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A Bilevel Game-Theoretic Decision-Making Framework for Strategic Retailers
in Both Local and Wholesale Electricity Markets
Qiuyi Honga, Fanlin Mengb,∗, Jian Liuc, Rui Boc
aDepartment of Mathematical Sciences, University of Essex, Colchester, CO4 3SQ, UK
bAlliance Manchester Business School, University of Manchester, Manchester, M15 6PB, UK
cDepartment of Electrical and Computer Engineering, Missouri University of Science and Technology, Rol la, MO 65409, USA
Abstract
This paper proposes a bilevel game-theoretic model for multiple strategic retailers participating in both whole-
sale and local electricity markets while considering customers’ switching behaviors. At the upper level, each
retailer maximizes its own profit by making optimal pricing decisions in the retail market and bidding deci-
sions in the day-ahead wholesale (DAW) and local power exchange (LPE) markets. The interaction among
multiple strategic retailers is formulated using the Bertrand competition model. For the lower level, there are
three optimization problems. First, the welfare maximization problem is formulated for customers to model
their switching behaviors among different retailers. Second, a market-clearing problem is formulated for the
independent system operator (ISO) in the DAW market. Third, a novel LPE market is developed for retailers
to facilitate their power balancing. In addition, the bilevel multi-leader multi-follower Stackelberg game forms
an equilibrium problem with equilibrium constraints (EPEC) problem, which is solved by the diagonalization
algorithm. Numerical results demonstrate the feasibility and effectiveness of the EPEC model and the im-
portance of modeling customers’ switching behaviors. We corroborate that incentivizing customers’ switching
behaviors and increasing the number of retailers facilitates retail competition, which results in reducing strate-
gic retailers’ retail prices and profits. Moreover, the relationship between customers’ switching behaviors and
welfare is reflected by a balance between the electricity purchasing cost (i.e., electricity price) and the electricity
consumption level.
Keywords: Bilevel model, strategic retailers, customers switching behaviors, wholesale and local markets,
Bertrand competition, EPEC
Nomenclature
Abbreviations and Indices
DAW Day-ahead Wholesale.
LPE Local Power Exchange.
ISO Independent System Operator.
ESS Energy Storage System.
∗Corresponding author
Email address: fanlin.meng@manchester.ac.uk (Fanlin Meng)
Preprint submitted to Applied Energy November 29, 2022
DR Demand Response.
KKT Karush-Kuhn-Tucher.
MPEC Mathematical Programming with Equilibrium Constraints.
EPEC Equilibrium Problems with Equilibrium Constraints.
MIQP Mixed-Integer Quadratic Programming.
kIndex of the strategic retailer.
nIndex of all retailers.
mIndex of generators.
tIndex of time periods.
Sets
MSet of generators in the grid.
NSet of retailers in the grid.
TSet of scheduling hours.
Parameters
∆tDuration of each time period.
ϵkSelf loss of the ESS of the retailer k.
ηc
k, ηd
kCharging and discharging efficiencies of the ESS of the retailer k.
ωn,j ,∀j=nSwitching coefficient among retailers.
ωt
n,n Self-elasticity of the retailer nat time t.
πbid,t
iThe electricity price the retailer ibought from the DAW market at time t.
πLP E,t
iThe electricity price the retailer ibought from the LPE market at time t.
πretail,t
iThe electricity price the retailer isold to customers at time t.
πbid,max
kMaximum bid price of the retailer k.
πbid,min
kMinimum bid price of the retailer k.
πretail,max
kMaximum retail price of the retailer k.
πretail,min
kMinimum retail price of the retailer k.
ckOperation and maintenance cost of the retailer k.
cmThe production cost of the generator m.
2
Emin
k, Emax
kMinimum and maximum energy level of ESS of the retailer k.
pc,min
k, pc,max
kMinimum and maximum charging power of ESS of the retailer k.
pd,min
k, pd,max
kMinimum and maximum discharging power of ESS of the retailer k.
qmin
m, qmax
mMinimum and maximum electricity volume that the generator msold to the DAW market.
qLP E,max,t
n,in Maximum electricity volume the retailer nbought from other retailers in LPE market.
qLP E,max,t
n,out Maximum electricity volume the retailer nsold to the other retailers in LPE market.
qbid,max,t
nMaximum electricity volume that the retailer nbought from the DAW market.
qbid,min,t
nMinimum electricity volume that the retailer nbought from the DAW market.
Variables
γc,t
k, γd,t
kCharging and discharging status of the ESS of the retailer k.
λLP E,t LPE market-clearing price at time t.
λtDAW market-clearing price at time t.
πbid,t
kBid price of the retailer kat time t.
πretail,t
kRetail price of the retailer kat time t.
Et
kEnergy level of the ESS of the retailer kat time t.
pc,t
k, pd,t
kCharging and discharging power of the ESS of the retailer kat time t.
qt
mElectricity volume that the generator msold to the DAW market at time t.
qbid,t
nElectricity volume that the retailer nbought from the DAW market at time t.
qLP E,t
nElectricity volume that retailer nbought from other retailers (if positive), or sold to other retailers
(if negative) in the LPE market.
qretail,t
nElectricity volume the retailer nsold in the retail market at time t.
1. Introduction
1.1. Background
Strategic bidding and offering are important research problems for both wholesale and local electricity
markets where market participants attempt to maximize their own profits or minimize their costs by choosing
optimal strategies. Many existing studies address along the direction, but mainly focus on the decision-making
problem of electricity producers (e.g., generators). This is due to the fact that previously only electricity
producers typically act as price-makers in the wholesale electricity markets [1–3]. However, with the development
of smart grids and demand response (DR) management, the role of market players such as energy retailers has
3
been changing. Traditionally, energy retailers act as price-takers in the wholesale market while offering fixed
retail prices to their customers. With the increasing demand-side flexibility empowered by the penetration of
distributed energy resources (DERs) such as electric vehicles, energy storage systems (ESS), photovoltaic, and
DR programs [4, 5], energy retailers are now better positioned to make strategic bidding in the wholesale and
local electricity markets and offer more flexible retail pricing decisions such as dynamic pricing to end customers
[6, 7].
1.2. Literature Review
The decision-making of participants in hierarchical systems (e.g., electricity markets) is often modeled as
a bilevel optimization problem or Stackelberg game [8][9]. In bilevel models for electricity markets, strategic
participants (e.g., electricity generators and retailers) either maximize their profits or minimize their costs at
the upper level. The lower level usually consists of a market-clearing problem solved by ISO or a customer-
side energy management problem. The standard approach to solving the bilevel models is reformulating it
as a single-level mixed-integer program by applying Karush-Kuhn-Tucker (KKT) conditions to the lower level
problem. There are numerous existing studies along this direction. For instance, in [10], a scenario-based
bilevel model has been applied to a large consumer’s profit maximization problem where the wholesale market-
clearing problem is considered at the lower level, and a heuristic method is introduced to solve one mathematical
programming with equilibrium constraints (MPEC) per scenario. [11] proposes a customized pricing framework
for retailers for different residential users. The pricing framework is modeled as bilevel program where retailers
purchase electricity from wholesale markets and compete for the market share. Although the bilevel models
considering retailers, system operators, or generators are prevailing, there are increasing attentions paid to
other market participants such as DR aggregators and microgrids. For instance, [12] introduces multi-energy
players as aggregators to maximize their profits and mitigate their operational risks. The problem is modeled
as a bilevel problem and interpreted as an MPEC problem. [13] focuses on the reserve management problem of
the EV aggregator. The upper level of the bilevel model is formulated as the profit maximization problem of
the EV aggregator. The lower level represents optimal charging/discharging decisions of EV owners. An exact
and finite decomposition algorithm is proposed to solve the problem in an iterative manner. [14] proposes a
bilevel program for EV aggregators from a different perspective. Instead of maximizing profit at the upper level,
charging cost minimization is formulated. The lower level represents the DAW market-clearing problem. [15]
develops a single-leader multi-follower game model where the market operator acts as the player at the upper
level and smart grid entities at the lower level aim to optimally schedule their own renewable energy resources,
energy storage, and DR resources. Likewise, [16] develops a bilevel model for microgrids to achieve optimal
bidding strategy, in which the lower level is distributed energy market’s clearing problem and the upper level
represents the optimal scheduling problem for a microgrid. [17] constructs a bilevel Stackelberg competition
model to investigate the interaction between regulated and merchant storage investment. A merchant profit
maximization problem is modeled at the upper level, while an overall system cost minimization problem is
formulated at the lower level. [18] proposes a stochastic bilevel framework to model the interactions between
a wind power producer at the upper level, and EV and DR aggregators at the lower level. The wind power
producer is also formulated to achieve optimal bidding decisions in the competitive wholesale markets.
From an economics point of view, existing studies on strategic bidding and offering problems can be classified
based on whether the market participants are price-makers or price-takers [2]. If the market participants have
4
relatively large-scale and flexible loads or supplies, they can be considered as price-makers. Along this direction,
[6] develops a short-term planning model of a price-maker retailer with flexible power demand participating
in the DAW electricity market. [19] develops a new scenario-based stochastic optimization model for price-
maker economic bidding in both day-ahead and real-time markets where a DR program with time-shiftable
load is adopted to create load flexibility. [14] proposes an optimal bidding strategy for a large-scale plug-
in electric vehicle (PEV) aggregator. The upper level represents the charging cost minimization of the PEV
aggregator, whereas the market-clearing problem is formulated at the lower level. In contrast, if the market
participants are small-scale or have inelastic loads or supplies, they usually act as price-takers. Along this
direction, [4] formulates a stochastic mixed-integer linear program to obtain an optimal bidding strategy for a
DERs aggregator participating in the day-ahead market where the market-clearing prices are given by different
scenarios. In [15], the lower level of the bilevel program represents multiple smart grids’ optimal scheduling
problems, whereas the ISO clears the day-ahead market at the upper level. [20] takes both price-maker and
price-taker positions into consideration. Specifically, the DR aggregator acts as a price-taker and a price-maker
in the day-ahead and real-time market, respectively.
Decision-making of multiple retailers has also been studied in the literature either through a single-level
model or a bilevel model. For the former, [21] addresses the portfolio optimization model of retailers, which
involves a risk-return optimization method based on the Markowitz theory. [22] proposes a multistage stochastic
optimization approach to capture the uncertainties of electricity loads and prices for retailers’ contract portfolios
which account for their risk preferences. For the latter, [23] proposes a bilevel multi-leader multi-follower game
to investigate the benefit of aggregation of prosumers to revenue generation in wholesale and retail markets in
which aggregated prosumers act as retailers (leaders) and end-users act as followers. [24] considers strategic
firms as leaders in the upper level problem, whereas electricity and natural gas market operators act as followers
in the lower level. [25] presents a dynamic pricing framework for electricity and gas utility companies in the
coupled retail electricity and natural gas markets by developing a two-leader multi-follower bilevel model.
In particular, the electricity and gas utility companies acting as leaders serve energies to the integrated DR
aggregators which are followers at the lower level. The competition among multi-energy retailers in the presence
of integrated DR prosumers is formulated as a multi-leader-follower bilevel game in [26]. Lastly, [27] considers
an EPEC framework to model the interaction among generation companies, microgrids, and load aggregators
participating in the wholesale and distribution network electricity markets. In this paper, we study multiple
strategic retailers as price-makers participating in both wholesale and local/regional energy markets within the
bilevel decision making framework.
Existing studies can be further categorized based on whether market players participate in multiple levels
of markets (e.g., wholesale vs. local/ retail) simultaneously. Most studies, however, are often based on a single
electricity market, such as day-ahead market [1, 3, 4, 6, 10, 14, 15, 29] or retail market [2, 7, 28, 33, 36]. There
are also a few studies focus on analyzing interactions among market participants in the wholesale (i.e., day-ahead
and real-time) electricity markets [11, 19, 20]. Only a few studies in the literature consider multiple levels of
markets simultaneously, such as wholesale and retail markets [12, 27, 34, 37]. For instance, the aggregator in
[12] participates in both the wholesale and local energy markets. [34] proposes a framework that can optimize
the strategy of a distribution company owning DERs and ESS in the wholesale and retail energy markets. In
this paper, we also consider multiple levels of electricity markets (i.e., wholesale and local markets). Apart
from the conventional retail market, we develop a novel local/regional energy exchange market named the LPE
5
Table 1: Literature classification. ✓: Yes; ✗: No; – : Not applicable.
Literature Bilevel model Price maker Multi-market Multi-leader Customer behavior
[23, 27] [25, 26] ✓ ✓ ✓ ✓ ✗
[2, 3, 10, 13, 14, 16, 28] ✓ ✓ ✗ ✗ ✗
[1, 24, 29] ✓ ✓ ✗ ✓ ✗
[15] ✓ ✗ ✗ ✗ ✗
[4] ✗✗✗–✗
[6, 30, 31][22] ✗✓✗–✗
[21] ✗ ✓ ✓ –✗
[12, 32] ✓ ✓ ✓ ✗ ✗
[33] ✓ ✓ ✗ ✗ ✓
[7] ✗✗✗–✓
[11, 20, 34][35] ✓ ✓ ✓ ✗ ✓
[19, 36] ✗ ✓ ✓ –✓
[37] ✗ ✓ ✓ ✓ ✓
market for retailers. In the literature, studying the local energy market typically focuses on modeling the
operation of emerging market participants such as prosumers, DERs aggregators, and microgrids [30, 31]. For
instance, in [30], a local power exchange center is developed where a novel clustering algorithm is developed
to cluster prosumers trading in the local energy market geographically. Another local energy exchange market
design for energy trading among energy storage unit’s owners is studied in [31], where a novel local energy
exchange market-clearing approach is proposed based on double auctions. However, modeling the established
and traditional role of energy retailers in the local market is much less studied. In this paper, we propose a
LPE market for strategic retailers equipped with energy storage to manage their supply and demand deviation.
Compared to the papers mentioned above, the uniqueness of our proposed LPE market lies in that: 1) The
participants in the LPE market are strategic retailers equipped with energy storage and arbitrage opportunities;
2) retailers in the LPE market can buy/ sell electricity from/ to other retailers; 3) the LPE market provides
a platform for retailers to balance their supply and demand deviation in a local level market. This new local
market for energy retailers will complement existing local energy markets to better facilitate the management
of local and distribution energy systems.
In addition to the strategic decision-making problem of multiple retailers in multiple levels of electricity
markets, customers’ switching behaviors are also modeled in this paper. There are only a few existing studies
that address along this direction. For instance, [37] considers customers’ switching behaviors in the retail market
where a single-level model is proposed to maximize the profit of strategic retailers. [35] presents a decision-
making framework for an electricity retailer considering the rational response of consumers under the competitive
environment. The retailer is considered as a price-taker in the day-ahead market, and the rival retailers’ selling
prices are assumed to be given. The switching behaviors of consumers are modeled as the switching cost for
the hesitation of consumers to switch contracts between retailers. [33] adopts utility functions to model three
categories of DR customers based on their sensitivity to retail prices from low, semi, to high flexibility. It
should be noted that modeling customers’ switching behaviors for the strategic offering of multiple retailers is
6
particularly crucial to capture the switching decisions of customers among different retailers, the implications
and impacts on retailers’ strategic decisions, and the market operations. To the best of our knowledge, there
is no existing research tackles this problem while considering the hierarchical nature of multiple competitive
price-maker retailers and customers.
The above reviewed literature is summarized in Table 1. To fill the research gap following the above
analysis, we propose a bilevel game-theoretic framework to model the multiple retailers’ (as price-makers)
optimal decision-making problems when participating in both wholesale and local markets with customers’
switching behaviors considered.
1.3. Contributions
The contributions of this paper are summarized as follows:
•We propose a novel bilevel model to formulate strategic behaviors of multiple retailers as price-makers
participating in both DAW and local markets. The proposed bilevel model consisting of multiple retailers,
multiple electricity markets, and customers’ abilities to switch to different retailers is particularly important to
model practical scenarios. To the best of our knowledge, this is the first work from the bilevel game-theoretic
perspective to investigate the problem for multiple retailers considering customers’ switching behaviors and
market share.
•The bilevel problem with a single retailer is firstly reformulated into an MPEC problem by deriving KKT
conditions from lower level problems. To overcome the non-convexity in the resulting MPEC problem intro-
duced by the bilinear terms and complementarity slackness constraints, linearization methods are conducted,
which leads to a tractable mixed-integer quadratic programming (MIQP) problem. In addition, the Bertrand
competition model is adopted to model the interaction among strategic retailers, which is formulated as an
EPEC problem and solved by the diagonalization algorithm.
•Comprehensive numerical results are provided to verify the feasibility and effectiveness of the proposed
EPEC model and diagonalization algorithm. In addition, the effects of customers’ switching behaviors and
the number of retailers in the markets on the strategic retailers’ optimal decisions are extensively studied.
Specifically, increasing customers’ switching behaviors and the number of retailers promotes retail competition,
which negatively correlated to strategic retailers’ equilibrium retail prices and profits. The relationship between
customers’ switching behaviors and their welfare is also elaborated.
1.4. Paper Organization
The remainder of this paper is organized as follows. The proposed bilevel model of a single retailer is
developed in Section 2. Section 3 discusses the methodologies for reformulating the bilevel model into an MIQP
model. Furthermore, the diagonalization algorithm for solving the EPEC problem with multiple retailers is
also proposed in this section. Numerical results are presented and discussed in detail in Section 4. Section 5
concludes this paper.
2. Bilevel Game-Theoretic Model
This section proposes a bilevel optimization problem for a strategic retailer who maximizes its profit. Specif-
ically, the strategic retailer participates in DAW and local markets (i.e., retail and LPE markets). The detailed
description of the proposed bilevel model is presented in Section 2.1. Furthermore, the upper and lower level
7
problems of the bilevel model are introduced and analyzed in Section 2.2 and 2.3, respectively. Consequently,
the complete bilevel model is formulated in Section 2.4.
2.1. Model Description
The proposed bilevel model with a single retailer can be interpreted as a single-leader multi-follower game
where the strategic retailer acts as the leader, whereas customers, ISO, and the LPE market operator are
followers. In particular, the strategic retailer optimizes the ESS management and pricing decisions (i.e., retail
prices in the retail market, bid prices in the DAW market, and bid/offer prices in the LPE market) at the
upper level. Subsequently, customers react to the optimal load demand at the lower level based on their
welfare. Market operators clear their corresponding markets (i.e., DAW and LPE markets) and send their
cleared electricity volume back to the strategic retailer. The structure of the proposed bilevel model is shown
in Figure 1. Specifically, the strategic retailer kmaximizes its profit at the upper level by setting its strategies
when participating in all three electricity markets. These strategies include its retail prices πretail,t
kin the
retail market, its bid prices in the DAW market πbid,t
k, its bid/offer prices in the LPE market πLP E,t
kand its
ESS charging/discharging volume pc,t
e/pd,t
e. Subsequently, there are three lower level problems. The first lower
level problem describes customers’ welfare maximization problem. The welfare function is formulated as the
difference between customers’ utility and their cost of purchasing electricity [38]. The market share function
of the retailer k, as opposed to other retailers participating in the retail market, is derived after reformulating
the problem, which can be embedded directly into the upper level problem as a constraint. The ISO’s DAW
market-clearing problem is constructed as the second lower level problem. The ISO receives the bid prices and
electricity load demand from retailers, and offer prices and generation capacities from generators to clear the
DAW market. As a result, generators receive the volume of electricity that needs to be produced in each time
period, while retailers receive the volume of electricity allocated to each of them. The market-clearing price
of the DAW market can also be obtained. The third lower level problem represents the LPE market-clearing
problem, where the volume of electricity that each retailer needs to buy or sell is optimized. The market-clearing
price of the LPE market can be derived simultaneously.
2.2. Upper Level Problem
The upper level problem aims to maximize the profit of the strategic retailer kparticipating in the retail,
DAW, and LPE markets. We assume that all three markets are operated on an hourly basis and scheduled on
the same time horizon T={1, ..., T }[19, 34]. It is also assumed that the retailer kowns the ESS, which aims
to facilitate its energy operations. Mathematically, the upper level problem is modeled as follows:
Maximize
Ξupper X
t∈T (πretail,t
kqretail,t
k−λtqbid,t
k−ck(pc,t
k+pd,t
k)∆t−λLP E,t qLP E,t
k)(1a)
Subject to:
πretail,min
k≤πretail,t
k≤πretail,max
k,∀t∈ T (1b)
πbid,min
k≤πbid,t
k≤πbid,max
k,∀t∈ T (1c)
πLP E,min
k≤πLP E,t
k≤πLP E,max
k,∀t∈ T (1d)
Et+1
k=Et
k+ηc
kpc,t
k∆t−1
ηd
k
pd,t
k∆t−ϵk∆t, ∀t∈ T (1e)
8
Strategic retailer
Bid/offer in
LPE market
Bid in
DAW
market
Retail price
Electricity volume
& market clearing
price in LPE
market
Electricity volume
& market clearing
price in DAW
market
Retail market
share
DAW market clearing process
Suppliers'
offers
Other retailers'
bids
DAW
market
clearing
price
Electricity
volume in
DAW
market
Other retailers'
electricity
volume in
DAW market
Suppliers'
electricity
volume
LPE market clearing process
Other retailers' bids/offers
LPE market
clearing price
Electricity
volume in
LPE
market
Other retailers'
electricity
volume in LPE
market
Customers' welfare
maximization process
Other retailers'
retail prices
Retail market
share
Other retailers'
retail market
share
Lower level problem 1Lower level problem 2Lower level problem 3
Upper level problem
ESS
management
Figure 1: Bilevel model structure.
Emin
k≤Et
k≤Emax
k,∀t∈ T (1f)
E1
k=ET+1
k(1g)
γc,t
kpc,min
k≤pc,t
k≤γc,t
kpc,max
k,∀t∈ T (1h)
γd,t
kpd,min
k≤pd,t
k≤γd,t
kpd,max
k,∀t∈ T (1i)
γc,t
k+γd,t
k≤1,∀t∈ T (1j)
γc,t
k, γd,t
k∈ {0,1},∀t∈ T (1k)
qbid,t
k+pd,t
k∆t+qLP E,t
k=qretail,t
k+pc,t
k∆t, ∀t∈ T (1l)
The decision variables of the upper level problem are Ξupper ={πretail,t
k, πbid,t
k, πLP E,t
k, pc,t
k, pd,t
k, Et
k, γc,t
k, γd,t
k,∀t∈
T }.
The upper level objective function (1a) denotes the overall profit that the strategic retailer kcan obtain. It
consists of the revenue made in the retail market, the cost of purchasing electricity in the DAW market, the cost
of operating the ESS, and the revenue or cost made in the LPE market. (1b)-(1d) constrain the pricing decisions
of the retailer in the three markets, respectively. We define the operating constraints for the ESS following [39]
9
[40]. In particular, (1e) represents the time-varying energy level of ESS. (1f), (1h) and (1i) ensure the energy
level, charging and discharging power of the ESS at each time period follow the operational limitations. (1g)
makes sure that by the end of the scheduling hours, the energy level of the retailer is equivalent to the initial
energy level. (1j) and (1k) ensure the ESS can only be in either charging or discharging state in a time period.
(1l) represents the retailer’s power balance constraint at each time period.
2.3. Lower Level Problems
The lower level of the proposed bilevel model consists of three different optimization problems: customers’
welfare maximization problem and market-clearing problems of the DAW and LPE markets, respectively. It
should be noted that we model aggregated customers’ welfare and behavior from the perspective of retailers
to reflect customers’ switching behaviors among different retailers. In addition, we follow [3, 12, 35, 41] in
formulating the market-clearing problems by omitting the loss of direct current power flow and line congestion
in transmission (i.e., DAW market) and distribution (i.e., LPE market) networks. Such a modeling choice will
improve the computational tractability and also allow us to focus on studying the strategic behaviors of retailers
in different electricity markets.
2.3.1. Customers welfare maximization
In the first lower level problem, customers’ satisfaction is considered and modeled as the utility function
from microeconomics [42]. Following [37] [43], the utility function can be formulated as follows:
U(qretail,t) = X
n∈N
αt
nqretail,t
n−1
2 X
n∈N
βt
nqretail,t2
n+X
n∈N ,i∈N \{k}
βt
n,iqretail,t
nqretail,t
i!(2a)
where N={1, ..., N }represents a set of retailers in the markets. qr etail,t ∈ RNis a vector where each element
denotes the electricity demand of customers from each retailer at time t. Moreover, customers’ welfare is defined
as the difference between the utility of all customers and the electricity purchase cost [38], which is formulated
below:
Maximize
Ξlower1X
t∈T (U(qretail,t)−X
n∈N
qretail,t
nπretail,t
n)(2b)
where the decision variables of the customer’s welfare maximization problem are Ξlow er1={qretail,t
n,∀n∈
N,∀t∈ T }.
After deriving KKT optimality conditions from (2b), the market share function of each retailer is obtained
below, which can be directly embedded in the upper level optimization problem of the retailer as a constraint.
qretail,t
n(πretail,t) = X
j∈N
ωt
n,j αt
j−ωt
n,nπretail,t
n−X
j∈N \{n}
ωt
n,j πretail,t
j,∀n∈ N ,∀t∈ T (2c)
where πretail,t ∈ RNis a vector that each element denotes the electricity retail price of each retailer at time t.
The details of the derivation of (2c) can be found in Appendix A. In particular, elements along the main diagonal
of Ωt(taking into account the negative sign) could be used to indicate the self-elasticity of the corresponding
retailer’s pricing decisions on its own customers. For instance, when the magnitude of ωt
n,n becomes larger, it
causes the load of customers served by the retailer nto reduce given that the unit retail price πr etail,t
nincreases.
Furthermore, other off-diagonal elements of Ωt(taking into account the negative sign) could be used to indicate
10
cross-impact effects among retail prices of different retailers, which can be interpreted as switching coefficients
[37]. The switching coefficients indicate the impact on the retailer’s market share when other retailers change
their retail prices. A larger magnitude of the switching coefficient demonstrates a more significant impact on
other retailers’ retail price change to the retailer’s market share. From the customers’ perspective, (2c) implies
that customers can switch among different retailers based on their offered retail prices. Specifically, customers
prefer to switch to other retailers who offer lower retail prices when their subscribed retailer increases its retail
price. Moreover, Pj∈N ωt
n,j αt
jindicates the market share potential of the retailer n, which is not affected by
the price changes. It is also worth noting that (2c) indicates customers switch energy retailers at each time
period t(e.g. on hourly basis), which could be a viable business model in practice. This is because with the
development of information and communication technology and smart meter analytics, technical barriers to
automatic and smart switching among retailers will be ultimately removed [44][45]. In addition, the proposed
agile customer switching model could be modified and utilized to provide much-needed demand flexibility in
short notice to help with the demand and supply management (e.g. unexpected peak demand or excessive
renewable generation in some hours due to forecast uncertainty).
2.3.2. DAW Market-Clearing Problem
The ISO’s DAW market-clearing problem is formulated to minimize the social cost among all generators
and retailers participating in the DAW market [46]. Specifically, the bid prices πbid,t
kof the strategic retailer
kare treated as known parameters in the lower level problem. Furthermore, all generators are assumed to be
non-strategic since we focus on the strategic behaviors of retailers in this paper. The optimization problem is
therefore formulated below.
Minimize
Ξlower2X
t∈T (X
m∈M
qt
mcm− qbid,t
kπbid,t
k+X
i∈N \{k}
qbid,t
iπbid,t
i!) (3a)
Subject to:
qmin
m≤qt
m≤qmax
m:µt
m, µt
m,∀m∈ M,∀t∈ T (3b)
qbid,min,t
k≤qbid,t
k≤qbid,max,t
k:ζt
k, ζt
k,∀t∈ T (3c)
qbid,min,t
i≤qbid,t
i≤qbid,max,t
i:ζt
i, ζt
i,∀i∈ N \ {k},∀t∈ T (3d)
qbid,t
k+X
i∈N \{k}
qbid,t
i−X
m∈M
qt
m= 0,:λt,∀t∈ T (3e)
where Ξlower2={qt
m, qbid,t
k, qbid,t
i,∀i∈ N \ {k},∀t∈ T } are the decision variables in this lower level problem.
Ξdual
lower2={µt
m, µt
m, ζt
k, ζt
k, ζt
i, ζt
i, λt,∀m∈ M,∀i∈ N \ {k},∀t∈ T } represents the set of dual variables of
corresponding constraints.
The objective function (3a) minimizes the social cost of the DAW market. The production level of each
generator is constrained in (3b). (3c) and (3d) constrain the demand level of strategic retailer kand other
retailers, respectively. (3e) represents the electricity supply and demand balance. Furthermore, the dual variable
λtin (3e) represents the market-clearing price of the DAW market.
2.3.3. LPE Market-Clearing Problem
The LPE market facilitates each retailer’s electricity supply and demand balance. The LPE market oper-
ator acts as a non-profit entity (the same role as the ISO) and clears the LPE market as the social welfare
11
maximization problem. The mathematical formulation is shown as follows:
Maximize
Ξlower3X
t∈T (πLP E,t
kqLP E,t
k+X
i∈N \{k}
πLP E,t
iqLP E,t
i)(4a)
Subject to:
−qLP E,max,t
k,out ≤qLP E,t
k≤qLP E,max,t
k,in :ψt
k,out, ψ t
k,in,∀t∈ T (4b)
−qLP E,max,t
i,out ≤qLP E,t
i≤qLP E,max,t
i,in :σt
i,out, σt
i,in,∀i∈ N \ {k},∀t∈ T (4c)
X
i∈N \{k}
qLP E,t
i+qLP E,t
k=0:λLP E,t ,∀t∈ T (4d)
where the decision variables are Ξlower3={qLP E,t
k, qLP E,t
i,∀i∈ N \ {k},∀t∈ T }. The dual variables of
corresponding constraints are denoted as Ξdual
lower3={ψt
k,out, ψ t
k,in, σ t
i,out, σt
i,in, λLP E ,t,∀i∈ N \ {k},∀t∈ T }.
The objective function (4a) maximizes the social welfare of the LPE market. (4b) and (4c) ensure the volume
of electricity that each retailer buys or sells in the LPE market is bounded. Finally, (4d) represents the power
balance constraint. The dual variable λLP E,t represents the market-clearing price of the LPE market.
2.4. Bilevel Model
After formulating both the upper and lower level problems, the proposed bilevel model for the strategic
retailer kcan be summarized as follows.
Ξupper ∈arg maximize
Ξupper
(1a) (5a)
Subject to:
Constraints (1b)-(1l) (5b)
Ξlower1∈arg maximize
Ξlower1
(2b) (5c)
Ξlower2, µt
m, µt
m, ζt
k, ζt
k, ζt
i, ζt
i, λt∈arg minimize
Ξlower2((3a)
Subject to:
Constraints(3b)-(3e)),∀m∈ M,∀i∈ N \ {k},∀t∈ T
(5d)
Ξlower3, ψt
k,out, ψ t
k,in, σ t
i,out, σt
i,in, λLP E ,t ∈arg maximize
Ξlower3((4a)
Subject to:
Constraints(4b)-(4d)),∀i∈ N \ {k},∀t∈ T
(5e)
(5a) and (5b) denote the strategies of the retailer kat the upper level. Furthermore, (5c)-(5e) represent the
reactions from the three electricity markets given by the upper level decisions, respectively. The bilevel model
forms a single-leader-multiple-follower Stackelberg game which can also be interpreted as an MPEC program
[47]. The methods to solve the MPEC problem are discussed in detail in the next section.
12
3. Solution Methods
The section illustrates the solution methods for both MPEC and EPEC problems. It first details the
treatment of the MPEC problem, which is linearized and reformulated to a MIQP problem. Furthermore,
the single leader MPEC model is extended to the multi-leader EPEC model, which can be solved by the
diagonalization algorithm.
3.1. MPEC Problem
The bilevel model can be transformed into a single-level MPEC problem by deriving KKT optimality condi-
tions for the lower level problems into a system of equations and inequalities. The transformed MPEC problem
is shown below:
Maximize
ΞMP E C
(1a) (6a)
Subject to:
Constraints (1b)-(1l), (2c) (6b)
cm−µt
m+µt
m−λt= 0,∀m∈ M,∀t∈ T (6c)
−πbid,t
k−ζt
k+ζt
k+λt= 0,∀t∈ T (6d)
−πbid,t
i−ζt
i+ζt
i+λt= 0,∀i∈ N \ {k},∀t∈ T (6e)
qbid,t
k+X
i∈N \{k}
qbid,t
i−X
m∈M
qt
m= 0,∀t∈ T (6f)
0≤(qt
m−qmin
m)⊥µt
m≥0,∀m∈ M,∀t∈ T (6g)
0≤(qmax
m−qt
m)⊥µt
m≥0,∀m∈ M,∀t∈ T (6h)
0≤(qbid,t
n−qbid,min,t
n)⊥ζt
n≥0,∀n∈ N ,∀t∈ T (6i)
0≤(qbid,max,t
n−qbid,t
n)⊥ζt
n≥0,∀n∈ N ,∀t∈ T (6j)
−πLP E,t
k−ψt
k,out +ψt
k,in +λLP E,t = 0,∀t∈ T (6k)
−πLP E,t
i−σt
i,out +σt
i,in +λLP E,t = 0,∀i∈ N \ {k},∀t∈ T (6l)
X
i∈N \{k}
qLP E,t
i+qLP E,t
k= 0,∀t∈ T (6m)
0≤ψt
k,out ⊥(qLP E,t
k−qLP E,max,t
k,out )≥0,∀t∈ T (6n)
0≤ψt
k,in ⊥(qLP E,max,t
k,in −qLP E,t
k)≥0,∀t∈ T (6o)
0≤σt
i,out ⊥(qLP E,t
i−qLP E,max,t
i,out )≥0,∀i∈ N \ {k},∀t∈ T (6p)
0≤σt
i,in ⊥(qLP E,max,t
i,in −qLP E,t
i)≥0,∀i∈ N \ {k},∀t∈ T (6q)
where the decision variables of the MPEC problem are ΞMP EC =nπretail,t
k, πbid,t
k, qretail,t
k, qretail,t
i, qbid,t
k, qbid,t
i, pc,t
k,
pd,t
k, Et
k, πLP E,t
k, qLP E,t
k, qLP E,t
i, qt
m, γc,t
k, γd,t
k, µt
m, µt
m, ζt
j, ζt
j, λt, ψt
k,out, ψ t
k,in, σ t
i,out, σt
i,in,∀t∈ T ,∀i∈ N \{k},∀m∈
M,∀j∈ N o.
(6a) denotes the objective function of the MPEC model. In the following constraints, (6b) represents a
collection of constraints from the upper level problem and retailers’ market share function. Equations (6c)-(6f)
13
and (6k)-(6m) are stationary conditions of the KKT optimality conditions. Moreover, (6g)-(6j) and (6n)-(6q)
represent the complementarity slackness.
3.2. Linearization of the MPEC Problem
The MPEC model above is non-convex and difficult to solve due to the existence of bilinear terms in the
objective function (6a) and complementarity slackness constraints (6g)-(6j) and (6n)-(6q). To overcome the
difficulties, we firstly deal with the bilinear terms in the upper level objective function (6a) through the strong
duality theorem [48]. Therefore, the objective function of the MPEC model becomes:
Φ = Pt∈T (Pm∈Mqt
mcm−µt
mqmin
m+µt
mqmax
m−Pj∈N \{k}πbid,t
jqbid,t
j+ζt
jqbid,min
j−ζt
jqbid,max
j+ckpc,t
k+
pd,t
k∆t−πretail,t
kPj∈N ωt
k,j αt
j+ωt
kπretail,t2
k+πretail,t
kPj∈N \{k}ωt
k,j πretail,t
j+Pi∈N \{k}σt
i,outqLP E ,max,t
i,out +
σt
i,inqLP E ,max,t
i,in −πLP E,t
iqLP E,t
i).
The details of the derivation of objective function Φ are provided in the Appendix B. Furthermore, Fortuny-
Amat transformation is used to linearize complementarity slackness by introducing additional binary variables
and a relatively large integer constant M[49]. The resulting linearized constraints of (6g)-(6j) and (6n)-(6q)
are shown in (7a)-(7j) and (7k)-(7t), respectively.
0≤µt
m≤ιt
mM, ∀m∈ M,∀t∈ T (7a)
0≤qt
m−qmin
m≤(1 −ιt
m)M, ∀m∈ M,∀t∈ T (7b)
0≤µt
m≤ιt
mM, ∀m∈ M,∀t∈ T (7c)
0≤qmax
m−qt
m≤(1 −ιt
m)M, ∀m∈ M,∀t∈ T (7d)
ιt
m, ιt
m∈ {0,1},∀m∈ M,∀t∈ T (7e)
0≤ζt
i≤ξt
iM, ∀i∈ N ,∀t∈ T (7f)
0≤qbid,t
i−qbid,min,t
i≤(1 −ξt
i)M, ∀i∈ N ,∀t∈ T (7g)
0≤ζt
i≤ξt
iM, ∀i∈ N ,∀t∈ T (7h)
0≤qbid,max,t
i−qbid,t
i≤(1 −ξt
i)M, ∀i∈ N ,∀t∈ T (7i)
ξt
m, ξt
m∈ {0,1},∀i∈ N ,∀t∈ T (7j)
0≤ψt
k,out ≤ρt
k,outM , ∀t∈ T (7k)
0≤qLP E,t
k+qLP E,max,t
k,out ≤(1 −ρt
k,out)M , ∀t∈ T (7l)
0≤ψt
k,in ≤ρt
k,inM , ∀t∈ T (7m)
0≤qLP E,max,t
k,in −qLP E,t
k≤(1 −ρt
k,in)M , ∀t∈ T (7n)
ρt
k,out, ρt
k,in ∈ {0,1},∀t∈ T (7o)
0≤σt
i,out ≤δt
i,outM , ∀i∈ N \ {k},∀t∈ T (7p)
0≤qLP E,t
i+qLP E,max,t
i,out ≤(1 −δt
i,out)M , ∀i∈ N \ {k},∀t∈ T (7q)
0≤σt
i,in ≤δt
i,inM , ∀i∈ N \ {k},∀t∈ T (7r)
0≤qLP E,max,t
i,in −qLP E,t
i≤(1 −δt
i,in)M , ∀i∈ N \ {k},∀t∈ T (7s)
δt
i,out, δt
i,in ∈ {0,1},∀i∈ N \ {k},∀t∈ T (7t)
14
LPE market Retail market
Multiple strategic retailers
Bids/offers
of all retailers
in LPE
market
Electricity
volume of all
retailers &
market clearing
price in LPE
market
Bids of all
retailers in
DAW
market
Electricity
volume of all
retailers &
market clearing
price in DAW
market
Retail
prices of all
retailers
Retail market
shares of all
retailers
Lower level
Upper level
DAW market
$
$
Figure 2: EPEC problem structure.
3.3. MIQP Problem
After the linearization, the MPEC model is reformulated into a MIQP problem and can be solved efficiently
using off-the-shelf solvers. The complete MIQP model is formulated as follows.
Minimize
ΞMI QP
Φ(8a)
Subject to:
Constraints (1b)-(1l), (2c), (6c)-(6f), (6k)-(6m), (7a)-(7t) (8b)
where ΞMI QP =nπretail,t
k, πbid,t
k, qretail,t
k, qretail,t
i, qbid,t
k, qbid,t
i, pc,t
k, pd,t
k, Et
k, πLP E,t
k, qLP E,t
k, qLP E,t
i, qt
m, γc,t
k, γd,t
k,
τt
in, τ t
out, µt
m, µt
m, ζt
j, ζt
j, ιt
m, ιt
m, ξt
j, ξt
j, λt, ψt
k,out, ψ t
k,in, σ t
i,out, σt
i,in, ρt
k,out, ρt
k,in, δ t
i,out, δt
i,in,∀t∈ T ,∀i∈ N \{k},∀m∈
M,∀j∈ N orepresents the set of decision variables of the MIQP model.
The objective function (8a) shapes a quadratic form with respect to πretail,t
k. Constraint (8b) consists of all
the constraints in the upper level problem, market share functions, KKT stationary conditions for the market-
clearing problems of the DAW and LPE markets, and the linearized complementarity slackness constraints.
3.4. EPEC Problem
The Bertrand competition model is utilized to extend the MIQP model from a single strategic retailer to
multiple strategic retailers. This results into a multi-leader multi-follower Stackelberg game and can be re-
15
Algorithm 1 Diagonalization algorithm
1: Initialization:
S0=nπretail,t
n, πbid,t
n, πLP E,t
n, pc,t
n, pd,t
n, Et
n, γc,t
n, γd,t
n,∀n∈ N ,∀t∈ T o;
maximum number of iterations Y; convergence criterion ϵ.
2: for y= 1 to Ydo
3: for i= 1 to Ndo
4: Solve strategic retailer i’s MIQP model assuming other retailers’ strategies as given parameters.
5: Update Sy
i;
6: end for
7: if ∥Sy
i−Sy−1
i∥ ≤ ϵ, ∀i∈ N then
8: The algorithm converges and terminates.
9: end if
10: if y=Ythen
11: The algorithm fails to converge and terminates.
12: end if
13: end for
formulated as an EPEC problem [47], which is illustrated in Figure 2. Although the retailers share complete
information among themselves in the theoretic setting of the EPEC problem, in practice, an independent market
agent (e.g. ISO for wholesale markets) can play such a role for sharing required information among retailers.
We adopt the diagonalization algorithm in [50] to tackle our formulated EPEC problem where the converged
strategies of strategic retailers represent a generalized Nash equilibrium. The diagonalization algorithm con-
sidered for solving our EPEC problem is outlined in Algorithm 1. In Step 1, the strategy set is initialized as
S0. The maximum iteration Yand convergence criterion ϵare also predefined. The main iteration procedure
of the diagonalization algorithm is shown in Steps 2 −13, which consists of an outer loop and an inner loop.
In particular, the outer loop controls the iteration of the algorithm. For each iteration of the outer loop, Steps
3−6 define the inner loop and aim to solve the MIQP problem for each strategic retailer sequentially with
the other retailers’ strategies as given parameters. The convergence of the algorithm is checked in Steps 7 −12
at each iteration of the outer loop. Specifically, in Steps 7 −9, if the difference between the retailers’ optimal
decisions of two adjacent iterations is less than ϵ, the algorithm converges and terminates with retailers’ optimal
decisions. However, in Steps 10 −12, if the algorithm reaches the maximum iteration Ywithout convergence,
it terminates and no optimal results are found.
4. Numerical Results
Numerical results are illustrated in this section to demonstrate the feasibility and effectiveness of the EPEC
model and the diagonalization algorithm. The effects of customers’ switching behaviors and the number of
strategic retailers on retail competition are discussed in detail. The proposed model is solved by Gurobi
Optimizer (version 9.5.2) using the branch and bound algorithm under Pyomo [51] on Windows 10 Enterprise
64-bit with 4 cores CPU at 3.6GHz and 16GB of RAM.
16
4.1. Experimental Setup
Data used in this section comes mainly from the PJM datasets [52], such as the initial retail and DAW
market bid prices for each retailer during the day. The initial LPE market bid/offer prices and the maximum
of cleared electricity volume are based on PJM real-time market bid prices and cleared electricity for each
retailer. We further calibrate the retailers’ maximum cleared electricity volume in the LPE market to be 5% of
the maximum bid load of retailers in the DAW market. The initial DAW market’s maximum bid load of each
retailer comes from the PJM DAW market bid load of different utility companies. In addition, the minimum and
maximum retail, DAW market bid, and LPE market bid/offer prices are all set to be $0/MWh and $300/MWh
respectively. The minimum bid load for the retailers in the DAW market is considered to be 0.1 MW following
PJM day-ahead wholesale market [53]. The maximum iteration Y= 30 and the termination criteria ϵ= 1 are
chosen for the diagonalization algorithm. The ESS-related parameters are modified based on [40]. In particular,
the initial ESS energy level is set to be 80M W h. The maximum and minimum charging and discharging
rates are 60MW and 2M W . The maximum and minimum ESS energy capacities are 200M W h and 30M W h.
The charging and discharging efficiencies are set to be 0.9. Lastly, the self-discharge rate ϵk= 0.002MW is
considered.
In this paper, we consider 24 time periods for the day starting from midnight. That is, each time period
represents an hour. In this case, 12 strategic retailers are considered in the proposed EPEC model. Furthermore,
the strategic retailers are classified into 3 groups based on their market share potential which the self-elasticity
coefficient ωt
n,n and parameter αt
nare assumed to be time-varying. Specifically, retailers 1 −4 are classified
into small market share group (group 1). Retailers 5 −8 belong to the medium market share group (group 2).
Lastly, retailers 9 −12 are in the large market share group (group 3). The input data of electricity prices and
volume, self-elasticity coefficient ωt
n,n, and αt
nfor each retailer can be found in Appendix C.1. Additionally,
we include 30 generators participating in the DAW market. The cost and maximum supply of each generator
are shown in Appendix C.2.
(a) Prices when switching coefficient is 0 (b) Prices when switching coefficient is - 4 (c) Prices when switching coefficient is - 7
Figure 3: Time-varying retail prices and market-clearing prices of LPE and DAW markets with different switching coefficients.
4.2. Illustrative Examples
In this section, illustrative examples are given to discuss the results of the EPEC model when switching
coefficients are set to be 0M W h/$, −4M W h/$, and −7M W h/$ respectively. The magnitude of the switching
coefficient represents the ability of customers to switch to other retailers and thus the competition level in the
retail market. A larger magnitude of the switching coefficient indicates more competition in the retail market.
17
Time-varying retail prices of each retailer and market-clearing prices of the LPE and DAW markets are shown
in Figure 3. It can be found that both the retail and market-clearing prices decrease from 1 am to around 5
am, then increase until around 5 pm and drop down again afterward, which follows customers’ demand during
the day.
Furthermore, the retail prices of all retailers are generally higher than the market-clearing prices of the
LPE and DAW markets but become closer to the market-clearing prices with the increase of magnitude of the
switching coefficient. This can be explained that with more competition in the retail market, it drives down the
retail prices and retailers’ profit margin becomes lower. In addition, the retail prices are typically higher when
the retailers have a larger market share (bigger retailers). This could be due to that retailers with large market
share have more flexibility in their pricing decisions without worrying losing customers.
It is also observed that the market-clearing prices of the LPE market are generally more volatile than the
market-clearing prices of the DAW market. This could be explained by the fact that the market size (market-
cleared electricity volume) of the LPE market is much smaller than the DAW market. Therefore, the unit
difference in customers’ demand has a more significant impact on the LPE market, which results in higher
volatility of its market-clearing prices.
(a) Retail prices at 5 am (b) Retail prices at 12 pm (c) Retail prices at 5 pm
Figure 4: Retail prices of retailers with different switching coefficients at different times of the day.
Figure 5: Percentage change in retail prices with different switching coefficients.
18
Figure 6: Profit of retailers with different switching coefficients.
4.3. Retail Prices and Profits
Figure 4 presents the equilibrium retail prices among all strategic retailers given by different switching
coefficients at 5 am, 12 pm, and 5 pm, respectively. It shows that when the magnitude of the switching
coefficient becomes larger, the retail prices among all retailers decrease dramatically. This is because retailers
would like to reduce their retail prices to prevent customer losses as customers are more capable of switching
their electricity retailers.
Moreover, the percentage changes in average retail prices of different retailer groups at 5 am, 12 pm, and
5 pm are shown against different switching coefficients in Figure 5. From the figure, we can find that with
the increase of the magnitude of the switching coefficient, average retail prices of all retailer groups decrease
consistently for different time periods. It should also be noted that when the magnitude is less than 4MWh/$,
there is not much difference in price changes among three retailer groups at different time periods. However,
following the continuing increase of the magnitude, the price changes differ in different retailer groups and
different time periods. For instance, the price changes in all three retailer groups at 5 am are much higher than
other time periods. In addition, the price change of small retailer group (e.g. group 1) is larger than large
retailer group (e.g. group 3). The above two phenomena enforces our findings that the switching coefficients
have a larger impact on prices of small retailers and low-demand time periods.
The impact of the switching coefficient on the profits of retailers is illustrated in Figure 6. Not surprisingly,
the retailers’ profits reduce significantly when increasing the magnitude of the switching coefficient. In addition,
although the profits of bigger retailers are usually higher, the profit difference among retailers tends to decrease
when the magnitude of the switching coefficient becomes larger (higher competition in the market). In other
words, a market with higher competition provides a healthier environment for small players/ entrants.
4.4. Customers’ Welfare
The relationship between the switching coefficient and customers’ welfare is displayed in Figure 7, which
reflects the balance between the customers’ utility (the amount of electricity consumed) and the electricity
purchase cost. In particular, there is a positive correlation between the magnitude of the switching coefficient
and customers’ welfare until it reaches the peak with the switching coefficient around −4M W h/$. Thereafter,
19
Figure 7: Customers’ welfare with different switching coefficients.
the customers’ welfare decreases drastically. Namely, compared to the situation of no switching behaviors being
considered, increasing the magnitude of the switching coefficient at a certain level can increase customers’ welfare
since it can cause the reduction of the retail price whilst keeping the retailers’ load supply at an acceptable
level. However, when the magnitude of the switching coefficient becomes sufficiently large, it discourages the
retailers from offering electricity supply since the smaller profit margin in return. In this regard, the customers
are provided less electricity by the retailers, which results in the reduction of the customers’ utility. Therefore,
it leads to the customers’ welfare losses.
(a) ESS result when switching coefficient is 0 (b) ESS result when switching coefficient is - 4 (c) ESS result when switching coefficient is - 7
Figure 8: ESS energy level, charging and discharging power of retailers in group 1.
(a) ESS result when switching coefficient is 0 (b) ESS result when switching coefficient is - 4 (c) ESS result when switching coefficient is - 7
Figure 9: ESS energy level, charging and discharging power of retailers in group 2.
20
(a) ESS result when switching coefficient is 0 (b) ESS result when switching coefficient is - 4 (c) ESS result when switching coefficient is - 7
Figure 10: ESS energy level, charging and discharging power of retailers in group 3.
4.5. ESS Result
This section discusses the ESS operation in the EPEC problem. Figure 8-10 show the ESS energy level,
charging, and discharging power of each retailer in different retailers’ market share groups, respectively. Par-
ticularly, Figure 8 (a) indicates the ESS result when there are no customers’ switching behaviors. Figure 8
(b) and (c) show the ESS results when the switching coefficients are −4MW h/$ and −7M W h/$. Notice that
the line plot in each figure denotes the ESS energy level, while the bar plot indicates the charging power (if
positive) and discharging power (if negative) of the ESS. We conclude that the retailers typically charge their
ESS when the DAW market-clearing price is low and discharge the ESS when the DAW market-clearing price
is high regardless of the corresponding market share and the value of the switching coefficient.
Moreover, by comparing the ESS results under different switching coefficients, we can find that each retailer’s
charging/discharging strategy within each market share group becomes similar when the magnitude of the
switching coefficient increases. The reason is that increasing the ability of customers’ switching behaviors
causes the convergence of the retailers’ optimal strategies, including the ESS operating decisions.
4.6. The Number of Retailers on the Retail Competition
This section discusses the effect of the number of strategic retailers on the retail competition where the
results are shown in Table 2. We consider three different cases with different number of retailers. All cases
have three retailer groups with different market share. To focus on the effect of the number of retailers, we do
not consider switching behaviours in these three cases. The parameter setup for cases 2 and 3 can be found
in Appendix C.3 and Appendix C.4, respectively. Compared to case 1, decreasing the number of retailers in
cases 2 and 3 can significantly reduce the competition among retailers, resulting into much higher daily average
retail prices in the larger market share group (e.g., group 3). Furthermore, the reduced retail competition surges
the retail prices in each group consistently. For instance, the retailer’s daily average retail price in group 3 of
case 3 is $299.86/MWh, which approaches the cap of the retail price ($300/MWh). In addition, reducing retail
competition causes the remarkable dilation of retailers’ profit in each group and the total profit in each case.
This is the result of the noticeably increased retail price and market power of the retailers in the absence of
competition.
21
Table 2: The effect of the number of retailers on the retail competition.
Retailer Average retail price ($/MWh) Retail price by group ($/MWh) Profit ($) Profit by group ($) Total profit ($)
Case 1
Group 1
1 106.26
105.64
1.04 ×107
1.12 ×
107
1.95 ×
108
2 104.05 1.11 ×107
3 105.45 1.14 ×107
4 106.80 1.18 ×107
Group 2
5 124.23
124.73
1.53 ×107
1.60 ×
107
6 123.35 1.57 ×107
7 125.29 1.62 ×107
8 126.04 1.69 ×107
Group 3
9 143.40
144.40
2.07 ×107
2.16 ×
107
10 143.65 2.13 ×107
11 144.33 2.18 ×107
12 146.20 2.25 ×107
Case 2
Group 1 1 141.38 143.77 3.73 ×1073.83 ×
107
3.20 ×
108
2 146.15 3.94 ×107
Group 2 3 168.23 171.16 5.16 ×1075.28 ×
107
4 174.09 5.41 ×107
Group 3 5 198.55 201.54 6.79 ×1076.90 ×
107
6 204.53 7.02 ×107
Case 3
Group 1 1 223.20 223.20 1.16 ×1081.16 ×108
4.71 ×
108
Group 2 2 268.61 268.61 1.56 ×1081.56 ×108
Group 3 3 299.86 299.86 1.99 ×1081.99 ×108
5. Conclusion
This paper proposes a bilevel game-theoretic framework for strategic retailers who aim to maximize their
profits by participating both DAW and local electricity markets. In terms of the proposed bilevel model,
customers’ welfare function and switching behaviors are considered in the lower level problem along with the
market-clearing problems for the DAW and local electricity markets, respectively. Furthermore, the proposed
model is formulated as an MPEC problem and then reformulated to a MIQP model. By extending the above
bilevel model from a single leader (one retailer) to multiple leaders (multiple retailers), a Bertrand competition
model is adopted to model the interactions among multiple leaders at the upper level. Finally, the resulting
multi-leader multi-follower Stackelberg game model is reformulated as an EPEC problem and solved by the di-
agonalization algorithm. Extensive numerical results are present to demonstrate the feasibility and effectiveness
of the proposed bilevel strategic decision-making framework and the effect of customers’ switching behaviors on
decision making and benefits of different market players (e.g. retailers and customers). In particular, results
show that incentivizing customers’ switching behaviors can decrease strategic retailers’ retail prices and profits.
However, switching may not always benefit customers’ welfare due to customers’ need of balance between the
electricity purchasing cost (i.e., electricity price) and the electricity consumption level. In addition, similar ESS
charging/discharging decisions among strategic retailers are observed when enhancing the customers’ switching
behaviors.
The work can be further developed in the following directions. First, the modelling of customers’ switching
behaviors among different retailers could be considered in enhancing existing demand response programs such
as load shifting and curtailment [9]. Although the proposed LPE market only considers retailers in this study,
it could be extended to include other emerging players such as variable renewable energy sources. In addition,
the effect of network congestion and locational marginal prices on main findings of this study is also worth
22
investigating. Moreover, the proposed bilevel strategic model could consider multi-energy scenarios involving
electricity, natural gas, and heat energy. Lastly, data-driven approaches can be employed to improve the
modeling process. For instance, customers’ switching behaviors can be learned from historical data through
machine learning methods.
Appendix A. Derivation of the market share function
The combination of (2a) and (2b) can derive an unconstrained minimization problem as follows:
Minimize
Ξlower1X
t∈T (1
2 X
n∈N
βt
nqretail,t2
n+X
n∈N ,i∈N \{k}
βt
n,iqretail,t
nqretail,t
i!+X
n∈N
qretail,t
nπretail,t
n−X
n∈N
αt
nqretail,t
n)
(9a)
The first order conditions of the objective function (9a) can be derived as:
βt
nqretail,t
n+X
n∈N ,i∈N \{n}
+βt
n,iqretail,t
i+πretail,t
n−αt
n= 0,∀n∈ N ,∀t∈ T (9b)
It can be reformulated to a compact form:
πretail,t =αt−Btqretail,t ,∀t∈ T (9c)
where αt∈ RNis a vector that each element represents a parameter of each retailer. Bt∈ RN×Nis a
symmetric strictly diagonally dominant matrix that each element in a row/column represents the parameter of
each retailer.
Let Ωtbe the inverse matrix of Bt, and (9c) can be reformulated as below:
qretail,t =Ωtαt−Ωtπt,∀t∈ T (9d)
where Ωt=
ωt
1,1... ωt
1,N
... ... ...
ωt
N,1... ωt
N,N
,∀t∈ T are all symmetric matrices. Therefore, the market share function of
each retailer can be derived as:
qretail,t
n=X
j∈N
ωt
n,j αt
j−ωt
n,nπretail,t
n−X
j∈N \{n}
ωt
n,j πretail,t
j,∀n∈ N ,∀t∈ T (9e)
which is equivalent to (2c).
Appendix B. Linearization of the objective function of MPEC
Appendix B.1. Reformulation of bilinear terms
The Lagrange function of the minimization problem (3a)-(3e) is formulated as follows.
L(Ξlower2|Ξdual
lower2) = X
t∈T (X
m∈M
qt
mcm− qbid,t
kπbid,t
k+X
i∈N \{k}
qbid,t
iπbid,t
i!)
+X
t∈T X
m∈M µt
mqmin
m−qt
m+µt
mqt
m−qmax
m!+X
t∈T X
i∈N ζt
iqbid,min,t
i
−qbid,t
i+ζt
iqt
i−qbid,max,t
i!+X
t∈T λtX
i∈N
qbid,t
i−X
m∈M
qt
m!
(10a)
23
Then, the dual program can be derived below:
Maximize
Ξdual
lower2X
t∈T X
m∈M µt
mqmin
m−µt
mqmax
m!+X
t∈T X
i∈N ζt
iqbid,min,t
i−ζt
iqbid,max,t
i!(10b)
Subject to:
cm−µt
m+µt
m−λt= 0,∀m∈ M,∀t∈ T (10c)
−πbid,t
i−ζt
i+ζt
i+λt= 0,∀i∈ N ,∀t∈ T (10d)
Since the primal program (3a)-(3e) is a linear program, the strong duality theorem holds. This indicates
that the value of the primal objective function (3a) is the same as the value of the dual objective function (10b).
Therefore, we can then obtain a system of equations:
Objective function (3a) = Objective function (10b) (10e)
Constraints (6d),(6e) (10f)
ζt
i(qbid,min,t
i−qbid,t
i) = 0,∀i∈ N ,∀t∈ T (10g)
ζt
i(qbid,t
i−qbid,max,t
i)=0,∀i∈ N ,∀t∈ T (10h)
After solving the system of equations (10e)-(10h), we can derive the equality below.
X
t∈T
λtqbid,t
k=X
t∈T X
m∈M (qt
mcm−µt
mqmin
m+µt
mqmax
m)−X
t∈T X
j∈N \{k}(πbid,t
jqbid,t
j+ζt
jqbid,min,t
j−ζt
jqbid,max,t
j)
(10i)
Analogously, the Lagrange function of the problem (4a)-(4d) is formulated as follows.
L(Ξlower3|Ξdual
lower3) = X
t∈T (πLP EM,t
kqLP EM,t
k+X
i∈N \{k}
πLP EM,t
iqLP EM,t
i)
+X
t∈T (ψt
k,outqLP E M,t
k+qLP EM,max,t
k+ψt
k,inqLP E M,max,t
k,in −qLP EM ,t
k)
+X
t∈T X
i∈N \{k}(σt
i,outqLP E M,t
i+qLP EM,max,t
i,out +σt
i,inqLP E M,max,t
i,in
−qLP EM,t
i)−X
t∈T (λLP EM,t X
i∈N \{k}
qLP EM,t
i+qLP EM,t
k)
(10j)
The dual program of (4a)-(4d) is derived below.
Minimize
Ξdual
lower3X
t∈T (qLP EM,max,t
k,out ψt
k,out +qLP EM ,max,t
k,in ψt
k,in)+X
t∈T X
i∈N \{k}(σt
i,outqLP E M,max,t
i,out +σt
i,inqLP E M,max,t
i,in )
(10k)
Subject to:
πLP EM,t
k+ψt
k,out −ψt
k,in −λLP EM ,t = 0,∀t∈ T (10l)
πLP EM,t
i+σt
i,out −σt
i,in −λLP EM,t ,∀i∈ N \ {k},∀t∈ T (10m)
24
The primal program (4a)-(4d) is also a linear program. Therefore, the strong duality theorem holds. A
system of equations can be derived as follows.
Objective function (4a) = Objective function (10k) (10n)
Constraint (10l)
ψt
k,out(qLP E M,t
k+qLP EM,max,t
k)=0,∀t∈ T (10o)
ψt
k,in(qLP E M,max,t
k,in −qLP EM ,t
k)=0,∀t∈ T (10p)
A solution of the system of equations (10n)-(10p),and (10l) is shown below.
X
t∈T
λLP EM,t qLP EM ,t
k=X
t∈T X
i∈N \{k}(σt
i,outqLP E M,max,t
i,out +σt
i,inqLP E M,max,t
i,in −πLP EM,t
iqLP EM,t
i)(10q)
Appendix B.2. Reformulation of objective function of MPEC
There are three bilinear terms in the objective function of the MPEC program, which are λtqbid,t
k,λLP EM,t qLP EM ,t
k
and πretail,t
kqretail,t
k. The first two bilinear terms are linearized in (10i) and (10q), respectively. The last bilinear
term can be linearized by substituting Pj∈N ωt
k,j αt
j−ωt
k,k πretail,t
k−Pj∈N \{k}ωt
k,j πretail,t
jfor qretail,t
kbased on
(2c).
After linearizing the bilinear terms, the final objective function of MPEC program is derived as follows.
Φ = X
t∈T (X
m∈Mqt
mcm−µt
mqmin
m+µt
mqmax
m−X
j∈N \{k}πbid,t
jqbid,t
j+ζt
jqbid,min
j
−ζt
jqbid,max
j+ckpc,t
k+pd,t
k∆t−πretail,t
kX
j∈N
ωt
k,j αt
j+ωt
kπretail,t2
k
+πretail,t
kX
j∈N \{k}
ωt
k,j πretail,t
j+X
i∈N \{k}σt
i,outqLP E M,max,t
i,out +σt
i,inqLP E M,max,t
i,in
−πLP EM,t
iqLP EM,t
i)
(10r)
Appendix C. Input data
Appendix C.1. Data in case 1
Table C.3: Initial retail prices of retailers in case 1 ($/MWh).
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 37.88 34.55 35.72 32.23 34.07 37.75 37.52 41.66 49.56 52.75 59.06 68.86 74.26 79.44 85.44 100.36 94.39 81.25 67.66 64.47 54.95 48.48 41.64 40.20
2 39.28 36.18 33.82 32.62 34.60 36.83 40.55 43.33 49.94 49.03 47.61 51.33 52.44 54.83 53.01 66.12 59.83 52.98 54.56 52.73 49.23 47.31 43.99 42.34
3 37.95 35.44 35.36 33.41 33.46 36.28 38.77 44.49 50.54 57.28 64.90 74.22 77.24 89.10 93.23 105.97 100.47 81.00 71.94 65.26 53.89 50.07 41.15 43.06
4 40.34 35.27 32.80 31.79 35.71 36.99 39.38 43.80 52.16 49.82 47.19 50.28 54.84 58.89 58.84 80.03 79.59 69.16 59.39 49.13 47.31 44.95 42.45 42.91
5 37.64 34.81 34.12 33.35 35.22 36.88 37.71 44.26 49.91 49.91 46.58 53.00 50.81 53.83 53.31 62.38 59.93 52.60 50.74 49.08 47.92 47.43 40.79 41.54
6 38.81 35.28 32.68 30.90 32.10 35.41 38.36 40.51 46.65 49.63 58.64 66.94 74.77 83.71 95.71 107.56 95.76 84.04 73.90 53.36 48.54 45.91 42.73 39.78
7 36.06 35.83 33.23 32.51 33.25 35.09 40.49 43.65 47.98 48.75 46.35 49.65 52.61 51.43 49.51 60.94 56.61 51.34 51.50 49.07 45.58 44.98 41.49 41.57
8 37.66 34.18 34.26 32.33 36.04 37.19 39.43 42.90 49.94 51.49 63.97 70.27 75.38 84.64 93.90 103.83 93.60 76.59 69.49 70.11 59.02 49.98 43.10 42.34
9 37.89 34.66 31.78 32.94 33.58 35.56 39.29 43.00 48.29 46.30 50.58 54.73 60.84 65.78 71.82 80.41 72.09 66.97 59.88 48.60 45.58 44.35 40.55 39.56
10 36.31 36.21 32.53 31.39 33.89 35.50 38.82 44.11 48.78 49.94 53.22 59.43 63.45 66.53 66.23 67.76 64.79 64.10 61.80 56.28 48.58 46.77 40.15 39.28
11 37.69 35.34 33.70 32.10 34.12 36.12 37.70 43.40 48.85 49.48 47.45 51.62 52.89 54.02 54.43 63.38 58.91 54.34 52.27 50.38 48.35 44.98 43.00 40.81
12 37.46 35.78 34.30 34.01 34.34 36.96 40.19 44.15 48.42 49.12 50.82 51.85 55.66 55.90 58.14 65.42 62.37 54.03 55.40 53.17 46.99 46.84 41.50 41.42
25
Table C.4: Initial DAW market bid prices of retailers in case 1 ($/MWh).
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 28.80 26.83 25.61 24.99 26.12 27.82 29.56 32.50 37.98 40.83 46.35 53.13 57.40 60.64 67.00 77.74 71.72 61.07 53.21 49.10 43.02 37.95 32.70 31.25
2 29.45 27.28 25.97 25.37 26.61 28.41 30.58 33.82 38.42 37.79 36.66 39.29 40.19 41.15 40.84 49.66 46.26 41.12 41.89 40.29 37.66 35.40 32.96 32.19
3 29.45 27.34 26.11 25.53 26.75 28.44 30.27 33.34 38.96 42.32 49.42 57.00 59.75 67.79 71.90 82.15 76.85 62.30 54.42 50.26 43.65 37.91 33.10 31.94
4 29.63 27.46 26.13 25.53 26.80 28.61 30.75 33.88 38.52 38.06 35.73 38.08 41.40 44.48 45.81 61.73 61.81 53.75 44.81 38.18 36.45 35.11 33.11 32.40
5 29.30 27.15 25.87 25.27 26.48 28.24 30.36 33.56 38.01 37.61 36.48 38.83 40.03 40.85 40.97 48.36 45.47 40.68 40.83 38.65 36.51 34.76 32.52 31.97
6 28.74 26.67 25.43 24.86 26.06 27.81 29.79 32.89 37.19 37.94 45.42 52.07 57.83 65.02 72.91 81.48 72.07 64.10 56.27 39.92 36.00 34.19 31.94 31.09
7 28.97 26.89 25.63 25.05 26.26 27.99 30.07 33.22 37.73 37.20 35.42 37.79 38.75 39.59 39.07 46.85 44.04 39.07 39.45 37.48 35.50 34.28 32.31 31.58
8 29.20 27.15 25.97 25.41 26.61 28.27 29.97 32.92 38.46 41.44 48.27 54.68 57.25 65.60 72.34 80.04 72.80 59.05 53.09 54.25 46.37 38.89 32.96 31.69
9 28.50 26.44 25.21 24.64 25.81 27.54 29.52 32.60 36.76 36.50 39.38 43.41 46.65 50.58 54.53 61.69 55.74 49.89 46.85 38.41 35.53 33.71 31.46 30.77
10 28.79 26.72 25.50 24.98 26.16 27.93 30.04 32.90 37.87 38.45 41.22 45.86 48.18 50.77 51.21 52.84 49.94 48.67 47.55 43.39 38.76 35.21 32.03 31.20
11 29.28 27.16 25.89 25.28 26.49 28.25 30.39 33.58 38.11 37.74 36.61 38.97 40.15 40.86 40.94 48.22 45.44 40.76 40.92 38.78 36.53 34.82 32.53 32.01
12 29.23 27.10 25.84 25.24 26.47 28.20 30.43 33.60 38.25 38.22 37.98 40.84 42.33 43.33 43.95 50.92 47.94 43.12 42.65 40.07 37.30 35.12 32.47 31.90
Table C.5: Initial LPE market bid/offer prices of retailers in case 1 ($/MWh).
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 30.63 28.06 26.38 25.95 27.23 28.75 29.94 33.99 34.99 35.55 39.75 40.37 47.26 47.82 50.22 117.52 70.51 56.30 44.02 40.82 46.75 43.10 32.50 31.92
2 31.24 28.55 26.84 26.43 27.74 29.17 30.36 34.31 34.80 33.85 39.53 36.07 41.90 37.68 36.92 47.14 41.88 36.28 38.11 38.17 41.81 41.49 32.16 31.94
3 31.19 28.53 26.81 26.39 27.73 29.26 30.50 34.57 35.51 36.22 40.21 41.62 47.42 47.69 43.67 104.86 59.44 45.74 42.10 41.54 48.01 43.81 32.44 32.21
4 31.52 28.77 27.03 26.61 27.96 29.47 30.62 34.48 34.98 34.06 39.87 36.27 42.39 37.77 37.31 46.94 42.61 271.33 38.61 38.57 42.11 42.03 32.46 32.24
5 30.97 28.35 26.69 26.33 27.64 29.03 30.18 34.03 34.40 33.65 39.11 35.91 41.39 37.81 36.48 47.78 41.08 37.77 37.61 37.87 41.63 40.99 31.78 31.69
6 30.26 27.71 26.09 25.72 27.01 28.42 29.57 33.44 33.98 33.28 38.57 35.56 40.99 37.43 35.68 46.85 39.83 38.59 36.73 37.16 40.87 39.92 31.14 31.01
7 30.67 28.07 26.42 26.02 27.30 28.71 29.86 33.72 34.18 33.28 38.77 35.32 41.02 36.88 35.92 44.67 40.18 41.09 37.08 37.27 40.84 40.71 31.60 31.43
8 31.01 28.38 26.67 26.26 27.60 29.14 30.35 34.38 35.41 36.75 39.97 43.51 49.21 51.43 46.08 129.88 66.77 46.94 43.35 42.59 50.28 43.68 32.27 32.03
9 30.06 27.57 25.95 25.58 26.84 28.23 29.35 33.12 33.63 32.96 38.17 35.31 40.45 37.22 35.12 46.26 39.02 37.68 36.33 36.88 40.63 39.61 30.88 30.83
10 30.24 27.74 26.13 25.79 27.06 28.33 29.63 33.57 34.21 34.17 38.83 37.92 39.01 41.69 38.18 71.84 47.25 40.06 38.48 38.54 43.64 41.24 31.53 31.42
11 31.00 28.39 26.75 26.37 27.68 29.08 30.26 34.09 34.44 33.69 39.11 35.98 41.39 37.88 36.54 48.37 41.21 37.94 37.57 37.83 41.66 41.07 31.87 31.78
12 31.00 28.40 26.76 26.38 27.70 29.09 30.33 34.22 34.64 34.05 39.35 36.61 41.75 39.01 37.50 54.56 43.35 39.15 38.13 38.21 42.36 41.37 31.98 31.87
Table C.6: Maximum LPE market bid/offer electricity volume of retailers in case 1 (MWh).
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 250 240 231 232 232 234 240 250 266 282 296 310 322 329 331 333 331 324 315 306 299 286 267 251
2 252 243 236 236 237 240 246 262 284 304 318 329 342 353 361 367 369 362 353 337 327 306 282 264
3 253 243 238 238 243 242 249 263 284 305 323 341 350 357 362 368 370 366 354 339 327 307 282 266
4 253 246 241 241 244 253 265 277 290 306 323 342 357 368 372 377 376 367 354 339 330 308 285 266
5 261 250 245 244 248 260 271 284 294 313 325 344 360 368 375 382 379 369 354 339 331 311 286 266
6 269 257 249 245 253 262 272 285 302 319 334 349 360 374 381 383 382 376 362 349 340 321 301 282
7 269 258 251 251 260 270 283 294 305 322 342 361 373 377 383 391 395 391 379 363 355 329 305 284
8 271 259 254 254 260 272 284 300 319 335 349 362 376 385 391 398 397 392 381 368 357 335 310 290
9 277 266 260 260 264 273 287 304 321 343 360 380 399 408 404 406 402 403 391 378 362 338 311 290
10 277 269 261 261 268 280 291 306 325 344 365 385 400 409 411 408 407 405 396 378 366 341 320 294
11 283 272 266 265 274 281 293 313 327 347 371 391 405 412 413 409 409 411 403 385 375 348 320 298
12 288 277 269 268 277 289 301 313 335 361 381 400 407 413 418 422 419 418 407 391 380 356 329 306
Table C.7: Maximum DAW market bid load of retailers in case 1 (MWh).
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 5184 4928 4739 4646 4645 4678 4866 5111 5338 5521 5667 5845 6056 6299 6602 6877 7039 6995 6744 6559 6352 5994 5607 5236
2 5191 4983 4827 4758 4757 4805 5082 5504 5956 6370 6760 7145 7440 7653 7822 7907 7923 7743 7482 7288 7070 6542 5982 5529
3 5220 4985 4871 4871 5037 5186 5371 5703 6111 6550 6882 7177 7485 7743 7912 8056 8099 7957 7639 7348 7076 6558 6029 5556
4 5443 5200 5047 5043 5161 5294 5651 5989 6267 6573 7007 7292 7542 7760 7996 8195 8290 8123 7767 7385 7107 6661 6090 5572
5 5611 5397 5239 5149 5187 5466 5852 6175 6441 6740 7062 7512 7936 8270 8422 8423 8333 8161 7968 7591 7203 6751 6327 5813
6 5871 5653 5517 5524 5692 5994 6452 6853 7180 7504 7837 8108 8320 8414 8563 8661 8646 8498 8048 7802 7500 6927 6334 5921
7 5897 5657 5525 5554 5753 6035 6473 6907 7326 7660 7958 8187 8409 8545 8656 8750 8745 8518 8122 7815 7576 7126 6630 6194
8 5927 5685 5537 5569 5765 6077 6492 6934 7339 7834 8204 8536 8846 8967 8810 8851 8999 8983 8772 8419 8048 7419 6796 6260
9 6109 5826 5652 5623 5785 6086 6569 7002 7434 7863 8378 8739 8889 9043 9210 9327 9362 9168 8776 8539 8270 7759 7096 6517
10 6178 5895 5709 5683 5848 6112 6570 7120 7569 8010 8423 8847 9225 9510 9700 9855 9889 9681 9320 8927 8476 7780 7184 6603
11 6310 6037 5862 5838 5983 6287 6710 7154 7659 8263 8799 9203 9521 9710 9863 10036 10077 9882 9398 8986 8572 7871 7203 6700
12 6335 6072 5903 5901 6086 6385 6865 7358 7854 8357 8835 9290 9654 9879 10087 10230 10294 10111 9611 9104 8602 7920 7264 6708
26
Table C.8: Alpha values of retailers in case 1.
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 144 133 124 119 115 112 117 123 132 143 153 163 171 178 183 187 189 186 183 177 168 163 153 142
2 148 134 126 119 117 117 120 127 134 144 155 165 175 182 186 189 189 188 184 183 173 165 154 146
3 147 138 128 123 118 118 122 130 138 149 158 168 176 183 188 191 194 192 187 184 176 168 159 149
4 153 140 133 125 121 121 125 132 140 152 162 169 181 189 193 195 195 196 192 188 180 172 162 152
5 175 167 156 150 147 144 148 157 164 177 185 194 205 212 218 220 219 221 215 211 204 194 186 176
6 179 167 160 154 148 149 152 161 169 179 188 199 206 214 218 223 224 222 219 213 205 198 189 179
7 184 173 162 157 151 152 156 164 171 181 192 201 211 216 221 225 226 228 223 217 209 202 191 180
8 186 175 167 160 157 155 158 167 174 185 195 205 213 219 225 227 231 228 225 221 213 206 194 186
9 211 199 192 185 182 182 187 193 200 211 222 231 240 244 251 254 256 255 252 246 240 230 220 210
10 215 205 198 189 184 185 188 196 203 215 225 234 242 250 254 258 259 259 254 251 244 232 225 215
11 218 208 199 192 189 187 194 200 207 217 229 237 247 253 260 263 263 261 257 254 245 238 228 218
12 220 212 204 196 192 193 197 200 214 223 232 241 251 257 261 268 267 267 262 259 251 242 231 224
Table C.9: Self-elasticity values of retailers in case 1.
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 130 122 117 113 113 110 116 117 123 128 133 133 138 140 144 143 144 143 143 141 137 136 133 129
2 127 120 116 111 110 111 111 115 120 125 129 133 135 139 142 142 142 141 140 140 136 132 130 128
3 124 119 114 110 107 108 110 114 119 125 128 132 133 138 139 139 140 139 139 138 136 129 126 123
4 122 117 112 106 106 107 108 112 116 122 126 129 132 134 136 138 138 139 137 134 131 128 126 122
5 114 109 103 101 98 99 101 104 107 114 118 120 124 127 127 130 131 129 129 125 124 119 119 114
6 112 106 103 100 94 97 100 102 105 113 115 118 121 125 127 127 129 126 127 125 122 119 116 113
7 111 105 100 97 94 95 96 97 106 112 114 117 120 122 124 124 125 125 125 122 120 117 115 110
8 110 101 97 95 95 92 94 97 104 108 112 115 117 121 123 125 124 123 123 121 118 115 113 108
9 103 96 91 85 86 85 88 91 95 102 104 110 109 111 113 115 116 118 115 112 112 107 103 102
10 99 93 90 85 85 84 87 89 92 101 102 106 108 111 113 115 115 116 113 110 108 105 101 100
11 96 91 87 82 80 82 84 86 93 96 100 104 104 111 112 111 115 113 112 109 107 102 101 97
12 95 91 85 81 80 79 82 86 90 96 98 101 104 109 109 112 110 110 110 108 105 102 99 95
Appendix C.2. Information of generators in DAW market
Table C.10: Information of generators in DAW market.
Information
Generator
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Cost ($/MWh) 10 12 15 17 20 23 25 27 30 34 36 38 40 45 46
Maximum supply (MWh) 5000 4350 3940 3460 5070 2810 5300 4250 4650 3910 3250 3500 4750 3000 5750
Information
Generator
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Cost ($/MWh) 48 51 53 56 60 65 68 70 74 76 78 80 84 88 90
Maximum supply (MWh) 2250 3460 3940 2290 1990 2600 3800 3000 2500 2000 1050 3860 4800 3900 3000
27
Appendix C.3. Data in case 2
Table C.11: Initial retail prices of retailers in case 2 ($/MWh).
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 38.77 38.38 33.33 34.95 33.91 32.60 32.75 32.67 33.96 33.96 36.60 37.59 37.46 40.12 42.48 43.63 48.24 49.18 54.56 47.28 58.28 44.30 69.34 51.51
2 39.41 39.63 35.56 36.69 33.49 34.37 32.71 32.48 34.73 34.12 38.24 36.95 40.15 38.62 45.22 42.06 50.61 49.65 48.19 53.38 46.85 62.54 50.74 71.40
3 38.85 36.90 35.52 34.66 33.40 33.48 34.03 32.24 37.17 34.47 36.86 35.44 39.12 37.87 44.47 41.68 50.57 47.26 56.06 46.03 63.88 51.07 74.42 56.89
4 38.30 38.42 36.01 37.20 32.46 33.77 32.14 31.68 36.83 35.75 35.34 36.09 39.91 36.92 44.00 43.16 50.29 47.24 50.59 50.23 46.39 52.83 50.22 60.54
5 37.61 38.15 36.12 33.35 33.75 31.75 31.93 33.33 34.28 34.04 36.42 37.57 40.17 38.72 43.92 43.18 51.82 48.80 48.58 48.71 46.98 47.73 50.75 49.04
6 36.79 36.68 35.91 37.70 34.44 33.94 33.34 32.98 35.53 35.08 35.92 35.53 38.09 39.87 41.53 43.46 49.69 49.75 51.31 51.13 58.76 48.01 67.64 52.61
Table C.12: Initial DAW market bid prices of retailers in case 2 ($/MWh).
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 28.80 28.97 26.83 26.89 25.61 25.63 24.99 25.05 26.12 26.26 27.82 27.99 29.56 30.07 32.50 33.22 37.98 37.73 40.83 37.20 46.35 35.42 53.13 37.79
2 29.45 29.20 27.28 27.15 25.97 25.97 25.37 25.41 26.61 26.61 28.41 28.27 30.58 29.97 33.82 32.92 38.42 38.46 37.79 41.44 36.66 48.27 39.29 54.68
3 29.45 28.50 27.34 26.44 26.11 25.21 25.53 24.64 26.75 25.81 28.44 27.54 30.27 29.52 33.34 32.60 38.96 36.76 42.32 36.50 49.42 39.38 57.00 43.41
4 29.63 28.79 27.46 26.72 26.13 25.50 25.53 24.98 26.80 26.16 28.61 27.93 30.75 30.04 33.88 32.90 38.52 37.87 38.06 38.45 35.73 41.22 38.08 45.86
5 29.30 29.28 27.15 27.16 25.87 25.89 25.27 25.28 26.48 26.49 28.24 28.25 30.36 30.39 33.56 33.58 38.01 38.11 37.61 37.74 36.48 36.61 38.83 38.97
6 28.74 29.23 26.67 27.10 25.43 25.84 24.86 25.24 26.06 26.47 27.81 28.20 29.79 30.43 32.89 33.60 37.19 38.25 37.94 38.22 45.42 37.98 52.07 40.84
Table C.13: Initial LPE market bid/offer prices of retailers in case 2 ($/MWh).
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 30.63 30.67 28.06 28.07 26.38 26.42 25.95 26.02 27.23 27.30 28.75 28.71 29.94 29.86 33.99 33.72 34.99 34.18 35.55 33.28 39.75 38.77 40.37 35.32
2 31.24 31.01 28.55 28.38 26.84 26.67 26.43 26.26 27.74 27.60 29.17 29.14 30.36 30.35 34.31 34.38 34.80 35.41 33.85 36.75 39.53 39.97 36.07 43.51
3 31.19 30.06 28.53 27.57 26.81 25.95 26.39 25.58 27.73 26.84 29.26 28.23 30.50 29.35 34.57 33.12 35.51 33.63 36.22 32.96 40.21 38.17 41.62 35.31
4 31.52 30.24 28.77 27.74 27.03 26.13 26.61 25.79 27.96 27.06 29.47 28.33 30.62 29.63 34.48 33.57 34.98 34.21 34.06 34.17 39.87 38.83 36.27 37.92
5 30.97 31.00 28.35 28.39 26.69 26.75 26.33 26.37 27.64 27.68 29.03 29.08 30.18 30.26 34.03 34.09 34.40 34.44 33.65 33.69 39.11 39.11 35.91 35.98
6 30.26 31.00 27.71 28.40 26.09 26.76 25.72 26.38 27.01 27.70 28.42 29.09 29.57 30.33 33.44 34.22 33.98 34.64 33.28 34.05 38.57 39.35 35.56 36.61
Table C.14: Maximum LPE market bid/offer electricity volume of retailers in case 2 (MWh).
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 1002 962 924 929 929 937 959 1000 1062 1128 1183 1241 1286 1316 1325 1332 1325 1296 1260 1226 1196 1145 1068 1005
2 1007 970 945 944 949 959 982 1049 1136 1214 1274 1314 1368 1412 1443 1468 1476 1450 1414 1347 1307 1226 1128 1057
3 1010 973 951 953 971 967 994 1051 1136 1218 1291 1365 1399 1430 1449 1471 1478 1463 1416 1357 1308 1227 1128 1063
4 1013 983 962 963 975 1012 1059 1109 1161 1222 1292 1367 1428 1470 1488 1507 1504 1467 1416 1357 1319 1232 1139 1064
5 1046 1001 978 978 992 1040 1084 1138 1178 1252 1300 1376 1439 1472 1502 1529 1517 1474 1418 1358 1322 1246 1145 1065
6 1074 1028 996 981 1012 1049 1087 1138 1207 1277 1336 1396 1441 1496 1523 1530 1529 1506 1449 1395 1361 1284 1204 1129
Table C.15: Maximum DAW market bid load of retailers in case 2 (MWh).
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 20735 19713 18956 18586 18582 18714 19464 20445 21350 22084 22668 23378 24225 25197 26409 27508 28154 27978 26977 26237 25408 23975 22428 20945
2 20764 19930 19306 19030 19027 19221 20328 22015 23826 25478 27038 28578 29760 30612 31288 31627 31690 30970 29928 29154 28280 26167 23929 22117
3 20881 19941 19483 19485 20149 20744 21486 22812 24443 26201 27528 28706 29939 30972 31646 32224 32397 31826 30555 29390 28304 26232 24117 22224
4 21773 20799 20186 20174 20645 21178 22604 23957 25069 26294 28027 29167 30166 31041 31986 32779 33161 32492 31067 29541 28428 26642 24360 22290
5 22442 21586 20957 20597 20750 21864 23409 24698 25765 26959 28247 30049 31744 33082 33690 33693 33333 32643 31872 30364 28811 27002 25310 23251
6 23484 22613 22068 22098 22769 23978 25807 27412 28722 30014 31350 32432 33278 33655 34252 34642 34585 33991 32191 31208 30000 27710 25336 23684
28
Table C.16: Alpha values of retailers in case 2.
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 242 230 222 216 212 210 215 221 230 240 250 261 269 275 281 284 287 284 281 274 266 260 251 239
2 252 238 231 224 222 222 224 231 239 248 260 270 279 286 290 294 294 292 289 287 278 269 259 251
3 293 284 274 269 264 264 269 276 284 295 304 314 322 329 334 337 340 338 333 330 322 314 305 295
4 307 294 287 279 275 276 279 287 294 307 316 324 335 343 347 349 350 351 346 342 334 326 316 306
5 354 346 335 329 327 323 327 336 343 356 364 373 384 391 397 399 398 400 395 390 383 374 365 355
6 367 355 348 342 337 337 340 349 357 367 376 387 394 402 407 411 412 410 408 401 393 387 377 367
Table C.17: Self-elasticity values of retailers in case 2.
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 128 120 115 111 111 108 114 115 121 126 131 131 136 138 142 140 142 141 140 139 135 134 130 127
2 123 116 111 107 106 107 107 111 116 121 125 129 131 135 138 138 138 137 135 136 132 128 126 124
3 112 107 102 98 95 96 98 102 107 113 116 120 121 126 127 127 128 128 128 126 124 118 114 111
4 108 103 98 92 92 93 94 98 102 108 112 115 118 121 122 124 124 125 124 121 118 114 112 108
5 98 93 88 86 83 84 85 89 92 99 103 105 109 111 112 115 116 114 114 110 109 104 103 99
6 95 89 86 83 77 80 83 85 88 96 98 101 104 108 110 110 112 109 110 108 105 102 99 96
Appendix C.4. Data in case 3
Table C.18: Initial retail prices of retailers in case 3 ($/MWh).
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 37.88 38.19 40.09 37.17 34.99 37.28 34.05 34.14 33.48 33.64 32.13 32.95 32.13 33.79 30.90 31.77 35.11 36.70 32.63 34.65 35.19 36.34 35.52 35.89
2 39.28 38.80 38.02 37.70 35.47 35.19 36.09 34.68 33.75 33.53 33.71 33.91 33.18 34.19 32.95 34.43 34.29 33.95 34.69 34.79 37.20 38.00 37.89 37.22
3 37.95 37.26 38.46 38.22 34.23 33.98 33.79 36.38 33.84 35.32 33.43 33.72 32.75 33.29 31.79 31.99 35.34 33.89 34.74 34.34 34.11 36.94 33.92 38.20
Table C.19: Initial DAW market bid prices of retailers in case 3 ($/MWh).
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 28.80 29.63 28.97 28.79 26.83 27.46 26.89 26.72 25.61 26.13 25.63 25.50 24.99 25.53 25.05 24.98 26.12 26.80 26.26 26.16 27.82 28.61 27.99 27.93
2 29.45 29.30 29.20 29.28 27.28 27.15 27.15 27.16 25.97 25.87 25.97 25.89 25.37 25.27 25.41 25.28 26.61 26.48 26.61 26.49 28.41 28.24 28.27 28.25
3 29.45 28.74 28.50 29.23 27.34 26.67 26.44 27.10 26.11 25.43 25.21 25.84 25.53 24.86 24.64 25.24 26.75 26.06 25.81 26.47 28.44 27.81 27.54 28.20
Table C.20: Initial LPE market bid/offer prices of retailers in case 3 ($/MWh).
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 30.63 31.52 30.67 30.24 28.06 28.77 28.07 27.74 26.38 27.03 26.42 26.13 25.95 26.61 26.02 25.79 27.23 27.96 27.30 27.06 28.75 29.47 28.71 28.33
2 31.24 30.97 31.01 31.00 28.55 28.35 28.38 28.39 26.84 26.69 26.67 26.75 26.43 26.33 26.26 26.37 27.74 27.64 27.60 27.68 29.17 29.03 29.14 29.08
3 31.19 30.26 30.06 31.00 28.53 27.71 27.57 28.40 26.81 26.09 25.95 26.76 26.39 25.72 25.58 26.38 27.73 27.01 26.84 27.70 29.26 28.42 28.23 29.09
Table C.21: Maximum LPE market bid/offer electricity volume of retailers in case 3 (MWh).
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 6260 6012 5777 5806 5806 5855 5995 6248 6639 7052 7392 7758 8038 8222 8282 8324 8282 8097 7876 7661 7472 7156 6674 6284
2 6296 6064 5905 5902 5928 5994 6140 6558 7098 7590 7960 8214 8547 8827 9018 9173 9228 9060 8837 8418 8170 7662 7050 6606
3 6314 6080 5946 5959 6072 6042 6215 6568 7102 7614 8070 8534 8744 8937 9056 9193 9239 9141 8848 8480 8174 7668 7052 6642
Table C.22: Maximum DAW market bid load of retailers in case 3 (MWh).
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 129595 123205 118475 116160 116135 116960 121650 127780 133440 138025 141675 146115 151405 157480 165055 171925 175965 174865 168605 163980 158800 149845 140175 130905
2 129775 124565 120665 118940 118920 120130 127050 137595 148910 159240 168990 178615 186000 191325 195550 197670 198065 193565 187050 182210 176750 163545 149555 138230
3 130505 124630 121770 121780 125930 129650 134285 142575 152770 163755 172050 179415 187120 193575 197790 201400 202480 198915 190970 183690 176900 163950 150730 138900
29
Table C.23: Alpha values of retailers in case 3.
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 411 399 391 385 381 379 384 390 399 409 419 430 438 444 450 453 456 453 450 443 435 429 420 408
2 503 489 481 474 472 472 474 481 489 498 510 520 530 537 540 544 544 543 539 537 528 519 509 501
3 599 590 580 575 570 569 574 581 590 601 610 619 628 635 640 642 645 644 639 636 627 620 611 600
Table C.24: Self-elasticity values of retailers in case 3.
Retailer
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
1 123 116 111 107 107 104 110 111 117 122 127 127 131 134 137 136 138 137 136 135 131 130 126 122
2 109 102 98 93 92 93 93 97 102 107 111 115 117 121 124 124 124 123 122 122 118 114 112 110
3 95 90 85 81 78 79 81 85 90 96 99 103 104 109 110 110 111 111 111 109 107 101 97 94
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