Multi-mode soft switching control for variable pitch of wind turbines based on T-S fuzzy weighted

2.1. The combined wind model

The wind has a strong randomness and mutability in the flow process, a model is required that can properly simulate the spatial effect of wind behavior, including gusting, rapid (ramp) changes, and background noise [17, 18].

The wind model chosen for this simulation is a four-component model and is defined by:

where, $v_b$ is the base wind velocity, m/s; $v_g$ is the gust wind component, m/s; $v_r$ is the ramp wind component, m/s; $v_n$ is the noise wind component, m/s; $v_w$ is the combined wind m/s.

These four components provide a reasonable flexibility for the study of one or a group of wind turbines.

The base wind velocity component can be described as:

where, $l$ and $s$ are the proportion and shape parameters of Weibull distribution; $\mathrm{\Gamma }(\cdot )$ is Gamma function.

The gust wind velocity component is given by:

where, $t_{1g}$ is the gust starting time; $t_g$ is the gust period time; $v_{\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}$ is the gust peak; $t$ is time.

The ramp wind velocity component is stated as:

where, $t_{1r}$ is the ramp start time; $t_{2r}$ is the ramp stop time; $t_r$ is the ramp period time; $v_{\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{x}}$ is the ramp peak.

The final wind velocity component is the random noise component, it can defined by:

where, ${\omega }_i=(i-1/2)\mathrm{\Delta }{\omega }_0\text{;}$$S_r\left({\omega }_i\right)=2\mu F^2\left|{\omega }_i\right|/{\pi }^2[1+(F{\omega }_i/\pi v_b{)}^2{]}^{4/3}\text{;}$${\phi }_i$ is a random variable with uniform probability density on the interval 0 to $2\pi$; $\mu$ is the mean speed of wind at reference height; $F$ is the turbulence scale; $\mathrm{\Delta }{\omega }_0$ is an accuracy frequency.

The wind model was installed in a program by adding a special function module to provide $v_w$ given the parameters of the four wind components.

2.2. PMSG Wind turbines model

The structure of direct-drive PMSG (permanent magnet synchronous generator) wind turbines is shown in Fig. 1. The wind turbines studied in this paper is a class of 2 MW horizontal axis and variable speed variable pitch wind turbines. The major components of this wind turbines model are the aerodynamic system, PMSG, generator-side converter, grid-side converter, generator-side controller, grid-side controller, pitch controller, yaw controller and wind turbine control system. Maximum wind tracking is carried out under the rated wind speed and the rotor speed of generator is controlled by the electromagnetic torque of the machine controller. The goal of pitch control above rated wind speed is to control pitch angle and regulate rotor speed to the rated power. To account for variations in wind, the pitch control response must be fast.

Fig. 1The structure of direct-drive PMSG wind turbines

2.3. Aerodynamic model

According to Batz’s law, all power cannot be absolutely captured by wind turbines while converting wind energy into mechanical energy. Yang et al. established the aerodynamic power absorbed by wind turbines [19], which is shown in:

where, $P_m$ is the aerodynamic power; $\rho$ is the air density; $R$ is the pitch radius; $v$ is the wind speed; $C_P$ is the power coefficient, which depend on pitch angle $\beta$ and tip speed ratio $\lambda$. The tip speed ratio is described as follows:

where, ${\omega }_f$ is the rotor speed.

$C_P$ is expressed as:

where, ${\lambda }_1=\frac{1}{\lambda +0.08\beta }-\frac{0.0035}{{\beta }^3+1}$.

The aerodynamic torque is also given by:

2.4. Pitch actuator model

The pitch angle control is usually applied to limit the output power of wind turbine in the medium or large wind turbines. The hydraulic device and motor drive are two types of actuators which are used to turn the blades of the wind turbine around their longitudinal axis. It can be simplified by one order inertial link. In the closed loop, an integrator, a pitch angle limiter, and a pitch angle rate limiter are included.

The dynamic behavior of the pitch servo is expressed as:

where, ${\tau }_b$ is a constant time, $\beta$ is the pitch angle, ${\beta }_{ref}$ is reference pitch angle.

The response of pitch control depends on the time constant of the pitch actuator, which is normally in a small range. Typically, the pitch angle ranges from 0 to 25 degrees and varies at the maximum rate of ±10°/s. The transient performance of the pitch control depends on the pitch rate. Hence, the variable range and the rate of change in the pitch angle have enormous influence on the performance of power output, these limits are not reached during the normal operation of wind turbine in order to decrease the risk of the fatigue damage.

3.1. The parameter design of fuzzy control

The fuzzy controller adopted two-dimensional fuzzy controller. It is assumed that the error is $E$, which is the difference of generator's speed and its rated speed, the change rate of error is $EC$. As the rotor speed of generator is higher than its rated rotor speed above rated speed, the error is positive, unilateral fuzzy control is adopted. Also, the rated speed of generator is 27.8 r/min = 2.91 rad/s, the basic domain of $E$ is [0, 8], the basic domain of $EC$ is [–10,10], assuming the fuzzy domain of $E$ is [0, 4], $EC$ is [–6, 6]. The pitch angle is limited to$$0°-25°, ignoring the negative, so assuming the basic domain of $U_{fuzzy}$ is [0, 30], $U_{fuzzy}$ is the output of fuzzy control, its fuzzy domain is [0, 5], consequently, the quantification factor of $E$ is $k_e=$ 4/8$=$ 1/2, the quantification factor of $EC$ is $k_{ec}=$ 6/10 $=$ 3/5, the quantification factor of $U_{fuzzy}$ is $k_u=$30/5 $=$ 6. The memberships of $E$, $EC$ and $U$ are all set to triangular form. Assuming the fuzzy subsets of $E$are {PB, PM, PS}, $EC$ are {NB, NM, NS, ZE, PS, PM, PB}, $U_{fuzzy}$ are {PB, PM, PS, ZE}.

The fuzzy control rules of $U_{fuzzy}$ are shown in Table 1.

3.2. The parameter design of RBFNN PID control

The radial basis function neural network is a three layers feed-forward neural network and has strong parameter online adjustment ability, which the input to the output are nonlinear and the hidden layer to the output layer is linear. The structure of RBF neural network is shown in Fig. 3.

In the structure of RBF neural network, $\mathbf{X}={\left[x_1,x_2\cdots ,x_n\right]}^T$ is the input vector, its radial basis vector is: $\mathbf{H}={\left[h_1,h_2,\cdots h_j,\cdots h_m\right]}^T$, $h_j$ is the Gaussian radial base function:

Fig. 3The structure of RBF neural network

The center vector of the $j$th node is: ${\mathbf{C}}_j=\left[c_{j1},c_{j2},\cdots ,c_{ji},\cdots ,c_{jn}\right]$, where, $i=\mathrm{1,2},\cdots ,n$.

The base width vector of the network is: $\mathbf{B}=\left[b_1,b_2,\cdots ,b_m\right]$, $b_j$ are the base width parameters of $j$ node, and it is positive. The weight vector of the network is: $\mathbf{W}=\left[w_1,w_2,\cdots ,w_j,\cdots w_m\right].$

The output of identification network is:

The performance indicator function of the identifier is:

According to the gradient descent, iterative algorithm of output weight, node center, the base width parameter is:

where, $\eta$ is the learning rate; $\alpha$ is the factor of momentum.

The arithmetic of Jacobian matrix is:

where, $x_1=\mathrm{\Delta }u\left(k\right)$.

Using the incremental PID, the control error is:

The three inputs of PID are:

The control arithmetic is:

The tuning index of neural network is:

The tuning of $k_p$, $k_i$ and $k_d$ employed the gradient descent algorithm:

where, $\partial y/\partial \mathrm{\Delta }u$ is the Jacobian information of the controlled object, which can be acquired from the identification network.

The simulation model of RBFNN PID is described in Fig. 4, RBFNN PID algorithm is fulfilled by programming S-Function of MATLAB using Eqs. (11-28). Setting the learning rate is $\eta =$ 0.25, the factor of momentum is $\alpha =$ 0.05, the number of nodes is $n=$ 6, the initial value of PID is [400, 5, 60], the initial weightings is $\mathbf{W}=$ [30, 40, 20].

Fig. 4The simulation model of RBFNN PID

3.3. The parameter design of PI control

PI control is a special case of PID control, the two parameters are $k_p=$ 40, $k_i=$ 20, which can bring about control system with the satisfactory steady state performance.

4.1. Design principle

The output membership of T-S fuzzy inference can be linear or constant, which is suitable for expressing the dynamic characteristics of complex systems [20, 21]. The goal of T-S fuzzy weighted soft switch is to smooth the outputs of three controllers, the membership of T-S fuzzy inference can be constant, which resulting the output of fuzzy inference is constant, three coefficients of three modes are obtained from T-S fuzzy inference. According to the state informations of control system, where there is need to switch among three modes in terms of the error of rotor speed and its rate change, T-S fuzzy inference method is applied, of which defuzzification employed the method of weighted average to attain the coefficients of three modes in the switching process, the final controlled quantity adopted the weighted sum method to weight the coefficients to the three modes to smooth the outputs of three controllers, consequently , the T-S fuzzy weighted soft switch can be achieved.

This switch method differs from traditional method who only switches in a certain threshold, introducing the deviation and its derivative, and a few moments before reach the threshold, the deviation is less than the setting threshold, if the change rate of the deviation is bigger, this method of the paper just to switch. In the same way, after the arrival of the deviation, the deviation is greater than the setting threshold, if the change rate of the deviation is smaller, the method used in this paper will not switch the mode. The specific switching time is realized by T-S fuzzy inference, which also makes full use of the advantage the intelligence of fuzzy control.

Assuming $M{D}_1$ is the fuzzy control mode; $M{D}_2$is the RBFNN PID control mode; $M{D}_3$ is the PI control mode; $w_a$, $w_b$ and $w_c$ are the outputs of T-S fuzzy inference, that are also the coefficients of fuzzy control mode, RBFNN PID control mode and PI control mode; $U_{fuzzy}$ is the output controlled quantity of fuzzy control; $U_{RBF}$ is the output controlled quantity of RBFNN PID control; $U_{PI}$ is the output controlled quantity of PI control; $U_{sum}$ is the synthetic output of T-S fuzzy weighted multi-mode soft switch.

The fuzzy control plays a major role in large error while RBFNN PID control and PI control take tiny effect, $w_a$ is the predominant coefficient; switching to the RBFNN PID in medium error, fuzzy control and PI control act feebly at this moment, $w_c$is the leading coefficient; in small error, PI control takes prime part while fuzzy control and RBFNN PID control attribute diminutively. Put $w_a$, $w_b$ and $w_c$ weighted to $U_{fuzzy}$, $U_{RBF}$ and $U_{PI}$, the T-S fuzzy weighted multi-mode soft switch will be come true, which include fuzzification, fuzzy inference, defuzzification and synthetic output four procedures.

4.2. Fuzzification

The error $E$ and its change rate $EC$ decided the modal division of T-S fuzzy weighted switch algorithm, which are used to confirm the coefficients of three modes of T-S inference. The domain and fuzzy domain of $E$ and $EC$ are set the same as fuzzy control. The T-S fuzzy inference system was built by using the fuzzy logical toolbox in MATLAB, as is shown in Fig. 5. Assuming the fuzzy subsets of $E$ are {PS, PW, PM, PB} and $EC$ are {NB, NS, ZE, PS, PB}. The memberships of $E$ and $EC$ are described in Fig. 6 and Fig. 7, the memberships of $w_a$, $w_b$ and $w_c$are the same which are two constants zero and one.

Fig. 5The T-S fuzzy inference system

Fig. 6The membership of E

Fig. 7The membership of EC

4.3. Fuzzy inference

The weights number distributions must be considered before fuzzy inference, the weights number distributions of $w_a$, $w_b$ and $w_c$ are described in Tables 3-5.

Assuming $M{D}_1$ is mode of fuzzy control, $M{D}_2$ is the mode of RBFNN PID control, $M{D}_3$ is the mode of PI control for the sake of accurate description the logic of fuzzy inference. The fuzzy subsets of MD are $\left\{M{D}_1,M{D}_2,M{D}_3\right\}$, the description of modal fuzzy rules is shown in Table 6.

Through the ruler viewer in fuzzy logical toolbox, obviously, $w_a$, $w_b$ and $w_c$ are satisfied to Eq. (30), as is shown in Fig. 8.

Fig. 8The ruler viewer

4.4. Defuzzification

The T-S fuzzy inference system has two defuzzification methods, that is the weighted average and the weighted sum, the weighted average method is adopted in the process of defuzzification. Assuming the output of rule $i$ is $u_i$, its weight is $w_i$, $m$ is the number of fuzzy ruler, so the total output is:

The coefficients of three modes $w_a$, $w_b$ and $w_c$ are calculated by the above equation. The output memberships of three modes of T-S fuzzy inference were set to two constants, which values are zero and one. After fuzzification and defuzzification, the coefficients of three modes, ($w_a$, $w_b$ and $w_c$) are all the accurate values between zero and one. At any given moment, one of the numbers $w_a$, $w_b$ and $w_c$ is larger, the other are relatively smaller under the given constraint condition:

4.5. Synthetic output

In the control process, put $w_a$, $w_b$ and $w_c$ weighted to $U_{fuzzy}$, $U_{RBF}$ and $U_{PI}$, the soft switch can be fulfilled. The synthetic output of controlled quantity employed the weighted sum method: