Undesired spaces in mathematics two column paper

Question

Problem Description

My equations are very long, they are not going from one column to the next, that is disturbing the layout of the paper, too much undesired space. What should I do? I tried \allowdisplayreaks. It is not working.

MWE:

\allowdisplaybreaks

\begin{equation} \tag{41} \label{41}
\begin{split}
\mathbb{E}\{\bigtriangleup{V}(\tilde{\mathrm{e}}_{i,k})\}&= \mathbb{E}\{ \displaystyle{\sum \limits_{i=1}^{N}}
\{\tilde{\mathrm{e}}_{i,k}^T\mathrm{S}_i^TP_i\mathrm{S}_i\tilde{\mathrm{e}}_{i,k}  +\tilde{\mathrm{e}}_{i,k}^T\mathrm{S}_i^TP_i\Phi_1\tilde{\mathrm{e}}_k \\&\mspace{20mu}+\tilde{\mathrm{e}}_{i,k}^T\mathrm{S}_i^TP_i\Psi(\tilde{\mathrm{e}}_{i,k}, \rho_k) 
\\&\mspace{20mu}- \tilde{\mathrm{e}}_{i,k}^T\mathrm{S}_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)
\\&\mspace{20mu}+\tilde{\mathrm{e}}_{i,k}^T\mathrm{S}_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k 
- \tilde{\mathrm{e}}_{i,k}^T\mathrm{S}_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} 
\\&\mspace{20mu}+ \tilde{\mathrm{e}}_{k}^{T}\Phi_1^{T}P_iS_i\tilde{\mathrm{e}}_{i,k}
+ \tilde{\mathrm{e}}_k^T\Phi_1^TP_i\Phi_1\tilde{\mathrm{e}}_k 
\\&\mspace{20mu}+ \tilde{\mathrm{e}}_k^T\Phi_1^TP_i\Psi(\tilde{\mathrm{e}}_{i,k}, \rho_k)
\\&\mspace{20mu}- \tilde{\mathrm{e}}_k^T\Phi_1^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k) 
\\&\mspace{20mu}+\tilde{\mathrm{e}}_k^T\Phi_1^TP_i\bar{\mathrm{B}}\mathrm{w}_k 
-\tilde{\mathrm{e}}_k^T\Phi_1^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k}  
\\&\mspace{20mu}+\Psi^T(\tilde{\mathrm{e}}_{i,k}, \rho_k)P_i\mathrm{S}_i\tilde{\mathrm{e}}_{i,k} 
\\&\mspace{20mu}+\Psi^T(\tilde{\mathrm{e}}_{i,k}, \rho_k)P_i\Phi_1\tilde{\mathrm{e}}_k
\\&\mspace{20mu}+\Psi^T(\tilde{\mathrm{e}}_{i,k}, \rho_k)P_i\Psi(\tilde{\mathrm{e}}_{i,k}, \rho_k) 
\\&\mspace{20mu}- \Psi^T(\tilde{\mathrm{e}}_{i,k}, \rho_k)P_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k)  
\\&\mspace{20mu}+\Psi^T(\tilde{\mathrm{e}}_{i,k}, \rho_k)P_i\bar{\mathrm{B}}\mathrm{w}_k 
\\&\mspace{20mu}-\Psi^T(\tilde{\mathrm{e}}_{i,k}, \rho_k)P_iL_i\mathrm{D}_i\mathrm{v}_{i,k}
\\&\mspace{20mu}-\zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\mathrm{S}_i\tilde{\mathrm{e}}_{i,k}  
\\&\mspace{20mu}-\zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\Phi_1\tilde{\mathrm{e}}_k 
\\&\mspace{20mu}-\zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\Psi(\tilde{\mathrm{e}}_{i,k}, \rho_k)
\\&\mspace{20mu}+\zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k) 
\\&\mspace{20mu}- \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k 
\\&\mspace{20mu}+ \zeta_i^T(\tilde{x}_{i,k}, \tau_k)L_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} 
\\&\mspace{20mu}+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\mathrm{S}_i\tilde{\mathrm{e}}_{i,k} 
+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\Phi_1\tilde{\mathrm{e}}_k  
\\&\mspace{20mu}+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\Psi(\tilde{\mathrm{e}}_{i,k}, \rho_k)
\\&\mspace{20mu}- \mathrm{w}_k^T\bar{\mathrm{B}}^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k) 
\\&\mspace{20mu}+ \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} 
\\&\mspace{20mu}-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\mathrm{S}_i\tilde{\mathrm{e}}_{i,k} 
-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\Phi_1\tilde{\mathrm{e}}_k  
\\&\mspace{20mu}-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\Psi(\tilde{\mathrm{e}}_{i,k}, \rho_k) 
\\&\mspace{20mu}+\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_iL_i\zeta_i(\tilde{x}_{i,k}, \tau_k) 
\\&\mspace{20mu}-\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}\} 
\\&\mspace{20mu}- \displaystyle{\sum \limits_{i=1}^{N}}\tilde{\mathrm{e}}_{i,k}^TP_i\tilde{\mathrm{e}}_{i,k}\}.
\end{split}
\end{equation}

Substituting (\ref{41}) into (\ref{40}) and manipulating it similar to (\ref{19}) leads to
\allowdisplaybreaks

\begin{equation*} \tag{42} \label{42}
\begin{split}
J&= \mathbb{E}\{\displaystyle{\sum \limits_{i=1}^{N}}
\{\tilde{\mathrm{e}}_{i,k}^T\mathrm{S}_i^TP_i\mathrm{S}_i\tilde{e}_{i,k}  +\tilde{\mathrm{e}}_{i,k}^T\mathrm{S}_i^TP_i\Phi_1\tilde{\mathrm{e}}_k  
+\tilde{\mathrm{e}}_{i,k}^T\mathrm{S}_i^TP_i\bar{\mathrm{B}}\mathrm{w}_k 
\\&\mspace{20mu}- \tilde{\mathrm{e}}_{i,k}^TS_i^TP_iL_i\mathrm{D}_i\mathrm{v}_{i,k} 
+ \tilde{\mathrm{e}}_{k}^{T}\Phi_1^{T}P_i\mathrm{S}_i\tilde{\mathrm{e}}_{i,k}
+ \tilde{\mathrm{e}}_k^T\Phi_1^TP_i  \Phi_1\tilde{\mathrm{e}}_k 
\\&\mspace{20mu}+\tilde{\mathrm{e}}_k^T\Phi_1^TP_i\bar{\mathrm{B}}\Phi_1\mathrm{w}_k 
-\tilde{\mathrm{e}}_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\tilde{\mathrm{e}}_{i,k} 
\\&\mspace{20mu}+ \mathrm{w}_k^T\bar{\mathrm{B}}^TP_i\Phi_1\tilde{\mathrm{e}}_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} 
\\&\mspace{20mu}-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\mathrm{S}_i\tilde{\mathrm{e}}_{i,k}
-\mathrm{v}_{i,k}\mathrm{D}_i^TL_i^TP_i\Phi_1\tilde{e}_k 
\\&\mspace{20mu}-\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} 
\\&\mspace{20mu}- \displaystyle{\sum \limits_{i=1}^{N}}\tilde{\mathrm{e}}_{i,k}^TP_i\tilde{\mathrm{e}}_{i,k}  + \sigma N\displaystyle{\sum \limits_{i=1}^{N}}\tilde{\mathrm{e}}_{i,k}^{T}\tilde{\mathrm{e}}_{i,k} - \sigma\displaystyle{\sum \limits_{i=1}^{N}}\tilde{\mathrm{e}}_k^{T}\tilde{\mathrm{e}}_k 
\\&\mspace{20mu}+ f'^{T}(x_k,\vartheta, \rho_k)P_if'(x_k,\vartheta, \rho_k)  
\\&\mspace{20mu}+ \lambda_{max}(L_i^TP_iL_i)\zeta_i^T(\tilde{x}_{i,k}, \tau_k)\zeta_i(\tilde{x}_{i,k}, \tau_k)\}  
\\&\mspace{20mu}+ \displaystyle{\sum \limits_{i=1}^{N}} \dfrac{1}{\xi N} \tilde{\mathrm{e}}_{i,k}^{T}\tilde{\mathrm{e}}_{i,k} 
- \gamma_1 \mathrm{w}_k^{T}\mathrm{w}_k 
- \displaystyle{\sum \limits_{i=1}^{N}} \gamma_2 \mathrm{v}_{i,k}^{T}\mathrm{v}_{i,k}\}.
\end{split}
\end{equation*}

Answer 1

The main problem you're facing is that the split environment does not allow column (and page) breaks. To allow column breaks, I would like to suggest you switch from a nested equation*/split setup to an align* setup. Then, place the \tag and \label directives in whichever line you think is the right one.

\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 2

A small variation of @Mico answer:

• used is \MoveEqLeft macro defined in the mathtools package
• a little bit are rearranged math terms

\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 3

Similarly to previous answers, this solution is based on align* environment. I applied a double indentation as one of the equation seems to have a inner part. Some parts in the other equation might need to be split with extra indentation in the twocolumn layout (see my comments). I also moved the equation number to the bottom according to Mico's suggestion.

A few points to OP

• Use \newcommand or \DeclareMathOperator to avoid repeated expressions and unnecessary clutter
• Use larger scaled brackets¹, such as \bigl\{...\bigr}, \Bigl\{...\Bigr\}, or
  \Biggl\{...\Biggr\} to
  • emphasise inner/outer parts of equations
  • fit adjacent expressions, etc.
• align*-like environments use display style by default, hence \displaystyle will become redundant.

Some additional comments to the code.
In the first equation, in order to reduce left spacing, I have enclose the first expression in \mathrlap{} (from mathtools) and add extra space afterwards (before the first &). The effect is an indentation is added to all subsequent lines.
This equation seems to have a large inner part. Therefore, a further indentation can be applied. If that is wrong or does not fit the article theme, the whole block of code can be substituted by the commented part (at the bottom).

In the second equation, a couple of lines have longer expressions and couldn't fit available space in the twocolumn layout. They are moved to subsequent lines with indentation to indicate continuation.
I find that \sum with limits occupies significant extra vertical space, which might not be desired when the sum is a part of the continuing line. In order to reduced the spacing, I combined two macros together: \smash{} with \vphantom{}. If this is not necessary, remove

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

or

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

and get rid of \smash{content} surrounding its argument; the content is still a part of the equation.

---

¹ - also works with other brackets, e.g. (...) and [...]

---

The final outcome

---

The code

\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}