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Net present value (€)
=====================
*indexed by: asset, cost_type, energy, node, period, technology, test_case*

### Description

The purpose of this KPI is to analyse the investments made during the pathway for the installation, repowering or 
decommissioning of capacities. It is used to analyse financial investment needs and their distribution over the pathway.

<blockquote class="calculation-hint">
  <h4>Calculation hint</h4>
    Although overnight investments costs are actualized in the objective function, they are displayed without any 
    actualization in this KPI. For instance, considering a pathway from 2020 to 2050 in 10-year time steps. If an 
    investment is made in 2030 to install a solar power plant, the entire investment cost will be displayed in 2030 in 
    this KPI and will not be actualized in 2020 euros.
</blockquote>

### Calculation
    
All the equations below are valid for any realization and are therefore implicitly indexed by test case. Let be 
$x_{a, c, p}$ the value returned by this KPI for a given asset $a$, cost type $c$ and period $p$.

#### KPI notations

##### Financial risks

To take into account the risk of financing an asset $a$, let us define a $premium_a$ by :

$$
premium_a = \frac{\sum_{y \in [0, D_a[} q^y}{\sum_{y \in [0, D_a[} q_a^y} = \frac{(1 - q_a) \cdot (1 - q^{D_a})}{(1 - q) \cdot (1 - q_a^{D_a})}
$$

With:

- $D_a$ : the lifetime of asset $a$
- $q_a$ = $\frac{1}{1 + R_a}$
- $q$ = $\frac{1}{1 + R}$
- $R_a$ : the discount rate parameter of asset $a$  
- $R$ : the global discount rate of considered pathway 

##### Side effects

If the lifetime of the plant expires after the end of the pathway horizon, the calculation takes into account a ratio,
which avoids side effects on final pathway periods. For a given asset $a$ and period $p$ starting at year $y_p$ this 
ratio is defined by:

$$
ratio_{a, y_p} = \frac{\sum_{y \in [y_p, y_P + D[} q^y}{\sum_{y \in [0, D_a[} q^y} = \frac{(1 - q^{y_P+D - y_p})}{(1 - q^{D_a})}
$$

With :

- $y_p$ : The first year of period $p$
- $y_P$ : The final year of considered pathway
- $D$ : The duration of a pathway step in years

#### Capacity investment costs

For $c=\text{Capacity investment costs}$ we have :

$$
x_{a, c, p} = 
\begin{cases}
    \quad oc\_cost_{a, p} \cdot add_{a, p} \cdot premium \text{ if } y_P - y_p > D \\
    \quad oc\_cost_{a, p} \cdot add_{a, p} \cdot premium \cdot ratio_{a, y_p} \text{ otherwise}
\end{cases}
$$

With :

- $oc\_cost_{a, p}$ : the overnight capacity cost parameter of asset $a$ during period $p$ in €/MW
- $add_{a, p}$ : the capacity of asset $a$ added during period $p$

#### Storage costs

For $c=\text{Storage investment costs}$ we have :

$$
x_{a, c, p} = 
\begin{cases}
    \quad osc\_cost_{a, p} \cdot add_{a, p} / dis\_times_{a, p} \cdot premium \text{ if } y_P - y_p > D \\
    \quad os\_cost_{a, p} \cdot add_{a, p} / dis\_times_{a, p} \cdot premium \cdot ratio_{a, y_p} \text{ otherwise}
\end{cases}
$$


With :

- $osc\_cost_{a, p}$ : the overnight storage capacity cost parameter of asset $a$ during period $p$ in €/MW
- $dis\_times_{a, p}$ : the discharge time parameter of asset $a$ during period $p$ in hours

<blockquote class="calculation-hint">
  <h4>Calculation hint</h4>
    For storage assets without DISCHARGE TIMES behaviour, no storage investment costs are taken into account and a 
    null value is returned as long as no optimization on stock is possible without this behaviour.
</blockquote>

#### Repowering costs

For $c=\text{Repowering costs}$ we have :

$$
x_{a, c, p} = 
\begin{cases}
    \quad roc\_cost_{a, p} \cdot repow_{a, p} \cdot premium \text{ if } y_P - y_p > D \\
    \quad roc\_cost_{a, p} \cdot repow_{a, p} \cdot premium \cdot ratio_{a, y_p} \text{ otherwise}
\end{cases}
$$

With :

- $roc\_cost_{a, p}$ : the repowering overnight capacity cost parameter of asset $a$ during period $p$ in €/MW
- $repow_{a, p}$ : the capacity of asset $a$ repowered during period $p$

#### Decommissioning costs

For $c=\text{Decommissioning costs}$ we have :

$$
x_{a, c, p} = 
\begin{cases}
    \quad dc\_cost_{a, p} \cdot less_{a, p} \cdot premium \text{ if } y_P - y_p > D \\
    \quad dc\_cost_{a, p} \cdot less_{a, p} \cdot premium \cdot ratio_{a, y_p} \text{ otherwise}
\end{cases}
$$

With :

- $dc\_cost_{a, p}$ : the decommissioning overnight capacity cost parameter of asset $a$ during period $p$ in €/MW
- $less_{a, p}$ : the capacity of asset $a$ decommissioned during period $p$


.. note::
   In the formula of the Overnight Investment Costs and Overnight Decommissioning Costs, in case of repowering, 
   $add_{a, p}$ (resp. $less_{a, p}$) is replaced by $add_{a, p} - repow_{a, p}$ (resp. $less_{a, p} - repow_{a, p}$)

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