Dissertation: Methodology (4 of 8) - Formulae (1)

B.    Formulae

This section outlines the formulae used to calculate the inputs (i.e. electricity demand, renewable generation patterns, traditional generation modes, and spinning reserve) and the model outputs (i.e. net GHG emissions, and GHG emissions attributed to ES). Please note that several of the equations used in this model make use of the random number generator in Microsoft Excel, which produces numbers between -1 and 1. In order to simplify the equations listed below the variable R_x will be used to denote any randomly generated number; however, the use of multiple R_x variables in one equation should be understood to denote multiple random numbers.

Based on the assumptions about electricity demand listed in the section above, the simulated electricity demand D was created using a combination of diurnal D_daily and weekly D_weekly fluctuations as well as fluctuations that represent a frequency shift D_shift similar to that described by Kempton et al (2008). Please see Equation 1 for a detailed view of demand simulation.

  (Equation 1)

Where:

;

with t being the 15-minute time interval;

on weekends or on weekdays;

And with M_day being the maximum demand for the day without including D_shift.

Interestingly, it was discovered that the random number generator created a set of numbers that were not evenly distributed about zero. Whereas in most cases this was not necessarily an issue, in the case of frequency shift, which Kempton et al (2008) explains should contain roughly as much demand for regulation up as regulation down, there was a need for a relatively even distribution. Thus, the bias, which was calculated to be proportionate to 234 for the particular set of random numbers in the model, was subtracted from the random numbers to provide an even distribution in the frequency shift.

By configuring D_curve as it is in the equation above, the daily demand fluctuations reach a minimum at the interval representing 5:00-5:15 a.m. and a maximum at the interval representing 5:00-5:15 p.m. This also means that daily demand fluctuations are about halfway up and halfway down at the interval of 11:00-11:15 a.m. and p.m. respectively. Meanwhile, D_daily is set up to offer a range of variability of up to 10% with regard to the daily maxima and minima on weekdays theoretically centered around 95% of what the inputs for maximum and minimum demands for the week (D_max and D_min respectively) are set to on the model’s user interface. Finally, D_weekly leaves the weekdays unchanged and reduces the demand on weekdays by approximately 10%, which is roughly consistent with the assumptions made in the previous section.

The amount of renewable energy generated on the simulated utility grid is essentially a function of two factors primary: the amount of generating capacity installed and the amount of natural energy (wind or solar rays) available to the system at any particular time. Østergaard (2008) indicates that for renewable energy it is important to consider the effects of the geographic aggregation of all renewable resources on a system. In order to approximate the nuances of a geographically dispersed system, the available renewable generating capacity (Wind_CAP and Solar_CAP) selected on the model’s user interface was divided evenly among 12 different energy ‘farms’ for both wind and solar. Equations 2 and 3 show that the total wind and solar generation (G_w and G_s respectively) are simply the sum of the electricity generated at each of the 12 farms.

  (Equation 2)

(Equation 3)

This segmentation allows for a level of overall wind and solar generation that is more stable than the generation at any particular farm. The individual wind and solar farms (represented as G_wi and G_si respectively) are each respectively governed by Equations 4 and 5 below.

Or

(Equation 4)

Where:

  and  ;

And or .

In the case of simulating wind-powered electricity generation, the user interface of the model allows the user to choose the average availability factor A_f (the proportion of the time that the wind turbines are available for electricity generation; i.e. when the turbines are not broken or under maintenance) and the average capacity factor C_f (the amount of electricity that is typically generated compared to the nominal capacity of the turbines) of the wind turbines connected to the grid.

The choice between G_wA and G_wB simply ensures that the model does not allow a wind farm to generate more than its nominal capacity during any particular interval. While G_wB allows for a random variation of electricity generation centered on the average amount of generation determined by the C_f, A_f, and Wind_CAP settings on the user interface. Meanwhile G_wC means that approximately 5% of the time the wind farm will generate no electricity, which represents intervals with a wind speed below the minimum speed required to turn the blades on the turbines.

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