1

Backstepping Control of aPhotovoltaic DC-AC Converter

Calvin M. P. Santana, MSc Student, IST, and J. Fernando A. Silva, Senior Member, IEEE

Abstract—A highly reliable, current-fed microinverter isproposed, without the traditional problematic electrolyticcapacitor, while simultaneously achieving a high nominal andEuropean Weighted Efficiency.

Two different control systems that guarantee maximumpower point tracking (MPPT) and DC-AC power control areimplemented and compared in order to evaluate each controlsystem in terms of power quality, efficiency, operation span anddynamic response performance to atmospheric or grid changes.

The first control system utilizes linear control laws tostabilize and control the nonlinear system and will be used as areference for comparison.

The second control system utilizes nonlinear control techniquesbased on Lyapunov’s second law, which include: backsteppingcontrol design and command filtering backstepping controldesign.

Index Terms— Microinverter, Electrolytic-less converter, Lin-ear Control, Nonlinear Control, Lyapunov’s Second Law, Back-stepping control design and Command Filtering Backstepping.

I. INTRODUCTION

TYPICALLY electrolytic capacitors are used as thedecoupling element due to their good capacitance

per volume ratio and low price. The use of electrolyticcapacitors makes it is easy and affordable to install very highcapacitances for decoupling between input and output of the1φ inverter. This comes with a price, a severe drawback ofthis type of capacitor is its limited lifetime.

Electrolytics are affected by ageing effects more thanother electronic components and are therefore a bottleneckfor inverter reliability and lifetime. Disadvantages includelow ripple current capability due to high equivalent seriesresistance and equivalent series inductance and high thermalresistances limiting the ability to get the heat out. Anothercharacteristic is that over time the liquid of electrolyteevaporates through the rubber seals of the capacitor,degrading the capacitance [1].Many efforts have been made towards replacing electrolyticcapacitors by film capacitors, due to this type having a muchhigher lifetime and can even be self-healing in case of minorisolation breakdowns. The disadvantage of film capacitors is

Calvin M. P. Santana is an Electrical and Computer Engineering MScstudent at Instituto Superior Tecnico, Universidade de Lisboa.

J. Fernando A. Silva is with Instituto de Engenharia de Sistemas eComputadores - Investigacao e Desenvolvimento (INESC-ID), Instituto Su-perior Tecnico, Universidade de Lisboa, 1649-004 Lisboa, Portugal (e-mail:fernandos@alfa.ist.utl.pt).

a low capacitance per volume ratio (≈ 20 times lower thanfor electrolytics) and a higher price. A direct replacement istherefore not feasible in terms of cost and size [1]. Fig. 1shows the ranges of capacitance per technology.

16/06/2016 https://upload.wikimedia.org/wikipedia/commons/9/93/Kondensatoren­Kap­Versus­Spg­English.svg

https://upload.wikimedia.org/wikipedia/commons/9/93/Kondensatoren­Kap­Versus­Spg­English.svg 1/1

1 kV

1 pF 1 nF 1 µF 1 MF1 kF1 mF 1 F

Capacitance

Voltage

10 kV

100 kV

100 V

10 V

1 V

Plastic filmcapacitors

Aluminumelectrolyticcapacitors

Powercapacitors

Ceramic capacitors

Double­layerand supercapacitors

Tantalum capacitors with solid electrolyte

Fig. 1. Capacitance/voltage ranges for each capacitor technology.

II. POWER DECOUPLING TECHNIQUES

Innovative power decoupling circuits can reduce the sizeof the required energy storage capacitor, thus improving theinverter lifetime, however, the power decoupling circuit willalso result in additional power losses, due to the powerflow through the decoupling circuit, consequently, reducingthe overall efficiency. There are two main power decouplingtechniques PV-Side Decoupling and DC-Link Decoupling.

A. PV-Side Decoupling

PV-side decoupling is the term used to describemicroinverters that have the decoupling reactive elementon the primary side of the transformer. This configurationis considered a single-stage converter, since there is nointermediate energy storage on the secondary side.

The primary side of the converter is responsible forguaranteeing MPPT of the PV module, increasing the voltagevia a step-up transformer (and a boost inductor if present) andgenerating a rectified sine waveform using a high frequencymodulation technique.

The secondary side of the converter is responsible forthe unfolding of the high-frequency sine waveform from theprimary side via an inverter which uses a line frequency

2

commutation. Since the output inverter has a low frequencycommutation the switching losses are decreased, which is oneof the benefits of this microinverter decoupling topology.

One of the main drawbacks of PV-side decoupling isthe low power density (W/m3) of the decoupling capacitor.This can be explained by the following two reasons:

• The power capacitor across the PV module terminalsresults in a very large capacitor since the allowablevoltage ripple must be kept to very low values (< 1%)in order to achieve high MPPT efficiency.

• The high current present in the primary side of the step-up transformer means that a higher capacitance value isneeded due to the capacitance being proportional to itscurrent.

B. DC-Link Decoupling

The DC-Link decoupling topology is considered a multi-stage converter typically with a DC-AC-DC-AC configuration,while the power conversion process can be divided into twoseparate stages: DC-DC and DC-AC conversions. For thismulti-stage converter the main power decoupling capacitor isplaced at the high-voltage secondary side.

The primary side of the transformer, or DC-DC converter, isresponsible for guaranteeing MPPT, amplifying the PV DCvoltage to a sufficient voltage level compatible with the gridmaximum voltage while simultaneously providing galvanicisolation. Isolated DC-DC converters can be classified as:

• Hard-switching: Less active/reactive components areneeded, but switching losses tend to be high due to theleakage inductance (of the transformer) being openedwhile its current is not equal to zero and/or by connectingit in series with other elements equivalent to currentsources.

• Soft-switching: Extra circuity elements are added whichimplement zero-voltage switching, zero-current switch-ing, or active clamp techniques to improve the efficiencyand overcome problems caused by the leakage induc-tance.

Any isolated DC-DC converter can be used to perform theMPPT. A solution to increase the reliability of the AC-modulecan pass by using an isolated current-fed DC-DC powerconverter since inductors have much longer lifetime whencompared to capacitors [1].

III. PROPOSED MICROINVERTER

The Active-Clamp Full-Bridge Boost Converter [2] is cho-sen as the isolated current-fed DC-DC converter stage due toits low number of components and easy control. The interfacebetween the DC-DC converter and the grid is a half-bridgeinverter since it further decreases the needed capacitance. Thecomplete microinverter is shown in Fig. 2.

PV

dcL

1S 2S

3S4ScC

LacL

1Q

2Q

1D

2D

1oC

2oCgv

pvV

pvI

gi

cS

Fig. 2. Proposed microinverter circuit.

A. Active-Clamp Full-Bridge Boost Converter

The clamp capacitor Cc clamps the voltage that appearsacross the full-bridge whenever full-bridge switches areturned off (transfer mode). The additional active switch isalways off whenever the converter is in current boost mode.

1) Active-Clamp Capacitor: The design of Cc is prosedin [2] and is based on the resonant tank formed by Cc andLλ. The resonance happens during the transfer mode wherea condition of resonance will be experienced in a tank circuitwhen the reactances of the capacitor and inductor are equalto each other. Because inductive reactance increases withincreasing frequency and the capacitive reactance decreaseswith increasing frequency, there will only be one frequencywhere these two reactances are equal, this premise will beused in the MPPT and is the reason why the active-clampfull-bridge should have a fixed PWM frequency controller.

2) Pulse Width Modulation: Because of the full-bridgepresent in the DC-DC converter, a 3-level PWM should beused. The PWM can be obtained using the logic circuit shownin Fig. 3.

AND

OR

OR

0

1Carrier

Auxiliary

Reference

0

1

NAND

1,4S

cS

2,3S

Fig. 3. 3-Level PWM logic circuit.

Where the reference waveform uc represents a varying quan-tity from the PI controller or backstepping controller, theauxiliary waveform is used to create a zero mean waveformand the carrier is a triangular waveform with [ucmin; ucmax].Analysing the relation between the main waveforms it can beshown that the duty cycle δ is given by:

δ = 1− ucucmax

(1)

IV. MAXIMUM POWER POINT TRACKING

As shown in Fig. 4 the nonlinear relation between currentand voltage (of PV cells) results in a unique global maximumpower point which can be achieved by imposing a specificvoltage Vmp or current Imp (MPPT).

3

Current [p.u.]0 0.2 0.4 0.6 0.8 1

[p.u

.]

0

0.2

0.4

0.6

0.8

1

I-V curve

Power curve

Vmppu

dPdI

< 0dPdI

> 0

MPP

Imppu Iccpu

Vcapu

Fig. 4. Characteristic curves of the PV module.

The premise that will be used to obtain MPPT will bebased on the derivative of the power which as evidenced inFig. 4 is equal to zero at MPP, monotonously positive beforeand monotonously negative after.

Since the proposed power converter has the current asa state variable, the derivative of the power used will be inorder to the current:

dP

dI=d(V I)

dI= V

dI

dI+ I

dV

dI= V + I

dV

dI(2)

Assuming that the sampling frequency (∆t - sampling time)of the current and voltage is much higher than the PWMfrequency of the power converter, we can assume that theincremental changes in voltage and current are quasi-linearand that the derivative of the power can be given by [3]:

dP

dI≈ V (t) + I

V (t)− V (t−∆t)

I(t)− I(t−∆t)(3)

This method is similar to the common used incrementalconductance method, but uses the incremental variation ofresistance (dV/dI) instead. The control objective is to obtainthe MPP which is equivalent to:

dP

dI= 0 (4)

To achieve the control objective, Eq. (4), the frequency ofthe power converter would tend to infinitive which is notplausible, meaning that a hysteresis band should be sized tomaintain power oscillations close to MPP.

To correctly size the hysteresis band it is useful to analysethe values of dP/dI close to MPP at different atmosphericconditions as shown in Fig. 5 and 6.

Once the hysteresis band [β, α] is defined, where β < 0 andα > 0, the MPPT strategy can be designed based on the signof the derivative power. The possible scenarios include:

•dP

dI> α⇒ I ↑⇒ Current boost mode.

• α <dP

dI< β ⇒ I →⇒ Maintain mode.

•dP

dI< β ⇒ I ↓⇒ Transfer mode.

Current [A]0 1 2 3 4 5 6 7 8 9

Pow

er [V

]

0

45

90

135

180

225

270

dP/d

I [W

/A]

-20

-10

0

10

20

30

40

-20

-10

0

10

20

30

40

-20

-10

0

10

20

30

40

-20

-10

0

10

20

30

40

α

β

Fig. 5. Curves for different levels of irradiance ([W/m2]: Solid – 1000, dash– 800, dash-dot – 600 and dot – 400) and constant cell temperature of 25◦C.

Current [A]0 1 2 3 4 5 6 7 8 9

Pow

er [V

]

0

45

90

135

180

225

270

dP/d

I [W

/A]

-20

-10

0

10

20

30

40

-20

-10

0

10

20

30

40

-20

-10

0

10

20

30

40

-20

-10

0

10

20

30

40

α

β

Fig. 6. Curves for different levels of cell temperature ([◦C]: Solid – 65, dash– 45, dash-dot – 25 and dot – 5) and constant irradinace of 1000W/m2.

A fixed hysteresis band control has the disadvantage that theswitching frequency varies which is particularly problematicfor the proposed converter. To obtain a MPPT with a constantfrequency, a PWM control should be used.

Since the proposed power converter has the current asa state variable, the reference current Ipvref = Imp canbe tracked using the fixed hysteresis band control and thefollowing equation, deduced from Fig. 2 and given by:

LdcdIpvdt

+ rLdcIpv = Vpv − γ nVo (5)

where γ = 0 in current boost mode, γ = 1 in transfer modeand γ being controlled by the hysteresis band of the PV powerderivative. By applying the Laplace transformation (LT) to Eq.(5) the integrator that allows to track Imp (virtual current) isobtained:

Imp = Ipvref =Vpv − γ nVosLdc + rLdc

(6)

This value will then be fed into a fixed frequency controller,solving the variable frequency problem, as shown in Fig. 7.

A. Linear PI Controller of MPPT

Instead of Eq. (5) being written in terms of the mode ofoperation (c. boost/transfer), it can be written in terms of its

4

dP

dIpvI

pvV 1

dc dcsL rPWM iS

cS

ccI

0

mpI

0

1

ConV

PI controller or

Backstepping

Fig. 7. Maximum power point tracking control of Imp (virtual current), notethat this control scheme could allow the use a sign comparator instead of ahysteresis band.

average value which is related to the duty-cycle δ as follows.

LdcdIpvdt

+ rLdcIpv = Vpv − nVo(1− δ)

= Vpv − nVouc

ucmax= Vpv −KDuc

(7)

This means that for small incremental variations the convertercan be represented using the following transfer function (TF).

Kc(s) ≈KD

1 + sTd(8)

Where KD denotes the equivalent gain and Td representsthe medium delay of the power converter when reacting toa change of uc which can be given by:

Td =Tpwm

2(9)

Using Eq. (7) and (8) the proportional gain Kp and integrationcoefficient Ki of the PI controller are sized based on theequivalent block diagram of the system shown in Fig. 8.

PIcu

pvV

i

mpI pvI

1D

d

K

sT

1

dcsL RpavV

Fig. 8. Block diagram of the feedback system of the MPPT current control.

Where the proportional integral controller (PI) is given by:

C(s) =1 + sTzsTp

= KP +Ki1

s(10)

and R by:

R = rLdc +Ro ≈ rLdc +V 2o

ηP rmpn2 (11)

A methodology to compute the PI gains is proposed in [4]and is described below. The zero of the PI controller shouldeliminate the low frequency dominant pole of the system asfollows:

1 + sTz = 1 + s(LdcR

)⇒ Tz =LdcR

(12)

By using the principle of linear superposition with Vpv(s) = 0and by substituting Tz by its value in Eq. (12), the closed-loopTF of system 8 is given by:

Imp(s)

Ipv(s)=

1

sTp

KD

1 + sTd

1

Rαi

1 +1

sTp

KD

1 + sTd

1

Rαi

=

KDαiTdTpR

s2 + s1

Td+KDαiTdTpR

(13)

where ξ and ωn denote the damping ratio and natural fre-quency of the system (Eq. (13)):

ω2n =

KDαiTdTpR

2ξω2n =

1

Td

(14)

which leads to:Tp =

4ξ2KDTdαiR

(15)

1) Anti-Windup: When a change in Imp occures, the outputuc of the PI controller might attain of the limits ucmax duringthe transient response and in this case the system operates asan open-loop (it winds up). An anti-windup based on Back-Calculation [4] should be used to avoid integrator windup.

B. Nonlinear Backstepping Control of MPPTAn alternative solution to compute the duty-cycle δ can

be a nonlinear backstepping approach [5], [6]. The first stepshould be to find the state variables x, the control input uand construct the systems strict-feedback form. The main statevariable of the full-bridge boost inverter is the current ipv ofthe Ldc inductor. The modulation used to control the inverteris a 3-level PWM that receives a reference uc responsible forgenerating the duty cycle. This means that uc is the controlinput u of the nonlinear inverter system. The systems strict-feedback form is given by:

Ldcd

dt(Ipv) = Vpv − rLdcIpv − nVo(1− δ)

⇔ d

dt(Ipv) = f1 + g1uc

(16)

with gi 6= 0 being a necessary condition. The backsteppingcontrol algorithm is applicable to a strict-feedback system ofany order, including one.

The process starts by defining the tracking error coordinates:

εIpv = Ipv − Imp (17)

With the change of coordinates (17) it is now possible to buildpositive quadratic control Lyapunov function of εIpv as:

V1 =1

2

(εIpv

)2(18)

The time derivative of (18) is given by:

d

dtV1 = εIpv

d εIpvdt

(19)

where the dynamics of the error coordinate is:d εIpvdt

= f1 + g1uc −d

dtImp (20)

Substituting (20) in (19) and by the application of Lyapunov’sSecond Method the system is exponentially stable if:

d V1dt

= εIpv [f1 + g1uc −d

dt(Imp)] = −c1(εIpv )2 (21)

where c1 is a positive design parameter. After some manipu-lation the control input u is obtained:

u = uc =1

g1(−c1εIpv − f1 +

d

dtImp) (22)

5

V. DC-AC INVERTER POWER CONTROL

The grid inverter is responsible for injecting the powerproduced by the PV module into the grid with a near unitypower factor. In order for the inverter to work correctly thevoltage at capacitors Co should be maintained at a higherlevel than the maximum grid voltage to be able to imposethe direction of power flow through inductor Lac:

VoREF>√

2Vgmax(23)

To guarantee that the voltage at the capacitors Co are main-tained at VoREF

while simultaneously injecting power into thegrid with near unity power factor two controllers should bedevised (i.e. an outer voltage loop and inner current loop).

A. Linear PI Controller of DC-Link

Applying Kirchhoff’s current law to each capacitor of theDC-link, it is obtained:

Co1d Vo1dt

= iD1− γgig (24)

Co2d Vo2dt

= iD2− (1− γg) ig (25)

where γg is given by:

γg =

{1, when : Q1 on0, when : Q2 off

(26)

Due to the fact that there are two degrees of freedom (ig; γg),the control of voltages Vo1 and Vo2 has to be done indirectlyby controlling the ig root mean square value (or amplitude)[4]. This is possible because the dynamic of Vo is muchslower when compared to the current ig dynamic.

Considering near unity power factor, electric powerconservation and Io being the average current from thePV module, the grid current RMS value is given by:

Pg = ηPo ⇔ IgRMS=

ηVoVgRMS

Io ≈ GIo (27)

It’s important to note that although there may be anequivalence in the average powers, it is not possible to equalthe instantaneous powers.

By assuming Co1 ≈ Co1 = Co and adding Eq. (24)and (25), the energy balance at the DC-AC inverter is:

2Cod Vodt

= (iD1+ iD2

)− ig (28)

Taking into account Eq. (27) and applying the LT to (28) it isobtained:

s 2CoVo(s) = [ID1(s) + ID2(s)]−GIo(s) (29)

Using Eq. (29) and assuming that the inverter has a delay TdVEq. (30), the control gains of the PI controller responsiblefor controlling the voltage of the DC-link capacitors can becomputed based on the block diagram shown in Fig. 9.

TdV = π

√LacCo√1− ξ2

(30)

PI1

2 os C

RMSgIoI

1DI

2DI

oVREFoV

1 d V

G

sT

v

Fig. 9. Block diagram of linearised voltage control.

Where the proportional integral controller (PI) is given by:

Cv(s) =1 + sTzVsTpV

= KPV+KiV

1

s(31)

By using the principle of linear superposition ID1 = ID2 = 0,the closed-loop TF of the system shown in Fig. 9 is given by:

VoREF(s)

Vo(s)=

1 + sTzVsTpV

−G1 + sTdV

1

s2Coαv

1 +1 + sTzVsTpV

−G1 + sTdV

1

s2Coαv

(32)

In order to compute TpV and TzV the polynomial rule b2k =a bk−1bk+1 (with a 6= 0 and bk being the coefficients) can beapplied to the denominator of polynomial (32), resulting in:TpV = −

a3T 2dVGαv

2CoTzV = a2TdV

(33)

1) Near Unity Power Factor Control: Once the grid currentIgRMS

is computed, it will be necessary to guarantee that theinverter injects the near sinusoidal current into the grid at nearunity power factor. A common used controller is the hysteresisband current control which controls the grid current by forcingit to follow a reference which is constructed based on IgRMS

and on the grid voltage vg frequency ω and phase θv , as shownbelow:

igREF=√

2IgRMSsin (ωt+ θv) (34)

In order to control the grid current ig it is necessary to knowhow its dynamic is affected by the inverter (actuator):

Lacd

dt(ig) = vpwm − vg = γpwmvo − vg (35)

where γpwm is given by:

γpwm =

1, when : Q1 on ⇒ dig

dt> 0

−1, when : Q2 off⇒ digdt

< 0

(36)

The switching action γpwm of the inverter keeps the currentig within the hysteresis band

[−εig ; εig

], by using Lyapunov’s

second method the following control principle can be used toachieve tracking of igREF

:

eig > +εig ⇒ igREF> ig ⇒ ig ↑⇒

digdt

> 0⇒ γpwm = 1

eig < −εig ⇒ igREF< ig ⇒ ig ↓⇒

digdt

< 0⇒ γpwm = −1

(37)

6

where:eig = igREF

− ig (38)

B. Nonlinear Backstepping Control of DC-Link

Once again the first step should be to find the statevariables x, the control input u and construct the systemsstrict-feedback form. The state variables of the output inverterare voltages Vo1 and Vo2 at each half-bridge capacitor andthe output current ig of the Lac inductor. The modulationused to control the inverter is a 2-level PWM that receives asinusoidal modulation index β as an input. This means thatβ is the control input u of the nonlinear inverter system.

The energy transfer through the DC-AC converter, consideringit is conservative and operates at unity power factor is givenby:

d

dt(1

2Co1V

2o1) +

d

dt(1

2Co2V

2o2) =iD1

Vo1 + iD2Vo2 − igvg

(39)

The stabilizing function must relate the system energy balanceto the amplitude value Ig to separate the converter dynamics[6]. This means that the control parameter IgRMS

for the innercurrent loop sinusoidal reference current (in phase with vg) isobtained based on:

d

dt(V 2o ) = f1 + g1IgRMS

(40)

with g1 6= 0 and Vo1 ≈ Vo2 = Vo. The expression that relatesthe control input u = β to the inverter model is given by:

Lacd

dt(ig) = vpwm − vg

⇔ d

dt(ig) = f2 + g2β

(41)

Finally, we can construct the systems strict-feedback form as:

d

dt(V 2o ) = f1 + g1IgRMS

(42)

d

dt(ig) = f2 + g2β (43)

with x = (x1, x2) = (V 2o , ig) ∈ R2 and gi 6= 0 ∀i ∈ {1, 2}.

The process starts by defining the tracking error coordinates:

εV 2o

= V 2o − V 2

oREF(44)

εig = ig − igREF(45)

With the change of coordinates (44) and (45) it is now possibleto build positive quadratic control Lyapunov Functions of εV 2

o

and εig as:

V1 =1

2(εV 2

o)2 (46)

V2 =1

2(εig )2 (47)

Note that (47) does not contain Vi−1 (i.e. not in an augmentedLyapunov function form). This can be explained by the fact ofthe tracking error εig not being directly related to α1 (Eq. 54).

The time derivatives of (46) and (47) are given by:

d

dtV1 = εV 2

o

d εV 2o

dt(48)

d

dtV2 = εig

d εigdt

(49)

where the dynamics of the error coordinate are:

d εV 2o

dt= f1 + g1IgRMS

− d

dtV 2oREF

(50)

and:d εigdt

= f2 + g2β

−√

2[ω cos (ωt+ θv)α1 + sin (ωt+ θv)d

dt(α1)]

(51)

substituting (50) in (48), (51) in (49) and by the application ofLyapunov’s Second Method the system is exponentially stable:

dV1dt

= εV 2o

[f1 + g1IgRMS− d

dt(V 2oREF

)] = −c2(εV 2o

)2 (52)

d V2dt

=εig [f2 + g2β −√

2(ω cos (ωt+ θv)α1+

sin (ωt+ θv)d

dt(α1))] = −c3(εig )2

(53)

for c2 > 0 and c3 > 0. After some manipulation of (52) and(53) the stabilizing virtual controls are obtained:

α1 = IgRMS=

1

g1(−c2εV 2

o− f1 +

d

dtV 2oREF

) (54)

u = β =1

g2(−c3εig − f2+

√2(ω cos (ωt+ θv)α1 + sin (ωt+ θv)

d

dt(α1)))

(55)

where α1 is part of the outer slow dynamics DC voltage loopand u the inner fast dynamics AC current loop.

C. Command Filtering Backstepping

Although the Backstepping design seems simple, theincrease in order dramatically complicates the computation ofthe time-derivative of the virtual control laws. This calculationcan be brought down to a differentiation by filtering using acommand filter (CF) [5] as shown in Fig. 10. The feedbackcontrol signal (54) is not directly applied, instead this signal iscorrected by the CF resulting in xi+1,c ≡ αi and xi+1,c ≡ αi.

where ζ and ωn are, respectively, the damping coefficient

2

n1

s

1

s

i 1,i cx

1,i cx

2 n

1fx

2fx

Fig. 10. Command filter (CF) block diagram [5], [7].

7

and the fixed natural frequency of the filter. Analysing Fig.10 it can be shown that the signal xi+1,c is obtained byusing integral method rather than a differential method whichattenuates the high frequency disturbances [7]. The CF TF isgiven by:

[xi+1,c

xi+1,c] =

ω2n

s2 + 2ζωns+ ω2n

[1s

]αi (56)

In order that xi+1,c and xi+1,c accurately track αi and αi ,the range of ci gains is restricted to be: ci ≤ ωn [7].

VI. SIMULATION AND PERFORMANCE ANALYSIS

In order to evaluate the performance of the control systems,a set of simulation scenarios will be performed, includingvarying atmospheric conditions, unbalanced initial voltagesbetween DC-link capacitors and grid disturbances using thedata available in Tables I and II. The controllers will bereferred to as: TCT, traditional control techniques, and CFBScommand filtering backstepping.

A. Varying Atmospheric Conditions

The most important feature of the microinverter controlis the MPPT and respective DC/AC voltage and currentcontrol. In order to evaluate the correctness of these controlsystems the PV module will be exposed to different levelsof irradiance, more specifically an irradiance variation of1000W/m2 to 100W/m2.

MPPT Control (Fig. 11): When the PV module is exposedto different levels of irradiance, it is expected to adjust itscurrent Ipv and consequent voltage Vpv in order to achieveMPPT. Fig. 11 proves that in both controllers the currentat the PV module Ipv is adjusted accordingly to the levelof irradiance. The TCT approach leads to a less successfulMPPT, this is seen in the form of a ”knee” (temporarynon-maximum point).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cur

rent

[A]

0

2

4

6

8

10

Ipv

Time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cur

rent

[A]

0

2

4

6

8

10

Ipv

Fig. 11. MPPT current control. Top Fig.) TCT. Bottom Fig.) CFBS.

DC-AC Voltage Control (Fig. 12) the voltage at eachcapacitor Coi should be maintained at a higher voltagethan the maximum expected grid voltage VgMAX

= 358 Vindependently of the power produced by the PV module. The

objective to maintain the voltage at the DC-link above themaximum expected maximum grid value, is achieved by bothcontrollers, although the CFBS obtains a better performancesince the overshoot and undershoot when in transient regimeis minimum.

Time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Vol

tage

[V]

0

100

200

300

400

500

VCo1

VCo2

Vo

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Vol

tage

[V]

0

100

200

300

400

500

VCo1

VCo2

Vo

Fig. 12. DC-AC Voltage Control. Top Fig.) TCT. Bottom Fig.) CFBS.

DC-AC Injected Current Control (Fig. 13): This controlsystem is responsible for injecting the power produced by thePV module into the grid by imposing a sinusoidal currentig with a nearly unity power factor. In this simulation, it ispossible to verify that both control systems do in fact injecta sinusoidal current ig into the grid for a wide range ofirradiance level while the amplitude of the current igREF

isprovided by the DC/AC voltage control and is used as partof the reference current. Lac is computed for a nominal igcurrent, so performance loss is expected for low ig levels.

• To evaluate the quality of the injected current a commonparameter used is the current total harmonic distortion(THD). As shown in Fig. 14 both control systems exhibitlow THD (< 10%) for a wide range of power levels, thisproves that the DC-AC injected current control is in factinjecting a near sinusoidal current as supposed. Anotherconclusion that can be made by analysing Fig. 14 is thatthe CFBS control has a lower THD for low level currentswhich indicates that it has a better performance than theTCT control.

• A successful power injection should exhibit a near unitypower factor (fp) since this means that the power con-verter is injecting mostly active power. As shown inFig. 15, once again, both control systems achieve thisobjective since they both display fp > 0.99 for a widerange of PV module output power levels.

• The total energy produced Epv and injected Eg canbe computed based on the power curves displayed inFig. 16. By applying a trapezoidal numerical integrationto each power curve the following produced/injectedenergies ([W s]) are obtained: TCT): Epv = 139.3 andEg = 130.0; CFBS): Epv = 146.1 and Eg = 136.4.Which represents a 4.7% increase in Epv and a 4.4%increase in Eg by the CFBS control approach.

In order for the converter to be a viable candidate for an ACmodule, it is necessary to exhibit a high Efficiency for a wide

8

Time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cur

rent

[A]

-2

-1

0

1

2

igigREF

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Cur

rent

[A]

-2

-1

0

1

2

igigREF

Fig. 13. DC-AC Current Control. Top Fig.) TCT. Bottom Fig.) CFBS.

Time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

[%]

0

0.1

0.2

0.3

0.4

0.5

KiTHD

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

[%]

0

0.1

0.2

0.3

0.4

0.5

KiTHD

Fig. 14. Injected current THD. Top Fig.) TCT. Bottom Fig.) CFBS.

range of power levels. Analysing Fig. 17 it can be shownthat the converter has a maximum steady-state efficiencyof ηTotal = 94.6 % and manages to maintain efficienciesηTotal ≥ 93.1 % ∀ Ppv ≥ 30 %P rmp using both controlapproaches.

B. European Weighted Efficiency

European Weighted Efficiency ηEU is commonly used tocompute the weighted average operating efficiency over ayearly power distribution, below we can find the ηEU of theCFBS approach since it has a better MPPT performance atlow irradiance levels.

ηEU = 0.20× η100% + 0.48× η50% + 0.10η30%

+ 0.13η20% + 0.06η10% + 0.03η5%

= 0.20× 93.2 + 0.48× 94.6 + 0.10× 94.0

+ 0.13× 93.5 + 0.06× 89.0 + 0.03× 0.55

= 91.0%

(57)

C. Unbalanced Initial Voltages

A robust control system should be able to start-up theconverter even if its capacitors are initially discharged, thisfeature is guaranteed as shown in Fig. 18. Another scenario

Time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.9

0.92

0.94

0.96

0.98

1

fp

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.9

0.92

0.94

0.96

0.98

1

fp

Fig. 15. Power factor f.p. Top Fig.) TCT. Bottom Fig.) CFBS.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pow

er [W

]

0

100

200

300

Ppv

Pg

Time [s]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pow

er [W

]0

100

200

300

Ppv

Pg

Fig. 16. Input and output power. Top Fig.) TCT. Bottom Fig.) CFBS.

that may occur is the existence of a much higher voltage inone of the capacitors when the converter is turned on (dueto less parasitic losses). The controller should be capable ofsteering the voltages (stabilize) in case of extreme unbalancedvoltages as shown in Fig. 19 to be considered robust.

D. Grid Disturbances

The control system should also be capable of maintaininga stable behaviour, even in the presence of temporary griddisturbances. There are three types of grid disturbances: (i)Frequency variations (ii) Voltage Sag and (iii) Voltage Swell.As shown in Fig. 20 and 21 in the presence of (ii) and (iii)grid disturbances both controllers are capable of maintaininga suitable behaviour. It is also possible to notice that thenonlinear controller maintains a near unity power factor fora wider range of disturbances (i.e. wider operation span).

VII. FINAL CONCLUSIONS

The research presented in this dissertation focused onpresenting a possible solution to remove the least reliablecomponent of the microinverter, the electrolytic capacitor, butnot without before explaining what causes the need for suchcapacitors. Due to the largest capacitor used in the proposed

9

Time [s]0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.5

0.6

0.7

0.8

0.9

2Total

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

0.6

0.7

0.8

0.9

2Total

Fig. 17. Microinverter Efficiency. Top Fig.) TCT. Bottom Fig.) CFBS.

Time [s]0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Vol

tage

[V]

0

200

400

600

VCo1

VCo2

Vo

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Vol

tage

[V]

0

200

400

600

VCo1

VCo2

Vo

Fig. 18. DC/AC voltage control when both capacitors are initially discharged.Top Fig.) TCT. Bottom Fig.) CFBS.

power converter being equal to 50µF the main objective wasachieved (Fig. 1).

The other objective was to implement two highly robustcontrollers, both using different theoretical concepts toachieve stability. Both control systems proved to be highlyrobust, however the nonlinear approach achieved betterindices of performance for every test performed withoutany trade-offs (i.e. overshoots or undershoots). This controlapproach converges to steady states much faster than thelinear control without the need of high overshoot currentswhile remaining robust, which indicates that it is the bettercontrol approach. Some trade-off’s of the nonlinear controlinclude:

• The need for more current measurements since thiscontrol does not use the superposition principle.

• Tedious gain selection since positive design gains cistrongly influence the dynamic of the closed loop.

In terms of the power converters performance, it has a maxi-mum efficiency of 94.6 %, which is a good value (maximumknown is 96 % [8]) for a PV microinverter since, as it is wellknown, microinverters have the least efficient converters (dueto the HF transformer). The converter also exhibits a good

Time [s]0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Vol

tage

[V]

0

200

400

600

800

VCo1

VCo2

Vo

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Vol

tage

[V]

0

200

400

600

800

VCo1

VCo2

Vo

Fig. 19. DC/AC voltage control when capacitors have different initialvoltages. Top Fig.) TCT. Bottom Fig.) CFBS.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Cur

rent

[A]

-4

-2

0

2

4

igV g100

Time [s]0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Vol

tage

[V]

0

200

400

600

VCo1

VCo2

Vo

VgRMS

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5C

urre

nt [A

]-4

-2

0

2

4

igV g100

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Vol

tage

[V]

0

200

400

600

VCo1

VCo2

Vo

VgRMS

!40% !30%!10%

!60% !50%

!40% !30%!10%

!50%!60%

Fig. 20. DC/AC control in the presence of a grid voltage sag. Top two Fig.)TCT approach. Bottom two Fig.) CFBS approach.

European Weighted Efficiency equal to ηEU = 91.0%.

REFERENCES

[1] Fritz Schimpf & Lars Norum, ”Effective use of filmcapacitors in single-phase PV-inverters by active powerdecoupling”, IECON - 36th Annual Conference on IEEEIndustrial Electronics Society, 7-10 Nov. 2010.

[2] En-Sung Park, Sung Jin Choi, J.M. Lee & B.H. Cho, Asoft-switching active-clamp scheme for isolated full-bridgeboost converter, Applied Power Electronics Conferenceand Exposition - Nineteenth Annual IEEE, 22-26 Feb.2004.

10

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Cur

rent

[A]

-5

0

5

igV g100

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Vol

tage

[V]

0

200

400

600

VCo1

VCo2

Vo

VgRMS

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Cur

rent

[A]

-5

0

5

igV g100

Time [s]0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Vol

tage

[V]

0

200

400

600

VCo1

VCo2

Vo

VgRMS

+50% +40% +30%+10%

+60%

+60% +50% +40% +30%+10%

Fig. 21. DC/AC control in the presence of a grid voltage swell. Top two Fig.)TCT approach. Bottom two Fig.) CFBS approach.

TABLE IMAIN DATA OF PV MODULE & PROPOSED POWER CONVERTER.

PV Module Power converter

P rMP 270Wp Transformer turns ratio n 0.125

V rMP 31.1W Leakage inductor Lλ 12 µH

IrMP 8.67A Intermediate inductor Ldc 6mH

V rca 38.2 V Active-clamp capacitor Cc 8.3 µF

Ircc 9.19A DC-Link capacitor Co 50 µF

NOCT 45±2 ◦C AC Inductor Lac 82.7mH

µV ca -118.4 mV/◦C

µIcc 4.9 mA/◦C

TABLE IIALL CONTROLLER DESIGN GAINS AND COMMAND FILTER PARAMETERS.

TCTMPPT control Kp =1.20 Ki =1959 Kw =48.49

TCTDC-Link control KpV =0.0078 KiV =0.1531

CFBSMPPT control c1 = 5000

CFBSDC/AC control c2 = 159 ωn = 2π50 ξ = 1 c3 = 5000

[3] Sonia Pinto, Conversores Comutados para EnergiasRenovaveis, Notes, Instituto Superior Tecnico, Universi-dade de Lisboa, 2014.

[4] Fernando Silva, ”ELECTRONICA INDUSTRIAL Semicon-dutores e Conversores de Potencia, 2nd Edition”, Gul-benkian, February 2013.

[5] Guilherme Trigo, ”Robust and Adaptive Nonlinear At-titude Control of a Spacecraft”, MSc Thesis, InstitutoSuperior Tecnico, Universidade de Lisboa, October 2011.

[6] Aranzazu D. Martin, J. M. Cano, J. Fernando A. Silva& Jesus R. Vazquez, ”Backstepping Control of SmartGrid-Connected Distributed Photovoltaic Power Suppliesfor Telecom Equipment”, IEEE Transactions on EnergyConversion, Volume: 30, Issue: 4, pp. 1496-1504, Dec.2015.

[7] Jay A. Farrell, Marios Polycarpou, Manu Sharma &Wenjie Dong, ”Command Filtered Backstepping”, IEEETransactions on Automatic Control, Volume: 54, Issue: 6,pp. 1391-1395, June 2009.

[8] Samir Kouro, Jose I. Leon, Dimitri Vinnikov & LeopoldoG. Franquelo, ”Grid-Connected Photovoltaic Systems: AnOverview of Recent Research and Emerging PV ConverterTechnology”, IEEE Industrial Electronics Magazine, Vol-ume:9, Issue: 1, pp. 47-61, March 2015.