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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.

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 [

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 [

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.

CARDAAV, the numerical tool used in this analysis, is based on an improved version of the “double-multiple streamtube” (DMS) model [

The two actuator disks model.

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

For the upwind interference factor

with

A similar set of equations is derived for the downwind interference factor

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 [

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

Comparison of CARDAAV results with experimental and other numerical predictions. (adapted from Paraschivoiu 2002) [

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)

On the other hand, for the pitch angle the following analytical expression was considered:

The genetic algorithm evolution strategy optimization package, GENIAL v1.1 [

As mentioned above, the coefficients

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 (

As the name indicates, the subroutine PITCH uses relation (

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

Characteristics of the rotor used in the study.

Parameter | Value |
---|---|

Rotor diameter | 6.0 m |

Rotor height | 6.0 m |

Blade length | 6.0 m |

Blade chord (constant) | 0.2 m |

Blade airfoil | NACA 0015 |

Number of blades | 2 |

Rotor ground clearance | 3 m |

The performance (power, power coefficient, Figures

Power versus wind speed for the 7 kW VAWT with fixed and variable pitch blades.

Power coefficient versus TSR for the 7 kW VAWT with fixed and variable pitch blades.

As can be observed on the graph of Figure

A good performance in winds above 11 m/s was also obtained with the variable pitch given by relation (

Moreover, the values that were used for the

At 125 rpm, the maximum of the power coefficient is attained for TSR

For all the optimizations that were carried out in the present study, the parameters which control the functioning of the three main modules of GENIAL were set to the values given in Table

Parameter settings used in GENIAL’s modules.

Evolution Module | |

Function | User function |

Worst fitness | 10E20 |

Problem type | Minimization |

Number of variables | 3 |

Number of evolutions | 1500 |

Population Module | |

Population size | 100 |

Reproduction model | Steady state without duplicates 10 |

Parent selection method | Stochastic tournament 0.6 |

Replacement method | Exponential ranking 0.3 |

Reproduction Module | |

Number of operators | 2 |

Uniform arithmetical crossover | 35.0 1.0 0.5 |

Uniform creep | 5.0 1.0 0.001 |

At the end of the optimization, for the optimization variables

Optimal variation of the blade pitch angle.

As one can notice by comparing the graphs in Figures

Power coefficient versus TSR for the 7 kW VAWT with fixed and optimized variable pitch.

However, the benefits of the optimized variation of the pitch do not extend far beyond a relatively narrow domain around this “point” of optimization. If the same pitch variation is used in off-design conditions, at other wind speeds (other TSRs), the turbine efficiency becomes lower than the one of the zero pitch version (Figure

Power versus wind speed for the 7 kW VAWT with fixed and optimized variable pitch.

Power versus wind speed for the 7 kW VAWT with fixed and optimized variable pitch (detail).

Therefore, the optimization of the pitch variation was also carried out at other wind speeds up to the rated wind. Indeed, since high winds are relatively rare, a loss in the turbine efficiency beyond the rated wind speed is not very penalizing in terms of annual energy production. Instead, by enhancing the turbine efficiency and its power output in the low-speed range, an increase of almost 30% in the annual energy production was obtained for the analyzed prototype. This is certainly an interesting gain, which might fully justify the supplementary costs related to the implementation of a variable pitch system. A Rayleigh distribution for the wind probability density and a 6 m/s mean wind speed were considered to calculate the turbine’s annual energy production.

An optimization tool, combining the CARDAAV code with a genetic algorithm-based optimizer, was developed and used to optimally define the variation of the blade pitch angle of an H-Darrieus (straight bladed) 7 kW wind turbine. The optimized pitch enhanced the turbine performance at the design point and in a relatively narrow wind speed domain around it. In “off-design” conditions the optimized pitch was detrimental to the turbine performance, thus imposing optimizations at other wind speeds below the rated wind. With a pitch variation optimized throughout the low wind domain, an increase of almost 30% in the annual energy production of the turbine could be obtained. This could justify the supplementary expenses related to the implementation of the variable pitch system.

Blade section drag coefficient

Blade section lift coefficient

Blade section normal force coefficient

Power coefficient

Blade section tangential force coefficient

Blade chord, m

Objective function

Number of blades

Rotor power, kW

Interference factor

Local induced velocity, m/s

Free stream wind velocity, m/s

Local tip-speed ratio,

Local angle of attack, deg

Angle between blade normal and equatorial plane, deg

Non-dimensional radius,

Azimuthal angle, deg

Blade pitch angle, rad

Optimization variables