SymDyn, six are chosen to represent the nonlinear plant model in this study; tower fore-aft deflection (Ï1), nacelle yaw (γ), rotor azimuth position (Ï), hub teeter.
AIAA-2002-0053
PERIODIC DISTURBANCE ACCOMMODATING CONTROL FOR SPEED REGULATION OF WIND TURBINES Karl Stol AIAA Student Member Mark Balas AIAA Associate Fellow Department of Aerospace Engineering Science University of Colorado at Boulder, Boulder, CO ABSTRACT Performance of a model-based periodic gain controller is presented using Disturbance Accommodating Control (DAC) techniques to estimate fluctuating wind disturbances. Operation is restricted to speed regulation using Independent Blade pitch Control (IBC) and by measuring only rotor angle and speed. The modeled turbine is a free-yaw, two-bladed, teetered rotor machine with simple blade and tower flexibility – 6 discrete degrees of freedom. A comparison is made to a time-invariant DAC controller, constructed by various approaches. Results indicate that the two controllers perform nearly identically, despite the inherent periodic dynamics in the system. The best time-invariant state estimator model was designed by freezing the periodic plant at a specified rotor azimuth position. INTRODUCTION Variable-speed wind turbines offer improved energy capture over constant speed machines due to the increased range of wind speeds that maximum power can be generated. Rotor speed must be regulated either to achieve maximum aerodynamic efficiency (often referred to as Region II) or to ensure mechanical limitations are not exceeded in high winds (Region III). (We define Region I as start-up, when there is insufficient wind to produce electrical power.) To control rotor speed one can choose to either regulate the generator back-torque or manipulate the blade pitch angles. The advantage of blade pitch actuation is that aerodynamic loads are controlled directly without undesirable transmission through the drive train and turbine structure. In this paper we deal solely with speed regulation by independent blade pitch control in Region III. Disturbance Accommodating Control (DAC) was originally developed by Johnson [1] as a model-based approach to reject persistent disturbances to a system. Copyright 2001 by the American Institute of Aeronautics and Astronautics Inc. and the American Society of Mechanical Engineers. All rights reserved.
The application of DAC to wind turbines has been made by the authors and their colleagues [2]-[3] to successfully estimate wind speed disturbances while regulating rotor speed. In these studies a single-state time-invariant turbine model was used. Research has also been published on new DAC theory applicable to Region II operation, called Disturbance Tracking Control [4]. The present study investigates the performance of time-varying DAC to capture the periodic nature of wind turbine dynamics. Periodic dynamics in a wind turbine arise from both structural and aerodynamic effects. A two-bladed rotor is asymmetric in terms of mass properties in the nonrotating reference frame. In particular, nacelle yaw motion is greatly affected, as the moment of inertia about the yaw axis is periodic with the rotation of the blades. A non-uniform wind field also contributes to periodic effects since each blade samples a changing wind velocity with each revolution. The dominant sources are wind shear and misaligned wind direction. The turbine model in this paper incorporates free-yaw, a two-bladed rotor, and vertical wind shear factor to capture the more significant contributions to periodic dynamics. Periodic control is essentially the use of time-varying feedback gains with a fixed time period. In a wind turbine the period would correspond to the time of one rotor revolution. While periodic control techniques have been applied to various fields, including helicopters [5] and spacecraft orbit optimization [6], wind turbines have received little attention. This is despite the fundamental periodicity in dynamics, as mentioned. Recently the authors published preliminary results on periodic control of wind turbines with fullstate feedback [7]. Results here indicate that periodic control performs no better than constant gain control in speed regulation using collective blade pitch. It has since been proposed that independent blade pitch control (IBC), as used in the helicopter field [5], would improve periodic control performance. This paper extends on previous work by incorporating periodic
state estimation, disturbance accommodating control, and IBC actuation.
Instead of modeling the generator in detail a constant generator torque is applied to the shaft, with a reaction torque on the nacelle. Turbine speed is then regulated by blade pitch alone.
The following sections describe the turbine modeling approach and the procedure for designing both periodic and time-invariant state-space controllers. Results are then presented to compare performance in speed regulation.
It is convenient to represent the entire nonlinear aeroelastic plant by the following vector equation.
where
Nonlinear Dynamics The modeled turbine is a generic downwind, free-yaw, direct-drive, teetered rotor machine with two blades. It is assumed that sensors on the shaft can relay accurate and noise-free information for the rotor position and speed.
Symbol dhh dt1 dn2 dh1 dh2 0
q 1 q
0 ,
(1)
1 2 , the d.o.f.' s, T
dq
, the angular velocities, dt d2 q q 2 , the angular accelerati ons, dt M is the familiar 66 mass matrix in terms of moments of inertia and angular positions, L is the vector function of applied aero loads, w is the horizontal wind speed, and = [1 2]T, the independent blade pitch angles.
SymDyn is used to formulate the structural dynamics, which models the flexibility of the tower and blades as rigid bodies with torsional joint springs [8]. Of the many degrees of freedom (d.o.f.’s) available to SymDyn, six are chosen to represent the nonlinear plant model in this study; tower fore-aft deflection (1), nacelle yaw (), rotor azimuth position (), hub teeter () and flap angle of each blade (1, 2). See Figure 1 for an illustration. Properties are chosen to resemble the AWT-26 horizontal-axis wind turbine, with its main geometric properties listed in Table 1. Description Hub height Height of tower fore-aft hinge Yaw axis to teeter joint distance Teeter to flap hinge distance Shaft to flap hinge distance Blade length Precone angle
M q q f q, q , L q, q , w,
MODEL DESCRIPTION
Eqn (1) is implemented in Matlab with Simulink, forming the „Nonlinear Plant (6 d.o.f.)‟ subsystem block in Figure 2. Linearized Model For control design the turbine model is reduced to 4 d.o.f. by eliminating tower fore-aft deflection (1) and hub teeter (). The yaw d.o.f. is kept because it contributes most to the periodicity of the system compared to tower and teeter motion. Evidence for this claim can be found in previous publications, [7] and [8]. A reduced order linear controller simplifies gain design calculations.
Value 25 m 9.3 m 2.4 m 0.4 m 5.8 m 13 m 7
Linearization is performed about an operating point in Region III, in which the turbine is aerodynamically stable and the blades are predominantly unstalled. We choose a wind speed of wop = 16 m/s and pitch angles of 1 op = 2 op = 15. The nominal rotor speed of the turbine of 57.5 rpm is then achieved on average by a constant generator torque of 13.4 kNm.
Table 1: Geometric properties of the turbine model
The structural model is coupled with aerodynamics using an interface to the AeroDyn code [9]. Loads are calculated at prescribed elements along each blade length, using blade-element momentum theory. The element loads are then summed and applied to the structure at the blade flap hinges, consistent with the SymDyn configuration. Models for dynamic inflow and dynamic stall in AeroDyn are not incorporated. A simple hub-referenced wind field is used. This consists of a horizontal wind speed at the hub-height, w, and a vertical distribution of wind speed governed by a power law, with exponent 0.2.
An assumption of linearization is that the nonlinear equation of motion (1) is satisfied at the operating point.
M(q op ) q op f (q op , q op , L (q op , q op , w op , op )) 0 (2) The trim or steady-state solution of (2) is periodic in time, with periodic equal to the time of one rotor
2
Q = CTC and R = I2x2,
revolution at the operating speed, T = 1.04 s. Except for azimuth position, the steady-state solution has only small variations in time and therefore mean values are adequate. We use
0
which weights the rotor azimuth error and rotor speed error states only. This results in the periodic gains shown in Figure 3, plotted against azimuth position instead of time. Closed-loop stability of (A(t) + B(t)G(t)) is guaranteed when (A(t), B(t)) is reachable. The test for reachability for a periodic system is described in [10] and is satisfied with our given system.
op t 2.4 2.4 T , q op 1.6 q op
q 0. op
op
T
0 0 , and
(3)
op 57.5 rpm) (
The disturbance gains, Gd(t), are calculated independently from the full-state feedback gains and are designed to minimize the effect of disturbance input to the plant. Assuming a step disturbance waveform (a worst case scenario for wind speed) we set
Recall that q = [ 1 2]T for the reduced control model. In familiar state-space form the linear plant is represented by x A( t ) x B( t ) u Bd ( t ) u d y Cx
(4)
Gd(t) = B(t)+ Bd(t)
where T x q q , the state vector containing angular
where B(t)+ = B(t)T (B(t) B(t)T)-1, the pseudoinverse of B(t).
positions and velocities as perturbations about the operating point (3), A(t) is the state matrix, B(t) is the control input matrix, Bd(t) is the wind disturbance input matrix, u = , the control input (blade pitch), ud = w, the disturbance input (wind speed), T , the measured output, and y
We cannot implement (5) in practice because the states of the plant and the wind input are not all measurable. The realizable control law is then
u G(t ) xˆ G d (t ) uˆ d
(7)
where xˆ and uˆ d are the state and disturbance estimates respectively.
0 1 0 0 0 0 0 0 C the output matrix. 0 0 0 0 0 1 0 0
Following standard state-estimator theory we use the available state information from the measured output, y, to reconstruct all states. DAC theory [1] requires a disturbance waveform generator to augment the statespace description of the plant. With dual periodic LQR techniques we solve for estimator gains, K (t ) , given the
Note that the linear matrices in (4) are periodic, with period T, due to the periodic nature of azimuth position at the operating point (3). They are formed by differentiation of the equation of motion (1); symbolically for the SymDyn structure and numerically for aerodynamics.
augmented system matrices (A(t ), C) , where
PERIODIC DAC DESIGN
A( t ) B d ( t ) A( t ) , C C 0 . 0 018
Consider the ideal periodic DAC control law as a superposition of full-state and disturbance rejection components: u* = G(t) x + Gd(t) ud
(6)
The chosen weightings on estimator states, QE, and output sensing, RE, are
(5) QE = diag([0 1 0 0 0 103 0 0 105]) and RE = I2x2.
The full-state gains, G(t), are designed for desirable transient behavior using optimal periodic control (or linear quadratic regulation, LQR) techniques [7,10]. Similar to the time invariant system case we choose a weighting for state regulation, Q, and a weighting for control usage, R. As our goal is rotor speed regulation we choose
The nonzero entries in QE correspond to state estimates for azimuth error, rotor speed error, and wind disturbance respectively. The augmented estimator system, given by (A(t ) K(t ) C) , is stable when
3
particular freezing angles are chosen. See [4] for details.
(A(t ), C) is observable. The observability test in [10] is
satisfied with our system.
To calculate constant full-state feedback gains, Gc, we investigate two approaches.
When implemented in Matlab with Simulink the block diagram for periodic DAC simulations is illustrated in Figure 2. Note that the azimuth signal, measured directly from the plant, is used to synchronize three essential components of the system; the full-state feedback gains (G(t) and Gd(t)), the estimator gains (K(t)), as well as the linear periodic plant model within the estimator subsystem.
1.
LQR of the LTI plant model. Standard calculation methods are available [11] using the same Q and R weightings from the periodic control design.
2.
Mean of periodic gains.
T
G c G( t ) dt
TIME-INVARIANT DAC DESIGN
0
The control system comprises both linear gains and a linear plant model for state estimation. The calculation of linear time-invariant (LTI) gains is a straightforward process, as there are many possible approaches contributed by the wealth of literature on the subject. In previous work a number of methods have been investigated, particularly for full-state feedback control [4,7]. The choice of an appropriate LTI plant model is less trivial. This is because it is not clear what the optimal choice should be in terms of performance. In this paper we explore the following two options.
The same approaches are available to calculate the constant estimator gains, K c , given disturbance augmentation of the state-space representation and the principle of duality. Therefore, for each plant model used (averaged or frozen) there are four combinations for calculating gains. RESULTS The wind disturbance input used in the simulations is based on actual data taken from the National Wind Technology Center in Colorado. The solid line in Figure 4 illustrates the 100-second sample. Recall that a vertical wind shear factor is also present.
1. Averaged Model
x a A a x a Ba u Bd a u d
(8)
where
A comparison of performance is made between the periodic controller and the various time-invariant controllers. All controllers are based on the linear 4 d.o.f. turbine model, while the simulation plant is nonlinear with 6 d.o.f.. Performance is measured by two metrics, RMS speed error and actuator duty cycle (ADC). Actuator duty cycle is related to pitching rate and measures the total angle pitched by the blades divided by the total simulation time. The maximum from the two blades is used. Generally, the better the speed regulation (lower RMS speed error) the higher the control usage (higher ADC). This is the typical design trade-off.
B( t ) dt , and B ( t ) dt . T
A a A( t ) dt , 0
T
Ba
0
T
Bd a
0
d
2. Frozen Model x f A f x f Bf u Bd f u d
(9)
where Af = A(tf), Bf = B(tf), and Bd f = Bd(tf),
Periodic DAC Simulation results for rotor speed and blade #1 pitch usage with the periodic controller are presented in Figure 5 and Figure 6 respectively (thin dark line). The pitch response follows the general trend of the wind speed, as expected. The pitch angle of blade #2 is similar in trend to the pitch of the first blade, except for a small variation over each rotation. This trend is shown in Figure 7 over a reduced time scale and is due
The freezing time, tf [0,T], is chosen to correspond to a rotor azimuth angle of 90 (blades horizontal). This selection is made on the basis of a parametric control study with full-state feedback, by comparing system performance at various freezing azimuth angles. Using an angle of 90 results in the best controller when acting on the linear periodic plant. Interestingly, it is possible to destabilize the linear periodic plant when
4
to the presence of the vertical wind shear. calculated performance metrics are RMS speed error: Actuator Duty Cycle:
attention on the results from the mean gain method – the case marked with an asterisk in the table.
The
0.230 rpm 2.341 deg/s
A comparison of the periodic and time-invariant DAC controllers reveals that there is no significant difference in speed regulation performance. Examination of either the performance metrics or response data (Figure 5 and Figure 6) suggests that time-invariant DAC may even be marginally more efficient, in terms of actuator usage. However, many more wind cases would be needed to confirm this hypothesis. While not plotted, the estimator traces for the time-invariant controller are very similar to those shown for periodic DAC. This suggests that the frozen plant model is adequate for estimating the states and disturbance input for a periodic plant.
Response plots for other selected states are shown in Figure 8 through Figure 11. With tower fore-aft deflection as an uncontrolled d.o.f. we notice that there is very little damping in the response, shown in Figure 8. The vibration mode exhibited is coupled with collective blade flap motion, shown in Figure 11, and has been identified to be the first mode to become unstable when gains are increased. The high yaw angle shown in Figure 9 is due to a number of factors; high mean pitch angles, wind shear, and free teeter motion – all of which exist here. Not surprisingly, the controller is unable to regulate the yaw state to the desired operating angle of -1.6, which is the steady-state yaw error used in the linearization (3). Teeter motion in Figure 10 displays the expected once-per-revolution oscillation that is due primarily to wind shear.
FURTHER DISCUSSION One drawback in the design of the frozen plant model is the need to perform a parametric study to find the optimum freezing azimuth angle. The process is essential, as it has been previously shown that freezing the periodic system at certain angles leads to instability. To its merit the freezing method does acknowledge the periodic nature of the system, whereas the averaging method performs blindly, with consequently less favorable results. While it is not immediately obvious, perhaps one could develop a more suitable direct method for constructing the LTI plant model. This would allow a search beyond the range of models provided by the freezing method.
Wind estimate capability is shown in Figure 4, with generally very good performance. This has been designed for, through the high estimator LQR weighting. Yaw angle estimation (the dotted line in Figure 9) is very poor, owing to the low observability of this state via azimuth position and rotor speed measurements alone. This compounds the poor yaw controllability issue. The final plot, Figure 11, illustrates good blade flap estimation properties.
A possible advantage of periodic control is that stability is always guaranteed when used with a linear periodic plant, even at high gains. This is a fundamental result of periodic optimal control theory. When applied to a nonlinear plant (as in this study), while stability is not guaranteed, one could expect that the periodic controller is less likely to destabilize the system. The biggest disadvantage of this controller is the additional issues of implementation. Periodic control requires knowledge of the rotor azimuth position at all times to synchronize feedback gains and the periodic estimator model. Therefore additional hardware, such as an optical sensor, would be required on the wind turbine to continuously measure angular position of the shaft.
Time-invariant DAC When the averaged model (Aa, Ba, Bd a) is used for state estimation the closed-loop response is either unstable or very poor wind and state estimation results. It is perhaps ironic that averaging theory in control is relatively well developed; yet it is apparently an inappropriate method for this turbine study. In contrast, use of the frozen model approach provides a remarkably good state estimator, as evident by the results in Table 2. For all the gain calculation methods examined, the RMS speed error and ADC are basically identical. For the remaining discussion we focus our
Wind and State Estimator Gain Calculation
LQR Mean of K(t)
Full-State Feedback Gain Calculation LQR Mean of G(t) RMS: 0.224 RMS: 0.224 ADC: 1.650 ADC: 1.648 RMS: 0.224 RMS: 0.225 * ADC: 1.648 ADC: 1.648
Table 2: Constant DAC performance results. (RMS speed error in rpm and ADC in deg/s)
5
REFERENCES
A time-invariant control system is more robust in the sense that there is no additional azimuth sensor required and is therefore less prone to malfunction. In this study we have assumed azimuth position has been measured, even in the time-invariant case. This has been done only so that a fair comparison can be made to periodic control. In actual implementation of a time-invariant controller one would measure only rotor speed. The additional technicality in this case is a resulting lack of observability of the disturbance. Removing the azimuth position state from the model solves this problem.
[1] Johnson, C.D., 1976, “Theory of Disturbance Accommodating Controllers,” Advances in Control and Dynamic Systems, 12, ed. C.T. Leondes. [2] Kendall, L., Balas, M., Lee, Y.J., and Fingersh, L.J., 1997, “Application of Proportional-Integral and Disturbance Accommodating Control of Variable Speed Variable Pitch Horizontal Axis Wind Turbines,” Wind Engineering, 21, pp. 21-38. [3] Stol, K., Rigney, B. and Balas, M., 2000, “Disturbance Accommodating Control of a Variable-Speed Turbine using a Symbolic Dynamics Structural Model,” Proc. 19th ASME Wind Energy Symposium, Reno, NV.
We have considered only speed regulation in Region III, where a constant rotor speed is desired. Operating in Region II with a periodic controller would involve added complexity since the period of rotation is continually changing with rotor speed. Here one could implement common gain-scheduling techniques and discretize the range of operating speeds. Disturbance Tracking Control, as mentioned in the introduction, could be employed instead of DAC for optimal energy capture.
[4] Balas, M., Lee, Y.J., and Kendall, L., 1996, “Disturbance Tracking Control Theory with Application to Horizontal Axis Wind Turbines,” Proc. 15th ASME Wind Energy Symposium, Reno, NV. [5] McKillip, R., 1984, “Periodic Control of the Individual-Blade-Control Helicopter Rotor,” Ph.D. thesis, MIT, Cambridge, MA.
CONCLUSIONS Performance was found to be nearly identical between periodic and time-invariant controllers for speed regulation. This conclusion is consistent with earlier published results concerning full-state feedback only. Now we find that a time-invariant plant model can estimate states and wind disturbance just as well as a periodic plant model. This is despite the inherent periodic nature of the two-bladed, free-yaw turbine that was analyzed.
[6] Jensen, K. E., Fahroo, F. and Ross, I. M., 1998, “Application of optimal periodic control theory to the orbit reboost problem,” Proc. AAS/AIAA Space Flight Mechanics Meeting, Univelt, Inc., San Diego, CA, pp. 935-945. [7] Stol, K. and Balas, M, 2001, “Full-state Feedback Control of a Variable-Speed Wind Turbine: A Comparison of Periodic and Constant Gains,” J. Solar Energy Engineering, ASME.
Designing the time-invariant state estimator by freezing the periodic plant at an azimuth angle of 90 proved to be an effective approach. Conversely, averaging the periodic plant resulted in a very poor estimator model.
[8] Stol, K., 2001, “Dynamics Modeling and Periodic Control of Horizontal-Axis Wind Turbines,” Ph.D. thesis, University of Colorado at Boulder, Boulder, CO.
The use of Independent Blade Control (IBC), as opposed to collective blade pitching, did not improve the relative performance of periodic control as expected. It is believed that the full potential of periodic IBC is not realized when applied to speed regulation only. Instead it may be more fruitful to explore different control objectives such as blade and tower load mitigation or yaw alignment. Here we may find performance problems that cannot be solved by a time-invariant controller.
[9] Hansen, A.C., 1996, Users Guide to the Wind Turbine Dynamics Computer Programs YawDyn and AeroDyn for ADAMS, Mechanical Engineering Department, University of Utah, Salt Lake City, Utah. [10] Bittanti, S., Laub, A.J., Willems, J.C. (eds.), 1991, The Riccati Equation, Springer Verlag, Berlin, pp.127-162. [11] Kwakernaak, H. and Sivan, R., 1972, Linear Optimal Control Systems, Wiley Interscience, New York.
ACKNOWLEDGEMENT This work was supported by the National Renewable Energy Laboratory (NREL), under contract number XCX-9-29204-04.
6
Nacelle
Tower
dh2 dhh
Hub dh1
Generator dn2
Blade
2
dt1
Figure 1: Schematic of the 6 d.o.f. SymDyn model
wind Wind gust Input
theta_op
Wind Gust
rotor speed
Operating Pitch
Blade Pitch
azimuth
y u
Nonlinear Plant (6 d.o.f.)
Subtract Operating Point
azimuth
Gd( t ) * ud^
ud^
u azimuth
Gx( t ) * x^
x^
y
State & Disturbance Estimator
Figure 2: Simulation block diagram for the implementation of periodic DAC in Simulink
7
0.8 0.75 0.7
Azimuth angle Rotor speed
Feedback Gain
0.08 0.06 0.04
Blade #1 flap angle Blade #2 flap angle
0.02 15
x 10
Yaw angle Yaw rate Blade #1 flap rate Blade #2 flap rate
-3
10 5 0 -5
45
0
90
135
180
225
270
315
360
Azimuth Pozsition [deg] Figure 3: Components of the periodic full-state feedback gain, G(t), for blade #1 pitch. Azimuth angle of zero degrees when blade #1 is in 12 o’clock position.
30
Wind speed [m/s]
25 20 15 10 5 0
0
10
20
30
40
50 Time [s]
60
70
80
90
Figure 4: Wind speed input (solid line) with wind estimate (dotted line) for the periodic DAC controller
8
100
Rotor speed [rpm]
58.5 58 57.5 57 56.5 56
0
10
20
30
40
50 Time [sec]
60
70
80
90
100
Figure 5: Rotor speed time response (Dark thin line: periodic DAC, light thick line: time-invariant DAC, dashed line: desired speed)
Blade #1 Pitch,1 [deg]
25 20 15 10 5 0
0
10
20
30
40
50 Time [sec]
60
70
80
90
100
Figure 6: Blade #1 pitch usage (Dark thin line: periodic DAC, light thick line: time-invariant DAC)
Blade Pitch, [deg]
19.5 Blade #1 pitch
19
Blade #2 pitch 18.5 18 17.5 17 16.5 15
15.5
16
16.5
17
17.5 Time [sec]
18
18.5
19
Figure 7: Blade pitch usage over a short time scale for the periodic DAC controller
9
19.5
20
Tower fore-aft angle [deg]
0.2 0.15 0.1 0.05 0 -0.05
0
10
20
30
40
50 Time [sec]
60
70
80
90
100
Figure 8: Tower fore-aft time response for the periodic DAC controller
5
Yaw angle [deg]
0 -5 -10 -15 -20 -25
0
10
20
30
40
50 Time [s]
60
70
80
90
100
Figure 9: Yaw angle response (solid line) with yaw estimate (dotted line) for the periodic DAC controller
3 Teeter angle [deg]
2 1 0 -1 -2 -3
0
10
20
30
40
50 Time [sec]
60
70
80
Figure 10: Hub teeter time response for the periodic DAC controller
10
90
100
Blade #1 flap angle [deg]
2
0
-2
-4
-6
0
10
20
30
40
50 Time [s]
60
70
80
90
100
Figure 11: Blade #1 flap response (solid line) with flap angle estimate (dotted line) for the periodic DAC controller
11
