## 4 Solving the Typical ORPD Problem by Using the Proposed ICOA Method  
### Population Initialization  
In order to carry out the optimal solution search program of the proposed ICOA method, the first step is to produce a set of initial solutions, called population. On the contrary to other metaheuristic algorithms, ICOA as well as COA have \(N_{pack}\) packs, and there are \(N_{coquote}\) particles in each. Thus, the initial population generation is executed for each pack as the following model.  
\[Sol_{d,p}=Sol_{\min}+\mu_{1}.(Sol_{\max}-Sol_{\min});\ \ d=1,\ \ldots,\ N_{ coquote}\ \&\ p=1,\ \ldots,\ N_{pack} \tag{25}\]  
In the equation above, \(Sol_{min}\) and \(Sol_{max}\) are the minimum and maximum values of control variables of the ORPD problem and can be represented as below:  
\[Sol_{\min}=\left\{V_{Gen,1,\min},\ \ldots,\ V_{Gen,NG,\min},\ TT_{1,\min},\ \ldots,\ TT_{NT,\min},\ Q_{cap,1,\min},\ \ldots,\ Q_{cap,NC,\min}\right\}^{T} \tag{26}\]  
\[Sol_{\max}=\left\{V_{Gen,1,\max},\ \ldots,\ V_{Gen,NG,\max},\ TT_{1,\max},\ \ldots,\ TT_{NT,\max},\ Q_{cap,1,\max},\ \ldots,\ Q_{cap,NC,\max}\right\}^{T} \tag{27}\]  
Finding solutions of the ORPD problem is also supported by using Matpower program for obtaining remaining variables, which are different from control variables available in solution \(Sol_{d,p}\). Each solution \(Sol_{d,p}\) is transferred to Matpower program for running, and then a set of dependent variables is obtained, consisting of apparent flow through each transmission line \(S_{branch,ij}\), reactive power output of generators \(Q_{Gen,\_has}\), and voltage of load buses \(V_{land,i}\). The three variables are imposed on inequality constraints (4), (8), and (9), and the three constraints are of course included in fitness function. However, other inequality constraints (5)-(7) together with two equality constraints (10) and (11) are not taken into account in fitness function. In fact, \(V_{Gen,\_has}\) and \(TT_{i}\) accompanied with \(Q_{cap,\_has}\) are control variables, which are considered in each social condition of coyotes. Thus, such variables' boundaries are always monitored and corrected within their limits, as shown in Section 4.3 below. In another side, equality constraints (10) and (11) have been taken into account in the Newton-Raphson method, which is run in Matpower program. Consequently, before calculating fitness function, such three dependent variables are checked and penalized as follows:  
\[Fine(Q_{Gen,i})=\left\{\begin{array}{ll}0&\text{if}\ \ Q_{Gen,\min}\leq Q_{Gen,i} \leq Q_{Gen,\max}\\ Q_{Gen,i}-Q_{Gen,\max}&\text{if}\ \ Q_{Gen,i}>Q_{Gen,\max}\\ Q_{Gen,\min}-Q_{Gen,i}&\text{else}\end{array}\right. \tag{28}\]  
\[Fine(V_{load,i})=\left\{\begin{array}{ll}0&\text{if}\ \ V_{load,\min}\ \leq V_{load,i}\leq V_{load,\max}\\ V_{load,i}-V_{load,\max}&\text{if}\ \ V_{load,j}>V_{load,\max}\\ V_{load,\min}-V_{load,i}&\text{else}\end{array}\right. \tag{29}\]  
\[Fine(S_{branch,ij})=\left\{\begin{array}{ll}0&\text{if}\ S_{branch,ij}\leq S _{branch,ij,\max}\\ S_{branch,ij}-S_{branch,ij,\max}&\text{else}\end{array}\right. \tag{30}\]  
As a result, fitness function for evaluating quality of solutions can be constructed by the following equation:  
\[Fitness_{d,p}=\text{Objective function}+\phi_{1}\left(\sum_{i=1}^{NG}Fine(Q_{Gen, i})\right)+\phi_{2}\left(\sum_{i=1}^{NL}Fine(V_{load,i})\right)+\phi_{3}\left( \sum_{i=1}^{NB}\sum_{\begin{array}{c}j=1\\ j\
eq i\end{array}}^{NB}Fine(S_{branch,ij})\right) \tag{31}\]After having fitness function, the solution with the lowest fitness function in each pack is set to local optimal solution \(Sol_{best,p}\), and the best solution among all packs is set to the global best solution \(Sol_{Gbest}\).