Journal of Control Science and Engineering
Volume 2018 (2018), Article ID 5342971, 11 pages
https://doi.org/10.1155/2018/5342971

Research Article

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

Hongchang Sun,¹ Yaozhen Han,² and Lvyuan Zhang³

¹Shandong Dawei International Architectural Design Co., Ltd., Jinan 250061, China
²School of Information Science and Electrical Engineering, Shandong Jiaotong University, Jinan, Shandong 250357, China
³Shandong Jinan Langchao Electronic Information Industry Group Corp. Co., Ltd., Jinan 250000, China

Correspondence should be addressed to Yaozhen Han; yzhan_sdjtu@163.com

Received 11 August 2017; Revised 20 November 2017; Accepted 5 December 2017; Published 29 January 2018

Academic Editor: Carlos-Andrés García

Copyright © 2018 Hongchang Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 1.5 MW DFIG wind turbine system.

1. 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 [10–12]. 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.

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 [14–19]. 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 second-order 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 second-order 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.

2. 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.

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

According to Betz’ Law, the captured mechanical power of wind turbine is

where is air density (kg/m³), represents blade radius (m), is wind speed (m/s), and is power conversion efficiency which is nonlinear function of tip speed ratio and pitch angle . Tip speed ratio is defined as

where is rotating speed (rad/s), is DFIG rotor speed, and is gear ratio.

Power conversion efficiency can be expressed as [23]

where , , , , , .

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. Reduced-order model of doubly fed wind turbine system can be deduced from formula (5) to formula (9):

where is network line voltage, . Then stator current is calculated as

Reactive power is expressed as

3. 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 related to optimal tip speed ratio 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.

Figure 2: The relation between power coefficient and tip speed ratio.

When wind turbine operates at maximum power point (, ), the corresponding torque is expressed as

where .

The objective of maximum wind power tracking control is further presented as achieving 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.

3.2. 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.

3.2.1. 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 and

where

Then there exist positive constants , , , , , satisfying

where . 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.

3.2.2. 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, and may be expressed as

where and mainly include fluctuation of , , , and satisfy

where is unknown upper bound.

Carrying out feedback control for system (15)

then

where and .

In practice, parameters , , , , vary slowly if they are influenced by uncertainties, and , , are also continuous, such that satisfies

where exists and is unknown.

Then aiming at formula (26), control law is designed to establish second-order sliding mode with respect to

where and are adaptive control gains.

can be regarded as an uncertain piecewise constant function. Then formulas (26) and (28) may be written as

is supposed to have uncertain upper bound ; meanwhile, control gain is bounded, meaning , ; then in any finite time interval

Thus, uncertain function possesses unknown upper bound; then

Second-order sliding mode control problem of system (26) is converted to design adaptive gain super-twisting algorithm for system (29) to achieve , 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 .

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 actual second-order sliding mode with respect to is established after finite time and , where , , , , are positive constant.

Proof. To bring in new state vector

Then formula (29) can be expressed as

Formula (34) is rewritten as matrix form

Considering and , then is expressed as

where ; then formula (35) is further denoted as

where .
It is observed that if in finite time, then is set up in finite time, and , are both established.
To choose Lyapunov function

where , , and , are positive constant. It is noted that matrix is always positive if and is arbitrary real number.
To calculate time derivative of Lyapunov function (38)

Considering formulas (35) and (37)

Considering and formula (36), Symmetric matrix is calculated as

where , .
To guarantee positive definiteness of matrix , is designed as

Then, if satisfies

matrix is positive definite, and minimum eigenvalue is satisfied.
In view of formulas (40), (42), and (43) being established, then

where , and

Then

where .
To combine formulas (39) and (46), then

where .
Considering inequation and formula (38), then

where . Then

One solution of formula (32) is

Hence, , are both bounded; namely, , . Then formula (49) is expressed as

where .
( Provided that , we have and . Then considering formula (32)

To choose and combine formula (42), then

then , and

Take note that increases according to . When formula (22) is satisfied, positive definiteness of matrix and formula (55) are guaranteed. Then after finite time , is established.
() To suppose , then according to formula (32), the formula

and formula (57) may become positive number:

It is worth noting that the expression under is effective in finite time, because as long as is valid, will increase according to , and then the other expression of formula (57) is effective.
The symbol in formula (52) becomes uncertain in view of formula (57). may be greater than due to the diminution of control gain and . Once is greater than , is established. Hence, reaches in finite time during adaptive process and then leaves this range and always stays in a bigger one , .
When is satisfied, can be estimated via formulas (29) and (32):

where and are the moment entering into and leaving . When

where and are the moment entering into and leaving . From formulas (58) and (59)

Namely, can enter into in finite time and stay here. Theorem 1 is proved.

4. 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 , , , , , , , , , , , , and . The choice of control parameters is according to formulas (19) to (21) and formula (32) and combines actual simulation experiment. The related fixed gain control parameters are , , , , , , and the adaptive gain parameters are , , , , , , , , , , , .

Case 1 (gradient wind). Gradient wind speed model can be described by formulas (61). To choose , and

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 second-order 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. Figure 9 is -axis rotor voltage and Figure 10 shows axis rotor current. Control gain regulation according to wind speed variation is shown in Figure 11.

Figure 3: Gradient wind speed.

Figure 4: Optimal torque tracking under fixed gain.

Figure 5: Reactive power regulation under fixed gain.

Figure 6: Optimal torque tracking under adaptive gain.

Figure 7: Reactive power regulation under adaptive gain.

Figure 8: Rotor voltage in -axis.

Figure 9: Rotor voltage in -axis.

Figure 10: Rotor current in -axis under adaptive gain.

Figure 11: Control gain under adaptive gain.

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 second-order sliding mode) method and ASOSM (adaptive second-order 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.

Figure 12: Random wind speed.

Figure 13: Optimal torque tracking.

Figure 14: Reactive power regulation.

Figure 15: Rotor voltage in -axis.

Figure 16: Rotor current in -axis.

5. 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. Then second-order sliding mode rotor voltage controllers are designed after detailed description of the two control objectives. Considering the circumstance of unknown uncertainty upper bound, adaptive gain second-order sliding mode rotor voltages controllers are designed, and finite time stability is proved in detail. Finally, simulation experiments under gradient wind and random wind verify that the proposed control strategy can achieve optimal torque tracking and reactive power regulation and is robust to parameter uncertainties such as network voltage, network frequency, rotor resistance, and mutual inductance. The proposed control method also makes rotor control voltage continuous and the control chattering is greatly restrained.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by National Natural Science Foundation of China under Grants 61375084 and 61773015, Doctoral Scientific Research Initial Foundation of Shandong Jiaotong University, Jinan Science and Technology Innovation Program of Shandong, China, under Grant 201401203, and Key Program of Natural Science Foundation of Shandong Province ZR2015QZ08.

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