By taking the ratio of a measure of an attribute with and without the hybrid WE/PEMFC system at the same loading conditions, some indices can be derived for any attribute as follows:

1) Minimizing BVI:

F 1 = M i n B V I % = | V B − V B , H | / V B (6)

where; V_B and V_b,H are the system bus voltage without and with the hybrid WE/ PEMFC system, respectively.

2) Minimizing ALI:

F 2 = M i n A L I % = ( A L S K , H / T A L s y s ) (7)

where, AL_sk,H is the APLs of section k of the EDS with considering the hybrid WE/PEMFC system. TAL_sys is the total active power losses (TAPLs) of the EDS without considering the hybrid WE/PEMFC system.

3) Minimizing RLI:

F 3 = M i n R L I % = ( R L S K , H / T R L s y s ) (8)

where, RL_sk,H is the RPLs of section k of the EDS with considering the hybrid WE/PEMFC system. TRL_sys is the total reactive power losses (TRPLs) of the EDS without considering the hybrid WE/PEMFC system.

The MOF is formulated as:

F = M i n M O F = W 1 ⋅ F 1 + W 2 ⋅ F 2 + W 3 ⋅ F 3 (9)

with 0 ≤ W i ≤ 1 , ∑ i = 1 3 W i = 1

where; W₁,W₂ and W₃ are the weighting factors (WFs) for BVI%, ALI% and RLI%, respectively. And can be obtained by trial and error.

The SOF/MOF are subjected to the following system constraints:

1) System bus voltage constraints

The voltage at each bus V_B must be within their minimum and maximum limits (V_bmin and V_bmax) as:

V B M I N ≤ V B ≤ V B M A X (10)

2) Total number of hybrid units (DGs) constraint

The number of DGs (N_DGs) must be less than or equal to the maximum permissible number of DGs (N_DGmax).

N D G s ≤ N D G m a x (11)

3) Active power flow constraint

The active power flow in the k^th line (AF^K) should be less than or equal to the maximum permissible active power flow (AF^Kmax).

A F K ≤ A F K m a x (12)

4) Reactive power flow constraint

The reactive power flow in line k (RF^K) should be less than or equal to the maximum permissible reactive power flow (RF^Kmax).

R F K ≤ R F K m a x (13)

CSA is a nature-inspired algorithm presented by Askarzadeh in 2016 [31], CSA mimics the behavior of Crow birds and their social interaction. Crows watch other birds and observe where they hide their food. Once the other birds leave, they steal their food. The Crows use their intelligence to hide their excess food in a hideout spot and restore this food when they need. If a Crow discovers another one is going after it, it will try to deceive that Crow and will go to another position. CSA can be implemented as follow [32] [33]:

● Initialization: the Crow in the folk represents a solution to the problem.

● If there is a solution space with dimension d has a Crow folk of N Crows, the position X of Crowi at iteration t can be expressed by the vector:

X i , t = [ X 1 i , t   X 2 i , t   X 3 i , t   ⋯   X d i , t ]

● Each Crow has a memory to store the best position of its stored food source.

● The memoryM of all Crows at iteration t for dimension d are initialized as follows:

M = [ m 11 t ⋯ m 1 d t ⋮ ⋱ ⋮ m N 1 t ⋯ m N d t ]

● Solution evaluation (fitness): each solution in the population is evaluated using objective function.

● Position update: to update the position of the Crow between Crows J and i, there are two cases:

Case 1: Crow J does not recognize that Crow i is going after it. Hence, Crow i will get close the storing place of Crow J.

Case 2: Crow J recognizes that Crow i is going after it, so, it will move to another position to deceive Crow i and to save its food.

X i , t + 1 = { X i , t + r a i × F l i , t × ( m j , t − X i , t )             r J ≥ A P J , t randomposition                                               otherwise

where; AP is the awareness factor and Fl^i,t is the flight length of Crow i at iteration t. The main reasons of using CSA are for easy implementation, few control parameters to adjust, fast convergence speed and high efficiency. To further enhance the performance of the classical CSA, BCSA is proposed. At each position, the solution considers values “0” or “1”, where, “0” represents the solution that is not selected and “1” represents the solution that is selected.

Jaya algorithm is one of the newly developed population based optimization method, which was proposed by Rao in 2016 [34]. The Jaya algorithm is simple in putting into use as a solution is updated only in a single phase by employing a single equation. The basic concept of the Jaya is based on the best and worst solution of objective function. Jaya needs only the common control parameters and does not need any algorithm clearly stated control parameters [35]. A new DJA is the optimization algorithm used here for finding the optimal allocation of the hybrid system problem. In this algorithm, the solution contains two phases, meanwhile the candidate locations are rounded about its real values and the number of candidate buses are determined by binary strategy. The number of the best locations, with their hybrid WE/PEMFC system, are updated by the DJA in each iterations. The algorithm is continued until the termination criterion is met.

The mechanics of PSO technique are inspired by the swarming of biological populations. In PSO technique, the particles change their positions with time [36]. The classical PSO is operated on real values, here the BPSO is proposed which is operated on [0, 1] values.

The optimal allocation procedure is illustrated below:

1) Input data: input the number of system buses, load demand at each bus, transmission lines’ impedance, control parameters of the used optimization technique.

2) Perform the initial load flow analysis using the BFS method to calculate the SFs, the VSF are arranged in ascending order for all buses of the study system. While, the ALSF and RLSF are arranged in descending order for all system lines. Thus, the candidate buses for inserting the hybrid WE/PEMFC system can be determined.

3) Implement one of the three proposed optimization techniques.

4) Perform the load flow analysis using BFS method again to obtain the voltage at each system bus, APLs and RPLs.

5) Calculate the fitness function of SOF/MOF for each proposed solution.

6) Repeat until the maximum number of iterations is reached or the solution is obtained.

7) The previous steps will be repeated with the other two proposed optimization techniques. Then the obtained results are compared with each other.

The average output of WT is given as follows [38]:

P a v e = exp [ − ( V c i n / C ) K ] − exp [ − ( V r / C ) K ] [ ( V r / C ) K − ( V c i n / C ) K ] − exp [ − ( V c o u t / C ) K ] ⋅ P r (14)

where; P_ave is the average power, P_r is the rated power, V_cin is the cut-in wind speed, V_r is the rated wind speed and V_cout is the cut-out wind speed, C and K are the scale and shape parameters of Weibull probability distribution function, respectively. The average output of the wind farm is given as follows:

P A , W F = P a v e × N D G s (15)

The unit energy cost (U_E.C) of WT is calculated dependent on the value of both the annual capital cost (AC.C) and the annual operation cost (AO.C). The U_E.C is given as:

U E . C = ( A C . C + A O . C ) / A E W T     $ / year (16)

where; AE_WT is the energy output of WT per year. The total unit energy cost of the wind farm (U_WF) is given as:

U W F = U E . C × N D G s + P L a n d     $ / year (17)

where; P_Land is the price of land used for the installation of the wind farm.

The amount of H₂ that needs to be supplied the FC to produce one kWh of electricity from the surplus power of wind farm can be obtained as follow [39]:

H 2 A m o u n t = ( 2 ∗ n r a t e ∗ 3600 ) / ( ( 237.2 n r a t e ) ∗ 10 − 3 ) gram / kWh (18)

where, n_rate is the rate of flow of H₂ into the cell in (mol/sec).

The electrolyzer size is determined by its production flow rate (H₂) as illustrated [41]:

H 2 r a t e = ( 60 ∗ E L ∗ n e ∗ g m ) / ( U G ∗ V c e l l ∗ n e m ∗ n m m ) (22)

where; H_2rate is the H₂ flow rate, E_L is the average electrical load in Wh/day, n_e is the number of electrons per second for 1 ampere, g_m is the molar mass of H₂ in g/mol, U_f is the utilization factor which takes 0.8, V_cell is the cell voltage, n_em is the number of electrons per each molecule of H₂ and n_mm is the number of molecules per H₂ mol. The electrolyzer size can be calculated by dividing the total amount of H₂ per year required from the PEMFC by the total windy hours per year as follow:

E S i z e = ( H 2 r a t e ∗ E L ∗ 365 ) / h W i n d (23)

where; h_wind is the total windy hours per year.

The volume of the H₂ tank can be determined from the ideal gas low as follow [41]:

V T = ( n ∗ R ∗ T ) / P (24)

where; V_T is the volume of the H₂ tank, n is the number of moles, R is the gas constant, T is the temperature in Kelvin and P is pressure.

The annual cost (AC) of PEMFC system can be calculated as follow:

where, C_O&M is the annual operating and maintenance costs of the PEMFC system, Y is the life time of the PEMFC system, C_C is the capital cost, C_Tank is the annual cost of H₂ tank, C_Electro is the annual cost of the electrolyzer, S_v is the annual salvage value, C_run is the annual running cost and i is the interest rate. The unit energy cost (C_OE) can be calculated by dividing the AC of the PEMFC system by the total generated energy, during the system cycle as follow:

where, E^e is the electrical generated energy in kWh.

Figure 3 includes four feeders, each feeder consists of four buses and five connecting cables, except the third feeder which includes only three buses and four connecting cables. This EDS is completely supplied from hybrid WE/PEMFC system that depends on WT which has rated of 3000 kW for each turbine combined with PEMFC stacks which has rated of 5 kW for each unit. Figure 4 and Figure 5 show the values of VSF, ALSF and RLSF, respectively. According to system buses ordering in these figures, the first seven buses of the bus-number are considered as the candidate buses for allocating the hybrid WE/PEMFC system.

Minimizing the SOF is performed using the three proposed optimization techniques without and with considering the SFs. Table 6 and Table 7 show the optimal allocating for the hybrid system, and the corresponding number of the hybrid units (DGs) using BCSA, DJA and BPSO techniques in year 2022 without and with considering SFs, respectively. While, Table 8 shows a comparison among the fitness values using the three proposed optimization techniques with and without the hybrid system for the SOF without and with considering the SFs in year 2022.

Table 8 illustrates the improvement in F₁ without and with considering the SFs using the three proposed optimization techniques in year 2022. From Table 8, the best technique without considering the SFs is DJA that gives improvement in bus voltage about 75%, then, BCSA improves the bus voltage by 72.3%, finally BPSO technique improves the bus voltage by 71.5%. Also, the best technique is

DJA when considering the SFs that improves the bus voltage about 97.5%, then, BCSA improves the bus voltage by 97.3%, finally BPSO technique improves the bus voltage by 94.7%. There is a clear improving in the value of F₁ when considering the effect of the SFs as F₁ is increased from 75% to 97.5% using DJA. Figure 6 and Figure 7 show the improvement in voltage profile using the three proposed optimization techniques without and with considering the SFs for F₁ as SOF, respectively.

Table 8 illustrates the saving in ALI without and with considering the SFs. From this Table, the best technique without considering the SFs is BCSA that gives the highest saving in active power losses (APLs) by 77.2%, then, DJA gives saving in APLs by 77%, finally BPSO technique gives saving in APLs of about 75.5%. Also, the best technique when considering the SFs is DJA that gives the highest saving in APLs by 79.27%, then, both BCSA and BPSO technique that give the same saving in APLs about 79.09%. There is a significant improvement in the value of F₂ when considering the SFs that the saving in APLs is increased from 77% to 79.27 using DJA.

Table 8 illustrates the saving in F₃ without and with considering the SFs using the three proposed optimization techniques, the best technique without considering the SFs is DJA that gives the highest saving in reactive power losses (RPLs) by 77.8% then, BCSA saves the RPLs by 77.12%, finally BPSO technique saves the RPLs by 76%. Also, the best technique, when considering the SFs, is DJA that gives the highest saving in RPLs by 79.55%, then, both BCSA and BPSO give the same saving in RPLs by 79.32%. There is a significant improving

in the value of F₃ after taking the SFs into consideration which the saving in RPLs is increased from 77.8% to 79.55% using DJA.

MOF are dependent on the values of WFs, which takes the value of W₁ as 0.5 to consider the voltage improvement is the first priority, while, the values of W₂ and W₃ take different values between [0 to 0.5 by step 0.05]. However, the values of W₂ and W₃ have taken as 0.45 and 0.05, respectively, to optimize the MOF using both BCSA and BPSO technique without and with considering the SFs. While, W₂ and W₃ are taken as 0.05 and 0.45, respectively, to optimize the MOF using DJA without and with considering the SFs. Table 9 shows the sitting and corresponding number of DGs using BCSA, DJA and BPSO technique using MOF without and with considering the SFs. A comparison of the fitness value using three proposed optimization techniques, with and without the hybrid WE/ PEMFC system using the MOF without and with considering SFs is shown in Table 10. From Table 10, DJA is the best technique without considering the SFs that minimizes the MOF from 67.07% (without inserting the hybrid system) to 18.07%, (after inserting the hybrid system), reduces the TAPLs from 1.2348 MW to 0.313 MW, and, reduces the TRPLs from 1.544 MVAR to 0.3812 MVAR. Also, DJA is the best technique with considering the SFs that minimizes the MOF from 67.07% (without inserting the hybrid system) to 17.22% (after inserting the hybrid system) reduces the TAPLs from 1.2348 MW to 0.2601 MW, and reduces the TRPLs from 1.5421 MVAR to 0.3231 MVAR. There is a significant minimization in the value of MOF with considering the effect of SFs, however, MOF is reduced from 18.07% to 17.22% using DJA.

Figure 8 and Figure 9 show the improvement in voltage profile using the three proposed optimization techniques without and with the effect of SFs when considering the MOF in year 2022.