Eng. Zahraa Ibrahim Mahdi Behind every lamp that lights up your home and every motor running in a factory lies a complex network of mathematical balances. Power systems and control systems are not just wires and machines; they are living mathematical entities governed by equations that ensure the world stays “lit” and stable. 1. The Language of Stability: Differential Equations In power engineering, electricity flow is never constant; consumers constantly switch devices on and off. Here, differential equations play a critical role: System Description: Differential equations describe the behavior of generators and motors during sudden disturbances. Angle Stability: Engineers rely on the Swing Equation to predict whether a generator will continue to rotate in synchrony with the grid or fail due to overload. 2. Mathematical Transforms: From Complexity to Simplicity Laplace and Fourier transforms are the magic tools of control engineers: Frequency Domain: Instead of solving complex differential equations in the time domain, these transforms move the problem into the frequency domain, where complex operations become simple algebraic ones. Transfer Function: This mathematical formula represents the relationship between system inputs and outputs, allowing engineers to design controllers that achieve desired performance with minimal error and fastest response. 3. Linear Algebra and Smart Grids As the world shifts toward renewable energy (solar and wind), power networks become more complex: Power Flow Analysis: Large matrices are used to solve thousands of instantaneous equations determining the energy passing through each line. Matrix Stability: Eigenvalues help identify weak points in the electrical grid before failures occur. 4. Control Systems and Feedback The goal of control is to make the system correct itself automatically. The famous mathematician Lyapunov laid the foundations for stability in such systems: PID Algorithm: The most widely used equation in factories, relying on proportional, integral, and derivative terms to precisely control motor speed or reactor temperature. Optimization: Mathematics helps determine the optimal operating point, maximizing power efficiency while minimizing fuel consumption. Al-Mustaqbal University — the first university in Iraq.