## 4 Numerical Application
### Case Study 2
By observing the figure, it clearly appears that the ABS/GA solution method allows identifying better solutions in a faster way, that practically means, in a lower number of iterations (i.e., individuals investigated). In fact, the best fitness function values in case of ABS/GA are always lower than those evaluated through the GA, thus meaning that the convergence criteria are met faster with the proposed approach. As stopping criterion, in our case study, we assumed that the procedure ended when the best fitness function maintained a constant value over ten consecutive iterations. With this stopping criterion, as shown in Figure 16, GA/ABS converged after analyzing  
Figure 11: Case Study 3: Line currents of the residential feeder (phase no.1) at hour 21.00 of the last typical day. Figure 12: Case Study 3: Voltage magnitude of the residential feeder at hour 21.00 of the last typical day. 260 individuals reaching a best fitness function value of $4.78 M. It is interesting to note that in the case of GA (case 2), after analyzing 260 individuals the best fitness function value is greater (i.e., $4.85 M). Moreover, in case 2, even with higher number of iterations, the GA provides worse solutions. In the case of Figure 16, the GA was stopped after having investigated 500 individuals, without meeting the stopping criterion and without obtaining a better solution, compared to the case 3 since the fitness function best value is still higher (i.e., $4.82 M). As a consequence of that, we can conclude that in the examined case GA/ABS allows identifying in a fast way a good solution with a drastically reduced computational time compared to the case of applying GA. We note also that the time requested to find the solution also decreases because in the case of GA/ABS the inner SPQ solution is requested to solve optimal load flow problems with a reduced number of DESSs of higher sizes, thus reducing the number of the optimization variables and so, requiring a lower computational time. In order to give an idea of the computational time of the simulations, by using a commercial personal computer (2.93 GHz processor, 4 GB RAM), the simulation time for each individual ranged from seconds to minutes depending on the number and size of DESSs corresponding to the individual. By increasing the number of DESSs, in fact, the optimization variables (i.e., stored energy, active and reactive power at each of the 24 h of the typical days) significantly increase. So high computational time is not discouraging if we consider that we are in a planning stage and that we used a detailed and complex model of the system under study to represent both the unbalances and the behavior of DESSs with the consequence that each GA individual implies many optimiz  
Figure 14: Case Study 3: Energy stored by the DESS at bus no.37 during the last typical day. Figure 13: Case Study 3: Active power of the DESS at bus no.37 during the last typical day. as obvious, the use of more sophisticated hardware and software arrangements will allow reducing dramatically the computation time. As a final consideration we outline that in order to check both the proposed procedure and the usefulness of adopting DESSs to face unbalances and PQ problems, numerous further simulations were performed, not reported here for the sake of conciseness. More specifically, further investigations were made by applying the procedure imposing (i) a higher value to the maximum battery's capacity that can be installed in the network; (ii) more stringent constraints to the PQ requirements. By analyzing the results, we observed that by allocating a higher number of DESSs further economic benefits can be obtained. In case of more stringent constraints on the PQ requirements, the proposed hybrid GA/ABS procedure still resulted more efficient than the GA in terms of computational time and accuracy of the results.