Problems with electrical outlet
I am having a lot of problems with the voltage where I live, having both brownouts and surges.


That could be a couple of different things at once.  Or it could be the same actual thing.   

Manassehkatz has floated the theory that both these things could be a lost neutral. That is a very serious defect that causes your two legs of power to "teeter-totter", one below 120V and the other above 120V, (but together adding up to 240V).  This alone would explain both brownouts and loss of equipment due to excessive voltage (which you can call "surges" if you really want to).  A lost neutral puts the two "legs" of service into a tug-of-war, the pulling force is based on current drawn on that leg.  The heavier draw leg gets the lower voltage.  

The way to spot a lost neutral is notice when voltage is far askew of 120V, and then, turn on a big load like a heater, hair dryer or microwave that's on a different circuit.  If the number moves more than a volt or two, you have a lost neutral. 

A lost neutral is a genuine, bona-fide power outage. Contact the power company and tell them you have a power outage.  If they ask you kindergarten-tier questions like "do your lights work", give them kindergarten-tier answers that tell them to roll the truck, like "No, they do not work" (properly, but keep that to yourself).  Power company operators are trained to make you go away, or to be more precise, solve your problem without rolling a truck.  


  One of the receptacles started to smoke so I replaced it. 


That wouldn't be caused by a lost neutral. That's caused by loose blades in a worn out socket, or poor wire connections in the usual place (correctly used backstabs being the #1 culprit, poor torquing of screws being the #2. NEC 2017 now requires you use a torque screwdriver, after science proved master electricians weren't any better at setting torque than their secretaries! Like literally, like not even 1% better.) 


  What I noticed was that contrary to other outlets in the house, this one has two black and two white wires but I did not find a ground wire. What I did find were two additional black wires taped together at the back of the outlet. 


And you leave that the heck alone.  You are in a dangerous place right now. You know a tiny little bit about wiring.  And you've developed a sophomoric tendency to "look down upon" the work of others.  When you see techniques you are not familiar with, you think "that's a defect".  Don't count on it. 

While you will surely find bad work, most of the time it will be proper technique that is new to you. 


  After I replaced the receptacle I decided to install a surge protector. It has two green lights. One says Protected and the other Grounded. Of course, it is not grounded.


Don't be so sure.  Many wiring methods ground via the metal box and the conduit or armored cable.  I have four buildings with EMT conduit in them. There is not a ground wire in any of the buildings.  I own 10 reels of #12 THHN wire, and none of it is green. Yet our grounds are tip-top, and have been put to the test! 

Switches can ground through their mounting screws.  Receps can ground through their mounting screws if certain rules are followed.  Otherwise, look for a hole in the back of the junction box tapped #10-32 (sometime it's being used by a nail or screw). Stick a 10-32 ground screw there, and pigtail that to the recep. 

So look closely at your installation. Especially given the dissimilar number of black and white wires, I suspect your installation is conduit of one kind or another. 


  Could one of those two black wires in the back be a ground wire?


Absolutely not. Grounds are green, yellow-green or bare, except in the former Soviet bloc, and they too have harmonized on yellow-green. 


  Why are there two? Why would someone just twist them together like that? How can I know for sure? I have a voltage tester and those two black wires seem to have current.  


Those wires are passing through, on their way to doing some other important job that doesn't involve this socket.  If you mess with them, you will find out why, because it will stop working! LOL! 

Wires passing by in the back of a junction box is something  you'll see a whole lot of, in any wiring method.  They should never be dismantled, especially not at the same time as everything else!  Everyone thinks they can put it back the way they found it, and everyone is wrong.  Allowing for curious noobs is why I use 10 colors of wire.  In my boxes, purple always goes to purple!