Dynamic Modeling of a ZETA Converter in DCM Applied to Low Power Renewable Sources

Renan Caron Viero and Fernando Soares dos Reis Power Electronics Laboratory of the PUCRS

Pontifical Catholic University of Rio Grande do SuI Porto Alegre, Brazil fdosreis@pucrs.br

Abstract- This paper presents the dynamic modeling of a ZETA converter in discontinuous conduction mode (DCM) applied to low power renewable sources using the generalized switch averaging technique. The whole system consists of a ZETA converter associated with a full bridge inverter operating at low frequency. Thus, the system works as a sinusoidal low power source. The ZETA converter plays the main role in this arrangement, producing a rectified sinusoidal current waveform synchronized with the power grid. Initially it is presented a brief analysis of the system, as well as the characteristics that make possible the dynamic analysis in DCM. The generalized switch averaging technique are discussed, and also the small-signal analysis of the ZETA converter. In order to validate the current study experimental and simulations results are presented.

I. INTRODUCTION

Nowadays the society is struggling increasingly to have no longer the word energy associated with terms such as pollution, creating new ecological forms of energy usage, raising the efficiency of the devices and improving the technology already known in power generation systems to reduce the emission of pollutants into the environment. The growing number of laws related to environmental care, mainly by encouraging the use and allocating more resources to renewable energy sources and power quality, confirm this fact [1].

Furthermore, the use of power factor correction regulators is becoming increasingly necessary in large systems, mainly because this approach does not limit the absorption of active power from the power grid and minimize problems caused by harmonic distortion [2].

The Zeta converter working in discontinuous conduction mode (DCM) has many applications in the areas previously mentioned [3 , 4]. However, until now, there is not a well­defined analytical mathematical model for this converter working in DCM. This fact makes the design of a closed­loop control for this converter an empirical and time consuming task, held in circuit simulation software.

978-1-4577-0541-01111$26.00 ©2011 IEEE 685

Typically, to connect low power renewable energy systems to the grid or to a load it is used a conventional step up DC-DC converter connected to a current or voltage source inverter. Due to the high frequency switching of both stages, this configuration reduces the overall system efficiency. In many cases the DC-DC converter is implemented using a Boost converter which, unfortunately, does not provide galvanic isolation. This paper presents the dynamic modeling of a system where the step-up converter is implemented by an isolated Zeta converter [5]. In this case, the ZETA converter has two functions: To provide electrical isolation and synthesize a rectified sinusoidal current waveform similar to the power grid. Once the inverter operates in low frequency (50/60 Hz) very low switching Power losses are expectable.

This paper presents a dynamic model for the ZETA converter working in DCM applied to low power renewable sources and is organized as follows: The principle of operation of the system is presented in Section 2. The technique employed to obtain the model is discussed in Section 3. The small-signal dynamic modeling is presented in Section 4. Sections 5 and 6 present the simulation results and experimental results respectively. The Section 7 concludes the paper.

II. PRINCIPLE OF OPERATION

A. Basic Topology In this section it is presented a modular system,

synchronized with the electric grid, able to inject low current harmonic content, providing galvanic insulation between the solar panels and the grid.

The proposed energy conversion system, composed of three stages is shown in Fig. 1.

- - - - - - - - - - - - - - - ~, - - - - - - - - - - - - - - - - - - - - - - - ,r - - - - - - - - - - - - - ~ , " I

-=::---.-~

Figure 1. Proposed energy conversion system.

The first stage consists of the solar panel, in parallel with a bulky capacitor Cpv. The second stage consists of an isolated Zeta converter operating in DCM. At this stage it is synthesized a sinusoidal current in absolute value ha in the inductor La' synchronized with the grid. The third stage consists of a low frequency full bridge inverter (a 180 degree phase inverter).

The main theoretical waveforms of the proposed arrangement are presented in Fig. 2.

Figure 2. Wave forms of each stage of the proposed system.

B. SimplifYing Assumptions The dynamic behavior and the stages of operation of this

system will be analyzed further from the following simplifying assumptions:

3.

• The transformer has an unitary ratio (Ns / Np = 1) and is represented by its magnetizing inductance Lm;

• The quasi-static approximations are assumed, hence the output voltage Va is considered constant within a high frequency switching period;

• The solar panels associated with the batteries are represented by the Vg voltage source;

• All semiconductors are considered ideal;

• The converter does not affect the power grid, and the grid may be represented by a simple equivalent resistive load R that represents the power absorbed by the power grid.

These simplifications lead to the topology shown in Fig.

+ VS'l,I- Vf:- + + VI., 11.> r -----ti-

0 + +

VLm. D VD Vo Vg iDf

Figure 3. System after simplifying assumptions.

C. Stages of Operation and Waveforms

In DCM the converter of Fig. 3 presents three stages of operation, represented schematically in Fig. 4. The main converter waveforms within a high frequency period are shown in Fig. 5(a). The main converter waveforms within a low frequency period are presented in Fig 5(b), considering the sinusoidal PWM modulation.

686

+ V,,, - + v .. - . U> ~

+

+ V,,, - + v .. - . ~ ~

E -is", S +

VLm.

+

Vg

+ V,,,,-

+ I ~'" S + VLm. R Vo

vg-rL-________ L-______ -L ________ ~ (c)

Figure 4. (a) First, (b) second and (c) third stages of operation of the DC­DC Zeta converter working in DCM.

iLIIl

(b)

Figure 5. Main converter waveforms : (a) high frequency period and (b) low frequency period.

D. Static Gain

The conduction time of the diode (tD) is constant when there is no variation of the load R in DCM [6]. So, the instantaneous static gain g(t) of the ZETA converter in DCM can be expressed by (1). This characteristic implies III a linear relationship between the input and output.

(1)

where

D = ~ = ~2Leq f 1 T R

(2)

d(t) is the instantaneous duty cycle of the converter, f is the switching frequency, L eq is the equivalent inductance of the converter and R is the load equivalent resistance.

It is possible to conclude that the control-to-output relationship of the converter is also linear, in the steady-state analysis.

III. GENERALIZED SWITCH AVERAGING TECHNIQUE

A. The Technique

The state-space averaging (SSA) technique was first presented in [7] , and previously used successfully in the analysis of the ZETA converter in continuous conduction mode (CCM) and for SEPIC converter in DCM [8][9]. This method can be extended to the DCM, but in this case it is less accurate, because the converter transfer function is dependent of the load [10]. Another aspect it is that only the coefficients of the matrices are averaged, and not necessarily the state variables themselves [11]. However, it is possible to use the Generalized Switch Averaging technique, where well-known models derived from CCM (including SSA model) may be used in DCM analysis, by including an additional switching network to the model through feedback [10]. Fig. 6 illustrates this concept. In Fig. 6, u(t) are the independent inputs of the converter. y(t) are the converter outputs. u,(t) are the inputs of the switch network. y,(t) is the output of the switch network. Finally, the input ult) are all the converter control inputs.

Switch Network

y,(t)

us(t)

u(t)

Time­invariant Network

Figure 6. Generalized Switch Averaging block diagram.

yet)

The main objective of this technique is to use an available time-invariant model obtained in CCM analysis to study power converters in DCM. To achieve this goal, the

687

control input d(t) in DCM must be substituted by an equivalent control input in CCM y,(t). This transformation will be presented in the next section of this paper.

B. Switch Network

The switch network method is based on the loss-free resistor model [12]. Fig. 7 illustrates this model.

i t i2 + +

VI v2

: Re(d)

- :

Figure 7. Switch Network. Loss-free resistor model.

The switch S is represented by the effective resistance of switch input port Re(d(t)) , as described by (3).

2L f Re(d(t))=~

d(t) (3)

The relationship between the steady-state gain and Re(d(t)) can be found applying (3) and (2) into (1).

d(t) R get) = ----n: = Re(d(t)) (4)

The diode D is represented by a dependent power source. In an ideal system the conservation of energy principle is applicable. Consequently, all the energy consumed by Re(d(t)) is delivered to the circuit through the power source, as described by (5).

(5)

It is possible to correlate the input and output parameters of the loss-free resistor model by the switch conversion ratio fJ(t).

f1 (VI (t) , i2(t) ,d(t)) = 1 i(t) 1 + Re(d(t))_2-

VI (t) (6)

Based on the conservation energy principle, a second (and equivalent) switch conversion ratio fJ(t) can be obtained.

. 1 f1 (V2 (t) , II (t) , d(t)) = i (t)

1 + Re(d(t))-I­v2 (t)

(7)

The variables v j, Vb if and i2 are respectively the currents at the switch (isw) and diode (iD) and the voltages at the switch (vsw) and diode (VD)'

fJ(t) can be used in any CCM model in place of d(t) to achieve the DCM model.

C. Time Invariant Network

In this paper, the Time-Invariant Network is a SSA model of the ZETA converter working in CCM [5] , but

instead of d(t) , the averaging model is obtained by means of fl(t). Considering the converter shown in Fig. 3, it can be represented in state-space by (8).

where

{x(t) = ~x(t) + ~ u(t)

y (t) = C x(t) + Eu(t)

A = A, fl(t) + A2 (1 - fl(t))

B = B, fl(t) + B2 (1 - fl(t))

C = C, fl(t) + C 2 (1 - fl(t))

E = E, fl(t) + E2 (1 - fl(t))

(3)

(4)

The state variables x(t) are the inductor currents (hm and h o) and the capacitor voltage (ve). In this model, u(t) contains the independent inputs of the power converter, such as the input voltage vl t) and, in some cases, the output current. fl(t) is the switch conversion ratio. The indexed matrices in (9) are the matrices that describe the behavior of the state variables and the output y(t) at the first and second stage of operation. These matrices will be presented further in the sequence of this paper.

The small-signal AC model is obtained by perturbation and linearization of (8). The system perturbations are presented in (10).

x(t) = X + x(t)

fl(t) = flo + ft(t)

u(t) = U + u(t)

yet) = Y + .Yet)

us(t)=Us +us(t)

ue(t) = Ue + ue(t)

(5)

The capital letters are the steady-state values of the variables. The lower case letters with a hat above them are small signal variations. Thus, (11) is the steady-state solution of the system and (12) is the small-signal AC model that contains the dynamic behavior of the converter in CCM.

where

fX=-£IBU

lY=(-C£IB+E)U

{i(t) = Ai(t) + Bu(t) + B D it(t)

Yet) = Ci(t) + Eu(t) + ED it(t)

A = Al flo +A2(1- flo)

B = BI flo + B2 (1- flo )

C = CI flo + C2 (1- flo )

E = EI flo + E2 (1- flo )

BD = (AI - A2)X + (BI - B2)U

ED = (CI - C2)X + (EI -E2)U

(6)

(7)

(8)

The parameter flo is obtained by evaluating fl(Vj ,i2,d) or fl(V2, hd) at the quiescent operation point.

(9)

688

The linearized switch conversion ratio is

ft(t) = kA (t) + ksus (t)

Where kc and ks are gains given by

k = dfl(U" UJI e dUe uc:Uc

us-us

k = dfl(U" UJI s d us=Us u,

uc=Uc

(10)

(11)

Assuming that the outputs of (12) are the switch network inputs, i.e. ,

us(t) = Cx(t)+Eu(t)+EDft(t)

Applying (17) into (15) leads to

(17)

" ) kC") kE " ) k " ) fl(t = s x(t + s u(t + e ue(t (18) l-ksEd l-ksEd l-ksEd

Finally, the dynamic behavior of the converter working in DCM can be obtained by substituting (18) in the first line of(12),

x(t) = ADCM x(t) + BDCM u(t) + BDDCM Uc (t)

Where

_ BDksC ADCM - A+---l-k E s d

_ BDksE BDCM - B+---l-k E s d

k B =B __ c_ DDCM D l-k E

s d

(19)

(20)

Note that the system does not have the switch conversion ratio at its input anymore because the real controllable input is the duty cycle, represented by Uc .

IV. DYNAMIC MODELING

It is necessary to define the system inputs and outputs. The converter of Fig. 3 has one control input and one independent input, as described in (21).

u(t) = [vg (t) ]

Uc (t) = [d(t) 1 (21)

The outputs of the converter (and the inputs of the switch network) are

yet) =u,(t) =[~ ] = [~J Substitution of (22) into (7) yields to:

. 1 fl (VD (t) , lsw (t) , d(t)) = i (t)

1 + Re(d(t))--""-----­vD (t)

(22)

(23)

Appling (23) to (14) leads to the quiescent switch conversion ratio for the ZETA converter.

D fJo = D + DI (24)

The gains kc and ks may be found applying (23) into (16).

k = 2DI c (D+DI)2

(25)

Re(D) DDI T

Vg (D+Dlf

DI2 (26) k =

$

Applying Kirchoffs law in the circuits of Fig. 4(a) and Fig. 4(b) it is possible to obtain the equations that describe the behavior of the system at each stage of operation. From the first stage results,

diLIn 1 --=-v

dt Lin g

diLo 1 (. ) --=- -RI -v +v dt L Lo C g o

dvC 1 . --=-1 dt C Lo

(27)

From the second stage results,

(28) dvC 1 . --=--1 dt C Lin

i = 0 sw

It is not necessary to modeling the third stage of operation, once in the context of this work, the time-invariant network is a SSA model of the ZETA converter working in CCM.

Writing (27) and (28) in matrix form, the indexed matrices are obtained. These matrices are presented in (29).

689

A -,-o o

o

o

o

0 0 -Yro -Xo Yc 0 0 XI/1

-Yro 0 0 0

B'=m

[

1 0 1 T [0 01 T c, = 1 0 C2 = 0 0 o -1 0 0

E,=[~l E2=[~l

(29)

By substituting the matrices of (29) and the quiescent switch conversion ratio (24) into (13), the averaged matrices are found. Appling (13), (25) and (26) into (20) leads to the dynamic model that represent the dynamic behavior of the converter in DCM.

v. SIMULATION RESULTS

The control-to-output dynamic model was obtained using an algorithm in the software MATLAB ™ and validated using the circuit of Fig. 8, on the power simulation software PSIMTM. Fig. 9 shows the bode diagram of the control-to­output transfer function at the operational point D = 0.5, considering the current in the output inductor as the converter output. Fig. 10 presents the step response of the obtained model and the simulated results from PSIMTM. The values of the components involved in the simulation are presented in Table I.

TABLE I. COMPONENT V ALVES

Component Definition Value

R Equivalent Load Resistance l30n

Lm Magnetizing Inductance 871lH

Lo Output Inductance 22mH

C Coupling Capacitance 680 nF

Vg Input Voltage 34 V

f Switching Frequency 20kHz

D Duty-cycle at operational point 47%

AC 0 DC 4.7 f 0

iii' ~ -20 " .E

130 "680n '22m ' 87u

- -- -

Figure 8. PSIM™ simulation circuit.

~ -40~--~----~~~~~~--~~~~~----~~

'" ::;:

-60~--~--~~--~~~-7----~ __ ~~----~~

" ~ -270 a.

-360

Frequency (Hz)

Figure 9. Bode diagram of the control-to-output system (Gid(s} ).

Figure 10. Step response of the system. d(t} = 0.4 7.

In order to validate the Bode diagram presented in Fig, 9, two test points were chosen and simulated using the circuit of Fig. 8. The selected test points are the unity gain frequency and the -90 0 phase frequency. The circuit was simulated using an AC sinusoidal source with a DC offset. The results of these simulations are shown in Fig. 11 and Fig. 12.

690

Figure 11. Unity gain frequency. d(t} = 0.47 + 0.3sen(27f1980t}.

Figure 12. -900 phase frequency. d(t} = 0.47 + 0.3sen(27f1 670t}.

VI. EXPERIMENTAL RESULTS

A prototype was designed and tested. A picture of the prototype is shown in Fig. 13. The step response of the prototype for d(t) = 0.47 is illustrated in Fig. 14.

Figure 13. Prototype.

The output inductor current was measured using a Hall Effect sensor, with a gain of approximately 4.8 VIA.

~ 0 2.6011

Figure 14 .. Prototipe step response. d(t) = 0.47.

VII. CONCLUSIONS

This paper presents a dynamic modeling of a ZETA converter in discontinuous conduction mode (DCM) applied to low power renewable sources using the generalized switch averaging technique.

In this work was developed an unpublished dynamic model of the Zeta converter working in DCM. The obtained results had shown that the proposed model is extremely accurate. It is therefore, useful for the development of the control strategies for the Zeta converter working in the DCM. The peak time and the current overshoot obtained by the proposed model had presented results nearly identical to simulated and experimental results. An AC analysis was performed in software PSIM™, this study shown that the Bode diagram generated from the proposed model is accurate. Finally, it is important to remark that the implementation of the model usmg the software MATLAB TM allows the development of a more precise control strategy.

691

ACKNOWLEDGMENT

The authors would like to thank the support of PUCRS (Pontificia Universidade Cat6lica do Rio Grande do SuI) and CAPES (Coordenac;ao de Aperfeic;oamento de Pessoal de Nivel Superior) which made possible this research.

REFERENCES

[I] Departamento de Estado dos EUA. "Soluyoes de Energia Limpa". eJoumal USA: Perspectivas Economicas. Vol 11. nO 2. 2006.

[2] J. A. Pomilio. "Pre-reguladores de Fator de Potencia". Publicayao FEE 03/95. 2007.

[3] H. F. M. Lopez, C. Zollmann, R. C. Viero and F. S. dos Reis. "Photovoltaic Panels Grid-Tied By a Zeta Converter". Brazilian Power Electronics Conference, pp. 1-6.2009.

[4] A. Peres, D. C. Martins and I. Barbi. "ZETA Converter Applied in Power Factor Correction". IEEE Power Electronics Specialists Conf., pp. 1152-1157. 1994.

[5] H. F. M. Lopez, R. C. Viero, C. Zollmann, L. L. Reckzielgel, R. Tonkoski, F. S. dos Reis. Low Power Solar System Grid-Tied With MPPT Based on Temperature Compensation. The 9th Annual Electrical Power and Energy Conference - EPEC. 2009.

[6] D. C. Martins, I. Barbi. "Eletronica de Potencia: Conversores CC-CC Basicos nao Isolados". 3th ed., Author's Edition. Fiorianopolis, SC. 2008.

[7] Middlebook, R. D. and Cuk, S. A General Unified Approach to Modeling Switching-Converter Power Stages. International Journal of Electronics, Vol. 42, pp. 521-550. 1977.

[8] Eng, V. and Bunlaksananusorn, C. Dynamic Modeling of a Zeta Converter with State-Space Averaging Technique. Proc. of ECTI­CON 2008, pp. 969-972. 2008.

[9] Eng, V. and Bunlaksananusom, C. Modeling of a SEPlC Converter Operatng in Discontinuous Conduction Mode. Proc. of ECTI-CON 2009, pp. 140-143. 2009.

[10] Erickson R. W. Fundamentals of Power Electronics. International Thomson Publishing. 1997.

[11] Sun, J; Mitchell, D. M; Greuel, M. F; Krein, P.T and Bass, R. M. Averaged modeling of PWM converters operating in discontinuous conduction mode. IEEE Trans. on Power Electronics. Vol. 16, No. 4. 2001.

[12] Singer, S. Realization of Loss-Free Resistive Elements. IEEE Trans. on Circuits and Systems. Vol. CAS-37, No. I, pp. 54-60. 1990.