You can't have that delay there, else has to be right after the if block {     }. There are also two semicolons missing after the digitalWrites, the if is capitalized (should be lowercase). You should also set the ledpin as an output in setup.

In hardware you would use a transistor to short the comparator output to zero while you activate the coil. (after a suitable resistor if needed).

analogRead has to be read in a variable like pot = analogread(0);
A0 is constant referring to the pin number, like "0" , it is not the voltage of that pin.

Code:

void loop(){
   int x;
   pinMode(logic, OUTPUT);
    x = analogRead(A0);
   analogWrite(logic, x / 4);
 //  pinMode(A0, INPUT);   //default is input already
  }

Some similar looking transformers I was looking at were 150mA current from either 9V or 18V (but not both). It is definitely in the 75 - 300mA range. (unless that's some sort of novelty coin)

I think the problem is the way you use the distance there. The PID would try to go UP to the limit (from zero) , and the moment it is over it, it would output a negative value to get back to it. However the library outputs 0 - 255 only, so you see zero. I am not sure if changing "DIRECT" to "reverse" will help. For testing you could make Kd=0 since that will be very sensitive to noise. You could make Kp zero, too. Hope you can fix it.

since it is unlikely 2 lcd screens failed in the same way I would guess one of the pins on the mega died probably. Too much current draw?

When you have no fan pulsein return zero, and the next line contains a divide by zero in this case.

the led is probably shorted out.

I would use 2 diodes at the battery / wallpack and use software sensors (resistor dividers) for the arduino to tell where the power is coming from. You could even use 2 wall transformers and 3 diodes and forget about the relays. A battery sensor could also tell you if the battery needs replacing. I am not sure what the 100 ohms resistor's purpose is. Also, what happens when the power comes back? Does it go into the generator? I am probably missing something.

I think the use of reset from the divider might create problems. Usually you need a stable supply and then apply reset, but if you conenct both the supply and reset to the same thing, you really are guessing the outcome, especially with mechanical connections like usb. I would use 2 rc filters, one with a longer constant for reset, but even that is a bad idea.

 I don't know what would happen if you power one vcc and not the other, it is a good question.

A fuel cell could periodically charge a rechargeable battery or supercapacitor , therfore neutralising each other's shortcomings. I have a feeling you will end up needing a lot of power. Suppose the rechargeble could stay charged for 10 years, then the fuel cell should be able to work 11 times. This seems doable to me. Or you could use a wet battery and just replace the electrolyte with fresh one periodically.

Maybe temperature gradient exploitation using peltier devices?

I was going to suggest a radioisotope generator, but I doubt you can get one.

Or , someone mentioned a turtle. Don't they live 100 years? How much food could a turtle possibly eat in 100 years?

Even if you power the synth with 30V you will still have problems controlling it with 0-5V from an arduino. Anyway, I was thinking, why use such a high supply, as far as I can tell the synth could be made on a much lower supply, even 5V, probably. It's mostly the TL072's opamps that should be replaced with something that works with lower power. The 3080, whatever that is, seems to work at 5V, and the transistors would need some lower resistors. Of course, I am not sure how much "black magic" goes into VCOs, but it would be an interesting project.

It is possible the mega is not registering a High logic level at 3.3V , ideally you want 5V input High level. Otherwise it can be caused by long wires or improper grounds. I am not sure why you can't use the same pin for both, I'm sure the current won't be exceeded. You could even power the ph module from an output pin set to high, I am sure that the 14mA current is within limits, but you should look it up.

For an opamp you would need an inverting circuit where you replace the virtual ground at the (+) input with a reference (pot) to remove the offset.  The gain should be 5 but you could make it adjustable, too. Some other factors to consider are the power supply, most op amps do not like 5V single supply power, so you need probably one of those fancy cmos ones. Then there is the output swing, that depends on your choice of power supply and op amp. You want to make sure it can swing to zero and it can reach 5V.

Another option would be to use a higher power supply for the potentiometer and use some diodes (or a zener diode) to remove the offset. The higher supply would act as an amplifier. But that could break the arduino if it gets in the analog pin, so this also presents problems.

Building a high voltage generator with no electronics knowledge is a very good way to accidentally die.