Modélisation et simulation des turbines éoliennes en vue du

Modélisation et simulation des turbines éoliennes
Masterand Florin Sebastian TUDOR
Professeur Dan STEFANOIU
Universite ”Politehnica” de Bucharest
Abstract
• The problem approached within this presentation is to
provide a useful and easy to use set of analytical
equations regarding the HAWTs dynamics, after being
expressed in input-output form, on automatic control
purposes.
• The equations are based here upon the causal series
of events that lead to the wind energy conversion.
• The HAWT model introduced next is a medium
complexity one, which tries to reach for a good tradeoff between operability and accuracy.
Contents
1. Introduction
2. Analytic and I/O modeling of a HAWT plant
a.
b.
c.
d.
The wind source model
The aerodynamics model
The mechanics model
The generator model
3. Simulation results
4. Conclusions
Introduction
• The research concerning modeling and control of renewable energy
production systems based on wind activity has known an impressive
development in the last years.
• The stimulus for the development of wind energy was the price of oil
and concern over limited fossil-fuel resources.
• Now, of course, the main driver for use of wind turbines to generate
electrical power is the very low CO2 emissions.
• Although there are exciting new developments, particularly in very
large wind turbines, and many challenges remain, there is a
considerable body of established knowledge concerning the science
and technology of wind turbines.
Introduction
General HAWT Model
Break
Propeller
Gear box
Slow
Fast
axis
axis
Electrical
generator
Hub
Coupling
Blade
Connection
to electrical
network
Pillar Nacelle
Gyration
mechanism
Introduction
A functional schemata
The wind source modeling
•
Stochastic time-frequency descriptions of wind generator and
filtering applied by the propeller.
•
Wind components:
– the average wind speed is a stepwise profile, with durations and heights
generated random, with normal distribution;
– stochastic disturbances.
•
Two phenomena are considered when generating the disturbances:
– Colored noise: von Karman adaptive filter
H N ( s) =
KvK (1 + bTvK s )
(1 + TvK s )(1 + aTvK s )
The wind source modeling
•
the impact between the propeller blades and the air flow is modeled
with two harmonic filters
–
–
the periodical effect of wind cutting is expressed by H1; H2 is used to estimate the part of the air flow that changes direction along the propeller plane. H h ,1 ( s) =
•
b1,0 + b1,1φs
1 + a1,1φs + a1,2 φ2 s 2
Thus the stochastic wind speed is
H h ,2 ( s ) =
b2,0 + b2,1φs
1 + a2,1φs + a2,2 φ2 s 2
The wind source modeling
Matlab / Simulink Diagram
The wind source modeling
•
Simulation results
Aerodynamics
•
One of the most interesting and realistic models is based on Blade
Element Momentum (BEM) Theory.
•
Two main hypotheses:
– blades are far enough to each other, so that the turbulent interferences can
be neglected;
– each blade can be decomposed into a number of elements (10 - 20) with
small length and approximately constant chord size.
•
After computing the local forces (lift and drag) applied by the wind
on each element, the overall forces are obtained by integration
(summation) along each blade.
Aerodynamics
BEM Theory
G
v
G
vr
∆r
φ/ 2
lc
G
tract Fss
Hub
⎛ v ⎞
ϕ ( r ) = arctan ⎜
⎟
⎝ ωss r ⎠
G
FL
inflow lift
FD sin ϕ JJJJG ϕ
ωss r
pitch β
π
−ϕ−β
2
attack
α=
G
FD
drag
FL sin ϕ
G
Ft thrust
HUB
FL cos ϕ
FD cos ϕ
wind
⎧
ρavr2 (r)
CL ( α(r)) lc (r)
⎪⎪FL (r) =
2
⎨
2
⎪F (r) = ρavr (r) C ( α(r)) l (r)
D
c
⎪⎩ D
2
⎧Ft ≡ FL sin ϕ − FD cos ϕ
⎨
⎩Fss ≡ FL cos ϕ + FD sin ϕ
Tss ( r ) = Fss ( r ) ⋅ r
⎧
∆Ft ,1 + ∆Ft , Ne ⎤
⎡ Ne
=
∆
∆
−
F
r
F
⎪ t
⎢∑ t , k
⎥
2
⎪
⎣ k =1
⎦
⎨
Ne
∆Ft ,1 + Ne ∆Ft , Ne ⎤
⎪
2⎡
⎥
⎪Tss = (∆r) ⎢∑k ∆Fss,k −
2
k
=
1
⎣
⎦
⎩
Aerodynamics
BEM Nonlinear Model
2
V
wind speed
4
dz/dt
Dyb
u
V2 (r)
rel
2
Q
5
d(zeta)/dt
u
Product1
-C-
2
-K-
Q*S
Lift Force F
L
l_c(r)
(vector)
rho/2
F * sin(φ)
L
1
om_ss
atan2
t
-K-
blade speed
r
Blade Length Elements
(vector)
Subtract1
Vectorial Gain
[dr/2 ; dr ; ... ; dr ; dr/2]
dr = R/Ne
Trigonometric
Function
Sum of
Elements1
2
Ft
F * cos(φ)
Product
L
C
3
beta(r)(rad)
-K-
F * cos(φ)
D
-C-
Bend Force
F (r)
φ(r)
β(r)
α(r)
L
Drag Force F
D
T (r)
ss
pi/2
const1
mod
lift polar
Add
C
2*pi
const
Math
Function
F * sin(φ)
D
D
cos
drag polar
sin
r
-KVectorial Gain
[dr/2 ; dr ; ... ; dr ; dr/2].
dr = R/Ne
Sum of
Elements2
Spin Moment
1
Tss
Mechanical Subsystem
•
Euler ‐ Lagrange equations:
d ⎛ ∂Ec
⎜
dt ⎜⎝ ∂q j
⎞ ∂Ec ∂E p ∂Ed
+
+
= Qj
⎟⎟ −
∂
∂
∂
q
q
q
j
j
j
⎠
q = [y t ζ θ r θ g ]
where generalized coordinates T
Q = [NFT NFT rb Tr − Tg ] generalized forces
T
•
After manipulations – two linear independent systems
x1 ≡ ⎣⎡ z ζ
⎧x i (t ) = Ai xi (t ) + Bi ui (t )
⎨
⎩yi (t ) = Ci xi (t )
•
T
z ζ ⎦⎤
x 2 ≡ [ θs
T
u1 ≡ Ft
u 2 ≡ ⎡⎣Tss
ω fs ⎤⎦
T
y1 ≡ ⎡⎣ z ζ ⎤⎦
y 2 ≡ ⎡⎣T fs
ωss ⎤⎦
Pitch actuator nonlinear model: H p (s) =
ωss ]
ω2p
β ∈ ⎡⎣ −2D ,30D ⎤⎦
s 2 + 2ζ p ω p s + ω2p
[ −10,10]
β∈
T
Electrical subsystem
•
Two types of generators have to be accounted:
– directly connected to the grid (with ON/OFF controller);
– indirectly connected to the grid through synchronizers and frequency controllers
(SCIG / DFIG).
•
In this paper: DFIG
•
Highly nonlinear model ! (induction generators…)
DFIG nonlinear model
•
Input:
{u
•
State: {i
•
Output: electro‐magnetic torque s ,d
s ,d
, u s , q , ur , d , u r , q }
, is , q , ir , d , ir ,q }
fs ≡ T fs − TG
JGω
TG ≡ 3 N p Lsr ( is , q ir , d − ir , q is , d ) / 2
•
DFIG is a 3‐phase machine i 3ϕ ≡ [i0
•
Park Transform: conversion to/from (d,q) coordinates
⎡⎣id
•
iq
T
0 ⎤⎦ ≡ i d ,q ≡ Pi 3ϕ
DFIG dynamics: iθ
i−θ ]
T
θ = 2π / 3
⎡cos ( ω fs t ) cos ( ω fs t − θ ) cos ( ω fs t + θ ) ⎤
⎥
2 ⎢⎢
P (t ) = sin ( ω fs t ) sin ( ω fs t − θ ) sin ( ω fs t + θ ) ⎥
3⎢
⎥
0.5
0.5
⎢ 0.5
⎥
⎣
⎦
⎧⎪x ≡ A ( ω fs , ωc ) x + Bu
⎨
⎪⎩ y ≡ 3 N p Lsr ( x2 x3 − x1 x4 ) / 2
Electrical subsystem
Simulink nonlinear model
2
3
om_fs
'
1
-K-
Tfs
ωfs
1/Jg
1
s
Integrator
T
ωc
Interpreted
MATLAB Fcn
P
Park's matrix
Interpreted
MATLAB Fcn
2
u_c
Saturation u_c
A *x
G
3
Saturation om_c
1
s
Product1
Matrix
Multiply
Integrator1
BG* u
sin
Gain1
DFIG_3U
Add
Park's Transform
om_c
G
Matrix
Multiply
x'
Clock
PG
ωfs
AG
Interpreted
MATLAB Fcn
Product
x
-K-
,
Gain
.
Add1
1
Saturation
TG
WINTUS at a glance
1
-30/pi
V
f(u)
Wind power
rot/min
3
8
om_ss
Tfs
2
Pw
1
1
u_c
z
Unit Delay
4
2
Tss
om_c
om_ss
V
Tss
V
T_f s
om_ss
Vm
Ft
beta(r)(rad)
Wind Generator
om_f s
dz/dt
3
beta_c
beta_c
Tf s
10
TG
TG
dz/dt
d(zeta)/dt
u_c
PG
Mechanics
Ft
beta
Pitch actuator
om_ss
Tss
-1
p
11
PG
d(zeta)/dt
om_c
Aerodynamics
om_f s
DFIG Model
5
6
7
d(zeta)/dt
dz/dt
Ft
-30/pi
rot/min.
9
om_fs
Simulation results
Wind speed
Wind spectrum
23
160
Measured wind speed
Mean wind speed
22
Wind speed [m/s]
SNR = 78.3527 dB
140
Spectral power of wind speed [dB]
22.5
21.5
21
20.5
120
100
80
60
40
20
0
20
-20
19.5
0
500
1000
1500
2000
2500
Time [s] (Ts = 0.2 s)
3000
3500
-40
4000
0
0.05
0.1
0.15
4
Slow shaft angular speed
14
22.4
22.38
0.2
0.25
0.3
0.35
Normalized Frequency
0.4
0.45
0.5
Slow shaft torque
x 10
12
22.36
10
22.32
Tss [Nm]
ωss [rot/min]
22.34
22.3
22.28
22.26
8
6
4
22.24
2
22.22
22.2
0
500
1000
1500
2000
Time [sec]
2500
3000
3500
0
0
500
1000
1500
2000
2500
Time [sec]
3000
3500
4000
-4
1
Fast shaft torque
Axial displacement speed of the nacelle
x 10
4000
3500
3000
T [Nm]
0
2000
fs
dz/dt [m/s]
2500
1500
1000
500
-1
0
500
1000
1500
2000
Time [sec]
2500
3000
3500
0
4000
0
500
1000
Thrust force
1500
2000
Time [sec]
2500
3000
3500
4000
3000
3500
4000
Generator torque
4000
5000
4000
3000
3000
2000
2000
T [Nm]
F [N]
1000
0
G
t
1000
-1000
0
-2000
-3000
-1000
-4000
-2000
-5000
0
500
1000
1500
2000
Time [sec]
2500
3000
3500
4000
0
500
1000
5
Fast shaft angular speed
5
900
1500
2000
Time [sec]
2500
Generated power
x 10
4
3
fs
G
P [W]
ω [rot/min]
895
2
1
890
0
885
0
500
1000
1500
2000
Time [sec]
2500
3000
3500
4000
-1
0
500
1000
1500
2000
Time [sec]
2500
3000
3500
4000
Conclusions
• We introduced WINTUS – a flexible and yet realistic simulator of usual HAWTs. • The design is modular and the analytical equations flow is based on the causality principle. The BEM Theory was successfully employed. • Although some functional features were not modeled (the azimuth orientation, the braking system or the gearbox), the simulator is sufficiently complex to allow connections to other systems, such as automatic controllers. Thank you ! ☺
Q & A ?
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