Nonlinear Intelligent Predictive Control for the Yaw System of Large-Scale Wind Turbines

Abstract

This chapter presents a nonlinear intelligent predictive control using multi-step prediction model for the electrical motor-based yaw system of an industrial wind turbine. The proposed method introduces a finite control set under constraints for the demanded yaw rate, predicts the multi-step yaw error using the control set element and the prediction wind directions, and employs an exhaustive search method to search the control output candidate giving the minimal value of the objective function. As the objective function is designed for a joint power and actuator usage optimization, the weighting factor in the objective function is optimally determined by the fuzzy regulator that is optimized by an intelligent algorithm. Finally, the proposed method is demonstrated by simulation tests using real wind direction data.

1. Introduction

With the increase in social demand, the scale of wind power generation continues to expand. The total installed capacity of global wind power in 2020 has reached 743GW, which means that wind turbines (WTs) are moving towards large-scale and high-capacity. Typical WT controls include pitch, torque, and yaw control, of which about 80% of the research is on the first two, while yaw control has received limited attention [1]. Meanwhile, with the large-scale development of WTs [2], problems such as power reduction and load increase caused by the yaw misalignment can no longer be ignored. According to the investigation result, the potential energy loss due to yaw misalignment is about 2.7% and the failure rate of yaw system accounts for approximately 12.5% of the total failure rate of WTs [3]. Therefore, there is an urgent need to improve the yaw control performance of WT.

Yaw control changes the direction of blade rotating surface by turning the nacelle horizontally. Traditional yaw control methods include Logic Control [4], PID, fuzzy PID [5], and so on. Yet, wind direction sensor suffers from the disturbance of rotor rotation, which makes it difficult to accurately measure the incoming wind direction. In order to avoid measurement error, some wind direction estimation methods are proposed, including the hill-climbing search algorithm [6] and so on. However, these improved methods have limited effect on large-scale WTs, and they are rarely applied in industry. In short, with the development of WT towards large-scale, traditional yaw control methods generally have shortcomings, which promote the development of new methods.

As the development of advanced prediction technologies like LiDAR [7], more recent research has concentrated on the advanced predictive controls [8]. Model predictive control (MPC) is a typical representative of predictive control, which has been proposed for the torque and yaw control of WTs, and has achieved good control performance [9, 10, 11]. The MPC for yaw system involves performance indicators such as energy capture efficiency and yaw actuator usage [12]. By adding weight coefficients, each performance indicator can be combined into a single objective function. Obviously, the setting of weight coefficient could influence the control performance. In order to find the connotative knowledge, potential regulations and methods, the Pareto optimization theory is used in [13] to explore and gives the suggestion that weight coefficients should be regulated according to the wind characteristics. However, how to effectively adjust the weight coefficients in real time remains unsolved.

Fuzzy logic (FL) is a potential solution to regulate the weight coefficient for model-predictive yaw control. FL is an abstraction of the approximate reasoning characteristics of human decision-making, which has been applied in many fields [14]. Yet, the excessive dependence of FL on expert experience leads to artificially set membership functions (MFs) and fuzzy rules that could limit the control performance. To this end, the optimization of FL is proposed to enhance its effect [15]. In summary, the advantages of FL and the potential room for optimization make it possible to effectively regulate the weight coefficient of MPC.

Motivated by above observations, in this study, the nonlinear MPC (NMPC) method with multi-step prediction models for the yaw control system is proposed. Specifically, an “ideal” NMPC controller that employs perfect previewed wind directions into the prediction model is used in this study and the NMPC problem is solved by using an exhaustive search method based on the sequential diagram. Further, a novel method of using the mind of FL to dynamically regulate the weight coefficient of the NMPC is proposed, which is called as fuzzy inference weight coefficient regulator (FIWR). Specifically, the fuzzy rules and MFs of FIWR are simultaneously optimized by an advanced intelligent algorithm, so as to fully exert the advantage of FIWR. By doing so, it is achieved the deep optimization of NMPC performance for yaw system.

2. Model and methodology

This study aims at proposing and studying the NMPC method for the yaw control system on a horizontal-axis WT. The yaw model for WTs will be introduced first, followed by the design of finite-set NMPC. On this basis, a weight coefficient regulator based on fuzzy inference is proposed, and the multi-objective optimization problem of fuzzy inference is summarized. Then a proposed solution strategy for this problem is introduced, and a multi-objective intelligent optimization algorithm is improved to solve it.

2.1 Yaw system modeling

The yaw system can be modeled according to the three types of yaw dynamics: rigid yaw, flexible yaw, and controlled yaw torque. Since the yaw rate of large WTs is very slow, the yaw rate is set to a fixed value of 0.5 deg./s. There are three yaw situations for WTs: no deflection (0 deg./s), clockwise deflection (0.5 deg./s) and counterclockwise deflection (−0.5 deg./s). In realistic operation, considering the safety requirement of the yaw actuator, the yaw rate at current moment is affected by the yaw rate at previous moment and meets following constraint:

$${\dot{\theta }}_{\mathit{np}}\left(k+1\right)=\left\{\begin{array}{cc}0.5deg/\mathrm{s}& \exists {\dot{\theta }}_{\mathit{np}}\left(k\right)\in \left\{0,0.5deg/\mathrm{s}\right\}\\ 0deg/\mathrm{s}& \exists {\dot{\theta }}_{\mathit{np}}\left(k\right)\in \left\{0,0.5,-0.5deg/\mathrm{s}\right\}\\ -0.5deg/\mathrm{s}& \exists {\dot{\theta }}_{\mathit{np}}\left(k\right)\in \left\{0,-0.5deg/\mathrm{s}\right\}\end{array}\right.$$E1

where ${\dot{\theta }}_{\mathit{np}}\left(k\right)$ is the yaw rate in k-th control period, and ${\dot{\theta }}_{\mathit{np}}\left(k+1\right)$ is the yaw rate in (k + 1)-th control period. According to Eq.(1), in a certain state at current moment, the control action of the system at next moment is an element in a finite set. This yaw action mode provides the basis for the finite-set NMPC.

2.2 Nonlinear model predictive control

The NMPC for the yaw system aims at maximizing energy capture through tracking the wind direction while avoiding over-usage of the yaw actuator. Accordingly, the yaw error and the yaw actuator usage are used to form the overall objective function. In the following, the proposed NMPC method will be specified in terms of the prediction model, the objective function, and the finite-set NMPC solver.

2.2.1 Multi-step prediction model

The primary control objective of the yaw control system is to minimize the yaw error. Thus, the yaw error ${\theta }_{\mathit{ye}}$ is selected as the state variable, and its one-step prediction model in the form of the discrete equation can be given as follows:

$${\theta }_{\mathit{ye}}\left(k+1\mid k\right)={\theta }_{\mathit{wd}}\left(k+1\mid k\right)-{\theta }_{\mathit{np}}\left(k+1\mid k\right)$$E2

where k is the k-th control period; ${\theta }_{\mathit{ye}}\left(k+1|k\right)$, ${\theta }_{\mathit{wd}}\left(k+1|k\right)$ and ${\theta }_{\mathit{np}}\left(k+1|k\right)$ are the next-step prediction values of yaw error, wind direction, and nacelle position, respectively.

Since the nacelle is rotated by the yaw control system at a certain yaw rate, the predicted nacelle position ${\theta }_{\mathit{np}}\left(k+1|k\right)$ can be obtained by:

$${\theta }_{\mathit{np}}\left(k+1\mid k\right)={\theta }_{\mathit{np}}\left(k\right)+{\dot{\theta }}_{\mathit{np}}\left(k+1\right)·T_s$$E3

where ${\theta }_{\mathit{np}}\left(k\right)$ is the nacelle position at the k-th control period, $T_s$ is the control period, and ${\dot{\theta }}_{\mathit{np}}\left(k+1\right)$ is the yaw rate at the (k + 1)-th control period.

By using Eq. (2) and (3), the m-step prediction model of the yaw error ${\theta }_{\mathit{ye}}\left(k+m|k\right)$ can be obtained by:

$${\theta }_{\mathit{ye}}\left(k+m\mid k\right)={\theta }_{\mathit{wd}}\left(k+m\mid k\right)-{\theta }_{\mathit{np}}\left(k\right)-\sum _{i=1}^m{\dot{\theta }}_{\mathit{np}}\left(k+i\right)·T_s$$E4

where ${\theta }_{\mathit{wd}}\left(k+m|k\right)$ is the variable that needs to be predicted, which can be predicted by LiDAR or some prediction methods.

2.2.2 Objective function

The first goal of yaw control is to improve the energy capture of the WT, and the second goal is to reduce the yaw actuator usage time. Considering that the energy capture of the WT has a cosine-squared relation to the yaw error, the two objectives can be express as:

$$E_{\mathit{cap}}=\sum _{i=k+1}^{k+m}\frac{1}{2}\rho A_r{C}_p{V}_0^3\mathrm{co}{\mathrm{s}}^2\left({\theta }_{\mathit{ye}}\left(i\right)\right)$$E5

$$t_{\mathit{yaw}}=\sum _{i=k+1}^{k+m}\left(\mid {\dot{\theta }}_{\mathit{np}}\left(i\right)\mid >0\right)·T_s$$E6

where $\rho$ is air density, $A_r$ is rotor area, $C_p$ is aerodynamic power coefficient, $V_0$ is the effective wind speed. Considering the dimensional difference, the two control objectives after normalization can be expressed as follows:

$$\xi =\frac{\left(E_{\mathit{\text{ideal}}}-E_{\mathit{cap}}\right)}{E_{\mathit{\text{ideal}}}}=1-\left(\sum _{n=k+1}^{k+m}\mathrm{co}{\mathrm{s}}^2\left({\theta }_{\mathit{ye}}\left(n\right)\right)\right)/m$$E7

$$\zeta =\frac{t_{\mathit{yaw}}}{t_{\mathit{tol}}}=\left(\sum _{n=k+1}^{k+m}\left(\mid {\dot{\theta }}_{\mathit{np}}\left(n\right)\mid >0\right)\right)/m$$E8

where $\xi$ is the energy capture loss ratio caused by the yaw error, $E_{\mathit{cap}}$ is the energy capture considering yaw error, $E_{\mathit{\text{ideal}}}$ is the energy capture in an ideal state; $\zeta$ is the yaw actuator usage ratio, $t_{\mathit{yaw}}$ is the yaw time, $t_{\mathit{tol}}$ is the running time of the WT.

By adding a weight coefficient between $\xi$ and $\zeta$, the objective function $\mathit{QF}$ of the NMPC can be written as:

$$\begin{array}{l}\mathit{QF}=\left(1-\omega \right)·\xi +\omega ·\zeta \\ =\left(1-\omega \right)·\left(1-\left(\sum _{n=k+1}^{k+m}\mathrm{co}{\mathrm{s}}^2\left({\theta }_{\mathit{ye}}\left(n\right)\right)\right)/m\right)\\ \begin{array}{ccc}& & \end{array}+\omega ·\left(\left(\sum _{n=k+1}^{k+m}\left(\mid {\dot{\theta }}_{\mathit{np}}\left(n\right)\mid >0\right)\right)/m\right)\end{array}$$E9

where $\omega$ is the weight coefficient, which is used to balance energy capture loss ratio $\xi$ and yaw actuator usage ratio $\zeta$.

2.2.3 Finite-set NMPC solver

So far, the NMPC problem for the yaw system has been formulated by Eq. (1)–(9), which is a nonlinear optimization issue under constraints. To facilitate the problem solver, the control horizon is selected to be equal to the prediction horizon. According to Eq. (1), under the finite prediction horizon, the control law of NMPC always belongs to a finite set. Thus, the designed optimal problem can be effectively solved by using an exhaustive search (ES) method.

Figure 1 illustrates the case of the NMPC with the three-step prediction model, from which the ES method is explained as follows:

Figure 1.

Sequential diagram of the ES for a three-step prediction model.

During the initialization period ($m=0$), the yaw control system is inactivated, and thus the total yaw state is 1.

• For $m=1$, because the current yaw rate is zero. Accordingly, there are three potential solutions and the total yaw states are 3.

• For $m=2$, the situation is different. Constrained by Eq. (1), there are only two candidate values in the case of ${\dot{\theta }}_{\mathit{np}}\left(m=1\right)=\pm 0.5deg/\mathrm{s}$, while there are three candidate values in the case of ${\dot{\theta }}_{\mathit{np}}\left(m=1\right)=0$. Consequently, along the sequential order, there are seven potential solutions and the total yaw states are 7.

• For $m=3$, the situation becomes slightly complicated but similar to the two-step prediction model. There are seventeen solutions for the three-step prediction model and the total yaw states are 17.

2.3 Intelligent fuzzy inference weight coefficient regulator

There is a contradiction between increasing energy capture and reducing yaw actuator usage. In Eq. (9), $\omega$ is used to balance $\xi$ and $\zeta$ in $\mathit{QF}$, so the choice of weight coefficient affects the performance of NMPC to a large extent. Even if an optimal constant $\omega$ is selected according to the Pareto theory, it cannot ensure that the best control performance is always provided. Therefore, the FIWR is proposed to dynamically adjust $\omega$ according to the predicted wind direction in each control period. Moreover, to better play the effect of FIWR and avoid the subjectivity of manual tuning, the intelligent optimization of FIWR is also necessary.

2.3.1 Design of FIWR

The proposed FIWR scheme is shown in Figure 2. This is a fuzzy inference system with two inputs and one output. Inputs ${\mathit{WD}}_{\mathit{av}}$ and ${\mathit{WD}}_{\mathit{std}}$ have three and five linguistic values, respectively, and output $\omega$ has five linguistic values. The initial membership function (IMF) adopts the equally divided triangular MF. Fuzzy inference adopts Mamdani-type algorithm and the center of area (COA) is utilized in defuzzification process.

Figure 2.

The scheme of FIWR.

2.3.1.1 Design of input/output

The yaw error is the core part of the input. Which directly determines the action of the yaw system. Therefore, the adjustment of $\omega$ takes the yaw error as a reference. Because the future yaw action is the control law to be solved, which is unknown, the predicted yaw error here refers to the difference between the predicted average wind direction of the (k + i)-th control period and the first sampled value of the nacelle position of the current k-th control period, expressed by ${\theta }_{\mathit{ye}}\left(k+i\to k\right)$, which denotes the difference between the future wind direction obtained by LiDAR and the nacelle position at the current moment.

Different from the error and error derivative method used by ordinary two-input fuzzy inference, the designed input of the FIWR is related to the statistical characteristics of the predicted yaw error. The two inputs are designed as the weighted average and standard deviation of yaw error in prediction horizon m respectively, named ${\mathit{WD}}_{\mathit{av}}$ and ${\mathit{WD}}_{\mathit{std}}$. The calculation of ${\mathit{WD}}_{\mathit{av}}$ is:

$$W{D}_{\mathit{av}}=\sum _{i=1}^m\left(m+1-i\right)·{\theta }_{\mathit{ye}}\left(k+i\to k\right)/\sum _{i=1}^{m}i$$E10

where $m$ is the prediction step, and ${\theta }_{\mathit{ye}}\left(k+i\to k\right)$ can be calculated by setting the yaw rate as zero in Eq. (4). Based on ${\mathit{WD}}_{\mathit{av}}$, ${\mathit{WD}}_{\mathit{std}}$ is calculated by:

$$W{D}_{\mathit{std}}=\sqrt{{\sum }_{i=1}^m{\left({\theta }_{\mathit{ye}}\left(k+i\to k\right)-W{D}_{\mathit{av}}\right)}^2/m}$$E11

In practice, the yaw error might be affected by some subtle factors, so moving average filter is presented to process the wind direction data. In this study, the filter value of each sample is the mean value of the N sample values in the sliding window. For wind direction, N usually takes 12.

2.3.1.2 Design of MFs and fuzzy rules

The IMFs corresponding to input ${\mathit{WD}}_{\mathit{av}}$, ${\mathit{WD}}_{\mathit{std}}$ and output $\omega$ are illustrated in Figure 3, respectively. The types of IMFs are all selected as sensitive and simple triangular MFs, with the bottom edge equal and overlapping with adjacent IMFs by 50%. The universe of discourse (UOD) of ${\mathit{WD}}_{\mathit{av}}$ and ${\mathit{WD}}_{\mathit{std}}$ is defined as [0 deg., 15 deg] and [0 deg., 25 deg] respectively. The best value range of $\omega$ is [0, 0.1], so the UOD of output is defined on [0, 0.1]. The linguistic values VS, S, M, L, and VL represent very small, small, medium, large, and very large, respectively. ${\mathit{WD}}_{\mathit{av}}$ and ${\mathit{WD}}_{\mathit{std}}$ are mapped to 3 and 5 linguistic values, respectively, so the fuzzy rule table will contain 15 different rules. The initial fuzzy rules of the proposed FIWR are listed in Table 1. The derivation of the fuzzy rules is based on the expert experience, that is, a larger yaw error and a smaller standard deviation will lead to the yaw action towards improving energy capture.

Figure 3.

Initial membership functions.

${\mathrm{WD}}_{\mathrm{std}}$		VS	S	M	L	VL
${\mathit{WD}}_{\mathit{av}}$	S	VL	L	M	M	S
	M	S	S	M	L	L
	L	VS	VS	S	L	VL

Table 1.

Initial fuzzy rules.

2.3.2 Intelligent optimization of FIWR

The advantage of fuzzy inference is that it can fully incorporate expert experience. However, when the expert experience is insufficient or wrong, the result of fuzzy inference will no longer be reliable; and the fuzzy relationship under complex input sometimes cannot be directly given by the expert experience. Therefore, the optimization of FIWR is proposed.

2.3.2.1 Fuzzy optimization problem formulation

The goal of the proposed FIWR is to reduce the yaw actuator usage and the energy capture loss, so the optimization problem can be expressed as:

$$\begin{array}{l}minF\left(x_{\mathit{\text{membership}}},x_{\mathit{\text{rule}}}\right)=\left(\frac{\int \mathit{\xi dt}}{\int \mathit{dt}},\frac{\int \mathit{\zeta dt}}{\int \mathit{dt}}\right)\\ s.t.\left\{\begin{array}{c}{x}_{\mathit{\text{membership}}}\in {\Omega }_m\\ x_{\mathit{\text{rule}}}\in {\Omega }_r\end{array}\right.\end{array}$$E12

where $x_{\mathit{\text{membership}}}$ and $x_{\mathit{\text{rule}}}$ is the optimization vector of MFs and fuzzy rules, respectively; ${\Omega }_m$ and ${\Omega }_r$ is the feasible regions of the two optimization vectors, respectively.

For ${\Omega }_m$, there are three kinds of constraints: the number constraint of MF, the type constraint of MF, and the position constraint of MF. As for this study, in order to simplify the optimization problem, the optimization variables corresponding to the first two constraints are fixed, that is, the number and type of MF do not need to be optimized. Assuming that the position of each MF is uniquely determined by a certain vertex of the triangle, the optimization dimension of MF is further reduced. Obviously, the optimization of position is subject to constraints, that is, a small linguistic value cannot exceed a large linguistic value and each MF must be changed within the UOD.

For ${\Omega }_r$, it is affected by the number of inputs and outputs. For the fuzzy inference using Mamdani model, the consequence of the fuzzy rule is a certain fuzzy set of output. If there are $s$ inputs and $h$ outputs in fuzzy inference, the feasible region of the fuzzy rule can be expressed as:

$${\Omega }_r=\mathit{\text{rule}}\left(\prod _{i=1}^h\mathit{nu}{m_i}^{\prod _{j=1}^s\mathit{nu}{m}_j}\right)$$E13

where ${\mathit{num}}_j$ is the number of linguistic values of j-th input, and ${\mathit{num}}_i$ is the number of linguistic values of i-th output.

2.3.2.2 Fuzzy optimization problem simplification

Although Eq. (12) has been simplified, it is still difficult to solve directly. For this complex problem, it is necessary to simplify the problem as much as possible on the premise of ensuring a certain solution accuracy. Therefore, a solution strategy is proposed to simplified the complex optimization problem with the purpose of quickly and reliably solving FIWR optimization parameters.

As mentioned earlier, the MFs are subject to order constraints that the MF with smaller linguistic value must be front of the MF with larger linguistic value. This constraint is to avoid repeated searches and ensure the logical accuracy of fuzzy inference. However, if the output MF is no longer constrained, the current linguistic value sequence of output can be used as a reference for the optimization of fuzzy rule. Therefore, the fuzzy rule is associated with MF, which can greatly reduce the complexity of optimization problem and ensure the search ability.

Specifically, the output linguistic value sequence after the sequence change can be expressed as:

$$\mathit{B}=\mathit{A}·\mathit{S}$$E14

where $\mathit{A}$ is the original sequence, and $\mathit{S}$ is the identity matrix after elementary transformation, called the transformation matrix.

For example, in a certain change, the output linguistic value sequence is transformed into [S M L VS VL], then it can be expressed as:

$$\left[\mathrm{S}\mathrm{M}\mathrm{L}\mathrm{VS}\mathrm{VL}\right]=\left[\mathrm{VS}\mathrm{S}\mathrm{M}\mathrm{L}\mathrm{VL}\right]\left[\begin{array}{ccccc}0& 0& 0& 1& 0\\ 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1\end{array}\right]$$E15

After each iteration, $\mathit{A}$ and $\mathit{B}$ are known, so $\mathit{S}$ can be calculated according to Eq. (14). Then the updated fuzzy rule table can be calculated according to $\mathit{S}$:

$$R_{\mathit{new}}\left[3\times 5\right]=R_{\mathit{old}}\left[3\times 5\right]·S\left[5\times 5\right]$$E16

where $R_{\mathit{new}}\left[3\times 5\right]$ and $R_{\mathit{old}}\left[3\times 5\right]$ are the fuzzy rule tables after and before the update, respectively, with a size of $3\times 5$. In order to ensure the exploitation of the solution process, each linguistic value in the fuzzy rule table is transformed correspondingly with a certain probability. For example, the S linguistic value is changed to VS with a small probability $p$. This probabilistic processing procedure can improve the search ability that find the optimal MF under the current fuzzy rule.

2.3.2.3 Improved AGA-MOPSO solver

Although Eq. (12) is simplified by the solution strategy, its objective function is complex, which makes it difficult to be solved by ordinary multi-objective optimization algorithms. Therefore, an improved multi-objective particle swarm optimization (MOPSO) algorithm based on adaptive grid algorithm MOPSO (AGA-MOPSO) is designed. AGA-MOSPO is an efficient variant of PSO for multi-objective problem. By combining with adaptive grid algorithm (AGA), it achieves nice balance between exploration and exploitation [16].

For a two-dimensional multi-objective problem, AGA-MOPSO first calculates the search ranges $\left[min{f}_1^k,max{f}_1^k\right]$ and $\left[min{f}_2^k,max{f}_2^k\right]$ of the objective space after k-th iteration, then calculates the grid number of the i-th particle according to the following equation:

$$\left(x_1^i,x_2^i\right)=\left(Int\left(\frac{f_1^i-min{f}_1^k}{\left(max{f}_1^k-min{f}_1^k\right)/M}\right)+1,Int\left(\frac{f_2^i-min{f}_2^k}{\left(max{f}_2^k-min{f}_2^k\right)/M}\right)+1\right)$$E17

where $x_1^i$ and $x_2^i$ is the grid numbers of the particles, $M$ is the number of grids, $f_1^i$ and $f_2^i$ are the fitness values of the two targets respectively, and $\mathit{Int}$ is rounding.

$M$ will be adaptively increased with the iteration to balance the computational cost and accuracy. The density information of each grid can be obtained according to the grid number. According to the density information, the global optimal particles are selected and the Archive set is clipped.

Considering the complexity of the optimization problem, the following improvements are proposed for the selection of the global optimal particle and the truncation of Archive set in AGA-MOPSO:

(1) First is to use two methods to determine the corresponding $g_{\mathit{\text{best}}}$ for each particle: 1) Select $g_{\mathit{\text{best}}}$ with the smallest grid density. This method focuses on the exploration of the search space to improve the ductility of Pareto front (PF). 2) Select the global optimal particle as $g_{\mathit{\text{best}}}$ according to the technique for order preference by similarity to an ideal solution (TOPSIS):

$$d=\frac{\sqrt{{\left(f_1-f_1^{\min }\right)}^2+{\left(f_2-f_2^{\min }\right)}^2}}{\sqrt{{\left(f_1-f_1^{\min }\right)}^2+{\left(f_2-f_2^{\min }\right)}^2}+\sqrt{{\left(f_1-f_1^{\max }\right)}^2+{\left(f_2-f_2^{\max }\right)}^2}}$$E18

where $d$ is the deviation between a certain point and the ideal point. The smaller the $d$, the smaller the deviation from the ideal point. This method focuses on the exploitation of the search space to make PF closer to the real optimal solution. In the early stage of algorithm, the probability of choosing $g_{\mathit{\text{best}}}$ by the first method is greater, so as to find as many non-dominated solutions as possible; in the later stage, the probability of choosing the second method is greater to approximate the true solution.

(2) Second is to truncate the Archive set by adaptive dynamic threshold. It is calculated using:

$$\mathit{Th}·M=C$$E19

where $\mathit{Th}$ is the threshold, and $C$ is a constant. When the number of particles in the grid exceeds $\mathit{Th}$, the grid is truncated; $\mathit{Th}$ is reduced following an increased $M$. This ensuring that the number of particles on the PF is relatively stable.

The main procedure of the improved AGA-MOPSO solver is shown in Figure 4.

Figure 4.

The flowchart of the improved AGA-MOPSO.

3. Validation and discussion

The experiment is based on MATLAB/SIMULINK. First, the optimized parameters of FIWR are obtained by the solution strategy that run on MATLAB. Then, the proposed FIWR-NMPC controller is simulated in SIMULINK. The common experimental parameters in Table 2 are set to the same value. The UOD is set to a uniform value [0, 10] to facilitate the handling of the constraints of optimization variables, so the universe conversion scale coefficients corresponding to ${\mathit{WD}}_{\mathit{av}}$, ${\mathit{WD}}_{\mathit{std}}$, and $\omega$ are 1.5, 2.5, and 0.01, respectively.

The wind direction data used in the experiment is from the actual wind direction data of a wind farm in operation for one day with a sampling period of 1 s, as shown in Figure 5(a). The wind direction after sliding average filtering is shown in Figure 5(b).

Figure 5.

24-hour wind direction used in experiment: (a) actual wind direction; (b) filtering wind direction.

3.1 Optimization results

The optimization results are analyzed and discussed first. Taking the case of prediction step m = 6 as an example, the optimization results of AGA-MOPSO are shown in Figure 6, where the horizontal axis is the yaw actuator usage ratio, and the vertical axis is the energy capture loss ratio. After 50 iterations, the particles finally converge to PF. The particles are evenly distributed in the search space near the PF, which indicate that this associated idea will not lead to the coupling optimization of PF and fuzzy rule in the search process.

Figure 6.

Iterative results.

The optimization results of the design variables are shown in Table A in Appendix. Considering space reasons, only ten points on PF are randomly selected to discuss. The simplification solution strategy associates fuzzy rule with MF to optimize, so as to reduce optimization complexity. The MF of $\omega$ does not strictly follow the linguistic value order. This random information is utilized to adjust the fuzzy rule, thereby simultaneously optimizing the fuzzy rule and the MF.

The prediction step $m$ in the proposed FIWR-NMPC is variable and has a greater impact on the performance of controller. Therefore, the influence of m on the optimization effect of FDWE is discussed. Figure 7 shows the optimization result under $m$ = 1–6. In Figure 11, the PF when $m$ =1 is significantly different from the PF when $m$ = 2–6; as $m$ increases, the yaw actuator usage ratio and energy capture loss ratio show a downward trend. This is because $m$ =1 is very short, and the predicted wind information is very few. But the actual wind direction greatly fluctuates due to the existence of turbulence, and the yaw system has a large time lag. So, it is difficult for the yaw system to track the change of the wind direction. When $m$ increases, the predicted wind direction information increases, and the controller can start yawing several control periods in advance to compensate for the time lag.

Figure 7.

Optimization results under different $m$.

Furthermore, in the enlarged part, the PF of $m$ =5 and $m$ =6 is very close, and the increase of $m$ gradually reduces the improvement effect of the FIWR-NMPC control performance. These results show that the increase in $m$ can provide better performance for FIWR-NMPC, but there is a limit to the performance improvement. Among the six FIWR-NMPCs with different $m$, the controller with $m$ =6 provides the best performance, which is very close to the ultimate performance.

3.2 Simulation results

A specific FIWR-NMPC controller and a baseline NMPC controller is designed based on the foregoing discussion. The MFs and fuzzy rules in FIWR are shown in Figure 8 and Table 3. The MFs and fuzzy rules are derived from the optimal solution obtained through TOPSIS. The remaining parameters of FIWR-NMPC and baseline NMPC are shown in Table 2, where $m$ =6.

Figure 8.

MFs used by FIWR.

AGA-MOPSO	Population size	Number of iterations	Number of grids
	100	50	[10 20]
FIWR	Deduction method	Defuzzification	UOD
	COA	Mamdani-type	[0 10]
NMPC	Sampling period(s)	Control period(s)	Prediction step
	1	30	[1 6]

Table 2.

Common parameters in the experiment.

${\mathrm{WD}}_{\mathrm{std}}$		VS	S	M	L	VL
${\mathit{WD}}_{\mathit{av}}$	S	VL	L	M	M	VL
	M	M	M	L	S	S
	L	VS	VS	S	S	VS

Table 3.

Fuzzy rules used by FIWR.

The simulation results are shown in Figure 9. Figure 9(a) and (b) respectively represent the energy capture loss ratio and the yaw actuator usage ratio of the WT within 24 hours. Compare with the baseline NMPC, the proposed FIWR-MPC increases the energy capture by about 0.3% while reducing the yaw actuator usage ratio by about 1%. This improvement benefits from the dynamically adjusted weight coefficient, that is, dynamically weighing the two control objectives based on the predicted wind information to determine the yaw action.

Figure 9.

Main performance of the two controllers: (a) energy capture loss ratio; (b) yaw actuator usage ratio.

4. Conclusions

In this study, an advanced nonlinear model predictive control solution including multi-step prediction models has been proposed and investigated for the yaw control system of a horizontal-axis wind turbine. The noticeable feature of the proposed solution is to use a finite control set under constraints for the possible demanded yaw rate, and thus the optimal control demand for the yaw system has been conveniently solved using an exhaustive search method based on the sequential diagram. On the other hand, the weighting coefficient in the objective function of NMPC has been dynamically tuned by employing the fuzzy inference regulator, as the proposed solution is designed for a joint energy capture and actuator usage optimization which is basically a two objective tradeoff that depends on the selection of the weighting factor. To give full play to the ability of the regulator, its parameter tuning is refined into an optimization problem and a solution strategy is designed to simplify it, and then the optimal fuzzy rules and membership functions are solved by the improved AGA-MOPSO algorithm. The final optimized FIWR-NMPC achieves deep optimization of wind turbine yaw performance. The important investigation findings include:

• Both fixed-weight NMPC and FIWR-NMPC can achieve higher energy capture with lower yaw time. NMPC can achieve 96.875% energy capture efficiency at 6.875% yaw time ratio, while FIWR-NMPC can achieve 97.052% energy capture efficiency at 5.792% yaw time ratio. FIWR-NMPC further improves the yaw performance by dynamically adjusting $\omega$ according to wind direction information.

• The fuzzy optimization problem simplified by the solution strategy can be solved reliably by AGA-MOPSO. The optimization method is promising to provide guidance for the design of fuzzy inference problem.

• Along with the extended prediction horizon of the NMPC, the energy capture performance is enhanced while maintaining the same yaw actuator usage, while the enhanced performance achieved by the NMPC is limited. The simulation results show that when the predicted horizon $m=6$, its influence on the control effect tends to be stable.

• Future work can be carried out from the following aspects:

• Verify algorithm performance in professional simulation software like Bladed to further improve the possibility of practical application.

• Another research focus can be focused on adaptive tuning of NMPC control parameters to further improve yaw performance.

• FIWR provides a promising direction for NMPC research. In addition to the yaw system, control systems with multiple contradictory targets can use this method for performance optimization. The application in other WT control systems could be considered in the future.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant 52177204; the Natural Science Foundation of Hunan Province (No. 2020JJ4744); the Innovation-Driven Project of Central South University (No. 2020CX031).

Conflict of interest

The authors declare no conflict of interest.

(a) Optimization results of MFs.

${{WD}}_{{av}}$			${{WD}}_{{stt}}$					$\omega$
S	M	L	VS	S	M	L	VL	VS	S	M	L	VL
7.8700	6.6634	2.9170	5.6587	2.9202	3.6664	7.4053	3.2371	3.9158	6.0601	7.5499	0.5157	8.4095
6.0657	7.8755	8.2295	8.9000	4.6107	5.4806	9.7508	3.6173	4.3579	3.0437	3.2677	0.0000	8.0467
6.1503	5.7286	8.3112	7.8680	1.0916	3.3298	5.8968	1.3362	6.4390	4.3686	3.7788	4.4940	5.8137
7.7184	9.6993	7.4724	8.5182	2.2270	5.5094	8.2835	4.7723	4.5060	5.1648	4.5060	1.4068	7.9818
5.0738	6.9428	9.3414	9.1123	2.4585	4.4399	8.0474	2.5071	5.5342	3.1609	3.2971	1.5578	6.9253
5.9604	7.1217	8.0914	8.2626	2.6369	4.7863	7.5243	2.8854	5.0664	3.3863	3.6186	1.9524	6.7971
5.6626	6.3262	7.9887	8.0156	1.9404	3.6519	6.5687	2.2624	5.3948	5.3088	3.5967	2.3074	6.4714
5.1838	7.6663	7.3318	7.9048	4.1286	6.1454	8.4669	3.0868	4.3406	2.4830	3.0608	0.7845	7.3532
6.9129	7.6576	7.2776	9.3193	2.6147	4.9671	8.1121	4.1930	3.9820	2.7621	4.6776	1.6381	7.6662
6.2102	7.5423	7.7496	8.9233	2.9329	5.1472	8.4352	3.7087	4.4784	2.7644	4.3071	1.1391	6.9978

(b) Optimization results of fuzzy rules.

${\mathrm{WD}}_{\mathrm{av}}$	S	M	L	S	M	L	S	M	L	S	M	L	S	M	L
${\mathit{WD}}_{\mathit{std}}$	VS	VS	VS	S	S	S	M	M	M	L	L	L	VL	VL	VL
1	VL	M	VS	L	S	VS	S	L	S	M	S	S	VS	S	VL
2	VS	L	VS	M	M	VL	M	L	S	L	M	S	VL	L	VS
3	VL	L	VL	L	L	VL	L	S	L	M	L	L	VS	L	VL
4	VL	S	VS	M	M	VL	S	M	M	M	M	M	VL	S	VS
5	VL	L	VS	S	L	VL	S	M	S	S	S	L	VS	S	VL
6	VL	S	VS	S	L	VL	L	S	M	S	M	S	VS	S	VL
7	VL	L	VL	M	M	VS	M	M	M	L	M	M	VS	S	VL
8	VS	L	VS	L	M	VL	S	S	S	L	L	L	VL	L	VS
9	VL	S	VS	S	M	VL	S	S	M	S	L	L	VS	L	VL
10	VL	L	VL	S	S	VS	M	L	L	L	M	L	VS	S	VL

Table A.

Design variable optimization results.