Based on a brief review on current harmonics generation mechanism for grid-connected inverter under distorted grid voltage, the harmonic disturbances and uncertain items are immersed into the original state-space differential equation of grid-connected inverter. A new algorithm of global current harmonic rejection based on nonlinear backstepping control with multivariable internal model principle is proposed for grid-connected inverter with exogenous disturbances and uncertainties. A type of multivariable internal model for a class of nonlinear harmonic disturbances is constructed. Based on application of backstepping control law of the nominal system, a multivariable adaptive state feedback controller combined with multivariable internal model and adaptive control law is designed to guarantee the closed-loop system globally uniformly bounded, which is proved by a constructed Lyapunov function. The presented algorithm extends rejection of nonlinear single-input systems to multivariable globally defined normal form, the correctness and effectiveness of which are verified by the simulation results.

Along with the strong demand of electric power energy, conventional fossil fuels are gradually reduced. The exploiting of renewable energy is an inevitable route for development of sustainable society. According to the planning of EU Commission, the renewable energy goal is achieving 20% of energy consumption from sustainable sources by 2020 [

Renewable energy sources include wind (onshore and offshore), solar PV, hydroenergy, biomass energy, geothermal energy, and tidal energy. Currently, two of main utilization approaches of renewable energy are off-grid power generation and connected to utility grid. In the past decade, the renewable energy sources connected to utility grid have achieved a considerable growth with the forceful needs of electric power energy [

The rejection solutions of current harmonics for grid-connected inverter mainly consist of two categories: hardware implementation and software programming. A simple and practical scheme of hardware implementation is adding filters to inverter’s output. The aim of adding filters is to compensate for the fluctuations of grid voltage. In [

For the disadvantages of the above methods, some scholars attempt to apply modern control theory to improve the capability of harmonics disturbance rejection. The nonlinear control methods, such as repetitive control [

The harmonics rejection algorithms are concerned primarily with the single-input field; the research of multiple-input field is rarely mentioned.

For an operating grid-connected system, the parameters of resistance and inductance for the system are often time varying and cannot be measured accurately. Hence, these proportional-integral-based controllers, which take no account of the time-varying characteristics of the parameters, cannot work very efficiently.

The parameter design of robust controller is relatively complicated and needs large amount of calculations.

In the paper, a multivariable state feedback control algorithm based on nonlinear backstepping control with multivariable internal model principle (MIMP) and adaptive control law (ACL) for grid-connected inverter is proposed to reject the harmonic disturbance produced by a class of exosystem and uncertainties related with system states caused by modeling perturbations, parameter uncertainty or actuator end disturbance. The main contributions in the paper are the following.

A new harmonic rejection algorithm based on nonlinear backstepping control with MIMP and ACL is proposed in the paper, which extends the disturbance rejection of the nonlinear single-input system to multivariable globally defined normal form in contrast with the previous literatures.

A new type of nonlinear multivariable internal model for a class of nonlinear harmonic disturbances and uncertain items related with system states is constructed.

The proposed algorithm is practically applied to reject current harmonic disturbances for grid-connected inverter under distorted grid voltage.

The paper is organized as the following: it starts with an introduction to the research status of current harmonics rejection for grid-connected inverter in Section

The model of grid-connected inverter can be shown in Figure

Modeling of grid-connected inverter.

In terms of Figure

Application of the park transformation to (

Rearrange (

In terms of Figure

Assume that the distorted voltage of the network contains the 5th and 7th harmonics. In stationary

Multiple synchronous rotating reference frames.

In rotating

In a similar way, the voltage equation of the 7th harmonics can be written as

Use

Considering disturbance-free and no modeling perturbations, the nominal system of (

There exist smooth functions

There exist a constant

Similar to the uncertain item

One of the research points in the paper is stability problem for multivariable input system. Currently, some literatures focus on the point of stability problem, but the focused attention of these literatures is the field of single input and single output [

For Assumption

For Assumption

Assumption

The problem solved in the paper can be described by the following theorem.

For any given compact set

Application of internal model principle (IMP) to reject the exogenous disturbance

In terms of Assumption

Define an auxiliary error vector as follows:

With the derivative of (

In comparison with the nominal system (

Apparently, designing of nonlinear internal model

Firstly, we define the errors as follows:

Derivative of

Derivative of

Now, from (

Hence, from (

For any positive number

If we choose

In terms of (

With the derivative of (

In terms of Assumption

According to Lemma

In terms of permanent establishment inequality

Suppose that

There exist positive definite matrices

To illustrate the performance of the present control algorithm, some numerical simulations are performed in the section. The whole simulation time is 5 seconds with the sampling interval 0.001 s. The simulation parameters are chosen as follows.

For the grid, the rms value of network voltage

Substituting the relevant parameters into the original state equation (

The network is immersed 3% 3rd harmonics to fundamental wave.

The exosystem matrices for exogenous disturbance

It is worth noticing that the rotating electrical angular velocity of the 3rd harmonics in stationary

In terms of (

Supposing that

we can obtain

Choosing

According to (

Hence, through the presented algorithm, the final internal models of disturbance and uncertain items are given by

Choosing

Let the initial condition of the simulation be

System states (output currents in

Control inputs in

Exogenous disturbances of

Estimating errors of

The network is immersed 1.5% 3rd harmonics and 1% 5th harmonics to fundamental wave.

The exosystem matrices for exogenous disturbance

The design of the nominal system and the verification of Assumption

Choosing

In terms of (

Hence, through the presented algorithm, the final internal models of disturbance and uncertain items are given by

Choosing

Let the initial condition of the simulation be

System states (output currents in

Control inputs in

Exogenous disturbances of

Estimating errors of

In this paper, a nonlinear backstepping control with multivariable adaptive internal model principle for grid-connected inverter is proposed to reject the harmonic disturbance produced by a class of exosystems under the existence of uncertain items related with system states. Due to the nonlinearity and multiple variables for the original system, a nonlinear and multivariable internal model is constructed. In addition, in order to compensate for the effect of bounded uncertain items, an adaptive control law is designed to realize the real-time estimation of the perturbation. Based on the backstepping control law of the nominal system, a state feedback controller combined with the multivariable internal model and the adaptive control law is designed. A Lyapunov function is constructed and theoretically proves that all the signals of the multivariable closed-loop system are global boundedness. The simulation results show that the proposed control algorithm can guarantee the closed-loop system asymptotically converge to expected references quickly and the designed internal model can produce the exogenous disturbances successfully.

Restricted by the actual problem of grid-connected inverter control under distorted grid voltage, the proposed algorithm in the research aims at rejection of harmonic disturbances, and without considering the nonharmonic disturbances. However, the nonharmonic disturbances may induce adverse impacts, for example, noise and precision reduction. The future research should extend the algorithm to reject the nonharmonic disturbances.

This research is supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant no. 20120036130001, the Fundamental Research Funds for the Central Universities of China under Grant no. 11MG40, and the Independent Research Funds of State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources of China under Grant no. 201209.

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