# A Neural Network Controller for Variable-Speed Variable-Pitch Wind Energy Conversion Systems Using Generalized Minimum Entropy Criterion

Authors: Mifeng Ren; Jianhua Zhang; Ye Tian; Guolian Hou
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper considers the neural network controller design problem for variable pitch wind energy conversion systems (WECS) with non-Gaussian wind speed disturbances in the stochastic distribution control framework. The approach here is used to directly model the unknown control law based on a fixed neural network (the number of layers and nodes in a neural network is fixed) without the need to construct a separate model for the WECS. In order to characterize the randomness of the WECS, a generalized minimum entropy criterion is established to train connection weights of the neural network. For the train purpose, both kernel density estimation method and sliding window technique are adopted to estimate the PDF of tracking error and entropies. Due to the unknown process dynamics, the gradient of the objective function in a gradient-descent-type algorithm is estimated using an incremental perturbation method. The proposed approach is illustrated on a simulated WECS with non-Gaussian wind speed.


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## Body

## 1. Introduction

With the rapid growth of the global wind industry, wind energy has become one of the most important renewable energy sources [1]. Wind energy conversion system (WECS) technology has undergone rapid development in response to the demands for increasing use of renewable energy [2]. WECSs present two operating modes according to how the wind turbine is connected to the grid. In the fixed-speed mode, the turbine is directly connected to the grid, fixing the rotational speed to the grid frequency. In the variable-speed mode, an electronic converter is inserted between the generator and the grid, or a doubly fed generator (DFIG) controlled by the rotor circuit is used. Thus, the rotational speed can change independently of the grid frequency. In this paper a variable-speed variable-pitch wind energy conversion system is considered. This combination aims to compensate the limitation of each strategy working independently and may improve the transient response and the overall performance.Control of WECSs is essentially important in terms of energy generation efficiency, power quality, and installation’s life. Nevertheless, due to the nonlinearity, uncertainty, and various disturbances that exist in WECSs, it is a challenging problem for controller design. Various control syntheses such as PI regulator [3, 4], optimal control in LQ [5], and LQG form [6] have been developed. These control strategies which use the pitch angle as a control input give acceptable results for rotor speed regulation but showed poor performances in power regulation. In [7], it was shown that the generator torque alone is able to regulate the electrical power in an acceptable way. However, it generates large variations of the rotor speed that are not desirable for the wind turbine structure. Most of the work reported ignores the multivariable nature of WECSs.Recently, the control of variable-speed variable-pitch WECSs’ operation has attracted a lot of attention. A PI controller in the power loop and a self-tuning regulator in the speed loop are proposed in [8]. Considering the nonlinear and time-varying characteristics of the WECS, advanced control theory and intelligent control scheme have been developed. In [9], sliding mode control is used to cope with system uncertainty and reduce mechanical efforts and chattering. Model predictive control (MPC) is discussed in [10, 11], where the constraints on pitch angle and performance specifications can be handled. A gain scheduled H-infinity controller is proposed in [12]. Fuzzy logic based control is an effective approach to address the problem of parameter uncertainties [13]. Based on adaptive subspace predictive control (SPC) method [14, 15], investigated the wind turbines control problem in the data-driven framework. However, all these control methods have not fully considered the possible random noises involved in the WECS.The LQG strategy has been shown to be effective in accommodating plant uncertainties and random disturbance in a systematic and straightforward way in energy conversion control for wind generating systems [16–18]. LQG synthesis design method is based on linear model, and the noises and disturbance are assumed to be Gaussian. However, there are two main problems for WECSs: (1) since it is subjected to nonlinearity and random noises, the accurate model is very complex and even cannot be built; (2) the practical random signal from measurement device and disturbances from wind speed are often of non-Gaussian nature. In this case, the LQG control method may not achieve satisfactory performance. As such, in this study, the entropy of tracking error is employed to characterize the randomness of non-Gaussian WECSs.Based on the minimum error entropy principle, the stochastic systems with non-Gaussian disturbances can be well controlled [19, 20] in the data-driven framework. Since equations governing the system dynamics are unknown, it is very difficult or even impossible to obtain the gradient of the proposed performance function, which is one of the important steps in this method. Therefore, the neural network model is proposed to approximate the unknown nonlinear dynamics firstly, and then the gradient-descent-type control law can be obtained [21]. Without assuming or constructing a separate model for the unknown process dynamics [22, 23], use a neural network controller to directly regulate stochastic systems and the gradient of the objective function is estimated by using the simultaneous perturbation stochastic approximation approach. Motivated by this idea, a neural network controller is proposed for WECSs, where the non-Gaussianity of wind speed disturbances is taken into full account and the control problem is solved in the stochastic distribution control framework. Since the accurate model of WECS is difficult to be established, the neural network controller is designed based on the measurable input and output data according to the generalized minimum entropy principle. An incremental perturbation method is adopted to estimate the corresponding gradient in a gradient-descent-type algorithm. Simulation results show that the proposed minimum error entropy (MEE) control strategy can effectively reduce the influence of the non-Gaussian disturbances from the wind speed.


## 2. Problem Formulation

### 2.1. System Description and Modeling

A model of the entire WECS can be structured as several interconnected subsystems as shown in Figure1. The aerodynamic subsystem describes the transformation of kinetic energy stored in the wind into mechanical power via the wind turbine rotor. The drive train subsystem represents the mechanical parts that transfer the aerodynamic torque on the blades to the generator shaft. The pitch actuator subsystem models the pitch control system that controls the pitch angle of the wind turbine’s blades. Finally, the electrical subsystem describes the electric generator, the power electronic converters, and the generator control system.Figure 1
A typical variable-speed variable-pitch WECS.In general, the generator control system is based on a field-oriented vector control strategy where the machine variables are expressed in a synchronously rotating reference frame. Vector control stems from decoupled flux-current and torque-current control in AC drives.In Figure1, the input signals coming from the turbine control system are the generator torque set point Tg* and the desired pitch angle βd. The measured outputs are assumed to be the generator speed ωg and the generator power Pg. The wind speed v is the disturbance signal affecting the WECS.Denotex=[x1x2x3x4x5]T=[ωtωgTtwTgβ]T, u=[u1u2]T=[Tg*βd]T, and y=[y1y2]T=[ωgPg]T. Based on the aforementioned works, the state-space representation of the WECS concerned in this study can be written as [24]
(1)x˙1=-iJtx3+1JtCp(λ,β)λ12ρπR3v2,x˙2=1Jgx3-1Jgx4,x˙3=ks·i·x1-ks·x2-(i2·BsJt+BsJg)·x3+i·BsJt·Cp(λ,β)x1·v32·ρπR2+BsJg·x4,x˙4=-1τg·x4+1τg·u1,x˙5=-1τx5+1τu2,y1=x2,y2=x2·x4.


### 2.2. Control Problem Description

Designing an effective control system for the WECS is not an easy task. The system variables must be regulated in the presence of severe fluctuations in the input turbine powerPt caused by erratic variations in the wind speed. Fluctuations in Pt can lead to harmful effects on the system [8]. Large variations in the drive train torsional torque Ttw can occur, thus reducing the life time of the mechanical parts of the system. Input power fluctuations can result in electric power fluctuations supplied to the grid. This, in turn, can cause voltage flicker problems and a reduction in the power quality.Based on the analysis above, the main control objectives in the full load regime are to regulate both the generator powerPg and the generator speed ωg at their rated values Pg,rat and ωg,rat, respectively. These objectives can be achieved by manipulating the desired pitch angle βd and/or the generator torque set point Tg*. This can be inferred from (2), where the aerodynamic power Pt extracted from the wind is determined by the power coefficient, Cp(λ,β). This coefficient can be interpreted as a variable gain controlled by λ and β:
(2)Pt=Cp(λ,β)12ρπR2v3,
where λ≜ωtR/v is the tip speed ratio and ωt is the rotor speed of the wind turbine.Thus, in the full load regime, to regulate the power at its rated value, the power coefficient should be reduced by increasingβ, decreasing λ, or changing both variables. Consequently, manipulating the pitch angle results in deviations in the power extracted by the wind turbine and, indirectly, induces deviations in the turbine speed via the drive train dynamics. Similarly, the generator torque can affect the turbine speed through the drive train dynamics and can be used for controlling the power extracted by the wind turbine by controlling λ.Different from the decentralized control strategy in [25, 26], the multivariable stochastic control approach, shown in Figure 2, is adopted here to establish the WECS control purpose.Figure 2
The multivariable control scheme.The control problem here is solved in the stochastic distribution control framework, and the WECS in (1) is firstly discretized as
(3)xk+1=f(xk,uk,vk),yk=h(xk),
where xk=[x1k⋯x5k]T∈R5, yk=[y1ky2k]T∈R2 are system state and output vectors, respectively. uk=[u1ku2k]T∈R2 is the control signal applied to the plant. f(·) and h(·) are the system state and output dynamics, respectively. vk∈R is a non-Gaussian bounded random variable with known PDF γvk.Denote the set point vector of generator speed and generator power byr=[r1r2]T=[ωgrefPgref]T, and then the tracking error ek=[e1ke2k]T is
(4)ek=rk-yk=g(xk,rk).The control problem for WECS can be expressed as follows: find proper generator torque set pointTg* and the desired pitch angle βd such that the generator speed ωg and the generator power Pg can track their set points ωgref and Pgref as closely as possible in the presence of non-Gaussian wind speed disturbance v.From the above presentation, even though the model of WECS is established under a lot of assumptions and simplifications, the obtained model equations (1)–(4) are still very complex. However, with the development of technology, the input and output data can be easily measured. Therefore, the controller for WECS is designed in the data-driven framework in this paper.


## 2.1. System Description and Modeling

A model of the entire WECS can be structured as several interconnected subsystems as shown in Figure1. The aerodynamic subsystem describes the transformation of kinetic energy stored in the wind into mechanical power via the wind turbine rotor. The drive train subsystem represents the mechanical parts that transfer the aerodynamic torque on the blades to the generator shaft. The pitch actuator subsystem models the pitch control system that controls the pitch angle of the wind turbine’s blades. Finally, the electrical subsystem describes the electric generator, the power electronic converters, and the generator control system.Figure 1
A typical variable-speed variable-pitch WECS.In general, the generator control system is based on a field-oriented vector control strategy where the machine variables are expressed in a synchronously rotating reference frame. Vector control stems from decoupled flux-current and torque-current control in AC drives.In Figure1, the input signals coming from the turbine control system are the generator torque set point Tg* and the desired pitch angle βd. The measured outputs are assumed to be the generator speed ωg and the generator power Pg. The wind speed v is the disturbance signal affecting the WECS.Denotex=[x1x2x3x4x5]T=[ωtωgTtwTgβ]T, u=[u1u2]T=[Tg*βd]T, and y=[y1y2]T=[ωgPg]T. Based on the aforementioned works, the state-space representation of the WECS concerned in this study can be written as [24]
(1)x˙1=-iJtx3+1JtCp(λ,β)λ12ρπR3v2,x˙2=1Jgx3-1Jgx4,x˙3=ks·i·x1-ks·x2-(i2·BsJt+BsJg)·x3+i·BsJt·Cp(λ,β)x1·v32·ρπR2+BsJg·x4,x˙4=-1τg·x4+1τg·u1,x˙5=-1τx5+1τu2,y1=x2,y2=x2·x4.


## 2.2. Control Problem Description

Designing an effective control system for the WECS is not an easy task. The system variables must be regulated in the presence of severe fluctuations in the input turbine powerPt caused by erratic variations in the wind speed. Fluctuations in Pt can lead to harmful effects on the system [8]. Large variations in the drive train torsional torque Ttw can occur, thus reducing the life time of the mechanical parts of the system. Input power fluctuations can result in electric power fluctuations supplied to the grid. This, in turn, can cause voltage flicker problems and a reduction in the power quality.Based on the analysis above, the main control objectives in the full load regime are to regulate both the generator powerPg and the generator speed ωg at their rated values Pg,rat and ωg,rat, respectively. These objectives can be achieved by manipulating the desired pitch angle βd and/or the generator torque set point Tg*. This can be inferred from (2), where the aerodynamic power Pt extracted from the wind is determined by the power coefficient, Cp(λ,β). This coefficient can be interpreted as a variable gain controlled by λ and β:
(2)Pt=Cp(λ,β)12ρπR2v3,
where λ≜ωtR/v is the tip speed ratio and ωt is the rotor speed of the wind turbine.Thus, in the full load regime, to regulate the power at its rated value, the power coefficient should be reduced by increasingβ, decreasing λ, or changing both variables. Consequently, manipulating the pitch angle results in deviations in the power extracted by the wind turbine and, indirectly, induces deviations in the turbine speed via the drive train dynamics. Similarly, the generator torque can affect the turbine speed through the drive train dynamics and can be used for controlling the power extracted by the wind turbine by controlling λ.Different from the decentralized control strategy in [25, 26], the multivariable stochastic control approach, shown in Figure 2, is adopted here to establish the WECS control purpose.Figure 2
The multivariable control scheme.The control problem here is solved in the stochastic distribution control framework, and the WECS in (1) is firstly discretized as
(3)xk+1=f(xk,uk,vk),yk=h(xk),
where xk=[x1k⋯x5k]T∈R5, yk=[y1ky2k]T∈R2 are system state and output vectors, respectively. uk=[u1ku2k]T∈R2 is the control signal applied to the plant. f(·) and h(·) are the system state and output dynamics, respectively. vk∈R is a non-Gaussian bounded random variable with known PDF γvk.Denote the set point vector of generator speed and generator power byr=[r1r2]T=[ωgrefPgref]T, and then the tracking error ek=[e1ke2k]T is
(4)ek=rk-yk=g(xk,rk).The control problem for WECS can be expressed as follows: find proper generator torque set pointTg* and the desired pitch angle βd such that the generator speed ωg and the generator power Pg can track their set points ωgref and Pgref as closely as possible in the presence of non-Gaussian wind speed disturbance v.From the above presentation, even though the model of WECS is established under a lot of assumptions and simplifications, the obtained model equations (1)–(4) are still very complex. However, with the development of technology, the input and output data can be easily measured. Therefore, the controller for WECS is designed in the data-driven framework in this paper.


## 3. Neural Network Controller Design

### 3.1. Formulation of Objective Function

Since the wind speedv is non-Gaussian, the tracking error in (4) is probably non-Gaussian. And it is well known that all the randomness information can be characterized by the whole PDF. The main purposes of controller design are twofold: to make the PDF of tracking error follow a narrow and sharp Gaussian-like distribution and to drive the tracking error approaching to zero. Therefore, the chosen objective functions should obey the above two principles.It is noted that entropy is a general measure of randomness. Minimum entropy of tracking error corresponds to a sharp and narrow PDF; this means that the randomness of the tracking error is minimized. Thus the objective function is(5)J1=H(e1k)+H(e2k)+H(e1k,e2k),
where eik∈[aei,bei](i=1,2) (aei and bei are lower and upper bounds of the tracking error ei,i=1,2), H(eik)=-log⁡∫aeibeiγeik2(τi)dτi is the quadratic Renyi’s entropy [27] of each tracking error eik(i=1,2), and H(e1k,e2k)=-log⁡∫ae1be1∫ae2be2γe1k,e2k2(τ1,τ2)dτ1dτ2 is the joint entropy of tracking errors e1k and e2k, where γ(·) stands for the PDF of corresponding random variable.On the other hand, mean value can reflect the magnitude of the tracking error, which calls for another performance function(6)J2=E(e1k2)+E(e2k2),
where E(eik2)=∫aeibeiτ2γeik(τi)dτi is the mean value of each tracking error eik(i=1,2).To design the optimal controller for WECS, two objective functions (5) and (6) should be minimized simultaneously. And the constrained control energy also should be considered. In this paper, the weighting method is used by forming a linear combination of the objectives:
(7)J(uk)=∑i=12(RiH(eik)+Ri+2E(eik2))+R5H(e1k,e2k)+12ukTR6uk,
where Ri(i=1,2,…,6) are corresponding weights.Remark 1.
Weights in performance index (7) denote the different relative importance of different objectives. The value of the weights usually could be only decided by try-and-error method, based on engineering experiences, repeating simulations, and other information. By parametrically varying the weights in the combined single objective function (7), we could get different optimal control inputs, which are called Pareto optimal solutions. In this paper, repeating simulation approach is used to decide weight values to obtain tradeoff optimal control inputs.


### 3.2. Nonparametric Estimation of Objective Function

In this section, two nonparametric estimation approaches are proposed to estimate the objective function (7).(
1) Kernel Density Estimation Method. The PDF of the tracking errors can be estimated from sample sequences {eik(1),eik(2),…,eik(N)}(i=1,2) using Kernel density estimation (KDE) method:(8a)γeik(τi)≅γ^eik(τi)=1N∑j=1Nκ(τi-eik(j),σi2)(i=1,2),(8b)γe1ke2k(τ1,τ2)≅γ^e1ke2k(τ1,τ2)=1N∑j=1Nκ(τ1-e1k(j),σ12)κ(τ2-e2k(j),σ22),where κ(x,σ2)=(1/2πσ)exp⁡(-x2/2σ2) is the Gaussian Kernel function with standard deviation σ.According to definitions, the estimation of quadratic Renyi’s entropy and mean value can be formulated as(9a)H(eik)≅-log⁡1N2∑j=1N∑l=1Nκ(eik(j)-eik(l),2σi2)(i=1,2),(9b)H(e1k,e2k)≅-log⁡1N2∑j=1N∑l=1Nκ(e1k(j)-e1k(l),2σ12)κ(e2k(j)-e2k(l),2σ22),(9c)E(eik2)≅1N∑j=1N(eik(j))2(i=1,2).Substituting (9a)–(9c) into (7), the single objective function then can be obtained.Remark 2.
Kernel density estimation method is verified to be an effective PDF estimation approach. However,N samples are generated at each instant k, which requires mass memory and results in heavy computing burden.(
2) Sliding Window Technique. To enhance the computing efficiency, an alternative method called sliding window technique is proposed here.At instantk, quadratic Renyi’s entropy can be rewritten as(10a)H(eik)=-log⁡∫aeibeiγeik2(τi)dτi=-log⁡(E[γeik(τi)]),(i=1,2),(10b)H(e1k,e2k)=-log⁡∫ae1be1∫ae2be2γe1ke2k2(τ1,τ2)dτ2dτ1,=-log⁡(E[γe1ke2k(τ1,τ2)]).Drop the expectation and use the most current sample of tracking error in the PDF to obtain the following stochastic estimate for entropies:(11a)H(eik)=-log⁡(E[γeik(τi)])≈-log⁡(γeik(τi)),(11b)H(e1k,e2k)≈-log⁡(E[γe1ke2k(τ1,τ2)])=-log⁡(γe1ke2k(τ1,τ2)),where eik(i=1,2) denote the most recent samples of tracking error at instant k. Next, the “sliding window” technology is employed to estimate the (joint) PDF of tracking error over the most recent L samples {ei,k-L,ei,k-L+1,…,ei,k-1}. When k<L, the needed data can be complemented according to the history data of the system. Then, the estimation of (joint) PDF is given by(12a)γeik(eik)=1L∑j=k-Lk-1κ(eik-eij,σi),(i=1,2),(12b)γe1ke2k(e1k,e2k)=1L∑j=k-Lk-1κ(e1k-e1j,σ1)κ(e2k-e2j,σ2).Thus the stochastic estimate of the (joint) entropy of tracking error at instantk becomes(13a)H(eik)=-log⁡(1L∑j=k-Lk-1κ(eik-eij,σi)),(i=1,2),(13b)H(e1k,e2k)=-log⁡1L∑j=k-Lk-1κ(e1k-e1j,σ1)κ(e2k-e2j,σ2).


### 3.3. Neural Network Controller

Based on the known informationIk={yk,yk-1,…,yk-M;uk-1,uk-2,…,uk-N;rk} (M and N are numbers of previous measurements and controls), a neural network with fixed number of layers and nodes is used in this paper to directly model the resulting unknown control law without the need to construct a separate model for the unknown WECS dynamics. Here, the output of the NN will correspond to the value of the control uk. Associated with the NN producing uk will be a vector of connection weights θk∈Rp that should be trained. Therefore, the control problem in this paper is equivalent to finding θk to minimize the performance index (7). And after the optimal θk has been found, the optimal control uk would be the output of neural network.As a result, in theory, the training ofθk can be obtained by minimizing Jk to give
(14)θk=θk-1-αk∂Jk∂θk|θk=θk-1.However, the gradient-descent-type algorithms are not generally feasible in the model-free setting here. As such, the above algorithm can only be regarded as a guideline for the theoretical inside into the training scheme. Motivated by the method in [21], the following steps should be employed.(1)
Set the current sample time ask-1, and fix θk-10=[θ1,k-10θ2,k-10⋯θp,k-10]T.(2)
According to the fixed neural network used in this paper, the control inputuk-1 can be calculated.(3)
Calculate the performance function asJ(θk-10) according to (7).(4)
Disturbθk-10 by Δθk-1 and use the new θ-value θk-1new=θk-10+Δθk-1 to generate another control input uk-1new.(5)
Calculate the related performance function asJ(θk-10+Δθk-1) using (7).(6)
Formulate the required gradient vector in (14):
(15)∂Jk∂θk|θk=θk-10≈J(θk-10+Δθk)-J(θk-10)Δθk.(7)
Calculateθk based on (14).Note that every time when the dataset is generated, the vectors{Ik} need to be updated as well.Based on the obtainedθk, the optimal control input can be calculated easily.Remark 3.
The estimation of gradient vector(∂Jk/∂θk)|θk=θk-1 in (14) can be obtained by using the simultaneous perturbation stochastic approximation approach in [22]. And the convergence of this method also can be found in [23].


## 3.1. Formulation of Objective Function

Since the wind speedv is non-Gaussian, the tracking error in (4) is probably non-Gaussian. And it is well known that all the randomness information can be characterized by the whole PDF. The main purposes of controller design are twofold: to make the PDF of tracking error follow a narrow and sharp Gaussian-like distribution and to drive the tracking error approaching to zero. Therefore, the chosen objective functions should obey the above two principles.It is noted that entropy is a general measure of randomness. Minimum entropy of tracking error corresponds to a sharp and narrow PDF; this means that the randomness of the tracking error is minimized. Thus the objective function is(5)J1=H(e1k)+H(e2k)+H(e1k,e2k),
where eik∈[aei,bei](i=1,2) (aei and bei are lower and upper bounds of the tracking error ei,i=1,2), H(eik)=-log⁡∫aeibeiγeik2(τi)dτi is the quadratic Renyi’s entropy [27] of each tracking error eik(i=1,2), and H(e1k,e2k)=-log⁡∫ae1be1∫ae2be2γe1k,e2k2(τ1,τ2)dτ1dτ2 is the joint entropy of tracking errors e1k and e2k, where γ(·) stands for the PDF of corresponding random variable.On the other hand, mean value can reflect the magnitude of the tracking error, which calls for another performance function(6)J2=E(e1k2)+E(e2k2),
where E(eik2)=∫aeibeiτ2γeik(τi)dτi is the mean value of each tracking error eik(i=1,2).To design the optimal controller for WECS, two objective functions (5) and (6) should be minimized simultaneously. And the constrained control energy also should be considered. In this paper, the weighting method is used by forming a linear combination of the objectives:
(7)J(uk)=∑i=12(RiH(eik)+Ri+2E(eik2))+R5H(e1k,e2k)+12ukTR6uk,
where Ri(i=1,2,…,6) are corresponding weights.Remark 1.
Weights in performance index (7) denote the different relative importance of different objectives. The value of the weights usually could be only decided by try-and-error method, based on engineering experiences, repeating simulations, and other information. By parametrically varying the weights in the combined single objective function (7), we could get different optimal control inputs, which are called Pareto optimal solutions. In this paper, repeating simulation approach is used to decide weight values to obtain tradeoff optimal control inputs.


## 3.2. Nonparametric Estimation of Objective Function

In this section, two nonparametric estimation approaches are proposed to estimate the objective function (7).(
1) Kernel Density Estimation Method. The PDF of the tracking errors can be estimated from sample sequences {eik(1),eik(2),…,eik(N)}(i=1,2) using Kernel density estimation (KDE) method:(8a)γeik(τi)≅γ^eik(τi)=1N∑j=1Nκ(τi-eik(j),σi2)(i=1,2),(8b)γe1ke2k(τ1,τ2)≅γ^e1ke2k(τ1,τ2)=1N∑j=1Nκ(τ1-e1k(j),σ12)κ(τ2-e2k(j),σ22),where κ(x,σ2)=(1/2πσ)exp⁡(-x2/2σ2) is the Gaussian Kernel function with standard deviation σ.According to definitions, the estimation of quadratic Renyi’s entropy and mean value can be formulated as(9a)H(eik)≅-log⁡1N2∑j=1N∑l=1Nκ(eik(j)-eik(l),2σi2)(i=1,2),(9b)H(e1k,e2k)≅-log⁡1N2∑j=1N∑l=1Nκ(e1k(j)-e1k(l),2σ12)κ(e2k(j)-e2k(l),2σ22),(9c)E(eik2)≅1N∑j=1N(eik(j))2(i=1,2).Substituting (9a)–(9c) into (7), the single objective function then can be obtained.Remark 2.
Kernel density estimation method is verified to be an effective PDF estimation approach. However,N samples are generated at each instant k, which requires mass memory and results in heavy computing burden.(
2) Sliding Window Technique. To enhance the computing efficiency, an alternative method called sliding window technique is proposed here.At instantk, quadratic Renyi’s entropy can be rewritten as(10a)H(eik)=-log⁡∫aeibeiγeik2(τi)dτi=-log⁡(E[γeik(τi)]),(i=1,2),(10b)H(e1k,e2k)=-log⁡∫ae1be1∫ae2be2γe1ke2k2(τ1,τ2)dτ2dτ1,=-log⁡(E[γe1ke2k(τ1,τ2)]).Drop the expectation and use the most current sample of tracking error in the PDF to obtain the following stochastic estimate for entropies:(11a)H(eik)=-log⁡(E[γeik(τi)])≈-log⁡(γeik(τi)),(11b)H(e1k,e2k)≈-log⁡(E[γe1ke2k(τ1,τ2)])=-log⁡(γe1ke2k(τ1,τ2)),where eik(i=1,2) denote the most recent samples of tracking error at instant k. Next, the “sliding window” technology is employed to estimate the (joint) PDF of tracking error over the most recent L samples {ei,k-L,ei,k-L+1,…,ei,k-1}. When k<L, the needed data can be complemented according to the history data of the system. Then, the estimation of (joint) PDF is given by(12a)γeik(eik)=1L∑j=k-Lk-1κ(eik-eij,σi),(i=1,2),(12b)γe1ke2k(e1k,e2k)=1L∑j=k-Lk-1κ(e1k-e1j,σ1)κ(e2k-e2j,σ2).Thus the stochastic estimate of the (joint) entropy of tracking error at instantk becomes(13a)H(eik)=-log⁡(1L∑j=k-Lk-1κ(eik-eij,σi)),(i=1,2),(13b)H(e1k,e2k)=-log⁡1L∑j=k-Lk-1κ(e1k-e1j,σ1)κ(e2k-e2j,σ2).


## 3.3. Neural Network Controller

Based on the known informationIk={yk,yk-1,…,yk-M;uk-1,uk-2,…,uk-N;rk} (M and N are numbers of previous measurements and controls), a neural network with fixed number of layers and nodes is used in this paper to directly model the resulting unknown control law without the need to construct a separate model for the unknown WECS dynamics. Here, the output of the NN will correspond to the value of the control uk. Associated with the NN producing uk will be a vector of connection weights θk∈Rp that should be trained. Therefore, the control problem in this paper is equivalent to finding θk to minimize the performance index (7). And after the optimal θk has been found, the optimal control uk would be the output of neural network.As a result, in theory, the training ofθk can be obtained by minimizing Jk to give
(14)θk=θk-1-αk∂Jk∂θk|θk=θk-1.However, the gradient-descent-type algorithms are not generally feasible in the model-free setting here. As such, the above algorithm can only be regarded as a guideline for the theoretical inside into the training scheme. Motivated by the method in [21], the following steps should be employed.(1)
Set the current sample time ask-1, and fix θk-10=[θ1,k-10θ2,k-10⋯θp,k-10]T.(2)
According to the fixed neural network used in this paper, the control inputuk-1 can be calculated.(3)
Calculate the performance function asJ(θk-10) according to (7).(4)
Disturbθk-10 by Δθk-1 and use the new θ-value θk-1new=θk-10+Δθk-1 to generate another control input uk-1new.(5)
Calculate the related performance function asJ(θk-10+Δθk-1) using (7).(6)
Formulate the required gradient vector in (14):
(15)∂Jk∂θk|θk=θk-10≈J(θk-10+Δθk)-J(θk-10)Δθk.(7)
Calculateθk based on (14).Note that every time when the dataset is generated, the vectors{Ik} need to be updated as well.Based on the obtainedθk, the optimal control input can be calculated easily.Remark 3.
The estimation of gradient vector(∂Jk/∂θk)|θk=θk-1 in (14) can be obtained by using the simultaneous perturbation stochastic approximation approach in [22]. And the convergence of this method also can be found in [23].


## 4. Simulation Results

In this section, the proposed control method is applied to a certain variable-speed variable-pitch WECS. The model in (1)–(4) is used for producing the measurements. Model parameters are shown in Table 1.Table 1
Parameters
Description
Numerical value
P
g
,
rat
Rated generator power
2.0 MW
τ
Time constant of the pitch system
0.1 s
τ
g
Time constant of the generator system
20 ms
J
g
Inertia of the generator
56.29 kg·m2
J
t
Inertia of the turbine
1.86
×
1
0
6
B
s
Shaft damping coefficients
212.2
ω
g
,
rat
Rated generator speed
220 rad/s
i
Gear ratio
74.38
R
Blade length of the wind turbine
33 m
ρ
Air density
1.25 kg/m3The simulation is carried out on the basis of the working condition withωg=150rad/s and Pg=1.5MW. The PDF of wind speed is given in Figure 3. The sampling period is Ts=1s, sliding window width is L=100, and forgetting factor is λ=0.0095. The weights in (7) are R1=0.1, R2=0.1, R3=0.001, R4=0.001, R5=10, and R6=[10-410-510-510-3]. The controller is modeled by using a NN with two layers, one of 20 nodes and one of 10 nodes [23]. The inputs to the controller include the current and most recent output, the most recent control, and the current set point, yielding a total of eight input nodes. Therefore, the total number of weights to be trained is 412.Figure 3
PDF of wind speed.The advantage of the proposed method is shown by comparing with PI controllers whose optimal PI parameters are tuned using the MATLAB NCD toolbox. Transfer functions of rotary speed controller and power controller areGC1(s)=0.0015+0.045/s and GC2(s)=0.095+0.058/s, respectively. The comparative results are shown in Figures 4, 5, 6, 7, 8, and 9.Figure 4
Responses of generator speed.Figure 5
Responses of generator power.Figure 6
Control inputs.Figure 7
Performance index.3D PDFs of tracking error: (a) generator speed; (b) generator power.
(a)
(b)PDFs of tracking error at typical instants: (a) generator speed; (b) generator power.
(a)
(b)From Figures4 and 5, it can be seen that, compared with the conventional PI control strategy, the proposed control method can make generator speed and power have smaller fluctuations. Small oscillation of the generator speed can reduce the mechanical load of wind turbines, corresponding to a reliable operation of the wind power system. On the other hand, small fluctuation of the generator power can guarantee more stable supply power and consumers’ required power quality. The variation of control inputs is presented in Figure 6. It is clear that the changes of electromagnetic torque and pitch angle are smooth. In Figure 7, the objective function (7) is decreasing with time and finally approaching a small value, and this means that the WECS can achieve a satisfying performance under the proposed control.The shapes of PDFs of generator speed and power in (8a) and (8b) become narrower and sharper along with sampling time using the proposed control method; this indicates that the WECS has a small uncertainty in its closed-loop operation. It also can be verified from Figure 9, in which the PDFs of generator speed and power at several typical instants are shown.The above simulation results illustrate that the proposed control approach can obtain better performance over the PI controller.


## 5. Conclusions

In this paper, a neural network controller design approach in the data-driven framework is proposed for the wind energy conversion system (WECS). The proposed method differs from previous results in minimum entropy control: this method avoids the construction of a system model and focuses directly on regulating the WECS via the construction of a closed-loop control algorithm based on a neural network with fixed structure. Since there are no assumption equations describing the WECS, it is not possible to calculate the gradient of the objective function for use in standard gradient-descent-type search algorithms. Therefore, an incremental perturbation approximation method is proposed to estimate the gradient. The proposed approach is applied to a certain WECS where the control objective is to track the target values for both generator speed and output power as closely as possible. Simulation results show that the proposed control method can achieve good performance.



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*Source: 412027-2014-08-12.xml*