng
ng
= ∑ Fi ( Pgi )
i =1
= ∑ ( ai Pgi2 + bi Pgi + ci )
ng
i =1
Subjected to: Active and reactive power balance in the network
Pgi − Pdi − Pi = 0
i = 1,2,…N
(3.30)
Qgi − Qdi − Qi = 0
i = 1,2,…Nb
(3.31)
Where Pgi, Qgi represents active and reactive power generations Pi, Qi represents active and
reactive power injections at bus i and Pdi, Qdi represents active and reactive power demands
at bus i, N is total number of buses and Nb is total number of load buses in the system.
39
Limits on active and reactive power generations on all generator buses:
Pgi ,min ≤ Pgi ≤ Pgi ,max
i = 1,2,…ng
(3.32)
Qgi ,min ≤ Qgi ≤ Qgi ,max
i = 1,2,…ng
(3.33)
Limits on voltage magnitudes and phase angles on all load buses:
Vi ,min ≤ Vi ≤ Vi ,max
i = 1,2,…Nb
(3.34)
δ i ,min ≤ δ i ≤ δ i ,max
i = 1,2,…Nb
(3.35)
Limits on line flows can be expressed either in MW, Amperes or MVA, if it is expressed in
MW then:
Pi j ,min ≤ Pi j ≤ Pi j ,max
i = 1,2,…Nl
(3.36)
Where Pij is the active power flow between buses i and j. Pij,min, Pij,max are corresponding
minimum and maximum limits, Nl is the total number of transmission lines.
The constraint optimization problem can be transformed into an unconstrained one by
augmenting the equality constraints of active and reactive power balance equations into the
objective function using Lagrange multipliers. The solution of this Lagrangian function
involves first order and second order partial derivates terms called the Jacobian and Hessian
matrices respectively. The complete solution of OPF using Hessian matrix by Newton’s
method is presented in [96].
3.8 Calculation of Bus Injections
The calculation the power injection at a bus requires basic power equation and the admittance
matrix Y. Apparent power at any node in the network is given by
Si = Vi I i*
= Pi + jQi
Where Si is the apparent power, Vi is the complex voltage and Ii is the complex current at bus
i. And ‘*’ represents complex conjugate.
40
For simplicity in calculations the above equation is rewritten as
Si∗ = Vi* I i
(3.37)
= Pi − jQi
Where Ii is the current flowing out at bus i, and is given as the sum of all the currents leaving
the bus. Using π equivalent model of transmission lines, it can be obtained as
N
I i = ∑ Yi jV j
(3.38)
j =1
Yij represents (i, j) element in the network admittance matrix, can be written in conductance
(G) and suseptance (B) form as Yi j = Gi j + jBi j . Thus,
 N

Pi − jQi = Vi ∠ − δ i  ∑ ( Gi j + jBi j )V j 
 j =1

On separating real and imaginary parts
Pi = Vi
∑ V (G
Qi = Vi
∑ V (G
N
j =1
j
ij
N
j =1
j
ij
cos (δ i − δ j ) + Bi j sin (δ i − δ j )
)
(3.39)
sin (δ i − δ j ) − Bi j cos (δ i − δ j )
)
(3.40)
Equation (3.39) and (3.40) represents real and reactive power injections respectively at bus i.
3.9 Calculation of Line Flows
Consider the π representation of a line connecting buses i and j shown in the Fig. 3.2. The
figure shows the bus i to be the transformer side bus, with the ratio 1: a. Hence, Vt = aVi . The
representation has a series admittance, yij and shunt admittances, ysi and ysj at the ends of the
line. The power from the bus i to bus j can thus be given as
41
Figure 3.2 Transmission Line π Model
Si j = Pi j + jQi j = ( aVi
)( I i )
∗
Si∗j = Pi j − jQi j = ( aVi ) I i
∗
And
I i = I i j + Isi
I i = ( aVi − V j ) yi j + ( aVi ) ysi
In polar form the equation becomes
Pi j − jQi j = aVi ∠ − δ i
( ( aV ∠δ − V ∠δ ) ( g
i
i
j
j
ij
+ jbi j ) + aVi ∠δ i ( gsi + jbsi )
)
On separating real and imaginary parts we arrive at the active and reactive power flows in the
line
Pi j = a 2 Vi
2
Qi j = −a 2 Vi
2
(
)
(3.41)
(
)
(3.42)
( gs + g ) − a V
i
V j gi j cos (δ i − δ j ) + bi j sin (δ i − δ j )
( bs + b ) − a V
i
V j g i j sin (δ i − δ j ) − bi j cos (δ i − δ j )
i
i
ij
ij
42
Chapter 4
4 Solar Thermal Power Plant
With the extended and over utilization, it has become obvious that fossil fuel
resources are fast depleting and that the fossil fuel era is gradually coming to an end. This is
in particularly true about coal, oil and natural gas resources. The large scale usage of fossil
fuels over the years also created harmful effects on the environment. The outcome of this is
the phenomenon of global warming which is a matter of great concern for future human life.
The situation triggered whole of the world in search of alternative sources to fulfil the energy
requirements. Now, as man embarks on this search, it is clear that he should do it well by
keeping the environmental concerns always in mind. Renewable energy sources like solar,
wind, tidal, geothermal are the alternatives that attracted the focus. To a significant level, the
extraction of energy from solar and wind is a success. The study here concentrates on the
usage of solar thermal energy for electricity production.
4.1 Solar Energy
Sun is a large sphere of very hot gases at a distance of 1.50X108 km from the Earth. It
is a very large inexhaustible source of energy. The heat is being generated by various fusion
reactions taking place. Brightness of sun varies from its centre to edges. Due to very large
distance between Earth and Sun, whatever the beam radiation falls on Earth is almost parallel.
The radiation from Sun has to pass through atmospheric layer and then reach the Earth
surface. The power intercepted by the Earth is many thousands times greater than the present
energy consumption rate from all commercial energy sources [97]. It is an environmentally
clean source of energy unlike fossil fuels. Thus, the Sun energy is a most promising source
that could supply all the present and future energy needs of world. The rate at which solar
energy arrives at the top of atmosphere in unit time on a unit area is called solar constant Isc.
According to National Aeronautics and Space Administration (NASA) the standard value for
this extra-terrestrial radiation of solar constant is 1353 W/m2 [98].
However, there are problems associate with its use, main one being the intermittent
nature of availability. The variation occurs because of the day-night and also seasonal cycles.
43
Apart from this major drawback, a huge reduction in available radiation for usage is effected
by the geometrical constraints of the Sun and Earth. Due to very long distance between Sun
and Earth the value of usable radiation accounts slightly higher than 1 kW/m2. This
consequently results in supply of low temperatures for the thermal applications. The use of
optical concentration devices enables thermal conversion at high temperatures applicable for
practical applications.
4.2 STPP
In a Solar Thermal Power Plant (STPP), solar energy is transferred to a thermal fluid
at an outlet temperature that is high enough to feed a heat engine or a turbine which in turn
drives a generator to produce electricity. It is therefore an essential requisite for STPPs to use
some kind of concentrating devices that raise the temperature of working fluid. Such a
technology is termed as Concentrated Solar Power (CSP). The CSP technologies are of
different kinds, but they use the same concept of focusing the weak solar radiation onto a
focal point to achieve high temperatures. They are Parabolic Trough Collectors (PTC), Linear
Fresnel Reflector Systems (LF), parabolic Dish/Engine systems (DE), and Power Towers or
Central Receiver Systems (PTS), commonly designed for a normal incident radiation of 800–
900 W/m2 [99].
All of the four concentrating solar technologies use large mirror areas and work under
real operating conditions. PTC and LF are two dimensional concentrating systems in which
the incoming solar radiation is concentrated onto a focal line by one-axis tracking mirrors as
shown in Fig.4.1. They are able to concentrate the solar radiation up to 30–80 times, heating
the thermal working fluid up to a temperature of 400°C. These are capable for power
conversion in the sizes of 30–80 MW, and are well suited for centralized power generation
with a Rankine steam turbine/generator cycle.
44
Figure 4.1 Schematic Diagrams of Parabolic Trough and Linear Fresnel CSP Systems
DE systems are small modular units with autonomous generation of electricity using
gas operated Stirling Engines located at the focal point. Dishes are parabolic concentrators
with high concentration ratios 1000–4000 and unit sizes of 5–25 kW.
PTS optics is more complex in design. The technique employs a solar receiver
mounted on top of a tower and sunlight is concentrated by means of a large paraboloid that is
discretised into a field of heliostats. Heliostats are individual concentrating mirrors that focus
the radiation onto the top of tower. These heliostats require two-axis tracking and
concentration factors are between 200–1000 and possible unit sizes are between 10 and 200
MW.
Figure 4.2 Schematic Diagrams of Dish type and Power Tower CSP Systems
With the raising interest towards clean energy resources many new renewable power
plants are coming up in recent times. Many opportunities are being open for CSP
45
technologies in bulk electricity production. The ability of CSP to provide power on demand
makes it stand out from other renewable energy technologies like PV or Wind. Even though
the Sun is an intermittent source of energy, CSP systems offer the advantage of being able to
run the plant continuously at a pre-defined load using thermal storage systems, naturally with
increased capital cost. In spite of the promising applications and environmental benefits, to
date only parabolic trough collectors based power plants have operated commercially. The
nine SEGS plants totalling 354 MW, built by LUZ in California in the 1980s and 1990s
employs the parabolic trough technology [100]. The plants have demonstrated excellent
performance in long run. The parabolic troughs, by far the most mature technology which has
been demonstrated commercially is preferred here for the study. The thesis designs a STPP
using the parabolic trough type concentrating technology without thermal storage system and
is explained below.
4.3 Parabolic Trough Collector
It is basically a parabolic trough shape mirrors arranged to reflect the direct solar
radiation incident on it. This optical subsystem in termed as concentrator and the absorber is
termed as receiver onto which the incident radiation is reflected. The receiver tube is located
exactly at the focal axis for the reflected radiation to fall on it. The concentrated radiation
heats the fluid that circulates through the receiver tube, thus transforming the solar radiation
into thermal energy in the form of the sensible heat of the fluid. Absorber tube is a steel pipe
surrounded by concentric glass tube with an annular gap to reduce the convective heat losses
from the inner hot steel pipe. In high performance collectors, the space between the absorber
tube and glass tube is evacuated. The two important device characteristics which influence
the collectors output are Aperture area and Concentration ratio. These are defined as:
Aperture area ‘Ac’: is the plane opening of the concentrator through which the solar
radiation fall. For parabolic trough collector it is characterized by width of the opening and
length of the collector. It is nothing but collector surface area.
Concentration ratio ‘C’: It is the ratio of the effective area of the aperture to the surface area
of the receiver Ar.
46
C=
Ac
Ar
The parabolic trough reflector is held by a steel support structure on pylons in the foundation.
Fig.4.3 shows a typical PTC.
Figure 4.3 Parabolic Trough Collector
Two PTC designs specially considered for large solar thermal power plants are the
LS-3 and EuroTrough (ET-100), both of which have a total length of 100 m and a width of
5.76 m, with back-silvered thick glass mirrors and vacuum absorber pipes. The work in this
thesis adopts the LS-3 type collector in computing the solar power output. In a typical PTC
power plant, in order to achieve the required nominal thermal output a number of collectors
are connected in series and parallel combinations. The number of collectors connected in
series in every row depends on the temperature increase to be achieved between the row inlet
and outlet. And number of parallel rows depends on the power output requirements.
47
4.4 Losses in PTC
The entire incident solar radiation on the aperture of parabolic trough will not be transferred
onto the receiver due to many loss factors. Typical losses in PTC can be mainly split into
three categories given below.
a. Optical losses
b. Thermal losses in receiver and its auxiliaries
c. Geometrical imperfection losses.
Geometrical losses are due to the imperfections in various accessories assembly while the
installation as well as manufacturing. These losses contribute a negligible measure. The other
two types of losses are significant and are explained below.
4.4.1 Optical Losses
These losses are associated with four behavioral parameters called Reflectivity ‘ρ’,
Intercept factor ‘γ’, Transmissivity ‘τ’ of the glass tube, and Absorptance ‘α’.
Reflectivity ‘ρ’: It is the ratio of radiation reaching onto the absorber tube and that of it
incident on collector. These are the losses purely brought up with reflector and absorber parts
of collector. Only a fraction of incident radiation is reflected to receiver tube due to formation
of dirt layer on the surface and the value will be always less than 1. Typical reflectivity values
of clean silvered glass mirrors are around 0.93. After washing the mirrors, their reflectivity
continuously decreases as dirt accumulates until the next washing.
Intercept factor ‘γ’: Full amount of direct solar radiation reflected by collector does not
intercept the absorber tube due to imperfections in reflectors and microscopic shape errors
during assembly. These errors cause reflection of some rays at wrong angles, and results them
in not intercepting the absorber tube. This loss is quantized by the intercept factor and is
typically 0.95 for a properly assembled collector.
Transmissivity ‘τ’: It is related to the absorber glass tube and is the ratio between the
radiation that passes through the glass tube and the total incident radiation on it after
reflection. The metal absorber tube is placed inside an outer glass tube in order to reduce
48
thermal losses to ambient. This outer layer of glass tube prevents the passage of radiation
reflected by the mirrors. The typical value is of 0.93.
Absorptance ‘α’: Is the parameter of the absorber selective coating that quantizes the
amount of energy absorbed by the steel absorber pipe, compared with the total radiation
reaching the outer wall of the steel pipe. This parameter is typically has the value of 0.95.
Product of these four parameters gives the optical efficiency (ηopt) of parabolic trough
collector. And the typical value ranges between 0.7 and 0.76. Thus,
η opt = ρ .γ .τ .α
(4.1)
4.4.2 Thermal Losses
Thermal Losses are due to radiation heat loss from the absorber pipe to ambient, and
convective and conductive heat losses from the absorber pipe to its outer glass tube. For
analysis the heat lost is expressed in terms of an overall loss coefficient UL. Total heat lost
‘ql’ depends upon loss coefficient and temperature difference between receiver and ambient
and is given in Eq.4.2.
ql = U L Ar (Tr − Ta )
(4.2)
Where Ar is the receiver area, Tr and Ta are the receiver and ambient air temperatures
respectively.
The heat loss coefficient UL depends on absorber pipe temperature and is found
experimentally by performing specific thermal loss tests with the PTC operating at several
temperatures within its typical working temperature range. A typical value of UL for absorber
tubes with vacuum in the space between the inner pipe and the outer glass tube is lower than
5 W/m2 K [99].
49
4.5 Thermal Analysis of PTC
The above set of looses occurring in PTC are of unavoidable nature and can only be
reduced to some extent by proper design and maintenance. The remainder of total incident
radiation is utilized in heating the working fluid flowing in receiver tubes. In order to
calculate the net heat collected from the incident radiation, a thermal analysis of the collector
is indeed required. An energy balance equation under steady state conditions yields the useful
heat energy collected. It is the difference between energy collected and that went in losses as
shown in Eq.4.3.
qu = IAcηopt − ql
(4.3)
Where qu stands for useful heat gain, Ac is the collector area in m2 and I is the solar radiation
per unit effective aperture area.
Equation (4.3) can be rewritten as under using the equation (4.2).
qu = IAcηopt − U L Ar (Tr − Ta )
U


qu = Ac  Iη opt − L (Tr − Ta ) 
C


(4.4)
C is the ratio of Ac/Ar and represents the collector concentration ratio. Though the above
expression for useful heat gain looks sensible, there is another factor which affects the net
heat gain. The removal of heat gained from the working fluid depends on its type, its specific
heat, and its heat transfer coefficient, flow rate inside the absorber tube. A factor which takes
into consideration of all the above is defined as heat removal factor and denoted as Fr. Thus
the net useful heat gain becomes:
U


qu = Fr Ac  Iηopt − L (Tr − Ta ) 
C


(4.5)
Collector efficiency is therefore the ratio of useful heat gain to that incident on the collector
aperture.
ηc =
qu
IAc
(4.6)
50
The above set of equations holds true only when the full solar radiation that is
incident on the collector is utilized in heat collection. The collector’s optical concentration
leads to two significant limitations on the practical use of solar radiation. Such limitations
occur due to the intrinsic characteristics of the radiation source. Solar radiation reaches the
earth surface in two different ways called beam and diffuse radiations and are explained in the
next section that follows. The diffuse solar radiation cannot be concentrated on the absorber
(energy spillage), therefore only direct solar radiation from the sun can only be used.
Moreover, in order to collect the full beam radiation continuously, costly mechanical devices
are required to track the sun. The thesis here assumes that PTC tracks the sun fully both for
the day variation and seasonal variations. Thus for the actual expressions of useful heat again
and efficiency, the total solar radiation I can be replaced with beam radiation Ib in equations
(4.5) and (4.6).
4.6 Solar Radiation
From the point of view of utilization of solar energy, it is the amount which falls on
Earth’s surface is of importance that is also called as Insolation or global radiation I. Due to
many atmospheric factors the amount of solar energy falling on Earth is very much different
from that of the extra-terrestrial radiation. It is reflected, absorbed and scattered while it
penetrates through the atmosphere. The radiation is reflected back into the atmosphere partly
by clouds and absorbed by the molecules present in the air. It is also scattered by droplets in
clouds, by atmospheric molecules and by dust particles. Based on these, the solar radiation
received on Earth is categorized into beam and diffuse radiations.
Beam radiation ‘Ib’: It is the one which produces shadow when interrupted by an opaque
object. Hence, it is the radiation that has not been scattered or absorbed and reaches the
ground directly from sun. It is also known as direct radiation.
Diffuse radiation ‘Id’: This is the radiation which changes its direction after being scattered
and reflected by atmospheric molecules. Because scattering takes place in all the directions,
this diffuse radiation comes from all parts of the sky and amounts to only 10-20% of the total
Insolaiton on a clear cloudless day.
51
Thus, the radiation I is partly direct and partly diffused. Total insolation falling on the surface
called as global radiation is the sum of both these direct and diffuse radiations.
4.7 Solar Radiation Geometry
In order to incorporate hard headed nature of solar energy, the work presented in this
thesis uses the actual radiation data for its computations [101]. The radiant energy available
at a location changes throughout the day and year due to varying weather patterns. The more
accurately the solar resources are known, the better will be the UC schedule obtained. The
data is considered from a book on ‘Solar Radiant Energy over India’ released by India
meteorological department, Ministry of earth sciences 2009. The data imparted from the book
is hourly global and diffuse radiations in MJ/m2. The measurement data given pertains to the
global radiation on horizontal surfaces. A solar parabolic trough power plant mainly utilizes
the direct or beam radiation falling on the aperture area for the collection of heat energy. This
necessitates the calculation of direct radiation from the given global and diffuse radiations.
The angle between incident radiation and the normal to the plane surface is called the
angle of incidence denoted by θ. The equivalent beam irradiance that heats up the area is the
radiation that is falling normal to the surface. This is proportional to the cosine of the angle of
incidence. Thus the global irradiance at a place can be written as:
I = I b cos θ + I d
(4.7)
For finding Ib the above equation can be rewritten as:
Ib =
I − Id
cos θ
(4.8)
For finding the direct beam radiation the angle of incidence is the main component
required to be found at the beginning. Angle of incidence varies throughout and depends on
the location and orientation of measuring surface. The angle θ is related by a general equation
to the latitude ϕl, declination δ, hour angle ω, slope β and the surface azimuth angle γ. These
are defined here first in short, and Figures 4.4 and 4.5 below help in giving clear
understanding.
52
Figure 4.4 Latitude, Hour angle and Declination
Figure 4.5 Incidence angle, Surface Azimuth angle and Slope
Latitude φl: The latitude of a location is the angle made by the radial line joining the
location to the center of the Earth with the projection of the line on the equatorial plane. By
convention, the latitude is measured as positive for the northern hemisphere. It can very from
-90° to +90°.
Declination δ: It is the angular distance of the sun’s rays north (or south) of the equator. It is
the angle between a line extending from the centre of the Sun and the centre of the Earth and
the projection of that line upon the Earth’s equatorial plane. It arises as direct consequence of
the fact that Earth rotates with a tilt around the Sun and would vary between +23.45° on June
53
21 to -23.45° on December 21. It is zero on the equinox days of March 21 and September 22.
It is calculated in degrees as shown below where n is the day of the year.
 360
( 284 + n )
 365

δ = 23.45sin 
(4.9)
Hour angle ‘ω’: It is an angular measure of time and is equivalent to 15° per hour. It varies
from +180° to -180° being positive in the morning and negative in the afternoon.
Surface azimuth angle ‘γ’: Is the angle made in the horizontal plane between the horizontal
line due south and the projection of the normal to the surface on the horizontal plane. The
angle is positive if the normal is east of south and negative if west of south.
Slope ‘β’: Is the angle made by the plane surface with the horizontal.
The equation which relates angle of incidence θ [97] is:
cos θ = sin φl ( sin δ cos β + cos δ cos γ cos ω sin β )
+ cos φl ( cos δ cos ω cos β − sin δ cos γ sin β )
(4.10)
+ cos δ sin γ sin ω sin β
The measurement data pertains to a horizontal plane and makes the value of slope β as zero.
By substituting β=0, the expression for θ becomes
cos θ = sin φl sin δ + cos φl cos δ cos ω
(4.11)
By employing the above expression for incidence angle θ, the hourly direct radiation can be
easily calculated by equation (4.8).
54
4.8 Example
Calculate the net heat gained in one hour and collector efficiency of a LS-3 type PTC for the
data given in Table 4.1. if situated at Hyderabad on January 3rd, 11AM. The global radiation
and diffuse radiation are 2.78MJ/m2 and 0.58MJ/m2 respectively on a horizontal surface. The
ambient temperature being 32°C and assume a heat removal factor of 0.93 and the overall
loss coefficient as 5W/m2K.
Sol: It is known that a parabolic trough concentrates only beam radiation onto the absorber
for thermal collection. So, it is required to calculate the beam radiation from the given global
and diffuse radiations. Thus, by (4.8) and (4.11)
Ib =
I g − Id
cos θ
cos θ = sin φl sin δ + cos φl cos δ cos ω
Latitude ϕl of Hyderabad is 17° 27’ this gives an angle of 17.45°.
Declination δ = 23.45sin 
360
( 284 + n ) since, it is on January 3rd, n becomes 3. Therefore,
 365

δ= -22.84°.
Hour angle ω =15°, since it is morning 11AM.
Therefore, on substituting above values in equations (4.8) and (4.11), we get Ib = 2.99 MJ/m2
this is nothing but 830.5 W/m2 instantaneous.
Now, the useful heat gained by the LS-3 type collector is
U


qu = Fr Ac  I bηopt − L (Tr − Ta ) 
C


5


= 0.93*545 ( 2.99*106 ) *0.8 − ( 390 − 32 ) 
26


=1212.35 MJ
ηc =
qu
I b Ac
= .7439
55
4.9 Electricity Generation from STPP
A typical solar thermal power plant uses Rankine cycle in steam generation and
expansion for the production of electricity. The ability of PTC in attaining high temperature
range of up to 400°C makes it possible to integrate itself with the Rankine power cycle. The
designed STPP in the work presented in this thesis also assumes a typical Rankine cycle
efficiency in calculation and does not model any of its behaviour as such. The plant design
capacity and the number of troughs required in generation are decided based upon the past
insolation levels recorded and the meteorological conditions. In the present work, a fixed
plant capacity is chosen and number of troughs required in generating the same amount is
decided based upon the maximum insolation available during a specific period. The plant
output will be at the desired level when the insolation level is at maximum and varies
accordingly. The power conversion methodology in STPP using PTC is explained in brief.
In any kind of thermal power plant there exists a heat source and a heat exchanger
which converts water into steam which is used in running a turbine, so also in STPP. There
are three stages in generation of electricity. Firstly, heat source is the solar field with PTCs,
which raises the temperature of working fluid (some kind of oil) flowing in the absorber tube,
high enough for further process. Once heated in the solar field, the working fluid enters the
second stage of steam generation. A steam generator is an oil–water heat exchanger where
the oil transfers its thermal energy to the water that is used to generate the superheated steam.
Third stage is the power conversion stage where the superheated steam passes through
turbine blades and expands to run a generator that is coupled to the turbine. Thus, the total
power conversion system is similar to that of in a conventional Rankine cycle power plant
except for the heat source.
Fig. 4.6 shows a schematic of STPP employing PTC with essential auxiliaries. In the
figure it is shown that there are few internal levels in a steam generator. They are:
Pre-heater : where water is preheated to a temperature close to evaporation.
Evaporator : where the preheated water is evaporated and converted into saturated steam.
Super-heater: the saturated steam produced in the evaporator is heated in the super-heater to
the temperature required by the steam turbine.
56
Description
Value
Description
Value
Aperture
Area
Length
Absorber tube Diameter
Reflectivity
Absorptivity
5.76m
545m2
99m
0.07m
0.94
0.96
Intercept Factor
Transmissivity
Optical Efficiency
Geometric concentration
Operating temperature
0.93
0.95
0.8
26
390°C
Table 4.1 Data of Luz System (LS-3) PTC
Figure 4.6 Solar Thermal Power Plant with Parabolic Trough Collector
The steam turbine is of two stage, for high-and low-pressure steams. The Steam
leaving the turbine at high-pressure stage goes to a reheater where its temperature rises before
entering the low-pressure turbine stage. After this stage, the steam is condensed using cooling
towers or air cooled condensers and the condensate goes to a water deaerator to remove
dissolved oxygen and gases dissolved in the water.
57
4.10 Example
Calculate the number of LS-3 type collectors required for the data given in previous example
for a plant capacity of 10MW. Assume the Rankine cycle and turbine-generator set
efficiencies as 0.3 and 0.9 respectively.
Sol: Electrical power output possible from one collector is:
PS = qu x Rankine efficiency x Turbine-generator efficiency
= __ x 0.3 x 0.9 = ____ kW
Number of collectors required = 10MW / output of one collector
= 10MW / __
58
Chapter 5
5 PSO and Implementation
5.1 Particle Swarm Optimization
Particle Swarm Optimization (PSO) is a computational intelligence technique
originated from the social behaviour of birds in a flock and fish schools. Birds fly in the sky
in certain patterns and each individual in the group will have knowledge of its neighbours in
following the pattern. While flying each individual optimizes its position accordingly and
always tries to follow the pattern and attain a best position without colliding the other. This is
evident when there is a change in their direction; every individual comes to know about it and
changes the direction suddenly. The other natural behaviour of the birds lies in finding food
and reaching there. Birds fly for food without any knowledge of its availability and place. But
in finding food a bird always uses past experience of its own and others. If a bird finds the
availability of food at a place it is obvious that many others come to know about this and
reach there soon by parting the knowledge. So, it is concluded that birds share information in
finding food and when they travel, affecting each other for their benefits.
This social-psychological behaviour of individuals in a swarm, which imitates the
success of others in attaining better position, gave the idea to use these doings in solving
optimization problems. Kennedy and Eberhart in 1995 proposed PSO for solving nonlinear
multidimensional problems by simulating the unpredictable nature of birds which changes
their direction suddenly with a regrouping in an optimal formulation [102]. PSO is purely a
population based iterative search technique which traverses the whole multidimensional
hyperspace for finding the optimal solution. Boundaries of the search space are dictated by
the constraints of the problem at hand.
In PSO an individual is referred as a particle. It is a feasible candidate solution to the
problem satisfying all the constraints. Many such particles are created in group randomly and
called as swarm or population that represents a bird’s flock. All these particles are ‘flown’
59
into the problem hyperspace in search of optimal solution. Initially particles are allowed to
move in search space with some randomly initiated velocities and latter they move according
to the velocities derived from individual’s current position and that of other particles in the
swarm. The objective function of the problem is solved using each particle and the evaluated
value is called the fitness of particle. Particles traverse the whole search space with the
experience of its own and what others have gone through. Experience here means the best
fitness value it has achieved so far and the corresponding location. The individual best fitness
value is stored as personal best or pbest and the best among all the particles is stored as
global best or gbest. The social interaction between particles causes them to learn from each
other and motivates to move towards better neighbors. Thus, the next movement of a particle
is decided by experience i.e. the new velocity of the particle is influenced by its best fitness
and neighbor’s fitness. Experiences are stored during search procedure as pbest and
gbest/lbest (depending upon the topology chosen) and assist the particles accordingly in
further search. The level of social interaction among particles depends on the topology design
[103]. Two such topologies widely in use are briefly discussed.
Star topology: In this, each particle is allowed to interact with all the remaining particles in
the swarm as shown in Fig.5.1. This allows the particles to get attracted towards the best
particle in the entire group. Particle with best fitness value in the group is called global best
or in short gbest.
Figure 5.1 Star and Ring Topologies
Ring topology: This allows a particle to communicate with its n immediate neighbours. If
n=2, a particle can communicate with its adjacent neighbours as shown in Fig.5.1. Here a
particle attempts to move closer to the best among its neighbours. The best particle among the
60
neighbours is called 4 or lbest. This topology helps a particle in moving towards
neighbourhood best as well as the swarm best.
5.2 PSO Algorithm
It is known that a particle in the swarm flies through hyperspace and alters its position
over the time iteratively, according to its own experience and that of its neighbours. Velocity
is the factor responsible for this and which reflects the social interaction. If xj represents
particle x in iteration j, it is modified for the next iteration or it can be said that it is moved to
a new location as shown, where vj+1 is the velocity term derived for j+1 iteration.
x j +1 = x j + v j +1
(5.1)
The PSO algorithm based on star topology is discussed here for a minimization problem. A
particle x flying in hyperspace has a velocity v. The best success attained by the particle is
stored as pbest and the best among all the particles in the swarm is stored as gbest.
Step1: Initialize the swarm or population Pop randomly of desired size, let K in the
hyperspace.
Pop = { x1 , x2 , x3 ,........xK }
Step 2: Calculate the fitness value of each particle f(xij).
Step 3: Compare the fitness of each particle with its own best attained thus far as illustrated
below

pbesti = f ( xij )
if
f ( xi ) < pbesti : 
xi , pbest = xij

else
: no change in pbest and xi , pbest
j
(5.2)
Step 4: Compare the fitness values of all particles and find gbest as shown

gbest = f ( xij )
if
f ( xi ) < gbest : 
xgbest = xij

else
: no change in gbest and xgbest
j
(5.3)
61
Step 5: Change the velocity of each particle for the next iteration as under, where w is inertia
weight, c1, c2 are constants, rand is random variable which assumes uniformly distributed
values between 0 and 1.
vij +1 = w * vij +c1*rand* ( xi , pbest − xij ) +c 2 *rand* ( xgbest − xij )
(5.4)
Step 6: Move each particle to a new position
xij +1 = xij + vij +1
(5.5)
Step 7: Repeat step 2 to 6 until convergence.
Inertia weight w: Controls the influence of previous velocity on the new velocity. Large
inertia weights cause larger exploration of search space, while smaller inertia weights focus
the search on a smaller region. Typical PSO starts with a maximum inertia weight wmax which
decreases over iterations to a minimum value wmin as shown.
w = wmax −
wmax − wmin
* it
itmax
(5.6)
Where it represents the current iteration count and itmax is the maximum iterations allowed.
Reference [104] gives the best values of wmax and wmin as 0.9 and 0.4 respectively for most of
the problems.
Convergence criteria: Defines the criterion to stop the iterative process. Usually one of the
criteria used is the execution of PSO algorithm for a fixed number of iterations. Alternatively,
a PSO algorithm can be terminated if the velocity changes are close to zero for all the
particles, in which case there will be no further changes in particle positions.
5.3 Binary Particle Swarm Optimization
The previous version of PSO algorithm works on real valued or continuous variables
and is not applicable to all real world problems where some pose discreteness in their nature.
Kennedy and Eberhart introduced a binary version of PSO called the Binary Particle Swarm
Optimization (BPSO) in their later work, where the particles in the search space can only take
the values of either 0 or 1 [105]. In BPSO also the personal best and global best are updated
62
as in continuous version. The major difference between BPSO and continuous version of
PSO is that the velocities of particles are rather defined in terms of probabilities, which
decides whether a bit takes the value of 1 or 0. Thus, the probabilistic nature of velocity must
be restricted within the range [0,1]. At first the velocities are found as in conventional PSO
but a function called sigmoid is introduced to map these real valued numbers of velocity to
the range [0,1].
Sigmoid function
:
Updation
:
sig =
1
j +1
(5.7)
1 + e − vik
 0
xikj +1 = 
 1
if rand (
if rand (
) ≥ sig
) < sig (5.8)
Where xikj represents kth element of ith particle in iteration j and vikj is the corresponding
velocity term.
5.4 Representation of UC as a BPSO Problem
When solving a problem using any kind of technique it is very important to represent
the problem in a form which the technique adopts. This stage of converting the problem into
the corresponding form adds many complications in the implementation procedure.
Sometimes, though the solution method looks easy and computationally inexpensive, the
effort required in expressing the problem in prescribed format may overlay the advantage.
PSO is a social psychological theory which is robust in solving problems featuring
non-linearity through adaptation. It can be easily implemented and has high-quality solution
with stable convergence characteristics. A conventional PSO algorithm cannot be
implemented as such in solving UC owing to dissimilarity in variable representation. UC
problem uses discrete variable representation for unit ON/OFF status where as PSO algorithm
uses real valued variables. The binary variable representation of unit ON/OFF status with 1 or
0 respectively in a UC problem is well treated by the discrete or binary version of PSO
discussed in previous section. A BPSO algorithm applied to UC problem is discussed in [60].
Before applying BPSO for solving a UC problem, some of the definitions used in this
thesis are introduced here.
63
Unit ON/OFF status: Binary ‘1’ represents ON and binary ‘0’ represents OFF status of a
unit.
Hour’s Combination: Represents the ON/OFF status of all units during a particular hour.
Particle: It is a candidate solution of the UC problem representing the entire 24 hour unit
commitment schedule in a day. It is a set of complete 24 individual hour’s combinations.
Population: A group of particles is called population.
Let there are 5 generating units to be scheduled for 24 hours in a day with the desired
BPSO population size of 50. Then an hour’s combination will be a row matrix of size (1X5)
with 0’s and 1’s as its entries representing individual unit status. A particle is a set of all such
24 hour’s combinations thus holds the dimension (24X5). The size of whole population to be
initialized and submitted to BPSO algorithm is (24X5X50).
5.5 Implementation
To apply PSO for solving any kind of problem with sound execution necessitates a
good initial population that is feasible in all respects. In addition it is also required to take
care of the feasibility when population gets updated during the iterative process. The effort of
using BPSO for UC problem demands feasibility by obeying to all of the unit and system
operational constraints. For a quick review the constraints are listed here in short.
ng
Power balance:
∑ Pg
i =1
i
= Pd + Ploss
ng
Minimum capacity committed:
∑ Pg
i =1
Minimum up/down time:
Unit generation limits:
Unit Ramp rate limits:
i ,max
≥ Pd + Presv + Ploss
Ti on ≥ Ti up
Ti off ≥ Ti down
Pgi ,min ≤ Pgi ≤ Pgi ,max
Pgit −1 ≤ Pgit ≤ Pgit −1 + Rupi
Pgit −1 ≥ Pgit ≥ Pgit −1 − Rdn i
64
(5.9)
(5.10)
(5.11)
(5.12)
(5.13)
5.5.1 Creation of Initial Population
Initial population with desired number of particles is created randomly. But all the
constraint restrictions which reduce the freedom and solution search space are considered in
the feasibility checks of a candidate solution. There may be numerous feasible solutions for
the problem which satisfies all limitations but only one will be optimum. The word random in
population creation is restricted only for the selection of few particles for BPSO
implementation out of the many feasible.
A particle is said to live up to every constraint only when it satisfies all of those,
during and in-between every hour of the commitment horizon. This implies that a particle has
to be built hour by hour fulfilling all the limitations. All the constraints are examined one
after the other for every hour’s combination. Before proceeding to know how the initial
population is created and maintain its feasibility during iterative procedure, some crucial
elements which influence the optimum UC solution and system behaviour are spotted and
their treatment in the thesis is discussed.
As highlighted in the previous sections transmission losses are very important system
behavioural attribute and must not be overlooked while computing the UC schedule. A
standard practice followed in existing literature is to treat these transmission losses as some
constant percentage of system loads. These losses gets updated to the actual values if an ED
problem with inclusion of losses or an OPF is solved during UC computation as secondary
optimization problem. Though losses are getting updated to new values by this, but still
remain ill valued and unrealistic in some sense. ED solved with inclusion of losses employs
transmission loss coefficients also called B-coefficients for the calculation of losses. These Bcoefficients will not resemble the actual system dynamic behaviour. This is because of the
reason that B-coefficients once derived are not updated dynamically according to the power
flows in transmission lines and are treated as constant throughout. However, OPF
computation will not result in any such ill feature.
In application of BPSO methodology for solving a UC problem, so far in literature
very few or to say least no attempts have been made for the inclusion of transmission losses.
Even if any such attempt is made, it is when updation process starts and particles fly in the
search space for optimum but not while initializing the population. Such a practice does not
guarantee a viable initial population. This has the danger of leading the algorithm to settle at
65
a point that does not satisfy both system losses and load. Reference [60] applied the BPSO
algorithm for solving UC as the first attempt and neglected transmission losses in the
procedure. In other works even if an effort is made to include losses it is restricted to an
arbitrary percentage value.
If a major aspect like system loss is ignored in the feasibility check of a particle, it
may result in an impractical candidate solution to the problem. The work presented in this
thesis attempts to include losses in a more realistic manner in the initial stages while creating
the population. To ensure that the initial population generated is also feasible according to
system losses, an Economic Dispatch with Losses (EDL) is run assuming ‘all’ generators are
available for dispatch during every hour of the commitment period. Therefore, the load
demand in every particular hour is dispatched among ‘all’ the units in system. This result in
an evenly distributed line flows and thus brings more practical scenario of losses inclusion.
The idea is seemingly better than assuming losses to be an arbitrary percentage of system’s
load. B-coefficients used for losses evaluation in this thesis are also not constant during every
hour. These loss coefficients are dynamically adjusted every hour according to the flows in
transmission lines.
Apart from the inclusion of transmission losses in generating feasible population,
some more major aspects like unit operational limitations should also be taken care. One such
limitation that required to be focused is the allowable instantaneous change in output power
of a unit, also called ramping rate. It is taken care in this thesis by performing an EDL for
every individual hour while initializing the population. Commitment for the coming next
hour is decided based on the previous hour units combination and their power outputs.
Performing EDL is to ensure that the practical limitations are enforced during the
initialization stages itself. However, when updation process starts an OPF is performed for
every individual hour. These ramping rates are considered both in initial population
generation and also when the particles update in search of optimal solution. A detailed
explanation of ramp rate enforcement is given latter.
Next significant necessity in an optimum UC schedule is the reliable operation of
power system. The amount of accessible spinning reserve in the system plays a remarkable
role in reliable operation. The work presented in this thesis adopts a standard practice that is
followed in general for spinning reserve requirements. Usually the level of spinning reserve
66
in a system is decided as a fixed percentage of the total load demand. The work here set a
measure of ten percentage of total system demand as the level of spinning reserve. Following
text explains the handling of different constraints in UC problem formulation.
5.5.1.1 Procedure followed in creating Initial Population
As a first step in population creation, the minimum generation that is needed to be
committed each hour should be determined by considering all the issues highlighted above.
Though the required generation so found will not give the actual generation needed but gives
a fairer value resembling practical picture.
a. Perform EDL for the given demand for each hour assuming all of the units are ON.
Sum up the outputs of all units to find the required generation which now includes
approximate losses.
b. Add 10% of load demand as the amount of spinning reserve level requirement.
NG
Thus, Net required generation = (10% of Pd ) + ∑ Pgi
i =1
(5.14)
With new net required generation obtained, the processes of building up a particle
hour by hour begins. At first a combination which suits the very first hour’s net required
generation is created at random according to the given initial status of units and minimum
up/down timings. This combination is a row array comprising 1’s and 0’s as its entries for all
the units indicating their ON/OFF status. As mentioned earlier, this combination next
undergoes feasibility check of minimum generation possible and unit ramping limits.
Minimum generation possible is to know whether the hour’s combination can meet net load
demand. Because, power output from any unit during a particular hour is dictated by its ramp
rates, the minimum possible generation for the combination has to be found considering unit
ramp rates.
Ramp limits are one of the most practical unit constraints which are imposed to keep
the change in generation between consecutive hours within feasible limits. Ramping limits of
a unit are maintained by actually restricting the power generation limits of a unit for that hour
according to its previous hour output [46], [94]. If a unit has to ramp down, its minimum
generation limit is changed and on the other hand if unit has to ramp up its maximum
67
generation limit is changed. Hence, the maximum and minimum generation limits of all the
units are changed dynamically but bounded within their corresponding original limits as
shown in Eq.(5.15).
{
= max {Pg
}
)}
t
t −1
Pg max,
+ Rupi Pg max,i )
i = min Pg max,i , ( Pg i
t
Pg min,
i
min, i
, ( Pgit −1 + Rdni Pg max,i
(5.15)
The viability test of whether combination meeting net load demand is carried out by summing
the altered maximum power outputs obtained above for the units which are ‘1’ i.e. turned
ON. If the sum is greater than or equal to load demand the combination is treated as feasible.
ng
i.e. if
∑ Pg
i =1
t
max, i
≥ Pd the combination is feasible else not.
The above process is repeated for each and every hour of the UC time horizon to
construct a feasible particle and on whole to initialize the population. If the time horizon of
UC schedule is ‘H’ hours and population size chosen is ‘K’, then the dimensions (H X NG),
(H X NG X K) holds for a particle and whole population.
Initialization process will be incomplete without assigning a velocity to each of the
entry in the population facilitating it to fly in the search space. Therefore, it requires a
velocity vector of same size as that of the population to be initialized, hence the dimension (H
X NG X K). The algorithm here initializes it randomly within the limits -0.5 and 0.5. These
limits restrict the element to fly too fast or too slow. This is necessary to traverse the whole
hyperspace thoroughly and helps in not getting trapped to any local minimum.
5.5.2 BPSO Updation
It is an iterative process that is crucial in the progress of algorithm towards optimum
solution and purely depended upon pbest and gbest values. It is worth mentioning here again
that the work in this thesis uses star topology of PSO. To find pbest and gbest in every
iteration it needs an evaluation function to be solved for all of the particles to find
corresponding fitness values. The evaluation function is the UC objective function discussed
earlier, which finds the total cost of generation to supply the load demand by each particle.
68
Cost of generation is the sum of production costs of each individual hour and the unit startup, shut down costs.
Total Cost = ∑∑ U it F ( Pg i ) + U it (1 − U it −1 ) STCi + U it −1 (1 − U it ) STDi 
(5.16)
F ( Pg i ) = ai Pg i2 + bi Pgi + ci
(5.17)
T
NG
t =1 i =1
Uit is the ith unit ON/OFF status (binary 1 if ON, binary 0 if OFF) during hour t, STCi is start
up cost and STDi is the shut down cost.
Evaluation of total cost calls for finding the hourly cost and start up and shut down
cost. Hourly cost is the sum of costs incurred by all generating units during that hour. It is
found by substituting the units power output in their respective cost functions. An OPF is
performed in the present work to find this hourly cost which is an optimum solution that is
secure and operationally well accepted.
5.5.3 OPF evaluation
OPF as well as EDL in the thesis are solved using a MATLAB function that is
available in optimization toolbox and is called ‘fmincon’. The function optimizes a user
defined non linear objective function and is provided with option of providing equality,
inequality constraints along with variable bounds. The function and OPF problem associated
details are as shown.
Minimization function is the total cost as shown in equation (5.16).
Independent variables and their bounds:
Pg min,i ≤ Pgi ≤ Pg max,i
i = all PV buses
Vmin,i ≤ Vi ≤ Vmax,i
i = all PQ buses
δ min,i ≤ δ i ≤ δ max,i
i = all PQ buses
tapmin,i ≤ tapi ≤ tapmax,i
i = all buses with tap transformers
φmin,i ≤ φi ≤ φmax,i
i = all buses with phase shifting transformers
69
Thus, the variable vector
T
T
T
T
T
X =   xPg  [ xV ] [ xδ ]  xtap   xφ  


X min ≤ X ≤ X max
T
Equality Constraints:
Pgi − Pd ,i − Pi = 0
i = all buses
Qgi − Qd ,i − Qi = 0
i = all PQ buses
Inequality Constraints:
Qg min,i ≤ Qg i ≤ Qmax,i
i = all PV buses
Plmin,i ≤ Pli ≤ Plmax,i
i = all transmission lines
tapmin,i ≤ tapi ≤ tapmax,i
i = all buses with tap transformers
φmin,i ≤ φi ≤ φmax,i
i = all buses with phase shifting transformers
The function returns the minimum cost along with the corresponding units’ Pgi
outputs.
OPF is performed for all of the hours to find costs and units power outputs. After
finding this, the particle fitness value will be the sum of all H hours’ costs and units’ start-up
and sunt down costs. A unit start-up cost is added to the fitness value if a unit is getting
turned ON, and the value depends on whether it hot start or cold start and similarly unit shutdown cost is added if a unit is getting turned OFF.
Fitness value so obtained is used to find pbest and the gbest. If it is a very first
iteration the evaluated fitness values itself becomes pbest for each particle. While finding a
pbest value its corresponding commitment (particle) is also stored. The best among all pbest
values is chosen as gbest and corresponding commitment is also stored. After finding all
particles’ best values and global best the PSO algorithm moves all of its particles from the
current positions to a new position in search space for optimum by deriving velocity terms
based on pbest and gbest values. These velocity terms are real valued and converted to binary
values using a sigmoid function. Binary values of velocity terms are added to each entry in
the particle to update it. Then the whole process repeats till the convergence criteria specified
is met.
70
A particle undergoes all the feasibility checks once it gets updated. If any of the
particle does not live up to any of the viability checks then the updated particle is infeasible
from the system operational view point. In such cases the violated particle has to be
reinitialized and allowed to fly in the hyperspace in order to maintain constant population
size. The work presented in this thesis follows a more logical and better method if any such
case arises. It is an assured fact that the pbest commitment schedule of any particle is always
secure even if it gets updated. This concept is used in thesis for reinitializing the infeasible
particle. The particle which violates a feasibility check is replaced by its own pbest. It is a
simple technique that avoids a whole new particle coming into the swarm through
reinitializing and automatically communicates the current state of the whole swarm to the
particle. The particle gets updated to a new value in the next iteration and follows the swarm
thereafter. This also results in speeding up the algorithm which otherwise would have slowed
down a little.
5.5.4 Details of Algorithm followed for a 24 hour UC schedule
Essential data:
1. System load curve over the time horizon during which UC schedule is desired.
2. Unit’s technical details such as maximum and minimum generation capacities,
minimum up/down timings, initial status whether the unit is ON/OFF along with the
corresponding duration in hours.
3. Unit’s production cost coefficients, start-up and shut-down costs.
4. Full system network data.
Load curve is the plot of forecasted load over 24 hour period. It is chosen to be a
stepped curve implying a constant load throughout an hour. The total hourly load is
distributed on to all buses based on corresponding bus load factors.
Step1: Sum up the loads at all buses to obtain total hourly demand.
Step2: In order to commit the units to cater primarily to system demand and in addition, the
system losses, a rough loss estimate is determined by performing economic dispatch with
71
losses employing B coefficients [2]. This is done by assuming all units to be available for
commitment over the entire UC planning period of 24 hours.
Losses obtained above will not represent the actual loading condition losses. It just gives an
idea regarding the same.
Step 3: The spinning reserve requirements of the system are considered to be 10% of the
system load. This is added to what was determined in step 2 to find the total MW generation
required to be committed.
In regard of the above steps, modified generation requirement is given as below:
Pd ( new ) = Pd + Ploss + Presv
Initial Population:
Step 4: Create a random initial population of desired size, say, K=50, i.e. ‘K’ number of 24
hour unit commitment schedules. Each particle of the population is (24 X NG). Create each
particle hour by hour at a time verifying whether it is a feasible combination satisfying the
hour’s net requirement and constraints like units minimum up/down times, ramp rates. If not,
generate hour’s combination again.
•
Requirement: (load + losses + spinning reserve) being met.
•
Constraint: Units’ minimum up/down times, ramp rate capability of generators.
The ramp rate capability constraint of the generating units is brought into picture as discussed
in earlier sections, i.e. by modifying their maximum and minimum capacities as:
{
= max { Pg
}
)}
t
t −1
Pg max,
+ Rupi Pg max,i )
i = min Pg max, i , ( Pg i
t
Pg min,
i
min, i
, ( Pg it −1 + Rdni Pg max,i
However, for the first hour, there is no such thing as previous hour to find the units
power outputs according to ramp rates. It uses the initial state information like base case load
flow. Thus, for every hour, capability of meeting (load + rough system losses + spinning
reserve) is verified by applying the new maximum and minimum generation capacities
obtained from ramp-rate constraints.
72
For the ease in computing, a judicious amount of ramp rates is chosen to be 40% of
the generator’s maximum capacity. The selected value is in perfect range of actual ramp rates
and is evident from the data given in Appendix B.
Unit Commitment
Step 5: OPF is run as a sub problem in UC determination to find the optimum generation cost
for the hour’s commitment. OPF admits for the new generation limits of committed units
obtained above.
An economic dispatch can be performed to provide a good initial guess for OPF run.
Criterion for Optimal Power Flow is, minimum production cost subjected to the equality and
inequality constraints listed in section 5.5.3.
Step 6: Repeat step 5 for every hour in a given particle and calculate the fitness value which
is the sum of production cost and transitional cost over 24 hours.
Step 7: Repeat steps 5 and 6 for every particle in the population. Each particle’s commitment
with least fitness value over the iterations is remembered as its pbest. The best among the
personal best of all particles is stored as gbest.
Step 8: Update the entire population using equations (5.4), (5.7) and (5.8).
Velocity and inertia weight are important parametric values of BPSO which govern the
particle movement. Instead of allowing the particle to fly in a large hyperspace, the maximum
and minimum values of velocity and weight are fixed at -0.5 and 0.5, 0.8 and 1.0,
respectively.
Subject each new particle to the requirement and constraints of steps 3 and 4. Chuck the
infeasible population and replace it with the particle’s pbest.
Step 9: Steps 5-8 are repeated till the convergence criterion is met.
Convergence criterion: The convergence criterion usually followed is a limit on maximum
iterations. A more logical criterion adopted is here. The iterations continue till no further
improvement in the gbest is evident for a fixed number of iterations. This will be a good
trade-off between speed and optimal requirements.
73
5.5.5 Simulation Results on IEEE Test Systems
The proposed methodology is implemented on standard IEEE 14 and 30 bus test systems.
The results of simulation studies are depicted in case 1 and case 2. The generator and line
data for the systems are furnished in Appendix B. The bounds on the voltage solution for all
load buses have been fixed to {0.9, 1.1} as it is seen as a practically reasonable operating
range.
5.5.5.1 Case 1: UC for standard IEEE 14 bus test system
Table 5.1 shows the 24 hour UC schedule for standard IEEE 14 bus test data given in
Appendix B. Results given in the table are self explanatory with hourly load demand, unit
status, and power output from each committed unit. Total cost of UC schedule along with
hourly production costs and total transitional cost are listed. In order to indicate the
effectiveness of proposed UC algorithm, the maximum and minimum load bus voltages
attained during every hour in the system are shown in the Table 5.2 that follows. The voltages
at the load buses in the system during 24 hour time period attained as high as 1.0751 PU and
as low as 1.0017 PU.
Hour
Unit Status
Load
Power Output (MW)
Cost($)
(MW)
1
2
3
6
8
1
2
3
6
8
X 10^3
1
181.30
1
0
1
1
1
68.91
0
53.38
27.82
32.49
0.9973
2
170.94
1
1
1
1
0
69.43
20.00
53.85
29.35
0
0.9359
3
150.22
1
1
0
1
0
79.65
31.40
0
41.63
0
0.8423
4
103.60
1
0
1
0
1
54.53
0
32.00
0
18.00
0.6244
5
129.50
1
1
1
0
1
51.20
20.00
37.07
0
22.30
0.7717
6
155.40
1
0
1
0
0
88.53
0
69.07
0
0
0.8129
7
181.30
1
0
1
0
0
104.1
0
80.00
0
0
0.9551
8
202.02
1
0
1
1
0
106.9
0
80.00
18.00
0
1.0852
9
212.38
1
0
1
1
1
88.94
0
71.48
36.00
18.00
1.1603
10
227.92
1
1
1
0
1
96.33
20.00
78.18
0
36.00
1.2390
11
230.51
1
1
1
1
1
79.70
31.23
64.13
18.000
39.53
1.2654
12
217.56
1
1
1
0
0
97.80
43.73
79.54
0
0
1.1585
13
207.20
1
1
1
0
1
86.53
35.87
69.41
0
18.00
1.1390
14
196.84
1
1
0
0
1
115.4
50.00
0
0
36.00
1.1219
15
227.92
1
1
0
1
1
120.1
50.00
0
18.00
45.00
1.3106
74
16
233.10
1
1
1
1
0
119.8
50.00
32.00
36.00
0
1.2938
17
220.15
1
1
0
1
1
121.1
50.00
0
45.00
18.00
1.2667
18
230.51
1
1
1
1
1
85.05
35.19
32.00
45.00
36.00
1.2788
19
243.46
1
0
1
1
1
91.62
0
64.00
45.00
45.00
1.3040
20
253.82
1
1
1
1
0
112.4
20.00
80.00
45.00
0
1.3677
21
259.00
1
1
1
1
1
88.53
37.47
72.83
45.00
18.00
1.4173
22
233.10
1
1
1
1
0
85.70
35.45
69.70
45.00
0
1.2410
23
225.33
1
1
1
1
0
82.69
33.35
66.86
45.00
0
1.2008
24
212.38
1
0
1
0
1
117.6
0
80.00
0
18.00
1.1683
Transitional Cost
2.7851
Total Cost
29.743
Table 5.1 UC for IEEE 14 Bus Test System
Hour
Vmax
Vmin
Hour
Vmax
Vmin
1
1.0714
1.0283
13
1.0543
1.0176
2
1.0655
1.0257
14
1.0619
1.0054
3
1.0648
1.0199
15
1.0709
1.0017
4
1.0588
1.0280
16
1.0587
1.0249
5
1.0576
1.0279
17
1.0696
1.0032
6
1.0372
1.0165
18
1.0717
1.0302
7
1.0360
1.0135
19
1.0751
1.0328
8
1.0621
1.0214
20
1.0579
1.0234
9
1.0704
1.0235
21
1.0697
1.0259
10
1.0577
1.0232
22
1.0602
1.0254
11
1.0708
1.0238
23
1.0598
1.0259
12
1.0314
1.0087
24
1.0514
1.0089
Table 5.2 Hourly Min. and Max. Load Bus Voltages for IEEE 14 Bus Test System
75
5.5.5.2. Case 2: UC for standard IEEE 30 bus test system
Table 5.3 shows the 24 hour UC schedule for standard IEEE 30 bus test data given in
Appendix B. The maximum and minimum load bus voltages attained during every hour in the
system are shown in the Table 5.4 that follows. The voltages at the load buses in the system
during 24 hour time period attained as high as 1.0579 PU and as low as 0.9705 PU.
Unit Status
Hour
Power Output(MW)
Cost($)
Load
1
2
5
8
11
13
1
2
5
8
11
13
X 10^3
1
166.0
1
1
0
1
1
1
106.1
31.72
0
10.00
10.00
12.00
0.4330
2
196.0
1
0
1
1
0
0
167.5
0
20.0
14.6
0
0
0.5350
3
229.0
1
0
1
1
0
1
181.0
0
21.41
22.05
0
12.00
0.6503
4
267.0
1
1
1
1
1
1
177.1
32.00
21.40
21.11
11.96
12.00
0.7474
5
283.4
1
1
0
1
1
0
194.7
53.36
0
31.72
15.63
0
0.8395
6
272.0
1
1
1
1
0
1
177.8
49.10
20.00
22.60
0
12.00
0.7645
7
246.0
1
1
1
0
0
1
174.0
48.13
21.11
0
0
12.00
0.6749
8
213.0
1
1
1
0
0
1
147.1
41.55
18.90
0
0
12.00
0.5593
9
192.0
1
1
1
0
0
1
130.2
37.44
17.54
0
0
12.00
0.4905
10
161.0
1
0
0
0
1
0
156.7
0
0
0
10.00
0
0.4380
11
147.0
1
0
1
1
1
1
102.4
0
15.26
10.00
10.00
12.00
0.3794
12
160.0
1
0
0
0
1
0
155.6
0
0
0
10.00
0
0.4345
13
170.0
1
0
0
0
0
0
177.6
0
0
0
0
0
0.4737
14
185.0
1
1
1
1
1
1
108.8
32.00
15.95
10.00
10.00
12.00
0.4734
15
208.0
1
1
0
1
1
1
142.0
40.38
0
10.00
10.00
12.00
0.5645
16
232.0
1
1
1
1
1
0
155.8
43.66
19.62
10.00
10.00
0
0.6223
17
246.0
1
0
1
1
1
0
193.3
0
22.44
24.04
14.67
0
0.7132
18
241.0
1
0
1
1
0
1
188.1
0
22.03
26.02
0
12.93
0.6946
19
236.0
1
1
1
1
0
1
164.7
32.00
20.28
14.50
0
12.00
0.6395
20
225.0
1
1
0
1
1
1
156.7
43.95
0
10.10
10.00
12.00
0.6221
21
204.0
1
0
1
0
1
1
168.4
0
20.00
0
10.10
12.00
0.5606
22
182.0
1
0
1
0
0
0
168.2
0
20.23
0
0
0
0.4885
23
161.0
1
0
0
1
1
0
146.0
0
0
10.00
10.00
0
0.4379
24
131.0
1
0
1
1
1
0
98.40
0
15.00
10.00
10.00
0
0.3279
Transitional Cost
2.9865
Total Cost
16.551
Table 5.3 UC for IEEE 30 Bus Test System
76
Hour
Vmax
Vmin
Hour
Vmax
Vmin
1
1.0573
1.0031
13
1.0204
0.9746
2
1.0324
1.0056
14
1.0552
1.0101
3
1.0467
1.0049
15
1.0553
0.9932
4
1.0499
0.9987
16
1.0486
1.0038
5
1.0388
0.9705
17
1.0452
0.9970
6
1.0433
0.9930
18
1.0455
1.0012
7
1.0401
0.9862
19
1.0472
1.0044
8
1.0460
0.9994
20
1.0531
0.9889
9
1.0496
1.0060
21
1.0542
1.0063
10
1.0551
0.9969
22
1.0296
0.9918
11
1.0579
1.0112
23
1.0542
0.9977
12
1.0555
0.9975
24
1.0504
1.0111
Table 5.4 Hourly Min. and Max. Load Bus Voltages for IEEE 30 Bus Test System
5.6. UC with STPP integration
In finding the UC schedule of conventional thermal units in the system, output of
STPP should be taken into account. STPP has to be added as an extra plant into the existing
system. For the sake of simulation and study the work here models one of the conventional
thermal units in the test system as a solar thermal unit and UC schedule is found for the
remaining. This is done so also to ensure that the main layout of standard system is not
disturbed with the addition of a new STPP. The methodology proposed would work equally
well with STPP added as an extra unit into the system or any existing conventional generator
being assigned as the solar unit irrespective of size. However, the main concern with the
option of assigning a conventional generator as STPP would remain the ability of remaining
generators to supply the load demand throughout the time horizon. This is owing to the
intermittent energy output from the STPP over the considered time frame. During the absence
or low output periods of solar generation, load demand has to be catered by the other units
reliably without violating any operational constraints. Thus, the choice of generating unit that
is to be substituted by the modelled STPP is required to be judicious. Without any loss of
generality, the lowest capacity unit in the test system is chosen for the same.
77
Further, the capacity of STPP modelled is same as that of the unit getting replaced and
treated as a generator bus in the OPF process. However, it is not considered to be voltage
controlled for the evident reason that the capacity of the STPP varies with every hour and
thus the reactive power required to support the constant voltage may not be sufficient. The
OPF formulation is given below with explanation of various constraints imposed and state
variable involved.
The active power injection P must be matched at every bus while the reactive power
injection Q is matched for all load buses and bus where STPP is situated because of the
unavailability of information regarding Qg. However, the reactive generation Qg is computed
for the generator buses (other than STPP bus) and checked for limit violations. The power
flow Pl on each line is also constrained to be within the specified limits.
Equality Constraints:
Bus active and reactive power injection matching
Pgi − Pd ,i − Pinj ,i = 0
Qgi − Qd ,i − Qinj ,i = 0
i = 1 to N
i = 1 to Nb
Inequality Constraints:
Reactive power generation limits and line flow limits
Qmin,i ≤ Qgi ≤ Qmax,i
i = 1 to ng
Plmin,i ≤ Pli ≤ Plmax,i
i = 1 to Nl
State variable vector and its bounds:
The active power generation is a variable for all generator buses except the STPP. The
voltage magnitudes V are variable for all load buses and the STPP, the load angle δ is a
variable for all buses except slack. Tap and ϕ are state variables for those lines having
transformers with tap and phase shifters. Each state variable is constrained to be within its
upper and lower bounds. The augmented state vector is given below.
T
T
T
T
T
X =   xPg  [ xV ] [ xδ ]  xtap   xφ  


X min ≤ X ≤ X max
T
78
Solar generation is possible only during day time when the incident radiation on
surface is greater than some useful value. In obtaining a 24 hour UC schedule by replacing
one existing conventional generator with solar thermal unit, there are three time intervals as
listed below. It has been observed that the output from STPP is in usable quantities from
09.00 to 17.00 hours. During early hours and evenings, the STPP output is too low to run the
unit and is taken as zero.
1. During 00.00 hrs to 09.00 hrs, only the conventional thermal units have to cater the
load.
2. 09.00 hrs to 17.00 hrs conventional units and solar thermal unit caters the load.
3. 17.00 hrs to 24.00 hrs again only conventional units caters the load.
5.6.1 Algorithm for 24 hr UC with STPP Integration
Apart from the data requirements mentioned for normal UC formulation, here hourly
solar radiation, temperatures are the extra information needed in finding the STPP output. To
utilize data from conditions close to what have existed, the proposed UC methodology has
considered the values of solar radiation and temperatures from the book by the metrological
department of Indian ministry for renewable energies [101].
Step 1: Depending upon the solar insolation level, the hourly expected output PS from STPP
is calculated. During early hours and evenings, the STPP output is too low to run the unit and
is taken as zero.
Step2: Sum up the loads at all buses to obtain the total hourly demand.
Step 3: In order to commit the units to cater primarily to system demand and in addition the
system losses, a rough loss estimate is determined by performing EDL employing B
coefficients [2]. This is done by assuming all units to be available for commitment over the
entire UC planning period of 24 hours.
Step 4: The spinning reserve requirements of the system are considered to be 10% of the
system load. This is added to what was determined in step 3 to find the total MW generation
required to be committed.
79
Step 5: Since one of the generators is treated as STPP, the calculated hourly fixed power
output of it is subtracted from the total MW generation requirement. Commitment schedule is
determined every hour among only the remaining (NG-1) generators to cater to this new net
generation requirement. In this regard the modified or new net generation requirement is
given as below
Pd ( new ) = Pd + Ploss + Presv − PS
Initial Population
Step 6: An initial population of desired size, K, with each particle of dimension T X (NG-1)
is generated. Each of the T X (NG-1) elements of the particle needs a velocity to update the
status of a unit, thus requiring a randomly initiated velocity matrix of the same size.
The initial population generated is checked for minimum generation requirements, unit
minimum up/down times and ramp rate limits as discussed in algorithm 5.5.4.
Unit Commitment
Step 7: OPF is run as a sub problem to determine the optimum generation cost for the hour’s
commitment. OPF admits the new generation limits according to ramp rates.
Step 8: Repeat step 7 for every hour in a given particle and calculate the fitness value. It is
the sum production cost and transition cost over 24 hours.
Step 9: Repeat steps 7 and 8 for every particle in the population and find pbest, gbest.
Step 10: Update the entire population using equations (5.4), (5.7) and (5.8)
Velocity and inertia weight are fixed at -0.5 and 0.5, 0.8 and 1.0, respectively.
Each new particle generated is checked for minimum generation requirements, unit minimum
up/down times and ramp rate limits. Chuck the infeasible population and replace it with the
particle’s pbest.
Step 11: Steps 7-10 are repeated till the convergence criterion is met. i.e., the iterations
continue till no further improvement in the gbest is evident over a given number of iterations.
80
5.6.2 Simulation Results on IEEE Test Systems
The proposed methodology is implemented on standard IEEE 14 and 30 bus test systems.
Results of simulation studies are depicted in case 3 and case 4. The line and generator data
for the systems are available in Appendix B. The voltage bounds are fixed at the same level
as given for case 1 and case 2 UC problems without STPP.
5.6.2.1 Case 3: UC for standard IEEE 14 bus test system with STPP integrated
As mentioned earlier, the lowest capacity unit in the system is modelled as STPP in
the UC formulation. In standard IEEE 14 bus system the unit at bus number 8 is of lowest
capacity of 45MW and is replaced as STPP of same capacity. Table 5.5 shows the UC
schedule for 24 hours along with hourly costs. Unit at bus number 8 modelled as STPP is ON
during the hours 09.00 to 17.00. The output during corresponding hours is also shown. The
maximum and minimum load bus voltages attained during the 24 hour time frame are shown
in Table 5.6. It can be observed that the maximum voltage is 1.0746 PU and minimum is
0.9902 PU which is bit lower than compared to what was obtained in simulation study of case
1 owing to the obvious reason of shortage in generation capacity when STPP is not in
operation.
Unit Status
Hour
(MW)
Power Output (MW)
Cost($)
Load
1
2
3
6
8
STPP
1
2
3
6
8
X10^3
STPP
1
181.30
1
1
0
1
0
96.64
43.37
0
45.00
0
1.0079
2
170.94
1
0
0
1
0
130.2
0
0
45.00
0
0.9654
3
150.22
1
1
0
1
0
87.83
20.00
0
45.00
0
0.8465
4
103.60
1
0
1
1
0
50.00
0
27.50
27.00
0
0.6090
5
129.50
1
0
1
0
0
74.41
0
56.72
0
0
0.6827
6
155.40
1
1
1
1
0
67.40
20.00
51.63
18.00
0
0.8653
7
181.30
1
1
1
0
0
83.70
33.86
66.31
0
0
0.9635
8
202.02
1
0
1
0
0
125.8
0
80.00
0
0
1.0828
9
212.38
1
1
1
0
1
96.45
20.00
77.90
0
20.78
1.0249
10
227.92
1
1
0
1
1
149.9
40.01
0
18.01
26.68
1.1905
11
230.51
1
1
1
1
1
92.65
4049
32.00
36.01
32.44
1.0877
12
217.56
1
0
0
1
1
144.6
0
0
45.11
35.90
1.0606
13
207.20
1
1
1
1
1
84.61
20.00
32.00
34.83
38.01
0.9411
81
14
196.84
1
1
0
0
1
122.5
40.00
0
0
38.92
0.8964
15
227.92
1
1
1
0
1
110.3
50.00
32.00
0
39.48
1.0403
16
233.10
1
0
1
1
1
119.3
0
64.00
18.00
34.91
1.0733
17
220.15
1
1
1
0
1
108.4
20.00
80.00
0
15.00
1.1028
18
230.51
1
1
1
1
0
96.78
40.00
79.05
18.00
0
1.2451
19
243.46
1
1
1
1
0
93.33
40.75
76.58
36.00
0
1.2999
20
253.82
1
0
1
1
0
133.0
0
80.00
45.00
0
1.3703
21
259.00
1
1
1
1
0
117.9
20.00
80.00
0
0
1.4000
22
233.10
1
1
1
0
0
117.5
40.00
80.00
0
0
1.2529
23
225.33
1
0
1
0
0
150.0
0
80.00
0
0
1.2428
24
212.38
1
1
1
0
0
116.1
20.00
80.00
0
0
1.1477
Transitional Cost
2.3748
Total Cost
27.742
Table 5.5 UC for IEEE 14 Bus Test System with STPP integrated
Hour
Vmax
Vmin
Hour
Vmax
Vmin
1
1.0639
1.0110
13
1.0746
1.0307
2
1.0626
1.0127
14
1.0614
1.0069
3
1.0646
1.0187
15
1.0481
1.0134
4
1.0677
1.0302
16
1.0725
1.0251
5
1.0443
1.0025
17
1.0419
1.0101
6
1.0640
1.0268
18
1.0583
1.0226
7
1.0353
1.0156
19
1.0571
1.0231
8
1.0326
0.9994
20
1.0551
1.0201
9
1.0413
1.0122
21
1.0554
1.0226
10
1.0569
1.0021
22
1.0232
0.9918
11
1.0740
1.0292
23
1.0300
0.9923
12
1.0553
0.9902
24
1.0278
1.0012
Table 5.6 Hourly Min. and Max. Load Bus Voltages for IEEE 14 Bus Test System with
STPP integrated
82
5.6.2.2. Case 4: UC for standard IEEE 30 bus test system with STPP integrated
The lowest capacity unit at bus number 11 in IEEE 30 bus system is modelled as
STPP in the UC formulation. The capacity of the lowest unit is 30MW and is replaced as
STPP of same capacity. Table 5.7 shows the UC schedule for 24 hours along with hourly
costs. Unit at bus number 11 modelled as STPP is ON during the hours 09.00 to 17.00. The
output during corresponding hours is also shown. The maximum and minimum load bus
voltages attained during the 24 hour time frame are shown in Table 5.8. It can be observed
that the maximum voltage is 1.0695 PU and minimum is 0.9528 PU which is bit lower than
compared to what was obtained in case study 2 due to the obvious reason of shortage in
generation capacity when STPP is not in operation.
Unit Status
Hour
Power Output(MW)
Cost($)
Load
1
2
5
8
11
STPP
13
1
2
5
8
11
STPP
13
x10^3
1
166.0
1
1
0
1
0
1
114.5
33.73
0
10.00
0
12.00
0.4302
2
196.0
1
0
1
1
0
0
167.5
0
20.00
14.69
0
0
0.5350
3
229.0
1
0
1
1
0
0
188.6
0
22.00
26.42
0
0
0.6546
4
267.0
1
0
1
1
0
0
200.0
0
41.11
35.00
0
0
0.8207
5
283.4
1
1
1
1
0
0
200.0
32.00
27.07
35.00
0
0
0.8207
6
272.0
1
1
0
1
0
0
196.9
53.88
0
33.41
0
0
0.8022
7
246.0
1
1
1
0
0
1
175.0
48.37
20.00
0
0
12.00
0.6750
8
213.0
1
1
0
0
0
1
163.8
45.67
0
0
0
12.00
0.5843
9
192.0
1
0
1
1
1
0
154.3
0
19.21
10.00
13.85
0
0.4736
10
161.0
1
0
1
1
1
0
120.6
0
16.56
10.00
17.79
0
0.3612
11
147.0
1
0
1
0
1
0
112.3
0
15.94
0
21.63
0
0.3039
12
160.0
1
0
1
0
1
0
122.8
0
16.75
0
23.93
0
0.3365
13
170.0
1
0
0
0
1
1
137.3
0
0
0
25.34
12.00
0.3850
14
185.0
1
0
1
1
1
0
135.5
0
17.80
10.00
25.95
0
0.4110
15
208.0
1
0
0
1
1
0
172.4
0
0
16.51
26.32
0
0.5124
16
232.0
1
1
0
0
1
0
186.6
32.00
0
0
23.27
0
0.5777
17
246.0
1
1
0
0
1
0
194.5
53.26
0
0
10.00
0
0.6740
18
241.0
1
1
0
0
0
0
199.5
54.52
0
0
0
0
0.6957
19
236.0
1
1
1
0
0
0
176.8
48.83
20.00
0
0
0
0.6432
20
225.0
1
0
1
0
0
1
196.5
0
22.63
0
0
14.84
0.6425
83
21
204.0
1
0
0
1
0
1
185.7
0
0
14.00
0
12.47
0.5893
22
182.0
1
1
0
0
0
0
157.1
32.00
0
0
0
0
0.4806
23
161.0
1
1
1
0
0
1
105.4
31.48
15.60
0
0
12.00
0.3954
24
131.0
1
0
1
0
0
1
106.2
0
15.45
0
0
12.00
0.3248
Transitional Cost
2.4455
Total Cost
15.575
Table 5.7 UC for IEEE 30 Bus Test System with STPP integrated
Hour
Vmax
Vmin
Hour
Vmax
Vmin
1
1.0554
1.0028
13
1.0695
1.0040
2
1.0324
1.0056
14
1.0451
1.0091
3
1.0297
0.9955
15
1.0564
0.9829
4
1.0272
0.9817
16
1.0480
0.9804
5
1.0246
0.9783
17
1.0413
0.9727
6
1.0227
0.9705
18
1.0003
0.9528
7
1.0400
0.9862
19
1.0137
0.9727
8
1.0430
0.9847
20
1.0405
0.9883
9
1.0462
1.0081
21
1.0472
0.9823
10
1.0447
1.0101
22
1.0265
0.9915
11
1.0629
1.0161
23
1.0555
1.0133
12
1.0648
1.0151
24
1.0644
1.0168
Table 5.8 Hourly Min. and Max. Load Bus Voltages for IEEE 30 Bus Test System with
STPP integrated
5.7 New Approach to Voltage Security Constrained UC
The solutions obtained so far have been obtained with voltage profiles governed by
hard limits imposed. Thus, the search space for the optimization algorithm is bound within
this range strictly. The solution obtained this way would work out on most of the occasions.
However, the operating range of voltage depends on the system. A usually stressed or
overloaded system is prone to have a few buses having their voltage dipping below the
generally considered lower bound. Similarly, small systems might tend to have voltage
hovering above the normally considered higher limit of 1.05 per unit that is usually assumed
to have a safe operation of system. A standard example is the IEEE 14 bus system which has
84
a few buses with voltage hovering above the 1.07 mark, which is way over 1.05. The same
simulation carried out in case 1 in section 5.5.5.1 was conducted with the voltage bounds
adjusted to {0.95, 1.05}. The problem failed to converge. This was due to the fact that the
voltages of a few buses on this system are of the order of 1.07 even at base case. A simple
base case load flow was conducted to assess the facts. It is already shown that the optimal
schedule in case 1 was obtained with the range fixed to {0.9, 1.1}. Further, the same voltage
profile considered safe for operation in a 14 bus system is not always conducive for a larger
or stressed system. The hard limits may have to be somewhat relaxed. In addition, it is not
always possible to know as to whether this relaxation is going to be unfairly high. The result
would then be a lesser secure solution even though more optimal because of the relaxation in
the hard limits.
The work presented in this thesis proposes a smarter way to handle the voltage
security concern by ensuring that the voltage limits are wide enough and at the same time,
ensuring that the voltage security is monitored. It employs the well-known concept of global
L-index for voltage security check on the solution obtained. This way, the need to be
concerned about hard limits on voltages is done away with. The simulation studies that
follow prove that this technique results in very acceptable voltage profiles finally. Also, the
L-index, which is a measure of the proximity to the operating limit and ensures a safe margin.
5.7.1 Voltage Security Margin: P-V Curve Fundamentals
Steady state voltage stability analysis evolved from simple power flow solutions in
the form of P-V curves. The curves use information no more than the quantities computed
from a simple power flow. Both active power P and reactive power Q are quadratic equations
in voltage magnitude V. The P-V curve imparts an idea about loadability of the system as
shown next.
85
PV Curve
The P-V curve plots P on the abscissa against V on the ordinate. Fig. 5.2 is a simple
illustration of the curve.
Figure 5.2 A Simple P-V Curve
The figure clearly marks out the stable and unstable operating regions. The stable
region ends where V drops beyond a level. It is also evident that there are two values of V for
every value of P. The higher value corresponds to the stable operation while the lower
corresponds to the opposite. The point where the curve enters the unstable region is called the
nose point or the knee of the curve. The power flow is known to diverge at and around this
critical operating point. The P-V curve asserts the fact that the system cannot be loaded
beyond a limit. When the active power demand in the system exceeds the knee or nose point,
the curve turns backward indicating progressive drop in voltage level. The lower region is the
unstable one which caters to only numerically possible values but not practically feasible
ones. The system cannot operate in this zone.
In the context of unit commitment, the final schedule obtained cannot resemble a
heavily loaded or stressed system. A cheaper UC solution may be possible within the
86
physical generating limits Pgmax and Pgmin. However, it might not be safe for the system and
the existing network to operate in this region. That is why, voltage monitoring is necessary. It
is only logical to not let the system reach a state as near to the nose point. Thus a margin can
always be specified with respect to the nose point and tracked. Fig.5.2 marks a region called
margin. This margin is used as a stability measure in the study. The linear distance in the
curve from the current loading point to the maximum loading point or nose point gives the
loading margin. The proposed UC algorithm attempts to maintain a user specified margin
dictated by the value specified for L-index.
5.7.2 L-Index Approach
The L-index was developed as a very convincing measure of the proximity of the
operating point to the point of instability [4].
Theory and Practical Sense
The power transmission system comprises generators, transmission lines and loads
where transmission lines are used to carry the power from the former to the latter. In the
process, power is also lost, known as losses, and the entire power is not transferred to the load
centres. Higher the fraction of losses, lower is the power transferred effectively. L-index
exploits this concept to obtain a measure of the proximity to the stability limit. This practical
concept can be found embedded in the mathematics formulated for the index.
The L-index is computed for every load bus in the system. If i denote load bus in the
system and j denotes generator bus, (5.18) gives the local L-index at the load bus i:
∑F
Li = 1 −
lj
Vj
j
(5.18)
Vi
Where the term Fij is given by (5.19):
 V l   Z ll
 g  =  gl
I  K
l
F lg   I 
 
Y gg   V g 
(5.19)
The basic network modelled by the admittance matrix Y looks like (5.20). Rearranging (5.20)
would give rise to (5.19).
87
 I l   Y ll
 g  =  gl
 I  Y
l
Y lg   V 
 
Y gg   V g 
(5.20)
From (5.19), (5.21) can be derived:
V l  =  Z ll   I l  +  F lg  V g 
(5.21)
Equation (5.21) is similar in structure to the voltage equation (5.22) in a single machine
system shown in Fig. 5.3.
V2 = − IZ + V1
(5.22)
I
Z
G
V1
V2
Load
Figure 5.3 Single Machine Load System
The fact that (5.21) and (5.22) are similar holds the key in the formulation of L-index. As the
loss component increases, the difference between the source and load voltage starts
increasing. It can be understood from (5.22) that if source voltage V1 equals load voltage V2,
there is practically no loss in the system. In other words, it is ideal. So, the multi-machine
equation for a load bus i looks something like (5.23) for the ideal case.
Vi = ∑ F i jV j
(5.23)
j
When this is reflected in (5.18), L-index becomes 0. So, zero is the ideal case of the system or
the most stable case. Whereas, the upper limit for the L-index or the measure of instability is
given by the value of L-index approaching to 1. This happens when
2Vi = ∑ F i jV j
(5.24)
j
So, when the load bus voltage drops to lesser than or equal to half the contribution from the
generators, the L-index becomes 1 or higher respectively.
88
Global L-index
The global L-index is used to determine which load bus in the system is the most susceptible
to voltage instability. The bus having the maximum L-index is the bus closest to the stability
limit.
5.7.3 Proposed UC Algorithm with L-index
Voltage magnitudes do not give any measure of loadability or rather security of
system as it varies with different systems. Instead of pondering over what range of voltage
limits would assure secure system, an indicator that continuously keeps watch would be a
practical solution. The work presented in this thesis also finds UC solutions by introducing
security of the system in a true sense using the concept of L-index. Since the critical nose
point voltage is unknown for a given system, voltages will not serve the purpose of judging
system security. Whereas an L-index measure, as explained just now, would always gives the
picture about proximity to the point of system collapse. For the system operator to ensure
adequate margin, the voltage security should be an integral part as early as in the UC stage.
Thus, the safety margin constraint with respect to bus voltages is appended to the
existing set of constraints in optimal power flow which is called by the UC procedure. The
actual value of L-index (Li) is calculated as pointed in section 5.7.2 for every load bus during
ever hour of the particle, and is restricted to the constraining value (L) while computing UC
schedule. The hard voltage limits on all the load buses in the system are relaxed while
enforcing the L-index constraint. The search space is now wider increasing the chances of the
optimality attained while the L-index constraint strictly monitors that the relaxation does not
lead to voltage security problems. The constraining value of L-index is fixed to 0.6 and can
be varied based upon the desired safety of margin in system operation. A lower L-index
guarantees higher margin of safety. The inequality constraint specified below is added to OPF
formulation which was detailed in section 5.5.3.
Li ≤ L
i = all load buses
However, the higher limit on the bus voltages might be a cause for concern because a lower
L-index (which is desired) might also mathematically arise from very high voltages.
However, this mathematical solution is practically infeasible if voltages rise to unreasonable
89
values like those hovering above 1.1 per unit. So, the higher limit is understandably fixed to
around 1.1 per unit.
The process of determining UC schedule remains the same as implemented previously on the
standard systems. The L-index based UC solution is found through simulations and are
shown in study cases 5-8. The UC schedule determined for standard IEEE 14 and 30 bus
systems employs the same generator and line data as specified in Appendix B.
5.7.3.1 Case 5: UC for IEEE 14 Bus Test System with L-index
Table 5.9 shows the 24 hour UC schedule for standard IEEE 14 bus test system with
L-index. In order to indicate the effectiveness of proposed UC algorithm with inclusion of Lindex and by relaxing the hard limits on voltages, the maximum and minimum load bus
voltages attained during every hour in the system are shown in the Table 5.10. Voltages are
relaxed between 0.6 PU to 1.1 PU. As mentioned earlier the upper limit is fixed at 1.1 PU to
ensure a practically feasible UC solution. It can be seen that the voltages at load buses in the
system during 24 hour time period attained as high as 1.0759 PU and as low as 0.9974 PU.
Though the load bus voltages are allowed to go to as low as 0.6 PU, the desired measure of
L-index restricts them to quite higher values as evident from Table 5.10. The load bus voltage
levels obtained show that they are very well in the range of feasible system operation.
Hour
Unit Status
Load
Power Output (MW)
Cost($)
(MW)
1
2
3
6
8
1
2
3
6
8
X10^3
1
181.30
0
1
1
1
1
0
33.47
64.54
43.90
39.99
0.9618
2
170.94
1
0
1
1
1
60.00
0
52.70
27.19
32.14
0.9506
3
150.22
1
1
1
1
0
62.84
20.00
47.64
21.20
0
0.8415
4
103.60
1
0
0
1
1
64.49
0
0
22.23
18.00
0.6392
5
129.50
1
0
1
1
1
58.06
0
32.00
14.45
25.99
0.7710
6
155.40
1
0
1
1
0
71.19
0
54.79
30.92
0
0.8321
7
181.30
1
0
1
0
0
104.1
0
80.00
0
0
0.9551
8
202.02
1
0
1
0
0
125.7
0
80.00
0
0
1.0823
9
212.38
1
0
1
1
0
117.6
0
80.00
18.00
0
1.1479
10
227.92
1
0
1
1
0
115.1
0
80.00
36.00
0
1.2155
11
230.51
1
1
1
0
1
116.1
20.00
80.00
0
18.00
1.2824
12
217.56
1
0
1
0
1
104.1
0
80.00
0
36.00
1.1681
13
207.20
1
0
1
1
1
83.40
0
66.35
18.00
41.16
1.1271
90
14
196.84
1
0
1
0
0
120.4
0
80.00
0
0
1.0496
15
227.92
1
1
1
0
1
113.4
20.00
80.88
0
18.00
1.2667
16
233.10
1
1
1
0
1
89.37
37.90
72.54
0
36.00
1.2584
17
220.15
1
0
1
1
1
88.59
0
71.11
18.00
44.29
1.1944
18
230.51
1
1
1
1
1
76.95
20.00
61.76
36.00
37.70
1.2616
19
243.46
1
0
1
1
1
86.76
0
70.44
45.00
43.23
1.3026
20
253.82
1
1
1
1
1
82.58
20.00
67.30
45.00
41.11
1.3782
21
259.00
1
0
1
1
1
94.07
0
77.19
45.00
45.00
1.3857
22
233.10
1
1
1
0
1
94.16
20.00
76.45
0
45.00
1.2638
23
225.33
1
0
1
0
1
102.8
0
80.00
0
45.00
1.2067
24
212.38
1
1
1
1
0
97.98
20.00
79.35
18.00
0
1.1559
Transitional Cost
2.7198
Total Cost
29.418
Table 5.9 UC for IEEE 14 Bus Test System with L-index
Vmax
Vmin
Hour
Vmax
Vmin
1
1.0779
1.0358
13
1.0704
1.0275
2
1.0710
1.0286
14
1.0325
0.9974
3
1.0651
1.0269
15
1.0529
1.0114
4
1.0737
1.0323
16
1.0535
1.0123
5
1.0722
1.0307
17
1.0700
1.0273
6
1.0631
1.0270
18
1.0709
1.0291
7
1.0360
1.0135
19
1.0704
1.0277
8
1.0353
1.0144
20
1.0705
1.0293
9
1.0659
1.0209
21
1.0697
1.0275
10
1.0626
1.0210
22
1.0541
1.0151
11
1.0659
1.0244
23
1.0530
1.0127
12
1.0546
1.0204
24
1.0582
1.0239
Hour
Table 5.10 Hourly Min. and Max. Load Bus Voltages for IEEE 14 Bus Test System with
L-index
91
5.7.3.2 Case 6: UC for IEEE 30 Bus Test System with L-index
Table 5.11 depicts the 24 hour UC schedule for standard IEEE 30 bus test system with
L-index. The maximum and minimum load bus voltages attained during every hour are
shown in the Table 5.12. The voltages at load buses in the system during 24 hour time period
attained as high as 1.0657 PU and as low as 0.9636 PU. The desired L-index measure of 0.6
restricted the lower voltage to 0.9636 PU.
Unit Status
Hour
Power Output(MW)
Cost
Load
1
2
5
8
11
13
1
2
5
8
11
13
($)x10^3
1
166.0
1
1
0
0
0
1
123.0
35.74
0
0
0
12.00
0.4274
2
196.0
1
1
1
0
0
0
143.1
40.55
18.51
0
0
0
0.5029
3
229.0
1
1
0
0
0
0
188.7
51.82
0
0
0
0
0.6487
4
267.0
1
1
0
1
0
1
196.2
53.70
0
14.00
0
15.10
0.7794
5
283.4
1
1
1
1
0
0
193.5
53.00
20.00
28.00
0
0
0.8121
6
272.0
1
1
1
1
1
1
170.9
47.41
20.98
18.46
11.00
12.00
0.7609
7
246.0
1
1
0
1
1
0
174.9
48.42
0
20.42
11.76
0
0.6990
8
213.0
1
1
0
1
1
1
146.4
41.43
0
10.00
10.00
12.00
0.5812
9
192.0
1
1
0
0
0
1
145.5
41.18
0
0
0
12.00
0.5117
10
161.0
1
0
1
0
1
1
125.7
0
16.97
0
10.00
12.00
0.4178
11
147.0
1
0
0
1
1
1
118.5
0
0
10.00
10.00
12.00
0.3953
12
160.0
1
0
0
1
1
0
144.9
0
0
10.00
10.00
0
0.4346
13
170.0
1
1
1
0
1
1
104.6
31.31
15.57
0
10.00
12.00
0.4250
14
185.0
1
1
0
0
0
1
139.4
39.71
0
0
0
12.00
0.4884
15
208.0
1
0
1
1
1
0
170.2
0
20.00
14.00
10.25
0
0.5748
16
232.0
1
0
1
0
1
1
190.3
0
22.16
0
14.40
13.50
0.6627
17
246.0
1
1
1
0
1
0
187.1
32.00
22.00
0
14.34
0
0.6800
18
241.0
1
1
1
1
1
1
153.5
43.12
19.50
10.00
10.00
12.00
0.6521
19
236.0
1
0
1
1
1
1
178.5
0
21.23
20.21
11.70
12.00
0.6721
20
225.0
1
0
1
0
0
1
196.5
0
22.63
0
0
14.84
0.6425
21
204.0
1
1
1
0
0
0
159.4
32.00
19.74
0
0
0
0.5321
22
182.0
1
1
1
0
0
1
122.1
35.51
16.90
0
0
12.00
0.4590
23
161.0
1
0
1
0
0
1
135.5
0
17.70
0
0
12.00
0.4168
24
131.0
1
0
1
0
1
1
96.30
0
15.00
0
10.00
12.00
0.3285
Transitional Cost
2.9930
Total Cost
16.498
Table 5.11 UC for IEEE 30 Bus Test System with L-index
92
The load bus voltage levels obtained are very well in the range of feasible system operation.
Hour
Vmax
Vmin
Hour
Vmax
Vmin
1
1.0538
1.0036
13
1.0634
1.0147
2
1.0259
0.9943
14
1.0494
0.9962
3
1.0049
0.9636
15
1.0505
1.0072
4
1.0426
0.9748
16
1.0486
0.9982
5
1.0243
0.9780
17
1.0395
0.9863
6
1.0492
0.9970
18
1.0519
1.0066
7
1.0449
0.9813
19
1.0514
1.0069
8
1.0545
0.9920
20
1.0405
0.9883
9
1.0478
0.9934
21
1.0234
0.9912
10
1.0640
1.0149
22
1.0515
1.0091
11
1.0570
1.0035
23
1.0542
1.0117
12
1.0541
0.9980
24
1.0657
1.0177
Table 5.12 Hourly Min. and Max. Load Bus Voltages for IEEE 30 Bus Test System with
L-index
5.7.3.3 Case 7: UC for STPP integrated IEEE 14 Bus Test System with L-index
The lowest capacity unit in the system at bus number 8 of 45 MW is modelled as
STPP of same capacity in the UC formulation with L-index. Table 5.13 shows the UC
schedule for 24 hours. Unit at bus number 8 modelled as STPP is ON during the hours 09.00
to 17.00. The output during corresponding hours is also shown. Voltages are relaxed between
0.6 PU to 1.1 PU. The upper limit is fixed at 1.1 PU to ensure a practically feasible UC
solution. The maximum and minimum load bus voltages attained during the 24 hour time
frame are shown in Table 5.14. It can be seen that the voltages at load buses in the system
attained as high as 1.0748 PU and as low as 0.9918 PU. Though the load bus voltages are
allowed to go to as low as 0.6 PU, the desired measure of L-index of 0.6 restricts them to
quite higher values.
93
Unit Status
Hour
(MW)
Power Output (MW)
Cost($)
Load
1
2
3
6
8
STPP
1
2
3
6
8
X10^3
STPP
1
181.30
1
1
0
1
0
96.64
43.37
0
45.00
0
1.0079
2
170.94
1
0
0
1
0
130.2
0
0
45.00
0
0.9654
3
150.22
1
1
0
1
0
87.83
20.00
0
45.00
0
0.8456
4
103.60
1
0
1
1
0
50.00
0
27.50
27.00
0
0.6090
5
129.50
1
0
1
0
0
74.41
0
56.72
0
0
0.6827
6
155.40
1
1
1
1
0
67.39
20.00
51.63
18.00
0
0.8653
7
181.30
1
1
1
0
0
83.69
33.86
66.31
0
0
0.9635
8
202.02
1
0
1
0
0
125.8
0
80.00
0
0
1.0828
9
212.38
1
1
1
0
1
96.45
20.00
77.90
0
20.78
1.0249
10
227.92
1
1
0
1
1
149.5
40.00
0
18.00
26.68
1.1905
11
230.51
1
1
1
1
1
92.65
40.49
32.00
36.00
32.44
1.0877
12
217.56
1
0
0
1
1
135.1
0
0
45.02
35.90
0.9970
13
207.20
1
1
1
1
1
78.92
20.00
32.00
40.36
38.01
0.9402
14
196.84
1
1
0
0
1
122.58
40.00
0
0
38.92
0.8964
15
227.92
1
1
1
0
1
110.3
50.00
32.00
0
39.48
1.0403
16
233.10
1
0
1
1
1
119.2
0
64.00
18.00
34.91
1.0733
17
220.15
1
1
1
0
1
108.4
20.00
80.00
0
15.00
1.1028
18
230.51
1
1
1
1
0
96.78
40.00
79.05
18.00
0
1.2451
19
243.46
1
1
1
1
0
93.33
40.75
76.58
36.00
0
1.2999
20
253.82
1
0
1
1
0
133.0
0
80.00
45.00
0
1.3703
21
259.00
1
1
1
1
0
117.9
20.00
80.00
45.00
0
1.4000
22
233.10
1
1
1
0
0
175.1
40.00
80.00
0
0
1.2529
23
225.33
1
0
1
0
0
149.6
0
80.00
0
0
1.2401
24
212.38
1
1
1
0
0
161.1
20.00
80.00
0
0
1.1477
Transitional Cost
2.4190
Total Cost
27.750
Table 5.13 UC for STPP integrated IEEE 14 Bus Test System with L-index
Hour
Vmax
Vmin
Hour
Vmax
Vmin
1
1.0639
1.0110
13
1.0748
1.0311
2
1.0626
1.0127
14
1.0614
1.0069
3
1.0646
1.0187
15
1.0481
1.0134
4
1.0677
1.0302
16
1.0725
1.0251
94
5
1.0443
1.0225
17
1.0419
1.0101
6
1.0640
1.0268
18
1.0583
1.0226
7
1.0353
1.0156
19
1.0571
1.0231
8
1.0326
0.9994
20
1.0551
1.0201
9
1.0413
1.0122
21
1.0554
1.0226
10
1.0569
1.0021
22
1.0232
0.9918
11
1.0740
1.0292
23
1.0312
0.9921
12
1.0677
1.0075
24
1.0278
1.0012
Table 5.14 Hourly Min. and Max. Load Bus Voltages for STPP integrated IEEE 14 Bus
Test System with L-index
5.7.3.4 Case 8: UC for STPP integrated IEEE 30 Bus Test System with L-index
The lowest capacity unit in the system at bus number 11 of 30 MW is modelled as
STPP of same capacity in the UC formulation with L-index. Table 5.15 shows the UC
schedule for 24 hours. Voltages are relaxed between 0.6 PU to 1.1 PU. The upper limit is
fixed at 1.1 PU to ensure a practically feasible UC solution. The maximum and minimum
load bus voltages attained during the 24 hour time frame with desired measure of L-index of
0.6 are shown in Table 5.16. It can be seen that the voltages at load buses in the system
attained as high as 1.0689 PU and as low as 0.9668 PU. The load bus voltage levels obtained
shows that they are very well in the range of feasible system operation.
Unit Status
Hour
Power Output (MW)
Load
(MW)
Cost($)
1
2
5
8
11
STPP
13
1
2
5
8
11
STPP
13
x 10^3
1
166.0
1
1
1
1
0
0
110.9
32.80
16.04
10.00
0
0
0.4097
2
196.0
1
0
1
1
0
0
167.4
0
20.20
14.61
0
0
0.5350
3
229.0
1
0
0
1
0
1
195.5
0
0
28.61
0
14.26
0.6820
4
267.0
1
1
1
1
0
0
194.1
32.00
20.00
30.93
0
0
0.7571
5
283.4
1
1
0
1
0
1
195.1
53.46
0
32.20
0
14.81
0.8398
6
272.0
1
0
1
1
0
1
200.0
0
20.00
35.00
0
26.66
0.8167
7
246.0
1
0
0
1
0
1
200.0
0
0
35.00
0
21.17
0.7486
8
213.0
1
0
1
1
0
1
166.3
0
20.00
21.00
0
12.00
0.5930
9
192.0
1
1
1
0
1
1
121.7
32.00
16.90
0
13.85
12.00
0.4474
10
161.0
1
1
0
0
1
1
103.6
31.12
0
0
17.79
12.00
0.3586
11
147.0
1
1
1
0
1
1
75.92
24.52
15.00
0
21.63
12.00
0.2955
95
12
160.0
1
1
0
1
1
1
89.35
27.75
0
10.00
23.93
12.00
0.3436
13
170.0
1
1
0
0
1
0
115.0
33.83
0
0
25.34
0
0.3590
14
185.0
1
1
0
0
1
0
127.4
36.81
0
0
25.95
0
0.4039
15
208.0
1
1
1
0
1
1
121.9
35.48
17.00
0
26.32
12.00
0.4584
16
232.0
1
1
1
1
1
0
145.2
41.09
18.82
10.00
23.27
0
0.5453
17
246.0
1
1
0
0
1
1
183.3
50.50
0
0
10.00
12.73
0.6681
18
241.0
1
1
1
0
0
1
170.6
47.29
20.00
0
0
12.00
0.6569
19
236.0
1
1
0
1
0
1
171.5
47.59
0
14.00
0
12.00
0.6631
20
225.0
1
1
0
0
0
0
185.1
50.91
0
0
0
0
0.6333
21
204.0
1
1
1
0
0
0
149.6
42.13
19.04
0
0
0
0.5298
22
182.0
1
0
0
0
0
0
191.0
0
0
0
0
0
0.5190
23
161.0
1
0
1
1
0
0
137.4
0
17.81
10.00
0
0
0.4167
24
131.0
1
1
0
1
0
1
84.91
26.67
0
10.00
0
12.00
0.3289
13.0094
Transitional Cost
2.3396
Total Cost
15.349
Table 5.15 UC for STPP integrated IEEE 30 Bus Test System with L-index
Hour
Vmax
Vmin
Hour
Vmax
Vmin
1
1.0418
1.0094
13
1.0677
1.0056
2
1.0324
1.0096
14
1.0633
0.9990
3
1.0442
0.9730
15
1.0650
1.0122
4
1.0282
0.9849
16
1.0502
1.0051
5
1.0403
0.9701
17
1.0521
0.9785
6
1.0428
0.9917
18
1.0409
0.9889
7
1.0425
0.9668
19
1.0460
0.9830
8
1.0484
1.0067
20
1.0067
0.9672
9
1.0653
1.0129
21
1.0233
0.9912
10
1.0689
1.0093
22
1.0146
0.9833
11
1.0647
1.0177
23
1.0413
1.0088
12
1.0582
1.0048
24
1.0601
1.0107
Table 5.16 Hourly Min. and Max. Load Bus Voltages for STPP integrated IEEE 30 Bus
Test System with L-index
96
Chapter 6
6. Summary and Conclusions
This chapter provides summary of the work presented in this thesis along with
discussions and conclusions on results obtained.
Unit Commitment is a complex combinatorial optimization problem in power
systems. A prior knowledge about the units to be committed among the available ones to
cater to the forecasted load demand not only reduces the cost of generation but also helps
system operators in its smooth functioning. Thus strategies have been developed to attend the
problem. But the problem of UC becomes more tedious with integration of renewable energy
sources into the power grid. With the increased attention towards harnessing renewable
sources for power production this happens to be an interesting task ahead of power engineers
today. Algorithms are required to be developed that solve the problem with significant
importance towards system security. The work presented in this thesis developed a unique
approach in solving UC problem under a significant energy feed from solar powered thermal
plant.
The thesis explains in detail the formulation of unit commitment problem and various
constraints involved in finding an optimal schedule. An overall optimal schedule is found
only by the optimal hourly commitments of units and dispatching the load among them
economically. In this regard the usually followed sub-problem of economic dispatch is
explained. A more composite sub-problem that takes into account some additional system
constraints called Optimal Power Flow is also explained. Clearly the difference between
classical ED and OPF is made. The problem of UC is treated as a main problem and a
subroutine of OPF is solved every hour for optimally dispatching the load among committed
units. The UC schedule obtained is secure with executions of full-fledged Optimal Power
Flow with additional constraints for every hour unlike conventional Economic Dispatch.
In order to incorporate the solar energy, a renewable energy source available in
abundance, a solar thermal power plant model is developed. The model uses parabolic trough
type technology which is a more mature and practically proven technology among the
concentrated solar power technologies for bulk power generation. The model considers the
97
significant factors that influence solar thermal generation. The true data of solar insolation
levels give the model a more virtual measure. The solar thermal power plant so modelled is
integrated into the grid. For the purpose of experimentation a conventional thermal unit in the
system is treated as solar powered unit in the thesis. This is done with the intension of not
disturbing the layout of standard systems. The installed capacity of modelled solar plant is the
same as that of the conventional thermal unit that is replaced to keep up the standard system.
The concept of Particle Swarm Optimization is described and its application towards
solving the UC problem is depicted. PSO is an effective stochastic optimization technique
with least number of parameters affecting its particles’ evolution and move towards optimal
solution. PSO is a population based search technique which consists of many individuals
called particles and are flown into problem hyperspace. The particles fly in search of optimal
solution moving to new positions based upon their past experience and that of its neighbours.
The UC schedule is obtained by creation of an initial population in which each particle
represents a candidate solution and needs to be optimized. That is, a particle represents a UC
schedule for the desired time frame. Owing to the binary representation of unit’s ON/OFF
status as binary 1 and binary 0, the discrete version of PSO called the Binary Particle Swarm
Optimization is employed in finding the UC schedule.
In the creation of initial PSO population for determining UC often transmission losses
are ignored which may yield unfaithful solutions. A rough idea about losses, which may
occur in respect to every hour’s load demand, has been made in the thesis by assuming all of
the units are ON. The assumption is seemingly better than taking losses as an arbitrary
percentage of system load demand.
The work presented in the thesis proposed an idea of incorporating system voltage
security in the process of finding UC schedule, which has not been addressed so far in any of
the previous works in the literature. Load demand on the system is increasing at a very fast
pace which on the other hand has not seen same raise in generation and transmission
capacities. This sometimes results in overloaded system that as a consequence endangering
the normal operation of the system. This is due to the fact that an overloaded system results in
quite low voltages and also has the danger of leading the system to a voltage collapse due to
cascade tripping.
98
A detailed explanation of voltage stability is given for a clear understanding of the
concept. The proposed work approaches the problem of including voltage security in
determining UC schedule by using what is known as global L-index. It gives the exact
estimate of the load bus in the system that is the most susceptible to voltage instability. The
bus having the maximum L-index is the bus closest to the stability limit. It is a widely used
measure to compute the distance of actual system current state from the stability limit. The
system operational safety is added to the constraints of OPF evaluation of UC procedure in
the form of L-index by relaxing hard limits on voltages of load buses. The usefulness of the
proposed approach while finding UC with L-index is clearly explained.
Finally, simulation results are illustrated for 8 cases for the standard test systems of
IEEE 14 and 30 buses. Results depict the effectiveness of proposed methodologies in
determining UC schedules for secure operation of solar energy integrated smart power grid.
With the current focus on integration of solar energy based power generation into the
grid, the work proposed in this thesis provides an effective solution for unit commitment for
the systems having solar power plants among their generation mix and will help Power
Utilities in their quest for adopting new solutions/tools while including the Solar energy in
their power generation portfolio.
99
Appendix A: Transmission Loss Coefficients
Total loss in the system is nothing but the difference between sum of all power
generations and system load. In other words it is just a summation of all the generations and
loads where loads are treated as negative generations. Thus, the total loss in the system is the
sum of the bus power i.e.
N
Pl + jQl = ∑ Si
i =1
(A.1)
N
= ∑V I
i =1
*
i i
Where, N represents total number of buses in a system. In general voltage can be found using
simple ohms law v = zi. Thus in a system the relation V = ZI holds good where V and I are
column vectors consisting complex values of individual bus voltages and currents
respectively. Whereas Z is symmetric matrix consisting of impedances between each
individual bus and other buses in the system and called the impedance matrix. Thus, voltage
at a particular bus it is written as
N
Vi = ∑ Z i j I j
(A.2)
j =1
Where Zij is the impedance between buses i and j (and is equal to Zji). And can be expanded
in terms of resistance R and reactance X as
Z i j = Ri j + jX i j
(A.3)
Thus,
N
N
Pl + jQl = ∑∑ Z i j I j I i*
i =1 j =1
N
N
(
Pl + jQl = ∑∑ Z i j I j cos θ j + j I j sin θ j
i =1 j =1
)( I
i
cos θ i − j I i sin θ i )
100
(A.4)
Where,
θ i = δ i − φi
δi is the voltage angle at ith bus
φi = tan −1
Qi
Pi
Equation (A.4) can be expanded as
Pl + jQl = ∑∑ Z i j I i I j ( cos θi cos θ j + sin θi sin θ j )
N
N
i =1 j =1
+ j ∑∑ Z i j I i I j ( cos θi sin θ j − sin θi cos θ j )
N
N
(A.5)
i =1 j =1
Since, Zij = Zji and
N
∑I
i =1
i
N
 N
N

cos θi  ∑ Z i j I j sin θ j  = ∑ I j cos θ j  ∑ Z ji I i sin θi 
 i =1

 j =1
 j =1
We get
Pl + jQl = ∑∑ Z i j I i I j ( cos θi cos θ j + sin θi sin θ j )
N
N
i =1 j =1
(A.6)
= ∑∑ ( Ri j + X i j ) I i I j cos (θi − θ j )
N
N
i =1 j =1
Therefore on separating real and imaginary parts,
Pl = ∑∑ Ri j I i I j cos (θi − θ j )
(A.7)
Ql = ∑∑ i j I i I j cos (θi − θ j )
(A.8)
N
N
i =1 j =1
N
N
i =1 j =1
It is known that current at bus can be given as
Ii =
Pi
Vi cos φi
(A.9)
101
or
Qi
Vi sin φi
Ii =
(A.10)
On substituting (A.9) in (A.7),
N
N
Pl = ∑∑ Ri j
i =1 j =1
Pi Pj
cos (θ i − θ j )
Vi V j
cos φi cos φ j
(A.11)
Equation A.11 can be rewritten as,
N
N
Pl = ∑∑ Pi Bi j Pj
(A.12)
i =1 j =1
Where,
Bi j =
Ri j
cos (θ i − θ j )
Vi V j
cos φi cos φ j
, called the B coefficient or active loss coefficient between buses i
and j.
Similarly on substituting (A.10) in (A.8),
N
N
Ql = ∑∑ X i j
i =1 j =1
Qi Q j
cos (θ i − θ j )
Vi V j
sin φi sin φ j
(A.13)
Equation (A.13) can be rewritten as,
N
N
Ql = ∑∑ Qi Ci j Q j
(A.14)
i =1 j =1
Ci j =
Xi j
cos (θ i − θ j )
Vi V j
sin φi sin φ j
, called the C coefficient or reactive loss coefficient between buses i
and j.
102
Appendix B: Standard IEEE 14 and 30 Bus Test
Systems Data
B.1 IEEE 14 Bus Test System Data
B.1.1 Unit Data
Min.
OFF
(hr)
Initial
Status
(hr)
+ve=On
-ve=Off
Cost Coefficients
Cold
Start
Hour
Shut
down
cost
Hot
Cold
b
c
Min.
ON
(hr)
0.00375
2.0
0
1
1
2
2
50
70
176
0.01750
1.75
0
1
1
3
2
60
74
187
-60
0.06250
1.0
0
1
1
2
1
30
110
113
30
-30
0.00834
3.25
0
1
1
3
1
85
50
267
30
-30
0.02500
3
0
1
1
2
1
52
72
180
Bus
No.
Pmax
(MW)
Pmin
(MW)
Qmax
(MVar)
Qmin
(MVar)
1
1
150
50
100
-100
2
2
50
20
50
-50
3
3
80
12
100
4
6
45
10
5
8
45
10
Unit
Start up
Costs
a
Table B.1 IEEE 14 Bus Unit Data
B.1.2 Line Data
Base MVA = 100
X(pu)
Full Line
Charging
Admittance
(pu)
Transformer
Tap Setting
Transformer
Phase Shifting
Angle
Flow
Limit
(MW)
0.0194
0.0592
0.0528
1
0
50
5
0.0540
0.2230
0.0492
1
0
65
2
3
0.0470
0.1980
0.0438
1
0
60
4
2
4
0.0581
0.1763
0.0374
1
0
60
5
2
5
0.0570
0.1739
0.0340
1
0
60
6
3
4
0.0670
0.1710
0.0346
1
0
60
7
4
5
0.0134
0.0421
0.0128
1
0
40
8
4
7
0.0000
0.2091
0.0000
0.978
0
65
9
4
9
0.0000
0.5562
0.0000
0.969
0
40
Line
No.
From
Bus
To
Bus
R (pu)
1
1
2
2
1
3
103
10
5
6
0.0000
0.2520
0.0000
0.932
0
65
11
6
11
0.0950
0.1989
0.0000
1
0
50
12
6
12
0.1229
0.2558
0.0000
1
0
50
13
6
13
0.0662
0.1303
0.0000
1
0
50
14
7
8
0.0000
0.1762
0.0000
1
0
50
15
7
9
0.0000
0.1100
0.0000
1
0
30
16
9
10
0.0318
0.0845
0.0000
1
0
50
17
9
14
0.1271
0.2704
0.0000
1
0
50
18
10
11
0.0821
0.1921
0.0000
1
0
50
19
12
13
0.2209
0.1999
0.0000
1
0
50
20
13
14
0.1709
0.3480
0.0000
1
0
50
Table B.2 IEEE 14 Bus Line Data
B.1.3 24 Hour Load Demand
Hour
MW
MVar
Hour
MW
MVar
1
181.30
51.45
13
207.20
58.80
2
170.94
48.51
14
196.84
55.86
3
150.22
42.63
15
227.92
64.68
4
103.60
29.40
16
233.10
66.15
5
129.50
36.75
17
220.15
62.48
6
155.40
44.10
18
230.51
65.42
7
181.30
51.45
19
243.46
69.09
8
202.03
57.33
20
253.82
72.03
9
212.38
60.27
21
259.00
73.50
10
227.92
64.68
22
233.10
66.15
11
230.51
65.42
23
225.33
63.95
12
217.56
61.74
24
212.38
60.27
Table B.3 IEEE 14 Bus 24 Hour Load Demand
104
B.1.4 Bus Load Factors
Load
At Bus
MW Load Factor
MVAR Load Factor
1
2
3
4
5
6
7
8
9
10
11
2
3
4
5
6
9
10
11
12
13
14
0.0838
0.3676
0.1846
0.0293
0.0432
0.1139
0.0347
0.0135
0.0236
0.0521
0.0575
0.1728
0.2585
-0.0531
0.0218
0.1020
0.2259
0.0789
0.0245
0.0218
0.0789
0.0680
Table B.4 Bus Load Factors
B.2 IEEE 30 Bus Test System Data
B.2.1 Unit Data
Cost Coefficients
Qmax
(MVar)
Qmin
(MVar)
Min.
ON
(hr)
Min.
OFF
(hr)
Start up
Costs
Cold
Start
Hour
Shut
down
cost
Bus
No.
Pmax
(MW)
1
1
200
50
200
-20
0.00375
2.0
0
1
1
2
2
50
70
176
2
2
80
20
100
-20
0.01750
1.75
0
2
2
3
2
60
74
187
3
5
50
15
80
-15
0.06250
1.0
0
1
1
2
1
30
110
113
4
8
35
10
60
-15
0.00834
3.25
0
1
2
3
1
85
50
267
5
11
30
10
50
-10
0.02500
3
0
2
1
2
1
52
72
180
6
13
40
12
60
-15
0.02500
3
0
1
1
2
1
30
40
113
Unit
Pmin
(MW)
Initial
Status
(hr)
+ve=On
-ve=Off
a
b
c
Table B.5 IEEE 30 Bus Unit Data
105
Hot
Cold
B.2.2 Line Data
Base MVA = 100
Line
No.
From
Bus
To
Bus
R (pu)
X(pu)
Full Line
Charging
Admittance (pu)
Transformer
Tap Setting
Transformer
Phase Shifting
Angle
Flow
Limit
(MW)
1
1
2
0.0192
0.0575
0.0528
1
0
30
2
1
3
0.0452
0.1852
0.0408
1
0
30
3
2
4
0.0570
0.1737
0.0368
1
0
30
4
3
4
0.0132
0.0379
0.0084
1
0
30
5
2
5
0.0472
0.1983
0.0418
1
0
30
6
2
6
0.0581
0.1763
0.0374
1
0
30
7
4
6
0.0119
0.0414
0.009
1
0
30
8
5
7
0.0460
0.1160
0.0204
1
0
30
9
6
7
0.0267
0.0820
0.017
1
0
30
10
6
8
0.0120
0.0420
0.009
1
0
30
11
6
9
0.0000
0.2080
0
0.978
0
30
12
6
10
0.0000
0.5560
0
0.969
0
30
13
9
11
0.0000
0.2080
0
1
0
30
14
9
10
0.0000
0.1100
0
1
0
30
15
4
12
0.0000
0.2560
0
0.932
0
65
16
12
13
0.0000
0.1400
0
1
0
65
17
12
14
0.1231
0.2559
0
1
0
32
18
12
15
0.0662
0.1304
0
1
0
32
19
12
16
0.0945
0.1987
0
1
0
32
20
14
15
0.2210
0.1997
0
1
0
16
21
16
17
0.0824
0.1923
0
1
0
16
22
15
18
0.1073
0.2185
0
1
0
16
23
18
19
0.0639
0.1292
0
1
0
16
24
19
20
0.0340
0.0680
0
1
0
32
106
25
10
20
0.0936
0.2090
0
1
0
32
26
10
17
0.0324
0.0845
0
1
0
32
27
10
21
0.0348
0.0749
0
1
0
30
28
10
22
0.0727
0.1499
0
1
0
30
29
21
22
0.0116
0.0236
0
1
0
30
30
15
23
0.1000
0.2020
0
1
0
16
31
22
24
0.1150
0.1790
0
1
0
30
32
23
24
0.1320
0.2700
0
1
0
16
33
24
25
0.1885
0.3292
0
1
0
30
34
25
26
0.2544
0.3800
0
1
0
30
35
25
27
0.1093
0.2087
0
1
0
30
36
28
27
0.0000
0.3960
0
0.968
0
30
37
27
29
0.2198
0.4153
0
1
0
30
38
27
30
0.3202
0.6027
0
1
0
30
39
29
30
0.2399
0.4533
0
1
0
30
40
8
28
0.0636
0.2000
0.0428
1
0
30
41
6
28
0.0169
0.0599
0.013
1
0
30
Table B.6 IEEE 30 Bus Line Data
B.2.3 24 Hour Load Demand
Hour
MW
MVAR
Hour
MW
MVAR
1
166.0
73.92
13
170.0
75.70
2
196.0
87.28
14
185.0
82.38
3
229.0
101.9
15
208.0
92.92
4
267.0
118.8
16
232.0
103.3
5
283.4
126.2
17
246.0
109.5
6
272.0
121.1
18
241.0
107.3
107
7
246.0
109.5
19
236.0
105.0
8
213.0
94.85
20
225.0
100.1
9
192.0
85.49
21
204.0
90.84
10
161.0
71.69
22
182.0
81.04
11
147.0
65.46
23
161.0
71.69
12
160.0
71.25
24
131.0
58.33
Table B.7 IEEE 30 Bus 24 Hour Load Demand
B.2.4 Bus Load Factors
Load
At Bus
MW Load Factor
MVAR Load Factor
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
19
20
21
2
3
4
5
7
8
10
12
14
15
16
17
18
19
20
21
23
26
29
30
0.0765
0.0084
0.0268
0.3323
0.0804
0.1058
0.0204
0.0395
0.0218
0.0289
0.0123
0.0317
0.0112
0.0335
0.0077
0.0617
0.0112
0.0123
0.0084
0.0374
0.1006
0.0095
0.0126
0.1505
0.0863
0.2377
0.0158
0.0594
0.0126
0.019
0.0142
0.0459
0.0071
0.0269
0.0055
0.0887
0.0126
0.0182
0.0071
0.0150
Table B.8 Bus Load Factors
108
Appendix C: Characteristics of Thermal Generating
Units
C.1 Input – Output Characteristics
Fundamental to the problem of generator economic operation of a thermal unit is its
corresponding input-output characteristics. A thermal generating unit is run by a single steam
driven turbine which is at high pressure produced by a boiler. The generator power output is
fed not only to the grid but also to its necessary auxiliary parts like cooling fans, feed water
pumps and so on. In defining the unit characteristics, we will talk about gross input versus net
output. This gross input to the unit can be measured in terms of dollars per hour or tons of
coal per hour or any other units. Thus, the characteristics of steam turbine generator are
defined according to the required heat input or the total cost. ‘H’ represents unit’ total heat
input in BTU per hour ( MBtu/h) and F represents the total input cost in cost per hour (C/h)
Figure C.1 shows the input-output characteristic of a steam unit in idealized form. The input
to the unit is shown on the ordinate either in terms of MBtu/h or cost per hour i.e. C/h. The
output is normally the net electrical output of the unit. The characteristic is idealized as a
smooth, convex curve.
Figure C.1 Input-Output Characteristics of a Steam Turbine Generator
109
Steam turbine generating units have several critical operating constraints. Generally,
the minimum load at which a unit can operate is influenced more by the steam generator
(boiler) and the regenerative cycle than by the turbine, which are generally caused by fuel
combustion stability and inherent steam generator design constraints. For example, most
supercritical units cannot operate below 30% of design capability which means a minimum
flow of 30% is required to cool the tubes in the furnace adequately.
C.2 Incremental Heat Rate Characteristics
This incremental heat rate characteristic is the slope of the input-output characteristic
(∆H/∆P or ∆F/∆P) which is as shown in Figure C.2. The ordinate represents Btu per KiloWatt
Hour and the abscissa represents the net power output of the unit in MegaWatts. This
characteristic is widely used in economic dispatching of the unit. Frequently this
characteristic is approximated by a sequence of straight-line segments.
Figure C.2 Incremental Heat Rate Characteristics of a Thermal Generator
C.3 Unit Heat Rate Characteristics
The last important characteristic of a steam unit is the unit (net) heat rate
characteristic shown in Figure C.3. This characteristic is H/P versus P. The unit heat rate
characteristic shows the heat input per kilowatt hour of output versus the megawatt output of
110
the unit. The operating range of efficiencies of a typical conventional steam turbine units are
between 30 and 35% .
Unit Heat Rate =
input in MBTU/hr
output in MW
Figure C.3 Unit Heat Rate Characteristics of a Thermal Generator
111
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Publications
1. "Unit Commitment Using Binary PSO for Solar Power Plant Integrated Smart Power
System”, published in the CD proceedings of IEEE Power & Energy Society General
Meeting (IEEE PESGM) 2013, Vancouver, British Columbia, Canada, July 21-25, 2013.
2. "A PSO based Unit Commitment strategy for Power Systems supported by Solar
Powered Thermal Units”, Advances in Intelligent Systems and Computing, Volume
201, pp. 531-542, 2012.
3. "Unit commitment with nature and biologically inspired computing", Proceedings
of IEEE PES Transmission and Distribution Conference and Exposition 2010 (IEEE PES
T&D 2010), New Orleans, Louisiana, USA, April 19-22, 2010.
4. "Sustainable Energy Plan for an Indian Village”, Proceedings of the IEEE PowerCon
2010 - Technological Innovations Making Power Grid Smarter, Hangzhou, China,
October 24-28, 2010.
5. “A BPSO Technique for Thermal Unit Commitment in Solar Integrated Power
Systems”, Accepted in International Conference on Power Systems Technology
(POWERCON 2012), Auckland, New Zealand, 30 Oct. - 2 Nov., 2012. The paper
couldn’t be presented so didn’t come in proceedings of conference.
Paper Presentation:
6. "A PSO based Unit Commitment strategy for Power Systems supported by Solar
Powered Thermal Units”, Seventh International Conference on Bio-Inspired
Computing: Theories and Applications (BIC-TA 2012), IIIT Gwalior, India, Dec 14-16,
2012.
119