Cost Minimizations and Performance Enhancements of Power Systems Using Spherical Prune Differential Evolution Algorithm Including Modal Analysis

: A novel application of the spherical prune differential evolution algorithm (SpDEA) to solve optimal power ﬂow (OPF) problems in electric power systems is presented. The SpDEA has several merits, such as its high convergence speed, low number of parameters to be designed, and low computational procedures. Four objectives, complete with their relevant operating constraints, are adopted to be optimized simultaneously. Various case studies of multiple objective scenarios are demonstrated under MATLAB environment. Static voltage stability index of lowest/weak bus using modal analysis is incorporated. The results generated by the SpDEA are investigated and compared to standard multi-objective differential evolution (MODE) to prove their viability. The best answer is chosen carefully among trade-off Pareto points by using the technique of fuzzy Pareto solution. Two power system networks such as IEEE 30-bus and 118-bus systems as large-scale optimization problems with 129 design control variables are utilized to point out the effectiveness of the SpDEA. The realized results among many independent runs indicate the robustness of the SpDEA-based approach on OPF methodology in optimizing the deﬁned objectives simultaneously.


Introduction
Till this moment, power networks remain one of the most complicated systems in industry due to many reasons, such as variation of generation and load demand, the inclusion of renewable energy systems, and storage devices. These power systems are entirely nonlinear systems, where many components including synchronous generators, transformers, transmission lines, and induction motors have a deep nonlinearity. The optimal operation of such components to achieve a concise economic target, emission minimization possibility, power loss reduction and other objectives under the power system constraints plays an important role in the power system operation and is named optimal power flow (OPF) [1][2][3][4][5]. This problem, as mentioned earlier, is consequently considered a significant nonlinear optimization problem. In this issue, some fitness functions can be sequentially solved under the system operating conditions, including fuel consumption cost (FC), pollution release rate, active power loss, reactive power loss, bus voltage declines and many more.

OPF Mathematical Representations
In this study, the optimization problem is expressed with four objectives for simultaneous optimization. The adopted objectives are depicted in (1)-(4): where TFC is the total FC of the generating units, P Gi is the active output generating power at ith bus, a i , b i , c i are FC coefficients of ith generating unit, N g defines the number of generators, TPL is the system total active power loss, TGP is the total generated power, TPD is total power demand, TVD PQ is the total voltage deviation of all PQ buses, V re f is the reference voltage magnitude (typically, V re f has a value of 1 per unit (pu)), N PQ defines the number of PQ buses, λ i specifies the magnitude of eigen value, and ∆Q mi and ∆V mi define the ith modal reactive power and voltage changes, respectively. It is worth mentioning that the stability of the mode i is based on λ i . The largest λ i indicates slight variations of the modal voltage due to reactive power change. Therefore, the least value of λ i is chosen as a fourth objective, which requires upgrading for a better power system voltage stability. A further derivation regarding eigenvalues and eigenvectors for the modal analysis can be found in [42].
The aforementioned optimization problem undergoes to set of equality/inequality limitations such as: t min k ≤ t k ≤ t max k , ∀ k ∈ N t (8) where P Di is the active demand power at bus i; G ij and B ij are the conductance and susceptance between bus i and j, respectively; δ ij is the power angle between bus i and j; Q Gi is the reactive power generation at bus i; Q Di is the load reactive demand power at bus i; P min Gi , P max Gi are the lower and higher limits of P Gi , respectively; Q min Gi , Q max Gi are the min/max limits of Q Gi , respectively; V min i , V max i are the min/max limits of |V i |, respectively; t min k , t max k are the min/max limits of tap settings of the kth transformer, respectively; N t defines the number of power transformers; S li is apparent power flow in ith line; S rated li specifies the rated line maximum transfer capacity; nbr defines the number of network branches; Q min ci , Q max ci are the min/max limits of Q ci , respectively; and N c defines the number of nominated buses for capacitive devices.
In this study, the vector-defined objectives are solved using the SpDEA and are based on the fuzzification of Pareto fuzzy optimal (PFO) solutions. In this issue, the objective function OF i is expressed by a fuzzy membership function µ i to normalize the values between 0 and 1 as expressed in (11) [23]: In addition, for each k-th Pareto solution, the normalized membership µ k is estimated by the formula depicted in (12): where Noj and M define the number of objectives and the number of PFO solutions, respectively. The best compromise solution is chosen for the minimum value of µ k .

Mathematical Model of SpDEA
The main three strategies for the DE procedure include mutation, crossover, and selection stage. Similar to other challenging optimization methods, at initial stage, a vector of random positions x (0) j,i for the population is created within the predefined boundaries, which can be expressed in (13): j,i = x j, min + r(. . .). x j, max − x j, min , ∀ i ∈ PS, ∀ j ∈ dim (13) where dim defines the number of decision variables, PS is the population-size, r(. . .) is a uniform random distribution function ∈ [0, 1], and x j, min and x j, max are min/max limits of the jth decision control variable, respectively. At this moment, the DE starts to mutate and to recombine the population to generate trial vectors x i . Then, DE implements a uniform crossover procedure to produce trial vectors x i . This is made by mixing x i and the target vector x i based on the crossover probability CR [0, 1], which is combined in a single formula as depicted in (14): (14) where g is the iteration counter, x a is a base-vector and x b and x c are the difference trial vectors that are picked up randomly, factors of a, b, c, and i are unequal and ∝ defines the scaling factor [0, 1]. Then, a move is made to the new position, and as a final point, the collection occurs where a tournament is seized between the target and trial vectors, and the one with the best fitness value is endorsed to pass to the next generation. Through this, agents of a new generation are better than that of the previous ones. The controlling parameters of the DE required for appropriate adaption by the users include ∝ and CR, along with those used with all other competing methods, such as the maximum iterations and PS. In common practice, the tuning of the DE-controlling factors is carried out by trial and error procedures to achieve a satisfactory performance of the DE algorithm.
This work cares by solving multiple objective optimization problems, such as the defined four objectives of OPF as stated in (1)- (4). SpDEA is applied to deal with such anticipated simultaneous objectives. Spherical pruning has been proposed by Reynoso-Meza [43,44] to improve diversity in the approximated PFO. Spherical coordinates are employed to partition the search space, and a selection of one solution is chosen in each spherical sector, evading congestion regions. SpDEA comprises actions to expand relevance applicability and to look effectively for a PFO approximation within the pertinency boundaries [45,46].
The principle motivation of the Sp is to investigate the offered solutions in the current PFO approximation using normalized spherical coordinates from a reference solution in the spherical sector. The general procedures of SpDEA are depicted in the flowchart shown in Figure 1. On the other hand, the detailed procedures of Sp mechanism are shown in Figure 2. The controlling parameters of the DE required for appropriate adaption by the users include ∝ and , along with those used with all other competing methods, such as the maximum iterations and PS. In common practice, the tuning of the DE-controlling factors is carried out by trial and error procedures to achieve a satisfactory performance of the DE algorithm.
This work cares by solving multiple objective optimization problems, such as the defined four objectives of OPF as stated in (1)- (4). SpDEA is applied to deal with such anticipated simultaneous objectives. Spherical pruning has been proposed by Reynoso-Meza [43,44] to improve diversity in the approximated PFO. Spherical coordinates are employed to partition the search space, and a selection of one solution is chosen in each spherical sector, evading congestion regions. SpDEA comprises actions to expand relevance applicability and to look effectively for a PFO approximation within the pertinency boundaries [45,46].
The principle motivation of the Sp is to investigate the offered solutions in the current PFO approximation using normalized spherical coordinates from a reference solution in the spherical sector. The general procedures of SpDEA are depicted in the flowchart shown in Figure 1. On the other hand, the detailed procedures of Sp mechanism are shown in Figure 2.   Further detailed explanation concerning the Sp definitions such as normalized spherical coordinates, sight range, spherical grid and sector, and so on, along with mathematical representations, can be obtained from [43,46]. The overall controlling parameters of SpDEA are ∝, CR, PS, and the number of arcs that should be tuned sufficiently for better performance. Fuzzy-based decision-making methodology [23] is used to choose the best compromise among trade-off PFO points for various scenarios.

Numerical Simulations, Scenarios, and Discussions
This current study focuses on solving the OPF problem in power systems from the multi-objective function perspective using the proposed SpDEA. The problem codes are built with MATLAB software [47]. The OPF problem is solved by different IEEE standard systems such as 30 and 118-bus systems. The main features of these two systems are depicted in Table 1, extracted from [1,15,23]. The aforementioned four objective functions are combined to form different multiobjective functions through the OPF problem. The summary of various vector objectives of the OPF problem is demonstrated in Table 2. The computer simulations are performed using a PC (Intel(R) Core(TM) i7 2.4 GHz µP, 16 GB RAM, and Windows 10 system). The controlling parameters that adjust the performance of the SpDEA are specified in Table 3. Further detailed explanation concerning the Sp definitions such as normalized spherical coordinates, sight range, spherical grid and sector, and so on, along with mathematical representations, can be obtained from [43,46]. The overall controlling parameters of SpDEA are ∝, CR, PS, and the number of arcs that should be tuned sufficiently for better performance. Fuzzy-based decision-making methodology [23] is used to choose the best compromise among trade-off PFO points for various scenarios.

Numerical Simulations, Scenarios, and Discussions
This current study focuses on solving the OPF problem in power systems from the multi-objective function perspective using the proposed SpDEA. The problem codes are built with MATLAB software [47]. The OPF problem is solved by different IEEE standard systems such as 30 and 118-bus systems. The main features of these two systems are depicted in Table 1, extracted from [1,15,23]. The aforementioned four objective functions are combined to form different multiobjective functions through the OPF problem. The summary of various vector objectives of the OPF problem is demonstrated in Table 2. The computer simulations are performed using a PC (Intel(R) Core(TM) i7 2.4 GHz µP, 16 GB RAM, and Windows 10 system). The controlling parameters that adjust the performance of the SpDEA are specified in Table 3.   It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limitations of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

Formulation
Case # Two objectives It is well-meaning to state that the power/load flow of the two systems un is carried out using the full Newton-Raphson method implemented under MA [48]. It may be useful to state here some penalties added to the objective fun produce a feasible solution. Among these penalties, the magnitude of bus volta line flow, and reactive power of generating units are proposed. The typical MAT to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. T tions of generating units' reactive power are controlled through N-R LF, as illu the next subsections. The studied cases of the OPF problem for these IEEE stand systems are demonstrated in the following subsections comprising necessary v comparisons, and discussions. It might be worth mentioning that all runs are on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

IEEE 30-Bus System
In this system, nine buses are nominated for capacitive shunt compensa numbers of {10, 12,15,17,20,21,23,24, and 29} [15,23]. The network data a strated in detail in [1,48]. The voltage magnitude of buses changes in the range the transformer taps settings lie in the range 90%-110% with a step of ±1.25%   It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limitations of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

Formulation
Case # / Two objectives It is well-meaning to state that the power/load flow of the two systems unde is carried out using the full Newton-Raphson method implemented under MATP [48]. It may be useful to state here some penalties added to the objective functio produce a feasible solution. Among these penalties, the magnitude of bus voltages, line flow, and reactive power of generating units are proposed. The typical MATLA to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The tions of generating units' reactive power are controlled through N-R LF, as illustr the next subsections. The studied cases of the OPF problem for these IEEE standard systems are demonstrated in the following subsections comprising necessary valid comparisons, and discussions. It might be worth mentioning that all runs are per on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

IEEE 30-Bus System
In this system, nine buses are nominated for capacitive shunt compensators numbers of {10, 12,15,17,20,21,23,24, and 29} [15,23]. The network data are d strated in detail in [1,48]. The voltage magnitude of buses changes in the range 90 the transformer taps settings lie in the range 90%-110% with a step of ±1.25%, a    It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limitations of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

IEEE 30-Bus System
In this system, nine buses are nominated for capacitive shunt compensators at bus numbers of {10, 12, 15, 17, 20, 21, 23, 24, and 29} [15,23]. The network data are demonstrated in detail in [1,48]. The voltage magnitude of buses changes in the range 90-110%, the transformer taps settings lie in the range 90%-110% with a step of ±1.25%, and the   It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limitations of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

IEEE 30-Bus System
In this system, nine buses are nominated for capacitive shunt compensators at bus numbers of {10, 12, 15, 17, 20, 21, 23, 24, and 29} [15,23]. The network data are demonstrated in detail in [1,48]. The voltage magnitude of buses changes in the range 90-110%, the transformer taps settings lie in the range 90%-110% with a step of ±1.25%, and the    It is well-meaning to state that the power/load flow of the tw is carried out using the full Newton-Raphson method implemen [48]. It may be useful to state here some penalties added to the produce a feasible solution. Among these penalties, the magnitud line flow, and reactive power of generating units are proposed. Th to fulfill the condition of bus voltage and line flow limits is shown tions of generating units' reactive power are controlled through N the next subsections. The studied cases of the OPF problem for the systems are demonstrated in the following subsections comprisin comparisons, and discussions. It might be worth mentioning that on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB instal

Formulation
Case # / Two objectives It is well-meaning to state that the power/load flow of the two systems unde is carried out using the full Newton-Raphson method implemented under MATP [48]. It may be useful to state here some penalties added to the objective functio produce a feasible solution. Among these penalties, the magnitude of bus voltages, line flow, and reactive power of generating units are proposed. The typical MATLA to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The tions of generating units' reactive power are controlled through N-R LF, as illustr the next subsections. The studied cases of the OPF problem for these IEEE standard systems are demonstrated in the following subsections comprising necessary valid comparisons, and discussions. It might be worth mentioning that all runs are per on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

IEEE 30-Bus System
In this system, nine buses are nominated for capacitive shunt compensators numbers of {10, 12, 15, 17, 20, 21, 23, 24, and 29} [15,23]. The network data are d strated in detail in [1,48]. The voltage magnitude of buses changes in the range 90 the transformer taps settings lie in the range 90%-110% with a step of ±1.25%, a    It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limitations of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

IEEE 30-Bus System
In this system, nine buses are nominated for capacitive shunt compensators at bus numbers of {10, 12, 15, 17, 20, 21, 23, 24, and 29} [15,23]. The network data are demonstrated in detail in [1,48]. The voltage magnitude of buses changes in the range 90-110%, the transformer taps settings lie in the range 90%-110% with a step of ±1.25%, and the   It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limitations of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

IEEE 30-Bus System
In this system, nine buses are nominated for capacitive shunt compensators at bus numbers of {10, 12, 15, 17, 20, 21, 23, 24, and 29} [15,23]. The network data are demonstrated in detail in [1,48]. The voltage magnitude of buses changes in the range 90-110%, the transformer taps settings lie in the range 90%-110% with a step of ±1.25%, and the   It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limitations of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

IEEE 30-Bus System
In this system, nine buses are nominated for capacitive shunt compensators at bus numbers of {10, 12, 15, 17, 20, 21, 23, 24, and 29} [15,23]. The network data are demonstrated in detail in [1,48]. The voltage magnitude of buses changes in the range 90-110%, the transformer taps settings lie in the range 90%-110% with a step of ±1.25%, and the   It is well-meaning to state that the power/load is carried out using the full Newton-Raphson metho [48]. It may be useful to state here some penalties a produce a feasible solution. Among these penalties, t line flow, and reactive power of generating units are to fulfill the condition of bus voltage and line flow li tions of generating units' reactive power are control the next subsections. The studied cases of the OPF pr systems are demonstrated in the following subsectio comparisons, and discussions. It might be worth me on a Laptop with Intel ® Core™ i7-7700HQ CPU with

IEEE 30-Bus System
In this system, nine buses are nominated for c numbers of {10, 12, 15, 17, 20, 21, 23, 24, and 29} [1 strated in detail in [1,48]. The voltage magnitude of the transformer taps settings lie in the range 90%-1    It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limitations of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

Formulation
Case # Two objectives  It is well-meaning to state that the power/load flow of the two systems un is carried out using the full Newton-Raphson method implemented under MA [48]. It may be useful to state here some penalties added to the objective fun produce a feasible solution. Among these penalties, the magnitude of bus volta line flow, and reactive power of generating units are proposed. The typical MAT to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. T tions of generating units' reactive power are controlled through N-R LF, as illu the next subsections. The studied cases of the OPF problem for these IEEE stand systems are demonstrated in the following subsections comprising necessary v comparisons, and discussions. It might be worth mentioning that all runs are on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

IEEE 30-Bus System
In this system, nine buses are nominated for capacitive shunt compensa numbers of {10, 12, 15, 17, 20, 21, 23, 24, and 29} [15,23]. The network data a strated in detail in [1,48]. The voltage magnitude of buses changes in the range the transformer taps settings lie in the range 90%-110% with a step of ±1.25%    It is well-meaning to state that the power/load flow of the tw is carried out using the full Newton-Raphson method implemen [48]. It may be useful to state here some penalties added to the produce a feasible solution. Among these penalties, the magnitud line flow, and reactive power of generating units are proposed. Th to fulfill the condition of bus voltage and line flow limits is shown tions of generating units' reactive power are controlled through N the next subsections. The studied cases of the OPF problem for the systems are demonstrated in the following subsections comprisin comparisons, and discussions. It might be worth mentioning that on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB instal

Formulation
Case # / Two objectives It is well-meaning to state that the power/load flow of the two systems unde is carried out using the full Newton-Raphson method implemented under MATP [48]. It may be useful to state here some penalties added to the objective functio produce a feasible solution. Among these penalties, the magnitude of bus voltages, line flow, and reactive power of generating units are proposed. The typical MATLA to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The tions of generating units' reactive power are controlled through N-R LF, as illustr the next subsections. The studied cases of the OPF problem for these IEEE standard systems are demonstrated in the following subsections comprising necessary valid comparisons, and discussions. It might be worth mentioning that all runs are per on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

IEEE 30-Bus System
In this system, nine buses are nominated for capacitive shunt compensators numbers of {10, 12, 15, 17, 20, 21, 23, 24, and 29} [15,23]. The network data are d strated in detail in [1,48]. The voltage magnitude of buses changes in the range 90 the transformer taps settings lie in the range 90%-110% with a step of ±1.25%, a    It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limitations of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

Formulation
Case # Two objectives  It is well-meaning to state that the power/load flow of the two systems un is carried out using the full Newton-Raphson method implemented under MA [48]. It may be useful to state here some penalties added to the objective fun produce a feasible solution. Among these penalties, the magnitude of bus volta line flow, and reactive power of generating units are proposed. The typical MAT to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. T tions of generating units' reactive power are controlled through N-R LF, as illu the next subsections. The studied cases of the OPF problem for these IEEE stand systems are demonstrated in the following subsections comprising necessary v comparisons, and discussions. It might be worth mentioning that all runs are on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

IEEE 30-Bus System
In this system, nine buses are nominated for capacitive shunt compensa numbers of {10, 12, 15, 17, 20, 21, 23, 24, and 29} [15,23]. The network data a strated in detail in [1,48]. The voltage magnitude of buses changes in the range the transformer taps settings lie in the range 90%-110% with a step of ±1.25%   It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limitations of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

IEEE 30-Bus System
In this system, nine buses are nominated for capacitive shunt compensators at bus numbers of {10, 12, 15, 17, 20, 21, 23, 24, and 29} [15,23]. The network data are demonstrated in detail in [1,48]. The voltage magnitude of buses changes in the range 90-110%, the transformer taps settings lie in the range 90%-110% with a step of ±1.25%, and the   It is well-meaning to state that the power/load is carried out using the full Newton-Raphson metho [48]. It may be useful to state here some penalties a produce a feasible solution. Among these penalties, t line flow, and reactive power of generating units are to fulfill the condition of bus voltage and line flow li tions of generating units' reactive power are control the next subsections. The studied cases of the OPF pr systems are demonstrated in the following subsectio comparisons, and discussions. It might be worth me on a Laptop with Intel ® Core™ i7-7700HQ CPU with     It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limitations of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

IEEE 30-Bus System
In this system, nine buses are nominated for capacitive shunt compensators at bus numbers of {10, 12, 15, 17, 20, 21, 23, 24, and 29} [15,23]. The network data are demonstrated in detail in [1,48]. The voltage magnitude of buses changes in the range 90-110%,   It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limitations of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

IEEE 30-Bus System
In this system, nine buses are nominated for capacitive shunt compensators at bus numbers of {10, 12, 15, 17, 20, 21, 23, 24, and 29} [15,23]. The network data are demonstrated in detail in [1,48]. The voltage magnitude of buses changes in the range 90-110%, Sustainability 2021, 13,8113    It is well-meaning to state that the power/load flow of the tw is carried out using the full Newton-Raphson method implemen [48]. It may be useful to state here some penalties added to the produce a feasible solution. Among these penalties, the magnitud line flow, and reactive power of generating units are proposed. Th to fulfill the condition of bus voltage and line flow limits is shown tions of generating units' reactive power are controlled through N the next subsections. The studied cases of the OPF problem for the systems are demonstrated in the following subsections comprisin comparisons, and discussions. It might be worth mentioning that on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB instal   It is well-meaning to state that the power/load is carried out using the full Newton-Raphson metho [48]. It may be useful to state here some penalties a produce a feasible solution. Among these penalties, t line flow, and reactive power of generating units are to fulfill the condition of bus voltage and line flow li tions of generating units' reactive power are control the next subsections. The studied cases of the OPF pr systems are demonstrated in the following subsectio comparisons, and discussions. It might be worth me on a Laptop with Intel ® Core™ i7-7700HQ CPU with     It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limitations of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

Formulation
Case # Two objectives  It is well-meaning to state that the power/load flow of the two systems un is carried out using the full Newton-Raphson method implemented under MA [48]. It may be useful to state here some penalties added to the objective fun produce a feasible solution. Among these penalties, the magnitude of bus volta line flow, and reactive power of generating units are proposed. The typical MAT to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. T tions of generating units' reactive power are controlled through N-R LF, as illu the next subsections. The studied cases of the OPF problem for these IEEE stand systems are demonstrated in the following subsections comprising necessary v comparisons, and discussions. It might be worth mentioning that all runs are on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.    It is well-meaning to state that the power/load flow of the tw is carried out using the full Newton-Raphson method implemen [48]. It may be useful to state here some penalties added to the produce a feasible solution. Among these penalties, the magnitud line flow, and reactive power of generating units are proposed. Th to fulfill the condition of bus voltage and line flow limits is shown tions of generating units' reactive power are controlled through N the next subsections. The studied cases of the OPF problem for the systems are demonstrated in the following subsections comprisin comparisons, and discussions. It might be worth mentioning that on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB instal    It is well-meaning to state that the power/load is carried out using the full Newton-Raphson metho [48]. It may be useful to state here some penalties a produce a feasible solution. Among these penalties, t line flow, and reactive power of generating units are to fulfill the condition of bus voltage and line flow li tions of generating units' reactive power are control the next subsections. The studied cases of the OPF pr systems are demonstrated in the following subsectio comparisons, and discussions. It might be worth me on a Laptop with Intel ® Core™ i7-7700HQ CPU with

IEEE 30-Bus System
In this system, nine buses are nominated for c numbers of {10, 12, 15, 17, 20, 21, 23, 24, and 29} [1 It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limitations of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.   It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limitations of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

IEEE 30-Bus System
In this system, nine buses are nominated for capacitive shunt compensators at bus numbers of {10, 12, 15, 17, 20, 21, 23, 24, and 29} [15,23]. The network data are demonstrated in detail in [1,48]. The voltage magnitude of buses changes in the range 90-110%, the transformer taps settings lie in the range 90%-110% with a step of ±1.25%, and the reactive power of the shunt compensator is 5 MVAr. The SpDEA solves the different multiple objectives that are formulated in Table 2. The optimal characteristics of the SpDEA are  Table 3. Many independent runs have been carried out to determine the optimal values of control variables. The optimal values of control variables and their corresponding fitness values are pointed out in Table 4. It may be noted that in the N-R LF using MATPOWER, the option of "pf.enforce_q_lims" is set to 1. As a result of activating this option, any generator's reactive power exceeds the min/max limits after running the N-R LF, and the corresponding bus is converted to a PQ-bus, with Q g . If the reference bus is converted to PQ-bus, the first remaining PV-bus will be used as the slack bus for the next iteration at the limit, and the case is re-run. The voltage magnitude at the bus will deviate from the specified value to satisfy the reactive power limit. In other words, the relevant operating constraints of the reactive power output from the generators are maintained within practical operating points. The corresponding generator's reactive power outputs are arranged in Table 5 of design settings for various objectives using the SpDEA. Figure 4a-c illustrate PFO solutions and best compromise value for cases 1-3, representing the anticipated bi-objective cases. On the other hand, Figure 5a-c illustrate PFO solutions and best compromise value for cases 4-6, representing the anticipated tri-objective cases. It is worthy of note here that these Pareto optimal solutions lie in an acceptable range of minimization of each simultaneous objective function. Moreover, the best compromise value is located at a proper value within the suggested PFO solutions.  Figure 4a-c illustrate PFO solutions and best compromise value for cases 1-3, representing the anticipated bi-objective cases. On the other hand, Figure 5a-c illustrate PFO solutions and best compromise value for cases 4-6, representing the anticipated tri-objective cases. It is worthy of note here that these Pareto optimal solutions lie in an acceptable range of minimization of each simultaneous objective function. Moreover, the best compromise value is located at a proper value within the suggested PFO solutions.   For a rational comparison, the SpDEA-based OPF results are compared with MODE and others, summarized in Table 6. It can be noted from this comparison that the best compromise values of OPF solutions using the SpDEA are very competitive, even though the TFC and TNL functions are contradictory. It reflects the proper design of the SpDEA to solve the multi-objective OPF problem in power systems. In addition to that, the average elapsed CPU times for various objectives' scenarios are mentioned in the last row of Table 4. It may be noted that bold font indicates the best results obtained so far. For a rational comparison, the SpDEA-based OPF results are compared with MODE and others, summarized in Table 6. It can be noted from this comparison that the best compromise values of OPF solutions using the SpDEA are very competitive, even though the TFC and TNL functions are contradictory. It reflects the proper design of the SpDEA to solve the multi-objective OPF problem in power systems. In addition to that, the average elapsed CPU times for various objectives' scenarios are mentioned in the last row of Table 4. It may be noted that bold font indicates the best results obtained so far.   Figure 3. The limitaunits' reactive power are controlled through N-R LF, as illustrated in s. The studied cases of the OPF problem for these IEEE standard power nstrated in the following subsections comprising necessary validations, discussions. It might be worth mentioning that all runs are performed Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

of 16
Scenarios of multiple objective representation of the OPF problem. is well-meaning to state that the power/load flow of the two systems under study d out using the full Newton-Raphson method implemented under MATPOWER may be useful to state here some penalties added to the objective function (s) to e a feasible solution. Among these penalties, the magnitude of bus voltages, power , and reactive power of generating units are proposed. The typical MATLAB code l the condition of bus voltage and line flow limits is shown in Figure 3. The limita-generating units' reactive power are controlled through N-R LF, as illustrated in t subsections. The studied cases of the OPF problem for these IEEE standard power s are demonstrated in the following subsections comprising necessary validations, isons, and discussions. It might be worth mentioning that all runs are performed ptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

Formulation
Case # / Two objectives 1 It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limita-tions of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

Formulation
Case # / Two objectives 1 It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limita-tions of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

Formulation
Case # / Two objectives 1 It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limita-tions of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

Formulation
Case # / Two objectives 1 It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limita-tions of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.   It is well-meaning to state that the power/load flow of the two systems und is carried out using the full Newton-Raphson method implemented under MAT [48]. It may be useful to state here some penalties added to the objective funct produce a feasible solution. Among these penalties, the magnitude of bus voltage line flow, and reactive power of generating units are proposed. The typical MATL to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. Th tions of generating units' reactive power are controlled through N-R LF, as illus the next subsections. The studied cases of the OPF problem for these IEEE standa systems are demonstrated in the following subsections comprising necessary va comparisons, and discussions. It might be worth mentioning that all runs are p on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.
MOALO [50] 826. 46    It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limita-tions of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

Formulation
Case # / Two objectives 1 It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limita-tions of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

Formulation
Case # / Two objectives 1 It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limita-tions of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

Formulation
Case # / Two objectives 1 It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limita-tions of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.   It is well-meaning to state that the power/load flow of the two systems und is carried out using the full Newton-Raphson method implemented under MAT [48]. It may be useful to state here some penalties added to the objective funct produce a feasible solution. Among these penalties, the magnitude of bus voltage line flow, and reactive power of generating units are proposed. The typical MATL to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. Th tions of generating units' reactive power are controlled through N-R LF, as illus the next subsections. The studied cases of the OPF problem for these IEEE standa systems are demonstrated in the following subsections comprising necessary va comparisons, and discussions. It might be worth mentioning that all runs are p on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.  ell-meaning to state that the power/load flow of the two systems under study ut using the full Newton-Raphson method implemented under MATPOWER y be useful to state here some penalties added to the objective function (s) to feasible solution. Among these penalties, the magnitude of bus voltages, power nd reactive power of generating units are proposed. The typical MATLAB code e condition of bus voltage and line flow limits is shown in Figure 3. The limita-nerating units' reactive power are controlled through N-R LF, as illustrated in bsections. The studied cases of the OPF problem for these IEEE standard power e demonstrated in the following subsections comprising necessary validations, ns, and discussions. It might be worth mentioning that all runs are performed p with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.
: not covered in this paper.

IEEE 118-Bus System
In this scenario, the SpDEA is applied to solve different vector objectives of the OPF problem for the standard 118-bus IEEE system [48] with the main features stated in Table 1. The capacitive compensating devices are installed on the bus numbers of {34, 44,45,46,48,74,79,82,83,105,107, and 110} and its maximum rated capacity is 30 MVAr [19,22,23]. The data are demonstrated in detail in Table 2. The generating voltage magnitude of buses changes in the range 90-110%, the tap settings lie in the range 90%-110%. This optimization problem involves 129 control variables, posing a large challenge to the proposed algorithm. The optimal characteristics of the proposed SpDEA for this test case are demonstrated in Table 7 (last column). The various multiple objectives that are formulated in Table 2 are solved using the proposed SpDEA. The best values of control variables and their best compromise settings are recorded. However, the data of all cases are huge, and it is incredibly difficult to write in a paper. Therefore, only one case (case 1) is proposed, as illustrated in Table 7. The best compromise Pareto records 140,700 $/h and 30.339 MW to the TFC and TNL of the network. Figure 6a illustrates PFO solutions and best compromise value for case 1. It can be noted here that the SPDEA selects the best compromise Pareto within the search space. It is worthy to note here that these PFO solutions lie in an acceptable range of minimization of each simultaneous objective function.
Moreover, Figure 6b,c point out the PFO solutions and their corresponding best compromise for the other cases that include bi-objective functions (cases 2-3). On the other hand, Figure 7a-c indicate the PFO solutions and their corresponding best compromise for three simultaneous objective functions (cases 4-6). The best compromise values for all these cases lie in acceptable ranges that can be confirmed. In addition, the SpDEA is applied to case 7 for solving the multi-objective function that contains all four objectives. In this issue, the best compromise Pareto records 150,718 $/h, 33.69 MW, 1.415 pu, and 0.2525 to the TFC, TNL of the system, TVD for PQ buses, and voltage stability index, respectively. Therefore, it is successfully applied to various multiple adopted scenarios of the OPF problem regarding spot-on large-scale power systems. Indeed, a fair comparison should be made between the proposed SpDEA-based OPF results and the MODE-based results to point out the flexibility, availability, and strength of the SpDEA-based OPF methodology. In this regard, Table 8 points out this detailed comparison for all seven cases. It is well-meaning here to say that the numerical results of the OPF problem using the SpDEA-based on OPF methodology are very similar to those obtained by the conventional MODE. The main merit of the proposed SpDEA is that it selects the best compromise among PFO solutions from several hundred Paretos within the search space. Moreover, the best compromise Paretos are within acceptable ranges. This property gives it a high possibility of reaching the optimal point in a quick way. This desired property is obviously illustrated in Figure 5. This reflects the proper design of the SpDEA to solve the multiple vector simultaneous objectives of the OPF problem in electric power networks.  Moreover, Figure 6b-c point out the PFO solutions and their corresponding best com promise for the other cases that include bi-objective functions (cases 2-3). On the oth hand, Figure 7a-c indicate the PFO solutions and their corresponding best compromi for three simultaneous objective functions (cases 4-6). The best compromise values for these cases lie in acceptable ranges that can be confirmed. In addition, the SpDEA is a plied to case 7 for solving the multi-objective function that contains all four objectives. this issue, the best compromise Pareto records 150,718 $/h, 33.69 MW, 1.415 pu, and 0.25 to the TFC, TNL of the system, TVD for PQ buses, and voltage stability index, respective Therefore, it is successfully applied to various multiple adopted scenarios of the OP problem regarding spot-on large-scale power systems.  Indeed, a fair comparison should be made between the proposed SpDEA-based OPF results and the MODE-based results to point out the flexibility, availability, and strength of the SpDEA-based OPF methodology. In this regard, Table 8 points out this detailed comparison for all seven cases. It is well-meaning here to say that the numerical results of the OPF problem using the SpDEA-based on OPF methodology are very similar to those obtained by the conventional MODE. The main merit of the proposed SpDEA is that it selects the best compromise among PFO solutions from several hundred Paretos within   ell-meaning to state that the power/load flow of the two systems under study out using the full Newton-Raphson method implemented under MATPOWER y be useful to state here some penalties added to the objective function (s) to feasible solution. Among these penalties, the magnitude of bus voltages, power nd reactive power of generating units are proposed. The typical MATLAB code e condition of bus voltage and line flow limits is shown in Figure 3. The limitanerating units' reactive power are controlled through N-R LF, as illustrated in bsections. The studied cases of the OPF problem for these IEEE standard power e demonstrated in the following subsections comprising necessary validations, ns, and discussions. It might be worth mentioning that all runs are performed p with Intel ® Core™ i7-7700HQ CPU with 16    ell-meaning to state that the power/load flow of the two systems under study out using the full Newton-Raphson method implemented under MATPOWER y be useful to state here some penalties added to the objective function (s) to feasible solution. Among these penalties, the magnitude of bus voltages, power nd reactive power of generating units are proposed. The typical MATLAB code e condition of bus voltage and line flow limits is shown in Figure 3. The limitanerating units' reactive power are controlled through N-R LF, as illustrated in bsections. The studied cases of the OPF problem for these IEEE standard power e demonstrated in the following subsections comprising necessary validations, ns, and discussions.   It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limitations of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

IEEE 30-Bus System
In this system, nine buses are nominated for capacitive shunt compensators at bus numbers of {10, 12, 15, 17, 20, 21, 23, 24, and 29} [15,23]. The network data are demonstrated in detail in [1,48]. The voltage magnitude of buses changes in the range 90-110%, the transformer taps settings lie in the range 90%-110% with a step of ±1.25%, and the ell-meaning to state that the power/load flow of the two systems under study out using the full Newton-Raphson method implemented under MATPOWER y be useful to state here some penalties added to the objective function (s) to feasible solution. Among these penalties, the magnitude of bus voltages, power nd reactive power of generating units are proposed. The typical MATLAB code e condition of bus voltage and line flow limits is shown in Figure 3. The limitanerating units' reactive power are controlled through N-R LF, as illustrated in bsections. The studied cases of the OPF problem for these IEEE standard power e demonstrated in the following subsections comprising necessary validations, ns, and discussions. It might be worth mentioning that all runs are performed p with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.   It is well-meaning to state that the power/load flow of the two systems under study is carried out using the full Newton-Raphson method implemented under MATPOWER [48]. It may be useful to state here some penalties added to the objective function (s) to produce a feasible solution. Among these penalties, the magnitude of bus voltages, power line flow, and reactive power of generating units are proposed. The typical MATLAB code to fulfill the condition of bus voltage and line flow limits is shown in Figure 3. The limitations of generating units' reactive power are controlled through N-R LF, as illustrated in the next subsections. The studied cases of the OPF problem for these IEEE standard power systems are demonstrated in the following subsections comprising necessary validations, comparisons, and discussions. It might be worth mentioning that all runs are performed on a Laptop with Intel ® Core™ i7-7700HQ CPU with 16 GB installed memory.

IEEE 30-Bus System
In this system, nine buses are nominated for capacitive shunt compensators at bus numbers of {10, 12, 15, 17, 20, 21, 23, 24, and 29} [15,23]. The network data are demonstrated in detail in [1,48]. The voltage magnitude of buses changes in the range 90-110%, 3  At last, it can be concluded that the SpDEA proves its viability in solving a large-scale conventional power system such as a standard IEEE 118-bus network. It is well-known strategically that many countries worldwide have planned to increase their shares of renewable energy generation from sources such as solar, wind, tidal, and many more, including energy storage facilities [51][52][53]. As a result, such penetrations to conventional power systems increase uncertainty. Therefore, it is important to extend the existing frameworks/methodologies to address this challenge. The later mentioned defines the future trend of our current work by incorporating system uncertainties and the variability of different types of renewable power sources and loads.

Conclusions
A novel application of the spherical prune differential evolution algorithm has been demonstrated to solve the OPF problem in electric power schemes to achieve simultaneous objectives under various scenarios. The OPF problem has been investigated with the wellknown IEEE standard 30-bus and 118-bus networks as a large-scale optimization problem with 129 design control variables. All constraints have been respected with no violations. MATPOWER has been used to implement the full load flow analysis of the networks under study using full Newton-Raphson method. PFO solutions are generated, and the best settings are carefully selected by using the technique of normalized fuzzifications. The best results for the IEEE 30-bus system with quad objectives of TFC, TPL, TVD PQ and 1/λ i are equal to 840.92 $/h, 5.89 MW, 9458 PU and 1.879, respectively. On the other hand, for the best results for the IEEE 118-bus system are equal to 150,718 ($/h), 33.6933 MW, 1.41465 PU and 0.252516 for the same quad objectives, respectively. The demonstrated numerical simulations using the proposed SpDEA-based OPF methodology have proved their high performance, effectiveness, and robustness for solving the OPF problem of power systems in comparison to others reported in the literature.
Since the shares of renewable power sources including energy storage facilities are booming, uncertainty is increasing. Therefore, it is important to extend the existing frameworks/methodologies to address this challenge. The later mentioned defines the future trend of our current work by incorporating system uncertainties and the variability of different types of renewable power sources and loads.

Institutional Review Board Statement:
The study did not involve humans or animals.

Informed Consent Statement:
The study did not involve humans.
Data Availability Statement: Not applicable.