A Systematic Review of Hyper-Heuristics in Power System Optimization: Theory, Applications, and Future Directions

Abstract

Modern power system optimization problems have become increasingly large-scale, non-convex, and highly constrained. Conventional optimization methods and metaheuristics struggle with these complexities, primarily due to premature convergence, extensive parameter-tuning requirements, and the limitations of the No Free Lunch theorem. To address these challenges, hyper-heuristics have emerged as high-level approaches that search the space of heuristics rather than the solution space directly, providing greater generality across different domains. Following the PRISMA 2020 guidelines, this systematic review searched Scopus, Web of Science, ScienceDirect, IEEE Xplore, Springer Nature Link, and Taylor & Francis Online for peer-reviewed studies published up to June 2026 on hyper-heuristics applied to power system optimization. Study eligibility was assessed based on topical relevance, and the findings were synthesized narratively. Out of 162 unique records screened, 21 studies were included in the final review without formal risk-of-bias restrictions. The findings indicate that while hyper-heuristics offer excellent efficiency, robustness, and solution quality, their overall application in power systems remains limited. Current research relies heavily on selection-based architectures, leaving significant gaps in computational time reporting and real-world grid testing. Future research should extend selection-based methods, explore wider problem variants, address underexplored power system optimization domains, and investigate the development of generation hyper-heuristics.

1. Introduction

The modernization of power networks has introduced complex characteristics to power generation, planning, and control operations. Modern power system optimization problems are typically characterized as large-scale, highly nonlinear, and non-convex, involving both discrete and continuous variables, as well as complex equality and inequality constraints. With the modern grid increasingly dominated by power electronics—such as inverter-based renewable generation and dynamic control devices—managing these systems efficiently has become even more challenging.

Historically, classical mathematical programming methods have been widely used to manage these systems. However, conventional optimization methods like Lagrangian relaxation (LR) [1], dynamic programming (DP) [2] and linear programming (LP) [3] often become trapped in local optima because they do not explore the solution space sufficiently [4]. They also suffer from an inability to handle discrete design variables and struggle to deal with multi-objective problems [5]. Because of these mathematical barriers, solving large-scale power system problems requires advanced computational approaches capable of escaping local optima to find promising regions with high-quality solutions without demanding unreasonable computational time or memory.

To overcome the limitations of traditional mathematical programming, researchers turned their focus to computational intelligence (CI) and metaheuristics—advanced algorithms inspired by natural phenomena and social behaviors. Metaheuristics have gained significant popularity due to their ability to escape from local optima by balancing two fundamental processes: exploitation and exploration [6]. Exploitation focuses on searching intensively around good solutions that have already been found, while exploration involves investigating diverse and less-visited areas of the solution space. Over the past few decades, well-established bio-inspired techniques such as the genetic algorithm (GA) [7,8], particle swarm optimization (PSO) [9,10], simulated annealing (SA) [11], differential evolution (DE) [12], artificial bee colony (ABC) [13], symbiotic organisms search (SOS) [4], grey wolf optimizer (GWO) [14], and bat algorithm (BA) [15], as well as newer methods like artificial rabbits optimization (ARO) [16], pelican optimization algorithm (POA) [17], and marine predators algorithm (MPA) [18], have demonstrated strong performance in power system optimization problems. These stochastic methods have been successfully employed in power system applications ranging from real-time operations, such as voltage control [19], to long-term decision-making tasks, such as distribution planning [20].

Despite their considerable success, the performance of these methods is highly dependent on parameter tuning, a problem-specific process that can be both time-consuming and expertise-intensive. Meanwhile, the exponential growth of proposed metaheuristic algorithms has led to an unsystematic “rush to heuristics” within the scientific community [21]. Hundreds of new algorithms and variants are continually introduced, often with questionable advancements and repeated mechanisms. Furthermore, the fundamental No Free Lunch (NFL) theorem [22] establishes that no single optimization algorithm is universally superior across all types of problems. An algorithm that performs exceptionally well on one specific power system task may perform poorly on another. As a result, the key challenge has evolved from simply designing new metaheuristics to determining which algorithms are best suited for specific conditions.

In response to these algorithmic challenges, the field of hyper-heuristics emerged. While traditional metaheuristics operate directly on the solution space of a problem, hyper-heuristics operate at a higher level. Specifically, they search through a space of heuristics to intelligently select, combine, or generate the most effective heuristics for a given optimization scenario. By automating algorithm design and selection, hyper-heuristics provide a flexible framework that can dynamically adapt to the complex constraints and nonlinear characteristics of modern power systems.

Although hyper-heuristics show strong potential for managing complex optimization tasks, their application in power systems remains a nascent research area. While existing literature surveys focus on standalone methods that search directly within the solution space, a critical knowledge gap remains regarding high-level heuristic search frameworks. This review fills that gap by providing the first systematic analysis dedicated to hyper-heuristics in power systems. There is currently a need to integrate and analyze existing knowledge, clarify the advantages of hyper-heuristics over established single or hybrid CI techniques, and identify unexplored areas in the literature. Therefore, this systematic review examines the theory, applications, and future perspectives of hyper-heuristics in power system optimization. To evaluate the current status and effectiveness of these applications, this study addresses two major research questions:

• What specific types of power system optimization problems are currently being addressed using hyper-heuristics? Answering this question helps identify the current landscape and scope of hyper-heuristic applications across different power system tasks, such as unit commitment (UC), economic dispatch (ED), and optimal power flow (OPF), while highlighting critical research gaps where these adaptive algorithms have not yet been applied, or have been applied only to a limited extent.
• How do hyper-heuristics impact the optimization of power system performance, and how effective are they compared to conventional optimization techniques, standard metaheuristics, or other well-established intelligent methods in this domain? Answering this provides empirical evidence regarding the capability of hyper-heuristics to enhance baseline system performance. It also evaluates their robustness, efficiency, and algorithmic generality compared to traditional mathematical methods, modern metaheuristics, and other intelligent techniques when handling highly complex, nonlinear optimization problems.

The remainder of the paper is structured as follows: Section 2 details the review methodology followed for selecting the relevant literature. Section 3 provides the necessary background context by evaluating recent reviews related to power system optimization algorithms. Section 4 introduces the fundamental concepts of hyper-heuristics and summarizes their main areas of application. Section 5 presents a detailed analysis of 21 papers examining the application of hyper-heuristics to power system optimization problems. Section 6 provides a discussion of the findings. Finally, Section 7 concludes the paper and outlines potential directions for future research in the field.

2. Review Methodology

To rigorously review and synthesize complex data regarding the applications of hyper-heuristics in power system optimization, this paper utilizes a systematic literature review (SLR) methodology based on the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) framework [23]. This standardized reporting guideline ensures a structured, transparent, and reproducible approach to the review and data extraction process.

2.1. PRISMA Compliance and Protocol Registration

This systematic review was conducted in accordance with the PRISMA 2020 guidelines [24]. The review steps, including database searching, study screening, data extraction, and qualitative synthesis, followed the structured framework outlined in this methodology section to ensure transparency and reproducibility. Because this review synthesizes computer science approaches and mathematical formulations within power systems, it does not meet the eligibility criteria for clinical or healthcare-related systematic review registries (such as PROSPERO). Consequently, a formal review protocol was not prospectively registered, and no external registration number was assigned.

2.2. Search Strategy

The initial search was conducted across six major scholarly databases: Scopus, Web of Science, ScienceDirect, IEEE Xplore, Springer Nature Link, and Taylor & Francis Online. All databases were last consulted and searched in June 2026. The search strings utilized to capture relevant literature are shown in

Table 1. Search sources and strings.

Database	Search in	String	Records
Scopus	Title-Abstract-Keywords	(“hyper heuristic” OR hyperheuristic) AND power	80
Web of Science	All fields	(“hyper heuristic” OR hyperheuristic) AND power	42
ScienceDirect	Title-Abstract-Keywords	(“hyper heuristic” OR hyperheuristic) AND power	17
IEEE Xplore	Title-Abstract-Keywords	(“hyper heuristic” OR hyperheuristic) AND power	45
Springer Nature Link	Title-Abstract-Body text	(“hyper heuristic” OR hyperheuristic) AND power	48
Taylor & Francis Online	Title-Abstract-Keywords	(“hyper heuristic” OR hyperheuristic) AND power	0

.

The general keyword “power” was intentionally selected as a root term to capture all relevant power engineering sub-domains while accommodating the limited number of hyper-heuristic studies in this field. Because “power” appears universally in the metadata of such studies, combining it with the specialized keyword “hyper-heuristic” ensured comprehensive coverage. To further confirm search completeness, supplementary verification queries were also conducted across the databases combining “hyper-heuristic” with specific domain terms including “smart grid,” “electricity market,” “microgrid,” “energy management,” “renewable integration,” and “power dispatch.” These additional queries yielded no further eligible studies that were not already captured or screened through the primary search. Due to the limited number of studies, the search did not restrict the publication year range. To ensure maximum coverage of the domain, conference papers and book chapters were included alongside peer-reviewed journal articles.

2.3. Study Selection Process

The screening and selection process was executed in distinct, sequential stages to systematically filter the literature. The primary screening of titles, abstracts, and full-text articles was performed by the first author. To minimize bias and ensure selection accuracy, any papers with inclusion doubts were independently reviewed and resolved through discussion and consensus with the supervisory co-authors. No automated screening tools were used.

A total of 232 references were generated from the searches. In addition, four articles were identified through backward snowballing, yielding a total of 236 initial records. After removing 74 duplicate records, 162 unique records remained. Initially, nine papers were excluded as either review/overview articles or conference abstracts. The remaining 153 papers were then screened based on title and abstract, excluding 128 out-of-scope studies (bringing total exclusions during title and abstract screening to 137) and leaving 25 articles for full-text assessment. From these, four papers were excluded: one did not answer the core research questions [25], one lacked an explicit focus on hyper-heuristic mechanisms [26], and two did not address clearly related topics [27,28]. This resulted in a final set of 21 included studies.

The flowchart of SLR in Figure 1 is adapted from the PRISMA framework [29], encompassing four main phases:

• Phase 1—Identification: Comprehensive search strings were executed across primary databases to identify relevant literature. To supplement the database searches, manual backward snowballing was applied to the reference lists of selected papers to reveal additional eligible studies. Duplicate records were then identified and removed to formulate the initial screening pool.
• Phase 2—Screening: Titles and abstracts were first evaluated according to the inclusion criteria. Subsequently, detailed full-text assessments for eligibility were performed on the retrieved reports using predetermined parameters.
• Phase 3—Inclusion: The final set of eligible papers was officially selected for the review. These included articles were then categorized into groups based on the objective function, decision variables, and the operational constraints characterizing each power system optimization problem.

2.4. Eligibility Criteria

To ensure an objective and transparent evaluation given the specialized nature of this domain, clear acceptance criteria were applied. To be included in this study, a paper had to fulfill all of the following requirements:

• Availability: The complete text is available and can be downloaded.
• Language: The paper is published in English.
• Conceptual boundary and scope: The study must explicitly focus on applying a true hyper-heuristic methodology to a power system optimization problem. To satisfy this conceptual boundary [30], the algorithm must feature a high-level management mechanism operating over a search space of low-level heuristics rather than searching the solution space directly. Eligible studies include selection-based hyper-heuristics (combining high-level selection mechanisms such as reinforcement learning, choice functions, or metaheuristics with low-level heuristic pools) and generative hyper-heuristics (such as genetic programming automatically creating or evolving search operators). Conversely, standard metaheuristics, self-adaptive metaheuristics that merely adjust low-level numerical parameters (e.g., adaptive inertia weights or mutation rates without selecting/generating heuristics), and hybrid metaheuristics lacking high-level heuristic selection rules are strictly excluded.
• Peer review: The article is peer-reviewed.

2.5. Data Collection and Extraction

Data extraction was systematically conducted by the first author, who reviewed each of the 21 included papers to gather relevant information directly addressing the two major research questions. To ensure reliability and minimize extraction errors, the extracted data items were subsequently reviewed and validated by the co-authors. No automated extraction tools were used, and no original authors were contacted for missing or unclear information. The following specific information was collected from each selected article:

• Optimization problem characteristics: Objective functions and decision variables of the problem.
• Algorithmic characteristics: Hyper-heuristic’s high-level mechanism and low-level heuristics.
• Test system characteristics: Dimensions, scale, and specific configurations.
• Performance metrics: Solution effectiveness, robustness, and comparison results which are discussed throughout the text of the review.

2.6. Study Quality Assessment

To evaluate the reliability and reporting quality of the included literature rather than merely summarizing it, an objective 3-point quality assessment (QA) framework was applied in accordance with established systematic review guidelines for computational engineering [31]:

• QA1 (Algorithmic reproducibility): Are the high-level hyper-heuristic mechanism, low-level heuristic pool, and control parameters explicitly specified?
• QA2 (Problem formulation clarity): Are the power system objective functions, decision variables, and operational constraints clearly defined?
• QA3 (Comparative rigor): Is the proposed algorithm evaluated against intelligent methods or conventional optimization techniques using clear performance metrics?

Each study was evaluated independently by the first author and cross-verified by the co-authors. All 21 included papers satisfied all three quality criteria (scoring “Yes” across all items), confirming high algorithmic reproducibility, mathematical clarity, and comparative rigor across the reviewed literature.

2.7. Data Synthesis

Due to the wide variety of algorithmic architectures and power system configurations across the literature, a quantitative meta-analysis was not feasible. Instead, a qualitative narrative synthesis approach was adopted.

The included studies are presented in Section 5 and categorized into three primary application domains: (1) power system operation, (2) specialized sub-domains, and (3) wind farm layout optimization (WFLO). The key findings, problem formulations, hyper-heuristic mechanisms, and comparative outcomes across all 21 reviewed studies are summarized into a single comprehensive synthesis table (

Table 2. Comparative analysis of literature reviews in power system optimization.

Reference	Primary Focus & Scope	Methods Reviewed	Literature Synthesis Gaps
[32]	Survey of hybrid bio-inspired CI techniques for traditional power system optimization problems	Neural network, Fuzzy system, and Population bio-inspired Hybrids; Memetic algorithms	Focuses on ad-hoc, pairwise hybridizations rather than automated hyper-heuristic selection/generation frameworks
[21]	Critical analysis of metaheuristic underlying principles, multi-objective formulations, metrics, and methodological pitfalls regarding the “rush to heuristics”	Trajectory, Population and multi-objective search, Hybrid metaheuristics, Multi-criteria decision-making tools, Benchmarking metrics	Critiques arbitrary metaheuristic creation conceptually, but lacks an SLR on hyper-heuristics in power systems
[5]	Metaheuristic search methods in smart grids, focusing on OPF, scheduling, and planning	Evolutionary computation, Swarm intelligence, Artificial immune systems, Non-population-based metaheuristics	Identifies parameter tuning limits but lacks a systematic review of hyper-heuristics for automated algorithm selection/generation
[33]	Survey of metaheuristic optimization algorithms applied to major power system problems	Trajectory, Evolutionary, Swarm, Physics-based, and Human-based algorithms	Reviews standalone metaheuristics; fails to address how hyper-heuristics overcome manual tuning and NFL limits
[34]	Applications of AI across power system operation, control, and planning	Neural/neuro-fuzzy networks, Fuzzy logic, Swarm/evolutionary algorithms Machine learning regression, Hybrid AI approaches	Surveys soft computing & ad-hoc hybrids, omitting automated hyperparameter tuning and high-level selection frameworks
[35]	Historical 50-year evolution of centralized and decentralized power system optimization across investment, operational planning, operations, control, and forecasting	Mathematical programming, Stochastic decomposition, Equilibrium modeling, Convex relaxations, Stochastic optimization, Statistical forecasting	Focuses on classical/convex operations research, omitting heuristic search algorithms and hyper-heuristic frameworks
[36]	Applications of AI and machine learning techniques across power system operation, control, distribution, and long-term planning horizons	Classical methods, Metaheuristics. Machine learning (supervised, unsupervised, and semi-supervised), Deep learning and deep reinforcement learning	Emphasizes predictive & individual ML/DL/metaheuristic models without synthesizing hyper-heuristic frameworks

). The reporting quality, algorithmic reproducibility, and comparative rigor of all included studies were systematically evaluated using the objective quality assessment framework detailed in Section 2.6.

3. Background

This section provides the context for the current study by examining prominent literature reviews published in this field over the last decade. By mapping the evolution of optimization methodologies in power systems, these studies highlight the progression from classical deterministic methods to a vast variety of metaheuristic and CI techniques. Examination of this evolution reveals a significant gap in the current literature: while standard metaheuristics have been extensively reviewed, references to hyper-heuristics are notably absent in the existing surveys. Specifically, existing surveys fail to address how hyper-heuristics can overcome the fundamental limitations of standard metaheuristics—such as the need for manual hyperparameter tuning and the constraints of the No Free Lunch (NFL) theorem. Furthermore, the current literature lacks a critical synthesis of how hyper-heuristics can be used to systematically select and generate optimal heuristic components for specific power system conditions, thereby offering a rigorous alternative to the unguided “rush to heuristics” currently seen in the field. The reviews discussed below show how the field has developed, ultimately leading to the identification of hyper-heuristics as an important methodology for future research. Table 2 presents a comparative analysis of existing surveys in power system optimization, highlighting their primary scope, reviewed methods, and key literature synthesis gaps.

The authors in [32] present a comprehensive survey focusing on the application of hybrid bio-inspired CI techniques for solving traditional power system optimization problems. The review analyzes combinations of popular methods such as PSO, GA, DE, and fuzzy logic applied to complex tasks like ED, load forecasting, and voltage stability. The authors emphasize that hybridization strategically combines the benefits of different techniques while mitigating their individual drawbacks to improve convergence and solution quality. Notably, when discussing future research directions, hyper-heuristics are introduced and proposed as an advanced methodology for addressing real-world power system problems.

In [21], the authors provide a critical examination of metaheuristic algorithms in power and energy systems. Rather than simply listing applications, their review focuses on the structural principles of metaheuristic construction—such as selection, elitism, and reinforcement—and points out methodological pitfalls in how these algorithms are frequently evaluated. To ensure that claims of algorithmic superiority are valid and reliable, they propose specific guidelines and robust statistics-based indicators for comparing solutions in both single- and multi-objective problems. This review introduces hyper-heuristics as a high-level design paradigm that explores a search space of reusable heuristic components to automatically select or generate effective solvers.

In [5], the authors review the use of metaheuristic search methods developed for smart grids. They categorize power system optimization tasks into three major classes: OPF, scheduling, and planning. The authors highlight that smart grid optimization often involves highly undesirable mathematical characteristics, including non-convexity, mixed types of design variables, and multiple conflicting objectives. To address these challenges, the review details how various metaheuristic categories, such as swarm intelligence and evolutionary computation methods, perform on common test grids, evaluating their multi-objective performance and solution accuracy relative to their computational complexity.

The authors in [33] conduct a broad review of metaheuristic optimization algorithms deployed for complex power system problems. The review systematically details the application of metaheuristics to problems such as optimizing power flow in transmission and distribution networks, reactive power dispatching, and combined economic and emission dispatching. Through this structure, they illustrate how metaheuristics search beyond local optima to efficiently handle the increasing uncertainties and constraints of modern grids, establishing them as an attractive topic for environmental sustainability.

The applications of artificial intelligence (AI) across power system operation, control, and planning are explored in [34]. This review extends beyond pure metaheuristics to include artificial neural networks (ANNs), adaptive neuro-fuzzy inference systems (ANFIS), and regression models, evaluating them alongside optimization algorithms like ant colony optimization (ACO) and ABC. It evaluates how these interconnected AI technologies are utilized to improve voltage profiles, predict equipment defects, minimize transmission losses, and manage frequency deviations to ensure the smooth, automated operation of modern power systems.

Providing a historical perspective, a recent review [35] covers fifty years of power system optimization. It examines the evolution of operations research methods across two distinct eras: the centralized, monopoly-driven frameworks spanning 1970 to 1990, and the decentralized, market-driven environments from 1990 to the present. The review describes the transition from classical linear LP and mixed-integer LP to advanced decomposition methods, stochastic programming, and equilibrium constraints. Across this timeline, the authors highlight the critical role of optimization techniques in addressing modern challenges such as renewable energy integration, uncertainty, and smart grid control. Notably, while the review maintains a primary focus on the classical operations research and mathematical programming perspective, it acknowledges the growing importance of intelligent methods like ANNs, deep neural networks (DNNs), and evolutionary algorithms over the last half-century.

Another recent review [36] examines the critical role of AI in designing resilient and adaptive power systems. The authors emphasize that while traditional optimization methods often struggle with nonlinear and uncertain system dynamics, AI-based techniques significantly enhance power system performance and reliable operations. The paper provides a comprehensive analysis of various optimization methodologies—including classical and metaheuristic approaches—applied across different facets of grid management, such as ED, UC, and transmission and distribution system planning. Furthermore, the study systematically reviews the deployment of diverse machine learning frameworks, such as deep learning, reinforcement learning, and ensemble learning, to solve complex operational challenges like real-time fault detection, transient stability assessment, and energy management in microgrids.

4. Principles of Hyper-Heuristics

Hyper-heuristics are high-level optimization methods that focus on selecting or generating techniques to efficiently solve complex problems. Unlike traditional heuristics (such as greedy algorithms and hill climbing) and metaheuristics which are designed to explore the solution space directly, hyper-heuristics operate at a higher level of abstraction, searching the space of heuristics. The main idea is to find the most appropriate algorithm or combination of algorithms to address a specific problem, rather than searching for an optimal solution directly. By separating the search mechanism from the solution space, more general methodologies are created that can effectively solve a wide range of hard computational problems.

The necessity for this high-level abstraction comes from the limitations imposed by the NFL theorem [22]. In optimization theory, the NFL theorem states that no single low-level optimization algorithm can outperform all others across all possible problem instances. In fields like power system optimization, problems are frequently characterized by high dimensionality, nonlinearity, and constrained search spaces. A metaheuristic that performs very well on one specific problem may completely fail on another problem or even on a different instance of the same problem. To achieve satisfactory performance, standard intelligent methods require extensive parameter tuning or manual hybridization.

4.1. Hyper-Heuristic Architecture

A typical hyper-heuristic relies on two core components: a high-level strategy and a set of low-level heuristics [37].

• High-level strategy (heuristic selection mechanism): This component acts as the supervisor, deciding which low-level heuristic to apply at each step of the search usually based on past performance, learning mechanisms, or other decision-making criteria.
• Low-level heuristics (heuristic pool): This set consists of domain-specific procedures that directly operate on problem solutions. The term “heuristic” is used generically and does not necessarily refer to a heuristic as conventionally defined. A low-level heuristic can range from a simple local search operator (such as a swap or insertion operator) to a complete optimization method (such as a metaheuristic).

Apart from the selection of low-level heuristics, the acceptance criterion is a very important component of the hyper-heuristic. It is the rule that determines whether to keep or reject the solution generated by the low-level heuristic based on factors like solution improvement or a predefined probability criterion. The interaction of these components is crucial, as it enables the high-level strategy to adapt, learn, and choose the most effective low-level heuristic or combination of heuristics based on the current state of the search process.

A fundamental characteristic of this architecture is the domain barrier [38], a conceptual boundary that separates the high-level strategy from the domain-specific low-level heuristics. The high-level strategy operates with no direct knowledge of the problem domain—for instance, in a power system it remains unaware of components like buses, transformers, or line limits. It receives non-domain-specific data, such as changes in the objective function value or parameters of the available low-level heuristics. Under this design, the low-level heuristics are solely responsible for managing the physical laws and operational constraints of the power grid—such as maintaining active and reactive power balances and enforcing line limits—while the high-level manager focuses strictly on guiding algorithm selection to navigate non-convex search spaces. The main motivation behind the domain barrier is generality. This isolation allows the exact same high-level mechanism to be applied across different problem domains without structural modifications. Figure 2 illustrates the general architecture of a hyper-heuristic.

4.2. Classification of Hyper-Heuristics

Hyper-heuristics can be classified based on two key characteristics: the nature of the heuristic space and the feedback used for learning mechanisms [37] as shown in Figure 3.

4.2.1. Nature of the Heuristic Space

According to how the heuristics are utilized, there are two main categories:

• Heuristic selection: Methodologies that choose from a predefined pool of existing low-level heuristics. In power system optimization, heuristic selection is commonly used, as many successful algorithms already exist for finding solutions in this field.
• Heuristic generation: Methodologies that generate new heuristics by combining or evolving existing algorithmic sub-components.

At a secondary level, these methods can be further categorized based on how they utilize solutions as follows:

• Constructive methods: They start with empty or partial solutions and iteratively add components until a complete solution is built.
• Perturbative methods: They start with a complete existing solution and iteratively modify or improve it by making small, local changes.

4.2.2. Learning Mechanisms

Depending on when and how the high-level strategy obtains knowledge for decision-making, hyper-heuristics use one of two learning types:

• Online learning: Learning occurs dynamically during the problem-solving process. The high-level strategy uses real-time feedback from the current search process to reinforce successful heuristics and reduce the influence of the less effective ones.
• Offline learning: Learning is performed prior to problem-solving. The hyper-heuristic is trained on historical data or representative problem instances to develop general rules or strategies that are later applied to new instances without further learning.

In addition to online and offline learning, hyper-heuristic frameworks can also operate under a no-learning scheme. In this approach, the high-level strategies that execute selection or acceptance decisions use fixed, deterministic, or purely stochastic rules (e.g., random selection or uniform choice functions) without updating their internal parameters based on historical search performance.

4.3. Applications of Hyper-Heuristics

Hyper-heuristics have a wide range of applications across domains that require solving complex optimization and decision-making problems where traditional heuristics and metaheuristics struggle to generalize. Common applications of hyper-heuristics include:

• Combinatorial optimization: Hyper-heuristics are widely used for general combinatorial optimization problems like the traveling salesman problem (TSP) [39], the knapsack problem [40], and the quadratic assignment problem (QAP) [41].
• Scheduling and timetabling: Hyper-heuristics have been extensively applied to problems like job shop scheduling [42], flow shop problem [43], examination timetabling [44], course timetabling [45], and nurse rostering [46].
• Logistics: Applications include vehicle routing problems [47], cutting stock problems [48], and bin packing [49].
• Data science and machine learning: Hyper-heuristics have been developed for tasks such as feature selection [50], clustering [51], and text classification [52].
• Other areas: Beyond these traditional areas, their application has been extended to other fields like financial forecasting [53], portfolio optimization [54], DNA sequencing [55], protein structure prediction [56], and failure prediction in industry [57].

Most of the applications mentioned above (such as scheduling, routing, and timetabling) are strictly discrete combinatorial problems. In contrast, power system optimization deals primarily with continuous search spaces—often mixed with discrete decision variables—described by nonlinear equations. The next section systematically reviews how these core hyper-heuristic concepts—such as selection mechanics, domain barriers, and learning rules—are applied to practical power grid operations, scheduling tasks, and planning problems.

5. Hyper-Heuristics on Power System Optimization

Power system optimization problems are inherently complex, highly nonlinear, and non-convex, involving multiple operational constraints such as thermal line limits, voltage stability requirements, and generator capacity limits. The integration of renewable energy sources and distributed energy resources further amplifies these constraints, transitioning traditional power networks into highly dynamic smart grids. Formulating these tasks involves handling continuous decision variables (e.g., generator power outputs and transformer tap ratios) and discrete variables (e.g., UC status variables).

To evaluate the application of hyper-heuristics across power grid challenges, our systematic literature search identified 21 primary studies. Among these, 11 papers focus on three central operational problems: UC, ED, and OPF. Another six studies address specialized tasks within highly specific sub-domains that usually sit outside macro-level generation and transmission flows. The remaining four studies investigate wind farm layout problems, which deal with the physical and geographic design of generation sites rather than power system network design.

The following subsections provide a comprehensive literature synthesis for each of these application areas.

5.1. Power System Operation

5.1.1. Unit Commitment

The UC is a fundamental task in power system management that deals with determining which generating units to turn on (commit), which to turn off (de-commit), and at what output level each committed unit should operate over a given time horizon, in order to minimize the total generating cost while satisfying a number of system constraints. It is separated into two main decisions:

• Commitment decision: This is a binary decision which specifies for each generator at each time step (such as hourly over a 24-h period) whether it should operate or remain offline. This decision is significant since switching units on or off incurs substantial costs.
• Dispatch decision: Once the set of committed units is established for a specific time period, the ED problem is addressed to determine the optimal power output for each active unit. This sub-problem aims to minimize the operating costs of the active units while satisfying the system demand. ED can also be considered a standalone problem and it will be analyzed further in the next subsection.

In UC problems, the main cost comes from fuel consumption. The objective function to be minimized is the total operating cost of the power system over a given time horizon (for example 24 h or one week). This total cost is a sum of several components for all generating units and all time periods [58] as described by the following equation:

$$J={\displaystyle \sum _{t=1}^T}{\displaystyle \sum _{i=1}^{NG}}[u_{i,t}·F_i\left(P_{i,t}\right)+{SU}_{i,t}+{SD}_{i,t}+u_{i,t}·M_i\left(P_{i,t}\right)+u_{i,t}·E_i\left(P_{i,t}\right)]$$

(1)

where:

• $NG$ is the number of generating units.
• $T$ is the planning horizon.
• $u_{i,t}$ is commitment status with binary values 0 (unit is off) and 1 (unit is on)
• $P_{i,t}$ is the power output from generator $i$ at hour $t$.
• $F_i\left(P_{i,t}\right)$ is the production (or fuel) cost which is usually computed by the quadratic expression $a_i·P_{i,t}^2+b_i·P_{i,t}+c_i$ where the $a_i$, $b_i, c_i$ are the fuel coefficients of the $i$-th generating unit.
• ${SU}_{i,t}$ is the start-up cost which corresponds to the fuel consumed when a unit starts up before becoming fully operational. It shows exponential behavior based on how long the unit had been offline. However, it is often approximated by a stepwise linear function.
• ${SD}_{i,t}$ is the shut-down cost which is incurred when a thermal unit is shut down. It is typically represented as a fixed cost.
• $M_i\left(P_{i,t}\right)$ is the maintenance cost. It is the additional operational cost that arises when a thermal unit operates for extended periods. It is typically expressed as a linear function of the unit’s power output. For simplicity, maintenance costs are often embedded within the total production cost.
• $E_i\left(P_{i,t}\right)$ is the financial penalty associated with pollutant emissions, and it follows a quadratic curve multiplied by the emission tax rate.

Being a mixed-integer nonlinear optimization problem [59], UC involves both binary (discrete) and continuous decision variables. For each generator and each time period, the binary variable is set to “1” if the unit is committed (on) and “0” if it is deactivated (off). The continuous variables represent the power output of each committed unit at each time period.

The UC problem is subject to several technical and system-level constraints [58]:

• Load demand constraint: The total power generated by all committed units must meet the forecasted electricity demand for every time period. It is represented by the following linear equation with continuous variables:

$${\displaystyle \sum _{i=1}^{NG}}{P}_{i,t}=D_t+L_t$$

(2)

where $D_t$ is system demand at hour $t$ and $L_t$ are the transmission losses.

• Generation capacity limits: The power output of each generator must be within its specified minimum and maximum operating limits. Linear inequalities are used with discrete and continuous variables:
• Reserve constraint: The system must provide extra available generation capacity reserved for unexpected events and sudden changes. It is a linear inequality involving both continuous and binary variables.
• Ramp rate limits: The change in power output of a thermal unit between consecutive time periods does not exceed its technical ramp-up or ramp-down limits. These constraints are linear and involve continuous variables.
• Minimum up/down time constraints: A generating unit must remain on or off for at least its specified minimum up or down time. These constraints are expressed using linear equations with binary variables.

In [60] Berberoğlu and Uyar propose a hyper-heuristic approach for four different power systems. The first three are benchmark test systems containing 10, 20, and 40 generators, each with 24-h time horizons. The fourth system is a real-world Turkish interconnected power system consisting of eight generators and an 8-h time horizon. The proposed method is applied to determine the binary decision variables that specify whether each generator is on or off. The power generation outputs for ED are evaluated using the lambda-iteration method.

The random permutation descent (RPD) method is applied (simply referred as “permutation descent”) as the selection mechanism. Within this method, a random permutation of low-level heuristics is created. At each iteration, the corresponding heuristic in the permutation is applied to the candidate solution. If a heuristic shows an improvement, it continues to be applied iteratively until no further enhancement can be obtained. The acceptance criterion for a new solution is limited to improving moves, meaning a candidate solution is accepted only if it is better than the current solution. Seven operators are used as low-level heuristics, which are separated into two categories: mutation operators and hill-climbers. The proposed hyper-heuristic method achieves strong results, outperforming all comparison methods (such as LR, GA, and DE variants) in all systems except for the small 10-unit system, where it is outperformed only slightly.

Berberoğlu and Uyar [61] also present a comparison of selection hyper-heuristics for the short-term electrical power generation scheduling (SEPGS) problem, an alternate name for UC. Six different selection mechanisms—simple random (SR), random descent (RD), random permutation (RP), RPD, greedy (GR), and choice function (CF)—are combined with four acceptance criteria: accept all moves (AM), accept only improving (OI) moves, accept improving and equal (IE) moves (also referred to as “improve or equal”), and great deluge (GD). Hence, 24 combinations are tested, corresponding to 24 distinct hyper-heuristics. Each one employs the same seven low-level heuristics as those used in [60]. These combinations are tested on two benchmark problems: one with 10 units and another with 20 units, both running over a 24-h time horizon. The study finds that the combination of RPD and accepting OI moves (RPD-OI) achieves the best results in both problem instances.

In [62], hyper-heuristic is used as a component of a framework rather than as a standalone approach. The framework, called HH-EDA2, combines Estimation of Distribution Algorithms (EDAs) and selection hyper-heuristics. The system used for the UC problem is a real-world instance that models the Turkish interconnected power network. It consists of eight generators over an 8-h time horizon. The UC problem is treated as a dynamic environment problem, where each hour represents a new environment. The solution approach is divided into two main phases:

• Offline learning phase: Eight training environments are created by adding random noise to the demand and reserve data from the system dataset. Then, a population-based incremental learning (PBIL) algorithm runs on the environments to learn a set of optimal probability vectors, which are stored for later use.
• Online learning phase: A dual-population approach is used. The first sub-population is sampled using the PBIL approach, while the second sub-population is sampled using a probability vector selected by a selection hyper-heuristic from the list of probability vectors learned offline. The following selection methods are tested: SR, RD, RP, RPD, reinforcement learning (RL), and ant-based selection (AbS). The hyper-heuristic does not employ a move acceptance criterion, which is equivalent to an accept AM strategy.

The goal is to find the optimal turn-on and turn-off schedule for power generation units to minimize total costs over the given time horizon. The experimental results on the UC problem show that HH-EDA2 outperforms several other approaches, including a previously known best method from the literature. The best overall results are obtained by the RP method.

5.1.2. Economic Dispatch

ED is one of the most widely studied optimization problems in the literature, aiming to determine the real power output of each generating unit (the decision variables) so that the total generation cost is minimized while satisfying a number of constraints. It was presented as a part of the UC problem, but it can also be treated as an independent task.

The individual generator cost functions are generally modeled as quadratic functions [63] that provide a good balance between modeling accuracy and computational simplicity. The objective function is typically a sum of the individual cost functions of all online generators:

$$F\left(P_i\right)={\displaystyle \sum _{i=1}^{NG}}(a_i·P_i^2+b_i·P_i+c_i)$$

(3)

where:

• $F\left(P_i\right)$ is the total generating cost.
• $P_i$ is the active power output of the $i$-th generator.
• $a_i$, $b_i$, $c_i$ are the fuel cost coefficients.

In more accurate and realistic models of the ED, the cost functions can be represented by more complex mathematical functions, such as expressions that include sinusoidal terms [63] or cubic polynomials [64].

The basic constraints of the ED problem are the following [65]:

• Power balance constraint: The total power generated by all units must be equal to the total power demand plus the total power losses in the transmission network. This is the single equality constraint and the most crucial one, as any imbalance can cause instability or a blackout. It is described by the following equation:

  $${\displaystyle \sum _{i=1}^{NG}}{P}_i=P_D+P_L$$

  (4)

  where $P_D$ is the total system load and $P_L$ are the total transmission losses.
• Generation capacity limits: Each generator must operate within its minimum and maximum power output limits. This can be expressed mathematically as:

  $$P_i^{\mathrm{m}\mathrm{i}\mathrm{n}}\le P_i\le P_i^{\mathrm{m}\mathrm{a}\mathrm{x}}$$

  (5)

Other notable constraints are:

• Ramp rate limits: The change in power output between two consecutive dispatch intervals must not exceed specified limits for each generator.
• Prohibited operating zones: Certain generators are not allowed to operate within specific ranges of power output due to technical or operational restrictions.
• Spinning reserve requirements: Connected generators are required to hold a portion of their available capacity unutilized as backup to ensure extra power is immediately available during sudden generator outages or load spikes.

Our review indicates that hyper-heuristic methodologies have been successfully deployed across three distinct formulations of the ED problem:

• Multi-objective environmental/economic dispatch: The earliest attempt reported in the literature where a hyper-heuristic is applied to solve a variation in ED appears in [66], namely the environmental/economic dispatch (EED). Unlike the single-objective ED problem, which typically focuses solely on minimizing total fuel cost, the EED problem is formulated as a nonlinear constrained multi-objective optimization model. It addresses the simultaneous minimization of two conflicting objectives: total fuel cost and pollutant emissions, thereby requiring a trade-off between economic and environmental performance. In the paper, the conventional polynomial fuel cost function is augmented with a sinusoidal term to account for the valve-point effect, as shown below:

$$F\left(P_i\right)={\displaystyle \sum _{i=1}^{NG}}[a_i·P_i^2+b_i·P_i+c_i+\left|e_i·\sin \left(f_i·\left(P_i^{\mathrm{m}\mathrm{i}\mathrm{n}}-P_i\right)\right)\right|]$$

(6)

where $e_i$, $f_i$ are fuel cost coefficients. The environmental objective considers the combined effects of sulfur dioxide (SO₂) and nitrogen oxides (NO_x) emissions. Total system emissions $E\left(P_i\right)$ across $n$ generating units are calculated using a combination of polynomial and exponential terms:

$$E\left(P_i\right)={\displaystyle \sum _{i=1}^{NG}}{(\alpha }_i·P_i^2+{\beta }_i·P_i+{\gamma }_i+{\eta }_i·\mathrm{e}\mathrm{x}\mathrm{p}({\lambda }_i·P_i))$$

(7)

where ${\alpha }_i$, ${\beta }_i$, ${\gamma }_i$, ${\eta }_i$, ${\lambda }_i$ are the emission cost coefficients for the $i$-th unit. The problem is subject to the two primary constraints described earlier.

The authors propose a high-level strategy named MOEA/D-HH_SW (MOEA/D-Hyper-Heuristic with Sliding Window), which extends the MOEA/D (Multi-objective Evolutionary Algorithm based on Decomposition) framework. MOEA/D applies a decomposition approach that transforms a multi-objective optimization problem into a number of single-objective sub-problems, each solved by an individual in a collaborative manner. MOEA/D-HH_SW combines this framework with an adaptive choice function enhanced with a sliding window to select among five DE operators during the search process. The proposed algorithm is tested on three systems with 6, 10, and 40 generating units. Results show that MOEA/D-HH_SW produces competitive and reliable solutions, outperforming traditional and state-of-the-art algorithms, especially in the largest test case with 40 units.

2.

Large-scale economic load dispatch: A hyper-heuristic for handling large-scale economic load dispatch (ELD) models (an alternative name for ED) is presented in [67]. Large-scale ELD models are characterized by a higher number of generating units (typically more than 40) and involve more complex operational constraints, making them considerably more challenging to solve than lower-dimensional ELD problems. In this study, in addition to the two basic constraints, ramp rate limits and prohibited operating zones are also taken into account. The proposed method, called hyper-heuristic large-scale global optimization (HH-LSGO), is based on the island model concept. It operates by managing five basic LSGO heuristics—random dynamic grouping, differential grouping, delta grouping, EDA-GA, and adaptive variable-size random grouping (AVS-RG)—which are responsible for problem decomposition. Each of these heuristics is paired with its own evolutionary algorithm on a separate “island”. HH-LSGO performs an online selection process where the performance of each “island” is evaluated. The hyper-heuristic then increases the population size of the most efficient islands and reduces the size of the less efficient ones, synthesizing an effective optimization algorithm during the problem-solving process. The approach is applied to a real-world ELD problem with 140 generators. HH-LSGO does not improve upon GA with multi-parent crossover (GA-MPC) and self-adaptive multi-operator DE (SAMODE) when looking strictly at the absolute lowest-cost schedule found in any single run. Proposed method is slightly outperformed by the competitors, but its strength lies in providing a much more stable and better average performance.

3.

Dynamic economic and environmental load dispatch (DEED): In [68], the authors propose a selection hyper-heuristic algorithm (SHHA) for solving the dynamic economic and environmental load dispatch (DEED) problem, which extends the static EED by considering multiple time periods. The DEED model in the study includes not only conventional thermal generators but also wind generators, photovoltaic generators, and energy storage. Beyond the two basic constraints, the model incorporates further operational constraints, including ramp rate limits, spinning reserve requirements, and energy storage state of charge (SOC) bounds. The low-level structure of the proposed method consists of three mutation operators—one derived from a genetic algorithm and two derived from differential evolution. The high-level structure manages the low-level operators by evaluating their performance and dynamically adjusting the probability of choosing each one. This is achieved using an evaluation strategy based on a sliding time window and a selection mechanism that employs the roulette method. The proposed SHHA is applied to four different power systems. The first two systems are classical models with 5 and 10 thermal units. The third system is a more complex IEEE 57-bus system that incorporates 7 thermal units, 3 wind generators, and 2 photovoltaic generators, while the fourth system adds energy storage to the components of the third case. The results show that the proposed selection hyper-heuristic is effective, outperforming other methods while also demonstrating fast convergence and greater robustness. In terms of computational execution speed, the proposed SHHA outpaces all other compared multi-objective methods and is slower than only two simpler, single-objective algorithms.

5.1.3. Optimal Power Flow

OPF is another fundamental issue in power system operation that seeks to determine the optimal operating point of a power system by finding the best settings for controllable variables (such as active and reactive power outputs of generators, bus voltages, transformer tap ratios, and shunt devices). The goal is to optimize specific objective functions while satisfying all power flow equations and operating limits. The most common objective functions to be minimized are the following [69]:

• Total generation cost: Similar to the previous problems, this is the most common objective. It typically involves a sum of the quadratic cost functions of all generators, but more complex functions can also be used.
• Total power loss: It is the sum of the power losses of all transmission lines, and it is formulated as a nonlinear function of the voltage magnitudes and angles, which are the state variables of the power system:

  $$P_{loss}=\sum _{\begin{array}{c}i,j\in NB\\ i<j\end{array}}{g}_{ij}·[V_i^2+V_j^2-2·V_i·V_j·\mathrm{c}\mathrm{o}\mathrm{s}({\theta }_i-{\theta }_j\left)\right]$$

  (8)

  where $NB$ is the number of buses; $g_{ij}$ is the conductance of the line connecting bus $i$ and $j$; $V_i$ and $V_j$ are the voltage magnitudes at bus $i$ and $j$; ${\theta }_i$ and ${\theta }_j$ are the voltage angles at buses $i$ and $j$.
• Total voltage deviation: It attempts to keep bus voltages as close as possible to their nominal values, which improves system stability and quality. It is formulated as the sum of either the absolute or the squared difference between the actual voltage magnitudes at each bus and their reference values:

  $$\begin{gathered} TVD={\displaystyle \sum _{i=1}^{NB}}\left|V_i-V_{\mathrm{r}\mathrm{e}\mathrm{f}}\right| \\ \mathrm{or} \\ TVD={\displaystyle \sum _{i=1}^{NB}}{\left(V_i-V_{\mathrm{r}\mathrm{e}\mathrm{f}}\right)}^2 \end{gathered}$$

  (9)

  where $V_{\mathrm{r}\mathrm{e}\mathrm{f}}$ is the target nominal voltage magnitude (usually set to 1.0 per unit).
• Pollutant emissions: It is the total amount of pollutants released from power generators. Similar to the cost function, the most common form of the emission objective function is a sum of quadratic functions, but more complex expressions may be employed.

OPF includes a set of variables which are divided into two main categories: control variables and state variables. Control variables are the decision variables, the inputs that can be adjusted to optimize the system. Examples of control variables are generator active power outputs, generator bus voltage magnitudes, and transformer tap settings. State variables are the outputs that describe the system’s operating condition, which are dependent on the control variables. Examples of state variables are generator reactive power outputs and load bus voltage magnitudes [69].

The main constraints of OPF are the following:

• Active and reactive power balance: At each bus in a power system, the difference between the active power generation and the active power load must equal the sum of active power flows to neighboring buses, as determined by the network power flow equations. These equations are nonlinear and depend on the voltage magnitudes and phase angles at the buses. The reactive power balance follows the same principle as active power balance. Both are expressed as follows:

  $$\begin{gathered} P_i^G-P_i^D=\sum _{j=1}^{NB}\left|V_i\right|·\left|V_j\right|·\left[G_{ij}·\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_i-{\theta }_j\right)+B_{ij}·\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_i-{\theta }_j\right)\right] \\ {Q}_i^G-Q_i^D=\sum _{j=1}^{NB}\left|V_i\right|·\left|V_j\right|·\left[G_{ij}·\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_i-{\theta }_j\right)-B_{ij}·\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_i-{\theta }_j\right)\right] \end{gathered}$$

  (10)

  where $P_i^G$, $Q_i^G$ are the active and reactive power generation respectively at bus $i$; $P_i^D$, $Q_i^D$ are the active and reactive load demand at bus $i$; $G_{ij}$ and $B_{ij}$ are the conductance and susceptance between buses $i$ and $j$.
• Generator active and reactive power limits: Each generator’s active and reactive power outputs must be within their minimum and maximum limits.
• Bus voltage magnitude limits: The voltage magnitude at every bus must be kept within an accepted range to ensure power quality and system stability.
• Transformer tap ratio limits: Tap changers on transformers can only be adjusted within their physical limits.
• Transmission line limits: The power flow on each transmission line must not exceed its thermal capacity to prevent overheating and potential damage. This is often expressed as an upper limit on the apparent power flowing through the line.
• Shunt compensator limits: The reactive power injected or absorbed by devices like capacitor banks and reactors must remain within the specified limits.

A specialized version of OPF is optimal reactive power dispatch (ORPD). Unlike OPF, which focuses on cost minimization, the main goal of ORPD is to manage the flow of reactive power to ensure system efficiency, security, and stability. The key difference is that in ORPD, the active power outputs of generators are held constant [70]. The most commonly considered objective functions in ORPD problems are total active power loss, voltage deviation, and voltage stability index [71].

Kanagasabai [72] presents a method based on Q-learning and a hyper-heuristic (QH) for the power loss reduction problem, which essentially involves the minimization of real power losses, voltage deviation, and the maximum L-index ($L_{\mathrm{m}\mathrm{a}\mathrm{x}}$) in ORPD. The L-index is a mathematical indicator calculated for each individual load bus to measure the system’s power permanency. Because a higher L-index indicates that a bus is more vulnerable and has worse stability, the third objective function in the paper is explicitly designed to minimize this maximum L-index. The QH method operates by choosing one of the following high-level selection methods at each iteration:

• Q-learning (QL): Utilizes a “Ql-bench” RL schema to select low-level algorithms based on Q-values updated dynamically with rewards or penalties from historical search performance.
• Hyper-heuristic (H): Updates the selection probabilities of the low-level algorithms based on their recent search success, measured by an “enhancement” metric that tracks solution improvements, and applies a cumulative probabilistic selector to choose the active operator.
• Random: Selects a low-level heuristic randomly.
• Sequential (seq): Selects the low-level heuristic in a predefined, fixed sequence.

Each of these methods then selects one of the following low-level heuristics to run: ABC, mobulidae optimization algorithm (MOA), enhanced salp swarm (ESS) algorithm, or the orcinus orca optimization (OOO) algorithm. The paper does not clarify how each high-level selection method is decided at every iteration. The proposed QH method is tested on the IEEE 30-bus system and provides superior performance compared to the individual low-level algorithms and a variety of other methods. The integrated QH method requires the most computational time in both cases, but it is only 7.39% to 17.76% higher than its standalone low-level components.

Sulaiman and Mustaffa [73] present a selection hyper-heuristic approach to minimize active power losses in the optimal OPF problem incorporating renewable energy sources. The framework utilizes a high-level selection and acceptance mechanism called exponential Monte Carlo with counter (EMCQ) to manage three low-level metaheuristics: GWO, barnacles mating optimizer (BMO), and whale optimization algorithm (WOA). At each iteration, if a selected low-level heuristic generates a candidate solution that is better than the current parent solution in the population, EMCQ automatically accepts it and maintains the active heuristic for the next step. If a non-improving solution is generated, EMCQ applies a dynamic probability density to decide whether to accept it. This threshold-based acceptance serves as a key exploratory feature, allowing the algorithm to temporarily accept worse solutions and escape local sub-optimal regions. If the non-improving move is rejected, a new low-level heuristic is selected completely at random. This methodology is evaluated on a modified IEEE 57-bus system comprising four thermal generators, one wind generator, one individual solar generator, and one combined solar-small hydro generator. Experimental results demonstrate that the proposed hyper-heuristic method consistently outperforms all three standalone low-level metaheuristics in terms of minimum power loss, average convergence, and search stability.

The same authors present a more advanced study of hyper-heuristic methodologies to solve OPF with three flexible alternating current transmission system (FACTS) devices and wind power integration [74]. The study aims to minimize two objectives: total transmission loss and total cost of power generation. The paper proposes two distinct hyper-heuristic approaches. One approach utilizes EMCQ, which was presented earlier, as its high-level mechanism. The second approach uses randomly select-OI, a simpler mechanism which selects a low-level heuristic at random and accepts the new solution if it results in an improvement. Both approaches select from the same set of low-level heuristics that consists of four metaheuristics: moth-flame optimizer (MFO), BMO, teaching-learning-based optimization (TLBO), and gradient-based optimizer (GBO). The proposed hyper-heuristics are applied to a modified IEEE 30-bus system. Both strategies achieve better solutions for both objectives compared to the individual metaheuristics. Specifically, the EMCQ hyper-heuristic achieves the best solution for power losses, while the OI-based strategy achieves the lowest generation cost.

In [75], Cong and Zhao propose a hyper-heuristic approach to solve the coordinated active-reactive power optimization problem, modeled as a multi-objective OPF variant. Two objective functions are minimized: total network transmission losses and node voltage offsets. Thompson sampling, a strategy that utilizes probabilistic learning and Bayesian estimation, is employed as the high-level strategy to select the most appropriate low-level heuristic at each iteration. The low-level heuristics are categorized by the type of variable they handle:

• For continuous variables (generator terminal voltage), the heuristics are a bird foraging search, random variation, and an animal-inspired odor concentration strategy.
• For discrete variables (taps of the on-load regulator transformer and the reactive power compensation device), the heuristics are a single-point crossover, a two-point crossover, and a partial fragment inversion strategy (noted as “reversal of footage” in the paper’s flowchart).

Three additional procedures are integrated to enhance search efficiency. First, an adaptive variation probability balances exploration and exploitation by dynamically scaling the variation rate based on the population’s current fitness range. Second, a Gaussian distribution random perturbation is applied to perform local search when a trigger condition is met. Finally, a Cauchy variation (referred to in the text as “Kersey variation”) is added to assign the population to new, distant regions to perform global exploration and prevent search stagnation. The algorithm is validated on an IEEE 30-bus system. The results show significant improvement in network losses and voltage levels compared to the system’s state before the optimization process. However, the paper does not provide a comparison against other algorithms.

A hyper-heuristic framework called Hyper-FDB-INFO was developed by Altun et al. [76] to solve the OPF problem in a power system that includes wind energy and FACTS devices. The authors improved the weighted mean of vectors (INFO) algorithm by incorporating the fitness-distance balance (FDB) method to enhance its exploration and exploitation capabilities, creating the FDB-INFO algorithm. In the proposed approach, linear population reduction success history-based adaptive differential evolution (LSHADE) is used as the high-level strategy, while chaotic maps (CMs), opposition-based learning (OBL) methods, and the population ratio are utilized in the low-level structure. LSHADE generates a population of individuals, where each individual represents a unique combination of these three low-level components. For each of these combinations, the INFO or FDB-INFO algorithm is run for a limited training stage of 50 iterations, and the best fitness value is assigned to the corresponding LSHADE individual. LSHADE then uses these fitness values to evolve the population, aiming to find the optimal combination. Upon completion of this training stage, the best combination is identified and used to generate the optimal initial population for the final INFO/FDB-INFO run. This marks the test stage where the algorithm runs for its maximum number of iterations.

The study evaluates the performance of the proposed framework against INFO, FDB-INFO, Hyper-INFO, and SHADE-SF (success history-based adaptation technique of DE with superiority of feasible solutions) on a modified IEEE 30-bus test system containing six FACTS devices and two wind generators. Four distinct case studies are investigated: the first minimizes total generation cost; the second minimizes active power transmission loss; the third minimizes both cost and loss via a gross cost equation; and the fourth minimizes the same gross cost equation while considering loading uncertainties modeled via a normal probability density function. The proposed Hyper-FDB-INFO algorithm consistently outperforms the competing methods across all evaluated cases, confirming its robustness and effectiveness in solving complex, real-world power system optimization problems.

5.2. Specialized Sub-Domains

5.2.1. Discharge Scheduling of Energy Storage Systems

Energy storage optimization is another vital issue in modern electricity distribution. As power systems integrate higher levels of renewable generation, such as wind and solar, energy storage systems play a critical role in power supply management [77]. In periods of excess generation, these systems allow the energy to be stored and re-dispatched when needed, reducing transmission losses and fossil fuel consumption.

Our search of the existing literature identified only a single study that applies a hyper-heuristic framework to solve an optimization problem involving energy storage systems. This unique research is ref. [78] and focuses on the discharge scheduling of energy storage systems problem (DSESP) to minimize utility grid consumption in MicroGrids using an evolutionary algorithm as hyper-heuristic.

In the context of this optimization problem, a MicroGrid (MG) is defined as a localized power system consisting of distributed energy resources, controllable loads, and energy storage system, connected to the centralized utility grid through a point of common coupling (PCC). While the utility grid provides a reliable backup supply, the primary objective of the MG is to manage its internal resources to minimize dependence on the utility grid.

The DSESP focuses on when and how stored energy should be released to achieve a specific goal within the MG. Given a set of load profiles and renewable generations over a specific time horizon, the problem aims to find the optimal scheduling vector that determines how much power the energy storage system should discharge at each period in order to minimize the total energy consumption from the utility grid. The goal is to find $D=(D_1,D_2,\dots ,D_Z)$ over periods $z$ such that the following objective function is minimized:

$$g\left(D\right)={\displaystyle \sum _{i=1}^Z}(L_i-G_i-D_i+C_i+O_i)$$

(11)

In this equation:

• $g\left(D\right)$ is the total energy consumption from the utility grid,
• $L_i$ is the load demand at time $i$,
• $G_i$ is the renewable energy generation at time $i$,
• $D_i$ is the power discharged by the battery to supply to the MG during period $i$. Battery can be discharged in a period $k$, if $G_k<L_k$.
• $C_i$ stands for the power used to charge the battery. In a specific period $k$ when $G_k>L_k$ battery is charged with a power $C_k=G_k-L_k$.
• $O_i$ accounts for the excess generation that occurs when the battery has almost reached its full capacity.

The problem is subjected to the following constraints:

• Interconnection assumption: The study assumes that power cannot be exported back to the utility grid.
• Grid dependency assumption: It is also assumed that ${\displaystyle {\sum }_{i=1}^Z}{L}_i>{\displaystyle {\sum }_{i=1}^Z}{G}_i$ which means that total energy demand over the entire period is greater than the total renewable energy produced. This constraint ensures that the system requires energy from the utility grid.
• Discharge limit: The maximum power that can be extracted from the battery during period $j$ is given by $LMG{R}_j=L_j-G_j$. It should be noted that discharging at the full $LMG{R}_j$ level is not the optimal energy management strategy.
• Energy storage limit: The battery is not charged once it reaches its SOC and is near its full capacity.

The proposed hyper-heuristic approach operates across two distinct levels: a global optimization technique (the high-level algorithm) that evolves a sequence of small heuristics (the low-level heuristics pool). The local search level consists of a pool of 26 low-level heuristics ($H_0$ to $H_{25}$) and each one of them represents a different percentage of $LMGR$. The discharge values are defined as follows:

• $H_0$: Discharges $0\% LMGR$ (no discharge)
• $H_1$ to $H_{25}$: Represent decreasing discharge percentages in 4% increments ($H_2=96\% LMGR$, $H_3=92\% LMGR$, …, $H_{25}=4\% LMGR$).

The high-level global search procedure is handled by an evolutionary algorithm. In this method, each solution is encoded as a vector of integer numbers from 0 to 25 and represents a sequential application of the low-level heuristics over the entire time horizon. For an hourly analysis over a one-week horizon, the encoding length is $Z=168$. The high-level algorithm applies four operators. First, a classical two-point crossover operator is used that forms two new offspring vectors. Then two mutation operators are applied to the offspring, each with a probability of 1%. One is a classical mutation which randomly selects elements within the offspring and substitutes them with new values while the secondary mutation mechanism operates by swapping elements within a vector. The fourth operator is a tournament selection mechanism that selects the parents for the next generation.

To evaluate the effectiveness of the hyper-heuristic, the study utilizes two real-world MG scenarios in Alcalá de Henares, Spain, analyzed over a one-week period ($Z=168$ h). Both experiments employ Vented Lead-Acid (VLA) batteries as the energy storage system. In scenario 1, the MG consists of a photovoltaic generator with 75 panels (245 Wp each) for a total installed power of 18,385 Wp, connected to the grid through a 15-kW inverter to support a load consisting of a 25-kW heating pump system for the garden’s offices and teaching buildings. Scenario 2 involves a 10-kW micro-wind turbine (WT) and a load representing standard household electrical consumption. The study evaluates system performance across three different conditions: energy consumption without an energy storage system, consumption with a non-optimal discharge strategy where the battery always discharges at 100% LMGR and consumption using the optimal scheduling of the proposed hyper-heuristic algorithm.

In Scenario 1, the non-optimal strategy reduces grid consumption by 37.47% compared to the condition without a battery. The hyper-heuristic further improves this performance, reaching a total reduction of 42.48%, which represents a 5.01% additional improvement over the non-optimal case. For Scenario 2, the non-optimal battery condition achieved a 66.37% reduction, while the optimal discharge schedule increased these savings to 72.10%, yielding a 5.73% additional reduction in consumption from the utility grid.

5.2.2. Demand Side Management of Plug-In Electric Vehicles

Demand side management (DSM) of plug-in electric vehicles (PEVs) is another challenge in modern power grids, focusing on the coordination of vehicle charging to assure grid stability and efficiency. As transport electrification expands, unmanaged PEV charging can overload grid infrastructure, leading to significant issues for the electric power system. By treating PEVs as flexible loads or virtual power plants, DSM strategies allow energy consumption to be shifted to off-peak periods or utilized as ancillary support through vehicle-to-grid (V2G) technology [79].

In the literature, there is one study that adopts DSM of PEVs with hyper-heuristics. Ref. [80] proposes a hybrid framework combining a decentralized multi-agent control system with a centralized hyper-heuristic optimizer for vehicles connected to a smart grid. Although the paper refers to a combined total cost and emission (TCE) objective equation, the problem is formally modeled as a multi-objective optimization (minimization) problem with two distinct objective functions to generate a non-dominated Pareto front:

• Total cost ($f_1$): The raw electricity generation costs and the costs associated with implementing the DSM program (where saving a unit of electricity is mathematically treated as equivalent to generation):

  $$f_1=\sum _i\sum _j{c}_{ij}·x_{ij}+\sum _j{SDSM}_j·{CDSM}_j$$

  (12)
• Total greenhouse gas emissions ($f_2$): Emissions from all active generator options:

  $$f_2=\sum _i\sum _j{x}_{ij}·{GHG}_i$$

  (13)

  where $i$ is the set of generators within the smart grid, $j$ is the set of PEVs, $c_{ij}$ is the electricity generation cost of resource $i$ when used by the $j$-th PEV, $x_{ij}$ is the optimal amount of energy from the $i$-th resource option for $j$-th PEV, $SDS{M}_j$ is the optimal amount of energy saved for the $j$-th PEV through the implemented DSM program, $CDS{M}_j$ is the electricity saving cost using the implemented DSM program for $j$-th PEV and $GH{G}_i$ is the greenhouse gas emissions of the $i$-th generator associated with the grid.

To ensure safe grid operation and meet user needs, the optimization model is subjected to the following constraints:

• Demand satisfaction: The sum of physical energy provided by all generators plus the shifted energy savings must meet the vehicle’s total demand $D_j$ as described below.

  $$\sum _i{x}_{ij}+SDS{M}_j\ge D_j$$

  (14)
• Generation source capacity: To guarantee physical feasibility, the total electricity drawn from any individual generation resource $i$ by all connected vehicles $j$ cannot exceed its physical supply capacity ($S_i)$:

  $$\sum _j{\displaystyle \frac{x_{ij}}{{\eta }_{ij}}}\le S_i$$

  (15)

  where ${\eta }_{ij}$ is the conversion efficiency associated with the $i$-th resource option for the $j$-th vehicle
• Saving limits: The optimal energy savings for each vehicle must fall within a minimum required target saving $SOB{J}_j$ and a realistic maximum limit ($P_j):$

  $$SOB{J}_j\le SDS{M}_j\le P_j$$

  (16)

The decision variables $x_{ij}$ and $SDS{M}_j$ must take non-negative values, expressed as $x_{ij}\ge 0$ and $SDS{M}_j\ge 0$.

The authors in [80] examine the problem of unmitigated PEV charging. In the absence of proper coordination, simultaneous PEV charging and discharging during peak periods can cause severe branch congestion, unpredictable system peaks, unacceptable voltage deviations, increase in system losses, and poor power quality. To manage this, the paper proposes a hybrid framework that integrates the following two components:

• A decentralized multi-agent system: This component is controlled by a market-based agent architecture where decisions are made through competitive bidding over 48 half-hourly trading periods. The architecture consists of several agents. PEV (consumer) agents optimize energy consumption for individual vehicles to maximize savings and user comfort. Source agents provide availability and pricing pairs for renewable energy resources. These interactions are coordinated by the auctioneer agent, which aggregates supply and demand data to determine a global equilibrium price. Finally, the optimizing agent executes the hyper-heuristic algorithm and communicates the resulting schedules to the various PEV agents. The optimization process occurs within the DSM Window, the idle timeframe between a vehicle completing its charge and its disconnection time. Within this window, the optimizing agent iteratively shifts charging blocks in 30-min intervals to identify the best charging schedule.
• A centralized multi-objective hyper-heuristic optimizer: The core of this component is a tabu search-based roulette wheel multi-objective hyper-heuristic. High-level strategy consists of a choice function that adaptively ranks the performance of three low-level heuristics which are not described in the paper. If a selected heuristic improves the objective value its rank is increased. If it fails to improve the solution, the rank is decreased. A tabu list acts as short-term memory that prevents heuristics with low performance from being chosen very soon. Since the problem is multi-objective, the performance of each heuristic is evaluated with respect to individual objectives (cost and emissions). At each iteration, a roulette wheel selection is used to choose which individual objective to focus on. This selection depends on how far an objective is from its ideal optimal value. Finally, the framework employs the AM acceptance strategy.

The proposed framework is evaluated on a test system consisting of 2516 domestic consumers, 296 small consumption firms, 150 medium consumption firms, and 4 large consumption firms. The PEV considered for simulation is a Chevy Volt with an 11-kWh battery capacity and the number of vehicles is set to 1000. The energy mix for the grid includes photovoltaic, wind, hydro, and geothermal sources. The results demonstrate that while uncontrolled charging increases the maximum peak load of the grid by nearly 6%, the proposed multi-objective decentralized approach achieves excellent improvements by shifting charging cycles into identified DSM windows. While the percentage of improvement is not summarized, the paper reports specific numerical optimal savings achieved through DSM for each sector. Furthermore, the decentralized architecture ensures that execution times remain nearly constant as the number of PEVs increases.

5.2.3. Reliability Optimization in Distribution Systems

Reliability optimization is a critical engineering and management discipline focused on finding the best balance between a system’s performance, its longevity, and the costs required to maintain it. At its core, the field aims to design systems that minimize the frequency and impact of failures.

In [81], the authors propose a hyper-heuristic model to solve the reliability optimization problem in medium voltage power distribution systems (MV PDS). The main task is determining the optimal location of sectioning/protection (s/p) equipment (fuses, disconnectors, and interrupters) to minimize the frequency and duration of power outages for customers. The following five conflicting objectives must be minimized simultaneously:

• System average interruption frequency index (SAIFI): Evaluates how often the average customer loses power per year and is calculated as:

  $$SAIFI={\displaystyle \frac{\sum N_i}{N_T}}$$

  (17)

  where $N_i$ is the number of customers experiencing a supply interruption and $N_T$ is the total number of customers in the system.
• System average interruption duration index (SAIDI): Evaluates the total annual time (in hours per year) that an average customer is without power and is calculated as:

  $$SAIDI={\displaystyle \frac{\sum N_i·T_i}{N_T}}$$

  (18)

  where $T_i$ is the interruption time of failure event $i$.
• Momentary average interruption frequency index (MAIFI): Tracks the average frequency of short-lived or momentary supply flickers and is calculated as:

  $$MAIFI={\displaystyle \frac{\sum N_{mi}}{N_T}}$$

  (19)

  where $N_{mi}$ is number of momentary customer supply interruptions.
• Expected energy not supplied (EENS): Measures the total amount of energy that fails to reach the system’s nodes due to interruptions, calculated as:

  $$EENS={\displaystyle \sum _{k=1}^{n{N}_T}}{p}_k·t_k$$

  (20)

  where $p_k$ is the expected power demanded at node $k$, $t_k$ is the time the node is interrupted and $n{N}_T$ is the total number of system nodes.
• Total annual cost (TAC): Evaluates the annual investment and maintenance cost for the protection hardware, calculated as:

  $$TAC=\sum A{C}_{S/P}+\sum A{C}_{SO\&M}$$

  (21)

  where $\sum A{C}_{S/P}$ is annual cost of sectioning/protection equipment and $\sum A{C}_{SO\&M}$ is the annual cost for operation and maintenance for this equipment.

The following constraints must be satisfied:

$$\begin{gathered} SAIFI\le SAIF{I}_{ALIM} \\ SAIDI\le SAID{I}_{ALIM} \\ MAIFI\le MAIF{I}_{ALIM} \end{gathered}$$

(22)

where $SAIF{I}_{ALIM}$, $SAID{I}_{ALIM}$ and $MAIF{I}_{ALIM}$ are the admissible limits for the respective indices defined by the regulator.

The high-level or reasoning part of the proposed hyper-heuristic is controlled by a back-propagation ANN (BPANN). The network receives 15 inputs based on swarm-intelligence principles (SIP): proximity, quality, diversity, stability, and adaptation. These inputs represent the maximum, average, and minimum fitness levels of the current swarm. The BPANN produces seven binary outputs. Five observe the satisfaction of SIP indices, while the final two form a binary code that selects one of four specific meta-heuristics for the next iteration. The algorithm chooses from a set of four multi-objective fuzzy PSO (FPSO) variants called X-FPSO: FPSO with constriction factor (CF FPSO), evolutionary FPSO (FEPSO), FEPSO with global-individual stochastic topology (FEPSO GIST) and FPSO with Inertial Decay (ID FPSO).

The study formulates the reliability optimization as a multi-objective optimization problem within a fuzzy domain, where each criterion (SAIDI, SAIFI, MAIFI, EENS, and TAC) is transformed into a fuzzy membership function to evaluate satisfaction levels between 0 and 1. To synthesize these competing objectives, the model uses a preference matrix to assign exponential weights, prioritizing reliability factors (like EENS) over economic factors (TAC). These five separate scores are integrated into a single fuzzy set using a specialized exponential Einstein product (EPE) t-norm. The hyper-heuristic explores the combinatorial search space of equipment configurations to identify the solution that maximizes this global satisfaction score. Finally, the model determines an intrinsic cost of failure (ICF) by analyzing the mathematical relationship (the partial derivative) between TAC and the reduction in EENS at that optimal point.

The proposed reasoning-based hyper-heuristic (BRHY) is applied through two distinct scenarios to a real MV PDS in Bariloche, Argentina consisting of a primary transformer station and 16 network sections with their respective sub-stations. In the first scenario, which focused on the optimal location of a single equipment type (disconnectors), the results of the BRHY are coincidental with those of traditional possibilistic dynamic programming (PDP). In the second scenario where three equipment types are taken into account (interrupters, disconnectors, and fuses) and PDP fails, the BRHY model successfully identifies a superior solution that prioritizes interrupters in high-demand sections and fuses in remaining branches. Comparing the two scenarios, the second one requires a higher TAC but achieves better results by reducing EENS and improving customer-based reliability indices like SAIFI, SAIDI, and MAIFI. Finally, this multi-equipment configuration results in a lower ICF, proving that the hyper-heuristic provides a more satisfactory balance between technical performance and economic cost than the single-equipment approach. The reported computational execution time of BRHY is exactly 0.842 min for both scenarios and the paper does not provide or compare the numerical execution time of PDP.

5.2.4. Controller Design

Controller design is a major issue in power systems engineering, aimed at establishing stable control laws that maintain voltage and frequency within strict operating limits. While it can be stated as an optimization problem, identifying a suitable cost function that considers all available control specifications -such as settling time, overshoot, or energy consumption- remains a challenge.

Although ref. [82] is positioned within this domain, it specifically addresses a controller selection and sequencing problem rather than a design task. The authors employ a sequence-based selection (SS) hyper-heuristic to handle power quality issues within microgrids caused by the intermittency and uncertainty of renewable energy sources. Integrating these sources often results in two specific technical failures: total harmonic distortion arising from nonlinear devices, and current unbalance caused by unbalanced loads, both of which deteriorate grid performance and can damage electrical components. While traditional robust controllers like the linear quadratic regulator (LQR) and proportional-integral (PI) are common, they are often too specialized and a controller tuned to minimize harmonic distortion may perform poorly against current unbalance and vice versa [82]. Thus, the optimization problem is not to design a new controller, but rather to identify the optimal switching sequence of existing specialized controllers to handle complex grid perturbations more effectively than a standalone architecture.

The system relies on two base controllers that act as low-level solvers: controller 1 (C1) and controller 2 (C2). C1 fuses a traditional PI controller with an LQR approach, and it is generally more effective at reducing harmonic distortion. C2 integrates a resonant controller (generated directly from PI) with the LQR approach, and it performs better at mitigating current unbalance in the grid. Before the hyper-heuristic is applied, both controllers are tuned using a genetic algorithm to minimize a cost function involving settling time and overshoot. The hyper-heuristic decides which controller is activated at specific points in time and organizes sequences such as $\left[A_1, A_2, \dots , A_f\right]$ where each element represents a specific controller ID. The hyper-heuristic executes this plan by following a schedule of timestamps, switching the control signal from one controller to the other.

However, the authors use handcrafted sequences for the simulation rather than proposing a specific automated high-level method to generate these schedules. To validate this approach, they define four specific sequence-based models (HH1 through HH4) that alternate between two base controllers, C1 and C2, at fixed time intervals. These models are tested across 13 experimental scenarios with different combinations of nonlinear loads and power quality events. The results demonstrate that even with manually designed sequences, the strategy is highly effective and outperforms the standalone controllers in the majority of the experiments.

5.2.5. Power Forecasting

In the modern energy landscape, the stability and efficiency of a power system are strongly linked to the accuracy of its forecasting. Accurate forecasting models serve as the fundamental foundation for critical operational tasks in power systems, including UC, ED and OPF. However, the effectiveness of these predictive models is highly sensitive to their initial parameter configurations.

In the existing literature, the studies [83,84] from the same authors utilize hyper-heuristics for this research topic. The problem addressed in the papers is the parameter determination for electricity consumption forecasting models. These models are mathematical tools used to predict future power demand based on historical data and help utility companies and energy authorities to schedule energy development, protect the environment and improve performance. Both papers identify four specific time-series models:

• Simple exponential smoothing (SES): This model is designed for data without trends or seasonal changes. It assigns higher weights to recent data points and lower weights to older ones, balancing them with a smoothing coefficient $a$. The basic formula of SES is:

  $$F_{t+1}=\alpha ·Y_t+\left(1-\alpha \right)·F_t$$

  (23)

  where $F_{t+1}$ is the forecasted electricity consumption value for time $t+1$, $Y_t$ is the actual observed electricity consumption value at time $t$, $F_t$ is the forecast at time $t$, and $a$ is the smoothing coefficient from 0 to 1 to balance the weights between new and old data.
• Double exponential smoothing (DES): It is suitable for data with trends but without seasonal variations. DES consists of two main components: the level and the trend as described by the following equations:

  $$\begin{gathered} S_t=\alpha ·Y_t+\left(1-a\right)·(S_{t-1}+T_{t-1}) \\ {T}_t=\beta ·{(S}_t-S_{t-1})+\left(1-\beta \right)·T_{t-1} \\ {F}_{t+k}=S_t+k·T_t \end{gathered}$$

  (24)

  where $S_t$ is the smoothed level value at time $t$, $T_t$ is the smoothed trend value at time $t$, $Y_t$ is the actual observed value at time $t$, $F_{t+k}$ is the forecast value at time $t + k$, $\alpha$ and $\beta$ are smoothing coefficients, both ranging from 0 to 1 for level and trend smoothing, respectively.
• Holt-Winters model: It is used for data that contain both trends and seasonal variations and it is formulated as follows:

  $$\begin{gathered} L_t=\alpha ·(Y_t-S_{t-m})+\left(1-a\right)·(L_{t-1}+T_{t-1}) \\ {T}_t=\beta ·(L_t-L_{t-1})+\left(1-\beta \right)·T_{t-1} \\ {S}_t=\gamma ·\left({Y_t-L}_t\right)+\left(1-\gamma \right)·S_{t-m} \\ {F}_{t+k}=L_t+k·T_t+S_{t+k-m} \end{gathered}$$

  (25)

  where $L_t$ is the smoothed level value at time $t$, $T_t$ is the smoothed trend value at time $t$, $S_t$ is the smoothed seasonal value at time $t$, $Y_t$ is the actual observed value at time $t$, $F_{t+k}$ is the forecast value at time $t+k$. $\alpha$, $\beta$ and γ are smoothing coefficients ranging from 0 to 1, and $m$ is the length of the seasonal period.
• Weight moving averaging (WMA): It assigns specific weights to different historical periods, with more recent observations typically given higher weights. This model is mathematically formulated as follows:

  $$WM{A}_t={\displaystyle \frac{{\displaystyle {\sum }_{i=0}^{n-1}}{w}_i·Y_{t-i}}{{\displaystyle {\sum }_{i=0}^{n-1}}{w}_i}}$$

  (26)

  where $WM{A}_t$ is the weighted moving average at time $t$, $Y_{t-i}$ is the actual observed value at time $t - i$, $w_i$ is the weight assigned to the observed value $Y_{t-i}$ and $n$ is the number of periods used in the calculation.

In [84] two additional models are tested:

• Auto regression (AR): It predicts future values of a variable using previous values known as lags. This model is effective when there is a strong correlation between past and presents values. The general form of this model is based on a linear expression as presented below:

  $$Y_t=c+{\displaystyle \sum _{i=1}^p}{\phi }_i·Y_{t-\mathrm{i}}+{ϵ}_t$$

  (27)

  where $p$ is the number of lags incorporated, $c$ is an optional constant term, ${\phi }_i$ are the coefficients of the model and ${ϵ}_t$ represents an error term.
• Long short-term memory (LSTM): It is a specialized architecture within recurrent neural networks (RNNs) specifically designed to process sequential data and overcome the limitations of traditional RNNs. It is highly effective at capturing long-term dependencies, making it valuable for tasks like time series forecasting, language modeling, and sequence prediction. To manage the flow of information, the LSTM uses a system of gates, specifically an input gate, a forget gate and an output gate. This gating mechanism acts as a filter that allows the network to selectively retain important past data and remove irrelevant information, enabling it to learn complex temporal relationships. During its training process, the LSTM continuously updates its internal parameters to improve its accuracy. Compared to previous models, LSTM uses significantly more complex equations which are described in [84]. The equation that dictates how the model updates its weights is:

  $$W=W-\eta ·\nabla L$$

  (28)

  where W represents the weight parameter, $\eta$ is the learning rate which controls how much the weights are adjusted and $\nabla L$ is the gradient of the loss function with respect to that weight parameter.

The smoothing coefficients $\alpha$, $\beta$, $\gamma$, the weights $w_i$, the coefficients ${\phi }_i$ and the learning rate $\eta$ are model parameters that are treated as decision variables during the optimization process. The accuracy of the prediction depends on the values assigned to these variables. If these parameters are set incorrectly, the model’s output will differ significantly from actual electricity consumption.

For the evaluation of prediction accuracy, authors use the mean absolute percentage error (MAPE) which acts as the objective function of the problem. It represents the average percentage difference between the values predicted by the model and the actual observed values, and it is calculated as follows:

$$MAPE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{t=1}^n}\left|{\displaystyle \frac{A_t-F_t}{A_t}}\right|·100\%$$

(29)

where $n$ is the number of observations, $A_t$ denotes the actual value at time t and $F_t$ denotes the forecast value at time $t$.

In ref. [83], the proposed hyper-heuristic is named GA-PB where genetic algorithm serves as the high-level procedure to adaptively adjust the parameters of the low-level heuristics. These low-level heuristics consist of search operators from PSO and BA, which are responsible for iteratively searching for the optimal smoothing coefficients of the forecasting models. Specifically, the velocity and the position update rules from PSO are used as low-level heuristic. From BA, the low-level heuristics comprise the frequency, velocity and position updates, and a local search via random flights.

The numerical experiments conducted in the study utilize electricity consumption data from 2000 to 2007 at Hokkaido, Japan and the United States. Compared to DE, PSO, and BA, the GA-PB algorithm demonstrates superior forecasting accuracy in the DES and Holt-Winters models for both datasets. In the cases of SES and WMA, the algorithm’s performance is equivalent to that of the other optimization methods. Among the time-series models, the Holt-Winters model consistently produces predictions that are closer to the actual data values. AR and LSTM models are not adopted in this study.

In ref. [84], the authors introduce an enhancement to this high-level strategy adding an auto-evolution mechanism, resulting in the AE-GAPB algorithm. In this strategy, the high-level GA dynamically adjusts its own crossover and mutation rates based on two factors: number of iterations and fitness (objective) value. Crossover and mutation rates decrease as the number of iterations increases and objective values improve. The low-level strategies are exactly the same as those stated earlier for [83].

The numerical experiments conducted in the study utilize both electricity consumption and renewable energy generation data from 2000 to 2015 (training on 2000–2011 and testing on 2012–2015) across three regions in Japan: Hokkaido, Kyushu, and Tohoku. Compared to a GA-PB (without auto-evolution), DE, PSO, BA, GWO, and the WOA, the AE-GAPB algorithm consistently demonstrates superior forecasting accuracy in the DES, Holt-Winters, and LSTM models across most datasets. In the cases of the simpler SES, WMA, and AR models, the algorithm’s performance is generally equivalent to that of the other optimization methods. Across the models, predictions for electricity consumption were generally more accurate than those for renewable energy generation, as renewable energy data is heavily influenced by unpredictable environmental factors.

5.3. Wind Farm Layout Optimization

WFLO is a crucial design and planning challenge in renewable energy systems that involves determining the most efficient physical arrangement of wind turbines within a designated area (wake effect). The primary objective is to maximize the total annual energy production while simultaneously reducing the financial and infrastructure costs associated with the farm’s construction. This survey identifies four studies that have successfully applied hyper-heuristics to the WFLO problem. Although these papers share the common aim of improving wind farm performance through high-level search strategies, they show significant diversity in their problem descriptions and objective functions apart from the optimization approaches.

5.3.1. Multi-Objective Wind Farm Layout Optimization

The first study [85] in this section addresses a variant of the WFLO problem with the aim of finding the best locations for a given number of wind turbines within a rectangular wind farm area. The problem is characterized by the wake effect where aerodynamic interactions between turbines reduce the energy production of those located downstream within the given wind farm area. The decision variables are the 2-D Cartesian coordinates for each of the $n$ turbines in the farm: $\left(X,Y\right)=\left[\left(x_1,y_1\right), \left(x_2,y_2\right),\dots , \left(x_n,y_n\right)\right]$, where $\left(x_i, y_i\right)$ is the position of the $i$-th turbine.

The problem can be described as the minimization of three conflicting objectives simultaneously:

• Inverse of energy production ($1/E\left(P\right)$): It is the total expected annual energy production ($E\left(P\right)$) by minimizing its reciprocal. The total energy production is calculated by combining the Park model and a sigmoid power curve to incorporate the efficiency losses caused by the wake effect.
• Minimum spanning tree (MST): It is the total cable length required to connect all turbines to each other, calculated using Prim’s MST algorithm.
• A: It is the used land area of the farm determined by the convex hull of all turbine locations which are computed using Graham’s Algorithm

The model incorporates two constraints:

• Wind farm boundary constraint: No turbine can be placed outside the given wind farm area.
• Minimum spacing constraint: The distance between any two turbines cannot be smaller than eight times the rotor radius.

In this paper three heuristic selection methods are employed combined with three move acceptance methods so nine high-level strategies are examined. Heuristic selection methods are random choice (RC), fixed sequence (FS) (a predefined permutation of low-level heuristics executed in order) and CF that assigns scores to low-level heuristics, based on their recent performance and the time since they were last used. The move acceptance methods include AM where all solutions are accepted (whether they improve the current solution or not), great deluge acceptance with D metric (GDA) where a worse solution is accepted if it is better than a threshold and best acceptance that rates current and new solutions and allows the best ones to survive. The high-level selection methods manage the following three multi-objective evolutionary algorithms (MOEAs) as low-level metaheuristics: non-dominated sorting genetic algorithm-II (NSGA-II), strength pareto evolutionary algorithm-2 (SPEA-2) and indicator-based evolutionary algorithm (IBEA).

The study utilized a real-world scenario to optimize the placement of 30 turbines within a 3 $\times$ 3 km² land area. The nine proposed hyper-heuristic frameworks are compared to the standalone MOEAs: NSGA-II, SPEA2, and IBEA. Performance was evaluated using the following quality indicators: hypervolume, uniform distribution, ratio of non-dominated individuals, and algorithm effort. In most cases, the hyper-heuristics outperform the standalone metaheuristics. In terms of hypervolume, all nine hyper-heuristic schemes outperform NSGA-II and SPEA2, while the RC-GDA hyper-heuristic achieves the best overall hypervolume performance. In addition, the results reveal a correlation between cable length and area usage minimization.

5.3.2. Optimal Wind Turbine Placement

In [86] the authors use the term “optimal wind turbine placement (OWTP)” to describe the problem and treat it as a binary optimization problem. The wind farm is divided into a $d\times d$ grid. The center of each grid square serves as a candidate location where a turbine can either be placed or left empty. A solution is represented as a binary matrix where a “1” indicates that a turbine is placed and a “0” indicates that the grid square is empty.

The paper uses two objective functions to evaluate the quality of solutions: the cost of energy (COE) which must be minimized and the efficiency of production (EOP) which is to be maximized. The COE is defined as the monetary cost per unit of energy produced and it is calculated by dividing the total purchase cost of the turbines by the total produced power. The EOP measures how well the turbine placement reduces the negative impact of the wake effect. It is expressed as a ratio between the power with and the power without the wake effect. The complex equations of COE and EOP are presented in [86].

The authors propose a hyper-heuristic framework that combines an ABC algorithm with simulated annealing. The framework is structured into two layers: a problem-dependent layer containing lower-level heuristics —specifically matrix-based row and column swaps, copies, and flips—and a problem-independent layer consisting of a heuristic selection method and a move acceptance method. Within the ABC algorithm, the heuristic selection method allows employed and onlooker bees to select the most effective low-level heuristics based on past performance, while the move acceptance method employs the Metropolis criterion from SA to decide whether to accept new layouts. In addition, scout bees try to replace the old solution that fails to improve.

The proposed method is evaluated in two test problems. In Problem A, there are 36 wind directions and 3 different wind speeds. Problem B has the same wind conditions but adds a land usage constraint, where certain parts of land cannot be used for turbine installation due to unstable terrain or high costs, such as in swamp areas. The experimental results demonstrate that for Problem A, the proposed method shows lower COE compared to two other standard methods and higher EOP than one of them. For Problem B, the algorithm avoids all restricted grids and places the 27 required turbines along the main (northwestern) wind direction.

5.3.3. Discrete Wind Farm Layout Optimization

A similar discrete approach to WFLO is followed in [87]. In this study, the site is partitioned into a number of cells where an array of Booleans represents the presence or absence of a wind turbine in each cell. The core objective is the minimization of COE, which balances the maximization of energy production with the minimization of layout, investment, and operational costs. The optimization framework evaluates these objectives by coupling the search algorithm with the WindFLO simulation platform. This platform encapsulates the COE evaluation equation alongside physical simulations that model wind speed, aerodynamic wake effects, and turbine power curves. The platform also accounts for site characteristics, turbine features, construction and maintenance costs, and cable and road optimization. Furthermore, the model enforces critical constraints, including safety spacing (requiring a minimum distance between neighboring turbines), topographic and human constraints (such as noise), and obstacle constraints where turbines cannot be placed in designated grid cells. Finally, assuming unidirectional wind, the site is structured as a square grid where the distance between adjacent points in a horizontal or vertical manner is fixed at exactly eight times the turbine radius.

The proposed hyper-heuristic employs a SR selection method to choose among seven low-level heuristics. This method incorporates a late acceptance (LA) strategy that compares the newly generated solution with a solution found $L$ iterations ago. Each of the seven low-level heuristics performs a modification of the farm layout array. The modifications applied to the layout array can be cell value inversion, cell swapping, random reconstruction, specific reconstruction, first improvement local search, row crossover, and column crossover.

The proposed LA selection hyper-heuristic was tested on five different scenarios and compared to a GA and an iterated local search approach (ILS). LA approach outperformed the GA in all five tested scenarios and achieved better results than the ILS in three scenarios.

5.3.4. Simultaneous Optimization of Turbine Placement and Cable Routing Within Offshore Wind Farms

In [88], WFLO consists of two primary optimization problems:

• Turbine placement problem: It aims to determine the optimal locations for turbines on a grid to maximize total power production. This process focuses heavily on the reduction in the wake effect, which occurs when upwind turbines disrupt wind flow and reduce the energy available to those downstream. Additionally, the placement must maintain a minimum separation distance between turbines.
• Cable routing problem: It involves finding the optimal power routing between turbines and the substation. This can be achieved by selecting the correct cable type, minimizing the power losses to resistivity in cables and minimizing the cost of cabling.

To solve these simultaneously, the paper employs an objective function that minimizes the ratio of total capital costs (cabling plus turbines) to the total net power produced, aiming for the lowest cost per unit of net power.

There are three primary decision variables. The first variable is turbine occupancy, a binary vector which indicates whether a specific grid position is occupied by a turbine (1) or remains empty (0). The second variable is a cabling binary matrix whose values denote whether an electrical cable exists (1) or does not exist (0) between two specific positions or between a turbine and the substation. The third decision variable is cable type selection where for every active connection in the layout, the model must select a specific cable from seven available types, which vary in material current capacity and cost per metre.

Finally, the model is bound by five critical constraints: a limited number of turbines can be placed on the farm; each cable must be capable of carrying the power transmitted between two points; every turbine must be connected to the substation; turbines cannot have split power outputs; and cables are forbidden from crossing each other.

Again, the hyper-heuristic is composed of two primary mechanisms: a selection method and a move acceptance criterion. Authors test various selection methods, including SR, SS, and four customized “best choice” variants that utilize real-time performance data to choose the most effective heuristic for a given state. These selection methods are combined with four move acceptance criteria: OI, IE, GD—all of which are deterministic—and SA which is stochastic. Selection is made from 10 low-level heuristics. The first four focus on turbine placement, while the remaining six deal exclusively with rearranging the cable routing.

In the study, three optimization models are applied to four specific problem instances based on the Borssele 4 windfarm located within the Dutch part of the North Sea. These models are: Model 1, which follows the traditional sequential process of optimizing turbine positions before addressing cabling; Model 2, a simultaneous approach that optimizes turbines and cables together starting from a randomized layout; and Model 3, another simultaneous method that begins with the pre-optimized turbine placement from the sequential model. The experimental results show that while simultaneous optimization of turbines and cabling is a feasible design approach, it does not provide a definitive statistical advantage over the traditional sequential method. Across the four tested instances, Model 2 achieved the best results by pairing SR with IE but the final results for all three models were extremely close, differing by less than 1%. This marginal difference occurs because the study relied on simplified assumptions such as straight-line cabling and fixed foundation costs.

6. Discussion

The systematic review of 21 studies indicates that the application of hyper-heuristics (HHs) to power system optimization remains an emerging, highly promising, yet relatively unexplored research domain. Notably, only nine of the reviewed papers are full journal articles, while the remaining 12 are conference proceedings or book chapters, underscoring the limited depth of literature currently available. While standard metaheuristics (e.g., GA, PSO and DE) have been deployed across hundreds of power grid studies, hyper-heuristics are rarely utilized.

The distribution of existing research reflects the centrality of core grid operations, with over half of the reviewed literature concentrated on classic power system operation problems: UC, ED, OPF. Beyond these core operational tasks, the literature addresses a small set of specialized sub-domains—each represented by only one or two studies—including battery discharge scheduling, electric vehicle demand-side management, distribution system reliability, controller tuning, and energy consumption forecasting. Finally, while WFLO represents a spatial planning task rather than a core power grid operational procedure, these studies were included in this review to provide a comprehensive evaluation of hyper-heuristic search mechanics in energy system applications.

To synthesize the literature reviewed in this section,

Table 3. Summary and comparison of reviewed studies.

Study	Optimization Problem	Objective Functions	High-Level Strategy	Low-Level Heuristics	Test Systems	Methods Compared to	Solution Improvement
[60]	UC	Generation cost	RPD with OI acceptance	7 operators (mutations, swap, hill climbers)	10, 20, & 40-unit (24-h), 8-unit Turkish system (8-h)	LR, GRA, GA, BDE, MA, ES, SMA	−0.0003% to 0.455%
[61]	SEPGS	Generation cost	24 combinations of 6 selections and 4 move acceptances	7 operators (mutations, swap, hill climbers)	10 & 20-unit (24-h)	Internal comparison of 24 hyper-heuristics	0–5.67% (the best combination compared to the best results of the other 23 combinations)
[62]	UC	Generation cost	SR, RD, RP, RPD, RL, AbS	Probability vectors	8-unit Turkish system with 8-h horizon	4 PBIL variants, Sentinel8, Known best	0.23% to 5.85%
[66]	EED	Fuel cost, Pollutant emissions	MOEA/D-HHSW	5 DE operators	6, 10, & 40-unit systems	14 methods (4 GA/ESs, 3 DEs, 2 TLBOs, 2 HSAs, PSO, BBO, GSA)	0.002% to 1.30%
[67]	ELD	Fuel cost	Online selective method using evolutionary algorithms within an island model concept (HH-LGSO)	Random dynamic grouping, Differential grouping, Delta grouping, EDA-GA, AVS-RG	Real-world 140-unit system	GA-MPC, SAMODE	−0.16% to 0.11%
[68]	DEED	Fuel cost, Pollutant Emissions	Selection mechanism with sliding time window	3 mutation operators (1 GA, 2 DE)	5, 10-unit systems, 2 IEEE 57-bus systems with PV and wind units-one adds storage	13 methods (10 metaheuristics and 3 conventional methods)	0.41% to 9.15%
[72]	Power loss reduction	Power loss, Voltage Deviation, Maximum L-index	QH	ABC, MOA, ESS, OOO	Standard IEEE 30-bus system	47 methods (6 internal components and 41 external algorithms)	0.079% to 21.48%
[73]	OPF	Power loss	EMCQ	GWO, BMO, WOA	IEEE 57-bus system with 4 thermal, 1 wind, 1 solar and 1 hydro-solar generator	GWO, BMO, WOA	0.096% to 2.66%
[74]	OPF	Power loss, Generation cost	EMCQ and randomly select-OI	MFO, BMO, TLBO, GBO	IEEE 30-bus system with 4 thermal, 2 wind generators and 6 FACTS devices	MFO, BMO, TLBO, GBO	0.024% to 9.1%
[75]	Active-reactive power optimization	Power loss, Voltage offset	Thompson sampling	8 search strategies (Foraging, Random, Odor search, 1/2-Point crossover, Inversion, Adaptive variation, Gaussian perturbation)	IEEE-30 bus system	Directly against the pre-optimized baseline state	30.01% to 38.77%
[76]	OPF	Generation cost, Power loss	LSHADE	Chaotic maps, Opposition-based learning methods, Population ratio	IEEE 30-bus system with 4 thermal, 2 wind generators and 6 FACTS devices	SHADE-SF, INFO, INFO-FDB, Hyper-INFO	0.000034% to 6.46%
[78]	DSESP	Energy consumption from the utility grid	Evolutionary algorithm	26 heuristics (H0 to H25), each representing a specific percentage of power to discharge from the battery	Two MGs in Alcalá, Spain (PV-powered Botanic Garden office, Micro-wind powered household)	No battery baseline, Non-optimized battery baseline	8.00% to 72.10%
[80]	DSM of PEVs	Generation cost, Greenhouse gas emissions	Tabu search with roulette wheel and AM acceptance	3 unnamed low-level heuristics	Test system with 2516 domestic consumers, 296 small consumption firms, 150 medium firms, 4 large firms and 1000 Chevy Volt PEVs	2 baseline models (No DSM, Centralized DSM)	2.20% to 6.82%
[81]	Reliability optimization problem in medium voltage power distribution	TAC, SAIFI, SAIDI, MAIFI, EENS	Reasoning-based Selection Function implemented via a BPANN	4 fuzzy PSO variants	Bariloche medium voltage grid in Bariloche Argentina (16 sections, 16 sub-stations)	PDP	1.92% to 13.28%
[82]	Power quality optimization	Harmonic distortion, Current unbalance	SS hyper-heuristics	Two base hybrid controllers pre-tuned by a GA	13 simulated electrical grid scenarios combining linear/nonlinear and balanced/unbalanced loads	Standalone C1/C2, Directly tuned controller, No control	0.02% to 11.74%
[83]	Electricity consumption forecasting	MAPE	GA-PB	Velocity and position updates from PSO; Frequency, velocity, position updates and local search from BA	Electricity consumption datasets from Hokkaido, Japan and the United States	PSO, BA, DE	0% to 36.07%
[84]	Electricity consumption forecasting	MAPE	AE-GAPB	Velocity and position updates from PSO; Frequency, velocity, position updates and local search from BA	Real datasets of renewable energy generation and electricity consumption from the Hokkaido, Kyushu and Tohoku regions of Japan	GA-PB, PSO, BA, DE, GWO, WOA	0% to 80.99%
[85]	Multi-objective wind farm layout optimization	Inverse of energy production, Cable length, Land area usage	9 combinations of selection (RC, FS and CF) and acceptance (AM, GDA and best acceptance) strategies	NSGA-II, SPEA2, IBEA	Real world wind scenario to a 3 × 3 km2 land area with 30 turbines (scaled from 20 to 100 turbines)	NSGA-II, SPEA2, IBEA	0% to 41.50%
[86]	Optimal wind turbine placement	COE, Efficiency of Production (maximize)	An ABC algorithm framework combined with probabilistic heuristic selection and the SA acceptance criterion.	Matrix improvement operators acting on grid rows/columns: Row-swap, Row-copy, Row-flip, Column-swap, Column-copy and Column-flip.	10 × 10 grid wind farm across Problem A (multiple wind directions/speeds) and Problem B (incorporating land usage constraints)	2 GAs	2.26% to 147.47%
[87]	Discrete wind farm layout optimization	COE	SR selection with LA strategy	7 operators: Cell value inversion, Cell swapping, Random/Specific reconstruction, First improvement local search, Row/Column crossover	Five problem scenarios from the WindFLO repository	ILS, GA	−0.01% to 4.86%
[88]	Simultaneous optimization of turbine placement and cable routing within offshore windfarms.	Total cabling and turbine costs relative to net power produced	SR, SS, and 4 “best choice” variants paired with OI, IE, GD and SA	10 defined operational heuristics: LLH1–LLH4 (turbine positions/movement), LLH5–LLH10 (cabling layout modifiers)	Four problem instances of varying sizes derived from the real-world Borssele 4 windfarm in the Dutch part of the North Sea	3 distinct structural models: sequential, simultaneous (random start), simultaneous (optimized start)	3.48% to 60.16%

provides a comprehensive comparison across all 21 studies. The table outlines each study alongside its specific optimization problem, objective functions, high-level strategy, low-level heuristics, test systems, comparative benchmarks, and the resulting improvement ratio range (where negative values indicate that the proposed hyper-heuristic was outperformed by competing methods/benchmarks).

6.1. Architectural Characteristics and Literature Gaps

The concentration of studies on a narrow subset of topics highlights major literature gaps. For instance, this review identified only three studies applying hyper-heuristics to ED, whereas surveys on standard metaheuristics cover hundreds of papers evaluating diverse problem variants. Consequently, several high-impact, complex optimization tasks remain unexamined, including combined heat and power economic dispatch (CHPED) and security-constrained economic dispatch (SCED). Furthermore, hyper-heuristic frameworks have not yet been applied to real-time control problems (e.g., voltage control, load frequency control or power electronics converter management), crucial long-term planning tasks (e.g., network reconfiguration or distributed generation allocation) or emerging AI-driven energy technologies (e.g., wireless power transfer coil design and model predictive control in motor drives).

From an architectural perspective, all reviewed papers exclusively employ selection hyper-heuristics, which operate as high-level managers selecting from a fixed pool of pre-existing low-level heuristics based on search performance. Generation hyper-heuristics—which automatically synthesize entirely new search operators using techniques like genetic programming—remain completely unexplored in this field. This absence stems from two main factors:

• Constraint handling complexity: Power system optimization problems are governed by highly non-linear equality constraints (e.g., active and reactive balance equations) and strict operational inequality limits (e.g., generator ramp rates and voltage bounds). Automatically generating new operators that reliably maintain solution feasibility while exploring such constrained landscapes presents extreme algorithmic complexity.
• Early-stage adoption: Given the nascent state of hyper-heuristics in power system engineering, researchers have prioritized selection frameworks that readily leverage established heuristics rather than generating new operators from scratch.

Nonetheless, generation hyper-heuristics represent a major future research direction. Currently, low-level heuristic pools in the literature range from simple neighborhood moves (e.g., single-point crossover, swap mutation) to complete metaheuristic algorithms (e.g., GA, DE, ABC and GWO).

This architectural gap is further underscored by the rapid emergence of data-driven AI paradigms—such as RL, graph neural networks (GNNs), transformers, large language models (LLMs), and foundation models—which present both competitive and complementary dynamics with hyper-heuristics. On one hand, deep RL paradigms may compete with hyper-heuristics in some sequential decision-making settings by learning policies directly from system states, potentially bypassing explicit heuristic selection or search. However, such data-driven policies often require additional mechanisms (e.g., safety layers or projection techniques) to guarantee hard operational constraints and reliable performance under out-of-distribution operating conditions. On the other hand, these paradigms offer powerful complementary synergies when integrated within hyper-heuristic frameworks: GNNs effectively capture the complex spatial topology of power networks, transformers handle temporal renewable variability, LLMs and foundation models show strong potential to assist in automated heuristic design, algorithm configuration and knowledge transfer, thereby providing a promising avenue to explore the as yet unexamined domain of generation hyper-heuristics in power system optimization. Ultimately, rather than rendering hyper-heuristics obsolete, these emerging paradigms point toward next-generation, neuro-symbolic frameworks that combine data-driven guidance with optimization frameworks capable of explicitly handling operational constraints.

6.2. Quantitative Bibliometric Analysis

Our quantitative aggregation of the 21 papers reveals several prominent trends that prove the research area’s scientific maturity:

• Temporal evolution and accelerating output: Publication activity has risen significantly over the last 15 years. After a pioneering foundational phase from 2010 to 2012, output has surged, peaking in 2023–2025 (holding 28.6% of all reviewed literature). This temporal trend indicates a highly active, expanding, and mature field that has shifted from basic single-objective models to self-adaptive and auto-evolutionary systems.
• Application frequency: The quantitative distribution of the reviewed literature reveals that traditional power system optimization remains the dominant application domain, accounting for over half of the corpus (52.38%, 11 papers). Specialized sub-domains represent the second most frequent category (28.57%, six papers) which extends hyper-heuristics into renewable/load forecasting, microgrid control and energy storage scheduling, decentralized demand-side management of PEVs and medium-voltage grid reliability. Lastly, WFLO and cabling problems constitute a highly specialized spatial-routing category 19.05.% demonstrating how hyper-heuristic search operators can be effectively tailored to complex geometric and wake-deficit constraints.
• Selection heuristics (high-level): The taxonomical distribution of high-level selection mechanisms across the reviewed literature reveals a clear divergence between non-learning baseline strategies, parameter-tuning metaheuristics, and advanced cognitive learning-based selectors. Random and permutation-based rules form the largest share (28.57%, 6 papers), bypassing active online tracking to minimize selection latency in tightly constrained scheduling and continuous OPF search spaces. High-level metaheuristics (23.81%, 5 papers) and score-based choice functions (23.81%, 5 papers) are equally prominent; the former utilize evolutionary algorithms to adaptively self-adapt population configurations offline, while the latter track real-time performance indicators to reward active operators online. The remaining portion of the corpus leverages advanced learning-based architectures, including reinforcement/Q-learning vector updates (9.52%, 2 papers), Thompson sampling (4.76%, 1 paper), back-propagation neural networks (4.76%, 1 paper), and sequence-based switching (4.76%, 1 paper), demonstrating a clear research progression toward highly cognitive and real-time adaptive grid coordination.

6.3. Cross Study Synthesis

A rigorous comparative analysis of the reviewed literature reveals clear relationships between specific power system problem characteristics and the design of hyper-heuristic selection, learning and move acceptance mechanisms.

6.3.1. Solution Quality

Solution quality across the reviewed power systems domains is fundamentally defined by an algorithm’s capability to balance conflicting technical and economic objectives while satisfying rigid physical and operational constraints. In discrete scheduling environments, such as UC/SEPGS [60,61], quality is determined by the minimization of total startup and fuel costs under heavily penalized generator minimum up/down times and load demands. Conversely, continuous-coordinate spatial problems, such as continuous WFLO [88], evaluate layout feasibility based on the minimization of the capital-cost-to-net-power ratio while mitigating aerodynamic wake deficits. Across these diverse formulations, traditional mathematical programming and standalone metaheuristics frequently suffer from premature convergence or scaling collapse in these non-convex, non-linear search spaces. Transitioning to selection hyper-heuristics consistently improves final solution quality by dynamically managing low-level search operators, thereby identifying superior localized configurations and wider, more stable Pareto-optimal frontiers.

In continuous multi-objective domains, solution quality is assessed not by a single objective value, but by the coverage, convergence, and diversity of the generated Pareto-optimal front. In multi-objective ED (EED and DEED), where fuel costs and gaseous emissions represent competing objectives, multi-objective selection hyper-heuristics like MOEA/D−HHSW consistently find best-compromise solutions that dominate traditional algorithms (such as MODE, MOPSO, and NSGA-II), especially as system dimensions scale up to 40 generators [76]. In continuous WFLO [85], selection hyper-heuristics evaluate front quality using metrics like hypervolume, uniform distribution (UD), and the ratio of non-dominated individuals (RNI), with schemes like RC-GDA demonstrating statistically superior coverage compared to individual metaheuristics. A crucial multi-objective insight in continuous WFLO is the strong positive linear correlation between inter-array cabling MST length and used land area, suggesting a structural redundancy where one of the objectives can be ignored or both can be merged to simplify the optimization process. Furthermore, in OPF with FACTS device allocation, hyper-heuristics such as the Hyper-FDB-INFO and EMCQ consistently achieve superior power loss and generating cost minimization over standard optimizers, ensuring all generator terminal voltages and reactive power limits remain strictly within physical boundaries [74].

In discrete combinatorial scheduling, where demand and up/down-time penalties are extremely high (typically 100,000) to filter out infeasible schedules, hyper-heuristics coupled with OI move acceptance maintain search trajectories tightly on feasible boundaries [61]. The permutation descent hyper-heuristic consistently matches or outperforms advanced single-objective optimizers and scales robustly without the performance degradation seen in traditional LR. For mixed-integer medium-voltage distribution systems, the authors in [81] demonstrated that when multiple protection device types (disconnectors, interrupters, fuses) are optimized concurrently, traditional possibilistic dynamic programming collapses due to combinatorial complexity. BRHY successfully bypasses this bottleneck, identifying a superior protection layout that increases global fuzzy satisfaction, reduces EENS and lowers the ICF [81].

In active distribution networks requiring high energy quality, sequence-based selection hyper-heuristics coordinate the switching between pre-tuned PI-LQR and resonant-LQR controllers, reducing total harmonic distortion and current unbalance factors by up to 11.74% more than standalone controllers across 13 unbalanced load scenarios [82]. In DSM of PEVs, selection hyper-heuristics shift charging cycles within idle windows to reduce grid peak demand and simultaneously minimize costs and emissions [80]. Furthermore, in electricity consumption and renewable forecasting, where prediction accuracy directly influences spinning reserve scheduling, the auto-evolutionary AE-GAPB framework dynamically updates hyper-parameters to find optimal model coefficients. This adaptive process minimizes prediction errors, achieving competitive MAPE ratios for the Holt-Winters model on the Hokkaido consumption dataset and for deep-learning LSTM models on the renewable generation dataset [84].

6.3.2. Convergence Behavior and Stagnation Avoidance

High-level selection mechanisms dynamically coordinate search trajectories to prevent premature stagnation and guide the population toward global optima. In non-stationary forecasting domains, auto-evolutionary hyper-heuristics (such as AE-GAPB) accelerate convergence by dynamically adapting high-level crossover and mutation rates in real-time based on elapsed iterations and fitness feedback. When prediction accuracy is low, these rates automatically scale up to inject population diversity and escape local sub-optimal traps; as iterations progress, the rates decrease to enable the search to concentrate on refining the optimal solution [84]. Similarly, in coordinated active-reactive power dispatch, Thompson sampling selection coupled with Gaussian random perturbations and Cauchy variations prevents late-stage search stagnation by stochastically exploring distant regions when early-stage target deviations occur [75]. In highly dynamic environments, hybrid offline-online learning frameworks utilize memory-based probability vectors or restart schemes to detect and track periodic variations, maintaining search adaptability even under high-frequency environmental changes [62].

The rate and final quality of this convergence are heavily governed by the interaction between the high-level move acceptance criteria and low-level search operators [62,88]. In continuous multi-objective environments, deterministic strategies like best acceptance—which actively filters out duplicate solutions that can degrade performance—and great deluge acceptance driven by hypervolume metrics maintain a steady convergence trajectory while preserving Pareto front diversity [85]. Conversely, in discrete combinatorial spaces with highly penalized operational constraints, deterministic OI move acceptance is preferred to restrict search trajectories strictly to feasible boundaries, outperforming stochastic SA or exploratory AM schemes [88]. For highly non-convex continuous domains like OPF with FACTS device sizing, EMCQ move acceptance utilizes a temperature-free probability density and a non-improving move counter to escape local minima [74]. This stochastically flexible coordination allows the hyper-heuristic to achieve robust convergence profiles, typically enabling the power networks to converge smoothly within 200 iterations [73,74].

6.3.3. Computational Complexity

The computational complexity of a selection hyper-heuristic framework is defined by population size, problem dimensionality or number of decision variables, number of low-level heuristics, and the execution cost of the objective evaluation functions ($C_{of}$). Across different application domains, the physical nature of $C_{of}$ represents the primary execution bottleneck, while the high-level heuristic selection logic—such as Q-learning [72] or Thompson sampling [75]—is computationally lightweight. In continuous non-linear domains like OPF and ORPD, evaluating each candidate solution requires solving non-linear AC load flow equations via numerical math engines (e.g., executing the MATPOWER toolbox) [73,74], which results in a heavy, execution-intensive $C_{of}$ bottleneck. Conversely, in 2D spatial domains such as continuous WFLO, the objective evaluation is dominated by geometric and graph-theoretic calculations, including cumulative aerodynamic wake deficit modeling and MST cable routing [85]. When optimizing turbine positioning and cabling simultaneously, the computational complexity scales exponentially due to concurrent coordinate and topological modifications, forcing researchers to restrict search budgets to fixed iteration limits (typically 10,000 to 20,000 iterations) or to utilize sequential models to isolate cabling operators and reduce search space dimensionality.

A key computational trade-off in hyper-heuristic design lies in dividing complexity between offline learning/training and online real-time execution. In smart grid control and protection coordination domains, researchers frequently shift the computational burden offline to achieve high-speed online performance. For example, in reliability optimization of medium-voltage distribution systems [81], a BPANN is trained offline over swarm intelligence principles to act as the selection function. While this offline training phase is highly complex, the online selection runs instantaneously by executing simple forward-pass neural operations. Similarly, in microgrid power quality control, sequence-based selection hyper-heuristics switch between pre-tuned controllers with zero online optimization overhead, meeting the instantaneous millisecond-level response times required by active distribution grids [82]. In contrast, in non-stationary forecasting domains, the auto-evolutionary AE-GAPB framework dynamically tunes the hyperparameters of low-level algorithms in real-time [84]. While this adaptive process avoids search stagnation, evaluating the objective function $C_{of}$ requires training deep-learning LSTM models or exponential smoothing models on multi-regional time-series datasets, creating a heavy online execution cost that cannot be pre-computed. In discrete scheduling environments like the UCP, HH-EDA2 balance these paradigms by executing an offline training phase to extract a list of high-quality probability vectors, which the online selection hyper-heuristic then rapidly samples and resets to adapt to real-time, non-stationary grid demand shifts [62].

6.3.4. Scalability to Large-Scale Power Systems

As power system networks expand in size and complexity, standard metaheuristics often suffer from the curse of dimensionality. Selection hyper-heuristics mitigate this by decomposing the decision space or dynamically coordinating operators. In continuous high-dimensional ED, variable-grouping schemes (HH-LSGO) successfully optimize massive 140-unit grids [67], while sliding-window methods (MOEA/D-HHSW) maintain tight Pareto fronts from 6 up to 40 generators [66]. Similarly, in discrete UC scheduling, permutation descent hyper-heuristics scale consistently up to 40 units and real-world national grids without requiring manual parameter retuning [60].

This scalable performance extends to multi-objective OPF and spatial wind farm layouts. Under mixed-integer OPF and reactive power environments, Thompson sampling and Q-learning hyper-heuristics optimize up to 27 control variables across 30-bus and 57-bus test grids under fluctuating loads. In continuous 2D spatial layouts, selection hyper-heuristics successfully scale to coordinate turbine placement and cable routing across massive projects like the 283-position Borssele 4 North Sea offshore farm, maintaining stable Pareto fronts across varying turbine densities and terrain constraint zones [88].

In active distribution grids and DSM, hyper-heuristics resolve temporal and multi-agent coordination issues at scale. For instance, decentralized PEV charging frameworks smooth peak demand across 2966 consumers and 1000 electric vehicles while maintaining near-constant execution times [80]. For microgrids, sequence-based and battery discharge hyper-heuristics coordinate long-term scheduling and power quality over 168-h horizons, while fuzzy reasoning models optimize equipment placement across multi-section systems [78]. Finally, in forecasting and non-stationary environments, the auto-evolutionary AE-GAPB framework scales robustly across three major Japanese utility zones (Hokkaido, Kyushu, Tohoku) by dynamically adjusting hyperparameters to fit regional renewable generations without site-specific manual tuning [84].

6.3.5. Statistical Stability and Operational Robustness

Selection hyper-heuristics significantly enhance statistical search stability, consistently yielding narrow variances across multiple independent runs compared to traditional standalone metaheuristics. In OPF, the Hyper-FDB-INFO achieves exceptionally tight standard deviations of 0.0383 for cost minimization and 0.0147 for transmission loss reduction [76], while the QH [72] and Thompson sampling [75] frameworks stabilize convergence margins on modified IEEE systems. Similarly, MOEA/D-HHSW [66] produces highly consistent empirical attainment functions in multi-objective dispatch, and the coevolutionary HH-LSGO [67] secures low-variance, stable results for high-dimensional 140-generator systems. In scheduling and dynamic load-dispatch environments, these high-level learning mechanisms—such as permutation descent and elite non-dominated sorting—safeguard the search trajectory against local entrapment, delivering robust and reproducible solutions across varying seed initializations.

Beyond statistical stability, hyper-heuristics provide high operational robustness when managing active power networks plagued by severe renewable intermittency and load unbalance. In medium-voltage reliability optimization, the BRHY fuzzy reasoning-based model utilizes a BPANN selector to monitor swarm intelligence indicators, adaptively switching protection devices to secure a global satisfaction level of 0.891 under severe network outages [81]. Under highly unbalanced electrical grid events, sequence-based selection hyper-heuristics switch between pre-tuned controllers to reduce total harmonic distortion and current unbalance by up to 11.74% more than standalone controllers [82]. This operational flexibility is further bolstered in dynamic pricing and load-shifting setups; HH-EDA2 [62] utilize offline-learned probability vectors to adapt to cyclic and high-frequency demand shifts, while sliding performance windows and auto-evolutionary parameter-tuning models dynamically scale crossover and mutation rates to track volatile wind-solar generation curves and fluctuating consumer loads.

6.4. Evaluation Benchmarks and Practical Limitations

In terms of overall performance, hyper-heuristics are highly efficient, robust, and capable of outperforming standalone metaheuristics. However, their practical adoption in power engineering remains limited. This slow uptake is largely driven by the widespread familiarity with established tools like ANNs and standard metaheuristics, whose parameter-tuning strategies and behaviors are well understood by domain practitioners. Additionally, hyper-heuristics present greater design complexity due to their two-level architecture requiring simultaneous management of high-level selection logic and low-level heuristic pools. Furthermore, a critical reporting gap exists across the 21 reviewed papers: while authors routinely highlight solution quality, only 3 studies [68,72,81] explicitly report direct computational times, preventing rigorous quantitative comparisons of computational efficiency against metaheuristics.

To understand why computational complexity remains a major barrier to real-world deployment, the physical and mathematical bottlenecks of power system simulation must be examined. The primary computational bottleneck in grid-level hyper-heuristic execution is the extreme cost of evaluating the objective function. In OPF and reactive power routing, evaluating a single candidate solution requires running non-linear alternating current (AC) power flow equations to verify voltage compliance and physical security limits. Similarly, in WFLO and cabling management, calculating cumulative turbine wake deficits and routing complex inter-array cable networks using graph-theoretic algorithms like Prim’s or Kruskal’s MST represent NP-hard geometric and routing problems that scale exponentially with farm size. Furthermore, integrating high-level cognitive learning selectors, such as model-free Q-learning [72] or BPANN [81], or online auto-evolutionary hyperparameter tuning [83,84], significantly increases algorithmic complexity and offline/online training overhead, making real-time, millisecond-level scheduling on physical edge-grid hardware computationally prohibitive.

The evaluation benchmarks used in the reviewed literature reveal a significant limitation: of the 21 reviewed studies, only nine evaluate real-world system data, while the majority rely on standard synthetic benchmark networks (such as IEEE 14, 30, and 118-bus topologies). Furthermore, existing real-world applications remain confined to small-scale isolates, such as localized segments of the Turkish network, individual microgrids in Spain, a single distribution feeder in Argentina, or the Borssele 4 wind farm. While benchmark instances are valuable for baseline validation, they provide a simplified representation that does not fully reflect the operational complexities encountered when scaling to industrial grid datasets.

Transitioning hyper-heuristics to realistic power system models is hindered by several key domain challenges. First, real grid datasets incorporate complex market constraints, such as non-convex market-clearing rules, locational marginal pricing, generator ramp limits, and multi-settlement bidding structures that simple cost functions omit. Second, realistic grid simulations demand rigorous security assessment, where evaluating dynamic N-1 and N-2 contingency constraints across large physical network topologies introduces massive computational overhead during solution search. Third, real operational data introduces severe renewable uncertainty, requiring hyper-heuristics to transition from static, deterministic models to dynamic online adaptation under stochastic wind and solar profile fluctuations. Fourth, modeling cyber-physical integration requires simulating telemetry latency, sensor noise, packet loss, and physical edge-device memory bounds that idealized test cases ignore. Finally, strict real-time computational requirements—such as sub-second control horizons for dynamic microgrid regulation—create a fundamental tension with the extreme evaluation costs of non-linear AC power flow equations on large-scale industrial grid datasets.

To resolve these reporting gaps, eliminate hardware-dependency issues, and enable fair cross-study comparisons in future literature, we recommend that future grid hyper-heuristic studies adopt a standardized benchmarking framework comprising four metrics:

• Algorithm effort ($AE$): To measure search speed independently of specific hardware architectures, future studies should report algorithm effort, defined as the total number of objective function evaluations executed per CPU second:

  $$AE={\displaystyle \frac{\mathrm{Total} \mathrm{Solution} \mathrm{Evaluations}}{\mathrm{CPU} \mathrm{Execution} \mathrm{Time} }}$$

  (30)
• Computational specifications: Authors must explicitly report the CPU model, clock speed, core counts, RAM capacity, operating system, and software tools (e.g., MATLAB or Python) to ground the recorded execution times.
• Statistical search rigor: Reporting solely “best-run” values should be avoided. Instead, studies must report the mean, worst, and standard deviation (std) of final objective values and CPU execution times over a minimum of 20 to 30 independent trials, verified using non-parametric statistical significance tests (such as Wilcoxon Rank-Sum, ANOVA, or Tukey HSD tests at a 95% confidence level).
• Control latency margin: In active microgrid or substation operation, control decisions must be calculated within strict physical time limits ($\Delta T_{\mathrm{limit}}$)—such as 100 ms for controller switching or 1 s for demand response. The control latency margin ($CLM$) quantifies the algorithm’s time safety buffer relative to that physical grid deadline:

  $$CLM=\left({\displaystyle \frac{\Delta T_{limit}-T_{execution}}{\Delta T_{limit}}}\right)\times 100\%$$

  (31)

A high positive percentage proves the hyper-heuristic is robust enough to handle communication delays and grid transient states on physical controller hardware whereas a negative value mathematically establishes real-time physical infeasibility.

6.5. Review Limitations

Finally, certain methodological restrictions of this systematic review must be acknowledged alongside the literature gaps:

• Database and Publication Scope: The search strategy was confined to specific academic databases and restricted to English-language peer-reviewed journals, conference papers, and book chapters. This introduces potential indexing and language biases, potentially excluding relevant technical reports or non-English studies.
• Terminology Constraints: Because search strings explicitly required “hyper-heuristic” terms, relevant frameworks utilizing alternative terminology—such as “adaptive parameter control” or “hybrid metaheuristics”—may have been omitted.
• Sample size and conference inclusion: The relatively small number of available primary studies (21 papers) limits the generalizability of broad trends. Furthermore, the inclusion of conference proceedings—while necessary to capture early emerging trends in a nascent field—introduces variability in peer-review depth and experimental reporting completeness compared to full-length journal articles.
• Synthesis approach and metric incomparability: A formal quantitative meta-analysis was not feasible due to heterogeneous mathematical problem formulations, non-standardized test grid scales, and missing reporting metrics (such as execution times). Furthermore, even where performance metrics such as computational time, convergence iteration counts, or function evaluations were reported, direct cross-study quantitative comparison remains fundamentally invalid. These metrics are inherently hardware-dependent (varying by processor architecture, RAM, and compiler optimizations), software-dependent (e.g., MATPOWER vs. custom power flow solvers), and tied to the specific algorithmic mechanics of each approach. Consequently, a qualitative narrative synthesis was conducted. While this analysis successfully highlights critical algorithmic mechanisms, operational trade-offs, and domain mappings, it cannot statistically aggregate numerical performance metrics to quantify exact effect sizes or universal algorithmic superiority across application domains.

7. Conclusions

This paper presented a systematic review of hyper-heuristics in power system optimization by analyzing 21 primary studies in the literature. The literature indicates that by shifting optimization from direct solution space exploration to high-level algorithm selection, hyper-heuristics exhibit promising potential for managing non-convex and constrained power system problems. However, the field is still in its early stages and requires further structural exploration. Most applications focus on only a few central operational tasks and use selection-based approaches exclusively, while generation hyper-heuristics remain unexplored because creating new search tools that keep solutions valid under strict power grid rules is algorithmically difficult. Furthermore, hyper-heuristics’ widespread adoption remains limited due to high-level design complexities, a general lack of reported computational execution times across existing studies, and a strong reliance on established standalone metaheuristics.

To support the wider adoption of hyper-heuristics in practical power system applications, several key research challenges must be prioritized rather than relying on small-scale synthetic benchmarks. Future studies should investigate a broader range of selection-based hyper-heuristic methods and gradually explore generation hyper-heuristics capable of automatically creating new optimization operators while incorporating mechanisms to maintain feasibility with respect to nonlinear grid constraints. Beyond conventional grid-level applications and high-level system operations, future research should also explore low-level power electronics applications, such as dynamic parameter tuning of converter control loops, component design optimization (e.g., AI-driven magnetic couplers and wireless power transfer coils), model predictive control in motor drives, and real-time energy management in inverter-dominated microgrids.

In addition, research must evaluate scalability on large-scale, real-world power systems to examine if hyper-heuristics can effectively handle practical operational obstacles within a reasonable time period. While this review provides a structured qualitative baseline, its scope is bounded by its focus on English-language literature, explicit hyper-heuristic terminology, and the inability to perform a quantitative meta-analysis due to heterogeneous benchmark reporting across current studies.

Looking forward, as power system optimization evolves toward complex, data-driven environments, the integration of advanced machine learning architectures presents a major frontier. While predictive models and representation learning offer powerful capabilities for forecasting and adaptive decision-making, hyper-heuristics remain essential for flexible algorithm selection and constraint-aware search. Future research should prioritize hybrid integration, leveraging data-driven guidance to enhance computational efficiency while relying on hyper-heuristics to guarantee operational reliability.

Achieving this vision requires advancing hyper-heuristics beyond isolated offline optimization studies toward practical, deployable decision-support and control frameworks powered by broader enabling paradigms:

• Online learning and automated operator generation: Coupling online learning with automated heuristic generation to synthesize problem-specific search operators in real time as grid topologies evolve.
• Explainability (XAI): Formulating transparent high-level selection policies that provide an audit trail for utility operators to interpret decision logic during critical operational states.
• Decentralization and digital twin environments: Embedding multi-agent hyper-heuristic architectures within real-time grid digital twins to enable decentralized control (e.g., across microgrids) and perform stochastic, uncertainty-aware optimization under high renewable volatility.