In the wind farm power reference dispatch section, the power reference dispatch controller regulates all the WT active outputs of the wind farm. The controller aims to minimize the fatigue load of WT and track the active power allocated by the frequency adjustment controller. Typically, the sampling time of the wind farm controller is in seconds [
28]. Therefore, the rapid dynamics of generators and pitch actuators can be ignored [
37]. In addition, shaft torsion and tower point oscillations are ignored to reduce the complexity of the model.
The fatigue loads of WTs can be divided into two parts: one is aerodynamic loads and gravity loads (external), and the other is structural loads (internal) [
38]. In this paper, the fatigue loads mainly focus on the loads of the drive train due to the torsion of the shaft and the loads of the tower structure due to the tower deflection. Compared with static loads, the dynamic stress causing structural damage of WTs is a much bigger issue. By reducing the fluctuations of low-speed shaft torque
Ts and thrust force
Ft, the related fatigue loads can be reduced.
can be represented by a combination of
and
when the drive train and tower structure loads are considered.
4.1. Improved Model of Fatigue Load Sensitivity
The WT (NREL 5 MW) developed by the National Renewable Energy Laboratory (NREL) is used in this paper [
39,
40]. The oscillations in the shaft torsion and tower nodding are disregarded, the fluctuations of wind speed are ignored based on the previous report [
30]. In order to better optimize the calculation, the equations of fatigue load sensitivity are re-derived. In the process of the optimization dispatch, the redefined fatigue load sensitivity will be used. The active power of the WT output is still controlled by adjusting the pitch angle and torque.
The equivalent mass
Jt of drive train system is described by [
41]
where
Jr is the rotor mass;
Jg is the generator mass;
ηg is the gear box ratio.
The low-shaft motion equation is described by
where
ωr is the measured rotor speed;
Trot is the aerodynamic torque;
Tg is the generator torque;
The measured generator speed
ωg is filtered by a low-pass filter and the filtered speed
ωf is
where
τf is time constant of the filter of
ωg.
According to the deviation of
ωf from generator rated speed
ωg-rated, pitch angle reference
θref can be obtained by the PI controller.
where
kp is the proportional gain;
ki is the integral gain;
ka is a function of
θref, defined by
. Where
ka1 and
ka2 are the constants.
By defining
(8) is transformed into
According to the motion equation of shaft torque
Ts,
, and
can be described by
where
ωg is the measured generator speed;
B is the main shaft viscous friction coefficient.
According to (11) and (12),
can be expressed by
The time of the operating point is assumed to be
k. The wind speed
v is a variable that can be estimated or measured [
42]. In this study,
v is estimated. The value at
t =
k is
v0 and is assumed to be constant over a short control period. The measured power output, generator speed, filtered speed, and pitch angle are defined as
Pg0,
ωg0,
ωf0, and
θ0 at
t =
k, respectively. The
Trot and
Tg at
t =
k can be defined as
Trot0 and
Tg0, respectively.
Based on (6), (7), (10), and (13), the incremental form can be obtained as
The aerodynamic torque
Trot is calculated by
where
R is the length of the blade;
ρ is the air density;
v is the wind speed on the rotor;
Cp is the power coefficient;
λ is the tip speed ratio, defined by
; In order to simplify the expression,
Psim is defined by
.
According to (18), Δ
Trot can be calculated by
where
Cp is described in a lookup table derived from the inputs
λ and
θ, as is shown in
Table 2. Where
n and
m are the corresponding rows and columns, respectively.
The generator torque reference
Tg_ref is filtered by a low-pass filter and the generator torque
Tg is derived by,
where
τg is the time constant of the filter of
Tg_ref.
Tg_ref is calculated by
According to (23) and (24), Δ
Tg and Δ
Tg_ref can be calculated by
where Δ
t is the control cycle.
According to (14)–(17), the continuous state space model for WT is formulated as
where
, and the state space matrices are
These matrices change every other dispatch cycle. Then, the continuous state space model is discretized with the sampling period
ts, which is
with
Substituting (6) and (11), the shaft torque
Ts can be calculated by
Based on (19) and (25), (30) can be transformed into
with
In order to simplify the expressions,
and
are defined by
Therefore, the fatigue load sensitivity of the drive train can be expressed as
According to [
43], The tower dynamics is not included in the simplified WT model. According to, it is assumed the tower base overturning moment
Mt can be approximately derived by
where
H is the tower height.
Ft is thrust force.
The thrust force
Ft is calculated by
where
Ct is the thrust coefficient.
Similar to (34) and (35),
Therefore, the fatigue load sensitivity of the tower structure can be expressed as
The fatigue load sensitivities considered in the paper are dynamic loads causing structural damage. The equation shown in (33) and (42) are not equations for calculating a fatigue load, but equations for calculating the change in the shaft torque Δ
Ms and the tower bending moment Δ
Mt associated with the fatigue load. The parameters in the equation are changing at different times. By reducing Δ
Ms and Δ
Mt, the corresponding fatigue load can be reduced. The calculations are carried out by this law of effect. Reductions on the changes of the moments Δ
Ms and Δ
Mt are highly corelated to reductions in the damage equivalent fatigue load [
44]. Δ
Ms and Δ
Mt are related to changes in power reference. Δ
Ms and Δ
Mt can be reduced by properly distributing the active power. Δ
Ms and Δ
Mt are reduced for each sampling period, so the fatigue loads are reduced.
4.2. Cost Function and Constraints
In
Figure 1, every individual WT is equipped with an exclusive control system that can follow the power references provided by the frequency adjustment controller.
The controller minimizes the variation of shaft torque
Ts and thrust force
Ft to reduce the fatigue load. Accordingly, the cost function is expressed as
For the convenience of calculation, the equivalent calculation formula is expressed as
where
ξ is the weight coefficient.
The constraints are expressed by
where
is the demanded power of WF;
is the maximum available power of WT-
i, it can be estimated by
where
is the rated power of WT;
vnac is the nacelle wind speed of WT. The corresponding data of
Cp and
Ct for this study can be accessed in the wind turbine model of SimWindFarm. The plot of
Cp(
λ,
θ) and
Ct(
λ,
θ) based on the lookup table is shown in
Appendix A.
The presented problems can be expressed as standard quadratic programming (QP) issues [
45,
46]. It can be effectively solved by a commercial solver. This optimization problem can be solved by different optimization methods. Group intelligent algorithms such as particle swarm optimization (PSO) and genetic algorithm (GA) have good search performance for solving complex problems, but the calculation time is long and is not suitable for real-time online optimization [
47,
48,
49,
50]. The programming algorithm has a good ability to search for non-complex solving problems, and the calculation time is short, which is suitable for online optimization. The model proposed in this paper belongs to the non-complex online solution model.
Then, the matrix
H and the matrix
f of QP can be expressed as
If the available power of WF does not meet the demand, the following procedure is available.