A novel MOGNDO algorithm for security-constrained optimal power flow problems

Citation: Pandya, S.B.; Visumathi, J.; Mahdal, M.; Mahanta, T.K.; Jangir, P. A Novel MOGNDO Algorithm for Security-Constrained Optimal Power Flow Problems. Electronics 2022,11, 3825. https://doi.org/10.3390/ electronics11223825 Academic Editor: Dario Di Cara Received: 26 September 2022 Accepted: 16 November 2022 Published: 21 November 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). electronics Article A Novel MOGNDO Algorithm for Security-Constrained Optimal Power Flow Problems Sundaram B. Pandya 1, James Visumathi 2, Miroslav Mahdal 3,* , Tapan K. Mahanta 4and Pradeep Jangir 5 1Department of Electrical Engineering, Shri K.J. Polytechnic, Bharuch 392 001, India 2Department of Computer Science and Engineering, Vel Tech Rangarajan Dr Sangunthala R&D Institute of Science and Technology, Chennai 600 062, India 3Department of Control Systems and Instrumentation, Faculty of Mechanical Engineering, VSB-Technical University of Ostrava, 17. Listopadu 2172/15, 708 00 Ostrava, Czech Republic 4School of Mechanical Engineering, Vellore Institute of Technology, Chennai 600 127, India 5Rajasthan Rajya Vidyut Prasaran Nigam, Losal, Sikar 332 025, India *Correspondence: miroslav[email protected] Abstract: The current research investigates a new and unique Multi-Objective Generalized Normal Distribution Optimization (MOGNDO) algorithm for solving large-scale Optimal Power Flow (OPF) problems of complex power systems, including renewable energy sources and Flexible AC Transmission Systems (FACTS). A recently reported single-objective generalized normal distribution optimization algorithm is transformed into the MOGNDO algorithm using the nondominated sorting and crowding distancing mechanisms. The OPF problem gets even more challenging when sources of renewable energy are integrated into the grid system, which are unreliable and fluctuating. FACTS devices are also being used more frequently in contemporary power networks to assist in reducing network demand and congestion. In this study, a stochastic wind power source was used with different FACTS devices, including a static VAR compensator, a thyristordriven series compensator, and a thyristor—driven phase shifter, together with an IEEE-30 bus system. Positions and ratings of the FACTS devices can be intended to reduce the system’s overall fuel cost. Weibull probability density curves were used to highlight the stochastic character of the wind energy source. The best compromise solutions were obtained using a fuzzy decision-making approach. The results obtained on a modified IEEE-30 bus system were compared with other well-known optimization algorithms, and the obtained results proved that MOGNDO has improved convergence, diversity, and spread behavior across PFs. Keywords: FACTS controller; MO-OPF; meta-heuristics; probability density function; stochastic; WTGS 1. Introduction Constraint-based optimization problems with multiple objectives are the most prevalent type. In contrast to single-objective optimization problems, multi-objective optimization problems have a wide variety of optimal solutions. The PF is an assortment of perfect responses [ 1 , 2 ]. A multi-objective optimization approach must be able to locate solutions that are uniform in the generated PFs and are workable optimum solutions to address multi-objective problems [ 3 ]. Multi-objective optimization approaches are challenged by the simultaneous achievement of these many objectives [ 4 ]. MH algorithms are typically tested on simpler, well-known optimization scenarios. However, unlike classic search problems, engineering design tasks can have different specifications. Modifying and developing the algorithm for them is the most effective way to optimize for them. The realm of application of multi-objective optimization algorithms is quite vast, ranging from machining processes [5,6], to vehicle routing [7], to optimizing AI systems [8]. Power systems researchers have been seeking solutions to the OPF challenges for many decades. One issue with managing power systems and making plans for modern electrical Electronics 2022,11, 3825. https://doi.org/10.3390/electronics11223825 https://www.mdpi.com/journal/electronics Electronics 2022,11, 3825 2 of 34 energy networks is working with systems that use non-conventional sources of energy. Ullah et al. [ 9 ] developed a hybrid phasor particle swarm optimization and gravitational search algorithm to address the OPF problem in the wind and solar energy systems that are connected to electrical power grids, while accounting for the control variables (PPSOGSA). For the OPF problem with wind and solar systems, the developed PPSOGSA algorithm produced outstanding and helpful results. In Elattar’s research, the OPF problem was principally modelled mathematically using a combined heat and power system with stochastic wind energy. On an IEEE 30-bus test system under various test situations, the suggested method was assessed. The formulation of the OPF problem covering energy sources and the suggested approach to solve it produced effective answers in contrast to preexisting algorithms [ 10 ] that were used to address similar problems. Anongpun et al. [ 11 ] used IEEE 30and 118-bus test systems to study the use of enhanced particle swarm optimization (PSO) to solve a multi-objective OPF problem using a wind energy system that combined chaotic mutation and stochastic weights. When compared to the other algorithms included in their study, the suggested method produced better results. Salkuti [ 12 ] used the glowworm swarm optimization method to offer a solution to a multi-objective OPF problem requiring a modern electrical energy system that utilized wind energy. On IEEE 30and 300-bus test systems, the methodology was examined in many operational scenarios. The simulation findings indicated that the proposed methodology might offer an alternative. A flower pollination algorithm was used by Kathiravan et al. [ 13 ] to address the OPF problem using coal-based, wind, and solar energy systems. In a variety of test situations, the authors used their method to test systems for the IEEE 30-bus and Indian utility 30-bus. Duman et al. [ 14 ] used differential evolutionary particle swarm optimization to address the OPF problem (DEEPSO) with manageable wind and solar (PV) energy sources. IEEE 30-, 57-, and 118-bus test systems were used to evaluate the DEEPSO technique to explore the issue under various objective functions. DEEPSO produced better simulation results when compared to the other optimization approaches that were looked at [ 14 ]. They recommended using FACTS devices such as a thyristor-controlled phase shifter (TCPS) and a thyristor-controlled series capacitor (TCSC) to solve the OPF problem. To account for the uncertainties associated with wind energy installation, they used chaotic maps and a modified version of the PSOGSA (particle swarm optimization and gravitational search algorithm). The method presented [ 15 ] appears to be a potential approach for a solution based on the findings of simulations. Biswas et al. [ 16 ] solved the OPF challenge, which incorporated coal-based, wind, solar, and small-hydro energy sources coupled to IEEE 30bus test systems, by running multiple rounds of the multi-objective evolutionary algorithm. The constrained multi-objective population extremal optimization (CMOPEO) technique was used to handle the wind and solar-integrated OPF problem by Chen et al. [ 17 ], who also tested the method on an IEEE 30-bus for different scenarios. Additionally, research has been done on the evolutionary particle swarm optimization (EPSO) [ 18 ], the hybrid differential evolution and symbiotic organisms search algorithm (HMICA-SQP) [ 19 ], the success-history-based adaptation of differential evolution with superiority of feasible solutions (SHADE-SF) [ 20 ], and the hybrid modified imperialist competitive algorithm and sequential quadratic programming algorithm (HMICA-SQP) [ 21 ]. Pandya and Jariwala [ 22 ] addressed single and multi-objective OPF issues by integrating with various sustainable energy sources using recently developed metaheuristics algorithms. Biswas et al. [ 23 ] analyzed the integration of three FACTS devices, wind turbines, and coal-fired power plants. The success history-based adaptive differential evolution (SHADE) method was used to conduct the investigation. According to the “No Free Lunch (NFL)” theorem [ 24 ], no metaheuristic can solve every issue that occurs in real-world situations. This theorem has opened the door to the creation of both novel metaheuristic techniques and improvements to existing ones. When using the Generalized Normal Distribution Optimization method [ 25 ] in multiobjective optimization scenarios, several things need to be taken into account. The initial problem in multi-objective generalized normal distribution optimization is balancing con- Electronics 2022,11, 3825 3 of 34 vergence and divergence archives. The algorithm’s generated solution set is filtered according to a certain quality metric, and the non-dominating solution set is kept in separate, external archives. In the literature, there are numerous recommendations for constructing archives that could be employed in the multi-objective Generalized Normal Distribution Optimization method. The pre-defined maximum size archives are widely employed, since more non-dominating solutions can emerge quickly. The algorithm’s ability to begin with the straightforward determination of the required population size and the terminal condition is the most noticeable feature of the GNDO. The location of the person is automatically changed by the generalized normal distribution function (GNDO), which has a simple construction. The benefits and drawbacks of the GNDO algorithm are as follows: • It offers a faster and smoother convergence, especially for difficult problems, and it strikes the perfect balance between exploration and exploitation. •Local minima are less likely to become entangled in relaxed convergence. •Effortlessly simple, adaptable, and simple to use • The traditional GNDO may have issues with convergence trends or become stuck in narrow, deceptive optima for challenging optimization tasks, such as high-dimensional and multimodal problems. Currently, both conventional and non-conventional energy sources require more studies. The current body of research recommends using coal-based plus wind and FACTS devices, combined with single and multi-objective optimum power flow (MOOPF) problems. The conventional IEEE 30-bus network has been altered to include non-conventional sources for research purposes. Using Weibull PDF, non-conventional units’ stochastic behaviors are calculated. The generating cost is suitably adjusted to account for reserve cost if these stochastic units are over-estimated and adjusted for penalty cost in the case that they are underestimated. Using the Generalized Normal Distribution Optimization method, Pareto solution clusters are discovered for the multi-objective problem. The following is a list of the contributions made by this study: 1. This work focuses on the mathematical modelling of the single and multiple-objective OPF issue modelled, which takes into account both conventional units and nonconventional sources of energy units, as well as FACTS devices. 2. The appropriate probability density functions (PDFs) are modeled in the second stage to describe the wind power plants’ random behavior. 3. Stochastic non-conventional sources of energy sources are among the single and multiple-objective OPF issues for which the Non-Dominated Sorting Generalized Normal Distribution Optimization (NSGNDO) technique is used to develop solutions. 4. Studies and performance evaluations of the MOGNDO algorithm using empirical comparisons are conducted. The notion of the mathematical models for coal-based power, wind power, and FACTS devices is presented in Section 2of the study. An explanation of the objectives that need to be optimized is included in Section 3. Section 4provides an explanation and illustrations of the multi-objective GNDO technique. Section 5presents numerical results and discussion, and Section 6provides concluding remarks. 2. Mathematical Representations The case studies presented here restructure the original IEEE 30-bus test apparatus. The modified approach incorporates wind turbines and FACTS devices, and is listed in Table 1. The equipment used for the analysis is depicted in Figure 1. The placement and ratings of FACTS devices are depicted in the diagram with dotted lines because they have been optimized. The section below provides information on the costs of traditional coal-based production facilities and plants using non-conventional sources of energy. Electronics 2022,11, 3825 4 of 34 Table 1. Test system key features for analysis. Particulars Quantity Details Total buses 30 [23] Total branches 41 [23] Coal-based generators (TG1; TG2; TG3; TG4) 4Buses: 1 (swing), 2, 8 and 13 Wind generators (WG1; WG2) 2Bus-5 and Bus-11 Tap changing transformers 4Branches: 11, 12, 15 and 36 SVC 2 Optimal bus and rating derived TCSC 2 Optimal branch position and rating derived TCPS 2 Optimal placement and rating derived Demand - 283.4 MW, 126.2 MVAr Electronics 2022, 11, x FOR PEER REVIEW 4 of 37 2. Mathematical Representations The case studies presented here restructure the original IEEE 30-bus test apparatus. The modified approach incorporates wind turbines and FACTS devices, and is listed in Table 1. The equipment used for the analysis is depicted in Figure 1. The placement and ratings of FACTS devices are depicted in the diagram with dotted lines because they have been optimized. The section below provides information on the costs of traditional coal-based production facilities and plants using non-conventional sources of energy. Table 1. Test system key features for analysis. Particulars Quantity Details Total buses 30 [23] Total branches 41 [23] Coal-based generators (TG1; TG2; TG3; TG4) 4 Buses: 1 (swing), 2, 8 and 13 Wind generators (WG1; WG2) 2 Bus-5 and Bus-11 Tap changing transformers 4 Branches: 11, 12, 15 and 36 SVC 2 Optimal bus and rating derived TCSC 2 Optimal branch position and rating derived TCPS 2 Optimal placement and rating derived Demand - 283.4 MW, 126.2 MVAr Figure 1. Adapted IEEE 30-bus scheme with power units and FACTS policies [23]. 2.1. Cost of Coal-Based Power Units The generalized quadratic equation for the calculation of generation cost is expressed in (1) in $/h [23]: Figure 1. Adapted IEEE 30-bus scheme with power units and FACTS policies [23]. 2.1. Cost of Coal-Based Power Units The generalized quadratic equation for the calculation of generation cost is expressed in (1) in $/h [23]: CT0(PTG)=∑NTG i=1ai+biPTGi +ciP2 TGi (1) For a more practical case, the valve point effect included: CT(PTG)=∑NTG i=1ai+biPTGi +ciP2 TGi +di×sinei×Pmin TGi −PTGi(2) The values of both coal-based price constants and emanation constants with various scenarios are shown in [23]. 2.2. Toxic Gas Emanation Polluted gases are released by using coal-based plants. So, toxic gas emanations in tons per hour can be determined as (in ton/h): F2 =t,E=∑NTG i=1hαi+βiPTGi +γiP2 TGi×0.01 +ωie(µiPTGi)i(3) Electronics 2022,11, 3825 5 of 34 The toxic gas emanation constants of coal-based power plants are taken from [22]. 2.3. Direct Cost of Stochastic Non-Conventional Sources Plants It is particularly challenging to integrate non-conventional energy sources into the power grid since they are stochastic. The independent system operator (ISO) is responsible for managing these non-conventional energy sources. Due to this, the private operator must contract with the grid or ISO for a specific quantity of planned power. The scheduled electricity must be maintained by the ISO scheduled power. If these non-conventional sources are unable to maintain the planned power, the ISO is liable for the absence of power. So, if a need arises, there are spinning reserve requirements. This spinning reserve increases costs for the ISO, and this circumstance is known as an overestimation of non-conventional sources. Conversely, if non-conventional sources formed more energy than was planned, it might go to waste, due to underuse. Therefore, the ISO must accept the penalty charge. The scheduled power cost, the overestimation cost caused by the spinning reserve, and the penalized cost caused by the underestimation are the three costs related to electricity. The direct cost linked to wind farms is demonstrated with the Pws scheduled power from the same sources as: Cw(Pws)=gwPws (4) 2.4. Indeterminate Non-Conventional Sources of Wind Power Cost Due to the erratic nature of wind, the wind farm occasionally produces less energy than expected. This means that if demand increases, it needs the spinning reserve to maintain the agreed-upon amount of scheduled power. It is sometimes feasible that the real power generated by wind farms won’t be enough to meet demand and will have lower values. Such power is referred to as exaggerated power by an ambiguous resource. To control this kind of uncertainty and provide end users with a reliable power source, the network ISO operates spinning reserves. The price of hiring a backup generator to supply the overestimated power is known as the reserve cost. Reserve cost for the wind unit is formulated by: CRw(Pws −Pwav)=KRw(Pws −Pwav)=KRw ZPws 0(Pws −pw)fw(pw)dpw(5) The possibility exists that the wind farm will generate more power than is required, which is the opposite of the overestimation scenario. Underestimated power is the term used to describe such a situation. If there is no provision for managing the output power from coal-based units, the excess power will be lost. Regarding the extra power, the ISO needs to be penalized. The penalty charge for the wind unit is given by: CPw(Pwav −Pws)=KPw(Pwav −Pws)=KPw ZPwr Pws (pw−Pws)fw(pw)dpw(6) 2.5. Uncertainty Models of Stochastic Wind Units In the redesigned IEEE-30, the wind power generating units installed at buses 5 and 11, which were originally thermal generators, were replaced. It should be noted that, as for a comparison point of view with the published reference article [ 23 ], in this paper, the thermal DGs were also replaced with the wind turbines. This will ensure compatibility of the results obtained by the proposed algorithm to the already published research article [ 23 ]. The scale (c) and shape (k) constants for the proposed Weibull model are detailed in Table 2. Electronics 2022,11, 3825 6 of 34 Table 2. PDF constants of wind power plants [23]. Windfarm No. of Turbines Rated Power Weibull PDF Parameters Cost Constants Direct Reserve Penalty WG5 (bus 5) 25 75 c= 9, k= 2 1.60 3.0 1.50 WG11 (bus 11) 20 60 c= 10, k= 2 1.75 3.0 1.50 The Weibull curve and wind frequency distributions in Figure 2(for the bus 5 wind plant) and Figure 3(for the bus 11 wind plant) were produced using 8000 Monte-Carlo settings. The standard provided explains the need for wind turbine design and specifies the maximum turbulence class IA that is confirmed to operate at the highest yearly average wind velocity of 10 m/s at hub height. The formed shape (k) and scale (c ) parameters of wind farms are given particular attention, because their highest Weibull mean value is fixed at around 10. It is commonly known that the wind speed distribution follows the Weibull PDF curve. Electronics 2022, 11, x FOR PEER REVIEW 7 of 37 Figure 2. Weibull PDF (bus 5). Figure 3. Weibull PDF (bus 11). 2.6. Average Power Calculation for Wind Plants The combined outputs of the 25 turbines in the farm are taken as the wind unit connected at bus 5. Every turbine has a 3 MW output rating. The wind velocity affects the wind turbine’s precise output, which varies. We used the following equations to express turbine output power in terms of wind velocity (v) [23]: 𝑝𝑤(𝑣)={0, for 𝑣〈𝑣𝑖𝑛 𝑎𝑛𝑑 𝑣〉𝑣𝑜𝑢𝑡 𝑝𝑤𝑟(𝑣−𝑣𝑖𝑛 𝑣𝑟−𝑣𝑖𝑛) for 𝑣𝑖𝑛⩽𝑣⩽𝑣𝑟 𝑝𝑤𝑟 for 𝑣𝑟<𝑣⩽𝑣𝑜𝑢𝑡 (10) The Enercon E82-E4 design specification is referred to for the 3 MW wind turbine. The various speeds are 𝑣𝑖𝑛 = 3 m/s, 𝑣𝑟 = 16 m/s, and 𝑣𝑜𝑢𝑡 = 25 m/s. Figure 2. Weibull PDF (bus 5). Electronics 2022, 11, x FOR PEER REVIEW 7 of 37 Figure 2. Weibull PDF (bus 5). Figure 3. Weibull PDF (bus 11). 2.6. Average Power Calculation for Wind Plants The combined outputs of the 25 turbines in the farm are taken as the wind unit connected at bus 5. Every turbine has a 3 MW output rating. The wind velocity affects the wind turbine’s precise output, which varies. We used the following equations to express turbine output power in terms of wind velocity (v) [23]: 𝑝𝑤(𝑣)={0, for 𝑣〈𝑣𝑖𝑛 𝑎𝑛𝑑 𝑣〉𝑣𝑜𝑢𝑡 𝑝𝑤𝑟(𝑣−𝑣𝑖𝑛 𝑣𝑟−𝑣𝑖𝑛) for 𝑣𝑖𝑛⩽𝑣⩽𝑣𝑟 𝑝𝑤𝑟 for 𝑣𝑟<𝑣⩽𝑣𝑜𝑢𝑡 (10) The Enercon E82-E4 design specification is referred to for the 3 MW wind turbine. The various speeds are 𝑣𝑖𝑛 = 3 m/s, 𝑣𝑟 = 16 m/s, and 𝑣𝑜𝑢𝑡 = 25 m/s. Figure 3. Weibull PDF (bus 11). Electronics 2022,11, 3825 7 of 34 The following formula can be used to calculate the probability of wind velocity v, in m/s, pursuing the Weibull PDF with shape factor (k) and scale factor (c) [23]: fv(v) = k cv c(k−1)e−(v c)kf or 0<v<∞(7) The Weibull distribution’s mean is given as follows [23]: Mwbl =c∗Γ1+k−1(8) and the gamma function Γ(x)is expressed in Equation (9): Γ(x) = Z∞ 0e−ttx−1dt (9) 2.6. Average Power Calculation for Wind Plants The combined outputs of the 25 turbines in the farm are taken as the wind unit connected at bus 5. Every turbine has a 3 MW output rating. The wind velocity affects the wind turbine’s precise output, which varies. We used the following equations to express turbine output power in terms of wind velocity (v) [23]: pw(v) =      0, for vhvin and vivout pwrv−vin vr−vin for vin ≤v≤vr pwr for vr<v≤vout (10) The Enercon E82-E4 design specification is referred to for the 3 MW wind turbine. The various speeds are vin = 3 m/s, vr= 16 m/s, and vout = 25 m/s. 2.7. Wind Power Probabilities Calculation In certain ranges of wind speeds, uncertain wind generation is noticeable. The generated power would be 0 if the wind speed was greater than or less than the cut-out speed or cut-in speed. The turbine thereby produces the specified amount of power within the range of the rated and cut-out wind speeds. These are possible ways to describe the likelihood of these areas [23]: fw(pw){pw=0}=1−exp−vin ck+exp−vout ck(11) fw(pw){pw=pwr}=exp−vr ck−exp−vout ck(12) Between the cut-in velocity and the rated velocity of the wind, the wind production remains constant. The following can be used to express the likelihood of the continuous zone [23]: fw(pw)=β(vr−vin) αβ∗pwr vin +pw pwr (vr−vin)β−1 exp − vin +pw pwr (vr−vin) α!β (13) 2.8. Thyristor-Controlled Series Compensator (TCSC) Modeling The basic circuitry of the TCSC is depicted in Figure 4. It consists of a fixed series capacitor (XC) and a reactor (XL) operated by a thyristor. For the TCSC to function as a variable capacitive reactance, reactance XC<XL is taken into consideration. By varying the firing angle ( α ) of the thyristors, the inductive reactance is changed, and for high values of inductive reactance, the least corresponding capacitive reactance is produced (Open circuit Electronics 2022,11, 3825 8 of 34 Inductive branch). As a result, the TCSC’s effective reactance with constant capacitive reactance XCand variable inductive reactance XL(α)can be written as [23]: XTCSC(α) = XCXL(α) XL(α)−XC =−jXC(14) Electronics 2022, 11, x FOR PEER REVIEW 8 of 37 2.7. Wind Power Probabilities Calculation In certain ranges of wind speeds, uncertain wind generation is noticeable. The generated power would be 0 if the wind speed was greater than or less than the cut-out speed or cut-in speed. The turbine thereby produces the specified amount of power within the range of the rated and cut-out wind speeds. These are possible ways to describe the likelihood of these areas [23]: 𝑓𝑤(𝑝𝑤){𝑝𝑤=0}=1−exp[−(𝑣𝑖𝑛 𝑐)𝑘]+exp[−(𝑣𝑜𝑢𝑡 𝑐)𝑘] (11) 𝑓𝑤(𝑝𝑤){𝑝𝑤=𝑝𝑤𝑟}=exp[−(𝑣𝑟 𝑐)𝑘]−exp[−(𝑣𝑜𝑢𝑡 𝑐)𝑘] (12) Between the cut-in velocity and the rated velocity of the wind, the wind production remains constant. The following can be used to express the likelihood of the continuous zone [23]: 𝑓𝑤(𝑝𝑤)=𝛽(𝑣𝑟−𝑣𝑖𝑛) 𝛼𝛽∗𝑝𝑤𝑟 [𝑣𝑖𝑛+𝑝𝑤 𝑝𝑤𝑟(𝑣𝑟−𝑣𝑖𝑛)]𝛽−1exp[−(𝑣𝑖𝑛+𝑝𝑤 𝑝𝑤𝑟(𝑣𝑟−𝑣𝑖𝑛) 𝛼)𝛽] (13) 2.8. Thyristor-Controlled Series Compensator (TCSC) Modeling The basic circuitry of the TCSC is depicted in Figure 4. It consists of a fixed series capacitor (XC) and a reactor (XL) operated by a thyristor. For the TCSC to function as a variable capacitive reactance, reactance 𝑋𝐶<𝑋𝐿 is taken into consideration. By varying the firing angle (𝛼) of the thyristors, the inductive reactance is changed, and for high values of inductive reactance, the least corresponding capacitive reactance is produced (Open circuit Inductive branch). As a result, the TCSC’s effective reactance with constant capacitive reactance 𝑋𝐶 and variable inductive reactance 𝑋𝐿(𝛼) can be written as [23]: 𝑋𝑇𝐶𝑆𝐶(𝛼)=𝑋𝐶𝑋𝐿(𝛼) 𝑋𝐿(𝛼)−𝑋𝐶=−𝑗𝑋𝐶 (14) Figure 4. Basic Structure and model of TCSC [23]. The TCSC static model, which is situated in the path between buses m and n, is shown in Figure 4. Following the TCSC’s integration (described as a variable capacitive reactance mode), the transmission line’s adjusted reactance (𝑋𝑒𝑞) is given by [23]: 𝑋𝑒𝑞=𝑋𝑚𝑛−𝑋𝑇𝐶𝑆𝐶=(1−𝜏)𝑋𝑚𝑛 (15) where 𝜏=𝑋𝑇𝐶𝑆𝐶 𝑋𝑚𝑛 (16) Figure 4. Basic Structure and model of TCSC [23]. The TCSC static model, which is situated in the path between buses mand n, is shown in Figure 4. Following the TCSC’s integration (described as a variable capacitive reactance mode), the transmission line’s adjusted reactance (Xeq) is given by [23]: Xeq =Xmn −XTCSC =(1−τ)Xmn (15) where τ=XTCSC Xmn (16) The power flow equations of the line incorporating the TCSC are written as [23]: Pmn =V2 mgmn −VmVngmn cos(δm−δn). −VmVnbmn sin(δm−δn)(17) Qmn =−V2 mbmn −VmVngmn sin(δm−δn). +VmVnbmn cos(δm−δn)(18) Pnm =V2 ngmn −VmVngmn cos(δm−δn). +VmVnbmn sin(δm−δn)(19) Qnm =−V2 nbmn +VmVngmn sin(δm−δn). +VmVnbmn cos(δm−δn)(20) where gmn =rmn r2 mn + (xmn −xc)2(21) bmn =−xmn −xc r2 mn + (xmn −xc)2(22) Electronics 2022,11, 3825 9 of 34 2.9. Model of Thyristor-Controlled Phase Shifter (TCPS) Figure 5displays the model of the TCPS placed between the line that connects buses m and n . The power flow equations of the line can be expressed as below, assuming that is the phase shift angle φis introduced by the TCPS: Pmn =V2 mgmn cos2φ−VmVn cos φ[gmn cos(δm−δn+φ). +bmn sin(δm−δn+φ)] (23) Qmn =−V2 mbmn cos2φ−VmVn cos φ[gmn sin(δm−δn+φ). −bmn cos(δm−δn+φ)](24) Pnm =V2 ngmn −VmVn cos φ[gmn cos(δm−δn+φ). −bmn sin(δm−δn+ϕ)](25) Qnm =−V2 nbmn +VmVn cos φ[gmn sin(δm−δn+φ). +bmn cos(δm−δn+ϕ)](26) Electronics 2022, 11, x FOR PEER REVIEW 9 of 37 The power flow equations of the line incorporating the TCSC are written as [23]: P𝑚𝑛=𝑉𝑚2𝑔𝑚𝑛−𝑉𝑚𝑉𝑛𝑔𝑚𝑛cos(𝛿𝑚−𝛿𝑛) −𝑉𝑚𝑉𝑛𝑏𝑚𝑛sin(𝛿𝑚−𝛿𝑛) (17) 𝑄𝑚𝑛=−𝑉𝑚2𝑏𝑚𝑛−𝑉𝑚𝑉𝑛𝑔𝑚𝑛sin(𝛿𝑚−𝛿𝑛) +𝑉𝑚𝑉𝑛𝑏𝑚𝑛cos(𝛿𝑚−𝛿𝑛) (18) P𝑛𝑚=𝑉𝑛2𝑔𝑚𝑛−𝑉𝑚𝑉𝑛𝑔𝑚𝑛cos(𝛿𝑚−𝛿𝑛) +𝑉𝑚𝑉𝑛𝑏𝑚𝑛sin(𝛿𝑚−𝛿𝑛) (19) 𝑄𝑛𝑚=−𝑉𝑛2𝑏𝑚𝑛+𝑉𝑚𝑉𝑛𝑔𝑚𝑛sin(𝛿𝑚−𝛿𝑛) +𝑉𝑚𝑉𝑛𝑏𝑚𝑛cos(𝛿𝑚−𝛿𝑛) (20) where 𝑔𝑚𝑛=𝑟𝑚𝑛 𝑟𝑚𝑛 2+(𝑥𝑚𝑛−𝑥𝑐)2 (21) 𝑏𝑚𝑛=− 𝑥𝑚𝑛−𝑥𝑐 𝑟𝑚𝑛 2+(𝑥𝑚𝑛−𝑥𝑐)2 (22) 2.9. Model of Thyristor-Controlled Phase Shifter (TCPS) Figure 5 displays the model of the TCPS placed between the line that connects buses 𝑚 and 𝑛. The power flow equations of the line can be expressed as below, assuming that is the phase shift angle 𝜙 is introduced by the TCPS: Figure 5. Model of TCP [23]. P𝑚𝑛=𝑉𝑚 2𝑔𝑚𝑛 cos2𝜙−𝑉𝑚𝑉𝑛 cos𝜙[𝑔𝑚𝑛cos(𝛿𝑚−𝛿𝑛+𝜙) +𝑏𝑚𝑛sin(𝛿𝑚−𝛿𝑛+𝜙)] (23) 𝑄𝑚𝑛=−𝑉𝑚 2𝑏𝑚𝑛 cos2𝜙−𝑉𝑚𝑉𝑛 cos𝜙[𝑔𝑚𝑛sin(𝛿𝑚−𝛿𝑛+𝜙) −𝑏𝑚𝑛cos(𝛿𝑚−𝛿𝑛+𝜙)] (24) P𝑛𝑚=𝑉𝑛2𝑔𝑚𝑛−𝑉𝑚𝑉𝑛 cos𝜙[𝑔𝑚𝑛cos(𝛿𝑚−𝛿𝑛+𝜙) −𝑏𝑚𝑛sin(𝛿𝑚−𝛿𝑛+𝜑)] (25) 𝑄𝑛𝑚=−𝑉𝑛2𝑏𝑚𝑛+𝑉𝑚𝑉𝑛 cos𝜙[𝑔𝑚𝑛sin(𝛿𝑚−𝛿𝑛+𝜙) +𝑏𝑚𝑛cos(𝛿𝑚−𝛿𝑛+𝜑)] (26) The inserted actual and reactive power of the TCPS at bus 𝑚 and 𝑛 is [23]: P𝑚𝑠=−𝑔𝑚𝑛𝑉𝑚2tan2𝜙−𝑉𝑚𝑉𝑛tan𝜙[𝑔𝑚𝑛sin(𝛿𝑚−𝛿𝑛) −𝑏𝑚𝑛cos(𝛿𝑚−𝛿𝑛)] (27) Figure 5. Model of TCP [23]. The inserted actual and reactive power of the TCPS at bus mand nis [23]: Pms =−gmnV2 mtan2φ−VmVntan φ[gmn sin(δm−δn). −bmn cos(δm−δn)](27) Qms =bmnV2 mtan2φ+VmVntan φ[gmn cos(δm−δn). +bmn sin(δm−δn)](28) Pns =−VmVntan φ[gmn sin(δm−δn)+bmn cos(δm−δn)] (29) Qns =−VmVntan φ[gmn cos(δm−δn)−bmn sin(δm−δn)] (30) 2.10. Model of Static VAR Compensator (SVC) The basic circuit architecture and the SVC model are depicted in Figure 6. It is made up of a thyristor-controlled reactor ( XL=ωL) and a fixed capacitor ( XC= 1 /ωC) . By changing the thyristor firing angle (α) , the reactance can be changed. The equivalent susceptibility is computed as: Beq =BL(α) + BC(31) where BL(α) = −1 ωL1−2α π,Bc=ω×C(32) The reactive power offered by the SVC can be expressed in terms within the context of power flow: QSVC =−V2 m·BSVC (33) Electronics 2022,11, 3825 16 of 34 Electronics 2022, 11, x FOR PEER REVIEW 16 of 37 Definition 3: Pareto Optimal Set All Pareto optimal solution sets are called the Pareto set, and are expressed as follows: 𝑃𝑠={𝑥,𝑦∈𝑋 | ∃𝐹(𝑦)≻𝐹(𝑥)} (67) Definition 4: Pareto Optimal Front The Pareto optimal front is a collective of Pareto optimal solutions in the Pareto optimal set, as shown in (68): 𝑃𝑓={𝐹(𝑥)|𝑥∈𝑃𝑠} (68) Any multi-objective optimization issue must be solved using the Pareto optimum set during the multi-objective optimization process. The search space (set of dominated solutions) and objective space (set of non-dominated solutions) are depicted in Figure 9. The Pareto optimum front describes the interaction between the objective space and search space. Figure 9. Objective space and search space in the multi-objective optimization problem. 4.6. Multi-Objective Generalized Normal Distribution Optimization (MOGNDO) The proposed MOGNDO algorithm optimizer uses both the crowding distance (CD) mechanism and the elitist non-dominated sorting (NDS) method. The NDS consists of the following stages: • Locating the non-dominated solution is the first step. • The second step is the use of the NDS strategy. • Performing non-dominated ranking (NDR) calculations on all non-dominated solutions. Between two fronts, the NDR process takes place. The first front’s solutions offer a “0” index because no solutions are dominated by them, but at least one solution from the first front dominates the second front’s solutions. A solution’s NDR is equal to the number of solutions that predominate it. The CD process is used to keep the created solutions diverse. The following is a definition of the CD mechanism: 𝐶𝐷𝑗𝑖 =𝑓𝑜𝑏𝑗𝑗𝑖+1−𝑓𝑜𝑏𝑗𝑗𝑖−1 𝑓𝑜𝑏𝑗𝑗𝑚𝑎𝑥−𝑓𝑜𝑏𝑗𝑗𝑚𝑖𝑛 (69) where 𝑓𝑜𝑏𝑗𝑗𝑚𝑎𝑥 and 𝑓𝑜𝑏𝑗𝑗𝑚𝑖𝑛are the maximum and minimum values of 𝑗th objective function. The diagrammatic illustration of an NDS-based approach is illustrated in Figure 10. Figure 9. Objective space and search space in the multi-objective optimization problem. 4.6. Multi-Objective Generalized Normal Distribution Optimization (MOGNDO) The proposed MOGNDO algorithm optimizer uses both the crowding distance (CD) mechanism and the elitist non-dominated sorting (NDS) method. The NDS consists of the following stages: •Locating the non-dominated solution is the first step. •The second step is the use of the NDS strategy. • Performing non-dominated ranking (NDR) calculations on all non-dominated solutions. Between two fronts, the NDR process takes place. The first front’s solutions offer a “0” index because no solutions are dominated by them, but at least one solution from the first front dominates the second front’s solutions. A solution’s NDR is equal to the number of solutions that predominate it. The CD process is used to keep the created solutions diverse. The following is a definition of the CD mechanism: CDi j=f obji+1 j−f obji−1 j f objmax j−f objmin j (69) where f objmax j and f objmin j are the maximum and minimum values of jth objective function. The diagrammatic illustration of an NDS-based approach is illustrated in Figure 10. The MOGNDO algorithm’s pseudocode is displayed in Algorithm 1. The MOGNDO method begins by specifying the necessary inputs, such as population size (N p ), termination criteria, the maximum number of generations, and the maximum number of iterations (Maxit). Then, each objective function in the objective space vector Ffor P o is evaluated using a randomly generated parent population P o in the feasible search space region S. Thirdly, P o is subjected to the elitist-based CD and NDS. Fourthly, P o is merged with a fresh population of P j to create a population, P i . This P i is sorted using the CD and NDR data, as well as elitist non-dominance. To establish a new parent population, the best N p options are evaluated. The process is then repeated until the termination criteria are met. MOGNDO’s flowchart is displayed in Figure 11. Algorithm 1: Pseudocode of Multi-objective Generalized Normal Distribution Optimization (MOGNDO). Step 1: Initially Generate population (Po) randomly in solution space (S) Step 2: Evaluate objective space (F) for the generated population (Po) Step 3: Sort the based on the elitist non-dominated sort method and find the non-dominated rank (NDR) and fronts Step 4: Compute crowding distance (CD) for each front Step 5: Update solutions (Pj) Step 6: Merge Poand Pjto create Pi=PoUPj Step 7: For Piperform Step 2 Step 8: Based on NDR and CD sort Pi Step 9: Replace Powith Pifor Npfirst members of Pi Electronics 2022,11, 3825 17 of 34 Electronics 2022, 11, x FOR PEER REVIEW 17 of 37 Figure 10. The procedure of the non-dominated sorting approach. The MOGNDO algorithm’s pseudocode is displayed in Algorithm 1. The MOGNDO method begins by specifying the necessary inputs, such as population size (Np), termination criteria, the maximum number of generations, and the maximum number of iterations (Maxit). Then, each objective function in the objective space vector F for Po is evaluated using a randomly generated parent population Po in the feasible search space region S. Thirdly, Po is subjected to the elitist-based CD and NDS. Fourthly, Po is merged with a fresh population of Pj to create a population, Pi. This Pi is sorted using the CD and NDR data, as well as elitist non-dominance. To establish a new parent population, the best Np options are evaluated. The process is then repeated until the termination criteria are met. MOGNDO’s flowchart is displayed in Figure 11. Algorithm 1: Pseudocode of Multi-objective Generalized Normal Distribution Optimization (MOGNDO). Step 1: Initially Generate population (Po) randomly in solution space (S) Step 2: Evaluate objective space (F) for the generated population (Po) Step 3: Sort the based on the elitist non-dominated sort method and find the nondominated rank (NDR) and fronts Step 4: Compute crowding distance (CD) for each front Step 5: Update solutions (Pj) Step 6: Merge Po and Pj to create Pi=Po U Pj Step 7: For Pi perform Step 2 Step 8: Based on NDR and CD sort Pi Step 9: Replace Po with Pi for Np first members of Pi Figure 10. The procedure of the non-dominated sorting approach. Electronics 2022, 11, x FOR PEER REVIEW 18 of 37 Figure 11. Flowchart of MOGNDO algorithm. 4.7. Constraint Handling Approach The majority of engineering design issues in the actual world are multi-objective and highly nonlinearly constrained. To solve constrained MOPs, managing all constraints within their bounds is crucial. A static penalty technique is used in the MOGNDO algorithm because it transforms a constrained problem into an unconstrained problem, despite the literature survey giving various constrained handling approaches. This approach adds a significant penalty to the relevant goal function if a constraint is broken. The following is a presentation of the static penalty system: 𝑓𝑗(𝑋)=𝑓𝑗(𝑋)+∑𝑝𝑖=1 𝑃𝑖max{𝑔𝑖(𝑋),0}+∑𝑁𝐶 𝑖=𝑝 𝑃𝑖max{|ℎ𝑖(𝑋)|−𝛿,0} (70) where 𝑓𝑗(𝑋),𝑗=1,2…𝑛 is the objective function to be optimized (here minimized), 𝑋= {𝑥1,𝑥2,…𝑥𝑚} are design variables, 𝑔𝑖(𝑋)⩽0,𝑖=1,2…𝑝 are inequality constraints, ℎ𝑖(𝑋)=0,𝑖=𝑝+1…𝑁𝐶 are equality constraints, and 𝛿 is the tolerance in equality constraints. 4.8. Fuzzy Approach for the Multi-Objective Problem The fuzzy membership approach can be used in multi-objective functions to identify the best compromising outcome out of all the non-inferior results. The fuzzy membership function 𝜇𝑓𝑖uses a fuzzy membership function to keep track of the minimum 𝑓𝑖𝑚𝑖𝑛 and maximum 𝑓𝑖𝑚𝑎𝑥 values for each objective aim. Now, the membership function of the 𝑖𝑡ℎ𝑡ℎ𝑒 objective is given as: 𝜇𝑓𝑖= { 1 𝑓𝑖≤ 𝑓𝑖𝑚𝑖𝑛 𝑓𝑖𝑚𝑎𝑥−𝑓𝑖 𝑓𝑖𝑚𝑎𝑥−𝑓𝑖𝑚𝑖𝑛𝑓𝑖𝑚𝑖𝑛<𝑓𝑖< 0 𝑓𝑖≥ 𝑓𝑖𝑚𝑎𝑥𝑓𝑖𝑚𝑎𝑥 (71) The standards of membership functions lie on the measure of (0–1) and display in the way that satisfies the function 𝑓𝑖. Later, the decision-making function 𝜇𝑘 should be calculated as follows: Figure 11. Flowchart of MOGNDO algorithm. 4.7. Constraint Handling Approach The majority of engineering design issues in the actual world are multi-objective and highly nonlinearly constrained. To solve constrained MOPs, managing all constraints within their bounds is crucial. A static penalty technique is used in the MOGNDO algorithm Electronics 2022,11, 3825 18 of 34 because it transforms a constrained problem into an unconstrained problem, despite the literature survey giving various constrained handling approaches. This approach adds a significant penalty to the relevant goal function if a constraint is broken. The following is a presentation of the static penalty system: fj(X) = fj(X) + ∑p i=1Pimax{gi(X), 0}+∑NC i=pPimax{|hi(X)|−δ, 0}(70) where fj(X) , j= 1, 2 . . . n is the objective function to be optimized (here minimized), X={x1,x2, . . . xm} are design variables, gi(X)≤ 0, i= 1, 2 . . . p are inequality constraints, hi(X) = 0, i=p+ 1 . . . NC are equality constraints, and δ is the tolerance in equality constraints. 4.8. Fuzzy Approach for the Multi-Objective Problem The fuzzy membership approach can be used in multi-objective functions to identify the best compromising outcome out of all the non-inferior results. The fuzzy membership function µfi uses a fuzzy membership function to keep track of the minimum fmin i and maximum fmax i values for each objective aim. Now, the membership function of the ith the objective is given as: µfi=     1fi≤fmin i fmax i−fi fmax i−fmin i fmin i<fi<fmax i 0fi≥fmax i (71) The standards of membership functions lie on the measure of (0–1) and display in the way that satisfies the function fi . Later, the decision-making function µk should be calculated as follows: µk=∑N i=1µk fi ∑M k=1∑N i=1µk fi (72) For non-inferior findings, the decision-making function can also be thought of as the normalized membership function, which displays the ordering of the undominated results. The end outcome is regarded as the best attainable compromise among all PFs, with a maximum value of maximum nµk:k=1, 2, 3 . . . . . . Mo. 5. Simulation Results, Analysis, and Comparative Study This section discusses the outcomes of the MOGNDO algorithm, which optimized the optimal power flow with non-conventional and FACTS device problems with control variables. The initialization of the algorithm’s population size, archive size, the maximum number of iterations, and boundary condition for optimal power flow problems all came first. To identify the best optimal tradeoff points between multiple objective functions, the MOGNDO algorithm was then used to obtain the initial position and objective function values. Optimal power flow with non-conventional sources and FACTS devices were used to apply the MOGNDO algorithm’s performance, which was initially verified on eight unconstrained multi-objective problems. On a computer with 4 GB of RAM and a 3.20 GHz clock speed, the simulation was run using the MATLAB program. The benchmark functions for each unconstrained test were solved using 10 separate runs. The population size was set to 30, the maximum number of iterations was set to 100, and the archive size was set to 30 when the control parameters for the proposed MOGWO algorithm were first set. The performance measures for the MOGNDO algorithm, including Generational Distance (GD), Inversion Generational Distance (IGD), Spacing Metrics (SP), Diversity Metrics (DM), and Spread Metrics (SD), are covered in this section. Electronics 2022,11, 3825 19 of 34 5.1. MOGNDO Results for Test Benchmark Problems Before tackling real-world issues, the MOGNDO was used to evaluate the performance of the benchmark unconstraint test function provided in [ 26 ]. Eight benchmark unconstrained test functions—ZDT1, ZDT2, ZDT3, ZDT4, ZDT6, KURSAVE, SCHAFFER-1, and SCHAFFER-2 (Figure 12) were taken into account, and a thorough simulation was performed using the MOGNDO technique. Any algorithm’s control parameters are crucial to the resolution of the optimization problem. As a result, the number of populations was decided after conducting a comparative analysis that took into account various population sizes, while holding all other variables constant. Following careful consideration, the population size, maximum iterations, and archive size were chosen as 30, 100, and 30, respectively, for the unconstrained test benchmark functions. The MOGNDO algorithm’s performance was evaluated using performance metrics, such as Generational Distance (GD), Inversion Generational Distance (IGD), Spacing Metrics (SP), Diversity Metrics (DM), and Spread Metrics (SD), for convergence measurement. Tables 3–7demonstrate that MOGNDO could achieve the best outcomes for all performance metrics, including Generational Distance (GD), Inversion Generational Distance (IGD), Spacing Metrics (SP), Diversity Metrics (DM), and Spread Metrics (SD), which cover convergence and solution accuracy. It follows that the suggested MOGNDO can provide the best convergence on all benchmark functions. The outcomes (archive solutions) of all eight test benchmark issues are displayed in Figures 1–5. As can be shown, the MOGNDO method was capable of approximating the PF. By comparing the PF estimations, it can also be seen that the suggested MOGNDO could provide acceptable performance. Thus, it was determined that the MOGNDO algorithm is more suitable for the stochastic OPF problem with three FACTS devices and wind power plants. Table 3. Results of GDMETRICS on test functions. TEST FUNCTIONS Minimum Average Median Maximum Std Dev ZDT-1 0.00014795 0.00025051 0.00025043 0.00033133 6.658 ×10−5 ZDT-2 0.00015293 0.00016665 0.00016915 0.00018271 8.9928 ×10−6 ZDT-3 0.00048314 0.00059601 0.00057943 0.00077354 9.9396 ×10−5 ZDT-4 0.00012104 0.00022808 0.00020834 0.0004758 9.4401 ×10−5 ZDT-6 9.084 ×10−50.046963 0.00011316 0.22777 0.084189 KURSAVE 0.00079741 0.0014452 0.0012905 0.0025211 0.00051722 SCHAFFER-1 0.00033607 0.00040992 0.00042203 0.00047897 4.714 ×10−5 SCHAFFER-2 4.3236 ×10−57.2824 ×10−55.2463 ×10−50.00016939 4.0743 ×10−5 Table 4. Results of IGD METRICS on test functions. TEST FUNCTIONS Minimum Average Median Maximum Std Dev ZDT-1 0.00086623 0.00098862 0.0010348 0.0010718 8.6005 ×10−5 ZDT-2 0.00086578 0.00098848 0.00094837 0.0012358 0.00012678 ZDT-3 0.001211 0.0020746 0.0013386 0.0077205 0.0020015 ZDT-4 0.00078066 0.00089518 0.00087008 0.0012276 0.00012621 ZDT-6 0.0004207 0.00045525 0.00044688 0.00049748 2.8268 ×10−5 KURSAVE 0.00054436 0.00062399 0.00057845 0.00085408 9.8135 ×10−5 SCHAFFER-1 0.0013154 0.0015246 0.0015027 0.0017064 0.0001302 SCHAFFER-2 0.00038808 0.00042594 0.00042586 0.00044933 1.8173 ×10−5 Electronics 2022,11, 3825 20 of 34 Electronics 2022, 11, x FOR PEER REVIEW 20 of 37 Figure 12. Best Pareto optimal front obtained of unconstrained test functions by MOGNDO algorithm. Electronics 2022,11, 3825 21 of 34 Table 5. Results of SPACINGMETRICS on test functions. TEST FUNCTIONS Minimum Average Median Maximum Std Dev ZDT-1 0.055513 0.068922 0.068773 0.078086 0.0087192 ZDT-2 0.055269 0.066982 0.06225 0.082706 0.010406 ZDT-3 0.23708 0.30199 0.30489 0.37195 0.038816 ZDT-4 0.055461 0.079673 0.082798 0.0984 0.012829 ZDT-6 0.060275 0.36467 0.082672 1.2984 0.50689 KURSAVE 1.7123 2.0817 2.0541 2.3807 0.20644 SCHAFFER-1 0.47734 0.6444 0.65722 0.78476 0.086121 SCHAFFER-2 4.3184 6.228 6.1458 8.1797 0.99173 Table 6. Results of DIVERSITYMETRICS on test functions. TEST FUNCTIONS Minimum Average Median Maximum Std Dev ZDT-1 0.39774 0.4726 0.46959 0.53772 0.060856 ZDT-2 0.34666 0.4369 0.44215 0.57139 0.069382 ZDT-3 0.42402 0.52122 0.50625 0.64813 0.07401 ZDT-4 0.3035 0.37573 0.36822 0.43095 0.043003 ZDT-6 0.36148 0.6907 0.45236 1.3455 0.42847 KURSAVE 0.28774 0.3839 0.39009 0.45763 0.050257 SCHAFFER-1 0.28774 0.3839 0.39009 0.45763 0.050257 SCHAFFER-2 0.9161 0.95607 0.95992 1.0033 0.030768 Table 7. Results of SPREAD METRICS on test functions. TEST FUNCTIONS Minimum Average Median Maximum Std Dev ZDT-1 0.38649 0.45794 0.45049 0.5251 0.059019 ZDT-2 0.33511 0.42459 0.42842 0.56709 0.070091 ZDT-3 0.5604 0.64503 0.64749 0.73814 0.066426 ZDT-4 0.30364 0.36741 0.36204 0.4213 0.040601 ZDT-6 0.35291 0.67963 0.4479 1.3293 0.42341 KURSAVE 0.37729 0.43491 0.43448 0.4872 0.030263 SCHAFFER-1 0.27113 0.37375 0.37832 0.44441 0.051198 SCHAFFER-2 0.56431 0.61293 0.61561 0.65022 0.02504 5.2. Multi-Objectives OPF Problem with Wind Power Plants and Three FACTS Devices The GNDO algorithm was used to address the stochastic OPF problem with wind power plants and three FACTS devices in this study. The solution to the optimum power flow problem was evaluated in parallel using newly created algorithms, such as the MultiVerse Optimization (MVO), the Sine-Cosine Algorithm (SCA) [ 27 ], the Grey Wolf Optimization (GWO), the Moth Fame Optimization (MFO), the Ant Lion Optimization (ALO) [ 28 ], and Ion Motion Algorithms (IMA) [ 29 ]. The proposed approach was demonstrated using a modified IEEE-30 bus infrastructure with wind power plants and FACTS devices. Table 1 lists the major characteristics of the customized IEEE-30 bus framework. The following are two scenarios: •Scenario-1 (Solo objective OPF with wind power plants and FACTS devices) •Scenario-2 (Multi-objective OPF with wind power plants and FACTS devices) As shown in Table 8, there were a total of thirteen different test scenarios to evaluate. In this section, the results of case studies using various metaheuristics methodologies are tabulated and presented. The first six case studies are for single-objective optimization, while the latter seven are multi-objective optimization problems that include non-conventional sources of energy resources, as well as optimal FACTS device sizes and locations. The search agent value was set to 40, and each algorithm underwent 500 iterations of analysis. Please refer to the original research for a detailed discussion of those procedures. Table 4 shows the parameter settings for these methods. Electronics 2022,11, 3825 22 of 34 Table 8. Summary of case studies for adapted IEEE-30 bus test system. Test System Case # Single and Multi-Objectives Functions Adapted IEEE 30-bus test system Case # 1 TFC includes FACTS devices, wind farms, and coal-based plants Case # 2 Reduction of total toxic gas emanations with the use of coal-based, wind, and FACTS technologies. Case # 3 Minimization of APL in FACTS devices, wind farms, and coal-based plants. Case # 4 Minimization of the total voltage variation using coal-based, wind, and FACTS devices. Case # 5 Voltage stability improvement in coal-based, wind, and FACTS equipment. Case # 6 Total Gross Generation Cost, includes FACTS devices, wind farms, and coal-based plants. Case # 7 Minimizing TFCs and toxic gas emanations while using non-conventional sources of energy and FACTS devices Case # 8 TFC and APL Minimization Including non-conventional sources and FACTS Devices Case # 9 TFC and VSI minimization including non-conventional sources and FACTS devices Case # 10 Total Gross Generation Cost and voltage deviation minimization with non-conventional sources and FACTS devices Case # 11 TFC, Toxic gas emanation, and APL minimization together with non-conventional sources and FACTS devices Case # 12 TFC, APL, and VSI minimization including non-conventional sources and FACTS devices Case # 13 TFC, Toxic gas emanation, APL, and voltage deviation minimization including non-conventional sources and FACTS devices 5.3. Scenario-1 (Single Objective OPF with Wind Power Plants and FACTS Devices) With the use of GNDO, MVO, ALO, SCA, and IMO methods, all of the objective goals indicated in the mathematical formulation were simultaneously handled as solo objective optimization issues. The limitations of all control variables, as well as proper FACTS device locations and sizing, are listed below. From case 1 to case 6, the outcomes of objective functions are tabulated in Tables 9–11, with the best minimum values containing five different recent techniques. Table 9. Single objectives simulation results for case 1 and case 2. Control & State Variables Min Max Case-1 Case-2 GNDO MVO ALO SCA IMO GNDO MVO ALO SCA IMO PTG2 20 80 41.427 40.311 40.960 35.458 30.881 46.634 46.639 46.634 48.357 46.726 PWG5 0 75 49.815 49.459 49.077 41.719 54.226 74.818 74.934 71.362 75.000 74.507 PTG8 10 35 10.000 10.352 13.038 15.172 14.361 35.000 35.000 35.000 35.000 35.000 PWG11 0 60 40.799 42.135 39.362 47.561 36.329 52.365 51.215 54.905 60.000 48.761 PTG13 12 40 12.002 12.000 12.000 14.321 16.648 40.000 40.000 40.000 40.000 40.000 V1 0.95 1.1 1.091 1.100 1.100 1.019 1.100 1.090 0.997 1.100 0.997 1.100 V2 0.95 1.1 1.075 1.090 1.091 0.992 1.100 1.080 1.013 1.078 0.950 1.100 V5 0.95 1.1 1.049 1.071 1.072 0.978 1.100 1.067 1.037 0.975 1.100 1.100 V8 0.95 1.1 1.046 1.076 1.079 0.950 1.100 0.962 1.100 1.100 1.100 1.100 V11 0.95 1.1 1.100 1.100 1.071 1.047 1.100 1.073 1.082 1.100 0.950 1.100 V13 0.95 1.1 1.069 1.062 1.038 0.950 1.100 1.087 0.970 1.047 1.100 1.100 T11 0.9 1.1 1.037 1.007 1.059 0.959 1.090 0.977 0.938 1.020 1.100 1.100 T12 0.9 1.1 0.997 1.087 1.084 0.900 1.090 0.943 1.055 1.073 0.965 1.100 T15 0.9 1.1 1.027 1.099 1.095 0.938 1.090 1.023 1.063 1.080 0.900 1.100 T36 0.9 1.1 0.957 1.024 1.087 0.900 1.090 1.017 1.024 1.054 0.900 1.100 SVC1 Location - - 24 27 6 19 22 27 11 21 25 29 SVC2 Location - - 7 27 10 9 15 6 20 30 25 30 SVC1 Rating −10 10 10.000 9.893 −6.541 3.882 1.740 −3.380 9.274 8.786 10.000 10.000 SVC2 Rating −10 10 5.764 −6.277 1.209 5.482 −3.094 −9.559 0.639 8.550 −10.000 4.480 TCSC1 Location - - 15 39 10 3 14 37 38 32 34 33 TCSC2 Location - - 12 29 20 4 32 36 24 37 41 39 TCSC1 Rating 0 0.5 0.491 0.219 0.218 0.000 0.452 0.110 0.196 0.497 0.000 0.500 TCSC2 Rating 0 0.5 0.496 0.239 0.440 0.000 0.492 0.454 0.060 0.485 0.193 0.500 TCPS1 Location - - 14 16 34 15 14 38 14 26 40 40 TCPS2 Location - - 16 22 19 1 30 35 5 32 1 41 TCPS1 Rating −5 5 2.714 4.503 −3.990 3.917 1.235 4.787 −0.175 0.926 5.000 5.000 TCPS2 Rating −5 5 1.705 3.761 −4.281 1.150 0.113 4.095 1.148 4.860 −3.963 4.566 TFC ($/h) 806.999 808.030 809.449 818.654 814.865 - - - - - Emission (Ton/h) - - - - - 0.138 0.138 0.138 0.138 0.138 Electronics 2022,11, 3825 23 of 34 Table 10. Single objectives simulation results for case 3 and case 4. Control & State Variables Min Max Case-3 Case-4 GNDO MVO ALO SCA IMO GNDO MVO ALO SCA IMO PTG2 20 80 69.005 75.900 79.745 80.000 79.644 79.721 78.271 28.577 49.606 34.817 PWG5 0 75 75.000 74.966 75.000 73.825 74.667 47.504 11.236 13.419 0.000 47.506 PTG8 10 35 34.999 34.656 35.000 30.726 34.844 25.357 32.736 16.328 35.000 34.441 PWG11 0 60 59.999 58.422 60.000 50.635 59.748 26.234 20.292 4.602 0.000 55.391 PTG13 12 40 39.976 31.239 39.823 31.706 39.832 26.916 32.996 24.669 24.060 39.576 V1 0.95 1.1 1.036 1.100 1.098 1.098 1.089 1.008 0.958 0.967 0.950 0.955 V2 0.95 1.1 1.037 1.100 1.100 1.091 1.089 1.032 1.054 1.051 1.072 1.052 V5 0.95 1.1 1.027 1.090 1.091 1.100 1.089 1.010 1.016 1.017 0.960 0.955 V8 0.95 1.1 1.030 1.092 1.096 1.100 1.090 1.025 0.992 1.013 1.024 1.015 V11 0.95 1.1 1.100 1.099 1.100 1.100 1.089 0.950 1.069 1.009 1.100 1.004 V13 0.95 1.1 1.100 1.100 1.077 0.990 1.089 1.004 1.067 1.084 1.043 1.038 T11 0.9 1.1 1.012 1.047 1.046 1.030 1.090 0.938 1.063 0.949 1.017 0.905 T12 0.9 1.1 0.903 0.903 1.045 1.100 1.090 0.907 0.903 0.904 0.912 0.986 T15 0.9 1.1 0.998 1.042 1.100 1.018 1.090 0.945 1.048 1.065 1.044 1.069 T36 0.9 1.1 0.935 0.983 1.068 1.100 1.090 0.934 0.930 0.936 0.957 0.936 SVC1 Location - - 18 12 27 11 27 19 24 28 19 19 SVC2 Location - - 24 15 30 16 30 10 14 16 21 23 SVC1 Rating −10 10 4.901 6.871 3.951 6.535 9.956 8.082 8.769 −1.898 4.969 9.593 SVC2 Rating −10 10 10.000 5.448 5.238 −0.936 4.219 5.421 −2.342 −5.291 −1.465 9.547 TCSC1 Location - - 34 5 40 3 40 14 13 2 2 40 TCSC2 Location - - 11 40 41 2 41 18 25 9 5 29 TCSC1 Rating 0 0.5 0.494 0.107 0.500 0.000 0.471 0.353 0.300 0.075 0.027 0.500 TCSC2 Rating 0 0.5 0.500 0.198 0.497 0.000 0.498 0.499 0.391 0.087 0.000 0.486 TCPS1 Location - - 16 12 40 4 33 19 38 3 1 37 TCPS2 Location - - 19 15 41 1 34 15 41 5 5 36 TCPS1 Rating −5 5 1.744 −0.898 −3.378 0.433 4.979 −4.997 3.754 −1.638 5.000 2.660 TCPS2 Rating −5 5 0.257 4.748 −3.441 1.249 0.946 0.767 0.550 −3.874 0.128 −1.703 APL (MW) 1.647 1.735 1.686 2.482 1.880 - - - - - Voltage Deviation (p.u) - - - - - 0.124 0.150 0.177 0.227 0.165 Table 11. Single objectives simulation results for case 5 and case 6. Control & State Variables Min Max Case-5 Case-6 GNDO MVO ALO SCA IMO GNDO MVO ALO SCA IMO PTG2 20 80 78.161 28.765 76.487 20.000 74.363 44.894 47.802 55.203 20.000 56.164 PWG5 0 75 75.000 16.020 74.240 0.000 67.677 74.998 74.564 71.256 75.000 68.555 PTG8 10 35 35.000 34.855 34.405 10.000 32.815 35.000 32.893 33.271 35.000 30.826 PWG11 0 60 54.355 0.000 55.983 49.379 14.173 59.006 58.030 50.102 60.000 58.732 PTG13 12 40 12.001 28.840 38.080 16.659 35.961 21.583 22.222 32.274 19.540 23.229 V1 0.95 1.1 1.100 1.100 1.100 1.100 1.097 1.046 1.100 1.100 1.100 1.100 V2 0.95 1.1 1.100 1.100 1.100 1.100 1.097 1.042 1.097 1.098 1.100 1.100 V5 0.95 1.1 1.100 1.100 1.100 1.100 1.097 1.032 1.087 1.087 1.100 1.100 V8 0.95 1.1 1.100 1.100 1.100 1.100 1.097 1.035 1.092 1.091 1.100 1.098 V11 0.95 1.1 1.100 1.100 1.100 1.100 1.097 1.098 1.100 1.100 1.100 1.100 V13 0.95 1.1 1.100 1.100 1.100 1.100 1.097 1.035 1.100 1.083 1.100 1.098 T11 0.9 1.1 0.905 0.910 0.990 1.100 1.089 1.077 1.054 1.007 1.100 1.084 T12 0.9 1.1 0.904 0.909 0.990 0.900 1.089 0.901 0.903 1.089 1.100 1.084 T15 0.9 1.1 0.901 0.900 0.930 0.900 1.089 1.065 1.053 1.081 1.100 1.100 T36 0.9 1.1 0.901 0.909 0.910 0.900 0.910 0.985 1.001 1.033 1.100 1.087 SVC1 Location - - 10 29 29 29 26 24 28 26 15 24 SVC2 Location - - 29 30 30 30 26 13 14 26 3 10 SVC1 Rating −10 10 9.999 4.260 7.322 10.000 9.312 9.999 −3.639 3.886 10.000 8.490 SVC2 Rating −10 10 9.999 2.982 9.608 9.747 9.622 3.844 −1.487 3.411 −0.065 7.591 TCSC1 Location - - 38 38 38 24 36 16 7 39 1 18 TCSC2 Location - - 15 36 40 1 38 19 29 34 3 31 TCSC1 Rating 0 0.5 0.500 0.490 0.500 0.002 0.499 0.500 0.468 0.470 0.002 0.500 TCSC2 Rating 0 0.5 0.500 0.325 0.475 0.013 0.499 0.013 0.490 0.355 0.000 0.147 TCPS1 Location - - 36 4 33 3 38 14 4 35 31 30 TCPS2 Location - - 41 17 38 1 39 4 2 25 11 33 TCPS1 Rating −5 5 −4.999 3.788 4.884 −5.000 4.540 3.173 0.486 4.591 2.761 1.494 TCPS2 Rating −5 5 −4.998 1.993 4.772 −1.126 4.697 −0.508 −1.527 −0.901 5.000 2.191 VSI 0.096 0.100 0.096 0.108 0.102 - - - - - Total Gross Fuel Cost ($/h) - - - - - 1120.996 1125.970 1138.357 1187.287 1148.359 The overall fuel cost with GNDO, which included the two non-conventional sources of power plants and optimal placement of FACTS devices, was 806.999 $/h, which was the best in comparison with the other cited algorithm shown in Table 9. The reductions in TFC in comparison with MVO, ALO, SCA, IMO, SHADE-SF, DE-SF, ABC-SF, PSO-SF, FPA-SF, and MSA-SF were 1.031 $/h, 1.031 $/h, 2.45 $/h, 11.655 $/h 7.866 $/h, 0.0176 $/h, 0.4917 $/h, 0.4 $/h, 1.2553 $/h, 3.398 $/h, and 1.0403 $/h, respectively. This demonstrated the GNDO algorithm’s superiority over other cited metaheuristics algorithms. Electronics 2022,11, 3825 24 of 34 Figure 13 illustrates the convergence traits of the TFC minimization. Similar convergence traits for APL, voltage deviations, and VSI are depicted in Figures 14–16. Figure 17 also displays a comparison of the fuel cost decrease with various algorithms. In example 2, the GNDO method resulted in a pollutant gas emission of 0.138 tons per hour. In instance 3, the APL of the various transmission lines using the GNDO approach was 1.647 MW. The APL was 0.088 MW, 0.039 MW, 0.835 MW, 0.233 MW, 0.0997 MW, 0.0997 MW, 0.2598 MW, 0.2494 MW, 0.6127 MW, and 0.4972 MW less compared to MVO, ALO, SCA, IMO, SHADESF, DE-SF, ABC-SF, PSO-SF, and MSA-SF, respectively. A crucial factor for the grid’s ability to operate reliably was the voltage divergence of each bus from 1.0 per unit. Therefore, in scenario 4, the moth flame algorithm produced the lowest voltage variation (0.124 p.u), making it the best of the five optimization methods. The VSI, sometimes referred to as the L max index, varied between zero (no load) and one (voltage collapse). Therefore, in case 5, 0.096 was the lowest value for the L max index. In scenario 6, the overall gross fuel cost using the GNDO method was 1120.996 dollars per hour. It is interesting to note here that in Table 12, the total gross fuel cost of the proposed GNDO was more than the SHADE-SF, which further enforces the narrative of the “No free lunch theorem,” which states that no algorithm gives the best result in every problem. Electronics 2022, 11, x FOR PEER REVIEW 26 of 37 index. In scenario 6, the overall gross fuel cost using the GNDO method was 1120.996 dollars per hour. It is interesting to note here that in Table 12, the total gross fuel cost of the proposed GNDO was more than the SHADE-SF, which further enforces the narrative of the “No free lunch theorem,” which states that no algorithm gives the best result in every problem. Figure 13. Convergence characteristics of TFC minimization. Figure 14. Convergence characteristics of APL minimization. Figure 13. Convergence characteristics of TFC minimization. Electronics 2022, 11, x FOR PEER REVIEW 26 of 37 index. In scenario 6, the overall gross fuel cost using the GNDO method was 1120.996 dollars per hour. It is interesting to note here that in Table 12, the total gross fuel cost of the proposed GNDO was more than the SHADE-SF, which further enforces the narrative of the “No free lunch theorem,” which states that no algorithm gives the best result in every problem. Figure 13. Convergence characteristics of TFC minimization. Figure 14. Convergence characteristics of APL minimization. Figure 14. Convergence characteristics of APL minimization. Electronics 2022,11, 3825 25 of 34 Electronics 2022, 11, x FOR PEER REVIEW 27 of 37 Figure 15. Convergence characteristics of voltage deviation minimization. Figure 16. Convergence characteristics of VSI minimization. Table 12. Comparison of the simulation results for single objectives. Objectives Functions GNDO MVO ALO SCA IMO SHADE-SF DE-SF ABC-SF PSO-SF FPA-SF MSA-SF Total F.C ($/h) 806.999 808.030 809.449 818.654 814.865 807.0166 807.4907 807.399 808.2543 810.397 808.0393 Emission (T/h) 0.138 0.138 0.138 0.138 0.138 - - - - - - Ploss (MW) 1.647 1.735 1.686 2.482 1.880 1.7467 1.7467 1.9068 1.8964 2.2597 2.1442 V.D (p.u) 0.124 0.150 0.177 0.227 0.165 - - - - - - Lmax 0.096 0.100 0.096 0.108 0.102 - - - - - - Total Gross F.C ($/h) 1120.996 1125.970 1138.357 1187.287 1148.359 1104.077 1113.676 1116.365 1118.601 1164.719 1122.331 Figure 15. Convergence characteristics of voltage deviation minimization. Electronics 2022, 11, x FOR PEER REVIEW 27 of 37 Figure 15. Convergence characteristics of voltage deviation minimization. Figure 16. Convergence characteristics of VSI minimization. Table 12. Comparison of the simulation results for single objectives. Objectives Functions GNDO MVO ALO SCA IMO SHADE-SF DE-SF ABC-SF PSO-SF FPA-SF MSA-SF Total F.C ($/h) 806.999 808.030 809.449 818.654 814.865 807.0166 807.4907 807.399 808.2543 810.397 808.0393 Emission (T/h) 0.138 0.138 0.138 0.138 0.138 - - - - - - Ploss (MW) 1.647 1.735 1.686 2.482 1.880 1.7467 1.7467 1.9068 1.8964 2.2597 2.1442 V.D (p.u) 0.124 0.150 0.177 0.227 0.165 - - - - - - Lmax 0.096 0.100 0.096 0.108 0.102 - - - - - - Total Gross F.C ($/h) 1120.996 1125.970 1138.357 1187.287 1148.359 1104.077 1113.676 1116.365 1118.601 1164.719 1122.331 Figure 16. Convergence characteristics of VSI minimization. Electronics 2022, 11, x FOR PEER REVIEW 28 of 37 Figure 17. Comparison of the TFC reduction ($/h) with other algorithms. Figures 18 and 19 provide comparison charts with the minimizing of APL and overall fuel cost. The results of the simulations were compared to those of the most recent algorithms, including MVO, ALO, SCA, IMO, and other mentioned optimization approaches. It was found that the proposed method of Generalized Normal Distribution Optimization methodology produced superior results. Figure 18. Comparison chart for TFC minimization with different algorithms. 1.031 2.45 11.655 7.866 0.0176 0.4917 0.4 1.2553 3.398 1.0403 Total cost reduction ($/hr) in comparision with GNDO algorithm. MVO ALO SCA IMO SHADE-SF 800 802 804 806 808 810 812 814 816 818 820 806.999 808.03 809.449 818.654 814.865 807.0166 807.4907807.399 808.2543 810.397 808.0393 TFC Different Meta-Heuristics Algorithms Comparison of Total Fuel Cost with different MAs GNDO MVO ALO SCA IMO SHADE-SF DE-SF ABC-SF PSO-SF FPA-SF MSA-SF Figure 17. Comparison of the TFC reduction ($/h) with other algorithms. Electronics 2022,11, 3825 32 of 34 solutions for each situation involving optimal power flow. All of the results point to the suggested technique’s significant advantage in obtaining the best solutions to OPF issues with one or more objectives. Finally, it was shown that by integrating wind farms with FACTS devices utilizing a non-dominated sorting technique, MOGNDO could be successfully employed to address small and large optimal power flow challenges. Based on the extensive analysis of the proposed MOGNDO, the following can be summarized as its advantages— • Randomization in MOGNDO includes the diversity of the Pareto front being enhanced, since all solutions in the first dominated front will have an equal chance of being selected, and multi-objectives are made uniformly significant while performing local exploration. •MOGNDO can deal with large-scale search spaces and is less dependent on problem characteristics. Moreover, these algorithms are capable of estimating multiple points in the search domain simultaneously, due to their population-based nature. • MOGNDO strikes a good balance between exploitation and exploration, providing powerful searchability for finding the optimum solution • MOGNDO is superior in terms of the balance of diversity and convergence, the distribution of PF, and better convergence. Author Contributions: Conceptualization, S.B.P., J.V. and M.M.; Methodology, S.B.P., J.V., M.M. and T.K.M.; Software, T.K.M. and P.J.; Validation, S.B.P., M.M. and T.K.M.; Formal analysis, J.V., M.M. and T.K.M.; Investigation, S.B.P.; Resources, P.J.; Data curation, J.V. and P.J.; Writing—original draft, S.B.P., J.V., M.M., T.K.M. and P.J.; Writing—review & editing, S.B.P.; Visualization, T.K.M. and P.J.; Supervision, S.B.P.; Funding acquisition, M.M. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by the project SP2022/60 Applied Research in the Area of Machines and Process Control, which is supported by the Ministry of Education, Youth and Sports, in the Czech Republic. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available in the article. Conflicts of Interest: The authors declare no conflict of interest. Abbreviations Acronyms OPF Optimal Power Flow Mas Meta Heuristics Algorithms MOGNDO Multi-Objective Generalized Normal Distribution Optimization TG Thermal Generating unit WG Wind Generation ISO Independent System Operator PDF Probability Density Function BCS Best Compromise Solution MOMFO Multi-Objective Moth Flame Optimization MOOPF Multi-Objective Optimal Power Flow SHADE-SF Success History-based Adaptive Differential Evolution using Superiority of Feasible solutions method DE-SF Differential Evolution using Superiority of Feasible solutions method ABC-SF Artificial Bee Colony using Superiority of Feasible solutions method PSO-SF Particle Swarm Optimization using Superiority of Feasible solutions method Electronics 2022,11, 3825 33 of 34 FPA-SF Flower Pollination Algorithm using Superiority of Feasible solutions method MSA-SF Moth Swarm Algorithm using Superiority of Feasible solutions method TFC Total Fuel Cost APL Active power Loss VSI Voltage Stability Index Nomenclature ai,bi,ci,eiand diPrice constants for ith coal-based power plants. αi,βi,γi,ωiand µiToxic gas emanation constants concerning the ith coal-based units. gwDirect cost constant Pws Scheduled power of the wind unit. KRw Reserve cost coefficient regarding wind unit KPw Penalty cost coefficient of wind unit Pws Accessible power from the wind unit Pwr Specified output power from the wind unit fw(pw)Wind energy probability density function for the wind unit. vin,vrand vout Cut-in, rated, and cut-out wind velocity of the turbine respectively pwr Rated value of the generated output of the wind turbine τDegree of series compensation Xmn Line inductive reactance linking buses mand n Rmn Resistance of the line linking buses mand n Vmand VnBus voltage magnitudes linking buses mand n. δmand δnPhase angles of the linking buses mand n gmn and bmn Conductance and susceptance of the line linking buses mand n. Npq Number of load (PQ) buses vipu voltage level of ith bus. 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