Objective Function¶

Formulation¶

The model minimizes the total value of the designated objective effect \(k^*\) (the one with is_objective=True).

Single-period¶

Without periods, the objective is simply:

\[ \min \; \Phi_{k^*} \]

Multi-period¶

With periods \(p \in \mathcal{P}\), the objective separates recurring and one-time domains with independent per-effect weights:

\[ \min \; \sum_{p \in \mathcal{P}} \omega^{\text{periodic}}_{k^*,p} \cdot \left(\sum_t \Phi_{k^*,t,p}^{\text{temporal}} \cdot w_t + \Phi_{k^*,p}^{\text{periodic}}\right) + \omega^{\text{once}}_{k^*,p} \cdot \Phi_{k^*,p}^{\text{once}} \]

\(\omega^{\text{periodic}}_{k,p}\) defaults to the global period_weights (inferred from gaps between period labels, or explicit). Override per effect via Effect.period_weights_periodic.
\(\omega^{\text{once}}_{k,p}\) defaults to 1 (no scaling). Override per effect via Effect.period_weights_once.

This allows different weighting strategies per effect (e.g., NPV discounting for costs, flat weights for emissions).

Total effect¶

The total effect per period combines all three domains:

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

The temporal domain accumulates flow contributions, running costs, startup costs, and cross-effect contributions per timestep:

\[ \Phi_{k,t}^{\text{temporal}} = \underbrace{\sum_{f} c_{f,k,t} \cdot P_{f,t} \cdot \Delta t_t}_{\text{flow}} + \underbrace{\sum_{f} r_{f,k,t} \cdot \sigma_{f,t} \cdot \Delta t_t}_{\text{running}} + \underbrace{\sum_{f} u_{f,k,t} \cdot \tau^+_{f,t}}_{\text{startup}} + \underbrace{\sum_{j} \alpha_{k,j,t} \cdot \Phi_{j,t}^{\text{temporal}}}_{\text{cross-effect}} \]

The periodic domain accumulates recurring sizing costs, fixed costs, and cross-effect contributions:

\[ \Phi_k^{\text{periodic}} = \underbrace{\sum_{f} \gamma_{f,k} \cdot S_f + \sum_{f} \phi_{f,k} \cdot y_f + \sum_{s} \gamma_{s,k} \cdot S_s + \sum_{s} \phi_{s,k} \cdot y_s}_{\text{direct sizing costs}} + \underbrace{\sum_{j} \alpha_{k,j} \cdot \Phi_j^{\text{periodic}}}_{\text{cross-effect}} \]

The once domain is reserved for one-time costs (e.g., investment CAPEX) that should not be scaled by period weights. Currently constrained to zero.

See Sizing, Status, and Effects for full formulations of each term.

Parameters¶

Symbol	Description	Reference
\(k^*\)	Objective effect	`Effect.is_objective = True`
\(c_{f,k,t}\)	Effect coefficient per flow-hour	`Flow.effects_per_flow_hour`
\(P_{f,t}\)	Flow rate variable	`flow--rate[flow, time]`
\(\Delta t_t\)	Timestep duration	dt
\(w_t\)	Timestep weight	weights
\(\omega^{\text{periodic}}_{k,p}\)	Period weight for recurring domain	`Effect.period_weights_periodic` (fallback: `Dims.period_weights`)
\(\omega^{\text{once}}_{k,p}\)	Period weight for one-time domain	`Effect.period_weights_once` (default: 1)
\(\Phi_{k,t(,p)}^{\text{temporal}}\)	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)]`

See Notation for the full symbol table.

Examples¶

Single-period¶

Consider a gas boiler over 3 timesteps (\(\Delta t = 1\,\text{h}\), \(w = 1\)):

\(t\)	\(P_{\text{gas},t}\) (MW)	\(c_{\text{gas,cost}}\) (€/MWh)	\(\Phi_{\text{cost},t}^{\text{temporal}}\) (€)
1	2.0	30	\(30 \times 2.0 \times 1 = 60\)
2	3.0	30	\(30 \times 3.0 \times 1 = 90\)
3	1.5	30	\(30 \times 1.5 \times 1 = 45\)

Total cost: \(\Phi_{\text{cost}} = \sum_t \Phi_{\text{cost},t}^{\text{temporal}} = 60 + 90 + 45 = 195\,\text{€}\)

Multi-period¶

Same system with periods=[2020, 2025], period_weights=[5, 5].

Each period has the same 3 timesteps with per-period cost = 30 €. The objective becomes:

\[ \sum_p \omega_p \cdot \Phi_{\text{cost},p} = 5 \times 30 + 5 \times 30 = 300\,\text{€} \]

The optimizer finds the \(P_{f,t(,p)}\) values that minimize the (period-weighted) \(\Phi_{k^*}\) subject to all constraints.