Notation and conventions¶

Every page in this section speaks the same small language. Skim this page now so the symbols look familiar, and come back whenever one of them doesn't. Each symbol means exactly one thing, on every page.

Sets and indices¶

Everything odys models lives in a few sets. A lowercase letter picks one element:

Time starts at 0, like in the code: the first step is \(t = 0\) and the last one is \(t = T - 1\).

Indices always come in the same order: asset, scenario, time. So \(p_{g,s,t}\) reads "the power of generator \(g\) in scenario \(s\) at step \(t\)". A symbol without \(s\) is the same in every scenario, and a symbol without \(t\) is the same at every step.

One more letter: \(\tau\) only ever appears as the counter inside a sum over a window of steps (the minimum up and down times on the Generators page).

Symbol convention¶

You can tell what a symbol is from how it looks:

The only decisions written in Greek are the two helper variables of the risk term, \(\zeta\) and \(\xi_s\), on the Objective page. They follow the paper that introduced them.

Superscripts only say what role a symbol plays, never a limit: ch and dis (charge, discharge), buy and sell, start and stop, up and down, net, trip, trade, fix (fixed), sw (switchable), init and final (the run's initial and final conditions), var (variable), deg (degradation) and val (value).

Each page ends with a table that maps its symbols to the odys attributes they come from, with units.

Sign convention¶

Every asset and market connects to one bus, and power into the bus counts as positive. So supply is positive and demand is negative:

One exception to keep in mind: a battery's net power \(p^{net}_{k,s,t} = p^{ch}_{k,s,t} - p^{dis}_{k,s,t}\) is positive while it charges. That's the battery's own point of view, and it's what the results report as net_power.

The optimizer maximizes profit. Costs enter the profit with a minus sign, and revenues with a plus sign.

Time and boundary conventions¶

odys cuts the run into \(T\) steps of equal length. \(\Delta t\) is the length of one step in hours (timestep), so a step of 15 minutes has \(\Delta t = 0.25\). Within a step, power is constant. The energy of a step is power times \(\Delta t\): 10 MW for a 15-minute step is 2.5 MWh.

States describe the end of a step. \(\mathrm{soc}_{k,s,t}\) is the state of charge once step \(t\) is over:

  soc_init        soc_0        soc_1                soc_{T-1}
     |   step 0     |   step 1   |   ...   | step T-1 |
     ^                                                ^
  initial condition                     final condition, if set

The run's boundary conditions fill in what lies outside the horizon:

• Before step 0, the initial conditions stand in for "the previous step". A generator's previous output is its initial output \(P^{init}_g\), it was on if \(P^{init}_g > 0\), and it had been in that state for \(N^{init}_g\) steps. A battery starts from \(\mathrm{soc}^{init}_k\).
• After step \(T - 1\), a final condition, when you set one, is an equality: the generator's output at the last step is exactly \(P^{final}_g\), and the battery's state of charge is exactly \(\mathrm{soc}^{final}_k\). Without a final condition, the last step is free. Final conditions as bounds instead of exact values are proposed in #163.
• The initial and final conditions are the same in every scenario.

Durations become whole numbers of steps: minimum up and down times, a generator's time in its initial state, and trip departures and returns. odys rejects a duration that doesn't fit the step length. With 15-minute steps, a 2-hour minimum up time is 8 steps, and a 10-minute one is refused.

A battery loses a share \(\delta_k\) of its charge per hour (self_discharge_rate), applied to the state of charge of the previous step as \((1 - \delta_k \Delta t)\). With \(\delta_k = 0.01\) and hourly steps, a full battery keeps 99 % of its charge from one step to the next.

Initial-condition defaults¶

You don't always have to say where an asset starts:

Final conditions are always optional.

Reading a constraint¶

Every constraint on these pages tells the same little story. First a heading and one plain sentence, so you know what it's about before any math. Then the equation, and "Read aloud", the same idea said the way you'd explain it to a colleague. "Why it looks like this" is for when you wonder why it's written that way, and what would go wrong without it. "Good to know" collects the details that are easy to miss.

Each constraint ends with a short line like this one:

Added: always · Binds: when a vehicle charges at its charger's rating, or isn't connected · Name: charger_power_limit_constraint

Think of it as the constraint's ID card. Added says when odys puts it in the model at all: most rules are always there, but some only appear when you ask for them, like a final state. Binds says when it actually shapes the plan. A constraint binds when the plan sits right against it, so that loosening it a little would let the optimizer do better. The rest of the time it's there but has no effect. Some lines say "never on its own": those rules are already covered by another one, and the note tells you which. Name is what odys calls the constraint inside the model, so you can find it in model.constraints when you want to look at it more closely, or when you're working out why a model is infeasible.

Scenarios and decision stages¶

A scenario is one possible future: its prices, demand and available capacity. Scenario \(s\) happens with probability \(\pi_s\), and the probabilities add up to 1.

Most decisions adapt to each scenario, as if the optimizer knew which future will happen. The only exceptions are the trades on stage-fixed markets: they are decided once (non-anticipativity), before the future is known, and are the same in every scenario. That's the reading rule from above at work: a symbol with \(s\) can differ between scenarios, and one without \(s\) cannot.

With a single scenario, \(\pi_s = 1\), and every equation in this section is the plain deterministic model. There is no separate deterministic version to read.

The first symbol you'll meet everywhere is \(s\). Where does it come from? See Scenarios.