Darth Vader's TIE Advanced x1 was a starfighter used by Darth Vader during the Galactic Civil War, leading Black Squadron, and flown by him during the Battle of Yavin. It was a modified early prototype of the TIE Advanced x1 line, and in fact was the first of the limited production.
The TIE Advanced x1 featured a greater speed than the TIE/LN. It was armed with heavy twin blaster cannons on fixed mounts, and was also capable of carrying secondary weapons such as cluster missiles. In addition, unlike most TIE models, the TIE Advanced x1 was equipped with both a hyperdrive and a deflector shield generator and even a much more effective solar ionization reactor and stronger hull.
The Advanced x1 had more cockpit capacity than other TIEs, thus catering to the larger-than-average figure of Darth Vader. It was built to Vader's specifications by Santhe/Sienar Technologies at the Sienar Advanced Projects Laboratory.
Unlike the class standard, Vader's TIE Advanced apparently possessed a life-support system. 
Upon the completion of Darth Vader's TIE Advanced x1 by Sienar Fleet Systems, Marketing Relations Manager Taed Augustus personally presented the new vessel to the Dark Lord of the Sith. Unfortunately Taed was far too familiar towards the Dark Lord, referring to him as "Darth" more than once. Vader, unamused by the SFS official's lack of respect, choked Augustus to death mid-presentation.
While on Kamino in 1 BBY, Darth Vader was personally overseeing the creation of the clones of his late apprentice Galen Marek; the Dark Lord's customized TIE fighter served as his means of transportation to Timira City Cloning Facility where the Marek clones were located. Before Vader could execute the latest imperfect clone who failed to meet his expectations, the clone broke free and slaughtered all of the Imperial machines and stormtroopers that stood in his way. Vader, knowing that the clone sought to escape from Kamino, attempted to intercept the clone, but failed to prevent the clone's theft of the Dark Lord's own TIE fighter.
Utilizing Vader's TIE Advanced x1, the clone traveled to the Imperial-occupied world of Cato Neimoidia. Since the TIE fighter was registered as Darth Vader's personal vessel, Baron Merillion Tarko was under the belief that the Dark Lord was paying an unexpected visit, and thus sought to personally welcome Vader's arrival at the landing platform with a stormtrooper escort. To his surprise, however, the TIE fighter's pilot was unknown to him and quickly sought to execute the clone when it became apparent that he sought to rescue General Rahm Kota; the Imperials' attempts utterly failed and resulted in the deaths of many Imperials, including Tarko.
Soon after the events on Kamino and Cato Neimoidia, Vader later retrieved his ship, which later saw action during the Battle of Yavin in 0 BBY. During the battle over the first Death Star, it was escorted by two handpicked Imperial pilots flying standard TIE/LN starfighters. After destroying many of the Rebel fighters, it spun out of control when one of Vader's Black Squadron wingmen collided with it in attempting to avoid cannon fire from the Millennium Falcon. Following the destruction of the Death Star, the Sith Lord made a hyperspace jump to a planet, Vaal, where he could land safely. His ship was damaged when it was hit by small asteroids in Vaal's orbit, and Vader was forced to make a crash-landing on the planet. Upon his return to the Empire, Vader gave orders that his craft be recovered for repair and that it was to be docked in his personal ship, the Executor.
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