A Geometric Approach for the Design of MIMO Sliding Controllers. Application to a Wind Driven Double Output Induction Generator

Abstract

This paper presents a systematic methodology to design controllers for a general class of nonlinear MIMO systems affine in the control in the presence of bounded uncertainties and disturbances. The proposed design method is developed through a theoretical framework based on the combination of a geometric approach and sliding mode techniques. The resulting robust control law guarantees finite time convergence, while chattering reduction is attained by utilising the minimum discontinuous action required to ensure disturbance rejection. The proposed methodology is applied to the control of a grid connected wind energy generation system based on a double output induction generator

Full text

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A Geometric Approach for the Design of MIMO Sliding Controllers.
Application to a Wind Driven Double Output Induction Generator
F. ValenciagaP.F. Puleston S.K. Spurgeon
Abstract—This paper presents a systematic methodology to
design controllers for a general class of nonlinear MIMO sys-
tems affine in the control in the presence of bounded uncertain-
ties and disturbances. The proposed design method is developed
through a theoretical framework based on the combination
of a geometric approach and sliding mode techniques. The
resulting robust control law guarantees finite time convergence,
while chattering reduction is attained by utilising the minimum
discontinuous action required to ensure disturbance rejection.
The proposed methodology is applied to the control of a grid
connected wind energy generation system based on a double
output induction generator.
I. INTRODUCTION
Environmental, economic, renewable portfolio standards,
fuel diversity, competitiveness and other international and
local elements have combined to create an increasing in-
terest in developing wind energy generation facilities. In
this expanding context it has been established that, together
with improvements in materials, power electronics and blade
design, the incorporation of advanced control systems into
wind energy generation systems (WECS) is one of the major
technological advances [1]. It has translated into the reduced
cost of energy turning the development of high-efficiency
control strategies for WECS into an essential R&D activity
[2]. At present, this control challenge can be tackled thanks
to the cost reduction and improvement of power semicon-
ductors, AC drives and microprocessors. In spite of the extra
initial investment, the inclusion of electrical control permits a
higher degree of flexibility and more complex objectives can
be achieved, particularly in variable speed operation control.
With such technological improvements, classical controllers
for WECS can be updated by the development of more
efficient strategies based on modern control techniques such
as: Fuzzy Logic Control, Robust Control, Adaptive Control,
etc. Among them, Sliding Mode (SM) Control emerges as
a particularly suitable option to deal with electronically
controlled variable speed operating WECS. This control
technique has proved to be very robust with respect to
system parameter variations and external disturbances [3] [4].
Recent examples of SM control utilised for WECS control
applications appear in [5] and [6].
This work was supported by the National University of La Plata (UNLP),
CONICET and ANPCyT.
F. Valenciaga and P.F Puleston are with the LEICI, Faculty of Engineer-
ing, University of La Plata, C.C.91, C.P.1900, Argentina and CONICET.
S.K. Spurgeon is with Control and Instrumentation Research Group,
Department of Engineering, University of Leicester, U.K.
II. MIMO SM CONTROL DESIGN METHOD
Consider the class of MIMO nonlinear systems repre-
sented by the following nominal dynamic model:
˙ x = f(x) + G(x)u = f(x) +
m
?
i=1
gi(x)ui
(1)
where f and the giare smooth vector fields locally defined
in an open neighbourhood X of ?n.
A. Step 1: Selection of the sliding manifold
In SM control of MIMO systems of the form (1), desired
performance is specified by selecting m individual sliding
variables si(x), which in turn define one (n-m)-dimensional
combined sliding manifold:
S =
m
?
i=1
si=
m
?
i=1
{x ∈ X : si(x) = 0}
(2)
This combined manifold is the intersection of the m smooth
(n-1)-dimensional manifolds defined by si= 0 and will be
referred to as the ‘intersection manifold S’. It is assumed
that the MIMO system (1) has relative degree [1,···1]1xm
with respect to the sliding variables si. In addition, it is also
assumed that the system is hyperbolically minimum phase,
i.e. the zero dynamics is exponentially stable [7].
From here, the task of developing a robust control design
method that guarantees finite-time local-attractivity towards
S will be addressed systematically. For the sake of clarity, the
proposed control law will be presented as the combination
of three terms, each one of them related to a specific control
action, i.e. drift cancellation, defining the characteristics of
the reaching mode and prescribing robustness:
u(x) = uI(x) + uII(x) + uIII(x)
(3)
where uI(x), uII(x) and uIII(x) vectors ∈ ?m.
The design of these control terms will be achieved by
combining Lyapunov’s theory with Variable Structure control
and geometric concepts. The computation of the expressions
for the control vectors uIand uIIwill be rather simple. The
following Lyapunov function will be considered:
V (x) =1
2STS
(4)
˙V (x) = S
∂S
∂x˙ x = ST∂S
∂x(f(x) + G(x)u)
(5)
with S = [s1, s2,...sm]Tand∂S
∂x= [∂s1
∂x,...,∂sm
∂x]
T
mxn.
Proceedings of the 2006 International Workshop on Variable Structure Systems
Alghero, Italy, June 5-7, 2006
MonE.1
1-4244-0208-5/06/$20 ©2006 IEEE
T
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Subsequently, with the help of a geometric approach,
particular attention will be paid to the design of the discon-
tinuous control term uIII, in order to ensure both: i) robust-
ness to internal and external bounded disturbances, and ii)
chattering reduction, by utilising the minimum discontinuous
action required for effective disturbance rejection.
B. Step 2: Design of uI. Drift Cancellation
The first control vector uI prescribes the control action
required to maintain the system on the intersection manifold
S assuming ideal operating conditions, that is: i) neither
disturbances nor uncertainties exist, and ii) the system has
already reached the intersection manifold (i.e. S = 0). Under
these assumptions, it is straightforward to determine from (5)
that the required control action to force˙V (x) = 0 is:
?∂S
Replacing (6) in (5) it is straightforward to see that uIzeroes
˙V (x), ensuring the invariance condition holds on S. Note that
the relative degree [1,···1]1xmrequired in (2) is necessary
for the existence of uI.
A geometric interpretation is provided below, given that
it will be necessary for the subsequent design of uIII in
subsection II-D. Figure 1 displays a schematic representa-
tion of the intersection manifold S and its orthogonal m-
dimensional subspace S⊥. A natural basis of this orthogonal
subspace is constituted by the normalised gradient vectors
(ˇ
∂x
2
uI(x) = −
∂xG(x)
?−1?∂S
∂xf(x)
?
(6)
∂si
∂x=
??∂si
??−1
g1uI1
∂si
∂x) of the m individual surfaces si.
??2
??2
??2
??2
??2
??2
??2
??2
f
^
s1
x
g1uI1
^
s1
x
g1uI1
^
s2
x
g2uI2
^
s1
x
g2uI2
g2uI2
^
s2
x
f
^
s2
x
f
f
^
s1
x
^
s2
x
S
S
??2
Fig. 1.Geometric interpretation of the drift cancellation by uI
From a geometric viewpoint, the action of the control
vector uI implies the cancellation of f⊥, i.e. the orthogonal
projection of the drift field vector f(x) onto the orthogonal
subspace S⊥. This cancellation can be easily attained by
orthogonally projecting f⊥onto the basis vectors
then designing uI to cancel each one of these projections.
Note that the i-th projection of f⊥can be straightforwardly
computed from f(x) as:
????
of f(x).
ˇ
∂x, and
∂si
ˇ
∂si
∂xf(x) =
∂si
∂x
????
−1
2
Lfsi
(7)
where Lfsidenotes the Lie derivative of siin the direction
On the other hand, it is straightforward to see that the
addition of the orthogonal projections of the m column
vectors of G(x) (scaled by uI) would cancel the projection
of f⊥onto
ˇ
∂x, if and only if the following equation holds:
∂si
m
?
j=1
??∂si
∂x
??−1
2
LgjsiuIj= −
????
∂si
∂x
????
−1
2
Lfsi
(8)
Then, considering the cancellation of f⊥onto the m basis
vectors
⎡
⎣
which, in fact, is a explicit version of (6).
ˇ
∂x, it can be shown that:
∂si
⎢⎢⎢
Lg1s1
Lg1s2
...
Lg1sm
···
···
...
···
Lgms1
Lgms2
...
Lgmsm
⎤
⎦
⎥⎥⎥
⎡
⎣
⎢⎢⎢
uI1
uI2
...
uIm
⎤
⎦= −
⎥⎥⎥
⎡
⎣
⎢⎢⎢
Lfs1
Lfs2
...
Lfsm
⎤
⎦
⎥⎥⎥
(9)
C. Step 3: Design of uII. Reaching Mode
Recall that for the design of uIideal conditions have been
assumed. However, if the second condition is violated, that
is if the system is not initially on S = 0, it is necessary
to add a second control vector term (u = uI + uII) to
guarantee reaching of the intersection manifold S (i.e. to
guarantee˙V (x) < 0). It can be inferred from (5) that several
control laws can be designed to fulfill˙V (x) < 0, which
will determine different reaching characteristics, for example,
convergence speed proportional to distance to S, a constant
convergence speed, etc. One simple choice is:
?∂S
where Γ is a positive definite matrix. Moreover, the simplest
design for Γ is a constant positive definite diagonal matrix.
Assuming the latter, the time derivative of the Lyapunov
function is:
˙V (x) = −
uII= −
∂xG(x)
?−1
Γ S
(10)
m
?
i=1
γis2
i
(11)
where the m γi parameters are positive design values that
correspond to the diagonal elements of Γ.
D. Step 4: Design of uIII. Robustness
Up to this point the undisturbed system has been con-
sidered for the purposes of design. Now, the uncertain-
ties and disturbances are taken into account by incorpo-
rating in the undisturbed (nominal) system (1) a vector
ξ
=[ξ1,ξ2,...ξn]Tof unknown disturbances (lumping
external disturbances, internal sources of uncertainty, etc) :
˙ x = f(x) + G(x) u + ξ
(12)
To ensure the controller can deal with such uncertainties
in a robust fashion, a third control vector term (uIII)
must be synthesised. For this, it is reasonable to expect
that in many practical engineering applications the elements
of the disturbance vector can be independently bounded.
This means that, after off-line computation, a vector of
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known bounds ξM=[ξM1, ξM2,... ξMn]Tis available for
the control design. The actual disturbance vector is in fact
unknown, but it can be assumed that it is constrained within
a disturbance bound hypercuboid in ?n.
Using this information, the control component uIII must
be designed to deliver the minimum possible control effort
necessary to cope with the worst possible case of distur-
bances, in order to avoid unnecessary chattering. Therefore,
the control uIII must be able to cancel the largest possible
projection of ξ onto the m-dimensional orthogonal subspace
S⊥. Its orthogonal projection onto the basis vectorsˇ
be bounded by:
????
modulus among the orthogonal projections of the hyper-
cuboid semi-diagonals ontoˇ
in ?3is shown in Figure 2. The white hexagon represents the
orthogonal projection of the uncertainty hypercuboid, while
its vertexes correspond to the projections of the semidiago-
nals). Then, following a reasoning similar to the one used to
derive uI, the expression for the control vector responsible
for system robustness is:
?∂S
where Λ = diag
∂x
.
∂si
∂xcan
ˇ ∂S
∂x
????ξM=
? ???
ˇ
∂s1
∂xT
??? ξM, ...,
???
ˇ
∂sm
∂xT
??? ξM
?T
(13)
where the i-th coordinate of (13) corresponds to the largest
∂si
∂x( a schematic representation
uIII(x) = −
????
∂xG(x)
??? ξM
?−1
Λ sgn(S)
(14)
ˇ
∂S
?
g1uIII1
^
s1
x
g1uIII1
^
s2
x
g2uIII2
^
s1
x
g2uIII2
^
s2
x
^
s1
x
^
s2
x
S
S
^
s1
x?M
^
s2
x?M
??2
??2
Fig. 2. Geometric interpretation of the robust design of uIII
E. Final Control Law
The final control law is obtained by combining the expres-
sions for uI, uIIand uIII. Furthermore, the applicability of
the proposed control law can be easily extended to control
applications where the sliding variables present an explicit
dependance on time (i.e. si(x,t)). If this is the case, equation
(5) becomes:
˙V (x) = ST[∂S
∂x(f(x) + G(x)u) +∂S
∂t]
(15)
and the final expression for the proposed control law is:
u = −?∂S
∂xG(x)?−1?∂S
∂xf(x) +∂S
∂t+ ΓS + Λsgn(S)?
(16)
Summarising, the proposed design method presents a num-
ber of valuable features: it provides a systematic approach for
designing a control capable to cope with an extensive class of
practical applications (those that can be modelled as MIMO
affine non linear systems and that accept a certain degree of
discontinuous control action); the resultant control is robust
in the presence of bounded disturbances and uncertainties;
it guarantees the reaching mode to the intersection manifold
and, while preserving the characteristic robustness of discon-
tinuous control, chattering reduction is accomplished thanks
to the design of uIII to deliver the minimum necessary
control effort to counteract any ξ bounded by ξM.
III. DYNAMIC MODEL OF A WECS BASED ON A DOIG
The proposed method developed in Section II will be
applied to the design of a robust MIMO controller for a
wind energy conversion system based on a double output
induction generator (DOIG). The dynamic model of the
WECS-DOIG is first presented. The stator of the DOIG
Fig. 3.Schematic Representation of the WECS-DOIG
is directly connected to the utility grid, while the rotor
windings are linked to the grid through a fractionally rated
bidirectional PWM converter (see Figure 3). The electrical
angular speeds of the stator (or grid) and the rotor are ω and
ωr, respectively. Then, two d-q reference frames rotating at ω
and ωrcan be utilised to describe the dynamic equations of
the DOIG. Selecting the q-axis of the stator reference frame
locked to the stator voltage (vqs = VL and vds = 0), the
DOIG dynamics are given by [8]:
˙ ϕqs= −Rsiqs− ωϕds+ VL
˙ ϕds= −Rsids+ ωϕqs
˙ ϕqr= −Rriqr− (ω − ωr)ϕdr+ vqr
˙ ϕdr= −Rridr+ (ω − ωr)ϕqr+ vdr
ϕqs= Lsiqs+ Lmiqr
(17)
(18)
(19)
(20)
(21)
ϕds= Lsids+ Lmidr
(22)
ϕqr= Lriqr+ Lmiqs
(23)
ϕdr= Lridr+ Lmids
(24)
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Te=3
2PLm(iqrids− idriqs)
(25)
where Rsand Rrare the stator and rotor winding resistances,
Ls and Lr are the self-inductances and Lm is the mutual
inductance. The vqs, vds, vqr and vdr are the quadrature-
direct components of the stator and rotor maximum winding
voltages, iqs, ids, iqrand idrare the current components, and
ϕqs, ϕds, ϕqrand ϕdrare the concatenated flux components.
Te> 0 is the generator torque, with P the pair of poles.
The mechanical dynamic equation of the WECS-DOIG
can be obtained by applying Newton’s Second Law to the
rotating parts. Neglecting the friction terms:
˙ ωrm=
˙ ωr
P
=1
J(Tt− Te)
(26)
where J is the combined inertia of the rotating parts, ωrm=
ωr
Pis the mechanical rotation speed and Tt is the turbine
torque produced by the blades. The turbine torque is:
Tt=1
2ρ π r3Ct(λ) v2
where ρ is the air density, r is the blade length, v is the wind
speed, Ct(λ) is the turbine torque coefficient and λ =r ωrm
is the tip speed ratio. The torque coefficient is a nonlinear
function of λ that can be expressed as follows [9]:
(27)
v
Ct=a
λ(b
λ− 1)e−c
λ
(28)
with a, b and c positive constants that depend on the
characteristics of the turbine under consideration.
It should be noted that the turbine power efficiency is
represented by another function of λ, called the power
coefficient Cp(λ) = Ct(λ)λ. It presents a unique maximum
at λ = λopt, where the turbine extracts the maximum energy
from the wind. Note then that the rotation speed for optimum
conversion depends on the wind speed (ωrm opt=
Operating algebraically with equations (17)-(27), the dy-
namic model of the system can be expressed as the nonlinear
affine structure:
⎡
⎣
where the states are x = [iqs,ids,iqr,idr,ωr]T, the control
inputs are the rotor voltages u = [vqr,vdr]Tand
f1=?RsLriqs−?ωL2
eq
λopt v
r
).
˙ x = f(x) + Gu =
⎢⎢⎢⎢
f1(x)
f2(x)
f3(x)
f4(x)
f5(x)
⎤
⎦
⎥⎥⎥⎥
+
⎡
⎣
⎢⎢⎢⎢
g11
g21
g31
g41
g51
g12
g22
g32
g42
g52
⎤
⎦
⎥⎥⎥⎥
u
(29)
eq+ ωrL2
m
?ids− RrLmiqr+ LrVL
−ωrLmLridr)/L2
f2=?ωL2
eqiqs+ ωrL2
−RrLmidr)/L2
miqs+ RsLrids+ ωrLmLriqr
eq
f3= (−RsLmiqs+ ωrLsLmids+ RrLsiqr− LmVL
−?(ω − ωr)LsLr− ωL2
m
?idr
?/L2
eq
f4= (−ωrLsLmiqs− RsLmids+ RrLsidr
+?(ω − ωr)LsLr− ωL2
f5=
J P
m
?iqr
?/L2
eq
1
?
Tt−3
2PLm(iqrids− idriqs)
⎡
⎣
eq= LsLr− L2
IV. ROBUST MIMO SLIDING MODE CONTROL FOR A
WECS-DOIG
?
⎤
⎦
G =
1
L2
eq
⎢⎢⎢⎢
m.
−Lm
0
Ls
0
0
0
−Lm
0
Ls
0
⎥⎥⎥⎥
with L2
In this section the proposed control design method is ap-
plied to the WECS-DOIG modelled by (29). Using equation
(2), two simultaneous control objectives will be pursued by
embedding them into the intersection manifold S.
The first one is the most widely held control objective for
a WECS connected to the utility grid, that is wind energy
conversion maximisation. The fulfilment of this objective
implies controlling the turbine rotation speed to operate at
its optimal tip speed ratio (λ = λopt) despite the variations
of the wind speed. One way to attain such conversion
efficiency maximisation without resorting to anemometers
is to compute a torque reference (Tref(ωr)), which defines
the locus of maximum conversion efficiency in the turbine
characteristic plane [6]. The expression for Tref(ωr) can be
straightforwardly obtained by evaluating the turbine torque
expression (27) at its optimum tip speed ratio λ = λopt.
Then, the expression for the torque reference is [6]:
Tref= Koptω2
r=ρπr5Ct(λopt)
2λ2
optP2
ω2
r
(30)
To track such a torque reference is equivalent to tracking
a filtered optimum rotation speed, neglecting the fluctuation
provoked by the high frequency components of the equiva-
lent wind. Therefore, utilising (30), the objective of energy
conversion maximisation can be set through the individual
sliding manifold determined by:
s1(x) = Koptω2
r−3
2PLm(iqrids− idriqs)
(31)
To complete the construction of the intersection manifold
S, a second control objective must be defined. In this sense,
one interesting option is primary reactive power tracking.
For instance, flexible control of the reactive power enables
minimisation of the copper losses of the DOIG. Such an
additional control objective can be expressed through the
following sliding variable:
s2= Qref−3
with Qref the reactive power reference. The expression
for the reference for machine losses minimisation can be
2P (vqsids− vdsiqs) = Qref−3
2PVLids (32)
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straightforwardly obtained by differentiating the overall cop-
per losses with respect to the DOIG currents [10]:
Qref=3
2P
ω LsRrϕ2
L2
ds
mRs+ L2
sRr
(33)
Both control objectives, wind energy conversion maximi-
sation and copper losses minimisation, can be successfully
addressed by applying the proposed method to the WECS-
DOIG. The terms of the resultant control law are:
⎡
+iqsf4+4Koptωrf5
??
+
VL(RsL2
uI=
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
L2
ϕds((idrf1− iqrf2− idsf3
eq
3PLm
?
??
−ϕqs
2ωL2
sRrϕds
m+RrL2
2ωLsRrϕds
m+RrL2
VLLm(RsL2
s)−
s)f4
1
Lm
?
f2
−L2
eq
??
2ωL2
sRrϕds
m+RrL2
2ωLsRrϕds
VL(RsL2
?
−2L2
VLLm(RsL2
+
⎡
s)−
s)f4
1
Lm
?
f2
m+RrL2
?
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
(34)
uII=
⎣
2L2
3ϕds
eq
γ1s1
PLm−ϕqsγ2s2
eqγ2s2
3PLmVL
PLmVL
?
⎤
⎦
(35)
uIII=
⎡
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
L2
ϕds((|idr|ξ1M+ |iqr|ξ2M+ |ids|ξ3M
+|iqs|ξ4M+4Koptωrξ5M
−ϕqs
+
VL(RsL2
eq
3PLm
sRrϕds
m+RrL2
?
sgn(s1)
????
2ωL2
VLLm(RsL2
2ωLsRr|ϕds|
m+RrL2
s)−
?
1
Lm
???ξ2M
s)ξ4M
sgn(s2)
?
−L2
eq
+
????
2ωL2
sRrϕds
m+RrL2
VLLm(RsL2
2ωLsRr|ϕds|
VL(RsL2
s)−
?
1
Lm
???ξ2M
m+RrL2
s)ξ4M
sgn(s2)
⎤
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
(36)
⎦
where the factors fiwhere defined in (29).
To conclude, note that the full order model of the DOIG
has been considered for the design of (34)-(36). However
further assumptions can allow model reduction and a consid-
erable simplification of the final control law. Effectively, as-
suming that the resistive voltage drop on the stator windings
is negligible and that the DOIG stator is directly connected
to a grid of constant voltage and frequency, then it can be
said that ϕqs= 0 and ˙ ϕds= 0. Therefore, the terms of the
final control law are reduced to:
⎡
Lm
⎡
−2L2
⎡
Lm
uI=
⎣
L2
Lm
eq
?
f1+4Koptωrf5
?
2L2
eqγ1s1
3PLmϕds
eqγ2s2
3LmVL
3Pϕds
Qref
3VL
?
L2
eq
f2−2
˙
?
⎤
⎤
⎦
(37)
uII=
⎣
⎦
?
(38)
uIII=
⎣
L2
Lm
eq
?
ξ1M+4Koptωrξ5M
L2
eqξ2M
3Pϕds
sgn(s2)
sgn(s1)
⎤
⎦
(39)
which in certain applications could be more suitable for
implementation.
V. SIMULATION RESULTS
The purpose of this section is to illustrate the efficiency of
the proposed WECS-DOIG MIMO controller over a wide op-
erating range, dealing with model uncertainties and external
disturbances. A 50 HP DOIG-WECS has been considered. To
assess the controller performance under realistic conditions
of operation, simulation tests have been conducted assuming
highly variable wind speeds. The simulated profile used for
the present case is depicted in Figure 4.a.
During the design and analysis phases, the controller
performance has been evaluated through a number of tests
using different values for the elements of Γ and manifold
disturbance signals. In particular,the simulations presented in
this section have been performed with γ1= 25 and γ2= 25.
In this test, the effects of disturbances and uncertainties have
been considered by incorporating disturbance signals from
time t=60s until t=240s (during the first 60 seconds the sys-
tem is undisturbed). The time evolution of the disturbances
(ξφ) of the dynamics of flux components ϕqs, ϕds, ϕqrand
ϕdr is shown in Figure 4.b. They translate into the distur-
bance signals that affect the dynamic equations of the DOIG
currents (i.e. the first four components of the disturbance
vector ξ). For visual presentation, the flux disturbances have
been normalised by assuming a maximum uncertainty of
10% in the stator and rotor winding resistances. In addition,
the effect of disturbances/uncertainties in the mechanical
dynamics (i.e. the fifth element of ξ) has been taken into
account by including an unmodelled friction term (depicted
in Figure 4.c).
Off line computation shows that ξ can be bounded by
ξM = [480;350;485;360;6]T. Once ξMis determined,
050100150200
8
10
12
14
16
t [s]
v [m/s]
050100 150200
−100
−50
0
50
100
t [s]
ξφ [%]
ξφ1
ξφ2
ξφ3
ξφ4
050 100150200
0
2
4
6
t [s]
ξ5 [Nm]
Fig. 4. (a) Wind Speed Profile (b) Disturbance Signals of the Concatenated
Flux, (c) Torque disturbance
119

Page 6

050 100150 200
−10
0
10
20
t [s]
s1 [Nm]
050 100150 200
−2000
0
2000
4000
6000
8000
t [s]
s2 [VAr]
Fig. 5.Evolution of the Sliding Variables (a) s1, (b) s2
050 100 150200
0
1
2
3x 10
4
t [s]
PIdealw ; Pref ; Pg[W]
PIdeal w
Pe
Pref
050 100150200
1.8
2
2.2
2.4
x 10
4
t [s]
Qref ; Qs [VAr]
Qs
Qref
Fig. 6.(a) Active Power, (b) Stator Reactive Power
the control elements (34)-(36) can be computed. To better
illustrate the operation of the proposed controller, only terms
uI and uII are active during the first 150 seconds of the
simulation. At t=150s, the term responsible for the controller
robustness (uIII) is connected. The evolution of the sliding
variables s1 and s2 (equations (31) and (32), respectively)
can be appreciated in figures 5.a and 5.b. Note that sliding
mode operation is lost in the period 60-150s. However, as
soon as uIII is activated, the sliding mode is successfully
reestablished.
The efficiency of the proposed WECS-DOIG controller to
fulfil simultaneously both control objectives, wind energy
conversion maximisation and copper losses minimisation,
can be confirmed through Figure 6. Figure 6.a shows (dashed
line) the theoretical power available in the wind PIdeal W
(which includes the high frequency components of the wind),
the reference power (corresponding to the torque reference
(33)) and the actual power extracted from the wind Pe(thick
line). It can be observed that excellent tracking is achieved
except for the period t=60-150s, during which disturbances
have already appeared but uIII is disconnected. Figure 6.b
displays the reactive power reference (33) and the stator
reactive power of the DOIG (thick line). Similar satisfactory
tracking features can be appreciated when the complete
control law is operational.
Finally, Figure 7 presents the rotor voltage components
or control signals. It is interesting to note the effect of the
050100150200
−400
−300
−200
−100
0
t [s]
uqr[V]
uI+uII
uI+uII+uIII
050 100150200
0
5
10
15
20
t [s]
udr[V]
uI+uII
uI+uII+uIII
Fig. 7.Control Signals (a) uqr, (b) udr
incorporation of uIII from t=150s.
In addition to the example presented in this section,
extensive and comprehensive simulation analysis has been
conducted utilising a wide range of variable wind speeds and
many different uncertainty/disturbance signals. In every test,
the proposed WECS-DOIG MIMO controller demonstrates
good performance, proving its robustness with respect to
torque disturbances and model parameter uncertainties.
VI. CONCLUSION
In this paper geometric techniques and sliding mode
methods have been harnessed to develop a robust control
framework appropriate for MIMO uncertain systems affine
in the control. The methodology has been utilised for control
of a wind-driven double-output induction generator and
excellent simulation results have been obtained. On going
work will seek to incorporate a second order sliding mode
control component into the framework to alleviate chattering
completely.
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