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One step ahead adaptive control technique for wind systems
~
Energy Convers. Mgmt Vol. 39, No. 5/6, pp. 399-413, 1998
Pergamon
PII: S0196-8904(97)00019-8
ONE
STEP AHEAD
ADAPTIVE
FOR
WIND
© 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0196-8904/98 $19.00 + 0.00
CONTROL
TECHNIQUE
SYSTEMS
A. DADONE, L. DAMBROSIO and B. FORTUNATO Istituto di Macchine ed Energetica, Politecnico di Bari, Bari, Italy
(Received 2 July 1996)
Abstract--The control of a wind system considered as an isolated source of power and composed of a horizontal-axis wind-turbine connected to an induction generator is analyzed. Appropriate mathematical models for both the horizontal-axis wind-turbine and the induction generator are used. The one-step-ahead adaptive control technique is presented and used to regulate the wind system. A sensitivity analysis of the induction generator performances with respect to control and disturbance variables is presented in order to evaluate the control flexibility. The results of three control problems are finally shown in order to prove the reliability of the suggested control technique. © 1997 Elsevier Science Ltd. Wind system
Control
Adaptive
One step ahead
1. I N T R O D U C T I O N
Wind energy is a free source of energy which can be converted into both mechanical and electrical energy by means of a wind system. When used to generate electrical energy, the wind system is basically composed of a wind turbine, an electrical generator and a control system. The wind turbine converts the wind energy into mechanical energy which is used by the electrical generator to produce electrical energy. The control system counteracts the disturbance effects in order to minimize the differences between the actual and the reference values of the output voltage and frequency. In the present paper, a horizontal-axis wind-turbine and an induction electrical generator will be considered. Appropriate mathematical models are obviously necessary to simulate the static and dynamic behavior of both components of the wind system. In principle, the flow about a horizontal-axis wind-turbine is three-dimensional, viscous and unsteady. Its direct simulation would be too complicated, while rather simple models are generally appropriate for a numerical simulation of a controlled system. The existing mathematical models to simulate wind turbines can be classified into momentum, vortex and finite difference models. The momentum models approximate the wind turbine effects by means of actuator surfaces and subdivides the flow domain into many streamtubes (see e.g. Ref. [I]). The vortex models represent the turbine blade effects by means of distributed bound vortices (see e.g. Ref. [2]). The finite difference models approximate the rotor effects by means of an actuator surface and solve the fluid-dynamic governing equations by means of finite volume or finite difference methods (see e.g. Ref. [3]). The momentum models are generally simpler, and they will be here considered to simulate the wind turbine. An appropriate mathematical model for the induction electrical generator can be found in many books [4] where the differential equations governing the dynamical behavior are reported. Reference [5] suggests inserting a variable capacitor and using it as the first control variable, the rotor winding resistance being the second one. These control variables are used in Ref. [5] to perform a steady-state control analysis of an induction electrical generator. In the present paper, a complete dynamical model of both the rotor and the stator winding will be presented, and only the rotor winding resistance will be considered as a control variable. The stator winding capacitor will not be used as a control variable, but it will be preserved and used to ensure the self excitation of the electrical generator [6]. 399
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DADONE et al.:
ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS
Classical controls of wind systems can be found in many papers with reference to open loop schemes [7] as well as to closed loop ones [8, 9]. An innovative technique which tries to anticipate the dynamical behavior of the controlled system is the one-step-ahead adaptive control [10] which has been successfully applied to helicopters [11]. Such a control technique linearly estimates the future status of the controlled system at time t + At on the basis of the system status at time t and of the previous dynamical behavior of the system itself. Therefore, it can be applied to nonlinear and time varying systems and automatically retunes itself whenever the control conditions change due to disturbance actions. The aim of the present paper is to extend the one step ahead adaptive technique to control a wind system. The problem we intend to solve is stated in the following paragraph. 2. PROBLEM STATEMENT The considered wind system is made of a three-blade horizontal-axis wind-turbine and a three phase induction electrical generator, connected by means of a gear box. The gear box may present either a fixed or a variable transmission ratio. In the present paper, the considered wind system is supposed to be an isolated electrical power generation station. The general layout of the control system is shown in Fig. 1. The block System represents the wind system to be controlled. In general, such a block represents the physical system, while in the present paper, it will be numerically simulated by means of an appropriate mathematical model. The block Parameter Estimator is an essential block of the one-step-ahead control system. On the basis of an appropriate time sequence of input and output data, the parameter estimator evaluates a suitable linear model of the system which will change at every considered time step. The block Design Calculations records the desired control targets, i.e. the desired time history of the system outputs. The block Control Law determines the appropriate system inputs on the basis of the actual system outputs and of a comparison between the target outputs (recorded by the design calculation block) and the data estimated by the parameter estimator block. The target of the control system here considered is to regulate the system outputs, represented by the frequency and the voltage generated by the electrical induction machine. In the following paragraphs, the mathematical models pertaining to the wind turbine and the induction electrical generator are first outlined, and a sensitivity analysis of the induction generator
Inputs
Outputs
System
' "~"~"
Piremeter elt;;'hlk~.
~.
Design cllculltlons
Control
Fig. 1. Block diagram of adaptive control system.
,,~
DADONE et al.: ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS
401
performances is presented in order to evaluate its control flexibility. Then, the one-step-ahead adaptive control technique is presented. Finally, the customized control technique is applied to regulate the wind system. A preliminary control study will consider the regulation of the output shaft rotational speed of the wind-turbine connected to the gear box in the presence of abrupt changes of wind velocity and load torque. A second preliminary control study will analyze the regulation of the output voltage of the electrical generator alone in the presence of rapid changes of shaft rotational speed and electrical load. Finally, the complete wind system will be considered, and the regulation of the output voltage and frequency in the presence of abrupt changes of the wind speed and of the electrical load will be studied. 3. M A T H E M A T I C A L M O D E L OF THE H O R I Z O N T A L - A X I S W I N D T U R B I N E
The mathematical model adopted to simulate the dynamical behavior of the horizontal-axis wind-turbine is based on streamtube discretization [1]. It relies on the momentum law in both the axial and the tangential directions and on the aerodynamic characteristics of the blade airfoil. The thrust and the torque acting on a blade element dr at distance r from the center of the rotor are given by [1]:
dT = p'rc "r"a'(1-a)2"V2°'CL'~(lsln q~ \ dQ = p.~z.r2.a.(l_a,)2.(f~.r)2.CL.~
+ ~__[L.tanCo,).dr
COS2tp
( 1 CD 1 ) CL tan ~b
"dr
(1)
where c, and A and .4b are the blade chord, the rotor area and the total blade area, while p, V0, a, a' and q~ are the air density, the wind velocity, the axial and tangential induced velocity coefficients and the inflow angle, respectively. Moreover, fl, CL and Co represent the rotor angular speed and the lift and drag coefficients of the airfoil, respectively. Finally, a = A b / A is the solidity ratio. Equation (1) allows one to determine the wind turbine torque, QT, and power, P:
QT =
_•Rrnax dO
(2)
d Rmin
P = f~'QT,
(3)
w i t h Rmin, Rmax being the internal and external rotor radii, respectively. Equation (3) allows one
to evaluate the wind turbine power coefficient, Cp: P
Cv - 1/2.p.A.V 3
(4)
which is representative of the wind turbine performance. In the present work, a three-blade horizontal-axis wind-turbine will be considered. The external (Rmax) and internal (Rm~n)rotor radii are 5 and 0.5 m, respectively. The nominal conditions are: wind velocity equal to 10 m/s; power equal to 22 kW; tangential velocity ratio A equal to 7 (A = 0.5"~'Dmax/Vo). The rotor blades are twisted according to maximum power and maximum airfoil efficiency criteria for the outer section of the blade and according to the constant chord criterion for its inner section. As a consequence, appropriate radial distributions of the pitch angle fl and of the chord e have been determined. The considered airfoil profile is a NACA 0012; the analytical expressions for the corresponding lift, CL and drag, CD, coefficients have been taken from Ref. [12]. As a characteristic of the considered wind-turbine, Fig. 2 shows plots of the power coefficient, Cp, vs the tangential velocity ratio, A. Each curve corresponds to a different value of the collective pitch/30.
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DADONE et al.:
ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS
0.50 -
Cp
13--O
0.40
0.30
0.20
I~X2 0.10
0.00
ft,=aS
"" 0.00
2.00
4.00
8.00
8.00
10.00
A
Fig. 2. Power coefficient vs the tangential velocity ratio for different values of the collective pitch.
As far as the gear box connected to the wind turbine is concerned, it will be represented either by a constant or by a second order transfer function. In some of the considered control applications, its gain will be constant, while in some other applications, such a gain will be a control variable.
4. M A T H E M A T I C A L
MODEL
OF THE
SELF-EXCITED
INDUCTION
GENERATOR
The wind turbine drives a three phase self-excited induction generator. Since the stator and rotor windings of the induction generator, together with the external load, are balanced, a per-phase analysis can be adopted [4]. According to [4], a fixed axis reference system, dq, can be used to determine the governing equations of both the stator and rotor windings: •
_
dill,,
di2d
Vjd = rl " hd W L~ " - - ~ " r LM" d l
Ulq =
•
- dilq_di2q rj'ilq+L~"-~TLM'---~
di2d
dild
di2q
dilq
.
.
0 = r2' t2d+ L2"--d-/-+ LM"--~-+p" Cog"(L2"Z2q+ LM"hq)
0 = r2"i2q + L2"--~- + LM "---~--p .09g. (L2" i2d + LM'i~d)
(5)
where rt and r2 are the stator and rotor internal resistances, respectively, while L~ and L2 are the corresponding inductances• Moreover, LM, ogg and p represent the mutual inductance, the rotor angular speed and the number of polar pairs, respectively. The voltages V~d, V~q and the currents i~d and i~q do not have a direct physical meaning, although they are related to the phase voltages and currents, as will be shown later.
DADONE et al.: ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS 403 The external load being represented by a capacity, C, an inductance, L, and a resistance, R, the corresponding governing equations in the dq system of reference are: UId : L "diLd dt L"
dDld
diLq dt
L~lq = rid
C" dt = - R--~L- -
C "dUlq: - u l q dt
RE
lid - - iLd
iiq -
iLq
(6)
The dynamic equilibrium of the moving parts of both the wind turbine and the induction generator can be expressed as: .
d-t -
"
.
.
.
= J" [aT-- 3 "p'(tld'/2q - - / I q "/2d)]
(7)
where J, z and t represent the moment of inertia of all the rotating parts referred to the electrical generator shaft, the transmission ratio of the gear box and the time, respectively, while QE is the rotor electromagnetic torque given by: Q E = 3 "p" (ild' i2q - - ilq" i2d)
(8)
Equations (5) -(7) represent a set of nine equations in the nine unknowns ild, iZq, 6d, i2q, /)ld, V,q, iLd, kq, and cos, which can be solved provided that the wind turbine torque is computed by means of equation (2). Finally, the phase voltages can be obtained from the unphysical voltages V~d,V,q, by means of the following relations: lfla ~ UId
1
,/3
1
,,/3
Vb =--~'V~d+ 2 "V~q re=--~'Uid-- 2 'Vlq
(9)
At this point, a steady-state model of the induction generator has been derived from the dynamic model (equations (5), (6) and (9)) by means of the phasor approach. Such a model has been used to perform a sensitivity analysis of the system in order to evaluate the sensitivity of the output voltage and frequency to changes of the external disturbances (load resistance and inductance) and of the manipulating variables (self exciting capacity and rotor resistance). Such a sensitivity analysis showed that the self-exciting capacity has a relevant influence on both output variables, while the influence of the rotor resistance is relatively small. Moreover, it has been determined that both the external disturbances significantly affect the output variables. The results of such a study are extensively reported in Ref. [6]. For the sake of conciseness, only the most relevant results, referring to the influence of the rotor resistance, are here plotted in Fig. 3, which shows the percentage variations of both output voltage, A V%, and frequency, Aft~o, vs the percentage variation of the rotor resistance, Ar2%. 5. O N E - S T E P - A H E A D ADAPTIVE CONTROL TECHNIQUE
As sketched in Fig. 1, and already presented in paragraph 2, the one-step-ahead adaptive control technique [10, 13] combines the parameter estimation algorithm with the control scheme. In the ECM 39/5-6--B A"
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DADONE et al.:
ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS
4.00
%
0.00
-
~f
-4.00
-8.00
I
I
I
I
I
-40.00
0.00
40.00
80.00
120.00
Ar2 %
Fig. 3. Percentage variations of output voltage and frequency vs the percentage variation of the rotor resistance.
following paragraphs, it will be first shown how to estimate a linearized model of the wind system on the basis of the previous sequence o f the wind system inputs and outputs• The least squares algorithm will be used for an on-line estimation of the linearized model parameters. Then, the control scheme will be applied to the linearized estimated model of the controlled system. 5. I. Least squares parameter estimation algorithm
The wind system can be generally represented in the state variable notations as: = f ( X , U)
(10)
where X and U represent the vectors of the controlled variables and of the control variables, respectively:
X =
X2
U =
U~
(11)
Let us now consider a Taylor series expansion of equation (10) at X = 0, U = U0, truncated at the first-order: = C.X+D.AU
(12)
where:
C =
~Xi
~X2
....
OX.
OXI
OX2
....
OX.
Of. OXj
Of. OX2
....
Of. ~X.
D
=
OU1
OU2
....
OU.
OUl
OU2
....
OU.
....
OU.
of.
of.
OUl
OU2
of.
(13)
D A D O N E et al.:
O N E STEP A H E A D A D A P T I V E C O N T R O L T E C H N I Q U E F O R W I N D SYSTEMS
405
At this point, we consider the time interval between (t - At) and t, and we approximate the time derivatives at the left hand side of equation (12) by means of forward finite differences, thus obtaining the following relation: X, = A,_a,.X,_A, + B,_a,'AU,_a,
(14)
where A and B can be easily related to C and D, while indexes (t - At) and t refer to the variable values at time (t - At) and t, respectively. The classical control systems can evaluate A and B on the basis of the controlled system mathematical or empirical models, e.g. transfer functions. The present technique does not require a previous knowledge of such relationships. On the contrary, it evaluates the values of A and B on the basis of the input and output data referring to an appropriate past time window. The least squares algorithm in recursive form is used as the on-line parameter estimator of A and B because of its simplicity and very high convergence rate. In order to apply the least squares algorithm, equation (14) is rewritten in the following form: X, = (9,_a,'tl),_a,
(15)
where (9,_a, = [A, _ a, B, _ A,]
F X,-a,
~,_a, =
1
(16)
The least squares algorithm considers X, and ~, _ a, as known terms and gives appropriate relations to estimate the unknown terms O,_~,. The following recursive formulae [10] are used to obtain O', _ A,, which represents a reasonable estimation of (9,_ a, (in the following the apex will be omitted for the sake of simplicity):
PI- 3a"tl"-2a'
(91_a, = Oi-2a, + 1 + ~,r_2~,'Pi_3A,'~,_2a,
[X,,,-A,- O1- 2A,'~,-2A,I
(17)
where i Pi-3at =
P,-4a,
-
T P , - 4 a , (llt-3at (~t_3at'Pi_4A, 1 q- ([~t_3a,.pt_4at.lljt_3a,T i
In equations (17) and (18), O~_A, denotes the ith row of the matrix O,_~,, while the covariance matrix of O1_ Ar and X~., is the ith state variable.
(18)
Pi-3a, indicates
5.2. One-step-ahead control scheme The one-step-ahead adaptive control technique considers the system linearized mathematical model given by: X, = A,_~,'X,_~, + B,_~,'Ur_A,
(19)
X, = O,_~,'@,_a,
(20)
or, alternatively, by
where A,_a, and B,_~,, obtained by means of equations (17) and (18), represent an estimation of the corresponding matrices in equation (14). Similarly, O,_a, represents an estimation of the corresponding matrix in equation (15). The target of the one-step-ahead adaptive control is to minimize the error tracking defined as: ~, = X, - X*
(21)
where X* denotes the desired output at time t. Taking into account equations (20) and (21), the error tracking pertaining to the time t + At can be expressed as: ~,+A, = O,'~,--X*+A,
(22)
406 DADONE et
al.:
ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS
The control system has to annihilate the error tracking. Accordingly, the following relation can be obtained from equation (22): O,.~,=X*+~,
(23)
Taking into account equation (16), equation (23) can be expressed as: X*+ A, = A,. X, + B,.U,
(24)
The one-step-ahead adaptive control assumes that the parameter matrices A,_ A, and B,_ ~, remain unchanged from the time t - At to the time t, i.e.: A, = A,_ ~, B, = B,_ A,
(25)
Since the matrices A,_ A, and B,_ a, are estimated by means of equations (17) and (18), the only unknown in equation (24) is represented by the vector of control variables U,: U, = B,~ A,'(X*+A,- A,_a,'X,)
(26)
Equation (26) provides the control laws, i.e. the wind system input variables required to warrant that the state variables follow the desired time sequence X*. Such a requirement is exactly satisfied for linear systems. For the present nonlinear system, the previous requirement is only approximately fulfilled, the error tracking being dependent on the time step At. As a consequence, the matrices A,_ a, and B,_ A, must be continuously updated at each time step. Equation (26) also allows one to infer that the number of wind system outputs (state variables) must be equal to the number of wind system inputs (control variables). 6. RESULTS In all the computed applications, we have considered a controlled system starting from rest. We have also assumed that the control system reaches a steady-state condition after a starting time interval. During such a steady-state time period, we have hypothesized that appropriate disturbances perturb the system status. The control system acts during the starting time interval in order to warrant an appropriate time sequence of the system status. On the contrary, during the steady-state time period, the control system has to counteract the disturbance effects in order to preserve the desired steady-state conditions. In a preliminary application, the present control technique has been applied to the horizontal-axis wind turbine and to the connected gear box, considered as a zero-order system. The aimed target was to control the output angular speed. Two different cases have been considered. In the first one, a variable gear box has been analyzed, the control variable being its transmission ratio, z. In the second one, a constant gear box has been considered, the blade collective pitch, fl0, being the control variable. In both cases, we have considered abrupt changes of wind velocity and load torque as system disturbances. In particular, we have assumed that the wind velocity instantaneously changes from 10 to 11 m/s at time t = 400 s, and instantaneously returns to the original value at t = 600 s. As far as the load torque disturbance is concerned, we have assumed an instantaneous change from 127.5 to 100 Nm at t = 700 s and an equal opposite variation at t = 900 s. For the first considered case, Fig. 4 shows the time variations of the output angular speed, ~os, while Fig. 5 presents the time variations of the transmission ratio, z. For the second considered case, Fig. 6 outlines the time variation of the output angular speed, o9~, while Fig. 7 presents the time variations of the blade collective pitch, fl0. In all the considered figures, the marked lines close to the abscissa indicate that the disturbances are affecting the system during the corresponding time intervals. Moreover, the dotted lines in Figs 4 and 6 refer to the target variations of the output angular speed, while the solid lines represent the corresponding actual variations. A quick glance to such figures allows one to state that the actual and the target angular speeds are practically coincident during the considered time interval. In particular, the disturbance effects appear to be effectively counteracted by the present control system. On the contrary, such effects are clearly visible in Figs 5 and 7, and correspond to abrupt changes of the transmission ratio and of the blade collective pitch. They represent how the one-step-ahead adaptive control system neutralizes the disturbance effects.
DADONE et al.:
ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS
407
160.00
% [tad/s] 120.00
60.00
40.00
...... 0.00
l
0.00
I
l
200.00
I 400.00
i 800.00
i 800.00
Actual Target I 1000.00
Time Is]
Fig. 4. First preliminary problem. First case. Time history of the output angular speed of the gear box.
In a second preliminary application, the present control technique has been applied to the electrical induction generator. The aimed target has been to control the output voltage, while the control variable has been represented by the rotor resistance, r2. We have considered rapid changes of the rotor angular speed and of the resistance load as system disturbances. In particular, we have assumed that the rotor angular speed linearly changes from 157.5 to 145.5 rad/s in the time interval t = 60-64 s. Then, we have supposed that it remains unchanged in the time interval t = 64-76 s and, finally, linearly recovers the original value in the time interval t = 76-80 s. As far as the
10.00
17 8.00
6.00
4.00
2.00
0.00
t
0.00
200.00
400.00
600.00 Time
800.00
1000.00
[s]
Fig. 5. First preliminary problem. First case. Time history of the transmission ratio of the gear box (control variable).
408
DADONE et al.: 160.00
ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS
-
% [racVs] 120.00
80.00
40,00
, 0.00
..... ......
~ ,
0.00
I 200.00
'
=
'
Actual Target
=
400.00
~
600.00
I
eoo.oo
1000.00
Time [s] Fig. 6 First preliminary problem. Second case. Time history of the output angular speed of the gear box.
resistance load disturbance is concerned, we have assumed an instantaneous change from 9 to 20 at t = 100 s and an equal opposite variation at t = 120 s. Figure 8 shows the time variations of the output voltage, V, while Fig. 9 presents the time variations of the rotor resistance, r2. Again, the disturbance actions are indicated by marked lines, while dotted and solid lines represent the target and the actual voltage variations, respectively. The actual and the target output voltages in Fig. 8 are practically coincident for most of the considered time interval. Two significant spikes in the actual voltage plot denote its sensitivity to the instantaneous changes of the resistance load. Moreover, at the beginning of the starting period, significant differences between the actual and
20.00
-
16.00
-
12.00
-
8.00
-
4.00
-
I
0.00
I
I
0.00
200.00
I
400.00
600.00
i
t I
800.00
I
1000.00
Time [s] Fig. 7. First preliminary problem. Second case. Time history of the blade collective pitch (control variable).
DADONE
et al.:
ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS 409
150.00 V
.....
[volt]
~00.00 -
50.00
-
00.00
-
50.00
-
Actual Target
. . . .
0.00
z
0.00
I
I
40.00
I
'
8o.oo Time [s]
I
'
I
12o.oo
18o.oo
Fig. 8. Second preliminary problem. Time history of the output voltage. the target voltage plots can be observed. Such differences can be explained as follows. At the very beginning of the starting period, the frequency is very low because of the corresponding very low values of the angular speed. As a consequence, the time interval between two consecutive measurements of the voltage amplitude are rather large. Such a situation causes some uncertainties referring to the definition of the voltage amplitude variations, so that the control system cannot work properly. This same phenomenon causes large oscillations of the rotor resistance, r2, which can be observed in Fig. 9, at the very beginning of the starting period. In the same figure, abrupt changes of the control variable can be also observed as a consequence of the corresponding
8.00
-
1"2 [ohm]
"
6.00
-
4.00
-
2.00
-
0.00
s
-
0.00
I
40.00
'
l
80.00 Time [s]
'
~
120.00
I
160.00
Fig. 9. Second preliminary problem. Time history of the rotor resistance (control variable).
410
DADONE et al.:
ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS
variations of the resistance load. On the contrary, no practical effects of the rapid changes of the rotor angular speed can be noticed in Figs 8 and 9. The final considered control problem refers to the complete wind system. In order to introduce a more realistic representation of the gear box, in the present application, such a component has been simulated by a second order transfer function: z(s) =
z0
.s+
co~ +2"6 co.
(27) 1
where 6 = 2 and co, = 3.162 rad/s. The considered transfer function corresponds to an overdamped system with a raising time equal to Ts = 8.23 s. The aimed target was to control the output voltage and frequency. The control variables have been represented by the rotor resistance, r2, and the static gain of the gear box, To. We have considered abrupt changes of wind velocity and resistance load as system disturbances. In particular, we have assumed that the wind velocity instantaneously changes from 10 to 11 m/s at time t = 150 s, and instantaneously returns to the original value at t = 180 s. As far as the resistance load disturbance is concerned, we have assumed an instantaneous change from 9 to 15 f~ at t = 200 s and an equal opposite variation at t = 230 s. Figure 10 shows the time variations of the output frequency, f, while Fig. 11 refers to the output voltage, V. In such figures, dotted and solid lines represent again the target and the actual frequency and voltage variations, respectively. Moreover, Fig. 12 shows the time variations of the first control variable, i.e. the rotor resistance, r2. Finally, Fig. 13 presents the time history of both the static gain of the gear box, z0, (dotted line) and the instantaneous value of the gear box transmission ratio, r (solid line). In Figs 10-13, the disturbance actions are again indicated by marked lines. The actual and the target output variables in Figs 10 and 11 are practically coincident for most of the considered time interval. Two significant spikes in the actual controlled variable plots denote their sensitivity to the instantaneous changes of the resistance load, while no practical effects of the instantaneous changes of the wind velocity can be noticed. Moreover, significant differences between the actual and the target output variable plots can be observed at the beginning of the starting period. Such differences are due to the initial problems of the control system caused by the already explained phenomenon, which also determines the initially oscillating plot of the rotor resistance in Fig. 12. Figure 13 shows some significant differences between the static gain of the
80.00
f [Hz]
40.00
g
20.00 -
Actual
I t
Target
. . . . .
i I i I
0.00
I
0.00
40.00
I
I
80.00
I
I
I
120.00
!
!
l
160.00
200.00
240.00
Time [s] Fig. 10. Complete configuration. Time history of the output frequency.
I
|
280.00
DADONE et al.:
ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS
411
300.00
V [volt]
!
200.00
100.00
,
~ - -
Actual
0.00 0.00
40.0o
60.00
120.00 Time
160.00 [s]
2oo.0o
24o.00
28o.oo
Fig. 11. Complete configuration. Time history of the output voltage.
gear box, z0, and its instantaneous transmission ratio, z. No evidence of such differences can be observed in the output variable plots (Figs l0 and 11). Such an important characteristic proves the adaptive nature of the one-step-ahead control technique, which automatically tunes itself in order to take into account all the nonlinearities and time-variances of the system under control. In particular, it adaptively annihilates the time delays of the system components. Therefore, we can state that the one-step-ahead adaptive control system has the property of forcing a physical system characterized by inherently relevant time delays to promptly react to external disturbances.
8.00 7
r' t
[ohm]
ii
4.00
2.00
o.oo 0.00
--'1"--
--T--
--T--"~ ~
~
~
40.00
80.00
120.00 160.00 T i m e Is]
200.00
240.00
Fig. 12. Complete configuration. Time history of the rotor resistance (control variable).
280.00
412 DADONE et al.: ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS r6.00 -
2.00 -
Iu it uI Is
II II
Ill
: 4.00 - ~
• •
' o.oo
q_
•
pill t
0.00
S °°°°°°S
I 4o.oo
'
I eo.oo
i
I ' u 12o.oo 16o.oo T i m e [s']
u 200.00
I 240.00
i
I 280.00
Fig. 13. Complete configuration. Time history of the static gain of the gear box and of the instantaneous value of the transmission ratio (control variable). 7. C O N C L U S I O N S In the present paper, it has been proposed to use the one-step-ahead control technique to regulate a wind system designed as an isolated electrical power generation station. The wind system is composed of a horizontal-axis three blade wind turbine connected to an electrical induction generator through a gear box. The wind system has been numerically simulated and the mathematical models of its components have been presented. The one-step-ahead control system is formed by a parameter estimator block, a design calculation block, and a control law block. The mathematical models of each one of the previous blocks have been presented in order to outline how the one-step-ahead technique evaluates the control law starting from an on-line estimation of the controlled system, which is performed by its parameter estimator block. Two preliminary control studies have been performed. The first one controlled the output shaft angular speed of the gear box connected to the wind turbine. The second one referred to the control of the output voltage of the electrical generator. The complete wind system has been finally considered, and the control of its output voltage frequency and amplitude has been shown. In all the considered control cases, the one-step-ahead control system has proved its ability to control adequately the output variables under the actions of relevant and abrupt disturbances. Also proven is the adaptive characteristic of such a control system, which automatically tunes itself in order to take into account all the nonlinearities and time-variances of the system under control. In particular, it has the property to force a physical system, characterized by inherently relevant time delays, to promptly react to external disturbances. Acknowledgements--This research has been financially supported by the Italian Agency MURST (Ministero dell'UniversitA e della Ricerca Scientifica e Tecnologica), under a MURST 40% contract.
REFERENCES
1. Wilson, R. E. and Lissaman, P. B. S., Applied Aerodynamics of Wind Power Machines, Oregon State University, 1974. 2. Brown, C. Wind Energy Conversion, Proceedings of the 13th BWEA Wind Energy Conference, eds D. C. Quarton and V. C. Fenton, 1991, pp. 255-259. 3. Chviaropoulos, P. and Papailiou, K. D., Proceedings of the European Community Wind Energy Conference, H. S. Stephens and Associates, Bedford, England, 1988, pp. 258-264. 4. Fitzgerald, A. E., Kingsley, C. and Kusko, A., Electric Machinery, McGraw-Hill, New York, 1983.
DADONE et al.: ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS
413
5. Fortunato, B., Maione, B. and Palmisano, N., Proceedings of the Flowers Worm Energy Research Symposium '94, eds E. Carnevale, G. Manfrida, F. Martelli, 1994, pp. 867-876. 6. Dambrosio, L., Controllo Adattativo One-Step-Ahead di un sistema eolico. PhD Dissertation, Istituto di Macchine ed Energetica, Politecnico di Bari, 1994. 7. Steinbuch, M., European Wind Association Conference and Exhibition, Rome, October 1986. 8. Madsen, P. H., European Wind Association Conference, Herning, Denmark, June 1988. 9. Sandhu, D. and Dias, S., European Wind Association Conference, Herning, Denmark, June 1988. 10. Goodwin, G. C. and Sin, K. S., Adaptive, Filtering, Prediction and Control, Prentice-Hall, Englewood Cliffs. New Jersey, 1984. 11. Dambrosio, L., Development and application of one-step-ahead adaptive control system for helicopters. MS Thesis, Department of Aerospace Engineering, Mississippi State University, 1994. 12. Prouty, R. W., Helicopter Performance, Stability and Control. PWS Engineering, Boston, 1986. 13. Bryson, A. E. and Ho Yu-Chi, Applied Optimal Control--Optimization, Estimation and Control, Hemisphere, New York, 1975.
Energy Convers. Mgmt Vol. 39, No. 5/6, pp. 399-413, 1998
Pergamon
PII: S0196-8904(97)00019-8
ONE
STEP AHEAD
ADAPTIVE
FOR
WIND
© 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0196-8904/98 $19.00 + 0.00
CONTROL
TECHNIQUE
SYSTEMS
A. DADONE, L. DAMBROSIO and B. FORTUNATO Istituto di Macchine ed Energetica, Politecnico di Bari, Bari, Italy
(Received 2 July 1996)
Abstract--The control of a wind system considered as an isolated source of power and composed of a horizontal-axis wind-turbine connected to an induction generator is analyzed. Appropriate mathematical models for both the horizontal-axis wind-turbine and the induction generator are used. The one-step-ahead adaptive control technique is presented and used to regulate the wind system. A sensitivity analysis of the induction generator performances with respect to control and disturbance variables is presented in order to evaluate the control flexibility. The results of three control problems are finally shown in order to prove the reliability of the suggested control technique. © 1997 Elsevier Science Ltd. Wind system
Control
Adaptive
One step ahead
1. I N T R O D U C T I O N
Wind energy is a free source of energy which can be converted into both mechanical and electrical energy by means of a wind system. When used to generate electrical energy, the wind system is basically composed of a wind turbine, an electrical generator and a control system. The wind turbine converts the wind energy into mechanical energy which is used by the electrical generator to produce electrical energy. The control system counteracts the disturbance effects in order to minimize the differences between the actual and the reference values of the output voltage and frequency. In the present paper, a horizontal-axis wind-turbine and an induction electrical generator will be considered. Appropriate mathematical models are obviously necessary to simulate the static and dynamic behavior of both components of the wind system. In principle, the flow about a horizontal-axis wind-turbine is three-dimensional, viscous and unsteady. Its direct simulation would be too complicated, while rather simple models are generally appropriate for a numerical simulation of a controlled system. The existing mathematical models to simulate wind turbines can be classified into momentum, vortex and finite difference models. The momentum models approximate the wind turbine effects by means of actuator surfaces and subdivides the flow domain into many streamtubes (see e.g. Ref. [I]). The vortex models represent the turbine blade effects by means of distributed bound vortices (see e.g. Ref. [2]). The finite difference models approximate the rotor effects by means of an actuator surface and solve the fluid-dynamic governing equations by means of finite volume or finite difference methods (see e.g. Ref. [3]). The momentum models are generally simpler, and they will be here considered to simulate the wind turbine. An appropriate mathematical model for the induction electrical generator can be found in many books [4] where the differential equations governing the dynamical behavior are reported. Reference [5] suggests inserting a variable capacitor and using it as the first control variable, the rotor winding resistance being the second one. These control variables are used in Ref. [5] to perform a steady-state control analysis of an induction electrical generator. In the present paper, a complete dynamical model of both the rotor and the stator winding will be presented, and only the rotor winding resistance will be considered as a control variable. The stator winding capacitor will not be used as a control variable, but it will be preserved and used to ensure the self excitation of the electrical generator [6]. 399
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DADONE et al.:
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Classical controls of wind systems can be found in many papers with reference to open loop schemes [7] as well as to closed loop ones [8, 9]. An innovative technique which tries to anticipate the dynamical behavior of the controlled system is the one-step-ahead adaptive control [10] which has been successfully applied to helicopters [11]. Such a control technique linearly estimates the future status of the controlled system at time t + At on the basis of the system status at time t and of the previous dynamical behavior of the system itself. Therefore, it can be applied to nonlinear and time varying systems and automatically retunes itself whenever the control conditions change due to disturbance actions. The aim of the present paper is to extend the one step ahead adaptive technique to control a wind system. The problem we intend to solve is stated in the following paragraph. 2. PROBLEM STATEMENT The considered wind system is made of a three-blade horizontal-axis wind-turbine and a three phase induction electrical generator, connected by means of a gear box. The gear box may present either a fixed or a variable transmission ratio. In the present paper, the considered wind system is supposed to be an isolated electrical power generation station. The general layout of the control system is shown in Fig. 1. The block System represents the wind system to be controlled. In general, such a block represents the physical system, while in the present paper, it will be numerically simulated by means of an appropriate mathematical model. The block Parameter Estimator is an essential block of the one-step-ahead control system. On the basis of an appropriate time sequence of input and output data, the parameter estimator evaluates a suitable linear model of the system which will change at every considered time step. The block Design Calculations records the desired control targets, i.e. the desired time history of the system outputs. The block Control Law determines the appropriate system inputs on the basis of the actual system outputs and of a comparison between the target outputs (recorded by the design calculation block) and the data estimated by the parameter estimator block. The target of the control system here considered is to regulate the system outputs, represented by the frequency and the voltage generated by the electrical induction machine. In the following paragraphs, the mathematical models pertaining to the wind turbine and the induction electrical generator are first outlined, and a sensitivity analysis of the induction generator
Inputs
Outputs
System
' "~"~"
Piremeter elt;;'hlk~.
~.
Design cllculltlons
Control
Fig. 1. Block diagram of adaptive control system.
,,~
DADONE et al.: ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS
401
performances is presented in order to evaluate its control flexibility. Then, the one-step-ahead adaptive control technique is presented. Finally, the customized control technique is applied to regulate the wind system. A preliminary control study will consider the regulation of the output shaft rotational speed of the wind-turbine connected to the gear box in the presence of abrupt changes of wind velocity and load torque. A second preliminary control study will analyze the regulation of the output voltage of the electrical generator alone in the presence of rapid changes of shaft rotational speed and electrical load. Finally, the complete wind system will be considered, and the regulation of the output voltage and frequency in the presence of abrupt changes of the wind speed and of the electrical load will be studied. 3. M A T H E M A T I C A L M O D E L OF THE H O R I Z O N T A L - A X I S W I N D T U R B I N E
The mathematical model adopted to simulate the dynamical behavior of the horizontal-axis wind-turbine is based on streamtube discretization [1]. It relies on the momentum law in both the axial and the tangential directions and on the aerodynamic characteristics of the blade airfoil. The thrust and the torque acting on a blade element dr at distance r from the center of the rotor are given by [1]:
dT = p'rc "r"a'(1-a)2"V2°'CL'~(lsln q~ \ dQ = p.~z.r2.a.(l_a,)2.(f~.r)2.CL.~
+ ~__[L.tanCo,).dr
COS2tp
( 1 CD 1 ) CL tan ~b
"dr
(1)
where c, and A and .4b are the blade chord, the rotor area and the total blade area, while p, V0, a, a' and q~ are the air density, the wind velocity, the axial and tangential induced velocity coefficients and the inflow angle, respectively. Moreover, fl, CL and Co represent the rotor angular speed and the lift and drag coefficients of the airfoil, respectively. Finally, a = A b / A is the solidity ratio. Equation (1) allows one to determine the wind turbine torque, QT, and power, P:
QT =
_•Rrnax dO
(2)
d Rmin
P = f~'QT,
(3)
w i t h Rmin, Rmax being the internal and external rotor radii, respectively. Equation (3) allows one
to evaluate the wind turbine power coefficient, Cp: P
Cv - 1/2.p.A.V 3
(4)
which is representative of the wind turbine performance. In the present work, a three-blade horizontal-axis wind-turbine will be considered. The external (Rmax) and internal (Rm~n)rotor radii are 5 and 0.5 m, respectively. The nominal conditions are: wind velocity equal to 10 m/s; power equal to 22 kW; tangential velocity ratio A equal to 7 (A = 0.5"~'Dmax/Vo). The rotor blades are twisted according to maximum power and maximum airfoil efficiency criteria for the outer section of the blade and according to the constant chord criterion for its inner section. As a consequence, appropriate radial distributions of the pitch angle fl and of the chord e have been determined. The considered airfoil profile is a NACA 0012; the analytical expressions for the corresponding lift, CL and drag, CD, coefficients have been taken from Ref. [12]. As a characteristic of the considered wind-turbine, Fig. 2 shows plots of the power coefficient, Cp, vs the tangential velocity ratio, A. Each curve corresponds to a different value of the collective pitch/30.
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ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS
0.50 -
Cp
13--O
0.40
0.30
0.20
I~X2 0.10
0.00
ft,=aS
"" 0.00
2.00
4.00
8.00
8.00
10.00
A
Fig. 2. Power coefficient vs the tangential velocity ratio for different values of the collective pitch.
As far as the gear box connected to the wind turbine is concerned, it will be represented either by a constant or by a second order transfer function. In some of the considered control applications, its gain will be constant, while in some other applications, such a gain will be a control variable.
4. M A T H E M A T I C A L
MODEL
OF THE
SELF-EXCITED
INDUCTION
GENERATOR
The wind turbine drives a three phase self-excited induction generator. Since the stator and rotor windings of the induction generator, together with the external load, are balanced, a per-phase analysis can be adopted [4]. According to [4], a fixed axis reference system, dq, can be used to determine the governing equations of both the stator and rotor windings: •
_
dill,,
di2d
Vjd = rl " hd W L~ " - - ~ " r LM" d l
Ulq =
•
- dilq_di2q rj'ilq+L~"-~TLM'---~
di2d
dild
di2q
dilq
.
.
0 = r2' t2d+ L2"--d-/-+ LM"--~-+p" Cog"(L2"Z2q+ LM"hq)
0 = r2"i2q + L2"--~- + LM "---~--p .09g. (L2" i2d + LM'i~d)
(5)
where rt and r2 are the stator and rotor internal resistances, respectively, while L~ and L2 are the corresponding inductances• Moreover, LM, ogg and p represent the mutual inductance, the rotor angular speed and the number of polar pairs, respectively. The voltages V~d, V~q and the currents i~d and i~q do not have a direct physical meaning, although they are related to the phase voltages and currents, as will be shown later.
DADONE et al.: ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS 403 The external load being represented by a capacity, C, an inductance, L, and a resistance, R, the corresponding governing equations in the dq system of reference are: UId : L "diLd dt L"
dDld
diLq dt
L~lq = rid
C" dt = - R--~L- -
C "dUlq: - u l q dt
RE
lid - - iLd
iiq -
iLq
(6)
The dynamic equilibrium of the moving parts of both the wind turbine and the induction generator can be expressed as: .
d-t -
"
.
.
.
= J" [aT-- 3 "p'(tld'/2q - - / I q "/2d)]
(7)
where J, z and t represent the moment of inertia of all the rotating parts referred to the electrical generator shaft, the transmission ratio of the gear box and the time, respectively, while QE is the rotor electromagnetic torque given by: Q E = 3 "p" (ild' i2q - - ilq" i2d)
(8)
Equations (5) -(7) represent a set of nine equations in the nine unknowns ild, iZq, 6d, i2q, /)ld, V,q, iLd, kq, and cos, which can be solved provided that the wind turbine torque is computed by means of equation (2). Finally, the phase voltages can be obtained from the unphysical voltages V~d,V,q, by means of the following relations: lfla ~ UId
1
,/3
1
,,/3
Vb =--~'V~d+ 2 "V~q re=--~'Uid-- 2 'Vlq
(9)
At this point, a steady-state model of the induction generator has been derived from the dynamic model (equations (5), (6) and (9)) by means of the phasor approach. Such a model has been used to perform a sensitivity analysis of the system in order to evaluate the sensitivity of the output voltage and frequency to changes of the external disturbances (load resistance and inductance) and of the manipulating variables (self exciting capacity and rotor resistance). Such a sensitivity analysis showed that the self-exciting capacity has a relevant influence on both output variables, while the influence of the rotor resistance is relatively small. Moreover, it has been determined that both the external disturbances significantly affect the output variables. The results of such a study are extensively reported in Ref. [6]. For the sake of conciseness, only the most relevant results, referring to the influence of the rotor resistance, are here plotted in Fig. 3, which shows the percentage variations of both output voltage, A V%, and frequency, Aft~o, vs the percentage variation of the rotor resistance, Ar2%. 5. O N E - S T E P - A H E A D ADAPTIVE CONTROL TECHNIQUE
As sketched in Fig. 1, and already presented in paragraph 2, the one-step-ahead adaptive control technique [10, 13] combines the parameter estimation algorithm with the control scheme. In the ECM 39/5-6--B A"
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ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS
4.00
%
0.00
-
~f
-4.00
-8.00
I
I
I
I
I
-40.00
0.00
40.00
80.00
120.00
Ar2 %
Fig. 3. Percentage variations of output voltage and frequency vs the percentage variation of the rotor resistance.
following paragraphs, it will be first shown how to estimate a linearized model of the wind system on the basis of the previous sequence o f the wind system inputs and outputs• The least squares algorithm will be used for an on-line estimation of the linearized model parameters. Then, the control scheme will be applied to the linearized estimated model of the controlled system. 5. I. Least squares parameter estimation algorithm
The wind system can be generally represented in the state variable notations as: = f ( X , U)
(10)
where X and U represent the vectors of the controlled variables and of the control variables, respectively:
X =
X2
U =
U~
(11)
Let us now consider a Taylor series expansion of equation (10) at X = 0, U = U0, truncated at the first-order: = C.X+D.AU
(12)
where:
C =
~Xi
~X2
....
OX.
OXI
OX2
....
OX.
Of. OXj
Of. OX2
....
Of. ~X.
D
=
OU1
OU2
....
OU.
OUl
OU2
....
OU.
....
OU.
of.
of.
OUl
OU2
of.
(13)
D A D O N E et al.:
O N E STEP A H E A D A D A P T I V E C O N T R O L T E C H N I Q U E F O R W I N D SYSTEMS
405
At this point, we consider the time interval between (t - At) and t, and we approximate the time derivatives at the left hand side of equation (12) by means of forward finite differences, thus obtaining the following relation: X, = A,_a,.X,_A, + B,_a,'AU,_a,
(14)
where A and B can be easily related to C and D, while indexes (t - At) and t refer to the variable values at time (t - At) and t, respectively. The classical control systems can evaluate A and B on the basis of the controlled system mathematical or empirical models, e.g. transfer functions. The present technique does not require a previous knowledge of such relationships. On the contrary, it evaluates the values of A and B on the basis of the input and output data referring to an appropriate past time window. The least squares algorithm in recursive form is used as the on-line parameter estimator of A and B because of its simplicity and very high convergence rate. In order to apply the least squares algorithm, equation (14) is rewritten in the following form: X, = (9,_a,'tl),_a,
(15)
where (9,_a, = [A, _ a, B, _ A,]
F X,-a,
~,_a, =
1
(16)
The least squares algorithm considers X, and ~, _ a, as known terms and gives appropriate relations to estimate the unknown terms O,_~,. The following recursive formulae [10] are used to obtain O', _ A,, which represents a reasonable estimation of (9,_ a, (in the following the apex will be omitted for the sake of simplicity):
PI- 3a"tl"-2a'
(91_a, = Oi-2a, + 1 + ~,r_2~,'Pi_3A,'~,_2a,
[X,,,-A,- O1- 2A,'~,-2A,I
(17)
where i Pi-3at =
P,-4a,
-
T P , - 4 a , (llt-3at (~t_3at'Pi_4A, 1 q- ([~t_3a,.pt_4at.lljt_3a,T i
In equations (17) and (18), O~_A, denotes the ith row of the matrix O,_~,, while the covariance matrix of O1_ Ar and X~., is the ith state variable.
(18)
Pi-3a, indicates
5.2. One-step-ahead control scheme The one-step-ahead adaptive control technique considers the system linearized mathematical model given by: X, = A,_~,'X,_~, + B,_~,'Ur_A,
(19)
X, = O,_~,'@,_a,
(20)
or, alternatively, by
where A,_a, and B,_~,, obtained by means of equations (17) and (18), represent an estimation of the corresponding matrices in equation (14). Similarly, O,_a, represents an estimation of the corresponding matrix in equation (15). The target of the one-step-ahead adaptive control is to minimize the error tracking defined as: ~, = X, - X*
(21)
where X* denotes the desired output at time t. Taking into account equations (20) and (21), the error tracking pertaining to the time t + At can be expressed as: ~,+A, = O,'~,--X*+A,
(22)
406 DADONE et
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ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS
The control system has to annihilate the error tracking. Accordingly, the following relation can be obtained from equation (22): O,.~,=X*+~,
(23)
Taking into account equation (16), equation (23) can be expressed as: X*+ A, = A,. X, + B,.U,
(24)
The one-step-ahead adaptive control assumes that the parameter matrices A,_ A, and B,_ ~, remain unchanged from the time t - At to the time t, i.e.: A, = A,_ ~, B, = B,_ A,
(25)
Since the matrices A,_ A, and B,_ a, are estimated by means of equations (17) and (18), the only unknown in equation (24) is represented by the vector of control variables U,: U, = B,~ A,'(X*+A,- A,_a,'X,)
(26)
Equation (26) provides the control laws, i.e. the wind system input variables required to warrant that the state variables follow the desired time sequence X*. Such a requirement is exactly satisfied for linear systems. For the present nonlinear system, the previous requirement is only approximately fulfilled, the error tracking being dependent on the time step At. As a consequence, the matrices A,_ a, and B,_ A, must be continuously updated at each time step. Equation (26) also allows one to infer that the number of wind system outputs (state variables) must be equal to the number of wind system inputs (control variables). 6. RESULTS In all the computed applications, we have considered a controlled system starting from rest. We have also assumed that the control system reaches a steady-state condition after a starting time interval. During such a steady-state time period, we have hypothesized that appropriate disturbances perturb the system status. The control system acts during the starting time interval in order to warrant an appropriate time sequence of the system status. On the contrary, during the steady-state time period, the control system has to counteract the disturbance effects in order to preserve the desired steady-state conditions. In a preliminary application, the present control technique has been applied to the horizontal-axis wind turbine and to the connected gear box, considered as a zero-order system. The aimed target was to control the output angular speed. Two different cases have been considered. In the first one, a variable gear box has been analyzed, the control variable being its transmission ratio, z. In the second one, a constant gear box has been considered, the blade collective pitch, fl0, being the control variable. In both cases, we have considered abrupt changes of wind velocity and load torque as system disturbances. In particular, we have assumed that the wind velocity instantaneously changes from 10 to 11 m/s at time t = 400 s, and instantaneously returns to the original value at t = 600 s. As far as the load torque disturbance is concerned, we have assumed an instantaneous change from 127.5 to 100 Nm at t = 700 s and an equal opposite variation at t = 900 s. For the first considered case, Fig. 4 shows the time variations of the output angular speed, ~os, while Fig. 5 presents the time variations of the transmission ratio, z. For the second considered case, Fig. 6 outlines the time variation of the output angular speed, o9~, while Fig. 7 presents the time variations of the blade collective pitch, fl0. In all the considered figures, the marked lines close to the abscissa indicate that the disturbances are affecting the system during the corresponding time intervals. Moreover, the dotted lines in Figs 4 and 6 refer to the target variations of the output angular speed, while the solid lines represent the corresponding actual variations. A quick glance to such figures allows one to state that the actual and the target angular speeds are practically coincident during the considered time interval. In particular, the disturbance effects appear to be effectively counteracted by the present control system. On the contrary, such effects are clearly visible in Figs 5 and 7, and correspond to abrupt changes of the transmission ratio and of the blade collective pitch. They represent how the one-step-ahead adaptive control system neutralizes the disturbance effects.
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160.00
% [tad/s] 120.00
60.00
40.00
...... 0.00
l
0.00
I
l
200.00
I 400.00
i 800.00
i 800.00
Actual Target I 1000.00
Time Is]
Fig. 4. First preliminary problem. First case. Time history of the output angular speed of the gear box.
In a second preliminary application, the present control technique has been applied to the electrical induction generator. The aimed target has been to control the output voltage, while the control variable has been represented by the rotor resistance, r2. We have considered rapid changes of the rotor angular speed and of the resistance load as system disturbances. In particular, we have assumed that the rotor angular speed linearly changes from 157.5 to 145.5 rad/s in the time interval t = 60-64 s. Then, we have supposed that it remains unchanged in the time interval t = 64-76 s and, finally, linearly recovers the original value in the time interval t = 76-80 s. As far as the
10.00
17 8.00
6.00
4.00
2.00
0.00
t
0.00
200.00
400.00
600.00 Time
800.00
1000.00
[s]
Fig. 5. First preliminary problem. First case. Time history of the transmission ratio of the gear box (control variable).
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ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS
-
% [racVs] 120.00
80.00
40,00
, 0.00
..... ......
~ ,
0.00
I 200.00
'
=
'
Actual Target
=
400.00
~
600.00
I
eoo.oo
1000.00
Time [s] Fig. 6 First preliminary problem. Second case. Time history of the output angular speed of the gear box.
resistance load disturbance is concerned, we have assumed an instantaneous change from 9 to 20 at t = 100 s and an equal opposite variation at t = 120 s. Figure 8 shows the time variations of the output voltage, V, while Fig. 9 presents the time variations of the rotor resistance, r2. Again, the disturbance actions are indicated by marked lines, while dotted and solid lines represent the target and the actual voltage variations, respectively. The actual and the target output voltages in Fig. 8 are practically coincident for most of the considered time interval. Two significant spikes in the actual voltage plot denote its sensitivity to the instantaneous changes of the resistance load. Moreover, at the beginning of the starting period, significant differences between the actual and
20.00
-
16.00
-
12.00
-
8.00
-
4.00
-
I
0.00
I
I
0.00
200.00
I
400.00
600.00
i
t I
800.00
I
1000.00
Time [s] Fig. 7. First preliminary problem. Second case. Time history of the blade collective pitch (control variable).
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ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS 409
150.00 V
.....
[volt]
~00.00 -
50.00
-
00.00
-
50.00
-
Actual Target
. . . .
0.00
z
0.00
I
I
40.00
I
'
8o.oo Time [s]
I
'
I
12o.oo
18o.oo
Fig. 8. Second preliminary problem. Time history of the output voltage. the target voltage plots can be observed. Such differences can be explained as follows. At the very beginning of the starting period, the frequency is very low because of the corresponding very low values of the angular speed. As a consequence, the time interval between two consecutive measurements of the voltage amplitude are rather large. Such a situation causes some uncertainties referring to the definition of the voltage amplitude variations, so that the control system cannot work properly. This same phenomenon causes large oscillations of the rotor resistance, r2, which can be observed in Fig. 9, at the very beginning of the starting period. In the same figure, abrupt changes of the control variable can be also observed as a consequence of the corresponding
8.00
-
1"2 [ohm]
"
6.00
-
4.00
-
2.00
-
0.00
s
-
0.00
I
40.00
'
l
80.00 Time [s]
'
~
120.00
I
160.00
Fig. 9. Second preliminary problem. Time history of the rotor resistance (control variable).
410
DADONE et al.:
ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS
variations of the resistance load. On the contrary, no practical effects of the rapid changes of the rotor angular speed can be noticed in Figs 8 and 9. The final considered control problem refers to the complete wind system. In order to introduce a more realistic representation of the gear box, in the present application, such a component has been simulated by a second order transfer function: z(s) =
z0
.s+
co~ +2"6 co.
(27) 1
where 6 = 2 and co, = 3.162 rad/s. The considered transfer function corresponds to an overdamped system with a raising time equal to Ts = 8.23 s. The aimed target was to control the output voltage and frequency. The control variables have been represented by the rotor resistance, r2, and the static gain of the gear box, To. We have considered abrupt changes of wind velocity and resistance load as system disturbances. In particular, we have assumed that the wind velocity instantaneously changes from 10 to 11 m/s at time t = 150 s, and instantaneously returns to the original value at t = 180 s. As far as the resistance load disturbance is concerned, we have assumed an instantaneous change from 9 to 15 f~ at t = 200 s and an equal opposite variation at t = 230 s. Figure 10 shows the time variations of the output frequency, f, while Fig. 11 refers to the output voltage, V. In such figures, dotted and solid lines represent again the target and the actual frequency and voltage variations, respectively. Moreover, Fig. 12 shows the time variations of the first control variable, i.e. the rotor resistance, r2. Finally, Fig. 13 presents the time history of both the static gain of the gear box, z0, (dotted line) and the instantaneous value of the gear box transmission ratio, r (solid line). In Figs 10-13, the disturbance actions are again indicated by marked lines. The actual and the target output variables in Figs 10 and 11 are practically coincident for most of the considered time interval. Two significant spikes in the actual controlled variable plots denote their sensitivity to the instantaneous changes of the resistance load, while no practical effects of the instantaneous changes of the wind velocity can be noticed. Moreover, significant differences between the actual and the target output variable plots can be observed at the beginning of the starting period. Such differences are due to the initial problems of the control system caused by the already explained phenomenon, which also determines the initially oscillating plot of the rotor resistance in Fig. 12. Figure 13 shows some significant differences between the static gain of the
80.00
f [Hz]
40.00
g
20.00 -
Actual
I t
Target
. . . . .
i I i I
0.00
I
0.00
40.00
I
I
80.00
I
I
I
120.00
!
!
l
160.00
200.00
240.00
Time [s] Fig. 10. Complete configuration. Time history of the output frequency.
I
|
280.00
DADONE et al.:
ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS
411
300.00
V [volt]
!
200.00
100.00
,
~ - -
Actual
0.00 0.00
40.0o
60.00
120.00 Time
160.00 [s]
2oo.0o
24o.00
28o.oo
Fig. 11. Complete configuration. Time history of the output voltage.
gear box, z0, and its instantaneous transmission ratio, z. No evidence of such differences can be observed in the output variable plots (Figs l0 and 11). Such an important characteristic proves the adaptive nature of the one-step-ahead control technique, which automatically tunes itself in order to take into account all the nonlinearities and time-variances of the system under control. In particular, it adaptively annihilates the time delays of the system components. Therefore, we can state that the one-step-ahead adaptive control system has the property of forcing a physical system characterized by inherently relevant time delays to promptly react to external disturbances.
8.00 7
r' t
[ohm]
ii
4.00
2.00
o.oo 0.00
--'1"--
--T--
--T--"~ ~
~
~
40.00
80.00
120.00 160.00 T i m e Is]
200.00
240.00
Fig. 12. Complete configuration. Time history of the rotor resistance (control variable).
280.00
412 DADONE et al.: ONE STEP AHEAD ADAPTIVE CONTROL TECHNIQUE FOR WIND SYSTEMS r6.00 -
2.00 -
Iu it uI Is
II II
Ill
: 4.00 - ~
• •
' o.oo
q_
•
pill t
0.00
S °°°°°°S
I 4o.oo
'
I eo.oo
i
I ' u 12o.oo 16o.oo T i m e [s']
u 200.00
I 240.00
i
I 280.00
Fig. 13. Complete configuration. Time history of the static gain of the gear box and of the instantaneous value of the transmission ratio (control variable). 7. C O N C L U S I O N S In the present paper, it has been proposed to use the one-step-ahead control technique to regulate a wind system designed as an isolated electrical power generation station. The wind system is composed of a horizontal-axis three blade wind turbine connected to an electrical induction generator through a gear box. The wind system has been numerically simulated and the mathematical models of its components have been presented. The one-step-ahead control system is formed by a parameter estimator block, a design calculation block, and a control law block. The mathematical models of each one of the previous blocks have been presented in order to outline how the one-step-ahead technique evaluates the control law starting from an on-line estimation of the controlled system, which is performed by its parameter estimator block. Two preliminary control studies have been performed. The first one controlled the output shaft angular speed of the gear box connected to the wind turbine. The second one referred to the control of the output voltage of the electrical generator. The complete wind system has been finally considered, and the control of its output voltage frequency and amplitude has been shown. In all the considered control cases, the one-step-ahead control system has proved its ability to control adequately the output variables under the actions of relevant and abrupt disturbances. Also proven is the adaptive characteristic of such a control system, which automatically tunes itself in order to take into account all the nonlinearities and time-variances of the system under control. In particular, it has the property to force a physical system, characterized by inherently relevant time delays, to promptly react to external disturbances. Acknowledgements--This research has been financially supported by the Italian Agency MURST (Ministero dell'UniversitA e della Ricerca Scientifica e Tecnologica), under a MURST 40% contract.
REFERENCES
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5. Fortunato, B., Maione, B. and Palmisano, N., Proceedings of the Flowers Worm Energy Research Symposium '94, eds E. Carnevale, G. Manfrida, F. Martelli, 1994, pp. 867-876. 6. Dambrosio, L., Controllo Adattativo One-Step-Ahead di un sistema eolico. PhD Dissertation, Istituto di Macchine ed Energetica, Politecnico di Bari, 1994. 7. Steinbuch, M., European Wind Association Conference and Exhibition, Rome, October 1986. 8. Madsen, P. H., European Wind Association Conference, Herning, Denmark, June 1988. 9. Sandhu, D. and Dias, S., European Wind Association Conference, Herning, Denmark, June 1988. 10. Goodwin, G. C. and Sin, K. S., Adaptive, Filtering, Prediction and Control, Prentice-Hall, Englewood Cliffs. New Jersey, 1984. 11. Dambrosio, L., Development and application of one-step-ahead adaptive control system for helicopters. MS Thesis, Department of Aerospace Engineering, Mississippi State University, 1994. 12. Prouty, R. W., Helicopter Performance, Stability and Control. PWS Engineering, Boston, 1986. 13. Bryson, A. E. and Ho Yu-Chi, Applied Optimal Control--Optimization, Estimation and Control, Hemisphere, New York, 1975.