Add documentation for commodity price calculations

docs/SUMMARY.md

@@ -12,6 +12,7 @@
 - [Model Description](model/README.md)
   - [Dispatch Optimisation](model/dispatch_optimisation.md)
   - [Investment Appraisal](model/investment.md)
+  - [Commodity Prices](model/prices.md)
   - [Model Diagrams](model/model_diagrams.md)
 - [Glossary](glossary.md)
 - [Developer Guide](developer_guide/README.md)

docs/model/prices.md

@@ -0,0 +1,239 @@
+# Commodity Prices
+
+This section describes how commodity prices are calculated in MUSE2 at the end of each
+milestone year iteration. The pricing algorithm operates on the results of the dispatch
+optimisation, translating flows, capacities, and solver dual values (shadow prices) into commodity
+market prices.
+
+## Pricing Strategies
+
+Each commodity is configured with a specific `pricing_strategy` via [`commodities.csv`].
+The options are:
+
+- **`marginal`**: Prices are set to the marginal cost of the highest-cost active asset producing the
+commodity.
+- **`marginal_average`**: Prices are set to the output-weighted average marginal cost across all
+active assets.
+- **`full`**: Prices are set to the full cost (marginal cost + annual fixed/capital costs) of the
+highest-cost active asset producing the commodity.
+- **`shadow`**: Prices are taken directly from the shadow prices (dual values) of the commodity
+balance constraints in the dispatch optimisation.
+- **`scarcity`**: Prices are set to the shadow price plus the highest activity dual of the assets
+producing the commodity in that region and time slice.
+- **`unpriced`**: No prices are calculated for the commodity.
+
+### Cost Calculations
+
+Cost-based prices (`marginal`, `marginal_average`, `full`, `full_average`) are calculated based on
+process cost parameters, asset capacities and dispatch flows. For assets producing multiple output
+commodities (e.g. an asset producing both oil and gas), generic activity costs and fixed costs are
+shared between outputs according to output flow coefficients.
+
+#### Generic Activity Cost
+
+The generic activity cost comprises all operating expenditures, input purchases, and flow costs/levies
+not associated with specific SED/SVD outputs:
+\[
+\text{GenericActivityCost} = \text{VariableOperatingCost} + \text{InputPurchases} + \sum \text{GenericFlowCosts}
+\]
+
+This is shared equally over all SED/SVD outputs in proportion to their output coefficients to
+compute a generic cost per unit of output flow:
+\\[
+\text{GenericCostPerOutput} = \frac{\text{GenericActivityCost}}{\sum_{c \in \text{SED, SVD}} \text{OutputCoefficient}_c}
+\\]
+
+#### Marginal Cost
+
+The marginal cost of an output commodity \\( c \\) is the sum of the generic cost per output and any
+commodity-specific costs (e.g., production levies, flow costs):
+\\[
+\text{MarginalCost}_c = \text{GenericCostPerOutput} + \text{SpecificCostPerOutput}_c
+\\]
+
+#### Full Cost
+
+The full cost includes the marginal cost plus annualised capital and fixed operating costs.
+
+Annualised capital costs are calculated using the [Capital Recovery Factor] (CRF):
+\\[
+\text{AnnualCapitalCostPerCapacity} = \text{TotalCapitalCostPerCapacity} \times \text{CRF}
+\\]
+
+The annualised fixed cost (AFC) is:
+\\[
+\text{AFC} = (\text{AnnualCapitalCostPerCapacity} + \text{AnnualFixedOpex}) \times \text{TotalCapacity}
+\\]
+
+This is divided by annual activity to get a cost per unit of activity:
+\\[
+\text{AnnualFixedCostPerActivity} = \frac{\text{AFC}}{\text{AnnualActivity}}
+\\]
+
+This is divided again by the sum of SED/SVD output coefficients to get a cost per unit output:
+\\[
+\text{AnnualFixedCostPerOutput} = \frac{\text{AnnualFixedCostPerActivity}}{\sum_{c \in \text{SED,
+SVD}} \text{OutputCoefficient}_c}
+\\]
+
+*Note: this only works if all output commodities are measured in the same energy units (e.g. PJ).
+For this reason, MUSE disallows processes that have output commodities with differing units.*
+
+The final full cost of output commodity \\( c \\) is:
+\\[
+\text{FullCost}_c = \text{MarginalCost}_c + \text{AnnualFixedCostPerOutput}
+\\]
+
+### Candidate asset fallback
+
+For cost-based strategies (`marginal`, `marginal_average`, `full`, `full_average`), if no active
+producers exist in the current year, prices are calculated by looking at candidate assets for
+investment in the following milestone year. Prices are set by the candidate asset with the lowest
+marginal/full cost, calculated assuming full utilisation (i.e. assuming annual activity equals the
+maximum annual activity limit).
+
+In the final milestone year, prices for commodities with no active producer are omitted.
+
+### Time Slice Aggregation
+
+For commodities defined at coarser time slice levels (e.g. seasonal or annual), prices are
+calculated by weighting asset costs across time slices by activity to yield a flat price for the
+season/year. For candidates, or assets with zero activity across the selection, upper activity
+limits are used as weights.
+
+## Price Calculation Order
+
+Prices are calculated sequentially starting from upstream commodities (e.g. raw fuels) and moving
+downstream to intermediate energy carriers and final service demands.
+
+This sequence is the **reverse** of the investment order. This ensures that the prices of input
+commodities (from upstream markets) are calculated and known before they are consumed as inputs by
+downstream processes evaluating marginal or full cost pricing.
+
+## Circularities
+
+When the energy system includes circular commodity flows (e.g., electricity producing hydrogen,
+which then generates electricity), the resulting cyclic dependencies are resolved as follows:
+
+1. **Initial Seeding:** Cyclic markets are seeded with their shadow prices.
+2. **Sequential Evaluation:** Commodities in the cycle are evaluated in sequence, starting with
+those furthest upstream relative to the rest of the system (i.e., furthest from the commodities
+downstream of the SCC), and working down. Marginal cost calculations use newly updated prices for
+input commodities already evaluated in the sequence, and fall back to the seeded shadow prices for
+any not yet evaluated, which may occur due to circular dependencies.
+<!-- Currently a hidden option, TBD whether we want to open this up: -->
+<!-- 2. **Iterative Refinement:** An iterative loop runs for a fixed number of iterations,
+re-evaluating each market in the cycle in reverse order (starting with those
+furthest from commodities downstream of the SCC) to propagate the feedback effects through the
+circular markets. -->
+
+<!-- markdownlint-disable MD024 -->
+## Example
+
+The following example demonstrates the calculation of full-cost prices for two commodities using the
+`full` pricing strategy.
+
+### GASPRD
+
+In this scenario, GASPRD is produced by a single GASDRV asset. This asset produces 1 unit of GASPRD
+per unit of activity, along with 5.113 units of CO₂ emissions. Suppose an annual utilisation
+of 0.20786471743 (based on dispatch results).
+
+#### Marginal Cost
+
+The asset incurs a CO₂ levy (0.04/unit) and a variable operating cost of 2.0:
+
+\\[
+\text{MarginalCost}_{\text{GASPRD}} = ( 5.113 \times 0.04) + 2.0 = 2.20452
+\\]
+
+#### Full Cost
+
+Using a lifetime of 25 years, a discount rate of 10%, annual fixed costs of 0.3 per unit capacity,
+and a total capital cost of 10 per unit capacity:
+
+\\[
+\text{CRF} = 0.11016807219
+\\]
+
+\\[
+\text{AnnualFixedCostPerCapacity} = (10 \times 0.11016807219 + 0.3) = 1.4016807219
+\\]
+
+Accounting for annual utilisation:
+
+\\[
+\text{AnnualFixedCostPerActivity} = \frac{1.4016807219}{0.20786471743} = 6.7432354044
+\\]
+
+Accounting for the output coefficient:
+
+\\[
+\text{AnnualFixedCostPerOutput} = \frac{6.7432354044}{1} = 6.7432354044
+\\]
+
+The full cost is:
+
+\\[
+\text{FullCost}_{\text{GASPRD}} = 2.20452 + 6.7432354044 = 8.9477554044
+\\]
+
+### GASNAT
+
+Following on from the above scenario, GASNAT is produced by a single GASPRC asset, which consumes
+GASPRD as an input fuel. This asset consumes 1.05 units of GASPRD and produces 1 unit of GASNAT
+per unit of activity, along with 2.5565 units of CO₂ emissions. Suppose an annual utilisation
+of 0.2094885671 (based on dispatch results).
+
+#### Marginal Cost
+
+The asset incurs a CO₂ levy (0.04/unit), a variable operating cost of 0.5 and fuel purchases:
+
+\\[
+\text{CO2Levy} = 2.5565 \times 0.04 = 0.10226
+\\]
+
+\\[
+\text{InputPurchases} = 8.9477554044 \times 1.05 = 9.39514317462
+\\]
+
+\\[
+\text{MarginalCost}_{\text{GASNAT}} = 0.10226 + 0.5 + 9.39514317462 = 9.99740317462
+\\]
+
+#### Full Cost
+
+Using the same lifetime and discount rate, along with annual fixed costs of 0.21 per unit capacity,
+and a total capital cost of 7 per unit capacity:
+
+\\[
+\text{CRF} = 0.11016807219
+\\]
+
+\\[
+\text{AnnualFixedCostPerCapacity} = (7 \times 0.11016807219 + 0.21) = 0.98117650533
+\\]
+
+Accounting for annual utilisation:
+
+\\[
+\text{AnnualFixedCostPerActivity} = \frac{0.98117650533}{0.2094885671} = 4.6836756722
+\\]
+
+Accounting for the output coefficient:
+
+\[
+\text{AnnualFixedCostPerOutput} = \frac{4.6836756722}{1} = 4.6836756722
+\]
+
+The full cost is:
+
+\\[
+\text{FullCost}_{\text{GASNAT}} = 9.99740317462 + 4.6836756722 = 14.6810788468
+\\]
+
+This example illustrates the sequential nature of price calculation: the GASPRD price is
+calculated first and then used as an input cost when calculating the GASNAT price.
+
+[Capital Recovery Factor]: https://homerenergy.com/products/pro/docs/latest/capital_recovery_factor.html
+[`commodities.csv`]: ../file_formats/input_files.md#commoditiescsv