A quick search for plumbing heat tape leads to Frost King products, which are sold in various lengths. Power consumption is listed as 7 watts/ft (120 V). Like Bob said, it makes a lot of sense to insulate the pipes with the heat tape inside the
, even insulate/close off the underside of your platform (just watch out for critters nesting...). Without knowing how many feet of heat cable you need, the level of your insulation, or the temps at different times, any number for total power used will only be an estimate.
For instance, assume 20 ft of cable (20*7=140 watts). Assume a lot of insulation (around pipes & enclosed under yurt platform), figure the heat cable kicks on 2 hours (intermittently) a day: 140*2=280 watt*hours. Your solar system might be collecting 1200 watts/hr during the winter, figure 4 hours good light: 1200*4=4800 watt*hrs (insolation numbers suggest 1/3 of your summer output, neglecting clouds). So you'd use 280/4800 or 6% of your solar bank each day--easily sustainable (if nothing goes wrong, like snow covering your solar panels).
Now assume you didn't insulate the pipes or enclose the
and the heat cable runs 14 hrs/day. 140*14=1960 watt*hrs. Now you're using 41% of your solar bank each day. Or if it's going 24 hrs/day: 3360 watt*hrs, or 70% of your daily solar intake. Not much wiggle room there!
Again a lot of variables here (cable lengths, insulation levels, winter temps, winter solar output), but they give you a rough idea of what could be going on. If you decide to go with electric heat cables, that is.