#### Abstract

This paper analyzed and discussed the nonlinear PI/PD control for wind energy conversion system (WECS) pitch controller design. The nonlinear PI/PD controller consists of a classical PI/PD controller and a nonlinear gain table. In this paper, the detailed design procedures of the controller including not only the selection of control parameters but also the formulation of nonlinear gain table were given. In order to verify the effectiveness and correctness of the algorithm, a detailed 2 MW simulation test platform based on MATLAB/SIMULINK environment was established. Meanwhile, some valuable conclusions were also obtained. The presented contents of this article have the reference value and guiding significance for practical engineering application.

#### 1. Introduction

With the global climate problem increasingly prominent and nonrenewable energy sources being largely consumed, the exploration and utilization of renewable energy sources such as wind energy and solar energy have been paid to a high attention. As the fastest growing renewable energy sources in the recent years [1–4], wind generation is the most prevalent in coastal regions spanning temperate and boreal climates. Countries such as China, USA, Denmark, and Canada possess significant wind power potential due to their high average wind velocities [5–8]. Hence, the research and development about wind power have much practical value nowadays. Compared with the constant speed-constant frequency power system, the greatest advantage of variable speed-constant frequency wind power system widely used is the ability to get access to the maximum energy conversion. Mainstream variable speed-constant frequency power generation models include doubly-fed induction generator (DFIG) and permanent magnet synchronous generator (PMSG). Compared with DFIG, PMSG has many superior characteristics such as more efficient performance, higher reliability, and wider speed control range and is gradually becoming the first choice [9]. Therefore, the WECS based on PMSG was selected for research in this paper.

As Figure 1 shows, according to the wind speed, the working region of WECS typically can be divided into two regions, namely, partial-load region that has wind speed below the rated wind speed and full-load region that has wind speed above the rated wind speed. In the partial-load region, the control goal of the wind power system is generally to capture the maximum wind energy in order to achieve the maximum economic benefit. At this time, the partial-load region can also be called the maximum power point tracking (MPPT) region. In the full-load region, the variable propeller is particularly important. At this time, the generator-output power is limited at the rated value by the pitch control since the capacity of the generator and converter is limited [10, 11]. Furthermore, pitch control is also needed in some special working conditions such as the limited power control under rated wind speed, the low-voltage crossing in power grid, power grid needs to be injected into the inertia, and lighten the wind turbine loads. Meanwhile, it is pointed out that pitch control can also smoothen the power [12]. This paper focuses on the pitch control above the rated wind speed.

Figure 2 illustrates the pitch-controlled system. Obviously, Figure 2(a) depicts the blades action under pitch-controlled, and Figure 2(b) shows the variable-pitch drive system in wind turbine hub. The variable-pitch drive system consists of pitch drive cabinet, variable-pitch drive, emergency power supply, pitch bearing and variable-pitch lubrication systems. The pitch control system and communication system are in the pitch drive cabinet. In general, the pitch controller and master controller are connected by the optical fiber communication. The PROFIBUS or CANopen protocols are generally used in this communication. Usually, the master controller sends the pitch instructions to the pitch controller and gets some important information from it at the same time. When the instructions are obtained, the pitch controller starts to control the variable-pitch drive-operating system.

**(a)**

**(b)**

At present, the research on variable-pitch control mainly includes two main categories: linear control and nonlinear control. The research of the linear controller mainly has PI/PD control, *H*∞ controller with a linear matrix inequality approach [13], linear quadratic Gaussian LQR control [14], and generalized predictive control (GPC) [15]. The nonlinear control mainly contains LPV control [16] and some intelligent control methods such as fuzzy control [17]. Because wind turbine is a multi-input multi-output system containing strongly nonlinear dynamics, the effect of nonlinear control is better than the linear control. Taking into account the complexity of the algorithm, the most applied controllers in engineering is still PI/PD control. Therefore, the design of the variable propeller with the nonlinear PI/PD controller considering the nonlinear characteristics of the wind turbine has become the key. So, this article takes this as the focus of the study and summarizes some valuable conclusions. And this study can also provide the necessary reference and guidance for the follow-up engineering practice.

The structure of this paper is as follows: the dynamic model and structure of WECS is briefly introduced in Section 2. Section 3 explains the controller design in details, including the controller parameter tuning and the calculation of nonlinear gain. In Section 4, the validation of the nonlinear PI/PD controller is surveyed using the MATLAB/Simulink simulator. Finally, Section 5 provides some useful conclusions.

#### 2. Mathematical Model of WECS

WECS is mainly constitutive of wind turbine, transmission chain, permanent magnet synchronous generator, generator-side converter, DC bus, and the grid-side inverter. Its structure is shown in Figure 3.

##### 2.1. Wind Turbine Model

Based on the Baez theory, the aerodynamic power, and torque of the wind turbine are [18–22]where is the air density, is the radius of wind wheel, is the wind speed, is the pitch angle, is the speed of wind turbine, is the tip speed ratio (TSR), and is defined as the wind energy conversion coefficient. A 2 MW baseline wind turbine-power coefficient is shown in Figure 4. Obviously, is the function of and .

The tip-speed ratio is given by

##### 2.2. Dynamic Model of PMSG

If PMSG was supposed to be an ideal motor, it should meet the following criteria:(1)three-phase stators are symmetry.(2)induced electromotive force is sinusoidal.(3)there is no damping winding on the rotor(4)magnetic saturation of iron core, vortex, and hysteresis loss can be neglected.(5)electronic conductivity of permanent magnet material is zero.

Under this assumption, the mathematical model of PMSG [23, 24] is

Electromagnetic torque is given bywhere and are the *d*-axis and *q*-axis stator reluctance, is the stator resistance, is the permanent flux, is pole pairs, is the speed of PMSG, and are *d*-axis and *q*-axis currents, respectively, and and are *d*-axis and *q*-axis voltages, respectively.

Because nonsalient PMSG meets , the mathematical model of PMSG has coupling terms such as , according to Equation (3).

GivenThe mathematical model of PMSG could be rewritten as

The above formula also can be rewritten into a transfer function .

Figure 5 shows the inner control of current. The closed loop transfer function of the current or torque can be obtained by Figure 5. And the closed loop transfer function also could be taken as a first-order inertial system.where is the inertial time constant and is the command signal of .

**(a)**

**(b)**

Equation (8) shows that the dynamic model of PMSG could be equivalent to a first-order inertial system by current feedforward decoupling control and first-order tuning for PI parameters. Usually, when , we can leave out the inertial time delay of PMSG.

##### 2.3. Dynamics of Blade Pitch Actuator

There are two kinds of drives in the blade pitch actuator at present. They are the motor drive and hydraulic drive. Compared with the hydraulic drive, motor drive has the characteristics of lower cost and higher reliability and has been widely used. Actual motor drive is shown in Figure 6.

The collecting variable-pitch strategy was used. According to Figure 7, the blade dynamics is expressed as

By leaving out the inertia time delay of the motor drive, the drive torque of motor is assumed as

The blade pitch actuator dynamic can be given by Figure 8 where is the command of and the PD parameters of the blade pitch actuator are and .

If the delay coefficient meetsthe blade pitch actuator dynamic model can be simplified as

Obviously, the inertia delay coefficient, in Equations (12) and (13) can be reduced by increasing PD parameters. When and take the appropriate value, the inertia delay can also be left out. Actually, because the value of the parameters , , and is unknown, the value of the control parameters and can only depend on experience.

##### 2.4. Dynamic Model of Transmission Chain

Figure 9 shows the generator cabin internal components of high-speed permanent magnet wind turbine. It is clear that its transmission chain mainly consists of spindle (or low-speed shaft), gearbox, and high-speed shaft. The dynamics of the transmission chain is characterized by Figure 10.

According to Figure 10, the dynamics of the transmission chain is derived as

The gearbox ratio is defined as

From Equations (14) and (15), the dynamic model of the transmission chain can be simplified aswhere

#### 3. Design of Controller

When the wind speed is above the rated wind , the system needs pitch control to unload the power and enable the speed to approach the rated speed . If the inertia delay and are neglected by adjusting control parameters, the electromagnetic torque and pitch angle can be approximated as their command values and .where is the Laplace factor.

By linearizing Equations (1) and (18), the small signal values arewhereSuperscripts ∼ and — refer to the small signal value and mean value, respectively. is high-order infinitesimal.

Meanwhile, Equation (16) can also be rewritten as

By eliminating the mean value of the upper equation and Equations (19) and (22) can be obtained.

Due to , can increase the inertia of the system. In general, and are not used at the same time because they have the opposite effect on the system, namely, either PI control or PD control can be used in pitch control.

##### 3.1. Design of Nonlinear PI Controller

When , Equation (22) could be rewritten as

By comparing with the typical second-order system, we get

Equation (22) can be obtained.

Figure 11 shows the torque characteristics of WECS. Obviously, there are two equilibrium points A and B in the system. For equilibrium point B, if there is a perturbation increasing the rotor speed, then PMSG torque will be over wind turbine torque. After the perturbation is eliminated, rotor speed can decrease. By the same method, we can also confirm the system will be back to point B by the assumption that a little perturbation reduced the rotor speed. Hence, the working point B is stable. It could be confirmed that the equilibrium point A is unstable by this perturbation observation method. Therefore, only the point B is a stable equilibrium point, and the range of stability is . The slope of wind turbine torque characteristic, is and the slope of PMSG torque characteristic, is . When the system works at the point B, we can get Equation (26) by Figure 11.

Generally, the damping is unknown. From Equation (26), the damping ratio in Equation (25) meets

Given andthe damping ratio and PI parameters will meet and . Therefore, the parameters and are given by

By the assumption of , Equation (29) could be rewritten as

##### 3.2. Design of Nonlinear PD Controller

Inertia has the ability to prevent frequency mutation in the system. Low frequency oscillation and rotor oscillation of the system can be suppressed by increasing inertia. Therefore, PD control-providing the virtual inertia for the system, is often applied to the small inertia systems and situations that need to be injected into inertia.

Similarly, when , Equation (22) also could be rewritten as

If in Equation (31) is neglected, the transfer function is given by

Givenand , the parameter iswhere is the bandwidth of .

In general, we keep and (or and ) fixed and make as a nonlinear gain table. Meanwhile, the limit of the variable pitch rate and angle is taken into account, and pitch control based on nonlinear PI or PD controller is shown in Figure 12.

##### 3.3. Nonlinear Gain Table

Obviously, all of the wind speed , rotor speed of wind turbine , and pitch angle can have effect on the nonlinear gain by Equation (20). But, the sensitivity of the system to the pitch angle is far greater than other factors generally. When the system works at the rated working point and , the relationship between and is

In general, the nonlinear gain only depends on the pitch angle . The calculation process of nonlinear gain is shown in Figure 13, and the parameters of WECS are shown in Table 1.

In practice, the function can only be obtained by data fitting, and these data for fitting are generated by Bladed Software. In order to verify the correctness of the algorithm shown in Figure 8, an empirical formula of is used.

The partial derivative of the upper formula is

The truth value of could be calculated by Equations (20) and (37). Table 2 shows the calculated and truth values of nonlinear gain at different wind speed. From Table 2, the calculated value of is very close to the truth value of . Therefore, the algorithm shown in Figure 13 is valid. The minimum of is 2.21° above the rated wind, and the relationship between and is shown in Figure 14,. Nonlinear gain is

#### 4. Validation of Algorithm by Simulation

In this section, simulations are carried out with MATLAB/Simulink environment. And the block diagram of WECS is shown in Figure 15. In order to verify the correctness and effectiveness of nonlinear PI and PD pitch control, a detailed 2 MW simulation test platform of WECS was constructed. The simulation parameters of the system are shown in Table 1, and the function is the empirical formula.

The step-change wind shown in Figure 16(a) is used to simulate the actual wind condition. And the response of PI control is shown in Figure 17. Given , the PI parameters *K*_{P} and *K*_{I} were calculated by Equation (30). Meanwhile, simulation results of PD control under step-change wind shown in Figure 16(b) are illustrated in Figure 18. Given and , the PD parameters *K*_{P} and *K*_{D} were calculated by Equation (34).

**(a)**

**(b)**

**(a)**

**(b)**

**(c)**

**(d)**

**(a)**

**(b)**

**(c)**

**(d)**

Figures 17(a) and 18(a) show the electromagnetic power curves. It is clear that the output power of WECS is maintained at 2 MW, and the power fluctuation with nonlinear gain control is obviously less than the one with fixed parameters control. The generator rotor speed curves are shown in Figures 17(b) and 18(b). Obviously, the speed steady-state errors of PI control is less than the speed steady-state errors of PD control. Figures 17(c) and 18(c) describe the pitch angle curve. Compared with the PI control, the actual value of the pitch angle under PD control failed to achieve the value in Table 2. However, the pitch control is generally used as a coarse control. Therefore, the small pitch angle errors and rotor speed errors of PD control are acceptable. By Figures 17(d) and 18(d), the stator three-phase currents could be observed. The simulation results also indicate that the power of WECS is more sensitive to pitch control than the rotor speed, when the wind speed changes.

In Figure 19(a), the wind speed decays at step change. Under this wind speed, the responses of WECS are shown in Figures 19(b)–19(f). Meanwhile, simulation results under gradual change wind speed shown in Figure 20(a) are given by Figures 20(b)–20(f). Different from other responses curves such as power, torque, and rotor speed, the changes in the pitch angle are not smooth. This is because the rate of pitch angle change is limited at ±7°/s.

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

By thesis comparisons among classical PI control algorithm, classical PD control algorithm, nonlinear PI control algorithm, and nonlinear PD control algorithm, the similar conclusions could be summed up. Compared with the PD control, PI control has high tracking accuracy and the pitch angle has changed into the right place in Table 2. Under the nonlinear PI/PD control, the output power fluctuate is small. And the output power is very sensitive to the change in pitch angle under different wind conditions. Furthermore, the merits of nonlinear PI/PD control algorithms have been clearly revealed in Figures 19 and 20.

#### 5. Conclusion

In this research, the design procedure of a pitch controller with nonlinear PI/PD control is given in detail, including the selection of PI/PD parameters and the calculation of nonlinear gain. After that, a simulation test platform for a two-megawatt WECS was built. Finally, the effectiveness and correctness of the algorithm was verified by this test platform, and some useful conclusions were summed up:(i)The output power of WECS is the most sensitive to the change in the pitch angle.(ii)Compared with the fixed parameters control, the nonlinear control with nonlinear gain can make the power fluctuate smaller.(iii)The precision of speed tracking is high, and the pitch angle can change into the right place under the PI control, compared with the PD control.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The research was supported by Key R&D projects in Shaanxi (Grant no. 2017GY-061).