## 3 Methodology
### Stability Analysis
During the exploration phase, division and multiplication operations facilitate a global search. These operators, due to their high discreteness, are apt for global exploration. The exploration method is described by (20):  
\[ \begin{array}{l}MOA\left(Z_{d}^{n}(Iter+1)\right)=\\ \left\{\begin{array}{l}Z_{B}(Iter)/(MP+\epsilon)\times((U_{d}-L_{d})\times \sigma_{1}+L_{d}),\ \ \gamma_{2}<0.5\\ Z_{B}(Iter)\times MOP\times((U_{d}-L_{d})\times\sigma_{1}+L_{d}),\ \ \gamma_{2}\geq 0.5\end{array}\right.,\end{array}  \tag{20}\]  
where \(Z_{d}^{n}(Iter+1)\) is the updated position of the \(n\)th solution in the \(d\)th dimension, and \(Z_{B}(Iter)\) is the position of the current best solution. \(\epsilon\), \(U_{d}\), and \(L_{d}\) represent the minimum integer, maximum and minimum values in the \(d\)th dimension. \(\sigma_{1}\) signifies a control parameter for the search phase, typically set at 0.499. \(\gamma_{2}\) is a random number in [0, 1]. The _MOP_ calculation is expressed in (21), with the sensitivity parameter set as follows:  
\[ \begin{array}{l}MOP(Iter)=1-\frac{Iter^{\frac{1}{\alpha}}}{Max_{Iter}^{ \frac{1}{\alpha}}},\end{array}  \tag{21}\]  
In the development phase, addition and subtraction operations focus on local development. These operators, with their low discreteness, enable in-depth local development. The development method is outlined in (22):  
\[ \begin{array}{l}Z_{d}^{n}(Iter+1)=\begin{cases}Z_{B}(Iter)-MOP\times((U_{d}- L_{d})\times\delta_{1}+L_{d}),\ \gamma_{3}<0.5\\ Z_{B}(Iter)+MOP\times((U_{d}-L_{d})\times\delta_{1}+L_{d}),\ \gamma_{3}\geq 0.5^{ \prime}\end{cases}\end{array}  \tag{22}\]  
where \(\gamma_{3}\) is a random number in [0, 1]. \(\delta_{1}\) is a design parameter that varies with each iteration, thus helping to break out of local optima. During the actual calibration process, the initialisation of AOA with random numbers resulted in prolonged optimisation times, reduced accuracy, and difficulties in escaping local minima or boundary values. To overcome these challenges, this paper incorporates chaotic mapping to enhance the randomness, sensitivity, and exploration capabilities of the algorithm, specifically targeting the issues arising from random number initialisation. However, the circular chaotic mapping did not yield uniformly distributed initial solutions. Consequently, an improvement to the circular chaotic mapping is introduced. The original circular chaotic mapping is outlined in (23). \[ \begin{array}{l}Y_{k+1}^{n}=mod(Z_{k}^{n}+0.2-\left(\frac{0.5}{0.2\pi} \text{sin}(2\pi\cdot x_{n})\right),1),\end{array}  \tag{23}\]  
To ensure a more uniform distribution of chaotic values, we propose an improved version of the circular chaotic mapping, detailed in (24). This improvement ensures that chaotic values are more evenly spread across the entire range, while maintaining the explorative and stable properties of the original mapping. \[ \begin{array}{l}Y_{k+1}^{n}=mod(3.85Z_{k}^{n}+0.4-\left(\frac{0.7}{3.85\pi} sin(3.85\pi\cdot x_{n})\right),1),\end{array}  \tag{24}\]where \(Y_{k+1}^{n}\) are chaotic sequences for the interval [0, 1]. \(n=1,2,3\ldots,N\) and \(k=1,2,3\ldots,D-1\). An inverse mapping is then performed to obtain the initial position of the population, as shown in Equation (25). \[Z_{d}^{n}=Y_{k+1}^{n}\times(U(N,D)-L(N,D))+L(N,D), \tag{25}\]  
Moreover, the _MOA_ value is crucial in determining whether the AOA transitions into global exploration or local development phases. A high _MOA_ value enhances local development but may impede global exploration, potentially leading to the algorithm becoming trapped in local optima. Conversely, a low _MOA_ value amplifies global exploration at the expense of local development, thereby extending optimisation times.