Csdrhrt

documentclass[journal]{IEEEtran}

usepackage{eqnarray,amsmath}

usepackage{amssymb}

usepackage{mathtools}

begin{document}

begin{align} maxlimits_{bm{P}^{(k)}} & R_k(bm{P}^{(k)})=sumlimits_{jin c^{(k)}}text{log}_2left(1+frac{p_j^{(k)}g_{j,j}^{(k)}}{sigma_N^2+sumlimits_{j'=1,j'neq j}^{vert mathcal{C}^{(k)}vert}p_{j'}^{(k)}h_{j',j}^{(k)}+p_{b}^{(k)}h_{b,j}^{(k)}}right) nonumber\ text{s.t}. &C_{1}:0le p_{b}^{(k)}leq P_{b}^{max}nonumber\ &C_{2}:0leq p_{j}^{(k)}leq P_{d}^{max},~~forall jin c^{(k)}nonumber\ &C_{3}:text{log}_2(1+frac{p_b^{(k)}g_{b,n}^{(k)}}{sigma_N^2+sumlimits_{jin c^{(k)}}p_j^{(k)}h_{j,b}^{(k)}})geq R_{b,min}label{powerprob}end{align}

end{document}

Some suggestions:

• Divide the material into two parts, and place the first part in a multlined environment and the second in an aligned environment. Place this material in a gather enrivonment.




• For the long denominator, use substack to introduce a line break in the lower limit of summation, and use a smashoperator directive to "smash" the space around the summation symbol.




• Place the "such that" string in an intertext directive.




• Optional: To limit the size of the outer parentheses, encase the smashoperator{...} material in a smash{...} wrapper. In general, doing this might be somewhat problematic. Here, though, there's no problem as the material that comes below is doesn't protrude much to the right.

documentclass[journal]{IEEEtran}

usepackage{amssymb,mathtools,bm}

begin{document}

begin{gather}label{powerprob}

begin{multlined}

max_{bm{P}^{(k)}} R_k(bm{P}^{(k)})\[-3ex]

=sum_{jin c^{(k)}} log_2 left( 1+frac{p_j^{(k)}g_{j,j}^{(k)}}{

sigma_N^2 +

smash[b]{smashoperator{

sumlimits_{substack{j'=1,\j'neq jhfill}}^

{lvertmathcal{C}^{(k)}rvert}}},

p_{j'}^{(k)}h_{j',j}^{(k)}

+p_{b}^{(k)}h_{b,j}^{(k)}}right)

end{multlined} notag \

intertext{such that}

begin{aligned}[b]

&C_{1}colon 0leq p_{b}^{(k)}leq P_{b}^{max}  \[1ex]

&C_{2}colon 0leq p_{j}^{(k)}leq P_{d}^{max}quad forall jin c^{(k)}  \[1ex]

&C_{3}colon log_2left(

1+frac{p_b^{(k)}g_{b,n}^{(k)}}{ sigma_N^2+sum_{jin c^{(k)}} p_j^{(k)}h_{j,b}^{(k)}}

right)geq R_{b,min}

end{aligned}

end{gather}

end{document}

You can use cuted package and use strip envrionment to achieve it:

documentclass[journal]{IEEEtran}

usepackage{mathtools,lipsum}

usepackage{cuted}

%https://tex.stackexchange.com/questions/255086/how-to-put-a-long-equation-spanning-two-columns-while-using-ieeetran

begin{document}

    lipsum

    begin{strip}

        begin{align}

         maxlimits_{{P}^{(k)}} & R_k({P}^{(k)})=sumlimits_{jin c^{(k)}}text{log}_2left(1+frac{p_j^{(k)}g_{j,j}^{(k)}}{sigma_N^2+sumlimits_{j'=1,j'neq j}^{vert mathcal{C}^{(k)}vert}p_{j'}^{(k)}h_{j',j}^{(k)}+p_{b}^{(k)}h_{b,j}^{(k)}}right) nonumber\ text{s.t}. &C_{1}:0le p_{b}^{(k)}leq P_{b}^{max}nonumber\ &C_{2}:0leq p_{j}^{(k)}leq P_{d}^{max},~~forall jin c^{(k)}nonumber\ &C_{3}:text{log}_2(1+frac{p_b^{(k)}g_{b,n}^{(k)}}{sigma_N^2+sumlimits_{jin c^{(k)}}p_j^{(k)}h_{j,b}^{(k)}})geq R_{b,min}label{powerprob}

        end{align}

    end{strip}

lipsum

end{document}

which would give you

PS: I did not know what you mean by bm, hence, I removed it.