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1 Copyright © 2012 by ASME
Proceedings of 2012 ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference
August 12-15, 2012, Chicago, IL, USA
DETC2012-70426
POLICY DESIGN FOR SUSTAINABLE ENERGY SYSTEMS CONSIDERING MULTIPLE OBJECTIVES AND INCOMPLETE PREFERENCES
Bryant D. Hawthorne Jitesh H. Panchal*
School of Mechanical and Materials Engineering Washington State University Pullman, WA, USA 99164
*Corresponding Author, Email: panchal@wsu.edu
ABSTRACT The focus of this paper is on policy design problems
related to large scale complex systems such as the decentralized
energy infrastructure. In such systems, the policy affects the
technical decisions made by stakeholders (e.g., energy
producers), and the stakeholders are coordinated by market
mechanisms. The decentralized decisions of the stakeholders
affect the sustainability of the overall system. Hence,
appropriate design of policies is an important aspect of
achieving sustainability. The state-of-the-art computational
approach to policy design problem is to model them as bilevel
programs, specifically mathematical programs with equilibrium
constraints. However, this approach is limited to single-
objective policy design problems and is based on the
assumption that the policy designer has complete information
of the stakeholders’ preferences. In this paper, we take a step
towards addressing these two limitations. We present a
formulation based on the integration of multi-objective
mathematical programs with equilibrium constraints with
games with vector payoffs, and Nash equilibra of games with
incomplete preferences. The formulation, along with a simple
solution approach, is presented using an illustrative example
from the design of feed-in-tariff (FIT) policy with two
stakeholders. The contributions of this paper include a
mathematical formulation of the FIT policy, the extension of
computational policy design problems to multiple objectives,
and the consideration of incomplete preferences of
stakeholders.
Keywords: Sustainability, energy policy, feed-in-tariff
policy, Game theory, Nash equilibrium, market models
Nomenclature
Policy payment
Duration of policy
Quantity produced by the i’th stakeholder
Market quantity minus quantity produced by the
i’th stakeholder
Total market quantity
Normalization factor for quantity
Market constants
Net present value
Maximum net present value
Operation and maintenance cost
Capital investment
Maximum capital investment
Policy cost
Maximum policy cost
Market price
Cost of electricity
Set of quantities satisfying the equilibrium
constraints
Market demand
Discount rate
Policy maker’s weight for quantity
Policy maker’s cost preference
Weight for stakeholder 1’s net present value
Weight for stakeholder 1’s capital investment
Weight for stakeholder 2’s net present value
Weight for stakeholder 2’s Capital Investment
1. INTRODUCTION – POLICY DESIGN FOR SUSTAINABLE ENERGY SYSTEMS
Traditionally, the emphasis in the engineering design
research has been on problems where the design space is
entirely under the control of designers. However, there is an
increasing importance of large-scale complex systems whose
designs are not directly controlled by designers, but emerge out
of the independent decisions of self-interested stakeholders
coordinated through market-based mechanisms. Consider the
example of a smart electric grid, a large-scale complex system
consisting of a wide range of decision makers including
consumers, utilities, micro-grid operators, and the other
participants of the distribution infrastructure. The distribution
infrastructure includes distribution lines and cables,
transformers, control devices, distributed generation devices,
2 Copyright © 2012 by ASME
and consumer loads. For such large-scale complex systems, the
stakeholders such as energy producers, distributers, and utilities
independently make technical decisions within rules and
regulations to meet their objectives of system performance,
reliability, security and load demand while maximizing their
profits. The decisions of the stakeholders can be influenced by
designing appropriate policies, provision of incentives and
development of standards, thereby affecting the design of the
entire system. Examples of such policy decisions include
renewable portfolio standards, carbon taxes, and incentives for
using specific technologies. Policy decisions such as the
incentives for the adoption of renewable energy technology
affect the technical decisions about the kinds of generation
technologies adopted by stakeholders. The technologies chosen
by these stakeholders affect the technical (e.g., system
reliability and security), social (jobs and economic
development), economic (e.g., cost to consumers), and
environmental (e.g., emissions) performance. Hence, these
policy decisions have an impact on the overall sustainability of
large-scale complex systems.
Policy design problems share a number of similarities
with multi-disciplinary engineering design problems including
the presence of multiple stakeholders, multiple objectives, and
the underlying decision-making nature. However, existing
hierarchical design approaches are not directly applicable to
policy design problems because of two fundamental differences
between traditional engineering design problems and policy
design problems: a) the nature of the stakeholders’ objectives,
and b) coordination mechanisms between stakeholders.
Existing design approaches are developed for systems such as
automobiles and aircrafts that are hierarchically coordinated
towards achieving organizational goals.
On the other hand, the stakeholders in the policy design
problems have their own objectives and make decisions in a
self-interested manner. The individual stakeholders’ goals may
or may not be aligned with the policy-makers’ objectives.
Additionally, coordination between stakeholders in a policy
design problem is generally through market-based mechanisms.
Hence, there is a need to model the market-based interactions
between stakeholders. These unique features of policy design
problems call for new design approaches for policy design
problems.
Existing research on computational policy analysis and
design models the design problem as a multilevel decision-
making problem with policy decisions representing higher-level
problems and stakeholders’ technical decisions as lower-level
problems. The multi-level problems are converted into
mathematical programming problems with equilibrium
constraints representing the outcomes of interactions between
stakeholders. A detailed literature review is provided in Section
2. While there are a number of limitations of existing
computational approaches for policy design, our focus in this
paper is on addressing the following two limitations:
a) the policy design problems are modeled as single-
objective problems, and
b) the assumption that policy-makers have complete
knowledge of the stakeholders’ preferences.
In this paper, we present a formulation and a simple
solution approach for addressing these two limitations of
approaches for policy design. The formulation is based on an
extension of the well known mathematical programs with
equilibrium constraints (MPECs), games with vector payoffs,
and Nash equilibra of games with incomplete preferences. We
present the approach using an illustrative example from the
design of feed-in-tariff (FIT) policy. The example problem is
limited to two stakeholders to retain the ability to plot the
decisions and rational reaction sets on 2D plots. The key
contributions of this paper include extension of computational
policy design problems to multiple objectives, the consideration
of incomplete preferences of stakeholders, a mathematical
formulation of the FIT policy.
The paper is organized as follows. In the following
section, a detailed review of the literature is presented. An
overview of the FIT policy is presented in Section 3. The
mathematical tools used in the proposed approach, including
MPEC and games with vector payoffs, are presented in Section
4. Mathematical formulation of the multi-objective FIT policy
with incomplete preferences is presented in Section 5. Results
of the illustrative example are presented in Section 6.
Discussion of limitations and future research opportunities are
presented in Section 7.
2. REVIEW OF RELEVANT LITERATURE
2.1. Modeling stakeholder decisions using non-cooperative games
The natural framework for analyzing systems that involve
multiple independent decision-makers is non-cooperative game
theory [1]. Non-cooperative games have been used in
engineering design, primarily as a way to represent
decentralized design scenarios [2, 3] where designers are
modeled as decision-makers. The designers’ decisions are in
equilibrium if none of the designers can unilaterally improve
their payoff by changing their own decisions. This equilibrium
is referred to as the Nash equilibrium. Current research on non-
cooperative game theory within engineering design is focused
on determining the Nash equilibria and their stability
properties.
One of the widely adopted approaches for finding Nash
equilibria is based on formulating the problem as a
complementarity problem [4, 5] and using the first order
necessary conditions of optimality of the individual
stakeholders’ decisions. The complementarity problem is a
special case of a variational inequality problem [5-7]. The
complementarity models for Nash equilibria have been used in
a number of applications related to modeling of markets [8].
For example, Hobbs [9] presents a model of bilateral markets
with imperfect competition between electricity producers using
linear complementarity models. Gabriel and co-authors [10-13]
model a natural gas equilibrium model with different types of
market participants including producers, storage operators,
pipeline operators, marketers, and consumers.
2.2. Policy-design as a bilevel problem While modeling of equilibria between stakeholders in a
market is an important problem, the goal from a policy-design
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standpoint is to design the Nash equilibria by influencing
stakeholders’ decisions. This problem of designing equilibria
can be viewed as a higher-level problem with design variables
(e.g., incentives or penalties) that can be used to modify the
Nash equilibria of the lower-level equilibrium problem. These
design problems represent a special class of bilevel programs
[14]. Within mathematical programming literature, such bilevel
problems are called mathematical programs with equilibrium
constraints (MPECs) [15]. Ye [16, 17] presents necessary and
sufficient conditions for optimality for bilevel programs and
MPECs.
MPECs are challenging because the optimality conditions
in the lower level problems lead to combinatorial issues, and
the potential lack of convexity and/or closedness of the feasible
region [15]. A number of specialized algorithms have been
developed to address these challenges of MPECs [15, 18-20].
Examples of the algorithms include piecewise sequential
quadratic programming, penalty interior-point algorithm,
implicit function based approaches, and smooth non-linear
programs [18]. Some of these algorithms are implemented in
commercial platforms for optimization such as GAMS and
Matlab [21]. Applications of MPEC include electricity markets
[22, 23], highway tax policy design [24], and critical
infrastructure planning [25].
2.3. Gap in the literature The existing work on designing policies using bilevel
programming techniques has two main limitations. The first
limitation is that the problems are modeled as single-objective
problems. Recently, there have been some efforts within the
mathematical programming area on extending the MPEC
formulation to multi-objective optimization problems with
equilibrium constraints (MOPEC) [26, 27]. The current work in
that direction is focused on deriving the necessary conditions
for optimality [28, 29]. However, such formulations have not
yet been utilized for policy design problems. In this paper, we
discuss the application of MOPEC to problems involving
policy design for sustainability.
The second limitation of the existing literature is that it is
assumed that the higher level policy designer has complete
knowledge of the preferences of the lever-level decision
makers. The common assumption is that the stakeholders are
profit-maximizing firms and their only objective is to maximize
their profits. However, the stakeholders may have multiple
objectives. Consider an example of a policy decision at the
federal level, which affects the decisions made by local (or
state) policy makers. In this case, the federal policy design is
the upper-level problem and the local policy design is the lower
level problem in MPEC, whose goal is not simply profit
maximization. Even the profit maximizing firms have
objectives that cannot be directly quantified in terms of profit.
Examples of such objectives include service quality, brand
recognition (through reduced green-house gas emissions), and
community service. Even in cases where the multi-objective
nature is acknowledged, it is implicitly assumed that the policy
decision-maker knows the stakeholders’ tradeoffs in advance,
allowing the lower-level decisions to be modeled as single-
objective optimization problems. Hence, it is assumed that the
stakeholders’ objectives can be combined into a single objective
function (such as a utility function). This single objective
function satisfies the completeness axiom of vonNeumann and
Morgenstern’s utility theory [30] and can be used to compare
all alternatives. However, this assumption can be invalid in
three scenarios which are particularly relevant in real policy
design problems [31, 32]. First, the policy decision maker may
not have complete information about the preferences of the
individual decision makers. Second, the lower-level decision
makers may represent groups of individuals (e.g., committees),
leading to incomplete social preferences. Third, the decision
makers may be indecisive, and hence, unable to rank all
combinations of alternatives in a multi-objective scenario [33].
To address the incomplete nature of preferences,
vonNeumann and Morgenstern’s utility theory has been
extended to utility theory with incomplete preferences [34-37].
While existing work on utility theory with incomplete
preferences is focused on modeling the decisions, there is
limited understanding of strategic interactions between players
with incomplete preferences. Bade [38] shows that the Nash
equilibria for any game with incomplete preferences can be
characterized in terms of certain derived games with complete
preferences. Additionally, if the players’ preferences are
concave, the Nash equilibria can be determined from derived
complete games by a simple linear procedure. The author [38]
discusses the Nash equilibrium of a game where a) each
decision maker has multiple objectives, b) decision makers are
able to rank alternatives based on each objective individually,
and c) the decision makers are unable to make tradeoffs among
different objectives. Such games are also referred to as games
with vector payoffs [39] or multi-objective games [40, 41]. The
equilibria of games with vector payoff are referred to as Pareto
equilibria [42, 43].
In this paper, we consider the multi-objective nature of the
policy design problem and the incomplete preferences of
stakeholders. We illustrate a framework based on MOPEC and
games with vector payoffs using Feed-In-Tariff policy design
problem.
3. SUSTAINABLE ENERGY INFRASTRUCTURE AND THE FEED-IN-TARIFF POLICY
3.1. Energy policy and the interplay between policy design and engineering design
With the increasing use of small-scale energy generation
from renewable sources and increasing deregulation of the
energy sector, an alternative paradigm of energy generation and
distribution is emerging and leading towards “smart grid
architecture”. In a decentralized infrastructure, different
stakeholders act as decision makers, and the overall system-
level performance is dependent on the individual decisions. For
example, the consumers can play an active role as energy
producers for actively managing their demand. They make
decisions on a) which technologies to invest in, b) how much
energy to generate, c) how much energy to buy and from
whom, d) how much energy to sell and e) how much to
participate in actively managing their load demand [44]. Other
stakeholders include power producers (e.g., utility companies),
grid operators, transmission companies (TRANSCO),
distribution companies (DISTCO) and regulators (e.g.,
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government and other regulating authorities). The decisions
made by different stakeholders are often conflicting in nature.
Examples of the goals and decisions of different stakeholders
are shown in Figure 1. Based on the decisions made by the
individual stakeholders, the system reaches an equilibrium,
which defines its overall behavior. The individual decisions can
be directed through a number of mechanisms such as policy
tools, incentives (e.g., tax breaks), penalties (e.g., tariffs),
markets rules, and laws. The corresponding equilibria can be
changed through these mechanisms.
Figure 1 – Decision-makers within a decentralized
energy infrastructure
The policy design problem is driven by a number of
social, environmental, technical, and economic objectives [45].
The technical objectives include replacing fossil fuel generating
plants with renewables, minimization of system losses,
maintaining required stability/ security/ reliability, avoiding
unbalance conditions, meeting power quality requirements,
peak shaving, targeting high efficiency systems, innovation and
early adoption of technologies and meeting the energy needs.
The environmental objectives are minimization of green house
gas emissions and hazardous materials. Economic objectives
include minimization of policy costs and ratepayer impact.
Social objectives include job creation, economic development,
meeting long term energy requirements, policy transparency,
fairness and quality of life.
To achieve these objectives, different policies can be
adopted at the federal, state, local, and utility levels. The policy
options include incentives to investment, guidelines for energy
conservation, taxation and other public policy techniques [46,
47]. Specific examples include emission taxes, incentives to
non-polluters and renewable energy, incentive for demand
response, emission cap-and-trade systems, emission intensity
standards and regulations, and alternative allocations of
emission rights to regions and sectors. In several counties,
including Germany and Spain, one of the mechanisms which
has been particularly successful in addressing environmental,
reliability, and security issues associated with decentralized
energy has been feed-in-tariff (FIT) policies [45]. FIT policies
are discussed in the following section.
3.2. Overview of feed-in-tariff (FIT) policies A feed-in-tariff is an energy supply policy that offers a
guarantee of payments to renewable energy (RE) developers for
the electricity they produce [48]. The objectives of these
policies are to motivate the deployment of RE technologies and
to increase renewable generation while reducing dependencies
on fossil-fuels. FIT programs support decentralized
infrastructure and motivate individuals along with companies to
invest in renewable energy technologies. FIT can be designed
by the utilities or the state government. Moreover, FIT can be
designed to work in conjunction with other US state policies
such as renewable portfolio standards (RPSs) and net-metering,
and federal policies such as the Production Tax Credit (PTC)
and the Investment Tax Credit (ITC).
Figure 2 - Illustration of the FIT policy design problem
The design of FIT programs can be categorized into two
classes - market independent design, and market dependent
design (see Figure 2). In the market independent design, the
investors are paid a fixed price per unit electricity produced,
independent of the market price. Different variations of the
market independent design include fixed price with full/partial
inflation adjustment, front-end loaded design, and spot market
gap model. In the market dependent FIT policy design, the
payment depends on the market price of electricity. Variations
of the market dependent design include premium price model
(fixed premium on top of the market price), variable premium
model (includes caps and floors), and percentage of retail price.
The price can be based on factors such as cost of generation,
value of the system (to the utility, consumers, etc.), and
auction-based price discovery. Additionally, the payment
differentiation can be based on technology and fuel type,
project size, resource quality, and location. Based on the design
of the policy, the investors (individuals or companies) decide to
invest in different technologies to maximize their objectives.
The resulting increase in generation capacity affects the market
equilibrium and the corresponding energy prices.
FIT is chosen in this paper because a) there is a clear need
to design the FIT policy for different scenarios, b) there are a
number of well known design options, c) data is available for
different variations of the FIT policy implemented in Europe,
and d) FIT is gaining popularity within the US. In the following
section, we provide an overview of the mathematical tools used
in this paper.
Government:
Federal
State, Local
Consumers:
Commercial
Residential
Micro-grids,
DNO and
DISTCO
Utilities,
GENCO and
TRANSCO
Grid Operators:
ISOs, TSOs
Goals: Minimize costs
Decisions: Technologies, demand response (e.g., load
shifting, load shedding, peak
shaving)
Goals: Social (safety+), Economic,
Technical (reliability, security, power quality, reduced losses),
Environmental, stability of energy
pricesDecisions: Policies, taxes, incentives,
rebates, loans, laws, grants, etc.
Goals: Reliable service,
low cost, load balancingDecisions: generating
station, price per KW
Goals: minimize cost, power quality,
self-relianceDecisions: Combinations of
technologies, sizing, operational factors
Goals: Reliability, power-
quality, security, profitDecisions: grid expansion
investments
Market
DynamicsFeed-In-Tariff (FIT) Policy Design
• Premium price model
• Variable premium model
• Percentage of retail price
• Price Differentiation
Energy Prices
Policy Goals
Market Equilibrium
(Cournot competition)
…
Investor 1 Investor 2 Investor n
Maximize: 1 Maximize: 2 Maximize: n
Subject to:
h1 ≤ 0
g1 = 0
Subject to:
h1 ≤ 0
g1 = 0
Subject to:
h1 ≤ 0
g1 = 0
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4. MATHEMATICAL MODELS AND TOOLS USED The decentralized decisions of the stakeholders can be
modeled as a non-cooperative game [1, 49] between players
(i.e., stakeholders) trying to achieve their own objectives. The
outcome of the non-cooperative game is defined in terms of the
Nash equilibrium. In the case of FIT policy, the equilibrium
point is determined by the payment to the investors. Hence, the
goal of the policy designer is to choose the payment (i.e., the
parameters of the policy parameters) such that the equilibrium
point represents the best system performance from a global
standpoint.
Within game theory, such interactions between decision
makers are referred to as the Stackelberg game [50] where one
of the players (in this case the policy maker) moves first and
the rest of the players (the stakeholders) move based on the
decision made by the player moving first. The Stackelberg
game can be mathematically modeled as a bi-level optimization
problem with policy design as the upper optimization problem
and the stakeholders’ decisions as the lower-level Nash
equilibrium problem [51]. Mathematical Programs with
Equilibrium Constraints (MPEC) is a mathematical framework
for such bi-level problems with higher level optimization
problem and lower level equilibrium problem. An overview of
the MPEC framework is provided in Section 4.1. In Section
4.2, we discuss games with vector payoffs that are used for
modeling the incomplete preferences of stakeholders.
4.1. Mathematical programming with equilibrium constraints (MPEC)
MPEC is a type of constrained nonlinear programming
problem where some of the constraints are defined as
parametric variational inequality or complementarity system
[15]. These constraints arise from some equilibrium condition
within the system, and hence, are called equilibrium constraints
[52]. MPEC is applicable to a variety of problems in
engineering such as optimal design of mechanical structures,
network design, motion-planning of robots, facility location,
and equilibrium problems in economics. Examples of problems
of economic equilibrium where MPEC has been used include
maximizing revenue from tolls on a traffic system [53], optimal
taxation [54], and demand adjustment problems [55].
Mathematically, a MPEC problem can be represented using two
sets of variables, and . Here, belongs to the upper-level
problem and solves the lower-level equilibrium problem. The
solution of depends on the value of chosen for the upper-
level problem. The overall objective function is
minimized.
Subject to: , and
(2)
(3)
where is the joint feasible region of and ; and
is a set of variational inequalities that represent the equilibrium
problem. The function represents a system-level
function that quantifies the goodness of the solution. The set
corresponds to the feasible Nash space. The Nash
equilibrium point can be formulated as a variational inequality
using the first order necessary conditions for optimality such as
Karush–Kuhn–Tucker (KKT) conditions [56]. Solving the
MPEC problems is challenging because of the non-linearities in
the problem, non-convex feasible space, combinatorial nature
of constraints, disjointed feasible space, and multi-valued
nature of the lower equilibrium problem [15]. Significant
research efforts have been devoted to developing efficient
algorithms for solving MPEC problems. These include
piecewise sequential quadratic programming (PSQP) [57],
penalty interior-point algorithm (PIPA) [15], implicit function-
based approaches [58], and smooth sequential quadratic
programming [18]. Recently, few efforts has been carried out
by Ye [28], Mordukhovich [26], and Bao et al. [59] on deriving
the necessary conditions for optimality of multi-objective
problems with equilibrium constraints (MOPEC).
4.2. Games with vector payoffs Games within which players can have several possibly
conflicting objectives are called “games with vector payoffs” or
“multi-objective games” [41, 42]. Due to the multi-objective
nature of the decisions, the concepts of rational reactions and
Nash equilibria need to be generalized. In the games with single
objectives, a rational reaction (best reply) of a player to other
players’ decisions is a point that maximizes his/her payoff. In
the case of games with vector payoffs, a player’s rational
reaction to the decisions of other players is a set of Pareto
optimal solutions, also referred to as Pareto Best Replies. In
single-objective games, the points of intersection of the rational
reaction sets are the Nash equilibria. In games with vector
payoff, the concept of Nash equilibrium is replaced by the
concept of Pareto equilibria defined as the pairs of strategies
which are Pareto best replies to one another [40, 43].
In this paper, we use games with vector payoffs to
represent situations where the policy designer knows the
multiple objective functions of the players but does not have
complete information about the players’ preferences for
tradeoffs. In such situations, the policy designer cannot assign a
single utility function to each player. Under certain conditions,
games with vector payoffs can be reduced to games with single
objectives by combining each player’s objectives using
Archimedean weighting scheme. The reduced single objective
game is called a tradeoff game [42] or a derived game with
complete information. In this paper, we convert the games with
vector payoffs into multiple tradeoff games and solve the
corresponding MOPECs to generate uncertainty bounds. The
approach is discussed in detail in Section 5.
5. FORMULATION OF THE FIT POLICY DESIGN PROBLEM FOR SUSTAINABLE ENERGY SYSTEMS
The formulation of upper-level policy decision involves
the identification of policy objectives, design variables, and
constraints. There are a number of objectives that can be
considered while designing FIT policy (see Table 1). As
mentioned in Section 3.2, there are two classes of FIT policy –
market independent and market dependent. In this paper we use
the market-dependent design, specifically the premium price
model to illustrate the approach. The lower-level stakeholders
are modeled as profit-maximizing entities who invest in
different technologies based on the incentives determined by
the policymakers. Examples of design variables include the
types and sizes of different generation technologies to invest in.
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The stakeholder objectives are calculated in terms of the net-
present value (NPV) of the all the expenses (capital investments
and maintenance costs) and the payments received for the
energy generated. One of the objectives of the stakeholders is to
minimize NPV. Table 1 - The objectives and design variables of the
policy design problem
FIT Policy Design Objectives [48]
Economic: job creation, economic development, economic
transformation, stabilization of electricity prices, lower electricity
prices, grow economy, revitalize rural areas, attract new
investment, develop community ownership, develop future export
opportunities
Environmental: clean air benefits, greenhouse gas emission
reduction, preserve environmentally sensitive areas, manage
waste streams, reduce exposure to carbon legislation
Energy Security: secure abundant energy supply, reduce long-term
price volatility, reduce dependence on natural gas, promote a
resilient system
Renewable Energy (RE) Objectives: rapid RE deployment,
technological innovation, drive cost reductions, meet renewable
portfolio standards, reduce fossil fuel consumption, stimulate
green energy economy, barriers to renewable development
Design Variables
Alternatives: 1) Premium price model: price per kWh; 2) Variable
premium price model: premium price in access of the market
price; 3) Percentage of retail value: % in access of the market
price.
Payment differentiation based on technology and fuel type, project
size, resource quality, resource quantity, and location. Other
design variables include tariff degression, and inflation
adjustment.
5.1. Formulation of an illustrative FIT problem To illustrate the approach for accounting for multiple
policy design objectives and incomplete knowledge of
stakeholders’ preferences, we present a simple example with
two stakeholders. An overview of the decisions of the policy
designer and each stakeholder is provided in Table 2. The
policy designer has two objectives: a) to maximize the total
quantity of energy generated by all the stakeholders ( ), and b)
to minimize the cost of implementing the policy ( ). The first
objective is related to the policy goal of renewable energy
penetration through the design of incentives and the second
objective is an economic goal associated with most policies.
The stakeholders’ decisions are driven by two objectives:
maximization of the net-present value of their investment and
the minimization of capital investment. The assumptions made
in this model are listed in Section 5.1.1.
5.1.1. Assumptions The problem formulation presented in this section is based
on the following assumptions:
1) There is only one policy maker – In a real policy design
problem, there are a number of entities involved in the
policy making process. Different entities may have
different objectives. An entity (such as environmental
protection agencies) may have an objective to reduce CO2
emissions, while another may be more interested in the
economic impacts of a policy. Here, we consider that one
policy maker is responsible for satisfying all the objectives
and has complete control of the design variables.
Table 2 – Overview of the decisions made by the policy designer and the stakeholders
Policy Designer
Objectives:
Maximize total quantity generated by all stakeholders,
Minimize the policy cost,
Decision Variables:
Premium price per unit energy generated,
Time period for which policy is implemented,
Each stakeholder’s decision
Objectives:
Maximize the net present value,
Minimize the capital investment, Decision Variables:
Quantity of energy to be generated,
2) There are two stakeholders – We consider only two
stakeholders because it helps us in visualizing the rational
reactions and the design space on 2D plots. An actual
policy design problem generally consists of a large number
of stakeholders. The model will be extended to n-
stakeholders in the future.
3) Fixed premium-price FIT model payment option – As
discussed in Section 3.2 there are many different models of
FIT policies. In our model we assume a FIT policy where
the payment is a fixed premium price ( ). In other words,
the investors are paid some set amount, , above the
market price of electricity.
4) The policy-maker can only control two decision variables –
We set the policy designer’s decision variables as the
premium-price of the policy, , and the duration of the
policy, . These two variables are the primary design
variables in a premium price model. In a real FIT design
scenario, the premium price may be different for different
types of technologies, and different size of generation
facilities installed by the stakeholders.
5) Stakeholders can only control one decision variable –
Stakeholders can only control the quantity of electricity
being generated ( ). Based on the duration of the policy,
the premium-price and the market demand, each
stakeholder decides upon the quantity to generate.
6) The policy options are not dynamic – In many FIT policies
the policy options (decision variables) can vary with time.
For example, the premium price may be high during the
initial period to encourage investment but may be reduced
over time. Alternatively, the premium price may increase
with time to account for inflation. In this paper, we assume
that is fixed for the duration of the policy.
7) Incomplete information about preferences – It is assumed
that the policy maker has complete information of the
stakeholder’s individual objectives (maximization of net-
present value and minimization of capital investment).
However, the policy maker does not have complete
information about how each stakeholder makes tradeoffs
among the different objectives.
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8) Electricity market is modeled using Cornout Nash
equilibrium – In order to simplify how stakeholders
respond to each other’s investments, we assume that the
electricity market is modeled as Nash equilibrium of a
Cournot competition game.
5.1.2. Simple model of the electricity market The electricity market modeling literature [60] consists of
two types of models for market equilibrium arising from profit
maximizing participants: the Cournot equilibrium [61] and
supply function equilibrium (SFE) [22]. Both concepts are
based on the Nash equilibrium, but differ in the decision
makers’ variables. In Cournot equilibrium model, the
participants compete in quantity of energy produced, whereas
in the supply function equilibrium model, the participants
compete both in quantity and price.
Consider a simple example of two producers deciding on
their production quantities and [62]. Assume that the
market price is determined by the overall quantity produced ( )
through a linear function: where A is a
constant. The profit of each firm is given by: where is the cost of production for
player . The resulting Nash equilibrium is given by:
Cournot equilibrium is more flexible and tractable
because it results in a set of algebraic equations for the Nash
equilibrium whereas SFE results in a set of differential
equations [60]. Hence, the Cournot equilibrium model has
attracted significant attention from the electricity market
modeling community [9]. For a given price and duration of a
policy, the stakeholders decide the quantities , which
correspond to the Cournot equilibrium value based on the
market and stakeholder preferences. The policy makers can
achieve their objectives by designing the price and duration.
5.1.3. Details of the decisions made by stakeholders and the policy maker
The objectives, design variables, and constraints for the
policy maker and the stakeholders are shown in Table 3. First,
we consider the objectives of the stakeholders. The first
objective is the net present value (NPV), which is the time
series of cash inflow and outflow. The second objective is
minimization of capital investment ( ). As previously stated,
the policy maker has control over two variables, , the
premium-price of the policy (which is constant during the
duration of policy implementation), and , the duration of the
policy. We assume that the market price takes the form,
, where . Here, and are
two constants based on the energy market. For the simplified
case of two stakeholders, . We assume the
cash inflow for stakeholder is and the outflow is
the operation and maintenance cost ( ) along with capital
investment ( ), see eqn. (1). Since capital investment is only
made at the start of the investment this payment is not
reoccurring and is not affected for varying time. Assuming that
is the discount rate, the net present value for stakeholder is:
(1)
The design variable for each stakeholder is the quantity,
, which is constrained to be positive. A negative value of
would indicate that instead of generating energy, the
stakeholder purchases it from the other stakeholder. However,
we do not consider that scenario in this paper. Table 3 – Multi-objective policy design problem with
multi-objective decisions of stakeholders. This is the most general problem
Upper-level policy design problem
Objectives:
Minimize
Maximize
Design variables:
,
Constraints:
$/KWh
years
Lower-level stakeholders’ problems
Stakeholder 1 Stakeholder 2
Objectives:
Design variable:
Constraints:
Objectives:
Design variable:
Constraints:
The policy maker can control , the non-variable
premium-price of the policy, and , the duration of the policy.
Based on these values, and the preferences from the
stakeholders, the stakeholders reach equilibrium of the quantity
being generated. If these preferences are known, the policy
maker can maximize his/her own payoff based on these
decisions. We assume that the policy maker has two objectives.
The first objective is to minimize the policy cost (PC). The
second objective is to maximize the total quantity of generation
( ). We constrain the design variables where the premium-
price of the policy is less than 0.2 $/KWh and the duration of
the policy is less than or equal to twenty one years.
5.2. Approach for policy decision making under incomplete preferences of stakeholders
The approach adopted in this paper for solving the policy
design problem listed in Table 3 is shown in Figure 3. There is
a lack of algorithms for solving the general problem presented
in the previous section. Hence, we follow a simple six step
process. Step 1 is the formulation of the decisions of the policy
decision maker and the stakeholders, as discussed in Section
5.1.3. The characteristics of this general problem are that the
policy designer has multiple objectives and has incomplete
information about the stakeholders’ preferences. We assume
that the policy designer knows the different objectives of the
stakeholders but has incomplete information about how they
8 Copyright © 2012 by ASME
tradeoff different objectives. For this problem, the lower level
decisions correspond to a game with incomplete preferences.
Figure 3 – Approach for solving the policy design
problem for multiple policy objectives under incomplete preferences of stakeholders.
As discussed in Section 2.3, games with incomplete
preferences can be analyzed by deriving games with complete
preferences. In Step 2, we derive a set of games with complete
preferences where each stakeholder’s payoff is given by . For
each derived game, it is assumed that the payoff of the
stakeholders is completely known. The resulting bilevel
problem in Step 2 has multiple objectives at the higher level
and a single objective for each stakeholder at the lower level.
The derived bilevel problem is a MOPEC. The formulation for
the FIT problem is shown in Table 4. Table 4 – Derived Problem 1: MOPEC formulation of
the policy design problem with weighted objectives of the stakeholders
Upper-level policy design problem
Objectives:
Minimize
Maximize
Design variables:
,
Constraints:
$/KWh
years
Lower-level stakeholders’ problems
Stakeholder 1 Stakeholder 2
Objective:
Maximize
Design variable:
Constraints:
Objective:
Maximize
Design variable:
Constraints:
Since the incomplete knowledge is assumed to be about
tradeoffs between objectives, we use an Archimedean
combination of the different objectives to define the payoffs,
, of the stakeholders in the games with complete information
n. The payoff functions are listed in equations (2) and (3)
below. and are used to normalize the net
present value and capital investment. A set of games with
complete information are obtained by choosing different values
of the weights ( and ).
(2)
(3)
where, and .
Step 3 of the approach is to derive a set of MPECs from
the MOPEC. This is achieved by taking weighted combinations
of the policy designer’s objectives. The derived problem is
shown in Table 5. The payoff of the policy designer is
calculated as shown in Eq. (4). and are used
to normalize the values of total quantity and policy cost.
Different values of weights ( and , with ) result in different MPEC problems in Step 3.
(4)
Step 4 involves solving the derived MPEC problems using
existing algorithms such as variations of NLP algorithms, and
interior-point algorithms. The solution of the MPEC represents
a single condition with given tradeoff between the policy
designer’s objectives and complete information about
stakeholders’ preferences. Some of these solutions within the
set of all MPECs derived from a MOPEC (Step 2) are Pareto
dominant. These Pareto dominant solutions are identified in
Step 5. The resulting sets of solutions represent solutions of a
MOPEC. The sets of solutions to the MOPECs derived from
the original policy design problem correspond to the
uncertainty in the solution of the original design problem due to
the incomplete preferences. Table 5 – Derived Problem 2: MPEC formulation of the
policy design problem with weighted objectives of the stakeholders
Upper-level policy design problem
Objective:
Maximize
Design variables:
,
Constraints:
$/KWh
years
Lower-level stakeholders’ problems
Stakeholder 1 Stakeholder 2
Objective:
Maximize
Design variable:
Constraints:
Objective:
Maximize
Design variable:
Constraints:
1. Original Design Problem
2. Derived Problem 1: MOPEC
3. Derived Problem 2: MPEC 4. Solution of MPEC
5. Solution of MOPEC
6. Solution of the Original Policy
Design ProblemUpper level:
• Minimize PC• Maximize Qtot
Lower level:
• Maximize NPV• Minimize CI
Upper level:
• Minimize PC• Maximize Qtot
Lower level:
• Maximize (i(NPVi,CIi))
Upper level:
• Maximize (PC,Qtot)
Lower level:
• Maximize (i(NPVi,CIi))
Upper level:
• , T
Lower level:
• q1, q2
Policy designer’s objectives
under incomplete preferences of stakeholders
Derivation of games with
complete preferences
Weighted combination of
upper-level objectives
Identification of Pareto-
dominant solutions
Solution
Upper level:
• Pareto sets of policy designs (, T)
Lower level:
• Sets of design variables (q1, q2)
Uncertainty due to
incomplete preferences
Upper level:
• Pareto sets of policy designs with uncertainty (, T)
Lower level:
• Sets of design variables (q1, q2)
Policy design with
uncertainty resulting from incomplete preferences
9 Copyright © 2012 by ASME
6. RESULTS FOR THE ILLUSTRATIVE PROBLEM In this section, we present the results for the illustrative
FIT policy design problem. As discussed in Section 5.2, the
design problem with incomplete preferences is converted into
multiple MOPECs and each MOPEC is converted into multiple
MPECs. In Section 6.1, we discuss the results of a single
MOPEC where the upper level designer has multiple objectives
and the lower level stakeholders are associated with a single
objective function ( ) which is a linear combination of the two
objectives with pre-defined weights ( ). Section
6.2 contains a discussion of the results from Step 5 in Figure 3.
In Section 6.2, the results of the original design problem, based
on Step 6 are presented. There, the outcomes and the associated
uncertainty resulting from the incomplete preferences are
presented. The values of the parameters used for the results
discussed in this section are shown in Table 6. Table 6 – Values of parameters used in this model
Parameter Value
1.03 x106 KWh
0.69 $
5x10-7 $/KWh
3.5 x106 $
0.1 $/kWh
1.04 x105 $
0.13 $
2.08 x105 $
0.06
For the MPEC shown in Table 5, the Nash equilibrium
condition (i.e., the equilibrium constraint for the upper-level
problem) can be derived using the first-order optimality
conditions for the stakeholders’ decisions. It is assumed that the
payoff functions for both stakeholders are equivalent. The
resulting quantity of generation for designer is:
(5)
where is for , and for .
By solving the equilibrium conditions for the quantities,
the equilibrium constraint is derived as:
(6)
To demonstrate the feasible ranges of equilibrium
quantities ( ), we show the region within which Nash
equilibria can possibly lie for $/KWh and
in Figure 4. The shaded regions represent the
feasible region for the upper-level problem, as constrained by
the equilibrium constraint. Four different levels of uncertainty
of stakeholders’ preferences are shown. Figure 4(a) shows
nearly complete uncertainty with and both within the
range [0.1 1.0]. Figure 4(b) represents a reduced uncertainty
about and where both weights are within the range
[0.2 0.4]. Figure 4(c) shows another scenario of reduced
uncertainty about and . Here, the weights are in the
range [0.7 0.8]. Note that despite the reduction in the width of
the interval representing uncertain payoffs, the size of the
region within which the equilibrium can lie is larger than in
Figure 4(b). The uncertainty in the preferences of the
stakeholders may not be uniform across stakeholders, as shown
in Figure 4(d), where is within [0.7 0.8] and is
within [0.6 0.7]. In this case, the space within which the
equilibrium points lie is not symmetric.
Figure 4 – A comparison of regions where Nash
equilibria can lie for different levels of uncertainty in stakeholder preferences
6.1. Results for the policy design with precise knowledge of stakeholder preferences
Consider a derived MOPEC in Step 2 where the
stakeholders are only interested in maximizing their net present
value ( ). This refers to the condition where and . Since the derived
MOPEC assumes complete knowledge of preferences of
stakeholder, the policy maker can choose the decision variables
which maximize the policy-level objectives. In this case, Eq.
(5) reduces to Eq. (7), and Eq. (6) reduces to Eq. (8). The
resulting quantity for stakeholder is:
(7)
where,
=
This equilibrium condition relates a stakeholder’s decision
( ) to the other stakeholder’s decision ( ), and is also referred
to as the rational reaction set (RRS). This condition results in
0 1 2 3 4 5 6 7 8
x 105
0
1
2
3
4
5
6
7
8x 10
5
q1
q2
0 1 2 3 4 5 6 7 8
x 105
0
1
2
3
4
5
6
7
8x 10
5
q1
q2
0 1 2 3 4 5 6 7 8
x 105
0
1
2
3
4
5
6
7
8x 10
5
q1
q2
0 1 2 3 4 5 6 7 8
x 105
0
1
2
3
4
5
6
7
8x 10
5
q1
q2
(a) (b)
(c) (d)
10 Copyright © 2012 by ASME
two equations in and . On solving the two equations, the
quantity produced by designer is determined to be:
(8)
Using Eq. (8), the equilibrium quantities of production by
different stakeholders can be determined for different values of
Δ and T.
Two different policy maker preferences and the
corresponding rational reaction sets of the stakeholders are
shown in Figure 5. The two policy designers’ preferences are
the two extreme conditions: a) maximization of overall quantity
only without any cost considerations, and b) minimization of
cost without any consideration to the quantity produced. The
intersection of the rational reaction sets is the equilibrium
quantities produced by two stakeholders.
Figure 5 – Rational reaction sets for the two
stakeholders under different preferences of policy maker.
Figure 6 displays the set of equilibria attained for different
tradeoffs between policy design objectives and the
corresponding values of policy design variables and the policy
objectives. If the policy designer is only concerned with the
cost of the policy ( .0, .0) the design variables
will be selected to minimize this cost resulting in a low quantity
of generation (each stakeholder will generate 298,854 KWh).
The optimum values of decision variables in this scenario is Δ =
0.0 $/KWh, T = 0.0 years, which results in low cost and low
generation.
If these preferences are equal ( , ) a
moderate quantity will be generated (each stakeholder will
generate 385,757 KWh). In this case, the optimum values of the
decision variables are Δ = 0.0 $/KWh, T = 21 years resulting in
moderate quantity of generation yet a low cost of the policy.
The policy will exist yet will not pay any premium-price yet
will maximize the duration of the policy. In this case, the
stakeholders will receive payments based on the market value
only. No payment will be received in addition to the market
value.
If the policy maker is only concerned with the quantity of
energy being produced ( , ), the production
quantities will be chosen to maximize this value (in this case
each stakeholder will generate 519,090 KWh). In this case, the
optimum values of the variables are Δ = 0.2 $/KWh, T = 21
years, which results in a high level of generation at a high
policy cost.
6.2. Results for multiobjective policy design with incomplete preferences
In this section, we present the results of the policy design
problem with incomplete preferences. The rational reaction sets
for three preference scenarios for stakeholders are shown in
Figure 7. For each case, the policy maker’s preference is fixed
at and the rational reaction sets are evaluated for the
values of and that maximize the policy maker’s payoff .
The stakeholders’ preferences and corresponding equilibrium
values of quantities are also shown in Figure 7. Since the policy
maker’s preference is chosen to be greater for maximizing the
quantity of generation ( ), Scenario 2 will maximize
this payoff with = 944459 KWh. However, the policy
maker does not have control over the preferences of the
stakeholders and cannot guarantee what his/her payoff will be.
This shows that for a fixed preference at the policy level, there
is significant uncertainty in the equilibrium point if the
knowledge about the preferences at the lower level is
incomplete.
While the impact of incomplete preferences on the
location of the Nash equilibrium is important, the important
question from the policy maker’s standpoint is - what is the best
decision for the policy maker to make? Although the
preferences may not be completely known at the stakeholder
level, the policy maker still has control over the decision
variables. In Figure 8, we show the best decisions from the
policy maker’s standpoint, and their impacts, under different
preference scenarios. In Figure 8(a) it is assumed that both
and are uncertain but are known to be within the range
[0.7 0.8]. For this uncertainty in stakeholder preferences, four
preference conditions for the policy decision ( ) are considered. It was also found that when
= 0.25, 0.39 or 0.5, the plots are identical. In Figure 8(b), it
is assumed that uncertainty in and is lower and both
weights are within the range [0.79 0.8]. Although the regions
where the equilibria may lie are still large, the corresponding
policy design variables have either no unceratinty or
significantly smaller uncertainty. For example, in Figure 8(a)
the region corresponding to has a large range of
quantity values. Although this region is large, the policy
maker’s best decision is to maximize both the premium-price
and the duration of the policy. Therefore, despite the
uncertainty in stakeholder preferences there is no uncertainty in
design of the policy.
In Figure 8(c), the uncertainty in and is
assumed to be between [0.4 0.5]. Comparing this with Figure
8(a), it is observed that despite the same amount of uncertainty
in the preferences, the uncertainty in the equilibrium values is
significantly greater. In this case, the uncertainty in the choice
of decision variables is also greater. This highlights the effects
of the values of stakeholder preferences, in addition to the
extent of uncertainty.
0 1 2 3 4 5 6 7 8 9 10
x 105
0
1
2
3
4
5
6
7
8
9
10x 10
5
q1
q2
Quantity Only
w11 = 1.0
w12 = 0.0
Cost Only
w11 = 0.0
w12 = 1.0
11 Copyright © 2012 by ASME
The results indicate that although there may be high
uncertainty in the quantities generated at market equilibria, the
policy maker may only have small uncertainty in the decision
variables. This is primarily due to the decrease in uncertainty
when the equilibria are mapped into the design space. We
envision that in some policy design problems, the uncertainty
may also increase when mapping the equilibria to design space.
The results are sensitive to the market parameters ( ). By
changing these parameters slightly we observed significant
changes in the equilibrium quantities. Therefore, in a real FIT
policy design, the market parameters need to be calibrated for
the specific market under consideration. The results are more
sensitive to the premium price and less sensitive to the duration
of the policy. It was found that the premium-price of the policy
motivated stakeholders to generate higher amounts of
electricity. Although changing the duration of the policy had an
effect, it was found that in most cases the duration of the policy
was maximized. The complexity of the problem increases with
the consideration of more stakeholders. Since we considered
only two stakeholders, it was relatively easy to determine
closed form solutions of equilibrium quantities. In order to
consider multiple decision makers, techniques such as agent-
based modeling may be employed.
7. CLOSING REMARKS In this paper, we present a computational approach for
policy design problems in sustainable energy systems such as
the decentralized energy infrastructure. The specific focus is on
policy design problems with multiple objectives and incomplete
knowledge of preferences of the stakeholders. The lack of
knowledge of the lower-level decision maker’s payoffs is one
of the reasons for the failure of policies. Assumptions can be
made as to what the objectives are and how these objectives are
weighted but they may not be accurate. This inaccuracy can
have detrimental effects creating an expensive non-successful
FIT policy. This is typically the case as to why FIT policies fail
[63]. Hence, there is a need to develop approaches to account
for uncertainty resulting from the lack of complete information
about stakeholder’s payoffs. In this paper, a specific class of
incomplete preferences is addressed. It is assumed that the
policy maker has knowledge about the different objectives of
the stakeholders but has incomplete knowledge about how the
stakeholders tradeoff different objectives.
Figure 6 – The set of equilibria attained for different tradeoffs between policy-design objectives. If the policy maker is
only concerned with the cost of the policy this will result in low quantities of energy produced by both stakeholders. However, if the policy maker is more concerned with the quantity of generation the policy maker is willing to spend more on
the policy resulting in an increased quantity of generation from both stakeholders.
Figure 7 – Rational reaction sets of three cases with corresponding equilibria. The preferences at the stakeholder level
have significant influence on the equilibrium points. The incompleteness of knowledge about stakeholders’ preferences results in significant uncertainty in the quantity of generation.
2.5 3 3.5 4 4.5 5 5.5
x 105
2.5
3
3.5
4
4.5
5
5.5x 10
5
q1
q2
5 6 7 8 9 10 11
x 105
0
0.5
1
1.5
2
2.5x 10
5
Quantity (Qtot
)
Polic
y C
ost
(PC
)
0 5 10 15 20 250
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Time (T)
Delta (
)
0 2 4 6 8
x 105
0
1
2
3
4
5
6
7
8x 10
5
q1
q2
w211 = 0.5
w212 = 0.7
w211 = 0.9
w212 = 0.8
w211 = 0.7
w212 = 0.6
6.5 7 7.5 8 8.5 9 9.5
x 105
1.3
1.4
1.5
1.6
1.7
1.8
1.9x 10
5
Quantity (Qtot
)
Polic
y C
ost
(PC
)
Qtot
= 667404 KWh
PC = $133480
Qtot
= 944456 KWh
PC = $188891
Qtot
= 753918 KWh
PC = $150783
0 5 10 15 200
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Time (T)
Delta (
)
T = 21 years
= 0.2 $/KWh
T = 21 years
= 0.2 $/KWh
T = 21 years
= 0.2 $/KWh
12 Copyright © 2012 by ASME
The approach presented in this paper is based on multi-
objective mathematical programs with equilibrium constraints
(MOPECs), games with vector payoffs, and Nash equilibra of
games with incomplete preferences. The primary contributions
in this paper are mathematical formulation of the FIT policy,
the extension of computational policy design problems to
multiple objectives, and the consideration of incomplete
preferences of stakeholders for policy design problems. The
consideration of incomplete preferences is important for
research on design under uncertainty because the existing
design literature is primarily focused on uncertainty about
physical phenomena but incomplete knowledge of preferences
have received relatively lower attention.
While the motivation in this paper is that the policy
designer has incomplete knowledge about the stakeholders’
payoffs, the approach can be used in two other situations also:
a) the stakeholders may themselves not know what their
preferences for tradeoffs are, b) the preferences of the
stakeholders may represent group preferences. The second
situation is common in many policy decisions because the
problem is not only bilevel in nature – it is indeed multilevel in
nature, as shown in the Figure 1. The proposed approach has
a)
b)
c) Figure 8 – A mapping between the equilibrium points at the stakeholder level, the policy outputs, and the decision
variables. Although the uncertainty of the equilibria may be high, the choice of the decision variables may be low.
0 1 2 3 4 5 6
x 105
0
1
2
3
4
5
6x 10
5
q1
q2
0 2 4 6 8 10
x 105
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
5
Quantity (Qtot
)
Polic
y C
ost
(PC
)
0 5 10 15 20 250
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Time (T)
Delta (
)
w11 = 0.0
w11 = 0.25
w11 = 0.75
w11 = 1.0
0 1 2 3 4 5
x 105
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 10
5
q1
q2
0 2 4 6 8 10
x 105
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
5
Quantity (Qtot
)
Polic
y C
ost
(PC
)
0 5 10 15 20 250
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Time (T)
Delta (
)
w11 = 0.0
w11 = 0.25
w11 = 0.75
w11 = 1.0
0 1 2 3 4
x 105
0
0.5
1
1.5
2
2.5
3
3.5
4x 10
5
q1
q2
0 1 2 3 4 5 6
x 105
0
2
4
6
8
10
12x 10
4
Quantity (Qtot
)
Polic
y C
ost
(PC
)
0 5 10 15 20 250
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Time (T)
Delta (
)
w11 = 0.0
w11 = 0.25
w11 = 0.75
w11 = 1.0
13 Copyright © 2012 by ASME
applications to other bilevel problems within engineering
design research such as design for market systems [64, 65], fuel
efficiency and emission policy [66, 67], and plug-in hybrid
charging patterns [68]. All these problems are generally multi-
objective in nature and require the knowledge of preferences of
the stakeholders whose interactions result in market equilibria.
The proposed approach has limitations due to the
assumptions made in this paper. First, it is assumed that the
market behavior can be defined in terms of the Nash
equilibrium. This is a common assumption made in the energy
market modeling literature. However, in reality, the market is a
dynamic system. Additionally, the decisions are not generally
made by all stakeholders at the same time. The decisions may
be made sequentially. Second, the approach presented in this
paper is based on the assumption that the lower-level decisions
can be converted into equilibrium constraints in the closed
form. However, as the decisions of the stakeholders become
more complex, deriving the equilibrium constraints in closed
form may not be feasible. Finally, we do not consider stability
of equilibria in this paper. The market equilibria for the
problem presented in this paper happen to be stable for the
ranges of decision variables considered. In a general case, the
stability of the equilibria may change by changing the policy
design variables. Price stability is an important aspect for the
engineering design of distributed energy systems within smart
electric grid. One of the goals of the policy design problem is to
ensure the stability of the equilibrium. Integrating the stability
considerations in the policy design problem is a challenge
especially given the different possible stability problems such
as price stability and voltage stability.
The illustrative example presented in this paper is also
highly simplified. In the example, we consider only two energy
producers whose quantity of generation is determined by the
equilibrium. However, in practice, these energy producers are
also required to meet local energy demands. The demand
fluctuates with time, and the local producers can also purchase
energy from central generation stations. The example presented
in this paper is not based on specific RE technologies. One of
the characteristics of these RE technologies is that their output
is uncertain. In a holistic policy design framework, it is
important to account for this uncertainty. The example is also
based on the assumption that both stakeholders enter the market
and make a decision at the same time. However, in practice,
different stakeholders may enter the market at different times.
Hence, the decisions are made at different time-steps with
different amount of available information. These limitations
clearly indicate the significant potential for further research in
this direction.
8. ACKNOWLEDGEMENTS The authors gratefully acknowledge the financial support from
the National Science Foundation through the CAREER grant #
0954447.
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