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APPLICATION OF A NOVEL OPTIMIZATION
TECHNIQUE TO PRODUCE MAXIMALLY DIFFERENT
ENERGY FUTURES
Joseph F. DeCarolis, Assistant Professor
Department of Civil, Construction, and Environmental Engineering
North Carolina State University
[email protected]
Motivation
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
Aggressive climate policy will bring about fundamental
changes in the way energy is produced and consumed

Energy-related decisions with long-lived consequences
must be made today with the best possible
information

Energy-focused optimization models have emerged as
an important tool to explore different energy futures
using a structured and self-consistent set of
assumptions
The Challenge of Uncertainty
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Energy and integrated assessment models are used to
determine what could or should happen in the future
Addressing large future uncertainties a critical
challenge for energy modelers
Must address 2 types of uncertainty:
 Structural: imperfect and incomplete set of
equations describing the system being modeled
 Parametric: imperfect knowledge of model inputs
The Conventional Approach
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To deal with structural uncertainty, build more complex
models that account for additional processes or effects
→ Add additional objectives, constraints, or processes
to address unmodeled issues
→ Increasing complexity then makes parametric
sensitivity analysis more difficult
→ Run a few detailed scenarios
 Many large models contribute relatively little insight
about alternative ways to structure and solve the
problem at hand (Morgan and Henrion, 1990)
Limitations of Energy Models
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Accurate predictions by energy models extending over several decades
would require both accurate model structure and precise specification
of inputs
Large and irreducible uncertainties preclude this possibility
Poor performance of past predictions provide validation
Better approach would systematically flex models in order to stretch our
thinking, challenge preconceptions, and suggest creative solutions
Rethinking the Role of Optimization Models
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Insights from Brill (1979) are remarkably prescient with regard to
energy modeling today
Models are always a simplification of reality, particularly in complex
planning problems
Rather than burden models with additional objectives and
complexity in an effort to obtain “the answer”, generate nearoptimal alternatives that facilitate comparison
Approach recognizes that model’s optimal solution is likely to be
inaccurate due to structural uncertainty
How Optimal is the “Optimal” Solution?
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Objective 2
Non-inferior frontier
Consider an optimization
model that only includes
Objective 1 and leaves
Objective 2 unmodeled.
The true optimum is
within the feasible,
suboptimal region of the
model’s solution space.
Viable alterative solutions
exist within the model’s
feasible region.
Objective 1
Example adopted from Brill et al. (1990).
Modeling to Generate Alternatives
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Need a method to explore an optimization model’s
feasible region → “Modeling to Generate Alternatives”†
MGA generates alternative solutions that are maximally
different in decision space but perform well with
respect to modeled objectives
The resultant MGA solutions provide modelers and
decision-makers with a set of alternatives for further
evaluation
†Brill
(1979), Brill et al. (1982), Brill et al. (1990)
Hop-Skip-Jump (HSJ) MGA
Brill et al. (1982)
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Steps:
1.
Obtain an initial optimal solution by any method
2.
Add a user-specified amount of slack to the value of the
objective function
3.
Encode the adjusted objection function value as an
additional upper bound constraint
4.
Formulate a new objective function that minimizes the
decision variables that appeared in the previous solutions
5.
Iterate the re-formulated optimization
6.
Terminate the MGA procedure when no significant
changes to decision variables are observed in the
solutions
HSJ MGA
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Mathematical formulation
min p   x k
kK
s.t. f j (x)  Tj j
xX
where:
K represents the set of indices of decision
variables with nonzero values in the
previous solutions

fj x
is the jth objective function
Tj is the target specified for the jth modeled
objective
X is the set of feasible solution vectors
Interpretation of MGA Solutions
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MGA solutions can be interpreted as equally plausible
alternatives to the model’s optimal solution given that
structural uncertainty exists
Question: Is there a way to reproduce the MGA
solution using the original model formulation?
Affirmative answer suggests a linkage between MGA
and parametric sensitivity analysis of the original
model.
A Simple MGA Example
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Original Formulation
x2
Minimize : c1 x1  c2 x2 ; c1  c2
1
Subject to : x1  x2  1
x 1 , x2  0
slack
First MGA Iteration
Minimize : x 1
1
x1
Subject to : x1  x2  1
c1x 1  c2 x2  c1  slack
x 1 , x2  0
Application to a Simple Electric Sector Model
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What effect might a cap-and-trade proposal have on the electric
sector?
Cap-and-trade (H.R. 2454) pending in Congress; 83 percent
reduction in CO2e emissions by 2050
Suppose this cap applied only to the electric sector without offsets
Assume new capacity must be installed to replace all existing
fossil-based plants and meet growing demand
The Electric Sector Challenge
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Objective is to deploy
new generating
capacity in the form
of wedges, using the
approach outlined in
Pacala and Socolow
(2004)
Model optimizes the
number and size of
wedges
The projection of business-as-usual electric sector electricity generation (TWh) and CO2 emissions is based on a linear
extrapolation of CO2 emissions projected in the Annual Energy Outlook 2009 reference case (EIA, 2009a).
Model Formulation
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Minimize: C 
n
c x
i
i 1
n
Subject to:
i
a x
 Pav
a x
 Pav baseload
a x
 Pav nonbaseload
i 1
bB
pP
i i
b b
p p
e x
f F
f
f
 E2050
Where:
• xi is the technology-specific
installed capacity
• B is the set of baseload
technologies
• N is the set of non-baseload
technologies
• F is the set of technologies
emitting CO2 emissions
• ai is the technology-specific
capacity factor
• P is total average power
production
Technology Costs
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The cost coefficients in the objective function represent the
2050 annual cost associated with each technology:
r
ci ($/kWyr)  capital cost
 fixed O & M
T
1  (1  r )


Fuel Cost
1 GJ 
 8760  capacity factor   variable O & M 


Efficiency 278 kWh 

13 different energy technologies included in the model
Parameters in brackets are drawn directly from the U.S. EIA’s Assumption to the
Annual Energy Outlook 2009
Assumed lifetime (T) is 40 years, discount rate (r) is 10% for all technologies
Wedges Under the CO2 Constraint
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Original Solution
1st MGA Solution
Slack set to 25% of the minimum cost in MGA iteration
MGA Iterations in Carbon Constrained Case
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2050 results
Slack = 25%
Upper bound
constraint on cost
binding in all
MGA iterations
Other MGA Objectives
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Identifying Robust Options
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Installed capacity across all BAU and CO2-constrained runs
Conclusions
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Large future uncertainties preclude accurate predictions
over several decades with energy optimization models
Such models are most useful when used to stimulate
creative thought about possible solutions
MGA provides a way to explore the feasible region to
generate solutions that are maximally different in
decision space, but perform well with respect to
modeled objectives
Conclusions (continued)
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Simple electric sector model developed to illustrate
MGA utility; application to a more sophisticated
MARKAL model underway
Modeling is an art; when to increase model complexity
and when to rely on MGA is a subjective judgment by
the modeler.
Thanks for your time!
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I’d like to acknowledge Downey Brill and Ranji
Ranjithan for very useful discussions regarding the
use of MGA techniques