Msc_Thesis_4th_July_2011.
Probabilistic Power Flow Analysis for a Given Network
In Case of Wind Power Penetration
Master of Science Thesis
Fatih Abanoz
(4039521)
July 2011
Delft University of Technology
Faculty of Electrical Engineering, Mathematics and Computer Science
Electrical Power Systems
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This thesis is approved by:
Supervisor: Dr.Ir. Madeleine Gibescu
And the committe members:
Prof. Dr.L.van der Sluis, Delft University of Technology, The Netherlands
Dr.Ir. Madeleine Gibescu, Delft University of Technology, The Netherlands
Dr.Ir. Sander Meijer, Delft University of Technology, The Netherlands
Ir. Bart Tuinema,Delft University of Technology, The Netherlands
And was in cooperation with Stedin B.V.
In Rotterdam, The Netherlands
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To my family and beloved ones
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TABLE OF CONTENTS
CHAPTER 1 –Introduction ..................................................................................................................... 1
1.1.
Introduction ................................................................................................................................. 1
1.2.
Conventional Generation of Electricity................................................................................... 1
1.3.
Main Problems of Using Conventional Sources...................................................................... 2
1.3.1 Depletion of fossil fuels.......................................................................................................... 2
1.3.2 Environmental Concerns ........................................................................................................ 3
1.3.3. Energy Security ..................................................................................................................... 3
1.4.
Renewable Energy Sources ..................................................................................................... 3
1.5.
Types of Electricity Generation Methods from Renewable Energy Sources .......................... 5
1.5.1 Combined Heat and Power (CHP) Plants............................................................................... 5
1.5.2 Geothermal Power Plants ....................................................................................................... 6
1.5.3 Biomass Power Plants ............................................................................................................ 7
1.5.4 Fuel Cells................................................................................................................................ 8
1.5.5 Solar Energy ........................................................................................................................... 8
1.5.6 Wind Energy......................................................................................................................... 10
1.6.
Power System Network Structures............................................................................................ 12
1.6.1 Passive Power Network............................................................................................................ 12
1.6.2 Active Power Network ............................................................................................................. 13
1.7.
Goal of the Thesis...................................................................................................................... 15
1.8.
Outline of Thesis ....................................................................................................................... 16
CHAPTER 2 - Mathematical Background for Probabilistic Load Flow Analysis .............................. 17
2.1. Introduction ................................................................................................................................ 17
2.2. Fundamental Concepts ........................................................................................................... 18
2.3. General Introduction to Monte Carlo Simulation................................................................... 24
CHAPTER 3 – Modeling ...................................................................................................................... 26
3.1. Wind Farm Modeling ................................................................................................................. 26
3.1.1 Wind Distribution................................................................................................................. 26
3.1.2 Selection of Appropriate Wind Turbine ............................................................................... 28
3.1.3 The wind speed – Wind Power Relationship........................................................................ 28
3.1.4 Correlation Factor..................................................................................................................... 33
3.2 Load Modeling ............................................................................................................................ 36
3.3. Power System Network .............................................................................................................. 42
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CHAPTER 4 – Case Studies ................................................................................................................. 46
4.2 Effect of Using Different Kinds of Voltage Control Methods in the System ............................. 48
4.2.1 Voltage Droop Control under 9MW of wind power penetration ......................................... 49
4.3 Effect of change in the value for the reference bus voltage ........................................................ 57
4.3.1 Total Current Distribution ........................................................................................................ 57
4.3.2 Total Reactive Power ........................................................................................................... 60
4.3.3 Total Resistive Loss ............................................................................................................. 63
4.4 Effect of Island Mode Operation in the System ......................................................................... 66
4.4.2) Increase In Power Capability of the Synchronous Generators............................................ 68
5.1 Conclusion and recommendations for future research .................................................................... 70
REFERENCES...................................................................................................................................... 73
APPENDIX ........................................................................................................................................... 75
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LIST OF FIGURES
Figure 1 - World Electricity Generation by Fuel (TWh) [2] ................................................................................... 2
Figure 2 - Fuel Shares of Electricity Generation [3] ............................................................................................... 2
Figure 3 - Global Fossil Fuel CO2 Emissions [4] .................................................................................................... 3
Figure 4- Operation of CHP plant [8] ..................................................................................................................... 6
Figure 5 – Typical Geothermal Plant [9]................................................................................................................. 7
Figure 6 - Biomass Power Plant [11] ...................................................................................................................... 7
Figure 7 - Typical Fuel Cell [12] ............................................................................................................................ 8
Figure 8 - Solar Cell [15] ........................................................................................................................................ 9
Figure 9-Wind Turbine [17].................................................................................................................................. 10
Figure 10 - Wind Turbine Evolution over Years [18] ........................................................................................... 11
Figure 11 – Simplified Vertical Operated Power System ..................................................................................... 12
Figure 12 – Active Power Network....................................................................................................................... 13
Figure 13 - Goeree Overflakkee Region [24]........................................................................................................ 15
Figure 14 – The method of sampling of a random variable [14]........................................................................... 21
Figure 15- Different correlation coefficients of two random variables ................................................................. 23
Figure 16 - Example of different Gaussian distributions ...................................................................................... 23
Figure 17 - Example of different Weibull distributions ........................................................................................ 24
Figure 18 – Monte Carlo Simulation Flowchart ................................................................................................... 25
Figure 19 - Location of Stavenisse wind station ................................................................................................... 26
Figure 20 - The wind speed data measured by wind station in 2004 .................................................................... 27
Figure 21- An example of the wind speed - wind power relation ......................................................................... 29
Figure 22 - Active power generation by four different types of wind turbine models .......................................... 30
Figure 23 - The wind speed at hub height of V – 112-wind turbine ..................................................................... 31
Figure 24 - PDF and CDF of the wind speed at hub height of V - 112................................................................. 32
Figure 25 - Real the wind speed and the one modeled in Matlab ......................................................................... 33
Figure 26 - The wind speed relationship between two wind turbines ................................................................... 34
Figure 27 - Active power generation by independent wind turbines .................................................................... 35
Figure 28 - Active power generation by dependent wind turbines........................................................................ 36
Figure 29 - National Load Profile for the Netherlands in 2023 in histogram........................................................ 37
Figure 30 - Predicted load data for the year 2023 ................................................................................................. 38
Figure 31 - Load modeled in Matlab..................................................................................................................... 39
Figure 32 – Distribution of two correlated loads and their histograms ................................................................. 39
Figure 33 – Distribution of three correlated loads ................................................................................................ 40
Figure 34 – Steps of modeling load ...................................................................................................................... 42
Figure 35 - The given power system network ....................................................................................................... 43
Figure 36- Total active and reactive power demand in the system ....................................................................... 44
Figure 37 - Monte Carlo simulation for the thesis ................................................................................................ 46
Figure 38 – Voltage distributions for the nodes in case of 9 MW of wind power penetration.............................. 49
Figure 39 - Voltage data measured in Middelharnis and Windmolenpark............................................................ 50
Figure 40 – Voltage distributions for the nodes .................................................................................................... 51
Figure 41 – Comparison of voltage control methods ............................................................................................ 55
Figure 42 – Distribution data of the current injected by the wind farm in case of wind power penetration levels
under 1.05 pu reference bus voltage ............................................................................................................. 58
Figure 43 – Distribution data of the current injected by the wind farm in case of different levels of wind power
penetration under 1.00 pu reference bus voltage .......................................................................................... 59
Figure 44 – Total reactive power distributions for the wind farm in case of different levels of wind power
penetration under 1.05 pu reference bus voltage .......................................................................................... 61
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Figure 45 – Total reactive distributions for the wind farm in case of different levels of wind power penetration
under 1.00 pu reference bus voltage ............................................................................................................. 62
Figure 46 – Total resistive loss distributions in case of different levels of wind power penetration under 1.05 pu
reference bus voltage.................................................................................................................................... 63
Figure 47 – Total resistive loss distributions in case of different levels of wind power penetration under 1.00 pu
reference bus voltage.................................................................................................................................... 65
Figure 48 – Schematic diagram of the given network........................................................................................... 66
Figure 49 – Voltage distributions at the nodes in case of 6MW wind power penetration in island mode operation
for the first scenario...................................................................................................................................... 67
Figure 50 – Generated power distributions of the wind farm (a) and the synchronous generators (b) ................. 68
Figure 51 – Voltage distributions at the nodes in case of 6MW of wind power penetration in island mode
operation for the second scenario ................................................................................................................. 69
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LIST OF TABLES
Table 1 - Shares of Renewable Energy in Electricity Generation (%) [7] .............................................................. 5
Table 2 - Different wind turbine models from Vestas Company .......................................................................... 28
Table 3 - Detailed data about four different types of wind turbine ....................................................................... 31
Table 4 - Total generation and consumption of power in the given network........................................................ 43
Table 5- Statistical data for total active and reactive power demand in the system .............................................. 44
Table 6 – Statistical data of voltage distributions with Voltage Droop Control.................................................... 52
Table 7 – Statistical data for voltage distributions with Cos( φ )voltage control with a leading power factor of
0.95............................................................................................................................................................... 52
Table 8 - Statistical data for voltage distributions with Cos( φ )voltage control with a lagging power factor of
0.90............................................................................................................................................................... 53
Table 9 – Statistical data of the current distributions for the wind farm in case of different levels of wind power
penetration under 1.05 pu reference bus voltage .......................................................................................... 58
Table 10 - Statistical data of the current distributions for the wind farm in case of different levels of wind power
penetration under 1.00 pu reference bus voltage .......................................................................................... 60
Table 11 – Statistical data of the reactive power distribution for the wind farm in case of different levels of
wind power penetration levels under 1.05 pu reference bus voltage ........................................................... 62
Table 12 – Statistical data of the total resistive loss distributions in case of different levels of wind power
penetration under 1.05 pu reference bus voltage .......................................................................................... 64
Table 13– Statistical data of the total resistive loss distributions in case of different levels of wind power
penetration under 1.00 pu reference bus voltage .......................................................................................... 65
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ACKNOWLEDGEMENT
This thesis has been turned in at the Department of Electrical Power Engineering at Delf University of
Technology (TUDelft) as part of the requirements in order to achive the degree of Master of Science
within the field of Electrical Power Engineering. The work is conducted at the Department of
Electrical Power Engineering at TUDelft and in collaboration with Stedin B.V, which is one of the
biggest gas and electricity network operators in the Netherlands.
I thank Stedin B.V. for giving me the greatest opportunity to realize my dreams of pursuing my
university education as a Msc student in the Netherlands by providing its financial support for my
whole education in TUDelft.
I would like to thank my supervisors, specialist Innovatie Asset Management from Stedin B.V., Phd
researcher Ir.Bart Tuinema TUDelft for their invaluable support and guidance throughout the thesis.
I would like to express my deep gratitude to Dr.Ir.M.Gibescu from TUDelft for her supervision,
valuable discussions and never – ending encouragement, patience and support. Many thanks
I shall express my deepest gratitude to my parents without whose unconditional, selfless and
perpetual supports I would not be even close to where I am now. Words would not be able to
convey the true degree of my appreciations and thanks for their dedications and surpassing the
paternal responsibilities.
Last but not least, all my benevolent friends and people whose help sustained me with hope
and passion during my studies in the Netherlands have to be particularly thanked – the
working atmosphere is very important for me and I have been provided with a good one.
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CHAPTER 1 –Introduction
1.1. Introduction
The main function of electrical power systems is to deliver electrical energy to the loads such as small
and large customers within the limits of desired safety and reliability as well as in economical way. In
a typical electrical power system, firstly, the generation of electricity through many sources is
achieved then the generated electricity is transmitted along the distances to reach customers such as
residents and industry so that they use the electricity to meet their own needs. The operation of power
systems actually depends on the relationship of equality of consumption and production of electricity
at the same time. This is mainly due to the effect of non – storability of electricity by nature in large
scale during the operation of the electrical power system in a network. Therefore, this leads to
consideration of control of power system by means of load and generation of the power during the
operation. Uncertainty of the power consumption as well as the power production makes the system
difficult to control.
In case of electrical power penetration due to the renewable energy sources, new challenges and
opportunities start to appear in the field of design and control of electrical power generation systems
for future. Especially, in Europe the production of electricity from wind has gained an increasing
importance therefore, the penetration of wind power in European electrical distribution networks has
been considerable issue in terms of the system stability [1]. One of the most significant issues to deal
with is keeping the voltage in a desired level in a network with increasing amount of wind power
penetration.
1.2.
Conventional Generation of Electricity
This type of generation method is the most common one used in today’s world in order to generate
electricity. Over 85 % of the total electrical energy is produced by means of conventional generation.
The conventional generation totally relies on fossil fuels such as coal, oil, natural gas, and nuclear
sources, which takes millions of years to be formed in the Earth’s crust. In order to generate
electricity, power plants use this type of natural sources. These power plants (thermal plants) are so
big constructions that hundreds of MW of electricity are produced. In thermal plants, fossil fuels are
burned to generate steam from the water. Afterwards, the mechanical energy stored in steam is used to
turn the rotor of the generators resulting in electrical generation as an output. The main reasons why
conventional generation is still being used is due to the economical advantage in terms of power
generation. The cost of using fossil fuels to generate electricity is the least among other types of
sources. Besides the cheapest price spent in order to obtain fossil fuels to burn, another advantage of
using generation plants based on fossil fuels is having less risk of main failures during the operation.
This is due to the fact that the electric industry based on using conventional resources are already
mature enough so that there are worldwide equipment manufacturers, which can support hardware
supply as well as the technical information so easily.
Using conventional sources for electricity production in fact brings many problems with it although it
is widely used all over the world. These problems are simply depletion of fossil fuels, environmental
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concerns, energy security and possible major financial problems in the future. Some of the problems
are stated as follows:
1.3.
Main Problems of Using Conventional Sources
1.3.1 Depletion of fossil fuels
In today’s world, humankind is facing the danger of depletion of natural resources such as oil, coal
and natural gas. In spite the fact that there is not an exact time for conventional resources to be
completely finished, many researchers agree with around 100 years for the lifetime of natural sources.
This consumption rate is increasing higher and higher along the years. As shown in figure 1, most of
the electricity production comes from burning of fossil fuels with an increasing rate.
Figure 1 - World Electricity Generation by Fuel (TWh) [2]
The evolution of the world electricity generation from fossil fuels has been drastically changed in the
direction of consuming more and more fossil fuels compared to the penetration rate of the renewable
sources. This is obviously shown in figure 2.
Figure 2 - Fuel Shares of Electricity Generation [3]
While the total electricity production in the world was 6.116 Terawatt (TWh) in 1973 and most of it
was coming from coal, oil and gas, in 2007 the total electricity production which was 19.771 TWh and
use of coal, natural gas, nuclear power increased in a significant way whereas use of oil decreased
considerably. In figure 2, there is one more remark that has to be taken into account, which is the
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change in rate of renewable energy sources (RES) labeled red color (Other**) sharing world’s total
electricity generation. After around two decades, there is a small amount of increase of RES. Since the
rate of the consumption of fossil fuels is rocket speed and these sources are not finite, use of RES with
conventional sources in the short term seems the best idea and after all fossil fuels are consumed, only
the solution will become RES for the humankind to survive.
1.3.2 Environmental Concerns
One of the major consequences of burning fossil fuels besides depleting them is to emit pollutant gases
so called greenhouse gases to the atmosphere resulting in thickening the ozone layer and increasing the
global temperature. As a result, climate changes all over the world, natural disasters frequently appear
causing big scale migrations and lack of food worldwide. After great industry revolution in Europe,
the dramatic increase of CO2 concentration in the atmosphere shown in figure 3 has been reached at
dangerous limits agreed in consensus.
Figure 3 - Global Fossil Fuel CO2 Emissions [4]
In today’s world, environmental policies may be the most important factor for the demand of
penetration of renewable energy sources in the market.
1.3.3. Energy Security
Even though the definition of energy security can be various among many people, here in this study
definition can simply be made as easy access of primary energy sources for consumption of energy. In
modern economies, availability of cheap energy has been always an important issue to consider. In
today’s world, as the consumption rate of primary energy sources increases so fast, threats to energy
security start to have dominant effects in energy policy, political stability, and price of fuels, standard
living costs, and race between big countries, which demand energy over energy supplies, and
economical and technological dependence on primary energy supplies [5]. Due to issues mentioned
above, there has been a considerable shift from using conventional energy sources to the renewable
energy sources as an alternative option.
1.4.
Renewable Energy Sources
Renewable energy refers to energy sources, which can be replenished by itself during a natural process
in spite of the fact that it is consumed during the utilization. This can be also known as infinite energy
sources. RES captures its own energy from a natural process. The common known sources of
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renewable energy are lights coming from sun so called solar energy, wind, wave power, biomass,
geothermal power, hydrogen derived from natural resources and hydropower [6]. Except geothermal
power, most of renewable energy sources derive from simply solar energy. Energy obtained by
biomass is produced by related chemical reactions called photosynthesis. Wind energy is obtained by
wind flow due to temperature and pressure difference of locations. Solar energy is derived from
energy packages existed in sunlight, which is known as photons.
Even though there are many differences between renewable energy sources and conventional energy
sources, there are actually two main properties of renewable energy sources, which are geographically
distributed and they cannot be controlled. All over the world, in spite the fact that there is abundant
supply of renewable energy, because those sources are spanned through in large distances, the overall
energy density of RES is low. Therefore, in order to harness the power of those sources, small scale of
converters should be used in many places in the power system. The desired impact of the renewable
energy sources can be obtained only by increasing the number and the efficiency of converters in areas
where renewable energy is expected to be harnessed. Since the level of penetration of electrical power
changes because of the renewable sources, there is voltage variation problem in those networks.
Therefore, in order to keep the voltage stability in a power system, many control methods should be
discussed when the renewable energy sources in large scales are being introduced. Another drawback,
which RES suffers is the non – controllability characteristic by nature. Since the power output from
renewable energy sources rely on nature, the ability of controlling the power coming from those
sources becomes difficult to manage. For instance, the wind power is always dependent on the wind
speed in which the wind speed at a specific location is not always constant and varies in a large scale
over time. This makes again voltage variation along the operation of power system network. As a
result, this feature of renewable energy sources force people who are interested in, to study out new
solutions to keep the voltage stability as much as possible.
Although the total conventional energy sources throughout the world has been drastically consumed
and the renewable energy sources seem as future promising energy sources in the world’s developed
countries energy policy agendas, in several industrialized countries incentives to improve utilization of
renewable energy sources are rather low than it should be. There is little improvement, which
especially EU -15 countries has taken. Detailed information, which shows that little progression, is
shown in table 1.
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Table 1 - Shares of Renewable Energy in Electricity Generation (%) [7]
During the period 1990 to 2003, it is evident that there is a 42% increase in renewable energy supply
in EU. However, only two countries Denmark and Spain have achieved the goal of expansion of use of
renewable energy sources greater than what main EU renewable energy policy aims. This is also
proved by IEA since it claims by 2001, only 4 countries out of 85% percent of its member countries
use only wind and solar power rather than other important types of renewable energy sources such as
geothermal and biomass.
1.5.
Types of Electricity Generation Methods from Renewable Energy
Sources
The increase of serious problems faced by societies and governments on the global scale due to the use
of conventional energy supplies such as coal, oil and natural gas force policy makers, government
constitutions, universities and non – government organizations to look for another option which is
simply called harnessing electricity power from available renewable energy sources. This notion of
using renewable energy sources results in wider range of research, more incentives, and efficiency
improvement of electricity generation systems implemented to extract the electricity as much as
possible. Today, actually there are many possibilities in order to use renewable energy sources to meet
the needs such as using combined heat and power (CHP) cogeneration, micro - turbine generator, solar
cells, biomass, wind turbine and new emerging technologies such as fuel cells. Thanks to advance
improvements in the field of power electronics and other technologies, the efficiency of use of non –
conventional generation has been increasing all over the world. The main types of generation systems
are mentioned in the following sections.
1.5.1 Combined Heat and Power (CHP) Plants
Combined heat and power plant is simply used to produce electricity as a main goal and heat as a by –
product shown in figure 4. The integration of production in terms of electricity and usable heat in one
single unit enables higher efficiency rates of using CHP plants. In today’s technology, the efficiency
of using CHP plants can reach at 80 %, which is rather high when compared to the efficiency of using
typical thermal plants, which is measured at around 40%.
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Figure 4- Operation of CHP plant [8]
In today’s CHP systems, dominantly well – known power generation technologies such as steam, gas
turbines, micro-turbines as well as reciprocating engines in the production of electricity are being
used. Even though using CHP plants is efficient, in large scales of CHP generation relies on burning
natural gas as a fossil fuel, which causes environmental concerns.
1.5.2 Geothermal Power Plants
Geothermal power plants use thermal energy stored in Earth’s crust through chemical and physical
reactions. Exploitation of energy stored in the crust as a form of hot water and steam or within dry
rocks enables the power plant to produce electricity shown in figure 5. Geothermal energy is often
considered as a form of renewable energy without any producing any pollutant material. However, the
lifetime of geothermal plant located on a specific area can change from 10 – 100 years depending on
the geological feature of that location. Therefore, in that respect some claim that the geothermal
energy can be depleted and cannot be considered as a renewable energy source. Benefits of geothermal
energy can be as follows:
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Produce no pollutants or in any case, pollutant can be deployed underground
No need to pay for fuel cost
Requirement for minimal land as a construction site
Lifetime can reach up to 100 years to generate electricity
Abundant potential availability all over the world as a renewable energy source
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Figure 5 – Typical Geothermal Plant [9]
Today, the use of geothermal energy is extended through many countries with many reasons such as
electricity production, industry as well as agriculture. According to sources, estimation of using
geothermal energy by means of electricity generation is around 11 GW, with a largest capacity in USA
(3086 MW), Philippines as well as Indonesia [10].
1.5.3 Biomass Power Plants
Biomass is a kind of renewable energy source in which biological material and living organisms such
as animal waste (manure), sewage, tree, etc is used for general purposes. A typical benefit of using
biomass is to produce electrical energy with the help of biomass power plants shown in figure 6.
Figure 6 - Biomass Power Plant [11]
There are in fact many benefits of using biomass power plants in order to produce electricity. These
benefits can be as follows:
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Less Air Pollution: Biomass power plants produce less air pollutants such as NOx and SO2
gases compared to its fossil fuel counterparts.
Reduction in animal, food and municipal wastes: Since main fuels of biomass power plants
are animal, food and municipal waste in order to generate electricity, during the operation of a
typical biomass power plant, those wastes are used as a return of benefit.
Besides those benefits of using biomass power plants, it also introduces many opportunities in terms of
economical benefits such as creation of new jobs as well as social benefits such as helping society to
have conscious thinking about renewable energy sources.
1.5.4 Fuel Cells
A fuel cell is a device like battery, which generates electricity from chemical reactions that take place
inside the cell. In order to convert chemical energy into useful electrical energy, hydrogen (H2) is
combined with oxygen (O2) from air resulting in electron flow or simply electricity, heat and water
vapor as a by – product shown in figure 7.
Figure 7 - Typical Fuel Cell [12]
The source of hydrogen can be obtained in two ways: Hydrogen fuel itself and so called fuel converter
that takes hydrocarbon fuels such as methanol, natural gas and gasoline and enriches as a form of
hydrogen gas [13]. Since the electricity generated from one single fuel cell is low, a number of fuel
cells connected in series must be used. Moreover, the output current form produced by the fuel cell is
direct and converters can be used in order to make it into alternating current. All fuel cells include two
electrodes that are anode (positive) and cathode (negative) shown in figure 7. Electrolyte is
sandwiched between these two electrodes. In today’s technology, there are different types of fuel
cells, which differ in size, weight, cost, efficiency and operating temperature for use of specific
purposes. Proton Exchange Membrane (PEM), Molten Carbonate Fuel Cell (MCFC), and Alkaline
Fuel Cell (AFC) are best-known types of fuel cells. The efficiency of fuel cells can reach up to 80 %
[14]. Fuel cells seem as the promising new technology in many fields such as automobile industry and
electrical industry.
1.5.5 Solar Energy
Solar energy is one of the most promising sources of energy in order to produce electricity in today’s
world. It offers abundant quantity of resources all over the world. The device, which converts photons
stored in sunlight into useful way of generating electricity, is called solar cell or photovoltaic cell.
Typical structure of solar cell is shown in figure 8.
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Figure 8 - Solar Cell [15]
In a typical solar cell, there are two or more than two semiconductor layers, which have specific
physical properties. Arrangement of layers are adjusted in such a way that in case of coming sunlight,
energy packages so called photons in the sunlight cause excitation of electrons resulting in crossing
the junction layer(s). At the end of the process, the direct current of electricity is produced. Since the
power output obtained by a single unit of cell is rather low, a stack of photocells connected in series is
preferred to get desired amount of power. Besides this, in order to use electricity generated by
photocells in applications in which AC electricity is required, dc to ac converters should be used.
Thanks to advanced techniques in power electronics, the number of applications being used is
increasing higher and higher. While looking at the number of types of solar cells, it can vary
throughout the market for specific purposes. However, Crystalline Si cells (C-Si), thin – film solar
cells and organic solar cells are the ones, which are best known in solar cell world. As there are
different types of solar cells, the efficiency related to those types of solar cells also vary. For instance,
the efficiency of a typical Crystalline Si solar cell is around 25 % while it is around 20% for thin –
film solar cells. However, the best efficiency obtained from solar cell, which was 42.4 %, was
achieved [16]. Even though the cost of generating electricity by using PV cells is rather high, the
capital cost of PV cells or modules per watt has been declined over two decades. Incentives support
from private sector as well as governments throughout the world will help the capital cost of using
solar cell to reduce at a desired level. There are many advantages of using solar cell generation
systems in order to generate useful electricity. Some of them are shown in the following:
Abundant available source: Since the source of solar energy is directly the sun, there is no worry
about possibility of depletion of the reserve.
Clean way of generating electricity: During the operation of solar cell in terms of involving
chemical reactions in order to produce electricity, by – product after the process is only small amount
of heat which is quite advantages compared to fossil fuel burning plants.
Maintenance and lifetime of solar panels: In case of installation of solar panels, there is only set up
cost to deal with and there are less moving parts, which make it quite compact. Besides this, lifetime
of solar cells can be about 30 years, which is quite essential especially for customers and in terms of
reliability.
Besides many advantages of harnessing solar power with solar cells, there are some important issues
regarding solar energy. There is a fact that solar energy is fully dependent on weather, which means
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that the rate of cloudiness on a specific area determines in fact how much solar energy can be obtained
during time. Therefore, simply to say, solar energy is a kind of stochastic generation of power as well
as wind energy.
1.5.6 Wind Energy
Wind energy is a form of promising renewable energy simply due to activity of airflow (wind) over an
area. Electricity obtained from wind energy is achieved by wind turbine shown in figure 9. To explain
in more detail how wind turbine generates electricity, one should understand basic mechanical and
electrical knowledge. One of key parts of wind turbine is its rotor blades, which can vary from one to
three. The function of rotor blade is simply to capture the wind as a mechanical energy. Since the wind
has enough kinetic energy so that rotor blades can be turned. The other key part of a typical wind
turbine is the generator, which converts mechanical energy by turning of rotor blades and delivering it
by the gearbox, into electrical energy. As shown in figure 9, total space in which gearbox and
generator are located is called as nacelle - the heart of wind turbine.
Figure 9-Wind Turbine [17]
During the operation of a typical wind turbine, it produces electrical power, which can be expressed as
follows [14].
= ×
where;
×
×
(1.11)
×
P: Electrical Power Output (W)
: Power Coefficient
: The wind speed (m/s)
ρ: Density of air (kgm-3)
: Area which rotor blades cover (m2)
By looking at the electrical power equation (1.11), it can be easily shown that the wind speed has the
most dominant role in case of the generation of electrical output power.
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Figure 10 - Wind Turbine Evolution over Years [18]
As shown in figure 10, throughout around 3 decades, there is a tremendous increase of possible
capacity of electrical power generated by a wind turbine as the height of related wind turbine
increases. This in fact shows how the wind energy industry is growing up and increase of installed
wind energy over years. According to the global wind report 2010 by GWEC [19], around 194.4 GW
of electrical energy is provided in terms of the wind energy all over the world. Besides, addition of
around 30 GW of electrical energy provided by wind energy is achieved in every year globally. There
are some facts showing how the wind energy is beneficial [18] as follows:
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At the end of year 2009, results showed that total installed wind capacity all over Europe
could meet the needs of 4.8 % of European total electrical energy.
By the help of wind energy in Europe, according to the results, around 106 million tones of
CO2 was avoided which also means taking 25% of cars in EU off the road.
In 2009, there was a money saving up to 6 billion € per year by cutting down fossil fuel needs.
Hundreds of thousands of people were employed due to the wind energy industry in EU in
2009.
Besides many benefits introduced by the wind energy, there are some important considerations
regarding about production of electricity from wind energy. Since, wind energy is completely
dependent on the wind speed in a location and the wind speed changes, electrical power obtained
during the operation of a typical wind turbine also changes. This is simply due to stochastic behavior
of wind energy just like solar energy. Therefore, in case of wind power penetration in a power
network, there is always voltage variation, which targets the problem of voltage stability along the
network operation over time. This is why many voltage control methods are introduced regarding the
wind power penetration.
Besides those renewable energy sources explained above, there are also some renewable sources such
as hydropower, which simply means converting mechanical energy of flowing water through dams
into electrical power by the help of generators and some derivations of biomass such as bio - fuel as
well as hydrogen technology. Besides this, the research on tidal and wave energy to generate
electricity is still going on with a high pace.
~ 11 ~
1.6. Power System Network Structures
1.6.1 Passive Power Network
In today’s world, besides the traditional vertical operated power system network (passive network), in
case of power penetration due to the renewable energy sources, there is also horizontally operated
power network (active network). In passive network shown in figure 11, the direction of the electrical
power flow is from top to bottom.
HV Network
Power Generation
69 kV – 1100 kV
MV Network
10 kV – 69 kV
LV Network
120V/240 V – 10 kV
Active Power Flow
Direction
Primary Distribution
Network
Secondary Distribution
Network
Figure 11 – Simplified Vertical Operated Power System
In a typical passive network, the power generation is achieved by using conventional energy sources
therefore, generation plants are thermal, nuclear and large – scale hydro dams. These plants are located
so far away from cities or locations where customers need electricity. The reasons are location of
primary energy sources to cut fuel transportation costs and avoiding high risks of air pollution from
people. Besides this, since the distance between power generation plants and last customer is so high,
around 11% of total electrical energy is lost due to thermal resistance during the operation. On the
other hand, this type of structure provides dispatchable generation and a reliable system operation
[20]. Since the voltage level at the end of synchronous generator stator bars is low, transformers are
being used to step up the voltage in a desired level to eliminate resistance losses as much as possible.
In this part of the network value of voltage changes from 69 kV to even 1100 kV. In medium voltage
level section, voltage is stepped down to levels of 10 kV. Another name for this section is primary
distribution network. The last part of the system is called secondary distribution network that is used
to deliver electrical power generated by plants to the last customers such as residents of a city. The
main characteristic of using conventional energy sources in this type of network is the easiness of
controlling primary energy sources. This simply means that the voltage can be under control.
~ 12 ~
1.6.2 Active Power Network
In horizontal power system or active power systems, besides using conventional sources, there is also
penetration of electrical power due to the renewable energy sources such as solar, combined heat
plants and wind power in the network. This type of power system is seen in figure 12.
Transmission Network or High Voltage Level Network
Direction of Active
Power Flow
Injection of Active
Power of Renewable
Primary Distribution
Network
Injection of Active
Power of Renewable
Source
Direction of Active
Power Flow
Injection of Active
Power of Renewable
Source
Secondary Distribution
Network
Injection of Active
Power of Renewable
Source
Loads
Figure 12 – Active Power Network
Unlike the traditional power system shown in figure 11, the electrical power produced by renewable
sources is injected into the primary and secondary distribution networks. This is perhaps the most
~ 13 ~
important difference between traditional one and active power network. In an active power network,
active power flow is bidirectional instead of being uni - directional power flow in a passive network.
Due to social, environmental and political efforts that support renewable source penetration in an
electrical network, active networks are being taken into consideration in many ways such as
protection, voltage stability as well as reliability. There are many advantages of using active network
electrical systems such as grid support, reduction of burning fossil fuels resulting in positive
environmental and economical impacts. For instance, in some locations in which voltage is not at
desired level, the voltage level can be increased in case of wind power generation at a certain extent.
This makes the system more stable and reliable as well. Besides this, since there is active power
injection because of renewable sources at medium and low voltage level networks, less fossil fuels
will be consumed which result in less amount of greenhouse gas emissions to the atmosphere and
economical savings in terms of converting free power into electrical power. Moreover, since the active
power from those renewable sources is harnessed at close distances to the customers, there will be
fewer amounts of resistive power loss resulting in increase of overall efficiency in the system.
Even though the transition from active network to passive network has advantages over existence of
passive networks, there are serious issues to be dealt with it such as keeping the system frequency (50
Hz or 60 Hz), the need for sophisticated protection methods, voltage quality and the impact of existing
grid structure [21]. These problems can be addressed as follows:




System Frequency [22]: Since there is no regulation strict by the network operator to keep the
system frequency, there can be considerable frequency changes during the operation or
frequency converters can be used as an alternative solution, which means extra money to pay.
Voltage Level [22]: The amount of power penetration of renewable source into the network has
an important role in terms of keeping the voltage stability especially at the nodes where
renewable source is connected. Due to high amount of power penetration and bi – directional
power flow, there can be voltage instability and possibility of harmonics in voltage.
Reactive Power [22]: Since small and medium size asynchronous generators are being used in
applications of distributed generation, there is no reactive power injected to the grid, which
has an essential effect in voltage stability issue. Therefore, special power electronic
equipments are needed to regulate reactive power flow during the operation of distributed
generation.
Impact of existing grid [22]: Penetration of distributed generation in a power system can have
a negative impact on especially weak parts of the grid resulting in a higher probability of
sudden voltage dips and increase of fault level.
~ 14 ~
1.7. Goal of the Thesis
The installation of wind power generation in the Netherlands has been increased drastically over years
and there are many on - going projects regarding harnessing wind energy and converting it into useful
electrical energy.
One of the largest gas and electrical network operators, Stedin B.V. [23] has decided to install wind
farm in Goeree Overflakkee region, Zeeland in the south of the Netherlands. The location is shown in
figure 13 and is pointed it out with a red arrow.
Figure 13 - Goeree Overflakkee Region [24]
The idea is that residents living in that region suffer voltage instability problem for a long time since
the current grid is not strong enough to meet the total load demand and does not reach at desired
reliability level. Therefore, the company has a plan that in case of installation of wind farms, the direct
advantage of using the wind energy in order to get sufficient electrical power will be obtained. This
will also enable a more reliable supply of electricity for the people living in that island. In order to
obtain desired results regarding the system stability and reliability when the wind power is injected
into the existing grid, there are many important steps to be taken into account.
The wind power penetration puts many serious problems on the table such as voltage and frequency
stability of the conventional power grid due to the uncertainty of the output power of wind during
operation of wind farm(s). Several serious technical problems are faced during the system operation,
such as over-voltage or under-voltages at the points of connection at steady state, protection
malfunctions, increase in short circuit levels and the power quality problems. However, in this study
frequency stability will not be discussed and assume that there is no problem with the frequency
stability. Voltage stability in steady – state operation of the system will be focused on throughout the
study as a main goal.
One of the most essential steps is the probabilistic power flow analysis since the wind power shows
stochastic behavior over time. By the help of probabilistic power flow analysis tool, one can have a
better understanding of total system behavior in terms of studying at variations of bus voltages, bus
phase angles, direction and value of active and reactive power flows as well as current and resistive
losses according to the different levels of wind power penetration in the system. In this study, it is
aimed that one can find clear answers to questions stated as follows:
~ 15 ~




To what extent of the wind power can be injected into the network while keeping the voltage
in stability?
What is the best voltage control method in case of the wind power penetration at a certain
amount?
What are the percentages of over - voltages and under - voltages in comparison with the total
number of the voltage samples during the operation of wind farm depending on the
penetration of different levels of wind power in the system?
What is the impact of island mode operation on the system in case of a certain amount of
wind power penetration?
1.8. Outline of Thesis
The thesis is organized as follows:




In the 2nd chapter of the thesis, the mathematical background for latter chapters is discussed so
that the reader can have a better understanding of the topics mentioned in those chapters.
Several equations and definitions mostly based on topics in probability and the statistics are
given. Literature survey about statistics based on the electrical power systems is discussed.
Besides this, the reason why Monte Carlo simulation is required for the thesis is also
explained as an introduction and the general information about it, is given as well.
In the 3rd chapter, modeling of the loads and the wind farm that is expected to be deployed in,
Goeree Overflakkee region is introduced in a way that several methods, which enable how to
model wind farm as well as the loads in the system, are explained extensively. Besides this,
general information about the part of the electrical network operated by Stedin B.V. is also
provided.
In the 4th chapter, several case studies are introduced in case of different levels of wind power
penetration in the system. These case studies are discussed in three different categories. First
case study is about the comparison of the voltage control methods in terms of keeping the
voltage in a desired interval as much as possible in case of different levels of wind power
penetration. The second case study is about the effect of different amount of wind power
penetration on the system depending on the consideration of two different reference bus
voltages. The probability distribution of the total current injection from the wind farm, the
probability distribution of the total reactive power absorbed and injected by the wind farm and
lastly the probability distribution of the total resistive loss in the whole system regarding
different levels of wind power penetration is given as well. The last case study investigates the
impact of the island mode operation in the electrical distribution network when there is a
certain amount of wind power injection into the system. Two scenarios in this case study are
discussed.
In the 5th chapter, conclusions and recommendations for the future research are made.
~ 16 ~
CHAPTER 2 - Mathematical Background for Probabilistic Load Flow
Analysis
2.1. Introduction
In today’s electrical power networks, deterministic power flow analysis is being used to calculate the
power flows, voltages, currents etc. This type of analysis is based on system variables, which are
known in a deterministic way. For instance, during the operation of a power system, the total power
consumption as well as total power generation in the system is measured. Nevertheless, the penetration
of distributed generation with an increasing speed in electrical power systems creates the problem of
uncertainty. This uncertainty comes from the fact that the active power production of many renewable
sources rely on what the nature rules over. For instance, the wind power always depends on the wind
speed of a region. However, the wind speed is totally dependent on natural phenomena existing in the
nature such as temperature, humidity, atmospheric pressure, etc. Thus, the wind speed cannot be
predicted precisely for future power generation in a power system. Since the stability of a power
system is kept by balancing the power generation as well as the power consumption during the
operation of the system, injection of wind power to the system creates a problem of predicting future
operating conditions for a typical power system when it comes to power generation. Besides the
unpredictability of the wind power in the system, there is also unpredictability problem for load
behavior since loads are also changing time to time. However, the uncertainty of the load behavior in a
system is not as high as much compared to the uncertainty of the wind power penetration.
Load flow computations play a vital role in terms of studying the performance of a power system. In a
basic sense, the computation is based on the assessment of magnitudes of bus voltages and angles as
well as the reactive and active power flows according to the given input power in the system.
However, in case of penetration of distributed generation in the system, the inputs start to be a random
variable as the reason is explained above. While looking at the general characteristic of a load – flow
computation, it is based on the solution of nonlinear algebraic equations in order to find voltage
magnitudes as well as angles according to a given power injection to the system.
In order to solve the uncertainty problem faced in an electrical power network including distributed
generation, probabilistic load flow (PLF) can be used as a tool. PLF method introduces studying with
experimental samples for the loads and distributed generation units such as wind farms in a system.
These experimental samples are obtained by using probabilistic methods, which will be discussed in
the later parts of the thesis.
The application of probabilistic load flow analysis tools has been studied for only around three
decades although the electric industry is around 120 years. Albert and Nitu [25] wrote one of the first
books, which was about the applications of probabilistic methods in power system studies, in 1968.
Afterwards, many researches were held and several papers were written. For instance, Allen and
Borkoswka [26] studied on a simplified version of probabilistic load flow. In this study, two things are
introduced: The electric power system is introduced with a dc power network; therefore, reactive
power is not considered (1). Real parts of the loads are assumed as independent random variables (2).
Later on, studies about AC network took place. R.N. Allan and Al – Shakarchi proposed two methods
in order to find probability curves of bus voltages, angles, generated active and reactive power flows
by using linearization techniques of ac load flow analysis [27].
~ 17 ~
2.2. Fundamental Concepts
After introducing the need for probabilistic load flow analysis, its characteristics as well as small
review of studies took place. It is now time to give some essential definitions of concepts in
probability [28] and the statistics, which will help to understand the issues in later chapters of the
thesis.
Random Variable: A random variable is a number x (ζ) assigned at every outcome ζ of an
experiment. The resulting function must satisfy the following two conditions but is otherwise
arbitrary.


The set { ≤ } is an event for every x
The probability events of the events { = ∞} and { = − ∞} equals zero
Mean or Expected Value (Central Value): The mean or expected value is defined as weighted
average of the possible values of the related random variable in which weights are probabilities.
Expected value of a distribution function gives an idea of the location of the distribution or density
function.
If X is considered as a discrete random variable, the mean value named as E(X) is given by
in which
( )=∑ (
×
)
(2.1)
xi: value of given random variable
pi: probability of that value to occur
Variance and Standard Deviation: Variance (VAR) is defined as a measure of how widely
distribution function spreads out. Standard Deviation denoted by σ, is the square root of variance of
the related distribution.
If X is considered as a discrete random variable, the variance of the random variable X is defined as
Var (X) = E [(X-E(X)) 2]
(2.2)
σx =
(2.3)
So , standard deviation is calculated as
( )
Cumulative Distribution Function: The cumulative distribution function (CDF) of the random
variable x is defined as:
F(
) = P( ≤
)
(2.4)
The properties of cumulative distribution function are given as:








(+∞) = 1 , (−∞) = 0
F(x1) < F(x2) if x1 < x2
F(x) = 0 if F(x0) = 0 for every x
P(X>x) = 1 – F(X)
F(x) is continuous in the right, F(x+) = F(x)
P(x1<x<x2) = F(x2) – F(x1)
P(x=x0) = F(x0) – F(x0-)
P( ≤ ≤ ) = F(x2) – F(x1-)
~ 18 ~
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
Probability Density Function: Probability density function (PDF) is defined as;
( )=
(2.12)
Any function having the properties, which are mentioned above can be use as a probability distribution
function of a random variable. The probability distribution function can be obtained in any kinds of
problem related to discrete random variables. In case of distributed generation involved in electrical
power systems, the probability distribution of random variables such as injected power and load
distribution play an important role to understand the operation of a power system clearly. Some of the
most well - known probability density functions that are also used extensively in later chapters of the
thesis are discussed below.
Joint Probability Distribution Function: In the study of statistics, if there are given random
variables such as X and Y, the joint probability of those random variables is defined as the probability
of the events took place for both X and Y. If there are two random variables to be considered, it is
called as bi variate distribution however if there are more than two random variables, the general term
is used as multivariate distribution.
If both random variables are discrete, then the joint probability mass function is given as:
( =
( =
= ) = ( = | = ). ( = )
(2.13)
and
= ) = ( = | = ). ( = )
(2.14)
The probabilities of these random variables must satisfy the following:
∑ ∑ ( =
0≤ ( =
= )=1
= )≤1
(2.15)
(2.16)
= )
(2.17)
The joint cumulative distribution function of the bi variate random variables (X, Y) is
( , )=∑ ∑
( =
In order to agree with the axioms of probability, the joint cumulative distribution function must satisfy
the following:
(−∞, −∞) = 0
(2.18)
( , −∞) = 0
(2.20)
(−∞, ) = 0
(∞, ∞) = 1
(∞, ) =
( , ∞) =
( )
( )
( , ) is nonnegative and non decreasing function of x and y
~ 19 ~
(2.19)
(2.21)
(2.22)
(2.23)
Marginal Distribution: If the random variables (X, Y) each have their own probability distributions,
the marginal distribution of a random variable is just its own distribution while another random
variable is being ignored. The reason why this concept is introduced is the fact that even though the
joint probability distribution of the random variables (X, Y) is known, their individual probability
distributions cannot be known. However, it becomes possible to find out these individual distribution
functions from their joint distribution function in such a way that if these two random variables are in
discrete form then the marginal distribution of X can be found as the following:
( = )=∑
( =
= )
(2.24)
Similarly, marginal distribution of Y can be found as the following:
( = )=∑
( =
= )
(2.25)
Covariance, Correlation, and Independence: Covariance of two random variables describe to what
extent these two random variables are dependent on each other.
Covariance (COV) is defined as follows:
( , )= (
)−
( ) ( )
(2.26)
If X and Y are statistically independent, then covariance of these two variables (Cov(X, Y)) is zero or
they are uncorrelated. Then, the formula becomes as:
(
)= ( ) ( )
(2.27)
If Cov(X, Y) is large and positive, the values of X and Y tend to both large or both small relative to
their respective means. On the other hand, the values of X tend to be large when the values of Y are
small and vice versa.
Therefore, the Cov(X, Y) is a measure of the degree of linear relationship between the variables X and
Y. Therefore, in practice it is more convenient to use the correlation coefficient (product moment
correlation, linear or Pearson correlation), which is defined as follows:
=
( , )
(2.28)
can change in a range between – 1 and +1. When is 1,
X and Y is perfectly linear. For instance, in figure 15, distributions of two random variables
having a specified correlation coefficient are given.
Depending on the formula, values of
Copula
One of the most popular methods in order to model variables with a degree of dependency is copula.
Abe Sklar [29] firstly used the term of copula. The general idea behind using a copula is to separate
the marginal distribution of each variable from the dependence (correlation) structure. In order to
explain how a copula works, the cumulative distribution function of a variable to generate a random
variable is used. In order to achieve that, a uni - variate distribution is simulated by sampling from a
uniform U (0, 1). Then, by treating this as an observation of the variables CDF, it is possible to obtain
a sample from a PDF. This process is given in figure 14.
~ 20 ~
Figure 14 – The method of sampling of a random variable [14]
Sklar’s theorem [30] states that for a given multivariate distribution function and their
relevant marginal distributions, there exists a copula function relates them.
The most commonly used copula functions are the Gumbel, Gaussian, Archimedean and t –
copula. In this chapter, only t – copula is studied in detail since t – copula is widely used for
modeling the loads in the next chapter. However, one can refer to the book (An Introduction
to Copulas) written by Roger B. Nelson to get more information about copulas.
t- Copula
It is one of the most well known copulas in the elliptical copula family. Throughout the
explanation of t – copula, mathematical calculations are obtained by referring to the paper
written by Stefano Demarta & Alexander J. McNeil et al. (2004). t copula comes from the
multivariate t distribution. Mashal, Zeevi, and Breymann et al. (2003) showed that the
empirical fit of the t – copula is usually better than the Gaussian copula, which is the
dependent structure of multivariate normal distribution.
Multivariate t – distribution
Multivariate t distribution is a common form of the univariate t – distribution of two or more
variables. It is a distribution for random vectors of correlated variables, each element of which
has a univariate t distribution. In the same way as the univariate t distribution can be
constructed by dividing a standard univariate normal random variable by the square root of a
univariate chi-square random variable, the multivariate t distribution can be constructed by
dividing a multivariate normal random vector having zero mean and unit variances by a
univariate chi-square random variable.
~ 21 ~
The multivariate t distribution is parameterized with a correlation matrix, P, and a positive
scalar degrees of freedom parameter, ν. ν is analogous to the degrees of freedom parameter of
a univariate t distribution. The off-diagonal elements of P contain the correlations between
variables. Note that when P is the identity matrix, variables are uncorrelated; however, they
are not independent. The multivariate t distribution is often used as a substitute for the
multivariate normal distribution in situations where it is known that the marginal distributions
of the individual variables have fatter tails than the normal [31].
The probability density function of the d-dimensional multivariate t distribution is given by:
= ( , , )=
|∑|
Г(
Г
) (
)
)
Г
(1 +
(2.29)
)
While the marginal distributions are being standardized, the copula remains invariant. It
means that the copula of a td (v, u, ∑) is identical to td (v, 0, P) distribution where P is defined
as the correlation matrix by multiple random variables. ∑ is called as the dispersion matrix or
the absolute value of the determinant of the d – dimensional matrix. Then the copula is given
as the following:
where
,
( )= ∫
(
)
…∫
(
Г
(
) | |
(1 +
)
(2.30)
: distribution function of v degrees of freedom
: inverse of the standard
distribution
C: Copula
P: Correlation matrix
ud: mean vector of d dimensional random vector of X (X1, X2…Xd)
Г = Gamma function
In order to produce correlated multivariate distributions by using t – copula, a small script as a
m- file in Matlab environment is used. Firstly, as seen in equation 2.29, by using random
generator in Matlab for “x” values and using the probability density function, distribution data
with a specified size for “x” values are obtained with the given correlation matrix and number
of samples. This process is done for all distribution data in the sample. In this study, since the
total number of sample data is selected as 15000 for only one load distribution data, total
matrix becomes 35 × 15000 in the system. The reason why 15000 is selected is that generally
10000 samples are enough for Monte Carlo analysis for load flow power analysis in the
systems and there is value (s) converged to a number after the simulations.
In equation 2.30, for each distribution data, integration process is done resulting in correlated
distribution data in the all sample. This integration also means that finding the cumulative
distribution functions of each correlated load data in the system. This process is also called as
empirical cumulative distribution fit.
~ 22 ~
=1
2
2
0
0
-2
-4
-4
-2
-2
0
x1
=0.25
2
-4
-4
4
-2
0
x1
=0
2
4
-2
0
x1
2
4
4
2
2
0
0
y1
y1
4
-2
-4
-4
=0.5
4
y1
y1
4
-2
-2
0
x1
2
-4
-4
4
Figure 15- Different correlation coefficients of two random variables
Normal (Gaussian) Distribution: Normal or Gaussian probability density function includes two
different parameters, which are mean (µ) and standard deviation (σ). The function is defined as
follows:
( )=√
× ×
×
(
×
)
(2.31)
In figure 16, normal distributions with different mean and standard deviation values are
shown.
0.7
1=0, 1 = 1
0.6
2=0.5,  2 = 0.75
0.5
f(x)
0.4
0.3
0.2
0.1
0
-4
-3
-2
-1
0
x
1
2
3
Figure 16 - Example of different Gaussian distributions
~ 23 ~
4
Weibull Distribution: It is commonly used in modeling the wind speed distribution over a region
where wind power is expected to be harnessed. Weibull probability distribution function is based on
two parameters, which are shape ( ) and scale ( ) parameter. The distribution function is defined as:
( | , )=
( )
(2.32)
exp (−( ) )
In figure 17, weibull distributions with different shape and scale parameters are shown.
0.12
1=10,1=1.5
2=8,2=2
0.1
f(x)
0.08
0.06
0.04
0.02
0
0
2
4
6
8
10
x
12
14
16
18
20
Figure 17 - Example of different Weibull distributions
2.3. General Introduction to Monte Carlo Simulation
In today’s world, problems of complex modeling and application of specific numerical methods to
obtain power flow solutions are being faced in modern electrical power systems. While distributed
generation takes its own place in a system, in order to approach load flow solutions, the use of
probabilistic methods becomes greatly difficult. In such cases, Monte Carlo simulation can be
regarded as an alternative option. Monte Carlo simulation is a repeated process of generating
deterministic solutions to a given problem. Besides this, each solution is linked to a set of
deterministic values of the related random variable. The main feature of Monte Carlo simulation is a
generation of random numbers from probability distributions that describe related random variables.
One of the most important aspects of Monte Carlo simulation is the number of repetitions of the
process. Since generally thousands of repetitions are required using Monte Carlo simulation, its
application to complex problems can be costly in terms of time consumption. However, thanks to
advanced computers, use of Monte Carlo simulation has extended to not only solutions of load flow
~ 24 ~
problems in electrical systems but also to many other fields such as finance, biology, business and
telecommunications.
Although Monte Carlo is used in various fields, there are core steps to be taken into account while
performing Monte Carlo simulations. These steps are given as follows [32].
START
Define Boundaries of the Input Values
Generation of Inputs from Probability
Distribution Functions
Deterministic Computation
Store Results
END
Figure 18 – Monte Carlo Simulation Flowchart
In the first step, boundaries of the input variables such as the wind speed in case of wind power
penetration are defined. Afterwards, samples related to input variables are generated depending on
their probability distribution functions. The number of samples, which are produced, varies depending
on the problem complexity. Then, in order to obtain the results, deterministic computations are
performed. These computations can change problem to problem. For instance, if the problem is about
computing load flow in a power system, one of the computations will be probably based on ac Newton
– Raphson load flow method, which is explained thoroughly in Appendix B.
~ 25 ~
CHAPTER 3 – Modeling
3.1. Wind Farm Modeling
In order to analyze the effect of wind power penetration in a power system as clear as
possible, firstly the fundamental goal, which is the wind farm modeling should be achieved.
There are actually many issues, which enable to model the wind farm such as wind
distribution for that region, type of single wind turbine in the wind farm, correlation factor
between wind turbines and finally the expected power delivered by the wind farm. These are
the most important steps taken into account during the study.
3.1.1 Wind Distribution
The wind speed distribution data is so vital that only logical estimation of the output power
from an operating wind farm can be obtained. Because the wind speed is the main fuel of
wind power and shows stochastic behavior, modeling of the wind speed plays a very
important role in the case of wind farm modeling. In this study, considering possible
implementation of the wind farm connected to Windmolenpark, related to the wind
distribution data is taken from Stavenisse wind station [33] shown in figure 19. This data
measured in 2004 is provided by National Meteorological Institute of the Netherlands
(KNMI) and shown in figure 20.
Figure 19 - Location of Stavenisse wind station
In this wind speed collection of data, the wind speed is measured in each of hour for each of
month throughout all year of 2004. Therefore, the number of the wind speed data is 8736.
Besides this, according to information provided by KNMI, the wind speed data is taken at
height of 16.5 m and with a roughness coefficient* of 0.002. Therefore, in order to obtain the
possible wind speed data related to at any height, the reader needs to use well-known wind
shear formula taken from Danish Wind Industry Association [34] shown as below:
~ 26 ~
*
Roughness coefficient: It is a coefficient of describing the wind speed reduction due to
physical obstacles of a terrain.
Wind Shear Formula:
(ℎ ) =
×
In which,
(
(
)
(3.1)
)
h: Height which desired the wind speed is calculated (m)
Vref: Reference the wind speed (m/s)
zo: Roughness of surface of terrain
href: Reference height (m)
Depending on the wind speed data regarding figure 20, one can easily see that the wind speed changes
from a minimum value of 0 m/s and maximum value of 22.5 m/s as the time passes in 2004.
Moreover, 22 hours in total year of 2004, there is no wind speed, resulting in no wind power output.
Since there is always variation in the wind speed according to changes in weather, wind power
generation varies as well. This is why the wind power generation behaves stochastically during the
operation.
Hourly Wind Speed Data in 2004
25
Wind speed (m/s)
20
15
10
5
0
0
1000
2000
3000
4000
5000
Hour
6000
7000
8000
Figure 20 - The wind speed data measured by wind station in 2004
~ 27 ~
9000
As shown in wind shear formula, one can actually see that the wind speed has a logarithmic
relationship with the height of a tower. As the height of the tower increases, the wind speed at that
height increases logarithmically as well. Therefore, this formula enables us to find an appropriate wind
turbine in terms of the generation of active wind power from a wind farm in a most efficient way.
3.1.2 Selection of Appropriate Wind Turbine
A typical wind farm operating in power system comprises of several wind turbines, which are same
type in most of the time. Therefore, if one wants to investigate the effect of generation of active power
by wind farm, the selection of suitable wind turbine takes an important place. There are several types
of wind turbines from several manufacturers all over the world. Besides this, each wind turbine has its
own characteristic operational features. However, in order to select the most appropriate wind turbine
for this study, main features of a wind turbine will be taken into consideration. These features are such
as the rated power that the wind turbine can provide during its operation, cut – in, rated speed, cut –
out speed and hub height of that wind turbine. These all features for the wind turbine have dominant
roles in order to get operating efficiency as well as possible.
Since there are many different types of wind turbines in the market, the selection of the wind turbine
can be achieved by looking at its operating efficiency in terms of the active power production
regarding the wind speed distribution data taken in 2004. In this study, several wind turbines provided
by Vestas Company [35] are compared as shown in table 2.
WIND TURBINE MODELS PROVIDED BY VESTAS
Main Features
V-60
V-80
V-90
V-112
60
80
105
119
Power
0.85
2
3
3.075
Cut – in Speed
(m/s)
3.5
4
3.5
3
Rated
(m/s)
Speed
13
16
15
13
Cut – out Speed
(m/s)
20
25
25
25
Hub Height (m)
Rated
(MW)
Table 2 - Different wind turbine models from Vestas Company
3.1.3 The wind speed – Wind Power Relationship
There is a quite close well-known relationship between the wind speed measured at the height of wind
turbine and wind power generated by related wind turbine. This relationship is shown in equation 3.1.
Wind turbine is operating in such a way that it produces no power below cut – in speed. Below cut – in
speed, there is no enough torque exerted by the wind to turn the rotor blades of the wind turbine to
generate electrical power. While wind begins to flow at a speed higher than cut – in, electrical power
starts to increase as well. This relation can be expressed linearly between cut – in speed and rated
speed. However, the wind turbine has a certain limit of generating electricity depending on the wind
speed. This limit is called rated electrical power output and the speed at that power is called as rated
~ 28 ~
operational speed of the wind turbine. As wind flows between rated speed and cut – out speed,
delivered output power is forced to be constant. In case of any speed greater than cut – out speed the
design of wind turbine is arranged in such a way that electrical production by the wind turbine is shut
down by using its own control system to prevent detrimental effects. As a result, the wind turbine does
not allow producing electrical power.
Figure 21- An example of the wind speed - wind power relation
In order to describe the mathematical relationship between the wind speed and output electrical power,
one can refer to the following power output function. In this mathematical function,
P(V ) =
where,
⎧
⎪
0
⎨P
⎪ 0
⎩
×P
0≤V <V
V ≤V <V
V ≤V <V
V ≥V
(3.2)
P (Vt): Output power depending on actual the wind speed (MW)
Pr: Rated wind power (MW)
Vt: Actual wind speed (m/s)
Vci: Cut – in wind speed (m/s)
Vr: Rated wind speed (m/s)
Besides this function, dynamic properties of wind turbine are not included because of considering
stable condition. Moreover, due to little contribution in output power, temperature as well as air
density of the weather is not inserted into the function.
~ 29 ~
Depending on the relationship between the wind speed and the wind power both graphically and
mathematically , it is now time to investigate the operating efficiencies of different types of wind
turbines by comparing the hours in which suggested wind turbines do not generate active power. Four
different wind power distribution data from single unit of specified wind turbine can be shown in
figure 22.
4000
Number of Sample
Number of Sample
3000
2000
1000
3000
2000
1000
0
0
0
0.5
1
Active Power Produced by V - 60 (MW)
0
0.5
1
1.5
2
Active Power Produced by V - 80 (MW)
3000
Number of Sample
Number of Sample
3000
2000
1000
2000
1000
0
0
1
2
3
Active Power Produced by V - 90 (MW)
0
0
1
2
3
Active Power Produced by V - 112 (MW)
Figure 22 - Active power generation by four different types of wind turbine models
In figure 22, different types of single wind turbine generates wind power with each different
distributions. Depending on the wind speed distribution measured at the height of wind turbines and
other characteristic operational features shown in table 3, one can see that V – 112 produces its
maximum power with a maximum total number of operation hours throughout all year of 2004. On the
other hand, V – 80 produces zero power with a maximum total number of operation hours in whole
year of 2004. Detailed necessary data for each different wind turbine is shown in table 3. Regarding
data, one can see that selection of wind turbine V – 112 is the most appropriate due to its better
performance compared to other types of wind turbines. To illustrate, it has the best operational
efficiency with 41.73 %, it delivers maximum power (3.075 MW) within maximum operational hours
in year 2004. Besides this, wind energy obtained from V -112 is the largest amount of energy, which is
around 4 times larger than the wind energy produced by V – 60. On the other hand, selection of V –
112 costs more than any other types of wind turbines due to its large size in terms of its nacelle, height
and rotor blades. However, in this study, these economical issues are not taken into account and only
selection is made depending on the data shown in table 3.
*
=
(
~ 30 ~
)
(
)
× 100
(3.3)
Data
V – 60
V – 80
V – 90
V - 112
The wind speed
Below Cut – in Speed
in Hours
1385
2411
1183
1183
The wind speed
Above Cut – out
Speed in Hours
26
3
3
4
Delivered
Maximum Power in
Hours
576
194
318
806
Delivered Zero
Power in Hours
1411
2414
1186
1187
Operational
Efficiency (%)
34.34
26.47
32.12
41.73
Table 3 - Detailed data about four different types of wind turbine
After obtaining enough information related to the selection of the appropriate wind turbine,
now it is time to look at the wind speed distribution at the height of hub of that selected wind
turbine. Since the hub height of the wind turbine is 119m, the wind speed distribution
according to the data taken from Stavenisse wind station in the region can be seen in figure
23. Regarding wind distribution data, highest possibility of the wind speed (10.07%) is 5.15
m/s and as the the wind speed increases higher, possibility of having the wind speed decreases
lower and lower. Probability density as well as cumulative density function of the the wind
speed is shown in figure 24.
1400
1200
Number of Sample
1000
800
600
400
200
0
0
5
10
15
20
25
Wind Speed at Hub Height of V - 112 (119 m)
Figure 23 - The wind speed at hub height of V – 112-wind turbine
~ 31 ~
30
Probability Density Function
Cumulative Density Function
0.2
0.15
0.1
0.05
0
0
5
10
15
20
25
Wind Speed at the Hub Height of V - 112 (119 m)
30
0
5
10
15
20
25
Wind Speed at the Hub Height of V - 112 (119 m)
30
1
0.5
0
Figure 24 - PDF and CDF of the wind speed at hub height of V - 112
As it can be seen in figure 23, the the wind speed distribution follows in fact the weibull distribution.
According to the weibull distribution formula which is introduced in Eqs. 2.32 in order to find the
probabilistic density function of a data sample, there are two parameters that play an important role in
the distribution. These are shape and scale parameters.
In order to find the values of scale and shape parameters of the wind speed distribution, Matlab can be
used. By using Matlab, scale and shape parameters are found as 8.03 and 1.88 respectively. After
obtaining scale and shape parameters, thanks to Matlab again, the only thing left is to model the wind
speed distribution is to use proper weibull functions in Matlab environment. In this study, 15000
samples of wind distribution data are taken into account. Real wind speed and the wind speed
distribution depending on the hub height of V – 112 modeled in Matlab environment is seen in figure
25. There is slight difference between real the wind speed distribution and one modeled in
Matlab. Therefore, studies will be based on this the wind speed distribution along the thesis.
After obtaining the wind speed distribution and the relationship between output power with
the wind speed, one can calculate the output power from one single wind turbine as shown in
figure 22. According to the output power delivered by one single wind turbine, it seems that
the power data distribution behaves in a way that there are two densest bars. While one of the
densest bars is around minimum, the other densest bar is around maximum. The reason is due
to stochastic behavior as well as linear relationship between the wind speed and active power
output of the wind turbine,
~ 32 ~
Number of Sample
1500
1000
500
0
0
5
10
15
20
Real Wind Speed (m/s)
25
30
0
5
10
15
20
Wind Speed Modeled in Matlab (m/s)
25
30
Number of Sample
1500
1000
500
0
Figure 25 - Real the wind speed and the one modeled in Matlab
which is shown in figure 21 and Eqs. 3.2. On the other hand, although it is not seen clearly, the
stochastic property of the wind speed reflects on the distribution of the output power especially in case
of obtaining the aggregation output power produced by the wind farm. Besides this, not only several
parameters influence the output power distribution, but also the correlation factor has a vital influence
on determining the shape of the distribution of the total power produced by the wind farm.
3.1.4 Correlation Factor
The concept of independence is fundamental in probability theory and has been widely used
in the stochastic modeling of power systems. It can be said that if two events such as A and B
are independent, there is no any influence of any of events on the other one in terms of
probability. Even though the correlation factor has a noticeable importance while modeling
wind farm, it is generally underestimated in several researches resulting in wrong results and
understandings. Correlation or dependence factor plays important role especially in case of
aggregation of output power from several wind turbines. The term “correlation” is generally
used for relationship of at least two units, which are operating. Here, in this report dependence
factor is considered for wind turbines in order to obtain output power from wind farm in
which several wind turbines exist. To illustrate, total power output is simply different in case
there are simply wind turbines operating independently from each other than in case wind
turbines operating dependently. In order to give an example for dependence factor, figure 26
can help to have a clear understanding. It is remarkable to realize that correlation factor (rho)
can simply change the characteristic property of the relationship between two wind speed
distributions. As far as the value of rho is increasing up to 1, the relationship between the
samples is becoming linear. On the other hand, when rho is small, there is no any linearity
between two data samples, which can also mean that they are independent from each other.
~ 33 ~
25
Correlation factor (rho) = 0.98
25
20
Wind Speed for Wind Turbine 2
Wind Speed for Wind Turbine 2
20
15
10
15
10
5
0
Correlation factor (rho) = 0.01
5
0
10
20
30
Wind Speed for Wind Turbine 1
0
0
10
20
30
Wind Speed for Wind Turbine 1
Figure 26 - The wind speed relationship between two wind turbines
The reason why dependence is so essential when considering the operation of wind farms
actually lies on the fact that since there are several wind turbines functioning in the wind
farm, the distribution of the total output power can change considerably. This can be shown as
an example by comparing output power distributions produced by wind turbines in figure 27
and 28.
~ 34 ~
1500
1000
500
1500
Number of Sample
Number of Sample
2000
1000
500
0
0
0
1
2
3
4
Total Active Power from 1 WT (MW)
1000
500
1500
Number of Sample
Number of Sample
1500
0
5
10
15
Total Active Power from 4 WTs (MW)
1000
0
0
10
20
30
Total Active Power from 10 WTs (MW)
500
0
0
20
40
60
Total Active Power from 20 WTs (MW)
Figure 27 - Active power generation by independent wind turbines
As can be seen in figure 27, in case of independent operation of wind turbines or rho is almost zero,
when the total number of wind turbines in the wind farm is increasing, the distribution of the total
output power produced by the wind farm starts to behave like much more Gaussian distribution. This
can be explained by Central Limit Theorem*.
In case of wind turbines operating dependently which also means that they are operating under more
or less same the wind speed, distribution of total output power does not look like the distribution in
figure 27 however it looks like the multiplication of total number of wind turbines with distribution of
power produced by only one single wind turbine. On the other hand, due to its stochastic property of
wind power generation, density of power distribution becomes close to zero and maximum power
output. This can be also shown in figure 28.
*
Central Limit Theorem is one of the most fundamental theorems in probability that briefly says that if
the sums of “S” mutually independent random variables increase, the distribution function of total
“S” starts to behave as normal distribution function. For more information, notes of Hans Fischer,”
The Central Limit Theorem from Laplace to Cauchy: Changes in Stochastic Objectives and in
Analytical Methods” can be referred.
~ 35 ~
2000
Number of Sample
Number of Sample
2000
1500
1000
500
1500
1000
500
0
0
0
1
2
3
4
Total Active Power from 1 WT (MW)
2000
Number of Sample
Number of Sample
2000
0
5
10
15
Total Active Power from 4 WTs (MW)
1500
1000
500
1500
1000
0
0
10
20
30
40
Total Active Power from 10 WTs (MW)
500
0
0
20
40
60
80
Total Active Power from 20 WTs (MW)
Figure 28 - Active power generation by dependent wind turbines
In order to investigate the impact of stochastic generation of the wind farm in a power network,
modeling the correlated wind turbines plays an important part. Here in this study, there are two
different cases, which result in two different conclusions. However, in many cases, in order to model a
wind farm in the power network, the latter part should be taken into consideration. It should be
reminded that in a typical wind farm consisting of several wind turbines, operating wind speed of each
wind turbine is almost same and the dependence factor is close to around one.
The reason why it is not exactly one is that since there are many wind turbines located in a wind farm,
at least one wind turbine can possibly block the wind in a way that the other wind turbine operates
with a different the wind speed. Therefore, throughout the studies mentioned in this report, correlation
factor (rho) is taken as 0.98 by considering the wind turbines are located in such a way that maximum
efficiency can be obtained. While considering how to obtain the maximum efficiency from wind farm
in terms of the production of electrical power, besides the decision of location of wind turbines, there
is also one more parameter to be taken into account. It is actually the relationship between the average
the wind speed of related region and the rated speed of the wind turbine. If the average wind speed is
close to the rated speed of the wind turbine itself, the density of data distribution will be close to
maximum power, which the wind farm can produce. Therefore, by choosing the right wind turbine
based on the previous observation, the efficiency of wind farm can be obtained as much as it can be
possible.
3.2 Load Modeling
In power system analysis, load modeling especially in case of penetration of distributed generation is
so important that modeling the load distribution with a clear understanding minimizes the problems
such as variations of load and generation at a time, errors in measurement and forecast. In the case of
~ 36 ~
studying probabilistic power flow analysis by using Monte Carlo simulation method for this thesis,
initially, wind modeling was introduced and now load modeling is introduced.
In this thesis, in order to achieve load modeling as well as possible, the total load distribution of all the
Netherlands predicted in year 2023 is taken as a base. This load distribution is shown in figure 29. The
total number of load sample is 34944 since each load data is taken in every 15 minutes resulting in 96
load data points in a day.
3500
3000
Number od Sample
2500
2000
1500
1000
500
0
1
1.2
1.4
1.6
1.8
2
2.2
Total Load in the Netherlands in 2023 (MW)
2.4
2.6
4
x 10
Figure 29 - National Load Profile for the Netherlands in 2023 in histogram
Relying upon figure 29, the power consumption in the Netherlands, is never zero and lowest
consumption rate is around 45 %. Besides this, average consumption is around 72% for overall the
Netherlands. Therefore, in order to model load distribution in the study, this data distribution is well
defined and accurate.
~ 37 ~
4
2.6
x 10
Total Active Load Consumption (MW)
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0
0.5
1
1.5
2
2.5
Load Data Points in 2023
3
3.5
4
x 10
Figure 30 - Predicted load data for the year 2023
In figure 30, the load profile over for the year of 2023 is shown. Results in figure 30 show that the
total load behavior changes as time passes meaning no certainty in actual load behavior. Moreover, the
lowest total electricity consumption is seen during summer months due to less need of heating for
residents and the highest electricity consumption is seen during winter probably because of energy
needed for heating.
After introducing the national load profile data predicted in 2023 for the Netherlands, it is time to
model appropriate load distribution in Matlab for the study. In order to achieve this, firstly the
distribution of the national load profile is scaled into unit form. Afterwards, by using the averaging
method in Matlab and considering the scaled national load profile data, a new load data distribution in
the power system network, which is introduced in figure 31, is obtained. This new load data
distribution is considered as a raw data, which will be processed later in order to obtain load data
distributions for other loads in the system. The reason why this raw data is scaled into unit is to enable
obtaining other loads in unit form so that after some process to get the load distribution for other loads,
final load data distributions can be obtained by simply multiplying scaled load data distribution with
the maximum value of the loads. This kind of consideration enables the user to obtain realistic results
after performing the probabilistic ac load flow simulations. Furthermore, the number of samples of a
single load as well as the active power samples produced by a single wind turbine modeled in Matlab
is taken as 15000. Scaled national load profile in 2023 for the Netherlands and one single load
modeled in Matlab based on the scaled national load profile is shown in figure 31.
~ 38 ~
Number of Sample
4000
3000
2000
1000
0
0.4
0.5
0.6
0.7
0.8
Load Predicted in 2023 in Unit
0.9
1
0.5
0.6
0.7
0.8
Load in Unit Modeled in Matlab
0.9
1
Number of Sample
1500
1000
500
0
0.4
Figure 31 - Load modeled in Matlab
Although the number of samples is around halved for the case of raw load data distribution modeled in
Matlab, it seems that both of load data distributions follow more or less the same pattern by averaging
technique. This helps to get proper results after probabilistic load flow analysis.
1
0.9
Load 25
0.8
0.7
0.6
0.5
0.5
0.6
0.7
Load 1
0.8
0.9
Figure 32 – Distribution of two correlated loads and their histograms
~ 39 ~
1
After introducing the national load profile and load data distribution modeled in Matlab environment,
it is time to talk about how to correlate all loads in the electrical power network. First, in order to
obtain the correlated load data distributions, the question of which method is convenient to get those
correlated data should be answered. In this study, using t – copula function used for multivariate data
distributions is chosen because of its fast convergence time and reliable data output in terms of getting
data distributions with any desired correlation factor. The detailed information of using t – copula
function is explained in the previous chapter. Throughout this study, the correlation coefficient for all
loads in the system is taken as 0.85. The reason is behind of the fact there are around 20.000 residents
living and the behavior of electricity consumption by many residents is thought to vary with little
difference and in the coming future, the penetration of electrical vehicles such as trams is considered
for that island resulting in bigger variation of total electricity consumption behavior for the whole
island.
In figure 32, the load data distributions of two loads (Load 1 and Load 25) are given. These loads are
distributed in such a way that the correlation coefficient is 0.85. In order to model these two loads
correlated to each other, t – copula function for bi - variate distribution is used.
1
0.9
Load 35
0.8
0.7
0.6
0.5
0.4
1
1
0.8
0.8
0.6
Load 15
0.6
0.4
0.4
Load 1
Figure 33 – Distribution of three correlated loads
In figure 33, the relationship of three different loads in the given network (Load 1, Load 15 and Load
35) are shown in terms of their degree of dependencies on each other in three-dimensional form
instead of the given relationship of bi-variate random variables in figure 32 (Here, these random
variables are known as loads). As stated earlier, the correlation factor for all loads in the network is
taken as 0.85. Therefore, this example shows that the data for each load in the system can be
distributed in a way that the load is correlated with every other load with a certain degree of
correlation factor by using t – copula function for multivariate random variable distributions in Matlab
~ 40 ~
environment. Since the number of the loads in the system is 35, the size of the correlation matrix is 35
× 35. As a result, a load matrix named as X can be obtained with a size of 35 × 15000 for the
simulation to be performed.
Coincidence Factor
According to US Department of Energy, the coincidence factor is defined as t he ratio of the
coincident of maximum demand or two or more loads to the sum of their non - coincident maximum
demand for a given period; the reciprocal of the diversity factor, and is always less than or equal to
one. Besides this, Erkki Lakervi & E. J. Holmes defined the coincidence factor as the ratio of the
simultaneous maximum demand of a group of load points to the sum of the maximum demand of the
individual loads. The inverse of the coincidence factor is called as diversity factor.
Kim Mason et al. (2004) defined the coincidence factor mathematically as in the following:
×
= 1−
and
=
In which;
×
(3.2.1)
(3.2.2)
×
i: individual customer
c: the number of the total customers in a group
α is a parameter that results from the regression as follows:
log(1 −
)=
×
(3.2.3)
The reason why the coincidence factor is so important especially in the design of the electrical
distribution networks lies behind the fact that the information on the individual load data points is
often limited with only annual unit consumption figures at low voltage being known. The power flow
through each section of the network is influenced by the disposition and loading of each node point,
and by the system losses. The numbers and ratings of customers’ equipment appliances and therefore
their maximum possible demands are known. However, in order to perform load flow computations
especially for MW and LV networks it is necessary to apply values of all the loads, which will result
in too high a value for the total current flows, and hence the overall voltage drop will occur, if the
loads do not peak at the same time. It is necessary to lower the rated electrical capability of each
individual load so that the summation of the individual loads equals to the simultaneous maximum
demand of the group of the loads. This is achieved by introducing a coincidence factor. For more
information about the coincidence factor, the reader can refer to the study of Bary, Constantine [36]
and Shaofeng Xie, Qunzhan Li, Liping Zhao [37].
Throughout the thesis, the coincident factor for the loads is taken as 0.80 in the given network relying
on the data provided by the company.
~ 41 ~
So far, all necessary things in order to model the loads correlated to each other with a desired
coefficient factor is explained. Moreover, these all methods can be summarized in figure 34.
START
Obtain suitable load data from a source
Scale the load data into unit form
Use averaging technique
Use t – copula method
Obtain correlated load matrix
Multiply the load matrix with
coincidence factor
END
Figure 34 – Steps of modeling load
3.3. Power System Network
In this section, general information for the network provided by Stedin B.V. is given. As it is seen in
figure 35, the network consists of one reference bus which is shown as voltage source with 1 per unit
reference voltage. Besides this, by the help of using two 52.5 kV – 13 kV, 25 MVA step – down
transformers, two common rails are obtained in the network. During normal operation, these feeders
are seperated from each other . Therefore, as shown in figure 35, there are in fact two different
network sections (A and B), which are powered by two different rails. In part A, it can be considered
in a way that there is one long string and along that string, one wind farm and one small size
generation unit located at the end of the string are expected to operate. During normal operation of the
system, although the string can be fed from the national grid, the voltages at the nodes are close to
~ 42 ~
critical limits in terms of having possibility of under voltages. The second thing to say is that in the
coming future, there will be penetration of electrical vehicles in the region so that electricity
consumption will increase. Considering these kind of problems, it can be said that using wind farm as
an electrical power source can be beneficial for part A. On the other hand, in part B, the generation
unit is so strong that there is no case that voltage is seen below minimum limit accepted as 0.95 pu for
any node in whole power system takes place. Generally, voltages at nodes in part B are around 1.00
pu. Therefore, throughout this thesis, only the wind farm operation for part A will be considered.The
data considering the total power consumption and production for parts (A and B) is shown in table 4.
B
A
Figure 35 - The given power system network
Consequently, there is no need to install wind farm for that area. However, in case of increase
of load in the future, there can be a possibility of having wind farm installation. Total active
and reactive power consumption is 21.65 MW and 8.72 MVA respectively while total active
and reactive power generation takes places as 3.2 MW and 0.963 MVA in part A. Small size
local thermal plants in the area generate specified amount of power.
Generation
Consumption
Active(MW) Reactive(MVAR) Active(MW) Reactive(MVAR)
Part A
3.2
0.963
21.65
8.72
Part B
2.4
0.7
2.5458
2.91
Total
5.6
1.6630
24.20
11.63
Table 4 - Total generation and consumption of power in the given network
~ 43 ~
For part B, load consumption compared to part A, is rather smaller and total active and reactive power
consumption by loads is 2.55 MW and 2.91 MVA while total active and reactive power production is
seen as 2.4 MW and 0.7 MVA respectively. Depending on table 4, the ratio between active power
generation and consumption is 23.14 % excluding wind power penetration. Furthermore, relying upon
what already explained in load modeling , data distribution of total amount of active and reactive
power consumption in the system, in figure 36, is obtained by multiplying maximum values of total
active and reactive load power (24.20 MW and 11.63 MVAR respectively) with the unity load
distribution shown in figure 36.
Number of Sample
1500
1000
500
0
10
15
20
Total Active Power in the System (MW)
25
Number of Sample
1500
1000
500
0
4
5
6
7
8
9
10
Total Reactive Power in the System (MVAR)
11
12
Figure 36- Total active and reactive power demand in the system
By looking at data distribution with a number of 15000 samples for total active as well as reactive
power in the system, detailed information at table 5 can be obtained.
Total Active Power
Consumption (MW)
Total Reactive Power
Consumption (MVAR)
Minimum Value
10.3492
4.9736
Maximum Value
24.2000
11.63
Mean Value
16.4133
7.8879
Standard Deviation
2.7203
1.3073
Table 5- Statistical data for total active and reactive power demand in the system
~ 44 ~
As shown in figure 36 and table 5 as well, the difference between minimum and maximum value for
total active power consumption is around 14 MW, which in facts shows that there is a large variation
in data distribution of total active power consumption. This is due to the general characteristic
behavior of loads in the system and surely enables the user to obtain much more reliable simulation
data such as voltage distribution about the operation of the system. Besides this, even though power
factor of individual load is different compared to others, overall power factor for the system is 0.90.
.
~ 45 ~
CHAPTER 4 – Case Studies
In this chapter, after giving necessary information especially about wind farm and load modeling
discussed in previous chapter, it is now time to bring all knowledge together to study further. Many
different case studies are performed in Matlab environment to investigate the effect of different levels
of wind power penetration in the power system mainly to look for voltage stability. In these
simulations, total numbers of samples for wind and load data are taken as 15000, which are enough to
have reliable information for the given network since in case studies values of the distribution data
converge to a number. After the simulations are performed in Matlab, results were also compared to
the simulation results obtained in power system software, Vision used by Stedin B.V. In all scenarios,
the difference between the results obtained in Matlab and Vision is significantly small so that
programming necessary m – files for this study gives high accuracy and trustworthy simulation
outputs. In the following flow – chart, general steps taken during the study is shown in figure 37.
START
Initial Values
MONTE CARLO SIMULATION




Wind Data
Load Data
Network Data
Generation of Inputs Randomly from
Probability Distribution Functions
Newton Rhapson AC Load Flow Method
+
Voltage Control Method
For Loop i=1:15000
Save Output Data
FINISH
Figure 37 - Monte Carlo simulation for the thesis
~ 46 ~
As shown in the flowchart diagram, there are four fundamental steps to be taken. In the first step,
necessary initial values required for the simulation is obtained. These initial values are load and wind
distribution data as well as network parameters such as impedance, admittance, maximum value of
loads and structure of the network. Afterwards, in order to study the system in a probabilistic manner,
by the help of Monte Carlo simulation, a number of samples from both wind and load distribution data
is generated in a way that statistical analysis about the results can be performed. In this study, the
number of samples is selected as 15000, which is aimed to get enough and reliable simulation outputs.
After having new wind and load distribution data, load flow analysis in the system can be performed
using Newton – Rhapson method (explained in appendix B), which is commonly used in academic
institutions as well as industry. Besides using Newton – Rhapson method, there are several different
kinds of voltage control methods regarding the wind farm operation, are also introduced to investigate
their impacts on the system operation by means of voltage stability in case of wind power injection to
the system. Each sample from load and wind distribution data is inserted until the number of iteration
reaches up to 15000 in the ‘for loop’. After each iteration in the loop, the desired output data is saved
in a way that they can be plotted and studied statistically. These output data throughout the study are
voltages of specific nodes. These nodes are selected in a way that they are considered as important in
terms of controlling the voltage stability. These nodes are in main string of part A of the system shown
in figure 35. They are separated into two different groups. In the first group, voltage distribution data
is obtained for nodes labeled as “Middelharnis, Watergatseweg, Windmolenpark and Down”. In the
second group, voltage data is obtained for the nodes named as “Down2, Down3, Down4 and Down5”.
The reason why 8 nodes are selected on the main string is the desire of monitoring the main string in
terms of voltage stability so that necessary actions can be taken in any case of undesired situation.
Therefore, main string, which the wind farm is planned to be deployed and plays a dominant role, is
somehow the heart of that given network. Besides obtaining information about voltage distributions,
the total reactive power flow in the wind farm, total current flowing from wind farm and total resistive
power losses in the system will also be studied statistically. The help of these data regarding different
variables in the power system such as voltages and currents aims that one will understand the behavior
of overall system in case of different levels of wind power penetration. In this chapter, the operation of
the given network in case of different penetration levels of wind power is investigated in three main
categories stated as follows:



Effect of using different kinds of voltage control methods in the system
Effect of change in value of reference bus voltage in the system
Effect of island mode operation in the system
In distributed generation, term of “penetration level” is used commonly, which actually enables the
one to have a clear understanding of how much percent of wind power injected into the system. In this
study, this term is defined as a function of total distributed generation production, which is the wind
power, over total active power consumption in the system: This definition can be formulated as
follows:
(%) =
∑
∑
× 100
(4.1)
Throughout the study, there are four different levels such as 9MW, 12 MW, 15 MW and 18MW of
wind power penetration are taken into account.
~ 47 ~
4.2 Effect of Using Different Kinds of Voltage Control Methods in the System
Since the voltage control issue has played a very important role in terms of voltage stability in the
system operation, this section is devoted to the investigation of different kinds of voltage control
methods in order to have an understanding of their impacts on the system. These voltage control
methods are voltage droop control, Cos (φ) voltage control with a leading power factor of 0.95 and
Cos (φ) voltage control with a lagging power factor of 0.90. The reason why those voltage control
methods are selected for the study is the fact that they are also used in Vision software program so that
one can check the validity of simulation results obtained in Matlab with the results obtained in Vision.
The method of voltage droop control is widely used in especially synchronous generator operations.
The output voltage at the terminals of the generator is controlled in a way that the reactive power
required can be determined within the limits of Q min and Qmax. If Qgenerator lies between the limits of
minimum and maximum of reactive power, then Ugenerator lies between the minimum and maximum
voltages. Furthermore, in this study as well as other case studies, depending on the general
characteristic information about the synchronous generators used in wind turbines, 10% of U/Q droop
is selected with a reference voltage (Uref) of 1.00 pu.
In load flow calculations, a synchronous generator with Cos (φ) control method is represented as a
negative load of constant power so that:
(4.2)
= −
and
or
= −
×
(
= +
×
(
) )
(
( ) )
( )
(
)
)
)
(4.3)
(4.4)
As shown in the formula for Cos (φ) voltage control method, reactive power injected or absorbed by
the generator is directly dependent on the magnitude of the active power generated by the wind
turbine. Consequently, in case of any wind power generation in the system, there will not be reactive
power generation by the wind farm as well resulting in considerable impact on the voltages in the
network. Moreover, not only the amount of reactive power is important but also direction of the
reactive power flow is also significant in terms of the voltage stability in the system. For instance, a
generator supplying reactive power to the system will cause an increase in voltage at the grid. This is
achieved by having leading power factor during the operation of the generator. In this case, generator
starts to act as a capacitor source. On the other hand, if the generator takes reactive power from the
grid, voltage at that grid tends to be lower. This is done by having lagging power factor. In that case,
the generator acts as an inductive load.
Besides the general information given about voltage controlling methods, there are also some other
important things to mention such as reference bus voltage value as well as the amount of wind power
penetration in the system. In this section, the reference bus voltage is taken as 1.05 pu.
~ 48 ~
4.2.1 Voltage Droop Control under 9MW of wind power penetration
200
0
0.96 0.97 0.98 0.99
1
Voltage in pu at Middelharnis
400
200
0
0.98
1
1.02
Voltage in pu at Windmolenpark
Number of Voltage Sample
400
Number of Voltage Sample
Number of Voltage Sample
Number of Voltage Sample
In this case, the distribution data of voltages at different nodes in the main string in part A of the given
power network has been obtained with 9MW penetration level of wind power. In order to supply this
amount of wind power, three V-112 wind turbines are used. In figure 38, these voltage distribution
data have been shown.
600
400
200
0
0.975 0.98 0.985 0.99
Voltage in pu at Watergatseweg
400
200
0
0.97
0.98
0.99
Voltage in pu at Down
1
Figure 38 – Voltage distributions for the nodes in case of 9 MW of wind power penetration
As shown in figure 38, it can be said that voltages for all nodes have distributed between 0.95 pu and
1.05 pu, which is actually the desired interval in terms of keeping voltage stability in a typical
distribution network. Furthermore, since the wind power obtained by wind farm fluctuates
considerably, the fluctuation of voltage at that wind farm named as Windmolenpark is seen as well.
While minimum voltage at that bus is 0.9656 pu, the maximum voltage that is seen as 1.0354 pu.
Moreover, mean value of that voltage distribution is 0.9986 pu and the standard deviation of total
related voltage distribution is seen as 0.0176 pu. These statistical data for the voltage distribution at
node Windmolenpark and more data regarding for other voltage distributions in the system are given
in table 6 as well. Besides this, although the reference bus voltage is selected as 1.05 pu, the minimum
and the maximum value of the voltages seen at node Middelharnis, are 0.9568 pu and 1.0039 pu
respectively. At the first glance, voltage distribution at Middelharnis is expected to be most frequent
around 1.05 pu. However, due to the direct influence of wind power generated at wind farm to voltage
distribution at Middelharnis, the minimum and maximum voltages seen at Middelharnis are rather low
compared to expected ones. For instance, in case of 9 MW of wind power penetration, since the active
power generated by the wind farm reaches at its maximum, significant amount of reactive power is
absorbed by the wind farm if either voltage droop or lagging Cos (φ) voltage control method is used.
As a result, although there is small amount of active power injected from the slack bus through step –
down transformer to the node Middelharnis for the demand of the load connected at that node
~ 49 ~
(Middelharnis), huge amount of reactive power from the slack bus starts to flow into the node
Middelharnis resulting in increase of total apparent power flowing from the slack bus. This results in
high amount of current flowing into the node Middelharnis through the step – down transformer
connected between the slack bus and Middelharnis. If the transformer is simply thought to be a
reactance (resistance and inductance), due to high amount of current, there is a voltage drop across the
reactance in a way that high amount of voltage samples below than 0.95 pu is seen.Moreover, the
relationship between voltages at node Middelharnis and Windmolenpark is seen in figure 39.
1.05
Middelharnis
Windmolenpark
1.04
1.03
Voltage in pu
1.02
1.01
1
0.99
0.98
0.97
0.96
0
10
20
30
40
50
Sample
60
70
80
90
100
Figure 39 - Voltage data measured in Middelharnis and Windmolenpark
According to figure 39, sample space with a number of 100 samples is obtained to give a clear
understanding of the relationship in terms of voltages between Middelharnis and Windmolenpark. The
indirect relationship between two voltage data is seen in a way that when the voltage at
Windmolenpark increases due to the increase for wind power, voltage measured in Middelharnis
decreases as well. This is actually due to the fact that increase of amount of the wind power leads to
less need of active power flowing the reference bus through the step down transformer (connected
between reference bus and node Middelharnis), to meet the demand of the main load (connected to
first common rail or Middelharnis). Furthermore, even though there is high amount of voltage
fluctuations for some voltage samples regarding the node Windmolenpark due to behavior of the wind
power, the voltage fluctuation seen at the node of Middelharnis is not so much because of the effect of
connection to the reference bus through the step down transformer. Since the reference bus voltage is
kept at fixed value, it tries to keep the voltage seen at Middelharnis not changing so much compared to
the higher voltage fluctuations for the node of Windmolenpark.
Relying on the voltage relationship between two nodes in figure 39, the power flow starts to appear as
bi - directional in case of wind power penetration at a certain amount. When the voltage at
Middelharnis is bigger than the voltage at wind farm, the active power starts to flow from
~ 50 ~
200
0
0.97
0.98
0.99
Voltage in pu at Down2
1
400
200
0
0.97
0.98
0.99
Voltage in pu at Down4
1
Number of Voltage Sample
400
Number of Voltage Sample
Number of Voltage Sample
Number of Voltage Sample
Middelharnis to Windmolenpark, on the other hand, when the voltage at the node in which wind farm
is connected, is bigger than the voltage at Middelharnis, direction of active power flow starts to appear
from wind farm to Middelharnis. Therefore, this can be considered as a good example of having bi –
directional power flows in active distribution networks instead of uni – directional power flow seen in
the conventional electrical power systems.
400
200
0
0.97
0.98
0.99
Voltage in pu at Down3
1
400
200
0
0.97
0.98
0.99
1
Voltage in pu at Down5
Figure 40 – Voltage distributions for the nodes
After having voltage distributions for the first group of nodes, the voltage distribution data for other
nodes such as “Down2, Down3, Down4 and Down5” are given in figure 40 as well. All voltages in all
sub – figures are distributed in a way that they are kept in the desired interval in terms of voltage
stability. Moreover, since the distance between node Down2 and node Down5 is around 3.66 km
resulting in low resistance and there are little amount of loads connected along that distance, all
voltage distributions in figure 40 are more or less similar to the voltage distribution for the node Down
given in figure 38. For this reason, in the following case studies voltage distribution data for the nodes
given in figure 40 will not be presented. Moreover, even though there is a high wind power
penetration in the system compared to the amount of loads especially starting from node Down and
node Down5, the shape of voltage distribution of those nodes are expected to be more or less similar to
the shape of voltage distribution at Windmolenpark at the first glance. However, the shape of the
voltage distributions for those nodes is similar to Gaussian distribution with a mean value around
0.9847 and the standard deviation value around 0.0064. The voltages are distributed in a smaller
interval compared to data distribution seen at Windmolenpark due to the fact that voltage distributions
at those nodes (Down, Down2, Down3, Down4 and Down5) are shaped by the load distribution data
mainly.
~ 51 ~
In table 6, the statistical analysis of the voltage distributions for the specified nodes are given in case
of 9MW of wind power penetration under voltage droop control method. None of the voltage samples
in any voltage distribution data is either bigger than 1.05 pu or lower than 0.95 pu during the operation
of the system. Therefore, the system is said to be in safe mode in terms of voltage stability
V in pu
Middelharnis Watergatseweg Windmolenpark
Down
Vmin
0.9568
0.9710
0.9656
0.9670
Vmax
1.0039
0.9932
1.0354
1.0006
Vmean
0.9849
0.9843
0.9986
0.9847
Vstd
0.0100
0.0038
0.0176
0.0063
V < 0.95 pu (%)
0
0
0
0
V > 1.05 pu (%)
0
0
0
0
Table 6 – Statistical data of voltage distributions with Voltage Droop Control
Depending on the data given in table 6, voltage samples are distributed in the largest interval for the
node Windmolenpark compared to the other nodes. The reason is that except for the wind power
generation at Windmolenpark, voltage distributions at other nodes are mainly affected by the load
distribution data since there are only loads connected to those nodes.
Cos( φ ) Control Setting – Power factor: 0.95 Lead – 1.05 PU – 9MW
V in pu
Middelharnis Watergatseweg Windmolenpark
Down
Vmin
0.9700
0.9521
0.9230
0.9330
Vmax
1.0074
1.0356
1.1115
1.0646
Vmean
0.9904
0.9926
1.0109
0.9953
Vstd
0.0065
0.0192
0.0499
0.0326
V < 0.95 pu (%)
0
0
12.39
5.35
V > 1.05 pu (%)
0
0
24.79
6.19
Table 7 – Statistical data for voltage distributions with Cos( φ )voltage control with a leading power factor of 0.95
In table 7, data for all voltage distributions at specified nodes is given under Cos (φ) voltage control
method with a leading power factor of 0.95. In this case, as mentioned earlier the wind farm starts to
~ 52 ~
behave as capacitor in a way that it pumps reactive power into the grid causing an addition of increase
in voltage at the grid side.
In comparison with the voltage droop control method, there are two nodes suffering from the voltage
stability problem significantly. For the node in which the wind farm is connected, there are voltage
samples either lower than 0.95 pu or bigger than 1.05 pu during the operation of the system. In
addition, there are 1858 voltage samples lower than 0.95 pu and there are 3718 voltage samples bigger
than 1.05 pu resulting in total number of 5577 over 15000 voltage samples with 37.18%, which are out
of the desired voltage interval (0.95 pu – 1.05 pu).
The reason why there are many voltage samples less than 0.95 pu is due to the direct relationship
between the generated active and reactive power by the wind farm. Although a leading power factor
means injection of reactive power into the system and addition of increase in voltage at the grid side,
depending on the distribution data of the wind power there are active power samples, which are zero.
As a result, there are reactive power samples, which are zero as well. Furthermore, there are many
voltage samples greater than 1.05 pu in the system and even maximum voltage sample can reach up to
1.11 pu due to the fact that increase in active wind power leads increase in the voltage at the node.
Besides this, the reactive power injection has also an addition to the voltage increase. Therefore, there
are two main factors, which enforce the increase of voltage at the grid side.
Cos( φ ) Control Method – Power factor: 0.90 - Lag – 1.05 PU-9MW
V in pu
Middelharnis
Watergatseweg Windmolenpark
Down
Vmin
0.9458
0.9519
0.9230
0.9330
Vmax
0.9950
0.9828
1.0246
0.9914
Vmean
0.9750
0.9693
0.9743
0.9649
Vstd
0.0095
0.0055
0.0217
0.0100
V < 0.95 pu (%)
0.23
0
15.66
7.67
V > 1.05 pu (%)
0
0
0
0
Table 8 - Statistical data for voltage distributions with Cos( φ )voltage control with a lagging power factor of 0.90
For the last node, named Down, there are 803 voltage samples lower than 0.95 pu and 929 voltage
samples bigger than 1.05 pu with a total number of 1732 over 15000 voltage samples and 11.54%
which are out of the desired voltage interval.
As a consequence, using the Cos (φ) voltage control method with a leading power factor of 0.95
compared to voltage droop control method is rather unreliable in terms of having many voltage
samples out of the desired voltage range in case of 9MW wind power injection into the system.
In table 8, data for all voltage distributions at specified nodes is given with Cos (φ) voltage control
method with a lagging power factor of 0.90. Unlike the previous method, here in this case the wind
farm behaves as an inductive load, which means consuming reactive power from the grid. In
comparison with the voltage droop control method, there are all nodes suffering from the voltage
stability problem significantly. However, unlike the previous Cos (φ) voltage control method, none of
the voltage samples is bigger than 1.05 pu. For the node of Middelharnis, there are only 35 over 15000
~ 53 ~
voltage samples lower than 0.95 pu with 0.23% of percentage. For the node Watergatseweg, all
voltage samples are in the desired level. Besides this, for the node in which wind farm is connected,
the situation is the most severe one compared to other nodes, the number of voltage samples lower
than 0.95 pu is 2349 with a percentage of 15.66. Finally, for the last node named Down, there are
1151 voltage samples lower than 0.95 pu with a percentage of 7.67% which are out of the desired
voltage interval.
Compared to the minimum voltage values obtained in the previous control method, there is a
significant number of voltage samples lower than 0.95 pu with lagging control method. The reason is
because of the relationship between the generated active and reactive power by the wind farm. On the
other hand, there are no any voltage samples higher than 1.05 pu even the wind farm produces its own
maximum amount of wind power. In this case, taking the reactive power from the grid with the direct
proportion of the generated active wind power but in the negative side prevents obtaining higher
amount of voltages, which are exceeding the maximum desired voltage level due to the higher amount
of wind power.
As a consequence, using Cos (φ) voltage control method with a lagging power factor of 0.90 for the
permanent synchronous generators of the operating wind turbines compared to the voltage droop
control method is rather unreliable method system depending on the results shown in table 8 in case of
9MW of wind power injection into the system. On the contrary, using Cos (φ) voltage control method
with a lagging power factor of 0.90 gives less number of total voltage samples out of the voltage
interval compared to using Cos (φ) voltage control method with a leading power factor of 0.95.
In figure 41, the number of the total undesired voltage samples including the number of voltage
samples lower than 0.95 pu and higher than 1.05 pu according to four different levels of wind power
penetration (9MW,12MW,15MW and 18MW) and three different voltage control methods regarding
for the nodes (Middelharnis, Windmolenpark, Watergatseweg and Down) are given. For the node of
Middelharnis, using voltage droop control up to 15MW of wind power penetration (including 9MW
and 12 MW) satisfies the voltage stability. However, starting from 15 MW of wind power, voltage
droop control causes a number of undesired voltage samples. For instance, the percentage of those
unwanted voltage samples reaches at around 22% in case of 18 MW of wind power injection into the
system. Besides this, using Cos (φ) voltage control method with a leading power factor of 0.95 gives
the most satisfactory results of having voltage samples regardless of any wind power penetration
levels. Moreover, Cos (φ) voltage control method with a lagging power factor of 0.90 is the worst
voltage control method considering the increasing number of unwanted voltage samples with the
increase of wind power penetration level. The number of undesired voltage samples start to appear
when 9 MW of wind power penetrates and furthermore the percentage of those voltage samples
reaches at around 26% of the overall generated voltage samples in case of 18 MW of wind power
penetration. For the node Middelharnis, using the voltage droop and (φ) voltage control method with a
lagging power factor of 0.90 is the worst voltage control method give unsatisfactory results in terms of
the higher amount of undesired voltage samples starting from 12 MW of wind power level. When the
amount of wind power increases, the percentage rate of those unwanted voltage samples increase
considerably. For instance, in case of 18 MW of wind power penetration, since the active power
generated by the wind farm reaches at its maximum, significant amount of reactive power is absorbed
by the wind farm if either voltage droop or lagging Cos (φ) voltage control method is used. As a result,
although there is small amount of active power injected from the slack bus through step – down
transformer to the node Middelharnis for the demand of the load connected at that node
(Middelharnis), huge amount of reactive power from the slack bus starts to flow into the node
Middelharnis resulting in increase of total apparent power flowing from the slack bus. This results in
high amount of current flowing into the node Middelharnis through the step – down transformer
~ 54 ~
connected between the slack bus and Middelharnis. If the transformer is simply thought to be a
reactance (resistance and inductance), due to high amount of current, there is a voltage drop across the
reactance in a way that high amount of voltage samples below than 0.95 pu is seen.
For the node Watergatseweg, using voltage droop control seems the best option since around 10% of
the total number of the voltage samples are unwanted in case of only 18 MW of wind power
penetration. Moreover, those unwanted samples are only seen in case of 18 MW of wind power
100
(%)
100
90
90
80
80
70
70
Middelharnis
60
50
40
40
30
30
20
20
10
10
0
0
9MW
100
12MW
15MW
18MW
(%)
(%)
100
Windmolenpark
80
12MW
15MW
18MW
(%)
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
9MW
12MW
15MW
Down
80
70


9MW
90
90

Watergatseweg
60
50
(%)
(%)
18MW
9MW
12MW
15MW
18MW
Voltage Droop Control Method
Cos (φ) voltage control method with a leading power factor of 0.95
Cos (φ) voltage control method with a lagging power factor of 0.90
Figure 41 – Comparison of voltage control methods
penetration. On the other hand, the situation is worse for the other voltage control methods. For
instance, while the wind power level starts to increase from 9 MW to 18 MW, the number of unwanted
~ 55 ~
voltage samples seen at Watergatseweg starts to increase dramatically if Cos (φ) voltage control
method with a leading power factor of 0.95 is used. Furthermore, using the last option of voltage
control method produces around 5% in only situation when the total amount of maximum wind power
reaches at 15 MW.
As mentioned above, there are all types of voltage samples in terms of three different voltage control
methods, are seen. The reason is the fact that since Watergatseweg is between the node Middelharnis
and Windmolenpark, the distribution of those unwanted voltage samples at Watergatseweg are
influenced by the voltage distributions of Middelharnis and Windmolenpark.
For the node Windmolenpark in which the wind farm is located, using voltage droop control again
seems the best option since there is no unwanted voltage samples regardless of what the penetration of
wind power level is. On the other hand, using the Cos (φ) voltage control method with a leading power
factor of 0.95 is the worst option among all since even if there is 9MW of wind penetration level, the
percentage of undesired voltage samples reaches more than 36% which is considerably high.
Furthermore, while the wind penetration level is increasing, there is a gradual increase of the number
of those voltage samples. For instance, when the penetration level is 18 MW, the percentage becomes
more than 60%. In comparision with the Cos (φ) voltage control method with a leading power factor
of 0.95, using Cos (φ) voltage control method with a lagging power factor of 0.90 seems more
advantageous since the percentage of those unwanted voltage samples is around 15% regardless of
different wind power penetration levels. However, still this method is not suggested to be used when it
is compared to the voltage droop control method.
For the last node which is Down, using voltage droop control gives the best results in terms of having
all voltage samples in the desired interval no matter how the wind power penetration level increases.
However, using Cos (φ) voltage control method with a leading power factor of 0.95 creates many
unwanted voltage samples with an increasing rate proportion to the increase of the maximum amount
of wind power injection into the system. For instance, when the maximum amount of wind power is
18 MW, the percentage of those unwanted samples reaches at around 42% meaning that during the
operation of the system, possibility of the undesired voltage to occur is really high. Therefore, this
method is the worst voltage control among all. Lastly, using Cos (φ) voltage control method with a
lagging power factor of 0.90 creates the most frequent unwanted voltage samples in case of 9MW of
wind power penetration and the number of those samples decrease as the wind power penetration level
increases. For instance, the percentage is around 8.00% when the maximum wind power is 9MW and
it drops to 5% when the penetration level becomes twice.
Depending on the data of total unwanted voltage samples for all specified nodes, except the node
Middelharnis, using the voltage droop control is the best option among all the control methods in
terms of keeping the voltages in a desired level as much as possible. Therefore, in latter case studies
while investigating the impact of different wind power penetration levels on the network, the voltage
droop control will be used. Moreover, the direct effect of the wind power generation at the node,
Windmolenpark is seen at the other nodes such as Watergatseweg and Down in terms of the
relationship between the undesired voltage samples and the wind power level. By using any kind of
Cos (φ) voltage control method whether the power factor is lagging or leading, the behaviour of the
number of unwanted voltage samples due to the change in the wind power penetration levels is same
however only the number of those voltage samples vary. On the other hand, that relationship is
completely different at the node, Middelharnis due to the fact that it is connected to the reference bus
through a step – down transformer.
~ 56 ~
4.3 Effect of change in the value for the reference bus voltage
In a power system, there are three types of buses to be classified. If there is a bus in which only a
generator is connected, that bus is called as generator bus, bus with only a load or several loads
connected is called a load bus and finally there is also reference bus. For the reference bus, any other
terms such as swing or slack bus can be used as well in the literature.
For a reference bus, all of the values such as voltages in the power flow simulation are computed with
respect to the reference bus. The reference bus in the system enables the user to set to any arbitrary
value for the voltage as well as phase angle. Although the voltage of the reference bus is usually at
1.00 pu, in this case study simulations will also be performed with a 1.05 pu of reference bus voltage.
Since the angle difference betweeen two sources that rules the active and reactive power flow between
them, the particular angle of the reference bus is not important. However, it sets the reference aganist
which the angles of all of the other bus voltages are measured. Therefore, the angle of the reference
bus is selected as 0o. As a consequence, according the definition of the reference bus and explanation
about it, the magnitude of the voltage as well as the phase angle of this bus is known.
In this case study, the effect of the change in the value of the reference bus voltage is investigated.
Those reference bus voltage values are taken as 1.00 pu and 1.05 pu along this case study. The
simulation outputs such as the distribution data of the current injected by the wind farm into the grid,
the distribution data of the total reactive absorbed and injected by the wind farm and the total resistive
power loss distribution data are obtained depending on the different levels of wind power penetration
(9 MW, 12 MW, 15 MW and 18 MW) in the system. Furthermore, as mentioned earlier, the
simulations are performed by using the voltage droop control method.
4.3.1 Total Current Distribution
In case of the different penetration levels of the wind power into the system, the total current injected
by the wind farm to the power network changes as well. In figure 42, the relationship between
different wind power levels and the total current produced by wind farm at specific generated power
level is given when the reference bus voltage value is taken as 1.05 pu. The first thing to notice
depending on current distribution data considering figure 42 is that the total current distribution is
shifting to the right as wind power produced by the wind farm increases which means the possibility
of having large amount of currents increase due to the high increase in wind power level. The second
thing to say is that for the current distribution data regarding all wind power penetration levels, there
are two most frequent distributions, which take place in each side (right and left) in each sub – figures.
One of the most frequent data or the data, which has the highest possibility to exist, is around 100 –
120 A in the left side of the whole distribution and valid for all current distributions. The other most
frequent data is the maximum current, which the wind farm can produce. The reason why the other
highest probability occurs at the maximum level is directly due to wind power distribution shown in
figure 22 in chapter 3. Besides this, for all current distributions, probability of having current around
120 A is higher than probability of having maximum current. The reason is again the wind power
distribution for V – 112. Therefore, it can be said that the distribution of current is more or less similar
to the wind power distribution of a single wind turbine.
~ 57 ~
Number of Sample
Number of Sample
1000
500
1000
500
0
0
100 200 300 400 500
Total Current Amplitude (A) - 12MW
Number of Sample
Number of Sample
100
200
300
400
Total Current Amplitude (A) - 9MW
1000
500
1000
0
200
400
600
Total Current Amplitude (A) - 15MW
500
0
200
400
600
800
Total Current Amplitude (A) - 18MW
Figure 42 – Distribution data of the current injected by the wind farm in case of wind power penetration levels under
1.05 pu reference bus voltage
As shown in table 9, although the total minimum current magnitude is more or less same as the
maximum amount of the wind power increases, there is a high variation of the total current
magnitudes at their maximum values.
Maximum Amount of Generated Wind Power
Current
Magnitude (A)
9 MW
12 MW
15 MW
18 MW
Cmin
63
68
70
72
Cmax
426
576
730
876
Cmean
197.0312
257.2899
318.9140
381.1292
Cstd
108.8077
156.6288
205.5938
254.4460
Table 9 – Statistical data of the current distributions for the wind farm in case of different levels of wind
power penetration under 1.05 pu reference bus voltage
Moreover, besides the increase of maximum current values, mean and standard values of current
distributions increase as well due to increase in wind power, which means that current distributions
move to the right.
Since total current magnitude is given in table 9, if one wants to look at current magnitude flowing
from a single wind turbine, that total current magnitude should be divided by the number of total wind
turbines used in the wind farm at a specified amount of the wind power. For instance, in case of 18
~ 58 ~
MW wind power generation, one single wind turbine generates 12 A with the minimum current
magnitude as well as 146 A with the maximum current amplitude. Actually, there are synchronous
generators designed in such a way that their insulation capabilities enable flow of higher currents than
146 A. However, the value of total amount of current magnitude should be taken into account
seriously since the existing grid is weak and there are many cables including the cable, which connects
the wind farm and actual grid, suffers from incapability of carrying existed active power. This
concludes in higher amount of the active power loss called as the resistive losses and it results in
decrease of overall efficiency of the network during the operation. Therefore, the current distribution
enables the one designing a power network, which can stand on higher levels of the wind power
penetration with a clear understanding.
Number of Sample
Number of Sample
On the other hand when the reference bus voltage is 1.00 pu, all results for all current data show
different distributions as shown in figure 43.
1000
500
1000
500
0
0
100 200 300 400 500
Total Current Amplitude (A) - 12MW
1000
500
Number of Sample
Number of Sample
100
200
300
400
Total Current Amplitude (A) - 9MW
1000
0
200
400
600
Total Current Amplitude (A) - 15MW
500
0
200
400
600
800
Total Current Amplitude (A) - 18MW
Figure 43 – Distribution data of the current injected by the wind farm in case of different levels of wind power
penetration under 1.00 pu reference bus voltage
In figure 43, relationship between different wind power levels and total current produced by wind
farms at that generated power level is given. The first thing to notice depending on current
distributions is that total current distribution is shifting to the right as wind power produced by the
wind farm increases which means the possibility of having large amount of currents increase due to
wind power just like the previous study when the reference voltage is 1.05 pu.
In this case, the current distributions are shifted right further than the previous case. The reason is the
fact that when the reference bus voltage is kept in 1.00 pu, the voltage in Middelharnis is kept below
than 1.00 pu initially resulting in higher amount of currents even there is less wind power injection to
the system resulting in larger voltage difference. This is better shown in table 10. The minimum values
of all current distributions are around 1.3 times than the minimum values of current distributions in the
~ 59 ~
previous study. On the other hand, when looking at the maximum current values in the distributions,
especially in case of 15 MW and 18 MW wind power penetration, the maximum current which a
single generator can supply is around 145 A just like the previous study. Consequently, the change in
value of reference bus voltage from 1.05 pu to 1.00 pu results in only higher minimum value of the
current distribution in the system.
Maximum Amount of Generated Wind Power
Current
Magnitude (A)
9 MW
12 MW
15 MW
18 MW
Cmin
81
92
95
96
Cmax
420
572
720
870
Cmean
205.1400
263.8744
324.0787
385.1792
Cstd
100.2832
146.9006
195.2421
243.9162
Table 10 - Statistical data of the current distributions for the wind farm in case of different levels of wind power
penetration under 1.00 pu reference bus voltage
As shown in table 10, although the total minimum current magnitude is more or less similar even wind
power increases, there is a high variation of total current magnitudes at their maximum values as
levels of wind power increase. Moreover, besides the increase of maximum current values, mean and
standard values of current distributions increase as well due to increase in wind power.
4.3.2 Total Reactive Power
In figure 44, the relationship between different wind power levels and total reactive power flow from
the wind farm at specific generated power level is given in case of reference voltage bus with 1.05 pu.
In this case, regarding the total reactive power distribution, the first thing to notice is that total
distribution for all distributions is shifting to the left since the amount of wind power produced by the
wind farm increases. This results in higher possibility of total reactive power flow in the negative side.
The second point to mention is that in case of 15 MW and 18 MW wind power penetration, the
amount of reactive power taken from the grid by a single wind turbine is same and around – 1.5
MVAR. In case of different penetration levels of wind power into the system, regulation of reactive
power going into wind farm as well as injected into the grid from the wind farm helps to keep the
voltage stability in the system. In these case studies, as mentioned before, in order to regulate the
amount of reactive power as well as flow of direction, voltage droop method is used. In figure 44, the
relationship between different wind power levels and amount of total reactive power at that generated
wind power level is given. In addition, the direction of reactive power flow is seen as well. Here in this
figure, positive direction means that reactive power flows into the grid resulting in capacitive behavior
whereas in case of negative direction, reactive power is consumed by the wind farm, resulting in
inductive behavior. Regarding the total reactive power distribution, the first thing to notice is that total
distribution for all cases is shifting to the left as amount of wind power produced by the wind farm
increases. This results in higher possibility of total reactive power flow in the negative side. The
reason is the fact that increase of active power production by the wind farm causes voltage increase in
at the grid. Therefore, in order to keep the voltage at a desired interval, one trick is to make reactive
power flowing negative direction, which results in helping the voltage not increasing higher and
~ 60 ~
higher. On the other hand, in case of no generation of wind power or less generation of wind power
than desired level, the direction of main power flow changes and starts to flow from node
Middelharnis to the wind farm, in this case, wind farm starts to behave as a load instead of generator,
which consumes active power. However, in order to keep the voltage at a desired level, wind farm
starts to have a capacitive behavior in a way that it pumps reactive power to the grid.
400
200
Number of Sample
Number of Sample
The second point to mention is that in case of 12 MW, 15 MW and 18 MW wind power penetration,
the amount of reactive taken from the grid by a single wind turbine is same and around – 1.5 MVAR.
The reason why the wind farm consumes same amount of maximum reactive power in the negative
side is that each synchronous generator reaches at its own maximum capability of consuming reactive
power regardless of increase in wind power generation in the system.
400
200
0
0
-2
0
2
Total Reactive Power (MVAR) - 9MW
-4
-2
0
2
Total Reactive Power (MVAR) - 12MW
400
200
Number of Sample
Number of Sample
600
400
200
0
-6
-4
-2
0
2
4
Total Reactive Power (MVAR) - 15MW
0
-8 -6 -4 -2
0
2
4
Total Reactive Power (MVAR) - 18MW
Figure 44 – Total reactive power distributions for the wind farm in case of different levels of wind power penetration
under 1.05 pu reference bus voltage
As the total reactive power is given in table 11, if one wants to look at reactive power which is
injected into the grid or consumed by single wind turbine, that total reactive power should be divided
by the number of total wind turbines used in the wind farm at a specified wind power level. As the
possible maximum amount of wind power increases, the maximum of reactive power injected to the
grid increases as well, however the rate of increase in maximum reactive power is lower than the rate
of increase in active power produced by the wind farm. Another point is about the fact that increase of
wind power penetration leads total reactive power distribution to the left. This is seen by looking at the
change in mean values of distributions in the positive side. On the other hand, standard values of the
distributions increase as the amount of wind power increases as well.
~ 61 ~
Maximum Amount of Generated Wind Power
Reactive Power
(MVAR)
9 MW
12 MW
15 MW
18 MW
Qmin
-3.6330
-5.6520
-7.5000
-9.0000
Qmax
3.5310
3.9960
4.3450
4.6140
Qmean
0.1436
-0.5871
-1.3825
-2.1877
Qstd
1.8033
2.5617
3.2813
3.9432
Table 11 – Statistical data of the reactive power distribution for the wind farm in case of different levels of
wind power penetration levels under 1.05 pu reference bus voltage
When the reference bus voltage changes into 1.00 pu, the results for the all total reactive power data
show different distributions as shown in figure 45.
400
200
Number of Sample
Number of Sample
600
400
200
0
0
-2
0
2
4
Total Reactive Power (MVAR) - 9MW
400
200
600
Number of Sample
Number of Sample
600
-4
-2
0
2
4
Total Reactive Power (MVAR) - 12MW
400
200
0
-5
0
5
Total Reactive Power (MVAR) - 15MW
0
-5
0
5
Total Reactive Power (MVAR) - 18MW
Figure 45 – Total reactive distributions for the wind farm in case of different levels of wind power penetration under
1.00 pu reference bus voltage
In figure 45, the relationship between different wind power levels and the total reactive power flow
from the wind farm at specific wind power level is given in detail. The first thing to notice depending
on reactive power distributions is that total reactive distribution is shifting to the right as wind power
produced by the wind farm increases. This means the possibility of having large amount of reactive
power increase in each side (negative and positive) when the reference voltage is 1.00 pu. The reason
is behind the fact that the increase of the active power production by the wind farm causes voltage
increase in at the grid and when the reference bus voltage is 1.00 pu, this leads to have larger voltage
~ 62 ~
difference between Middelharnis and Windmolenpark. Therefore, in order to keep the voltage at a
desired interval, one way is used to make reactive power flowing negative direction, which results in
helping the voltage not increasing higher and higher. On the other hand, in case of no generation of the
wind power or less generation of wind power than desired level, the direction of main power flow
changes and starts to flow from node Middelharnis to the wind farm, in this case, wind farm starts to
behave as a load instead of generator, which consumes active power. However, in order to keep the
voltage at a desired level, the wind farm starts to have a capacitive behavior in a way that it pumps
reactive power to the grid.
4.3.3 Total Resistive Loss
1000
500
0
Number of Sample
Number of Sample
1500
2000
1500
1000
500
0
2000
1500
1000
500
0
0.2 0.4
0.6 0.8
1
Resistive Loss (MW) - 9 MW
Number of Sample
Number of Sample
In case of different penetration levels of wind power into the system, total resistive loss in the power
network changes as well. In figure 46, the relationship between different wind power levels and total
resistive loss at that generated wind power level is given. The first thing to notice depending on
resistive loss distributions is that total resistive loss distribution is shifting to the right as wind power
produced by the wind farm increases which means that possibility of having large amount of resistive
loss due to wind power injection. Second thing to say is that for all resistive loss distributions, the
distribution is seen as the most frequent around 0.2 MW of power loss. This is also valid for all other
resistive loss distributions.
0.5
1
1.5
2
Resistive Loss (MW) - 12 MW
2000
1000
0
1
2
3
Resistive Loss (MW) - 15 MW
1
2
3
4
Resistive Loss (MW) - 18 MW
Figure 46 – Total resistive loss distributions in case of different levels of wind power penetration under 1.05 pu
reference bus voltage
The reason why the highest probability occurs at the minimum level regardless of increase in wind
power is directly due to wind power distribution since value of highest density distribution of wind
power samples exist when there is no wind power generation in the system. On the other hand,
increase of wind power level leads to increase of maximum value of resistive loss. Starting from 9MW
~ 63 ~
wind power penetration, as the wind power increases, it starts to have two highest resistive loss
densities in the distributions. This is clearly seen in the case of 18 MW wind power penetration.
Third point to mention is that although there is a number of wind power samples equal to zero, there is
still around 0.20 MW resistive loss in the power system. However, in case of 18 MW wind power
penetration resistive losses can reach up to even 4.90 MW resulting in lowest efficiency for the system
operation. Increasing wind power means the increase of the current pumped into the grid because of
wind farm operation. However, excessive wind power is lost as resistive loss through distribution
cables resulting in overloading problems in the system. This overloading problem is so serious that it
can damage all entire system during the operation if the grid structure is weak.
According to simulation outputs in terms of overloading problem, starting from 9 MW of wind power
penetration, all power cables between node Middelharnis and Windmolenpark face this kind of
problem. This result is actually so severe that the one who is responsible for the planning and design
of the network should consider these results in order to prevent failure of the system and can have an
idea of how much the wind power should be injected to the system without any modifications for the
existed grid structure.
Detailed information regarding resistive power loss distributions in case of different levels of wind
power production in the system are given in table 12.
Maximum Amount of Generated Wind Power
Resistive Power
Loss (MW)
9 MW
12 MW
15 MW
18 MW
Lossmin
0.1648
0.1655
0.1673
0.1678
Lossmax
1.1237
2.1041
3.3877
4.8994
Lossmean
0.3920
0.6116
0.9275
1.3351
Lossstd
0.2437
0.5379
0.9429
1.4482
Table 12 – Statistical data of the total resistive loss distributions in case of different levels of
wind power penetration under 1.05 pu reference bus voltage
According to data shown in table 12, firstly to say is that minimum resistive loss is more or less
similar for all distributions regardless of increase in wind power penetration in the system. On the
other hand, as expected while amount of generated wind power increases, maximum value of resistive
loss distribution increases as well as mean and standard deviation values of resistive loss distributions.
This shows that due to increase in wind power injection, distribution of resistive loss moves to the
right.
When the reference bus voltage is taken as 1.00 pu in the system, all distributions of the total resistive
loss in the system changes differently compared to the previous case. These distributions are shown in
figure 47.
~ 64 ~
Number of Sample
Number of Sample
1500
1000
500
2000
1500
1000
500
2000
1500
1000
500
0
0
Number of Sample
Number of Sample
0
0.2
0.4
0.6
0.8
1
Resistive Loss (MW) - 9MW
0.5
1
1.5
2
Resistive Loss (MW) - 12MW
2000
1000
0
1
2
3
Resistive Loss (MW) - 15MW
1
2
3
4
Resistive Loss (MW) - 18MW
Figure 47 – Total resistive loss distributions in case of different levels of wind power penetration under 1.00 pu
reference bus voltage
Looking at the total resistive distributions shown in figure 47 and statistical data given in table 13,
even though the reference voltage is reduced to 1.00 pu, there is a slightly difference of the minimum
as well as maximum values of the resistive power distribution for each of the wind power penetration
level. Actually, this may be explained best by comparing the current distributions shown in figure 42
and figure 43 and their statistical data given in table 9 and table 10. Although the reference bus voltage
changes from 1.05 pu to 1.00 pu, there is a negligible difference in minimum and maximum values of
the current distributions. This reflects in the total resistive loss distribution as well.
Maximum Amount of Generated Wind Power
Resistive Power
Loss (MW)
9 MW
12 MW
15 MW
18 MW
Lossmin
0.1707
0.1730
0.1757
0.1824
Lossmax
1.0819
2.0362
3.3164
4.8980
Lossmean
0.3945
0.6049
0.9098
1.3063
Lossstd
0.2274
0.5097
0.9044
1.4019
Table 13– Statistical data of the total resistive loss distributions in case of different levels of
wind power penetration under 1.00 pu reference bus voltage
~ 65 ~
4.4 Effect of Island Mode Operation in the System
In this case study, the impact of the possible island mode operation is investigated in case of different
amounts of wind power penetration in the given electrical network. In this case study, the island mode
operation refers to the situation in which the main cable (labeled in green) between the node
Middelharnis or common feeder (labeled as RAIL 1) and the node Watergatseweg is disconnected due
to an assumption of a possible fault seen at that cable. Therefore, there are three sources of active
power generation left along the main string labeled in red. These three power sources are the wind
farm and two synchronous generators (labeled as SG1 and SG2) connected at the last part of the string
as shown in figure 48. Besides this, there is no electrical connection between the node in which the
synchronous generators are connected and another common feeder labeled as RAIL2. Therefore,
RAIL2 meets the power demand of the loads connected along the string labeled in blue.
Figure 48 – Schematic diagram of the given network
The maximum total active power consumption including coincidence factors for all the loads
connected along the string (colored in red) equals to 6.03 MW while the maximum total reactive
consumption is 2.62 MVAR. Therefore, in order to satisfy load demands for that string, the total
power production by the wind farm and two synchronous generators in the string must reach at the
maximum load consumption level. However, the capability of delivering active power for a single
synchronous generator is 0.4 MW so that there is 0.8 MW of active power generation due to the total
synchronous generators in the red string according to the data provided by the company. Hence, the
power produced by the generators is so small compared to the total load demand for red string.
Moreover, the stochastic behavior of the wind farm in terms of its active power distribution over time
~ 66 ~
gives serious problems. Maybe the most significant problem to tackle related to the wind power is the
percentage of zero active power samples, which reaches up to 15%. As a result, regardless of what the
maximum amount of active power that the wind farm achieves, the system suffers significant portion
of severe under - voltage samples considering only the operation of the wind farm as well as two
synchronous generators. By taking these problems explained above into account, two options can be
used to satisfy all load demands along the string colored in red. One of the options that can be
considered is to connect the cable between the node in which two synchronous generators are
connected and common feeder labeled as RAIL2 via the string colored in blue as shown in figure 48.
Consequently, even though there is no wind power generation in the system, all loads along the red
string can be satisfied with the power coming from the common feeder labeled as RAIL2 so that the
system becomes more reliable. Another option which is used how to satisfy all load demands along the
red string, may be the increase in power delivering capabilities of two synchronous generators up to
6.5 MW which is the summation of their total power. Using this second option enables no need of
connecting the cable between the node in which the generators are connected and RAIL2.
Furthermore, the reason why the total power of the synchronous generators is taken as maximum
amount of 6.5 MW, is because of the total amount of the load demand along the red string in the
electrical network. Therefore, in case of no wind power generation in the system, all load demand can
be satisfied by those generators. Moreover, there is one thing to mention that in order to use this option
as an alternative approach, total power generation including the wind power and the power produced
by the generators should equal to 6.5 MW of maximum level. Therefore, this option is assumed in a
way that the amount of power generation by the generators equals to subtraction of the amount of the
wind power from 6.5 MW. For instance, if the amount of power generated by the wind farm is X, then
the amount of the power that the generators should produce, equals to 6.5 – X. Otherwise, if the
amount of power generated by the generators (6.5 MW) is kept constant along the operation, the
addition of the wind power into the system will increase the possibility of having over – voltage
samples which is truly undesired situation.
400
200
0
N u m b e r o f S a m p le
N u m b e r o f S a m p le
600
1500
1000
600
400
200
500
0
0.98
0.99
1
V o lt a g e in p u a t W a t e rg a t s e w e g
N u m b e r o f S a m p le
N u m b e r o f S a m p le
4.4.1) Connection of RAIL2
0
0.975 0.98 0.985 0.99 0.995
V o lt a g e in p u a t D o w n
0.995
1
1.005
V o lt a g e in p u a t W in d m o le n p a rk
600
400
200
0
0.99
1
1.01
V o lt a g e in p u a t D o w n 5
Figure 49 – Voltage distributions at the nodes in case of 6MW wind power penetration in island mode operation for
the first scenario
~ 67 ~
In figure 49, the voltage distributions for the nodes such as Watergatseweg, Windmolenpark, Down
and Down5 are shown in case of wind power penetration with a maximum level of 6MW in the
network. In this scenario, as explained previously only the cable between the node in which the
generators are connected and the blue string is connected so that the power can flow from the common
feeder labeled as RAIL2 into the loads connected along the red string.
Furthermore, there is no any voltage sample which is either lower than 0.95 pu or higher than 1.05 pu
for all voltage distributions at the nodes. Therefore, the connection of the cable makes the system
operating in a reliable way in terms of having no any undesired voltage samples.
Regarding for the shape of the voltage distributions seen at the nodes, except the node
Windmolenpark, all voltage distributions follow more or less the same pattern and the pattern for those
distributions can be considered as Gaussian distribution. This reason may be due to the fact that there
is only load connected to those nodes and the shape of the load distribution has a direct influence on
the voltage distributions for those nodes. On the other hand, for the node in which the wind farm is
connected, the voltage distribution in fact includes two different Gaussian distributions with their
different mean and standard values because of the direct influence of wind power distribution on the
voltage distribution.
4.4.2) Increase In Power Capability of the Synchronous Generators
Number of Sample
As already explained previously, in this situation there are only wind farm and synchronous generators
used in order to provide sufficient power for the loads connected along the red string. Besides this, the
number of wind turbines used in this situation is only two resulting in maximum 6MW of possible
wind power. The distribution of the power produced by the wind farm as well as the sychronous
generators during the operation of the system is shown in figure 50.
2000
1500
1000
500
Number of Sample
0
0
1
2
3
4
5
Total power production by the wind farm (MW)
6
2000
1500
1000
500
0
1
2
3
4
5
6
Total power production by the synchronous generators (MW)
Figure 50 – Generated power distributions of the wind farm (a) and the synchronous generators (b)
~ 68 ~
In comparison with the power distribution of the wind farm, the power distribution of the generators
follows the same pattern whereas the location of the most frequent data in each sub – figures are
opposite due to the assumption of the behavior of the power delivery by the generators which is
explained previously.
In figure 51, the voltage distributions for the nodes such as Watergatseweg, Windmolenpark, Down
and Down5 are shown in case of wind power penetration with a maximum level of 6MW in the
network. Furthermore, all voltage samples in each voltage distributions regarding for the nodes are in
desired interval so that using this option enables the system operating reliably in terms of having no
unwanted voltage samples for all nodes in the red string.
1500
Number of sample
Number of sample
600
400
200
1000
800
600
400
200
0
0
1.01
1.015
1.02
Voltage in pu at Windmolenpark
Number of sample
Number of sample
0
0.975 0.98 0.985 0.99 0.995
Voltage in pu at Watergatseweg
500
0.98 0.99
1
1.01 1.02
Voltage in pu at Down
600
400
200
0
0.98
1
1.02
Voltage in pu at Down5
Figure 51 – Voltage distributions at the nodes in case of 6MW of wind power penetration in island mode operation for
the second scenario
Starting from the node named as Down5, the shape of the voltage distributions appears to be
differently. For instance, when it comes to the distribution of voltage at the node Windmolenpark,
there are two different Gaussian distributions due to the reflection of the wind power distribution on
the voltage distribution in case of 6MW of wind power injection into the given network.
~ 69 ~
5.1 Conclusion and recommendations for future research
In this thesis, the probabilistic power flow analysis for the given network, which is located in Goeree
Overflakkee Island, Zeeland in the South of the Netherlands was studied extensively. Firstly,
providing the mathematical background of the thesis is provided for the reader so that, the reader can
have a clear understanding of what is going on in the later chapters of the thesis. Furthermore, a short
introduction to Monte Carlo simulation analysis, which is the one of the fundamental tools throughout
the thesis, is given. Besides the given mathematical background for the thesis and a description of
Monte Carlo simulation, the crucial steps for modeling the loads and the wind farm, which are the core
elements for the network, are explained in detail. While modeling the wind farm, the importance of
selection of the most efficient wind turbine on the market depending on its power output over the year
related to the data of the wind speed distribution for the region in which the wind farm is expected to
be located, is mentioned. Afterwards, the term of correlation factor is introduced in order to emphasize
the relationship between the wind power output of the dependent wind turbines with a certain defined
correlation coefficient and independent wind turbines in a wind farm during the operation of the
electrical network. After combining all the steps related to the design of wind farm, the simulation
results of the output wind power in certain amount of penetration levels were obtained in Matlab
environment. Another section in that chapter was devoted to modeling of the load. Considering the
load modeling, there are many important things to think about such as evaluation of the load data
profile from a reliable source, the methods about how to correlate all loads to each other with any
desired correlation coefficient in the given network and the importance of coincidence factor
especially for the distribution networks. Just like the wind power output, simulation results for the
correlated loads with a certain correlation factor are obtained again in Matlab. Moreover, the general
information of the given network structure and the total consumption as well as total generation
excluding the wind farm operation, which is operated by Stedin B.V is also provided. In the fourth
chapter of the thesis, three main case studies are performed in order to analyze mainly the effect of
different wind power penetration levels in the system with varying network parameters such as the
reference bus voltage values. These case studies are briefly introduced as the following:



Investigation of the impact of using different kinds of voltage control methods in the system in
case of different wind power penetration levels.
Investigation of the effect of different reference bus voltage values in the system in case of
different wind power penetration levels.
Investigation of the effect of island mode operation in the system
Before going through the explanations of what have been done in the case studies in chapter four,
Monte Carlo analysis for the latter case studies is clarified systematically instead of the general
introduction made in the second chapter of the thesis.
In the first case study, three different types of voltage control methods for the operation of wind farm,
which are completely integrated into Vision software package, are introduced and the selection of the
voltage droop control is made based on having the largest number of desired voltage samples
considering the different amount of wind power injections into the network. The voltage distributions
are obtained at the specific nodes that are strategically important by means of monitoring the effect of
the wind power during the operation of the wind farm. Besides obtaining the voltage distribution data
for different nodes, statistical analysis of those were also performed in order to understand the
distributions more clearly.
~ 70 ~
The results regarding for the first case study reveal that selection of voltage control method for the
wind farm is so significant in a way that the number of undesired voltage samples can vary for
different nodes in case of different levels of wind power penetration in the system.
In the second case study, mainly the effect of change in the value of the reference bus voltage is
introduced in terms of investigating the behavior of the total amount of the current flowing from the
wind farm, total reactive power absorbed and injected by the wind farm and lastly the total resistive
loss in the overall system. These all are performed under four different levels of the wind power
penetration and different values of the reference bus voltages. Besides obtaining their data
distributions in Matlab, the statistical analysis of the distributions were also performed.
For the distribution data of the current injected by the wind farm into the grid, when the possible
amount of wind power in the system increases, the possibility of obtaining higher currents increase as
well. This situation is same for both of two different voltages of the reference bus in the system.
Moreover, when the voltage of the reference bus voltage is increased up to 1.05 pu from 1.00 pu, the
current injected by the wind farm in a specified level of wind power, slightly decreases. This decrease
is due to only 5% increase of reference bus voltage. This small increase in the reference bus voltage
reduces the difference between the available maximum voltage of the node in which the wind farm is
connected at a certain level and the node of Middelharnis resulting in lower amount of currents
generated by the wind farm. The reason why the current distribution data is important relies on the fact
that the behavior of the current distribution data can guide the network designer in terms of selection
of the suitable power cables, protection elements as well as possible of overloading effects due to the
current injection by the wind farm.
When the amount of wind power increases, the possibility of the reactive power absorbed by the wind
farm increases as well in order to keep the voltage in a desired level as much as possible while using
the method of the voltage droop control. Besides this, it can also be claimed that the behavior of the
reactive power distribution is directly affected by the selection of the voltage control methods. Since
each voltage control method produces different reactive power distribution and reactive power is
significantly important on the control of the voltage, the designer of the network has to consider it very
seriously.
The behavior of the distribution data for the total resistive loss distribution data is explained in a way
that as the wind power level increases, the possibility of obtaining higher resistive power losses in the
system increase significantly resulting in decrease of the efficiency of the total system considerably.
The resistive loss distribution data can give a clear notion for the network designer in a way that how
the network can handle with different levels of wind power penetration considering the overloading
effect for the power cables as well as transformers, can be understood in terms of analyzing the total
power loss in the system. In this thesis, considering total resistive losses in the system as well as
number of undesired voltage samples for all nodes, it can be suggested that allowing 9 MW and 12
MW of wind power injection into the system is a good option for the proper operation of the network
in case of wind power penetration.
In the third and last case study of the thesis, the island mode operation is considered since the higher
possibility of its existence in distribution networks when the wind power penetrates as well. Therefore,
after some assumptions made in the given network, such as the connection as well as the
disconnection of the specific cables and increase in the capability of the delivering power of the
synchronous generators, simulation results were obtained and studied extensively in case of maximum
6MW wind power penetration . According to the results obtained in this case study, wind operation
with island mode holds some requirements in the system in a way that during island mode operation if
generators cannot supply active power to the system, the system fails even there is wind farm
~ 71 ~
operating. Therefore, during island mode operation, using only wind farm is not the option, which also
means that there should be always support from the national grid or generators along the wind farm
operation in the system.
Throughout the thesis, maybe the most important goal here is to give a feeling to the reader that the
probabilistic approach provides more reliable and explicit results about the network operation in terms
of behavior of its own variables such as voltages, currents, power flows etc. in case of wind power
penetration as a DG source. Therefore, the probabilistic approach takes an advantage on deterministic
approach of solving the variables depending on the rigid inputs in the electrical network especially in
case of DG penetration in the system.
As a recommendation, this study can be extended in order to investigate not only the effect of the wind
power but also other variable types of DG sources such as mainly the solar power. Moreover, the
effect of using different types of DG sources together in the system can be analyzed in detail. This has
significance since the structure of the electrical networks will be shaped due to the high penetration
level of the DG source in recent future. Therefore, this study and other studies are aimed to have a
contribution in designing and planning such electrical networks by providing reliable and clear results
as much as possible.
~ 72 ~
REFERENCES
1. Leweson, Janos Hethey & Sofie. Probabilistic Analysis of Reactive Power Control Strategies for
Wind Farms. July 2008.
2. Agency, International Energy, Key World Energy Statistics, pg: 6, 2009.
3. Agency, International Energy, Key World Energy Statistics, pg: 24, 2009.
4. Global Warming: Energy, Environmental Pollution, and the Impact of Power Electronics . Bose, B.
1, Univ. of Tennessee, Knoxville, TN, USA : Industrial Electronics Magazine, IEEE , 25 March 2010,
Vol. 4.
5. Professor Michael Wesley, Director of the Griffith Asia Institute at Griffith University. Power
plays: Energy and Australia's security . s.l. : Australian Strategic Policy Institute (ASPI), Thursday, 11
October 2007.
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7. Sustainable energy development: performance and prospects. Jefferson, Michael. Bedfordshire
MK44 1BB, UK : Science Direct, 3 November 2005.
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International Market . s.l. : Geothermal Energy Association, May 2010 .
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12. http://running_on_alcohol.tripod.com/id30.html. [Online]
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Environmental Protection Agency.
14. Papaefthymiou, Georgios. Integration of Stochastic Generation in Power Systems. Delft, The
Netherlands : Delft University of Technology, 13 june 2006.
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Efficiencies. March 2011.
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Energy.
19. International, the Global Wind energy council & Greenpeace. Global Wind Energy Outlook .
2010.
20. Modeling the stochastic dependencies in a probabilistic load flow including wind generation.
Rodrigues, P.R., et al. Valencia : IEEE, 28-30 Sept. 2009.
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21. The existed problems and possible solutions of micro-grid based on distributed generation . Wu,
C.X. Wen, F.S. Lou, Y.L. Autom. Sch. of Hangzhou, DIANZI Univ., Hangzhou : IEEE, 6-9 April
2008 .
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Que. : IEEE, 16 October 2006 .
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25. Bibliography on composite system reliability. Schilling, M.T. Billington, R. Leite da Silva, A.M.
El-Kady, M.A. 3, Ontario Hydro, Toronto, Ont. : IEEE, Aug 1989 , Vol. 4.
26. Probabilistic load flow. Borkowska, Allen & B. s.l. : IEEE Trans. Power App. Syst.1974, Vol.
PAS93.
27. Probabilistic ac load flow. Alshakarchi, R. N. Allan and M. R. G. s.l. : Proc. Inst. Elect. Eng.,
Vol. 123.
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McGraw-Hill, 4th Edition.
29. Fonctions de répartition à n dimensions et leurs marges. Sklar, A. Paris : Inst. Statist. Univ. Paris,
1959.
30. An introduction to copulas. Nelsen, R. B. New York : Springer, 1999.
31. http://www.mathworks.com/help/toolbox/stats/brn2ivz-93.html. [Online] MATLAB.
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Institute of the Netherlands (KNMI).
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[Online]
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Xie, Qunzhan Li, Liping Zhao. s.l. : IEEE, 20-25 June 2004 , Vol. 2.
~ 74 ~
APPENDIX A
Detailed data for V 112 – 3.0 MW Wind Turbine
Operational data
Rated power: 3,075 kW
Cut-in the wind speed: 3 m/s
Rated the wind speed: 12 m/s
Cut-out the wind speed: 25 m/s
Re cut-in the wind speed:23 m/s
Wind class: IEC IIA and IEC IIIA
Operating temperature range: standard range -20°C to 40°C, low temperature option -30°C to
40°C
Rotor
Rotor diameter: 112 m
Swept area: 9,852 m2
Nominal revolutions: 12.8 rpm
Operational interval: 6.2 to 17.7 rpm
Air brake: full blade feathering with 3 pitch cylinders
Electrical
Frequency: 50 Hz/60 Hz
Converter type: full scale converter
Generator type: permanent magnet generator
Gearbox
Type: 4-stage planetary/helical
Power regulation
Pitch regulated with variable speed
~ 75 ~
APPENDIX B
Newton – Raphson Power – Flow Solution Method
In order to use the Newton – Raphson method for the solution of the steady state power - flow
equations for the given network, firstly bus voltages and line admittances in the polar form must be
expressed for a × size of the matrix as the following:
The voltage at a typical bus i
= | |∠
The typical line admittance
=
is
∠
= | |(cos
=
cos
(B.1)
+ sin )
+ sin
=
(B.2)
+
where,
Gij: Conductance between the line i and line j
Bij: Susceptance between the line i and line j
The net current flowing into the network at bus i in terms of the elements Y in of Ybus is given as:
=
+
+ …
(B.3)
=
If Pi and Qi is considered as the net active and reactive power entering the bus i, then the complex
conjugate of the power at that bus i is defined as follows:
−
=
∗
(B.4)
Equations (B.1) and (B.2) are inserted to obtain as follows:
−
=
|
|∠(
+
−
)
The above equation can be separated into active and reactive parts as follows:
~ 76 ~
(B.5)
=
|
=−
| cos(
|
+
| sin(
−
+
(B.6)
)
−
(B.7)
)
When n equals to i in the equations (B.6) and (B.7) and the related terms are separated from the
summations, the following equations are obtained:
=| |
+
|
| cos(
+
−
)
(B.8)
and
= −| |
−
|
| sin(
+
−
)
(B.9)
By writing the power mismatches for the typical load bus i,
=| |
+
|
| cos(
+
−
)
(B.10)
Mismatch equations for active and reactive power for a typical load bus i can be written as follows:
and
∆
=
,
−
For real power Pi,
∆
=
,
−
∆
+
∆
For reactive power Qi
=
+
∆
| |
+ ⋯+
(B.11)
,
∆| | + ⋯ +
~ 77 ~
(B.12)
,
∆
|
+
|
∆|
| |
|
∆| |
(B.13)
∆
=
∆
+
∆
+
| |
+ ⋯+
∆
∆| | + ⋯ +
|
|
+
∆|
| |
|
∆| |
Each non - slack bus of the system includes two equations like those for ∆
mismatch equations into vector – matrix form results in:
⎡
⎢⎛
⋮
⎢⎜
⎢
⎢⎝
⎢
⎢
⎢⎛ ⋮
⎢⎜
⎢
⎣⎝
Sjj
⋯
⋯
⋯
⋯
| |
| |
⎛
⋮
⎜
| |
| |
⎝
⎞
⋮
⎟
⎠
| |
| |
⎞ ⎛
⋮
⋮
⎟ ⎜
| |
| |
⎠ ⎝
⋯
|
|
⋯
|
|
⋯
⋯
|
|
|
|
⋮
⋮
| ⎤ ∆
⎞⎥
⎡
⎟⎥ ⎢ ⋮
⎥ ∆
|⎠⎥ ⎢ ∆|
⎢
⎥ |
⎢
| ⎥⎢ ⋮
⎞⎥ ∆|
⎟⎥ ⎢ |
⎣
⎥
|⎠⎦
|
|
|
|
jj
Jacobian Matrix
(B.15)
and ∆ , bringing all the
⎤
∆
⎥ ⎡ ⋮
|⎥ ⎢∆
⎥ = ⎢∆
|
⎥ ⎢
⋮
⎥ ⎢
|
∆
⎥ ⎣
|⎦
⎤
⎥
⎥
⎥
⎥
⎦
jj
(B.16)
Corrections Mismatches
For the iterations used in the Newton – Raphson load flow method,
(
)
and
(
)
=
( )
=
+∆
( )
( )
+∆
( )
(B.17)
=
( )
(1 +
| sin(
+
−
∆
( )
( )
)
(B.18)
For the core elements of the Jacobian matrix,
For J11,
= −|
~ 78 ~
)
(B.19)
=
|
| sin(
= −|
| cos(
+
−
)
(B.20)
For J21,
=
|
+
| cos(
−
(B.21)
)
+
−
)
(B.22)
+
−
)
(B.23)
For J12,
| |
| |=|
| |
For J22,
| |
| cos(
| |=
| | = −|
| |
| sin(
| |=
(B.24)
+| |
−| |
~ 79 ~
+
−
)
(B.25)
(B.26)
APPENDIX C
Correlation Matrix for the Loads in the System
In this section, the correlation matrix used in order to model correlated loads in the thesis starting from
Load 1 to Load 10 is given as the following:
1
0.84
0.8394
0.8416
0.8414
0.8408
0.8387
0.837
0.8367
0.8378
0.84
1
0.8382
0.8427
0.8414
0.8416
0.8443
0.843
0.8387
0.8416
0.8394
0.8382
1
0.8396
0.8388
0.8387
0.8411
0.8393
0.8369
0.8348
0.8416
0.8427
0.8396
1
0.8402
0.8428
0.8403
0.8426
0.8439
0.8434
0.8414
0.8414
0.8388
0.8402
1
0.8423
0.8411
0.8381
0.8439
0.8399
0.8408
0.8416
0.8387
0.8428
0.8423
1
0.8387
0.8368
0.8451
0.841
0.8387
0.8443
0.8411
0.8403
0.8411
0.8387
1
0.8434
0.8418
0.8409
0.837
0.843
0.8393
0.8426
0.8381
0.8368
0.8434
1
0.8402
0.8402
0.8367
0.8387
0.8369
0.8439
0.8439
0.8451
0.8418
0.8402
1
0.8413
0.8378
0.8416
0.8348
0.8434
0.8399
0.841
0.8409
0.8402
0.8413
1
*Due to any request, the total matrix can be provided.
~ 80 ~
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