Adaptive fuzzy-sliding control with fuzzy-sliding switching for three-phase active power filter

Introduction

A variety of non-linear and time-varying electronic devices bring power quality problems to the power system such as low power factor, waveform distortion, surges, phase distortion problems and so on. An active power filter (APF) is useful for power system harmonic suppression and reactive current compensation. The basic principle of APF is to produce compensation current, which is of the same amplitude and opposite phase with the harmonic currents, to eliminate the unexpected harmonic currents. Therefore, it is widely used in many applications to compensate for the harmful harmonic currents produced by non-linear loads on industrial, commercial and residential equipments.

Over the past few years, fuzzy control has been extensively applied because of its model-free approach. Wang (1994) demonstrated that an arbitrary function can be approximated with arbitrary accuracy using a fuzzy system on a compact domain by proposing a universal approximation theorem. Guo and Woo (2004) proposed an adaptive fuzzy-sliding mode controller for a robot manipulator. Yoo and Ham (2000) developed an adaptive controller for a robot manipulator using a fuzzy compensator. Wang et al. (2001) derived an indirect adaptive fuzzy-sliding mode control using a fuzzy switching approach. In this paper, a fuzzy controller will be investigated to approximate non-linear dynamic systems such as an APF, since it is very hard to establish an accurate mathematical model for APF and a classical linear controller cannot achieve an ideal current tracking performance. There are many current tracking control methods, such as single cycle control, hysteresis current control, space vector control, sliding mode control, deadbeat control, repetitive control, predictive control, fuzzy control, adaptive control, iterative learning control and artificial neural network control. Singh et al. (2007) presented a simple fuzzy logic-based robust APF for harmonics minimization under a random load variation. Bhende et al. (2006) developed a TS-fuzzy-controller for load compensation of an APF. Komucugil and Kukrer (2006) proposed a new control strategy for single-phase shunt APFs using a Lyapunov function. Rahmani et al. (2010) introduced an experimental design of a non-linear control technique for three-phase shunt APF. Luo et al. (2009) derived an adaptive fuzzy dividing frequency-control method to minimize the capacity of a hybrid APF with an injection circuit. Kumar and Mahajan (2009) summarized soft computing techniques for the current control of an APF. Chang and Chang (2004) proposed a novel reference compensation current strategy for shunt APF control. Shyu et al. (2008) developed a model reference adaptive control design for a shunt APF system. Matas et al. (2008) showed a feedback linearization approach of a single-phase APF via sliding mode control. Hua et al. (2009) described control analysis of an APF using a Lyapunov candidate. Lu and Xia (2008) applied a simple adaptive fuzzy control into the single-phase APF but the asymptotical Lyapunov stability cannot be guaranteed. Montero et al. (2007) compared different control strategies for shunt APF in three-phase four-wire systems. Valdez et al. (2009) designed an adaptive controller for a shunt APF in the presence of a dynamic load and line impedance. Marconi et al. (2007) developed a robust non-linear controller to compensate for harmonic current for a shunt APF. However, systematic stability analysis and controller design of the adaptive fuzzy controller have not been investigated for APF. Therefore, it is necessary to utilize the adaptive fuzzy control scheme to improve the current tracking and filtering performance. In this paper, a novel adaptive fuzzy control with a fuzzy-sliding term is developed to improve the current tracking performance and guarantee the Lyapunov stability of the close-loop system. The control strategy proposed here has the following advantages:

Principle of active power filter

The shunt APF can be considered the most basic structure of APF. This paper mainly studies the most widely used parallel-voltage type of APF. In the practical application, the three-phase is the most widely used shunt APF because of its excellent performance characteristics and simplicity in implementation; therefore the three-phase, three-wire system will be investigated and the dynamics of three-phase APF will be described in this section.

In practical operation, an APF is equivalent to a flow control current source. The whole APF system consists of three sections – the harmonic current detection module, the current tracking control module and the compensation current generating circuit. The harmonic current detection module usually uses instantaneous reactive power theory based on the rapid detection of harmonic current. A three-phase, three-wire APF produces compensation currents with a three bridge-arm circuit. In order to eliminate the harmonic components in the currents from the power supply, the compensation circuit produces compensation currents that have same amplitude and opposite phase with the harmonic currents.

The block diagram of the three-phase, three-wire active power system is given in Figure 1. The principle of the APF is to detect voltage and current of the compensation object, obtain command signal i c * of the compensation current by using the operation circuit for instruction current, then obtain compensation current i c by a PWM generator in order to offset the harmonic current and achieve the ideal source current.

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Figure 1.

Block diagram for main circuit of active power filter (APF).

The mathematical model of the APF can be described in the following steps. According to circuit theory and Kirchhoff’s theorem, we can obtain following state equations:

{ v 1 = L c d i 1 d t + R c i 1 + v 1 M + v M N v 2 = L c d i 2 d t + R c i 2 + v 2 M + v M N v 3 = L c d i 3 d t + R c i 3 + v 3 M + v M N

(1)

The parameters L c and R c are the inductance and resistance of the APF, respectively. V _MN is the voltage between M and N points.

By summing the three equations in (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 obtains:

v MN = - 1 3 ∑ 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. We can define c k as

c k = { 1 , if S k is On and S k + 3 is Off 0 , i f S k is Off and S k + 3 is On

(3)

where k = 1 , 2 , 3 .

Hence, by writing v kM = c k v dc , then (1) becomes

{ d i 1 d t = − R c L c i 1 + v 1 L c − v d c L c ( c 1 − 1 3 ∑ m = 1 3 c m ) d i 2 d t = − R c L c i 2 + v 2 L c − v d c L c ( c 2 − 1 3 ∑ m = 1 3 c m ) d i 3 d t = − R c L c i 3 + v 3 L c − v d c L c ( c 3 − 1 3 ∑ m = 1 3 c m )

(4)

Adaptive fuzzy control

In this section, an adaptive fuzzy-sliding control is derived and Lyapunov analysis is implemented to guarantee the asymptotic stability of the closed-loop system. An adaptive fuzzy control method based on the sliding-mode control is proposed to approximate the unknown equivalent control and sliding term. First a fuzzy logic system is introduced.

Fuzzy controller

A fuzzy controller is composed of the following four elements: a fuzzifier, some fuzzy IF-THEN rules, a fuzzy inference engine and a defuzzifier. The fuzzy inference engine uses the fuzzy IF-THEN rules to perform a mapping from an input linguistic vector x = ( x 1 , x 2 , ⋯ , x n ) T ⊂ R n to an output variable y ⊂ R . The fuzzy rule base consists of a collection of fuzzy IF-THEN rules, which can be expressed as

R l : If x 1 is A 1 l and … x n is A n l , then y is B l

where A i l and B l are fuzzy sets, and l = 1 , … , M , M denotes the number of fuzzy IF-THEN rules. The singleton fuzzifier mapping is adopted; x i and y have the same kind of member functions that are all Gaussian membership functions defined as

μ A i l ( x i ) = exp ( − ( x i − ψ i ) 2 2 σ i 2 )

(5)

where ψ i and σ i are the centre and width of the ith fuzzy set A i l , respectively.

From the knowledge of the fuzzy systems, the output of the fuzzy system can be expressed using the centre-average defuzzifier, product inference and singleton fuzzifier.

y ( x ) = ∑ l = 1 M h l ( Π i = 1 n μ A i l ( x i ) ) ∑ l = 1 M ( Π i = 1 n μ A i l ( x i ) ) = θ T ξ ( x )

(6)

where μ A i l ( x i ) is the membership function value of the fuzzy variable x i , θ T = ( h 1 , h 2 , ⋯ , h M ) is adaptive parameter vector and ξ ( x ) = ( ξ 1 ( x ) , ξ 2 ( x ) , ⋯ , ξ M ( x ) ) T is the vector of the fuzzy basis functions.

Adaptive fuzzy controller

How to construct adaptive fuzzy-sliding control is derived in next steps. The block diagram of adaptive fuzzy-sliding control system for the APF is shown in Figure 2. Systematic stability analysis is performed in the design of the proposed adaptive fuzzy-sliding control. The detailed design procedure of the adaptive fuzzy-sliding control system can be described in the following steps.

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Figure 2.

Adaptive fuzzy-sliding control block for active power filter (APF).

We can transform the dynamic model of (4) into the following form:

x · = f ( x ) + bu + d

(7)

where x = [ i 1 i 2 i 3 ] , f ( x ) = - R c L c i k + v k L c , k = 1 , 2 , 3 and b = - v dc L c , because v dc is not constant, so b is unknown. d is an unknown disturbance, and the model of (4) is proposed in an ideal situation, so in order to demonstrate that the control system has strong robustness, we add the disturbance d to the model, | d ( t ) | ≤ D , D > 0 . The control target is to make currents x to track the given reference current signal x m . The tracking error is defined as e = x − x m and the sliding function is designed as

s ( t ) = λ e

(8)

where λ is a positive constant.

We choose the control law as

u * = 1 b [ − f ( x ) + x · m − u sw ]

(9)

where η > 0

Substituting (9) into (7) yields:

s · ( t ) = − λ η sgn ( s ) + d

(10)

Then s ( t ) · s · ( t ) = − λ η | s | + sd ≤ − λ η | s | + | s | | d | ≤ − λ η | s | + | s | D = − | s | ( λ η − D )

If we choose η ≥ D λ , then s ( t ) · s · ( t ) ≤ 0 shows that the asymptotical stability can be guaranteed.

If f ( x ) and b are unknown, the control law (9) cannot be directly implemented. However, fuzzy system f ^ and h ^ can be utilized to approximate f and η sgn ( s ) .

The fuzzy controller can be expressed as

u = 1 b [ − f ^ ( x ) + x · m − h ^ ( s ) ]

(11)

where f ^ ( x | θ f ) = θ f T ξ ( x ) , h ^ ( s | θ h ) = θ h T ϕ ( s ) , ξ ( x ) and ϕ ( s ) are fuzzy basis functions, θ f T and θ h T can be updated by the following adaptive laws:

θ · f = r 1 λ s ξ ( x )

(12)

θ · h = r 2 λ s ϕ ( s )

(13)

where r 1 and r 2 are positive constants. Then we will give the proof for the adaptive laws in (12) and (13).

Theorem. A feedback controller u = 1 b [ − f ^ ( x ) + x · m − h ^ ( s ) ] and adaptive laws (12) and (13) for adjusting parameters vector θ f ( t ) and θ h ( t ) are designed to satisfy that the closed-loop system must be globally stable in the sense that all variables, x ( t ) , θ f ( t ) , θ h ( t ) and u , must be uniformly stable and the current tracking error e ( t ) should be as small as possible.

Proof. Define optimal parameter vector:

θ f * = arg min θ f ∈ Ω f [ sup | f ^ ( x | θ f x ∈ R n ) − f ( x ) | ]

(14)

θ h * = arg min θ h ∈ Ω h [ sup | h ^ ( s | θ h x ∈ R n ) − u s w | ]

(15)

where Ω f and Ω h are assemble for θ f and θ h .

Define fuzzy approximation error ω :

ω = f ( x ) − f ^ ( x | θ h * )

(16)

where ω is bounded by positive constant ω max that is

| ω | ≤ ω max

(17)

Then the derivative of sliding surface can be derived as

s · = λ e · = λ ( x · − x · m ) = λ [ f ( x ) + b u + d − x · m ] = λ [ f ( x ) − f ^ ( x ) − h ^ ( s | θ h ) + d ] = λ [ f ^ ( x | θ f * ) − f ^ ( x ) − h ^ ( s | θ h ) + ω + h ^ ( s | θ h * ) − h ^ ( s | θ h * ) + d ] = λ [ φ f T ξ ( x ) + φ h T ϕ ( s ) + w − h ^ ( s | θ h * ) + d ]

(18)

where φ f = θ f − θ f * , φ h = θ h − θ h * .

Define Lyapunov function candidate:

V = 1 2 ( s 2 + 1 r 1 φ f T φ f + 1 r 2 φ h T φ h )

(19)

Then the derivative of V becomes

V · = s s · + 1 r 1 φ f T φ · f + 1 r 2 φ h T φ · h = s λ [ φ f T ξ ( x ) + φ h T ϕ ( s ) + w − h ^ ( s | θ h * ) + d ] + 1 r 1 φ f T φ · f + 1 r 2 φ h T φ · h = s λ φ f T ξ ( x ) + 1 r 1 φ f T φ · f + s λ φ h T ϕ ( s ) + 1 r 2 φ h T φ · h + λ s ω + λ s d − λ s h ^ ( s | θ h * )

(20)

Because h ^ ( s | θ h * ) = η Δ sgn ( s ) , η Δ = D + η , | d ( t ) | ≤ D and D > 0 , thus

V · = 1 r 1 φ f T [ φ · f + r 1 λ s ξ ( x ) ] + 1 r 2 φ h T [ φ · h + r 2 λ s ϕ ( s ) ] + λ [ s ω + s d − ( D + η ) | s | ] ≤ 1 r 1 φ f T [ φ · f + r 1 λ s ξ ( x ) ] + 1 r 2 φ h T [ φ · h + r 2 λ s ϕ ( s ) ] + λ [ s ω − η | s | ]

(21)

Substituting (12), (13) into (21) yields

V · ≤ λ ( s ω − η | s | ) ≤ λ ( | s | | ω | − η | s | ) = − λ | s | ( η − | ω | )

(22)

Assume the approximation error ω is bounded as | ω | ≤ ω max , if we choose η ≥ ω max , then V · ≤ 0 . This implies that V · is negative semi-definite, and s ( t ) , θ ~ f and θ ~ h are bounded. Since V ( 0 ) is bounded, and V ( t ) is non-increasing and bounded function, then we have ∫ 0 t | s | ( η − | w | ) d τ ≤ V ( 0 ) − V ( t ) ≤ ∞ , i.e. s ∈ L 1 . From (18), it can be known that s · ( t ) is bounded. If s · ( t ) is bounded, then s ( t ) is uniformly continuous. According to the Corollary of Barbalat’s Lemma, we have lim t → ∞ s ( t ) = 0 . Then lim t → ∞ e ( t ) = 0 .

Simulation study

The performance of the proposed adaptive fuzzy-sliding control will be testified using the Matlab/Simulink package with SimPower Toolbox. Simulation results are presented to verify the effectiveness of the proposed adaptive fuzzy-sliding control.

We choose six membership functions as

μ = exp { − [ x + 4 − 2 ( i − 1 ) ] 2 } , i = 1 , … , 6

where i is number of fuzzy rules, i = 6 , which means that there are 36 fuzzy rules to approximate the unknown system. The selected 36 fuzzy rules can cover the entire space and approximate any system states and non-linear functions. The membership function is shown in Figure 3. The three membership functions over the interval [−3,3] for sliding function s are selected as

μ NM ( s ) = 1 1 + exp ( 5 ( s + 3 ) )

μ ZO ( s ) = exp ( − s 2 )

μ PM ( s ) = 1 1 + exp ( 5 ( s − 3 ) )

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Figure 3.

Membership function degree of x.

The initial values of fuzzy parameters are chosen randomly in the interval, and the vector of fuzzy basis functions were constructed by (6).

Sliding function is chosen as s = λ e , where λ = 100 , adaptive gain r 1 = 10000 , r 2 = 1000 ,

θ f = [ θ fa θ fb θ fc ] = [ θ fa 1 θ fa 2 θ fa 3 θ fa 4 θ fa 5 θ fa 6 θ fb 1 θ fb 2 θ fb 3 θ fb 4 θ fb 5 θ fb 6 θ fc 1 θ fc 2 θ fc 3 θ fc 4 θ fc 5 θ fc 6 ] θ h = [ θ ha θ hb θ hc ] = [ θ ha 1 θ ha 2 θ ha 3 θ hb 1 θ hb 2 θ hb 3 θ hc 1 θ hc 2 θ hc 3 ]

V s 1 = V s 2 = V s 3 = 110 V , f = 50 Hz . The inductance in the circuit of APF is 10 mH and the capacitance is 100 µF. The non-linear load branch is a rectifier bridge connecting parallel RC load, where R=15 Ω, C=5e-3 F. PI control is adopted for DC voltage in the APF and the parameters of the PI controller are chosen as k p = 0 . 005 and k i = 0 . 02 to achieve satisfactory performance. The difference between the DC capacitor voltage and reference voltage is the control input for PI control; then the control input is added to the DC component of instantaneous active current in order to obtain the reference current, and the compensation current includes the active component of fundamental current, which leads to energy exchange between AC side and DC side. In this way, the DC capacitor voltage can follow the reference voltage.

A phase source current before and after APF works is shown in Figure 4. Current harmonic analysis for the first two circles and last two cycles are depicted in Figures 5 and 6. When t = 0 . 04 s , the switch of compensation circuit is closed and the APF begins to work. It can be seen there are many harmonics before 0.04 s, the source currents recover to steady state after a half cycle at about 0.01 s, THD is 24.71% before harmonic compensation and it decreases to 1.59% after harmonic compensation, which is within the limit of the harmonic standard of IEEE of 5%. It can be observed that the supply current is close to a sinusoidal wave and it remains in phase with the supply voltage, demonstrating that the APF performs well in the steady state operation.

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Figure 4.

A phase source current.

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Figure 5.

Current harmonic analysis for the first two circles.

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Figure 6.

Current harmonic analysis for the last two circles.

Instruction current and compensation current are drawn in Figure 7, and the compensation current tracking error is depicted in Figure 8. It can be observed that the compensation current can track the instruction current well, which demonstrates that the proposed adaptive fuzzy-sliding control can guarantee asymptotic state tracking. Thus the harmonic current can be effectively compensated and the harmonic distortion of the source current can be reduced. Adaptive parameters of θ f and θ h are depicted in Figures 9 and 10. It is demonstrated that the parameters of the proposed adaptive fuzzy-sliding controller converge to stable constant values. As can be seen from Figure 11, the DC capacitor voltage is stable using PI controller. It is noted that the DC voltage is zero before 0.04 s, because the initial capacitor voltage is not set, and there is no energy exchange between the AC side and DC side before APF works; then the DC capacitor voltage cannot change from 0 to 1000.

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Figure 7.

Instructions current and compensation current.

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Figure 8.

Compensation current tracking error.

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Figure 9.

Adaptive law θ f .

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Figure 10.

Adaptive law θ h .

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Figure 11.

DC capacitor voltage.

In order to demonstrate that the adaptive fuzzy-sliding control system has strong robustness in the presence of parameter variation, an APF with the parameter variation is testified. As shown in Table 1, the THD is still in the normal range with the parameter variation. It can be concluded that the adaptive fuzzy-sliding control system has good robustness to the parameter uncertainties.

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Table 1.

Performance for variation in filter inductance and DC capacitor using adaptive fuzzy sliding control.

L (mH)	C (µF)	THD (%)
10	100	1.59
10	200	1.58
10	1000	2.32
8	100	1.53
5	100	1.66

THD, total harmonic distortion.

In order to demonstrate that the adaptive fuzzy-sliding control system can achieve better performance than the conventional method, APF using hysteresis control is also testified. It can be seen from Table 2 that the adaptive fuzzy-sliding control system has better robustness compared with the conventional method.

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Table 2.

Performance for variation in filter inductance and DC capacitor using hysteresis controller.

L (mH)	C (µF)	THD (%)
10	100	1.72
10	200	1.78
10	1000	4.15
8	100	1.78
5	100	2.05

THD, total harmonic distortion.

In summary, the current tracking, THD performance and the control performance and robustness to external disturbance can be improved with the proposed adaptive fuzzy-sliding controller.

Conclusion

An improved adaptive fuzzy-sliding control system has been applied to the three-phase APF in this paper. Universal approximation properties of the fuzzy system are employed to approximate the unknown equivalent control and sliding mode control. The parameters of the fuzzy system in both the fuzzy control part of the unknown equivalent control and the fuzzy-sliding mode part can be adaptively updated based on the Lyapunov analysis. The asymptotic stability of the closed-loop system can be guaranteed with the proposed adaptive fuzzy control strategy with a fuzzy switch term. The proposed adaptive fuzzy-sliding controller can make the compensation current follow the instruction current, and effectively eliminate the reactive and harmonic components of the load current. The designed APF control system has superior harmonic suppression performance and yields an improved THD performance. Simulation results demonstrated the excellent dynamic performance, asymptotic stability and strong robustness with the proposed APF control system.