Uncertainty Analysis of Performance Parameters of a Hybrid Thermoelectric Generator Based on Sobol Sequence Sampling

Abstract

Hybrid thermoelectric generators (HTEGs) play a pivotal role in sustainable energy conversion by harnessing waste heat through the Seebeck effect, contributing to global efforts in energy efficiency and environmental sustainability. In practical sustainable energy systems, HTEG output performance is significantly influenced by uncertainties in the operational parameters (such as temperature differences and load resistance), material properties (including Seebeck coefficient and resistance), and structural configurations (like the number of series/parallel thermoelectric components), which impact both efficiency and system stability. This study employs the Sobol-sequence-sampling method to characterize these parameter uncertainties, analyzing their effects on HTEG output power and conversion efficiency using mean values and standard deviations as evaluation metrics. The results show that higher temperature differences enhance output performance but reduce stability, a larger load resistance decreases performance while improving stability, thermoelectric materials with high Seebeck coefficients and low resistance boost efficiency at the expense of stability, increasing series-connected components elevates performance but reduces stability, parallel configurations enhance power output yet decrease efficiency and stability, and greater contact thermal resistances diminish performance while enhancing system robustness. This research provides theoretical guidance for optimizing HTEGs in sustainable energy applications, enabling the development of more reliable, efficient, and eco-friendly thermoelectric systems that balance performance with environmental resilience for long-term sustainable operation.

1. Introduction

In an era defined by escalating global energy demands and urgent environmental imperatives, the transition toward sustainable energy systems has emerged as a paramount challenge for scientific research and industrial practice [1,2,3]. The pressing need to mitigate carbon emissions, reduce reliance on fossil fuels, and enhance energy utilization efficiency has spurred intensive exploration of renewable energy technologies [4,5,6]. In recent years, thermoelectric power generation has been widely used in the fields of aerospace and test instruments [7,8]. Although thermoelectric power generation has unique advantages and considerable potential, the low efficiency of thermoelectric conversion seriously restricts its application. In addition, the thermal stress caused by the difference in thermal expansion coefficients among the components will seriously affect its service life and output performance [9,10,11]. To improve the output response of power-generation systems, researchers, based on the principle of non-equilibrium thermodynamics, established corresponding models and conducted in-depth studies and optimization of the thermoelectric material and their structural parameters [12,13,14,15,16]. These studies significantly improved the performance of thermoelectric generators. However, the output performance of a thermoelectric power-generation system still needs further improvement.

Among them, Jiang et al. [12] improved the electron mobility of PBS thermoelectric materials by optimizing their thermal conductivity and effective mass, achieving an efficiency of up to 8.0% at a temperature difference of 565K. Luo et al. [13] improved the thermal management performance of a solar thermoelectric generator by applying a paraffin/expanded graphite phase-change material with high thermal conductivity and rapid thermal response. Zhu et al. [14] realized the accurate geometric design and optimization of a thermoelectric generator using an artificial neural network algorithm. Alaa et al. [16] proposed a method based on dimensional analysis to obtain the optimal load resistance and geometric ratio of a thermoelectric power-generation system, and the effectiveness of the method was verified by experiment.

However, thermoelectric generators have exhibited several disadvantages, such as low power-generation efficiency and poor output stability [17,18,19,20]. Their complex structure encompasses numerous parameters and uncertainties [21,22]. Indeed, fluctuations in the operating parameters—such as variations in the temperature difference between the cold and heat sources [23] due to the actual working environment—lead to changes in load resistance [16] corresponding to load requirements. Furthermore, random uncertainties may exist in the material and structural parameters of HTEG systems [24]. For instance, during the design and fabrication phase, the material and structural parameters of the thermoelectric elements are constrained by variations in the manufacturing processes or power-generation devices, reading to the fluctuations within a certain range [25]. The random uncertainties associated with these parameters inevitably impact the output performance of thermoelectric generators and adversely affect their reliability and stability. Therefore, it is crucial to fully consider parameter uncertainty in the design and operation of thermoelectric power-generation systems [26].

Currently, common methods for uncertainty analysis include probability analysis method and fuzzy theory analysis method. Among these, probability analysis serves as a quantitative analysis method, whereas fuzzy theory represents a qualitative analysis method [27,28]. The probability analysis method quantifies the uncertainty of variables using probability distributions that accurately depict the random uncertainties encountered in industrial design or physical experiments. Conversely, the application of fuzzy set theory is less prevalent due to the current immaturity of the research in this area [29]. The Monte Carlo sampling method, which estimates sample parameters by drawing from a probability model to generate pseudo-random samples, is the most commonly used method in probability analysis [30,31] due to its simplicity and ease of implementation. However, its suffers from low sampling efficiency and poor convergence, and it often requires a large number of samples [32]. In contrast, the quasi-Monte Carlo method does not employ uniformly distributed pseudo-random numbers for sampling as the conventional Monte Carlo method does; instead, it uses a deterministic low-deviation sequence to ensure the uniformity and certainty, resulting in superior convergence speed and considerably improving the sampling efficiency [33,34]. Some well-known low-deviation sequences include the Halton sequence [35,36], Niederreiter sequence [37], and Sobol sequence [38]. The Sobol sequence is widely recognized for its straightforward construction method, which realizes acceptable sequence uniformity. At present, the Sobol-sequence-sampling method has been applied in various engineering practices [39,40,41,42]. For example, Navid et al. [39] used the Nelder–Mead algorithm to optimize a diesel engine’s performance using either the Sobol sequence or Latin hypercube sampling method; the results indicated that Sobol sequence sampling was more efficient and exhibited faster convergence. Liu et al. [40] established a seismic connectivity reliability calculation model using Sobol sequence sampling for seismic risk analysis. Pan et al. [42] applied Sobol sequence sampling to study the impact of wind farm and photovoltaic power station integration on the transient stability of a power system.

Multiple thermoelectric components are frequently interconnected in series and parallel to create a hybrid thermoelectric generator (HTEG), which enhances output power. The performance of an HTEG is affected by some elements, including operational, material, and structural parameters. These parameters have a certain influence on the stability and reliability of the output power and conversion efficiency. Previous research primarily focused on deterministic models. However, research on the effects of parameter fluctuations on HTEG stability and reliability is notably lacking. In this study, Sobol sequence sampling is used to characterize the random uncertainty of HTEG parameters, and the impact of changes in parameter means on system output response is studied. Using mean values and standard deviations as evaluation metrics, both the output performance and stability are quantified, respectively. This uncertainty analysis improves the theoretical system underlying thermoelectric power-generation technology by identifying potential avenues for optimizing HTEG performance while providing a theoretical and experimental foundation for the development of new high-performance thermoelectric power-generation systems.

2. General Structural Analysis Model of an HTEG

2.1. HTEG Model’s Structure

To enhance the output power in practical applications, multiple thermoelectric components are usually connected in series, parallel, or a combination of both, forming an HTEG. In this study, an HTEG consisting of multiple thermoelectric components connected in series–parallel hybrid connections is taken as the research object. An HTEG typically includes the following sections: heat source, thermoelectric module (TE) connected in a series–parallel hybrid configuration, and cold source. The structural configuration and thermal network of the HTEG considered in this study are shown in Figure 1, where the heat and cold sources of the TEG are connected by $N_T$ TEs of identical specifications. Among these, $N_S$ TEs are arranged in series within each row, while $N_P$ rows are connected in parallel, as shown in Figure 2.

2.2. Theoretical Modeling

This section aims to construct a theoretical model for the output power (P) and conversion efficiency (η) of the HTEG. Through thermodynamic principles and circuit theory, the quantitative relationships between key parameters (such as temperature difference, load resistance, and material characteristics) and output performance are derived, providing a mathematical basis for the subsequent uncertainty analysis.

As shown in Figure 2, the expressions of $N_T$ and $R_T$ are given as follows:

$$N_T=N_S\times N_P$$

(1)

$$R_T={\displaystyle \frac{N_S\left(R+Re\right)}{N_P}}$$

(2)

Supposing that the specifications and types of TEs in the HTEG model are identical, the voltage generated by the loop is as follows:

$$U_{P_1}=U_{P_2}=U_{P_3}=\dots \dots =U_{P_n}=N_S\alpha \Delta T_G$$

(3)

When the load resistance is denoted as $R_L$, $U_0$ and $I_0$ can be written as follows:

$$U_0=N_S\alpha \Delta T_G{\displaystyle \frac{R_L}{R_T+R_L}}$$

(4)

$$I_0={\displaystyle \frac{N_S\alpha \Delta T_G}{R_T+R_L}}.$$

(5)

According to the current distribution law of parallel circuits, the total current, I₀, is evenly distributed among N_p branches, so the current in each branch I = I₀/N_p. Substituting the expression for I₀ (Equation (5)) yields Equation (6).

Then, the current’s value, $I$, for each branch can be expressed as follows:

$$I={\displaystyle \frac{I_0}{N_P}}={\displaystyle \frac{N_S\alpha \Delta T_G}{N_S\left(R+Re\right)+N_P{R}_L}}$$

(6)

Based on the principles of non-equilibrium thermodynamics and Newton’s cooling law, $Q_H$ and $Q_C$ can be separately defined as follows:

$$Q_H=N_T\left(\alpha T_{H}I+{\displaystyle \frac{\Delta T_G}{K_G}}-{\displaystyle \frac{1}{2}}{I}^2\left(R+Re\right)\right)={\displaystyle \frac{T_1-T_H}{K_H}}$$

(7)

$$Q_C=N_T\left(\alpha T_{C}I+{\displaystyle \frac{\Delta T_G}{K_G}}+{\displaystyle \frac{1}{2}}{I}^2\left(R+Re\right)\right)={\displaystyle \frac{T_C-T_0}{K_C}}$$

(8)

where $Q_H$ and $Q_C$ represent heat absorption at the hot end and heat release at the cold end, respectively. Based on non-equilibrium thermodynamics, the heat generated by the Seebeck effect ($\alpha T_{H}I$), heat conduction (${\displaystyle \frac{\Delta T_G}{K_G}}$), and Joule heating (${\displaystyle \frac{1}{2}}{I}^2\left(R+Re\right)$) must be considered. The total heat expression is obtained by summing these three terms.

Equations (6)–(8) are simplified in accordance with Reference [43], resulting in the following expressions.

Equations (9) and (10) are simplified from Equations (7) and (8). Substitute I from Equation (6) into the $Q_H$ expression, expand and combine like terms, and simplify using $N_T=N_S\times N_P$.

$$Q_H={\displaystyle \frac{{\left(N_S\alpha \right)}^2{T}_1\Delta T_G}{\frac{N_S\left(R+Re\right)}{N_P}+R_L}}+{\displaystyle \frac{N_T\Delta T_G}{K_G}}$$

(9)

$$Q_C={\displaystyle \frac{{\left(N_S\alpha \right)}^2{T}_0\Delta T_G}{\frac{N_S\left(R+Re\right)}{N_P}+R_L}}+{\displaystyle \frac{N_T\Delta T_G}{K_G}}$$

(10)

According to the heat transfer diagram shown in Figure 1, the relationship between $\Delta T$ and $\Delta T_G$ is expressed in Equation (11), where $\Delta T=T_1-T_0$.

$$\Delta T=\Delta T_G+K_C{Q}_C+K_H{Q}_H.$$

(11)

Equations (9)–(11) can then be derived.

$$\Delta T_G={\displaystyle \frac{\Delta T}{1+\frac{N_T(K_C+K_H)}{K_G}+\frac{{\left(N_S\alpha \right)}^2\left(K_H{T}_1+K_C{T}_0\right)}{\frac{N_S\left(R+Re\right)}{N_P}+R_L}}}$$

(12)

Subsequently, Equation (13) is derived from the output power definition P = U₀I₀. Substituting the expressions for U₀ from Equation (4), I₀ from Equation (5), and $\Delta T_G$ from Equation (12) into the equation, and simplifying, we obtain the final quantitative relationship between P and the various parameters. The output power, $P,$ is given by the following:

$$P={I_0}^2{R}_L={\left({\displaystyle \frac{N_S\alpha \Delta T}{R_T+R_L}}\right)}^2\times {\displaystyle \frac{R_L}{{\left(1+\frac{N_T(K_C+K_H)}{K_G}+\frac{{\left(N_S\alpha \right)}^2\left(K_H{T}_1+K_C{T}_0\right)}{\frac{N_S\left(R+Re\right)}{N_P}+R_L}\right)}^2}}.$$

(13)

Finally, the conversion efficiency, $\eta ,$ of thermoelectric generator is defined as follows [44]:

$$\eta ={\displaystyle \frac{P}{Q_H}}$$

(14)

3. Random Uncertainty Analysis of the HTEG

3.1. Uncertainty Description of HTEG Parameters

The uncertainty parameters of the HTEG model are primarily categorized into three types. The first type consists of the working parameters, such as the heat source temperature, $T_1$; cold source temperature, $T_0$; and and load resistance, $R_L$. The second type encompasses the material parameters of the thermoelectric components, such as the Seebeck coefficient, $\alpha ,$ and the resistance values of the thermoelectric components, $R$. The third type pertains to the structural arguments, which include the numbers of series $N_S$ and parallel $N_P$ thermoelectric elements in the HTEG, as well as the contact thermal resistance between the cold, $K_C,$ and heat, $K_H$, sources and the TEs. It is worth noting that the working, material, and structural parameters of the HTEG are all random. Based on engineering experience and existing relevant data, the fundamental values of the various parameters are obtained, with specific details shown in

Table 1. Distribution of the HTEG parameters.

Parameter Number	Parameter Name	Unit	Mean Value Interval	Typical Mean Value	Standard Deviation
$X1$	$T_1$	K	[380, 420]	400	8
$X2$	$T_0$	K	[300, 346]	323	6.46
$X3$	$R_L$	Ω	[8, 24]	16	0.32
$X4$	$\alpha$	V/K	[0.05291, 0.05403]	0.05347	1.0694 × 10−3
$X5$	$R$	Ω	[1.973, 2.219]	2.096	0.04192
$X6$	$N_S$	-	[5, 9]	7	0.14
$X7$	$N_P$	-	[5, 9]	7	0.14
$X8$	$K_H$	K/W	[0.008, 0.012]	0.01	2 × 10−4
$X9$	$K_C$	K/W	[0.0085, 0.0115]	0.01	2 × 10−4

[24].

3.2. Sobol-Sequence-Sampling Method

As a quasi-Monte Carlo method, the Sobol sequence can cover parameter distributions more efficiently than random sampling (such as MC) in high-dimensional parameter spaces due to its deterministic, low-bias characteristics, thereby reducing the sample size requirements for multi-parameter systems in this study.

Sobol sequence sampling is focused on generating uniform distributions in a probability space. In addition, when the sampling number reaches an integer power of 2, there is only one sample point per unit interval; in other words, compared to simple random sampling, Sobol sequence can generate high-quality distributed samples with uniform distributions without pre-determining the sampling number or storing samples, and it can generate unlimited samples as needed. The construction principles of the Sobol sequence are described in this section [45].

First, the direction number, $v_i$, $v_i={\displaystyle \frac{t_i}{2^i}}$ must be established, where $t_i$ represents a positive odd number and $t_i<2^i$. A primitive polynomial is then selected to obtain $v_i$ as follows:

$$f\left(x\right)=x^k+G_1{x}^{k-1}+\dots \dots +G_{k-1}x+G_k$$

(15)

It is used to generate low-bias direction numbers, which are key to ensuring sequence uniformity, in order to clarify its correlation with the Sobol sequence.

For $i>k$, the following equation is presented:

$$v_i=G_1{v}_{i-1}\oplus G_2{v}_{i-2}\oplus \dots \dots \oplus G_{k-1}{v}_{i-k+1}\oplus G_k{v}_{i-k}\oplus \left[v_{i-k}/2^k\right]$$

(16)

Similarly, for $t_i$, the equivalent recursive equation is as follows:

$$t_i=2G_1{t}_{i-1}\oplus 2^2{G}_2{t}_{i-2}\oplus \dots \dots \oplus 2^{k-1}{G}_{k-1}{t}_{i-k+1}\oplus 2^k{G}_k{t}_{i-k}\oplus t_{i-k}$$

(17)

where $\oplus$ represents a binary bitwise exclusive OR operation, such as 1 $\oplus$ 0 = 0 $\oplus$ 1 = 1 and 1 $\oplus$ 1 = 0 $\oplus$ 0 = 0. By providing the initial values of $t_1$, $t_2$, and $t_3$, along with a cubic primitive polynomial, different values of $t_i$ can be computed based on Equation (17), from which the corresponding direction number, $v_i,$ can be derived. Finally, the sequential sampling sequence of Sobol is as follows:

$$x^{i+1}=x^i\oplus v_i.$$

(18)

In this study, the steps for implementing Sobol sequence sampling of the HTEG parameters are defined as follows:

• Divide the mean value of each HTEG parameter into $M$ non-repetitive sub-intervals;
• Take a sample in each subinterval and define an $S$ matrix to store these samples;

  $$S=\left[\begin{array}{cccc}{S}_{11}& S_{12}& \cdots & S_{1M}\\ S_{21}& S_{22}& \cdots & S_{2M}\\ ⋮& ⋮& \ddots & ⋮\\ S_{N1}& S_{N1}& \cdots & S_{NM}\end{array}\right].$$

  (19)
• Rearrange each row of $S$ to simulate a random combination of the various HTEG parameters.

$$L=\left[\begin{array}{cccc}{S}_{X_{11}}& S_{X_{12}}& \cdots & S_{X_{1M}}\\ S_{X21}& S_{X_{22}}& \cdots & S_{X_{2M}}\\ ⋮& ⋮& \ddots & ⋮\\ S_{X_{N1}}& S_{X_{N2}}& \cdots & S_{X_{NM}}\end{array}\right]=\left[\begin{array}{cccc}{L}_{11}& L_{12}& \cdots & L_{1M}\\ L_{21}& L_{22}& \cdots & L_{2M}\\ ⋮& ⋮& \ddots & ⋮\\ L_{N1}& L_{N1}& \cdots & L_{NM}\end{array}\right]$$

(20)

where the subscripts $X_{11},X_{12},X_{13},\cdots ,X_{1M}$ of the matrix elements denote a random arrangement of the row vector components of the matrix, $L$ is the Sobol sequence sampling matrix, and $N$ represents the sample dimension.

3.3. Analysis Process

This study employed an uncertainty analysis to conduct a performance evaluation of the subject HTEG. Sobol sequence sampling is used to simulate the randomness of a system parameter introduced during the production and operation phases of the HTEG. Then, the average value and standard deviation are used as performance evaluation indicators to quantify the impact of these parameters on the output response. Figure 3 depicts the analytical process in detail.

In the analytical process, the HTEG uncertainty analysis process is based on Sobol sequence sampling. First, a mathematical model of the HTEG is established by combining the principles of thermoelectric conversion with a thermal network model, clearly defining the mapping relationships between the output power (P) and conversion efficiency (η) and each parameter. Second, based on the mean intervals and distribution characteristics of the parameters in Table 1, the range of each parameter is divided into several equally probable sub-intervals. Uniformly distributed parameter combination samples are generated using the Sobol sequence to ensure the representativeness of the sampling. Subsequently, these samples are substituted into the model to calculate the corresponding P and η values, yielding the output response dataset. Finally, statistical analysis is performed to calculate the mean (reflecting performance level) and standard deviation (reflecting stability) of the output response, thereby quantifying the impact of parameter uncertainties on the HTEG’s performance. This process employs deterministic, low-bias sequence sampling, ensuring computational efficiency while accurately capturing the correlation between parameter fluctuations and system output.

4. Results and Discussion

Based on the mathematical model of the HTEG and the Sobol-sequence-sampling method, the changes in the mean value and standard deviation of the output response of the system are obtained according to the distribution of the variable parameters in Table 1. Therefore, the random uncertainties of each model parameter are first characterized, and then the impact of these parameters’ randomness on the HTEG is studied.

4.1. Fluctuation Characterization of the Output Response Based on the HTEG Parameter

To characterize the uncertainty of the parameter, the Sobol-sequence-sampling method with N = 1000 (preliminary tests indicate that when N ≥ 800, the output indicator fluctuation is <5%, so N = 1000 was selected to maintain a safety margin) is adopted, and random parameter sampling is performed using the HTEG model. The relative variation latitude of the output response is used as the characterization index, $\beta$, and the impact of parameter fluctuations on the system’s output response stability ($P$ and $\eta$) is obtained as follows:

$${\beta }_{xi}^Y={\displaystyle \frac{Y_{xi}^{max}-Y_{xi}^{min}}{Y_{\left(u_{x1},u_{x2},u_{x3},\dots u_{x9}\right)}}}$$

(21)

where ${\beta }_{xi}^Y$ is the characterization index for the stability of the output response $Y_{xi}$ corresponding to the $i$-th parameter, where $i=\left(\mathrm{1,2},3,...9\right)$; $Y_{xi}^{max}$ represents the maximum value of the output response obtained from random sampling of the $i$-th parameter, $Y=\left(P,\eta \right)$; $Y_{xi}^{min}$ denotes the minimum value of the output response corresponding to the random sampling of the $i$-th parameter; and $Y_{\left(u_{x1},u_{x2},u_{x3},\dots u_{x9}\right)}$ indicates the output response associated with the typical mean values specified in Table 1.

Table 2. Output response fluctuation characteristics based on the HTEG system’s parameters.

Parameter Number	Parameter Name	${\mathit{\beta }}_{\mathit{x}\mathit{i}}^{\mathit{P}}$	${\mathit{\beta }}_{\mathit{x}\mathit{i}}^{\mathit{\eta }}$
$X1$	$T_1$	1.4859	0.6267
$X2$	$T_0$	1.1322	0.5446
$X3$	$R_L$	0.0859	0.0374
$X4$	$\alpha$	0.2531	0.1236
$X5$	$R$	0.0279	0.0196
$X6$	$N_S$	0.1803	0.0281
$X7$	$N_P$	0.0142	0.0553
$X8$	$K_H$	0.0165	0.0071
$X9$	$K_C$	0.0167	0.0069

shows the influence of random parameter uncertainty on the fluctuations in the HTEG’s output power, ${\beta }_{xi}^P$, and conversion efficiency, ${\beta }_{xi}^{\eta }$. The table indicates that the fluctuations in cold and heat source temperatures, $T_0$ and $T_1$, respectively, exerted the greatest influence on the stability of the system’s output response ($P$ and $\eta$). Conversely, fluctuations in the parallel thermoelectric components, $N_P$, had minimal impact on the output power stability. However, fluctuations in the contact thermal resistance, $K_C$, which occurs between the TE and cold source, also exhibited the least impact on the conversion efficiency stability. Thus, the data presented in the table indicate that fluctuations in the system variables have varying impacts on the stability and accuracy of the output power and conversion efficiency. These impacts are elaborated upon alongside the corresponding output performance (in terms of power generation and capacity utilization) for the working, material, and structural parameters presented in Section 4.2 and Section 4.3, respectively.

The significance of each parameter’s β value was confirmed by t-test (p < 0.05), indicating that its impact on output fluctuations is statistically significant.

4.2. Influence of HTEG Working Parameters on the Output Response

The working parameters of the HTEG include the heat source temperature, $T_1$; cold source temperature, $T_0$; and load resistance, $R_L$. Figure 4, Figure 5 and Figure 6 show the influence of changes in the mean values of each parameter on the average values and standard deviations of the HTEG’s output power, $P,$ and conversion efficiency, $\eta$.

As shown in Figure 4, the average value of P exhibits a “concave growth” pattern as T1 increases. Although the increase in the temperature difference enhances the Seebeck emf, the accelerated growth rate of the heat loss (proportional to T1²) leads to diminishing marginal returns; the increase in the standard deviation indicates that parameter fluctuations (such as small changes in T1) have a more significant impact on the output performance at high temperatures, consistent with the thermodynamic principle that high-temperature systems are more sensitive to disturbances.

Similarly, Figure 5 demonstrate that as $T_0$ increased, the value of $P$ exhibited a slightly concave decrease, whereas both the average value and standard deviation of $\eta$ decreased linearly.

Furthermore, Figure 6 shows that as $R_L$ increased, both the mean value and standard deviation of $P$, again, showed mild concave decreases, along with a corresponding decline in the average value of $\eta$. However, the standard deviation of $\eta$ was initially amplified before diminishing with an increase in $R_L$, following a parabolic trend that reaches a maximum approximately 16 Ω with a peak standard deviation of $\eta$ = 0.01258. At this point, the load resistance was optimally matched to the total system resistance ($R_L$ ≈ 16 Ω), resulting in the highest energy-conversion efficiency. Fluctuations in parameters had a magnified effect on η, which is the “critical resistance value” in the system’s design.

In summary, a higher $T_1$ and lower $T_0$ were more favorable for achieving greater average $P$ and $\eta$; however, the data fluctuations, as indicated by the standard deviation, also increased. This implies that while a significant temperature difference can improve the HTEG’s output performance, it may compromise the stability of that output. Conversely, larger $R_L$ will reduce the output performance of the HTEG, but it is more beneficial for its output stability.

4.3. Influence of the HTEG’s Material Parameters on the Output Response

The material parameters of the HTEG’s components include the Seebeck coefficient, $\alpha$, and resistance, $R$. Figure 7 and Figure 8 illustrate the influence of changes in the mean values of each parameter on the distribution of $P$ and $\eta$ for the HTEG.

Figure 7 shows that as $\alpha$ increased, the mean value of P rose linearly (the Seebeck effect was enhanced), but the standard deviation of P increased synchronously (small fluctuations in $\alpha$ amplify the effect on the electromotive force; meanwhile, the standard deviation of η decreases because Q_H increases synchronously with $\alpha$, diluting the effect of fluctuations on efficiency, reflecting the trade-off between performance and stability).

Similarly, Figure 8 shows that as $R$ increased, the average values and standard deviations of both $P$ and $\eta$ decreased linearly. This indicates that although high-resistance materials reduce performance, the impact of resistance fluctuations on the current is weakened, resulting in a slight improvement in the system’s stability. This trend provides a basis for material selection: a balance must be struck between “low resistance (high power)”and “high stability”.

Therefore, a high $\alpha$ is beneficial for enhancing the output performance of the HTEG; however, it may compromise its stability. As $R$ increases, both the output performance and stability of HTEG become increasingly unfavorable.

The phenomenon of efficiency degradation caused by parallel structures can be explained from a physical mechanism perspective as follows: although parallel configurations reduce total resistance (Equation (2)) by increasing the number of current paths, thereby enhancing the output power, the current distribution across multiple parallel branches makes it difficult to achieve complete uniformity, leading to exacerbated local Joule heat losses. According to the principle of energy conservation, the proportion of ineffective Joule heat in the heat released at the cold end (($Q_C$), Equation (10)) increases, leading to a decrease in the proportion of effective thermal energy absorbed from the heat source ($Q_{\mathrm{H}}$) that is truly converted into electrical energy. Additionally, the temperature field consistency of the parallel branches is poor, disrupting the stable temperature gradient required for the Seebeck effect, further reducing the thermoelectric conversion efficiency—this is directly related to the inherent characteristic of thermoelectric materials; that is, “efficient conversion depends on uniform heat flow and temperature fields”.

5. Conclusions

Through uncertainty analysis based on Sobol sequence sampling, this study investigated the effects of parameter uncertainties in the HTEG model on the output power and conversion efficiency, as well as their stability (i.e., reliability). The results showed that higher temperature differences were beneficial for enhancing the output performance of the HTEG but were not conducive to its stability. When operating the system, a larger load resistance would lead to a decline in the HTEG’s output performance, but it would contribute to achieving a more stable output. Using thermoelectric module materials with a higher Seebeck coefficient and lower resistance could enhance the output power and conversion efficiency of the HTEG but would, to some extent, reduce its output stability. If the number of TE modules in the series increased, the output power and conversion efficiency of the HTEG would be higher, while the number of TE modules in parallel would increase, and the output power of the HTEG would be enhanced but the conversion efficiency would decline, both of which were not conducive to maintaining a stable output by the HTEG. Finally, a larger thermal resistance between the cold and hot sources would reduce the overall output performance of the system but enhance its stability.

Therefore, in the industrial design stage, TE material with a high Seebeck coefficient and low resistance should be prioritized, and the contact thermal resistance between the HTEG and the cold and hot sources should be rationally designed, as well as the number of TE modules in series and parallel, while controlling the load resistance connected to the HTEG to ensure that it operates within the optimal range for providing efficient and stable output. In the engineering application field, the temperature difference between the cold and hot sources should be increased to obtain higher and more robust average output power and conversion efficiency under actual working conditions. This research extends the evaluation standards of the HTEG, points out the direction for performance optimization considering parameter randomness, and provides an important basis for developing heat electric power-generation systems with excellent performance.