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I need to place the following bulk of equation to left alignment with keeping the equation number in the middle, however using flalign require set option on/off by line for equation number which at last will not be in the middle.

 begin{flalign}

    ell_1(bm{phi})& =sum_{i=1}^{N-N^*} sum_{j=2}^{d_i-1} ln(1- 

    bm{P}_{ij}),&nonumber\

    ell_2(bm{phi})& =sum_{i=1}^{N-N^*} ln,bm{P}_{id_i},&

 end{flalign}

On the other hand, when using aligned with equation, equation number will be in the middle but the whole bulk of equation moved to center.

begin{equation}

 begin{aligned}

    ell_1(bm{phi})& =sum_{i=1}^{N-N^*} sum_{j=2}^{d_i-1} ln(1- 

    bm{P}_{ij}),\

    ell_2(bm{phi})& =sum_{i=1}^{N-N^*} ln,bm{P}_{id_i},

 end{aligned}

end{equation}

How can I keep the equation left aligned with equation number in the middle.

It's not clear why a particular equation should be left aligned. If you want all equations to be left aligned, use the fleqn option.

Anyway, here's a way with flalign.

Function names such as “ln”, “sin”, “exp” should be typed in with a backslash: ln and so on; this will give the correct spacing, besides using upright font shape.

documentclass{article}

usepackage{amsmath,bm}



usepackage[nopar]{lipsum} % for context



begin{document}



lipsum[4]

begin{flalign}

& begin{aligned}

    ell_1(bm{phi})& =sum_{i=1}^{N-N^*} sum_{j=2}^{d_i-1} ln(1-bm{P}_{ij}),\

    ell_2(bm{phi})& =sum_{i=1}^{N-N^*} lnbm{P}_{id_i},

 end{aligned}

&&

end{flalign}

lipsum[4]



end{document}

Do note that the left-hand shift of the first set of equations is not caused by the use of flalign, per se. Instead, the shift is caused by the presence of additional & symbols.

To shift a nested equation/aligned system to the left, all you need to do is insert the instruction

hspace*{6cm}null  % choose offset amount suitably

between end{aligned} and end{equation}.

Oh, and do please write ln rather than just ln to denote the natural logarithm operator.

documentclass{article}

usepackage{amsmath,bm}

begin{document}

begin{equation}

begin{aligned}

   ell_1(bm{phi}) &= sum_{i=1}^{N-N^*} sum_{j=2}^{d_i-1} ln(1-bm{P}_{ij}),\

   ell_2(bm{phi}) &= sum_{i=1}^{N-N^*} lnbm{P}_{id_i},

end{aligned}hspace*{6cm}null  % choose offset amount suitably

end{equation}

end{document} 

You can use the fleqn environment from nccmath, which makes equations in the environment start at a fixed distance of the left margin via an optional argument (default 0pt):

documentclass{article}

usepackage{showframe}

renewcommand{ShowFrameLinethickness}{0.3pt}

usepackage{amsmath, bm, nccmath}



begin{document}



begin{fleqn}[1em]

  begin{equation}

 begin{aligned}

    ell_1(bm{phi})& =sum_{i=1}^{N-N^*} sum_{j=2}^{d_i-1} ln(1-

    bm{P}_{ij}),\

    ell_2(bm{phi})& =sum_{i=1}^{N-N^*} ln,bm{P}_{id_i},

 end{aligned}

end{equation}

end{fleqn}



end{document} 

You could, alternatively use, usepackage{empheq}. Here I have added a picture of the differents environment (see the table) that you can use:

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

documentclass{article}

usepackage{empheq}

usepackage{amsmath}

usepackage{bm}

begin{document}



begin{empheq}{flalign}

  begin{aligned}

    ell_1(bm{phi})& =sum_{i=1}^{N-N^*} sum_{j=2}^{d_i-1} ln(1- 

    bm{P}_{ij}),\

    ell_2(bm{phi})& =sum_{i=1}^{N-N^*} ln,bm{P}_{id_i},

  end{aligned}&&

end{empheq}





end{document}