ELECTRIC POWER GENERATION EXPANSION
IN DEREGULATED MARKETS
Except where reference in made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee. This
dissertation does not include proprietary or classified information.
_______________________________
P?nar Kaymaz
Certificate of Approval:
______________________________ ______________________________
Chan S. Park Jorge Valenzuela, Chair
Professor Associate Professor
Industrial and Systems Engineering Industrial and Systems Engineering
______________________________ ______________________________
Robert L. Bulfin Joe F. Pittman
Technology Management Professor Interim Dean
Industrial and Systems Engineering Graduate School
ELECTRIC POWER GENERATION EXPANSION
IN DEREGULATED MARKETS
P?nar Kaymaz
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
August 4, 2007
iii
ELECTRIC POWER GENERATION EXPANSION
IN DEREGULATED MARKETS
P?nar Kaymaz
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon request of individuals or institutions and at their expense.
The author reserves all publication rights.
_____________________________
Signature of Author
_____________________________
Date of Graduation
iv
VITA
P?nar Kaymaz, daughter of Ayten and Bahattin Kaymaz, was born on January 1,
1979, in Izmir, Turkey. She graduated from Istanbul Ataturk High School of Science in
1996. She attended Galatasaray University, Istanbul where she earned her Bachelor of
Science Degree in Industrial Engineering in 2001, and Master of Science Degree in
Industrial Engineering in 2003. She enrolled in Graduate School at Auburn University, in
August 2003.
v
DISSERTATION ABSTRACT
ELECTRIC POWER GENERATION EXPANSION
IN DEREGULATED MARKETS
P?nar Kaymaz
Doctor of Philosophy, August 4, 2007
(M.S., Galatasaray University, Turkey, 2003)
(B.S., Galatasaray University, Turkey, 2001)
107 Typed Pages
Directed by Jorge Valenzuela
The generation expansion problem involves increasing electric power generation
capacity in an existing power network. In competitive environment, power producers,
distributors, and consumers all make their own decisions with the aim of maximizing
their own utilities from power transactions. Therefore, models that incorporate a
behavioral structure in decision making of power producers are needed to analyze market
operation and to make better decisions.
In this dissertation, the main purpose is to develop generation expansion models for
deregulated power markets using game theoretic approaches in order to make accurate
analysis of generation capacity under transmission constraints in the power network and
uncertainties in the power markets. Three new generation expansion models are
vi
introduced. The first model is a generation expansion model that incorporates the features
of the transmission grid to the investment model to study the interaction between
competition and network transfer capabilities in capacity expansion. The other two
models are generation capacity expansion models under competition that incorporate the
uncertainty on electricity demand and fuel costs. Numerical examples and analysis on the
use of the models are given using a 5-bus and a 24-bus power network.
vii
ACKNOWLEDGEMENTS
First, I would like to thank my advisor Dr. Jorge Valenzuela for his invaluable
guidance, and support all along my doctoral education. I would also like to thank Dr.
Chan Park and Dr. S. Mark Halpin for being great mentors for me throughout my
research, and Dr. Robert L. Bulfin for his suggestions and opinions. Next, I would like to
thank all my friends who made my years in Auburn memorable.
Finally, I would like to state my endless appreciation to my parents Ayten and
Bahattin, my sisters Ay?e ?zlem, Lale Didem, Zeynep, and my brother Mehmet Murat
Kaymaz for their continuous support and patience all along my life and education.
viii
Style manual or journal used Bibliography conforms to those of Institute of Electrical
and Electronics Engineers (IEEE) Transactions
Computer software used AMPL
CPLEX
MATLAB
Microsoft Office Excel
Microsoft Office Word
ix
TABLE OF CONTENTS
LIST OF TABLES xi
LIST OF FIGURES xiii
NOMENCLATURE xiv
CHAPTER 1. INTRODUCTION 1
CHAPTER 2. REVIEW OF LITERATURE AND BACKGROUND 5
2.1. Literature Review 5
2.2. Background 9
2.2.1. Consumer Demand 10
2.2.2. Cost of New Generation 12
2.2.3. Transmission System Owner 13
2.2.4. Market Clearing 16
2.2.5. Spot Market Dispatch 17
2.2.6. Optimization Theory 18
2.2.6.1. Complementarity 18
2.2.6.2. Karush-Kuhn-Tucker Conditions 20
2.2.6.3. Quadratic Programming 21
CHAPTER 3. TRANSMISSION CONGESTION AND COMPETITION ON
POWER GENERATION EXPANSION
23
x
3.1. Power Producer Model 23
3.2. Equilibrium model 26
3.3. Numerical Example and Analysis 31
3.3.1. 5-Bus System 31
3.3.2. IEEE 24-Bus System 36
3.4. Effects of Expansion Decisions on the Network Congestion 40
3.5. Conclusion 43
CHAPTER 4. POWER GENERATION EXPANSION IN DEREGULATED
MARKETS UNDER UNCERTAINTY
44
4.1. Uncertainty in Power Demand 45
4.2. Uncertainty in the Fuel Costs 46
4.3. Power Generation Expansion Under Uncertainty ? Model 1 47
4.4. Power Generation Expansion Under Uncertainty ? Model 2 54
4.5. Numerical Example and Analysis 60
4.6. Evaluation of Expansion Decisions with Cournot Competition Model 68
4.7. Conclusion 75
CHAPTER 5. CONCLUSION AND FURTHER RESEARCH DIRECTIONS 77
REFERENCES 80
APPENDIX 85
xi
LIST OF TABLES
Table 3.1. Line Data For 5-Bus System 32
Table 3.2. Generator Data 32
Table 3.3. Cournot Equilibrium Solution 34
Table 3.4. Perfect Competition Equilibrium Solution 35
Table 3.5. 24-Bus Network Data 38
Table 3.6. 24-Bus System Data 38
Table 3.7. 24-Bus Cournot Expansion Solution (MW) 39
Table 3.8. 24-Bus Cournot Equilibrium Solution ($) 40
Table 3.9. Load Curtailments (MWh) 41
Table 3.10. Network Economic performance ($) 41
Table 3.11. Prices ($/MWh) 42
Table 4.1. Price intercept of inverse demand functions ($/MWh) 61
Table 4.2. Slope of inverse demand functions ($/MWh2) 62
Table 4.3. Different fuel types and capacities in the 24-Bus System (MW) 63
Table 4.4. Fuel price change parameters 64
Table 4.5. Operational costs of existing generators in the 24-Bus System ($/MWh) 64
Table 4.6. Operational costs of new generators in 24-Bus System ($/MWh) 65
Table 4.7. Investment costs of new generators in the 24-Bus System ($/MWh) 66
xii
Table 4.8. Capacity expansion comparison (MW) 67
Table 4.9. Economic analysis comparison ($) 68
Table 4.10. Comparison of the producer profits ($) 70
Table 4.11. Cournot economic analysis ($) 71
Table 4.12. Cournot Prices ($/MWh) 72
Table 4.13. Total amount paid by the consumer at each node ($) 73
Table 4.14. Total consumer benefit at each node ($) 74
Table 4.15. Total transmission price collected at each node ($) 75
Table A1. 24-Bus System PTDF data 86
Table A2. Capacity expansion solution of Model 1 (MW) 89
Table A3. Capacity expansion solution of Model 2 (MW) 90
Table A4. Capacity expansion solution of the Certainty model (MW) 91
xiii
LIST OF FIGURES
Fig. 2.1. Restructured market operation 10
Fig. 2.2. Consumer demand curve 11
Fig. 3.1. Equilibrium Procedure 27
Fig. 3.2. 5-bus, 4-generator power system 32
Fig. 3.3. Components of the social welfare 36
Fig. 3.4. 24-bus system network 37
Fig. 4.1. Demand states 46
xiv
NOMENCLATURE
Parameters
L : Set of lines, ?ij L
? : Set of buses, ?k ?
P : Set of producers, p, ?q P
T : Set of states of Nature
F : Set of flowgates
fF : Set of transmission lines of flowgate f
U : Set of fuel types, ?u U
0
pkG : Current capacity of producer p at node k
max
pkG : Maximum new capacity installation of producer p at node k
0
pkC : Cost of power generation from existing capacity of producer p at node k
pkC : Cost of power generation of producer p at node k
pkI : Capital investment cost for capacity expansion of firm p at node k
ijT : Thermal limit on line i-j
fF : Maximum capacity of the flowgate f
k
ijPTDF : Power transfer distribution factor of node k for link ij
xv
u
pkG Upper bound on generation at real-time for producer p at node k
l
pkG Lower bound on generation at real-time for producer p at node k
kk ?? , : Vertical intercept and the slope of the inverse demand function at node k
T : Number of states
tukC : Cost of power generation at state t from fuel type u at node k
0
tukC :
Cost of power generation from existing capacity at state t of producer p at
node k
pukI : Capital investment cost for expansion of firm p on fuel type u at node k
tktk ?? , :
Vertical intercept and the slope of the inverse demand function at state t at
node k
0
pukG Existing capacity limit of producer p from fuel type u at node k
max
pukG Maximum capacity installation of producer p from fuel type u at node k
Decision Variables
0
pkg :
Amount of power generated at node k from existing capacity of producer
p
pkg : Amount of power generated at node k from new capacity of producer p
pkG : Total amount generated at real-time by producer p at node k
0
pks : Amount of demand met at node k from existing capacity of producer p
pks : Amount of demand met at node k from new capacity of producer p
xvi
kD : Demand at node k for
kP : Price of power at node k
ky : Amount of power transmitted by the grid operator from hub to node k
k? : Congestion based transmission price at node k
0
tpukg :
Amount of power generated at state t from existing capacity of producer p
from fuel type u at node k
tpukg :
Amount of power generated at state t from new capacity of producer p
from fuel type u at node k
n
pukg : Amount of new capacity of producer p from fuel type u at node k
tpks : Amount of demand met by producer p at state t at node k
tkD : Demand for at state t at node k
tkP : Price of power at state t at node k
tky :
Amount of power transmitted by the grid operator at state t from hub to
node k
tk? : Congestion based transmission price at state t at node k
)( kyE : Expected transmission quantity at node k
)(pER : Expected revenue of producer p
)(pEC : Expected cost of producer p
ECR : Expected congestion revenue for transmission owner
1
CHAPTER 1
INTRODUCTION
The electric power industry throughout the world has been undergoing restructuring
in many power markets, which includes major markets in the United States such as
Pennsylvania-New Jersey-Maryland Power Market (PJM), California Independent
System Operator (CAISO), and New York Independent System Operator (NYISO).
Restructuring has also many examples in Europe, Africa, and Latin America.
The main aim of the introduction of deregulation to the electricity industry is to
obtain more liberalized and therefore more efficient power markets. In competitive
environment, power producers, distributors, and consumers all make their own decisions
with the aim of optimizing their own objectives from power transactions. For that
purpose, these market agents have to consider the actions of other market players in their
operational and planning decisions. Thus, planning decisions such as addition of new
generation capacity is expected to be highly influenced by competition.
The generation expansion problem involves increasing electric power generation
capacity in an existing power network. In traditional (vertically integrated) power
markets, the expansion decisions are made by the central operator of the power network,
considering all existing generation and available transmission capacity in the network as
well as the electricity demand. As the central operator decides which generator to operate
2
to meet the demand, dispatches the power, and decides on the selling price of power, the
regulated model of generation capacity expansion can be easily formulated with a least-
cost structure of the operation and investment costs of capacity expansion over the whole
system. The model then becomes a minimization problem subject to operational
constraints. In deregulated markets, the generators are not owned by a central operator.
The electric power companies decide on the power quantities for their generation as well
as expanding their generation capacities. With the restructuring, the generation
companies now can make their own decisions in order to maximize their profits. While
these companies maximize their profits, they need to consider that the market operates
under a competitive structure with the existence of other market players. That is, the
decisions and behaviors of other generation companies operating in the market are
equally important for a power producer in making decisions as the demand quantities or
transmission availability. Therefore, models that incorporate a behavioral structure in
decision making of power producers are needed to analyze market operation and to make
better decisions. Game theoretic approaches are appropriate to study these kinds of
behavioral decision structures in power markets as they provide tool for modeling
competitive markets. Generation expansion under competition remains one important
topic to be explored using game theoretic approaches as very few research projects are
now emerging in this area.
In this research, the main purpose is to develop generation expansion models for
deregulated power markets, which incorporate the new power market structure. The term
?new power structure? refers to the changes in power trading, demand?s price sensitivity,
transmission system?s functioning, and system operation. In order to build a generation
3
expansion model for deregulated power markets, it is necessary to incorporate the above
elements in the decision model. This research?s purpose is to construct capacity
investment models under competition using game theoretic approaches with the aim of
making accurate analysis of generation capacity and its effect on power market and
network operation. With this purpose, two main factors in power markets are considered
while modeling capacity investments: transmission network and uncertainty in power
markets.
The first important factor that is considered in this dissertation is the power
transmission network in power generation expansion. Capacity expansion may have
several effects on the market?s situation. One of these is the effect on the power
network?s operation. Increased generation capacity in a power network may encounter
transmission congestion in the transmission network as a result of excess regional
capacity investments. Expansion, which actually aims to increase the investors bargaining
power, may in fact have a negative impact on generator companies? profits as a result of
the increased transmission congestion and decreased ability to transfer power to demand
locations. The first model that is introduced in this dissertation is a generation expansion
model that incorporates the features of the transmission grid to the investment model to
study the interaction between competition and network transfer capabilities in capacity
expansion.
The second important factor that is considered in this dissertation is mitigating the
risk involved in the decision of capacity expansion in restructured power markets. Since
generation capacity expansion is an investment decision that needs to be made with the
current information available to the investor generation companies, there is an uncertainty
4
involved in this decision, which brings risk to the investment decisions. Therefore, the
generation expansion models, either aiming to analyze generation capacity or the network
operation, need to incorporate the uncertainty involved in the power markets. Two
models are introduced in this dissertation for generation capacity expansion under
competition and uncertainty in electricity demand and fuel cost.
The remainder of this dissertation is organized as follows: In chapter 2, a review of
related research currently available and power market concepts, preliminary information
and assumptions used for formulating the restructured power markets in this dissertation
are explained. Chapter 3 describes the generation expansion framework under
transmission constraints. Chapter 4 describes the generation expansion models under
uncertainty, competition, and network constraints. In chapter 5, conclusions are
presented.
5
CHAPTER 2
REVIEW OF LITERATURE AND BACKGROUND
2.1. Literature Review
The generation expansion problem involves increasing electric power generation
capacity in an existing power network. Generation expansion decisions involve the
location and the amount of new capacity to be built. Generation expansion in the
regulated power market structure aims to optimize the total generation capacity in the
whole power network, considering the future electricity demand. As in the regulated
markets the generators are owned by the system operator, one decision model that
minimizes the total costs to the entire generator set is sufficient. This approach is known
as the least-cost investment approach and has been extensively studied in centralized
electricity markets. The generation expansion model in this case is called the traditional
and static generation expansion planning model.
Among the major objectives of the traditional generation expansion planning are the
minimization of investment and operating costs of capacity, and meeting the demand
criteria [1]. This approach has been used with different optimization techniques such as
stochastic programming [2], nonlinear programming [3], heuristic optimization methods
as genetic algorithms [4], [5], and evolutionary programming [6]. In addition, multiple
6
criteria decision making approaches were applied to the generation expansion problem
[7], [8]. In the static model approach the decisions are based on inelastic fixed demand
quantities. In restructured markets, one important change in production cost modeling is
the introduction of the elasticity of demand [9], which refers to the decentralization of the
decisions after deregulation.
With the deregulation, the generation companies are owned by independent power
producers, transmission network is owned by independent entities, and the market
involves competition of the suppliers. Models that incorporate the competition of
generators have been introduced to the power market literature. These models mainly
focus on the decision of bidding strategies of the generator companies under competition
using game theoretic approaches. Demonstrating the usefulness of the game theoretic
approaches is amongst the most important research needs in electricity resource planning
[10]. Finding the equilibrium solution for market games is desirable for market
participants and regulators [11]. According to [12], searching for the Nash-equilibrium is
important as it provides participants? best strategies in different market conditions.
Recent examples of research on bidding strategy modeling using game-theoretic
approaches involve [13], [14], and [15]. In addition these kinds of approaches have been
used for environmental problems in deregulated markets [16] and modeling the network
effects on strategic bidding [17] and [18]. However, few models have been proposed for
studying the capacity investment decisions in oligopolistic power markets. Generation
capacity expansion remains one important topic to be explored using these approaches as
very few research projects are now emerging in this area.
7
In power markets, restructuring has been initiated in order to introduce competition
among power producers to obtain more liberalized and therefore more efficient power
markets. In other words, the expectation is that the power markets will benefit from the
competition of generation companies with decreasing market power and power prices.
Under this market structure, the generators are competing in an oligopoly in order to
maximize their profit objectives. Although the power market restructuring had the aim to
obtain perfectly competitive markets in which generators bid under perfect competition
for prices at competitive levels, in the power system literature, most of the time, perfect
competition is referred to as unrealistic as it is a very strong assumption [19]. Perfect
competition is used widely in the literature for comparison purposes [20].
Chuang et al. [21] have proposed a single-period Cournot competition model that
incorporates plant capacity and energy balance constraints. Murphy and Smeers [19],
[22] have introduced three new models concerning capacity expansion under competition
in electricity markets. The open-loop Cournot model given in [19] is based on a single-
stage decision process where the capacity expansion and production decisions are made
at the same stage. The other two models refer to the two-stage (closed-loop) and three-
stage decision processes in which the decisions are also based on Cournot competition.
These two models are more realistic. However, they are also more difficult to solve and
the equilibrium solution is not guaranteed. Neither of these models considers the
transmission network in the decision framework to prevent additional complexity.
Adding new generation capacity to a power network may have positive or negative
impacts depending on where the new capacity will be installed and the amount of such
capacity. Shahidehpour et al. [23] have pointed out that generation companies?
8
investment decisions may challenge the transmission system security. Therefore, it is
important for transmission owners and operators to make investments in transmission to
overcome future conditions and demand growth in the power network. The first model
that is presented aims to contribute to the lack of models in the literature that incorporate
the transmission grid in a power system subject to competitive generation expansion
decisions using game theoretic approaches. The competition is based on Cournot which is
known as the ?competition on quantities?. The assumption of Cournot competition is
based on three major facts:
a) although power companies may compete on price in the short term, they also need
to make long-term commitments to capacity, which supports adopting competition on
quantities, i.e. the Cournot model
b) power plant owners may reduce generation to deliberately induce higher profits, a
feature of Cournot competition
c) the Cournot model can render simple analytical expressions that can be easy to
manipulate.
Another important factor in power markets that may effect the investment decisions is
the uncertainty in market parameters on which the investment decisions are based. As the
investment decisions are made today in consideration of future market conditions, a level
of risk is involved in these decisions that needs to be incorporated for more realistic
models and investment decisions. Two important sources of risk in power markets are the
demand uncertainty and fuel cost uncertainty. In regulated power markets, these kinds of
9
risks were generally passed to the consumers by reflecting the uncertainties onto the
power prices [19]. In deregulated power markets, the need for the power prices to stay in
competitive levels gives rise to the need for considering these factors as part of the
decision models in order to obtain a complete investment analysis. These kinds of
investment models in deregulated markets can create investment plans for producers that
can mitigate the above mentioned risk factors and also may help power prices to stay in
competitive levels.
In [19] and [22], capacity expansion under Cournot competition is studied with
several models. However, in these models instead of accounting for the uncertainty in
power markets, the risks are considered with increased discount rates with respect to the
risk-free rate. Although, in [19] it has been noted that representation of uncertainty, such
as found in [24] is necessary for capacity investment decisions. The second model that is
introduced in this dissertation aims to fill this research need with a generation expansion
model under competition which incorporates the uncertainty in power demand and
uncertainty in fuel costs with network constraints. Therefore, the second model adds up
on the first model of generation expansion under transmission network constraints with
the purpose of creating a risk mitigating model by considering the sources of uncertainty
in the power markets.
2.2. Background
Electricity markets involve suppliers and consumers that bid for power generation
and consumption. Many electricity markets throughout the world include some form of
10
supplier bidding in which suppliers submit MW outputs, along with the associated prices
[25]. All markets based on spot pricing consist of supplier bidding for electricity, in terms
of power quantities or prices, which allows the consumers to react to power prices [25].
As represented in figure 2.1, several power producers in the power markets compete to
meet the power demand under the constraints of the power network.
Fig. 2.1. Restructured market operation
2.2.1. Consumer Demand
In traditional production cost models, an inelastic demand is assumed and this
demand has to be met with a penalty for unserved load [9]. To model the capacity
expansion in deregulated markets, at first it is needed to identify the behavior of
consumers. In this study, a linear inverse demand function is assumed as it is a good
approximation of a nonlinear demand curve in the neighborhood of the equilibrium, and
it simplifies the mathematics of the models [19]. In figure 2.2, the curve shows a general
11
form of a linear inverse demand function that is used in this dissertation and indicates the
price responsiveness of the consumers.
Fig. 2.2. Consumer demand curve
It can be observed that the power price and the demand quantity are inversely
proportional. The reason for this linear function is to simplify the mathematics of the
framework and to ensure that a unique equilibrium solution exists. According to the
demand function illustrated, the price at node k is defined in (2.1).
kkkk DP ?? ?= (2.1)
The demand function is used in the producer decision model to calculate its revenue
defined as the price multiplied by the quantity sold. With a linear demand curve, the
benefit function of the consumers (the integral of the demand function) can be calculated
12
as in (2.2) [20], [25].
2
2
1benefitConsumer
kkkk DD ?? ?= (2.2)
This benefit function is used in the equilibrium model to compute the total social
welfare defined as the benefit to the consumers minus the cost to the generators [20]. The
demand bids may vary for each customer. The variation may be depending on a single
parameter which affects both the intercept and the slope as in [25], or it may be assumed
that only the intercept of the curves is under manipulation as in [26]. In the numerical
examples in this research, only the variation on the intercept is considered.
2.2.2. Cost of New Generation
Regarding the cost of new generation, two types of cost need to be considered:
operating and capital costs. An operating cost is incurred every time the power plant is
operated and it includes costs such as maintenance and fuel costs. The capital cost, on the
other hand, is non-recurrent and it involves the initial investment and the salvage value at
the time of disposal of the power plant. The operating cost is usually expressed in $/MWh
and the capital cost is expressed in $/MW. To write these two quantities in similar
measurement units, the capacity pricing method suggested by Stoft in [27] is adopted. In
this method, the total capital cost is annualized over the life of the plant using an
appropriate discount rate, and then divided by the number of operating hours per year.
13
The investment decision is based on long term power purchases, which are made
bilaterally between consumers and producers, to meet the base load. Therefore, the new
capacity is expected to be fully used in order to meet the resultant load. Under the
assumption that the plant will operate at full-capacity (a capacity factor equal to one), the
cost of new generation can be represented by investment and operation terms both
expressed in $/MWh. As suggested in [27], in this research, the cost of new capacity is
represented with the per unit annualized cost of investment, which is also followed in
[19] and [22]. The cost of investment of the power generators are given per unit values
with constant marginal cost of operation.
2.2.3. Transmission System Owner
A power transmission system model usually includes constraints such as thermal
limits on transmission lines and power balances at each node of the network. Capacity
expansion may encounter congestion in the transmission network by constrained single-
line limits as well as flowgate transfer capabilities. In [28], flowgates are defined as
bottlenecks in transmission lines where congestion takes place during certain operation
conditions. In addition to single-line constraints, flowgates may be used for stability
purposes to control the reliability of the network. A flowgate is a group of lines in the
network that has limited total flow capability in its lines. Flowgates may be used for
thermal or voltage based reasons in a power network. Flowgates may also be imposed to
limit the flows within interregional coordinated markets, which help to control the
regional operations.
14
Flowgates represent the available transfer capability through a congested region
including more than one transmission line of the network. The basic idea is to assign a
portion of the flowgate limit to each operating generator in the constrained area using a
linear combination of the PTDF matrix elements. With these PTDFs, a single inequality
constraint based on flowgate total can be added to the power dispatch model in a manner
exactly analogous to that used when mitigating thermal constraints on single elements
[29]. The inequality constraint is of the form shown in (2.3) where yk is the amount of
power transmission to node k, and ij is summed over all lines that make up the flowgate
to get a flowgate total. This total must be kept less than some maximum limiting value,
Ff.
F
fF
????
?
??
?
?? ?
?
fFyPTDF f
ij k
k
k
ij * (2.3)
Flowgate constraints, like other transmission related constraints such as thermal limits
of transmission lines, are for securing the transmission network operation and therefore
they should be included in the Transmission System Owner's decision model. Thus,
transmission line and flowgate constraints can both be modeled mathematically by
making use of Power Transfer Distribution Factors (PTDFs) and be included in the
decision models that consider the transmission network.
Transmission System Owner (TSO) decides what amount of power needs to be
transmitted to each node of the network using the transmission lines, while safeguarding
the single line and flowgate limits. In order to model the TSO, the approach proposed in
15
[13] is used. In this approach the transmission owner is assumed to behave under the
Bertrand competition assumption, where it cannot control the power prices. The revenue
of the TSO is the amount of total congestion rent that is collected from the network for
transmitting power through the congested regions. With this aim, the objective function
and the constraints given in [13] are used to formulate the equilibrium model for
generation expansion explained in the proceeding section.
The TSO is modeled based on [13] with the following profit maximizing objective:
k
k
k yMax?? (2.4)
subject to
thermal capacity constraints:
)( +???? ijij
k
k
k
ij ijTyPTDF ?L (2.5)
)( ?????? ijij
k
k
k
ij ijTyPTDF ?L (2.6)
and the flowgate constraints:
)(* ff
ij k
k
k
ij fFyPTDF
f
?F
F
????
?
??
?
?? ?
?
(2.7)
In this model, the coefficient k? denotes the congestion based transmission prices that
need to be paid for transmitting power from hub to node k which is the dual price of the
market clearing constraint written for that node. TSO is paid k? to get power from node k
16
to the hub and pays k? to convey power from the hub to node k. The TSO model includes
the constraints associated with both thermal and network stability. The thermal
constraints are represented using the PTDFs of each line following the framework in [29].
This framework coincides with the network stability modeling that is explained above
and used in the TSO model, as both the thermal and flowgate stability consideration of
the network are represented in terms of generation PTDF amounts.
2.2.4. Market Clearing
Given that the power transmission amounts are decided by the Transmission System
Owner and the generation and sales quantities are decided by the power producers, then
these quantites need to be balanced for every node at the power network with market
clearing constraints given below.
)()()( 00 k
p
k
p
pkpkpkpk kyggss ?BP P ??=+?+? ?
? ?
(2.8)
The variables 0pkg and pkg denote the generation quantities from existing and new
capacity installed of producer p at node k, whereas 0pks and pks denote the power sales of
producer p to node k from existing and new capacity. The balancing of the power
quantities is performed by an Independent System Operator in order to maintain feasible
operation of the power system. In addition, this constraint provides the marginal cost of
power transmission to each node k from the hub, ?k. This marginal cost is the cost that the
17
power producer pays for transferring power from the hub and receives for transferring
power to the hub [13]. The cost of transmission is used in formulating the profit
maximization objective of the power producer.
2.2.5. Spot Market Dispatch
In the spot market, an independent central system operator ensures network reliability
and sets the prices for the energy and transmission rights using an Optimal Power Flow
(OPF) model. With a flow-based structure, the OPF can be represented by the following
model.
?
kp
pkpkyG GCMin
kpk ,,
(2.9)
subject to
balance constraint:
)(0
,
?=???
k
k
kp
pk DG (2.10)
thermal capacity constraints:
)( +???? ijij
k
k
k
ij ijTyPTDF ?L (2.11)
)( ?????? ijij
k
k
k
ij ijTyPTDF ?L (2.12)
flowgate constraints:
)( ff
ij k
k
k
ij fFyPTDF
f
?F
F
????
?
??
?
?? ?
?
(2.13)
18
generation capacity limits:
u
pkpk
l
pk GGG ?? (2.14)
and nonnegativity constraints:
0?pkG (2.15)
In this formulation, the objective is minimizing the total generation cost (dispatch
cost). The constraint (2.10) is the balance of total generation and total load in the system,
and the constraints (2.11)-(2.13) are the power network feasibility constraints. Equation
(2.14) is the upper and lower bounds on the generation quantity. The electricity prices in
the real-time market are calculated using the Locational Marginal Pricing (LMP) concept.
This model provides one dual price (?) for the system balance equation, which represents
the LMP of the hub. The LMPs for other nodes can be calculated by using the dual prices
of the single circuit and flowgate limits, which correspond to the congestion prices
associated with those limits.
2.2.6. Optimization Theory
2.2.6.1. Complementarity
A complementarity condition between a nonnegative variable xi and a function )(xgi
of a vector of variables }{ ixx = can be defined as in [20]:
19
0?ix (2.16)
0)( ?xgi (2.17)
0)( =xgx ii (2.18)
This can also be expressed as 0)(0 ??? xgx ii . A complementarity problem is to
find the x such that 0)(0 ??? xgx , where { })()( xgxg i= . Specifically, given a set of
functions )(xg , the complementarity problem is to find a vector x such that
0?x (2.19)
0)( ?xg (2.20)
0)( =xgxT (2.21)
If all the )(xgi are linear functions, then complementarity problem is referred to as a
linear complementarity problem [20]. Mixed linear complemetarity problem is a more
general form of a linear complementarity problem with additional equality constraints.
Let y be a second vector of variables, and ),( yxh be a vector valued function. An
example of a mixed linear complementarity problem (MLCP) can be stated as finding x
and y such that [20]:
0?x (2.22)
0)( ?xg (2.23)
20
0)( =xgxT (2.24)
and 0),( =yxh . (2.25)
2.2.6.2. Karush-Kuhn-Tucker Conditions
Now let us assume a constrained optimization problem in order to visualize how the
market equilibrium conditions are obtained using the first order necessary conditions for
an optimizer vector. Consider the following optimization problem:
),(Max yxf (2.26)
subject to
0),( ?yxg (2.27)
0),( =yxh (2.28)
0?x (2.29)
The first order, Karush-Kuhn-Tucker (KKT) conditions for this problem are written
by using complementarity and equality constraints. Assuming that ? and ? are the dual
variables associated with the inequality and equality constraints respectively, the KKT
conditions are [13]:
for x: 0)0(,0,0 =??????????????????? xhxgxfxxxhxgxf ???? (2.30)
21
for y: 0=???????? yhygyf ?? (2.31)
for ? : 0),(,0,0),( =?? yxgyxg ?? (2.32)
for ? : 0),( =yxh (2.33)
This formulation of the KKT conditions will be used to obtain the market equilibrium
conditions in the power generations expansion model in the following chapters.
2.2.6.3. Quadratic Programming
Quadratic programming deals with the problem of maximizing (or minimizing) a
quadratic objective function over a polyhedral feasible region [30]. The mixed linear
complementarity problems provide a natural setting for the Karsuh-Kuhn-Tucker
conditions of a quadratic program with general equality and inequality constraints [30].
Consider the quadratic program:
xcQxx TT +21Max (2.34)
subject to
bAx ? (2.35)
dCx = (2.36)
0?x (2.37)
22
with Q a symmetric square matrix. The KKT conditions for a locally optimal solution of
the above quadratic program are written as follows:
for x: 0=??+ ?? TT CAcQx (2.38)
for ? : 0)(,0,0 =???? bAxbAx ?? (2.39)
for ? : 0=?dCx (2.40)
These conditions are an example of a mixed linear complementarity problem where
? and ? are the dual variables associated with the inequality and equality constraints
respectively. These are the necessary conditions for x to be a local optimizer of the
problem. In addition, in the case where Q is a negative semi-definite matrix, then the
objective function of the quadratic program is convex, and thus the KKT conditions are
sufficient for x to be the global optimal solution [30].
23
CHAPTER 3
TRANSMISSION CONGESTION AND COMPETITION ON
POWER GENERATION EXPANSION
In this section, a generation expansion model that incorporates the features of the
transmission grid is developed to study the interaction between competition and network
transfer capabilities in capacity expansion. In this framework, a DC-approximation model
that includes network constraints is used to represent the transmission grid while a
Cournot oligopolistic market model is used to represent competition. Hobbs [13]
proposed a Nash-equilibrium modeling approach to strategic bidding in deregulated
markets under network constraints. The same approach is undertaken in this research to
find the equilibrium solution under network consideration for generation capacity
expansion. The proposed model is a single-stage deterministic Cournot capacity
expansion model where the decision to invest and operate are made at the same stage
with bilateral contracts.
3.1. Power Producer Model
Generators have two basic decisions concerning the amount of power trading for
maximizing their profit: Amount of power sales to each node and amount of capacity
24
expansion. In order to model the generation expansion decision problem, an open-loop
Cournot approach similar to [19] is undertaken. In this approach, the producers are
assumed to make long-term power agreements while they make decisions on capacity
expansion. That is, the decision on where and how much to expand is given together with
how much to offer into the power market. Firms decide on their output and capacity
expansion in Cournot behavior. This approach helps integrate the two decisions in one
producer profit maximization model and also represents the imperfect competition. The
generation expansion model for the producers is written by undertaking the strategic
bidding approach in [13], which is a Cournot competition model that maximizes the
profit of generation firms while deciding on the optimum power sales of each firm. The
power producer model proposed in this paper is an investment model that uses this
approach, but it assumes that each power producer may have existing generation capacity
and is able to expand capacity at any node of the power network. Hobbs?s model is
extended by including capacity expansion decision variables and investment cost terms to
the power producer profit maximization model. Unlike [19] and [22], this model
distinguishes between existing and new capacity of the power producers so that new
entrants are modeled considering the current conditions of the competitors. Adding new
nodes to the system has not been included, as this issue needs to be studied together with
the transmission expansion problem.
In deregulated power markets, the amount of demand supplied from each power
producer in the network may vary. However, the demand at one node is equal to the
amount that is sold to that node by different producers. The plant owners receive the price
obtained from the inverse demand function (2.1) per MW sold and pay for the operation,
25
transmission and capital investment costs. The resulting producer?s objective is the
revenue from sales minus transmission cost, operating cost and investment cost. The
decision model of a producer p ( P??p ) is as follows:
( )( ) ( )
???
??
???
??+?++++? ??
k
pkpkpk
k
pkpk
k
pk
k
pkpkpkpkkpkpk
k
pkpkpkpkkk
gIgCgC
ggssssssss
00
00000 ][Max ???
(3.1)
subject to
generator capacity constraints:
00
pkpk Gg ? B??k )(
0
pk? (3.2)
max
pkpk Gg ? B??k )( pk? (3.3)
firm energy balance:
?? =
k
pk
k
pk gs
00 )( 0
p? (3.4)
?? =
k
pk
k
pk gs )( p? (3.5)
and non-negativity constraints:
0,,, 00 ?? pkpkpkpk ggss (3.6)
In this model, the variable 0pks and pks denote the sales of producer p at node k,
whereas 0pks? and pks? represent the sales of the remaining producers, from their existing
and new capacity respectively. The variables given in parenthesis correspond to the dual
variables of the related constraints. For instance, the variable 0p? can be interpreted as the
26
marginal cost of production of the existing capacity of producer p. The new capacity
variables pkg and pks are introduced to the problem in [13] with additional constraints,
which are maximum capacity installation constraints (3.3) and sales from the new
capacity constraints (3.5). Furthermore, the objective function (3.1) includes the cost
terms pk
k
pkgC? and ?
k
pkpkgI , which represent the new capacity operating costs and
investment costs, respectively. The model optimizes the sales and generation variables
from existing and new capacity while deciding on the capacity expansion amount.
3.2. Equilibrium model
The power producers need to maximize their own profit maximizing objectives while
deciding on the capacity expansion quantities. Therefore, each power producer needs to
solve the Power Producer Model given in the previous section iteratively until an
equilibrium solution is reached among power producers. Transmission System Owner
solves its own profit maximizing problem and then the outcomes of these models are
balanced in the market clearing constraint by the ISO as explained in chapter 2. The
framework of the generation expansion equilibrium under transmission constraints is
represented in figure 3.1 based on the bidding procedure in [13].
27
Fig. 3.1. Equilibrium Procedure
A solution that is optimal for each producer, transmission system owner, and the
market clearing optimization models is considered an equilibrium solution. A Nash-
equilibrium capacity solution is the quantities of expansion and power sales that optimize
every power producer?s profit model. Thus, the equilibrium solution needs to satisfy the
optimality conditions of the producer models written for each power producer. The
Karush-Kuhn-Tucker (KKT) optimality conditions of the producer model for each
producer are written as follows.
KKTs related to the Power Producer Model:
for 0pkg : ,0,0 0000 ??+?? pkppkpkk gC ??? 0)( 0000 =+?? ppkpkkpk Cg ??? (3.7)
for pkg : 0,0 ??+??? pkppkpkpkk gIC ??? , (3.8)
Producer?s model
Independent System Operator
Producer?s modelProducer?s model
Producer?s modelCournot Transmission Owner?s Model
Sales
Prices Transmission
amounts
Expansion
Demand
28
0)( =+??? ppkpkpkkpk ICg ???
for 0p? : ? ?=
k k
pkpk gs
00 (3.9)
for p? : ? ?=
k k
pkpk gs (3.10)
for 0pk? : 0)(,0, 000000 =??? pkpkpkpkpkpk GgGg ?? (3.11)
for pk? : 0)(,0, =??? pkpkpkpkpkpk GgGg ?? (3.12)
for 0pks : ( )[ ] ,0,02 0000 ????+++? ?? pkpkpkpkpkpkkk sssss ???? (3.13)
( )[ ] 0)2( 0000 =??+++? ?? pkpkpkpkpkkkpk sssss ????
for pks : ( )[ ] ,0,02 00 ????+++? ?? pkpkpkpkpkpkkk sssss ???? (3.14)
( )[ ] 0)2( 00 =??+++? ?? pkpkpkpkpkkkpi sssss ????
The solution also needs to satisfy the optimality conditions of the Transmission
System Owner and the ISO models simultaneously for market equilibrium. Applying the
same framework the KKT optimality conditions of the TSO and ISO can be obtained as
follows.
KKTs related to the transmission owner?s model:
for ky : ? =??+ +?
ij
f
k
ijij
k
ijij
k
ijk PTDFPTDFPTDF 0???? (3.15)
for ?ij? : ,0, ??? ?? ij
k
ijk
k
ij TyPTDF ? ? =+
?
k
ijk
k
ijij TyPTDF 0)(? (3.16)
29
for +ij? : ,0, ?? +? ij
k
ijk
k
ij TyPTDF ? ? =?
+
k
ijk
k
ijij TyPTDF 0)(? (3.17)
for f? : ,0, ???? ffk
ij k
k
ij FyPTDF ? 0)( =??? fk
ij k
k
ijf FyPTDF? (3.18)
KKT related to the market clearing:
for k? : ? ?
? ?
=+?+
P Pp
k
p
pkpkpkpk yggss )()(
00 (3.19)
By implementing the procedure presented in [13], a single equivalent optimization
model is built using the social welfare concept and the constraints of the optimization
models of all market participants. This procedure consists of transforming the Linear
Complementarity Constraints that represents the equilibrium solution into a Quadratic
Programming model, called the equilibrium model. The resulting equilibrium model has
the same Karush-Kuhn-Tucker (KKT) optimality conditions as the combination of the
KKT conditions of the producer model, transmission system owner model, and a market
clearing constraint. Thus, the equilibrium generation expansion solution can be obtained
by solving the following equilibrium model.
( ) ( ) ( )
)](
2][2[Max
00
20200
pkpkpkpkpk
p
pk
p
pkpk
k
p
pkpk
k
p
pkpkk
k
gIgCgC
ssssss
++?
+?+?+
?
????
??? PPP
???
(3.20)
subject to
30
00
pkpk Gg ? BP ??? kp , )(
0
pk? (3.21)
max
pkpk Gg ? BP ??? kp , )( pk? (3.22)
?? =
k
pk
k
pk gs
00 P??p )( 0
p? (3.23)
?? =
k
pk
k
pk gs P??p )( p? (3.24)
)( +???? ijij
k
k
k
ij ijTyPTDF ?L (3.25)
)( ?????? ijij
k
k
k
ij ijTyPTDF ?L (3.26)
)( ff
ij k
k
k
ij fFyPTDF
f
?F
F
????
?
??
?
?? ?
?
(3.27)
)()()( 00 k
p
k
p
pkpkpkpk kyggss ?BP P ??=+?+? ?
? ?
(3.28)
0,,, 00 ?? pkpkpkpk ggss (3.29)
Equation (3.28) that includes the amount of power transmitted ky is defined as the
market clearing condition, and is explained in more detail in [13] and in chapter 2. It
balances the difference between the power produced and demanded at one node to the
amount of power transmitted to that node by the transmission system operator. The model
optimizes the amount of power transmitted, sales and generation variables from existing
and new capacity.
Similar to Hobbs? model, the concavity of the objective function assures a unique
solution if a solution exists. The objective function of the equilibrium model represents
the total welfare of the system, which is the benefit to the consumers minus the Cournot
31
producers? effect on the benefit and the operation and investment cost to the producers.
For more detail the reader should refer to [20]. The above Quadratic Programming model
can be solved using a commercial solver engine.
In the case of perfect competition, the market players do not act as if they can alter
the market prices. The objective for the perfect competition model is to maximize the
social welfare which can be calculated as the benefit to the consumers minus the cost to
the producer, and does not include the price alteration term used in the Cournot objective
[20]. The perfect competition model is used for comparison purposes.
3.3. Numerical Example and Analysis
To study the interaction of the transmission constraints and competition on optimal
generation expansion decisions, two different examples are used: the 5-bus, 4-generator
power system [31] and the IEEE Reliability Test System [32]. The commercial package
CPLEX is used to solve the models of both systems.
3.3.1. 5-Bus System
For illustrative purposes, the model is first applied to a small power network based on
[31] and shown in figure 3.2. The generator and line data for the 5-bus system are given
in table 3.1 and 3.2. As this system is easier to illustrate, detailed analysis including
perfect competition among the generators and monopoly cases are given using this
example.
32
Bus 4
Bus 1
Bus 2
Bus 3
Bus 5
Fig. 3.2. 5-bus, 4-generator power system based on [31]
Table 3.1. Line Data for 5-Bus System
1 2 3 4 5
1-2 0.0138 0.2941 -0.5294 -0.3529 -0.1765 0
2-3 0.0138 0.2941 0.4706 -0.3529 -0.1765 0
4-3 0.0138 -0.2941 -0.4706 -0.6471 0.1765 0
5-4 0.0414 -0.5147 -0.5735 -0.6324 -0.6912 0
4-1 0.0552 -0.2206 -0.1029 0.0147 0.1324 0
5-1 0.0690 -0.4853 -0.4265 -0.3676 -0.3088 0
PTDFijkLine Reactance
(per unit)
Table 3.2. Generator Data
Generator Capacity (MW) Cost ($/MWh)
Bus 1 110 14
Bus 3 520 30
Bus 4 200 25
Bus 5 600 10
In this system, the flows on lines 5-1, 4-1, and 4-3 have been selected as the lines that
constitute a transmission flowgate. The total flow over these three lines is limited to 670
33
MW. There are no flow constraints on single lines. The PTDF values for the transmission
lines of this are calculated using the MATLAB software and are given in table 3.1.
The demand locations are at nodes 1, 2, and 3 of the network and the inverse demand
functions of the three demand nodes are defined by ii DP 04.080?= . As the focus of this
dissertation is on the effect of transmission constraints, the same function is used to
represent the demand at each node. The existence of two producers (duopoly), namely
producer p and q, is considered. Producer p owns the generators at buses 1 and 3, while
producer q owns the generators at buses 4 and 5. These buses are selected such that the
producers are separated by the transmission flowgate. This is done in order to see the
effect on the producers at the different sides of the flowgate (demand side and non-
demand side). It is assumed that the producers may expand their capacity at nodes where
they currently own generators. The capital cost per hour of adding extra capacity is
assumed to be $25 per MW at each node, while the operating costs are given in table 3.2.
The problem is solved using the Cournot generation expansion model explained in
chapter 3.2. The expansion solution obtained is given in table 3.3. The notation WO/TC
(without transmission constraints) and W/TC (with transmission constraints) indicates the
presence or not of transmission constraints in the model. These results indicate that
producer p (which is at the demand side of the flowgate limit) expands more when the
flowgate limit is active, and the expansion is at the same location in both cases. On the
other hand, producer q at the non-demand side of the flowgate limit does not expand any
of its plant capacities when the flowgate constraint is present, independently of the type
of market competition. The profits of the firms for one hour are also given in table 3.3.
Without the transmission constraint, producer q has a much higher profit than producer p.
34
However, the transmission constraint changes the market conditions by decreasing the
producer q?s profit and increasing p?s profit in all cases. These profit figures indicate the
relative effect of a constrained power network on different power producers. Using these
numbers, additional decisions such as transmission capacity investments can be made.
Table 3.3. Cournot Equilibrium Solution
W/O TC W/ TC W/O TC W/ TC
1 295 572
3 0 0
4 0 0
5 425 0
Firm Node Expansion (MW)
p 18,838 26,710
q 37,008 14,985
Profit ($)
In addition to the Cournot solution, for comparison purposes, the 5-bus system is also
solved with the perfect competition equilibrium model. The results are shown in table
3.4. By comparing tables 3.3 and 3.4, it can be seen that the expansion decisions depend
also on the type of market competition. Producer q at the non-demand side of the
flowgate limit does not expand any of its plant capacities when the flowgate constraint is
present, independently of the type of market competition as it cannot sell more because of
the transmission limits. Another observation is that under Cournot competition the
optimal expansion quantities are lower than in perfect competition as a result of the
tendency to withhold production capacity of the Cournot suppliers. As expected, the
profits of the producers are higher with the Cournot competition than with the perfect
competition. Under both competition structures, the transmission limits prevent q to sell
more to the demand side, which decreases q?s profit highly while increasing p?s profit to
35
some extent.
Table 3.4. Perfect Competition Equilibrium Solution
W/O TC W/ TC W/O TC W/ TC
1 0 1775
3 0 0
4 0 0
5 1945 0q 17,000 9,000
Firm Node Expansion (MW) Profit ($)
p 4,910 7,430
Social welfare can be calculated as the sum of the producer profits, consumer surplus,
and the TSO revenue. In figure 3.3, the components that build up the social welfare
amounts in each case are given. The social welfare of the monopoly case, assuming that
all power plants are owned by a single company, is shown also in this figure. Under
monopoly, the producer has a better control on the market and therefore has the ability to
make higher profits. The consideration of a flowgate decreases the total producer profit in
all cases, while increasing the TSO revenue. The increase in total producer profit is more
remarkable in this case while the model is solved under Cournot competition. In addition,
the decrease in the consumer surplus is more significant for the case when the monopoly
assumption is applied.
36
0
20
40
60
80
100
WO/TC W/TC WO/TC W/TC WO/TC W/TC
So
cia
l W
elf
are
(1
04
$
)
Total Producer Profit Consumer Surplus Congestion Rent
Cournot
Monopoly
Cournot
Duopoly
Perfect
Competition
Fig. 3.3. Components of the social welfare
3.3.2. IEEE 24-Bus System
To illustrate the Cournot equilibrium results in a larger system, the model is applied
to the IEEE Reliability Test System [32]. The 24-bus system has 38 transmission lines
with 4 double lines. For simplicity, the double lines have been merged into single lines
with adjusted reactance and thermal limits. As the purpose of this example is to study the
effect of transmission congestion, one forth of the flow limits of the lines given in [32] is
used. The network is illustrated in figure 3.4 and line reactances and limits for the 34
lines are given in table 3.5. The PTDF values for the lines are calculated using MATLAB
software and are given for every node in Appendix A1.
37
Fig. 3.4. 24-bus system network
The demand functions are defined by using the suggested peak demand values in
[32]. The demand intercept is defined by multiplying the peak demand values at each
node by 3, which is selected arbitrarily. The same slope for the inverse demand functions
is used for all demand nodes and it is selected as 0.15. The generator and demand
locations are the same as the original system. Two different firms, p and q are defined.
The operating costs for the existing and new generators are assumed to be 20 $/MWh and
$15 $/MWh respectively, whereas the capital cost per hour of adding extra capacity is
$30 per MW at each node. The generator ownership, capacities, and the price intercepts
of the inverse demand functions are given for each node in table 3.6.
38
Table 3.5. 24-Bus Network Data
1-2 0.014 43.75 11-13 0.048 125
1-3 0.211 43.75 11-14 0.042 125
1-5 0.085 43.75 12-13 0.048 125
2-4 0.127 43.75 12-23 0.097 125
2-6 0.192 43.75 13-23 0.086 125
3-9 0.119 43.75 14-16 0.049 125
3-24 0.084 100 15-16 0.017 125
4-9 0.104 43.75 15-21 0.024 250
5-10 0.088 43.75 15-24 0.052 125
6-10 0.061 43.75 16-17 0.026 125
7-8 0.061 43.75 16-19 0.023 125
8-9 0.165 43.75 17-18 0.014 125
8-10 0.165 43.75 17-22 0.105 125
9-11 0.084 100 18-21 0.013 250
9-12 0.084 100 19-20 0.020 250
10-11 0.084 100 20-23 0.011 250
10-12 0.084 100 21-22 0.068 125
Reactance
(per unit)
Limit
(MW)Line
Reactance
(per unit)
Limit
(MW) Line
Table 3.6. 24-Bus System Data
p q p q
1 192 0 48.6 13 0 591 119.3
2 192 0 43.65 14 0 0 87.3
3 0 0 81 15 0 215 142.7
4 0 0 33.3 16 0 155 45
5 0 0 31.95 17 0 0 0
6 0 0 61.2 18 0 400 149.9
7 300 0 56.25 19 0 0 81.45
8 0 0 76.95 20 0 0 57.6
9 0 0 78.75 21 0 400 0
10 0 0 87.75 22 0 300 0
11 0 0 0 23 660 0 0
12 0 0 0 24 0 0 0
Capacity? Node ?Node Capacity
39
The expansion solutions obtained by solving the Cournot model are given in table
3.7. These results indicate that the existence of the transmission limits have a more
important effect on the expansion decision of producer p than producer q under Cournot
competition. When the results are observed, it can be seen that producer p?s decision is
highly affected when the transmission limits are inserted to the model. On the other hand,
producer q does not need to expand under Cournot competition without transmission
limits, and it needs to expand a small quantity when limits are included.
Table 3.7. 24-Bus Cournot Expansion Solution (MW)
Producer WO/ TC W/ TC
p 244.8 664.5
q 0 55.7
The profits of the firms and other economic analysis information are given in table
3.8. When the profits are observed, it can be said that the existence of the transmission
constraints in the analysis decreases the profits of both firms. In other words, the
transmission limits diminish the producers? ability to manipulate the prices by the
quantities they sell. As a result, the total consumer surplus value (see table 3.8) is higher,
but the total welfare is lower.
40
Table 3.8. 24-Bus Cournot Equilibrium Solution ($)
WO/ TC W/ TC
p 59,078 57,104
q 83,706 72,324
0 21,115
52,536 55,597
179,403 170,786
Metric
Profit
Congestion rent
Consumer surplus
Total Welfare
3.4. Effects of Expansion Decisions on the Network Congestion
In this section, an analysis of how the generation expansion decisions could impact
the congestion of the network is conveyed. The case where transmission constraints are
considered in the expansion decisions and the case where these constraints are neglected
are examined. Since the electricity market balances at real-time and long-term power
purchases are financially binding economic contracts, the schedules will liquidate in the
spot market.
In order to evaluate the performance of the expansion decisions on the network, the
dispatch model (2.9)-(2.15) is solved using the expanded capacity solutions obtained
from the Cournot equilibrium model of the 24-Bus system example. Notice that the
dispatch model is always solved including the transmission constraints regardless of the
expansion model used (WO/TC or W/TC). When the expansion decisions are obtained
including the transmission constraint, the dispatch model gives feasible solutions.
However, when the transmission constraints are excluded in the expansion planning, the
dispatch model could not find a feasible solution. In order to obtain a feasible power
dispatch, the equilibrium load is curtailed. The load curtailments necessary to make the
41
model feasible are given in table 3.9.
Table 3.9. Load Curtailments (MWh)
Node Unmet Demand Node Unmet Demand
1 0 13 0
2 0 14 152.81
3 81.72 15 0
4 0 16 79.02
5 12.78 17 0
6 88.90 18 0
7 0 19 35.37
8 77.57 20 0
9 77.65 21 0
10 91.19 22 0
11 0 23 0
12 0 24 0
After solving the dispatch model using the new capacities obtained by the expansion
models WO/TC and W/TC, the congestion rents, consumer benefits, and consumer losses
are calculated for each solution dispatch. Table 3.10 gives the results for both cases. The
consumer benefits have been calculated using (2.2). The consumer loss is the difference
of consumer benefits before and after the load curtailments.
Table 3.10. Network Economic performance ($)
WO/ TC W/ TC
26,686 0
239,403 276,807
35,032 0
Consumer benefit
Consumer loss
Metric
Congestion rent
42
In order to observe the effects on market prices, averages of the LMPs for both
expansion solutions are shown in table 3.11. The weighted averages LMPs have been
calculated by multiplying the LMPs at each node by the ratio of the load at that node
relative to the total system load.
Table 3.11. Prices ($/MWh)
LMP WO/ TC W/ TC
Average 32.80 20
Weighted Avg. 34.03 20
In table 3.10, when expansion decisions are made using the model W/TC the
congestion rent resulted to be zero, i.e. there is no congestion since it has been relieved
with new generation. Furthermore, the model W/TC provides a higher consumer?s benefit
than the model WO/TC. The reason is that in the latter case the load needed to be
curtailed. From table 3.11, the decision model W/TC produces lower average spot prices
than the model WO/TC.
This example demonstrates that the inclusion of transmission network constraints in
Cournot decision models for expanding generation is crucial. Since when they are
neglected from the decision process, the expansion decisions can lead to obligatory load
curtailments and induce higher congestion charges and higher spot prices.
43
3.5. Conclusion
A methodology for analyzing the effect of system congestion and Cournot
competition on generation expansion decisions in deregulated electricity markets is
proposed. The presented approach considers the behavioral models of the power market
players and the transmission network in the generation expansion modeling. By using the
suggested model, effects of the network transmission constraints on the generation
expansion decision under perfect and imperfect competition are evaluated. Transmission
constraints represented in terms of PTDFs are included in modeling generation expansion
decisions under Cournot competition. It has been shown that transmission constraints
may affect the outcome of competition by limiting expansion decisions of producers
according to where the generators are located relative to the transmission limits. It has
also been shown that if transmission constraints are not considered in planning decisions,
the consumers benefit may diminish due to higher prices and higher congestion charges.
44
CHAPTER 4
POWER GENERATION EXPANSION IN DEREGULATED MARKETS
UNDER UNCERTAINTY
The generation expansion problem involves increasing electric power generation
capacity in an existing power network. Capacity expansion decisions, like in other
investment decision problems, are subject to risk due to future conditions of the system.
To be able to cope with the uncertainties in the investment problem, the uncertainty in the
information available to the decision makers needs to be integrated into the investment
models. In this section, models that integrate the uncertainty in power markets to the
decision of generation expansion are introduced. The aim of the models is to mitigate the
risk involved in the decision of capacity expansion in restructured power markets. Based
on the power market modeling principals and assumptions described in chapter 2 and the
generation expansion framework described in chapter 3, two new generation expansion
models have been developed.
The first model developed for generation expansion under uncertainty in restructured
markets is an optimization model that considers the uncertainty of the information from
the producer point of view. In this new generation expansion model, the producers have
additional information on the sources of uncertainty that could have an effect on
45
generation expansion decisions. Thus, the generation expansion equilibrium model is
built considering this information.
The second model developed for generation expansion under uncertainty considers
the uncertainty to all decision makers in the market. In this generation expansion model,
all market participants have the same information on the data.
In both of the above models, the capacity expansion is represented as a Cournot
competition game whereas the Transmission Owner is assumed to be a Bertrand decision
maker. These assumptions comply with the generation expansion model in the previous
section. Two sources of uncertainty are considered in this research: Uncertainty in power
demand and uncertainty in fuel costs. These two factors have high volatility and directly
affect the firm profits.
4.1. Uncertainty in Power Demand
According to [19] and [24], uncertain demand can be formulated using a set of
demand states with associated probabilities. In order to model the uncertainty in the
demand, possible load scenarios are considered in different demand states. Different
linear demand functions as explained in chapter 2 are used for all demand states in this
formulation. As shown in figure 4.1, these demand functions may have different
intercepts and slopes for all demand states depending on the price sensitivity of the load
at the corresponding network node. At every demand state, the price function is defined
by a different price intercept ?t and demand slope ?t. where t denotes a particular state of
nature.
46
MWh
$/MWh
?tk
?tk
Fig. 4.1. Demand states
In this framework, the inverse demand function for node k at state t can be written as
in (4.1). The inverse demand function is used in the producer decision model to calculate
its expected revenue from the power sales over the different states.
tktktktk DP ?? ?= (4.1)
4.2. Uncertainty in the Fuel Costs
The uncertainty in the fuel costs has much importance as the operational costs of the
power generators are directly affected from changes in fuel costs. To formulate the
uncertainty of fuel costs, T states of fuel costs are used. At each state, the operation cost
of a power generator is affected from the change in fuel cost. Therefore, T states of the
operational cost are included and written as:
47
???? +=
u k
tpuktuk
u k
tpuktuk gCgC
00Cost lOperationa (4.2)
where the first term relates to the existing capacity available and the second term to the
new generation units that will be available after the capacity investment of firm p is
implemented. The summation is for each fuel type at each node. The subscript t denotes
the index of the state.
4.3. Power Generation Expansion Under Uncertainty ? Model 1
Considering the different states of nature, the objective of the power producer is to
maximize its expected profit. The expected profit is calculated by the difference of the
expected revenue and expected cost terms, minus the total investment cost of the
producer.
In this model, equal probability of occurrence is assumed for all possible states of
power demand and fuel costs. Therefore, by using the nodal power price definition given
in chapter 3 and adapting it for the uncertainty model, the expected revenue for producer
p under uncertainty can be written as follows:
( )[ ]( )?? ?? ??
?
?
???
? ?
?
??
?
? ???+?=
?
t k u
tpuk
u
tpuktpkktpkpkttpktktk ggssssTp
01)( ER ??? (4.3)
In the revenue calculation for the producers, the congestion rent that needs to be paid
by producer p needs to be taken into account, thus the transmission prices that are defined
48
by the ISO model are needed. In this model, the transmission price for each node k? is
independent of the states and is defined by the market clearing constraint. In this
formulation the variable tpks denotes the sales of producer p at node k for the realization
of state t, pkts ? refers to the sales of producers other than producer p at state t and node k.
The variables 0tpukg and tpukg denote the generation at state t, of producer p, from fuel
type u, at node k, from existing and new capacities respectively. Similarly, by using the
operational cost formulation explained previously, the expected total generation cost for
producer p from its existing capacity in the network and its new capacity that will be
available after the investment takes place can be formulated as below:
?? ?? ?????? +=
t k u
tpuktuk
u
tpuktuk gCgCTp
001)( EC (4.4)
The coefficients 0tukC and tukC correspond to the operational cost at state t, for
generation unit of fuel type u, at node k, from existing and new capacities. The producer
model?s objective is to maximize the expected profit subject to the capacity constraints
and balance constraints, which needs to be written for each state of demand and fuel cost.
The decision model of a producer p ( P??p ) is as follows:
49
( )[ ]( )
??
?? ?? ??
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
??
?
? ???+?
?
u k
n
pukpuk
t k
u
tpuktuk
u
tpuktuk
u
tpuk
u
tpuktpkktpkpkttpktktk
gI
gCgC
ggssss
T 00
0
1Max ???
(4.5)
00
puktpuk Gg ? T??t U??u B??k )(
0
tpuk? (4.6)
n
puktpuk gg ? T??t U??u B??k )( tpuk? (4.7)
max
puk
n
puk Gg ? U??u B??k )(
n
puk? (4.8)
? ??? ?????? +=
k u
tpuk
u
tpuk
k
tpk ggs
0 T??t )(
tp? (4.9)
0,,, 0 ?? npuktpuktpuktpk gggs (4.10)
The objective function of the producer model represents the difference of the total
expected revenue for producer p and the total investment cost of all new fuel type
capacity investments of producer p which is given by ??
u k
n
pukpukgI . As in chapter 3, the
variables given in parenthesis correspond to the dual variables of the related constraints.
Here, a new capacity investment variable npukg is introduced with additional constraints,
which are maximum capacity installation constraints given by equation (4.8). Notice that
these constraints are independent of the realization of a specific state. In addition, in this
model the generation quantity from the new capacity pukg is limited with the variable
n
pukg denoting the investment amount, which is represented in (4.7). The model optimizes
the sales and generation variables from existing and new capacity while deciding on the
50
capacity expansion amounts.
In addition to the power producers, the TSO?s model and the market clearing
constraints need to be modified under the given state structure. The associated TSO
model has an objective of maximizing the congestion rent by deciding on the expected
transmission quantities, which are defined as )( kyE . The existence of different states of
demand and fuel cost are not taken into consideration from the TSO viewpoint. The TSO
model is then formulated as:
?
k
kk yE )(Max ? (4.11)
)()( +???? ijij
k
k
k
ij ijTyEPTDF ?L (4.12)
)()( ?????? ijij
k
k
k
ij ijTyEPTDF ?L (4.13)
)()( ff
ij k
k
k
ij fFyEPTDF
f
?F
F
????
?
??
?
?? ?
?
(4.14)
The Independent System Operator, on the other hand, balances the expected value of
total power sales to each node, the expected value of the generation quantities from
existing and new capacities, and the transmission values for each node. This balance is
written for every node in the power network. The market clearing constraint becomes:
)()()(11 0 k
p
k
p u
tpuk
u
tpuktpk kyEggTsT ?BP P ??=+?? ? ??
? ?
(4.15)
51
The solution of this problem needs to satisfy the optimality conditions of all three
models for all the market participants as explained previously. The same framework
explained by figure 3.1 needs to be satisfied by an equilibrium solution to the given three
models if a solution exists. The equilibrium solution of these models can be found with a
quadratic programming approach as done in chapter 3. In the quadratic programming
approach, a quadratic programming model that has the same KKT conditions as the KKT
conditions of the models for the different market participants combined is defined. These
models include the models written for all the power producers, transmission network
owner, and independent system operator. The KKT conditions of the generation
expansion problem under uncertainty for each of these entities are given below:
KKTs related to the Power Producer Model:
for tpks : [ ]( ) ,0,021 ????+? ? tpktpkpkttpktktk sssT ???? (4.16)
[ ]( ) 0)21( =??+? ? tpkpkttpktktktpk ssTs ????
for 0tpukg : ( ) ,0,01 000 ???+? tpuktpuktptukk gCT ??? (4.17)
( ) 0)1( 000 =?+? tpuktptukktpk CTg ???
for tpukg : ( ) ,0,01 ???+? tpuktpuktptukk gCT ??? (4.18)
( ) 0)1(0 =?+? tpuktptukktpk CTg ???
52
for npukg : ,0,0 ???+? ? npuknpuk
t
tpukpuk gI ?? (4.19)
0)( =?+? ? npuk
t
tpukpuk
n
pk Ig ??
for tp? : ? ??? ?
?
??
?
? +=
k u
tpuk
u
tpuk
k
tpk ggs
0 (4.20)
for 0tpuk? : 0)(,0, 000000 =??? puktpuktpuktpukpuktpuk GgGg ?? (4.21)
for tpuk? : 0)(,0, =??? npuktpuktpuktpuknpuktpuk gggg ?? (4.22)
for npuk? : 0)(,0, maxmax =??? puknpuknpuknpukpuknpuk GgGg ?? (4.23)
KKTs related to the transmission owner?s model:
for )( kyE : ??? =??+ +?
ij
f
k
ij
ij
ij
k
ij
ij
ij
k
ijk PTDFPTDFPTDFE 0)( ???? (4.24)
for )( ?ijE ? : ,0,)( ??? ?? ij
k
ijk
k
ij TyEPTDF ? ? =+
?
k
ijk
k
ijij TyEPTDF 0))((? (4.25)
for )( +ijE ? : ,0,)( ?? +? ij
k
ijk
k
ij TyEPTDF ? ? =?
+
k
ijk
k
ijij TyEPTDF 0))((? (4.26)
for )( fE ? : ,0,)( ???? ffk
ij k
k
ij FyEPTDF ? (4.27)
0))(( =??? fk
ij k
k
ijf FyEPTDF?
53
KKT related to the market clearing:
for k? : ? ? ??
? ?
=+?
P Pp kp u tpuku tpuktpk
yEggTsT )()(11 0 (4.28)
The KKT conditions of the generation expansion under uncertainty problem (4.16)-
(4.28) form a mixed linear complementarity problem with both equality constraints and
complementarity constraints. These KKT conditions are equivalent to that of the below
generation expansion equilibrium model, which is obtained by adapting the equilibrium
model in chapter 3 for the uncertainty problem. The objective function represents the
total welfare of the system under uncertainty, which is the benefit to the consumers minus
the Cournot producers? effect on the benefit and the operation and investment cost to the
producers. The constraints are equal to the producer model constraints written for all
producers (4.30)-(4.33) and (4.38), TSO model constraints (4.34)-(4.36), and the ISO
constraint (4.37). The equilibrium model for the power generation expansion under
uncertainty ? Model 1 is defined by (4.29)-(4.38).
( )
???
?? ???????
?
?
?
?
?
?
?
?
?
?????
?
?
???
??
p u k
n
pukpuk
t k p u
tpuktuk
p u
tpuktuk
p
tpk
tk
p
tpk
tk
p
tpktk
gI
gCgCsssT 002
2
22
1Max ???
(4.29)
00
puktpuk Gg ? T??t P??p U??u B??k )(
0
tpuk? (4.30)
n
puktpuk gg ? T??t P??p U??u B??k )( tpuk? (4.31)
54
max
puk
n
puk Gg ? U??u B??k )(
n
puk? (4.32)
? ??? ?????? +=
k u
tpuk
u
tpuk
k
tpk ggs
0 T??t P??p )(
tp? (4.33)
)()( +???? ijij
k
k
k
ij ijTyEPTDF ?L (4.34)
)()( ?????? ijij
k
k
k
ij ijTyEPTDF ?L (4.35)
)()( ff
ij k
k
k
ij fFyEPTDF
f
?F
F
????
?
??
?
?? ?
?
(4.36)
)()()(11 0 k
p
k
p u
tpuk
u
tpuktpk kyEggTsT ?BP P ??=+?? ? ??
? ?
(4.37)
0,,, 0 ?? npuktpuktpuktpk gggs (4.38)
Similar to the Cournot equilibrium model in the previous section, the concavity of the
objective function assures a unique solution. The above Quadratic Programming model
can be solved using a commercial quadratic programming solver.
4.4. Power Generation Expansion under Uncertainty ? Model 2
In the previous model, the different states of demand and fuel costs were only
considered in the Cournot capacity expansion game of the producers. In the power
generation expansion under uncertainty ? Model 2 the uncertainty in demand and fuel
costs are brought to the transmission and market clearing models. With consideration of
different states of nature in the TSO and ISO models, the power producer model is
55
modified with the new objective calculated by the different transmission prices for each
state t, which are the dual prices of the modified market clearing constraint. The expected
profit, similar to Model 1, can be calculated by the difference of the expected revenue
and expected cost terms, minus the total investment cost of producer p. Once again, equal
probability of occurrence is assumed for all possible states of power demand and fuel
costs. Therefore, with state dependent transmission prices tk? , the expected revenue for
producer p for Model 2 can be written as follows:
( )[ ]( )?? ?? ??
?
?
???
? ?
?
??
?
? ???+?=
?
t k u
tpuk
u
tpuktpktktpkpkttpktktk ggssssTp
01)( ER ??? (4.39)
The producer model?s objective is to maximize the expected profit for producer p
subject to the capacity constraints and balance constraints, which needs to be written for
each state of demand and fuel cost. The decision model of a producer p ( P??p ) is as
follows:
( )[ ]( )
??
?? ?? ??
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
??
?
? ???+?
?
u k
n
pukpuk
t k
u
tpuktuk
u
tpuktuk
u
tpuk
u
tpuktpktktpkpkttpktktk
gI
gCgC
ggssss
T 00
0
1Max ???
(4.40)
00
puktpuk Gg ? T??t U??u B??k )(
0
tpuk? (4.41)
n
puktpuk gg ? T??t U??u B??k )( tpuk? (4.42)
56
max
puk
n
puk Gg ? U??u B??k )(
n
puk? (4.43)
? ??? ?????? +=
k u
tpuk
u
tpuk
k
tpk ggs
0 T??t )(
tp? (4.44)
0,,, 0 ?? npuktpuktpuktpk gggs (4.45)
In Model 2, the availability of uncertain information on states of demand quantities
and fuel costs is considered in the TSO model and the ISO model in addition to the power
producer game. Therefore, the TSO?s objective function is modified under the given state
structure. The associated TSO model has an objective of maximizing the expected
congestion rent which is calculated, considering T equal probability states, as:
??=
t k
tktkyTECR ?
1 (4.46)
Therefore, the new TSO model can be formulated as below:
??
t k
tktkyT ?
1Max (4.47)
)( +?????? tijij
k
tk
k
ij tijTyPTDF ?T L (4.48)
)( ???????? tijij
k
tk
k
ij tijTyPTDF ?T L (4.49)
)(* tff
ij k
tk
k
ij tfFyPTDF
f
?T F
F
??????
?
??
?
?? ?
?
(4.50)
57
Since there are T different demand and fuel cost states, the balance equation needs to
be written according to these states. The balance of sales, generation, and transmission
values at each node should be satisfied at all different states of nature. Therefore, T
different balance equations for the ISO model are written and the market clearing
constraint becomes:
)()( 0 tk
p
tk
p u
tpuk
u
tpuktpk tkyggs ?T BP P ????=+?? ? ??
? ?
(4.51)
To find the equilibrium solution of these three models, yet again the quadratic
programming approach is applied. With this aim, the KKT conditions of the generation
expansion problem under uncertainty for each of market participants are given below:
KKTs related to the Power Producer Model:
for tpks : [ ]( ) ,0,021 ????+? ? tpktptkpkttpktktk sssT ???? (4.52)
[ ]( ) 0)21( =??+? ? tptkpkttpktktktpk ssTs ????
for 0tpukg : ( ) ,0,01 000 ???+? tpuktpuktptuktk gCT ??? (4.53)
( ) 0)1( 000 =?+? tpuktptuktktpk CTg ???
for tpukg : ( ) ,0,01 ???+? tpuktpuktptuktk gCT ??? (4.54)
58
( ) 0)1(0 =?+? tpuktptuktktpk CTg ???
for npukg : ,0,0 ???+? ? npuknpuk
t
tpukpuk gI ?? (4.55)
0)( =?+? ? npuk
t
tpukpuk
n
pk Ig ??
for tp? : ? ??? ?
?
??
?
? +=
k u
tpuk
u
tpuk
k
tpk ggs
0 (4.56)
for 0tpuk? : 0)(,0, 000000 =??? puktpuktpuktpukpuktpuk GgGg ?? (4.57)
for tpuk? : 0)(,0, =??? npuktpuktpuktpuknpuktpuk gggg ?? (4.58)
for npuk? : 0)(,0, maxmax =??? puknpuknpuknpukpuknpuk GgGg ?? (4.59)
KKTs related to the transmission owner?s model:
for tky : ??? =??+ +?
ij
tf
k
ij
ij
tij
k
ij
ij
tij
k
ijtk PTDFPTDFPTDFT 0
1 ???? (4.60)
for ?tij? : ,0, ??? ?? tij
k
ijtk
k
ij TyPTDF ? ? =+
?
k
ijtk
k
ijtij TyPTDF 0)(? (4.61)
for +tij? : ,0, ?? +? tij
k
ijtk
k
ij TyPTDF ? ? =?
+
k
ijtk
k
ijtij TyPTDF 0)(? (4.62)
for tf? : ,0, ???? tfftk
ij k
k
ij FyPTDF ? 0)( =??? ftk
ij k
k
ijtf FyPTDF? (4.63)
59
KKT related to the market clearing:
for tk? : ? ? ??
? ?
=+?
P Pp
tk
p u
tpuk
u
tpuktpk yggs )(
0 (4.64)
The KKT conditions of this problem also form a mixed linear complementarity
problem with both equality constraints and complementarity constraints. Power
generation expansion under uncertainty ? Model 2 that is solved in this research is the
equivalent quadratic programming problem, which is defined by (4.65)-(4.74).
( )
???
?? ???????
?
?
?
?
?
?
?
?
?
?????
?
?
???
??
p u k
n
pukpuk
t k p u
tpuktuk
p u
tpuktuk
p
tpk
tk
p
tpk
tk
p
tpktk
gI
gCgCsssT 002
2
22
1Max ???
(4.65)
00
puktpuk Gg ? T??t P??p U??u B??k )(
0
tpuk? (4.66)
n
puktpuk gg ? T??t P??p U??u B??k )( tpuk? (4.67)
max
puk
n
puk Gg ? U??u B??k )(
n
puk? (4.68)
? ??? ?????? +=
k u
tpuk
u
tpuk
k
tpk ggs
0 T??t P??p )(
tp? (4.69)
)( +?????? tijij
k
tk
k
ij tijTyPTDF ?T L (4.70)
)( ???????? tijij
k
tk
k
ij tijTyPTDF ?T L (4.71)
60
)(* tff
ij k
tk
k
ij tfFyPTDF
f
?T F
F
??????
?
??
?
?? ?
?
(4.72)
)(1)(1 0 tktk
p u
tpuk
u
tpuk
p
tpk tkyTggsT ?T BPP ????=??
?
?
???
? +? ? ???
??
(4.73)
0,,, 0 ?? npuktpuktpuktpk gggs (4.74)
Notice that in this model, unlike Model 1, the transmission quantity variable tky has
two indexes. It depends on both the state realized and the nodes of the network. Thus,
constraints (4.70)-(4.73) are revised according to the new TSO and ISO models. Notice
that the constraints are again written for all the power producers and that this quadratic
program needs to be solved only once to find the equilibrium solution.
4.5. Numerical Example and Analysis
To study the effect of uncertainties on optimal generation expansion decisions under
competition, the IEEE Reliability Test System [32] is used. The MOSEK solver available
in NEOS server [33], [34], [35] is used to solve the Generation Expansion Equilibrium
Model 1 and Model 2.
As detailed in chapter 3, the 24-bus system has 34 transmission lines with the
merging of the double lines into a single line. The network line reactances and limits for
the 34 lines are given in table 3.5. The PTDF values are given in Appendix A1 and the
full network is described in figure 3.4.
61
The demand functions are defined by using the suggested peak demand values in
[32]. The base price intercept which is assumed to be the current price intercept for each
demand location is defined proportional to the peak demand values at each node. The
yearly growth in power demand is assumed to be uniformly distributed between 1% and
3%, and thus the price intercept of the inverse demand functions are assumed to be
uniformly distributed within this interval. The existing generator locations and demand
locations are the same as the original system. Ten possible states of nature with equal
probability of occurrence are assumed. The demand parameters are generated using the
random numbers generated with help of the Microsoft Excel software. The current price
intercepts, the price intercepts for ten states and the average of the states are shown in
table 4.1.
Table 4.1. Price intercept of inverse demand functions ($/MWh)
1 2 3 4 5 6 7 8 9 10
1 72.9 84.1 83.5 89.5 91.1 83.6 85.4 93.4 83.1 87.2 89.2 87.0
2 65.5 75.6 75.0 80.4 81.8 75.1 76.7 83.9 74.6 78.3 80.1 78.2
3 121.5 140.2 139.2 149.1 151.8 139.3 142.4 155.7 138.5 145.3 148.7 145.0
4 50.0 57.7 57.2 61.3 62.4 57.3 58.5 64.0 56.9 59.7 61.1 59.6
5 47.9 55.3 54.9 58.8 59.9 55.0 56.2 61.4 54.6 57.3 58.7 57.2
6 91.8 106.0 105.1 112.7 114.7 105.3 107.6 117.6 104.6 109.8 112.4 109.6
7 84.4 97.4 96.6 103.6 105.4 96.7 98.9 108.1 96.2 100.9 103.3 100.7
8 115.4 133.2 132.2 141.7 144.2 132.4 135.3 147.9 131.6 138.0 141.3 137.8
9 118.1 136.3 135.3 145.0 147.6 135.4 138.4 151.3 134.6 141.3 144.6 141.0
10 131.6 151.9 150.7 161.6 164.5 150.9 154.3 168.6 150.0 157.4 161.1 157.1
13 178.9 206.5 204.9 219.6 223.5 205.1 209.6 229.2 203.9 213.9 219.0 213.5
14 131.0 151.1 150.0 160.7 163.6 150.2 153.5 167.8 149.3 156.6 160.3 156.3
15 214.0 247.0 245.1 262.6 267.3 245.4 250.8 274.1 243.9 255.9 261.9 255.4
16 67.5 77.9 77.3 82.9 84.3 77.4 79.1 86.5 76.9 80.7 82.6 80.6
18 224.8 259.4 257.4 275.9 280.8 257.7 263.4 288.0 256.2 268.8 275.1 268.3
19 122.2 141.0 139.9 150.0 152.6 140.1 143.2 156.5 139.3 146.1 149.6 145.8
20 86.4 99.7 99.0 106.0 107.9 99.1 101.3 110.7 98.5 103.3 105.8 103.1
Node State AvgBase
62
As in the previous example in chapter 3, the same slope for the inverse demand
functions is used for all demand nodes and it is selected as one of the three values 0.09,
0.15, or 0.18 ($/MWh2) to have different levels of price sensitivity over the demand
states. The slopes of the inverse demand functions for the ten states of nature and the
average of the states are shown in table 4.2.
Table 4.2. Slope of inverse demand functions ($/MWh2)
1 2 3 4 5 6 7 8 9 10 Avg
1 0.15 0.15 0.09 0.09 0.15 0.15 0.18 0.18 0.15 0.15 0.14
2 0.15 0.15 0.09 0.09 0.15 0.15 0.18 0.18 0.15 0.15 0.14
3 0.15 0.15 0.09 0.09 0.15 0.15 0.18 0.18 0.15 0.15 0.14
4 0.15 0.15 0.09 0.09 0.15 0.15 0.18 0.18 0.15 0.15 0.14
5 0.15 0.15 0.09 0.09 0.15 0.15 0.18 0.18 0.15 0.15 0.14
6 0.15 0.15 0.09 0.09 0.15 0.15 0.18 0.18 0.15 0.15 0.14
7 0.15 0.15 0.09 0.09 0.15 0.15 0.18 0.18 0.15 0.15 0.14
8 0.15 0.15 0.09 0.09 0.15 0.15 0.18 0.18 0.15 0.15 0.14
9 0.15 0.15 0.09 0.09 0.15 0.15 0.18 0.18 0.15 0.15 0.14
10 0.15 0.15 0.09 0.09 0.15 0.15 0.18 0.18 0.15 0.15 0.14
13 0.15 0.15 0.09 0.09 0.15 0.15 0.18 0.18 0.15 0.15 0.14
14 0.15 0.15 0.09 0.09 0.15 0.15 0.18 0.18 0.15 0.15 0.14
15 0.15 0.15 0.09 0.09 0.15 0.15 0.18 0.18 0.15 0.15 0.14
16 0.15 0.15 0.09 0.09 0.15 0.15 0.18 0.18 0.15 0.15 0.14
18 0.15 0.15 0.09 0.09 0.15 0.15 0.18 0.18 0.15 0.15 0.14
19 0.15 0.15 0.09 0.09 0.15 0.15 0.18 0.18 0.15 0.15 0.14
20 0.15 0.15 0.09 0.09 0.15 0.15 0.18 0.18 0.15 0.15 0.14
StateNode
As for the different fuel types, different fuel types existing in the system were not
considered in the deterministic equilibrium model in chapter 3 in order to be able to focus
on the transmission constraints and keep the network simple. However, in the uncertainty
model, one of the uncertainty sources is the fuel cost, which can have a huge effect on the
generator operational costs. For that reason, the existing units in the network are
63
separated according to their fuel types based on [32]. The existing generators in the 24-
bus network with their fuel types are given in table 4.3. In this problem, two different
producers, p and q are studied. Producer p is assumed to own the existing generators at
nodes 1 through 13, whereas the producer q is assumed to own the generators at nodes 14
through 24. However each producer is able to invest unlimited capacity at any node in the
network.
Table 4.3. Different fuel types and capacities in the 24-Bus System (MW)
Oil Hydro Coal Nuclear
1 40 152
2 40 152
7 300
13 591
15 60 155
16 155
18 400
21 400
22 300
23 660
Node Fuel Type
To consider the fuel cost uncertainty, fuel price growth is assumed normally
distributed with the means taken from the Annual Energy Outlook 2007 official statistics
of the US Government [36] for the future of power markets. In addition, it is assumed
that two types of existing units are affected from the fuel cost uncertainty, namely coal
and oil units. In addition, the capacity investments are assumed to be made in one of the
coal or gas fuel types. The price of coal and oil is expected to increase with the associate
means, whereas the price of gas is expected to decrease. The parameters of the fuel cost
64
price growth are given in table 4.4.
Table 4.4. Fuel price change parameters
Mean Std
Coal 4.575 0.15
Oil 19.13 0.25
Gas -32.78 0.23
Fuel
type
Growth
As the operational costs of the power generators are directly affected by the changes
in fuel costs, the operational cost data is generated based on the above fuel cost data. The
Microsoft Excel software is used to construct tables 4.5 and 4.6. These tables show the
operational cost of existing and new generators depending on the location in the power
network.
.
Table 4.5. Operational costs of existing generators in the 24-Bus System ($/MWh)
1 2 3 4 5 6 7 8 9 10
Coal
1 13.24 13.84 13.86 13.84 13.81 13.84 13.84 13.82 13.85 13.85 13.83 13.84
2 13.24 13.84 13.86 13.84 13.81 13.84 13.84 13.82 13.85 13.85 13.83 13.84
15 10.42 10.89 10.91 10.89 10.87 10.89 10.89 10.88 10.90 10.90 10.88 10.89
16 10.42 10.89 10.91 10.89 10.87 10.89 10.89 10.88 10.90 10.90 10.88 10.89
23 10.52 11 11.01 10.99 10.97 11 10.99 10.98 11 11 10.99 10.99
Oil
1 32.36 38.52 38.61 38.51 38.41 38.52 38.51 38.45 38.57 38.57 38.47 38.51
2 32.57 38.77 38.86 38.76 38.66 38.77 38.76 38.70 38.82 38.82 38.72 38.76
7 20.7 24.64 24.70 24.63 24.57 24.64 24.63 24.60 24.67 24.67 24.61 24.64
13 20.47 24.37 24.42 24.36 24.30 24.37 24.36 24.32 24.40 24.40 24.34 24.36
15 25.95 30.89 30.96 30.88 30.80 30.89 30.88 30.84 30.93 30.93 30.85 30.89
Hydro
22 10 10 10 10 10 10 10 10 10 10 10 10
Oil
18 5.46 5.46 5.46 5.46 5.46 5.46 5.46 5.46 5.46 5.46 5.46 5.46
21 5.46 5.46 5.46 5.46 5.46 5.46 5.46 5.46 5.46 5.46 5.46 5.46
Current AvgNode Segment
65
Table 4.6. Operational costs of new generators in 24-Bus System ($/MWh)
1 2 3 4 5 6 7 8 9 10
Coal
1 10 10.45 10.45 10.48 10.47 10.45 10.45 10.45 10.48 10.45 10.47 10.46
2 10 10.45 10.45 10.48 10.47 10.45 10.45 10.45 10.48 10.45 10.47 10.46
3 10 10.45 10.45 10.48 10.47 10.45 10.45 10.45 10.48 10.45 10.47 10.46
4 10 10.45 10.45 10.48 10.47 10.45 10.45 10.45 10.48 10.45 10.47 10.46
5 10 10.45 10.45 10.48 10.47 10.45 10.45 10.45 10.48 10.45 10.47 10.46
6 10 10.45 10.45 10.48 10.47 10.45 10.45 10.45 10.48 10.45 10.47 10.46
7 10 10.45 10.45 10.48 10.47 10.45 10.45 10.45 10.48 10.45 10.47 10.46
8 10 10.45 10.45 10.48 10.47 10.45 10.45 10.45 10.48 10.45 10.47 10.46
9 16.1 16.83 16.83 16.88 16.86 16.83 16.83 16.83 16.87 16.83 16.86 16.84
10 16.1 16.83 16.83 16.88 16.86 16.83 16.83 16.83 16.87 16.83 16.86 16.84
11 16.1 16.83 16.83 16.88 16.86 16.83 16.83 16.83 16.87 16.83 16.86 16.84
12 16.1 16.83 16.83 16.88 16.86 16.83 16.83 16.83 16.87 16.83 16.86 16.84
13 10 10.45 10.45 10.48 10.47 10.45 10.45 10.45 10.48 10.45 10.47 10.46
14 10 10.45 10.45 10.48 10.47 10.45 10.45 10.45 10.48 10.45 10.47 10.46
15 10 10.45 10.45 10.48 10.47 10.45 10.45 10.45 10.48 10.45 10.47 10.46
16 10 10.45 10.45 10.48 10.47 10.45 10.45 10.45 10.48 10.45 10.47 10.46
17 10 10.45 10.45 10.48 10.47 10.45 10.45 10.45 10.48 10.45 10.47 10.46
18 17.35 18.13 18.13 18.19 18.17 18.13 18.14 18.14 18.19 18.14 18.17 18.15
19 10 10.45 10.45 10.48 10.47 10.45 10.45 10.45 10.48 10.45 10.47 10.46
20 10 10.45 10.45 10.48 10.47 10.45 10.45 10.45 10.48 10.45 10.47 10.46
21 17.35 18.13 18.13 18.19 18.17 18.13 18.14 18.14 18.19 18.14 18.17 18.15
22 10 10.45 10.45 10.48 10.47 10.45 10.45 10.45 10.48 10.45 10.47 10.46
23 10 10.45 10.45 10.48 10.47 10.45 10.45 10.45 10.48 10.45 10.47 10.46
24 10 10.45 10.45 10.48 10.47 10.45 10.45 10.45 10.48 10.45 10.47 10.46
Gas
1 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
2 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
3 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
4 30.1 20.25 20.19 20.23 20.16 20.33 20.18 20.10 20.13 20.17 20.27 20.20
5 30.1 20.25 20.19 20.23 20.16 20.33 20.18 20.10 20.13 20.17 20.27 20.20
6 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
7 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
8 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
9 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
10 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
11 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
12 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
13 45.8 30.81 30.72 30.78 30.67 30.93 30.70 30.59 30.63 30.69 30.84 30.74
14 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
15 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
16 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
17 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
18 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
19 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
20 45.8 30.81 30.72 30.78 30.67 30.93 30.70 30.59 30.63 30.69 30.84 30.74
21 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
22 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
23 45.8 30.81 30.72 30.78 30.67 30.93 30.70 30.59 30.63 30.69 30.84 30.74
24 35 23.54 23.48 23.52 23.44 23.64 23.46 23.38 23.41 23.45 23.57 23.49
AvgNode Current Segment
66
As stated previously, the investments can be made on two types of units which are
coal and gas. These fuel types are selected as they reflect two cases of generator unit
investments. Coal units have larger investment costs and smaller operational costs. Gas
units on the other hand have relatively smaller investment costs and larger operational
costs. Therefore, coal units can be an example of base-load generation units, whereas gas
units are similar to the peak-load generation units. In table 4.7, the investment costs of
these unit types are shown.
Table 4.7. Investment costs of new generators in the 24-Bus System ($/MWh)
Node Coal Gas Node Coal Gas
1 12.21 4.62 13 12.21 7.2
2 12.21 4.62 14 12.21 4.62
3 12.21 4.62 15 12.21 4.62
4 12.21 3.5 16 12.21 4.62
5 12.21 3.5 17 12.21 4.62
6 12.21 4.62 18 20.1 4.62
7 12.21 4.62 19 12.21 4.62
8 12.21 4.62 20 12.21 7.2
9 18.5 4.62 21 20.1 4.62
10 18.5 4.62 22 12.21 4.62
11 18.5 4.62 23 12.21 7.2
12 18.5 4.62 24 12.21 4.62
The capacity expansion solutions of both producers in both fuel types are obtained by
solving the Generation Expansion under Uncertainty models with the ten states. The
capacity expansion quantities resulting from Model 1 and Model 2 are given in table 4.8.
According to these results, Model 1 and Model 2 give similar results in terms of total
capacity expansion. The results indicate that the producer q chooses to invest in similar
67
capacity quantities to both coal and gas units; however producer p invests mostly in coal
type units. The reason for that is the lower need of producer q for base-load units.
Producer q already owns less expensive units such as hydro and nuclear units, which
have relatively smaller generation costs. Thus, peak-load generators may be more
profitable for producer q after investing a certain amount of capacity on coal units.
Table 4.8. Capacity expansion comparison (MW)
Coal Gas Total Coal Gas Total Coal Gas Total
p 3157 1874 5031 3375 1454 4829 3553 528 4081
q 2344 1812 4156 2105 2121 4226 2192 394 2587
Total 5501 3686 9187 5480 3574 9054 5746 922 6668
Producer Certainty ModelGen. Exp. Model 2Gen. Exp. Model 1
The results of the Generation Expansion problem are compared as in [24] with the
Certainty game (known also as the Expected Value Solution) in order to see the effect of
modeling the uncertainties in generation expansion. For the Certainty game solution, the
averages of the demand quantity and operational costs (given in Avg column of the data
tables) are used over all the states to solve the equilibrium problems described. It can be
observed that in the Certainty game, the investment on gas units are less than the
generation expansion under uncertainty solution of Model 1 and Model 2 for both
producers, whereas the coal investments are not that affected. This shows that when the
Certainty values are used in the model the results indicate less need in peak-load units as
high and low demand states are combined in an average demand state. The capacity
expansion solution results with their locations in the network for all three models are
given in Appendix A2-A4.
68
The profits of the firms and other economic analysis information are given in table
4.9. With the generation expansion model under uncertainty Model 1 and Model 2, it can
be observed that Model 1 produces higher producer profits. Since the information on the
states is not considered in the TSO model, this is reflected on the producer profits. Model
1 and Model 2 producer profits are both larger than the profits obtained in the Certainty
approach. This shows that the uncertainty game can produce more profitable results for
both of the producer players. Similarly, the higher consumer benefit, and total welfare
values given in table 4.9 indicate that the uncertainty model produces more beneficial
results.
Table 4.9. Economic analysis comparison ($)
p
q
Total
226,105 215,875
252,555
16,278 21,067
Consumer Benefit
Congestion Rent
1,107,236 1,050,929
Metric
478,660
241,779
Certainty Model
226,005
248,398
696,957
457,654
Gen. Exp. Model 2Gen. Exp. Model 1
1,102,707
691,255
21,487
Total Welfare
Profit
474,404
667,626
4.6. Evaluation of Expansion Decisions with Cournot Competition Model
In this section, a comparative analysis of the generation expansion decisions made
using the different models is conveyed. Since Model 2 is constructed with the probability
of occurrence of each state of nature, the decisions of all market players are provided by
the model for each state of nature. That is, by solving Model 2 not only the expansion
69
decisions are obtained, but also the sales of the producers, the transmission quantities,
and the prices at each node for all the possible states. However, in Model 1 and the
Certainty game, the expansion decisions need to be evaluated considering the possible
states of nature in order to obtain the market operation decisions of all players. That is, all
possible scenarios sales, transmission amounts, and power prices should be calculated
again using the bidding under competition model. With this aim, the Cournot competition
model presented in [13] is used. This equilibrium model is rewritten using the notation of
this dissertation and given in (4.75)-(4.81). The subscript h indicates the generating
facility index at a certain node.
( ) ( ) ]2][2[Max 22 ?????? ???
??? p h
pkhpkh
p
pk
k
p
pk
k
p
pkk
k
gCsss
PPP
??? (4.75)
subject to
max
pkhpkh Gg ? HBP ???? hkp ,, )( pkh? (4.76)
??? =
k h
pkh
k
pk ss P??p )( p? (4.77)
)( +???? ijij
k
k
k
ij ijTyPTDF ?L (4.78)
)( ?????? ijij
k
k
k
ij ijTyPTDF ?L (4.79)
)( k
p p
k
h
pkhpk kygs ?BP P H ??=?? ? ?
? ? ?
(4.80)
0, ?? pkhpk gs (4.81)
70
In order to evaluate the performance of the expansion decisions on the network, the
Cournot equilibrium model is solved using the expanded capacity solutions obtained
from the generation expansion equilibrium Model 1 and the Certainty game of the 24-Bus
system example and compared to the results obtained from Model 2. The Cournot
competition model is solved using the data of all 10 possible states taking the capacity
expansion solutions of Model 1 and the Certainty game under consideration separately.
Afterwards, the results obtained from the different states are averaged and compared to
those obtained by solving Model 2. These results are represented in tables 4.10-4.15.
Table 4.10. Comparison of the producer profits ($)
p
q
Total
283,848
Profit
552,401
274,699
557,728551,571
Certainty ModelGen. Exp. Model 2
273,880 277,702270,015
281,555
Gen. Exp. Mod. 1
The economic analysis results for the producers are shown in table 4.10. According to
these results it is observed that when the expansion decisions are obtained considering the
different states of nature, generation expansion Model 2 produces the highest total
producer profit.
After solving the Cournot model using the new capacities obtained by the expansion
models, the congestion rents, consumer benefits, and total welfare are also calculated for
each Cournot solution. The results for all three cases are represented in table 4.11. It can
be indicated that the proposed generation expansion model under uncertainty Model 2 is
beneficial as higher consumer benefit and higher overall system welfare can be obtained
71
by using this equilibrium capacity investment model. In addition, it is worth mentioning
that Model 1 and the Certainty Model also produces much higher congestion rents.
Table 4.11. Cournot economic analysis ($)
Gen. Exp. Model 1 Gen. Exp. Model 2Metric Certainty Model
1,053,276
36,608
752,944
33,287 21,487
767,333 774,579
Consumer Benefit
Congestion Rent
Total Welfare
1,066,620 1,102,707
For a more through analysis of the results on the system and the power
consumers, the power prices at every load location, the amounts that are paid by the
consumers and the consumer benefits at the demand nodes, and the transmission revenues
collected for each node of the network are given in tables 4.12-4.15. The Cournot prices
given in table 4.12 give an idea of the different result provided by the three models.
Although in certain demand nodes Model 2 results in higher power prices as a result of
the network constraints, in general terms it can be concluded that lower prices can be
obtained with the expansion solution found by Model 2. This also explains the reason
why the total consumer benefit in table 4.11 is higher for Model 2.
72
Table 4.12. Cournot prices ($/MWh)
1
2
3
4
5
6
7
8
9
10
13
14
15
16
18
19
20
63.50 63.72 66.63
49.59 49.49 50.10
116.66 108.17 119.10
105.36 100.25 107.14
41.37 41.97 43.81
86.03 86.28 81.67
65.55 67.21 67.04
68.91 65.74 70.14
74.64 71.11 76.37
46.14 48.68 45.19
63.80 61.04 62.68
32.45 34.18 34.52
51.31 51.64 52.68
63.55 63.46 65.54
33.26 34.99 35.18
44.51 44.12 43.94
41.35 41.16 41.33
Node Gen. Exp. Model 1 Gen. Exp. Model 2 Certainty Model
Table 4.13 represents the total amounts that need to be paid by each load location in
the network. These amounts show the quantities paid to receive the optimal amount of
power for the three models. However, these amounts are not significant for comparison
as the power quantities purchased by the consumers are different for all three models.
Therefore, it is reasonable that the total amounts paid by the consumers are also different.
73
Table 4.13. Total amount paid by the consumer at each node ($)
1
2
3
4
5
6
7
8
9
10
13
14
15
16
18
19
20
Total
Node Gen. Exp. Model 1 Gen. Exp. Model 2 Certainty Model
13,522 13,658 13,562
10,801 10,878 10,913
36,686 37,995 35,014
6,404
5,951 5,483 5,896
6,495 6,046
20,957 21,627 20,486
18,608 18,681 18,241
32,190
35,373 36,402 33,603
33,610 34,311
44,120 45,117 41,484
81,383 81,471 75,641
41,508
113,147 115,458 106,341
43,219 44,173
10,536 11,368 10,424
127,422 128,534 118,073
35,858
18,053 19,025 18,199
37,624 38,430
657,508 668,659 623,837
In order to be able compare the consumer viewpoint to the three models, consumer
benefit at each load location is calculated and given in table 4.14. Consumer benefit is
defined as the willingness of the load to pay for a certain quantity of power. Therefore,
the consumer benefit better represents the actual advantage of the load from the power
transactions instead of the total amount paid. Depending on the nodal power prices,
Model 2 results in lower consumer benefits in a few demand locations, however; as the
prices obtained by Model 2 are lower in most nodes the consumer benefits are higher,
which is reflected in the total consumer benefits. The consumer surplus values given in
table 4.14 show the difference between what the consumer is willing to pay for the power
and what they actually pay. It is again observed that the highest consumer surplus is
74
obtained by the expansion solution of Model 2.
Table 4.14. Total consumer benefit at each node ($)
1
2
3
4
5
6
7
8
9
10
13
14
15
16
18
19
20
Total
Consumer Surplus 409,111 434,048 429,439
67,941 71,634 66,037
29,402 28,441 29,623
1,102,707 1,053,2761,066,620
Node Gen. Exp. Model 1 Gen. Exp. Model 2 Certainty Model
19,840 20,109 20,045
15,528 15,654 15,484
58,508
9,093 8,158 8,073
59,851 61,565
8,243 7,327 7,253
32,780 33,392 32,223
53,086
53,543 56,636 52,360
52,685 55,129
140,315 140,261 143,229
72,164 72,471 71,345
189,837
15,960 16,549 15,148
191,858 202,284
208,008 221,698 203,462
61,349 62,325 59,225
28,059 29,075 28,336
Finally, how the transmission revenue collected from each node of the network is
analyzed. The transmission revenues collected from each node are given in table 4.15. It
can be observed that these quantities show much difference depending on which model is
used for finding the generation expansion solution. All three models result in a different
dispatch of power through the transmission lines. Thus, the transmission revenue
collected for each node is different for all three models with Model 2 providing the least
congestion-based transmission revenue.
75
Table 4.15. Total transmission price collected at each node ($)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Total
516 26 1,303
330 -158 755
3,106 8
33,287 21,487 36,608
0 0 0
5,976 5,715 8,591
1,847 2,745 4,324
7,179 7,687 10,602
112 25 0
0 -75 0
0 0 0
2,981 1,792 2,273
3,069 1,790 1,868
-464 -2 119
533 278 574
67 4 99
563 79 -11
264 48 133
-43 11 -159
12 15 160
-66 -59 535
Node Gen. Exp. Model 1 Gen. Exp. Model 2 Certainty Model
1,860
125 -7 552
1,160 0 0
6,511 1,573 3,945
-492 -10 -916
4.7. Conclusion
Two new models for finding the generation expansion decisions in deregulated
electricity markets under Cournot competition and uncertainty are proposed. The
presented models consider the behavioral models of the power market players and the
uncertainty in the generation expansion modeling. By using the suggested generation
76
expansion models for a given power system additional information can be added in the
generation expansion modeling process with the utilization of states of nature of demand
quantities and fuel costs. Comparing these models to the Certainty game solution, it has
been shown that modeling the uncertainties in capacity expansion may provide higher
benefits for the market participants and the overall system welfare. It has also been
shown that considering the states in all market participants? models may results in much
lower congestion rents in Cournot equilibrium.
77
CHAPTER 5
CONCLUSION AND FURTHER RESEARCH DIRECTIONS
In this research, models of generation capacity investments in deregulated power
markets were introduced. The first model aimed to analyze the effect of transmission
congestion and Cournot competition on generation expansion decisions in deregulated
electricity markets. Additionaly, two new models were introduced for finding the
generation expansion decisions in deregulated electricity markets under Cournot
competition and uncertainty. The models presented considered the behavioral models of
the power market players and the transmission network in the generation expansion
modeling. In addition, uncertainties in power demand and fuel costs were taken into
consideration in the second model.
By using the suggested models, the effects of the network transmission constraints on
the generation expansion decision can be analyzed under perfect and imperfect
competition. In addition, how decisions are affected by uncertainty can be evaluated with
the help of the generation expansion under uncertainty models.
In the model presented in chapter 3, transmission constraints represented in terms of
PTDFs were included in modeling generation expansion decisions under Cournot
competition. It has been shown that transmission constraints may affect the outcome of
competition by limiting expansion decisions of producers according to where the
78
generators are located relative to the transmission limits. It has also been shown that if
transmission constraints are not considered in planning decisions, the consumer?s benefit
may diminish due to higher prices and higher congestion charges.
In the models presented in chapter 4, the behavioral models of the power market
players and the uncertainty in the generation expansion modeling were both taken into
account. By using the suggested generation expansion models for a given power system
additional information has been added in the generation expansion modeling process with
the utilization of states of nature of demand quantities and fuel costs. Comparing the
generation expansion models under uncertainty and the Certainty game solutions, it has
been shown that modeling the uncertainties in capacity expansion provides higher total
benefits for the market participants and the overall system welfare. It has also been
shown that considering the states in all market participants? models may result in much
lower congestion rents in Cournot equilibrium.
For future research, the sensitivity analysis concept in quadratic programming can be
considered. The equilibrium models presented in this research are convex quadratic
programs and sensitivity analysis on parameters of the models can be conveyed using
optimization theory. In recent theory, it has been proved that in a quadratic program,
sensitivity analysis on the coefficients of the right-hand-side vector and the objective
function results in a partially quadratic objective value function [37]. This property will
be used in further research to obtain a mapping of demand and fuel cost parameters on
the objective value function.
Moreover, in this dissertation transmission expansion has not been taken into
consideration to avoid additional complexity. The transmission network has been
79
accounted for using the current network. In future research, models that relate generation
expansion and transmission expansion decision in deregulated markets will be studied.
Furthermore, in this research the uncertainties in power demand and fuel costs are
modeled using equal probability states. Using probability distributions in competitive
models result in very complicated models, especially under generation capacity and
network constraints. One additional area of research in the future could be to study
modeling generation expansion under deregulation using probability distribution
functions of the sources of uncertainties.
80
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85
APPENDIX
86
Table A1. 24-Bus System PTDF data
Link 1 2 3 4 5 6 7 8
1-2 0.380 -0.563 0.041 -0.304 0.171 -0.171 -0.070 -0.070
1-3 0.360 0.337 -0.085 0.211 0.273 0.220 0.145 0.145
1-5 0.260 0.226 0.044 0.093 -0.445 -0.049 -0.075 -0.075
2-4 0.224 0.252 0.013 -0.385 0.135 0.093 -0.022 -0.022
2-6 0.157 0.185 0.028 0.081 0.036 -0.264 -0.048 -0.048
3-9 -0.161 -0.177 0.181 -0.284 -0.203 -0.230 -0.309 -0.309
3-24 0.521 0.514 0.735 0.495 0.476 0.450 0.454 0.454
4-9 0.224 0.252 0.013 0.615 0.135 0.093 -0.022 -0.022
5-10 0.260 0.226 0.044 0.093 0.556 -0.049 -0.075 -0.075
6-10 0.157 0.185 0.028 0.081 0.036 0.736 -0.048 -0.048
7-8 0 0 0 0 0 0 1 0
8-9 0.036 0.034 -0.012 -0.016 0.067 0.084 0.521 0.521
8-10 -0.036 -0.034 0.012 0.016 -0.067 -0.084 0.479 0.479
9-11 0.070 0.076 0.102 0.180 0.022 -0.003 0.119 0.119
9-12 0.028 0.033 0.079 0.136 -0.024 -0.051 0.071 0.071
10-11 0.212 0.210 0.054 0.117 0.285 0.326 0.202 0.202
10-12 0.170 0.167 0.031 0.073 0.239 0.278 0.154 0.154
11-13 0.006 0.006 0.004 0.007 0.007 0.007 0.007 0.007
11-14 0.275 0.279 0.153 0.291 0.301 0.316 0.314 0.314
12-13 0.080 0.082 0.045 0.085 0.088 0.092 0.092 0.092
12-23 0.117 0.119 0.065 0.124 0.128 0.135 0.134 0.134
13-23 0.087 0.088 0.048 0.091 0.095 0.099 0.099 0.099
14-16 0.275 0.279 0.153 0.291 0.301 0.316 0.314 0.314
15-16 -0.413 -0.419 -0.229 -0.436 -0.451 -0.474 -0.470 -0.470
15-21 -0.066 -0.067 -0.037 -0.070 -0.073 -0.076 -0.076 -0.076
15-24 0.479 0.486 0.265 0.506 0.524 0.550 0.546 0.546
16-17 0.066 0.067 0.037 0.070 0.073 0.076 0.076 0.076
16-19 -0.204 -0.207 -0.113 -0.215 -0.223 -0.234 -0.232 -0.232
17-18 0.054 0.055 0.030 0.057 0.059 0.062 0.061 0.061
17-22 0.013 0.013 0.007 0.013 0.014 0.014 0.014 0.014
18-21 0.054 0.055 0.030 0.057 0.059 0.062 0.061 0.061
19-20 -0.204 -0.207 -0.113 -0.215 -0.223 -0.234 -0.232 -0.232
20-23 -0.204 -0.207 -0.113 -0.215 -0.223 -0.234 -0.232 -0.232
21-22 -0.013 -0.013 -0.007 -0.013 -0.014 -0.014 -0.014 -0.014
87
Table A1. 24-Bus System PTDF data ? Continued 1
Link 9 10 11 12 13 14 15 16
1-2 -0.093 -0.047 -0.058 -0.062 -0.058 -0.046 -0.025 -0.033
1-3 0.108 0.183 0.120 0.128 0.120 0.096 0.053 0.068
1-5 -0.015 -0.136 -0.063 -0.066 -0.062 -0.050 -0.027 -0.035
2-4 -0.087 0.043 -0.018 -0.020 -0.018 -0.015 -0.008 -0.010
2-6 -0.005 -0.090 -0.040 -0.042 -0.039 -0.032 -0.017 -0.022
3-9 -0.371 -0.247 -0.256 -0.272 -0.255 -0.204 -0.112 -0.144
3-24 0.479 0.430 0.376 0.400 0.374 0.300 0.164 0.211
4-9 -0.087 0.043 -0.018 -0.020 -0.018 -0.015 -0.008 -0.010
5-10 -0.015 -0.136 -0.063 -0.066 -0.062 -0.050 -0.027 -0.035
6-10 -0.005 -0.090 -0.040 -0.042 -0.039 -0.032 -0.017 -0.022
7-8 0 0 0 0 0 0 0 0
8-9 -0.057 0.099 0.017 0.019 0.017 0.014 0.008 0.010
8-10 0.057 -0.099 -0.017 -0.019 -0.017 -0.014 -0.008 -0.010
9-11 0.265 -0.028 -0.215 0.004 -0.096 -0.153 -0.063 -0.081
9-12 0.220 -0.078 -0.042 -0.276 -0.160 -0.052 -0.049 -0.063
10-11 0.041 0.362 -0.146 0.077 -0.027 -0.098 -0.033 -0.043
10-12 -0.005 0.312 0.027 -0.203 -0.091 0.003 -0.019 -0.024
11-13 0.007 0.008 0.216 -0.200 -0.431 0.115 -0.002 -0.003
11-14 0.300 0.328 0.423 0.281 0.308 -0.367 -0.094 -0.122
12-13 0.088 0.096 -0.088 0.293 -0.318 -0.064 -0.028 -0.035
12-23 0.128 0.139 0.072 0.227 0.067 0.015 -0.040 -0.052
13-23 0.094 0.103 0.129 0.093 0.250 0.052 -0.030 -0.038
14-16 0.300 0.328 0.423 0.281 0.308 0.633 -0.094 -0.122
15-16 -0.449 -0.491 -0.537 -0.517 -0.539 -0.603 0.142 -0.679
15-21 -0.072 -0.079 -0.086 -0.083 -0.087 -0.097 0.023 -0.109
15-24 0.522 0.570 0.624 0.601 0.626 0.700 0.836 0.789
16-17 0.072 0.079 0.086 0.083 0.087 0.097 -0.023 0.109
16-19 -0.222 -0.242 -0.201 -0.320 -0.318 -0.067 0.070 0.090
17-18 0.059 0.064 0.070 0.068 0.070 0.079 -0.019 0.089
17-22 0.014 0.015 0.016 0.016 0.016 0.018 -0.004 0.021
18-21 0.059 0.064 0.070 0.068 0.070 0.079 -0.019 0.089
19-20 -0.222 -0.242 -0.201 -0.320 -0.318 -0.067 0.070 0.090
20-23 -0.222 -0.242 -0.201 -0.320 -0.318 -0.067 0.070 0.090
21-22 -0.014 -0.015 -0.016 -0.016 -0.016 -0.018 0.004 -0.021
88
Table A1. 24-Bus System PTDF data ? Continued 2
Link 17 18 19 20 21 22 23 24
1-2 -0.031 -0.030 -0.037 -0.046 -0.029 -0.029 -0.050 0
1-3 0.064 0.062 0.078 0.095 0.059 0.061 0.104 0
1-5 -0.033 -0.032 -0.040 -0.049 -0.031 -0.032 -0.054 0
2-4 -0.010 -0.010 -0.012 -0.014 -0.009 -0.009 -0.016 0
2-6 -0.021 -0.021 -0.026 -0.031 -0.020 -0.020 -0.034 0
3-9 -0.136 -0.133 -0.165 -0.201 -0.126 -0.130 -0.221 0
3-24 0.200 0.195 0.242 0.296 0.186 0.191 0.325 0
4-9 -0.010 -0.010 -0.012 -0.014 -0.009 -0.009 -0.016 0
5-10 -0.033 -0.032 -0.040 -0.049 -0.031 -0.032 -0.054 0
6-10 -0.021 -0.021 -0.026 -0.031 -0.020 -0.020 -0.034 0
7-8 0 0 0 0 0 0 0 0
8-9 0.009 0.009 0.011 0.014 0.009 0.009 0.015 0
8-10 -0.009 -0.009 -0.011 -0.014 -0.009 -0.009 -0.015 0
9-11 -0.077 -0.075 -0.076 -0.066 -0.072 -0.074 -0.060 0
9-12 -0.060 -0.058 -0.090 -0.136 -0.055 -0.057 -0.161 0
10-11 -0.041 -0.040 -0.031 -0.012 -0.038 -0.039 -0.001 0
10-12 -0.023 -0.022 -0.046 -0.082 -0.021 -0.022 -0.102 0
11-13 -0.003 -0.003 -0.060 -0.157 -0.002 -0.003 -0.210 0
11-14 -0.115 -0.112 -0.047 0.080 -0.107 -0.110 0.149 0
12-13 -0.034 -0.033 -0.034 -0.033 -0.031 -0.032 -0.032 0
12-23 -0.049 -0.048 -0.101 -0.186 -0.045 -0.047 -0.232 0
13-23 -0.036 -0.035 -0.094 -0.190 -0.034 -0.035 -0.242 0
14-16 -0.115 -0.112 -0.047 0.080 -0.107 -0.110 0.149 0
15-16 -0.482 -0.393 -0.653 -0.607 -0.232 -0.330 -0.582 0
15-21 -0.318 -0.412 -0.105 -0.098 -0.582 -0.479 -0.094 0
15-24 0.800 0.805 0.758 0.704 0.814 0.809 0.675 0
16-17 -0.682 -0.588 0.105 0.098 -0.418 -0.521 0.094 0
16-19 0.085 0.083 -0.805 -0.625 0.079 0.081 -0.527 0
17-18 0.258 -0.598 0.085 0.079 -0.339 -0.105 0.076 0
17-22 0.060 0.010 0.020 0.018 -0.079 -0.416 0.018 0
18-21 0.258 0.402 0.085 0.079 -0.339 -0.105 0.076 0
19-20 0.085 0.083 0.195 -0.625 0.079 0.081 -0.527 0
20-23 0.085 0.083 0.195 0.375 0.079 0.081 -0.527 0
21-22 -0.060 -0.010 -0.020 -0.018 0.079 -0.584 -0.018 0
89
Table A2. Capacity expansion solution of Model 1 (MW)
Fuel Type
Node p q p q
1 63 0 59 0
2 3 0 3 0
3 333 0 246 0
4 0 493 0 824
5 0 482 0 797
6 266 0 210 0
7 181 0 154 0
8 359 0 260 0
9 0 207 0 58
10 0 289 0 66
11 0 0 0 0
12 0 0 0 0
13 529 0 331 0
14 543 0 353 0
15 290 0 219 0
16 0 0 0 0
17 88 0 81 0
18 0 403 0 67
19 217 0 177 0
20 178 0 151 0
21 0 0 0 0
22 0 0 0 0
23 0 0 0 0
24 107 0 97 0
Total 3157 1874 2344 1812
Coal Gas
90
Table A3. Capacity expansion solution of Model 2 (MW)
Fuel Type
Node p q p q
1 97 82 0 0
2 89 76 0 0
3 344 207 17 18
4 103 86 19 20
5 87 75 24 25
6 246 167 3 3
7 166 126 0 0
8 417 232 50 58
9 0 0 237 301
10 0 0 291 370
11 0 0 0 0
12 0 0 16 15
13 561 265 0 0
14 498 253 0 0
15 335 200 194 405
16 117 96 0 0
17 0 0 0 0
18 0 0 441 626
19 155 119 161 279
20 160 122 0 0
21 0 0 0 0
22 0 0 0 0
23 0 0 0 0
24 0 0 0 0
Total 3375 2105 1454 2121
Coal Gas
91
Table A4. Capacity expansion solution of the Certainty model (MW)
Fuel Type
Node p q p q
1 134 110 0 0
2 53 49 0 0
3 316 204 0 0
4 105 90 0 0
5 85 75 0 0
6 248 175 0 0
7 208 155 0 0
8 388 230 0 0
9 0 0 103 88
10 0 0 148 118
11 0 0 0 0
12 0 0 0 0
13 645 290 0 0
14 580 288 0 0
15 254 175 0 0
16 46 43 0 0
17 0 0 0 0
18 0 0 277 188
19 121 98 0 0
20 226 147 0 0
21 0 0 0 0
22 0 0 0 0
23 0 0 0 0
24 142 62 0 0
Total 3553 2192 528 394
GasCoal