 Open Access
 Total Downloads : 14
 Authors : Ranjan Pramanik
 Paper ID : IJERTCONV3IS20053
 Volume & Issue : ISNCESR – 2015 (Volume 3 – Issue 20)
 Published (First Online): 24042018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Mathematical Modelling of NonIsolated BiDirectional DCDC 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 nonconventional/conventional sources as well as from energy storage element to provide uninterruptable power supply to the load. The bidirectional converter placed in between a DC voltage source and a battery to allow energy transfer. The converter connected with high voltage DCbus and also the converter backed up with battery. Using the bidirectional dcdc 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 halfbridge bidirectional dcdc converter is studied for hybrid vehicle technology. The state space formulation of the bi directional dcdc 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.
KeywordsBidirectional DCDC Converter, Energy management system,State space modelling, SmallSignal Analysis Hybrid electric vehicle etc.

INTRODUCTION
Using a bidirectional dcdc converter along with low voltage energy storage for the highvoltage dc bus has been a prominent option for hybrid electric storage technology. Its have huge applications on hybrid electric vehicles and with nonconventional 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.
Asinglebidirectional dcdc converter can replace two uni directional converters. A single bidirectional dcdc converter is capable to flow the power in opposite directions and provides the functionality of two unidirectional converters in a single converter unit. The converter is required to draw power from the high voltage dcbus side to charge up the battery, and when the condition arrives it will to draw the power from battery to boost up the bus [24].
Here, a brief review and simulation of developed non isolated bidirectional dcdc power converter for hybrid storage technology application is presented. The bidirectional
dcdc power stage model is derived with the statespace averaging method. This derived model is validated by comparing between controltoinductor 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.

Circuit topology and its Power Stage Modelling
The objective of dcdc conversion is to convert a source voltage to a nearconstant output voltage under disturbances at the source voltage and load. A dcdc converter must provide a regulated dc output voltage under such condition like, input Voltage conditions, varying load, as well as converter component values. A bidirectional dcdc converter topology is a combination of buck and boost converter. A bi directional dcdc 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 statespace averaging. After that the controllers aredesigned. The combined smallsignal model generates all the transfer functions required for design purposes.
As discussed, a nonisolated bidirectional dcdc converter technology is to combine a buck and a boost converter in a halfbridge configuration. When charging the battery, this converter working as a buck converter, it operates in voltage stepdown mode during the battery discharging its working as a boost converter; it operates in voltage stepup mode. Fig. 1 shows a nonisolated half bridge bidirectional dcdc power converter circuit topology. The bi directional dcdc converter is placed in between highvoltage and lowvoltage sources to allow energy transfer. This kind of power converters use in many applications like in hybrid vehicles, in aerospace etc.

Stepup mode
This operation mode applies when battery discharges the power to the load to the connected DC bus. Converter operates in voltage stepup mode. Switch S2 remain ON and the switch S1 OFF. On during stepup mode bidirectional converter equivalent circuit shown in Fig. 3.
Fig. 1 Basic circuit of the proposed bidirectional dcdc converter


STATE SPACE FORMULATION
The mathematical models for the nonisolated bi directional dcdc converter have been developed for both the stepdown and stepup mode operation in the continuous current conduction mode. Statespace formulation method is employed for the modelling of the bidirectional dcdc converter with the following assumptions [11].
A. Stepdown mode
In stepdown mode operation battery is charging. During his stepdown mode converter switch S1 remain ON and the switch S2 OFF. There are three energy storage components
Fig. 3 Converter circuit in stepup 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 stepdown 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 statespace 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


SMALLSIGNAL 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 statespace 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 (1d)(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 cycletoinductor 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 statespace 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 highside voltage transfer function become

IL D iL
(
(
vector u.
V C
. )
C d
1 1 1

Linearization
d S
1
C1Rd
The perturbed statespace averaged model is nonlinear. By Taylor series expansion and under the assumption of smallsignal operation, linearization is done around the points
The control to lowside voltage transfer function become
1
(X, D, u), and nonlinear terms of higher orders are cancelled, i.e., departures from the steadystate values are negligible
V 2
d
C i
L
L
2 .
1 d
compared to the steadystate 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 cycletoinductor current transferfunction behaves like a standard secondorder 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

Smallsignal 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 highside voltage transfer function
Bode Diagram
50
cycle has been derived and the Bode plot is shown in Figure 7.
Magnitude (dB)
Magnitude (dB)
0
50
100
90
Phase (deg)
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 lowside voltage transfer function
Frequency (Hz)
Fig 7 Bode plot for duty cycletoinductor currenttransfer function in boost mode condition.
G(s)
V1 (C2Rbs 1)
RbC2s(Rp Ls) Rp Ls Rb
Bode Diagram
60
Magnitude (dB)
Magnitude (dB)
When the highside voltage Vdis zero, Rd is treated as a 40
resistive load and Rb is negligible, duty cycletoinductor
current transfer function is simplified into a standard second 20
order boost converter model. The Statespace equation
Phase (deg)
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)


RESULTS AND DISCUSSIONS
The transfer function which is required to form the dynamic model of the converter for control purposes is the duty cycletoinductor 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 cycletoinductor 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 secondorder buck converter model; the stability analysis is done through Bode plot, shown in Figure 8.

CONCLUSION AND FUTURE WORK
The analysis, state space formulation, controller design and the simulation of the nonisolated bidirectional DCDC 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 smallsignal 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 cycletoinductor current transfer function
Using MATLAB, the Bode plot has been done for this duty cycletoinductor 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 nonisolated 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 bidirectional dcdc converter and to maintains the voltage label at a standard value on the different operations mode.
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