Document [original]

Jan Hendrik Carstens, Clemens Gühmann

Maximum power point controller for

thermoelectric generators to support a

vehicle power supply

Article, Postprint version

This version is available at https://depositonce.tu-berlin.de/.

Suggested Citation

Carstens, Jan Hendrik; Gühmann, Clemens: Maximum power point controller for thermoelectric

generators to support a vehicle power supply. - Materials today : roceedings. - ISSN: 2214-7853. - 2

(2015), 2. - pp. 790–803. - DOI:10.1016/j.matpr.2015.05.099. (Postprint version is cited, page numbers

differ.)

This work is licensed under a Creative Commons BY-NC-ND

4.0 License. For more information see

https://creativecommons.org/licenses/by-nc-nd/4.0/.

Maximum Power Point Controller for Thermoelectric Generators to

Support a Vehicle Power Supply

Jan Hendrik Carstensa,* , Clemens Gühmanna

aDepartment of Energy and Automation Technology, Chair of Electronic Measurement and Diagnostic Technology, TU Berlin, Einsteinufer 17,

10587, Berlin, Germany

Abstract

The growing mobility increases the world-wide fuel consumption. Yet the amount of fossil fuel is limited and the environmental

burden is increasing dramatically as well. Many governments have enacted laws to regulate and reduce the fuel consumption as

well as the CO2 emissions of combustion engines. An idea to save fuel and to reduce the environmental burden is to use

thermoelectric generators (TEGs) to recover the waste heat of the exhaust gas and convert into electric energy in automotive

applications. For the linking of TEGs to the vehicle is power supply, a DC-DC converter can be used. To support a wide range of

TEGs with different electric parameters, the control of DC-DC converter must be robust. Further, the control should track the

maximum power point (MPP) of the TEG for an efficient power recovery. This paper presents a digital cascade controller for a

boost-buck converter that charges a vehicle battery and supplies the load. To model and analyze the discontinuous converter, the

state-space-averaging (SSA) is used. The tracking of the MPP is realized with a gradient algorithm and an input current control.

An adaptive step size algorithm reduces the conversion time of the maximum power point tracking algorithm (MPPT).

Experiments verified the controller design and the efficiency of the MPPT.

Keywords: DC-DC converter; Maximum Power Point Tracking; Thermoelectric Generator; Digital Control

Nomenclature

area

BA,

control plant polynomials

controller

control plant

TSR ,,

controller polynomials

number of modules

power

switch

temperature at hot-side, cold-side

duty cycle

gain margin

current

* Corresponding author. Tel.: +49-30 314-21168; fax: +49-30 314-22120.

E-mail address: jan.h.carstens@tu-berlin.de

First publication, doi: 10.1016/j.matpr.2015.05.099 by Elsevier, in Materials Today: Proceedings

length

voltage

switching frequency

temperature coefficient

εγδβα

,,,,

parameters

phase margin

Seebeck coefficient

electrical conductance

1. Introduction

Today, mobility increases world-wide and is accompanied by a rising fuel consumption. This trend is in conflict with

the limitation of fossil fuels. Additionally, the environmental burden is increasing dramatically. In consequence,

many governments have enacted laws to reduce the CO2 emissions of combustion engines. The resulting

requirements represent a formidable challenge for research and development. Conventional internal combustion

engines (ICEs) convert chemical energy, stored in fossil fuel, into mechanical energy. During this process the main

part of the energy (about 50%) is dissipated as heat [1].

The recovery of the waste heat has a high potential to increase the efficiency of an ICE and consequently to reduce

the fossil fuel consumption and to reduce toxic emissions. Especially the unused exhaust gas waste heat can be

converted into electric power through a thermoelectric generator (TEG) [1,2]. A TEG consists of couples of n- and

p-doted semiconductors. The Seebeck-effect causes a TEG to generate electrical power when a temperature gradient

is applied. The maximum power of a TEG depends on the material characteristics of the semiconductors in the TEG

as well as the geometric parameters of the couples [3,5]. These parameters and materials can be selected in relation

to various optimization criteria.

This includes the temperature range of materials, the desired electrical rated power and the back pressure of the ICE

[1,4,5,6]. This implies that TEGs must be designed and optimized for individual applications. The result is that the

electrical characteristics can vary in a wide range of a TEG.

For the electrical linking of a TEG to a vehicle power supply, a DC-DC converter is necessary, if the voltage

amplitudes of source and load are different. Furthermore, a DC-DC converter works in different modes. The first is

the maximum power point tracking (MPPT) and the second is the trickle charging control (TCC) of the battery

voltage. The MPPT is used to search and match the maximum power point of the TEG, regardless of the temperature

gradient at the TEG. The result is that the alternator of the automobile needs to generate leas power. This increases

the efficiency of the ICE. When the battery is charged and the power of the TEG is higher than the necessary load

power, the trickle charge controller is active.

A MPPT is based on a gradient search method and is used in particular for photovoltaic cells. This method can be

applied to a TEG as well. An overview of different search algorithms is presented by Esram [7]. Beside the

algorithms, the implementation of a MPPT can be divided in two general concepts: Control with and without a

feedback signal [8,9,10,11].

The implementation of a feedback control loop requires a detailed system model of the converter, the load and the

TEG to ensure a stable closed loop dynamic. However, a closed loop system can compensate disturbances and a

desired closed loop dynamic behavior can be designed with the control parameters. In contrast to that, a MPPT

without a feedback control – also called direct control – is not able to compensate feedback signals like transient

oscillations which result in a higher conversion time or in a stagnation of the algorithm.

Nevertheless, the problem in designing a feedback control is the necessary knowledge of the control plants.

Especially because the control plant dynamic depends on the electrical parameters of the TEG. Presented control

structures [10,12] are designed based on the assumption, that the parameters of the TEG are nearly constant and well

known. With this assumption the use of the converter is limited to a small number of TEGs. In contrast to that, the

introduction of a robustness criterion allows to design the control parameters to compensate wide variations of the

electrical parameters from a TEG.

In this paper a boost-buck converter with a feedback controller for MPPT and a control of the trickle charge voltage

of the battery (see Fig. 1) is presented. The control design includes a robustness criterion to ensure a stable closed

loop dynamic. The result is a regulated DC-DC converter, which is able to support a wide range of TEGs with

different electrical parameters and different voltage levels of a vehicle power supply. In this context, this paper is

organized as follows: The brief overview of the characteristic from a TEG is presented in section 2. In section 3 the

model of the boost-buck converter is described. The digital control structure of the DC-DC converter is shown in

section 4. Experimental results are presented and discussed in section 5. Finally the methods and results are

summarized in section 6.

2. Thermoelectric Generator

A thermoelectric generator consists of one or more thermoelectric modules (TEMs), which are used in an

application. The TEM is based on a series connection of couples of thermoelectric elements (TEs). A TE is an n- and

p-doted semiconductor, with an electrical connection. Due to a temperature gradient of the TE, heat flows from the

hot to the cold side. The result is that the electrons and holes of the semiconductors diffuse to the cold side and a

voltage potential can be measured at the TE. This physical principle is called the Seebeck-effect, which describes the

voltage

u∆

in relation to the temperature gradient

)( c

T−

and Seebeck coefficent

Fig. 1. Overview of control structure from boost-buck converter. The alternator is neglected the vehicle power supply.

))((),( chnpte TTTu −−=∆

ρρρ

(1)

The resistor

of the couple depends on the geometry, the material parameters and the temperature

)1(/),( TAlTr

rrte

ασσ

+=∆

(2)

where

is the electrical conductance,

is the temperature coefficient,

is the length and

is the area of a

couple. The design parameters allow a wide variation of the electrical parameters. A detailed overview of the

electrical conductance, in relation to the semiconductors is presented in Rowe [13].

Generally, the voltage of a TE is between 50 μV/K and 300 μV/K [13]. For practical use, TEs are connected in series

to a TEM. The voltage

tem

and the resistor

tem

of a module yield to

∑

tetem iuu

)(

(3)

∑

tetem irr

)(

(4)

where

is the number of the TEs in a module. With the assumption that the temperature is homogeneously

distributed over a TEM and the materials are ideal, the electric equivalent circuit of a TEM can be illustrated as an

ideal voltage source with an internal resistor (see Fig. 2). The relation of the electrical characteristics of a TEM to

the temperature gradient is presented in Fig. 3. This example shows also the electrical power

( )

temintem

iuP =

curve. It

can be clearly seen that only one maximum power point exists at a given temperature gradient. In this point, the

output power of the TEM has the maximum.

3. Modeling of Boost-Buck converter

For the linking of the TEM to the on-board power supply, a boost-buck converter is selected. The advanteges are a

high efficiency and the possibility to generate a variable output voltage [14].

The physical principle of a DC-DC converter is to charge and discharge electric storage elements like inductors or

capacitors. The duration of the charging and discharging is controlled through electric switching elements, like

MOSFETs. As a result, the magnitude of current and voltage of storage elements can be manipulated. The boost-

buck converter includes four MOSFETs, where

21 /SS

and

/SS

are synchronized (see Fig. 4). This means, if

/SS

is switched on,

/SS

is off and vice versa. The MOSFETs are controlled with a pulse-width-modulation

Fig. 2. TEM Module with equivalent electric circuit.

signal (PWM) with a constant switching frequency

. Caused by the PWM, the converter is a non-linear time-

variant system in relation to the duty cycle

. The modeling of such a system can be complex. An alternative model

approach for a converter is the state-space-averaging (SSA) [15]. The idea of the SSA is to average the period

changing circuit over a switching period

fT /1=

. The result is a continuous time-invariant model. This method is

accurate for frequencies up to

[15]. The time derivations of voltages and currents in the boost-buck converter

are:

temLC

Cuiu

321 11

ααα

++=

(5)

2161326154321

1211

ddidudidiuuui

LCLLtemCCL

ββββββββ

−−+++++=

(6)

23121

211

didii

LLL

γγγ

++=

(7)

21625243

321

122

ididieui

LLLblCL

δδδδδδ

+++++=

(8)

blCL

Ceui

321

εεε

++=

(9)

where

are the duty cycles of

21 /SS

/SS

and

εδγβα

,,,,

are system parameters (see Appendix). The

derivatives of the states

[ ]

32211

,,,,

CLCLC

uiuiu=x

are nonlinear. For the following analysis of the converter the

averaged values are substituted by means of a DC and a small AC value [12]. This implies a linearization of (5) – (9)

and results to the small signal state space model for the converter at the operation point

BuAxx+=



(10)

Fig. 3. Voltage and current characteristics of a simulated PbTe TEM at different temperatures (gray line) and the power curves (black line).

where

is the state matrix,

is the input matrix and

[ ]

21,,, ddeu bltem

the input vector. All states can be

measured and are filtered with a second-order low pass. The cutoff frequency is 1 kHz.

Fig. 4. Electric circuit of boost-buck converter with TEM and load.

4. Digital Control Structure

The design of the boost-buck converter is supposed to meet three requirements: The first is the maximum power

point tracking (MPPT), the second is the trickle charge controller (TCC) of the load and the third is the robustness

and stability of the controllers. An overview of the control structure is shown in Fig. 1. The MPPT is based on a

gradient search algorithm, which uses the input current and voltage of the TEM to detect the actual power. This is

realized with the boost converter. The buffer capacitor

is used to decouple the two converters. A cascade

controller regulates the voltage magnitude of 2

and the output current 2L

to supply the load and charge the

battery. In case, the output voltage

is reaching the maximum charging voltage of the battery, the MPPT will be

disabled and the trickle voltage charge controller is activated. The controllers are implemented on a microcontroller.

In the next subsections the digital control structure is presented, in relation to a robustness criterion. Additionally,

the MPPT algorithm and TCC are presented.

4.1. Control design

In general, the discrete time transfer function from input signal

to output signal

is defined as

)(

)( 1

−

−== zA

plant

(11)

where

and

are the plant polynomials. The closed loop of a feedback control can be described by

)(

)()(

)(

−

−−

−

== zA

zBzT

(12)

where

is the reference signal,

the pre-filter polynomial of the controller and

)()()()()(

11111 −−−−−

+= zBzRzAzSzA

(13)

is the characteristic polynomial of the closed loop dynamic, in relation to the control polynomials

and

. The

detailed digital control structure is shown in Fig. 5. In relation to (5) - (10) the transfer function of the boost-buck

converter follow to

)(

)( 1

22 −

−

−=zd

zG C

ud C

(14)

)(

)( 1

22 −

−

−=zd

zG L

id L

(15)

)(

−

=zd

(16)

)(

)( 1

−

−=zd

zG L

(17)

The control plants are sensitive to changes of the series resistance of the TEM. In Fig. 6 the frequency response of

the control plant is given for different values of

tem

. It can be noted that the gains of

31 C

and

22 L

are

significantly sensitive to

tem

. For small values of the resistance, the DC gain shows a maximum and decreases for

higher values. However,

22 C

and

11 L

seem to be smoother, in contrast to

31 C

and

22 L

but also an

influence of the parameter variation from the resistance can be detected. Hence, in all bode plots it can be seen that

the control plants depend on the resistor value of the TEM. If these influences are not taken into account during the

design of control parameters, the stability of the closed loop cannot be guaranteed.

In the context of the analytic results for control parameter design, the robustness criterions gain

and phase

margin

of the open loop

)()(

)( 11

−−

−=zSzA

zRzB

zGol

(18)

can be specified [16]. A widely used approach is to define

6≥

dB and

29≥

°, which must satisfy all values of

Fig. 5. Digital control loops of boost-buck converter.

tem

. For this reason, the nominal resistance value of 0.1 Ω is selected, because the DC gains of

22 L

and

31 C

are maximal at this value. For higher values of the resistance, the gain and phase margin increase with a

positive effect on the robustness. This characteristic is vice versa for

11 L

and

22 C

, but the variation can be

compensated from the controller with the selected robustness criterion. In context to this criterion, the control

polynomials of the digital controller

and

must be designed. The control parameterization is

based on a two-degree of freedom design [16]. The desired closed loop system is chosen as a second order system













−

zG dcl

)(

(19)

)(

dcl sDs

ωω

(20)

where

is the natural frequency,

is the damping factor and

is the discrete time transformation.

For the current control dynamic it is necessary to select an adequate bandwidth to compensate disturbances like load

changes of the vehicle power supply and to reduce the convergence time of the desired input current of the MPPT.

In contrast to that, the voltage controller can be designed with a smaller bandwidth to compensate for lower

frequency control errors. The selected closed loop parameters of the controllers are presented in Table 1.

Table 1. Design parameters for controllers

Controller

[kHz]

Rise time [ms]

3.14 0.95 1

0.628 0.95 5

1.57 0.95 2.5

4.2. Maximum Power Point Tracking

In Fig. 3 the quality characteristic of a TEM and the power curves in relation to different temperature gradients are

presented. In general the electric characteristics and the temperature at the TEM are unknown. The idea is to search

and track the maximum power point by using a gradient algorithm. A gradient algorithm has the advantage, that not

prior knowledge is necessary. In [17] a hill climbing algorithm (HC) for a boost-buck converter is presented.

However, the HC algorithm uses a direct control of the MOSFETs without a feedback signal of the state signals.

The problem is that signal feedbacks or disturbances can result in oscillations. In the case, that the HC recovers

enough energy to charge the buffer capacitor

, the cascade controller increases the output current

and the

voltage of the capacitor

drops down and results in a feedback signal at the signals

and

. If the HC samples

the disturbance signals, the algorithm interprets this as a power change of the TEM. The algorithm reduces the duty-

cycle and the input power drops down to zero. A new start of the algorithm can result in an oscillation during the

start phase.

An alternative MPPT algorithm is the perturb and observe method (P&O) [7]. This algorithm uses the same gradient

search method as the HC, however the P&O estimates only one new desired operation point of the input current

This desired current is the reference signal of the input current controller

. In contrast to the direct control of the

MOSFETs, the controller compensates disturbances of the feedback signal. The desired input current

calculated in relation to the actual input power

)(kP

tem

and compared with the previous power

)1( −kP

tem

. If the

difference

)1()()( −−=∆ kPkPkP temtem

is negative, the desired input current

is decreased. If

0)( >∆ kP

, the

desired input current is increased. The MPP is reached if

0)( =∆ kP

. The algorithm for P&O yields to

[ ]

)()()1( 11 kPsignkiki rLrL ∆+=+

(21)

[ ]











<∆−

=∆

>∆+

=∆

0,1

0,0

0,1

:)(

kPsign

where

is a constant step size. The selection of the step size influences the performance time of the P&O. For a

large

the P&O converges faster to the MPP. However, the algorithm can oscillate around the exact MPP. In

contrast, the MPPT converges slower for a smaller

, nevertheless the tracked power is closer on the MPP. An

adaption of the step size combines the fast convergence time and the accuracy of a P&O:











∆

≤

∆

≤+

∆











∆

)(

)(2

)(

),1(

)(2

)(

),(2

)(

)1(

satk

(23)

The step size reduces for a small gradient of

)(kP∆

and vice versa. The saturation is a boundary of the adaption size.

4.3. Trickle Charge Controller

The TCC is activated in case the load voltage reaches the maximum battery voltage

. The idea of the presented

control structure (see Fig. 5) is to disable the MPPT and use the controller

to regulate the output voltage on the

desired voltage

. For the selection of MPPT and TCC a switch

is used. The activation function of

defined as











≥

+−

−

oCo

uuu

,21

(24)

where

−

is the defined low voltage magnitude to restart the MPPT.

5. Experiments

In section 3 and 4 the modeling and the control structure are presented. To verify the theoretical design, an

experimental device is used. The TEM is simulated with a constant voltage source and a series resistor. The load is a

12.5 V 72 Ah lead acid battery. The converter is controlled by a 320F28069 microcontroller from Texas

Instruments, with a sampling frequency of 10 kHz. The hardware specifications of the converter are presented in

Table 2.

Fig. 6. Bodeplot of control plants.

11 L

is presented in (1) and (3),

31 C

is presented in (2) and (4),

22 L

is presented in (5) and (7) and

22 C

is presented in (6) and (8).

5.1. Verification of Control Design

In relation to the control requirements from Table 1 and the parameters of the converter from Table 2, the controllers

and

are designed for a nominal value of

30=

tem

V and

1.0=

tem

Ω. To verify the feedback

controllers, the step response of each closed loop is measured (see Fig. 7). Each measured output signal of the

control loop matches the desired dynamic of the control design. The deviations of the measurements are caused by

noise and quantization effects of the microcontroller.

Table 2. Parameter of converter

Ωmr

2.43

ΩmrC2.1

Ωmr

8.0

4.22

Vebl 5.12

Ωmr

7.17

Ωmr

1.17

Ωmrbl 100

4321

,,, SSSS

Infineon IPP80N06S2L-07 ,

= 100 kHz

Fig. 7. Step responses of the control loops. The red dashed line is the reference signal. The black line is the desired output signal and grey the

measurement. (1) shows the controlled signal of

, (2) the signal of

, (3) the signal of

and (4)

5.2. Verification of Maximum Power Point Tracking

In this experiment, the TEM is simulated with nominal values

15=

tem

V and

1.3=

tem

Ω at the beginning. To verify

the convergence and adaption, the resistor and the voltage are changed.

In Fig. 8 the signals of the converter and the simulated TEM are shown. After 1.5 s the MPPT is enabled and the

algorithm starts to calculate the desired reference input current, which is regulated with the input current

controller

. After approximately 1 s the MPP is reached and the algorithm has converged. This can also directly

be recognized from the measurement, because

temin uu 2/1=

in this case. At 3.7 s the resistor changes to 1.8 Ω. The

reaction of the MPPT has a dead time of nearly 300 ms before an adaption of the new MPP can be observed. The

dead time is the result of the adaptive P&O algorithm (23), because the step size is reduced in relation to the

previously small power difference

P∆

. After the detection of the power change, the step size must increase. This

dead time can also be observed when the input changes to

30=

tem

However, at the moment when the input voltage increases, a peak of value at the input current can be detected. This

disturbance is compensated by the input controller. Without the control, it would lead to an undesired increase of the

current signal. Finally, the estimated gradient of the input power could be incorrect.

Nevertheless, the P&O algorithm converges in all three cases to the MPP with a maximum converge time of 2 s.

Fig. 9 shows the detailed calculation and adaption of the desired input current

of the MPPT algorithm for the

time spans (a), (b) and (c) from Fig. 8 in relation to the actual power curves of the TEM.

5.3. Verification of Trickle Charge Controller

In this experiment, the battery was charged to 13.3 V. At the beginning of the experiment

15=

tem

V and

Ω= 1.3

tem

are selected and the MPPT is active (see Fig. 10). The resistor value changes from 3.1 Ω to 1.8 Ω at 40

ms and the MPPT increases the input current and also the input power. As a result, the output voltage increases up to

the selected maximum battery voltage

6.13=

V. At this time, the TCC is activated and the desired voltage

4.13

V is regulated. To match this voltage, the input current is reduced to 2.1 A. At 500 ms the resistor

changes to

Ω= 1.3

tem

and the input power is not sufficient to hold on the trickle voltage. As a consequence, the

output voltage decreases and the TCC increases the input current to match the desired battery voltage. At the point

that the output voltage drops down to

3.13=

−

V, the MPPT is reactivated.

Fig. 10. Verification of TCC algorithm. The grey line is

tem

, the blue line is

and the red line is the input current

tem

at the top

ill t ti I th b tt i h th t t lt ith th t d t t f th

Fig. 8. Verification of MPPT algorithm. The grey line is

tem

, the blue line is

, the grey dashed line is the output voltage

and the red

line is the input current

tem

Fig. 9. Step size adaption of MPPT algorithm for the time spans (a), (b) and (c) from Fig. 8.

6. Conclusions

In this paper a control concept for a boost-buck converter is presented. A variation of the electrical characteristics of

the TEGs is analysed in the frequency domain. The series resistance of the TEG influences the state dynamic of the

control plants. The operation point for the state space model is selected for the resistor value, which caused the

highest DC gain of the plants. Based on this selected operation point, the linear digital controllers are designed. A

robustness criterion is defined to ensure the stability of the closed loop dynamic for a variation of the electrical

parameters of the TEG.

The presented MPPT is based on a perturbe and observation algorithm, which determines a desired input current. In

contrast to a direct control algorithm, an oscillation at the start phase of the algorithm or disturbances can be

compensated. The perturbed and observation algorithm is extended with a variable step size to reduce the oscillation

at the MPP and to adapt the step size in relation to the power gradient of the actual power from the TEG. An

overvoltage protection for the battery is realized with a trickle charging control. In this case, the MPPT algorithm is

disabled and the desired trickle charge voltage is controlled with a cascade voltage controller. The control and

algorithms are proven in experiments.

The next steps are to analyse and optimize the electrical wiring of explicit TEMs for a DC-DC converter to optimize

the power efficiency. Furthermore a TEG prototype with DC-DC converter should be integrated and verified at an

exhaust gas system of internal combustion engine.

Acknowledgements

The project “Thermoelektrische Generatoren 2020” (03X3553E) is supported by the German ministry of education

and research (BMBF). The authors would like to thank Texas Instruments for the support through their innovative

products.

Appendix

)(

temCtem

rrCr +

−=−==

(25)

)(

)()()(

2122111

Ctem

temdsCtemdsCtemCLCtem

rrL

rrrrrrrrrrr

++++++

(26)

)(

Ctem

tem

rrL

(27)

LrC

−=−=

(28)

)(

Ctem

rrL

(29)

)( 122

rrr

dsdsC

−+

(30)

321

=−=−=

γγγ

(31)

3333223

(

blC

bldsCdsblCLblLC

rrL

rrrrrrrrrr

++++

−=

(32)

)(

blC

rrL

−=−=

δδ

(33)

342

rrr

dsdsC

−+

−=

(34)

65 2

=−=

δδ

(35)

)(

blCbl

rrCr +

−=−=−=

(36)

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