Robust Control of Wind Turbines by Using Singular Perturbation Method and Linear Parameter Varying Model

The maximum power point tracking problem of variable-speed wind turbine systems is studied in this paper. The wind conversion systems contain both mechanical part and electromagnetic part, which means the systems have time scale property. The wind turbine systems are modeled using singular perturbation methodology. A linear parameter varying (LPV) model is developed to approximate the nonlinear singularly perturbed model. Then stability and robust properties of the open-loop linear singularly perturbed system are analyzed using linear matrix inequalities (LMIs). An algorithm of designing a stabilizing state-feedback controller is proposed which can guarantee the robust property of the closed-loop system. Two numerical examples are provided to demonstrate the effectiveness of the control scheme proposed.


Introduction
Wind energy has become one of the fastest-growing energies during the last two decades [1][2][3].Because the wind turbines are large, flexible structures operating in noisy environment can offer abundant energy to mankind, and control problems are necessary to be studied to improve the reliability and conversion efficiency of the wind energy conversion systems (WECS).
Generally, wind turbines can be classified into four categories by [4], namely, fixed-speed fixed-pitch turbines, fixedspeed variable-pitch turbines, variable-speed fixed-pitch turbines, and variable-speed variable-pitch turbines.Compared to fixed-speed wind turbines, variable-speed ones can capture more energy and achieve better dynamic loads alleviation and fewer grid connection power peaks [4,5].
Even though a wind energy conversion system can operate over a wide range of wind speeds, basically it is only active in two regions including partial load and full load [2].In the partial load region, the control objective is to track the desired rotor speed corresponding to varying wind speed () and maintain optimal tip-speed ratio [2,6].In the full load region, the generation goal is to limit the generated power to avoid overloading [4,7].Many researchers focus their interest on the control problems in the partial load region.In [8], an adaptive control scheme was developed for partial load region control of a variable-speed wind turbine, and the stability properties of the adaptive controller and the rotor speed were analyzed.Adaptive control approaches for maximizing power capture were also studied in [9,10].In [11], a two-mass model and a wind speed estimator were used to propose a nonlinear controller, which was aimed to optimize the wind power capture.A high-order sliding mode controller is designed based on a high gain observer to optimize the capture wind energy by tracking the optimal torque in [12,13].
The LPV reformulation of gain scheduling has become very popular during the last decades [4].Compared with switching linear time invariant (LTI) controllers, LPV method can achieve better stability property.The LPV model was firstly introduced by Shamma and Athans [14] and then has been widely applied to wind turbine systems [15][16][17][18].
As known to all, the wind conversion systems include mechanical part and electromagnetic part, and the time scales of mechanical dynamics (slow) and electrical dynamics (fast) are quite different from each other [2,19].So the wind conversion systems have two-time-scale property, which (2) The stability and robust property of the singularly perturbed LPV systems are proved under certain conditions.Besides, an algorithm is presented to design a robust stabilizing control scheme for the singularly perturbed LPV system.Using LMI technology, the stability and robust property are proved for the closed-loop system.
The rest of this paper is organized as follows.In Section 2, the nonlinear model of the WECS is presented which involves the dynamic characteristics of the mechanical part and electrical part.Then, a singular perturbation parameter is extracted which leads to a singularly perturbed nonlinear model.The nonlinear singularly perturbed model is reformed into a singularly perturbed LPV model that contains a family of LTI models.Later, Lyapunov theory is utilized to prove  that the tracking error between the rotor speed   () and the desired rotor speed  ref () decays to zero in Section 3. Furthermore, for a given constant  > 0, ‖   ‖ <  is proved.In Section 4, the conditions in LMI forms are given to design a robust stabilizing parameter-dependent controller.Section 5 presents an algorithm to design a parameterdependent controller.In Section 6, two numerical examples are given to illustrate the effectiveness of the results obtained.Finally, conclusions are drawn in Section 7.

System Description
In this section, the mathematical model of the wind turbine systems with permanent magnet synchronous generators is developed.
Because the electrical part of the system changes much faster than the mechanical part, namely, the states of the wind energy conversion systems have two different time scales.A singular perturbation parameter is extracted to obtain the singularly perturbed nonlinear model.The following LPV technique is used to linearize the singularly perturbed nonlinear system.
Our consideration is focused on partial load.In the partial load region, the wind rotor speed is adjusted to maintain the optimal tip-speed ratio as the wind speed changes.For this control purpose, only aerodynamics, drive train dynamics, and generator dynamics are taken into account.
Commonly, the aerodynamic torque   is given as follows [24,25]: where  is the air density,  is the wind speed,  is the radius of the wind rotor plane, and power coefficient   () is approximated by a second-order polynomial of tip-speed ratio  [4] as below: where  max is the maximum power coefficient,  max is the optimal tip-speed ratio corresponding to  max , and  is defined as where   is the wind rotor speed.
The drive train block has the model as below [2,26]: where   is the generator speed,   is the internal torque,   is the wind rotor inertia,   is the generator inertia,   is the stiffness coefficient of the high-speed shaft (the generator shaft),   is the damping coefficient of the high-speed shaft (the generator shaft),  is the gearbox ratio, and  is the gearbox efficiency.Then, the generator dynamics are modeled as [2,25] where   is the generator electromagnetic torque,   ,   ,   and   ,   ,   are the  and  components of the stator current, inductance, and voltage, respectively,   is the stator resistance,  is the number of pole pairs, and   is the flux.By combining (4) and ( 5), the complete nonlinear model of the wind energy conversion system is obtained.Next, we will introduce the singular perturbation method.
Considering the order of magnitude of   and   , we define the so-called singular perturbation parameter  = 1 × 10 −2 and obtain where Now the singularly perturbed nonlinear system is obtained by uniting ( 4) and (6).
And now we are ready to derive the linear model using LPV method.Choose an operating point  1 = [ω  V ω î î ]  and linearize the nonlinear parts in the singularly perturbed nonlinear system at point  1 : where By substituting ( 7) into ( 4) and ( 6), we have the following singularly perturbed linear system at the operating point  1 : where Furthermore, note , and rewrite (15) as where Then, by appropriately choosing operating points   , is a convex polytope with   being vertices,  = 1, . . ., 5. Note that the LPV model (17), with   () and  V () approximated by (11) and (12), is affine in the parameters.Namely, there exist scalars , , , and  such that   () =  +  and  V () =  + .Hence, it is easy to see that, for any  ∈ Θ, there exists a set of positive numbers   > 0,  = 1, . . ., 5, such that where Therefore, for any  ∈ Θ, we have derived the singularly perturbed LPV model as below: Remark 1.For the details of skills to choose operating points appropriately, please refer to [4].
Remark 2. The wind conversion system considered here contains both mechanical and electrical parts which are of different time scales.Singularly perturbed model is developed of this system and LPV method is used to reform the model.The combination of singular perturbation theory and LPV method is novel for the maximum power point tracking problem.

Stability Analysis of Open-Loop System
In this paper, the objective is to maintain the optimal tipspeed ratio by adjusting wind rotor speed as the wind speed changes.So the operating points are chosen such that the tipspeed ratio is optimal.If the states  and  in (21) decay to zero, it means that the errors between the actual states and the desired states tend to zero.In this case, the wind turbine runs to extract all the available power.
And, this section will analyze the stability of the singularly perturbed system (21) when the control input  = 0 based on LMI technique.
When () = 0, from ( 21) we can have Before the main result, Schur Complement Lemma is given below.(
Firstly, the asymptotic stability of the system is proved under conditions in Theorem 4 when the disturbance () is zero.Derive () with the respect to  along the trajectory of (22), with () = 0, and we obtain From the LMI (25), it is not difficult to have the following inequality: And, by adding the weight values (i.e.,   ) of   to (29), we can get Therefore, Hence, the system of ( 22) at point  is asymptotically stable when () = 0.
Next, the robust property of system (22) will be proved when the disturbance () ̸ = 0.Because   > 0, using the LMI (25) it can be obtained that Then by using ℎ   twice, LMI (33) can be transformed into inequality below: Derive () along the trajectory of ( 22) and use the inequality (34); Ẇ() becomes Based on the asymptotic stability proved at the first part of this proof, we have (∞) = 0. Assume (0) = 0, and integrate both sides of (35) from  = 0 to  = ∞; we can have Then, (36) can be rearranged to have Therefore, as can be seen from the inequality above, Now, it is easy to see that ‖(  − ()) −1  1 ()‖ <  is satisfied.This completes the proof.
Remark 5.Even though Theorem 4 can guarantee the stability and robust property of system (22), the LMI ( 25) is dependent on small parameter , so it might be singular and difficult to solve.
Proof.Define a Lyapunov function as below: where  = [    ]  and  = ∑ 5 =1     .According to condition (39), we have As a consequence, () > 0 is satisfied.Then, the time-derivative of () along the solution of ( 21) is given by The following part is similar to the proof of Theorem 4 and is consequently omitted here.

Controller Design
In this section, a robust state-feedback controller is designed for system (21), and the stability property of the closed-loop system is analyzed.
Since the coefficients of system (21) depend on , it is reasonable to design a controller whose feedback gain matrix also depends on .
As can be seen from ( 20) and ( 22), at any point  ∈ Θ, the dynamic equation of the nonlinear singularly perturbed system can be expressed as a weight-sum of the dynamic equations at the vertices   of Θ,  = 1, . . ., 5. Therefore, we will design controllers for the LTI systems operating at the vertices   of Θ,  = 1, . . ., 5, and use a weight-sum of the controllers at vertices as the control input to the system at point  ∈ Θ.
Then, for the nonlinear singularly perturbed system (21) at , the state-feedback controller is as follows: where   > 0 and ∑ 5 =1   = 1.Applying (47) to system (21), the closed-loop system is obtained as below: The properties of closed-loop system of (48) are analyzed in Theorem 8. To prove Theorem 8, the following lemma from [27] is needed.

Algorithm of Synthesis
In order to clarify the whole process of designing a parameterdependent controller for the original nonlinear system (4) and ( 5), the following algorithm is presented.
Step 6.The control gain matrix at time   is obtained as below: Step 7. Apply the controller (62) to the original nonlinear system (4) and ( 5).
Remark 10.Since the operating point  = [ω  V ω î î ]  involves wind speed , Θ implies the range of the wind speed within which our algorithm is effective.Remark 11.In [12,13], a high-order sliding mode control strategy is proposed based on a high gain observer to optimize the maximum power point tracking problem of wind energy conversion system.Their method presents chatteringfree, behavior, finite reaching time and robustness.However, high-order controller and observer need more online computation.In this paper, the wind energy conversion system is modeled using singular perturbation theory which considers the generator dynamics as fast subsystem and the drive train block as slow subsystem.This method combines the singular perturbation methodology and LPV model for the first time to solve the MPPT problem.

Numerical Examples
In this paper, the goal is to track the desired wind rotor speed and maintain the optimal tip-speed ratio when the wind speed changes.The simulation study is performed to verify the effectiveness of the proposed control algorithm.
The two examples consider the CART 3-blades wind turbine taken from [20] as an objective.The parameters of the wind turbine are given in Table 1.The experiment is carried out on the MATLAB FAST5 software which is developed by the American National Renewable Energy Laboratory.
Example 12. Consider a wind turbine system with the parameters depicted in Table 1.The wind speed is assumed as a constant 9 m/s in this example.The tracking result controlled by the algorithm developed in Section 5 is compared with that of the optimal torque (OT) method, and the compared rotor tracking results are shown in Figure 2.
It can be seen that the tracking error between actual rotor speed controlled by the algorithm presented here and the desired rotor speed decays to zero after a small overshoot.But  the tracking error between the desired rotor speed and the rotor speed controlled by the optimal torque method can not decay to zero with time, which means a static error exists.So this example illustrates that the algorithm developed in this paper is more effective than the optimal torque method.
Example 13.In this example, a turbulence of 600 seconds by using TurbSim software is applied.This turbulence satisfies the IEC-61400-1 standard and the turbulence is shown in Figure 3.
The rotor speeds controlled by the algorithm presented in this paper and the optimal torque method are shown in Figure 4.It is obvious that the rotor speed controlled by the  ∞ algorithm can track the desired rotor speed much better than that of the optimal torque method.
Furthermore, the wind power capture efficiency and energy conversion efficiency controlled by this novel method and optimal torque method are compared in Table 2.It can be seen from Table 2 that the  ∞ method developed can obtain better wind power capture efficiency and energy conversion efficiency.

Conclusions
This paper extended the continuous-time infinite horizon nonlinear quadratic optimal control problem of NSPSs to discrete-time version with the weight matrices dependent on the states in the cost function.For a class of discretetime NSPSs in this paper, we used the theory of singular perturbations and time scales to decouple the original highorder NSPS into order-reduced slow and fast (boundary layer) subsystems.Then, via the state-dependent Riccati equation, suboptimal controllers for the two subsystems are designed with the weight matrices varying with states   in the cost functions.A composite controller consisting of two suboptimal controllers is developed for the original system.It is proved that the equilibrium point of the original closed-loop system with a composite controller is locally asymptotically stable.In the end, an example is given to show the effectiveness of the results obtained.

Figure 1 :
Figure 1: Control configuration of wind turbine system.

Table 1 :
Parameters of wind turbine.

Table 2 :
Compared power conversion efficiency.