Predefined time backstepping control of photovoltaic grid connected inverter

The large-scale grid connection of photovoltaic (PV) systems creates great hazards to the operation of power equipment and has a great impact on the stability and reliability of power systems. To achieve stable control and flexible management of PV grid connection, a predefined time adaptive backstepping control method based on the predefined boundary is proposed. Based on the predefined boundary time-varying function, the user’s predefined boundary design of the control signal is accomplished to ensure the boundedness of the global signal, which is independent of the initial state of the system. The controller with predefined boundaries is designed based on backstepping control, and the estimation and compensation of perturbations are attained based on adaptive control. The global predefined time stability of the predefined boundary of the closed-loop system is certified using the Lyapunov function. Simulation studies attest to the performance of the predefined time control strategy.


Introduction
With the transformation of energy structure and the requirements of low-carbon environmental protection, photovoltaic power generation technology has been vigorously developed, which alleviates the contradiction between environmental protection and resource demand, and also puts forward new requirements on the level of the electric power technology [1] .Photovoltaic power generation has randomness and volatility, working in grid-connected mode will have a certain impact on the stability and reliability of the power system [2] .Subject to the development trend of large-scale grid-connected photovoltaic power generation, the power coordination and control problems of the power system have become more complex, and the transformation and upgrading of power technology is imminent, which requires effective control and management of PV grid-connected [3] .
To realize the effective control of PV grid-connected stabilization time and help the flexible management of grid-connected stabilization time, the realization method of stabilization time control is discussed [4] .Based on Lyapunov stability, the finite time stability principle allows a small set of systems converging to the origin in a finite time, but the upper limit of the stabilization time of the system varies with it due to the different initial states, which is an uncertain quantity [5] .The fixed-time stabilization theory ensures that the upper limit of the stabilization time of the system is fixed, and its upper limit expression can be formulated as a complex expression for the design parameters and is independent of the initial conditions [6] .It is difficult for these two control methods to obtain a direct relationship between the parameters and the upper limit of convergence time for direct design.To solve this problem, the predefined time control theory is proposed [7] .The predefined time control can accurately express the upper limit of stabilization time in parameters, which can be preset by the user so that the system can complete stability control in a predefined time.
Considering the predefined time control theory, an adaptive backstepping control method based on predefined boundaries is proposed for PV inverters.Firstly, considering the non-exact model part of the system and external perturbations, the disturbance part is added to the inverter model, and the inverter controller is devised based on the backstepping framework.The predefined time function is affiliated with the controller design process to control the predefined bound of the system, and the piecewise function is used to avoid the singularity of the control process.Secondly, the adaptive control method is used to estimate and compensate for the disturbance, which reduces the dependence on accurate model information.Finally, the controller realizes that the user can design the predefined time and achieve the predefined time stability control, which is independent of the initial state, but the choice of time T should be designed according to the controller parameters and actual requirements.

The structure of the PV system
The topology of the PV system consists of a PV array, a DC-DC converter, a voltage source converter (VSC), and a transformer as shown in Figure 1.The PV power generation is affected by solar irradiation and temperature, and the conductivity increment method is used to accomplish the maximum power point tracking (MPPT) control, the inverter converts the DC power to three-phase AC power, which is connected to the grid through filters and transformers.For the angular frequency U of the system, the AC voltage is obtained by a phase-locked loop (PLL) for Park transformation.
Figure 1 is the structure diagram of the PV system, the power electronic device and its parameters information are given, and setting the change of illumination and temperature.

The dynamic model of PV inverter
The low-frequency mathematical model is imported for the inverter.Ignoring the effect of system impedance and other factors, the inverter DC side current relationship is: where pvout i is the output current of the DC-DC converter.
where ( ) i d t is a disturbance of the system that satisfies ( ) and i D \ is the upper bound of the disturbance.

Prerequisite knowledge
For the PV inverter model, controllers d W and q W with predefined boundaries are brought to the second- order model of the D-axis and the first-order model of the Q-axis, respectively.The system can achieve the stability of a predefined time controller with the same user-defined time.The conditions for predefined time stability and Lyapunov stability are discussed: Lemma 1. [8] For a system ( , ( , )) x , if there exists a parameter vector x that satisfies ( ) x t a d at p t T t , the system is predefined time bounded for arbitrarily 0 x , predefined time p T and positive constant a .
Lemma 2. [9] A predetermined timescale function is defined with positive constant D and E , initial time 0 t , and time T that satisfying p T T t : For scalars 0 b !, 0 c ! and the user-defined constant 0 K !, the function ( ) V x is satisfied with Then the system x is bounded in predefined time and its predefined boundary is / K D , which means / V K D d .Lemma 3. [10] For smooth positive time-varying functionX and e \ , the following inequality satisfies ( , ) 0 e e sg e X X !t and 2 2 ( , ) / sg e e e X X , where 0 ( )

Design of predefined time adaptive backstepping control
In the design of the predefined time adaptive backstepping controller with predefined boundaries, the tracking errors on the global signals are designed as: where d y is the reference signal for the D-axis, 1 D is the virtual controller, and q y is the reference signal for the Q-axis.For 1,2,3 i , the normal number upper bound i D is estimated and compensated using adaptive control with an estimate of i D and an adaptive error of i i i D D D .The adaptive rates with predefined bounds are led using the parameter The perturbation is expressed as an estimate 1 D .The virtual controller 1 D is designed with predefined boundaries as:

Global stability analysis
Defining the Lyapunov function as , the derivation of 1 V is: With the inequality: By substituting Equations ( 5)-( 7), Equation ( 10) is simplified as: Defining the Lyapunov function as , and simplifying it using Equations ( 7)-( 9) and Equation (11), it is obtained that: Defining the Lyapunov function of the Q-axis as , and simplifying it using Equations ( 7)-( 9), it is obtained that: Discussing the stability of the global Lyapunov function V V V V , it is obtained by simplification that: where    Analysing the three-phase voltage and current at 0.1 T s , we can see that when the illumination and amplitude change at 0.6 s~1.8 s in Figure 1, the voltage remains stable and the current changes as shown in Figure 6.Three-phase voltage harmonics in Figure 7 meet the requirements of grid connection.

Conclusion
For the predefined time control of grid-connected inverter for photovoltaic power generation, this paper achieves the predefined time control of the photovoltaic inverter by designing the predetermined timescale function.Based on the framework of backstepping control, the virtual controller and the controller of the DQ-axis are drawn to finish the predefined time control under the predefined boundary of global signals.For the disturbance part of the system, adaptive control laws with predefined boundaries are introduced for disturbance estimation and compensation.By using backstepping and adaptive control to ensure that all signals are bounded at the user's set time, the tracking error is gained to converge within the user-defined range.The stability of the system is substantiated by Lyapunov theory and the performance of the controller is verified by simulation.

Figure 1 .
Figure 1.The structure diagram of the PV system.
the designed constants, and ( ) e I is a segmented function to eliminate the singularity in the process of control, which satisfies: the above equations, it follows Lemma 2 that the tracking errors 1 e and 3 e are bounded in a predefined time T , and the errors converge into an invariant set Simulations are performed and analysed with the reference signals 500 simulation debugging, the backstepping control parameters are chosen as 1 130 k

Figure 2 .Figure 3 .Figure 4 .
Figure 2. DC voltage ‫ݑ‬ ௗ at ܶ = ‫.ݏ1.0‬ Figures2-4show the tracking effect of the predefined time backstepping control based on the predefined boundary on the DC bus voltage.The preset time T is selected for 0.1 s, 0.2 s and 0.3 s, and the controller fulfills the effective tracking of the DC voltage reference signal within the predefined time.Figure5is the output power of the PV system.