Passivity and higher-order sliding mode control for doubly-fed induction generation wind power system

To simplify the model of DGIG and improve the robustness, the variable speed and constant frequency (VSCF) DFIG wind power generation systems from the perspective of energy and robustness were studied. It was based on passivity and sliding mode control (SMC) theory. Firstly, the model of DFIG based on Euler-Lagrange (EL) equations was established, so the original DFIG wind turbine system is divided into electrical and mechanical subsystems. For two subsystems, the electrical subsystem was designed. In contrast, the mechanical subsystem is an energy-consuming system and is stable through selecting suitable design parameters. Hence, the system control algorithm is simplified. A high-order terminal SMC was designed to get a faster speed and better robustness for the speed loop. Simulation results show that the proposed controller is simple and effective. It can guarantee the wind power system outputs constant frequency. The motor speed can also track the reference speed quickly, so the performance of the wind turbine system is improved.


Introduction
With the depletion of traditional fossil fuels and the pollution caused by their combustion, the development and utilization of renewable energy, such as wind energy, is of great significance.The development of various wind power generation technologies is on the ascendant; the DFIG is used to achieve variable speed and constant frequency, and its advantages have been widely applied [1][2][3].
DFIG is a nonlinear, multi-variable, and strong coupling system.The traditional vector control method has the disadvantage of relying on the system parameters, and it is easy to produce current distortion.The uncertainty of wind speed will even cause instability.At present, the emerging PBC method [4][5][6] shows superior control performance.It takes global stability as the control objective and the passive system balance as the design criterion to find out the 'reactive power' that does not affect the stability, forcing the total energy to track the expected energy function and making the state variables asymptotically converge to the set value [7].This method only pays attention to the natural attributes reflected by the physical characteristics of the system.It does not depend on the design method of the exact linearization of the object model.It can effectively simplify the design of the controller and improve the robustness of the system.The variable speed constant frequency DFIG is analyzed from the coordinate system in [8], and a good control effect is obtained.However, a large number of rotation transformations are needed in practical applications, and it is complicated to realize with a digital controller.In [9], the d-q coordinate system is used to analyze the system in depth, and an adaptive controller is designed to track the time-varying wind speed.However, due to the poor robustness of the outer loop using the PI controller, it is easy to be disturbed.This paper starts with the mathematical model of DFIG.It combines the passive control method of DFIG with the high-order non-singular terminal sliding mode control.Then, it applies it to the DFIG system.Firstly, the passivity model and description method of doubly-fed variable speed constant frequency wind power generation system are established.Then, by introducing a high-order sliding mode controller into the passivity control method, the robustness of the system is effectively improved.It can solve the problems of divergence singularity and PI control links vulnerable to external interference in other control methods.Finally, the effectiveness of the designed controller is verified by simulation experiments.

Passivity of DFIG
In the two-phase synchronous rotating coordinate system, the electrical and mechanical subsystems of DFIG can be represented as follows [6]. () In order to obtain the passive model of DFIG, Equation ( 1) is transformed into EL form [7]: ( ) where

M
On the premise of ignoring the capacitance effect, the electrical subsystem function of the motor can be defined as T 1 2 H = q Dq .Deriving the above equation to time, we can get T H = q Dq .By substituting Equation ( 4), we get TT () the 'q T Cq' term does not affect the energy change and the system stability.Therefore, it is not necessary to offset this part of the nonlinear factors.
The process can be considered as configuring the system's reactive power.Integrating the two sides of Equation ( 7), we can get . The left of the equation is the increment of the electrical subsystem energy, and the right of the equation is the power supply motor energy.If [usd usq urd urq] T is used as the input of the electrical subsystem and [isd isq ird irq] T is used as the output, the electrical subsystem of DFIG is strictly passive.
Assuming that the motor shaft is rigid, the energy function is  2), and the integral on both sides of the equation is obtained: The left of the equation is the energy increment of the mechanical subsystem, and the right of the equation is the input.If (Te-TL) is used as the input and m is used as the output, the mechanical subsystem of DFIG is strictly passive.
A feedback interconnection of the electrical and mechanical passive subsystems can be used to repress the DFIG.Based on the passivity principle [10], the DFIG system is strictly passive, as shown in Figure 1.
Through decomposition, the mechanical subsystem can be regarded as a passive interference term, and only the electrical subsystem can be treated as a controlled system.

Design of torque controller
Assuming that the expected output torque is Te*, and the stator flux linkage is ψs, then the output torque of the system in the two-phase rotating coordinate system can be expressed as: In order to realize the asymptotic vector control of stator flux linkage and the asymptotic tracking of electromagnetic torque, the control objective is set as follows: (1) Electromagnetic torque asymptotic tracking: .For this reason, the tracking error is described as e=q-q* , and the error equation of the system is obtained from Equation ( 4 The control objectives of the flux asymptotic vector control can be obtained by: The desired value is set as a constant value.
In this way, i * sd can be determined by the set stator side reactive power, and then the other three currents can be obtained by the given torque command value.Finally, from 0  = , the equations of the controller urd and urq can be obtained: In order to make the control system strictly passive, the dynamic response of the system is improved, and the sensitivity of parameter changes is reduced.The damping coefficients k1 and k2 are added to Equation (11).

Design of speed controller
Since the passivity-based controller can asymptotically track the time-varying torque, the goal of the speed controller is to accurately track the given speed signal.It is completely robust to parameter perturbations such as external load disturbances and frictional resistance and outputs a smooth torque reference signal Te*.Due to the traditional control strategy such as PID, it is hard to satisfy the requirements of high precision.In this paper, the second-order terminal non-singular sliding mode control strategy is used to realize the speed control of the outer ring.The given velocity signal is ω*.It is assumed that ω* is smooth enough.The error state is defined as * e  =− .Therefore, we have: In this paper, the sliding mode hypersurface design is as follows: where p>0, p and q are positive odd numbers and 12 pq  .By designing an appropriate sliding mode control law, the second-order sliding mode surface can converge to zero in finite time so that the error state e , e , and e finally asymptotically reach zero.The structure of the whole control system is shown in Figure 2.

Simulation
MATLAB/Simulink is used to verify the effectiveness of the designed controller.

=25
 , p=5, and q=3.In order to verify the dynamic and static performance of the PBC method of the designed DFIG control system, the initial speed is set to 636 r/min.After entering the steady state, the speed suddenly increases to the synchronous speed of 1000 r/min at t=5 s.The PI adjustment and sliding mode adjustment processes are compared.The experimental results are shown in Figures 3-6

Conclusion
The variable speed constant frequency doubly-fed wind power generation system adopts the passivity control scheme to ensure the global stability of the system.In this paper, the passivity and the secondorder terminal sliding mode are combined to derive the control strategy for controlling the DFIG.Compared with the traditional control method, the control method has many advantages, such as a simple algorithm, good control response, and control robustness.The simulation results show that the control method has a faster response speed, smaller overshoot, and better robustness than the traditional PI control method when wind speed changes.
function Hd=1/2e T De is chosen, its derivative is

Figure 6 .
Figure 6.The local amplification diagram of rotor voltage waveform under sliding mode control.
If controlling the reactive power of the DFIG stator side is zero, namely * 0