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Chapter -2
DDyynnaammiicc MMooddeell ooff BBuucckk CCoonnvveerrtteerr wwiitthh CCoonnvveennttiioonnaall PPIIDD
CCoonnttrroolllleerr
2.1 Introduction
The DC -DC power converters are extensively utilized in SMPS and are a critical component
of DG system, particularly for sustainable energy system based distributed generators. The
objective of this chapter is to design a PV fed buck regulator with feedback control by using
classical Proportional plus integral plus derivative controller. PID controller maintains a
constant output voltage by varying the input voltage and load. SSA technique is use d for
getting a linear ized model of this power converter and the controller gain parameters are
determined by Ziegler -Nichol‘s tuning method. The mathematical analysis and dynamic
model of DC -DC converters can be analyzed by numerical or analytical method s. In
numerical methods , vari ous algorithms are used to give quantitative results and in analytical
methods, analytic al expressions are provided to exhibit the operation and performance of the
power converters. The most popular method is the small -signal model analysis, which include
three techniques such as circuit averaging, state -space averaging or PWM switch mode lling.
In this chapter the linearized model of convert er is formed by SSA technique and close d loop
control is applied to stat e space average mo del of a DC-DC C onverter. The PV module
consists of number of PV arrays and each PV array consists of many solar cells. The entire
PV module depends on solar irradiance and cell te mperature. Since PV module has nonlinear
characteristics, it is necessary to model and design for PV sy stem applications. In this chapter
the PV module is designed at standard temperature and insolation i.e 25oC and 1000 W att
(peak)/ m2 respectively. This chapter is divided into three main parts :
(a) First is the modelling part , where the dynamic model of buck converter is established
based on it‘s on and off states , and the State space averaging technique is applied.
(b) Second one deals with the modelling of solar PV module in a Simulink environment
which is fed to the converter as input .
(c) Third one is the control mechanism applicable to the buck regulato r to regulate the
output voltage for different input and load conditions.
The parameters of the PID controller are obtained by the Zeigler -Nichols method. The
advantage of PID controller is it gives faster response, less overshoot and less oscillation.

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2.2 Operation of Buck Converter
The buck regulator is a switched mode DC -DC power converter whose voltage output is less
than the input voltage, so it is otherwise known as step down converter. The converter uses
one inductor and one capacitor as energy storing elements .Though the converter contains L
& C, the system becomes non -linear. Therefore the non -linear sys tem is characterized as
linear model within certain range and time -period by using the SSA technique which is a very
common solution. Figure : 2.1 shows the ideal Buck converter operating in continuous
conduction mode.
S L

C

Figure: 2.1 Circuit scheme of DC -DC Buck Converter

2.2.1 Switch on mode
During the switch S is ON, inductor accumulates the energy i.e V Li ton and the current and
voltage equations are given by equation ( 2.1).
CRv
Ci
dtdvLv
Lv
dtdi
o L co in L
 ,
(2.1)

2.2.2 Switch off mode

During the switch S is OFF, diode gets ON and it provides a path for current flow, the
stored energy in the inductor will dissipate through load resistance .The current and
voltage equations are given by equation (2 .2)
CRv
Ci
dtdvLv
dtdi
c L cc L

(2.2)

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Here the i L and v C are considered to be two states of the system. The important use of the
buck converter is to regulat e DC power supplies and control the speed of DC motor . The
output voltage of the converter varies linearly with the control voltage. It varies between zero
and V in. The output voltage fluctuation can be minimized by means of a LPF or by designin g
a suitable controller. For steady state analysis the output capacitor value is assumed to be
very large for getting fixed DC output voltage. The aver age inductor current is the same as
the output current because the average capacitor current at steady state is zero.
o LII
(2.3)
) ( ) (on s o ono in tTv tv v 
(2.4)
dTt
vv
son
ino
(2.5)
―d‖ represents the duty ratio.
2.3 Control of Switch -Mode DC Power Supplies
The output of switch mode power supplies (SMPS) are to be regulated within a specified
tolerance line to get the de sired response by varying output load and input voltage. The
converter output voltage (V o) is compared with the reference output voltage (V oref) to
generate an error. T he error goes to the controller to adjust the duty cycle of the power
converter. State space avera ging technique is used for converting the non linear ized model to
linear ized one and it is lin earized around a steady state DC operating point [100] .
2.3.1 State Space Averaging Technique
The objecti ve of this technique is to derive a control transfer function Vo(s)/d(s) where Vo is
the output voltage and‗d‘ is the duty ratio. In this work the converter operates in continuous
conduction mode (CCM). In CCM there are two stages of operation: ON state and OFF state.
During each stage a set of linear equations are described by means of state variables. Here i L
and v C are considered as two states of the system. To derive an average model of the power
converter over one switching cycle, the equations corresponding to ON state and OFF state
are time weighted and averaged as given in [18 ] and t he resulting equations are obtained in
equation ( 2.6) and ( 2.7).
invd BdBxd AdAx )]1( [ )]1( [2 1 2 1.

(2.6)

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xd CdC vo )]1( [2 1 (2.7)
The transfer functions with respect to input voltage and with respect to duty ratio are given by
the equation ( 2.8) and ( 2.9) respectively.
DBA SICsvsv
ino1) ()()(
(2.8)
xCC vBBxAA ASICsdsv
in ) (]) () [(] [)()(
2 1 2 1 2 11 0
(2.9)
The averaged model of buck regulator is designed by taking a weighted average of the
equations ( 2.1) and (2.2) and is expressed as
) 1() 1() 1(
2 12 12 1
d CdCCd BdBBd AdAA


The state space model of equation ( 2.1) and ( 2.2) can be written as
0 1/ 1
11
0L
L
in
C CdiLi dtvLv dv
C RCdt        
(2.10)
in
CL
CL
vvi
RCCL
dtdvdtdi













00
1 110
(2.11)
The output equation is same in both the states and is given by equation ( 2.12) and (2.13) .
C ovv
(2.12)


CL
oviv 10
(2.13)
So the averaged matrices of the converter is obtained as below
 ]0[ ,10 ,
0,1 110
 










 D CLd
B
RCCLA

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The input voltage to the buck converter is 12 volt and it is operated with a frequen cy of 55
kHz and duty ratio of 0.4. T he desired output voltage must be 5 volt. Switching losses can be
determined by knowing the drain current and on state drain to source resistance of the
MOSFET but this is not the objective. Switching losses is directly proportional to switching
frequency. Using the data sheet values [52] and equations ( 2.8) and ( 2.9), the transfer
functions of the buck converter for input voltage and duty cycle are as ( 2.14) and (2.15).
7 27
1033.3 2501033.1
)()(
s s svsv
ino
(2.14)
7 28
1033.3 250104
)()(
s s sdsvo
(2.15)

2.4 D esign of the Proposed Solar PV System
The proposed PV module consists of 20 solar cells in series along with the Simulink -PS
converter and PS – Simulink converter. The open circuit voltage (v oc) and the short circuit
current (I sc) of each solar cell is found to be 0.6 volt and 4.75 ampere s respectively. The
module provides a photovoltaic voltage o f 12 volt which is fed to the converter directly as
input at standard temperature and irradiation i.e. 1000 Watt ( peak)/m2 and 25°C .The
photovolta ic voltage (V pv) can be varied by varying the insolation level. Fig ure: 2. 2 represent
the PV module in Simulink which yields the required V pv of 12 volts.

Figure : 2.2 PV Modules in S imulink
The voltage output of the Buck regulator w ithout controller is depicted in Figure: 2.3 and it is
found to be 4.8 volt, which is nearly equal to the 5 volt according to [52].

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Figure: 2 .3 Output voltage of open loop Buck Converter without controller

Figure : 2.4 Bode diagram of open loo p buck converter

Figure: 2. 4 show the Bode characteristic of the converter without controller. The frequency
response of the open loop system is absolutely stable with Gain Margin (GM) of Infinite
dB and a Phase Margin (PM) of 0.7454 degree with ga in crossover frequency of
2.08×104rad/sec. Phase Margin, of the open loop system is very low hence less robust to
parameter variations.

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2.5 Design of the Controller
The main aim of designing a controller is to make the system stable and to make Vo
constant in spite of the pe rturbations in V in and Io. The control law regulates the output
voltage thereby reducing the transients and tracking of the set point value as in [101]. The
Vo is regulated by a feed back loop between V o and d . Figure: 2.5 illustrates the closed loop
design o f buck regulator where the voltage output of the converter is matched with the
desired voltage , and the error produced is passed through the control transfer function
block to produce the control signal using a PWM technique. The control signal is fed to the
gate of the conver ter switch.

Figure : 2.5 closed loop design with subsystem

2.5.1 Tuning of PID controller using Zeigler – Nichols method
The controller gains are calculated from Zeigler – Nichols method of controller tuning
based on Critical gain ( Kcr) and Critical time Period (T cr) as in [2] . The critical time period
is calculated from the step response plot where the output exhibits sustained oscillations.
Based on K cr and T cr values, the proportional gain (K p), integral time (T i) and the
derivative time (T d) are calculated and the controller is tuned according to table 2.1. The
transfe r function of PID controller is obtained by equation ( 2.16)
)11()()(
d
ipasTsTKsEsE
(2.16)

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Table 2.1: Zeigler -Nichol‘s Table

Controller Proportional
gain(K p) Integral time(T i) Derivative time(T d)
P+I+D 0.6K cr Tcr/2 Tcr/8

The t ransfer function of PID controller is obtained as ( 2.17) in the s- domain.

ss ssGc67. 1666 01338.0 10 144.0)(23 
(2.17)

The output voltage waveform and frequency response plot of the converter using controller is
depicted in Figure: 2.6 and 2.7 respectively. It is observed that the oscillations died out very
fast within 0.03 second.

Figure : 2.6 Output Voltage of Converter using PID controller

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Figure : 2.7 Bode Diagram of plant with PID Controller

Bode plot is the graphical representation of any system in the frequency domain to deter mine
stability . Positive p hase margin gives a stable syste m and negative phase margin gives an
unstable system . The PID controller increases the phase margin to 90.2 degrees as compared
to 0.745 degree of the open loop plant.
2.5.2 Effects of PID controller on time domain specificat ion
The main purpose of designing a controller in conver ter system is to reduce the maximum
overshoot and to decrease the settling time so that desired output voltage is obtained after a
few seconds. The formula for calculating peak oversho ot and settling time is given by
equation (2.18)
nsp
ttctc tcM
4100)()( )(%max

(2.18)
Where ξ is the damping ratio and
n is the natural frequency of oscillation. The steady state
output value of the Buck converter is 5 volt and the peak output voltage for Buck Converter
is found to be 7.4 volt. Hence the PID controller reduces the peak overshoot to 84% which is
less as compared to the converter without controller. The settling time is calculated from the
response curve within a 2% tolerance ban d. Steady -state error is 0.05 without controller and it
reduces to zero by using a PID controller which indicates absolute tracking of the set point
value. PID controller set tles the system very fast and with a negligible margin. Table -2.2
shows the compar ison of time domain specifications between a plant without controller and

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plant with controller.

Table 2.2: Time domain parameters

Time domain specification Plant without controller Plant with controller
Rise time 0.005 seconds 0.005 seconds
Peak time 0.005 seconds 0.009 seconds
Peak overshoot 92% 84%
Settling time 0.035 seconds 0.02 seconds
steady state error 0.05 0

2.5.3 Analysis of sensitivity and complementary sensitivity function
Sensitivity and c omplementary sensitivity functions play a key role in designing of feedback
control ler. Sensitivity analysis is the study of how far the uncertainty in the output of
a mathematical model can be approximated to different sources of uncertainty in its inputs. It
is otherwi se termed as uncertainty investigations . The design parameters for the controller are
chosen in such a way that the closed loop system is not sensitive to parameter variations. The
best way to characterize sensitivity is through the nominal sensitivity peak. The sensitivity
function(S) relates to the disturbance rejec ting property and c omplementary sensitivity
function (T) provide s a measure of set-point tracking performance, and the equations are
given by ( 2.19) and ( 2.20).
CGGS11
(2.19)
CC
GGGGT1
(2.20 )

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Figure: 2.8 Singular value diagram of open loop plant

Figure: 2.9 Singular value diagram of closed loop plant
The sensitivity and complementary sensitivity fun ctions are plotted using MATLAB code
and the results are displayed in Figures: 2.8 and 2.9. Sensitivity and c omplementary
sensitivity for open loop system is 40.9dB which is considerably reduced to 3.06 ×10-13 and
6.02dB respectively. Thus the output remains constant irrespective of the perturbations in
the load resistance.Figure: 2.10 and 2.11 g ives the output voltage for change in load
resistance from 20Ω to 1kΩ and change in insolation level also. The PV module has been
modeled and different parameters of PV modules are simulated at standard temperature and
irradiance condition shown in figure 2.2. Figure 2.11 shows the voltage output for different
insolation level.

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Figure : 2.10 Output voltages for load resistance variation

Figure : 2.11Output voltage for variation in insolation

From the above figures it can be observed that output voltag e remains constant always either
changing the insolation level or changing the load resistance .
2.6 Conclusion
In this chapter a PV module is designed for an output voltage of 12 volt wh ich is fed to the
Buck regulator and closed loop control by PID controller is obtained by using Zeigler Nichols
method for maintaining output voltage constant irrespective of the input and load variation.
From sensitivity analysis it is concluded that closed loop system is less sensitive to variation
in input voltage and load resistance. Again , it was concluded that output volta ge always
remains constant within a very short interval of time by varying input voltage. The proper
functioning of the controller is also validated i n this chapte r.