ELECTRIC POWER BIDDING MODELS
FOR COMPETITIVE 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 wit my advisory committee. This
dissertation does not include proprietary or classified information.
___________________________
Ahmet D. Yucekaya
Certificate of Approval:
____________________________ ____________________________
Chan S. Park Jorge Valenzuela, Chair
Professor Associate Professor
Industrial and Systems Engineering Industrial and Systems Engineering
____________________________ ____________________________
Gerry V. Dozier George T. Flowers
Professor Dean
Computer Science Graduate School
ELECTRIC POWER BIDDING MODELS
FOR COMPETITIVE MARKETS
Ahmet D. Yucekaya
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
December 19, 2008
iii
ELECTRIC POWER BIDDING MODELS
FOR COMPETIVE MARKETS
Ahmet D. Yucekaya
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
Ahmet D. Yucekaya, son of Husne and Mehmet Yucekaya, was born on December
2, 1978, in Kahramanmaras, Turkey. He attended Istanbul Technical University, Istanbul
where he earned his Bachelor of Science Degree in Industrial Engineering in 2000, and
Master of Science Degree in Strategy Development in 2004. He has worked as a
consultant and a Planning manager in Industry during 2000 and 2004. He enrolled in
Graduate School at University of Pittsburgh in August 2004 where he earned his Master
of Science Degree in Industrial Engineering in 2005. He then enrolled in Graduate School
at Auburn University in August 2005.
v
DISSERTATION ABSTRACT
ELECTRIC POWER BIDDING MODELS
FOR COMPETIVE MARKETS
Ahmet D. Yucekaya
Doctor of Philosophy, December 19, 2008
(M.S., University of Pittsburgh, 2005)
(M.S., Istanbul Technical University, Turkey, 2004)
(B.S., Istanbul Technical University, Turkey, 2000)
121 Typed Pages
Directed by Jorge Valenzuela
Profit maximization for power companies is highly related to the bidding strategies
used. In order to sell electricity at high prices and maximize profit, power companies
need suitable bidding models that consider power operating constraints and price
uncertainty within the market. Therefore, models that include price uncertainty are
needed to analyze the market and to make better bidding decisions.
vi
In this dissertation, the main objective is to develop bidding models for electric power
generators and wholesale power suppliers under price uncertainty. A quadratic
programming model and a nonlinear programming model were developed to find a
solution to the bidding problem. However, in these models the computational time
increases exponentially as the size of the problem increases. To overcome this limitation,
two particle swarm optimization models are developed. The first method uses a
conventional particle swarm optimization technique to find bid prices and quantities
under the rules of a competitive power market. The second method uses a decomposition
technique in conjunction with the particle swarm optimization approach. In addition, a
spreadsheet based simulation algorithm is developed to evaluate a bid offer under given
price samples. It is shown that for nonlinear cost functions particle swarm optimization
solutions provide higher expected profits than marginal cost based bidding. A model to
find an equilibrium solution in competitive power markets for power suppliers bidding
into day-ahead market under forecasted demand is also developed.
vii
ACKNOWLEDGEMENTS
First, I would like to thank Dr. Jorge Valenzuela, my advisor and mentor, for his
invaluable guidance and support throughout this dissertation. I also would like to thank
Dr. Chan S. Park and Dr. Gerry Dozier for their invaluable suggestions. Thanks also to
Dr. Latif Kalin for being my out-side reader. Finally, I would like to express my endless
appreciation to my parents Husne and Mehmet Yucekaya and to my other family
members, for their continuous support and motivation all along my life.
viii
Style manual or journal used Bibliography conforms to those of Institute of Electrical
and Electronics Engineers (IEEE) Transactions
Computer software used AMPL
CPLEX 7.0
MATLAB
MINITAB
Microsoft Visual Studio C/C++
Microsoft Office Excel
Microsoft Office Word
ix
TABLE OF CONTENTS
LIST OF TABLES xii
LIST OF FIGURES xv
NOMENCLATURE xvii
CHAPTER 1. INTRODUCTION 1
CHAPTER 2. REVIEW OF LITERATURE AND BACKGROUND 5
2.1 Literature Review 5
2.2 Background 8
2.2.1 Auction Theory 8
2.2.2 Market Design 10
2.2.3 Locational Marginal Pricing 12
2.2.4 Cost of Generation 13
2.2.5 PJM Power Market 14
2.2.6 Bidding into Market 15
2.2.7 Power Market Equilibrium 19
CHAPTER 3. SPREADSHEET BASED SIMULATION FOR 22
OFFER EVALUATION
3.1 Bid Simulator 22
3.2 Simulation Functionality 23
3.3 Numerical Example 28
x
3.4 Conclusion 31
CHAPTER 4. ELECTRIC POWER BIDDING UNDER PRICE UNCERTAINITY 32
4.1 Price Uncertainty 32
4.2 The bidding model 34
4.3 Quadratic Programming Model 37
4.4 Nonlinear Programming Model 37
4.5 Marginal Cost Biding 39
4.6 Numerical Example and Analysis 40
4.6.1 Quadratic Programming 40
4.6.2 Nonlinear Programming 40
4.6.3 Marginal Cost Bidding 42
4.7 Conclusion 42
CHAPTER 5. PARTICLE SWARM OPTIMIZATION FOR BIDDING INTO 44
MARKET UNDER PRICE UNCERTAINITY
5.1 Rationale for Heuristics Approach 44
5.2 Particle Swarm Optimization 44
5.3 Conventional Particle Swarm Optimization 47
5.4 Decomposition Based Particle Swarm Optimization 49
5.5 dBPSO Genetic Representation 50
5.6 Numerical Example and Analysis 52
5.6.1 Comparison Analysis of cPSO and Marginal 54
Cost based Bidding
5.6.2. Comparison Analysis of cPSO and dBPSO 56
xi
5.6.3. Converging Process of dBPSOB and dBPSOQ 59
5.6.4 Impact of the order of dBPSOB and dBPSOQ 61
5.6 Conclusion 63
CHAPTER 6. AGENT BASED PARTICLE SWARM OPTIMIZATION FOR 65
SUPPLY FUNCTION EQUILIBRIUM
6.1 Introduction 65
6.2 Supply Function Equilibrium Model 65
6.3 Agent Based Modeling and Simulation 69
6.4 Agent based particle swarm optimization model 72
6.5 Numerical Example and Analysis 78
6.5.1 ABPSO Experiment with Duopoly 78
6.5.2 ABPSO Experiment with m=5 firms 83
6.5.3 ABPSO Experiment with m=10 firms 89
6.6 Conclusion 93
CHAPTER 7. CONCLUSION AND FUTURE RESEARCH 94
REFERENCES 96
xii
LIST OF TABLES
Table 2.1. Comparison of offer types 17
Table 2.2. Bid components in offer types 18
Table 3.1. An example bidding solution to the problem 29
Table 3.2. Descriptive statistics for daily profits 29
Table 4.1. Price sample for the example problem 40
Table 4.2. Optimum solution to the problem 40
Table 4.3. Price samples used in the problem 41
Table 4.4. Optimum solution to the problem 41
Table 4.5. Bid prices and quantities for marginal cost bidding 42
Table 5.1. Day-ahead market price samples for PSO 53
Table 5.2. Quantity-price solutions for GEN-1 and GEN-2 55
Table 5.3. Parameter set for the experiment 57
Table 5.4. Statistical summary of the results of the cPSO solutions 57
Table 5.5. Bid prices and quantities for best solutions of cPSO 57
Table 5.6. ANOVA test results for cPSO 58
Table 5.7. Statistical summary of dBPSO solutions 58
Table 5.8. Bid prices and quantities for best solutions of dBPSO 58
Table 5.9. ANOVA test results for dBPSO 59
Table 5.10. ANOVA test results for method comparisons 59
Table 5.11. Statistical summary of dBPSO solutions 63
xiii
Table 6.1. Day-ahead demand for next day 78
Table 6.2. Cost functions and capacities of the firms (m=2) 78
Table 6.3. Equilibrium prices for the first stopping rule (m=2) 79
Table 6.4. Results found in the first stopping rule (m=2) 79
Table 6.5. Equilibrium prices for the second stopping rule (m=2) 80
Table 6.6. Results found in the second stopping rule (m=2) 81
Table 6.7. Equilibrium prices for the third stopping rule (m=2) 81
Table 6.8. Results found in the third stopping rule (m=2) 82
Table 6.9.Overview of the results found in each method (m=2) 82
Table 6.10. Profit increases in each rule (m=2) 83
Table 6.11. Cost functions and market capacities of the firms for (m=5) 83
Table 6.12. Equilibrium prices for first stopping rule (m=5) 84
Table 6.13. Results found for firm 1, 2 and 3 in first stopping rule (m=5) 84
Table 6.14. Results found for firm 4 and 5 in first stopping rule (m=5) 85
Table 6.15. Equilibrium prices for second stopping rule (m=5) 85
Table 6.16. Results found for firm 1, 2 and 3 in second stopping rule (m=5) 86
Table 6.17. Results found for firm 4 and 5 in second stopping rule (m=5) 86
Table 6.18. Equilibrium prices for third stopping rule (m=5) 87
Table 6.19. Results found for firm 1, 2 and 3 in third stopping condition (m=5) 87
Table 6.20. Results found for firm 4 and 5 in third stopping condition (m=5) 88
Table 6.21. Overview of the results found in each method (m=5) 88
Table 6.22. Profit increases in each rule (m=5) 89
Table 6.23. Cost functions and market capacities of the firms (m=10) 89
xiv
Table 6.24. Equilibrium prices for second stopping rule (m=10) 90
Table 6.25. Results found for firm 1, 2 and 3 (m=10) 90
Table 6.26. Results found for firm 4, 5 and 6 (m=10) 91
Table 6.27. Results found for firm 7 and 8 (m=10) 91
Table 6.28. Results found for firm 9 and 10 (m=10) 92
Table 6.29. Unit?s profits (m=10) 92
Table 6.30. Profit increases in rule 2 (m=10) 92
xv
LIST OF FIGURES
Figure 2.1. Services in a typical SMD 10
Figure 2.2. Electric power system 11
Figure 2.3. A generator?s offer curve in the PJM day-ahead market 16
Figure 3.1. Pseudo code of the simulation model 23
Figure 3.2. Bid Simulator main screen 24
Figure 3.3. Bid Simulator menu 24
Figure 3.4. Bid Simulator simulation and price generation 25
Figure 3.5. Bid Simulator hourly profits 26
Figure 3.6. Bid Simulator daily profits 26
Figure 3.7. Bid Simulator output analysis 28
Figure 3.8. Expected hourly price graph from generated prices 29
Figure 3.9. Expected hourly profits based on generated prices 30
Figure 3.10. Histogram of daily profits 30
Figure 3.11. Cumulative histogram of daily profits 31
Figure 4.1. Day-ahead and real time prices for February 29, 2008 33
Figure 4.2. Bid prices and power quantity intervals 35
Figure 5.1. Pseudo code for the PSO method 45
Figure 5.2. Pseudo code for the cPSO method 48
Figure 5.3. Pseudo code for the dBPSO method 51
Figure 5.4. Hourly day-ahead market price samples 53
xvi
Figure 5.5a. Cost function of GEN-1 54
Figure 5.5b. Cost function of GEN-2 54
Figure 5.6. Percentage of increase of cPSO solution in relation to MC 56
Figure 5.7. Converging process of dBPSOB and dBPSOQ to the solution (C=200) 60
Figure 5.8. Converging process of dBPSOB and dBPSOQ to the solution (C=100) 60
Figure 5.9. Converging process of dBPSOB and dBPSOQ to the solution (C=50) 61
Figure 5.10. Converging process of dBPSOB and dBPSOQ to the solution (C=200) 62
Figure 5.11. Converging process of dBPSOB and dBPSOQ to the solution (C=100) 62
Figure 5.12. Converging process of dBPSOB and dBPSOQ to the solution (C=50) 63
Figure 6.1. Equilibrium process in day-ahead power market 66
Figure 6.2. Agent based modeling and simulation 69
Figure 6.3. General flow of a typical ABMS 70
Figure 6.4. Components of ABMS method 73
Figure 6.5. The flow of the ABPSO 74
xvii
NOMENCLATURE
Parameters
i Block number of bidding offer
j Firms in the market
t Time of bidding period
m Number of firms
r Iteration number
N Number of blocks in bidding offer
T Time period that the bid is valid for
R Number of rounds or iterations
Bmax Price Cap or Maximum allowable bid price in the market
Qmax Maximum generation capacity of the generator (MWh)
Qjmax Maximum market capacity of the firm j (MWh)
a1 No-load cost
a2 Linear cost coefficient
a3 Quadratic cost coefficient
aj1 No-load cost for firm j
aj2 Linear cost coefficient for firm j
aj3 Quadratic cost coefficient for firm j
Dt Demand at time t
?j Profit function of firm j
xviii
Pt Market price at time t
k
tP Market clearing price at time t of scenario k ($/MWh)
r
tP Market clearing price at time t of iteration r ($/MWh)
C(q) Cost of generating q MWh ($/MWh)
?1 Stopping condition value for first rule
?2 Stopping condition value for second rule
?3 Stopping condition value for third rule
C Cycle value for particle swarm optimization analysis
R Number of replication for particle swarm optimization analysis
Decision variables
qjt(Pt) Dispatched bid quantity of firm j at time t at market price Pt
q-jt(Pt) Cumulative Bid quantity of competitor?s of firm j at time t at market price Pt
?qi Bid energy amount increase of block i (MWh)
bi Bid price of block i ($/MWh)
qt Total energy generated at hour t (MWh)
?qji Bid energy amount increase of block i of firm j (MWh)
bji Bid price of block i of firm j ($/MWh)
?t Stopping condition criterion value for first rule
t
?
? Stopping condition criterion value for second rule
tw? Stopping condition criterion value for third rule
1
CHAPTER 1
INTRODUCTION
After the 90s, many countries changed the economics of their electricity markets
from monopolies to oligopolies in an effort to increase competition. One of the main
market competition structures used in the new deregulated environments is the poolco
[1]. A poolco market is a central auction that brings regional buyers and sellers together.
All competitive power generators (supply) and buyers (demand) are required to submit
blocks of energy amounts and corresponding prices they are willing to receive from or
pay to the pool. The prices and quantities submitted by the market participants are
binding obligations as they require financial commitments to the market. Once all the
supply and demand bids have been submitted and the bidding period ends, an
Independent System Operator (ISO) ranks these quantity-price offers based on the least-
cost for selling bids and the highest price for buying bids. The ISO then matches the
selling bids with buying offers such that the highest offers are matched with the lowest
selling bids. The point that supply meets demand determines the market clearing price
(MCP).
Perfect competition and oligopoly are two models of interest in the deregulation
of the electricity market. Under a perfect competition model, power suppliers are
expected to bid their marginal costs. For generating units with a nonlinear cost function
(such as coal and gas fired units), the marginal cost depends on the quantity of electricity
2
produced by the unit. This implies the necessity of knowing the amount of energy that
will be offered to the market before the marginal production cost can be computed.
However, if the electricity market is not perfectly competitive, a power supplier may
increase benefits by bidding a price higher than its marginal production costs [2]. This
behavior is called strategic bidding [3]. This strategy imposes the risk of producing no
profit at all if the bid price is too high. There is a risk that the supplier?s bid might be
placed in jeopardy. Thus, the strategic bidding problem (SBP) is to determine proper sizes
and bid prices such as to maximize expected profits.
In both situations described above, the MCP plays an important role in the SBP
since it determines what blocks will be selected by the market clearing mechanism. The
MCP is the result of a complex interaction among producers and consumers. When power
suppliers have the ability to affect the MCP by altering its power output, oligopoly
models such as Cournot [4] or supply function equilibrium [5] are usually adopted.
These models can incorporate detailed economic information about the system, but they
are difficult to solve and they present theoretical problems related to the lack of
equilibrium or the existence of multiple equilibriums. In addition, these models require
suppliers? cost data, which may not be openly available. An alternative approach,
suggested in [6], is to assume that the future values of the MCP are actually unknown by
the market participants since the interaction of processes that governs the MCP is too
complex to model. Thus, the MCP can be modeled as a random variable to represent the
complexity and uncertainty involved in current electricity markets. The advantage of this
modeling approach is that it allows the inclusion of the MCP as an exogenous variable to
the SBP. In [6], it has been shown that when the MCP is assumed to be exogenous each
3
generating unit can be considered separately. Nevertheless, this modeling approach can
be considered as an approximation to the game theoretic method. Its accuracy will
depend in part on whether there is a chance that the generating unit that is bidding into
the market will be the unit that will set the MCP.
In this dissertation, bidding models for two power producer behaviors are
developed. The first set of bidding models assumes that power producers cannot
influence the price, i.e. they behave as price takers; while the second set assumes that
power producers influence the prices with their bids. One contribution of this dissertation
is to model the SBP and find a solution using quadratic and nonlinear programming given
that the power producer has imperfect price estimations. However, an optimal solution
can only be obtained within a reasonable computational time for a limited number of
price scenarios. A second contribution is the development of a spreadsheet based
simulation algorithm to evaluate bids and help power companies with their decision
analysis. A third contribution is the application of heuristic approaches to solve the SBP.
Two models are developed to demonstrate the effectiveness of particle swarm
optimization method to solve the SBP. A fourth contribution is the application of agent-
based simulation method for finding bids when power producers compete for fixed
demand. Numerical results show that the agent based simulation model is capable of
finding an equilibrium solution for each power supplier.
The remainder of this dissertation is organized as follows: Chapter 2 provides a
review of related research currently available about auction designs, bidding processes,
pricing and market equilibrium. Chapter 3 provides details about the developed
spreadsheet based simulation method and its application on bid evaluation. Chapter 4
4
describes the model formulation of the SBP and implemented solution techniques.
Chapter 5 provides a description of the conventional particle swarm optimization (cPSO)
and decomposition based particle swarm optimization (dBPSO) methods. Chapter 6
describes the formulation and model of the proposed market equilibrium model and
developed solution technique. The conclusion and implications for future research may
be found in Chapter 7.
5
CHAPTER 2
REVIEW OF LITERATURE AND BACKGROUND
2.1 Literature Review
Since the 1980s, much effort has been made to restructure the traditional
monopoly of the power industry with the objectives of introducing fair competition and
improving economic efficiency [7]. The electricity supply industry has unique features
such as a limited number of producers, large investment size (which poses a barrier to
entering the market), transmission constraints (which are obstacles for consumers to
effectively reach many generators), and transmission losses (which discourage consumers
from purchasing power from distant suppliers) [7].These features force market players to
be more aggressive on their bidding strategies; it also makes them construct models that
carefully consider their constraints and the uncertainty of the market price.
The deregulated electricity market usually has a few generators (market suppliers)
that usually dominate the market. This makes the market seem more similar to an
oligopoly. In such oligopolistic markets, an individual generator can exercise market
power and manipulate the market price via its strategic bidding behavior [8]. Companies
have to determine bidding strategies so that they can profit even if they are price takers
and do not dominate the market.
6
There are several approaches to analyze and develop bidding models. The bidding
strategies used in the market are discussed in [9] which contains a literature survey of the
current approaches to the bidding problem. The performance of a power market is
measured by the common term ?social welfare?. Social welfare is the benefit of a
commodity to society: to both customers and suppliers.
The game theory approach is commonly used in the literature to model market
participant?s behaviors [10]. The approach assumes that each market player tries to
maximize its profit. The behavior of the market player is affected by other players?
behavior. Several methods used in modeling bidding strategies are explained in [11] as
they compare the game theory approach with the conjectural variation based method. In
both approaches, each firm in the market rationally tries to maximize its profit while
considering the reactions of its competitors. They show that firms can increase their profit
by using conjectural variation based method and the equilibrium found corresponds to
Nash equilibrium.
Several solution methods have been used to solve the bidding problem. In [12], a
genetic algorithm is used to solve the bidding problem. Although the solution obtained is
a heuristic one, it could be used by a company in its daily bidding process. The authors
explain the process of bidding and how the equilibrium price is determined. They
construct their model based on the assumption of exogenous prices. In [13], an
optimization tool to determine a bidding curve for the Ontario Power Market was
developed. The authors used different scenarios of market prices and load. The decision
process for the generator is based on the probability distribution of forecasted prices. The
model chooses the block of the curve (the price and the corresponding quantity) to be
7
submitted in order to maximize expected profits. The model assumes exogenous prices
and includes operational constraints such as ramp-up limits, start-up limits and minimum
up-down times. In [14], bids are represented as quadratic functions of power levels. The
model optimizes the parameters of each function during a two-phase process. In the first
phase, the ISO minimizes the total system cost in which the parameters for other
generators are known. In the second phase, solutions are plugged into the generator?s
model. The Lagrangian relaxation procedure is used to solve the expected cost
minimization problem.
In [15], the authors assume that suppliers bid linear supply functions; the
coefficients of the functions are chosen for each supplier in such a way that the expected
profit is maximized subject to the behavior of one?s rivals. They formulate a stochastic
optimization model and use a Monte Carlo based method to tune the parameters of the
function. They also include the level of information known for each generator in a
symmetric and asymmetric market. In [16], [17] and [18] bidding strategies are
developed for price taker generating units.
The papers mentioned above deal with bidding problems that consist of bids of
one or two blocks. The models developed in this dissertation consider that companies
submit up to ten blocks in their bids and include multiple price scenarios. These
additional features increase the computational time required to solve these problems. In
addition, in some papers the bids are modeled as a price-quantity function. Thus, the
optimization consists of finding the coefficients of this function. In contrast, the price-
quantity is function free in this dissertation.
8
2.2 Background
In order to better analyze the SBP, one should understand the fundamentals of
auction theory, power market, electricity pricing, cost of electricity production and bid
structures. The fundamentals might not be exactly the same for all power markets, but
they are similar. Auction based markets are also used in some other markets such as the
stock exchange, agricultural wholesale, and goods wholesale on the Internet. The solution
approaches developed in this dissertation could also be applied to such markets.
2.2.1 Auction Theory
Auction theory deals with how participants of an endeavor behave in auction
markets where game-theoretic behaviors are involved. The objective of an auction is
generally determined by an operator. It can be the maximization of the outcome (revenue)
like government licenses, or it can be the minimization of the cost like public service by
giving equal market opportunity to each competing player. The players, rules, outcomes
and payoffs of the auction along with their mission might change depending on the
objective, but the same essence of competition remains [19]. An auction design requires
careful research and experiments on efficiency, optimum and equilibrium bidding
strategies and revenue so that an effective auction can be created and manipulations can
be eliminated. It is important that each player does not have perfect information about
their competitors? bid. The operator is supposed to provide a confidential environment to
provide equal opportunity for players [19].
9
Typical auctions are classified by designs (rules). Some examples are first-price
sealed-bid auctions, second-price sealed-bid auctions, open ascending-bid auctions
(English auctions) and open descending-bid auctions (Dutch auctions). In first-price
sealed-bid auctions, bids are submitted simultaneously in sealed envelopes by all bidders
to the operator. The individual with the highest bid wins and pays the proposed amount.
In second-price sealed-bid auctions, bids are submitted in sealed envelopes
simultaneously. The individual with the highest bid wins and pays the amount equal to
the second highest bid. In open ascending-bid auctions, the price is steadily raised by the
operator. Some players drop out as the price becomes too high and the last player wins
the auction at the current price. This is more common in revenue maximizing auctions. In
open descending-bid auctions, the price starts at a relatively high level and is steadily
decreased by the operator until one player is willing to accept the offer. This approach is
more common in cost minimization auctions which often are an issue for public service
providers. Many auctions are hybrids of these four types. Since Electric power auctions
aims to minimize the cost of electricity provided to the market, they are a hybrid of open
descending-bid auctions and first-price sealed-bid auctions [20]. The success of a
competetive market is related to the design of the auction mechanism used [19].
Auctions can also be classified as single-round auctions and multi-round auctions.
In single-round auctions, sell bids are matched with buy bids to reach equilibrium at
once. On the other hand, in multi-round auctions bidders are asked to update their offers
at each iteration/round so that a more effective equilibrium can be reached [21].
10
2.2.2 Market Design
Federal Energy Regulatory Commission proposed a design on July 31, 2002 titled
?Standard Market Design? (SMD) for the standardization of electric power markets in
USA [22], [23]. The major power markets in the US such as PJM, New England
(NEEPOL), New York (NYPX), and Midwest ISO (MISO) are variants of the SMD [19].
The Electric Reliability Council of Texas (ERCOT) and California power market
(CAISO) are in the process of implementing SMD rules as well.
The objective of a typical SMD is to develop a market structure that brings
together the physical system and the economic financial operations. This is achieved by
defining the roles and the interaction of system components. SMD also deals with the
system governance, market operations, risk management, market monitoring and conflict
resolutions that might occur among the members [23], [24]. Figure 2.1 illustrates general
services involved in SMD [24].
ENERGY
Settlements,
Auctions,
Dispatches
MARKET
MONITORING
Price Caps,
Market mitigation
INVESTMENT
Transmission,
Generation
Expansion
TRANMISSION
Congestion,
Transmission,
Hedging
ANCILLARY
SERVICES
Auction
Structure
RETAIL
SERVICES
Demand,
Customers
Figure 2.1. Services in a typical SMD
11
An electric power system consists of four main parts: generation, transmission,
distribution and customers. Figure 2.2 shows the flow of electricity from generation to
customers [24]. SMD governs the processes through scheduling coordinator (SC), power
exchange (PX), independent system operator (ISO), transmission owner (TO) and load
serving entity (LSE). The role of the ISO might slightly change in each SMD, but the
major role remains the same.
Competition
Generation
Transmission
Distribution
Demand
Federally
Regulated (FERC)
State Regulated
Customers
Figure 2.2. Electric power system
The ISO is a neutral entity responsible for maintaining the instantaneous balance
of the grid system [24], [25]. It performs its function by controlling the dispatch of
flexible plants to ensure that loads match resources available to the system. It also
coordinates the day-ahead market and real-time balancing market and monitors
compliance with all regional operating and reliability standards. The members of the ISO
12
are power suppliers, wholesale power customers and, transmission line owners. The day-
ahead market and real-time market are used to equate supply and demand based upon the
sell and buy bids submitted by the members [25]. In a typical SMD, pricing is handled by
a nodal mechanism [24].
2.2.3 Locational Marginal Pricing
Locational Marginal Pricing (LMP) is a market-pricing method that is used to
manage the efficient use of the transmission system when congestion occurs between
source and sink. In a SMD, congestion occurs when restrictions such as capacity of the
line and losses prevent the economic or least expensive supply of energy from serving the
demand. It also means that, if the system was entirely unconstrained and there were no
losses, all the LMPs would be same and it would reflect only the energy price [26], [27].
In an SMD, after offers and bids are submitted and the market is settled, the
LMPs are usually calculated at three types of locations, at the node, the load zone and the
hub. Nodes are the places on the system where generators inject power into the system or
demand (load) withdraws from the system. Each node is connected to one or more buses,
which are specific components of the power system at which generators, loads or the
transmission system are connected. Prices are made up of three components energy,
congestion and losses. The energy component is the cost to serve the next increment of
demand at a specific location or node which should be produced by the least expensive
generating unit in the system that still has available capacity [26], [28]. If the
transmission network is congested, the next increment of energy cannot be delivered from
the least expensive unit on the system because it would violate transmission operating
13
criteria or cause overloading. The extra cost which is called congestion cost is calculated
at a node as the difference between the energy component of the price and the cost of the
providing the additional more expensive energy that can be delivered at that location
[24],[25]. Losses occur during the transmission of power from one location to another
and incurred costs are also included to the calculation.
Generators are paid nodal LMPs and assured by market rules to recover their
costs. Load zones consist of aggregations of nodes in a region. SMD requires a load to
pay the price calculated for that particular zone. The prices calculated for each zone are a
load-weighted average of the nodal prices located within each zone. They are less volatile
than nodal prices since they are aggregated from nodes that reflect the true cost for
delivering power by different location [25].
2.2.4 Cost of Generation
The cost of the energy produced by the generating unit depends on the amount of
fuel consumed and is typically approximated by a quadratic cost
function 2321)( qaqaaqC ++= ($/MWh), where q is the amount of energy generated in
one hour [27],[29],[[30]. The coefficient a1 represents the fixed cost or no-load cost for
each hour. This cost includes the labor and the cost of non-direct goods necessary to
produce power for that hour. The value a2 represents the linear cost which is proportional
to the amount of power produced. The parameter a3 is the quadratic cost coefficient and it
is related with the amount of fuel used to produce electricity [29].
14
Generators use a single cost function when they bid into market but it is also
possible to combine number of cost functions and get an approximated cost function.
Most of the time this approach is used by firms which have several generators and prefer
to bid into the market using portfolio-based cost functions. Generators also have start-up
costs, minimum-load operating costs and minimum up-down constraints. These are also
used by the ISO when the bids are evaluated [31].
2.2.5 PJM Power Market
The PJM interconnection is a federally regulated and nonprofit organization that
manages the transmission of wholesale electricity in Pennsylvania, New Jersey, Maryland
(PJM), Delaware, Illinois, Indiana, Kentucky, Michigan, North Carolina, Ohio,
Tennessee, Virginia, West Virginia and the District of Columbia, involving more than 51
million people. Its dispatching capacity is more than 164,000 MW [32]. PJM?s members,
totaling more than 450, include power generators, transmission owners, electricity
distributors, power marketers and large consumers. PJM?s role as a federally regulated
RTO means that it acts independently and impartially in managing the regional
transmission system and the wholesale electricity market.
The PJM energy market includes two markets ? day-ahead and real-time markets.
In addition to these markets, there is daily capacity market, Monthly capacity market,
fixed transmission rights (FTRs) auction market, regulation market and spinning reserve
market. In the day-ahead market, bilateral transaction schedules, generator offers, and
consumer demands are submitted twelve hours before the actual delivery of electricity.
15
Each installed capacity in the day-ahead market has an obligation to submit an offer to
the market even if it is unavailable or in outage [24], [25]. All participants must submit
bids offers until 12:00 p.m. for the next operating day. The ISO evaluates the bid offers
between 12:00-4:00 p.m. No offer is accepted during this time. PJM announces the
accepted bids at 4:00 p.m. Non-winning participants have the chance to modify their bids
until 6:00 p.m. Demand bids also follow the same process for the day-ahead market [32].
Based on these offers and demands, market clearing prices are determined for
each hour of the next operating day. The day-ahead market is considered a forward
market because the formation of the generation and consumption is determined the day
before the operating day [25]. On the other hand, the real-time energy market balances
the deviations occurred in the day-ahead market and the actual generation. Unlike the
day-ahead market, market clearing prices in the real-time market are calculated every 5
minutes based on the actual system operations. The methods developed in this
dissertation assume that the bids are submitted to the PJM power market.
2.2.6 Bidding into the PJM Market
A generator offer for the PJM market is composed of two components, the price
and quantity of electricity that a supplier is willing to generate. Offers are submitted in
blocks of price quantity pairs. PJM allows submitting at the most ten blocks for a
generator offer [32]. Figure 2.3 illustrates a valid offer curve in PJM power market.
16
0
50
100
150
200
250
300
0 50 100 150 200 250 300
Quantity (MWh)
Pr
ice
($
/M
W
h)
Figure 2.3. A generator?s offer curve in the PJM day-ahead market
Each generating unit also submits its minimum run time, minimum down time,
no-load costs and start-up costs to the PJM market [32].
PJM runs the ?two-settlement? software to determine the hourly commitment
schedules and the LMPs. Generating units that have minimum run times that exceed 24
hours are asked by PJM to submit binding offer prices for the next seven hours. PJM
supports mainly three offer types: Cost-capped offers, historic LMP based offers and
market-based (or price-based) offers. Each unit submits its operating constraints in its
profiles however its usage differs for each offer type [32].
Cost-based offer in the PJM market consists of the incremental operating cost of
the generation resource, plus a 10% and, plus variable operations and maintenance costs.
The production cost method is the method of capping. Generators use no-load cost and
start-up costs as classified hot, intermediate and cold in their offers. Generator offers
consist of three parts, offers for energy ($/MWh), start-up ($/day) and no-load ($/hr).
17
Historic LMP capped offers are determined by calculating the average LMP at the
generation bus during all hours over the past six months in which the resource was
dispatched above minimum. Generators also use start-up cost and no-load cost in their
offers. Table 2.1 shows the summary of the offer characteristics [32].
Table 2.1. Comparison of offer types
Offer Component Cost-Capped Offers LMP-capped offers Market-based offers
Energy offer
($/MWh)
Production cost, Average historic
LMP
No cap, support
up to 10 blocks
Start-up cost
($/day)
Production cost,
plus 10%
Production cost,
plus 10%
No cap
No-load cost
($/hr)
Production cost,
plus 10%
Production cost,
plus 10%
No cap
Price-capped offers are offers that are not necessarily capped with costs. Firms bid
on the market price and get paid the price determined with the respective auction
procedure. The start-up cost and no-load cost are still submitted to the PJM but their
usage is not same as the former two offer types. However, cost components for cost-
capped offers and historic LMP-capped offers directly impact their selection chance since
if cost recovery is not possible for those generator, ISO chooses cheaper offers. Figure
2.7 summarizes the characteristics of three types of offers including differences [32].
18
Table 2.2. Bid components in offer types
Component Cost & LMP-capped offers Market-based offers
Start-up Daily- 3 types Optional- every 6 months
No-load Yes Optional- every 6 months
Maintain Minimum Yes No
Cooling Requirement Basic Optional- every 6 months
Incremental Cost ? Energy from min to max ? Energy from 0 to max
Maximum Offer Based on cap methodology $1000/MWh
Notice that in Table 2.2 there is a cap for start-up and no-load costs for cost-
capped offers, while there is no-cap for market-based offers. This is because market-
based offers bid on the price while cost based offers bid on their cost. PJM audits the
costs if it is necessary. The unit?s offer type is initially set by the generator owner to
indicate whether the unit is to be scheduled as a market-based offer or a cost-based offer.
PJM also imposes the rule if the generator once chooses market-based offer and it cannot
switch to another offer type [32].
SMDs usually use uniform-price auctions and pay-as-bid auctions to govern the
market mechanism. After bids are submitted and the market is settled by the ISO, all
dispatched generators in the uniform price auction are paid the market price where as
they got paid their bid price in pay-as-bid auction. The selection process for winning
generators and the equilibrium price are the same for both designs with the difference that
the generators would make different revenues.
19
The pay-as-bid auction method motivates the lower cost generators to bid too high
to increase the market price. However, in uniform price auctions generators need to bid
lower than market price in order to be selected. Thus, it is accepted that uniform price
auctions generally end up with lower market prices because of price pressure [31].
2.2.7 Power Market Equilibrium
SMD aims to increase competition and hence it is a good place where suppliers
and consumers meet under the supervision of an ISO and economic fundamentals. The
balancing of supply and demand is always crucial in an economic market. However it is
vital for an electricity market since the lack of electricity when needed can cause very
costly consequences. After wholesale power suppliers bid sell offers and wholesale power
customers bid demand offers into the market, the next step is to find an equilibrium point
where supply and demand meet. The price at this point not only sets the market clearing
price but also sets the market clearing quantity that makes social welfare an optimum. A
power market is said to be in equilibrium if i) all suppliers maximize their return at the
given market clearing price and market determined dispatch schedule ii) all consumers
maximize their utility at the given price and schedule iii) the total supply equals total
demand [33].
Equilibrium approaches for power markets are constructed on the behavior of
system participants, which involve the game theory approach. In [34], authors classify the
modeling approaches of power market problems. They use exogenous price and demand-
price function methods to model optimization problems for a single firm.
20
Simulation models that consider all firms behaviors? are classified into equilibrium
models and agent-based models. There have also been efforts to model power market
equilibrium as Cournot, Bertrand and Supply Function Equilibrium (SFE). The SFE has
been of most use in power market modeling [31], [35].
Cournot models have not been found satisfactory for the power markets since
quantity produced by each player, the decision variable, is not responsive to the effects of
price sensitivity. The cournot model also has the assumption that the residual demand is
elastic. But it is not considered as an issue for electricity markets [31], [35]. Cournot
models also expect that player?s output will not change the rival?s output, but the offered
quantity along with its price by the market participant actually affect the rival?s expected
revenue in the market. On the other hand, Bertrand model which has the market price as a
decision variable requires each player to build its strategy on expected market price.
Bertrand leads to perfect competition if there were player with unlimited capacity. But it
is known that as the market converges to equilibrium the players with small capacities
will be out of the game. This strategy also does not cover all market?s issues [31], [35].
Klemperer and Meyer first developed SFE [36] and showed that each firm can
express its decisions in terms of a quantity and a price in the absence of certainty and
having an idea about competitors? strategic variables. It is after that SFE was applied to
power markets by Green and Newbery showing that if a firm expose its decision tool in a
form of supply function indication prices at which it is willing to offer various quantities
to the market for a given demand curve, it can expect in general a greater profit in return
[37]. SFE is more accurate comparing to former mentioned approaches since it reflects
the bidding rules in SMD where players submit price-quantity offers as decision
21
variables. In [38], authors analyze SFE applications by assuming that the functions
provided are linear and there are strategic players in the market with price and capacity
cap. They show that SFE will be more effective than the other approaches in terms of
representation and reaching to the equilibrium. In [39], authors work on SFE by modeling
the market players as non-degreasing supply function providers and competing in a game.
In [40], authors analyze the market power by modeling the equilibrium for large scale
power systems. They briefly explain the equilibrium approaches and show that SFE can
fit best to the power market. In [41], it is showed that if there are multiple players with
identical marginal costs and asymmetric capacities, a unique piece-wise symmetric
supply function exists. The authors show that small players will be eliminated at some
point and larger players will use this advantage in their supply functions. In [31], authors
analyze the equilibrium models in terms of transmission network, generator cost function
and operating characteristics, bidding, demand and uncertainty. These are evaluated under
the umbrella of economic, physical and commercial modeling.
22
CHAPTER 3
SPREADSHEET BASED SIMULATION FOR ELECTRIC
POWER OFFER EVALUATION
3.1 Bid Simulator
In order to evaluate a bidding strategy for given market price scenarios, a
simulation method needs to be developed. The simulation method should include
different price samples and should be able to work for different cost functions. We
develop a simulation model called Bid Simulator in Excel that includes all parameters
needed to evaluate a bidding curve. Bid Simulator is a spreadsheet based simulation
model to assess the value of a given bid using market price samples. It provides
supportive statistical outputs to the decision maker.
The simulation model includes market price scenarios and calculates hourly
profits according to the market prices. The pseudo code for the simulation is given in
Figure 3.1. If the market price at a particular hour is larger or equal to any given price
bid, the supplier would sale power. Otherwise it would not sell power at that hour.
23
Figure 3.1. Pseudo code of the simulation model
3.2 Simulation Functionality
The model is implemented in Excel. The simulation spreadsheet consists of five
modules: main screen, simulation, hourly profits, daily profits and output analysis. Figure
3.2 and Figure 3.3 show the main screen and menu for the model respectively.
Determine bid prices and quantities
Generate N Price Samples each for 24 hours
For each N
For each hour
For each block in bidding curve
If bidding price <= Market price
Calculate hourly profit
Else
Hourly profit=0
Next block
Next hour
Daily Profit = Sum (Hourly profit)
Next sample
Average Profit = Sum (Daily Profit )/N
24
Figure 3.2. Bid Simulator main screen
Figure 3.3. Bid Simulator menu
25
The simulation model generates up to 1000 market price samples for 24 hour, i.e.
a total of 24000 market prices. The price sampling can be defined in terms of a
probabilistic distribution of interest before the simulation. Figure 3.4 shows the
simulation model. The results shown are for a generator whose cost function is
20.004245)( qqqc += and maximum capacity 300 MW. The market prices are
generated using a normal distribution with mean equal to day-ahead prices of April 20th ,
2008 of the PJM market and standard deviation 2 $/MWh.
Figure 3.4. Bid Simulator simulation and price generation
For the hourly profits module, a given bid is evaluated for each hour using the
sampled prices. Figure 3.5 shows the hourly profits.
26
Figure 3.5. Bid Simulator hourly profits
Daily profits are calculated for each day using hourly profits in the daily profits
module of the package, i.e. 1000 profits. Figure 3.6 shows the hourly profits.
Figure 3.6. Bid Simulator daily profits
27
In the output analysis module, the descriptive statistics are calculated for the
hourly generated prices, hourly profits and daily profits. Figure 3.7 shows the output
analysis. The output analysis includes the following components.
Statistics on hourly generated prices:
-Expected market price for each hour
- Expected hourly market price graph
Statistics on hourly profits:
- Expected hourly profit
- Expected hourly profit graph
Statistics on daily profits:
- Minimum daily profit
- Maximum daily profit
- Expected daily profit
- Standard deviation of daily profits
- Variance of daily profits
- 5% and 95% percentile of daily profits
- 5% confidence interval on mean
- Probabilistic distribution of profits
- Histogram of probability density function (PDF)
- Histogram of cdf (cumulative distribution function)
28
Figure 3.7. Bid Simulator output analysis
3.3 Numerical Example
We evaluate an offer using the Bid Simulator and we compute the probabilistic
distribution of profits. The descriptive statistics and graphs mentioned above indicate to
the decision maker how good the bidding solution is. We evaluate the bid given in Table
3.1. Figure 3.8 gives expected hourly prices and Figure 3.9 gives expected hourly profits.
Table 3.2 gives the descriptive statistics. A 90% confidence interval mean profit lies
between $55,934 and the $56,207. Figure 3.10 gives the plot of the histogram and Figure
3.11 gives the cumulative distribution.
29
Table 3.1. An example bidding solution to the problem
Block 1 2 3 4 5 6 7 8 9 10
bi 45.12 45.50 45.75 46.00 46.26 46.51 46.76 47.01 47.26 47.52
qi 30 60 90 120 150 180 210 240 270 300
0
10
20
30
40
50
60
70
80
90
100
1 3 5 7 9 11 13 15 17 19 21 23
Hour
Price ($/MWh)
Figure 3.8. Expected hourly price graph from generated prices
Table 3.2. Descriptive Statistics for Daily Profits
Maximum Profit ($) 64,066.00
Minimum Profit ($) 49,248.06
Expected Profit ($) 56,070.85
Standard Deviation 2,204.29
Variation 4,858,884.50
95% Percentile ($) 59,652.06
5% Percentile ($) 52,335.05
30
0
2000
4000
6000
8000
10000
12000
14000
16000
1 3 5 7 9 11 13 15 17 19 21 23
Hour
Profit($)
Figure 3.9. Expected hourly profits based on generated prices
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
47,248 54,657 62,066
Probability
Profits ($)
Figure 3.10. Histogram of daily profits
31
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
47,248 54,657 62,066
Profits($)
Cumulative Probability
Figure 3.11. Cumulative histogram of daily profits
3.4 Conclusion
In this chapter, the use of simulation to evaluate a bidding curve was explained.
Its effectiveness was showed using a numerical example. A decision maker can use the
simulator to estimate expected profit and variation under the market price uncertainty.
The tool also gives expected profit and price for each hour which can be used as an input
in unit scheduling problems. Especially if an efficient market price forecasting method is
available, Bid Simulator method can be used to help decision maker to reach more
accurate results. Different offers can be compared to further analyze the sensitivity. The
Bid Simulator will be used in next chapters to verify the results found in the analysis.
32
CHAPTER 4
ELECTRIC POWER BIDDING UNDER PRICE UNCERTAINITY
4.1 Price Uncertainty
Electricity is generally accepted as different from other commodities. It is not
storable and its demand is instantaneous so it must be produced and used in real time.
These unique characteristics of electricity and necessity of real time balance create a need
for coordinated markets. As explained in previous chapters, LMP create diversified
market prices by location. The price is strongly load-dependent, highly volatile, seasonal
and consumption dependent [16], [17], [42]. The parameters are stochastic which gives a
stochastic behavior to the electricity price [43]. Energy consumption, fuel costs,
availability of fuels, equipment capacity and market participants? behavior are stochastic
[31], [44],[49]. Figure 4.1 shows an example of day-ahead market prices and real time
market prices for one day in the PJM power market.
33
0
20
40
60
80
100
120
140
160
180
200
0 2 4 6 8 10 12 14 16 18 20 22 24
Hour
Ma
rk
et
Pr
ice
($
/M
wh
)
Day Ahead
Real Time
Figure 4.1. Day-ahead and real time prices for February 29, 2008
In the literature there are many models that are used to forecast market clearing
prices. In [45], the authors develop wavelets and multivariate time series based price
modeling. They analyze the market price data statistically to determine the model
parameters. In [46], the authors use artificial neural networks to model the market price
changes. Neural networks are useful to reflect nonlinear changes that are difficult to
predict otherwise. Automatic dynamic harmonic regression model is used in [47] to
handle the regression among market prices. In [48], the authors develop an algorithm to
calculate the mean and variance of the electricity market price. They also give a
stochastic method for load estimation.
The bidding models that consider the market price exogenous usually include the
electricity price as an input. Market price forecasting methods can be used to determine
the prices used in the model. As an alternative, electricity prices can be generated based
on a probability distribution function.
34
The market price scenarios then can be included into the model and the developed
bidding strategy is the offer that maximizes the expected profit across all scenarios.
4.2 The bidding model
We consider an electric power producer with a set of generating units and assume
that the producer wishes to submit an offer curve to the day-ahead market for each of its
units. We assume that the producer is a price-taker market player that obtains its revenues
by selling power at the market clearing prices of the PJM pool. That is, if the power
supplier produces electricity with its generating units at a particular hour, it is then
willing to take the price prevailing in the market at that hour. We also assume a lack of
market power, i.e. the power supplier does not perceive its decisions as affecting market
prices.
The SBP is formulated under the following additional assumptions:
i) An offer or bid, which consists of N price-quantity blocks at the most, needs to be
determined for each generator separately.
ii) The market clearing prices are considered exogenous to the model, i.e. they are not
affected by the bidding decisions of the unit for which the model is being solved.
iii) The market clearing price at each hour is considered a random variable whose
probability distribution has known parameters.
iv) The offer is determined before the market closes at 12 noon and is valid for the next
twenty four hours, starting at 12 midnight the same day.
35
In finding an optimal offer curve for a particular generator (see Figure 4.2), there
are N pairs of decision variables ib and iq? (i=1,..,N) that need to be determined. Figure
4.2 shows the relations of ib and iq? (i=1,..,N) in the PJM market. The
variable iq? denotes the amount of energy increase to get the bid price ib for delivery at
any hour of the next day. These values are represented by the vectors?q and b,
respectively. If the market clearing price at hour t is equal to or higher than the offered
price ib , then all energy blocks offered at this price or lower are accepted by the market
operator. Thus, the total energy to be produced at time t and sold to the market at a price
Pt is given by:
?
=
?=
)(
1
tPI
i
it qq , where tjt PbjPI ?= such that Max )( for t=1..T; i=1..I(Pt) (4.1)
Figure 4.2. Bid prices and power quantity intervals
36
The objective function is total profit (revenue minus generation cost) over a 24-
hour period. The revenue during hour t is obtained from selling the quantity stipulated
under the offer (only if the generator is dispatched) at the market price, Pt ($/MWh). The
cost includes those of producing the energy. As mentioned above, Pt is a random variable,
and therefore the total profit over a period of T hours is also a random variable. We
assume that K samples of the hourly prices are available and they have equal probability
of occurrence. We denote the price at hour t of sample k as Ptk. Thus, the objective is to
maximize the expected profit over the time period T (usually 24 hours). The bidding
problem, which is called ),P( b?q , is stated as follows:
])([1E[Profit] Max),P(
1 1,
??
= =
?==
K
k
T
t
k
t
k
t
k
t qCqPKb?qb?q for k=1..K; t=1..T; (4.2)
Subject to the following constraints:
max
1
Qq
N
i
i ???
=
(4.3)
max0 Bb
i ?? for i=1..N. (4.4)
max0 Qq
i ??? for i=1..N. (4.5)
?
=
?=
)(
1
ktPI
i
i
k
t qq , ktikt PbiPI ?= such that Max )( for k=1..K; t=1..T; i=1..I(Pt
k). (4.6)
2
321 )()(
k
t
k
t
k
t qaqaaqC ++= for k=1..K; t=1..T. (4.7)
37
4.3 Quadratic Programming Model
Mathematical Programming is one way to find an optimal solution to the bidding
problem. However, it can solve relatively small sized problems. By setting the number of
samples to the one and the number of maximum bidding blocks equal to the number of
hours of the time horizon, a quadratic programming model can be formulated and solved
using a commercial software package such as Cplex. Notice that when the market price
consists of one sample and the number of blocks is equal to the number of hours, the
optimal bidding price of a block of power is equal to one of the market prices. Therefore,
the bidding problem ),P( b?q reduces to the following mathematical representation:
Max Z = ?
=
??
T
t
tttt qaqaqP
1
2
32 ][ for t=1,..,N (4.8)
Subject to
?
=
?=
t
i
it qq
1
for t=1,..,N (4.9)
max
1
Qq
N
i
i ???
=
for i=1,..,N (4.10)
0?? iq for i=1,..,N (4.11)
4.4 Nonlinear Programming Model
Nonlinear Programming (NLP) is the process of solving a problem that includes
equalities, inequalities, constraints and an objective function some of which is nonlinear.
38
The process finds a set of unknown real variables that makes the objective function
maximized or minimized.
The bidding model is formulated as follows:
])([1E[Profit] Max),P(
1 1,
??
= =
?==
K
k
T
t
k
t
k
t
k
t qCqPKb?qb?q (4.2)
Subject to the following constraints:
max
1
Qq
N
i
i ???
=
(4.3)
i
k
t
k
t bPiMz ?? 001.1)( for i = 1..N ; t = 1..T ; k = 1..K. (4.12)
i
k
t
k
t bPizM ??? )1)(( for i = 1..N ; t = 1..T ; k = 1..K. (4.13)
0)1()( ?+? iziz ktkt for i = 1..N ; t = 1..T ; k = 1..K. (4.14)
1+? ii bb for i=1..N. (4.15)
max0 Bb
i ?? for i=1..N. (4.4)
max0 Qq
i ??? for i=1..N. (4.5)
ii qMr ???1 for i=1..N. (4.16)
)1(max ii rQq ??? for i=1..N. (4.17)
1+? ii rr for i=1..N. (4.18)
for i = 1..N ; t = 1..T ; k = 1..K. (4.19)
and (4.7).
?
=
?=
N
i
i
k
t
k
t qizq
1
)(
39
Note that M is a large number here, and z and r are binary variables. When we
model the problem in AMPL and solve it using the NEOS solver MINLP, an optimal
solution can be found for a limited number of price samples.
4.5 Marginal Cost Bidding
A power producer also can submit its marginal cost of production as its bid offer.
As a matter of fact, in a perfectly competitive market it is expected that each player
submits its marginal costs. To do so, a power producer could offer, for each generating
unit, one energy block consisting of the maximum capacity and price equal to the
marginal cost of producing this amount. Alternatively, the power supplier could split the
maximum capacity into N blocks of identical size and offer them at prices equal to the
marginal costs of producing each block. PJM accepts a maximum of ten energy blocks in
its daily bidding process, so maximum capacity can be split into 10 blocks and marginal
cost of these quantities can be offered to the market.
])([1E[Profit] ),P(
1 1
??
= =
?==
K
k
T
t
k
t
k
t
k
t qCqPKb?q for k=1..K; t=1..T; (4.2)
Subject to the following constraints:
N
Qq
i
max
=? for i=1..N. (4.20)
k
ti qaab 32 2+= for k=1..K; t=1..T; i=1..N. (4.21)
(4.4), (4.6) and (4.7).
40
4.6 Numerical Example and Analysis
We now present results obtained with the developed models. We use CPLEX for
solving the quadratic programming model, NEOS MINLP for solving the nonlinear
programming problem.
4.6.1 Quadratic Programming Model
In order to solve the quadratic programming problem we use the generator used in
Chapter 3 whose cost function is 20.004245)( qqqc += and maximum capacity 300
MW. The time horizon is set to ten hours. The ten hourly market prices are given in Table
4.1.
Table 4.1. Price sample for the example problem
Hour 1 2 3 4 5 6 7 8 9 10
Price($/MWh) 35.20 36.80 39.30 45.00 45.50 45.90 46.10 46.80 47.10 50.20
After solving the above model using Cplex 7.0, the optimal profit is found to be
$1772.48. The optimal solution to the problem is given in Table 4.2.
Table 4.2. Optimum solution to the problem
Block 1 2 3 4 5 6
bi ($/Mwh) 45.50 45.90 46.10 46.80 47.10 50.20
qi (Mwh) 59.52 107.14 130.95 214.28 250.00 300.00
4.6.2 Nonlinear Programming Model
In order to solve the bidding problem under market price uncertainty, we use the
same generator used in section 4.6.1 but the problem is solved for three day price
41
samples. The problem was coded in AMPL and submitted to the one of the NEOS Servers
MINLP to get a solution. Table 4.3 gives the price samples used in the model and Table
4.4 provides the optimum solution.
Table 4.3. Price samples used in the problem
Hour 1 2 3
1 49.21 49.58 47.14
2 47.56 46.78 46.52
3 50.44 40.68 46.1
4 43.48 44.59 46.59
5 41.97 46.87 46.16
6 43.78 48.8 49.83
7 47.25 44.27 45.56
8 57.28 54.37 55.42
9 57.56 53.37 53.29
10 58.59 52.87 55.02
11 51.66 48.85 46.39
12 45.54 48.36 49.36
13 40.14 38.55 42.55
14 37.39 40.44 38.28
15 37.52 42.54 38.82
16 39.45 34.4 37.2
17 40.64 44.53 40.38
18 53.45 53.41 53.71
19 79.44 77.53 75.92
20 72.7 75.02 74.5
21 71.83 70.09 67.66
22 60.39 62.78 64.41
23 48.94 50.19 50.53
24 40.8 44.97 45.43
Table 4.4. Optimum solution to the problem
Block 1 2 3 4 5 6 7 8 9
bi ($/MWh) 45.01 45.54 46.10 46.16 46.39 46.52 46.56 46.87 47.56
qi (MWh) 51.19 65.47 130.94 138.08 165.46 180.93 211.88 222.59 261.28
42
The objective function of the optimum solution was $44,779.83. However, it takes
about 5 hours to solve the problem with 3 price samples and 300 Mwh capacity. If we
increase the capacity to 1500 Mwh and try to solve the problem with same price samples,
we could not find an optimal solution after 24-hour run. Results show that it is not likely
to solve the problem with more than 3 price samples.
4.6.3 Marginal Cost Bidding
The Marginal cost bidding model requires splitting the maximum capacity into
equal block sizes. We solve the problem using the same generator with the price samples
given in Table 4-3. Table 4.5 gives the bid prices and quantities for marginal cost bidding.
Table 4.5. Bid prices and quantities for marginal cost bidding
Block 1 2 3 4 5 6 7 8 9 10
bi ($/MWh) 45.25 45.50 45.75 46.00 46.26 46.51 46.76 47.01 47.26 47.52
qi (MWh) 30 60 90 120 150 180 210 240 270 300
Marginal cost model is evaluated for both price samples given in Table 4.1 and
4.3. The profit found for 10-hour price sample is $1766.58 where the optimum solution is
$1,772.48. The profit found in for 3 day price samples is $44,750.86 where the optimum
solution is $44,779.83.
4.7 Conclusion
In this chapter the strategic bidding model was defined and possible solution
approaches and their limitations were explained. One can solve a problem with 10 hours
of market prices using quadratic programming. We also showed that it is possible to find
43
an optimum solution using 3 days price samples for a generator with 300 Mw capacity. It
is clear that a more effective method is needed to solve the problems with more than 3
days price samples and more generating capacity. The solution method should require
low computational time since the bidding process is done daily. It might be that in same
cases there are not much profit difference between bidding the marginal cost and
optimum solution. We will show in Chapter 5 that this is not always the case.
44
CHAPTER 5
PARTICLE SWARM OPTIMIZATION FOR BIDDING
INTO MARKET UNDER PRICE UNCERTAINITY
5.1 Rationale for Heuristics Approach
Heuristic approaches are commonly used when it is not possible to find optimum
solutions to the problems, or when it takes much computational time to find optimum
solutions. The approaches are generally accepted as easy to implement, easy to apply to
the problems and require less computational time. They do not guarantee an optimum
solution though. But in cases where an optimum solution and a good solution don?t make
much difference, heuristics approaches are preferred because of the less effort that they
require. It is possible to find a solution using more than 3 days price samples using
heuristics approaches whereas nonlinear programming could not find an optimum
solution. We will show in this chapter that it is possible to find a good solution for every
generator regardless of generator capacity. It will also be showed that the required
computational time is dramatically less than the nonlinear programming method.
5.2 Particle Swarm Optimization
Particle Swarm Optimization (PSO) is a computation technique, introduced by
Kennedy and Eberhart in 1995 [50], which was inspired by social behavior of bird
45
flocking or fish schooling. Like genetic and evolutionary algorithms, PSO is a
population-based search method, i.e. it moves from a set of points (particles? positions) to
another set of points. The particles move through a D-dimensional space and each
particle has a velocity that acts as an operator to obtain a new set of individuals. The
particles adjust their movements depending on both their own experience and the
population?s experience.
The following pseudo code describes the PSO approach:
Figure 5.1. Pseudo code for the PSO method
(The symbol ? denotes the multiplication of two vectors component by component.)
46
At each iteration a particle moves in a direction computed from its best visited
position and the best visited position of all the particles in its neighborhood. Among the
several variants of PSO, the global variant considers the neighborhood as the whole
population, called the swarm, which enables the global sharing of information. The basic
elements of the PSO technique are particle, population, velocity, inertia weight,
individual best, global, learning coefficients, and stopping criteria best [51]. These are
briefly discussed below.
I. Particle, Xj(t): a particle j represents an m-dimensional vector candidate solution. The
value of m is determined by the number of decision variables. At time t the particle j
can be described as Xj(t) = [x1,j,?, xm,j] where the x components are the decision
variables. A value of xi,j denotes the position of particle j in the ith coordinate in the
search space, i.e. the value of the ith decision variable in the candidate solution j.
II. Population, POP(t): The population is a set of n particles at any given time t and it can
be represented as POP(t)=[ X1(t) ?, Xn(t)].
III. Velocity, Vj(t): The velocity of moving particles at time t represented by an
m-dimensional vector Vj(t) = [v1,j,?, vm,j].
IV. Inertia weight, w: The parameter that controls and directs the impact of the previous
velocities on the current velocity. If the inertia weight is large, the search becomes more
global, while for smaller w the search becomes more local.
V. Individual best, )(* tjX : When a particle flies through the search space it compares its
fitness value at the current position to the best fitness value so far. The best position
visited by the particle i at time t is denoted by ],...,[)( *,*,1* jmjj xxt =X .
47
VI. Global best, G(t): Represents the best position that gives the best fitness among all
individual best positions achieved so far. It is defined by G(t) = [g1,?, gm].
VII. Learning factors, c1 and c2: these coefficients help particles to accelerate towards better
areas of the solution space.
VIII. Stopping Criteria: represent the conditions for which the search process will terminate
and lead to a result.
5.3 Conventional Particle Swarm Optimization
The conventional PSO approach (cPSO) is used for solving the whole problem.
The population size is set to ? particles. Decision variables bi and iq? are evaluated
using the price scenarios in fitness function below.
])([1]Profit[EMax
1 1
??
= =
?=
K
k
T
t
k
t
k
t
k
t qCqPK for k=1..K; t=1..T; (5.1)
Subject to:
max0 Bb
i ?? for i=1..N. (5.2)
maxQqk
t ? for k=1..K; t=1..T (5.3)
?
=
?=
)(
1
ktPI
i
i
k
t qq , ktikt PbiPI ?= such that Max )( for k=1..K; t=1..T; i=1..I(Pt
k). (5.4)
2
321 )()(
k
t
k
t
k
t qaqaaqC ++= for k=1..K; t=1..T. (5.5)
1+? ii bb for i=1..N (5.6)
48
)(penalty]))(()([1))(),((P of Fitness
1 1
jjqCjqPKjj
K
k
T
t
k
t
k
t
k
t +?= ??
= =
?qb (5.7)
Where
??
??? >?>= ?
=
Otherwise 0
)(or )( if )(penalty
1
maxmax
N
i
ii QjqBjbMj (5.8)
Notice that the function penalty (j) is used to penalize the objective function due
to the violation of the maximum bid price and/or the maximum available capacity of the
generating unit. Penalty functions are generally used to eliminate solutions that violate
constraints. The following pseudo code describes the main procedure:
Figure 5.2. Pseudo code for the cPSO method
Main Procedure
Randomly generate q? , b
Set q?=*?q and b=*b
While Run < NRuns
Run PSO and obtain solution?q, b
If fitness of P(?q,b) > fitness of P( *?q , *b ) then
Set ?q?q =* and bb =*
Endif
EndWhile
Output *?q and *b as the best solution of P
49
5.4 Decomposition Based Particle Swarm Optimization
The decomposition-based PSO (dBPSO) consists of separating P into two sub-
problems. One of the sub-problems assumes that the values of the decision variables bi
are known for each i. This sub-problem is called PQ. Using the given values of bi?s and
price scenarios, the following variable is computed as in (5.4):
k
ti
k
t PbiPI ?= such that Max )( for k=1..K; t=1..T. (5.9)
Thus, the formulation of PQ is as follows:
])([1]Profit[EMax
1 1
??
= =
?=
K
k
T
t
k
t
k
t
k
t qCqPK?q for k=1..K; t=1..T; (5.10)
Subject to: (5.3), (5.4), (5.5) and
0?? iq for i=1..N (5.11)
The second sub-problem assumes that the values of the decision variables iq? are
known for each i. The objective of this optimization problem is to find the values of bi
that maximize the firm?s profits. This sub-problem is called PB.
])([1]Profit[EMax
1 1
??
= =
?=
K
k
T
t
k
t
k
t
k
t qCqPKb for k=1..K; t=1..T; (5.12)
Subject to: (5.2), (5.3), (5.4), (5.5), and (5.6) constraints.
50
5.5 dBPSO Genetic Representation
The dBPSO is used for solving both sub-problems, PQ and PB. The sub-problem
PB is solved first using the algorithm dBPSOB and its solution is then used to solve PQ
using the algorithm dBPSOQ. Then the new solution of PQ is used to re-solve PB. This
process is applied successively until no improvement is observed after two iterations. For
both problems, the population size is set to ? particles. To initialize the population of
dBPSOQ, we proceed as follows: we first sample ?q1 as the amount at which the marginal
production cost is equal to the given bid price b1 plus a random uniformly distributed
quantity in the interval [-Q, Q], where Q is a user defined parameter. Thus,
),(2 )(
3
21
1 QQUa
abq ?+?=? (5.13)
and all other values ?qi (for i=2,..,N) are sampled as
)1,0(2 )(
3
1 U
a
bbq ii
i
??=? . for i=2,..,N (5.14)
Similarly, we use the marginal cost function to initialize the population of dBPSOB.
),(2 1321 BBUqaab ?+?+= (5.15)
)1,0(2 3 Uqab ii ?= for i=2,..,N (5.16)
Since the objective functions of P, PQ, and PB are identical, we define the same
fitness function for each problem. When b(j) and )(j?q denote a particular solution j
(again a particle is not necessarily feasible), the fitness function is calculated as in (5.4),
(5.5), (5.7) and (5.8).
51
Notice that the same penalty function penalty (j) is used to penalize the objective
function due to the violation of the maximum bid price and/or the maximum available
capacity of the generating unit. The following pseudo code describes the main procedure:
Figure 5.3. Pseudo code for the dBPSO method
In this procedure, the initial values of ?qare randomly selected from a uniform
distribution as follows:
?q1 = U(0, Qmax/N) (5.17)
?qi = U(qi-1, Qmax) ? q i-1, where ?
=
?=
i
k
ki qq
1
for i=2,..,N (5.18)
Main Procedure
Set 0?q =* and 0b =*
Randomly generate ?q
NoImprovement = 0
While NoImprovement < 2
Run dBPSOB and obtain solution b
Using this b as input, run dBPSOQ and obtain solution ?q
If fitness of P(?q,b) > fitness of P( *?q , *b ) then
Set ?q?q =* and bb =*
NoImprovement = 0
Else
NoImprovement = NoImprovement + 1
Endif
EndWhile
Output *?q and *b as the best solution of P
52
5.6 Numerical Example and Analysis
The cPSO and dBPSO methods were coded in C. We run the methods using c1=2,
c2=2, w=0.5 [52], and ?=30 [51]. We compute the quantity-price offers for two different
generators say GEN-1 and GEN-2. The maximum capacity of GEN-1 is 400 MW and its
cost function is equal to 56.52q + 0.0139q2 ($/MWh). This cost function is obtained by
multiplying the heat-input function of the generator given in [29] by the current fuel cost
of oil-fired units (7.2 $/MBtu) [52]. Similarly, the maximum capacity of GEN-2 is 600
MW and its cost function is equal to 43.2q + 0.108q2 ($/MWh). This cost function is
obtained by multiplying the function of the generator given in [53] by 7.2 as well. In both
cases, the aim is to have the marginal cost of the unit within the range of the sampled
prices. Otherwise, the solution of the SBP would be trivial.
In market price generation procedure, we first choose the date of May 17th, 2007,
arbitrarily, and use the PJM day-ahead market prices to produce twelve 24-hour price
scenarios. At each hour twelve prices are generated by sampling values from a normal
distribution with mean equal to the price at that hour of May 17th and a standard deviation
equal to 4 ($/MWh). For example, for hour 1 AM twelve prices are sampled from a
normal distribution with mean 27.14 ($/MWh) and standard deviation 4. The price
scenarios for all twenty four hours are given in Table 5.1. To show the variability among
samples and the hourly fluctuations in the market price, we plot the prices in Figure 5.4.
53
Table 5.1. Day-ahead market price samples for PSO
Sample
Hour 1 2 3 4 5 6 7 8 9 10 11 12
1 29.36 26.54 29.47 25.30 25.09 31.33 24.26 27.05 23.79 25.17 28.28 26.59
2 24.62 24.04 24.26 24.74 26.44 31.88 28.00 26.45 21.09 24.37 24.28 24.87
3 26.09 21.12 25.66 20.57 20.29 21.28 18.79 20.80 21.50 23.20 21.96 22.31
4 21.04 16.80 19.94 21.71 23.34 21.25 22.54 21.38 19.46 21.54 18.82 20.58
5 24.16 24.30 21.24 18.89 22.24 20.51 18.20 21.63 18.25 22.61 25.04 24.29
6 24.89 23.56 25.21 26.88 25.16 25.70 25.15 27.25 25.60 23.45 27.76 23.77
7 36.87 33.02 33.47 34.77 33.24 36.83 31.67 34.86 32.51 31.01 35.13 35.82
8 43.27 46.09 44.46 44.99 44.43 44.75 46.92 43.09 47.58 46.53 47.04 48.78
9 50.33 48.31 45.03 47.86 47.85 45.83 48.64 48.90 48.45 48.01 47.12 47.92
10 50.09 51.61 54.90 54.44 52.08 55.02 52.34 51.52 54.34 50.53 50.23 50.53
11 58.65 54.42 56.90 58.72 57.59 52.62 56.27 56.17 58.57 56.40 53.20 56.75
12 58.97 57.09 57.96 57.88 56.81 57.02 58.60 56.35 54.99 63.38 56.13 59.27
13 62.51 55.20 59.87 56.92 59.85 59.02 54.83 57.97 56.97 58.22 57.82 59.38
14 60.64 63.48 61.66 55.30 60.85 58.96 58.68 60.22 58.82 64.52 58.21 61.70
15 59.63 59.17 62.39 60.45 59.99 60.60 59.86 60.68 63.71 62.35 59.85 59.16
16 56.70 58.07 59.67 58.83 61.24 61.37 56.58 61.36 60.53 58.92 63.00 61.66
17 65.77 61.57 65.12 63.23 64.56 67.54 65.22 64.65 62.40 61.77 62.87 61.79
18 58.31 59.13 59.22 57.99 64.11 56.78 56.24 57.55 59.78 55.91 57.63 58.52
19 49.60 48.38 50.14 50.82 48.86 49.97 52.34 52.97 52.01 50.92 53.74 53.00
20 47.06 48.48 49.13 48.45 47.35 49.46 49.05 48.82 48.19 48.52 48.74 48.76
21 59.66 61.85 55.34 58.47 60.59 58.58 58.35 57.67 59.93 61.16 59.85 62.18
22 61.14 55.29 59.71 56.65 61.09 56.54 58.43 58.51 63.87 60.14 54.70 62.94
23 34.77 36.46 33.50 34.29 37.06 35.50 29.80 32.79 36.34 35.31 35.58 36.44
24 19.95 28.45 24.52 26.64 23.69 26.28 29.70 30.15 26.51 26.80 29.55 27.38
Figure 5.4. Hourly day-ahead market price samples
54
5.6.1 Comparison Analysis of cPSO and Marginal Cost based Bidding
We first compare the results provided by the cPSO and the marginal cost method
(MC). The MC method consists of dividing the maximum capacity into ten blocks of
equal size and then offering them at prices equal to their marginal cost. We plotted GEN-
1 and GEN-2 in Figure 5.5a and Figure 5.5b respectively. We compute the quantity-price
offer for two different generators say GEN-1 and GEN-2. Notice that although the cost
function of GEN-1 is quadratic, the curve is approximately linear in the range 0 to 400
MWh; whereas, the curve of GEN-2 is clearly quadratic. In addition, we assume that each
generator bids into the PJM market (Bmax =999) where the maximum number of blocks is
ten (N=10).
Figure 4.5.a. Cost function of GEN-1 Figure 4.5.b. Cost function of GEN-2
We run the cPSO algorithm using 10 replications and 400 cycles, and choose the
solution with the highest fitness value. The MC and cPSO solutions are given in Table
5.2. For GEN-1, the expected profit of the MC solution is $2,766. The cPSO provides a
55
solution with a slightly higher expected profit of $2,819. For GEN-2, the expected profit
of the MC solution is $5,525, whereas cPSO provides a solution with a higher expected
profit of $6,619.
Table 5.2. Quantity-price solutions for GEN-1 and GEN-2
Block GEN-1- MC GEN-1- cPSO GEN-2 ?MC GEN-2- cPSO
i
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
1 57.64 40 57.37 68.43 56.16 60 40.10 24.55
2 58.75 80 59.20 124.17 69.12 120 52.75 64.29
3 59.87 120 60.83 177.19 82.08 180 59.55 86.30
4 60.99 160 62.02 220.51 95.04 240 98.79 179.08
5 62.11 200 63.05 258.47 108.00 300 98.89 479.77
6 63.22 240 64.41 294.94 120.96 360 99.20 480.27
7 64.34 280 65.69 312.82 133.92 420 99.30 480.77
8 65.46 320 65.79 332.54 146.88 480 99.49 481.27
9 66.58 360 66.68 396.18 159.84 540 155.02 481.77
10 67.64 400 67.64 400.00 172.80 600 172.80 600.00
The cPSO approach is more effective for GEN-2 than for GEN-1. The small
difference in profit for GEN-1 seems to be due to the small value of the quadratic
coefficient of its cost function. To show that, we change the quadratic coefficient a3 of
GEN-1 between 0.0139 and 0.1139 in steps of 0.01 and compute the quantity-price
solutions for each of these values using the MC and cPSO methods. We plot the
difference of profits in relation to the MC solution in Figure 5.6 below.
56
0
2
4
6
8
10
12
14
16
18
0 0.02 0.04 0.06 0.08 0.1 0.12
Quadratic Cost Coefficient (a3)
Pr
of
it
di
ffe
re
nc
e (
%
)
Figure 5.6. Percentage of increase of cPSO solution in relation to MC
5.6.2. Comparison Analysis of cPSO and dBPSO
For the comparison analysis of cPSO and dBPSO, we use GEN-2 and the same
price samples and PSO parameters. To compare the performance of both methods in
terms of the fitness value, we use three different combinations of replications and cycles
(see Table 5.3). Notice that the dBPSO algorithm evaluates two fitness functions at each
iteration, which we refer to as a ?cycle?, while cPSO evaluates one function at each
cycle. We compare both methods using different numbers of cycles and replications. A
replication is a complete run of the algorithm using a different starting seed for
generating the random numbers. We select three combinations of numbers of cycles and
replications to reach a total of 4,000 function evaluations in each PSO. These values are
given in Table 5.3. To assure that these combinations are indeed different settings, we
conduct an ANOVA test for cPSO and dBPSO. This procedure allows us to properly
compare both approaches under similar parameters and diverse setting.
57
Table 5.3: Parameter set for the experiment
Parameter cPSO dBPSO
Number of Replication (R) 10 20 40 10 20 40
Number of Cycles (C ) 400 200 100 200 100 50
Total Function Evaluations 4000 4000 4000 4000 4000 4000
First, we run the cPSO for each combination. In Table 5.4 I give the averages,
standard deviations, and the best fitness values. The bid prices and quantities that provide
the best fitness are given in Table 5.5. Second, we conduct the one-way ANOVA test on
means to determine whether there is a statistical difference among these results. The
values of the ANOVA table are given in Table 5.6. The results of the test show that with
a p-value=0.15 and level of significance ?=0.05 there is no statistical evidence to show
that the setting conditions are different. This result indicates that cPSO is robust with
regard to changes to R and C.
Table 5.4. Statistical summary of the results of the cPSO solutions
Parameter set Average fitness($) Standard deviation ($) Best fitness ($)
R=10, C=400 6,346.84 143.08 6,530.02
R=20, C=200 6,304.85 125.24 6,529.35
R=40, C=100 6,315.40 127.89 6,524.05
Table 5.5. Bid prices and quantities for best solutions of cPSO
R=10, C=400 R=20, C=200 R=40, C=100
Block i bi ($/MWh) qi (MWh) bi ($/MWh) qi (MWh) bi ($/MWh) qi (MWh)
1 46.72 31.57 46.89 31.60 45.86 27.66
2 54.75 75.58 54.80 75.58 54.70 75.69
3 114.76 109.02 107.84 348.03 77.03 290.36
4 136.21 298.12 131.37 430.19 77.13 386.62
5 136.31 432.95 166.70 448.01 80.01 417.34
6 136.41 477.58 166.80 448.51 80.11 481.06
7 136.51 478.08 166.90 449.01 80.39 503.23
8 136.61 480.47 167.00 449.51 80.49 503.73
9 136.71 480.97 167.10 450.01 80.59 504.23
10 172.80 600.00 172.80 600.00 172.80 600.00
58
Table 5.6. ANOVA test results for cPSO
Source of Variation Sum of Square DF Mean Square F0 P-Value
Factor 11959 2 5980 0.36 0.701
Error 1120190 67 16719
Total 1132149 69
Similarly, we run dBPSO for each of the setting conditions. In Table 5.7, I show
the averages, standard deviations, and the best fitness values. The bid prices and
quantities that provide the best fitness are given in Table 5.8. The same ANOVA test on
means was applied to the experimental data of dBPSO. The results of the statistical test
are given in Table 5.9. This ANOVA table also shows no evidence of statistical
difference among the setting conditions with a level of significance ?=0.05 and p-
value=0.488. dBPSO is also robust with regard to changes to its parameters.
Table 5.7. Statistical summary of dBPSO solutions
Parameter Set Average fitness ($) Standard Deviation ($) Best fitness ($)
R=10, C=200 6,607.18 64.91 6,691.43
R=20, C=100 6,630.04 52.10 6,701.11
R=40, C= 50 6,612.69 61.29 6,689.76
Table 5.8: Bid prices and quantities for best solutions of dBPSO
R=10, C=200 R=20, C=100 R=40, C=50
Block
i
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
1 46.45 27.71 46.17 27.72 46.20 27.02
2 52.48 57.18 52.48 57.18 52.29 56.36
3 57.54 74.03 57.18 71.46 57.46 73.54
4 57.64 74.53 59.84 81.74 61.08 89.17
5 61.57 91.34 59.94 82.24 61.18 89.67
6 61.67 91.84 60.04 82.74 61.28 90.17
7 61.77 92.34 62.85 96.58 61.38 90.67
8 61.87 92.84 62.95 97.08 61.48 91.17
9 61.97 93.34 63.05 97.58 61.58 91.67
10 172.80 600.00 172.80 600.00 172.80 600.00
59
Table 5.9. ANOVA test results for dBPSO
Source of Variation Sum of Square Df Mean Square F0 P-Value
Factor 5,106 2 2,553 0.72 0.488
Error 236,012 67 3,523
Total 241,118 69
Using an analysis of means and a t-test, we test whether the mean of the fitness
values of dBPSO is greater than that of cPSO. The results are given in Table 5.10. It
shows strong evidence with p=0.00 and confidence level ?=0.05 that dBPSO provides
better solutions than cPSO on the average.
Table 5.10. ANOVA test results for dBPSO for method comparisons
Sample Sample Size Mean St. Deviation T-Value P-Value
dBPSO 70 6616.9 59.10 16.57 0.00
cPSO 70 6316.9 128.10
Difference 70 300.00 151.50
5.6.3. Converging Process of dBPSOB and dBPSOQ
To illustrate the converging process of the two sub problems, dBPSOB and
dBPSOQ, of the dBPSO approach, we plot the evolution of the fitness function of each
sub problem with respect to the number of iterations. In Figures 5.7 through 5.9, we show
the evolution of the fitness value for the number of cycles equal to 200, 100, and 50,
respectively. Notice that in the three plots the fitness value of the dBPSOB starts at a low
value, whereas the dBPSOQ starts at a higher value. This is because we begin by solving
dBPSOB using randomly selected bi values, while dBPSOQ uses ?qi values that are
determined by dBPSOB. Also notice that for the three cases the two sub-problems
converge before the 25th iteration.
60
-1000
1000
3000
5000
7000
0 5 10 15 20 25
Cycle (C)
Pr
ofi
t($
)
dBPSOQ
dBPSOB
Figure 5.7: Converging process of dBPSOB and dBPSOQ to the solution (C=200)
-1000
1000
3000
5000
7000
0 5 10 15 20 25
Cycle (C)
Pr
ofi
t($
)
dBPSOQ
dBPSOB
Figure 5.8: Converging process of dBPSOB and dBPSOQ to the solution (C=100)
61
-1000
1000
3000
5000
7000
0 5 10 15 20 25
Cycle (C)
Pr
ofi
t($
)
dBPSOQ
dBPSOB
Figure 5.9: Converging process of dBPSOB and dBPSOQ to the solution (C=50)
5.6.4 Impact of the order of dBPSOB and dBPSOQ
In order to analyze the impact of the order of solving dBPSOB and dBPSOQ, we
solve the problem using dBPSOB first and then dBPSOQ. In this procedure, the initial
values of b are randomly selected from a uniform distribution as follows:
b1 = U(0, Bmax/N) (5.19)
bi = U(bi-1, Bmax) ? b i-1, for i=2,..,N (5.20)
In Figures 13 through 15, we show the evolution of the fitness value for number of cycles
equal to 200, 100, and 50, respectively.
62
-1000
1000
3000
5000
7000
0 5 10 15 20 25
Cycle (C)
Pr
of
it
($
)
dBPSOQ
dBPSOB
Figure 5.10: Converging process of dBPSOB and dBPSOQ to the solution (C=200)
-1000
1000
3000
5000
7000
0 5 10 15 20 25
Cycle (C)
Pr
of
it
($
)
dBPSOQ
dBPSOB
Figure 5.11: Converging process of dBPSOB and dBPSOQ to the solution (C=100)
63
-1000
1000
3000
5000
7000
0 5 10 15 20 25
Cycle (C)
Pr
of
it
($
)
dBPSOQ
dBPSOB
Figure 5.12: Converging process of dBPSOB and dBPSOQ to the solution (C=50)
Similarly to the former case of dBPSO, in the three plots the fitness value of the
dBPSOQ starts at a low value, whereas the dBPSOB starts at a higher value. This is
because we begin by solving dBPSOQ using randomly selected ?qi values, while dBPSOB
uses bi values that are determined by dBPSOQ. Notice that for the three cases the two
sub-problems converge again before the 25th iteration. Table 5.11 gives a summary of the
results. After comparing the plots and Table 5.11 with Table 5.7, the order of the sub-
problems is not relevant for solving the problem.
Table 5.11: Statistical summary of dBPSO solutions
Parameter Set Average fitness ($) Standard Deviation ($) Best fitness ($)
R=10, C=200 6,615.90 73.83 6,688.56
R=20, C=100 6,635.57 55.67 6,688.72
R=40, C= 50 6,614.85 69.28 6,689.48
5.7 Conclusion
Results showed that PSO outperforms the MC method on determining price-
quantity pairs that will be submitted to the day-ahead market. The percentage of
64
improvement is accentuated when the quadratic coefficient of the generator?s cost
function is significant. In terms of time and experimentation burden, both models took
three minutes on average to find a solution. All three experiments of dBPSO showed that
we need less than 25 iterations to obtain a good solution. The results also showed that
dBPSO give much better solutions than cPSO. This tells us that the decomposition
technique can be applied to problems that have two or more decision variable sets. The
problems can be decomposed into two or more parts, and one or more decision sets can
be used as components of the solution of the other parts of the problem. This process
continues until no improvement is observed. The model discussed here can be further
improved by including a forecasting technique for the market prices. If the characteristics
of the generating unit and constraints change, the model and solution approach still may
remain valid.
65
CHAPTER 6
AGENT BASED PARTICLE SWARM OPTIMIZATION FOR SUPPLY
FUNCTION EQUILIBRIUM
6.1 Introduction
In the previous chapters, we developed models for an individual generator which
submits a bidding curve to the PJM Day-ahead market. We developed two PSO methods
to find a heuristic solution to the problem. The models did not include the behavior of all
market participants. In this chapter, we include the strategic change in competitors?
behavior for a particular generator. The model assumes that the strategy employed by one
player is affected by the others? behavior. Game theory and agent based models are two
ways to represent this market interaction. We develop an agent based simulation method
to simulate the behaviors of all firms in the market. We combine the dBPSO approach
and agent based model to compute an equilibrium solution.
6.2 Supply Function Equilibrium Model
This model assumes that there are m participants in a power pool who may be
referred to as power suppliers. These power suppliers may either have individual
generators or even a portfolio of generators. All these participants bid into the day-ahead
market and aims to maximize their profit by using bidding strategies that represent their
expectations best. The ISO collects the buy bids simultaneously and it starts the security
66
constrained optimum dispatch algorithm to set the equilibrium for the market. The ISO
sorts the sell offers starting from minimum price offers to more expensive ones and sorts
the buy offers starting from the maximum price offers to less expensive ones. The ISO
sets the equilibrium market price where the aggregated supply and demand meet. Figure
6.1 shows the equilibrium process in the day-ahead market [54]. In this figure, supply
represents the power quantities offered and demand represents the respective buy offers;
bij and qij represent the offered bid price and the power quantity respectively. Notice that
the day-ahead equilibrium price is used for uniform bid auctions, and the winners will be
paid the MCP.
bij
Equilibrium
Pt
Market
Price ($/Mwh)
Power Allocation
(Bid Quantities Mwh)
Demand
Supply
qij
(Bid Prices)
The model is formulated under the following additional assumptions:
Figure 6.1. Equilibrium process in day-ahead power market
67
i. An offer or bid, which consists of N price-quantity blocks at the most, needs
to be determined for each firm separately.
ii. Each firm can build its strategy based on separate resources or on its portfolio
of power resources.
iii. The equilibrium for each day is determined before the market closes at noon
and is valid for the next twenty four hours, starting at midnight the same day.
iv. Demand can be forecasted and is known for the analysis
v. The transmission constraints are not included in the model.
vi. Equilibrium of interest occurs in a single-round auction market.
In finding an equilibrium, there are Nxm pairs of decision variables bji and ?qji
(j=1,?M) (i=1,..,N) for each firm j that need to be determined.. The variable ?qji denotes
the amount of energy increase in firm j in block i, to get the bid price bji for delivery at
any hour of the next day. Total energy to be produced at time t and sold to the market at a
price Pt by the firm j is given by:
?
=
?=
)(
1
)(
tPI
i
jitjt qPq , tjit PbiPI ?= such that Max )( for j=1...m; t=1...T; i=1..I(Pt) (6.1)
In equilibrium, it is accepted that firms cannot make more profit by bidding other
than their current bid. Also the cost of dispatching is minimized for the system operator at
this state [33].
68
The maximum profit for the firm j in equilibrium can be expressed as:
?
=
?=?=
T
t
tjttjttjj PqCPqP
jj 1
j, ))(()( Max),P( b?qb?q for j=1...m; t=1..T; (6.2)
Subject to the following constraints [55]:
))(),(())(),(( *** tjttjtjtjttjtj PqPqPqPq ?? ??? for j=1...m; t=1...T; (6.3)
max
1
j
N
i
ji Qq ???
=
for i=1..N; j=1...m; (6.4)
max0 Bb
ji ?? for j=1...m; i=1...N; (6.5)
max0
jji Qq ??? for j=1...m; i=1...N; (6.6)
?
=
?=
)(
1
)(
tPI
i
jitjt qPq , (6.7)
Where tjit PbiPI ?= such that Max )( for j=1..m; t=1..T; i=1...N;
2
321 ))(()())(( tjtjtjtjjtjt PqaPqaaPqC ++= for j=1..m; t=1..T; (6.8)
The equilibrium in an economic system requires supply and demand to be equal.
As equilibrium constraint, the total amount of power generated is equal to total supply as
given below:
t
m
j
tjt DPq =?
=1
)( t=1..T; (6.9)
69
6.3 Agent Based Modeling and Simulation
Agent-Based Modeling and Simulation (ABMS) is a computational approach to
model economic systems which have interacting components or dynamic agents. Agents
usually interact among themselves and between environments by updating themselves
sequentially rather than simultaneously. In building an ABMS, the definition of agents
and their interaction environment are crucial. Figure 6.2 shows the ABMS building
process [56].
Figure 6.2. Agent based modeling and simulation
70
ABMS is applied to very complex systems where interdependencies are difficult
to capture and traditional models are hard to apply. ABMS replaces its framework with an
individual agent?s behavioral rules that are updated over time [57]. ABMS is a descriptive
method which aims to model the behavior of agents rather than optimality. ABMS models
are useful in economics models. In a micro-economic point of view, agents assume that
1) they behave in a rational manner that aims to optimize their well-defined objectives 2)
they have identical characteristics that make them alike 3) they will have decreasing
marginal utility as the number of agents increase 4) long-run equilibrium of the system is
of primary interest to the model [58], [59]. Figure 6.3 shows the process of a typical
ABMS [57], [58], [59].
Figure 6.3. General flow of a typical ABMS
71
ABMS, like many other heuristic methods, searches for the best feasible solution
by an updating process. The process is evolutionary, in which updating, learning and
convergence are involved. ABMS has been used in flow management of evacuation,
traffic, stock market, strategic simulation of market, organizational design, and in other
areas where players dynamically move [56], [58], [60]. In [61], an agent-based model is
developed for a supply chain in which the flow of a commodity finds a way between
factories, distributors, wholesalers, retailers and customers. The goal of supply chain
agents is to minimize their cost on their way. These are the research assumptions that are
made for the agent based modeling to work. In [57], authors develop an agent based
model for modeling the human immune system. Authors in [62] address the agent-based
modeling approach in financial markets by explaining trading behaviors of firms.
In [54], [59] authors develop an agent-based simulation approach for modeling
the day-ahead power trading in the US wholesale power market. The model is developed
for wholesale power suppliers, individual power generators and wholesale power
consumers that are bidding into the day-ahead market. Their hypothesis is that agent-
based approach is as good as other approaches like neural networks. They show the
effectiveness of the model using data provided in PJM west. Agents bid into the market
and they update their bidding strategies in each run of the simulation based on a learning
factor until the equilibrium is reached.
In [63], an agent-based model was developed for the wholesale electricity market,
operating in a short-term environment under capacity conditions and double auction
rules. A simulation model that uses a multi-adaptive agent model for generators bidding
in the UK power market was developed in [64]. It shows that agents learn bidding
72
strategies in a manner similar to their behavior in real world. In [65], authors use an agent
based simulation model to show how companies use market power during the electricity
market bidding process. They compare the production cost bidding with the bidding
strategies based on physical and economic withholding, including congestion
management.
6.4 Agent based particle swarm optimization model
It was demonstrated in Chapter 5 that the dBPSO method is an effective way to
find a solution for an individual generator. Now let?s assume that m firms compete in the
day-ahead market using dBPSO. The objective of this chapter is to show that the model
can reach an equilibrium point at which each player?s payoff is maximized and the
equilibrium conditions are met using ABMS and dBPSO. Each firm participating in the
day-ahead market is modeled as an agent. Figure 6.4 shows the details for the ABMS
method with dBPSO applied. Each agent has a unique cost function, a capacity and pairs
of quantity-price bids as attributes. The agents? interaction occurs in the pool where
offered quantities and corresponding prices are submitted. Agents aim to allocate their
price-quantity offers in a way that their profit is maximized. In other words, their
interaction occurs based on power quantities and offered prices.
73
Figure 6.4: Components of ABMS method
The model assumes that the day-ahead market demand is known and the final
objective is to reach equilibrium for the next day. The model starts with an initial price
scenario and given demand for 24 hours. It runs until the equilibrium is reached. Each
agent?s objective is to maximize its profit by bidding into the environment using dBPSO.
Figure 6.5 shows a flow chart of the Agent Based Particle Swarm Optimization
(ABPSO).
74
Figure 6.5: The flow of the ABPSO
75
Notice that each agent in the model is a price-taker while bidding into market. The
cleared prices that are set at each iteration are the result of bidding strategies submitted
by each agent. Since the analysis is for a single-round auction market, an agent can use
the simulation to test the behavior of its rivals. Agents update their price and quantity
bids at each iteration until the equilibrium is reached. The particular agents simulating the
model would be able to observe the behavior of their competitors during this process.
This iterative process ensures that each generator minimizes its risk of cost recovery and
infeasibility of its offer.
The definition of stopping criteria is important in ABSM since it shows whether
the equilibrium is reached or not. In [21], authors propose two kinds of stopping criteria.
First criterion stops the equilibrium process when the prices are too low. This is because
the demand is not sufficiently covered at this price since some units will be eliminated
due to low prices. At this point, iteration goes 2 steps back and defines the point as
equilibrium point. The second criterion calculates demand-weighted average price and
total financial loss for all players and it compares the values with those of the previous
ones. When the current average prices are higher than those of the previous two iterations
the process stops.
The submission of the same or similar strategies at each iteration could lead to
similar market prices. It might indicate that market price is converging and strategies
submitted are reaching equilibrium. However, we need to verify that this state actually
satisfies equilibrium conditions. We evaluate three new stopping rules to end the agent-
based process in order to find the best stopping rule that satisfies equilibrium conditions
most of the time. In the first stopping rule, we calculate the percentage of differences of
76
resulting market prices and previous iterations? market prices for each hour. When the
absolute value percentage of market price differences for the 24 hours are less than a
value ?1 the process stops. It can be represented mathematically as:
1
1||
???=?
?
r
t
r
t
r
tr
t P
PP t=1..T. (6.10)
In the second stopping rule, we calculate the average of differences of resulting
market prices and previous iterations? market prices for each hour. At each iteration,
current market prices are compared with previous market prices. When the absolute
average value of market price differences for 24 hours are less than a small value ?2 the
process stops. It can also be represented mathematically as:
2
1
1 ||1 ???=? ?
=
?
T
t
r
t
r
t
r PP
T (6.11)
In the third stopping rule, we calculate the weight of each hour?s demand with
regard to the total demand for 24 hours. At each iteration, we multiply this percentage for
each hour with the price difference percentage found in the first stopping condition.
When the average value of this calculation is less than a small value ?3 the process stops.
The mathematical representation is:
77
3
1
1
1
||1 ???=?
?
?
=
?
=
T
t
t
r
t
t
r
t
r
t
T
t
r
DP
DPP
Tw (6.12)
Notice that at each iteration, agents could have available strategies from which
they choose the best strategy for themselves. This strategy is selected based on
interactions with other agents and interaction with the environment. The interaction with
the environment occurs in a way that total supply of all agents should be equal to total
demand. The amount of power allocated to an agent affects other agents. The interaction
between agents occurs based on the offer prices and offer quantities. If the offer of an
agent is selected, the agent can either maintain this strategy or update it in the next
iteration in order to get better results. This process continues until the price difference in
the two iterations is so low that it might lead to convergence.
In order to test whether the results of the experiments actually satisfy the
equilibrium conditions, we define conditions for equilibrium.
I. Supply should be equal to demand as an essential rule of equilibrium.
II. In equilibrium, all suppliers maximize their return at the given prices. Also according to
the Nash equilibrium, if a player deviates from its strategy it will loose in the long run
[38], [39], [40]. We will let each firm behave separately and run the dBPSO to get a
separate bidding strategy and profit. The results found in dBPSO will be compared with
the ABPSO results for each firm. It is expected that firms find a close strategy in
dBPSO to the strategy found in equilibrium. If the difference is less than 1%, we will
accept this as a satisfied condition.
78
6.5 Numerical Example and Analysis
We coded the model in C and used the same dBPSO parameters we used in
Chapter 5. We are more interested in the supply side bidding rather than demand side
bidding. The demand for next day is known and is given in Table 6.1.
Table 6.1. Day-Ahead demand for next day
Hour Demand(Mwh) Hour Demand(Mwh)
1 3115 13 4510
2 3711 14 5142
3 3346 15 3424
4 3771 16 3287
5 3298 17 4501
6 4266 18 5236
7 4117 19 5790
8 5176 20 6084
9 5751 21 6561
10 6513 22 6411
11 6280 23 4411
12 4472 24 4664
6.5.1 ABPSO Experiment with Duopoly
We start with a duopoly SMD that has m=2 firms competing. The cost functions
of the firms and their capacities are given in Table 6.2.
Table 6.2. Cost functions and market capacities of the firms for m=2
Firm(m) a1 a2 a3 Unit
1 0 41.73 0.0063 8085
2 0 47.25 0.0057 6281
79
We start with the first stopping condition set to the value ?1 = 1%, in other words,
the percentage price difference for each of the 24 hours is less than 1%. The equilibrium
prices found are given in Table 6.3.
Table 6.3. Equilibrium prices for first stopping rule (m=2)
Hour Price($/Mwh) Hour Price($/Mwh)
1 77.15 13 77.15
2 77.15 14 77.15
3 77.15 15 77.15
4 77.15 16 77.15
5 77.15 17 77.15
6 77.15 18 77.15
7 77.15 19 80.29
8 77.15 20 80.29
9 80.29 21 83.98
10 83.68 22 83.68
11 83.68 23 77.15
12 77.15 24 77.15
Table 6.4 shows the bids found for each firm at the equilibrium and their profits.
Table 6.4. Results found in first stopping rule (m=2)
Firm-1 Firm-2
Block i
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
1 70.79 2856.23 77.15 2642.64
2 70.89 2861.75 77.42 2887.47
3 70.99 2862.25 83.68 3200.27
4 80.29 3254.57 83.78 3200.77
5 80.47 3338.97 83.88 3218.77
6 84.57 3450.30 83.98 3247.07
7 86.50 3503.35 84.08 3254.22
8 86.60 3507.39 84.18 3254.72
9 86.70 3507.89 84.28 3255.22
10 143.60 8085.00 118.85 6281.00
80
?1 = 1% was chosen because values smaller than 1% increase the computational
time and in some cases the stopping condition could not be reached in a reasonable time.
In the second stopping rule we set ?2 = $0.25. In other words, the average price
difference for the 24 hours should be less than $0.25. The values less than $0.25 also
increase computational time and many times return no results. The equilibrium prices
found are given in Table 6.5.
Table 6.5. Equilibrium prices for second stopping rule (m=2)
Hour Price($/Mwh) Hour Price($/Mwh)
1 77.15 13 77.15
2 77.15 14 77.15
3 77.15 15 77.15
4 77.15 16 77.15
5 77.15 17 77.15
6 77.15 18 77.15
7 77.15 19 80.29
8 77.15 20 80.29
9 80.29 21 83.98
10 83.68 22 83.68
11 83.68 23 77.15
12 77.15 24 77.15
Table 6.6 shows the bids submitted by each firm at the equilibrium and firms? profits.
81
Table 6.6. Results found in second stopping rule (m=2)
Firm-1 Firm-2
Block i
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
1 70.79 2856.23 77.15 2642.64
2 70.89 2861.75 77.42 2887.47
3 70.99 2862.25 83.68 3200.27
4 80.29 3254.57 83.78 3200.77
5 80.47 3338.97 83.88 3218.77
6 84.57 3450.30 83.98 3247.07
7 86.50 3503.35 84.08 3254.22
8 86.60 3507.39 84.18 3254.72
9 86.70 3507.89 84.28 3255.22
10 143.60 8085.00 118.85 6281.00
In the third stopping rule we set ?3 =1%. In other words, the load weighted
average for the 24 hours should be less than 1%.The equilibrium prices found are given
in Table 6.7.
Table 6.7. Equilibrium prices for the third stopping rule (m=2)
Hour Price($/Mwh) Hour Price($/Mwh)
1 78.45 13 78.45
2 78.45 14 78.45
3 78.45 15 78.45
4 78.45 16 78.45
5 78.45 17 78.45
6 78.45 18 78.45
7 78.45 19 79.77
8 78.45 20 79.77
9 79.77 21 84.17
10 84.17 22 82.6
11 82.6 23 78.45
12 78.45 24 78.45
Table 6.8 gives the bids submitted by each firm at the equilibrium and profits.
82
Table 6.8: Results found in third stopping condition (m=2)
Firm-1 Firm-2
Block i
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
1 58.01 1343.58 78.45 2744.24
2 78.32 2955.50 78.59 2795.03
3 79.77 3318.26 82.60 3168.58
4 93.90 3832.65 84.17 3247.16
5 94.00 3840.36 84.27 3275.26
6 94.38 3840.86 84.37 3275.76
7 94.48 3841.36 84.47 3276.26
8 94.58 3841.86 84.57 3276.76
9 94.68 3863.55 84.67 3277.26
10 143.60 8085.00 118.85 6281.00
Using the equilibrium prices found, we run dBPSO to find a solution for each
firm separately. The summary of the results are given in Table 6.9.
Table 6.9: Overview of the results found in each method (m=2)
Method Firm-1 Profit ($) Firm-2 Profit ($)
First stopping rule 1,302,689 1,043,974
Second stopping rule 1,302,689 1,043,974
Third stopping rule 1,355,414 1,094,031
All three stopping rules satisfy the balance condition, which is supply equals
demand. The second and third stopping rules generally require close computational times
which is around 15 minutes. Notice that rule-1 and rule-2 return the same values, i.e.,
they both satisfy each rule. We let each firm develop a separate strategy. To do so, we use
the equilibrium prices found in each rule and run dBPSO. Table 6.10 shows the
percentage of profit increase in dBPSO comparing with the equilibrium solution.
83
Results show that three rules give similar profit increases which are smaller than
acceptable level 1%. However, the profits found in rule-1 are almost same with the
ABPSO solution.
Table 6.10: Profit increases in each rule (m=2)
Rule 1 Rule 2 Rule 3
Firm 1 0.08% 0.08% 0.15%
Firm 2 0.00% 0.00% 0.01%
6.5.2 ABPSO experiment with m=5
We now suppose that there is a SMD that has m=5 firms computing. The cost
functions of the firms and their capacities are given in Table 6.11.
Table 6.11. Cost functions and market capacities of the firms for (m=5)
Firm(m) a1 a2 a3 Unit
1 0 47.273 0.0074 3450
2 0 45.18 0.0048 1600
3 0 44.76 0.0066 2935
4 0 45.35 0.0087 4950
5 0 46.72 0.0061 2281
We set the ?1 = 1% and the find the results with firs stopping condition. The
equilibrium prices found are given in Table 6.12 and the results for each firm are given in
Table 6.13 and Table 6.14.
84
Table 6.12. Equilibrium prices for first stopping rule (m=5)
Hour Price($/Mwh) Hour Price($/Mwh)
1 56.24 13 57.83
2 56.30 14 59.18
3 56.24 15 56.24
4 56.30 16 56.24
5 56.24 17 57.83
6 57.17 18 60.46
7 56.30 19 63.11
8 60.46 20 63.21
9 63.11 21 63.90
10 63.51 22 63.24
11 63.24 23 57.83
12 57.83 24 58.36
Table 6.13. Results found for firm 1, 2 and 3 in first stopping rule (m=5)
Firm-1 Firm-2 Firm-3
Block i
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
1 54.76 636.21 52.77 1090.48 53.57 922.97
2 57.83 829.31 54.67 1090.98 60.46 1365.98
3 63.11 1061.94 55.88 1199.05 64.78 1610.22
4 63.21 1082.94 58.36 1397.38 64.88 1617.56
5 63.31 1083.44 58.86 1500.82 64.98 1618.06
6 63.41 1088.85 58.96 1501.32 65.08 1627.13
7 63.51 1101.42 59.06 1576.97 65.18 1627.63
8 63.61 1105.74 59.16 1577.47 65.28 1648.61
9 63.71 1128.88 59.26 1595.29 65.38 1649.11
10 98.33 3450.00 60.54 1600.00 83.50 2935.00
85
Table 6.14. Results found for firm 4 and 5 in first stopping rule (m=5)
Firm-4 Firm-5
Block i
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
1 56.24 690.41 56.30 788.31
2 63.24 1031.96 56.40 814.21
3 63.34 1061.27 57.17 908.10
4 64.42 1112.65 59.18 1128.05
5 64.68 1181.62 63.16 1359.23
6 65.25 1182.12 63.33 1383.06
7 65.35 1182.62 63.43 1386.21
8 65.45 1185.39 63.53 1386.75
9 65.55 1185.89 63.90 1414.27
10 131.48 4950.00 74.55 2281.00
In the second stopping rule we keep the same value and set the ?2 = $0.25. The
equilibrium prices found are given in Table 6.15. Table 6.16 and Table 6.17 below show
the bids submitted by each firm at the equilibrium.
Table 6.15. Equilibrium prices for second stopping rule (m=5)
Hour Price($/Mwh) Hour Price($/Mwh)
1 54.61 13 57.69
2 54.95 14 59.65
3 54.61 15 54.61
4 56.16 16 54.61
5 54.61 17 57.69
6 57.29 18 60.54
7 57.29 19 61.86
8 59.65 20 62.18
9 61.86 21 63.95
10 63.75 22 63.75
11 62.67 23 57.69
12 57.69 24 57.89
86
Table 6.16. Results found for firm 1, 2 and 3 in second stopping rule (m=5)
Table 6.17. Results found for firm 4 and 5 in second stopping rule (m=5)
Firm-4 Firm-5
Block i
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
1 46.40 532.01 54.61 663.56
2 50.27 604.27 54.71 685.44
3 56.16 764.99 57.29 928.55
4 61.86 967.41 58.93 1078.69
5 63.72 967.91 61.41 1287.72
6 63.82 994.97 63.75 1396.49
7 63.93 1009.70 63.85 1396.99
8 64.03 1057.48 63.95 1400.27
9 64.13 1057.98 64.10 1414.97
10 131.48 4950.00 74.55 2281.00
For the third stopping rule we also go with the same stopping value and set ?3 =
1%. The equilibrium prices found are given in Table 6.18.
Firm-1 Firm-2 Firm-3
Block i
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
1 53.21 527.86 53.12 835.03 50.71 781.89
2 56.61 769.26 53.28 877.46 57.69 1052.93
3 61.25 1001.89 54.07 982.34 62.18 1338.63
4 61.35 1006.83 54.95 1113.52 62.67 1433.25
5 61.87 1007.33 57.89 1330.21 62.77 1439.68
6 61.97 1007.83 57.99 1332.43 67.37 1463.38
7 62.60 1040.61 58.13 1390.31 67.47 1474.14
8 62.70 1064.30 59.65 1511.66 67.57 1475.78
9 63.17 1114.08 59.75 1569.86 67.67 1479.80
10 98.33 3450.00 60.54 1600.00 83.50 2935.00
87
Table 6.18. Equilibrium prices for third stopping rule (m=5)
Hour Price($/Mwh) Hour Price($/Mwh)
1 54.07 13 58.75
2 54.07 14 60.85
3 54.07 15 54.07
4 54.10 16 54.07
5 54.07 17 58.75
6 57.36 18 60.85
7 57.13 19 62.23
8 60.85 20 62.43
9 62.23 21 63.88
10 62.76 22 62.43
11 62.43 23 57.36
12 58.75 24 60.54
Table 6.19 and Table 6.20 below show the bids submitted by each firm at the
equilibrium with third stopping rule. The summary of the results are given in Table 6.21.
Table 6.19. Results found for firm 1, 2 and 3 in third stopping condition (m=5)
Firm-1 Firm-2 Firm-3
Block i
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
1 54.12 623.77 55.47 1075.35 55.24 814.10
2 59.82 934.01 55.67 1224.07 55.88 1010.89
3 63.23 1084.28 58.32 1377.30 60.30 1237.79
4 63.33 1088.76 58.42 1377.80 63.27 1404.40
5 63.43 1095.11 58.52 1378.30 63.37 1404.90
6 63.56 1095.61 58.62 1378.80 63.47 1412.75
7 63.66 1115.29 58.72 1379.30 63.59 1418.85
8 63.76 1115.79 58.82 1460.73 63.69 1432.33
9 63.86 1116.29 58.92 1461.23 63.79 1440.80
10 98.33 3450.00 60.54 1600.00 83.50 2935.00
88
Table 6.20. Results found for firm 4 and 5 in third stopping condition (m=5)
Firm-4 Firm-5
Block i
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
1 54.38 641.24 46.25 0.50
2 59.40 894.78 54.62 802.07
3 62.08 965.90 60.51 1134.85
4 62.33 998.53 60.61 1159.17
5 62.67 999.03 60.71 1170.19
6 62.77 999.53 61.58 1263.43
7 63.30 1034.49 63.29 1367.16
8 63.40 1042.20 63.39 1367.66
9 63.50 1059.84 63.49 1398.41
10 131.48 4950.00 74.55 2281.00
Table 6.21. Overview of the results found in each method (m=5)
Firm-1
Profit($)
Firm-2
Profit ($)
Firm-3
Profit($)
Firm-4
Profit($)
Firm-5
Profit($)
First rule 121,337 252,427 195,799 133,962 141,900
Second rule 112,390 235,851 182,632 127,743 149,483
Third rule 110,166 240,275 182,493 126,768 149,629
In terms of computational time, again first stopping rule is the most time
consuming. The second and third stopping rules took around 35 minutes. Again we let
each firm develop a separate strategy. Table 6.22 shows the percentage of profit increase
in dBPSO comparing with the equilibrium solution.
89
Table 6.22. Profit increases in each rule (m=5)
6.5.3 ABPSO Experiment with m=10
We now suppose that there is a SMD that has m=10 firms competing. Based on
the results found for m=2 and m=5 it is better to use rule-2 as stopping rule. The cost
functions of the firms and their capacities are given in Table 6.23 below.
Table 6.23. Cost functions and market capacities of the firms (m=10)
Rule 1 Rule 2 Rule 3
Firm 1 0.46% 0.14% 4.28%
Firm 2 0.17% 0.40% 0.11%
Firm 3 0.51% 0.85% 2.20%
Firm 4 2.87% 0.72% 2.30%
Firm 5 13.51% 0.06% 1.54%
Firm(m) a1 a2 a3 Capacity
1 0 46.18000 0.00477 500
2 0 32.95070 0.002357 600
3 0 42.40000 0.004664 250
4 0 40.12000 0.004364 1100
5 0 41.75679 0.003896 585
6 0 46.26748 0.007919 3000
7 0 42.71000 0.016201 1528
8 0 44.68000 0.017737 2000
9 0 42.45727 0.006044 403
10 0 43.19774 0.006153 4400
90
For given ?2 = $0.45, the equilibrium prices found are given in Table 6.24. Table
6.25 through Table 6.28 below show the bids submitted by each firm at the equilibrium.
The summary of the results and profit increase percentages are given in Table 6.29 and
table 6.30 respectively.
Table 6.24:.Equilibrium prices for second stopping rule (m=10)
Hour Price($/MWh) Hour Price($/MWh)
1 47.18 13 50.28
2 48.56 14 53.26
3 47.18 15 47.18
4 48.56 16 47.18
5 47.18 17 50.28
6 49.04 18 53.26
7 49.03 19 57.32
8 53.26 20 57.33
9 57.32 21 59.15
10 59.08 22 58.94
11 57.42 23 49.92
12 50.18 24 51.57
Table 6.25. Results found for firm 1, 2 and 3 (m=10
Firm-1 Firm-2 Firm-3
Block i
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
1 46.90 117.39 20.91 0.50 14.03 1.96
2 48.56 283.36 22.02 85.45 14.13 3.18
3 49.92 394.32 22.27 99.43 14.23 3.68
4 50.18 425.57 22.37 114.02 14.33 12.13
5 50.28 455.75 22.55 122.16 14.43 23.87
6 50.38 456.25 22.65 124.67 14.53 33.95
7 50.48 478.10 22.75 145.18 14.63 41.91
8 50.58 478.78 22.85 149.52 14.73 51.55
9 50.68 479.28 22.95 152.46 14.83 56.99
10 50.95 500.00 35.78 600.00 44.73 250.00
91
Table 6.26. Results found for Firm 4, 5 and 6 (m=10)
Table 6.27. Results found for firm 7 and 8 (m=10)
Firm-7 Firm-8
Block i
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
1 46.71 187.75 44.57 116.03
2 52.99 347.34 51.57 261.76
3 56.30 451.20 54.82 390.04
4 58.18 479.35 59.08 452.94
5 58.28 479.85 59.19 453.44
6 58.38 480.35 59.29 453.94
7 58.48 480.85 59.39 456.96
8 58.58 502.14 59.49 457.46
9 58.68 502.64 59.59 457.96
10 92.22 1528.00 115.63 2000.00
Firm-4 Firm-5 Firm-6
Block i
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
1 47.18 809.78 24.65 0.50 49.03 363.19
2 47.28 825.15 27.70 343.13 57.33 784.18
3 48.32 973.01 27.95 368.73 58.94 866.18
4 49.04 1028.94 28.07 383.73 59.04 878.95
5 49.14 1040.43 28.17 385.00 59.14 889.98
6 49.24 1042.02 28.27 396.61 59.24 913.65
7 49.34 1047.96 28.37 400.68 59.34 914.15
8 49.44 1057.33 28.47 411.19 59.44 921.36
9 49.54 1057.83 28.57 423.03 59.54 921.86
10 49.72 1100.00 46.32 585.00 90.37 3000.00
92
Table 6.28. Results found for Firm 9 and 10 (m=10)
Firm-9 Firm-10
Block i
bi
($/MWh)
qi
(MWh)
bi
($/MWh)
qi
(MWh)
1 46.41 343.69 45.34 454.83
2 46.51 347.94 53.26 875.22
3 46.61 351.35 57.32 1149.35
4 46.71 354.55 57.42 1216.91
5 46.85 368.00 57.52 1222.98
6 46.95 373.18 58.85 1255.57
7 47.05 383.41 58.95 1256.07
8 47.15 392.15 59.05 1258.21
9 47.25 399.01 59.15 1284.19
10 47.33 403.00 97.34 4400.00
Table 6.29. Unit?s profits (m=10)
Unit Profit ($) Unit Profit ($)
1 46,867 6 38,681
2 254,940 7 38,759
3 51,019 8 24,115
4 190,774 9 69,412
5 112,786 10 93,351
Table 6.30. Profit increases in rule 2 (m=10)
Rule 2
Firm 1 0.05%
Firm 2 0.00%
Firm 3 0.00%
Firm 4 0.01%
Firm 5 0.00%
Firm 6 0.96%
Firm 7 0.30%
Firm 8 0.97%
Firm 9 0.00%
Firm 10 0.37%
93
6.6 Conclusion
In this chapter, we showed that ABMS with dBPSO applied can be used to
simulate the bidding process and to find a nash equilibrium solution. The defined
stopping conditions and their results confirm the equilibrium conditions. In terms of
computational time, relatively not much time is required to reach to the results. The
model can further be applied to the real market if anyhow real operational cost and
demand data is provided.
94
CHAPTER 7
CONCLUSION AND FUTURE RESEARCH
In this research, models of electric power bidding in power markets were
introduced. The fundamentals of a market design were described and related current
literature was discussed. In order to evaluate a given bid, a spreadsheet based simulation
algorithm was developed. The results found in the numerical examples were verified
using the Bid Simulator. It was shown that the nonlinear and quadratic programming
models were able to give an optimal solution for a limited number of price samples.
However, due to the stochastic nature of market prices more price samples needed to be
included. This limitation was overcome using two particle swarm optimization models.
From experimental results, both approaches did not carry much computational time
burden.
Based on the statistical analysis, the decomposition based approach gave better
results than the conventional particle swarm optimization. The bids obtained were
compared with the marginal cost bidding method. The comparison showed that the
quadratic term of the cost function plays an important role in determining the bid
strategy. An additional method that models the bidding behaviors of power suppliers for a
fixed demand was developed. The method uses the decomposition based particle swarm
optimization and agent based simulation. Three stopping conditions to find the Nash
95
equilibrium were tested. The results found were analyzed using the equilibrium
conditions needed for a competitive power market.
Although the models described cover the fundamental process of bidding, they
can be improved in future research. One improvement to the models is to include
transmission constraints and congestion. Thus, the new models could be used to analyze
the effect of transmission constraints on market prices as well as on bidding behaviors.
Another improvement is to include operational constraints such as minimum up-down
times, start-up costs and ramp-up limits of generating units. The models could provide a
more realistic competition environment.
Another extension could be considering operating reserve and contracts in the
power market. Firms might choose to sell their power with fixed contracts or can be part
of the operating reserve market. Thus, the bidding model could include these additional
markets.
The models developed in this dissertation consider uniform price auctions.
However, there are markets that use the pay-as bid price auction mechanism. Therefore,
new bidding models can be developed that include the pay-as bid auction type. Also, each
power market has their own rules for the bidding process including number of blocks and
the time period that the bid is valid for. The developed models can further be applied to
those markets and perform economic comparisons.
96
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