CO₂-equivalent Model Explanation

Introduction

• Global warming impact is the focus
• What is CO₂ eq
• Explain that we take the point of view of the consumer
• Explain the problem of the chain of imports potentially being infinite

Methodology

All the electricity that is consumed somewhere must have been produced somewhere else. This gives us a balance equation, that can be applied to each zone.

Therefore, for the $i$-th country, the following equality holds about electricity:

$$ consumption_i = production_i + import_i - export_i - transmission_i $$

We will now assume that transmission costs are negligible. Each term of the equation has an associated carbon intensity, which we will describe:

• The carbon intensity of the consumption, denoted by $x_i$, is the one we are seeking to determine.
• The carbon intensity of imports is the the carbon intensity of the zone exporting that power. If that zone is identified by the letter $j$, then carbon intensity of power imported from $j$ to $i$ is $x_j$. Therefore, the carbon intensity of exports from contry $i$ is naturally $x_i$.
• If production is split up by production mode (i.e. power plant type), then each production mode $m$ has an associated intensity $I_{i,m}$ depending on its type of power plant.

This enables us to write an updated balance equation for the carbon flow taking place in the $i$-th country, having imports from $j$, exports to $k$ and production modes $m$

$$ x_i \cdot consumption_i = \sum_m I_{i,m} \cdot production_{i,m} + \sum_j x_j \cdot import_{i,j} - \sum_k x_i \cdot export_{i,k} $$

Note that because in this simplified model $consumption = production + import - export$, exports cancel out, and the equation reduces to

$$ x_i \left(\sum_m production_{i,m} + \sum_j import_{i,j} \right) = \sum_m I_{i,m} \cdot production_i + \sum_j x_j \cdot import_{i,j}$$

which is neatly renamed to

$$ x_i \left(production_i + import_i \right) = \sum_m I_{i,m} \cdot production_i + \sum_j x_j \cdot import_{i,j}$$

Writing out all equations for all countries, one obtains a linear system of equations, where we wish to solve for $x$:

$$ \begin{bmatrix} production_{1} + import_{1} & -import_{1,2} & \cdots & -import_{1,n} \\ -import_{2,1} & production_{2} + import_{2} & \cdots & -import_{2,n}\\ \vdots & \vdots & \ddots & \vdots\\ -import_{n,1} & -import_{n,2} & \cdots & production_{n} + import_{n} \end{bmatrix} \begin{bmatrix}x_1\\\vdots\\x_n\end{bmatrix} = \begin{bmatrix} \sum_m I_{1,m} \cdot production_1\\ \vdots\\ \sum_m I_{n,m} \cdot production_n \end{bmatrix}$$

where $import_{i,j}$ represents the amount of import to $i$-th country from $j$-th country. The matrix is sparse, and represents the connectivity of the zones.

Solving this linear system gives the carbon intensities $x_i$ of each zone.

Special cases

- Import from missing countries

• Storage