# An Energy-Distance-Balanced Cluster Routing Protocol for Smart Microgrid IoT Sensing Networks
## 3 Framework Architectures
### Dynamic Cluster Radius Equation
The energy for a cluster head to receive data from its nodes is  
\[E_{R}=(\pi\rho R_{i}^{2}-1)mE_{\text{RX}} \tag{13}\]  
Thus, the total energy consumption for the cluster, \(E_{\text{Cluster}}\), is  
\[E_{\text{Cluster}} =E_{\text{To\_Head}}+E_{R}+E_{A}+E_{\text{To\_BS}}\] \[=m\rho\pi R_{i}^{2}\bigg{(}\frac{1}{2}\varepsilon R_{i}^{2}+E_{ \text{PX}}+E_{\text{RX}}+\frac{\varepsilon D_{i}^{2}+E_{\text{PX}}-E_{\text{ RX}}}{\pi\rho R_{i}^{2}}+E_{\text{DA}}\bigg{)} \tag{14}\]  
The average energy consumption per node, \(\overline{E_{\text{Node}}}\), is calculated as  
\[\overline{E_{\text{Node}}}=\frac{E_{\text{Cluster}}}{\rho\pi R_{i}^{2}}=m \Bigg{(}\frac{1}{2}\varepsilon R_{i}^{2}+E_{\text{PX}}+E_{\text{RX}}+\frac{ \varepsilon D_{i}^{2}+E_{\text{PX}}-E_{\text{RX}}}{\pi\rho R_{i}^{2}}+E_{ \text{DA}}\Bigg{)} \tag{15}\]  
To minimize the energy consumption of the nodes, the optimal cluster radius is determined by  
\[R_{i}=\sqrt[4]{\frac{2(\varepsilon D_{i}^{2}+E_{\text{PX}}-E_{\text{RX}})}{ \varepsilon\rho\pi}} \tag{16}\]In sensor networks, especially those utilized in the IoT, efficient energy management is crucial for prolonging network lifespan and ensuring operational stability. The detailed analysis of energy components within a network cluster provides significant insights into potential areas for optimization and efficiency improvements. Each component described in the equations plays a specific role in the overall energy dynamics of the cluster:  
* Energy to Base Station (\(E_{T_{Q}\_BS}\)): This component highlights the energy required for direct communication from the cluster head to the base station. It is heavily influenced by the distance to the base station and the size of the data packet, emphasizing the importance of strategic placement of nodes and base stations to minimize transmission energy. * Energy for Data Aggregation (\(E_{A}\)): Data aggregation by the cluster head reduces the volume of data transmissions, thereby conserving energy. The energy cost depends on the node density and the cluster radius, which dictates that effective clustering and optimal node distribution are key for minimizing this energy component. Energy to Cluster Head (\(E_{T_{Q}\_Head}\)): This calculates the energy used by all nodes in transmitting data to the cluster head. The formula integrates over the cluster radius, suggesting that smaller cluster sizes can reduce energy use, balancing the trade-off between cluster size and number of transmissions. * Node Density (\(\rho\)): Node density directly affects other calculations and represents how crowded a network area is. Higher densities can lead to higher energy consumption in data aggregation and reception, which must be managed by adjusting cluster sizes and head selection strategies. * Energy for Reception (\(E_{R}\)): This component addresses the energy the cluster head expends in receiving data from the nodes. It is dependent on the number of nodes and highlights the energy costs associated with high-density deployments. * Total Cluster Energy (\(E_{Cluster}\)) and Average Energy per Node (\(\overline{E_{Node}}\)): These summaries provide overall metrics for evaluating the efficiency of the cluster configuration. They combine all individual energy expenditures into a holistic view, allowing for the assessment of current energy management strategies and the identification of potential improvements. * Optimal Cluster Radius (\(R_{i}\)): Determining the optimal cluster radius based on energy parameters and node density is crucial for minimizing overall energy consumption. This calculation serves as a guideline for setting up clusters in a way that balances communication costs and operational efficiency. This analysis underscores the interconnectedness of various energy components and their collective impact on the network's energy efficiency. It highlights the need for a comprehensive approach to network design and management, focusing on optimizing each aspect of energy expenditure to enhance the sustainability and functionality of IoT networks.