Model Predictive Control Using Preview

Measurements From LIDAR �y

Jason Laksz Lucy Y. Pao x Eric Simley { Alan Wright k Neil Kelley ��

Bonnie Jonkman yy

Light detection and ranging (LIDAR) systems are able to measure conditions at a dis-tance in front of wind turbines and are therefore suited to providing preview informationof wind disturbances before they impact the turbine blades. In this study, a time-varyingmodel predictive controller is developed that uses preview measurements of wind speedsapproaching the turbine. Performance of the controller is evaluated using ideal, undistortedmeasurements at positions that rotate with the turbine blade and measurements obtainedat the same locations, but including distortion characteristic of LIDAR systems. Usingthese measurements, the model predictive controller is simulated in turbulent wind con-ditions and its performance is compared against previously designed, linear-time-invariantH1 preview controllers and industry standard controllers. Surprisingly, even though theLIDAR distortions produce signi�cant measurement error, controller performance is foundto surpass that obtained using individual-pitch feedback-only controllers without preview.In previous studies, errors introduced arti�cially, but of the same order of magnitude, wereshown to degrade the performance of preview control so that it is worse than using feed-back only. In this study, we also incorporate a simple error model to compensate the e�ectof LIDAR induced error, but �nd that it does not improve performance.

Nomenclature

1P once per a revolution,LIDAR light detection and ranging,MPC model predictive control,MBC multi-blade coordinates,DEL damage equivalent load,1

CP collective-pitch feedback-only controller,IP independent-pitch, feedback only controller,LTI-OFBK preview linear time-invariant output-feedback controller,MPC-SFBK preview model predictive controller with nominal state-feedback,Nprev number of sample hits of preview at wind measurement location,Npr number of sample hits of preview used by nominal feedback,Nmpc number of sample hits of preview used by MPC algorithm,

�This work was supported in part by the US National Renewable Energy Laboratory and the US National Science Foundation(NSF Grant CMMI-0700877).yEmployees of the Midwest Research Institute under Contract No. DE-AC36-99GO10337 with the U.S. Dept. of Energy

have authored this work. The United States Government retains, and the publisher, by accepting the article for publication,acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, worldwide license to publish orreproduce the published form of this work, or allow others to do so, for the United States Government purposes.zDoctoral Candidate, Dept. of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, CO, Student

Member AIAA.xRichard and Joy Dorf Professor, Dept. of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder,

CO, Member AIAA.{Graduate Student, Dept. of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, CO, Student

Member AIAA.kSenior Engineer, NREL, Golden, Colorado, Member AIAA.��Principal Scientist, NREL, Golden, Colorado, Member AIAA.yySenior Scientist, NREL, Golden, Colorado, Member AIAA.

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49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition4 - 7 January 2011, Orlando, Florida

AIAA 2011-813

Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Ts sample period for controller operation and preview measurements,mr(k) vector of out-of-plane bending moments at blade roots,!g(k) generator speed,p(k) vector of blade pitches at sample hit k,pr(k) vector of blade pitch rates at sample hit k,r(k) vector of reference (preview wind speeds) inputs at sample hit k,rf vector of reference (preview wind speeds) at sample hit k = Nmpc + 1,wt(k) vector of known/previewed wind speeds at the blades at sample hit k,u(k) vector of commands from the MPC algorithm at sample hit k,cmpc(k) vector of MPC pitch commands at sample hit k,ui(k) MPC control input to integral of generator speed error at sample hit k,u1p(k) vector of MPC control inputs to 1P dynamics at sample hit k,wt(k) vector of blade horizontal wind speeds at sample hit k,xs(k) generalized system state vector at sample hit k,xf generalized system state vector at sample hit k = Nmpc + 1,ys(k) generalized system performance output vector,yp(k) vector of generalized system outputs with pos. constr.,ypd(k) vector of generalized system outputs with pos. constr. and direct feed-through,yn(k) vector of generalized system outputs with neg. constr.,ynd(k) vector of generalized system outputs with neg. constr. and direct feed-through,Ydmax vector of upper limits for constr. outputs with direct feed-through,Ydmin vector of lower limits for constr. outputs with direct feed-through,Ymax vector of upper limits for constr. outputs,Ymin vector of lower limits for constr. outputs,x(k) x-coordinates of blades,y(k) y-coordinates of blades,z(k) z-coordinates of blades,wx wind velocity component in the x-coordinate directionwy wind velocity component in the y-coordinate directionwz wind velocity component in the z-coordinate directionxmbc(k) MBC equivalent of blade x-coordinates,ymbc(k) MBC equivalent of blade y-coordinates,zmbc(k) MBC equivalent of blade z-coordinates,� rotor azimuth,�prev rotor azimuth expected after elapsed preview time,T (�) transformation from non-rotating MBC equivalent coordinates to rotating blade coordinates,A state-to-state coe�. matrix,Bu input-to-state coe�. matrix,Br reference-to-state coe�. matrix,C state-to-performance output coe�. matrix,Cpp state-to-pitch coe�. matrix,Cpr state-to-pitch rate coe�. matrix,Cp state-to-output coe�. matrix for pos. constr. outputs,Cpd state-to-output coe�. matrix for pos. constr. outputs having direct feed-through,Cn state-to-output coe�. matrix for neg. constr. outputs,Cnd state-to-output coe�. matrix for neg. constr. outputs having direct feed-through,Dta control-to-output coe�. matrix for generator speed, bending moments, and pitch rates,Du control-to-performance output coe�. matrix,Dpru control-to-pitch rate coe�. matrix,Dpu control-to-output coe�. matrix for pos. constr. outputs,Dpdu control-to-output coe�. matrix for pos. constr. outputs having direct feed-through,Dnu control-to-output coe�. matrix for neg. constr. outputs,Dndu control-to-output coe�. matrix for neg. constr. outputs having direct feed-through,Dr reference-to-performance output coe�. matrix,Dpr reference-to-output coe�. matrix for pos. constr. outputs,

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Dpdr reference-to-output coe�. matrix for pos. constr. outputs having direct feed-through,Dnr reference-to-output coe�. matrix for neg. constr. outputs,Dndr reference-to-output coe�. matrix for neg. constr. outputs having direct feed-through,F LIDAR distance of the focal point from the LIDAR,R distance from LIDAR to a point on the ray (line-of-sight) to the focal point,W (R) LIDAR range weighting function,RR LIDAR Rayleigh range,KN normalizing constant for the LIDAR range weighting function,vlos detected wind velocity along the LIDAR line-of-sight, angle between wind direction and LIDAR line-of-sight,� angle between the LIDAR line-of-sight and the x-coordinate direction,�RMS RMS value of wind velocity component in the y � z (vertical) plane,�err RMS error of the LIDAR measurement due to directional bias

I. Introduction

The stochastic nature of the wind resource and the high initial capital cost and increasing, structural exibility of utility-scale turbines motivate the adoption of advanced instrumentation and measurementtechnologies. One of the most attractive technologies is LIDAR (light detection and ranging) that has theability to make real-time measurements of wind conditions local to individual turbines. In addition, increasingthe computational ability of the turbine controller requires an ever smaller percentage of the overall turbinecost, while potentially signi�cantly increasing the life expectancy through improved load mitigation. Thesefactors make advanced techniques such as model predictive control (MPC) increasingly attractive.

Model predictive control is essentially repeated optimization of the control inputs to plan future actionsover a �nite time-span into the future (referred to as the horizon). This optimization is updated as the stateof the system is observed over time and the most recent, optimized control is applied to the inputs of thesystem. MPC has traditionally been viewed as computationally expensive and time consuming. However,MPC has been found to be extremely robust and e�ective in practice. Particularly in the process industry,2

where the time scale of dynamics is slow compared to the computation time required, MPC has becomestandard and often the method of choice. The notion that MPC is too computationally intensive is alsobecoming outdated. Recent results3 using a Linux PC, report that with a simpli�ed optimization schemein application to problems with as many as thirty states, eight inputs, and a preview/control horizon of 30samples, the control can update on average every 26 ms (38.5 Hz){ a larger problem operating faster thanthe problem and sample rate used in this study. Experimental results have also reported operation at ratesup to 25K Hz.4

MPC has been investigated in application to wind turbine control in previous work. The range encom-passes operation of energy systems with intermittent resources,5 local control of the individual turbine inconsideration of power quality and generator torque limitations,6 as well as coordination of power qualitywith mitigation of drive train loading.7 This study extends the existing literature through the incorporationof preview measurements and exploration of the e�ects of distortions typical of LIDAR. In addition, thisstudy emphasizes mitigation of blade-root bending moments{ a goal that has not been the focus of previouswork.

A. Past Results Using Preview Control

A previous study8 demonstrates that large improvements in blade-root ap loads can be obtained whenpreview measurements are accurate. However, the controllers designed were linear time-invariant usingoutput-feedback (LTI-OFBK) that are prone to integral wind up. This is a de�ciency that presents itselfwhen wind speeds become low while the speed set point is held steady as shown in Fig. 1. In this case,integral control on speed error will cause the blade pitch to saturate at the low end; then, as wind speedsincrease away from the minimum, there is a delayed response in the average pitch level until the integrator\unwinds". As depicted in Fig. 1, the controller fails to mitigate blade loads during actuator saturation.Normally, LTI controls require additional anti-wind up features to prevent this type of response.

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Figure 1. Worst case ap moment exhibited by a LTI preview controller during a simulation where there isa particularly di�cult wind speed drop: (1st/top plot) wind speeds going to a minimum and then recovering;(2nd plot) blade pitch intermittently saturates between 55 and 74 sec; (3rd plot) shows blade-root momentpeaking during pitch saturations between 67 and 73 sec. The MPC controller is tuned to allow larger under-speed conditions{ alleviating integral wind up{ and mitigates the spikes in blade ap completely.

B. Motivation for Application of Model Predictive Control

MPC techniques address actuator limitations and other hard constraints directly as part of the optimizationprocess. Indeed, as is evident in Fig. 1, MPC’s ability to obey explicit constraints on actuation while avoidingintegral wind up is a key motivation for adopting this approach. Furthermore, the applied constraints canbe asymmetrical. For example, we investigate a less aggressive regulation on speed errors while adding ahard constraint on over-speed.

In addition, preview measurements and models of measurement error can be incorporated into MPC ina natural way. At each sample step, an entire sequence of controls is computed over the duration of thepreview/control horizon and this, in e�ect, anticipates integral wind-up that may develop during actuatorsaturation. With the actuator saturated (if need be), the response of the plant is \predicted" using aplant model and future commands are adjusted/optimized accordingly. Block diagrams of standard MPCarchitectures are depicted in Figure 2. As in previous applications of MPC to wind turbine operation,7 weincorporate a nominal state feedback control as part of the overall approach. In order to separate stateestimation issues from the estimation of wind components, we utilize the actual plant states as in Fig. 2a,but compute estimates of wind components not accurately previewed, using an observer as in Fig. 2b. A

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(a) Standard MPC w/Full State Feedback (b) Standard MPC w/Observer

Figure 2. Standard MPC con�gurations- in all cases, the MPC block has knowledge of plant control inputsand references. The plant model is \generalized" in the sense that it computes tracking errors, perturbations,etc. from plant I/O and references and includes any augmented dynamics such as error integrators and perfor-mance weighting. The MPC algorithm views any nominal feedback as part of the plant. In (a) The MPC blockhas access to the generalized plant state and in (b) the state is derived from output measurements using anobserver (measurements used by the observer may or may not be the same as performance and/or constrainedoutputs).

more detailed representation of the resulting con�guration can be viewed in Fig. 3. The MPC algorithmutilizes a model (denoted as \MPC PLANT" in Fig. 2 and \MPC Turbine" in Fig. 3) that includes thenominal state feedback and the addition of any other dynamics as part of a generalized turbine model.

C. Study Overview

In this study, we investigate the performance of an MPC controller designed for a single operating point inabove rated wind conditions (Region 3) where over-speed regulation is paramount for safe operation and theother main objective is load mitigation. We focus on out-of-plane blade- ap mitigation, but the approachcan be extended to encompass multiple loads in a straight forward manner. The next section provides anoverview of the MPC con�guration used in this study. We obtain results without hard constraints on turbinespeed and compare them to those obtained with an over-speed constraint in place. In addition, we comparecontroller performance when using idealized wind measurements to performance obtained when using windmeasurements distorted by the volumetric sampling and directional bias of LIDAR. Finally, we consider theuse of an observer to estimate the error induced by LIDAR distortions. The approach for this estimation issimilar to that used in disturbance accommodating control as investigated in previous studies9 wherein theobserver is augmented with dynamics representing persistent wind components at the blades, but that arenot present (or distorted) in the wind preview measurements. Heuristically, these components are estimatedfrom that part of the turbine response that does not correspond with the measured wind.

After setting up our application of MPC in Section II, the remainder of the paper is organized as follows.The wind conditions and measurement models are presented in Section III and we provide detail on theturbine model used for simulation and the simulation cases in Section IV. Simulation results are presented inSection V and we conclude with discussion of the e�ectiveness of the proposed method and plans for futurework in Section VI.

II. Overview of Model Predictive Control

MPC is a repeated application of �nite horizon optimization in discrete time based on a system model.At each sample hit, system measurements (ideally a complete measurement of the system state) are usedto determine initial conditions for the model. This model is \simulated" to optimize a sequence of controlinputs and �nd the minimum of a prescribed cost. For regulation about a single operating point (such asour Region 3 operating point), a single linear model of the turbine is utilized both in prediction and in cost

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Figure 3. The MPC implementation used in this study: when using ideal measurements, the observer block isnot present; when using LIDAR measurements, the con�guration is simulated without the observer block andthen again with the observer to estimate the LIDAR measurement errors. In all cases, the nominal feedbackdesign includes integral control on speed error and 1P (once-per-revolution) control on each blade-root bendingmoment.

evaluation.

A. MPC Quadratic Cost Function

The objective is to minimize the sum square of perturbations in generator speed, blade-root bending mo-ments, pitch rates, and control e�ort as applied to the augmented dynamics (see Fig. 3), all collected into ageneralized performance vector ys(k):

ys(k) =

2666664!g(k)

mr(k)

pr(k)

ui(k)

u1p(k)

3777775 =

2666664generator speed perturbation

blade-root bending moment perturbations

pitch rates

MPC control input to speed error integrator

MPC control inputs to 1P dynamics

3777775 : (1)

We adopt the convention that at each sample hit the present time index is always k = 0. Also, in ourimplementation, the MPC control u(0) for the present sample hit has been determined previously andcannot be changed. Hence, the presence of the \1=z" block in Fig. 3 denoting the fact that at each samplehit we optimize controls u(k) for k = 1 to k = Nmpc and not the present control. So, the cost objective is

f0(u; x) =1

2(Cxf +Drrf )

TQf (Cxf +Drrf ) +

1

2

NmpcXk=1

ys(k)TQys(k); (2)

where the performance output ys(k) and system state xs(k) are related to the controls u(k) and referencer(k) according to the linear state-space model

x(k + 1) =Axs(k) +Buu(k) +Brr(k) (3a)

ys(k) =Cxs(k) +Duu(k) +Drr(k); : (3b)

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The state xs(k) incorporates the modeled turbine state and the states of all augmented integral and 1P(once-per-revolution) dynamics (e.g., Fig. 3). This also means that from the perspective of the MPC cost,the state at the next sample hit

x(1) = Ax(0) +Buu(0) +Brr(0) (4)

is predetermined as a function of the known inputs and new state measurement at k = 0. This has someminor implications relative to the satisfaction of output constraints as discussed in the next subsections.

The term Q in Eq. (2) is a diagonal matrix of positive weights, although more general, positive-de�nitematrices may be used. It is desired that the �nal state xf , x(Nmpc + 1) make the �nal output yf ,Cxf+Drrf small in the absence of any additional control input and given the �nal reference rf , r(Nmpc+1).

The quantities speci�c to each blade are root bending moment, pitch rate, and the controls to theaugmented dynamics driven by the blade responses and are denoted, respectively, as

mr(k) =

264mr1(k)

mr2(k)

mr3(k)

375 ; (5)

pr(k) =

264pr1(k)

pr2(k)

pr3(k)

375 ; (6)

u1p(k) =

264u1p1(k)

u1p2(k)

u1p3(k)

375 : (7)

We note that the MPC control u(k) = [cmpc(k)T ui(k) u1p(k)T ]T contains the vector of MPC pitchcommands cmpc(k) and the MPC commands ui(k) and u1p(k) that are part of the performance vector ys(k).So, in Eq. (3b), Du is of the form

Du =

"Dta 0

0 I

#(8)

where I is an identity matrix of dimension 4 and Dta is the feed-through from pitch command to generatorspeed, blade bending moments and pitch rates. In our models Dta has full column rank (each of the threepitch commands contributes exclusively to one of the three pitch rates);8 this, combined with the presenceof the identity in Du, makes the quadratic optimization well posed:

DTuQDu > 0: (9)

In this application, preview measurements of wind serve as the reference r(k), and subsequently this termcollects together all measured wind speeds over the preview horizon of Npr samples utilized by the nominalstate-feedback:

r(k) =hwt(k)T wt(k + 1)T : : : wt(k +Npr � 1)T

iT; (10)

wt(k) =

264wt1(k)

wt2(k)

wt3(k)

375 : (11)

The term wt(k) is the vector of known/previewed wind encountered at the turbine blades at time k. Notethat this requires that the preview horizon for optimization be longer than the amount of preview utilizedby the nominal feedback: Nmpc > Npr.

B. MPC Constraints for Quadratic Optimization

In this study, actuator pitch angles p(k) and rates pr(k) are always subject to minimum and maximumconstraints and a generator over-speed constraint is used in some simulations as summarized in Section IV.When the generator constraint is used, quadratic optimization only occurs when this constraint has beenfound feasible during the pre-optimization described in the next subsection.

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In our model (essentially identical to the one used in our prior work8), pitch rate has direct feed-throughfrom the command input while pitch angle does not,

pr(k) =Cprxs(k) +Dpruu(k) (12a)

p(k) =Cppxs(k); (12b)

where the vector of blade pitch angles is p(k) = [p1(k) p2(k) p3(k)]T . The D-matrix Dpru mapping controlto pitch-rate turns out to be onto and therefore pitch rate can be guaranteed feasible over the MPC opti-mization/preview horizon starting from k = 1. The same cannot be said for blade pitch angles which mustbe specially handled in order to avoid a pre-optimization for these variables. As a result, constraints arepartitioned into those with direct feed-through and that can be a�ected by the control at k = 1, and thoseoutputs without direct feed-through. De�ne

ypd(1) =pr(1) (13a)

ynd(1) =pr(1) (13b)

yp(k) =

264!g(k)

p(k)

pr(k)

375 (13c)

yn(k) =

"p(k)

pr(k)

#; (13d)

(13e)

then the applied constraints may be written as

ypd(1) = Cpdx(1) +Dpduu(1) +Dpdrr(1) � Ydmax (14a)

ynd(1) = Cndx(1) +Dnduu(1) +Dndrr(1) � Ydmin (14b)

yp(k) = Cpxs(k) +Dpuu(k) +Dprr(k) � Ymax (14c)

yn(k) = Cnxs(k) +Dnuu(k) +Dnrr(k) � Ymin; (14d)

where \� & �" imply element-wise comparisons. The matrices in Eq. (14) are composed from rows inthe matrices of Eq. (3b) or Eq. (12) as appropriate. Note that we only require that generator speed !g(k)satisfy a maximum constraint.

With the exception of pitch angle, this partitioning prevents infeasibility at k = 1. However, theoretically,in the event that pitch angle p(1) is infeasible, then there may not be a command u(1) which satis�es rateconstraints (that are applied at k = 1) and makes pitch feasible at k = 2 at which time pitch constraints areapplied. In this case, our code handles the event as an exception and halts simulation{ this never occurred.Simulation starts with pitch squarely in the feasible region and apparently the linearized model is accurateenough that at k = 1, the actual pitch never falls so far out of the feasible range that a rate constrainedcommand cannot correct the situation. In an actual implementation (not just simulation), some provisionfor this exception would have to be made.

So, when using the over-speed constraint, the constrained optimization problem is posed as

min(u;x)

f0(u; x) =1

2(Cxf +Drrf )

TQf (Cxf +Drrf ) +

1

2

NmpcXk=1

ys(k)TQys(k); (15a)

subject to: x(k + 1) = Axs(k) +Buu(k) +Brr(k); k 2 [1; Nmpc]; (15b)

ys(k) = Cxs(k) +Duu(k) +Drr(k); k 2 [1; Nmpc]; (15c)"ypd(1)� Ydmax�ynd(1) + Ydmin

#� 0; (15d)"

yp(k)� Ymax�yn(k) + Ymin

#� 0; k 2 [2; Nmpc] (15e)

wherein the use of Eq. (14) is implicit. When simulations do not use the generator over-speed constraint,the �rst row of Eq. (14c) is simply omitted.

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C. Pre-optimization to Determine Feasibility

Inequality constrained optimization requires that the constraints be feasible and that the initial control se-quence used to start optimization be such that constraints are satis�ed. In practice, minimum and maximumconstraints are grouped into two main categories. Actuator constraints generally apply to command inputs,or possibly to outputs of actuator dynamics that can otherwise be guaranteed feasible. Output constraintsgenerally apply to plant/system outputs that may not have direct feed-through (the \D-matrix" coe�cientsare zero) from command inputs and are dependent on the system state and reference inputs in such a waythat they are not guaranteed feasible.

When such output constraints are used (e.g., the over-speed constraint on !g(k)), it is necessary toprovide for a pre-optimization wherein controls are found that make the constrained outputs as small aspossible. If a control sequence is found that brings the constrained outputs within their prescribed limits,then it is used as the initial control in optimizing the quadratic objective of the previous section. Otherwise,the output constraint is infeasible, and the �nal control sequence obtained during pre-optimization is used,skipping optimization of the quadratic objective.

In our application, the constraint on generator speed !g(k) is not guaranteed feasible, in contrast withpitch and pitch rate constraints which we �nd can always be satis�ed. So, when the over-speed constraint isused, we perform the pre-optimization

min(u;x;s)

f(u; x; s) = 0Tx+ 0Tu+ s (16a)

subject to: x(k + 1) = Axs(k) +Buu(k) +Brr(k); k 2 [1; Nmpc] (16b)

!g(k) = C!xs(k) +D!uu(k) +D!rr(k); k 2 [1; Nmpc] (16c)"ypd(1)� Ydmax�ynd(1) + Ydmin

#� 0 (16d)

2666664�s

�s� !g(k)

�s+ !g(k)

yp(k)� Ymax�yn(k) + Ymin

3777775 � 0; k 2 [2; Nmpc]; (16e)

(16f)

wherein !g(k) is dropped from the de�nition of yp(k). The s term in Eq. (16) is being used as a slackvariable; controls are found that allow s to be as small as possible, but constrained to be positive (via the1st line in Eq. (16e)). Since the function f(u; x; s) is linear in its arguments, it is referred to as the \linearobjective". This strategy is summarized in the ow chart of Fig. 4.

III. Wind Conditions and Measurements

We use the turbulent in ow simulator TurbSim10 developed at the National Wind Technology Center(NWTC) to generate wind �elds for simulation. We have found that one set of atmospheric parameters inparticular produces the widest range of conditions in which to simulate the controller and turbine. Theseconditions are summarized in Table 1. TurbSim is used to generate an ensemble of 31 �elds characteristicof these conditions and each controller con�guration is simulated using each of the resulting wind �elds asthe in ow for the turbine. Preview measurements for use by the controllers are obtained from these �eldsas described in the next subsections.

Details on the wind measurement techniques are provided in Sections III-A and III-B. Here, we summarizethe error between the preview measurement and the wind actually encountered at the 75% span of each blade.When using ideal (without distortion) measurements, the baseline LTI-OFBK controller uses a preview ofwind speed 0.45 s out in front of the turbine while the MPC controller uses measurements 1.0 s out. Thesource of error in these measurements arises from the inability to know exactly the rotor position after theelapsed preview time, and hence, there is an error in the location at which the measurement is taken. WhenLIDAR is used, the measurements are taken 2.0 s out so that this e�ect is magni�ed, but the LTI-OFBKand MPC controllers still only use 0.45 s and 1.0 s, respectively, of the available measurements. As explained

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Figure 4. MPC with output constraints: at each sample hit, the initial control sequence is evaluated to see ifgenerator speed stays below a 3% limit. If not, then a pre-optimization is performed to �nd a control sequencethat satis�es the over-speed limit. If one cannot be found, then the control minimizing generator speed is usedat the next sample hit; if a control satisfying the over-speed constraint is found, then it is used as the initialguess during optimization of the quadratic cost.

Table 1. Meteorological input parameters for TurbSim: a three-dimensional wind vector is generated over a31x31 point grid in the vertical y � z plane, centered so that it encompasses the rotor disk. Over time thegrid is sampled at 20 Hz for a total duration of 630 seconds. The wind pro�le within the grid is varied by thevertical stability parameter Ritl and the mean friction velocity (shearing stress) u�D; a power law variation ofthe vertical wind speed pro�le is speci�ed by the listed shear exponent �0.

in Section III-B, LIDAR also introduces a directional bias and an averaging of wind speeds over a sampledvolume of air that further increases the preview error. Histograms of the errors for each measurementcon�guration are displayed in Figure 5.

The worst case error magnitude becomes slightly larger as the preview time is increased from 0.45 s to1.0 s, but increases substantially when the measurements are distorted by LIDAR and the preview distanceis increased to 2.0 s. As discussed in the results of Section V, the error in the LIDAR measurements appearsto have a large once-per-rev (1P) component. An observer augmented with dynamics generating a persistent1P wind disturbance was added to the MPC con�guration (see Fig. 3) in order to estimate this componentof the LIDAR distortion; the output of these dynamics is then added back into the preview measurementsto try and adjust for this error. As can be observed in the lower right histogram of Fig. 5, this actuallyincreases the root-mean-square of the preview error. Also shown in the lower two plots is the histogram oferrors obtained using interpolated measurements as described in our prior work.8 These errors are on thesame order of magnitude and were found to degrade the performance of the LTI-OFBK preview control tothe point that it is inferior to that obtained using independent pitch (IP) feedback-only.

A. Ideal Preview Wind Measurements

During simulation, preview measurements are obtain at locations ahead of the turbine at 75% blade spanas explained in our earlier work8 and summarized here. Relative to the tower base, denote the position byhorizontal (positive down wind), perpendicular (positive to the left when facing the turbine from upwind),and vertical (positive up) coordinates x, y, and z, respectively. Then the horizontal wind speed at anarbitrary position is denoted by w(x; y; z). Measurements ahead of the turbine are computed from bladepositions as follows.

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Figure 5. Normalized histograms of preview measurement errors: (top-left) ideal-rotating measurements givethe tightest error distribution; (top-right) at 1.0 sec preview, the error distribution spreads slightly due toinaccuracy in predicting blade position; (bottom-left) using LIDAR introduces signi�cant error mostly asa result of directional bias, but compared to interpolating wind speed at blade positions from stationary(\Stat.") measurements, it is 3 times more accurate (in terms of standard deviation); (bottom-right) addingin an estimate of the LIDAR error from an observer increases the error standard deviation by about 10%.

At at each sample hit k, denote the present rotor position by �. Then, the present x, y, and z coordinatesof each turbine blade are transformed to their equivalent multi-blade (MBC)11 coordinates xmbc, ymbc, andzmbc, e.g.,

xmbc(k) = T�1(�)

264x1(k)

x2(k)

x3(k)

375 ; (17)

where [x1(k); x2(k); x3(k)] is the corresponding x-location of each blade at 75% span and

T (�) =

2641 cos(�) sin(�)

1 cos(� + 2�3 ) sin(� + 2�

3 )

1 cos(� + 4�3 ) sin(� + 4�

3 )

375 (18)

is the inverse MBC transformation (the naming convention is backwards from the notation). Let the desiredpreview time be Nprev samples. Then, if the present rotor speed is !, after Nprev samples the expectedposition of the rotor is

�prev = � +NprevTs!; (19)

where Ts is the sample period. The 75% blade span positions then are interpolated according to

x(k +Nprev) =T (�prev)xmbc(k); (20a)

y(k +Nprev) =T (�prev)ymbc(k); (20b)

z(k +Nprev) =T (�prev)zmbc(k): (20c)

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If the average wind speed is w0 m/s, then wind at a distance xprev = NprevTsw0 ahead of the turbine willarrive after Nprev samples. So, preview measurements [wt1 wt2 wt3] of horizontal wind speed for each bladeare obtained from the TurbSim wind �eld according to264wt1(k +Nprev)

wt2(k +Nprev)

wt3(k +Nprev)

375 =

264w ( x1(k +Nprev)� xprev; y1(k +Nprev); z1(k +Nprev) )

w ( x2(k +Nprev)� xprev; y2(k +Nprev); z2(k +Nprev) )

w ( x3(k +Nprev)� xprev; y3(k +Nprev); z3(k +Nprev) )

375 : (21)

B. LIDAR Preview Wind Measurements

When using a LIDAR model, the preview measurement positions [x(k +Nprev) y(k +Nprev) z(k +Nprev)]computed in the previous section determine the focal point for the LIDAR measurement. As discussed ina companion paper,12 we assume a hub-mounted continuous-wave LIDAR and set the focal point for theinstrument to this position for each required measurement. In our prior studies8 based on the three-bladed,controls advanced research turbine (CART3) at the National Wind Technology Center, it was found thatafter more than about 0.5 s the blade load mitigation performance of preview control did not substantiallyincrease. However, as we describe in this section, the distortion/error inherent in LIDAR measurementsincreases with the tangent of the measurement angle from horizontal and this becomes signi�cant at anglesless than 45 deg so that there is a strong motivation to focus the LIDAR as close to horizontal as possible.

Figure 6. A hub mounted LIDAR takes measurements 2 s out in front of the turbine at 75% blade span.Assuming 18 m/s wind speed and a 20 m blade length, this corresponds to a distance of 36 m and a measurementangle of about 23 deg.

This motivates the use of longer preview times, since the blade is typically most sensitive to wind speeds atspans greater than 50%,13 and aiming the LIDAR at small angles from horizontal requires larger distancesto reach the corresponding blade span. As depicted in Fig. 6, we choose a 2 sec preview distance thatcorresponds to 36 m at the average wind speed of 18 m/s used here, and with the 75% span for the CART3at 15 m from the hub, this corresponds to a measurement angle of 23 deg. This allows us to evaluatethe performance of preview control measuring at a distance ahead of the turbine where there is still someaccuracy in predicting the rotor position. However, as explained below, this geometry still su�ers from twosigni�cant distortions that are inherent to the optical properties of continuous wave LIDAR.

Continuous wave LIDAR determines the line-of-sight wind speed at a speci�c location by focusing thelaser beam at that position in space. Rather than only detecting the wind speed at the intended point, a focaldistance F away from the LIDAR, wind speed values along the entire laser beam are averaged accordingto what is called the \range weighting function," W (R), to yield the detected value. The general e�ectof range weighting is the low-pass �ltering of the true wind speed. As focal length increases, more highfrequency wind information is lost in the measurement. The line-of-sight wind speed measurement due torange weighting at a focal distance F is given by

vlos(F ) =

Z 1�1

vr(R)W (R)dR (22)

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where vr(R) is the line-of-sight velocity at a range R along the laser beam14f. The range weighting functionfor a focal distance F is given by

W (R) =KN

R2 + (1� RF )2R2

R

(23)

where RR is the Rayleigh range and KN is a normalizing constant so that the entire range weighting functionintegrates to 1. The Rayleigh range is a value determined by the laser wavelength and beam width. Forthe commercially available ZephIR Doppler LIDAR system modeled here (http://www.naturalpower.com/zephir-300-wind-LIDAR), RR is approximately 1,570 meters.

When the LIDAR is pointing directly in the upwind x-direction, the line-of-sight velocity is the desiredvalue, which will be called wx. When the LIDAR is not pointing in the x direction, the measured line-of-sightwind velocity contains contributions from the y and z components of wind as well. Determining the desiredvelocity from the measured value in this case, biases the result to the extent that these latter componentsare large relative to the x component. When the measured line-of-sight is at an angle relative to theinstantaneous wind direction, the line of sight velocity is

vlos =q

w2x + w2

y + w2z cos (24)

where wx, wy, and wz are the x, y, and z components of wind speed, respectively. In this work it is assumedthat wy and wz are zero, since they are, on average, much smaller than wx, as long as the yaw error of theturbine is also small. Given this assumption, an estimation of the desired component wx is

bwx =vloscos�

(25)

where � is the angle between the LIDAR line-of-sight and the x direction.An analysis of errors caused by measurement angle12 has shown that for a transverse wind speed (per-

pendicular to the desired wx component) with magnitudeq

w2y + w2

z equal to � and a uniformly distributed

random direction in the y-z plane, the root mean square wind speed measurement error is given by

�err =�RMS tan�p

2(26)

where �RMS is the RMS value of the transverse wind speed magnitude. The most important aspect of thisformula is that in the absence of range weighting, measurement errors will scale with tan�.

IV. Turbine Model and Simulation Cases

The turbine used in this study is based on the CART3 and is modeled in the turbine simulation codeFAST15 also developed at NWTC. FAST provides a full non-linear, aero-elastic simulation of the turbine andwill compute numerically linearized models at desired operating points. The structural modes and operatingpoints modeled in this study are summarized in Table 2. As discussed in our prior work,8 the linearizedmodel from FAST is scaled so that individual point measurements of wind at 75% blade-span can be usedas disturbance inputs to the model.

During simulation all available degrees of freedom (DOF) provided by FAST are utilized except for teeterand those related to o�shore operation. Doing so adds a second blade ap mode, an edge-wise blade apmode, and two modes each for tower sway in the fore-aft and side-to-side directions, and a single yaw mode(yaw actuation is not used, but this latter DOF approximates a side-to-side compliance in the drive train).

When LIDAR measurements are used, they are focused at 36 m out in front of the turbine so that theyprovide 2.0 s of preview. However, the preview controllers themselves do not utilize all the available preview.With ideal measurements, the preview distance is set to provide only as much preview as is required by thecontrollers. As summarized in Table 3, the LTI-OFBK controller only uses 0.45 s of the available preview.The nominal state-feedback used in the MPC controller also uses 0.45 s of preview, but the MPC algorithmitself uses 1.0 s of preview.

As indicated in Table 3, the LTI-OFBK controller is simulated with and without the LIDAR modeldistortions. Using the ideal measurements, the MPC controller is simulated once with and once withoutthe over-speed constraint on rotor speed. When using the LIDAR measurements the MPC controller is

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Table 2. Set points for turbine linearization and MPC operation. The center section lists the operatingpoints used during linearization and the resulting modes; the right section lists the rated turbine values andthe maximum and minimum limits used for MPC.

again simulated twice{ the second simulation is used to investigate the bene�t of (or lack thereof) addingestimation of preview errors from the turbine response via an observer as in Fig. 3. In analyzing the results,this arrangement separates the degradation in performance caused by adding the over-speed constraint fromthe possible bene�t of adding a measurement error estimation technique.

The MPC performance is compared against collective pitch (CP) and individual pitch (IP) controllers thatuse only feedback and no preview measurements. For the CP feedback-only controller the objective is speedregulation. For all other controllers, the objective is both blade load mitigation and speed regulation. Also, inall controllers except the CP feedback-only design, the additional feedback from measurements of the blade-root bending moments is providing a means to adjust the pitch of each of the three blades individually. TheCP feedback-only controller uses only a measurement of generator speed error that provides no informationas to how the blades might be pitched individually.

Also provided are the results obtained using an LTI-OFBK preview controller that is an MBC basedH1 design from our prior work.8 In addition, our prior work discusses the constrained, linear-matrixinequality technique used to design the preview state-feedback gains that make up the nominal feedback inthe implementation of the MPC controller.

Table 3. Preview times used by controllers and measurement con�gurations: the controllers always use thesame amount of preview, but the measurement scheme may take measurements further out so that the systemhas additional preview that is unused by the controller.

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(a) Blade-root Bending Moments using Ideal Measurements (b) Blade-root Bending Moments using LIDAR Measurements

Figure 7. Cyclic loads observed over all 31 in ows used for simulation: in both �gures, results obtainedusing CP, IP, and LTI-OFBK are presented. The LTI-OFBK results shown in both plots are obtained usingideal measurements (solid magenta) and measurements interpolated from wind speeds at stationary locations(dashed magenta). (a) Using ideal-rotating measurements, MPC (red) proves to have signi�cantly smaller peakloads (OUTLIERS) while being comparable to LTI-OFBK preview control (magenta), but adding an over-speedconstraint (gold) degrades performance slightly. (b) Using LIDAR measurements degrades performance of allcontrollers, but MPC still has signi�cantly lower peak loads, and adding an estimate of LIDAR error basedon turbine response to the preview measurements degrades MPC performance further; in this case, previewcontroller performance is still signi�cantly better than that obtained using interpolated wind speeds (dashedmagenta) even though interpolated measurement errors are on the same order of magnitude as those producedby LIDAR.

V. Simulation Results

A. Blade Load Mitigation

The dynamics of the tower and its loads were not considered in the controller designs, so we examine theresults for blade load �rst. The cyclic loads encountered over the course of all 31 wind in ows are presentedin Fig. 7 for each control and preview con�guration. The percentage of all load cycles of a given size isplotted along the y-axis and the size of the load cycle is plotted along the x-axis. Curves to the left andlower represent better performance, consistent with smaller loads and less cycles. The CP controller (blue)has the largest percentage of loads over 400 kN-m. Adding individual pitch feedback from blade-root bendingmoments (IP, green) decreases average blade loads by 20% (c.f. Table 4) and adding preview (LTI-OFBK,magenta) provides about another 30% load reduction compared with IP. These three cases are plotted forreference in both the plot displaying results using ideal measurements (Fig. 7a) and the plot displayingresults using LIDAR (Fig. 7b). Further, presented in both cases is the load curve obtained using threestationary point measurements (without LIDAR distortion) that are then interpolated to wind speeds atblade locations for use by the LTI-OFBK controller (as in previous work8); in this case, the load performanceof the LTI-OFBK preview controller degrades (dashed magenta) to become worse than that of IP withoutpreview.

This degradation occurs, because the error in preview wind speeds increases signi�cantly when interpo-lating from stationary point measurements (see Fig. 5) so that the maximum error is about 6 m/s. However,this performance degradation does not occur if the LTI-OFBK preview controller uses LIDAR distortedmeasurements (’*-’ magenta, Fig. 7b), even though the maximum measurement error of the LIDAR previewmeasurements (about 5 m/s) is on the same order of magnitude as the interpolated preview measure-ments. So, it appears preview control performance has a greater dependence on the RMS error of previewmeasurements{ as shown in Fig. 5, the RMS error of rotating LIDAR measurements is nearly three times

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smaller than interpolated measurements. Preview control may be insensitive to speci�c features character-istic of LIDAR distortion, or which may be prevalent to the wind �elds used here, but a characterizationof such features is not a topic of this study. What is apparent at this point, is that the angle at whichthe LIDAR measurements are made can be quite large (nearly 23 degrees as is used here) and much of theadvantages of preview control are retained.

These results are made more precise in Table 4. The improvement in load performance of the LTI-OFBKcontroller relative to IP feedback-only is about the same independent of the use of LIDAR, and whetherconsidering the maximum cyclic load or the average of damage-equivalent-loads (DEL)1 observed over all 31in ows. In the former metric, maximum load decreases by 8% and in the latter metric load decreases by atleast 22%, independent of whether the measurements are ideal or LIDAR distorted.

Also evident in Table 4 is the fact that the main advantage in using MPC at a single operating point,is in avoiding the maximum loads that can be produced by the LTI-OFBK controller during blade pitchsaturation. The peak load using MPC is at least 25% better than that generated by the LTI-OFBK controlleras long as neither the over-speed constraint nor the error estimate is used. Without the over-speed constraint,the MPC controller su�ers from signi�cant over-speed faults as are evident in Fig. 8 (red). Including thehard constraint on over-speed is e�ective (Fig. 8, gold), but its use reduces the advantage of using MPCin preventing large loads by about 7%. Careful inspection of Fig. 8 shows that the actual rotor speed stillviolates the 3% limit imposed by the MPC controller. If the actual speed needs to be guaranteed, then thelimit must be tightened to account for modeling error, and doing so will further increase the associated costin terms of load mitigation.

B. Observer Based LIDAR Distortion Estimates

In this subsection, we consider the results obtained when adding an estimate of the LIDAR error through theuse of an observer. The approach is somewhat ad-hoc, but is based on the fact that the majority of the errorin the LIDAR measurements centers on the 1P frequency. This is evident in Fig. 9 which shows the errorin the LIDAR measurements and an estimate of them obtained from the observer; the top plot shows timedomain waveforms from one simulation and the bottom plot shows the amplitude spectrum averaged acrossall 31 in ows. Clearly, the frequency content generated from the observer estimate is quite accurate (theactual error and the observer output waveforms are essentially on top of each other). Unfortunately, the lowpassed wave forms in the top plot show that phasing of the 1P content from the observer is often incorrectand the modulation of the amplitude also lags behind. However, the results are promising enough that wesuspect a more methodical approach to the characterization of LIDAR errors and their estimation based onturbine measurements might still be fruitful. In the ad-hoc approach used here, the design of the observeris not known to be optimal in any sense; it is simply a pole-placement design that guarantees damping of atleast 0.75 and increases the speed of all modes by at least 50%.

C. Other Metrics

Finally, in order to be complete, results across all 31 in ows for speed, power, and tower loads are providedin the box plots of Figures 10 and 11. The LTI-OFBK design has more emphasis on speed regulation than

Table 4. Load metrics: \+" denotes absolute maximum in kN-m; \*" denotes average over each blade andover each of the 31 in ows simulated. Percent improvement (load reduction) is shown for each controller; theIP performance is rated relative to CP; all other controllers are rated relative to IP.

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Figure 8. Speed regulation for a speci�c in ow case (AR4 S11). The LTI-OFBK controller (magenta) hastight speed regulation in order to prevent over-speed faults (which can occur anyway due to integral windup);the MPC controller (red) is tuned to provide less speed regulation, but does not su�er from integral wind-up;the over-speed faults of the MPC controller are mitigated with the use of a 3% over-speed constraint (gold).

do the MPC designs and this is evident in the size of the box (the extent of which depicts +/- 25%) in theplots of both the speed (Fig. 10a) and power (Fig. 10b) results. Also evident is the e�ectiveness of theover-speed limit in keeping the outliers below the 3% limit (the gold box in the \IDEAL" section has a 95percentile dot below the black solid line representing the 3% limit in Figures 10a and 10b). MPC also showsslightly improved (lower) tower loads relative to the baseline feedback-only designs in that the quartiles havea smaller spread (as long as the observer estimate of LIDAR error is not used). In terms of extreme/outliertower loads, MPC appears to have done no harm, but there was no consideration of these dynamics in thedesign of any of the controllers.

VI. Conclusions and Future Work

MPC has been demonstrated to be an e�ective method for load mitigation in highly variable wind con-ditions during which pitch saturation is likely to occur. It explicitly satis�es constraints on pitch magnitudeand rate while avoiding integral windup. In the event that wind speeds momentarily drop while the speedset point is held constant, MPC with preview can recover without the excessive loads or the over-speed faultsthat may typically be exhibited by LTI preview control approaches. This is due in part to the fact that withMPC, nominal speed regulation can be relaxed while limiting over-speed faults with application of a hardconstraint on maximum speed. However, it was also demonstrated that the application of hard constraintswill typically degrade load mitigation performance.

Further, we have found that preview control is robust to the distortions typical of LIDAR measurementsystems in turbulent conditions, in the presence of shear, and even at fairly large measurement angles.Measuring wind speeds with a LIDAR directional bias of 23 deg degrades controller performance by only8% in terms of average DEL. Relative to an IP feedback-only controller, preview using LIDAR still providesan improvement in damage equivalent load of 32%, 25%, and 29% when using MPC without an over-speed

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Figure 9. Estimation of LIDAR measurement errors: the MPC controller is augmented with an observer toestimate LIDAR errors from the turbine response. The estimates have accurate frequency content (bottomplot), but in the time domain (top plot), they su�er from phase o�sets and a modulation of amplitude thatlags the actual errors.

constraint, MPC with an over-speed constraint, and a LTI preview controller, respectively.There is, however, a de�ciency in the implementation of MPC used here. It was necessary to provide the

MPC algorithm inputs ui(k) and u1p(k) to the augmented dynamics (see Fig. 3) that are normally used toobtain asymptotic rejection of constant speed errors and 1P bending moments at the blade roots. This wasdone to mitigate integral windup, but the manner in which these controls have been used essentially removesthe asymptotic rejection properties for which the additional dynamics were originally added. It is certainlypossible to implement MPC so that it provides o�set free regulation16 and it may be possible to extend themethod to also provide asymptotic rejection of 1P disturbances as well. This is a topic for future work. Itmay turn out that the 1P objective is easily achieved using o�set free MPC formulations if the controller isimplemented in the multi-blade coordinate framework.11

There is also a large body of work that remains in characterizing the optimal preview distance for a givenLIDAR system. We just scratched the surface here and found that preview systems are surprisingly robustto LIDAR distortion. However, the trade o� between preview time and directional bias (preview close to theminimum time required implies more directional bias) is most likely dependent on atmospheric conditions.Moreover, this brings to the fore the issue of simulating with \frozen" wind �elds as is common practice.Obviously, the wind in ow is going to evolve en route from the measurement location to the turbine andin this regard, less preview time would seem to be an advantage. In future work, we hope to extend ourinvestigations to explore the e�ect of wind evolution on the trade o� between preview time and directionalbias.

References

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(a) Speed Regulation (b) Generator Power Output

Figure 10. Box plots for speed regulation and average power using data from all 31 in ows: boxes have linesat the lower quartile, median, and upper quartile values. A solid red line shows the mean (the median is aplotted with a black dotted line). The whiskers extend to the 10th and 90th percentiles. Large black dotsindicate the extent of the 5th and 95th percentiles.

(a) Tower Fore-Aft Bending Moments at Base (b) Tower Side-to-Side Bending Moments at Base

Figure 11. Box plots for tower loads using data from all 31 in ows.

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12E. Simley, R. J. B. Pao, L.and Frehlich, and N. Kelley, \Analysis of wind speed measurements using coherent lidar forwind preview control," in Proc. 49th AIAA/ASME Wind Energy Symposium, Orlando, FL, Jan. 2011.

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2007, pp. 1245 {1250.

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