Overall Adaptive Controller Design of PMSG Under Whole Wind Speed Range: A Perturbation Compensation Based Approach

Abstract

This paper proposes an adaptive overall control strategy of the permanent magnet synchronous generator-based wind energy conversion system (WECS) in the whole wind speed range. For the machine side, the maximum power point tracking (MPPT) operation is realized by stator current and mechanical rotation speed control under below-rated wind speeds. Under above-rated wind speeds, the extracted wind power is limited via pitch control. For the grid side, the reactive and active power injected into grid is regulated by DC-Link voltage and grid current control loop. In addition, under grid voltage dips, the pitch control is employed for limiting grid current and maintaining the DC-Link voltage around its rated value. The fault ride-through capability (FRTC) can be enhanced. The overall control strategy is based on perturbation estimation technique. A designed observer is used for estimating the perturbation term including all system nonlinearities, uncertainties and disturbances, so as to compensate the real perturbation. Then, an adaptive control for the original nonlinear system can be realized. The effectiveness of the proposed overall control strategy is verified by applying the strategy to a 2-MW WECS in MATLAB/Simulink. The results show that, compared with the feedback linearizing control (FLC) strategy and conventional vector control (VC) strategy, the proposed perturbation observer based adaptive control (PO-AC) strategy realizes the control objectives without knowing full state information and accurate system model, and improves the robustness of the WECS parameter uncertainties and FRTC.

1. Introduction

Recent years, renewable energy sources have attracted a lot attention, in which wind energy is the fastest growing and most competitive source so far, because it is clean, safe and exhaustless [1,2,3]. Recently, the world market share of wind energy conversion systems (WECSs) is dominated by the variable-speed wind turbines (VSWTs) due to their merits, e.g., high power quality and efficiency [4,5]. Moreover, the VSWTs can control output power and wind turbine (WT) speed for the reduction of stresses and load on blades and tower. Nowadays, doubly fed induction generators (DFIGs) based VSWTs occupy a large market share of the WECSs with the advantages such as large operation region, small capability of power electronic devices and full decoupled control of reactive and active power [6,7]. However, a gearbox is required for coupling the generator to the WT in the WECS based on DFIG, which increases failure rate and maintenance expenses [8,9,10]. Recently, permanent magnet synchronous generators (PMSGs) based WECSs are without gearboxes, and possess merits e.g., self-excitation, fast dynamic response, low noise, and high efficiency [11,12,13,14,15,16,17]. Usually, an AC-DC-AC converter system connects a PMSG to the power grid in the WECS, which is shown in Figure 1.

In the WECS, it consists two voltage-source converters (VSCs), i.e., a grid-side converter (GSC) and a machine-side converter (MSC). The GSC is responsible for maintaining the DC-Link voltage and delivering active power to the grid, whereas the objective of the MSC is controlling generating power of the generator. The well regulated DC-Link voltage can decouple the control of the GSC and MSC. In addition, for the control of the MSC, the maximum wind power can be extracted through using speed control or torque control under below-rated wind speeds [18,19,20,21]. Meanwhile pitch control methods have been proposed to limit the extracted wind power under above-rated wind speeds [22,23,24]. However, the works of control strategies based on whole wind speed range for WTs are relatively few. In this paper, an overall control strategy for the WECS in the whole wind speed range including mechanical rotation speed control at below-rated speeds and pitch control at rated power area, and the GSC control, has been considered.

In addition, the grid integration of the WECSs has been enforced more and more regulations with sharp increment of wind energy penetrating into the grid [25,26,27,28,29]. According to the requirements of the grid codes, the wind farms should provide fault ride-through capability (FRTC) during or after grid faults. To meet these requirements, the WECSs are required to remain connected to the grid during grid faults, and supply the generating power to the grid when the grid faults are cleared. A stable operation is important for the grid and WECS. Additional protection devices, e.g., active crowbars and damping resistor, have been installed in PMSG and DFIG based WECSs for enhancing the FRTC of WECSs [30,31,32,33]. Although the FRTC of the WECSs has been enhanced, the cost of the whole system has been increased by installing the additional protection devices. Another effective way of enhancing the FRTC of WECSs is improving or redesigning the control algorithms of the VSCs. It can avoid installing extra equipment, and the power rating of the protection devices can be reduced [34,35,36,37,38,39,40]. Numerous designed controllers have been studied for the control of the GSC, i.e., feedback linearisation control (FLC), conventional vector control with proportional-integral (VC) loops, sliding mode control, and perturbation observer based adaptive control (PO-AC) [40,41,42,43,44,45,46]. Although these advanced control methods have provided satisfactory performances of the GSC and enhanced the FRTC of the WECSs based on the PMSGs, all these works assumed that a current source can simply replace the generator output. The reduction of generator output is simultaneous and proportional to the decrement of the grid voltage. In addition, the dynamics of the machine-side during the tests of different voltage dips have not been considered in [39,40,41] or have been simply simulated by varying current in [45]. However, a real WT cannot be simply replaced by a current source. In this paper, a pitch control strategy is employed for assisting the WECS operation under grid voltage dips. Without additional devices, the pitch control strategy can be applied in reducing the extracted wind power by changing pitch angle once grid voltage dips is detected.

To investigate the performances of the permanent magnet synchronous generator based wind turbine (PMSG-WT) under normal and fault operation conditions, an overall adaptive control approach for the WECS has been proposed. This paper integrates the author’s previous works reported in [45,47]. Under normal operation condition, for MSC control, the maximum power point tracking (MPPT) operation is realized by mechanical rotation speed control and stator current control under below-rated wind speeds. The proposed MPPT control method in [47] is employed in this paper. It requires mechanical rotation speed to track its optimal value. Under above-rated wind speeds, the extracted wind power is limited by mechanical rotation speed control and stator current control. The mechanical rotation speed and stator currents are required to track their rated values, respectively. For GSC control, the reactive and active power injected into grid is regulated by using grid current control and DC-Link voltage control loop designed in [45]. Under the grid voltage dips, the GSC control strategy adopts the same control strategy as that in normal operation. For the MSC control strategy, only pitch control strategy is applied. The pitch control is employed to assist in enhancing the FRTC of the WECS. The extracted wind power can be reduced through increasing pitch angle. The proposed control strategy provides satisfactory performances in the different operation conditions, e.g., ramp wind speeds and random wind speeds, and enhances the FRTC of the WECS during grid voltage dips.

The remaining parts of this paper is organized as follows. The model of the WECS based on PMSG is presented in Section 2. Section 3 presents the design of the overall PO-AC. In Section 4, simulation studies are carried out for verifying the effectiveness of the proposed PO-AC in comparing with the FLC and VC. Finally, conclusions of this work are presented in Section 5.

2. System Model

The WECS considered in this paper is shown in Figure 1. This section presents the models of the machine and grid sides.

2.1. Model of Machine Side

For the machine side, the state-space model of the PMSG-WT is given as [48]:

$$\dot{x}=f\left(x\right)+b_1\left(x\right)u_1+b_2\left(x\right)u_2+b_3\left(x\right)u_3$$

(1)

where

$$\begin{array}{ccc}f\left(x\right)=\left[\begin{array}{c}{f}_1\left(x\right)\\ f_2\left(x\right)\\ f_3\left(x\right)\\ f_4\left(x\right)\end{array}\right]& =& \left[\begin{array}{c}-\frac{\beta }{{\tau }_{\beta }}\\ -\frac{R_{\mathrm{s}}}{L_{\mathrm{md}}}{i}_{\mathrm{md}}+\frac{{\omega }_{\mathrm{e}}{L}_{\mathrm{mq}}}{L_{\mathrm{md}}}{i}_{\mathrm{mq}}\\ -\frac{R_{\mathrm{s}}}{L_{\mathrm{mq}}}{i}_{\mathrm{mq}}-\frac{1}{L_{\mathrm{mq}}}{\omega }_{\mathrm{e}}(L_{\mathrm{md}}{i}_{\mathrm{md}}+K_e)\\ \frac{1}{J_{\mathrm{tot}}}(T_{\mathrm{e}}-T_{\mathrm{m}})\end{array}\right],\hfill \\ b_1\left(x\right)& =& {[-\frac{\beta }{{\tau }_{\beta }}000]}^{\mathrm{T}},\hfill \\ b_2\left(x\right)& =& {\left[0\frac{1}{L_{\mathrm{md}}}00\right]}^{\mathrm{T}},\hfill \\ b_3\left(x\right)& =& {\left[00\frac{1}{L_{\mathrm{mq}}}0\right]}^{\mathrm{T}},\hfill \\ x& =& {[x_1,x_2,x_3,x_4]}^T={\left[\beta i_{\mathrm{md}}{i}_{\mathrm{mq}}{\omega }_{\mathrm{m}}\right]}^{\mathrm{T}},\hfill \\ u& =& {[u_1,u_2,u_3]}^T={[{\beta }_{\mathrm{r}},V_{\mathrm{md}},V_{\mathrm{mq}}]}^T,\hfill \\ y& =& {[y_1,y_2,y_3]}^T={[h_1\left(x\right),h_2\left(x\right),h_3\left(x\right)]}^T={[{\omega }_{\mathrm{m}},i_{\mathrm{md}},i_{\mathrm{mq}}]}^T\hfill \end{array}$$

where $x\in R^4$, $u\in R^3$ and $y\in R^3$ are state vector, input vector and output vector, respectively; $f\left(x\right)$, $b\left(x\right)$ and $h\left(x\right)$ are smooth vector fields. $\beta$ and ${\beta }_{\mathrm{r}}$ are pitch angle and pitch angle reference, ${\tau }_{\beta }$ is the actuator time constant, $V_{\mathrm{md}}$ and $V_{\mathrm{mq}}$ are the $\mathrm{d},\mathrm{q}$ axis stator voltages, $i_{\mathrm{md}}$ and $i_{\mathrm{mq}}$ are the $\mathrm{d},\mathrm{q}$ axis stator currents, $L_{\mathrm{md}}$ and $L_{\mathrm{mq}}$ are $\mathrm{d},\mathrm{q}$ axis stator inductances, $R_{\mathrm{s}}$ is the stator resistance, p is the number of pole pairs, $K_e$ is the field flux given by the magnet, $J_{\mathrm{tot}}$ is the total inertia of the drive train, ${\omega }_{\mathrm{m}}$ and ${\omega }_{\mathrm{e}}(=p{\omega }_{\mathrm{m}})$ are the mechanical and electrical rotation speed, respectively, and $T_{\mathrm{e}}$ and $T_{\mathrm{m}}$ are the WT electromagnetic and mechanical torque, respectively.

The electromagnetic torque is expressed as:

$$\begin{array}{c}{T}_{\mathrm{e}}=p[(L_{\mathrm{md}}-L_{\mathrm{mq}})i_{\mathrm{md}}{i}_{\mathrm{mq}}+i_{\mathrm{mq}}{K}_e]\hfill \end{array}$$

(2)

Under below-rated wind speed, its aim is extracting the maximum wind power, which requires that the maximum power coefficient $C_{p\max }$ should be achieved under time-varying wind speeds. To maintain $C_p$ at its maximum value $C_{p\max }$, the tip speed ratio $\lambda$ is required to keep at its optimal value ${\lambda }_{\mathrm{opt}}$ with a fixed pitch angle $\beta$.

$$C_{p\max }=C_{\mathrm{p}}\left({\lambda }_{\mathrm{opt}}\right)$$

(3)

To obtain the optimal value of tip speed ratio ${\lambda }_{\mathrm{opt}}$, the mechanical rotation speed ${\omega }_{\mathrm{m}}$ is required to track its optimal reference ${\omega }_{\mathrm{m}}^{*}$ as

$${\omega }_{\mathrm{m}}^{*}=\frac{V}{R}{\lambda }_{\mathrm{opt}}$$

(4)

Under above-rated wind speeds, to maintain the extracted wind power at rated power in normal operation, it requires the corresponding pitch angle should be achieved, which in turn requires both the mechanical rotation speed ${\omega }_{\mathrm{m}}$ and the mechanical torque $T_{\mathrm{m}}$ should be kept at their rated values, respectively. The rated mechanical torque $T_{\mathrm{m}}$ is achieved when the electromagnetic torque $T_{\mathrm{e}}$ can track its rated reference $T_{\mathrm{er}}$ and the ${\omega }_{\mathrm{m}}$ is kept at it rated value. From (2), it can be seen that the electromagnetic torque $T_{\mathrm{e}}$ can be maintained at $T_{\mathrm{er}}$ if the q-axis stator current $i_{\mathrm{mq}}$ can track its rated reference $i_{\mathrm{mq}}^{*}$ and $i_{\mathrm{md}}$ is kept at 0.

Note that, under the grid voltage dips, the pitch control is employed to reduce the extracted wind power for maintaining the DC-Link voltage $V_{\mathrm{dc}}$ at its rated value.

2.2. Model of Grid Side

The state-space model of the GSC of the WECS is expressed as [39]

$$\dot{x}=f\left(x\right)+b_4\left(x\right)u_4+b_5\left(x\right)u_5$$

(5)

where

$$\begin{array}{ccc}f\left(x\right)=\left[\begin{array}{c}{f}_5\left(x\right)\\ f_6\left(x\right)\\ f_7\left(x\right)\end{array}\right]& =& \left[\begin{array}{c}\frac{1}{L_{\mathrm{g}}}{E}_{\mathrm{gd}}-\frac{R_{\mathrm{g}}}{L_{\mathrm{g}}}{i}_{\mathrm{gd}}+\omega i_{\mathrm{gq}}\\ \frac{1}{L_{\mathrm{g}}}{E}_{\mathrm{gq}}-\frac{R_{\mathrm{g}}}{L_{\mathrm{g}}}{i}_{\mathrm{gq}}-\omega i_{\mathrm{gd}}\\ \frac{3E_{\mathrm{gd}}{i}_{\mathrm{gd}}}{2C{V}_{\mathrm{dc}}}-\frac{i_{\mathrm{dc}2}}{C}\end{array}\right],\hfill \\ b_4\left(x\right)& =& {[-\frac{1}{L_{\mathrm{g}}}00]}^{\mathrm{T}},\hfill \\ b_5\left(x\right)& =& {[0-\frac{1}{L_{\mathrm{g}}}0]}^{\mathrm{T}},\hfill \\ x& =& {[x_5,x_6,x_7]}^T={\left[i_{\mathrm{gd}}{i}_{\mathrm{gq}}{V}_{\mathrm{dc}}\right]}^{\mathrm{T}},\hfill \\ u& =& {[u_4,u_5]}^T={[V_{\mathrm{gd}},V_{\mathrm{gq}}]}^T,\hfill \\ y& =& {[y_4,y_5]}^T={[h_4\left(x\right),h_5\left(x\right)]}^T={[i_{\mathrm{gq}},V_{\mathrm{dc}}]}^T\hfill \end{array}$$

where $x\in R^3$, $y\in R^2$ and $u\in R^2$ are state, output and input vectors, respectively; $f\left(x\right)$, $b\left(x\right)$ and $h\left(x\right)$ are smooth vector fields. $i_{\mathrm{gd},\mathrm{q}}$ are the grid current in the $\mathrm{d},\mathrm{q}$ axis, $E_{\mathrm{gd},\mathrm{q}}$ are the grid voltage in the $\mathrm{d},\mathrm{q}$ axis, $V_{\mathrm{gd},\mathrm{q}}$ are the GSC voltage in the $\mathrm{d},\mathrm{q}$ axis, $L_{\mathrm{g}}$ and $R_{\mathrm{g}}$ are equivalent inductance and resistance between the high voltage terminals of the grid-connected transformer and GSC, C is capacitance of DC-Link and $\omega$ is grid voltage’s angular speed, $i_{\mathrm{dc}1.2}$ are the grid-side and machine-side DC-Link current, respectively, $i_{\mathrm{dc}}$ is the current of the DC-Link capacitor, and $V_{\mathrm{dc}}$ is the voltage of the DC-Link capacitor.

When the WECSs encounter the grid voltage dips without protection measures, the DC-Link voltage cannot well regulated. It will result in the sharp increment of the grid-side current, which may exceed its safe boundary. For the power grid, the varying power injected into the power grid can be seen as disturbance. The objective of the GSC is to enhance the FRTC of the WECS and decouple the control of reactive and active power injected into the power grid.

3. Adaptive Overall Control Approach of the WECS

The brief control block diagram for the proposed control approach of the WECS is shown in Figure 2. For the GSC, the adaptive control approach proposed in author’s previous work [45] has been employed for maintaining the DC-Link voltage at its rated value. For the MSC, under below-rated wind speeds, its objective is capturing maximum wind power. The proposed adaptive control approach has been proposed in author’s previous work [47]. Under above-rated wind speeds, its aim is to limit the extracted wind power through adaptive pitch control. In addition, the adaptive pitch control is employed for limiting extracted wind power under grid voltage dips. The adaptive pitch control is introduced firstly. Then, the control strategy of the WECS operating under normal operation and grid voltage dips will be introduced.

3.1. Adaptive Pitch Control

3.1.1. Design for Proposed Strategy

This section presents the design of the perturbation observer-based PO-AC. For each subsystem in the PMSG-WT, a lumped perturbation term is defined to contain disturbances, subsystem nonlinearities, and interactions between subsystems. Observers are employed for estimating each subsystem state and perturbation term. The estimation processes are implemented via introducing fictitious states. The proposed adaptive control design is illustrated in the following subsections.

3.1.2. States and Perturbation Observer

A canonical control form of studied system is considered as follows

$$\left\{\begin{array}{ccc}{\dot{x}}_1\hfill & =\hfill & x_2\hfill \\ \hfill & ⋮\hfill \\ {\dot{x}}_n\hfill & =\hfill & f\left(x\right)+b\left(x\right)u\hfill \\ y\hfill & =\hfill & x_1\hfill \end{array}\right.$$

(6)

and a fictitious state is defined to express the system perturbation, i.e., $x_{n+1}=\mathsf{\Psi }$, thus the state equation becomes

$$\left\{\begin{array}{ccc}{\dot{x}}_1\hfill & =\hfill & x_2\hfill \\ \hfill & ⋮\hfill \\ {\dot{x}}_n\hfill & =\hfill & x_{n+1}+b\left(0\right)u\hfill \\ {\dot{x}}_{n+1}\hfill & =\hfill & \dot{\mathsf{\Psi }}(·)\hfill \\ y\hfill & =\hfill & x_1\hfill \end{array}\right.$$

(7)

where $\mathsf{\Psi }=f\left(x\right)+\left[b\right(x)-b(0\left)\right]u$, $b\left(0\right)$ is the nominal value of $b\left(x\right)$. In the system (7), these assumptions are made as follows [49,50].

A.1.

$b\left(0\right)$ is chosen to meet: $\left|b\right(x)/b(0)-1|\le \theta <1$, where $\theta$ is a positive constant.

A.2.

The function $\mathsf{\Psi }(x,u,t):R^n\times R\times R^{+}\to R$ and $\dot{\mathsf{\Psi }}(x,u,t):R^n\times R\times R^{+}\to R$ are locally Lipschitz in their arguments in the domain of interest and are globally bounded in x:

$$\begin{array}{c}\hfill \mid \mathsf{\Psi }(x,u,t)\mid \le {\gamma }_1,\mid \dot{\mathsf{\Psi }}(x,u,t)\mid \le {\gamma }_2\end{array}$$

(8)

where ${\gamma }_1$ and ${\gamma }_2$ are positive constants. Besides, $\mathsf{\Psi }(0,0,0)=0$ and $\dot{\mathsf{\Psi }}(0,0,0)=0$. Assumption $A.2$ can assure that the origin is an equilibrium point in the open-loop system.

The perturbations can be estimated by employing the high-gain observer proposed in [49]. $b\left(x\right)$ and $f\left(x\right)$ are unknown continuous functions under the assumptions $A.1$ and $A.2$. State $x_1=y$ is assumed to be available. Therefore, a (n+1)th-order high-gain observer can be designed as

$$\left\{\begin{array}{ccc}{\dot{\widehat{x}}}_1\hfill & =\hfill & {\widehat{x}}_2+h_1(y-{\widehat{x}}_1)\hfill \\ \hfill & \dots \hfill \\ {\dot{\widehat{x}}}_n\hfill & =\hfill & {\widehat{x}}_{n+1}+h_n(y-{\widehat{x}}_1)+b\left(0\right)u\hfill \\ {\dot{\widehat{x}}}_{n+1}\hfill & =\hfill & h_{n+1}(y-{\widehat{x}}_1),\hfill \end{array}\right.$$

(9)

where ${\widehat{x}}_k,k=1.2\cdots n+1$ are the estimations of $x_k,k=1.2\cdots n+1$, respectively, and $h_{ij}={\alpha }_{ij}/{ϵ}^i$, $i=\mathrm{1.2.3}$; $j=1,\cdots ,r_i+1$ are the observer gains, $0<ϵ\ll 1$ is a specified positive constant and parameters ${\alpha }_{ij}$ are chosen so that the roots of

$$s^{n+1}+{\alpha }_{i1}{s}^n+\cdots +{\alpha }_{i{r}_i}s+{\alpha }_{i(r_i+1)}=0$$

(10)

are in the open left-half complex plan.

3.1.3. Adaptive Control for PMSG-WT

The proposed adaptive control adopts perturbation estimation term $\mathsf{\Psi }(·)$ for the compensations of the actual system perturbation and the adaptive feedback linearizing control can be achieved. The detail system model and accurate system parameters are required in conventional FLC design. More details can be found in previous work [45,47,49]. The estimation of x is denoted as $\widehat{x}$ in this paper.

For the WT, the objective system output is selected as $y_{\mathrm{WT}}=y_1={\omega }_{\mathrm{m}}$, and ${\omega }_{\mathrm{m}}^{*}$ is the reference value of mechanical rotation speed ${\omega }_{\mathrm{m}}$. According to Equation (7), differentiate $y_1$ until the control input appears explicitly, it yields

$$\begin{array}{cc}\hfill {\ddot{y}}_1& ={\mathsf{\Psi }}_1+b1\left(0\right)u_1\hfill \end{array}$$

(11)

where $b1.0)$ is nominal value of $b1\left(x\right)=\frac{AE\beta }{{\tau }_{\beta }}[-0.088e^{-12.5\tau }-\frac{0.08V^2}{F}+\frac{0.105{\beta }^2}{{(1+{\beta }^3)}^2}]$, and $A=\frac{\rho \pi R^2{V}^3}{2{\omega }_{\mathrm{m}}},E=(39.27-319\tau +1.1\beta )e^{-12.5\tau },F={\omega }_{\mathrm{m}}R+0.08\beta V$, and $\tau =\frac{1}{\lambda +0.08\beta }-\frac{0.035}{{\beta }^3+1}.$

According to Equations (6) and (7), the perturbation yields as follows

$$\begin{array}{c}\hfill \begin{array}{ccc}{\mathsf{\Psi }}_1\left(x\right)& =& (b1\left(x\right)-b1\left(0\right))u_1+A[-\frac{C_{\mathrm{p}}}{{\omega }_{\mathrm{m}}}-\frac{RV}{F^2}E]\frac{\mathrm{d}{\omega }_{\mathrm{m}}}{\mathrm{d}t}\hfill \\ & & -\frac{AE\beta }{{\tau }_{\beta }}[-0.088e^{-12.5\tau }-\frac{0.08V^2}{F}+\frac{0.105{\beta }^2}{{(1+{\beta }^3)}^2}]\frac{\mathrm{d}\beta }{\mathrm{d}t}\hfill \end{array}\end{array}$$

According Equation (9), a third-order states and perturbation observer (SPO) proposed in [49] is employed for estimating ${\mathsf{\Psi }}_1$ as

$$\left\{\begin{array}{c}{\dot{\widehat{\omega }}}_{\mathrm{m}}=\frac{{\alpha }_{11}}{{ϵ}_1}({\omega }_{\mathrm{m}}-{\widehat{\omega }}_{\mathrm{m}})+{\dot{\widehat{\omega }}}_{\mathrm{m}}\hfill \\ {\ddot{\widehat{\omega }}}_{\mathrm{m}}={\widehat{\mathsf{\Psi }}}_1+\frac{{\alpha }_{12}}{{ϵ}_1^2}({\omega }_{\mathrm{m}}-{\widehat{\omega }}_{\mathrm{m}})+b1\left(0\right)u_1\hfill \\ {\dot{\widehat{\mathsf{\Psi }}}}_1=\frac{{\alpha }_{13}}{{ϵ}_1^3}({\omega }_{\mathrm{m}}-{\widehat{\omega }}_{\mathrm{m}})\hfill \end{array}\right.$$

(12)

where ${\widehat{\omega }}_{\mathrm{m}}$, ${\dot{\widehat{\omega }}}_{\mathrm{m}}$, ${\ddot{\widehat{\omega }}}_{\mathrm{m}}$ and ${\dot{\widehat{\mathsf{\Psi }}}}_1$ are the estimations of ${\omega }_{\mathrm{m}}$, ${\dot{\omega }}_{\mathrm{m}}$, ${\ddot{\omega }}_{\mathrm{m}}$ and ${\dot{\mathsf{\Psi }}}_1$, respectively, and ${\alpha }_{11}$, ${\alpha }_{12}$ and ${\alpha }_{13}$ are the positive constants and $0<{ϵ}_1\ll 1$.

By using the estimated perturbations to compensate the actual perturbations, the control laws of the WT can be obtained as following:

$$\begin{array}{c}{u}_1=b1{\left(0\right)}^{-1}[-{\widehat{\mathsf{\Psi }}}_1-k_{11}({\omega }_{\mathrm{m}}-{\omega }_{\mathrm{m}}^{*})-k_{12}({\dot{\widehat{\omega }}}_{\mathrm{m}}-{\dot{\omega }}_{\mathrm{m}}^{*})+{\ddot{\omega }}_{\mathrm{m}}^{*}]\hfill \end{array}$$

(13)

where $k_{11}$ and $k_{12}$ are positive gains, and the poles of the closed-loop system are in the left-half plane (LHP).

For the PMSG, the system outputs are chosen as $y_{\mathrm{PMSG}}={[y_2,y_3]}^{\mathrm{T}}={[i_{\mathrm{md}},i_{\mathrm{mq}}]}^{\mathrm{T}}$, and $i_{\mathrm{md}}^{*}$ and $i_{\mathrm{mq}}^{*}$ are the reference values of stator currents $i_{\mathrm{md}}$ and $i_{\mathrm{mq}}$, respectively. According to Equation (7), differentiate $y_{\mathrm{PMSG}}$ until the control input appears explicitly, it expresses as

$$\begin{array}{cc}\hfill {\dot{y}}_2& ={\mathsf{\Psi }}_2+b_2\left(0\right)u_2\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\dot{y}}_3& ={\mathsf{\Psi }}_3+b_3\left(0\right)u_3\hfill \end{array}$$

(15)

where $b_2\left(0\right)$ and $b_3\left(0\right)$ are nominal values of $b_2\left(x\right)=\frac{1}{L_{\mathrm{md}}}$, and $b_3\left(x\right)=\frac{1}{L_{\mathrm{mq}}}$, respectively.

According to Equations (6) and (7), the perturbations are expressed as follows

$$\begin{array}{ccc}\hfill {\mathsf{\Psi }}_2& =& \frac{1}{L_{\mathrm{md}}}(-i_{\mathrm{md}}{R}_{\mathrm{s}}+{\omega }_{\mathrm{e}}{L}_{\mathrm{mq}}{i}_{\mathrm{mq}})+(b_2\left(x\right)-b_2\left(0\right))u_2\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill {\mathsf{\Psi }}_3& =& -\frac{R_{\mathrm{s}}}{L_{\mathrm{mq}}}{i}_{\mathrm{mq}}-\frac{1}{L_{\mathrm{mq}}}{\omega }_{\mathrm{e}}(L_{\mathrm{md}}{i}_{\mathrm{md}}+K_e)+(b_3\left(x\right)-b_3\left(0\right))u_3\hfill \end{array}$$

(17)

According Equation (9), two second-order perturbation observers (POs) for ${\mathsf{\Psi }}_2$ and ${\mathsf{\Psi }}_3$ estimation are designed, respectively:

$$\left\{\begin{array}{c}{\dot{\widehat{i}}}_{\mathrm{md}}={\widehat{\mathsf{\Psi }}}_2+\frac{{\alpha }_{21}}{{ϵ}_2}(i_{\mathrm{md}}-{\widehat{i}}_{\mathrm{md}})+b_2\left(0\right)u_2\hfill \\ {\dot{\widehat{\mathsf{\Psi }}}}_2=\frac{{\alpha }_{22}}{{ϵ}_2^2}(i_{\mathrm{md}}-{\widehat{i}}_{\mathrm{md}})\hfill \end{array}\right.$$

(18)

$$\left\{\begin{array}{c}{\dot{\widehat{i}}}_{\mathrm{mq}}={\widehat{\mathsf{\Psi }}}_3+\frac{{\alpha }_{31}}{{ϵ}_2}(i_{\mathrm{mq}}-{\widehat{i}}_{\mathrm{mq}})+b_3\left(0\right)u_3\hfill \\ {\dot{\widehat{\mathsf{\Psi }}}}_3=\frac{{\alpha }_{32}}{{ϵ}_2^2}(i_{\mathrm{mq}}-{\widehat{i}}_{\mathrm{mq}})\hfill \end{array}\right.$$

(19)

where ${\widehat{i}}_{\mathrm{md}}$, ${\dot{\widehat{i}}}_{\mathrm{md}}$, ${\dot{\widehat{\mathsf{\Psi }}}}_2$, ${\widehat{i}}_{\mathrm{mq}}$, ${\dot{\widehat{i}}}_{\mathrm{mq}}$ and ${\dot{\widehat{\mathsf{\Psi }}}}_3$ are the estimations of $i_{\mathrm{md}}$, ${\dot{i}}_{\mathrm{md}}$, ${\dot{\mathsf{\Psi }}}_2$, $i_{\mathrm{mq}}$, ${\dot{i}}_{\mathrm{mq}}$ and ${\dot{\mathsf{\Psi }}}_3$, respectively, and ${\alpha }_{21}$, ${\alpha }_{22}$, ${\alpha }_{31}$ and ${\alpha }_{32}$ are the positive constants.

The adaptive control of the PMSG employing the perturbation estimation is designed as

$$\left\{\begin{array}{c}{u}_2=b_2{\left(0\right)}^{-1}(-{\widehat{\mathsf{\Psi }}}_2-k_{21}({\widehat{i}}_{\mathrm{md}}-i_{\mathrm{md}}^{*})+{\dot{i}}_{\mathrm{md}}^{*})\hfill \\ u_3=b_3{\left(0\right)}^{-1}(-{\widehat{\mathsf{\Psi }}}_3-k_{31}({\widehat{i}}_{\mathrm{mq}}-i_{\mathrm{mq}}^{*})+{\dot{i}}_{\mathrm{mq}}^{*})\hfill \end{array}\right.$$

(20)

where $k_{21}$ and $k_{31}$ are positive feedback control gains, hence put the poles of the closed-loop system in the LHP.

Remark 1.

The adaption laws are developed based on quadratic Lyapunov function. The performance of adaptive control can be enhanced by using non-quadratic Lyapunov functions [44,51].

3.2. Normal Operation

In normal operation, for the MSC, the MPPT algorithm reported in [47] is applied in extracting the maximum wind power under below-rated wind speeds. The mechanical rotation speed ${\omega }_{\mathrm{m}}$ is required to track its optimal reference ${\omega }_{\mathrm{mopt}}$, while the d-axis stator current reference $i_{\mathrm{mdr}}$ is 0. When the wind speeds are above the rated speed, the adaptive pitch control is employed to limit the captured power from wind. The extracted wind power should be maintained at its rated value. It in turn requires both the mechanical torque $T_{\mathrm{m}}$ and mechanical rotation speed ${\omega }_{\mathrm{m}}$ to keep at their rated values, respectively.

For the GSC, the active power $P_{\mathrm{m}}$ generated by the PMSG is first transferred to DC-Link, then from DC-Link to grid. Two VSCs are controlled separately, and the dynamics of the PMSG-WT and power grid are decoupled via the DC-Link when DC-Link voltage $V_{\mathrm{dc}}$ is always maintained at its reference $V_{\mathrm{dc}}^{*}$. To regulate the DC-Link voltage $V_{\mathrm{dc}}$, an adaptive DC-Link voltage $V_{\mathrm{dc}}$ control strategy has been developed in [45]. In addition, the reference of the grid current in the q-axis $i_{\mathrm{gq}}^{*}$ is 0, grid voltage in the d-axis $E_{\mathrm{gd}}$ is rated voltage, and the grid voltage in the q-axis $E_{\mathrm{gq}}$ is 0. The injected grid reactive power $Q_{\mathrm{g}}$ should be around 0 when GSC controller can realize its control objectives.

3.3. Operation Under Grid Voltage Dips

Under the grid voltage dips, the GSC adopts the same control strategy as normal operation. Both the reactive and active current of the GSC required by the fault ride-through of the grid codes have not been applied in GSC controller. The DC-Link voltage $V_{\mathrm{dc}}$ is required to be kept at its rated value. It will not absorb excess active power. For the MSC, the generating active power is still delivering to the grid during the grid voltage $E_{\mathrm{gd}}$ dips. The generated active power $P_{\mathrm{m}}$ is the same as the injected grid active power $P_{\mathrm{g}}$ when MSC controller does not take any control action. The grid current $i_{\mathrm{gd}}$ will increase rapidly when the DC-Link voltage $V_{\mathrm{dc}}$ is well regulated under the grid voltage dips. This may damage VSCs or other power electronic devices. Hence, the active power exported by the WECS should be in proportion to the level of the retained grid voltage. The damage of the GSC caused by the large inrush current during the grid voltage dips can be prevented [25,29]. When the grid voltage dips are detected, the generated active power $P_{\mathrm{m}}$ should reduce simultaneously [39]. For this reason, pitch control is employed to assist in reducing generated active power $P_{\mathrm{m}}$ in this paper. When the pitch control is applied in MSC under grid voltage dips, the mechanical rotation speed reference ${\omega }_{\mathrm{m}}^{*}\left(\mathrm{FRTC}\right)$ is the same as normal operation. However, the electromagnetic torque reference $T_{\mathrm{er}}$ is in proportion to the level of the retained grid voltage shown in (22).

$$T_{\mathrm{er}}=\frac{E_{\mathrm{gd}}\left(t\right)}{E_{\mathrm{gd}}\left(0\right)}{T}_{{\mathrm{er}}_{\mathrm{normal}}}$$

(21)

where $T_{\mathrm{er}}$ is the electromagnetic torque reference applied in pitch control when the grid voltage dips; $E_{\mathrm{gd}}\left(t\right)$ is the value of the grid voltage after the grid voltage dips; $E_{\mathrm{gd}}\left(0\right)$ is the rated grid voltage; and $T_{{\mathrm{er}}_{\mathrm{normal}}}$ is the electromagnetic torque in normal operation.

According to (2), the q-axis stator current reference $i_{\mathrm{mq}}^{*}\left(\mathrm{FRTC}\right)$ applied in pitch control is $\frac{E_{\mathrm{gd}}\left(t\right)}{E_{\mathrm{gd}}\left(0\right)p{K}_e}{T}_{{\mathrm{er}}_{\mathrm{normal}}}$.

The overall control structure of adaptive control-based GSC and MSC for the WECS based on PMSG is shown in Figure 3.

4. Simulation Results

To verify the effectiveness of the proposed adaptive control approach, a series of simulations studies are carried out, and the VC and FLC are employed for comparisons. A 2-MW WECS is investigated. The parameters of the PMSG-WT system are $\rho$ = 1.205 $\mathrm{kg}/{\mathrm{m}}^3$, R = 39 m, rated wind speed $V_{\mathrm{r}}$ = 12 m/s, $J_{\mathrm{tot}}=1\times {10}^4$$\mathrm{kg}·{\mathrm{m}}^2$, $K_e$ = 136.25 $\mathrm{V}.\mathrm{s}/\mathrm{rad}$, p = 11, $L_{\mathrm{md}}=5.5\times {10}^{-3}$ H, $L_{\mathrm{mq}}=3.75\times {10}^{-3}$ H, and $R_{\mathrm{s}}=5\times {10}^{-5}$$\Omega$; the parameters of the grid are $L_{\mathrm{g}}=6.31\times {10}^{-5}$ H, $C=0.06$ F, and $R_{\mathrm{g}}=1.98\times {10}^{-3}$$\Omega$.

The parameters of adaptive controller for the GSC are the same as listed in [45]. For the MSC, the parameters of adaptive controller aiming to extract the maximum wind power under below-rated wind speeds are the same as listed in [47]. The adaptive pitch controller parameters are listed in

Table 1. Parameters of adaptive pitch control.

Parameters of the adaptive controller
Gains of observer (12)	${\alpha }_{11}=50$, ${\alpha }_{12}=1.875\times {10}^3$, ${\alpha }_{13}=1.5625\times {10}^4$, ${ϵ}_1=0.02$
Gains of observer (19)	${\alpha }_{21}=4\times {10}^2$, ${\alpha }_{22}=4\times {10}^4$, ${ϵ}_2=0.01$
Gains of observer (20)	${\alpha }_{31}=4\times {10}^2$, ${\alpha }_{32}=4\times {10}^4$, ${ϵ}_3=0.01$
Gains of linear controller	$k_{11}=4\times {10}^2$, $k_{12}=40$, $k_{21}=1.6\times {10}^2$, $k_{31}=1.6\times {10}^2$

. Note that the FLC uses the same controller parameters as the PO-AC, moreover the FLC uses the full state measurements except $\frac{\mathrm{d}V}{dt}$ and the exact parameters.

4.1. Ramp Wind Speed

The responses of the WT to ramp wind speed is shown in Figure 4. Wind speed is shown in Figure 4a. Under the above-rated wind speeds, among these three controllers, the worst tracking performance of the mechanical rotation speed ${\omega }_{\mathrm{m}}$ with the longest recovery time and biggest overshoots is achieved by the VC. The maximum tracking error ($\frac{{\omega }_{\mathrm{m}}-{\omega }_{\mathrm{mr}}}{{\omega }_{\mathrm{mr}}}\times 100\%$) is more than $15\%$ under the VC. The tracking error of the ${\omega }_{\mathrm{m}}$ still exists under the FLC. The best tracking performance is provided by the PO-AC. Under the below-rated wind speeds, both the FLC and PO-AC provide satisfactory tracking performances of the ${\omega }_{\mathrm{m}}$. However, the VC still has tracking error of the ${\omega }_{\mathrm{m}}$. It can be explained that the VC is designed based on one specific operation point and cannot provide optimal performance in the whole operation region. The FLC exists tracking error under the above-rated wind speeds, because the $\frac{\mathrm{d}V}{\mathrm{d}t}$ is unknown in the FLC design. Under the below-rated wind speeds, the FLC knows the full state measurements and accurate system parameters. Hence, high performance is achieved by the FLC.

Under the above-rated wind speeds, the pitch angle ${\beta }_{\mathrm{r}}$ changes with the varying wind speeds to limit the extracted wind power. Under the below-rated wind speeds, the pitch angle ${\beta }_{\mathrm{r}}$ is constant. The response of the ${\beta }_{\mathrm{r}}$ is shown in Figure 4d. When the wind speed decreases from 14 $\mathrm{m}/\mathrm{s}$ to 12 $\mathrm{m}/\mathrm{s}$, the pitch angle reference ${\beta }_{\mathrm{r}}$ value also decreases. It is because that the captured wind power decreases with the decreasing wind speed when the pitch angle reference ${\beta }_{\mathrm{r}}$ does not change. To keep the captured wind power at rated power, the pitch angle should decrease for increasing the power coefficient $C_{\mathrm{p}}$. Under the above-rated wind speeds, comparing with the FLC or PO-AC, the VC requires more time to reach the expected ${\beta }_{\mathrm{r}}$. Figure 4e–h show the responses of the power coefficient $C_{\mathrm{p}}$, $T_{\mathrm{m}}$ and $T_{\mathrm{e}}$ to ramp wind speed. In Figure 4e, the power coefficient $C_{\mathrm{p}}$ increases with decreasing pitch angle reference ${\beta }_{\mathrm{r}}$. The expected wind power can be achieved by the PO-AC even when wind speed varies, that neither the VC nor FLC can provide, as shown in Figure 4h.

The responses of the PMSG to ramp wind speed are shown in Figure 5. Figure 5a,b show that larger overshoots and longer recovery time of the stator voltages $V_{\mathrm{md},\mathrm{q}}$ are obtained by the VC comparing with the FLC or PO-AC. In addition, the PO-AC obtains the best performance. The stator current $i_{\mathrm{md}}$ can be well tracked by these three controllers shown in Figure 5c. Figure 5d shows the response of $i_{\mathrm{mq}}$. The active and reactive power of the PMSG ($P_{\mathrm{m}}$, $Q_{\mathrm{m}}$) are shown in Figure 5e,f, respectively. It can be seen from Figure 5e that, when the wind speed is above-rated wind speeds, the rated electrical power can be always achieved by the PMSG under the PO-AC. Under below-rated wind speed, the generating power by the PMSG changes with varying wind speed.

Figure 6 shows the responses of the grid to ramp wind speed. The q-axis current $i_{\mathrm{gq}}$ can be almost maintained around 0 by these three controllers shown in Figure 6a. To decouple the control of GSC and MSG, the DC-Link voltage $V_{\mathrm{dc}}$ should be maintained at 1500 V. It can be seen from Figure 6b that, the VC has larger overshoots and longer recovery time of the DC-Link voltage $V_{\mathrm{dc}}$. The injected active power $P_{\mathrm{g}}$ and reactive power $Q_{\mathrm{g}}$ of the grid are shown in Figure 6c,d, respectively. Figure 6c,d show that the decoupled control for reactive and active power of grid can be realised.

During the whole operating period, the PO-AC can always keep consistent responses of $P_{\mathrm{m}}$, $Q_{\mathrm{m}}$, $P_{\mathrm{g}}$ and $Q_{\mathrm{g}}$. The performances of the VC and FLC are affected by the varying wind speed. Note that the FLC uses the full state feedback except $\frac{\mathrm{d}V}{dt}$ under above-rated wind speeds and extract parameters of the system, but the PO-AC is without requiring the details of the system information and extract parameters.

The estimates of the states and perturbations are shown in Figure 7. It shows that both the states and perturbations can be well estimated by the designed observers. The estimated perturbations including nonlinearities, uncertainties and disturbance are used for compensations of the real perturbations.

4.2. Random Wind Speed

The responses of the WT to random wind speed are shown in Figure 8. Wind speed is shown in Figure 8a. When wind speed is time-varying, neither the VC nor FLC can provide satisfactory tracking performances of the mechanical rotation speed ${\omega }_{\mathrm{m}}$ shown in Figure 8b. The PO-AC always keeps ${\omega }_{\mathrm{m}}$ around reference mechanical rotation speed. It can be seen from Figure 8c that, the maximum tracking errors ($\frac{{\omega }_{\mathrm{m}}-{\omega }_{\mathrm{mr}}}{{\omega }_{\mathrm{mr}}}\times 100\%$) are approximately $13\%$ and $5\%$ under the VC and FLC, respectively. The pitch angle reference ${\beta }_{\mathrm{r}}$ and power coefficient $C_{\mathrm{p}}$ should change with the time-varying wind speeds under above-rated wind speeds, and keep constant under below-rated wind speeds. Figure 8d,e show that, the corresponding pitch angle reference ${\beta }_{\mathrm{r}}$ and power coefficient $C_{\mathrm{p}}$ can be achieved under time-vayring wind speeds. Figure 8f–h show the responses of mechanical torque $T_{\mathrm{m}}$, electromagnetic $T_{\mathrm{e}}$, and mechanical power $P_{\mathrm{w}}$, respectively.

Figure 9 shows the responses of the PMSG to random wind speed. During the whole operating period, the PO-AC can always keep consistent responses of $P_{\mathrm{m}}$ and $Q_{\mathrm{m}}$. The performances of the VC and FLC are affected by the time-varying wind speeds. In Figure 10, the rated value of the DC-Link voltage $V_{\mathrm{dc}}$ can be achieved, and the control for reactive and active power of grid is well decoupled. In Figure 11, it shows that the observers can provide satisfactory estimations of the states and perturbations for compensating the real perturbations.

4.3. FRTC Compliance With the Pitch Control

During grid voltage dips, if the extracted wind power does not change, the grid current $i_{\mathrm{gd}}$ increases sharply under well control of the DC-Link voltage $V_{\mathrm{dc}}$. The fast increasing grid current $i_{\mathrm{gd}}$ may damage the devices. If the extracted wind power can be reduced immediately as grid voltage dips, the increasing grid current $i_{\mathrm{gd}}$ can be reduced. To reduce the wind power captured by the WT, the power coefficient is decreased by using pitch control. By using this control strategy, it can assist the WECS in protecting the devices and improving the FRTC of the WECS.

When the grid voltage dips are detected, the pitch control is applied to reduce the captured wind power. When pitch control strategy is applied, the mechanical rotational speed ${\omega }_{\mathrm{m}}$ should keep constant, and the mechanical torque $T_{\mathrm{m}}$ is reduced to the required value. However, the mechanical rotational speed ${\omega }_{\mathrm{m}}$ increases when pitch control strategy is applied to reduce the captured wind power, as shown Figure 12c. It is because that the pitch angle rate is limited in the range $\pm 20$ degree/s under the emergency condition [52,53]. From Figure 12d, it can be seen that the pitch angle reference ${\beta }_{\mathrm{r}}$ rate is much higher than $\pm 20$ degree/s under the FLC and PO-AC. Hence, the real pitch angle $\beta$ applied to the WT cannot be the same as the ${\beta }_{\mathrm{r}}$ due to pitch angle rate limit. It in turn results in the increment of the ${\omega }_{\mathrm{m}}$. The excess wind power is stored in the rotor inertia. Figure 12e shows the power coefficient $C_{\mathrm{p}}$ can be reduced by varying pitch angle. The decrement of the $C_{\mathrm{p}}$ is much larger under the FLC and PO-AC than the VC. It can be explained that the VC has much slower response of the $\beta$ compared to the FLC or PO-AC. If the $\beta$ cannot decrease quickly, the $C_{\mathrm{p}}$ cannot decrease immediately. Both the $T_{\mathrm{m}}$ and $T_{\mathrm{e}}$ have fast decrements under the FLC and PO-AC, as shown in Figure 12f,g. However, the VC has smaller reduction and longer recovery time. Figure 12h shows both the PO-AC and FLC have larger reduction of captured wind power $P_{\mathrm{w}}$ than the VC under grid voltage dips.

The responses of the PMSG to constant wind speed with protection under grid voltage dips are shown in Figure 13. When the grid voltage dips, the electromagnetic torque reference $T_{\mathrm{er}}$ is reduced proportionally to the grid voltage dips to keep balance between the generating power $P_{\mathrm{m}}$ and injected grid power $P_{\mathrm{g}}$. It in turn limits the increasing grid current $i_{\mathrm{gd}}$ for protecting the devices. If the stator current $i_{\mathrm{md}}$ is well controlled, the $T_{\mathrm{e}}$ is proportional to the $i_{\mathrm{mq}}$. However, it can be seen from Figure 13d that, the $i_{\mathrm{mq}}$ cannot track its reference under the VC. Both the FLC and PO-AC provide satisfactory tracking performance of $i_{\mathrm{mq}}$. Figure 13e shows the active generating power $P_{\mathrm{m}}$ has bigger reduction with shorter recovery time under the FLC and PO-AC than that under the VC.

Figure 14shows the responses of the grid under grid voltage dips. It can be seen from Figure 14that, the grid current $i_{\mathrm{gd}}$ achieve smaller current under the FLC and PO-AC than that achieved under the VC when grid voltage dips occur. When pitch control strategy is applied under grid voltage dips, the $i_{\mathrm{gd}}$ can be reduced to approximately 1300A under the FLC and PO-AC, as shown in Figure 14. However, the VC does not limit the $i_{\mathrm{gd}}$ significantly as FLC or PO-AC. Compared with the VC, the DC-Link voltage $V_{\mathrm{dc}}$ has smaller overshoots and shorter recovery time under the FLC and PO-AC. When pitch control strategy is applied, the injected active power of grid $P_{\mathrm{g}}$ has bigger reduction under the FLC and PO-AC compared to the VC. The FRTC of the WECS can be enhanced by using the pitch control strategy based on the FLC or PO-AC. In Figure 15, it shows that the estimations of the states and perturbations can be well estimated to compensate the real perturbations.

In addition, the control performances of these three controllers via integral of absolute error (IAE) in different simulation scenarios are shown in

Table 2. IAE indices of different controllers in different scenarios.

Simulation Scenarios	Variables	Controllers
		VC	FLC	PO-AC
Ramp wind speed	$IA{E}_{{\omega }_{\mathrm{m}}}\left(\mathrm{rad}\right)$	0.5116	0.09338	$0.01746$
	$IA{E}_{i_{\mathrm{mq}}}\left(\mathrm{A}.\mathrm{s}\right)$	$10.46$	$13.25$	$11.15$
	$IA{E}_{V_{\mathrm{dc}}}\left(\mathrm{V}.\mathrm{s}\right)$	$0.3654$	$7.616\times {10}^{-3}$	$2.521\times {10}^{-3}$
Random wind speed	$IA{E}_{{\omega }_{\mathrm{m}}}\left(\mathrm{rad}\right)$	0.814	0.2949	$0.06717$
	$IA{E}_{i_{\mathrm{mq}}}\left(\mathrm{A}.\mathrm{s}\right)$	$41.84$	$10.84$	$9.173$
	$IA{E}_{V_{\mathrm{dc}}}\left(\mathrm{V}.\mathrm{s}\right)$	$1.054$	$0.02976$	$0.01186$
FRTC compliance with the pitch control	$IA{E}_{{\omega }_{\mathrm{m}}}\left(\mathrm{rad}\right)$	0.04794	0.02485	$0.02611$
	$IA{E}_{i_{\mathrm{mq}}}\left(\mathrm{A}.\mathrm{s}\right)$	$30.19$	$0.2657$	$0.393$
	$IA{E}_{V_{\mathrm{dc}}}\left(\mathrm{V}.\mathrm{s}\right)$	0.1299	0.01892	0.01419

. Here, $IA{E}_x={\int }_0^T|x-x^{*}|$. The reference value of the variable x is $x^{*}$. The simulation time T is set as 15s, 10s and 1s in these three simulation scenarios, respectively. It can be seen from Table 2 that, in first and second simulation scenarios, the $IA{E}_{{\omega }_{\mathrm{m}}}$ and $IA{E}_{V_{\mathrm{dc}}}$ achieves the smallest values under the PO-AC. In the FRTC compliance with the pitch control simlation scenario, both the PO-AC and FLC provide much smaller $IA{E}_{{\omega }_{\mathrm{m}}}$, $IA{E}_{i_{\mathrm{gq}}}$ and $IA{E}_{V_{\mathrm{dc}}}$ than those achieved by the VC.

5. Conclusions

This paper has proposed a PO-AC for the GSC and MSC of the WECS based on PMSG. The objectives of different operation conditions can be achieved by using the PO-AC, e.g., extract the maximum wind power under the below-rated wind speeds, limit the extracted wind power under the above-rated wind speeds, and regulate the DC-Link voltage and decouple the reactive and active power injected into the grid. It provides better performances of the WECS than the VC designed based on optimal operating point and FLC requiring full state measurements and accurate system model. In addition, the pitch control strategy has been employed to assist in reducing the extracted wind power for enhancing the FRTC of the WECS. In the further work, the adaptive control strategy based on grid code requirement can be applied for improving the FRTC of WECSs.