Sliding mode control of a shunt hybrid active power filter based on the inverse system method

Electrical Power and Energy Systems 57 (2014) 39–48

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Sliding mode control of a shunt hybrid active power filter based on the inverse system method Wei Lu ⇑, Chunwen Li, Changbo Xu Department of Automation, Tsinghua University, Beijing 100084, China

a r t i c l e

i n f o

Article history: Received 17 January 2013 Received in revised form 12 September 2013 Accepted 25 November 2013

Keywords: Harmonics Hybrid active power filter Sliding mode control Inverse system method Zero dynamics

a b s t r a c t In this paper, an inverse system method based sliding mode control strategy is proposed for the shunt hybrid active power filter (SHAPF) to enhance the harmonic elimination performance. Based on the inverse system method, the d-axis and q-axis current dynamics of the SHAPF system are firstly linearized and decoupled into two pseudolinear subsystems. Then a sliding mode controller is designed to reject the influence of load changes and system parameter mismatches on the system stability and performance. It is proved that the current dynamics are exponentially stabilized at their reference states by the controller. Moreover, the stability condition of the zero dynamics of the SHAPF system is presented, showing that the zero dynamics can be bounded by adding an appropriate DC component to the reference of the q-axis current dynamics. Furthermore, a proportional-integral (PI) controller is employed to facilitate the calculation of the DC component. Simulation and experimental results demonstrate the effectiveness and reliability of the SHAPF with the proposed control strategy. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction The widespread application of power electrical devices (e.g., diode rectifiers) has increased the harmonic pollution in modern power transmission/distribution systems. The harmonics generated by nonlinear loads can cause additional power losses, interfere with nearby communication networks and disturb sensitive loads [1,2]. Therefore, many international standards such as IEEE 519-1992 and IEC 61000-3-2 have been recommended to limit the harmonic pollution. Traditionally, low-cost passive power filters (PPFs) with high efficiencies were widely used to eliminate the harmonics. However, the bulky PPFs only provide fixed harmonic compensation and they detune with age [3]. These drawbacks can be overcome by the power converter based active power filters (APFs), but they are usually expensive and have high operating losses [4–8]. For the sake of improving the compensation performance and reducing the cost of the APFs, a number of topologies of hybrid active power filters (HAPFs) have been proposed [9–15]. Peng et al. proposed a HAPF system combining a series APF and a shunt PPF [9]. In this system, the APF endured high load currents works as a ‘‘harmonic isolator’’ between the source and the nonlinear load. A novel

⇑ Corresponding author. Tel.: +86 10 62799024, mobile: +86 15210958618; fax: +86 10 62795356. E-mail addresses: [email protected] (W. Lu), [email protected]. edu.cn (C. Li), [email protected] (C. Xu). 0142-0615/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2013.11.044

topology is proposed in [10], where the APF is connect in series with a C-type PPF. However, an additional power supply is needed to support the DC-link capacitor. Ref. [11] presented a combined system of many PPFs connected in series with an APF via a matching transformer. This topology might not be preferable since many passive components are required. In particular, a novel shunt hybrid active power filter (SHAPF), where three tuned PPFs are connected in series with a small-rated APF without any matching transformers, has attracted much attention [12–15]. Since the source voltage is applied across the PPF, the required rating of the APF can be substantially reduced. Furthermore, no additional output filters are needed to suppress the switching ripples produced by the power converter. The control strategy is important to enhance the harmonic elimination performance of the SHAPF. Many control strategies have been proposed for the SHAPF. In [13], a linear feedback-feedforward controller is designed for the SHAPF. Because the dynamic model of the SHAPF system contains multiplication terms of the control inputs and the state variables, it is not easy to achieve both satisfactory steady-state and transient-state performances with the linear control strategy. To deal with the nonlinear characteristic of the SHAPF, a sliding mode controller was presented in [14], which has the property of robustness against load changes and system parametric uncertainties. But the steady-state errors may still be nonzero due to the absence of integrators in the closed loop system. In [15], a Lyapunov function based control strategy is developed to globally stabilize the SHAPF system. Unfortunately, owing to the difficulty in estimating the ripple component of the

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W. Lu et al. / Electrical Power and Energy Systems 57 (2014) 39–48

DC-link capacitor voltage, the obtained controller is an approximate one. In this paper, an inverse system method based sliding mode control strategy is proposed for the SHAPF. The inverse system method is one of the linearization and decoupling (L&D) methods, which does not need complicated coordinate transformation compared with the differential geometric L&D based methods [16–21]. For the control of power converters [18,19] and moment gyros [20,21], it has exhibited desirable steady and dynamic performances. Here we decouple the nonlinearity of the SHAPF system with the inverse system method, such that the d-axis and q-axis current dynamics of the SHAPF system can be regulated independently. In addition, the sliding mode control [22–27] is applied to the decoupled pseudolinear system to reject the influence of external disturbances and system parameter mismatches. Since the SHAPF system has three internal dynamic variables, the internal stability is analyzed based on the derived stability condition of the zero dynamics. Moreover, to avoid the difficulty in calculating the DC component from the stability condition, a PI controller over the square of the DC-link capacitor voltage is adopted. This paper is organized as follows. In Section 2, the nonlinear mathematic model of the SHAPF system in the d–q reference frames is described. A novel control strategy combining the sliding mode control and the inverse system method is presented in Section 3. The stability of the SHAPF including the internal and external dynamics is analyzed in Section 4. Simulations for testing the effectiveness and reliability of the proposed control strategy are conducted in Section 5. Experimental results of a laboratory prototype are presented in Section 6. Finally, conclusions are given in Section 7.

2. The SHAPF model The topology of the SHAPF is shown in Fig. 1. A small-rated APF using a voltage-source power inverter is directly connected in series with three tuned PPFs. Since the source voltage is taken by the PPF, the required ratings of the inverter and DC-link capacitor voltage are much smaller than those of a stand-alone shunt APF. The three-phase diode bridge rectifier with RL loads is considered as a nonlinear load. In this figure, vSj, vLj, vCj, iSj, iLj and iFj, j = a, b, c, represent the three-phase source voltage, the point of common coupling (PCC) voltage, the PPF capacitor voltage, the source current, the load current and the compensating current, respectively. Cdc and vdc are the capacitance of the DC-link capacitor and the voltage across the capacitor. LF, CF and RF represent the inductance, the capacitance and the resistance of the PPF, respectively.

The dynamic model of the SHAPF under the synchronous rotating d–q reference frame can be expressed by the following differential equations [15]:

8 _ LF iFd ¼ RF iFd þ xLF iFq  v Cd  ud v dc þ v Ld > > > > > > < LF i_Fq ¼ RF iFq  xLF iFd  v Cq  uq v dc þ v Lq ; C F v_ Cd ¼ iFd þ xC F v Cq > > > > _ C v ¼ i  x C v > F Cq Fq F Cd > : C dc v_ dc ¼ ud iFd þ uq iFq

ð1Þ

where iFd and iFq denote the d–q axis compensating currents, vCd and vCq are the d–q axis PPF capacitor voltages, vLd and vLq represent the d–q axis PCC voltages, ud and uq are the d–q axis duty ratio functions, and x is the source angle frequency of the source voltage. To facilitate the controller design, the SHAPF system model can be formally rewritten as follows:



x_ ¼ f ðxÞ þ gðxÞu y ¼ hðxÞ

ð2Þ

;

where x = [iFd, iFq, vCd, vCq, vdc]T stands for the system state vector, the vector u = [ud, uq]T is taken as the system control variables, the vector y = [y1, y2]T = [iFd, iFq]T represents the system outputs. The functions f(x), g(x) and h(x), respectively, are given as 3 ðRF iFd þ xLF iFq  v Cd þ v Ld Þ=LF 7 6 6 ðRF iFq  xLF iFd  v Cq þ v Lq Þ=LF 7 7 6 7; f ðxÞ ¼ 6 ði þ x C v Þ=C F Cq F Fd 7 6 7 6 ðiFq  xC F v Cd Þ=C F 5 4 0   iFd and hðxÞ ¼ : iFq 2

2 6 6 6 gðxÞ ¼ 6 6 6 4

v dc =LF 0 0 0 iFd =C dc

0

3

v dc =LF 7 7 7 7 0 7 7 0 5 iFq =C dc

It should be noted that the obtained multi-input multi-output (MIMO) system model (2) is affine nonlinear due to the multiplication terms of the state variables and the control variables. In addition, the state variables are strongly coupled to each other. These two problems can be properly handled by the inverse system method, which aims at directly finding the relationship between the control variables and the system outputs. 3. Controller design In this section, the synthesis of the sliding mode controller based on the inverse system method for the SHAPF is presented. 3.1. L&D of the SHAPF For the L&D of the SHAPF system with the inverse system method, we firstly prove the invertibility of the system according to the Interactor algorithm [16]. Based on the system model (2), we differentiate the output vector y with respect to the time until the control variables ud and uq appear explicitly, which leads to the following equation:

JðuÞ ¼



y_ 1 y_ 2



 ¼

ðRF iFd þ xLF iFq  v Cd þ v Ld  v dc ud Þ=LF ðRF iFq  xLF iFd  v Cq þ v Lq  v dc uq Þ=LF

 :

ð3Þ

The Jacobi matrix of J(u) with respect to the control vector u can be calculated as follows:

 v dc =LF @JðuÞ ¼ @uT 0

Fig. 1. Topology of the SHAPF.

0 v dc =LF

 :

ð4Þ

Because the DC-link capacitor voltage vdc is positive in the operation range, oJ(u)/ouT is nonsingular. Moreover, the relative degree vector of the system is a = [a1, a2]T = [1, 1]T, and a1 + a2 = 2 is strictly less than the system order n = 5. Therefore, there are two

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W. Lu et al. / Electrical Power and Energy Systems 57 (2014) 39–48

observable external dynamic variables and three unobservable internal dynamic variables, respectively. According to the inverse system theory [16], the SHAPF system (2) is invertible. Define the new input variables









u1 y_ 1 ¼ : u2 y_ 2

ð5Þ

Substituting (5) into (3), we can construct an inversion of the SHAPF system based on the state feedback as follows:



ud ¼ ðv Ld  RF iFd þ xLF iFq  v Cd  LF u1 Þ=v dc uq ¼ ðv Lq  RF iFq  xLF iFd  v Cq  LF u2 Þ=v dc

ð6Þ

:

If we cascade the inverse system with the original system (2), two decoupled pseudolinear subsystems both with transfer function G(s) = 1/s are formed, as shown in Fig. 2. It can be seen that the nonlinearity of the system is canceled by the nonlinear feedback law (6), and the system outputs iFd and iFq are independently controlled by the new inputs u1 and u2, respectively.

Fig. 2. Control block diagram of the SHAPF.

the manifold (10) is attractive and invariant. Additionally, the external   dynamics of iFd and iFq converge to iFd andiFq exponentially, respectively. Proof. We select a Lyapunov function as

VðxÞ ¼ 3.2. Sliding mode controller design For the pseudolinear subsystem, the steady-state error of the system output may be nonzero under the condition of external disturbances and system parameter variations. To synthesize a robust SHAPF system, a sliding mode controller is designed based on the linearized model (5). Taking into account the control objective of the SHAPF system is to force the compensating currents iFd and iFq to track their refer  ence values iFd and iFq , we design a sliding mode surface with integral actions [24,25] as

S¼



Sd Sq

"

 ¼

 iFd  iFq

 iFd þ  iFq þ

Rt

 a1 0 ðiFd Rt  a2 0 ðiFq

 iFd Þds  iFq Þds



"

_i þ ðk11 þ u1 Fd ¼ _i þ ðk21 þ u2 Fq

 a1 ÞðiFd  a2 ÞðiFq

 _ VðxÞ ¼ S T S_ ¼ Sd ½ði_Fd  i_Fd Þ þ a1 ðiFd  iFd Þ þ Sq ½ð_iFq  i_Fq Þ 

þ a2 ðiFq  iFq Þ:

  _ VðxÞ ¼ Sd ½ði_Fd  u1 Þ þ a1 ðiFd  iFd Þ þ Sq ½ði_Fq  u2 Þ þ a2 ðiFq

 iFq Þ 

 iFd Þ þ k12 signðSd Þ  iFq Þ þ k22 signðSq Þ

 k22 signðSq Þ 



;

ð8Þ

Fq

4. Stability analysis In this section, we will analyze the stability of the external and internal dynamics of the SHAPF system with the proposed control strategy. For the stability analysis of the system external dynamics of iFd and iFq, we define a sliding mode manifold as below:

S ¼ ½Sd ; Sq T ¼ 0:

ð10Þ

k12 >

 iFd Þj  iFq Þj

;



ð11Þ

ð14Þ

_ < 0, which implies According to (11), we can see from (14) that VðxÞ the attractiveness of the manifold (10). Moreover, since limS!0 ST S_ ¼ 0, the manifold is invariant. When the sliding mode occurs on the manifold (10), then

S ¼ S_ ¼ 0:

ð15Þ

Thus, the tracking behavior of the system external dynamics is governed by the following equations:

(

_i  i_Fd ¼ a1 ði  iFd Þ Fd Fd : _i  i_Fq ¼ a2 ði  iFq Þ Fq Fq

ð16Þ

Because a1 and a2 are positive constants, iFd and iFq exponentially   converges to iFd and iFq , respectively. The stability of the internal dynamics is usually determined by the zero dynamics where the tracking error of the external dynamics approaches to zero [28–31]. Generally, the three-phase balanced load current in the d–q axis can be expressed as

8 N X > > > ILdk cosðkxt þ wLdk Þ > iLd ¼ ILd0 þ > < k¼6;12;18::: > > > > > : iLq ¼ ILq0 þ

Theorem 1. Under the control inputs defined in (8), with

k22 >

 k22 jSq j ¼ ½k12  jk11 ðiFd  iFd ÞjjSd j  ½k22  jk21 ðiFq  iFq ÞjjSq j:

The control block diagram of the proposed control strategy is illustrated in Fig. 2.

 jk11 ðiFd  jk21 ðiFq



6 jk11 ðiFd  iFd ÞjjSd j  k12 jSd j þ jk21 ðiFq  iFq ÞjjSq j

#

ð9Þ

(



¼ Sd ½k11 ðiFd  iFd Þ  k12 signðSd Þ þ Sq ½k21 ðiFq  iFq Þ

ð7Þ

;

 ud ¼ ½v Ld  LF i_Fd  RF iFd þ xLF iFq  v Cd  LF ðk11 þ a1 ÞðiFd  iFd Þ  LF k12 signðSd Þ=v dc :   _ uq ¼ ½v Lq  LF i  RF iFq  xLF iFd  v Cq  LF ðk21 þ a2 Þði  iFq Þ  LF k22 signðSq Þ=v dc Fq

ð13Þ

Substituting (5) and (8) into (13), we can get

#

where k11 and k21 are constant gains, k12 and k22 are the switching gains, and sign() is the switching function. Substituting (8) into (6), the nonlinear control law can be directly derived: (

ð12Þ

h whose time-derivative is

where a1 and a2 are positive constants. Moreover, the inputs u1 and u2 are designed as



1 T S S; 2

N X

;

ð17Þ

ILqk cosðkxt þ wLqk Þ

k¼6;12;18:::

where ILd0, ILq0, ILdk, ILqk, wLdk and wLqk are constants. Due to the nature of the tuned PPF, there is a fixed amount of fundamental current

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W. Lu et al. / Electrical Power and Energy Systems 57 (2014) 39–48

flowed into the SHAPF. Moreover, the active and reactive parts of the current in the d–q axis are

(

pffiffiffi 3V cos hF =jZ F j pffiffiffi m ; ¼ 3V m sin hF =jZ F j

IFd0 ¼ IFq0

ð18Þ

where Vm is the root mean square (RMS) value of the source voltage, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jZ F j ¼ R2F þ ðxLF  1=xC F Þ2 and hF = arctan [(1/xCF  xLF)/RF] are the amplitude and impedance angle of the PPF impedance, respectively. To eliminate the harmonic component of the load current, the reference compensating current of the SHAPF can be written as

8 N X >  > > i ¼ I  ILdk cosðkxt þ wLdk Þ > Fd0 Fd > < k¼6;12;18... > >  > > > : iFq ¼ IFq0 þ Iex 

N X

;

ð19Þ

Obviously, the zero dynamics vCd and vCq with period T = p/3x; are stable. Since the zero dynamics vdc in (22) is always positive in the operation range, we apply a nonlinear transformation g ¼ v 2dc to it, then we can get h    i g_ ¼ iFd v Ld  LF _iFd  RF iFd  v Cd þ iFq v Lq  LF i_Fq  RF iFq  v Cq =ð2C dc Þ:

ð24Þ pffiffiffi For the three-phase balanced power system, v Ld ¼ 3V m and   vLq = 0. Considering (19) and (23), the term iFd LF i_Fd  v Cd    þiFq v Lq  LF _iFq  v Cq is periodic with the period T. Hence, the average value of g_ in a period is

1 T

Z

T

g_ dt ¼

0

ILqk cosðkxt þ wLqk Þ

1 2TC dc

¼

k¼6;12;18...

where the constant Iex represents the additional fundamental reactive current absorbed by the SHAPF. The following theorem shows the relationship between the additional current Iex and the stability of the zero dynamics. Theorem 2. With the reference output (19), the zero dynamics are instable when Iex = 0. The zero dynamics are stable when

8 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u > N X > u > > > IFq0 P t12 ðI2Ldk þ I2Lqk Þ > > < k¼6;12;18... vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : u > N > X u2 > > tI  1 > I ¼ I þ ðI2Ldk þ I2Lqk Þ > ex Fq0 Fq0 2 > :

ð20Þ

k¼6;12;18...

Proof. In term of the system model (2), the internal dynamics are described by:

8 > < v_ Cd ¼ ðiFd þ xC F v Cq Þ=C F v_ Cq ¼ ðiFq  xC F v Cd Þ=C F : > :_ v dc ¼ ðiFd ud þ iFq uq Þ=C dc

ð21Þ

h   As the external dynamics of iFd and iFq approach to iFd and iFq , substituting the control law (9) into (21), the zero dynamics are given by: 8 v_ Cd ¼ ðiFd þ xC F v Cq Þ=C F > > < v_ Cq ¼ ðiFq  xC F v Cd Þ=C F h i: > > : v_ dc ¼ 1 i ðv Ld  LF i_  RF i  v Cd Þ þ i ðv Lq  LF i_  RF i  v Cq Þ Fd Fd Fq Fq Fq Fd C v

Z

T



2

2

iFd v Ld  RF iFd  RF iFq dt

0

" # N   RF 1 X I2ex þ 2IFq0 Iex þ I2Ldk þ I2Lqk : 2 k¼6;12;18... 2C dc

Since RF, ILdk and ILqk are not zero, when Iex = 0, the following relation holds:

1 T

Z

T

g_ dt ¼ 

0

N   X RF I2 þ I2Lqk < 0: 4C dc k¼6;12;18... Ldk

1 T

Z

T

g_ dt ¼ 0:

ð27Þ

0

This means that g is also periodic with the period T. Moreover, if the DC-link capacitor Cdc is large, the zero dynamics vdc can be bounded in a small neighborhood. Consequently, the stability of the zero dynamics is guaranteed. Based on Theorem 2, we can see that the stability of the zero dynamics can be secured by adding an appropriate fundamental reactive current to the q-axis reference compensating current. Since a precise estimation of the parameter in (20) is difficult, it is not easy to directly compute the current Iex from (20). From the relationship between g and vdc, it is shown that g will be decreased if no additional fundamental reactive current is injected to the SHAPF. Therefore a PI controller is employed to regulate the current Iex by controlling g. That is

Iex ¼ kp ðg  g Þ þ ki

Z

t

ðg  g Þds:

0

ð22Þ

Taking account of (19), the analytic solutions of the zero dynamics

vCd and vCq are

ð23Þ

k¼6;12;18...

where

 "

Ak Bk ^k A ^ Bk

 ¼ # ¼

"

1 2

1k

"

1 1k

2

kILdk cos wLdk þ ILqk sin wLqk kILdk sin wLdk  ILqk cos wLqk kILqk cos wLqk  ILdk sin wLdk kILqk sin wLqk þ ILdk cos wLdk

ð26Þ

This implies that the transformed variable g decays oscillatingly. Because of the positive correlation between the zero dynamics vdc and g, the instability of the system zero dynamics is obtained. When Iex – 0, substituting (20) into (25), we can get

dc dc

8 N X > > > ðAk sin kxt þ Bk cos kxtÞ > v Cd ¼ x1C F ½IFq0 þ Iex þ > < k¼6;12;18... " # ; N > > > v ¼ 1 I þ X ðA ^ ^ > > Fd0 k sin kxt þ Bk cos kxtÞ : Cq xC F

ð25Þ

# and # : Fig. 3. System control structure of the SHAPF.

ð28Þ

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W. Lu et al. / Electrical Power and Energy Systems 57 (2014) 39–48

350

v Labc /V

v La v Lb

0

v Lc -350 0.06

0.08

0.1

0.12

0.14

0.16

0.18

70

iLabc /A

iLa iLb

0

iLc -70 0.06

0.08

0.1

0.12

0.14

0.16

0.18

30

iFabc /A

iFa iFb

0

iFc -30 0.06

0.08

0.1

0.12

0.14

0.16

0.18

70

iSabc /A

iSa iSb

0

iSc -70 0.06

0.08

0.1

0.12

0.14

0.16

0.18

3

ieabc /A

iea ieb

0

iec -3 0.06

0.08

0.1

0.12

0.14

0.16

0.18

450

v Cabc /V

v Ca v Cb

0

v Cc -450 0.06

0.08

0.1

0.08

0.1

0.12

0.14

0.16

0.18

0.12

0.14

0.16

0.18

v dc /V

161 160 159 0.06

Time (s) Fig. 4. Steady-state performance of the SHAPF.

60

60

iLa /A

iSa/A

Fundamental(50Hz)=54.99,THD=24.43%

40

20

20 0

Fundamental(50Hz)=53.87,THD=1.74%

40

0 0

5

10

15

20

25

0

5

10

15

Harmonic order

Harmonic order

(a)

(b)

20

25

Fig. 5. Harmonic spectra of the load current and the source current in phase-a: (a) Harmonic spectrum of the load current in phase-a. (b) Harmonic spectrum of the source current in phase-a.

44

W. Lu et al. / Electrical Power and Energy Systems 57 (2014) 39–48

350

v Labc /V

v La v Lb

0

v Lc -350 0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

120

iLabc /A

iLa iLb

0

iLc -120 0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

50

iFabc /A

iFa iFb

0

iFc -50 0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

120

iSabc /A

iSa iSb

0

iSc -120 0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

10

ieabc /A

iea ieb

0

iec -10 0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

450

v Cabc /V

v Ca v Cb

0

v Cc -450 0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

v dc /V

163 160 157 0.16

Time (s) Fig. 6. Transient-state performance of the SHAPF.

5. Simulation results

3.5

RF

THD%

3

LF

2.5

CF 2 1.5 1 -30

-20

-10

0

10

20

30

Mismatches of prameters R F, LF and CF/% Fig. 7. The THD of source current under PPF parameter mismatches.

where g ¼ V 2dc is the reference value of g, kp and ki are the proportional and integral gains of the controller, respectively. Note that the above PI controller is usually considered as the outer control loop of the SHAPF [15]. This paper gives a new insight into the PI controller from the stability of the system zero dynamics.

To verify the effectiveness of the proposed control strategy for the SHAPF, we simulate the system under Matlab/Simulink environment. The system parameters selected for the simulation are given in Appendix A. The system control structure of the SHAPF is shown in Fig. 3. The total harmonic distortion (THD) of the source currents is taken to evaluate the harmonic elimination  performance. Moreover, the error current iej ¼ iFj  iFj is defined to evaluate the tracking performance between the compensating  current iFj and its reference iFj . The performance of the SHAPF with the proposed controller is analyzed in the following cases. 5.1. Steady-state performance of the SHAPF In this case, the three-phase rectifier L1 is connected at the PCC as the nonlinear load. The steady-state harmonic elimination performance of the SHAPF with the proposed control strategy is shown in Fig. 4. The waveforms of the PCC voltage, the load

45

W. Lu et al. / Electrical Power and Energy Systems 57 (2014) 39–48

350

v Labc /V

v La v Lb

0

v Lc -350 0.06

0.08

0.1

0.12

0.14

0.16

0.18

70

iLabc /A

iLa iLb

0

iLc -70 0.06

0.08

0.1

0.12

0.14

0.16

0.18

30

iFabc /A

iFa iFb

0

iFc -30 0.06

0.08

0.1

0.12

0.14

0.16

0.18

70

iSabc /A

iSa iSb

0

iSc -70 0.06

0.08

0.1

0.12

0.14

0.16

0.18

ieabc /A

6 iea ieb

0

iec -6 0.06

0.08

0.1

0.12

0.14

0.16

0.18

v Cabc /V

450 v Ca v Cb

0

v Cc -450 0.06

0.08

0.1

0.08

0.1

0.12

0.14

0.16

0.18

0.12

0.14

0.16

0.18

v dc /V

163 160 157 0.06

Time (s) Fig. 8. Steady-state performance of the SHAPF with the linear control strategy.

current, the compensating current, the source current, the error current, the PPF capacitor voltage and the DC-link capacitor voltage are displayed. The harmonic spectra of the phase-a load current and source current after compensation are shown in Fig. 5a and b. It can be clearly seen that the three-phase load current is seriously distorted with the nonlinear load. When the control strategy is applied to the SHAPF, three-phase source current becomes balanced and sinusoidal. In the steady-state, the error current is limited to 2 A. The THD of the source current can be reduced from 24.43% to 1.74% by compensation, well below the requirement of the IEEE 519 standard (5%). The PPF capacitor voltage is periodic and bounded, which is consistent with the stability analysis of the zero dynamics. The DC-link capacitor voltage is adequately controlled around its reference value Vdc with a very slight ripple (within 1V), showing that the proposed control strategy is effective.

variation is created by connecting and disconnecting the threephase rectifier L2 at t = 0.2 s and t = 0.28 s, respectively, under the condition of the load L1 connecting at the PCC. The corresponding waveforms are shown in Fig. 6. The simulation results show that the proposed controller is able to force the compensating current to track its reference promptly when the nonlinear load changes. After a transient period of one cycle, the source current is almost sinusoidal with its THD of around 1.80%. Due to the sudden increase and decrease of the nonlinear load, the DC-link capacitor voltage fluctuates correspondingly, which takes about one cycle to reach its reference value Vdc. Furthermore, the voltage overshoot of the DC-link capacitor voltage dynamics is only 2% of Vdc in the transient-state. This verifies that the stabilizability of the SHAPF under time-varying nonlinear loads.

5.2. Transient-state performance of the SHAPF

5.3. Robustness to PPF parameter mismatches

To study the transient-state performance of the SHAPF, a step increase and decrease of the nonlinear load is considered. The load

The tuned PPFs are important components of the SHAPF. To evaluate the robustness of the proposed control strategy, a series

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W. Lu et al. / Electrical Power and Energy Systems 57 (2014) 39–48

350

v Labc /V

v La v Lb

0

v Lc

-350 0.06

0.08

0.1

0.12

0.14

0.16

0.18

70

iLabc /A

iLa iLb

0

iLc -70 0.06

0.08

0.1

0.12

0.14

0.16

0.18

30

iFabc /A

iFa iFb

0

iFc -30 0.06

0.08

0.1

0.12

0.14

0.16

0.18

iSabc /A

70 iSa iSb

0

iSc

-70 0.06

0.08

0.1

0.12

0.14

0.16

0.18

ieabc /A

6 iea ieb

0

iec -6 0.06

0.08

0.1

0.12

0.14

0.16

0.18

v Cabc /V

450 v Ca v Cb

0

-450 0.06

v Cc 0.08

0.1

0.08

0.1

0.12

0.14

0.16

0.18

0.12

0.14

0.16

0.18

v dc /V

163 160 157 0.06

Time (s) Fig. 9. Steady-state performance of the SHAPF with the sliding mode control strategy.

of simulations of the filter resistance RF, inductance LF and capacitance CF variations are conducted. The mismatches of the parameters are in the range of ±30% with respect to the rated values. The relationships between the parameter mismatches and the THD of the source current are presented in Fig. 7. It can be seen from the results that the THD of the source current remains within 3.5% in the conditions of PPF parameter mismatches. The results also show that too small LF or CF will degrade the harmonic elimination performance for the switching harmonic ripple generated by the SHAPF injecting into the power system. Moreover, the variation of RF has no evident effect on the SHAPF performance. Hence, we can conclude that the proposed control strategy is robust to the parameter mismatches of the PPF.

simulation are chosen as the same set in Appendix A. The SHAPF performances with the latter two control strategies are shown in Figs. 8 and 9, respectively. The simulation results show that the two control strategies are effective for the SHAPF to eliminate the harmonics. In the steady-state, the error current can be limited to 6 A and 3.5 A with the two control strategies, respectively. Additionally, the harmonic analysis of the source current after compensation is given in Fig. 10, where the THD of the source current with the two control strategies can be decreased within 5% after compensation, but this is not as good as that obtained by the proposed control strategy. Moreover, the ripple of the DC-link capacitor voltage with the linear control strategy is much greater compared with Figs. 4 and 9. Although we can properly select the parameter of the linear controller, it is difficult to simultaneously attain satisfactory performance of the source current and the DC-link capacitor voltage.

5.4. Control strategy comparison 6. Experimental results The performance is compared between the proposed control strategy, the linear control strategy presented in [13] and the sliding mode control strategy presented in [14]. The system parameters for

The proposed control strategy has been experimentally tested on a 50-kVA SHAPF laboratory prototype. In the prototype, a

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W. Lu et al. / Electrical Power and Energy Systems 57 (2014) 39–48

60

60

iSa/A

iSa/A

Fundamental(50Hz)=54.81,THD=4.58%

40 20 0

0

5

10

15

20

25

Fundamental(50Hz)=53.83,THD=2.93%

40 20 0

0

5

10

15

Harmonic order

Harmonic order

(a)

(b)

20

25

Fig. 10. Harmonic spectra of the source current in phase-a: (a) Linear control strategy. (b) Sliding mode control strategy.

Fig. 11. Prototype of the SHAPF: (a) Voltage source inverter. (b) PPF. (c) Control board.

Fig. 12. Performance of the SHAPF with proposed control strategy: (a) PCC voltage and load current. (b) PCC voltage and source current after compensation.

voltage source inverter with three SKM100GB063D IGBT modules is used as the main circuit of the SHAPF. The PPF is tuned at 7th order harmonics, and its parameters are as same as those presented in Appendix A. A low-cost FPGA chip EP2C20Q240 is used on the control board to implement the proposed control strategy. In this paper, the PCC voltage is measured downstream through an ac transformer with a 1/60 transformation ratio. The load and source currents are transformed into the voltage signals with proper sensing circuitries. The scale of the current to the voltage transformation is 300A/0.353V. Then the transformed signals are measured by a two-channel TDS1012 oscilloscope. Fig. 11 shows the SHAFP prototype, which consists of the voltage source inverter, the PPF and the FPGA based control board. Fig. 12 shows the waveforms of the PPC voltage and the source current before and after compensation with a 20 A nonlinear load. From Fig. 12a, it can be seen that the source current is highly distorted in the connection of the nonlinear load. The PCC voltage is

slightly distorted due to the current harmonics. The THD of the source current before compensation is about 25%. When the SHAPF is adopted, the source current becomes sinusoidal as presented in Fig. 12b. The THD of the source current can be reduced to 3% by compensation, well below the requirement of the IEEE 519 standard (5%). The experimental results demonstrate the effectiveness of the SHAPF with the proposed control strategy.

7. Conclusion A novel control strategy has been proposed for the SHAPF by combining the sliding mode control and the inverse system method. The SHAPF are linearized and decoupled into two pseudolinear subsystems by the inverse system method. Then a sliding mode controller is designed for the subsystems to reject the influence of external disturbances and system parameter mismatches. The stability of the SHAPF system with the proposed control strategy

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has been carefully analyzed. It is proved that the external dynamics converge to their references without any steady-state errors, and the zero dynamics can be stabilized by controlling the square of the DC-link capacitor voltage. The performance of the proposed strategy has been studied in various simulation conditions. It is shown that the SHAPF has satisfactory performance in reducing harmonics. The proposed control strategy exhibits higher performances compared with the linear control strategy and the sliding mode control strategy. In addition, a laboratory prototype has been built and tested, which further supports the effectiveness of the proposed control strategy. Appendix A. System parameters for simulation – – – – – – – – – –

RMS value and frequency of source voltage (Vm, f): 220 V, 50 Hz. Source impedance (LS, RS): 0.01 mH, 0.002 O. PPF parameters (CF, LF, RF): 100 lF, 2 mH, 0.2 O DC-link capacitor (Cdc): 10000 lF DC-link reference voltage (Vdc): 160 V Frequency of PWM: 6 kHz Three-phase rectifier L1: 10 O, 5 mH Three-phase rectifier L2: 15 O, 5 mH Parameters of the PI controller (kp, ki): 0.01 A/V, 0.005 A/V s Parameters of the controller (a1, a2, k11, k12, k21, k22): 1,000,000, 1,000,000, 500,000, 1,000,000, 500,000, 1,000,000

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