# A Sea Horse Optimization-Based Approach for PEM Fuel Cell Model Parameter Estimation
## 3 Sea Horse Optimization Algorithm (SHO)
### Movement Behavior  
1. Condition: In their natural habitats, seahorses demonstrate unique movement patterns to explore food sources and investigate new living areas. To mathematically simulate this natural motion, the current best solution in the search space is considered as \(X_{\text{elite}}\). This reference point guides the movement of the entire population toward optimal solutions within the search domain. \[X_{\text{new}}^{1}(t+1)=X_{i}(t)+\text{Levy}(\lambda)((X_{\text{elite}}(t)-X_ {i}(t))\times x\times y\times z+X_{\text{elite}}(t)),\quad r_{1}>0\] (12) This initial behavior is inspired by the spiral motion of seahorses as they create vortices while swimming in the ocean and plays a critical role in enabling local exploitation. When the standard random variable \(r_{1}\) exceeds a predefined threshold, seahorses imitate spiral trajectories, progressing steadily toward the elite solution. This movement behavior is further enhanced by Levy flight patterns [33], which dynamically adjust step sizes, allowing the algorithm to explore diverse regions in early iterations and avoid premature convergence to local optima. Simultaneously, the spiral turning angle is continuously adjusted to expand the area around local optima, thereby enabling more detailed exploration. Mathematically, this spiral motion is described by Equation (12) using the three-dimensional coordinates \((x,y,z)\) that define the updated position of the seahorse. The spiral length \(\rho\) and turning angle \(\theta\) define the geometry of the motion, while the Levy flight mechanism introduces stochastic variation. Together, these components allow seahorses to dynamically adapt their positions in the search space, emulating purposeful swimming behavior and maintaining an effective balance between exploration and exploitation [33].
2. Condition: Among the movement behaviors exhibited by seahorses is Brownian motion, which is characterized by random drifting along with ocean currents [34]. This motion is activated when the value of \(r_{1}\) is less than a predefined threshold, corresponding to the exploration phase of the algorithm. In this context, the SHO algorithm employs this behavior to prevent entrapment in local extrema and to enable a more extensive search of the solution space. Brownian motion enhances the exploration capability by simulating a new movement length for the seahorses, thereby promoting broader search dynamics. Mathematically, the updated position expression for this movement is as in Equation (13): \[X_{\text{new}}^{1}(t+1)=X_{i}(t)+\text{rand}\cdot l\cdot\beta_{t}\cdot(X_{i}(t) -\beta_{t}\cdot X_{\text{elite}})\quad\text{for }r_{1}\leq 0\] (13)
* \(l\) is a constant coefficient.
* \(\beta_{t}\) is the Brownian motion random step size coefficient, which follows a standard normal distribution and represents a randomly generated value.  
In conclusion, the movement phase of the seahorses can be summarized based on two distinct behavioral models:  
* Spiral Motion: When \(r_{1}>0\), the seahorses moves toward the elite solution through a spiral trajectory.
* Brownian Motion: When \(r_{1}\leq 0\), it randomly drifts with ocean currents to explore new regions.  
These two behavior mechanisms simulate seahorses' adaptive behavior under uncertain environmental conditions and effectively balance exploration and exploitation dynamics in the algorithm.