# Postprocessing 

## Socio-economic surplus

_The socio-economic surplus_ (SES) is calculated as the sum of consumer surplus ($ \textit{CS} $ ) and producer surplus ( $ \textit{PS} $), as illustrated below.
![Sketch socio-economic welfare calculation](market-cross.png)

In our calculation of SES we also calculate the difference in water filling ( $ WF $ ) and congestion rent on transmission corridor ( $ \textit{T} $ ), which is considered a part of the consumer surplus. 


### Mathematical description
The following mathematical description divides the calculation of socioeconomic  welfare in different parts. Each of the defined assets / technologies provides a function to calculate its profit margin $\Pi(k)$ at time step $k$. The sum over all profit margins is the socio-economic welfare.


- $p(k)$ - production of a unit at time step $k$
- $\lambda(a,k)$ - power price in the respective area $a$

#### Thermal production

For all thermal plants and all timesteps $ k $:

$$ 
\textit{CS} \mathrel{+}= p(k) * [\lambda(a,k)-MC(k)] 
$$

Where  $MC(k)$ is the marginal production cost.

#### Intermittent RES - wind, solar

For all wind parks, solar parks and all timesteps $ k $:

$$ \textit{PS}_\textit{res}(k) \mathrel{+}= p(k) * \lambda(a,k) $$

#### Market steps

For all market steps and all timesteps $ k $:

$$ 
\textit{PS} \mathrel{+}= d(k) * [\lambda(a,k) - \textit{Cost}(k)], \textit{if } d(k) > 0.0
$$


$$
\textit{CS} \mathrel{+}= d(k) * [\textit{Cost}(k) - \lambda(a,k)], \textit{if } d(k) < 0.0
$$

Where $ Cost(k)$ is the cost of the market step.

#### Firm consumption (incl. price elasticity)

The consumer surplus is also adjusted by price elasticity defined with the segments $x \in \mathbb{X}$.

The segments have:
- a defined capacity $C_{e}(a, x,k)$
- a defined consumption $c_{e}(a, x,k)$
- and a cost $\textit{Cost}_{e}(a, x,k)$ 

For all price elastic steps and all timesteps $ k $:

$$ 
\textit{CS} \mathrel{+}= [C_{e}(a, x, k) - c_{e}(a, x, k)] * [\textit{Cost}(a, x, k) - \lambda(a,k)], \textit{if } C(k) > 0.0
$$

$$
\textit{CS} \mathrel{-}= c(x, k) * [\textit{Cost}(x, k) - \lambda(a,k)], \textit{if } C(k) < 0.0
$$

The consumer surplus for firm demand $ D(a, k) $ is curtailed by price elasticity and consumed rationing:

$$ 
\textit{D}_{cur}(a, k) = D(a, k) - R_{consumption}(a, k) - \sum_{x \in X} C_{e}(a, x, k)
$$

For all $ C_{e(a, x, k)} > 0 $.

Thus, for all areas and all timesteps $ k $:

$$ 
\textit{CS} \mathrel{+}= D_{cur}(a, k)  * [R_{cost}(a, k) - \lambda(a, k)]
$$

#### Battery
- $dis(k)$ - discharge of battery
- $chg(k)$ - charge of battery

For all batteries and all timesteps $ k $:

$$ \textit{PS} \mathrel{+}= (dis(k) - chg(k))*\lambda_{a,k}$$


#### Congestion rent on transmission corridors

The surplus for a transmission corridor, or congestion rent is calculated by the price difference of the connected areas and the flow. Prices in both ends of the line are defined as:
- $\lambda(a_{to},k)$
- $\lambda(a_{fr},k)$

The exchange on a transmission corridor from area $a_{fr}$ to $a_{to}$ is defined as $ exc(k) $, and the surplus is split between the two areas:

$$
T(a_{to})) \mathrel{+}= 0.5*exc(a_{to}, a_{fr}, k) * (\lambda_{a_{to},k} - \lambda_{a_{fr},k}) 
$$

$$
T(a_{fr})) \mathrel{+}= 0.5*exc(a_{to}, a_{fr}, k) * (\lambda_{a_{to},k} - \lambda_{a_{fr},k}) 
$$ 

If losses are defined for transmission, these need to be accounted for. Losses are distributed  on both sides of the line. Losses on the corridor are defined as $loss(k)$

Both sides:

$$ 
T(a_{to}) \mathrel{+}= loss(k) * 0.5 * (\lambda(a_{to},k) + \lambda(a_{fr},k)) 
$$

$$ 
T(a_{fr}) \mathrel{+}= loss(k) * 0.5 * (\lambda(a_{to},k) + \lambda(a_{fr},k)) 
$$

#### Hydro power production

For all hydro power plants and all timesteps $ k $:

$$ 
\textit{PS} \mathrel{+}= p(k) * \lambda(a,k)
$$

#### Water filling difference

For the calculation of the profit margin for hydro power a correction for the change in reservoir filling needs to be done. This is done by integrating over the reservoir filling and water value for each reservoir as:


$$
\textit{WF} = \int_0^X res(x, k_N) * WV(x, k_N) \, dx - \int_0^X res(x, k_0) * WV(x, k_0) \, dx
$$

Where $WV(x,k)$ is water value for module for all levels $x$, $res(x, k)$ is the amount of water at level $x$. The reservoir volume is split into $ X $ number of levels, and each level is of size $ (V_\textit{max} - V_\textit{min}) / (X - 1) $, where $ V_\textit{max} $ and $ V_\textit{min} $ is the user provided maximum and minimum limits for the reservoir.



#### Total 

Total socio-economic surplus is thus  $ \textit{CS} + \textit{PS} + \textit{WF} + T $ 

### Code example

See the API refrence for more [details](trident_extras.post_processing.socio_economic_surplus.calc_socio_economic_surplus).

```python
# Required structures
run_config: RunConfig
logical_model: LogicalModel
input_ds: DataStore
result_ds: DataStore

# Perform a simulation
# ...

from trident_extras.post_processing import calc_socio_economic_surplus
import json

report = calc_socio_economic_surplus(run_config, logical_model, input_ds, result_ds)
print(report.to_ascii_table())
print(json.dumps(report.to_dict()))

```