Effects, Penalty & Objective

Effects

Effects are used to allocate things like costs, emissions, or other "effects" occuring in the system. These arise from so called Shares, which originate from Elements like Flows.

Example:

Flows have an attribute called effects_per_flow_hour, defining the effect amount of per flow hour. Assiziated effects could be: - costs - given in [€/kWh]... - ...or emissions - given in [kg/kWh]. - Effects are allocated seperatly for investments and operation.

Shares to Effects

\[ \label{eq:Share_invest} s_{l \rightarrow e, \text{inv}} = \sum_{v \in \mathcal{V}_{l, \text{inv}}} v \cdot \text a_{v \rightarrow e} \]

\[ \label{eq:Share_operation} s_{l \rightarrow e, \text{op}}(\text{t}_i) = \sum_{v \in \mathcal{V}_{l,\text{op}}} v(\text{t}_i) \cdot \text a_{v \rightarrow e}(\text{t}_i) \]

With:

$\text{t}_i$ being the time step
$\mathcal{V_l}$ being the set of all optimization variables of element $e$
$\mathcal{V}_{l, \text{inv}}$ being the set of all optimization variables of element $e$ related to investment
$\mathcal{V}_{l, \text{op}}$ being the set of all optimization variables of element $e$ related to operation
$v$ being an optimization variable of the element $l$
$v(\text{t}_i)$ being an optimization variable of the element $l$ at timestep $\text{t}_i$
$\text a_{v \rightarrow e}$ being the factor between the optimization variable $v$ to effect $e$
$\text a_{v \rightarrow e}(\text{t}_i)$ being the factor between the optimization variable $v$ to effect $e$ for timestep $\text{t}_i$
$s_{l \rightarrow e, \text{inv}}$ being the share of element $l$ to the investment part of effect $e$
$s_{l \rightarrow e, \text{op}}(\text{t}_i)$ being the share of element $l$ to the operation part of effect $e$

Shares between different Effects

Furthermore, the Effect $x$ can contribute a share to another Effect ${e} \in \mathcal{E}\backslash x$. This share is defined by the factor $\text r_{x \rightarrow e}$.

For example, the Effect "CO$_2$ emissions" (unit: kg) can cause an additional share to Effect "monetary costs" (unit: €). In this case, the factor $\text a_{x \rightarrow e}$ is the specific CO$_2$ price in €/kg. However, circular references have to be avoided.

The overall sum of investment shares of an Effect $e$ is given by $\eqref{Effect_invest}$

\[ \label{eq:Effect_invest} E_{e, \text{inv}} = \sum_{l \in \mathcal{L}} s_{l \rightarrow e,\text{inv}} + \sum_{x \in \mathcal{E}\backslash e} E_{x, \text{inv}} \cdot \text{r}_{x \rightarrow e,\text{inv}} \]

The overall sum of operation shares is given by $\eqref{eq:Effect_Operation}$

\[ \label{eq:Effect_Operation} E_{e, \text{op}}(\text{t}_{i}) = \sum_{l \in \mathcal{L}} s_{l \rightarrow e, \text{op}}(\text{t}_i) + \sum_{x \in \mathcal{E}\backslash e} E_{x, \text{op}}(\text{t}_i) \cdot \text{r}_{x \rightarrow {e},\text{op}}(\text{t}_i) \]

and totals to $\eqref{eq:Effect_Operation_total}$ $$\label{eq:Effect_Operation_total} E_{e,\text{op},\text{tot}} = \sum_{i=1}^n E_{e,\text{op}}(\text{t}_{i}) $$

With:

$\mathcal{L}$ being the set of all elements in the FlowSystem
$\mathcal{E}$ being the set of all effects in the FlowSystem
$\text r_{x \rightarrow e, \text{inv}}$ being the factor between the operation part of Effect $x$ and Effect $e$
$\text r_{x \rightarrow e, \text{op}}(\text{t}_i)$ being the factor between the invest part of Effect $x$ and Effect $e$
$\text{t}_i$ being the time step
$s_{l \rightarrow e, \text{inv}}$ being the share of element $l$ to the investment part of effect $e$
$s_{l \rightarrow e, \text{op}}(\text{t}_i)$ being the share of element $l$ to the operation part of effect $e$

The total of an effect $E_{e}$ is given as $\eqref{eq:Effect_Total}$

\[ \label{eq:Effect_Total} E_{e} = E_{\text{inv},e} +E_{\text{op},\text{tot},e} \]

Constraining Effects

For each variable $v \in \{ E_{e,\text{inv}}, E_{e,\text{op},\text{tot}}, E_e\}$, a lower bound $v^\text{L}$ and upper bound $v^\text{U}$ can be defined as

\[ \label{eq:Bounds_Single} \text v^\text{L} \leq v \leq \text v^\text{U} \]

Furthermore, bounds for the operational shares can be set for each time step

\[ \label{eq:Bounds_Time_Steps} \text E_{e,\text{op}}^\text{L}(\text{t}_i) \leq E_{e,\text{op}}(\text{t}_i) \leq \text E_{e,\text{op}}^\text{U}(\text{t}_i) \]

Penalty

Additionally to the user defined Effects, a Penalty $\Phi$ is part of every FlixOpt Model. Its used to prevent unsolvable problems and simplify troubleshooting. Shares to the penalty can originate from every Element and are constructed similarly to $\eqref{Share_invest}$ and $\eqref{Share_operation}$.

\[ \label{eq:Penalty} \Phi = \sum_{l \in \mathcal{L}} \left( s_{l \rightarrow \Phi} +\sum_{\text{t}_i \in \mathcal{T}} s_{l \rightarrow \Phi}(\text{t}_{i}) \right) \]

With:

$\mathcal{L}$ being the set of all elements in the FlowSystem
$\mathcal{T}$ being the set of all timesteps
$s_{l \rightarrow \Phi}$ being the share of element $l$ to the penalty

At the moment, penalties only occur in Buses

Objective

The optimization objective of a FlixOpt Model is defined as $\eqref{eq:Objective}$ $$ \label{eq:Objective} \min(E_{\Omega} + \Phi) $$

With:

$\Omega$ being the chosen Objective Effect (see $\eqref{eq:Effect_Total}$)
$\Phi$ being the Penalty

This approach allows for a multi-criteria optimization using both... - ... the Weigted SumMethod, as the chosen Objective Effect can incorporate other Effects. - ... the ($\epsilon$-constraint method) by constraining effects.