Error minimized LO modeling of electric vehicle integrated off-grid microgrids using Taylor-Laurent series expansion and BBO based optimization under stability and steady state constraints

Keeping the Lights On in Remote Places

Remote communities and islands increasingly rely on small, self-contained power networks known as off-grid microgrids. These systems juggle electricity from solar panels, wind turbines, biofuel generators, batteries, and even parked electric cars acting as mobile batteries. Managing all of this in real time is a complex balancing act: the power must be steady and stable even as sunshine, wind, and charging needs constantly change. The paper summarized here introduces a way to dramatically simplify how engineers model these microgrids, making them easier to design and control without sacrificing reliability.

Why Small Power Networks Matter

Off-grid microgrids promise clean, reliable power in places far from traditional utility lines or where the main grid is weak. They combine many different devices: solar photovoltaic panels, wind turbines, biodiesel and biogas generators, microturbines, battery storage, and electric vehicles that can both draw from and feed into the system. Together, these pieces create a flexible but intricate network whose behavior is described mathematically by a “higher-order” model with many dynamic components. Such detailed models are accurate but can be slow to simulate and hard to use when designing controllers that keep voltage and frequency within safe limits.

Turning a Complicated System into a Simple One

The authors tackle this problem by reducing a seventh‑order mathematical description of an electric-vehicle-integrated off-grid microgrid to a much simpler second‑order model. In everyday terms, they compress a long, detailed recipe into a short version that still tastes the same. Their key idea is to look at the system from two complementary viewpoints. One viewpoint focuses on what happens slowly over time, such as how the system settles after a disturbance; the other focuses on fast, sharp changes, such as brief swings in power or voltage. Mathematically, these two viewpoints are captured using Taylor and Laurent series expansions, which expand the system’s behavior around low and high frequencies.

Letting a Virtual Bear Hunt for the Best Match

To make the simplified model mimic the original as closely as possible, the authors turn the task into an optimization problem: they define a “fitness” measure that quantifies the mismatch between the complex and simplified models from both slow and fast perspectives. This measure combines three separate error terms derived from the Taylor and Laurent expansions. They then let a computer search for the best set of parameters for the simple model using a nature-inspired method called the Brown Bear Optimization algorithm. This algorithm, modeled loosely on how brown bears mark and sniff tracks, explores many possible parameter combinations and gradually homes in on those that minimize the error.

Stability and Accuracy as Non‑Negotiable Rules

While the optimization is running, it must obey two strict rules. First, the simplified model must have zero steady‑state error, meaning that after any disturbance it settles to exactly the same final value as the original system. Second, it must meet a classical stability requirement known as the Hurwitz criterion, which ensures the model does not drift or oscillate uncontrollably. Under these constraints, the Brown Bear Optimization algorithm produces a second‑order model that matches the original microgrid’s key behavior in both time and frequency domains. When the authors compare their model with others created by more traditional reduction techniques, their approach consistently yields smaller errors in standard performance measures and better agreement in step, impulse, and frequency‑response plots.

What This Means for Future Microgrids

For non‑specialists, the takeaway is that the authors have designed a highly compact mathematical stand‑in for a very complex off‑grid microgrid that includes electric vehicles. This reduced model is accurate enough to reproduce the original system’s most important dynamics, yet simple enough to speed up simulations and make controller design more practical. In the long run, such tools can help engineers design cleaner, more reliable microgrids for remote communities and critical facilities, enabling better use of renewable energy and electric vehicles without compromising stability or power quality.

Citation: Chaudhary, R., Singh, V.P., Mathur, A. et al. Error minimized LO modeling of electric vehicle integrated off-grid microgrids using Taylor-Laurent series expansion and BBO based optimization under stability and steady state constraints. Sci Rep 16, 13561 (2026). https://doi.org/10.1038/s41598-026-43306-0

Keywords: off-grid microgrid, electric vehicles, model reduction, renewable energy, optimization algorithm