## 1 Introduction
### TEG Modeling  
The thermoelectric generation building block consists of thermocouples that are connected in a series configuration. The main objective of the thermoelectric generator is to convert the thermal energy to electrical energy based on the Seebeck effect, i.e., the temperature difference at the junctions between two different metals results in a voltage. The TEG module structure with an equivalent electrical circuit is shown in Figure 2. The electrical equivalent of a thermoelectric generator consists of a voltage source with its internal resistance. If the difference in the temperature between the hot and cold junctions is constant, the open circuit voltage of a particular module is provided as follows:  
\[V_{TEG}=s_{p-n}\;\Delta T=s_{p-n}\;(T_{h}-T_{c}) \tag{1}\]  
where \(V_{TEG}\) is the open circuit voltage, \(s_{p-n}\) is the Seebeck co-efficient, and \(\Delta T\) is the variation in temperature between the hot junction \(T_{h}\) and the cold junction \(T_{c}\). The Seebeck co-efficient depends upon the type of material used for TEG. The Seebeck coefficient can be determined as follows:  
\[s_{p-n}=n_{t}\;\big{(}s_{p}-s_{n}\big{)} \tag{2}\]  
where \(n_{t}\) are the number of thermocouples, and \(s_{p}\) and \(s_{n}\) indicate the Seebeck co-efficient for p-type and n-type thermocouples. The voltage, current, and output power of the thermoelectric generator can be mathematically expressed as:  
\[V_{TEG}=\left(\frac{\big{(}s_{p}-s_{n}\big{)}\times(T_{h}-T_{c})\times r_{L}}{ r_{L}-r_{TEG}}\right)\times n_{t} \tag{3}\]  
\[I=\frac{\big{(}s_{p}-s_{n}\big{)}\times(T_{h}-T_{c})}{r_{L}-r_{TEG}} \tag{4}\]  
\[P_{TEG}=r_{L}\times\frac{\big{(}s_{p}-s_{n}\big{)}^{2}\times(T_{h}-T_{c})^{2}} {\big{(}r_{L}-r_{TEG}\big{)}^{2}}\times n_{t} \tag{5}\]  
where \(r_{L}\) represents the load resistance in \(\Omega\) and \(r_{TEG}\) represents the internal resistance of TEG. The currently studied TEG model is rated at 25 W with 40 mm \(\times\) 40 mm based on Bi2 Te3 material. The used centralized TEG structure is shown in Figure 3 and the IV and PV curve under uniform and non-uniform temperature distribution is shown in Figure 4.  
The MPP of a TEG system is the operating point at which the generator can produce the maximum amount of power for a given temperature gradient. The accurate tracking of the MPP is critical for achieving the maximum efficiency and performance of the TEG system. If the TEG system is not able to accurately track the MPP, the generator may not be able to operate at its maximum power output, thus resulting in reduced efficiency and performance. Furthermore, TEG systems often have non-uniform temperature gradients due to various factors such as uneven heating or cooling, variations in material properties, and environmental conditions. These non-uniform temperature gradients can significantly affect the accuracy of the MPP tracking in the TEG system. If the MPP tracking is not accurate in a TEG system with non-uniform temperature gradients, it can lead to inefficient energy conversion and reduced power output. Therefore, it is essential to accurately track the MPP in a TEG system, especially when there are non-uniform temperature gradients present. This can be achieved by using advanced control algorithms and feedback mechanisms that can adjust the TEG system's operating conditions in real-time to maintain the system at its MPP. Accurate MPP tracking in TEG systems with non-uniform temperature gradients can result in improved efficiency, increased power output, and reduced energy waste, causing TEG systems to be a more viable and sustainable option for energy conversion.