### Application of Linear Quadratic Regulator to Control Directly Power for DFIG Wind Turbine This paper designs a new control law to control directly the power output of a doubly fed induction generator wind turbine

Application of Linear Quadratic Regulator to Control Directly Power for DFIG Wind Turbine
This paper designs a new control law to control directly the power output of a doubly fed induction generator wind turbine. The proposed control law is based on Linear Quadratic Regulation for both grid side and rotor side converter; the rotor side converter’s controller is to get the maximum power output of the wind turbine and reference reactive power in the stator winding side while the grid side converter’s controller aims to remain a constant voltage on the DC link and a constant reactive power output in the grid side converter. The proposed method is evaluated by simulating a 1.5-MW doubly fed induction generator wind turbine in MATLAB/Simulink environment. From simulation results, we can be seen that by using the suggested controllers, the errors between the reference values and actual values are smaller than that of a proportional–integral controller. Therefore, the doubly fed induction generator wind turbine with proposed controllers has a good performance.

Keywords: Direct power control; doubly-fed induction generator; LQR-based control; wind turbine
1. Introduction
Generally, a doubly fed induction generator (DFIG) has been used popularly in wind turbines, especially in large-scale wind turbine and large wind farms. The main reason is that the DFIG wind turbine can work in variable speed to extract maximum power from wind energy; it is interfaced to the connected grid through a partial scale back-to-back converter, approximate 30% of DFIG capacity 1, 2. However, since DFIG has two output sides including the stator side and the rotor side, the control of DFIG wind turbine is quite complicated.
Until now, many control laws have been proposed for DFIG wind turbine with different control purposes. A conventional controller using PI control has been well-known in literature 2, 3. This controller normally consists of an internal loop and an external loop; the external loop is to adjust power/speed/torque by determining the desired rotor current from errors between the actual power/speed/torque and their references; the internal loop is to adjust rotor current to the desired rotor current. However, determining parameters for PI controller is not easy, in some case, these need to be adapted 4. Sliding mode control was also introduced to replace PI controllers and it can track reference value quite good; however, for the use of the sliding mode control, chattering problem must be considered carefully; moreover, it is hard to be implemented in practice because of multiplying measurement noises by differential equations. Intelligent controller using Fuzzy, neural network… were also suggested 5, 6, 7. For the Fuzzy based controller, propose fuzzy sets and fuzzy laws must be defined carefully while the neural network-based controller requires a training process. A simple controller which is based Linear Quadratic Regulator (LQR) was proposed for DFIG 8, 9, 10, 11. In 8, LQR was implemented to design the pitch controller of DFIG wind turbine; LQR or advanced LQR-based controllers were also designed to adjust the rotor flux 9 and rotor current 10, 11 of DFIG In 10 and 11, two control loops were used, the outerloop is to calculate directly the reference rotor current from the reference power and the interloop adjusts the rotor current following the output of the outer loop by LQR.
In this paper, we are going to implement LQR for directly controlling the power output in the stator side via the rotor side converter, the DC voltage on the DC link and the constant reactive power output in the grid side via the grid side converter. This controller is verified by simulating a 1.5MW DFIG wind turbine in Matlab/simulink. Simulation results are compared to the case of a PI controller.

2. Notation
Constants:
?Air density,
?sSpeed of stator flux in DFIG,
Rs and LsResistance and inductance of the stator winding,
Rr and LrResistance and inductance of the rotor winding,
LmMutual inductance,
Rf and LfResistance and inductance of filter at the grid side converter,
InIdentify matrix n×n.

Variables:
VwWind speed,
?rRotor speed,
CpPower coefficient of wind turbine,
PmMechanical power,
? Pitch angle,
? the tip speed ratio,
vs=vsdvsqT Voltage at the stator winding terminal in dq frame,
vr=vrdvrqT Voltage at the rotor winding terminal in dq frame,
is=isdisqTCurrent in the stator winding in dq frame,
ir=irdirqTCurrent in the rotor winding in dq frame,
vg=vgdvgqT Voltage at the AC side of the grid side converter in dq frame,
ig=igdigqTCurrent output of the grid side converter in dq frame,
Ps and PrActive power output in the stator and rotor side,
QsReactive power output in the stator side,
Sg ,Pg and QgApparent, active and reactive power output in the grid side.

3. DFIG-Wind Turbine
A DFIG wind turbine 1, 12 is interfaced to an alternating current grid through a partial back-to-back converter which consists of a rotor side converter (RSC), a grid side converter (GSC) and a DC link, as shown in Figure 1. Normally, the GSC is interfaced to the power system through a filter.

Figure 1: Configuration of DFIG wind turbine connecting to a grid.

3.1. Wind turbine
Mechanical power on the wind turbine shaft at wind speed Vw is computed as:
where, Cp(?,?) depends on both ? and ? which is determined by
Practically, as ? is kept at constant, Cp has a maximum point at a specific ?. When Vw is over the rated value, Pm will be over the rated value so we need to increase ? to reduce Cp such that we can remain Pmin its permissible operation range. By contrast, ? is kept at a minimum value and we adjust ?r to obtain maximum power coefficient. Suppose that as ?=0, Cp approaches to Cpmax at ?=?opt, Pm automatically becomes the maximum mechanical power Pmax:
3.2. DFIG
In the dq frame, the DFIG can be described as 1:
We suppose that the stator winding’s resistance can be ignored, i.e., Rs=0, the stator flux is kept at a constant, and the d-axis of the dq-frame is aligned with the stator flux vector. Hence, we can get 12
3.3 Back-to-back converter interfacing to the connected grid
In the dq frame which d axis is aligned with the stator voltage, the back-to-back-converter interfacing the connected grid through a filter was described in 12:
From (29), we can obtain (22).

4. Controller design
4.1. RSC controller
The control objective of RSC is in order that the wind turbine operates on the locus of maximum power point and unity power factor in the stator side. To obtain maximum power output, Pe should be equal to Pmppt and from (18) we can obtain the reference value of Ps as
From (30)-(66), we have control diagram of DFIG wind turbine as Figure 2.

Figure 2: Control system of DFIG wind turbine
5. Verification
To evaluate the efficiency of the proposed controllers, in this research, a DFIG wind turbine is simulated in Matlab/simulink. The DFIG-wind turbine is controlled such that it can generate maximum power and supply to the connected grid with unity power factor. In this verification, two cases of wind speed, constant and varying wind speed, are considered. Simulation results of the DFIG wind turbine using the proposed controllers are compared to that using PI-controllers. In this research, we use the DFIG wind turbine with parameters as Table 1.

Table 1: Parameters of DFIG wind turbine
Parameters Symbol Value in pu
The inductance of rotor winding Lr3.056
The inductance of stator winding Ls3.071
The ultual inductance Lm2.9
The resistance of rotor winding Rr0.005
Stator voltage Vs1
The resistance of filter Rf0.015
The inductance of filter Lf0.015
We choose
Qra=100I2?0.01I2, Rra=10-4I2, Kwr=02×2 .

Qga=I4, Rga=I2, Kwg=02×2. .

From these parameters, we can get
KPr=-999.995I2, KIr=1-0.00030.00031, Krref=-10000.3-0.3-1000,KPg=-1, KIg=0.9999-0.01480.01480.9999, Krref=-1.0150.015-0.015-1.015.5.1. Constant wind speed
In this subsection, suppose that wind speed profile at the wind turbine is shown in Figure 3. Simulation results are demonstrated in Figure 4 and Figure 5
Figure 3: Constant wind speed
Figure 4: A comparison of the errors of controllers as the wind speed is constant.

As can be seen from Figure 4, the errors of the proposed controller are very small, less than 1% for both the rotor side and grid side controller. It means the active and reactive power in the stator side are satisfied the reference values, Ps?Psref=?skopt?r2 and Qs?Qsref=0 . Likely, the derivative of Vdc2 on the DC link and the reactive power in the grid side are almost equal to zero. Comparing to the case of the PI controller, the proposed controllers have a better performance.
Figure 5: Performance of DFIG wind turbine with the proposed controllers for constant wind speed.

From Figure 5, we can see that the rotor speed follows the variation of the wind speed. However, due to no wind speed sensor is used, at the 60s the rotor speed responses slowly than the wind speed change. Thanks to the proposed controllers, the active power supplying to the connected grid by the DFIG wind turbine reaches to rated value as the wind speed at rated speed 12m/s. Both reactive power in the stator side and the grid side are equal to zero so the reactive power supplying to the connected grid is zero. It means the DFIG-wind turbine exchange to the connected grid with unity power factor.
5.2. Variable wind speed
In this section, we suppose that the wind profile varies as Figure 6. Simulation results are shown in Figure 7 and Figure 8. Figure 7 indicates that the errors, (Psref-Ps) and (Qsref-Qs), of the proposed controller applying to the rotor side converter is smaller than that of the PI control while the errors of the proposed controller applying to the grid side converter are comparable to that of the PI controller. Thanks for the proposed controllers, the DFIG wind turbine has a good performance as Figure 8.

Figure 6: Variable wind speed profile

Figure 7: A comparison of the errors of controllers for variable wind speed
Figure 8: Performance of DFIG wind turbine with the proposed controllers for variable wind speed.

5. Conclusion
This paper proposed LQR based controllers for the DFIG wind turbine. The RSC’s controller is to control directly power in the stator side in order to get the maximum power output of the wind turbine and the reference reactive power in the stator winding side. The GSC’s controller aims to remain a constant voltage on the DC link and a constant reactive power output in the grid side converter. Simulation results depicted that with the suggested controllers, the errors between the reference values and actual values are smaller comparing to that of a PI controller and the DFIG wind turbine with proposed controllers has a good performance.