，*，，

1.College of Energy and Power Engineering，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China；2.Department of Electromechanics，Ufa State Aviation Technical University，Ufa，Russia

Abstract: A robust control strategy using the second-order integral sliding mode control（SOISMC）based on the variable speed grey wolf optimization（VGWO）is proposed. The aim is to maximize the wind power extraction of wind turbine. Firstly，according to the uncertainty model of wind turbine，a SOISMC torque controller with fast convergence speed，strong robustness and effective chattering reduction is designed，which ensures that the torque controller can effectively track the reference speed. Secondly，given the strong local search ability of the grey wolf optimization （GWO） and the fast convergence speed and strong global search ability of the particle swarm optimization（PSO），the speed component of PSO is introduced into GWO，and VGWO with fast convergence speed，high solution accuracy and strong global search ability is used to optimize the parameters of wind turbine torque controller. Finally，the simulation is implemented based on Simulink/SimPowerSystem. The results demonstrate the effectiveness of the proposed strategy under both external disturbance and model uncertainty.

Key words：integral sliding mode；second-order sliding mode；maximum power point tracking；optimization algorithm；wind turbine

0 Introduction

Energy is particularly important in the rapid de?velopment of society. At present，the proportion of non-renewable energy power generation is very large，causing environmental pollution and global warming. Therefore，it is of significance to develop clean and sustainable energy［1］. In recent years，wind energy，as a clean and environmentally friend?ly renewable energy，has been developed rapidly［2］.Wind energy is a kind of uncontrollable energy be?cause of its randomness and instability. Wind tur?bines feature complex nonlinear dynamics under the influence of stochastic wind disturbance and gyro?scopic load，and are not so easy to control. At pres?ent，studies in wind power generation mainly in?clude maximum power point tracking（MPPT）and reduction of mechanical stresses［3］.

The wind turbine has strong nonlinearity and uncertainty，so it is important to design a controller with strong robustness and desired dynamic charac?teristics. Robust control for wind turbine includes ro?bust adaptive control［4］，fuzzy control［5］，model pre?dictive control［6］，back-stepping control［7］，and slid?ing mode control（SMC）［8］. Among these robust control methods，SMC has relatively convenient im?plementation and strong robustness against external disturbance and model uncertainty. To overcome the shortcomings of quasi-continuous sliding mode in convergence speed and smoothness of control ac?tion［9］，a virtual control was proposed to increase the relative order. Compared with original algorithms，faster convergence speed and smoother control ac?tion can be achieved. Saravanakumar et al.［10］de?signed a wind turbine controller with fast dynamic response and high steady-state tracking accuracy to improve the utilization of wind energy. Hu et al.［11］designed a SMC based on disturbance observer（DOB）. The parameter uncertainty and wind speed randomness of wind turbine were estimated by DOB to achieve maximum wind power extraction.At the same time，to improve the utilization rate of wind energy，the actual rotor speed of wind turbine should accurately track the reference rotor speed.Nonlinear terminal sliding mode control（TSMC）can meet the requirement，but it generates chatter?ing，which is unfavorable to the operation of wind turbines［12］. At present，the methods of eliminating chattering mainly include the quasi-sliding mode method，the boundary layer method，the reaching law method， and the high-order sliding mode（HOSMC）［13］. The quasi-sliding mode method，the boundary layer method and the reaching law can at?tenuate chattering to some extent，but they lose the invariance of SMC.HOSMC not only retains the ad?vantages of traditional SMC，but also attenuates chattering and achieves finite time convergence.

HOSMC can be divided into the second-order sliding mode control（SOSMC）and the arbitrary or?der SMC. The SOSMC is widely used because of its simple structure and less required information.The most common methods are the twisting algo?rithm，the sub-optimal algorithm，the prescribed convergence law， and the super twisting algo?rithm［14］. Among them，the twisting algorithm is the earliest proposed second-order sliding mode algo?rithm. Its convergence trajectory features“twist?ed”，and the convergence process inevitably chat?ters. The prescribed convergence law is to use the control law based on symbolic function in the tradi?tional first-order SMC to make the system converge and keep on a preset sliding surface，and then make the system converge to the second-order sliding sur?face in a finite time，and finally realize the SOSMC.Super twisting algorithm is the only one that does not need to calculate the derivatives of sliding mode variables and has continuous output among four SOSMC algorithms. Therefore，the super twisting algorithm has attracted the most extensive attention.A second-order sliding mode controller was pro?posed to reduce the fatigue load during the operation of wind turbine［15］. Abolvafaei et al.［16］designed a second-order fast terminal sliding mode controller by combining TSMC and PID. The better tracking effect could be guaranteed by using the PID sliding mode surface. Moreover，the model uncertainty and external disturbance were not considered，which made it difficult to highlight the robustness advan?tages of SMC. And a strategy of designing speed controllers based on the quasi-continuous HOSMC was proposed to ensure that the wind turbine works well in different wind modes. The influence of exter?nal disturbance and parameter uncertainty was con?sidered［17］. In this paper，the presented second-order integral sliding mode control（SOISMC）strategy eliminates steady-state tracking error in integral slid?ing mode （ISMC） and reduces chattering in HOSMC. The controller has a fast convergence speed，strong robustness and effective chattering re?duction.

Due to various uncertainties in the actual sys?tem，the existing SOSMC method can deal with the problem of system uncertainty，but the premise is that the upper bound of the system uncertainty must be estimated in advance before the parameters are selected. However，in practical engineering design，it is unrealistic to know the bounds of various uncer?tainties of the system，which leads to the difficulty in parameter selection. Besides，the SOISMC can effectively reduce chattering，but the number of ad?justable parameters will increase with the increase of order. It is extremely difficult to manually adjust the controller parameters. As far，some conventional gradient-based optimizations such as Newton and in?terior point methods have been proposed. But these optimization methods may fail to obtain the optimal parameters due to their high dependence of the accu?rate system model. Hence，several heuristic algo?rithms have been developed to overcome the above challenges，such as the particle swarm optimization（PSO） and the grey wolf optimization（GWO）.These optimization algorithms can achieve an effi?cient global search with a lower dependence of the accurate system model［18］. Typical swarm intelli?gence optimization algorithms include PSO［19］，GWO［20］and the artificial bee colony（ABC）［21］.Heuristic optimization algorithms have been widely used to solve the problems of function optimization and clustering optimization because of its simple structure and few adjusting parameters. Bekakra et al.［22］used PSO to adjust the controller parameters to ensure the maximum wind energy extraction.The results show that the PSO can obtain better control effects than manual adjustment parameters.And differential evolution（DE）algorithm is used to improve the doubly fed induction generator perfor?mance under disturbance［23］. Recently，GWO，in?spired by the leadership and hunting methods of the grey wolves in nature，has been widely used. In GWO，the location of the prey is the solution of the corresponding problem. And it is shown that GWO is superior to PSO and genetic algorithms in search?ing global optimal solution［24］. GWO has a fast con?vergence speed and strong local search ability. But there are some problems such as insufficient global search ability，low solution accuracy and slow con?vergence speed in the later stage of optimization［20］.The main advantages of PSO are strong global search ability，simple principle and fast convergence speed. The main disadvantages are that the local search ability of the algorithm is poor and the search accuracy is low［19］. These algorithms have their own advantages. Combining the advantages of different algorithms can achieve better results. A hybrid algo?rithm of GWO and DE was proposed［25］，which used GWO to improve the local search ability and used DE to improve the global search ability. The presented variable speed grey wolf optimization（VGWO）combines the strong local search ability of GWO and the fast convergence speed and strong global search ability of PSO，and realizes the param?eter optimization of wind turbine torque controller.

1 Mathematical Model of Wind Turbine

In general，a wind turbine is mainly composed of an aerator（including blades，a pitch system，a hub and a yaw system），a drive-train（including main shaft bearings，the main shaft and a gearbox）and a generator，corresponding to aerodynamic，me?chanical and electrical components，respectively.Fig.1 shows the structure of a wind turbine.

Fig.1 Typical structure of a wind turbine[26]

The work state of a wind turbine can be divid?ed into four parts，as shown in Fig.2. The wind tur?bine is in shutdown state in stage 1 and stage 4.Stage 2 is the MPPT stage. MPPT can be realized by controlling the rotation speed of the wind turbine rotor. Stage 3 is a constant power operation stage.Considering the mechanical condition of a wind tur?bine，constant power operation is realized by adjust?ing the pitch angle. This paper mainly studies that the wind turbine operates in the maximum wind en?ergy capture stage（stage 2，as shown in Fig.2）.

Fig.2 Power curve of wind turbine

1.1 Aerodynamics model

Eq.（1）gives the nonlinear function for wind energy capture

wherePis the output power，Cpthe power coeffi?cient of the wind turbine，vthe wind speed，ρthe air density，λthe tip speed ratio，Rthe wind turbine rotor radius，ωrthe rotor speed，andTathe aerody?namic torque.

λis defined as

From Eq.（2a），we can obtain

whereλoptdenotes the best tip speed ratio，andthe reference rotor speed.

TheCpcan be expressed as follows［27］

According to Eqs.（2，3），the change ofωrandvwill cause the change ofλandCp. Fig.3 shows the relationship amongθ，λandCp. With the constant pitch angleθ=0°，MPPT can be achieved by changingλandωr. As long as a wind turbine meetsλ=λopt，the wind turbine can operate withCpmax.

Fig.3 Power coefficient curve of wind turbine

1.2 Drive train model

Fig.4 shows a dynamic wind turbine model.The model includes a rotor and a generator，which are modeled by two inertial models. The two-mass model is suitable for the analysis of the dynamic characteristics of wind turbines［3，28］. In Fig.4，ωgis the generator speed andngthe transmission ratio，JrandJgare the rotor inertia and the generator inertia ，respectively，DrandDgare the rotor external damp?ing coefficient and the generator external damping coefficient， respectively，Teis the generator torque，andKdt，Ddtare the torsional stiffness and the torsional damping，respectively.

Fig.4 Model of wind turbine

Assuming the gearbox is ideal， we havengωr(t) =ωg(t).It gives

then we have

whereu=Te.

1.3 Uncertain model of wind turbine

The parametersJtandDthave variations ΔJtand ΔDt，respectively. Given the parameter varia?tions，the model of the wind turbine is defined as

where ΔA，ΔBand Δdrepresent the uncertainties ofA，Bandd，respectively.

Letg(t) =ΔAωr+ΔBu+d+Δd，which rep?resents the lumped disturbance on subsystems Eq.（6a）.It can be re-written as

Assumption 1 The lumped disturbancesg(t)are continuous and satisfywhereDis known positive constants.

2 Design of SOISMC

With the wind speed changing，the rotor speed of the wind turbine can be adjusted by controlling the generator torque，so that the rotor speed of the wind turbine can track the reference rotor speed.The control schematic is shown in Fig.5.

Fig.5 Principle of wind turbine torque controller

2.1 Design of sliding mode surface

The sliding mode surface is designed based on rotor speed tracking errore，and the integral term is added to eliminate the steady-state error. The firstorder sliding mode surface is as follows

whereαis a positive constant，and

The second-order sliding mode surface is as fol?lows

whereγandβare positive numbers，which can be chosen and utilized to guarantee the convergence speed of the rotor speed tracking error.

Hence，the sliding mode manifold is

2.2 Design of torque controller

The derivation ofσis obtained

The derivation of Eq.（8a）can be obtained as follows

The derivation ofsis obtained

By substituting Eq.（8）into Eq.（9），we have

The control lawuis as follows

Overlook the uncertainty and disturbance，the equivalent control lawueqis as follows

The switching control lawuswis as follows

where bothks1andks2are position numbers.

2.3 Stability analysis

Theorem 1The wind turbine subsystem giv?en by Eq.（6）converges asymptotically under the SOISMC of Eq.（7），with sliding mode controller of Eq.（11），if Eq.（12）is held.

ProofChoose the following Lyapunov func?tion

The derivation of Lyapunov function is ob?tained

According to Assumption 1，it gives

Fromks1＞0 andks2＞D，we have

The proof is completed.

3 Parameter Optimization Based on VGWO

Combining the advantages of different algo?rithms to construct a new hybrid algorithm is an im?portant research direction of current algorithm im?provement. In this paper，VGWO with a fast con?vergence speed，high solution accuracy and strong global search ability is proposed，which combines strong local search ability of GWO and fast conver?gence speed and strong global search ability of PSO.The VGWO introduces the velocity component of PSO into GWO. The velocity component includes inertia，social and perceptual components. The ve?locity component in GWO can not only avoid prema?turity but also accelerate the convergence speed.The SOISMC control structure for the wind turbine is shown in Fig.6.

Fig.6 The proposed control structure for the wind turbine

3.1 Principle of GWO

GWO is a new heuristic algorithm，which imi?tates the dominance and hunting style of grey wolves in nature. According to the fitness value，we consider the position of alpha（αGWO）as the optimal solution. Consequently， the positions of beta（βGWO）and delta（δGWO）are considered as the sec?ond and the third best solutions，respectively. The position of omega（ωGWO）are assumed to be candi?date solutions. In GWO，the optimization is guided byαGWO，βGWO，δGWOandωGWO. TheωGWOwolves follow the other three wolves. The specific steps of the VGWO algorithm are as follows wherekindicates the current iteration，DGWOthe dis?tance between grey wolf and prey，Xpthe position vector of the prey，Xthe position vector of current grey wolf，ais linearly decreased in [0，2]，AGWOandCGWOare the coefficient vectors，r1，r2the ran?dom vectors in[0，1]；X1，X2，X3the distance vec?tors ofωGWOrelative toαGWO，βGWO，andδGWO，re?spectively；A1，A2，A3，C1，C2，C3the coefficients..

3.2 Improvement of GWO

The particles of PSO can memorize the best position they found and the distance and direction of flight are determined by the velocity of each particle.The speed can be adjusted dynamically based on the flight experience of itself and its companions. There?fore，PSO has a faster convergence speed and stron?ger global search capability.

whereζis the inertial factor，vithe gray wolf speed，xithe current position of the grey wolf， andc1=c2=c3=0.5 are the learning factors.

The pseudo code of the VGWO algorithm is shown in Table 1，whereXα，XβandXδrepre?sent the positions ofαGWO，βGWOandδGWO，respec?tively. The size of the grey wolf population is set to 30 and the maximum number of iterations is set to 100.

Table 1 Pseudo code of the VGWO algorithm

To optimize the parameters by using perfor?mance indexes，the fitness function of the integral absolute error（IAE）is designed as

4 Simulation and Analysis

The proposed control structure for the wind tur?bine is shown in Fig.6. The simulation in this paper includes three aspects. Firstly，the optimization re?sults based on PSO，GWO and VGWO optimiza?tion algorithms are given. Secondly，the simulation is carried out under SOISMC. Finally，considering the model uncertainty and external disturbance，the proposed SOISMC and ISMC are compared and simulated. The wind turbine parameters are given in Table 2.

The wind speed simulation is shown in Fig.7.Wind speed model includes basic wind，random wind，gust and gradual wind. The basic wind is 7 m/s，the holding time is 400 s and the sampling time of the random wind is 2 s.

Fig.7 Wind speed

Table 2 Parameters of wind turbine

4.1 Comparison of optimization algorithms

Fig.8 is the 3D graph of universal standard functions. In this paper，the function is used to test the performance of PSO，GWO and VGWO briefly.

Fig.9 shows the performance comparison of PSO，GWO and VGWO. It can be seen that the op?timal value of VGWO is the minimum，and its ad?justment time is shorter than that of GWO.The sim?ulation result shows that VGWO combines the fast convergence speed of PSO and the effectiveness of GWO. Thus，it has a better optimization perfor?mance.

Fig.8 Universal standard functions

Fig.9 Performance comparison of PSO, GWO and VGWO

4.2 Robustness simulation

In order to show the robustness of the present?ed method in this paper，the following two cases are considered.

Case 1 ΔA=0，ΔB=0，Δd=0.

Case 2 ΔA=10%A， ΔB=10%B， Δd=sin(πt/125).

To illustrate the validity，we conduct a compar?ative study between the proposed SOISMC and ISMC.The expression of ISMC is as follows where ISMC controller parameters are shown in Ta?ble 3.

Table 3 Optimization results based on VGWO

（1）Case 1

Fig.10 shows the comparison of fitness func?tion for SOISMC and ISMC based on the VGWO optimization algorithm. It can be seen that the fit?ness function value based on SOISMC is smaller than that of ISMC. The result shows that SOISMC has a better control effect than ISMC. Fig.11 shows that the rotor speed based on the proposed SO?ISMC has higher tracking accuracy，whereeωis the rotor speed error. Fig.12 and Fig.13 show that the proposed SOISMC can effectively reduce chatter?ing，which reduces the fatigue load of generator and prolong the service life. Fig.14 and Fig.15 show that the proposed SOISMC has higherCpandP，whereis the reference power coefficient of the wind tur?bine.

Fig.10 Comparison of fitness function

Fig.11 Rotor speed for Case 1

The data in Figs.11—13 are further processed and then shown in Tables 4—6，respectively.Min(?)，Max(?)，Mean(?) and STDEV(?) represent the minimal，the maximal，the mean and the standard deviation of the corresponding variables，respective?ly. It can be seen from the tables that the standard deviation of SOISMC is less than that of ISMC. It shows that the operation of the wind turbine with SOISMC control strategy is more stable.

Fig.13 Aerodynamic torque for Case 1

Fig.14 Power coefficient for Case 1

Fig.15 Power for Case 1

Table 4 Comparison of rotor speed error for Case 1

Table 5 Comparison of generator torque for Case 1

Table 6 Comparison of aerodynamic torque for Case 1

（2）Case 2

Figs.16—20 show the case in which the same model uncertainty 10% and external disturbance Δd=sin(πt/125) are considered. Compared with Case 1，the system uncertainty is considered in Case 2. ISMC control performance deteriorates compared with SOISMC. It shows that SOISMC can effectively weaken system chattering and im?prove the convergence speed. The conclusion is sim?ilar to Case 1. The data in Figs.16—18 are further processed and then shown in Tables 7—9，respec?tively. It can be seen from the tables that SOISMC has stronger robustness to external disturbance and model uncertainty.

Fig.16 Rotor speed for Case 2

5 Conclusions

Based on the model uncertainty and external disturbance，a robust control method based on VG?WO for wind turbines is proposed. The VGWO al?gorithm is used to adjust the parameters of the slid?ing mode controller. The proposed method is proved mathematically and simulated in Matlab/Simulink，which provides a strong theoretical back?ground for the application of the SOISMC method for wind turbines. Simulation results show that SO?ISMC has a fast convergence speed and strong ro?bustness，and it can effectively weaken system chat?tering compared with ISMC. Besides，VGWO has a fast convergence speed，high accuracy and strong global search ability. The presented wind turbine torque controller effectively realizes maximum wind power extraction for wind turbines，which meets the control requirements. In the future study，the upper bound of disturbance should be estimated by the adaptive law，and the actuator fault should be con?sidered.

Fig.17 Generator torque for Case 2

Fig.18 Aerodynamic torque for Case 2

Fig.19 Power coefficient for Case 2

Fig.20 Power for Case 2

Table 7 Comparison of rotor speed error for Case 2

Table 8 Comparison of generator torque for Case 2

Table 9 Comparison of aerodynamic torque for Case 2