MODELING OF VARIABLE ENERGY RESOURCES IN POWER SYSTEM
RELIABILITY ASSESSMENT
By
Samer Sulaeman

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Electrical Engineering-Doctor of Philosophy
2017

ABSTRACT
MODELING OF VARIABLE ENERGY RESOURCES IN POWER SYSTEM
RELIABILITY ASSESSMENT
By
Samer Sulaeman
In recent years, the reliability evaluation of power systems has been receiving increasing
attention. This is largely because of the ongoing changes in generation portfolios and environmental constraints. In the assessment of the increasing penetration levels of variable
energy resources (VERs), such as wind turbine generators (WTGs) and photovoltaic (PV)
systems, reliability methods associated with improved modeling and related phenomena have
an essential role. One important factor the planning of wind power and PV systems projects
is to measure the contribution of such resources to the adequacy of the power system. For
power systems, the intermittent nature of VERs introduces a new level of complexity of modeling such resources using traditional reliability methods used for conventional generators. In
contrast to conventional generators, VERs differ in terms of capability and dispatchability.
Also, the fuel source of VERs cannot always be made available and controlled. This is due
to the fact that the outputs of VERs depend mainly upon the availability of the input (wind
or solar irradiation), and their outputs show a high degree of correlation. Therefore, the
output of individual generators in the same farm cannot be modeled independently in the
probabilistic reliability studies. The work presented in thesis develops an analytical method
to model the output of large wind farms and PV systems for power system reliability assessment. Further, a reliability model of VERs that can be used for power system reliability
assessment including VERs is developed. The solution of the aforementioned problems is
addressed in this work by separately modeling the independent mechanical failures of gener-

ation units and the dependency on the input, and then convolving the two distributions. The
resulting model includes both probability and frequency distributions of the power output
of the VERs. The proposed method reduces the complexity of modeling the VERs, as it
considers the effects of input dependency, correlation, and system component failures on reliability assessment. The resulting model is used to evaluate the reliability of test systems in
the presence of VERs in several case studies. In power system adequacy assessment, capacity
value is used to measure the contribution of VERs to the overall system adequacy. Iterative
methods are traditionally used to calculate the capacity value of VERs. However, iterative
methods are computationally demanding, especially for large systems. In this work, a direct,
non-iterative method is developed to calculate the capacity value of VERs while considering
the effect of the input uncertainty and the failure of generation units. The method developed in this work is based on augmenting the cumulative distribution function (CDF) of
the generation margin (prior to adding VERs) to include the output power of VERs. From
the augmented CDF of the generation margin, the capacity value is analytically determined
without performing iterations. The presented method reduces modeling complexity and the
computational burden associated with calculating the capacity value of VERs.

Copyright by
SAMER SULAEMAN
2017

My Mother, wife and my four little children

v

ACKNOWLEDGMENTS

First and foremost, I would like to express my sincerest and deepest appreciation to
my advisor, Professor Joydeep Mitra. It has been an honor to be his Ph.D. student. He
has taught me and inspired me, both consciously and unconsciously, how to be a part of a
research team. Notably, for creating the research environment in which I have performed
my graduate studies. I appreciate his contributions of time, ideas and interminable support
to make my Ph.D. experience productive and stimulating. The joy and enthusiasm he has
for his research group was contagious and motivational for me, even during tough times in
the Ph.D. pursuit. I am greatly fortunate to have him as my advisor and mentor.
I would also like to thank Professor Subir Biswas, Professor Fang Z. Peng, Professor
Shlomo Levental and Professor Jongeun Choi for serving on my committee and their valuable
suggestions and comments. A word of gratitude goes to Professor Tim Hogan and Professor
Kalyanmoy Deb for their support to complete my work. I would like to thank Dr. Mohammed
Ben-Idris for his contribution, comments, suggestions, and collaboration.
I sincerely acknowledge the funding sources that helped me to complete my Ph.D. work:
the Libyan North American Program and the Ministry of Higher Education and Scientific
Research-Libya, and the Electrical and Computer Engineering Department at Michigan State
University. I would also like to thank my friends at Michigan State University and the
research group at the ERISE lab.
Most of all, I appreciate the support and encouragement of my family: my mother, my
wife, and my sisters.

vi

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

Chapter 1 Introduction . . . .
1.1 Motivation . . . . . . . . .
1.2 Challenges . . . . . . . . .
1.3 Organization of the Thesis

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Chapter 2 Background and Literature Review . . . . . . . . . . .
2.1 Modeling the Output of Wind Farms in Power System Adequacy .
2.2 Modeling the Output of PV Systems in Power System Adequacy .
2.3 Capacity Value . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Sizing the Energy Storage . . . . . . . . . . . . . . . . . . . . . .

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Chapter 3 Models and Methods . . . . . .
3.1 Power System Adequacy Assessment . .
3.2 Analytical Methods . . . . . . . . . . . .
3.2.1 Generation Model . . . . . . . . .
3.2.2 Load Model . . . . . . . . . . . .
3.2.3 Generation Reserve Model . . . .
3.2.4 Calculation of Reliability Indices
3.2.5 Simulation Methods . . . . . . .

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Chapter 4 Comprehensive Models of Variable Resources . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Wind Power . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Attributes of Wind Power Output . . . . . . . . . . . .
4.2.1.1 Wind Turbine Output Power Curve . . . . . .
4.2.1.2 Correlation Between Turbine Outputs . . . .
4.2.1.3 Correlation with Other Stochastic Elements .
4.3 Reliability Modeling of Wind Farms . . . . . . . . . . . . . . .
4.3.1 Modeling of Forced Outages of WTGs . . . . . . . . .
4.3.2 Clustering of WTG Outputs . . . . . . . . . . . . . . .
4.3.3 Combining Variability and Failure Models . . . . . . .
4.4 Model Validation and Test Results . . . . . . . . . . . . . . .
4.4.1 Test System . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Case Studies . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2.1 Reliability Analysis–Base Case . . . . . . . .
4.4.2.2 Reliability Analysis–After Adding Wind Farm

vii

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4.4.2.3

4.5

4.6
4.7

Reliability Analysis–After Adding Wind Farm A Using Monte
Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2.4 Effect of Correlation with Load . . . . . . . . . . . . . . . .
Photovoltaic (PV) Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Attributes of PV System . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.2 Grid Level PV Farm Structure . . . . . . . . . . . . . . . . . . . . . .
4.5.3 Output Power of PV Systems . . . . . . . . . . . . . . . . . . . . . .
4.5.4 Attributes of PV System Components . . . . . . . . . . . . . . . . .
4.5.5 Reliability Modeling of Major Components of PV Systems . . . . . .
4.5.5.1 Power electronic circuit components . . . . . . . . . . . . .
4.5.5.2 Reliability of PV Panels: . . . . . . . . . . . . . . . . . . . .
Reliability Modeling of PV system . . . . . . . . . . . . . . . . . . . . . . . .
Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 5 Capacity Value of Variable Energy Resources . . . . . . . . . . .
5.1 Reliability-Based Methods used to Calculate the capacity Value of VERs . .
5.1.1 Effective Load Carrying Capability . . . . . . . . . . . . . . . . . . .
5.1.2 Equivalent Conventional Firm Power (ECFP) . . . . . . . . . . . . .
5.1.3 Equivalent Conventional Power (ECP) . . . . . . . . . . . . . . . . .
5.1.4 Approximation Methods . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 A Direct Method to Calculate the Capacity Value of VERs . . . . . . . . . .
5.3 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Capacity Value of Wind Farm–Forced Outages and Wake Effect . . .
5.3.1.1 Capacity Value of Wind Power Considering the Forced Outages of WTGs . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1.2 Capacity Value of Wind Power Considering the Wake Effect
5.3.2 Effect of Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Capacity Value of VERs Versus Penetration Levels . . . . . . . . . .
5.3.4 Validation of Proposed Method . . . . . . . . . . . . . . . . . . . . .

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Chapter 6 Quantification of Storage Necessary to Firm Up Wind Generation
6.1 Energy Storage System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Energy Storage Sizing . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.2 Energy Storage Operation . . . . . . . . . . . . . . . . . . . . . . . .
6.1.3 Approach for Reliability-Based Storage Sizing . . . . . . . . . . . . .
6.2 Network Modeling And Reliability Evaluation . . . . . . . . . . . . . . . . .
6.2.1 System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Calculation of Reliability Indices . . . . . . . . . . . . . . . . . . . .
6.2.2.1 Mean down time (MDT) . . . . . . . . . . . . . . . . . . . .
6.2.2.2 Loss of load probability (LOLP) . . . . . . . . . . . . . . .
6.2.2.3 Loss of load frequency (LOLF) . . . . . . . . . . . . . . . .
6.2.3 Expected demand not supplied (EDNS) . . . . . . . . . . . . . . . . .
6.2.4 Stopping Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Wind Farm Locations and System Reliability . . . . . . . . . . . . .

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viii

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6.3.2

Storage System Augmentation for Different Wind Data Sets . . . . .

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Chapter 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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BIBLIOGRAPHY

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ix

LIST OF TABLES

Table 4.1

Wind Turbine Reliability Data Used in the Test Cases . . . . . . .

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Table 4.2

Reliability Model: Farm Output Bands and Probabilities . . . . . .

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Table 4.3

Reliability Model: Farm Output States and Probabilities . . . . . .

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Table 4.4

Reliability Indices of the IEEE RTS (Base Case) . . . . . . . . . . .

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Table 4.5

Annual Reliability Indices of the IEEE RTS After Adding Farm A
Considering Different Values of Turbine FOR . . . . . . . . . . . . .

44

Annual Reliability Indices of the IEEE RTS After Adding Farm A
Considering Different Values of Turbine FOR, Using Monte Carlo
Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

Table 4.7

Reliability Indices of the IEEE RTS After Adding Farm B . . . . .

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Table 4.8

Reliability Indices of the IEEE RTS After Adding Farm C . . . . .

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Table 4.9

Reliability Indices of the IEEE RTS With 100 MW PV Farm . . . .

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Table 5.1

Reliability Indices and Capacity Value–150 MW Wind Farm A Considering the Effect of Different FORs of WTGs . . . . . . . . . . . .

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Reliability Indices and Capacity Value–150 MW Wind Farm A Considering the Wake Effect and Different FORs of WTGs . . . . . . .

65

Table 5.3

Reliability Indices and Capacity Value–150 MW Wind Farm B . . .

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Table 6.1

System Reliability Indices for IEEE RTS with 200 MW Wind Power
Added at Different Buses: Data Set A . . . . . . . . . . . . . . . . .

84

System Reliability Indices for IEEE RTS with 200 MW Wind Power
Added at Different Buses: Data Set B . . . . . . . . . . . . . . . . .

85

System Reliability Indices for IEEE RTS with 200 MW Wind Power
Added at Different Buses: Data Set C . . . . . . . . . . . . . . . . .

86

System Reliability Indices for IEEE RTS with Wind Power and Energy Storage at Different Buses: Data Set A . . . . . . . . . . . . .

87

Table 4.6

Table 5.2

Table 6.2

Table 6.3

Table 6.4

x

Table 6.5

Table 6.6

System Reliability Indices for IEEE-RTS with Wind Power and Energy Storage at Different Buses: Data Set B . . . . . . . . . . . . .

88

System Reliability Indices for IEEE RTS with Wind Power and Energy Storage at Different Buses: Data Set C . . . . . . . . . . . . .

88

xi

LIST OF FIGURES

Figure 3.1

Adequacy assessment methods applied for power system . . . . . .

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Figure 3.2

Adequacy assessment of power system hierarchical levels . . . . . .

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Figure 3.3

Generation reliability evaluation process . . . . . . . . . . . . . . .

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Figure 3.4

Two-state model of a generating unit

. . . . . . . . . . . . . . . .

18

A typical cumulative probability distribution function of generation
reserve margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

Figure 4.1

Model combining wind variability and turbine failures . . . . . . . .

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Figure 4.2

Bands of the output power and wind speed distribution . . . . . . .

35

Figure 4.3

The combined outage due to transition of wind power (horizontal
transitions) and failures of wind turbines (vertical transitions) . . .

36

Figure 4.4

Single line diagram for IEEE-RTS . . . . . . . . . . . . . . . . . . .

40

Figure 4.5

Model combining input variability and system component failures .

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Figure 5.1

Cumulative distribution function of an augmented margin before and
after adding VERs . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

Figure 5.2

Capacity value of wind power as a function of penetration level . .

67

Figure 5.3

Capacity value of wind power using the proposed method . . . . . .

68

Figure 5.4

Capacity value of wind power as function of peak load increase using
iterative method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

General procedure for reliability-based storage sizing . . . . . . . .

77

Figure 3.5

Figure 6.1

xii

Chapter 1
Introduction
In recent years, the drive to bring about technological and regulatory changes that concern
energy, natural resources and climate change has gathered significant momentum. Out of
the numerous changes that the power grid is undergoing, perhaps the most transformative is
the inclusion of VERs such as wind and photovoltaic (PV) systems. Despite their numerous
advantages, the inclusion of variable energy resources (VERs) in the steady-state analysis of
power systems such as power system reliability assessment, for instance, has also brought up
a number of technological and operational issues that need to be addressed. The input uncertainty and the output fluctuation of VERs bring concerns into the power system planning
and operation especially with increasing VERs penetration levels. In response to the growing
need for a proper method to model the output of VERs with high accuracy and reduction
of modeling complexity of VERs in power system reliability assessment, the work reported
in this thesis develops a reliability model of VERs for power system reliability studies.

1.1

Motivation

In response to the rapid growth of the global economy, environmental concerns, economic
incentives of fossil fueled generating units, and global warming awareness, the movement towards renewable energy resources has gained significant momentum. Despite the advantages
of such resources, the new reality of input uncertainty and output fluctuation has introduced

1

another layer of complexity to the planning and operation of power systems. From a reliability point of view, the major problem and barrier to effectively utilizing the intermittent
resources is the fact that they are intermittent and stochastic in nature. For instance, a wind
farm comprises a multitude of wind turbine generators (WTGs) spread out over a geographic
area. The operating characteristics of wind turbines differ from those of traditional power
plants in terms of their capability and dispatchability. Moreover, in a wind farm, all wind
turbines are driven by the same input (wind). Output depends upon the variable input and
output correlation of wind turbines, which increases modeling complexity. In addition to
the dependency on the input and the correlation between generation units, failure of WTGs
adds more challenges and complexity to the application of reliability assessment methods
that are applicable to conventional generators.
The grid-connected PV systems are constructed from a large number of components
such as power electronic circuits and solar panels. The functionality of such components
is highly dependent upon the operating and ambient conditions. The operating conditions
and environmental factors may lead to an overall reduction in output power and the failure
of system components. Given the large number of components involved in PV systems and
the intermittency of the source (solar), modeling of these systems in reliability evaluation
studies is a complex task due to the dependency of output power on an intermittent source
(solar) and the availability of a large number of system components. In addition, in a PV
farm, PV panels are placed adjacent to each other, and all PV panels will be subjected to the
same solar radiation and experience approximately the same operational and environmental
factors that may affect the output of the farm. Failure of wind farm generation units and PV
systems have a significant impact on available power. Several studies show that the operation
and environmental factors could lead to different failure modes of VERs [1–8]. In addition,
2

when input is not available, generators would be unavailable to generate power. The effect
of input shortage in this case can be regarded as forced outage rates, which are much higher
when considering the intermittency of the input. In general, Mote Carlo simulation is usually
adopted for a reliability model of variable energy resources, Mote Carlo simulation able to
capture a wide range of modeling detail and system dependencies. However, Mote Carlo
simulation becomes time consuming, especially when implemented in a large and complex
system. Further, solution accuracy depends mainly on the number of sampling performed.
The purpose of this research is to construct a model for a cluster of output power for wind
farms and PV systems. For power system adequacy assessment, Monte Carlo simulation
is normally not considered appropriate for model building. A direct analytical method is
considered a better approach for building reliability models. This work develop a model
that combines a cluster of variable resources into a single equivalent multi-state unit that
captures their correlation, output variability, and forced outages that can be incorporated
into traditional probabilistic planning methods without further regard to the correlation
between generating units within the farm.

1.2

Challenges

The challenges that were addressed in this thesis are summarized as follows:
A. One of the challenges imposed by the increasing penetration of wind generation and PV
systems is how to model the output power of such resources, considering the operational
characteristics and failures of generation units.
B. The intermittent nature and uncertainty of variable energy resources introduces complexity to applying traditional reliability methods applied to conventional generators.
3

C. The operating characteristics of variable energy resources differ from those of traditional power plants in terms of their capability and dispatchability.
D. Variable energy resources are contracted over a geographic area in the form of farms.
These farms comprise a multitude of generation units (wind turbines or PV panels)
spread out over a geographic area, and their output shows correlation. Therefore, the
output of individual units cannot be modeled as independent random variables.
E. For a wind farm, the output power of wind turbines in a particular geographical location
is highly dependent on the wind speed regime at that particular site. In addition, wind
speed experiences variation all the time, and the dependency on geographical location
and seasonality increases the complexity of providing a generalized model that describes
the correlation between wind turbines output at a wind farm location.
F. Capacity value is used as a measure of the capacity contribution of intermittent resources in power system capacity planning. An evaluation process in this case is carried
out to make sure that the system reliability will be maintained when the VERs are
introduced to the power grid. The input variability, failure of VERs, and correlation is
a matter of modeling complexity in evaluating the capacity value of VERs. Therefore,
in the assessment of such resources in the traditional effective load carrying capability calculation method, the output of a wind farm or PV system is considered a load
modifier. The capacity credit in this case may not be realistically represented.
G. Iterative methods are traditionally used to calculate the capacity value of VERs. However, iterative methods are computationally expensive and time consuming, especially
for large and complex systems. Therefore, developing a direct, non-iterative method
that considers the effect of the input intermittency, forced outages of VERs, and corre4

lation of the output of VERs with system load would present a more computationally
efficient alternative to the system planner.
H. Although the integration of wind power has several advantages, it poses several technical challenges such as variability and uncertainty of wind speed and failures of wind
turbine generators (WTGs) which may deteriorate the reliability of power systems.
One of the most practical solutions to mitigate these drawbacks is the use of energy
storage systems. However, variability and uncertainty of wind speed and failures of
WTGs not only deteriorate the reliability of power system but also complicate the
process of determining suitable sizes of energy storage systems for a specific reliability
target.

1.3

Organization of the Thesis

The remainder of this thesis is organized as follows: Chapter 2 presents a literature review
of the methods used in power system adequacy assessment in presence of VERs. Chapter
3 discuses the reliability method used for power system adequacy assessment. Chapter 4
presents the development of an analytical method to model the outputs of VERs in power
system adequacy assessment. Chapter 5 discuses the reliability methods used for calculating
the capacity value of VERs and presents the development of a new method to calculate the
capacity value of VERs. Chapter 6 presents an implementation of an analytical approach to
quantify the size of energy storage to firm up wind power. Chapter 7 summaries the findings
and provides a general conclusion about the methods that have been developed in this work.

5

Chapter 2
Background and Literature Review

2.1

Modeling the Output of Wind Farms in Power System Adequacy

The reliability assessment of systems that integrate wind power and associated technical
aspects such as adequacy assessment and the determination of wind farm capacity values
has been amply addressed in the literature [9–14]. In [15] a discrete state algorithm is adopted
to model the reliability of unconventional energy resources. The hourly modified generation
model of unconventional generators based on their generation capacity is combined with
conventional generators to determine the hourly loss of load expectation and the frequency
of capacity deficiency. In [16], load and wind are treated as correlated random variables.
Monte Carlo simulation is used to evaluate system reliability. The time correlation between
load and wind power is considered in the clustering process by assigning for each load value
a specific value of wind power that occurs in the same time-frame.
Several methods have been proposed for analytical reliability modeling of wind farms to
include the effect of forced outages of WTGs. For instance, a multi-state capacity outage
probability model has been proposed in [17]. However, the frequency and duration indices
in power system reliability assessment cannot be calculated using this model. A more comprehensive model that takes into account frequency and duration indices has been proposed

6

in [18]. This model combines the wind speed model and the WTG reliability model using
a multi-layer concept. In this model, the wind speed is classified into a finite number of
states, and WTGs are modeled as two states, up and down. A birth-death Markov chain is
used to model wind speed transitions. However, in reality, wind speed transitions cannot be
assumed to transit smoothly to the next neighboring state through the birth-death process
unless the sampling time is close to zero. Therefore, if the model is to be used for wind
speed, some transitions between wind speed will be lost, and the steady state probabilities
will not be correct. Also, two failure rates (corresponding to normal and extreme wind
speeds) for WTGs are considered, which require duplicating the model although the effect of
failures due to extreme wind speeds is very low, as concluded in [18]. In fact, if extreme wind
speeds are considered higher than the cut-out speed of WTGs, these wind speeds correspond
to zero output power, regardless of whether or not the state of the WTGs is up or down.
Therefore, the computation burden can be significantly reduced if forced outages of WTGs
due to extreme wind speeds are aggregated with the state of zero output power. Further, for
a wind farm consisting of n wind speed states and m wind turbines, there will be (2m ∗ n)
possible states. Therefore, for large wind farms, the size of the full Markov matrix would be
extremely large. This adds a complexity to the modeling process and computation burden
due to the large number of system states [14, 19–21].
A method to determine an equivalent capacity of wind farms based on Monte Carlo
simulation is presented in [22]. Monte Carlo simulation has also been used to evaluate the
reliability indices of wind power in [23–25]. Although Monte Carlo simulation can be used for
the evaluation of complex systems, analytical methods using system data and characteristics
are considered more suited for the construction of reliability models. However, the purpose of
this research is to build a model for a cluster of wind turbines, a direct analytical method is
7

considered to offer a better approach for building reliability models. In this work, therefore,
Monte Carlo simulation is only used for model validation.

2.2

Modeling the Output of PV Systems in Power System Adequacy

In the literature, most of the work presented on the reliability aspects of PV systems has
examined the performance of the major power electronic components and circuits, such as
capacitors, IGBTs, or the entire inverter, to assess the reliability of PV systems [26–28]. The
power electronic circuits deliver the output power of PV panels to the load or to the main
grid. Reliability assessment of the power electronic circuits of grid-connected PV systems for
different typologies have been addressed in the literature [29–31]. In the evaluation process,
the mean time to failure (MTTF) has been used as an indicator for system reliability; the
overall reliability of a circuit has been calculated by summing the failure rates (the failure
rate is the reciprocal of the MTTF) of the components [32–34]. However, this is true only
if the failure of a component leads to an overall failure of the power electronic circuit. In
some cases such as multilevel inverters, a redundancy exists in power electronic topologies
and their desired function is not totally affected by a component failure [35, 36]. The Monte
Carlo simulation technique has been used in [37] for reliability assessment of a hybrid system
of wind and PV systems. A combination of deterministic and probabilistic techniques has
been used in the reliability evaluation of these systems. In [38], the hourly mean solar
radiation data has been used as the main input and a Monte Carlo simulation technique was
implemented to evaluate system reliability. In [39], an analytical method has been used to
model the availability of the PV systems in the reliability assessment. In [40], reliability of a
8

wind and solar system has been evaluated using Monte Carlo simulation technique. In [41],
a methodology for PV system reliability and economic analysis has been introduced. The
proposed method is used to model the reliability of a PV system that could be utilized to
estimate life-cycle energy costs.
Grid-connected PV systems are constructed from a large number of components, including power electronic circuits and solar panels. The performance of such components is
highly dependent upon their operation and ambient conditions. Operational and environmental factors may lead to an overall reduction in the output power and the failure of system
components [42,43]. Given the large number of components involved in PV systems and the
intermittency of the source (solar), the modeling of these systems in reliability studies is a
complex task. From the reliability point of view, the output model of PV systems should be
able to capture the effect of the input variability and the failures of system components.

2.3

Capacity Value

Capacity value is used as a quantitative measure to determine the capacity of conventional
units that can be replaced by renewable generation, which enhances system reliability or
at least provides the same reliability target prior to adding these resources to the system.
In determining the capacity value of variable energy resources, the effective load carrying
capability (ELCC) approach has been favorably used among the available criteria [44]. The
ELCC approach is based on calculating the amount of load increase that would result in
reliability indices equal to system indices prior to adding the VERs. The common practice
of the existing methods is the use of the reliability indices such as the LOLE (loss of load
expectation) index as a criterion to determine the capacity value of VERs; other reliability

9

indices such as the LOLF (loss of load frequency) index are rarely considered. In addition,
the impact of forced outages of wind farm generating units on the output is usually ignored.
Several methods have been introduced in the literature to calculate the capacity value
of VERs [44–47]. These methods can be classified into intelligent search methods [48],
simulation methods [49–57], and analytical methods [58–63]. Further, these methods can be
classified into iterative methods [48–60] and non-iterative methods [61–63]. Calculation of the
capacity value of VERs based on iterative methods is deemed to be computationally intensive
due to the requirement of repetitive calculations. On the other hand, non-iterative methods
are direct and computationally efficient; however, some approximations are usually involved.
For instance, the method presented in [61] uses a graphical method that is based on the work
presented in [64] to estimate the capacity value of wind power. The graphical method is based
on approximating the change in the LOLE index versus load increase using an exponential
function. Although the graphical method is computationally efficient, this approximation
could lead to considerable errors [63]. The method presented in [62], the Z-method, uses the
probability distribution of the generation margin to calculate the capacity value of VERs.
However, this method assumes that the probability distribution of the generation margin
preserves its shape even after adding wind power [65]. Also, the methods presented in [61,62]
ignore the correlation between demand and wind power [65]. The non-iterative analytical
method presented in [63] uses the concept of joint probability distribution to calculate the
ELCC and the capacity value of VERs considering the correlation between the load and the
available output power of VERs. However, this method requires a pre-specified value for
the LOLE index to be used to calculate the ELCC before and after adding VERs. Further,
the accuracy of estimating the ELCC of the system relies on the chosen number of the
“coincidental demand-generation levels,” whose optimal value is difficult to determine, as
10

the authors have concluded [63].
Iterative methods are commonly used to calculate the capacity value of VERs. However,
iterative methods are computationally expensive and time consuming especially for large
and complex systems. In addition, determining the optimum sizes of VERs in terms of
their capacity values, has not been given much attention; using iterative methods would
be extremely complex and computationally expensive as they require several runs for every
candidate-installed capacity. Therefore, developing a direct and non-iterative method that
considers the effect of the input intermittency, forced outages of VERs generation units,
and correlation of the output of VERs with the system load would reduce the computation
burden and enable the planner to determine optimum penetration levels of VERs.

2.4

Sizing the Energy Storage

The integration of energy storage systems with intermittent sources has become a practical
solution for overcoming these challenges, largely because of the rapid growth in deployment
and improved technology of energy storage systems.
In the literature, several authors have discussed the economic aspects of energy storage
systems in the electricity market [66–68]. In [69], dynamic programming with time-shift
is used to determine the optimal size of battery storage in utility load-leveling operations
applied to the Kansas City Power and Light system. The results showed that the optimal
size of battery storage varied with the daily and seasonal load variations. Reference [70]
uses Benders decomposition to determine the optimal location and size of a compressed
air energy storage system. The optimal size of the energy storage is evaluated based on the
capital investment in the energy storage system that leads to daily operating cost reductions.

11

Also, according to [70], the trade-off between the capital cost of the energy storage and the
daily system operating cost becomes insignificant as the energy storage investment increases.
This can be attributed in part to the high variation of wind power and the high cost and
limited operating lifespan of energy storage systems [67,71,72]. Due to these factors, several
management and control systems have been proposed to optimize their operation [73–75].
In [76], a predictive control method is proposed for controlling the charging and discharging
of a battery storage system; the authors concluded that the size of the battery storage system
needed to mitigate the hourly variation of wind power and achieve minimum operating costs
should be around 75% of the nameplate capacity of the wind farm.
In practice, different types and sizes of energy storage systems have been deployed for
different applications such as energy management, backup power, load leveling, frequency
regulation, voltage support, and grid stabilization [77,78]. However, large scale adoption and
deployment of energy storage technologies is still limited, largely due to the aforementioned
factors [72, 79]. In spite of the fact that the increased generation from renewable sources
reduces overall generating costs, emissions, and consumption of fossil fuels, the intermittency and uncertainty of the output raise several concerns, including peak load capability
and system adequacy [80–85]. Moreover, maintaining the efficiency and reliability of a grid,
especially with high penetration of variable generation, has become a challenging task. Several studies have shown that high penetration levels of wind power could significantly affect
power system quality and security due to its intermittency [80, 83].
The cost of an energy storage system is a major factor in the planning of energy storage
system projects. In the literature, several authors have summarized and discussed the cost
of integrating energy storage systems [86–88]. These studies show that the cost of an energy
storage system depends on several factors, including energy storage size and the types of
12

technology used. One of the practical and promising solutions to the problem of energy
storage sizing was proposed by Mitra [89, 90]. The proposed method in [89, 90] was used to
quantify the size of energy storage systems analytically, in terms of both power and energy
capacity, to meet a reliability target.

13

Chapter 3
Models and Methods

3.1

Power System Adequacy Assessment

Power system reliability evaluation is important for studying the current system to identify
weak points in the system, determining what enforcement is needed to meet future demand
and planning for new reliable power system, i.e., network expansion. Power system reliability
assessment is vital to avoid economic and social losses resulting from power outages.
In general, reliability is used to measure the ability of the system to perform its designated
functions under the conditions within which it was designed to operate. For power systems,
the system reliability assessment is aimed to measure the system ability to deliver electricity
to all points of utilization at acceptable standards and in the amount desired. Power system
adequacy assessment methods that are used for generation systems can basically be classified
into deterministic and probabilistic [91–93], these methods are described in Figure 3.1. The
common criterion for generation adequacy assessment using deterministic method, is to set
generation margins equal to a fixed percentage of the forecast peak demand and operating
generation margins sufficient to supply the most likely contingencies. The disadvantages of
these methods are that they do not account for the stochastic events and the forced outages
of generating units, and uncertainty in load demand. On the other hand, the assessment of
generation system adequacy using probabilistic methods encounter for the stochastic events
and the forced outages of generating units, and uncertainty in load demand [91–93].
14

Power System Adequacy
Assessment

Probabilistic Approach

Simulation Methods (Monte
Carlo Simulation )

Deterministic Approach

Analytical Methods

Reserve Margin

Loss of Largest Unit

Figure 3.1 Adequacy assessment methods applied for power system
Adequacy assessment methods in power systems are mainly applied to three different
hierarchical levels [91, 92]. Due to the complexity of power system, the studies of different
hierarchical levels are analyzed separately [91]. At hierarchical level I (HL-I), the total
system generation is examined to determine its adequacy to meet the demand. This is
usually termed as generating capacity reliability evaluation [91]. In the HL-I studies, the
transmission lines are considered highly reliable and able to transfer the generated power
to the all load points. On the other hand, in the hierarchical level II (HL-II) studies, the
adequacy analysis is usually termed composite system evaluation. HL-II studies can be used
to assess the adequacy of an existing or proposed system including the impact of various
reinforcement alternatives at both the generation and transmission levels. The objective
of the hierarchical level III (HL-III) studies is to obtain suitable adequacy indices at the
consumer load points (i.e. distribution level). The hierarchical levels of system adequacy
assessment are described in Figure 3.2.

15

Generation Adequacy
Assessment

Composite System
Reliability Assessment

Reliability of Distribution
System

Generation Facilities

Hierarchical Level I
HL-I

Transmission Facilities

Hierarchical Level II
HL-II

Distribution Facilities

Hierarchical Level III
HL-III

Figure 3.2 Adequacy assessment of power system hierarchical levels

3.2

Analytical Methods

Adequacy analysis of power systems essentially consists of identification and evaluation of
failure states, states in which the power system cannot satisfy load demand. According
to [91, 92], system adequacy defined as a measure of sufficient facilities within the system
to deliver an aggregate electric power and energy requirements without violating system
operational constraints. System adequacy is associated with static conditions which do
not include system disturbances. The well-known unit addition algorithm is considered a
practical method for large-system adequacy analysis. This is a recursive algorithm that
starts with the distribution of one unit and successively convolves with it the distributions
of the remaining units, one unit at a time. This technique can be used for two-state or
multi-state unit and provides a fast technique for building capacity models [93]. Generation
adequacy assessment is generally used to evaluate system reliability indices for short-term
and long-term generation capacity planning. In this work, the discrete convolution method
16

is used in evaluating the reliability of the system by convolving the discrete probability
distributions of generating units to construct a capacity outage probability and frequency
table (COPAFT) [93]. In reliability analysis, generation and load models are needed to
construct a generation reserve margin of the system from which the reliability indices are
calculated.
The reliability assessment using analytical method, the generating system model used for
generation capacity adequacy assessment is known as capacity outage probability table and
frequency table (COPAFT). In this process, the generation and load models are convolved
to form an appropriate risk model where the element of interest is the risk of generation
capacity less than the load. In short, adequacy evaluation of generation systems consists of
four general steps as shown in Figure 3.3 [91, 92].

Generation Model

Load Model

Evaluation of Generation
Adequacy

Reliability Indices

Figure 3.3 Generation reliability evaluation process

17

3.2.1

Generation Model

For a system composed of N generating units with known forced outage rates (FORs), the
COPAFT is built using the unit addition algorithm. This is a recursive algorithm that starts
with the distribution of one unit and successively convolves it with the distributions of the
remaining units, one unit at a time. Considering failures of generating units as independent
events, the probability of failure of generation unit can be modeled as Markovian components
with two states, up and down states with known failure and repair rates, λ and µ respectively
(λ is the failure transition rate from an up state to a down state, and µ is the repair transition
rate from a down state to an up state) as shown in Figure 3.4.


Unit Up

Unit Down


Figure 3.4 Two-state model of a generating unit

Based on the FOR (forced outage rate is the failure probability) of generation units, the
COPAFT can be built using the unit addition algorithm. This is a recursive algorithm that
starts with the distribution of one unit and successively convolves with it the distributions
of the remaining units, one unit at a time. The FOR is defined as follows [93].

F OR =

r
λ
=
λ+µ
m+r

(3.1)

where m is the mean time to failure (M T T F ): M T T F = λ1 , r is the mean time to
repair(M T T R): M T T R = µ1 .
In general, when adding a new unit of capacity C, the cumulative probability and fre-

18

quency of an outage of X MW can be determined by the following expression: [93, 94].

P (X) = P̄ (X)(1 − q) + P̄ (X − C)q,

(3.2)

where P (X) and P̄ (X) denote the cumulative probabilities of the capacity outage state
before and after the units are added respectively. Equation (3.2) is initialized as follows.

P (X) =





1, if X ≤ 0



0, otherwise,

The cumulative probability P (X) for a forced outage of X MW or greater can be in
general calculated using (3.3).

P (X) =

X

P (Xi ),

∀ Xi ≤ X,

(3.3)

i

where
P̄ (X) = 1,

∀ X ≤ C,

In the case of multi-state generating units, (3.2) can be modified as follows.

P (X) =

n
X

P (i) × P̄ (X − Ci ).

(3.4)

i

The cumulative frequency of capacity outage of X MW can also be calculated using the
same approach using (3.5).

F (X) = Fi (1 − q) + Fj q + (Pj − Pi )qµ,

19

(3.5)

where q and µ are respectively the probability of failure and repair rate of the new added
unit; Pi , Pj , Fi , and Fj are determined from the old COPAFT (prior to adding the new unit);
i is the index of the existing capacity outage state, Ci = X, and j is the index of existing
capacity outage state, Cj , such that Cj = X − C. Fi and Fj are cumulative frequencies of
states i and j respectively.

3.2.2

Load Model

The load model is usually expressed in the form of probability and frequency distribution of
the random variable that represents system load [93,95]. The load model can be constructed
by scanning the hourly load data of the system over the time period of study, usually one
year. In general, the load model can be built in terms of load level Li with its commutative
probability and frequency as follows [93, 95].

H(L ≥ Li )
,
T

(3.6)

Γ (L < Li → L ≥ Li )
,
T

(3.7)

PL (L ≥ Li ) =

FL (L ≥ Li ) =

where H(L ≥ Li ) is the number of hours that the hourly load is greater than or equal to
the load level Li , T is the number of hours in the interval, PL (L ≥ Li ) is the commutative
probability of the load level Li , Γ (L < Li → L ≥ Li ) is the number of transitions from
(L < Li ) to (L ≥ Li ), and FL (L ≥ Li ) is the commutative frequency of Li .

20

3.2.3

Generation Reserve Model

The generation reserve margin has been used to estimate the reliability indices. The generation reserve margin model consists of the probability and frequency of a random variable M ,
which represents the difference between the available generation and load [93, 95]. Though
the rigorous derivation of the generation reserve margin model will not be reproduced here,
some expressions will be presented and explained by means of simple arguments to derive
expressions for analytically determining the capacity value of VERs. The reserve margin
model can be expressed in terms of generation and load model as follows [93, 95].

M = CC − CO − L,

(3.8)

where CO is the capacity outage, L is the system load, and CC is the net available generation
capacity for commitment which can be expressed as follows.

CC = Ci − COP ,

(3.9)

where Ci is the installed capacity, COP is the capacity on planned outage.
For each generation reserve margin level, Mi , the cumulative probability and cumulative
frequency can be calculated using (3.10) and (3.11) respectively.
nG
X





P (M ≤ Mi ) =
PG C j − PG C j + 1
j=1



× PL CC − Cj − Mi ,

21

(3.10)

F (M ≤ Mi ) =

nG
X








FG Cj − FG Cj + 1 PL mij

j=1

+ [PG (Cj ) − PG (Cj + 1)]FL (mij ).

(3.11)

where nG is the number of states in the generation model (the COPAFT), Cj is the capacity
outage level, mij = CC − Cj − Mi , and PG (•) and FG (•) are the cumulative probability and
frequency of the generation model respectively.

3.2.4

Calculation of Reliability Indices

The indices that have been most commonly used in power system reliability analysis are
defined as follows:
A. Loss of Load Probability (LOLP):
A Loss of Load (LOL) event is one in which a system is unable to meet its total demand.
The LOLP index is the probability of encountering one or more LOL events during a
given time period. In general, the LOLP is given by (3.12) [93].

LOLP =

n
X

P (LOLi ),

(3.12)

i=1

where LOLi is the ith LOL event, P (LOLi ) is its probability, and n is the total number
of LOL events encountered during the time period of study.
For the purpose of illustration, consider the CDF of a typical generation reserve margin
as shown in Figure 3.5. The LOLP index can be calculated from the generation reserve
margin, which is the probability of the smallest margin level as indicated in Figure

22

3.5 [93]. Mathematically it can be calculated as follows.

LOLP = P (−∆M ),

(3.13)

where ∆M is the increment at which the probabilities of the margin are computed.

P (M)
P (0)

P (- ΔM)
P (- 2ΔM)
Probability Distribution
Function

P (- 3ΔM)

0

- ΔM

M

- 2ΔM

- 3ΔM

- L+ΔM

-L

P(-L)

Figure 3.5 A typical cumulative probability distribution function of generation reserve margin

B. Expected Power Not Supplied (EPNS):
The EPNS index is the expected (average) demand that the system is unable to serve as
a result of LOL events during a given time period. It is calculated as the weighted sum
of the demands curtailed during the LOL events, the weights being the probabilities of
the corresponding LOL events. It is expressed in MW/year or GW/year. The EPNS

23

index is equal to the area under the probability distribution curve over the domain of
M ≤ 0 as shown in Figure 3.5 [93]. This area can be approximated by the sum of the
areas of the rectangles shown in 3.5, which is given by (3.14) [93].

EP N S = (∆M )[

−L
X

M =0

1
P (M ) − (P (0) + P (−L))],
2

(3.14)

where −L is the smallest (largest in magnitude) negative margin considered.

C. Loss of Load Expectation (LOLE):
The LOLE index is the expected number of LOL hours during a given time period.
It is expressed in hr/year. LOLE index can be directly calculated by multiplying the
LOLP index by the number of hours in a year as expressed in (3.15) [93].

LOLE = LOLP × 8760.

(3.15)

D. Loss of Load Frequency (LOLF):
The LOLF index is the expected frequency of encountering one or more LOL events
during a given time period. The LOLF index can be calculated using (3.16) [93].

LOLF = F (−∆M ).

(3.16)

E. Loss of Load Duration (LOLD):
LOLD index identifying the average deficiency of encountering one or more LOL events

24

during a given time period.The LOLD index can be calculated using (3.17) [93].

LOLD =

LOLP
.
LOLF

(3.17)

These reliability indices are used to assess the reliability performance of a generation system.
In the evaluation process, the generation model is convoluted with generation model to obtain
system indices. The process of convolving the generation model and load model is explained
in [94].

3.2.5

Simulation Methods

Mote Carlo simulation able to capture a wide range of modeling detail and system dependencies,especially when systems are too complex to solve with analytical methods. The
operation characteristics of VERs, such as correlation, input fluctuation, and the random
failure of generating units, can be mimicked by sequential simulation. Monte Carlo simulation is a flexible technique, applicable to simulate the operation behavior of complex systems,
and can easily be implemented to determine reliability indices. On the other hand, Monte
Carlo simulation becomes time consuming, especially when implemented in a large and complex system. Further, the solution accuracy depends mainly on the number of sampling
performed to produce the output [96, 97]. However, the purpose of this research is to construct a model for a cluster of VERs, and Monte Carlo simulation is normally not considered
appropriate for model building. A direct, analytical method is considered a better approach
for building reliability models. In this work, therefore, Monte Carlo is used only for model
validation.

25

Chapter 4
Comprehensive Models of Variable
Resources

4.1

Introduction

In this work an analytical method is described for constructing a reliability model of a wind
farm and a PV system, an analytical method is used to construct a generation model for
both wind farms and PV system. The generation model that represents the output power of
VERs is denoted in this work in terms of COPAFT. A COPAFT represents a combination of
capacity levels with the probabilities of existence and cumulative probabilities to represent
the probability distribution function of the outage states.

4.2

Wind Power

In this work an analytical method is described for constructing a reliability model of a wind
farm, considering turbine correlation, output variability, and forced outages. The resulting
model resembles a multi-state generating unit with probability and frequency distributions
of the discrete capacity states, expressed in the form of a capacity outage probability and
frequency table (COPAFT) [98,99]. This affords the advantage that now the entire wind farm
can be treated as a single, equivalent generating unit that can be incorporated in traditional

26

probabilistic planning methods, without further regard to the correlation between turbines
within the farm. In constructing the proposed wind farm model, several distinctive and
innovative approaches are utilized. These contributions are summarized below and described
further in section 4.3.
1. The number of states is reduced without compromising the model accuracy. This is
achieved by first converting the time series wind speed data into turbine output power
and then clustering the output power data, which is less dispersed and easier to cluster.
The method considers transition of output power between all clusters, not just adjacent
clusters (unlike a birth-death process).
2. The model is comprehensive in that it captures (a) wind variability, (b) turbine outages, and (c) WTG output correlations. This is achieved by separately modeling the
independent outages of wind turbines and the dependency on wind speed, and then
convolving the two distributions. The overall approach is depicted in Figure 4.1.
3. A generalized approach is developed for calculating not only the probabilities but also
the frequencies within the multi-state convolution process described above, where the
tracking of transitions between so many states is fairly complex.

4.2.1

Attributes of Wind Power Output

In power system reliability assessment, several wind power models have been utilized to
calculate the output power of wind farms. For instance, the actual value and the mean
value of the recorded wind speed data at certain wind farm locations have been utilized to
estimate the outputs of WTGs [100, 101]. The output of wind turbines are known to vary
with wind speed and the design characteristics of wind generators. This section describes the
27

characteristics of wind power in terms of WTG output power and the correlation between
the output of WTGs and changes in the wind speed across the farm.

4.2.1.1

Wind Turbine Output Power Curve

Wind turbine power curve provides a quantitative relationship between wind speed and the
output power. It describes the operational characteristics of a WTG. The output power that
can be extracted from wind turbines can be calculated as follows [102].

1
P = Cp ρAv 3 ,
2

(4.1)

where P is the output power (watts), ρ is the air density (kg/m3 ), v is the wind speed
(m/sec), A is the swept area of the turbine (m2 ), and Cp is the power coefficient.
The output power curve combines (4.1) with the physical constrains in the system. The
output power curve including the physical constraints can be expressed as follows.

P =






0,
if v < vcut-in








 12 ρACp v 3 , ifvcut-in ≤ v < vr




Pr ,








0,

,

(4.2)

if vr ≤ v < vcut-out
if vcut-out ≤ v

where vcut-in is the designed cut-in speed, vcut-out is the designed cut-out speed, vr is the
rated speed and Pr is the rated power of the wind turbine.

28

4.2.1.2

Correlation Between Turbine Outputs

In the literature on power system reliability modeling, several studies have investigated the
effect of the correlation between the output power of WTGs and the change in the wind speed
within the same farm [103–109]. The degree of the correlation depends on several factors such
as geographic location, separation distances between WTGs, terrain and number of WTGs
on the wind farm and the influence of the resulting to wake effect and turbulence [110–112].
These correlations also vary with time scales (the frequency of wind speed observations can
be intervals of 5 minutes, 10 minutes, hourly, etc.). Thus, determining the exact degree of
correlation between individual WTGs and the change in the wind speed within the same
farm is a challenging and complex task particularly for large wind farms [113]. However, it
has been reported that WTGs within a wind farm are approximately driven by same values
of wind speed and the power outputs of WTGs show consistent behavior among individual
WTGs with the change in the wind speed on the entire farm [105–109]. Despite individual
variation of WTGs output, it can be safely assumed that there is an average behavior among
wind turbines without excessive deviation across the farm. This simultaneously captures the
correlation across the farm.

4.2.1.3

Correlation with Other Stochastic Elements

Correlations have also been observed between wind production and other stochastic elements
that are considered in power system analyses. For instance, [114] reports negative correlations
between wind and solar production, and [115] reports positive, negative, and negligible
correlations between wind production and system loads depending on time scales, time of day,
and seasons. While some researchers [14, 50] have modeled the correlation with load, others
[116] have postulated that the correlation is negligible. Moreover, the impact of correlation
29

is also affected by such factors as differences in scale (sizes of wind or solar installations
and load), geographic location, and electrical distances between points of insertion. Another
challenge in modeling correlation between wind production and system load is that most
of these models treat wind injections as negative loads, inherently assuming that all energy
that can be produced by a wind farm is completely accepted by the system. Despite wind
production being designated as “must take” in many regulatory regimes, it is well known
that prevailing system conditions such as low loads or network congestion often necessitates
spillage of wind generation. In view of the above reasons, as well as the fact that the stated
intent of the work reported here focuses on the construction of a reliability model of the
production of a wind farm, this work does not attempt to model correlations between wind
production and other stochastic elements. It does, however, evaluate some of the impacts in
the case studies.

4.3

Reliability Modeling of Wind Farms

In this work an analytical method is described for constructing a reliability model of a wind
farm, considering turbine correlation, output variability, and forced outages. The resulting
model resembles a multi-state generating unit with probability and frequency distributions
of the discrete capacity states, expressed in the form of a capacity outage probability and
frequency table (COPAFT) [117]. This affords the advantage that now the entire wind farm
can be treated as a single, equivalent generating unit that can be incorporated in traditional
probabilistic planning methods, without further regard to the correlation between turbines
within the farm. In constructing the proposed wind farm model, several distinctive and
innovative approaches are utilized. These contributions are summarized below and described

30

further
• The number of states is reduced without compromising the model accuracy. This is
achieved by first converting the time series wind speed data into turbine output power
and then clustering the output power data, which is less dispersed and easier to cluster.
The method considers transition of output power between all clusters, not just adjacent
clusters (unlike a birth-death process).
• The model is comprehensive in that it captures (a) wind variability, (b) turbine outages, and (c) WTG output correlations. This is achieved by separately modeling the
independent outages of wind turbines and the dependency on wind speed, and then
convolving the two distributions. The overall approach is depicted in Figure 4.1.
• A generalized approach is developed for calculating not only the probabilities but also
the frequencies within the multi-state convolution process described above, where the
tracking of transitions between so many states is fairly complex.

4.3.1

Modeling of Forced Outages of WTGs

In the assessment of the effect of wind power on power system reliability, failures of WTGs
have a significant impact on the available power [118–122]. Several studies show that the size
of turbines, operation and environmental factors could lead to different failure modes of wind
turbines. Failures of WTGs usually occur due to ageing, worn out parts or manufacturing
defects [118–121]. Considering failures of WTGs as independent events, the probability of
failure of WTG can be modeled as Markovian components with two states, up and down
states with known failure and repair rates, λ and µ respectively (λ is the failure transition
rate from an up state to a down state, and µ is the repair transition rate from a down state
to an up state). Based on the FOR (forced outage rate is the failure probability) of turbines,
31

Powe r (MW )

Speed

Wi nd spee d (m /Sec .)

Time

Hourly Wind Speed
Data

Turbine Output
Power Curves

Turbine Outage
Model

Convolution
Over Bands

Bands of the Output
Power

Turbine Failures
Data

Combined
Outage Model

Reliability
Assessment

Figure 4.1 Model combining wind variability and turbine failures
the COPAFTf of wind turbines can be built using the unit addition algorithm. This is a
recursive algorithm that starts with the distribution of one unit (a WTG in this case) and
successively convolves with it the distributions of the remaining units, one unit at a time. In
general, when adding a new unit of capacity C, forced outage rate q and failure frequency
µ, the cumulative probability and frequency of an outage state X can be determined by the
following expressions [94].

P (X) = P̄ (X)(1 − q) + P̄ (X − C)q

(4.3)

where P (X) and P̄ (X) denote the cumulative probabilities of the capacity outage state
before and after the units are added.
Equation (4.3) is initialized as follows.

32

P (X) =





1, if X ≤ 0



0, otherwise,

The cumulative probability P (X) for a forced outage of X MW or greater can be in
general represented by (4.4).

P (X) =

X

P (Xi ),

∀ Xi ≤ X,

(4.4)

i

where

P̄ (X) = 1,

∀X≤C

In the case of multi-state generating units, (4.3) can be modified as follows.

P (X) =

n
X

Pi × P̄ (X − Ci ).

(4.5)

i=1

The cumulative frequency F (X) of capacity outage X can also be calculated using the
same approach, as shown in (4.6).

F (X) = Fi (1 − q) + Fj q + (Pj − Pi )qµ,

(4.6)

where Pi , Pj , Fi , and Fj are determined from the old COPAFT; i is the index of the existing
capacity outage state Ci = X and j is the index of existing capacity outage state Cj such
that Cj = X − C; Fi and Fj are cumulative frequencies of states i and j respectively.

33

4.3.2

Clustering of WTG Outputs

The first step in the proposed method consists of converting the wind speed time-series
data into output power data using (4.2). The values of wind power that correspond to
the observed wind data are clustered into several bands (states) based on their similarities.
In the clustered wind power, every observation belongs to one and only one band based on
minimum Euclidean distances between the observed data points. By converting the observed
wind speed into power using power curve characteristics of WTGs, several dissimilar and
often different observed wind speed points (that result in similar output powers) will be
grouped in the same bands. For example, wind speeds that range from 0 (m/sec) to the
cut-in speed of the WTG and the wind speeds range from the cut-out speed of WTG to
the maximum observed value are grouped in one band, i.e., the band that have zero output
power which is denoted band 1 in Figure 4.2. Also, wind speeds that range from the rated
speed of WTG to the cut-out speed of WTG are grouped in one band (band n in Figure
4.2).

4.3.3

Combining Variability and Failure Models

A capacity outage probability and frequency table for a wind farm can be constructed by
combining the COPAFTf that represents the rated capacities and failures of WTGs (section
4.3.1) and the clustered wind power table (section 4.3.2). The COPAFTf is convolved over
each wind power band to construct a COPAFT for the entire wind farm (COPAFTc ).
This section explains the combination procedure of these two tables. Figure 4.3 shows the
process of combining the outages of wind power due to (1) wind power transitions between
bands and (2) failures of WTGs. Horizontal transitions represent the transitions due to the

34

Cut–in
speed

band n

band n-1

band 4

band 3

band 2

band 1

Power (MW)

Rated
power

band 1

Wind speed
distribution

Wind speed (m/sec.)

Cut–out
speed

Figure 4.2 Bands of the output power and wind speed distribution
change in the wind power between bands and vertical transitions represent transitions due
to failures of WTGs. The new outage states due to wind power band transition and failures
can be obtained by scanning through all bands in both vertical and horizontal directions. In
convolving the states of the (COPAFTf ) of wind turbines over each band, the capacity outage
state i in the (COPAFTc ) with Co ≥ X occurs in both vertical and horizontal directions.
The procedure of the proposed method of clustering wind power into bands, modeling
failures of WTGs and building the generation model (COPAFTc ) for a wind farm is summarized in the following steps:
1. From the observed wind speed data and the output power curve of WTGs, determine the
output power time series of the wind farm.
2. Use nearest centroid sorting to cluster the output power into bands; then calculate the
cluster (band) means, cluster probabilities, and inter-cluster transition rates.
35

Band 1
0

Band 2
0

Band 3
0

...

Band n
0

C  0
O
CO  C11
CO  C 21

C21

C31

...

C n1

C12

C 22

C 32

...

Cn2

C 13

C 23

C33

...

...

...
C 1m

Cn3

C2m

C 3m

...

C nm

CO due to turbine failures

C11

CO due to transition between wind speed bands
Figure 4.3 The combined outage due to transition of wind power (horizontal transitions)
and failures of wind turbines (vertical transitions)
3. Build the COPAFTf of WTG using the method presented in section 4.3.1.
4. Convolve the COPAFTf with the probability and frequency of each wind power band.
The failure and repair rates of the WTGs remain the same in each band. Only the output
power changes from band to band due to the effect of wind power transition from stat
from state k to state k + 1.
5. The available capacity and capacity outage for state i in COPAFTf that exists in band k
can be calculated using (4.7) and (4.8) respectively.

Ai
× Wk ,
CR

(4.7)

Oi,k = CR − Mi,k ,

(4.8)

Mi,k =

36

where Mi,k is the new available capacity in state i of the COPAFTf being in band k, Ai
is the available capacity in state i of the COPAFT, CR is the total installed capacity of
wind farms, Wk is the available wind power in band k and Oi,k is the new outage capacity
of state i of the COPAFTf being in band k.
6. The probability of the new outage state is given by

Pi,k =

X

PW (k) × PM (i) ,

(4.9)

where Pi,k is the probability of outage capacity of state i being in band k, PW (k) is the
band probability, and PM (i) is the COPAFTf state outage probability.
7. The cumulative outage probability of an outage state with Co ≥ X can be obtained by
scanning through all bands. Whenever an outage state with outage capacity of Co ≥ X
is encountered, its probability is accumulated as follows.

P (Co ≥ X) =

n X
m
X

Pi,k ,

(4.10)

i=1 k=1

where n is the number of COPAFTf states and m is the number of bands.
8. Scan through all states in the COPAFTc and combine the duplicated states.
9. The transition rates of wind power between bands from state i in band (k) to state
(i + 1) in band (k + 1) is given by (4.11). The exact transition rates between bands are
considered [14].
λ=

n(i→i+1)
D

,

(4.11)

where n(i→i+1) counts the number of wind power transitions between bands from state
37

(wpi < wp) to (wp ≥ wpi ) and D is the time period of study.
10. The frequency of wind power of band (k) can be calculated by

fW (k) = PW (k) × λ,

(4.12)

11. The cumulative outage frequencies of encountering an outage state with an outage capacity of (Co ≥ X) due to the transition of COPAFTf states between wind power bands can
be calculated by (4.13).
F (Co ≥ X) = Fk + FW T G ,

(4.13)

where
FW T G =

X

PW (k) × FM (i)

(4.14)

where FM (i) is the cumulative outage frequency of ith outage state and,

Fk =

X

PM (i) × fW (k) .

(4.15)

As a result, the reliability model of wind farms can be represented by single generation unit
with multiple states.

4.4

Model Validation and Test Results

The resulting COPAFTc of a wind farm can now be treated as the reliability model of a
single equivalent unit with multiple states [94], for purposes of generation adequacy analysis
or composite reliability analysis. In this work, we have evaluated the most common indices
that have been used in power system reliability analysis to test the proposed method. In
38

this work, we have evaluated the most common indices that have been used in power system
reliability analysis to test the proposed method. These indices are LOLP, EDNS, LOEE,
LOLE and LOLF.

4.4.1

Test System

The proposed method is applied on the IEEE RTS [123]. This system has been extensively
used in the literature for purposes of benchmarking. The IEEE RTS consists of 32 generating
units with total generation capacity of 3405 MW and total peak load of 2850 MW. The single
line diagram of this test system is shown in Figure 4.4.

39

BUS 18

BUS 17

BUS 21

BUS 22
BUS 23

BUS 16

BUS 19

BUS 20

BUS 14
BUS 15

synchronous
condenser
BUS 13

230kV

BUS 11

BUS 24

BUS 3

BUS 9

BUS 12

BUS 10
cable

138kV

BUS 6

BUS 4
BUS 5
BUS 8

cable
BUS 2

BUS 1

Figure 4.4 Single line diagram for IEEE-RTS

40

BUS 7

4.4.2

Case Studies

In this case studies, a typical wind turbine characteristic is assumed (type Vesta V90-3.0
MW [124]); rated power: 3 MW; rated, cut-in and cut-out speeds: (15 m/sec., 3.5 m/sec., 25
m/sec.). In the cases reported here, the effect of different values of WTG failure probabilities
were studied, and the reliability data used are shown in Table 4.1. The failure rates and
mean time to failure (MTTF) shown here lie in the ranges for actual WTGs, as reported
in [125]. Repair times are calculated from the FORs and failure rates of WTGs and are
expressed as mean time to repair (MTTR)
Table 4.1 Wind Turbine Reliability Data Used in the Test Cases
FORs
0.04
0.08
0.12

λ (failure/yr)
2.43333
2.72050
3.61983

MTTF (hour)
3600
3220
2420

MTTR (hour)
150
280
330

Three wind data sets are used: data set A [126], data set B [127], and data set C [128].
The correlation coefficients between these data sets and the load of the IEEE RTS are as
follows: (1) −0.002 for data set A, (2) 0.0686 for data set B, and (3) −0.1059 for data
set C. Whereas data set A and the load are almost uncorrelated, data set B has positive
correlation and data set C has negative correlation with the load. A wind farm consisting of
50 WTGs (150 MW total capacity) is added to the IEEE RTS. In the case studies reported
below, this farm is assumed to be subjected to each of the three wind speed data sets.
For convenience, these are referred to as Farm A, Farm B, and Farm C. In each case, the
outputs of the corresponding wind farm are grouped into ten clusters. As an illustration,
Table 4.2 shows the mean output of each band, individual and cumulative probabilities, and
the cumulative frequencies of transition across bands of wind output for the Farm A. The
COPAFTf of WTGs is built using the reliability data corresponding to FOR of 0.04 shown
41

in Table 4.1, and then convolved with the distributions of the farm output power bands
(variability model) using the procedure described in section 4.3.1 to produce the combined
COPAFT, i.e., COPAFTc . Table 4.3 is the COPAFTc of Farm A. The COPAFTc shows the
available capacity Ci , individual and cumulative probabilities, and the cumulative frequency
of each wind output state i.
Table 4.2 Reliability Model: Farm Output Bands and Probabilities
No.
1
2
3
4
5
6
7
8
9
10

Output
Exact State
Power (MW) Probability
0
0.13653
8.59
0.17648
20.55
0.13025
40.58
0.14486
58.04
0.08482
74.64
0.03721
94.37
0.05993
117.15
0.02819
125.55
0.03025
149.23
0.17146

Cumulative
Probability
1
0.86347
0.68698
0.55673
0.41187
0.32705
0.28984
0.22991
0.20171
0.17146

Cumulative
Frequency
0
0.1862
0.3892
0.4226
0.2978
0.3158
0.2836
0.2396
0.1860
0.1301

Generation adequacy or composite system reliability assessment can be performed by
using a multi-state model such as that shown in Table 4.3 to represent the output of the
wind farm. In the next section, the following case studies are presented: (1) reliability
analysis including forced outages of WTGs, (2) the effect of correlation between wind power
and system load on reliability indices, and (3) determination of capacity values of wind farms.

4.4.2.1

Reliability Analysis–Base Case

In this section, the reliability indices of the IEEE RTS are calculated prior to adding the wind
farm. These indices are calculated using discrete convolution and Monte Carlo simulation.
The results are shown in Table 4.4 where the second row shows the reliability indices of the

42

Table 4.3 Reliability Model: Farm Output States and Probabilities
No.
i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

Output
Exact State
Power (MW) Probability
Ci
P {C = Ci }
0
0.003133
2.984
0.013057
5.968
0.026658
8.953
0.035544
11.937
0.034804
14.922
0.026683
17.906
0.016676
20.891
0.008735
23.875
0.014417
29.844
0.013514
34.415
0.013085
39.101
0.033351
43.787
0.017900
47.751
0.008530
71.903
0.023100
90.714
0.037809
99.316
0.081800
129.085
0.160200
140.974
0.162100
149.223
0.268900

Cumulative
Probability
P {C ≥ Ci }
1
0.9968662
0.9838091
0.9571506
0.9216061
0.8868021
0.8601190
0.8434421
0.8347065
0.8202890
0.8067750
0.7936900
0.7603390
0.7424390
0.7339090
0.7108090
0.6730000
0.5912000
0.4310000
0.2689000

Cumulative
Frequency
F {C ≥ Ci }
0
0.0037882
0.0080676
0.0112011
0.0114028
0.0090757
0.0058807
0.0031896
0.0035518
0.0085116
0.0207453
0.0349233
0.0286773
0.0074431
0.1055822
0.0648995
0.1159003
0.2455493
0.1444312
0.1188310

IEEE RTS as calculated using Monte Carlo simulation.
Table 4.4 Reliability Indices of the IEEE RTS (Base Case)
Method
DC
MCS

LOLP

LOLE
LOLF
LOLD
EDNS
LOEE
hr/yr
occ./yr
yr
MW/yr MWh/yr
0.001069 9.36881 2.01600 4.64723 0.1348396 1181.195
0.001070 9.37407 2.01800 4.64522 0.1349867 1182.483

DC: Discrete convolution; MCS: Monte Carlo simulation

4.4.2.2

Reliability Analysis–After Adding Wind Farm A

In this case study, the IEEE RTS with Farm A added is used to test the proposed method
and to investigate the effect of forced outages of WTGs on the reliability indices. The rated
43

capacity of the wind farm is 150 MW. The reliability indices are evaluated for different values
of FOR as shown in Table 4.5. As expected, ignoring the WTG failures in adequacy studies
produces optimistic results.
Table 4.5 Annual Reliability Indices of the IEEE RTS After Adding Farm A Considering
Different Values of Turbine FOR
Method
0.04
0.08
0.12

4.4.2.3

LOLP
0.0007664
0.0007756
0.0007852

LOLE
hr/yr
6.713700
6.794491
6.878366

LOLF
occ./yr
1.603389
1.626876
1.645477

LOLD
yr
4.187400
4.176403
4.180165

EDNS
LOEE
MW/yr MWh/yr
0.093696 820.778
0.094894 831.270
0.096148 842.254

Reliability Analysis–After Adding Wind Farm A Using Monte Carlo
Simulation

In this case study, Monte Carlo simulation is also performed on the same system (with Farm
A), and the results are very close to those calculated using the proposed method as shown
in Table 4.6. This observation is not surprising since the wind speeds for Farm A are almost
uncorrelated with the load.
Table 4.6 Annual Reliability Indices of the IEEE RTS After Adding Farm A Considering
Different Values of Turbine FOR, Using Monte Carlo Simulation
Method
0.04
0.08
0.12

4.4.2.4

LOLP
0.0007642
0.0007748
0.0007876

LOLE
hr/yr
6.694428
6.786822
6.899390

LOLF
occ./yr
1.603279
1.619107
1.647222

LOLD
yr
4.175438
4.173034
4.188492

EDNS
LOEE
MW/yr MWh/yr
0.093653 820.401
0.094841 830.808
0.096248 843.130

Effect of Correlation with Load

To investigate the effect of correlation between wind speed and load, the results of adding
Farms B and C to the IEEE RTS are studied. The reliability indices, as calculated analyti44

cally by convolving the multi-state farm model with the generation model of the IEEE RTS
are compared with those obtained from Monte Carlo simulation. (For simplicity, turbine
failures are neglected, as they have no effect on correlation with load.) Since the Monte
Carlo method uses a sequential simulation that tracks hourly load and wind production,
thereby intrinsically capturing their correlation, it enables us to assess the error introduced
by ignoring the correlation in the proposed analytical method. Table 4.7 shows the reliability
indices after adding Farm B to the IEEE RTS. The results in Table 4.7 show that for Farm
B, where there is a positive correlation between wind speed and load, the proposed method
produces pessimistic results with an average error of −2.17%.
Table 4.7 Reliability Indices of the IEEE RTS After Adding Farm B
Method

LOLP

LOLE
LOLF
EDNS
hr/yr
occ./yr MW/yr
Proposed 0.00067 5.90528 1.41762 0.080620
MCS
0.00065 5.69402 1.33438 0.080138

LOEE
MWh/yr
706.248
702.011

Table 4.8 shows the reliability indices after adding Farm C to the IEEE RTS. These
results show that for Farm C, where there is a negative correlation between wind speed and
load, the proposed method produces optimistic results with an average error of 2.07%.
Table 4.8 Reliability Indices of the IEEE RTS After Adding Farm C
Method

LOLP

LOLE
LOLF
EDNS
LOEE
hr/yr
occ./yr
MW/yr MWh/yr
Proposed 0.0007998 7.005971 1.558705 0.0974247 853.4403
MCS
0.0008132 7.124022 1.598659 0.0994838 871.4785

The above results show that the proposed method is effective in building an accurate
reliability model of a wind farm, but small errors are introduced in adequacy analyses due to
external correlations. However, where the benefits of being able to use an equivalent model
in analytical studies outweigh those of detailed Monte Carlo simulation, as well as for the
45

reasons discussed in section 4.2.1.3, the proposed method offers the advantage of producing
an accurate and reusable model.

4.5

Photovoltaic (PV) Systems

Different topologies of PV systems have been addressed in literature [129, 130, 130]. In general, topologies of PV systems are mainly classified based on the arrangement of PV panels
and DC/DC and DC/AC converters. These arrangements are usually set up to enhance the
system efficiency and reliability. Grid-connected PV systems are usually constructed from
a large number of components such as power electronic circuits and solar panels. The reliability of such components is highly dependent upon the operation and ambient conditions.
Operational and environmental factors may lead to overall reduction in the output power
and failure of system components [30, 131, 132]. Modeling of these systems in power system
reliability is a complex task due to the dependency of the output power on the intermittent
source (solar radiation) and the availability of a large number of system components. An analytical method to construct a capacity outage probability and frequency table (COPAFT)
that captures both the intermittency of the input source and component failures can be built
using the same approach used for wind power as explained in section 4.3.

4.5.1

Attributes of PV System

In PV systems, power electronic systems are considered major components. These component play an important role and high efficient solutions to power conversion. However, the
large number of power electronic devices and the failure of power electronic circuits leads to
more challenges into the evaluation process of the reliability of the overall systems. One of

46

the concerns related to reliability lies in the power semiconductor devices and electrolytic
capacitors that are the most vulnerable links. Most of power electronic converters are not
equipped with redundancy. Therefore, any fault that occurs to the components or subsystems of the system will lead to shutdown of the system. From reliability point of view, such
components should be considered in evaluating the adequacy of PV systems. Due to large
number of system components of PV systems, only the reliability of major components have
been considered [30, 131, 132]. For power system adequacy, when PV power is introduced to
the main grid, its necessary to investigate the effect of all system components.

4.5.2

Grid Level PV Farm Structure

In general, a PV farm is constructed from a large number of PV panels arranged in arrays.
The PV arrays are constructed from PV panels which in turn are connected to the main grid
through DC/AC inverters. A set of N panels in each array is called sub-array. PV panels are
constructed from a large number of solar cells in series configurations. At the higher level,
the output of the PV system is then connected to the grid through a power transformer
[13], [15]. The amount of the output power of a PV farm at a certain location depends on
the input availability (mainly solar radiation) and the availability of system components. In
a PV farm, since PV panels are placed adjacent to each other, all PV panels are assumed
to have the same solar radiation, and approximately experience the same operational and
environmental factors that may affect the output of the farm.

4.5.3

Output Power of PV Systems

The maximum output power ratings of PV systems are provided by the manufacturer and
usually are expressed in peak-watt (WP ). The currentvoltage characteristics (I − V char47

acteristics) under the standard test conditions (the radiation level of 1 kW/m2 is given for
temperature of 25◦ C) can be calculated using the following relationships [133]:

I = s [Isc + KI (Tc − 25)] ,

(4.16)

V = Voc − KV Tc ,

(4.17)

where s is the radiation level, Isc is the short circuit current, KI is the short circuit current
temperature coefficient in A/◦ C, Voc is the open circuit voltage, KV is the open circuit
voltage temperature coefficient in V /◦ C and Tc is the cell temperature in ◦ C which can be
expressed as follows [133]:

Tc = Ta + s

Tno − 20
0.8


,

(4.18)

where Ta is the ambient temperature and (Tno ) is the nominal operating temperature of the
cell (◦ C).
The output power (Ppv ) for a given radiation level, ambient temperature and the currentvoltage characteristics can be calculated using the following relationships [133]:

Ppv = N × F F × I × V,

(4.19)

where N is the number of panels and F F is the fill factor, which depends on the module
characteristics, and can be expressed as follows [133]:

FF =

Vmpp Impp
,
Voc Isc

where Vmpp and Impp are the current and voltage at the maximum power point.

48

(4.20)

4.5.4

Attributes of PV System Components

In a PV farm, power electronic devices play an important role. Power electronic circuits
deliver the output power of PV panels to the load or to the grid. Reliability assessment
of power electronic circuits of large-scale grid-connected PV systems for different topologies
have been addressed in [129, 132, 134]. The major components that contribute to system
failure are power electronic circuits. In the literature, the mean time to failure (MTTF)
has been used as an indicator for system unavailability; the overall reliability of the circuit
is calculated by summing the failure rates (the failure rate is reciprocal of the MTTF) of
the components [4, 5, 135]. However, this is true if a failure of a component leads to overall
failure of the power electronic circuit. In some cases such as multilevel inverters, redundancy
exists in power electronic topologies and their desired function is not totally affected by a
component failure [136]. In this case, there are several possible different paths that can
still produce an acceptable output voltage level. In such configurations, the overall failure
cannot be assumed by summing the failure rates of the components. Analytical techniques
can be used to estimate system reliability such as reliability network analysis, minimal cut
set, minimal path set and state enumeration techniques [137]. In general, the reliability
function, R(t), indicates the probability that the system will operate without failures over a
time interval [0, t]. The failure rate of system component during the useful lifetime is usually
considered independent of time. The reliability of system components can be expressed as
an exponential function of their constant failure rate during the useful lifetime as follows.

R(t) = e−λi t ,
where λi is the constant failure rate of a power electronic component.

49

(4.21)

The mean time to repair (MTTR) has been considered small enough and can be neglected
[138]. This could be a valid assumption for a small system or a system with a few components
where the time required to repair or replace a component is small comparing to MTTF.
However, for large systems, neglecting the repair time could produce optimistic results.
Therefore, the MTTR should be included in modeling the output of PV systems. System
availability and unavailability as functions of MTTF and MTTR are described by (4.22) and
(4.23).

p=

µ
,
λ+µ

q =1−p=

(4.22)

λ
,
λ+µ

(4.23)

where p is the probability of component availability, q is the probability of component failure
and µ is the repair rate (reciprocal of M T T R).
For a system with N components connected in series, system availability (pssys ) can be
calculated by (4.24).
pssys

=

N
Y

pi ,

(4.24)

i=1

where pi is the probability of availability of component i.
For parallel configuration, there might be a tolerance for the desired output, and the
reliability of the system is the probability of all possible successful paths that can provide
the desired output. system availability (pssys ) can be calculated by (4.25).

p

psys = 1 − 

N
Y


(1 − pi ) ,

(4.25)

i=1

In the cases of series-parallel reliability configuration, it is useful to use network reduction
50

techniques (NRT). The availability and unavailability can then be calculated by combining
all series and parallel configurations to a equivalent unit using (4.24) and (4.25).

4.5.5

Reliability Modeling of Major Components of PV Systems

In the PV systems, the major components that contribute to system failure are power electronic circuits. In power electronic systems, components that contribute predominantly to
system failure are IGBTs/MOSFETs, diodes and capacitors. The reliability performance of
these components are affected by environmental and stress factors. One of the documents
that provides reliability data and modeling of power electronic components is the military
handbook, MILHDBK-217F [139]. According to the MILHDBK-217F, the failure rates of
these components are described as shown below.

4.5.5.1

Power electronic circuit components

Since the MILHDBK-217F military does not provide data for IGBTs, it was assumed that
the estimation of failure rates of MOSFETs are applicable for IGBTs.
For switches (MOSFET/IGBTS), the failure rate is described by:

λIGBT = λIGBT πT πQ πE
b

Failures/106 hours,

(4.26)

where λIGBT is base failure rate of the IGBT, πT temperature factor, πQ quality factor
b

and πE environmental factor.
Similarly, failure rates of diodes are given by:

λdiode = λdiode πT πS πC πQ πE
b

51

Failures/106 hours,

(4.27)

where λdiode is base failure rate of the diode, πS voltage stress factor, πC contact construcb
tion factor and πT , πQ and πE are as for IGBTs.
Capacitor failure rate is given by

λcp = λcpb πT πC πV πSR πQ πE Failures/106 hours,

(4.28)

where λcpb is base failure rate of the capacitor, πV voltage stress factor, πSR series resistance
factor and πT , πC , πQ and πE are as for IGBTs.

4.5.5.2

Reliability of PV Panels:

Reliability modeling of large systems such as PV systems with a large number of system
components could result in a large number of system states and increase modeling complexity.
Therefore, the equivalent MTTF and MTTR for subsystems are needed to drive an equivalent
failure rate λ and repair rate µ. The availability of system components can be modeled as a
Markovian with two states, “up” and “down” with known failure and repair rates. The PV
panels are constructed from n parallel branches; each branch is constructed by connecting m
series solar cells. The equivalent failure rate of a branch can be calculated by summing the
failure rates of the m series solar cells. The failure of a solar branch leads to the failure of
the PV panel. Thus, the availability and unavailability of PV panels can also be calculated
using (4.22) and (4.23).

4.6

Reliability Modeling of PV system

In order to build a generation model for the entire PV system that consider both the effect
of input variability and system components failures, two modeling steps are taken to build
52

a generation model for PV system: (1) modeling the dependency of the output on the
variability of the solar radiation and (2) the outage due to failure of system components
(failure rate). These two models are then combined to produce a single generation model
that can be viewed as a single source with multi-state. The blocks that constituted the
model are depicted in Figure 4.5.

Figure 4.5 Model combining input variability and system component failures

The failures of the components a PV system are assumed independent events; availabilities of system components can also be modeled as Markovian components with two states,
“up” and “down”. The assessment of the availability of the system output is addressed by
separately modeling the independent failures of system components and the dependency on
the input availability. The equivalent availability and unavailability of system components
can be estimated by calculating the equivalent failure and repair rates. For each array, the
53

MTTF and MTTR for M components can be obtained. Each array can then be represented
by equivalent MTTF and MTTR. The capacity outage probability and frequency table can
be constructed for a PV farm with N arrays by adding new units sequentially to the table
with the capacity model building approach explained in section 4.3. Based on the FOR of
system arrays, the COPAFT of PV system can be built using the unit addition algorithm.
This is a recursive algorithm that starts with the distribution of one unit (a PV array in this
case) and successively convolves with it the distributions of the remaining units, one unit at
a time. The steps of combining the viability model with the outage model are the same as
explained in section 4.3.

4.7

Case Studies

The proposed method is applied on the IEEE RTS [140]. The parameters and structure of
the PV system were taken from [39]. In this work, the solar radiation data for Santa Maria,
CA region were taken from [127]. The data obtained from [127] for one year were utilized
to estimate the output power of the PV system. In this work, the failure rates of system
components were assumed as follows: for both DC/AC inverters and DC/DC converter are
0.000027 failures per hour, and for solar panels are 0.000011 failures per hour [141]. The
repair rates of the DC/AC inverters and DC/DC converter were assumed 0.1 repair per hour.
For the PV panel, the repair rate was assumed 0.2 repair per hour.
To validate the results, the reliability indices of IEEE RTS are calculated using Monte
Carlo simulation and the results are shown in Table 4.9.

54

Table 4.9 Reliability Indices of the IEEE RTS With 100 MW PV Farm

Method

EDNS

LOEE

LOLF

LOLE

(MW/yr)

(MWh/yr)

occ./yr

(hr/yr)

2.01600

9.3688

1.73463

6.9453

LOLP

Base Case
DC

0.001069

0.1348396

1181.195

PV Power Only
Proposed

0.000795

0.097248

849.554

PV Power with Inclusion of (FOR)
Proposed

0.0008259

0.1030553

900.2912

1.74285

7.2157

MCS

0.0008356

0.1030920

900.7016

1.74397

7.2998

DC: Discrete convolution; MCS: Monte Carlo simulation

55

Chapter 5
Capacity Value of Variable Energy
Resources
Capacity value of VERs (or capacity credit) is an important metric used to assess the contribution of VERs to the adequacy of power system. The effective load carrying capability
(ELCC) approach is commonly used to measure the capacity value of generating units [64].
The ELCC of a generating unit is measured by the amount of system load increase that
gives the same LOLE index of the system (prior to adding a new generating unit) [64]. Traditionally, iterative process is required to measure the capacity value of a generating unit.
In the process of calculating the capacity value, the system load is incremented across all
hours using an iterative process by a step of ∆L, and reliability metrics such loss of load
probability (LOLP) or loss of energy expectation (LOLE) are recalculated for every iteration
until the predefined reliability target is met (i.e. the reliability target of the system before
adding the new generation unit). However, this process is computationally expensive and
time consuming especially for large and complex systems. In this work, direct, non-iterative
method is developed to calculate the capacity value of VERs.

56

5.1

Reliability-Based Methods used to Calculate the
capacity Value of VERs

Reliability based methods are considered the most robust in the assessment of the contribution of VERs to the generation adequacy assessment [142–144]. Reliability based methods
use the reliability indices such as LOLP or LOLE as a criterion to determine the capacity
value of VERs. There are several methods to calculate the capacity value of VERs using
ELCC. These methods include; 1) effective load carrying capability, 2) equivalent conventional power, and 3) equivalent firm power. These methods and other methods, approximation methods, are briefly discussed in the following sections.

5.1.1

Effective Load Carrying Capability

The calculation of the capacity value of VERs using ELCC method requires building a
generation model, COPAFT, and load model as explained in section 3.2. These two model
are convolved to calculate system reliability indices. When VERs are added to the system,
the COPFAT is reconstructed and convolved with system load model. The system load is
then increased by a step of ∆L across all hours and the system indices such as LOLE index
is recalculated until the same LOLE index prior adding the new generator is obtained. The
capacity value of the addition generating unit is calculated as follows.

P
CV =

∆L
× 100%,
Gc

(5.1)

where the CV is the capacity value of the new generating unit, Gc is the nameplate capacity
of the new generating unit.

57

5.1.2

Equivalent Conventional Firm Power (ECFP)

The capacity value of VERs is defined as the capacity of a benchmark generation unit that
resulting the same reliability indices when the VERs is added to the system [44, 45]. In this
process, the benchmark unit is assumed to be perfectly reliable. The steps of calculating the
capacity value using this approach are summarized as follows.
1. Add the output of VERs to the system.
2. Build COPAFT with VERS.
3. Convolve generation model with load model.
4. Calculate the LOLE index of the system.
5. Replace the VERs with a dispatchable unit (benchmark) with a capacity equal to the
installed capacity of VERs.
6. Recalculate the LOLE index of the system.
7. Gradually reduce the capacity of the benchmark unit.
8. Repeat the step 7 until the LOLE index of the system after adding VERS is achieved.
9. The capacity of the benchmark unit that procedures the same LOLE index after adding
VERS is the capacity value of VERs.

5.1.3

Equivalent Conventional Power (ECP)

The capacity value of VERs are calculated using the same steps as ECFP, except that the
benchmark unit is assumed not perfectly reliable. The FOR of the benchmark unit is usually

58

chosen to be small. The main difference between the ELCC approach and ECFP and ECP
is that in the ECFP and ECP methods, the distribution of distribution of available capacity
in a given hour change whereas ELCC method does not [44, 45, 145].

5.1.4

Approximation Methods

Calculation of the capacity value of VERs based on iterative methods tends to be computationally intensive due to the requirement of repetitive calculations. Several authors
have proposed different methods to calculate the capacity value of VERs using non-iterative
methods. Non-iterative methods are direct and computationally efficient, but often involve
approximations. For instance, the method presented in [61] uses a graphical method to
estimate the capacity value of wind power. The graphical method is based on using an exponential function to approximate the change in LOLE (loss of load expectation) resulting
from an increase in load. Although the graphical method is computationally efficient, this
approximation could lead to considerable errors [63]. The method presented in [62], the
Z-method, uses the probability distribution of the generation reserve margin to calculate the
capacity value of VERs. However, this method assumes that the probability distribution
of the generation margin preserves its shape even after adding wind power [65]. Also, the
methods presented in [61, 62] ignore the correlation between demand and wind power [65].
The non-iterative analytical method presented in [63] uses the concept of joint probability
distribution to calculate the capacity value of VERs using effective load carrying capability
(ELCC) approach considering the correlation between the load and the available output
power of VERs. However, this method requires a pre-specified value for the LOLE index to
be used to calculate the ELCC before and after adding VERs. Furthermore, as stated by
the authors themselves [63], the accuracy of estimating the ELCC of the system relies on
59

the chosen number of “i.e. coincidental demand-generation levels”, and the optimal number
of such levels that yields the best accuracy is difficult to determine.

5.2

A Direct Method to Calculate the Capacity Value
of VERs

The ELCC is traditionally estimated using iterative processes where the system load across
all hours is linearly incremented by a certain amount, usually a constant step, until the
targeted reliability index reaches a specified level. However, this process is computationally
expensive and time consuming especially for large systems. In the process of calculating
the PLCC, if VERs are added to the system as load modifiers, the system load model is
modified but the generation model remains unchanged. On the other hand, if VERs are
added to the system as generating units, the generation model is modified and the load
model remains unchanged. However, in the adequacy assessment, both approaches have
been used to calculate the capacity value of VERs. After adding VERs, the generation
reserve margin increases which leads to improvement in system reliability.
In this work, a direct method is proposed to calculate the capacity value of VERs using
the ELCC as a criterion. The proposed method is based on the fact that the capacity value
of VERs is bounded between 0 MW and their installed capacity (rated capacity). Also, it is
assumed that the increase in system load is evenly distributed on the load profile, as has been
commonly used [44, 45, 64]. Therefore, the capacity value can be calculated by augmenting
the generation reserve margin by the maximum capacity of VERs. The augmented reserve
margin model can be expressed in terms of generation, load, and VER models using (3.8)

60

and (3.9) as follows.
Mau = M + CV ,

(5.2)

where M is the generation reserve margin prior to adding VERs and CV is the available
capacity of VERs.
The CDF of the augmented margin, CDFau , is created in same manner as creating a CDF
for the original margin. However, the range of the augmented reserve margin is from −L
to CR rather than ranging from −L to 0 as in the original system (prior to adding VERs),
where CR is the maximum capacity of the VER. A typical CDF of an augmented margin is
shown in Fig. 5.1. The CDF of the generation reserve margin of the original system crosses
the zero level at P (0). Since the capacity value is defined as the amount of load increase
(after adding VERs) that would return a reliability index to its initial value (prior to adding
VERs), the capacity value of a VER can be determined by projecting P (0) to the augmented
margin. In other words, the capacity value of a VER can be directly calculated by projecting
the point at which the CDF of the augmented margin meets the level of reliability index
before adding VERS to the system as shown in Fig. 5.1. Mathematically, the capacity value
of VERs can be determined as follows.

Capacity Value = Mau {X : CDFau (X) = P (0)},

(5.3)

where CDFau (X) is the cumulative probability of X MW of the CDF of the augmented
reserve margin.
The general procedure of the proposed method for calculating the capacity value is given
as follows.
1. Construct probability distributions of conventional generators, loads, and output power
61

Capacity
value

P (M)
P (0)

Augmented CDF
P (- ΔM)
CDF after adding
VER power

P (- 2ΔM)
P (- 3ΔM)

- 2ΔM
- ΔM

M

- 3ΔM

-L
- L+ΔM

CDF of the original
system

0

P(-L)
CR

Figure 5.1 Cumulative distribution function of an augmented margin before and after adding
VERs
of VERs with and without including their forced outage rates.
2. Use discrete convolution to determine the margins for: (1) the original system, (2) the
system after adding the output power of VERs, and (3) the system after adding the
output power of VERs and considering their forced outage rates.
3. Adjust the new margins (after adding the power of VERs) by augmenting these margins
by the rated installed capacity of the VERs.
4. Determine the reliability indices of the original system. Determine the points at which
the new margins (after adding the power of VERs) intersect with these indices. These
points represent the capacity value.

62

5.3

Case Studies

The proposed method is applied on the IEEE reliability test system (IEEE RTS) [123]. The
output of VERs is modeled as described in section 4.3. In this paper, a wind farm with
150 MW (50 WTGs × 3 MW) capacity is added to the system assuming that all WTGs
are identical. In the case studies reported below, two wind data sets are used: data set
A [128] and data set B [146]. For convenience, these are referred to as Farm A and Farm B.
The reason of choosing these data sets is to determine the effect of the correlation between
the output power of wind farms and load profile on the reliability indices and the capacity
values. The correlation coefficients between these data sets and the load profile of the IEEE
RTS are as follows: −0.1059 for Farm A and 0.0686 for Farm B.
A typical wind turbine characteristic is used (type Vesta V90-3.0 MW [124]; rated power:
3 MW; rated, cut-in, and cut-out speeds: 15 m/sec, 3.5 m/sec, and 25 m/sec respectively).
In the cases reported here, the impact of different values of forced outage rates of WTGs
and wake effect on the capacity values of wind farms are studied. The reliability data of
WTGs are given in Table 4.1. The failure rates and mean time to failure (MTTF) shown
here lie in the ranges for actual WTGs, as reported in [125]. Repair times are calculated
from the FORs and failure rates of WTGs and are expressed in terms of mean time to repair
(MTTR).

5.3.1

Capacity Value of Wind Farm–Forced Outages and Wake
Effect

In this section, the well-known reliability indices and the capacity value of the wind Farm A
are calculated. Grid scale wind farms are constructed from a large number of WTGs, and

63

the variation of the output of WTGs within the farm depends on several factors including
the wake effect [14]. Due to the wake effect, the downstream (with respect to the direction
of the wind) WTGs generate less power than the upstream WTGs because they are exposed
to lower wind speed. Several methods have been proposed in the literature to model the
wake effect and to evaluate the resulting losses in the output power of wind farms [147, 148].
In this work, system advisor model (SAM) [149] is used to consider the output variation of
WTGs due to the wake effect in evaluating the capacity value of wind power. Two factors
are considered in calculating the capacity value and reliability indices as following: (1) forced
outages of WTGs and (2) wake effect .

5.3.1.1

Capacity Value of Wind Power Considering the Forced Outages of
WTGs

In this case study, the impact of forced outages of WTGs on the capacity value of wind
power is evaluated by considering different FORs of WTGs. In this work, the capacity value
of wind power is evaluated for four different case studies. The forced outage rates of WTGs
are considered as follows. Case 1 : 0.0, Case 2 : 0.04, Case 3 : 0.08, Case 4 : 0.12. In all of
these cases, the wake effect is ignored. System indices and capacity values of wind power for
the these cases are given in Table 5.1. For each case study, the capacity value is calculated
by taking the LOLE and the LOLF indices as criteria as shown in the last two columns of
Table 5.1. The first row of Table 5.1 represents the base case (prior to adding wind power).

5.3.1.2

Capacity Value of Wind Power Considering the Wake Effect

In this case study, the capacity value of wind power is evaluated considering the wake effect
with different values for the forced outage rates of WTGs. Four case studies are performed

64

Table 5.1 Reliability Indices and Capacity Value–150 MW Wind Farm A Considering the
Effect of Different FORs of WTGs
LOLP
Base
Case
Case
Case
Case

case
1
2
3
4

0.001072
0.000799
0.000821
0.000830
0.000839

EPNS
(MW)
0.13520
0.09573
0.09721
0.09862
0.09990

LOLF
(occ./yr)
2.01582
1.55870
1.61347
1.63284
1.64435

LOLE
(h/yr)
9.3681
7.0059
7.1969
7.2778
7.3549

Capacity
LOLE
–
26.67
25.33
23.33
22.67

Value (%)
LOLF
–
24.67
24.00
22.00
20.67

considering the wake effect with different forced outage rates of WTGs as follows. Case 1 :
0.0, Case 2 : 0.04, Case 3 : 0.08, and Case 4 : 0.12. System reliability indices and capacity
values of wind power for the four cases are given in Table 5.2. For each case study, the
capacity value is calculated by taking the LOLE and the LOLF indices as criteria as shown
in the last two columns of Table 5.2.
Table 5.2 Reliability Indices and Capacity Value–150 MW Wind Farm A Considering the
Wake Effect and Different FORs of WTGs
LOLP
Case
Case
Case
Case

1
2
3
4

0.000992
0.001001
0.001011
0.001021

EPNS
LOLF
(MW) (occ./yr)
0.123285 1.91430
0.124471 1.92040
0.125668 1.93530
0.126872 1.94617

LOLE
(h/yr)
8.6945
8.7704
8.8538
8.9176

Capacity Value (%)
LOLE
LOLF
20.00
18.67
18.66
17.68
18.00
16.67
17.00
15.33

As it is expected, the results given in Table 5.1 and Table 5.2 show that the addition
of a wind farm improves the reliability of the system. They also show that as the value of
FOR increases, the reliability and the capacity value of the system deteriorates. In addition,
when both wake effect and forced outages of WTGs are considered, the capacity value of
wind power decreases.

65

5.3.2

Effect of Correlation

To investigate the effect of correlation between wind speed and load demand on the capacity
value of wind power, wind Farm B is added to the IEEE RTS and the results are compared
with case of wind farm A. For simplicity and to avoid repetition, only the forced outage
rate of 0.08 is used in this section to calculate reliability indices and the capacity value of
wind Farm B. Three case studies are considered as follows. Case 1 : ignore both wake effect
and forced outages of WTGs, Case 2 : consider wake effect and ignore the forced outages
of WTGs, Case 3 : consider both wake effect and forced outages of WTGs. The results of
these case studies are given in Table 5.3.
Table 5.3 Reliability Indices and Capacity Value–150 MW Wind Farm B
LOLP
Case 1
Case 2
Case 3

0.000673
0.000757
0.000915

EPNS
LOLF
(MW) (occ./yr)
0.080620 1.41762
0.093128 1.54557
0.113578 1.84227

LOLE
(h/yr)
5.9053s
6.6349
8.0132

Capacity Value (%)
LOLE
LOLF
32.00
26.00
30.66
25.33
28.66
22.00

To show the effect of the correlation between the output power of wind farms and the
load profile on the reliability indices and capacity values, compare Case 1 of Table 5.3 with
Case 1 of Table 5.1, Case 2 of Table 5.3 with Case 1 of Table 5.2, and Case 3 of Table 5.3 with
Case 3 of Table 5.2. By comparing these results, it can be seen that a positive correlation
between wind speed and load profile contributes to a higher capacity value of wind power
and improvement in system reliability.

5.3.3

Capacity Value of VERs Versus Penetration Levels

The proposed method can also be used to easily determine the optimal capacity to install at
a location selected for a wind farm. Since the proposed method requires only one solution for
66

every candidate installed capacity, it is easy to determine the optimal capacity value. In this
work, data of Farm B is used and the capacity values are calculated for different penetration
levels as shown in Fig 5.2.
Four case studies are implemented as follows: (1) using the LOLP index as a criterion and
neglecting the forced outages, (2) using the LOLP index as a criterion and considering the
forced outages, (3) using the LOLF index as a criterion and neglecting the forced outages,
and (4) using the LOLF index as a criterion and considering the forced outages. It can be
seen from the curves of Fig. 5.2 that as the penetration level increases, the capacity value
of wind power also increases and then starts decaying as the penetration level is further
increased. This analysis allows the planner to determine the penetration level that results
in a maximum capacity value to allocate the cost of a project.

30

Considering
LOLE

(%)(%)
value
Capacity
value
Capacity

25

Considering LOLE
with FOR

20

Considering
LOLF

15
10

Considering LOLF
with FOR

55

0

0

100

200

300

400

500

Wind
power
penetration
(MW)
Wind
power
penetration
(MW)

Figure 5.2 Capacity value of wind power as a function of penetration level

67

5.3.4

Validation of Proposed Method

The proposed method is validated by comparing the results obtained by the method with
those obtained using an iterative method. The two methods are used to calculate the capacity
value of wind Farm A for Case 1 and Case 2 (with and without considering the forced outages
of WTGs and neglecting the wake effect). Using the proposed method, the results of Case
1 and Case 2 are shown in Fig. 5.3. The capacity value of the wind farm is 40 MW and 38
MW in cases of neglecting and considering the forced outages of WTGs respectively.
The iterative ELCC method is used to calculate the capacity value of wind farm A by
increasing the loading level of the system by ∆L until the reliability level (prior to adding
wind power) is reached, which is in this case the LOLE index. The results are shown in Fig.
5.4 which are identical to those calculated using the proposed method.

Cumulative Probability

Adding wind power
Adding wind power &
considering FOR of wind
turbine
40
38
Original system

Margin (MW)

Figure 5.3 Capacity value of wind power using the proposed method

68

12
Base case
Add wind power
Add wind power considering FOR
of wind turbines

11
LOLE (hr/yr)

10
9

38 MW

8
40 MW

7
6
5
0

10

20
30
40
50
Peak load increase in MW

60

Figure 5.4 Capacity value of wind power as function of peak load increase using iterative
method

It can be seen from Fig. 5.3 that the proposed method is able to calculate the capacity
value using only one iteration compared to the capacity value calculated using the iterative
method shown in Fig. 5.4. It is also clear that the capacity value of wind power is less when
the FORs of WTGs are considered which is not surprising.

69

Chapter 6
Quantification of Storage Necessary
to Firm Up Wind Generation
The integration of energy storage systems with intermittent sources has become a practical solution to overcome these challenges. This is largely because of the rapid growth in
deployment and improved technology of energy storage systems. A method for quantifying
the size of energy storage systems to meet specified reliability targets was proposed by Mitra in [89, 90]. This work extends this method to quantify the size of the required energy
storage to firm up wind power and improve system reliability to a specific target. In this
work, the effect of the input variation (wind speed) on the output power of wind turbine
generators (WTGs), forced outages of generating units including WTGs, and other factors
(e.g., transmission capacity and power quality constrains) are included. Correlation between
wind generation and load demand is also considered. Taking into consideration the above
factors, the proposed method determines the amount of storage required to firm up the
generation from the wind farm that is added, so as to provide the same reliability level as
a conventional (dispatchable) generating unit of the same nameplate capacity as the wind
farm. System reliability enhancement, in terms of reliability indices, is also assessed with
respect to wind farm location and the energy storage. Metrics commonly used for reporting
bulk power system reliability are utilized in this work: loss of load probability (LOLP), expected demand not supplied (EDNS), loss of load frequency (LOLF), and mean down time
70

(MDT). However, this paper addresses the matter of size only; it is up to the planner to
select a suitable technology based on cost and other considerations.

6.1

Energy Storage System

This section presents a method for determining the size of the energy storage that is required to increase the available energy at wind farm locations. Due to transmission line
capacity limits and operational constraints, and the uncertainty associated with the overall
energy production of wind farms, the output power of a wind farm that is available to the
system could be lower than its nameplate capacity. In fact the output variations depend
upon several factors such as geographic location, number of WTGs on the farm, differences
between WTGs, turbulence and wake effects, and terrain effects [111, 150]. The accuracy
of modeling the output power of WTGs also varies with time scales (the frequency of wind
speed observation intervals can be 5 minutes, 10 minutes, hourly, etc.). Further, it has been
reported that the error in estimating annual energy production of wind power can be up
to 12% [150, 151]. Moreover, failures of WTGs have a significant impact on the available
power. Thus, in determining the energy storage size, such uncertainty factors should also be
considered.

6.1.1

Energy Storage Sizing

The proposed method of quantifying the size of the energy storage is based on previous
work presented in [89, 90]. Consider a wind-integrated power system that provides supply of
availability ρ0 to the system load. Now consider that part of the load curtailment, PL , that
can be directly attributed to the variability of wind power. This quantity can be determined

71

from the difference between the nameplate capacity and the capacity value of the wind farm.
The capacity value is the amount of load the wind farm can reliably support, given the
variability of wind. Firming the wind output consists of adding a storage system that is
sufficient to increase the availability of supply to ρ1 , which is what would be available if
there were firm (dispatchable) generation instead of wind. Thus, the power capacity of the
required storage unit should be at least PL . The energy capacity can then be determined as
follows.
Define the unavailability reduction ratio α as [89]:

α=

1 − ρ1
.
1 − ρ0

(6.1)

The unavailability reduction ratio can be understood thus: suppose that ρ0 = 0.999 and
that it is required to increase the availability by an additional “9,” i.e., to ρ1 = 0.9999; then,
α = 0.1.
Now assume that the storage system that will improve the system reliability to ρ1 can
sustain a load of PL for time tA . Then, service interruption occurs when the grid supply is
down and the storage has been depleted. The probability of this event can be described by
(6.2).

P {Ls } = P {{R > tA } ∩ Ls̄ }
= P {R > tA | Ls̄ }P {Ls̄ }
!
Z ∞
=
fR (r)dr P {Ls̄ }.
tA

The variables in (6.2) are defined as follows [89].

72

(6.2)

Ls̄ : event that the load is curtailed in the absence of storage;
Ls : event that the load is curtailed in the presence of storage;
R: random variable representing the down time (outage duration);
fR (r): probability density function of R.
In (6.2), P {Ls } is clearly 1 − ρ1 , and P {Ls̄ } = 1 − ρ0 . From (6.1) and (6.2), it is clear
that
Z ∞
tA

fR (r)dr = α.

(6.3)

The solution to (6.3) can be obtained analytically for simple systems [89], but for more
complex systems it is more convenient to use (6.4), which has been shown to be equivalent
to (6.3) [90].
ZtA
rfR (r)dr = (1 − α)r̄

(6.4)

0

The solution to (6.4) can be easily obtained from interruption time statistics generated by
using sequential Monte Carlo simulation. Suppose the interruption durations without the
storage system are arranged in order of increasing magnitude. If r̄ is the mean interruption
duration (given by the mean of all interruption durations), then that time for which the
mean of all equal and shorter interruptions is closest to (1 − α)r̄ gives the estimate of tA .
Equation (6.4) represents the basic relation that quantifies the required energy capacity
of the storage system. However, the storage unit itself may not be perfectly reliable. In
order to compensate for this, the storage device should have an energy capacity that enables
it to provide the required power (PL ) for a period of time (tS ) that is given by (6.5) [89, 90].
t
tS = A ,
AS
73

(6.5)

where AS is the availability of the storage system. Therefore, the power capacity of the
selected storage unit should be at least PL , and the energy capacity should be at least PL tS .
In wind power planning projects, the errors associated with wind power prediction can
lead to significant challenges for system planners and operators. Also, wind speed varies
with both time and location. For these reasons, wind integration studies are generally
conducted prior to the development phase of wind power projects to assess these impacts.
With regard to storage system planning projects, several authors have discussed the effect
of wind power prediction error on sizing energy storage systems [151–154]. Therefore, the
available statistical data of wind power may not accurately reflect the long term variation of
wind speed [150,151,155]. Thus, in the planning phase, additional power may be required to
compensate for the long term error associated with wind energy production and estimation.
In this work, the additional power of energy storage system PESS can be calculated as
follows.
PESS = PL (1 + γ),

(6.6)

where γ is the anticipated long term error associated with wind energy production and
estimation.

6.1.2

Energy Storage Operation

In order to evaluate the effect of adding an energy storage device with a rated power of PESS
on the reliability of the system, the following operation constraints are considered [70]:

i (t) ≤ P i
0 ≤ Pdis
r

for t ∈ [0, T ], i = 1, 2, . . . , Nb ,

74

(6.7)

i (t) is the discharging power of the energy storage
where T is the period of the study, Pdis

device at bus i at time t in MW, and Pri is the rated power of the energy storage system at
bus i in MW.
i (t) ≤ P i
0 ≤ Pchr
r

for t ∈ [0, T ], i = 1, 2, . . . , Nb ,

(6.8)

i (t) is the charging power of the energy storage system at bus i at time t in MW.
where Pchr

The inequalities (6.7) and (6.8) constrain the charging and discharging power to remain
within the power rating of the energy storage system. The energy constraints of the storage
system are imposed by (6.9).

0 ≤ Eni (t) ≤ Enir

for t ∈ [0, T ], i = 1, 2, . . . , Nb ,

(6.9)

where Eni (t) and Enir are the energy state of charge and the energy rating of the storage
system respectively at bus i in MWh. The energy state of charge of the storage system can
be expressed as follows.

Eni (t + 1) = Eni (t) −

1
ηdis

i (t) + η
i
Pdis
chr Pchr (t),

where ηchr and ηdis are the charging and discharging efficiencies of the storage system respectively.
In the literature, several authors have proposed different operation strategies for energy
storage systems [69, 156, 157]. In this work, time-shift technique is applied as described
in [69]. The state of charge and discharge of the storage system is represented by means
of logic states (i.e., 1 for discharging and 0 for charging). The discharging states represent
the hourly down-time statistics of wind power. In the simulation process, the available wind

75

power is evaluated based on the temporal resolution of wind data and the discharging and
charging capacities are determined accordingly.

6.1.3

Approach for Reliability-Based Storage Sizing

Fig. 6.1 provides an overview of the steps employed in the proposed approach. Reliability
evaluation is performed by seeking dispatch solutions that minimize system curtailment,
considering wind power uncertainty, forced outages of generation units including WTGs,
and system operations constraints (power balance, generation and transmission capacites,
and power quality constraints). Details of this model are provided in the next section.

6.2

Network Modeling And Reliability Evaluation

In composite reliability evaluation studies, repetitive solutions of an optimization problem
with an objective function of minimum load curtailment are performed. This section describes the formulation and incorporation of the objective function of minimum load curtailment using nonlinear programming and the AC power flow model. Sequential Monte Carlo
simulation is used to emulate the behavior of the system and estimate system reliability indices. This consists of sequential assessments of the reliability of the states that the system
assumes in successive time steps over the planning horizon. For each such state, the objective is to determine a dispatch that minimizes the load curtailment, subject to the equality
constraints of power balance, the inequality constraints of equipment capacity and power
quality, and the availability of system components. In this section, the details of system
modeling and reliability evaluation are described.

76

Start

Read generation, load, transmission, and wind power data

For each candidate bus i:

Add wind power and determine its capacity value and
system reliability indices

Form the capacity value and mean downtime (MDT),
determine the power and energy ratings of storage
system

Add the storage system and calculate the reliability
indices

Are all candidate buses evaluated?

i=i+1

No

Yes
End

Figure 6.1 General procedure for reliability-based storage sizing

6.2.1

System Modeling

For each hour, the system state is defined by the component states and capacities. The
output of wind power is determined by the corresponding wind turbine states and hourly
77

output power. Then a feasible dispatch is sought by solving the following minimization
problem [158].

Loss of Load = min 



N

b
X

Ci 

(6.10)

i=1

subject to
P (V, δ) − PD + C = 0
Q(V, δ) − QD + CQ = 0
min ≤ P ≤ P max
PG
G
G
max
Qmin
G ≤ QG ≤ QG

(6.11)

V min ≤ V ≤ V max
|F (V, δ)| ≤ F max
δ

unrestricted

where Ci is the load curtailment at bus i, C is the vector of load curtailments (Nb × 1), CQ is
the vector of reactive load curtailments (Nb × 1), V is the vector of bus voltage magnitudes
(Nb × 1), δ is the vector of bus voltage angles (Nb × 1), PD and QD are the vectors of real and
reactive power loads (Nb × 1), PG and QG are the vectors of real and reactive power outputs

 min max min
of the generators Ng × 1 , PG
, PG , QG and Qmax
are the vectors of real and reactive
G


power limits of the generators Ng × 1 , V max and V min are the vectors of maximum and
minimum allowed voltage magnitudes (Nb × 1), F (V, δ) is the vector of power flows in the
lines (Nt × 1), F max is the vector of power rating limits of the transmission lines (Nt × 1),
and P (V, δ) and Q(V, δ) are the vectors of real and reactive power injections (Nb × 1). In
the foregoing, Nb is the number of buses, Nt is the number of transmission lines, and Ng is

78

the number of generators.
The above model implies that for any encountered scenario (generation and transmission
availability and load state) power will be routed through the network in such a manner as
to minimize the system load curtailment.

6.2.2

Calculation of Reliability Indices

In order to capture interruption times and temporal relationships such as state of charge of
the storage system, all indices are determined from sequential Monte Carlo simulation [159].

6.2.2.1

Mean down time (MDT)

The MDT is the average interruption duration, denoted by r̄ earlier in this section. By
definition,
Z ∞
MDT = r̄ =
0

rfR (r)dr

(6.12)

where the variables are as defined in (6.2). From the simulation, MDT is estimated using

MDT = E [r̂] ;

r̂ =

Nc
1 X
i
Tdn
Nc

(6.13)

i=1

i is the duration
where E [·] is the expectation operator, r̂ is the estimator of MDT, Tdn

of ith interruption encountered during the sequential simulation, and Nc is the number
i and an interruption period
of cycles simulated. A cycle consists of a service period Tup
i ; the ith cycle time T i equals T i + T i . The total period of simulation T is given by
Tdn
c
up
dn


i + Ti .
T = Nc Tc = Nc Tup
dn

79

6.2.2.2

Loss of load probability (LOLP)

The LOLP index can be estimated as follows.
Nc
1X
i
Tdn
Π=
T

LOLP = E [Π] ;

(6.14)

i=1

where Π is the estimator of LOLP and the other variables are as defined above.

6.2.2.3

Loss of load frequency (LOLF)

The LOLF index gives the frequency of service outage and can be estimated as follows.

LOLF = E [Φ] ;

Φ=

Nc
T

(6.15)

where Φ is the estimator of LOLF and the other variables are as defined above. It should
be noted that the unit of LOLF is failures per unit time; hence, if time is tracked in hours,
(6.15) will yield LOLF in f/h, and may need to be converted into f/y, which is the customary
unit for expressing LOLF. Also, LOLF is related to LOLP and MDT as follows.

LOLF =

6.2.3

LOLP
MDT

(6.16)

Expected demand not supplied (EDNS)

The EDNS index is the sum of the products of probabilities of failure states and the corresponding load curtailments which can be calculated as follows.

EDNS =

X

P {xi } × C{xi }

xi ∈Xf

80

(6.17)

where P {xi } and C{xi } are the probability of occurrence of state xi and the system load
curtailment in state xi , and Xf is the set of failure states.
Using sequential simulation, EDNS is estimated from

h i
EDNS = E dˆ ;

NC
1X
i C{x }
ˆ
Tdn
d=
i
T

(6.18)

i=1

where dˆ is the estimator of EDNS and C{xi } is the minimum curtailment obtained from the
solution of (6.10) for the prevailing state.

6.2.4

Stopping Criterion

In using Monte Carlo simulation to estimate power system reliability indices, a convergence
criterion should be applied to stop the algorithm if there is not much change in the reliability
indices. In this work, the stopping criterion is applied on the reliability indices as follows.
p
Var(ρNc )

 ≤ ε,
COV =
E ρNc

(6.19)

where COV is the coefficient of variation, Var(·) is the variance function, ρNc is the value
of the estimate of the reliability index of interest (such as LOLP or EDNS) at the end of Nc
cycles, and ε is a pre-specified tolerance.
At intervals of several cycles, the COV is calculated. If this amount is less than or equal
to the specified tolerance ε, the algorithm is terminated; otherwise, the simulation continues.

81

6.3

Case Studies

The proposed formulation is applied on the IEEE reliability test system (IEEE-RTS) [140].
The single line diagram of this test system is shown in Fig. 4.4. The load profile of the
IEEE-RTS is used to calculate the hourly load of each bus in the system for a year. For the
base case of the IEEE-RTS, the LOLP, LOLF, and EDNS indices are 0.001538, 2.695 f/y,
and 0.20151 MW/y respectively.
In this study, a wind farm with 200 MW capacity is added to the system at the load buses
assuming that all WTGs are identical (100 WTGs with nameplate capacity of 2 MW each).
Actual observed wind speed data sets are used to calculate the output of WTGs. In the
following case studies, three wind speed data sets are used to evaluate the effects of different
correlation between the wind power and the load profile, and different temporal resolutions
on the system reliability indices: data set A [126], data set B [160], and data set C [128].
The wind speed data sets are utilized to calculate the output of WTGs using equations (4.1)
and (4.2).
A typical wind turbine characteristic is assumed (type Vesta V90-2.0 MW [124]). The
reliability data of the WTGs are as follows: The mean time to failure (MTTF) and mean
time to repair (MTTR) are 3600 h and 150 h respectively [81]. The MTTF lies in the ranges
of actual WTGs, as reported in [125]. The MTTR is calculated from the forced outage rates
and MTTF of WTGs. The case studies are summarized as follows:
1. Evaluating the effect of the temporal resolutions of the wind speed on system reliability:
Due to unavailability of wind speed data for the same site with different temporal resolutions, the effects of correlation and temporal resolution on the system reliability indices
are evaluated using three different wind speed data sets for different sites. Three different

82

case studies are considered. These case studies are not intended to compare the results;
rather, they illustrate the impact of considering such factors on the planning decisions.
In an actual planning study, comparisons should be conducted for wind speed data with
different temporal resolutions, at the site under consideration.
2. Energy storage sizing: The proposed method (as explained in section 6.1) is used to
determine the power and energy capacities required to firm up the generation from the
wind farm that is added, so as to provide the same reliability level as a conventional
(dispatchable) generating unit of the same nameplate capacity as the wind farm.

6.3.1

Wind Farm Locations and System Reliability

The selection of wind farm location usually depends on several factors such as wind availability, access to the grid, economic aspects, and reliability enhancement of the grid. In this
work, the wind farm candidate locations are evaluated based on how much reliability benefit
they can bring to the system i.e., by the extent to which a wind farm at a candidate location
reduces the system EDNS. The following three cases are studied:
1. Case study 1: IEEE-RTS reliability indices with wind farm, assuming wind speed data
set A.
2. Case study 2: IEEE-RTS reliability indices with wind farm, assuming wind speed data
set B.
3. Case study 3: IEEE-RTS reliability indices with wind farm, assuming wind speed data
set C.
The temporal resolutions of the wind speed data sets are as follows: (1) wind speed data
set A has a temporal resolution of one hour, (2) wind speed data set B has a temporal
83

resolution of 5 minutes, and (3) wind speed data set C has a temporal resolution of 10
minutes. The correlation coefficients between these data sets and the load profile of the
IEEE-RTS are as follows: (1) −0.002 for data set A, (2) 0.0686 for data set B, and (3)
−0.1059 for data set C. Whereas data set A and the load are almost uncorrelated, data set
B has positive correlation and data set C has negative correlation with the load. For each
case study, the wind power is connected to the load buses and then the reliability indices
are calculated. System reliability indices for different wind locations are listed in Table 6.1,
Table 6.2, and Table 6.3. Buses 21–24 do not have load and are not considered.
The entries in the tables may be understood thus. Each row represents a separate study
wherein a 200 MW wind farm as described above is added to the IEEE-RTS at the indicated
bus, and the system EDNS, LOLP, LOLF and MDT are evaluated.
Table 6.1 System Reliability Indices for IEEE RTS with 200 MW Wind Power Added at
Different Buses: Data Set A
Wind Power
at Bus No.
1
2
3
4
5
6
7
8
9
10
13
14
15
16
18
19
20

LOLP
0.001054
0.001071
0.001114
0.001122
0.001143
0.000988
0.001517
0.001201
0.001038
0.001025
0.000950
0.001161
0.001185
0.001011
0.000938
0.001032
0.001209

EDNS
(MW/y)
0.135182
0.137305
0.142770
0.143889
0.147287
0.125983
0.198447
0.155500
0.133087
0.131022
0.121135
0.149592
0.153112
0.128986
0.119248
0.132050
0.156707
84

LOLF
(f/y)
3.079
3.120
3.228
3.240
3.285
2.892
2.639
3.467
3.042
3.012
2.770
3.341
3.418
2.955
2.737
3.020
3.496

MDT
(h)
3.00000
3.00641
3.02261
3.03302
3.04749
2.99239
5.03714
3.03375
2.99047
2.98240
3.00578
3.04280
3.03686
2.99695
3.00110
2.99238
3.02918

Table 6.2 System Reliability Indices for IEEE RTS with 200 MW Wind Power Added at
Different Buses: Data Set B
Wind Power
at Bus No.
1
2
3
4
5
6
7
8
9
10
13
14
15
16
18
19
20

LOLP
0.000991
0.001007
0.001047
0.001055
0.001078
0.000919
0.001496
0.001137
0.000971
0.000954
0.000887
0.001094
0.001120
0.000937
0.000876
0.000963
0.001147

EDNS
(MW/y)
0.120629
0.122628
0.127766
0.128817
0.132020
0.112048
0.195424
0.139781
0.118663
0.116734
0.107531
0.134194
0.137520
0.114838
0.105768
0.117694
0.140924

LOLF
(f/y)
1.960
1.974
2.036
2.054
2.083
1.814
2.601
2.204
1.915
1.878
1.773
2.122
2.157
1.860
1.749
1.894
2.223

MDT
(h)
4.42908
4.46960
4.50344
4.50049
4.53337
4.43826
5.03883
4.51906
4.44386
4.45101
4.38240
4.51602
4.54845
4.41344
4.38536
4.45565
4.51867

The best candidate buses are determined by tracking system reliability improvement with
respect to wind power location. Of the indices reported, the EDNS most effectively reflects
the extent to which customers are affected by system outages. Therefore, the EDNS index
is chosen to determine the best candidate buses to connect the wind farm at. The candidate
buses are ranked based on the improvement of the system EDNS index (highest to smallest).
In general, system reliability indices improve when wind power is added to the system in
every case study which is not surprising. Overall, the system is found to benefit the most
from installing the 200 MW wind farm at bus 18, 13, 6, 10 or 9.

85

Table 6.3 System Reliability Indices for IEEE RTS with 200 MW Wind Power Added at
Different Buses: Data Set C
Wind Power
at Bus No.
1
2
3
4
5
6
7
8
9
10
13
14
15
16
18
19
20

6.3.2

LOLP
0.001188
0.001207
0.001252
0.001262
0.001290
0.001115
0.001525
0.001359
0.001170
0.001152
0.001080
0.001306
0.001337
0.001136
0.001067
0.001160
0.001370

EDNS
(MW/y)
0.151188
0.153582
0.159732
0.160989
0.164816
0.140846
0.199972
0.174072
0.148830
0.146509
0.135374
0.167412
0.171378
0.144221
0.133227
0.147665
0.175437

LOLF
(f/y)
2.171
2.205
2.261
2.272
2.311
2.047
2.657
2.468
2.138
2.117
1.969
2.351
2.413
2.091
1.940
2.121
2.491

MDT
(h)
4.79364
4.79410
4.84918
4.86576
4.89139
4.77040
5.02747
4.82253
4.79373
4.76665
4.80599
4.86729
4.85454
4.75849
4.81907
4.79208
4.81614

Storage System Augmentation for Different Wind Data Sets

The size of the energy storage PESS and the time tS are calculated as explained in Section 6.1.
A prediction error of γ = 5% is assumed for the annual energy production of the wind farm.
The unavailability reduction ratio α is determined for each instance in the following manner.
The wind farm is replaced by a conventional unit with the same nameplate capacity as the
wind farm, i.e., 200 MW. Typical reliability characteristics are assumed for the conventional
unit. Then, the system LOLP is calculated using the conventional unit instead of the wind
farm, at the same location. This LOLP is deemed the target LOLP, since it is intended to
use storage to firm the wind supply by increasing its reliability to the same level as that of a
conventional unit. Now α is determined by dividing the target LOLP by the LOLP obtained

86

from wind injection.
Then, the energy storage system is added to the grid at the candidate buses in presence of
wind power. The storage system is sized as described in section 6.1.1. The power capacities
PESS of the storage systems for data sets A, B and C are found to be 153.3 MW, 150.15
MW and 157.5 MW; these values differ because a higher positive correlation between wind
and load requires a lower PESS . Then using α as calculated above, tA is determined as
described in section 6.1.1. The storage system is assumed to have 0.99 availability. From
tA and AS = 0.99, tS can be determined. But since the simulation is performed in hourly
increments, tS is rounded of to the nearest hour. The energy capacity of the storage system
is given by PESS tS .
The charging and discharging efficiencies of the storage are assumed 90% and 87.5%
respectively [70]. During the simulation, the storage system logic described in section 6.1.2
is implemented. For the three case studies, the PESS , target LOLP, α, tS , and the LOLP
achieved with storage integration at each of the five selected locations are shown in Tables
6.4, 6.5 and 6.6.
Table 6.4 System Reliability Indices for IEEE RTS with Wind Power and Energy Storage
at Different Buses: Data Set A
Wind Power
at Bus No.
18
13
6
10
9

Target
LOLP
0.00036
0.00039
0.00044
0.00053
0.00057

PESS
(MW)
153.3
153.3
153.3
153.3
153.3

87

α
0.38
0.41
0.45
0.52
0.55

tS
(h)
4
4
3
3
3

LOLP
with Storage
0.0003726
0.0003808
0.0004153
0.0005445
0.0005845

Table 6.5 System Reliability Indices for IEEE-RTS with Wind Power and Energy Storage
at Different Buses: Data Set B
Wind Power
at Bus No.
18
13
6
10
9

Target
LOLP
0.00036
0.00039
0.00044
0.00053
0.00057

PESS
(MW)
150.15
150.15
150.15
150.15
150.15

α
0.41
0.43
0.48
0.56
0.59

tS
(h)
6
5
4
3
3

LOLP
with Storage
0.0003434
0.0003856
0.0004208
0.0005217
0.0005369

Table 6.6 System Reliability Indices for IEEE RTS with Wind Power and Energy Storage
at Different Buses: Data Set C
Wind Power

Target

PESS

at Bus No.

LOLP

(MW)

18

0.00036

157.5

13

0.00039

6

α

tS

LOLP

(h)

with Storage

0.34

8

0.0003686

157.5

0.36

6

0.0004154

0.00044

157.5

0.40

6

0.0004300

10

0.00053

157.5

0.46

5

0.0005922

9

0.00057

157.5

0.49

4

0.0005836

As evident from Tables 6.4, 6.5 and 6.6, the proposed method was effective in firming the
reliability of the wind farm using appropriate amounts of storage. The LOLP with storage
was improved to the target LOLP, i.e., the same level as that of a conventional unit with
the same nameplate capacity as the wind farm. It is also evident from these results how
the storage capacity, both in MW and MWh, differ based on factors such as correlation
between wind speed and load, and grid access, i.e., strength of interconnection to the grid.
The results shown also validate the proposed methodology.

88

Chapter 7
Conclusions
This chapter summaries the findings and provides a general conclusion about the methods
that have been developed in this work.

7.1

Conclusion

This work presented an analytical method of constructing reliability models of VERs for
planning studies involving wind and PV system integration. The proposed method uses a
recursive technique to build a generation model that considers input variability, correlation
between VERs outputs, and forced outages of VERS. An illustrative reliability model was
constructed using real input data and real. The effectiveness of the proposed method was
demonstrated on a well-known test system. A comparison of several reliability indices (LOLP,
LOLF, EDNS, LOLE and LOEE), with and without consideration of VER forced outages
was provided in each case. The results obtained by the proposed method were validated
using Monte Carlo simulation. The work presented here focused on the development of the
reliability model of wind farms and PV systems; it did not attempt to capture correlations
with other stochastic elements such as system load, for reasons described in section 4.2.1.3.
The results of the case studies indicate that in the case of wind power, neglecting this
correlation introduces small errors in some cases, but the advantages of an accurate and
reusable model that can be used in analytical methods of adequacy assessment or capacity

89

valuation may overshadow these errors. In applications where it is deemed necessary to
consider such correlation, the approach described in [100] may be used. However, for the
purpose of constructing an accurate reliability model of a wind farm and PV system, the
proposed method was shown to be effective.
Further, this work has introduced a direct method to calculate the capacity value of VERs
without performing iterations. The proposed method is based on augmenting the generation
reserve margin to include the output power of VERs. The intermittency of wind, forced
outages of WTGs, and correlation between WTGs on the farm were all considered. The
proposed method was applied on IEEE RTS and the results provided. The proposed method
reduces modeling complexity and the computational burden associated with calculating the
capacity value of VERs using iterative methods. Capacity values estimated using an iterative method corroborated those obtained from the proposed method, thereby validating the
method.
Also, a method for quantifying the amount of energy storage required to firm up wind
power is presented in this work. Several case studies were performed to evaluate the reliability
indices and determine the sizes and locations of the energy storage systems. Preferred
locations for wind injection were also evaluated by comparing the enhancement of system
reliability indices at the candidate buses. Sequential Monte Carlo simulation was used to
emulate the stochastic behavior of wind power and forced outages of WTGs in calculating
the reliability indices and determining the sizes of energy storage systems. In this work, three
wind speed data sets were used. These data sets were utilized to study the effect of correlation
between load demand and wind power, and of temporal resolutions on determining the size of
the energy storage. The system reliability indices and the rated power of the energy storage
systems were calculated for three case studies using the three data sets. These case studies
90

were not intended to compare the results; they only demonstrate the effect of wind speed
temporal resolution and correlation with the load on system reliability and determining
size of the energy storage system. The results validated the proposed methodology and
illustrated the dependence of the required sizes of storage systems on wind characteristics
and wind farm location. Although the IEEE-RTS is considered a highly reliable system, some
transmission lines have limited capacities. Therefore, placing wind farms at some buses may
not improve system reliability. For example, placing a wind farm at bus 7 would not enhance
the reliability of the system because the line connecting bus 7 with the rest of the grid cannot
carry the available power due to the line capacity limit. On the other hand, other buses such
as 13, 19, and 18 are connected with more than one transmission line to the system and they
have higher loads compared to other load buses. In this regard, the reliability indices of the
system are compared for different locations of wind farms and energy storage systems. These
indices are dependent upon wind farm and energy storage system characteristics, as well as
the connectivity with the rest of the system. These observations also indicate the importance
of considering the influence of wind speed measurement resolutions and correlation with the
load in evaluating the size of energy storage in planning projects. System reliability indices
before and after installing the energy storage are also evaluated. From the EDNS indices it
appears that bus 18 is the best location to site the wind farm.

7.2

Future Work

The work developed in this thesis could be extended further considering the following points:
1. The method presented in chapter 4 can be extended to construct a unified outage
model of a clustered output of variable resources.
91

2. The method presented in chapter 5 is intended to calculate the capacity value of VERs
in generation adequacy assessment utilizing a discrete convolution method. However,
the method can be extended further for composite systems as well.
3. Because of the ongoing changes in generation portfolios, the integration of energy
storage systems has become an important part in power system as it plays a major
role in overcoming the challenges posed by the integration of variable resources. The
method presented in this work can be extended to model both variable resources and
energy storage systems in power system adequacy assessment.
4. One of the most important factors in power system planning projects is the evaluation
of variable resources penetration levels. The method presented in chapter 5 can also
be used to determine the optimal penetration levels of VERs.

92

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