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Vittal.pdf
Performance and Reliability Analysis of Wind Turbines using
Monte Carlo Methods based on System Transport Theory
Sameer Vittal1 and Michel Teboul2
2
1
Reliability Engineering
GE Energy, Greenville, SC
Clockwork Solutions Ltd.
Austin, TX
ABSTRACT
Reliability and performance assessments of wind turbine systems are particularly challenging as
they operate in highly stochastic, non-linear, coupled, multidisciplinary environments. The
traditional approach has been to decouple performance from reliability and analyze them
separately, which results in sub-optimal design and operational practices. In this paper, a method
of jointly simulating both the performance and reliability of wind turbines is presented. The
approach is based on system simulation using novel Monte Carlo algorithms derived from system
transport theory (SPARTM technology), a method originally developed for nuclear physics
applications. In the representative wind turbine case study discussed in this paper, both machine
availability and energy produced is simulated as a function of basic weather variables like wind
speeds, turbulence intensity and design intent. In addition, statistical confidence bounds on energy
and availability are also calculated for a full twenty year life.
problem; and is based on a Monte Carlo approach
derived from system transport theory of nuclear physics
[Dubi, 2000]. The full paper will include a detailed
description of system transport theory as applied to the
reliability analysis of mechanical systems along with
numerical implementation. In this extended abstract, a
brief description of the theory is provided in subsequent
sections.
INTRODUCTION & OVERVIEW
Wind Turbine systems are rapidly becoming an
economically viable source of renewable energy. A key
element in making wind energy both a technical and
commercial success is the ability to develop accurate
and computationally efficient modeling and simulation
platforms which serve as the basis for machine design
and performance optimization. Two key elements of
wind turbine technology are turbine performance and
availability. Turbine performance (energy produced) is a
function of design variables and a highly stochastic
operating environment. Machine availability is a function
of system reliability, and is impacted by design,
operating environment and maintenance considerations.
Hence, the wind turbine simulation problem includes
elements of probabilistic design, multi-state reliability
theory, multidisciplinary optimization as well as
traditional fields like engineering and operations
research. Hence, any modeling framework will have to
include elements of all these subjects.
MULTI-STATE RELIABILITY ANALYSIS USING
SYSTEM TRANSPORT THEORY
Historically, reliability theory has been based on a
binary approach, where a system can exist in two states
– an “up” state where the system is completely
operational and working at full performance; and a
“down” state where the system has failed. The
probability of a system existing in the “up” state is
characterized by the reliability, R(t), which is the
probability of the system being operational at time ‘t’, as
well as system availability, A(t|k), which is the probability
of the system being operational at time ‘t’ given that it
has seen ‘k’ failures in the past. It is clear that R(t) refers
to system survival before the first failure, and A(t|k)
refers to system survival for repairable systems, i.e. R(t)
is the special case of A(t|0). In reality, complex systems
exist in multiple degraded states, which is studied under
the emerging discipline of Multi-State reliability theory
[Lisnianski, 2003].
In recent years, researchers have recognized the
benefits of incorporating both reliability and performance
in a unified mathematical model, giving rise to the
emerging field of “performability” analysis [Trivedi,
2001]. For wind turbines, “performability” analysis has
applications in developing design specifications, in
choosing wind farm sites, establishing maintenance and
logistics protocols and in modeling power performance
and equipment availability guarantees. This paper deals
with a wind turbine case study analyzed using a new,
unified approach to the wind turbine “performability”
There are two main approaches for modeling multistate problems for systems with non-exponential failure
and repair distributions, (E.g. most mechanical systems)
– Markovian Models, and a more general approach,
which is System Transport Theory. Variations of Markov
approaches include
Semi-Markov or Generalized
Markov theory [Bolch, et al, 1998]. Markov-based
approaches work best when the failure and repair rates
Copyright 2004 by S. Vittal (Member- AIAA) and M. Teboul.
Published by the American Institute for Aeronautics &
Astronautics with permission
1
American Institute for Aeronautics and Astronautics
are exponential – for non-exponential systems, both
complexity and computational costs increase.
A more recent, generalized approach is based on
system transport theory, and was developed by Dubi
[Dubi, 1995]. In the Markov model approach, the focus
is on calculating the probability of a part/system being
any state ‘ i ’ at time ‘t’. In system transport theory, the
focus is on three events – the probability of a part
entering a state ‘ i ’ at time ‘t’; the probability of it
remaining in that state ‘ i ’ for some time ‘t’ to ‘t1’, and
the probability of it transitioning to some other state ‘j’ at
time ‘t1’. In system transport theory the problem is
formulated as a set of generalized state equations which
are solved using specialized Monte Carlo algorithms
developed for this purpose [Fritz, 2001].
The system transport equations can be explained
with a simple example. Consider a single component
system which can enter into ‘n’ states. Let ∆t be some
time interval of interest, and N i ( t , ∆t ) be the number of
times the component enters state ‘ i ’ in the time interval
[t , t + ∆t ] . The event density ψ i (t ) is then defined as
shown in Eq. (1) below.
ψ i ( t ) = lim
∆t → 0
N i ( t , ∆t )
The event density fulfills the system transport theory
equation of nuclear physics, hence the solutions to the
transport theory problem can be used in estimating the
probability of the component being in a given state. In
addition, the solution can be extended to calculate the
energy produced in a given state, leading to
performance calculations as well.
For some time point, t > t ′ , the probability that the
system will enter state ‘ i ’ at time ‘ t ′ ’ and will remain in
that state until time ‘ t ′ ’ is given by the term
ψ i ( t ) dt ′Ri ( t − t ′ ) , where,
(
Ri ( t ) = 1 − Fi ( t ) = exp − ∫ zi ( x ) dx
0
)
(2)
in reliability theory.
Let Pi ( t ) denote the probability that the system is in
state ‘i’ at time ‘t’. It can be expressed in terms of the
event density, as shown in Eq. (3).
(3)
i
t
∫ ψ ( t ′) R ( t − t ′) dt ′
0
i
To obtain the equation for
i
(4)
ψ i (t) , we consider two
time points, t and t ′ , where t > t ′ . Then the number of
system transitions into state ‘ j ’ in dt ′ at time t ′ is given
ψ j ( t ′ ) dt ′
by
.
A
fraction
of
this,
ψ j ( t ′ ) R j ( t − t ′ ) dt ′ will remain in state ‘ j ’ upto time ‘ t ’
and a fraction , ψ j ( t ′ ) R j ( t − t ′ ) z ji ( t − t ′ ) dt ′ , will enter
state ‘ i ’ at time ‘ t ’ per unit time. Integrating over time,
and summing over all the states leads to the total
contribution to ψ i ( t ) as shown in Eq (5).
ψ i ( t ) = Pi 0δ ( t )
t
+∑ ∫ ψ j ( t ′ ) R j ( t − t ′ ) z ji ( t − t ′ ) dt ′
j =1
j ≠i
(5)
0
The term Pi 0δ ( t ) accounts for entries into state ‘ i ’
at time ‘ t = 0 ’, and δ ( t ) is the dirac delta function. Eq
(5) comprises of a set of ‘n’ simultaneous integral
equations, the solution of which will provide the
probabilities of interest. These are the system transport
equations of the process, and they “control” the transfer
of a component/system into different states at various
time points.
It has been shown [Dubi, 1995] that Eq (5) follows
the basic relations of a random walk process. In fact,
any process that follows a Markovian or Non-Markovian
random walk process will automatically be a solution of
a transport equation – this is what makes this approach
flexible and powerful.
For numerical implementation, the set of integral
equations described in Eq (5) can be written in vectormatrix form shown in Eq (6)
( )
In Eq (2), Ri ( t ) is the probability that a state transfer
will not occur, analogous to the reliability cumulative
density function in classical reliability theory. The term
zi ( t ) is the rate function, analogous to the hazard rate
i
0
A ( t ) = Pi ( t ) =
n
From Eq. (1) , for some infinitesimally small time, the
probability that the system will enter state ‘ i ’ around
time t ′ is given by ψ i (t ′)dt ′ .
t
∫ ψ ( t ′) R ( t − t ′) dt ′
Note that for the special case of a two-state single
component system, if the operational state is denoted
by “1”, then the probability of being in state “1” is the
availability, written as shown in Eq. (3),
(1)
∆t
t
Pi ( t ) =
(
)
ψ (P) = f 1 P + ∫ψ (P′)K P′,P dP′
T
(6)
Equation 6 is the form fitting nuclear particles. In the
case of systems transport it takes a slightly different
form shown in equation 6a
t
ψ(B,t) = f 1(B,t) + ∑ ∫ ψ(B',t ')K(B',t ';B,t)dt '
B' 0
with
P = (B, t)
2
American Institute for Aeronautics and Astronautics
(6a)
Note that ψ (P) is the event density for state vector
( )
P at time t , f 1 P is the first state probability, i.e. the
probability that the first state of the system will be Pi ,
(
data has been scaled appropriately. This does not in
any way impact the modeling methodology discussed in
this paper, not does it affect the validity of the final
conclusions.
)
ψ (P′) is the event density at time t ′ and K P,P′ is
called the state transition kernel. The state transition
kernel is critical to the solution of system transport
problems, and in the nuclear physics literature, it is
referred to as the transport kernel, itself the product of
two other matrices called the free flight kernel and the
collision kernel. It is to be noted that equation 6a does
not refer to the most general case It is actually related to
the confined case of a single system with a given
number of states. Te most general case involves a more
elaborate state vector containing both the states of all
the components, the time of entries of all the
components into their respective states and the “birth
times” of each component in its state. This results in a
rather elaborate equation which captures all the
possible features of systems including any type of
interaction between components, or the system and its
environment. Additional details are beyond the scope of
this abstract, and can be found in Reference [7] [Dubi,
1986].
The system transport theory approach for reliability
analysis and system simulation has been implemented
in the SPAR software program, originally developed by
Dubi and currently available from Clockwork Solutions.
SPAR basically solves the most general transport
equation by sampling the “nest event” from the transport
kernel. The unique features of SPAR involve a detailed
treatment of aging and interactions of various kinds
along with efficient sampling algorithms and tally
procedures. The concept of “Logical points” in SPAR
allows the user to intervene in the transport process and
define unlimited complexities in the operation of the
system using a simple and efficient language. This
means , among other things, that one can create
detailed high resolution models and treat both discrete
and continuous events processes.
The wind turbine performability analysis project has
been implemented in SPARTM, by developing custom
code around the core software engine. Details are
discussed in subsequent sections.
Figure 1: Typical land-based wind turbine
(representative purposes only)
A reliability block diagram of the complete structure
is fairly complex, comprising of the primary systems like
blade, main bearings, drivetrain and generator in series
with controls and other structural elements having
redundancy designed into them. A simplified reliability
block diagram for the system can be represented by
four key sub-systems in a series configuration, as
shown in Figure 2. The main elements chosen for
analysis are blades, gearbox, controls and generator.
Gearbox
Generator.
Blades
Controls
AVAILABILITY & PERFORMANCE MODELING OF
WIND TURBINES
Figure 1 shows a typical 3-bladed horizontal-axis
wind turbine, designed for land-based operation.
(Picture courtesy GE Wind). This type of design is
extremely common, where three composite blades drive
a rotor, which in turn drives a gearbox which is coupled
to a generator. The entire drivetrain, along with blade
controls and electronics is housed in a nacelle, mounted
on top of the tower. To protect proprietary information,
only a generic, hypothetical wind turbine case study is
discussed in this paper. All machine and environmental
Figure 2: Simplified Reliability Block Diagram
The life distribution of each of these components is
assumed to be a standard two-parameter Weibull
distribution, of the form shown in Eq. (7)
  t β 
F ( t ) = 1 − exp  − 
 
  η ( X )  


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American Institute for Aeronautics and Astronautics
(7)
Here, F ( t ) is the CDF of the failure distribution at
time ‘t’, β is the Weibull slope (shape parameter) and
η ( X ) is the characteristic life (scale parameter) of the
distribution, and is a function of a vector of life-limiting
variables, X = { X1,…, X m } . It is important to note that
the characteristic life is not a constant, but depends on a
number of variables like wind speed, turbulence
intensity, capacity factor, temperature, etc which are
random variables. Relationships between η ( X ) and
these variables can be obtained in many ways – the
most popular ones are physics-based (usually from
some kind of probabilistic fracture mechanics code),
purely empirical (statistical methods like proportional
hazards, Weibull-Log Linear models, etc) approaches or
some kind of analytic-empirical hybrid model. In this
paper, a standard statistical model based on Weibullproportional hazards is used. This particular form,
described in Eq. (8) is also called a Weibull-regression
or Weibull-Log Linear model. The transfer function
between η ( X ) and X = { X1,…, X m } is as per in Eq. (8).
ln η ( X )  = α 0 + α1 X1 + α 2 X 2 + α 3 X 3
(8)
Where, X1 = Wind Velocity (m/s)
X 2 = Capacity Factor
X 3 = Temperature (Centigrade)
and α 0 ,α1,α 2 ,α 3 are the coefficients of the WeibullLog Linear model, obtained from field data, using any
standard statistical analysis program like Minitab.
Values of the Log Linear coefficients for the
gearbox, generator, blades and controls are detailed in
Table 1.
Gearbox
Generator
Blades
Controls
spare parts to a site and carry out repair actions during
winter (if there is snow or icing). In this paper, downtime
distributions include crane access time (for land based
machines), part replacement time and other logistics
delays. Note that in addition to the temperature based
variation, the time to repair each type of component has
it’s own distribution. The total time to restore a failed
turbine back to an operational state is modeled as a
lognormal distribution, with both parameters (lognormal
mean and lognormal standard deviation) functions of
site temperature as shown in the PDF in Eq. (9) and
CDF in Eq. (10) below.
f (t ) =
 1  ln(t ) − µ ( X ) 2 
3
exp  − 
  (9)
 2  σ ( X 3 )
2π σ ( X 3 ) t
 

1
F (t ) =
t
∫ f ( t )dt
(10)
0
Here, µ ( X 3 ) and σ ( X 3 ) are the lognormal mean
and standard deviation respectively, which are functions
of the site temperature, X 3 . The values of the repair
distribution parameters are detailed in Table 2.
Note that the cut-off temperature that separates a
“summer repair distribution” from a “winter repair”
distribution is chosen as 12C. In reality, the downtime
distributions can be functions of several variables like
site location, wind speed, wave state (for offshore
turbines), season, physical size of the spare part,
location of the spare pool etc. Although the logistics
distribution has been highly simplified in this case study,
the SPAR program was capable of handling fairly
complex repair logic with out any significant increase in
computational burden.
Component
Condition
µ ( X3 )
σ ( X3 )
Gearbox
X3 > 12C
4.453
0.4724
Gearbox
X3 ≤ 12C
5.369
0.4724
Generator
X3 > 12C
4.093
0.6064
β
2.6395
3.1814
1.8757
1.2640
α0
11.7062
12.1763
12.2596
12.8252
Generator
X3 ≤ 12C
5.093
0.5742
α1
0.00762
-0.02551
-0.00593
0.02848
Blades
X3 > 12C
5.323
0.1655
α2
-1.6672
-1.1269
-0.9489
-2.4701
Blades
X3 ≤ 12C
5.807
0.1980
α3
0.003483
0.007930
0.007453
0.001433
Controls
X3 > 12C
3.858
0.3246
Controls
X3 ≤ 12C
4.768
0.1421
Table 1: Weibull-Log Linear Model Coefficients
Using the Weibull Log Linear model, a more
accurate failure distribution can be calculated, which
takes into account the extreme weather and electrical
power variations that drive turbine life.
Turbine availability is a function of both the failure
distribution as well as repair distribution. In addition, the
time to repair (unplanned outage) is a function of
weather variables like temperature – for example, in
North America and Europe, it usually takes longer to get
Table 2: Repair distribution parameters
Once the reliability and repair logic has been
defined, the availability models can be built. The next
step is to develop the performability logic and link it to
the availability/reliability module.
The power produced in a wind turbine can be
expressed as shown in Eq. (11)
4
American Institute for Aeronautics and Astronautics
P=
1
ρ ⋅ V 3 ⋅ A ⋅ Cp
2
(11)
Where P is the power produced (W), ρ is the Air
density (Kg/m3), V is the wind speed (m/s), A is the
turbine rotor Area (m2) and Cp is the power coefficient.
The power coefficient is itself the product of the
mechanical efficiency (usually driven by the gearbox,
with typical values of 0.95 to 0.97), the electrical
efficiency (due to generator and electrical circuits
losses, typical values are 0.97-0.98) and the
aerodynamic efficiency. Aerodynamic efficiency is
governed by the Betz law, which states that the
maximum possible theoretical efficiency is 16/27 ~ 0.59.
A graph of power produced Vs wind speed is commonly
used to assess the performance of the given wind
turbine design in a known environment. Another
important performance parameter is the “capacity
factor”, which is the ratio of actual energy produced
(kW-hr) in a time ‘t’ (hr) to the maximum theoretical
power that could be produced in that period. For
example, if a machine rated 500kW produced 150kW in
an hour, then the capacity factor is (150 x 1)/(500 x 1) =
0.3. In this paper, the capacity factor ( X 2 ) for the
hypothetical machine is defined as the ratio of the power
produced at a given wind speed P ( X1 ) , to the
maximum rating, Pmax , as shown in Eq (12) below.
X2 =
P ( X1 )
(12)
Pmax
The variation of power produced with wind speed
(the “power curve”) is shown in Figure 3.
Assumed Power Curve
Another measure of turbine performance is the
annual energy production. Usually, for a given wind farm
site, measurements of wind speed (mean and standard
deviation) are made every 10 minutes for at least a
year, and the resulting wind speed data is modeled as a
Weibull distribution with PDF as shown in Eq (13).
f (v ) = ( β v ηv )(v ηv )
β
exp  − (v ηv ) v 


(13)
Where f (v ) is the probability density function of the
wind speed distribution, v is the wind speed, β v is the
weibull shape parameter and ηv is the scale parameter.
If P (v ) be the power curve function (from Figure 3) ,
N0 be the number of hours in a year (~8765), Vin be the
cut-in wind speed and Vout be the cut-out wind speed,
then the annual energy production, E year , can be
written as shown in Eq (14).
Vout
E year = N0
∫ P (v )f (v ) dv
(14)
Vin
It is important to note that Eq (14) does not
incorporate any reliability information, nor does it
account for month-to-month variations in the wind speed
distribution. It is, at best, an “averaged” power
production estimate, which can vary widely in a short
operating period. Using the simulation approach
described in the full paper, it is possible to estimate the
energy produced as a function of wind speed as well as
availability - in addition, confidence intervals on power
produced can also be obtained. This information can be
used to construct power distributions for any part of the
year, develop seasonal power guarantees, estimate
downtime distributions, etc. Some preliminary results
are discussed in the next section.
SIMULATION OUTPUT & RESULTS
2500
1000
The full paper will include details of the simulation
logic flowchart, reports, etc. Some sample results are
shown below. One useful measure of maintainability is
the CDF of the downtime distribution, as shown in
Figure 4. This is useful in assessing the expected costs
for repair, in logistics planning, in establishing spare
pool locations, etc. The wide variation in downtime can
be attributed to the effect of weather on repair time.
500
The expected number of component failures over a
20-year design life (172500 hrs) are shown in Table 3.
2000
Power, kW
βv −1
1500
Part
0
0.0
5.0
10.0
15.0
Wind Speed (m/s)
Figure 3: Power produced Vs wind speed
(representative example only)
Gearbox
Blades
Generator
Controls
Failures
in 20 yrs
Unavailability
Sensitivity
4.45
21.5
3.2
7.65
2.09E-01
6.11E-01
6.51E-02
1.15E-01
Table 3: Component Failures & Sensitivity
5
American Institute for Aeronautics and Astronautics
sensitivity, i.e. they are the most important components
that drive availability. Generators have the lowest
sensitivity, and are the “best parts” from an availability
perspective. Again, we reiterate that these results are
valid for the hypothetical case study only, and cannot be
generalized to any operational wind turbine.
Turbine Downtime Distribution (CDF)
100%
90%
80%
The simulation outputs also include a time series of
energy produced at each hour over a 20-year life. This
has been “averaged” into an equivalent annual time
series as shown in Figure 5. In addition to the mean
value, confidence intervals can also be calculated. A
time-series of machine availability is shown in Figure 6.
This includes an estimate of interval availability for each
time point as well as a lower 95% confidence level. The
time series can be used in establishing “threshold
levels” for power production and availability guarantees,
and in setting product performance specifications.
Overall, the performability simulation approach provides
results at a level of detail and granularity that is not
available with other tools and methods.
Probability (%)
70%
60%
50%
40%
30%
20%
10%
0%
0.0
200.0
400.0
600.0
800.0
1000.0
Downtime (hr)
Figure 4: Turbine downtime CDF from simulation
From table 3, it is clear that the blades have the
maximum expected failures, and spares should be
planned accordingly. The unavailability sensitivity is a
parameter that helps identify the “high value” single
point failures, i.e. the components which are both likely
to fail and stay failed for a longer time relative to the
remaining parts. Identifying these parts is critical to
developing effective repair and spare strategies. In
addition, baseline warranty and life-cycle costs can be
estimated over the expected design life. For the
example problem, blades have the highest unavailability
It is also possible to develop monthly distributions on
power produced. These turned out to be (mostly) normal
distributions, where the standard deviations were similar
but the mean values shifted depending on the season.
Figure 7 shows the PDF’s of the monthly power
distributions, and Figure 8 shows the CDF’s of the same
distributions.
Time Series - Turbine Availability
100.00%
95.00%
Availability, %
90.00%
85.00%
80.00%
Lower Confidence Interval
75.00%
Interval Availability
70.00%
480
14880
29280
43680
58080
72480
86880
101280 115680 130080 144480 158880 173280
Time, hrs
Figure 6 - Turbine availability time series, 20 year simulation output
6
American Institute for Aeronautics and Astronautics
approach will be an additional tool in uncertainty
modeling and optimization in multidisciplinary systems.
Power Distribution (kW) - Monthly Comparison
3.00E-3
REFERENCES
[1] Trivedi, K.S., Probability and Statistics with
Reliability, Queuing and Computer Science
Applications, 2nd Ed. Wiley-Interscience, 2001
2.40E-3
[2] Dubi, A., Monte Carlo Applications in Systems
Engineering, John Wiley & Sons, 2000
Probability
1.80E-3
1.20E-3
[3] Lisnianski A., and Levitin, G., Multi-State System
Reliability:
Assessment,
Optimization
and
Applications, World Scientific Publishing, 2003
6.00E-4
[4] Dubi, A. and Gurvitz, N., “A note on the analysis of
systems with time dependent failure rates”, Annals
of Nuclear Energy, 22(3/4), pp. 215-248, 1995
0
500.00
900.00
1300.00
1700.00
Power Produced
2100.00
2500.00
Figure 7: Comparison of monthly power (PDF)
Power Distribution (kW) - Monthly Comparison
1.00
[5] Bolch, G, et al., Queuing Networks and Markov
Chains : Modeling and Performance Evaluation with
Computer Science Applications, John Wiley &
Sons, NY, 1998
[6] Fritz, A., and Bertsche, B., “Algorithms for the
Reliability and Availability Simulation of Mechanical
Systems”, European Safety and Reliability
International Conference (ESREL2001), Torino,
Italy. Sep-16-20, 2001
[7] Dubi, A., “Monte Carlo Calculations for Nuclear
Handbook
of
Nuclear
Reactor
Reactors”,
Calculations, (Ed. Y. Ronen), CRC Press, 1986
0.60
[8] Vittal S. and Hajela P., “A State Transition
Approach to Reliability Based Design and
AIAA/ASME/
Optimization”,
Proc.
of
44th
ASCE/AHS/ASC SDM Conference, Norfolk,
Virginia, April 2003.
Probability
0.80
0.40
[9] S. Vittal and P. Hajela, “Probabilistic Design using
Empirical Distributions”, Proc. 44th AIAA/ASME/
ASCE/AHS/ASC SDM Conference, Norfolk,
Virginia, April 2003
0.20
0
0
600.00
1200.00
1800.00
Power Produced
2400.00
3000.00
Figure 8: Comparison of monthly power (CDF)
These charts can be used to establish upper and
lower specification limits on monthly energy production
– quantities that are very useful to electric utilities who
need this kind of data for planning their generation
schedules.
To summarize, concepts of system transport theory
have been used to develop a performability model of
extreme-weather wind turbines. Results indicate the
model works well and captures critical reliability and
performance drivers at a level of detail and fidelity not
seen in other tools. In addition, this approach provides a
novel, unified framework for the performability based
optimization of mechanical systems. It is hoped that this
Disclaimer & Acknowledgement: The views
expressed in this paper are those of the authors only
and do not necessarily reflect the views of the General
Electric Company, Ben-Gurion University or Clockwork
Solutions. The authors would like to acknowledge the
assistance provided by Marshall Smith and Barrett
Lepore of Clockwork Solutions in modeling and
simulation using SPARTM. The authors would like to
express their gratitude to Prof. Arie Dubi of Ben Gurion
University who developed the core SPAR technology,
and pioneered the use of system transport theory for
reliability and performability applications. Prof. Dubi’s
comments and review contributed significantly to this
paper.
7
American Institute for Aeronautics and Astronautics
Time Series - Average Hourly Energy Produced
2500
Energy, kW-Hr
2000
1500
1000
500
0
1
730
1459
2188
2917
3646
4375
5104
5833
6562
Time, Hr
Figure 5 : Turbine energy time series
8
American Institute for Aeronautics and Astronautics
7291
8020
8749
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