- Email: [email protected]
Wake flow model of wind turbine using particle simulation
Renewable Energy 41 (2012) 185e190
Contents lists available at SciVerse ScienceDirect
Renewable Energy journal homepage: www.elsevier.com/locate/renene
Wake flow model of wind turbine using particle simulation M.X. Song, K. Chen, Z.Y. He, X. Zhang* Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 July 2011 Accepted 19 October 2011 Available online 21 November 2011
A particle model for calculating turbine wake flow during the optimization of wind farm micro-siting is presented. This model treats the wake flow as virtual particles generated by the turbine rotor. Based on the pre-calculated flow field of empty wind farm, the motions of the particles are simulated. The wake flow effect of velocity decrement is obtained from the density of the particles. On flat terrain, the proposed particle model fits the experimental data better than the previous linear model. On complex terrain, since the particle simulation is based on numerical result of flow field, the particle model can be coupled with optimization algorithm of generating turbine layout. Ó 2011 Elsevier Ltd. All rights reserved.
Keywords: Wind power Micro-siting Wake flow Computational fluid dynamics
1. Introduction As a kind of clean and renewable energy source, wind energy is of great quantity all over the world. But when converting wind power into electricity using wind turbines, wake flow is generated due to the extraction of kinetic energy and the disturbance of rotating blades. In order to improve the efficiency of a wind farm, the positions of turbines must be carefully chosen so that each turbine will not be significantly influenced by others’ wake flow. This procedure is called micro-siting. The previous studies mainly concentrated on the calculations or experiments on wake flow, and the optimization approaches on micro-siting. Among the optimization approaches, genetic algorithm (GA) is the most commonly used method. Mosetti [1], Grady [2], Wan [3] et al, have done detailed analysis on GA. Most of these approaches employ the linear model for calculating turbine wake flow, which is suggested by Jensen in 1983 [4]. The Jensen model is an approximate description of the velocity distribution within the wake flow area, with experiential constants according to experimental measurements. Jensen model treats the wake flow as a conical area, in which the velocity only depends on the distance downstream from the turbine center. This model calculates the wake flow very fast on flat terrain. It is utilized during the GA optimization to evaluate the total power output for each layout. The assumption in Jensen model is that the wind speed is uniform all over the wind farm, so that the wake flow maintains its regular shape. But for those terrains with higher slopes, although some
* Corresponding author. E-mail address: [email protected] (X. Zhang). 0960-1481/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2011.10.016
geometric modifications can be applied to Jensen model, it is still unable to reflect the actual complex wake flow precisely. Especially, if a large number of vortexes exist, wake flow could be entrained into them and lose its conical shape totally. On the other hand, detailed calculation of computational fluid dynamics (CFD) in fine grid system is an accurate way for air flow simulation on complex terrain. Uchida et al. applied large eddy simulation (LES) of CFD to calculate the air flow over complex terrain and in urban area [5,6]. Their results demonstrate that the CFD method is effective for this problem. Zahle et al. [7] and Troldborg et al. [8] respectively used Reynolds averaged numerical simulation (RANS) and LES to calculate the flow field around rotating turbine blades. In order to achieve sufficient accuracy for both large scale of complex terrain and placed turbines, grids must be fine enough to represent the disturbance of turbine rotor to the incoming air flow, and to resolve the vortex motions behind it. So these calculations require a large quantity of grids adapted to the blade surfaces, which takes long time for the iteration to converge. Besides, LES solves the unsteady equations, which takes much longer time to obtain averaged solution. Furthermore, the optimization approaches need to alter the layout repeatedly in order to search for optimal solution, the flow field calculation will be necessary to be performed for thousands of times, which definitely makes the calculation time unacceptable. Therefore, a simplified model for calculating turbine wake flow with both speed and adaptiveness to complex terrain is needed. In this paper, a particle tracking model is introduced to calculate wake flow. This model uses the concentration of a kind of virtual matter to represent the intensity of wake flow. The virtual matter has been studied in previous researches [9], where it is simulated by solving convective and diffusive equation of scalar. Unlike the
186
M.X. Song et al. / Renewable Energy 41 (2012) 185e190
1. Production: Particles are generated at a certain rate within the round area of each turbine rotor. The rate is noted as p, which indicates the number of particles generated in unit area in unit time period. 2. Convection: Local velocity for each particle is interpolated based on pre-calculated flow field. The convective displacement is added to the particle position as:
Table 1 Turbine properties. Property
Value
Hub height Rotor diameter Thrust coefficient (CT)
60 m 40 m 0.88
previous way, the particle model treats the development of wake flow as the motion of multiple virtual particles generated by the rotating turbine blades. It implements the Lagrange concept of tracking each particle’s position and velocity, and by counting the numbers, the effect of wake flow is obtained. Since the particles’ convective motions are fully controlled by their local velocities, the shape and intensity of wake flow automatically deform according to the flow field. The local characteristics of wake flow are not only determined by incoming wind or start point, but also changes along stream line. By using this method, it is only necessary to perform a pre-calculation of CFD of an empty wind farm where no turbines are placed. For each different layout, the simulation of particles runs just based on the pre-calculated flow field without new grid system adapted to turbine blades. It significantly reduces the difficulty of the CFD calculation, and save much time in optimization. 2. Calculation of wake flow 2.1. Method and steps The most significant effect of wake flow is the decrement of local velocity magnitude. In the proposed particle tracking method, each slight amount of decrement effect is packed into a virtual particle. The number of particles in a certain area will reflect the intensity of wake flow that affects the local velocity. The steps of calculation are: 1. Simulate the production, motion and disappearance of the particles. 2. Calculate the relative density of virtual particles in a statistical way. 3. Transform the density of particles into the decrement of velocity. In step 1, the flow field of the wind farm with no turbines must be obtained first. On flat terrain, it could still be assumed that the wind speed and direction is uniform all over the field. On complex terrain, CFD calculation is necessary to be performed before the particle simulation. Hundreds of time steps are simulated to let the wake flow fully develop. There are four actions in each time step:
1 0.95 0.9 0.85 0.8 exp num Jensen
0.75 0.7
3. Diffusion: A Gaussian distributed random displacement is added to each particle’s position as:
Dx ¼ surDt
0
0.5
1
1.5
2
2.5
(2)
where s is the ratio of intensity of diffusion and convection, r is the Gaussian distributed random number related to the diffusion intensity in a time step. 4. Disappearance: each particle disappears at a certain probability g, which we called the attenuation factor In step 2, the velocity after the influences of wake flow at each other turbine needs to be determined. Each turbine has a corresponding volume for particle counting. The volume locates at the center of the turbine rotor, with the same scale of the turbine rotor. The density of particles that stay in the corresponding volume approximately represents the intensity of the wake effect to the turbine. In order to obtain a stable statistical result for density of particles, it is necessary to simulate several more time steps as in step 1, meanwhile in each simulated time step, particles in each corresponding volume are counted. The relative density of particles is defined to be 1 at their generation place, where the number of particles in unit volume is:
n0 ¼
pAt p ¼ Au0 t u0
(3)
where A is the area of turbine rotor, t is a tiny time duration, u0 is the local velocity at the turbine rotor from the original precalculated flow field. So relative density in a turbine’s corresponding volume can be calculated by c ¼ n/n0, where n is the counted number of particles in the corresponding volume. In step 3, the relative density is transformed into the effect of velocity decrement by the following linear transformation:
b
1.1 1.05
(1)
where u is the local velocity at the position of the particle.
non-dimensional variable
non-dimensional variable
a
Dx ¼ uDt
1 0.95 0.9 0.85 0.8 0.75
num Jensen 0
0.5
1
r/D
x
7 5d Fig. 1. Velocity profile with inlet velocity at 11.52 m/s.
1.5 r/D
x
10d
2
2.5
M.X. Song et al. / Renewable Energy 41 (2012) 185e190
187
Fig. 2. Layouts for single wind direction.
u0 ¼ uð1 bcÞ
(4)
where u is the original velocity from pre-calculated flow field, u0 is the equivalent velocity after the influences of wake flow, c is the local relative density from step 2, b is a coefficient that needs to be determined. 2.2. Parameters In previous description, parameters p and Dt only affect the accuracy and time cost of simulation. They should be chosen with considerations of the requirement and computer capabilities. Parameter b is deduced by maintaining consistency with Jensen linear model. In Jensen model, velocity with wake flow influences is:
" u0 ¼ u 1
2a ð1 þ ax=r1 Þ2
# (5)
where a is the entrainment constant, x is the distance to the turbine center in the downstream direction, r1 is related to the radius of the turbine rotor, and a is the axial induction factor that can be calculated from
CT ¼ 4að1 aÞ
(6)
where CT is the thrust coefficient of the turbine. The further away from the turbine in downstream direction, the larger the affected area becomes, and the weaker the wake flow becomes. In present study, we take a typical turbine as in Grady’s paper, the properties are listed in Table 1. Because the relative density of particles is defined to be 1 at its generation place, we introduce x ¼ 0 into Eq. (5) to calculate the velocity right at the turbine rotor. Our parameter b is deduced to be 0.65 to fit Jensen model. Parameters s and g are determined by fit the numerical results to Taylor’s experimental data [10]. In this case, one single turbine on flat terrain with uniform velocity inlet is calculated. Velocity
Fig. 3. Layouts for multiple wind directions.
188
M.X. Song et al. / Renewable Energy 41 (2012) 185e190
Fig. 4. Distributions of relative density of wake particles.
profiles under three different inlet velocities (8.5 m/s, 9.56 m/s, 11.52 m/s) at three different distances downstream (2.5d, 6d, 7.5d, d is the diameter of the turbine rotor) are compared to the experimental data and Jensen model, and velocity profiles at 10d downstream are compared to Jensen model only. The parameters are finally chosen as in Eq. (7) to archive good fit. Two typical comparisons of velocity profiles are shown in Fig. 1.
s ¼ 0:3; g ¼ 0:005
(7)
In Fig. 1, the horizontal axis represents the non-dimensional distance to the center line of the turbine rotor in the radial direction, while the vertical axis is the non-dimensional velocity. The curve labeled num is the numerical result from the particle simulation, the dashed curve labeled Jensen is the result calculated using Jensen linear model, and the dots are the experimental results of Taylor. According to the comparisons, it is seen from Fig. 1 (a) that the result from our particle simulation fits the experimental data better. The Jensen model underestimates the wake effect around the center region, and overestimates it in the region further away from the center. Taylor’s data does not contain the ones at 10d downstream, so our result is compared only to Jensen model, as shown in Fig. 1(b). In average, the results from two methods are consistent. The comparisons under other circumstances show the same conclusion.
Jensen model and the present particle model are both applied respectively for the four layouts. For Layout 1 and 2 that are corresponding to the single wind direction case, the calculated results of wake flow by particle model are visualized in Fig. 4. The characteristics of wake flow by particle model are similar to those by Jensen model. The total power outputs for the four layouts are calculated respectively by the two wake models. Based on the velocity calculated from the wake model, the total power output (kW) of a wind farm is approximately calculated by:
P ¼
N X i¼1
0:3u3i
(8)
where N is the number of turbines in the wind farm, ui is the local velocity at the ith turbine with the wake effect. The results are listed in Table 2. For Layout 1 and 2, since the wind direction is precisely along the axis, most turbines affected by wake flow in at the center line of the wake area. According to Fig. 1, Jensen model underestimates the wake effect, which produces larger total power output for the entire wind farm. For Layout 3 and 4, as multiple wind directions exist, the averaged differences of estimation of wake effect is less significant. The total power outputs by particle model are close to those by Jensen model. In general speaking, it is concluded that the present particle model has consistency with the Jensen linear model on flat terrain.
3. Applications
3.2. Complex terrain
3.1. Flat terrain
Assuming the terrain to be flat is a simplification for calculation, but not suitable for most situations in reality. Mountains produce local acceleration of air flow, concentrating the kinetic energy. Many wind farms are built on complex terrain because of this
We apply the proposed particle model in evaluating the existed layout of wind farm, and compare the results to those calculated using Jensen model. Four typical layouts obtained using GA from the previous studies of micro-siting are chosen for evaluation. Fig. 2 shows the two layouts optimized for single wind direction, where incoming wind is along the positive direction of x-axis with speed at 12 m/s Fig. 3 shows the two layouts optimized for 36 wind directions, distributed uniformly with intervals of 10 . Inlet speeds are 12 m/s for all directions. The turbines are all with the same type as their properties listed in Table 1.
Table 2 Calculated total power output (kW).
Jensen model Particle model Relative difference
Layout 1
Layout 2
Layout 3
Layout 4
12 354 11 552 6.5%
14 312 13 292 7.1%
9268.4 9292.8 0.26%
17 241 17 486 1.4%
M.X. Song et al. / Renewable Energy 41 (2012) 185e190
0
40
40
Y
80
46 00
80
80
0
40
14 0
-500 1 44 00
0 104 11220 80
4 60 0
60 20
120
60
60
0
20
10
-500
0
80
80
120
80 0 4
60
0
80 0 1 2 0 1 68 0 00 16 0
80
0 16 0 80 12 60
0
80
4400
60
-1000
60 80 8 0
500 0
12
60
20
60
12
10
0
60
60 40 40 60 80
80
40 80
60 80 60
0
12 01 0 0
80
40 0 6 10
80
12
00 8 60
80 8 0 80
0
40
10
40 60
120
100
4 04 0
20
100
0
140
40
10
10
10
60
14 0
60 80 80
160
60 80
60
advantage. We calculated a simple case with a typical complex terrain to see the distortion of wake flow on complex terrain. The chosen typical terrain contains only a smooth hill with the height of 120 m, which is twice the hub height of the turbine. A turbine is placed 100 m away from the center of the hill in the upstream direction. As required by the particle model, CFD calculation is performed. The parameters of CFD calculation are listed in Table 3. The flow field on the center vertical cross section is shown in Fig. 5 (a). The incoming air flow is along the horizontal axis from left to right. The filled contour represents the magnitude of velocity, while the curves with arrows are streamlines. The local acceleration is obviously noticed around the top of the hill, where the velocity magnitude reaches 16 m/s. The air flow is separated from the surface of the hill after it crosses the top, and a vortex with the scale similar to the hill itself is generated behind the hill. The particle model is then applied to calculate the wake flow. The relative density of particles on the same cross section is shown in Fig. 5 (b). The filled contour indicates the intensity of the wake flow. The center of the wake region is at the same height of the turbine hub when it is generated. As it develops, the air flow is extruded by the rising ground, so the wake flow becomes closer to the ground surface. After it crosses the top of the hill, due to the existence of the vortex, the wake flow would travel a long distance before it drops to the original height. Meanwhile, the shape of the wake flow calculated using Jensen model is also plotted in Fig. 5 (b). Jensen model would totally ignore the wake flow above the top of the hill, and obtain wake flow effect behind the hill where there should be no wake flow. Using Jensen model on complex terrain would bring additional errors to the evaluation of layouts when calculating the power output of turbines. For a fixed layout, when the terrain is altered from flat one to complex one, due to the local acceleration and deceleration of air flow and the distorted wake flow, the power output of each turbine could either increase or decrease, depending on how the air flow is changed by terrain geometry and where the turbine is placed. So the total power output usually does not change much from when it
4400
1800 m 600 m 400 m 12 m/s 90 30 30
1000
0
Value
Domain length (streamwise) Domain width Domain height Inlet velocity Grid numbers
60
10
Parameter
60
60
Table 3 Parameters of CFD calculation.
189
40
-1000 500
1000
X Fig. 6. Contour of relative altitude of the complex terrain.
is on flat terrain. For example, if we apply Layout 2 in Fig. 2 (b) to the terrain in Fig. 6, the total power output calculated using particle model is 13131 kW (the original on flat terrain is 13292 kW). The difference is only 1.2%. But for the turbines whose power outputs are decreased, since the wind field contains regions with velocities higher than the inlet one, it is possible to relocate them to better positions so that the total power output could be additionally increased. Therefore, the optimized layout from optimization algorithms on flat terrain is probably not optimal on complex terrain. It is necessary to couple the present particle model with optimization algorithm to generate new layout. A lazy greedy optimization algorithm proposed by Zhang et al. [11] is employed to test the proposed particle model on complex terrain. This algorithm produces similar or even better layout than the one from GA, and is 600 times faster than GA in the specific case of their studies. The greedy algorithm starts from an empty layout. The steps are: 1. Construct a list containing the possible positions for turbines. Each element corresponding to a position has an evaluation
Fig. 5. Numerical results of a typical complex terrain.
190
M.X. Song et al. / Renewable Energy 41 (2012) 185e190
60
0
60
120
60
40 40
80
60
14
40
40
0
80
-500
0
0
12 23 0
3 20
26 100
14
-1000 -1000
80
10
9
100
80
8060 12 0 10 0
15 80
2
80 60
0
60
40
80
20
60
80 60
0 1216 0 0
80
80
60
7
29
100
16
1 6
80
19
40
18
60
100
13 100
60
100
0 40 6
21 10200 1
14
0
1 00
25 60 40
Y
14 120 100
24
27
10
12
5
20
0 60
60
80
2880
12 60 10 40
-500
4
40
80
17
60
140 80
500
4. Conclusions A model for calculating turbine wake flow using particle simulation is proposed in this paper. The model has the following advantages:
22
0
500
30
60
10
11
80
8 120
40
40
40 0 6
40
60 100
0
60
0 100 60 8
10
80 00
1000
80
1000
X Fig. 7. Optimal layout on complex terrain.
value, representing the contribution to the total power output if a turbine is placed there. 2. Test all the elements, calculate their evaluation values. Keep the list in descending order according to the evaluation value. 3. Place a turbine at the corresponding position of the top element in the list, and remove the element from the list. 4. If the turbine number reaches the given limit, then the optimal solution is obtained. Otherwise, go to Step 2. In the ordinary greedy algorithm, a large number of remained elements have to be tested in Step 2. But in the lazy greedy algorithm, as proved in Zhang’s paper [11], the problem of wind turbine positioning has the submodular property. As a result, if it is only needed to find the element with the maximum evaluation value, the list does not need to be fully updated in Step 2. This advantage greatly reduces the number of elements that are to be tested, therefore greatly reduce the number of times of running the wake flow model. In order to compare the result to the one on flat terrain, turbine number is fixed at 30, and the grid numbers are 10 10. The obtained optimal layout is shown in Fig. 7. The total power output of the layout in Fig. 7 is 17,728 kW, increased by 35% comparing to Layout 2, whose power output is 13,131 kW. The layout is totally different from the optimal layout on flat terrain.
1. On flat terrain, the result from the particle model is closer to the experimental data than the Jensen linear model. In application of evaluating power outputs of layouts, the particle model produces consistent result to the Jensen linear model. 2. On complex terrain, the model reflect the characteristics of turbine wake flow more accurately than the previous linear model. 3. The model calculates wake flow effect based on pre-calculated flow field of empty wind farm. It is not required to perform CFD calculations respectively for every different turbine layouts. It can be applied with optimization algorithms for micro-siting. The application together with lazy greedy algorithm shows that the presented particle model is effective in micro-siting on complex terrain. Acknowledgements This research is supported by the National High-Tech R&D Program (863 Program) of China (No.2007AA05Z426). References [1] Mosetti G, Poloni C, Diviacco B. Optimization of wind turbine positioning in large windfarms by means of a genetic algorithm. Journal of Wind Engineering and Industrial Aerodynamics 1994;51:105e16. [2] Grady SA, Hussaini MY, Abdullah MM. Placement of wind turbines using genetic algorithms. Renewable Energy 2005;30:259e70. [3] Wan C, Wang J, Yang G, Li X, Zhang X. Optimal micro-siting of wind turbines by genetic algorithms based on improved wind and turbine models. In: Joint 48th IEEE conference on decision and control and 28th Chinese control conference, Shanghai, P.R.China; 2009. p. 5092e6. [4] Jensen NO. A note on wind generator interaction. DK-4000 Roskilde, Denmark: Tech. Rep., RisøNational Laboratory; 1993. [5] Uchida T, Ohya Y. Large-eddy simulation of turbulent airflow over complex terrain. Journal of Wind Engineering and Industrial Aerodynamics 2003;91: 219e29. [6] Uchida T, Ohya Y. Micro-siting technique for wind turbine generators by using large-eddy simulation. Journal of Wind Engineering and Industrial Aerodynamics 2008;96:2121e38. [7] Zahle F, Sørensen NN. On the influence of far-wake resolution on wind turbine flow simulations. Journal of Physics: Conference Series. 75(012042). [8] Troldborg N, Sorensen JN, Mikkelsen R. Numerical simulations of wake characteristics of a wind turbine in uniform inflow. Wind Energy 2010;13: 86e99. [9] Chen K, Song M, Zhang X, Wang J. Mass diffusive wake model for micro-siting optimization of wind farm. In: Proceedings of seventh international conference on flow dynamics, Sendai, Miyagi, Japan; 2010. p. 144e5. [10] Kasmi AE, Masson C. An extended k- ε model for turbulent flow through horizontal-axis wind turbines. Journal of Wind Engineering and Industrial Aerodynamics 2008;96:103e22. [11] Zhang C, Hou G, Wang J. A fast algorithm based on the submodular property for optimization of wind turbine positioning. Renewable Energy 2011;36: 2951e8.
Contents lists available at SciVerse ScienceDirect
Renewable Energy journal homepage: www.elsevier.com/locate/renene
Wake flow model of wind turbine using particle simulation M.X. Song, K. Chen, Z.Y. He, X. Zhang* Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 July 2011 Accepted 19 October 2011 Available online 21 November 2011
A particle model for calculating turbine wake flow during the optimization of wind farm micro-siting is presented. This model treats the wake flow as virtual particles generated by the turbine rotor. Based on the pre-calculated flow field of empty wind farm, the motions of the particles are simulated. The wake flow effect of velocity decrement is obtained from the density of the particles. On flat terrain, the proposed particle model fits the experimental data better than the previous linear model. On complex terrain, since the particle simulation is based on numerical result of flow field, the particle model can be coupled with optimization algorithm of generating turbine layout. Ó 2011 Elsevier Ltd. All rights reserved.
Keywords: Wind power Micro-siting Wake flow Computational fluid dynamics
1. Introduction As a kind of clean and renewable energy source, wind energy is of great quantity all over the world. But when converting wind power into electricity using wind turbines, wake flow is generated due to the extraction of kinetic energy and the disturbance of rotating blades. In order to improve the efficiency of a wind farm, the positions of turbines must be carefully chosen so that each turbine will not be significantly influenced by others’ wake flow. This procedure is called micro-siting. The previous studies mainly concentrated on the calculations or experiments on wake flow, and the optimization approaches on micro-siting. Among the optimization approaches, genetic algorithm (GA) is the most commonly used method. Mosetti [1], Grady [2], Wan [3] et al, have done detailed analysis on GA. Most of these approaches employ the linear model for calculating turbine wake flow, which is suggested by Jensen in 1983 [4]. The Jensen model is an approximate description of the velocity distribution within the wake flow area, with experiential constants according to experimental measurements. Jensen model treats the wake flow as a conical area, in which the velocity only depends on the distance downstream from the turbine center. This model calculates the wake flow very fast on flat terrain. It is utilized during the GA optimization to evaluate the total power output for each layout. The assumption in Jensen model is that the wind speed is uniform all over the wind farm, so that the wake flow maintains its regular shape. But for those terrains with higher slopes, although some
* Corresponding author. E-mail address: [email protected] (X. Zhang). 0960-1481/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2011.10.016
geometric modifications can be applied to Jensen model, it is still unable to reflect the actual complex wake flow precisely. Especially, if a large number of vortexes exist, wake flow could be entrained into them and lose its conical shape totally. On the other hand, detailed calculation of computational fluid dynamics (CFD) in fine grid system is an accurate way for air flow simulation on complex terrain. Uchida et al. applied large eddy simulation (LES) of CFD to calculate the air flow over complex terrain and in urban area [5,6]. Their results demonstrate that the CFD method is effective for this problem. Zahle et al. [7] and Troldborg et al. [8] respectively used Reynolds averaged numerical simulation (RANS) and LES to calculate the flow field around rotating turbine blades. In order to achieve sufficient accuracy for both large scale of complex terrain and placed turbines, grids must be fine enough to represent the disturbance of turbine rotor to the incoming air flow, and to resolve the vortex motions behind it. So these calculations require a large quantity of grids adapted to the blade surfaces, which takes long time for the iteration to converge. Besides, LES solves the unsteady equations, which takes much longer time to obtain averaged solution. Furthermore, the optimization approaches need to alter the layout repeatedly in order to search for optimal solution, the flow field calculation will be necessary to be performed for thousands of times, which definitely makes the calculation time unacceptable. Therefore, a simplified model for calculating turbine wake flow with both speed and adaptiveness to complex terrain is needed. In this paper, a particle tracking model is introduced to calculate wake flow. This model uses the concentration of a kind of virtual matter to represent the intensity of wake flow. The virtual matter has been studied in previous researches [9], where it is simulated by solving convective and diffusive equation of scalar. Unlike the
186
M.X. Song et al. / Renewable Energy 41 (2012) 185e190
1. Production: Particles are generated at a certain rate within the round area of each turbine rotor. The rate is noted as p, which indicates the number of particles generated in unit area in unit time period. 2. Convection: Local velocity for each particle is interpolated based on pre-calculated flow field. The convective displacement is added to the particle position as:
Table 1 Turbine properties. Property
Value
Hub height Rotor diameter Thrust coefficient (CT)
60 m 40 m 0.88
previous way, the particle model treats the development of wake flow as the motion of multiple virtual particles generated by the rotating turbine blades. It implements the Lagrange concept of tracking each particle’s position and velocity, and by counting the numbers, the effect of wake flow is obtained. Since the particles’ convective motions are fully controlled by their local velocities, the shape and intensity of wake flow automatically deform according to the flow field. The local characteristics of wake flow are not only determined by incoming wind or start point, but also changes along stream line. By using this method, it is only necessary to perform a pre-calculation of CFD of an empty wind farm where no turbines are placed. For each different layout, the simulation of particles runs just based on the pre-calculated flow field without new grid system adapted to turbine blades. It significantly reduces the difficulty of the CFD calculation, and save much time in optimization. 2. Calculation of wake flow 2.1. Method and steps The most significant effect of wake flow is the decrement of local velocity magnitude. In the proposed particle tracking method, each slight amount of decrement effect is packed into a virtual particle. The number of particles in a certain area will reflect the intensity of wake flow that affects the local velocity. The steps of calculation are: 1. Simulate the production, motion and disappearance of the particles. 2. Calculate the relative density of virtual particles in a statistical way. 3. Transform the density of particles into the decrement of velocity. In step 1, the flow field of the wind farm with no turbines must be obtained first. On flat terrain, it could still be assumed that the wind speed and direction is uniform all over the field. On complex terrain, CFD calculation is necessary to be performed before the particle simulation. Hundreds of time steps are simulated to let the wake flow fully develop. There are four actions in each time step:
1 0.95 0.9 0.85 0.8 exp num Jensen
0.75 0.7
3. Diffusion: A Gaussian distributed random displacement is added to each particle’s position as:
Dx ¼ surDt
0
0.5
1
1.5
2
2.5
(2)
where s is the ratio of intensity of diffusion and convection, r is the Gaussian distributed random number related to the diffusion intensity in a time step. 4. Disappearance: each particle disappears at a certain probability g, which we called the attenuation factor In step 2, the velocity after the influences of wake flow at each other turbine needs to be determined. Each turbine has a corresponding volume for particle counting. The volume locates at the center of the turbine rotor, with the same scale of the turbine rotor. The density of particles that stay in the corresponding volume approximately represents the intensity of the wake effect to the turbine. In order to obtain a stable statistical result for density of particles, it is necessary to simulate several more time steps as in step 1, meanwhile in each simulated time step, particles in each corresponding volume are counted. The relative density of particles is defined to be 1 at their generation place, where the number of particles in unit volume is:
n0 ¼
pAt p ¼ Au0 t u0
(3)
where A is the area of turbine rotor, t is a tiny time duration, u0 is the local velocity at the turbine rotor from the original precalculated flow field. So relative density in a turbine’s corresponding volume can be calculated by c ¼ n/n0, where n is the counted number of particles in the corresponding volume. In step 3, the relative density is transformed into the effect of velocity decrement by the following linear transformation:
b
1.1 1.05
(1)
where u is the local velocity at the position of the particle.
non-dimensional variable
non-dimensional variable
a
Dx ¼ uDt
1 0.95 0.9 0.85 0.8 0.75
num Jensen 0
0.5
1
r/D
x
7 5d Fig. 1. Velocity profile with inlet velocity at 11.52 m/s.
1.5 r/D
x
10d
2
2.5
M.X. Song et al. / Renewable Energy 41 (2012) 185e190
187
Fig. 2. Layouts for single wind direction.
u0 ¼ uð1 bcÞ
(4)
where u is the original velocity from pre-calculated flow field, u0 is the equivalent velocity after the influences of wake flow, c is the local relative density from step 2, b is a coefficient that needs to be determined. 2.2. Parameters In previous description, parameters p and Dt only affect the accuracy and time cost of simulation. They should be chosen with considerations of the requirement and computer capabilities. Parameter b is deduced by maintaining consistency with Jensen linear model. In Jensen model, velocity with wake flow influences is:
" u0 ¼ u 1
2a ð1 þ ax=r1 Þ2
# (5)
where a is the entrainment constant, x is the distance to the turbine center in the downstream direction, r1 is related to the radius of the turbine rotor, and a is the axial induction factor that can be calculated from
CT ¼ 4að1 aÞ
(6)
where CT is the thrust coefficient of the turbine. The further away from the turbine in downstream direction, the larger the affected area becomes, and the weaker the wake flow becomes. In present study, we take a typical turbine as in Grady’s paper, the properties are listed in Table 1. Because the relative density of particles is defined to be 1 at its generation place, we introduce x ¼ 0 into Eq. (5) to calculate the velocity right at the turbine rotor. Our parameter b is deduced to be 0.65 to fit Jensen model. Parameters s and g are determined by fit the numerical results to Taylor’s experimental data [10]. In this case, one single turbine on flat terrain with uniform velocity inlet is calculated. Velocity
Fig. 3. Layouts for multiple wind directions.
188
M.X. Song et al. / Renewable Energy 41 (2012) 185e190
Fig. 4. Distributions of relative density of wake particles.
profiles under three different inlet velocities (8.5 m/s, 9.56 m/s, 11.52 m/s) at three different distances downstream (2.5d, 6d, 7.5d, d is the diameter of the turbine rotor) are compared to the experimental data and Jensen model, and velocity profiles at 10d downstream are compared to Jensen model only. The parameters are finally chosen as in Eq. (7) to archive good fit. Two typical comparisons of velocity profiles are shown in Fig. 1.
s ¼ 0:3; g ¼ 0:005
(7)
In Fig. 1, the horizontal axis represents the non-dimensional distance to the center line of the turbine rotor in the radial direction, while the vertical axis is the non-dimensional velocity. The curve labeled num is the numerical result from the particle simulation, the dashed curve labeled Jensen is the result calculated using Jensen linear model, and the dots are the experimental results of Taylor. According to the comparisons, it is seen from Fig. 1 (a) that the result from our particle simulation fits the experimental data better. The Jensen model underestimates the wake effect around the center region, and overestimates it in the region further away from the center. Taylor’s data does not contain the ones at 10d downstream, so our result is compared only to Jensen model, as shown in Fig. 1(b). In average, the results from two methods are consistent. The comparisons under other circumstances show the same conclusion.
Jensen model and the present particle model are both applied respectively for the four layouts. For Layout 1 and 2 that are corresponding to the single wind direction case, the calculated results of wake flow by particle model are visualized in Fig. 4. The characteristics of wake flow by particle model are similar to those by Jensen model. The total power outputs for the four layouts are calculated respectively by the two wake models. Based on the velocity calculated from the wake model, the total power output (kW) of a wind farm is approximately calculated by:
P ¼
N X i¼1
0:3u3i
(8)
where N is the number of turbines in the wind farm, ui is the local velocity at the ith turbine with the wake effect. The results are listed in Table 2. For Layout 1 and 2, since the wind direction is precisely along the axis, most turbines affected by wake flow in at the center line of the wake area. According to Fig. 1, Jensen model underestimates the wake effect, which produces larger total power output for the entire wind farm. For Layout 3 and 4, as multiple wind directions exist, the averaged differences of estimation of wake effect is less significant. The total power outputs by particle model are close to those by Jensen model. In general speaking, it is concluded that the present particle model has consistency with the Jensen linear model on flat terrain.
3. Applications
3.2. Complex terrain
3.1. Flat terrain
Assuming the terrain to be flat is a simplification for calculation, but not suitable for most situations in reality. Mountains produce local acceleration of air flow, concentrating the kinetic energy. Many wind farms are built on complex terrain because of this
We apply the proposed particle model in evaluating the existed layout of wind farm, and compare the results to those calculated using Jensen model. Four typical layouts obtained using GA from the previous studies of micro-siting are chosen for evaluation. Fig. 2 shows the two layouts optimized for single wind direction, where incoming wind is along the positive direction of x-axis with speed at 12 m/s Fig. 3 shows the two layouts optimized for 36 wind directions, distributed uniformly with intervals of 10 . Inlet speeds are 12 m/s for all directions. The turbines are all with the same type as their properties listed in Table 1.
Table 2 Calculated total power output (kW).
Jensen model Particle model Relative difference
Layout 1
Layout 2
Layout 3
Layout 4
12 354 11 552 6.5%
14 312 13 292 7.1%
9268.4 9292.8 0.26%
17 241 17 486 1.4%
M.X. Song et al. / Renewable Energy 41 (2012) 185e190
0
40
40
Y
80
46 00
80
80
0
40
14 0
-500 1 44 00
0 104 11220 80
4 60 0
60 20
120
60
60
0
20
10
-500
0
80
80
120
80 0 4
60
0
80 0 1 2 0 1 68 0 00 16 0
80
0 16 0 80 12 60
0
80
4400
60
-1000
60 80 8 0
500 0
12
60
20
60
12
10
0
60
60 40 40 60 80
80
40 80
60 80 60
0
12 01 0 0
80
40 0 6 10
80
12
00 8 60
80 8 0 80
0
40
10
40 60
120
100
4 04 0
20
100
0
140
40
10
10
10
60
14 0
60 80 80
160
60 80
60
advantage. We calculated a simple case with a typical complex terrain to see the distortion of wake flow on complex terrain. The chosen typical terrain contains only a smooth hill with the height of 120 m, which is twice the hub height of the turbine. A turbine is placed 100 m away from the center of the hill in the upstream direction. As required by the particle model, CFD calculation is performed. The parameters of CFD calculation are listed in Table 3. The flow field on the center vertical cross section is shown in Fig. 5 (a). The incoming air flow is along the horizontal axis from left to right. The filled contour represents the magnitude of velocity, while the curves with arrows are streamlines. The local acceleration is obviously noticed around the top of the hill, where the velocity magnitude reaches 16 m/s. The air flow is separated from the surface of the hill after it crosses the top, and a vortex with the scale similar to the hill itself is generated behind the hill. The particle model is then applied to calculate the wake flow. The relative density of particles on the same cross section is shown in Fig. 5 (b). The filled contour indicates the intensity of the wake flow. The center of the wake region is at the same height of the turbine hub when it is generated. As it develops, the air flow is extruded by the rising ground, so the wake flow becomes closer to the ground surface. After it crosses the top of the hill, due to the existence of the vortex, the wake flow would travel a long distance before it drops to the original height. Meanwhile, the shape of the wake flow calculated using Jensen model is also plotted in Fig. 5 (b). Jensen model would totally ignore the wake flow above the top of the hill, and obtain wake flow effect behind the hill where there should be no wake flow. Using Jensen model on complex terrain would bring additional errors to the evaluation of layouts when calculating the power output of turbines. For a fixed layout, when the terrain is altered from flat one to complex one, due to the local acceleration and deceleration of air flow and the distorted wake flow, the power output of each turbine could either increase or decrease, depending on how the air flow is changed by terrain geometry and where the turbine is placed. So the total power output usually does not change much from when it
4400
1800 m 600 m 400 m 12 m/s 90 30 30
1000
0
Value
Domain length (streamwise) Domain width Domain height Inlet velocity Grid numbers
60
10
Parameter
60
60
Table 3 Parameters of CFD calculation.
189
40
-1000 500
1000
X Fig. 6. Contour of relative altitude of the complex terrain.
is on flat terrain. For example, if we apply Layout 2 in Fig. 2 (b) to the terrain in Fig. 6, the total power output calculated using particle model is 13131 kW (the original on flat terrain is 13292 kW). The difference is only 1.2%. But for the turbines whose power outputs are decreased, since the wind field contains regions with velocities higher than the inlet one, it is possible to relocate them to better positions so that the total power output could be additionally increased. Therefore, the optimized layout from optimization algorithms on flat terrain is probably not optimal on complex terrain. It is necessary to couple the present particle model with optimization algorithm to generate new layout. A lazy greedy optimization algorithm proposed by Zhang et al. [11] is employed to test the proposed particle model on complex terrain. This algorithm produces similar or even better layout than the one from GA, and is 600 times faster than GA in the specific case of their studies. The greedy algorithm starts from an empty layout. The steps are: 1. Construct a list containing the possible positions for turbines. Each element corresponding to a position has an evaluation
Fig. 5. Numerical results of a typical complex terrain.
190
M.X. Song et al. / Renewable Energy 41 (2012) 185e190
60
0
60
120
60
40 40
80
60
14
40
40
0
80
-500
0
0
12 23 0
3 20
26 100
14
-1000 -1000
80
10
9
100
80
8060 12 0 10 0
15 80
2
80 60
0
60
40
80
20
60
80 60
0 1216 0 0
80
80
60
7
29
100
16
1 6
80
19
40
18
60
100
13 100
60
100
0 40 6
21 10200 1
14
0
1 00
25 60 40
Y
14 120 100
24
27
10
12
5
20
0 60
60
80
2880
12 60 10 40
-500
4
40
80
17
60
140 80
500
4. Conclusions A model for calculating turbine wake flow using particle simulation is proposed in this paper. The model has the following advantages:
22
0
500
30
60
10
11
80
8 120
40
40
40 0 6
40
60 100
0
60
0 100 60 8
10
80 00
1000
80
1000
X Fig. 7. Optimal layout on complex terrain.
value, representing the contribution to the total power output if a turbine is placed there. 2. Test all the elements, calculate their evaluation values. Keep the list in descending order according to the evaluation value. 3. Place a turbine at the corresponding position of the top element in the list, and remove the element from the list. 4. If the turbine number reaches the given limit, then the optimal solution is obtained. Otherwise, go to Step 2. In the ordinary greedy algorithm, a large number of remained elements have to be tested in Step 2. But in the lazy greedy algorithm, as proved in Zhang’s paper [11], the problem of wind turbine positioning has the submodular property. As a result, if it is only needed to find the element with the maximum evaluation value, the list does not need to be fully updated in Step 2. This advantage greatly reduces the number of elements that are to be tested, therefore greatly reduce the number of times of running the wake flow model. In order to compare the result to the one on flat terrain, turbine number is fixed at 30, and the grid numbers are 10 10. The obtained optimal layout is shown in Fig. 7. The total power output of the layout in Fig. 7 is 17,728 kW, increased by 35% comparing to Layout 2, whose power output is 13,131 kW. The layout is totally different from the optimal layout on flat terrain.
1. On flat terrain, the result from the particle model is closer to the experimental data than the Jensen linear model. In application of evaluating power outputs of layouts, the particle model produces consistent result to the Jensen linear model. 2. On complex terrain, the model reflect the characteristics of turbine wake flow more accurately than the previous linear model. 3. The model calculates wake flow effect based on pre-calculated flow field of empty wind farm. It is not required to perform CFD calculations respectively for every different turbine layouts. It can be applied with optimization algorithms for micro-siting. The application together with lazy greedy algorithm shows that the presented particle model is effective in micro-siting on complex terrain. Acknowledgements This research is supported by the National High-Tech R&D Program (863 Program) of China (No.2007AA05Z426). References [1] Mosetti G, Poloni C, Diviacco B. Optimization of wind turbine positioning in large windfarms by means of a genetic algorithm. Journal of Wind Engineering and Industrial Aerodynamics 1994;51:105e16. [2] Grady SA, Hussaini MY, Abdullah MM. Placement of wind turbines using genetic algorithms. Renewable Energy 2005;30:259e70. [3] Wan C, Wang J, Yang G, Li X, Zhang X. Optimal micro-siting of wind turbines by genetic algorithms based on improved wind and turbine models. In: Joint 48th IEEE conference on decision and control and 28th Chinese control conference, Shanghai, P.R.China; 2009. p. 5092e6. [4] Jensen NO. A note on wind generator interaction. DK-4000 Roskilde, Denmark: Tech. Rep., RisøNational Laboratory; 1993. [5] Uchida T, Ohya Y. Large-eddy simulation of turbulent airflow over complex terrain. Journal of Wind Engineering and Industrial Aerodynamics 2003;91: 219e29. [6] Uchida T, Ohya Y. Micro-siting technique for wind turbine generators by using large-eddy simulation. Journal of Wind Engineering and Industrial Aerodynamics 2008;96:2121e38. [7] Zahle F, Sørensen NN. On the influence of far-wake resolution on wind turbine flow simulations. Journal of Physics: Conference Series. 75(012042). [8] Troldborg N, Sorensen JN, Mikkelsen R. Numerical simulations of wake characteristics of a wind turbine in uniform inflow. Wind Energy 2010;13: 86e99. [9] Chen K, Song M, Zhang X, Wang J. Mass diffusive wake model for micro-siting optimization of wind farm. In: Proceedings of seventh international conference on flow dynamics, Sendai, Miyagi, Japan; 2010. p. 144e5. [10] Kasmi AE, Masson C. An extended k- ε model for turbulent flow through horizontal-axis wind turbines. Journal of Wind Engineering and Industrial Aerodynamics 2008;96:103e22. [11] Zhang C, Hou G, Wang J. A fast algorithm based on the submodular property for optimization of wind turbine positioning. Renewable Energy 2011;36: 2951e8.