Power Flow Control of a Doubly-fed
Induction Machine Coupled to a Flywheel
Carles Batlle, Arnau Dòria-Cerezo
Universitat Politècnica de Catalunya
Romeo Ortega
LSS - Supélec
Power Flow Control of a Doubly-fed Induction Machine Coupled to a Flywheel – p. 1/14
System
System DFIM+Flywheel
in
Goals:
To supply the required
power to the load when
it demands more than
the grid can give, and
to store mechanical
energy in the flywheel
when the load doesn’t
need all the grid power,
with a high power factor.
il
vs
isr
Power
Network
Local
load
Ideal transformer
is
ω
ir
Control
inputs
B2B
converter
Flywheel
DFIM
Local source
Power Flow Control of a Doubly-fed Induction Machine Coupled to a Flywheel – p. 2/14
System
System DFIM+Flywheel
ir
B2B converter → ideal controllable
voltage source.
Control inputs vr
is
+
+
DFIM
vr
-
-
System outputs
Generator Pn and Qn
Motor ω and Qn
vs
in
il
+
Zl
vn
-
Br
τe ω
Jm
Power Flow Control of a Doubly-fed Induction Machine Coupled to a Flywheel – p. 3/14
Port-controlled Hamiltonian Systems
General Description

 ż = (J (z) − R(z))(∇H(z))T + g(z)u

y = g T (z)(∇H(z))T
where
z are the Hamiltonian variables,
H(z) is the Hamiltonian function (or energy function),
J = −J T is the interconnection matrix,
R = RT ≥ 0 is the dissipation matrix,
g is the connection matrix with the power ports and
u, y are the port variables (uy is power).
Power Flow Control of a Doubly-fed Induction Machine Coupled to a Flywheel – p. 4/14
Port-controlled Hamiltonian Systems
PCHS DFIM Description
Using dq-transformations
z
T

→
= [λs , λr , Jm ω] ∈ R
−ωs Ls J2

J (z) = 
 −ωs Lsr J2
O1×2
ż = (J (z) − R(z))(∇H(z))T + g(z)u
5
u
T
= [vs , vr ] ∈ R
−ωs Lsr J2
O2×1
−(ωs − ω)Lr J2
Lsr J2 is
Lsr iT
s J2
0

J2 = 
0
1
H=
−1
0



4

g=







I2 = 

R=

1
0
0
1
I2
O2
O2
I2
O1×2
O1×2
Rs I2
O2




O2×1
O2
Rr I2
O2×1
O1×2
O1×2
Br






1 2
1 T −1
z
ze L ze +
2
2Jm m
Power Flow Control of a Doubly-fed Induction Machine Coupled to a Flywheel – p. 5/14
PCHS Stability
Energy function bounded from below (H(z) > c), then
Ḣ(z) = ∇H(z)ż = ∇H(z)(J (z) − R(z))(∇H(z))T + ∇H(z)g(z)u
with J (z) = −J (z)T and y = g(z)T ∇H(z), yields
Ḣ(z) = −∇H(z)R(z)(∇H(z))T + uy
with R(z) = R(z)T ≥ 0 and if u = 0, the system is asymptotically stable,
Ḣ(z) ≤ 0.
Hamiltonian function H(z) is a Lyapunov function.
Power Flow Control of a Doubly-fed Induction Machine Coupled to a Flywheel – p. 6/14
IDA-PBC
Interconnection and Damping Assignment Passivity-based Control(R. Ortega et al., Automatica, 2002)
Interconnection control: ΣC + ΣI + Σ = PCHS
fC
ΣC
ΣC Controller
ΣI Interconnection
Σ System
f
+
eC
-
ΣI
+
e
-
Σ
system behaves as
ż = (Jd − Rd )(∇Hd )T
Hd (z) > c has a local minimum at z ∗
Jd = −JdT and Rd = RTd ≥ 0.
We have to solve the matching equation
(J − R)(∇H)T + g(z)u = (Jd − Rd )(∇Hd )T
Power Flow Control of a Doubly-fed Induction Machine Coupled to a Flywheel – p. 7/14
DFIM IDA-PBC
Hd = 12 (ze − ze∗ )T L−1 (ze − ze∗ ) +
We choose
Jd = J + Ja
with

O2
Ja =  O2
O1×2
O2
O2
T
Jrm

O2×1
−Jrm 
0
1
2Jm (zm
∗ 2
− zm
)
Rd = R + Ra

O2
O2
rI2
Ra =  O2
O1×2 O1×2

O2×1
O2×1 
0
The control law
vr = vr∗ − (ω − ω ∗ )(Lr J2 i∗r + Jrm ) − Lsr ω ∗ J2 (is − i∗s ) − rI2 (ir − i∗r )
Jrm =
)
(ir −i∗
Lsr |ir −i∗r|2 (is
r
− i∗s )T J2 i∗r
Singularity of Jrm at z ∗ !!!
Power Flow Control of a Doubly-fed Induction Machine Coupled to a Flywheel – p. 8/14
DFIM IDA-PBC

O2
Ra =  O2
O1×2
O2
rI2
O1×2

O2×1
O2×1 
ξ(z)
1
∗
(τ
− τe (ze ))
ξ(z) =
e
ω − ω∗
control law yields
vr = vr∗ − (ω − ω ∗ )(Lr J2 i∗r + Jrm ) − Lsr ω ∗ J2 (is − i∗s ) − rI2 (ir − i∗r )
Jrm = Lsr J2 is
Power Flow Control of a Doubly-fed Induction Machine Coupled to a Flywheel – p. 9/14
DFIM IDA-PBC
Stability
Dynamical system ż = (Jd − Rd )(∇Hd )T ,


Jd − Rd = 
−ωs Ls J2 − Rs I2
−ωs Lsr J2
O2×1
−ωs Lsr J2
−(ωs − ω)Lr J2 − (Rr + r)I2
O2×1
O1×2
O1×2
−Br − ξ(z)



can be seen as
Electrical Sist.

 że = f1 (ze , zm ) = . . . = (J − R)(∇H)T ⇒ Stable!!!
Mechanical Sist.  żm = −Bzm + h(ze )
where ze ∈ R4 , zm ∈ R, B > 0 and h is a continuous function. We assume that the
∗
system has the equilibrium points ze∗ , zm
where limt→+∞ ze (t) = ze∗ , ∀zm (t). Then
∗
.
limt→+∞ zm (t) = zm
Power Flow Control of a Doubly-fed Induction Machine Coupled to a Flywheel – p. 10/14
Energy Management
DFIM Power flow study
∗
∗
∗
∗T ∗
Pr ≡ i∗T
v
=
B
ω
(ω
−
ω
)
+
R
i
r
s
r
r
r
r ir
∗
∗
∗
Qr ≡ i∗T
r J2 vr = (ω − ωs )f (Qs , ω )
Optimal velocity ω ∗ = ωs ⇒ Pr ∼ 0 and Qr = 0.
Control Action
Pn∗ < Pl
True
True
False
False
|ω − ωs | < ǫ
True
False
True
False
Mode
Generator
Generator
Stand-by
Storage
References
Pn∗ = PnM AX and Q∗n = 0
Pn∗ = PnM AX and Q∗n = 0
ω ∗ = ωs and Q∗n = 0
Pn∗ = PnM AX and Q∗n = 0
Power Flow Control of a Doubly-fed Induction Machine Coupled to a Flywheel – p. 11/14
Simulations - Resistive Load - R
Stator voltage (Vsa ) and network current (ina )
Network (Pn ) and load (Pl ) active powers
4
Stator voltage and network current (R−load)
Network and Load Active Power (R−load)
x 10
3
300
2.5
200
2
100
Vna[V],ina[A]
Pn, Pl[W]
Pn
Pl
1.5
vsa
ina
0
−100
1
−200
0.5
−300
0
0
0.5
1
1.5
2
2.5
time[s]
3
3.5
4
4.5
5
1
1.05
1.1
1.15
time[s]
Angular speed (ω )
Angular speed (R−load)
320
315
w[rad/s]
310
305
300
295
290
0
0.5
1
1.5
2
2.5
time[s]
3
3.5
4
4.5
5
Power Flow Control of a Doubly-fed Induction Machine Coupled to a Flywheel – p. 12/14
Simulations - Inductive Load - RL
Network (Pn ) and load (Pl ) active powers
4
Network and Load Active Power (RL−load)
x 10
3
Stator voltage (Vsa ) and network (ina ) and
load current (ila )
Pn
Pl
Stator voltage and network and load currents (RL−load)
300
vsa
ina
ila
2.5
200
100
1.5
vsa[V],ina[A],ila[A]
Pn, Pl[W]
2
1
0
−100
0.5
−200
0
5
5.5
6
6.5
7
7.5
time[s]
8
8.5
9
9.5
10
−300
6
Angular speed (ω )
6.05
6.1
6.15
time[s]
Angular speed (RL−load)
320
315
w[rad/s]
310
305
300
295
290
5
5.5
6
6.5
7
7.5
time[s]
8
8.5
9
9.5
10
Power Flow Control of a Doubly-fed Induction Machine Coupled to a Flywheel – p. 13/14
Conclusions
Conclusions
IDA-PBC technique for a DFIM.
The system provides the active power required by the load, and
compensates the reactive power of the whole system.
Work in progress
More robust control version.
Experimental validation.
Power Flow Control of a Doubly-fed Induction Machine Coupled to a Flywheel – p. 14/14