Appendix A — Derivations

Equation 2.2 shows us the options we have to reduce emissions \(E\) associated with energy use:

\[ \text{GHG Emissions } E = F \beta = \frac{S \, e_P \, \beta}{\eta_D} \]

where \(F\) is the final energy demand, \(\beta\) is the emissions intensity of that final energy, \(e_P\) is the energy intensity of the passive system, and \(\eta_D\) is the efficiency of the conversion device.

Equation 2.3 is found as:

\[\begin{align} \frac{\partial E}{\partial \beta} &= F \\ % \int_{E_0}^{E_0 + \Delta E} \mathrm{d}E &= \int_{\beta_0}^{\beta_1} F \mathrm{d} \beta \\ \Delta E &= F \Delta \beta \end{align}\]

Equation 2.4 is found as:

\[\begin{align} \frac{\partial E}{\partial \eta} &= -\frac{S \, e_P \, \beta}{\eta_D} \frac{1}{\eta_D} = -\frac{E}{\eta_D} \\ \int_{E_0}^{E_0 + \Delta E} \frac{\mathrm{d}E}{E} &= -\int_{\eta_0}^{\eta_1} \frac{\mathrm{d} \eta_D}{\eta_D} \\ \frac{E_0 + \Delta E}{E_0} &= \frac{\eta_0}{\eta_1} \\ \Delta E &= E_0 \left( \frac{\eta_0}{\eta_1} - 1 \right) \\ &= F \beta \left( \frac{\eta_0}{\eta_1} - 1 \right) \end{align}\]

Equation 2.5 is found as:

\[\begin{align} \frac{\partial E}{\partial e} &= \frac{S \, \beta}{\eta_D} = \frac{S \, e_P \, \beta}{\eta_D} \frac{1}{e_P} = \frac{E}{e_P} \\ \int_{E_0}^{E_0 + \Delta E} \frac{\mathrm{d}E}{E} &= \int_{e_0}^{e_1} \frac{\mathrm{d} E_P}{e_P} \\ \frac{E_0 + \Delta E}{E_0} &= \frac{e_1}{e_0} \\ \Delta E &= E_0 \left( \frac{e_1}{e_0} - 1 \right) \\ &= F \beta \left( \frac{e_1}{e_0} - 1 \right) \end{align}\]