Adaptive Sliding Mode Control of Single-Phase Shunt Active Power Filter

Abstract

This paper presents a thorough study of the adaptive sliding mode technique with application to single-phase shunt active power filter (APF). Based on the basic principle of single-phase shunt APF, the approximate dynamic model is derived. A model reference adaptive sliding mode control algorithm is proposed to implement the harmonic compensation for the single-phase shunt APF. This method will use the tracking error of harmonic and APF current as the control input and adopt the tracking error of reference model and APF output as the control objects of adaptive sliding mode. In the reference current track loop, a novel adaptive sliding mode controller is implemented to tracking the reference currents, thus improving harmonic treating performance. Simulation results demonstrate the satisfactory control performance and rapid compensation ability of the proposed control approach under different conditions of the nonlinear load current distortion and the mutation load, respectively.

1. Introduction

With the widely used single-phase electric devices and increased high power electric appliance, it becomes more and more obvious that the quality of power supply drops and power factor reduces because of nonlinear factors. Since power electronic device and nonlinear load seriously damage the power quality, they have become the main harmonic pollution source of power network. APF could compensate the harmonics generated by the load current through injecting compensation current to the grid, having the advantages of high controllability and fast response. It not only can compensate harmonics, but also can inhibit the flicker and compensate reactive power; therefore, it is an effective approach to suppress the harmonic pollution.

In recent years, the research and design of APF have made great progress, and a large number of successful APF products have been put into market. Along with the rapid development of precision, the speed and reliability in hardware equipment, high performance algorithm, and real time control can be realized. The models of APF have been established using various methods, and the behavior of reference signal tracking has been improved using advanced control approaches. Rahmani et al. [1] presented a nonlinear control technique with experimental design for three-phase shunt APF. Singh et al. [2] designed a simple fuzzy logic-based robust APF to minimize the harmonics for wide range of variation of load current under stochastic conditions. Bhende et al. [3] proposed TS-fuzzy-controlled APF for load compensation. Montero et al. [4] compared some control strategies for shunt APFs in three-phase four-wire systems. Matas et al. [5] succeeded in linearizing the mathematical model of APF with feedback linearization method. Hua et al. [6] and Komucugil and Kukrer [7] used Lyapunov function to design some new control strategies for single-phase shunt APFs. Chang and Shee [8] proposed novel reference compensation current strategy for shunt APF control. Pereira et al. [9] derived new strategies with adaptive filters in APFs. Marconi et al. [10] proposed robust nonlinear control of shunt active filters for harmonic current compensation. APF control methods based on the adaptive algorithm have been proposed by some researchers [11–15]. Asiminoaei et al. [11] derived adaptive compensation scheme of reactive power for APF. Luo et al. [12] developed hybrid APF based on the adaptive fuzzy dividing frequency control method. Ribeiro et al. [13] presented a robust adaptive control strategy for power factor correction, harmonic compensation, and balancing of nonlinear loads in APFs. Valdez et al. [14] designed an adaptive controller for shunt active filter in the presence of a dynamic load and the line impedance. Shyu et al. [15] proposed a model reference adaptive controller to control the circuit and improve the current and reduce the current harmonics by using the approximate dynamic model of single phase shunt APF. Different control methods and harmonic suppression approaches for APF have been investigated [16, 17].

In the presence of model uncertainties and external disturbance, sliding mode control is necessary to incorporate into the adaptive control system since sliding mode control is a robust control technique which has many attractive features such as robustness to parameter variations and insensitivity to disturbance. Adaptive sliding mode control has the advantages of combining the robustness of variable structure methods with the tracking capability of adaptive control. However, systematic stability analysis and controller design of the adaptive sliding mode control with application to single-phase shunt APF have not been found in the literature; therefore, it is necessary to adopt adaptive sliding mode control that can on-line adjust the control parameter vector combined with the great robustness of sliding mode control for the harmonic suppression of single-phase shunt APF.

This paper will expand the model reference adaptive control (MRAC) and incorporate the sliding mode control into the adaptive system to design the adaptive sliding mode control algorithm and apply to the single-phase shunt APF. The contribution of this paper can be summarized as the following.(1) A novel adaptive sliding mode control is proposed in reference current tracking to reduce the tracking error. The designed APF has superior harmonic treating performance and minimizes the harmonics for wide range of variation of load current under different nonlinear load; therefore, an improved THD performance can be achieved with the proposed control scheme.(2) It is the first time that adaptive sliding mode control is applied to the APF. The advantage of using adaptive controller for the shunt APF with sliding mode technique is that it has better harmonic treating performance and will improve the robustness of the APF under the nonlinear loads. (3) This paper systematically and deeply studies the adaptive control and sliding mode technique with application to APF, comprehensively uses the adaptive control, sliding mode control with the APF, thereby significantly reducing the APF’s sensitivity to the nonlinear load and disturbance, and improving the robust performance. The APF control system is designed to make the compensation current track the command signal in real time, thereby eliminating the harmonics, improving the electric energy quality, and enhancing the security of the power transmission and distribution and power grid. Therefore, this research has great theoretical value and application potentials.

The paper is organized as follows. In Section 2, the basic principle of single-phase shunt APF and the approximate dynamic model are introduced. In Section 3, an adaptive sliding mode controller for shunt APF is designed and Lyapunov stability is established. Simulation results are presented in Section 4. Concluding remarks are summarized in Section 5.

2. Dynamic Model of APF

In this section, the basic principle of single-phase shunt APF and the approximate dynamic model are introduced [15]. Main circuit of shunt APF in parallel with the load connected to the network is shown in Figure 1. Currently, this is the most basic form of APF and the most widely used. In this way, the APF is equivalent to current generator, and the current value is the compensation current.

Figure 1 

Principle diagram of SAPF.

As shown in Figure 2, APF connected in parallel with the load could cancel the harmonic components in the line current (𝑖_𝑆) so that the current flowing into and from the power line is sinusoidal and in phase with the power line voltage. In other words, the compensating current (𝑖_𝐿) is injected into the line to force the line current (𝑖_𝑆) to become sinusoidal wave and to achieve a unity power factor.

Figure 2 

Single phase SAPF and nonlinear loads in parallel.

The current of the shunt APF can be expressed as 𝑖_𝑆=𝑖_𝑂+𝑖_𝐿, (2.1) where 𝑖_𝑂 is the nonlinear load current.

The operation of the APF can be divided into two modes, and its four switches have a switching frequency of 𝑓_𝑆. In mode 1, 𝑄₂ and 𝑄₃ are turned ON, while 𝑄₁ and 𝑄₄ are turned OFF when 0<𝑡<𝐷𝑇_𝑆, where 𝑇_𝑆=1/𝑓_𝑆 is the switching period and 𝐷=𝑇_ON/𝑇_𝑆 is the duty ratio. In mode 2, the switching states of four switches in mode 1 are reversed when 𝐷𝑇_𝑆<𝑡<𝑇_𝑆.

By observing the equivalent circuits that are shown in Figure 3, one circuit shows the inductor voltage and current during one switching cycle when 𝑣_𝑆>0, which are expressed by the following: 𝑣_𝐿(𝑡)=𝑣_𝑆+𝑣_𝐶𝑖_𝐿(𝑡)=𝑖_𝐿(0)+1𝐿∫^𝐷𝑇𝑆₀𝑣_𝑆+𝑣_𝐶𝑑𝑡for0≤𝑡≤𝐷𝑇_𝑆,𝑣_𝐿(𝑡)=𝑣_𝑆−𝑣_𝐶𝑖_𝐿(𝑡)=𝑖_𝐿𝐷𝑇_𝑆+1𝐿∫^𝑇𝑆_𝐷𝑇𝑆𝑣_𝑆−𝑣_𝐶𝑑𝑡for𝐷𝑇_𝑆≤𝑡≤𝑇_𝑆. (2.2) Figures 3(a) and 3(b) show the equivalent circuits at one switching cycle. Define the switching function 𝑆_𝑛 of each switch as 𝑆_𝑛=1whenQ_𝑛isturnedON,0whenQ_𝑛isturnedOFF, (2.3) where 𝑛=1−4 is denoted as the switch number.

(a)

(b)

Figure 3 

Equivalent circuit of APF (a) equivalent circuit when 0<𝑡<𝐷𝑇𝑆 and (b) equivalent circuit when 𝐷𝑇𝑆<𝑡<𝑇𝑆.

Therefore, the state equations of the inductor current and the capacitor voltage are written by Kirchhoff’s laws 𝑖^⋅_𝐿=1𝐿𝑣_𝑆−𝑆𝑣_𝐶,_⋅𝑣_𝐶=1𝐶𝑆𝑖_𝐿−𝑣_𝐶𝑅_𝐿, (2.4) where 𝑆=(𝑆₁−𝑆₂). In the following expressions, 𝑥₁(𝑡) and 𝑥₂(𝑡) represent the state variables of the average values of the inductor current and the capacitor voltage over a switching period, respectively. Substituting 𝑆=1 and 𝑆=−1 into (2.4) yields 𝑥₁(𝑡)=1𝑇_𝑆^{𝑡+𝑇𝑆}_𝑡𝑖_𝐿(𝜏)𝑥𝑑𝜏,₂(𝑡)=1𝑇_𝑆^{𝑡+𝑇𝑆}_𝑡𝑣_𝐶(𝜏)𝑑𝜏. (2.5) The average state-space model of the converter can be written as 𝑑𝑥₁=𝑣𝑑𝑡^𝑆+𝑣_𝐶𝐿𝑣𝑢+^𝑆−𝑣_𝐶𝐿(1−𝑢),𝑑𝑥₂=𝑑𝑡−𝑖_𝐿𝐶−𝑣_𝐶𝑅_𝐿𝐶𝑖𝑢+^𝐿𝐶−𝑣_𝐶𝑅_𝐿𝐶(1−𝑢), (2.6) where 𝑢 is the duty ratio, which can take any value between 0 and 1. Rearranging (2.6) yields 𝑑𝑥₁=()𝑥𝑑𝑡2𝑢−1₂𝐿+𝑣_𝑆𝐿,𝑑𝑥₂=()𝑥𝑑𝑡1−2𝑢₁𝐶−𝑥₂𝑅_𝐿𝐶. (2.7) Consequently, the dynamic behavior of the APF can be described by the following state-space model: 𝑥̇=𝐹𝑥+𝐺𝑥𝑢+𝐸𝑤, (2.8) where 𝑥=𝑖_𝐿𝑣_𝐶^𝑇0,𝐹=−1𝐿1𝐶−1𝑅_𝐿𝐶02,𝐺=−2𝐿𝐶01,𝐸=𝐿0^𝑇,𝑤=𝑣_𝑆. (2.9) Now, consider the bilinear state equation (2.8). If 𝑥=𝑥₀ and 𝑢=𝑢₀ suffice to 𝑓𝑥₀,𝑢₀=𝐹𝑥₀+𝐺𝑥₀𝑢₀+𝐸𝑤(𝑡)=0, (2.10)(𝑥₀,𝑢₀) is called its equilibrium or operating point. Let 0=𝑓𝑥₀,𝑢₀,𝑤,𝑡=𝐹𝑥₀+𝐺𝑥₀𝑢₀+𝐸𝑤(𝑡). (2.11) We can expand the right-hand side of (2.11) into a Taylor series about (𝑥₀,𝑢₀) and then neglect the high-order terms so that 𝑥̇=𝑓𝑥₀,𝑢₀+𝜕𝑓𝜕𝑥^𝑥=𝑥0_𝑢=𝑢0𝑥−𝑥₀+𝜕𝑓𝜕𝑢^𝑥=𝑥0_𝑢=𝑢0𝑢−𝑢₀. (2.12) Moreover, since our interest is on the trajectories near (𝑥₀,𝑢₀), let 𝑥_𝛿=𝑥−𝑥₀, 𝑢_𝛿=𝑢−𝑢₀, and we have the following: 𝑥^⋅_𝛿=𝐹+𝐺𝑢₀𝑥_𝛿+𝐺𝑥₀𝑢_𝛿≡𝐴_𝑝𝑥_𝛿+𝐵_𝑝𝑢_𝛿, (2.13) where 𝐴_𝑝=𝐹+𝐺𝑢₀ and 𝐵_𝑝=𝐺𝑥₀.

The bilinear state equation (2.10) can be approximately described by a linear state equation of the form (2.13).

Considering the equilibrium of bilinear state equation (2.13), the equilibrium values of the inductor current and the capacitor voltage can be obtained as 𝑥₀₂=𝑣_𝑆1−2𝑢₀,𝑥₀₁=𝑥₀₂𝑅_𝐿1−2𝑢₀, (2.14) where 𝑥₀₂ and 𝑥₀₁ are the equilibrium values of 𝑣_𝐶 and 𝑖_𝐿, respectively. The average value of the duty ratio 𝑢₀ can be obtained from (2.15) as 𝑢₀=12𝑣1−^𝑆𝑥₀₂. (2.15) It should be noted that by setting the DC voltage and controlling the duty ratio, we could control the inductor current, and thus the APF can work.

3. Design of Adaptive Sliding Mode Controller

In this section, a detailed study of the shunt APF with parameter uncertainties is proposed. A new adaptive sliding mode control strategy for shunt APF using a proportional sliding surface is proposed, and an adaptive sliding law to overcome the parameter uncertainties is also derived. The block diagram of the designed adaptive sliding mode controller for APF is shown in Figure 4.

Figure 4 

block diagram of adaptive sliding mode control approach.

The goal of APF control is to design an adaptive sliding mode controller so that the APF output trajectory can track the reference model.

Rewriting the linear state equation (2.13), we will obtain the following: 𝑋^•_𝛿=𝐴_𝑃𝑋_𝛿+𝐵_𝑃𝑢_𝛿. (3.1) Consider the system (3.1) with parametric uncertainties, that is, the linearization errors such as high-order terms in Taylor series (2.12), 𝑋^•_𝛿=𝐴_𝑃+Δ𝐴_𝑃𝑋_𝛿(𝑡)+𝐵_𝑃+Δ𝐵_𝑃𝑢_𝛿+𝑓_𝑑, (3.2) where Δ𝐴_𝑃 is the unknown parameter uncertainties of the matrix 𝐴_𝑃, Δ𝐵_𝑃 is the unknown parameter uncertainties of the matrix 𝐵_𝑃, and 𝑓_𝑑 is external disturbances such as nonlinear loads in the APF.

The reference model is defined as 𝑋^•_𝑚=𝐴_𝑚𝑋_𝑚+𝐵_𝑚𝑟, (3.3) where 𝐴_𝑚, 𝐵_𝑚 are known constant matrices of a reference model and 𝑟 is reference input.

We make the following assumptions.

Assumption 3.1. There exist unknown matrices of appropriate dimensions 𝐷, 𝐺 such that Δ𝐴_𝑃(𝑡)=𝐵_𝑃𝐷(𝑡)+Δ𝐴_𝑃(𝑡), Δ𝐵_𝑃(𝑡)=𝐵_𝑃𝐺(𝑡)+Δ𝐵_𝑃(𝑡), where 𝐵_𝑃𝐷(𝑡) and 𝐵_𝑃𝐺(𝑡) are matched uncertainty, Δ𝐴_𝑃(𝑡) and Δ𝐵_𝑃(𝑡) are unmatched uncertainty. From this assumptions, (3.2) can be rewritten as 𝑋^•_𝛿=𝐴_𝑃𝑋_𝛿(𝑡)+𝐵_𝑃𝑢_𝛿+𝐵_𝑃𝑓_𝑚+𝑓_𝑢, (3.4) where 𝑓_𝑚(𝑡,𝑋_𝛿,𝑢_𝛿) represents the matched lumped uncertainty and 𝑓_𝑢(𝑡,𝑋_𝛿,𝑢_𝛿) represents the unmatched lumped uncertainty, respectively, which is given by 𝑓_𝑚𝑡,𝑋_𝛿,𝑢_𝛿=𝐷(𝑡)𝑋_𝛿(𝑡)+𝐺(𝑡)𝑢_𝛿,𝑓_𝑢𝑡,𝑋_𝛿,𝑢_𝛿=Δ𝐴_𝑃(𝑡)𝑋_𝛿(𝑡)+Δ𝐵_𝑃(𝑡)𝑢_𝛿. (3.5)

Assumption 3.2. The matched and unmatched lumped uncertainty 𝑓_𝑚 and 𝑓_𝑢 are bounded such as ‖𝑓_𝑚(𝑡,𝑋_𝛿,𝑢_𝛿)‖≤𝛼_𝑚 and ‖𝑓_𝑢(𝑡,𝑋_𝛿,𝑢_𝛿)‖≤𝛼_𝑢, where 𝛼_𝑚, 𝛼_𝑢 are known positive constants.

Assumption 3.3. There exist constant matrices 𝐾^∗ and 𝜃^∗ such that the following matching conditions 𝐴_𝑃+𝐵_𝑃𝐾^∗𝑇=𝐴_𝑚 and 𝐵_𝑃𝜃^∗𝑇=𝐵_𝑚 can always be satisfied.
Then, the adaptive sliding mode controller will be designed.

The tracking error and its derivative are 𝑒=𝑋_𝛿−𝑋_𝑚,_•𝑒=𝐴_𝑚𝑒+𝐴_𝑃−𝐴_𝑚𝑋_𝛿+𝐵_𝑃𝑢_𝛿−𝐵_𝑚𝑟+𝐵_𝑃𝑓_𝑚+𝑓_𝑢. (3.6)

The proportional sliding surface is defined as 𝑠(𝑡)=𝜆𝑒, (3.7) where 𝜆 is a constant matrix satisfying that 𝜆𝐵_𝑃 is a nonsingular diagonal matrix.

The derivative of the sliding surface is 𝑠^•=𝜆𝑒^•=𝜆𝐴_𝑚𝑒+𝜆𝐴_𝑃−𝐴_𝑚𝑋_𝛿+𝜆𝐵_𝑃𝑢_𝛿−𝜆𝐵_𝑚𝑟+𝜆𝐵_𝑃𝑓_𝑚+𝜆𝑓_𝑢. (3.8)

Setting 𝑠^•=0 to solve equivalent control 𝑢_𝑒𝑞 gives 𝑢_𝑒𝑞=−𝜆𝐵_𝑃⁻¹𝜆𝐴_𝑚𝑒−𝜆𝐵_𝑃⁻¹𝜆𝐴_𝑃−𝐴_𝑚𝑋_𝛿+𝜆𝐵_𝑃⁻¹𝜆𝐵_𝑚𝑟−𝑓_𝑚−𝜆𝐵_𝑃⁻¹𝜆𝑓. (3.9)

From Assumption 3.3 and (3.9) can be rewritten as 𝑢_𝑒𝑞=−𝜆𝐵_𝑃⁻¹𝜆𝐴_𝑚𝑒+𝐾^∗𝑇𝑋_𝛿+𝜃^∗𝑇𝑟−𝑓_𝑚−𝜆𝐵_𝑃⁻¹𝜆𝑓. (3.10)

Since the equivalent control 𝑢_𝑒𝑞 (3.10) gives the controller structure of what we should propose and the unknown disturbance term 𝑓_𝑚 and 𝑓_𝑢 can be dealt with the sliding mode terms, the control signal 𝑢_𝛿 can be proposed as 𝑢_𝛿=−𝜆𝐵_𝑃⁻¹𝜆𝐴_𝑚𝑒+𝐾^∗𝑇𝑋_𝛿+𝜃^∗𝑇𝑟−𝜌𝜆𝐵_𝑃⁻¹𝑠‖𝑠‖, (3.11) where 𝜌 is constant, 𝑠/‖𝑠‖ is the sliding mode unit control signal.

The adaptive sliding mode version of control input is 𝑢_𝛿(𝑡)=−𝜆𝐵_𝑃⁻¹𝜆𝐴_𝑚𝑒(𝑡)+𝐾^𝑇(𝑡)𝑋_𝛿(𝑡)+𝜃^𝑇(𝑡)𝑟(𝑡)−𝜌𝜆𝐵_𝑃⁻¹𝑠‖𝑠‖, (3.12) where 𝐾(𝑡) is the estimate of 𝐾^∗, 𝜃(𝑡) is the estimate of 𝜃^∗.

Define the estimation error as 𝐾(𝑡)=𝐾(𝑡)−𝐾^∗,𝜃(𝑡)=𝜃(𝑡)−𝜃^∗. (3.13) Substituting (3.13) into (3.12) yields 𝑢_𝛿(𝑡)=−𝜆𝐵_𝑃⁻¹𝜆𝐴_𝑚𝑒(𝑡)+𝐾^𝑇(𝑡)+𝐾^∗𝑇𝑋_𝛿(𝑡)+𝜃^𝑇(𝑡)+𝜃^∗𝑇𝑟(𝑡)−𝜌𝜆𝐵_𝑃⁻¹𝑠‖𝑠‖. (3.14) From (3.14) and Assumption 3.3, rewrite (3.3) as follows 𝑋^•_𝛿(𝑡)=−𝐵_𝑃𝜆𝐵_𝑃¹𝜆𝐴_𝑚𝑒(𝑡)+𝐵_𝑃𝐾^𝑇(𝑡)𝑋_𝛿(𝑡)+𝐴_𝑚𝑋_𝛿(𝑡)+𝐵_𝑃𝜃^𝑇(𝑡)𝑟(𝑡)+𝐵_𝑚𝑟(𝑡)−𝐵_𝑃𝜌𝜆𝐵_𝑃⁻¹𝑠‖𝑠‖+𝐵_𝑃𝑓_𝑚+𝑓_𝑢. (3.15) Then, we have the derivative of the tracking error equation: 𝑒^•(𝑡)=−𝐵_𝑃𝜆𝐵_𝑃⁻¹𝜆𝐴_𝑚𝑒(𝑡)+𝐵_𝑃𝐾^𝑇(𝑡)𝑋_𝛿(𝑡)+𝐵_𝑃𝜃^𝑇(𝑡)𝑟(𝑡)+𝐴_𝑚𝑒(𝑡)−𝐵_𝑃𝜌𝜆𝐵_𝑃⁻¹𝑠‖𝑠‖+𝐵_𝑃𝑓_𝑚+𝑓_𝑢=𝐼−𝐵_𝑃𝜆𝐵_𝑃⁻¹𝜆𝐴_𝑚𝑒(𝑡)+𝐵_𝑃𝐾^𝑇(𝑡)𝑋_𝛿(𝑡)+𝐵_𝑃𝜃^𝑇(𝑡)𝑟(𝑡)−𝐵_𝑃𝜌𝜆𝐵_𝑃⁻¹𝑠‖𝑠‖+𝐵_𝑃𝑓_𝑚+𝑓_𝑢, (3.16) and the derivative of 𝑠(𝑡) is 𝑠^•(𝑡)=𝜆𝐵_𝑃𝐾^𝑇(𝑡)𝑋_𝛿(𝑡)+𝜆𝐵_𝑃𝜃^𝑇(𝑡)𝑟(𝑡)𝑠−𝜌‖𝑠‖+𝜆𝐵_𝑃𝑓_𝑚+𝜆𝑓_𝑢. (3.17) Define a Lyapunov function 1𝑉=2𝑠^𝑇1𝑠+2tr𝐾𝑀⁻¹𝐾^𝑇+12tr𝜃𝑁⁻¹𝜃^𝑇, (3.18) where 𝑀, 𝑁 are positive definite matrix, tr denoting the trace of a square matrix.

Differentiating 𝑉 with respect to time yields 𝑉^•=𝑠𝑠^𝑇•+tr𝐾𝑀^•−1𝐾^𝑇+tr𝜃𝑁^•−1𝜃^𝑇=−𝜌‖𝑠‖+𝑠^𝑇𝜆𝐵_𝑃𝑓_𝑚+𝑠^𝑇𝜆𝑓_𝑢+𝑠^𝑇𝜆𝐵_𝑃𝐾^𝑇(𝑡)𝑋_𝛿(𝑡)+tr𝐾𝑀^•−1𝐾^𝑇+𝑠^𝑇𝜆𝐵_𝑃𝜃^𝑇(𝑡)𝑟(𝑡)+tr𝜃𝑁^•−1𝜃^𝑇. (3.19) For 𝑠^𝑇𝜆𝐵_𝑃𝐾^𝑇(𝑡)𝑋_𝛿(𝑡) and 𝑠^𝑇𝜆𝐵_𝑃𝜃^𝑇(𝑡)𝑟(𝑡) are scalar, from the properties of matrix trace 𝑥^𝑇𝐴𝑥=tr(𝑥𝑥^𝑇𝐴), tr(𝐴)=tr(𝐴^𝑇) there have 𝑠^𝑇𝜆𝐵_𝑃𝐾^𝑇(𝑡)𝑋_𝛿(𝑡)=tr𝑋_𝛿(𝑡)𝑠^𝑇𝜆𝐵_𝑃𝐾^𝑇(𝑡)=tr𝐾(𝑡)𝐵_𝑃^𝑇𝜆^𝑇𝑠𝑋_𝛿^𝑇(𝑡,𝑠_𝑇)𝜆𝐵_𝑃𝜃^𝑇(𝑡)𝑟(𝑡)=tr𝑟(𝑡)𝑠^𝑇𝜆𝐵_𝑃𝜃^𝑇(𝑡)=tr𝜃(𝑡)𝐵_𝑃^𝑇𝜆^𝑇𝑠𝑟^𝑇(𝑡). (3.20) To make 𝑉^•≤0, we choose the adaptive laws as 𝐾^•𝑇(𝑡)=𝐾^•𝑇(𝑡)=−𝑀𝐵_𝑃^𝑇𝜆^𝑇𝑠𝑋_𝛿^𝑇(𝑡),_•𝜃^𝑇(𝑡)=𝜃^•𝑇(𝑡)=−𝑁𝐵_𝑃^𝑇𝜆^𝑇𝑠𝑟^𝑇(𝑡). (3.21) This adaptive sliding law yields 𝑉̇=−𝜌‖𝑠‖+𝑠^𝑇𝜆𝐵_𝑃𝑓_𝑚+𝑠^𝑇𝜆𝑓_𝑢≤−𝜌‖𝑠‖+‖𝑠‖𝜆𝐵_𝑃𝑓_𝑚+‖𝑠‖‖𝜆‖𝑓_𝑢≤−𝜌‖𝑠‖+‖𝑠‖𝜆𝐵_𝑃𝛼_𝑚+‖𝑠‖‖𝜆‖𝛼_𝑢≤−‖𝑠‖𝜌−𝜆𝐵_𝑃𝛼_𝑚−‖𝜆‖𝛼_𝑢≤0, (3.22) with 𝜌≥‖𝜆𝐵_𝑃‖𝛼_𝑚+‖𝜆‖𝛼_𝑢+𝜂, where 𝜂 is a positive constant and 𝑉^⋅ becomes negative semi-definite, that is, 𝑉^⋅≤−𝜂‖𝑠‖. According to Barbalat’s lemma [18], it can be proved that 𝑠(𝑡) will asymptotically converge to zero, lim_𝑡→∞𝑠(𝑡)=0. Consequently, 𝑒(𝑡) will asymptotically converge to zero, lim_𝑡→∞𝑒(𝑡)=0.

4. Simulation Analysis

In this section, the single-phase shunt APF using adaptive sliding mode control is implemented with MATLAB/SIMPOWER Toolbox. The goal of adaptive sliding mode control is to make the APF output current track the detected harmonic current. Simulations on APF system will verify the control effects of the proposed adaptive sliding approach.

First of all, the nonlinear load of simulation model is described. Nonlinear load: a load branch is rectifier bridge connecting parallel RC load, 𝑅=15 Ω, 𝐶=5𝑒−3𝐹. When 0–0.4 s, a load works, the total harmonic distortion (THD) of nonlinear load current is 45.82%; When 0.4–0.8 s, the breaker is switched on and another load branch which is the same as first one is injected, accompanied by a disturbance load (interference frequency 𝑓=1000 Hz, square wave 𝑇=0.001 s, pulse width 50%, parallel disturbance load 𝐿=0.2 H, and 𝑅=20 Ω), and the total harmonic distortion (THD) of nonlinear load current is 40.12%. The current mainly contains 3rd and 7th times, odd harmonic. The parameters of the proposed adaptive sliding mode controller are chosen as follows: 𝐴_𝑀=[−49.6−351.8;5190.21], 𝐵_𝑀=[7400;−8.6], 𝐶_𝑀=[10;01], 𝐷_𝑀=[0;0], 𝑀=5𝑒−7, 𝑁=5𝑒−5, 𝜌=200, and 𝜆=[0.040.05]. As DC voltage 𝑉_𝐶=600 V, it can be calculated that 𝑥₀=[0.11573;600], 𝑢₀=0.24077. According to the APF circuit model, it is calculated that 𝐴_𝑝=[0−86.4;518.5−0.1], 𝐵_𝑝=[200000;−231.5].

Because the rectifier bridge and nonlinear load result in harmonic and reactive currents, serious distortions of the circuit current show up. Figure 5 draws the current waveform before and after the nonlinear load change. Mutation distortions of current have serious effects on the power system and other electrical equipments that should be compensated and eliminated.

Figure 5 

Nonlinear load current (before and after load change).

From the adaptive harmonic detection output 𝑖_ℎ and the APF current 𝑖_𝐿 obtained from measuring module, we can get the input signal of adaptive sliding mode controller (which is also the reference model input signal) 𝑟=𝑖_ℎ−𝑖_𝐿+𝑖_PI, where 𝑖_PI is the compensation current of voltage PI control for APF DC side. Single-phase harmonic detection method is used, where the input has only sine wave signal, implying that the detection result 𝑖_ℎ contains the harmonic and reactive current.

According to the approximation process of APF dynamic model, near the equilibrium point (𝑥₀,𝑢₀), the simulation variables 𝑥_𝛿=𝑥−𝑥₀, 𝑢_𝛿=𝑢−𝑢₀, so we can get 𝑥_𝛿, 𝑢 from 𝑥_𝛿=𝑥−𝑥₀, 𝑢=𝑢_𝛿+𝑢₀, respectively.

The input waveform of adaptive sliding mode controller (i.e., input waveform of the reference model) is drawn in Figure 6. The controller not only can compensate and eliminate the harmonic, but also can compensate the DC voltage and make it stay stable in /near a predetermined value to improve the compensation effect.

Figure 6 

Input of adaptive sliding mode controller.

The output of the adaptive sliding mode controller is shown in Figure 7. The output 𝑢 of the controller compares with triangular waves which have similar amplitude with 𝑢, then PWM generator output is the standard of four-phase control pulse and can fully reflect the control function of controller output to the APF. PWM signal is generated using triangular wave comparison method, where the triangular wave amplitude is ±1 and frequency is 1000 Hz.

Figure 7 

Output of adaptive sliding mode controller.

By controlling on or off of the four thyristors in the full-controlled bridge, four phase PWM pulse control signal will control the charging and discharging process of the DC side capacitor 𝐶, thereby producing the inductor current 𝑖_𝐿, that is, APF compensating current. The parameters of APF main circuit: 𝐿=0.006 H inductor, capacitor 𝐶=0.001 F, 𝑅=10 KΩ, thyristors using the IGBT/Diode module, a default parameter.

Figure 8 compares between the harmonic and APF current, where the red curve represents APF output current (inductor current) and blue curve represents detected harmonic current. Compared with the detected harmonic current, the APF output current has equal amplitude but opposite direction. The APF output current is injected into the nonlinear load current, then the harmonics and reactive current caused by nonlinear loads can be eliminated, thus the harmonic current compensation can be achieved. After 0.4 s, the load and interference are increased, and the APF current still can quickly track the harmonic current although there may have some errors between harmonic current and the APF output current even if the APF works stable. The errors mainly contain compensation current from PI control circuit of the DC voltage for the DC voltage compensation of APF.

Figure 8 

Harmonic and APF current.

Simulation of the PI control signal is shown in Figure 9. The initial moment due to the DC voltage is zero, and the output of PI controller is a big compensation signal, so that the DC voltage can quickly arrive at the setting value. After 0.4 s, the load and interference are increased, the harmonic is increased, and the DC voltage is reduced; the PI control output could adjust in time, then quickly recover normal working state.

Figure 9 

Output of DC voltage PI control.

Figure 10 shows that, after the circuit starts to work, the APF capacitor voltage quickly rises to more than 500 V and basically stays at a setting value near 600 V, rapidly adjusted by the PI control of DC voltage. The load changed at 0.4 s, and the DC voltage could be quickly stabilized to the set value also.

Figure 10 

APF DC voltage.

Figure 10 plots the power current after APF has been compensated, where the Figure 11(b) zooms in on 0.4 s when the loads change. It can be seen from Figure 11 that the current waveform has been improved and adjusted rapidly to approximate standard sine wave within 0.08 s after the circuit started to work and load changed. Compared with the nonlinear load current waveform in Figure 5, it can be seen that before compensation the load current has serious distortion and after compensation the current quality has been obviously improved. Thus, the proposed purposes of nonlinear load current compensation through APF can be successfully achieved.

(a)

(b)

Figure 11 

(a) Power current (0–0.8 s). (b) Power current (0.35–0.55 s) (nonlinear load changes in 0.4 seconds).

Through the FFT Analysis tools of SIMLINK powergui module, the load current and the power current are acquired.

Figure 12 is the nonlinear load current waveform and its harmonic analysis. The serious waveform distortions can be observed. FFT analysis shows that nonlinear harmonic current is mainly 3rd and 7th harmonic interference and with a certain degree of 5th and higher harmonic interference. Power current waveform after APF is shown in Figure 13. After compensation, the 3rd harmonic is greatly reduced, the 7th harmonic is almost eliminated, all other higher harmonics are reduced, and the power current is approximate to sine wave. The compensated current THD is 3.5%, more than the national standard level of 5%. After load changed, nonlinear load current THD changed from 45.81% to 40.12% this is due to the percentage of nonlinear part load (bridge, capacitance, and inductance) relative to the linear portion (resistance) decreased. But we can still see the obvious distortion of nonlinear load current.

Figure 12 

Nonlinear load current analysis (before load change).

Figure 13 

Power current after APF compensation (before load changes).

Figure 14 shows the current after load changed, the same bridge and nonlinear load (with certain high frequency interference) paralleled in the circuit. It can be seen that although the nonlinear load and the load current increase, the current waveform distortion is relatively minor as of 40.12% THD. The current waveform after APF compensation is shown in Figure 15. After compensation, the quality of power current is greatly improved with THD reaching at 3.28%; it demonstrates good compensation capability of APF using adaptive sliding mode controller. Because of the increase of load disturbance and the changes of currents, the compensated current has some minor glitches, but the power current quality remains within the national standard.

Figure 14 

Nonlinear load current (after load change).

Figure 15 

Power Current after APF Compensation (after load changes).

The control parameters 𝐾, 𝑒 in the adaptive sliding mode controller are analyzed in the following steps. From Figures 16 and 17, it can be seen that the parameters can be adjusted to a stable value soon and the stability of the control system can be maintained, and the parameters 𝐾 and 𝜃 can converge to constant and stay in stable state within 0.1 s, that is, within 4-5 circuit cycles.

(a)

(b)

Figure 16 

Controller parameters 𝐾.

Figure 17 

Controller parameter theta.

The current error between model reference current item and APF inductance current and the voltage error between model reference voltage item and APF DC voltage are drawn in Figures 18 and 19, respectively. The tracking error 𝑒=[^{𝑒𝑖𝑒𝑢}]^𝑇 as parameters of adjustment process exist within the adaptive sliding mode process. It will decrease in a certain way and maintain in a certain range finally. The simulation is based on the actual APF model with nonlinear load and model uncertainties. Because of the complexity of the circuits and the approximation of the APF model, the parameters of APF model have some uncertainties. The tracking error can be quickly adjusted to dozens of magnitude of dimensionless parameters and to be maintained in a certain range.

Figure 18 

Current error between model reference current item and APF inductance current.

Figure 19 

Voltage error between model reference voltage item and APF DC voltage.

It can be observed from Figures 20 and 21 that adaptive sliding mode control has better harmonic compensation performance that adaptive control, the index of THD, has been reduced with the adaptive sliding control.

Figure 20 

APF DC voltage using adaptive control.

Figure 21 

APF DC voltage using adaptive sliding mode control.

Compared the waveforms before and after APF works, it can be seen from Table 1 that the adaptive sliding mode control not only has good compensation effect, but also has quick compensation ability. At the same time, the controller could optimize the adaptive parameters to make it more reasonable and make it possible for the investigation of the intelligent control. Simulation and analysis verified the feasibility of adaptive sliding mode control theory and showed good control effect in single-phase SAPF application.

Table 1 

Comparison of THD after compensation.

5. Conclusion

This paper studied the principle and dynamic model of single-phase shunt APF and proposed a new adaptive sliding mode control algorithm. The simulation results proved that for nonlinear load current the adaptive sliding mode controller has successful compensation effect; that is, it can compensate the most harmonic current and eliminate certain reactive current, which can recover sine wave from severely detuned current waveform and improve the power factor. From Figure 9, we can see that the APF current can quickly track the harmonic current, thus to achieve the harmonic compensation. The reference currents tracking behavior has been improved, and the power supply harmonic current has been reduced with novel adaptive sliding mode control. The proposed control system has the satisfactory adaptive and robust ability in the presence of the changing disturbances and nonlinear loads.

Acknowledgments

The authors thank the anonymous reviewer for useful comments that improved the quality of the paper. This work is partially supported by National Science Foundation of China under Grant no. 61074056, Natural Science Foundation of Jiangsu Province under Grant no. BK2010201, Scientific Research Foundation of High-Level Innovation, and Entrepreneurship Plan of Jiangsu Province. They also would like to thank the Fundamental Research Funds for the Central Universities under Grant no. 2012B06714.