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

Buildings are responsible for an enormous share of global electricity demand, and the way their air conditioners, lights, and heating systems are scheduled can make the difference between a grid under strain and one running smoothly. For years, engineers have tried to optimize these schedules with evolutionary algorithms and reinforcement learning, but the problem becomes brutally hard when dozens of buildings share a single electrical network and must be coordinated simultaneously. A new study published in Discover Artificial Intelligence proposes a strikingly different approach: instead of searching step by step for the best schedule, a generative artificial intelligence model learns to paint one directly out of pure noise, guided by a map of how the buildings are physically wired together.

The framework, called DMCBO, short for Diffusion-driven Multi-objective Collaborative Optimization for Building Clusters, was developed by Yalan Zheng of Zhengzhou Technical College and Taixiang Yin of North China University of Water Resources and Electric Power. Their central insight is that the joint scheduling of equipment settings and load distribution across an entire cluster of buildings can be treated as a high-dimensional data sample, much like an image. Diffusion models, the same family of generative techniques behind modern text-to-image systems, work by progressively corrupting data with Gaussian noise and then learning to reverse that corruption. The researchers reformulated multi-objective building optimization as exactly this kind of conditional denoising process: starting from random noise, the model strips away noise step by step until a complete, Pareto-optimal collaborative schedule for every piece of equipment in every building emerges.

Technically, the forward diffusion process is a fixed Markov chain that adds tiny amounts of Gaussian noise to real scheduling data over one hundred steps, with the noise coefficient increasing linearly from 0.0001 to 0.02 until the original data becomes indistinguishable from standard Gaussian noise. A neural network is then trained to predict the noise that was added at each step, given the noisy schedule and a conditional encoding vector describing the current situation. Once trained, the model can generate fresh solutions at inference time by sampling pure noise and running the reverse process, refining it over one hundred denoising steps into a schedule tailored to the prevailing weather, occupancy, and grid conditions. Because this generation is a gradient-based learning process rather than an iterative search, it avoids the local optima traps that plague traditional multi-objective evolutionary algorithms and the reward oscillations that destabilize deep reinforcement learning.

What makes the system genuinely cluster-aware is its use of graph neural networks. The researchers model the building cluster as a graph mapped onto a radial distribution microgrid topology, where nodes represent buildings and edges encode physical constraints such as maximum line thermal capacities and transformer limits. A two-layer graph convolutional network passes messages between neighboring buildings, fusing spatial-topological dependencies and network constraints into a single graph embedding. This embedding is concatenated with an adjustable preference weight vector to form the condition that steers the denoising trajectory. In effect, when the model generates a control strategy for one building, it simultaneously perceives the state of its neighbors and the limits of the shared electrical infrastructure, suppressing systemic coordination errors from 0.18 down to 0.06 in the team’s experiments.

The multi-objective aspect is handled through an elegant scalarization trick. Each objective, such as energy consumption, operating cost, or thermal comfort, is normalized to a dimensionless value between zero and one using Min-Max normalization based on historical operational bounds. The normalized objectives are then blended into a single scalar condition via a weight vector whose components sum to one. During training, these weights are sampled from a uniform Dirichlet distribution, exposing the model to the entire space of possible preferences so it learns the distribution of the full Pareto front. At deployment, a decision maker can slide the weight vector like an adjustment knob, instantly generating schedules that emphasize energy saving, comfort, or a balanced compromise, all within the same trained model.

The denoising backbone itself is a U-Net adapted to process one- and two-dimensional building scheduling tensors. Diffusion time steps are embedded through sinusoidal positional encodings, while the graph-derived condition is injected into every residual block via Adaptive Layer Normalization. A bottleneck layer with multi-head self-attention captures long-range dependencies across the entire daily scheduling horizon, and symmetric skip connections preserve high-resolution information during reconstruction. The authors trained the model for 300 epochs on dual NVIDIA Tesla V100 GPUs, and the loss curves dropped from 0.75 to below 0.03 with training and validation curves closely coinciding, indicating no overfitting. As the reverse diffusion step count decreased from 90 to 10 during analysis, the noise-prediction error distribution became sharply concentrated at zero, demonstrating extremely high denoising precision at the end of generation.

Validation was carried out on the open-access Building Data Genome Project 2 dataset, which supplied real hourly weather records, operational schedules, and annual electricity, cooling, and heating load profiles across diverse climate zones. Because that dataset provides metered energy data rather than physical building models, the team used United States Department of Energy Commercial Reference Buildings as templates and calibrated them in EnergyPlus, a high-fidelity thermodynamic simulation platform, to create a digital twin consistent with the real data. All results were benchmarked against a rule-based control strategy with static thermostat setpoints of 25 degrees Celsius in summer and 22 degrees in winter, and every experiment was repeated ten independent times to confirm statistical significance.

The headline numbers are impressive. In high-demand summer conditions, DMCBO achieved an average energy-saving rate of 28.6 percent and cut peak load by 38.5 percent, shaving the afternoon peak from roughly 560 kilowatts down to about 440 kilowatts and saving approximately 142.80 US dollars per day. Cumulative daily power savings exceeded 1,000 kilowatt-hours, compared with 400 kilowatt-hours for a traditional collaborative strategy and just 120 kilowatt-hours without coordination. In winter, peak shaving and valley filling reached about 39 and 42 percent respectively, far above the 22 and 25 percent of conventional methods. When the solution sets were projected into objective space, DMCBO dominated the baselines: it reached a hypervolume indicator of 0.95 within 12,000 function evaluations, versus 0.85 for NSGA-II and 0.70 for MOPSO, while keeping operating costs between 140 and 160 dollars against more than 210 dollars for deep reinforcement learning and 260 to 300 dollars for a GAN-based generative approach.

Robustness and scalability proved equally compelling. Under injected stochastic disturbances mimicking sudden weather changes and occupancy swings, the framework confined power fluctuations to a narrow band of plus or minus 5 kilowatts, whereas the deep reinforcement learning baseline swung between minus 15 and plus 20 kilowatts. Scaling from 5 to 50 buildings, the global energy-saving rate actually rose from 28.5 to 31.2 percent, because larger clusters unlock more load-complementary opportunities that the graph encoder can safely exploit within encoded grid boundaries. Computation time grew to 9.8 seconds at the 50-building scale, but that remains comfortably within the 15-minute control horizon used by real building management systems, and the hypervolume index slipped only slightly to 0.88. Ablation experiments confirmed that both the graph neural network module and the choice of 100 diffusion steps, which yields a 50-millisecond inference latency, were essential to this balance of speed, accuracy, and stability.

The authors are candid about limitations. The framework has so far been validated only in simulation, and EnergyPlus simplifies sub-minute mechanical actuation latencies that real HVAC hardware would introduce; they propose cascading PID loops and predictive safety margins to absorb those delays in practice. Data quality, generalization to residential buildings and extreme climates, the computational cost of scaling to city-sized networks, and the black-box nature of the generative process all remain open challenges, with explainable AI and transfer learning flagged as future work. Still, the architecture is designed to slot into standard building management systems as a supervisory layer, transmitting setpoints over protocols such as BACnet or Modbus. If the simulation gains survive contact with physical buildings, the idea of generating optimal energy schedules from noise, guided by a graph of the grid itself, could become a powerful new tool in the push toward decarbonized cities.

Subject of Research: Conditional diffusion models with graph neural networks for multi-objective energy optimization in building clusters

Article Title: Diffusion model-driven multi-objective collaborative optimization for building energy management using graph neural networks

Article References: Zheng, Y., & Yin, T. (2026). Diffusion model-driven multi-objective collaborative optimization for building energy management using graph neural networks. Discover Artificial Intelligence, 6(1), Article 1383. https://doi.org/10.1007/s44163-026-02140-z

Image Credits: AI Generated

DOI: 10.1007/s44163-026-02140-z

Keywords: diffusion models, graph neural networks, building energy management, multi-objective optimization, EnergyPlus, smart buildings, Pareto optimization, deep learning, energy efficiency, HVAC control, microgrids, generative AI

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