# Mathematical Modelling of Non-Isolated Bi-Directional DC-DC Converter on Transients and Steady State Response

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#### Mathematical Modelling of Non-Isolated Bi-Directional DC-DC Converter on Transients and Steady State Response

Ranjan Pramanik

Department of Electrical and Electronics Engineering O P Jindal Institute of Technology

Raigarh, CG, India

AbstractEnergy management strategy is gaining much popularity because now the load is taking energy from different non-conventional/conventional sources as well as from energy storage element to provide uninterruptable power supply to the load. The bi-directional converter placed in between a DC voltage source and a battery to allow energy transfer. The converter connected with high voltage DC-bus and also the converter backed up with battery. Using the bi-directional dc-dc converter operation modes (Buck and Boost) this battery will be charged and discharged as per the suitable condition for uninterrupted power supply to the load. In this paper, a non- isolated half-bridge bi-directional dc-dc converter is studied for hybrid vehicle technology. The state space formulation of the bi- directional dc-dc converter in ideal case as well as with parasitics in different modes of operations is derived. The averaging and linearization technique is applied to get the small signal model of the converter.. To verify the methodology, converter model is developed in MATLAB/SIMULINK environment.

Keywords-Bi-directional DC-DC Converter, Energy management system,State space modelling, Small-Signal Analysis Hybrid electric vehicle etc.

1. INTRODUCTION

Using a bi-directional dcdc converter along with low- voltage energy storage for the high-voltage dc bus has been a prominent option for hybrid electric storage technology. Its have huge applications on hybrid electric vehicles and with non-conventional energy sources like fuel cell or photovoltaic cell etc. On this kind of hybrid storage technology, there is need to create an energy management strategy because to maintain continuous operation which provides uninterruptable power supply to the load with a backup planning [1].This topology improves the performance of the system; also it reduces the size and the cost of the system.

Asinglebi-directional dc-dc converter can replace two uni- directional converters. A single bidirectional dc-dc converter is capable to flow the power in opposite directions and provides the functionality of two uni-directional converters in a single converter unit. The converter is required to draw power from the high voltage dc-bus side to charge up the battery, and when the condition arrives it will to draw the power from battery to boost up the bus [2-4].

Here, a brief review and simulation of developed non- isolated bi-directional dc-dc power converter for hybrid storage technology application is presented. The bi-directional

dc-dc power stage model is derived with the state-space averaging method. This derived model is validated by comparing between control-to-inductor current transfer function from the simulation results and the derived mathematical model. This power stage model can be used under different operating modes of the bidirectional converter.

1. Circuit topology and its Power Stage Modelling

The objective of dc-dc conversion is to convert a source voltage to a near-constant output voltage under disturbances at the source voltage and load. A dc-dc converter must provide a regulated dc output voltage under such condition like, input Voltage conditions, varying load, as well as converter component values. A bi-directional dc-dc converter topology is a combination of buck and boost converter. A bi- directional dc-dc converter consists with some basic functional blocks like, the power stage (plant), the modulator, and the controller. Here a proposed modelling method is used based on modelling of each component individually, and then combining them to a complete model. The power stage was modelled using state-space averaging. After that the controllers aredesigned. The combined small-signal model generates all the transfer functions required for design purposes.

As discussed, a non-isolated bi-directional dc-dc converter technology is to combine a buck and a boost converter in a half-bridge configuration. When charging the battery, this converter working as a buck converter, it operates in voltage step-down mode during the battery discharging its working as a boost converter; it operates in voltage step-up mode. Fig. 1 shows a non-isolated half bridge bi-directional dc-dc power converter circuit topology. The bi- directional dc-dc converter is placed in between high-voltage and low-voltage sources to allow energy transfer. This kind of power converters use in many applications like in hybrid vehicles, in aerospace etc.

2. Step-up mode

This operation mode applies when battery discharges the power to the load to the connected DC bus. Converter operates in voltage step-up mode. Switch S2 remain ON and the switch S1 OFF. On during step-up mode bi-directional converter equivalent circuit shown in Fig. 3.

Fig. 1 Basic circuit of the proposed bi-directional dc-dc converter

2. STATE SPACE FORMULATION

The mathematical models for the non-isolated bi- directional dc-dc converter have been developed for both the step-down and step-up mode operation in the continuous current conduction mode. State-space formulation method is employed for the modelling of the bi-directional dc-dc converter with the following assumptions [11].

A. Step-down mode

In step-down mode operation battery is charging. During his step-down mode converter switch S1 remain ON and the switch S2 OFF. There are three energy storage components

Fig. 3 Converter circuit in step-up mode

According to the above circuit when the switch S2 is only on, the following equations are derived by using KVL and KCL formula.

diL RPiL V2

high side capacitor voltage, low side capacitor voltage, inductor current.

dt

dV1

dt

L

V1 RdC1

L

• Vd RdC1

dV2 iL Vb V2

dt CL RbC2 RbC2

From the above equations

R p 1

0

IL L

V

V

L IL 0 0

V 0 1

0 V 0

1 Vb

1 R C

1

R C

V

d 1 V

d 1

d

2 1 1 2 1

Fig. 2 Converter circuit in step-down mode

The converter equivalent circuit represent in Fig. 2.

C2

0

R bC

2

R

0

bC2

According to the above circuit when the switch S2 is only on, the following equations are derived by using KVL and KCL

So, from the above equation the state space average dc model become

formula.

R p D

1

di R i V

L L

L IL 0 0

L P L 2

D

0

1

1

1

0 V 0

1 Vb

dt L L

C1

1

RdC1

V

V

1 2 1

RdC1 Vd

dV1 V1

• Vd

C2

0

R bC2

0

R bC2

dt RdC1

RdC1

The state-space averaged ac model become

dV2 iL Vb V2

R p D 1 1

0

0

dt C R C R C

iL L L

L iL

L IL

L b 2 b 2

d V

D

1

0 V

0 0V d

dt 1 C R C

1 C

1

From the above equations

V 1 d 1

V 1

V

2

1

1

0

2 0

0 0 2

• R p 1

1

C2

R bC

2

IL L

L L IL 0 0

3. SMALL-SIGNAL ANALYSIS OF THE SYSTEM

V 1

1

0 V 0

1 Vb

1 C

R C 1

R C V

V 1

d 1 V

d 1

d

Signal ac analysis for the different modes of the bi-

2 1

C2

0 1 2

R bC2

1

0

R bC2

directional converter operation under current mode control

and also derives the transfer functions which describing the converter characteristics.

The state variables of the above system are the capacitor

x(s)

u(s)

[B1D B2D]

inv[sI (A1D A2D)]

voltages and the inductor current. Therefore by considering ideal switching, the following two sets of state-space equations can be derived for each circuit state:

When switch S1 on during d(t) period

The above expression is basically used to analyze the bidirectional converters dynamic behavior. Equation need to put in to a standard state space form.

p

p

1

x A x(t) B u(t)

R D

S 0 0 L L

1 1

L 0 L 0

1 1 iL

0 S

0 D 1

C R C

0 . 1 0 0

C

When switch S2 on during (1-d)(t) period

d 0 0 S 1

1

d 1 1 0 2 0 0

x A

x(t) B u(t) C 0 R C

2 2 2

b 1

From the above equation, system, where t is the switching period,

x [A1x(t) B1u(t)]dt [A2x(t) B2u(t)][1 d(t)]

So the duty cycle-to-inductor current transfer function

Where

d(t) (D d)

And [1 d(t)] (D d)

Now by substituting the perturbations terms the equations becomes

 Rd Rb L( H) C1(F) C2(F) FSW(KHZ) Rdson(milioh om) V1(volt) V2(volt) Rlp(milio hom) 15 8 7 20 20 70 36 42 14 36
 Rd Rb L( H) C1(F) C2(F) FSW(KHZ) Rdson(milioh om) V1(volt) V2(volt) Rlp(milio hom) 15 8 7 20 20 70 36 42 14 36

become

iL S2.a.bV1 S(a.V1 b.V1 a.D.IL.R d ) V1 D.IL.R d

X x [A1(X x) B1(U u)](D d) [A2(X x) B2(U u)](D d)

d S3.a.b.L S2 (a.L b.L a.b.R ) S(L a.R b.R a.D2.R b.R ) R R D2.R

p

p

p

p

p d b b p d

The perturbed state-space description in above equation

becomes nonlinear due to the presence of x and d. The duty cycle is the control input, not being an element in the input

The control to high-side voltage transfer function become

• IL D iL

(

(

vector u.

V C

. )

C d

1 1 1

1. Linearization

d S

1

C1Rd

The perturbed state-space averaged model is nonlinear. By Taylor series expansion and under the assumption of small-signal operation, linearization is done around the points

The control to low-side voltage transfer function become

1

(X, D, u), and nonlinear terms of higher orders are cancelled, i.e., departures from the steady-state values are negligible

V 2

d

C i

L

L

2 .

1 d

compared to the steady-state values themselves [14],

S

C2R b

So it can be say

x.d 0 u.d 0 x.d 0

When the low side voltage Vb is zero, Rb is treated as a resistive load and Rd is negligible, the model derived in duty cycle-to-inductor current transferfunction behaves like a standard second-order buck converter model. The Buck Mode with Resistive Load Converter Mode state space equation become,

u.d 0

Now in linear approximation of the state space equations representing the averaged state space model

x (A1D A2D)x (B1D B2D)u

40

Magnitude (dB)

Magnitude (dB)

20

0

-20

-40

-60

Bode Diagram

• [(A X A X) (B U B

U)]d

270

1 2 1 2

225

2. Small-signal transfer function

Taking the Laplace transform of equation with zero initial condition, we getting the following equation

180

135

90

2

10

Phase (deg)

Phase (deg)

4 6

10 10

Frequency (Hz)

Fig. 4 Bode plot of the control to high-side voltage transfer function

Bode Diagram

50

cycle has been derived and the Bode plot is shown in Figure 7.

0

-50

-100

90

#### Phase (deg)

0

-90

80

Magnitude (dB)

Magnitude (dB)

60

40

20

0

-20

Phase (deg)

Phase (deg)

0

-45

Bode Diagram

-180

2

10

4 6

10 10

Frequency (Hz)

-90

2

10

4 6

10 10

Fig. 5 Bode plot of the control to low-side voltage transfer function

Frequency (Hz)

Fig 7 Bode plot for duty cycle-to-inductor currenttransfer function in boost mode condition.

G(s)

V1 (C2Rbs 1)

RbC2s(Rp Ls) Rp Ls Rb

Bode Diagram

60

#### Magnitude (dB)

When the high-side voltage Vdis zero, Rd is treated as a 40

resistive load and Rb is negligible, duty cycle-to-inductor

current transfer function is simplified into a standard second- 20

order boost converter model. The State-space equation

#### Phase (deg)

90

90

become, 0

1 V1

ILD 45

G(s) =

(s +

C1Rb

) –

L LC1

2

0

-45

(s + 1 )(s + Rp ) + D

-90

C1Rb L LC1

2 4 6

10 10 10

Frequency (Hz)

4. RESULTS AND DISCUSSIONS

The transfer function which is required to form the dynamic model of the converter for control purposes is the duty cycle-to-inductor current transfer function. Taking the inductor curren as the output variable, the transfer function to the duty cycle has been derived.

Bode Diagram

Magnitude (dB)

Magnitude (dB)

60

40

20

0

90

Phase (deg)

Phase (deg)

45

0

-45

-90

Fig. 8 Bode plot for duty cycle-to-inductor current transfer function in buck mode condition.

In Buck Mode with Resistive Load Converter Model, the low side voltage Vd is zero, Rb is treated as a resistive load and the high side resistance Rd is negligible. That behaves like a standard second-order buck converter model; the stability analysis is done through Bode plot, shown in Figure 8.

5. CONCLUSION AND FUTURE WORK

The analysis, state space formulation, controller design and the simulation of the non-isolated bi-directional DC-DC converter were examined. The converter topology is analyzed with state space formulation in different modes of operations and then by using averaging and linearization process the small-signal models of the converter derived. So as a

2 3

10 10

4 5 6

10 10 10

Frequency (Hz)

Fig. 6 Bode plot for duty cycle-to-inductor current transfer function

Using MATLAB, the Bode plot has been done for this duty cycle-to-inductor current transfer function to analyze the stability of the system. The Bode plot for is shown in Figure

6. Note the RHP zero does not appear in this transfer function, and it is inherently stable. For boost resistive load, Rd indicates a resistive load, where Vd does not exist. Battery internal resistance Rb is as small. Here Rb and C2 are negligible. At low frequency of less than kHz, the equivalent circuit is simplified. Taking the inductor current as the output variable at theboost mode, the transfer function tothe duty

conclusion, the project objective is to mathematical modelling of a non-isolated bidirectional converter on transients and steady state responsefor energy management system is done and system stability has been analysis through the MATLAB/SIMULATION. The Future work will be designing the controller circuit both the buck and boost mode for the bi-directional dc-dc converter and to maintains the voltage label at a standard value on the different operations mode.

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