When there's a gap in the Run Period, does EnergyPlus reinitialize the simulation?

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Not directly related, but if what you are interested in is reducing EnergyPlus runtime and you trust the methodology of splitting up an annual simulation temporally and then summing (and potentially scaling) the results, you could give Autodesk's EnergyPlus Cloud service a try. That's what they do.

Julien's response answers the bulk of your question but regarding the approach of using several short run periods with EnergyPlus to shorten run time has been considered for code compliance tools such as COM-Check by PNNL and included with CBECC-Com by the California Energy Commission because it provides a remarkably good indicator of energy usage and uses only about a quarter of the time.

PNNL has documented this approach in the report:

PNNL-19961 QuickSim Analysis Approach

If you supply several discrete run periods in the RunPeriod object, such as January 1 to 7 and July 1 to 7 for example, EnergyPlus will launch a warmup period before the start of each period, in order to initialize the loads and temperature values (see why in the "Engineering Reference manual" below)

Input/Output Reference Manual

What this warmup period is is explained in the Input/Output reference guide under the Building object:

Warmup Convergence

The following two fields [Loads convergence torelance value and Temperature convergence value] along with the minimum and maximum number of warmup days (also in this object) define the user specified criteria for when EnergyPlus will “converge” at each environment (each sizing period or run period set as Yes in the SimulationControl object).

EnergyPlus “runs” the first day of the environment (starting with a set of hard-coded initial conditions) until the loads/temperature convergence tolerance values are satisfied (next two fields) or until it reaches “maximum number of warmup days”. Note that setting the convergence tolerance values too loose will cause the program to be satisifed too early and you may not get the results you expect from the actual simulation

You can specify the maximum and minimum number of warmup days in the Building object, as well as the loads and temperature convergence tolerance values. The default values are:

• Minimum Number of Warmup Days = 6
• Maximum Number of Warmup Days= 25

I suggest you read the Input/Output reference guide that comes with EnergyPlus, or you can access it online (Building and Runperiod objects). It has more information on the warmup convergence and explains why they chose the defaults of 6 and 25 for example.

Engineering Reference Manual

If you want to read further, the Engineering Reference manual is where you should look. It notably has a section called "Warmup Convergence" that should be of interest too. One interesting extract:

Since everything in EnergyPlus is based on the foundation of the loads simulation, it stands to reason that any inaccuracies in the loads calculation will result in inaccuracies of similar or larger magnitude in the HVAC calculations. In the presumably limited cases where convergence was not truly achieved before the actual simulation began, it is unknown how much error would be introduced into the results. While simulations that last longer (annual vs. design day) will hopefully have any initial condition problems balanced by the shear number of days in the simulation, shorter simulations — particularly those used for sizing — could result in relatively large errors.

Effect on runtime

Whether it gets significantly faster to only run a sequence of 7 days each month depends on the complexity of your model and your chosen timestep. With a fairly complex model at 6 timesteps per hour, the time it takes to warmup will very likely be less than the time it takes to simulate the remaining 3000 timesteps in each month.