Maximum Power Point Tracking of DFIG with DC-Based Converter System Using Coordinated Feedback Linearization Control

This paper presents a coordinated feedback linearization strategy (CFLS) for DC-based doubly-fed induction generator (DFIG) system to track the maximum power point. The stator and rotor of DFIG are connected to DC grid directly by two voltage source converters. Compared with a traditional DFIG system, the DC-based DFIG system has more system inputs and coupling, which increases the difficulty of vector control strategy. Accordingly, CFLS is proposed to make DFIG operate at the maximum power point (MPP), and two aspects are improved: first a single-loop control is adopted to make DFIG operate steady and accurate under coordinated the control ofRSCandSSC. Second systemcontrol laws are obtainedby the feedback linearization strategy that achieves DC-basedDFIG system decoupling fully during theMPPT and system control. Simulations are carried out the comparison between CFLS and conventional vector control (VC), and it shows that the control performance of CFLS is superior.


Introduction
In recent years, the technology of wind power generation has developed rapidly, due to its freely available and renewable resource.Variable speed constant frequency doubly fed induction generator (DFIG) is often selected in wind power generation [1].Compared with the traditional AC transmission, the DC transmission is more economical and stable for the long distance high voltage transmission [2].The traditional DFIG stator windings are connected to an AC grid by transformer, and the rotor windings is connected to the AC grid by back to back converters.Therefore, the traditional DFIG is used into the DC grid and additional converter will be needed, which is bound to increase the cost [3].
Accordingly, the new converter system is adopted for DFIG in DC grid.A diode-based stator converter interfaces a DFIG with a DC grid, which has the advantages of low cost and simple structure [4][5][6][7].But harmonic may reach from 5.97% to 11.66% [8].An IGBT-based converters system consisting of a rotor side converter (RSC) and a stator side converter (SSC) connects rotor and stator with DC grid, respectively [9][10][11][12][13].This structure has the advantages that it can reduce the current harmonic effectively and regulate the stator flux and current flexibly according to the needs of the system, not limited to AC grid.However, the above improved structure results in that the state variables and the system outputs of DFIG with DC-based converters system are twice that of the traditional system, which greatly increases the complexity and coupling of the system and also increases the control difficulty of the system.Therefore, the traditional vector control (VC), which is based on approximate linear model, is difficult to achieve the global optimal control requirements of the new system [14].Reference [9] adopts indirect air gap flux linkage orientation strategy to control DFIG in a DC grid, but air gap flux linkage is not suitable for measurement.Model predictive control for DFIG in a DC grid is adopted by [13], which requires high precision of system model.
Nowadays, Feedback linearization control (FLC) has been widely used in power electronics and power systems [15][16][17][18][19][20][21].It adopts a dual-loop control strategy of FLC and PI to control DFIG in an AC grid in [22].The advantage of   this method is simple to calculate, but the disadvantage is that could not achieve full system decoupling.Article [23] adopts rotor speed and stator reactive power of DFIG in an AC grid as state variables for FLC, which computationally complex and does not select direct system parameters to controller which affects control accuracy [24].This paper designs a CFLS for DC-based DFIG to achieve maximum power point tracking (MPPT).This control strategy employs stator flux, rotor speed, and rotor current as system inputs, applies FLC to make up a single-loop control, and achieves complete decoupling of the system, which attains coordinated optimal control performance between the SSC and the RSC.This proposed control strategy has better control accuracy and tracking speed than the traditional VC strategy.
The paper is organized as follows.In Section 2, the model of DFIG based on DC grid is introduced.In Section 3, the maximum wind power is achieved by designing CFLS.
In Section 4, simulation studies are evaluate the control performance of the proposed control strategy on a DCbased DFIG system.Finally there is conclusion of this paper.

Modeling of DFIG System
2.1.System Configuration.DFIG with its DC-based converters system in a DC grid is shown in Figure 1 [12].The stator and the rotor of DFIG are connected to a DC grid through SSC and RSC.The stator voltage and frequency are completely unrestricted by the power grid.SSC can adjust stator voltage and frequency to control stator current and compensate stator reactive power.RSC can adjust rotor voltage to control rotor flux.The du/dt filter inductors are connected to stator and rotor, respectively, to prevent sharp voltage caused by converters and smooth currents.

DFIG Model Description.
In this paper, DFIG model is adopted according to the motor direction in a d-q synchronous frame that can be expressed as [24,25] where

MPPT Control Strategy.
Usually the maximum kinetic power captured from the wind by a wind turbine is expressed as follows [26]: where  is air density,   is radius of wind turbine,   is wind speed, and   is maximum power coefficient.
To capture the maximum wind power, the power coefficient   should maintain maximum   at any wind speed  within the operating range.Maximum   is achieved by maintaining the tip speed ratio  equal to optimal value   and the pitch angle  at a fixed value.
The tip speed ratio  indicates the state of the wind wheel under different wind speeds, as When  is equal to   , the optimal reference  *  is In this paper, the pitch angle is  = 0 ∘ , the optimal tip speed ratio is   = 9.7, and the maximum power coefficient is   = 0.4642.  is shown in the Figure 2.

CFLS of DFIG.
From ( 1)-( 3), the DFIG model can be derived in the form where This is a multi-input multioutput (MIMO) system.An approach to obtain the inputCoutput linearization of the MIMO system is to differentiate the output   of the system until the inputs   appear, assuming that the corresponding relation degree   is the smallest integer such that at least one of the inputs explicitly appears in [27] where denotes the   th-order derivative of   .In (8),  is 5 dimensional state phase quantity,   () ( = 1, 2, . . ., 5) is 5 smooth vector field, and   () ( = 1, 2, 3, 4) is 4 smooth vector field.Each output   has a   , and by calculating that is 1, 1, 1 and 2. The system relation degree is  = 1 + 1 + 1 + 2 = 5 = ; therefore, the system has no nontrivial zero dynamics, which can be linearized by feedback linearization.According to (10) and   of each output   , the Lie Derivative of ℎ with along  and the Lie Derivative of   ℎ() with along () are obtained, and they are According to (8) and (11), the system can be described in the following matrix form: where There is det(()) = 2 2 / 2 ̸ = 0, so the inverse matrix () −1 exists.A new input variable is defined for input-output feedback linearization, as The conversion relation between the original input variable and the new input variable is as follows: where .Thus, the inputoutput mapping of ( 8) can be simplified as (14), which realizes the linear decoupling between the system output and input variables.point is moved to the origin, and the input variable can be redesigned into ( 14) In the controller design we can choice the parameters   and   ( = 1, 2, 3, 4) in ( 15) and ( 16) to ensure the convergence and stability of the   ( = 1, 2, 3, 4).Now we have derived that the partial states  1  2  3  4 track the setting reference points.Therefore, from the above analysis, the whole control scheme is shown in Figure 3.

Reference Points of Controller.
The rotor flux and stator current are the control targets; the method determined in this paper is that, when a stator-oriented flux frame is adopted with its vector direction aligned with the q-axis, from (2), the stator flux reference value ( *  ) and its d-q components are given by where   is generator rated voltage amplitude,  1 is synchronous speed.In this paper,   = 1.0 pu,  1 = 1.0 pu.Substituting ( 17) into (2), the rotor current are obtained: Substituting ( 14) into (19), that can be derived as From ( 15), the states  1  2  4  5 are convergent to  *   *   *   *  , and further from ( 17), (18), and (7) know that their values are 0 − 1 − 0.34 0.09  , so that dynamics (20) can be approximated as In this paper, the rated wind speed of wind turbine is    = 13 m/s, and the variable wind speed range is focus on 6-14 m/s.And the high frequency components in the wind speed measurement   may cause undesired noise; therefore the measured wind speed is passed through lowpass filter to attenuate its effect [28].It is assumed that the time (s) The above can be known, the state variable  3 (  ) is bounded, when  → ∞.

Simulation Results
In this section, simulation results are carried out in MAT-LAB/SIMULINK, to verify the coordinated optimal control performance in a wide range operating conditions.There are two tests performed: case one shows the accuracy of CFLS compare with the VC under the ramp-change wind and case two is the system that operates at random-change wind to show the tracking speed of controller.The DFIG system parameters in the following simulation are listed in Appendix A. As shown in Figure 5, the CFLS can well capture the maximum power coefficient   value, but the VC needs long time to catch   .It means that the wind turbine maintains the maximum output power under the proposed control strategy.
Figure 6 shows the system output DC-voltage   that connected to a stable DC-grid.It can be seen that the DC voltage remains stable under the control of CFLS and VC, when the wind ramp changes.
The rotor currents and voltages (  ,   ), and the stator currents and voltages (  ,   ) can be seen that the DFIG is operated at the rated value while under the CFLS in Figure 7.   and   are sine wave with frequency of 60 Hz, and their amplitudes are 1.0 pu and 0.8 pu.  is sine wave with amplitude of 0.85 pu.  is the RSC output pulse wave, and its amplitude is 0.7 pu.

Figure 1 :Figure 2 :
Figure 1: DFIG with its DC-based converter system in a DC grid.

Figure 6 :
Figure 6:   at the ramp-change wind speed.
Stator voltages   under CFLS

Figure 7 :
Figure 7: Performance response to the ramp-change wind speed.

Figure 10 :
Figure 10:   at the random-change wind speed.

4 (Figure 11 :
Figure 11: The output control signals response to the random-change wind speed.

4. 1 .
Ramp-Change Wind.Figures 4-7 shows the performance of a DFIG in DC-grid controlled by the VC and CFLS under the ramp-change wind.The wind speed condition is depicted in Figure 4(a).The wind speed rises from 7 m/s to 14 m/s and then remains 14 m/s.It can be seen from Figure 4(b) that the optimal reference rotor speed  *  is well tracked by the CFLS, while the response of the VC with larger overshoots is slower.Figures 4(c), 4(e), and 4(f) show that the tracking references performance of VC and CFLS and the tracking accuracy of CFLS are better than that of VC.The state variable  3 (  ) is shown in Figure 4(d), as it keep stable in both VC and CFLS under ramp-change wind.

4. 2 .
Random-Change Wind.The performance of a DC-based DFIG system controlled by the VC and CFLS under randomchange wind is shown in Figures 8-11.The wind speed

)
(8).Stability Analysis of StateVariable  3 .From the error equation (16), we obtain that the partial states  1  2  4  5 are bounded are converge to the reference point.Next we will analyze the boundedness of the state  3 .From(8), the equation about  3 is extracted