## 2 WOA and VMD Algorithms
### WOA Algorithm  
While optimization algorithm simulates hunting behavior of humpback whales, and is applied to solve complex optimization problems [16]. The calculation process mainly includes three parts: surrounding prey, attacking prey, and searching prey.  
(1) Surrounding prey The hunting process is able to determine the optimal prey position and surround it. The mathematical model of this process is shown as follow.  
\[\overset{\rightarrow}{D}=\left|\overset{\rightarrow}{C}\overset{\rightarrow}{X} ^{*}-\overset{\rightarrow}{X}(t)\right| \tag{5}\]  
\[\overset{\rightarrow}{X}(t+1)=\overset{\rightarrow}{X}^{*}(t)-\overset{ \rightarrow}{A}\overset{\rightarrow}{D} \tag{6}\]  
where \(t\) is the number of iterations; \(\overset{\rightarrow}{X}^{*}\) is the vector of prey position; \(\overset{\rightarrow}{X}\) is the whale position vector; \(\overset{\rightarrow}{D}\) is the hunting distance; \(\overset{\rightarrow}{A}\) and \(\overset{\rightarrow}{C}\) are coefficient vectors which are expressed as:  
\[\overset{\rightarrow}{A}=2 ag\overset{\rightarrow}{r}-a \tag{7}\]  
\[\overset{\rightarrow}{C}=2g\overset{\rightarrow}{r} \tag{8}\]  
where \(a\) is a convergence factor, and its value is [0, 2]; \(\overset{\rightarrow}{r}\) is the random vector of [0, 1].  
(2) Attacking prey  
Two methods are used to simulate the bubble-net attacking strategy of whales. The mathematical model is shown as follow.  
(1) Shrinking Encirclement  
This behavior can be realized by reducing \(a\) in Equation (7), and it should be noted that \(\overset{\rightarrow}{A}\) also decreases as \(a\) decreases.  
(2) Updating position  
With the spiral motion close to the prey, the mathematical model is shown as follow.  
\[\overset{\rightarrow}{X}(t+1)=\overset{\rightarrow}{D}e^{ll}\cos(2\pi l)+ \overset{\rightarrow}{X}^{*}(t) \tag{9}\]  
\[\overset{\rightarrow}{D}^{\prime}=\left|\overset{\rightarrow}{X}^{*}- \overset{\rightarrow}{X}(t)\right| \tag{10}\]  
where \(\overset{\rightarrow}{D}^{\prime}\) is the best solution; \(b\) is the helical shape parameter; \(l\) is the random number of [0, 1].  
In order to simulate these two behaviors of humpback whale hunting more accurately, the probability of contraction encirclement and spiral movement is considered to be 50%. Therefore, constructing Equation (11) to update the position of whales:  
\[\overset{\rightarrow}{X}(t+1)=\left\{\begin{array}{cc}\overset{\rightarrow }{X}^{*}(t)-\overset{\rightarrow}{A}\overset{\rightarrow}{D}&p<0.5\\ \overset{\rightarrow}{D}e^{ll}\cos(2\pi l)+\overset{\rightarrow}{X}^{*}(t)&p \geq 0.5\end{array}\right. \tag{11}\]  
where \(p\) is the random number.  
(3) Searching prey  
The search for preys is a random search that uses the relative position of the whale and does not depend on the location of the prey. The mathematical model is  
\[\overset{\rightarrow}{D}=\left|\overset{\rightarrow}{C}\overset{\rightarrow }{X}_{rand}(t)-\overset{\rightarrow}{X}(t)\right| \tag{12}\]  
\[\overset{\rightarrow}{X}(t+1)=\overset{\rightarrow}{X}_{rand}(t)-\overset{ \rightarrow}{A}\overset{\rightarrow}{D} \tag{13}\]  
where \(\overset{\rightarrow}{X}_{rand}(t)\) is a randomly determined whale location.  
According to the above mathematical model, when \(p>0.5\), the whales attack the prey. Otherwise the whales are in the search or siege phase. \(\left|A\right|\geq 1\) is a random search algorithm to update the location of the whale, and \(\left|A\right|<1\) refers to the stage of surrounding the prey after the maximum number of iterations to stop calculation.