The core idea of the research is sensorless voltage estimation, meaning the reconstruction of phase voltages in a three-phase inverter from signals that are already available in any drive system, without dedicated voltage measurement hardware on the motor terminals. The team built their estimator on a rich experimental dataset of approximately 235,000 samples, collected under both steady-state and transient operating conditions. Rather than relying on a single measured quantity, the framework ingests a vector of features that any industrial drive can provide cheaply: the DC-link voltage, the input voltage, the motor speed, the phase currents, and the pulse-width modulation duty ratios that dictate how the inverter switches its transistors. From these inputs, the model learns to infer the actual phase voltages being applied to the machine, capturing the nonlinear behavior of the inverter that simple analytical formulas tend to miss.

The machine learning architecture at the center of the work is a stacking ensemble, a technique in which multiple diverse models are trained first and a second-level learner then combines their predictions to produce a final output that is more accurate than any individual member. Stacking works because different algorithms make different kinds of mistakes; a decision tree might err on one region of the operating envelope while a neural network errs on another, and a well-trained combiner learns when to trust which base model. In this study, the pool of candidate base learners included Decision Tree, Random Forest, Gradient Boosting, XGBoost, and neural network models. Among these individual models, XGBoost, a gradient-boosted tree algorithm known for its efficiency on tabular data, demonstrated superior performance, achieving a root mean square error of 2.97 volts, a mean absolute error of 2.05 volts, and a coefficient of determination of 0.999, outperforming all of its rivals in the comparison.

What distinguishes this work from prior applications of ensemble learning is the way the ensemble itself is tuned. The hyperparameters that govern a stacking ensemble, such as the settings of each base learner and the configuration of the meta-learner, form a vast search space that manual tuning or grid search cannot explore efficiently. The researchers turned to Particle Swarm Optimization, a bio-inspired algorithm that mimics the social behavior of bird flocks and fish schools. In PSO, a population of candidate solutions, called particles, flies through the hyperparameter space, each particle remembering the best position it has personally found and being attracted toward the best position discovered by the whole swarm. This balance between individual memory and collective knowledge allows the swarm to converge on promising regions of the search space without exhaustive enumeration.

The optimization setup in the study was deliberately lean. The authors employed a swarm of just 7 particles over a maximum of 20 iterations, with early stopping triggered after 5 non-improving iterations to avoid wasting computational effort once convergence had effectively been reached. This economy matters in practice, because hyperparameter optimization of ensembles can otherwise become prohibitively expensive, particularly for industrial teams without access to large computing clusters. The result of this swarm-guided search was a final optimized ensemble that improved upon even the strong XGBoost baseline, achieving a root mean square error of 2.87 volts, a mean absolute error of 2.04 volts, and a coefficient of determination of 0.999. In other words, the ensemble explained essentially all of the variance in the voltage data while keeping its typical error to a few volts.

Accuracy on aggregate metrics is one thing, but industrial control systems care about worst-case behavior across the full operating envelope. To probe this, the researchers evaluated the optimized ensemble across seventeen test cases spanning a voltage range of 28 to 360 volts, covering everything from low-voltage operation near the bottom of the range to the high voltages typical of demanding industrial drives. Across all seventeen cases, prediction errors remained consistently below 1.6 percent. That level of relative accuracy, sustained from the gentlest to the most extreme operating points, is what makes the framework credible for real deployment rather than merely impressive on a benchmark. It suggests the model has genuinely learned the underlying physics of the inverter rather than memorizing a narrow slice of operating conditions.

The significance of this performance becomes clearer when contrasted with conventional model-based estimation approaches. Classical observers and Kalman-style estimators require precise knowledge of motor parameters, and their accuracy collapses when those parameters deviate from the values assumed in the model, as inevitably happens with temperature rise, magnetic saturation, and aging. The data-driven framework, by contrast, captures nonlinear inverter dynamics and operational variability directly from data, absorbing the messy realities of hardware behavior into its learned mapping. Because it requires no additional sensors, it also avoids the cost, wiring complexity, and reliability concerns that come with instrumentation on power electronic systems, making it attractive for retrofitting into existing drives that were never designed with voltage sensing in mind.

The potential applications extend across the industrial landscape. Accurate, sensorless voltage estimates feed directly into real-time control loops, improving the quality of torque and speed regulation in motor drives. They also enable condition monitoring, since deviations between expected and inferred voltages can flag developing faults in the inverter or the machine before they escalate into failures. The authors frame the work as a contribution toward Sustainable Development Goal 9, Industry, Innovation and Infrastructure, arguing that intelligent, reliable, and cost-effective power electronic systems of this kind underpin the modernization of industrial infrastructure. A drive that monitors its own electrical state with software alone is cheaper to build, simpler to maintain, and less likely to fail unexpectedly than one dependent on physical sensing chains.

The study also offers a template for how bio-inspired optimization and machine learning can combine in engineering domains where data is abundant but physical modeling is hard. The modest swarm size and short iteration budget show that meaningful hyperparameter optimization does not require enormous computational resources, and the stacking architecture demonstrates that combining heterogeneous learners can extract accuracy that no single model reaches alone. The dataset itself draws on a publicly available benchmark together with experimental test bench data, and the authors note that further details can be made available from the corresponding author upon reasonable request, an openness that supports replication and extension by other groups.

As factories pursue greater automation and energy efficiency, the demand for software-defined sensing, in which algorithms replace hardware in measuring quantities that were once the exclusive province of physical instruments, will only grow. This research shows that with roughly a quarter of a million training samples, a carefully stacked ensemble, and a small flock of optimizing particles, the phase voltages of an industrial inverter can be inferred with sub-2-percent error across a thirteen-fold voltage range, without a single additional sensor. For engineers designing the next generation of motor drives, that is a compelling demonstration that the swarm, and the data, can see what the sensors used to.

Subject of Research: Sensorless phase voltage estimation in industrial three-phase inverter drives using a PSO-optimized stacking ensemble of machine learning models

Article Title: PSO-optimized ensemble learning framework for high-accuracy sensorless voltage estimation in industrial inverter drives

Article References: Nayak, P. K., Panda, K. P., Swain, D., & Panda, J. K. (2026). PSO-optimized ensemble learning framework for high-accuracy sensorless voltage estimation in industrial inverter drives. Complex & Intelligent Systems. https://doi.org/10.1007/s40747-026-02470-6

Image Credits: AI Generated

DOI: 10.1007/s40747-026-02470-6

Keywords: sensorless voltage estimation, ensemble learning, Particle Swarm Optimization, industrial inverter drives, XGBoost, stacking ensemble, machine learning, power electronics, condition monitoring, hyperparameter optimization, three-phase inverter, SDG 9