Is there any formulae to calculate whether the sun will depress 18 degrees or more below the horizon on a certain day / location?
An approximation of the Sun’s declination for a certain date is:
sin δ = 0.39795 ⋅ cos [ 0.98563 ⋅ ( N – 173 ) ]
where N is the number of days since January 1.
That, combined with the formula given by @user21, viz.:
$ a_{min} = \lvert \phi + \delta \rvert - 90° $
will give you the dates where the Sun is at least 18° under the horizon.
(Oddly enough, a simple instrument, invented by the Greeks almost 2,000 years ago and perfected by the Muslims over the next few centuries, the astrolabe, would give you the answer almost instantly, without having to do any calculations. Below is a photo of one I made in brass.)