Improved Lyrebird optimization for multi microgrid sectionalizing and cost efficient scheduling of distributed generation

Abstract

The rising energy demand, substantial transmission and distribution losses, and inconsistent power quality in remote regions highlight the urgent need for innovative solutions to ensure a stable electricity supply. Microgrids (MGs), integrated with distributed generation (DG), offer a promising approach to address these challenges by enabling localized power generation, improved grid flexibility, and enhanced reliability. This paper introduces the Improved Lyrebird Optimization Algorithm (ILOA) for optimal sectionalizing and scheduling of multi-microgrid systems, aiming to minimize generation costs and active power losses while ensuring system reliability. To enhance search efficiency, ILOA incorporates the Levy Flight technique for local search, which introduces adaptive step sizes with long-distance jumps, improving the exploration-exploitation balance. Unlike conventional local search strategies that rely on fixed step sizes, Levy Flight prevents premature convergence by allowing the algorithm to escape local optima and explore the solution space more effectively. Additionally, a chaotic sine map is integrated to enhance global search capability, ensuring better diversity and superior optimization performance compared to traditional algorithms. Simulation studies are conducted on a modified 33-bus distribution system segmented into three independent microgrids. The algorithm is evaluated under single-objective scenarios (cost and loss minimization) and a multi-objective optimization framework combining both objectives. In single-objective optimization, ILOA achieves a generation cost of $19,254.64/hr with 0.7118 kW of power loss, demonstrating marginal improvements over the standard Lyrebird Optimization Algorithm and significant gains over Genetic Algorithm (GA) and Jaya Algorithm (JAYA). In multi-objective optimization, ILOA surpasses competing methods by achieving a generation cost of $89,792.18/hr and 10.26 kW of power loss. The optimization results indicate that, for the IEEE-33 bus system without considering EIR, the proposed ILOA algorithm achieves savings of approximately 0.0014%, 0.0041%, and 0.657% in operation costs compared to LOA, JAYA, and GA, respectively, when MG-1, MG-2, and MG-3 are operational. The analysis of real power loss reduction demonstrates that, in the IEEE-33 bus system without considering EIR, the proposed ILOA algorithm effectively minimizes power loss by approximately 0.692%, 1.696%, and 1.962% in comparison to LOA, JAYA, and GA, respectively, under the operational conditions of MG-1, MG-2, and MG-3. Additionally, reliability constraints based on the Energy Index of Reliability (EIR) are effectively incorporated, further validating the robustness of the proposed approach. Considering EIR, the real power loss analysis for the IEEE-33 bus system highlights that the proposed ILOA algorithm achieves a reduction of approximately 1.319%, 2.069%, and 2.134% in comparison to LOA, JAYA, and GA, respectively, under the operational scenario where MG-1, MG-2, and MG-3 are active. The results confirm that ILOA is a highly efficient and reliable solution for distributed generation scheduling and multi-microgrid sectionalizing, showcasing its potential for real-world applications such as dynamic economic dispatch and demand response integration in smart grid systems.

Introduction

The contemporary energy scenery is evolving, with a modification towards decentralized energy systems. Microgrids, small-scale power networks encompassing various distributed energy resources, play a pivotal role in this transformation^1,2. However, managing these interconnected microgrids poses challenges due to their diverse energy sources, demand fluctuations, and the necessity for constant optimization^3,4. The integration of distributed generation (DG) and microgrids into power systems has garnered significant attention for enhancing grid reliability, reducing emissions, and achieving energy independence. However, the complex interplay of sectionalizing multi-microgrids and optimally scheduling DG units necessitates advanced optimization techniques. Traditional methods often struggle to balance competing objectives such as cost minimization, emission reduction, and reliability improvement. Multi-microgrid systems represent a paradigm shift in the way we conceive, design, and operate power networks⁵. As an evolution of conventional microgrids, which are localized grids capable of operating self-reliantly or in combination with the utility, multi-microgrid systems expand this concept by interconnecting multiple microgrids into a more complex network⁶. Multi-microgrid systems are designed to improve the resilience and flexibility of power distribution⁷. By interconnecting several microgrids, these systems can achieve higher reliability, allowing for more dynamic and robust energy management⁸. Unlike standalone microgrids, multi-microgrid systems emphasize interconnectedness. These systems enable collaboration among individual microgrids, allowing for shared resources, energy trading, and mutual support during contingencies or peak demand periods⁹. Scaling up the concept of microgrids, multi-microgrid systems cater to larger geographical areas, diverse energy sources, and varied load demands¹⁰. Optimization algorithms play a crucial role in managing multiple objectives such as cost minimization, emission reduction, and system stability across interconnected microgrids¹¹. Leveraging renewable energy sources is a cornerstone of multi-microgrid systems. These systems facilitate the incorporation of various non-conventional energy resources, for instance wind, photovoltaic and hydro-electric, across multiple microgrids, enabling a more sustainable and greener energy matrix¹². Effective communication and control systems are essential for the seamless operation and coordination of multi-microgrid systems¹³. Advanced control strategies, including hierarchical control architectures and decentralized control mechanisms, are implemented for efficient management¹⁴. Developing sophisticated optimization and scheduling algorithms is critical for achieving the best utilization of resources in multi-microgrid systems. These algorithms consider multiple objectives, uncertainties in renewable energy generation, load variations, and grid stability constraints¹⁵. Enhancing grid resilience and reliability is a significant challenge. Multi-microgrid systems need to address issues related to grid stability, fault management, and rapid response to disturbances to ensure uninterrupted power supply¹⁶. Establishment of suitable regulatory frameworks and market structures is necessary to facilitate energy trading, incentivize efficient energy management strategies, and encourage participation from diverse stakeholders in multi-microgrid systems¹⁷. The integration of optimization techniques within multi-microgrid systems is indispensable, serving as a linchpin for efficient, resilient, and sustainable energy management¹⁸. These techniques address the complexities inherent in managing multiple interconnected microgrids, ensuring optimal utilization of resources, and meeting various operational objectives. Multi-microgrid systems entail juggling multiple objectives, including cost minimization, emission reduction, reliability enhancement, and grid stability¹⁹. Optimization techniques provide the framework to balance these conflicting objectives effectively. Optimizing resource allocation, such as distributed generation sources, storage systems, and load management, is essential. These techniques enable the allocation of resources efficiently across interconnected microgrids to meet demand while minimizing costs²⁰. The intermittent nature of renewable energy sources adds complexity. Optimization techniques accommodate uncertainties in renewable generation, ensuring optimal scheduling of renewables while maintaining system stability²¹. The long-term multi-objective optimization of renewable distributed generation (DG) power ratings and battery energy storage system (BESS) energy and power ratings in a grid-connected microgrid was carried out using the fuzzified Grey Wolf Optimizer²². A comprehensive approach was implemented to optimize the sizing of renewable DGs and BESS in grid-connected microgrids. The optimization framework incorporated multiple objectives, including minimizing total annual costs, emissions, and energy losses, while maximizing annualized benefits by deferring network upgrades. The day-ahead scheduling of microgrids was formulated as a multi-objective optimization problem, considering wind turbines, solar photovoltaics, and energy storage systems (EES)²³. This was achieved using game theory and a deep learning neural network (DL-NN) forecasting model, which integrated the Wind-Corrected Moving Average (WCMA) technique to predict wind speed, solar radiation, and load demand with improved accuracy.

Recent advancements in microgrid optimization have significantly emphasized cost-efficiency, emission reduction, and system reliability. Paul et al.²⁴ developed a quantum particle swarm optimization framework for optimizing sustainable energy management in grid-connected microgrids. Their approach simultaneously minimized cost and emissions, integrating renewable sources into a multi-objective optimization setup that addressed environmental and economic performance trade-offs. This study emphasized the role of intelligent algorithms in handling the conflicting objectives of operational cost and carbon footprint in energy systems. Phommixay et al.²⁵ presented a comprehensive review of cost optimization techniques for microgrids using particle swarm optimization (PSO). They highlighted the evolution of PSO variants and their application in economic dispatch, load scheduling, and system cost reduction. The study underscored the need for more adaptable and hybrid strategies to manage system uncertainties, particularly in renewable-integrated microgrids. Singh et al.²⁶ introduced a greedy rat swarm optimization algorithm coupled with price-elastic demand response to enhance economic and environmental efficiency in microgrid operation. Their approach incorporated real-time pricing signals, enabling responsive load management and improved system-level optimization through behavioral modeling of demand-side participants. Nguyen and Crow²⁷ proposed a stochastic optimization strategy for microgrids with renewable resources, incorporating battery degradation cost into the scheduling problem. Their model addressed the operational uncertainties of wind and solar power while also accounting for lifecycle cost implications of energy storage systems, providing a more holistic economic framework for microgrid planning. In a related study, Singh et al.²⁸ developed a hybrid demand-side policy for microgrid scheduling that balanced economic and emission objectives. By integrating demand-side management with generation scheduling, their model achieved enhanced flexibility in dispatch decisions, especially under variable generation and load conditions. Selvaraj et al.²⁹ applied the crow search algorithm for real-time power scheduling in distribution systems to improve microgrid performance. Their metaheuristic strategy focused on dynamic adaptability and convergence speed, demonstrating promising results for real-time operation under changing demand profiles and resource availability. Garcia-Torres et al.³⁰ examined stochastic optimization of microgrids incorporating hybrid energy storage systems and energy forecast uncertainties. Their study emphasized the significance of accounting for forecast deviations in both demand and renewable generation, which affect the optimal dispatch of storage units for grid flexibility services. Nadimuthu et al.³¹ explored the feasibility of renewable energy-based microgrids with vehicle-to-grid (V2G) technology in smart village applications. Their case study from India highlighted how V2G integration can support energy balancing and storage in isolated and rural microgrid setups, demonstrating social and technical viability. Singh et al.³² proposed a machine learning-based framework for energy management and forecasting in grid-connected microgrids. By leveraging predictive models, their approach improved the scheduling accuracy of distributed energy sources, enabling proactive optimization under uncertain operating conditions. Ott et al.³³ developed a mixed-integer linear programming (MILP) model for restoration planning in multi-microgrid distribution networks. Their framework supported system reconfiguration and restoration following faults, focusing on operational resilience and supply continuity in complex distributed environments. Karthik et al.³⁴ introduced a chaotic self-adaptive sine cosine algorithm for solving microgrid optimal scheduling problems. The chaotic adaptation enhanced the algorithm’s ability to escape local optima and maintain search diversity, making it suitable for multi-objective scenarios involving generation cost and power quality metrics. Abdalla et al.³⁵ examined optimized economic operation of microgrids that integrate combined cooling, heating, and power (CCHP) systems along with hybrid energy storage. Their model facilitated comprehensive energy flow management and improved economic efficiency in multiservice microgrid applications. Artis et al.³⁶ proposed a seismic-resilient planning framework for distribution networks with renewable-based multi-microgrids. Their multi-level strategy incorporated structural and operational planning to enhance grid survivability under seismic disturbances, addressing the critical aspect of infrastructure resilience. Rajagopalan et al.³⁷ developed a multi-objective energy management model for microgrids integrated with electric vehicles, using an iterative map-based crystal structure optimization algorithm. Their approach addressed the operational complexity introduced by mobile storage and dynamic charging demands, while optimizing for cost, emissions, and load balancing. Arefifar et al.³⁸ investigated the controllability of voltage and current in multi-microgrid smart distribution systems. Their analysis provided insights into hierarchical control strategies and dynamic coordination among interconnected microgrids, enabling improved system stability and regulation. Malik et al.³⁹ presented an intelligent multi-stage optimization approach for community-based microgrids under the multi-microgrid paradigm. Their method layered multiple decision levels—generation, storage, and load scheduling—achieving fine-grained control over distributed assets in community-scale applications. He et al.⁴⁰ proposed an improved genetic algorithm for economic scheduling in multi-microgrid systems. The enhancements in population diversity and crossover strategy led to better convergence and more robust solutions for cost optimization across interconnected grids.

Table 1 presents a comprehensive review of the literature on Multi-Objective Optimal Scheduling of Multi-Microgrids. The review highlights that real power loss is often overlooked as an objective function, despite its impact on system efficiency. This underscores the need for a more comprehensive optimization framework that integrates real power loss for improved practical applicability. The Improved Lyrebird Optimization Algorithm (ILOA) addresses this gap by analyzing seven case studies, demonstrating its ability to enhance system performance. These findings reinforce the importance of considering real power loss and diverse operational conditions to improve the reliability and efficiency of multi-microgrid systems.

Table 1 Literature review on multi-objective optimal scheduling of multi-microgrids.

The growing adoption of multi-microgrid systems in modern power grids, driven by the demand for resilience, sustainability, and flexibility, has introduced new challenges in distributed generation (DG) scheduling. Interconnected microgrids must balance conflicting objectives, such as minimizing generation costs, reducing power losses, and maintaining system stability under dynamic load conditions. Traditional optimization methods, including Genetic Algorithm (GA) and Jaya Algorithm (JAYA), often struggle with slow convergence, suboptimal solution quality, and limited scalability in large, complex systems. A major limitation is their tendency for premature convergence, restricting their ability to fully explore the solution space.

Additionally, the increasing integration of renewable energy sources introduces variability and uncertainty, complicating stable and cost-effective energy management. Given these challenges, advanced optimization techniques are essential to effectively balance exploration and exploitation, address renewable energy uncertainties, and optimize multi-objective functions in large-scale interconnected microgrid systems.

To enhance the local search efficiency of the Improved Lyrebird Optimization Algorithm (ILOA), the Levy Flight technique was incorporated due to its ability to balance exploration and exploitation effectively. Conventional local search strategies, such as Gaussian-based random walks, often struggle with premature convergence and getting trapped in local optima, limiting their effectiveness in complex optimization problems. In contrast, Levy Flight introduces adaptive step sizes with occasional long-distance jumps, allowing the algorithm to explore the solution space more efficiently while maintaining precise searches in promising regions. This characteristic helps ILOA navigate multi-modal search spaces, leading to faster convergence, improved solution accuracy, and enhanced robustness. By leveraging Levy Flight, ILOA achieves a well-optimized trade-off between global and local search, resulting in higher-quality solutions in both single-objective and multi-objective optimization problems. The effectiveness of this approach is demonstrated through comparative analysis, where ILOA consistently outperforms traditional optimization techniques in multi-microgrid scheduling tasks.

The key contributions of this paper are as follows:

• Integrated Levy Flight into the Lyrebird Optimization Algorithm (LOA), significantly improving local search efficiency and accelerating convergence to optimal solutions in multi-microgrid systems.

• Introduced a chaotic sine map to enhance global exploration, ensuring better diversity in the search space and reducing the likelihood of premature convergence.

• Proposed and applied the Improved Lyrebird Optimization Algorithm (ILOA) to a multi-microgrid test system, addressing both single-objective (generation cost minimization) and multi-objective (generation cost and power loss reduction) optimization problems, with a focus on real-world microgrid applications.

• Conducted a thorough comparison of ILOA with traditional algorithms like LOA, Genetic Algorithm (GA), and Jaya Algorithm (JAYA), demonstrating its superior performance in convergence speed, solution quality, and computational efficiency in large-scale systems.

• Validated the robustness of ILOA through extensive simulations, showing its ability to effectively balance multiple conflicting objectives, handle renewable energy uncertainties, and maintain system stability in dynamic microgrid environments.

• Highlighted ILOA’s potential for real-world applications in smart grid systems, particularly in dynamic economic dispatch and demand response integration, positioning the algorithm as a practical solution for future energy management challenges.

The manuscript is organized as follows:"Introduction"section introduces the problem of optimizing distributed generation in multi-microgrid systems, providing the motivation and background for the study."Sectionalization of microgrid distribution system"section presents the sectionalization of microgrid distribution systems, explaining how the system operates under normal and fault conditions, and the process of creating self-sufficient microgrids."Mathematical formulation of multi-objective optimal scheduling of distributed generators"section outlines the mathematical formulation of the multi-objective optimal scheduling problem, presenting the objective functions for cost minimization and power loss reduction, as well as the constraints involved."Improved lyrebird optimization algorithm"section details the proposed Improved Lyrebird Optimization Algorithm (ILOA), explaining its structure, including the integration of Levy Flight for local search enhancement and chaotic sine map for better global exploration."Simulation results and discussion"section presents the simulation results and discussions, comparing the performance of ILOA with traditional algorithms such as LOA, GA, and JAYA, and highlighting its superior performance in both convergence speed and solution quality."Conclusion and directions for future research" section concludes the paper, summarizing the key findings, and suggestsfuture directions for research and the real-world application of ILOA in smart grid systems.

Sectionalization of microgrid distribution system

The sectionalization process of a multi-microgrid system ensures a continuous power supply under both normal and faulty conditions through a self-healing mechanism that autonomously isolates faults while maintaining stable operation in unaffected areas. The primary goal of this approach is to maximize power delivery to consumers by dynamically reconfiguring the network in response to system conditions. During normal operation, control variables such as micro-source allocations across the distribution network are optimized to achieve specific objectives, including minimization of operating costs, reduction of system losses, and voltage deviation control. These objectives can be addressed individually or in combination to enhance overall system performance. The system maintains a radial topology, ensuring stability and effective protection coordination.

When a fault occurs in any microgrid section, the Microgrid Central Controller (MGCC) detects and isolates the affected region using real-time monitoring data. The faulted section is then disconnected from the rest of the system by opening the tie-line static switches, ensuring that power flow is maintained in the non-affected microgrids. If a fault occurs in a single microgrid (e.g., MG-1), it is isolated from MG-2 and MG-3, allowing the unaffected microgrids to continue operating independently. In the case of a multi-area fault, all impacted areas are disconnected, ensuring that only the healthy microgrids remain operational. Upon sectionalization, each microgrid operates independently in islanded mode, supplying its local loads using available distributed generation (DG) resources. The MGCC plays a crucial role in ensuring that each microgrid maintains self-sufficiency in supply and demand while optimizing energy distribution.

Following sectionalization, the distributed generation units are dynamically rescheduled to optimize power supply within the operational microgrids. The optimization process continues to consider the original objective functions, ensuring reliable and cost-effective energy distribution, efficient power balancing, and system resilience under fault conditions. Once the fault is cleared, the system gradually transitions back to its normal state by reclosing the tie-line switches, with the MGCC ensuring the smooth reintegration of previously disconnected microgrids, preventing power surges or instability. This sectionalization strategy enhances grid resilience by minimizing service disruptions, reducing downtime, and ensuring a reliable power supply to the maximum number of consumers. This approach follows established methodologies from prior studies (such as [Ref.52]) while incorporating modifications tailored to our test system.

Mathematical formulation of multi-objective optimal scheduling of distributed generators

The problem formulation of multi-objective optimal scheduling of Distributed Generators (DGs) in a Distribution System entails a nuanced approach aimed at balancing various competing objectives. At its core, this challenge revolves around achieving efficient energy generation and distribution while minimizing operational costs and real power loss. At its core, this challenge revolves around achieving efficient energy generation and distribution while minimizing operational costs and active power loss⁶⁹. In this complex scenario, the distribution system is divided into multi-microgrids, each representing a distinct section with its own set of DGs and loads. The primary objectives to be optimized are the operational costs associated with running the DGs and the reduction of real power loss within each microgrid. To address these objectives, a multi-objective optimization framework is employed. This involves formulating mathematical models that simultaneously optimize the operation of DGs to minimize costs and mitigate real power losses. The formulation process typically involves defining objective functions that quantify the operational costs and real power loss within each microgrid. Constraints are then imposed to ensure the feasibility of solutions, considering factors such as power balance, voltage limits, and DG capacity constraints.

Mitigation of operation cost

The operation costs objective function aims to minimize the expenses associated with running the DGs within the distribution system. This encompasses various factors for instance generation costs, maintenance costs, and operational overheads incurred in managing the generation units. By optimizing the scheduling of DGs, the research seeks to devise strategies that effectively reduce these operation costs, thereby enhancing the economic efficiency of the system. Here, the generation costs for all units are modelled as second-order quadratic equations, where the cost is a function of the active power generated by each unit. The objective function for minimizing these costs is formulated as the summation of the quadratic cost models for each generating unit, articulated as⁴¹:

$$F\left( {P_{g} } \right) = \mathop \sum \limits_{j = 1}^{k} (x_{j} + y_{j} P_{gj} + z_{j} P_{gj}^{2} )$$

(1)

Here ${x}_{j}$, ${y}_{j}$​, and ${z}_{j}$ represent the operational cost coefficients of the $jth$ generating unit. The variable $k$ denotes the total number of committed online generators.

Mitigation of real power loss

Minimizing real power loss, the energy dissipated during electricity flow, is vital for enhancing system efficiency and reliability. The objective function for real power loss focuses on reducing losses through strategic DG scheduling, voltage profile optimization, and network congestion mitigation^41,70.

$$F(P_{loss} ) = \mathop \sum \limits_{n = 1}^{NL} g_{n} \left[ {V_{j}^{2} + V_{k}^{2} - 2V_{j} V_{k} \cos \left( {\delta_{j} - \delta_{k} } \right)} \right]{ }$$

(2)

Here, ${g}_{n}$ represents the conductance of the $nth$ transmission line connecting bus $j$ to bus $k$. Additionally, $NL$ signifies the total number of transmission lines.

Constraints ensuring power balance

Power balance constraints enforce the fundamental principle that total power generation must equal total power consumption within each microgrid. These constraints ensure that the energy produced by DGs matches the energy demand from consumers, maintaining system stability and reliability. Neglecting to meet power balance constraints can result in voltage fluctuations, deviations in frequency, and general instability across the grid. Given that the network operates as a radial system, featuring numerous buses and loads within every feeder, it is essential to account for losses in transmission within the system⁴¹.

$$\mathop \sum \limits_{i = 1}^{m} \mathop \sum \limits_{j = 1}^{{N_{G} }} P_{Gi,j} = \mathop \sum \limits_{i = 1}^{m} P_{i,demand} + \mathop \sum \limits_{i = 1}^{m} P_{i,loss}$$

(3)

$$\mathop \sum \limits_{i = 1}^{m} \mathop \sum \limits_{j = 1}^{{N_{G} }} Q_{Gi,j} = \mathop \sum \limits_{i = 1}^{m} Q_{i,demand} + \mathop \sum \limits_{i = 1}^{m} Q_{i,loss}$$

(4)

Here ${P}_{Gi,j}$ and ${Q}_{Gi,j}$ represent the active and reactive power generated by the $jth$ generating unit at bus $i$ respectively. The variables ${P}_{i,demand}$ and ${Q}_{i,demand}$ represent the active and reactive power demands at bus $i$ respectively. Similarly ${P}_{i,loss}$ and ${Q}_{i,loss}$ denote the active and reactive power losses in the system at bus $i$. The term ${N}_{G}$ refers to the total number of generating units, while $m$ represents the total number of buses in the system. These equations ensure that the total generated power meets the system’s load demand while accounting for power losses.

Constraints on generation capacity

Generation capacity constraints limit the maximum amount of power that each DG unit can produce within a given time period. These constraints are essential for preventing overloading of generation units and ensuring that their operation remains within safe operating limits. By adhering to generation capacity constraints, the optimization algorithm can prevent the generation units from operating beyond their rated capacities, thereby safeguarding equipment integrity and reliability. The constraints to ensure power balance are indeed necessary, as they ensure that the total generation from distributed generation (DG) units and other sources matches the total load demand and losses in the network. This is critical for maintaining stable operation and avoiding issues like overloading, under-voltage, or unbalanced power flows. Without these constraints, the optimization results may be infeasible or lead to unstable network operation.

The active power generation output of every generating unit should be controlled within specified minimum and maximum boundaries⁴¹.

$$P_{gimin} \le P_{gi} \le P_{gimax}$$

(5)

$P_{gi}$ signifies the active power output of $ith$ generating unit while the maximum and minimum active power output are characterized as $P_{gimax} ,$ $P_{gimin}$ for the $ith$ generating unit⁷¹.

$$Q_{gimin} \le Q_{gi} \le Q_{gimax}$$

(6)

Here ${P}_{gimin}$ and ${P}_{gimax}$ represent the minimum and maximum active power operational bounds of unit ${\prime}j{\prime}$ within MG ${\prime}i$', respectively. Similarly ${Q}_{gimin}$ and ${Q}_{gimax}$ denote the minimum and maximum reactive power operational bounds of unit ${\prime}j{\prime}$ within MG ${\prime}i{\prime}$.

Constraints on bus voltages

Bus voltage constraints dictate the permissible voltage levels at various nodes or buses within the distribution network. Maintaining voltage within acceptable limits is crucial for ensuring the proper functioning of electrical equipment and appliances connected to the grid. Violation of bus voltage constraints can result in equipment damage, inefficient operation, and voltage instability. By enforcing bus voltage constraints, the optimization algorithm ensures that the voltage profile across the distribution system remains within specified limits, thus safeguarding the reliability and quality of power supply to consumers⁷².

$$V_{gimin} \le V_{gi} \le V_{gimax}$$

(7)

The above constraint ensures that the voltage magnitude ${V}_{gi}$ at the generating unit remains within the specified lower ${V}_{gimin}$ and upper ${V}_{gimax}$ limits. This maintains system stability and prevents voltage fluctuations that could impact the reliability and efficiency of power distribution.

Energy index of reliability (EIR)

The Energy Index of Reliability (EIR), represented by ($\xi$), is used as a constraint to assess the reliability of the power supply, indicating the number of customers impacted by supply disruptions. This index measures the dependability of load power delivery within the system by the collective operation of generators. A higher EIR value implies a lower likelihood of customers experiencing interruptions. The EIR is influenced by the Forced Outage Rate (FOR) of the $jth$ generator ($\Lambda$) and its output power (${P}_{j}$). The Forced Outage Rate reflects the probability of a generator failing to meet the required load demand. The mathematical formulation for calculating EIR is provided in Eq. (3.8) as referenced in^19,73.

$$EIR\left( \xi \right) = 1 - \left( {\frac{{\mathop \sum \nolimits_{j = 1}^{{N_{G} }} \Lambda_{j} P_{j} }}{{\mathop \sum \nolimits_{j = 1}^{{N_{G} }} P_{j} )}}} \right)$$

(8)

Here ${\Lambda }_{j}$ and ${P}_{j}$ represents the forced outage rate and generated output power of $jth$ generating unit correspondingly.

Formulation of multi‑objective optimal scheduling problem

The devising of the multi-objective optimal scheduling problem is presented as follows:

In this context, $F\left({P}_{g}\right)$ represents the objective function aimed at minimizing generation costs, while ${F(P}_{loss)}$ targets the reduction of active power loss, as described in Eq. (2). Various methods exist for tackling multi-objective optimization problems, including the weighted sum methodology⁷⁴, evolutionary algorithms⁷⁵, and the ε-constraint method⁷⁶. This paper employs the weighted sum approach to address the multi-objective optimal scheduling problem. In this approach, different weights are assigned to the conflicting objectives to generate multiple sets of Pareto optimal solutions. The optimal compromise solution is then selected from these sets based on the weights. By introducing a price penalty factor through $h$, the multi-objective problem is transformed into a single-objective optimization problem, as depicted in Eq. (8). The process for calculating the value of $h$ is detailed in⁷⁷.

In this methodology, the weighting factor ${w}_{1}$ and ${w}_{2}$ indicates the relative importance of each objective function. When ${w}_{1}$ is set to 1 and ${w}_{2}$ is set to 0, the focus is on minimizing generation costs. When ${w}_{1}$ is set to 0 and ${w}_{2}$ is set to 1 the emphasis shifts to minimizing active power loss. For multi-objective optimal scheduling, ${w}_{1}$ and ${w}_{2}$ are gradually varied from 1 to 0, generating a compromise solution at every step.

The multi-objective function minimization using the weighted sum method is defined as follows⁷⁸:

$$F\left( T \right) = w_{1} *F\left( {P_{g} } \right) + h*w_{2} *F\left( {P_{loss} } \right)$$

(9)

where ${w}_{1}+{w}_{2}=1$

A value for ${w}_{1}$ and ${w}_{2}$ at 0.5 signifies an equal balance between the generation cost and active power loss functions.

Determination of the optimal compromise solution with fuzzy logic

Prior to making a decision, it is essential to determine the most balanced solution from the set of optimal alternatives. The best compromise solution (BCS) is identified using the fuzzy membership methodology where a decrease in ${w}_{1}$ leads to an increase in generation costs and lessening in active power loss. The fuzzy membership approach is employed to identify this ideal compromise⁷⁸. In the $jth$ fitness function, the value ${f}_{j}$ for individual k is represented by a membership function ${\mu }_{j }^{k}$ which incorporates the inherent uncertainty in the decision maker’s judgment, as detailed below⁷⁸:

$${\mu }_{j}^{k}=\left\{\begin{array}{c}1 { f}_{j }\le {f}_{j}^{min} \\ \frac{{f}_{j}^{max}-{f}_{j} }{{f}_{j}^{max}- {f}_{j}^{min} } {f}_{j}^{min}<{f}_{j}<{f}_{j}^{max} \\ 0 {f}_{j} \ge {f}_{j}^{max} \end{array}\right.$$

(10)

Here, ${f}_{j}^{max}$ represents the highest value of the jth fitness function, while ${f}_{j}^{min}$ denotes its lowest value among the non-dominated solutions. The standardized membership function ${\mu }^{k}$ is then computed for every non-dominated solution $k$ as follows⁷⁸:

$$\mu^{k} = \frac{{\mathop \sum \nolimits_{j = 1}^{N} \mu_{j}^{k} }}{{\mathop \sum \nolimits_{k = 1}^{r} \mathop \sum \nolimits_{j = 1}^{N} \mu_{j}^{k} }}$$

(11)

In this context, $r$ symbolizes the overall number of non-dominated solutions. The optimum compromise solution is determined by selecting the one with the maximum value of ${\mu }^{k}.$

To determine the best compromise solution (BCS) from the complete set of Pareto optimal solutions, the min–max criterion⁷⁹ is applied as follows:

$${\text{ max}}\left( {min_{j} \left( {f_{r} } \right)} \right)$$

(12)

This implies that the solution with the highest value of ${min}_{j}\left({f}_{r}\right)$ is considered the best compromise solution. In this study, for objective functions (1) and (2), the normalized fitness values are represented as follows⁸⁰:

$$P_{g} = f_{1} = \frac{{P_{g} - P_{g}^{max} }}{{P_{g}^{min} - P_{g}^{max} }}$$

(13)

$$P_{loss,pu} = f_{2} = \frac{{P_{loss} - P_{loss}^{max} }}{{P_{loss}^{min} - P_{loss}^{max} }}$$

(14)

Improved lyrebird optimization algorithm

The Lyrebird Optimization Algorithm (LOA) is a population-based metaheuristic technique inspired by the adaptive behaviors of lyrebirds in nature⁸¹. When faced with threats, lyrebirds either flee rapidly or remain motionless in a concealed location, demonstrating an effective exploration–exploitation balance. In LOA, each individual represents a lyrebird, forming a population that iteratively searches for optimal solutions.

To enhance LOA’s performance, the Improved Lyrebird Optimization Algorithm (ILOA) integrates Levy Flight and a chaotic sine map. Levy Flight enhances exploitation, enabling a more efficient local search and faster convergence, while the chaotic sine map improves exploration, increasing search diversity and reducing premature convergence. Each lyrebird, acting as an agent, determines decision parameters based on its location in the search space. The population is represented as a matrix, where each vector corresponds to a decision variable, with initial positions set randomly as defined by Eq. (16).

$$X = \left[ {\begin{array}{*{20}l} {X_{1} } \\ \vdots \\ {\begin{array}{*{20}l} {X_{j} } \\ \vdots \\ {X_{n} } \\ \end{array} } \\ \end{array} } \right]_{N \times m} = \left[ {\begin{array}{*{20}l} {x_{1,1} } \\ \vdots \\ {\begin{array}{*{20}l} {x_{i,1} } \\ \vdots \\ {x_{N,1} } \\ \end{array} } \\ \end{array} \begin{array}{*{20}l} \ldots \\ \ddots \\ {\begin{array}{*{20}l} \ldots \\ {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} \\ \ldots \\ \end{array} } \\ \end{array} \begin{array}{*{20}l} {x_{1,j} } \\ \vdots \\ {\begin{array}{*{20}l} {x_{i,j} } \\ \vdots \\ {x_{N,j} } \\ \end{array} } \\ \end{array} \begin{array}{*{20}l} \ldots \\ {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} \\ {\begin{array}{*{20}l} \ldots \\ \ddots \\ \ldots \\ \end{array} } \\ \end{array} \begin{array}{*{20}l} {x_{1,m} } \\ \vdots \\ {\begin{array}{*{20}l} {x_{i,m} } \\ \vdots \\ {x_{N,m} } \\ \end{array} } \\ \end{array} } \right]_{N \times m}$$

(15)

$$x_{i,d} = lb_{d} + r \cdot \left( {ub_{d} - lb_{d} } \right)$$

(16)

In this context, $X$ represents the ILOA population matrix, where ${X}_{I}$ denotes the $ith$ ILOA member (candidate solution). Each ${X}_{I}$ represents the $dth$ dimension of the search space where $N$ is the number of lyrebirds, $m$ is the total number of decision variables, $r$ is a random number within the interval $[\text{0,1}]$ and ${lb}_{d}$ and ${ub}_{d}$ denote the lower and upper bounds of the $dth$ decision parameter correspondingly.

Every ILOA member serves as a candidate solution to the problem, and for every member, the objective function of the problem can be computed. Consequently, for every population member, a corresponding value for the objective function is obtained. These objective function values, equal in number to the size of population, can be organized into a vector representation, as per Eq. (17), indicating the set of evaluated objective function values for the problem⁸¹.

$$F = \left[ {\begin{array}{*{20}l} {F_{1} } \\ \vdots \\ {\begin{array}{*{20}l} {F_{i} } \\ \vdots \\ {F_{N} } \\ \end{array} } \\ \end{array} } \right]_{N \times 1} = \left[ {\begin{array}{*{20}l} {F\left( {X_{1} } \right)} \\ \vdots \\ {\begin{array}{*{20}l} {F\left( {X_{i} } \right)} \\ \vdots \\ {F\left( {X_{N} } \right)} \\ \end{array} } \\ \end{array} } \right]_{N \times 1}$$

(17)

In this context, $F$ represents the vector of fitness function evaluations, with ${F}_{i}$ denoting the evaluation of the objective function using the $ith$ ILOA member. These evaluations serve as a measure of the quality of candidate solutions. The optimal solution corresponds to the best evaluated objective function value (associated with the best ILOA member), while the poorest solution corresponds to the worst evaluated objective function value (linked to the worst ILOA member). Additionally, since the lyrebirds’positions in the problem-solving space is adjusted in each iteration and the finest candidate solution must be revised depending on a comparison of objective function values.

Mathematical modeling approach for ILOA

In the proposed ILOA methodology, the adjustment of population member positions occurs iteratively, guided by the mathematical emulation of lyrebird behavior in response to perceived threats. This modeling incorporates two distinct phases: (i) escape and (ii) concealment, mirroring the decision-making process observed in lyrebirds facing danger.

Within the ILOA framework, the decision-making process of lyrebirds, whether to employ escape or concealment strategies when confronted with danger, is replicated using Eq. (18). Equation (18) in the ILOA framework represents the decision-making mechanism inspired by the behavior of lyrebirds when responding to danger. Specifically:

The decision to either escape or conceal is determined by a randomly generated number ${r}_{p}$ within the range [0, 1]. Consequently, the position update of each ILOA member is determined solely by either the escape or concealment phase. If ${r}_{p}\le 0.5$, the position update is governed by Stage-I, corresponding to the“escape”strategy. Otherwise, the position update follows Stage-II, corresponding to the“concealment”strategy. This mechanism mimics how lyrebirds dynamically choose their response based on situational cues. Within the optimization process, these two stages represent different position update strategies tailored to exploration (escape) and exploitation (concealment), ensuring a balanced search process for optimal solutions.

$$X_{i} = \left\{ {\begin{array}{*{20}l} {Based \,on\, Stage - I, } \hfill & {\quad r_{p} \le 0.5} \hfill \\ {Based \,on\, Stage - II, } \hfill & {\quad else,} \hfill \\ \end{array} } \right.$$

(18)

Exploration stage

During this stage of ILOA, the adjustment of population member positions within the search space is depending on simulating the lyrebird’s evasive maneuvers from a perilous location to safer zones. The transition of the lyrebird to these secure regions results in substantial alterations to its position, facilitating the exploration of diverse regions within the problem-solving space. This underscores ILOA’s capacity for global exploration.

In the design of ILOA, each member identifies safer areas by considering the loci of other population associates with superior fitness function values. Consequently, Eq. (19) can be utilized to determine the set of safe zones for each ILOA member⁸¹.

$$SA_{i} = \{ X_{k} :F_{k} < F_{i} \;{\text{and}}\;k \ne i,{\text{where}}\;i = \left\{ {1,2, \ldots ,N} \right\}\;{\text{and}}\;k \in \left\{ {1,2, \ldots ,N} \right\}$$

(19)

In this context, ${SA}_{i}$ denotes the set of secure zones for the ith lyrebird, while ${X}_{k}$ represents the $kth$ row of the $X$ matrix, where $X$ has a better fitness function value (i.e., ${F}_{k}$) compared to the $ith$ ILOA associate (i.e., ${F}_{k}$ < ${F}_{i}$).

Within the ILOA framework, it is presumed that the lyrebird arbitrarily selects one of these safe zones for evasion. Following the modeling of lyrebird transposition in this stage, an updated location is computed for each ILOA member by applying Eq. (20). Subsequently, if this new location leads to an enhancement in the fitness function value, it supplants the earlier location of the equivalent associate as per Eq. (15).

$$x_{i,j}^{P1} = x_{i,j} + r_{i,j} .\left( {SSA_{i,j} - I_{i,j} .x_{i,j} } \right)$$

(20)

$$X_{i} = \left\{ {\begin{array}{*{20}l} {X_{i}^{P1} , } \hfill & {F_{i}^{P1} \le F_{i} ,} \hfill \\ {X_{i} , } \hfill & {else,} \hfill \\ \end{array} } \right.$$

(21)

In this context, ${SSA}_{i}$ represents the chosen secure zone for the $ith$ lyrebird, where ${SSA}_{i}$, denotes its $jth$ dimension. ${X}_{i}^{P1}$ represents the newly calculated position for the $ith$ lyrebird depending on the escape strategy of the suggested ILOA, with ${X}_{i}^{P1}$ representing its $jth$ dimension. FiP1 corresponds to its objective function value and ${I}_{i,j}$ are randomly selected as either 1 or 2⁸¹.

The indiscriminate number in Eq. (20) can be computed utilizing a sine map, with the preliminary values of ${C}_{t}$ and $a$ set to 0.36 and 2.8, respectively^82,83. The sine map introduces a chaotic behavior in the sequence generation, enhancing the algorithm’s exploration capability and preventing premature convergence. By iterating through Eq. (22), the sequence of ${C}_{t}$ maintains a non-linear and dynamic progression, improving the diversity of solutions in the optimization process.

$$C_{t + 1} = \frac{a}{4}\sin \left( {\pi C_{t} } \right), 0 < a < 4$$

(22)

where $t$ is the existing iteration number.

Exploitation stage

In the course of this phase of ILOA, the population member’s position within the exploration space is adjusted according to the lyrebird’s hiding strategy, aiming to seek refuge in nearby secure areas. This strategy involves meticulously surveying the surrounding environment and taking incremental steps to find an optimal hiding spot, resulting in minor adjustments to the lyrebird’s position. This characteristic highlights ILOA’s proficiency in local exploitation.

In the design of ILOA, the movement of each member towards a nearby suitable hiding area is modeled, and an updated position is computed for every associate using Eq. (23). If this new position enhances the fitness function value, it swaps the preceding location of the respective associate as per Eq. (26).

In this phase, the Levy flight methodology is used to modify the position of the overall finest component^84,85. Known for its exploratory capabilities, the Levy flight technique is also connected with restricted search^86,87.

$$x_{i,j}^{P2} = x_{i,j} + \left( {1 - 2 \cdot Levy\left( \lambda \right)} \right) \cdot \frac{{ub_{j} - lb_{j} }}{t}$$

(23)

$$Levy\left( \lambda \right) = 0.01 \frac{{r_{5} \sigma }}{{\left| {r_{6} } \right|^{{\frac{1}{\beta }}} }}$$

(24)

where $\sigma$ is determined as:

$$\sigma = \left[ {{{\Gamma \left( {1 + \lambda } \right){\text{sin}}\left( {\pi \frac{\lambda }{2}} \right)} \mathord{\left/ {\vphantom {{\Gamma \left( {1 + \lambda } \right){\text{sin}}\left( {\pi \frac{\lambda }{2}} \right)} {\left( {\Gamma \left( {\frac{1 + \lambda }{2}} \right)\lambda \left[ {2^{{\frac{{\left( {\lambda - 1} \right)}}{2}}} } \right]} \right)}}} \right. \kern-0pt} {\left( {\Gamma \left( {\frac{1 + \lambda }{2}} \right)\lambda \left[ {2^{{\frac{{\left( {\lambda - 1} \right)}}{2}}} } \right]} \right)}}} \right]^{1/\lambda }$$

(25)

where $\Gamma \left(\text{x}\right)=(x-1)!$,, ${r}_{5}$ represents the ${r}_{6}$ random numbers in the range [0,1], and $1<\beta \le 2$,. In this research, a persistent value of (β = 1.5) is applied. $\text{Levy}(\uplambda )$ relates to the step length realized by the Levy distribution, which has infinite mean and variance for 1 < λ < 3. $\uplambda$ is the distribution factor, and $\Gamma (.)$ signifies the gamma distribution function.

$$X_{i} = \left\{ {\begin{array}{*{20}l} {X_{i}^{P2} , } \hfill & {F_{i}^{P2} \le F_{i} ,} \hfill \\ {X_{i} , } \hfill & {else} \hfill \\ \end{array} } \right.$$

(26)

In this context, ${X}_{i}^{P2}$ represents the newly calculated position for the $ith$ lyrebird depending on the hiding approach of the suggested ILOA, where ${X}_{i}^{P2}$ denotes its $jth$ dimension. ${F}_{i}^{P2}$ corresponds to its objective function value. Additionally, $t$ denotes the iteration counter.

Iterative process for implementing the ILOA algorithm

After revising the positions of all lyrebirds, the principal iteration of ILOA concludes. Subsequently, the algorithm progresses to the next iteration, where the ILOA population update process, guided by Eqs. (11)–(19), persists until the final iteration. The finest candidate solution is revised and stored during each iteration. Upon the full execution of ILOA, the finest candidate solution accumulated throughout the algorithm’s iterations is outputted as the problem solution.

The procedural workflow for implementing the ILOA algorithm is outlined below:

• i. Input problem information: Gather details such as the fitness function, constraints, and decision parameters.

• ii. Set population and iteration parameters: Determine the number of population associates (lyrebirds) and the total iterations necessary for solving the problem.

• iii. Initial population generation: Randomly generate the initial population of lyrebirds and evaluate each lyrebird using the objective function.

• iv. Start iterative process: Begin with the first iteration.

• v. Update lyrebird positions: Update the locus of the main lyrebird in the problem-solving space. This update considers two strategies, chosen randomly with equal probability depending on Eq. (4):

• vi. If the escape approach is chosen, update the position using Eqs. (5)–(7).

• vii. If the hide strategy is chosen, update the position using Eqs. (8) and (9).

• viii. Update positions for all lyrebirds: Repeat the position update process for all lyrebirds in the population, similar to the first lyrebird.

• ix. Complete iteration: Once all lyrebirds’positions are updated, complete the current iteration. Save the best candidate solution based on the objective function evaluations during this iteration.

• x. Proceed to the next iteration: Repeat the lyrebird position update process iteratively until the final iteration is reached.

• xi. Finalize algorithm execution: After completing all iterations, identify and output the finest solution attained through the algorithm’s execution as the elucidation to the specified problem.

This concludes the implementation of the ILOA algorithm, providing the optimal solution based on the specified problem parameters and constraints.

Figure 1 illustrates a systematic flowchart representing the optimization process for power system operation, focusing on balancing generation costs, minimizing losses, and ensuring voltage stability. It integrates load flow analysis, candidate evaluation, and iterative updates to refine solutions based on fitness metrics. The flowchart effectively visualizes the decision-making process, highlighting convergence checks and scenario-specific objective weighting to achieve an optimal configuration. This structured approach ensures efficient handling of computational tasks and adaptable implementation across various case studies.

Fig. 1

Flowchart illustrating the optimal scheduling of microgrids using the Improved lyrebird optimization algorithm across diverse scenarios and case studies.

Evaluation of the proposed ILOA algorithm

To assess the effectiveness of the proposed ILOA algorithm, it is implemented in MATLAB R2023 A and tested on five standard benchmark functions. Its performance is compared against LOA, SCA, FSAPSO, KH, GA, DE, PSO, CLPSO, ICLPSO, FBCLPSO and FBICLPSO algorithms. The results demonstrate that ILOA outperforms all competing methods in terms of the best solution, mean solution, and standard deviation across all benchmark functions, as presented in Table 2.

Table 2 Evaluation of the proposed ILOA algorithm on benchmark test functions.

Simulation results and discussion

The test system utilized in this research is the standard IEEE 33-bus distribution network, with input data obtained from Ref⁴¹. It is separated into three independent microgrids, while preserving the radial configuration of the system. During the creation of these microgrids, specific modifications were made to the existing 33-bus system, as detailed below. The allocation of active and reactive power loads for each area is also based on the data from Ref⁴¹.

Altered 33-bus distribution test system and microgrid realization

The 33-bus distribution system is partitioned into three microgrids, designated as MG-1, MG-2, and MG-3⁵⁵. The specifics of line status, including reactance and resistance, are obtained from Ref⁴¹. for both scenarios: mitigation of generation cost and mitigation of active power loss.

To examine the suggested ILOA algorithm, the subsequent conventions are made:

• The distributed generators (DGs) used in this study are dispatchable, and their locations remain fixed.

• Isolation and tie-line connections can be established using a static switch.

The population size is fixed as 80, and the maximum no. of iterations is 200. Depending on these optimization attributes, two case studies are implemented to achieve optimal operation of the microgrid, namely cost minimization and real power loss minimization. The subsequent case studies are also examined:

Multiple areas fault:

Case-1: MG-1 is currently operational.

Case-2: MG-2 is currently operational.

Case-3: MG-3 is currently operational.

Single area fault:

Case-4: MG-1 & MG-2 are currently operational.

Case-5: MG-2 & MG-3 are currently operational.

Case-6: MG-1 & MG-3 are currently operational.

Not any fault:

Case-7: MG-1, MG-2 & MG-3 are currently operational.

Table 3 provides the percentage contributions of real power (P) and reactive power (Q) loads from different microgrids (MG1, MG2, MG3) and their combinations. MG1 has the smallest contribution, while MG3 contributes the largest share. The table also shows combined contributions from multiple microgrids, such as MG1 & MG2, MG2 & MG3, and MG1 & MG3. When all three microgrids operate together, they account for 100% of both real and reactive power. This information is essential for understanding how loads are distributed across the system. Table 4 outlines the line parameters (resistance R and reactance X in per-unit) for specific bus connections under various microgrid configurations. It also indicates whether certain lines are opened or closed for different scenarios, such as MG1, MG2, MG3, and their combinations. For example, Line 22 (from bus 3 to 23) remains open in many configurations. This data is crucial for analyzing the flexibility and reliability of the system under different microgrid operations. Table 5 presents the placement of distributed generators (DGs) in the 33-bus system for each microgrid. Microgrid-1 has DGs on buses 1, 2, and 20; Microgrid-2 has DGs on buses 3, 7, and 18; and Microgrid-3 has DGs on buses 23, 26, and 30. The strategic placement of DGs ensures optimal power generation and efficient energy distribution across the system. Table 6 provides the cost coefficients and operational constraints for each generator in the system. These coefficients are used to determine the generation costs. Additionally, the table specifies the minimum and maximum generation limits for each generator. The data for the 33-bus distribution system⁴¹ is provided in Table 7. Table 7 provides detailed information on line impedances and connected loads for the 33-bus system. It includes the resistance (R) and reactance (X) of each line in per-unit and the real (P) and reactive (Q) power loads connected to the buses. This data is fundamental for power flow analysis and optimizing system performance.

Table 3 Area-wise distribution of real and reactive power load percentages⁴¹.

Table 4 Line parameters of closed/opened lines for 33-bus system for various cases⁴¹.

Table 5 Placement of distributed generators (DGs) in a 33-bus system⁴¹.

Table 6 Cost coefficients for generators in the 33-bus system⁴¹.

Table 7 Data for the 33-bus system⁴¹.

Single and multi-objective optimization of generation cost and real power loss without EIR

Scenario-I (mitigation of generation cost)

In this scenario, the fitness function was focused exclusively on cost reduction. The operating cost coefficients for each distributed generator (DG) in the 33-bus distribution system were obtained from Ref⁴¹. For the minimization of generation cost, when the weighting factor w is fixed as 1, the minimum generation cost attained is 19,254.64 $/hr, with a corresponding real power loss of 0.7118 kW for Case-1. Similarly, the optimal generation cost and corresponding real power loss were determined for Case-2 through Case-7. Table 8 displays the optimal power generated by various distributed generators (DGs) for minimizing generation cost using the ILOA. Different generator units are activated based on system requirements, with some cases excluding certain generators to optimize cost and minimize losses. Case_1 has the lowest power losses and the lowest cost, indicating a minimal load scenario. Case_5 experiences the highest losses and the highest operational cost, suggesting a high-demand scenario. Cases with higher power demand (e.g., Case_7) show increased generation costs and losses, requiring multiple generators to meet load demand efficiently. Table 9 reveals that the ILOA algorithm yields better generation cost results, with values of 19,254.64 $/hr, 70,900.83 $/hr, 97,915.95 $/hr, 89,443.86 $/hr, 168,662.74 $/hr, 115,061.66 $/hr and 187,645.94 $/hr for cases 1 through 7, respectively. In comparison, the generation costs obtained using the LOA are 19,255.52 $/hr, 70,901.79 $/hr, 97,917.64 $/hr, 89,445.27 $/hr, 168,664.36 $/hr, 115,063.75 $/hr, and 187,647.63 $/hr for the same cases. In every case study, ILOA achieves the lowest operating cost compared to LOA, JAYA, and GA, making it a cost-effective choice for power system operators, particularly under high-load conditions. The results indicate that ILOA becomes increasingly efficient in reducing operational costs as system size grows. Figure 2 illustrates the convergence behavior of ILOA and LOA for Case-7, showing their progression toward the optimal solution. ILOA converges significantly faster, reaching the optimal value within 26 iterations, whereas LOA requires more iterations and exhibits fluctuations, reflecting instability in its optimization path. These oscillations indicate a less efficient trajectory, making LOA slower and less reliable in achieving convergence. In contrast, ILOA maintains a smooth and consistent search path, demonstrating superior exploration and exploitation capabilities that enable it to locate the global optimum more effectively. Figure 3 further highlights ILOA’s advantages, confirming its faster, steadier, and more reliable convergence, making it a more robust optimization approach than standard LOA.

Table 8 optimum values for different case studies in mitigating generation cost for a 33-bus system using the ILOA algorithm.

Table 9 Assessment of optimization results for the mitigation of generation cost for various case studies.

Fig. 2

Convergence characteristics for the mitigation of generation cost for Case_7.

Fig. 3

Convergence characteristics for the mitigation of active power loss for Case_7.

Scenario-II (mitigation of active power loss)

In this state, the objective function considered is solely the mitigation of active power loss. It is presumed that the accessible DGs are dispatchable with stable locations. To minimize active power loss, when the weighting factor w is fixed as 0, the lowest active power loss achieved is 0.5846 kW, with a corresponding generation cost of 19,256.84 $/hr for Case-1. Likewise, the optimal real power loss and corresponding generation cost were calculated for Cases 2 through 7. The losses for different case studies, as designated above, are presented in Table 10 for the ILOA algorithm applied to the 33-bus distribution system. Power generation is dynamically adjusted based on system demand, ensuring optimal loss reduction. Case_1 shows the best performance with the lowest active power loss of 0.5846 kW. This suggests that the ILOA algorithm is effective in reducing losses and achieving cost-efficient operations. Case_2 and Case_3 demonstrate higher losses, at 8.5169 kW and 32.6285 kW, respectively, but still outperform the other algorithms in terms of minimizing power loss. Case_4, Case_5, Case_6, and Case_7 all show varying degrees of active power loss with ILOA achieving relatively better results compared to the other methods in most cases.

Table 10 Optimum values for different case studies in mitigating active power loss for a 33-bus test system using the ILOA algorithm.

Table 10 provides the scheduled output power for each distributed generator (DG), along with the system’s active and reactive power losses and the overall generation cost. From Table 11, it is observed that the ILOA approach achieves minimum losses of 0.5846 kW, 8.5169 kW, 32.6285 kW, 10.25763 kW, 52.41465 kW, 32.6145 kW, and 70.4914 kW for cases I through VII, respectively. In contrast, the LOA results are 0.6219 kW, 9.0291 kW, 33.1028 kW, 11.3049 kW, 53.0127 kW, 33.1453 kW, and 70.9827 kW for the same cases. The convergence characteristics depicted in Fig. 3 vividly illustrate the superior performance of the Improved Lion Optimization Algorithm (ILOA) in reducing power losses when compared to the conventional Lion Optimization Algorithm (LOA). Moreover Fig. 3 highlights that the ILOA achieves a more significant reduction in active power loss, underscoring its enhanced optimization capabilities. Furthermore, the convergence curve of the proposed ILOA exhibits a smoother and more rapid descent toward the optimal solution in comparison to the LOA. This indicates that the ILOA not only accelerates the convergence process but also ensures greater stability in the optimization trajectory, thereby demonstrating its efficiency and robustness in minimizing active power losses.

Table 11 Comparison of optimization results for the mitigation of active power loss for various case studies.

Scenario-III (mitigation of generation cost and active power loss)

In this scenario, the optimization of generation cost and reduction of active power loss is considered as a multi-objective problem. Table 12 presents the optimal power generated by various distributed generators (DGs) to mitigate both generation cost and active power loss using the Improved Lyrebird Optimization Algorithm (ILOA) for all the case studies considered. The optimal trade-off between the two objectives was achieved by fine-tuning the weighting factor $w$ from 1 to 0. This table also serves as a foundation for evaluating the finest compromise solution, which aims to balance both minimizing generation costs and active power losses across various operational scenarios.

Table 12 Optimal results for different case studies in reducing generation cost and active power loss for a 33-bus test system using the ILOA algorithm.

Table 13 illustrates that the ILOA provides the most favourable compromise solution. This indicates that as the system size grows, the ILOA proves to be more efficient in reducing operational costs.

Table 13 Assessment of finest compromise solution for the alleviation of generation cost and active power loss for various case studies.

Figures 4 and 5 provide a comprehensive visual representation of the Pareto fronts obtained for the multi-objective optimization problem, focusing on the simultaneous minimization of generation cost and active power loss reduction. These figures illustrate the comparative performance of both the Improved Lion Optimization Algorithm (ILOA) and the standard Lion Optimization Algorithm (LOA) for Case_4 and Case_7, respectively. A close examination of the Pareto fronts reveals that the ILOA consistently identifies superior trade-off solutions, positioning itself more favourably than the LOA. The optimal points on the Pareto front demonstrate that the ILOA not only surpasses the LOA in achieving lower costs and reduced power losses but also exhibits better solution distribution and diversity. The well-spread, non-dominated solutions offered by the ILOA confirm its robustness and effectiveness in handling the optimization problem. Moreover, these findings underscore the feasibility and reliability of the ILOA in optimizing power distribution within the modified IEEE 33-bus system. By providing a more comprehensive and balanced set of optimal solutions, the ILOA ensures that decision-makers can select the most suitable operational conditions based on system requirements, further validating its superiority over conventional approaches.

Fig. 4

Pareto front distribution for generation cost and emission mitigation in Case_4.

Fig. 5

Pareto front distribution for generation cost and emission mitigation in Case_7.

Single and multi-objective optimization of generation cost and real power loss with EIR

Table 14 presents the placement and Forced Outage Rate (FOR) of Distributed Generators (DGs) in each microgrid. It outlines the specific buses where DGs are positioned in Microgrid-1 (MG1), Microgrid-2 (MG2), and Microgrid-3 (MG3), along with their associated FOR values. This information is crucial for understanding the reliability and operational constraints of DGs within each microgrid.

Table 14 Placement and forced outage rate (FOR) of distributed generators in each microgrid⁷¹.

Scenario-I (mitigation of generation cost)

Table 15 provides a comprehensive analysis of Scenario 1, emphasizing the minimization of operating costs while ensuring an Energy Index of Reliability (EIR) of at least 0.97. It provides variables and their corresponding values across seven case studies, showcasing the results of optimization efforts to minimize operational costs under this scenario. In this context, the fitness function was designed with a primary focus on minimizing operational costs. The operating cost coefficients for each distributed generator (DG) within the 33-bus distribution system were derived from Ref⁴¹. To reduce generation costs, the weighting factors ${w}_{1}$ and ${w}_{2}$ were assigned values of 1 and 0, respectively. Under this condition, the minimum generation cost achieved was 19,254.64 $/hr, with an associated real power loss of 0.6487 kW for Case_1. Similarly, the optimal generation cost and corresponding real power losses were calculated for Cases 2 through 7. Furthermore, Table 15 outlines the optimal power outputs of various DGs aimed at minimizing generation costs using the Improved Lyrebird Optimization Algorithm (ILOA). Additionally, Table 15 demonstrates that the ILOA consistently outperforms in terms of generation cost efficiency. The results obtained using the ILOA for Cases 1 through 7 are 19,255.39 $/hr, 87,174.21 $/hr, 110,297.19 $/hr, 102,014.03 $/hr, 196,721.79 $/hr, 125,375.57 $/hr, and 213,370.91 $/hr, respectively. In contrast, the generation costs obtained using the standard Lyrebird Optimization Algorithm (LOA) are 19,256.62 $/hr, 87,175.93 $/hr, 110,300.41 $/hr, 102,017.62 $/hr, 196,724.28 $/hr, 125,379.4 $7/hr, and 213,375.18 $/hr for the same cases. The comparison shows that as system size increases, ILOA significantly outperforms LOA in reducing operational costs. Generation cost minimization was successfully achieved across Cases 1 to 7, while maintaining an Energy Index of Reliability (EIR) of at least 0.97 in every scenario. This demonstrates that ILOA not only lowers costs but also ensures system reliability, guaranteeing stable and secure operation. By keeping EIR at or above 0.97, the optimization strategy effectively balances cost reduction with reliability, ensuring that cost-saving measures do not compromise system stability. As EIR reflects the system’s ability to deliver power reliably, maintaining this threshold confirms that economic benefits are achieved without sacrificing performance. Table 16 presents a detailed evaluation of generation cost minimization across different case studies, comparing ILOA with LOA, JAYA, and GA. The results highlight ILOA’s superior efficiency, making it a more effective solution for optimizing both economic and operational performance.

Table 15 Scenario-I: operating cost minimization with EIR ($\Lambda$) ≥ 0.97.

Table 16 Assessment of optimization results for the mitigation of generation cost for various case studies.

Scenario-II (mitigation of active power loss)

Table 17 focuses on Scenario-II, which aims to minimize active power loss with an EIR greater than or equal to 0.97. It lists the variables and their respective values for the seven case studies, demonstrating the optimization results achieved under this scenario. In this scenario, the objective function is exclusively focused on minimizing active power loss. It is assumed that the available distributed generators (DGs) are dispatchable and have fixed locations. The weighting factors ${w}_{1}$ and ${w}_{2}$ were set to 0 and 1, respectively, in order to minimize active power loss. Under these conditions, the minimum active power loss achieved for Case 1 is 0.6759 kW, accompanied by a corresponding generation cost of $19,256.84 per hour. Similarly, the optimal active power loss and the associated generation costs were determined for Cases 2 through 7. Table 17 presents the active power losses for the various case studies computed using the Improved Lyrebird Optimization Algorithm (ILOA) applied to the 33-bus distribution system. Moreover Table 16 provides detailed information on the scheduled power outputs for each DG, as well as the system’s active and reactive power losses and the overall generation costs. The results in Table 17 demonstrate that the ILOA achieves minimal active power losses of 0.6759 kW, 13.893 kW, 42.7795 kW, 14.2931 kW, 55.0349 kW, 37.9977 kW, and 73.8054 kW for Cases 1 through 7, respectively. In comparison, the corresponding results obtained using the Lyrebird Optimization Algorithm (LOA) are 0.6883 kW, 14.591 kW, 43.3917 kW, 14.8704 kW, 55.7019 kW, 38.8627 kW, and 74.7915 kW. Active power loss minimization was successfully achieved across Cases 1 to 7, while maintaining an Energy Index of Reliability (EIR) of at least 0.97. This ensures that operational efficiency does not compromise system reliability, demonstrating the robustness of the optimization approach. By sustaining an EIR of 0.97 or higher, the system achieves significant power loss reduction while preserving high reliability standards, effectively managing complex power distribution challenges. Table 18 provides a comparative analysis of active power loss minimization results, evaluating the performance of ILOA against LOA, JAYA, and GA. The results highlight ILOA’s superior efficiency in reducing power losses, making it a more effective optimization technique compared to conventional methods.

Table 17 Scenario-II: Active power loss minimization with EIR ($\Lambda$) ≥ 0.97.

Table 18 Comparison of optimization results for the mitigation of active power loss for various case studies.

Scenario-III (mitigation of generation cost and active power loss)

Table 19 presents Scenario-III, which addresses the combined objectives of minimizing operating costs and active power losses while ensuring an EIR of 0.97 or higher. It provides detailed results for various variables and their outcomes across seven case studies, highlighting the trade-offs and benefits of addressing both objectives simultaneously. In this scenario, the optimization process is formulated as a multi-objective problem, aiming to minimize both generation cost and active power loss. This approach ensures a balanced trade-off between economic and technical objectives. The optimization is carried out under the condition that the Energy Index of Reliability (EIR) remains greater than 0.97 for all cases, thereby maintaining a high level of reliability in the system. Table 19 provides detailed insights into the optimal power outputs of the distributed generators (DGs) achieved using the Improved Lyrebird Optimization Algorithm (ILOA). The optimal balance between the two objectives was achieved by setting the weighting factors ${w}_{1}$ and ${w}_{2}$ both to 0.5, ensuring the best compromise solution. The Improved Lyrebird Optimization Algorithm (ILOA) produced generation costs of 19,254.78$/hr, 89,388.26 $/hr, 110,383.43 $/hr, 106,527.38 $/hr, 201,843.27 $/hr, 127,333.58 $/hr, and 227,641.59 $/hr, with corresponding real power losses of 0.6141 kW, 13.7381 kW, 40.6321 kW, 14.1616 kW, 53.7482 kW, 38.6456 kW, and 69.8214 kW for Cases 1 through 7, respectively. Table 20 presents the results for all case studies, demonstrating ILOA’s effectiveness in reducing both generation costs and active power losses simultaneously. This dual-objective optimization ensures efficient system operation while maintaining reliability constraints, highlighting ILOA’s robustness in addressing complex power distribution challenges. The results indicate that ILOA’s efficiency improves as system size increases, consistently delivering optimal compromise solutions. Additionally, the Table 20 provides a comparative analysis of ILOA, LOA, and JAYA, showcasing their respective strengths and limitations, with ILOA demonstrating superior optimization performance. ILOA outperforms LOA due to its advanced mechanisms, specifically the chaotic sine map and Levy Flight, which significantly enhance its search capabilities. The chaotic sine map introduces non-linear, dynamic behavior, enabling broader exploration of the solution space while preventing the algorithm from getting trapped in local optima. This controlled randomness improves search diversity, ensuring a more effective and extensive search process. Meanwhile, Levy Flight enhances exploration by allowing larger, adaptive jumps, facilitating the discovery of more optimal solutions. Inspired by natural foraging behavior, this technique enables ILOA to efficiently navigate multi-modal search spaces, improving both solution quality and convergence speed. By integrating these two mechanisms, ILOA achieves a more balanced and robust optimization process, outperforming standard LOA in solving complex, multi-objective problems. These enhancements make ILOA a highly effective approach for optimizing large-scale power distribution systems compared to traditional algorithms.

Table 19 Scenario 3: Operating cost and active power loss minimization with EIR ($\Lambda$) ≥ 0.97.

Table 20 Assessment of finest compromise solution for the alleviation of generation cost and active power loss for various case studies.

Analysis of results

A comprehensive analysis of distributed generation (DG) scheduling is presented, utilizing various optimization algorithms within an altered 33-bus electrical distribution system. The study is organized across multiple case studies and divided into two main optimization strategies: single-objective optimization and multi-objective optimization. Table 21 and 22 offer an extensive dataset, examining operating costs and real power losses across different scenarios, both with and without the Enhanced Index of Reliability (EIR) criterion. Table 20 showcases the performance of the proposed Improved Lyrebird Optimization Algorithm (ILOA) alongside the Lyrebird Optimization Algorithm (LOA), JAYA, and Genetic Algorithm (GA) across seven distinct case studies. These results are compared for scenarios with and without the inclusion of EIR as a scheduling criterion. The EIR values for optimal scheduling without considering the EIR criterion have been derived based on the DG power outputs from the test results summarized in Tables 15, 16, 17, 18, 19 and 20. Furthermore, Table 21 highlights that incorporating the EIR criterion into the scheduling process ensures that DGs are optimally scheduled to meet the reliability requirement, in addition to achieving the desired minimization objectives. This approach enhances system performance by simultaneously addressing reliability and operational efficiency.

Table 21 Comparison of EIR values for optimal dg scheduling across different scenarios in the altered 33-bus system with and without the reliability criterion (single-objective optimization).

Table 22 Comparison of EIR values for optimal DG Scheduling across different scenarios in the altered 33-bus system with and without the reliability criterion (multi-objective optimization).

The results are divided into two scenarios: without EIR and with EIR. Across all cases, the inclusion of EIR results in higher operating costs, indicating that reliability considerations introduce additional operational expenses. The operating cost varies significantly across the different cases. Among the algorithms, ILOA consistently yields lower operating costs compared to LOA, JAYA, and GA, making it a potentially more cost-effective choice for DG scheduling. A similar trend is observed for real power loss minimization, where the inclusion of EIR generally leads to higher losses. The real power loss values vary across the cases, with Case_1 showing the lowest losses (around 0.58 kW without EIR and 0.67 kW with EIR), whereas Case_7 has significantly higher losses (70.49 kW without EIR and 73.80 kW with EIR). Among the optimization algorithms, GA and JAYA tend to exhibit slightly higher power losses compared to ILOA and LOA, although the differences are marginal in certain cases.

Table 22 extends the analysis to a multi-objective framework, considering a compromise between operating cost and real power loss. The evaluation follows the same seven-case structure, comparing the four optimization techniques with and without EIR. Without EIR, the operating costs are significantly lower across all cases compared to the EIR-included scenarios. In higher complexity cases like Case_5 and Case_7, the cost increase is more substantial. For example, in Case_7, the cost rises from 1,87,892.37 $/hr (ILOA) without EIR to 2,27,642 $/hr with EIR, indicating a major impact of reliability considerations. The real power loss values also increase when EIR is considered, though the magnitude of increase varies across cases and algorithms. For instance, in Case_1, the real power loss remains relatively low at 0.66037 kW (ILOA) without EIR, but with EIR, it is slightly reduced to 0.6141 kW. This suggests that in some cases, EIR can actually help optimize power loss while still increasing costs. However, in higher load cases like Case_7, the power loss increases from 71.07 kW to 69.82 kW with EIR, showing that reliability considerations do not always lead to higher losses, but often create a trade-off between cost and loss performance.

Across both single and multi-objective optimization frameworks, including EIR consistently raises operational costs, reflecting the additional constraints imposed by reliability. While most cases show an increase in power loss with EIR, some cases (such as Case_1) exhibit reduced power loss, highlighting non-linear interactions between DG scheduling, reliability, and power flow optimization. Unlike single-objective optimization, where either cost or loss is minimized independently, the multi-objective approach balances the two, resulting in compromise solutions that reflect real-world trade-offs. ILOA consistently achieves lower operating costs compared to LOA, JAYA, and GA, making it an optimal choice for economic DG scheduling. In many scenarios, ILOA maintains lower real power losses, ensuring higher energy efficiency. Unlike traditional optimization methods that prioritize global solutions, ILOA achieves a more balanced approach between cost efficiency and power loss reduction.

Conclusion and directions for future research

This paper proposed the Improved Lyrebird Optimization Algorithm (ILOA) as a robust and efficient solution for the optimal sectionalizing and scheduling of multi-microgrid systems.

• The algorithm effectively minimized generation costs and active power losses while addressing reliability constraints, such as the Energy Index of Reliability (EIR), and ensured stable system performance with renewable energy integration.

• By integrating advanced mechanisms like Levy Flight for enhanced local search and a chaotic sine map for improved global exploration, ILOA achieved faster convergence and superior optimization results compared to conventional algorithms like the Genetic Algorithm (GA), Jaya Algorithm (JAYA), and the original Lyrebird Optimization Algorithm (LOA).

• Simulation results on a modified 33-bus distribution system, segmented into three independent microgrids, demonstrated the practical applicability of ILOA in both single-objective and multi-objective optimization scenarios.

• In single-objective cases, the algorithm achieved notable improvements in generation cost and active power loss reduction. In multi-objective optimization, it balanced these objectives more effectively than competing methods, further validating its robustness and effectiveness.

• For the IEEE-33 bus system under multi-objective optimization without considering EIR, the proposed ILOA algorithm significantly enhances system performance by reducing generation cost by approximately 0.1062%, 1.0822%, and 1.5318% compared to LOA, JAYA, and GA, respectively. Additionally, ILOA lowers active power loss by around 0.5968%, 1.942%, and 1.3891% relative to LOA, JAYA, and GA, respectively, under the operational scenario of Case-7. These results highlight the effectiveness of ILOA in optimizing both economic and technical parameters in power system operation.

• For the IEEE-33 bus system with considering EIR, the proposed ILOA algorithm achieves generation cost savings of approximately 0.0057% and 0.0214% compared to LOA and JAYA, respectively. Additionally, ILOA demonstrates a notable reduction in active power loss by 4.07% and 8.47% compared to LOA and JAYA, respectively, under the operational scenario of Case-7. These findings further validate the effectiveness of ILOA in optimizing economic and technical performance in power systems.

• While the results highlighted the significant potential of ILOA, certain limitations remained. The scalability of the algorithm to larger, more complex systems and its adaptability to dynamic and uncertain grid conditions warranted further exploration.

• Additionally, the incorporation of constraint-handling mechanisms, such as reliability indices and advanced forecasting techniques for renewable energy sources, could have enhanced its robustness.

Future work could focus on leveraging real-time grid data and integrating machine learning techniques to improve decision-making under uncertainty. Exploring hybrid frameworks combining ILOA with methods like game theory or reinforcement learning could extend its application to more complex objectives, including energy storage management, demand response, and fault detection in multi-microgrid systems. The ILOA showcased its capability as an efficient and reliable method for optimizing distributed generation scheduling and sectionalizing multi-microgrid systems, highlighting its promise as a key enabler for future advancements in smart grid applications.