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                <term xml:lang="en">Stability analysis</term>
                <term xml:lang="en">Temperature control</term>
                <term xml:lang="en">PI α D controller design</term>
                <term xml:lang="en">Harris Hawks optimization (HHO)</term>
                <term xml:lang="en">Electric furnace</term>
                <term xml:lang="en">Electric furnace Harris Hawks optimization (HHO) PI α D controller design Temperature control Stability analysis</term>
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              <p>This study introduces the development and implementation of an advanced Proportional-Integral-Derivative (PID) control strategy, termed the Proportional Fractionalized Integral Derivative (PI α D) controller, aimed at enhancing transient dynamics, frequency response, and robustness in the regulation of electric furnace temperature. A key feature of the proposed controller design is the incorporation of the Harris Hawks Optimization (HHO) algorithm, employed to optimally tune the controller parameters. The selection of HHO is justified by its superior global search capability, fast convergence, and effectiveness in avoiding local minima, making it well-suited for addressing the complex, nonlinear characteristics of electric furnace systems. The suggested PI α D controller is used for the first time in electric furnace applications, providing a novel enhancement to traditional PID controllers by incorporating a fractional-order element. The controller's efficacy is evaluated through stringent simulations encompassing step reference alterations, load disturbances, and continuous random setpoints. Compared to classical PID, PID Acceleration (PIDA), and Real PID with second-order derivative (RPIDD²) controllers, the proposed PI α D controller exhibits superior performance, achieving the fastest rise time (1.65 s), shortest settling time (3.43 s), and lowest overshoot (0.12%). It also provides the best robustness trade-off, with high gain and phase margins, the largest bandwidth, and the lowest error indices. Frequency-domain analysis further confirms its enhanced disturbance rejection and stability, underscoring the suitability of the proposed controller and HHO for accurate, reliable, and energy-efficient temperature regulation in nonlinear industrial systems.</p>
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