Multi-objective Salp Swarm Algorithm Research Articles

Coordination of PSS and Multiple FACTS-POD to Improve Stability and Operation Economy of Wind-thermal-bundled Power System

Aims and Background: Focusing on the low-frequency oscillation and system power loss problem in wind-thermal-bundled (WTB) transmission systems and the interaction problem due to different controllers, this study aimed to improve the low-frequency oscillation characteristics of WTB transmission system while ensuring the economy of operation and suppressing the adverse interaction between controllers. Methods: For this purpose, a coordination and optimization strategy for a power system stabilizer (PSS), static synchronous series compensator with additional power oscillation damping controller (SSSC-POD), and static synchronous compensator with additional power oscillation damping controller (STATCOM-POD) is proposed based on multi-objective salp swarm algorithm (MSSA). The controller regulation characteristics are taken into account in the coordination method. Results: Several designed scenarios, including changing the transmission power of the tie line and increasing wind power output, are considered in the IEEE 4-machine 2-area to test the proposed method. The power flow analysis, characteristic root analysis, and time-domain simulation are used to analyze the simulation results. Conclusion: Simulation results demonstrate that the proposed approach can effectively suppress the low-frequency oscillation of the WTB system while reducing its net loss. The application in engineering issues for MSSA is supplemented.

UAV-Enabled Secure Communications via Collaborative Beamforming With Imperfect Eavesdropper Information

Unmanned aerial vehicles (UAVs) are playing a pivotal role in wireless networks due to their high mobility and on-demand deployment advantages. However, the UAV-enabled communications are susceptible to be wiretapped by eavesdroppers due to the strong line-of-sight (LoS) dominated air-ground channel. In this paper, we consider a UAV-enabled secure communication scenario, in which a group of UAVs form a UAV-enabled virtual antenna array (UVAA) to transmit information towards the remote base stations (BSs) via collaborative beamforming (CB), while multiple known and unknown eavesdroppers aiming to wiretap the information. Specifically, a secure communication multi-objective optimization problem (SCMOP) is formulated to achieve the maximization of the worst-case secrecy rate, the minimization of the maximum sidelobe level (SLL) as well as the minimization of the flight energy consumption of UAVs by obtaining optimal locations and excitation current weights concerning the UAVs as well as determining an optimal receiver BS that can achieve superior communication performance. To solve the formulated SCMOP which is demonstrated to be non-convex and NP-hard, an improved multi-objective salp swarm algorithm (IMSSA) with several specific operating factors is proposed. Simulations results demonstrate that the proposed IMSSA can deal with the formulated SCMOP effectively and outperforms other benchmark strategies. Moreover, the multi-hop relay is introduced to verify the reasonability of the UVAA system, and two benchmark schemes of the formulated SCMOP are introduced to demonstrate the necessity of the formulated SCMOP. In addition, the performance of the UVAA system under certain unexpected circumstances is estimated. Finally, experimental implementation is conducted by using a Raspberry Pi and the results demonstrate the practicality of the proposed CB-based secure communication approach in real-world scenarios.

Coverage Optimization of WSNs Based on Enhanced Multi-Objective Salp Swarm Algorithm

In complex two-dimensional monitoring environments, how to enhance network efficiency and network lifespan while utilizing limited energy resources, and ensuring that wireless sensor networks achieve the required partial coverage of the monitoring area, are the challenges of optimizing coverage in wireless sensor networks.With the premise of ensuring connectivity in the target network area, an enhanced multi-objective salp swarm algorithm based on non-dominated sorting (EMSSA) is proposed in this paper, by jointly optimizing network coverage, node utilization, and network energy balance objectives. Firstly, the logistic chaotic mapping is used to maintain the diversity of the initial salp swarm population. Secondly, to balance global and local search capabilities, a new dynamic convergence factor is introduced. Finally, to escape local optima more effectively, a follower updating strategy is implemented to reduce the blind following of followers while retaining superior individual information. The effectiveness of the strategy is validated through comparative experiments on ZDT and DTLZ test functions, and the proposed algorithm is applied to coverage optimization in WSNs in complex environments. The results demonstrate that the algorithm can adjust coverage thresholds according to different application requirements, providing various effective coverage optimization configurations. With the same preset requirements for partial coverage achieved, both network efficiency and lifespan have been significantly improved.

Parameter estimation of the hydraulic turbine regulator system using multi-objective lion algorithm

Accurate modeling of electric power generating unit and its hydraulic turbine regulation systems provides support for the speed controller synthesis and stability analysis. It is however a difficult task due to the presence of many non-linear factors in this system. an approach to estimate the parameters of hydraulic turbine regulatory system models is to derive the physical representation of each component and, through simulation, to compare to compare their models, outputs with real data obtained from a hydroelectric plant located in Brazil. The objective of this paper is to find the best values that will represent the system under study as a whole. This problem can be seen as an optimization problem. To find its feasible and optimal solution, this work proposes a new metaheuristics multi-objective based on the Lion Algorithm (LA), called the Multi-Objective Lion Algorithm (MOLA), and its application in the estimation of parameters of the system under study. In addition, the new metaheuristic proposed is validated by using a set of benchmark cases. The results have demonstrated that MOLA outperforms or at least performs similarly to Multi-objective Grey Wolf Optimizer (MOGWO), Multiple Objective Particle Swarm Optimization (MOPSO), Multi-objective Salp Swarm Algorithm (MSSA), Multiobjective Evolutionary Algorithm Based on Decomposition (MOEA/D), and Non-dominated Sorting Genetic Algorithm III (NSGA-III) in the optimization of multi-objective benchmark functions. These results, suggest that the proposed MOLA algorithm works efficiently.

Multi Objective Salp Swarm based Energy Efficient Routing Protocol for Heterogeneous Wireless Networks

Routing is a persistent concern in wireless sensor networks (WSNs), as getting data from sources to destinations can be a tricky task. Challenges include safeguarding the data being transferred, ensuring network longevity, and preserving energy in harsh environmental conditions. Consequently, this study delves into the suitability of using multi-objective swarm optimization to route heterogeneous WSNs in the hope of mitigating these issues while boosting the speed and accuracy of data transmission. In order to achieve better performance in terms of load balancing and reducing energy expenditure, the MOSSA-BA algorithm was developed. This algorithm combines the Multi-Objective Salp Swarm Algorithm (MOSSA) with the exploiting strategy of the artificial bee colony (BA) in the neighbourhood of Salps. Inspired by the SEP and EDEEC protocols, the integrated solutions of MOSSA-BA were used to route two and three levels of heterogeneous networks. The embedded solutions provided outstanding performance in regards to FND, HND, LND, percentage of remaining energy, and the number of packages delivered to the base station. Compared to SEP, EDEEC, and other competitors based on MOSSA and a modified multi-objective particle swarm optimization (MOPSO), the MOSSA-BA-based protocols demonstrated energy-saving percentages of more than 34% in medium-sized areas of interest and over 22% in large-sized areas of detection.

Proposed energy efficient clustering and routing for wireless sensor network

Wireless sensor network (WSN) is considered a growing research field that includes numerous sensor nodes used to gather, process, and broadcast information. Energy efficiency is considered one of the challenging tasks in the WSN. The clustering and routing are considered capable approaches to solve the issues of energy efficiency and enhance the network’s lifetime. In this research, the multi-objective-energy based black widow optimization algorithm (M-EBWOA) is proposed to perform the cluster-based routing over the WSN. The M-EBWOA-based optimal cluster head discovery is used to assure an energy-aware routing over the WSN. The main goal of this M-EBWOA is to minimize the energy consumed by the nodes while improving the data delivery of the WSN. The performance of the M-EBWOA is analyzed as alive and dead nodes, dissipated energy, packets sent to base station, and life expectancy. The existing research such as low-energy adaptive clustering hierarchy (LEACH), hybrid grey wolf optimizer-based sunflower optimization (HGWSFO), genetic algorithm-particle swarm optimization (GA-PSO), and energy-centric multi-objective Salp Swarm algorithm (ECMOSSA) are used to evaluate the efficiency of M-EBWOA. The alive nodes of the M-EBWOA are 100 for 2,500 rounds, which is higher than the LEACH, HGWSFO, GA-PSO, and ECMOSSA.

Optimal Control Strategy for Ship Cabin’s Active Chilled Beam System Using Improved Multi-Objective Salp Swarm Algorithm

Heating, ventilation, and air conditioning (HVAC) systems are the second largest energy consumers on cruise ships after the propulsion system. As a kind of HVAC system, active chilled beam (ACB) systems have been widely used for cabin service due to their performance of energy efficiency and good thermal comfort. However, conventional control strategies for ships’ ACB systems are not intelligent enough and cannot balance energy consumption and cabin comfort during the voyages of ships. This study developed an optimal control strategy for cabins’ ACB systems. First, a simulation environment considering dynamic conditions is established in TRNSYS. Second, an artificial neural network model is utilized to predict the energy consumption of the ACB system, while the predicted percentage dissatisfied is adopted to represent cabins’ thermal discomfort. Third, an improved multi-objective salp swarm algorithm is proposed to dynamically minimize both energy consumption and thermal discomfort. A TRNSYS–MATLAB co-simulation testbed is established to simulate the cabins served by an ACB system on a small cruise ship navigating from Hong Kong to Shanghai for validation tests and a comparison study. Compared to the conventional strategies, the proposed strategy can achieve a maximum energy savings of 12% while maintaining a predicted mean vote index less than 0.5, meeting the comfort requirements set by ASHRAE.

Evaluation and multi-objective salp swarm optimization of a new solid oxide fuel cell hybrid system integrated with an alkali metal thermal electric converter/absorption power cycle

This research introduces a new combination of an alkali metal thermal electric converter (AMTEC) and an absorption power cycle (APC) to boost the performance of the solid oxide fuel cell (SOFC) by recovering the high-quality heat released from the SOFC for further electricity production. The performance criteria of the proposed SOFC/AMTEC/APC hybrid system are mathematically derived using thermodynamic, exergoeconomic, and exergoenvironment-based carbon footprint analysis. Extensive parametric assessment is carried out to evaluate the effect of substantial design parameters on the system's performance criteria. Furthermore, the multi-objective salp swarm algorithm (MSSA) with decision-making techniques is performed to ascertain the hybrid system's maximum power density and minimum cost and environmental impact rates densities. Findings depict the effectiveness and merit of the proposed system with 81% power density improvement compared with the related SOFC-based system. Moreover, the system yields the maximum power density of 9589.37 W/m2, and the corresponding energy and exergy efficiencies are obtained by 55.2% and 57.74%, indicating 59.26% and 53.76% improvements relative to the single SOFC. Furthermore, the minimum cost and CO2 emission rate densities are achieved by 4.26 $/h m2 and 3.32 kgCO2/h m2, respectively, with 31.73% and 42.3% enhancements relative to the base point.

Optimal reservoir operation using the improved multi-step-ahead time-varying hedging rule under climate and land-use changes

ABSTRACT This study aims to optimize Yeywa Hydropower Reservoir (YHR) operation using the multi-step-ahead time-varying hedging (TVH) rule under climate change (CC) and land-use change (LUC) to improve summer power generation. The performance of three multi-objective algorithms – the Multi-objective Salp Swarm Algorithm (MOSSA), the Multi-objective Antlion Optimizer (MOALO) and the Non-dominated Sorting Whale Optimization Algorithm (NSOWA) – are compared. MOSSA provides the best solutions with a higher mean hypervolume in a shorter computation time, and it is utilized to optimize the TVH rules for four periods: monthly (TVH-1), quarterly (TVH-3), half-yearly (TVH-6) and yearly (TVH-12). The six-month-ahead TVH-6 rule and the five-month-ahead TVH-6 rule generate the highest summer power for the historical period (2011–2020) and the future period (2020–2059), respectively. The future decadal power generation is expected to be higher than the historical power generation. The future TVH-6 rule is more reliable and it has lower water deficits than the historical YHR operation rule.

Efficient, economical and energy-saving multi-workflow scheduling in hybrid cloud

Benefiting from the flexible, scalable and secure environment, hybrid cloud can overcome the shortage of limited resources in private cloud to simultaneously execute large-scale scientific workflows. In hybrid cloud, privacy-sensitive tasks are not allowed to be executed on public resources, while non-sensitive tasks are unrestricted. As an NP-Complete problem, it is extraordinarily challenging to schedule multiple workflows efficiently, economically and energy-savingly under quality-of-service constraints. This paper models the hybrid-cloud-based privacy-aware multi-workflow scheduling as a tri-objective optimization problem that optimizes workflow-oriented total tardiness, private-cloud-oriented total energy consumption, and public-cloud-oriented total monetary cost. To the best of authors’ knowledge, few studies have been conducted on the tri-objective privacy-aware multi-workflow scheduling in hybrid cloud (PMWS-HC). To solve this problem, we dissect various factors involved during task scheduling and devise a novel Heuristic Scheduling Algorithm based on 9 Factors (HSA9Fs), which dynamically selects the workflows and tasks to be scheduled, and the corresponding VMs to execute them. To optimize the three conflicting objectives simultaneously, we propose a nested algorithm called MSIA, which first employs a Multi-objective Salp swarm algorithm to explore for the Pareto solutions, and then uses an Iterative greedy Algorithm to perform a refined search on individuals to obtain high-quality solutions. Extensive Medium-Small-Scale and Large-Scale simulation experiments show that both HSA9Fs and MSIA outperform state-of-the-art scheduling algorithms in several multi-objective performance metrics.

An innovative optimal 4E solar-biomass waste polygeneration system for power, methanol, and freshwater production

Modern polygeneration systems have increased the flexibility of production in energy systems. In this paper, a polygeneration system for producing power, freshwater, and methanol using solar and waste resources is defined. By using waste heat recovery in Brayton, Steam Rankine, organic Rankine, and Kalina systems, power has been produced in the proposed system. Methanol is also provided by using CO2 capture from exhaust flue gases and hydrogen produced by Proton Exchange Membrane electrolyzer. The integration of microbial fuel cells, multi-effect distillation, and reverse osmosis desalination has been used for muddy seawater recovery and freshwater production. The proposed system has been analyzed using energy, exergy, exergeoeconomic, and exergoenvironmental (4E) analyses for three biomass fuels. The Multi-Objective Dragonfly Algorithm, Multi-Objective Thermal Exchange Optimization, Multi-objective Salp Swarm Algorithm, and Multi-Objective Water Cycle Algorithm have been used for the optimization of the whole system. In the end, the studied system has been modeled hourly. In this research, machine learning has been used to facilitate modeling the desalination sector, CO2 capture sector, and methanol production sector. Also, it has played a significant role in reducing optimization run time. The modeling results show that Municipal solid waste is the most economical fuel for the study system with a payback of 4.28 years. Date palm waste has also caused fewer environmental impacts than other fuels. The comparison of the results of optimization indicates that Multi-objective Salp Swarm Algorithm is the fastest method for system optimization and can lead to 4.04% and 7.16% improvement in the system cost and environmental impact rate. In general, the energy and exergy efficiencies of the system in the optimal state are calculated as 29.25%, and 23.59%, respectively. The total optimal production of electricity, methanol, and freshwater in the whole system is calculated at 62.86 MW, 1820.78 kg/h, and 63.88 kg/s, respectively. The dynamic analysis of the proposed system also showed that the production of methanol throughout the year will be higher using Municipal solid waste fuel than other fuels.

DV-Hop Algorithm Based on Multi-Objective Salp Swarm Algorithm Optimization.

The localization of sensor nodes is an important problem in wireless sensor networks. The DV-Hop algorithm is a typical range-free algorithm, but the localization accuracy is not high. To further improve the localization accuracy, this paper designs a DV-Hop algorithm based on multi-objective salp swarm optimization. Firstly, hop counts in the DV-Hop algorithm are subdivided, and the average hop distance is corrected based on the minimum mean-square error criterion and weighting. Secondly, the traditional single-objective optimization model is transformed into a multi-objective optimization model. Then, in the third stage of DV-Hop, the improved multi-objective salp swarm algorithm is used to estimate the node coordinates. Finally, the proposed algorithm is compared with three improved DV-Hop algorithms in two topologies. Compared with DV-Hop, The localization errors of the proposed algorithm are reduced by 50.79% and 56.79% in the two topology environments with different communication radii. The localization errors of different node numbers are decreased by 38.27% and 56.79%. The maximum reductions in localization errors are 38.44% and 56.79% for different anchor node numbers. Based on different regions, the maximum reductions in localization errors are 56.75% and 56.79%. The simulation results show that the accuracy of the proposed algorithm is better than that of DV-Hop, GWO-DV-Hop, SSA-DV-Hop, and ISSA-DV-Hop algorithms.

Optimal capacity configuration of the wind-storage combined frequency regulation system considering secondary frequency drop

With wind power integrated into the power system on a large scale, the system has become vulnerable to the frequency stability issue. The battery energy storage system (BESS) is considered the key solution to improving the system frequency regulation performance due to its fast response ability. Furthermore, the construction of wind-storage combined frequency regulation systems has been developed for many years, in which the optimal capacity configuration of the wind-storage system is getting more attention. However, the secondary frequency drop (SFD) caused by wind turbines (WTs) participating in primary frequency regulation (PFR) is neglected in most existing capacity configurations, which is worthy of further study. In this paper, the optimal capacity of the wind-storage combined frequency regulation system is studied from the perspective of SFD. The time-domain expressions of two-stage system frequency response considering SFD are derived based on the wind-storage combined frequency regulation model. Next, considering the technical and economic characteristics of wind-storage combined frequency regulation, an optimization model of the energy storage capacity configuration is established with the objective of minimizing the sum of the maximum frequency deviations in two stages and the energy storage cost. The optimization model is solved by the multi-objective salp swarm algorithm (MSSA) to obtain the setting value of wind-storage combined frequency regulation parameters and the optimal energy storage capacity. The effectiveness of the proposed method is verified in MATLAB. The simulation results show that the proposed model can effectively improve the frequency regulation effect of the system and ensure the optimal capacity configuration with better economy.

Multi‐objective salp swarm algorithm‐based fractional order fuzzy precompensated PDPI controller for frequency regulation of hybrid power system

AbstractSignificant number of renewable sources when integrated with distributed generations and energy storage components form a sustainable power system called hybrid power system (HPS). But the stochastic nature of the sources and continuous variation of load cause serious issues in the power systems like frequency instability. Our work proposes a novel approach for load frequency control (LFC) of the proposed system. For the work, both frequency deviation and control signal are used to give the input signal to energy storage components. A new fuzzy‐based controller called fractional order fuzzy precompensated PDPI (FO‐FPPDPI) controller for LFC of the mentioned HPS is suggested in this paper. The controller is optimized by a nature inspired multi‐objective salp swarm algorithm (MSSA). The comparison of the proposed approach with the classical PID controller and Fuzzy PID (FPID) controller is carried out by taking several system operating conditions. Robustness of the FO‐FPPDPI controller is also validated for system parameter variations and the superiority of the proposed control strategy over PID and FPID controllers is validated for all the situations. Finally, the control strategy is realized in OPAL‐RT to measure its fidelity with the simulation results.

Multi-objective capacity optimization configuration of independent wind-photovoltaic- hydrogen-battery system based on improved MOSSA algorithm

Under the background of large-scale and rapid development of renewable energy, in order to improve the economic benefit of the system and ensure the reliability of the system, this paper introduces hydrogen production and energy storage into the independent wind/photovoltaic/hydrogen/storage capacity optimization configuration method. In order to minimize the total planning cost and self-sufficiency rate of system power supply, a capacity optimization allocation model was established. On the basis of this model, the Multi-objective Salp Swarm Algorithm (MOSSA) was improved in Algorithm structure, Tent chaotic mapping was introduced to initialize the population, and the positions of leaders and followers were updated based on the adaptive spiral search strategy. The validity of the algorithm is verified by test function. Finally, the specific example is solved by MATLAB programming, and the improved multi-objective Salp Swarm Algorithm (IMOSSA) is used to obtain the capacity configuration scheme, which provides reference for the optimization design of independent photovoltaic hydrogen storage system.

Multivariable short-term electricity price forecasting using artificial intelligence and multi-input multi-output scheme

In deregulated power markets, electricity price forecasting is the most valuable tool. However, with inherent electricity price characteristics, such as high frequency and volatility, constructing an electricity price forecasting model remains a difficult task for decision-makers and scholars. Accurate electricity price point forecasting (PF) can guide market participants in maximizing benefits. Moreover, appropriate interval forecasting (IF) can provide further information based on PF. Accordingly, a novel electricity price multi-bi-forecasting system using multivariable and multi-input multi-output structures is formulated. The system has three stages: data preprocessing, combination forecasting, and performance evaluation. The data preprocessing stage removes and smooths the high-frequency electricity price and load data. Because the load series has a more regular cycle and smoother fluctuation than the electricity price series, two variables, electricity price and load, are employed for forecasting using a multivariable data arrangement rolling forecast mechanism. In addition, a multi-input multi-output structure is utilized by three member models (back propagation, bidirectional long short-term memory, and gated recurrent unit) to derive PF and IF results concurrently. The final results are obtained using a combined strategy based on the multi-objective salp swarm algorithm. Finally, three experiments are conducted in the Australian electricity market to evaluate the proposed system quantitatively. Results show that the designed system has superior ability in forecasting electricity price and practical application in real situations.

Multiobjective Salp Swarm Algorithm Approach for Transmission Congestion Management

In the newly emerged electric supply industry, the profit maximizing tendency of market participants has developed the problem of transmission congestion as the most crucial issue. This paper proposes a multiobjective salp swarm algorithm (MOSSA) approach for transmission congestion management (CM), implementing demand side management activities. For this, demand response (DR) and distributed generation (DG) have been employed. For willingly reducing the demand, demand response has been called by providing appropriate financial incentives that supports in releasing the congestion over critical lines. Distributed generation implementing wind plant as renewable independent power producer (RIPP) has also been included in order to reduce the load curtailment of responsive customers to manage transmission congestion. The proposed incentive-based demand response and distributed generation approach of CM, has been framed with various strategies employing different thermal limits over transmission lines and has resulted into significant reduction in congestion and in-turn improvement of transmission reliability margin. Diversity has been obtained in multiobjective optimization by taking two and three objective functions, respectively (minimization of overall operational cost, CO2 emission, and line loading). The by-products of the proposed algorithm for multiobjective optimization are minimized demand reduction, optimum size, and location of DG. To examine the proposed approach, it has been implemented on IEEE 30-bus system and a bigger power system IEEE 118-bus system; as well as the proposed technique of MOSSA has been compared and found better than reported methods and two other meta heuristic algorithms (multiobjective modified sperm swarm optimization and multiobjective adoptive rat swarm optimization).

Multi-period dynamic multi-objective emergency material distribution model under uncertain demand

This paper proposes a multi-period dynamic emergency material distribution (EMD) model under uncertain demand by considering the disaster degree and demand of different affected areas to improve the reasonability of disaster relief material distribution in post-disaster emergencies. The objective is to minimize the unsatisfied demand, materials distribution cost, and the risk of choosing the path to distribute materials. The model also focuses on the heterogeneity of emergency materials, the dynamics of demand, and the diversity of transportation tools. In addition, the demands and transportation times are described by fuzzy numbers to better fit with the real situation. To solve the problem we propose a hybrid multi-objective salp swarm algorithm and sine cosine algorithm (HMSSA-SCA). Furthermore, numerical examples of different scenarios and scales are solved and compared with the multi-objective salp swarm algorithm (MSSA), multi-objective evolutionary decomposition algorithm for emergency resource scheduling (MOEA/D-mdERS), non-dominated sorting genetic algorithm II (NSGA-II), and strength pareto evolutionary algorithm 2 (SPEA2) to verify the performance of the HMSSA-SCA algorithm. Finally, the floods and landslides disaster event in 2011 in Rio de Janeiro state, Brazil is used to verify the effectiveness and accuracy of the model and algorithm in the real case.

MSSA-DEED: A Multi-Objective Salp Swarm Algorithm for Solving Dynamic Economic Emission Dispatch Problems

Due to the rising cost of fuel, increased demand for energy, and the stresses of environmental issues, dynamic economic emission dispatch (DEED), which is the most precise mode for actual dispatching conditions, has been a significant study topic in current years. In this article, the higher dimensional, deeply correlated, non-convex, and non-linear multi-objective DEED problem is designated, involving both the fuel costs and emissions objectives simultaneously. In addition, the valve point effect, transmission loss, as well as the ramping rate, are considered. The Salp Swarm Algorithm (SSA) is a well-established meta-heuristic that was inspired by the foraging behavior of salps in deep oceans and has proved to be beneficial in estimating the global optima for many optimization problems. The objective of this article is to evaluate the performance of the multi-objective Salp Swarm Algorithm (MSSA) for obtaining the optimal dispatching schemes. Furthermore, the fuzzy decision-making (FDM) approach is employed to achieve the best compromise solution (BCS). In order to confirm the efficacy of the MSSA, the IEEE 30-bus six-unit power system, standard 39-bus ten-unit New England power system, and IEEE 118-bus fourteen-unit power system were considered as three studied cases. The obtained results proved the strength and supremacy of the MSSA compared with two well-known algorithms, the multi-objective grasshopper optimization algorithm (MOGOA) and the multi-objective ant lion optimizer (MALO), and other reported methods. The BCS of the proposed MSSA for the six-unit power system was USD 25,727.57 and 5.94564 Ib, while the BCS was 2.520778 × USD 106 and 3.05994 × 105 lb for the ten-unit power system, and was 1.29200 × USD106 and 98.1415 Ib for the 14 generating units. Comparisons with the other well-known methods revealed the superiority of the proposed MSSA and confirmed its potential for solving other power systems’ multi-objective optimization problems.

Hybrid multi-objective Harris Hawk optimization algorithm based on elite non-dominated sorting and grid index mechanism

In order to find Pareto optimal solution set uniformly distributed along all objectives, a Hybrid Multi-Objective Harris Hawk Optimization Algorithm (H-MOHHO) was proposed based on elite non-dominated sorting and grid indexing mechanism. In order to maintain and improve the coverage of Pareto optimal solution, a method combining the two terms is adopted to obtain the optimal Pareto optimal solution set. Firstly, a non-dominated ranking mechanism based on elite was used to assign rank and sum to select the best solution set, and then the archived grid index mechanism with update mechanism was used to select the final solution set. This hybrid structure can not only obtain the optimal Pareto solution set but also keep the diversity of the population and improve the effectiveness of solving multi-objective optimization problems. In order to verify the performance of the proposed H-MOHHO algorithm, 22 test functions and 4 multi-objective engineering problems are used for simulation, and four performance indexes are compared with Multi-Objective Particle Swarm Optimization (MOPSO), Non-dominated Sorting Genetic Algorithm II (NSGA-II), Multi-Objective Ant Lion Optimizer (MOALO), Multi-Objective Salp Swarm Algorithm (MSSA) and Multi-Objective Dragonfly Algorithm (MODA). Experimental results show that the proposed H-MOHHO algorithm has better competitiveness and applicability.