How to identify the seasonality of a timeseries from the Periodogram? I need to identify seasonality/ periodicity of a dataset so as to develop an ARMAX model. This is what the original time-series looks like

I have plotted the periodogram of the dataset. 
Ps: I used the first difference of the original time series so as to remove  the trend from the original time series . This what the periodogram looks like
I also removed the linear trend from the data and estimated the periodogram. Below is what it looks like

I also estimated periodogram of the raw data set without any differencing. Clearly, it shows there is some trend in the data.
I am using following R-code to generate it

A <- read.xlsx("Interpolation_Data.xlsx")
y <- A[,2]
x <- as.numeric(A[,4])
diff.y <- diff(y)
diff.x <- diff(x)
Fs<- 1/60
n=length(diff.y)
xdft <-fft(diff.y)
xdft <- xdft[1:(n/2+1)]
psdx <- (1/(Fs*n))* abs(xdft)^2
psdx[c(-1,-n)] <- 2*psdx[c(-1,-n)]
f <- seq(0,Fs/2,by = Fs/n)
plot(f,10*log10(psdx),type='l')

This is the link to data set
 A: Well, the periodogram after taking first differences doesn't indicate any clear periodicity. However, be aware that taking first differences amplifies high-frequency components, which should appear in the power spectrum as a quadratic trend, and in the log-power spectrum as a log-shaped trend (which is roughly compatible with what we see here).
My recommendations:
– First compute the periodogram without any preprocessing.
– Then, remove the linear trend, but not by using first differences but by fitting a line by regression and subtracting the result from the data (there should also be a "detrend" function in R).
– Use a proper spectral estimation function. I'm not familiar with R, but I'm sure it contains an implementation of Welch's modified periodogram method.

Update for the updated question:
The time series exhibits a dominant period of roughly 360 samples, which for a sampling rate of 1 per minute means 360 minutes. The dominant frequency should therefore be about 0.0028 min$^{-1}$. This seems to be consistent with the periodogram after subtracted trend. Try zooming into the low-frequency range to more precisely determine the peak location. Superimposed to that is a secondary oscillation of 180 samples period or 0.0056 min$^{-1}$ frequency.
Analyzing your data myself I get a main peak at 0.0029 min$^{-1}$ and a secondary peak at 0.0059 min$^{-1}$ (factor 2, the first harmonic). The secondary peak is smaller by a factor of about 36, or by 16 dB. I can share my code, but I am using Matlab, not R.

Btw., the frequency scale on your plots does not appear to be consistent; sometimes it goes up to 0.008, sometimes to 0.5? I think the former gives the frequency in Hz and the latter in min$^{-1}$.
