## 8 Load Optimization and Scheduling
### Scheduling Algorithm
#### 8.1.1 Population Evaluation Functions  
To evaluate the performance of the WDO algorithm, different population evaluation functions (e.g., sphere, Ackley, Rastrigin, rotated hyper ellipsoid, step function) can be used and are defined \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Symbol** & **Description** & **Symbol** & **Description** \\ \hline \(v_{new}\) & new velocity & \(v\) & velocity vector of air particles \\ \(v_{old}\) & current velocity & \(F_{c}\) & Coriolis force \\ \(x_{curr}\) & current position & \(\Omega\) & Earth rotation \\ \(x_{opt}\) & optimal position & \(\rho\) & density of air particles \\ \(P_{old}\) & pressure at current location & \(V_{g}\) & vertical force on air particles \\ \(P_{opt}\) & optimal pressure & \(F_{sig}\) & sigmoid function \\ \(\Omega\times v\) & Coriolis force & \(p_{gb}\) & local best position \\ \(\alpha\) & constant in update position & \(\delta_{V}\) & volume of air \\ \(R\) & universal gas constant & \(F_{sphere}\) & sphere function \\ \(v\) & velocity of air particles & \(\omega\) & inertia factor \\ \(F_{g}\) & gravitational force of the Earth & \(n\) & total No. of air particles (Equation (23)) \\ \(g\) & gravitational acceleration & \(T\) & temperature \\ \(\Delta t\) & unit step time & \(v_{i,n}\) & velocity of the \(i\)-th particle in _n_-dimensions \\ \(c_{1}andc_{2}\) & weights for local and global positions & \(p_{lb}\) & local best position \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Symbols used in the wind-driven optimization (WDO) and PSO algorithms. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & **Evaluation Function** & **No. of Iterations** & **Global Pressure/\(p_{gb}\)** & **Converse** \\ \hline WDO & Sphere & 204 & 0 & Yes \\ WDO & Step & 500 & 0.6616 & No \\ PSO & Sphere & 500 & \(9.069^{-6}\) & No \\ PSO & Step & 370 & 0 & Yes \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Convergence comparison of the WDO and PSO algorithms. in [47,48]. Each function has different properties and parameter ranges based on which performance is measured. In our proposed scheme, we use a modified sphere function as defined in [47] for the evaluation of initial random population generation. Moreover, the convergence time of the WDO algorithm using sphere and step functions is evaluated and compared to the PSO algorithm (Table 3). The sphere function given in [47] is:  
\[F_{sphere}=\sum_{t=1}^{t_{n}}\sum_{i=1}^{n}\big{[}(x_{(t,i)}-2)\big{]}^{2},\; \forall\;x_{i}\in[-1,+1] \tag{24}\]  
The function used in our algorithm can be written as:  
\[F_{sphere}=\sum_{t=1}^{t_{n}}\sum_{i=1}^{n}x_{(t,i)})^{2},\;\forall\;x_{i}\in[ -1,+1] \tag{25}\]  
The step function can be defined as:  
\[F_{step}=\sum_{t=1}^{t_{n}}\sum_{i=1}^{n}\big{[}(x_{(t,i)}+0.5)^{2}\big{]},\; \forall\;x_{i}\in[-1,+1] \tag{26}\]  
where \(F_{sphere}\) and \(x_{i}\) denote the global optimal value and position of the air particle, respectively. These two functions are evaluated under the parameter values given in Tables 4 and 5. Alternatively, PSO is also a widely-used heuristic algorithm, because of its low complexity and ease of implementation, as compared to WDO. One of the major reasons for the low complexity is due to the lesser number of variables involved, as shown in Table 5. In PSO, population is initialized randomly, which moves in the n-dimensional search space \(X_{(i,t)}=[x_{(i_{1},t_{1})},x_{(i_{2},t_{2})},x_{(i_{3},t_{3})}.....x_{(i_{n},t_{n})}]\) with velocity \(V_{i,t}=[v_{(i_{1},t_{1})},v_{(i_{2},t_{2})},v_{(i_{3},t_{3})}.....v_{(i_{n},t_{n})}]\). After initialization, particles move with some randomness in order to avoid premature convergence, as shown in Equation (32). The velocity \(v_{(i,n)}\) of each particle is updated using the following equation [47].