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For example, it seems switching from some high wattage headlights to much lower wattage LED's frees up more of the alternator's power capacity. But, does this mean the alternator (or other electrical components like the voltage regulator) will start to run hotter? (to dissipate excess power via heat?)

Or, does the decrease in overall electrical load simply get handled by just easing up the load on the engine making it easier to spin the alternator in the first place?

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Your second thought is the answer. It's all about the load and how the regulator deals with it. As the demand goes down, so does heat and less of a power demand upon the engine.

EDIT: (To provide more context.)

To put more emphasis upon what I'm saying, I'll reference this article which talks about how the alternator and regulator work:

Lights, ignition, and accessories use power from the electrical system. Every time we switch an accessory ON, more power is drawn from the system. Voltage (electrical pressure) drops as power is drawn from the system, and then the voltage regulator causes the alternator to make more current.

If you were going in the opposite direction of what you're asking and drawing more from the system, the voltage regulator makes the alternator work harder. Therefore, if you demand less of a draw on the system, the regulator isn't going to tell the alternator to kick in as often to do its job.

In the electrical system, the voltage regulator “turns the alternator ON,” or “turns OFF the alternator” as needed to maintain voltage at the proper level. And in the air compressor system the pressure regulator stops and starts the compressor as needed to maintain the proper level of pressure.

The useful electrical system will require an alternator that can produce an average of more output than we use, and the regulator will limit system voltage to the safe level we need. ... Alternators make heat as a by-product of making electrical power, and the more power they supply the more heat they make. Some models of alternators can stand to put out a much higher percentage of their gross output rating than others, during extended periods of operation.

Since we know that increased load on an alternator will make heat, the bi-product of drawing less from the system is the alternator will run cooler because it's not having to generate as much power over a given amount of time.

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    +1, but the regulator isn't involved here. It's an intrinsic feature of any electrical machine that current depends on mechanical load and vice versa. The regulator addresses voltage, which depends on speed (and therefore, engine speed).
    – Janka
    Commented Jan 24, 2018 at 1:00
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    @Paulster2: The article you are citing is wrong – that misconception unfortunately has spread widely. The reason why there is a regulator is the voltage an electrical machine delivers depends on speed. So, if you had your engine running at 1000rpm and the alternator produced 12V, it would produce roughly 36V at 3000rpm. That's not what you want, you want about 12V independent of engine speed. That's what the regulator in the alternator is for. It cannot regulate small voltage differences of about 1V, which may happen with a flat battery and lots of power draw.
    – Janka
    Commented Jan 24, 2018 at 11:04
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    It also doesn't turn the alternator ON/OFF (1940ies regulators did this, mechanically) but controls the excitation of the alternator and thus, the output voltage.
    – Janka
    Commented Jan 24, 2018 at 11:05
  • @Janka AND more output voltage means a voltage difference between battery and alternator output, which by definition causes a current, which inreases the mechical load and the load on the engine.
    – SteveRacer
    Commented Jan 25, 2018 at 3:19
  • The current through the alternator could even stay the same, also the torque needed, and it was still additional power drawn, because of the voltage. Or in mechanical terms, because of the higher speed at which the needed torque is applied.
    – Janka
    Commented Jan 25, 2018 at 10:40

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