International Journal of Machine Learning and Cybernetics https://doi.org/10.1007/s13042-018-0792-y

ORIGINAL ARTICLE

Adaptive fuzzy-neural-network based on RBFNN control for active power filter Juntao Fei1,2 · Tengteng Wang1,2 Received: 5 May 2016 / Accepted: 9 February 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract In this paper, an adaptive fuzzy-neural-network (FNN) control scheme based on a radial basis function (RBF) neural network (NN) is proposed to enhance the performance of a shunt active power filter (APF). APF can efficiently eliminate harmonic contamination and improve the power factor compared with traditional passive filter. The proposed approach gives a RBF NN control scheme, which is utilized on the approximation of a nonlinear function in APF dynamic model, the weights of the RBF NN are adjusted online according to adaptive law from the Lyapunov stability analysis. In addition, adaptive fuzzyneural-network systems is employed to compensate the neural approximation error and eliminate the existing chattering, enhancing the robust performance of the system. Simulation results confirm the effectiveness of the proposed controller, demonstrating that APF with the proposed method has strong robustness and the outstanding compensation performance. Keywords  Radial basis function (RBF) · Fuzzy-neural-network control (FNN) · Adaptive control · Active power filter

1 Introduction In recent years, power electronic technology is widely used in electrical power systems. However, the distortion of power quality has become a serious problem with the increasing number of nonlinear loads of power electronic devices in electrical power systems. Traditional passive power filter is gradually replaced by shunt active power filters (SAPF) because of its bad dynamic performance in eliminating both current distortion and reactive power. APF can efficiently eliminate harmonic contamination and improve the power factor compared with traditional passive filter. The selection of intelligent control techniques plays an important role in the performance of the APF. In addition to traditional current control methods including hysteresis control, experts have put forward many new control strategies to improve APF dynamic performance, such as sliding mode control, neural network control, fuzzy control, and adaptive control. Ruanmakok et al. [1] presented a control of shunt power * Juntao Fei [email protected] 1



College of IoT Engineering, Hohai University, Changzhou 213022, China



Jiangsu Key Lab. of Power Transmission and Distribution Equipment Technology, Changzhou, China

2

filter using a sliding mode controller. Narongrit et al. [2] designed a fuzzy controller for a shunt active power filter without the experience of specialists. Bandal [3] described two control stratde control (SMC) and proportional-integral control for a three-phase four-wire shunt APF. Fei et al. [4] proposed a model reference adaptive fuzzy controller (MRAFC) to improve the power performance for a shunt APF. Hou et al. [5] designed an adaptive fuzzy back stepping controller to ensure proper tracegies: sliding moking of the reference current, and impose a desired dynamic behavior, giving robustness and insensitivity to parameter variations. Neural network (NN) is utilized on the approximation of nonlinear function in APF dynamic model in order to improve the robustness of the control system. Li et al. [6] use a RBF NN with an additional linear neuron for fitting experimental stopping power data to a simple empirical formula. Liu et al. [7] proposed an adaptive RBFNN which combinated with fuzzy sliding mode control to eliminate the reaching phase. Van et al. [8] presented an adaptive trajectory tracking neural network control using RBF neural network with robust compensator to achieve the highprecision position tracking for an n-link robot manipulator. Tabatabaei et al. [9] described an adaptive neural network controller for a class of uncertain nonlinear switched systems. A fuzzy sliding mode controller based on RBF neural network controller for a three link robot system was

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proposed in [10]. It is easy to discover in [11–17] that NN was suitable to control uncertain nonlinear dynamical systems. The fuzzy neural network combines fuzzy logic based on the experience of human experts and fast nonlinear learning ability of RBF neural network, which can approach a nonlinear system with unknown parameters fastly. Fuzzy neural network control has made practical success in industrial processes and other fields. Xue et al. [18] presented a fault detection and accommodation method based on fuzzy neural networks for nonlinear systems. Lin et al. [19] developed a fuzzy neural network control system with adaptive algorithm to control permanent magnet motor drive system. Wen et al. [20] presented an adaptive fuzzy neural network control scheme for a class of uncertain multi-input multi-output (MIMO) nonlinear systems. Lin et al. [21] used an adaptive backstepping fuzzy neural network (ABFNN) controller for a permanent magnet synchronous motor (PMSM) drive system. Lin et al. [22] proposed an adaptive fuzzy neural network (AFNN) strategy to control the position of the mover of a permanent magnet linear synchronous motor servodrive system. El-Sousy [23] employed an adaptive hybrid control system based on the computed torque control for permanent-magnet servo drive. Lin et al. [24] proposed a wavelet Petri fuzzy neural network (WPFNN) controller to control squirrel-cage induction generator system. Fei et al. [25] proposed an adaptive control using global fast terminal sliding mode control and fuzzy-neural-network for a micro-electro-mechanical systems vibratory gyroscope. Hsu et al. [26] presented a dynamic RBF network with a constructive learning to tackle this problem of a trade-off between the approximation performance of RBF network and the number of hidden neurons.Some real-life applications of contemporary soft computing techniques in different fields have been shown in [27–29]. Li et al. [30, 31] developed adaptive fuzzy output-feedback dynamic surface control schems for nonlinear systems. Xu et al. proposed composite neural control strategies for strictfeedback systems [32, 33] and flexible hypersonic flight vehicle [34]. Wang Xu et al. [35–37] presented some novel RBFNN schemes and their applications. As the relevant publications in this field, the aims of this paper is to improve the accuracy of harmonic compensation, however, almost all of the aforementioned articles lack of the discussion of the influence of the uncertainty of the model parameters and system nonlinear characteristics on controller accuracy and system performance. In this paper, an adaptive FNN based on RBF NN control scheme for APF is proposed to thoroughly solve the drawbacks mentioned above. Adaptive sliding mode control with the FNN guarantees that the designed system could reach the sliding surface and converge to equilibrium point

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asymptotically. The proposed control strategy has the following advantages: (1) The RBF NN based on the sliding mode control is utilized on the approximation of the nonlinear function in APF dynamic model in order to improve the robustness of the control system. In order to compensate the neural approximation error and eliminate the existing chattering, the sliding gain is adjusted by adaptive fuzzy-neural-network systems, enhancing the robust performance. (2) The key property of this method is that the weights of the RBF NN and adaptive fuzzy-neural-network parameter are adjusted online by adaptive laws, to ensure the state hitting the sliding surface and sliding along it and guarantee the asymptotic stability of the system. This paper is organized as follows. In Sect. 2, the dynamics of APF is established. In Sect. 3, a sliding mode control and RBF NN control are proposed. An adaptive fuzzy-neural-network based on RBF NN control scheme are given in Sect. 4. Simulation results are presented in Sect. 5. Finally, conclusions are provided in Sect. 6.

2 Dynamic model of active power filter The system structure of a three-phase shunt APF 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. According to Fig. 1, we can establish dynamic model of three-phase shunt active power filter. According to circuit theory and Kirchhoff’s voltage law, we can get

⎧ v1 = Lc di1 + Rc i1 + v1M + vMN dt ⎪ di2 ⎨ v2 = Lc dt + Rc i2 + v2M + vMN , ⎪ v = L di3 + R i + v + v c dt c 3 3M MN ⎩ 3

(1)

where, Lc and Rc are the inductance and resistance of the APF respectively, vMN is the voltage between point M and N. Assuming that the AC supply voltages are balanced, we have:

International Journal of Machine Learning and Cybernetics Fig. 1  System structure of three-phase shunt APF

9V

1

/V

LV

L/

Y

9V

LV

9V

LV

L/

Y

Nonlinear loads

L/

Y /F

LGF

5F 6

6

6

L



9GF

L

&

L

 6

6

6

Y0

Y0

Y0

0

vMN = −

3 1∑ v . 3 m=1 mM

(2)

Define the switching function ck  , which denotes the ON/OFF status of the devices in the two legs of the IGBT Bridge. { 1, if Sk is On and Sk+3 is Off , ck = (3) 0, if Sk is Off and Sk+3 is On where, k = 1, 2, 3. Because of vkM = Ck vdc , then (1) becomes

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

� di1 dt

=

R v − Lc i1 + L1 c c R v − Lc i2 + L2 c c

−

di2 dt

=

di3 dt

= − Lc i3 + L3 −

R

c

−

v

c

vdc Lc vdc Lc

c1 − �

1 3

c2 −

1 3

�

vdc Lc

c3 −

1 3

∑ 3

3 ∑

cm

m=1 3 ∑

3 1∑ c , 3 m=1 m

R

v

c Rc

c v2

c

c

= − Lc i1 + L1 − = − L i2 + L − =

v R − Lc i3 + L3 c c

−

vdc d Lc 1 vdc d Lc 2 vdc d Lc 3

Define

�

Putting (8) into (7), we have:

�

(4)

cm

(7)

.

x = ik . ẋ = i̇ k

(8)

ẋ = i̇ k = −

Rc v v i + k − dc dk . Lc k L c Lc

(9)

Rc 1 dvk 1 dvdc i̇ + − d Lc k Lc dt Lc dt k ) ( R2c Rc Rc 1 dvk 1 dvdc = 2 ik − 2 vk + dk . v − + Lc dt Lc Lc Lc2 dc Lc dt

m=1

ẍ = −

We can define dk as:

dk = ck −

di1 dt di2 dt di3 dt

⎧ ⎪ ⎨ ⎪ ⎩

�

.

(6)

Then (4) becomes

{

cm

m=1

⎡ 2 −1 −1 ⎤⎡ c1 ⎤ ⎡ d1 ⎤ ⎢ d2 ⎥ = 1 ⎢ −1 2 −1 ⎥⎢ c2 ⎥. ⎥⎢ ⎥ ⎢ ⎥ 3⎢ ⎣ −1 −1 2 ⎦⎣ c3 ⎦ ⎣ d3 ⎦

(5)

which shows that dk depends on the switching function ck , which is nonlinear term of the system. From (5) and the eight permissible switching states of the IGBT, we can obtain that

(10)

Then (10) becomes (11)

ẍ = fa (x) + Mu, where,

fa (x) =

R2c

i Lc2 k

−

Rc dv v + 1 k  , Lc2 k Lc dt

M=

Rc v Lc2 dc

−

1 dvdc  , Lc dt

u = dk . In the following section, based on the dynamic model of active power filter, we will discuss three control strategies,

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International Journal of Machine Learning and Cybernetics

which are sliding mode controller, adaptive RBFNN controller and adaptive fuzzy-neural-network gain controller.

3 Adaptive RBFNN controller 3.1 RBF neural network structure

We consider the first Lyapunov function candidate as

RBF NN has the universal approximation property that states that any suffciently smooth function can be approximated by a suitable large network for all inputs in a compact set and the resulting function reconstruction error is bounded. It can meet the requirements of real-time control of APF because it can avoid local minima problems, accelerate learning speed. The RBFNN structure employed in our work is shown as Fig. 2. Where x(t) is the input variable, 𝜔(t) is the real-time weights of RBF NN, 𝜙i (x) = [𝜙1 (x), 𝜙2 (x) ⋯ 𝜙n (x)]T are Gaussian functions.

( ‖ ) x − ci ‖ ‖ ‖ 𝜙i (x) = exp − b2

) 1 1 1 1 1 1 1 1( ṡ = ë + 𝜆 e= ẍ d + 𝜆ė − ẍ ̇ ẍ d − ẍ + 𝜆 e= ̇ M M M M M M M M ) 1 1 1( ẍ + 𝜆ė − fa (x) − u= (̈xd + 𝜆ė − fa (x)) − u. = M d M M (17)

(12)

i = 1, … , n,

where, ci is the center of number i neurons, bi is width of number i neurons. The output of RBF NN is defined as: (13)

f̂ (x, t) = 𝜔̂ T ∗ 𝜙(x).

3.2 Design of RBF neural network controller

(14)

And the derivative of tracking error is:

(15)

ė = ẋ d − x. ̇ The sliding surface is:

1 2 s , 2M

(18)

where M is a postive constant. And the derivative of V̇ 1 is:

( ) 1 1 1 V̇ 1 = sT ṡ = sT ë + 𝜆 ė M M M ( ) 1 1 T 1 =s ẍ d − ẍ + 𝜆 ė M M M ( ) 1 1 T 1 ẍ d + 𝜆 ė − fa (x) − u . =s M M M

s = ė + 𝜆e. (16) Then derivative of sliding surface as in (16) can be written as:

(19)

Then, the nonlinear functions f is defined as

f =

1 1 1 ẍ + 𝜆 ė − fa (x). M d M M

(20)

Putting (20) into (19) yields

V̇ 1 = sT (f − u). For the V1′ ≤ 0 , the controller can be designed as

According to the mathematical model of APF (11), we set reference current as xd , sliding surface as s , a positive constant as 𝜆 = diag(𝜆1 , 𝜆2 , … , 𝜆n ), (𝜆i > 0). The tracking error e is defined as

e = xd − x.

V1 =

u=f +K

s + Ka s, ‖s‖

(21)

(22)

where, K and Ka are positive constants. Putting (22) into (21) yields � � s V̇ 1 = sT f − f − K − Ka s = − Ka sT s − K‖s‖ ≤ 0. ‖s‖ (23) The system global asymptotic stability can be establiahsed. Taking into account the good nonlinear approximation capability of NN, a RBF NN f̂ is used to approach the nonlinear functions f  , defined as (24) The approximated system model can be described as

f̂ = 𝜔̂ T 𝜙(x). x1 x2 xn Input layer

Fig. 2  RBF NN structure

13

φ1 φ2. ..

φn Hidden layer

ω1

ω2 ωn

(25) where, 𝜀 is the approximation error vector, |𝜀| ≤ 𝜀N  , 𝜀N is a small positve constant, 𝜔∗ is the optimal weight vector of the RBF NN. Putting (24) into (22), then we can get the controller of RBF NN as

f = 𝜔∗T 𝜑(x) + 𝜀,

Σ

fˆ Output layer

u = 𝜔̂ T 𝜙(x) + K

s + Ka s. ‖s‖

(26)

International Journal of Machine Learning and Cybernetics

Defining a second Lyapunov function candidate V2:

V2 =

1 T 1 s s + tr(𝜔̃ T 𝜇−1 𝜔), ̃ 2M 2

f11 e11

(27)

f1k

f1 j

where, 𝜔̃ is the estimation error of weight vector, 𝜔̃ = 𝜔∗ − 𝜔̂  , 𝜇 is a positive constant. The derivative of V2 is:

( ) 1 V̇ 2 = sT ṡ − tr 𝜔̃ T 𝜇−1 𝜔̂̇ M ( ) ( ) = s 𝜔∗T 𝜑(x) + 𝜀 − u − tr 𝜔̃ T 𝜇−1 𝜔̂̇ .

f 21 e1i

f 2k

(28)

f2 j f 31

Substituting (26) into (28) yields: � � � � ̇V2 = sT 𝜔∗T 𝜑(x) + 𝜀 − 𝜔̂ T 𝜑(x) − K s − Ka s − tr 𝜔̃ T 𝜇−1 𝜔̂̇ ‖s‖ � � � � s = −Ka sT s + sT (𝜔∗T − 𝜔̂ T )𝜑(x) + sT 𝜀 − K − tr 𝜔̃ T 𝜇−1 𝜔̂̇ ‖s‖ � � = −Ka sT s + sT 𝜔̃ T 𝜑(x) + sT 𝜀 − K‖s‖ − tr 𝜔̃ T 𝜇−1 𝜔̂̇ � � �� = −Ka sT s + sT 𝜀 − K‖s‖ + sT 𝜔̃ T 𝜑(x) − tr 𝜔̃ T 𝜇−1 𝜔̂̇ .

(29)

Choose an adaptive mechanism: (30)

𝜔̂̇ = 𝜇𝜙(x)sT . Putting (30) into (29) leads to:

V̇ 2 = −Ka s s + s 𝜀 − K‖s‖ ≤ −Ka ‖s‖ + ‖𝜀‖‖s‖ − K‖s‖ T

T

2

≤ −Ka ‖s‖2 + 𝜀N ‖s‖ − K‖s‖ = −Ka ‖s‖2 − ‖s‖(K − 𝜀N ). (31) If K ≥ 𝜀N  , V̇ 2 ≤ 0 . According to Barbalart lemma, s will converge to zero as time goes to infinity. Then it can be concluded that with control law (26) and adaptive law (30), the asymptotic stability of the closed-loop system can be guaranteed .

4 Design of adaptive fuzzy‑neural‑network controller 4.1 Adaptive fuzzy‑neural‑network structure Because the fuzzy-neural-network combines fuzzy logic based on the experience of human experts and fast nonlinear learning ability of RBF neural network, it can approach nonlinear system with unknown parameters fastly. The fuzzyneural-network structure employed in our work is shown as Fig. 3. The layers function can be described as follows: 1. Input layer input variable spreads to the next level. 2. The membership layer represents the input values with the following Gaussian membership functions:

f 3k

e1n

f3 j

.. .. .. .. ω

ϕ1 Π

k ji

.. ..

Input Layer Membership Layer

. .

Σ

ϕk

. .

Π

Σ

. .

Σ

. .

ϕj

Π

y1

yo

U FNNSMC

ym

W

Rule Layer

Output Layer

Fig. 3  The structure of a four-layer FNN

( ( ) )2 fij = exp − xi − cij ∕bj 2 ,

(32)

where, cij and bj are the mean and standard deviation of the Gaussian function on the jth term of the ith input linguistic variable to the node of the layer. 3. The rule layer implements the fuzzy inference mechanism. The output of this layer is given as

𝜑̄ j (x) =

N ∏

(33)

fij ,

j=1

where, N = nodes.

∏n i=1

Ni , Ni is ith input membership layer

4. The output layer

∑N yl =

N Wlj ⋅ 𝜑̄ j (x) � Wlj ⋅ 𝜑j (x), = ∑N 𝜑̄ j=1 j=1 j (x)

j=1

l = 1, 2, … m. (34)

4.2 Design of adaptive fuzzy neural network controller From (26), the chattering on the sliding surface is cased by the constant value of y and the discontinuous function sgn(s) . Taking into account the good approximation capability of the fuzzy neural network, a fuzzy neural network

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International Journal of Machine Learning and Cybernetics

controller is employed to adjust the sliding control gain. s Let the sliding mode controller K ‖s‖ be replaced by the output of fuzzy neural network y . The block diagram of the adaptive fuzzy neural network control scheme is shown in Fig. 4. The control law of (24) is assumed to take the following form: (35) ∗ where, W is the optimal weight vector, 𝜀b is the approximation error, ||𝜀b || ≤ 𝜀̄ , 𝜀̄ is a small positive constant. y is the estimated value of y∗ , we can get:

Apparently, we should keep W a constant when the system ̃̇ = −W ̂̇ ,then differentiatconverges, so we have Ẇ i = 0 and W ing V3 with respect to time gets: ( ( )) V̇ 3 = −Ka sT s + sT (𝜀 − y) + sT 𝜔̃ T 𝜙(x) + tr 𝜔̃ T 𝜇−1 𝜔̃̇

+

3 (T T ( T −1 )) 1 ∑ ̃ TW ̃̇ . ̇ + s 𝜔̃ 𝜙(x) + tr 𝜔̃ 𝜇 𝜔̃ + W 2𝜂 i=1 i i

y∗ = W ∗T 𝜑(s) + 𝜀b ,

(36) ∗ W ̃ where, W is the estimated value of W  , is the estimation error of weight vector T

y = W 𝜑(s),

̃ = W ∗ − W. (37) W The adaptive fuzzy neural network controller can be designed as: (38) To overcome the impacts of the unknown parameters and amplitude varitions for the controller accuracy in the APF system, the adaptive law of adaptive fuzzy neural network controller is designed as:

3 1 ∑ ̃ T ̃̇ W W = −Ka sT s + sT 𝜀 − sT W T 𝜑(s) 2𝜂 i=1 i i

Substituting adaptive law (39) into (41) yields: 3 ∑ ( ) ̃ T 𝜑(s) + 1 ̃ T 𝜂(𝜑i (s)s + 𝜎Wi ) W V̇ 3 = −Ka sT s + sT 𝜀 − sT W 𝜂 i=1 i

= − Ka sT s + sT 𝜀 +

) 1 ∑ ̃ T( ̃ T 𝜑i (s) W 𝜂(𝜑i (s)s + 𝜎Wi ) − 𝜂sT W i 𝜂 i=1 i

= − Ka sT s + sT 𝜀 +

) 1 ∑ ̃ T( ̃ T 𝜑i (s) + 𝜎Wi W 𝜂𝜑i (s)sT − 𝜂sT W i 𝜂 i=1 i

3

3

u = 𝜔̂ T 𝜙(x) + y + Ka s.

(

)

̃̇ i = 𝜂 𝜑i (s)sT + 𝜎Wi , W

(39)

where, Wi ∈ ℜN×1  ; 𝜂 > 0  , 𝜎 > 0  , 𝜎Wi can improve the robustness of the controller. Theorem 1  The APF system with the control law (38) and adaptive law (39) can guarantee the asymptotic stability of the closed-loop Proof  Define a Lyapunov function candidate as

1 T 1 1 ∑ ̃T ̃ W W. ̃ + s s + tr(𝜔̃ T 𝜇−1 𝜔) 2M 2 2𝜂 i=1 i i 3

V3 =

Fig. 4  Block diagram of an adaptive fuzzy-neural-network based on RBFNN control scheme

13

(40)

(41)

= − Ka sT s + 𝜎

3 ∑

̃ T Wi + sT 𝜀. W i

i=1

(42) ∗ 2 ̃ T W ≤ −0.5�W ̃ �2 According to inequality W � � + 0.5‖W ‖  , ‖sT 𝜀‖ ≤ 0.5sT s + 0.5𝜀̄ 2 , (42) becomes: ‖ ‖ ) ( 1 T 𝜎 ̃ ‖2 W + 𝜍 ∗, V̇ 3 ≤ − y − s s− ‖ (43) 2 2‖ ‖

̃ ‖2 1 2 where, 0.5 < y ; 𝜍 ∗ = 𝜎2 ‖ ‖W ‖ + 2 𝜀̄  , we can get: V̇ 3 ≤ −𝜇V2 + 𝜍 ∗ ,

(44) where, 𝜇 = min(2y − 1, 𝜂𝜎)  , (44) meets the following conditions:

V̇ 3 ≤

) ] ( 3 [ ∗ ∑ 𝜍 ∗ −𝜇t 𝜍 , e + V2 (0) − 𝜇 𝜅 i=1

(45)

International Journal of Machine Learning and Cybernetics Table 1  System parameters Supply voltage and frequency Vs1 = Vs2 = Vs3 = 220V, f = 50Hz Nonlinear load R = 20 Ω , L = 2mH Active power filter parameters Lc = 10mH  , Rc = 0.1Ω , C = 5000𝜇Fvdcref = 1000V fsw = 10KHz Switching frequency

̃ are bounded. Since V3 (0) is bounded and where, s(t) and W(t) bounded and non-increasing, we can conclude that V3 (t) is � � √ ∗ Ωs = s ∶ �s� ≤ 2𝜍 ∕𝜇  . According to the Barbalart lemma, s(t) will asymptotically converge to zero, limt→∞ s(t) = 0 . It can be concluded e(t) will asymptotically converge to zero, limt→∞ e(t) = 0 . Thus, according to the adaptive mechanism in (39), we can know that the overall system is asymptotically stable.

5 Simulation study In order to verify the feasibility and advantage of the proposed control scheme, we verify it using Matlab/Simulink package with SimPower Toolbox.APF system parameters are shown in Table 1.

Fig. 5  The grid current waveform without APF

Sliding parameter is 𝜆 = 100, 000 . Adaptive parameters 𝜂 = 500, 000  , 𝜎 = 1 . The number of hidden layer’s nodes in RBF network node = 5, the centric vector c = [15 7.5 0 − 7.5 − 15] and base width vector b = [1 1 1 1 1]T  . Initial weights of network are adjusted to zero. When, the switch of compensation circuit is closed and APF begins to work. To verify the effectiveness and robustness of the proposed APF control strategy, nonlinear loads increases totally two times during the entire process and nonlinear load is at the time of 0.1 s. Besides, multiple comparisons are implemented in the simulation process between proposed controller and hysteresis controller. Figure 5 shows the grid current waveform without APF. It can be seen that there exists severe distortion of the grid current waveform due to the effects of nonlinear load. Figure 6 is the grid current waveform with the adaptive FNN based on RBFNN for current control of APF. From Fig. 6, grid current distortion has been significantly improved with the proposed scheme. Figure 7 is instruction current and compensation current using adaptive FNN based on RBFNN and Fig. 8 shows the compensation current’s tracking error. It is seen that the compensation current can track the instruction current properly using the proposed controller and trackingerror keeps within a reasonable range. Figure 9 is the waveform of direct current (DC) side voltage.

100 80 60 40

is(A)

20 0 -20 -40 -60 -80 -100 0

Fig. 6  The grid current waveform using adaptive FNN based on RBF NN

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

100 80 60 40

iL(A)

20 0 -20 -40 -60 -80 -100 0

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International Journal of Machine Learning and Cybernetics

Fig. 7  Instruction current and compensation current using adaptive FNN based on RBF NN

50

icref

40

ic

30

icref and ic(A)

20 10 0 -10 -20 -30 -40 -50 0

Fig. 8  Compensation current’s tracking error using adaptive FNN based on RBF NN

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.1

0.12

0.14

0.16

0.18

0.2

50 40 30

error(A)

20 10 0 -10 -20 -30 -40 -50 0

0.04

0.06

0.08

900

vref

800

vdc

700

vref and vdc(V)

Fig. 9  The waveform of dc side voltage. a The harmonic spectrum of load current without controller in t = 0 s. b The harmonic spectrum of load current without controller in t = 0.06 s. c The harmonic spectrum of load current without controller in t = 0.16 s

0.02

600 500 400 300 200 100 0 -100 0

0.02

0.04

Figure 10 shows the harmonic spectrum of load current in different times, showing that nonlinear loads generate many harmonics in grid current. In Fig. 10, (a) is the grid current spectrogram without APF in t = 0 s, where total harmonic distortion (THD) = 24.22%, (b) is the grid current spectrogram without APF in t = 0.06 s, where THD = 24.30%, (c) is the grid current spectrogram without APF in t = 0.16 s, where THD = 21.72%. Figure  11 draws the harmonic

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0.06

0.08

0.1

Time(s)

0.12

0.14

0.16

0.18

0.2

spectrum of source current after harmonics compensation using the adaptive FNN based on RBF NN in different times. At the end of this section, for the purpose of demonstrating the superiority of the proposed controller over conventional hysteresis control, comparison between them is also given in Table 2. To facilitate observation and comparison, the data of Fig. 11 is enumerated in Table 2. From Fig. 11 and Table 2, it is obvious to discover that with the increasing of nonlinear

International Journal of Machine Learning and Cybernetics

Fundamental (50Hz) = 31.95 , THD= 24.22%

Mag (% of Fundamental)

20

15

10

5

0

0

200

400

600

800

1000

Frequency (Hz)

1200

1400

1600

1800

2000

(a) The harmonic spectrum of load current without controller in t=0s Fundamental (50Hz) = 31.94 , THD= 24.30%

Mag (% of Fundamental)

20

15

10

5

0

0

200

400

600

800

1000

Frequency (Hz)

1200

1400

1600

1800

2000

(b) The harmonic spectrum of load current without controller in t=0.06s Fundamental (50Hz) = 61.56 , THD= 21.72%

Mag (% of Fundamental)

20 18 16 14 12 10 8 6 4 2 0

0

200

400

600

800

1000

Frequency (Hz)

1200

1400

1600

1800

2000

(c) The harmonic spectrum of load current without controller in t=0.16s Fig. 10  The harmonic spectrum of load current without controller in different times

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International Journal of Machine Learning and Cybernetics

Fundamental (50Hz) = 31.95 , THD= 24.22%

Mag (% of Fundamental)

20

15

10

5

0

0

200

400

600

800

1000

Frequency (Hz)

1200

1400

1600

1800

2000

(a) The harmonic spectrum of source current after harmonics compensation using adaptive FNN based on RBF NN in t=0s Fundamental (50Hz) = 32.56 , THD= 1.72%

Mag (% of Fundamental)

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

200

400

600

800

1000

Frequency (Hz)

1200

1400

1600

1800

2000

(b) The harmonic spectrum of source current after harmonics compensation using adaptive FNN based on RBF NN in t=0.06s Fundamental (50Hz) = 61.54 , THD= 1.52%

Mag (% of Fundamental)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

200

400

600

800

1000

Frequency (Hz)

1200

1400

1600

1800

2000

(c) The harmonic spectrum of source current after harmonics compensation using adaptive FNN based on RBF NN in t=0.16s Fig. 11  The harmonic spectrum of source current after harmonics compensation using adaptive FNN based on RBF NN in different times

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International Journal of Machine Learning and Cybernetics Table 2  THD values in different simulation time Different time

THD (%) Adaptive FNN based on RBF HysNN teresis control

0 0.06 s 0.16 s

24.22 1.72 1.52

24.22 2.18 1.64

load impact, the adaptive FNN based on RBFNN controller for APF illustrates better tracking performance, suitability and robustness at different simulation stages than the one based on hysteresis control.

6 Conclusion In this paper, an adaptive FNN based on RBFNN is applied in three-phase shunt APF successfully. We establish the mathematical model of APF and use sliding mode controller to ensure the robustness of the system. Then, RBF NN is used to approximate the nonlinear function in APF dynamic model and the sliding mode gain is adjusted by adaptive fuzzy-neuralnetwork systems to compensate the neural approximation error and eliminate the existing chattering. The parameters of these approaches can be adaptively updated based on the Lyapunov analysis. The simulation results illustrate that the APF system based on the proposed method has the outstanding compensation performance and strong robustness. However, it is not an in-depth study for the harmonic detection. Because of accurate harmonic detection can effectively improve the compensation performance of APF system, and the proposed harmonic detection method has some delay in phase due to the use of filtering algorithm. Therefore, the next step of this research is to improve the accuracy of harmonic detection. Acknowledgements  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. BK20171198, the Fundamental Research Funds for the Central Universities under Grant No. 2017B20014, 2017B21214.

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