Improved Pied Kingfisher Optimization Algorithm for optimal scheduling of microgrids with hybrid energy storage

Abstract

With the increasing penetration of renewable energy sources, microgrids have become an effective platform for integrating distributed generation and energy storage. However, renewable intermittency and the integration of hybrid energy storage systems lead to a highly nonlinear and nonconvex scheduling problem, for which conventional deterministic solvers can be computationally demanding and may struggle with nonconvexity and coupled constraints. To address this challenge, this paper proposes an Improved Pied Kingfisher Optimization algorithm (IPKO) for microgrid optimal scheduling. The proposed IPKO integrates four complementary enhancement strategies, including Chebyshev-chaotic initialization, multi-source guidance with adaptive perturbation, differential–spiral development, and periodic dynamic adjustment, to improve population diversity, search adaptability, and convergence stability. A grid-connected microgrid scheduling model with a hybrid energy storage system comprising battery storage and hydrogen storage is further developed, with demand response modeled as time-of-use price-driven load shifting. Simulation results demonstrate that the IPKO-based scheduling framework consistently reduces both economic and environmental costs. Compared with representative metaheuristic algorithms over multiple independent runs, IPKO achieves about 4–10% lower mean daily operating cost with significantly reduced solution dispersion, confirming its superior solution quality and robustness for complex microgrid scheduling problems.

Introduction

In recent years, microgrids (MGs) have attracted growing attention as an effective paradigm for integrating distributed energy resources, such as solar and wind power, together with energy storage systems, in the context of the global energy transition^1–3. By coordinating diverse energy sources at the local level, microgrids provide a promising approach to increasing renewable energy utilization and reducing reliance on conventional fossil-based generation. However, the inherent intermittency and uncertainty of renewable energy sources pose significant challenges to the secure, economical, and stable operation of microgrid systems^4,5. In particular, mismatches between renewable generation and load demand may lead to inefficient resource utilization and increased operating costs if not properly managed^6–8. To address these challenges, microgrid scheduling and optimization play a critical role in coordinating power generation, storage, and consumption across different time intervals, thereby enabling efficient resource allocation and ensuring system reliability^9,10.

With the increasing penetration of renewable energy sources (RES), particularly wind and solar power, energy storage technologies have become essential for mitigating the mismatch between generation and demand^11,12. Conventional battery energy storage systems (BESS) are widely deployed due to their fast response and high efficiency. Li et al.¹³ presented a battery management system design for grid energy storage with real-time validation, while Mancarella et al.¹⁴ provided an editorial overview of BESS for net-zero power systems and markets, underscoring its growing role in system flexibility. However, limited energy density and self-discharge characteristics constrain their suitability for long-term energy storage, especially in microgrids with high renewable penetration^15–17.

To overcome these limitations, hydrogen energy storage (HES) has emerged as a promising complementary solution^18–20. By converting surplus electrical energy into hydrogen through electrolysis and enabling long-term storage with high energy density, hydrogen-based systems provide additional operational flexibility for renewable-rich microgrids^21–23. In recent years, increasing attention has been devoted to hydrogen-integrated microgrids, where HES is often combined with BESS to exploit their complementary characteristics across different time scales. Qi et al.²⁴ investigated hybrid hydrogen–battery storage configurations and highlighted their time-scale complementarities, while Abdelghany et al.²⁵ demonstrated that coordinated scheduling of battery and hydrogen storage improves economic performance and power balance in grid-connected wind–solar microgrids. Abomazid et al.²⁶ further showed that integrating battery storage with photovoltaic-based hydrogen systems helps reduce hydrogen production fluctuations and operational costs.

Despite these advances, most existing studies primarily focus on component-level coordination or local energy management strategies^27,28, whereas system-level optimal scheduling of hydrogen-integrated microgrids under multiple practical operational constraints has not been sufficiently investigated.

In microgrid scheduling studies, relying solely on supply-side regulation is often insufficient to cope with renewable intermittency and demand variability. As a result, demand response (DR) is commonly introduced as an auxiliary mechanism to enhance operational flexibility^29–32. In particular, price-based DR driven by time-of-use (TOU) tariffs enables partial load shifting over time, thereby alleviating peak demand and reducing reliance on grid dependence^33,34. For instance, Nikzad and Samimi³⁵ incorporated price-based DR into a stochastic bi-level scheduling framework, while Hwang Goh et al.³⁶ and Bolurian et al.³⁷ demonstrated that moderate load shifting contributes to improved economic performance in grid-connected microgrids.

Microgrid scheduling coordinates heterogeneous resources under practical constraints and is inherently nonlinear and nonconvex due to coupled device dynamics such as ramping behavior and storage state evolution, time-coupled constraints, and mixed decision variables^38–40. These intrinsic characteristics significantly complicate the optimization landscape and limit the effectiveness of conventional gradient-based optimization methods^41,42. Consequently, increasing attention has been directed toward metaheuristic optimization approaches, which are better suited for exploring complex and highly constrained scheduling spaces^43,44.

Beyond classical baselines such as genetic algorithms, particle swarm optimization, and differential evolution that have been widely used for microgrid optimal scheduling^45,46, recent studies have increasingly explored newer metaheuristics to improve robustness under nonlinear and time-coupled constraints. Abdelsalam et al.⁴⁷ developed a Harris hawks optimization (HHO)–based scheduling strategy and demonstrated its effectiveness in handling nonlinear operating constraints in microgrids. Yadav et al.⁴⁸ investigated grey wolf optimization for isolated microgrid scheduling and reported improved techno-economic performance under practical operational limits. Datta et al.⁴⁹ compared multiple metaheuristic approaches for managing multi-microgrid systems under adverse operating conditions. In addition, Teo et al.⁵⁰ applied NSGA-II to the optimal scheduling of a grid-connected microgrid, demonstrating the applicability of metaheuristic optimization methods to microgrid energy management problems. Fathy et al.⁵¹ introduced the honey badger algorithm to microgrid scheduling and verified its capability to balance exploration and exploitation for complex energy management tasks. Karthik et al.⁵² further formulated a multi-objective microgrid scheduling problem and applied a sine–cosine-based optimizer to obtain a set of trade-off solutions across conflicting objectives. Barua et al.⁵³ proposed a Lévy-flight-enhanced arithmetic optimization approach, highlighting continued interest in improving convergence behavior and solution quality for microgrid operation.

Despite the progress achieved by metaheuristic-based microgrid scheduling studies, performance is still frequently constrained by search efficiency, convergence stability, and robustness, especially in microgrids with hybrid energy storage and time-coupled operating constraints. To address these issues, this paper proposes an improved Pied Kingfisher Optimization algorithm (IPKO) and applies it to the optimal scheduling of a microgrid equipped with a hybrid energy storage system under TOU price-based demand response.

Despite the progress achieved by metaheuristic-based microgrid scheduling studies, existing algorithms may still face challenges in solving scheduling problems involving hybrid energy storage systems and time-coupled operational constraints. In such problems, the coordinated operation of BESS and HES, renewable generation, controllable units, grid exchange, and demand response leads to a highly nonlinear and nonconvex search space. As a result, conventional metaheuristic algorithms may suffer from insufficient population diversity, premature convergence, or unstable convergence behavior. Therefore, improving the exploration–exploitation balance and convergence robustness remains necessary for hybrid-energy-storage microgrid scheduling.

The main contributions of this paper are summarized as follows:

1. An improved Pied Kingfisher Optimization algorithm (IPKO) is developed by integrating four enhancement strategies, including Chebyshev-chaotic initialization, dual-random individual guidance with Lévy-based mixed perturbation, differential–spiral fusion development mechanism, and periodic dynamic adjustment with hybrid perturbation. These strategies jointly enhance population diversity, search adaptability, and convergence robustness for highly nonlinear and nonconvex scheduling problems.

2. A comprehensive microgrid scheduling model is established by incorporating a hybrid energy storage system comprising HES and BESS, with price-based demand response modeled using time-of-use pricing. The proposed framework enables coordinated operation of multiple energy resources and provides sufficient flexibility for optimal scheduling under renewable generation variability.

3. Extensive simulation demonstrate that the proposed IPKO-based scheduling strategy significantly reduces both economic and environmental costs. Across multiple independent runs, IPKO achieves about 4–10% lower mean daily operating cost than representative metaheuristic algorithms, while exhibiting substantially reduced solution variability, thereby confirming its superior solution quality and robustness for the studied microgrid scheduling problem.

The remainder of this paper is organized as follows: “Microgrid system description and modeling” section introduces the studied grid-connected microgrid with hybrid energy storage, and establishes the mathematical models. The “Optimization formulation for microgrid scheduling” section formulates the microgrid scheduling problem with economic and environmental objectives under operational constraints, incorporating price-based demand response. The “Improved Pied Kingfisher Optimization Algorithm” section details the proposed IPKO and its four improvement strategies and workflow. The “Performance evaluation of the proposed IPKO” section evaluates IPKO through ablation studies, search behavior analysis, and benchmark comparisons on the CEC2005, CEC2017, and CEC2022 test suites. The “Case study analysis” section investigates the grid-connected microgrid scheduling problem and evaluates the proposed IPKO-based solution through scheduling results and comparative cost analysis. The “Conclusion” section summarizes the main contributions and outlines future work.

Microgrid system description and modeling

Microgrid system configuration

The configuration of the studied microgrid is shown in Fig. 1. The system is interconnected with the main grid through an AC–DC interface converter, enabling bidirectional power exchange. The microgrid comprises renewable sources, including photovoltaic (PV) generation and wind turbines (WT), as well as dispatchable units such as a diesel engine (DE) and a micro gas turbine (MT). In addition, a hydrogen energy system (HES)—including an electrolyzer (EL), a hydrogen storage tank, and a fuel cell (FC)—together with a battery energy storage system (BESS), is integrated. All components are connected to the common bus through appropriate power electronic converters to support coordinated microgrid operation.

Fig. 1.

Configuration of the microgrid system.

Mathematical models of microgrid components

Photovoltaic generation model

The output power of the photovoltaic unit is influenced by solar irradiance and cell temperature. Let PPV t denote the PV output power at time t. The PV generation model is given by:

1

where PPV STC is the rated PV power under standard test conditions (STC), κ_pv is the derating factor accounting for losses, G_t and G_STC denote the actual and standard solar irradiance, respectively, and η_T is the temperature coefficient of the PV module. T_t and T_STC represent the PV cell temperature at time t and under STC.

Wind turbine generation model

The output power of a wind turbine is primarily determined by the wind speed and can be approximated using a piecewise wind speed–power characteristic curve. Let PWT t denote the wind turbine output power at time t, and let v_t represent the corresponding wind speed. The wind power model is expressed as:

2

where v_ci, v_r, and v_co denote the cut-in, rated, and cut-out wind speeds, respectively, and PWT r is the rated power of the wind turbine. The coefficients a_w, b_w, c_w, and d_w are obtained by fitting the turbine power curve within the partial-load region.

Dispatchable generation cost models

In the studied microgrid, controllable generation units, including the DE and the MT, are employed to rapidly respond to load fluctuations and compensate for the intermittency of renewable energy sources. Their operating costs are mainly determined by fuel consumption characteristics and output power levels.

• Diesel engine fuel cost model

The fuel cost of the diesel engine is commonly represented by a quadratic function of its power output. Let PDE tdenote the electrical power generated by the diesel engine at time t. The corresponding fuel cost is expressed as:

3

where µ₁, µ₂, and µ₃ are cost coefficients reflecting the fuel consumption characteristics of the diesel engine.

• Micro gas turbine fuel cost model.

For the micro gas turbine, the fuel cost is related to both the output power and the operating efficiency. Let PMT tdenote the electrical output power of the micro gas turbine at time t. The corresponding fuel cost is given by:

4

where χ is the unit price of natural gas, H_g denotes the lower heating value of natural gas, and ξ_t represents the generation efficiency of the micro gas turbine at time t.

The efficiency of the MT is modeled as a cubic polynomial of the normalized output power:

5

where PMT ref is the rated power of the micro gas turbine, and a₁–a₄ are empirical coefficients obtained from efficiency curve fitting.

Battery energy storage system model

To enhance the operational flexibility of the microgrid, a BESS is incorporated to balance power fluctuations and provide short-term energy buffering. For modeling simplicity, the terminal voltage of the BESS is assumed to remain constant during charging and discharging processes, and the energy loss inside the battery is characterized by a self-discharge factor and charge–discharge efficiency.

Let EB t and EB t + 1 denote the stored energy of the BESS at time intervals t and t + 1, respectively. The charging and discharging processes of the BESS are modeled by the following discrete-time equations:

6

where PB, c t and PB, d t represent the charging and discharging power of the BESS, respectively. σ denotes the self-discharge rate, η_c and η_d represent the charging and discharging efficiencies, and Δt denotes the scheduling time step.

The state of charge (SOC) of the BESS is updated according to

7

where EB max denotes the rated energy capacity of the BESS. Accordingly, SOC_t+1 represents the state of charge at time interval t + 1, which reflects the dynamic evolution of the energy storage state during charging and discharging operations.

Hydrogen energy system model

To realize long-term energy storage and enhance multi-energy coupling capability, a HES consisting of an electrolyzer, a hydrogen storage tank, and a fuel cell is incorporated into the microgrid. The HES enables bidirectional energy conversion between electricity and hydrogen and is modeled using an energy balance formulation.

Let H_t denote the hydrogen energy stored in the tank at time interval t. Considering different operating modes, the hydrogen state evolution is expressed as follows:

8

where PEL t and PFC t denote the electrical power consumed by the electrolyzer and the electrical power generated by the fuel cell, respectively. λ_h represents the hydrogen leakage rate per time interval, η_el is the electricity-to-hydrogen conversion efficiency, and η_fc denotes the hydrogen-to-electricity conversion efficiency.

Optimization formulation for microgrid scheduling

Objective function

Economic operating cost

In the microgrid scheduling problem, the total operating cost consists of multiple cost components arising from the operation of controllable generation units, power exchange with the utility grid, unit start-up and shutdown actions, and hydrogen energy system operation. These cost components jointly reflect the economic performance of the microgrid under different operating conditions.

Accordingly, the total operating cost over the scheduling horizon can be expressed as:

9

where Cgen t, CRES t, Cgrid t, and CH₂ tdenote the controllable generation operating cost, renewable generation operating cost including curtailment penalty, electricity trading cost, and hydrogen-related cost at time interval t, respectively. Each cost component is detailed as follows.

• Generation operating cost

The controllable generation units considered in this study include the DE, MT, and FC. Let Pi t denote the active power output of controllable unit i∈H at time interval t, where H= {DE, MT, FC}. The operating cost of controllable generation units mainly consists of fuel consumption cost and operation and maintenance cost. The fuel costs of the diesel engine and micro gas turbine are denoted by FDE t and FMT t, respectively, which have been formulated in the previous subsection. Since the fuel cell generates electricity by consuming stored hydrogen, its operating cost is primarily associated with operation and maintenance expenses.

Accordingly, the total operating cost of controllable generation units at time interval t can be compactly expressed as:

10

where κ_i denotes the operation and maintenance cost coefficient of unit i. For the fuel cell unit, the fuel cost term satisfies FFC t = 0. For the fuel cell, the hydrogen consumption cost is accounted for in CH₂ t; therefore, FFC t is set to zero in Cgen t.

• Renewable generation operating cost and curtailment penalty

In addition to controllable generation units, the operating cost model further accounts for the operating and curtailment-related costs of renewable energy sources (RES), including PV and WT units. Although RES do not incur fuel consumption, their operation involves maintenance-related costs that are commonly modeled as being proportional to their actual power outputs.

Let Ppv t and Pwt t denote the actual output power of the PV and WT units at time interval t, respectively. The renewable generation operating cost is expressed as:

11

where c_pv and c_wt denote the operation and maintenance cost coefficients of the PV and WT units, respectively.

Furthermore, to promote the local utilization of RES and discourage excessive curtailment, penalty costs associated with curtailed PV and WT are introduced. Let Ppv* t and Pwt* t denote the available PV and WT power before curtailment at time interval t. The renewable energy curtailment penalty cost is formulated as:

12

where ς_pv and ς_wt represent the penalty coefficients for PV and wind power curtailment, respectively.

Accordingly, the total operating cost associated with renewable energy sources at time interval t is given by:

13

• Electricity trading cost

In addition to the operating cost of local controllable generation units, the microgrid may exchange power with the utility grid to maintain power balance. The electricity trading cost is determined by the amount of electricity purchased from or sold to the main grid. To explicitly characterize the asymmetric pricing mechanism, the grid power exchange is decomposed into electricity purchase and selling components. Let P_{gb, t} and P_{gs, t} denote the purchased and sold electricity at time interval t, respectively. The electricity trading cost at time interval t is then expressed as:

14

where Jbuy t and Jsell t represent the electricity purchase and selling prices at time interval t, respectively. In practical grid trading, Jsell t is usually lower than Jbuy t, reflecting the asymmetric pricing mechanism between electricity purchase and sale.

• Hydrogen energy cost

To reflect the operational characteristics of hydrogen energy system, an electrolyzer start-up cost is further considered in the scheduling model. This cost is incurred only when the electrolyzer is switched from off to on between two consecutive time intervals. The electrolyzer start-up cost is given by:

15

where [x]⁺=max(x,0), Cst EL and U_{EL, t} represent the start-up cost coefficient and operating status of the electrolyzer, respectively.

In addition, the hydrogen energy system introduces additional operating costs associated with hydrogen production and storage. Specifically, hydrogen is produced via the electrolyzer by converting electrical energy into chemical energy, which incurs a non-electricity production cost, while the stored hydrogen in the tank leads to a storage-related cost proportional to the hydrogen inventory level. By jointly considering the hydrogen production and storage processes, the total hydrogen-related cost at time interval t is modeled as:

16

Here, cprod H₂ denotes the unit non-electricity cost of hydrogen production, cstor H₂ represents the hydrogen storage cost coefficient, P_el,t is the electrolyzer input power, η_el is the electrolyzer efficiency, and H_t denotes the stored hydrogen energy at time interval t.

Environmental and pollution mitigation cost

In addition to economic operating costs, environmental impact is also an important consideration in microgrid scheduling. The operation of power generation units and electricity exchange with the utility grid inevitably leads to the emission of various atmospheric pollutants, which incur corresponding environmental and pollution mitigation costs. The environmental and pollution mitigation cost over the scheduling horizon is formulated as:

17

where C^env denotes the environmental and pollution mitigation cost, χ_j(t) denotes the aggregated emission level of pollutant j at time interval t, representing the overall intensity of pollutant emissions during that interval. The aggregated emission level χ_j (t) is calculated as:

18

Here, β CO₂, β_CO, β SO₂, and β NOx are the treatment cost coefficients corresponding to CO₂, CO, SO₂, and NO_x emissions, respectively; P_{i, t} represents the power supplied by emission-related source i at time interval t; ρ_{i, j} is the emission coefficient of pollutant type j from source i, and K is the number of emission-related energy sources.

Overall objective function

The overall objective of the microgrid is defined as the sum of the economic operating cost and the environmental and pollution mitigation cost. Accordingly, the objective of the scheduling problem is formulated as:

19

Constraints

Micro-source output and ramping constraints

To ensure the secure and feasible operation of the microgrid, the active power outputs of all micro-sources are subject to technical operating constraints. These constraints include output capacity limits and ramping rate limits, which reflect both physical limitations and operational flexibility.

• Output power constraints

For all generation and conversion units, including PV, WT, DE, MT, FC, and EL, the active power output at time interval t is constrained by its minimum and maximum allowable limits:

20

where P_{i, t} denotes the active power of unit i at time interval t, and Pmin i and Pmax i represent its minimum and maximum operating limits, respectively.

• Ramping power constraints

To capture the dynamic operational characteristics of controllable generation and conversion units, ramping rate constraints are imposed on the diesel engine, micro gas turbine, fuel cell, and the electrolyzer. These constraints limit the rate of change in output power between two consecutive time intervals:

21

where RR k and RF k denote the ramp-up and ramp-down limits of unit k, respectively, and Δt is the scheduling time interval.

Microgrid–utility grid power exchange constraint

To maintain power balance and ensure secure operation, the microgrid is allowed to exchange power with the utility grid through the point of common coupling. However, the power exchanged with the main grid is constrained by the capacity limits of the interconnection equipment and distribution lines.

Let Pgrid t denote the power exchanged between the microgrid and the utility grid at time interval t, where a positive value indicates power import from the grid and a negative value represents power export. The grid power exchange is restricted within predefined bounds as follows:

22

Here, Pmin g and Pmax g represent the minimum and maximum allowable grid exchange power, respectively, which are determined by the rated capacity of the grid connection and operational requirements.

Power balance constraint

To ensure secure and stable operation of the microgrid, the total electricity supply must satisfy the system demand at each scheduling interval. The power balance constraint is formulated by explicitly considering the contributions of all generation units, the bidirectional BESS, the electrolyzer, and the power exchanged with the utility grid.

Specifically, the power balance at time interval t is expressed as:

23

where PL t denotes the total electrical load demand, PEL t represents the power consumption of the electrolyzer, and PB, c t and PB, d t denote the charging and discharging power of the BESS, respectively. The set F includes all electricity generation units within the microgrid, namely PV, WT, DE, MT, and FC. Pgrid t denotes the power exchanged with the utility grid, where a positive value indicates power purchase and a negative value indicates power export.

Hybrid energy storage constraints

To ensure reliable operation of the microgrid, the constraints of energy storage systems are considered. Both the BESS and the HES are included, while different state and power variables are adopted to reflect their distinct physical characteristics.

For the BESS, the stored electrical energy is constrained by:

24

and its charging and discharging operations satisfy:

25

Similarly, the hydrogen energy stored in the tank is limited by:

26

with mutually exclusive hydrogen production and consumption processes described as:

27

Here, PEL t and PFC t denote the hydrogen-related conversion power of the electrolyzer and the fuel cell, respectively. To enable cyclic operation of hybrid storage systems over the scheduling horizon, the terminal storage states at the end of the last scheduling interval are constrained to return to their predefined reserve levels:

28

where EB 0 and H₀ are preset reserve levels that serve as the initial states for the next day.

Demand response model

To enhance the operational flexibility of the microgrid and mitigate peak demand, demand response driven by time-of-use (TOU) electricity pricing is incorporated into the scheduling framework. This mechanism reshapes the original load profile by shifting a portion of flexible load among different time periods in response to TOU electricity prices.

Let Dbase tdenote the baseline load at hour t. The actual load after applying demand response is expressed as:

29

where ΔD_t represents the net load adjustment induced by price-based demand shifting.

The proportion of load that can be shifted is assumed to depend on the electricity price difference between time periods and is modeled using a logistic function:

30

where Δτ denotes the TOU electricity price difference and α, β, δ, and γ are empirical parameters characterizing the price sensitivity of load shifting. In this logistic formulation, α controls the adjustable range of the transferable load ratio, β determines the price-difference threshold for significant load shifting, γ reflects the response sensitivity to the TOU price difference, and δ denotes the basic response offset. This formulation captures the saturation effect of price-driven load shifting, where the transferable load ratio increases with the price difference but remains bounded.

Based on the transferable load ratio, the load adjustment at each hour is determined by the corresponding price segment and is formulated as:

31

where‾D_g,‾D_f and‾D_P denote the average baseline loads of the valley, flat, and peak price segments, respectively; Τ_g, T_f, and T_p are the corresponding sets of hours for these segments, and ω(.) ∈ [0,1] are weighting coefficients reflecting the practical acceptance level of load shifting in each segment.

Through the above formulation, part of the load is shifted from high-price periods to low-price periods while maintaining the overall energy consumption approximately unchanged. The resulting demand response–adjusted load profile is subsequently incorporated into the microgrid power balance constraints and optimization process, thereby introducing additional demand-side flexibility for the proposed IPKO-based optimal scheduling strategy.

Improved Pied Kingfisher Optimization Algorithm

Mechanism and limitations of the original PKO algorithm

The Pied Kingfisher Optimization (PKO) algorithm is a nature-inspired metaheuristic motivated by the hovering, perching, underwater-diving, and cooperative hunting behaviors of pied kingfishers. Its search process follows a biologically inspired sequence: individuals first perform wide-range global exploration through perching–hovering movements, then conduct targeted local exploitation by diving into promising regions, and finally engage in a symbiotic cultivation process that facilitates information exchange and enables escape from local optima. Through this behavioral flow, PKO establishes a functional balance among exploration, exploitation, and local-escape capability.

Despite these advantages, the original PKO algorithm exhibits several structural limitations when applied to complex optimization problems. Random population initialization may result in insufficient diversity, weakening early-stage exploration. Moreover, the search dynamics rely on a limited set of individual interactions, which can constrain global information exchange and lead to the population toward inefficient search trajectories. In addition, local exploitation can become overly aggressive as the search space narrows, increasing the risk of premature convergence. Furthermore, the evolutionary process follows a relatively fixed rhythm, which limits its ability to adapt to different optimization stages or dynamically changing landscapes. As a result, PKO may exhibit unstable search behavior and reduced robustness, particularly in complex and high-dimensional optimization problems. To address these limitations, the improvement strategies incorporated in the proposed IPKO are presented in Sect.  4.2.

The proposed IPKO algorithm

Chebyshev chaotic mapping-based initialization strategy

In most population-based optimization algorithms, the quality of the initial population strongly affects global search capability and subsequent convergence behavior. Conventional random initialization methods typically sample individuals uniformly at random, which may result in uneven coverage of the solution space and insufficient population diversity, thereby degrading the efficiency of subsequent exploration and exploitation. To alleviate this issue, the proposed IPKO incorporates a Chebyshev chaotic mapping mechanism to enhance the ergodicity of the initial population distribution.

The Chebyshev chaotic sequence is generated as:

32

where a is the control parameter. When a ≥ 2, the mapping exhibits chaotic behavior; in this study, a = 4 is adopted. For each dimension j, the initial value x_1,j is independently drawn from a uniform distribution U (0,1). Here, n denotes the sequence index of the Chebyshev map. For each dimension j, the chaotic values {x_n,j}N n = 1 are generated by iterating Eq. (32) with n = 1, 2, …, N-1, where N is the population size.

The chaotic sequence is then mapped onto the feasible search space [lb_j, ub_j] of the j-th dimension as:

33

where X_{i, j} denotes the initial position of the i-th individual in the j-th dimension, and lb_j, ub_j are the corresponding lower and upper bounds. The absolute operation maps the chaotic values from [− 1,1] to [0,1], after which they are linearly scaled to [lb_j, ub_j]. For each dimension j, the first N chaotic values {x_1,j, …, x_N,j} are used to initialize the j-th components of the population. Specifically, the value allocated to the i-th individual is denoted by x_i,j (i.e., x_i,j= x_n,j with n = i), and it is mapped to the feasible interval [lb_j, ub_j] via Eq. (33) to obtain X_i,j.

By leveraging the ergodicity of the Chebyshev map, the initial population becomes more broadly distributed across the solution space, effectively avoiding local clustering caused by purely random initialization. This strategy increases population diversity and provides richer starting points for the subsequent exploration and exploitation phases, thereby accelerating convergence and improving overall search accuracy. As illustrated in Fig. 2, the Chebyshev chaotic sequence and its histogram demonstrate broad coverage of the numerical space and a high-entropy distribution.

Fig. 2.

Chebyshev chaotic sequence characteristics. (a) scatter distribution. (b) histogram.

Dual-random individual guidance and Lévy mixed perturbation strategy

In the original PKO, the search update relies on the interaction between an individual and only one randomly selected companion. Although this single-source difference information provides basic exploratory capability, it leads to a directional bias and restricts the ability to cover a wider search region. Specifically, two limitations arise. First, relying solely on the difference with one random individual constrains the search trajectory to the local distribution of surrounding individuals, making the algorithm prone to blind spots in high-dimensional spaces where the global landscape is more complex. Second, the lack of effective integration of information from the current global best solution causes the update direction to lose global guidance, resulting in blind or inefficient exploration; as iterations proceed, population diversity decays and individuals easily fall into local optima.

Furthermore, the disturbance-generation mechanism in the original PKO does not account for the dynamic balance between exploration and exploitation. Without an adaptive perturbation mechanism that adjusts its behavior over iterations, the algorithm struggles to shift appropriately between long-range global exploration and fine-grained local refinement, limiting its performance in complex search environments.

To address these issues, the proposed IPKO incorporates a dual-random individual guidance mechanism together with a Lévy mixed perturbation strategy. By integrating information from two independently and randomly selected individuals and combining it with Lévy-based long-tailed perturbations, the algorithm enhances multi-source guidance, increases search diversity, and improves the adaptability and robustness of the exploration process.

The updated position is expressed as:

34

where Xnew idenotes the updated position of the i-th individual; X^* denotes the current global best position. The mapped values X_{i, j} form the initial population matrix X, where the i-th individual is denoted by the position vector X_i∈ℝ^D in the subsequent update equations. X_rk denotes an individual randomly selected from the current population. The indices rk are independently re-sampled at each iteration and for each update step, and thus do not correspond to fixed or shared individuals across different update equations. The binary switching variable s ∈ {0,1}is activated with a small probability (about 0.05) to introduce occasional extra disturbance, and ξ ∼ N (0,1) is implemented as a scalar Gaussian perturbation shared across dimensions. ⊙ is defined as the element-wise multiplication operator between vectors.

The Lévy step vector in D dimensions is generated by the Mantegna method as:

35

where β = 1.5 denotes the characteristic parameter of Lévy flights, σ is the scale parameter computed accordingly. The absolute value, power, and division are applied element-wise.

By combining multi-source guidance and Lévy-based perturbations, this strategy improves the adaptability of the search process, expands the feasible search region, and enhances convergence stability. It effectively avoids premature clustering and strengthens the algorithm’s ability to balance global exploration and local refinement.

Differential–spiral fusion development strategy

In the original PKO, the development-stage update mechanism is primarily guided by a single information source and constructs its update direction solely based on the current best solution. This single-source guidance restricts the diversity of directional information and confines the search trajectory to a narrow region around the best position. Moreover, the linear difference terms used in traditional strategies may not sufficiently match the nonlinear distribution of promising regions during the development phase, leading to incomplete coverage of potential optimal areas.

To address these limitations, the proposed IPKO introduces a differential–spiral fusion development strategy that enhances multi-source information utilization and improves adaptability to nonlinear search landscapes. By adding a pairwise difference term between two randomly selected individuals (X_r3 − X_r4) alongside the core term (X^* − X_i), the strategy embeds richer multi-source spatial information into the update direction and constructs a multi-source local information interaction component, compensating for the deficiency of relying on a single information source in the original development mechanism and enhancing the directional diversity of local search. This fusion expands the variety of potential search directions, prevents premature convergence near the best solution, and strengthens the robustness of the development process.

To further enhance nonlinear search capability, a spiral perturbation mechanism is integrated into the update equation. The spiral term generates a dimension-wise helical perturbation around the best position using a randomly angular vector and radius-scaling vector k. This mechanism provides structured and fine-grained local refinement while maintaining stochastic adaptability, enabling both precision and flexibility during the development phase.

The update equation of this strategy is expressed as:

36

where the spiral perturbation is defined as:

37

where the parameter F denotes a stochastic contraction–expansion factor drawn from F∼U (0,1), which regulates the relative influence of the differential and spiral components. The variable θ denotes a random angular vector whose elements are independently sampled from U (0,2π). The factor k from acts as a dimension-wise radius-scaling term that controls the amplitude of the spiral perturbation, thereby enabling both near-field and relatively larger-range refinement. The term spiral represents the nonlinear helical perturbation used to achieve fine-grained local exploitation.

Overall, the differential–spiral fusion strategy combines multi-source differential information with nonlinear helical perturbations to enhance directional diversity and improve the precision and stability of the developmental search stage. This design strengthens local refinement capability while maintaining adaptability, ultimately improving convergence accuracy and robustness during later iterations.

Periodic dynamic adjustment and hybrid perturbation symbiosis strategy

In the original PKO, the symbiotic stage provides limited and occasionally triggered positional updates, which may lead to search stagnation and inefficiency, especially in later iterations when population diversity decreases and individuals tend to cluster prematurely. Moreover, this fixed update pattern cannot accommodate the dynamic requirement of alternating “exploration–exploitation” intensities at different evolutionary stages. Even when updates are triggered, the strategy still relies on the absolute difference between two random individuals, resulting in a single-source information exchange pattern that is insufficient to stimulate effective cooperative search within the population.

To address these issues, the proposed IPKO introduces a periodic dynamic adjustment mechanism combined with a hybrid perturbation symbiosis strategy. This approach extends the idea of multi-source fusion by incorporating periodic modulation factors to dynamically regulate perturbation intensity and step size throughout the evolutionary process. By exploiting the sinusoidal periodicity, the search step size oscillates rhythmically between strong and weak phases, thereby broadening exploration in early iterations and gradually reducing perturbation magnitude in later iterations to promote convergence.

Furthermore, a contraction–expansion differential term together with a Lévy-flight-based perturbation is adopted. This design avoids over-reliance on a single absolute difference and enhances robustness against search stagnation. By additionally constructing a pairwise-difference term from two randomly selected individuals, the update rule injects cooperative information from multiple sources, preventing isolated search trajectories and improving the overall adaptability of the symbiotic stage.

Specifically, when the proposed symbiosis update is activated, two randomly controlled update forms are defined as:

38

where the variable ε denotes a stochastic expansion–contraction factor drawn from ε∼ U (0,1), which controls the magnitude of the differential perturbation. The parameter G denotes a fixed contraction factor, regulating the strength of Lévy-flight-based perturbation. The variable t denotes the current iteration index, and T denotes the maximum number of iterations. The term sin(2πt/T) denotes the periodic modulation factor that dynamically adjusts the search intensity within the range [− 1,1], while the coefficient c denotes the amplitude-scaling factor controlling the influence of the periodic modulation on step length.

Overall, this strategy integrates periodic dynamic modulation with hybrid perturbation and multi-source differential symbiosis, enhancing intra-population cooperative search and improving the stability and convergence performance of IPKO in complex optimization scenarios.

Overall procedure of the proposed IPKO algorithm

The overall procedure of the proposed IPKO algorithm is illustrated in Fig. 3. As shown in Fig. 3, the algorithm begins with population initialization based on Chebyshev chaotic mapping, followed by fitness evaluation and determination of the initial global best solution. At each iteration, the decay factor and predatory efficiency are updated to regulate the search dynamics. The population then proceeds to the exploration–exploitation module, after which candidate solutions are evaluated with boundary handling and fitness computation, and both individual solutions and the global best are updated using a greedy selection rule. Subsequently, a commensal symbiosis stage is executed, with a PE-controlled periodic-modulated hybrid-perturbation update is applied to strengthen population interaction and alleviate stagnation. The iteration continues until the stopping criterion is met.

Fig. 3.

Overall flowchart of the IPKO algorithm.

In the implementation, a projection-based boundary handling rule is adopted after each position update. For each decision variable, if the updated value exceeds its predefined lower or upper bound, it is directly projected to the nearest feasible boundary, it is corrected as:

39

This ensures that all candidate solutions remain within the allowable search space before fitness evaluation.

Figure 4 presents the exploration–exploitation module, where individuals are assigned to exploration or exploitation according to probabilistic rules. Global exploration is enhanced using dual random-individual guidance combined with Lévy-based perturbations, whereas local exploitation is refined through a differential–spiral fusion strategy. Figure 5 shows the individual evaluation and global-update subroutine, which consistently performs boundary handling, fitness comparison, individual updating, and global best maintenance throughout the optimization process. To improve the reproducibility of the proposed algorithm, the main parameter settings used in IPKO are summarized in Table 1.

Fig. 4.

Flowchart of the exploration–exploitation module.

Fig. 5.

Flowchart of the individual evaluation and global update subroutine.

Table 1.

Main parameter settings of the proposed IPKO algorithm.

Parameters	Symbol	Values
Population size	N	30
Maximum iterations	T	1000
Chebyshev chaotic control parameter	a	4
Lévy flight characteristic parameter	β	1.5
Contraction factor for Lévy-based perturbation	G	0.5
Amplitude-scaling coefficient	c	0.2

Performance evaluation of the proposed IPKO

Ablation study on the CEC2005 classical benchmark

To investigate the contribution of each proposed improvement strategy to the overall performance of IPKO, an ablation study was conducted on the CEC2005 benchmark suite. The CEC2005 benchmark functions and their settings, including the dimension, search range, and global optimum, are summarized in Table 2.

Table 2.

Summary of the CEC2005 classical benchmark functions and the corresponding parameter settings.

Function	Function name	Dimension	Search range	Global optimum
F1	Sphere	30	[-100, 100]	0
F2	Schwefel 2.22	30	[-10, 10]	0
F3	Schwefel 1.2	30	[-100, 100]	0
F4	Schwefel 2.21	30	[-100, 100]	0
F5	Rosenbrock	30	[-30, 30]	0
F6	Step	30	[-100, 100]	0
F7	Quartic with Noise	30	[-1.28, 1.28]	0
F8	Schwefel 2.26	30	[-500, 500]	-12569.5
F9	Rastrigin	30	[-5.12, 5.12]	0
F10	Ackley	30	[-32, 32]	0
F11	Griewank	30	[-600, 600]	0
F12	Penalized 1	30	[-50, 50]	0
F13	Penalized 2	30	[-50, 50]	0
F14	Shekel’s Foxholes	2	[-65.536, 65.536]	0.998003837
F15	Kowalik’s Function	4	[-5, 5]	0.000307486
F16	Six-Hump Camel-Back	2	[-5, 5]	-1.031628453
F17	Branin Function	2	x1∈[-5,10], x2∈ [0,15]	0.397887358
F18	Goldstein–Price Function	2	[-2, 2]	3
F19	Hartmann 3-D Function	3	[0, 1]	-3.862782148
F20	Hartmann 6-D Function	6	[0, 1]	-3.322368012
F21	Shekel Function (m = 5)	4	[0, 10]	-10.15319968
F22	Shekel Function (m = 7)	4	[0, 10]	-10.40294057
F23	Shekel Function (m = 10)	4	[0, 10]	-10.53640982

In this study, four variant algorithms were constructed by independently incorporating the improvement strategies introduced in Sects. 4.2.1–4.2.4 into the original PKO framework. These variants are denoted as PKO-CCI, PKO-DRL, PKO-DSF, and PKO-PDH, each corresponding to one individual enhancement strategy. By comparing these variants with the complete IPKO algorithm, this ablation study quantitatively evaluates the individual effectiveness of each strategy and clarifies their respective roles in enhancing convergence accuracy and robustness.

Table 3 reports the mean objective value obtained for each function, and the overall performance is further assessed using the Friedman ranking. The results show that each single-strategy variant achieves performance improvements over the original PKO on several benchmark functions, indicating that the proposed strategies are individually beneficial. However, their performance gains are limited and vary across different function types. Overall, IPKO achieves the best Friedman rank and attains superior or competitive results on most functions. These results demonstrate that the proposed improvement strategies complement each other, and their joint integration within IPKO leads to a more balanced search behavior and enhanced optimization performance, which cannot be fully achieved by any single strategy alone.

Table 3.

Ablation study results of IPKO and its single-strategy variants on the CEC2005 classical benchmark functions. Best values are highlighted in bold.

Function	IPKO	PKO	PKO-CCI	PKO-DRL	PKO-DSF	PKO-PDH
F1	8.45E-14	5.62E-08	1.23E-07	3.55E-07	2.93E-10	1.14E-11
F2	3.11E-08	2.03E-06	1.69E-06	4.47E-06	7.32E-07	2.73E-08
F3	0.094518	136.8434	320.8075	257.8308	1.693099	2.902159
F4	0.130521	0.853508	1.154123	0.699582	0.46049	0.18261
F5	11.37719	26.47601	26.70137	9.239563	27.58962	24.21295
F6	1.78E-11	1.32E-05	1.35E-05	4.63E-05	1.47E-08	4.38E-08
F7	0.011601	0.017278	0.011602	0.01716	0.014383	0.012325
F8	-7374.07	-6847.84	-6919.7	-7194.98	-6759.96	-7373.66
F9	22.66308	14.68482	9.230039	14.24872	27.62828	17.93449
F10	13.11513	12.63095	18.61262	12.61089	6.647146	11.9425
F11	0.003696	0.006728	0.003696	0.007472	0.006235	0.004183
F12	0.003456	0.006929	0.059902	0.049852	0.010367	0.003456
F13	0.000366	0.007046	0.006725	0.00384	0.003997	0.000367
F14	0.998004	0.998004	0.998004	0.998004	0.998004	0.998004
F15	0.000552	0.000929	0.00076	0.000866	0.000918	0.000826
F16	-1.03163	-1.03163	-1.03163	-1.03163	-1.03163	-1.03163
F17	0.397887	0.397887	0.397887	0.397887	0.397887	0.397887
F18	3	3	3	3	3	3
F19	-3.86278	-3.86278	-3.86278	-3.86278	-3.86278	-3.86278
F20	-3.29029	-3.30218	-3.2784	-3.30218	-3.26651	-3.322
F21	-10.1532	-8.96367	-8.62532	-9.81333	-9.81333	-9.98327
F22	-10.4029	-10.2258	-9.87141	-10.4029	-10.4029	-10.2258
F23	-10.5364	-10.5364	-10.5364	-10.3561	-10.5364	-10.3561
Friedman average ranks	2.260870	4.173913	4.152174	3.913043	3.673913	2.826087
Final ranks	1	6	5	4	3	2

Search behavior analysis of IPKO on the CEC2005 benchmark

To examine the search behavior of the proposed IPKO algorithm, four representative benchmark functions from the CEC2005 test suite, namely F1, F6, F13, and F19, are selected to analyze population evolution during the optimization process. These functions cover different search landscape characteristics and are investigated using parameter-space distributions, population search histories, and first-dimension trajectories. In addition, the evolution of exploration and exploitation proportions is examined to illustrate how IPKO adaptively regulates its search behavior over iterations.

As shown in Fig. 6, the parameter-space distributions and contour plots indicate that IPKO effectively drives the population toward the global optimum while mitigating excessive trapping in local optima. The population search histories further show a progressive concentration of individuals around the optimal region as iterations proceed, indicating a stable convergence process. Meanwhile, the first-dimension trajectory exhibits pronounced fluctuations in the early stage, followed by reduced oscillations in later iterations, suggesting a clear transition from global exploration to local exploitation.

Fig. 6.

Visualization of the population search behavior of IPKO on representative CEC2005 benchmark functions. (a) F1. (b) F6. (c) F13. (d) F19. Each subfigure shows the function landscape, population distribution, and first-dimension trajectory.

The exploration and exploitation proportions in Fig. 7 are computed based on the population diversity defined in Eqs. (39)–(41). It can be observed that IPKO maintains a relatively high exploration level in the early iterations, which gradually decreases as exploitation becomes dominant in later stages. This adaptive transition indicates that the IPKO algorithm can balance global exploration and local exploitation throughout the optimization process, thereby enhancing convergence stability.

Fig. 7.

Evolution of exploration and exploitation proportions of IPKO during the optimization process. (a) F1. (b) F6. (c) F13. (d) F19.

40

41

42

Convergence and robustness analysis on the CEC2017 benchmark

Convergence performance analysis

To evaluate the convergence behavior of the proposed IPKO algorithm, experiments were conducted on six representative benchmark functions from the CEC2017 test suite, including unimodal F1, multimodal F10, hybrid F12 and F14, and composition functions F21 and F26. The convergence performance of IPKO was compared with the following algorithms: PKO, IVY, PSO, SSA, GOOSE, WOA, and DBO.

As shown in Fig. 8, IPKO achieves faster convergence and lower final fitness values across most test functions. On the unimodal function F1, IPKO converges rapidly and exhibits a stable descent. On the multimodal and hybrid functions, IPKO maintains sustained improvement across iterations, which helps mitigate premature convergence. For the more challenging composition functions, several comparison algorithms stagnate at relatively high fitness levels, whereas IPKO continues to refine solutions and reaches lower fitness values. Overall, these results indicate that IPKO provides stable convergence behavior, strong search capability, and improved resistance to stagnation and premature convergence across diverse optimization scenarios.

Fig. 8.

Convergence curves of IPKO and comparison algorithms on representative CEC2017 benchmark functions. (a) F1. (b) F10. (c) F12. (d) F14. (e) F21. (f) F26.

Sensitivity to problem dimensions

To further examine the scalability of the proposed IPKO algorithm, additional experiments were performed on the CEC2017 benchmark functions under different problem dimensions, with D ∈ {30, 50, 100}. The corresponding statistical results are reported in Tables 4, 5 and 6. These results are used to assess how the optimization performance of IPKO and the comparison algorithms evolves as the problem dimensionality increases and the search space expands.

Table 4.

Mean objective values and Friedman ranks of IPKO and competing algorithms on the CEC2017 benchmark functions (D = 30). Best values are highlighted in bold.

Function	IPKO	PKO	IVY	PSO	SSA	GOOSE	WOA	DBO
F1	2573.134	101612.4	1.02E + 08	9.88E + 08	4012.054	9448.698	1.55E + 09	51,001,564
F2	6.97E + 11	2.04E + 25	2.17E + 34	6.03E + 31	2.34E + 11	2.03E + 18	3.4E + 33	3.77E + 33
F3	2505.498	92107.51	65230.13	28018.82	28964.19	54862.15	243,214	78148.82
F4	482.7266	496.3242	512.4028	698.4681	495.5152	502.0727	833.8604	639.7273
F5	554.3296	572.8907	697.3141	611.9233	756.4094	884.197	821.1655	738.7013
F6	600.1572	601.1041	620.731	608.993	646.41	674.8135	677.6194	651.2345
F7	800.1361	819.5056	1245.208	827.3787	1174.931	2222.245	1299.015	1007.335
F8	860.4885	874.9442	955.9129	907.0564	983.0964	1089.894	1046.97	1011.592
F9	1015.834	1038.457	5091.71	2135.297	5256.394	8219.181	10061.63	5437.374
F10	4658.228	5045.366	5448.177	4435.126	5282.781	5821.507	6956.229	6285.338
F11	1162.655	1245.457	1650.35	1264.472	1305.513	1321.698	7361.651	1533.903
F12	53540.13	1,426,248	44,069,780	67,039,577	465959.1	4,254,606	3.21E + 08	64,814,698
F13	4615.311	45185.32	47,773,215	2,558,015	23527.07	111052.5	2,849,179	3,423,216
F14	1520.92	67382.55	1,348,114	40368.48	36177.26	29036.74	2,221,593	139509.7
F15	1991.991	21105.5	2,929,880	9394.404	14655.02	55890.86	1,564,610	71416.09
F16	2226.107	2434.182	2994.037	2622.199	2931.73	3578.009	4258.757	3378.431
F17	1900.114	2077.825	2696.155	2139.672	2564.901	2923.477	2791.464	2640.306
F18	4731.091	667077.7	1,276,866	1,035,012	346697.1	438,266	9,293,418	3,385,319
F19	2029.279	22680.3	19,283,340	6,012,271	13350.45	783950.6	12,392,074	2,772,432
F20	2304.227	2447.918	2783.844	2453.639	2729.826	3057.349	2859.942	2702.791
F21	2357.172	2373.752	2391.934	2416.522	2520.477	2677.738	2618.625	2560.215
F22	5555.938	5898.838	5285.85	4199.741	6577.097	7573.815	7769.205	4652.841
F23	2706.118	2723.442	2800.641	2934.855	2905.061	3543.871	3097.568	2973.295
F24	2882.154	2902.21	3012.589	3109.508	3076.043	3680.201	3213.877	3180.146
F25	2887.403	2897.534	2916.414	2906.963	2894.646	2916.826	3108.238	2955.434
F26	4326.655	4544.023	6960.081	4910.736	6410.204	10435.08	7917.72	6909.787
F27	3209.757	3213.117	3358.403	3275.273	3258.797	4168.501	3462.324	3331.959
F28	3221.348	3258.385	3288.397	3350.955	3212.707	3256.927	3596.692	3580.838
F29	3569.053	3722.436	4373.063	3737.469	4154.296	5065.683	5347.585	4520.648
F30	13914.05	16158.19	76805.32	72157.37	12045.97	2,111,315	30,966,850	3,764,413
Friedman average ranks	1.233333	2.933333	5.366667	3.900000	3.433333	5.900000	7.466667	5.766667
Final ranks	1	2	5	4	3	7	8	6

Table 5.

Mean objective values and Friedman ranks of IPKO and competing algorithms on the CEC2017 benchmark functions (D = 50). Best values are highlighted in bold.

Function	IPKO	PKO	IVY	PSO	SSA	GOOSE	WOA	DBO
F1	4305.752	5,940,742	22,935,786	5.18E + 09	3438.237	55110.04	8.78E + 09	1.72E + 09
F2	1.52E + 31	1.59E + 58	1.26E + 78	1.21E + 64	2.42E + 26	1.34E + 39	1.15E + 76	9.6E + 65
F3	41307.36	304,715	315516.1	151982.7	163247.6	228524.7	267150.4	215,674
F4	567.9154	598.4975	747.3119	1043.158	544.6317	567.5518	2259.116	1306.979
F5	644.9106	680.1401	853.0492	744.5216	878.5154	1146.514	1118.374	958.4375
F6	604.6351	605.1772	637.9909	628.9127	661.5487	681.6104	692.0121	666.961
F7	959.0314	989.6807	1700.343	1059.808	1706.784	3872.669	1873.831	1349.716
F8	942.713	976.1757	1201.33	1064.349	1197.962	1404.553	1359.642	1286.762
F9	2203.302	4051.126	13009.23	15916.89	13072.05	20446.43	31559.68	22502.07
F10	7346.815	8434.594	8281.643	6730.051	8369.334	9225.142	12385.38	11146.65
F11	1257.865	1986.641	2900.808	1485.841	1349.755	1510.606	5131.567	2936.99
F12	1,582,959	10,588,644	1.68E + 08	2.97E + 09	6,161,539	26,120,510	1.45E + 09	3.58E + 08
F13	8310.115	15450.59	2.18E + 08	7.91E + 08	28129.74	174645.5	1.4E + 08	42,137,384
F14	1811.977	769062.1	3,667,185	416,979	206,471	138143.1	5,136,188	2,398,845
F15	6481.351	13279.59	17,458,172	7,136,519	19735.98	47179.83	17,006,042	6,238,766
F16	2920.028	3212.303	3769.545	3329.727	3872.974	4632.544	5850.763	4612.419
F17	2628.753	2894.336	3575.835	3147.977	3535.008	3719.935	4397.545	4309.259
F18	92821.94	3,305,047	4,635,098	3,180,537	1,742,628	1,595,923	35,935,458	7,228,786
F19	7869.488	15362.38	1.47E + 08	205874.6	21654.77	1,553,155	9,683,320	7,310,471
F20	2862.112	3146.098	3502.069	3053.714	3492.609	3887.928	3861.735	3694.782
F21	2424.935	2448.358	2544.173	2584.432	2811.234	3064.101	3014.984	2873.798
F22	9334.012	9927.562	10111.24	9197.954	10297.39	10620.69	14019.27	11913.23
F23	2867.327	2900.254	3092.744	3368.716	3285.791	4313.936	3801.724	3564.653
F24	3034.956	3052.876	3295.178	3591.097	3540.922	4378.776	3785.359	3660.235
F25	3062.484	3127.027	3130.64	3182.914	3071.91	3100.241	4146.901	3480.916
F26	5388.846	5403.539	10718.73	6702.128	6997.246	14528.37	14481.32	10198.09
F27	3389.233	3472.487	3967.722	3733.161	3658.303	6050.401	4654.558	3972.823
F28	3321.913	3481.38	3440.55	4061.457	3326.329	3544.884	5239.415	6042.749
F29	3950.986	4506.187	5311.175	4606.352	5000.307	6474.96	9124.369	6309.152
F30	1,300,578	1,591,527	10,551,331	3,862,964	1,653,031	50,921,128	2.58E + 08	42,925,048
Friedman average ranks	1.200000	3.000000	5.100000	4.133333	3.400000	5.666667	7.400000	6.100000
Final ranks	1	2	5	4	3	6	8	7

Table 6.

Mean objective values and Friedman ranks of IPKO and competing algorithms on the CEC2017 benchmark functions (D = 100). Best values are highlighted in bold.

Function	IPKO	PKO	IVY	PSO	SSA	GOOSE	WOA	DBO
F1	15,464,593	1.2E + 09	7.62E + 09	1.63E + 10	4,145,912	2.32E + 09	6.47E + 10	5.37E + 10
F2	4.15E + 95	7.5E + 147	1E + 177	2.6E + 132	3.26E + 98	6.9E + 115	1.8E + 171	1.7E + 158
F3	258056.3	927,938	584544.8	529358.7	630696.4	731943.5	944677.5	459354.7
F4	831.1748	1246.513	1431.643	2527.425	810.3213	1175.225	11803.4	10812.5
F5	950.6106	1119.876	1346.651	1272.919	1376.937	1784.852	1881.446	1589.182
F6	631.4656	633.7702	655.2661	652.1568	663.9832	680.4833	698.898	675.7048
F7	1587.882	1799.226	3143.936	1925.982	3189.762	7993.153	3702.247	2816.445
F8	1221.665	1366.233	1796.45	1585.245	1823.788	2408.589	2283.974	2094.017
F9	15005.34	21200.21	24060.14	62819.35	23813.28	41762.37	71110.48	69372.82
F10	16223.23	18054.26	16001.8	16636.91	16526.02	16876.96	28271.8	26096.51
F11	6847.417	112387.1	108715.2	29562.67	21363.36	54105.73	230056.4	172574.7
F12	32,793,766	2.83E + 08	5.61E + 08	9.85E + 09	55,352,532	3.17E + 08	1.38E + 10	3.4E + 09
F13	7644.705	48488.99	2.36E + 08	9.95E + 08	30200.28	99086.97	6.35E + 08	1.97E + 08
F14	304,237	5,230,305	4,292,920	4,241,453	921587.2	1,232,419	16,117,234	10,440,517
F15	4240.336	6990.093	73,404,807	5.09E + 08	15329.71	59959.71	89,816,582	33,404,550
F16	5008.322	5623.064	6359.098	5844.524	6343.323	8228	14942.96	8934.29
F17	4310.665	4978.078	5425.965	5867.879	5955.585	6160.034	16865.74	8374.073
F18	537573.7	8,129,940	2,838,298	7,711,601	1,732,191	1,653,861	12,546,699	19,588,724
F19	6914.327	12300.78	2.52E + 08	2.91E + 08	9302.85	6,320,300	1.51E + 08	47,390,799
F20	4650.996	5215.337	5664.595	5168.49	6053.497	6386.056	6971.064	6735.789
F21	2725.388	2813.01	3000.711	3353.291	3688.49	4406.428	4349.228	3957.324
F22	18547.68	20481.07	20169.15	19942.6	19702.53	19655.55	30789.36	25963.93
F23	3176.616	3317.23	3662.325	4737.893	4306.598	5959.058	5110.922	4746.478
F24	3730.672	3882.427	4617.025	5884.609	5271.126	7030.059	6424.529	6232.27
F25	3527.087	3948.51	4035.546	4429.709	3441.046	3689.639	8073.519	8087.118
F26	10870.95	11869.76	25083.51	19382.07	21880.4	29410.97	37400.68	25462.5
F27	3502.022	3574.751	4246.888	4119.02	3790.757	6636.927	5654.335	4577.965
F28	4103.905	4510.929	3803.684	6613.378	3547.582	5138.251	11205.9	13731.23
F29	6161.414	7499.703	8219.567	7486.16	7655.302	10892.79	18303.02	11011.18
F30	39479.5	1,155,078	2,446,788	9.33E + 08	234636.7	50,600,116	1.18E + 09	1.02E + 08
Friedman average ranks	1.200000	3.366667	4.466667	4.633333	3.233333	5.300000	7.533333	6.266667
Final ranks	1	3	4	5	2	6	8	7

As the problem dimensionality increases, the optimization task becomes progressively more challenging for all algorithms. As indicated by the results in Tables 4, 5 and 6, IPKO achieves the best or highly competitive performance on most benchmark functions in low-dimensional cases, reflecting efficient convergence and effective exploitation. When the dimensionality increases, the average fitness values of several comparison algorithms grow rapidly on hybrid and composition functions, implying reduced search efficiency in high-dimensional spaces. By contrast, IPKO exhibits a relatively moderate performance degradation and consistently maintains the best Friedman ranking across different dimensions, demonstrating more stable convergence characteristics and stronger robustness to dimensional scaling.

Benchmark comparison with recent algorithms on the CEC2022

To further validate the optimization performance of the proposed IPKO, comparative experiments were conducted on 12 benchmark functions from the CEC2022 test suite. IPKO was evaluated against several recent and competitive optimization algorithms, including PKO, DE⁵⁴, AL-SHADE⁵⁵, IMODE⁵⁶, LSHADE-SPACMA⁵⁷, LSHADE-cnEpSin⁵⁸, and EBOwithCMA⁵⁹. The mean fitness values over independent runs, together with the Friedman statistical ranking, are summarized in Table 7.

Table 7.

Mean objective values on CEC2022 test functions F1 to F12 and Friedman average ranks of competing algorithms. Best values are highlighted in bold.

Function	IPKO	PKO	DE54	AL-SHADE55	IMODE56	LSHADE-SPACMA57	LSHADE-cnEpSin58	EBOwithCMA59
F1	300	306.2958	2138.169	300	300	625.9253041	300.0474711	550.8409218
F2	405.364	406.8893	408.0117	406.6785	402.8463	404.2694622	405.4525571	405.7240478
F3	600	600.0008	600	600.0021	600.3202	600.0016771	600	600.0000299
F4	807.9597	812.4038	820.9369	804.4238	809.2531	804.8434112	816.1667319	806.0153236
F5	900	900	900	900.033	900.3088	900.1348134	900.000019	900.0575916
F6	1807.796	4755.088	3916.591	1812.617	1802.8	1815.931901	1809.913922	1803.5434
F7	2015.064	2017.405	2004.597	2007.799	2003.526	2004.202085	2003.800098	2004.288939
F8	2212.126	2220.591	2218.184	2215.378	2214.794	2214.516505	2214.717318	2215.028708
F9	2529.284	2529.284	2529.284	2529.284	2529.284	2529.284383	2529.284384	2529.284383
F10	2500.367	2500.404	2490.6	2542.981	2507.545	2510.934728	2504.079935	2521.78209
F11	2750	2800	2887.718	2890	2810.145	2825.449308	2901.12481	2860.001531
F12	2861.403	2861.882	2862.385	2862.641	2862.598	2864.57679	2863.814203	2863.694253
Friedman average ranks	2.333333	5.000000	5.166667	5.166667	4.125000	4.750000	4.708333	4.750000
Final ranks	1	6	7	7	2	4	3	4

These competitors were selected because they represent strong and widely used baselines in recent CEC benchmark studies, covering both differential-evolution variants and hybrid CMA-ES–based designs, and thus provide a rigorous reference for assessing convergence accuracy and robustness.

As reported in Table 7, IPKO achieves the best overall Friedman ranking among all compared algorithms, indicating superior and consistent performance across the CEC2022 benchmark set. Although some advanced DE-based variants, such as IMODE and SHADE-based algorithms, demonstrate competitive results on several functions, IPKO attains lower average fitness values on the majority of test problems. These results suggest that IPKO offers a more robust search capability and well-balanced performance when tackling complex and diverse optimization landscapes, further confirming the effectiveness of the proposed multi-stage improvement strategies embedded in IPKO.

Case study analysis

Microgrid parameter settings

To evaluate the performance of the improved IPKO algorithm in microgrid optimal scheduling, IPKO is employed to solve the grid-connected microgrid scheduling problem. The key parameters of the distributed generation units are listed in Table 8, and the pollutant emission treatment costs and related parameters are summarized in Table 9. In addition, the time-of-use electricity purchase and sale prices at the microgrid location are provided in Table 10, from which the valley, flat, and peak periods used in the demand response model are determined.

Table 8.

Operating limits and cost parameters of distributed power sources used in the microgrid optimization.

Power source	Minimum output (kW)	Maximum output (kW)	Fuel cost (CNY/kW)	Maintenance cost (CNY/kW)
PV	0	605	0	0.0096
WT	0	624	0	0.045
DE	0	650	0.251	0.055
MT	0	650	0.26	0.128
EL	0	500	0	0.05
FC	0	300	0	0.0293
BESS	-250	250	0	0.066

Table 9.

Pollutant treatment costs and emission factors used in the microgrid optimization model.

Gas species	Treatment cost (CNY/kg)	Emission factor of DE (g/kWh)	Emission factor of MT (g/kWh)	Emission factor of grid (g/kWh)
CO	10.29	0.03	0.047	0.043
CO2	0.21	649	724	889
SO2	14.842	0.206	0.004	1.8
NOx	62.964	3.898	0.2	1.6

Table 10.

Time-of-use electricity purchase and sale prices adopted in the microgrid model.

Time-of-use period	Time interval	Purchase price (CNY/kWh)	Sale price (CNY/kWh)
Peak	08:00–12:00; 17:00–21:00	1.0347	1.0012
Flat	12:00–17:00; 21:00–00:00	0.6068	0.5822
Valley	00:00–08:00	0.2589	0.2437

Simulation results and performance analysis

Figure 9 depicts the typical-day day-ahead forecasted PV and WT generation profiles used for the microgrid scheduling case. PV generation exhibits a clear diurnal pattern with a daytime peak, whereas WT generation shows higher variability due to wind intermittency. Given such variability, optimization-based scheduling is essential to effectively coordinate renewable generation with system demand and flexible resources, thereby enhancing operational efficiency and system reliability.

Fig. 9.

Photovoltaic (PV) and wind turbine (WT) generation forecast curves over a typical day.

Figure 10 compares the load profiles before and after introducing price-based demand response under time-of-use electricity pricing, showing that load demand is shifted from high-price periods to low-price periods, resulting in peak shaving and valley filling. With demand response, the load profile becomes more balanced, reducing the need for expensive conventional generation during peak hours. These results indicate that price-based demand response provides supplementary load-shaping capability and facilitates improved coordination between renewable generation and system demand.

Fig. 10.

Load profiles before and after introducing price-based demand response under time-of-use electricity pricing.

Figure 11a and b present the microgrid scheduling results obtained by the IPKO algorithm without and with price-based demand response under time-of-use electricity pricing, respectively. The introduction of demand response in Fig. 11b results in a more balanced load profile, reducing the need for expensive conventional generation during peak hours. Overall, the IPKO-based strategy achieves coordinated operation among wind power, photovoltaic generation, energy storage systems, and conventional generation units while satisfying the load demand, resulting in a well-matched load profile and stable system operation.

Fig. 11.

Microgrid optimized scheduling results based on the IPKO algorithm without and with price-based demand response. (a) Without price-based demand response. (b) With price-based demand response.

During 0–8 h, wind power and grid import are preferentially utilized to supply the load, while surplus energy is stored in the BESS or converted into hydrogen. In 8–12 h, as wind power decreases and PV generation increases, grid import is reduced and excess renewable energy is allocated to hydrogen production to mitigate renewable curtailment. In these periods, the BESS mainly acts as a short-term buffer to absorb rapid renewable power fluctuations, while the HES converts surplus renewable electricity into hydrogen for longer-term energy storage. During 12–17 h, the fuel cell and BESS are dispatched to reduce the output of high-cost conventional generation units, such as gas turbines and diesel generators. In the 17–21 h peak period, the load demand is met by a coordinated combination of fuel cell generation, BESS discharging, and grid import. Finally, during 21–24 h, the energy storage systems are recharged to restore their predefined reserve levels, completing the daily energy cycle.

The energy variations of the BESS and hydrogen storage shown in Fig. 12 further confirm the effectiveness of the proposed IPKO-based scheduling strategy in achieving efficient energy management and stable multi-energy operation. The dashed line indicates the predefined reserve level imposed by the daily cycling constraint. By leveraging the complementary buffering capability of the BESS and the hydrogen energy chain, the strategy absorbs surplus renewable generation and mitigates curtailment, thereby improving renewable energy utilization. Specifically, the BESS responds quickly to intra-day power fluctuations and peak-load changes, while the HES provides a slower but larger-capacity energy-shifting pathway through the electrolyzer, hydrogen tank, and fuel cell. This complementary operation allows short-term power imbalance and longer-term renewable surplus to be handled simultaneously. Notably, the coordinated dynamics of the BESS and HES ensure that both energy states return to their predefined reserve levels by the end of the day, which serve as the initial states for the next day, satisfying the daily cycling requirement.

Fig. 12.

Energy state variations of the battery energy storage system and hydrogen storage system.

Table 11 shows that the cost composition of the microgrid varies significantly under different operating strategies. Without scheduling and relying entirely on grid power, the total cost reaches 23520.44, including an economic cost of 16325.08 and an environmental cost of 7195.36. After applying the proposed IPKO-based scheduling strategy, the total cost is substantially reduced to 17999.24, with the economic and environmental costs decreasing to 12742.16 and 5093.84, respectively. By explicitly accounting for renewable curtailment penalties, the IPKO-based framework achieves an effective balance between cost reduction and renewable energy utilization, corresponding to a curtailment rate of 12.60%. When demand response under time-of-use pricing is additionally enabled, the total cost is further reduced to 16855.24, and the curtailment rate decreases to 10.64%, indicating that price-driven load shifting offers additional performance improvements when combined with supply-side coordination. Overall, the results confirm the effectiveness of the proposed IPKO-based scheduling strategy in enhancing cost efficiency, environmental performance, and renewable energy utilization.

Table 11.

Cost comparison of the microgrid under different IPKO-based scheduling strategies.

Cost component	Grid-only operation	IPKO-based scheduling	IPKO-based scheduling with price-based demand response
Economic cost	16325.08	12742.16	11408.02
Environmental cost	7195.36	5093.84	4933.87
Renewable curtailment penalty cost	/	163.24	137.88
Renewable curtailment rate (%)	/	12.60%	10.64%
Total cost	23520.44	17999.24	16855.24

To further evaluate the solution quality and run-to-run stability of different optimization algorithms, the microgrid optimal scheduling problem was solved independently 30 times using the IPKO, PKO, SSA, PSO, and IVY algorithms. The resulting daily operating costs are illustrated in Fig. 13a, and the corresponding statistical indicators, including the minimum, mean, standard deviation, and maximum values, are summarized in Table 12.

Fig. 13.

Comparison of daily operating cost distributions obtained by different optimization algorithms over 30 independent runs. (a) Run-by-run costs across 30 trials. (b) Boxplot of cost distributions.

Table 12.

Statistical comparison of daily operating costs obtained by different optimization algorithms.

Algorithm	Minimum	Mean	Standard deviation	Maximum
IPKO	16855.24	17175.14467	315.8203	18110.5
PKO	16925.3	18699.37333	738.7043	20825.6
SSA	16912.6	17938.94333	751.3651	20888.5
PSO	17175.8	18167.89	545.6701	19325.6
IVY	17325.2	19093.62333	1081.1913	21345.5

As shown in Table 12, the IPKO algorithm achieves the lowest average daily operating cost among all methods, with reductions of 8.15%, 4.26%, 5.46%, and 10.05% compared with the other four algorithms, indicating superior solution quality. Moreover, IPKO exhibits the smallest standard deviation across repeated runs, with reductions of 57.25%, 57.97%, 42.12%, and 70.79%, respectively, demonstrating markedly improved solution consistency and robustness.

The dispersion patterns in Fig. 13a further support this observation, showing that IPKO produces a more concentrated cost distribution over multiple independent executions. In contrast, the competing methods exhibit larger run-to-run fluctuations and occasional high-cost outliers, with pronounced spikes mainly observed for IVY and PKO. The boxplot in Fig. 13b further confirms this trend by showing a lower median cost and a smaller interquartile range for IPKO. These differences can be attributed to the higher sensitivity of the competing methods to random initialization and their less effective balance between global exploration and local exploitation, which can cause some trials to be trapped in suboptimal regions or experience stagnation. By contrast, the proposed IPKO incorporates enhanced population diversity and more robust search dynamics, enabling more reliable convergence and consistently lower operating costs with reduced variability. Overall, these results demonstrate that IPKO outperforms the competing methods in terms of both solution quality and robustness for the microgrid optimal scheduling problem.

Conclusion

This work proposes the IPKO algorithm to enhance population diversity, search adaptability, and convergence stability for highly nonlinear and nonconvex optimization problems, with a particular focus on microgrid scheduling applications. By integrating four complementary enhancement strategies, including chaotic initialization, multi-source guidance with adaptive perturbation, differential and spiral development, and periodic dynamic adjustment, the proposed IPKO achieves an improved exploration–exploitation balance and more robust performance, as verified on benchmark test suites.

The proposed IPKO is further applied to the optimal scheduling of a grid-connected microgrid with hybrid energy storage, comprising a hydrogen energy storage system and a battery energy storage system, where price-based demand response is modeled using time-of-use pricing. Simulation results show that the IPKO-based framework consistently reduces economic and environmental costs. Compared with representative metaheuristic algorithms, it achieves about 4–10% lower average daily operating cost with reduced variability, confirming its superior solution quality and robustness. Overall, the proposed framework provides an effective metaheuristic solution for nonconvex scheduling of microgrids with hybrid energy storage.

Although the proposed framework demonstrates strong performance for the studied scheduling problem, some aspects still deserve further investigation. The current case study is mainly based on deterministic renewable generation and load profiles, and future work will incorporate uncertainty representations for PV, wind, and load demand to enhance the practical applicability and robustness of the scheduling framework. In addition, since the present study focuses on a typical grid-connected microgrid with hybrid energy storage, extending the proposed IPKO-based approach to larger-scale microgrids with more diverse resources will be considered to further evaluate its scalability and generalization capability. Battery degradation modeling can also be incorporated as a lifecycle-related cost term in future long-term scheduling studies.

Author contributions

Q.L. (Qisheng Liu) conceptualized the research idea, wrote the main manuscript, and implemented the simulations. C.C. (Changxi Chen), L.W. (Liang Wang), and W.L. (Weixiang Lei) conducted the literature review and performed the statistical analysis. H.O. (Honglin Ouyang) and M.X. (Muxuan Xiao) provided guidance and assisted with the final revision of the manuscript. All authors reviewed and approved the final version of the manuscript.

Funding

This work was supported by the National Key Research and Development Program of China (2022YFB2404800).

Data availability

The datasets generated and/or analyzed during the current study are not publicly available but are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.