## 3 DC Bus Voltage Control Strategy
### Improved Control Strategy Combined with the Variable Step-Size Algorithm
#### 3.3.3 The Variable Step-Size Algorithm Based on the Arctangent Function  
Although the specific principles of variable step-size algorithms differ, they all adhere to a core adjustment strategy. Generally, a larger step-size is employed in the early stages of iteration or when the error is significant to accelerate convergence. In the steady state or when the error has been minimized, a fixed step-size factor is applied to reduce the mean square error. This paper proposes the use of the arctangent function to create a nonlinear relationship between the error and the step-size. The arctangent function adaptive filter combined with linear active disturbance rejection control (AVALADRC) demonstrates better convergence, stability, and noise adaptability compared to the S-function and instantaneous error variable step-size algorithms. Due to the bounded and smooth output of the arctangent function, it provides rapid convergence when the error is large and gradually decreases the step-size as the error reduces, thereby avoiding the risk of over-adjustment. Moreover, the nonlinear characteristics of the arctangent function make the step-size adjustment more flexible, capable of adapting to different error environments and noise variations, thus improving the stability and robustness of the system. In contrast, the S-function may converge slowly under some extreme conditions, and the instantaneous error method is overly sensitive to error fluctuations, potentially leading to unstable behavior. The improved method introduces three adjustable parameters to account for different noise conditions. The modified step-size coefficient is represented as:  
\[\mu(n)=b\bigg{(}\frac{3}{\left(1+\exp\left(-a|\arctan(e(n))|\right)^{c}\right) }-1\bigg{)} \tag{21}\]  
The parameters \(a\), \(b\), and \(c\) in the variable step-size adaptive algorithm influence different characteristics of the algorithm. Parameter \(a\) controls the algorithm's sensitivity to the error; increasing \(a\) accelerates the convergence speed but may increase the steady-state error. Parameter \(b\) affects the upper limit of the maximum step-size; increasing \(b\) can accelerate convergence, but if \(b\) is too large, it may lead to an increase in the error. Parameter \(c\) mainly adjusts the smoothness of the algorithm; increasing \(c\) enhances the convergence speed and smoothens the bottom of the response, but it also increases computational complexity. In this paper, \(a=0.97\), \(b=0.003\), and \(c=4\).  
The variable step-size algorithm in Equation (21) ensures fast convergence while avoiding oscillations caused by excessive step-size adjustments. Compared to the instantaneous error-based algorithm in Equation (19), the nonlinear characteristics of the arctangent function in Equation (21) accelerate convergence when the error is large and smoothly adjust the step-size when the error is small, thereby improving control accuracy. Moreover, in contrast to the S-function algorithm in Equation (20), the algorithm in Equation (21) demonstrates better stability and robustness under noisy conditions or when error fluctuations are significant. Its three adjustable parameters allow the algorithm to flexibly adapt to different operating environments and error conditions, resulting in superior control performance in various scenarios. Therefore, in this study, we consider the variable step-size algorithm in Equation (21) to be the optimal choice.