Modeling Considerations for the Long-Term …...Modeling Considerations for the Long-Term Generation and Transmission Expansion Power System Planning Problem Elliott J. Mitchell-Colgan

Modeling Considerations for the Long-Term Generation andTransmission 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 TransmissionExpansion 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 PLtare 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

bmij (θmi − θmj ) = PmGi − PDi ∀i ⊂ n ∩m ⊂ C

bij(θi − θj) ≤ PkMAX ∀(i, j) ∈ (LC ∪ LE)

bmij (θmi − θmj ) ≤ 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 ≤ ymaxi , ∀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

aiP2i + biPi + ci

subject to

P = B−1θ(∑i∈G

Pi

)−

(∑i∈D

di

)= 0(∑

i∈B

Pi + ci − di

)−∑

k∈TLi

Pik = 0, ∀i ∈ B

Pmini ≤ Pi ≤ Pmaxi , ∀i ∈ CG

ywPminw ≤ Pw ≤ ywPmaxw , ∀w ∈ WF

− Pminij ≤ Pij ≤ Pmaxij , ∀i ∈ TL

− wkPmaxk ≤ Pk ≤ wkPmaxk , ∀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−1θ(∑i∈G

Pi

)−

(∑i∈D

di

)= 0(∑

i∈B

Pi + ci − di

)−∑

k∈TLi

Pik = 0, ∀i ∈ B

Pmini ≤ Pi ≤ Pmaxi , ∀i ∈ CG

ywPminw ≤ Pw ≤ ywPmaxw , ∀w ∈ WF

− Pminij ≤ Pij ≤ Pmaxij , ∀i ∈ TL

− wkPmaxk ≤ Pk ≤ wkPmaxk , ∀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)

√√√√ 1N(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 col

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Modeling Considerations for the Long-Term …...Modeling Considerations for the Long-Term Generation and Transmission Expansion Power System Planning Problem Elliott J. Mitchell-Colgan