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My laptop is capable of running most games on high settings at a decent frame rate. However, if the power cable gets unplugged while I am playing, the game immediately starts slowing down, even if I am using the high performance power plan.

Why is that? Is the battery unable to keep up with the power demands of the GPU? Is there any way to fix this?

Running a high-performance GPU at full speed while running on battery power can damage the battery or require more power than the battery can safely supply.

• High-performance mobile GPUs can require significant amounts of power to operate at full speed. The GTX 765M requires 75 watts while top-of-the-line mobile GPUs like the GTX 780M and GTX 980M can consume up to 122 watts.
• The GPU is not the only power-hungry part in a laptop. A modern Intel performance mobile CPU typically draws about 47 watts at full power. In addition, you need to power other system components, such as the display, disk, and USB peripherals. When you add it all up, you might need anywhere from 140 to 200 watts to operate a gaming laptop under full load (depending on your system configuration).
• A typical battery in a gaming laptop can store about 60 to 80 watt-hours of energy. Most Li-Ion batteries are not designed to be discharged faster than twice their watt-hours rating per hour (2C) and continuous discharge at rates exceeding 1C can significantly reduce the life of the battery. Continuously pulling 150 watts or more from a typical 77 watt-hours battery is not a great idea. Your battery could overheat and fail or even catch fire. While it is likely the battery’s own protection circuitry would shut down the battery if overloaded or overheated, a device should never subject its battery to an unsafe load at any time during operation.
• To avoid overloading the battery, the GPU will typically throttle to a lower clock speed. The GTX 780M on my personal laptop will not run faster than about 400 Mhz when running on battery power. Lower clock speeds reduce power consumption not only by having transistors switch slower, but also by allowing lower voltages. Keep in mind that power consumption and dissipation scale with the square of voltage.