Modeling and Control of Variable-Speed Wind-Turbine

Drive-System Dynamics P. Novak, T. Ekelund, I. Jovik, and B. Schmidtbauer

en designing control for variable-speed wind turbines, one w deals with highly resonant, non-linear dynamic systems subject to random excitation, i.e.. wind turbulence. This requires good knowledge of the dynamics to be controlled, particularly when combined with the increasingly common “soft” concept of lightweight. flexible constructional components; i t creates cost advantages compared to more material-consuming rigid con- structions. but also results in low frequency structural eigenfre- quencies, some of which may appear in the bandwidth of closed-loop operation.

For this article, system-identification experiments have been performed on an existing 400 kW, variable-speed, horizontal- axis wind turbine using various identification schemes. The identification results have then provided numerical values of the parameters in a physical model of the drive system. The acquired model has been used for design and evaluation of a number of linear and non-linear control schemes for wind-turbine speed regulation.

Introduction The simplest wind-turbine configuration is the uncontrolled

(stall-regulated), where the turbine speed is constant due to the generator being directly connected to the fixed-frequency utility grid. With this arrangement it is not possible to affect the amount of power delivered by the turbine to the generator; it is directly given by the wind variations. Since the available aerodynamic power is proportional to the cube of the wind speed, only a fraction of the obtainable energy is captured in extremely strong wind; it would not be economically viable to design plants capable of converting maximum power at these occasions. In a stall-regulated machine the power is naturally limited by the decreased aerodynamic efficiency in stall.

The benefits of incorporating power control in a wind turbine are increased energy capture and reduction of dynamic loads. Actuators also provide flexibility to adjust the operating point; the safety margins of the design can be reduced. The most frequently used method to actively control the power is to adjust the pitch angle of the turbine blades. In this case, as in the uncontrolled plant, the generator is directly connected to the grid.

Parts of this nrutcvYal M‘ere preseriterl ut the I994 IEEE Cor$erence o t i Control Applicatioris. P. N o w k , T. Ekelurid, and B. Schmidthuuer ure wzith the Coritrol Eti<g. Lab.. Chulnro.s Uni\lei-sity clf’Technology, Sweden. 1. Jovik is vi. i th the Dept. ofTec.hriology, University ofBoras. Snvderi. This M . O I ~ M’US spcirrsord I?? the Swedish National Board f o r Industrial uric1 Ter ,hnicnl Dewlop ie t i t (NUTEK).

28 0272- 1708/95/$04.000 199SlEEE I E E E Control Systems

? t QR + DTD 4 GD 4 W

w AD QA QE

Fig. 2 . Default phxsical niodrl of dri\w(rairi. JTJG = turbine and cgenerato,- inertias. K.y.B.y = shaft c,oniplitrnc.e urid daniping.

By connecting the generator via an AC/DC/AC link it becomes possible to regulate the power by changing the turbine speed. The advantages with speed control, [ I ] , compared to pitch con- trol, are: added compliance (an increase in the aerodynamic load leads to acceleration instead of load increase), improved energy capture. and less aerodynamic noise, A drawback of the variable- speed concept is the need for a more complicated electrical system including frequency-conversion equipment, cf. [2]. This implies additional investment costs and certain energy (conver- sion) losses, and i t impairs the quality of the output power: therefore, continued technical progress in this area is vital for the success of the variable speed concept.

Today most variable-speed machines are also equipped with control of the pitch. but there is an increased attention directed toward fixed pitch variable speed plants. However, it still re- mains to show. in practice, that i t is possible, using this concept. to keep the power variations acceptable in strong wind.

One prerequisite of a well designed control system is a reliable drive system model of a reasonable complexity. The importance of accurate modeling is accentuated when the wind turbine, as in our case, is constructed with a flexible or soft concept, resulting in lightly damped structural resonances in the frequency range of closed-loop operation, cf. [ 2 ] .

This article describes physical modeling, system identifica- tion (performed on measurement data from a 400 kW full-scale wind turbine), and control design for variable speed, fixed pitch, horizontal axis wind turbines.

Physical Modeling The physical model of the drive-systems dynamics. defined

in Fig. 1, incorporates three sub-models, presented below.

Drive-Train Dynamics The drive train includes the turbine and generator inertias and

the main shaft connecting the two, The model order is determined by the number of rotating masses (or interconnecting compli- ances) one chooses to assign; see [3] for an analysis. We have chosen the least complex resonant system, described in Fig. 2. assuming an ideal gearbox and reducing all quantities to the primary (low-speed) side. The choice of model order is based on the a-priori knowledge that the system has a dominating funda- mental resonant mode and that we wish to keep model complex- ity low. The gear-box inertia is typically much smaller (here, approximately a factor of 1/30) than the generator inertia, which means it will have no dynamic influence for lower frequencies; it is thus not specifically modeled but can be assumed included in the generator inertia. For the drive system model is also included the interaction between the drive train dynamics and the aerodynamics and generator dynamics, respectively. We get for the drive train dynamics"2:

where ./T is the turbine inertia. Jc; is the generator inertia. Ks is the shaft compliance, B,y is the shaft damping, and Q is the shaft torque.

Aerodynamics The aerodynamics block in Fig. 1 includes the variable torque

coefficient function cy(A) (see Fig. 3), which is a turbine specific function defining the ability to convert kinetic energy of the moving air to mechanical torque (QA). The 1-to- 1 corresponding power coefficient function is +(A) = hc,(h); it is the maximum of the +(h)-function that one wishes to track during partial load operation (see further Section 5) . Note that these functions are of one variable (A), named the rip-speed ratio and defined as the ratio between the speed of the turbine tip and the wind speed. We get:

(2) where p is the air density, R is the rotor radius, A is the rotor disc area, and M' is the wind speed. where QA becomes a non-linear function of two variables (M, and w). A first-order linearization of (2) yields:

'SI units are a u m e d throughout the paper. 'The time argument will be left out if given by context

(3)

August I995

where A denotes deviations from the operating point (O.P.) values (Q/\o: ~ ' 0 , ?uj, etc.) while ' denotes a derivative w.r.t. h. Also, y < 0 represents the t 7 o I " U l (attached flow) region, and y > 0 the stall (separated flow) region; as can be seen from (3) these two regions coincide with (,;, < 0 and c;l > 0 , respectively, also

depicted in Fig. 3. Stall is induced at high incidence angles, which is equivalent to low values of the tip-speed ratio; then the inertia forces of the moving air inhibit the air stream to follow the curvature of the blade and the air boundary layer separates. The result of a stalled blade is a sudden and significant decrease

tipspeed mtio

Fig. 3. The used ~ , ~ ( h ) - und cl,(h)-c,urves. (The former up-scaled.) The h-range is dilided into t ~ v regions: h > h(.rrt being the normal region, and h < h, ,.if the stall rq ion (unstable); hop corresponds to nmrimul poM'er coey-icient.

in lift force. For wind turbines this creates a corresponding drop in the aerodynamic torque, which is why it is often used forpower limitation at high wind speeds.

One can now relate (3) to Fig. 1. by noting that y [Nm/(rad/s)] denotes the relocity feedhac.k coefficient from the drive train to the aerodynamics. The parameter CI is an 0.P.-dependent scaling parameter converting the wind disturbance into a torque distur- bance.

Generator Dynamics Although there exists a specific generator configuration for

the plant (induction generator with frequency converter), this topic will initially be addressed more generally here, in order to analyze differences in dynamics for alternative configurations.

Assuming an induction generator (without frequency con- verter) there will be, equivalently to the effect from the aerody- namics, velocity feedback on the generator side via the quasi-stationary relation:'

'Valid when the generator-torque transient is much faster than the fundamen- tal dripe-train dynamics.

Fig. 4 . Block (.>tute i~uritiblei diagrtrm; linearized ph!sicul model assuming un indiiction ,qenc'ratoi:

where w.\ is the synchronous speed. This results in the block diagram of Fig. 4. For induction generators, the generator speed has a very stiff dynamic connection to the synchronous speed, i.e., the parameter p [Nm/(rad/s)] is large: in the limit as p + M the turbine inertia J r and the shaft compliance Ks together form a resonance frequency:

with low damping provided Bs is small (which is true for a standard shaft without any additional damping).

If we use a synchroiiou.s generator, there will be no velocity feedback but instead a rotor position feedback, p, since the generator torque is nom given by the relation:

QE =p((jr -e , )=CL(o,q -m.,) (6)

where w, is the rotor-position angle. This means that p + p/s and the block diagram of Fig. 4

remains principally unaltered. On the other hand. using either generator type equipped with

frequency conversion, the combined action of generator and converter causes the feedback of the rotor velocity or position, respectively, to be eliminated. Le., we get p = 0 or p = 0; this means the converter makes the generator torque independent of the system dynamics, u hich causes the generator-torque magni- tude always to be equal to its reference value. In this case the system will again have a lightly damped resonance: the reso- nance frequency will be higher than without conversion:

(7 )

It can be shown that for a specific intermediate value of p = pc damping is maximi7ed [4]. For higher and lower values of p either of the two resonances mentioned above will appear (Fig. 5) . A possibility to modify p is by the use of feedback. In the induction generator case. p will be modified to p' = p/( 1 + k ) by using proportional feedback k of the mechanical or generator torque. Equivalently. we may use feedback k of the generator speed in the case of frequency conversion to achieve an artificial

3 0 IEEE Control Systems

I 1 r------I

Fig. 5 . Frequency functions (amplitude) of m,(s)/QR(s) (a) and Q(s)IQA(s) (b). (Direction of arrow indicates increasing j3 values.)

p’ = k , see [4]. Thus, in either case, damping will be increased with increasing feedback gain, up to a certain point correspond- ing to when p’ = pc.

As a footnote to the above, one can question the relevance of using the quasi-stationary relations ((4) and (6)). There are results at least for the induction generator case, [5 ] , that indicate that these relations are inaccurate already at frequencies as low as 1-3 Hz for comparable machines. Since we are dealing with a frequency converted system and therefore not using the men- tioned induction-generator models, we do not pursue this ques- tion; however, we would recommend to do so in the case of a configuration with the generator directly connected to the grid (Le., fixed speed operation).

Drive System Model The following state-space model results from Fig. 4. (see also

(l)) , assuming p = 0:

where wl(t) is the wind disturbance (torque), wz(r) is the meas- urement noise, and the matrices A, B, C are given by:

where Ores is the resonance frequency and Oar is the anti-reso- nance frequency, recognizing the resonance and anti-resonance frequencies as equivalent to (5) and (7), respectively.

If we assume small values of yin (lo), the modification of the dynamics compared to the simpler (11) amount to (cf. [6]):

the pure integration in (1 1) is replaced by a real pole a =

the relative damping of the complex pole (and zero) is

while the changes in the resonance frequency will be insignifi- cant. Since y is negative for operation below stall, the real pole a will be stable in this region.

For a chosen wind turbine we can calculate y as a function of h from (3) given an estimation of the current cp(k)-function. For the turbine in question, these calculations yield Iv < 90,000 [Nm/(rad/s)] in stall (provided5 w < 25 m/s) and Iv < 25,000 in the normal region (provided w < 15). The corresponding values for the real pole are Id < 0.45 and la1 < 0.13, respectively, (For details in the calculations, see [6]) . Since this is significantly slower than the resonant dynamics of the drive system the dynamic influence of yshould be low, i.e. there will be only weak aerodynamic feedback.

~ / ( J T + Jc)

increased4

Identification of a Discrete-Time Model System identification was performed on real measurement

data from a full scale wind turbine operating in closed loop as described below. The turbine is equipped with a wide range of sensors and a registration equipment that stores all measured signals on tape, enabling subsequent data processing.

which gives the following transfer function from QR to 0,:

If we furthermore assume operation in the stall transition region, equivalent to making y = 0, we get instead:

Experiment Description The input and output signals used for identification were

generator torque QE and generator speed a,, respectively. The closed-loop setup (Fig. 6) was due to safety reasons-however, as seen from the figure it was possible to add an external input signal, ux, to the controller output. The closed-loop response to

-

Fig. 6. Experiment layout, block diagram: AD = aerodynamics, DTD = drive train dynamics, PI = PI-controller, Wref= speed setpoint, QE = gen. torque (reference), u, = additional input.

4Explaining the expression aerodynamic damping. 5Proportional to the mean wind speed.

August 1995 31

10' SOLID = ARMAX

DOT = STATE-SPACE

1 o-2~ 1 oo 1 o1

freauencv raws 1 0'

Fig. 7 . Data registrations: Closed-loop step response (a), and sequence of data for identi3cation (b); top: output (os), bottom: input (QE) including ux; signals are scaled.

a step in ux indicated a clearly under-damped closed-loop reso- nance frequency of about 3 Hz (as can be seen from Fig. 7.(a)). Due to loose controller action6, the identified closed-loop reso- nance frequency could be presumed to be close to the open-loop resonance frequency. Based on this, the signal ux, subsequently used to provide excitation in the frequency band of up to around 3 Hz, was chosen a PRBS7 with clock interval Tc = 0.24 sec, added to the controller output u. Data series (Fig. 7.(b)) were registered during partial-load operation (mean wind speed less than 5 m/s) with a sampling frequency of 20 Hz.

Non-Parametric Method (Spectral Analysis) Spectral analysis could not be performed directly on the

registered data due to the obvious correlation between input and disturbances via the output feedback. One can, however, use indirect identification (see [7]) using an alternative input, uncor- related with the disturbances. In our case it was appropriate to use the PRBS input, ux. Denoting the transfer function from -QE to W, as G(s), Le., the same transfer function as in Section 2, we get from Fig. 6.:

(12) where ̂ signifies estimates

Fig. 8 . Black box (solid), discretized state-space (dotted), and spectral-analysis models, frequency amplitude functions (amplitude).

where @(io) is the spectral density function and G(im) is the frequency function of -wg(s)/QE(s).

The result can be seen in Fig. 8. The fundamental resonance peak appears clearly at a frequency below 3 Hz, as expected close to the closed-loop resonance frequency.

%e controller parameter values were, however, unknown. 7Pseudo-random binary input.

Discrete-Time Parametric Model The MATLAB System Identification Toolbox was used to

perform identification of a discrete-time black box model, Le., G(q) and H(q) in:

For low orders, the Output-Error structure (as well as ARX) gave poor parameter convergence in accordance with the fact that the closed-loop setup destroys the well-known asymptotic convergence of the Output-Error structure, cf. [8]. An ARMAX' model of third order was chosen (Fig. 8), yielding:

A(q) = I - 2.27q-' + 2.19q-2 -0.91q", B q ) 0.26q-' -0.45q-' + 0.24q-3, C(q) = 1 - I .46q- I = + 0.78q-'

G(q ) = B(q//A(q), H(q) = C(q)/A(q) (14)

corresponding to continuous poles (pi), zeros (zi), eigenfrequen- cies (mi), and relative dampings (ci) as:

p1 = -0.35

6 , = 0.046 w,, = 16.7

p2,3 = -0.77 k i16.7 +

6 , = 0.054 W, = 9.21

(15) z , , ~ = -0.49 k i9.20 +

Note that, according to what was expected, the real pole is significantly slower than the rest of the identified dynamics.

*In the Box-Jenkins structure the poles of G(q) and H(q) showed close to identical, motivating the choice of ARMAX.

32 IEEE Control Systems

Continuous-Time Parametric Model The wide class of methods estimating parameters in continu-

ous-time models can be divided into two subclasses, viz. indirect and direct methods. The idea behind the former is that a discrete- time model of the system is first estimated and then a continu- ous-time model is deduced from it, cf. [9]. Direct methods use a procedure where the continuous-time parameters are directly estimated without using a discrete-time model as a first step. The purpose of this is often to estimate also the physical parameters, cf. [IO] and [l l] . Both methods have been used here and the results are presented below.

Indirect Estimation of Physical Parameters In order to estimate values of the physical parameters a

continuous model of the same structure as in (10) was desirable. Using the MATLAB matched method [12], the following trans- fer function was obtained:

5 . 4 6 ~ ~ + 5.39s + 464 Lx bis3-j Cais3-i, 3

‘(’) = s3 + 1 . 9 1 ~ ~ + 28 Is + 98.9 i=l i=O (16)

where a0 = 1 .Having hereby verified the assigned physical model structure (cf. (IO)), the following step was to estimate the pa- rameters JT, JG, Ks, Bs, and y by equating the coefficients of (10) and (16), using p = 0. This gives us five unknowns and six equations yielding an overdetermined, non-linear system of equations. We separate this into two parts, neglecting the second-order damping term (see 1, below). This yields two linear systems of equations, easily solved by using the following steps:

1. Calculation of JT, JG, and Ks from the three equations that do not contain any damping terms, provided we neglect the term B f l in the &-equation. This yields the unique solution:

2. Calculation of Bs and y for the remaining overdetermined, linear system of equations by the least-squares method:

-JG JT + JG where W=-[-:s 1 A ]

JT JG

To be able to compare the from (18) resulting parameter vector with the following a-priori values:

[ J T ~ u ~ J G C U ~ KscUlJ = [ I .8e5 2.3e4 1 Se7J (19)

we must scale the former’, a natural scaling factor being the one between the sum of the inertias. This leads to the following identification-based parameters lo:

[JTJG Ks Bs YJ = [ I .4e5 6.1 e4 I .2e7 6.0e4 -7.1 e41 (20)

It is of specific interest to compare the ratio between the inertias, uniquely determined by the ratio between the two reso- nant frequencies, Le., W2/wi (see (ll)), and therefore highly significant for the system dynamics: JT/JG = ( w d w ~ ) ~ - 1 = 2.3 for identified parameters; this is to be compared to the value 7.8 for a-priori data. The plausible explanation to this difference lies in the problem formulation itself. We are dealing with a distrib- uted dynamic system which in the default model is aggregated into a model of third order, assigning two inertias to describe the low-frequent behavior of the system. These two inertias have then been assumed to represent, respectively, the turbine and generator inertias, whereas the compliance represents the main shaft. In reality, the correct partitioning of the inertia is, however, not known beforehand; if there are, e.g., individual construction elements of low stiffness in the system, they might completely alter the low-frequency behavior. For the identified parameter values it is notable that 22% of the a-priori-estimated turbine inertia has been transferred to the generator, indicating that the fundamental resonance arises not in the main shaft, but rather in the turbine blades oscillating against the hub and the rest of the drive train (the S-mode, Fig. 9). This means that J ~ i n the model, instead of being the total turbine inertia, rather represents the outer part of the blades, while JG in the model represents the generator, shaft, gearbox, turbine hub, and the inner part of the blades. (Since the inertia of the blades is highly distributed, the exact partitioning into inner and outer parts is not obvious but also not significant for our purpose.)

Also, the value of y is clearly outside the pre-calculated range (being -9e3 at 5 m/s) and corresponds to a higher aerodynamic damping than was anticipated; this can be explained by un- modeled viscous damping (e.g., friction in bearings) that has “spilled over” into the identified y value. Also, note that the identified y value is of limited use since it stems from an assumed linear aerodynamics model which is 0.P.-dependent, Le., a func- tion of both kind speed and the current value of h during data registration; to get a more use- ful estimate of y, one would have to perform several longer identification sessions and then by statistical methods ar- rive at yas a function of O.P.

Direct Identification of Physical Parameters

Here we will consider a di- rect method which can gener- ally be applied to both SISO and MIMO systems. By as- signing a model structure given by the physical model in state-space form (8), discretiz- ing it, assuming zero-order

J BLADE

NACELLE

HUB

TOWER

Fig. 9 . Frontal view of schematic wind power plant illustrating the edgewise (i.e. in the rotor-disc plane) S-mode.

9The registered input-output data were individually scaled by unknown factors yielding an incorrect static gain. “A-posteriori checking of the assumption: I B d = 4e8 << 2e 12 = Ks(JT+ JG)

August 2995 33

hold on the input, and applying the prediction error method, we arrive at a model with explicit physical parameters. For details of the method see [ 131 and [ 141.

With the experiment design identical to the one described in Section 3 we obtained the following values of the estimated parameter vector 8 along with the standard deviations (Ti:

6 f 3 G i =

(1.4 f 0.2)e5 (5.8 f 0.5)e4

(1.2 k O.l)e7

(1.7 k 1.5)e4 (-4.1 k 3.7)e4

As can be seen, there is a significant uncertainty in the two parameter estimates connected with the damping, while the inertias and the stiffness are more accurately estimated; this is the same effect as was observed for the black box model.

One important point worth noting using this method is the relevance of making proper initial guesses, 80, of the parameter vector. In general, the required accuracy of the initial guesses chiefly depends on the signal-to-noise ratio (SNR) of the meas- ured signals [15]. In this particular experiment the resulting parameter values showed to be rather insensitive to different choices of 00 and the obtained values (21) are in good accordance with the ones obtained by indirect estimation (20). Reasons for this are an acceptable SNR and a relatively well known physical model structure. The somewhat divergent value of the relative damping < = 0.015 indicates a very lightly damped system; in Fig. 8 this lower damping is recognized as a higher resonance peak compared to the black box model frequency function. However, apart from the fact that the damping estimate is in either case rather inaccurate, for control purposes this discrep- ancy is not an issue since the system can be regarded as approxi- mately undamped.

Control Design and Evaluation Since we are treating partial load operation, the objectives of

the control are to maximize the captured energy while restricting the dynamic loads in the drive train. Maximal energy production is achieved by keeping the tip-speed ratio constant, at the peak of the cp(h)-curve (Fig. 3). Therefore the turbine speed must track the wind-speed variations, which creates large torque variations. Hence there is a contradiction between the two aims, and it is necessary to choose a suitable compromise. The tradeoff mainly depends on three things: the flatness of the cp(h)-curve around its maximum, the moment of inertia of the turbine, and the bandwidth of the wind turbulence, 131. The two control objec- tives are taken into account by minimizing the criterion:

J = E{ a. 6P + ; (Q: + Q; )} where a is a positive design parameter that reflects the relative importance between the two aims. In the first term 6P is the mean power-loss, due to not keeping optimal tip speed ratio, and the second term the sum of the torque variances of the shaft and generator. This criterion can be stated as a linear quadratic

function of the state and control variables by using a quadratic approximation of the cp(h)-curve (see [16]). For the design, we will use the dynamic model from the preceding section.

Model Modification The first step in this section is to convert to normalized

models, variables and parameters; this simplifies the controller design and interpretation of results. The state variables will be normalized, indicated with top bars, w.r.t. an operating point (O.P.); note that the A operator that should precede the linearized variables will henceforth be omitted.

The second step is to include the wind speed (turbulence) as a fourth state, driven by a continuous-time, first-order model-with time constant Tw-an approximation of standard, more complex models (Kaimal, von Karman a.0.). The time constant is 0.P.-dependent and given by a relation between the turbine-hub height and the mean wind speed (see [3]).

Normalizing the linear aerodynamic-torque model (3) yields:

where the value of 7 equals -1 when operating at optimal h, which will be the O.P. value used for control design. This yields the following complete state-space model of the drive system:

x = A a + B u + n ' , y = C x + w *

where wl(t),wz(t) are white noises with spectral densities Rwand R2, respectively, with the matrices A, B, C given as:

l o 0 0 - l / T , J

Above, all lowercase parameters equal the corresponding uppercase parameters times a factor n = WQo, the subscript 0 indicating O.P. values.

Controllers In this section six different control schemes, three linear and

three non-linear, are compared. In all of the cases it is assumed that the only available measurement is the generator speed, and the control signal being the generator torque. Some of the schemes require knowledge of additional variables, e.g. the state vector. In these cases a stationary Kalman filter is included in order to estimate these variables based on the generator speed measurement. The schemes and their appellations are:

1. LQG: Linear-quadratic gaussian controller (Fig. IO). The state vector has to be estimated, since the generator speed is the only measured variable.

34 IEEE Control Systems

Wind speed Gen.

torque LQgain 1 + 7 Piant t

(4 Gen. , KTman 4 Veed

~- filter 4

Wind Gen. speed 4 Plant

Gen 1 speed

- 1 torque ’ Speed ref.

(bl h LPor HPLP + &

Fig. 10. The LQG controller (a) , controllers LP and HPLP (b).

Toroue Gen. speed

~

Wind speed Track. G e n .

~ error torque v - ~

y x 2 + K , + -+ P 7 b Plant + -4 I

1 Low pass

A

(b) Gen. ~ filter speed

~

Kalman 4 - * Torque estim. filter t ~

~

Fig. 11. The non-linear TS controller (a) and TQ controller (b).

0.34

0.331

0.32

0.31

2 0.31 0

% 0.29 (I)

8

-

90.28L c

0.27

0.26 1.5

LINEAR I R2=le-7’rw

DOT = LO

SOLID = LQG

= HPLP

0 = LP

2 2.5 3 3.5 4 4.5 5 mean power loss (normalized)

I

I

I

5.5 1 0 . ~

Fig. 12. Analytically calculated performance curves for the three linear controllers 1-3; also shown is the ideal noise-free LQ case. One marker type for each controller type; increasing a values to the le) in the figure.

2. LP: Proportional (P-) controller with first-order low-pass

3 . HPLP: P-controller with non-minimum-phase high-pass filter (Fig. lo), numerically optimized for Equation (22).

and low-pass filters (Fig. 10):

K( l - T,s) G(s ) = -

(1 + T2s)2

where K, T I , and T2 are numerically optimized. 4.7’s: “Tracking controller, speed” (Fig. 11). The idea behind

this non-linear scheme was presented in [I]; it is based on the observation that keeping optimal tip-speed ratio can be reformu- lated as a problem in the torque-speed plane. The torque is estimated by a Kalman filter and the reference speed becomes:

The time constant of the low-pass filter of Fig. 11 is used as an extra design parameter. The generator speed is controlled with a simple proportional controller to track the speed reference.

5 . TQ: “Tracking controller, torque.” It is based on the same idea as the previous scheme, but instead uses a torque reference derived from the measured speed:

Also here an extra design parameter is included viaa low-pass filter. The inner loop still uses a P-controller as in 4 above.

6. w2: In this case the generator torque is set directly to the right-hand side of (28), cf. [17], i.e., as in Fig. 11 but without the inner loop. This is a simple (one-degree-of-freedom) special case of the TQ controller with the bandwidth determined by the stationary relation (28).

A main feature of the non-linear controllers is thus to provide an outer reference value loop, thereby enabling operation over a range of mean wind speeds (as opposed to the linear controllers).

Linear Control Design For the linear analysis, the controllers 1-3 above have been

compared by numerical optimization of the criterion (22), calcu- lated from the corresponding Lyapunov equation. The mean wind speed was 10.5 m/s, the turbulence intensity 0.15 (cf. [2] using a hub height of 40 m) and the measurement noise spectral density a factor lo-’ times the turbulence-induced disturbance, based on assuming a pulse tachometer with 5,000 pulses/rev.

The results in terms of mean power loss and torque standard deviation (the latter is the square root of the second term in (22)) are shown in Fig. 12. Here it is seen how the controller perform- ances diverge as the design parameter a increases, Le., the closer we try to track optimal h. The proper choice of controller, based on the above figure, is therefore dependent on how high a torque variation one can accept-the higher the value, the stronger the motivation to use a more complex controller.

August 1995 35

1 2

SOLID=LQG

DOT=HPLP

Fig. 13. Control signal response to a wind speed step.

The rationale behind controller 3 ) being non-minimum phase is that it thereby imitates the LQG controller behavior, as can be seen in Fig. 13. The physical interpretation of the phenomenon is that, in order to more closely follow wind variations, the LQG and HPLP controllers initially reduce the generator torque; hereby the turbine is allowed to accelerate to track the time-vari- able, optimal rotational speed, before the generator torque is allowed to increase to its new steady-state value. The simplerLP controller cannot give this behavior, which explains its poorer performance for high values of a.

Stability Considerations It is known that the stability margins decrease with increasing

feedback gain. Here it corresponds to increasing a, Le., using larger penalty for deviations from the optimal tip-speed ratio. In the schemes where a Kalman filter is used, (i.e. LQG, TS, and TQ), the feedback gain is also affected by the observer gain. Therefore the spectral density matrices used in the filter design should be well balanced.

In LQG design it is common to improve the stability margins using loop-transfer recovery, LTR. This has been incorporated here by adding a fictitious disturbance term to the element of the state spectral density matrix associated with the control input, cf. [18]. This method proves to work, to some extent, also for the HPLP-controller design, and has been used for the high-gain HPLP controllers presented.

In order to examine the stability margins the system has to be linearized. Such an investigation may not be sufficient, since it is only valid in a limited region about the O.P. If the excitation from the wind turbulence is large enough the system is forced too far away from the O.P. As an example, consider a controller with small or zero feedback gain (constant generator torque), which is optimal when minimizing load variations. Since the linearized open-loop system is stable, one would not expect any problems with stability. However, if the aerodynamic torque is not large enough to match the reaction torque of the generator, due to a temporary reduction in wind speed, the rotational speed will decrease. If the wind speed maintains its low value, the turbine speed and tip-speed ratio become too low, and the power coefficient drops rapidly when it enters the stall region. The

turbine is then retarded even faster and it will not be able to retum to the desired O.P. even if the wind speed eventually increases (Fig. 14). This may not be damaging, but it will require a re-start of the turbine. It should, however, be easy to avoid this scenario, by introducing a lower limit for the rotational speed.

In the tracking-controller schemes the tracking error is calcu- lated as the difference between two feedback loops, formed by the torque estimate and the generator speed, respectively. A suitable design parameter was introduced by adding a low-pass filter in the torque loop. It is essential that the filter is placed there and not in the speed loop; the speed feedback secures the stability, whereas the turbine-torque feedback by itself desta- bilizes the system.

Non-Linear Control Design and Simulations The original, non-linear system has been simulated using

each of the six control schemes described above with different sets of controller parameters. All simulations used the same realizations of the wind speed and measurement and control noise; the duration of the simulation was five minutes. The results are shown in Fig. 15.

The main reason for the difference between LQG and LQ performance is the inaccuracy of the wind speed estimation; it is based on a linearized aerodynamic model whereas the simulation model is non-linear. Nevertheless, the results show that the LQG

0.6

0.4

0.2

0 20 40 60 80 1M) 120 140 160 180 Time (s)

Fig. 14. Simulation showing an unstable situation caused by using an exceedingly low power loss penal0 (a); notice that the generator torque, u , maintains a high value throughout the sequence.

. < TQ I 0.5

TS

5

U $ 0.4

01

m HPLP e P

,2 LP I 0.3 * * LQ 8

.* L e - * LOG

02 1 2 3 4 5 6

I i n 4 Mean power loss (Normalized)

Fig. 15. Performance comparison via simulations.

36 IEEE Control Systems

performance is quite competitive with that of the non-linear controllers, which indicates a satisfactory performance robust- ness, since the wind variations cause the system to operate well outside the O.P.

The results of the LP and HPLP filters show that for the high-a region (high penalty on power loss) they are outper- formed by most of the other schemes. Here it is obviously not

struction elements (in our case a teetered hub) nor the data on materials (e.g., the turbine-blade elasticity) are available with any degree of accuracy. The use of system identification to get accurate numerical estimates of key parameters is thus often a prerequisite for an appropriate control system design.

The evaluation of different controllers for the specific plant (regarding partial load operation) show that when there is a

sufficient to use a constant ref- erence speed; the high-a per- formance would doubtless be improved if the tip-speed ratio could be used as a reference sig- nal-which is actually the case in the LQG scheme-but this would require the use of an ob- server. Another possibility, es- pecially for larger wind-speed variations, is to use gain-switch- ing. In the lower-a region (high penalty on control effort, yield- ing looser controller action), the non-linear analysis confirms the results from the linear analysis, viz. that the two simpler control- lers, particularly the HPLP, are quite competitive.

The TS and TQ schemes were simulated using the same Kal- man filter as in the LQG case. The design parameters are the controller gain and low-pass fil- ter time constant. In the plotted simulations, the gain is set at a fixed value while the time con- stant is varied. The controllers are seen to perform very simi- larly; the performance is also very similar to that of the LQG controller. It should be stressed that in order to perform as well

The 400 kW experimental wind turbine.

in the high-a region as indicated by Fig. 15 the assumed low measurement noise must be valid; therefore the quality of the transducers becomes an essential feature when increasing the demands on control performance. This topic is further addressed in [6 ] .

The last controller, w2, is interesting only for relatively low penalty on power loss, as can be seen from the results in Fig. 15. In the simulations a first-order low-pass filter was included and its time constant was varied, the leftmost point on the curve corresponding to the apparently best choice.

Conclusions The process parameter of the greatest importance for control-

ler design and closed-loop performance is the fundamental reso- nance frequency. It is interesting to note that calculations of this frequency, based on construction data and materials properties, were in error by a factor of two. This is not an isolated aberration but rather typical for numerical calculations based on a-priori models, since neither the compliance properties of crucial con-

strong aim to penalize power loss, the evaluated simpler linear controllers are outper- formed by non-linear con- trollers. This is primarily due to the simplicity of the for- mer controllers-that they use a constant reference speed-and secondarily to the non-linearity introduced by the cp(h)-curve. The LQG controller, on the other hand, performs as well as the non- linear controllers, suggest- ing that a linear controller might still be adequate for this problem. The main ad- vantage of the non-linear controller, that it can cope with variations in mean wind speed, however remains. If linear quadratic control is to be used, this problem will have to be solved, e.g., by using gain switching. Also the non-linear controllers would gain from this con- cept, since although there is an automatic adjustment of the current speedhorque ref- erence via the non-linear part of the controller scheme, there is still an 0.P.-depend-

ent optimal choice of the parameter values in the linear p-art of these controllers.

Acknowledgments The authors gratefully acknowledge the assistance, during the

experiments, of T. Lyrner and S. Strandberg, both involved in the NWP 400 wind turbine project. We also want to express our thanks for the cooperation of VATTENFALL, who run the wind power station at Lyse, Sweden.

References [ l ] W.E. Leithead, “Variable Speed Operation-Does It Have Any Advan- tages?” Wind Eng., vol. 13, no. 6, pp. 302-314, 1989. [2] L.L. Freris, “Wind Energy Conversion Systems,” Prentice Hall Intema- tional, pp. 131-138 and 189-205, 1990. [3] T. Ekelund, Control of Variable Speed Wind Turbines in a Broad Range of Wind Speeds, Control Eng. Lab., Chalmers Univ. of Techn., Goteborg, Techn. Rep. No. 172L, 1994.

August 1995 37

[4] B. Schmidtbauer, Basdynamik och Kompenseringsmetoderf~r Vindtur- binKenerator, Control Eng. Lab., Chalmers Univ. of Techn., Goteborg, Rep. NO. R-94/0002, 1994. [SI T. Thiringer, Measurements and Modelling of Low-Frequency Distur- bances in Induction Machines, Dept. OfElectncal Machines and PowerElecbon- ics, chalmers Univ. of Technology, Goteborg, Techn. Rep. No. 151L, 1993. [6] P. Novak, Modeling, Identijication and Control of Variable-Speed Wind Turbines, Coniml Eng. Lab., Chahers UNv. of Techn., Goteborg, m preparation.

Peter Novak received the M.S. degree in mechanical engineering from Chalmers University of Technology (CTH), Goteborg, in 1989 and is currently pursuing the Ph.D. degree in control engineering at CTH. His re- search interests include modeling and control of dy- namic systems, discrete-time estimation, and applications to the wmd-energy field.

[7] R. Johansson, System Modeling and Identification, Prentice-Hall Inter- national, pp. 146-151, 1991. [8] L. Ljung, System Identification-Theory for the User, Prentice-Hall, Englewood Cliffs, N.J., pp. 69-80 and 345-346, 1987. [9] H. Unbehauen and G.P. Rao, “Continuous-Time Approaches to System Identification-A Survey,” Aufomatica, vol. 26, pp. 23-35, 1990. [IO] S. Dasgupta and B.D.O. Anderson, “Physically Based Parametrizations forDesigning Adaptive Algorithms,” Automatica, vol. 24, no. 2, pp. 217-225, 1987. [ l l ] R. Isermann, “Schatzung physikalischer Parameter fur dynamische Prozesse,” Automatisierungstechi~, vol. 39.9-10 R Oldenburg Verlag, 1991. [12] L. Ljung,SystemIdentijication ~serj.Guide,TheMathWorks,Inc., 1991. [ 131 P. Novak, I. Jovik, and B. Schmidtbauer, “Modeling and Identification of Drive-System Dynamics in a Variable-Speed Wind Turbine,” Proc. ZEEE 3rd Conf. on Control Appl., vol. 1, pp. 233-238, 1994. [14] T. Kailath, “An Innovations Approach to Least-Squares Estima- tion-Part 1: Linear Filtering in Additive White Noise,” IEEE Trans. on Autom. Control, vol. AC-13, pp. 646-655,1968. [ 151 I. Jovik and B. Lennartson, Identification of Physical Parameters in Mechnical Structures, Control Eng. Lab., Chalmers Univ. of Technology, in preparation. [16] T. Ekelund and B. Schmidtbauer, “Tradeoff Between Energy Capture and Dynamic Loads in a Variable Speed Wmd Turbines,” Proc. IFAC 12th World Congress, vol. 7, pp. 521-524, 1993. [ 17) J. Emst and W. Leonhard, Optimization of the Energy Output of Variable Speed Wind Turbines, Proc. WindPower ‘85, vol. 1, pp. 183-188,1985. [18] B.D.O. Anderson and J.B. Moore, Optimal Control, Prentice-Hall Inc., N.J., pp. 228-250, 1989.

Thommy Ekelund graduated with an M.S. in engineer- ing physics from Chalmers University of Technology in 1990. Since 1991 he has worked as a Ph.D. student at the Control Engineering Laboratory at Chalmers. His research deals with modeling and control of wind tur- bines, characterized by the non-linear and periodically time-varying behavior.

Inge Jovik received his M.S. degree in electrical engi- neering from Chalmers University of Technology (CTH), Goteborg, in 1987. Since 1988 he has worked as a Ph.D. student at the Control Engineering Labora- tory at Chalmers. His research deals mainly with iden- tification and modeling of mechanical applications as well as robust control of MIMO systems. Since 1991 he has worked as a lecturer at the University of Boraas.

Ben@ Schmidtbauer received his basic engineering degrees from Chalmers University of Technology (Engi- neering Physics) in 1961 and M.I.T. (Aeronautics and Astronautics) in 1965. He has been employed by the Saab Scania company, working with systems and control in the aerospace field. He received his Ph.D. in Control from Chalmers in 1973, where he is currently associate profes- sor. His research interests are mainly within the field of electro-mechanical control applications.

Sampled Data

A Compendium of Ground Rules for Engineers

When you don’t know what you are doing, do it NEATLY. When you don’t know what you are talking about, say it LOUD. Experiments must be reproducible. They should all fail the same way. First draw your curves, then plot the data. Experience is directly proportional to the equipment ruined. A record of data is essential. It indicates you have been working. To study a subject best, understand it thoroughly before you start. In case of doubt, make it sound convincing. Talk fast and authoritatively. Do not believe in miracles; rely on them. Teamwork is essential. It allows you to blame someone else. Remember, an ounce of image is worth a pound of performance.

-Taken from Rettip’s Guide for Engineers, Scientists, Etc.

38 IEEE Control Systems