Date: 202003
Local time of day: 0601-1200
Location: ZZZ.ARTCC, US
Flight conditions: VMC
Light: Dusk
Aircraft: Commercial Fixed Wing
Operator: Air Carrier
Operating under: Part 121
Flight plan: IFR
Mission: Passenger
Flight phase: Climb
Airspace: Class A ZZZ
ATC / advisory: Center ZZZ
Reporter organization: Air Carrier
Reporter function: Pilot Not Flying; Captain
Reporter qualification: Flight Crew Instrument; Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP)
Human factors: Distraction; Situational Awareness; Workload
Second reporter function: Enroute
Anomaly: Conflict Airborne Conflict
When detected: In-flight
Result: General None Reported / Taken
Contributing factors: Aircraft; Environment - Non Weather Related; Human Factors; Procedure
Primary problem: Environment - Non Weather Related

Narrative: This flight did happen some time ago so I do apologize if the story has a few holes in it. We were on a flight climbing out to go to our destination. The flight was relatively uneventful. It was a gorgeous VFR day. I don't remember about the initial altitudes that center gave us but I do remember leveling off a few times before the RA event happened. We were over Lake X when the RA happened. ZZZ Center was busier than ever; I do remember the radios being very congested during the climb out. I think finally after a couple minutes we got to climb again; we received Clearance to go to flight level 200; or 20;000 feet. I think the Controller even asked us to verify 20;000 feet. We read it back to him correctly and we continued the climb. I was the Captain and pilot not flying. I was watching my First Officer fly the leg; and in fact doing a great job. I did notice a target on our TCAS. It was another aircraft. I think if I remember right ATC even pointed him out to us; we had contact. ATC said roger. As the target got closer; it turned into a RA; a resolution advisory. My first officer took manual control of the airplane; and proceeded to do the maneuver. We did eventually level off at 20;000 feet as we followed our RA. I think what actually happened since we were very light loaded and had a very high climb rate that the two planes thought we were going to collide; it is actually very common in the [type] jets. Yes we were leveling off but the rate of climb in our plane was abnormally high because we were light on fuel and passengers because of the COVID-19 outbreak. So that triggered the RA to go off. That's why the monitor vertical speed was given and not actually a climb or decent was given to us from the system. There was a report of an altitude deviation of which in my professional opinion [I] doubt happened. Our altimeters were off but were still well with in the 200 feet tolerance for RVSM flights. We did concur with air traffic control 2 times the needed altitude with correct responses. After the level off at 20;000 feet. We told ATC we were cleared of conflict. After word he asked us for assigned altitudes that was given to us and we said 20;000 feet. After the RA; we were cleared to 25;000 feet. Cause of the event is that the rate of climb in our plane and rate of climb of the opposing plane was very high. Causing the computer to think that there was going to be a collision; but in fact since we were leveling off at 20;000 feet which meant there was never a threat. It happens to aircraft going in and out of ZZZ all the time; you will get a RA chirp; or an actual RA when you turn base to final or on the last segments on a arrival because of the departures. Another factor could be that all three altimeters are not going to be perfect; like a simulated flight in a sim. Yes; they are off but there still well within the tolerance needed for a safe flight. Second; it's a busy airspace. ATC needs to better manage their airspace; hire more people or something. Because time and time again they give wrong clearances to the wrong aircraft. Better planning by air traffic control and by pilots alike. Safety is paramount in these days of aviation; now more than ever. Verifying altitudes; speeds and headings are an absolute must because technology obviously has its flaws. Technology does not replace actual pilots and their flying skills.


Second narrative: Aircraft X was departing from ZZZ going to ZZZ1. I was step climbing the aircraft up underneath Aircraft Y; which was a departure off of ZZZ2 to ZZZ3. After I had observed Aircraft Y leaving FL200 I climbed Aircraft X to FL200 and cleared them direct to a fix in the same clearance. I was sure to emphasize FL200 in my clearance and I observed them respond with 200 as well. My D-sider at the time was doing coordination with another sector so I do not believe they heard the clearance and read-back. At the time; we were figuring out (with the D-side position who was training) what to do with an aircraft at 120 going over Lake X and how to miss approach airspace. While the volume of traffic was not high; the complexity of us figuring out what to do with the low and slow aircraft did take up a good amount of our attention. As Aircraft X made the turn to ZZZZZ and climbed to FL200 I told myself to make sure to call traffic with the Aircraft Y because they would be close. However as we decided what to do with the other aircraft; I failed to call the traffic for the Aircraft X and Aircraft Y. After telling the slow aircraft what our plan was; Aircraft X immediately checked in saying they were responding to an RA and were leveling at FL200. I immediately called the traffic for both AC involved. The Aircraft Y did not get an RA. The Conflict Alert did not go off; but we observed the Aircraft X go to FL201 and then back down to FL200 in the next hit. Aircraft Y was level at FL210 the whole time. When I asked Aircraft X what altitude was assigned to them; they replied FL250. However; when I looked back at the tapes; to me; FL200 was clearly assigned and read back by Aircraft X. This probably could have been avoided if I had called the traffic for the Aircraft X ahead of time and just made sure they were climbing to FL200. Another issue is I gave a climb clearance and a direct to fix in the same clearance. Though I was clear; this has the potential to lead to issues; which obviously in this case it did. Going forward; making ABSOLUTELY certain the aircraft has the right altitude in a situation like this is the solution.


Synopsis: Pilot and Controllers reported an altitude overshoot that caused an airborne conflict.