A. Fatigue Analysis of the WT
A. WT 的疲劳分析

Due to the complex structure of WTs, fatigue load analysis generally focuses on the main structure. According to the results of [13], the drivetrain is one of the failure-prone structures and has a greater impact on the WT which has a greater impact on the WT. And the damage of the tower will bring the most serious economic loss. Therefore, in this paper, the fatigue loads of the drivetrain and tower are selected for analysis and modeling. The model schematic of the transmission system and the tower is shown in Fig. 1.
由于水轮机结构复杂，疲劳载荷分析一般集中在主体结构上。根据结果 [13] ，传动系统是容易发生故障的结构之一，对WT影响较大。而铁塔的损坏将带来最严重的经济损失。因此，本文选择传动系统和塔架的疲劳载荷进行分析和建模。输电系统及铁塔模型示意图如图所示 Fig. 1 。

Fig. 1.  图 1.

The causes for the fatigue load of the drivetrain and tower.
传动系统和塔架疲劳载荷的原因。

Show All

1) Drivetrain 1) 传动系统

The drivetrain can usually be simplified to the two-mass model shown in Fig. 1. The unbalanced torque on both sides is the cause of the fatigue load. The characteristics of the drivetrain can be expressed as follow:

⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪Jmω˙r=Tm−TsTs=Kspψ+Kviψ˙Jeω˙e=Ts/Ngear−Teψ˙=ωr−ωe/Ngear(1)

View Source \begin{equation*} \left\{ \begin{array}{l} {J}_{\mathrm{m}}{{\dot{\omega }}}_{\mathrm{r}} = {T}_{\mathrm{m}} - {T}_{\mathrm{s}}\\ {T}_{\mathrm{s}} = {K}_{\text{sp}}\psi + {K}_{\text{vi}}\dot{\psi }\\ {J}_{\mathrm{e}}{{\dot{\omega }}}_{\mathrm{e}} = {T}_{\mathrm{s}}/{N}_{\text{gear}} - {T}_{\mathrm{e}}\\ \dot{\psi } = {\omega }_{\mathrm{r}} - {\omega }_{\mathrm{e}}/{N}_{\text{gear}} \end{array} \right. \tag{1} \end{equation*}
传动系统通常可以简化为如图所示的两质量模型 Fig. 1 。两侧扭矩不平衡是产生疲劳载荷的原因。传动系统的特性可表示为：

⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪Jmω˙r=Tm−TsTs=Kspψ+Kviψ˙Jeω˙e=Ts/Ngear−Teψ˙=ωr−ωe/Ngear(1)

View Source \begin{equation*} \left\{ \begin{array}{l} {J}_{\mathrm{m}}{{\dot{\omega }}}_{\mathrm{r}} = {T}_{\mathrm{m}} - {T}_{\mathrm{s}}\\ {T}_{\mathrm{s}} = {K}_{\text{sp}}\psi + {K}_{\text{vi}}\dot{\psi }\\ {J}_{\mathrm{e}}{{\dot{\omega }}}_{\mathrm{e}} = {T}_{\mathrm{s}}/{N}_{\text{gear}} - {T}_{\mathrm{e}}\\ \dot{\psi } = {\omega }_{\mathrm{r}} - {\omega }_{\mathrm{e}}/{N}_{\text{gear}} \end{array} \right. \tag{1} \end{equation*}

2) Tower 2) 塔

The main source of tower damage is the aerodynamic thrust F_t driving the rotation of the turbine, which is related to the wind speed v, pitch angle β, and tip-speed ratio λ. Since the mechanism of the tower is similar to the cantilever arm shown in Fig. 1, researchers usually consider the fatigue load of the tower base as the fatigue load of the whole tower and use the tower base bending moment to evaluate the fatigue load of the tower. The expressions for the force and bending moment of the tower base are shown in (2) and (3).

{Ft=0.5πρR2Ct(λ,β)v2λ=ωrR/vMt=Ft⋅h(2)(3)

View Source \begin{align*} &\left\{ {\begin{array}{l} {{F}_{\mathrm{t}} = 0.5\pi \rho {R}^2{C}_{\mathrm{t}}(\lambda,\beta){v}^2}\\ {\lambda = {\omega }_{\mathrm{r}}R/v} \end{array}} \right. \tag{2}\\ &{M}_{\mathrm{t}} = {F}_{\mathrm{t}} \cdot h \tag{3} \end{align*}
塔筒损坏的主要来源是驱动涡轮机旋转的气动_推力Ft ，它与风速v 、桨距角β和叶尖速比λ有关。由于塔的机构类似于图中所示的悬臂 Fig. 1 研究人员通常将塔基的疲劳载荷视为整个塔的疲劳载荷，并利用塔基弯矩来评估塔的疲劳载荷。塔底受力和弯矩表达式如(2)和(3)所示。

{Ft=0.5πρR2Ct(λ,β)v2λ=ωrR/vMt=Ft⋅h(2)(3)

View Source \begin{align*} &\left\{ {\begin{array}{l} {{F}_{\mathrm{t}} = 0.5\pi \rho {R}^2{C}_{\mathrm{t}}(\lambda,\beta){v}^2}\\ {\lambda = {\omega }_{\mathrm{r}}R/v} \end{array}} \right. \tag{2}\\ &{M}_{\mathrm{t}} = {F}_{\mathrm{t}} \cdot h \tag{3} \end{align*}

B. DEL Calculation Method
B. DEL 计算方法

DEL is defined as the amplitude of a sinusoidal stress that produces the same damage as the original signal at a constant frequency f for a time T and is calculated as follows:

View Source \begin{equation*} DEL = {\left({\sum\limits_{j = 1}^M {\frac{{\sigma _j^m{n}_j}}{{Tf}}} } \right)}^{\frac{1}{m}} \tag{4} \end{equation*}

DEL 定义为在恒定频率f下持续时间T产生与原始信号相同损伤的正弦应力的幅值，计算公式如下：

DEL=(∑j=1MσmjnjTf)1m(4)

View Source \begin{equation*} DEL = {\left({\sum\limits_{j = 1}^M {\frac{{\sigma _j^m{n}_j}}{{Tf}}} } \right)}^{\frac{1}{m}} \tag{4} \end{equation*}

In terms of parameter selection, Tf = 1, and m = 4.
参数选择方面， Tf =1， m =4。

Based on the definition in (4), the calculation of DEL requires specific information about the original signal. It is necessary to convert the complex load history into a load reversal set that affects fatigue based on the rain flow counting method. And calculate DEL based on the Palmgren-Miner method [26].
根据(4)中的定义，DEL的计算需要原始信号的特定信息。需要基于雨流计数法将复杂的荷载历史转换为影响疲劳的荷载反转集。并基于Palmgren-Miner方法计算DEL [26] 。

In the DEL calculation of WT mechanical structures, DEL can be calculated based on bending moments rather than stresses because the main shaft, tower, etc. can be simplified to a bending uniform section, that is, the fluctuations of T_s and M_t can be used to calculate the DEL [27].
在WT机械结构的DEL计算中，可以根据弯矩而不是应力来计算DEL，因为主轴、塔架等可以简化为弯曲均匀截面，即T _s和M _t的波动可以用于计算 DEL [27] 。

According to (4), the calculation of DEL depends on the accurate measurement of T_s and M_t. And the calculation process of rain flow counting is complicated. These conditions make it difficult to apply DEL to the online control of WTs.
根据式(4)，DEL的计算取决于T _s和M _t的准确测量。而且雨流计数的计算过程比较复杂。这些条件使得DEL难以应用于WT的在线控制。

A. Parameter Selection For DNN
A. DNN 的参数选择

In order to use a DNN to calculate DEL, this subsection begins by selecting the parameters related to DEL as the input to the DNN. The input should be easily measurable, not conventional torque data. According to (1)–(4), the easily measurable parameters related to DEL of the drivetrain and tower include wind speed, rotor speed, pitch angle and active power. Therefore, this section constructs the input of DNN based on the above parameters as shown in Table I.
为了使用 DNN 计算 DEL，本小节首先选择与 DEL 相关的参数作为 DNN 的输入。输入应该易于测量，而不是传统的扭矩数据。根据（1）-（4），与传动系统和塔架的DEL相关的易于测量的参数包括风速、转子速度、桨距角和有功功率。因此，本节根据上述参数构造DNN的输入，如下所示 Table I 。

TABLE I Nine Selected Indicators for the Input of DNN
表1 DNN输入的九个选定指标

According to the definition of DEL, the magnitude and fluctuation of pressure are two important dimensions that affect DEL. Therefore, the metrics constructed in this paper are in the form of mean and standard deviation. In practical application, the above nine indicators can be obtained from the data in the SCADA system by simple calculation, which is easier to obtain compared to torque.
根据DEL的定义，压力的大小和波动是影响DEL的两个重要维度。因此，本文构建的指标采用均值和标准差的形式。在实际应用中，上述九个指标可以通过简单的计算从SCADA系统中的数据中获得，相对于扭矩来说更容易获得。

B. Data Preparation For DNN
B. DNN 的数据准备

In order to characterize the coupling between the above parameters and DEL, a Monte Carlo experiment based on SimWindFarm [28] is designed in this subsection to generate the operational data. The experiment was conducted on a simulated WT, the NREL 5MW, which is widely used in research in this field and developed by the National Renewable Energy Laboratory. Parameters such as wind speed and power have been artificially designed in order to simulate different working conditions. The WT parameters and experimental parameters are set as shown in Tables II and III.
为了表征上述参数与DEL之间的耦合性，基于SimWindFarm进行了蒙特卡罗实验 [28] 本小节设计用于生成操作数据。该实验是在模拟WT（NREL 5MW）上进行的，该模型由国家可再生能源实验室开发，广泛应用于该领域的研究。风速、功率等参数经过人为设计，以模拟不同的工况。 WT参数和实验参数设置如图 Tables II 和 III 。

TABLE II NREL 5MW WT Parameters
表 II NREL 5MW WT 参数

TABLE III Setting of Experimental Conditions for WTs
表III WT的实验条件设置

In Table I, the mean of wind speed, turbulence intensity, and mean of power are preset by SimWindFarm. A total of 480 sets of working conditions are set at equal intervals. The standard deviation of power and power fluctuations cannot be preset. In this paper, a range of variable random numbers are added to the power set values in the simulation, and there are a total of 100 sets of random numbers. Therefore, 48000 different working conditions were constructed for the experiments in this paper. A total of 48000 sets of data were obtained for training DNN. In the acquisition of experimental data, each group of data is calculated for 300 s with a sampling interval of 1 s. That is, the mean, variance and other parameters are found for the data of 300 time points.
在 Table I 、风速平均值、湍流强度平均值和功率平均值由 SimWindFarm 预设。等间隔设置工况共480组。功率和功率波动的标准差无法预设。本文在仿真中将一系列可变随机数添加到幂集值中，总共有100组随机数。因此，本文构建了48000个不同的工况进行实验。总共获得了48000组数据用于训练DNN。实验数据的获取时，每组数据计算300 s，采样间隔1 s。即对300个时间点的数据求均值、方差等参数。

The above nine indicators are obtained from the equations in Table I. DEL_Ts and DEL_Mt are calculated by the MCrunch code [26]. DEL is also calculated as the other parameters mentioned before, using the data sampled over 300 s to obtain. After data collection, the Pearson correlation coefficient (PCC) are shown in Table IV.
上述九个指标由式子求得： Table I 。 DEL _Ts和DEL _Mt由 MCrunch 代码计算 [26] 。 DEL 的计算方法也与前面提到的其他参数一样，使用 300 s 内采样的数据来获得。数据收集后，皮尔逊相关系数（PCC）显示在 Table IV 。

TABLE IV PCC of DEL and the Selected Indicators
表 IV DEL 的 PCC 和所选指标

The numbers in Table IV indicate the correlation between the data. The higher the number, the higher the correlation. Negative (positive) numbers indicate negative (positive) correlation [30]. In this paper, the six sets of data with the highest correlation with DEL_Ts and DEL_Mt are selected as input, DEL_Ts and DEL_Mt as output for DNN training. Six sets of data with higher correlation with DEL_Ts are I, P¯e, std(Pe), std(ΔPe), std(β) and std(ωr). Six sets of data with higher correlation with DEL_Mt are I, v¯, P¯e, std(Pe), std(ΔPe), std(β), ω¯r.
中的数字 Table IV 表明数据之间的相关性。数字越大，相关性越高。负（正）数表示负（正）相关 [30] 。本文选择与DEL _Ts和DEL _Mt相关性最高的六组数据作为输入，DEL _Ts和DEL _Mt作为输出进行DNN训练。与DEL _T相关性较高的六组数据是I ， P¯e , std(Pe) , std(ΔPe) , std(β) 和 std(ωr) 。与DEL _Mt相关性较高的六组数据是I ， v¯ , P¯e , std(Pe) , std(ΔPe) , std(β) , ω¯r 。

C. Model Structure of DNN
C. DNN 的模型结构

DNN includes an input layer, several hidden layers and an output layer, as shown in Fig. 2. The first layer is the input layer, the last layer is the output layer, and the middle layers are the hidden layers. All neurons between adjacent layers are interconnected. There are forward transmission and backward transmission in DNN. Forward transmission initially builds a neural network, and backward transmission optimizes the neural network.
DNN 包括一个输入层、几个隐藏层和一个输出层，如图所示 Fig. 2 。第一层是输入层，最后一层是输出层，中间层是隐藏层。相邻层之间的所有神经元都是互连的。 DNN中有前向传输和后向传输。前向传输最初构建神经网络，后向传输优化神经网络。

Fig. 2.  图 2.

Schematic diagram of DNN.
DNN 示意图。

Show All

1) Forward Transmission 1）前向传输

Each two neurons of adjacent layers have different weight coefficients bias ε vectors b between. In (5)–(6), ε and b are used for data transfer from the input layer to the output layer.

χliφli=∑j=1Nl−1εlijφl−1j+bli=Φ(εlijφl−1j+bli)(5)(6)

View Source \begin{align*} \chi _i^l &= \sum\limits_{j = 1}^{{N}_{l - 1}} {\varepsilon _{ij}^l\varphi _j^{l - 1}} + b_i^l \tag{5}\\ \varphi _i^l &= \Phi (\varepsilon _{ij}^l\varphi _j^{l - 1} + b_i^l) \tag{6} \end{align*}
相邻层的每两个神经元具有不同的权系数偏差 ε 之间的向量b 。在(5)-(6)中， ε 和b用于从输入层到输出层的数据传输。

χliφli=∑j=1Nl−1εlijφl−1j+bli=Φ(εlijφl−1j+bli)(5)(6)

View Source \begin{align*} \chi _i^l &= \sum\limits_{j = 1}^{{N}_{l - 1}} {\varepsilon _{ij}^l\varphi _j^{l - 1}} + b_i^l \tag{5}\\ \varphi _i^l &= \Phi (\varepsilon _{ij}^l\varphi _j^{l - 1} + b_i^l) \tag{6} \end{align*}

2) Back Transmission 2) 回传

The purpose of backpropagation is to complete the update of the weight coefficient ε and the bias vector b in forward propagation by rational iterative optimization of the loss function. The loss function in this paper is defined as follows:

J=12∥∥φL−Y∥∥22(7)

View Source \begin{equation*} J = \frac{1}{2}\left\| {\boldsymbol{\varphi }_{}^L - \boldsymbol{Y}} \right\|_2^2 \tag{7} \end{equation*}
反向传播的目的是完成权重系数的更新 ε 以及通过损失函数的理性迭代优化得到的前向传播中的偏置向量b 。本文的损失函数定义如下：

J=12∥∥φL−Y∥∥22(7)

View Source \begin{equation*} J = \frac{1}{2}\left\| {\boldsymbol{\varphi }_{}^L - \boldsymbol{Y}} \right\|_2^2 \tag{7} \end{equation*}

To complete the update of the weight coefficient ε and the bias vector b, the following calculations are performed:

ςl∂J∂ωl∂J∂bl=∂J∂χL=(φL−Y)⊗Φ′(χL)=ςl(φL−1)T=ςl(8)(9)(10)

View Source \begin{align*} \varsigma _{}^l &= \frac{{\partial J}}{{\partial \chi _{}^L}} = (\boldsymbol{\varphi }_{}^L - \boldsymbol{Y}) \otimes {\Phi }^{\prime}(\chi _{}^L) \tag{8}\\ \frac{{\partial J}}{{\partial \omega _{}^l}} &= \varsigma _{}^l{(\boldsymbol{\varphi }_{}^{L{\rm{ - }}1})}^{\mathrm{T}} \tag{9}\\ \frac{{\partial J}}{{\partial b_{}^l}}& = \varsigma _{}^l \tag{10} \end{align*} where ςl is the partial derivative of the loss function of layer l.
完成权重系数的更新 ε 和偏置向量b ，执行以下计算：

ςl∂J∂ωl∂J∂bl=∂J∂χL=(φL−Y)⊗Φ′(χL)=ςl(φL−1)T=ςl(8)(9)(10)

View Source \begin{align*} \varsigma _{}^l &= \frac{{\partial J}}{{\partial \chi _{}^L}} = (\boldsymbol{\varphi }_{}^L - \boldsymbol{Y}) \otimes {\Phi }^{\prime}(\chi _{}^L) \tag{8}\\ \frac{{\partial J}}{{\partial \omega _{}^l}} &= \varsigma _{}^l{(\boldsymbol{\varphi }_{}^{L{\rm{ - }}1})}^{\mathrm{T}} \tag{9}\\ \frac{{\partial J}}{{\partial b_{}^l}}& = \varsigma _{}^l \tag{10} \end{align*} 在哪里 ςl 是第l层损失函数的偏导数。

The updates of ε and b are shown in (9)–(10). However, in the training of the DNN, the parameters may deviate from their set ranges when the depth of the network increases. To solve the problem, the Adam algorithm is used in this paper to calculate the exponentially weighted average of the gradient while training. The obtained gradient values are used to update the parameters. The related algorithm can be found in [31].
的更新 ε 和b如(9)-(10)所示。然而，在DNN的训练中，随着网络深度的增加，参数可能会偏离设定的范围。为了解决这个问题，本文使用Adam算法在训练时计算梯度的指数加权平均值。获得的梯度值用于更新参数。相关算法可以参见 [31] 。

Based on the above data and DNN, a fast calculation model for DEL can be obtained, as shown in following formula:

DELTs=DNNTs(I,P¯e,std(Pe),std(ΔPe),std(β),std(ωr))DELMt=DNNMt(I,v¯,P¯e,std(Pe),std(ΔPe), std(β),ω¯r)(11)(12)

View Source \begin{align*} &DE{L}_{\text{Ts}} = DN{N}_{\text{Ts}}(I,{\bar{P}}_{\mathrm{e}},\text{std}({P}_{\mathrm{e}}),\text{std}(\Delta {P}_{\mathrm{e}}),\text{std}(\beta),\text{std}({\omega }_{\mathrm{r}})) \tag{11}\\ &DE{L}_{\text{Mt}} = DN{N}_{\text{Mt}}(I,\bar{v},{\bar{P}}_{\mathrm{e}},\text{std}({P}_{\mathrm{e}}),\text{std}(\Delta {P}_{\mathrm{e}}),\ \text{std}(\beta),{\bar{\omega }}_{\mathrm{r}}) \tag{12} \end{align*}
基于以上数据和DNN，可以得到DEL的快速计算模型，如下式所示：

DELTs=DNNTs(I,P¯e,std(Pe),std(ΔPe),std(β),std(ωr))DELMt=DNNMt(I,v¯,P¯e,std(Pe),std(ΔPe), std(β),ω¯r)(11)(12)

View Source \begin{align*} &DE{L}_{\text{Ts}} = DN{N}_{\text{Ts}}(I,{\bar{P}}_{\mathrm{e}},\text{std}({P}_{\mathrm{e}}),\text{std}(\Delta {P}_{\mathrm{e}}),\text{std}(\beta),\text{std}({\omega }_{\mathrm{r}})) \tag{11}\\ &DE{L}_{\text{Mt}} = DN{N}_{\text{Mt}}(I,\bar{v},{\bar{P}}_{\mathrm{e}},\text{std}({P}_{\mathrm{e}}),\text{std}(\Delta {P}_{\mathrm{e}}),\ \text{std}(\beta),{\bar{\omega }}_{\mathrm{r}}) \tag{12} \end{align*}

A. Objective Function Settings

The optimization algorithm in this section is used for active power control. Therefore, the power requirements of the WF and the upper and lower limits of each WT need to be satisfied while achieving the objective of fatigue load suppression. Without loss of generality, it can be formulated as a minimization problem as follows:

View Source \begin{align*} &\text{minimize}:\quad \sum\limits_{i = 1}^{{N}_3} {DE{L}_{{\rm{Ts,}}i}|{P}_{{\rm{e,}}i}} + \sum\limits_{i = 1}^{{N}_3} {{\eta }_i \cdot DE{L}_{{\rm{Mt,}}i}|{P}_{{\rm{e,}}i}} \tag{13}\\ &\text{subject}\;\text{to}:\quad {P}_{\text{wf}} = \sum\limits_{i = 1}^{{N}_3} {{P}_{{\rm{e,}}i}} \tag{14}\\ &\qquad\qquad\qquad\quad P_{{\rm{e,}}i}^{\min } \leq {P}_{{\rm{e,}}i} \leq P_{{\rm{e,}}i}^{\max } \tag{15} \end{align*}

本节中的优化算法用于有功功率控制。因此，在达到抑制疲劳载荷的目的的同时，需要满足WF的功率要求以及每个WT的上下限。不失一般性，它可以表述为一个最小化问题如下：

minimize:∑i=1N3DELTs,i|Pe,i+∑i=1N3ηi⋅DELMt,i|Pe,isubjectto:Pwf=∑i=1N3Pe,iPmine,i≤Pe,i≤Pmaxe,i(13)(14)(15)

View Source \begin{align*} &\text{minimize}:\quad \sum\limits_{i = 1}^{{N}_3} {DE{L}_{{\rm{Ts,}}i}|{P}_{{\rm{e,}}i}} + \sum\limits_{i = 1}^{{N}_3} {{\eta }_i \cdot DE{L}_{{\rm{Mt,}}i}|{P}_{{\rm{e,}}i}} \tag{13}\\ &\text{subject}\;\text{to}:\quad {P}_{\text{wf}} = \sum\limits_{i = 1}^{{N}_3} {{P}_{{\rm{e,}}i}} \tag{14}\\ &\qquad\qquad\qquad\quad P_{{\rm{e,}}i}^{\min } \leq {P}_{{\rm{e,}}i} \leq P_{{\rm{e,}}i}^{\max } \tag{15} \end{align*} 在哪里 DELTs,i|Pe,i 和 DELMt,i|Pe,i 是当第i个WT的有功功率为时的DEL _Ts和DEL _Mt Pe,i 。以功率作为DEL寻求的决策变量，在满足电网需求的同时，实时为每个WT分配最佳参考值。

To deal with the constraints in (14) in the optimization process, this paper adds it to the objective function as a penalty function by adding a penalty coefficient γ. Thus, the objective function form is changed to the following formula.

View Source \begin{align*} &\min {\rm{ }}\sum\limits_{i = 1}^{{N}_3} {DE{L}_{{\rm{Ts,}}i}|{P}_{{\rm{e,}}i}} + \sum\limits_{i = 1}^{{N}_3} {{\eta }_i \cdot DE{L}_{{\rm{Mt,}}i}|{P}_{{\rm{e,}}i}} \\ & \quad + \gamma \cdot \left| {{P}_{\text{wf}} - \sum\limits_{i = 1}^{{N}_3} {{P}_{{\rm{e,}}i}} } \right| \tag{16} \end{align*}

为了在优化过程中处理式(14)中的约束，本文通过添加惩罚系数将其作为惩罚函数添加到目标函数中 γ 。因此，目标函数形式改为以下公式。

min∑i=1N3DELTs,i|Pe,i+∑i=1N3ηi⋅DELMt,i|Pe,i+γ⋅∣∣∣∣Pwf−∑i=1N3Pe,i∣∣∣∣(16)

View Source \begin{align*} &\min {\rm{ }}\sum\limits_{i = 1}^{{N}_3} {DE{L}_{{\rm{Ts,}}i}|{P}_{{\rm{e,}}i}} + \sum\limits_{i = 1}^{{N}_3} {{\eta }_i \cdot DE{L}_{{\rm{Mt,}}i}|{P}_{{\rm{e,}}i}} \\ & \quad + \gamma \cdot \left| {{P}_{\text{wf}} - \sum\limits_{i = 1}^{{N}_3} {{P}_{{\rm{e,}}i}} } \right| \tag{16} \end{align*}

The upper and lower bound constraints in (15) are easy to handle in the optimization algorithm.
(15) 中的上限和下限约束在优化算法中很容易处理。

In the above model, the WF output power needs to meet the power grid control requirements (e.g., frequency regulation), while minimize the aggregate of DEL of all WTs in the WF, through optimizing the power command value of each WT under different working conditions.
上述模型中，WF输出功率需要满足电网控制要求（如调频），同时通过优化各WT在不同工况下的功率指令值，最小化WF内所有WT的DEL总和。

B. Improved MOGWO B.改进的MOGWO

The problems mentioned above are non-linear and non-convex. Considering the application effects of multiple optimization algorithms, a MOGWO algorithm [22] is improved and applied in this section. The principle of MOGWO can be found in [22].
上述问题是非线性和非凸的。考虑多种优化算法的应用效果，提出MOGWO算法 [22] 本节对此进行了改进和应用。 MOGWO的原理可以参见 [22] 。

1) Optimization of the Initial Population
1）初始种群的优化

The improvement of the initial population uniformity is also an important way to improve global convergence, avoid falling into local optimal and reduce negative optimization [32]. The traditional MOGWO uses a random function to generate the initial population, which has poor population uniformity. In this section, the chaotic sequence [33] is used to generate the initial population to improve the diversity and traversal of the population. To obtain better uniformity, this section uses the chaotic sequence obtained by mixing Logistic and Chebyshev. The ratio of the two mappings is set to 0.5.

⎧⎩⎨⎪⎪⎪⎪Inik+1=0.5Lok+1+0.5∣∣Chk+1∣∣Lok+1=δ⋅Lok⋅(1−Lok)Chk+1=cos(2arccos(Chk))(17)

View Source \begin{equation*} \left\{ {\begin{array}{l} {In{i}^{k + 1} = 0.5L{o}^{k + 1} + 0.5\left| {C{h}^{k + 1}} \right|}\\ {L{o}^{k + 1} = \delta \cdot L{o}^k \cdot (1 - L{o}^k)}\\ {C{h}^{k + 1} = \cos (2\arccos (C{h}^k))} \end{array}} \right. \tag{17} \end{equation*}
初始种群均匀性的提高也是提高全局收敛性、避免陷入局部最优、减少负优化的重要途径 [32] 。传统的MOGWO采用随机函数生成初始种群，种群均匀性较差。在本节中，混沌序列 [33] 用于生成初始种群，以提高种群的多样性和遍历性。为了获得更好的均匀性，本节使用混合Logistic和Chebyshev获得的混沌序列。两个映射的比率设置为0.5。

⎧⎩⎨⎪⎪⎪⎪Inik+1=0.5Lok+1+0.5∣∣Chk+1∣∣Lok+1=δ⋅Lok⋅(1−Lok)Chk+1=cos(2arccos(Chk))(17)

View Source \begin{equation*} \left\{ {\begin{array}{l} {In{i}^{k + 1} = 0.5L{o}^{k + 1} + 0.5\left| {C{h}^{k + 1}} \right|}\\ {L{o}^{k + 1} = \delta \cdot L{o}^k \cdot (1 - L{o}^k)}\\ {C{h}^{k + 1} = \cos (2\arccos (C{h}^k))} \end{array}} \right. \tag{17} \end{equation*}

Where δ is a certain coefficient, the value range is [04].
其中δ为某个系数，取值范围为[04]。

2) Optimization of the Weight
2）权重优化

In optimization, DEL_Ts and DEL_Mt are not in the same order of magnitude. Therefore, an appropriate weighting factor between them is needed to make the optimization more balanced. The Pareto front is used to optimize the weights in this section. After the Pareto solution set is obtained, the weight coefficients are obtained as follows:

ηiFiwt,j=∑j=1MFiMt,j/∑j=1MFiTs,j=FiMt,j+ηi⋅FiTs,j(18)(19)

View Source \begin{align*} {\eta }_i &= \sum\limits_{j = 1}^M {F_{{\rm{Mt, }}j}^i} /\sum\limits_{j = 1}^M {F_{{\rm{Ts, }}j}^i} \tag{18}\\ F_{{\rm{wt, }}j}^i &= F_{{\rm{Mt, }}j}^i + {\eta }_i \cdot F_{{\rm{Ts, }}j}^i \tag{19} \end{align*} where η_i is mentioned in (16), F_Mt,j is the j-th solution of DEL_Mt in the Pareto solution set, F_Ts,j is the j-th solution of DEL_Ts in the Pareto solution set. Fiwt,j is the j-th solution of the i-th WT's DEL. In this paper, the solution with the smallest Fiwt,j in the Pareto solution set is taken as the optimal solution.
在优化中， DEL _Ts和DEL _Mt不在同一数量级。因此，它们之间需要一个适当的权重因子，使优化更加平衡。 Pareto 前沿用于优化本节中的权重。得到Pareto解集后，得到权重系数如下：

ηiFiwt,j=∑j=1MFiMt,j/∑j=1MFiTs,j=FiMt,j+ηi⋅FiTs,j(18)(19)

View Source \begin{align*} {\eta }_i &= \sum\limits_{j = 1}^M {F_{{\rm{Mt, }}j}^i} /\sum\limits_{j = 1}^M {F_{{\rm{Ts, }}j}^i} \tag{18}\\ F_{{\rm{wt, }}j}^i &= F_{{\rm{Mt, }}j}^i + {\eta }_i \cdot F_{{\rm{Ts, }}j}^i \tag{19} \end{align*} 其中，(16)中提到的η _i ， F _{Mt ,j}是DEL _Mt在 Pareto 解集中的第 j 个解， FTs _,j是DEL _Ts在 Pareto 解集中的第 j 个解。 Fiwt,j 是第 i 个WT 的 DEL 的第 j 个解。本文采用最小的解决方案 Fiwt,j 取 Pareto 解集中的最优解。