<font face="Arial, Verdana">I changed everything in my setup, except mechanical components (because they are the same as before the problem). i.e.:</font>
<font face="Arial, Verdana">- PC (from Desctop to Notebook) + the host software (different versions of Repetier Host - from v.1.06 to v.1.53)</font>
<font face="Arial, Verdana">- USB cable</font>
<font face="Arial, Verdana">- Arduino Mega 2560 + few times new firmware (Repetier 0.92.3)</font>
<font face="Arial, Verdana">- RAMPS 1.4</font>
<font face="Arial, Verdana">- Y driver (A4988)</font>
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<font face="Arial, Verdana">In addition I made some changes in my firmware too, especially STEPPER_HIGH_DELAY 1 (from "0")</font>
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<font face="Arial, Verdana">I checked many times the current from the drivers and theirs (and motor's) temperatures. They are as follow:</font>
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<font face="Arial, Verdana">Axis - Driver - Motor</font>
<font face="Arial, Verdana">"Y" - 32.7°C - 31.5°C</font>
<font face="Arial, Verdana">"X" - 28.0°C - 32.4°C</font>
<font face="Arial, Verdana">"Z1" - 35.5°C - 34.7°C</font>
<font face="Arial, Verdana">"Z2" - 28.5°C - 32.4°C</font>
<font face="Arial, Verdana">"E" - 28.0°C - 31.8°C</font>
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<font face="Arial, Verdana">All temperatures were measured with digital infrared thermometer and the ambient temperature was 27.2°C. </font>
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<font face="Arial, Verdana">The Vref of the "Y" axis was 0.8V but the motor has left some steps. That's why I raised the Vref to 1.00 V and now it works smoothly. The temperatures quoted above are with Vref = 1.00 V. !!! Before the problem, Vref was even more high - 1.52 V, and I had NO any problems for more than a year !!!</font>
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<font face="Arial, Verdana">The last, that can be changed is the firmware type. </font>
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<font face="Arial, Verdana">Thank's to all</font>