The closest attempt to permanent buoyant tethered platforms implemented so far has been the air force radars suspended from aerostats at 4.5km altitude along the southern US border. These were developed in the 1980s before PV was a viable source of electricity so the radars they carry are powered from diesel generators. Sufficient fuel is carried aloft to sustain about two weeks of operation. Given that constraint, permanent uninterrupted operation was not achievable.
However, even without the energy supply constraint, the simple physics of the environment of the troposphere also make permanent tethered platforms beyond what is achievable with current technology. Troposphere wind speed can exceed 100m/s and air density is around 1kg/m3. A tethered platform that can station keep under these conditions has to be a minimum of about 500 meters long and sustain wind pressure exceeding 2000 Pascals. There are additional environmental constraints including strong vertical wind shear in thunderstorms and rain, snow and ice. The combination of the size required and the forces a lightweight structure of this size can sustain make this impossible to achieve under steady state conditions, never mind the chaos of stormy conditions.
The radar aerostats are about 80 meters long and can therefore operate in a wind of around 30m/s before being blown over. This wind speed also produces forces within the range that the super pressure body can sustain mechanically. Higher winds or bad weather cause the platforms to be winched down. This highlights another aspect of tethered aerostats. Even winched down they are exposed to damaging winds and accidental impact with the ground or other objects, wind born debris etc. If the weather changes suddenly they may not be winched down in time and suffer damage. All these operational conditions add up to aerostats having a tough and limited average lifetime.
The Stratosphere at 20km has a maximum wind speed of 50m/s and a density of about .09kg/m3. There is no water and there are no storms. This operational environment means a platform theoretical minimum length is around 150 meters and the maximum wind forces are around 150 Pascals, or more than an order of magnitude less than the troposphere. This along with the benign stable weather free environment mean that real tethered platforms of about 300 meters in length can station keep at 20km altitude.