Analytical Methods

Analytical Methods#

This page describes the analytical and mathematical methods available in GreenBubble beyond the core LP. Each method is optional and controlled by its own configuration block.

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Stochastic optimisation#

GreenBubble can solve a two-stage stochastic linear programme across multiple price/weather scenarios (e.g. different market years).

Motivation

Capacity expansion decisions must be made before knowing which price year will materialise. A stochastic formulation captures this uncertainty by coupling all scenarios in a single LP, ensuring that the first-stage capacity decisions are feasible and optimal across all scenarios.

Formulation

  • First stage — capacity variables (p_nom) are shared across all scenarios; they represent investment decisions made before uncertainty resolves.

  • Second stage — dispatch variables (p(t)) are scenario-specific; the optimiser can dispatch differently in each scenario year.

  • Objective — minimise expected total cost: \(\sum_s \pi_s \cdot C_s(x, y_s)\), where \(\pi_s\) is the probability of scenario s, x are first-stage capacity decisions, and y_s are second-stage dispatch decisions.

The coupled network is built by scripts/create_stoch_scenarios.py: scenario-specific time series are attached as extra snapshots in separate PyPSA sub-networks that share the same capacity variables.

EVPI

When stochastic.EVPI: true, an additional deterministic solve is run for each scenario individually (perfect-information runs). The Expected Value of Perfect Information is then:

\(\text{EVPI} = \text{EV(perfect info)} - \text{EV(stochastic solution)}\)

A large EVPI indicates that the uncertainty in scenario realisations strongly influences the optimal design.

Configuration — see stochastic for all parameters.

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Rolling horizon dispatch#

See the dedicated guide: Rolling Horizon Dispatch Optimisation.

The rolling horizon method runs a dispatch-only optimisation on a capacity-fixed network. It is used to evaluate how a designed plant performs in a different weather or price year, or to obtain high-resolution dispatch profiles for a given design.

The year is divided into overlapping windows of horizon hours, solved sequentially. The overlap parameter avoids end-of-window artefacts by carrying forward the storage state from the overlapping portion.

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RFNBO compliance#

The EU Renewable Fuels of Non-Biological Origin (RFNBO) delegated acts impose requirements on electrolytic hydrogen to be certified as renewable. GreenBubble implements two constraint modes, controlled by rfnbos_dict.limit:

Mode

Constraint applied

price

Electrolysis can only operate when the electricity spot price is below price_threshold (€/MWh) — a proxy for high-renewable-share hours

emissions

Electrolysis is restricted to hours when the grid emission intensity is below emission_threshold (tCO₂/MWh) — direct temporal correlation

unlimited

No RFNBO constraint; electrolysis can draw from the grid freely

disconnected

Electrolysis is fully disconnected from the grid; must be covered entirely by on-site renewables

The price and emission time series are downloaded per year from Energi Data Service.

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Shadow prices and internal market#

The dual variables of the bus balance constraints in the LP are the shadow prices (marginal values) of each energy carrier at each hour. In economic terms they represent the internal market clearing price: what it would cost to produce one additional MWh (or tonne) of that carrier at that moment.

GreenBubble extracts and exports shadow prices for all internal buses (electricity, H₂, CO₂, biomethane, methanol, heat levels). These are used in the economic analysis module (scripts/plots.py) to:

  • Decompose the system value created at each node

  • Identify binding constraints (hours where prices spike)

  • Assess the implicit cross-subsidies between co-located partners

The list of buses to analyse is configured in plots_config.default.yaml under plotting.bus_list_mp.

See Economic Analysis of Results for interpretation and post-processing.

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Shapley value cost allocation#

Note

Full documentation coming soon.

When multiple industrial partners share infrastructure (grid connection, storage, compression), the total system cost is lower than the sum of each partner building independently. The Shapley value from cooperative game theory provides a fair allocation of these savings.

GreenBubble computes the Shapley value by:

  1. Enumerating all coalitions (subsets) of partners

  2. Running a separate optimisation for each coalition (with only the corresponding n_flags active)

  3. Computing marginal contributions and averaging over all orderings

The number of optimisation runs grows as \(2^N\) in the number of partners N, so this is typically run for N ≤ 5. Results are used to allocate shared infrastructure costs fairly among partners.