There are four variability options for EV charging in HOMER Grid, but it is not explained how those parameters are calculated or how they interact together. So, I would appreciate if you could an equation or explanation for each parameter.  

The charge duration for each session is drawn from a modified normal distribution with the following parameters: 

Mean (hr): "Mean time connected" input in hours 

Standard deviation (hr): Charge duration variability (%) multiplied by the mean time connected 

It is modified in that all draws more than one standard deviation from the mean are moved to be plus or minus 1 SD from the mean (whichever is closer to the initial draw). So you can be sure that the charge duration will always fall within the following range: 

Max = (1+variability/100)*mean 

Min = (1-variability/100)*mean  

Therefore: 

Charge duration variability (%): the standard deviation of the charge duration distribution, as a percent of the mean. 

Mean time connected (hr): The mean of the random distribution used to set the charge duration for each session. 

For each charging session, the vehicle type is selected randomly according to the weights entered in the table (Proportion of EV Population). The "Required charge energy (%)" below corresponds to the selected vehicle. The charge energy supplied in each charging session is drawn from a similarly modified normal distribution with the following parameters: 

Mean (kWh) = Required charge energy (kWh) / Mean time connected * session duration 

Standard dev = Charge energy variability (%) times the mean as calculated above 

(Limited to +/- 1 SD) 

The extra complexity here (as opposed to just using the "Required charge energy" as the mean) is to reduce the fraction of sessions that are bounded by the max power constraints. We use the Required charge energy and mean time connected inputs to calculate a mean power in kW and then apply the random variation to this power, not to the overall energy. This makes it easier to choose settings that never require max power to reach the required energy for any sessions. 

If the max power is needed to supply a session (i.e. max power = 50 kW; energy = 200 kWh; duration = 4 hr) then the "deferrable" behavior is lost and the charger must supply the max power constantly. This can ruin demand charge reduction for the whole month with one session. By design, we assumed that the deferrable charger is used in situations where the total load needs to be managed, and so users of the charger would be constrained from forcing a max power session. When you input the "mean time connected" and "required charge energy" you also define the mean charge power: 

Mean charge power (per vehicle type): Required charge energy / mean time connected 

Max charge power (per vehicle type): mean * (1 + variability/100) 

You can ensure that the system is always able to manage the load (rather than being pinned at constant power) by ensuring that this max value is less than the charger and vehicle max power constraints. If you want the charger to supply a constant power level rather than the more complex behavior of the deferrable charger, use the "on-demand" charger type. 

Next, the day-to-day and timestep variability are used to determine the number of charging sessions each hour. Day-to-day variability shifts the load up or down from day to day. Timestep variability shifts the load between one timestep and the next. At the start of each day, a number is drawn from a normal distribution with mean = 0.0 and SD = 1.0, modified such that the area below -1 SD is moved to the point at -1 SD, and the area above 1 SD is moved to the point at 1 SD. This is the "daily noise factor". For each hour, another number is drawn from the same distribution, the "timestep noise factor". The number of visits for each hour is then calculated as follows: 

Frequency = (1 + Day-to-day variability/100*daily noise factor + Timestep variability/100*timestep noise factor) * hourly average sessions * scale factor 

where the "hourly average sessions" is the number from the user-specified usage profile table corresponding to the day and time, and the scale factor is the ratio of the profile's daily average and the specified "Scaled average sessions/day" input. The calculated frequency is not a whole number. A single random draw is made from a uniform distribution to determine whether to round up or round down. The decision is weighted based on the decimal value -- a frequency of 5.2, for example, would have a 20% chance of rounding up to 6, and an 80% change of rounding down to 5. This ensures the average tends towards your specified average values. 

Finally, the start minute for each session is set randomly with an integer in the range [0, 59]. The energy required during each timestep of the simulation is proportional to the average power during that hour and the fraction of the timestep that overlaps with the session. If more EVs arrive than available charging plugs, EVs will wait for the next available charger.