数学双栏试卷中不需要的空格

Question 1

你面临的主要问题是split环境不是允许分栏（和分页）。要允许分栏，我建议您从嵌套的equation*/split设置切换到align*设置。然后，将\tag和\label指令放在您认为正确的行中。

\documentclass[journal]{IEEEtran} 

\usepackage{amssymb,amsmath}
\newcommand\tildeE{\tilde{\mathrm{e}}} % 61 [!] occurrences...
\allowdisplaybreaks 

%\usepackage{newtxmath} % optional: Times Roman math font

\begin{document}

\begin{align*} 
&\mathbb{E}\{\bigtriangleup{V}(\tildeE_{i,k})\}
= \mathbb{E} \biggl\{ \sum_{i=1}^{N}
\Bigl[ \tildeE_{i,k}^T\mathrm{S}_i^TP_i\mathrm{S}_i\tildeE_{i,k}  
  +\tildeE_{i,k}^T\mathrm{S}_i^TP_i\Phi_1\tildeE_k 
\\&\quad+\tildeE_{i,k}^T\mathrm{S}_i^TP_i\Psi(\tildeE_{i,k}, \rho_k) 
\\&\quad- \tildeE_{i,k}^T\mathrm{S}_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
\\&\quad+\tildeE_{i,k}^T\mathrm{S}_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k 
  - \tildeE_{i,k}^T\mathrm{S}_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} 
\\&\quad+ \tildeE_{k}^{T}\Phi_1^{T}P_iS_i\tildeE_{i,k}
  + \tildeE_k^T\Phi_1^TP_i\Phi_1\tildeE_k 
\\&\quad+ \tildeE_k^T\Phi_1^TP_i\Psi(\tildeE_{i,k}, \rho_k)
\\&\quad- \tildeE_k^T\Phi_1^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k) 
\\&\quad+\tildeE_k^T\Phi_1^TP_i\bar{\mathrm{B}}\mathrm{w}_k 
  -\tildeE_k^T\Phi_1^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}  
\\&\quad+\Psi^T(\tildeE_{i,k}, \rho_k)P_i\mathrm{S}_i\tildeE_{i,k} 
\\&\quad+\Psi^T(\tildeE_{i,k}, \rho_k)P_i\Phi_1\tildeE_k
\\&\quad+\Psi^T(\tildeE_{i,k}, \rho_k)P_i\Psi(\tildeE_{i,k}, \rho_k) 
\\&\quad- \Psi^T(\tildeE_{i,k}, \rho_k)P_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)  
\\&\quad+\Psi^T(\tildeE_{i,k}, \rho_k)P_i\bar{\mathrm{B}}\mathrm{w}_k 
\\&\quad-\Psi^T(\tildeE_{i,k}, \rho_k)P_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
\\&\quad-\zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\mathrm{S}_i\tildeE_{i,k}  
\\&\quad-\zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\Phi_1\tildeE_k 
\\&\quad-\zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\Psi(\tildeE_{i,k}, \rho_k)
\\&\quad+\zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k) 
\\&\quad- \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k 
\\&\quad+ \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} 
\\&\quad+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\mathrm{S}_i\tildeE_{i,k} 
  + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\Phi_1\tildeE_k  
\\&\quad+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\Psi(\tildeE_{i,k}, \rho_k)
\\&\quad- \mathrm{w}_k^T\bar{\mathrm{B}}^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k) 
\\&\quad+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\bar{\mathrm{B}}\mathrm{w}_k  
  -\mathrm{w}_k^T\bar{\mathrm{B}}^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} 
\\&\quad-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\mathrm{S}_i\tildeE_{i,k} 
  -\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\Phi_1\tildeE_k  
\\&\quad-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\Psi(\tildeE_{i,k}, \rho_k) 
\\&\quad+\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k) 
\\&\quad-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k 
  +\mathrm{v}_{i,k}D_i^TL_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} \smash{\Bigr]} 
\\&\quad- \sum_{i=1}^{N}\tildeE_{i,k}^TP_i\tildeE_{i,k} \biggr\}.
\tag{41} \label{41}
\end{align*}

Substituting equation \eqref{41} into \eqref{40} and manipulating it 
similarly to \eqref{19} leads to
\begin{align*} 
&J = \mathbb{E} \biggl\{ \sum_{i=1}^{N}
\Bigl[ \tildeE_{i,k}^T\mathrm{S}_i^TP_i\mathrm{S}_i\tilde{e}_{i,k}  +\tildeE_{i,k}^T\mathrm{S}_i^TP_i\Phi_1\tildeE_k  
+\tildeE_{i,k}^T\mathrm{S}_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k 
\\&\quad- \tildeE_{i,k}^TS_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} 
  + \tildeE_{k}^{T}\Phi_1^{T}P_i\mathrm{S}_i\tildeE_{i,k}
  + \tildeE_k^T\Phi_1^TP_i  \Phi_1\tildeE_k 
\\&\quad+\tildeE_k^T\Phi_1^TP_i\bar{\mathrm{B}}\Phi_1\mathrm{w}_k 
  -\tildeE_k^T\Phi_1^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}   
  + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\mathrm{S}_i\tildeE_{i,k} 
\\&\quad+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\Phi_1\tildeE_k  
  + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\bar{\mathrm{B}}\mathrm{w}_k 
  - \mathrm{w}_k^T\bar{\mathrm{B}}^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} 
\\&\quad-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\mathrm{S}_i\tildeE_{i,k}
  -\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\Phi_1\tilde{e}_k 
\\&\quad-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\bar{\mathrm{B}}w_k 
  +\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} 
\\&\quad- \sum_{i=1}^{N}\tildeE_{i,k}^TP_i\tildeE_{i,k}  
  + \sigma N\sum_{i=1}^{N}\tildeE_{i,k}^{T}\tildeE_{i,k} 
  - \sigma\sum_{i=1}^{N}\tildeE_k^{T}\tildeE_k 
\\&\quad+ f'^{T}(x_k,\vartheta, \rho_k)P_if'(x_k,\vartheta, \rho_k)  
\\&\quad+ \lambda_{\max}(L_i^TP_iL_i) \zeta_i^T(\tilde{x}_{i,k}, \tau_k) \zeta_i(\tilde{x}_{i,k}, \tau_k) \Bigr]
\\&\quad+ \sum_{i=1}^{N} \dfrac{1}{\xi N} \tildeE_{i,k}^{T}\tildeE_{i,k} 
  - \gamma_1 \mathrm{w}_k^{T}\mathrm{w}_k 
  - \sum_{i=1}^{N} \gamma_2 \mathrm{v}_{i,k}^{T}\mathrm{v}_{i,k} \biggr{\}}.
  \tag{42} \label{42}
\end{align*} 
\end{document}

Answer

你面临的主要问题是split环境不是允许分栏（和分页）。要允许分栏，我建议您从嵌套的equation*/split设置切换到align*设置。然后，将\tag和\label指令放在您认为正确的行中。

\documentclass[journal]{IEEEtran} 

\usepackage{amssymb,amsmath}
\newcommand\tildeE{\tilde{\mathrm{e}}} % 61 [!] occurrences...
\allowdisplaybreaks 

%\usepackage{newtxmath} % optional: Times Roman math font

\begin{document}

\begin{align*} 
&\mathbb{E}\{\bigtriangleup{V}(\tildeE_{i,k})\}
= \mathbb{E} \biggl\{ \sum_{i=1}^{N}
\Bigl[ \tildeE_{i,k}^T\mathrm{S}_i^TP_i\mathrm{S}_i\tildeE_{i,k}  
  +\tildeE_{i,k}^T\mathrm{S}_i^TP_i\Phi_1\tildeE_k 
\\&\quad+\tildeE_{i,k}^T\mathrm{S}_i^TP_i\Psi(\tildeE_{i,k}, \rho_k) 
\\&\quad- \tildeE_{i,k}^T\mathrm{S}_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
\\&\quad+\tildeE_{i,k}^T\mathrm{S}_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k 
  - \tildeE_{i,k}^T\mathrm{S}_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} 
\\&\quad+ \tildeE_{k}^{T}\Phi_1^{T}P_iS_i\tildeE_{i,k}
  + \tildeE_k^T\Phi_1^TP_i\Phi_1\tildeE_k 
\\&\quad+ \tildeE_k^T\Phi_1^TP_i\Psi(\tildeE_{i,k}, \rho_k)
\\&\quad- \tildeE_k^T\Phi_1^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k) 
\\&\quad+\tildeE_k^T\Phi_1^TP_i\bar{\mathrm{B}}\mathrm{w}_k 
  -\tildeE_k^T\Phi_1^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}  
\\&\quad+\Psi^T(\tildeE_{i,k}, \rho_k)P_i\mathrm{S}_i\tildeE_{i,k} 
\\&\quad+\Psi^T(\tildeE_{i,k}, \rho_k)P_i\Phi_1\tildeE_k
\\&\quad+\Psi^T(\tildeE_{i,k}, \rho_k)P_i\Psi(\tildeE_{i,k}, \rho_k) 
\\&\quad- \Psi^T(\tildeE_{i,k}, \rho_k)P_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)  
\\&\quad+\Psi^T(\tildeE_{i,k}, \rho_k)P_i\bar{\mathrm{B}}\mathrm{w}_k 
\\&\quad-\Psi^T(\tildeE_{i,k}, \rho_k)P_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
\\&\quad-\zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\mathrm{S}_i\tildeE_{i,k}  
\\&\quad-\zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\Phi_1\tildeE_k 
\\&\quad-\zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\Psi(\tildeE_{i,k}, \rho_k)
\\&\quad+\zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k) 
\\&\quad- \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k 
\\&\quad+ \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} 
\\&\quad+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\mathrm{S}_i\tildeE_{i,k} 
  + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\Phi_1\tildeE_k  
\\&\quad+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\Psi(\tildeE_{i,k}, \rho_k)
\\&\quad- \mathrm{w}_k^T\bar{\mathrm{B}}^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k) 
\\&\quad+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\bar{\mathrm{B}}\mathrm{w}_k  
  -\mathrm{w}_k^T\bar{\mathrm{B}}^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} 
\\&\quad-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\mathrm{S}_i\tildeE_{i,k} 
  -\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\Phi_1\tildeE_k  
\\&\quad-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\Psi(\tildeE_{i,k}, \rho_k) 
\\&\quad+\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k) 
\\&\quad-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k 
  +\mathrm{v}_{i,k}D_i^TL_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} \smash{\Bigr]} 
\\&\quad- \sum_{i=1}^{N}\tildeE_{i,k}^TP_i\tildeE_{i,k} \biggr\}.
\tag{41} \label{41}
\end{align*}

Substituting equation \eqref{41} into \eqref{40} and manipulating it 
similarly to \eqref{19} leads to
\begin{align*} 
&J = \mathbb{E} \biggl\{ \sum_{i=1}^{N}
\Bigl[ \tildeE_{i,k}^T\mathrm{S}_i^TP_i\mathrm{S}_i\tilde{e}_{i,k}  +\tildeE_{i,k}^T\mathrm{S}_i^TP_i\Phi_1\tildeE_k  
+\tildeE_{i,k}^T\mathrm{S}_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k 
\\&\quad- \tildeE_{i,k}^TS_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} 
  + \tildeE_{k}^{T}\Phi_1^{T}P_i\mathrm{S}_i\tildeE_{i,k}
  + \tildeE_k^T\Phi_1^TP_i  \Phi_1\tildeE_k 
\\&\quad+\tildeE_k^T\Phi_1^TP_i\bar{\mathrm{B}}\Phi_1\mathrm{w}_k 
  -\tildeE_k^T\Phi_1^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}   
  + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\mathrm{S}_i\tildeE_{i,k} 
\\&\quad+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\Phi_1\tildeE_k  
  + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\bar{\mathrm{B}}\mathrm{w}_k 
  - \mathrm{w}_k^T\bar{\mathrm{B}}^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} 
\\&\quad-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\mathrm{S}_i\tildeE_{i,k}
  -\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\Phi_1\tilde{e}_k 
\\&\quad-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\bar{\mathrm{B}}w_k 
  +\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} 
\\&\quad- \sum_{i=1}^{N}\tildeE_{i,k}^TP_i\tildeE_{i,k}  
  + \sigma N\sum_{i=1}^{N}\tildeE_{i,k}^{T}\tildeE_{i,k} 
  - \sigma\sum_{i=1}^{N}\tildeE_k^{T}\tildeE_k 
\\&\quad+ f'^{T}(x_k,\vartheta, \rho_k)P_if'(x_k,\vartheta, \rho_k)  
\\&\quad+ \lambda_{\max}(L_i^TP_iL_i) \zeta_i^T(\tilde{x}_{i,k}, \tau_k) \zeta_i(\tilde{x}_{i,k}, \tau_k) \Bigr]
\\&\quad+ \sum_{i=1}^{N} \dfrac{1}{\xi N} \tildeE_{i,k}^{T}\tildeE_{i,k} 
  - \gamma_1 \mathrm{w}_k^{T}\mathrm{w}_k 
  - \sum_{i=1}^{N} \gamma_2 \mathrm{v}_{i,k}^{T}\mathrm{v}_{i,k} \biggr{\}}.
  \tag{42} \label{42}
\end{align*} 
\end{document}

Question 2

@Mico 答案的一个小变化：

使用的是包\MoveEqLeft中定义的宏mathtools
有一点是重新排列的数学术语

\documentclass[journal]{IEEEtran}

\usepackage{amssymb,
            mathtools}
\newcommand\tildeE{\tilde{\mathrm{e}}} % 61 [!] occurrences...
\allowdisplaybreaks
    \def\arraystretch{2}

\usepackage{lipsum}

\begin{document}
\lipsum[1-2]
    \begin{align*}
    \MoveEqLeft[1]
\mathbb{E}\{\Delta{V}(\tildeE_{i,k})\}
    = \mathbb{E} \biggl\{ \sum_{i=1}^{N}\Bigl[ \tilde{E}_{i,k}^T\mathrm{S}_i^T P_i \mathrm{S}_i \tildeE_{i,k}
        + \tilde{e}_{i,k}^T \mathrm{S}_i^T P_i\Phi_1\tildeE_k                       \\
    & + \tildeE_{i,k}^T\mathrm{S}_i^T P_i\Psi(\tildeE_{i,k}, \rho_k)          
        - \tildeE_{i,k}^T\mathrm{S}_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)       \\   
    & + \tildeE_{i,k}^T\mathrm{S}_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k
        - \tildeE_{i,k}^T\mathrm{S}_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}       
        + \tildeE_{k}^{T}\Phi_1^{T}P_iS_i\tildeE_{i,k}                              \\
    & + \tildeE_{k}^{T}\Phi_1^{T}P_iS_i\tildeE_{i,k}
        + \tildeE_k^T\Phi_1^TP_i\Phi_1\tildeE_k
        + \tildeE_k^T\Phi_1^TP_i\Psi(\tildeE_{i,k}, \rho_k)                         \\
    & - \tildeE_k^T\Phi_1^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
        + \tildeE_k^T\Phi_1^TP_i\bar{\mathrm{B}}\mathrm{w}_k                        \\                                    
    &   - \tildeE_k^T\Phi_1^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}                  
        + \Psi^T(\tildeE_{i,k}, \rho_k)P_i\mathrm{S}_i\tildeE_{i,k}                 \\              
    & + \Psi^T(\tildeE_{i,k}, \rho_k)P_i\Phi_1\tildeE_k
        + \Psi^T(\tildeE_{i,k}, \rho_k)P_i\Psi(\tildeE_{i,k}, \rho_k)               \\
    & - \Psi^T(\tildeE_{i,k}, \rho_k)P_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
        + \Psi^T(\tildeE_{i,k}, \rho_k)P_i\bar{\mathrm{B}}\mathrm{w}_k              \\
    & - \Psi^T(\tildeE_{i,k}, \rho_k)P_iL_i\mathrm{D}_i\mathrm{v}_{i,k}         
        - \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\mathrm{S}_i\tildeE_{i,k}       \\
    & -\zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\Phi_1\tildeE_k                
        - \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\Psi(\tildeE_{i,k}, \rho_k)     \\
    & + \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
        - \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k    \\
    & + \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
        + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\mathrm{S}_i\tildeE_{i,k}              \\
    & + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\Phi_1\tildeE_k
        + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\Psi(\tildeE_{i,k}, \rho_k             \\            
    & - \mathrm{w}_k^T\bar{\mathrm{B}}^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
        + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\bar{\mathrm{B}}\mathrm{w}_k           \\
    & - \mathrm{w}_k^T\bar{\mathrm{B}}^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
        - \mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\mathrm{S}_i\tildeE_{i,k}           \\
    &   - \mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\Phi_1\tildeE_k
        - \mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\Psi(\tildeE_{i,k}, \rho_k)         \\
    & + \mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
        - \mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k        \\
    & + \mathrm{v}_{i,k}D_i^TL_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} \smash{\Bigr]}
        - \sum_{i=1}^{N}\tildeE_{i,k}^TP_i\tildeE_{i,k} \biggr\}.
                    \tag{41} \label{41}
\end{align*}
Substituting equation \eqref{41} into \eqref{40} and manipulating it
similarly to \eqref{19} leads to
    \begin{align*}
    \MoveEqLeft[1]
J = \mathbb{E} \biggl\{ \sum_{i=1}^{N} \Bigl[ \tildeE_{i,k}^T \mathrm{S}_i^TP_i \mathrm{S}_i\tilde{e}_{i,k}
        + \tildeE_{i,k}^T\mathrm{S}_i^TP_i\Phi_1\tildeE_k
        + \tildeE_{i,k}^T\mathrm{S}_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k  \\
    & - \tildeE_{i,k}^TS_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
        + \tildeE_{k}^{T}\Phi_1^{T}P_i\mathrm{S}_i\tildeE_{i,k}
        + \tildeE_k^T\Phi_1^TP_i  \Phi_1\tildeE_k                       \\
    & + \tildeE_k^T\Phi_1^TP_i\bar{\mathrm{B}}\Phi_1\mathrm{w}_k
        - \tildeE_k^T\Phi_1^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
        + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\mathrm{S}_i\tildeE_{i,k}
\\
    & + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\Phi_1\tildeE_k
          + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\bar{\mathrm{B}}\mathrm{w}_k
          - \mathrm{w}_k^T\bar{\mathrm{B}}^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
\\
    & - \mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\mathrm{S}_i\tildeE_{i,k}
        - \mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\Phi_1\tilde{e}_k
\\
    & - \mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\bar{\mathrm{B}}w_k
        + \mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
\\
    & - \sum_{i=1}^{N}\tildeE_{i,k}^TP_i\tildeE_{i,k}
        + \sigma N\sum_{i=1}^{N}\tildeE_{i,k}^{T}\tildeE_{i,k}
  - \sigma\sum_{i=1}^{N}\tildeE_k^{T}\tildeE_k
\\
    & + f'^{T}(x_k,\vartheta, \rho_k)P_if'(x_k,\vartheta, \rho_k)
\\
    & + \lambda_{\max}(L_i^TP_iL_i) \zeta_i^T(\tilde{x}_{i,k}, \tau_k) \zeta_i(\tilde{x}_{i,k}, \tau_k) \Bigr]
\\
    & + \sum_{i=1}^{N} \dfrac{1}{\xi N} \tildeE_{i,k}^{T}\tildeE_{i,k}
          - \gamma_1 \mathrm{w}_k^{T}\mathrm{w}_k
          - \sum_{i=1}^{N} \gamma_2 \mathrm{v}_{i,k}^{T}\mathrm{v}_{i,k} \biggr\}.
  \tag{42} \label{42}
\end{align*}
\lipsum[3-7]
\end{document}

Answer

@Mico 答案的一个小变化：

使用的是包\MoveEqLeft中定义的宏mathtools
有一点是重新排列的数学术语

\documentclass[journal]{IEEEtran}

\usepackage{amssymb,
            mathtools}
\newcommand\tildeE{\tilde{\mathrm{e}}} % 61 [!] occurrences...
\allowdisplaybreaks
    \def\arraystretch{2}

\usepackage{lipsum}

\begin{document}
\lipsum[1-2]
    \begin{align*}
    \MoveEqLeft[1]
\mathbb{E}\{\Delta{V}(\tildeE_{i,k})\}
    = \mathbb{E} \biggl\{ \sum_{i=1}^{N}\Bigl[ \tilde{E}_{i,k}^T\mathrm{S}_i^T P_i \mathrm{S}_i \tildeE_{i,k}
        + \tilde{e}_{i,k}^T \mathrm{S}_i^T P_i\Phi_1\tildeE_k                       \\
    & + \tildeE_{i,k}^T\mathrm{S}_i^T P_i\Psi(\tildeE_{i,k}, \rho_k)          
        - \tildeE_{i,k}^T\mathrm{S}_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)       \\   
    & + \tildeE_{i,k}^T\mathrm{S}_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k
        - \tildeE_{i,k}^T\mathrm{S}_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}       
        + \tildeE_{k}^{T}\Phi_1^{T}P_iS_i\tildeE_{i,k}                              \\
    & + \tildeE_{k}^{T}\Phi_1^{T}P_iS_i\tildeE_{i,k}
        + \tildeE_k^T\Phi_1^TP_i\Phi_1\tildeE_k
        + \tildeE_k^T\Phi_1^TP_i\Psi(\tildeE_{i,k}, \rho_k)                         \\
    & - \tildeE_k^T\Phi_1^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
        + \tildeE_k^T\Phi_1^TP_i\bar{\mathrm{B}}\mathrm{w}_k                        \\                                    
    &   - \tildeE_k^T\Phi_1^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}                  
        + \Psi^T(\tildeE_{i,k}, \rho_k)P_i\mathrm{S}_i\tildeE_{i,k}                 \\              
    & + \Psi^T(\tildeE_{i,k}, \rho_k)P_i\Phi_1\tildeE_k
        + \Psi^T(\tildeE_{i,k}, \rho_k)P_i\Psi(\tildeE_{i,k}, \rho_k)               \\
    & - \Psi^T(\tildeE_{i,k}, \rho_k)P_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
        + \Psi^T(\tildeE_{i,k}, \rho_k)P_i\bar{\mathrm{B}}\mathrm{w}_k              \\
    & - \Psi^T(\tildeE_{i,k}, \rho_k)P_iL_i\mathrm{D}_i\mathrm{v}_{i,k}         
        - \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\mathrm{S}_i\tildeE_{i,k}       \\
    & -\zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\Phi_1\tildeE_k                
        - \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\Psi(\tildeE_{i,k}, \rho_k)     \\
    & + \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
        - \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k    \\
    & + \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
        + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\mathrm{S}_i\tildeE_{i,k}              \\
    & + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\Phi_1\tildeE_k
        + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\Psi(\tildeE_{i,k}, \rho_k             \\            
    & - \mathrm{w}_k^T\bar{\mathrm{B}}^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
        + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\bar{\mathrm{B}}\mathrm{w}_k           \\
    & - \mathrm{w}_k^T\bar{\mathrm{B}}^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
        - \mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\mathrm{S}_i\tildeE_{i,k}           \\
    &   - \mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\Phi_1\tildeE_k
        - \mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\Psi(\tildeE_{i,k}, \rho_k)         \\
    & + \mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
        - \mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k        \\
    & + \mathrm{v}_{i,k}D_i^TL_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} \smash{\Bigr]}
        - \sum_{i=1}^{N}\tildeE_{i,k}^TP_i\tildeE_{i,k} \biggr\}.
                    \tag{41} \label{41}
\end{align*}
Substituting equation \eqref{41} into \eqref{40} and manipulating it
similarly to \eqref{19} leads to
    \begin{align*}
    \MoveEqLeft[1]
J = \mathbb{E} \biggl\{ \sum_{i=1}^{N} \Bigl[ \tildeE_{i,k}^T \mathrm{S}_i^TP_i \mathrm{S}_i\tilde{e}_{i,k}
        + \tildeE_{i,k}^T\mathrm{S}_i^TP_i\Phi_1\tildeE_k
        + \tildeE_{i,k}^T\mathrm{S}_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k  \\
    & - \tildeE_{i,k}^TS_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
        + \tildeE_{k}^{T}\Phi_1^{T}P_i\mathrm{S}_i\tildeE_{i,k}
        + \tildeE_k^T\Phi_1^TP_i  \Phi_1\tildeE_k                       \\
    & + \tildeE_k^T\Phi_1^TP_i\bar{\mathrm{B}}\Phi_1\mathrm{w}_k
        - \tildeE_k^T\Phi_1^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
        + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\mathrm{S}_i\tildeE_{i,k}
\\
    & + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\Phi_1\tildeE_k
          + \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\bar{\mathrm{B}}\mathrm{w}_k
          - \mathrm{w}_k^T\bar{\mathrm{B}}^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
\\
    & - \mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\mathrm{S}_i\tildeE_{i,k}
        - \mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\Phi_1\tilde{e}_k
\\
    & - \mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\bar{\mathrm{B}}w_k
        + \mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
\\
    & - \sum_{i=1}^{N}\tildeE_{i,k}^TP_i\tildeE_{i,k}
        + \sigma N\sum_{i=1}^{N}\tildeE_{i,k}^{T}\tildeE_{i,k}
  - \sigma\sum_{i=1}^{N}\tildeE_k^{T}\tildeE_k
\\
    & + f'^{T}(x_k,\vartheta, \rho_k)P_if'(x_k,\vartheta, \rho_k)
\\
    & + \lambda_{\max}(L_i^TP_iL_i) \zeta_i^T(\tilde{x}_{i,k}, \tau_k) \zeta_i(\tilde{x}_{i,k}, \tau_k) \Bigr]
\\
    & + \sum_{i=1}^{N} \dfrac{1}{\xi N} \tildeE_{i,k}^{T}\tildeE_{i,k}
          - \gamma_1 \mathrm{w}_k^{T}\mathrm{w}_k
          - \sum_{i=1}^{N} \gamma_2 \mathrm{v}_{i,k}^{T}\mathrm{v}_{i,k} \biggr\}.
  \tag{42} \label{42}
\end{align*}
\lipsum[3-7]
\end{document}

Question 3

与之前的答案类似，此解决方案基于align*环境。我应用了双缩进，因为其中一个方程似乎有一个内部部分。另一个方程中的某些部分可能需要在双列布局中使用额外的缩进进行拆分（请参阅我的评论）。我还根据以下情况将方程编号移到了底部Mico 的建议。

对原帖的几点看法

使用\newcommand或\DeclareMathOperator避免重复表达和不必要的混乱
使用更大比例的括号¹，例如\bigl\{...\bigr}，\Bigl\{...\Bigr\}或
\Biggl\{...\Biggr\}
- 强调方程式的内部/外部
- 适合相邻的表达式等
align*类似环境默认使用显示样式，因此\displaystyle会变得多余。

对代码进行一些额外的注释。
在第一个等式中，为了减少左间距，我将第一个表达式括在\mathrlap{}（从mathtools）中，并在后面（第一个之前&）添加额外的空格。效果是向所有后续行添加缩进。
这个等式似乎有一个很大的内部部分。因此，可以应用进一步的缩进。如果这是错误的或不符合文章主题，可以用注释部分（在底部）替换整个代码块。

在第二个等式中，有几行表达式较长，无法容纳布局中的可用空间twocolumn。它们被移动到后续行，并带有缩进以表示延续。
我发现\sum使用限制会占用大量额外的垂直空间，当总和是后续行的一部分时，这可能是不希望的。为了减少间距，我将两个宏组合在一起：\smash{}使用\vphantom{}。如果没有必要，请删除

\vphantom{\sum \limits_{i=1}}

或者

\vphantom{\sum \limits^{N}}

并摆脱\smash{content}围绕它的论点；content仍然是等式的一部分。

¹- 也适用于其他括号，例如(...)和[...]

最终结果

代码

\documentclass[journal]{IEEEtran}
\usepackage{amssymb}
\usepackage{mathtools}
\usepackage{kantlipsum}

    
\newlength\flinesep \setlength\flinesep{1em}
\newlength\nlinesep \setlength\nlinesep{4em}

\allowdisplaybreaks
% \setlength\jot{6pt}   % extra line spacing in equations

\DeclareMathOperator{\E}{\mathbb{E}}
\newcommand{\e}{\mathrm{e}}
\newcommand{\te}{\tilde{\e}}
\newcommand\oB{\mathrm{B}}
\newcommand\oD{\mathrm{D}}
\newcommand\oS{\mathrm{S}}
\newcommand\ov{\mathrm{S}}
\newcommand\ow{\mathrm{S}}


\begin{document}
\section{First section}
\kant*[1-2]
\begin{align*}
    \mathrlap{\E \Bigl\{\bigtriangleup{V}(\te_{i,k})\Bigr\}
        = \E\Biggl\{\sum \limits_{i=1}^{N}
        \Bigl\{\te_{i,k}^T\oS_i^TP_i\oS_i\te_{i,k}}
    \\&&&+ \te_{i,k}^T\oS_i^TP_i\Phi_1\te_k
    \\&&&+ \te_{i,k}^T\oS_i^TP_i\Psi(\te_{i,k}, \rho_k) 
    \\&&&- \te_{i,k}^T\oS_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
    \\&&&+ \te_{i,k}^T\oS_i^TP_i\bar{\oB}\ow_k 
    - \te_{i,k}^T\oS_i^TP_iL_i\oD_i\ov_{i,k}
    \\&&&+ \te_{k}^{T}\Phi_1^{T}P_iS_i\te_{i,k}
        + \te_k^T\Phi_1^TP_i\Phi_1\te_k 
    \\&&&+ \te_k^T\Phi_1^TP_i\Psi(\te_{i,k}, \rho_k)
    \\&&&- \te_k^T\Phi_1^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k) 
    \\&&&+ \te_k^T\Phi_1^TP_i\bar{\oB}\ow_k
        -\te_k^T\Phi_1^TP_iL_i\oD_i\ov_{i,k}  
    \\&&&+ \Psi^T(\te_{i,k}, \rho_k)P_i\oS_i\te_{i,k} 
    \\&&&+ \Psi^T(\te_{i,k}, \rho_k)P_i\Phi_1\te_k
    \\&&&+ \Psi^T(\te_{i,k}, \rho_k)P_i\Psi(\te_{i,k}, \rho_k) 
    \\&&&- \Psi^T(\te_{i,k}, \rho_k)P_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)  
    \\&&&+ \Psi^T(\te_{i,k}, \rho_k)P_i\bar{\oB}\ow_k 
    \\&&&- \Psi^T(\te_{i,k}, \rho_k)P_iL_i\oD_i\ov_{i,k}
    \\&&&- \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\oS_i\te_{i,k}  
    \\&&&- \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\Phi_1\te_k 
    \\&&&- \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\Psi(\te_{i,k}, \rho_k)
    \\&&&+ \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k) 
    \\&&&- \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\bar{\oB}\ow_k 
    \\&&&+ \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_iL_i\oD_i\ov_{i,k} 
    \\&&&+ \ow_k^T\bar{\oB}^TP_i\oS_i\te_{i,k} 
        + \ow_k^T\bar{\oB}^TP_i\Phi_1\te_k  
    \\&&&+ \ow_k^T\bar{\oB}^TP_i\Psi(\te_{i,k}, \rho_k)
    \\&&&- \ow_k^T\bar{\oB}^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k) 
    \\&&&+ \ow_k^T\bar{\oB}^TP_i\bar{\oB}\ow_k  
    -\ow_k^T\bar{\oB}^TP_iL_i\oD_i\ov_{i,k} 
    \\&&&- \ov_{i,k}\oD_i^TL_i^TP_i\oS_i\te_{i,k} 
        -\ov_{i,k}\oD_i^TL_i^TP_i\Phi_1\te_k  
    \\&&&- \ov_{i,k}\oD_i^TL_i^TP_i\Psi(\te_{i,k}, \rho_k) 
    \\&&&+ \ov_{i,k}\oD_i^TL_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k) 
    \\&&&- \ov_{i,k}\oD_i^TL_i^TP_i\bar{\oB}\ow_k 
        +\ov_{i,k}D_i^TL_i^TP_iL_i\oD_i\ov_{i,k} \Bigr\}
    \\& \hspace{\flinesep}
        -\mathrlap{\sum \limits_{i=1}^{N}\te_{i,k}^TP_i\te_{i,k}\Biggr\}.}
     \tag{41} \label{41}
\end{align*}

Substituting (\ref{41}) into (\ref{40}) and manipulating it similar to (\ref{19}) leads to
\begin{align*}
    \mathrlap{J = \E \Biggl\{\sum \limits_{i=1}^{N}
        \{\te_{i,k}^T\oS_i^TP_i\oS_i\tilde{e}_{i,k}}
        \hspace{\flinesep}
    \\& + \te_{i,k}^T\oS_i^TP_i\Phi_1\te_k  
        + \te_{i,k}^T\oS_i^TP_i\bar{\oB}\ow_k
    \\&- \te_{i,k}^TS_i^TP_iL_i\oD_i\ov_{i,k} 
        \\&\hspace{\nlinesep} + \te_{k}^{T}\Phi_1^{T}P_i\oS_i\te_{i,k}
                     + \te_k^T\Phi_1^TP_i  \Phi_1\te_k 
    \\&+\te_k^T\Phi_1^TP_i\bar{\oB}\Phi_1\ow_k 
        \\&\hspace{\nlinesep} - \te_k^T\Phi_1^TP_iL_i\oD_i\ov_{i,k}
                     + \ow_k^T\bar{\oB}^TP_i\oS_i\te_{i,k} 
    \\&+ \ow_k^T\bar{\oB}^TP_i\Phi_1\te_k  
        \\&\hspace{\nlinesep} + \ow_k^T\bar{\oB}^TP_i\bar{\oB}\ow_k 
                     - \ow_k^T\bar{\oB}^TP_iL_i\oD_i\ov_{i,k} 
    \\&-\ov_{i,k}\oD_i^TL_i^TP_i\oS_i\te_{i,k}
        - \ov_{i,k}\oD_i^TL_i^TP_i\Phi_1\tilde{e}_k 
    \\&-\ov_{i,k}\oD_i^TL_i^TP_i\bar{\oB}w_k 
        + \ov_{i,k}\oD_i^TL_i^TP_iL_i\oD_i\ov_{i,k} 
    \\&- \vphantom{\sum \limits^{N}}
         \smash{\sum \limits_{i=1}^{N}\te_{i,k}^TP_i\te_{i,k}}
        \\&\hspace{\nlinesep} + \vphantom{\sum \limits_{i=1}}
                            \smash{\sigma N \sum \limits_{i=1}^{N}\te_{i,k}^{T}\te_{i,k}}
                     -  \vphantom{\sum \limits_{i=1}}
                            \smash{\sigma\sum \limits_{i=1}^{N}\te_k^{T}\te_k}
    \\&+ f'^{T}(x_k,\vartheta, \rho_k)P_if'(x_k,\vartheta, \rho_k)  
    \\&+ \lambda_{max}(L_i^TP_iL_i)\zeta_i^T(\tilde{x}_{i,k}, \tau_k)\zeta_i(\tilde{x}_{i,k}, \tau_k)\}  
    \\&+ \vphantom{\sum \limits^{N}}
            \smash{\sum \limits_{i=1}^{N} \dfrac{1}{\xi N} \te_{i,k}^{T}\te_{i,k}}
        \\&\hspace{\nlinesep} - \vphantom{\sum \limits_{i=1}}
                            \smash{\gamma_1 \ow_k^{T}\ow_k} 
                     - \vphantom{\sum \limits_{i=1}}
                            \smash{\sum \limits_{i=1}^{N} \gamma_2 \ov_{i,k}^{T}\ov_{i,k}\Biggr\}}.
     \tag{42} \label{42}
\end{align*}

\kant[1]
\end{document}

Answer