Adaptive Backstepping Control of Three-Phase Four-Wire Shunt Active Power Filters for Energy Quality improvement

DOI 10.1007/s40313-015-0221-3

Adaptive Backstepping Control of Three-Phase Four-Wire Shunt Active Power Filters for Energy Quality improvement

A. Ait Chihab¹ · H. Ouadi¹ · F. Giri² · K. El Majdoub²

Received: 13 May 2015 / Revised: 5 September 2015 / Accepted: 30 October 2015

© Brazilian Society for Automatics–SBA 2015

Abstract

The increasing use of nonlinear loads entails seri- ous electrical energy quality problem in power grids. In this paper, the energy quality issue is dealt with by using three- phase four-leg shunt active power filters (SAPFs). A new model that describes the whole controlled system including the SAPF and associated nonlinear and unbalanced loads is developed. The model also accounts for the electrical grid line impedance. The control objective is threefold: to make up for the harmonics and the reactive currents absorbed by the loads; to cancel the neutral current; and to regulate the inverter DC capacitor voltage. Based on the new model, a nonlinear controller is designed, using the backstepping technique. Moreover, the controller is made adaptive for com- pensating the uncertainty on the switching loss power. The performances of the proposed adaptive controller are for- mally analysed using tools from the Lyapunov stability and averaging theory, and their supremacy with respect to stan- dard control solutions is illustrated through simulation.

Keywords

Four-leg SAPF · Harmonic and reactive currents · Unbalanced loads · Adaptive control · Lyapunov stability · Average performance analysis

1 Introduction

Power grids are generally three-phase four-wire structure.

The increasing use of nonlinear unbalanced loads in indus-

B

A. Ait Chihab

abdou.chihab15@gmail.com

1 PMMAT Lab, Faculty of sciences, University Hassan II, Casablanca, Morocco

2 GREYC Lab, UMR CNRS, University of Caen, Caen, France

trial and residential applications (such as switching power supplies, motor drivers and single- and three-phase rectifiers) causes serious power quality problem. Indeed, these loads contribute to the generation of higher-order harmonics and reactive power consumption. Furthermore, unbalanced loads cause excessive neutral current. These problems are likely to bring several harmful effects, e.g. distortion of the volt- age waveform at the point of common coupling (PCC), and overheating of the neutral conductor, transformers and distri- bution lines. On the other side, the presence of neutral current makes the synchronization with voltage network harder, in application requiring such synchronization (Sreenivasarao et al. 2012; Davi et al. 2014; Marcelo et al. 2014).

Recently, APFs have received a great deal of atten- tion for their ability to handle harmonics, reactive currents and unbalanced loads. These filters exist in various con- figurations that are arranged in parallel, serial and hybrid (serial–parallel) topologies, single- and three-phase struc- tures (Ramon et al. 2009; Jinn et al. 2012; Ucar et al. 2011;

Izzeldin et al. 2010; Garcesa et al. 2012). For industrial use, the most widely implemented topology is the shunt configu- ration (El-Habrouk et al. 2000). The principle of shunt active power filter (SAPF) is to cancel the harmonic and reactive currents as well as the neutral current (generated by disturb- ing loads) by injecting a compensation current at the PCC (Fig. 1).

Over the last decade, a great deal of interest has been

paid to the problem of controlling SAPFs. However, most of

the solutions offered are limited to the three-phase three-leg

SAPFs (Zaveri and Chudasama 2012). The point is that the

three-leg SAPFs are only useful in balanced-load applica-

tions, and as such they cannot be used to cancel the current

in the neutral line. To overcome this drawback, the four-leg

APFs have been suggested. This topology features a high

capability of compensating reactive and harmonic currents

Nonlinear and single-phase

loads.

Four -leg SAPF N

N

Fig. 1 Four-leg shunt active power filter associated with nonlinear and unbalanced load

and balancing phase currents, thus improving power qual- ity (Ucar and Engin 2008; Acordi et al. 2012; Mahnia et al.

2014). Nowadays, this filter configuration is considered as one of the most effective solutions to deal with energy quality issues caused by nonlinear and single-phase loads connected to distribution network.

Various control strategies have been proposed to improve power quality using four-leg SAPFs. The first control approach consists in using hysteresis operators or fuzzy logic control technical (Chakphed et al. 2002; Ceni et al. 2014;

Ranjeeta and Anup 2014; Fahmy et al. 2014). The con- trollers pertaining to this approach are not model based, and, consequently, their performances are not formally proved (using, e.g. stability analysis tools); they are generally illus- trated by simulations. The second control approach consists in using linear controllers (Benchouiaa et al. 2014). The problem with this approach is that optimal performances are not guaranteed except at the neighbourhood of the operat- ing point because of the nonlinear nature of the controlled system. The third category includes nonlinear controllers, designed on the basis of accurate nonlinear models of the system. In this regard, different control design techniques have been used including passivity, sliding mode, Lyapunov design and backstepping (Liang and Qiu 2009; Juming et al. 2006; Bin and Tong 2012; Benhabib and Saadate 2005; Salmerón and Herrera 2009). A common limitation of all proposed nonlinear controllers is that they only con- sist in current loops, designed to meet the objective of compensating harmonic and neutral currents. Without an

explicit voltage regulation loop, the DC voltage regulation objective cannot be achieved in general operation condi- tions.

Besides, all previously proposed nonlinear controllers were designed, assuming that grid line impedances and inverter switching losses were negligible. The point is that, in real-life applications, grid line impedances cause voltage deformation at the PCC while inverter switching losses cause DC bus voltage drop.

The present paper is devoted to the control of energy sys- tems that involve four-leg SAPFs with nonlinear unbalanced loads. Specifically, the study seeks to achieve the following control requirements simultaneously:

i) Compensation of harmonic and reactive currents absorbed by the loads.

ii) Cancellation of neutral current caused by unbalanced loads.

iii) Regulation of the inverter DC capacitor voltage.

iv) Estimation of the unknown parameter, namely the inverter switching loss.

To this end, a nonlinear controller is developed on the basis of the exact nonlinear model, using backstepping and Lya- punov design techniques. The controller features a two-loop cascade structure. The inner loop involves a current regulator designed to cope with harmonics compensation and neu- tral current cancellation. The outer-loop voltage controller regulates the DC capacitor voltage. Furthermore, this con- troller is provided with parameter adaptation capability to compensate for the uncertainty that prevails on the switch- ing losses in the inverter. It is formally shown, using tools from Lyapunov stability and system averaging theory, that all control objectives are actually achieved. This theoretical result is confirmed by numerical simulations that emphasize additional controller features (e.g. controller robustness to load change and uncertainty). Moreover, the simulation study shows that the negligence of the nonlinear nature of the filter in the controller design may lead to drastic deterioration of the closed-loop performance. This is demonstrated through a comparative study with previously proposed controllers, e.g.

PI controller.

In the light of the above description, it is evident that the new nonlinear adaptive controller enjoys several features.

Specifically, the present control design model accounts for the grid line impedance while those used in previous con- trollers do not. The new controller includes a physical DC voltage regulator loop while previous controllers do not.

Furthermore, the former is adaptive; it performs an explicit compensation of the power switching losses in the inverter while previous controllers do not.

Unlike the author’s conference paper (Ait Chihab et al.

2014), the present paper includes a number of extensions.

For example, the analysis of the closed-loop control per- formances is presently made without resorting to the prior assumption that the harmonics currents injected by the fil- ter are neglected in the DC bus state equation. Compared with most existing controllers, the present simulation study provides much richer information.

The paper is organized as follows: first, we describe the basic structure of the four-leg SAPF modelling. In Sect. 2, we determine the reference signals of all con- trolled currents (harmonics, reactive and neutral). In Sect. 3, we formulate the control problem. In Sect. 4, we design and analyse the controller design and analysis. In Sect. 5, we present the simulation results. Section 6 concludes the paper.

2 Modelling of Three-Phase Four-Wire SAPF This section aims to develop a control-oriented model, for the four-leg SAPF under study. The power system struc- ture is depicted in Fig. 1. It shows the power grid feeds nonlinear and unbalanced loads. Where the filter is con- nected on the PCC through a filtering inductor (L

f

, R

f

);

this reduces the circulation of the harmonics currents gen- erated by the inverter. The four-leg full-bridge inverter has an energy storage capacitor C

dc

, placed at the DC side, for producing a reactive and harmonic currents to compensate undesirable components propagate in the net- work. The DC–AC inverter operates in accordance with the well-known pulse width modulation (PWM) principle (Tse and Chow 2000).

The notations in Table 1 are used to describe the system model.

Applying the usual electric laws to the four-leg shunt APF considering the line impedance, one easily gets:

⎡

⎢ ⎢

⎣ v

f a

v

f b

v

f c

v

f n

⎤

⎥ ⎥

⎦ −

⎡

⎢ ⎢

⎣ v

Ca

v

Cb

v

Cc

v

Cn

⎤

⎥ ⎥

⎦ = L

f

d dt

⎡

⎢ ⎢

⎣ i

f a

i

f b

i

f c

i

f n

⎤

⎥ ⎥

⎦ + R

f

⎡

⎢ ⎢

⎣ i

f a

i

f b

i

f c

i

f n

⎤

⎥ ⎥

⎦ (1)

where:

⎡

⎢ ⎢

⎣ v

Ca

v

Cb

v

Cc

v

Cn

⎤

⎥ ⎥

⎦ =

⎡

⎢ ⎢

⎣ v

sa

v

sb

v

sc

v

sn

⎤

⎥ ⎥

⎦ − R

s

⎡

⎢ ⎢

⎣ i

sa

i

sb

i

sc

i

sn

⎤

⎥ ⎥

⎦ + L

s

d dt

⎡

⎢ ⎢

⎣ i

sa

i

sb

i

sc

i

sn

⎤

⎥ ⎥

⎦ (2)

Let’s denote by:

W

₍t)

=

⎡

⎢ ⎢

⎣ W

1

W

2

W

3

W

4

⎤

⎥ ⎥

⎦ = R

s

⎡

⎢ ⎢

⎣ i

sa

i

sb

i

sc

i

sn

⎤

⎥ ⎥

⎦ + L

s

d dt

⎡

⎢ ⎢

⎣ i

sa

i

sb

i

sc

i

sn

⎤

⎥ ⎥

⎦ (3)

Then, the four-leg SAPF equations become:

L

f

d dt

⎡

⎢ ⎢

⎣ i

f a

i

f b

i

f c

i

f n

⎤

⎥ ⎥

⎦ = −R

f

⎡

⎢ ⎢

⎣ i

f a

i

f b

i

f c

i

f n

⎤

⎥ ⎥

⎦ +

⎡

⎢ ⎢

⎣ v

f a

v

f b

v

f c

v

f n

⎤

⎥ ⎥

⎦

−

⎛

⎜ ⎜

⎝

⎡

⎢ ⎢

⎣ v

sa

v

sb

v

sc

v

sn

⎤

⎥ ⎥

⎦ − W

₍t)

⎞

⎟ ⎟

⎠ (4)

Table 1 System model

notations vCa, vCb, vCc, vCn PCC to neutral potential voltages

vSa, vSb, vSc, vSn Power source to neutral potential voltages

i_sa,i_sb,i_sc,i_sn Power source currents

vf a, vf b, vf c, vf n AC output filter to neutral potential voltage

i_{f a},i_{f b},i_{f c},i_{f n} AC inverter currents

i_la,i_lb,i_lc Load current

i_{f a},i_{f b},i_{f c},i_{f n} Fundamental AC inverter current

i_{f a},i_{f b},i_{f c},i_{f n} Harmonics AC inverter current

u_a,u_b,u_c,u_n PWM inverter control voltage

(vα, vβ, v0) PCC voltage inα-β-0 coordinate

(i_lα,i_lβ,i_l0) Load current inα-β-0 coordinate

vdc DC bus voltage

L_f Decoupling filter induction

R_f Decoupling filter resistor

C_dc DC bus capacitor

R_dc C_dcLeak resistance

ωs Network frequency

On the other hand, we recall that the output voltages of the DC–AC inverter are given by Ait Chihab et al. (2014):

⎡

⎢ ⎢

⎣ v

f a

v

f b

v

f c

v

f n

⎤

⎥ ⎥

⎦ = V

dc

4

⎡

⎢ ⎢

⎣

3 −1 −1 −1

−1 3 −1 −1

−1 −1 3 −1

−1 −1 −1 3

⎤

⎥ ⎥

⎦

⎡

⎢ ⎢

⎣ μ

a

μ

b

μ

c

μ

n

⎤

⎥ ⎥

⎦ (5)

where the inverter switching functions μ

i

(i = a, b, c or n) are defined by:

μ

i

=

1 if S

i1

is ON; S

i2

is OFF

−1 if S

i1

is OFF; S

i2

is ON

Introducing (5) in (4), the filter equations in a-b-c coordinates become:

d dt

⎡

⎢ ⎢

⎣ i

f a

i

f b

i

f c

i

f n

⎤

⎥ ⎥

⎦ = −R

f

L

f

⎡

⎢ ⎢

⎣ i

f a

i

f b

i

f c

i

f n

⎤

⎥ ⎥

⎦

− v

dc

L

f

⎡

⎢ ⎢

⎣ μ

a

μ

b

μ

c

μ

n

⎤

⎥ ⎥

⎦ − 1 L

f

⎛

⎜ ⎜

⎝

⎡

⎢ ⎢

⎣ v

sa

v

sb

v

sc

v

sn

⎤

⎥ ⎥

⎦ − W

₍t)

⎞

⎟ ⎟

⎠ (6)

The system (6) is useful for building up an accurate simula- tor of the four-leg SAPF. However, it cannot be considered in the control design as it involves a binary control input, namely (μ

a

, μ

b

, μ

c

, μ

n

) . This kind of difficulty is generally coped with by resorting to average models. Signal averaging is performed over cutting intervals (Krein et al. 1990).

The obtained average model is the following:

d dt

⎡

⎢ ⎢

⎣ x

1

x

2

x

3

x

4

⎤

⎥ ⎥

⎦ = − R

f

L

f

⎡

⎢ ⎢

⎣ x

1

x

2

x

3

x

4

⎤

⎥ ⎥

⎦

+ V

dc

L

f

⎡

⎢ ⎢

⎣ u

a

u

b

u

c

u

n

⎤

⎥ ⎥

⎦ − 1 L

f

⎛

⎜ ⎜

⎝

⎡

⎢ ⎢

⎣ v

sa

v

sb

v

sc

v

sn

⎤

⎥ ⎥

⎦ − W

₍t)

⎞

⎟ ⎟

⎠ (7)

where x

1

, x

2

, x

3

, x

4

, V

dc

, u

a

, u

b

, u

c

and u

d

denote the average values, over cutting periods, of the signals i

f a

, i

f a

, i

f a

, i

f a

, v

dc

, μ

a

, μ

b

, μ

c

andμ

d

. In (7), the mean value (u

a

, u

b

, u

c

and u

d

) of (μ

a

, μ

b

, μ

c

and μ

d

) turns out to be the system control input.

To carry out the DC bus voltage control, the system mod- elling must be supplemented with a fifth equation describing the energy stored in the capacitor E

dc

=

¹₂

C

dc

v

_dc²

. To this end, consider the power (P

DC

) at the DC bus:

P

DC

= P

net

− P

Rdc

− P

sc

(8)

where

P

net

: power supplied by the electrical network to maintain in charge the DC bus capacitor.

P

Rdc

: Joule losses in the leakage resistance

P

sc

: switching inverter losses (considered as unknown parameter),

It is readily checked that:

P

net

= v

f a

i

f a

+ v

f b

i

f b

+ v

f c

i

f c

+ v

f n

i

f n

(9) P

Rdc

= v

_dc²

R

DC

(10) P

DC

= ˙ E

dc

= d

dt 1

2 C

dc

v

dc²

(11) Substituting (9)–(11) in (8), one obtains:

E ˙

dc

=

v

f a

i

f a

+ v

f b

i

f b

+ v

f c

i

f c

+ v

f n

i

f n

− v

²_dc

R

DC

− P

sc

(12)

Now using (5), Eq. (12) becomes:

E ˙

dc

= v

dc

μ

a

i

f a

+ μ

b

i

f b

+ μ

c

i

f c

+ μ

n

i

f n

− v

_dc²

R

DC

− P

sc

(13)

Equation (13) cannot be based upon in the control design as it involves a binary control input, namely (μ

a

, μ

b

, μ

c

, μ

n

) . By resorting to average model, one obtains:

˙ x

5

=

2x

5

C

dc

(u

a

x

1

+ u

b

x

2

+ u

c

x

3

+ u

n

x

4

)

− 2 C

dc

R

dc

x

5

− P

sc

(14)

where x

5

denotes the average value, over cutting periods, of the energy stored E

dc

. By regrouping Eqs. (7) and (14), we construct the average state model of the four-leg SAPF:

⎡

⎢ ⎢

⎣

˙ x

1

˙ x

2

˙ x

3

˙ x

4

⎤

⎥ ⎥

⎦ = − R

f

L

f

⎡

⎢ ⎢

⎣ x

1

x

2

x

3

x

4

⎤

⎥ ⎥

⎦ − 1 L

f

2x

5

C

dc

⎡

⎢ ⎢

⎣ u

a

u

b

u

c

u

n

⎤

⎥ ⎥

⎦

− 1 L

f

⎛

⎜ ⎜

⎝

⎡

⎢ ⎢

⎣ v

sa

v

sb

v

sc

v

sn

⎤

⎥ ⎥

⎦ − W

₍t)

⎞

⎟ ⎟

⎠ (15)

˙ x

5

=

2x

5

C

dc

(u

a

x

1

+ u

b

x

2

+ u

c

x

3

+ u

n

x

4

)

− 2 C

dc

R

dc

x

5

− P

sc

(16)

3 Currents References Extraction

Asymmetric load currents can be decomposed into three components: positive, negative and zero sequences. In order to identify and then extract the latter, we make use of the so- called instantaneous power technique (Hyosung and Akagi 1999), which enjoys a good compromise between accuracy and computational complexity. The use of p-q-r reference frames leads to extract the one instantaneous active power P and two reactive powers q

q

and q

r

; the principle of this technique is shown in Fig. 2.

In the α-β-0 coordinate, the instantaneous real load power (P ), the imaginary power (q ) and the zero sequence power ( P

0

) are given by Ucar and Engin (2008):

⎡

⎣ P

0

P q

⎤

⎦ =

⎡

⎣ v

0

0 0 0 v

α

v

β

0 −v

_β

v

_α

⎤

⎦

⎡

⎣ i

l0

i

lα

i

lβ

⎤

⎦ (17)

To obtain the currents references, the instantaneous currents are transformed from α-β -0 coordinate to p-q -r coordinate as Eq. (18) shows:

⎡

⎣ i

p

i

q

i

r

⎤

⎦ = 1 v

0αβ

⎡

⎢ ⎣

v

0

v

_α

v

_β

0 −

^v0_v^αβ_αβ^{vβ v0}_v^αβ_αβ^vα

v

_αβ

−

^v_v⁰_αβ^vα

−

^v_v⁰_αβ^vβ

⎤

⎥ ⎦

⎡

⎣ i

l0

i

lα

i

lβ

⎤

⎦ (18)

where v

0αβ

=

v

²₀

+ v

_α²

+ v

_β²

, v

_αβ

= v

²_α

+ v

_β²

To construct the current references compensation in p-q -r coordinate, from (18) we get:

i

_cp^∗

= i

p

i

_cq^∗

= i

q

i

_cr^∗

= i

r

+ i

p

v

0

v

_αβ

+ + p-q-r to --0 +

p-q-r Transformation (18) --0 To a-b-c

a-b-c To --0 a-b-c To --0

Fig. 2 The principle of thep-q-rtheory

Using the inverse transformation from the p-q -r to α-β -0 and then to a -b-c coordinates, one easily gets:

⎡

⎣ i

_ca^∗

i

_cb^∗

i

_cc^∗

⎤

⎦ = 1 v

0αβ

⎡

⎢ ⎢

⎣

√1

2

1 0

√1

2

−

¹₂ ^√₂³

√1

2

−

¹₂

−

^√₂³

⎤

⎥ ⎥

⎦

⎡

⎣ i

_cp^∗

i

_cq^∗

i

_cr^∗

⎤

⎦ (19)

The neutral current reference is determined as a sum of the three lines current reference by the following expression:

i

_cn^∗

= i

^∗_ca

+ i

_cb^∗

+ i

^∗_cc

(20)

4 Control Design

4.1 Control Objective Reformulation

We aim to achieve the following control objectives:

– Controlling the filter current (i

f a

, i

f b

and i

f c

) so that the loads reactive and harmonic currents are well compen- sated for.

– Controlling the fourth leg of the SAPF to eliminate the neutral line current.

– Regulating the DC bus voltage (V

dc

) to maintain the capacitor charge at a suitable level so that the filter oper- ates properly.

– Estimating the unknown model parameter, namely the inverter switching loss.

In order to satisfy the control objectives simultaneously, the system model comes with five output variables (x

1

, x

2

, x

3

, x

4

and x

5

); however, only four control inputs are available, namely ( u

a

, u

b

, u

c

, u

n

) . This is coped with by considering a cascade control strategy, involving two loops.

The inner loop is designed to meet the power quality requirements, including the control of the neutral line current.

However, the outer-loop control aims to regulate the DC bus voltage. The virtual control signals generated by that loop, denoted i

f a

, i

f b

, i

f c

, and i

f n

, serve as the desired fundamen- tal components of the output current filter. These components are augmented with harmonic and reactive currents compo- nents, next expressed as i

^∗_{f a}

, i

^∗_{f b}

, i

^∗_{f c}

, i

^∗_{f n}

, to constitute the final AC currents references, x

₁^∗

, x

^∗₂

, x

₃^∗

and x

₄^∗

. The proposed control strategy is illustrated by (Fig. 3).

4.2 Inner Control Loop Design

The inner loop is designed to minimise the current tracking errors:

z

1

= x

1

− x

₁^∗

z

2

= x

2

− x

₂^∗

Fig. 3 The block scheme of the nonlinear cascade control strategy

+

Power Network N.L and unbalanced

loads

Cascade controller

Inner loop

Outer loop +

Currents reference Computation (19-20)

Voltage Controller (35-39)

References Generation Controlled System

+ -

+

Current Controller (22)

4 Leg SAPF (15-16)

z

3

= x

3

− x

₃^∗

z

4

= x

4

− x

₄^∗

To this end, the dynamics of these errors need to be deter- mined. It follows, using the model Eqs. (15), that the errors derivatives undergo the following equations:

⎡

⎢ ⎢

⎣

˙ z

1

˙ z

2

˙ z

3

˙ z

4

⎤

⎥ ⎥

⎦ = −R

f

L

f

⎡

⎢ ⎢

⎣ x

1

x

2

x

3

x

4

⎤

⎥ ⎥

⎦ − 1 L

f

2x

5

C

dc

⎡

⎢ ⎢

⎣ u

a

u

b

u

c

u

n

⎤

⎥ ⎥

⎦

− 1 L

f

⎛

⎜ ⎜

⎝

⎡

⎢ ⎢

⎣ v

sa

v

sb

v

sc

v

sn

⎤

⎥ ⎥

⎦ − W

₍t)

⎞

⎟ ⎟

⎠ −

⎡

⎢ ⎢

⎣

˙ x

₁^∗

˙ x

₂^∗

˙ x

₃^∗

˙ x

₄^∗

⎤

⎥ ⎥

⎦ (21)

To ensure the asymptotic stability of the equilibrium (z

1

, z

2

, z

3

, z

4

) = (0, 0, 0, 0), Eq. (21) suggests that the con- trol inputs ( u

a

, u

b

, u

c

, u

n

) be chosen so that

⎡

⎢ ⎢

⎣ u

a

u

b

u

c

u

n

⎤

⎥ ⎥

⎦ =

C

dc

2x

5

⎛

⎜ ⎜

⎝ R

f

⎡

⎢ ⎢

⎣ x

1

x

2

x

3

x

4

⎤

⎥ ⎥

⎦ +

⎛

⎜ ⎜

⎝

⎡

⎢ ⎢

⎣ v

sa

v

sb

v

sc

v

sn

⎤

⎥ ⎥

⎦ − W

₍t)

⎞

⎟ ⎟

⎠

+ L

f

⎡

⎢ ⎢

⎣

˙ x

₁^∗

˙ x

₂^∗

˙ x

₃^∗

˙ x

₄^∗

⎤

⎥ ⎥

⎦ − L

f

⎡

⎢ ⎢

⎣ c

1

z

1

c

2

z

2

c

3

z

3

c

4

z

4

⎤

⎥ ⎥

⎦

⎞

⎟ ⎟

⎠ (22)

If one combines Eqs. (22) and (21), one gets the following equations that describe the inner closed loop:

⎡

⎢ ⎢

⎣

˙ z

1

˙ z

2

˙ z

3

˙ z

4

⎤

⎥ ⎥

⎦ = −

⎡

⎢ ⎢

⎣ c

1

z

1

c

2

z

2

c

3

z

3

c

4

z

4

⎤

⎥ ⎥

⎦ (23)

It readily follows that:

z

1

(t ) = z

1

(0) e

^−c1t

, z

2

(t) = z

2

(0) e

^−c2t

z

3

(t ) = z

3

(0) e

^−c3t

, z

4

(t) = z

4

(0) e

^−c4t

(24) This shows that the errors are globally exponentially van- ishing, that is the objective of cancelling the load current harmonics, load current reactive component and the neutral line current is well established.

4.3 Outer Control Loop Design

The outer loop aims at making the voltage tracking error

z

5

= x

5

− x

₅^∗

(25)

as small as possible, where x

₅^∗

is the reference value of the DC bus voltage. Without loss of generality, it is assumed that x

₅^∗

as well as its first time-derivative are known and bounded. Fur- thermore, by respecting the notations presented in Table 1, the AC filter currents verify:

⎡

⎢ ⎢

⎣ i

f a

i

f b

i

f c

i

f n

⎤

⎥ ⎥

⎦ =

⎡

⎢ ⎢

⎣ i

f a

i

f b

i

f c

i

f n

⎤

⎥ ⎥

⎦ +

⎡

⎢ ⎢

⎣ i

f a

i

f b

i

f c

i

f n

⎤

⎥ ⎥

⎦ (26)

By introducing (5) and (26) into (14), the DC side equation

becomes:

˙ x

5

=

v

f a

v

f b

v

f c

v

f n

⎡

⎢ ⎢

⎣ i

f a

i

f b

i

f c

i

f n

⎤

⎥ ⎥

⎦

+

v

f a

v

f b

v

f c

v

f n

⎡

⎢ ⎢

⎣ i

f a

i

f b

i

f c

i

f n

⎤

⎥ ⎥

⎦ − v

_dc²

R

DC

− P

sc

(27)

Then, using (25) and (27), time derivative of the error voltage (25) can be written as:

˙

z

5

= (v

f a

i

f a

+ v

f b

i

f b

+ v

f c

i

f c

+ v

f n

i

f n

) − 2x

5

C

dc

R

dc

−P

sc

− ˙ x

₅^∗

+ f (z

1

, z

2

, z

3

, z

4

, t) (28) where

f (.) = i

f a

R

f

x

1

+ v

Sa

+ W

1

+ L

f

x ˙

₁^∗

− L

f

c

1

z

1

+ i

f b

R

f

x

2

+ v

Sb

+ W

2

+ L

f

x ˙

₂^∗

− L

f

c

2

z

2

+ i

f c

R

f

x

3

+ v

Sc

+ W

3

+ L

f

x ˙

₃^∗

− L

f

c

3

z

3

+ i

f n

R

f

x

4

+ v

Sn

+ W

4

+ L

f

x ˙

^∗₄

− L

f

c

4

z

4

(29)

Note that the quantities (x

₁^∗

, x

₂^∗

, x

₃^∗

, x

₄^∗_,

x

i

f a

, i

f b

, i

f c

, i

f n

, W

1

, W

2

, W

3

, W

4

) are zero-mean periodic with period 2π/ω

s

(or 2π/kω

s

for some integer k).

The switching loss power ( P

sc

) is seen as an unknown parameter in (28). In reality, the latter mainly depends on the load, which is assumed to undergo a piecewise constant variation. On the other hand, the quantity v

f a

i

f a

+ v

f b

i

f b

+ v

f c

i

f c

+ v

f n

i

f n

stands in (28) as a virtual control. Interest- ingly, this quantity is nothing other than the electric network power, denoted P

net

, transmitted to control the voltage DC bus. Indeed, the power balance Eq. (8) shows that P

net

equals:

P

net

=

v

f a

v

f b

v

f c

v

f n

i

f a

i

f b

i

f c

i

f n

T

= v

f a

i

f a

+ v

f b

i

f b

+ v

f c

i

f c

+ v

f n

i

f n

(30) In order to obtain a stabilizing control law of the error sys- tem (28), let us introduce the following Lyapunov function candidate:

V = z

₁²

2 + z

²₂

2 + z

²₃

2 + z

²₄

2 + z

²₅

2 + 1

2γ P ˜

_sc²

(31) where P ˆ

sc

denotes the online estimate of P

sc

and P ˜

sc

= P

sc

− ˆ P

sc

is the corresponding estimation error; γ is a positive design parameter. Deriving V along (28) and (23) yields:

V ˙ = −c

1

z

₁²

− c

2

z

²₂

− c

3

z

²₃

− c

4

z

₄²

+ z

5

P

net

− 2x

5

C

dc

R

dc

− P

sc

− ˙ x

₅^∗

+ f ( z

1

, z

2

, z

3

, z

4

, t )

+ 1

γ P ˜

sc

P ˙˜

sc

(32) V ˙ = −c

1

z

²₁

− c

2

z

²₂

− c

3

z

₃²

− c

4

z

²₄

+ z

5

P

net

− 2x

5

C

dc

R

dc

− ˆ P

sc

− ˙ x

₅^∗

+ ˜ P

sc

1

γ P ˙˜

sc

− z

5

+ z

5

f (z

1

, z

2

, z

3

, z

4

, t) (33)

Equation (33) suggests the following control law:

P

net

= −c

5

z

5

+ 2x

5

C

dc

R

dc

+ ˆ P

sc

+ ˙ x

₅^∗

(34) and the following parameter adaptation law:

P ˙˜

sc

= γ z

5

(35)

In fact, substituting (34) and (35) in (33) yields:

V ˙ = − c

1

z

²₁

− c

2

z

²₂

− c

3

z

₃²

− c

4

z

²₄

− c

5

z

²₅

+ z

5

f ( z

1

, z

2

, z

3

, z

4

, t ) (36) The outer closed-loop equation is obtained by substituting (34) in (28):

˙

z

5

= −c

5

z

5

− ˜ P

sc

+ f (z

1

, z

2

, z

3

, z

4

, t ) (37) Now, as P

net

is a virtual control input defined in (30), we make use of Eq. (34) to obtain:

⎡

⎢ ⎢

⎣ i

f a

i

f b

i

f c

i

f n

⎤

⎥ ⎥

⎦ = P

net

v

²_f0

+ v

²_fα

+ v

²_fβ

⎡

⎢ ⎢

⎣

v

f a

0 0 0 0 v

f b

0 0 0 0 v

f c

0 0 0 0 v

f n

⎤

⎥ ⎥

⎦ (38)

Substituting (34) with (38), one finds the following expres- sion of the fundamental current references:

⎡

⎢ ⎢

⎣ i

f a

i

f b

i

f c

i

f n

⎤

⎥ ⎥

⎦ = 1

v

²_{f a}

+ v

²_{f b}

+ v

²_{f c}

+ v

²_{f n}

×

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣ v

f a

−c

5

z

5

+

_Cdc^2x_R⁵_dc

+ ˆ P

sc

+ ˙ x

₅^∗

v

f b

−c

5

z

5

+

_Cdc^2x_R⁵_dc

+ ˆ P

sc

+ ˙ x

₅^∗

v

f c

−c

5

z

5

+

_Cdc^2x_R⁵_dc

+ ˆ P

sc

+ ˙ x

₅^∗

v

f n

−c

5

z

5

+

_Cdc^2x_R⁵_dc

+ ˆ P

sc

+ ˙ x

₅^∗

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦ (39)

Remark 1 Although the (inner and outer) control laws (22)

and (39) involve a division by x

5

and (v

²_{f a}

+v

²_{f b}

+v

²_{f c}

+v

²_{f n}

),

respectively, this entails no singularity in practice. Indeed, the

active filter cannot work if these quantities are zero. In other

words, the latter are nonzero as long as the active filter is carrying non-identical null currents.

The adaptive outer regulator designed, thus, includes the parameter adaptation law (35) and the control law (39). In the following theorem, it is shown that the control objectives are achievable (in the mean) with an accuracy that depends, among others, on the network frequency ω

s

. The following notations are needed to formulate the results.

ε = 1 ω

s

, A

0

=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

− c

1

0 0 0 0 0

0 −c

2

0 0 0 0

0 0 −c

3

0 0 0

0 0 0 − c

4

0 0

0 0 0 0 −c

5

−1

0 0 0 0 γ 0

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

Theorem

Consider the closed-loop system composed of the four-leg SAPF represented by the model (15)–(16), and the adaptive controllers consisting of:

• The cascade regulator including the (inner) current con- trol defined by (22), and the (outer) adaptive voltage control defined by ( 35) and (39).

• The current references estimation described by (19)–(20)

1) The closed-loop system is described in the error coordi- nates

z

1

z

2

z

3

z

4

z

5

P ˜

sc

T

by the following equations:

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎣

˙ z

1

˙ z

2

˙ z

3

˙ z

4

˙ z

5

P ˙˜

sc

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎦ = A

0

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣ z

1

z

2

z

3

z

4

z

5

P ˜

sc

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦ +

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

0 0 0 0

f ( z

1

, z

2

, z

3

, z

4

, t ) 0

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

2) The current tracking errors ( z

1

, z

2

, z

3

, z

4

) vanish exponentially fast.

3) Let the design parameters (c

5

, γ ) satisfy the follow- ing inequality:

c

5

> 2 √

γ (40)

Then, there exists a positive real ε

^∗

such that if ε ≤ ε

^∗

, the tracking errors z

5

and the parameter estimation error P ˜

sc

are harmonic signals continuously depending on ε, i.e.

z

5

= z

5

( t , ε) and P ˜

sc

= ˜ P

sc

( t , ε) . Furthermore, if ε → 0, then the tracking error z

5

and the parameter estimation error P ˜

sc

asymptotically vanish.

Proof Part 1 is readily established by putting together Eqs. (23, 37) and Part 2 is a direct consequence of (24).

To establish Part 3, introduce the augmented state vector Z =

z

1

z

2

z

3

z

4

z

5

P ˜

sc

T

which, by Part 2, undergoes the following state equation:

Z ˙ (t ) = ϕ ( t, Z ) (41)

where

ϕ ( t , Z ) =

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

−c

1

z

1

−c

2

z

2

−c

3

z

3

−c

4

z

4

− c

5

z

5

− ˜ P

sc

+ f ( z

1

, z

2

, z

3

, z

4

, t ) γ z

5

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

(42)

The stability of the time-varying system (41)–(42) will now be analysed using the averaging theory Khalil (2003). To this end, introduce the time-scale change τ = ω

s

t. It is readily seen from (41) that ∅(τ) = Z (τ/ω

s

) undergoes the differen- tial equation:

˙∅(τ) = εΨ (∅, τ) (43)

with:

Ψ (∅, τ) =

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

−c

1

∅

1

− c

2

∅

2

−c

3

∅

3

−c

4

∅

4

− c

5

∅

5

− ˜ P

sc

+ g (∅

1

, ∅

2

, ∅

3

, ∅

4

, τ) γ ∅

5

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦ (44)

and g (∅

1

, ∅

2

, ∅

3

, ∅

4

, τ) = f (∅

1

, ∅

2

, ∅

3

, ∅

4

, τ/ω

s

) since f (∅

1

, ∅

2

, ∅

3

, ∅

4

, t) is periodic in t with period

²_ω^π

s

, it fol- lows that g (∅

1

, ∅

2

, ∅

3

, ∅

4

, τ) is periodic in τ with period 2 π .

Now, let us introduce the averaged function:

Ψ

0

(∅

0

) = li m

ε→0

1 2π

_2π

0

Ψ (∅, τ) d τ It follows from (44) that:

Ψ

0

(∅

0

) =

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎢

⎣

− c

1

∅

1.0

−c

2

∅

2.0

−c

3

∅

3.0

− c

4

∅

4.0

−c

5

∅

5.0

− ∅

6.0

γ ∅

6.0

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎥

⎦

(45)

where ∅

i.0

(i = 1 . . . 6) refers to the components of ∅

0

. In order to get stability results regarding the system of interest (43), it is sufficient (with averaging theory) to analyse the following averaged system:

˙∅

0

= εΨ

0

(∅

0

) = ε A

0

∅

0

(46)

Notice that the time-invariant system (46) is linear and has a

unique equilibrium at:

∅

^∗0

= [ 0 0 0 0 0 0 ]

^T

(47) It is readily checked that the given values of the matrix A

0

are the zeros of the following characteristic polynomial:

P (θ) =(θ + c

1

) (θ +c

2

) (θ + c

3

) (θ +c

4

)

θ

²

+c

5

θ + γ (48) Clearly, all zeroes of the polynomial (48) have negative real parts if condition (40) is fulfilled. Then, the matrix A

0

turns out to be Hurwitz and the equilibrium ∅

0

= ∅

^∗₀

exponentially stable. Then, invoking the averaging theory Khalil (2003, Theorem 10.4), one concludes that there exists a ε

^∗

> 0 such that for ε < ε

^∗

, the differential (43) has a harmonic solution ∅ (τ) = ∅ (t , ε) , that continuously depends on ε.

Moreover, one has li m

ε→∞

∅ ( t , ε) = ∅

^∗₀

This, yields, due to (47), that Theorem is established

5 Simulation and Discussion of Results

5.1 Simulation Protocol

The simulations are performed on MATLAB/SIMULINK environment. According to Fig. 3, the three-phase four-wire power distribution is connected to nonlinear and unbal- anced loads. Indeed, the considered load is constituted by an AC/DC inverter (with a RC load) associated with mono- phase unbalanced loads. The load and filter characteristics are summarized in Table 2. On the other hand, the inverter operates according to the PWM principle with a switching frequency of 25 kHz.

In this section, the new cascade regulator (referred to as nonlinear controller: NLC) including the current control loop (22) and the adaptive voltage control loop (35) and (39) is evaluated. To this end, its performances are compared with

Table 2 Shunt APF and loads

characteristics Parameters Values

Active power filter

L_f 0.022 H

C_dc 1000µF

R_f 0.2

V_dc 500 V

Rectifier-load

C_l 1 mF

R_l 100/20

Single loads

R₁ 100

R₂ 200

R₃ 300

0.85 0.9 0.95 1 1.05 1.1

0 10 20 30 40 50 60 70 80 90 100 110

R(t)

t(s)

Fig. 4 Load resistance variation profile in()

0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05

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

il(t)

t(s)

Fig. 5 Load current in (A) in time domain

0 200 400 600 800 1000 1200 1400 1600

0 2 4 6 8 10 12

Frequency analysis

Fs(t)

Hz

Fig. 6 Load current in (A) in frequency domain

the simpler linear regulator (referred to as proportional inte- gral controller: PIC).

To compare the performances of the two regulators in

different operation points, the load resistance (R

l

) is made

variable, according to the profile described in Fig. 4. The

resulting unbalanced current is illustrated in Fig. 5. The har-

monic load current spectra and the corresponding THD value

are, respectively, presented in Fig. 6 and Table 3. Moreover,

Fig. 7 presents the resulting neutral current. Obviously, it is

not zero, since the load is unbalanced.

Table 3 The load current THD

In % Phase a Phase b Phase c

50.19 43.58 47.85

0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05

-15 -10 -5 0 5 10 15

in(t)

t(s)

Fig. 7 Neutral current in Vs time in (s)

Table 4 PIC parameters Current regulator K p₁ 0.5 T i₁ 0.1

0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05

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

if(t)

t(s) 0.998 1 1.002 1.004

-12 -10 -8 -6

Zoom In

Fig. 8 Harmonic current tracking performances of PIC

5.2 PI Controller Performances Evaluation

For simplicity, a linear PI controller (denoted PIC and defined by (K

p

(1 + T

i

s) / T

i

s)) is presently considered within the simulated experimental setup of Fig. 3. The PIC parame- ters have been tuned based on several system step responses.

Table 4 summarizes the adopted PIC parameters, which have proved to be convenient:

Figure 8 shows that, for a small value of the load current amplitude (t ∈ [0s 1s], R

1

= 100 ), the inner loop (for PIC) ensures a perfect asymptotic tracking of the current ref- erence signal. However, the tracking performances degrade when the filter operates in the vicinity of the defined operat- ing point for t ∈ [1s 2s]. This degradation appears clearly on the shape of generated network current in spite of the intro- duction of the filter (see Fig. 9). In fact for a small value of

0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05

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

is(t)

t(s)

Fig. 9 Three-phase line current with PIC (in time domain)

0 200 400 600 800 1000 1200 1400 1600

0 2 4 6 8 10 12

Frequency analysis

Fs(t)

t(s)

Fig. 10 Line current with PIC in frequency domain Table 5 The obtained line

current THD value with PI controller In %

Phase a Phase b Phase c

30.18 38.84 32.93

the load current amplitude (t ∈ [0 1s]), the corresponding network current is practically sinusoidal, but for t ∈ [1s 2s], the shape of this current is largely distorted. The harmonic grid current spectra and the corresponding THD value are, respectively, presented in Fig. 10 and Table 5. They confirm that the line current is still rich in current harmonics.

The neutral current tracking performance is plotted in Fig. 11, showing that for t ∈ [0s 1s] the neutral current pursuit is perfect, but this is not the case for t ∈ [1s 2s].

Indeed, Fig. 12 shows that the compensation of neutral cur- rent is insured when the filter operating point is in the vicinity of the one defined for t ∈ [0s 1s]. But nothing is guaranteed outside this neighbourhood, this is the case for t ∈ [ 1s 2s ] . Figure 12 confirms that with the PIC, the filter is unable to cancel the neutral current for any load connected to the net- work.

5.3 Nonlinear Controller Performances Evaluation

We now evaluate the nonlinear controller, where the inner and

the outer loops are implemented using Eqs. (22) and (35, 39),