My understanding is that eink only uses power on refresh; if you don’t change the image, the pigments remain where they are and draw no power.
if batteries are an option, e-ink is a super-low-power option because it uses 100 milliwatts or maybe 10 milliwatts at a cellphone size, while a conventional backlit color lcd uses 1000. but to run off solar panels, indoors, my power budget for the whole zorzpad is 1 milliwatt, and the screen can only have a fraction of that