# An Adaptive 24 GHz PSO-Based Optimized VCO in Next-Generation Wireless Sensor Networks
## 3 The Proposed Adaptive PSO Algorithm  
In our work concerning the proposed adaptive PSO, we modify the existing algorithm in the following way: in a D-dimensional space, the \(j\)th particle is denoted as \(V_{j}=(v_{j1},v_{2},\ldots,v_{jD})\). The highest fitness value is \(C_{j}=(c_{j1},c_{j2},\ldots,c_{jD})\), and \(\beta\) refers to the index of the best particle within the entire population of \(j\) particles. \(S_{j}=(s_{j1},s_{j2},\ldots,s_{jD})\) is the velocity (i.e., the rate of change in position) for the \(j\)th particle.  
The particle adjustments in the traditional PSO are provided in (1).  
\[s_{jd}=s_{jd}+\alpha_{1}rand_{1}()(c_{1d}-v_{jd})+\alpha_{2}rand_{2 }()(c_{gd}-v_{jd})\] \[s_{jd}=S_{\max},if\quad s_{jd}>S_{\max}\] \[s_{jd}=-S_{\max},if\quad s_{jd}<-S_{\max}\] \[v_{jd}=v_{jd}+s_{jd} \tag{1}\]  
In contrast, the proposed adaptive PSO algorithm modifies the velocity update rule to incorporate an adaptive scaling factor. This factor, which is gradually reduced over time, adjusts the velocity limits dynamically to speed up convergence and improve the exploration-exploitation balance. Specifically, as the algorithm progresses, the velocity limit Vmax decreases, promoting faster convergence and reducing the number of objective function evaluations required. Furthermore, the adaptive PSO algorithm incorporates a tunable scaling factor controlled by a positive constant h, which adapts based on current generation. This parameter allows the algorithm to adjust more effectively across different phases of the search process, providing greater flexibility and improving performance in diverse optimization scenarios.  
Figure 3: (**left**) Traditional PSO. (**right**) Proposed adaptive PSO.  
In the adaptive PSO algorithm, \(\alpha_{1}\), \(\alpha_{2}\) is a positive constant, and \(rand_{1}()\) and \(rand_{2}()\) are random functions whose values lie within the range of [0, 1]. Constants \(\alpha_{1}\) and \(\alpha_{2}\) are used in the algorithm, while Smax is a parameter that limits the rate of change in the particle's position, acting as a system parameter in the PSO algorithm.  
In a search method based on population, a larger search space is anticipated in the initial stages; therefore, a diverse population is beneficial to guide the system towards the global optima. As the search progresses, however, the search size is expected to decrease to refine the solution. To address this, the introduction of a multiplying factor within the traditional PSO algorithm is considered effective in accelerating convergence.  
The adaptive PSO algorithm works similarly to the traditional PSO algorithm, with the addition of a multiplying factor \((1-(t/T)H)\) that is incorporated to update the \(S_{max}\) parameter in (2). In this multiplying factor, \(t\) is the number of generations completed so far, \(T\) is the maximum number of generations, and H is the constant term using the generate-and-test method.  
The adaptive PSO algorithm, as explained in (2), evolves with a gradually shrinking search space, which facilitates more efficient convergence towards the optimal solution.  
\[s_{jd}=s_{jd}+\alpha_{1}rand_{1}()(c_{1d}-v_{jd})+\alpha_{2}rand_{ 2}()(c_{gd}-v_{jd})\] \[s_{jd}=(1-t/T)^{h}S_{\max},if\quad s_{jd}>(1-t/T)^{h}S_{\max}\] \[s_{jd}=-(1-t/T)^{h}S_{\max},if\quad s_{jd}<-(1-t/T)^{H}S_{\max}\enspace. \tag{2}\] \[v_{jd}=v_{jd}+s_{jd}\]