Evaluation of performances of analytical wind turbine wake models using wind farm data

Due to limited suitable area and economical reasons such as lower transportation, installation and maintenance costs, wind turbines are grouped into wind farms with a limited areas. Wind turbines in a wind farm produce less power than their standalone power productions. The reason of less power production is wake effect. Downwind turbines in the wake effect of upwind turbines experience lower wind speed and higher turbulence intensity. Lower wind speed and higher turbulence intensity cause less power production and shorter turbine lifetime, respectively. Power loss due to wake effect can be in the order of 20 %. Loss of power production are based on many reasons such as local wind regime, topographical characteristics and turbine aerodynamics. The best wind farm layout should be optimized to maximize the power production of wind farm. For wind farm layout optimization, two different wake models, analytical and computational, have been developed. Due to computational efficiency, analytical wake models are preferred for wind farm layout optimization. However, these analytical wake models still must be validated with measurement data from wind-tunnel experiments or wind farms to ensure that their predictions are correlated well with measurement data. There are many analytical wind turbine wake models used in the wind power industry such as Jensen model, Frandsen model, Larsen model, Ishihara model and etc. In the first step of this thesis, basic four different analytical wake models, standard Jensen, new Jensen, Frandsen and Larsen models, were used to obtain their characteristics such as initial wake diameter prediction, its development with distance and recovery speed of wake velocity deficit with distance using the data of small-scale wind turbine and wind measurement mast from Meteorological Park at Istanbul Technical University. In the first step of this thesis, these models were also used to obtain three different wake effect condition (full wake, partial wake and no wake condition) for varied wind direction in the case of hypothetical downwind wind turbine in wake of upwind turbine. In the second step of thesis, both these basic analytical models and analytical models assisted with WindSim software based on Computational Fluid Dynamics (CFD) technology were tested to evaluate their performances in a complex terrain. WindSim include three different wake models which are standart Jensen, Larsen and Ishihara models. In this thesis, only standard Jensen and Larsen models were selected as analytical model assisted by CFD. For this performance evaluation, wind data from Çatalca Wind Farm on complex terrain were used. Secondstep is the primary objective of this thesis. In this step five different single wake cases were tested. Standard Jensen analytical wake model is one of the oldest analytical wake models. This model assumes that wake behind the turbine expands linearly with distance and velocity deficit is only function of downwind distance of upwind turbine and has uniform profile. Standard Jensen model neglects contribution from tip vortex and assumes fully turbulent wake, and thus, this model is not usable for near wake predictions. Equation of wake velocity is as follows U_w=U_inf*(1-2*a/(1+k_w*(x/r_1))^2) Here, x is downwind distance, U_inf unperturbed free stream velocity, is characteristic downwind rotor radius that represents expanding wake diameter immediately behind the turbine, and is axial flow induction factor. Equation of axial flow induction factor, depends on thrust coefficient C_T , and characteristic downwind rotor radius are, respectively, as follows a=(1-sqrt(1-C_T))/2 r_1=r_r*sqrt((1-a)*(1-2*a)) Here, r_r is rotor radius of turbine. Wake expands linearly with a coefficient called wake decay constant, k_w. Wake decay constant and the spread wake radius are, respectively, as follows k_w=1/2*ln(z_h/z_0) r_w=k_w*x+r_r Here, z is wind turbine hub height and z_w is roughness length of interested terrain. New Jensen analytical wake model was developed based on standard Jensen model. This model uses cosine-shape wake velocity profile instead of uniform profile in standard Jensen model. Also, this model uses a wake decay parameter, that depends on both atmospheric turbulence and turbine-generated turbulence, instead of a wake decay constant, that depends only on atmospheric turbulence, in standard Jensen model. In new Jensen model, wake velocity profile is as follows U_w=(U_inf-U^star)*cos(pi*r/r_x+pi)+U^star Here, U^star is wake velocity profile in standard Jensen model, U_inf is unperturbed free stream velocity. r is radial distance from wake centerline. This model represent a new effective turbulence model, that includes contributions of both atmospheric (ambient) and turbine-generated turbulences, and equation of that model is as follows I_wake=K_n*C_T/(x/D)+I_0 Herre constant K_n is assumed to be 0.4. D is rotor diameter and I_0 is ambient turbulent intensity. Equation of wake decay parameter is as follows k_wake=k_0*I_wake/I_0 Here, as subscript 'wake' indicates wake flow area, subscript '0' indicates free stream area. The parameters that have subscript '0', is estimated from standard Jensen model. Finally, estimated wake decay parameter is put into standard Jensen model instead of original wake decay constant. Third model used in this thesis is Frandsen model. Frandsen model was developed to analyze wake behavior for entire wind farm rather than wake behavior in individual turbines. In this model, there are three different wake regimes. First regime assumes no mutual interaction of single or multiple wakes with neighbor wakes. Second regime assumes two neighbor wakes interacts and wake expansion is limited only in vertical direction. Third regime assumes wake flow is balanced with planetary boundary layer. This last regime is seen for infinitely large wind farms. This model has uniform wake velocity profile. For single wake case, wake velocity is as follows U_w=1/2*U_inf+(or)-sqrt(1/4-1/2*U_inf*A_r/A_w*C_T Here, A_r is swept area of rotor and A_w is area of wake. Wake diameter is D_w=D_r*(beta^(k/2) + alfa^star*x)^1/k Here, wake expansion parameter beta, is as follows beta=(1+sqrt(1-C_T))/2*sqrt(1-C_T)=(D_eff/D)^2 where coefficients k and alfa^star are, respectively, 2 and order of 10*k_w. k_w is the wake decay constant used in standard Jensen model. Last analytical wake model used in the thesis is Larsen model. This model is based on Prandtl's turbulent boundary layer equations. A self-similar wind profile is assumed and Prandtl's mixing length theory is used to obtain closed-form solution. Flow is axisymmetric due to assumptions of incompressible and steady flow and no-slip condition. Both first order and second order approximate solutions were represented by Larsen. Second order approximate solution can solve double dips in the near wake wind profile. Equation of wake deficit (Delta(U))_1 and wake radius r_w are,respectively, represented below (Delta(U))_1=-U_inf/9*(C_T*A*(x+x_0)^-2)^(1/3)*(r^(3/2)*(3*c_1^(2)*C_T*A*(x+x_0))^(-1/2)-(35/(2*pi))^(2/10)*(3*c_1^2)^1/5)^2 r_w=(35/(2*pi))^(1/5)*(C_T*A*(x+x_0))^1/3 where c_1 is related to Prandtl's mixing length and x_0 is position of rotor according to applied coordinate system. These constants are, respectively, represented below c_1=(D_eff/2)^(5/2)*(105/(2*pi)^(-1/2)*(C_T*A*x_0)^5/6 x_0=(9.5*D)/((2*R_9.5/D_eff)^3-1) Here, D_eff is effective rotor radius and its formula is as follows D_eff=D*sqrt((1+sqrt(1-C_T))/(2*sqrt(1-C_T)) Wake radius of upwind turbine at distance of 9.5 rotor diameter R_9.5 is estimated in following equation R_9.5=0.5*(R_nb+min(z_h, R_nb)) Here, R_nb is R_nb=max(1.08*D,1.08*D+21.7*D*(I_a-0.05)) Here, I_a represents ambient turbulent intensity. In this thesis analytical wind turbine models were applied with aid of WindSim software based on CFD technology. WindSim is a modern and developed wind farm design tool supported with CFD technology. In simulation of wind field in WindSim, Reynolds-averaged Navier-Stokes equations is solved with finite volume method using Phoenics solver that is a general purpose CFD code. RANS equations are given below : dU_i/dx_i=0 U_j*dU_i/dx_j=-1/rho*dp/dx_i+d(mu*(dU_i/dx_j+dU_j/dx_i)-ReynoldAvg(u_i*u_j))/dx_j Here, first equation and second equation represent continuity equation and momentum equation, respectively. U, x, p, rho, mu represent velocity, pressure, density and kinematic viscosity. Subscripts i and j represent unit vectors. The term ReynoldAvg(u_i*u_j) represent Reynolds stresses and its formula is given below. ReynoldAvg(u_i*u_j)=mu_T*(dU_i/dx_j+dU_j/dx_i)+2/3*delta_ij*k Here, mu_T , k and delta represent turbulent viscosity, turbulent kinetic energy and Kronecker delta function. WindSim include 6 different modules that are terrain, wind field, objects, results, wind resources and energy. Terrain module allows to establish the numerical model based on height and roughness data. Wind field module provides calculation of numerical wind fields. In the objects module wind turbines and climatology data are placed. Results module provides to analyze wind fields. Wind Resources module allows to couple the numerical wind fields with climatology data by statistical means to provide wind resource map . Energy module calculates annual energy production of a wind farm, including wake losses. For results in the first step of the thesis which is to represent characteristics of five different basic analytical wind turbine wake models, Larsen and Frandsen models predicted larger wake diameter than other two wake models initially, but the largest wake diameter prediction and the largest wake diameter increase with distance were predicted by Larsen model. After a specific downwind distance, both Jensen models predicted larger wake diameter than Frandsen model. Although, recovery of wake was predicted faster by new Jensen model, lowest normalized wake velocity deficit for all downwind distance was predicted by Larsen model. In the case of model predicitions of three different wake effect conditions for varied wind direction, Larsen model predicted only full wake till a specific downwind distane beyond far wake, and also predicted largest full wake width for all downwind distances. New Jensen model predicted larger full and partial wake width for all downwind distance than standard Jensen model. Although Frandsen model predicted larger full and partial wake width tha both Jensen models, both Jensen model predicted larger full and partial wake diameter than Frandsen model after some downwind distance. For results in the second step which is primary objective of thesis and evaluations of performances of both basic and CFD assited analytical wind turbine wake models in complex terrain using measurement data, some measured normalized downwind turbine wake velocity data is greater than 1 due to complex terrain. This situation causes complications for validation of wake models especially basic analytical wake models. As a result of all tests for basic analytical wake models , no clear-cut result which model has the best accuracy to predict normalized wake velocity. All model results were close each other. For consideration of every single wake tests separately, while models have poor performance for some single wake cases, they have good performances for others. For consideration of cumulative performance of each model, new Jensen model, standard Jensen model, Frandsen model and Larsen model had mean absolute error of 9.7 % , 9.9 %, 9.4 % and 9.1 %, respectively. Although Frandsen model had second best performance. If Analytical wake models assisted by CFD, that are standard Jensen and Larsen models are evaluated separately with both basic standard Jensen and Larsen models and other basic analytical wake models (new Jensen and Frandsen models) for each single wake case, these models assisted by CFD generally exhibited more poor performance. For consideration of cumulative performance of each model assisted by CFD, standard Jensen and Larsn models assisted by CFD had mean absolute error of 14.3 % and 12.7 %, respectively. Although CFD has better accuracy to solve wind field in a complex terrain, the reason of poor performance of analytical wake models assisted by CFD by comparing with basic analytical wake models in the complex terrain may be diffrences between wind direction prediction of analytical wake model assited by CFD and measurement data of wind direction for downwind turbine. In the application of analytical wake models assisted by CFD for each single wake case, wind measurement data of upwind turbine was used for wind input data for upwind turbine inserted in WindSim. These models predicted wind directions close to input data given for upwind turbine. Therefore differences of 10-20 degree in wind direction occured between analytical wake models asssited by CFD and measurement for downwind turbine. This means that downwind turbine experiences lower wind speed due to larger wake effect, especially for full wake condition. For basic analytical wake models, measurement data of wind direction and measurement data of both wind direction and wind speed were used for downwind turbine and upwind turbine, respectively.