3.1. Modeling of Wake Superposition Effect
Almost all wind farms have wake effects, which refer to the influence of rotating
mechanical components, blades, and tower structures generated during the operation
of WT on wind speed and direction in the wind field. When a WT operates, its rotating
blades deprive the air of kinetic energy and provide electricity by converting it
into mechanical energy. During this process, the WT generates an area called the wake
region, where the wind speed is lower than the surrounding area and the wind direction
may also change. The wake effect can affect the layout and operational efficiency
of wind farms, leading to mutual influence between turbines. Especially for those
located downwind, changes in wind speed and direction can affect their power generation
capacity. Meanwhile, the wake effect can also have a negative impact on the safety
of other equipment and structures in the wind farm, such as vibration and corrosion
of WTs. To reduce the impact of wake effects on wind farms, the wind farm layout and
the WT arrangement need to be carefully designed. Researchers adopt meteorological
data, fluid dynamics simulations, and field observations to minimize wake effects
and improve the overall efficiency of wind farms. In addition, some advanced control
strategies and technologies have also been applied to WT to adjust the blade angle
and the interaction between generator units, reducing the impact of wake effects.
Despite the wake effect, wind power remains an important clean energy source due to
the richness and renewability of wind energy. The optimization of wake effect is one
of the research hotspots in the wind power industry. The airflow model and actual
diagram of the WT are shown in Fig. 1.
Fig. 1. The air-flow model and actual diagram of the wind turbine.
In Fig. 1(a), $V$, $V_1$ and $V_2$ are the wind speeds at the WT, upstream and downstream of the
WT, respectively. $S$, $S_1$ and $S_2$ are the cross-sectional areas of air flow at
the WT, upstream and downstream of the WT, respectively. Fig. 1(b) shows the real-life image of WT. The interior of WT includes wind and direction sensing
system, rotating blades, alternator, and power distribution unit [16]. WT rotation is driven by the kinetic energy generated by the air-flow, and the air
density remains unchanged. According to the law of conservation of momentum, it is
shown in Eq. (1).
In Eq. (1), $F$ is the axial thrust exerted on the WT. The wind energy utilization coefficient
$C_P$ and the thrust coefficient $C_T$ are shown in Eq. (2).
In Eq. (2), $a$ is the axial induction factor. Due to the simplicity, speed, and high computational
efficiency of the Jensen wake model, it is chosen as the wake model. According to
the Bates limit theory, the relationship between $V_1$, and $V_2$, and $C_T$ is shown
in Eq. (3).
When calculating the captured wind speed $V_j$ of the WT, the wake region of the upstream
WT may contain downstream WT, resulting in wake superposition effects. The wake superposition
has an impact on the captured wind speed and power generation efficiency of the downstream
WT. The wake superposition is shown in Fig. 2.
Fig. 2. Schematic diagram of wake superposition.
In Fig. 2, $j$ is the unit number. The radius of the WT is $R_0$. $RX_{i,j}$ is the wake radius
of the $i$-th upstream at WT $j$. $X_{i,j}$ is the distance from the upstream $WT_i$
following the wind direction to the downstream $WT_j$. $A_{i,j}^{shadow}$ is the shadow
area of the wake for the upstream $i$-th at WT $j$. $A_0$ is the area swept by the
WT rotor. The effect $RX_{i,j}$ of wake attenuation factor $k$ is shown in Eq. (4).
The captured wind speed of downstream WT is shown in Eq. (5).
In Eq. (5), $n$ represents the number of units. For centralized optimization control of wind
farms, optimization control can help wind farms achieve GMP status. The goal $P_{GMP}$
is the GMP of all WTs, as shown in Eq. (6).
The expression relationship of the wake model is shown in Eq. (7).
In Eq. (7), $R$ is the weight factor of the total change in pitch angle. The capture power of
the WT is shown in Eq. (8).
The upper and lower limits of rotational speed and pitch angle are shown in Eq. (9).
The power generation of wind farms is affected by wind speed. Because wind speed is
time-varying, the output power of wind farms also fluctuates. This fluctuation may
pose certain challenges to the stability and reliability of the power grid. Through
flexible active power regulation, wind farms can adjust the output power to meet the
power grid needs. When the power grid requires additional power, wind farms can increase
the output power. When the power grid needs to reduce power, wind farms can reduce
output power. This flexibility can help maintain balance in the power grid and reduce
reliance on other power generation equipment. To achieve active power control of wind
farms and prolong service life, the speed and pitch angle are combined together. In
different wind speeds, the unified optimization objective is shown in Eq. (10).
In equation (10), $\beta_{j0}$ is the initial pitch angle of the unit. $E_{j0}(\omega_{j0})$ is the
initial kinetic energy of the unit. $Q$ is the weight factor of the total change in
kinetic energy. The steady-state power output of the wind farm is shown in Eq. (11).
In Eq. (11), $d\%$ is the load shedding rate. $P_{j0}$ is the initial power of the unit. The
load shedding rate limits the wind farm power, which can ensure that the power output
of the wind farm does not exceed its rated capacity or the carrying capacity. When
$(1+d\%) \sum_{j=1}^{n} P_{j0} > P_{GMP}$, the output power of the wind farm is $P_{GMP}$.
In objective function $J$, kinetic energy optimization refers to controlling the $d\%$
of WT to reasonably absorb or release kinetic energy. According to Eq. (10), when $Q < 0$, $Q < 0$, and $d\% < 0$, $P_{GMP}$ is higher than the rated power $P_n$.
The kinetic energy reaches the maximum absorption. When $Q < 0$, $d\% > 0$, and $P_{GMP}
< P_n$, the kinetic energy reaches the maximum release [17].
3.2. Wind Farm Active Power Optimization Algorithm Combining K-Means Clustering and
Improved Particle Swarm Optimization
Due to the nonlinear effects of wake models and WT capture power, the wind farm optimization
algorithm is implemented based on an improved PSO algorithm. However, the continuous
expansion of wind farms will greatly increase computational cost and complexity, posing
challenges to real-time control of wind farms. In order to reduce computational burden,
parallel processing is chosen for real-time control in large-scale wind farms. Therefore,
based on the PSO, the particle velocity and position obtained through second-order
oscillation improvement are shown in Eq. (12)
[18,
19].
In Eq. (12), $i$ is the number in the particle swarm. $d$ is the particle dimension. $t$ is the
current number of iterations. $\mu$ is the inertia weight. $r_1$ and $r_2$ are random
numbers uniformly distributed in $[0,1]$. $c_1$ is the self-learning factor. $c_2$
is the social learning factor, generally taking 1.5. $P_{besti}$ is its extreme value.
$G_{best}$ is the global peak. $\xi_1$, $\xi_2$, $\xi_3$ and $\xi_4$ are expressed
in Eq. (13).
In Eq. (13), $t_{\max}$ is the maximum number of iterations. After adopting second-order oscillation,
it plays different roles in different optimization stages of the algorithm. The second-order
oscillation in the early stage has strong global search ability. The reason is that
it introduces oscillatory behavior, causing the system to oscillate continuously in
the search space, which covers a wider solution space. This global search ability
is beneficial for the initial stage algorithm to quickly explore the solution space
and find potential optimization directions. The second-order oscillation in the later
stage can enhance the local search ability and achieve asymptotic convergence. As
the algorithm approaches the optimal solution, the impact of oscillation link will
gradually weaken. The oscillation range of the system is reduced, allowing the algorithm
to search more accurately in local areas. This local search ability can help the algorithm
gradually approach the optimal solution in the later stages, improving the convergence
performance [20,
21]. The wind power generation system based on the improved algorithm is shown in Fig. 3.
Fig. 3. Wind power generation system based on improved algorithm.
In Fig. 3, wind farm control mainly includes centralized control and local control of WT. After
the algorithm is improved, the power grid can provide feedback on the status of wind
farms. To implement the optimization algorithm for wind farms, it is necessary to
first initialize the velocity and position of each particle $C_p$. Then the fitness
value of each particle is calculated and the optimal value is recorded. After updating
the particle velocity and position, the penalty weight and constraint penalty value
for each particle are calculated to update the historical and global optima. Whether
the iteration has ended is judged. If not, the fitness value of each particle is recalculated
and the optimal value is recorded. If the iteration is completed, the optimized speed
and pitch angle are output and distributed to the wind farm. The current status $\omega_{j0}$,
$\beta_{j0}$, and $P_j$ of the wind farm are obtained. Based on grid frequency and
dispatch instructions, the weight values in the objective function $J$ in Eq. (10) are determined. The objective function and fitness function in the optimization algorithm
are clearly defined. The values in the power constraint are determined. The constraint
function is clearly defined. For inertia weights, linear decrement is used [22], as shown in Eq. (14).
In Eq. (14), $\mu_{\max}$ and $\mu_{\min}$ are the maximum and minimum values of inertia weight,
respectively. Due to the increased wind farm scale, the used WT adds, increasing the
computational complexity of PSO. To avoid the influence of calculations on speed control,
K-means clustering is used to reduce the number of control parameters. The initial
clustering reference value selects $K$ data points in the dataset, which is characterized
by randomness. After that, the Euclidean distance between the data point and the reference
value is calculated. The data points are assigned to group $K$ according to the distance.
Finally, the reference value of each data point in the group is calculated repeatedly.
When the reference value is the maximum number of iterations or no longer changes,
the calculation ends. The relative position and wind direction of WT is shown in Eq.
(15).
The wake size is determined by $\Omega_{i,j}$, and its distribution is consistent.
Therefore, the same control parameters can be used between the same group, which reduces
the amount of calculation and improves the optimization speed. The contour coefficient
method is selected to determine the $K$ value in the study. $m$ is the number of samples.
$i$ is the sample point, as shown in Eq. (16).
In Eq. (16), $a(i)$ is the average distance between $i$ and the remaining samples of each group.
$b(i)$ is the minimum value of the average distance between $i$ and the remaining
samples. The values of $S$ and $K$ corresponding to different cluster numbers are
the best cluster number. Namely, the cluster number corresponding to the maximum value
of $S$ is selected [23].