Welfare (€)
======

*indexed by: node, test_case*

### Description

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The KPI computes the welfare of each node, i.e., the sum of its consumer surplus, its producer surplus, the 
border exchange surplus, and half of the congestion rent for power transmission lines connected to the node.

### Calculation

To compute the welfare associated to a node $n$, all asset costs associated to this node are subtracted to the sum of 
asset revenues associated to this node. This view is therefore an aggregation by node of [Asset surplus KPI](asset_surplus.md)

Let be $x_{n}$ the welfare of node $n$.
We can then express $x_{n}$ as:

$$
x_{n} = \sum_{a \in A_n} as_{a, n} = \sum_{a \in A_n} (ar_{a, n} - ac_{a, n})
$$

With :

- $as_{a, n}$ : the asset surplus of asset $a$ at node $n$
- $ar_{a, n}$ : the asset revenues of asset $a$ at node $n$
- $ac_{a, n}$ : the asset costs of asset $a$ at node $n$

For more detailed explanations on how [asset surplus](asset_surplus.md), [asset revenues](asset_revenues.md) and [asset costs](asset_costs.md) are computed 
please consider looking at associated documentation pages.

*Global variables and parameters notations definitions can be consulted [here](../notations.md).*

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