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Steady-state optimum resistive load control for wind-driven permanent magnet alternators
AppliedEnergy12(1982)317-325
STEADY-STATE OPTIMUM RESISTIVE LOAD CONTROL FOR WIND-DRIVEN PERMANENT MAGNET ALTERNATORS
1
A.M.
DE PAOR
Department of Electrical Engineering, University College, Dublin, 2, Ireland
ABSTRACT
Two approaches are explored to the problem of controlling a resistive load on a winddriven permanent magnet alternator in such a way as to maximise load power dissipation. Thefirst shows how to establish the explicit dependence of load resistance on windspeed and/or turbine angular velocity, using mathematical models of a class recently proposed by Power. 3 The second shows how to compute a curve of set-point values of tip-speed ratio versus windspeed, working from a given turbine characteristic of power coefficient versus tip-speed ratio.
INTRODUCTION
An application of wind power which is currently of research interest 1'2 and which has also been developed commercially in Denmark, the UK and the USA, is electrical resistance heating via high efficiency permanent magnet alternators, with storage where necessary provided by tanks of water or brick-filled domestic storage heaters. This is technically much simpler than other electrical applications, since precise voltage and frequency controls are not necessary. However, in order to maximise power transfer to the load, some degree of control is required, and this can be applied either to the wind turbine (e.g. via blade pitch control) or to the load resistance (as is done on a switched basis in at least one commercial UK machine-the Trimble Mill). The purpose of this paper is to explore the latter possibility theoretically, using the basic premises underlying mathematical models produced by Power. 3'4 These are that the torque generated by a wind turbine'is a function of the two variables, windspeed, V, and angular velocity, f~; and that the power 317 Applied Energy 0306-2619/82/0012-0317/$02"75 © Applied Science Publishers Ltd, England, 1982 Printed in Great Britain
318
A . M . DE PAOR
coefficient (as defined in reference 3 and again below) is a function only of the dimensionless group: Dr) X(l) 2V which is the tip-speed ratio. In eqn. (1) a slight change in notation has been made from reference 3: D/2 has been introduced for the radius of the area swept out by the mill, the symbol r being used here to denote electrical resistance. The control strategies derived below are obtained on the basis of steady-state optimisation: power delivered to the load is maximised during periods of constant mill angular velocity. The performance will, therefore, be sub-optimal during transients. However, the author has found that application of optimal control theory under general dynamic conditions leads to violent control actions which would be difficult to implement and which would impose extreme mechanical shocks on the turbine. Two viewpoints on load control are developed below. In the first, a prescription for varying the load resistance as a function of Vand/or D is derived. In the second it is shown that the control problem may be viewed as that of establishing a target or 'set point' tip-speed ratio for each windspeed and a feedback controller could then be used to vary the load current in such a way as to achieve that set point. The second approach can be implemented from a graph of power coefficient versus tip-speed ratio for the turbine, which need not be modelled analytically. It may, therefore, be more convenient to apply in practice.
LOAD RESISTANCE CONTROL
The system considered is shown in Fig. 1. The only loss mechanism represented explicitly in the alternator is its internal resistance, r. Windage, friction and iron losses, which depend only on f~, may, if not negligible, be incorporated in the turbine torque generation function g(V, f~). The final term in the Euler homogeneous function proposed by Power 3 is a2.f~ 2 with a2, < 0 and this is also a reasonable approximation for the non-resistive losses in the alternator. The quantity p, which has no place in this analysis, is the number of pole pairs on the alternator. No gearbox is r
v ~
t
~
~
(LoAo)
\ ,~v,.tt, I \
=. l - / ~ k,.a. lie (pakl.)
Fig. i.
Schematic of the system.
LOAD CONTROL FOR WIND-DRIVEN MAGNET ALTERNATORS
319
shown between the turbine and the alternator, in line with the practice now evolving of coupling large diameter, custom designed, muitipole alternators directly to wind turbines. A further item not shown on Fig. i is the leakage inductance, L, of the alternator windings. The implication is that either the associated reactance, p~L, is negligible over the range of operating frequencies or that it is tuned out by a series capacitive reactance (1/pDC). To accomplish this at all frequencies of interest would require varying C with ~ in accordance with the law: C-
1
(2)
p2~2L
Piecewise constant approximations to this are possible using electronic switching, but capacitive tuning appears to be unnecessary with the alternators now being produced for wind turbine use. The power delivered to the load resistance, R, is: k2~'] 2
P
(r + R) 2 R
(3)
Assuming that the machine is not accelerating, the torque generated by the turbine is balanced by the electromagnetic torque of the alternator: k2~ g( V, f2) -
(4)
r+R
The optimisation problem is to choose R so that P is maximised for any Vsubject to the constraint imposed by eqn. (4). The simplest way to solve this is the Lagrange multiplier 5 approach in which the Hamiltonian: H
(r + R)z
R+2
{ g(V, ~)
- rk:~ -~j
(5)
is formed, and the conditions obtained by setting:
t~H OH O--R-= 8 ~ = 0
(6)
are investigated. The former gives:
(7) and the latter:
2Rk2~
{ Og
0=(r+R)------5+2
~-D
k2 } r+R
(8)
320
A.M.
DE P A O R
Substituting eqn. (7) into eqn. (8) leads, after rearrangement, to: k2
R= r- -
(9)
Og/O~
As an illustration of eqn. (9) we consider a model of type 1 for g( V, f~) (using the terminology of reference 3): (10)
g(V,~~) -~ a V 2 -{- b V ~ - c ~ 2
with a _>0 and c > 0. This gives: k2 R=r-~
(ll)
2cf~ - b V
The special case b = 0 is of some practical interest. It corresponds to tip-speed ratio, X,,, at maximum power coefficient being l/x/3 = 0.557 times the tip-speed ratio, Xo, under no load. The iauthor has noticed that several two-bladed wind turbines with untwisted aerofoil blades tested by Kloeffler and Sitz 6 correspond closely to this condition. In such a case the control strategy given by eqn. (11) simplifies enormously, since there is no explicit dependence on Vand since fl can be measured very simply from the frequency of the generated voltage. Equation (9), as illustrated by eqn. (1 l), generally results in a control law which depends explicitly on Vand fl. However, it may be implemented entirely in terms of one quantity or the other, at the expense of some additional computation. If eqn. (9) is substituted into eqn. (4) there results: •g 2r egg g( V, f~) + f~ 0 ~ = ~k g( V, f~) 0~
(12(a))
which may be rearranged as: ~v
c ~ {rig(V, n ) } = - U
"
cgta}
The point about the rearrangement in eqn. (12(b)) is that, when g( V, f~) is in the class proposed by Power, 3 the left-hand side and the quantities in brackets on the righthand side are functions only of: f~ z =-V
(13)
Thus, eqn. (12(b)) defines z as an implicit function of V. This is a multivalued function, but it is easy to deduce that, along the branch of interest, z moves smoothly from the value 2Xm/D to the value 2 X o / D as Vgoes from zero to oo. This makes the use of eqn. (12(b)) quite straightforward, since, for computational
LOAD CONTROL FOR WIND-DRIVEN MAGNET ALTERNATORS
321
purposes, z can be scanned in the indicated range and corresponding values o f V calculated very readily. We now illustrate the above observations and the use o f eqn. (12(b)).
Example With:
g ( V , ~ ) = a V 2 + bV~-'2 - cf'~ 2 the functions involved in eqn. (12) are: 1 -V5 c ~ {f~g( V, f~) } = a + 2bz - 3cz ~
g(V, ~) 1 Og Vz
-- - - = (a + bz - czZ)(b - 2cz) " V 0~
Dividing t h r o u g h by the coefficients of the highest powers of z now gives eqn. (12) in the form :
(z z
2b 3,'
z-
a)2crV( +
~cc
zz
3-~7-
bz
a)(
c
z-
b) 2cc
= 0
(13(a))
Equation (13(a)) is in the f o r m needed for a root locus study 7 to show how the roots move as the parameter 2crV/3k 2 is varied in the range 0 to oo. The root locus is shown on Fig. 2, which has been drawn for b > O. The 'poles', Px and P2, which are starting points for root motion, are the roots of the polynomial:
zz
2b
a )
Z-3
The value o f the right-hand one, after some manipulation, reduces to:p~ = 2Xm/D. The 'zeros', z 1 and z 2, are the roots of the polynomial:
and the right-hand one of these has the value z I = 2 X o / D . The truth o f these assertions can be established f r o m the analysis given in reference 3. Imz
X
L
X
Zz
P,
_
2(: Fig. 2. Root locus ofeqn. (13).
Z
-~
Re
z
322
A . M . DE PAOR
It is noted that as Vgoes from 0 to oo, one of the roots moves smoothly from Px to zl : this is the branch which solves the m a x i m u m power transfer problem. Having demonstrated the existence of this branch, it is unnecessarily cumbersome to compute z versus V by repeated factorisation of the cubic equation (eqn. (13)). Rather, one would rearrange eqn. (13) as: -3k 2
(z2
2b
a)
V=
(14)
Z-cZ
c I\
2cJ
and use this to compute V as a function of z in the range Pl < z < zl. This is the essence of the 'calibration' procedure used in applications of root locus theory to feedback system design, v
TIP-SPEED RATIO CONTROL
The previous section concluded by illustrating how an analytical model of g( V, f~) could be used to relate the variable z = ~/V to V, in order that the load resistance control law in eqn. (9) could be expressed in terms of Vor f2 alone. The variable z is, of course, very closely related to the tip-speed ratio X = D9/2 V, and this realisation motivated an attempt to reformulate the control law in terms which would enable it to be implemented from the familiar curve of power coefficient C o versus X. It will be recalled 3 that C o is a dimensionless quantity which relates the power harvested by the wind turbine to that available in the area of wind intercepted by it:
g( V, ~)ff2 Cp- ½pAV3
(15)
F r o m eqns (15) and (1) it follows that:
g( V, f~) = ¼pADV2. C~ X
(16)
This leads to:
• =¼pADV2 ~
c?f~
[xdCo-cl
=~pAD2VL
dX X2
(17)
LOAD CONTROL FOR WIND-DRIVEN MAGNET ALTERNATORS
323
When eqns (16) and (17) are substituted into eqn. (12(a)) then: X dCp
rpAD 2
dX Cp [ x dCp~ _
4k2
Cp1
=ftX)
(18)
The attraction ofeqn. (18) is thatf(X) can be computed numerically from a given experimental curve of Cp versus X: there is no need to go through the process of modelling the torque of the wind turbine by a mathematical model g(V, fl). Once the graph off(X) versus X has been prepared, eqn. (18) allows the desired value, which we shall denote by X d to be determined as a function of V. A typical qualitative sketch off(X) versus Xis shown in Fig. 3, along with the corresponding sketch of Xd versus V. It would be a straightforward matter to store the relationship of X d to V in a microcomputer, providing instrumentation to measure the actual tip-speed ratio, X, and form the error Xd -- X. This would then be used to adjust the load current via,
IXm IXo I
___.L
f
r/oAD a" v
Xd
1
K
Xm
Fig. 3.
Generation of Xd versus V from
Cpversus 1".
324
A . M . DE PAOR
for example, a PI controller 7 and a series regulator (which would form part of the load) in such a way as to keep the error in the tip-speed ratio close to zero in the face of the random variations i n V. In this approach, the load resistance variation described by eqn. (9) would be imposed implicitly rather than explicitly. It is worth pointing out that if the internal resistance of the alternator were zero we would get X a = X,,, and so the controller would always try to hold the turbine at its optimum tip speed ratio. This would give dCp/dX = O, Cp = Cp,, and X = X,, in eqn. (17), and the resistance control law could then be expressed in either of the forms:
R
8k2Cpm 1 pAD2XZm V
(19)
16kZCpm 1 pAD3X f~
(20)
or:
R
These control laws can be derived independently by elementary reasoning. The latter form is interesting, as it seems likely to be the basis for the controller on the Trimble Mill, in which, as the mill speed rises to a certain value, the heater elements are grouped by a switching controller into progressively lower resistance configurations.
DISCUSSION
In this paper an optimum control problem has been posed for a particular application of wind turbines, and formal solutions have been given. The actual details of implementing these solutions--whether by varying a load resistance in accordance with V and/or fL or controlling the load current in order to achieve a target tip-speed ratio--have only been hinted at. Modern electronic technology provides many ways in which suitable controllers could be constructed. The problem treated here is not, of course, the whole story: once rated load power has been reached, the control law must be changed and/or some over-riding mechanism must come into play. Centrifugal governors actuating spoilers or end flaps are reasonably common, and their use in conjunction with a change in control law to hold the load resistance or current constant beyond a certain windspeed would achieve the desired result. The other common expedient is to use a tailvane actuated by a pressure plate to turn the mill progressively out of the wind at high wind speeds. The permanent magnet alternator is only one of many electrical machines which have been used or proposed for use with wind turbines. Each machine gives rise to distinctive problems of dynamics and control, and each has its own best application.
LOAD CONTROL FOR WIND-DRIVEN MAGNET ALTERNATORS
325
REFERENCES !. L. L. FREmS,H. BOLTON,L. K. BUEHRINGand V. C. NICODEMOU,A low Cost wind energy conversion system for heating of domestic premises. Future Energy Concepts (IEE Conference Publication No. 171), (1979), pp. 282-5. 2. H. R. BOLTON and V. C. NICODEMOU, Permanent magnet alternators for small wind systems, Proceedings of the BWEA Conference, Cranfield, April, 1979, pp. 165-80. 3. H. M. POWER,A simulation model for wind turbines, Applied Energy, 6 (1980), pp. 395-9. 4. H. M. POWER,Analytical solution of a simulation model for wind turbines, Applied Energy, 9 (198 l), pp. 311-16. 5. A. P. SAGE, Optimum systems control, Prentice-Hall, 1968. 6. R.G. KLOEFFLERand E. L. SITZ, Electric Energy from the Winds, Kansas State College Engineering Experiment Station, Bulletin No. 52, 1946. 7. H. M. POWERand R. J. SIMPSON, Introduction to dynamics and control, McGraw-Hill (UK), 1978.
STEADY-STATE OPTIMUM RESISTIVE LOAD CONTROL FOR WIND-DRIVEN PERMANENT MAGNET ALTERNATORS
1
A.M.
DE PAOR
Department of Electrical Engineering, University College, Dublin, 2, Ireland
ABSTRACT
Two approaches are explored to the problem of controlling a resistive load on a winddriven permanent magnet alternator in such a way as to maximise load power dissipation. Thefirst shows how to establish the explicit dependence of load resistance on windspeed and/or turbine angular velocity, using mathematical models of a class recently proposed by Power. 3 The second shows how to compute a curve of set-point values of tip-speed ratio versus windspeed, working from a given turbine characteristic of power coefficient versus tip-speed ratio.
INTRODUCTION
An application of wind power which is currently of research interest 1'2 and which has also been developed commercially in Denmark, the UK and the USA, is electrical resistance heating via high efficiency permanent magnet alternators, with storage where necessary provided by tanks of water or brick-filled domestic storage heaters. This is technically much simpler than other electrical applications, since precise voltage and frequency controls are not necessary. However, in order to maximise power transfer to the load, some degree of control is required, and this can be applied either to the wind turbine (e.g. via blade pitch control) or to the load resistance (as is done on a switched basis in at least one commercial UK machine-the Trimble Mill). The purpose of this paper is to explore the latter possibility theoretically, using the basic premises underlying mathematical models produced by Power. 3'4 These are that the torque generated by a wind turbine'is a function of the two variables, windspeed, V, and angular velocity, f~; and that the power 317 Applied Energy 0306-2619/82/0012-0317/$02"75 © Applied Science Publishers Ltd, England, 1982 Printed in Great Britain
318
A . M . DE PAOR
coefficient (as defined in reference 3 and again below) is a function only of the dimensionless group: Dr) X(l) 2V which is the tip-speed ratio. In eqn. (1) a slight change in notation has been made from reference 3: D/2 has been introduced for the radius of the area swept out by the mill, the symbol r being used here to denote electrical resistance. The control strategies derived below are obtained on the basis of steady-state optimisation: power delivered to the load is maximised during periods of constant mill angular velocity. The performance will, therefore, be sub-optimal during transients. However, the author has found that application of optimal control theory under general dynamic conditions leads to violent control actions which would be difficult to implement and which would impose extreme mechanical shocks on the turbine. Two viewpoints on load control are developed below. In the first, a prescription for varying the load resistance as a function of Vand/or D is derived. In the second it is shown that the control problem may be viewed as that of establishing a target or 'set point' tip-speed ratio for each windspeed and a feedback controller could then be used to vary the load current in such a way as to achieve that set point. The second approach can be implemented from a graph of power coefficient versus tip-speed ratio for the turbine, which need not be modelled analytically. It may, therefore, be more convenient to apply in practice.
LOAD RESISTANCE CONTROL
The system considered is shown in Fig. 1. The only loss mechanism represented explicitly in the alternator is its internal resistance, r. Windage, friction and iron losses, which depend only on f~, may, if not negligible, be incorporated in the turbine torque generation function g(V, f~). The final term in the Euler homogeneous function proposed by Power 3 is a2.f~ 2 with a2, < 0 and this is also a reasonable approximation for the non-resistive losses in the alternator. The quantity p, which has no place in this analysis, is the number of pole pairs on the alternator. No gearbox is r
v ~
t
~
~
(LoAo)
\ ,~v,.tt, I \
=. l - / ~ k,.a. lie (pakl.)
Fig. i.
Schematic of the system.
LOAD CONTROL FOR WIND-DRIVEN MAGNET ALTERNATORS
319
shown between the turbine and the alternator, in line with the practice now evolving of coupling large diameter, custom designed, muitipole alternators directly to wind turbines. A further item not shown on Fig. i is the leakage inductance, L, of the alternator windings. The implication is that either the associated reactance, p~L, is negligible over the range of operating frequencies or that it is tuned out by a series capacitive reactance (1/pDC). To accomplish this at all frequencies of interest would require varying C with ~ in accordance with the law: C-
1
(2)
p2~2L
Piecewise constant approximations to this are possible using electronic switching, but capacitive tuning appears to be unnecessary with the alternators now being produced for wind turbine use. The power delivered to the load resistance, R, is: k2~'] 2
P
(r + R) 2 R
(3)
Assuming that the machine is not accelerating, the torque generated by the turbine is balanced by the electromagnetic torque of the alternator: k2~ g( V, f2) -
(4)
r+R
The optimisation problem is to choose R so that P is maximised for any Vsubject to the constraint imposed by eqn. (4). The simplest way to solve this is the Lagrange multiplier 5 approach in which the Hamiltonian: H
(r + R)z
R+2
{ g(V, ~)
- rk:~ -~j
(5)
is formed, and the conditions obtained by setting:
t~H OH O--R-= 8 ~ = 0
(6)
are investigated. The former gives:
(7) and the latter:
2Rk2~
{ Og
0=(r+R)------5+2
~-D
k2 } r+R
(8)
320
A.M.
DE P A O R
Substituting eqn. (7) into eqn. (8) leads, after rearrangement, to: k2
R= r- -
(9)
Og/O~
As an illustration of eqn. (9) we consider a model of type 1 for g( V, f~) (using the terminology of reference 3): (10)
g(V,~~) -~ a V 2 -{- b V ~ - c ~ 2
with a _>0 and c > 0. This gives: k2 R=r-~
(ll)
2cf~ - b V
The special case b = 0 is of some practical interest. It corresponds to tip-speed ratio, X,,, at maximum power coefficient being l/x/3 = 0.557 times the tip-speed ratio, Xo, under no load. The iauthor has noticed that several two-bladed wind turbines with untwisted aerofoil blades tested by Kloeffler and Sitz 6 correspond closely to this condition. In such a case the control strategy given by eqn. (11) simplifies enormously, since there is no explicit dependence on Vand since fl can be measured very simply from the frequency of the generated voltage. Equation (9), as illustrated by eqn. (1 l), generally results in a control law which depends explicitly on Vand fl. However, it may be implemented entirely in terms of one quantity or the other, at the expense of some additional computation. If eqn. (9) is substituted into eqn. (4) there results: •g 2r egg g( V, f~) + f~ 0 ~ = ~k g( V, f~) 0~
(12(a))
which may be rearranged as: ~v
c ~ {rig(V, n ) } = - U
"
cgta}
The point about the rearrangement in eqn. (12(b)) is that, when g( V, f~) is in the class proposed by Power, 3 the left-hand side and the quantities in brackets on the righthand side are functions only of: f~ z =-V
(13)
Thus, eqn. (12(b)) defines z as an implicit function of V. This is a multivalued function, but it is easy to deduce that, along the branch of interest, z moves smoothly from the value 2Xm/D to the value 2 X o / D as Vgoes from zero to oo. This makes the use of eqn. (12(b)) quite straightforward, since, for computational
LOAD CONTROL FOR WIND-DRIVEN MAGNET ALTERNATORS
321
purposes, z can be scanned in the indicated range and corresponding values o f V calculated very readily. We now illustrate the above observations and the use o f eqn. (12(b)).
Example With:
g ( V , ~ ) = a V 2 + bV~-'2 - cf'~ 2 the functions involved in eqn. (12) are: 1 -V5 c ~ {f~g( V, f~) } = a + 2bz - 3cz ~
g(V, ~) 1 Og Vz
-- - - = (a + bz - czZ)(b - 2cz) " V 0~
Dividing t h r o u g h by the coefficients of the highest powers of z now gives eqn. (12) in the form :
(z z
2b 3,'
z-
a)2crV( +
~cc
zz
3-~7-
bz
a)(
c
z-
b) 2cc
= 0
(13(a))
Equation (13(a)) is in the f o r m needed for a root locus study 7 to show how the roots move as the parameter 2crV/3k 2 is varied in the range 0 to oo. The root locus is shown on Fig. 2, which has been drawn for b > O. The 'poles', Px and P2, which are starting points for root motion, are the roots of the polynomial:
zz
2b
a )
Z-3
The value o f the right-hand one, after some manipulation, reduces to:p~ = 2Xm/D. The 'zeros', z 1 and z 2, are the roots of the polynomial:
and the right-hand one of these has the value z I = 2 X o / D . The truth o f these assertions can be established f r o m the analysis given in reference 3. Imz
X
L
X
Zz
P,
_
2(: Fig. 2. Root locus ofeqn. (13).
Z
-~
Re
z
322
A . M . DE PAOR
It is noted that as Vgoes from 0 to oo, one of the roots moves smoothly from Px to zl : this is the branch which solves the m a x i m u m power transfer problem. Having demonstrated the existence of this branch, it is unnecessarily cumbersome to compute z versus V by repeated factorisation of the cubic equation (eqn. (13)). Rather, one would rearrange eqn. (13) as: -3k 2
(z2
2b
a)
V=
(14)
Z-cZ
c I\
2cJ
and use this to compute V as a function of z in the range Pl < z < zl. This is the essence of the 'calibration' procedure used in applications of root locus theory to feedback system design, v
TIP-SPEED RATIO CONTROL
The previous section concluded by illustrating how an analytical model of g( V, f~) could be used to relate the variable z = ~/V to V, in order that the load resistance control law in eqn. (9) could be expressed in terms of Vor f2 alone. The variable z is, of course, very closely related to the tip-speed ratio X = D9/2 V, and this realisation motivated an attempt to reformulate the control law in terms which would enable it to be implemented from the familiar curve of power coefficient C o versus X. It will be recalled 3 that C o is a dimensionless quantity which relates the power harvested by the wind turbine to that available in the area of wind intercepted by it:
g( V, ~)ff2 Cp- ½pAV3
(15)
F r o m eqns (15) and (1) it follows that:
g( V, f~) = ¼pADV2. C~ X
(16)
This leads to:
• =¼pADV2 ~
c?f~
[xdCo-cl
=~pAD2VL
dX X2
(17)
LOAD CONTROL FOR WIND-DRIVEN MAGNET ALTERNATORS
323
When eqns (16) and (17) are substituted into eqn. (12(a)) then: X dCp
rpAD 2
dX Cp [ x dCp~ _
4k2
Cp1
=ftX)
(18)
The attraction ofeqn. (18) is thatf(X) can be computed numerically from a given experimental curve of Cp versus X: there is no need to go through the process of modelling the torque of the wind turbine by a mathematical model g(V, fl). Once the graph off(X) versus X has been prepared, eqn. (18) allows the desired value, which we shall denote by X d to be determined as a function of V. A typical qualitative sketch off(X) versus Xis shown in Fig. 3, along with the corresponding sketch of Xd versus V. It would be a straightforward matter to store the relationship of X d to V in a microcomputer, providing instrumentation to measure the actual tip-speed ratio, X, and form the error Xd -- X. This would then be used to adjust the load current via,
IXm IXo I
___.L
f
r/oAD a" v
Xd
1
K
Xm
Fig. 3.
Generation of Xd versus V from
Cpversus 1".
324
A . M . DE PAOR
for example, a PI controller 7 and a series regulator (which would form part of the load) in such a way as to keep the error in the tip-speed ratio close to zero in the face of the random variations i n V. In this approach, the load resistance variation described by eqn. (9) would be imposed implicitly rather than explicitly. It is worth pointing out that if the internal resistance of the alternator were zero we would get X a = X,,, and so the controller would always try to hold the turbine at its optimum tip speed ratio. This would give dCp/dX = O, Cp = Cp,, and X = X,, in eqn. (17), and the resistance control law could then be expressed in either of the forms:
R
8k2Cpm 1 pAD2XZm V
(19)
16kZCpm 1 pAD3X f~
(20)
or:
R
These control laws can be derived independently by elementary reasoning. The latter form is interesting, as it seems likely to be the basis for the controller on the Trimble Mill, in which, as the mill speed rises to a certain value, the heater elements are grouped by a switching controller into progressively lower resistance configurations.
DISCUSSION
In this paper an optimum control problem has been posed for a particular application of wind turbines, and formal solutions have been given. The actual details of implementing these solutions--whether by varying a load resistance in accordance with V and/or fL or controlling the load current in order to achieve a target tip-speed ratio--have only been hinted at. Modern electronic technology provides many ways in which suitable controllers could be constructed. The problem treated here is not, of course, the whole story: once rated load power has been reached, the control law must be changed and/or some over-riding mechanism must come into play. Centrifugal governors actuating spoilers or end flaps are reasonably common, and their use in conjunction with a change in control law to hold the load resistance or current constant beyond a certain windspeed would achieve the desired result. The other common expedient is to use a tailvane actuated by a pressure plate to turn the mill progressively out of the wind at high wind speeds. The permanent magnet alternator is only one of many electrical machines which have been used or proposed for use with wind turbines. Each machine gives rise to distinctive problems of dynamics and control, and each has its own best application.
LOAD CONTROL FOR WIND-DRIVEN MAGNET ALTERNATORS
325
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