Control Vsc Converter

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MODELLING AND CONTROL OF A DC

TO DC SWITCHED MODE CONVERTER

Gonzlez Muoz, Rubn

El Mariachet Carreo, JordiMOSIC, January 2014


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CONTENTS

LIST OF FIGURES ............................................................................................................................ 3

LIST OF TABLES .............................................................................................................................. 4

1. OVERVIEW ............................................................................................................................. 5

2. METHODOLOGY .................................................................................................................... 6

3. THEORETICAL ANALYSIS ........................................................................................................ 6

4. SWITCHED MODE CONVERTER (SMC) CONTROL ................................................................ 10

5. STUDY OF THE SYSTEM WITHOUT CHANGES IN LOAD ....................................................... 12

6. STUDY OF THE SYSTEM WITH ABRUPT CHANGES IN LOAD ................................................ 19

7. START UP AND TRANSIENT OPERATION ............................................................................. 26

8. CONCLUSSIONS ................................................................................................................... 31


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LIST OF TABLES

Table 1: Design specifications for the DC to DC switch-mode converter ..................................... 5

Table 2: Values for voltage source and passive elements of the circuit ..................................... 13

Table 3: Values for the elements in the control loop.................................................................. 14

Table 4: Plot statistics on Vo during the steady-state ................................................................. 15

Table 5: Plot statistics on iL during the steay-state .................................................................... 16

Table 6: Plot statistics on the switching frequency during the steay-state ................................ 17

Table 7: Plot statistics on Vo during the steady-state ................................................................. 22

Table 8: Plot statistics on iL during the steay-state .................................................................... 23

Table 9: Plot statistics on the switching frequency during the steay-state ................................ 24


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1. OVERVIEWIn this work, a control based in sliding modewill be designed for a DC to DC switch-

mode converter. The converter must work in steady state and the transient operation

(including start-up) will be studied. Moreover, the response to external disturbances will

be studied.

The design specifications are the following:

Figure 1: DC to DC switch-mode converter

Parameter Value Units

Input voltage, Vin 24 V

Output voltage, Vout 96 V

Output power 500 W

Switching frequency 100 kHz

Inductor current ripple 20 %

Capacitor voltage ripple 1 %

Table 1: Design specifications for the DC to DC switch-mode converter


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2. METHODOLOGYThe methodology of this work is structured in three phases:

Definition of modeland controlstrategies. Design of the control strategies. System verification throughsimulation.

During the process, an iteration of these three stages is expected, and some design

alternatives may appear.

For modelling the requested system, simulation software will be used. There, only ideal

components will be implemented, specifically sources and switches.

The employed software is the following:Matlab 7.10.0.499 (R2010a)

32-bit (win32)

These are the system specifications:Intel(R) Core (TM)2 CPU T5500 @1.66GHz 1.67GHz

RAM 2.00 GB

System Type: 32-bit Operating System

Operating System:Microsoft Windows 8

3. THEORETICAL ANALYSISThe system to control shown in figure 1 is modelled as follows in MATLAB:

Figure 2: MATLAB model for the circuit in figure 1


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The system may show different topologies when switch Q1 (ideal switch) changes its

state being controlled through a signal u(t). Two different phases are distinguished

depending on the state of the circuit: there is a pre-charge stage, which corresponds to

an initial transient state, and then there is a steady state stage. Simultaneously both

states give respectively two different equivalent circuit models for each value of the

control variable u(t).

State A: pre-charge stage with u(t) = 0

The first equivalent circuit depicts a scenario where D1 and D2 diodes are directly

biased, being represented as a short circuit. Therefore, the load formed by C2 and Rc is

pre-charged when switch Q1 is open.

Figure 3: Equivalent circuit for pre-charge initial state with u(t) = 0


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State B: pre-charge stage with u(t) = 1

The second model shows the equivalent circuit when still pre-charging and Q1 is

switched to close. The anode voltage of D1 is still higher than its cathode voltage. Thenthis diode can be represented as a short circuit. Nevertheless, the voltage at the anode of

D2 becomes lower than at its cathode, leaving this diode as an equivalent open circuit.

As a result, the circuit is divided into two parts. At the left, the voltage generator feeds

C1 and L, which are left in parallel. At the right, the load formed by the parallel of C2

and Rc is left unconnected to some source, and so it starts to discharge.

Figure 4: Equivalent circuit for pre-charge initial state with u(t) = 1

State C: steady-state stage with u(t) = 1

At this equivalent circuit, the system has reached a steady state providing a boost effect.

The diode D2 gets stuck at ON state during the steady state of the circuit. At this stage,

the control variable controls not only the state of Q1, but also the state of D1.

In this case, Q1 is switched to be a short circuit and D1 becomes an open circuit. The

source supplied current finds an easier way to be driven through the short circuited

switch. As a result, the correction at the load voltage deviation is performed by

discharge.


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Figure 5: Equivalent circuit for steady state with u(t) = 1

State D: steady-state stage with u(t) = 0

In this case, Q1 is switched to be an open circuit and D1 becomes a short circuit. The

source supplied current is driven through the inductance L and the capacitance C1 to the

load formed by the capacitance C2 and the resistor. As a result, the correction at the

load voltage deviation is performed by a charge of it.

Having reached a steady state, the system behavior will only change alternativelybetween states C and D.

Figure 6: Equivalent circuit for steady state with u(t) = 0


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In this system, first of all we will need to consider the variables that could intervene in a

sliding mode controlthrough a hysteretic comparator. The design of the control circuit

will be shown in the next section.

4. SWITCHED MODE CONVERTER (SMC) CONTROLIn order to implement the control of this circuit, due to some similarities it must be seen

as a particular kind of boost DC to DC switch-mode converter. From the design

specifications it can be appreciated that the voltage expected at the output is greater than

the voltage supplied by the source of the circuit. A boost effect is required in to raise the

output voltage, and so the sliding surface must be designed with this purpose.

The control through sliding surfacesmeans that we have to impose a desired dynamicbehavior on the main state variable of the system. A typical control of this type of

circuits relays on the current going through the inductor. The control surface to develop

starts at this point:

Using this surface is necessary to guarantee switching between the two models of

equivalent linear circuit (states C and D) presented at steady stage.

Vo

iL

iref

Circuit

Eq. 1

Equilibrium 2

Equilibrium 1

Circuit

Eq. 2

Figure 7: Switching of generic circuit model depending on the position (iL, Vo)

The variable depicted as Iref in this case can be obtained using a typical PIDcontrol.

The error introduced in this device is obtained through the difference between the

desired voltage at the load and its value. The design specifications indicate that the

target voltage is set to Vo = 96V, so as a consequence the reference for this control will

be Vref = 96V.

In our case, as the system equations are as follows:


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And the dynamic behavior of the current using the PID, where the differential term is

set to zero (rarely used in power electronics control), will be imposed as:

( ) ( )

As a result, thesliding surfacefor the control circuit is expressed as follows:

( ) ( )

So, it means to apply this control structure to the system:

Figure 8: Feedback control circuit proposed to generate u(t)

In the figure above the PID block is fed by the error between Vref and Vo, aproportional factor is introduced for the inductor current, and a hystereticrelayis used

to generate and maintain the discrete signal u(t).

We have to point that K is a negative value, for the current, and the constants

proportional and integrator from the PID have been chosen through an iterative method.


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5. STUDY OF THE SYSTEM WITHOUT CHANGES IN LOADFirst of all, there must be established a stable control system that fits the requested

specifications at the beginning (Table 1). When there are no disturbances neither

changes in the load, the proposed design is the following:

Figure 9: System with a constant value of the load resistor

As expected, current through the resistor and Vo are giving a calculated value of power

Pout = 502.12 W at steady state. See Fig 9. to check this data:


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Figure 10: Voltage and current at the load, and current at the inductor (iL)

The following values have been applied to the components of the circuit:

Component Value

L 1e-3 HC1 10e-6 FC2 100e-6 FRc 18.43 Ven 24 V

Table 2: Values for voltage source and passive elements of the circuit

The values of the passive components L, C1 and C2 are taken from other basic circuits

based in boost DC to DC switch mode converters. These values satisfy the ripple

percentage conditions expressed in the premises of this design. Ripple values on

voltage and current are adjusted using an alternative control circuit based in a fix

frequency pulse generator instead of the proposed control circuit in order to get a

measure of these magnitudes. Nevertheless, in this document the target ripple is

demonstrated with the designed control circuit.

The methodology used to get the ripple is the following: the MATLAB vector

containing the rippled variable under study (current at the inductor or voltage at the load

capacitor) is truncated in order to suppress the transient. After plotting the steady state

section of this variable, the statistics are enabled at the plot window. The main statistical

values are represented there. This is an example:


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1. Need to get the ripple of Vo. First get the size of this sample vector withsize.2. Then, Vo is plotted without depending on time in order to see which samples

contain the steady state.

3. V1 vector is defined by truncating Vo at its steady state samples.4. V1 is plotted, and so theData Statisticsare enabled at plot window.5. Taking the values max, min and average the ripple can be calculated.

The value of Rc is fix and it is obtained through calculus with the design premises in

Table 1:

The control loop receives the following values:

Controlelement

PID Inductor currentgain

Relay limits

Values Kp = -1e-18

Ki = 250

Kd = 0

K = -2.5 Switch on: 0.125

Switch off: -0.125

Output when on: 1

Output when off: 0

Table 3: Values for the elements in the control loop

The PID part of the control commonly does not use a differential term for control

purposes. The proportional part increases the speed of the system, but in this case is

almost considered null. The integral term is adjusted manually and through this value

the transient response of the system can be controlled. In this case, it has been left in

order not to have overshoot.

The inductor current gainneeds to be negative, given the current model of sliding mode

surface.

The relay limits provide the value of the switching frequency. This magnitude is

measured and calculated through a particular block in MATLAB which has beendesigned and implemented by Professor M. Castilla: a frequency meter. Switch onand

switch offvalues are adjusted manually in order to get the desired switching frequency.

Statement premises check

A. Output power

As shown in Fig.10, the output values in steady state are:


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B. Voltage ripple at the capacitor C2 less than 1% (Fig. 11)

The sample vector containing the voltage value at C2 is composed of 200001 samples,

as well as the rest of the variables of this simulation. By visual plot inspection, it can be

seen that the transient is before the sample 50000. So, samples between 50000 and200000 are taken to get the ripple for the components of the circuit without disturbances

at the load.

The following table is taken from the Data Statisticsoption for the steady state of the

voltage at C2:

time Vo

min 1 95.8

max 5e+004 96.22

mean 2.5e+004 96median 2.5e+004 95.99

mode 1 95.8

std 1.443e+004 0.113

range 5e+004 0.4121

Table 4: Plot statistics on Vo during the steady-state

As result, the ripple measured at steady-state conditions is:

Figure 11: Voltage ripple at capacitor C2 with fixed load conditions


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C. Current ripple at the inductor L less than 20% (Fig. 12)


current at L:

time iLmin 1 16.11

max 5e+004 16.3

mean 2.5e+004 16.21

median 2.5e+004 16.21

mode 1 16.11

std 1.443e+004 0.05167

range 5e+004 0.1888

Table 5: Plot statistics on iL during the steay-state


Figure 12: Current variations of the inductor (inductor ripple at fixed load conditions)

D. Switching Frequency

Monitoring the u(t) signal, and with the aid of data cursors, we can observe that the

switching frequency fits is exactly the requested value.


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To calculate the switching frequency, three periods are considered from the plot in order

to have an accurate value of the period. These are the calculus:

Figure 13: Signal u(t) scoped directly from the control loop

If the frequency meter is used, we obtain after statistical analysis of the steady state

samples offreqsignal that the value of the frequency is near to 100 kHz. The results of

the block depicted as frequency meter are given in kHz:

time Sw. freq

min 1 90.91

max 5e+004 142.9mean 2.5e+004 104.2

median 2.5e+004 100

mode 1 100

std 1.443e+004 13.05

range 5e+004 51.95

Table 6: Plot statistics on the switching frequency during the steay-state


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Figure 14: Frequency plot obtained through Professor M. Castilla frequency meter

Figure 15: Expanded region of the measured frequency caption


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6. STUDY OF THE SYSTEM WITH ABRUPT CHANGES IN LOADIn this section, an abrupt change in the load will be introduced by means of connecting

in parallel an extra resistorRcxwith a value which is the double:

In order to prove the proper working of the system in steady state, the resistor Rcxwill

be connected at t = 0.1s after system start-up. To do so, the new resistor is added in

parallel to the load through a serial switch controlled by a step function. Further 0.1s

will be given to the system in order to let it achieve the steady state.

The equivalent load resistor for t > 0.1s results in:

And so the power dissipated at this load increases:

We expect a transient response with a temporary decay and a damping effect

compensated thanks to theKivalue.

The complete model in MATLAB is shown in fig. 16:

Figure 16: System with a disturbance on the equivalent load resistor


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As all the parameters of the system are exactly the same than in MATLAB model

described in the previous section, and considering only the changes introduced in the

load after the step function switches at t = 0.1s, we obtain the following voltage and

current behavior at the load point:

Figure 17: Voltage and current at the disturbed load, and current at the inductor (iL)

The green line represents the current in the original Rc = 18.46. The power

consumption at this resistor barely changes, as Vo = 95.98V and ic = 5.22A.

The magenta line represents the total current through the load, before and after the

addition of the Rcx in parallel. There can be observed that after the step, suffers an

important increase. This increase implies that the total power supplied by the source

will be higher:


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Figure 18: Plot of the load resistor and equivalent load resistor currents

Statement premises check

A. Output power

As shown in Fig.10, the output values in steady state are (consider just original Rc):

B. Voltage ripple at the capacitor C2 less than 1% (Fig. 19)

Due to the fact that before the load disturbances the circuit remains the same as in the

previous experiment (and so the ripples before this event), in this section the ripples will

be calculated after the disturbance happens. By visual plot inspection, it can be seen that

the second transient starts at sample 100000 and is assured to be finished at sample

150000. Then, this set of samples is taken to get the ripple for the components of the

circuit with disturbances at the load.


voltage at C2:


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time Vo

min 1 95.71

max 5e+004 96.32

mean 2.5e+004 96

median 2.5e+004 95.99

mode 1 95.71

std 1.443e+004 0.1709

range 5e+004 0.6182

Table 7: Plot statistics on Vo during the steady-state


Figure 19: Voltage ripple at capacitor C2 with disturbed load conditions

C. Current ripple at the inductor L less than 20% (Fig. 20)


current at L:


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time iL

min 1 24.14

max 5e+004 24.33

mean 2.5e+004 24.24

median 2.5e+004 24.24

mode 1 24.14

std 1.443e+004 0.05232

range 5e+004 0.1897

Table 8: Plot statistics on iL during the steay-state


Figure 20: Current variations of the inductor (inductor ripple with disturbed load)

D. Switching Frequency

Monitoring the u(t) signal, and with the aid of data cursors, we can observe that the

switching frequency fits is exactly the requested value. Given that the behavior is

exactly the same as in the prior experiment before disturbances in the load; the steady

state after disturbance introduction will be the range of time under study.


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To calculate the switching frequency, three periods are considered from the plot in order

to have an accurate value of the period. These are the calculus:

Figure 21: Signal u(t) scoped directly from the control loop

If the frequency meter is used, we obtain after statistical analysis of the steady state

samples offreqsignal that the value of the frequency is near to 100 kHz. The results of

the block depicted as frequency meter are given in kHz:

time Sw. freq

min 1 90.91

max 5e+004 142.9

mean 2.5e+004 102.6median 2.5e+004 100

mode 1 100

std 1.443e+004 11.86

range 5e+004 51.95

Table 9: Plot statistics on the switching frequency during the steay-state


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Figure 22: Frequency plot obtained through Professor M. Castilla frequency meter


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7. START UP AND TRANSIENT OPERATIONThe transient response of the system can be adjusted to an optimal behavior through the

gain coefficients of the system control part. The variables that can control the transient

response are the gain Kapplied to the inductor current and the PID coefficients Kp,Ki

and Kd. In order to obtain a proper value of these parameters, the system has been

adjusted iteratively aided with MATLAB. As a result, overshoot and over damping

effects have been minimized.

At the present section the plots a group of results varying two of the parameters

commented above are represented: the inductor current gain K and the integral

coefficientKiof the PID. For control purposes, Kp coefficient is rarely used and Kd can

be considered cero or almost cero in the design of the current control due to the fact that

an increase of the speed of reaction is not required.

A. Variations introduced with K (inductor gain)

The following three plots represent the variations introduced by the variation of K gain

that accompanies the inductor current (iL) term in the control part. Plots of the output

power (Po), current at inductor (iL) and output voltage (Vo) are considered respectively.

Each color represents a different value of K: light bluecorresponds to K = 1, dark blue

corresponds to K = 2.5, and greencorresponds to K = 3.

Figure 23: Influence of current gain K in transient at load active power


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Figure 24: Influence of current gain K in transient at inductor current iL

Figure 25: Influence of current gain K in transient at load voltage Vo

The predominance of the inductor current term in the locked-loop control is

demonstrated through the fact that with a slight variation ofKthe system stability at the


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transients degenerates. In the figures above, an overdamp effect can be observed at

start-up and in the transient at t=0.1 (s) when the value of K increases over 2.5, and also

an overshoot effect occurs when the value of K decreases under 2.5. Precisely, for a

value of K = 2.5the control provides a critically damped response at the transients.

B. Variations introduced with Ki (integral term of PID)

The following three plots represent the variations introduced by the variation of Ki gain

of the integral term in the PID control part. Plots of the output power (Po), current at

inductor (iL) and output voltage (Vo) are considered respectively. Each color represents

a different value of Ki: light bluecorresponds to Ki = 100, dark bluecorresponds to

Ki = 250, greencorresponds to K = 500, andred corresponds toK = 750.

Figure 26: Influence of Ki coefficient of PID in transient at load active power


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Figure 27: Influence of Ki coefficient of PID in transient at inductor current iL

Figure 28: Influence of Ki coefficient of PID in transient at load voltage Vo

The influence of the Ki factor is only appreciated when keeping Kp at a low value near

to zero. Then, Ki can be modified iteratively in order to get an optimal response at Ki =

250.


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As Ki variations go on, it can be observed on figures that the system stability is

degenerating again. Particularly, if Ki takes values under Ki = 250, a delay both in the

start-up and transient regions can be appreciated. An overshoot effect at start-up and in

the transient at t=0.1 (s) can be appreciated when the value of Ki increases over 250, as

well as an overdamp effect occurs when the value of K decreases under 250. Precisely,

for a value of Ki = 250 the control provides a critically damped response at the

transients.


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8. CONCLUSSIONSThe result of this work shows that the DC to DC converter control is achieved through a

sliding mode controlmodel. The suitable states of the circuit need to be studied first of

all in order to determine if a circuit switching is possible and how the switching should

be controlled.

Having seen some particularities of the circuit, the main idea that remains is that it must

be treated as a boost switched DC to DC converter, and so the control loop must be

designed under this premise. This implies the development of a model of control based

on asliding surfacewhich is well-known, and raging from this point the main behavior

of the circuit is set.

Once the main value of the output behaved as expected, the next step was adjusting the

circuital parameters in order to fit the design premises: optimize transient damping or

overshoot, and adjust ripples and the switching frequency of the control variables.

These adjustments have been based on iterative methods aided with MATLAB

simulations.

Two main simulations have been performed in order to check the validity of the

designed circuit and control. The first of them consisted on the design premises imposed

on this circuit, which considered a stable load. The second experiment considers the

same conditions but with a disturbance on the load. At half simulation, the equivalent

resistor load is reduced by putting in parallel another resistor. It is observed that the

expected output voltage is maintained after a transient decay, but the supplied currenthas increased, increasing the output power too. As a result, it must be indicated that in

the practice the design of this circuit is only possible for just a determined value of the

load due to the inherent variability of the power supplied (some devices could be

damaged).

This exercise has been presented to us as a tool to start and develop our knowledge

about circuit design, control and modelling in the field of power circuits. We have

enhanced our skills in the analysis of this type of circuits, as well as some techniques to

design and develop them in a practical way without a deep theoretical base. Among

them, we must highlight the indications provided by Professor M. Castilla, who hasmanaged to teach us these techniques, including the handling of MATLAB-Simulink in

this particular environment.

At the end, we can conclude that we have been able to organize ourselves working as a

team in order to prepare and develop the periodical lab sessions and to compose the

current report.

Control Vsc Converter

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