Every time a power grid operator decides how much electricity each generator should produce, a quiet balancing act takes place. Burning less fuel saves money, but it often means allowing more pollution, and cutting emissions usually drives costs up. Engineers have long searched for a way to satisfy both goals at once, a challenge known as combined economic emission dispatch, or CEED. Now, a team of researchers has turned to one of the ocean’s most graceful feeders, the manta ray, for inspiration, and the result is an optimization algorithm that outperforms a crowded field of competitors on both cost and cleanliness.

The study, published in the journal Results in Engineering by Yu-Feng Sun, Jie-Sheng Wang and colleagues, describes an enhanced version of the Manta Ray Foraging Optimization algorithm, or MRFO. The original algorithm, first introduced a few years ago, mimics the cooperative hunting strategies that manta rays use to harvest plankton. Because plankton drifts unpredictably with seasons and weather, manta rays have evolved three distinct foraging behaviors: chain foraging, in which animals line up in arcs so that each one catches food missed by the one ahead; spiral foraging, where the group coils around a dense patch of food; and somersaulting, in which individuals flip back and forth around a promising spot. Translated into mathematics, these behaviors become rules that move a population of candidate solutions around a search space, gradually converging on the best answer.

What makes the CEED problem so stubborn is its mathematical character. The relationship between a generator’s output and its fuel consumption follows a cubic curve, and the emissions of carbon dioxide, nitrogen oxides and sulfur dioxide each follow their own cubic functions. The combined objective is nonlinear, nonconvex and discontinuous, riddled with local minima that can trap classical techniques such as lambda iteration or gradient-based methods. To make matters harder, the problem is genuinely multi-objective, so the researchers converted it into a single objective by applying price penalty factors that fold the cost of each pollutant into one total figure in dollars per hour.

Constraint handling is where many algorithms stumble. Traditional approaches add a fixed penalty whenever a solution violates the power balance requirement, the equation demanding that total generation minus transmission losses equals demand. But a fixed penalty is a blunt instrument. Set it too low and the algorithm happily returns infeasible schedules; set it too high and the search becomes so restricted that it cannot explore properly. Finding the right value through trial and simulation is time-consuming and inefficient. The team’s answer was to make the penalty itself dynamic, drawing on five different mathematical functions: sinusoidal, hyperbolic tangent, arctangent, linear and hyperbolic secant. Each function is raised to an order that changes during the run, so the penalty strength adapts to both the size of the violation and the stage of the optimization. Early on, mild penalties preserve exploration; later, penalties tighten to push solutions toward feasibility.

The second major innovation targets the heart of the MRFO algorithm. In the original design, a parameter called C governs the switch between exploration, casting the net wide across the search space, and exploitation, homing in on the best solution found so far. This parameter rises monotonically from zero to one, a predictable march that can leave the algorithm poorly balanced at certain stages. The researchers replaced it with a chaotic switching factor, drawn from nine well-known chaotic maps including Chebyshev, logistic, tent, sine, Gauss and singer maps, combined with an exponential adjustment function. The result is a parameter that still trends upward but oscillates along the way, periodically reopening the search space and preventing the population from settling prematurely into a local optimum.

To find out which chaotic map worked best, the team ran all nine enhanced variants, named C1MRFO through C9MRFO, through the CEC-2022 benchmark suite of twelve test functions, using a population of thirty and five hundred iterations averaged over thirty independent runs. The Chebyshev-based variant, C1MRFO, emerged as the overall winner, ranking first among the nine and beating the original MRFO, which finished last. Wilcoxon rank-sum tests confirmed that most of the improvements were statistically significant, with p-values below the conventional 0.05 threshold. When C1MRFO was pitted against a roster of state-of-the-art metaheuristics, including the dhole optimization algorithm, evolutionary mating algorithm, dung beetle optimizer, snow geese algorithm, particle swarm optimization, grey wolf optimizer, sparrow search algorithm and marine predators algorithm, it achieved the best final rank, taking first place on many of the test functions while keeping computational times competitive.

The real test, however, came when the algorithm was applied to actual dispatch problems. The researchers simulated a six-unit power system under four load demands of 150, 175, 200 and 225 megawatts. Across all four scenarios, C1MRFO produced the lowest comprehensive total cost, which combines fuel cost with the weighted penalties for carbon dioxide, nitrogen oxide and sulfur dioxide emissions. At 150 megawatts, for example, it reached a total cost of about 10,137 dollars per hour, edging out the grey wolf optimizer and clearly beating algorithms such as the arithmetic optimization algorithm, Harris hawks optimization, whale optimization algorithm, sine cosine algorithm, particle swarm optimization and the lambda iteration method. The pattern held at every load level, with the margin over the weakest classical method growing to thousands of dollars per hour at 225 megawatts.

The dynamic penalty functions delivered a further, striking gain. When the sinusoidal penalty function was applied with the parameter order set to dy5, the total cost at 150 megawatts dropped from roughly 10,137 to 9,065.71 dollars per hour, and similar reductions appeared at the higher loads, reaching 14,621.45 dollars per hour at 225 megawatts. Comparing the five penalty shapes, the sinusoidal function consistently produced the lowest minimum costs, beating the hyperbolic tangent, arctangent, linear and hyperbolic secant alternatives by margins ranging from hundreds to nearly two thousand dollars per hour depending on the load. Notably, when the penalty order was fixed rather than dynamic, performance was worst across the board, confirming that the adaptivity of the penalty, not merely its presence, is what matters.

To verify that each improvement contributes on its own, the team conducted an ablation study on a much larger twenty-unit system with a total demand of 2,500 megawatts. The results formed a clear hierarchy: the original MRFO achieved 61,835.16 dollars per hour, the chaos-enhanced C1MRFO reached 61,826.11, and the fully improved version combining chaos with the optimal sinusoidal dynamic penalty, sin(dy5)MRFO, achieved the best figure of 61,807.36 dollars per hour. All three outperformed the phasor particle swarm optimization benchmark, which came in at 61,895.57 dollars per hour. The stepwise improvement demonstrates that the two enhancements are not redundant but superimpose their benefits, and that the framework scales to systems far larger than the six-unit case.

The authors are careful to note where the work goes next. The current approach converts the multi-objective problem into a single objective through price penalty factors, but future versions could adopt Pareto-based multi-objective optimization to generate full sets of non-dominated trade-off solutions. The test cases also assume deterministic demand, so extending the algorithm to stochastic dispatch with wind and solar integration is another goal, as is combining the dynamic penalty with feasibility-first constraint-handling strategies. For now, the study offers a compelling demonstration that borrowing the foraging logic of a filter-feeding fish, perturbing it with chaos, and teaching it to calibrate its own punishments can squeeze meaningful savings and emission reductions out of one of power engineering’s hardest scheduling problems.

Subject of Research: An improved manta ray foraging optimization algorithm with chaotic switching factors and dynamic penalty functions for solving the combined economic emission dispatch problem in power systems.

Article Title: Manta ray foraging optimization algorithm with dynamic penalty function and chaos switching factor to solve combined economic emission dispatch problem

Article References: Sun, Y.-F., Wang, J.-S., Zhang, X.-Y., Li, Y.-X., Guan, X.-Y., & Liu, X. (2026). Manta ray foraging optimization algorithm with dynamic penalty function and chaos switching factor to solve combined economic emission dispatch problem. Results in Engineering, 32, Article 113383. https://doi.org/10.1016/j.rineng.2026.113383

Image Credits: AI Generated

DOI: Not provided

Keywords: manta ray foraging optimization, combined economic emission dispatch, metaheuristic algorithms, chaotic maps, dynamic penalty function, power system optimization, emission reduction, fuel cost minimization, CEC-2022 benchmarks, swarm intelligence, constraint handling, Results in Engineering

Some of the most promising locations to search for possible alien megastructures, known as Dyson spheres, may not be stars like the Sun, but rather the galaxy’s smallest and dimmest stars.

Physicist Freeman Dyson introduced the idea of a Dyson sphere in 1960, describing it as a shell or, more realistically, a swarm of orbiting collectors that could capture most of a star’s energy. This concept has interested scientists searching for signs of advanced technology, but the question of how such a structure would appear in astronomical observations remained unknown.

Now, a new study, available on the arXiv preprint server, suggests that one of the coldest objects in a stellar system could, in principle, be an artificial megastructure rather than a natural astronomical object.

Amirnezam Amiri, a physicist at the University of Arkansas, explored the question of Dyson spheres by modeling these hypothetical megastructures around two often-overlooked types of stars: red M-dwarfs and white dwarfs. Amiri credits discussions with Harvard astronomer and founder of the Galileo Project, Avi Loeb, with helping shape the study.

Small, Dim, and Built to Last

Red dwarfs make up about 70% of the stars in the Milky Way. These stars burn slowly and can remain stable for trillions of years, much longer than the universe has even existed so far. White dwarfs offer a different benefit. As the dense remnants of stars like the Sun, they shrink to about 1% of their original size. This means a Dyson swarm could orbit much closer, requiring less material and a smaller structure than one built around a star like the Sun.

Amiri’s study explores how astronomers could detect such structures. Both red dwarfs and white dwarfs are dim enough that the infrared heat emitted by a Dyson swarm would stand out more clearly against the star’s own light.

Reading the Wreckage on a Star Chart

Astronomers use the Hertzsprung-Russell diagram to classify stars by temperature and brightness, and this tool is central to Amiri’s analysis. A Dyson swarm would not destroy a star’s energy, but would capture its light and re-emit it as heat at much longer wavelengths. 

In the new model, a star with a Dyson swarm would appear on the H-R diagram at the same luminosity but shifted dramatically toward the cool end. For instance, a red dwarf at about 3,000 K could appear surrounded by a shell as cold as 50 K, a difference of about two orders of magnitude. No known stars naturally occupy this region of the H-R diagram.

Identifying Factors

Dust and distant galaxies can also produce strong infrared emissions, so astronomers cannot identify a Dyson swarm using temperature alone. Amiri’s model points to a more complex set of clues. A true Dyson swarm would not show the silicate dust features seen in normal dusty disks, so its spectrum would be unusually smooth for an object with strong infrared emission.

Additionally, if the swarm consists of many separate collectors rather than a solid shell, there would be gaps that cause irregular, non-stellar flickering as the pieces move. Each of these signs can have natural causes, but finding them together is rare and hard to explain without considering artificial structures.

Future Observations

Astronomers can carry out this search using existing telescopes. The James Webb Space Telescope’s infrared instruments are already designed to detect faint, cold signals like these. The Vera C. Rubin Observatory’s new sky survey and the Nancy Grace Roman Space Telescope, which is expected to launch later this year, will add to these efforts. Together, they will generate the kinds of observations needed to search for the signatures described in the study.

Possible candidates for these future observations have already been identified. In 2024, Project Hephaistos identified seven potential Dyson sphere candidates among about 5 million cataloged stars, all linked to red dwarfs. One was ruled out after it was shown that a background black hole caused the observed signal. Several candidate Dyson sphere systems still remain under investigation. Amiri does not claim that any of these are actual Dyson spheres, and the paper makes this clear. Instead, the study offers a better way to tell true anomalies apart from dusty disks before using valuable telescope time. If a Dyson swarm surrounds a star and reradiates its energy at temperatures unlike those of natural stars, upcoming infrared surveys may finally be able to detect one.

Austin Burgess is a writer and researcher with a background in sales, marketing, and data analytics. He holds an MBA, a Bachelor of Science in Business Administration, and a data analytics certification. His work focuses on breaking scientific developments, with an emphasis on emerging biology, cognitive neuroscience, and archaeological discoveries.