Wind Turbine Micrositing: Comparison of Finite Difference Method and Computational Fluid Dynamics

Transcription

1 IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 1, No 1, January 01 ISSN (Online): Wind Turbine Micrositing: Comparison of Finite Difference Metod and Computational Fluid Dynamics Samina Rajper 1 and Imran J. Amin 1 Department of Computing, SZABIST Karaci, Sind, Pakistan Department of Computing, SZABIST Karaci, Sind, Pakistan Abstract For smoot and optimal operation of wind turbines te location of wind turbines in wind farm is critical. Parameters tat need to be considered for micrositing of wind turbines are topograpic effect and wind effect. Te location under consideration for wind farm is Garo, Sind, Pakistan. Several tecniques are being researced for finding te most optimal location for wind turbines. Tese tecniques are based on linear and nonlinear matematical models. In tis paper wind pressure distribution and its effect on wind turbine on te wind farm are considered. Tis study is conducted to compare te matematical model; Finite Difference Metod wit a computational fluid dynamics software results. Finally te results of two tecniques are compared for micrositing of wind turbines and found tat finite difference metod is not applicable for wind turbine micrositing. Keywords: Wind farm, Finite Difference Metod, Wind turbine micrositing, wind pressure, computational fluid dynamics. 1. Introduction Arrenius gave an idea about a century ago tat emission of carbon dioxide from te fossil fuels will be te reason of Eart warming [1]. At present, te globe is looking for te alternate solutions instead of expensive and polluted fossil fuel or any oter medium tat discarge CO. Unfortunately, Pakistan depends upon fossil fuels for producing electricity, wic is costly and polluted source of energy. However, fossil fuel is one of te major imports of Pakistan to meet te energy requirements of country, as 6.1% of country s total electricity production is obtained using fossil fuels []. Wind energy is used as te main source of producing electricity in many countries, suc as Germany, Spain, United States, Cina, India []. At present, Wind turbine tecnology is te most mature and pollution free renewable tecnology available [, 5]; because wind exists everywere on te eart and it as been used as a source of mecanical energy for many decades, furtermore tere is a tecnological capacity also [5].Tis study is conducted to optimize wind turbine micrositing. For tis purpose, a researc is being conducted to study wind pressure distribution and its effect on wind turbines, comparison between two tecniques, i.e. computational fluid dynamics software VIZIFLOW.0 and discretized form of Laplace equation is made. Te main objective of wind turbine micrositing is to gain te maximum result (net revenue) and to minimize te overall cost for energy. Systematical approac to wind turbine placement is a tecnical issue and was first been proposed by Moseti et al. and gradually worked and been improved by Grady et al.; genetic algoritm was used to find te optimal solution of te problem [6, 7]. Wereas, Computational fluid Dynamics tecniques based on linear and nonlinear equations were used for te optimal placement of wind turbines in complex terrain [8]. Te present study deals wit te analysis of Garo wind corridor wic is te main site for installation of wind farm to meet te requirements of energy in Pakistan. Garo, is a small town near te coast of Sind, Pakistan. It is 1,06 Km long coastline, average wind speed is more tan 7. m/s in Garo and te estimated wind potential from tis site is 50,000 MW [9, 10]. Te site is sown in figure 1.. Metodology Te metodology tat is used to conduct tis study is based on te comparison of two models to analyze te results of wind pressure distribution and its effects on te wind turbine on wind farm. Te experiment Design can be viewed torougly using figure. Te purpose of adopting tis metodology is to study te two models and analyze te results of te proposed matematical model tat weter it can be applicable for te problem or not. Suppose te terrain is a square grid and is supposed to be Copyrigt (c) 01 International Journal of Computer Science Issues. All Rigts Reserved.

2 IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 1, No 1, January 01 ISSN (Online): divided into possible locations. To examine te concept only two Wind Turbines are supposed to be placed. Te Diameter of wind turbine is supposed as 00 m. Terefore, te total area can be subdivided into kmxkm as sown in figure. As te widt supposed for eac square grid/cell is good enoug, te wake of a column of wind turbine sould not affect anoter. As done in [11]. It is observed tat 5D square grid size satisfies te Rule of Tumb spacing requirements for all vertical and orizontal directions [11]. Only one case is considered if te wind speed is constant at 10m/s and te direction of wind is from left to rigt, for te examining of Laplace equation. Same case is also examined using computational fluid dynamics software VIZIFLOW.0. Fig. 00 m eac cell for a square grid of kmxkm. Discretized form of Laplace Equation Te Laplace Equation is a second order partial differential equation (PDE). It is used in different areas of science and engineering for example in electricity, fluid mecanics etc. wic is + =0 (1) Fig. 1 Depicts te under consideration site Garo, Tatta, Sind. Fig. Represent te Metodology to compare two Tecniques. Wen te selection of a matematical model is made, te next step is to select a suitable discretization metod, i.e., a metod of approximating te differential equations by a system of algebraic equations for te variables at some set of discrete locations in space and time. In a finite difference metod, approximations for te derivatives at te grid points must be selected. Taylor series expansion is used to obtain approximations to te first and second derivatives of te variables wit respect to te coordinates. Wen necessary, tese metods are also used to obtain variable values at locations oter tan grid points. Here te Laplace equation is derived until and unless to te set of Linear i-e simultaneous equations [1]. A formal basis for developing finite difference approximation to derivative is by te use of Taylor series expansion. Because Taylor series expansion of a function f(x) is about a point x in te forward (positive x) and backward(negative x) directions given, similarly, for a point y upward (positive) and downward (negative) directions. Assume tat tere are tree points on te X-axis separated by a distance let. Consider tat I, i-1 and i+1 are center, left and rigt sides of te distance. Similarly, for Y-axis j, j-1 and j+1 are center, downward and upward points. By applying Taylor expansions [1]. 5 ( i 1) i i i i O( ) x x! x! x! () Copyrigt (c) 01 International Journal of Computer Science Issues. All Rigts Reserved.

3 IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 1, No 1, January 01 ISSN (Online): i i i O( ) ( i1) i x!! x x x! () i sows tat te derivative is computed at te point i. 5 ( i1) i i O( ) ( i 1) i x x 1 ( i1) i ( i1) i x O ( ) () (5) Subtracting equation (1) from equation (), te resultant equation will be: i i o x x 5 i1 i1 ( ) Or ( i1) ( i1) i O( ) x (6) (7) Applying same procedure to Y-axis, finally anoter equation will be: ( j1) j ( j1) j y ( j1) ( j1) j O( ) y O( ) (8) (9) By combining equations (7) and (9) te resultant equation for x and y will be: ) i, j x y ( i1), j i, j ( i1), j i,( j1) i, j i,( j1) (10) After substitution of equation (9) in Laplace equation te resultant equation will be: 1 i, j ( 1), ( 1),,( 1),( 1) i j i ji j i j (11) Or U ij, U( i1), ju( i1), jui,( j1) Ui,( j1) (1) Assume tat te x-axis and y-axis points i-e (i, j) are superposed ten te resultant will be a 5- point pattern sown in figure. Fig. sows te position of points at x-axis and y-axis Te Discretized form of Laplace equation is used to study te wind pressure distribution and its effect on te wind turbine on te wind farm. Here, te equation (1) sows tat tere are four variables represent te wind pressure on wind turbine and at different points as well to study te effects of wind pressure distribution. Based on equation 1 [1], air pressure distribution and its effects on wind turbine on te wind farm are considered. Initially, te effects are proven on MS Excel using Laplace equation ten te Laplace equation s discretized form is used for finite solution. Te computed results based on discretized form of Laplace equation using MS Excel are depicted in table 1 and table. Te air flow rate is constantly assumed as 10 m/s and using te discretized form of Laplace equation, te results are studied. It is represented in table 1 and table tat cells D5, C6,D6,E6,D7 and M10,L11,M11,N11,M1 are supposed as 0 because te concept is to be examined tat if te wind flows at constant speed and in te same direction on te wind turbine, wat are wind pressure distribution effects. Te mentioned cells of Excel seet are te representation of wind turbine virtual existence. Suppose tat cells D5,C6,D6,E6,D7 are wind turbine called a and cells M10,L11,M11,N11,M1 are named as b.it is depicted tat results are satisfactory at wind turbine a but te results are not satisfactory wen te same wind speed is applied to wind turbine b. As te wind pressure is gradually distributed and be slowed down if a wind turbine is tere. Te wind pressure at cells O11, N1 and N10 cannot be raised after it gets down to 0. Figure 7 is te MS-Excel Seet results wic are depicted by Table 1 and. Figure 5 is te grapical representation of te results depicted earlier and represented by table 1 and table. Tere are two wind turbines are represented in figure 5. Te wind pressure is assumed to be from left to rigt as sown in figure 5. It is sown in te figure 5 tat te peaks are te ig pressure points in te wind turbine. Table depicts te computed results of wind turbine a and b. Later te effect of air pressure in matematical model using finite difference metod is compared wit computational fluid dynamics software discussed in section 5. Copyrigt (c) 01 International Journal of Computer Science Issues. All Rigts Reserved.

4 IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 1, No 1, January 01 ISSN (Online): Computational fluid Dynamics A computational fluid dynamics software is used to compare te results of matematical model wit te results of computational fluid dynamics. Te gauge teory of tis software functions on te principles of two basic equations of fluid flow. a) Te continuity equation A1v1=Av (1) Wic means tat te volume flow rate is constant, so tat a smaller area for te fluid to flow will result in a iger velocity. b) Bernoulli s equation Tis equation relates te pressure, flow velocity, fluid density and fluid eigt. P1 + ρgy1 + ½ρv1 = P + ρgy + ½ρv (1) Te second term on eac side of te equation represents te potential energy due to te eigt of te fluid, but since te eigt is constant in tis cfd, tese terms cancel and te equation reduces to: P1 + ½ ρv1 = P + ½ρv (15) P is te pressure, v te velocity and ρ te density of te fluid. To find te pressure P te equation is rearranged: P=½ρ(v v1 )-P1 (16) Te wind flow is supposed as from left to rigt again. Table is depicting te tested values for different attributes used for computational fluid dynamics software. Table 5 depicts wind pressure distribution effect is satisfactory at wind turbine a if te reading is taken at all points i.e. up, down, left and rigt of te turbine and te same procedure of taking readings is applied at wind turbine b. Te computed results are not muc different from wind turbine a and wind turbine b. Te grapical representation of computed results using computational fluid dynamics software based on te discussed equation above as equation (1) and (16) is depicted using figure 6. Te figure 6 also represents te same scenario wit two wind turbines a and b, te blue color around te circles (virtual representation of wind turbines a and b) sows te ig wind pressure. Table 1: Results of wind speed pressure using Discretized form of Laplace equation on MS Excel. A B C D E F G A-Initial Wind Speed 10m/sec Table : Depicts te continued results of wind speed pressure using Discretized form of Laplace Equation on MS Excel. H I J K L M N O Copyrigt (c) 01 International Journal of Computer Science Issues. All Rigts Reserved.

5 IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 1, No 1, January 01 ISSN (Online): Table.Sows te Computed Results using Discretized form of Laplace Equation. Wind-Turbine Position Flow Rate a UP 1.5 Down.51 Left 5.61 Rigt 0.9 b UP 0.11 Down 0.06 Fig. 5 Wind turbines effect on Air pressure using Finite Difference Metod Left 0. Rigt 0.00 Table. Data for computational fluid Dynamics Test WT Flow Rate Pressur e Densit y Diameter Dist b/w WT 10 m/s 1 1 kg/ 0,0m,m 10 m m WT- Wind turbines Table 5.Sows te Computed Results using computational fluid Dynamics. Wind-Turbine Position Flow Rate a UP 8. Down 6. Left.1 Rigt.91 Fig. 6 Wind turbines effect on Air pressure using Computational fluid dynamics model (Two wind Turbines- a, b, Blue color sows ig wind pressure) b UP 6.79 Down 6.0 Left.65 Rigt.5 Copyrigt (c) 01 International Journal of Computer Science Issues. All Rigts Reserved.

6 IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 1, No 1, January 01 ISSN (Online): equation in tis study was initially tested using Excel seet so it is suggested tat it sould be tested using Genetic algoritm to see te results. Because te Genetic algoritm tecnique operate on te coding of te parameter set rater tan te parameter itself. Fig. 7 resulted Values on MS-EXCEL SHEET after using Laplace equation (Data is already given in Tables 1 and ) 5. Discussion In tis paper, te study of wind turbine micro-siting using Finite Difference Metod is represented. In tis study wind pressure distribution and its effect on wind turbine on te wind farm are considered. Initially Laplace equation s discretized form is used for finite solution. First te concept is proven on MS. Excel. Later te effect of air pressure in matematical model using finite difference metod is compared wit computational fluid dynamics software. Te computational fluid Dynamics software is based on anoter equation. Te purpose was to study te wind air pressure distribution effects using two tecniques. Wile using Tecnique 1; te result is depicted tat if te wind speed is constant at 10m/s and direction is from left to rigt, te wind pressure distribution effect is computed at wind turbine a and wind turbine b are totally different from te results computed using te Tecnique. Te concept tat te wind pressure will be less or even zero at te wind turbines and it gradually be raised to ig pressure as it goes farter from wind turbines. Tis concept is proven positively at computational fluid dynamics owever, tis concept cannot be proved at Laplace equation. Using te Laplace equation, it is observed tat wen te wind pressure goes to zero it cannot be raised at te points were te wind pressure raised using computational fluid dynamics. Terefore, it is concluded tat tis model is not successful in comparison of computational fluid dynamics software (VIZIFLOW.0)based on te continuity equation and Bernoullis equation Furtermore, Wind turbine placement is a discrete problem, terefore, classical metods are more complicated for optimization of tis problem [1]. Tis study depicts tat Matematical model based on Laplace References [1] I. Martin, et al, Advanced Tecnology Pats to Global Climate Stability: Energy for a Greenouse Planet a review in SCIENCE 98, 1, 00 [] Electrical Energy Sector Overview, Alternate Energy Development Board, Government of Pakistan, 010 ttp:// [] M. A. Amed, F. Amed and M. W. Aktar, wind caracteristics and wind power potential for soutern coasts of sind, pakistan, Journal of Basic and Applied Sciences, 6 (), , 010 [] A. Houri, Effects of Funds Diversion from Fuel Purcasing to Wind Energy Installations: A Case Study [5] J. F. Manwell, et al., Wind Energy Explained : Teory, design and Application Second edition, Willey, publised in 009 [6] C. Wan, et al., Optimal Micro-Siting of Wind Turbines by Genetic Algoritms Based on Improved Wind and Turbine Models, Joint 8t IEEE Conference on Decision and Control [7] S.A. Grady et al., Placement of wind turbines using genetic algoritms, Renewable Energy 0 (005) [8] J.M.L.M. Palma, Linear and nonlinear models in wind resource assessment and wind turbine micro-siting in complex terrain, Journal of Wind Engineering and Industrial Aerodynamics, Volume 96, Issue 1, December 008, Pages 08-6 [9] Saifulla, and A. Allauddin, Opportunities & Callenges toscaling up wind power in Pakistan, Presentation in Quantum Leap in Wind Power in Asia Structured Consultation, June 1, 010 [10] C. Qamar-uz-Zaman et al., Establisment of commercial wind power plant of 18 MW at Garo, Tecnical Report No. PMD-11/00, pakistan meteorological department sector -8/, p. O. Box 11, Islamabad [11] G. Marmidis, et al., Optimal placement of wind turbines in a wind park usingmonte Carlo simulation, Renewable Energy (008) [1] K. Ambar, Finite Difference Metod for te Solution of Laplace Equation, Department of Aerospace Engineering Iowa State University [1] A. Emami, P. Nogre, New approac on optimization in placement of wind turbines witinwind farm by genetic algoritms, Renewable Energy 5 (010) Copyrigt (c) 01 International Journal of Computer Science Issues. All Rigts Reserved.