Concepts

Time and units

Prices are a series on a regular UTC grid; each timestamp labels the start of its interval of length \(\Delta t\) hours. Power is in MW, energy in MWh and prices in currency per MWh. Days are UTC days.

Battery physics

A battery has a power rating \(P\) (MW), a rated energy \(E_{rated}\) (MWh), charge and discharge efficiencies \(\eta_c, \eta_d\), and a state-of-charge window \([SOC_{min}, SOC_{max}]\) expressed as fractions of usable energy.

For charge \(c_t\) and discharge \(d_t\) (MW) in interval \(t\), stored energy evolves as

\[E_{t+1} = E_t + \eta_c\, c_t\, \Delta t - \frac{d_t\, \Delta t}{\eta_d}\]

Each interval adds its own throughput to the equivalent full cycle count:

\[N \mathrel{+}= \frac{(c_t + d_t)\,\Delta t}{2 E_{rated}}\]

State of health is computed from lifetime totals, so it never compounds:

\[SOH = 1 - k_{cycle}\, N - k_{calendar}\, \text{age in years}\]

Usable energy is \(E_{rated} \cdot SOH\). Ageing is applied at the end of every UTC day, including days that are not simulated.

Dispatch

Each day is planned by maximising forecast revenue less an optional degradation cost \(\kappa\) per MWh of throughput:

\[\max \sum_t \hat p_t\, (d_t - c_t)\, \Delta t - \kappa \sum_t (c_t + d_t)\, \Delta t\]

With a carbon price \(\lambda\) (currency per tCO2), the planning price becomes \(\hat p_t + \lambda \hat e_t / 1000\) for planned carbon intensity \(\hat e_t\) in gCO2/kWh, so importing is dearer and exporting worth more when the grid is carbon-intensive. Reported revenue stays at market prices; see System analysis.

subject to the energy balance above for \(t = 0 \dots T-1\) with \(E_0\) fixed, and:

• \(0 \le c_t \le P\, u_t\) and \(0 \le d_t \le P\,(1 - u_t)\) with \(u_t \in \{0, 1\}\), so the battery never charges and discharges at once (an LP would, to burn energy at negative prices);
• \(SOC_{min}\, E_{usable} \le E_t \le SOC_{max}\, E_{usable}\);
• \(E_T \ge E_{target}\), by default the energy the day started with;
• optionally, \(\sum_t (c_t + d_t)\, \Delta t \le 2 N_{max}\, E_{usable}\).

The model is solved with HiGHS; a day takes a few milliseconds.

Backtest timing

For each day \(D\):

1. The forecaster receives prices strictly before the decision time, lead_hours before the start of \(D\) (default 12, i.e. noon on \(D-1\)).
2. The optimiser plans \(D\) on the forecast, together with the next lookahead_days days of forecasts if set (all made at the same decision time); the end-energy target then applies at the end of the last planned day.
3. Only \(D\)'s part of the plan is applied through the battery physics and valued at actual prices.
4. The battery state carries into \(D+1\), which is planned afresh.

Near the end of the data, or before a gap, the lookahead shortens to the days that can be forecast. With lookahead_days: 0 (the default) each day is planned alone and must end with the energy it started with.

Days with missing prices, or for which the forecaster has no usable history, are skipped and listed with the reason. Scenarios that are not perfect foresight also run a perfect-foresight benchmark on the same days.

Forecasters

Method	Description
perfect_foresight	The actual prices; the upper bound.
naive_last_week	The same interval seven days earlier, falling back to fourteen.
noisy_foresight	Actual prices plus seeded Gaussian error of a chosen spread.
gradient_boosting	Gradient-boosted trees on calendar features and 1/2/7/14-day lags (needs the forecast extra).

The gradient-boosting model masks any lag at or after the decision time in exactly the same way during training and forecasting, and is trained only on a window that ends before the backtest starts.

Metrics

Metric	Definition
Revenue	\(\sum_t p_t\,(d_t - c_t)\,\Delta t\) at actual prices
Revenue per MW-year	Revenue \(/\,P \times 365\,/\) days simulated
Captured spread	Revenue per MWh discharged
Equivalent cycles	Lifetime \(N\) at the end of the run
Forecast MAE / RMSE	Error of the forecast against actual prices
Capture ratio	Revenue / perfect-foresight revenue on common days

Renewable plants

A plant has a technology, an installed capacity \(C\) and an annual degradation rate \(r\). Its available output follows the OPSD capacity factor \(cf_t\):

\[g_t = C \cdot cf_t \cdot (1 - r)^{\text{age}}\]

Captured price

The captured price is what the market paid per MWh the plant actually produced; the capture rate compares it with the time-weighted (baseload) average:

\[\bar p_{cap} = \frac{\sum_t p_t\, g_t}{\sum_t g_t} \qquad \text{capture rate} = \frac{\bar p_{cap}}{\frac{1}{T}\sum_t p_t}\]

A capture rate below 100% means the technology tends to produce when prices are low; falling capture rates as a technology's share grows are known as cannibalisation. openenergy capture reports this per year, together with capacity factor and the share of output produced at negative prices.

Co-located sites

A plant and a battery can share one grid connection with an export limit \(X\), an import limit \(I\) and an optional premium \(\pi\) per MWh of renewable output delivered (ROC- or CfD-like support).

flowchart LR
    G[Plant output] -->|y| Grid
    G -->|s| B[Battery]
    G -->|k| C[Curtailed]
    Grid -->|m| B
    B -->|d| Grid

Each interval, output \(G_t = y_t + s_t + k_t\) goes to the grid, to the battery, or is curtailed; the battery charges \(c_t = s_t + m_t\) from the plant and the grid (\(m_t \le I\)), and net export \(x_t = y_t + d_t - m_t \le X\). The daily plan maximises

\[\sum_t \left[\hat p_t\, x_t + \pi\,(y_t + s_t)\right]\Delta t - \kappa \sum_t (c_t + d_t)\,\Delta t\]

with the battery constraints above. Output is curtailed when the price falls below \(-\pi\) or the connection is full and the battery cannot absorb it.

Every site run is compared with the same plant and battery on separate connections: the plant exports up to its own capacity and the battery trades on its own. The difference is the co-location value; it is negative when a shared connection is too small, and should be weighed against the connection capacity saved.

Upper bound

Plans use the actual plant output (perfect generation foresight) and forecast prices. Co-location results are therefore an upper bound.