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Advances in Environmental Biology, 6(9): 2564-2573, 2012
ISSN 1995-0756
This is a refereed journal and all articles are professionally screened and reviewed
ORIGINAL ARTICLE
LQR Optimum Control for Wind Energy Conversion system
1
E. Nekooei, 2S.M. Mousavi Badjani, 2M.R. Alizadeh Pahlavani
1
Mapna Power Plants Construction & Development Co., Mapna-MD1, 15 Africa Ave., Tehran 1518614411,
Iran.
2
Malek-Ashtar University of Technology (MUT), Shabanlo St., Lavizan, Tehran, Iran.
E. Nekooei, S.M. Mousavi Badjani, M.R. Alizadeh Pahlavani; LQR Optimum Control for Wind
Energy Conversion system
ABSTRACT
In this paper, a grid-connected wind energy converter system including a Matrix Converter (MC), a
Squirrel-Cage Induction Generator (SCIG), a gear box and a horizontal axes wind turbine is proposed. An
overall dynamical model for the proposed wind turbine system is considered. The dynamic behaviour of the
model is investigated by simulating the response of the overall model to step changes in selected input variables.
Moreover, a linear model of the system is developed at a typical operating point, and a LQR optimum control is
designed for the system. The nonlinear and linear governing equations of the system are implemented using
MATLAB. The obtained results show that the settling time of closed loop system using LQR step response is
0.002 sec., steady state error is zero and overshoot is zero too, which is highly acceptable for performance goals
of this system.
Key words: Wind turbine, MC, LQR
Introduction
Wind is one of the most abundant renewable
sources of the energy in the nature. The economical
and environmental advantages offered by wind
energy are the most important reasons why electrical
systems based on wind energy are receiving
widespread global attention [1]. The SCIG is a lowcost and robust machine and the industry has a lot of
experience in dealing with it. In addition, the
induction generator has high reliability and the right
mechanical properties for wind turbines, such as slip
and a certain degree of overload capability. These are
the main reasons for choosing this type of generator
in the proposal system [2].
As a power electronic converter, a MC is used.
MC interfaces the SCIG with the grid and
implements shaft speed control. In the proposed
system, the MC input is connected to the grid and the
generator terminals are connected to the MC output.
MC is highly controllable and allows independent
control of the output voltage magnitude, frequency,
and phase angle, as well as input power factor.
In this paper first, MC dynamical model is
described. And in sec.IV dynamical model of wind
turbine system is proposed. Linearized model
described in sec.V and LQR design is proposed in
sec.VI and at last the simulation results are taken in
sec.VII.
Wind Energy Conversion System(WECS):
Wind energy conversion system composed of
wind turbine blades, a SCIG, a MC and
corresponding control system. Fig.1 shows the block
diagram of basic components of WECS. The
functional objective of this system is converting the
wind kinetic energy into electric power and injecting
this electric power into a utility grid.
Fig. 1: Block diagram of a WECS.
Corresponding Author
E. Nekooei, Mapna Power Plants Construction & Development Co., Mapna-MD1, 15 Africa Ave.,
Tehran 1518614411, Iran.
E-mail: [email protected]
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Adv. Environ. Biol., 6(9): 2564-2573, 2012
MC Structure and Dynamic Model:
The structure of a MC is shown in Fig.2 and the
parameters of the circuit are defined in Table1 [3-5].
The MC connects the three phase ac voltages on
input side to the three phase voltages on output side
by 3x3 matrixes of bi-directional switches.
Fig. 2: A WECS circuit including MC.
Table 1: Parameters of the MC circuit.
Parameters
Descriptions
Three phase MC input voltages and currents
Three phase MC output voltages and currents
Three phase input voltages sources
Three phase output voltages sources
Input source resistance per phase
Input source plus filter inductance per phase
Input filter capacitor per phase per phase
Output source resistance per phase
Output source inductance
Switch between phase and
For deriving of dynamical model first the switching
method should be chosen. Switching function is as
(1):
Where
is the on period of the switches in a
switching sequence time. The following two
restrictions apply to the duty cycle:
(1)
The (2) constraint applies to the switching function:
(2)
This constraint comes from this fact that MC is
supplied from a voltage source and generally feeds
an inductive load. Connecting more than one input
phase to one output phase causes a short circuit
between two input sources. Also, disconnecting all
input phases from an output phase causes an open
circuit. Therefore 27 allowable combinations from
512 states can exit. For transfer all quantity to one
side and consider the same frequency for all quantity,
we define a duty cycle as (3):
(3)
By considering the duty cycles of the switches,
the matrix D (modulation matrix) is as follows:
So, the output voltages and input currents of the
matrix converter can be written as:
(4)
Where
and
are a set of sinusoidal
input voltages and input currents. The transformation
matrix D should satisfy the following three
requirements:
(a) Restrictions on duty cycles
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(b) Sinusoidal output voltages with controllable
frequency, magnitude and phase angle.
(c) Sinusoidal input current,
(d) Desired input displacement power factor.
The following low- frequency transformation
matrices are two basic solutions that individually
satisfy the requirements (a), (b) and (c)
(5)
Where
: Output frequency of the MC, : output voltage
angle of the MC and is MC output to input voltage
gain. The combination of the two transformation
matrices
provide
controllability
of
input
displacement power factor.
(6)
In (6), is a constant parameter varying between 0
and 1.
All equations transferred to qdo frame because
of investigating of dynamical behaviour of the model
is simple. so, By KVL and KCL laws, the state space
qdo model of matrix converter system can be written
as Eq. (7):
(7)
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Dynamical Model of Wind Turbine System:
The power coefficient
analytically according to:
is
approximated
The mechanical power and torque on the wind
turbine shaft are given by 8, 9 equations respectively
[6-17]:
(11)
(8)
(9)
Where:
: Mechanical power extracted from turbine rotor,
: Mechanical torque extracted from turbine rotor,
: Area covered by the rotor,
: Performance coefficient,
: Air density,
: Tip speed ratio,
: Rotor blade pitch angle,
: Angular speed of turbine shaft,
: Wind velocity,
: Turbine rotor radius.
The power coefficient is related to tip speed
ration and blade pitch angle and changes with
different values of the pitch angle, but the efficiency
and also for maximum power
is obtained for
point tracking. In this paper, it is assumed that the
rotor pitch angle is fixed and equal to zero.
The mechanical model of turbine can be
considered as Fig.3 the driven train converts the
aerodynamic torque on the low-speed shaft to the
torque on the high-speed shaft. The dynamic of the
driven train are described by the following three
differential equations:
(12)
(13)
The blade tip speed ratio is defined as Eq. (10):
(10)
(14)
In the above equations:
And
Fig. 3: Mechanical model of wind turbine shaft
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A commonly-used induction machine model is based
on the flux linkages. The dynamic equivalent circuit
of the induction machine in qdo frame is illustrated
in Fig.4
Fig. 4: qdo equivalent circuit of an induction machine.
In order to model the gearbox, it is only needed to
consider that the gearbox torque can simply be
transferred to the low speed shaft by a multiplication.
So, for gearbox of Fig.1, one can be write:
(15)
So, the overall model of the system in state space qdo
frame can be written as follow:
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
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Where:
,
,
Linearized model performance evaluation:
Table2. Simulation results are shown in simulation
results section.
In order to evaluate the linearization error at an
operating point, the linearized model responses to
step changes in some inputs are compared with those
of nonlinear model. The operating point is shown in
LQR Design:
First, an integrator is added to the wind turbine
system, the augmented system is shown in Fig. 5
Table 2: State variables at the operating point for linearization.
-8.45
2.5
1632.3
-2.16
26.45
109.5
1591.1
378.5
0.004
9.45
1632.5
Fig. 5: Augmented system.
The wind turbine model has the input vector
, the state vector contains eleven
, the output
variables, the output vector is
and the tracing error
reference is
defined as:
, so, the state space model of
the augmented system as follows:
(29)
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Fig. 6 shows the closed loop system with
regulator state-feedback. The closed loop state space
model is as Eq. 30:
Fig. 6: Closed loop system with state feedback.
(30)
The aim of the controller design is to determine
the gain of the controller, to make the augmented
system stable and result in a proper
undershoot/overshoot and settling time of the closed
loop time domain responses.
The cost function used in the design of an LQR
controller, is defined as:
(31)
~ ~
~
For the closed loop feedback system if ( A, B ) is controllable and (Q, A) is observable, then the control
gain can be calculated from the formula:
(32)
Where P is a unique symmetric positive semi- definite solution of the algebraic riccati equation:
(33)
(34)
Simulation Results:
Parameters of the system under study are shown
in Table 3. Fig.7, 8 shows the system response to a
step decrease of 1 HZ in the MC output frequency
Table 3: Parameters of the system under study.
Grid
MC
1
for linearized and nonlinear models. In the next
simulation the system response to a 10% step
changes in wind velocity for linearized and nonlinear
model are shown.
Wind turbine
Induction Generator
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Active Output Pow er
[p.u]
3
Nonlinear
Linear
2
1
0
0.2
0.4
0.6
0.8
Time[sec]
1
1.2
Fig. 7: Active output power response to 1 frequency decrease in 0.3 sec.
Reactive Output pow er
[p.u]
1
Nonlinear
Linear
0.5
0
0.2
1.2
1
0.8
0.6
Time[sec]
0.4
Fig. 8: Reactive output power response to 1 frequency decrease in 0.3sec.
As Fig.7, 8 shows with 1 frequency decrease, we have approximately 2 p.u increase in active and reactive
power.
Active Output Pow er
[p.u]
1
Nonlinear
Linear
0.8
0.6
0.4
0.4
0.6
0.8
Time[sec]
1
1.2
Reactive Output Pow er
[p.u]
0.42
Nonlinear
Linear
0.4
0.38
0.2
0.4
0.6
0.8
Time[sec]
1
1.2
Fig. 9: Active and reactive power response to 10% decrease of wind velocity in 0.5 sec.
And also Fig.10 shows active power response to 50% decrease in parameter a.
Active Output Pow er
[p.u]
2
1
Nonlinear
0
Linear
-1
0.4
0.6
0.8
Time[sec]
1
Fig.10: Active power response to 50% decrease of parameter a in 0.7 sec.
1.2
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The simulation results reveal that the linear model is
an accurate approximation of the nonlinear model.
By tuning Q, R, we can achieve an appropriate
controller for the system. For this system Q and R
matrix and also the active and reactive power step
responses are as follows:
Step Response of Active and Reactive Powers
[p.u]
1.5
1
0.5
0
0
0.05
0.1
0.15
0.2
0.15
0.2
Time[sec]
30
[p.u]
20
10
0
-10
0
0.05
0.1
Time[sec]
Fig. 11: System (linearized model) with LQR controller: step response of the active and reactive powers.
As Fig.11 shows the settling time is less than
0.002sec.
2.
Conclusions
In this paper, an overall dynamical model of
wind energy conversion system is proposed. A linear
model of it is derived and evaluation of this linear
model investigated with nonlinear model. A closed
loop feedback controller, based on the linear model,
is designed. The open loop system is locally stable
around the operating point, and the controller design
objective is to track a class of desired active and
reactive powers delivered to the grid. A state
feedback controller is adopted for the WECS.
In the state feedback controller, since the goal is
to achieve zero steady state error for the family of
step reference signals, first an integrator is added
after the system. The combination of the system with
the integrator forms the augmented system. Then, the
state feedback controller is designed based on the
LQR method.
3.
4.
5.
6.
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