Odds are that you have quite enough P and K in your pile without adding more. Ashes are alkaline and raise the pH of the pile. Might be good for a slight bit if you have lots of acidic leaves (i.e., oak and some others) but add very little...it can affect decomp if pH gets too high. Not necessarily stop decomp, but certainly slow it for the long run.
hortster

The stuff we usually put in our compost pile: We have alot of pine needles and maple leaves...grass clippings and some kitchen waste (cofffee grinds..egg shells..vegetable scraps..etc...) some straw and chicken manure when we clean out the chicken coop...weeds...ripped up cardboard.
I just made a strawberry bed from using the compost I got this spring from the stuff that composted last summer and over winter.
I planted about 50 strawberry plants into a new raised bed.

The ash from wood pellets will have much more Calcium Carbonate (lime) then Potash, like any other wood ash.

And it will not remove nitrogen like excess browns would. However, in my experience a high pH combined with high nitrogen (lots of greens) can allow nitrogen to be lost as ammonia gas. In other words, a fresh pile heavy on grass clippings with too much ash added will smell like ammonia. That's the aroma of nutrients being lost. Ask me how I know - been there, done that. :-] Just take it easy with the ashes.

I thought it was a good sign when the pile got steamy...that it was getting hot enough to destroy weed seeds and decompose faster. Thats what usually happens when we mow the lawn and add the grass clippings.

I get grass clippings from a spot where an office-building grounds crew dumps them. What looks very much like steam from them when dug is usually apparently a cloud of mold or fungal spores. On frosty mornings a bit may be "steam" but the cloud is there even in hot weather.

Kimm is absolutely right on the chemistry of wood ashes. The ash starts out mainly as calcium oxide, which when it becomes damp becomes calcium hydroxide, and thereafter absorbs carbon dioxide from the atmosphere or "air" in the compost to become calcium carbonate, chemically the same as simple garden lime.

The steam, the vapor, you see emanating from a hot compost pile is excess moisture, not necessarily nutrients. It takes a balance of material (C:N ratio close to 30:1), moisture (not too much), and enough Nitrogen to get the bacteria that will digest that material working hard enough to generate that heat. Steam emanating from a compost pile is not a bad thing.
However, a too wet compost pile excludes air the bacteria need so they cannot work. A too dry compost pile also keeps the bacteria from working. A Carbon to Nitrogen ratio out of wack also prevents them from working as well. Balance, but not necessarily equality is what is needed.

All I meant about nutrient loss was that when your pile smells like ammonia you are losing nitrogen to the air. Not that steam is a bad thing. It certainly isn't in a well balanced pile.

I thought it was a good sign when the pile got steamy...that it was getting hot enough to destroy weed seeds and decompose faster. Thats what usually happens when we mow the lawn and add the grass clippings.

Absolutely. In the instance I referred to, I either had too much grass or too much ashes in the pile, or both.