| H-Darrieus Wind Turbine with Blade Pitch Control Abstract A procedure for computing the optimal variation of the blades ' pitch angle of an H-Darrieus wind turbine that maximizes its torque at given operational conditions is proposed and presented along with the results obtained on a 7 kW prototype . The CARDAAV code , based on the “ Double-Multiple Streamtube ” model developed by the first author , is used to determine the performances of the straight-bladed vertical axis wind turbine . This was coupled with a genetic algorithm optimizer . The azimuthal variation of the blades ' pitch angle is modeled with an analytical function whose coefficients are used as variables in the optimization process . Two types of variations were considered for the pitch angle : a simple sinusoidal one and one which is more general , relating closely the blades ' pitch to the local flow conditions along their circular path . A gain of almost 30 % in the annual energy production was obtained with the polynomial optimal pitch control . 1. Introduction Following the 1973 energy crisis , large-scale research and development programs were initiated , directed toward finding replacement solutions to the limited fossil fuel reserves . Wind energy was given , along with photovoltaic , solar , hydroelectric , biomass , and other resources , particular attention as a renewable and environmentally friendly energy alternative . Its technological progress has been spectacular , especially in the last ten years and , due to its steady growth in competitiveness , wind power developed into a mainstream energy source in many countries worldwide . At the global scale , over 74000 MW of wind power are already installed , and current estimates indicate that by 2030 wind energy could cover as much as 29 % of world’s electricity needs . In the wind power domain two main technologies were considered as having the necessary potential for a viable development : the Horizontal Axis Wind Turbine ( HAWT ) and the Darrieus-type ( lift-based ) Vertical Axis Wind Turbine ( VAWT ) . A number of features have made HAWT to be preferred and become the dominant design type , especially in the utility-scale ( large and very large turbines ) segment . But , in certain conditions ( sites with highly turbulent wind like in the mountains or in urban environment ) , VAWTs seem to offer a better solution for the wind energy harnessing . If , through further and well-targeted research , increased attention is paid to the known VAWT drawbacks ( a somewhat less overall efficiency than the one of an HAWT , difficult/impossible self-starting , lower output due to operation closer to the ground , higher level of vibration caused by the inherent torque ripple and dynamic stall of the blades ) , at least in the “ small wind ” domain the VAWT design might become a major player . Among the most important problems that are now under study in the VAWT technology , the “ variable pitch ” for the H-Darrieus turbines is regarded as a promising solution for the alleviation of the negative effects of the blades dynamic stall ( efficiency loss , vibration ) , improvement of the rotor’s self starting qualities , and torque ripple smoothing [ 1–3 ] . At École Polytechnique de Montreal , Canada , in the wind energy research the major effort is devoted toward the development and improvement of the performance prediction of VAWTs [ 4 ] . The variable pitch is also included in the current research subjects in this domain , and results were already published [ 8 ] , obtained from the analysis that was carried out to determine if the performance of a VAWT , in terms of the power output , could be improved by simulating the operation of the blade just below stall . The present study aimed at determining if a more general cyclical pitch variation can be determined , so as to maximize the performance of an H-Darrieus ( straight bladed ) vertical axis wind turbine at given operation conditions . In order to perform this investigation , an optimization package was set up to serve in the determination of the optimal variation of the blade’s pitch angle for a small two-bladed VAWT . The paper presents first the main components of the numerical tool that was developed . Then , the results of an optimization case are discussed . Finally , several conclusions are formulated at the end of the paper . 2. The Optmization Tool Since the local flow parameters on the blades vary along their circular path and differ quite significantly between the upwind and the downwind parts of the rotor , an optimization procedure had to be employed to determine the best law of variation of the blades ’ pitch angle . In the present study a tool for numerical optimization was set up by coupling the code CARDAAV , which computes the flow through and the performances of a VAWT , to an optimizer based on the genetic algorithm method . These ( main ) components of the optimization package are briefly presented in the following sections , along with the objective function , its variables , and the constraints that were imposed on their value during the optimization process . 2.1 . The CARDAAV Code CARDAAV , the numerical tool used in this analysis , is based on an improved version of the “ double-multiple streamtube ” ( DMS ) model [ 4 ] . This model considers a partition of the rotor in streamtubes and treats each of the two blade elements defined by a given streamtube as an actuator disk . Figure 1 illustrates such a streamtube and shows the values of the velocity of the flow at a number of key stations along it . Disk 1 represents the upwind blade element while disk 2 represents the downwind blade element . The actuator disk theory is based on the momentum conservation ; therefore , the velocities of the wind must be known in order to compute the force acting on the disks . The different values of the velocity ( see notations in Figure 1 and relations ( 1 ) depend on the incoming ( “ undisturbed ” ) wind velocity and on the interference factors 𝐮 and 𝐮 ( 1 ) To determine the interference factors , a second set of equations is used . Those equations are derived from the blade element theory [ 4 ] , which equates , in each streamtube , the normal forces acting on the upwind and downwind blade elements to the forces acting upon the actuator disks . To compute the normal and tangential forces , the blade element theory is applied , and the lift ( 𝐶 𝑙 ) and drag ( 𝐶 𝑑 ) coefficients , obtained from the airfoil data , are used . For the upwind interference factor 𝑢 , the following expression , relating it to the azimuthal angle 𝜃 , is obtained : 𝑢 ( 𝜃 ) = 𝐾 𝐾( 2 ) with 𝐾 = 8 𝜋 𝑟 ( 3 ) A similar set of equations is derived for the downwind interference factor 𝑢 ′ . An interference factor equal to 1 is assumed at the beginning of the iterative process . Once the force given by the blade element theory equates the one given by the actuator disk theory , the convergence is achieved and the upwind and downwind velocities are obtained . Then the torque and the mechanical power are computed . CARDAAV has the capability to analyze several predefined or user-defined rotor shapes with straight or curved blades ( parabola , catenary , ideal and modified troposkien , and Sandia shape ) . The code requires three main sets of input data , giving the geometry definition of the wind turbine ( diameter , height , blade section airfoil , blade shape , etc. ) , the operational conditions ( wind velocity , rotational speed , atmospheric conditions ) and the main control parameters ( convergence criterion , computation of the secondary effects , and the effect of dynamic stall ) . The software includes several dynamic stall semiempirical models : Gormont [ 5 ] and its variations ( Strickland , Paraschivoiu , and Berg ) and the one based on the indicial method [ 4 ] . Dynamic stall results in increased peak aerodynamic torque and affects the structural fatigue of the Darrieus turbine . This effect significantly impacts the drive-train generator sizing and system reliability . The dynamic stall used in this study was the Berg version of the Gormont model , because it was found out to be the best correlated with the experimental studies carried out on similar rotor configurations as those used in the present investigation . CARDAAV is also able to account for the so-called “ secondary effects , ” such as those due to the rotating central tower , struts , and spoilers . CARDAAV has made it possible to design , analyze , and build more efficiently and at lower costs wind energy systems such as the Darrieus-type VAWT . The code is used to determine , at specified operational conditions , aero-dynamic forces and power output of VAWTs of any blade geometry . Wind speed can vary with height above ground according to a power law . The program output consists of the local-induced velocities , the local Reynolds numbers and angle of attack , the blade loads , and the azimuthal torque and power coefficient data . Each of these is calculated separately for the upwind and downwind halves of the rotor . The numerical models used by the program have been validated for different Darrieus-type VAWTs , through comparison with experimental data obtained from laboratory tests ( wind or water tunnels ) or from field tests , thus making CARDAAV a very attractive and efficient design and analysis tool . In Figure 2 the power output of the SANDIA 17 m wind turbine computed with CARDAAV is compared with experiments and results provided by other numerical codes . 𝐴 𝑀 is an empirical constant used to correct 𝐶 𝑙 and 𝐶 𝑑 for dynamic stall effects [ 6 ] . 2.2 . The Optimizer To search for the best pitch variation law , an optimization strategy was adopted , namely , one that uses a genetic algorithm ( GA ) method . At the beginning of the optimization process , a genetic algorithm randomly selects an initial “ population ” composed of “ individuals ” , which are solutions of the analyzed problem computed for particular , randomly selected , values of the optimization variables . Three operations are typically performed by the genetic algorithms on the analyzed “ population ” : “ selection ” ( choice of the “ individuals ” for the next generation , according to a “ survival of the fittest ” criterion ) , “ crossover ” ( operation which allows information exchange between the “ individuals ” by swapping parts of the parameter vector in an attempt to get better “ individuals ” ) and “ mutation ” ( operation which introduces new or prematurely lost information in the form of random changes applied to randomly chosen vector components ) . Like in any optimization study , an “ objective function ” had to be defined . In this case the inverse of the rotor power , for given conditions of operation ( wind speed , rotational speed ) 1 𝐹 = 𝑃 ( 4 ) was used as optimization function 𝐹 . On the other hand , for the pitch angle the following analytical expression was considered : 𝜏 = 𝑥 ( 5 ) For different values given to the optimization variables 𝑥 , the variation of the blade pitch angle 𝜏 with the angle of azimuth 𝜃 will be different . Since the local angle of attack α of a blade element 𝛼 = s i n ( 6 ) contains the pitch angle , the later influences the aerodynamic characteristics , the torque , and ultimately the power output of the rotor . Hence , through relations ( 5 ) and ( 6 ) an indirect link is established between the objective function ( 4 ) and the optimization variables 𝑥 , which control the variation of 𝜏 . The genetic algorithm evolution strategy optimization package , GENIAL v1.1 [ 7 ] , was employed to minimize the objective function . This code includes three main modules , which perform the above mentioned operations . A number of parameters are available in each module to control its functioning during the optimum search process . As mentioned above , the coefficients 𝑥 of the pitch variation function ( 5 ) were used as optimization variables . Their values were subjected to certain constraints , to avoid any possible mathematical invalidity and to keep the pitch angle within reasonable or practical ( technically feasible ) limits . 2.3 . The Optimization Package To set up the optimization package , including GENIAL and CARDAAV as principal components , a main program ( MAIN ) and a new subroutine ( PITCH ) had to be coded . When the program is launched , MAIN reads some of the parameters that control the optimization process , namely , those that are frequently changed ( size of the “ population”—number of “ individuals ” , number of evaluations , constraints to be set on the optimization variables ) . These parameters have to be provided through the keyboard when a new optimization is initiated . Then , MAIN calls the optimizer ( GENIAL ) , which takes control and carries through the optimization process . For each combination of the optimization variables , defining a distinct ‘ individual ” , GENIAL calls ( using a “ system function ” ) PITCH then CARDAAV , which performs the analysis of the VAWT for that specific variation of the blades ’ pitch angle . With the turbine power , calculated by CARDAAV , the objective function ( 4 ) is determined and , based on it , the “ fitness function ” which characterizes that “ individual ” is obtained . Depending on its “ fitness ” , an “ individual ” can be maintained or eliminated from the “ population ” during the optimization process . Selection , crossover , and mutation are used to advance the “ population ” from one optimization cycle to the next until the optimum or the predefined limit of evolutions is reached . At the end of the optimization , MAIN outputs the optimal values of the 𝑥 variables ( as found through optimization ) , which define the best variation law for the pitch angle . Corresponding to this , the performance characteristics of the turbine are computed and stored in a file for postprocessing purposes . As the name indicates , the subroutine PITCH uses relation ( 5 ) to calculate the pitch angle over the entire circular trajectory of the blades , based on the values of 𝑥 received from GENIAL . It stores the azimuthal and the pitch angles in a file where CARDAAV seeks this information before performing each new analysis . 3. Results This study was carried out on an H-Darrieus VAWT , having two constant-chord blades with an NACA 0015 airfoil cross section . It is a small , 7 kW rated power prototype , its rotor having the ( main ) geometrical characteristics given in Table 1. The performance ( power , power coefficient , Figures 3 and 4 ) of this turbine was computed with CARDAAV in several situations : without taking into account the influence of the dynamic stall , by applying the dynamic stall correction , with a blade pitch that had a sinusoidal variation and with a blade pitch that varied according to relation ( 5 ) in which : 𝑥 . These calculations , as well as all the other that are presented in this paper , have been performed for a constant rotational speed of 125 rpm and at one or several different wind speed values . |