Notation¶

This page defines the canonical symbols used throughout the mathematical formulation. Each symbol maps to a specific field or variable in the code.

Sets & Indices¶

Symbol	Description	Code
\(t \in \mathcal{T}\)	Timesteps	`time` dimension
\(p \in \mathcal{P}\)	Periods (multi-period only)	`period` dimension
\(f \in \mathcal{F}\)	Flows	`flow` dimension
\(b \in \mathcal{B}\)	Buses	`bus` dimension
\(s \in \mathcal{S}\)	Storages	`storage` dimension
\(k \in \mathcal{K}\)	Effects (cost, CO₂, …)	`effect` dimension
\(j \in \mathcal{K}\)	Source effect (cross-effect)	`source_effect` dimension

Variables¶

Symbol	Code	Domain	Unit	Description
\(P_{f,t(,p)}\)	`flow--rate[flow, time(, period)]`	\(\geq 0\)	MW	Flow rate
\(E_{s,t(,p)}\)	`storage--level[storage, time(, period)]`	\(\geq 0\)	MWh	Stored energy
\(\Phi_{k,t(,p)}^{\text{temporal}}\)	`effect--temporal[effect, time(, period)]`	\(\mathbb{R}\)	varies	Temporal (per-timestep) effect
\(\Phi_{k(,p)}^{\text{lump}}\)	`effect--lump[effect(, period)]`	\(\mathbb{R}\)	varies	Lump (sizing + one-time) effect
\(\Phi_{k(,p)}\)	`effect--total[effect(, period)]`	\(\mathbb{R}\)	varies	Total effect per period
\(S_{f(,p)}\)	`flow--size[flow(, period)]`	\(\geq 0\)	MW	Invested flow capacity
\(y_{f(,p)}\)	`flow--size_indicator[flow(, period)]`	\(\{0, 1\}\)	—	Binary invest indicator (flow)
\(S_{s(,p)}\)	`storage--capacity[storage(, period)]`	\(\geq 0\)	MWh	Invested storage capacity
\(y_{s(,p)}\)	`storage--size_indicator[storage(, period)]`	\(\{0, 1\}\)	—	Binary invest indicator (storage)
\(\sigma_{f,t(,p)}\)	`flow--on[flow, time(, period)]`	\(\{0, 1\}\)	—	On/off indicator
\(\tau^+_{f,t(,p)}\)	`flow--startup[flow, time(, period)]`	\(\{0, 1\}\)	—	Startup event indicator
\(\tau^-_{f,t(,p)}\)	`flow--shutdown[flow, time(, period)]`	\(\{0, 1\}\)	—	Shutdown event indicator
\(\mathrm{D}^{\text{up}}_{f,t(,p)}\)	`uptime[flow, time(, period)]`	\(\geq 0\)	h	Consecutive uptime
\(\mathrm{D}^{\text{down}}_{f,t(,p)}\)	`downtime[flow, time(, period)]`	\(\geq 0\)	h	Consecutive downtime

Indexing Convention¶

Variables (the table above) are shown with all dimensions they live on. There's no broadcasting for decision variables — every \((\text{flow}, \text{time}, \ldots)\) cell is its own variable.

Optional dims — currently just period — exist in some models and not others. We mark them in parentheses, e.g. \(P_{f,t(,p)}\): the period subscript is present in multi-period models, absent in single-period ones. To keep formulas readable, constraint derivations that aren't specifically about multi-period dynamics drop the parenthetical and treat \(p\) as implicit context — the constraint applies pointwise across whichever optional dims happen to be in the model.

Parameters are different. Symbol subscripts show only the indices the formulation structurally requires — the dims a constraint iterates over and that change its meaning. Most parameters also broadcast over additional dims in the API; we don't decorate the symbols with every dim they could take, because that turns formulas into pyramids of subscripts.

The broadcast hierarchy:

Period (\(p\)) — almost every parameter accepts this: even "scalar" things like \(\bar{\mathrm{P}}_f\), \(\mathrm{D}^{\text{up,min}}\), or \(\mathrm{S}^-\) become \(\bar{\mathrm{P}}_{f,p}\) etc. in multi-period models. Variables already reflect this in the table above (e.g. \(P_{f,t(,p)}\)).
Time (\(t\)) — only fields typed Variate accept this: bounds \(\underline{\mathrm{p}}_{f,t}, \bar{\mathrm{p}}_{f,t}\), profiles \(\pi_{f,t}\), efficiencies \(\eta^c_s, \eta^d_s\), losses \(\delta_s\), conversion coefficients \(\mathrm{a}_{f,i}\), effect / running / startup costs \(\mathrm{c}_{f,k,t}, \mathrm{r}_{f,k,t}, \mathrm{u}_{f,k,t}\), cross-effects \(\alpha_{k,j,t}\).
Build period (\(p_b\)) — at-build investment coefficients only: \(\gamma^{\text{build}}_{f,k}, \phi^{\text{build}}_{f,k}\), …

When a formula shows a \(t\) subscript on a parameter, it's because time variation is structurally meaningful for that constraint (e.g. a fixed profile is meaningless without time). Parameters without a \(t\) subscript may still be time-varying when the API field accepts it — the constraint just reads it pointwise. The same applies for \(p\): omit unless we're discussing multi-period dynamics specifically.

Parameters¶

Symbol	Code	Domain	Unit	Description
\(\bar{\mathrm{P}}_f\)	`Flow.size`	\(\geq 0\) or \(\infty\)	MW	Nominal capacity
\(\underline{\mathrm{p}}_{f,t}\)	`Flow.relative_minimum`	\([0, 1]\)	—	Relative lower bound
\(\bar{\mathrm{p}}_{f,t}\)	`Flow.relative_maximum`	\([0, 1]\)	—	Relative upper bound
\(\pi_{f,t}\)	`Flow.fixed_relative_profile`	\([0, 1]\)	—	Fixed profile
\(\mathrm{c}_{f,k,t}\)	`Flow.effects_per_flow_hour`	\(\mathbb{R}\)	varies	Effect coefficient per flow-hour
\(\bar{\mathrm{E}}_s\)	`Storage.capacity`	\(\geq 0\)	MWh	Storage capacity
\(\eta^{\text{c}}_s\)	`Storage.eta_charge`	\((0, 1]\)	—	Charging efficiency
\(\eta^{\text{d}}_s\)	`Storage.eta_discharge`	\((0, 1]\)	—	Discharging efficiency
\(\delta_s\)	`Storage.relative_loss_per_hour`	\([0, 1]\)	1/h	Self-discharge rate
\(\underline{\mathrm{e}}_s\)	`Storage.relative_minimum_level`	\([0, 1]\)	—	Relative min SOC
\(\bar{\mathrm{e}}_s\)	`Storage.relative_maximum_level`	\([0, 1]\)	—	Relative max SOC
\(\mathrm{a}_{f,i}\)	`Converter.conversion_factors`	\(\mathbb{R}\)	—	Conversion coefficient (per flow, per equation)
\(\alpha_{k,j}\)	`Effect.contribution_from`	\(\mathbb{R}\)	varies	Cross-effect factor (scalar)
\(\alpha_{k,j,t}\)	`Effect.contribution_from` (Variate)	\(\mathbb{R}\)	varies	Cross-effect factor (time-varying; lump uses time-mean)
\(\bar{\Phi}_k\)	`Effect.maximum`	\(\mathbb{R}\)	varies	Maximum aggregate (weighted sum across periods)
\(\underline{\Phi}_k\)	`Effect.minimum`	\(\mathbb{R}\)	varies	Minimum aggregate (weighted sum across periods)
\(\bar{\Phi}_k^{\text{per period}}\)	`Effect.maximum_per_period`	\(\mathbb{R}\)	varies	Maximum per period
\(\underline{\Phi}_k^{\text{per period}}\)	`Effect.minimum_per_period`	\(\mathbb{R}\)	varies	Minimum per period
\(\bar{\Phi}_{k,t}^{\text{per hour}}\)	`Effect.maximum_per_hour`	\(\mathbb{R}\)	varies/h	Maximum per hour (rate, scaled by \(\Delta t_t\))
\(\underline{\Phi}_{k,t}^{\text{per hour}}\)	`Effect.minimum_per_hour`	\(\mathbb{R}\)	varies/h	Minimum per hour (rate, scaled by \(\Delta t_t\))
\(\mathrm{S}^-\)	`Sizing.min_size`	\(\geq 0\)	MW or MWh	Minimum invested size (flow or storage)
\(\mathrm{S}^+\)	`Sizing.max_size`	\(\geq 0\)	MW or MWh	Maximum invested size (flow or storage)
\(\gamma_{f,k}\), \(\gamma_{s,k}\)	`Sizing.effects_per_size`	\(\mathbb{R}\)	varies	Per-size investment cost (flow or storage; one-time, sized)
\(\phi_{f,k}\), \(\phi_{s,k}\)	`Sizing.effects_fixed`	\(\mathbb{R}\)	varies	Fixed investment cost (flow or storage; one-time, sized)
\(\gamma^{\text{build}}_{f,k}\)	`Investment.effects_per_size_at_build`	\(\mathbb{R}\)	varies	Per-size CAPEX charged in the build period
\(\phi^{\text{build}}_{f,k}\)	`Investment.effects_fixed_at_build`	\(\mathbb{R}\)	varies	Fixed CAPEX charged in the build period
\(\gamma^{\text{rec}}_{f,k}\)	`Investment.effects_per_size_recurring`	\(\mathbb{R}\)	varies	Recurring per-size cost (each active period)
\(\phi^{\text{rec}}_{f,k}\)	`Investment.effects_fixed_recurring`	\(\mathbb{R}\)	varies	Recurring fixed cost (each active period)
\(\mathrm{D}^{\text{up,min}}\)	`Status.min_uptime`	\(\geq 0\)	h	Minimum consecutive uptime
\(\mathrm{D}^{\text{up,max}}\)	`Status.max_uptime`	\(\geq 0\)	h	Maximum consecutive uptime
\(\mathrm{D}^{\text{down,min}}\)	`Status.min_downtime`	\(\geq 0\)	h	Minimum consecutive downtime
\(\mathrm{D}^{\text{down,max}}\)	`Status.max_downtime`	\(\geq 0\)	h	Maximum consecutive downtime
\(\mathrm{r}_{f,k,t}\)	`Status.effects_per_running_hour`	\(\mathbb{R}\)	varies	Running cost coefficient
\(\mathrm{u}_{f,k,t}\)	`Status.effects_per_startup`	\(\mathbb{R}\)	varies	Startup cost coefficient
\(\mathrm{w}_t\)	weights	\(> 0\)	—	Timestep weight
\(\Delta t_t\)	dt	\(> 0\)	h	Timestep duration
\(\omega_p\)	`optimize(period_weights=...)`	\(> 0\)	—	Global period weight (multi-period only)
\(\omega_{k,p}\)	`Effect.period_weights`	\(> 0\)	—	Per-effect period weight

Naming Conventions¶

Italic letters denote decision variables; upright (\mathrm{}) letters denote parameters — following ISO 80000-2. So \(P_{f,t}\) (italic) is the flow rate variable, while \(\bar{\mathrm{P}}_f\) (upright with overbar) is the parameter that bounds it.

Convention	Meaning	Example
Italic (default math mode)	Decision variables	\(P\) (flow rate), \(E\) (stored energy), \(S\) (size)
Upright (`\mathrm{}`)	Parameters	\(\mathrm{c}_{f,k,t}\) (cost coefficient), \(\bar{\mathrm{P}}_f\) (capacity), \(\mathrm{a}_{f,i}\) (conversion coefficient)
Overbar / underbar	Bound (paired with the variable's letter)	\(\bar{\mathrm{P}}\) (upper bound on \(P\)), \(\underline{\mathrm{P}}\) (lower bound)
Lowercase variant	Relative bound (fraction of size/capacity)	\(\bar{\mathrm{p}}_{f,t}\), \(\underline{\mathrm{p}}_{f,t}\) (fraction of \(\bar{\mathrm{P}}_f\)); \(\bar{\mathrm{e}}_s\), \(\underline{\mathrm{e}}_s\) (fraction of \(\bar{\mathrm{E}}_s\))
Greek	Parameters with established physical meaning	\(\eta\) (efficiency), \(\delta\) (loss), \(\pi\) (profile), \(\gamma\) (per-size cost)
Subscripts	Indexing	\(f\) (flow), \(t\) (time), \(s\) (storage), \(b\) (bus), \(k\) (effect), \(j\) (source effect)
Superscripts	Qualification	\(\eta^{\text{c}}\) (charge), \(\eta^{\text{d}}\) (discharge)

Greek letters render with their conventional shape regardless of \mathrm{}, so we use them as-is for parameters without further wrapping.