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Design of Pareto optimal CO2 cap-and-trade policies for deregulated electricity networks
h i g h l i g h t s
A mathematical–statistical model for designing Pareto optimal CO2 cap-and-trade policies.
The model fills a gap in the current literature that primarily supports cap-and-trade policy evaluation but not policy design.
Pareto optimal policies accommodate conflicting goals of the market constituents.
Electricity demand-price sensitivity and social cost of carbon have significant influence on the cap-and-trade policies.
Higher demand-price sensitivity increases the influence of penalty and social cost of carbon on reducing carbon emissions.
a r t i c l e i n f o a b s t r a c t
Article history: Received 15 August 2013 Received in revised form 2 January 2014 Accepted 4 January 2014
Keywords: Electricity networks Cap-and-trade Game theory MPEC/EPEC
Among the CO2 emission reduction programs, cap-and-trade (C&T) is one of the most used policies. Economic studies have
shown that C&T policies for electricity networks, while reducing emissions, will likely increase price and decrease consumption
of electricity. This paper presents a two layer mathematical– statistical model to develop Pareto optimal designs for CO2 cap-
and-trade policies. The bottom layer finds, for a given C&T policy, equilibrium bidding strategies of the competing generators
while maximizing social welfare via a DC optimal power flow (DC-OPF) model. We refer to this layer as policy evaluation.
The top layer (called policy optimization) involves design of Pareto optimal C&T policies over a planning horizon. The
performance measures that are considered for the purpose of design are social welfare and the corresponding system marginal
price (MP), CO2 emissions, and electricity consumption level.
2014 Elsevier Ltd. All rights reserved.
1. Introduction
A major part of the total CO2 emissions come from the electricity
production sector, e.g., 40% in the U.S. ([1]). In 2009, 70% of the electricity
was produced from fossil fuel such as gas, coal, and petroleum ([2]). In 2005,
the European Union Emissions Trading System (EU ETS) launched a cap-
and-trade system that seeks to reduce the greenhouse gas (GHG) emissions
by 21% by 2020 from the 2005 level. Currently, the EU ETS is the largest
emission market in the world [3], and according to the European Commission
[4], at least 20% of its budget for 2014–2020 will be spent on climate-related
projects and policies. In the United States, as well as in the EU, different
regulations have been discussed to cut CO2 emissions such as carbon tax,
renewable portfolio standards (RPS), and capand-trade programs (C&T). In
the northeastern U.S., the Regional Greenhouse Gas Initiative (RGGI) has
already implemented a C&T program through a nine state collaborative effort,
which seeks to cut the CO2 emissions by 10% by 2018. Recently the
California Air Resources Board adopted a C&T program, held its first auction
on November 2012, and started operation on January 2013. It is a part of
California’s historic climate change law (AB 32) that will reduce the carbon
pollution to 1990 levels by 2020. Other countries who have already
implemented C&T programs include New Zealand, Japan, The Netherlands,
and Australia.
C&T policies implemented in the past for greenhouse gases had resulted in
increase in cost for households, since the generators passed the emissions
reduction costs to the consumers [5]. Linn [6] studied the economic impact of
C&T policies for nitrogen oxides (NOx) on deregulated firms. The study showed
that the effect of reduction of NOx emissions may not increase electricity prices
to a level which could fully compensate firms for their compliance costs,
particularly for coal generators. This lead them to reduce their expected profit by
as much as $25 billions. Chen et al. [7] developed a mathematical model to
examine the ability of larger producers in an electricity market under NOx C&T
policy to manipulate both the electricity and emission allowances markets. With
regards to CO2 cap-and-trade policies, Ruth et al. [8] examined the economic and
energy impacts that the state of Maryland will have by joining the RGGI
initiative. They identified several issues that are important to the acceptance and
effectiveness of CO2 C&T programs, such as rules for allowances distribution
and subsidies for energy efficiency programs. Bird et al. [9] claimed that, while
renewable energy will likely benefit from carbon cap-andtrade programs, C&T
372
can also impact the ability of renewable energy generation to affect overall CO2
emissions levels. They summarized the key issues for markets that are emerging
under CO2 C&T policies and also the policy design options to allow renewable
energy generation to impact the emissions level. For further reading about
economic impacts of CO2 C&T, we refer the readers to Goettle and Fawcett [10]
and Parmesano and Kury [11]. Linares et al. [12] have studied the impact of CO2
C&T regulation and green certificates on power generation expansion model.
They have shown that when a C&T policy is considered, locational marginal
prices (LMPs) increase, emissions decrease, and installed green generation
capacity increases. Limpaitoon et al. [2] studied the impact of C&T regulation
and the interaction of demand elasticity, transmission network, market structure,
and strategic behavior for an oligopoly electricity market (in the state of
California). They concluded that GHG regulations will affect the system
operations and market outcomes by increasing electricity prices (for both
oligopoly and perfect competition scenarios), and reducing both GHG emissions
and energy consumption. Their results also suggest that the interaction between
CO2 C&T regulations, market structure, and congestion might lead to potential
abuses of market power and create more congestion, which will limit nuclear
access to the market. This generates higher demand for emissions permit, which
in turn increases permit prices. Fullerton and Metcalf [13] showed how certain
environmental policies reduce profit under monopoly, raise prices, and reduce
welfare profit. [14] presented a game-theoretic model for developing joint
bidding strategies in C&T allowances and electricity markets for competing
generators. Rocha et al. [15] examined the impact of C&T policies on generation
capacity investment.
Some of the current research concerning C&T policies is focused on the issue
of development and analysis of allowance allocation mechanisms [16–18], and
suggestions for policy effectiveness [8,9,19]. However, the open literature does
not offer a methodology for design of CO2 C&T policies that takes into account
a range of conflicting measures of performance that appeal to different market
stakeholders. For example, a consumer’s concern is price increase [6,8,13], a
generator’s concern is profit/revenue reduction [2,13], and a policy maker’s
concerns include adequate emissions reduction and sustaining electricity
consumption necessary to support economic growth [9,12,13]. There does not
appear to be a consensus in the literature about economic impact that can be
expected under C&T programs, neither is there an agreement about the choice of
the C&T parameters and their values [20]. This paper attempts to fill the above
gaps by presenting a 2-layer mathematical–statistical model to design Pareto
optimal CO2 C&T policies for deregulated electricity networks. This paper
presents an elaborate sensitivity analysis of selected policy parameters (initial
allowance cap, cap reduction rate, violation penalty) and network parameters
(congestion, social cost of carbon, and demand-price sensitivity of the
consumers).
In the bottom layer of the 2-layer model, the strategic bidding behavior of the
competing generators is formulated as a bi-level mathematical model. The upper
level model focuses on maximizing overall generator profit by bidding to the
independent system operator (ISO) in the allowance and electricity markets. The
lower level model focuses on social welfare maximization (or, social cost
minimization) while meeting the network and policy constraints via a DC-OPF
model. Each bi-level optimization problem is reformulated as an mathematical
problem with equilibrium constraints (MPEC). Equilibrium bidding strategies
among the competing generators are obtained by solving the set of MPECs as an
equilibrium problem with equilibrium constraints (EPEC). Thus, the bottom
layer of the 2-layer model essentially evaluates the impact of a given C&T policy
on a network by obtaining the performance measures including electricity price,
emissions level, and consumption level. The top layer model, using as input the
results of the bottom layer model, develops regression equations for different
network performance measures using the tools of analysis of variance (ANOVA).
The regression equations relate the performance measures to the parameters of
the C&T policy and the network. These equations are used in forming a multi-
objective optimization model, solution of which yields the Pareto optimal CO2
C&T policy designs.
The rest of the paper is organized as follow. In Section 2 we introduce the
elements of a C&T policy and discuss social cost of carbon. Section 3 presents
the complete 2-layer model-based methodology that obtains the Pareto
optimal C&T policies. Section 4 demonstrates the application of our
methodology on a sample network. Section 5 provides the concluding
remarks.
2. Cap-and-trade and the social cost of carbon
Cap-and-trade is a market based mechanism that can be used to regulate
the GHG emissions. The following are some of the primary features of a cap-
and-trade mechanism.
Point of regulation: Different approaches to regulate emissions in the
electricity markets have been proposed and implemented. They range
from regulating far upstream at the point of sale of fossil fuels to far
downstream at the point of purchase of manufactured products and energy
by ultimate consumers [21]. The upstream approach sets an emissions cap
on producers of raw material that contains GHG (e.g., coal, gas, or
petroleum), whereas the downstream approach regulates the direct
producers of GHG [22].
Allowance distribution: In an Emissions Trading System (ETS), one of the
major concerns is how to distribute the allowances, as both the initial as
well as the continuing distribution strategies have a significant influence
on the final market equilibrium. Under a downstream regulation, several
allowance allocation/ distribution mechanisms have been studied in recent
years. Most commonly discussed mechanisms in the literature are free
allocation and auction based allocation [18,17]. Free allocation of
allowances is often based either on historical emissions (known as
grandfathering) or on energy input/product output (known as
benchmarking) [23]. An allocation mechanism based on equal per capita
cumulative emissions was presented in [24]. RGGI has implemented an
auction based model with a combination of uniform and discriminatory
pricing strategies. In the EU ETS, during the first and second trading
periods, most of the allowances were given freely according to historical
emissions. In the third trading period, free allowances allocation is
scheduled to be progressively replaced by auctioning through 2020 [23].
Cap stringency: This is a common feature in all C&T policies and refers to
the rate of reduction of allowance cap. The cap reduction rate varies
among markets due to network (market) intrinsic characteristics, policy
decisions, external economic variables, and other C&T parameter values
[25]. For instance, EU ETS implemented a 1.74% linear cap reduction for
2012– 2020 and beyond ([26]), while RGGI implemented a fixed cap for
2009–2014 and a subsequent 2.5% reduction until 2018.
Banking: It allows generators to save allowances for future periods. Some
markets also allow secondary trading of excess allowances at the end of
each period. Banking rules could vary among different markets. EU ETS
permitted allowance banking during phase I of its operation, but did not
permit allowances to be carried over into phase II. From phase II and on,
unlimited banking and borrowing is allowed [25]. The RGGI allows
banking with no restrictions from current compliance periods into the
future.
Penalty: Fossil fuel generators must procure allowances commensurate with
their emissions level or pay a penalty for any shortfall. EU ETS introduced
a monetary penalty (40 euros during phase I and 100 euros for phase II),
whereas California’s policy require four additional allowances for each
shortfall. IETA (International Emissions Trading Association) advises
against using a non-compliance penalty by additional withdrawal of
allowances. Instead, IETA recommends adopting a more traditional fixed
monetary fine similar to those entered in the SO2 and NOx emissions
markets in the U.S. [27].
Other features of a cap-and-trade mechanism include initial cap size,
safety value, revenue recycling and cost containment mechanisms [28,29].
In this paper, we consider a downstream regulation for C&T. The
electricity generators are assume to acquire allowances by competing in an
allowance market that is settled via discriminatory auction. The generators
submit price-quantity bids that are arranged in a decreasing price order by the
auctioneer. The price paid by each generator selected to received allowances
is its own bid price [17]. As presented in our model in Section 3.2, the
independent system operator (ISO) obtains the allowance allocation together
with generation dispatch by incorporating the allowances auction model
within the OPF model.
2.1. Social cost of carbon
The fiscal impact of CO2 emissions on the environment and society is
often referred to as the social cost of carbon (SCC). The most common means
that are used to characterize SCC include marginal social cost of emissions
and shadow prices of policies (e.g., cap-and-trade). Mandell [32] presents
various definitions and estimates of SCC. The most common approach used
in the literature defines the marginal social cost of carbon as ‘‘the cost to the
society for each additional unit of carbon (in the form of CO2) into the
atmosphere.’’ However, as the carbon remains in the atmosphere, it is difficult
to estimate the cost of the carbon in the future. This motivates to redefine the
SCC as ‘‘the present value of the monetized damage caused by each period
of emitting one extra ton of CO2 today as compared to the baseline.’’ Another
challenge for assessing SCC is in estimating the discount rate of the future
social cost of CO2.
The estimation of the SCC will always suffer from uncertainty,
speculation, and lack of information [30]. This can be noticed, for example,
in (1) estimating future emissions of greenhouse gases, (2) monetizing the
effects of past and future emissions on the climate system, (3) assessing the
impact of changes in climate on the physical and biological environment, and
(4) translating environmental impacts into economic damages. Therefore, any
effort to quantify and monetize the harms associated with climate change is
bound to raise serious questions of science, economics, and ethics, and thus
should be viewed as provisional [30]. Avato et al. [31] consider the carbon
emissions as one of the main barriers to the development and deployment of
clean energy technologies. They argue that emissions (a negative externality)
is not valued and therefore not included into investment decisions by energy
providers. Hence, by considering the monetary effects of emissions, i.e., the
SCC, an increase in development and deployment of green technologies can
be achieved.
Mandell [32] summarized the result of 211 SCC estimates from 47 different
studies using an integrated assessment model (IAM). The SCC was estimated to
have a mean value of €19:70=tCO2, a median of €5:45=tCO2 and an estimated
range from €1:24=tCO2 to €451=tCO2. Using the shadow price approach, the
estimated value of SCC has a range of €32=tCO2 to €205=tCO2 among the
countries in the European Union. Hope [33] estimated SCC considering two
scenarios: low emissions case, and a business as usual (BAU) emissions scenario.
This study concluded that the median SCC for the BAU scenario is $100=tCO2
with a range of $10=tCO2—$270=tCO2, and a median of $50=tCO2 for the low
emission scenario with a range of $5=tCO2 $130=tCO2. The U.S. government,
through the interagency working group (IWG), calculated the cost imposed on
the global society by each additional ton of CO2. They included health impact,
economic dislocation, agricultural changes and other effects that climate change
can impose on humanity. They estimated the SCC to have a range from $5=tCO2
to $65=tCO2. The IWG suggests setting the SCC to $21=tCO2. The IWG also
propose to utilize a discount rate of 2.5– 5% to address future SCC [34].
3. A model for developing Pareto optimal C&T policies
In this section, we present the complete methodology for designing Pareto
optimal cap-and-trade policies for a fixed planning horizon. The methodology
comprises a 2-layer model and a detailed solution approach.
3.1. A 2-layer model
The mathematical–statistical model for obtaining Pareto optimal C&T policy
designs has two broad layers, which we call the top and the bottom layers.
The bottom layer (also referred to in this paper as the policy evaluation layer)
involves obtaining equilibrium bidding strategies of the competing generators as
a function of the C&T and network parameters, while maximizing social welfare
via a DC-OPF. The network performance measures (emissions, electricity price,
and consumption level) corresponding to the equilibrium bidding strategy, for a
given C&T policy, are fed to the top layer (policy optimization layer) that obtains
the Pareto optimal C&T policy designs. The top layer model first performs an
analysis of variance (ANOVA) using the C&T and network parameters as
factors, and the values of the network performance measures from the bottom
layer as the responses. The ANOVA results are used to formulate regression
equations for the network performance measures as functions of the C&T and
network parameters. The Pareto optimal policy designs are then obtained by
optimizing the regression equations via a multi-objective optimization model.
The 2-layer model is given as follows.
Top layerf Optimize fðc;r;p;b;p;lÞ
8>< Maximize giðai;xiÞ; 8i 2 I ð1Þ Bottom layer s:t: Maximize WðQ;HÞ
>: s:t: Network and policy constraints:
In the top layer, fðc; r; p; b;p; lÞ represents the objective function of the multi- objective optimization model, where c is the cap size representing the maximum
tons of CO2 allowed per year (tCO2=yr), r is the yearly cap reduction rate (% of
c) which goes into effect after a preset number of initial years, p is the penalty
for exceeding the emissions limit per allocated allowances ($=tCO2), b represents
the demand price sensitivity (slope of the demand curve), p is the social cost of
carbon ($=tCO2), and l denotes the vector of line capacities (MW h) in the
network (a determinant for network congestion). The bottom layer is formulated,
for each generator i 2 I ¼ f1; 2;...; ng, as a bi-level optimization model, where
giðai;xiÞ denotes the profit of generator i, and ai (intercept of supply function)
and xi (allowance price) are the generator bids in the electricity and allowances
markets, respectively. WðQ;HÞ denotes the social welfare objective function for
the DC optimal power flow problem, where Q ¼ ðq1;...; qi;...; qnÞ is the electricity
dispatch and H ¼ ðh1;...;hi;...;hnÞ is the allowances allocation. For each generator
i, the bi-level model is solved as a mathematical problem with equilibrium
constraints (MPEC). The equilibrium bidding strategies for all generators in both
electricity and allowances markets are obtained by solving an equilibrium
problem with equilibrium constraints (EPEC).
A schematic of the solution methodology for the 2-layer model is shown in
Fig. 1. The methodology begins by designing a factorial experiment with all C&T
and network related parameters as factors at two or more levels, and enumerating
all possible factor level combinations [box (1)]. Factor level combinations, one
at a time, while yet to be evaluated [box (2)], are forwarded as input to the bottom
layer [box (3)]. In the bottom layer, after the cap is adjusted, MPEC/EPEC are
solved to obtain equilibrium bidding strategies of the generators for each year of
the planning horizon T [boxes (6) and (7)]. The cap is considered to remain
constant for
the first d years [box (4)] and decreasing thereafter [box (5)]. For a given factor
level combination, once the equilibrium strategies are obtained for each year of
the planning horizon T, the performance measures are updated [box (8)]. Once
all years of the planning horizon are considered [box (9)], performance measures
for the complete horizon for the current factor level combination are sent to
ANOVA [box (10)] and the next combination is drawn for evaluation. When all
factor combinations are exhausted [box (2)], ANOVA is performed [box (10)].
Regression equations (one for each measure of performance) are developed using
ANOVA results and are sent as input for the multi-objective optimization [box
(11)], which yields the Pareto optimal C&T policies. In what follows, we present
the bi-level optimization model and discuss its solution using the MPEC
approach. Thereafter, we explain how EPEC approach is used in obtaining the
equilibrium bidding strategies for all generators in the electricity and allowance
markets.
3.2. A bi-level model for joint allowance and electricity settlement
We adopt the bi-level framework that is commonly used in modeling
generator bidding behavior in deregulated electricity markets [35–37]. We
374
extend the objective functions of both levels by incorporating emissions
penalty cost in the upper level (see (2)) and both allowances revenue and
social cost of carbon in the lower level (see (6)). We also expand the
constraints set in the lower level (DC-OPF) to accommodate a C&T policy,
which are explained later. The solution of the modified DC-OPF yields both
the electricity dispatch and the allowances allocation.
We consider that the generators and the consumers bid with their linear
supply and demand functions, respectively. ISO determines energy dispatch
and allowances allocation by maximizing social welfare while satisfying
network and C&T constraints. The flow in the network is controlled by the
Kirchhoff’s law represented by the power transfer distribution factors
(PTDFs). Capacity limit constraints on the lines are also considered. The
supply cost of the generators, indexed by i, are assumed to be quadratic
convex functions given by CiðqiÞ ¼ aiqi þ biq 2
i , where ai and bi represent the
intercept and the slope of the supply function, respectively. Consumers,
indexed by j, are considered to have negative benefit functions given by
DjðqjÞ ¼ djqj bjqj 2 ; qj 0 [38]. Generators are paid at their marginal cost,
dC @
i q
ðq i
iÞ ¼ ai þ 2biqi. Therefore, the profit of a generator i is given as ðai þ
2biqiÞqi ðAiqi þ Biqi 2 Þ, where Ai and Bi are the true cost parameters of
generator i estimated through a standard Brownian motion model as follows.
For any year t; Ati is obtained as A t i ¼ sAti1 þ rAti1h, where s represent a trend
parameter, r represent the standard deviation of the process, and h is a standard normal random variable. Our DC-OPF model is similar to that
presented by Hu and Ralph [38], which showed that the use of the above
functions ensure a unique solution. For some other variants of the OPF model,
readers are referred to Berry et al. [39], Borenstein et al. [40] and Limpaitoon
et al. [2].
Without loss of generality, we assume that the generators pass on the cost
of allowance to the consumers by modifying (shifting up) the intercept (ai) of
their supply functions as a^i ¼ ai þ cixi, where ci is the CO2 emissions factor
and xi is the allowance bid (cost) of generator i. Hence, the cost of allowance
(xi) does not explicitly appear in the generator’s (upper level) objective
function. Note that, shifting of the supply function results in a new equilibrium
point, where consumption (dispatch) is reduced and electricity price is
increased (see Fig. 2). Hence, xi implicitly impacts the generator’s profit
through quantity dispatch qi and excess emissions penalty paid by the
generators.
Social welfare is commonly considered in the literature to be total benefit
to the consumers DjðÞ minus total cost to the generators CiðÞ. Since
allowances cost is passed onto the consumers, we include the cost of
allowance as part of generators cost (i.e., use a^i as the intercept of the supply
cost function). We also add the allowance revenue to social welfare based on
the assumption that it is recycled back as credits to consumers (to mitigate
economic impacts of increased electricity prices) and subsidies to green
generators. Revenue recycling has been widely discussed and recommended
by many economists [e.g., 41–43]. We also subtract from the welfare function
the social cost of the emissions, which is obtained by adding for all generators
the product of the emissions quantities and the social cost of carbon.
Let I ¼ f1;...; ng be the set of generator, and J ¼ f1;...; mg be the set of
consumers on the network. Let U ¼ ð/1;...;/nÞ be the vector of generators’
electricity bids, where /i ¼ ðai;biÞ represents the bid of generator i, and X ¼
ðx1;...;xnÞ be the vector of allowance bids. Let Q ¼ ðq1;...; qNÞ, where N ¼ jðI
[ JÞj, be the electricity dispatch response vector obtained from the DC-OPF.
Let C ¼ ðc1;...;cnÞ be the vector of CO2 emissions factor ðtCO2=MW hÞ of
the generators. Then the yearly emissions of generator i is given by ci qi. Let
H ¼ ðh1;...;hnÞ be the vector of yearly allowance allocation to the generators.
Therefore, given the supply function bids and the allowance bids of all other
generators, Ui and Xi, respectively, the bi-level optimization model for
generator i is given as follows.
Maxai;xi ðai þ 2biqiÞqi ðAiqi þ Biq2i Þ pðciqi hiÞ ð2Þ
s:t: ai 2 ½A;A; ð3Þ
xi 2 ½W;W; ð4Þ
ai þ cixi 6 dj; 8j; j ¼ 1;...;J; " # X X X X Q;H ¼ Max DjðqÞ CiðqÞ þ xihi qicip
q;h j i i i
ð5Þ
ð6Þ
X
s:t Cl 6 qiukl 6 Cl; l ¼ 1;...;L; ðl ;þl Þ k2ði;jÞ
ð7Þ
X X qi þ qj ¼ 0; ðlÞ i j
ð8Þ
X
c hi P 0; ðkÞ i
ð9Þ
ciqi hi P 0; 8i 2 I; ðqiÞ ð10Þ
qi Rlo P 0; 8i 2 I; ðsiÞ ð11Þ
Rup qi P 0; 8i 2 I; ðtiÞ ð12Þ
qi P 0; 8i 2 I; ðpiÞ ð13Þ
qj P 0; 8j 2 J; ðjiÞ ð14Þ
hi P 0; 8i 2 I: ðfiÞ ð15Þ
Fig. 1. A schematic of the solution methodology for the two-layer model.
In the formulation, the elements within parentheses in constraints (7)–(15)
represent the corresponding dual variables or shadow prices. Our attention is
focused primarily on l, which represents the system marginal energy price (cost) of the network [2]. Constraints (3) and (4) incorporate the bounds for
electricity and allowance bids, respectively. Constraint (5) ensures that the
supply function intersects with the demand curve. Objective function (6)
represent the social welfare where CiðqÞ is the cost to the generators obtained
using the shifted supply function (a^i;bi), DjðqÞ is the benefit to the
consumers, and p denotes to the SCC. Flow con-
)
Fig. 2. Electricity market equilibrium with supply function shifted by allowance cost.
straints are given in (7), where Cl is the line capacity and ukl is the PTDF for node
k and line l. Energy balance is maintained by constraint (8). Constraint (9)
ensures that the allocation of allowances does not violate the cap. Since we do
not consider banking of allowances, constraint Eq. (10) ensures that a generator
is not allocated with more allowances than the emissions emitted. Maximum and
minimum production level for each generator are controlled by constraints (11)
and (12) respectively. Finally, (13)– (15) are non-negativity constraints for
electricity dispatch and allowance allocation.
3.3. Model solution for equilibrium bidding strategies: A MPEC/EPEC
approach
Bi-level optimization models have been widely studied in the literature
[44,45]. Bi-level models include two mathematical programs, where one serves
as a constraint for the other. For the lower level problem, with a convex objective
function and non-empty feasible set, the first order necessary conditions for a
solution to be optimal are given (under some regularity conditions) by the Karush
Kuhn Tucker (KKT) equations. Hence, replacing the lower level problem in
Section 3.2 by the set of optimality conditions, yields what is known as a
mathematical program with equilibrium constraints (MPEC). Further details on
the MPEC models can be found in [46–48].
In the other hand, an equilibrium program with equilibrium constraint
(EPEC) is defined as a game, EPEC ¼ ðf MPECÞgn1, among competing
generators. Typically, MPECs have non-convex feasible sets (due to the
complementarity constraints), therefore the resulting games are likely to have
non-convex feasible strategy sets. Since a global equilibrium for this type of
games is difficult to identify, a local Nash equilibrium concept is presented in
[38]. The local Nash equilibrium is comprised of stationary points of the MPEC
problem for each player i in the game EPEC ¼ ðf MPECÞgn1. Lack of knowledge
of other players’ exact strategies, which might also be changing with time, along
with other uncertainties may minimize the value of the effort required to seek
global optimal strategies.
Literature presents different strategies to solve EPECs, of which linear and
non-linear complementarity (LCP/NCP) formulation [49] and diagonalization
methods [50,51] are the most discussed. In this paper we consider a
diagonalization method algorithm as presented in [52]. In the diagonalization
method, MPECs for all generators are solved, for which we use a regularization
method in conjunction with NLP solvers [53,54]. For other techniques to solve
MPECs, readers are refers to [55,47].
3.4. Multi-objective optimization for Pareto optimal designs
In a complex system with multiple performance measures and the set of
significant design factors that affect them, the nature of the relationship could
vary widely among the performance measures. That is, a set of factor values
(levels) that are optimal for a particular performance measure may not be optimal
for other measures. For example, the C&T and network parameters that
minimizes CO2 emissions may not yield lowest electricity prices, when both
reduced emissions and lower prices may be among the priorities. Therefore, a
design approach that can balances among multiple priorities must be considered.
Multi-objective optimization models, that yield Pareto fronts, provide such an
approach.
A Pareto front is a set of points representing factor level combinations where
all points are Pareto efficient. A Pareto efficient point indicates that no measure
of performance can be further improved without worsening one or more of the
other performance measures. Similarly, for any point outside of the Pareto front,
by moving the point onto the front, one or more measures can be improved
without worsening the others. In order to obtain these points, we formulate the
multi-objective optimization problem (16). We used the NSGA-II genetic
algorithm [57] to solve the model and obtain the Pareto envelope. Equations CL
(consumption level), TE (total emissions), and AMP (average marginal price) are
explained in detail in Section 4.3.1.
Min TEðc;r;p;b;p;lÞ;
Max CLðc;r;p;b;p;lÞ; ð16Þ Min AMPðc;r;p;b;p;lÞ; s:t: bounds on c;r;p;b;p;l:
4. C&T policy development for a sample network
In this section, we demonstrate, using a sample electricity network, how our
model can be used to develop Pareto optimal C&T policies. We conduct the
numerical study in two phases. We first implement the bottom layer of the model
for evaluating a number of different ad hoc C&T policy scenarios, as shown in
Table 1. In the ad hoc policies, penalty and the cap vary within their
corresponding ranges (per RGGI), and SCC varies according to the IWG
recommendations (see Section 2). The above variations coupled with different
levels of demand-price sensitivity (slope of the demand curve) are captured as
the three main scenarios SN1, SN2, and SN3 (see Table 1). Samples of results
obtained by evaluating these scenarios are presented to demonstrate the impact
of both C&T policy and network parameters. In phase 2, we implement the
complete two layer model. We first develop a factorial experiment and conduct
ANOVA. Response surfaces derived from ANOVA are used to obtain Pareto
optimal C&T policies via multi-objective optimization model.
Scenario designation Demand (b) (slope) Penalty (p) ð$=tCO2Þ SCC (p) ð$=tCO2Þ Cap (c) (tCO2)
SN1 0.025 10–50 0–21 80–120
SN2 0.05 10–50 0–21 80–120
SN3 0.075 10–50 0–21 80–120
376
Table 1 Ad hoc C&T
policies.
4.1. Sample electricity network
We consider a sample network with 4-nodes and 5-lines (see Fig. 3).
Similar sample networks were considered in numerical studies in [58–62].
The network operates under a CO2 C&T policy with three generation nodes
and one load node. The network has two fossil fuel (coal) generators
(GENCO1 and GENCO3) and one green generator (GENCO2). The green
generator does not participate in the allowance auction. Emissions factor for
both coal generators is assume to be one (i.e., 1 ton of CO2 emission per MW
h of electricity production). Allowances are distributed among competing
generators using a discriminatory (pay-as-bid) auction pricing strategy. The
cap is considered to decrease at a yearly rate of 2.5% starting the sixth year of
implementation (this is similar to the policy implemented by the RGGI). Line
capacities for C1 and C4 are set to 80 MW h, while the rest of the lines have
a capacity of 120 MW h. The model is implemented for a thirty year planning
horizon. Demand is considered to increase at a yearly rate of 1.1%, which is
implemented in our model by raising the intercept of the demand function.
For supply functions, we consider that the generators bid only on the
intercept parameter a. Intercept parameter Ai of the true cost function of the
generators is estimated using a Brownian motion model (BMM) as explained
in Section 3.2. The trend parameter (s) of the BMM model is set to 0.0059 for
green generator and 0.0662 for coal generators. The standard deviation (r) of the BMM model is set to 0.081 for green generator and 0.0714 for coal
generators. The above numerical values were obtained from the Electric
Power Annual Data 2009 ([63]). For year t ¼ 1; A1i is set to 2011 cost given
in the above data. For all generators, we set the value of Bi to 0.05.
4.2. Analysis of ad hoc C&T policies
We implemented the policy evaluation part of our model in GAMS using
MPEC and CONOPT3 solvers. For each scenario in Table 1, we obtained the
equilibrium bidding strategies for the complete planning horizon. For each
year of the planning horizon, we recorded performance measures such as
production levels of each generator (sum of which is the consumption level),
marginal electricity price, generator profits, emission levels, and market share
of the green generator. These yearly measures are used in calculating the
network performance measures for the complete horizon.
4.2.1. Impact of C&T and network parameters on performance
measures
In this section we examine the effect (sensitivity) of some of the policy
and network parameters on the performance measures. A more
comprehensive analysis of the impact of the C&T and network parameters on
the network performance is conducted in Section 4.3.1 using the analysis of
variance (ANOVA) technique.
Tables 2–4 summarize the performance measures for various ad hoc C&T
policies belonging to SN1, SN2, and SN3 with a cap of 80 tCO2. It can be
observed that a higher sensitivity in the price reduces production (or
consumption) and emission levels. Social cost of carbon further contributes to
this reduction. Along with the reduction of emissions, the percentage of
market share of green
Fig. 3. A 4-node sample network for numerical study.
sources is in general higher when consumers are more sensitive, and their
profits are slightly reduced. Electricity prices suffer a significant reduction of
almost $10=MW h on average.
Table 2 Performance of ad hoc C&T policies belonging to SN1 (demand slope 0.025).
4.2.1.1. Generators behavior. Fig. 4 shows all three generators’ yearly production
levels for the scenario SN1 under two different penalty levels ð$30=tCO2 and
$50=tCO2) and zero SCC. Recall that we considered the emissions factor c ¼ 1,
which means each MW h of coal based electricity production results in one unit
(ton) of CO2 emission. Hence, sum of the production levels of GENCO1 and
GENCO3 represents the total emission in the network. GENCO2 is the green
generator with zero emissions. The thicker solid lines in the figures represent the
total emissions cap for the network.
In the low penalty cost scenario (p ¼ $30=tCO2), as the demand grows over
the years and the cap reduces, the coal generators still find it profitable to increase
their share of production (above green generation) while paying more in
penalties. Whereas, in the high penalty cost scenario (p ¼ $50=tCO2), the coal
generators control their bids to significantly lower their production. In fact, till
year 22, the combined emissions from GENCO 1 and 3 remain below the cap.
The green generator maintains a high level of production.
10 30 50 10 30 50
Production (MW h) 8021 8023 7483 8026 7397 5893
Emissions (tCO2) 5633 5641 4038 5636 4018 2057
Market share (%) 32 36 45 38 55 57
AMP ($/MW h) 66.7 67.9 67.4 67.8 68.2 69.8
Profit ($) 129,436 128,290 165,259 128,470 167,445 203,593
Table 3 Performance of ad hoc C&T policies belonging to SN2 (demand slope 0.05).
SCC-0 SCC- 21
10 30 50 10 30 50
Production (MW h) 7742 7617 6155 7738 6305 5942
Emissions (tCO2) 5094 5060 2580 4917 2415 2016
Market share (%) 34 34 58 36 62 66
AMP ($/MW h) 55.6 57.01 60.98 56.39 60.04 62.15
Profit ($) 103,740 97,776 155,773 1,067,733 161,307 176,520
Table 4 Performance of ad hoc C&T policies belonging to SN3 (demand slope 0.075).
SCC-0 SCC- 21
10 30 50 10 30 50
Production (MW h) 7983 6754 5575 6672 5694 5641
Emissions (tCO2) 5772 3472 1905 3362 2052 1982 Market share (%) 28 49 66 50 64 65 AMP ($/MW h) 42.37 48.19 54.47 48.65 53.82 54.12 Profit ($) 64,193 103,425 142,079 107,056 139,902 141,322
0
40
80
120
160
22 1 28 25 19 16 13 10 7 4
Year
Production level (p=$30/tCO 2 , SCC=0)
GENCO1 GENCO2 GENCO3 cap
0
40
80
120
160
25 19 1 22 4 7 10 13 16 28
Year
Production level (p=$50/tCO 2 , SCC=0)
GENCO1 GENCO2 GENCO3 ca p
378
A higher cost of penalty causes the total electricity consumption over the horizon
to fall from 8256 MW h to 6213 MW h (a 23% reduction) and the total emissions
for the horizon to reduce from 5635 tCO2 to 2795 tCO2 (a 50% reduction). The
share of green generation in the two penalty levels are 32% and 49%,
respectively. The average marginal electricity price rises from $52:7= MW h to
$60:8= MW h (a 15.4% increase).
The penalty level choice varies for different markets. California’s C&T set a
penalty of 4 allowances per ton not covered by allowances, and the reserve
allowance price was set to $10.71 in 2013, which is set to increase to $11.34 for
2014 [64]. In the case of EU ETS, the penalty was set to 40 euros in the phase I,
and increased to 100 euros during phase II [25]. The effectiveness of penalty is
affected by allowance prices and quantity (cap), among others. For example, if
allowance prices drop, as was the case in phase I in the EU ETS, a low level of
penalty will unlikely generate emissions reduction since generators can either
afford to buy allowances, and/or pay a penalty for non-compliance.
4.2.1.2. Generation and emission levels. Fig. 5 shows the effect of SCC on total
coal-based generation (or, equivalently, total emissions) as well as total green
generation over the horizon for increasing values of penalty cost and demand-
price sensitivity. As expected, at higher values of penalty and demand-price
sensitivity, the coalbased generation decreases. At higher SCC, the reduction in
coalbased generation is sharper (plot b). Also at higher SCC and penalty costs,
the effect of increased demand-price sensitivity is reduced (plot b). As far as
green generation is concerned, it can be observed (plot a) that higher demand-
price sensitivity compounds the effect of penalty in increasing generation. This
effect is further pronounced when SCC is higher (plot b).
4.2.1.3. Electricity prices. Fig. 6 depicts the average marginal prices under the
same parameter combinations as in Fig. 5. It can be seen that higher penalty
produces higher price, which increases further with increased SCC, as
expected. We also note that, higher demand-price sensitivity significantly
lowers the price at all levels of penalty and SCC.
4.2.1.4. Generators market share. Fig. 7 shows the market share of green and
coal based energy, aggregated over the complete horizon. As expected, an
increase in penalty (from $10=CO2 to $50=tCO2) raises the share of green
energy from 32% to 45%. The increase in green energy is further noticeable
when the SCC is included. By considering SCC (Plot (b)) the share of green
based energy achieves a 57% versus a 43% of coal share for a penalty of
$50=tCO2. Note that adding a SCC produces the share of green based energy
to be higher than coal based energy share (which is not the case when SCC =
0). As mentioned in Section 2, SCC could accelerate the transition to green
technologies. Results presented in this section helps to quantify the assertion.
(a) Total production/emissions SCC=$0/tCO2 (b) Total production/emissions SCC=$21/tCO2
Fig. 5. Impact of penalty and demand-price sensitivity on CO2 emissions (for different values of SCC).
Fig. 6. Impact of penalty and demand-price sensitivity on average marginal price of electricity (for different values of SCC).
Fig. 7. Impact of penalty and SCC on market share of green generation. Fig. 8. Generator profits for different penalties and SCC.
4.2.1.5. Generators profit. Fig. 8(a) shows the cumulative generator profits
(coal generators are combined as one) for different penalty levels and zero
SCC. When higher penalty for non-compliance is considered, coal generators
see their profit to be reduced (approximately 43%). Coal based generators
reduce their electricity production in order to maintain emissions according
with the allowances allocated. It was also observed in Figs. 5 and 7 that the
increase of penalty reduces the emissions (and hence production) and coal
market share, resulting in lower profits for coal based generators. Green
generators observe an increase in profit as we consider higher penalties
(approximately 21%). It can be observed that including SCC (Plot (b)) further
Penalty ($/tCO 2 )
Coal-0.025
Coal-0.05
Coal-0.075
Green-0.025
Green-0.05
Green-0.075
Penalty ($/tCO 2 )
Coal-0.025
Coal-0.05
Coal-0.075
Green-0.025
Green-0.05
Green-0.075
increases the cumulative profit of the green generator. These results show that
C&T policies will likely have conflicting implications. On one hand, the
environment and society may benefit from a reduction in emissions and
increase in green power. On the other hand, coal-based generators do not have
incentives to participate in the market because of profit loses. This is one of
the major concerns, for example in the U.S., where fossil fuel energy sources
are still a major portion of the energy portfolio.
4.3. Development of Pareto optimal C&T policies
In this section, we first develop response surface equations for each
performance measure using a factorial design and ANOVA. We then
demonstrate how those equations can be used to obtain Pareto optimal C&T
designs.
4.3.1. Performance response surface generation
In order to obtain the response surfaces for the network performance
measures, we adopt a designed factorial experiment comprising six factors:
penalty, cap size, cap reduction rate, SCC, demand-price sensitivity, and line
capacities (congestion). We consider each factor at three levels resulting in a
36 factorial experiment. The numerical levels of the factors are given in Table
5. All 729 factor combinations were evaluated using the bottom layer model,
and the resulting performance measures were used to conduct ANOVA (with
type I error a ¼ 0:05) utilizing the R software (version 2.15.1). Results from
ANOVA conducted separately for each of the three performance measures
(total CO2 emissions (TE), total electricity consumption (CL), and average
marginal price
Table 5 Factor levels for 36 experiment.
1
0 Cap reduction (r) (%) 2.5 5 7.5 Initial cap size (c) 80 120 160 Penalty (p) 10 30 50 Demand-price sensitivity (b) 0.075 0.05 0.025 Congestion (l) 120 100 80
SCC ðpÞ 0 21 42
(AMP)) are presented below. For more detailed information about factorial
design of experiments and ANOVA, we refer the readers to [56].
4.3.1.1. Total CO2 Emissions Level: TE. When TE was used as the response
variable, all main factors and some two level interactions were found to be
statistically significant. Using the significant factors and interactions, a second
order regression model was developed. The model has a multiple R-squared
value of 0.839 and an adjusted R-squared value of 0.833.
TE ¼ 3696:63 1186:03p 989:66p þ 618:84b þ 683:56c
322:92l 316:37r þ 270:66p2 þ 353:8p2 253:34b2 þ 648:22pp þ
340:81cp þ 256:36pc 219:83bl 217:25pb þ 194:49pl þ 184:02lp
142:89rp 141:85pr 128:33bc 124:11bp þ 259:39p2p 312:67p2p2
þ 161:21pp2 150:1cp2 þ 234:56p2b2 þ 61:19cr:
Factor effect plots for the six significant main factors are presented in Fig. 9.
All the factors exhibit approximately linear behavior over the three levels,
indicating that their impact on TE are either higher the better or lower the better.
4.3.1.2. Total Consumption Level: CL. As in the case for TE, all six main factors
and some two level interactions were found to be statistically significant. A
second order regression model incorporating the significant factors and
interactions was developed. The model has a multiple R-squared value of 0.842
and an adjusted R-squared value of 0.835. Plots of the factor effects at various
levels are presented in Fig. 10. Non-linearity in the factor effects can be observed
to be a bit higher than in the case of TE, which can also be seen in the regression
equation that has more non-linear terms.
CL ¼ 7315:88 1237:44p þ 689:52b 758:85p þ 425:21c
263:04l 521:34p2 344:16b2 232:83r þ 86:63l2 þ 84:57c2 þ 67:39p2
þ 304:15pp 216:74bl þ 167:3pl 166:88bp þ 157:77pc þ 156:99bp
þ 146:67lp þ 128:88cp þ 128:26cr þ 292:63p2b2 91:97pr þ
112:51pp2 111:2cp2 60:14rp þ 144:98rc2 þ 55:24cl þ 95:2bl2 þ
92:89cl2 52:3bc þ 90:43pl2 þ 84:05rl2 þ 244:67p2p 89:2cp2:
4.3.1.3. Average Marginal System Price: AMP. For AMP as the response
variable, the only main factor that was not statistically significant was the cap
reduction rate. Some of the two level interactions were significant. The
regression model has a multiple R-squared value of 0.637 and an adjusted R-
squared value of
380
0.621. Lower R-squared value is, perhaps, indicative of the fact that the factor
effects have a higher order (>2) of non-linearity (see Fig. 11), which could not
be captured from a 3-level factorial experiment.
AMP ¼ 55:1717 þ 6:809p þ 2:9673p2 þ 1:6172p þ 0:9829p2 þ 5:2204b þ
3:6239b2 1:1398c 0:3786l 3:2884l2 0:2442r 0:6737r2 0:3548c2
0:8224pp 1:0853bp þ 0:8359rp þ 0:7443lp 3:5386p2b2 þ
2:4959p2l2 2:0189p2p þ 2:3979p2b þ 1:2676p2r 0:7284pc
0:7782pl 0:8208br 1:1743cr 0:7409lr 1:3341cr2 1:2651rc2
1:3376cl:
4.4. Design of Pareto optimal C&T policies
We first demonstrate the need for Pareto optimal designs to simultaneously
accommodate multiple performance measures.
Table 6 Performance measures of C&T policies optimized for individual measures.
Optimized measure Consumption (MW h)
Emissions ðtCO2Þ AMP ($/MW h)
CL 8950 7272 35.8
TE 5606 765 54.7
AMP 7747 4274 21.9
Thereafter, we discuss how to obtain the Pareto optimal C&T policy designs.
The need for Pareto designs is motivated by the results presented in Table
6, which shows that a C&T policy optimized for a particular performance
measure tends to produce poor outcomes for the other measures. Notice from
the table that, a design optimized for TE yields a consumption level of 5606
MW h whereas the design optimized for CL has a consumption level of 8950
MW h. Similarly, a design optimized for CL yields an AMP of $35.8/MW h,
which is much higher than $21.9/MW h which is attained by a design
optimizing AMP.
We formulated a multi-objective optimization model, as presented earlier
in (16), using the equations developed above for TE, CL, and AMP. Of the
six variables in these equations, SCC (p) and demand-price sensitivity (b) are not controlled by a C&T policy maker, and hence each of those variables were
considered at three different fixed values, resulting in nine possible
combinations. For each of these nine combinations, the multi-objective model
was solved and the optimal settings for the other four variables (cap reduction
rate, cap size, penalty, and congestion level) were obtained.
Before presenting the complete Pareto envelopes for all three measures,
parts (a), (b), and (c) of Fig. 12 show the Pareto fronts for each pair of
performance measures. Each front is plotted for three combinations of coded
values of ðp; bÞ (0,1), (0,0), and (0,1). It may be noted that, for a fixed SCC value of 21 (coded value of 0), lower values of demand-price sensitivity result
in higher Par-
Table 7 Consider the scenario with p ¼ 0 and b = 1. Say that it is desired
Factor effects plot for consumption (a) Factor effects plot for consumption (b)
Factor levels Factor levels
Fig. 10. Factor effect plots for consumption level.
Fig. 11. Factor effect plots for average electricity marginal price.
p b CL TE AMP
Penalty Demand Slope SCC
Cap reduction Cap size Line capacity
Average values of performance measures from Pareto envelopes. to have an aggregated
consumption level above 9000 MW h and the total emissions below 6000 tCO2
over the complete planning horizon. It can be seen from the Table 8 that the
Pareto efficient designs 85, 86, 87, 90, and 91 are those that meet both the
consumption and emissions conditions. Since AMP is lower the better, the
Pareto efficient design with the lowest AMP should be chosen. The design 86
happens to be the one with lowest AMP of $56.17/ MW h, for which the actual
consumption level is 9017.7 MW h and the emissions is 5211.57 tCO2. The
values of the C&T design parameters for the Pareto efficient point 86 are p ¼
$28:2=tCO2; r ¼ 0:075, c ¼ 159:05 tCO2, and l = 120 MW h.
eto fronts and therefore increases the levels of consumption, emis-
5. Concluding remarks
sions, and AMP.
We obtained the Pareto envelopes (considering all three mea-
In this paper, we have developed a mathematical–statistical
sures) for all nine combinations of p and b. For each envelope, model that allows us to obtain Pareto optimal C&T policies. we computed the average values of the measures over all the Par-
The model has two broad layers. The bottom (policy evaluation)
eto efficient points. In the solution of the multi-objective model, layer evaluates the impact of C&T and network parameters on we chose to obtain 100 Pareto efficient points. The results are prethe performance measures of an electricity network. The top
sented in Table 7. As observed earlier in Fig. 12, for fixed value of p, (policy optimization) layer obtains the Pareto optimal designs
decrease in demand-price sensitivity (from 1 to 1) results in using the results of the bottom layer. The existing literature, increase in CL, TE, and AMP. For a fixed value of b, increase in containing both empirical studies [8] and mathematical models SCC results in decreases in CL and TE and an increase in AMP.
[2,12], helps to effectively evaluate the impact of given C&T pol-
To illustrate further, we present in Fig. 13 two complete Pareto icies. Our research extends the literature from evaluation of C&T
envelopes for two arbitrarily chosen combination of ðp; bÞ of (0,0) policies to design of Pareto optimal policies that accommodate and (0,1). It can be observed that Pareto efficient designs that mindifferent interests of the network constituents (e.g., higher con- imize emissions yield lower consumption levels and higher aversumption, lower emissions, lower electricity prices). We have age marginal prices. Allowing higher emissions results in demonstrated that a Pareto envelope generated by our model increase in consumption and decrease in price. This shows that can serve as an useful tool for policy makers to select alternative stricter emissions policies will increase electricity prices. For sixty
C&T policies satisfying various interests of the electricity net-
of the one hundred efficient points on the Pareto envelope (0,1), work constituents. Results presented in the paper also examine the corresponding values of the four C&T design parameters are the sensitivity of important factors affecting electricity markets shown in Table 8. We picked a subset of the designs (instead of such as social cost of carbon and demand-price sensitivity. Elec- all 100) for reasons of space. In what follows, we describe how tricity generators can benefit from this model by using the bot- an envelope can be used to select an appropriate design for a tom layer to assess impact of given C&T policies on their bidding C&T policy. strategies and capacity expansion planning.
Table 8
Pareto C&T designs for p=0 and b = 1.
1 1 7125.03 3921.19 43.79
1 0 7850.56 4788.07 50.18
1 1 8505.91 5689.47 57.51
0 1 6314.70 2907.43 50.17
0 0 7425.11 3760.65 52.41
0 1 7774.23 4007.63 61.64
0 1 4942.74 2620.05 57.53
0 0 5888.57 2883.47 63.02
0 1 6699.43 3232.42 67.95
382
p r c 1 Penalty Cap red. Cap Lines cap Consumption Emissions AMP
1 l.00 l.00 l.00 0.71 50.00 0.075 80.00 85.80 6137.88 1780.92 69.25
2 l.00 l.00 0.91 0.65 50.00 0.075 83.76 87.06 6145.92 1870.79 69.36 3 l.00 l.00 l.00 0.90 50.00 0.075 80.00 82.04 6148.38 1715.37 68.14 4 l.00 l.00 l.00 0.99 50.00 0.075 80.00 80.16 6160.64 1682.59 67.51 5 l.00 0.75 l.00 0.96 50.00 0.069 80.00 80.78 6181.69 1823.88 67.59 6 l.00 l.00 l.00 0.21 50.00 0.075 80.00 95.84 6201.12 1955.74 71.04 7 0.91 0.96 l.00 0.90 48.12 0.074 80.00 82.04 6213.98 1779.46 67.14 8 l.00 l.00 0.51 0.71 50.00 0.075 99.45 85.80 6225.41 2132.34 67.85 9 0.97 l.00 0.51 0.77 49.37 0.075 99.45 84.55 6254.46 2125.61 67.14
10 l.00 l.00 l.00 0.07 50.00 0.075 80.00 101.49 6295.03 2054.07 71.32 11 l.00 0.91 0.92 0.07 50.00 0.073 83.14 101.49 6313.72 2159.18 71.11 12 l.00 0.97 0.72 0.ll 50.00 0.074 91.29 102.12 6327.84 2284.80 70.88 13 l.00 0.94 0.31 0.96 50.00 0.073 107.61 80.78 6363.68 2222.25 65.33 14 l.00 0.40 l.00 0.07 50.00 0.060 80.00 101.49 6401.94 2363.68 70.67 15 l.00 0.91 0.25 0.07 50.00 0.073 110.12 101.49 6438.88 2642.75 69.64 16 l.00 0.00 0.96 0.07 50.00 0.050 81.57 101.49 6485.80 2601.45 70.45 17 l.00 l.00 0.06 0.ll 50.00 0.075 117.65 102.12 6490.29 2745.16 68.95 18 0.91 0.49 0.28 0.90 48.12 0.062 108.86 82.04 6528.29 2518.58 65.14 19 l.00 0.91 0.45 0.48 50.00 0.073 101.96 109.65 6587.54 2638.59 69.98 20 l.00 0.97 0.72 0.61 50.00 0.074 91.29 112.16 6607.42 2459.61 70.16 21 l.00 0.50 0.03 0.07 50.00 0.062 118.90 101.49 6654.26 2987.76 69.12 22 0.98 0.91 0.45 0.58 49.69 0.073 101.96 111.53 6664.47 2682.77 69.59 23 l.00 0.91 0.28 0.58 50.00 0.073 108.86 111.53 6698.67 2795.07 69.40 24 l.00 0.97 0.29 0.99 50.00 0.074 131.45 80.16 6713.10 2625.98 61.75 25 l.00 0.91 0.01 0.48 50.00 0.073 119.53 109.65 6735.35 2953.48 68.80 26 l.00 0.91 0.03 0.58 50.00 0.073 118.90 111.53 6792.65 2975.0l 68.69 27 l.00 l.00 0.51 0.97 50.00 0.075 140.24 80.63 6862.97 2779.06 60.34 28 l.00 0.78 0.63 0.10 50.00 0.070 145.25 101.96 6938.0l 3333.37 66.18 29 0.00 0.97 0.75 0.90 29.92 0.074 90.04 82.04 6974.37 2813.35 61.37 30 l.00 0.50 0.65 0.07 50.00 0.062 145.88 101.33 7012.55 3451.77 66.88 71 0.07 l.00 l.00 0.60 28.67 0.075 160.00 112.00 8476.50 4769.34 57.23 72 0.79 0.79 0.64 0.37 14.24 0.070 94.43 107.45 8521.54 5134.39 64.49 73 0.25 l.00 0.62 0.85 24.90 0.075 144.94 117.02 8618.33 4954.14 58.49 74 0.76 0.95 l.00 0.51 14.86 0.074 80.00 110.27 8631.73 5000.01 63.85 75 0.76 0.93 0.72 0.61 14.86 0.073 91.29 112.16 8655.31 5165.07 63.57 76 0.35 l.00 0.56 0.85 23.02 0.075 142.43 117.02 8675.17 5064.49 58.84 77 0.65 0.94 0.49 0.58 17.06 0.073 139.61 111.53 8699.10 5354.30 61.09 78 0.79 0.79 0.99 0.ll 14.24 0.070 159.69 102.12 8755.75 5451.45 60.71 79 0.76 0.58 0.15 0.51 14.86 0.036 114.20 110.27 8794.46 5618.38 63.46 80 0.69 0.96 0.56 0.92 16.12 0.074 97.57 118.43 8823.88 5281.57 61.67 81 0.76 0.93 0.75 0.51 14.86 0.073 149.96 110.27 8882.61 5587.20 60.74 82 0.79 0.94 0.71 0.57 14.24 0.073 148.39 111.37 8938.72 5662.54 60.94 83 0.76 0.43 0.76 0.60 14.86 0.061 150.27 112.00 8985.68 5730.99 62.43 84 0.76 0.43 0.76 0.61 14.86 0.061 150.27 112.16 8992.19 5736.41 62.40 85 0.24 0.75 0.98 0.99 25.22 0.069 159.06 119.84 9006.27 5262.84 57.73 86 0.24 l.00 0.98 l.00 25.22 0.075 159.06 120.00 9017.70 5211.57 56.17 87 0.99 0.99 0.33 0.75 10.16 0.075 106.98 114.98 9066.58 5915.05 64.11 88 0.99 0.99 0.27 0.75 10.16 0.075 130.82 114.98 9142.15 6042.52 63.04 89 0.99 0.84 0.27 0.75 10.16 0.071 130.82 114.98 9155.30 6067.47 63.28 90 0.76 l.00 0.74 0.85 14.86 0.075 149.65 117.02 9187.42 5805.73 59.44 91 0.99 0.93 0.88 0.61 10.16 0.073 155.29 112.16 9254.73 6077.10 61.47 92 0.96 0.13 0.76 0.73 10.78 0.053 150.27 114.51 9285.63 6189.26 63.63 93 0.69 0.96 0.88 l.00 16.12 0.074 155.29 120.00 9371.63 5859.68 58.04 94 0.93 0.93 0.77 0.95 11.41 0.073 150.90 119.06 9478.67 6196.06 60.20 95 0.98 0.87 0.78 0.98 10.47 0.028 151.22 119.69 9513.49 6539.19 61.98 96 0.93 0.93 0.76 l.00 11.41 0.073 150.27 120.00 9521.13 6226.28 60.02 97 0.98 0.87 0.98 0.98 10.47 0.028 159.06 119.53 9525.84 6554.68 62.46 98 0.99 0.01 0.99 0.94 10.16 0.050 159.69 118.75 9571.73 6450.68 62.98 99 0.99 0.01 0.96 l.00 10.16 0.050 158.43 120.00 9620.80 6492.06 62.63
100 0.99 0.99 0.99 l.00 10.16 0.075 159.69 120.00 9716.60 6381.46 59.18
Computational challenges associated with our model are limited to the
bottom layer (in particular, solving the EPEC problem), especially when a
network has a large number of generators [38]. The number of time periods in
the planning horizon also adds to the computational burden as EPEC needs to be
solved for each period. However, the computation time for the top layer model
is independent of the size of the network, number of generators, and the time
periods of the planning horizon.
Acknowledgements
Authors acknowledge the useful comments and suggestions provided by
the reviewers which have significantly improved the quality of the
manuscript.
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