# Appendix B - Developing Code of Federal Regulations Averaging Equations to Fit the Canadian Regulatory Framework

This appendix presents the equations intended for use in the averaging provisions of the planned regulations. The averaging calculations in the planned Regulations are intended to be the same as those found in the *U.S. Code of Federal Regulations (CFR)*. However it was necessary to modify some of the U.S. equations to be consistent with the Canadian regulatory framework which is modeled on the one developed for the *On-Road Vehicle and Engine Emission Regulations*^{29}. An approximation of some terms used in the U.S. averaging provisions for outboard and personal watercraft engines was introduced to simplify the equations.

Section 720 of Part 1051 of the CFR^{30} presents the calculation of average emission level and emission credits for a company's fleet of snowmobiles, off-highway motorcycles, or all-terrain vehicles as

[1]

[2]

*TOT*represents the number of models of vehicles in the fleet;*FEL*represents the family emission limit for the_{i}*i*vehicle model (for HC, HC+NOx, CO or evaporative permeation, as the case may be) in g/kW-hr, g/km, or g/m^{th}^{2}/day;*sales*represents the number of vehicles of model_{i}*i*in the fleet;*UL*represents the useful life of the_{i}*i*model of vehicle in hr or km;^{th}*average standard*represents the applicable emission standard (for HC, HC+NOx, CO or evaporative permeation, as the case may be) in g/kW-hr, g/km, or g/m^{2}/day.

In equation [2], the applicable emission standard is obtained by looking up in the emission standard table applicable to the model year and the type of vehicle in the fleet (i.e., snowmobile, off-highway motorcycle or all-terrain vehicle). For vehicles that have standards in g/kW-hr and a useful life in km, the useful life is converted to kW-hr based on the maximum power output and an assumed vehicle speed of 30 km/hr

[3]

For evaporative permeation standards expressed in g/m^{2}/day, the useful life expressed in years is multiplied by 365.24 days.

Section 207 of Part 91 of the CFR presents the calculation of emission credits of the *i ^{th}* model of outboard or personal watercraft engines in a company's fleet as

[4]

Equation [4] can be rearranged by removing from the summation some of the variables that are independent of *t* to yield

[6]

When focusing on the summation in equation [6],

[7]

it can be observed that the value *X _{i}* depends only on the type (outboard or personal watercraft) and power of the

*i*engine model. The values of all other variables in equation [4] are set by these two values. Figure B-1 shows a plot of the value of

^{th}*X*as a function of engine power for outboards and personal watercrafts. The data points represents the values calculated using equation [7] and the lines shows a curve fitting equation that can be used to approximate equation [7]:

[8]

*X _{i} = ~ K_{i} x P_{i}^{(ni)}*

where *K _{i}* and

*n*are 136.6 hours-kW

_{i}^{0.14}and 0.86 if the

*i*engine model is an outboard and 114 hours and 1.0 if it is a personal watercraft.

^{th}Substituting equation [8] into [6] yields

[9]

*credits _{i} = ( STD_{i} - FEL_{i} ) x sales_{i} x K_{i} x P_{i}^{(ni)}*

which can be applied to the *company's* entire fleet using a summation

[10]

where *TOT* represents the number of different engine models in the fleet.

Equation [10] now needs to be further modified to be consistent with the regulatory framework based on the *On-Road Vehicle and Engine Emission Regulations*. The averaging equations for outboard and personal watercrafts and for recreational vehicles will also be manipulated to have the same format. The names of the variables will be changed to a one-letter format to make it easier to incorporate the equations in a bilingual text.

The planned regulatory framework requires the fleet average standard and the fleet average emission values. Equation [1] gives the fleet average emission value for a fleet of snowmobiles, all-terrain vehicles or off-highway motorcycles. By inspection, equation [1] can be generalized to all forms of vehicles and engines covered by the planned regulations

[11]

The fleet average standard value required by the planned regulatory framework can be obtained by simply looking up a value in the appropriate table for a fleet of snowmobiles, off-highway motorcycles and all-terrain vehicles. However, the fleet average standard for a fleet of outboard and personal watercraft engines requires a calculation as the emission standard varies with the power of the engine models in the fleet. This value can be calculated in a manner similar to equation [11]

[13]

*A*represents the sales-weighted fleet average standard;*W*represents the applicable emission standards to the_{i}*i*engine model in the fleet in g/kW-hr; and^{th}- the other variables are as previously defined.

The last requirement of the planned regulatory framework is the calculations of each fleet emission credits which can obtained by modifying equation [2] as

[14]

where *A* is calculated using equation [13] for a fleet of outboards and personal watercraft or is obtained by lookup in the appropriate standard table.

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