Nope. The amount of power we can transmit is defined by power = current x voltage, so to carry more electricity, we simply raise current or voltage. On the other hand, the amount of power lost when transmitted through a wire is governed by the equation power = current squared x resistance. Thicker lines would lower the amount of resistance of the wire, which would make the wire more efficient. Lower resistance means less energy lost in transit as heat, so you would get more power out of it. It’s not realistic to make lines thicker and thicker to accommodate more electricity though, so we usually reduce current instead. To transmit more power while lowering current (remember power = current x voltage), voltage is increased.
I have to disagree with the above answer, to an extent. The future of our power grid structure demands that we answer increasing electric demand with the ability to furnish an increased supply. While you can raise the voltage to increase the power provided, the equation above presumes the stability of the transmitting circuit. You can’t exactly just tamper with the voltage and current and expect increased power, because electric circuits do have a carrying capacity.
Given those stability considerations, physical attributes of the existing grid must be updated. While I was unable to find data specifying the diameter of the thicker lines needed to support higher voltages, I know that we would at least need larger, taller electrical transmission towers and correspondingly broader right of ways for power lines.
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