Objective Function¶

Formulation¶

The model minimizes the total value of the designated objective effect \(k^*\) (specified via optimize(objective_effects='cost')).

Single-period¶

Without periods, the objective is simply:

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

Multi-period¶

With periods \(p \in \mathcal{P}\), the objective weights each period's total effect:

\[ \min \; \sum_{p \in \mathcal{P}} \omega_{k^*,p} \cdot \left(\sum_t \Phi_{k^*,t,p}^{\text{temporal}} \cdot \mathrm{w}_t + \Phi_{k^*,p}^{\text{lump}}\right) \]

\(\omega_{k,p}\) defaults to the global period_weights (inferred from gaps between period labels, or explicit). Override per effect via Effect.period_weights.

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 both domains:

\[ \Phi_{k(,p)} = \sum_{t \in \mathcal{T}} \Phi_{k,t(,p)}^{\text{temporal}} \cdot \mathrm{w}_t + \Phi_{k(,p)}^{\text{lump}} \]

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

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

The lump domain accumulates sizing costs, fixed costs, one-time costs, and cross-effect contributions:

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

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

Parameters¶

Symbol	Description	Reference
\(k^*\)	Objective effect	`optimize(objective_effects='cost')`
\(\mathrm{c}_{f,k,t}\)	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
\(\mathrm{w}_t\)	Timestep weight	weights
\(\omega_{k,p}\)	Period weight	`Effect.period_weights` (fallback: `optimize(period_weights=...)`)
\(\Phi_{k,t(,p)}^{\text{temporal}}\)	Temporal (per-timestep) effect variable	`effect--temporal[effect, time(, period)]`
\(\Phi_{k(,p)}^{\text{lump}}\)	Lump effect variable (sizing + one-time costs)	`effect--lump[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)	\(\mathrm{c}_{\text{gas,cost}}\) (€/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.