AN EXPERIMENTAL ACFT PLT VIOLATED A TFR INVOLVING A PWR PLANT.

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I DEPARTED ZZZ ARPT FOR A LOCAL FLT UNDER FAR 91 VISUAL FLT RULES (VFR); AT APPROX 5-8 MI E OF THE ARPT; ATC CALLED ME AND ASKED ME TO RECYCLE MY XPONDER. I ACKNOWLEDGED THE CALL AND RECYCLED. DURING THIS TIME; THERE IS A NATIONAL ALERT FOR RESTR FLTS INTO ANY PWR PLANTS. A SECOND CALL WAS RECEIVED FROM ATC APPROX 1 MIN LATER ASKED ME TO RECYCLE MY XPONDER AGAIN. I PROCEEDED WITH THE REQUEST AND TOLD HER I WAS STILL AT A LOW ALTITUDE AND THAT I NEEDED TO CLIMB FOR BETTER RECEPTION. DURING THIS TIME I WAS DISTRACTED BY THE XPONDER PROB AND THE WX CONDITION AROUND THE AREA. MY FLT PLAN WAS TO FLY E; THEN SE IN ORDER TO CLR THE 10 MI LIMIT PLACED AT YYY PLANT WHICH IS APPROX 15 MI SE OF THE ARPT. A THIRD CALL WAS RECEIVED FROM ATC ASKING IF I WAS AWARE OF THE DISTANCE LIMIT FROM YYY I TOLD HER I WAS AWARE OF THE 10 MI LIMIT. SHE DID NOT RESPOND BACK WITH ANY COMMENTS. I ASSUMED THE XPONDER PROB WAS CORRECTED AND RESUMED WITH THE FLT. UPON CLRNC TO LAND AT ZZZ; I WAS TOLD TO CONTACT ZZZ APCH. ZZZ APCH TOLD ME I WAS FLYING WITHOUT MODE C; THAT I HAD PENETRATED RESTR AREA AT YYY. I FEEL CONTRIBUTING FACTORS ARE: 1) DISTRACTION WITH TROUBLESHOOTING XPONDER PROB. 2) WX CONDITION LIMITING AIR TO GND VISIBILITY DUE TO LOW CLOUD COVERAGE; UNABLE TO LOCATE LANDMARKS FOR REFERENCE THEREFORE CREATING ERROR IN DISTANCE JUDGEMENT. 3) LACK OF FEEDBACK FROM ATC THAT COULD HAVE PREVENTED THE PROB. DURING ALL THIS TIME I WAS MONITORING ATC FREQ AND ATIS.