Novel model and maximum power tracking algorithm for thermoelectric generators operated under constant heat flux

Applied Energy 256 (2019) 113930

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Novel model and maximum power tracking algorithm for thermoelectric generators operated under constant heat flux

T

Marcos Compadre Torrecillaa,b, , Andrea Montecuccoc, Jonathan Sivitera,c, Andrew R. Knoxa,c, Andrew Strainb ⁎

a

School of Engineering, University of Glasgow, G12 8LT, UK Clyde Space Ltd., 5B Skypark 5, 45 Finnieston Street, G3 8JU, UK c Thermoelectric Conversion Systems Ltd, UK b

HIGHLIGHTS

maximum power point of a TEG under constant heat flux differs from constant ΔT. • The transients impact the tracking accuracy of maximum power point algorithms. • Thermal new proposed algorithm generates more power than other state-of-the-art methods. • The performance of the new algorithm for different heat flux profiles is shown. • The • Maximum power occurs when the voltage is higher than half the open-circuit voltage. ARTICLE INFO

ABSTRACT

Keywords: Thermoelectric MPPT Constant heat Transient response Modeling Boost converter

Thermoelectric generators (TEGs) are solid-state devices used to convert heat into electricity. The use of TEGs in waste heat recovery systems offers a source of sustainable electricity, which helps to reduce emissions to the environment. Optimization of the electrical operating point of TEGs is important to improve the overall efficiency of TEG systems. Previous literature focused mostly on characterizing the maximum power point (MPP) of TEGs when operating at constant temperature difference. However, in most practical applications TEGs operate under constant or limited heat conditions. In fact, in waste heat recovery systems the amount of thermal energy is limited. This work presents a new simplified TEG model that simulates the dynamic response of the TEG systems with a good degree of accuracy. With the aid of this TEG model, power and control electronics have been designed to operate a TEG system, with limited input heat flux, at its optimum load. The control architecture is based on the perturb and observe maximum power point tracking (MPPT) technique and modified to take into consideration the thermal transient response of the TEG. A boost dc-dc converter is used to step-up the TEG voltage to 28 V for connection to an eight-cell Lithium-Ion battery. A microcontroller implements the control algorithm that drives the power converter. Experimental results show that the proposed algorithm outperforms two state-of-the-art algorithms (standard perturb and observe and fractional open-circuit) by 1.14% and 2.08%, respectively, when the TEG operates under constant heat flux.

1. Introduction Thermoelectric generators have been used for many years to convert heat into electricity. Their high robustness and lack of moving parts makes them long-lasting devices that are suitable for harvesting waste heat into electricity. The figure of merit, denoted as ZT, is one of the parameters that are often used to evaluate the efficiency of thermoelectric devices. ZT is a dimensionless performance parameter used to

⁎

compare materials with good thermoelectric properties [1]. In the past twenty years advances in thermoelectric research have brought materials with values of ZT close to 2.5 [2]. Higher values of ZT lead to higher thermoelectric efficiency, which is defined as the ratio of the electrical output current to the heat energy absorbed by the hot junction. Previous research articles have proposed the integration of TEGs in different applications, like burning stoves that use heat transfer to pre-

Corresponding author at: School of Engineering, University of Glasgow, G12 8LT, UK and Clyde Space Ltd., 5B Skypark 5, 45 Finnieston Street, G3 8JU, UK. E-mail address: [email protected] (M. Compadre Torrecilla).

https://doi.org/10.1016/j.apenergy.2019.113930 Received 5 February 2019; Received in revised form 16 September 2019; Accepted 20 September 2019 0306-2619/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature Α A CTEG ΔT k ΚEFF KTEG L Pout,el

Q QH Rint TC TH τTEG Vload VOC Wel Wth θTEG

Seebeck coefficient [V/K] surface area of the TEG [m2] TEG system equivalent thermal capacitance [J/K] temperature gradient across the TEG [oC] thermal conductivity of the TEG pellets [W/m K] effective thermal conductance of the TEG [W/K] thermal conductance of the TEG @ zero current [W/K] length of the pellets [m] electrical output power of the TEG [Wel]

quality factor input heat flux [Wth] equivalent series resistance of the TEG [Ω] cold side temperature [oC] hot side temperature [oC] Tthermal time constant of the TEG system [s] output voltage of the TEG [V] instantaneous open-circuit voltage of the TEG [V] Watt electric (electric power) Watt thermal (heat flux) TEG system equivalent thermal resistance [K/W]

algorithm for different sampling frequencies and different temperature change rates (in °C/s). The results reported in [27,28] show that, although P&O has tracking problems at low sampling rates (2.5 Hz) and rapidly changing conditions (5 °C/s), the overall efficiency was only less than 1% lower than that with a P&O sampling frequency of 10 Hz. The TEG is simulated using a solar array simulator (SAS) and it does not take into consideration the thermal response typical of a TEG driven by a limited heat source. In all the aforementioned cases either the model of the TEG used, or the algorithms employed, are only valid for situations where the TEG operates under constant temperature difference conditions, which requires an unlimited source of heat. With a limited source of heat a specific temperature gradient across the TEG might not be always reached when operating at the MPP. For instance, in the case of an exhaust gas application, several studies have demonstrated that the amount of available heat, in great part given by the driving conditions, it is not enough to obtain the temperature gradient desired across the TEG. [29] shows a simulation of a TEG operated under constant heat conditions, where the Seebeck coefficient and the internal resistance of the TEG are modeled as second order equations. The results show that the MPP under these conditions is higher than half of the open-circuit voltage. Experimental results of the conclusions presented in [29] are presented in [30] as well as the effect of the transient response of a TEG to load step changes. The generation of local maxima can deceive the MPP and a relatively fast P&O sampling time can prevent the TEG from operating from the “true” MPP produced under constant heat conditions. The most important characteristic is that the MPPT sampling period found in the literature is in the range between few milliseconds to few seconds. A comparison of different MPPT methods is presented in [31], where an adaptive compass search technique is used and proves to perform better than several other algorithms, P&O amongst others. In this article, it can be seen that the MPPT sampling frequency used for the different algorithms is equal to 10 ms. This article will show that, in order to accurately track the MPP of a TEG operating under constant heat flux, the MPPT sampling frequency has to be of the same order of magnitude as the thermal time constant of the TEG system. As a matter of fact, P&O was developed for solar cells, which respond almost instantaneously to variations of the input power source; i.e., solar irradiation. TEGs’ response is composed of a fast change in the open-circuit voltage and in a slow variation of the internal resistance. Also, the electrical operating point further affects the thermal behavior and hence the open-circuit voltage and the internal resistance [30]. It is important to track both responses, and a slower evolution of the P&O will aid tracking of the thermal effects. This article presents a new TEG model and a new algorithm with outstanding MPPT performance in limited heat systems. The MPPT sampling frequency is adjusted to the thermal transient response of the TEG, which has a direct impact on the performance of the MPPT algorithm. Simulation results show a comparison between the proposed MPPT algorithm and the conventional P&O and FOV algorithms. Furthermore, experimental results confirm the superior tracking

heat water for household use [3], cooking stoves for rural areas [4,5], vehicle exhaust systems [6,7], combustion chambers [8] and harvesters for geothermal power generation [9]. From a system perspective, there are several aspects that can be improved and will increase the overall efficiency of the TEG system. The power conversion stage covers the power converter topology, the control circuit and the MPPT technique and it has a direct impact on the overall efficiency of the system. The simplest and most efficient architecture is a direct connection from the TEG to a battery [10,11], but this technique has the disadvantage that it can only be used when the open-circuit voltage of the TEG is higher than the battery voltage. Another big disadvantage is that the battery voltage sets the operating point of the TEG; hence the TEG voltage cannot be changed as it is done with MPPT techniques. MPPT techniques are used to obtain the maximum available energy from the TEG, and the MPPT accuracy is defined as the ratio of the power delivered by the TEG to the maximum theoretical power. Different types of MPPT algorithms have been proposed for TEGs and most of them are based on open-circuit voltage (fractional open-circuit voltage, FOV) measurements or hill-climbing techniques, like perturb and observe (P&O) and incremental conductance (INC). FOV is used in [12] to control the MPP of a TEG connected to a boost-cascaded-buck converter. A SEPIC converter is used using the same technique in [13], while [14,15] show the technique applied to a low-power boost converter. Montecucco et al. [16] present a circuit to eliminate the influence of the input capacitance of a power converter during the opencircuit measurement of the TEG voltage. An equivalent method to measuring the open-circuit voltage of the TEG is to measure the shortcircuit current, as shown in [17]. As with FOV, the P&O technique has been applied to different power conversion topologies like boost-cascaded-buck [18], interleaved [19] and conventional [20] boost converters, amongst others. In [19] a P&O based algorithm was used to generate power from an array of TEGs and it concludes that a simple P&O could provide erroneous judgment if the thermal environment is not stable. The main difference when using P& O, with respect to FOV, is the need for a digital processing unit that processes the output power and compares it against previous samples. Some P&O algorithms make use of an adaptive step [21] to improve the speed and accuracy with respect to conventional P&O. This is the case even in low voltage and low power applications [22]. INC is very similar to P&O (a comparison between both methods can be found in [23]), although it does not require the use of a digital processing unit [24]. For this reason, a control circuit that implements INC can be designed with low-power consumption components that could be more suitable for low-power applications [25]. The work presented in [25] shows a low power consumption INC based algorithm that does not require the calculation of the derivative of the output current with respect to the output voltage. [25] reports a theoretical mismatch of less than 1.1% between the TEG MPP and the measured operating points in the range of 1–100 °C. The TEG is modeled as a power supply and a series resistor. Another example of INC is presented in [26], where the sampling frequency to perform MPPT is in the order of few milliseconds. [27] shows the MPP tracking efficiency of a standard P&O 2

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performance and the improved stability with respect to the conventional P&O. 2. Characterization of TEGs under constant heat flux As described by Min and Yatim [32], the effective thermal conductance Keff of a TEG varies depending on the electrical current passing through it. Keff can be expressed in terms of the heat flux flowing through the TEG, QH, and the temperature difference across it, ΔT, as shown in Eq. (1).

QH QH = T TH TC

K eff =

(1)

where TH and TC are the hot and cold side temperatures, respectively. From the one-dimensional heat conduction equation [33], Eq. (2), and neglecting the Thomson effect, the relationship between Keff and the current through the TEG is found. This relationship is shown in Eq. (3).

kA T QH = + TH I L QH = T

K eff =

kA T L

= KTEG + I

1 Rint I 2 2

+ TH I

Vload +

1 R I2 2 int

T (TH + TC ) 2 T

Fig. 1. Power and voltage plotted against output current for the GM250-24110-12 when operated under a constant thermal flux equal to 150 Wth.

Rint = d T 2 + e T + f

The polynomial coefficients represent the degree of dependency of the parameters α and Rint with temperature. The intercepts, c and f, represent the values of α and Rint at 0 K. The values of b and e represent the linear dependency of α and Rint with temperature, whereas the values of a and d represent the curvature of α and Rint. when they are plotted against temperature. With Eqs. (8) and (9) being second order polynomials, Eqs. (6) and (7) are also second order polynomials. Fig. 1 shows the maximum power point, MPP, under steady-state conditions being equal to 5.14 Wel. When the TEG operates at the MPP, the output voltage is 7.9 V and the instantaneous open-circuit voltage is 13.8 V. Defining the parameter β = Vload/Vin [30], it can be observed that the MPP corresponds to a value of β greater than 0.5 (β = 0.57) as opposed to β = 0.5, which is the case when the TEG operates under constant temperature gradient.

(2)

=

kA + L

TH I T

Rint I 2 2 T (3)

where k is the thermal conductivity, A is the area of the TEG, L is the thickness of the device, is the Seebeck coefficient, Rint the internal resistance of the TEG, Vload the voltage at the output terminals and I the generated electrical current. The thermal conductance of the TEG in open-circuit conditions is equal to KTEG = kA/L. From Eq. (3) it can be seen that the effective thermal conductance increases with the output current, hence the temperature difference across the TEG decreases. This phenomenon is due to the Peltier effect, which is considered a parasitic effect in power generation and highlights an important difference in the operation of a TEG under constant heat flux as opposed to operation under constant temperature gradient. Assuming a fixed temperature on the cold side of the TEG, and knowing the output current, the temperature difference can be obtained solving Eq. (2) for ΔT, and it is shown in Eq. (4). With the value of ΔT the output power can be calculated using Eq. (5). Eq. (6) is used to calculate the instantaneous open-circuit voltage and Eq. (7) the voltage at the output of the TEG. 1

T=

QH + 2 Rint I 2

Pout , el = I [(

VOC =

Vload =

T)

(Rint I )]

Several TEG models have been presented in the literature. Some of them are not accurate for applications where the TEG operates under constant heat flux like the models presented in [34,35], which are only valid for TEGs that operate under constant temperature gradient. Others, like those presented by [36,37] are more complex and require the accurate knowledge of system parameters not related to the TEG model, like the heat capacity, convection coefficient and thermal resistance of the different elements of the system. A more accurate and comprehensive model is presented in [38], where the overall TEG system architecture is modeled. This model, however, can also be used for constant heat flux applications. A complex model also requires more computational resources and it often means that the model is very slow. The simulation speed is an important aspect when the model has to be

(4) (5) (6)

T

T

3. Model of a thermoelectric generator operating under constant heat flux

TC I

KTEG + I

IRint

(9)

(7)

VOC is the instantaneous open-circuit voltage of the TEG and Pout,el is the electrical output power of the TEG. With Eqs. (5) and (6) it is possible to plot the output power and instantaneous open-circuit voltage against the output current. Fig. 1 shows the results for the GM250241-10-12 TEG1 where the electrical power output from the TEG, the output voltage Vload and the instantaneous open-circuit voltage VOC are plotted. The Seebeck coefficient, α, and the internal resistance, Rint, are temperature dependent parameters that can be approximated by the second order polynomial equations shown in Eqs. (8) and (9), with a, b, c, d, e and f being real coefficients.

= a T2 + b T + c 1

(8) Fig. 2. Simplified diagram of the TEG setup (a) and equivalent circuit of the thermal model (b).

European Thermodynamics – www.europeanthermodynamics.com. 3

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Fig. 3. Transient response used to obtain the time constant for (a) 75 Wth and (b) 300 Wth.

used along with other models, like the power conversion stage, to reproduce several seconds of operation of the overall system. Eqs. (4)–(9) represent the operating points of a TEG and, therefore, they are evaluated on every simulation step. The system dynamics, influenced by the equivalent thermal capacitance and resistance of the TEG system, are simulated using a first-order transfer function that represents the equivalent thermal impedance of the overall system. Fig. 2 shows a simplified diagram of the TEG system and the equivalent thermal circuit used to implement the thermal dynamics of the system, where θTEG and CTEG are the thermal resistance and capacitance of the thermoelectric system, respectively. The thermal circuit of Fig. 2(b) implements the system delay to temperature changes using the difference between two consecutive values of steady-state temperature gradient across the TEG as the input to the circuit; that is, ΔTSS = ΔTSS(j) − ΔTSS(j − 1). Based on this circuit model, the first order equation that models the temperature dynamics of the system can be obtained, and shown in Eq. (10), where τTEG is the thermal time constant of the system. 1

TCTEG (S ) = TSS

(

TEG CTEG

s s+

1 TEG CTEG

1

= TSS

(

TEG

s s+

1 TEG

)

+

)

TSS , j

+

the 63.2% figure, is 3.09%. Based on the previous observations, the value of the thermal time constant of the system can be approximated to 115 s, without great loss of accuracy. The model of the thermal time constant of the TEG system using the first order equation, Eq. (10), will be verified in the time domain. The time domain representation of Eq. (10) is shown in Eq. (11) for the temperature difference ΔT and Eq. (12) for the TEG output power.

TTEG (t ) = TSS, j PTEG (t ) = PTEG, j

s

+ TSS e

+ (PTEG, j

t

(11)

TEG

PTEG, j 1) e

t

TEG

(12)

Fig. 4 shows the dynamic response of the TEG when the current is stepped, in discrete values, from open-circuit to short-circuit [30]. The step response of the TEG is used to characterize the dynamic behavior of the TEG. The encircled portions of temperature gradient ΔT and the TEG power shown in Fig. 4 are used to check Eqs. (11) and (12) for the case of decreasing exponentials. The theoretical decreasing exponentials curves are plotted against the measured curves in Fig. 5(a) and (b) for TEG power and ΔT, respectively. The encircled portions of temperature gradient ΔT and the TEG power shown in Fig. 6 are used to check Eqs. (11) and (12) for the case of increasing exponentials. The theoretical increasing exponentials curves are plotted against the measured curves in Fig. 7(a) and (b) for TEG power and ΔT, respectively. The model of the TEG is developed in Matlab/Simulink®2 and it reproduces, with a good degree of accuracy, the dynamic response of a real TEG. The Simulink model of the TEG is shown in Fig. 8(a), where the main block “TEG Equations” contains a script that processes Eqs. (4)–(9) sequentially. The inputs to the block are the heat flux QH, the cold side temperature TC, the temperature difference from the previous iteration, TO, and the output current of the TEG, IOUT. The outputs are the temperature gradient, Delta T, the output power, Power, the load voltage, Vout, and the instantaneous open circuit voltage, Voc. Fig. 8(b) shows the Simulink model used to characterize the TEG. The main block “TEG” contains the block shown in Fig. 8(a). The block “Resistor” simulates a load and provides the output current that is fedback into the TEG block. The characterization model contains three blocks, “Temp_TEG”, “Power_TEG” and “Volt_TEG” that take external

1

s

TSS , j

1

1

1

(10)

The values of θTEG and CTEG can be found experimentally from a step response of the TEG. The thermal time constant corresponds to the time it takes the temperature gradient across the TEG to reach 63.2% of the difference between the initial and the final value of temperature. For 75 Wth, see Fig. 3(a), τTEG = 114.43 s. The time constant can also be obtained by measuring the time it takes the temperature across the TEG to reach the steady-state value. The temperature gradient across the TEG is considered to be at steady-state; that is, the temperature gradient has reached 98.16% of the final value, after 464.17 s, so 4τTEG = 464.17 s. The value of the time constant is therefore one fourth of that value; that is, τTEG = 116.04 s. The error produced between these two values, with respect to the 63.2% figure, is 1.4%. The same exercise can be done for 300 Wth, a zoomed-in view of which is shown in Fig. 3(b). Using 63.2% of the final steady-state value, the time constant is found to be 114.18 s. On the other hand, using the 98.16% value for four times the value of the time constant, the value is 117.65 s. The error produced between these two figures, with respect to

2

4

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Fig. 4. Portions of the power (a) and ΔT (b) (encircled areas) used to check the modeled thermal delay for decreasing exponential curves.

Fig. 5. Decreasing theoretical exponential curves plotted against experimental waveforms of Power (a) and ΔT (b).

data saved in a matrix format. This data is taken from real measurements on actual TEGs and it is fed into the simulation to provide a method of direct comparison between the simulation and the real performance of the TEG. The output current is taken as an input parameter and, the voltage will change according to the temperature gradient. Conversely, the TEG voltage could be used as an input to the system, treating the output current as a changing output variable. In order to verify the accuracy of the simulated TEG, a test is performed using a real TEG and the results are compared against the results from the simulation. The TEG used for the verification of the model is the monTETM, from Thermoelectric Conversion Systems3, and the parameters used by the Simulink model are taken from the manufacturer’s datasheet and are shown in Table 1. First, the TEG is placed in the test setup described in [30] with an input thermal power of 300 Wth, as shown in Table 1. The TEG is connected to an electronic load and the current is stepped, from zero amperes up to the short-circuit current, in steps of 500 mA. The time between current steps is 600 s. The hardware equipment used to drive the TEG and measure the different TEG parameters is shown in Table 2. The results are shown in Fig. 9 where the correlation between the model and the TEG is shown. The maximum steady-state power and voltage errors, between the model and the TEG, are 1.8% and 1.54%, 3

respectively. The maximum steady-state temperature error is 9.31% with respect to the measured temperature across the TEG. It is important to note that the model provides the temperature between the hot and cold plates of the TEG, whereas the measurements are not taken exactly at the plates. This is because the thermocouples are inside the hot and cold blocks and, although they are very close to the TEG plates, they are not in perfect contact with them. The model and the TEG show very good correlation with a very small error, and the dynamic behavior of the TEG is replicated which is important in order to evaluate MPPT techniques when the TEG operates under constant heat flux. 4. Averaged model of the boost converter The TEG is used to charge a 28 V battery, which corresponds to the nominal voltage of eight series Li-Ion battery cells. The TEG will output a voltage that is lower than the battery voltage for heat fluxes lower than 500 Wth thus the selected converter topology is the boost, or stepup, converter, whose circuit is represented in Fig. 10(a). A model of the boost converter is implemented in Simulink and it will be interfaced with the model of the TEG for system optimization. A switched model [39–43] reproduces the behavior of the converter and contains all the switching information produced by the effect of the pulsed width modulation (PWM). The switching information is, in many cases, not very useful and it renders the model very slow which is

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Fig. 6. Portions of the power (a) and ΔT (b) (encircled areas) used to check the modeled thermal delay for increasing exponential curves.

Fig. 7. Increasing theoretical exponential curves plotted against experimental waveforms of Power (a) and ΔT (b).

Fig. 8. Simulink model of the TEG (a) and TEG characterization model (b).

undesirable for simulations involving long thermal time constants. Instead, an averaged model [44], (Fig. 10b), is developed based on the steady-state converter equations shown in Eqs. (13)–(16). vL(t) and iL(t) are the voltage and current in the inductor, respectively; vC(t) and iC(t) are the voltage and current in the input capacitor, respectively; Ri is the internal resistance of the TEG; Vi (or VTEG) is the input voltage, corresponding to the open-circuit voltage of the TEG; and D is the duty-cycle of the converter, defined as the ratio between the on and the off time of the power switch.

Table 1 Parameters of the monTETM used in the simulation. Parameter

Value

Units

α Rint TC QH θTEG

0.0517 −0.000006 T2 + 0.0068 T + 1.6033 310 300 1.03

V/K Ω K Wth K/W

6

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Table 2 Equipment used to test the TEG. Equipment

Model

Power Supply Unit Electronic Load Data Acquisition Unit Digital Multimeter Digital Multimeter

Agilent Agilent Agilent Agilent Agilent

Function N5750 N3300A 34972A 34410A 34405A

Drive the heater in the heater block Provide load changes on the TEG Acquisition of the hot and cold side temperatures Acquisition of the output current from the TEG Acquisition of the output voltage of the TEG

diL (t ) = vc (t ) dt

(1

dvC (t ) = iRi (t ) dt

iL (t )

vL (t ) = L

i c (t ) = C iRi (t ) =

vi (t )

VC =1 VO

D ) Vo

iL (t ) =

vC (t ) =

vc (t )

1 C

1 L

vc (t )

iRi (t )

(1

D) Vo dt (13)

iL (t ) dt

(14) (15)

Ri D

(16)

The block diagram of the averaged boost converter (Eqs. (13)–(16)) implemented in Simulink, is shown in Fig. 11. By varying the duty cycle of the boost converter the algorithm changes the operating point of the TEG to find the MPP. The boost converter must not exhibit an oscillatory behavior when the duty-cycle is changed. The duty-cycle to input voltage transfer function for the input-controlled boost converter is shown in Eq. (17) and it is important that the transfer function does not exhibit high resonant peak to avoid such oscillations.

Gvd (S ) =

Vi (S ) = d (S )

Vo L

LCS 2 + R S + 1

(17)

The quality factor of the Gvd (S) transfer function, Q [45], which defines the amount of resonance in the response of the system, is evaluated around a DC operating point of the power converter, which is selected based on the input thermal power under which the TEG will operate. The TEG is intended to operate at the MPP in the range of 200–300 Wth, and the values of ΔT, Vin (being the instantaneous opencircuit voltage of the TEG) and Rint are obtained from the TEG model and are shown in Table 3. The DC operating point used to evaluate the transfer function Gvd (s) in the frequency domain corresponds to the average of the values obtain between 200 and 300 Wth; that is, a value of Rint = 2.75 Ω. The values of the boost inductor, L, and input capacitance, Cin, are calculated for a low value of resonant peak and are also shown in Table 3. The Bode plot of Gvd (S) is shown in Fig. 12(a) and the input voltage changes with duty-cycle in Fig. 12(b), where a step-change in the duty-

Fig. 9. Simulink model and TEG results plotted on the same graphs to verify the accuracy of the model. (a) Power curves (b) Voltage curves (c) Temperature curves.

Fig. 10. Boost converter (a) Circuit diagram (b) Averaged model.

7

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Fig. 11. Simulink model of the averaged boost converter.

the voltage is perturbed depending on the TEG power. When the TEG power measured at instant “i” is higher than the power measured at instant “i-1”, the voltage is perturbed in the same direction it was done at instant “i-1”, otherwise the direction of the perturbation is changed. If at any point the calculated target voltage is lower than 50% or higher than 60% of the instantaneous open-circuit voltage, the voltage is set to 52% or 58%, respectively. The step size used in the P&O, stated as “step” in the flow chart, is proportional to the instantaneous open-circuit voltage with a factor of proportionality equal to VNEXT = 0.015VOC. Derivative adaptive steps increase the theoretical speed and accuracy of the MPPT algorithm; however, in real digital MPPT implementations there are the limitations imposed by different factors like the numerical stability and discretization and quantization errors [46]. For instance, when using a step equal to dP/dV, when the converter operates in the vicinity of the MPP, the perturbations are very small hence dV and dP are also very small. A quantization error that makes dV smaller than dP might lead to an excessive step that would send the next operating point far from the MPP. When the converter operates in P&O mode the power delivered by the TEG is monitored and the operating point changed after 4 system thermal time constants, that is, after 4τTEG. By doing so, the actual value of steady-state power is evaluated. In order to track the changes in the input heat flux and quickly respond to these variations, the open-circuit voltage is monitored constantly after each perturbation step. First, after two system thermal time constants, 2τTEG, and then it is monitored continuously until the next perturbation step. If the difference in the instantaneous open-circuit voltage is greater than 0.1 V then the operating point is

Table 3 TEG parameters used in the model. Parameter

Value

L Cin VO ΔT QH Vin @ MPP Rint

270 μH 4.7 μF 28 V 244–163.7 °C 300–200 Wth 12.61–8.46 V 3–2.5 Ω

cycle between 50% and 75% is applied. With an output voltage of 28 V, the input voltage changes between 14 V and 7 V. It can be observed that the converter exhibits a well-damped response even under a large variation of the duty-cycle. 5. Maximum power point tracking algorithm The proposed MPPT algorithm takes into consideration the “long” thermal time constant of typical TEG systems and it adapts the perturbation step dynamically based on the steady-state value of the instantaneous open-circuit voltage. The proposed algorithm builds on previous literature that explored how in constant heat TEG systems the MPP is found with TEG voltages between 50 and 60% of the instantaneous open-circuit voltage [29,30]. The flow chart of the proposed algorithm is shown in Fig. 13. The first operating point is set to 50% of the open-circuit voltage and the subsequent operating points are set using a conventional P&O whereby

Fig. 12. Bode plot of the Gvd(s) transfer function of the power converter (a). Input voltage change due to a 25% duty-cycle step (b). 8

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Fig. 13. Flow Chart of the new proposed MPPT algorithm.

Fig. 14. Simulink block diagram of the overall MPPT system.

immediately changed to 0.5VOC, because this means that the input heat flux has changed. The power converter regulates the input voltage using a control loop that monitors the TEG voltage every 5 ms and compares it to the target voltage VNEXT. The operating point is varied accordingly, increasing or decreasing the duty-cycle by the minimum duty-cycle step. If the output voltage (battery voltage) reaches the maximum charge voltage of the

battery, also known as End of Charge voltage, the power converter starts decreasing the duty cycle until it becomes 0% and the converter delivers zero energy to the battery, i.e. the battery is fully charged. The model of the overall system is shown in Fig. 14. The “From Spreadsheet” block, in red, reads the parameters from an external spreadsheet that contains the input power profile. The input power sequence is defined as follows: 100 Wth for 0 < t < 19.8 s, 200 Wth for 9

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Fig. 15. Simulation results of the power generation of the conventional P&O, FOV and the proposed algorithm for (a) 100 Wth, (b) 200 Wth and (c) 300 Wth. Table 4 Values of power generation, total energy and ratio between average TEG voltage and instantaneous open-circuit voltage (β) in steady-state conditions obtained from the TEG model. QH = 100 Wth

QH = 200 Wth

QH = 300 Wth

Algorithm

PAVG

β

PAVG

β

PAVG

β

Total Energy

Conventional P&O FOV Proposed Algorithm

1.98 Wel 1.99 Wel 2.03 Wel

0.506 0.495 0.558

6.68 Wel 6.70 Wel 6.83 Wel

0.490 0.496 0.574

13.18 Wel 13.23 Wel 13.46 Wel

0.492 0.494 0.562

325.53 J 326.74 J 332.05 J

Fig. 17. TEG voltage change due to a 25% duty-cycle step.

“MPPT” block, in green, contains the proposed MPPT algorithm. The inputs “vin” and “iin” take the voltage and current readings for the TEG and these inputs are filtered using a first order low-pass filter block. These filter blocks implement the low-pass RC filters used in the actual hardware. Four blocks (FracVoc, PO, FracVoc_V and PO_V) are used to plot the results of the proposed algorithm along with the results obtained using conventional P&O and conventional FOV. The monTETM generator comprises 196 pairs of pellets, three of which are not connected to the main string of pellets and are used for open-circuit and temperature sensing. The output voltage of these pellets is proportional to the voltage of the main string of pellets by a

Fig. 16. Hardware implementation of the power converter.

19.8 s ≤ t < 30.3 s and 300 Wth for 30.3 s ≤ t < 40 s. The maximum steady-state power levels that the monTETM will deliver for these three power levels are 2.04 Wel, 6.81 Wel and 13.45 Wel, respectively. The “TEG” block, in cyan, is the TEG model which has the output Vout connected to the Vin input of the boost model (orange block). The 10

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Table 5 Values of TEG power, voltage and beta parameter at 100 Wth, 200 Wth and 300 Wth. Parameter

QH = 100 Wth

QH = 200 Wth

QH = 300 Wth

PMPP VTEG β

2.028 Wel 2.37 V 0.57

7.527 Wel 5.15 V 0.585

13.84 Wel 7.1 V 0.55

Fig. 19. Power generated by the TEG for an input heat flux equal to 100 Wth.

Fig. 20. Power generated by the TEG for an input heat flux equal to 200 Wth.

Fig. 21. Power generated by the TEG for an input heat flux equal to 300 Wth.

temperature on the cold side of the TEG, and the “VO” block, in yellow, is a constant value that represents the battery voltage at the output of the boost converter. The thermal time constant of the TEG model, as well as the sampling time of the algorithms, have been decreased by a factor of 100 so that the simulation time is also decreased by the same factor. This is done to accelerate the simulation time. The regulation period, on the other hand, is not decreased in order to simulate the dynamics of the power converter. Three simulations are run. The first one is performed using conventional FOV, where the input voltage is set to 50% of the instantaneous open-circuit voltage every 10 ms. The second simulation is performed using conventional P&O with a sampling time of 10 ms. The maximum step size is equal to 0.015VOC. The third simulation uses the proposed algorithm in this article in which the samples for P&O are

Fig. 18. Power and beta curves for (a) 100 Wth input power, (b) 200 Wth input power and (c) 300 Wth input power.

factor of 1/63. The voltage reading from these pellets can be used to sense the instantaneous open-circuit voltage of the TEG while working at load without having to disrupt the operation of the power converter. The Simulink model of the monTETM also implements this output and it is fed into the MPPT algorithm block to obtain the reading of the instantaneous open-circuit voltage. Finally, the “Tcold” block, in blue, is a constant that is used to fix the 11

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Table 6 Values of power generation, total energy and ratio between average TEG voltage and instantaneous open-circuit voltage (β) in steady-state conditions obtained from the experimental results. QH = 100 Wth

QH = 200 Wth

QH = 300 Wth

Algorithm

PAVG

β

PAVG

β

PAVG

β

Total Energy

Conventional P&O FOV Proposed Algorithm

1.97 Wel 1.95 Wel 2.01 Wel

0.516 0.499 0.564

7.38 Wel 7.31 Wel 7.48 Wel

0.514 0.499 0.577

13.84 Wel 13.72 Wel 13.90 Wel

0.533 0.500 0.570

84.41 kJ 83.63 kJ 85.37 kJ

changes in the duty-cycle, from 50% to 75%, is shown in Fig. 17. The voltage exhibits the same damped response depicted in Fig. 12(b). The TEG is placed in the test rig described in [30]. The control of the test rig accounts for the losses to ambient and regulates the supply power to the hot block in order to keep constant heat throughout the test. The TEG power is measured using a data-logger that measures the TEG voltage and current with a sampling period of 7 s. The battery is simulated using a power supply set to 28 V and an electronic load connected in parallel with the power supply. A first test is performed to find the real MPP of the TEG for the three levels of input thermal power; that is, 100 Wth, 200 Wth and 300 Wth. Beyond 300 Wth the selected TEG approaches the absolute maximum hot side temperature. The results are shown in Fig. 18, where the TEG power is plotted against the output voltage in the vicinity of the MPP. The maximum power point, the load voltage and the parameter β are presented in Table 5. The power converter implements all three algorithms and the results are presented. In FOV, the algorithm monitors the input open-circuit voltage every 5 s and regulates the TEG voltage to 50% of that value. For the conventional P&O, the input power is sampled every 200 ms and the TEG voltage is increased or decreased, depending on whether the TEG power increases or decreases, by a factor of 0.015 times the value of the sampled open-circuit voltage. If the sampled power is higher than it was in the previous iteration, the step is applied with the same sign. If, on the other hand, the sampled power is lower than it was in the previous iteration, the step is applied inverting the sign. Each algorithm has been tested with an input power of 100 Wth, 200 Wth and 300 Wth. The results are shown in Figs. 19, 20 and 21, respectively. The average power, total energy and the β parameters are shown in Table 6. The proposed algorithm generates more power from the TEG than FOV and the conventional P&O. For QH = 100 Wth, the proposed algorithm generates 3.07% and 2.03% more power than FOV and conventional P&O, respectively. For QH = 200 Wth, the proposed algorithm generates 2.32% and 1.35% more power than FOV and conventional P& O, respectively. Finally, for QH = 300 Wth, the proposed algorithm generates 1.31% and 0.43% more power than FOV and conventional P& O, respectively. It is clear that the proposed algorithm is more effective at lower input thermal power, and the gain over the other two algorithms decreases with input thermal power. The algorithm has also been tested for cases where the input heat flux changes linearly. The ramp profiles are shown in Fig. 22 and Table 7. The results show that the for the case of the fastest ramp, 225 mWth/ s, the “fast” P&O algorithm produces more energy, during the rising ramp, than the other two algorithms. For the slowest ramp, 50 mWth/s, the low MPPT frequency of the proposed algorithm has less effect on the MPP tracking capability and it therefore produces more energy than the other two algorithms. For the 150 mWth/s ramp, “fast” P&O generates more energy than the proposed algorithm. In both cases, where the “fast” P&O algorithm generates more energy than the new proposed one, the extra energy is less than 1%. During the decreasing ramps the proposed MPPT algorithm is capable of generating more power than the other two algorithms. The energy generated by each algorithm during the ramps is shown

Fig. 22. Heat flux input thermal profile used to simulate the input power ramp transients. Table 7 Heat flux parameters used in the ramp test. Parameter

1st ramp

2nd ramp

3rd ramp

QH1 QH2 tUP tDOWN Rate

100 Wth 250 Wth 1000 s 1000 s 150 mWth/s

100 Wth 250 Wth 3000 s 3000 s 50 mWth/s

75 Wth 330 Wth 1000 s 1000 s 225 mWth/s

taken every 500 ms, which is about five times the value of the thermal time constant of the system. The regulation loop monitors the input voltage and regulates to the value of the target voltage every 5 ms. For each simulation three levels of input thermal power are used: 100 Wth, 200 Wth and 300 Wth. The total simulation time is 45 s and the results are shown in Fig. 15. In all cases the proposed algorithm generates more power than the conventional P&O and the FOV algorithms. The average values of generated power as well as total energy are presented in Table 4. The operating voltage of the TEG with respect to the instantaneous open-circuit voltage is also shown in Table 4 using the parameter β, which is defined as VTEG/VOC. From the values of β it is observed that the proposed algorithm operates at a point that is higher than 50% of the instantaneous open circuit voltage and generates more power than the other two algorithms that operate at around 50% of VOC. Fig. 15 also shows the effect of the regulation loop. As the voltage is changed to track the target voltage, the power also changes. 6. Hardware implementation and test results The proposed algorithm will be implemented using a microcontroller PIC16F1788 from Microchip4. This microcontroller provides all the modules required for the implementation of the proposed algorithm and the control of the power converter. The proposed algorithm, described in the flow chart in Fig. 13, has been programmed in the microcontroller. An LCD screen is used to print the input and output voltages and currents as well as the input and output power. The power converter is shown in Fig. 16 and the voltage response to 4

www.microchip.com. 12

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Table 8 Energy generated by all three algorithms during increasing and decreasing ramp changes in input heat flux (simulation results). 1st Ramp – 150 mWth/s

Rising ramp EOUT (J) Falling ramp EOUT (J)

2nd Ramp – 50 mWth/s

3rd Ramp – 225 mWth/s

New

“Fast” P&O

FOV

New

“Fast” P&O

FOV

New

“Fast” P&O

FOV

4957.40 7089.40

4960.88 7004.45

4918.33 6994.12

16811.67 19294.24

16667.37 19103.44

16516.96 18979.26

6034.80 9627.11

6079.25 9626.78

5984.15 9561.35

The data in bold shows which algorithm, of all three, generates more energy for each input heat flux profile.

in Table 8. All three algorithms have been tested against step changes in the input heat flux. The results are shown in Fig. 24. Fig. 24(a) shows the response of all three algorithms to the step in input heat flux from 75 Wth to 250 Wth whereas Fig. 24(b) shows the response of all three algorithms to the step in input heat flux from 250 Wth to 150 Wth. The new algorithm and FOV perform in a very similar manner, whereas the conventional P&O algorithm losses its tracking capability during these transients. The new algorithm is capable of detecting the change in the

input heat flux and react, in a fast manner, to the changes. The main limitation of the new algorithm is that it will only detect the transient at t = 2τTEG after the previous perturbation of the operating point, otherwise it could interpret a normal perturbation as a change in the input heat flux. The abrupt changes and spikes shown in Figs. 23 and 24 correspond to the transient response of the TEG, shown in Fig. 4(a) and (b), Fig. 6(a) and (b) and extensively explained in [30].

Fig. 23. Power generated by the TEG (all three algorithms) for the ramp profiles specified in Table 7 and Fig. 22.

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Fig. 24. Zoomed in view of the power generation of the TEG to the step transients performed in the test of the TEG (a) from 75 Wth to 250 Wth and (b) from 250 Wth to 150 Wth.

7. Conclusions

algorithm. For decreasing linear changes of input heat flux, the proposed algorithm performs better than the other two for slow and fast changes. Based on the experimental results presented in this article, for a TEG system with a thermoelectric area of 10 m2 operating under 300 Wth, a power processing unit operating the proposed algorithm in this article would produce 375 Wel more than a system using conventional P&O, and 1.13 kWel more than a system using FOV.

In real TEG systems, in which temperatures are not maintained constant, the effective thermal conductance of a TEG increases with the output current thus decreasing the temperature gradient across the TEG. These TEG systems are considered to operate under constant heat flux as opposed to constant temperature difference. A close examination of the power curve of a TEG operating under these conditions shows that the MPP is not located at half of the open-circuit voltage, which is the MPP when the TEG operates under constant temperature difference. Defining the parameter β as the ratio of the TEG voltage to the opencircuit voltage, VTEG/VOC, the MPP of a TEG that operates under constant heat flux is found at β > 0.5. A Simulink model of the TEG that considers the thermal time constant of the TEG has been proposed to design TEG systems operating under constant heat flux. The thermal time constant of the TEG can be calculated from the step response, which simplifies the calculations required to obtain an accurate model. This TEG model is used in conjunction with the average signal model of the power converter to optimize the MPPT control system. The simulation results show that only when the thermal time constant of the TEG system is taken into consideration, the control system is capable of operating the TEG at the “real” MPP. Building on these simulations, a new MPPT algorithm is proposed in this article. It is based on the P&O method but it differs in the sampling speed of the electrical parameters. The proposed algorithm waits for a time equal to four thermal time constants before the next value of power is sampled, as opposed to the faster sampling speed of the conventional P&O that is typically in the order of 100 ms to few seconds. This algorithm has been optimized compared to the one presented in [30] in that it has been optimized for cases when there are changes in the input heat flux. It also implements an adaptive step that increases the tracking speed of the algorithm. The algorithm performs a mechanism to detect changes in the input heat flux and reacts without having to wait for a time equal to four thermal time constants. A TEG has been precisely characterized in the vicinity of its MPP for input thermal powers of 100, 200 and 300 Wth. The MPP is consistently found at β > 0.5. A dc-dc power converter has been operated using three different algorithms, FOV, conventional P&O and a new proposed algorithm. The proposed algorithm performs better than the other two algorithms for constant values of input heat flux. For step changes in the input heat flux, the proposed algorithm performs as well as FOV, so long as the step change occurs at a time equal to 2τTEG after the previous perturbation of the operating point. For increasing linear changes in the input heat flux, the proposed algorithm performs better with slow changes. However, as the speed of change of input heat flux increases, the “fast” P&O algorithm becomes more effective than the proposed

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