Maximum Wind Power Tracking of Doubly Fed Wind Turbine System Based on Adaptive Gain Second-Order Sliding Mode

This paper proposes an adaptive gain second-order sliding mode control strategy to track optimal electromagnetic torque and regulate reactive power of doubly fed wind turbine system. Firstly, wind turbine aerodynamic characteristics and doubly fed induction generator (DFIG) modeling are presented. Then, electromagnetic torque error and reactive power error are chosen as sliding variables, and fixed gain super-twisting sliding mode control scheme is designed. Considering that uncertainty upper bound is unknown and is hard to be estimated in actual doubly fed wind turbine system, a gain scheduled law is proposed to compel control parameters variation according to uncertainty upper bound real-time. Adaptive gain second-order sliding mode rotor voltage control method is constructed in detail and finite time stability of doubly fed wind turbine control system is strictly proved.The superiority and robustness of the proposed control scheme are finally evaluated on a 1.5MWDFIGwind turbine system.


Introduction
In the 21st century, as human population grows and nonregeneration energy is rapidly consumed, climate warming and environmental degradation problems become increasingly serious.Renewable energy such as wind energy, hydroenergy, and nuclear energy is highly favored.Particularly, wind energy, as one of the clean renewable energy sources, is attracting more and more attention [1].The newly installed capacity of wind power reaches 54.6 GW in 2016 [2].
Among the many adopted wind power technologies, doubly fed induction generator (DFIG) has been widely used in the wind power industry and dominated the market due to its excellent operating performances such as light weight, small size, highly cost effective, and small capacity of converter.In order to enhance generating efficiency, wind energy conversion optimization is one of the key issues in the DFIG wind power generation control system [3,4].
Many scholars study various control methods for optimizing wind energy conversion [5].However, due to the randomness of wind, nonlinearity and uncertainty of generator set, it is hard to achieve wind energy capture rapidly and accurately via linear control theory.Nonlinear control techniques such as fuzzy logic [6], neural networks [7], feedback linearization [8], and backstepping control [9] are widely employed to achieve optimal conversion of wind power system.Despite the recognized advantage of using nonlinear controllers to cope with nonlinear systems, many of these techniques produce control laws with a rather high computational burden.These calculi commonly depend on the system states and also on several model parameters having the secondary effect of reducing control robustness.Other controllers, like fuzzy and neural networks methods, present the drawback of neither being systematic in the design stage nor providing a rigorous stability proof.
Sliding mode control is a nonlinear robust control method and has already successfully applied to doubly fed wind turbine system [10][11][12].However, as an on-off control, control effect of conventional first-order sliding mode is discontinuous and chattering phenomenon is serious.These drawbacks inevitably increase mechanical and electrical wear and shorten service life.

Journal of Control Science and Engineering
Higher-order sliding mode can cancel the requirement of compelling relative degree to be one, enhance sliding accuracy, attenuate chattering phenomenon, and make control effective continuous [13].Super-twisting algorithm, called higher-order sliding mode of the second order, has been adopted to achieve maximum wind power tracking and reactive power regulation [14][15][16][17][18][19].However, control gains for these schemes cannot be real-time adjusted along with variation of system uncertainty.The rotor current or rotor voltage is conservatively chosen, and this will aggravate control chattering.Paper [20] proposes an adaptive secondorder sliding mode control strategy to maximize the energy production simultaneously reducing the mechanical stress on the shaft.Yet, preliminary results of this proposal are applied to a simple single-input single-output topology and based on a unidirectional DFIG.Paper [21] presents a novel adaptive higher-order sliding mode control strategy for DFIG energy conversion system.Adaptive natures of the gains formulate a zero steady state transient performance of the system gifted by a reduced settling time.However, the system cannot converge in finite time.Evangelista et al. [22] achieve adaptive secondorder sliding mode control for doubly fed wind turbine in finite time based on receding horizon adaptation time windows algorithm.Yet, the selection of adaptation period parameter should be kept in mind while tuning the whole control system.
In this context, to face the stringent specifications of maximum wind power tracking and reactive power control problems of doubly fed wind turbine system where the uncertainties are subjected to fast variations, this paper proposes a new adaptive control gain second-order sliding mode scheme.It is not necessary to know the upper bound of uncertainty in advance by choosing suitable sliding function and designing adaptive gain second-order sliding mode controller.Control parameters can be adjusted according to variation of uncertainty upper bound.The control objectives of maximum wind power tracking and reactive power regulation are achieved by controlling rotor voltage, and the output chattering is greatly restrained.
This paper is organized as follows.Section 2 presents system modeling of doubly fed wind turbine.The detailed fixed gain and adaptive gain control strategies are explained in Section 3. The proposed control strategy is validated in MATLAB environment which is illustrated in Section 4.

Doubly Fed Wind Turbine System Modeling
Structure diagram of doubly fed wind turbine system is shown in Figure 1, which mainly includes wind turbine, gearbox, DFIG, and converter.Firstly, wind energy is converted to mechanical energy via wind turbine, and then mechanical energy is converted to electric energy by means of DFIG.DFIG stator is directly connected to grid, and rotor is by way of converter.DFIG can feed into the grid via stator and rotor under hypersynchronous mode.
Wind turbine mechanical power can also be defined as Then, wind turbine mechanical torque is where   (, ) =   (, )/ is wind turbine torque coefficient.Dynamic characteristics of doubly fed wind turbine system can be accurately described via fifth-order differential equations [24], four of which are expressed by flux linkage under  −  synchronous rotating coordinate axis: where In formulas ( 6) and (7), subscripts  and  denote stator and rotor physical quantities. and  are physical quantities of -axis and -axis., , ,  denote voltage, current, resistance, and flux linkage, respectively,   is power grid angular frequency,   and   are self-inductance of stator winding and rotor winding, and   is mutual inductance of stator and rotor.
The fifth differential equation is kinematic equation: where  represents inertia of the whole rotating part and   is DFIG electromagnetic torque: where  is number of pole pairs.For achieving decoupling control, -axis under synchronous rotating coordinate system is oriented to stator flux linkage vector and stator resistance is neglected.Reducedorder model of doubly fed wind turbine system can be deduced from formula (5) to formula (9): where Then stator current is calculated as Reactive power is expressed as

Adaptive Gain Higher-Order Sliding Mode Control Strategy
3.1.Control Objective Description.In the constant power zone, the objective is to adjust pitch angle to maintain wind turbine system operating at rated power, while, in the partial load zone, the objective is to track maximum wind energy.Control objectives of this paper are to achieve maximum wind power tracking and reactive power regulation according to grid demand and tolerance range of wind power system.The pitch angle  is set as zero when wind turbine system operates at the partial load zone.Then wind turbine power coefficient   is only related to tip speed ratio .As is shown in Figure 2, there is always  max related to optimal tip speed ratio  opt for certain wind turbine.In other words, wind turbine can only operate under specific revolving speed to achieve maximum wind energy conversion efficiency for specific wind speed.
When wind turbine operates at maximum power point (  =  max ,  =  opt ), the corresponding torque is expressed as where  0 =  5  max /2 3   3 opt .The objective of maximum wind power tracking control is further presented as achieving   =  opt for any wind speed at partial load zone.Considering power regulation ability enhancement and power system stability improvement, the other control objective is reactive power regulation.The choice of reactive power reference value mainly relies on certain evaluation function to attain the optimal one.

Controller Design.
For doubly fed wind turbine system (10), this section firstly presents standard super-twisting second-order sliding mode design method.Then considering unknown uncertainty upper bound and system wear reduction, adaptive gain second-order sliding mode control scheme is detaily designed to achieve maximum wind power tracking and reactive power regulation.Finite time stability of the closed-loop system is also strictly analyzed and proved.

Design Procedure under Fixed Control Gain.
For tracking maximum wind power and regulating reactive power, the sliding variables are chosen as To calculate first-order time derivative of  1 and  2 where where  = 1, 2. To apply standard super-twisting second-order sliding mode control algorithm According to sufficient condition for finite time stability of super-twisting algorithm [25,26], when control parameters satisfy second-order sliding mode with respect to   can be established in finite time, and optimal torque tracking and reactive power regulation are achieved.
It is observed that uncertainty upper bounds are needed when designing rotor voltage second-order sliding mode controller, yet the upper bounds are usually unknown in practical wind power system and hard to be estimated.Overestimation of these upper bounds may lead to conservative choice of controller parameters which will produce more control effective, aggravate control chattering, and shorten service cycle of wind turbines.

Adaptive Gain Second-Order Sliding Mode Control
Design.Adaptive control gain can deal with unknown uncertainty upper bound problem and further restrain rotor voltage chattering.Here   is redesigned as an example because the system has been decoupled, and design procedure of   is similar to   .
Due to the existence of parameter uncertainty and external disturbance,  1 (,   ,   ,   ) and  1 (,   ,   ,   ) may be expressed as where Δ 1 and Δ 1 mainly include fluctuation of   ,   ,   ,   and satisfy where  1 is unknown upper bound.
Carrying out feedback control for system (15) then where ã = Δ where  1 exists and is unknown.Then aiming at formula (26), control law V  is designed to establish second-order sliding mode with respect to  1 where   and   are adaptive control gains.b can be regarded as an uncertain piecewise constant function.Then formulas ( 26) and (28) may be written as Second-order sliding mode control problem of system ( 26) is converted to design adaptive gain super-twisting algorithm for system (29) to achieve  1 → 0, ṡ 1 → 0 in finite time under conditional formulas ( 24), ( 27), (30), and (31).
Theorem 1 provides adaptive law of control gain   ,   , and guarantees establishment of second-order sliding mode with respect to  1 .
Theorem 1.In order to achieve optimal torque tracking of doubly fed wind power system, -axis rotor voltage   is designed as formula (25), auxiliary control law is formula (28), and adaptive control gain   ,   are designed as Then formula (29) can be expressed as Formula ( 34) is rewritten as matrix form where 0 <  2 ≤ 2 3 ; then formula (35) is further denoted as where It is observed that if  1 ,  2 → 0 in finite time, then  1 , ṡ 1 → 0 is set up in finite time, and To choose Lyapunov function where To calculate time derivative of Lyapunov function ( 38) Considering formulas ( 35) and ( 37) Considering | ȧ  | ≤  1 and formula (36), Symmetric matrix Θ is calculated as where To guarantee positive definiteness of matrix Θ,   is designed as Then, if   satisfies matrix Θ is positive definite, and minimum eigenvalue  min (Θ) ≥ 2  is satisfied.

Case Studies
Simulation verification of the proposed control scheme is carried out based on MATLAB platform.Nominal parameters of doubly fed wind turbine system are   = 260 rad/s,   = 460√2/3 V,   = 35.Case 1 (gradient wind).Gradient wind speed model can be described by formulas (61).To choose  1 = 6 s,  2 = 12 s and max  = 4 ) . (61) Figure 3 shows gradient wind speed variation.The curves of optimal tracking torque and reactive power regulation under fixed control gain scheme are shown in Figures 4 and 5.It is observed that the two control objectives can be achieved, but the uncertainty upper bound is not easy to measure in actual wind power system.Therefore, adaptive gain secondorder sliding mode control scheme is put into effect.Maximum wind power tracking and reactive power regulation effect can also be achieved under adaptive control gain scheme as shown in Figures 6 and 7. Figure 8 shows rotor control voltages which indicate chattering reduction.is -axis rotor voltage and Figure 10 shows  −  axis rotor current.Control gain regulation according to wind speed variation is shown in Figure 11.
Case 2 (random wind).The random wind speed variation is shown in Figure 12.Considering system uncertainties, coefficients of mutual induction   and rotor resistance fluctuate ± 10%.Grid voltage and grid frequency fluctuation are ±10 and ±20%, respectively.For highlighting the controller performances, the proposed AGSOSM (adaptive gain secondorder sliding mode) method and ASOSM (adaptive secondorder sliding mode) controller [25] are, respectively, executed on the wind turbine system.The related ASOSM controller parameters are referred to in [25].Figure 13 shows that the actual electromagnetic torque can rapidly track desired value, and Figure 14 shows that reactive power regulation can still   satisfy the demand under random wind under the two control schemes.However, the proposed scheme can achieve higher control accuracy.As is shown in Figure 15, the rotor control voltage chattering is greatly alleviated under the proposed control method.Rotor current under -axis is shown in Figure 16.

Conclusions
For achieving maximum wind power tracking and reactive power regulation of doubly fed wind turbine system, this paper proposed an adaptive gain second-order sliding mode control scheme.Firstly, the system model is established.

Figure 1 :
Figure 1: Schematic diagram of doubly feed wind turbine system.

Figure 2 :
Figure 2: The relation between power coefficient and tip speed ratio.
the proposed AGSOSM T e under ASOSM
Proof.To bring in new state vector z and  *  ,  *  are positive constant.It is noted that matrix P is always positive if   > 0 and   is arbitrary real number.