I can say that back in the 80's I built a neat controller. It could control current, voltage, power, temperature effectively at the same time. It had the ability to limit dp/dt It also was able to detect a shorted TC on startup and it had recipies.

I would advise talking to Eurotherm to see what they can offer. One of their controllers: http://www.eurotherm.com/products/controllers/single-loop/2400/2404/ you can specify an output change limit.

Their power modules have gotten a lot more sophisticated now than what I was used to. I am somewhat familiar with this controller series.

Their power controllers are a little more sophisticated now as well. They always did support current limit and phase angle firing.

When I convinced management to go to DC applied to tantalum low voltage heaters we got really good life times.

So, out of the box, the controller can incorporate soft-start by just limiting the output change in %/unit time.

Have you considered using triacs with soft start control between the contactor and the heating elements? Or perhaps triac driven elements that are always on and driven by commands from the controller to maintain a constant temperature?

Using a solid state relay there are temperature controllers that have a ramp feature that will slow the heat up time. The feature let you set the length of time until it reaches set temperature. This will accomplish what you wish to do.

You could try a solid-state relay controller (you may have to build one at this current level using a couple of high current single phase solid-state relays connected to a control interface to make it compatible with your digital controller signal).

Alternatively you could try a variable speed drive, and as a bonus it should interface directly to your digital controller. Bit expensive thou compared to a solid-state relay controller, but an option.

Another option may be to ramp up the supply voltage with a variac or multi-tapped transformer (although even more expensive than the variable speed drive option).

Yes, there are solid state heat controllers that work similarly to a soft starter. The wave form is switched on/off at a different point in the cycle, though. I think you can get ramp, soak, and PID control features, too.

What you need is a power controller like this one:

http://compare.ebay.com/like/290693487949?var=lv<yp=AllFixedPriceItemTypes&var=sbar&_lwgsi=y&cbt=y

This type of device can use an analog input signal from a temperature controller (http://www.instrumart.com/products/21724/watlow-ez-zone-pm-temperature-controller) to control the rate that power to the heating elements is increased thus controlling the rate that temperature in the kiln increases. So you can have the kiln reach say 500 degrees in 2 hours or in 20 hours based on what you set the temperature controller to do.