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The OA controller is turning off the air flow. The minimum and maximum outdoor air flow rates are autosized. Note also that the OA controller input fields stop at the "Minimum Outdoor Air Schedule Name" field and there are more inputs that can be used to control outdoor air.

Controller:OutdoorAir,
  Data Center OA Controller,  !- Name
  autosize,                   !- Minimum Outdoor Air Flow Rate {m3/s}
  autosize,                   !- Maximum Outdoor Air Flow Rate {m3/s}
  FixedMinimum,               !- Minimum Limit Type
  MinOA_Sched;                !- Minimum Outdoor Air Schedule Name

The result of sizing the autosized inputs yields a 0 minimum (I used Miami weather so your results may be different).

Component Sizing Information, Controller:OutdoorAir, DATA CENTER OA CONTROLLER, Maximum Outdoor Air Flow Rate [m3/s], 6.83914
Component Sizing Information, Controller:OutdoorAir, DATA CENTER OA CONTROLLER, Minimum Outdoor Air Flow Rate [m3/s], 0.00000

The Minimum Outdoor Air Schedule is constant at 1, however, 1 * 0 = 0.

Schedule:Compact,
  MinOA_Sched,             !- Name
  Fraction,                !- Schedule Type Limits Name
  Through: 12/31,          !- Field 1
  For: AllDays,            !- Field 2
  Until: 24:00,            !- Field 3
  1;                       !- Field 4

If you change the OA controller inputs to place the min OA schedule name in the "Minimum Fraction of Outdoor Air Schedule Name" input field, the evap cooler will operate all year.

Controller:OutdoorAir,
  Data Center OA Controller,  !- Name
  OARelief Node,           !- Relief Air Outlet Node Name
  Supply Equipment Inlet Node,  !- Return Air Node Name
  Unitary Package Node,    !- Mixed Air Node Name
  OAInlet Node,            !- Actuator Node Name
  autosize,                !- Minimum Outdoor Air Flow Rate {m3/s}
  autosize,                !- Maximum Outdoor Air Flow Rate {m3/s}
  DifferentialDryBulb,     !- Economizer Control Type
  ModulateFlow,            !- Economizer Control Action Type
  28,                      !- Economizer Maximum Limit Dry-Bulb Temperature {C}
  64000,                   !- Economizer Maximum Limit Enthalpy {J/kg}
  ,                        !- Economizer Maximum Limit Dewpoint Temperature {C}
  ,                        !- Electronic Enthalpy Limit Curve Name
  0,                       !- Economizer Minimum Limit Dry-Bulb Temperature {C}
  NoLockout,               !- Lockout Type
  FixedMinimum,            !- Minimum Limit Type
  ,                        !- Minimum Outdoor Air Schedule Name
  MinOA_Sched;             !- Minimum Fraction of Outdoor Air Schedule Name