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

3 Copyright © 2012 by ASME

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.,

4 Copyright © 2012 by ASME

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.

6 Copyright © 2012 by ASME

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.

9. REFERENCES [1] Nash, J., 1951, "Non-Cooperative Games," The Annals

of Mathematics, 54(2), pp. 286-295.

[2] Marston, M. and Mistree, F., 2000, "Game-Based

Design: A Game Theoretic Extension to Decision-

Based Design," in ASME DETC, Design Theory and

Methodology Conference, Baltimore, MD. Paper

Number: DETC2000/DTM-14578.

[3] Chanron, V. and Lewis, K., 2006, "A Study of

Convergence in Decentralized Design Processes,"

Research in Engineering Design, 16, pp. 133-145.

[4] Ferris, M. C. and Pang, J. S., 1997, "Engineering and

Economic Applications of Complementarity

Problems," SIAM Review, 39(4), pp. 669-713.

[5] Facchinei, F. and Pang, J.-S., 2003, Finite-

Dimensional Variational Inequalities and

Complementarity Problems I, New York: Springer.

[6] Billups, S. C. and Murty, K. G., 2000,

"Complementarity Problems," Journal of

Computational and Applied Mathematics, 124(1-2),

pp. 303-318.

[7] Nabetani, K., Tseng, P., and Fukushima, M., 2011,

"Parametrized Variational Inequality Approaches to

Generalized Nash Equilibrium Problems with Shared

Constraints," Computational Optimization and

Applications, 48(3), pp. 423-452.

[8] Harker, P. T. and Pang, J.-S., 1990, "Finite-

Dimensional Variational Inequality and Nonlinear

Complementarity Problems: A Survey of Theory,

Algorithms and Applications," Mathematical

Programming, 48(1-3), pp. 161-220.

[9] Hobbs, B. F., 2001, "Linear Complementarity Models

of Nash-Cournot Competition in Bilateral and Poolco

Power Markets," IEEE Transactions on Power

Systems, 16(2), pp. 194 - 202.

[10] Gabriel, S. A., Kiet, S., and Zhuang, J., 2005, "A

Mixed Complementarity-Based Equilibrium Model of

Natural Gas Markets," Operations Research, 53(5),

pp. 799-818.

[11] Gabriel, S. A., Zhuang, J., and Kiet, S., 2005, "A

Large-Scale Linear Complementarity Model of the

North American Natural Gas Market," Energy

Economics, 27(4), pp. 639-665.

[12] Gabriel, S. A., Zhuang, J., and Egging, R. G., 2009,

"Solving Stochastic Complementarity Problems in

Energy Market Modeling Using Scenario Reduction,"

European Journal of Operational Research, 197(3),

pp. 1028-1040.

[13] Zhuang, J. and Gabriel, S. A., 2008, "A

Complementarity Model for Solving Stochastic

Natural Gas Market Equilibria," Energy Economics,

30(1), pp. 113-147.

[14] Colson, B., Marcotte, P., and Savard, G., 2005,

"Bilevel Programming: A Survey," 4OR: A Quarterly

Journal of Operations Research, 3(2), pp. 87-107.

[15] Luo, Z.-Q., Pang, J.-S., and Ralph, D., 1996,

Mathematical Programs with Equilibrium Constraints,

Cambridge: Cambridge University Press.

[16] Ye, J. J., 2005, "Necessary and Sufficient Optimality

Conditions for Mathematical Programs with

Equilibrium Constraints," Journal of Mathematical

Analysis and Applications, 307(1), pp. 350-369.

[17] Ye, J. J., 1999, "Optimality Conditions for

Optimization Problems with Complementarity

14 Copyright © 2012 by ASME

Constraints," SIAM Journal on Optimization, 9(2), pp.

374-387.

[18] Pieper, H., 2001, "Algorithms for Mathematical

Programs with Equilibrium Constraints with

Applications to Deregulated Energy Markets,"

Department of Management Science and Engineering,

Stanford University, Stanford, California.

[19] Benson, H. Y., Sen, A., Shanno, D. F., and Vanderbei,

R. J., 2006, "Interior-Point Algorithms, Penalty

Methods and Equilibrium Problems," Computational

Optimization and Applications, 34(2), pp. 155-182.

[20] Fletcher, R. and Leyer, S., 2004, "Solving

Mathematical Programs with Complementarity

Constraints as Nonlinear Programs," Optimization

Methods Software, 19(1), pp. 15-40.

[21] Dirkse, S. P. and Ferris, M. C., 1999, "Modeling and

Solution Environments for Mpec: Gams and Matlab,"

in Reformulation: Nonsmooth, Piecewise Smooth,

Semismooth and Smoothing Methods, (M. Fukushima

and L. Qi, Editors), Kluwer Academic Publishers:

Dordrecht, The Netherlands, pp. 127-148.

[22] Hobbs, B. F., Metzler, C. B., and Pang, J.-S., 2000,

"Strategic Gaming Analysis for Electric Power

Systems: An Mpec Approach " IEEE Transactions on

Power Systems, 15(2), pp. 638 - 645.

[23] Gabriel, S. A. and Leuthold, F. U., 2010, "Solving

Discretely-Constrained Mpec Problems with

Applications in Electric Power Markets," Energy

Economics, 32(1), pp. 3-14.

[24] Labbé, M., Marcotte, P., and Savard, G., 1998, "A

Bilevel Model of Taxation and Its Application to

Optimal Highway Pricing," Management Science,

44(12), pp. 1608-1622.

[25] Scaparra, M. P. and Church, R. L., 2008, "A Bilevel

Mixed-Integer Program for Critical Infrastructure

Protection Planning," Comput. Oper. Res., 35(6), pp.

1905-1923.

[26] Mordukhovich, B. S., 2009, "Multiobjective

Optimization Problems with Equilibrium Constraints,"

Mathematical Programming, 117(1-2), pp. 331-354.

[27] Ye, J. J. and Zhu, Q. J., 2003, "Multiobjective

Optimization Problem with Variational Inequality

Constraints," Mathematical Programming, 96(1), pp.

139-160.

[28] Ye, J. J., 2011, "Necessary Optimality Conditions for

Multiobjective Bilevel Programs," Mathematics of

Operations Research, 36(1), pp. 165-184.

[29] Bao, T., Gupta, P., and Mordukhovich, B., 2007,

"Necessary Conditions in Multiobjective Optimization

with Equilibrium Constraints," Journal of

Optimization Theory and Applications, 135(2), pp.

179-203.

[30] Von Neumann, J. and Morgenstern, O., 1947, The

Theory of Games and Economic Behavior, Princeton,

NJ: Princeton University Press.

[31] Aumann, R. J., 1962, "Utility Theory without the

Completeness Axiom," Econometrica, 30(3), pp. 445-

462.

[32] Ok, E. A., 2002, "Utility Representation of an

Incomplete Preference Relation," Journal of Economic

Theory, 104(2), pp. 429-449.

[33] Rabin, M., 1998, "Psychology and Economics,"

Journal of Economic Literature, 36(1), pp. 11-46.

[34] Dubra, J., Maccheroni, F., and Ok, E. A., 2004,

"Expected Utility Theory without the Completeness

Axiom," Journal of Economic Theory, 115(1), pp.

118-133.

[35] Ok, E. A., Ortoleva, P., and Riella, G., in press,

"Incomplete Preferences under Uncertainty:

Indecisiveness in Beliefs Vs. Tastes," Econometrica.

[36] Nau, R., 2006, "The Shape of Incomplete

Preferences," Annals of Statistics, 34(5), pp. 2430-

2448.

[37] Mandler, M., 2005, "Incomplete Preferences and

Rational Intransitivity of Choice," Games and

Economic Behavior, 50(2), pp. 255-277.

[38] Bade, S., 2005, "Nash Equilibrium in Games with

Incomplete Preferences," Economic Theory, 26(2), pp.

309-332.

[39] Somasundaram, K. and Baras, J., 2008, "Pareto Nash

Replies for Multi-Objective Games," Institute for

Systems Research, University of Maryland, Report

Number: TR_2009-4.

[40] Zhao, J., 1991, "The Equilibria of a Multiple Objective

Game," International Journal of Game Theory, 20, pp.

171-182.

[41] Borm, P., Vermeulen, D., and Voorneveld, M., 2003,

"The Structure of the Set of Equilibria for Two Person

Multicriteria Games," European Journal of

Operational Research, 148(3), pp. 480-493.

[42] Borm, P. E. M., Tijs, S. H., and Aarssen, J. C. M. v. d.,

1988, "Pareto Equilibria in Multiobjective Games,"

Methods of Operations Research 60, pp. 303-312.

[43] Krieger, T., 2003, "On Pareto Equilibria in Vector-

Valued Extensive Form Games " Mathematical

Methods of Operations Research, 58(3), pp. 449-458.

[44] Chicco, G. and Mancarella, P., 2009, "Distributed

Multi-Generation: A Comprehensive View,"

Renewable and Sustainable Energy Reviews, 13(3),

pp. 535-551.

[45] Couture, T. D., Cory, K., Kreycik, C., and Williams,

E., 2010, "A Policymaker’s Guide to Feed-in Tariff

Policy Design," National Renewable Energy

Laboratory, Report Number: NREL/TP-6A2-44849.

[46] Larsen, C. and Rogers, H., National Summary Report

on State Financial and Regulatory Incentives for

Renewable Energy, Report by Interstate Renewable

Energy Council, North Carolina State University,

2000; Web Link:

http://www.dsireusa.org/documents/PolicyPublications

/final_report_all.pdf.

[47] Smith, D. W., Nordhaus, R. R., Roberts, T. C., Fidler,

S., Anderson, J., Danish, K., Agnew, R., and Chupka,

M., Designing a Climate Friendly Energy Policy:

Options for the near Term, 2002; Web Link:

http://www.pewclimate.org/docUploads/energy_policy

.pdf.

15 Copyright © 2012 by ASME

[48] Couture, T. and Cory, K., 2009, "State Clean Energy

Policies Analysis (Scepa) Project: An Analysis of

Renewable Energy Feed-in Tariffs in the United

States," National Renewable Energy Laboratory,

Report Number: NREL/TP-6A2-45551.

[49] Myerson, R. B., 1991, Game Theory: Analysis of

Conflict, Cambridge, MA: Harvard University Press.

[50] von Stackelberg, H., 2011, Market Structure and

Equilibrium (Translated from German Language),

New York: Springer.

[51] Fudenberg, D. and Tirole, J., 1993, Game Theory,

Cambridge, MA: MIT Press.

[52] Dempe, S., 2002, Foundations of Bilevel

Programming, Dordrecht, The Netherlands: Kluwer

Academic Publishers.

[53] Patriksson, M. and Rockafellar, R. T., 2002, "A

Mathematical Model and Descent Algorithm for

Bilevel Traffic Management," Transportation Science,

36(3), pp. 271–291.

[54] Light, M. K., 1999, "Optimal Taxation and

Transboundary Pollution (Working Paper),"

Department of Economics, University of Colorado,

Report Number: 99-22.

[55] Chen, Y. and Florian, M., 1994, "O-D Demand

Adjustment Problem with Congestion: Part I. Model

Analysis and Optimality Conditions," Centre de

Recherche sur les Transports, Université de Montréal,

Report Number: CRT-94-56.

[56] Bertsekas, D. P., 1999, Nonlinear Programming:

Second Edition, UK: Athena Scientific.

[57] Luo, Z.-Q., Pang, J.-S., and Ralph, D., 1998,

"Piecewise Sequential Quadratic Programming for

Mathematical Programs with Nonlinear

Complementarity Constraints," in Multilevel

Optimization: Algorithms and Applications, Kluvwer

Academic Publisher: Dordrecht, pp. 209-229.

[58] Outrata, J., Kocvara, M., and Zowe, J., 1998,

Nonsmooth Approach to Optimization Problems with

Equilibrium (Nonconvex Optimization and Its

Applications), Dordrecht: Kluwer Academic

Publishers.

[59] Bao, T. Q., Gupta, P., and Mordukhovich, B. S., 2007,

"Necessary Conditions in Multiobjective Optimization

with Equilibrium Constraints," Journal of

Optimization Theory and Applications, 135(2), pp.

179-203.

[60] entosa, M., a llo, Á., Ramos, A., and Rivier, M.,

2005, "Electricity Market Modeling Trends," Energy

Policy, 33(7), pp. 897-913.

[61] Hobbs, B., Rothkopf, M., O'Neil, R., and Chao, H.,

2001, The Next Generation of Unit Commitment

Models, Boston: Kluwer Academic Publishers.

[62] Aliprantis, C. D. and Chakrabarti, S. K., 1999, Games

and Decision Making, New York: Oxford University

Press.

[63] Grace, R., Rickerson, W., and Corfee, K., 2008,

"California Feed-in Tariff Design and Policy Options,"

California Energy Commission, Report Number: CEC-

300-2008-009D.

[64] Shiau, C.-S. N. and Michalek, J. J., 2009, "Should

Designers Worry About Market Systems?," Journal of

Mechancial Design, 131, pp. 011011(1-9).

[65] Shiau, C.-S. N. and Michalek, J. J., 2009, "Optimal

Product Design under Price Competition," Journal of

Mechanical Design, 131(7), pp. 071003-10.

[66] Michalek, J. J., Papalambros, P. Y., and Skerlos, S. J.,

2004, "A Study of Fuel Efficiency and Emission

Policy Impact on Optimal Vehicle Design Decisions,"

Journal of Mechancial Design, 236, pp. 1062-1070.

[67] Shiau, C.-S. N., Michalek, J. J., and Hendrickson, C.

T., 2009, "A Structural Analysis of Vehicle Design

Responses to Corporate Average Fuel Economy

Policy," Transportation Research Part A, 43, pp. 814-

828.

[68] Shiau, C.-S. N., Samaras, C., Hauffe, R., and

Michalek, J. J., 2009, "Impact of Battery Weight and

Charging Patterns on the Economic and

Environmental Benefits of Plug-in Hybrid Vehicles,"

Energy Policy, 37, pp. 2653-2663.