Effect Tracking & Bounding¶

Overview¶

Effects represent quantities that are tracked across the optimization horizon (e.g., cost, CO₂ emissions, primary energy). One effect is designated as the objective to minimize.

Effects are split into three domains based on how they vary over time and how they are weighted in multi-period optimization:

Domain	Dims	What goes here	Multi-period weighting
Temporal	`(effect, time)`	Flow costs, running costs, startup costs — anything that varies per timestep	Summed over time (× \(w_t\)), then weighted like periodic
Periodic	`(effect,)`	Recurring costs that repeat each period — sizing costs, fixed annual O&M	Weighted by \(\omega^{\text{periodic}}_{k,p}\) (defaults to global `period_weights`)
Once	`(effect,)`	One-time costs at a point in time — CAPEX, decommissioning	Weighted by \(\omega^{\text{once}}_{k,p}\) (defaults to 1, no scaling)

The key distinction: periodic costs are assumed to recur across the gap between periods (e.g., annual O&M for 5 years), while once costs happen at a specific point (e.g., an investment decision in 2025). This matters because their period weights differ — recurring costs scale with duration, one-time costs typically don't (or use discount factors instead).

All domains support cross-effect chains via contribution_from.

In multi-period mode, all variables gain an optional period dimension. See Objective for how the domains are weighted in the objective.

Temporal Domain¶

Each effect accumulates contributions from all flows at each timestep:

\[ \Phi_{k,t}^{\text{temporal}} = \underbrace{\sum_{f \in \mathcal{F}} c_{f,k,t} \cdot P_{f,t} \cdot \Delta t_t}_{\text{direct flow contributions}} + \underbrace{\sum_{j \in \mathcal{K}} \alpha_{k,j,t} \cdot \Phi_{j,t}^{\text{temporal}}}_{\text{cross-effect contributions}} \quad \forall \, k, t \]

The coefficient \(c_{f,k,t}\) specifies how much of effect \(k\) is produced per flow-hour of flow \(f\) (e.g., €/MWh for cost, kg/MWh for emissions).

The cross-effect factor \(\alpha_{k,j,t}\) can be time-varying (contribution_from_per_hour) or constant (contribution_from). Because \(\Phi_{k,t}^{\text{temporal}}\) is a variable, the solver resolves multi-level chains (e.g., PE → CO₂ → cost) automatically.

Periodic Domain¶

Sizing costs and fixed costs (not time-varying) are accumulated per effect:

\[ \Phi_k^{\text{periodic}} = \underbrace{\Phi_k^{\text{invest,direct}}}_{\text{direct sizing costs}} + \underbrace{\sum_{j \in \mathcal{K}} \alpha_{k,j} \cdot \Phi_j^{\text{periodic}}}_{\text{cross-effect contributions}} \quad \forall \, k \]

where the direct investment term is:

\[ \Phi_k^{\text{invest,direct}} = \sum_{f} \gamma_{f,k} \cdot S_f + \sum_{f} \phi_{f,k} \cdot y_f + \sum_{s} \gamma_{s,k} \cdot S_s + \sum_{s} \phi_{s,k} \cdot y_s \]

Because \(\Phi_k^{\text{periodic}}\) is a variable (not an expression), the solver resolves multi-level chains correctly: if PE has sizing costs and CO₂ depends on PE and cost depends on CO₂, the chain propagates through the periodic domain just as it does through the temporal domain.

Cross-Effect Contributions¶

An effect can include a weighted fraction of another effect's value via contribution_from. This enables patterns like carbon pricing (CO₂ → cost) or transitive chains (PE → CO₂ → cost).

The scalar factor \(\alpha_{k,j}\) from contribution_from applies to both domains. The time-varying factor from contribution_from_per_hour overrides the temporal factor only.

Validation¶

Self-references (\(\alpha_{k,k}\)) and circular dependencies (\(k \to j \to \cdots \to k\)) are rejected at build time to prevent singular systems.

Once Domain¶

One-time costs that should not be scaled by period weights (e.g., investment CAPEX, decommissioning costs). Currently constrained to zero — a placeholder for future investment modelling:

\[ \Phi_{k(,p)}^{\text{once}} = 0 \quad \forall \, k \]

Total Aggregation¶

The total effect combines all three domains:

\[ \Phi_{k(,p)} = \sum_{t \in \mathcal{T}} \Phi_{k,t(,p)}^{\text{temporal}} \cdot w_t + \Phi_{k(,p)}^{\text{periodic}} + \Phi_{k(,p)}^{\text{once}} \quad \forall \, k \in \mathcal{K} \]

Weights \(w_t\) allow scaling timesteps (e.g., a representative week scaled to a year).

Total Bounds¶

Upper and lower bounds on the total effect over the entire horizon:

\[ \underline{\Phi}_k \leq \Phi_k \leq \bar{\Phi}_k \]

This is useful for emission caps or budget constraints.

Per-Timestep Bounds¶

Bounds on the effect value at each timestep:

\[ \underline{\Phi}_{k,t} \leq \Phi_{k,t} \leq \bar{\Phi}_{k,t} \quad \forall \, t \in \mathcal{T} \]

This enforces per-hour limits (e.g., maximum hourly emissions).

Parameters¶

Symbol	Description	Reference
\(\Phi_{k,t(,p)}^{\text{temporal}}\)	Per-timestep effect variable	`effect_temporal[effect, time(, period)]`
\(\Phi_{k(,p)}^{\text{periodic}}\)	Periodic effect variable (recurring costs)	`effect_periodic[effect(, period)]`
\(\Phi_{k(,p)}^{\text{once}}\)	One-time effect variable	`effect_once[effect(, period)]`
\(\Phi_{k(,p)}\)	Total effect variable	`effect_total[effect(, period)]`
\(c_{f,k,t}\)	Effect coefficient per flow-hour	`Flow.effects_per_flow_hour`
\(\alpha_{k,j,t}\)	Cross-effect contribution factor (per hour)	`Effect.contribution_from_per_hour`
\(\alpha_{k,j}\)	Cross-effect contribution factor (scalar)	`Effect.contribution_from`
\(P_{f,t}\)	Flow rate variable	`flow_rate[flow, time]`
\(\Delta t_t\)	Timestep duration	dt
\(w_t\)	Timestep weight	weights
\(\bar{\Phi}_k\)	Maximum total	`Effect.maximum_total`
\(\underline{\Phi}_k\)	Minimum total	`Effect.minimum_total`
\(\bar{\Phi}_{k,t}\)	Maximum per hour	`Effect.maximum_per_hour`
\(\underline{\Phi}_{k,t}\)	Minimum per hour	`Effect.minimum_per_hour`

See Notation for the full symbol table.

Examples¶

Direct effects¶

A system with two effects — cost (objective) and CO₂ (capped at 1000 kg):

effects = [
    Effect("cost", unit="€", is_objective=True),
    Effect("CO2", unit="kg", maximum_total=1000),
]

A gas flow with both effect coefficients:

gas_flow = Flow("gas", bus="gas_bus", effects_per_flow_hour={"cost": 30, "CO2": 0.2})

At timestep \(t\) with \(P_{\text{gas},t} = 5\) MW and \(\Delta t = 1\) h:

\(\Phi_{\text{cost},t} = 30 \times 5 \times 1 = 150\) €
\(\Phi_{\text{CO₂},t} = 0.2 \times 5 \times 1 = 1.0\) kg

Carbon pricing via `contribution_from`¶

CO₂ priced at 50 €/t into the cost effect:

effects = [
    Effect("cost", is_objective=True, contribution_from={"co2": 50}),
    Effect("co2", unit="kg"),
]

With \(\alpha_{\text{cost,co2}} = 50\), the per-timestep cost becomes:

\[ \Phi_{\text{cost},t} = c_{\text{cost}} \cdot P_t \cdot \Delta t + 50 \cdot \Phi_{\text{co2},t} \]

The CO₂ total itself is not affected — contribution_from is one-directional.