Effects, Penalty & Objective

Effects

Effects are used to quantify system-wide impacts like costs, emissions, or resource consumption. These arise from shares contributed by Elements such as Flows, Storage, and other components.

Example:

Flows have an attribute effects_per_flow_hour that defines the effect contribution per flow-hour: - Costs (€/kWh) - Emissions (kg CO₂/kWh) - Primary energy consumption (kWh_primary/kWh)

Effects are categorized into two domains:

Temporal effects - Time-dependent contributions (e.g., operational costs, hourly emissions)
Periodic effects - Time-independent contributions (e.g., investment costs, fixed annual fees)

Multi-Dimensional Effects

The formulations below are written with time index $\text{t}_i$ only, but automatically expand when periods and/or scenarios are present.

When the FlowSystem has additional dimensions (see Dimensions):

Temporal effects are indexed by all present dimensions: $E_{e,\text{temp}}(\text{t}_i, y, s)$
Periodic effects are indexed by period only (scenario-independent within a period): $E_{e,\text{per}}(y)$
Effects are aggregated with dimension weights in the objective function

For complete details on how dimensions affect effects and the objective, see Dimensions.

Effect Formulation

Shares from Elements

Each element $l$ contributes shares to effect $e$ in both temporal and periodic domains:

Periodic shares (time-independent): $$ \label{eq:Share_periodic} s_{l \rightarrow e, \text{per}} = \sum_{v \in \mathcal{V}{l, \text{per}}} v \cdot \text{a} $$

Temporal shares (time-dependent): $$ \label{eq:Share_temporal} s_{l \rightarrow e, \text{temp}}(\text{t}i) = \sum{l,\text{temp}}} v(\text{t}_i) \cdot \text{a}_i) $$}(\text{t

Where:

$\text{t}_i$ is the time step
$\mathcal{V}_l$ is the set of all optimization variables of element $l$
$\mathcal{V}_{l, \text{per}}$ is the subset of periodic (investment-related) variables
$\mathcal{V}_{l, \text{temp}}$ is the subset of temporal (operational) variables
$v$ is an optimization variable
$v(\text{t}_i)$ is the variable value at timestep $\text{t}_i$
$\text{a}_{v \rightarrow e}$ is the effect factor (e.g., €/kW for investment, €/kWh for operation)
$s_{l \rightarrow e, \text{per}}$ is the periodic share of element $l$ to effect $e$
$s_{l \rightarrow e, \text{temp}}(\text{t}_i)$ is the temporal share of element $l$ to effect $e$

Examples: - Periodic share: Investment cost = $\text{size} \cdot \text{specific\_cost}$ (€/kW) - Temporal share: Operational cost = $\text{flow\_rate}(\text{t}_i) \cdot \text{price}(\text{t}_i)$ (€/kWh)

Cross-Effect Contributions

Effects can contribute shares to other effects, enabling relationships like carbon pricing or resource accounting.

An effect $x$ can contribute to another effect $e \in \mathcal{E}\backslash x$ via conversion factors:

Example: CO₂ emissions (kg) → Monetary costs (€) - Effect $x$: "CO₂ emissions" (unit: kg) - Effect $e$: "costs" (unit: €) - Factor $\text{r}_{x \rightarrow e}$: CO₂ price (€/kg)

Note: Circular references must be avoided.

Total Effect Calculation

Periodic effects aggregate element shares and cross-effect contributions:

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

Temporal effects at each timestep:

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

Total temporal effects (sum over all timesteps):

\[\label{eq:Effect_temporal_total} E_{e,\text{temp},\text{tot}} = \sum_{i=1}^n E_{e,\text{temp}}(\text{t}_{i}) \]

Total effect (combining both domains):

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

Where:

$\mathcal{L}$ is the set of all elements in the FlowSystem
$\mathcal{E}$ is the set of all effects
$\text{r}_{x \rightarrow e, \text{per}}$ is the periodic conversion factor from effect $x$ to effect $e$
$\text{r}_{x \rightarrow e, \text{temp}}(\text{t}_i)$ is the temporal conversion factor

Constraining Effects

Effects can be bounded to enforce limits on costs, emissions, or other impacts:

Total bounds (apply to $E_{e,\text{per}}$, $E_{e,\text{temp},\text{tot}}$, or $E_e$):

\[ \label{eq:Bounds_Total} E^\text{L} \leq E \leq E^\text{U} \]

Temporal bounds per timestep:

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

Implementation: See Effect parameters: - minimum_temporal, maximum_temporal - Total temporal bounds - minimum_per_hour, maximum_per_hour - Hourly temporal bounds - minimum_periodic, maximum_periodic - Periodic bounds - minimum_total, maximum_total - Combined total bounds

Penalty

In addition to user-defined Effects, every FlixOpt model includes a Penalty term $\Phi$ to: - Prevent infeasible problems - Simplify troubleshooting by allowing constraint violations with high cost

Penalty shares originate from elements, similar to effect shares:

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

Where:

$\mathcal{L}$ is the set of all elements
$\mathcal{T}$ is the set of all timesteps
$s_{l \rightarrow \Phi}$ is the penalty share from element $l$

Current usage: Penalties primarily occur in Buses via the excess_penalty_per_flow_hour parameter, which allows nodal imbalances at a high cost.

Objective Function

The optimization objective minimizes the chosen effect plus any penalties:

\[ \label{eq:Objective} \min \left( E_{\Omega} + \Phi \right) \]

Where:

$E_{\Omega}$ is the chosen objective effect (see $\eqref{eq:Effect_Total}$)
$\Phi$ is the penalty term

One effect must be designated as the objective via is_objective=True.

Multi-Criteria Optimization

This formulation supports multiple optimization approaches:

1. Weighted Sum Method - The objective effect can incorporate other effects via cross-effect factors - Example: Minimize costs while including carbon pricing: $\text{CO}_2 \rightarrow \text{costs}$

2. ε-Constraint Method - Optimize one effect while constraining others - Example: Minimize costs subject to $\text{CO}_2 \leq 1000$ kg

Objective with Multiple Dimensions

When the FlowSystem includes periods and/or scenarios (see Dimensions), the objective aggregates effects across all dimensions using weights.

Time Only (Base Case)

\[ \min \quad E_{\Omega} + \Phi = \sum_{\text{t}_i \in \mathcal{T}} E_{\Omega,\text{temp}}(\text{t}_i) + E_{\Omega,\text{per}} + \Phi \]

Where: - Temporal effects sum over time: $\sum_{\text{t}_i} E_{\Omega,\text{temp}}(\text{t}_i)$ - Periodic effects are constant: $E_{\Omega,\text{per}}$ - Penalty sums over time: $\Phi = \sum_{\text{t}_i} \Phi(\text{t}_i)$

Time + Scenario

\[ \min \quad \sum_{s \in \mathcal{S}} w_s \cdot \left( E_{\Omega}(s) + \Phi(s) \right) \]

Where: - $\mathcal{S}$ is the set of scenarios - $w_s$ is the weight for scenario $s$ (typically scenario probability) - Periodic effects are shared across scenarios: $E_{\Omega,\text{per}}$ (same for all $s$) - Temporal effects are scenario-specific: $E_{\Omega,\text{temp}}(s) = \sum_{\text{t}_i} E_{\Omega,\text{temp}}(\text{t}_i, s)$ - Penalties are scenario-specific: $\Phi(s) = \sum_{\text{t}_i} \Phi(\text{t}_i, s)$

Interpretation: - Investment decisions (periodic) made once, used across all scenarios - Operations (temporal) differ by scenario - Objective balances expected value across scenarios

Time + Period

\[ \min \quad \sum_{y \in \mathcal{Y}} w_y \cdot \left( E_{\Omega}(y) + \Phi(y) \right) \]

Where: - $\mathcal{Y}$ is the set of periods (e.g., years) - $w_y$ is the weight for period $y$ (typically annual discount factor) - Each period $y$ has independent periodic and temporal effects - Each period $y$ has independent investment and operational decisions

Time + Period + Scenario (Full Multi-Dimensional)

\[ \min \quad \sum_{y \in \mathcal{Y}} \left[ w_y \cdot E_{\Omega,\text{per}}(y) + \sum_{s \in \mathcal{S}} w_{y,s} \cdot \left( E_{\Omega,\text{temp}}(y,s) + \Phi(y,s) \right) \right] \]

Where: - $\mathcal{S}$ is the set of scenarios - $\mathcal{Y}$ is the set of periods - $w_y$ is the period weight (for periodic effects) - $w_{y,s}$ is the combined period-scenario weight (for temporal effects) - Periodic effects $E_{\Omega,\text{per}}(y)$ are period-specific but scenario-independent - Temporal effects $E_{\Omega,\text{temp}}(y,s) = \sum_{\text{t}_i} E_{\Omega,\text{temp}}(\text{t}_i, y, s)$ are fully indexed - Penalties $\Phi(y,s)$ are fully indexed

Key Principle: - Scenarios and periods are operationally independent (no energy/resource exchange) - Coupled only through the weighted objective function - Periodic effects within a period are shared across all scenarios (investment made once per period) - Temporal effects are independent per scenario (different operations under different conditions)

Summary

Concept	Formulation	Time Dependency	Dimension Indexing
Temporal share	$s_{l \rightarrow e, \text{temp}}(\text{t}_i)$	Time-dependent	$(t, y, s)$ when present
Periodic share	$s_{l \rightarrow e, \text{per}}$	Time-independent	$(y)$ when periods present
Total temporal effect	$E_{e,\text{temp},\text{tot}} = \sum_{\text{t}_i} E_{e,\text{temp}}(\text{t}_i)$	Sum over time	Depends on dimensions
Total periodic effect	$E_{e,\text{per}}$	Constant	$(y)$ when periods present
Total effect	$E_e = E_{e,\text{per}} + E_{e,\text{temp},\text{tot}}$	Combined	Depends on dimensions
Objective	$\min(E_{\Omega} + \Phi)$	With weights when multi-dimensional	See formulations above

Concept	Formulation	Time Dependency	Dimension Indexing
Temporal share	\(s_{l \rightarrow e, \text{temp}}(\text{t}_i)\)	Time-dependent	\((t, y, s)\) when present
Periodic share	\(s_{l \rightarrow e, \text{per}}\)	Time-independent	\((y)\) when periods present
Total temporal effect	\(E_{e,\text{temp},\text{tot}} = \sum_{\text{t}_i} E_{e,\text{temp}}(\text{t}_i)\)	Sum over time	Depends on dimensions
Total periodic effect	\(E_{e,\text{per}}\)	Constant	\((y)\) when periods present
Total effect	\(E_e = E_{e,\text{per}} + E_{e,\text{temp},\text{tot}}\)	Combined	Depends on dimensions
Objective	\(\min(E_{\Omega} + \Phi)\)	With weights when multi-dimensional	See formulations above

Effects, Penalty & Objective

Effects

Multi-Dimensional Effects

Effect Formulation

Shares from Elements

Cross-Effect Contributions

Total Effect Calculation

Constraining Effects

Penalty

Objective Function

Multi-Criteria Optimization

Objective with Multiple Dimensions

Time Only (Base Case)

Time + Scenario

Time + Period

Time + Period + Scenario (Full Multi-Dimensional)

Summary

See Also