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The statement "My timeseries loads change every T=600s so I thought maybe I should use that as the Ti term for all the parts of the system" indicates a misunderstanding. The time constant of the controller, Ti, needs to be appropriate for the dynamics of the feedback control loop to which the time series load is just an input, or a disturbance. The value of Ti is not dependent on how fast your time series change. Generally, the default values for Ti (and k) as entered for the template models in the Buildings library should work or may need only minor tuning.
I suggest you check for one of your 150 buildings whether they are appropriate, tune them if needed, and these settings should typically then also work for the other buildings because of the way the dynamic response of HVAC equipment is configured in the Buildings library.