![implemented. A formalized representation and a comparison of some of the most
common game theory models will be provided with some conceptual examples.
In addition, some newly proposed approaches for strategic bidding modeling based
on the complex systems techniques such as Multi Agent systems and Complex
Networks will be mentioned and some related references provided.
Keywords Electricity markets • Game theory • Network constraints • Strategic
bidding
1 Introduction
The electric power industry has over the years been dominated by large state-owned
monopolies that had an overall authority over all the activities in generation,
transmission and distribution of power within their jurisdiction. Chile is often
considered as the first Country to introduce liberalization in the electricity sector
in 1982. Regulatory reforms of the industry in the United States started in 1978
with the passage of the Public Utility Regulatory Policies Act; regulatory reform
was accelerated over the latter half of the 1990s with the advent of the open access
transmission regime in 1996, the subsequent formation of several large regional
spot markets, later, regional transmission organizations e.g. PJM, 1997, CAISO,
1998, Midwest 2002, etc. [1]. The first initial steps of liberalization of electricity
markets in Scandinavia started in Norway in 1990. Through the subsequent steps of
development through market expansion, the Nordic market became the world’s first
multi-national market that was quite well-functioning [2]. Since 1996 the genera-
tion, distribution and supply of electricity in eastern and southern Australian states
has been amalgamated under the National Electricity Market and in 2009 the
Australian Energy Market Operator (AEMO) has been established. Australia took
the forefront of energy industry reform worldwide, one of the first countries to
establish highly competitive and transparent electricity markets underpinned by
strong governance structures [3]. In UK, The Electricity Pool of England and Wales
was created in 1990 to balance electricity supply and demand, acting as a clearing
house between generation and wholesale. In March 2001 the electricity pool is
replaced by the New Electricity Trading Arrangements (NETA). On 1 April 2005,
British Electricity Trading and Transmission Arrangements (BETTA) is introduced
to replace the NETA in England and Wales, and the separate arrangements that
existed in Scotland and the British Grid System Agreement, to create fully-compet-
itive, British-wide wholesale market for the trading of electricity generation [4].
The justification for introducing competition in the electricity sectors is that in a
monopoly it is not possible to achieve, no matter which market rules are designed,
two important objectives at the same time: to hold down prices to marginal costs
and to maximize efficiency [5]. On the contrary, in a competitive market those two
objectives may be reached by a proper market design and the specification of a
proper set of rules. In this respect, the goal of electric industry restructuring is to
4 E. Bompard and Y. Ma](https://image.slidesharecdn.com/1148042-250526014811-8fd922cc/85/Handbook-Of-Networks-In-Power-Systems-I-1st-Edition-Ettore-Bompard-18-320.jpg)
![achieve a better, more efficient allocation of resources by increasing the role of
market forces, and simultaneously decreasing the role of regulation. The main
objectives of the reforms are achieved through a clear separation between produc-
tion and sale of electricity and the operation of electric power grid. Based on the
market reform, many market based roles have been penetrating into the electricity
industry in progress with a slight relaxation of the obligation to serve the loads that
has been segregated and assigned to various entities. Among the new roles energy
producers and retailers, brokers, independent system operator (ISO) or transmis-
sion system operator (TSO) are the most popular entities existing in real electricity
markets around the world.
There are two reference paradigms proposed for electricity markets: pool and
bilateral. Pool electricity markets coordinate the selling and buying activities
through a centralized market place administrated by third entity that may coincide
with the ISO, whilst the bilateral markets the transactions to be contracted between
the seller and buyer directly on a private basis. In the pool paradigm the market
optimum is reached a central decision making run after collecting the offers from
the producers and the bids from the customers while in the bilateral paradigm the
decision making process is distributed among various sellers and buyer that meet in
the marketplace.
Electricity markets are pretty different from other commodity markets mainly
due to the physical constraints related to the network structure that may impact
the market performance. The network constraints and the special features of the
electricity provide the market players an opportunity to behave strategically,
gaming the market, which is very specific this context and cannot be found in
other commodity markets. Strategic bidding behaviours of electricity producers are
widely studied in open literature [6, 7].
Game theory is popularly used in investigating the strategic bidding interactions
between the electricity producers. Models based on various games such as Bertrand
[8, 9], Cournot [10–12], Stackelberg [13, 14], Supply Function Equilibrium
[15–17], as well as conjectural supply function [18, 19] have been proposed. By
taking into account the network constraints, the strategic bidding behaviors analysis
based on game theory models usually involves a bi-level optimization problem
modeled as a mathematical program with equilibrium constraints (MPEC) [20, 21].
The lower level problem of market clearing with the consideration of the network
constraints, i.e. the equilibrium constraints, is inserted in the upper level problem of
the maximization of the producer surplus [22–26]. There has been an intensive
research for efficient solution methods for the MPEC problem, with proposed
solution schemes ranging from specific analytic algorithms to heuristics procedures
[27–32].
However, the strategic interactions among participants in today’s electricity
markets can be very complicated, due to various aspects such as supply and demand
uncertainties, unit commitment arrangement, multi-rounds auctions in both energy
and ancillary service markets, which are not conveniently modeled by game theory
techniques. An alternative efficient approach for analyzing the strategic bidding and
decision support of the market participants is provided by the multi agent system
Models of Strategic Bidding in Electricity Markets 5](https://image.slidesharecdn.com/1148042-250526014811-8fd922cc/85/Handbook-Of-Networks-In-Power-Systems-I-1st-Edition-Ettore-Bompard-19-320.jpg)
![environment [33]. Based on the computational economics approach, several auton-
omous adaptive agent models have been proposed, including those created by
Anthony J. Bagnall and George D. Smith,[34], Athina C. Tellidou and Anastasios
G. Bakirtzis [35], Sun J. and Tesfatsion L. [36], as well as Isabel Praca and Carlos
Ramos [37]. Recently, the application of evolutionary complex network has been as
well proposed by Ettore B. and Ma Y.C. [38] for modeling the bilateral electricity
markets. Stable network structures that can be used to anticipate possible bilateral
transactions in the real market place are developed by the improving path rule of
evolutionary complex network principles. Different stable network structures can
produce the same maximum value of global utility, reflecting the complex and
disordered individual behavior has self-organization properties that produce the
highest market efficiency in terms of the social welfare without the need for a
centralized decision-making authority.
In this chapter, we first outline the basic features of electricity as a commodity
and of electricity markets recalling the basic metrics used for assessing the market
performance. Then we propose a formalized representation of some of the most
common game theory models by taking into account network constraints. A com-
parison study on those game theory models through a unified conceptual example is
investigated, which has not been discussed in open literature so far. Such compari-
son study provides a quantitative assessment on the electricity market performance
affected by the different strategic gaming behaviors of the electricity producers.
Notations
N: Number of network buses
Nl: Number of network lines
NG: Number of generators
ND: Number of demand consumers
n: Index of the network bus set N
N ¼ [1 2,,. . ., n 1, n, n +1, . . ., N]
l: Index of the network line set ℒ
ℒ ¼ [1 2,,. . ., l 1, l, l +1, . . ., Nl]
g: Index of the electricity producer set G
G ¼ [1, 2, . . ., g–1, g, g + 1, . . ., NG]
d: Index of the electricity consumer set D
D ¼ [1, 2, . . ., d–1, d, d + 1, . . ., ND]
a/b Intercept ($/MW) and slope ($/MW/MW) parameters of the marginal cost curves of
the electricity producers, dim (a) ¼ dim (b) ¼ NG 1 (am
, bm
refer to marginal
cost curve parameters)
e/h Intercept ($/MW) and slope ($/MW/MW) parameters of the demand curve of the
electricity consumers, dim (e) ¼ dim (h) ¼ ND 1
p/q Power production and demand vector, MW, dim(p) ¼ NG 1, dim (q) ¼ ND 1
IG/ID All-one-element vector for producers/consumers, dim (IG) ¼ NG 1, dim
(ID) ¼ ND 1
P +
/P
Upper and lower production limits of the producers, MW, dim (P +
) ¼ dim (P
) ¼ NG 1
H Diagonal matrix formed by the vector h, dim (H) ¼ ND ND
Bm
Diagonal matrix formed by the vector bm
, dim (Bm
) ¼ NG NG
(continued)
6 E. Bompard and Y. Ma](https://image.slidesharecdn.com/1148042-250526014811-8fd922cc/85/Handbook-Of-Networks-In-Power-Systems-I-1st-Edition-Ettore-Bompard-20-320.jpg)
![2.1 Power Systems Operation
Power systems are composed a transmission system with buses, to which are
connected the generators and the loads, interconnected by lines and transformers
over the meshed network where the power injected by the generators is delivered to
load centers.
The operation of the system should satisfy a set of boundary conditions which
can be addressed by a set of equality and inequality constraints.The equality
constraints assure power balance between the electricity generation and load
demand while inequality constraints define a feasible operation region represented
by the line transmission capacity limits, system frequency and bus voltage ranges. If
all the equality and inequality constraints are satisfied the system is operated in its
normal state. The inequality constraints are satisfied with certain security margins
in terms of generation spinning reserve and/or transmission capacity reserve, etc. If
the reserve margin, due to some disturbances, is reduced, the system enters into the
alert state in which the constraints are still satisfied. Preventive control then takes
place to secure again proper security margins. If the preventive control fails or the
disturbance was severe enough the system goes into the emergency state in which
the inequality constraints are not; emergency control actions such as cutting of
faults, rerouting of generation, excitation control, fast-valving, generation tripping,
generation run-back, HVDC (high voltage dc) modulation, and load shedding are
undertaken. If emergency control fails the system will go into the extremis state
with cascading outages and system islanding. From this state the system operator
to push the system to restorative state, in which the system, matching again
generation and load, is driven back to the normal state [39].
Mathematically, the steady-state power system operation can be expressed as:
s ¼ fðxÞ ¼ 0 8 n 2 N (1)
gðxÞ 0 (2)
where s ¼ sP + jsQ is the vector of complex power (real and reactive) injection
at each bus n, n ∈N, f (·) is the vector of functions that express the complex power
transferred over the lines connected to each bus and x is the vector of unknown
phasor voltages (magnitudes and angles) at each bus; the group of inequities g(·)
represent the network constraints.
The group of Eq. 1 represents the so called AC power flow model where real and
reactive power determined by both the voltage magnitude and voltage angle at each
bus. The real power balance enforced at each bus is to keep the electricity frequency
at the expected value. Any unbalance of the real power between the electricity
generated and consumed causes the frequency drifting to a new value. The reactive
power balance at each bus is to govern the voltage magnitude, i.e. to generate
capacitive reactive power to restore the voltage magnitude from low value to high
value or to absorb the inductive reactive power to reverse the process.
8 E. Bompard and Y. Ma](https://image.slidesharecdn.com/1148042-250526014811-8fd922cc/85/Handbook-Of-Networks-In-Power-Systems-I-1st-Edition-Ettore-Bompard-22-320.jpg)
![• Line flow limits. Lines have some limitation on the maximum power they can
transmit due to thermal, voltage drop and stability limits. If the power flow of the
branch from bus 1 to bus 2 reaches its limit, the system should re-route the power
through other lines to be kept feasible.
2.3 Electricity Markets
The energy market in most countries is organized as a day-ahead market (DA)
where the electricity energy transactions are cleared for each hour of the next day.
In day-ahead market, demand is forecasted for each trading interval, e.g. 1 h period,
24 h ahead and offers and bids are received from the market participants. The
market clearing is conducted by an independent body, which may coincide with the
Independent System Operator (ISO), to match demand and supply. In the pool
models the dispatching of injected and withdrawn power quantities are assigned
considering the transmission limits and providing, as a by-product, the locational
marginal prices (LMPs), such as in PJM Interconnection, in order to capture the
network impacts on the market clearing [40].
Gen.1 Ld.1 ; Gen.2 Ld.2 ;
0.2 p.u. 0.2 p.u.
Economic transactions
Gen.2 Ld.1
0.2 p.u.
G
G
0.1517+j0.0794
Gen.1
Gen.2
Ld.2
Ld.1
0.4 + j0.1315 p.u.
0.2024+j0.1104
Z Z
0.1511+j0.0765
0.05074+j0.031 Z
Z
0.05067+j0.0307
0.1493+j0.01 0.1498+j0.0122
0.2489+j0.055
0.2502+j0.0616
0.4 + j0.0738 p.u.
0.2+ j0.0406 p.u.
Bus1
Z = 0.02 + j0.1 p.u.
Physical power flows
V: 1 / 0
V: 0.9959 /- 0.2562 deg. V: 1 / 0.5914 deg.
V: 0.9891/ -0.7867 deg.
Bus2
Bus3 Bus4
p.u. : per unit of system base value
j: imaginary unit of the complex number
Fig. 1 Network impacts on electric power transactions
10 E. Bompard and Y. Ma](https://image.slidesharecdn.com/1148042-250526014811-8fd922cc/85/Handbook-Of-Networks-In-Power-Systems-I-1st-Edition-Ettore-Bompard-24-320.jpg)
![Several related markets concur to make the electricity transactions possible,
including ancillary service market and transmission right market. Ancillary market
is organized to acquire on the market all the services needed for the operation of the
power system as scheduling, system control and dispatch, reactive and voltage
support, regulation and frequency response, energy imbalance, spinning reserves
and supplemental reserves. Transmission right market is designed for auctioning the
right to assure the availability of transfer capability over the network or edge against
the risk of change in transmission cost due to the scarcity of transfer capability.
There are two types of transmission rights: physical transmission rights (PTRs) and
financial transmission rights (FTRs). However, PTRs are gradually replaced by
FTRs, defined upon the locational marginal prices, due to its superiorities over
PTRs by opening the network access to all the market participants. Several of the
restructured US electricity markets have already experimented with styles of the
transmission rights markets in the last decade [41].
Strategic bidding behaviors are extensively investigated in the day-ahead energy
market whilst, due to their functional complexity, ancillary markets attract more
technical concerns than the economic behaviors of the market participants. There-
fore, in this chapter we study the strategic bidding behaviors of the electricity
producers in the day-ahead market clearing process.
2.4 Reference Paradigms of the Electricity Markets
A major objective of electricity deregulation is to achieve a workably competitive
wholesale market. Wholesale electricity markets are organized with several gener-
ation companies that compete to sell their electricity in a centralized pool and/or
through bilateral contracts with large electricity buyers [42–44]. The transac-
tions among sellers and buyers can take place in an organized market (“power
exchange”) that collects all the offers from the generators and the bids from the
loads and performs a centralized market clearing compatible with the network
constraints, as in the “pool paradigm”; otherwise sellers and buyers arrange, on a
bilateral basis, their own transactions, submitting afterward to an ISO to check their
feasibility in terms of the network constraints as in the “bilateral paradigm.”
A vast majority of studies on electricity markets to date either explicitly or
implicitly assume a centralized auction process, administered by a pool, through
which generators sell energy to consumers. A growing number of studies typically
assume a decentralized trading process by which generators sell to consumers
bilaterally through power exchanges or arbitragers [38, 45, 46].
2.4.1 Pool Markets
Many of the restructuring experiences (e.g., in the UK, Argentina, Chile, Australia)
have been based on pool trading with centralized coordination in the Power
Models of Strategic Bidding in Electricity Markets 11](https://image.slidesharecdn.com/1148042-250526014811-8fd922cc/85/Handbook-Of-Networks-In-Power-Systems-I-1st-Edition-Ettore-Bompard-25-320.jpg)
![Exchange (PX) by an Independent System Operator (ISO). Examples include
Australia, Argentina, the PJM Interconnection and the New England Power Pool.
In the pool market all producers sell into a pool run by an independent entity.
The ISO has also the responsibility for system reliability and transmission
congestion management. Producers’ physical sales of power and energy trades
are all within the pool. The pool is the only buyer (for the producers) and the
only seller (for the electricity consumers). The ISO holds central auction in which
each generator bids different prices for different quantities (from specific plants or
as a portfolio) or an offer curve for the trading period; for example, for each hour of
the following day. Based on the bids and the considered demand quantities, the ISO
uses a security based dispatch process to set the market price and the generation
quantities [42–44].
2.4.2 Bilateral Markets
Under bilateral trading model buyers and sellers individually contract with
each other for power quantities at negotiated prices, terms, and conditions. All
transactions must be announced to the ISO, which analyzes all the trades in each
period and determines, without discrimination, which ones are infeasible for grid
security constraints. The ISO does not need to know the prices and demand side
bids may co-exist with the generation bids. Generally, the bilateral types of markets
are split into two markets named forward contract market (PX) and spot market
(ISO). Two steps are needed to arrange the bilateral trading:
• Step 1: Dispatch without network constraints;
• Step 2: Re-dispatch with consideration of transmission constraints based on the
adjustment offers (demands) from generators and loads, in case that the dispatch
would lead to network constraints violations.
Gen. offers
Load demands
Network
structure
Network
constraints
Market
clearing
quantities
prices
Max
objective
ISO
Ld. 1
POOL
Gen. G
Gen. g
Gen. 1
…
…
…
…
Ld. d Ld. D
$ MWh
Fig. 2 Pool type market
12 E. Bompard and Y. Ma](https://image.slidesharecdn.com/1148042-250526014811-8fd922cc/85/Handbook-Of-Networks-In-Power-Systems-I-1st-Edition-Ettore-Bompard-26-320.jpg)
![which states that quantity demanded falls as a price rises, Fig. 7. The degree of such
price responsiveness is called demand elasticity. Under regulation, electricity
demand was considered inelastic (fixed amount with no price responsiveness) and
new capacity was built to cover the projected demand to minimize investment plus
operation costs. Under deregulation, the consumers’ demands for electricity are
encouraged to be price responsive to enhance the wholesale market efficiency and
system reliability [47, 48]. Strategic behaviors from the supply side are more
evident when the demand elasticity of the electricity consumers is low. Improving
demand elasticity plays a positive contribution in mitigating the strategic bidding
behaviors, pushing the uncompetitive electricity market performance towards to a
high level competitive one [49].
3.2 Market Equilibrium
A market is a real or virtual environment in which buyers and sellers interact to
exchange goods, services or commodities; the outcome of a market, from a macro-
scopic viewpoint is the quantity and the price of the good traded.
Given the supply and demand curves, the electricity market is cleared at the
equilibrium at which the market clearing price is established such that the amount
of goods or services sought by buyers is equal to the amount of goods or services
produced by sellers. Graphically, the market equilibrium is represented by the
intersection of the supply curve and the demand curve, point E in Fig. 4. The
equilibrium is established in a price-quantity adjustment process where incremental
quantity of electricity is supplied at the price that demand customers willing to buy,
i.e. the supplier offer price is lower than the demand bid price. Equivalently, the
market clearing can be expressed mathematically as
A
E
Price
Q
Quantity
λN
demand curve
supply curve
B
Fig. 4 Electricity supply and demand curves
14 E. Bompard and Y. Ma](https://image.slidesharecdn.com/1148042-250526014811-8fd922cc/85/Handbook-Of-Networks-In-Power-Systems-I-1st-Edition-Ettore-Bompard-28-320.jpg)
![Max 1=2 qT
Hq þ qT
e 1=2 pT
Bp þ pT
a
(3)
s:t: ID
T
q IG
T
p ¼ 0 (4)
The market clearing price, lN, in Fig. 4. is the Lagrange multiplier value of the
optimization problem (19)–(20) and can be expressed as
lN ¼
IT
GB1
a IT
DH1
e
IT
DH1
ID IT
GB1
IG
(5)
The market clearing quantities are
pg ¼ ðlN agÞ=bg 8g 2 G (6)
qd ¼ ðlN edÞ=hd 8d 2 D (7)
In Eqs. 3–7, the vector of a and the diagonal matrix B indicate the parameters of
the supply curves of the electricity producers. Those parameters are not necessarily
equal to the parameters of the marginal cost curves, i.e. am and Bm. The strategic
bidding behaviour of the electricity producers can be represented by choosing the
parameters of a and/or B to achieve the economic goals in the market clearing.
3.3 Levels of Competition in the Markets
According to the competition level, market is classified as perfect one or imperfect.
In a perfect competitive market, all sellers and buyers are “price-takers” who
assume that their own production and purchase decisions do not affect the
market price [50]. According to the classic economic theory, a price-taking firm
that wishes to maximize its profits would bid the products at its own marginal cost
[51]. Each supplier submits the marginal cost as supply function and the social
surplus, SS
, is maximized in the market clearing, as mentioned in the last
section. Price-taking behaviors will lead to the most efficient market operation
characterized by the least cost of the production while meeting the demand of the
consumers. In reality, perfect competition is difficult to be implemented in which-
ever real market due to the strict conditions need to be satisfied, such as a
large number of price-taking producers with a very small market share produce
homogeneous and perfectly substitutable products. Nevertheless it can serve as a
reference case to identify the market power behaviors in a real implemented market
[22, 52, 53].
The opposite situation to perfect competition is monopoly in which just one
producer faces all the market demand. Monopoly market is thought to have no
competition. Perfect competition and monopoly represent the two extreme cases of
Models of Strategic Bidding in Electricity Markets 15](https://image.slidesharecdn.com/1148042-250526014811-8fd922cc/85/Handbook-Of-Networks-In-Power-Systems-I-1st-Edition-Ettore-Bompard-29-320.jpg)
![The price distortion may be then measured by
Kl ¼ ðl lN
Þ=lN
(17)
For the index for quantity, the general form (28) is adapted as
KP ¼ ðIG
T
p IG
T
p
Þ=IG
T
p
(18)
4 Modeling Strategic Bidding Under Network Constraints
in Pool Model
The analysis of strategic bidding behaviors seeks to answer basic questions; includ-
ing how a firm exercises strategic biddings and to what extent the strategic bidding
behavior affects equilibrium quantities, prices, and market efficiency. Such assess-
ment, at least implicitly, requires a comparison of observations of real world market
prices and quantities to the comparable values of the variables that the perfectly
competitive model predicts.
Game theory provides an efficient tool to model the strategic bidding behaviors
of the market participants. The solution of the game models is to derive the market
clearing results at Nash Equilibrium (NE) by which the distortion of the perfect
competition can be predicted and measured for assessing the strategic bidding
impacts. The key point in formulating the game models to represent the strategic
bidding behavior is a bi-level programming problem where the market clearing is
inserted, as subject conditions, into the individual Producer Surplus Maximization
(PSM) problem [11, 24–26, 49]. The market clearing is to maximize the system
surplus taking into account the physical constraints of electricity networks
represented by the power flow model. According to the Karush-Kuhn-Tucker
(KKT) conditions, the market clearing is transformed into a group of equality and
inequality constraints of the individual PSM problem. The optimal variable derived
from the individual PSM problem is the optimal strategies that the producer will
submit to the market coordinator.
With the individual PSM problems, the Nash Equilibrium is derived at the point
no player can be benefited from changing his/her strategy when his/her competitors
do not. Very often, the individual PSM problem can be quite complicated with the
large size of the electricity network to be considered. The derivation of the NE and
the unique/existence of the NE are general concerns in many related references [22,
23, 49, 54]. The unique/existence NE may be guaranteed in an analytical way under
simple systems while for the complex systems, numerical approaches is usually
employed and ex-post check of the NE is needed.
Models of Strategic Bidding in Electricity Markets 19](https://image.slidesharecdn.com/1148042-250526014811-8fd922cc/85/Handbook-Of-Networks-In-Power-Systems-I-1st-Edition-Ettore-Bompard-33-320.jpg)
![4.1 Electricity Network Model
In the pool operated electricity markets, the Independent System Operator (ISO) is
responsible the aggregate offers from the supply side and the aggregate demands
from the demand side for a specified time interval, usually 1 h.
Due to the peculiarities of the electricity transmission, the transactions must be
settled according to the physical conditions of the electricity network and different
nodal prices may arise when the flow limits are binding. In addressing the network
features of the electricity transactions, the DC power flow model is popularly
employed in studying the strategic interactions in the competitive electricity
markets [22, 26, 49]. DC power flow provides a fairly good approximation of the
AC power flow in terms of the real power flow due to the fact of the large ratio of X/
R and invariant of the voltage magnitude of the power transmission network where
wholesale electricity transaction is accommodated. The characteristics of the DC
power flow is summarized as
• Reactive power balance at each bus n, fn
Q
, is disregarded
fn
Q
¼ 0 8n 2 N (19)
• Lossless network
Pl
loss
¼ 0 8l 2 L (20)
• A group of linear expressions of the real power flow of the branches, fl
P
, in
terms of the bus voltage angles
fl
P
¼ Ylu 8l 2 L (21)
Where Yl is the row vector of the branch matrix Y (Nl Nb1) with the positive
admittance value at the element of row index l (line l) and column index of from-
bus f and negative admittance value of to-bus t; u is a column vector of the bus
voltage angles. Both of the Y and u matrix are formulated without the elements
related to the reference bus.
In terms of the net injection of the active power at each bus, the real power flow
of the branches can be expressed as
fl
P
¼ Yy ¼ Y B1
ðp qÞ ¼ Jðp qÞ (22)
Where, J ¼ Y B 1
, is called power transfer distribution factor (PTDF) matrix,
J, Nl (N1); matrix B is the admittance matrix, (N1) (N1), and elements
of the column and row related to the reference bus are not included; p q is vector
of the net injection of the active power at each bus.
20 E. Bompard and Y. Ma](https://image.slidesharecdn.com/1148042-250526014811-8fd922cc/85/Handbook-Of-Networks-In-Power-Systems-I-1st-Edition-Ettore-Bompard-34-320.jpg)
![affect their results in the market. Market power analysis in terms of the strategic
bidding behaviours can be gained from the use of game-theoretic models through
simulating the competition between a given set of competitors in a well-specified
market environment, taking into account the network constraints that may provide
additional possibilities of market power arising that are very specific of this in
electricity markets [52, 55].
According to the classification of the strategic variables, there are three types of
game models which are bidding in price, bidding in quantity and supply function
bidding models.
The price bidding models include Bertrand and Forchheimer models. However,
taking into account network constraints for the analysis of hourly electricity
markets, so far there no literature using price bidding game models as an efficient
tool. Another reason for the Bertrand model has not been the focus in the literature
would be that Bertrand model might correspond to perfect competition case [56].
The quantity bidding game models include Cournot, Stackelberg and Conjec-
tural Supply Function (CSF) models. An essential assumption of the former two
models is that the individual player’s own output decision will not have an effect on
the decisions of its competitors, i.e. the optimal offer quantities are assumed fixed in
formulating the optimal strategy of the considered player. As for the CSF model,
the basic assumption is that the output of the other competitors can be estimated to
change in an expected way with respect to the output decision of the considered
player. Since the strategic variable is the quantity of the electricity transacted, those
game models do not give meaningful equilibrium when price elasticity of the
demand curve is low (the demand quantity is fixed with the zero value of the
price elasticity).
The supply function bidding models choose a strategic supply function from the
marginal cost curve with the aim of maximizing individual producer surplus.
Different the quantity bidding game models, for the optimal strategy formulation
of one producer, the given strategies are the supply functions of other competitors
but the production quantities. The dispatched quantities of the competitors are
determined by the supply function of the considered producer through current
decision making process.
Taking into account the strategic bidding behaviors of the electricity suppliers,
the quantity, p and the price at the reference bus, lN, become the variables with
respect to the strategic variables, r (NG 1), that are p(r) and lN (r) in problem
(32)(40). According to the strategic variable of the different game modes, the of
the lN (r) and p(r) are summarized in Table 2.
The information given in Table constitutes the core part of the game theory
models in a formalized way. Such formulations will be used in the conceptual
examples to obtain the market results under game theory models.
Models of Strategic Bidding in Electricity Markets 23](https://image.slidesharecdn.com/1148042-250526014811-8fd922cc/85/Handbook-Of-Networks-In-Power-Systems-I-1st-Edition-Ettore-Bompard-37-320.jpg)
![Table 2 Price at the reference bus and production of the players according to different game
models
Model Strategy
variable rg
Formulation
Cournot pg 8 g∈ G
lN ¼
IT
Gp0
þ pg þ IT
DH1
½JT
Dðmþ
m
Þ þ e
IT
DH1
ID
where, p0
¼ [p1
0
,. . ., pg–1
0
, 0, pg+1
0
,. . ., pNG]T
, pi
0
(i∈G, i 6¼ g) is the
optimal offered quantity derived from the last move of the producer i,
pg is the decision variable in the optimal problem
Stackelberg pg
l
8 g∈
Gl G lN ¼
IT
FpF
þ IT
LP0L
þ pL
g þ IT
DH1
½JT
Dðmþ
m
Þ þ e
IT
DH1
ID
where P0 L
¼ [P1
0
,. . ., Pi – 1
0
, 0, Pi+ 1
0
,. . ., PNGl
0
]T
, Pi
0L
(i∈ G l G,
i 6¼ g∈ G l, G l is the set of the leader producers and NGl is the number
of the leaders) is the optimal offered quantity of the leader i, pg
L
is the
decision variable in the optimal problem; pF
is the vector of the
optimal production quantities of the followers
If the slope parameter of the marginal cost curve of the follower producers
are identical, b1
m
¼ b2
m
¼,. . ., bi
m
¼,. . ., ¼ bNGf
m
¼ bm
, (i ∈
Gf G, G f is set of the follower producers and G f G l ¼ G, NGf is
the number of the followers), the optimal production quantities of the
followers can be derived from the first order rule of the producer
surplus maximization as
IT
FpF ¼
ðIT
DH1
IDÞðIT
Fam
FÞ NGf ðpl þ IT
LP0
L þ IT
DH1
eÞ
NGf þ 1 bmIT
DH1
ID
Conjecture supply
function (CSF)
pg, 8 g∈G
lN ¼
pg þIT
DH1
½JT
Dðmþ
m
ÞþerT
g JT
Gðmþ
m
ÞþIT
Gpg1
rT
g Lg1
g
IT
DH1
ID IT
Grg
where
rg ¼ [r1,g, r2,g, . . ., rg–1,g, 0, rg+1, g, rNG, g]T
p g1
¼ [p1
g–1
, p2
g–1
, . . ., pg–1
g–1
, 0, pg+1
g–1
, . . ., pNG
g–1
]T
Lg
g–1
¼ [L1
g–1
, . . ., L g–1
g–1
, 0, L g +1
g–1
, . . ., L NG
g–1
]T
pi
g–1
and Li
g–1
are the dispatched quantity and the nodal price of producer
i derived from the last move of the producer g1, respectively. rg,i, 8g,
i∈G, i 6¼ g, represents the assumed rate of change in competitor
supply per unit price. The CSF function is
pi ¼ pi
g–1
+ rg, i (li Li
g1
) 8i∈G, i 6¼ g
SFE Intercept ag 8 g∈G
lN ¼
IT
GðBm
Þ1
½JT
Gðmþ
m
Þ þ a0
IT
DH1
½JT
Dðmþ
m
Þ þ e
IT
GðBm
Þ1
IG IT
DH1
ID
The supply function
is ag + bg
m
pg
pg ¼
lN JT
g ðmþ
m
Þ ag
bm
g
where a0
¼ [a1
0
, a2
0
,. . ., ag1
0
, ag, ag+1
0
, . . ., aNG
0
]T
; ai
0
, i 6¼ g, 8 i, g∈ G, is
the obtained value derived from the last move of producer i
SFE slope bg 8 g∈G
lN ¼
IT
GðB0
Þ
1
½JT
Gðmþ
m
Þ þ am
IT
DH1
½JT
Dðmþ
m
Þ þ e
IT
GðB0
Þ
1
IG IT
DH1
ID
The supply function
is ag
m
+ bg pg
pg ¼
lN JT
g ðmþ
m
Þ am
g
bg
where B0
is the diagonal matrix formulated with the vector of [b1
0
, . . .,
bg1
0
, bg, bg+1
0
, . . ., bNG
0
]T
; bi
0
, i 6¼ g, 8 i, g∈ G, is the obtained value
derived from the last move of producer i
(continued)
24 E. Bompard and Y. Ma](https://image.slidesharecdn.com/1148042-250526014811-8fd922cc/85/Handbook-Of-Networks-In-Power-Systems-I-1st-Edition-Ettore-Bompard-38-320.jpg)
![4.5 Nash Equilibrium and Search Methods
The scope of the game is to get the optimal strategy of each market player derived
at the Nash Equilibrium (NE). In terms of solving techniques for the NE, there
are two different to be considered: best response functions and iterative search
algorithm. The former is an analytic approach that can be used to analyze simple
duopoly games without network consideration while the latter is a numerical
approach that is suitable for studying muli-player games with network
consideration.
4.5.1 Best Response Functions
Best response function is an analytic approach to obtain the oligopoly market
equilibrium. The idea behind the best response function approach is that all the
players get the maximum surplus concurrently. Mathematically, define the best
response function of producer g, in terms of strategies of all the producers, r, as the
first derivative of his/her producer surplus with respect to his/her own strategic
variable, rg, as
BgðrÞ :¼ @Sg
G
ðrÞ=@rg ¼ 0 ; 8g 2 G (41)
Based on, (34) we have
BgðrÞ ¼ ðlNðrÞ Jg
T
ðmþ
m
ÞÞ@pgðrÞ=@rg þ ð@lNðrÞ=@rgÞpgðrÞ
ag
m
bg
m
pgðrÞ; 8g 2 G (42)
The Nash equilibrium is derived by solving the group of the best response
functions simultaneously, i.e. the intersection point of the best response functions.
This approach is suitable for simple problems especially in duopoly game
models (two players) and no power system operation considered. In case of network
constraints to be considered, a specific branch power flow state can be integrated
Table 2 (continued)
Model Strategy
variable rg
Formulation
SFE k multiplier kg 8 g∈G
lN ¼
IT
GK½JT
Gðmþ
m
Þ þ IT
GðBm
Þ1
am
þ IT
DH1
½JT
Dðmþ
m
Þ þ e
IT
GKiG IT
DH1
iD
The supply
function is
kg(ag
m
+ bg
m
pg)
pg ¼
lN JT
g ðmþ
m
Þ kgam
g
kgbm
g
where K is a diagonal matrix formulated by the vector [1/(k1
0
b1
m
), . . ., 1/
(kg–1
0
bg–1
m
), 1/(kg bg
m
), 1/(kg+1
0
bg+1
m
), . . ., 1/(kNG
0
bNG
m
)]T
; ki
0
, i 6¼ g,
8 i, g∈ G, is the obtained value derived from the last move of
producer i
Models of Strategic Bidding in Electricity Markets 25](https://image.slidesharecdn.com/1148042-250526014811-8fd922cc/85/Handbook-Of-Networks-In-Power-Systems-I-1st-Edition-Ettore-Bompard-39-320.jpg)
![or congested in the negative direction (1). Therefore, each player, in choosing a
move, should consider 3z
alternatives if there are z lines to be considered in the
power system network. In each state, the complementary relationships can be
determined. For instance, given the state of the three lines of interest as 1, 1, 0,
(37)(39) are transformed as
F3 J3ðp qÞ F3 (46)
J1ðp qÞ ¼ F1 (47)
J2ðp qÞ ¼ F2 (48)
m1
þ
0; m2
þ
0; m3
þ
¼ 0
Where the subscript of 1, 2, 3 in (46)–(47) are the indices of the three considered
lines.
The PSM problem is formulated by model of (32)–(40) and the price at the
reference bus lN and the production of the players, pg, listed in Table 2. The
iteration process is terminated by either the Nash Equilibrium derived by checking
the maximal derivation from the last move strategy, smaller than a gate value of f,
or by exceeding the predefined iteration steps. The latter case the Nash Equilibrium
has not been found.
An important note is that the unique/existence of the NE is a general concern in
many related references [12, 17, 22, 23, 25, 26]. So far the unique NE can be
guaranteed under simple test system by using the best response functions in
analytical way [16, 49]. As for the complicate game problems where iterative
search algorithm is used, the existence of equilibrium can not be guaranteed
analytically and ex-post check is needed [22].
4.6 Conceptual Examples
In this section, two examples are designed for comparison study of different game
theory models in assessing the strategic bidding behaviors of the electricity
producers. The first example is a duopoly game in an electricity market modeled
by a simple three-bus power system whilst the is a relative complicated one
with multi-players in the electricity market represented by a large size power
system.
4.6.1 Test Case1: A Conceptual Application for Duopoly Game
The three-bus test system is shown in Fig. 8. The system is characterized by two
generators and one load respectively connected at the bus 1, 2 and 3 (reference bus),
28 E. Bompard and Y. Ma](https://image.slidesharecdn.com/1148042-250526014811-8fd922cc/85/Handbook-Of-Networks-In-Power-Systems-I-1st-Edition-Ettore-Bompard-42-320.jpg)
Handbook Of Networks In Power Systems I 1st Edition Ettore Bompard
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- 6. Energy Systems Series Editor: Panos M. Pardalos, University of Florida, USA For further volumes: http://www.springer.com/series/8368
- 7. .
- 8. Alexey Sorokin l Steffen Rebennack l Panos M. Pardalos l Niko A. Iliadis l Mario V.F. Pereira Editors Handbook of Networks in Power Systems I
- 9. Editors Alexey Sorokin University of Florida Industrial and Systems Engineering Weil Hall 303 32611 Gainesville Florida USA sorokin@ufl.edu Steffen Rebennack Colorado School of Mines Division of Economics and Business Engineering Hall 15th Street 816 80401 Golden Colorado USA srebennack@mines.edu Panos M. Pardalos University of Florida Dept. Industrial & Systems Engineering Weil Hall 303 32611-6595 Gainesville Florida USA pardalos@ufl.edu Mario V.F. Pereira Centro Empresarial Rio Praia de Botafogo -A-Botafogo 2281701 22250-040 Rio de Janeiro Rio de Janeiro Brazil mario@psr-inc.com Niko A. Iliadis EnerCoRD - Energy Consulting Research & Development Plastira Street 4 171 21 Athens Nea Smyrni Greece niko.iliadis@enercord.com ISSN 1867-8998 e-ISSN 1867-9005 ISBN 978-3-642-23192-6 e-ISBN 978-3-642-23193-3 DOI 10.1007/978-3-642-23193-3 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2012930379 # Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
- 10. Handbook of Networks in Power Systems: Optimization, Modeling, Simulation and Economic Aspects This handbook is a continuation of our efforts to gather state-of-the-art research on power systems topics in Operations Research. Specifically, this handbook focuses on aspects of power system networks optimization and is, as such, a specialization of the broader “Handbook of Power Systems I & II,” published by Springer in 2010. For decades, power systems have been playing an important role in humanity. Industrialization has made energy consumption an inevitable part of daily life. Due to our dependence on fuel sources and our large demand for energy, power systems have become interdependent networks rather than remaining independent energy producers. Such dependence has revealed many potential economic and operational chal- lenges with energy usage and the need for scientific research in this area. In addition to fundamental difficulties arising in power systems operation, the industry has experienced significant economic changes; specifically, the power industry has transformed from being controlled by government monopolies to becoming deregu- lated in many countries. Such substantial changes have brought new challenges in that many market participants maximize their own profit. The challenges mentioned above are categorized in this book according to network type: Electricity Network, Gas Network, and Network Interactions. Electricity Networks constitute the largest and most varied section of the hand- book. Electricity has become an inevitable component of human life. An over- whelming human dependence on electricity presents the challenge of determining a reliable and secure energy supply. The deregulation of the electricity sector in many countries introduces financial aspects such as forecasting electricity prices, deter- mining future investments and increasing the efficiency of the current power grid through network expansion and transmission switching. The Gas Networks section of the book addresses the problem of modeling gas flow, based on the type of gas, through a pipeline network. The section describes the v
- 11. problem of long-term network expansion as well as the optimal location of network supplies. Deregulation of the gas sector is becoming common in many countries. The deregulation presents new decisions to the gas industry including determining optimal market dispatch and nodal prices. Network Interactions are common in power systems. This section of the book addresses the interaction between gas and electricity networks. The development of natural gas fired power plants has significantly increased interdependence between these two types of networks. This handbook is divided into two volumes. The first volume focuses solely on electricity networks, while the second volume covers gas networks, and network interactions. We thank all contributors and anonymous referees for their expertise in providing constructive comments, which helped to improve the quality of this volume. Furthermore, we thank the publisher for helping to produce this handbook. Alexey Sorokin Steffen Rebennack Panos M. Pardalos Niko A. Iliadis Mario V.F. Pereira vi Handbook of Networks in Power Systems
- 12. Contents Part I Electricity Network Models of Strategic Bidding in Electricity Markets Under Network Constraints ........................................................... 3 Ettore Bompard and Yuchao Ma Optimization-Based Bidding in Day-Ahead Electricity Auction Markets: A Review of Models for Power Producers ............. 41 Roy H. Kwon and Daniel Frances Finding Joint Bidding Strategies for Day-Ahead Electricity and Related Markets .......................................................... 61 Patricio Rocha and Tapas K. Das Short-Term Electricity Market Prices: A Review of Characteristics and Forecasting Methods ................................ 89 Hamid Zareipour Forecasting Prices in Electricity Markets: Needs, Tools and Limitations ............................................................... 123 H. A. Gil, C. Gómez-Quiles, A. Gómez-Expósito, and J. Riquelme Santos ECOTOOL: A general MATLAB Forecasting Toolbox with Applications to Electricity Markets ................................... 151 Diego J. Pedregal, Javier Contreras, and Agustín A. Sánchez de la Nieta vii
- 13. Electricity Markets Simulation: MASCEM Contributions to the Challenging Reality ................................................... 173 Zita A. Vale, Hugo Morais, Tiago Pinto, Isabel Praça, and Carlos Ramos Differentiated Reliability Pricing Model for Customers of Distribution Grids ......................................................... 213 Arturas Klementavicius and Virginijus Radziukynas Compromise Scheduling of Bilateral Contracts in Electricity Market Environment ......................................... 241 Sergey I. Palamarchuk Equilibrium Predictions in Wholesale Electricity Markets ............... 263 Talat S. Genc The Economic Impact of Demand-Response Programs on Power Systems. A Survey of the State of the Art ...................... 281 Adela Conchado and Pedro Linares Investment Timing, Capacity Sizing, and Technology Choice of Power Plants ...................................................... 303 Ryuta Takashima, Afzal S. Siddiqui, and Shoji Nakada Real Options Approach as a Decision-Making Tool for Project Investments: The Case of Wind Power Generation .......... 323 José I. Muñoz, Javier Contreras, Javier Caamaño, and Pedro F. Correia Electric Interconnections in the Andes Community: Threats and Opportunities .................................................. 345 Enzo Sauma, Samuel Jerardino, Carlos Barria, Rodrigo Marambio, Alberto Brugman, and José Mejía Planning Long-Term Network Expansion in Electric Energy Systems in Multi-area Settings ..................................... 367 José A. Aguado, Sebastián de la Torre, Javier Contreras, and Álvaro Martínez Algorithms and Models for Transmission Expansion Planning .......... 395 Alexey Sorokin, Joseph Portela, and Panos M. Pardalos viii Contents
- 14. An Approximate Dynamic Programming Algorithm for the Allocation of High-Voltage Transformer Spares in the Electric Grid ............... 435 Johannes Enders, Warren B. Powell, and David Egan Decentralized Intelligence in Energy Efficient Power Systems ........... 467 Anke Weidlich, Harald Vogt, Wolfgang Krauss, Patrik Spiess, Marek Jawurek, Martin Johns, and Stamatis Karnouskos Realizing an Interoperable and Secure Smart Grid on a National Scale ........................................................... 487 George W. Arnold Power System Reliability Considerations in Energy Planning ........... 505 Panida Jirutitijaroen and Chanan Singh Flexible Transmission in the Smart Grid: Optimal Transmission Switching ...................................................... 523 Kory W. Hedman, Shmuel S. Oren, and Richard P. O’Neill Power System Ancillary Services ........................................... 555 Juan Carlos Galvis and Antonio Padilha Feltrin Index .......................................................................... 581 Contents ix
- 15. .
- 17. Models of Strategic Bidding in Electricity Markets Under Network Constraints Ettore Bompard and Yuchao Ma Abstract Starting from the nineties of the last century, competition has been introduced in the electricity industry around the world, as a tool to increase market efficiency and decrease prices. Electricity is a commodity that needs to be traded over a physical network with strict physical and operational constraints that cannot be found in other commodity markets. Present electricity markets may be better described in terms of oligopoly than of perfect competition from which they may be rather far. In an oligopoly market, the producer is a market player that shows strategic behavior, submitting offers higher than the marginal costs, as they under perfect competition, with the aim to maxi- mize its individual surpluses. The market clearing price, quantities and the market efficiency depending on the strategic interactions among producers must be taken into account in modeling competitive electricity markets. The network constraints provide very specific opportunities of exercising strate- gic behaviors to the market participants. Game theory provides a conceptual framework and analytical tool to model such a context. The modeling of electricity markets will be presented by discussing the tradi- tional Game Theory models, such as bertrand, cournot, conjecture supply function, supply function equilibrium, adapted to be able to capture, in determining the Nash equilibrium, the network structure of the system in which the market is E. Bompard (*) Department of Electrical Engineering, Polytechnic di Torino, Torino, Italy CERIS-CNR (Institute for Economic Research on Firms and Growth of the National Research Council), Moncalieri (TO), Italy e-mail: ettore.bompard@polito.it Y. Ma Department of Electronic & Electrical Engineering, University of Strathclyde, Glasgow, UK e-mail: yuchao.ma@eee.strath.ac.uk A. Sorokin et al. (eds.), Handbook of Networks in Power Systems I, Energy Systems, DOI 10.1007/978-3-642-23193-3_1, # Springer-Verlag Berlin Heidelberg 2012 3
- 18. implemented. A formalized representation and a comparison of some of the most common game theory models will be provided with some conceptual examples. In addition, some newly proposed approaches for strategic bidding modeling based on the complex systems techniques such as Multi Agent systems and Complex Networks will be mentioned and some related references provided. Keywords Electricity markets • Game theory • Network constraints • Strategic bidding 1 Introduction The electric power industry has over the years been dominated by large state-owned monopolies that had an overall authority over all the activities in generation, transmission and distribution of power within their jurisdiction. Chile is often considered as the first Country to introduce liberalization in the electricity sector in 1982. Regulatory reforms of the industry in the United States started in 1978 with the passage of the Public Utility Regulatory Policies Act; regulatory reform was accelerated over the latter half of the 1990s with the advent of the open access transmission regime in 1996, the subsequent formation of several large regional spot markets, later, regional transmission organizations e.g. PJM, 1997, CAISO, 1998, Midwest 2002, etc. [1]. The first initial steps of liberalization of electricity markets in Scandinavia started in Norway in 1990. Through the subsequent steps of development through market expansion, the Nordic market became the world’s first multi-national market that was quite well-functioning [2]. Since 1996 the genera- tion, distribution and supply of electricity in eastern and southern Australian states has been amalgamated under the National Electricity Market and in 2009 the Australian Energy Market Operator (AEMO) has been established. Australia took the forefront of energy industry reform worldwide, one of the first countries to establish highly competitive and transparent electricity markets underpinned by strong governance structures [3]. In UK, The Electricity Pool of England and Wales was created in 1990 to balance electricity supply and demand, acting as a clearing house between generation and wholesale. In March 2001 the electricity pool is replaced by the New Electricity Trading Arrangements (NETA). On 1 April 2005, British Electricity Trading and Transmission Arrangements (BETTA) is introduced to replace the NETA in England and Wales, and the separate arrangements that existed in Scotland and the British Grid System Agreement, to create fully-compet- itive, British-wide wholesale market for the trading of electricity generation [4]. The justification for introducing competition in the electricity sectors is that in a monopoly it is not possible to achieve, no matter which market rules are designed, two important objectives at the same time: to hold down prices to marginal costs and to maximize efficiency [5]. On the contrary, in a competitive market those two objectives may be reached by a proper market design and the specification of a proper set of rules. In this respect, the goal of electric industry restructuring is to 4 E. Bompard and Y. Ma
- 19. achieve a better, more efficient allocation of resources by increasing the role of market forces, and simultaneously decreasing the role of regulation. The main objectives of the reforms are achieved through a clear separation between produc- tion and sale of electricity and the operation of electric power grid. Based on the market reform, many market based roles have been penetrating into the electricity industry in progress with a slight relaxation of the obligation to serve the loads that has been segregated and assigned to various entities. Among the new roles energy producers and retailers, brokers, independent system operator (ISO) or transmis- sion system operator (TSO) are the most popular entities existing in real electricity markets around the world. There are two reference paradigms proposed for electricity markets: pool and bilateral. Pool electricity markets coordinate the selling and buying activities through a centralized market place administrated by third entity that may coincide with the ISO, whilst the bilateral markets the transactions to be contracted between the seller and buyer directly on a private basis. In the pool paradigm the market optimum is reached a central decision making run after collecting the offers from the producers and the bids from the customers while in the bilateral paradigm the decision making process is distributed among various sellers and buyer that meet in the marketplace. Electricity markets are pretty different from other commodity markets mainly due to the physical constraints related to the network structure that may impact the market performance. The network constraints and the special features of the electricity provide the market players an opportunity to behave strategically, gaming the market, which is very specific this context and cannot be found in other commodity markets. Strategic bidding behaviours of electricity producers are widely studied in open literature [6, 7]. Game theory is popularly used in investigating the strategic bidding interactions between the electricity producers. Models based on various games such as Bertrand [8, 9], Cournot [10–12], Stackelberg [13, 14], Supply Function Equilibrium [15–17], as well as conjectural supply function [18, 19] have been proposed. By taking into account the network constraints, the strategic bidding behaviors analysis based on game theory models usually involves a bi-level optimization problem modeled as a mathematical program with equilibrium constraints (MPEC) [20, 21]. The lower level problem of market clearing with the consideration of the network constraints, i.e. the equilibrium constraints, is inserted in the upper level problem of the maximization of the producer surplus [22–26]. There has been an intensive research for efficient solution methods for the MPEC problem, with proposed solution schemes ranging from specific analytic algorithms to heuristics procedures [27–32]. However, the strategic interactions among participants in today’s electricity markets can be very complicated, due to various aspects such as supply and demand uncertainties, unit commitment arrangement, multi-rounds auctions in both energy and ancillary service markets, which are not conveniently modeled by game theory techniques. An alternative efficient approach for analyzing the strategic bidding and decision support of the market participants is provided by the multi agent system Models of Strategic Bidding in Electricity Markets 5
- 20. environment [33]. Based on the computational economics approach, several auton- omous adaptive agent models have been proposed, including those created by Anthony J. Bagnall and George D. Smith,[34], Athina C. Tellidou and Anastasios G. Bakirtzis [35], Sun J. and Tesfatsion L. [36], as well as Isabel Praca and Carlos Ramos [37]. Recently, the application of evolutionary complex network has been as well proposed by Ettore B. and Ma Y.C. [38] for modeling the bilateral electricity markets. Stable network structures that can be used to anticipate possible bilateral transactions in the real market place are developed by the improving path rule of evolutionary complex network principles. Different stable network structures can produce the same maximum value of global utility, reflecting the complex and disordered individual behavior has self-organization properties that produce the highest market efficiency in terms of the social welfare without the need for a centralized decision-making authority. In this chapter, we first outline the basic features of electricity as a commodity and of electricity markets recalling the basic metrics used for assessing the market performance. Then we propose a formalized representation of some of the most common game theory models by taking into account network constraints. A com- parison study on those game theory models through a unified conceptual example is investigated, which has not been discussed in open literature so far. Such compari- son study provides a quantitative assessment on the electricity market performance affected by the different strategic gaming behaviors of the electricity producers. Notations N: Number of network buses Nl: Number of network lines NG: Number of generators ND: Number of demand consumers n: Index of the network bus set N N ¼ [1 2,,. . ., n 1, n, n +1, . . ., N] l: Index of the network line set ℒ ℒ ¼ [1 2,,. . ., l 1, l, l +1, . . ., Nl] g: Index of the electricity producer set G G ¼ [1, 2, . . ., g–1, g, g + 1, . . ., NG] d: Index of the electricity consumer set D D ¼ [1, 2, . . ., d–1, d, d + 1, . . ., ND] a/b Intercept ($/MW) and slope ($/MW/MW) parameters of the marginal cost curves of the electricity producers, dim (a) ¼ dim (b) ¼ NG 1 (am , bm refer to marginal cost curve parameters) e/h Intercept ($/MW) and slope ($/MW/MW) parameters of the demand curve of the electricity consumers, dim (e) ¼ dim (h) ¼ ND 1 p/q Power production and demand vector, MW, dim(p) ¼ NG 1, dim (q) ¼ ND 1 IG/ID All-one-element vector for producers/consumers, dim (IG) ¼ NG 1, dim (ID) ¼ ND 1 P + /P Upper and lower production limits of the producers, MW, dim (P + ) ¼ dim (P ) ¼ NG 1 H Diagonal matrix formed by the vector h, dim (H) ¼ ND ND Bm Diagonal matrix formed by the vector bm , dim (Bm ) ¼ NG NG (continued) 6 E. Bompard and Y. Ma
- 21. F Flow limits of the transmission lines, MW, dim(F) ¼ Nl 1 m+ /m– Lagrange multipliers corresponding to the inequality expressions of the line flow, $/MW, dim (m+ ) ¼ dim (m– ) ¼ dim (F) J Matrix of power transfer distribution factors, Nl N 1 JG T , JD T Generator and load buses rows of the transpose of J matrix, respectively, dim (JG T ) ¼ NG Nl, dim (JD T ) ¼ ND Nl lG, lD Nodal prices at the generator and load buses, $/MW, dim (lG) ¼ NG 1, dim (lD) ¼ ND 1 lN Price at the reference bus N, $/MW l Average price weighted by the quantity ($/MW), l ¼ (lG T p + lD T q)/ (IG T p + ID T q) Sg G Surplus of producer g, 8g∈G, ($) SG Total surplus of producers ($) Sd D Surplus of consumer d, 8d∈D, ($) SD Total surplus of consumers, ($) SM Merchandise surplus ($) SS Social surplus, SS ¼ SG + SD + SM , ($) Special operators m1 m2 Matrix multiplication of m1 and m2 v1 · v2 Element by element multiplication of vector v1 and vector v2 (v1) (v2) Element by element addition or subtraction of vector v1 and vector v2 (v1) (v2): Element by element inequality between vector v1 and vector v2 v1 ¼ v2: Element by element equality between vector v1 and vector v2 vT /mT : Transpose of vector v/matrix m m1 : Inverse of the matrix m 2 Electricity as a Commodity Liberalization has been introduced in many economic sectors such as air transpor- tation, telecommunication with the goal of achieving efficiency though competi- tion. Electricity is one of the last sectors in which liberalization and competition has been introduced. Those very specific features of electricity as a commodity, from one side, and its criticality to the society, from the other, need to be considered when switching from regulated monopoly to competition. The market modeling and simulation need to capture those specificities related to the physical constraints and network structure of the power systems. In this section, we first review the specific electric power system operation. Then, paradigms of the electricity markets are elaborated with two reference models, pool and bilateral markets whilst the poor model is used in this chapter to study the strategic bidding models in an electricity market. Models of Strategic Bidding in Electricity Markets 7
- 22. 2.1 Power Systems Operation Power systems are composed a transmission system with buses, to which are connected the generators and the loads, interconnected by lines and transformers over the meshed network where the power injected by the generators is delivered to load centers. The operation of the system should satisfy a set of boundary conditions which can be addressed by a set of equality and inequality constraints.The equality constraints assure power balance between the electricity generation and load demand while inequality constraints define a feasible operation region represented by the line transmission capacity limits, system frequency and bus voltage ranges. If all the equality and inequality constraints are satisfied the system is operated in its normal state. The inequality constraints are satisfied with certain security margins in terms of generation spinning reserve and/or transmission capacity reserve, etc. If the reserve margin, due to some disturbances, is reduced, the system enters into the alert state in which the constraints are still satisfied. Preventive control then takes place to secure again proper security margins. If the preventive control fails or the disturbance was severe enough the system goes into the emergency state in which the inequality constraints are not; emergency control actions such as cutting of faults, rerouting of generation, excitation control, fast-valving, generation tripping, generation run-back, HVDC (high voltage dc) modulation, and load shedding are undertaken. If emergency control fails the system will go into the extremis state with cascading outages and system islanding. From this state the system operator to push the system to restorative state, in which the system, matching again generation and load, is driven back to the normal state [39]. Mathematically, the steady-state power system operation can be expressed as: s ¼ fðxÞ ¼ 0 8 n 2 N (1) gðxÞ 0 (2) where s ¼ sP + jsQ is the vector of complex power (real and reactive) injection at each bus n, n ∈N, f (·) is the vector of functions that express the complex power transferred over the lines connected to each bus and x is the vector of unknown phasor voltages (magnitudes and angles) at each bus; the group of inequities g(·) represent the network constraints. The group of Eq. 1 represents the so called AC power flow model where real and reactive power determined by both the voltage magnitude and voltage angle at each bus. The real power balance enforced at each bus is to keep the electricity frequency at the expected value. Any unbalance of the real power between the electricity generated and consumed causes the frequency drifting to a new value. The reactive power balance at each bus is to govern the voltage magnitude, i.e. to generate capacitive reactive power to restore the voltage magnitude from low value to high value or to absorb the inductive reactive power to reverse the process. 8 E. Bompard and Y. Ma
- 23. The group inequalities (2) include the various limits with respect to the problem size to be considered. Generally, the voltage magnitude limits and real power flow limits of the network branches are the two typical conditions imposed to the power system for operating in a secure normal state. In addition, real and reactive power generation limits are incorporated to capture the production characteristics of the electricity generators. The model can be simplified with some assumptions that lead to a linearized model that will be introduced in Sect. 3. 2.2 Specific Features of Electricity Apart from its economic features such as the no-direct-storability, lack of good substitutes and inelastic demand, the technical peculiarities of the electricity as a commodity the electricity market a very specific one. Electricity can only be delivered by wires over a transmission and distribution networks to customers at the same time when the electricity is generated. Actual power flows of the network wires are governed by the Kirchhoff laws, which makes transmission of power different from the transportation of an ordinary commodity in a spatial market. In addition, network constraints need to be enforced on the trading activities and affect the market clearing results. The specific features of the electricity are summarized by using the example shown in Fig. 1 where all the parameters are in per unit of system base value. • Need for an instantaneous balance between power production and power con- sumption plus losses. In the example, the total power generated, 0.6024 þ j0.1842 p.u. is equal to the total power demanded, 0.6 + j 0.1721 p.u., plus total power loss, 0.0024 + j0.0121 p.u. • Power flow path depending on the system physical parameters. The paths followed by the power flows do not coincide with the contract paths of the economic transactions; almost all the lines, other than that connecting the generator and the load in the transaction, are involved. If the values of the line impedances change the power flows over the network change as well. • Transmission losses. The total real power generated exceeds the real power demanded by the loads of 0.0024 p.u. and that correspond to the losses on the transmission system. Gen 1, in this case, need to produce additional 0.0024 p.u. to balance the real power loss of the network. • Reactive power support. To allow for the transactions scheduled in terms of real power, a reactive power support, to balance the reactive power demanded by the loads and used by the transmission systems, needs to be provided by the generators. The total reactive power demanded by the loads is 0.1721 p.u. while those totally generated by the two generators is 0.1842 p.u.; the difference is the reactive power needed by the system to feasibly allow for the transaction (0.0121 p.u.). Models of Strategic Bidding in Electricity Markets 9
- 24. • Line flow limits. Lines have some limitation on the maximum power they can transmit due to thermal, voltage drop and stability limits. If the power flow of the branch from bus 1 to bus 2 reaches its limit, the system should re-route the power through other lines to be kept feasible. 2.3 Electricity Markets The energy market in most countries is organized as a day-ahead market (DA) where the electricity energy transactions are cleared for each hour of the next day. In day-ahead market, demand is forecasted for each trading interval, e.g. 1 h period, 24 h ahead and offers and bids are received from the market participants. The market clearing is conducted by an independent body, which may coincide with the Independent System Operator (ISO), to match demand and supply. In the pool models the dispatching of injected and withdrawn power quantities are assigned considering the transmission limits and providing, as a by-product, the locational marginal prices (LMPs), such as in PJM Interconnection, in order to capture the network impacts on the market clearing [40]. Gen.1 Ld.1 ; Gen.2 Ld.2 ; 0.2 p.u. 0.2 p.u. Economic transactions Gen.2 Ld.1 0.2 p.u. G G 0.1517+j0.0794 Gen.1 Gen.2 Ld.2 Ld.1 0.4 + j0.1315 p.u. 0.2024+j0.1104 Z Z 0.1511+j0.0765 0.05074+j0.031 Z Z 0.05067+j0.0307 0.1493+j0.01 0.1498+j0.0122 0.2489+j0.055 0.2502+j0.0616 0.4 + j0.0738 p.u. 0.2+ j0.0406 p.u. Bus1 Z = 0.02 + j0.1 p.u. Physical power flows V: 1 / 0 V: 0.9959 /- 0.2562 deg. V: 1 / 0.5914 deg. V: 0.9891/ -0.7867 deg. Bus2 Bus3 Bus4 p.u. : per unit of system base value j: imaginary unit of the complex number Fig. 1 Network impacts on electric power transactions 10 E. Bompard and Y. Ma
- 25. Several related markets concur to make the electricity transactions possible, including ancillary service market and transmission right market. Ancillary market is organized to acquire on the market all the services needed for the operation of the power system as scheduling, system control and dispatch, reactive and voltage support, regulation and frequency response, energy imbalance, spinning reserves and supplemental reserves. Transmission right market is designed for auctioning the right to assure the availability of transfer capability over the network or edge against the risk of change in transmission cost due to the scarcity of transfer capability. There are two types of transmission rights: physical transmission rights (PTRs) and financial transmission rights (FTRs). However, PTRs are gradually replaced by FTRs, defined upon the locational marginal prices, due to its superiorities over PTRs by opening the network access to all the market participants. Several of the restructured US electricity markets have already experimented with styles of the transmission rights markets in the last decade [41]. Strategic bidding behaviors are extensively investigated in the day-ahead energy market whilst, due to their functional complexity, ancillary markets attract more technical concerns than the economic behaviors of the market participants. There- fore, in this chapter we study the strategic bidding behaviors of the electricity producers in the day-ahead market clearing process. 2.4 Reference Paradigms of the Electricity Markets A major objective of electricity deregulation is to achieve a workably competitive wholesale market. Wholesale electricity markets are organized with several gener- ation companies that compete to sell their electricity in a centralized pool and/or through bilateral contracts with large electricity buyers [42–44]. The transac- tions among sellers and buyers can take place in an organized market (“power exchange”) that collects all the offers from the generators and the bids from the loads and performs a centralized market clearing compatible with the network constraints, as in the “pool paradigm”; otherwise sellers and buyers arrange, on a bilateral basis, their own transactions, submitting afterward to an ISO to check their feasibility in terms of the network constraints as in the “bilateral paradigm.” A vast majority of studies on electricity markets to date either explicitly or implicitly assume a centralized auction process, administered by a pool, through which generators sell energy to consumers. A growing number of studies typically assume a decentralized trading process by which generators sell to consumers bilaterally through power exchanges or arbitragers [38, 45, 46]. 2.4.1 Pool Markets Many of the restructuring experiences (e.g., in the UK, Argentina, Chile, Australia) have been based on pool trading with centralized coordination in the Power Models of Strategic Bidding in Electricity Markets 11
- 26. Exchange (PX) by an Independent System Operator (ISO). Examples include Australia, Argentina, the PJM Interconnection and the New England Power Pool. In the pool market all producers sell into a pool run by an independent entity. The ISO has also the responsibility for system reliability and transmission congestion management. Producers’ physical sales of power and energy trades are all within the pool. The pool is the only buyer (for the producers) and the only seller (for the electricity consumers). The ISO holds central auction in which each generator bids different prices for different quantities (from specific plants or as a portfolio) or an offer curve for the trading period; for example, for each hour of the following day. Based on the bids and the considered demand quantities, the ISO uses a security based dispatch process to set the market price and the generation quantities [42–44]. 2.4.2 Bilateral Markets Under bilateral trading model buyers and sellers individually contract with each other for power quantities at negotiated prices, terms, and conditions. All transactions must be announced to the ISO, which analyzes all the trades in each period and determines, without discrimination, which ones are infeasible for grid security constraints. The ISO does not need to know the prices and demand side bids may co-exist with the generation bids. Generally, the bilateral types of markets are split into two markets named forward contract market (PX) and spot market (ISO). Two steps are needed to arrange the bilateral trading: • Step 1: Dispatch without network constraints; • Step 2: Re-dispatch with consideration of transmission constraints based on the adjustment offers (demands) from generators and loads, in case that the dispatch would lead to network constraints violations. Gen. offers Load demands Network structure Network constraints Market clearing quantities prices Max objective ISO Ld. 1 POOL Gen. G Gen. g Gen. 1 … … … … Ld. d Ld. D $ MWh Fig. 2 Pool type market 12 E. Bompard and Y. Ma
- 27. 3 Market Equilibrium and Market Performance Like any commodity or, electrical energy can be bought and sold in an established market place. Bids, from buyers, and offers, from sellers, set the price in the electricity markets on the basis of principle of supply/demand intersection. The difference among electrical energy and other commodities is the delivery system and the technical features of the framework in which the market transactions are undertaken. 3.1 Format of the Electricity Supply and Demand Although it may be different with respect to the functional rules and scopes, most day-ahead markets consist of unbundled Generation, Transmission and Retailing sectors that are corresponding to the electricity suppliers, transmission organization and the electricity customers. A neutral entity called Independent System operator is introduced to undertake the role of the market coordinator. For the electricity suppliers, the supply curve is usually represented as an upward liner or stepwise function that expresses the quantity that all the sellers in a market are willing to sell as the function of price, as shown in Fig. 4. The increasing trend of the curve is explained by the fact that as the power offered raises more expensive unit need to be committed. It is important to note that the supply curve is not necessarily equal to the marginal cost curve, the additional cost incurred in produc- ing one extra unit of output, from which the strategic bidding behavior of the electricity suppliers is originated. The demand curve shows the relationship between the quantity demanded and the price of a commodity. All other factors held constant, almost all commodities obey the law of downward-sloping demand, Max objective Min adjustment costs Merit order based transactions Network structure Network constraints Adjustment Offers from Gens and Lds Security based redispatch Contracts between Gens Lds ISO PX PX Ld.1 Gen. 1 ISO PX $ PX MWh Ld. d Ld. D Gen. g Gen. G Forward market Spot market Bilateral $ Bilateral MWh Coordination … … … … Fig. 3 Bilateral type market Models of Strategic Bidding in Electricity Markets 13
- 28. which states that quantity demanded falls as a price rises, Fig. 7. The degree of such price responsiveness is called demand elasticity. Under regulation, electricity demand was considered inelastic (fixed amount with no price responsiveness) and new capacity was built to cover the projected demand to minimize investment plus operation costs. Under deregulation, the consumers’ demands for electricity are encouraged to be price responsive to enhance the wholesale market efficiency and system reliability [47, 48]. Strategic behaviors from the supply side are more evident when the demand elasticity of the electricity consumers is low. Improving demand elasticity plays a positive contribution in mitigating the strategic bidding behaviors, pushing the uncompetitive electricity market performance towards to a high level competitive one [49]. 3.2 Market Equilibrium A market is a real or virtual environment in which buyers and sellers interact to exchange goods, services or commodities; the outcome of a market, from a macro- scopic viewpoint is the quantity and the price of the good traded. Given the supply and demand curves, the electricity market is cleared at the equilibrium at which the market clearing price is established such that the amount of goods or services sought by buyers is equal to the amount of goods or services produced by sellers. Graphically, the market equilibrium is represented by the intersection of the supply curve and the demand curve, point E in Fig. 4. The equilibrium is established in a price-quantity adjustment process where incremental quantity of electricity is supplied at the price that demand customers willing to buy, i.e. the supplier offer price is lower than the demand bid price. Equivalently, the market clearing can be expressed mathematically as A E Price Q Quantity λN demand curve supply curve B Fig. 4 Electricity supply and demand curves 14 E. Bompard and Y. Ma
- 29. Max 1=2 qT Hq þ qT e 1=2 pT Bp þ pT a (3) s:t: ID T q IG T p ¼ 0 (4) The market clearing price, lN, in Fig. 4. is the Lagrange multiplier value of the optimization problem (19)–(20) and can be expressed as lN ¼ IT GB1 a IT DH1 e IT DH1 ID IT GB1 IG (5) The market clearing quantities are pg ¼ ðlN agÞ=bg 8g 2 G (6) qd ¼ ðlN edÞ=hd 8d 2 D (7) In Eqs. 3–7, the vector of a and the diagonal matrix B indicate the parameters of the supply curves of the electricity producers. Those parameters are not necessarily equal to the parameters of the marginal cost curves, i.e. am and Bm. The strategic bidding behaviour of the electricity producers can be represented by choosing the parameters of a and/or B to achieve the economic goals in the market clearing. 3.3 Levels of Competition in the Markets According to the competition level, market is classified as perfect one or imperfect. In a perfect competitive market, all sellers and buyers are “price-takers” who assume that their own production and purchase decisions do not affect the market price [50]. According to the classic economic theory, a price-taking firm that wishes to maximize its profits would bid the products at its own marginal cost [51]. Each supplier submits the marginal cost as supply function and the social surplus, SS , is maximized in the market clearing, as mentioned in the last section. Price-taking behaviors will lead to the most efficient market operation characterized by the least cost of the production while meeting the demand of the consumers. In reality, perfect competition is difficult to be implemented in which- ever real market due to the strict conditions need to be satisfied, such as a large number of price-taking producers with a very small market share produce homogeneous and perfectly substitutable products. Nevertheless it can serve as a reference case to identify the market power behaviors in a real implemented market [22, 52, 53]. The opposite situation to perfect competition is monopoly in which just one producer faces all the market demand. Monopoly market is thought to have no competition. Perfect competition and monopoly represent the two extreme cases of Models of Strategic Bidding in Electricity Markets 15
- 30. market structures. A more common case is the oligopoly in which the market is dominated by a small numbers of the sellers and the market equilibrium is in between the two preceding cases. Current electricity markets are oligopoly in generation competition where the electricity suppliers will adopt the strategic behaviors, different from the marginal cost curves shown in Fig. 5, striving to get the maximum profit, the area E0 lN 0 BC in Fig. 5. The main goal of the restructuring the electricity industry for the market regulators is to force the market toward perfect competition, from point E0 to point E, while monitoring continuously the distance from such a condition. Under imperfect competition, the market clearing produces the maximization not of the social surplus, area ABE in Fig. 5, but of the similar quantity that we denote as the system surplus, area AB0 E 0 in Fig. 5. The key part of studying the imperfect competition of the electricity markets is to derive the oligopoly equilibrium at which each individual electricity supplier’s optimal strategy is established. Although to obtain the market clearing results at oligopoly equilibrium is straightforward by inserting the optimal strategy of each electricity producer into the market clearing model (3) and (4), to derive the equilibrium of optimal strategies of the electricity suppliers is a complex problem where each producer will compete against other’s strategies. Such problem can be addressed by game theory models which will be discussed in detail in the next sections. 3.4 The Impacts of the Scarcity in the Transfer Capability Due to the capacity limits of the network lines, electricity transactions have to be settled according to the power flow constraints. The impacts of the scarcity of the transfer capability of the network lines on the electricity market performance can be briefly introduced by a simple two bus system, Fig. 6. One supplier located at bus 1 is to deliver the electricity to the consumer at bus 2. Without the network constraint; the transaction will be clearing at the point E, the left hand side of Fig. 6. The market clearing price is unique at which electricity supplied is equal to demanded. marginal cost A strategic offer E′ Price E Q′ Q Quantity λN′ B demand curve supply curve C λN B′ Fig. 5 Market clearing under strategic bidding of the supply side 16 E. Bompard and Y. Ma
- 31. If the real power flow limit, F12, is imposed, the market clearing process splits the unique market clearing price into supply price and demand price. The market clearing quantity of the electricity is limited at the real power flow limit, F12. Based on Fig. 6, the market performance can be summarized in Table 1. 3.5 Metrics for Assessing the Equilibrium and the Level of Competition Given the market clearing results of the price, lg and ld, and quantity, pg and qd of the electricity transactions, (6) and (7), the social surplus, SS , producer surplus of producer g, Sd G , consumer surplus demander d, Sd D , and merchandise surplus, SM , can be determined as SS ¼ 1=2 qT Hq þ qT e 1=2 pT Bmp pT am (8) Sg G ¼ lgpg 1=2 bg;m pg 2 ag;m pg 8 g 2 G (9) Bus1 MCQ = F12 0 demand supply SG SC SS = SC +SG Unconstrained (flow limit: ∞ MW) $/MW Price SS = SC +SG +SM MW Quantity SC SM SG demand supply Constrained (flow limit: F12 MW) $/MW Price A B λ2 λ E A B supply MCQ=F12 0 MW Quantity MCQ =Q Q MCQ =qE demand supply demand E E′ E′′ λ1 Bus2 Bus1 Bus2 Fig. 6 Market clearing without (left hand side) and with (right hand side) network constraints Table 1 Network congestion impacts on the market clearing Unconstrained network Constrained network Market clearing price, $/MW l (l ¼ l1 ¼ l2) Bus 1: l1 Bus 2: l2 Market clearing quantity, MW Q F12 Social surplus, SS , $ area ABE area AB E00 E0 Producer surplus, SG , $ area ElB area E00 l1 B Consumer surplus, SC , $ area ElA area E0 l2 A Merchandise surplus, SM , $ 0 area E0 l2l1 E00 Models of Strategic Bidding in Electricity Markets 17
- 32. Sd D ¼ ldqd 1=2hdqd 2 edqd 8 d 2 D (10) SM ¼ lD T q lG T p (11) According to (8)–(11) and Fig. 6, social surplus is defined by the difference between the total benefit to consumers minus the total cost of production. Producer surplus is the difference between the producer sales revenue and the producer variable cost while the consumer surplus is the difference between the amounts that a consumer would be willing to pay for a commodity and the amount actually paid. The merchandise surplus is non zero when the price charged for the buyers is not equal to the selling price of the suppliers, in which case the electricity market is cleared under congested network, as shown in the right part of the Fig. 6. With the market clearing results derived under the competition, comparison indices are introduced to assess the imperfect market performance against the reference case of the perfect competition market. We will use the superscript * for the values associated with perfect competition. 3.5.1 Efficiency and Allocation Indices The efficiency and allocation indices are expressed in per unit and have the general form as K ¼ ðS S Þ=S (12) This index can be used for the social surplus, the total producer surplus SG , and the total consumer surplus SD KS ¼ ðSS SS Þ=SS (13) KG ¼ ðSG SG Þ=SG (14) KD ¼ ðSD SD Þ=SD (15) 3.5.2 Price and Quantity Indices To have a reference value for the prices under congestion, we define the weighted average price as l ¼ ðlG T p þ lD T qÞ=ðIG T p þ ID T qÞ (16) 18 E. Bompard and Y. Ma
- 33. The price distortion may be then measured by Kl ¼ ðl lN Þ=lN (17) For the index for quantity, the general form (28) is adapted as KP ¼ ðIG T p IG T p Þ=IG T p (18) 4 Modeling Strategic Bidding Under Network Constraints in Pool Model The analysis of strategic bidding behaviors seeks to answer basic questions; includ- ing how a firm exercises strategic biddings and to what extent the strategic bidding behavior affects equilibrium quantities, prices, and market efficiency. Such assess- ment, at least implicitly, requires a comparison of observations of real world market prices and quantities to the comparable values of the variables that the perfectly competitive model predicts. Game theory provides an efficient tool to model the strategic bidding behaviors of the market participants. The solution of the game models is to derive the market clearing results at Nash Equilibrium (NE) by which the distortion of the perfect competition can be predicted and measured for assessing the strategic bidding impacts. The key point in formulating the game models to represent the strategic bidding behavior is a bi-level programming problem where the market clearing is inserted, as subject conditions, into the individual Producer Surplus Maximization (PSM) problem [11, 24–26, 49]. The market clearing is to maximize the system surplus taking into account the physical constraints of electricity networks represented by the power flow model. According to the Karush-Kuhn-Tucker (KKT) conditions, the market clearing is transformed into a group of equality and inequality constraints of the individual PSM problem. The optimal variable derived from the individual PSM problem is the optimal strategies that the producer will submit to the market coordinator. With the individual PSM problems, the Nash Equilibrium is derived at the point no player can be benefited from changing his/her strategy when his/her competitors do not. Very often, the individual PSM problem can be quite complicated with the large size of the electricity network to be considered. The derivation of the NE and the unique/existence of the NE are general concerns in many related references [22, 23, 49, 54]. The unique/existence NE may be guaranteed in an analytical way under simple systems while for the complex systems, numerical approaches is usually employed and ex-post check of the NE is needed. Models of Strategic Bidding in Electricity Markets 19
- 34. 4.1 Electricity Network Model In the pool operated electricity markets, the Independent System Operator (ISO) is responsible the aggregate offers from the supply side and the aggregate demands from the demand side for a specified time interval, usually 1 h. Due to the peculiarities of the electricity transmission, the transactions must be settled according to the physical conditions of the electricity network and different nodal prices may arise when the flow limits are binding. In addressing the network features of the electricity transactions, the DC power flow model is popularly employed in studying the strategic interactions in the competitive electricity markets [22, 26, 49]. DC power flow provides a fairly good approximation of the AC power flow in terms of the real power flow due to the fact of the large ratio of X/ R and invariant of the voltage magnitude of the power transmission network where wholesale electricity transaction is accommodated. The characteristics of the DC power flow is summarized as • Reactive power balance at each bus n, fn Q , is disregarded fn Q ¼ 0 8n 2 N (19) • Lossless network Pl loss ¼ 0 8l 2 L (20) • A group of linear expressions of the real power flow of the branches, fl P , in terms of the bus voltage angles fl P ¼ Ylu 8l 2 L (21) Where Yl is the row vector of the branch matrix Y (Nl Nb1) with the positive admittance value at the element of row index l (line l) and column index of from- bus f and negative admittance value of to-bus t; u is a column vector of the bus voltage angles. Both of the Y and u matrix are formulated without the elements related to the reference bus. In terms of the net injection of the active power at each bus, the real power flow of the branches can be expressed as fl P ¼ Yy ¼ Y B1 ðp qÞ ¼ Jðp qÞ (22) Where, J ¼ Y B 1 , is called power transfer distribution factor (PTDF) matrix, J, Nl (N1); matrix B is the admittance matrix, (N1) (N1), and elements of the column and row related to the reference bus are not included; p q is vector of the net injection of the active power at each bus. 20 E. Bompard and Y. Ma
- 35. 4.2 Market Clearing Model Under Network Constraints The market clearing can be modelled with an optimization problem subject to the electricity network constraints represented by the DC power flow, as max SS ¼ 1=2 qT Hq þ qT e 1=2 pT Bm p pT am (23) s.t. IG T p ID T q ¼ 0 (24) F Jðp qÞ F (25) The equality constraint (24) indicates the total power production and consump- tion balanced at the reference bus. The inequality constraints (25) represent the line flow limits. Note that the inequality and equality symbols in (24) and (25) represent the inequality and equality between two vectors. The solution of the above optimization problem provides the nodal price at bus n, as ln ¼ lN Jn T ðmþ m Þ 8n 2 N and n 6¼ N (26) Where, lN, is the price at the reference bus N, i.e., the Lagrange multiplier of the equality constraint (24); Jn T is the nth row vector of the transposed matrix J; m+ and m are Lagrange multipliers of the power flow inequities (25). The price at the reference bus Nb, lN is equal to lN ¼ IT GðBm Þ1 ½JT Gðmþ m Þ þ am IT DH1 ½JT Dðmþ m Þ þ e IT GðBmÞ1 IG IT DH1 ID (27) The nodal price differs from the price, lN, by the values of m+ /m .When network constraints are not considered, which means m + ¼ m ¼ 0, the prices at all buses are equal to lN that is the market clearing price derived in the (5). Equivalently, the problem (23)(25) is reduced to the problem (3)(4). 4.3 The Point of View of Each Producer The model of PSM is a bi-level mathematical programming problem in which the lower level of market clearing is taken into account to get the price and quantity values to compute the objective function for the upper level of producer surplus optimization problem. The mathematical model is generally expressed as Models of Strategic Bidding in Electricity Markets 21
- 36. Max Sg G 8g 2 G (28) s: t: max SS (29) s:t: IG T p ID T q ¼ 0 (30) F Jðp qÞ F (31) Due to the convexity property of the market clearing problem (quadratic pro- gramming with the DC flow model), KKT conditions outline the optimal solution and can be used as the constraint functions of the maximum producer surplus problem to get the optimal strategy. By using the KKT conditions of the market clearing, the above problem can be transformed into Max Sg G ¼ ðlN Jg T ðmþ m ÞÞ pg ðam þ bmpgÞ 8g 2 G (32) s.t. am þ Bmp ¼ lNIG JG T ðmþ m Þ (33) e þ Hq ¼ lNIG JG T ðmþ m Þ (34) IG T p ID T q ¼ 0 (35) F Jðp qÞ F (36) mþ ½Jðp qÞ F ¼ 0 (37) m ½Jðp qÞ þ F ¼ 0 (38) mþ 0; m 0; lN0 (39) P p Pþ (40) Where lN is the nodal price at the reference bus N. 4.4 Strategic Interaction Among Producers: Game-Theory Model for Oligopoly In the last 50 years, game theory provides an efficient tool to model the strategic interactions among individuals aware that the behaviour of their competitors can 22 E. Bompard and Y. Ma
- 37. affect their results in the market. Market power analysis in terms of the strategic bidding behaviours can be gained from the use of game-theoretic models through simulating the competition between a given set of competitors in a well-specified market environment, taking into account the network constraints that may provide additional possibilities of market power arising that are very specific of this in electricity markets [52, 55]. According to the classification of the strategic variables, there are three types of game models which are bidding in price, bidding in quantity and supply function bidding models. The price bidding models include Bertrand and Forchheimer models. However, taking into account network constraints for the analysis of hourly electricity markets, so far there no literature using price bidding game models as an efficient tool. Another reason for the Bertrand model has not been the focus in the literature would be that Bertrand model might correspond to perfect competition case [56]. The quantity bidding game models include Cournot, Stackelberg and Conjec- tural Supply Function (CSF) models. An essential assumption of the former two models is that the individual player’s own output decision will not have an effect on the decisions of its competitors, i.e. the optimal offer quantities are assumed fixed in formulating the optimal strategy of the considered player. As for the CSF model, the basic assumption is that the output of the other competitors can be estimated to change in an expected way with respect to the output decision of the considered player. Since the strategic variable is the quantity of the electricity transacted, those game models do not give meaningful equilibrium when price elasticity of the demand curve is low (the demand quantity is fixed with the zero value of the price elasticity). The supply function bidding models choose a strategic supply function from the marginal cost curve with the aim of maximizing individual producer surplus. Different the quantity bidding game models, for the optimal strategy formulation of one producer, the given strategies are the supply functions of other competitors but the production quantities. The dispatched quantities of the competitors are determined by the supply function of the considered producer through current decision making process. Taking into account the strategic bidding behaviors of the electricity suppliers, the quantity, p and the price at the reference bus, lN, become the variables with respect to the strategic variables, r (NG 1), that are p(r) and lN (r) in problem (32)(40). According to the strategic variable of the different game modes, the of the lN (r) and p(r) are summarized in Table 2. The information given in Table constitutes the core part of the game theory models in a formalized way. Such formulations will be used in the conceptual examples to obtain the market results under game theory models. Models of Strategic Bidding in Electricity Markets 23
- 38. Table 2 Price at the reference bus and production of the players according to different game models Model Strategy variable rg Formulation Cournot pg 8 g∈ G lN ¼ IT Gp0 þ pg þ IT DH1 ½JT Dðmþ m Þ þ e IT DH1 ID where, p0 ¼ [p1 0 ,. . ., pg–1 0 , 0, pg+1 0 ,. . ., pNG]T , pi 0 (i∈G, i 6¼ g) is the optimal offered quantity derived from the last move of the producer i, pg is the decision variable in the optimal problem Stackelberg pg l 8 g∈ Gl G lN ¼ IT FpF þ IT LP0L þ pL g þ IT DH1 ½JT Dðmþ m Þ þ e IT DH1 ID where P0 L ¼ [P1 0 ,. . ., Pi – 1 0 , 0, Pi+ 1 0 ,. . ., PNGl 0 ]T , Pi 0L (i∈ G l G, i 6¼ g∈ G l, G l is the set of the leader producers and NGl is the number of the leaders) is the optimal offered quantity of the leader i, pg L is the decision variable in the optimal problem; pF is the vector of the optimal production quantities of the followers If the slope parameter of the marginal cost curve of the follower producers are identical, b1 m ¼ b2 m ¼,. . ., bi m ¼,. . ., ¼ bNGf m ¼ bm , (i ∈ Gf G, G f is set of the follower producers and G f G l ¼ G, NGf is the number of the followers), the optimal production quantities of the followers can be derived from the first order rule of the producer surplus maximization as IT FpF ¼ ðIT DH1 IDÞðIT Fam FÞ NGf ðpl þ IT LP0 L þ IT DH1 eÞ NGf þ 1 bmIT DH1 ID Conjecture supply function (CSF) pg, 8 g∈G lN ¼ pg þIT DH1 ½JT Dðmþ m ÞþerT g JT Gðmþ m ÞþIT Gpg1 rT g Lg1 g IT DH1 ID IT Grg where rg ¼ [r1,g, r2,g, . . ., rg–1,g, 0, rg+1, g, rNG, g]T p g1 ¼ [p1 g–1 , p2 g–1 , . . ., pg–1 g–1 , 0, pg+1 g–1 , . . ., pNG g–1 ]T Lg g–1 ¼ [L1 g–1 , . . ., L g–1 g–1 , 0, L g +1 g–1 , . . ., L NG g–1 ]T pi g–1 and Li g–1 are the dispatched quantity and the nodal price of producer i derived from the last move of the producer g1, respectively. rg,i, 8g, i∈G, i 6¼ g, represents the assumed rate of change in competitor supply per unit price. The CSF function is pi ¼ pi g–1 + rg, i (li Li g1 ) 8i∈G, i 6¼ g SFE Intercept ag 8 g∈G lN ¼ IT GðBm Þ1 ½JT Gðmþ m Þ þ a0 IT DH1 ½JT Dðmþ m Þ þ e IT GðBm Þ1 IG IT DH1 ID The supply function is ag + bg m pg pg ¼ lN JT g ðmþ m Þ ag bm g where a0 ¼ [a1 0 , a2 0 ,. . ., ag1 0 , ag, ag+1 0 , . . ., aNG 0 ]T ; ai 0 , i 6¼ g, 8 i, g∈ G, is the obtained value derived from the last move of producer i SFE slope bg 8 g∈G lN ¼ IT GðB0 Þ 1 ½JT Gðmþ m Þ þ am IT DH1 ½JT Dðmþ m Þ þ e IT GðB0 Þ 1 IG IT DH1 ID The supply function is ag m + bg pg pg ¼ lN JT g ðmþ m Þ am g bg where B0 is the diagonal matrix formulated with the vector of [b1 0 , . . ., bg1 0 , bg, bg+1 0 , . . ., bNG 0 ]T ; bi 0 , i 6¼ g, 8 i, g∈ G, is the obtained value derived from the last move of producer i (continued) 24 E. Bompard and Y. Ma
- 39. 4.5 Nash Equilibrium and Search Methods The scope of the game is to get the optimal strategy of each market player derived at the Nash Equilibrium (NE). In terms of solving techniques for the NE, there are two different to be considered: best response functions and iterative search algorithm. The former is an analytic approach that can be used to analyze simple duopoly games without network consideration while the latter is a numerical approach that is suitable for studying muli-player games with network consideration. 4.5.1 Best Response Functions Best response function is an analytic approach to obtain the oligopoly market equilibrium. The idea behind the best response function approach is that all the players get the maximum surplus concurrently. Mathematically, define the best response function of producer g, in terms of strategies of all the producers, r, as the first derivative of his/her producer surplus with respect to his/her own strategic variable, rg, as BgðrÞ :¼ @Sg G ðrÞ=@rg ¼ 0 ; 8g 2 G (41) Based on, (34) we have BgðrÞ ¼ ðlNðrÞ Jg T ðmþ m ÞÞ@pgðrÞ=@rg þ ð@lNðrÞ=@rgÞpgðrÞ ag m bg m pgðrÞ; 8g 2 G (42) The Nash equilibrium is derived by solving the group of the best response functions simultaneously, i.e. the intersection point of the best response functions. This approach is suitable for simple problems especially in duopoly game models (two players) and no power system operation considered. In case of network constraints to be considered, a specific branch power flow state can be integrated Table 2 (continued) Model Strategy variable rg Formulation SFE k multiplier kg 8 g∈G lN ¼ IT GK½JT Gðmþ m Þ þ IT GðBm Þ1 am þ IT DH1 ½JT Dðmþ m Þ þ e IT GKiG IT DH1 iD The supply function is kg(ag m + bg m pg) pg ¼ lN JT g ðmþ m Þ kgam g kgbm g where K is a diagonal matrix formulated by the vector [1/(k1 0 b1 m ), . . ., 1/ (kg–1 0 bg–1 m ), 1/(kg bg m ), 1/(kg+1 0 bg+1 m ), . . ., 1/(kNG 0 bNG m )]T ; ki 0 , i 6¼ g, 8 i, g∈ G, is the obtained value derived from the last move of producer i Models of Strategic Bidding in Electricity Markets 25
- 40. into the (41). In this respect, the state of the line (not congested or congested in the two possible directions) can be determined by comparison of the line flow’s amplitude and direction, derived in the case of no network constraints, with the line flow limit. In duopoly markets where the power system operational constraints are repre- sented by one line power flow constraints, the NE of r1* and r2* can be derived from the following three steps. Step 1, Game solution without the network constraints B1ðr 1; r 1Þ ¼ 0 B2ðr 1; r 1Þ ¼ 0 mþ l ¼ m l ¼ 0 8 : (43) Step 2, If: Jl (p(r1*, r2*) q(r1*, r2*)) Fl Then B1ðr 1; r 2Þ ¼ 0 B2ðr 1; r 2Þ ¼ 0 Jlðpðr 1; r 2Þ qðr 1; r 2ÞÞ ¼ Fl mþ l 0; m l ¼ 0 8 : (44) Else if : Jl (p(r1*, r2*) q(r1*, r2*)) Fl Then B1ðr 1; r 2Þ ¼ 0 B2ðr 1; r 2Þ ¼ 0 Jlðpðr 1; r 2Þ qðr 1; r 2ÞÞ ¼ Fl mþ l ¼ 0; m l 0 8 : (45) Where Jl is the row vector of the matrix J, corresponding to the power transfer distribution factors for the selected branch l. 4.5.2 Iterative Search Algorithm For the iterative search algorithm, while keeping the competitor strategies as given the players solve the PSM problem one by one. The gaming behaviours of the market players converge to the equilibrium point through a process characterised by a set of moves. Each move is a PSM problem where the optimal strategies of the individual player is derived based on the updated optimal strategies of the competitors. NE is found when the changing process, in terms of optimal strategies of the market players, is stabilized, i.e. no player can improve his/her payoff by 26 E. Bompard and Y. Ma
- 41. changing his/her strategy if his/her competitors do not. The best response function approach, the iterative search algorithm is suitable for the multi player games with consideration of the power system network constraints. Fig. 7 the flow chart of NE derivation based on the iteration search algorithm. As shown the innermost iteration of the flow chart, Fig. 7, complementary terms of the power flows of the considered lines are managed in a way of traversing all the states that may be uncongested (0), congested in the positive direction (1), Maximize Sg S ,state according to (6.14) and (6.22) Input the parameters of the electricity producers, demand loads and network Max i | ρi, Iter – ρi, Iter–1 | φ Iter = Iter +1 ρi ′= ρi, Iter−1, ∀i≠g, i, g∈G state = state+1 Determine the complementary terms of the power flow (5.19) ~ (5.20) according to the rule based on (5.28)~(5.30) Max state Sg S ,state ρg,Iter = ρg, state * Last state? End Last producer? Last iteration? Nash Equilibrium Y N Y N Y N N Y Fig. 7 Flow chart of the iterative search of the Nash equilibrium Models of Strategic Bidding in Electricity Markets 27
- 42. or congested in the negative direction (1). Therefore, each player, in choosing a move, should consider 3z alternatives if there are z lines to be considered in the power system network. In each state, the complementary relationships can be determined. For instance, given the state of the three lines of interest as 1, 1, 0, (37)(39) are transformed as F3 J3ðp qÞ F3 (46) J1ðp qÞ ¼ F1 (47) J2ðp qÞ ¼ F2 (48) m1 þ 0; m2 þ 0; m3 þ ¼ 0 Where the subscript of 1, 2, 3 in (46)–(47) are the indices of the three considered lines. The PSM problem is formulated by model of (32)–(40) and the price at the reference bus lN and the production of the players, pg, listed in Table 2. The iteration process is terminated by either the Nash Equilibrium derived by checking the maximal derivation from the last move strategy, smaller than a gate value of f, or by exceeding the predefined iteration steps. The latter case the Nash Equilibrium has not been found. An important note is that the unique/existence of the NE is a general concern in many related references [12, 17, 22, 23, 25, 26]. So far the unique NE can be guaranteed under simple test system by using the best response functions in analytical way [16, 49]. As for the complicate game problems where iterative search algorithm is used, the existence of equilibrium can not be guaranteed analytically and ex-post check is needed [22]. 4.6 Conceptual Examples In this section, two examples are designed for comparison study of different game theory models in assessing the strategic bidding behaviors of the electricity producers. The first example is a duopoly game in an electricity market modeled by a simple three-bus power system whilst the is a relative complicated one with multi-players in the electricity market represented by a large size power system. 4.6.1 Test Case1: A Conceptual Application for Duopoly Game The three-bus test system is shown in Fig. 8. The system is characterized by two generators and one load respectively connected at the bus 1, 2 and 3 (reference bus), 28 E. Bompard and Y. Ma
- 43. respectively. The line from bus 2 to bus 3 has line flow limit of 10 MW, shown in red line in Fig. 8, while other lines have no line flow limits. Since no demand loads are located at bus 1 and 2, the line 3 can only be possibly congested the positive direction, from bus 2 to bus 3. The market clearing under perfect competitive without network constraints is summarized in Table 3. The market clearings under different game models, without and with the net- work constraints, are reported in Tables 4 and 5, respectively. Without network consideration, the nodal prices l1, l2 and l3 are identical, as shown in the Tables 3 and 4. The merchandise surplus aroused from the different nodal prices is equal to zero. With the introduction of the network constraints, the NEs of the game models can only be obtained with the power flow of line binding at its limit value. The reason is that the power flow at the market equilibriums without consideration of the network constraints is beyond the line flow limit, 10 MW. Due to the network congestion, the nodal prices, l1, l2 and l3, are different with each other, as shown in Table 5. Due to the strategic bidding behavior from the supply side, the market perfor- mance is deviated from the perfect competition case (Table 3) with the increased Bus 1 ~ Bus2 ~ Generator 1 Marginal cost:η1 = 18+0.12p1 η1 : $/MW, p1: MW Bus3 Generator 2 Marginal cost:η2 = 20+0.1p2 η2 : $/MW, p2: MW Load Demand:γ = 1200−2q γ: $/MW, q: MW X13 = 0.05 F13 = ∞ X12 = 0.06 F12 = ∞ X12 = 0.07 F23 = 10MW Fig. 8 Three buses test system Table 3 Reference case: market clearing under perfect competition without network constraints p1 p2 q l1 l2 l3 S1 G S2 G SC SM SS 31.4 17.7 49.1 21.8 21.8 21.8 60.2 16.2 2410.8 0 2487.2 Table 4 Market clearings under different models without network constraints Variables p1 p2 q l1 l2 l3 S1 G S2 G SC SM SS Models Cournot 16.9 16.1 33 53.9 53.9 53.9 590 533 1092 0 2214.7 Stackelberg 24.5 12.4 36.9 46.1 46.1 46.1 652.4 652.4 1365 0 2333.7 CSF (r12 ¼ r21 ¼ 1) 22.16 20.1 42.3 35.4 35.4 35.4 356.1 289.3 1789.3 0 2436.2 SFE-intercept 28.5 19.4 47.9 24.1 24.1 24.1 125.1 60.7 2299.2 0 2485 SFE-slope 23.9 20.7 44.6 30.8 30.8 30.8 271.3 201.3 1990.5 0 2463.1 SFE-k parameter 28.4 19.3 47.8 24.4 24.4 24.4 134 66.7 2284.2 0 2484.8 Models of Strategic Bidding in Electricity Markets 29
- 44. prices, by comparison of the values in column 5 and 7 in Tables 5 to the reference values in column 4 and 6 in Table 3, at the decreased transacted amount of the electricity, by comparison of the values in the column 4 in Tables 4 and 5 to the reference values in the column 3 in Table 3. The total producer surplus is changed from the reference value, the addition of the values in column 7 and 8 of Table 3, with a significant increment as shown Tables 4 and 5. However, the consumer surplus of demand side is decreased from the reference case; by comparison the values in column 10 in Table 3 to the values in column 10 in the Table 4 and column 11 in the Table 5. Another important observation is that the values in Table 4 and the values in Table 5, network congestion pushes the market performance toward lower level of efficiency. The market players can utilize the network constraints, very specific to the electricity market, to obtain an even higher producer surplus at higher nodal prices. 4.6.2 Test Case 2: IEEE57 Bus System for Multiple player’s Game In this test case, the IEEE57 bus system is used to model the network constraints. The IEEE57 bus system is composed of 7 generators, 50 demand loads and 80 transmission branches, as shown in the appendix Fig. 10. The seven generators are assumed as individual players to have strategic biddings. The parameters of the marginal cost curves of the generators and the demand curves of the loads are reported in the appendix Tables 9 and 10. The network structure parameters are listed in the appendix Table 11. The flow limits of the selected lines are shown as the red lines in the Fig. 10. Cournot model and SFE intercept model are run based on the iterative search method in this test case. To get the comparison indices, we consider the following four different cases: • C0 unconstrained market under perfect competition • C1 constrained market under perfect competition • C2 unconstrained market under strategic bidding • C3 constrained market under strategic bidding The optimal strategies of the electricity producers under Cournot and SFE_Intercept game modes of test case C2 and test case C3 are shown in Table 6. Table 5 Market clearings under different models with network constraints Variables p1 p2 q l1 l2 l3 l S1 G S2 G SC SM SS Models Cournot 17.2 8.5 25.7 54.6 37.9 68.5 58.8 612.8 148.5 663.1 499.2 1922.5 Stackelberg 25 5 30 46.6 30.5 60 52 677.5 51.3 900 482.5 2111.3 CSF (r12 ¼ r21 ¼ 1) 28.8 3.26 32.1 40.7 22.5 55.8 47.4 604 7.6 1030.4 545.7 2187.7 SFE-intercept 31.5 2 33.5 41.2 27.2 52.9 46.6 671.3 14.2 1125.6 420 2231 SFE-k parameter 31.6 1.98 33.6 41.5 28.1 52.8 46.7 684.7 15.8 1130.6 403.7 2235 30 E. Bompard and Y. Ma
- 45. For Cournot model, the optimal strategies are derived by the flow chart of Fig. where the equilibrium is attainted when the variation of the optimal quantities of each player is smaller than 1 MW. For SFE_intercept model, the equilibrium optimal strategies is attained when the variation of the optimal intercept of supply curve of each player is smaller than 0.001 $/MW. The market clearing results under the four test cases, C0 ~ C3, are reported in the Table 7. The market performance indices, introduced in the Sect. 3.5, are computed based on the market clearing results and are shown in Table 8. Under strategic bidding test cases, the market is cleared at the higher price with lower transacted quantities, causing the producer surplus increased at the expense of the decrement of the consumer surplus. Due to the surplus share transferred from the demand side to the producer side, the total social surplus is changed from the perfect case of C0 in a minor value. From Tables 7 and 8, one can find that the competition level of Cournot game model is lower than the SFE_intercept model, by comparison of the index values in row 3 to row 6 of Table 8. Such uncompetitive market level is more intensive when the network constraints taken into account, the market is cleared at even higher price and lower quantities in test case C3 that in test case C2, the Cournot model and SFE_intercept model. The network constraints can provide opportunities for the market players, very specific to the electricity market, to obtain even higher individual producer surplus. Table 7 Market clearing under different test cases Test case Game model Market clearing price Market clearing production Producer surplus $ Consumer surplus $ Merchandise surplus $ Social surplus $ $/MW MW C0 – 48.5 815.1 10821.0 32707.8 0 43529 C1 – 50.4 781.3 10793.5 30255.2 1560.7 42609 C2 Cournot 56.1 735.6 16306.5 26794.7 0 43101 SFE_intercept 50.7 792.5 12518.7 30967.2 0 43486 C3 Cournot 62.3 667 18870.6 22187.9 467.3 41526 SFE_intercept 55 747.2 14118.9 27621.5 0 41740 Table 6 Optimal strategies of the electricity producers at NE Test case Producers model G1 G2 G3 G4 G5 G6 G7 C2 Cournot (optimal quantity) 96.11 108.05 95.56 95.72 131.04 101.89 107.18 SFE_intercept (optimal intercept parameter of the supply curve) 22.643 25.144 26.269 26.681 24.111 28.976 21.031 C3 Cournot (optimal quantity) 91.95 95.55 71.26 25.94 139.28 123.81 119.18 SFE_intercept (optimal intercept parameter of the supply curve) 23.029 45.511 40.180 44.713 20 26 18 Models of Strategic Bidding in Electricity Markets 31
- 46. For the individual producer surplus, Fig. 9 the deviation percentage computed against the reference case of C0. When the network constraints are not considered, test case C2, all the producers can gain increased individual surplus both under Cournot and SFE_intercept model, the red and black bars shown in Fig. 9. How- ever, such case of increased surpluses of all the electricity producers due to strategic bidding behaviors is not applied to the case where the network constraints are considered, the blue and green bars shown in Fig. 9 for test case C3. There is producer G4, at bus 6, obtains decreased individual surplus under Cournot model while producer G2, at bus 3, gets decreased value under SFE_intercept model. The network constraints cause the electricity market to have geographic property that lend its self to benefit some producers at specific locations at the expense of other producers at other locations, the grey bars shown in Fig. 9. However, from the whole point of view of supply side, the strategic bidding behavior with the network constraints result in getting high producer surplus from the demand side, causing the market cleared at a uncompetitive level that against the deregulation goal of the electricity industry. Positive approaches, such as transmission network Table 8 Market performance indices (in percentage) under different test cases Test case Game models KG KD KS KP Kl C1 – 0.25 7.50 2.11 4.15 3.82 C2 Cournot 50.69 18.08 0.98 9.76 15.72 SFE_intercept 15.69 5.32 0.10 2.77 4.46 C3 Cournot 74.39 32.16 4.60 18.18 28.49 SFE_intercept 30.48 15.55 4.11 8.33 13.42 -100 -50 0 50 100 150 C1 C2-Cournot C2-SFE_intercept C3-Cournot C3-SFE_intercept % G1 G2 G3 G4 G6 G5 G7 Fig. 9 Individual producer surplus deviation from the reference case C0 32 E. Bompard and Y. Ma
- 47. reinforcement, introduction of more electricity producers and encouragement of demand side bidding, may be adopted to push the electricity markets toward a high competitive level one. 5 Conclusion For the specific features of the electricity industry, the present electricity markets may be better described in terms of oligopoly than of perfect competition from which they may be rather far. In an oligopoly, the producer as a market player adopts strategic behavior, offerring submitted different from the marginal cost, to obtain the maximization of the producer surplus. The producer surplus maximiza- tion problem is outlined by a bi-level mathematic programming problem where the optimal DC power flow model, the lower level optimization problem, is inserted as the subject conditions of the upper level programming problem. With the surplus maximization problem of each producer, the oligopoly compe- tition in the electricity markets can be addressed by the game theory models. The solution of the game model is to derive the Nash Equilibrium at which point no individual player can improve his/her payoff by changing his/her strategy if his/her competitors do not. The approaches to obtain the Nash Equilibrium are best response functions and iterative search algorithm. The former can be used to simple games while the latter is suitable for the complicated games. One shortcoming of the game theory models is the assumption of the rational market players, with the aim of maximization producer surplus, which is not always applicable to the real markets. The other disadvantage is the unique/existence of the Nash Equilibrium, especially when electricity network constraints are taken into account, can not be guaranteed for different markets. However, game theory models can provide some instructive results on the oligopoly electricity market performance if the Nash Equilibrium can be obtained by ex-post check. At the oligopoly equilibrium, social surplus is transferred from the consumer side to the supply side. The main effects of the strategic behavior from the supply side are remarkably presented in obtaining extra surplus from the consumer side with higher market clearing prices and production withdrawn. Due to the network constraints, the transmission network plays a major role in deter- mining the oligopoly equilibrium. With the consideration of network constraints, the market clearing price is higher and the cleared demand is lower than the corresponding values under unconstrained network. As for the producer surpluses, the network constraints provide some producers with opportunities to obtain higher values at the expense of consumer surplus, leading to a higher level of market inefficiency compared with the unconstrained electricity market. Models of Strategic Bidding in Electricity Markets 33
- 48. Appendix 1 2 4 5 6 26 18 19 20 23 21 24 22 25 27 30 28 31 52 53 54 55 32 34 33 35 36 37 38 8 7 57 40 56 42 43 41 11 46 47 48 44 45 15 13 14 17 50 51 10 9 16 12 ~ ~ ~ ~ 29 49 3 39 ~ ~ ~ ~ Generator Load Considered lines for flow congestion 28MW 20MW 30MW G1 G2 G3 G4 G5 G6 G7 Fig. 10 IEEE 57 bus system for test case 2 34 E. Bompard and Y. Ma
- 49. Table 10 Parameters of the electricity demand loads Bus Parameters of demand curve Bus Parameters of demand curve Bus Parameters of demand curve Bus Parameters of demand curve Intercept Slope Intercept Slope Intercept Slope Intercept Slope $/MW $/MW2 $/MW $/MW2 ($/MW) $/MW2 $/MW $/MW2 4 120 5 22 120 5.5 36 135 6 50 125 4.5 5 130 4.5 23 125 6.5 37 150 5.5 51 120 4 7 135 6 24 130 5.5 38 125 4.5 52 135 4.5 10 150 5.5 25 135 4 39 130 5 53 130 5 11 135 6 26 150 5.5 40 110 3.5 54 150 5.5 13 120 5 27 120 4 41 120 4.5 55 125 4.5 14 135 6 28 110 4 42 120 4 56 115 4.5 15 150 5.5 29 135 6 43 150 5.5 57 110 3.5 16 110 4 30 120 5.5 44 125 4.5 – – – 17 135 6.5 31 110 3.5 45 135 6 – – – 18 135 6 32 135 4.5 46 110 3.5 – – – 19 150 5.5 33 150 5.5 47 125 4.5 – – – 20 110 4 34 115 4.5 48 135 4.5 – – – 21 115 4.5 35 135 6.5 49 110 4.5 – – – Table 9 Parameters of the electricity producers Bus Number Producer Marginal cost curve parameters P+ MW P MW Intercept $/MW Slope $/MW2 1 G1 20 0.28 300 0 2 G2 22 0.22 150 0 3 G3 25 0.23 150 0 6 G4 24 0.24 100 0 8 G5 20 0.18 250 0 9 G6 26 0.2 250 0 12 G7 18 0.26 300 0 Models of Strategic Bidding in Electricity Markets 35
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- 54. Optimization-Based Bidding in Day-Ahead Electricity Auction Markets: A Review of Models for Power Producers Roy H. Kwon and Daniel Frances Abstract We review some mathematical programming models that capture the optimal bidding problem that power producers face in day-ahead electricity auction markets. The models consider both price-taking and non-price taking assumptions. The models include linear and non-linear integer programming models, mathemat- ical programs with equilibrium constraints, and stochastic programming models with recourse. Models are emphasized where the producer must self-schedule units and therefore must integrate optimal bidding with unit commitment decisions. We classify models according to whether competition from competing producers is directly incorporated in the model. Keywords Auctions • Bidding • Day-ahead electricity markets • Day-ahead markets • Mathematical programming • Unit commitment 1 Introduction The transformation from regulation to competition in power industries around the world have led to the development of markets for power. Day-ahead electricity markets are emerging as an important medium through which power is allocated in many de-regulated environments. A day-ahead electricity market is a short term hedge market that operates a day in advance of the actual physical delivery of power. In these environments, the generation decisions for the next day are in most cases the result of a double (two-sided) auction where producing (selling) and consuming (buying) agents submit a set of price-quantity curves (bids). The bids must be submitted by a deadline on the day before actual delivery of power. R.H. Kwon (*) • D. Frances Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON, Canada e-mail: rkwon@mie.utoronto.ca; frances@mie.utoronto.ca A. Sorokin et al. (eds.), Handbook of Networks in Power Systems I, Energy Systems, DOI 10.1007/978-3-642-23193-3_2, # Springer-Verlag Berlin Heidelberg 2012 41
- 55. Exploring the Variety of Random Documents with Different Content
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