Multi-Objective Optimization Framework for Optimal Power Flow Problem of Hybrid Power Systems Considering Security Constraints

The hybrid model of the power system infrastructure is an essential part of the sophisticated technology of the electrical network. Generally, for the Optimal Power Flow (OPF) problem, the power system with only thermal generators is considered. In traditional OPF problems, the fuel cost required to produce electrical energy is considered, and emissions are frequently neglected. Renewable Energy Sources (RESs) have received increasing attention due to various potential characteristics such as clean, diversity, and renewability. As a result, RESs are being integrated into the existing electrical grid at an increasing rate. The study in this paper proposes a techno-economic investigation into the single- and multi-objective OPF, coordinating with RESs, such as wind, PhotoVoltaic (PV), and small hydropower units with hybrid PV. Moreover, the probability density functions of Weibull, Lognormal, and Gumble have been used to predict the required power. A recently reported equilibrium optimizer and its multi-objective version are considered for handling OPF problems. The superior performance of the equilibrium optimizer is further verified with the results of both single- and multi-objective through comparative analysis with state-of-the-art counterparts, and the indications are that the suggested algorithm can find better optimal solutions in a smaller number of generations (iterations) with faster convergence and well distributed optimal Pareto front for multi-objective problems. The results are verified by employing an IEEE-30 bus hybrid power network, and performance comparisons are made among well-established algorithms. Simulation findings show that the suggested algorithm can achieve a reasonable compromise solution for different objectives.


I. INTRODUCTION
The Optimal Power Flow (OPF) has become recognized as one of the more challenging problems that must be handled in the planning and operation of modernized power systems. It is anticipated that the power system would need to be The associate editor coordinating the review of this manuscript and approving it for publication was Ruisheng Diao . run in the best possible condition to provide the highest possible levels of security and reliability. Essentially, the OPF is a multi-model, non-linear minimization problem with high-dimensional features and great computational complexity. Modern society's socioeconomic expansion has posed a great challenge to the electrical power grid. Energy consumption rises each day as the demand for customers grows exponentially. Meanwhile, rising electricity consumption VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ and the rapid depletion of natural resources could present major supply and demand management problems for utilities [1], [2], [3]. To address this problem, an attractive alternative is to use price-based load management programs to encourage users to alter existing consumption patterns or to deploy on-site distributed Renewable Energy Sources (RESs) to improve power network performance, dependability, reliability, and profitability. Integration of information and communication tools with grid systems pushes the development of conventional grids into smart grids [4], [5], [6]. In this regard, control design system components must be considered and incorporated into the OPF analysis. The OPF is an important strategy for calculating transmission and distribution system power losses as well as power generating costs. The most fundamental and crucial way of managing and controlling an electrical network is the OPF method. The primary purpose of OPF is to define the network's stable operating condition by achieving a certain goal while adhering to equality and inequality constraints [7], [8], [9]. Carpentier [10] was the first to introduce it, and it has been studied over the past half-century. The OPF's traditional objective has been to build coal-based generation systems that run on fossil fuels. Because of the growing usage of sustainable power in the power system, the assessment of the OPF is particularly important after incorporating the irregularity of these renewable units of energy. Researchers from entirely over the world have examined OPF pertaining to thermal-based power generation units [11], [12], [13], [14].
The OPF techniques are regarded as a fundamental and crucial resource for power system operators to assure the sustainability and stability of the system. The OPF algorithms are typically executed at defined time frames to optimize system operation to adjust the values of several dependent parameters to their optimal setting. Numerous researchers have utilized traditional OPF procedures that examine the Economic Environment Dispatch (EED) problem but cater to thermal-based electrical power generators, resulting in CO 2 emissions [15], [16]. The OPF problem is complicated from an economic and computational standpoint. In the context of an economic power network, load flow can be viewed as a problem of Economic Load Dispatch (ELD) to lower the generation costs. The ELD appears to be unable to deal with many operating constraints. The demand non-linearities, losses during the transmission, and the capacity of the generation units can add to the complexity of the load or power flow problem. During its operation, the OPF identifies the optimal values for several control variables, which adds to the computing complexity. The Multi-Objective Grey Wolf Optimizer (MOGWO) can find the Pareto optimal front of any shape. Finally, the result of a complex real-world Multi-Objective OPF (MOOPF) problem validates that the MOGWO algorithm can solve any kind of non-linear and complex problem with many constraints in unknown search space [17], [18], [19].
Meanwhile, special emphasis is being paid to incorporating renewable energy supplies into the grid to decarbonize the electric power system. Solar photovoltaic and wind deployments are continuously rising due to concerns about global warming due to climate change. On the alternative, fossil fuel-based power generation systems are becoming less adaptable. According to recent research, dynamic energy supplies such as solar and wind have characteristics that span several time scales, impacting various layers of power system regulation. Such results indicate that, as RES adoption grows, standard load flow analyses aren't adequate to assure dependability through efficient utilization of resources. It is also proven that, due to the high integration of RESs, operators have difficulties in handling power requirements and depend on user curtailment. Furthermore, the unpredictability and intermittent nature of fluctuating energy sources are expected to raise backup capacity needs, hence raising the incremental cost of electricity [20], [21], [22]. Different alternatives have already been dedicated to the OPF problems for the last few decades. To handle the hydrothermal OPF, the authors of [23] have utilized nonlinear and linear programming methods. The OPF with continuous and discrete optimization parameters was solved using a mixed Particle Swarm Optimization (PSO) algorithm [24]. For the OPF problem with integrated security restrictions, the authors of [25] presented a PSO with reconstructive operators. The authors of [26] have solved the OPF problem using PSO with aging challengers and leaders. The OPF problem was optimized using glowworm swarm optimization in [27], using the minimization of emission and generation cost as fitness functions. To handle the OPF effectively, the authors of [28] have devised an enhanced colliding bodies optimizer. The Differential Evolutionary (DE) algorithm for single-and multi-objective optimal power flow problems was given in Ref. [29]. For the OPF problem, the authors of [30] have presented an enhanced Artificial Bee Colony (ABC) algorithm based on orthogonal learning. To overcome the limited OPF challenge, the authors of [31] have proposed a unique moth swarm method. Upgraded selfadaptive DE with a mixed crossover algorithm was used to address multi-objective OPF problems with conflicting objectives [32]. To obtain OPF solutions, the authors in [33] and [34] used a DE algorithm combined with effective constraint handling strategies. The authors of [35] have presented an enhanced social spider optimizer for tackling the optimal power flow problem with a single objective. The authors of [36] suggested an adaptive perturbationguiding JAYA algorithm developed to deal with various single-objective optimal power flow problems. Although these algorithms have produced more satisfactory outcomes, the majority of such algorithms are based on conventional thermal-based power generation units and ignore other essential objectives, such as emission reduction of toxic gases, such as CO 2 , NO x, and SO x in ton/hr. Since the citations above consider traditional generation units, some articles now consider an electrical system combining wind and coal-based power units in their quest for the lowest generating prices. The authors of [37] presented the Gbest-directed ABC to increase OPF results reported in previous studies using the same experimental design. In the OPF problem, the authors of [38] proposed an Improved Bacteria Foraging Algorithm (IBFA) and a framework for a double-fed induction generator to emphasize restrictions on the ability to create VAR fuel. The STATCOM is another VAR power compensating device for a network with coalbased and wind units (static synchronous compensator). Ant colony optimization and IBFA were also used to tackle the OPF problem [39]. The authors of [40] suggested a method for assessing wind power prices. The difficulty of planning generators for ELD is common for a grid with both coalbased and wind units. The random nature of the wind power output was provided by [41]. Researchers in [42] also used the DFIG wind turbine model despite solving a comparable problem. The complex ELD problem, which included a wide variety of wind energy and risk reserve constraints, was discussed in [43]. The authors of [44] have incorporated the valve-point loading impact in the Dynamic ELD (DELD) structure of the producing unit and emanation. The optimal power flow scheduling approach is given in [45] for a single system with battery energy storage, Photovoltaic (PV), and gas-based power generation units. The authors of [46] defined pumped hydro storage as a storage alternative for a freestanding hybrid network comprising a wind turbine, solar PV unit, and a diesel generator. The problem was developed using three sources, as mentioned in ref. [47], in addition to a single objective DELD problem. The authors of [48] have used non-conventional sources such as coal-based and solar-wind power, as well as battery storage, to achieve optimum scheduling. Biswas et al. [34] established a framework for evaluating OPF challenges for solar, wind, and river units with various probability density functions.
Even though incorporating renewable energy resources into OPF is addressed in such literature, it should have been viewed as a starting point rather than an end. Numerous literature reviews examine a single optimization objective when addressing the OPF problem with renewable energy. Other objectives, especially those related to emissions, are significant and, therefore, should be considered. In conclusion, multi-objective OPF with renewable energy sources merits further investigation. The typical IEEE-30 bus system is enhanced in this research by incorporating probabilistic wind and solar energy. Sustainable energy generators, like traditional power generators, are now aiming to emphasize everyday research more. In addition to coal-based units, solar, wind, and small-hydro power generation units are considered in this paper, and the OPF problem is handled by both singleand multi-objective algorithms. For research purposes, these sustainable energy sources are being integrated into the standard IEEE 30-bus network. The Equilibrium Optimizer (EO) technique can be used to generate clusters of solutions for single-objective and multi-objective problems [49], [50]. The Multi-Objective Whale Optimization Algorithm (MOWOA) is developed by equipping the whale optimization algorithm with a crowding distance, an archive, and whales' position (according to ranking) selection method based on Pareto optimal dominance nature.
The MOWOA algorithm is first applied on 17 standard test functions (including eight unconstraint, five constraints, and four engineering design multi-objective problems) to prove its capability in terms of qualities and quantities showing numerical as well as convergence and coverage of Pareto optimal front with respect to true Pareto front. Then, after the MOWOA algorithm is applied to a real-world complex MOOPF problem, the algorithm is proved with a Summation-based Multi-Objective DE (SMODE) algorithm, Multi-Objective Symbiotic Search Algorithm (MOSOS), Multi-Objective Colliding Bodies Optimization (MOCBO) algorithm, Multi-Objective PSO (MOPSO), Non-Dominated Sorting Genetic Algorithm (NSGA-II), and other well-known algorithms in the field of multi-objective algorithms. A storage and leader selection strategy were then combined into a single objective Ion Motion Optimization (IMO) approach to solving multiobjective problems. A group of unconstrained, constrained, and engineering benchmark functions were employed to assess the performance of the Multi-Objective IMO (MOIMO) [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61]. The outcomes of MOIMO are compared with those of the Multi-Objective Dragonfly Algorithm (MODA), Multi-Objective Multi-Verse Optimizer (MOMVO), and Multi-Objective Ant Lion Optimizer (MOALO). It was evident that the MOIMO algorithm is very competent and modest in determining an exact estimation of the Pareto optimal front with uniform dispersal across considered objectives with minimum simulation time. The major contributions of this paper are as follows.
• Formulation of objective functions for OPF problem with single-and multi-objective to handle four objective functions, such as minimization of total fuel cost, voltage deviation, voltage stability index, and the active power loss • Development and application of single-and Multi-Objective Equilibrium Optimizer (MOEO) in OPF problem of hybrid power network considering the security constraints • Application of suitable Probability Density Functions (PDFs) to randomize the behavior of solar PV, wind, and small hydro plants • Validating the suggested algorithm on a modified IEEE-30 bus hybrid power system network • Comparative analysis is made among well-established algorithms The paper is organized as follows. Section 2 describes the mathematical models that present the uncertainty in solarwind-small hydro energy outputs in the context of the OPF problem. Section 3 clarifies the various objective functions. Section 4 describes the EO and MOEO algorithms and their application in the OPF problem. The quantitative findings and analysis are found in Section 5, while the concluding remarks are found in Section 6. VOLUME 10, 2022 FIGURE 1. Enhanced IEEE 30-bus hybrid wind-solar-small hydroelectric power plant system [34].

TABLE 1.
Key specification of the system of the hybrid network considered in this paper [34].

II. MATHEMATICAL MODEL
This paper considers the enhanced IEEE-30 bus network to validate the performance of the EO and MOEO when it handles the OPF problem. The power framework of the improved IEEE-30 bus system is shown in Fig. 1, where thermal units are located at buses 1, 2, and 8, one solar PV unit is located at bus 11, one wind unit is located at bus 5, and one small hydro with hybrid PV unit is located at bus 13. Table 1 summarizes a complete specification of the selected IEEE-30 bus power system. Table 2 shows the upper and lower bounds of the selected power system.

A. GENERATION COST OF THERMAL UNITS
As discussed in the literature, there are general optimization objectives, namely, minimizing the generating cost, the active power loss, and the voltage deviation. In classical coal-based power generation, the cost is computed as follows, allowing the valve-point loading effect.
where a i , b i , and c i denote the cost coefficients for i th generator, and d i and e i denote the cost constants for i th generator with a loading effect. The values of the emission and cost coefficients are presented in Table 3.

B. EMISSION
Minimizing emissions is an extremely important challenge. One main reason is that typical thermal power generation units release toxic gases, such as CO 2 , NO x , and SO x, into the atmosphere, leading to pollution. The emission in tons/hour can be computed as follows.
where, α i , β i ,γ i , ω i and µ i are the emission constants of the i th unit and the values of such coefficients are listed in Table 3.

C. DIRECT COSTS OF STOCHASTIC UNITS
The direct cost associated with the wind unit is mathematically modeled with the P ws scheduled power from the same sources. The associated cost is as follows.
where, g w signifies the direct cost coefficient and P ws signifies the scheduled wind power. Likewise, the direct cost associated with the photovoltaic unit is presented with the P ss scheduled power from the same sources. The associated cost is presented as follows.
where, h s signifies the direct cost coefficient and P ss signifies the scheduled PV power. A hybrid solar unit with a smallhydro unit is known as a third sustainable energy source.
So, the scheduled power consists of the summation of the hydropower and the PV output power. The direct costs for the PV and the small-hydro units are as follows.
where, P ss denotes the expected production from the hybrid plant, P ssh,s denotes the influence of the PV unit, and P ssh,h denotes the influence of the small-hydro unit. The direct variables of price h s and m s are similar to the previous discussion.

D. UNCERTAIN SUSTAINABLE WIND POWER COST
The standby cost for the wind unit is formulated as follows.
where, K Rw denotes the reserve cost coefficient for wind component, the wind power Probability Density Function (PDF) is represented by f w (p w ), and P w,s denotes the amount of power that is accessible from the same wind unit. The penalty cost for the same is given as follows.
where, P wr signifies the rated output from the wind unit and K Pw denotes the wind penalty cost coefficient.

E. UNCERTAIN SUSTAINABLE PV POWER COST
The standby cost of a solar PV unit can be written as follows.
where, K Rs denotes the standby price constant regarding solar PV units, P s,av denotes the amount of power that is accessible from the same PV unit, f s (P s,av < P s,s ) represents the probability of PV output shortage rate concerning scheduled output power P s,s , and E(P s,av < P s,s ) represents the expected PV output power lesser than P s,s . The penalty price for overrating PV units can be given as follows.   where, K Ps represents the coefficient of penalty price regarding SPV unit, f s (P s,av > P s,s ) represents the expected PV output in excess of the scheduled output, and E(P s,av > P s,s ) represents the expected PV output higher than P s,s .

F. UNCERTAIN SUSTAINABLE HYBRID SOLAR PV AND SMALL-HYDRO POWER COST
In this subsection, another sustainable generation network is a cluster of small-scale hydropower units, and solar PV units are considered. The water flow strength determines the generation of the small-hydro unit that is famous for accompanying Gumbel distribution [41]. Satisfying Eq. (8), the reserve cost for hybrid unit output is as follows.
where, K Rsh is the standby cost coefficient and is related to the hybrid unit and P shav is the actual accessible output from the system. The possibility of a shortage of hybrid system output from the scheduled power (P ssh ) is given by f sh (P shav < P ssh ) and the principle of distributed power below P ssh is E(P shav < P ssh ). After Eq. (9), the cost of penalty for underestimating the performance of the hybrid system is as follows. (11) where, K Psh is the penalty cost coefficient is related to the hybrid unit, f sh (P shav > P ssh ) is the possibility of the excess of hybrid system power from the scheduled power (P ssh ) , andE(P shav > P ssh ) is the expectancy of the hybrid system output above P ssh .

G. UNCERTAINTY OF STOCHASTIC SOLAR/WIND/SMALL-HYDRO UNITS
Wind speed distributions are represented using the Weibull Probability Density Function (PDF). The following is how the Weibull PDF was used by the authors of [34] to depict the probability of wind speed in meters per second. The data of the suggested Weibull scale (c) and shape (k) constants were presented in Table 4. The distributions of the Weibull curve and wind frequency are shown in Fig. 2 by taking 8000 steps from the Monte-Carlo simulation. The probability of Weibull PDF following wind velocity vm/s, including scale factor (c) and shape factor (k) could be  estimated as follows.
The Weibull distribution mean is reported as follows.
The expression for (x) is presented as follows.
The thermal plant at bus-11 of the standard IEEE-30 bus system is replaced with the PV unit. The power production of the PV unit is contingent on sun irradiation (G) that follows the Lognormal PDF [34]. The solar irradiation probability tracking Lognormal PDF has a standard deviation σ and mean µ can be presented as follows.
By replicating the Monte-Carlo setup with a set point of 8000, Figure 3 defines a frequency distribution and Lognormal distribution of solar irradiation. The mean of the Lognormal distribution is given as follows.
VOLUME 10, 2022  The possibility of water flow rate Q w , subsequent Gumbel PDF through position constant λ and scale constant γ has been written as follows.
The hybrid PV and hydropower unit are at bus-13 of the enhanced network, swapping the coal-based generator. Fig. 4 shows the solar PV PDF output and Lognormal scale obtainable for the PV located at bus-13. Figure 5 illustrates the strength of water mass flow distribution and the scale of Gumbel. To perform an 8000 Monte Carlo setup, both figures are generated using the basic standards of PDF parameters tabulated in Table 4.

H. POWER MODELS OF SOLAR PV, WIND, AND HYBRID PLANTS
The wind plant at bus-5 comprised the combined outputs of the 25 turbines. The power capacity of each turbine is 3 MW. Depending on the speed, the exact energy production of the wind turbine varies. In terms of wind velocity (v), the turbine output power is expressed as follows.
where, v out , v r , and v in , denotes the cut-out, rated, and cut-in speed of the wind turbine, respectively and p wr signifies the rated generated power of the turbine. The Enercon E82-E4 model datasheet is referenced for the 3MW wind turbine. The different speeds are v out = 25 meter/sec, v r = 16 meter/sec, and v in = 3 meter/sec. The solar irradiation P s (G) to transform the energy of the photovoltaic can be represented as follows.
where, R c denotes a particular irradiance position set as 120 W m 2 , G std denotes the solar irradiation in the typical environment fixed at 1000 W m 2 , and the PV unit's rated performance is P sr . The flow rate of water (Q w ) and effective pressure head (H w ) are calculated by the output of a small hydro plant. The efficiency of the small hydro plant can therefore be determined as follows.
where η stands for the efficiency of the generator-turbine set, g denotes the acceleration due to gravity, and ρ denotes the water concentration. To calculate the hydro output, the parameters are set at these values, η =0.85, ρ =1000 kg/m 3 , g = 9.81 m/s 2 , and H w =25 m. Table 5 shows the quantitative value of direct, penalty, and reserve cost factors for wind, PV, and small hybrid hydropower. The direct cost factors are set, so that wind energy is the most priced, trailed by PV power and hydroelectric power. The reserve cost coefficient is larger than the direct cost coefficient for sustaining spinning reserves. On the other hand, the penalty for not using the power available is less than the direct cost.

III. PROBLEM FORMULATION
The optimal VAR power and real power dispatch objectives are included in the OPF. The following objectives for OPF with PV, wind, and hybrid plants are included in this section. Entirely the fitness functions considered in this paper are minimization functions.

A. MINIMIZATION OF TOTAL FUEL COST (TFC)
The very first objective is the reduction of the total overall cost. Including coal-based unit prices and direct, reserve, and penalty prices for non-conventional resources is the expense of the entire generation. Therefore, the combined price for a coal-based power plant, a wind power plant, a solar photovoltaic power plant, and a hybrid solar and small hydropower plant is presented as follows.

B. VOLTAGE DEVIATION (VD) MINIMIZATION
The VD is a measure of voltage reliability of entire load buses in the system. The voltage variations of the PQ bus can be controlled starting at 1.0 for every unit, resulting in a more consistent voltage profile overall. The problem is formulated as follows.
where v i represents the voltage level of i th bus in p.u. and N pq presents the number of PQ buses.

C. MINIMIZATION OF ACTIVE POWER LOSS (APL)
Transmission line losses are caused by the transmission cables' due to their resistive nature. When electrical energy flows from one node to the other in a branch, it dissipates as heat. Whenever this loss is incurred, it impacts the node voltages, which may be explained in terms of the magnitude and angle of the node voltage. The third objective function for minimizing transmission network Active Power Loss (APL) can be represented as follows.
where, P Gi and P Di denote the generated power output and power distributed at i th bus, and NB denotes the number of buses.

D. VOLTAGE STABILITY INDEX (L-INDEX)
The L-index, which depicts each bus's voltage uniformity margins, is the most important metric for keeping the constant voltage at a tolerable level under typical conditions. L-index provides a scalar count for any PQ bus. The L-index remains between '0' (no load) and '1' (maximum load) (collapse of voltage). The total voltage collapse indication for the i th bus is as follows.
where Y 2 and Y 1 denote the sub-matrices of Y BUS . The fitness function to improve stability is described as follows.

E. EQUALITY CONSTRAINTS
The power flow expressions provide equality constraints, showing that both reactive and real power generated in a network necessity satisfy the system's power demands and losses.
where NB signifies the total buses, P Di and Q Di signify the real power and reactive power demand at i th bus, δ ij = δ i − δ j denotes the variance in phase angles of voltage among bus i and bus j, and P Gi and Q Gi are the active power and reactive power generation respectively of i th bus by either unit (coalbased or renewable) as appropriate. G ij displays the conductance and B ij signifies the susceptance among bus j and bus i, correspondingly.

F. INEQUALITY CONSTRAINTS
The functional limitations of systems, as well as the security limits of lines and PQ buses, constituted inequity constraints.
The following are the generator boundaries:

IV. SINGLE-AND MULTI-OBJECTIVE EQUILIBRIUM OPTIMIZER
This part describes the elementary notion of the singleobjective equilibrium optimizer and the Multi-Objective Equilibrium Optimizer (MOEO) formulation.

A. EQUILIBRIUM OPTIMIZER (EO)
Equilibrium Optimizer is a metaheuristic optimization tool that was reported by Faramarzi et al. [49]. The strategy is based on mass conservation balances for a particle entering and departing a regulated volume, to achieve equilibrium. The optimization procedure, likewise, tries to maintain the system's balance. There are three primary mathematical representations, i) initializing concentrations, ii) computing an equilibrium pool of candidates, and iii) updating concentrations. The mathematical model of the EO algorithm is discussed in this sub-section. The readers should read the base paper for comprehensive details [49]. Like entirely other metaheuristic algorithms, the initial random population solution of EO is generated using Eq. 40.
where n denotes the number of particles in the population, C initial i signifies the initial concentration vector, C min and C max are the lower and upper bounds, and rand i is a random number between [0,1]. The best solution is to metaphorically reach an equilibrium condition by calculating candidate solutions and the equilibrium pool. To reach the unknown equilibrium state, the EO uses five concentrations. The particle pool is made up of the average of the four best concentrations so far. The following are the pool's characteristics. Equation 42 is used to update the population solution in EO. (44) where Max_FEs denotes the total number of function evaluations, and FE is the current function evaluations.

FE Max_FEs
where r1, r2, and r are arbitrary numbers ∈ [0, 1], and r1 and r2 are factors that balance the exploitation and exploration phase of EO. The term sign ( − → r −0.5) regulates the exploitation and exploration directions. The rate − → G is a section that may progress the divergence procedure and − − → RCP controls if the control variable is utilized or not in the updating procedure.

B. MULTI-OBJECTIVE EQUILIBRIUM OPTIMIZER
In multiobjective problems, two or more objective functions are solved at the same time when entirely constraints are satisfied [51], [52], [53]. To describe the multi-objective OPF problem, several optimization approaches are used in the literature. From the previous works, it can be seen that many researchers have turned a multiobjective problem into a single-objective problem by combining the two conflicting objective works into a single-objective problem and then employing the weighting components approach. Furthermore, estimating the set of optimal tradeoffs and finding the best compromise solutions around each of the Pareto fronts may be the best method for determining the result of the multi-objective problem. The multi-objective problem should be written as follows.
Minimize : f i (u) , i = 1, 2, 3, . . . , N (49) Subjected to: g j (u) = 0, j = 1, 2, 3, . . . , M (50) where f i is the irrational fitness function, N indicates the total fitness function, M denotes the number of equality constraints, and K denotes the total inequality constraints. The non-dominated sorting strategy in multiobjective optimization might have two probabilities: one objective dominating the other or not. To put it another way, without losing the ability to generalize; Only if the given two requirements are met, does u 1 have the upper hand over u 2 .
If any of the preceding conditions are not met, the solution u 1 is no longer in control of u 2 . If u 1 outnumbers u 2 , it is referred to as the non-dominated solution. The pseudocode of the MOEO is presented in the Algorithm [50]. The maximum function evaluations and the population size N pop are control parameters of the MOEO algorithm. In addition, a random parent population P o is generated in search space S and every fitness function of the fitness vector F for P o is evaluated. P o is then subjected to non-dominated sorting and the determination of crowding distances. The MOEO method is then used to create the new population P j , which is subsequently converged with P o to form the combined population P i . The best N pop solutions are referred to as parent population based on non-domination sorting and calculated estimates of crowding distance and non-domination rank. This process is repeated until the maximum number of generations has been reached. It's worth noting that a similar method can be used in conjunction with the termination criteria based on the function's overall evaluations. The readers are highly recommended to read [50] for more comprehensive details. For a non-dominated solution, the membership function score for every fitness function is first determined as follows.   (55) where N obj is the number of the fitness functions and N d represents the number of non-dominated solutions. The value with the highest µ k value is the most compromised.

Algorithm Pseudocode of MOEO Algorithm
Step-1: Generate random population P o in the solution set S and fitness vector F for the generatedP o Step-2: Categorize the P o using non-dominated sorting process and compute the non-dominated rank and generate Pareto fronts Step-3: Determine the crowding distance for every Pareto front generated Step-4: Update the P j solutions by EO algorithm Step-5: To produce P i = P o ∪ P j , combine P o and P j Step-6: For P i , complete step-2 as per the rank and crowding distance sort P i Step-7: Replace P o with P i for first N pop of P i

V. RESULTS AND DISCUSSIONS
In this section, the validation of EO and MOEO on an enhanced IEEE-30 bus system is carried out, and the performance comparison is also made among well-established algorithms. There are two simulation scenarios considered in this section. Scenario 1 deals with the single-objective OPF problem, and Scenario 2 deals with the multi-objective OPF problem.
A.  This study uses the EO algorithm to address the probabilistic optimal power flow problem with three sustainable energy units: wind, solar, and hybrid small-hydropower. The features of the enhanced IEEE-30 bus network are listed in Table 1. Table 6 shows the entire test cases considered in this paper. The proposed strategy is developed using the MATLAB tool installed on a 3.4GHz Intel i5 CPU with 8 GB memory. The population size is set to 40, and each algorithm is run 10 times with 100 iterations. The EO algorithm is related to finding solutions for sustainable power generation units. The solution to the OPF problem is also verified and compared with the recently developed algorithms such as Ion Motion Optimizer (IMO) [54], Grey Wolf Optimizer (GWO) [55], and Harris Hawks Optimization (HHO) [56].
The obtained results for the case study from 1-5 are listed in Tables 7-9. Table 6 lists the decision vectors for case studies 1-2, including the best Total Fuel Cost (TFC) and the emission. The results obtained by entirely algorithms are also recorded in Table 7. Table 7 shows that the TFC value obtained by the EO is 891.651 $/hr, which is the best, followed by GWO, HHO, and IMO. Similarly, the value of emission obtained by EO is 0.092 Ton/hr, which is the best in addition to the GWO and IMO. The emission achieved by the HHO is higher than the selected algorithms. Table 8 lists the decision vectors for case studies 3-4, including the best APL and the VD values. The results obtained by selected algorithms are also recorded in Table 8. Table 8 shows that the APL value obtained by the EO is 1.866 MW, which is the best, followed by IMO, GWO, and HHO. Similarly, the value of VD obtained by EO is 0.284 p.u., which is best followed by GWO, HHO, and IMO. Table 9 lists the decision vectors for case study 5, including the best VSI (L-index) values. The results obtained by selected algorithms are also recorded in Table 9. From Table 9, it is noticed that the L-index value obtained by the EO is 0.133, which is similar to the selected algorithms.
Some of the traditional and well-established algorithms are called the Multi-Objective Evolutionary Algorithm with Superiority of Feasible solution constraint handling mechanism (MOEA/D-SF) and Summation-based Multi-Objective DE algorithm with Superiority of Feasible solution constraint handling mechanism (SMODE/SF) is applied for first case studies of OPF problem [34]. The consolidated results obtained by EO and other selected algorithms, including MOEA/D-SF and SMODE/SF for all case studies, are listed in Table 10. The result of TFC, including the sustainable power plants with EO, is 891.651 $/hr which is the best among the four algorithms. Also, reduced up to 1.352 $/hr in comparison with the MOEA/D-SF and 1.852 $/hr in VOLUME 10, 2022  comparison with the SMODE/SF, as seen in Table 10. The boldfaces in the tables indicate the best result.
The convergence curves of entire case studies are illustrated in Figs. 6-10. As consolidated in Table 9, the EO algorithm produces the best optimal results for all case studies of single-objective OPF problems. From the convergence curves, it is observed that the EO algorithm produces the best optimal results, in addition to quick converges than the selected algorithms. For most entirely case studies of singleobjective OPF, HHO produces the worst results, and the convergence speed is also poor than entire algorithms. Next to HHO, the IMO algorithm also produces worst results than the GWO and EO, apart from HHO. The results produced by the GWO are almost similar to EO for a few case studies; however, the convergence speed is slower than the EO algorithm. Therefore, from the discussions, it is discovered that the EO algorithm stood first, followed by GWO, IMO, and HHO.
Comparative analysis plots with respect to total fuel cost are shown in Fig. 11. While analyzing the simulated results of entire case studies, it is shown that the suggested EO algorithm provides the best optimal solutions.

B. SCENARIO-2
In this scenario, the MOEO algorithm was applied to optimize two, three, and four conflicting objectives simultaneously.
The non-dominated sorting EO algorithm is used in the multiobjective OPF problems to determine the archives of different objectives simultaneously. The Pareto front for the VOLUME 10, 2022   adapted IEEE 30-bus framework is obtained using 30 nondominate results. Similar to the previous test scenario, scenario 2 was executed 30 times to get fair results. Cases 6 to 8 are referred to as two objectives optimization problems, whereas cases 9 and 10 are referred to as three objectives optimization problems. In case 11, entirely four objectives are optimized at the same time. The fuzzy membership function is utilized to determine the best compromising response from the Pareto archives [57], [58]. Comparing the best Pareto fronts obtained by the four established multi-objective  algorithms are important. The optimal Pareto fronts generated by entirely algorithms are shown in Figs. 12(a-f). For cases 6 to 9, the best-compromised responses using the MOEO algorithm and other well-established algorithms, such as MOWOA [59], MOGWO [18], [60], and MOIMO [61], are shown in Table 11, and the best results are shown in boldface. The selected multi-objective algorithms are applied to generate the optimal Pareto fronts for case studies. The population size for entire algorithms is 30, and the maximum number of generations is 500. Other algorithmic parameters of entire algorithms are selected as per the respective base paper. The Pareto fronts obtained for Cases 6-11 are shown in Fig. 12. As seen in the Pareto fronts, MOEO has more diversity than other algorithms. In other words, the emission, TFC, APL, and VD values reached by MOEO is better than that of selected algorithms, which is true throughout trial runs.
Compliance with system constraints is a crucial part of the constraint optimization problem. In a single objective problem, dealing with constraints is challenging. The scenario is significantly more complicated for multiobjective problems, such as the OPF with stochastic renewables because the Pareto front contains numerous solutions necessary to be practicable. The Pareto front produced from every test run contains 30 non-dominated solutions, and algorithms provide feasible opportunities. However, it is not feasible to VOLUME 10, 2022   verify all solutions at this time, and we do focus on choosing selected solutions from specific test runs and monitoring the availability of important constraints. When an EO is combined with a non-dominated sorting and crowding distance mechanism, the search process of the MOEO can be systematically guided towards optimal global solutions.

VI. CONCLUSION
The research on OPF of hybrid power systems has become more significant in modern trends in the energy industry and environmental policy since it analyses sustainable and constantly replenished renewable resources. Therefore, this paper suggests an algorithm for locating Pareto's optimal solutions to the multiobjective OPF problem, which includes stochastic solar, wind, and hybrid small hydropower units. The renewable energy sources studied in this study have stochastic characteristics that are represented using suitable probability density functions. The non-convex, nonlinear, and multimodal multi-objective OPF problem is solved using the MOEO algorithm and formulated using EO, non-dominated sorting, and crowding distance mechanisms. The system's constraints, notably network security constraints, have been satisfied. The obtained results are compared with several well-established algorithms. The results reveal that the suggested MOEO algorithm obtained higher quality with more practicable solutions. Without a doubt, entirely of the results suggest that the proposed algorithm has an advantage in achieving optimal solutions to single-and multi-objective OPF problems.
In future studies, the MOEO algorithm may also be applied to other engineering problems, including cyber security, demand forecasting, motor design, optimal reactive power dispatch, compensator location, transformer design, load scheduling, distribution planning, and many more.