*5.2. CFA*

A meta-heuristic optimization algorithm called Franklin's and Coulomb's algorithm (CFA) has been created by Ghasemi et al. [41], is based on the theories of the Coulomb's and Franklin's law. For optimal outcomes, the CFA employs two separate theories. First is the Coulomb's Law, which is based on the attraction and repulsion of electrons. This phenomenon governs the interaction of two independent point charges separated by a certain distance. Second is the Franklin's Law, which is based on that every item has an equal quantity of positive and negative charges, according to this law. CFA's mathematical model is based on four steps: Initialization, Attraction/repulsion, Probabilistic ionization, and Probabilistic contact. The pseudo-code of the CFA technique is indicated in Algorithm 2.



### *5.3. STOA*

A novel bio-inspired optimization algorithm called Sooty Tern Optimization Algorithm (STOA) has been created by Dhiman and Kaur [42] to address the constraints of the industrial issues. The movement and attacking habits of the sea bird sooty tern in nature are the key motivations for modeling the STOA technique. STOA was validated using 44 benchmark test functions and compared it with nine well-known optimization techniques in terms of performance. The results of CEC 2005 and CEC 2015 standard test functions prove that the STOA is capable of addressing difficult and high dimensionality bound constrained actual situations. The pseudo-code of the STOA technique is indicated in Algorithm 3.

**Algorithm 3**. STOA [42]

1. Initialize the population Xp <sup>1</sup> = (X<sup>p</sup> <sup>1</sup> , Xp <sup>2</sup> , Xp <sup>3</sup> ,......, X<sup>p</sup> <sup>N</sup> within the limits Xmin <sup>i</sup> <sup>≤</sup> Xp <sup>i</sup> <sup>≤</sup> Xmax i

2. Initialize parameters DA and CB 3. Evaluate the fitness of whole population 4. Best search agent → Xbest 5. **While** (it < Maxit) 6. **for** (i = 1: N) **do** 7. Update the position of the current search agent 8. **end for** 9. Initialize parameters DA and CB 10. Evaluate the fitness of whole population 11. Update Xbest 12. it = it +1 13. **end while**

14. **return** Xbest

### *5.4. GWO*

The GWO has been proposed by Mirjalili et al. [43], it's a heuristic optimization technique created to find a candidate solution from a large solution space without requiring any explicit input parameters. Such qualities are ideal for dealing with nonlinear issues, such as controller parameter tweaking. Grey wolves' natural behavior and social structure in seeking prey served as inspiration for GWO. There is a hierarchical framework that governs each wolf pack. The alpha wolf, who heads the entire group, is the most formidable. In the absence of the alpha wolf, the second strongest wolf, called as the beta wolf, assumes leadership. The weaker wolves are the delta and omega wolves. The pseudo-code of the GWO technique is indicated in Algorithm 4.

#### **Algorithm 4**. GWO [43]


### **6. Results and Discussion**

In this work, a novel HBO technique is suggested to determine the optimal sizing of four alternatives off-grid hybrid system scenarios based on PV, WT, biomass, and battery units. These four scenarios of the hybrid system are namely PV/WT/biomass/Bat, PV/biomass/Bat, WT/biomass/Bat, and PV/WT/Bat. In order to validate the effectiveness of this HBO as a way to provide optimal reliability and least cost, the results achieved by the suggested algorithms are compared with other recent optimization techniques CFA, GWO and STOA. The control parameters used in the optimization process for each algorithm are listed in Appendix A.

Figure 8 presents the graphic form of the final values of the target function over the 50 executes for the four analyzed configurations scenarios utilizing the optimization techniques namely, HBO, CFA, GWO, and STOA. It can be noted that, the fitness values for the suggested HBO method in the four system cases were within a limited range, which demonstrated the stability of the suggested technique over the other techniques. Therefore, parametric and nonparametric metric values are superior using the HBO method compared to the rest of the optimization techniques.

Figure 9 displays the best optimal solution convergence curve for each scenario utilizing HBO, CFA, GWO, and STOA. For Case (1), the best solution achieved by using HBO technique which is 0.0643767 after 27 iterations, followed by CFA technique with 0.06437783 after 44 iterations. For Case (2), the best solution achieved by using HBO technique which is 0.0703404 after 49 iterations, followed by best solution achieved by CFA technique with 0.07034462 after 32 iterations. For Case (3), the best solution achieved by using HBO technique with 0.0705909 after 41 iterations, followed by best solution achieved by CFA technique with 0.0651240320 after 39 iterations. Finally for Case (4), the best solution achieved by using HBO technique with 0.151991724 after 41 iterations, followed by best solution achieved by CFA technique with 0.152001799 after 58 iterations. It can be noticed that the HBO method provides a good convergence characteristic over the other optimization algorithms CFA, GWO, and STOA in all suggested cases.

**Figure 8.** *Cont*.

**Figure 8.** End values of the fitness function for 50 executions using HBO, CFA, GWO, and STOA methods: case-1: PV/WT/Biomass/Bat system, case-2: PV/Biomass/Bat system, case-3: WT/Biomass/Bat system, case-4: PV/WT/ Bat system.

**Figure 9.** *Cont*.

**Figure 9.** The Convergence curves for 100 iterations using HBO, CFA, GWO, and STOA methods: case-1: PV/WT/Biomass/Bat system, case-2: PV/Biomass/Bat system, case-3: WT/Biomass/Bat system, case-4: PV/WT/ Bat system.

Tables 6–9 illustrate the results of the optimization properties for the four system scenarios proposed, which is based on many factors including the best value of the objective function, the decision variables (NPV, NWT, Ng and NBat), the COE, LPSP, and NPC of the suggested optimization algorithms (HBO, CFA, GWO and STOA).

**Table 6.** The optimization properties for the proposed hybrid system based on using HBO, CFA, GWO and STOA for an isolated PV/WT/Biomass/Bat.


**Table 7.** The optimization properties for the proposed hybrid system based on using HBO, CFA, GWO and STOA for an isolated PV/Biomass/Bat.


**Table 8.** The optimization properties for the proposed hybrid system based on using HBO, CFA, GWO and STOA for an isolated WT/Biomass/Bat.


**Table 9.** The optimization properties for the proposed hybrid system based on using HBO, CFA, GWO and STOA for an isolated PV/WT/Bat.


In Table 6, for the PV, WT, Biomass, and Bat system, the results indicate that the HBO has the best configuration by using 15 PV panels, 1 WTs, 2 biomass generators, and 400 batteries, achieving the least COE, and NPC with 0.121171\$/kWh and \$ 3,559,143, respectively. In Table 7, for the second system case based on PV, Biomass, and Bat, the results prove that the HBO has the best configuration by using 17 PV panels, 2 biomass generators, and 447 batteries, achieving the least COE, and NPC with 0.1311804\$/kWh and \$ 3,853,160, respectively.

While Table 8, for the WT, Biomass, and Bat system, the results prove that the STOA has the best configuration by using 1 WT, 2 biomass generators, and 375 batteries, achieving the least COE, and NPC with 0.1056732 \$/kWh and \$ 3,103,938, respectively. In Table 9, for the fourth system case based on PV, WT, and Bat, the results illustrate that the STOA has the best configuration by using 170 PV panels, 88 WTs, and 983 batteries, achieving the least COE, and NPC with 0.3324975\$/kWh and \$ 9,766,441, respectively.

By comparing the COE and NPC of the four suggested cases, it finds that Case-3 achieved the lowest COE and NPC, followed by the Case-1. Although the third scenario which based on WT/biomass/Bat units produces the minimum value of COE and NPC, but it is not the optimal and efficient system for use. As the design of this case is based on batteries and biomass generators only, which have the highest yearly sharing of the capital cost, operating and maintenance cost. While the first scenario which is consists of PV/WT/biomass/Bat units considered an appropriate solution with minimal investment cost for the suggested case study area.

Parametric and non-parametric statistical measurements were performed for a more accurate comparison between the four optimization methods (HBO, CFA, GWO and STOA) on the basis of the acquired values of the objective function across a hundred individual runs for all analyzed cases. Parametric measurements comprise the lower value (Min.), maximum value (Max.) and mean of the target function, whereas the nonparametric measurements contain the median, relative error (RA), mean absolute error (MAE), standard deviation (SD), and efficiency. The efficiency here referred to the ratio of the lower value to the mean value of the goal function. For all four system scenarios, the results for statistical metrics for HBO, CFA, GWO, and STOA are shown in Table 10. On the basis of the results obtained, the proposed HBO in each case proved the best sensitivity and stability results compared to other optimization methods.

**Table 10.** The statistical performance of the studied optimization algorithms for the four system cases.



**Table 10.** *Cont.*

Figure 10, illustrate the sensitivity analysis of studying the impact of the variation of the decision parameters on the stand-alone system objective functions, (a) COE, (b) NPC, (c) LPSP, (d) EXP. Where "0" on the x-axis refers to the nominal values of the sensitivity factors.

**Figure 10.** *Cont*.

**Figure 10.** *Cont*.

**Figure 10.** Sensitivity analysis of studying the influence of sizing parameters variation on the stand-alone system variables, (a) COE, (b) NPC, (c) LPSP, (d) EXP. (**A**) The influence of sizing change on the energy cost. (**B**) The influence of sizing change on the NPC. (**C**) The influence of sizing variation on the LPSP. (**D**) The influence of sizing variation on an excess of energy.

Figure 10A,B illustrate the effect on the COE and the NPC. As it can be noted that, at lower values of the specified parameters, both COE and NPC drop when the number of each PV panels, biomass generators, and batteries decreased. While, at a higher parameter values, the COE and NPC raise with increasing the number of each PV panels, biomass generators, and batteries. For the number of wind turbines, it can be noted that both the COE and NPC are nearly constant with the variation of the wind turbines number.

Figure 10C,D indicates that the chosen parameters has an effect on the system parameters, especially the number of the biomass generators.

Table 11, illustrate the yearly expenses breakdown of the hybrid system units and in turns show the system's NPC. The reader can notice that, for all suggested system cases the battery storage system has the highest yearly sharing of the capital cost compared to other system units. While the Biomass system has the highest operating and maintenance cost compared with other generating units in the suggested hybrid power system.


