## 4 Parallel Multi-Objective Energy Valley Optimizer  
### Basic EVO  
EVO searches the solution space by simulating the migration of particles from high-energy states to low-energy states in a physical system. [PERSON] et al. [28] and [PERSON] et al. [29] have shown that EVO has great potential for application in energy system scheduling.  
By the Energy Valley Optimizer, solution candidates (\(X_{i}\)) are considered as particles possessing varying degrees of stability within the search space, metaphorically regarded as a specific segment of the universe. The Neutron Enrichment Level (NEL) is determined by the fitness level of each particle (solution candidates) and dictates the stability level of each particle, as determined by the NEL.  
\[\text{SL}_{i}=\frac{\text{NEL}_{i}-\text{BS}}{\text{WS}-\text{BS}},i=1,2, \ldots,\text{n}. \tag{39}\]  
where BS and WS represent the particles exhibiting the highest and lowest stability levels within this universe, corresponding to the minimum and maximum values, respectively. The variable \(n\) represents the total number of solutions within the solution set.  
EVO uses Enrichment Bound to consider the differences between neutron-rich and neutron-poor particles. The calculation method for Enrichment Bound is as follows:  
\[\text{EB}=\frac{\sum_{i=1}^{n}\text{NEL}_{i}}{n},i=1,2,\ldots,n. \tag{40}\]  
EVO generates a random number within the range of \([0,1]\) to simulate the Stability Bound (SB) in the system. By comparing each particle's NEL and SL with EB and SB, the occurrence of alpha decay, beta decay, gamma decay, or random movement is determined. The equations for alpha decay, beta decay, gamma decay, and random movement to generate new solutions are as follows:  
\[\text{X}_{i}^{\text{Alpha}}=\text{X}_{i}\Big{(}\text{X}_{\text{BS}}\Big{(}x_{ i}^{j}\Big{)}\Big{)},\begin{cases}i=1,2,\ldots,n.\\ j=\text{Alpha Index}.\end{cases} \tag{41}\]  
\[\text{X}_{i}^{\text{New1},beta}=\text{X}_{i}+\frac{(\text{r}_{1}\times\text{X} _{\text{BS}}-\text{r}_{2}\times\text{X}_{\text{CP}})}{\text{SL}_{i}},i=1,2, \ldots,\text{n}. \tag{42}\]  
\[\text{X}_{i}^{\text{New2},beta}=\text{X}_{i}+\big{(}\text{r}_{3}\times\text{X} _{\text{BS}}-\text{r}_{4}\times\text{X}_{\text{N}\text{S}}\big{)},i=1,2, \ldots,\text{n}. \tag{43}\]  
\[\text{X}_{i}^{\text{Gamma}}=\text{X}_{i}\Big{(}\text{X}_{\text{N}\text{S}} \Big{(}x_{i}^{j}\Big{)}\Big{)},\begin{cases}i=1,2,\ldots,n.\\ j=\text{Gamma Index}.\end{cases} \tag{44}\]  
\[\text{X}_{i}^{\text{random movement}}=\text{X}_{i}+r,i=1,2,\ldots,\text{n}. \tag{45}\]  
where \(\text{X}_{i}^{\text{Alpha}}\), \(\text{X}_{i}^{\text{New1},beta}\), \(\text{X}_{i}^{\text{New2},beta}\), \(\text{X}_{i}^{\text{Gamma}}\), and \(\text{X}_{i}^{\text{random movement}}\) are resulting in new solutions; \(\text{X}_{i}\) is the \(t\)-th solution candidate at present. \(\text{x}_{i}^{j}\) denotes the value of the \(j\)-th dimension of the solution candidate \(\text{X}_{i}\); \(\text{X}_{\text{BS}}\) is the best SL solution currently; \(\text{X}_{\text{CP}}\) is the center of the current solution set; \(\text{X}_{\text{N}\text{S}}\) is an adjacent solution to the solution \(\text{X}_{i}\); \(\text{r}_{1}\), \(\text{r}_{2}\), \(\text{r}_{3}\), \(\text{r}_{4}\), and r are all random numbers in \([0,1]\); Alpha Index and Gamma Index are random numbers in \([1,d]\), where \(d\) is the dimension of the solution candidate. In EVO, alpha decay, beta decay, and gamma decay are used to exploit the solution space, while random movement is used to explore it.