Wind-turbine collective-pitch control via a fuzzy predictive algorithm

Renewable Energy 87 (2016) 298e306
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Renewable Energy
journal homepage: www.elsevier.com/locate/renene
Wind-turbine collective-pitch control via a fuzzy predictive algorithm
Ahmed Lasheen*, Abdel Latif Elshafei
Electric Power and Machines Department, Faculty of Engineering, Cairo University, Giza, Egypt
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 22 May 2015
Received in revised form
23 September 2015
Accepted 19 October 2015
Available online xxx
This paper proposes a new fuzzy predictive algorithm for collective pitch control of large wind turbines.
Collective pitch controllers operate in region three to harvest the rated power and maintain the rated
speed. The wind turbine model is represented by a TakagieSugeno (TeS) fuzzy model. The number of T
eS fuzzy rules is reduced based on a gap e metric criterion. A model predictive controller is designed
based on the fuzzy model taking into consideration the pitch actuator constraints. The proposed
controller is coupled with conventional PI controllers for individual pitch control so as to minimize the
moments on the turbine blades. A Kalman observer is designed to estimate the immeasurable states. The
performance of the proposed fuzzy-predictive controller is compared to a gain schedule PI controller and
a mixed H2/H∞ controller. Simulation results, based on a typical 5-MW offshore wind turbine,
demonstrate the superiority of the proposed fuzzy-predictive controller.
© 2015 Elsevier Ltd. All rights reserved.
Keywords:
Wind turbines
Collective pitch control
Predictive control
TeS fuzzy models
1. Introduction
The installed wind-energy capacity reached 336 GW in the mid
of 2014 [1]. The expected annual growth rate in 2014 is 13.5%. It was
12.8% in 2013. The steady growth of the installed wind power is due
to its economic and environmental advantages. Control systems
can play a pivotal role in enhancing the economic performance of
the wind energy systems [2]. This can be achieved by increasing
power extraction, alleviating mechanical stresses, and improving
power quality.
The operation modes of a wind turbine depend on wind speeds.
Typically, there are three main regions of operations. Region 1 is
defined by wind speeds up to the cut-in value. In this region, the
wind is utilized merely to accelerate the rotor for startup. Wind
speeds higher than the cut in value and lower than the rated value
define Region 2. In this region, the turbine should operate to extract
the maximum power [3]. Wind speeds higher than the rated value
and up to the cut-out value specify Region 3. Typically, wind turbines are controlled using decentralized controllers to take care of
the operations in regions 2 and 3. This study focuses on Region 3. In
Region 3, wind turbines are usually subject to undesirable high
structural loading. As a result, the control objectives are to reap
rated power, maintain rated speed, and alleviate the mechanical
* Corresponding author.
E-mail addresses: [email protected] (A. Lasheen), [email protected].
edu.eg (A.L. Elshafei).
http://dx.doi.org/10.1016/j.renene.2015.10.030
0960-1481/© 2015 Elsevier Ltd. All rights reserved.
loads. Designing a pitch controller, taking into consideration the
restrictions on the pitch angle limits and rates of change, is our
target. The pitch angle of a wind turbine is controlled individually
and collectively. The target of collective pitch control (CPC) is to
regulate the generator power at the rated value by maintaining the
rated generator speed. On the other hand, individual pitch control
(IPC) aims at attenuating the flap-wise moments on the blades [4].
Many researchers have focused on controlling variable-speed
variable-pitch wind turbines. In Ref. [5], an adaptive neural
network controller is introduced. The objective is to control all
operating regions of the turbine. In Ref. [6], the pitch angle is
controlled utilizing the artificial neural network. The operation of
the turbine is observed by utilizing the back propagation learning
algorithm. In Ref. [7], a CPC is designed based on mixed H2/H∞
control with pole placement. In Ref. [8], the objective of the pitch
control is to achieve different load reduction criteria. In Ref. [9], a
linear quadratic Gaussian controller is proposed. Both CPC and IPC
are designed based on a single linear model. The main disadvantage
in Refs. [5e9] is that the proposed algorithms do not take into account constraints on the pitch angles. This may lead to the wind up
phenomenon and significant degradation of performance if the
control signal hits the saturation limits.
In Refs. [10e12], different IPC strategies are discussed to reduce
the flicker emission on the turbine blades. In Ref. [13], a non-linear
pitch control strategy is proposed to damp the tower oscillations,
regulate the rotor speed, and compensate the phase-lag introduced
by the pitch actuator. In Ref. [14], pitch faults, due to pitch
A. Lasheen, A.L. Elshafei / Renewable Energy 87 (2016) 298e306
asymmetry and pitch implausibility, are analyzed. In Refs. [15,16],
the effects of short-duration wind variations on the output power
of a wind turbine, under pitch control, are studied.
Model Predictive Control (MPC) is a control algorithm that depends on a system's model for predicting the future output over a
selected horizon. At every sampling instant, an optimization
problem is solved on-line over the prediction horizon to get the
control action. Several authors have used MPC to implement
different wind turbine controllers. In Refs. [17e19], an MPC is used
to extract the maximum power from the wind turbine (region 2). In
Ref. [20], a multiple MPC strategy is represented for the full operating regions of the turbine. The generator torque and pitch angle
are controlled simultaneously to maximize energy capturing,
smooth the generator power, and reduce the transient loads. The
main disadvantage in Ref. [20] is the use of a large number of
models which increases the computational complexity of the algorithm. Abrupt switching between models could result in sluggish
transient responses. In this paper, a fuzzy model predictive algorithm for CPC is investigated. The fuzzy rule-base is reduced by
using a gap metric criterion. The fuzzy model is employed to
construct the predictor. Fuzzy models represent nonlinear mappings. So, they can effectively represent the nonlinearities of wind
turbine models. MPC is used to obtain the CPC action taking into
account the allowed pitch-actuator limits and rates of change.
There are four main contributions in this research. First, it introduces a gap metric measure as a tool to determine the number of
fuzzy rules that adequately model a system. Second, it utilizes a
fuzzy model to mimic the nonlinear behavior of wind turbines in
region 3. The resulting model has a simple form amenable to predict the future response. Third, the use of a predictive control algorithm allows us to include the input constraints explicitly while
deducing the optimal control action. Explicit inclusion of the constraints is a fundamental difference in predictive control design
compared to traditional PI controllers. Fourth, the proposed
controller significantly reduces the mechanical stresses applied to
the wind turbine. This has a definite economic benefit as it reduces
the maintenance cost.
This paper is organized as follows. In Section 2, the wind model is
discussed. The procedure to emulate the nonlinear behavior of a
wind turbine using a fuzzy model is proposed. A gap metric criterion
is applied to these linearized models to determine which of them
should be included. Furthermore, the procedure to write a simplified fuzzy model for design purposes is discussed. The design of MPC
for collective pitch control is discussed in Section 3. Simulation results, comparing the proposed controller to different controllers, are
depicted in Section 4. Conclusions are derived in Section 5.
2. Developing the fuzzy model
A large wind turbine can be modeled by up to 24 degrees of
Table 1
The degrees of freedom of a 3-blade horizontal-axis wind turbine.
Element
Number of DOF
Description
Blades
2
1
1
1
1
1
1
2
2
3
3
Flap modes per blade
Edge mode per blade
Yaw bearing
Generator azimuth
Shaft torsion
Rotor-furl hinge between nacelle and rotor
Tail-furl hinge between nacelle and tail
Fore-aft modes
Side-to-side modes
Translational modes (surge, sway, heave)
Rotational modes (roll, pitch, yaw)
Nacelle
Derive-train
Furl
Tower
Platform
299
freedom (DOF). Table 1 summarizes the main degrees of freedom of
a 3-blade horizontal-axis wind turbine [4]. Detailed wind-turbine
models are available via specialized software packages like
HAWC2 [21], FAST [4], and Cp-Lambda [22]. FAST (Fatigue, Aerodynamics, Structures and Turbulence) is employed in this work to
simulate the operation of a 5-MW, three-blade, variable-speed
variable-pitch offshore wind turbine. FAST is developed by the US
National Renewable Energy Laboratory. It is provided with a tuned
gain schedule PI controller for CPC. The gain schedule PI controller
supplied with FAST is considered the baseline controller and used
for comparisons with the proposed CPC controller. The 5-MW wind
turbine specifications are stated in Ref. [4].
For a CPC design, the enabled DOFs are the generator and the
drivetrain. The DOFs are considered as the dominant dynamics of
the turbine [4]. Although our control design is based on a reduced
order model that represents 2 DOF, the resulting controller will be
tested based on the full order nonlinear model. The states of the
design model are the perturbations in the drivetrain torsional
speed, drivetrain torsional displacement, and rotor speed.
This section discusses three main points:1) the linearized
models that cover the operating region, 2) the gap-metric measure
to select the effective linearized models, and 3) the procedure to
write a simplified fuzzy model.
2.1. FAST linearized models
In order to design a CPC, linearized models are derived. Each of
the resulting models relates the perturbations of the collective
pitch to those of the generator speed. FAST can produce these linear
models around a specific operating point in the form of (1).
Generator speed, azimuth angle of the rotor, the hub height wind
speed, and the pitch angle are the main variables that specify an
operating point. In Region 3, the generator speed should be at the
rated value. Linearized models are calculated at different azimuth
angles for a given wind speed. Then, an average model is derived,
for design purposes, using Multi-blade Coordinate Transformation
[23]. Moreover, at steady state, FAST will give a nominal pitch angle
that is associated with a given average wind speed. Hence, the main
variable that characterizes the linearized models is the wind speed.
The resulting model takes the standard state-space form
x_ ¼ Ac x þ Bc ucpc
y ¼ Cc x
(1)
where Ac, Bc, and Cc are constant matrices with appropriate
dimensions. ucpc and y are the perturbations in the collective pitch
and generator speed, respectively. x is the perturbation in the
states. The state vector, x, includes the perturbations in the drivetrain torsional speed, drivetrain torsional displacement, and rotor
speed. In a FAST model, the generator torque is based on the
generator speed. If the generator speed is greater than or equal to
the rated value, then the generator torque is constant at its rated
value. Otherwise, the generator torque is proportional to the square
of its speed while the generated power is proportional to the cube
of the speed. Hence, the generator speed is the fundamental
quantity that we measure and rely on to obtain our results [4].
2.2. The gap-metric concept
The gap metric criterion can be used to check the proximity of
linear models [24,25]. The idea is that the distance between the two
selected models should be larger than a prescribed level, otherwise,
one model is enough to describe them both.
In this section, a gap metric criterion is defined. Then, it is used
to determine the linearized models that can adequately represent
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A. Lasheen, A.L. Elshafei / Renewable Energy 87 (2016) 298e306
the wind turbine through region 3. The resulting linearized models
will form the consequents of the proposed fuzzy rules. This means
that the number of models suggested by the gap metric criterion
dictates the number of rules in the fuzzy system that models the
wind turbine. This concept is one of the main contributions of this
work. It avoids the rule explosion phenomenon which is a main
criticism of fuzzy systems. This leads to a limited number of rules so
the computational burden is reduced.
The gap metric, d(Gi,Gj), is a measure of the maximum difference
between the two transfer functions Gi and Gj [24]. For single-input
single-output systems, it is defined as
G
G
i
j
d Gi ; Gj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ Gi Gi 1 þ Gj Gj (2)
∞
where Gi and Gj are the conjugates of Gi and Gj, respectively. It is
noted that the infinite norm, k:k∞ , is simply the maximum amplitude of the frequency response of the associated transfer function.
In this work, the objective of the controller is to limit the power
and speed at their rated values. The rated values define the operating point which is used to derive the linearized models. The wind
speeds range from 12 m/s to 22 m/s. Six linearized models are
obtained in the previous range with a step of 2 m/s. The gap metric
criterion is used to measure the mutual distance between each pair
of models. The results are shown in Table 2.
In Table 2, G12 refers to the linearized model calculated at the
wind speed 12 m/s, G22 refers to the linearized model calculated at
the wind speed 22 m/s, and so on. The entries of Table 2 give the
gap-metric measures between the two relevant models. For
example, the gap-metric measure between the models G18 and G14
is 0.253. The values are calculated using the MATLAB command
‘gapmetric’. Theoretically, if the result is zero, this means that the
two models have the same dynamics. On the other hand, if the
result is one, this means that the two models are totally different in
the dynamics. The threshold value of the gap metric, at which two
models are considered as different, is a design parameter. It reflects
a compromise between the simplicity of a fuzzy model and the
computational complexity. Simulations can be used to guide our
choice of the gap metric threshold value. Fig. 2 demonstrates the
accuracy of the proposed fuzzy model.
2.3. A simplified TeS fuzzy model
The core of a fuzzy system is its rulebase. A fuzzy rulebase
consists of if-then rules. The antecedent of each rule assigns a linguistic value to each input variable. The linguistic value is defined
by a membership function selected by the designer. The consequent
of the rule, in a TakagieSugeno fuzzy system, is a mathematical
expression. The output of the fuzzy system is a weighted sum of its
consequents where the weights depend on the firing strength of
each rule. So, a fuzzy system represents a mapping function of its
input space to it output space. The nonlinearity of the fuzzy model
stems from the multiplication of the rule strength by the rule
Table 2
Gap metric between the linearized models.
G
G
G
G
G
G
12
14
16
18
20
22
consequent. By adjusting the shapes and parameters of the memberships, it is possible to shape the mapping from the input to
output of a fuzzy system.
Based on the Universal Approximation Theorem [26], the
nonlinear model of a wind turbine can be approximated by a
TakagieSugeno fuzzy model. This retains the accuracy of the model
and the simplicity of the representations. The main variable that
characterizes the linearized models is the wind speed. This motivates us to use the wind speed as the linguistic variable in the rules'
antecedents of the fuzzy rule-base that models a wind turbine.
Measurements of the wind speed are inaccurate due to its variations across the blades' swept area and due to the high level of
noise. The use of a fuzzy model is suitable as it can inherently
accommodate inaccurate wind measurements. The linearized
models, which are selected based on the gap criterion, form the
consequents of the fuzzy rules. The resulting overall fuzzy model
(after blending all rules) looks like a time varying linear model
which eases its use to calculate the future predictions of the output
and states.
Using TakagieSugeno fuzzy models [26], the discrete linear
models equivalent to (1), and the gap metric discussed above, it is
possible to emulate the wind turbine by a set of fuzzy rules. Each
rule represents a local model in the form:
Rule #j: If the wind speed is Vj Then
xðk þ 1Þ ¼ Adj xðkÞ þ Bdj ucpc ðkÞ
yðkÞ ¼ Cdj xðkÞ
where x is the state vector, k is the current instant, and j is the rule
index. ucpc and y are the perturbations in the collective pitch and
generator speed, respectively. Adj, Bdj, and Cdj are the discrete model
matrices that follow from discretizing (1).
The antecedent of the fuzzy rule (3) has one linguistic variable
which is the wind speed. The wind speed as a linguistic variable is
assigned a linguistic value Vj which is described by an appropriate
membership function. In our problem, there are three rules corresponding to the linearized models selected based on the gap metric
criterion. The models correspond to wind speeds 12 m/s, 16 m/s,
and 20 m/s. Each speed, Vj, is assigned a membership function, Nj,
as shown in Fig. 1. This allows a fuzzy representation of the wind
speeds. The advantage of using a fuzzy model is that it can
accommodate imprecise wind-speed measurements.
Once a wind-speed measurement is received, the fuzzy rules are
fired. This corresponds to calculating the rules' strength which is
obtained by direct substitution of the wind-speed measured value
into the membership function Vj, j ¼ 1, 2, 3. The resulting rule
strength is a weight between 0 and 1. The output of each rule is the
product of the rule strength and the rule consequent. The rule
consequent is nothing but the linearized model that appears in the
rule. Finally, the outputs of the rules are aggregated by adding the
outputs of the fired rules. So, our fuzzy model is a weighted sum of
the linearized models. The weight of each rule is time varying as it
depends on the wind measurement at each sample.
Based on the fuzzy model in (3), firing and aggregation of the
rules leads to
xðk þ 1Þ ¼
G 12
G 14
G 16
G 18
G 20
G 22
0
0.176
0.402
0.494
0.552
0.594
0.176
0
0.142
0.253
0.326
0.382
0.402
0.142
0
0.115
0.193
0.253
0.494
0.253
0.115
0
0.080
0.142
0.552
0.326
0.193
0.080
0
0.063
0.594
0.382
0.253
0.142
0.063
0
(3)
yðkÞ ¼
n
P
j¼1
n
P
j¼1
n
o
Nj ðkÞ Adj xðkÞ þ Bdj ucpc ðkÞ ;
n
o
Nj ðkÞ Cdj xðkÞ ;
(4)
where Nj(k) is the membership value calculated at the sampling
instant k and n is the number of fuzzy rules (n ¼ 3).
As stated in Subsection 2.2, our controller is designed based on
A. Lasheen, A.L. Elshafei / Renewable Energy 87 (2016) 298e306
301
Fig. 1. The membership functions assigned to (3).
linear models for prediction. As real industrial processes are
nonlinear, there is a trend is to rely on nonlinear models. On the
other hand, the utilization of a nonlinear model will increase the
complexity of the optimization problem [29]. This may limit the
utilization of nonlinear MPC in industrial applications. So, we
propose here the use of fuzzy models as a remedy. A fuzzy model
adopts linearized models in its consequents while the overall
model can well-approximate a nonlinear system.
The following subsections describe the control loop structure
and the optimization problem setup.
3.1. The pitch control loop
Fig. 2. The generator speed measured from FAST versus the generator speed calculated
based on the fuzzy model (a) full time (120 s) (b) 1 s interval.
the TeS fuzzy model which consists of three fuzzy rules.
Fig. 2 shows the generator speed measured from the nonlinear
FAST model and the generator speed calculated based on the TeS
fuzzy model. As shown in Fig. 2, the exact output and the calculated
output are very close to each other.
As shown in Fig. 3, the pitch control loop consists of three main
components. The first component is the individual pitch controller
(IPC). A PI controller is implemented as an IPC [30]. The main
objective is to reduce the flap-wise moment on the turbine blades.
The steady state pitch angle operating point is the second
component. It depends on the average wind speed. It can be
calculated through a lookup table. The collective pitch component
is the third component. An MPC is used to control the collective
pitch angle in order to keep the generator speed and power at the
rated values. The design of the CPC will be discussed in the
following subsection.
Consider Fig. 3. Let FM1,2,3 be the flap-wise moments on the
blades, wgen be the generator speed, and U be the control action
(pitch angle). The control action, U, can be calculated as
U ¼ uipc þ u0 þ ucpc
(5)
where uipc is the IPC action, u0 is the steady state control action, and
ucpc is the CPC action.
3.2. MPC for collective pitch control
3. Model predictive collective pitch control
The proposed controller uses the fuzzy model (4) to predict the
future outputs over a specific prediction horizon. At every sampling
instant, an optimization problem is solved on-line over the prediction horizon to obtain the control actions. According to the
receding horizon policy [27], the first control action only is
implemented. The whole optimization problem is solved again in
the next sampling interval. MPC has several benefits: 1) it can
control multivariable systems subject to constraints [28], 2) it is
effective in controlling non-minimum phase and unstable systems
[29].
It is observed that most industrial predictive controllers use
The main components of the proposed fuzzy MPC are the predictor, the optimization solver, and the state estimator. A brief
description of each component is given below.
3.2.1. The predictor
Let x(k) be the current state where k is the index of the current
sampling interval. The notation x(k þ j/k) refers to the predicted
value of the state j-step ahead given information at sample k. The
future control actions are ucpc(k), ucpc(k þ 1),…, ucpc(k þ Nu1),
where Nu is the control horizon. Let NP be the prediction horizon.
The future states predicted at k are
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A. Lasheen, A.L. Elshafei / Renewable Energy 87 (2016) 298e306
Fig. 3. The structure of a pitch control loop.
xðk þ 1=kÞ ¼ Ad xðkÞ þ Bd ucpc ðkÞ
xðk þ 2=kÞ ¼ A2d xðkÞ þ Ad Bd ucpc ðkÞ þ Bd ucpc ðk þ 1Þ
«
P
P 1
xðk þ NP =kÞ ¼ AN
xðkÞ þ Bd ucpc ðk þ NP 1Þ þ Ad Bd ucpc ðk þ NP 2Þ þ … þ AN
Bd ucpc ðkÞ
d
d
P
P
P
where Ad ¼ nj¼1 Nj ðkÞAdj , Bd ¼ nj¼1 Nj ðkÞBdj , Cd ¼ nj¼1 Nj ðkÞCdj .
Let the notation y(k þ j/k) refer to the predicted output j-steps
ahead giving information at the sample k. The predicted outputs are
3.2.2. The optimization problem
The predictive control main target is to find the control vector up
that brings the predicted outputs (8) as close as attainable to the
yðk þ 1=kÞ ¼ Cd Ad xðkÞ þ Cd Bd ucpc ðkÞ
yðk þ 2=kÞ ¼ Cd A2d xðkÞ þ Cd Ad Bd ucpc ðkÞ þ Cd Bd ucpc ðk þ 1Þ
«
P
P 1
yðk þ NP =k Þ ¼ Cd AN
xðkÞ þ Cd Bd ucpc ðk þ NP 1Þ þ Cd Ad Bd ucpc ðk þ NP 2Þ þ … þ Cd AN
Bd ucpc ðkÞ
d
d
Assume that Nu < NP, this leads to ucpc(k þ Nu),…,
ucpc(k þ Np1) ¼ ucpc(k þ Nu1). Hence, the output prediction can
be expressed as
Y ¼ FxðkÞ þ wup
(8)
3
2
3
Cd Ad
yðk þ 1=kÞ
6 Cd A2 7
6 yðk þ 2=kÞ 7
d 7
where
F¼ 6
5, w ¼ ½ w1 w2 G ,
4 « 5; Y ¼ 4
«
NP
yðk þ NP =kÞ
Cd Ad
3
2
Cd Bd
0
…
0
B
…
0
C
7
6 Cd Ad Bd
d d
7,
w1 ¼ 6
«
«
«
5
4
«
NP 2
NP Nu
NP 1
A
B
…
C
A
B
C
Bd
Cd Ad
d d
d
d d
d
3
2
0
0
…
0
6
«
«
«
0 7
7
6
T
…
…
« 7
Cd Bd
w2 ¼ 6
7, G ¼ ½ 1 / 1 ,
6
4
«
«
«
0 5
P Nu 1
Cd AN
Bd … Cd Ad Bd Cd Bd
2d
3
ucpc ðkÞ
4
5.
«
and up ¼
ucpc ðk þ Nu 1Þ
2
(6)
(7)
zero set-point in the least-squares sense. The set-point is zero since
our objective is to maintain zero deviations from the rated power
and speed. Hence, the MPC cost function can be written as
J ¼ Y T Y þ uTp Rs up
(9)
where Rs is a diagonal matrix that is used as a tuning parameter to
penalize high control actions.
The pitch angle of the wind turbine has a permissible range of
variations from 0 rad to 1.57 rad with a maximum rate of 0.139 rad/s
[4]. The input constraints can be written as
uipc ðkÞ u0 ðkÞ þ umin ucpc ðkÞ
(10)
uipc ðkÞ u0 ðkÞ þ umax ucpc ðkÞ
(11)
Duipc ðkÞ Du0 ðkÞ þ amin Ducpc ðkÞ
(12)
Duipc ðkÞ Du0 ðkÞ þ amax Ducpc ðkÞ
(13)
umin and umax are the minimum and maximum values of the pitch
A. Lasheen, A.L. Elshafei / Renewable Energy 87 (2016) 298e306
303
angle, respectively. amin and amax are the minimum and maximum
pitch angle rates of change, respectively. Note that, amin and amax
depend on the sampling interval (see Appendix A.1). D is the
backward difference operator.
To find the optimal sequence of control inputs, the following
optimization problem is posed
min :J ¼ ½FxðkÞT ½FxðkÞ þ 2uTp ðwÞT FxðkÞ þ uTp ðwÞT ðwÞ þ Rs up
up
(14)
Subject to the following constraints
Tup M; and ð4Þ
(15)
where T and M can be constructed from (10)e(13) (given in
Appendix A.1). Since the cost function is quadratic and the constraints are linear, the problem is a standard constrained quadratic
program. The Hildreth's quadratic programming procedure,
described in Ref. [27], is used to solve the optimization problem
(14)e(15) (see Appendix A.2).
3.2.3. The state estimator
In this work, some states are immeasurable. So, a Kalman filter is
implemented as a state observer. The Kalman filter used here is
based on a TeS fuzzy model. The development of the Kalman filter
assumes a stochastic model of the form
xðk þ 1Þ ¼ AxðkÞ þ BuðkÞ þ dðkÞ
yðkÞ ¼ CxðkÞ þ hðkÞ
(16)
A, B, and C are constant matrices of appropriate dimensions
based on a specific linearized model. d and h are stochastic signals
that have zero means. The covariances of d and h are respectively
defined as
o
n
E dðkÞdðtÞT ¼ Q dðk tÞ
n
o
E hðkÞhðtÞT ¼ Rdðk tÞA
(17)
where E is the expectation operator, d(kt) ¼ 1 if k ¼ t and
d(kt) ¼ 0 otherwise. The matrices Q and R are design parameters.
Typically Q is positive semi-definite and R is positive definite. The
objective of a Kalman filter (observer) is to calculate an estimate
b
x ðkÞ of x(k) such that the covariance of the state estimation error is
minimized. The state estimation is calculated using the fuzzy rules
in (18) and the membership function shown in Fig. 1.
Rule #j: If the wind speed is Vj. Then
b
x ðk þ 1Þ ¼ Adj b
x ðkÞ þ Bdj ucpc ðkÞ þ Kkj yðkÞ Cdj b
x ðkÞ
j ¼ 1; 2; 3
(18)
where j is the rule index and Kkj is the Kalman gain at rule number j.
Assuming the pair (Adj,Cdj) is detectable, the gain Kkj (given in
Appendix A.3) is calculated as
1
T
T
R þ Cdj Pj Cdj
KKj ¼ Adj Pj Cdj
where Pj is the solution of the algebraic Riccati equation
1
T
T
R þ Cdj Pj Cdj
Pj ¼ Adj Pj Pj Cdj
Cdj Pj ATdj þ Q
(19)
Fig. 4. Wind speed profile at the hub level.
4. Control implementation and simulation results
In this section, we show the details of implementing the proposed controller and compare its performance to the gain schedule
PI controller provided with FAST [4], and the mixed H2/H∞
controller in Ref. [7]. First, the real system is simulated based on a
24-DOF model and a practical wind profile. Second, off-line and online calculations, that are required to calculate the pitch angle
control action, are stated. Finally, numerical results that prove the
prevalence of the proposed controller are illustrated.
4.1. Real system simulations
In order to obtain practical results, two aspects are embedded.
First, all of the 24 DOF are enabled. Second, a stochastic wind speed
profile is applied to the wind turbine. The wind speed profiles are
generated using the TurbSim software package [31]. TurbSim produces a two-dimensional wind speed profile that covers the whole
turbine body including its tower. The applied wind speed profile at
the hub level used to obtain the simulation results is shown in
Fig. 4.
4.2. The control algorithm
The proposed CPC is a fuzzy MPC. The control algorithm consists
of off-line and on-line calculations as illustrated below.
Off-line calculations:
Obtain the continuous state space linear models (Ac, Bc,Cc) at
different operating points.
Obtain the discrete state space models (Adj, Bdj,Cdj) using an
appropriate sampling interval (12.5msec is recommended in
FAST [4]).
Construct the gap-metric table as in Table 2. Then, select the
minimum number of models that can represent significantly
different operating points.
Construct the fuzzy rulebase as in (3) and let the membership
functions be as in Fig. 1.
Calculate Kalman gains for each selected model in the fuzzy
rulebase.
On-line calculations at each sampling instant:
Based on the wind speed at the hub height, calculate the
normalized weight Nj. Hence, obtain the appropriate matrices
(Ad, Bd,Cd); see (4) and (6).
304
A. Lasheen, A.L. Elshafei / Renewable Energy 87 (2016) 298e306
Fig. 7. The control signal for the mixed H2/H∞ controller and the proposed controller:
(a) CPC control signal; (b) total pitch control signal on the first blade.
Fig. 5. A comparison between the proposed controller and PI controller: (a) generator
speed; (b) generator power; (c) flapwise moments.
Use Kalman filter to estimate the design model states.
Solve the optimization problem in (14)e(15) to obtain the
optimal trajectory of the collective pitch control action.
Only the first element of the optimal control sequence, obtained
from the above step, is applied to the plant.
The total control signal is calculated as in (5).
Note: the possibility of online model-predictive-control calculations is demonstrated in Refs. [32,33].
4.3. The numerical results
Fig. 6. A comparison between the proposed controller and mixed H2/H∞ controller:
(a) generator speed; (b) generator power; (c) flapwise moments.
In this section, a comparison between the proposed fuzzy MPC
controller and two different controllers is carried out. The controllers, to be compared with the proposed one, are a gain schedule
PI controller and the mixed H2/H∞ controller [7]. The results are
shown in Figs. 5e7. The speed regulation, electric power, flap-wise
moment, and the collective pitch control signals are compared in
Table 3. In Fig. 5, the first 10 s are omitted due to the sluggish
response of the PI controller. This is because the turbine abruptly
jumps from region 1 to region 3 which imposes a severe nonlinearity to the PI controller.
The comparison depends on the average values, the standard
deviations, and the maximum absolute values of the errors. It includes the generator power, flapwise moment, generator speed,
and CP control signal. The proposed fuzzy MPC improves the
standard deviation (reduces the fluctuation) of the generator speed
by 659% and 37.4% relative to the PI controller and mixed H2/H∞
controller, respectively. Moreover, it improves the standard deviation of generator power by 307% and 22.7% relative to the PI
controller and mixed H2/H∞ controller, respectively. Further, the
proposed fuzzy MPC improves the standard deviation of the flapwise moment by 132.5% and 15.8% relative to the PI controller and
mixed H2/H∞ controller, respectively. The proposed fuzzy MPC
improves the standard deviation of the collective pitch control
signal by 550%, and 52.3% relative to the PI controller, and mixed
H2/H∞ controller, respectively.
A. Lasheen, A.L. Elshafei / Renewable Energy 87 (2016) 298e306
305
Table 3
Data analysis of the results in Figs. 5e7.
Generator speed (rpm)
Electric power (KW)
Flapwise moment (KN.m)
Collective pitch action (rad/s)
Max (abs (error))
Mean
Std (error)
Max (abs (error))
Mean
Std (error)
Max
Mean
Std
Max
Mean
Std
The proposed fuzzy MPC improves the maximum absolute value
of the error signal of the generator speed by 434% and 34.3% relative to the PI controller and mixed H2/H∞ controller, respectively.
Moreover, it improves the maximum absolute error of the generator power relative to the PI controller and mixed H2/H∞ controller
by 95.8% and 33.7%, respectively. Further, it improves the maximum
absolute error of the flap-wise moment by 26.4% and 6.8% relative
to the PI controller and mixed H2/H∞ controller, respectively. Also,
it improves the maximum absolute value of the control signal by
543% and 47.6% relative to the PI controller and mixed H2/H∞
controller, respectively.
The proposed fuzzy MPC improves the average value of the
generator speed by 0.1% relative to the PI controller. It improves the
average power by 6.77% relative to the PI controller. The average
values of the proposed fuzzy MPC and the mixed H2/H∞ controller
are near to each other.
Note that, the wind speed variations may contain wind speeds
below and above the rated value. This means that, we need a
controller in region 2 to maximize the power and in region 3 to
maintain the power at the rated value. This problem can be solved
using decentralized controllers. However, we discuss here the
operation of the wind turbine in region 3. IF the wind speed value
goes below the rated value, the pitch controller will be automatically disconnected. As shown in Fig. 4, the stochastic wind speed
profile contains wind speeds below the rated value in the interval
between 125 s and 140 s the corresponding pitch angle control
action is indeed equal to zero as shown in Fig. 7. This means that the
pitch controller functions as it should.
From the above analysis, it is evident that the proposed fuzzy
MPC improves the overall performance of the wind turbine.
5. Conclusions
This paper has proposed a new design of a fuzzy predictive algorithm for collective pitch control. A fuzzy model has been used to
model a large 5-MW wind turbine. The size of the rulebase of the
fuzzy model has been significantly reduced using a gap-metric
criterion. Consequently, a model predictive controller has been
designed to act as a collective pitch controller. The proposed predictive controller can naturally accommodate hard constraints on
the pitch angles and their rates of change. Since the calculations
have required the system states, a Kalman observer has been
designed to estimate the immeasurable system states. The simulations have been carried out utlizing a FAST model for a 5-MW
offshore wind turbine. The simulation results have compared the
proposed fuzzy MPC to different control strategies including a gain
schedule PI controller and an H2/H∞ mixed controller. The results
of the comparison (shown in Table 3) have confirmed significant
enhancements in power harvest, mechanical load reduction, and
speed regulation.
Gain schedule PI
Mixed H2/H∞
Proposed controller
85.04
1170.6
28.03
1573.6
4537.5
570.88
7328
3287
1449
0.1878
0.0142
0.0423
21.382
1172.1
5.07
1074.8
4864
172.11
6193
4269
721.65
0.0431
0.0031
0.0099
15.92
1171.8
3.69
803.74
4867.1
140.25
5798
4271
623.08
0.0292
0.0031
0.0065
Appendix A
A.1. Parameters of equation (15)
3
uipc ðkÞ u0 ðkÞ þ umax
7
6
6 1 0 … 0 7
7
uipc ðkÞ þ u0 ðkÞ umin
6
6
7
7
T¼6
7; M ¼ 6
6 u ðk 1Þ Du ðkÞ Du ðkÞ þ amax 7 ,
4 1 0 … 05
5
4 cpc
0
ipc
1 0 … 0
ucpc ðk 1Þ þ Du ðkÞ þ Du0 ðkÞ amin
2
0 … 0
1
3
2
ipc
amax ¼ 0:139*Ts ; amin ¼ 0:139*Ts
where Ts is the sampling time.
A.2. Hildreth's quadratic programming procedure
The cost function (14) can be rewritten in the form of quadratic
cost function as following:
min: J ¼
up
1 T
u Eup þ uTp F
2 p
Subject to. Tup M
where E ¼ 2*ððwÞT ðwÞ þ Rs Þ; F ¼ 2*ððwÞT FxðkÞÞ
The primal problem is equivalent to:
max: min:
l0
up
1 T
u Eup þ uTp F þ lT Tup M
2 p
where the vector l contains the Lagrange multipliers. The minimization over up is unconstrained and is attained by:
up ¼ E1 F þ T T l
Substituting up in the primal problem, the dual problem is
written as:
max:
l0
1 T
1
l Hl lT K F T E1 F
2
2
where H¼ TE1TT, K ¼ MþTE1F
Thus, the dual is also a quadratic programming problem with l
as the decision variable. The dual problem is equivalent to:
min:
l0
1 T
1
l Hl þ lT K þ MT E1 M
2
2
306
A. Lasheen, A.L. Elshafei / Renewable Energy 87 (2016) 298e306
Note that the dual problem may be much easier to solve than the
primal problem because the constraints are simpler. The main steps
in Hildreth's algorithm to solve the dual problem are following:
Step 1: The objective function is regarded as a quadratic function
in li
Step 2: Adjust li to minimize the objective function. If this requires li < 0, set li ¼ 0
Step 3: Move to the next component liþ1 and repeat step 1 and
2.
If we consider one complete cycle through the components to be
one iteration taking the vector lm to lmþ1 the method can be
expressed explicitly as
[6]
[7]
[8]
[9]
[10]
[11]
[12]
mþ1
; wmþ1
lmþ1
¼
max
0;
w
i
i
i
2
3
i1
n
X
X
1 4
5
Ki þ
¼
hij lmþ1
þ
hij lm
j
j
hii
j¼1
j¼iþ1
where hij is the ijth element in the matrix H¼ TE1TT, and Ki is the ith
element in the vector
[13]
[14]
[15]
[16]
[17]
[18]
K ¼ M þ TE1 F
[19]
[20]
when the algorithm
up ¼ E1(F þ TTl*)
converges
to
l*, the solution is
[21]
[22]
A.3. Parameters of the Kalman's observer
Q ¼ 2, R ¼ 0.01.
[23]
[24]
T
Kk1 ¼ ½ 0:0001 0:0009 0:0018 ;
Kk2 ¼ ½ 0:0002 0:0010 0:0021 T ;
Kk3 ¼ ½ 0:0002 0:0010 0:0024 T ;
[25]
[26]
[27]
[28]
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