energies
Article
A Flexible Maximum Power Point Tracking Control
Strategy Considering Both Conversion Efficiency and
Power Fluctuation for Large-inertia Wind Turbines
Hongmin Meng *
, Tingting Yang, Ji-zhen Liu and Zhongwei Lin
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources,
North China Electric Power University, Beijing 102206, China; [email protected] (T.Y.);
[email protected] (J.-z.L.); [email protected] (Z.L.)
* Correspondence: [email protected]; Tel.: +86-010-61772965
Academic Editor: Frede Blaabjerg
Received: 13 June 2017; Accepted: 4 July 2017; Published: 6 July 2017
Abstract: In wind turbine control, maximum power point tracking (MPPT) control is the main control
mode for partial-load regimes. Efficiency potentiation of energy conversion and power smoothing are
both two important control objectives in partial-load regime. However, on the one hand, low power
fluctuation signifies inefficiency of energy conversion. On the other hand, enhancing efficiency may
increase output power fluctuation as well. Thus the two objectives are contradictory and difficult
to balance. This paper proposes a flexible MPPT control framework to improve the performance of
both conversion efficiency and power smoothing, by adaptively compensating the torque reference
value. The compensation was determined by a proposed model predictive control (MPC) method
with dynamic weights in the cost function, which improved control performance. The computational
burden of the MPC solver was reduced by transforming the cost function representation. Theoretical
analysis proved the good stability and robustness. Simulation results showed that the proposed
method not only kept efficiency at a high level, but also reduced power fluctuations as much as
possible. Therefore, the proposed method could improve wind farm profits and power grid reliability.
Keywords: wind turbine; maximum power point tracking; MPPT; conversion efficiency;
power smoothing; power fluctuation; multi-objective control; cost function
1. Introduction
With more and more attention being paid to environmental problems, wind energy has become
one of the most important renewable energy sources to develop and utilize [1,2]. Although wind
turbine control techniques have been developed for several decades, there are still many tough control
problems to be solved. Generally, wind turbine control is divided into two parts: partial-load regime
(below rated wind speed) and full-load regime (above rated wind speed). In full-load regime, power
smoothing is the main control objective, and it is unnecessary to consider energy conversion efficiency.
In partial-load regime, maximum power point tracking (MPPT) is the main operation mode [3,4]. The
control objectives in partial-load regime consist of achieving high wind energy conversion efficiency,
smoothing output power, diminishing mechanical loads, riding through fault conditions, etc. In these
objectives, efficiency potentiation and power smoothing have been studied in-depth separately [2,5,6].
However, on one hand, with the increase of turbine inertia, enhancing conversion efficiency leads to
higher fluctuations of generator power. On the other hand, power smoothing without energy storage
devices may cause efficiency loss. Although many power smoothing methods based on storage devices
are able to reduce this contradiction, the cost of energy storage devices is expensive. Therefore, it is
Energies 2017, 10, 939; doi:10.3390/en10070939
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Energies 2017, 10, 939
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essential to balance conversion efficiency and power smoothing in partial-load regime without energy
storage devices.
Balancing conversion efficiency and power smoothing is multi-objective optimization (MOO)
problem. A novel MPPT framework employing fuzzy inference system (FIS) was proposed in [3] to
consider these multiple objectives. The captured energy efficiency is enhanced, and power fluctuation
is suppressed as much as possible, but FIS requires a priori artificial experience, and it is complex
to establish fuzzy memberships. Other advanced control algorithms, such as adaptive control [7],
robust control [8] and nonlinear control [9], are also employed in wind energy conversion systems
(WECSs). Reference [7] proposed an adaptive MPPT control based on a network-based reinforcement
learning method to deal with the changes of operating environment. A robust nonlinear control is
presented in [8] to maximize energy conversion which allows maximization without exacting wind
turbine model knowledge. In [9], the dynamics aspects of the wind and aeroturbine are taken into
consideration by proposing nonlinear static and dynamic state feedback controllers. Although the
aforementioned advanced control methods are effective for MPPT control, they cannot handle the
problems with linear or nonlinear constraints. Model predictive control (MPC) is widely adopted in
engineering for multi-objective problems with linear or nonlinear constraints. It has been employed in
WECS for generator control [10,11], power converter control [12,13], wind-battery hybrid control [14]
and other multi-objective control [15]. Mechanical load and conversion efficiency are considered
in [16,17]. In [18,19], MPC is used to maximize energy conversion, mitigate drive train loads in both
partial-load and full-load regime. However, power smoothing is only considered in full-load regime.
In fact, it is essential to suppress extra fluctuations of generator power in partial-load regime. The
stability of MPC with receding horizon is guaranteed by employing a terminal quality constraint [20].
The modeling of local linearization state-space equations is important for MPC design. Mostafa [18]
presented a MPC method where the linear models are switched according to wind speed condition.
A fuzzy model- based multivariable predictive control approach is also proposed in [21] to improve
control performance. In the state-space models for the studies above, the wind speed within the
control horizon is often regarded as constant, and wind variations are omitted in the optimization
solving process. However, power fluctuations are closely related to the wind variations in current
control horizon, and reducing fluctuations requires taking wind speed variations into account. The
weights in the MPC value function play a key role in the trade-off between the multiple objectives.
Flexible weight tuning strategies can help improve control performances. A tuning approach based on
the computation of sensitivity tables is proposed for tradeoff in [22]. In [23], weights are adjusted in
response to the variable wind conditions and operational requirements by classifying different wind
speed scenarios.
In this paper, a balancing strategy for large-inertia WECSs is proposed based on a flexible
MPPT control framework employing a model predictive method. Energy conversion efficiency and
power smoothing are both considered under partial-load regime. Firstly, the relationship between
conversion efficiency, generator power smoothing and turbine inertia is analyzed. Conversion
efficiency potentiation may increase the generator power fluctuation. For a large inertia turbine,
increasing the same conversion efficiency brings much larger fluctuation than for a small inertia turbine.
Thus, it is necessary to balance and alleviate extra power fluctuation at no or very low conversion
efficiency cost. Secondly, a new control framework based on classical optimal torque control (OTC)
is presented to compensate the torque reference in order to regulate the dynamic performance of the
system by adjusting the compensation gain. For large-inertia wind turbines, different compensation
gains may lead to different conversion efficiencies, as well as power fluctuations. Then, thirdly,
to determine the compensation gain, the MPC method is proposed to solve the optimization problem
considering both conversion efficiency and power smoothing. The computational burden to solve the
optimization problem is reduced by transforming the cost function representation. Besides, dynamic
weights in the cost function are presented to improve the control performance. Benefiting from
the proposed control framework, the stability of the control system is guaranteed without any
Energies 2017, 10, 939
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and the stability is kept despite uncertain parameters and unmodeled dynamics in the MPC. The
Energies 2017, 10, 939
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proposed method not only maintains the energy conversion efficiency at a high level, but also
reduces power fluctuations as much as possible. Therefore, it can increase the economic profits of
terminal constraints, which are usually adopted in traditional open-loop MPC control methods.
wind farm operators and improve the reliability of power systems.
In addition, the proposed control has good robust performance, and the stability is kept despite
Theuncertain
remainder
of thisand
paper
is organized
asinfollows:
2 presents
model
descriptions
parameters
unmodeled
dynamics
the MPC.Section
The proposed
methodbrief
not only
maintains
used fortheour
state-space
and
Section
the relationship
energy
conversionmodel
efficiency
at asimulations.
high level, butInalso
reduces3,power
fluctuations as between
much as power
possible.
Therefore,
it
can
increase
the
economic
profits
of
wind
farm
operators
and
improve
the method
fluctuations, conversion efficiency and turbine inertia is analyzed. Details of the proposed
reliability
power systems.
are discussed
inof Section
4, including the proposed control framework, MPC implementation,
The remainder of this paper is organized as follows: Section 2 presents brief model descriptions
dynamic weights and stability and robustness analysis. Finally, simulation studies are carried out to
used for our state-space model and simulations. In Section 3, the relationship between power
verify itsfluctuations,
effectiveness
in Section
5. and
Section
presents
the conclusions
summarize
conversion
efficiency
turbine6 inertia
is analyzed.
Details of thethat
proposed
method arethe main
contributions
of this
paper.
discussed
in Section
4, including the proposed control framework, MPC implementation, dynamic
2. Brief
weights and stability and robustness analysis. Finally, simulation studies are carried out to verify its
effectiveness
in Section 5. Section 6 presents the conclusions that summarize the main contributions of
Model
Description
this paper.
2.1. Overall
WECS
Model
2. Brief
Model
Description
Figure
1 outlines
a Model
simplified WECS. The main components of a WECS consist of wind turbine
2.1. Overall
WECS
system (wind turbine model), transmission system (drive train model) and electrical system
Figure 1 outlines a simplified WECS. The main components of a WECS consist of wind
(generator
model).
windmodel),
turbine
converts system
wind (drive
power
into
captured
mechanical
turbine
systemFirstly,
(wind turbine
transmission
train
model)
and electrical
system power.
Secondly,
the transmission
system
the mechanical
captured
by thepower.
rotor to the
(generator
model). Firstly,
windtransmits
turbine converts
wind power power
into captured
mechanical
Secondly,
the transmission
system
transmitsisthe
mechanical
power
captured bysystem
the rotortotoincrease
the
generator
in the electrical
system.
A gearbox
included
in the
transmission
the
generator
in
the
electrical
system.
A
gearbox
is
included
in
the
transmission
system
to
increase
the
rotor speed to values more suitable for driving the generator. Note that gear box will be removed in
rotor speed to values more suitable for driving the generator. Note that gear box will be removed in a
a direct-driven
wind turbine. Thirdly, the generator transforms mechanical power into electrical
direct-driven wind turbine. Thirdly, the generator transforms mechanical power into electrical power.
power. Its
electrical
terminals
are connected
to system.
the power
system.
Note thatdevices,
power
electronics
Its electrical
terminals
are connected
to the power
Note that
power electronics
which
devices, are
which
areused
usually
usedgenerator
to control
generator
output
power,inare
not 1.illustrated in Figure 1.
usually
to control
output
power, are
not illustrated
Figure
Figure
1. Overall
schematicof
ofaa wind
wind energy
conversion
system.
Figure
1. Overall
schematic
energy
conversion
system.
2.2. Effective Wind Speed Model
2.2. Effective Wind Speed Model
Since effective wind speed contains the rotational sampling effect of the wind turbine, it can be
Since
effective
wind assessment
speed contains
the rotational
effect
the wind
used
for a thorough
of the designed
controllersampling
performance.
Theof
effective
windturbine,
speed canit can be
as aassessment
sum of two components
[18,24]: controller performance. The effective wind speed
used forbea modeled
thorough
of the designed
can be modeled as a sum of two components [18,24]:
Vw = Vmean + Vturb
Vw  Vmean  Vturb
(1)
(1)
where Vmean is an average wind speed component that varies slowly in the long-term time domain.
Vturb is a turbulent component that changes rapidly. The turbulent component is obtained by passing
white noise through three cascaded filters [25], HF(w), HSF(w) and HRSF(w). The first filter HF(w) is
Energies 2017, 10, 939
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where Vmean is an average wind speed component that varies slowly in the long-term time domain. Vturb
is a turbulent component that changes rapidly. The turbulent component is obtained by passing white
noise through three cascaded filters [25], HF (w), HSF (w) and HRSF (w). The first filter HF (w) is used to
obtain
turbulence
by approximating the von Karman spectrum or the Kaimal spectrum. The filters
Energiesthe
2017,
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HSF (w) and HRSF (w) are included to take account of the rotational sampling effect on the turbulence
sampling,
andfilter
amplifies
with
frequencies
close attenuates
to triple thethe
turbine
speed (3P
derived
from
HF (w).those
Thecomponents
filter HSF (w),
called
spatial filter,
high-frequency
frequency). of
Details
of effectiveThe
wind
speed
caneffect
be found
in [25]. sampling, and amplifies
components
the turbulence.
filter
HRSFcalculation
(w) models the
of rotational
those components with frequencies close to triple the turbine speed (3P frequency). Details of effective
2.3. Wind
Turbine
Modelcan be found in [25].
wind
speed
calculation
Wind energy is captured by the wind turbine, and the captured mechanical power Pwt can be
2.3. Wind Turbine Model
written as:
Wind energy is captured by the wind turbine, 2and
the captured mechanical power Pwt can be
Pwt  0.5  Rw Vw3 C p (  ,  )
(2)
written as:
2 3
Pwtradius
= 0.5ρπR
β)
(2)a
w V
w C p ( λ, rotor
where  is the air density, Rw is the
of the
turbine
and Vw is the wind speed. Cp is
powerρcoefficient
that denotes
theradius
wind of
energy
conversion
It iswind
the function
where
is the air density,
Rw is the
the turbine
rotor efficiency.
and Vw is the
speed. Cpofistip-speed
a power
ratio   that
pitch
angle
β. r conversion
is the rotorefficiency.
speed.
R / Vw andthe
coefficient
wind
energy
It is the function of tip-speed ratio
r w denotes
λ = ωr Rw /Vw and pitch angle β. ωr is the rotor speed.
2.4. Drive Train Model
2.4. Drive Train Model
The drive train model illustrated in Figure 1 can be equivalent to a single-mass shaft model
The drive
train turbine
model illustrated
in Figureside,
1 can
equivalent
to 2.
a single-mass
shaft
model
coupling
two parts:
side and generator
as be
shown
in Figure
Inertia J is the
equivalent
coupling
two
parts:
turbine
side
and
generator
side,
as
shown
in
Figure
2.
Inertia
J
is
the
equivalent
inertia of single mass that including both turbine rotor and generator rotor inertia. Ngear is the gear
inertia
massspeed,
that including
turbine
and on
generator
rotorof
inertia.
Ngear is the gear
ratio,
ratio, of
is rotor
Tm andboth
Te are
the rotor
torques
each side
the transmission
system.
rsingle
ωr is rotor speed, Tm and Te are the torques on each side of the transmission system. The equivalent
The equivalent values of inertia and generator torque need to take gear ratio into account.
values of inertia and generator torque need to take gear ratio into account.
N gear  Te
r
Tm
Figure
Figure2.2.One-mass
One-massdrive
drivetrain
trainmodel.
model.
Accordingto
toNewton's
Newton'ssecond
secondlaw
lawof
ofrotation,
rotation,the
thedrive
drivetrain
trainmodel
modelcan
canbe
beexpressed
expressedas:
as:
According
. J  T  N gear  Te
J ω r =r Tmm− Ngear
· Te
(3)
(3)
where J is the equivalence of the entire inertia, Ngear is the gear ratio, Tm is the mechanical torque, Te is
where
J is the equivalence
inertia,
Ngear is the
gearturbine
ratio, Trotor.
mechanical
torque,
Te
m is the
the generator
torque,  r. of
is the
the entire
angular
acceleration
of the
For
a direct-drive
wind
is the generator torque, ω r is the angular acceleration of the turbine rotor. For a direct-drive wind
turbine, Ngear = 1.
turbine, Ngear = 1.
2.5. Generator
GeneratorModel
Model
2.5.
Todecouple
decoupleactive
activeand
andreactive
reactivepower
power
control,
a d-q
synchronous
reference
frame
is employed
To
control,
a d-q
synchronous
reference
frame
is employed
to
to
express
the
voltage
and
flux
equations:
express the voltage and flux equations:

d ds
uds  rs ids  s qs 
dt

d qs

uqs  rs iqs  s ds 
dt


d
dr
u  r i  s  
s qr
 dr r dr
dt

d
(4)
Energies 2017, 10, 939
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
dψ

uds = −rs ids − ωs ψqs + dtds



Energies 2017, 10, 939
5 of 19
 u = −r i + ω ψ + dψqs
qs
s qs
s ds
dt
(4)
dψ

rr idr − sωs ψqr + dtdr

 udrds =
L
L
i
L
i







s
m
ds
m dr
dψ
 
uqr = rr iqr + sωs ψdr + dtqr
    Ls  Lm  iqs  Lm iqr
  qs
(5)

ψds = −( Lsσ + Lm )ids + Lm idr

  Lr  Lm  idr  Lm ids

 ψ dr=−(
Lsσ + Lm )iqs + Lm iqr
qs
(5)



 Lmm)iiqrdr+LmLimqsids


ψdr qr= −(LLrσr +



ψqr = −( Lrσ + Lm )iqr + Lm iqs
where s is the synchronous speed, u is the voltage, i is the current, r is the resistance,  is the
where ωs is the synchronous speed, u is the voltage, i is the current, r is the resistance, ψ is the flux,
flux, Lm is the mutual inductance, L is the leakage inductance. Subscripts d, q denote the d-axis
Lm is the mutual inductance, Lσ is the leakage inductance. Subscripts d, q denote the d-axis and q-axis,
and q-axis, respectively.
s and
denote thestator
generator
statorrespectively.
and rotor, respectively.
respectively.
Subscripts sSubscripts
and r denote
thergenerator
and rotor,
Turbine Characteristic
Characteristic
3. Analysis of Wind Turbine
the relation
relation between
between captured
captured
rr ,, the
Multiplying both sides of Equation (3) by the rotor speed ω
mechanical
and output
output electric
electric power
power P
Pee is
is expressed
expressed as:
as:
mechanical power
power P
Pwt
wt and
 Pwt −
P
e Pwt
PeP=
Protor
rotor
(6)
(6)
where Protor =J r . r . According to Equation (6), the captured Pwt is divided into two parts, most of the
where Protor
= J ω r ωr . According to Equation (6), the captured Pwt is divided into two parts, most
power
is converted
into into
the generator
power
Pe, Pthen
higher
variations
the
of
the power
is converted
the generator
power
higher
variationsofofPPwtwtresult
result in
in the
e , then
fluctuations
of
electric
power
P
e
.
The
rest
of
P
rotor
is
stored
as
rotational
kinetic
energy.
It
can
be
fluctuations of electric power Pe . The rest of Protor is stored as rotational kinetic energy. It can be
.
regulated by
bycontrolling
controlling
rotor
speed
and angular
acceleration
of ωthe
rotor  r in
regulated
thethe
rotor
speed
ωr andrangular
acceleration
of the rotor
r in coordination
to
reduce the to
fluctuation
of P
the
efficiency
and
Thus P
rotorcapability
supplies to
thebalance
capability
to balance
coordination
reduce the
fluctuation
of PeP[26,27].
e [26,27]. Thus
rotor supplies
power
smoothing.
efficiency
and power smoothing.
According to Equation(6), enhancing the efficiency may increase the fluctuations of the output
output
power.
thethe
relationship
between
turbine
inertia,inertia,
power fluctuations
and conversion
efficiency,
power. To
Tostudy
study
relationship
between
turbine
power fluctuations
and conversion
studies
are carried
on out
the MPPT
in [5], which
is ainflexible
way toischange
conversion
efficiency,
studies out
arebased
carried
based strategy
on the MPPT
strategy
[5], which
a flexible
way to
efficiency
by
tuning
a
single
control
parameter.
The
first
case
is
to
investigate
the
variation
trend
change conversion efficiency by tuning a single control parameter. The first case is to investigate
the
of
power trend
fluctuations
as the
conversion
is getting
higheris under
sameunder
turbine
variation
of power
fluctuations
as efficiency
the conversion
efficiency
gettingthe
higher
theinertia
same
conditions.
The results
in Figure
show that
a higher3 efficiency
leads
to larger
power fluctuations
with
turbine inertia
conditions.
The 3results
in Figure
show that
a higher
efficiency
leads to larger
the
same
turbine inertia.
second
case isinertia.
carriedThe
out second
to studycase
howisthe
power
fluctuations
change
power
fluctuations
with The
the same
turbine
carried
out
to study how
the
with
larger
turbine inertia
when
conversion
are kept
constant. The
results in are
Figure
4
power
fluctuations
change
withthe
larger
turbineefficiencies
inertia when
the conversion
efficiencies
kept
indicate
that,
to
achieve
the
same
conversion
efficiency,
a
wind
turbine
with
larger
inertia
would
cause
constant. The results in Figure 4 indicate that, to achieve the same conversion efficiency, a wind
higher
fluctuations.
turbinepower
with larger
inertia would cause higher power fluctuations.
J  60  10 5 kg  m 2
0.4
0.2
0
100
120
140
160
180
200
Figure 3.
3. Power
Power fluctuations
fluctuations with
with different
different conversion
Figure
conversion efficiency
efficiency in
in the
the same
same turbine.
turbine.
Energies 2017, 10, 939
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Energies 2017, 10, 939
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Energies 2017, 10, 939
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0.4
0.20.4
0.2
0
100
120
140
160
180
200
0
100
120
140
160
180
200
Figure 4. Power fluctuations with different turbine inertia under the same conversion efficiency.
Figure 4. Power fluctuations with different turbine inertia under the same conversion efficiency.
Figure
4. Power
with
inertiaanalyses.
under theFor
samedifferent
conversion
efficiency.
The
same
MPPTfluctuations
strategy in
[5]different
is usedturbine
for these
studies,
only one
control parameter
tuned
respectively
regulate
the conversion
It is improved
The same is
MPPT
strategy
in [5] istoused
for these
analyses. Forefficiency.
different studies,
only onebased
The same
MPPT
strategy
in [5]isisone
used
analyses.
For efficiency.
different
studies,
only one
control
on optimal
torque
control,
which
of for
thethese
most
widely
used
MPPT control
strategies
in
wind
control
parameter
is tuned
respectively
to
regulate
the
conversion
It is improved
based
parameter
is
tuned
respectively
to
regulate
the
conversion
efficiency.
It
is
improved
based
on
optimal
on
optimal
torque
control,
which
is
one
of
the
most
widely
used
MPPT
control
strategies
in
wind
turbine engineering for its simple implementation and low power fluctuations [28]. A novel
torque
control,
which isloop
one
mostinwidely
used MPPT
strategies
in wind
turbine
engineering
turbine
engineering
forof
simple
implementation
andcontrol
lowtorque
power
fluctuations
[28].to Aimprove
novel the
proportional
control
isitsthe
added
this classical
optimal
control
scheme
proportional
control
loop
is
added
in
this
classical
optimal
torque
control
scheme
to
improve
for
its simple
implementation
and lowThe
power
fluctuations
[28].
proportional
controlthe
loopthe
is
dynamic
performance
of the system.
proportional
gain
Kp Ainnovel
this control
loop determines
dynamic
performance
of the
system.
The proportional
gain Kpthe
in this
control
loop determines
the
added
in
this
classical
optimal
torque
control
scheme
to
improve
dynamic
performance
of
the
system.
wind energy conversion efficiency. By tuning the single control parameter (proportional gain Kp)
wind energy gain
conversionthis
efficiency.
By tuning
the single
control
parameter
(proportional
gainBy
Kptuning
)
The
proportional
control
loop
determines
the wind
energy
conversion
efficiency.
conversion
efficiencyKisp in
changed
flexibly.
It is not important
to
determine
which
MPPT strategy
is
conversion efficiency is changed flexibly. It is not important to determine which MPPT strategy is
the
single
control
parameter
(proportional
gain
K
)
conversion
efficiency
is
changed
flexibly.
It
is
not
p
adopted
for this study above because the variation trends of efficiency, fluctuations and inertia are
adopted for this study above because the variation trends of efficiency, fluctuations and inertia are
important
to determine
which MPPT strategy
is adopted
for this study
above strategy
because the variation
similar
for
control
foremploying
employing
MPPT
is that,
similar different
for different
controlstrategies.
strategies. The
The reason
reason for
thethe
MPPT
strategy in [5]inis[5]
that,
trends
of
efficiency,
fluctuations
and
inertia
are
similar
for
different
control
strategies.
The
reason
this strategy
is
flexible
to
change
conversion
efficiency
by
tuning
only
single
control
parameter.
this strategy is flexible to change conversion efficiency by tuning only single control parameter. for
employing
thethere
MPPT
strategy
[5] isofof
that,
thisstrategy
strategy
is flexible
to change
conversion
efficiency
by
Although
there
may
be
some
kinds
MPPT
based
generator
speed
could
smooth
Although
may
be
somein
kinds
MPPT
strategy
based
onon
generator
speed
thatthat
could
smooth
tuning
only
single
control
parameter.
Although
there
may
be
some
kinds
of
MPPT
strategy
based
on
output
power
utilizing
large
inertia,the
theconversion
conversion efficiency
decrease
significantly.
Because
output
power
utilizing
large
inertia,
efficiencywould
would
decrease
significantly.
Because
speed
is
controlled
to
smooth
output
power,
and
track
rapid
wind
variations.
This
generator
speed
that
could to
smooth
output
power
utilizing
inertia,
the
conversion
efficiency
would
rotorrotor
speed
is controlled
smooth
output
power,
anditlarge
itcannot
cannot
track
rapid
wind
variations.
This
defect
would
become
more
evident
with
larger
turbine
inertia.
Therefore,
for
the
same
MPPT
decrease
significantly.
Because
rotor
speed
is
controlled
to
smooth
output
power,
and
it
cannot
track
defect would become more evident with larger turbine inertia. Therefore, for the same MPPT
strategy,
is hard to
maintain
both
efficiency
fluctuation
unchanged
as turbine
inertia
rapid
wind
This
defect
would
becomeand
more
evident level
with
larger
turbine
inertia.
Therefore,
strategy,
it variations.
isithard
to maintain
both
efficiency
and
fluctuation
level
unchanged
as turbine
inertia
getting
larger.
In
general,
large-inertia
wind
turbine
brings
more
complex
problems
in
balancing
for
the same
MPPT
strategy,large-inertia
it is hard to wind
maintain
both brings
efficiency
andcomplex
fluctuation
level unchanged
as
getting
larger.
In general,
turbine
more
problems
in balancing
both power smoothing and conversion efficiency enhancement.
turbine
inertia
getting
larger.
In
general,
large-inertia
wind
turbine
brings
more
complex
problems
in
both power smoothing and conversion efficiency enhancement.
balancing
both power
smoothing and conversion efficiency enhancement.
4. Proposed
Method
4. Proposed Method
4. Proposed
Method
4.1 Description
of the Proposed Framework
4.1 Description
of thecontrol
Proposed
Framework
The MPPT
framework
employing a model predictive method is proposed to derive
4.1. Description
of the Proposed
Framework
generator
torque
reference
closed-loopasystem.
shown in method
Figure 5, is
theproposed
left side intothederive
Te* for aemploying
The MPPT
control
framework
modelAs
predictive
The MPPT control framework
employing a model predictive method is proposed to derive
*
grey area
is thereference
proposed method.
generator
torque
T for a closed-loop system. As shown in Figure 5, the left side in the
generator torque reference Te∗ efor a closed-loop system. As shown in Figure 5, the left side in the grey
grey is
area
the proposed
method.
area
theisproposed
method.
r
r
 r
r
 r
r
Te
Te
Te*
Teopt
Teopt
KTe

K
Kw
w
Tcomp
Tcomp
Te*
KTe
iqsref
 s Lm
iqs idr
Ls Lm
iqsref
uqc
uqs
uqc
uqs
 s Lm
iqs idr
udc
udr
uqsc
c
dr
uu
Ls Lm

Q ref
uqs

Q ref
dc
udr
uqsc
udrc
Figure 5. Proposed control framework with predictive method.
Figure 5.
5. Proposed
control framework
framework with
with predictive
predictive method.
method.
Figure
Proposed control
uqs
Energies 2017, 10, 939
7 of 19
Energies 2017, 10, 939
7 of 19
The right side of Figure 5 is the decoupling control in the d-q synchronous reference frame for
the generator side converter (GSC), which is employed to control the active power and reactive
The
right side of Figure
5 is the
decoupling
control
in the d-q of
synchronous
reference
frame for the
power
independently.
The active
power
is the main
component
output power
that influences
generator
side
converter
(GSC),
which
is
employed
to
control
the
active
power
and
reactive
power
frequency of the power system. In a normal operating situation, the value of reactive power
is
independently.
active
power
the two
mainmain
component
of output power(PI)
that
influences
usually kept at The
a low
level.
Thereis are
proportional-integral
control
loopsthe
infrequency
the GSC:
of
power
system.
In aloop
normal
situation,
valueloop.
of reactive
power
is usually
kept at a
thethe
active
power
control
andoperating
the reactive
powerthe
control
The active
power
is determined
low
level.
There
are
two
main
proportional-integral
(PI)
control
loops
in
the
GSC:
the
active
by iqs control and the reactive power is determined by ids control. Details of decoupling control power
can be
control
loop
reactive
control power
loop. The
active power
determined
iqs control
found in
[3].and
It isthe
noted
that power
the generator
discussed
in thisispaper
is activebypower,
andand
the
the
reactive
power
is
determined
by
i
control.
Details
of
decoupling
control
can
be
found
in
[3].
It
is
reactive power is set to zero.
ds
notedThe
thatproposed
the generator
power
discussed
in this 5paper
is active
power,
reactive
power
method
illustrated
in Figure
consists
of two
parts,and
Partthe
1 and
Part 2.
Part 1isisseta
to
zero. OTC MPPT calculator. OTC is one of the most widely employed MPPT strategies for its
classical
proposed
method
illustrated[3,28]
in Figure
consistsimplementation.
of two parts, Part
1 and
Part 2.
is a
goodThe
stability,
control
performance
and5simple
The
output
of Part
the 1OTC
classical
OTC
MPPT calculator.
OTC is one of the most widely employed MPPT strategies for its good
calculator
is expressed
as:
stability, control performance [3,28] and simple
implementation. The output of the OTC calculator is
Teopt  K opt   r2 / N gear
(7)
expressed as:
opt
T = Kopt · ωr2 /Ngear
(7)
3
. opt isethe optimum
tip-speed ratio. From(7), Teopt is a function of
where K opt  0.5  Rw5 C p max / opt
opt
5C
where
Kopt =Since
0.5ρπR
/λ3opt
. λopttoisslow
the optimum
tip-speedof
ratio.
(7), Te ωris
rotor speed.
large
inertia
leads
dynamic behavior
the From
rotor speed
, Taeoptfunction
varies
w pmax
opt
of
rotor
speed.
Since
large
inertia
leads
to
slow
dynamic
behavior
of
the
rotor
speed
ω
,
T
varies
r
e
variation.
As a
slowly without compensation Tcomp, and the control system does not track wind
slowly
compensation
Tcompof
, and
the is
control
system
does not track
variation.
As a
result, without
the conversion
efficiency
OTC
reduced
significantly,
butwind
the speed
capability
of power
result,
the
conversion
efficiency
of
OTC
is
reduced
significantly,
but
the
capability
of
power
smoothing
smoothing is improved apparently.
is improved
apparently.
Thus, Part
2 in Figure 5 is designed to take advantage of this and make up the disadvantages
Thus, above.
Part 2 inCompensation
Figure 5 is designed
takeisadvantage
this and make
up the
disadvantages
discussed
torque toTcomp
derived of
depending
on rotor
acceleration
and
discussed
above.
Compensation
torque
T
is
derived
depending
on
rotor
acceleration
and
comp
compensation gain:
compensation gain:
 r.
Tcomp=
=KKw  
(8)
Tcomp
(8)
w·ω
r
Then, the
the final
final generator
generator reference
reference is
is derived
derived after
after compensation:
compensation:
Then,
opt
Te* =Topt
T
Te∗ =
Tee − Tcomp
comp
(9)
(9)
Since . equals to zero when the wind turbine operates in steady state (Cp achieves a
Since ω rr equals to zero when the wind turbine operates in steady state (Cp achieves a maximum
maximum
the compensation
ofonly
Part activated
2 is only activated
during transient
processes
where
the
value),
the value),
compensation
of Part 2 is
during transient
processes
where the
system
w is regulated by a model predictive
system
deviates
from
its
steady-state
operation
point
(OP).
K
deviates from its steady-state operation point (OP). Kw is regulated by a model predictive method
method considering
both conversion
andsmoothing.
power smoothing.
considering
both conversion
efficiencyefficiency
and power
Figure 66 is
is the
the locus
locus of
of OPs
OPs for
for classical
classical OTC
OTC and
and proposed
proposed control
control framework,
framework, assuming
assuming that
that
Figure
wind
speed
steps
from
7
m/s
to
11
m/s.
Figure
6a,b
is
coincident,
and
the
only
difference
is
that
rotor
wind speed steps from 7 m/s to 11 m/s. Figure 6a,b is coincident, and the only difference is that rotor
acceleration is
is shown
shown in
in Figure
Figure 6b.
6b.
acceleration
Figure 6.
Figure
6. Moving
Moving locus
locus of
ofoperation
operationpoints,
points,(a)
(a)2D
2Dview;
view;(b)
(b)3D
3Dview
viewwith
withrotor
rotoracceleration
accelerationincluded.
included.
Energies 2017, 10, 939
Energies 2017, 10, 939
8 of 19
8 of 19
The
The blue
blue curve
curve is
is the
the locus
locus of
of classical
classical OTC,
OTC, the
the red
red curve
curve is
is the
the locus
locus of
of the
the proposed
proposed control
control
framework.
When
the
wind
speed
changes,
the
operation
points
step
from
A
to
B
,
and
B
,
respectively.
1
2 B1, and B2,
framework. When the wind speed changes, the operation points step from
A to
The
rotor acceleration
B2 is muchoflarger
than that
of than
B1 . This
that means
the OPthat
of the
respectively.
The rotorof
acceleration
B2 is much
larger
thatmeans
of B1. This
theproposed
OP of the
method
reaches
steady-state
point
C
faster
(time
=
3
s)
than
that
of
classical
OTC
method
(time
= 10 s).
proposed method reaches steady-state point C faster (time = 3 s) than that of classical OTC method
Therefore,
the
proposed
control
framework
improves
the
dynamic
performance
of
control
system,
(time = 10 s). Therefore, the proposed control framework improves the dynamic performance of
and
as asystem,
result, increases
the conversion
of the efficiency
wind turbine.
This
effect
all depends
on Kall
w.
control
and as a result,
increasesefficiency
the conversion
of the
wind
turbine.
This effect
A
larger
K
means
higher
efficiency.
However,
it
also
brings
higher
power
fluctuations.
Therefore,
it
is
depends won Kw. A larger Kw means higher efficiency. However, it also brings higher power
essential
to regulate
Kw reasonably
in order
to balance
efficiencyinenhancing
power
fluctuations.
Therefore,
it is essential
to regulate
Kw the
reasonably
order to and
balance
thesmoothing
efficiency
capabilities
according
the currentcapabilities
operating conditions.
paper, the
optimal conditions.
Kw is derived
a
enhancing and
powertosmoothing
according In
to this
the current
operating
Inby
this
model
predictive
method.
paper, the optimal Kw is derived by a model predictive method.
4.2.
4.2. Procedures
Procedures of
of MPC
MPC Implement
Implement
In
discussed
above,
thethe
model
predictive
solver
plays
a keya role
In the
theproposed
proposedcontrol
controlframework
framework
discussed
above,
model
predictive
solver
plays
key
in
determining
the
value
of
compensation
gain
K
.
MPC
is
widely
used
to
handle
multi-objective
w
role in determining the value of compensation
gain Kw. MPC is widely used to handle
control
problems
in multi-input
(MIMO)
systems.
The procedure
MPC
design is
multi-objective
control
problemsmulti-output
in multi-input
multi-output
(MIMO)
systems. of
The
procedure
of
as
follows.
MPC design is as follows.
•
•
•
The
The input,
input, output
output variables
variables and
and control
control signals
signals need
need to
to be
beclear.
clear. Linear
Linear dynamic
dynamic state-space
state-space
model
model also
also requires
requiresto
tobe
beestablished.
established.Simplification
Simplificationof
ofthe
themodel
modelisisnecessary.
necessary.
The
The future
future output
output sequence
sequence within
within the
the control
control horizon
horizon isispredicted.
predicted. The
The optimized
optimized control
control
T
T isis
sequence
. . , u(k
u(k ++N
Ncc−−1)]
1)]
derivedfrom
fromsolving
solvingaaconstrained
constrained problem expressed
sequenceU(k)
U(k)==[u(k),
[u(k),. …,
derived
expressed
as
as aa value
value function
function at
ateach
eachsampling
samplingtime.
time.
The
The first
first control
control signal
signal in
in the
thesequence
sequence is
is sent
sentto
tothe
thesystem.
system. The
The optimization
optimization will
will be
be repeated
repeated
and
and updated
updated with
with predictive
predictivehorizon
horizonrolling.
rolling.
The flow
flow chart
chart of
of this
this procedure
procedureisispresented
presentedin
inFigure
Figure7.7.
The
Figure 7.
7. Flow
Flow chart
chart of
of MPC
MPC implementation
implementation procedure.
procedure.
Figure
In classical
classicalMPC,
MPC,system
systemstability
stability
is guaranteed
by employing
terminal
quality
constraints.
In
is guaranteed
by employing
terminal
quality
constraints.
The
The optimization
is realized
by solving
a quadratic
programming
(QP) problem
line tocontrol
derive
optimization
is realized
by solving
a quadratic
programming
(QP) problem
on line on
to derive
control sequence,
whichabout
brings
aboutamount
a great amount
of calculation.
sequence,
which brings
a great
of calculation.
4.3. State-Space
State-Space Model
Model for
for Predictive
Predictive Control
Control
4.3.
Toreduce
reducethe
thecomputing
computingburden,
burden,simplification
simplificationof
ofthe
thedetailed
detailedmodel
model is
is necessary.
necessary.The
The generator
generator
To
model can
can be
be simplified
simplifiedas
asaafirst-order
first-ordermodel
modelto
torepresent
representthe
theelectrical
electricaldynamics
dynamics[18]:
[18]:
model
. T  11 T *  T
e
Te =
( T ∗e − Te e )
τee e


(10)
(10)
*
whereT ∗Teis
is the
generator
torque
reference,
is the equivalent
timeAccording
constant.toAccording
to
where
the
generator
torque
reference,
τe is theequivalent
time constant.
Equations (3)
e
e
Equations
(3)simplified
and (10), the
simplified
model Assuming
is nonlinear.
Assuming
is the
steady
and
(10), the
model
is nonlinear.
that
x = x + that
δx, x xisthe
under
a
x steady
 x , x state
state under a certain wind speed condition, and  x is the deviation from x , linearizing the
simplified Equations in (3) and (10) yields:
Energies 2017, 10, 939
9 of 19
certain wind speed condition, and δx is the deviation from x, linearizing the simplified Equations in
(3) and (10) yields:
(
.
Jδω r = Γ Tw δωr − Ngear · δTe + Γ Tb δβ + Γ Tv δVw
.
(11)
τe δ T e = δTe∗ − δTe
∂Tm ∂Tm m
, Γ Tv = ∂V
,
Γ
=
.
where Γ Tw = ∂T
Tb
∂ωr ∂β
w
(vw0 ,β 0 )
(ωr0 ,β 0 )
(ωr0 ,Vw0 )
Since this research focuses on the MPPT below rated wind speed, then the pitch angle is fixed, δβ
is set to zero.
The state-space model for predictive method is given as:
"

.

 X( t ) =


1
J Γ Tw
0
#
"
#
− 1J Ngear
0
X( t ) +
u(t) + 1J Γ Tv d(t)
1
− τ1e
τe
Y(t) = [0, 1]X(t)
(12)
where X = [ωr , Te ], u = T∗e , d = δVw .
Discretizing the continuous model in Equation (12), the state-space equation is expressed as:
(
X(k + 1) = AX(k) + Bu(k) + Ed(k)
Y(k) = CX(k)
(13)
The cost function for the predictive method is:

Jmin

 Nc

Nc

2
2
2 

= min ∑ ker (k + i )kM1 + ∑ k∆eP (k + i )kM2 + k∆Kw (k)kM3 

|
{z
}
 i =1
i =1
|
{z
} |
{z
}
Increment
E f f iciency
(14)
Fluctuations
.
.
max subject to ωrmin ≤ ωr (k + i ) ≤ ωrmax , ω r (k + i ) ≤ ω r , 0 ≤ Te (k + i ) ≤ Temax , 0 ≤
Pe (k + i ) ≤ Perated .
def
Here, Nc is the control horizon, kek2M = eT Me. M 1 , M 2 , M 3 are weights that denote different
control objectives, respectively. Jmin is the cost function in quadratic form required to be minimized by
the MPC solver, consisting of different control objectives. The first term in the equation represents the
opt
index of conversion efficiency, and rotor error er = ωr − ωr . The second term denotes the output
power fluctuations, ∆ePe = Pe (k ) − Pe (k − 1). In the third term, the increment of control law at each
sample time is substituted by the increment of compensation Kw , ∆Kw = Kw (k ) − Kw (k − 1).
Because of the proposed MPPT framework, the control law at every sample time is calculated
according to the special equation derived from Equation (9):
0
u(k) = (Kopt
− Kw · Γ Tw /J ) · δωr (k) + Ngear · Kw /J · δTe (k) − Kw · Γ Tv /J · δVw (k)
(15)
0 = 0.5ρπR 5 C
3
where Kopt
w
pmax / ( Ngear · λopt ).
In the proposed control, Kw is constant within the entire control horizon. Therefore, the solving
problem is simplified into computing the single parameter Kw (k) instead of computing u(k), . . . ,
u(k + Nc − 1), which is required in traditional MPC.
Ultra-short term prediction of wind speed is also required. Considering the emphasis is the MPPT
method, wind forecasting is not studied in this paper. A wind prediction method can be found in [29],
where the approximate precision of ultra-short term forecasting is up to more than 85%. This means
the uncertain dynamics of the predicted wind is small. Note that high precision of wind prediction is
not necessary in this method, because just the information of variation trends of wind speed is just
Energies 2017, 10, 939
10 of 19
Energies
2017, 10,is939
wind speed
just
10 of 19
enough to consider the power fluctuations. The impacts of uncertain predictive
wind on the control system are also discussed in Section 4.6.
enough
to consider
the
powerRegulation
fluctuations. The impacts of uncertain predictive wind on the control
4.4. Proposed
Dynamic
Weights
system are also discussed in Section 4.6.
For a large-inertia wind turbine, the energy conversion efficiency can be kept at a high level
4.4.
Proposed
Dynamic
Weights
Regulation
under
low wind
speed
fluctuations
[3]. Only when the wind speed varies rapidly, the efficiency
drops significantly, and this loss of conversion efficiency cannot be ignored. Therefore, different
For a large-inertia wind turbine, the energy conversion efficiency can be kept at a high level
control objectives should be emphasized under different wind speed conditions. In this paper, a
under low wind speed fluctuations [3]. Only when the wind speed varies rapidly, the efficiency drops
dynamic weights adjustment method is presented, rather than other preset weights methods. As
significantly, and this loss of conversion efficiency cannot be ignored. Therefore, different control
shown in Figure 8, after uniformization, the three weights M1 + M2 + M3 = 1. Weight M3 is fixed which
objectives should be emphasized under different wind speed conditions. In this paper, a dynamic
is designed to avoid oversize increments of the control law. Weights M1 and M2 are adjusted
weights adjustment method is presented, rather than other preset weights methods. As shown in
dynamically, and their sum is equal to (1 − M3). If the second weight M2 is determined, then
Figure 8, after uniformization, the three weights M1 + M2 + M3 = 1. Weight M3 is fixed which is
M1 = 1 − M2 − M3. The second weigh M2 is expressed as:
designed to avoid oversize increments of the control law. Weights M1 and M2 are adjusted dynamically,
and their sum is equal to (1 − M3 ). If the second weight
M  Fˆ M2 is determined, then M1 = 1 − M2 − M3 .
M2 = 2 v
(16)
The second weigh M2 is expressed as:
Fv
M2 · F̂v
M2 =
(16)
Nc
2
Fv
where Fv    V 2 w  k  i   V 2 w  k  i  1  denotes the
statistical wind speed fluctuations within
j 1 Nc
2
where Fv = ∑ V 2 w (k ˆ+ i ) − V 2 w (k + i − 1) denotes the statistical wind speed fluctuations within
the control jhorizon.
is
a
preset
threshold
of wind fluctuations. M 2 varies around the preset
F
=1
v
F̂v is on
the
control M
horizon.
a preset
threshold of Ignoring
wind fluctuations.
M2 varies
the preset
depending
wind fluctuations.
wind variations,
if M2 isaround
preset higher,
the
parameter
2
parameter
M
depending
on
wind
fluctuations.
Ignoring
wind
variations,
if
M
is
preset
higher,
2
2
effect of power smoothing becomes more evident.
the effect of power smoothing becomes more evident.
Figure 8.
8. Schematic
Schematic of
of dynamic
dynamic weights
weights in
in value
value function.
function.
Figure
Since conversion efficiency can be kept at a high level when the wind changes slowly, M2 is
Since conversion efficiency can be kept at a high level when the wind changes slowly, M2 is
increased to be inclined to smoothing the power. Similarly, if the wind fluctuation gets higher, the
increased to be inclined to smoothing the power. Similarly, if the wind fluctuation gets higher,
conversion efficiency declines significantly, so M1 has to be raised to enhance the efficiency.
the conversion efficiency declines significantly, so M1 has to be raised to enhance the efficiency.
4.5. Stability
Analysis
4.5.
Stability Analysis
Classical MPC
MPCisisan
anopen
openloop
loopcontrol
controlsystem.
system.The
Thestability
stability
control
system
is guaranteed
Classical
ofof
thethe
control
system
is guaranteed
by
by
terminal
quality
constraints.
In
this
paper,
the
control
law
of
U(k)
is
determined
by
both
the
terminal quality constraints. In this paper, the control law of U(k) is determined by both the proposed
proposed
control framework
MPC
solver. Therefore,
thesystem
controlof
system
of the proposed
control
framework
and MPCand
solver.
Therefore,
the control
the proposed
methodmethod
can be
can
be
guaranteed
so
long
as
the
control
framework
is
stable.
guaranteed so long as the control framework is stable.
The discrete
discrete system
system stability
stability of
of control
control framework
frameworkwith
withdifferent
differentKKw is
is discussed.
discussed. Ignoring
Ignoring wind
wind
The
w
speed
turbulences,
Equation
(15)
is
rewritten
as:
speed turbulences, Equation (15) is rewritten as:
u( k)  G  X ( k)
u( k ) = G · X( k )
  K w  Tw / J ), N gear  K w / J  .
where G  h( K opt
i

0 − K · Γ /J ), N
·
K
/J
where G = (Kopt
.
w
gear
w
Tw
Combining (13) and (17), the discrete state space equation is rewritten as:
Combining (13) and (17), the discrete state space equation is rewritten as:
(  X  k  1  AX  k 
X(k + 1) = A0 X(k )
Y  k   CX  k 
Y(k) = CX(k)
where A  [ A  B  G] .
where A0 = [A + B·G].
(17)
(17)
(18)
(18)
Energies
Energies2017,
2017,10,
10,939
939
11
11of
of19
19
Figure 9 shows the locus of system eigenvalues with Kw increasing from 0 to 70000. All the
Figure 9 shows the locus of system eigenvalues with Kw increasing from 0 to 70,000. All the
eigenvalues are located within the unit disk. According to the discrete system control theory [30], the
eigenvalues are located within the unit disk. According to the discrete system control theory [30],
control system is stable. With the increasing Kw, the eigenvalues move to the inner boundary of the
the control system is stable. With the increasing Kw , the eigenvalues move to the inner boundary of the
unit disk. If the gain Kw exceeds the limiting value, the eigenvalues would get out of the unit disk,
unit disk. If the gain Kw exceeds the limiting value, the eigenvalues would get out of the unit disk, and
and as a result, the stability of the control system cannot be ensured in this case and therefore, the
as a result, the stability of the control system cannot be ensured in this case and therefore, the stability
stability of the proposed method can be guaranteed, so long as Kw is limited within maximum value.
of the proposed method can be guaranteed, so long as Kw is limited within maximum value. This
This requirement can be satisfied by adding a limiter to ensure the reference torque to be non-negative.
requirement can be satisfied by adding a limiter to ensure the reference torque to be non-negative.
0.2
0.1
0
-0.1
-0.2
0.5
0.6
0.7
0.8
0.9
1
Figure9.9. Eigenvalue
Eigenvalue locus
locus of
ofproposed
proposedcontrol
controlsystem.
system.
Figure
4.6. Robustness
Robustness Analysis
Analysis
4.6.
Uncertainparameters,
parameters,including
includingwind
windprediction
predictionuncertainties,
uncertainties,and
andunmodeled
unmodeleddynamics,
dynamics, may
may
Uncertain

bring about
about uncertain
uncertain deviations
deviations of
the
eww.. Assume
K
bring
of MPC
MPC output
output K
Kwwfrom
fromthe
theoptimal
optimalvalue
value K
Assume δkk isisthe
maximal
maximaluncertain
uncertaindeviation
deviationof
ofMPC
MPCoutput,
output,then
thenthe
thebounds
boundsof
ofMPC
MPCoutput
outputisisexpressed
expressedas:
as:
±
K w=K
 δk
eKww ±
Kw
k
(19)
(19)

Therefore, the MPC output Kw at each sample time lies in the uncertain range [ K  , K
As
− w, ].
+
Therefore, the MPC output Kw at each sample time lies in the uncertain rangew [Kw
Kw
].
discussed
in in
Section
4.5,
w determines
the
location
of
closed-loop
eigenvalues.
As
discussed
Section
4.5,the
thevalue
valueKK
determines
the
location
of
closed-loop
eigenvalues.
w
Eigenvaluesare
arealso
alsocalled
calledpoles.
poles. The
Theclosed-loop
closed-loopeigenvalue
eigenvaluelocation
locationof
ofwithin
withinthis
thisuncertain
uncertainrange
range
Eigenvalues


−
+
[ Kww, ,Kw
] isillustrated
illustratedininFigure
Figure10.
10.All
Allthe
theeigenvalues
eigenvaluesare
aredistributed
distributedininthe
thecolored
coloredregion.
region. The
The
K ]w is
[K
− , K+ ]. If the uncertain range is widened too much,
colored
thethe
uncertain
range
[Kw
coloredregion
regiondepends
dependsonon
uncertain
range
[ K ww, K w ]. If the uncertain range is widened too
then
the
colored
region
would
overflow
the
stability
As a result,As
eigenvalues
may be located
much, then the colored region would overflow theboundary.
stability boundary.
a result, eigenvalues
may
outside
of
the
stability
area,
then
the
system
becomes
unstable.
Therefore,
to
improve
robustness,
be located outside of the stability area, then the system becomes unstable. Therefore, the
to improve
the
− , K + ] of MPC output requires to be limited within stable area. In practice, MPC
uncertain range [Kw


w
robustness, uncertain range [ K w , K w ] of MPC output requires to be limited within stable area.
In
− , K+ ]
is capable of dealing with optimization
problems with constraints. Thus uncertain range [Kw
w
practice,
MPC isby
capable
of dealing
with optimization
withparameters
constraints.and
Thus
uncertain
can
be restricted
MPC solver,
no matter
how serious problems
the uncertain
unmodeled


range [ K ware.
, KIn
] can be restricted by MPC solver, no matter how serious the uncertain parameters
w general, the proposed control has good robust performance, and the stability is kept
dynamics
and unmodeled
dynamics and
are. unmodeled
In general, dynamics
the proposed
control has good robust performance, and
from
uncertain parameters
in MPC.
the stability is kept from uncertain parameters and unmodeled dynamics in MPC.
Energies 2017, 10, 939
12 of 19
Energies 2017, 10, 939
12 of 19
K w
K w
Kw
Energies 2017, 10, 939
12 of 19
K w
K w
Kw
Figure 10. Closed-loop eigenvalue location within uncertain ranges of Kw.
Figure 10. Closed-loop eigenvalue location within uncertain ranges of Kw .
5. Simulation Studies
5. SimulationToStudies
verify the effectiveness of the proposed method, comprehensive simulation studies are
carried out based on the Wind Turbine Blockset Toolbox in Matlab/Simulink platform. The Blockset
2
Power spectra density
Power
spectra
((m/s)
/ Hz)density ((m/s)2 / Hz)
Figure 10. Closed-loop
location
within uncertain
ranges of Kw. simulation studies are
To verify
the effectiveness
of the eigenvalue
proposed
method,
comprehensive
was developed by RISØ National Laboratory and Aalborg University, and it has been used as a
carried out
based
on Studies
the tool
Wind
Turbine
in Matlab/Simulink
platform.
The Blockset
general
developer
for other
threeBlockset
simulationToolbox
tools, namely:
Saber, DIgSILENT and
HAWC [31].
5.
Simulation
The
Blockset
has
both
mechanical
and
electrical
model
libraries
that
are
accurate
enough,
especially
was developed by RISØ National Laboratory and Aalborg University, and it has been used as a
To verify models,
the effectiveness
of the Considering
proposed method,
comprehensive
simulation
studies
are
the generator
for this
study.
the focus
in this
paper,
the power
grid
that
general developer
tool for
other
three
simulation
tools,
namely:
Saber,
DIgSILENT
and
HAWC [31].
carried
out
based
on
the
Wind
Turbine
Blockset
Toolbox
in
Matlab/Simulink
platform.
The
Blockset
generator connects is assumed to be stable, and the power converters are regarded as average models.
The Blockset
has
both
mechanical
and
electrical
model
libraries
that
are
accurate
enough,
was developed
by RISØ
Laboratory
and Aalborg
University,
andwind
it has
been module
used as in
a especially
Effective wind
speedNational
time series
for simulations
are generated
by the
model
general
developer
tool
for
other
three
simulation
tools,
namely:
Saber,
DIgSILENT
and
HAWC
[31].
the generator
models,that
for takes
this study.
Considering
thethe
focus
in thisturbulences
paper, theinto
power
grid Itthat
the Blockset
the tower
shadow and
rotational
account.
is generator
The
Blockset
bothNational
mechanical
and electrical
model
that
are accurate
especially
developed
byhas
RISØ
Laboratory
based converters
on thelibraries
Kaimal
spectra.
The power
spectra
density
connects is
assumed
to
be
stable,
and
the power
are
regarded
as enough,
average
models.
the
generator wind
models, for
thisisstudy.
Considering
focus in
paper, the power
grid tothat
for generated
calculated
as shownthe
in Figure
11. this
Wind
is assumed
be
Effective
wind speedistime
timeseries
series
simulations
are
generated
bydirection
the wind
model
module
in the
generator
to befor
stable,
and the
power
converters
are regarded
average
models.
fixed, thusconnects
yaw angleassumed
is set to zero.
Wind shear
is not
taken
into account,
becauseasthe
effective
wind
Blockset that
takes
the
tower
shadow
andofthe
rotational
turbulences
into
account.
It is developed
by
wind
time series
simulations
arewind
generated
the
wind
model
module
in
speedEffective
is calculated
asspeed
an average
valuefor
the
fixed-point
speed by
over
the
whole
rotor.
the Blockset
that takes
theon
tower
shadow spectra.
and the rotational
turbulences
into account.
It is
RISØ National
Laboratory
based
the Kaimal
The power
spectra density
for generated
wind
by RISØ National Laboratory based on the Kaimal spectra. The power spectra density
time seriesdeveloped
is calculated
as shown in Figure 11. Wind direction is assumed to be fixed, thus yaw angle
for generated wind time series is calculated as shown in Figure 11. Wind direction is assumed to be
is set to zero.
Wind
shear
is is
not
account,
because
theaccount,
effective
windthe
speed
is calculated
as an
fixed, thus yaw angle
settaken
to zero.into
Wind
shear is not
taken into
because
effective
wind
average value
of
the
fixed-point
wind
speed
over
the
whole
rotor.
speed is calculated as an average value of the fixed-point wind speed over the whole rotor.
Figure 11. Power spectra density of wind speed.
The statistical approach of fluctuations is introduced in [1]. The main parameters in the
simulations are listed as follows, Prated = 2 MW, Rw = 40 m, Vw_rated = 11 m/s, Ngear = 83.531,
J = 56.27  10 5 kg  m 2 , Cpmax = 0.48,  opt =8.1 [3,31].
Figure 11. Power spectra density of wind speed.
Figure
11. Power spectra density of wind speed.
The statistical approach of fluctuations is introduced in [1]. The main parameters in the
simulationsapproach
are listed of
as fluctuations
follows, Prated is
= introduced
2 MW, Rw =in40[1].
m,The
Vw_rated
= 11
m/s, Ngear =in83.531,
The statistical
main
parameters
the simulations
J = 56.27  10 5 kg  m 2 , Cpmax = 0.48,  opt =8.1 [3,31].
are listed as follows, Prated = 2 MW, Rw = 40 m, Vw_rated = 11 m/s, Ngear = 83.531, J = 56.27 × 105 kg·m2 ,
Cpmax = 0.48, λopt = 8.1 [3,31].
5.1. Case 1
The first set of simulations is designed to investigate the control performance of the proposed
method when the wind fluctuations change from high level to low level. The variations of wind
fluctuations are achieved by changing the wind turbulence intensity [32].
method when the wind fluctuations change from high level to low level. The variations of wind
fluctuations are achieved by changing the wind turbulence intensity [32].
As shown in Figure 12a, the wind speed is divided into two regions. At the beginning
(Region I), the wind turbulence intensity is 20%, and the fluctuation is large. In Region II, the wind
Energies 2017, 10, 939
13 of 19
turbulence intensity drops to 8%. The proposed method is compared with a former method
presented recently which has high conversion efficiency [5]. To compare them with each other
shown
in Figure
12a, the wind
speed is
into two are
regions.
At the
beginning
(Region
I),
impartially,As
the
average
conversion
efficiency
ofdivided
two methods
tuned
to be
the same
in Region
I,
the wind turbulence intensity is 20%, and the fluctuation is large. In Region II, the wind turbulence
as shown
in Table 1. Parameters in (16) are, M =0.15 , Fˆ =0.0045 , M3 = 0.3. This simulation reveals
intensity drops to 8%. The proposed method is2 comparedv with a former method presented recently
two main
aspects
advantages.
Firstly,
incompare
Regionthem
I, the
has
which
has highofconversion
efficiency
[5]. To
withproposed
each other method
impartially,
thelower
averagepower
conversion
efficiency
of
two
methods
are
tuned
to
be
the
same
in
Region
I,
as
shown
in
Table 1.
fluctuations when the average efficiencies of the two methods are almost the same. Secondly,
in
Parameters
in
(16)
are,
M̄
=
0.15,
F̂
=
0.0045,
M
=
0.3.
This
simulation
reveals
two
main
aspects
of
v
2
3
Region II, when wind speed varies gently, the output power of the proposed method is much
advantages. Firstly, in Region I, the proposed method has lower power fluctuations when the average
smoother,
which is nearly one fifth of former method, as shown in Table 1 and Figure 12b. However,
efficiencies of the two methods are almost the same. Secondly, in Region II, when wind speed varies
the conversion
efficiency is also kept at a high level. Although the efficiency of the proposed method
gently, the output power of the proposed method is much smoother, which is nearly one fifth of former
is 0.0002
smaller
than former
as shown
in Table
this difference
is issoalso
tiny
that
method, as shown
in Tablemethod
1 and Figure
12b. However,
the1,conversion
efficiency
kept
at acan be
ignored.
It is
noted
that the
power fluctuation
of the
proposed
method
higher
thatasof the
high
level.
Although
the efficiency
of the proposed
method
is 0.0002
smaller is
than
formerthan
method
in Table
1, this=difference
is so tiny
thatThe
canreason
be ignored.
It isthe
noted
thatspeed
the power
fluctuation
former shown
method
near time
95 s in Figure
12b.
is that
wind
drops
significantly
of
the
proposed
method
is
higher
than
that
of
the
former
method
near
time
=
95
s
in
Figure
12b.tends
at this time. As a result, to avoid conversion efficiency dropping too much, control objective
The reason
is thatfrom
the wind
speed drops
significantly
at this time.
a result,
to avoid conversion
to restrain
efficiency
increasing,
as shown
in Figure
12d. As
This
also indicates
the proposed
efficiency dropping too much, control objective tends to restrain efficiency from increasing, as shown
method has the capability to adjust to different wind situations dynamically considering both
in Figure 12d. This also indicates the proposed method has the capability to adjust to different wind
conversion
efficiency and power smoothing.
situations dynamically considering both conversion efficiency and power smoothing.
12. Simulation results under low and high wind fluctuations, (a) wind speed; (b) generator
Figure Figure
12. Simulation
results under low and high wind fluctuations, (a) wind speed; (b) generator
power; (c) generator torque; (d) conversion efficiency; (e) rotor speed; (f) compensation gain.
power; (c) generator torque; (d) conversion efficiency; (e) rotor speed; (f) compensation gain.
Energies 2017, 10, 939
14 of 19
Table 1. Statistics of efficiency and fluctuations.
Energies 2017, 10, 939
Methods
Former
Proposed
5.2. Case 2
14 of 19
Region I
Region II
Table
of efficiency and fluctuations.
Average
Cp1. StatisticsFluctuations
Average Cp
Fluctuations
0.4783
1.884
Region
I × 1011
Methods0.4783
1.279
× 1011
Average Cp Fluctuations
Former
0.4783
1.884 × 1011
Proposed
0.4783
1.279 × 1011
0.4796 II
1.140 × 1010
Region
9
0.4794
Average Cp Fluctuations 2.541 × 10
10
0.4796
1.140 × 10
0.4794
2.541 × 109
Instead
5.2. Case of
2 separating the wind fluctuations into two levels as in Case 1, different wind fluctuation
conditions are all taken into account to verify the proposed method in depth. Since different parameters
Instead of separating the wind fluctuations into two levels as in Case 1, different wind
in control
systems
lead to are
different
control
it the
is essential
impartially
compare
fluctuation
conditions
all taken
into performances,
account to verify
proposedtomethod
in depth.
Sincethe
control
performance
of those
methods
under
thetosame
operation
conditions.
Therefore,
as shown
different
parameters
in control
systems
lead
different
control
performances,
it is essential
to in
Tableimpartially
2, to compare
the
capability
of
power
smoothing,
controller
parameters
are
tuned
to
guarantee
compare the control performance of those methods under the same operation conditions.
that two
methods
haveinalmost
same conversion
efficiency.
in (16) are
as follows,
Therefore,
as shown
Table 2,the
to compare
the capability
of powerParameters
smoothing, controller
parameters
M2 =are
0.15,
F̂v =
0.0045, M3that
= 0.3.
tuned
to guarantee
two methods have almost the same conversion efficiency. Parameters in
(16) are as follows, M 2 =0.15 , Fˆv =0.0045 , M3 = 0.3.
Table 2. Statistics of efficiency and fluctuations.
Table 2. Statistics of efficiency and fluctuations.
Method
Average Cp
Fluctuations
MethodFormer
Average
Cp
0.4785
5.806 Fluctuations
× 1011
Proposed
Former
Proposed
0.4785
0.4785
0.4785
5.041 ×5.806
1011 × 1011
5.041 × 1011
The simulation
results illustrated in Figure 13 show that the generator power of the proposed
The simulation results illustrated in Figure 13 show that the generator power of the proposed
method
is
smoother
than
thatthat
of the
former
method.
The
method is
method is smoother than
of the
former
method.
Thepower
powerfluctuation
fluctuation of
of the
the proposed
proposed method
reduced
by
13.18%
compared
with
the
former
method
within
the
entire
simulation
time.
is reduced by 13.18% compared with the former method within the entire simulation time.
Figure
13. Simulation
results
under
differentwind
windfluctuations,
fluctuations, (a)
generator
power;
Figure
13. Simulation
results
under
different
(a)wind
windspeed;
speed;(b)(b)
generator
power;
(c) enlarged
of generator
power;
conversionefficiency;
efficiency; (e)
(e) compensation
compensation gain.
(c) enlarged
viewview
of generator
power;
(d)(d)
conversion
gain.
Energies 2017, 10, 939
15 of 19
However, the conversion efficiency of the proposed method is not reduced. Figure 13c is the
Energies 2017, 10, 939
15 of 19
enlarged
view of generator power in the dashed region of Figure 13b. In general, this set of simulations
reveals that
the proposed method has better power smoothing capability than that of other efficiency
However, the conversion efficiency of the proposed method is not reduced. Figure 13c is the
enhancement
In addition,
thein
parameters
of region
the control
system
require
no changes
adjust
enlarged methods.
view of generator
power
the dashed
of Figure
13b.
In general,
this settoof
to thesimulations
variations reveals
of windthat
speed.
the proposed method has better power smoothing capability than that of
other efficiency enhancement methods. In addition, the parameters of the control system require no
5.3. Case
3 to adjust to the variations of wind speed.
changes
The third set of simulations is designed to prove the advantages of the proposed dynamic weights
5.3. Case 3
in Equation (14). The two compared methods have the same control framework proposed in this
The
third
set of simulations
designed
prove
the advantages
of the proposed
paper. The
only
difference
between is
them
is thattothe
weights
of the proposed
method dynamic
are changed
weights
in
Equation
(14).
The
two
compared
methods
have
the
same
control
framework
proposed
in
dynamically, and the weights of the compared one are constant. We respectively name the
two methods
this paper. The only difference between them is that the weights of the proposed method are
after proposed method with dynamical weights and proposed method with constant weights in the
changed dynamically, and the weights of the compared one are constant. We respectively name the
following analysis. The settings of weights are listed in Table 3.
two methods after proposed method with dynamical weights and proposed method with constant
weights in the following analysis. The settings of weights are listed in Table 3.
Table 3. Parameters in value function.
Table 3. Parameters in value function.
Proposed Method
Parameters
Proposed Method
Parameters
Dynamic weights
F̂v = 0.004, M2 = 0.2, M3 = 0.3
ˆ
0.004,
=0.2,
Dynamic
Constantweights
weights
MF1v = 0.5,
M2 M
= 20.2,
M3M=3 =0.3
0.3
Constant weights
M 1 =0.5, M 2 =0.2, M 3 =0.3
As shown in Figure 14a, the wind fluctuation changes gradually from low level to high level along
As shown in Figure 14a, the wind fluctuation changes gradually from low level to high level
with the time. According to [3], the conversion efficiency loss of a large-inertia wind turbine under low
along with the time. According to [3], the conversion efficiency loss of a large-inertia wind turbine
windunder
fluctuation
condition is small, but the loss increases significantly when the wind speed varies
low wind fluctuation condition is small, but the loss increases significantly when the wind
rapidly.
Thus,
when
the wind
varies
slowly,
is better
to reduce
Kwtotoreduce
utilizeKthe
large inertia to
speed varies rapidly.
Thus,speed
when the
wind
speed it
varies
slowly,
it is better
w to utilize the
smooth
output
power,
and
avoid
extra
fluctuations.
large inertia to smooth output power, and avoid extra fluctuations.
Figure
14. Comparison
results
withdifferent
differentweights
weights preset
(a)(a)
wind
speed;
(b) (b)
compensation
Figure
14. Comparison
results
with
presetmethods,
methods,
wind
speed;
compensation
gain;
(c)
generator
power;
(d)
conversion
efficiency;
(e)
enlarged
view
of
generator
power;
(f) enlarged
gain; (c) generator power; (d) conversion efficiency; (e) enlarged view of generator power;
(f) enlarged
view of conversion efficiency.
view of conversion efficiency.
Energies 2017, 10, 939
Energies 2017, 10, 939
16 of 19
16 of 19
As illustrated in Figure 14b, in the low wind fluctuation region (30–130 s), the Kw of the
proposed
methodin
with
dynamic
smaller
than that
of the
method
with
weights.
As illustrated
Figure
14b, inweights
the low is
wind
fluctuation
region
(30–130
s), the
Kwconstant
of the proposed
Therefore,
the
proposed
method
with
dynamic
weights
obtains
better
power
smoothing
performance
method with dynamic weights is smaller than that of the method with constant weights. Therefore,
with
almost method
no efficiency
losses. This
advantage
moresmoothing
evident when
the windwith
fluctuation
the
proposed
with dynamic
weights
obtainsbecomes
better power
performance
almost
becomes
lower,
such
as
in
the
very
low
fluctuation
region
(100–130
s),
where
the
generator
power
of
no efficiency losses. This advantage becomes more evident when the wind fluctuation becomes
lower,
the method
weights
is much
smoother,
but the
thegenerator
efficiencies
of the
methods
are
such
as in thewith
very dynamic
low fluctuation
region
(100–130
s), where
power
of two
the method
with
almost
the
same,
as
shown
in
Figures
14e,f.
dynamic weights is much smoother, but the efficiencies of the two methods are almost the same, as
Ininthe
high14e,f.
wind fluctuation region (160–190 s), the conversion efficiency would decline
shown
Figure
significantly.
Thus,
it is fluctuation
essential to increase
M1 and reduce
2 to enhance the efficiency. As a result,
In the high
wind
region (160–190
s), theMconversion
efficiency would decline
K
w of the proposed method with dynamic weights is raised, which is close to the proposed method
significantly. Thus, it is essential to increase M1 and reduce M2 to enhance the efficiency. As a
without
weights. method with dynamic weights is raised, which is close to the proposed
result,
Kwdynamic
of the proposed
Thewithout
advantages
above
benefit from the dynamic weights within the entire simulation time.
method
dynamic
weights.
Different
from
the
fixed
weights
(M1from
= 0.5,the
M2dynamic
= 0.2) in the
proposed
method
with constant
weights,
The advantages above benefit
weights
within
the entire
simulation
time.
the weights
in the
method
along
with
the
time
to get better
controlwith
performance.
Different
from
the proposed
fixed weights
(Mchange
=
0.5,
M
=
0.2)
in
the
proposed
method
constant
1
2
The variations
of weights
in the
proposed
method
with dynamic
weights
are illustrated
in Figure
15.
weights,
the weights
in the
proposed
method
change
along with
the time
to get better
control
The
vertical
axis
denotes
the
relative
proportion
of
each
weight
at
each
sample
time.
Here,
performance. The variations of weights in the proposed method with dynamic weights are illustrated
MFigure
1 + M2 + M3 = 1, M3 = 0.3. For instance, at time 170 s, the weight M1 is 0.33, M2 is 0.37, M3 is 0.3.
in
15. The vertical axis denotes the relative proportion of each weight at each sample time. Here,
Because
the
M3 is fixed, the sum of M1 and M2 is 0.7.
M1 + M2 +
Mweight
3 = 1, M3 = 0.3. For instance, at time 170 s, the weight M1 is 0.33, M2 is 0.37, M3 is 0.3.
Note
that
dynamic
may
cause
efficiency losses in some special situations.
Because the weight M3 isweights
fixed, the
sum
of Ma1 few
and limited
M2 is 0.7.
For Note
example,
as shownweights
in Figure
due
to the
step efficiency
size limitation
w, the efficiency of the
that dynamic
may14d,
cause
a few
limited
lossesof
in Ksome
special situations.
proposed
method
with
dynamic
weights
is
a
little
lower
than
that
of
the
compared
method
around
For example, as shown in Figure 14d, due to the step size limitation of Kw , the efficiency of
the proposed
time = with
135 sdynamic
when the
wind isfluctuation
changes
suddenly,
but this method
also means
a lower
method
weights
a little lower
than that
of the compared
around
time =power
135 s
fluctuation.
This
special
situation
can
also
be
alleviated
by
decreasing
the
weight
3 in
when the wind fluctuation changes suddenly, but this also means a lower power fluctuation.MThis
Equation
(14).
special situation can also be alleviated by decreasing the weight M in Equation (14).
Relative proportion of weights
3
1
0.5
0
40
60
80
100
120
140
160
180
Figure15.
15.Weights
Weightsvariations
variationsof
ofthe
theproposed
proposedmethod
methodwith
withdynamic
dynamicweights.
weights.
Figure
In general, the two compared methods based on proposed MPC approach both have high
In general, the two compared methods based on proposed MPC approach both have high
efficiency and low fluctuations, but the power smoothing performance of the proposed method with
efficiency and low fluctuations, but the power smoothing performance of the proposed method
dynamic weights is better than that of the method with constant weights at almost no cost of
with dynamic weights is better than that of the method with constant weights at almost no cost of
conversion efficiency.
conversion efficiency.
5.4. Computing Time
5.4. Computing Time
The execution time of MPC solver within one sample period is calculated to verify the capability
The execution time of MPC solver within one sample period is calculated to verify the capability
of real-time implementation. The environment information for the execution program is shown in
of real-time implementation. The environment information for the execution program is shown in
Table 4. MATLAB is set to run the program in single thread mode. As shown in Table 5, MPC solver
Table 4. MATLAB is set to run the program in single thread mode. As shown in Table 5, MPC solver
takes 0.1737 s at each sample time to execute the optimization problem, which is much shorter than
takes 0.1737 s at each sample time to execute the optimization problem, which is much shorter than the
the sample period 1 s. This indicates that the proposed MPC is fully capable of real-time
sample period 1 s. This indicates that the proposed MPC is fully capable of real-time implementation.
implementation. Furthermore, the fast development of embedded processors, such as digital signal
Furthermore, the fast development of embedded processors, such as digital signal processing (DSP)
processing (DSP) and field-programmable gate array (FPGA), supplies a practicable computing
platform for real-time execution [33].
Energies 2017, 10, 939
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and field-programmable gate array (FPGA), supplies a practicable computing platform for real-time
execution [33].
Table 4. Computing environment.
CPU
RAM
Operating system
Platform
Intel Core i3-2370M (2.4 GHz) (Single Core Mode)
4 GB
Window 7 (32 bit)
MATLAB R2012a
Table 5. Computing time of MPC solver.
Parameter
Value
Sample period T
Control horizon Nc
Solver
Computing time
1 (s)
5
Sequence Quadratic Program
0.1737 (s)
6. Conclusions
This paper presents a flexible MPPT control method to balance captured energy efficiency and
generator power smoothing for large-inertia WECSs. A new control framework is proposed by
adaptively compensating the torque reference value. The compensation was determined by the
proposed model predictive control approach with dynamic weights in the cost function, which
improved control performance. The computational burden of the MPC solver was reduced by
transforming the cost function representation. The main contributions of this paper are summarized
as follows:
•
•
•
•
•
Analysis reveals that a larger wind turbine inertia brings difficulties to balance energy conversion
efficiency and generator power smoothing. To achieve the same conversion efficiency, a larger
inertia causes higher power fluctuations.
The proposed method not only keeps the conversion efficiency at a high level, but also reduces
power fluctuations as much as possible. Emphasis of control objectives is regulated dynamically.
Conversion efficiency is attached importance when it is about to decline too much. Power
smoothing is taken more into account if the wind varies slowly, and efficiency is kept at a
high level.
Unlike the classical MPC used for wind turbine control, MPC in this paper is only employed to
determine the compensating gain. The system stability is totally based on the proposed control
framework, rather than the stability of MPC itself. Theoretical analysis proves that the control
system is stable, so long as the compensating gain in proposed control framework is kept within
the maximal value.
The robustness of the system is guaranteed by the proposed control, the stability avoids uncertain
parameters and unmodeled dynamics in the MPC.
Simulation studies are carried out to prove that the proposed method has good control
performance. The computing time of the MPC solver at each sampling time is short enough to
satisfy the requirements of real-time implementation in engineering.
In general, these good capabilities of the proposed method, such as high conversion efficiency
and low power fluctuations can increase the wind farm profits and improve power system reliability.
Acknowledgments: This work was supported in part by the National Basic Research Program of China
(973 Program) (Grant No. 2012CB215203), National Nature Science Foundation of China (Grant No. 51606033)
and Beijing Postgraduate Joint-supervision Cooperate Project.
Energies 2017, 10, 939
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Author Contributions: Hongmin Meng and Tingting Yang conceived the control strategy, designed and
performed simulations. Ji-zhen Liu provided key advices of proposed strategy. Zhongwei Lin analyzed the data.
Hongmin Meng wrote and revised the paper.
Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations
MPPT
MOO
FIS
WECS
MPC
OTC
GSC
PI
OP
MIMO
QP
Maximum power point tracking
Multi-objective optimization
Fuzzy inference system
Wind energy conversion system
Model predictive control
Optimal torque control
Generator side converter
Proportional-Integral
Operation point
Multi-input multi-output
Quadratic programming
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Simani, S. Overview of modelling and advanced control strategies for wind turbine systems. Energies 2015,
8, 12374. [CrossRef]
Phan, D.-C.; Yamamoto, S. Maximum energy output of a dfig wind turbine using an improved mppt-curve
method. Energies 2015, 8, 11718–11736. [CrossRef]
Liu, J.; Meng, H.; Hu, Y.; Lin, Z.; Wang, W. A novel mppt method for enhancing energy conversion efficiency
taking power smoothing into account. Energy Conv. Manag. 2015, 101, 738–748. [CrossRef]
Hussain, J.; Mishra, M.K. Adaptive maximum power point tracking control algorithm for wind energy
conversion systems. IEEE Trans. Energy Conv. 2016, 31, 697–705. [CrossRef]
Kim, K.H.; Van, T.L.; Lee, D.C.; Song, S.H.; Kim, E.H. Maximum output power tracking control in
variable-speed wind turbine systems considering rotor inertial power. IEEE Trans. Ind. Electron. 2013,
60, 3207–3217. [CrossRef]
Uehara, A.; Pratap, A.; Goya, T.; Senjyu, T.; Yona, A.; Urasaki, N.; Funabashi, T. A coordinated control
method to smooth wind power fluctuations of a pmsg-based wecs. IEEE Trans. Energy Conv. 2011, 26,
550–558. [CrossRef]
Wei, C.; Zhang, Z.; Qiao, W.; Qu, L. An adaptive network-based reinforcement learning method for MPPT
control of pmsg wind energy conversion systems. IEEE Trans. Power Electron. 2016, 31, 7837–7848. [CrossRef]
Iyasere, E.; Salah, M.H.; Dawson, D.M.; Wagner, J.R.; Tatlicioglu, E. Robust nonlinear control strategy to
maximize energy capture in a variable speed wind turbine with an internal induction generator. J. Control
Theory Appl. 2012, 10, 184–194. [CrossRef]
Boukhezzar, B.; Siguerdidjane, H. Nonlinear control of a variable-speed wind turbine using a two-mass
model. IEEE Trans. Energy Conv. 2011, 26, 149–162. [CrossRef]
Dawei, Z.; Lie, X.; Williams, B.W. Model-based predictive direct power control of doubly fed induction
generators. IEEE Trans. Power Electron. 2010, 25, 341–351. [CrossRef]
Sguarezi Filho, A.J.; de Oliveira Filho, M.E.; Filho, E.R. A predictive power control for wind energy.
IEEE Trans. Sustain. Energy 2011, 2, 97–105.
Calle-Prado, A.; Alepuz, S.; Bordonau, J.; Nicolas-Apruzzese, J.; Cortes, P.; Rodriguez, J. Model predictive
current control of grid-connected neutral-point-clamped converters to meet low-voltage ride-through
requirements. IEEE Trans. Ind. Electron. 2015, 62, 1503–1514. [CrossRef]
Yaramasu, V.; Wu, B. Predictive control of a three-level boost converter and an npc inverter for high-power
pmsg-based medium voltage wind energy conversion systems. IEEE Trans. Power Electron. 2014, 29,
5308–5322. [CrossRef]
Energies 2017, 10, 939
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
19 of 19
Kou, P.; Liang, D.; Gao, F.; Gao, L. Coordinated predictive control of dfig-based wind-battery hybrid systems:
Using non-gaussian wind power predictive distributions. IEEE Trans. Energy Conv. 2015, 30, 681–695.
[CrossRef]
Evans, M.A.; Cannon, M.; Kouvaritakis, B. Robust MPC tower damping for variable speed wind turbines.
IEEE Trans. Control Syst. Technol. 2015, 23, 290–296. [CrossRef]
Spencer, M.D.; Stol, K.A.; Unsworth, C.P.; Cater, J.E.; Norris, S.E. Model predictive control of a wind turbine
using short-term wind field predictions. Wind Energy 2013, 16, 417–434. [CrossRef]
Barlas, T.; van der Veen, G.; van Kuik, G. Model predictive control for wind turbines with distributed active
flaps: Incorporating inflow signals and actuator constraints. Wind Energy 2012, 15, 757–771. [CrossRef]
Soliman, M.; Malik, O.P.; Westwick, D.T. Multiple model predictive control for wind turbines with doubly
fed induction generators. IEEE Trans. Sustain. Energy 2011, 2, 215–225. [CrossRef]
Soliman, M.; Malik, O.P.; Westwick, D.T. Multiple model mimo predictive control for variable speed variable
pitch wind turbines. In Proceedings of the American Control Conference (ACC), Baltimore, MD, USA,
30 June–2 July 2010; pp. 2778–2784.
Mayne, D.Q.; Rawlings, J.B.; Rao, C.V.; Scokaert, P.O. Constrained model predictive control: Stability and
optimality. Automatica 2000, 36, 789–814. [CrossRef]
Bououden, S.; Chadli, M.; Filali, S.; El Hajjaji, A. Fuzzy model based multivariable predictive control of a
variable speed wind turbine: Lmi approach. Renew. Energy 2012, 37, 434–439. [CrossRef]
Jain, A.; Schildbach, G.; Fagiano, L.; Morari, M. On the design and tuning of linear model predictive control
for wind turbines. Renew. Energy 2015, 80, 664–673. [CrossRef]
Kusiak, A.; Li, W.; Song, Z. Dynamic control of wind turbines. Renew. Energy 2010, 35, 456–463. [CrossRef]
Can, H.; Fangxing, L.; Zhiqiang, J. Maximum power point tracking strategy for large-scale wind generation
systems considering wind turbine dynamics. IEEE Trans. Ind. Electron. 2015, 62, 2530–2539.
Bianchi, F.D.; Battista, H.D.; Mantz, R.J. Wind Turbine Control Systems. Principles, Modelling and Gain Scheduling
Design; Springer: London, UK, 2007.
Abedini, A.; Mandic, G.; Nasiri, A. Wind power smoothing using rotor inertia aimed at reducing grid
susceptibility. Int. J. Power Electron. 2008, 1, 227–247. [CrossRef]
Varzaneh, S.G.; Gharehpetian, G.B.; Abedi, M. Output power smoothing of variable speed wind farms using
rotor-inertia. Electr. Power Syst. Res. 2014, 116, 208–217. [CrossRef]
Nasiri, M.; Milimonfared, J.; Fathi, S. Modeling, analysis and comparison of tsr and otc methods for mppt and
power smoothing in permanent magnet synchronous generator-based wind turbines. Energy Conv. Manag.
2014, 86, 892–900. [CrossRef]
Pourmousavi Kani, S.A.; Ardehali, M.M. Very short-term wind speed prediction: A new artificial neural
network–markov chain model. Energy Conv. Manag. 2011, 52, 738–745. [CrossRef]
Zheng, A.; Morari, M. Stability of model predictive control with mixed constraints. IEEE Trans. Autom. Control
1995, 40, 1818–1823. [CrossRef]
Iov, F.; Hansen, A.D.; Blaabjerg, F. Wind Turbine Blockset in Matlab/Simulink; Institute of Energy Technology,
Aalborg University: Aalborg, Denmark, 2004.
Hansen, K.S.; Barthelmie, R.J.; Jensen, L.E.; Sommer, A. The impact of turbulence intensity and atmospheric
stability on power deficits due to wind turbine wakes at Horns Rev wind farm. Wind Energy 2012, 15,
183–196. [CrossRef]
Gulbudak, O.; Santi, E. Fpga-based model predictive controller for direct matrix converter. IEEE Trans.
Ind. Electron. 2016, 63, 4560–4570. [CrossRef]
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