J =
0
− 1
= −JT, eJx =
cos( x)
− sin( x)
M
1
0
sin( x)
cos( x)
= ( e−Jx) T.
From Dynamic Modeling to Experimentation of
Induction Motor Powered by Doubly-Fed Induction Generator by Passivity-Based Control
117
⎡
⎤
[ LrM( LsG + LsM) − L 2 ]
mM I 2
−LmGLrMe−JnGθG
−LmGLmMe−J( nGθG−nMθM)
L− 1( θ) = 1 ⎣
−
⎦
Δ
LmGLrMeJnGθG
LrGLrM I 2
LrGLmMeJnMθM
(9)
−LmGLmMeJ( nGθG−nMθM)
LrG LmMe−JnMθM
[ LrG( LsG + LsM) − L 2 ]
mG I 2
⎡
⎤
L
1 ⎢
11
L 12
L 13
T
⎥
Δ ⎣ L 12
L 22
L 23 ⎦
(10)
T
T
L 11
L 23
L 33
where
Δ = LrG[ LrM( LsG + LsM) − L 2 ] −
<
mM
LrML 2 mG
0
(11)
We recall that, due to physical considerations, R > 0, L( θ) = LT( θ) > 0 and L− 1( θ) =
L− 1 T( θ) > 0.
A state–space model of the (6–th order) electrical subsystem is finally obtained replacing (7)
in (6) as
Σ e : ˙ λ + RL( θ) − 1 λ = BvrG
(12)
The mechanical dynamics are obtained from Newton’s second law and are given by
Σ
¨
˙
m : Jmθ + Bmθ = τ − τL
(13)
where Jm = diag{JG, JM} > 0 is the mechanical inertia matrix, Bm = diag{BG, BM} ≥ 0
contains the damping coefficients, τL = [ τLG, τLM] T are the external torques, that we will assume constant in the sequel. The generated torques are calculated as usual from
∂
τ =
τG
τ
= − 1
λT[ L( θ)] − 1 λ .
(14)
M
2 ∂θ
From (7), we obtain the alternative expression
∂
τ = 12 ∂θ iTL( θ) i .
The following equivalent representations of the torques, that are obtained from direct
calculations using (7), (8) and (14), will be used in the sequel
⎡
⎤
−LmGiT Je−JnGθGi
rG
sG
τ = ⎣
⎦
(15)
−LmMiT JeJnMθMi
sG
r M
⎡
⎤
− nG
˙ λT
J( λ
R
sG M
sG M − LmMeJnMθM ir M)
= ⎢
sG + RsM
⎣
⎥
⎦
(16)
nM ˙ λT Jλ
R
r M
rM
r M
2.1 Modeling of the DFIG-IM in the stator frame of the two machines
It has been shown in (4) and (3) that the DFIG-IM is Blondel–Park transformable using the
following rotating matrix:
⎡
⎤
e( Jσ)
0
0
⎢
⎥
Rot( σ, θG, θM) = ⎣
0
e( J( σ+ nGθG))
0
⎦
(17)
0
0
e( J( σ+ nGθG−nMθM))
118
Electric Machines and Drives
where σ is an arbitrary angle.
The model of the DFIG-IM in the stator frame of the two machines is given by (see (4) and (3)
for in depth details):
⎧ ⎡
⎤ ⎡
⎤ ⎡
⎤ ⎡
⎤
⎪
⎪
˙
˙
⎪
λ
aI
θ
λ
I
⎪
rG
2 − nG G J
bI 2
0
rG
2
⎪ ⎢
⎥
⎨ ⎣ ˙ λ
⎣
˙
˙
⎦ ⎣
⎦ ⎣
⎦
sMG ⎦+
aI 2 − nGθG J
eI 2
−cI 2 + nMθMJ
λsMG
= I 2
vrG
˙
˙ θ
λ
0
⎪
λ
0
−dI 2
cI 2 − nM M J
(18)
⎪
r M
r M
⎪
⎪
⎪
⎩
JG ˙ ωG
+ BG
0
ωG
+
f λT
Jλ
sMG
rG
=
−τLG
JM ˙ ωM
0
BM
ωM
− f λT Jλ
−τ
sMG
r M
LM
or
⎧
⎡
⎤
⎡
⎤
⎡
⎤
⎪
⎪
˙
⎪
λ
i
I
⎪
rG
rG
2
⎪
⎢
⎥
⎨
⎣ ˙ λ
˙
˙
⎣
⎦
⎣
⎦
sMG ⎦ + ( R + LG nGθG + LMnMθM)
isM
=
I 2
vrG
˙
0
⎪
λ
i
(19)
⎪
r M
r M
⎪
⎪
⎪
⎩
JG ˙ ωG
+ BG
0
ωG
+
f λT
Jλ
sMG
rG
=
−τLG
JM ˙ ωM
0
BM
ωM
− f λT Jλ
−τ
sMG
r M
LM
λsMG corresponds to the total leakage flux of the two machines referred to the stators of the
machines.
LsMG represent the total leakage inductance.
with
⎧
⎡
⎤
⎪
⎪
R
⎪
rG I 2
0
0
⎪
⎪
⎪
⎣
⎦
⎨
R =
RrG I 2
( RsG + RsM) I 2 −RrMI 2
0
0
RrM I 2
⎡
⎤
⎡
⎤
⎪
⎪
⎪
−J J 0
0
0
0
⎪
⎪
⎪
⎣
⎦
⎣
⎦
⎩ LG = LMG
−J J 0
et LM = LMM
0
J
J
0
0
0
0
−J −J
with the positive parameters: a = RrGL− 1 , b = R
, c = R
, d = R
,
MG
rG L− 1
sMG
r M L− 1
MM
r M L− 1
sMG
e = ( RrG + RsG + RsM + RrM) L− 1 , f = L− 1 , and the following transformations: sMG
sMG
λrG = LmG eJnGθGλ
eJnGθG v
L
rG,
vrG = LmG
rG
rG
LrG
λrM = LmM eJnMθMλ
eJnGθG i
L
r M,
irG = LrG
rG
r M
LmG
3. Properties of the model
In this section, we derive some passivity and geometric properties of the model that will be
instrumental to carry out our controller design.
3.1 Passivity
An explicit power port representation of the DFIG interconnected to the IM is presented in
Fig.4
From Dynamic Modeling to Experimentation of
Induction Motor Powered by Doubly-Fed Induction Generator by Passivity-Based Control
119
vrG
irG
τ LG
− ∑
-w
G
G
i
isG
sM
vsM = vsG ∑
M
-τ
w
LM
M
Fig. 4. Explicit power port representation of the DFIG with IM.
Claim 1. The interconnection of the DFIG with the IM presented in the explicit power port
⎡
⎤
⎡
⎤
vrG
irG
representation in Fig.4 is a passive system 2 with the passive map ⎣
τ
⎦
⎣
⎦
LG
→
−ωG .
−τLM
ωM
⎡
⎤
⎡
⎤
vrG
irG
Proof. Consider the Fig. 4, Σ =
Σ G
⎣
⎦
⎣
⎦
Σ
is passive ⇒
τLG
→
−ωG
is passive ?
M
−τLM
ωM
For this purpose, we have to prove that
vT i
rG rG − τLGωG − τLMωM
≥ 0. We know that
each machine separately is passive (see [1])
Σ G is a passive system
⇔
vTrGirG − τLGωG + iTsMvsM ≥ 0
(20)
Σ M is a passive system
⇔
vTsGisG − τLMωM ≥ 0
(21)
where equation (1) has been used in (20) and (21). Let’s consider
d
vTrGirG − τLGωG + iTsMvsM ≥ 0
(22)
Using the energy conserving principle
iTsMvsM = − iT v
sG sG yields
d =
vTrGirG − τLGωG −
iTsGvsG
(23)
From (21) we have
− iTsGvsG ≤ − τLMωM
(24)
Finally (23) and (24) yields
vTrGirG − τLGωG − τLMωM ≥ d ≥ 0
(25)
Hence, the passivity of the DFIG interconnected to the IM is proven
2 Passive systems are defined here with no causality relation assumed among the port variables (13). This, more natural, definition is more suitable for applications where power flow (and not signal behaviour)
is the primary concern.
120
Electric Machines and Drives
4. Zero dynamics
For the IM speed control, we are interested in the internal behaviour of the system when the
motor torque τM is constant. In addition, for practical considerations, we are interested in the
control of the IM flux norm |λrM|, where | · | is the Euclidean norm.
For the study of the zero dynamics regarding these two outputs, we consider the DFIG-IM
model3 given by (18).
The control vrG is determined to obtain the desired equilibrium points of the DFIG-IM:
¨ θG = ¨ θM = 0,
˙ θG = ˙ θd =
=
G
Constant,
˙ θM = ˙ θdM Constant,
τLG = τLG 0 = Constant,
τLM = τLM 0 = Constant
λT λ
=
r M r M = β 2 d
Constant > 0
(26)
The IM mechanical dynamics show that the desired equilibrium points are obtained if:
τM = τd = τ
M
LM 0 = Constant
Hence:
f λT
=
sMG Jλr M = τM = τd
M
Constant
(27)
The equation (27) can also be expressed by replacing λsMG by its value given by the third line
of the electrical subsystem (18):
f
˙
T
λ
˙ θd
Jλ
d
r M + cλr M − nM M JλrM
r M = τM
Hence
f
˙ T
T
λ
˙ θd λT λ
˙ λ
˙ θd β 2 = τ
= cte
(28)
d
r M Jλr M − nM M r M r M
= fd rMJλrM − nM M
M = τd
M
The relative degrees of the outputs y 1 = β 2 = λT λ
Jλ
r M r M and y 2 = τM = f λT
sMG
r M, regarding
the input control vrG, are 2 and 1, respectively.
The zero dynamics of the mechanical subsystem (18) is stable, since the mechanical parameters
are positive.
Following on we will analyze