Thesis-MitchellColgan_LongTerm_PowerSystem_Planning

Modeling Considerations for the Long-Term Generation and
Transmission Expansion Power System Planning Problem
Elliott J. Mitchell-Colgan
Dissertation submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Master of Science
in
Electrical Engineering
Virgilio A. Centeno, Chair
Jaime De La Ree Lopez
James S. Thorp
December 4th, 2015
Blacksburg, Virginia
Keywords: Power System Planning, Optimization, Load Uncertainty

Modeling Considerations for the Long Term Generation and Transmission
Expansion Power System Planning Problem
Elliott J. Mitchell-Colgan
(ABSTRACT)
Judicious Power System Planning ensures the adequacy of infrastructure to support continu-
ous reliability and economy of power system operations. Planning processes have a long and
rather successful history in the United States, but the recent influx of unpredictable, non-
dispatchable generation such as Wind Energy Conversion Systems (WECS) necessitates the
re-evaluation of the merit of planning methodologies in the changing power system context.
Traditionally, planning has followed a logical progression through generation, transmission,
reactive power, and finally auxiliary system planning using expertise and ranking schemes.
However, it is challenging to incorporate all of the inherent dependencies between expansion
candidates’ system impacts using these schemes. Simulation based optimization provides a
systematic way to explore acceptable expansion plans and choose one or several ”best” plans
while considering those complex dependencies.
Using optimization to solve the minimum-cost, reliability-constrained Generation and Trans-
mission Expansion Problem (GTEP) is not a new concept, but the technology is not mature.
This work inspects: load uncertainty modeling; sequential (GEP then TEP) versus unified
(GTEP) models; and analyzes the impact on the methodologies achieved near-optimal plan.
A sensitivity simulation on the original system and final, upgraded system is performed.

Acknowledgments
The presented work benefited from the work of the Chetan Mishra, who programmed in
MATLAB the BPSO solver used in the outer optimization, the National Renewable Energy
Labs (NREL) who made publicly available the Eastern Interconnection Wind Dataset; the
wonderful OPF solvers of MATPOWER of Power Systems Engineering Research Center
(PSERC); the North American Electric Reliability Corporation and Transmission Owners
who made publicly available the Transmission Availablility Data System; and the IEEE and
members who made publicly available the IEEE 14 bus and Roy Billington Reliability Test
Systems and data.
I would also like to thank Dr. Virgilio Centeno, Dr. James Thorp, Dr. Jaime De La Ree,
Dr. Douglas Bish, and other professors of Virginia Tech with whom I’ve had the pleasure of
chatting. It is interesting and often even inspiring to hear their questions and comments in
meetings.
If I have seen further, it is by standing on the shoulders of giants.
iii

Contents
Chapter 1: Introduction 1
1.1 Introduction to Power System Planning . . . . . . . . . . . . . . . . . . . . . 1
1.2 Introduction to Wind Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Motivation and Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Organization of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Chapter 2: Background 9
2.0.1 Optimization and Power Systems . . . . . . . . . . . . . . . . . . . . 9
2.1 Power System Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Load Forecasting and Uncertainty . . . . . . . . . . . . . . . . . . . . 16
2.1.2 Generation Expansion Planning . . . . . . . . . . . . . . . . . . . . . 17
2.1.3 Transmission Expansion Planning . . . . . . . . . . . . . . . . . . . 20
2.2 Evaluating Reliability in Power Systems . . . . . . . . . . . . . . . . . . . . 22
2.3 Evaluating System Cost in Power Systems . . . . . . . . . . . . . . . . . . . 26
2.4 Operational Challenges with Wind Power . . . . . . . . . . . . . . . . . . . 27
2.5 State of the Art GTEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
iv

Chapter 3: Methodology 30
3.1 Optimization Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.1 Outer Optimization: Search for Candidate Upgrades . . . . . . . . . 31
3.1.2 Inner Optimization Layer: Evaluating Cost and Reliability . . . . . . 32
3.1.3 Capturing Load Uncertainty in the Optimization . . . . . . . . . . . 36
3.2 System Reliability Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.1 Reliability Models for Key Power System Components . . . . . . . . 39
3.3 Input Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.1 Cost Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.2 Cost and Impedance of Transmission Upgrades . . . . . . . . . . . . 46
3.3.3 System Load and Uncertainty . . . . . . . . . . . . . . . . . . . . . . 47
3.3.4 Pre-selection of Candidate Upgrades . . . . . . . . . . . . . . . . . . 50
3.4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4.1 Near-Optimal Expansion Plan . . . . . . . . . . . . . . . . . . . . . . 50
3.4.2 Comparison of Unified GTEP and Sequential GEP and TEP . . . . . 51
3.4.3 Sensitivity Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4.4 Load Uncertainty Simulation . . . . . . . . . . . . . . . . . . . . . . 53
Chapter 4: Results 55
4.1 Benchmarking Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Near-Optimal Expansion Plan Results . . . . . . . . . . . . . . . . . . . . . 57
4.3 Sequential GEP and TEP Results . . . . . . . . . . . . . . . . . . . . . . . . 60
v

4.4 Sensitivity Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4.1 Sensitivity About the Original System . . . . . . . . . . . . . . . . . 63
4.4.2 Sensitivity about the Upgraded System . . . . . . . . . . . . . . . . . 71
4.5 Load Uncertainty Simulation Results . . . . . . . . . . . . . . . . . . . . . . 75
4.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Bibliography 82
Appendix A: Input Data 91
5.1 Wind Farm Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2 Transmission Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Appendix B: Results 95
vi

List of Figures
1.1 A simplified, typical power system. Image from All Time Electrical [1] . . . 2
1.2 Levelized cost of energy by fuel in the United States in 2013. Image from
AWEA [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Installed wind capacity and cost over time in the United States. Image from
AWEA [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 A single variable nonlinear feasible region whose optimization presents a chal-
lenge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Purely Demonstrative algorithm for solving the DCOPF . . . . . . . . . . . 13
2.6 Capacity Outage Table for adequacy index calculation [3] . . . . . . . . . . . 24
2.7 Basic two-state model for Frequency and Duration Method reliability analysis.
[4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.8 Capacity and load history for the state duration method. Energy Not Served
is highlighted in black [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.9 The hierarchical levels of power system planning. . . . . . . . . . . . . . . . 26
3.10 An overview of the presented algorithm showing the outer and inner opti-
mization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.11 All Possible Generating States according to IEEE Std 762™-2006. . . . . . . 40
vii

3.12 Two-State base load reliability model. . . . . . . . . . . . . . . . . . . . . . . 41
3.13 The PJM load zones, 14 of which were used to generate the load uncertainty
set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.14 Convergence of the Monte Carlo reliability simulation using the random data.
The convergence criterion were met at t = 2,000,000 hours. LOLE values are
discarded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.15 The IEEE 14 bus system upgraded with the results of the GTEP. Arrows
depict reconductored lines. Green line depict new lines. No wind farms were
selected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.16 The selected candidate lines’ adjacent buses and binary number representing
whether the transmission corridor is new (1) or reconductored (0). PSO with
50 iterations and 10 particles was run. . . . . . . . . . . . . . . . . . . . . . 59
4.17 The IEEE 14 bus system upgraded with the results of the GTEP during the
sequential vs. unified GTEP experiment. Arrows depict reconductored lines.
Green line depict new lines. Wind farms are shown as green generators with
numbers depicting the number of turbines installed. . . . . . . . . . . . . . . 61
4.18 The selected wind farms for the GEP and unified GTEP. . . . . . . . . . . . 61
4.19 The selected candidate lines’ adjacent buses and binary number represent-
ing whether the transmission corridor is new (1) or reconductored (0) when
turbine cost was reduced. Selections for both the TEP and GTEP shown. . . 62
4.20 The sensitivity of system cost to perturbations in decision variables about
zero, plus no and all upgrade cases. Cost in USD is shown for the minimum
and maximum aggregate load in the uncertainty set and the forecasted load. 64
viii

4.21 The sensitivity of system HL2 LOLE to perturbations in decision variables
about zero, plus no and all upgrade cases. LOLE in hours per year is shown
for the minimum and maximum aggregate load in the uncertainty set and the
forecasted load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.22 The IEEE 14 bus system bus loads used in the sensitivity analysis, each in
MW and as a percent of the aggregate load. The minimum and maximum
loads are those in the 1000 load uncertainty set. The mean is the forecast load. 67
4.23 Case in which the system operating cost increases after transmission upgrade.
The green square indicates the upgraded line, and the red oval indicates the
congested line constraining operating cost. . . . . . . . . . . . . . . . . . . 68
4.24 Case in which upgrading transmission line 1-5 (green square) increases HL2
LOLE. Outages are shown with red x’s, congested lines shown with red ovals.
Bus load outage shown with blue triangle. . . . . . . . . . . . . . . . . . . . 70
4.25 Shown again: the IEEE 14 bus system upgraded with the results of the GTEP
during the sequential vs. unified GTEP experiment. Arrows depict recon-
ductored lines. Green line depict new lines. Wind farms are shown as green
generators with numbers depicting the number of turbines installed. . . . . . 72
4.26 The sensitivity of system cost to perturbations in decision variables about
zero, plus no and all upgrade cases. Cost in USD is shown for the minimum
and maximum aggregate load in the uncertainty set and the forecasted load.
”Remove” means the line was selected in the plan, so it will be taken out in
the sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
ix

4.27 The sensitivity of system HL2 LOLE to perturbations in decision variables
about zero, plus no and all upgrade cases. LOLE in hours per year is shown
for the minimum and maximum aggregate load in the uncertainty set and the
forecasted load. ”Remove” means the line was selected in the plan, so it will
be taken out in the sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.28 The impact of uncertainty set on system cost for the original system. The
expected system cost did not change after 400 samples. . . . . . . . . . . . 77
4.29 The impact of the uncertainty set on expected HL2 LOLE for the original sys-
tem. The expected LOLE over the uncertainty set and for the mean forecast
load are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.30 The impact of the uncertainty set on expected operating cost for the upgraded
system. The straight line is cost to meet the mean forecast load. . . . . . . 79
4.31 The impact of the uncertainty set on expected HL2 LOLE for the upgraded
system. The straight line is the LOLE for the mean forecast load. . . . . . . 80
5.32 Data for wind farms. Weibull parameters, wind speed parameters in m/s,
and the per-turbine MW rating. . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.33 Cost data for wind farms in USD per turbine. Turbine and land costs are in
the first column, followed by transmission intertie costs. . . . . . . . . . . . . 92
5.34 Reliability data for wind turbines in hours, including capacity in MW. . . . 92
5.35 Data for Candidate lines. Impedances in per unit. MTTF and MTTR in
hours. Capacity in MW. Cost in U.S. dollars. . . . . . . . . . . . . . . . . . 93
5.36 Data for pre-existing lines. Impedances in per unit. MTTF and MTTR in
hours. Capacity in MW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.37 Lengths for existing (estimated), and candidate lines in miles. . . . . . . . . 94
x

6.38 The optimal value after each iteration of PSO. 50 iterations with 10 particles
are run, and cost converged well before the optimization terminated. . . . . 95
6.39 An example of PSO convergence. There is one row per particle, where each
particle is the best particle for that iteration. The first three candidates are
wind farms, the final 22 are candidate lines. Wind farm variables are in the
order of the bus numbers at which they are installed starting with the lowest.
Candidate lines are in the same order and the candidate line information. 50
iterations with 10 particles are run, and results converged long before the last
iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
xi

Chapter 1: Introduction
This chapter introduces and motivates the proposed work. Power system planning and the
continued rise of wind power systems are discussed, and the organization of this document
is described.
1.1 Introduction to Power System Planning
Electricity has become a necessity. Society relies on electricity for: lighting; cooking; air
conditioning; the production of materials like steel; and more recently, powering of computers
and the infrastructure of the Internet. The electric power system, which is composed of any
infrastructure necessary to produce and deliver electricity to consumers, has therefore become
a backbone of developed nations. Figure 1.1 shows a simplified example of an electric power
system.
Because of the importance of the service it provides, the primary goal of the electric power
system is to provide reliable service. Because the benefits of electric power should be available
to all people, a secondary goal is to provide economic service. These are conflicting goals
because redundancy increases reliability, but power system infrastructure is costly. Judicious
planning is required to ensure that appropriate infrastructure exists to meet reliability goals
in a cost effective manner.
Electric power system reliability goals are always framed in terms of consistency in supplying
1

Elliott J. Mitchell-Colgan Chapter 1. Introduction 2
Figure 1.1: A simplified, typical power system. Image from All Time Electrical [1]
the demand. Load-driving factors including population growth, weather, and the advent of
new, electrically-powered technologies cause demand to fluctuate unpredictably on many
timescales from hour to hour to decade to decade. Thus, every planning process begins with
forecasting the demand over the planning horizon. In general, the longer the forecast, the
more uncertain the demand. Thus, the appropriate future infrastructure itself is, to a degree,
uncertain.
Postponing the investment decision to reduce future uncertainty is not always an acceptable
option. Economies of scale encourage the construction of large power plants with accordingly
long lead times (on the order of several years) [5] and large capital costs [6]. The installation
of a new transmission line can also take nearly a decade depending on the acquisition of
right-of-way and the physical length of the transmission line [7, 8]. For these reasons and
others (like uncertainty in availability and price of fuel commodities), utilities are required
to provide a long-term (5-15 year) plan to help ensure the continued reliable and economic
operation of the electrical grid [9]. Thus, long term power system planning is important to
power system operation and by extension society as a whole.

Current trends in the United States of population growth, decommissioning of coal plants
and aging equipment, as well as uncertainty inherent in load forecasting, and more recently,
operational challenges with the integration of non-dispatchable renewable energy conversion
systems suggest that additional generation and transmission facilities need to be installed
to maintain U.S. electric system reliability at acceptable levels. Thus generation, transmis-
sion, and reactive power expansion plans are developed and executed [10]. These studies
involve: the selection of technologies under consideration; selection of geographical locations
for development; assessment of additional capacity required; evaluation of system operational
impacts; mitigation of risk due to inherent economic and technical uncertainties associated
with the load; the provision of evidence of meeting future regulations (especially emissions
and green energy goals); and the final selection of an expansion plan [10, 9]. Key to evalua-
tion and comparison of infrastructure expansion options are the power system reliability and
economic analyses. Generation Expansion Plans, Regional Transmission Expansion Plans,
and Integrated Resource Plans (among others) detailing these and other analyses are de-
veloped and submitted to entities overseeing electric power systems and their adherence to
policy [11, 12, 13, 14, 10].
Considering all possibilities in large-scale power system planning is impractical. Thus, rank-
ing schemes and optimization theory provide mathematical tools to facilitate the expansion
plan selection process by enabling the formal and systematic comparison of potential strate-
gies. Though an optimization framework that simultaneously considers generation, trans-
mission, reactive power, and auxiliary device expansion planning would theoretically give a
better investment strategy, this problem lacks tractability with the computing power and
algorithms currently available. In addition, information exchange between generation and
transmission entities is restricted in deregulated power system environments. Historically,
these problems are solved sequentially in a logical progression.
However, as coal plants retire and the number of renewable energy conversion system inter-
connections increase, the planning practices of the past may not be sufficient for the future.
Changes to reserve planning and generation capacity credit evaluation practices have already

changed in some ISOs in the United States [15]. Furthermore, in the literature, unified gen-
eration and transmission expansion planning is becoming more prevalent [7, 16, 17, 18, 19].
Policy and improvements in technology drive the integration of non-dispatchable systems
like utility-scale solar and Wind Energy Conversion Systems (WECS). These systems are
inherently different than conventional fossil-fuel plants, and require different considerations
during power system planning [20].
This work intends to enhance the existing literature by developing expansion optimization
models to assess the importance of various model considerations for the expansion of Wind
Energy Conversion Systems (WECS) and transmission. For a deeper understanding of the
phenomena motivating the change to planning practice to come, the next section elaborates
on the recent rise of wind power.
1.2 Introduction to Wind Energy
Harnessing the energy in wind to provide services to humans is not a new concept. Even
before electric power systems, windmills were used to grind grains or pump water. For many
decades, wind systems have provided some electric power to off-grid consumers and the grid.
However, wind penetration has increased drastically in the last several decades in many coun-
tries around the world including the U.S. As an energy-dense resource with ever-decreasing
capital costs, wind receives much attention as a replacement for a portion of our conven-
tional fossil-fuel needs and as a means to meet green energy goals. The decrease in wind
energy costs can be understood from the following statistics: turbine costs account for 70%
of the costs of wind farm installations, the costs decreased almost $300/kW installed capacity
from 2009 to 2012 when prices were roughly $1940/kW. Wind Power Purchase Agreements
decreased from between $44/MWh and $99/MWh to $31/MWh and $84/MWh (with an
average levelized cost of $40/MWh) over the same time period [2]. Indeed, wind is becoming
an inexpensive option overall. See Figure 1.2.

Figure 1.2: Levelized cost of energy by fuel in the United States in 2013. Image from AWEA
[2]
Figure 1.3 below shows installed wind capacity in the United States over the last few decades.
While expansion of WECS offers many benefits, wind is a meteorological phenomenon largely
uncontrolled by humans. This fact in combination with electrical energy storage’s minor role
in current power system operations [6] means that wind integration introduces additional
uncertainty to the system energy balance. Uncertainty in the operation of power system was
traditionally dominated by consumer-controlled changes in demand and disturbances, now
wind-related uncertainties are becoming prominent in some grids. The related operational
challenges with capacity reserves, voltage fluctuations, protection systems, and markets faced
in Northern Europe, the Midwestern United States, and Texas make it clear that wind energy
requires additional operational considerations [21, 20, 22, 23]. Because of these challenges,
solar and wind energy systems have been hot-topics in power systems recently.
At low installed capacities relative to the load and dispatchable generation, wind systems
can be ignored in operations and planning. Thus, power system planning does not tradi-

Figure 1.3: Installed wind capacity and cost over time in the United States. Image from
AWEA [2]
tionally feature as much attention to wind energy’s operational impacts as is required in
today’s planning analyses [3]. However, today’s operational challenges with wind will only
be exacerbated in the future when there will likely be even higher wind power penetration
[2]. Though the industry is still learning practices to reliably integrate a high penetration of
WECS, there are excellent works demonstrating the impacts of WECS on system behavior
as well as possible techniques to mitigate negative impacts. Notably, the NREL Western
Wind and Solar Integration Study completed during 2010-2014 was so thorough and suc-
cessful that it warranted a similar study in the Eastern Interconnection with a scheduled
completion date of Winter 2015 [21, 20].
Though the analyses necessary in systems with wind energy are developing steadily, there
is much opportunity for research introducing these analyses into expansion optimization
frameworks. For example, interesting would be a study which analyzes the optimization
model considerations by impact on the optimal solution that offers best-practices for future
expansion planning frameworks.

1.3 Motivation and Objective
The goal of this work is to demonstrate the importance of two important modeling features
for the expansion of WECS via optimization. The use of load uncertainty modeled as an
uncertainty set will be compared with the use of a single forecast load. A sensitivity analysis
is performed around the initial system (future load with no expansions) and the final load
(future load with near-optimal solution). Finally, the sequential GEP and TEP will be
compared to the GTEP. The optimization itself results in an selection on the total-system-
cost-optimal number and location of candidate wind farms and transmission lines. Substation
expansion (aside from the new wind farm substations) is not considered. Transmission
systems are simultaneously expanded in order to maintain the hierarchical level two LOLE
at acceptable levels.
1.4 Organization of this Thesis
This thesis is organized as follows:
Chapter 1: Introduction
This chapter provides an overview of power systems, power system planning, the rise of Wind
Energy Conversion Systems, and the motivation and goal of the proposed methodology.
Chapter 2: Historical and State-of-the-Art Power System Planning
This chapter describes historical approaches to power system planning, the state of the art in
generation and transmission expansion optimization, and provides more detailed motivation
of the proposed methodology.

Chapter 3: Proposed Methodology
This chapter provides the details of the wind, load, economic, OPF, and adequacy models.
It includes the motivation behind, description of, and shortcomings of use of each model.
Chapter 4: Results
This chapter contains descriptions of the experiments run and data attained, as well as a
detailed discussion about their meaning and the utility of the proposed methodology.
Chapter 5: Conclusions and Future Work
This chapter summarizes the lessons learned from the creation of the proposed planning tool
and experimentation with it. The implications of the results for electric power utilities are
discussed, and improvements that can be made are suggested.

Chapter 2: Background
In the previous chapter, a power system’s purpose and impact on society was briefly de-
scribed, and long term expansion planning to incorporate wind energy conversion systems
was motivated. This chapter delves into some background necessary to refine the goals and
understand the methodology of this work. To that end, a brief overview of optimization is
provided first. Then, long term generation and transmission expansion planning (GTEP) is
discussed in the context of optimization. Reliability and cost evaluation of power systems
is then discussed, and remarks are made about the impact of increasing wind power pene-
tration on the GTEP processes. The background is concluded with the state of the art in
GTEP and a note about the contribution of this work.
2.0.1 Optimization and Power Systems
Optimization is an important branch of mathematics with widespread applications. Not
only do humans seek optimal resource allocations, paths that minimize distances between
locations, and so forth, many natural systems function in a manner that optimizes something.
For example, Water follows the path of least resistance (locally). Power flows on transmission
lines to minimize power losses (heat waste).
Formal optimization dates back hundreds of years to calculus methods called calculus of
variation. The Brachistochrone problem was posed in 1694 [24]. Iterative methods that can
be used to optimize functions were proposed by Euler, Newton, and others in the late 1600’s
9

Elliott J. Mitchell-Colgan Chapter 2. Background 10
and 1700’s. However, the linear and non-linear programming methods that are typically used
in GEP and TEP problems today were developed in the mid and late 1900’s. Building on
theory established by Kantorovich in 1939, Danzig created the Simplex Method in 1947 [25]
that optimizes linear cost functions over linear constraints using successive transformations
of variables. The Simplex Method, though conceived long ago, is still very competitive for
solving general linear programming problems. Karush-Kuhn-Tucker optimality conditions
(used to identify if a given solution is optimal or not) were an important development in-
troduced into the literature in 1951, though their first conception was over a decade earlier
[26]. Interior point methods, methods that search for the optimal while always satisfy-
ing the constraints, were developed in the 1980’s [27]. Convex relaxations (simplifications)
of non-convex feasible regions motivate iterative algorithms to solve challenging problems.
Optimization is still a growing field, and robust optimization, stochastic optimization, and
optimization with multiple objectives have been introduced to the theory recently [24].
As for current capabilities of mathematical programming, linear programming is well devel-
oped. Using a commercial Simplex-based solver, one can solve any linear program with the
exception of very large-scale problems. For linear programming problems, both optimality
of a solution and infeasibility can be proven [25].
In contrast, available nonlinear programming technology cannot reliably and efficiently solve
general optimization problems. However, non-linear programming can be broken down into
many different classes of problems, some of which can be reliably and efficiently solved for
small and medium scale problems. Convex conic programming is an example [28]. In convex
cases, the optimality of a solution can be proven. Many algorithms exist in the literature
for each type of currently recognized nonlinear programming problem. However, the algo-
rithms typically include relaxations and/or computations of derivatives, though derivative
free methods do exist [27, 28, 25, 29].
As an example, consider Figure 2.4 which depicts a highly nonlinear, cost function of one
variable to be maximized. The function does not appear continuously differentiable which

Figure 2.4: A single variable nonlinear feasible region whose optimization presents a challenge
poses challenges for derivative-based methods; and because of all the local optima, it is likely
that an optimization technique will fail to find the global optimal. We can solve this problem
visually, but constrained problems of high dimensionality are challenging to solve visually.
This exaggerated single-variable example unconstrained optimization problem demonstrates
the possible challenges associated with solving a problem with hundreds or thousands of
decision variables and constraints. In power systems, the AC Optimal Power Flow (ACOPF)
is still a field of research, though good solvers for simple ACOPF formulations exist [30].
Stochastic programming is programming involving randomness, either in the objective func-
tion, constraints, or solution method. Classical examples include portfolio optimization with
uncertain returns on investment and newspaper salesmen problems with uncertain future
demand. General stochastic programs are difficult to solve even if they contain few vari-
ables and constraints, but if the problem has structural properties like finitely many random
variable samples, separable, alpha-concave probabilistic constraint functions, and linear cost
and other deterministic constraints, the programming problem is convex and solutions can
be generated using mathematical programming techniques relatively easily [31].
In addition to mathematical programming techniques, there are heuristic techniques like
Particle Swarm Optimization, Genetic Algorithm, and Tabu Search for solving non-linear
optimization problems. These techniques only require the objective function and constraints

to be evaluable [25, 29], but sacrifice the nice convergence or optimality properties as many
mathematical programming techniques. This is because such techniques do not take ad-
vantage of the structure of the problem. Thus, they are applicable to problems with a
wide variety of feasible region structures and cost functions, but they present other sorts of
challenges. For example, they may be slower than other solvers for simple problems. Fur-
thermore, many such techniques may require modification (relaxation) of the optimization
problem of interest [32], and equality constraints may pose challenges to generating feasible
solutions [29]. However, in practice, meta-heuristic techniques are often powerful when it
comes to finding competitive solutions and incorporating complex (black-box) analyses into
the optimization problems [10].
In the context of solving industry problems, robust, efficient algorithms with convergence
properties are preferred with an understanding that relaxation of the problem may be nec-
essary [30]. That being said, heuristic techniques are gaining popularity in the literature
[33, 10].
Applications in Power System
In power systems research and industry today, many important decision-making processes
use optimization. Examples include: economic dispatch, unit commitment, optimal power
flow, minimum load curtailment, expansion plan selection, reactive power optimization, and
network reconfiguration [33]. Because of their importance in the proposed methodology,
optimal power flow (OPF) and minimum curtailment will be described.
The general OPF problem essentially meets the demand while optimizing all equipment
settings such as generator real power dispatch, voltage set-point, transformer tap settings,
switched capacitor settings, and so on. Constrained are load bus voltages, transmission line
flows, generator real and reactive powers, tap settings, capacity reserves, system security,
and so on. The optimal solution usually represents a lowest cost operation strategy for an
instance in time. Because the ISOs seeks to optimize cost [30], the OPF is a popular way to

1. Import system bus data, branch data, generator cost curves, line limit data, etc.
2. Select a set of generator real power settings
3. Solve the DC power flow, check for transmission constraint violations.
4. If the solution is feasible, compute the cost of operation
5. Select another set of generator real power settings
6. Repeat step 3 until stopping criteria met.
Figure 2.5: Purely Demonstrative algorithm for solving the DCOPF
evaluate the operating cost of a system in the literature [34].
Unfortunately, the OPF considering both real and reactive power currently faces solution
challenges in large systems [30], Thus, ACOPF has seen limited utility in practice [10,
33]. However, the linearized DCOPF model shows no convergence issues. The DCOPF
is commonly used in the literature today, and is the method used to evaluate the operation
costs in the proposed methodology.
Though the DC power flow is not iterative itself, the DCOPF is because multiple feasible
solutions must be compared. A purely educational algorithm is shown in Figure 2.5.
The system operating costs are assumed to be comprised of generator fuel costs, or in a
market environment, the cost of paying for generators. In either case, the calculation of the
operating costs is the computation of the cost to produce the set-point power.
Traditionally, a generator cost curve is either a piece-wise function or a second order poly-
nomial. Market bids may be a quadratic, piece-wise linear or step function. For computa-
tional ease, wind energy systems are often assumed to have zero operations cost, though the
author’s previous work shows an example implementation of penalizing mis-estimation of
available wind power adapted from [35]. It should be noted that if the generator cost curve

is linear (or piecewise linear), the DCOPF can be modeled as linear (and therefore convex).
If the generator cost curve is quadratic, the DCOPF can be modeled as a convex quadratic
program [27].
The minimum curtailment formulation considers many of the same modeling features as the
DCOPF. The goal of minimum curtailment is to find the best plan to shed load in order to
maintain system integrity, i.e. to preventing islanding and/or widespread blackout.
Load may be curtailed due to real power or reactive power deficiencies. Load curtailment to
alleviate real power deficiencies is called under-frequency load shedding. Load curtailment
to alleviate reactive power deficiencies is called under-voltage load shedding [36].
Minimum load curtailment optimization involves modeling generation, the electrical network,
and demand, system stability, and ideally includes the impacts of control actions which may
stabilize the power system. Generator capability curves and transmission thermal and/or
stability limits are modeled. As with the OPF, an AC power flow model would be more
realistic, and would enable under-frequency load shedding considerations, but degrades so-
lution reliability and time. Thus, the network is often linearized in the literature. Loads are
modeled with levels of criticality and have associated weights or functions in the objective
function. The loads could be shed in discrete steps, or a relaxed problem could be solved in
which load can be shed continuously [33].
A simple formulation is identical to the DCOPF except featuring curtailment as a controllable
real power injection and the objective function as the sum of the curtailment variables. Such
a formulation is linear (and therefore convex) [25].
Particle Swarm Optimization
One popular heuristic algorithm applied in power systems is the Particle Swarm Optimization
(PSO) [37]. This algorithm borrows from swarm behavior (such as the flocking of birds) as
a means to home-in on the optimal solution to a problem. The units of the swarm, particles,

each have a trajectory through the solution space.
In PSO, the trajectory of the particle is decided by three terms: an inertial term, a local
memory term, and a global memory term. The local memory term causes particles to
explore the solution space close to that particle’s best solution so far. The global memory
term causes particles to explore the solution space close to the best solution of all of the
particles so far. Random number factors applied to each term ensure that the particles can
explore the solution space adequately. The governing equation of PSO is given below [37].
Vx = V ′
x + 2 ∗ rand ∗ (pbestx − presentx) + 2 ∗ rand ∗ (gbestx − presentx)
Where Vx is the velocity of the particle, V ′
x is the previous velocity of the particle, pbest and
gbest are the best of that particle so far and the global best solution, and rand is a random
number. The 2s are weights that could in principle be any number, but the creator of the
method recommends a weight of 2 based on empirical tests [37]. A single iteration of PSO
ends when the position is updated. The new position of a particle in the solution space is
the old position plus the velocity.
Meta-heuristic techniques like PSO are popular because they are simple to implement and
ability to explore highly non-linear feasible regions. Because of the success of PSO in solving
other power systems problems, it was selected for a solution method in this methodology.
2.1 Power System Planning
Power system planning predates both computers and optimization. Thus, methodologies
have evolved from easily-performed, deterministic analyses with large safety factors into
complex, probabilistic, and simulation-based approaches both within and without an opti-
mization framework [3, 10].
Traditionally, long-term power system planning is performed in a logical sequence starting
with load forecasting, moving to generation expansion, transmission expansion, and finally

expansion of other supporting equipment. The load forecasting, generation, and expansion
stages are performed to meet real power goals. Reactive power concerns are accounted for
afterward [10, 38]. The load forecasting, generation and transmission expansion stages will
be explained in more detail.
As a power system’s primary goal is to meet the electrical load, load forecasting is essential
to understanding the requirements of the future power system. At each expansion stage, the
essential questions are which technology, where, when, and how much capacity to install to
facilitate the supplying of the electrical demand. As computing power has increased over
the last several decades, more complex analyses have enriched each of the planning stages,
but in practice, the sequence has generally remained unchanged [10]. Planning according to
this logical sequence is effective with the United States’ excellent electrical power reliability
standing as testimony, but changing regulatory landscape and generation mix in the United
States, and ever-increasing computing power suggest there is room for improvement to the
traditional practices [10, 39, 40].
2.1.1 Load Forecasting and Uncertainty
Long-term load forecasting is the first step of long term power system planning. Point-
estimates and confidence intervals are developed with the purpose of defining the require-
ments of the future power system infrastructure.
The demand for geographical regions called load zones are predicted using past load, weather,
population, and other data. Point-estimate forecasts for the peak load of each area and for
the area loads during the total system peak (non-coincident and coincident peak loads) are
computed [41]. However, the future load is uncertain. Thus, the infrastructure necessary to
carry out the mission of the power system for the future load is also uncertain. This inherent
limitation impacts the power system planner’s methodologies.
In order to account for uncertainties in the future load, confidence intervals on the peak

load are constructed and used to perform adequacy studies [42, 41]. Confidence interval
construction is a useful, traditional statistical method to attain a better understanding of
the merit of statistical estimates. Confidence intervals enable the quantification of bounds
within which a random variable (a future load) will lie up to a certain probability. They
are constructed using the definite integral of the random variable’s underlying probability
distribution [43].
Parametric and non-parametric methods exist to compute confidence intervals. In general,
parametric methods provide more information, but require the assumption of the under-
lying probability model for the random variable. Unfortunately, the mechanisms driving
long-term load uncertainty are complex, involving uncertainty in population growth, tech-
nological advancements, and to some extent even weather [42]. Perhaps it is even unlikely
that the distribution of the forecast error is time invariant. For example, the advent of the
consumer electronics caused load to increase much faster than planners of the antecedent
decade predicted, and today there exists a threat of shifting weather patterns. Thus, it is
difficult to generate confidence intervals in which we can have complete faith.
However, it’s arguable that a flawed confidence interval is better than a point estimate. Thus
confidence intervals are constructed with assumptions. For lack of a better assumption and
for the sake of simplicity, it is popular in the literature to approximate the error in the
load forecast as a Gaussian distributed random variable. This involves estimating sample
standard deviation using previous load forecast and actual load data. Another approach
used by PJM involves non-parametric confidence interval construction using Monte Carlo
load forecast simulation via samples of load-driving data [41]. Using either method, the final
output is a set of system loads to use as inputs to expansion planning process.
2.1.2 Generation Expansion Planning
Generation Expansion Planning (GEP) focuses on the selection of: fuel; plant type and
capacity; construction site; and the date of first interconnection with the grid. Traditional

methodologies determine the required capacity via reliability analysis. The generation re-
quired to meet a reliability index such as the Loss of Load Expectation (to be discussed later
in this chapter) is found. This analysis results in the capacity required during each planning
period. Then, production simulation and investment analyses are performed to select from
a number of different generation technologies in order to fulfill the needed capacity expan-
sion. The reliability and production simulation studies may or may not include transmission
network models [38].
The location of generation must also be chosen. As the number of possible plans grows
exponentially with the number of potential sites, exhaustive comparison of all possibilities
is nearly impossible in large systems. Thus, pre-selections are made by systematically eval-
uating possibilities with engineering judgment. It is important to note that not all locations
in the transmission network are appropriate for installation of an new generating facility.
For example, a new facility can not practically be installed in the middle of Washington DC
even though there exist several high voltage transmission substations. For computational
and practicality reasons, pre-selection of sites in an important step in the solution of the
GEP problem. This is good, because the problem without pre-selection of locations has
NB
solutions, where N is the number of possible upgrade decisions per bus and B is the
number of buses. Decisions can be made using optimization or ranking sites by cost, or
reliability [10, 38, 14]. Of the two options, optimization’s formal, automatic search proce-
dure is perhaps better suited to comparing the merit of GEP solutions including complex
system interdependencies of associated with installation of multiple generators. Thus, an
optimization method is used in this work.
Any optimization problem is comprised of an objective function to be minimized (or max-
imized), and constraints. The GEP problem often minimizes total system costs including
investment, operations and maintenance, fuel costs, and the cost of energy not served [10].
The constraints include reliability, reserve, emissions, fuel availability, pollution constraints,
and constraints on the power available from each plant. An example formulation is shown
below in Equation 2.1 (modified from [10]).

minimize Ctotal = Cinv + CO&M + Cfuel + CENS
subject to Cinv =
T∑
t=1
Ng
∑
i=1
aitPGiXit
Cfuel =
T∑
t=1
(
bet +
Ng
∑
i=1
bitENERGYitXit
)
CO&M =
T∑
t=1
Ng
∑
i=1
citPGiXit
CENS =
T∑
t=1
dtENSt
(1 + Rest/100) ∗ PLt =
Ng
∑
i=1
PGciXit + PGt ∀t = 1, . . . , T
LOLPt ≤ LOLPMAX ∀t = 1, . . . , T
FUELejt +
Ng
∑
i=1
FUELitENERGYitXij ∀t = 1, . . . , T
∀j ∈ Nf
POLejt +
Ng
∑
i=1
POLijENERGYitXit ∀t = 1, . . . , T
∀j ∈ Np
CinvXit ≤ CMAXt ∀t = 1, . . . , T
∀i ∈ GenTypes
ENERGYij ≤
Ng
∑
i=1
HrsPerY earCAPijXijt ∀t = 1, . . . , T
∀i ∈ GenTypes
(2.1)
Where Cinv, CO&M , Cfuel, CENS are the costs for investment, operation and maintenance,
fuel, and the energy not served, a,b,c,d are the respective linear cost coefficients, Rest is the

reserve at time t, PGci is the capacity of the candidate generator at location i, PGt and PLt
are the generation and load during period t, Xit is the investment binary decision variable for
location i during time t, LOLP is the loss of load probability, FUELejt and FUELjt are the
fuel of type j consumed by existing generators and the fuel type j consumed by plant type i,
POLejt is similar, except the pollution produced. The final two constraints ensure that the
money spent does not exceed the available investment capital and that the generators stay
within their operating limits.
After a formulation is selected, commercial solvers can be used to find optimal or near-
optimal solutions to the problem, i.e. produce good choices for where, when, and what type
of generators to install.
The formulation shown above is a Mixed-Integer Non-Linear Program MINLP. This is be-
cause it has both integer (Xij) and continuous (ENERGYit) decision variables, and non-
linear because the integer decision variables are multiplied by the continuous ones. It is a
combinatorial optimization problem with multiple time stages corresponding to dispatching
periods per year. Even though the implementation shown is fairly simple as it contains no
network constraints, only linear generator cost curves, no security constraints, unit com-
mitment, and so on, MINLPs such as this are among the hardest optimization problems
to solve. Fortunately, operations research has techniques to lessen the computational bur-
den for certain structures of problems. For example, dynamic programming can be used to
split up the problem into smaller subproblems, and branch and bound can help us eliminate
combinations of generations that cannot produce the optimal solution.
2.1.3 Transmission Expansion Planning
The basic question that the TEP problem attempts to answer is where, what type, and what
capacity of transmission conductors should be installed in order to enable the energy trans-
action between generators and load at the point of the transition from the bulk transmission
network to the distribution systems.

Traditional transmission expansion planning is composed of several screening processes with
increasing model complexity in order to narrow down candidate upgrades. Analysis pro-
gresses from power flow linearizations of a large transmission system model to identify cor-
ridors which have potential future flow violations to transient analyses on key corridors and
relay coordination. Along the way, AC power flow calculations and transmission adequacy
studies under different load scenarios are conducted in order to identify system weaknesses
[38].
The combinations of technologies, periods of investment, and locations make this optimiza-
tion problem very large. The pre-selection of appropriate sites is as important in the TEP
as in the GEP problem. Fortunately, it is usually unreasonable to build a transmission line
longer than a few hundred miles. For that reason alone, most combinations of buses are
impractical and end-points for a single transmission line. Without this practical limitation,
the TEP problem that would otherwise be greater than (B − 1)!T
in complexity, where B is
the number of buses and T is the number of conductor technologies to choose from.
The transmission expansion problem can be formulated as shown below in Equation 2.2,
taken from [10]. Unfortunately, the DC power flow precludes the ability to set voltage
constraints.
Minimize
∑
i∈L
CL(xi, Len(i))
subject to
∑
j=1
Bij(θi − θj) = PGi − PDi ∀i ∈ n
∑
j=1
bm
ij (θm
i − θm
j ) = Pm
Gi − PDi ∀i ⊂ n ∩ m ⊂ C
bij(θi − θj) ≤ PkMAX ∀(i, j) ∈ (LC ∪ LE)
bm
ij (θm
i − θm
j ) ≤ PkMAX ∀(i, j) ∈ (LC ∪ LE) ⊂ m ∈ C
(2.2)

Where CL is the cost of transmission investment as a function of the location and line length,
bij is the susceptance of transmission line k, θi is the voltage angle at bus i, PGi and PDi are
the real power generation and demand at each bus, PkMAX is the maximum capacity of the
transmission line, LC and LE are the sets of candidate and existing transmission lines, C is
the set of contingencies, and the superscript m indicates the quantity after contingency m
occurs.
It should be noted that the above two formulations (2.1) and (2.2) are each stand-alone. In
other words, the joint generation and transmission expansion problem has a different formu-
lation than either of the ones above. The motivation for a joint generation and transmission
expansion plan can be simply explained. Historically, the GEP is performed, and then used
as an input to the TEP. However, the transmission infrastructure could be the main factor
constraining the system cost and or reliability. Performing both analyses simultaneously
more systematically balances the marginal increase in reliability gained from investment in
the two classes of infrastructure.
Joined Generation and Transmission Expansion Planning has also been introduced to the
literature [16, 17, 18, 19]. Yet, most of the formulations in the literature focus on one facet of
planning, Perhaps because the size of the problems separately can pose practical challenges
finding optimal solutions in life-sized networks.
2.2 Evaluating Reliability in Power Systems
Power system planners attempt to ensure the electrical system meets federal, state, and
local regulations for both reliability and economy. Traditionally, deterministic methods have
been used in order to evaluate the reliability of generation and transmission systems. Thus
can power system planners attempt to ensure that the future system will be capable of
meeting the load even during credible contingencies. However, deterministic approaches,
whether analytical or simulation-based, cannot appropriately evaluate risk because they

ignore the probabilistic and stochastic nature of power systems. Expansion plans produced
using deterministic reliability approaches may be more expensive than necessary. Today,
power system planners use probabilistic indices like Loss of Load Expectation (LOLE) to
evaluate reliability [4].
Power system reliability can be of divided into two concepts: adequacy and security. A
power system is said to be adequate if there exists the necessary infrastructure to meet the
demand. That is, there is sufficient generation to meet the demand, and there is sufficient
transmission and distribution facilities to deliver the power to the loads without violating
any system constraints (voltage, frequency, etc.). A power system is said to be secure if it
will be able to sustain a disturbance and attain a new steady-state operating point within
the limits of operation, potentially through control actions [3]. In general, adequacy metrics
are easier to compute than security indices because adequacy can be evaluated using static
system models.
Reliability of the power system is evaluated using one or several adequacy and security
metrics such as Loss of Load Expectation or Expected Frequency of Load Curtailment, and
the N-1 security criterion. These metrics indicate the ability of an existing power system to
meet the load considering the possibility of failure in the components of the power system.
FERC mandates adequacy by stipulating an LOLE of 1 day in 10 years [44] or lower, and
security by stipulating that systems are continuously N-1 secure [45] [46].
There are a few basic ways to compute the LOLE metric. The most common way is the
state enumeration method using Forced Outage Rates of each components in the model. A
capacity outage table (see Figure 2.6) is computed for each of the system outage states with
a significant probability. Then, the capacity outage is convolved with an array probabilities
that each load occurs to enumerate all of the significant states of load and generation . The
probabilities of each state are computed, as well as the power not served during each of those
states. This method is straightforward and fast, but does not consider the temporal aspects
of forced outages [3].

Figure 2.6: Capacity Outage Table for adequacy index calculation [3]
The LOLE can also be calculated using the Frequency and Duration method. Using this
approach, generator state transition models like the one shown in Figure 2.7 are used. State
transition times are then sampled in order to construct a history of available generator
capacities (see Figure 2.8). With a load history, the excess capacity history can be generated,
and the duration and severity of outages can be computed. This method requires data that is
perhaps harder to collect, and also may be more computationally burdensome, but facilitates
the time series modeling of power system components like WECS [4].
Figure 2.7: Basic two-state model for Frequency and Duration Method reliability analysis.
[4].
The process of computing the LOLE using the Frequency and Duration method above is
essentially a Monte Carlo simulation. Random variables are sampled, and for each set of
samples, the metric of interest is computed. As the output of a Monte Carlo is a sample
expectation, a natural question is, ”how many samples are necessary to achieve a certain
accuracy?”. This is often a challenging question to answer because Monte Carlo simulation
is usually found in applications in which direct analytical evaluation is hard or impossible.
In practice, however, stopping criteria can be used to terminate simulations in a reasonable

Figure 2.8: Capacity and load history for the state duration method. Energy Not Served is
highlighted in black [4].
fashion. An example stopping criterion applied to LOLE calculation is a threshold on the
coefficient of variation of the estimated criterion [4]. A good stopping criterion will terminate
the simulation when the estimated value is acceptably close to the true value.
Power system reliability can also be broken down into several hierarchical levels to clarify
the model boundaries of the studies. Figure 2.9 shows the hierarchies. Hierarchical level
one involves planning using a system model comprised only of generation and the load. The
generation and load are connected at the same bus, and real-power adequacy metrics like
Loss of Load Expectation (LOLE) and Loss of Energy Expectation (LOEE) are computed.
Hierarchical level two includes the transmission system as well as the generation and load.
Reliability indices are computed for the composite system. Hierarchical level three includes
the distribution system as well as the generation and transmission system. Traditionally,
the distribution system is considered separately for computational ease, and total system
reliability is computed using load-point indices [4].
The GEP problem typically focuses on the hierarchical level 1 model. The TEP problem

Figure 2.9: The hierarchical levels of power system planning.
focuses on the hierarchical level 2 model, assuming that the generation is fixed [10, 3].
Therefore, the unified problem is computed on the hierarchical level 2 model. Though in the
literature system reliability is not always computed, realistic GTEP formulations must ensure
that system reliability is maintained within reasonable limits to reflect the legal obligations
to reliability industry members have.
2.3 Evaluating System Cost in Power Systems
An attempt is made to optimize power system operation. This involves the minimization of
the total production cost with the intent to give rate-payers a fair electrical price.
Electricity is considered to be a necessity. Thus, to make power affordable the production
cost, the cost to produce the real and/or reactive power required to meet the demand,
is minimized. In market-based operation under ISOs in the united states, this involves
generator bids and market clearing by an independent entity. In a regulated environment,
this involves more centralized decision-making. Whatever the case, comparing the cost-based
merit of two power systems involves minimizing the cost of the system over the lifetime of

the power system infrastructure [34].
In the literature, the total system cost is usually evaluated using a production cost model
or, for the network-constrained case, an ACOPF or DCOPF with either piece-wise linear
or parabolic generator cost curves. Though an ACOPF would be preferable, robust, fast
solvers are still a topic of research [30]. However, the DCOPF approximation is both fast
and reliable, making it popular in the literature [34, 47]. The DCOPF’s objective function
is the total cost of real power generation, and one DCOPF is run per load scenario in a
representative set of load scenarios. The sum of all of the production costs is considered to
be the total system operating cost over the period that the load represents. When introducing
investment capital costs as in the GEP and TEP problems, the operating cost as described
above may be multiplied by a factor to compute the system operating cost over the service
life of the investments.
Calculating a realistic absolute system cost may be rather difficult (as is the case with
absolute system reliability). Fortunately, for the purposes of the GEP and TEP, power
system costs must only be compared. It is generally assumed that errors in calculating each
system cost are in some sense canceled during the comparison [3].
2.4 Operational Challenges with Wind Power
Wind is not controllable by humans unlike the fossil fuel inputs of conventional generators.
Thus, wind turbines are not dispatchable. Furthermore, wind power output is variable on
many time-scales and unpredictable, as is the wind itself. These fundamental differences have
caused many operating challenges that must be considered in power system planning. Large
changes in the power output of a wind farm over short periods of time (wind power ramps)
have caused or nearly caused blackouts in Texas [22] and Germany [23, 48]. Forecasting
these ramps to prepare the system against stress is also a challenge [49, 50, 51]. These
ramps can also require different calibrating of frequency controls [52, 53]. NREL has shown

that wind also may increase the ancillary servics prices, volatility of energy prices, and place
require changes to market operations [54, 55]. The variable and unpredictable nature of wind
may cause a need for an increase in reserve requirements or dynamic reserve requirements
[56, 57, 58]. Wind power fluctuation can also cause rapid changes in voltage (flicker) in the
distribution system [59].
Models of wind power vary based on the phenomenon under study. For incorporation into
power flow (static) models like the OPF and load curtailment, wind speed is often drawn
as a sample from a Weibull random variable and then converted into wind turbine or wind
farm power output using key results from Bernoulli’s Equation from the study of fluid flows
[34]. Computation of power system adequacy metrics like LOLE can be performed using this
method [3]. Dynamic models require a more accurate wind speed time-series model such as
a sample of real wind data or simulated data from an ARMA process. Such models may
include turbine, generator mechanics, and power electronic controller models [60].
As regulatory incentives and economic viability of installing wind energy conversion systems
increase, a natural question is where should the wind farms be installed. In addition to local
considerations like merit of wind portfolio, system considerations like transmission infras-
tructure and accuracy of aggregate wind power forecasts also impact the merit of wind farm
sites. Thus, finding the optimal wind farm expansion may benefit from inclusion into the
expansion plan optimization formulations. From the author’s previous work shown in [34],
generation expansion to meet the future load in a network constrained optimization con-
text may produce unrealistic results without considering appropriate transmission upgrades.
Thus, wind farm expansion lends itself to modeling via GTEP optimization.
2.5 State of the Art GTEP
The previous literature shows several models that can be used to solve expansion planning
problems. NREL’s ReEDS [16] is perhaps the most comprehensive, considering conventional,

renewable resources, storage, and transmission planning with linear programming, but does
not perform AC power flow. Reserve requirements are computed for each time period based
on technology and reserve type. Reliability metrics are not computed internally, and the
planning period is set to two year periods. In [61], transmission security constraints and
unit commitment are included in a formulation that meets a desired wind energy penetra-
tion while minimizing investment. That optimization also selects from among two WECS
technologies. In [62], only the GEP problem is solved, but the AC power flow is used and
maintenance is scheduled, but reliability is not considered. In [63], renewable energy, stor-
age, and transmission are expanded, but with a focus on cost of electricity and emissions.
Roh et. al in [17] propose an optimization framework that considers the deregulated market
dynamics of generation companies and transmission companies. For additional review and
a list of available commercial optimizers, [7] is an excellent resource.
Though several interesting works exist demonstrating the influence of co-optimizing gener-
ation and transmission expansion, no optimization calculates the composite generation and
transmission system LOLE including wind generation. Furthermore, no study benchmarks
the impacts of including load uncertainty in the study, nor in the comparison of unified
GTEP and sequential GEP and TEP.

Chapter 3: Methodology
In the previous chapter, a brief background of load forecasting and uncertainty, expansion
planning, cost and reliability calculation, and wind modeling was discussed. In this chapter,
these concepts will be combined to depict the methodology through which this work attempts
to describe the importance of load uncertainty, impact of unification of the GEP and TEP,
and demonstrate a sensitivity analysis. Modeling decisions are justified and alternatives are
briefly discusses.
The chapter is organized as follows. First comes the description of the general idea of the
optimization framework and solution algorithms which is used to attain the results. Then,
specifics of the cost and reliability constraint are detailed. Finally, the procurement of load
and other input data are discussed.
3.1 Optimization Framework
The platform of this work is the optimization framework. It systematically searches for
a lowest cost (investment and operating cost) expansion plan that chooses location and
capacity of wind farms and location of transmission upgrades. Acceptable expansion plans
meet the load of the system within thermal and network constraints assuming no components
on outage, as well as a constraint on LOLE considering component outages.
The proposed algorithm contains two inter-related yet distinct optimization layers. The
30

Elliott J. Mitchell-Colgan Chapter 3. Methodology 31
outer optimization layer searches through the investment decisions. The inner optimization
evaluates the cost merit and reliability feasibility of each of the candidate solutions generated
by the outer optimization. Such a structure exists because the operating cost of a given
power system topology is controlled via an optimal power flow in industry, but the system
topology is precisely the object of desire in this work. An analogous situation applies for the
system reliability calculation using a minimum curtailment formulation. The specifics of the
optimization process are discussed below.
3.1.1 Outer Optimization: Search for Candidate Upgrades
The outer optimization systematically selects candidate expansion plans. That is, candidate
wind farms and transmission lines are selected. Excluding the sub-optimization problems,
it is a mixed integer linear program (MILP). The constraints are simple: there is an upper
limit to the number of turbines chosen for each site, and only pre-selected locations of both
wind farms and transmission lines are acceptable. However, the objective function must be
calculated by solving an optimization problem, and feasibility depends on the Monte Carlo
reliability simulation. Because of the complexities of the formulation, a heuristic optimiza-
tion technique is selected. More specifically, heuristic optimization techniques do not require
the computation of derivatives of the optimal values of the optimization subproblems with
respect to the candidate upgrade binary decision variables [29]. Particle Swarm Optimiza-
tion is used in this work because it has become particularly popular in the power systems
literature, though PSO is by no means the only choice. The outer optimization problem can
be formulated as is shown in Equation 3.3.

minimize Ctotal = CT−Lines + CWindFarms + Q(y, w, ζ)
subject to
0 ≤ yi ≤ ymax
i , ∀i ∈ WF
yi ∈ Z*, ∀i ∈ WF
wi ∈ {0, 1}, ∀i ∈ TL
(3.3)
Where CT−lines is the capital cost of transmission upgrades, CWindFarms is the capital cost
of wind farms, y and w are the integer variables associated with the decision to install wind
farms or upgrade transmission.
All of the system data including random variable data is generated before the algorithm
begins. Generating random data beforehand ensures that all systems are compared on even
grounds. Benchmarking studies are performed to ensure that the load, wind speed, and
generator outage datasets are large enough to estimate expected cost and reliability values.
3.1.2 Inner Optimization Layer: Evaluating Cost and Reliability
The inner optimization layer evaluates the expected cost and expected LOLE of the candidate
solution generated by the outer optimization. The cost and reliability evaluations will be
considered in the next sections.
Computation of the System Cost Function
The cost of a candidate system is computed using the DCOPF over a predetermined set
of load scenarios. The DCOPF formulation is a convex quadratic program. It has linear
constraints and the non-negative sum of traditional parabolic (convex) generator cost curves
[27]. Because load and wind are modeled as random variables, the cost is computed over
several load scenarios over several wind scenarios to achieve an expected value. There are no
component outages considered in the cost calculation. MATPOWER’s DCOPF solver using

the default algorithm is used to compute the optimal system cost [47] for each sample wind
speed and load value. A formulation of the optimization is shown in Equation 3.4.
minimize Q(y, w, ζ) =
∑
i∈CG
aiP2
i + biPi + ci
subject to
P = B−1
θ
(
∑
i∈G
Pi
)
−
(
∑
i∈D
di
)
= 0
(
∑
i∈B
Pi + ci − di
)
−
∑
k∈TLi
Pik = 0, ∀i ∈ B
Pmin
i ≤ Pi ≤ Pmax
i , ∀i ∈ CG
ywPmin
w ≤ Pw ≤ ywPmax
w , ∀w ∈ WF
− Pmin
ij ≤ Pij ≤ Pmax
ij , ∀i ∈ TL
− wkPmax
k ≤ Pk ≤ wkPmax
k , ∀k ∈ CTL
(3.4)
where the cost is a minimization of the thermal generators fuel costs (wind operation costs
are assumed negligible), the first constraint is the DC Power Flow equation, the second
constraint ensures the demand is equal to the load, the third constraint is Kirchoff’s Current
Law, the next four constraints bound the thermal and wind farm generation and transmission
line flows, and the final constraint places a lower bound on the reliability as explained below.
CG is the set of conventional generators, WF is the set of wind farms, TL is the set of
existing transmission lines, and CTL is the set of candidate transmission lines. It should
be noted that pre-processing performed by the author’s code ensures that the MATPOWER
case has the correct formulation including selected candidate transmission lines, and thus
no modification is required to the standard DCOPF as described in [47]. The wind farms
are modeled as PV buses with output determined by transformation of wind speeds samples
from Weibull random variables. A wind farm may reasonably be modeled as a PV bus
because wind farms can and do control voltage in the real system [64].

The cost of operation computed as above can be augmented and added to the investment
cost of the candidate solution to achieve the expected total system investment plus operating
costs. Because the operating cost is computed over a subset of hourly loads, it must be
weighted in order to be comparable to the investment cost [65]. The weight is computed
by dividing the expected operating life of the equipment in hours by the number of hourly
load cases. This is an approximation because the load drives the system operating costs in a
non-linear fashion, load grows year to year, and only one planning year was selected for this
methodology. A more thorough and much more computationally expensive approach would
be to model loads throughout the entire expected life of the selected candidate equipment.
The service life of the candidate equipment was selected to be 30 years according to recent
energy agreements of wind farms [66], though transmission lines can have significantly longer
service lives [67].
Computation for the System Reliability Constraint
The system reliability is featured in a constraint in the inner-optimization. As with any
reliability calculation one must know: 1) the state of the system components; and 2) how
to calculate the system outage given the system state [43]. Based on the optimization
algorithm, the topology of the system with no components on outage is fixed and known
in the inner-optimization. The outage histories of each of the components in the system
are generated through sampling of times to failure (TTF) and times to repair (TTR) as
in [4]. Equations 3.5 and 3.6 show the sampling equations where U1 and U2 are uniformly
distributed random number on the interval [0,1] (generated by MATLAB’s rand() function).
Though other distributions could be used to sample the TTF and TTR, because we desire
mean behaviour in this work, such extra effort is not necessary [4]. The reliability models of
the system components are described in Section 3.2.1.
TTF = MTTF ln(U1) (3.5)

TTR = MTTR ln(U2) (3.6)
Using the component outage histories generated, the state duration method is used to com-
pute the system LOLE. The load curtailment is computed for each system state (accounting
for islands). As shown in Equation 3.8, the estimated system LOLE is the sum of the
durations of the states for which there is any load on outage.
The computation of the system outage for a single system state is performed via load-
curtailment minimization. In this optimization, all committed generating units are scheduled
to meet the thermal and network constraints with the minimum curtailment of loads [33].
The LP load-shedding formulation implemented is shown in equation 3.7.
minimize
∑
i∈D
ci
subject to
P = B−1
θ
(
∑
i∈G
Pi
)
−
(
∑
i∈D
di
)
= 0
(
∑
i∈B
Pi + ci − di
)
−
∑
k∈TLi
Pik = 0, ∀i ∈ B
Pmin
i ≤ Pi ≤ Pmax
i , ∀i ∈ CG
ywPmin
w ≤ Pw ≤ ywPmax
w , ∀w ∈ WF
− Pmin
ij ≤ Pij ≤ Pmax
ij , ∀i ∈ TL
− wkPmax
k ≤ Pk ≤ wkPmax
k , ∀k ∈ CTL
(3.7)
where the formulation is almost exactly that of the DCOPF except that the cost objective
function is the minimization of load curtailment variables. It should be noted that the
formulation does not include the estimation and bounds on minimum frequency. This is
done for simplicity, and the interpretation is that this formulation finds the minimum load

to shed such that there exists a steady state solution to the linearized network. It is also
assumed that the IEEE 14 bus system has no demand resources.
While there are many system outage states, LPs in general are solved quickly and reliably by
commercial solvers [25]. Because load-shedding is a decision variable greater than or equal
to zero, a correctly modeled load-shedding minimization does not suffer from infeasibility
or unbounded-ness. Furthermore, system states can be intelligently pruned to eliminate
unnecessary computation as in [68], though such pruning is not performed in this work.
After the system outage for all system states for all islands has been computed, the estimate
of the total system LOLE can be computed via the Equation 3.8, taken from [4].
1
NS
∑
i∈S
ti (3.8)
where S is the set of states, NS is the number of states, ti is the duration of state i.
Once the total system reliability has been computed, it can be compared with the bound
attained by base-lining the original system with the original load.
The flow diagram in Figure 3.10 depicts the overview of the algorithm.
3.1.3 Capturing Load Uncertainty in the Optimization
The uncertainty in the future load impacts our investment decisions. In order to formally
include this uncertainty in the optimization framework, we use ideas from Stochastic Pro-
gramming. Generally speaking, this type of optimization searches for optimal decisions (i.e
expansion plans) that must be made before key information (i.e. future load) is known. The
uncertain information is modeled as a random variable of which samples can be taken [31].
The sampling of the random variable influences how risky our decisions are. For example,
samples with low system loads could be rejected resulting in a bias in the expansion plans
toward expansion plans that minimize cost and meet reliability for extreme future loads. In

Figure 3.10: An overview of the presented algorithm showing the outer and inner optimiza-
tion

this methodology, expectations are computed without bias toward over or under estimation
of the future load.
In the proposed methodology, the set of future load scenarios to consider in the inner opti-
mization’s DCOPF and minimum curtailment is uncertain. We capture the uncertainty by
taking samples from a multivariate Gaussian that considers the forecast error, thus building
a finite uncertainty set of data for which the DCOPF can be solved. It is stressed that this
uncertainty set is distinct from a set of loads built by selecting system loads at different times
during the year. Specifics are given in the load uncertainty section of the methodology.
3.2 System Reliability Constraint
The Bound on System Reliability
NERC established a ”One day in ten years” criterion for the LOLE of a system [69] using
estimated forced outage rates for critical system components [3]. However, adding the NERC
criterion as a constraint to the IEEE 14 Bus System may be somewhat unhelpful. For
example, it could be the case that the 14 Bus System is over-built and the ”One day in
ten years” criterion will never be violated even after the load is scaled according to a load
forecast. Thus, the notion of base-lining applied elsewhere in the industry is implemented to
attempt to find a bound on the system reliability that will appropriately limit the feasible
region such that this optimization problem is reasonable and solutions are interesting.
As the term base-lining implies, the original system is modeled and used to compute a
reliability value against which system models for the future can be compared. Thus, the
methodology for evaluating the system reliability mentioned below is used to compute the
bound for the system without load scaling or candidate upgrade implementation. The inter-
pretation of a bound (constraint) achieved by base-lining is that the system LOLE should
not degrade in the planning year of interest.

This notion of base-lining also has value in terms of specifying the length of the Monte Carlo
reliability simulation. The system LOLE can be estimated for the 14 Bus System for a range
of Monte Carlo simulation hours. When the system reliability is insensitive to an increase
in the simulation hours, it can be assumed that the number of simulation hours is adequate
to estimate the LOLE. Because the convergence time generally increases with the number
of random variables (i.e. the number of candidate upgrades selected), the convergence base-
lining is performed for the system with all candidates selected. The stopping criterion was
selected to be the smallest time that the coefficient of variance of the estimated LOLE falls
below 5%. In mathematical terms [4]:
1
E(LOLE)
1
N(N − 1)
N∑
i=1
(LOLEi − E(LOLE))2 ≤ .05
It was found that slightly under two million simulated hours were necessary to reach con-
vergence according to the criterion above with outage and wind datasets and all system
upgrades selected. Thus, two million hours was selected as the duration of Monte Carlo
reliability simulations in this work.
3.2.1 Reliability Models for Key Power System Components
In order to generate the outage histories necessary for the stage duration LOLE calcula-
tion in this methodology, a model of generation and transmission components are required.
These Markov Chain reliability models enable the sampling of times to failure and times to
repair in order to construct a time series of components maximum capacities that is repre-
sentative of real world behaviour on average. Both generators and the transmission system
are modeled with the classical two-state model. No bus failures are considered in this study.
Typical values for generator state transition rates are taken from the IEEE Reliability Test
System [70]. Typical values for transmission line state transition rates are taken from the
Transmission Availability Data System [71].

Generator Reliability Modeling
According to the IEEE Standard Definitions for Use in Reporting Electrical Generating Unit
Reliability, Availability, and Productivity [72], there are many states in which a generating
unit may reside. A diagram depicting the states is shown in Figure 3.11. This model’s
complexity is beyond the scope of this work, and collecting data for such a model may prove
cumbersome. Instead, behaviour of generators is simplified as described below.
Figure 3.11: All Possible Generating States according to IEEE Std 762™-2006.
In this work, it is assumed that all units are committed all of the time. This means that there
must be at least two states: ”unit up” with maximum rated capacity and ”unit down” with
zero capacity. In this work, derated states are not accounted for; however, including them
in the model for subsequent work would involve merely the modification of the base-load
generator Markov Chain models. Four-state peaking generator modeling is also possible,
although more involving because start-up and shut-down times must be identified according
to system need and restricted by generator minimum up and minimum down times [4].

Figure 3.12: Two-State base load reliability model.
A diagram of the two-state base-load reliability model used in this work is shown in Figure
3.12. The states are shown in blocks, and the transitions and their rates are shown using
arrows between states. The mean time to failure (MTTF) is λ and the mean time to repair
(MTTR) is µ.
In order to designate MTTF and MTTR values to the generators in the system, some
typical data must be acquired. These values depend on the machine size and technology,
but unfortunately, only the size of machines is known in the IEEE 14 bus system. Thus,
fuel types were assumed only according to size. The MTTF and MTTR values were chosen
from machines in the IEEE 1996 Reliability Test System without modification [70]. Another
potential datasource is the NERC Generator Availability Data System [73]. The appendix
shows the generator reliability data used in the study.
Transmission Line and Transformer Models
It is assumed that all transmission lines are single circuit, and that a transmission line can
either be in service or out of service. Thus, the transmission model is essentially identical
to the base-load generator shown in Figure 3.12. It should be noted that the MTTF and
MTTR of transmission lines generally differ drastically from base-load generators [71, 73].
This is no surprise because of the difference in complexity of construction and operation of
the two types of components.
Transmission line MTTF is dependent on the voltage level and length of the conductors, and

the TADS data lists MTTF per circuit mile for various voltage levels and initiating events.
For simplicity, in this study only element-initiated events are considered; that is, outages of
one branch do not force other branches out of service.
To compute the total MTTF of a transmission line, the length of the transmission line is
estimated using available IEEE 14 bus system data. In practice, an industry member could
likely procure the measured lengths of the existing and estimated lengths of candidate lines.
However, because the IEEE 14 bus is a fictitious system, the lengths must be estimated
or assumed. The 14 bus system is composed of two voltage levels, but unfortunately, the
voltage levels are only designated ”LV” and ”HV”. Thus, in this methodology, there is
assumed to be a constant series reactance per mile in ohms for both of the transmission
levels. Using the ACSR table supplied in the appendix of [67] and the assumption that each
corridors contains only one conductor per phase with a current capability of roughly 1000A,
the value .4Ω/mi was attained. To convert from per unit line reactance quantities supplied
in the system data, the MVA base of the system is used as supplied and voltage levels were
assumed. Reasonable voltage levels are searched for from the set of voltages common to the
United States by trial and error until reasonable transmission line lengths (much less than
100 miles) are computed. Finally voltage levels selected were 115kV on the generation side
and 65kV on the load side. In the authors opinion, these voltage levels are also consistent
with the 14 bus system topology. Indeed, other candidate voltage levels at transmission
voltage produced rather unreasonable estimates with some lines approximately 300 miles
in length. Thus, transmission line lengths were attained that can be used to compute the
MTTF for each transmission line for each voltage level in the system. Another means of
estimating transmission line length using a typical line geometry approach is explained in
[74].
The mathematical representation for the MTTF for non-transformer branches is as follows:
l =
Xpu
.4 Ω
mi
∗
V 2
base
Sbase

In contrast with MTTF, MTTR is less sensitive to transmission line length [71]. Thus, in
this study, each transmission line in the same voltage class is given the same MTTR.
One final assumption is made in order to designate an MTTF and MTTR for each transmis-
sion line in the system. Because the TADS only contains data for EHV transmission lines,
the same voltage levels used to estimate length could not be used to select reliability data.
MTTF per mile and MTTR from the TADS data were chosen from 230kV and 500kV levels.
This inconsistency should not cast doubt on comparison of model considerations, though it
does cast doubt on the merit of the proposed expansion plans as strategies to improve the
fictitious 14-bus system.
TADS lists transformer data by voltage class. Thus, consistent with the choice of TADS
data for transmission lines, 230kV to 500kV transformer reliability data were designated to
the 14 bus transformers.
The appendix shows the reliability data for existing and candidate lines used in the study.
Wind Farms
Wind farm reliability models are composed of a model for each wind turbine generator
(WTG) and a model for the transmission inter-tie, both of which use the classical two state
model. A wind farm may contain multiple WTGs, and a maximum capacity history is
created for each one.
The main difference between a wind farm and a group of conventional generator is that the
capacity of wind farms are dictated not only by rated power output, but also by available
wind. Thus, a wind farm power output time series itself is used as the history of wind farm
capacities. The history will be updated when generators are on outage. In this analysis, the
wind speeds for each wind farm are computed using samples from a Weibull distribution.
The Weibull distributions are fitted to real wind data from the NREL Eastern Wind Dataset
[75]. The wind turbine power output is computed using the key equation from fluid flow

studies below [65]
Pw =



0 s ≤ ci
s3−c3
i
c3
r−c3
i
ci < s < cr
Pr cr ≤ s ≤ co
where s is the wind speed of the turbine, ci, cr, co are the cut-in, rated, and cut-out wind
speeds respectively, and Pr is the rated power output of the turbine. This equation assumes
that the wind velocity is perpendicular to the turbine’s rotor swept area. In this study, the
wind farm power output is computed by scaling this value by the number of turbines in the
wind farm. Thus, natural wind speed variation over the wind farm geography and turbine
wake effects are neglected.
Finally, the power available to the system from the wind farm is limited according to the
failure states of the wind turbine generators (WTGs) and the transmission intertie. Trans-
former and collector reliabilities are neglected for simplicity, though incorporation of these
series elements would be a straightforward extension of WTG or transmission line failures
[43]. Should the transmission intertie be the factor limiting power injected into the grid,
wind power generator is curtailed.
In this study, it is assumed that the wind turbines have capacities and rated powers of 2MW,
cut-in, rated, and cut-out speeds of 5m/s, 20m/s and 30m/s. It is assumed that transmission
interties have capacities to carry 80% of the maximum capacity of the wind farm [76].
3.3 Input Data
The following sections describe how the load, cost data, candidate lines and wind farms, and
system data were procured for the study.

3.3.1 Cost Modeling
Conventional Generator and Wind Farm Costs and Parameters
In order to evaluate the merit of an investment decision, the life-cycle cost of the system
should be computed. In actuality, this is a very difficult problem involving uncertainties
inherent to load and fuel-commodity price forecasting, potential changes in reserve require-
ments, and potentially even changing market structures over the entire life of the system
components [13]. In this methodology, only a few key costs will be considered.
Conventional generators are modeled by their traditional quadratic cost expression aP2
+
bP + c where P is the real power produced at the terminals of the generator and a, b, and
c are parameters that can be found for a given power plant by fitting dollar-cost vs real
power produced curves [77]. The values for the generator cost and operating parameters are
pre-compiled in MATPOWER. Because the conventional generation already exists, there is
no need to estimate its installation costs in this methodology.
In general, wind farms have capital costs and operations costs. Capital costs are dominated
by the cost of the turbines themselves, with 70% of the costs comprised of the turbine costs
and the rest comprised of land and labor costs of installation. Thus, the cost of the wind
farm can be estimated given only the number of turbines installed. The average price of
wind turbines in 2012 was $1140/kW [66]. In the future, the total investment cost to install
a wind turbine will likely decrease, but orders for turbines would have to be made with
a significant lead time. Thus it is assumed that the 2012 prices approximate the costs of
turbines in this work. Using a linear wind turbine cost multiplier to calculate the cost of a
wind farm most likely results in the underestimation of costs of installing a wind farm with
a small number of turbines. Assuming wind turbines are not penalized for unpredictability
or non-dispatchability in the market, only maintenance factors into the operations costs of
the turbines. Wind farm operating costs are assumed to be negligible in this work.

3.3.2 Cost and Impedance of Transmission Upgrades
The cost of a transmission upgrade depends on many factors including length of the transmis-
sion lines, land acquisition requirements, ground slope, Earth foundation in the right of way,
and requirements for upgrading other substation equipment. It is clear that costs may differ
greatly for different types of projects. For example, a new transmission line between two
new substations may require permitting and land acquisition, construction of new towers, as
well as significant substation upgrades. However, the cost of reconductoring in an existing
circuit may be dominated by conductor length, dispatch and installation man-hours, and
costs associated with substation outage. [78, 79, 80]. The WECC transmission cost study
was used as a guideline to establish the costs of transmission lines. Cost per mile for 230kV
and 500kV voltage classes was used (consistent with MTTF and MTTR data collection), as
well as multipliers associated with projects shorter than 3 miles or between 3 and 10 miles.
For new transmission lines, right-of-way capital costs and multipliers for tower requirements
were also included.
A few simplifying assumptions were made when producing the costs of the candidate trans-
mission upgrades for the IEEE 14 bus system. All lines were assumed to be single circuit.
Note that this means that some line reconductoring costs may be underestimated if the
transmission corridor has two or more circuits.
Upgrades to existing lines were assumed to be reconductoring costs. Reconductoring the
lines is assumed to increase the capacity roughly 30%. It was assumed that transmission
parameters changed with the same percentages as upgrades from Condor to Grackle conduc-
tors. That is, reactance decreases to 93% of the original line (resistance, though not used
in this work, decreases to 66%) [67]. The two conductors were chosen to be consistent with
the reactance per mile in the transmission line length estimation, and provide significant
capacity increase without placing undue strain on the towers due to a conductor weight in-
crease. Unfortunately, even the 30% increase in capacity implied a 50% increase in conductor
weight, which is possibly unacceptable without upgrading or replacing towers.

New transmission corridors are considered to be single circuit with basic towers, terrain cost
multipliers close to one, and the same land-cost multiplier for each new corridor associated
with a middle point between urban and rural. The capacity and impedance parameters of
the new transmission corridor circuits are assumed to be similar to nearby existing lines of
the same estimated length and the same voltage level.
Note that the above assumptions are made because the real IEEE 14 bus system information
is not known. A utility could produce more realistic cost and impedance input parameters
using available estimates or data without needing to change the rest of the methodology.
Because data had to be fabricated for this fictitious system, a challenge arose. The costs
were computed using data from real transmission upgrades [80] and conductors [67] that
carry on the order of 1000A, but the capacities of the IEEE 14 bus branches are small.
Thus, there is possibly an imbalance between upgrade cost and impact in this methodology.
This may impact the selection of candidate transmission lines for cost reasons. However,
because all the transmission capital costs were created using the same methodology, the
appropriate mix of may be selected to improve reliability at the cheapest cost. This brings
to a close the discussion of the computation of capital costs of transmission upgrades.
Operating costs are not considered for transmission lines.
The resulting transmission upgrade costs are on the order of 10M USD. Specific values for
the costs can be seen in the appendix. These costs appear to be reasonable according to a
brief perusal of the PJM Transmission Construction Status database as of September 2015
[81].
3.3.3 System Load and Uncertainty
The future load uncertainty is represented in this Stochastic Optimization as an uncertainty
set. This set is a collection of system loads sampled from a random variable that in this
case represents the forecast mean and variance. Expected values of cost and reliability can

be computed using this set [31]. To develop the uncertainty set, the industry practice of
constructing confidence intervals on the future peak aggregate load (as well as a point-
estimate load for comparison) will be used [41].
In order to establish the boundaries of the uncertainty set, point forecast and the variance
of the forecast error are estimated. The mean 10 year load forecast and the current load
of PJM load zones [82] are used to compute load growths that are applied to buses in the
IEEE 14 bus system to achieve the mean forecast. The forecast error variance is equated
to PJM’s forecast error factor (FEF) of 1% per year multiplied by the study year (10) [15].
The standard deviation of .1 multiplied by the mean is selected. This requires past data
for forecast load over the study year of interest and the measured actual aggregate system
peak load. The load uncertainty set can be developed by allowing the load to vary between
its mean plus or minus some number of standard deviations [83]. In this methodology, 2
standard deviations are chosen in order to enhance the differences between using a point
forecast and an uncertainty set in the optimization framework. The author stresses the
difference between this set of bounds and the 95% confidence level (the load forecast error
is of unknown distribution).
With the information above can we generate all of the bus loads throughout the system by
taking a single sample from the multi-variate Gaussian we estimate. Sampling the multivari-
ate Gaussian is performed by MATLABs mvnrnd() function. A mathematical representation
of this process is shown below
G =
PPJM forecast
PPJM today
P14 bus forecast = G ∗ P14 bus today
σpjm = Cov(PD hourly PJM )
U = {u : u ∼ N(P14 bus forecast, σpjm) , ¯ui − 2σi ≤ ui ≤ ¯ui + 2σi ∀i = 1...14}
where U is the uncertainty set, composed of vectors of bus loads u, whose elements ui are

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