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Dynamic reconfiguration for TEG systems under heterogeneous temperature distribution via adaptive coordinated seeker

Abstract

A thermoelectric generation (TEG) system has the weakness of relatively low thermoelectric conversion efficiency caused by heterogeneous temperature distribution (HgTD). Dynamic reconfiguration is an effective technique to improve its overall energy efficiency under HgTD. Nevertheless, numerous combinations of electrical switches make dynamic reconfiguration a complex combinatorial optimization problem. This paper aims to design a novel adaptive coordinated seeker (ACS) based on an optimal configuration strategy for large-scale TEG systems with series–parallel connected modules under HgTDs. To properly balance global exploration and local exploitation, ACS is based on ‘divide-and-conquer’ parallel computing, which synthetically coordinates the local searching capability of tabu search (TS) and the global searching capability of a pelican optimization algorithm (POA) during iterations. In addition, an equivalent re-optimization strategy for a reconfiguration solution obtained by meta-heuristic algorithms (MhAs) is proposed to reduce redundant switching actions caused by the randomness of MhAs. Two case studies are carried out to assess the feasibility and superiority of ACS in comparison with the artificial bee colony algorithm, ant colony optimization, genetic algorithm, particle swarm optimization, simulated annealing algorithm, TS, and POA. Simulation results indicate that ACS can realize fast and stable dynamic reconfiguration of a TEG system under HgTDs. In addition, RTLAB platform-based hardware-in-the-loop experiments are carried out to further validate the hardware implementation feasibility.

Introduction

Coinciding with rapid economic growth, over the past few decades, global demand for electrical energy has increased dramatically [1,2,3]. However, traditional fossil fuels currently acting as the main source of electricity generation, e.g., oil, coal, natural gas [4], face ever-increasing depletion while causing severe environmental deterioration [5]. Hence, it is crucial to reform the current energy structure and utilization patterns. Renewable energy is envisaged as a revolutionary and promising candidate to ease the continued tension and pressure in the global energy crisis [6].

In addition to wind, hydro, solar, and hydrogen [7, 8], thermoelectric generation (TEG) also possesses remarkable application potential and value [9] because of its prominent advantages of simple structure, sturdiness, being noiseless, long service life, etc. TEG has been applied in some industrial production, such as in automobile engines [10], geothermal energy exploitation [11], natural gas boilers [12], solar thermoelectric cooling systems [13], combined heat and power generation [14], wearable devices [15], energy-autonomous sensors [16], etc.

However, the low energy conversion efficiency of TEG systems given unpredictable and ineluctable heterogeneous temperature distribution (HgTD) is the main obstacle that limits its larger scale and wider range of application [17]. To solve this issue, efficient thermoelectric materials and maximum power point tracking (MPPT) technology are considered as two promising solutions [18]. Unfortunately, the former is hard for large-scale applications because of its high cost and long research process [19].

In contrast, MPPT is a more feasible and practical solution. Thus far, a large variety of MPPT approaches have been exploited and employed to capture the global maximum power point (GMPP) of TEG systems under HgTD. Perturb and observe (P&O) [20] and incremental conductance (INC) [21] are two of the most commonly used MPPT strategies because of their simple operational mechanisms. Nevertheless, their performance tends to degrade when multiple local maximum power points (MLMPPs) [22] appear caused by HgTD. Thus, a series of meta-heuristic algorithm (MhAs)-based methods have been proposed to find the GMPP based on their superior global searching ability, e.g., the fast atom search optimizer (FASO) [4], interacted collective intelligence (ICI) [17], adaptive compass search [23], equilibrium optimizer (EO) [22], arithmetic optimization algorithm (AOA) [24], etc. In addition, a novel adaptive rapid neural optimization approach is designed in [25] to realize an available approximation of duty cycle and output power curves for centralized TEG systems through a generalized regression neural network. This significantly reduces power oscillations and energy losses.

It is noteworthy that the aforementioned MPPT methods are all accomplished by controlling the centralized TEG converter to match the impedances of the source and load [26]. This cannot improve the potential GMPP of TEG systems. To fundamentally address this issue, TEG system reconfiguration can be an alternative and effective tool. It is inspired by photovoltaic (PV) system reconfiguration in partial shade conditions (PSC). Specifically, reconfiguration strategies for TEG systems can be divided into two types that are similar to PV system reconfiguration [27], i.e., static and dynamic. Static reconfiguration disperses the temperature difference area by changing the physical position of TEG modules rather than changing electrical connection, while dynamic reconfiguration is capable of dynamically altering the electrical interconnection through switching matrices against various temperature variations. This method generally possesses higher flexibility and lower complexity.

To alleviate adverse effects (e.g., module mismatch, difficulty in tracking the maximum power point, extreme power loss, hot spots, etc. [28]) of partial shading caused by meteorological or geographic factors on PV systems, numerous intensive studies have been done to develop sophisticated technologies on the reconfiguration of PV systems. For instance, a dragonfly algorithm (DA)-based reconfiguration technique for PV systems is proposed in [29]. This is capable of achieving up to 22% output power improvement over conventional methods (e.g., electrical reconfiguration, Sudoku, etc.) under (3 × 3) and (9 × 9) arrays. Distinct from MhAs-based techniques, reconfiguration mechanisms enlightened by the magic square puzzle are conceived in [30, 31] to substantially reduce mismatch loss and enhance the GMPP of PV systems. These can offer an appropriate shade dispersion via relocation of PV modules. In addition, reference [32] applies reinforcement learning to the optimal reconfiguration of PV systems under PSC to provide frequency support for the power system.

However, there are not many current studies on the TEG system reconfiguration. In [10], a dynamic reconfiguration algorithm is proposed for a TEG array installed on vehicle radiators, one which can change interconnections (series or parallel) of TEG modules. Nevertheless, it assumes that the temperature difference between the TEG modules connected in parallel in the same row is similar, which is not always guaranteed in large-scale TEG systems. Reference [33] achieves TEG array reconfiguration via adjusting switches with wireless sensors. This is suitable for occasions where wiring is inconvenient.

Although using electrical switchgear is feasible, how to obtain an appropriate switch adjustment scheme is complicated and challenging since dynamic reconfiguration of TEG systems is a nonlinear, discrete, and constrained optimization problem. Furthermore, traditional approaches, e.g., branch and bound as well as enumeration methods, face the risk of combination explosion with increasing dimensionality. Therefore, traditional methods are normally not applicable to the reconfiguration of large-scale TEG systems. To address the above limitations, a novel adaptive coordinated seeker (ACS) is presented here for dynamic reconfiguration of TEG systems under different HgTDs.

The major contributions of this paper can be summarized as follows:

  • Dynamic reconfiguration of TEG systems under HgTDs is developed based on MhAs, for the first time.

  • ACS is first designed to coordinate the local searching capability of a tabu search (TS) and the global searching capability of a pelican optimization algorithm (POA). This can efficiently overcome drawbacks of traditional MhAs that are highly randomized and heavily dependent on certain parameters.

  • An equivalent re-optimization strategy (EROS) for reconfiguration solutions is developed to reduce unnecessary switching actions.

  • The practical and specific performance of an ACS is comprehensively evaluated under two typical scenarios, i.e., symmetrical and asymmetrical, via comparison with seven typical MhAs, i.e., artificial bee colony (ABC) algorithm, ant colony optimization (ACO), genetic algorithm (GA), particle swarm optimization (PSO), simulated annealing (SA) algorithm, TS, and POA.

The rest of this paper is structured as follows: TEG system modeling under HgTD is established in Sect. 2, while Sect. 3 introduces the optimization framework of the ACS. In Sect. 4, the execution process of TEG system reconfiguration and EROS are provided, respectively. Case studies, statistical analysis, and hardware-in-the-loop experiments are carried out in Sect. 5. Finally, main findings and conclusions are drawn in Sect. 6.

TEG system modelling

TEG module modelling

As illustrated in Fig. 1, a TEG module consists of two main components packaged together, i.e., two ceramic plates located on the cold and hot sides, respectively, while multiple thermoelectric couple units are connected in series with conductive metal strips between them. The thermoelectric couple unit is a fundamental and crucial element for a TEG module.

Fig. 1
figure 1

Schematic of TEG and equivalent circuit of TEG module

In general, the equivalent circuit of a TEG module is modeled as a voltage source Voc connected with a series resistance RTEG [4]. Here, the open-circuit voltage Voc (V) is directly determined by the temperature difference between the two ceramic plates as [34]:

$$V_{{{\text{oc}}}} = n_{{{\text{np}}}} \cdot \alpha_{{{\text{sb}}}} \cdot \Delta T = n_{{{\text{np}}}} \cdot \alpha_{{{\text{sb}}}} (T_{{{\text{hs}}}} - T_{{{\text{cs}}}} )$$
(1)

where nnp = 488 is the number of thermoelectric units, αsb is the Seeback coefficient (V/K), Ths (K) and Tcs (K) represent the temperature of the hot and cold sides, respectively.

The relationship between αsb and Thomson coefficient τ (V/K) is given by [22]:

$$\tau = T_{{{\text{av}}}} \cdot {{{\text{d}}\alpha_{{{\text{sb}}}} } \mathord{\left/ {\vphantom {{{\text{d}}\alpha_{{{\text{sb}}}} } {{\text{d}}T_{{{\text{av}}}} }}} \right. \kern-\nulldelimiterspace} {{\text{d}}T_{{{\text{av}}}} }}$$
(2)

where Tav (K) is the average value of Ths and Tcs.

For practical materials, τ is not zero. Thus, αsb can be deduced from (2) as [4]:

$$\alpha_{{{\text{sb}}}} (T_{{{\text{av}}}} ) = \alpha_{{{\text{sb0}}}} + \alpha_{{{\text{sb1}}}} \ln ({{T_{{{\text{av}}}} } \mathord{\left/ {\vphantom {{T_{{{\text{av}}}} } {T_{0} }}} \right. \kern-\nulldelimiterspace} {T_{0} }})$$
(3)

where αsb0 = 210 μV/K and αsb1 = 120 μV/K. These are the constant portion and variation rate of αsb, respectively [17, 21, 35]. T0 = 300 K denotes the temperature reference.

In addition, RTEG is affected by the operational temperature of the TEG module [36]. This can be described as:

$$R_{{{\text{TEG}}}} (T_{{{\text{av}}}} ) = - 2.597 + 0.014T_{{{\text{av}}}}$$
(4)

TEG system modelling under HgTD

In general, multiple TEG modules are interconnected in an SP electrical connection to obtain the expected voltage level and power density. As shown in Fig. 2, the series diodes are used to avoid circulating current between different strings (columns) within the TEG system when the total voltages of different strings mismatch, while the bypass diodes guarantee that the string with damaged TEG modules still operates normally.

Fig. 2
figure 2

Configuration of (N × M) TEG system with SP

According to the equivalent circuit of a single TEG module illustrated in Fig. 1, the total voltage Voc,m and total internal resistance Rm at the mth string can be expressed as:

$$V_{{{\text{oc}},m}} = \sum\limits_{n = 1}^{N} {(\beta_{n,m} } \cdot V_{{{\text{oc}},n,m}} )$$
(5)
$$R_{m} = \sum\limits_{n = 1}^{N} {(\beta_{n,m} } \cdot R_{{{\text{TEG}},n,m}} )$$
(6)

where N is the number of rows in the TEG system, βn,m stands for the state of the bypass diode in the nth row and the mth column. This is either 0 (forward conduction) or 1 (reverse cutoff). Voc,n,m and RTEG,n,m represent the open-circuit voltage and internal resistance of the TEG module in the nth row and the mth column, respectively.

Therefore, the mth string is equalized to a voltage resource Voc,m in series with an internal resistance Rm. Based on the Thevenin theorem, the total current Itotal and total resistance Rtotal of the TEG system can be obtained by:

$$I_{{{\text{total}}}} = \sum\limits_{m = 1}^{M} {(\gamma_{m} } \cdot {{V_{{{\text{oc}},m}} } \mathord{\left/ {\vphantom {{V_{{{\text{oc}},m}} } {R_{m} }}} \right. \kern-\nulldelimiterspace} {R_{m} }})$$
(7)
$$R_{{{\text{total}}}} = {1 \mathord{\left/ {\vphantom {1 {\sum\limits_{m = 1}^{M} ( {{\gamma_{m} } \mathord{\left/ {\vphantom {{\gamma_{m} } {R_{m} }}} \right. \kern-\nulldelimiterspace} {R_{m} }})}}} \right. \kern-\nulldelimiterspace} {\sum\limits_{m = 1}^{M} ( {{\gamma_{m} } \mathord{\left/ {\vphantom {{\gamma_{m} } {R_{m} }}} \right. \kern-\nulldelimiterspace} {R_{m} }})}}$$
(8)
$$\gamma_{m} = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {{\text{if}}\,V_{{{\text{oc}},m}} > V_{{\text{t}}} } \hfill \\ {0,} \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right.$$
(9)

where M is the number of columns in the TEG system, γm represents the state of the mth series diode, which is determined by the terminal voltage Vt of the TEG system and Voc,m.

A voltage vector [V0, V1, V2, …, Vk, …, VK] (1 ≤ K ≤ M) is acquired by sorting the unique values of [V0, Voc,1, Voc,2, …, Voc,m, …, Voc,M] in ascending order, while V0 = 0 V. According to the maximum power transfer theorem [37], the maximum output power Pmax,k (1 ≤ k ≤ K) can be written when Vt falls into the interval of (Vk-1, Vk], as:

$$P_{\max ,k} = (0.5I_{{{\text{total}}}} )^{2} R_{{{\text{total}}}}$$
(10)
$$P_{{{\text{GMPP}}}} = \max (P_{\max ,1} ,P_{\max ,2} ,...,P_{\max ,k} ,...,P_{\max ,K} )$$
(11)

where K stands for the conductive number of TEG strings.

Hence, the potential GMPP of a TEG system under HgTD can be calculated by Eq. (11) and approached by the aforementioned MPPT techniques.

Design of adaptive coordinated seeker

Tabu search

The TS [38, 39] developed by F. Glover is a widely applied and representative algorithm in combinatorial optimization. It has powerful local searching capability. Its fundamental principle is demonstrated in Table 1, in which neighbourhood structure is defined as the interchange of any two TEG modules within the same row of the TEG system.

Table 1 The basic framework of single execution process for TS

Pelican optimization algorithm (POA)

The swarm-based POA mimics the hunting process of the pelican [40], and has the merits of simple structure and easy implementation. POA contains two principal phases, i.e., moving towards prey and winging on the water surface, as described by (12) and (13), respectively. It should be noted that results are discretized using integer coding with an ascending rule [41], which makes POA applicable to the dynamic reconfiguration of TEG systems.

$$x_{i} (it + 1) = \left\{ {\begin{array}{*{20}l} {x_{i} (it) + r_{1} \cdot (x_{{\text{p}}} - I \cdot x_{i} (it)),} \hfill & {{\text{if}}\,F_{{\text{p}}} > F_{i} } \hfill \\ {x_{i} (it) + r_{1} \cdot (x_{i} (it) - x_{{\text{p}}} ),} \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right.$$
(12)
$$x_{i} (it + 1) = x_{i} (it) + 0.2(1 - {{it} \mathord{\left/ {\vphantom {{it} {It}}} \right. \kern-\nulldelimiterspace} {It}}) \cdot (2r_{2} - 1) \cdot x_{i} (it)$$
(13)

where xi(it) and xi(it + 1) indicate the status of the ith pelican in the itth and (it + 1)th iteration, respectively. xp represents a randomly selected prey from remaining pelicans, while r1 and r2 are both random values distributed in the interval of [0, 1]. It stands for the maximum iteration, while Fp and Fi are objective function values of xp and xi(it), respectively.

Adaptive coordinated seeker

Based on the idea of parallelism, the entire population is divided equally and randomly into two subpopulations to execute different operations separately. All individuals for the first subpopulation perform algorithm #1 in Table 1. The neighbourhood size of each individual is adaptively adjusted according to its objective function value, as indicated in (14). Therefore, the drawback of traditional TS with heavy dependence on starting solution and search efficiency can be effectively lessened.

$$NS_{j} = \left\{ {\begin{array}{*{20}l} {NS_{0} + NoSP - 1,} \hfill & {{\text{for}}\,{\text{best}}\,{\text{individual}}} \hfill \\ {NS_{0} - 1,} \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right.$$
(14)

where NSj is the neighbourhood size of the jth individual, with j = 1, 2, …, NoSP. NS0 is the initial neighbourhood size of each individual, and NoSP stands for the individual number per subpopulation, which is equal to NP/2.

For the second subpopulation, two different operations can be selected with the same probability, i.e., the discretized POA and TS without tabu. In addition, similar to the first subpopulation, the concept of neighbourhood search with adaptive adjustment is used for each individual of the second subpopulation.

To ensure communication between the two subpopulations, the top 20% of the row vectors with the largest difference between the optimal individuals of the two subpopulations are swapped in order after certain iterations (0.04It). For instance, the differences between the rows for the two solutions in Fig. 3 are 2/3 and 1, respectively. An effective updating mechanism [40] is adopted during the optimization of ACS to obtain a better solution. The overall ACS implementation procedure for the TEG system dynamic reconfiguration is provided in Fig. 4.

Fig. 3
figure 3

Coding for solution (2 × 3) and its configuration of the switching matrix

Fig. 4
figure 4

The overall implementation procedure of dynamic reconfiguration for a TEG system via ACS

As the discretized POA has strong global search ability, it can effectively avoid the problem of premature convergence. In contrast, the mutation mechanism is employed in [42] to improve the PSO for parameter optimization of a transient stability prediction model, which can ensure population diversity. TS is biased to local exploitation, which can reasonably prevent ACS from blind search. Similarly, reference [43] constructs the memetic algorithm using a binary differential evolution algorithm and TS to extract optimal feature subsets for transient stability assessment of the power system. This can retain valid information to the greatest extent.

Dynamic reconfiguration of TEG system via ACS

Coding and switching matrix

For dynamic reconfiguration, each solution represents one arrangement mode while variables are the column serial number of each module in the TEG system, and the change of module locations is achieved by switching matrices. As shown in Fig. 2, reconfiguration within the same column has no impact for PGMPP as the TEG modules in the same column are connected in series. Therefore, consecutive integer coding from 1 to M (column serial number of modules) is only adopted separately for each row of the TEG system. For instance, the initial solution and a possible solution, and their corresponding configuration of the switching matrices are shown in Fig. 3 for a (2 × 3) TEG system.

Objective function and constraints

Dynamic reconfiguration of the TEG system aims to maximize its potential GMPP, while a high-quality solution corresponds to a high potential GMPP. Therefore, the objective function is designed as:

$$x^{*} = \mathop {\text{argmax }}\limits_{{x^{*} \in X}} P_{{{\text{GMPP}}}}$$
(15)

where X means the feasible solution space and x* represents the optimal solution.

Moreover, dynamic reconfiguration is performed between different TEG modules in the same row. Thus, a feasible solution x should satisfy the following constraints:

$$\left\{ {\begin{array}{*{20}c} {x(n,m) \in \{ 1,2,...,M\} \, } \\ {\bigcup\limits_{m = 1}^{M} {x(n,m) = \{ 1,2,...,M\} \, } } \\ \end{array} } \right.$$
(16)

where x(n,m) is the variable value of the nth row and the mth column in the feasible solution x.

Equivalent re-optimization of reconfiguration solution

Reconfiguration solutions obtained by various MhAs may not be in an optimal form under the same PGMPP because of the inherent randomness of MhAs and the non-uniqueness of the solution. For example, assuming that the temperature conditions for modules (1,2) and (1,3) in Fig. 3 are identical, the two modules are completely equivalent according to (1) and (4). Thus, the switching states for modules (1,2) and (1,3) of the possible solution can also be consistent with the initial states. However, if the operation is performed directly following the possible solution, unnecessary switching actions will be caused. Therefore equivalent re-optimization for reconfiguration solutions obtained by various MhAs is essential and meaningful based on the temperature distribution of the TEG system. Its detailed procedure is tabulated in Table 2.

Table 2 The procedure of EROS for reconfiguration solution

Implementation process of dynamic reconfiguration for TEG system using ACS

Figure 4 explicitly demonstrates the overall implementation procedure of dynamic reconfiguration for the TEG system using ACS. rand stands for a random value distributed in the interval of [0, 1]. It is worth noting that the initial arrangement of the TEG system needs to be included in the population during random initialization to ensure an effective optimization. Additionally, a basic precondition for this work is that the operating temperatures of each module in the TEG system can be collected by measuring devices.

Case studies

To validate the dynamic reconfiguration performance of ACS for the TEG systems under various HgTDs, two typical scenarios are developed, i.e., symmetrical (15 × 15) and asymmetrical configuration (20 × 15). Furthermore, four typical temperature distributions are studied for each configuration, including diagonal, outer, inner, and random [28]. Seven conventional MhAs, e.g., ABC [44], ACO [45], GA [28], PSO [28], SA [41], TS [38, 39], and POA [40], are used for comparison with ACS. For a fair comparison, NP and It are uniformly set as (NP = 30, It = 500) and (NP = 30, It = 1000) under symmetrical and asymmetrical scenarios, respectively. All algorithms are independently executed 50 times to obtain the statistical results, while all simulations are executed in the MATLAB/Simulink 2019b environment with a personal computer with Intel(R) Core(TM) i5 CPU at 2.9 GHz and 16 GB of RAM, and ode45 (Dormand-Prince) with variable step chosen as the solver.

Symmetrical configuration

Figures 5 and 6 illustrate the initial and optimal temperature distributions acquired by ACS in a symmetrical TEG system (15 × 15) under four different scenarios. Boxplots of the maximum output power obtained by eight algorithms under the four different scenarios of the (15 × 15) TEG system are shown in Fig. 7. This reveals the statistical distribution of the maximum output power for each algorithm in 50 runs. Clearly ACS can obtain the highest average value of the maximum output power of the TEG system. In particular, the distribution range of the maximum output power acquired from ACS is the smallest in the outer, inner, and random conditions. Thus, it can be seen that ACS has superior performance in terms of convergence stability and expectation of the maximum output power.

Fig. 5
figure 5figure 5

Initial temperature distribution (°C) of four different scenarios for (15 × 15) TEG system: a, b diagonal, c, d outer, e, f inner, g, h random

Fig. 6
figure 6figure 6

Temperature distribution (°C) of four different scenarios for (15 × 15) optimized by ACS: a, b diagonal, c, d outer, e, f inner, g, h random

Fig. 7
figure 7

Boxplots of maximum output power obtained by eight algorithms under four different scenarios of a (15 × 15) TEG system: a diagonal, b outer, c inner, and d random

Figure 8 draws the convergence curves of various algorithms for the four different TEG systems, where ACS can converge within 350 iterations along with optimum output power under the four conditions. As depicted in Fig. 8b, c, although ABC has a faster convergence speed in the early iterations, it falls into a local optimum solution in the late stages, which further proves the superior global searching capability of ACS.

Fig. 8
figure 8

Convergence curves of eight algorithms under four different scenarios of (15 × 15) TEG system: a diagonal, b outer, c inner, and d random

The power-voltage (PV) and current–voltage (IV) curves of the TEG systems before and after optimization using ACS are shown in Fig. 9. It can be seen that the potential GMPPs of the TEG systems are notably enhanced by 3.4%, 4.5%, 2.7%, and 22.8% via ACS under diagonal, outer, inner, and random scenarios, respectively. In addition, the total voltage difference between the different strings in the TEG system is significant when the temperature is regionally and intensively distributed as shown in Fig. 5. Hence, MLMPPs phenomena are inevitably induced because of the presence of series diodes [4, 22, 23]. More specifically, the voltage discrepancy among different strings will be decreased on account of uniform temperature distributions in the row direction obtained by ACS as in Fig. 6, and thus MLMPPs phenomena of PV curves are significantly reduced, as shown in Fig. 9.

Fig. 9
figure 9

Output characteristics of (15 × 15) TEG system under four different scenarios before and after being optimized by ACS: a diagonal, b outer, c inner, and d random

Asymmetrical configuration

To further validate the superiority of ACS under various HgTDs, an asymmetrical TEG configuration (20 × 15) is designed and comprehensively evaluated. The initial and optimized temperature distributions based on ACS are shown in Figs. 10 and 11, respectively. It can be easily seen that temperatures in Fig. 11 are uniformly distributed in the row direction under all tested HgTDs.

Fig. 10
figure 10figure 10

Initial temperature distribution (°C) of four different scenarios for (20 × 15) TEG system: a, b diagonal, c, d outer, e, f inner, g, h random

Fig. 11
figure 11figure 11

Temperature distribution (°C) of four different scenarios for (20 × 15) optimized by ACS: a, b diagonal, c, d outer, e, f inner, g, h random

As delineated in Fig. 12, the distribution of the output power generated from 50 independent reconfigurations of the (20 × 15) TEG system by ACS is more centralized than others, where the acquired lower and upper bounds of output power are higher.

Fig. 12
figure 12

Convergence curves of eight algorithms under four different scenarios of (20 × 15) TEG system: a diagonal, b outer, c inner, and d random

It is seen in Fig. 13 that faster convergence and higher output power are achieved by ACS. In contrast, other algorithms, particularly ACO, PSO, and POA, tend to suffer from worse reconfiguration solutions with inferior convergence rates. Based on the above analysis and discussion, it is shown that ACS can rapidly provide reconfiguration solutions for TEG systems with high desired output power.

Fig. 13
figure 13

Convergence curves of eight algorithms under four different scenarios of a (20 × 15) TEG system: a diagonal, b outer, c inner, and d random

Figure 14 shows the output characteristics of the TEG system under the four different scenarios. This indicates that potential GMPPs corresponding to the optimal solutions are respectively boosted by 164.0 W, 122.8 W, 34.7 W, and 47.4 W, compared to those of the initial solutions. The output characteristics are also significantly smoothed via ACS, which significantly alleviates the burden and complexity of MPPT.

Fig. 14
figure 14

Output characteristics of (20 × 15) TEG system under four different scenarios before and after being optimized by ACS: a diagonal, b outer, c inner, and d random

Analysis and discussion for statistical results

Tables 3 and 4 provide statistical results obtained by the eight methods in 50 independent runs under the symmetrical and asymmetrical scenarios, respectively, where the optimal results are shown in bold. Although all methods can improve the output power of the TEG systems because of their original arrangements being used as the initial solutions of each method, ACS can harvest the highest output power and maximum average output power under all cases. In particular, in the symmetrical TEG system with random temperature distribution, the maximum output power of ACS is 103.1%, 116.3%, 114.3%, 110.7%, 108.4%, 104.7%, and 113.7% of that of ABC, ACO, GA, PSO, SA, TS, and POA, respectively. In addition, GMPPs of the optimized PV curves in Fig. 14 are 16.9 W, 23.1 W, 18.7 W, and 19.8 W lower than the best results obtained by ACS in Table 4 under the four scenarios, respectively. This result is mainly caused by the forward conduction losses on the diodes.

Table 3 Statistical results obtained by each algorithm under four different scenarios of a (15 × 15) TEG system
Table 4 Statistical results obtained by each algorithm in four different scenarios of a (20 × 15) TEG system

As illustrated in Tables 3 and 4, both ACS and TS can obtain reconfiguration solutions with similar maximum output power under two HgTDs (i.e., outer and inner), which indicates that they have similar performance in these scenarios. However, as mentioned above, ACS is superior to TS in terms of convergence speed and stability, as displayed in Figs. 7, 8, 12, and 13. Overall, it is evident that ACS is successfully applied to the dynamic reconfiguration of the TEG systems with high accuracy, fast convergence speed, and excellent stability.

Table 5 tabulates the action module number reduced by EROS for the optimal solution of each algorithm in both symmetrical and asymmetrical scenarios. It is apparent that EROS can noticeably reduce the number of action modules in all cases. Hence, unnecessary switching actions and reconfiguration costs can be considerably reduced.

Table 5 Action modules reduced by EROS for optimal solution of each algorithm in symmetrical and asymmetrical

RTLAB based hardware-in-the-loop experiments

RTLAB platform-based hardware-in-the-loop experiments are carried out to further validate the hardware implementation feasibility, as shown in Fig. 15, with a sampling frequency of 1 kHz. Figures 16 and 17 demonstrate the output characteristics of each TEG system under various HgTDs optimized by ACS. It can be seen from Figs. 16 and 17 that the responses of the simulation and experiment acquired by the two configurations under several HgTDs are largely similar.

Fig. 15
figure 15

Hardware-in-the-loop experiment using RTLAB platform

Fig. 16
figure 16

Output characteristics of (15 × 15) TEG system for simulation and experiment under various HgTDs optimized by ACS: a diagonal, b outer, c inner, and d random

Fig. 17
figure 17

Output characteristics of (20 × 15) TEG system for simulation and experiment under various HgTDs optimized by ACS: a diagonal, b outer, c inner, and d random

Conclusions

In this work, a novel ACS is designed to realize an effective and efficient dynamic reconfiguration of two different TEG systems, i.e., symmetrical and asymmetrical under various HgTDs. The main conclusions are:

  1. 1.

    An MhA-based dynamic reconfiguration strategy for large-scale TEG systems is presented and accomplished for the first time. It aims to re-use waste heat generated in daily life and industrial production.

  2. 2.

    Compared with traditional MhAs, ACS is able to execute a reasonable equilibrium between global exploration and local exploitation via simultaneous execution of TS and discretized POA. Therefore, ACS can accurately and rapidly converge to optimal reconfiguration solutions for TEG systems under various HgTDs with a higher probability.

  3. 3.

    Comprehensive case studies verify that ACS has a better performance in dynamic reconfiguration than other MhAs. In particular, the maximum output power of ACS is 103.1%, 116.3%, 114.3%, 110.7%, 108.4%, 104.7%, and 113.7% of that of ABC, ACO, GA, PSO, SA, TS, and POA under symmetrical scenario with random temperature distribution, respectively. The expected output power is enhanced by 34.6 W-247.8 W via ACS compared with that of initial solutions under eight HgTDs, which further validates that ACS can significantly improve the potential GMPP of TEG systems.

  4. 4.

    Statistical results show that EROS can significantly reduce the number of action modules required under various cases. This can boost the practicability and economy of TEG systems with low generation efficiency.

However, it is necessary to reasonably set the number of individuals and iterations of ACS according to the scale of the TEG system in practical application in order to obtain an acceptable computational cost. The completion of the dynamic reconfiguration of the TEG system relies highly on accurate and reliable temperature measurement equipment, which increases the cost. Therefore, the deployment of the thermometer and operator of ACS can be further optimized, while ACS can be extended to the reconfiguration of PV and PV-TEG systems in future studies.

Availability of data and materials

The datasets used or analyzed during the current study are available from the corresponding author on reasonable request.

Abbreviations

α sb :

Seeback coefficient (V/K)

α sb0 :

Constant portion of αsb (V/K)

α sb1 :

Variation rate of αsb (V/K)

β n , m :

State of bypass diode in the nth row and the mth column

γ m :

State of the mth series diode

τ :

Thomson coefficient (V/K)

it :

Current iterations of pelican optimization algorithm

It :

Maximum iterations of pelican optimization algorithm

I total :

Total current of TEG system (A)

M :

Number of columns in TEG system

N :

Number of rows in TEG system

NP :

Total number of individuals of each algorithm

n np :

Number of thermoelectric units

P GMPP :

Global maximum output power of TEG system (W)

R TEG :

Internal resistance of TEG system (Ω)

R TEG, n , m :

Internal resistance of TEG module in the nth row and the mth column (Ω)

R m :

Total internal resistance of the mth module (Ω)

R total :

Total resistance of TEG system (Ω)

T hs :

Temperature of hot side (K)

T cs :

Temperature of cold side (K)

T av :

Average value of Ths and Tcs (K)

T 0 :

Temperature reference (K)

V oc :

Open-circuit voltage of TEG module (V)

V oc, m :

Total open-circuit voltage of the mth module (V)

V oc, n , m :

Open-circuit voltage of TEG module in the nth row and the mth column (V)

V t :

Terminal voltage of TEG system (V)

X :

Feasible solution space

x :

A solution in X

x i :

Status of the ith pelican

x * :

Optimal solution

ACS:

Adaptive coordinated seeker

AOA:

Arithmetic optimization algorithm

ABC:

Artificial bee colony

ACO:

Ant colony optimization

DA:

Dragonfly algorithm

EO:

Equilibrium optimizer

EROS:

Equivalent re-optimization strategy

FASO:

Fast atom search optimizer

GMPP:

Global maximum power point

GA:

Genetic algorithm

HgTD:

Heterogeneous temperature distribution

INC:

Incremental conductance

ICI:

Interacted collective intelligence

MPPT:

Maximum power point tracking

MLMPPs:

Multiple local maximum power points

MhAs:

Meta-heuristic algorithms

P&O:

Perturb and observe

PV:

Photovoltaic

PSC:

Partial shading conditions

POA:

Pelican optimization algorithm

PSO:

Particle swarm optimization

SP:

Series–parallel

SA:

Simulated annealing

TS:

Tabu search

TEG:

Thermoelectric generation

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Acknowledgements

The authors appreciatively acknowledge the support of by National Natural Science Foundation of China (61963020).

Funding

National Natural Science Foundation of China (61963020).

Author information

Authors and Affiliations

Authors

Contributions

YC: Conceptualization, methodology, writing, and software; BY: Writing-review and editing; ZG: Investigation, and visualization; JW: Supervision, writing-review, and editing; MZ: Writing-reviewing, editing, and methodology; ZL: Experiment and software; TY: Supervision, project administration, resources, and data curation. All authors read and approved the final manuscript.

Authors' Information

Yijun Chen (1996–), male, Master Degree Candidate, Major in the renewable energy optimization and control.

Bo Yang (1988–), male, PHD and Professor, Major in the intelligent optimization and control of renewable energy and energy storage system.

Zhengxun Guo (1998–), male, Master Degree Candidate, Major in the renewable energy optimization and control.

Jingbo Wang (1996–), male, Master Degree, Major in the optimization operation of photovoltaic system.

Mengmeng Zhu (1986–), male, PHD Candidate and Engineer, Major in the voltage and current transformers.

Zilin Li (1988–), male, PHD, Major in the electric power electronic power system stability and operation.

Tao Yu (1974–), PHD and Professor, Major in the nonlinear control theory, optimization for complex power systems, and machine learning.

Corresponding author

Correspondence to Bo Yang.

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Chen, Y., Yang, B., Guo, Z. et al. Dynamic reconfiguration for TEG systems under heterogeneous temperature distribution via adaptive coordinated seeker. Prot Control Mod Power Syst 7, 38 (2022). https://doi.org/10.1186/s41601-022-00259-6

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  • DOI: https://doi.org/10.1186/s41601-022-00259-6

Keywords

  • Thermoelectric generation systems
  • Dynamic reconfiguration
  • Heterogeneous temperature distribution
  • Adaptive coordinated seeker