PO-09
DAC with LQR Control Design for Pitch Regulated Variable Speed Wind Turbine Raja M. Imran1, D. M. Akbar Hussain2 and Mohsen Soltani3 Department of Energy Technology, Aalborg University, Denmark
[email protected] ,
[email protected],
[email protected] Abstract - Disturbance Accommodation Control (DAC) is used to model and simulate a system with known disturbance waveform. This paper presents a control scheme to mitigate the effect of disturbances by using collective pitch control for the aboverated wind speed (Region III) for a variable speed wind turbine. We have used Linear Quadratic Regulator (LQR) to obtain full state feedback gain, disturbance feedback gain is calculated independently and then estimator gain is achieved by poleplacement technique in the DAC augmented plant model. The reduced order model (two-mass model) of wind turbine is used and 5MW National Renewable Energy Laboratory (NREL) wind turbine is used in this research. We have shown comparison of results relating to pitch angle, drive train torsion and generator speed obtained by a PID controller and DAC. Simulations are performed in MATLAB/Simulink. The results are compared with PID controller for a step wind and also for turbulent wind disturbance. DAC method shows better regulation in output power and less fatigue of drive train in the presence of pitch actuator limits. Proposed controller tested on wind turbine shows better robustness and stability as compared to PID. The paper describes practical experiences to develop a new DC power plant controller user interface based on humancentered design ideas.
I.
INTRODUCTION
Wind is a safe, ecological and environment friendly source of renewable energy. It has less aging than nuclear plant, less fuel emission and protects fossil fuel sources for future generations. Wind turbines are used to convert winds kinetic energy to electricity. Wind turbines are classified as variable speed and fixed speed. Variable speed wind turbines are common these days because they captures more wind power than constant speed turbine and it can be operated at its maximum power efficiency for a wide range of wind speed. Most failure of the wind turbine are associated with structural dynamics and as modern turbines are larger in size, so its component like blades, hub, low- and high-speed shaft, gearbox, generator, nacelle, and tower becomes very expensive. Various control schemes are used to regulate turbine power in the presence of turbulent wind and reduce dynamic loads acting on wind turbine. Wind turbine is a complex nonlinear system and its output power is directly proportional to the cube of wind speed. Therefore turbulence in wind produces fluctuation in the output power, fatigue of components (i.e., Gearbox, Generator etc.) that potentially reduces the life time. Sophisticated control schemes are
978-1-4799-3104-0/14/$31.00 ©2014 IEEE
required to achieve the control objectives because the above issue varies with change in wind speed. For above rated wind speed scenario, control objective is to regulate rotor speed at its rated value to get rated power and mitigate the effect of wind variation using pitch control for variable speed wind turbine. Various linear and nonlinear techniques have been used to meet the control objectives for the wind turbines operating at above-rated wind speed. Classical control based on Proportional-Integral-Derivative (PID) controller discussed in [1] is used to regulate rotor speed by using pitch mechanism to reduce wind disturbance effect in region III and other techniques are used in [2] and[3] to design a controller to reduce load on various components of the wind turbine. As wind turbine models are nonlinear, so [4] and [5] have used expert PID controller for better performance and to reduce vibration generated during its operation. Linear Quadratic Gaussian (LQG) is an optimal control technique designed to minimizing certain quadratic criterion depending on objectives and taking into account the stochastic nature of wind speed. LQG techniques is discussed in [6] and used on wind turbine to multiple objectives in [7], [8], [9], [10]. DAC is model based control approach to reject disturbance from a linear system. Similarly Periodic Disturbance Accommodation control techniques is discussed in [11] to reduce load, [12] is about disturbance tracking control theory and [13] shows optimal method based DAC scheme to cope on control objectives for wind turbine. In this paper, DAC control scheme is designed to reduce disturbance effect in above rated wind speed for 5MW NREL wind turbine. We have used optimal control theory to design full state feedback gain in the presence of actuator dynamics and also disturbance feedback gain is calculated independently. Pole-placement technique is used to estimate plant states as well as disturbance states from the disturbance accommodated model of the system. Reduced order two-mass model of wind turbine is used and then its linearized threestate model is generated at an operating point for the plant for above-rated wind speed. We have done simulation in MATLAB/ Simulink and finally results are analyzed in the presence of the pitch actuator dynamics. This paper is organized as follows; Section II is the modeling of wind turbine and Section III discuss the control theory. Section IV is the simulation and discussion about results is in Section V.
PO-09 II.
WIND TURBINE MODELING
Wind turbine is a complex nonlinear system but here we are using its reduced order two-mass model. Wind turbine has three main subsystem i.e., aerodynamics, mechanical (drive train) and electrical (generator). Aerodynamic power available on the swept area of wind turbine rotor is given by = (1) A is the swept area of the rotor blades, ρ is air density and is the wind speed. Wind turbine can extract a part of available power represented by ( , )
=
;
=
(2)
Where β is rotor blade pitch angle. Cp (λ, β) is power coefficient, it is nonlinear function of pitch angle and tip speed ratio λ. r is rotor angular rotational speed and R is the length of rotor blade. Drive train is transmission system which convert low rotational speed to high rotational speed to drive the generator as shown in Fig. 1. Its dynamics can be represented by the following differential equations and discussed in [10], [15]. = = =
−(
− +
)
− −
−
(3) +
−
−
(4) (5)
represents the inertia of the rotor and shaft, is damping and constant of the friction of low speed shaft. represent the stiffness and damping coefficients of the drive train respectively. The high-speed shaft is modeled by an inertia and a damper. , represents the sum of the inertia of the high-speed shaft, the gearbox, and the rotor of the represents the friction of the high-speed shaft generator. is the aerodynamic torque, is the generator bearings. is the gear ratio, is the efficiency of the drive torque, train. is generator speed and is the shaft torsion.
We have considered that low speed shaft bearings are frictionless i.e., B = 0, high speed shaft bearing are frictionless i.e., B = 0 and η = 1. Let x, u, u be the state, input and disturbance deviation from the operating point (x ∗ , u∗ , u ∗ ) respectively i.e., x = x − x ∗ , u = u − u∗ , ∗ u =u −u . =
=
;
= ;
(6)
=
Then the linearized three-state space two-mass model of drive train becomes − 0 0 −
= −
1
=
Where
1=
−
∗
+
0
1
0 0
, 2=
∗
+
, 3=
+
0
0
0
(7)
(8)
u
∗
A, B, C, B , D are state transition, control input, measured state, disturbance input and output gain matrices of the plant respectively. III.
CONTROLLER DESIGN
DAC is a control scheme to augment the known waveform disturbance with the states of the system, so that it becomes a part of the exogenous system. Then we can design an observer which can estimate the states of the plant as well as states of the disturbance. Disturbance feedback is used with state feedback to mitigate the effect of the disturbance. State space model of plant is of the form ( )=
( )+ ( )+ ( )= ( )
( )
State space model of the disturbance waveform is ( )= ( ) ( )= (t); (0) =
(9) (10)
(11) (12)
Where is state of the disturbance, is the initial state of the disturbance. F and are known matrices of disturbance waveform i.e., for step waveform; = 1, F = 0 and for ramp; 0 1 as discussed in [9]. θ = [1 0 ], F= 0 0 Figure 1: Two-Mass Model of Wind Turbine
The control law is = −(
( )+
̂ ( ))
(13)
PO-09 Where is state feedback matrix, is disturbance feedback matrix, is estimated state of the plant and ̂ is estimated state of the disturbance. If the system (A, B) is controllable then can be calculated on the basis of LQR control theory. LQR is full state feedback optimization problem with quadratic cost function. It is desired to minimize a performance index that represent optimal performance of a plant and it is (
=
( ) = ( )− ( ) ( )=( − Also,
( ) = (−
Weighing matrices Q symmetric positive semi definite and R = ≥0 , is symmetric positive definite matrix, i.e., = ≻ 0. Control Performance depends upon the selection of Q and R matrices, various techniques for their selection are discussed in [6] and [10]. Also is calculated independently to reject the effect of wind and to make closed loop system “disturbance free”. (
)
(14)
( )+
( )=
( ) − ( ) (15)
(16)
( ); (0) = 0
Disturbance state estimator is of the form ( )+
̂ =
( )=
( )− ( )
−
̂ ( ); ̂ (0) = 0
( )+(
) ( )+
(18)
−
(25) ( )
So
( )=( −
Where =
,
0
=
) ( )
(28)
0,
=
(29)
Let
(30)
If the disturbance augmented plant (A, C) is observable then we can design the state estimator for the disturbance augmented plant by proper selection of the poles of the system. So by arranging equation 12, equation 20, equation 21 and equation 22 in matrix form, we get the state space model of the DAC augmented plant 0 = ̂
0 0 0
−
− −
0
−
0 −
−
(19)
Simulink model of the DAC augmented plant is shown in Fig. 2.
( ) (20)
By using equation 13 in equation 15, we get ( )=
+
−
−
( )+(
−
) ̂ ( )
(21)
Putting equation 13 in equation 17, we get ̂ ( )=
−
+
̂ ( )
̂
+ 0 0 0
(31) ) ̂ ( )
̂ ( )+
( )−
−
(26)
(27)
, and are estimated input, disturbance and output respectively. is state estimation matrix and is disturbance estimation matrix. Now putting equation 13 in equation 9, we get =
(24)
( )
( )
( )=
(17)
Applying feedback control law to the plant =
( )− ̂ ( )
=
( )+
( )+
( )
Let
Plant state estimator is of the form =
(23)
) ( )+
( )=
)
+
=
( ) and ( ) be the plant and disturbance state estimator error respectively. Then by using equation 12, equation 20, equation 21 and equation 22, we can write
(22)
Figure 2: DAC Simulink Model
PO-09 IV. SIMULATION We have used 5MW wind turbine as research object. Its state space model is generated by using parameter values of the turbine discussed in [14] at operating point (12.1 rpm, 15.2 deg, 18 m/s) and is −1.23
119 −0.28 1
A= 1.80 × 10 −0.01
1.67 × 10 −24.47 0
= 0 −5.14 × 10 = 0
0.02
= 1
0 0
0 0
Figure 3: Comparison of the Drive Train Torsion for Step Wind
=0 Here we have considered disturbance as a step change in wind speed so F = 0 and = 1 . Full state feedback matrix is calculated by LQR by using weighting matrices for optimal performance and disturbance feedback matrix is calculated independently. =
0.5 0 0 0 0 0
0 0 , 0
= 0.14 −147
= 0.2
−114
−3.88
Observer gain is calculated by pole-placement by placing poles of disturbance augmented plant at −150 −160 −170 −180 = 658
752
Figure 4: Comparison of the Pitch Angle for Step
4.32 2.19 × 10
We have calculated the transfer function of the plant from the wind to output from DAC augmented plant model 2.07s (s + 139.6)(s + 0.68)(s2 + 1.58s + 197) ____________________________________________________ s (s + 180)(s + 170)(s + 160)(s + 150)(s + 0.6)(s2 + 1.58s + 197)
We have tuned PI controller to reduce the effect of wind disturbance at the output and its optimum parameters values are Kp = −0.40 and Ki = −0.15. Simulations are performed in MATLAB/Simulink with the actuator dynamics which Include pitch angle saturation and rate limit as mentioned in [14]. We have compared PI controller with DAC for step wind. Drive train torsion, input pitch angle and generator speed step response are shown in Fig. 3, Fig. 4 and Fig. 5 respectively.
Figure 5: Comparison of Generator Speed for Step Wind
PO-09 Then we have applied the turbulent wind as shown in Fig. 6 to the system and compared the results with PI controller. Results of the drive train torsion, pitch angle and generator speed are shown in Fig. 7, Fig. 8 and Fig. 9 respectively for the turbulent wind. It can be seen that our proposed controller shows less fluctuation in generator speed and less fatigue on drive train as compared to PI controller.
Figure 8: Comparison of Pitch Angle for Turbulent Wind
Figure 6: Turbulent Wind
Figure 9: Comparison of Generator Speed for Turbulent Wind
Figure 7: Comparison of Drive Train Torsion for Turbulent Wind
V. DISCUSSION We have used step wind to analyze the controller performance and then we have tested this controller with turbulent wind. This paper has presented a DAC controller design methodology with optimal control theory and its mathematical modeling. State feedback matrix is calculated for the using LQR. Although we have used trial and error scheme to adjust the values of weighting matrices in the
presence of actuator rate and saturation limits but we have tested scheme discussed in [6] and [10]. We have generated two-mass state space model of wind turbine for above rated wind speed and then DAC controller is implemented in MATLAB/Simulink. We have tuned PI controller for robust performance to mitigate the effect of disturbance and used MATLAB to tune it properly for optimum performance. Disturbance feedback matrix is calculated to mitigate effect of wind on the plant and make system disturbance free and to get better estimation of the wind speed. We have compared the results for a step wind and then for the turbulent wind. Results for step shows that there is less overshoot and better settling time of the proposed controller as compared to PID. Then results with turbulent wind shows less fluctuation in generator speed which will result stability of output power and less fatigue of drive train to increase its life time. The performance analysis of the system shows that our proposed controller have better power regulation and less fatigue of drive train as compared to PID controller for 5MW wind turbine in the presence of actuator dynamics.
PO-09 REFERENCES [1] M.M. Hand, and M.J. Balas, “Systematic Approach for PID Controller Design for Pitch-Regulated Variable-Speed Wind Turbines,” In Proc. ASME Wind Energy Symposium, Reno, Nevada, 12-15 January,1998. [2] E.A. Bossanyi, “The design of closed loop controllers for wind turbines,” Wind Energy 2000; 3:149163. [3] E. A. Bossanyi, “Wind Turbine Control for Load Reduction,” Wind Energy 2003; 6:229244. [4] Y. Xingjia, L. Yangming, X. Zuoxia, and Z. Chunming, “Active Vibration Control Strategy based on Expert PID Pitch control of Variable Speed Wind Turbine,” In Proc Int. Conf. Electrical Machines and Systems, 17-20 Oct. 2008. [5] X. Anjun, X. Hao, H. Shuju, and X. Honghua, “Pitch Control of Large Scale Wind Turbine Based on Expert PID Control,” In Proc. Int. Conf. Electronics, Communications and Control, 9-11 Sept. 2011. [6] B.D.O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods. New Jersey: Prentice-Hall, 1989. [7] Y. Xingjia, G. Changchun,X. Zuoxia, L. Yan, L.Shu, and W. Xiaodong, “Pitch Regulated LQG Controller Design for Variable Speed Wind Turbine,” In Proc. Int. Conf. Mechatronics and Automation, 9-12 Aug. 2009. [8] A. Kalbat, “Linear Quadratic Gaussian (LQG) control of Wind turbines,” In Proc. 3rd Int. Conf. on Electric Power and Energy Conversion Systems (EPECS), 2-4 Oct. 2013. [9] A.D. Wright., “Modern Control Design for Flexible Wind Turbines,“ NREL Report No. TP-500-35816, Golden, CO: National Renewable Energy Laboratory, 2004. [10] Raja M. Imran, D. M. Akbar Hussain and Zhe Chen, “LQG Controller Design for Pitch Regulated Variable Speed Wind Turbine,” In Proc. Of IEEE Int. Energy Conference (ENERGYCON), Dubrovnik, Croatia, 13- 16 May, 2014. [11] K.A. Stol, and M.J. Balas, “Periodic Disturbance Accomodation Control of blade load mitigation in WT,” Journal of Solar Energy Engineering, vol. 125, no. 379, 2003. [12] K.A. Stol, “Disturbance Tracking and Blade Load Control of Wind Turbines in Variable-Speed Operations,” In Proc. of ASME Wind Energy Symposium, Reno, Nevada, 6-9 Jan. 2003. [13] J. Li, H. Xu, L. Zhang, Zhuying, Shuju, and Hu, “Disturbance Accommodating LQR Method Based Pitch Control Strategy for wind turbines,” In Proc. 2nd Int. Symposium on Intelligent Information Technology Application, 20-22 Dec. 2008. [14] J. Jonkman, S. Butterfield, W. Musial, and G. Scott,”Definition of a 5MW Reference Wind Turbine for Offshore System Development”, Technical Report NREL/TP-500-38060,February 2009. [15] M.N. Soltani, T. Knudsen, M. Svenstrup, R. Wisniewski, P. Brath, R. Ortega, and K. Johnson, “Estimation of Rotor Effective Wind Speed:A Comparison,” IEEE Transactions on Control Systems Technology, Vol. 21, No. 4, July 2013.
