MATHEMATICAL MODEL OF WIND TURBINE IN GRID-OFF SYSTEM
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This paper presents a mathematical model of a wind turbine operating in a grid-off system, detailing the conversion of wind kinetic energy to mechanical energy and subsequently to electrical energy through a permanent magnet synchronous generator. The study includes equations governing the mechanical power, generator torque, and electrical output, emphasizing the significance of wind speed and other parameters in the model. The research supports project efforts aimed at improving the application of renewable energy technologies.
![Milan Jus / University Review, Vol. 14, 2020, No. 1, p. 10-13
1
MATHEMATICAL MODEL OF WIND TURBINE IN GRID-OFF SYSTEM
Milan JUS
Ing. Milan Jus, PhD., Faculty o Special Technology, Alexander Dubček University of Trenčín, Pri Parku 19, 911 06
Trenčín, Slovakia
Corresponding author E-mail address: milan.jus@tnuni.sk
Abstract
This paper deals with the construction of a mathematical model of a wind turbine, which is one of the sources in the Grid-Off
system.
Keywords: mathematical model, wind turbine, Grid-Off system, electric generator, wind conditions.
1 Introduction
As one of the power sources of the Grid-Off system is a wind turbine. It is advantageous to work with a
mathematical model for the need of experimental research. In Fig. 1 is a schematic connection of a wind turbine
to a container, which is a Grid-Off system. [1-4]
Fig. 1 Wind turbine connection - Grid-Off system
The process of producing electricity is, which in this case applies is as follows. The wind is carrying kinetic
energy. Basically, the flow of air mass. The wind causes the propeller blades to rotate, so the kinetic energy is
converted to mechanical energy. Subsequently, this mechanical energy is transformed into electrical energy by
means of a suitable generator.
The mass flow energy passing through the area S determines the power of the intact air flow
𝑃 =
1
2
𝜌𝑆𝑣3
(1)
where is air density ([] = kg/m3
), S is area ([S] = m2
) and v is velocity of flowing air ([v] = m/s).
The mechanical energy that can be obtained by converting the kinetic energy of the air flow corresponds to
the difference in power of the air flow upstream and downstream of the turbine, as shown in Fig. 2. The
mathematical expression is as follows:
𝑃 =
1
2
𝜌𝑆1 𝑣1
3
−
1
2
𝜌𝑆2 𝑣2
3
(2)
Fig. 2 Air flow through turbine
Based on the continuity equation is valid
Container
S1
S
S2
v2
v1
v](https://image.slidesharecdn.com/urv14iss-110to13jus-201119110451/85/MATHEMATICAL-MODEL-OF-WIND-TURBINE-IN-GRID-OFF-SYSTEM-1-320.jpg)
![Milan Jus / University Review, Vol. 14, 2020, No. 1, p. 10-13
2
𝜌𝑆1 𝑣1 = 𝜌𝑆2 𝑣2 = 𝜌𝑆3 𝑣3 (3)
We can then equation (2) using equation (3) shaped in the form
𝑃 =
1
2
𝜌𝑆1 𝑣1(𝑣1
2
− 𝑣2
2) (4)
The mechanical power of the wind turbine rotor can be expressed by relation (4) considering the momentum
conservation law and its mathematical expression is
𝑃 =
1
4
𝜌𝑆𝑣1
3
(1 −
𝑣2
2
𝑣1
2 +
𝑣2
𝑣1
−
𝑣2
3
𝑣1
3) =
1
2
(1 −
𝑣2
2
𝑣1
2 +
𝑣2
𝑣1
−
𝑣2
3
𝑣1
3)
⏟
𝐶 𝑝
1
2
𝜌𝑆𝑣1
3
⏟
𝑃 𝑤
(5)
where Cp is dimensionless ratio of the extractable power P to the kinetic power Pw available in the undistributed
stream. [4-5]
Maximum mechanical performance can only be achieved under certain conditions when the ratio v2/v1 is 1/3.
In this case, the theoretical value of 59. 26 % is reached. [6]
The ratio of the peripheral speed of the rotor blade tip to the speed of the intact air flow upstream of the
turbine is called turbine speed [7]
𝜆 =
𝜔 𝑟 𝑅 𝑡
𝑣1
(6)
Next, we express the relation for the calculation of the mechanical moment Tm of the turbine rotor
𝑇 𝑚 =
𝑃
𝜔 𝑟
=
𝐶 𝑝
1
2
𝜌𝑆𝑣1
3
𝜔 𝑟
=
1
2
𝜌𝑆
𝐶 𝑝 𝑣1
3
𝜆𝑣1
𝑅 𝑡
=
1
2
𝜌𝑆𝑣1
2
𝑅𝑡
𝐶 𝑝
𝜆⏟
𝐶 𝑞
=
1
2
𝜌𝑆𝑣1
2
𝑅𝑡 𝐶 𝑞 (7)
where Cq is turbine rotor torque coefficient. [8-9]
2 Synchronous generator
The transformation of mechanical energy into electrical energy is carried out by means of a wind turbine,
which is a synchronous generator with permanent magnets. The schematic diagram of the generator is shown in
Fig. 3.
Fig. 3 Transformation of mechanical energy into electrical energy
The wind turbine includes blades that rotate the shaft on which the rotor of the permanent magnet
synchronous generator is mounted. The rotor is formed by a permanent magnet, which causes a time-varying
magnetic field, which results in voltage generation in the stator winding.
A schematic representation of a synchronous generator with a rotor of a permeable magnet and a stator
formed by a winding is shown in Fig. 4.
GeneratorWind](https://image.slidesharecdn.com/urv14iss-110to13jus-201119110451/85/MATHEMATICAL-MODEL-OF-WIND-TURBINE-IN-GRID-OFF-SYSTEM-2-320.jpg)
![Milan Jus / University Review, Vol. 14, 2020, No. 1, p. 10-13
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Fig. 4 A schematic representation of a synchronous generator
Depending on the torque balance condition, the dynamic system applies
𝑇 𝑚 = 𝑇𝑔 + 𝐽 𝑎
𝑑𝜔 𝑟
𝑑𝑡
(8)
where Tm is the turbine torque produced by the wind on its blades, Tg is the braking torque of the generator with
the load acting against the mechanical torque of the turbine and Jt is the moment of inertia of the aggregate. If we
use the equation (7) and equation (8) we can adapt to the shape
𝑑𝜔 𝑟
𝑑𝑡
=
𝑇 𝑚−𝑇 𝑔
𝐽 𝑎
=
1
2
𝜌𝑆𝑣1
2 𝑅 𝑡 𝐶 𝑞−𝑇 𝑔
𝐽 𝑎
(9)
The permanent magnets in the rotor, by their rotation around the stator, induce the internal voltage of the
generator, for which
𝐸𝑔 = K 𝑝𝑚
𝜔 𝑟 𝑛 𝑝
2
(10)
where Kpm is constant, which expresses the magnetic field strength of the permanent magnets of the generator, 𝑛p
is the number of poles of the generator and 𝜔r is the angular velocity of the rotor. [10-13]
If X is the reactance of the generator coil (X = 𝜔gLs, 𝜔g = 𝑛p 𝜔r/2), and is angle between Ug and Eg, the
equation applies to the terminal voltage Ug
𝑼 𝑔 = 𝑬 𝑔 − j𝑋𝑰 𝑔 = |𝑬 𝑔|e 𝑗𝛿
− 𝑋|𝑰 𝑔|e 𝑗
𝜋
2 = 𝐸𝑔cos(𝛿) + j𝐸𝑔sin(𝛿) − j𝑋𝐼𝑔⏟
0
= 𝐸𝑔cos(𝛿) (11)
where
𝐸𝑔 sin(𝛿) = 𝑋𝐼𝑔 (12)
The active power of the generator can be expressed as
𝑃𝑔 =
3𝐸 𝑔 𝑈 𝑔
𝑋
sin(𝛿) (13)
Equation (13) using equation (11) and (10) can be modified to shape
𝑃𝑔 =
3𝐸 𝑔 𝐸 𝑔
𝑋
cos(𝛿) sin(𝛿)⏟
1
2
sin(2𝛿)
=
3𝐸 𝑔
2
2𝑋
sin(2𝛿) =
3(K 𝑝𝑚
𝜔 𝑟 𝑛 𝑝
2
)
2
2𝑋
sin(2𝛿) =
3
8𝑋
(K 𝑝𝑚 𝜔𝑟 𝑛 𝑝)
2
sin(2𝛿) (14)
If we consider equation (12), then equation (14) can be expressed as
𝑃𝑔 =
3𝐸 𝑔
2
𝑋
cos(𝛿) sin(𝛿) =
3(
𝑋𝐼 𝑔
sin(𝛿)
)
2
𝑋
cos(𝛿) sin(𝛿) =
3𝐼 𝑔
2 𝑋
tan(𝛿)
(15)
Generator moment Tg can be expressed depending on which equation we use, whether (15) or (14)
𝑇𝑔 =
𝑃 𝑔
𝜔 𝑟
=
3𝐼 𝑔
2 𝑋
𝜔 𝑟 tan(𝛿)
=
3𝐼 𝑔
2 𝐿 𝑠 𝑛 𝑝
2 tan(𝛿)
(16)
r
Permanent magnet (rotor)
Coil (stator)
Ls](https://image.slidesharecdn.com/urv14iss-110to13jus-201119110451/85/MATHEMATICAL-MODEL-OF-WIND-TURBINE-IN-GRID-OFF-SYSTEM-3-320.jpg)
![Milan Jus / University Review, Vol. 14, 2020, No. 1, p. 10-13
4
𝑇𝑔 =
𝑃 𝑔
𝜔 𝑟
=
3
8𝑋𝜔 𝑟
(K 𝑝𝑚 𝜔𝑟 𝑛 𝑝)
2
sin(2𝛿) (17)
Based on these mathematical expressions, it is possible to construct a mathematical model by considering the
particular parameters of a real wind turbine in the equations. The input parameter is the wind speed and the
electrical output generated. It is also possible to add a connected load or use a more sophisticated wind model.
Also in the model could be expressed regulation of wind turbine blades, possibly supplemented generator with
speed control (angular speed).
3 Conclusion
In this paper is analyzed generator of electric energy, which is formed by wind turbine. It's about using that
generator in the Grid-Off system. Mathematically, the continuity and continuity between the kinetic energy
conversion that is contained in the wind and converted to the mechanical energy of the turbine are expressed.
Subsequently, this mechanical energy is transformed into electrical energy by means of a synchronous permanent
magnet generator.
Acknowledgements
The paper was treated within the project no. ITMS 26220220083 "Research technological base for designing
applications for renewable energy in practice" EU operational program "Research and Development".
References
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[7] Hau, E.: Wind Turbines: Fundamentals, Technologies, Application, Economics. 2nd. Edition. Berlin:
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[8] Hansen, M.O.L.: Aerodynamics of Wind Turbines, 3rd ed., Routledge, 2015, ISBN 978-1138775077.
[9] Zuščák, J., Janíček, F.: Calculation of wind turbine power using Matlab. In ELITECH´16 [elektronický
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2016, CD-ROM, [5] s. ISBN 978-80-227-4561-1.
[10]Mellah, H., Hemsas, K. E.: Simulations Analysis with Comparative Study of a PMSG Performances for
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[13]Schaffarczyk, A.P.: Introduction to Wind Turbine Aerodynamics. Berlin: Springer, 2014. ISBN 978-3-642-
36408-2.](https://image.slidesharecdn.com/urv14iss-110to13jus-201119110451/85/MATHEMATICAL-MODEL-OF-WIND-TURBINE-IN-GRID-OFF-SYSTEM-4-320.jpg)
MATHEMATICAL MODEL OF WIND TURBINE IN GRID-OFF SYSTEM
- 1. Milan Jus / University Review, Vol. 14, 2020, No. 1, p. 10-13 1 MATHEMATICAL MODEL OF WIND TURBINE IN GRID-OFF SYSTEM Milan JUS Ing. Milan Jus, PhD., Faculty o Special Technology, Alexander Dubček University of Trenčín, Pri Parku 19, 911 06 Trenčín, Slovakia Corresponding author E-mail address: milan.jus@tnuni.sk Abstract This paper deals with the construction of a mathematical model of a wind turbine, which is one of the sources in the Grid-Off system. Keywords: mathematical model, wind turbine, Grid-Off system, electric generator, wind conditions. 1 Introduction As one of the power sources of the Grid-Off system is a wind turbine. It is advantageous to work with a mathematical model for the need of experimental research. In Fig. 1 is a schematic connection of a wind turbine to a container, which is a Grid-Off system. [1-4] Fig. 1 Wind turbine connection - Grid-Off system The process of producing electricity is, which in this case applies is as follows. The wind is carrying kinetic energy. Basically, the flow of air mass. The wind causes the propeller blades to rotate, so the kinetic energy is converted to mechanical energy. Subsequently, this mechanical energy is transformed into electrical energy by means of a suitable generator. The mass flow energy passing through the area S determines the power of the intact air flow 𝑃 = 1 2 𝜌𝑆𝑣3 (1) where is air density ([] = kg/m3 ), S is area ([S] = m2 ) and v is velocity of flowing air ([v] = m/s). The mechanical energy that can be obtained by converting the kinetic energy of the air flow corresponds to the difference in power of the air flow upstream and downstream of the turbine, as shown in Fig. 2. The mathematical expression is as follows: 𝑃 = 1 2 𝜌𝑆1 𝑣1 3 − 1 2 𝜌𝑆2 𝑣2 3 (2) Fig. 2 Air flow through turbine Based on the continuity equation is valid Container S1 S S2 v2 v1 v
- 2. Milan Jus / University Review, Vol. 14, 2020, No. 1, p. 10-13 2 𝜌𝑆1 𝑣1 = 𝜌𝑆2 𝑣2 = 𝜌𝑆3 𝑣3 (3) We can then equation (2) using equation (3) shaped in the form 𝑃 = 1 2 𝜌𝑆1 𝑣1(𝑣1 2 − 𝑣2 2) (4) The mechanical power of the wind turbine rotor can be expressed by relation (4) considering the momentum conservation law and its mathematical expression is 𝑃 = 1 4 𝜌𝑆𝑣1 3 (1 − 𝑣2 2 𝑣1 2 + 𝑣2 𝑣1 − 𝑣2 3 𝑣1 3) = 1 2 (1 − 𝑣2 2 𝑣1 2 + 𝑣2 𝑣1 − 𝑣2 3 𝑣1 3) ⏟ 𝐶 𝑝 1 2 𝜌𝑆𝑣1 3 ⏟ 𝑃 𝑤 (5) where Cp is dimensionless ratio of the extractable power P to the kinetic power Pw available in the undistributed stream. [4-5] Maximum mechanical performance can only be achieved under certain conditions when the ratio v2/v1 is 1/3. In this case, the theoretical value of 59. 26 % is reached. [6] The ratio of the peripheral speed of the rotor blade tip to the speed of the intact air flow upstream of the turbine is called turbine speed [7] 𝜆 = 𝜔 𝑟 𝑅 𝑡 𝑣1 (6) Next, we express the relation for the calculation of the mechanical moment Tm of the turbine rotor 𝑇 𝑚 = 𝑃 𝜔 𝑟 = 𝐶 𝑝 1 2 𝜌𝑆𝑣1 3 𝜔 𝑟 = 1 2 𝜌𝑆 𝐶 𝑝 𝑣1 3 𝜆𝑣1 𝑅 𝑡 = 1 2 𝜌𝑆𝑣1 2 𝑅𝑡 𝐶 𝑝 𝜆⏟ 𝐶 𝑞 = 1 2 𝜌𝑆𝑣1 2 𝑅𝑡 𝐶 𝑞 (7) where Cq is turbine rotor torque coefficient. [8-9] 2 Synchronous generator The transformation of mechanical energy into electrical energy is carried out by means of a wind turbine, which is a synchronous generator with permanent magnets. The schematic diagram of the generator is shown in Fig. 3. Fig. 3 Transformation of mechanical energy into electrical energy The wind turbine includes blades that rotate the shaft on which the rotor of the permanent magnet synchronous generator is mounted. The rotor is formed by a permanent magnet, which causes a time-varying magnetic field, which results in voltage generation in the stator winding. A schematic representation of a synchronous generator with a rotor of a permeable magnet and a stator formed by a winding is shown in Fig. 4. GeneratorWind
- 3. Milan Jus / University Review, Vol. 14, 2020, No. 1, p. 10-13 3 Fig. 4 A schematic representation of a synchronous generator Depending on the torque balance condition, the dynamic system applies 𝑇 𝑚 = 𝑇𝑔 + 𝐽 𝑎 𝑑𝜔 𝑟 𝑑𝑡 (8) where Tm is the turbine torque produced by the wind on its blades, Tg is the braking torque of the generator with the load acting against the mechanical torque of the turbine and Jt is the moment of inertia of the aggregate. If we use the equation (7) and equation (8) we can adapt to the shape 𝑑𝜔 𝑟 𝑑𝑡 = 𝑇 𝑚−𝑇 𝑔 𝐽 𝑎 = 1 2 𝜌𝑆𝑣1 2 𝑅 𝑡 𝐶 𝑞−𝑇 𝑔 𝐽 𝑎 (9) The permanent magnets in the rotor, by their rotation around the stator, induce the internal voltage of the generator, for which 𝐸𝑔 = K 𝑝𝑚 𝜔 𝑟 𝑛 𝑝 2 (10) where Kpm is constant, which expresses the magnetic field strength of the permanent magnets of the generator, 𝑛p is the number of poles of the generator and 𝜔r is the angular velocity of the rotor. [10-13] If X is the reactance of the generator coil (X = 𝜔gLs, 𝜔g = 𝑛p 𝜔r/2), and is angle between Ug and Eg, the equation applies to the terminal voltage Ug 𝑼 𝑔 = 𝑬 𝑔 − j𝑋𝑰 𝑔 = |𝑬 𝑔|e 𝑗𝛿 − 𝑋|𝑰 𝑔|e 𝑗 𝜋 2 = 𝐸𝑔cos(𝛿) + j𝐸𝑔sin(𝛿) − j𝑋𝐼𝑔⏟ 0 = 𝐸𝑔cos(𝛿) (11) where 𝐸𝑔 sin(𝛿) = 𝑋𝐼𝑔 (12) The active power of the generator can be expressed as 𝑃𝑔 = 3𝐸 𝑔 𝑈 𝑔 𝑋 sin(𝛿) (13) Equation (13) using equation (11) and (10) can be modified to shape 𝑃𝑔 = 3𝐸 𝑔 𝐸 𝑔 𝑋 cos(𝛿) sin(𝛿)⏟ 1 2 sin(2𝛿) = 3𝐸 𝑔 2 2𝑋 sin(2𝛿) = 3(K 𝑝𝑚 𝜔 𝑟 𝑛 𝑝 2 ) 2 2𝑋 sin(2𝛿) = 3 8𝑋 (K 𝑝𝑚 𝜔𝑟 𝑛 𝑝) 2 sin(2𝛿) (14) If we consider equation (12), then equation (14) can be expressed as 𝑃𝑔 = 3𝐸 𝑔 2 𝑋 cos(𝛿) sin(𝛿) = 3( 𝑋𝐼 𝑔 sin(𝛿) ) 2 𝑋 cos(𝛿) sin(𝛿) = 3𝐼 𝑔 2 𝑋 tan(𝛿) (15) Generator moment Tg can be expressed depending on which equation we use, whether (15) or (14) 𝑇𝑔 = 𝑃 𝑔 𝜔 𝑟 = 3𝐼 𝑔 2 𝑋 𝜔 𝑟 tan(𝛿) = 3𝐼 𝑔 2 𝐿 𝑠 𝑛 𝑝 2 tan(𝛿) (16) r Permanent magnet (rotor) Coil (stator) Ls
- 4. Milan Jus / University Review, Vol. 14, 2020, No. 1, p. 10-13 4 𝑇𝑔 = 𝑃 𝑔 𝜔 𝑟 = 3 8𝑋𝜔 𝑟 (K 𝑝𝑚 𝜔𝑟 𝑛 𝑝) 2 sin(2𝛿) (17) Based on these mathematical expressions, it is possible to construct a mathematical model by considering the particular parameters of a real wind turbine in the equations. The input parameter is the wind speed and the electrical output generated. It is also possible to add a connected load or use a more sophisticated wind model. Also in the model could be expressed regulation of wind turbine blades, possibly supplemented generator with speed control (angular speed). 3 Conclusion In this paper is analyzed generator of electric energy, which is formed by wind turbine. It's about using that generator in the Grid-Off system. Mathematically, the continuity and continuity between the kinetic energy conversion that is contained in the wind and converted to the mechanical energy of the turbine are expressed. Subsequently, this mechanical energy is transformed into electrical energy by means of a synchronous permanent magnet generator. Acknowledgements The paper was treated within the project no. ITMS 26220220083 "Research technological base for designing applications for renewable energy in practice" EU operational program "Research and Development". References [1] Jus, M.: Využitie termoelektrického efektu v grid-off fotovoltických systémoch. In: Výzbroj a technika ozbrojených síl 2019 : 25. medzinárodná vedecká konferencia. - Liptovský Mikuláš : Akadémia ozbrojených síl generála Milana Rastislava Štefánika, 2019. - ISBN 978-80-8040-585-4. - s.83-88. [2] Jus, M.: Thermoelectric generator model and simulation. In: Transfer 2019 : 20th International Scientific Conference. - Trenčín : TnUAD, 2019. - ISBN 978-80-8075-889-9. - s.71-78, CD ROM. [3] Jus, M., Balla, J.: Parametric model of thermoelectric generator. In: University review. - ISSN 1339-5017. - Roč.13, č.3(2019), s.40-45. [4] [02.03.2020], https://en.wikipedia.org/wiki/Betz%27s_law [5] Ragheb, M., Ragheb, A. M.: Wind turbines theory—The Betz equation and optimal rotor tip speed ratio, fundamental and advanced topics in wind power. In R. Carriveau (ed.), Fundamental and Advanced Topics in Wind Power, InTech, 2011. [6] Zuščák, J., Janíček, F., Kijan, V., Váry, M.: Dynamic behavior of self-excited induction generator in Off- grid operation under resistive load. In Power engineering 2018. Control of Power Systems 2018 : 13th International scientific conference. Tatranské Matliare, Slovakia. June 5-7, 2018. 1. vyd. Bratislava : Slovak University of Technology, 2018, S. 89-94. ISBN 978-80-89983-00-1. [7] Hau, E.: Wind Turbines: Fundamentals, Technologies, Application, Economics. 2nd. Edition. Berlin: Springer, 2006. ISBN 3-540-24240-6. [8] Hansen, M.O.L.: Aerodynamics of Wind Turbines, 3rd ed., Routledge, 2015, ISBN 978-1138775077. [9] Zuščák, J., Janíček, F.: Calculation of wind turbine power using Matlab. In ELITECH´16 [elektronický zdroj] : 18th Conference of doctoral students. Bratislava, Slovakia. June 8, 2016. 1. vyd. Bratislava : STU, 2016, CD-ROM, [5] s. ISBN 978-80-227-4561-1. [10]Mellah, H., Hemsas, K. E.: Simulations Analysis with Comparative Study of a PMSG Performances for Small WT Application by FEM International Journal of Energy Engineering e-ISSN: 2163-1905. [Online]. 2015, [cit. 03.03.2020]. Dostupné na: http://article.sapub.org/10.5923.j.ijee.20130302.03.html [11][03.03.2020], https://www.vutbr.cz/www_base/zav_prace_soubor_verejne.php?file_id=99990 [12]Heier, S.: Grid Integration of Wind Energy. 3rd. Edition. John Wiley & Sons Ltd., 2014. ISBN 978-1-119- 96294-6. [13]Schaffarczyk, A.P.: Introduction to Wind Turbine Aerodynamics. Berlin: Springer, 2014. ISBN 978-3-642- 36408-2.