T-S fuzzy model based adaptive fuzzy current tracking control of three-phase active power filter

Abstract

This paper proposed an adaptive fuzzy state feedback controller for a three-phase active power filter (APF) based on a Takagi-Sugeno (T-S) fuzzy model. The T-S fuzzy model is based on the APF dynamic model in the dq frame, which ensures that the current control dynamic and DC capacitor voltage dynamic become decoupled. By using the parallel distributed compensation (PDC), local linear state feedback controller could be designed for each T-S fuzzy sub-model. To deal with the parametric uncertainties and external disturbances, a parameter estimation scheme for tuning the parameters of T-S fuzzy model is designed and the parameters are adjusted online by the adaptive law derived from the Lyapunov stability analysis. Simulation studies demonstrate that the proposed control method exhibit excellent performance in both steady state and transient operation during balanced and unbalanced load, and source voltage.

Keywords

T-S fuzzy active power filter parameter estimation Lyapunov stability analysis

1 Introduction

In recent years, the distortion of power quality has become a serious problem with the increasing number of nonlinear loads in electrical power systems. With the development of high speed power switching devices, shunt active power filters (SAPF) instead of passive power filter become main harmonic mitigating tools because they can efficiently eliminate both current distortion and reactive power. The APF operates by injecting compensation current which is of the same magnitudes and opposite phases with the harmonic currents into the power system to eliminate harmonic contamination and improve the power factor. The performance of the APF is affected significantly by the selection of control techniques. Therefore, the choice of the control technique is very important for the achievement of a satisfactory APF performance. Various control methods have been commonly discussed to ensure expected performance. They can be classified into three main categories: reference generation, compensation current tracking control, DC capacitor voltage control [1]. Compared with conventional current control methods including hysteresis control, single cycle control, space vector control, and repetitive control, many new control strategies have been designed to improve the power quality, such as fuzzy control, neural network control, sliding mode control, and adaptive control due to their ability to handle complex problem at difficult situations. Shyu et al. [2] introduced a model reference adaptive control (MRAC) to improve line power factor and to reduce line current harmonics for a single-phase SAPF. Radzi et al. [3] proposed a combination of neural network and a bandless hysteresis controller for a switched capacitor active power filter to reduce line current harmonics. Luo et al. [4] designed a hybrid active power filter and proposed an adaptive fuzzy dividing frequency-control method for adjusting proportional-integral coefficients timely. Ventutini et al. [5] presented an adaptive current controller for active power filters to compensate selected load current harmonics. Fei et al. [6] proposed an adaptive fuzzy-sliding control to improve the dynamic performance of the three-phase active power filter.

It has been shown that the fuzzy-model-based representation proposed by Takagi and Sugeno [7], known as the T-S fuzzy model, is a successful scheme for coping with the nonlinear control system, and there have recently been many successful applications of the TS fuzzy model based approach to nonlinear control system [8 –11]. Unlike conventional modeling approaches, where a single model is utilized to express a overall nonlinear system, the T-S fuzzy modeling approach can be utilized as multi-model approach, in which a group of sub-models is designed to express the global behavior of the nonlinear control system. Each fuzzy rule in the T-S fuzzy control system can express a local linear model. Hence, the advantage of the controller synthesis for such a T-S fuzzy model is that the linear control methods can be utilized. In fact, in many cases, it is very difficult to obtain the accurate values of some system parameters because of parametric uncertainties and external disturbances. Moreover, the parametric uncertainties may destabilize the T-S fuzzy model based control system [12]. So an appropriate adaptive law is significant in estimating the parameters in the T-S fuzzy model as its parameters are unknown or varying because of the parametric uncertainties and external disturbances. Hence, the parametric estimation for the T-S fuzzy model is essential to the design of adaptive fuzzy controller [13, 14]. Since there exist system nonlinearities in APF system, it is necessary to utilize adaptive fuzzy controller to APF and use T-S model to represent such system nonlinearities. T-S fuzzy applied into APF have been presented for load compensation [15, 16], but they both focus on DC capacitor voltage control. Moreover, the proposed T-S fuzzy controller cannot guarantee the Lyapunov stability of the close-loop system [15], and they did not take into account that parametric uncertainties may destabilize the T-S fuzzy model based control system [16]. Thus it is necessary to adopt the adaptive fuzzy control scheme to approximate the nonlinear system and compensate parametric uncertainties and external disturbances in the control of APF using T-S fuzzy model for improving the tracking and compensation performance.

In this paper, the Lyapunov-based robust adaptive fuzzy control strategy is applied to the current tracking compensation of APF dynamic system using T-S fuzzy model. The paper integrates adaptive control and the nonlinear approximation of fuzzy control with T-S fuzzy model. Moreover, the proposed adaptive fuzzy T-S controller can guarantee the Lyapunov stability of the close-loop system and improve the current tracking performance in the presence of model uncertainties and external disturbances. The motivation of the control strategy proposed here can be emphasized as:

T-S fuzzy modeling approach is applied to a three-phase active power filter for current tracking control, and few works exploited this control technique in active power filter before. In this control technique, each fuzzy rule can be thought as a linear dynamic model, and all fuzzy rules combine the overall fuzzy model. This makes it possible that the linear control methods can be used in the APF system. So the proposed control scheme has important theoretical and practical significance for promoting the application of APF, improving total harmonic distortion (THD) and strengthening the quality of power supply.

There are no relevant research works which combine T-S fuzzy control, model reference adaptive control and apply these control methods to APF so far. This paper combines the model reference adaptive control, T-S fuzzy control and online parameter estimation. T-S Fuzzy controller is proposed to transform APF nonlinear system into linear dynamic model which may simplify the design of the controller. The choice of reference model can impose desired dynamic response to improve the effect of the controller. Online parameter estimation derived by solving the Lyaponov function which can guarantee the stability of the closed-loop system is employed to compensate the parametric uncertainties. Combination of these methods has a general sense and can be extended to other power electronic converter topology.

2 Principle of active power filter

The block diagram for three-phase shunt active power filter is shown in Fig. 1. The APF contains three sections, harmonic current detection module, control system and main circuit. The rapid detection of harmonic current based on instantaneous reactive power theory is most widely used in harmonic current detection module. The control system can be divided into two separate parts, namely the current control system to ensure the precise tracking of the reference current and the DC voltage regulator to achieve power balance between the DC side and AC side by regulating the DC voltage to its reference value. The main circuit which consists of power switching devices produces compensation currents according to the control signal from the control system.

The dynamic model of APF is proposed in the following steps. Applying Kirchhoff rules to this system yields the following circuit equations: ${\begin{matrix} v_{1} = L_{c} \frac{{di}_{1}}{dt} + R_{c} i_{1} + v_{1 M} + v_{MN} \\ v_{2} = L_{c} \frac{{di}_{2}}{dt} + R_{c} i_{2} + v_{2 M} + v_{MN} \\ v_{3} = L_{c} \frac{{di}_{3}}{dt} + R_{c} i_{3} + v_{3 M} + v_{MN} \end{matrix}$ (1)

The parameter of L_c and R_c are the inductance and resistance of the APF respectively, ν_MN is the voltage between point M and N.

By summing the three equations in Equation (1), taking into account the absence of the zero-sequence in the three wire system currents, and assuming that the AC supply voltages are balanced, one can obtain: $v_{MN} = - \frac{1}{3} \sum_{m = 1}^{3} v_{mM}$ (2)

The switching function c_k denotes the ON/OFF status of the devices in the two legs of the IGBT Bridge. c_k is defined as: $c_{k} = {\begin{matrix} 1, if S_{k} is On and S_{k + 3} is Off \\ 0, if S_{k} is Off and S_{k + 3} is On \end{matrix}$ (3)

Where k = 1, 2, 3.

Taking into account the relation v_kM = c_kv_dc, then Equation (1) becomes ${\begin{matrix} \frac{{di}_{1}}{dt} = - \frac{R_{c}}{L_{c}} i_{1} + \frac{v_{1}}{L_{c}} - \frac{v_{dc}}{L_{c}} (c_{1} - \frac{1}{3} \sum_{m = 1}^{3} c_{m}) \\ \frac{{di}_{2}}{dt} = - \frac{R_{c}}{L_{c}} i_{2} + \frac{v_{2}}{L_{c}} - \frac{v_{dc}}{L_{c}} (c_{2} - \frac{1}{3} \sum_{m = 1}^{3} c_{m}) \\ \frac{{di}_{3}}{dt} = - \frac{R_{c}}{L_{c}} i_{3} + \frac{v_{3}}{L_{c}} - \frac{v_{dc}}{L_{c}} (c_{3} - \frac{1}{3} \sum_{m = 1}^{3} c_{m}) \end{matrix}$ (4)

Then we define d_nk called the switching state function as $d_{nk} = {(c_{k} - \frac{1}{3} \sum_{m = 1}^{3} c_{m})}_{n}$ (5)

Equation (5) shows that d_nk depends on the switching function c_k. Moreover, based on Equation (5) and for the eight permissible switching states of the IGBT, one can obtain that $[\begin{matrix} d_{n 1} \\ d_{n 2} \\ d_{n 3} \end{matrix}] = \frac{1}{3} [\begin{matrix} 2 & - 1 & - 1 \\ - 1 & 2 & - 1 \\ - 1 & - 1 & 2 \end{matrix}] [\begin{matrix} c_{1} \\ c_{2} \\ c_{3} \end{matrix}]$ (6)

On the other hand, the following equation can be obtained for the DC side: $\frac{{dv}_{dc}}{dt} = \frac{1}{C} i_{dc} = \frac{1}{C} \sum_{m = 1}^{3} c_{m} i_{m}$ (7)

And it can be verified that $\sum_{m = 1}^{3} d_{nm} i_{m} = \sum_{m = 1}^{3} c_{m} i_{m}$ , so Equation (7) can be changed into: $\frac{{dv}_{dc}}{dt} = \frac{1}{C} \sum_{m = 1}^{3} d_{nm} i_{m}$ .

By using i₁ + i₂ + i₃ = 0, the following equation can be obtained: $\frac{{dv}_{dc}}{dt} = \frac{1}{C} (2 d_{n 1} + d_{n 2}) i_{1} + \frac{1}{C} (d_{n 1} + 2 d_{n 2}) i_{2}$ (8)

So, the model of APF in the ‘abc’ frame is given as follows: ${\begin{matrix} \frac{{di}_{1}}{dt} = - \frac{R_{c}}{L_{c}} i_{1} + \frac{v_{1}}{L_{c}} - \frac{v_{dc}}{L_{c}} d_{n 1} \\ \frac{{di}_{2}}{dt} = - \frac{R_{c}}{L_{c}} i_{2} + \frac{v_{2}}{L_{c}} - \frac{v_{dc}}{L_{c}} d_{n 2} \\ \frac{{dv}_{dc}}{dt} = \frac{1}{C} (2 d_{n 1} + d_{n 2}) i_{1} + \frac{1}{C} (d_{n 1} + 2 d_{n 2}) i_{2} \end{matrix}$ (9)

The model can be transformed into the synchronous ‘dq’ frame, which is given as: ${\begin{matrix} \frac{{di}_{d}}{dt} = - \frac{R_{c}}{L_{c}} i_{d} + \frac{v_{d}}{L_{c}} + ω i_{q} - \frac{v_{dc}}{L_{c}} d_{nd} \\ \frac{{di}_{q}}{dt} = - \frac{R_{c}}{L_{c}} i_{q} + \frac{v_{q}}{L_{c}} - ω i_{d} - \frac{v_{dc}}{L_{c}} d_{nq} \\ \frac{{dv}_{dc}}{dt} = \frac{1}{C} d_{nd} i_{d} + \frac{1}{C} d_{nq} i_{q} \end{matrix}$ (10)

The principle of operation of APF requires that the state variables must be controlled respectively, but there are only two independent inputs. One way to overcome this difficulty is to divide the model into two separate loops, namely the inner current dynamic and outer DC capacitor voltage dynamic. The interaction between the two loops can be avoided by adequately separating their respective dynamics. And the model represented by Equation (10) is nonlinear due to the existence of multiplication terms between the state variables {i_d, i_q, v_dc} and the inputs {d_nd, d_nq}. So it is necessary to utilize the T-S fuzzy based modeling approach to represent the nonlinear APF system so that the linear control approach which is simple design and implement can be used.

3 T-S fuzzy model based adaptive fuzzy controller

In this section, the T-S fuzzy model for APF is designed based on the nonlinear dynamic model. Moreover, by using PDC, the local state linear feedback controller is derived for each sub-model to make it possible that T-S fuzzy model has the same dynamic response with the reference model.

In general, DC capacitor voltage v_dc always equal to reference voltage by proper control strategy and the change of DC capacitor voltage is far less than the change of compensation current. So v_dc can be seen as constant in the design of current control system. In order to design the current controller, the first two equations in Equation (10) are considered. ${\begin{matrix} \frac{{di}_{d}}{dt} = - \frac{R_{c}}{L_{c}} i_{d} + \frac{v_{d}}{L_{c}} + ω i_{q} - \frac{v_{dc}}{L_{c}} d_{nd} \\ \frac{{di}_{q}}{dt} = - \frac{R_{c}}{L_{c}} i_{q} + \frac{v_{q}}{L_{c}} - ω i_{d} - \frac{v_{dc}}{L_{c}} d_{nq} \end{matrix}$ (11)

Define state $x = {[\begin{matrix} x_{1} & x_{2} \end{matrix}]}^{T} = {[\begin{matrix} i_{d} & i_{q} \end{matrix}]}^{T}$ and input $u = {[\begin{matrix} u_{1} & u_{2} \end{matrix}]}^{T} = {[\begin{matrix} d_{nd} & d_{nq} \end{matrix}]}^{T}$ , and based on (11), the T-S fuzzy model for APF can be established, which has the following form: $\begin{matrix} Rule i : IF x_{1} is about M_{i 1} and x_{2} is about M_{i 2}, \\ THEN \dot{x} = A_{i} x + B_{i} u i = 1, 2, 3 \end{matrix}$

The defuzzification fuzzy dynamic system can be expressed using center-average defuzzifier, product inference and singleton fuzzifier $\dot{x} = \frac{\sum_{i = 1}^{3} μ_{i} (η) [A_{i} x + B_{i} u]}{\sum_{i = 1}^{3} μ_{i} (η)}$ (12)

Where $A_{i} = [\begin{matrix} a_{11}^{i} & a_{12}^{i} \\ a_{21}^{i} & a_{22}^{i} \end{matrix}]$ , $B_{i} = [\begin{matrix} b_{11}^{i} & b_{12}^{i} \\ b_{21}^{i} & b_{22}^{i} \end{matrix}]$ , μ_i (η) = μ_i1 (x₁) μ_i2 (x₂), μ_i1 (x₁) and μ_i2 (x₂) are membership function values of the fuzzy variable x₁ and x₂ with respect to fuzzy set M_i1 and M_i2. As can be seen from Equation (11), there always exist $b_{11}^{i} = b_{22}^{i}, b_{12}^{i} = b_{21}^{i} = 0$ . For the sake of convenience in presentations, define $b_{11}^{i} = b_{22}^{i} = b^{i}, b_{12}^{i} = b_{21}^{i} = 0$ .

According to PDC, the local state linear feedback controllers for each linear sub-model can be designed. The fuzzy control rules have following forms: $\begin{matrix} Rule i : IF x_{1} is about M_{i 1} and x_{2} is about M_{i 2}, \\ THEN u = - K_{i} x + l_{i} r i = 1, 2, 3 \end{matrix}$

The controller can be expressed using center-average defuzzifier, product inference and singleton fuzzifier $u = \frac{\sum_{i = 1}^{3} μ_{i} (η) [- K_{i} x + l_{i} r]}{\sum_{i = 1}^{3} μ_{i} (η) b^{i}}$ (13)

Where K_i = A_i - A_m, A_m is based on the reference model which has the desired dynamic response: ${\dot{x}}_{m} = A_{m} x + B_{m} r$ (14)

Where $A_{m} = [\begin{matrix} a_{11}^{m} & a_{12}^{m} \\ a_{21}^{m} & a_{22}^{m} \end{matrix}]$ , $B_{m} = [\begin{matrix} b_{11}^{m} & b_{12}^{m} \\ b_{21}^{m} & b_{22}^{m} \end{matrix}]$ .

By using the controller Equation (13), the T-S fuzzy model Equation (12) can be transformed into the desire reference model Equation (14).

4 Parameter estimation

In the previous section, the T-S fuzzy model for APF and local linear state feedback controllers based on PDC are presented. Based on the analysis, in this section, the parametric estimation is applied to the adaptive fuzzy control taking into account that the parametric uncertainties and external disturbances always exist in practical application. The overall control structure is shown in Fig. 2. The detailed design procedure can be described in the followingsteps.

By using an appropriate adaptive law to estimate the parameters in the T-S fuzzy model as its parameters are unknown or varying because of the parametric uncertainties and external disturbances, the controller in Equation (14) is changed into: $u = \frac{\sum_{i = 1}^{3} μ_{i} (η) [- ({\hat{A}}_{i} - A_{m}) x + l_{i} r]}{\sum_{i = 1}^{3} μ_{i} (η) {\hat{b}}^{i}}$ (15)

Where ${\hat{A}}_{i} = [\begin{matrix} {\hat{a}}_{11}^{i} & {\hat{a}}_{12}^{i} \\ {\hat{a}}_{21}^{i} & {\hat{a}}_{22}^{i} \end{matrix}]$ , ${\hat{B}}_{i} = [\begin{matrix} {\hat{b}}^{i} & 0 \\ 0 & {\hat{b}}^{i} \end{matrix}]$ , ${\hat{A}}_{i}, {\hat{B}}_{i}$ are the estimates of A_i, B_i which are estimated by the adaptive laws.

To design the adaptive laws for the parameters of the T-S fuzzy model, the T-S fuzzy model can be changed into the following equation by considering the plant parameterization, $\dot{x} = A_{s} x + \frac{\sum_{i = 1}^{3} μ_{i} (η) [(A_{i} - A_{s}) x + B_{i} u]}{\sum_{i = 1}^{3} μ_{i} (η)}$ (16)

Where A_s is an arbitrary stable matrix.

Thus, the estimation model can be defined as: $\dot{ˆ} x = A_{s} \hat{x} + \frac{\sum_{i = 1}^{3} μ_{i} (η) [({\hat{A}}_{i} - A_{s}) x + {\hat{B}}_{i} u]}{\sum_{i = 1}^{3} μ_{i} (η)}$ (17)

Where $\hat{x}$ is the estimate of x.

By defining the estimation error $e = x - \hat{x}$ , one can obtain

$\begin{matrix} \dot{e} & = & A_{s} e - \frac{\sum_{i = 1}^{3} μ_{i} (η) {[\begin{matrix} {\tilde{a}}_{1 i} & {\tilde{a}}_{2 i} \end{matrix}]}^{T} x}{\sum_{i = 1}^{3} μ_{i} (η)} \\ - \frac{\sum_{i = 1}^{3} μ_{i} (η) {[\begin{matrix} {\tilde{b}}_{1 i} & {\tilde{b}}_{2 i} \end{matrix}]}^{T} u}{\sum_{i = 1}^{3} μ_{i} (η)} \end{matrix}$ (18)

Where ${\tilde{A}}_{i} {[\begin{matrix} {\tilde{a}}_{1 i} & \tilde{a} \end{matrix}]}^{T}$ , ${\tilde{a}}_{1 i} = {\hat{a}}_{1 i} - a_{1 i}$ , ${\hat{a}}_{1 i} = [\begin{matrix} {\hat{a}}_{11}^{i} & {\hat{a}}_{12}^{i} \end{matrix}]$ , $a_{1 i} = [\begin{matrix} a_{11}^{i} & a_{12}^{i} \end{matrix}]$ , ${\tilde{a}}_{2 i} = {\hat{a}}_{2 i} - a_{2 i}$ , ${\hat{a}}_{2 i} = [\begin{matrix} {\hat{a}}_{21}^{i} & {\hat{a}}_{22}^{i} \end{matrix}]$ , $a_{2 i} = [\begin{matrix} a_{21}^{i} & a_{22}^{i} \end{matrix}]$ , ${\tilde{B}}_{i} = {[\begin{matrix} {\tilde{b}}_{1 i} & \tilde{b} \end{matrix}]}^{T}$ , ${\tilde{b}}_{1 i} = {\hat{b}}_{1 i} - b_{1 i}$ , ${\hat{b}}_{1 i} = [\begin{matrix} {\hat{b}}_{11}^{i} & {\hat{b}}_{12}^{i} \end{matrix}]$ , $b_{1 i} = [\begin{matrix} b_{11}^{i} & b_{12}^{i} \end{matrix}]$ , ${\tilde{b}}_{2 i} = {\hat{b}}_{2 i} - b_{2 i}$ , ${\hat{b}}_{2 i} = [\begin{matrix} {\hat{b}}_{21}^{i} & {\hat{b}}_{22}^{i} \end{matrix}]$ , $b_{2 i} = [\begin{matrix} b_{21}^{i} & b_{22}^{i} \end{matrix}]$ .

The adaptive laws can be chosen as: ${\dot{\tilde{a}}}_{1 i} = m_{1 i} \frac{μ_{i} (η)}{\sum_{i = 1}^{3} μ_{i} (η)} p_{1}^{T} {ex}^{T}$ (19) ${\dot{\tilde{a}}}_{2 i} = m_{2 i} \frac{μ_{i} (η)}{\sum_{i = 1}^{3} μ_{i} (η)} p_{2}^{T} {ex}^{T}$ (20) ${\dot{\tilde{b}}}_{1 i} = n_{1 i} \frac{μ_{i} (η)}{\sum_{i = 1}^{3} μ_{i} (η)} p_{1}^{T} {eu}^{T}$ (21) ${\dot{\tilde{b}}}_{2 i} = n_{2 i} \frac{μ_{i} (η)}{\sum_{i = 1}^{3} μ_{i} (η)} p_{2}^{T} {eu}^{T}$ (22)

Where m_1i, m_2i, n_1i, n_2i > 0 are constant scalars.

Theorem. The adaptive laws can guarantee the asymptotic convergence of the estimation error between the T-S fuzzy model Equation (12) and the estimation model Equation (17), which makes both of them become the desired reference model Equation (14). Taking into account of that there is no approximation error between the original nonlinear model and its T-S fuzzy model, the original nonlinear model follows the desired reference model.

Then we will give the proof for the adaptive laws in Equations (19–22).

Proof. By choosing the following function as the Lyapunov function candidate,

$\begin{matrix} V & = & e^{T} Pe + \sum_{i = 1}^{3} \frac{{\tilde{a}}_{1 i} {\tilde{a}}_{1 i}^{T}}{m_{1 i}} + \sum_{i = 1}^{3} \frac{{\tilde{a}}_{2 i} {\tilde{a}}_{2 i}^{T}}{m_{1 i}} \\ + \sum_{i = 1}^{3} \frac{{\tilde{b}}_{1 i} {\tilde{b}}_{1 i}^{T}}{n_{2 i}} + \sum_{i = 1}^{3} \frac{{\tilde{b}}_{2 i} {\tilde{b}}_{2 i}^{T}}{n_{2 i}} \end{matrix}$ (23)

Where P = P^T > 0 is chosen as the solution of the Lyapunov equation A_sP + PA_s = - I.

The derivative of V with respect to time becomes

$\begin{matrix} \dot{V} & = & {\dot{e}}^{T} Pe + e^{T} P \dot{e} + 2 \sum_{i = 1}^{3} \frac{{\dot{\tilde{a}}}_{1 i} {\tilde{a}}_{1 i}^{T}}{m_{1 i}} + 2 \sum_{i = 1}^{3} \frac{{\dot{\tilde{a}}}_{2 i} {\tilde{a}}_{2 i}^{T}}{m_{2 i}} \\ + 2 \sum_{i = 1}^{3} \frac{{\dot{\tilde{b}}}_{1 i} {\tilde{b}}_{1 i}^{T}}{n_{1 i}} + 2 \sum_{i = 1}^{3} \frac{{\dot{\tilde{b}}}_{2 i} {\tilde{b}}_{2 i}^{T}}{n_{2 i}} \end{matrix}$ (24)

Where $P = [\begin{matrix} p_{1} & p_{2} \end{matrix}]$ .

Substituting Equations (19–22) into Equation (24) yields $\dot{V} = - e^{T} e \leq 0$ (25)

This implies that $\dot{V}$ is a negative definite, implying that e, ${\tilde{A}}_{i}$ and ${\tilde{B}}_{i}$ are all bounded. From Barbalat Lemma, it can be proved that e, ${\tilde{A}}_{i}$ and ${\tilde{B}}_{i}$ can asymptotically converge to zero as t→ ∞. Thus the asymptotic stability of the control system can be guaranteed.

5 Simulation study

In order to validate the correctness and advantage of the proposed control strategies, the proposed controller is carried out using Matlab/Simulink package with SimPower Toolbox. The system parameters used in the simulation are shown in Table 1. The main purpose of the simulation is to study the following five different aspects: 1) steady-state response for harmonic compensation; 2) dynamic response to load variations; 3) compensation of unbalanced load; 4) compensation of unbalanced source voltage; 5) control strategy comparison.

The plant parameters ${\hat{A}}_{i}$ , ${\hat{B}}_{i}$ are adjusted online by adaptive laws (19–22) where

m_1i = 1, m_2i = 1, n_1i = 1, n_2i = 1. A_s is chosen as $A_{s} = [\begin{matrix} - 1 & - 1 \\ 0 & - 1 \end{matrix}]$ . The initial values of the plant parameters were set as ${\hat{A}}_{i} (0) = 0.9 * A_{i}^{*}$ , ${\hat{B}}_{i} (0) = 0.9 * B_{i}^{*}$ where $A_{i}^{*}$ , $B_{i}^{*}$ are the true value. $A_{i}^{*}$ , $B_{i}^{*}$ are set as $A_{1}^{*} = (\begin{matrix} - 1355 & - 1659 \\ - 314.2 & - 10 \end{matrix})$ , $A_{2}^{*} = (\begin{matrix} - 10 & 314.2 \\ - 314.2 & - 10 \end{matrix})$ , $A_{3}^{*} = (\begin{matrix} 1335 & 1031 \\ - 314.2 & - 10 \end{matrix})$ , $B_{1}^{*} = B_{2}^{*} = B_{3}^{*} = [\begin{matrix} 100000 & 0 \\ 0 & 100000 \end{matrix}]$ .

With regard to the reference model, it must have desired system responses. Here, we design it as an overdamped system, damping ratio ζ = 1.5, rise time t_r = 0.1s, and natural frequency ω_n = 55rad/s, so the reference model can be obtained as $A_{m} = [\begin{matrix} - 165 & - 3025 \\ 1 & 0 \end{matrix}]$ . We choose three membership functions as μ = exp {- [(x + 15 - (i - 1) *15)/7] ²} , 1pti = 1, 2, 3, shown in Fig. 3. When t = 0.04s, the switch of compensation circuit is closed and APF begins to work. The source current is equal to the load current when the APF is not connected.

5.1 Steady-state response for harmonic compensation

The load current (only one phase current is represented for the clearness), source currents, compensating current, reference current and error current are shown in Fig. 4. It is clearly shown that there is a serious distortion of load current and the Total Harmonic Distortion (THD) is relatively high (22.24%). When the control strategy is applied to the APF, source current becomes balanced and sinusoidal. In the steady-state, the error current can be limited to 1A. The THD is reduced to 1.01%, well below the requirement of the IEEE 519 standard (5%). The results confirm the capability of the control strategy for APF to cancel the harmonics.

5.2 Dynamic responses to load variations

In practice, nonlinear loads are usually time-varying, so it is necessary to study the dynamic performance of the APF under the circumstance of variations in the nonlinear loads. The nonlinear load was subjected to 100% step increase at t = 0.1 s and 100% step decrease at t = 0.2 s in Fig. 5. It can be observed that the source current settles smoothly to a new steady state value within a half cycle,and from Fig. 5, one can notice that the proposed control method imposes a perfect dynamic performance to APF under the circumstance of a rapid change in the nonlinear load.

5.3 Compensation of unbalanced load

This test aims to evaluate the capability of the APF to compensate for unbalanced load. To carry out this test, a single-phase diode rectifier, followed by inductor L = 10 mH in series with a resistor R = 40 Ω is connected between phases ‘1’ and ‘2’ with the nonlinear loads. The relevant waveforms are shown in Fig. 6. We can see that the source current is sinusoidal and balanced after compensation even with unbalanced load. The THD of the source currents are reduced from 19.52%, 20.18% and 24.81% to 1.62%, 1.54% and 1.48% with proposed controller. The results demonstrate the superiority of proposed controller.

5.4 Compensation of unbalanced source voltage

In industrial applications, small unbalance is always present in source voltage which may cause significant unbalance in source current, so care should be taken in designing controller. The simulation results with the proposed controller under unbalanced source voltages are shown in Fig. 7. We can see that the source current is sinusoidal and balanced after compensation. The THD of the source currents are reduced from 20.71%, 26.12% and 28.39% to 1.46%, 1.73% and 1.58% with proposed controller. These results confirm the capability of the proposed control algorithm to balance the source current even with unbalanced supply voltage.

5.5 Control strategy comparison

In order to demonstrate the superiority of the proposed control strategy, simulation results with adaptive fuzzy sliding control (AFSC) and adaptive fuzzy backstepping control (AFBC) are presented in Figs. 8 and 9. Note that the structure of AFSC and AFBC is the same with [6] and [17] respectively and the gains in these controllers are chosen through trial-and-error to achieve satisfactory tracking performance. One can see that these two control strategies are effective for APF to eliminate the harmonics. The THD with these two control strategies can be decreased within 5% after compensation, but error current is both larger than that obtained by the proposed control strategy at t = 0.1 ∼ 0.2 s, showing that the system robustness to parametric uncertainties and external disturbances is not as good as that with the proposed control strategy.

6 Conclusion

In this paper, a T-S fuzzy model based adaptive fuzzy current tracking controller for the three-phase APF is presented. Based on PDC, local linear state feedback controller can be designed for each T-S fuzzy sub-model. By using the parameter estimator, the parameters of the T-S fuzzy model can be adaptively updated based on the Lyapunov analysis. The stability of the closed-loop system can be guaranteed with the proposed control strategy. Simulation results prove that the proposed control scheme gives high performance under the following different aspects: 1) steady-state response for harmonic compensation; 2) dynamic response to load variations; 3) compensation of unbalanced load; 4) compensation of unbalanced source voltage.

It is difficult to establish accurate mathematical model for APF because of its nonlinearity and coupling, so the proposed control strategy which possess the salient merit of model-free control has a great potential to be used in APF. Moreover, a parameter estimation scheme for tuning the parameters of T-S fuzzy model is designed to cope with the parametric uncertainties and external disturbances, and the adaptive laws for system parameters are derived from Lyapunov stability theorem to ensure the convergence as well as stable control performance. Simulation results also showed that proposed controller with the parameter estimator could cope with the varying or unknown plant model. To the best of the authors’ knowledge, no TS fuzzy model based major control design for APF with parametric uncertainties and external disturbances addressing the issues of parametric estimation and system stability assurance simultaneously has been reported. The novelty in the proposed algorithm is that a synthesis of TS fuzzy model based control design and parametric estimation is exploited for APF in order to achieve better control performance for the first time.

Compared with the previous research results in [6] and [17], the proposed control system can provide superior high-precision control performance to parametric uncertainties and external disturbances. Furthermore, the proposed controller has a general sense and also can be used in extensive applications, such as grid-connected converters, programmable ac power supply, and so on. Note that indeed, the nonlinear model of APF is not needed to design both the controller and the parameter estimator. So the proposed control technique has salient advantages for large multilevel converters such as five-level or seven-level converters which will require a more complex mathematical model. However, real time implementation may be difficult due to heavy computation burden if the update strategy of the parameters is complicated. In order to solve this problem, some investigations to reduce redundant or inefficient computation for improving online learning abilities can be developed in the future.

The proposed control scheme can be readily realized by hardware such as digital signal processor (DSP), which is already scheduled in the future works. A laboratory prototype has been built and tested in our laboratory. As soon as it is finished, more comprehensive theoretical and practical analyses supported by the experimental results will be available, which can further demonstrate its potential in industry application.

Footnotes

Acknowledgments

The authors thank the anonymous reviewers for their useful comments that improved the quality of the paper. This work is partially supported by National Science Foundation of China under Grant No. 61374100; Natural Science Foundation of Jiangsu Province under Grant No. BK20131136; The University Graduate Research and Innovation Projects of Jiangsu Province under Grant No. KYLX_0430; The Fundamental Research Funds for the Central Universities under Grant No. 2014B33214.

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