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Vibration control of wind turbine blade with minimum control cost is investigated. Realization of minimum control cost is based on pole placement with minimum-order observer (PPMO). The blade analysis employs a novel compromise method between the 2D airfoil analysis method and the 3D coupled blade body analysis method based on data fitting. It not only ensures certain accuracy, but also greatly improves the speed of calculation. The Wilson method, developed on the basis of the blade momentum theory, is adopted to optimize the structural parameters of the blade, with all parameters fitted as general model Sin6 (Sum of Sine) fitting curves. Also the aerodynamic coefficients based on data obtained by Xfoil software are fitted. Pole placement technology based on minimum-order observer is applied to control unstable vibrations of vertical bending and lateral bending with minimum control cost characterized by the energy consumption of the controller. The pole placement technology is a novel pole assignment technique based on self-poles derived from constant stable eigenvalues, which can effectively avoid the mismatch problems caused by pole selection. The superiority of PPMO can be apparently demonstrated by comparison of linear quadratic regulator (LQR). Analytical proof of the control accuracy and feasibility analysis of the physical realization of the PPMO algorithm are also investigated by experimental platform of hardware-in-the-loop simulation.

The flutter analyses of wind turbine blades usually include two cases: typical blade section analysis and the three-dimensional coupled blade body analysis. The former is widely used in quick study of flap/lag vibrations or bending/twist vibrations of single cross-section with a wide range of structural and external parameters [

Structural failures observed in recent years of large wind turbine blades have been mostly attributed to severe vibrations in flapwise and edgewise directions [

Since the cross-sections of different sizes experience different loads, the combined effects of different loads have far surpassed those of the same loads of the typical sections of the same sizes described in related references [

In present work, a novel compromise method (NCM) between the 2D typical section analysis method and the 3D coupled blade body analysis method is investigated to study vertical and lateral vibrations of blade tip. Then the analysis accuracy might be between the accuracies of the two structural model analysis methods due to the approximate compromise behavior. The NCM not only preserves the basic characteristics of typical section model, but also extends the integral behavior of blade body along the spanwise direction but avoids the decoupling problem of partial differential equations. More importantly, the vibration control of blade model based on NCM is more efficient and easier to be implemented in controller hardware. In addition, realization of minimum control cost is based on pole placement with minimum-order observer (PPMO), which can be depicted as pole placement technology based on minimum-order observer. The PPMO is applied to control unstable vibrations of vertical/lateral directions based on self-poles derived from constant stable eigenvalue analysis, which is exactly the advantage of the method NCM. At the same time, the minimum-order observer is applied in PPMO to estimate the unmeasurable variables.

It should be stated that the emphasis of present study is on the application of the control method, but not on the aerodynamic nonlinearities associated with dynamic stall in wind turbines [

Consider the blade structure and coordinate system in Figure

The blade structure and coordinate system.

Consider the single blade section in Figure

As the actual sectional position is different, the structural parameters of airfoil and blade body are also changed. In order to take full account of the stability performance, the blade structure is optimized by Wilson method [

The Wilson method is based on BEM theory. The method aims to maximize the utilization coefficient of wind energy. When designing the blade, it can take into account the influence of blade tip loss and airfoil lift-drag ratio on the optimum aerodynamic performance of the blade, so the designed blade has high wind energy conversion efficiency. Hence the incoming velocity is

The acquisition of

Assume a reasonable set of initial values of

The incoming angle

A new set of parameters of

The steps are repeated continuously until the two adjacent calculated values of

The final calculated values of

The optimized data of the key parameters and corresponding fitting curves.

According to the chord length formula [

Coefficients and errors of the fitted parameters.

Items | Parameters | |||||||
---|---|---|---|---|---|---|---|---|

| | | | | | | | |

| 0.5392 | 0.3246 | 5.561 | 5.33 | 21.66 | 1.401 | 0.8691 | 3.309 |

| 3.047 | 3.545 | 3.943 | 2.92 | 4.656 | 2.976 | 2.073 | 0.351 |

| −0.3229 | 0.9545 | −0.3818 | 1.233 | 0.2604 | 0.9264 | 0.04122 | 1.488 |

| 0.2553 | 0.7905 | 141.9 | 0.1832 | 25.49 | 0.921 | 0.307 | 0.715 |

| 6.71 | 9.677 | 8.576 | 18.32 | 8.564 | 4.827 | 4.013 | 2.037 |

| 0.692 | 0.5523 | −0.3956 | 1.246 | −0.2689 | 2.951 | −0.05805 | −1.58 |

| 0.077 | 0.6918 | 139.1 | 8.023 | 12.5 | 0.1057 | 0.1994 | 2.658 |

| 12.02 | 11.39 | 8.63 | 28.74 | 12.53 | 11.32 | 5.892 | 0.4024 |

| 0.8957 | 2.719 | 2.71 | −3.652 | 0.7943 | 2.427 | −0.05113 | −1.663 |

| 0.03021 | 0.1294 | 0.332 | 0.3119 | 4.097 | 0.03815 | 0.1706 | −0.01812 |

| 19.04 | 18.53 | 18.44 | 12.93 | 21.25 | 17.08 | 0.7129 | 2.987 |

| 0.2307 | 1.815 | −0.4086 | 2.527 | −1.891 | 2.236 | 0.8469 | −1.528 |

| 0.01978 | 0.08002 | 0.2582 | 4.864 | 2.297 | 0.009333 | 0.1377 | 0.01948 |

| 25.12 | 24.63 | 24.52 | 3.297 | 25.08 | 26.92 | 7.613 | 3.949 |

| 0.07678 | 1.495 | −1.307 | 4.787 | −1.4 | 5.012 | −0.1192 | 1.586 |

| 0.01366 | 0.0478 | 0.1401 | 8.004 | 0.3702 | −0.009578 | 0.08546 | 0.000341 |

| 31.36 | 30.07 | 30.59 | 28.82 | 36.38 | 34.03 | 9.298 | 4.816 |

| −0.1892 | 1.455 | −1.185 | −0.566 | −5.521 | 0.4797 | −0.1933 | −1.575 |

| 0.002921 | 0.01901 | 0.06146 | 0.09117 | 0.1135 | 0.000607 | 0.07078 | 0.004993 |

| 0.9755 | 0.9913 | 0.9896 | 0.9958 | 0.9997 | 0.9998 | 0.9956 | 0.9992 |

| 0.8368 | 0.9423 | 0.9304 | 0.9721 | 0.9982 | 0.9986 | 0.9926 | 0.9979 |

| 0.0312 | 0.07961 | 0.04526 | 0.01743 | 0.06151 | 0.01422 | 0.05321 | 0.02131 |

At the same time, the linear density

The fitting curves of the 4 parameters in Figure

After using graphical methods to evaluate the goodness of fitting, some items of the goodness-of-fit statistics for parametric models should be examined. The items include the sum of squares due to error (SSE),

It can be seen in Table

Aerodynamics characteristics of airfoil are the basis of aerodynamic performance design of blades. Meanwhile the acquisition of airfoil aerodynamic data is often the most difficult part of blade design. The accuracy and completeness of the aerodynamic data obtained are of great significance to the analysis of the dynamic behavior of the blade. The blade of the wind turbine usually operates at the angle of attack range of −180°~180° with a wide range of Reynolds number. Hence the aerodynamic coefficients of airfoils in corresponding ranges are also required. Considering the actual situation of the present design, it is preliminarily estimated that the angle of attack will change between −90°~90°. The acquisition steps of aerodynamic coefficients are as follows [

Firstly, the lift and drag coefficients within a range of small attack angles (−15°~15°) are obtained by using Xfoil software [

Using the worksheet “TableExtrap” in AirfoilPrep_v2p2 program that prepares the corresponding airfoil data for AeroDyn software [

The validity of the fitting processes of structural parameters is demonstrated by the error data displayed in Table

Figure

The curves of the aerodynamic lift coefficient

It should be emphasized that, in view of the actual situation of the design, only the aerodynamic coefficients were fitted to the angles of attack within the range of −90°~90°. If the large angles of attack and deep stall-induced situation are put into consideration, this design method can also be applied to the angles of attack within the range of −180°~180°.

The 3D coupled blade body analysis method fully considers all the cross-sections along the spanwise length, and the cross-section displacements have complex coupling behaviors. Its solution needs strip theoretical decomposition, aerodynamic decomposition, and decoupling using Galerkin method [

Some cases intended to highlight the effects of the damping ratios and wind velocities on the vibration and stability of different sectional radius

The real parts of flap eigenvalues of vertical direction and lag eigenvalues of lateral direction at different positions

As is known to all, the first step in the pole-placement design approach is to choose the locations of the desired closed-loop poles. The most frequently used approaches are to choose such poles based on experience and the quadratic optimal control approach. Otherwise, the system responses become very fast, with very large amplitudes; or the poles cannot balance the acceptable responses and the amount of control energy required. And the pole assignment technology based on self-poles derived from these stable eigenvalues can just overcome these shortcomings and save the trouble in finding the right poles.

Different from specifying only dominant closed-loop poles in the conventional design approach [

According to Ackermann’s formula [

During the pole-placement process, it is assumed that all state variables are available for feedback. In present study, velocity variables are unmeasurable for direct feedback and need to be observed. Notice that, in order to deal with velocity variables, one needs the derivative of displacement variables, which presents a difficulty due to the reason that differentiation amplifies noise. Hence it is important to estimate the velocity variables accurately. Because only two velocity variables need to be estimated, so the observer called minimum-order observer. Figure

The feedback system with a minimum-order observer.

The PPMO method exactly consists of the self-pole placement technology and the structure of minimum-order observer. Firstly, both the system and state variables

The closed-loop poles of the PPMO system in Figure

It should be emphasized that a necessary and sufficient condition for arbitrary pole placement is that the system be completely state controllable. In view of the constant stability of the homogeneous equation structure in (

Figures

The uncontrolled displacements (a) and those controlled by PPMO (b) under the condition of wind speed of

The uncontrolled displacements

The displacements controlled by PPMO method

Comparisons of control effects between PPMO and LQR

The linear quadratic regulator (LQR) is often used to analyze vibration control of rotating structure. In [

In order to verify that the PPMO control method can be commonly used at that given external wind velocities. The other cases of different wind speeds are investigated. Figure

Comparisons of control effects between PPMO and LQR under conditions of wind speeds of

Comparison under

Comparison under

In addition, for the linear control strategy, the control cost (control consumption of energy) often affects the implementation of the control measures seriously. Therefore, the magnitude of the controller itself needs further exploration. Figure

Amplitudes of PPMO controller and LQR controller versus different state variables.

From the point of view of the “control cost” of controller hardware mentioned above, another advantage of PPMO control over LQR control is that the implementation of LQR control requires full state-feedback, which will not only increase the number of the hardware sensors used for measurement, what is more, the measurement process itself of the state variables of velocities is a complex process.

In addition, the implementation of PPMO is an automatic process, including the configuration of self-poles which is also automatic. Performance of LQR control depends on the choice of weighting Q/R matrices which have no analytic solution. So the choice of Q/R matrices is artificial, which is another drawback of LQR control.

In view of the large wind power system being mostly controlled by PLC, some intelligent control algorithms cannot be implemented in PLC hardware because of their complexity. At present, most of wind turbine control system algorithms employ fuzzy control, which can be implemented in PLC system. However, the fuzzy algorithm itself is tedious. For the other neural network algorithm, although the numerical simulation of control effect is more ideal, because of the complex integration problem, it is difficult to be implemented in the PLC hardware [

In present study, a hardware-in-the-loop simulation platform is built by OPC technology [

The experimental system.

Figure

The vertical and lateral displacements without control (a) and those under PPMO control (b).

The vertical and lateral displacements without control

The vertical and lateral displacements under PPMO control

It should be noted that the data matching and parameter transmission planning of OPC technology are not discussed here, which is another specific and complex problem. What is more, the speed of PPMO control in experimental platform is faster, compared with the hardware implementation of conventional fuzzy control by OPC technology in [

Consider the controlled cases by PPMO in Figure

Still taking the cases of Figure

Responses based on the improved scheme.

It should be stated that this performance improvement is only a theoretical improvement, and it only fully combines the advantages of PPMO control and LQR control and does not consider the shortcomings of LQR control. Therefore, this scheme has some unavoidable drawbacks in control cost, automation, and hardware implementation of control algorithms as mentioned above. Its implementation can only be expected in hardware performance improvement and control cost reduction in the future.

In this study, vibration control for vertical bending and lateral bending based on pole placement with minimum-order observer is investigated by numerical simulation and hardware-in-the-loop simulation platform. Some concluding remarks can be drawn from the results:

Structural modeling is investigated based on a NCM method between the 2D typical section analysis method and the 3D coupled blade body analysis method, which implies a compromise result that is bound to occur, but it can greatly reduce the difficulty of processing. Aeroelastic equation is realized by fitting of structural parameters and aerodynamic coefficients based on six-order Sin6 model, which can greatly simplify the integral behavior along the spanwise length and, at the same time, reduce the system variables (completely eliminate the aerodynamic variables).

PPMO method combines the advantages of pole placement and minimum-order observation, which is much better than LQR control either from vibration amplitude or from control cost. Pole assignment is based on self-poles, which is the most essential feature of PPMO control. Otherwise, the control performance of the proposed method is greatly reduced due to the difficulty of finding poles and the matching problem of configuration poles.

Feasibility analysis of hardware implementation of control algorithm is verified by hardware-in-the-loop simulation platform. The platform consists of PLC controller system connected with HMI, MATLAB simulation environment, and the OPC link between the two structures. In the experiment, the running speed of the platform and the accuracy of the displacement display further demonstrate the advantages of PPMO control.

In view of the fact that both simulation and experiment are the analysis and discussion of the control theory itself, some future work needs to be further envisaged. Practical engineering application of PPMO control can be realized via external pitch motion which can be fed back to the aeroelastic system. The pitch motion can be driven by a hydraulic system. Hydraulic systems have a rich variety of components and are easy to be controlled and should be easier to realize pole placement.

The authors declare that there are no conflicts of interest regarding the publication of this article.

This work is supported by the National Natural Science Foundation of China (Grant no. 51675315) and the Graduate Innovation Project of SDUST (Grant no. SDKDYC180333).