Running the prototype and after doing some tuning (using Ziegler-Nichols open loop test), the following quite good-looking temperature curve was measured (the brown curve):

After approximately 10 minutes (from cold start), the temperature settles and remains really stable.
The temperature is actually within +/- 0.1 °C now! Compare this with the original thermostat.

This means the operator will always start brewing at the correct temperature.

The red curve shows the PID controller output (in %). It starts with 100% and then continously changes to compensate the temperature fluctuations. When the temperature has stabilized, the output averages at 7%. With this 1400W boiler machine it means just below 100W average power, i.e. 2.4 kWh/day.

The next step in this PID project is to develop an auto-tuning function so the controller can be optimized for any espresso machine.