  We report on the results of a search for radio transients between 115 and
190\,MHz with the LOw-Frequency ARray (LOFAR). Four fields have been monitored
with cadences between 15 minutes and several months. A total of 151 images were
obtained, giving a total survey area of 2275 deg$^2$. We analysed our data
using standard LOFAR tools and searched for radio transients using the LOFAR
Transients Pipeline (TraP). No credible radio transient candidate has been
detected; however, we are able to set upper limits on the surface density of
radio transient sources at low radio frequencies. We also show that
low-frequency radio surveys are more sensitive to steep-spectrum coherent
transient sources than GHz radio surveys. We used two new statistical methods
to determine the upper limits on the transient surface density. One is free of
assumptions on the flux distribution of the sources, while the other assumes a
power-law distribution in flux and sets more stringent constraints on the
transient surface density. Both of these methods provide better constraints
than the approach used in previous works. The best value for the upper limit we
can set for the transient surface density, using the method assuming a
power-law flux distribution, is 1.3$\cdot$10$^{-3}$ deg$^{-2}$ for transients
brighter than 0.3 Jy with a time-scale of 15 min, at a frequency of 150 MHz. We
also calculated for the first time upper limits for the transient surface
density for transients of different time-scales. We find that the results can
differ by orders of magnitude from previously reported, simplified estimates.
