In CFM56, I read that it is supposed to increase HPT performance but I don't know how it does that.
High Pressure Turbine Active Clearance Control (HPTACC) is a system that controls the temperature of the shroud of the high pressure turbine rotor (HPT). It can cool or heat the shroud support structure using bleed air from the 4th (relatively cool) or hot bleed air from the 9th stage to control the (tip) gap (or tip clearance) between the rotor (rotating part) and the shroud (stationary part).
Why take all the trouble of controlling the temperature?
Metals expand, but metal in the gas path (e.g. rotor blades) warm up much faster than the surrounding structures when power settings change. Differences in expansion from uneven heating and stresses from rotation (the higher the spool speed the higher the stress the higher the elongation), especially during transient operation (low to high or high to low power setting). This makes it very difficult (but not impossible, there are e.g. also seal types that use abradable honeycomb material in combination with knife edges) to have a single gap between the rotor and the shroud. If a single gap size would have been chosen it would be too big as it has to prevent rubbing at all times. If you can control the temperature of the shroud structure you can, based on measurements and design information, create a system that controls the gap between the shroud and the rotor blade by expanding and contracting the shroud structure.
The image below shows a typical impression of the radial position of the shroud (top orange line) of which the position is based on a displacement by thermal expansion (δth), and the tip of the HPT rotor blade (bottom displacement blue line) of which the displacement is based on thermal expansion and an elongation by rotation (δth + δm) as function of power setting. The tip clearance clearly is not constant but a function of power setting, active clearance control can be used to minimize the tip clearance by controlling the expansion of the shroud structure.
If you can keep the gap between the rotor blade and the shroud as small as possible, less air (which is a mixture of combustion products, cooling air from the high pressure stator vane and compressor discharge air) will pass over the top of the blades. The more air passes the rotor over the blade top (tip) the less will be available to extract work from to power the high pressure compressor (HPC), the less efficient your core will work.
The turbine blade clearance is the gap between the tip of a turbine rotor blade and the turbine shroud:
CFM56-7B HPTACC principle, adapted from Airbus engine systems manual
The blade is part of the rotor, and the shroud is part of the stators assembly. The length of a blade and the diameter of the shroud vary with the temperature of gas from the combustion chamber (which is on the left side above). However they don't vary exactly in the same proportion, therefore the tip clearance also varies.
When the gap is wider, a greater proportion of gas do not turn the turbine disk but escapes directly to the next stage, which in the engine shown (CFM56-7B) is the LPT since the HPT has only one rotor. This reduces the engine efficiency.
To prevent such loss the active clearance control (HPTACC) cools the turbine shroud with air from the compressor to reduces its diameter. Air is tapped at two levels in the HPC: 4th (colder) and 9th stage (hotter) and mixed to obtain the desired cooling air temperature.
This air in injected in 3 circular manifolds around the HPT (in light green above). These manifolds have many holes to diffuse air near the shroud. The shroud is cooled according to the ratio of 4th stage/9th stage air, the shroud diameter is reduced accordingly and the clearance too.
Cooling air temperature is controlled by the HPTACC valve:
CFM56-7B, HPTACC functional diagram, source Airbus engine systems manual