Model Framework¶

Mathematical Structure¶

The food-opt model is formulated as a mixed integer linear programming (MILP) problem that minimizes total system costs subject to constraints on production capacity, nutritional requirements, and environmental limits. Here, we give a high-level overview over the model structure; a complete listing of equations is outstanding.

Objective Function¶

The objective function minimizes total costs across multiple dimensions:

\[\min \sum_{i} c_i x_i\]

where \(x_i\) are decision variables and \(c_i\) are associated costs including:

Production costs: Crop and livestock production expenses
Trade costs: Transportation costs based on distance and product type
Environmental costs: Emissions priced at configured carbon price (USD/tCO₂-eq)
Health costs: Dietary risk factors valued using years of life lost (YLL) multiplied by the configured health.value_per_yll

Decision Variables¶

The model optimizes the following classes of decision variables:

Crop production (\(P_{c,r,w,k}\)): Production of crop \(c\) in region \(r\) with water supply \(w\) (irrigated/rainfed) and resource class \(k\)
Livestock production (\(L_{a,r,s}\)): Production of animal product \(a\) in region \(r\) using production system \(s\) (grazing/feed-based)
Food processing (\(F_{c,f,r}\)): Processing activities converting crop \(c\) to food product \(f\) in region \(r\)
Land allocation (\(A_{r,w,k}\)): Cropland area allocated in region \(r\), water supply \(w\), resource class \(k\)
Trade flows (\(T_{c,r,h}\), \(T_{c,h,h'}\)): Trade of commodity \(c\) between region \(r\) and hub \(h\), as well as between hubs \(h\) and \(h'\).
Food consumption (\(D_{f,r}\)): Per-capita consumption of food \(f\) in region \(r\)

Constraints¶

The model is subject to multiple constraint categories:

Production Constraints

Crop yields limiting crop production based on region and resource class
Livestock feed requirements and conversion efficiencies
Land availability limits by region and resource class
Water availability constraints by basin and growing season

Nutritional Constraints

Minimum macronutrient requirements (carbohydrates, protein, fat, calories)
Minimum food group consumption (whole grains, fruits, vegetables, etc.)
Per-capita dietary balance across the population

Processing and Trade Constraints

Crop-to-food conversion efficiencies
Hub-based trade network topology
Transport costs differentiated by commodity category
Non-tradable commodities (e.g., fodder crops)

Environmental Constraints

Optional limits on total greenhouse gas emissions
Optional limits on total nitrogen fertilizer application

PyPSA Implementation¶

The model is implemented using PyPSA (Python for Power System Analysis), which provides a flexible framework for optimizing flow networks with linear constraints. While PyPSA was originally designed for energy systems, its component-based structure maps naturally to food system flows.

Network Components¶

Carriers

Define the commodity types flowing through the network (e.g., crop_wheat, food_bread, nutrient_protein). Each carrier has an associated unit (tonnes, megacalories, etc.). See the PyPSA carriers documentation for more details.

Buses

Represent locations where commodities accumulate or are exchanged. Buses are typically defined per-country or per-region (e.g., crop_wheat_USA, land_class0_region42). See the PyPSA buses documentation for more details.

Links

Represent transformations or transport of commodities. See the PyPSA links documentation for more details. Links have:

bus0: Primary input bus
bus1: Primary output bus
efficiency: Conversion efficiency from bus0 → bus1
bus2, bus3, …: Additional input/output legs
efficiency2, efficiency3, …: Efficiencies for additional legs (positive = output, negative = input)

Examples:

Crop production link: inputs = land + water + fertilizer; output = crop
Food processing link: inputs = crops; output = food product
Trade link: input = crop in region A; output = crop in region B (with transport cost)

Stores

Represent resource availability or capacity limits. See the PyPSA stores documentation for more details. Key uses:

Land area available in each region/resource class
Water availability in each basin
Global fertilizer supply limits

Global Constraints

Enforce system-wide limits. See the PyPSA global constraints documentation for more details. Examples include:

Total fertilizer consumption
Total greenhouse gas emissions
Population-level nutritional requirements

Component Naming Conventions¶

The model uses systematic naming conventions to organize components:

Crops: crop_{crop_name}_{country_code}
Foods: food_{food_name}_{country_code}
Nutrients: nutrient_{nutrient_name}_{country_code}
Land: land_class{class_num}_{region_id}
Water: water_basin{basin_id}
Primary resources: primary_fertilizer, primary_water

Multi-Bus Links for Complex Processes¶

Many agricultural processes involve multiple inputs and outputs, which are represented as multi-bus links. For example, crop production:

bus0: Land (input, primary)
bus1: Crop product (output, primary)
bus2: Water (input, with negative efficiency2)
bus3: Fertilizer (input, with negative efficiency3)
bus4: CO₂ emissions (output, with positive efficiency4)

The efficiency parameters capture:

Crop yield per hectare (efficiency on bus0→bus1)
Water requirement per tonne of crop (efficiency2, negative)
Fertilizer requirement per tonne of crop (efficiency3, negative)
CO₂ emissions per tonne of crop (efficiency4, positive)

Resource Flow Structure¶

The model follows a hierarchical flow structure:

Primary resources → Land, water, fertilizer availability in each region
Crop production → Raw agricultural commodities produced on land
Animal production → Livestock products from grassland (grazing) or crops (feed-based)
Food processing → Conversion of crops to food products
Trade → Inter-regional flows via hub networks
Consumption → Aggregation to nutritional outcomes and dietary risk factors
Health impacts → DALYs from dietary exposures

Units and Conversions¶

The model uses consistent units throughout:

Mass

Land area: Mha (million hectares)
Crop/food production: tonnes (t) or megatonnes (Mt)
Nutritional mass (protein, etc.): grams/person/day → Mt/year

Energy

Nutritional energy (calories): kcal/person/day → Mcal (megacalories)/year

Emissions

Greenhouse gases: tCO₂-eq (tonnes CO₂-equivalent)
Conversion factors: CH₄ (28 GWP100), N₂O (265 GWP100)

Water

Water use: km³ (cubic kilometers) or Mm³ (million cubic meters)

Economic

Costs: USD (various sub-units: USD/tonne, USD/km, USD/tCO₂-eq)

Key conversion factors used in the code (workflow/scripts/build_model.py):

TONNE_TO_MEGATONNE = 1e-6
KCAL_TO_MCAL = 1e-6
KCAL_PER_100G_TO_MCAL_PER_TONNE = 1e-2
DAYS_PER_YEAR = 365

Solver Configuration¶

The model supports multiple LP solvers:

HiGHS (default, open-source): Fast interior-point method, suitable for large problems
Gurobi (commercial): Often faster for very large problems, supports advanced solver options

Solver selection and options are configured in config/default.yaml:

solving:
  solver: highs
  # solver: gurobi
  options_gurobi:
    LogToConsole: 0
    OutputFlag: 1
    Method: 2
    MIPGap: 0.001  # target 0.1% relative optimality gap
  options_highs:
    solver: "ipm"
    mip_rel_gap: 0.001  # align relative gap with gurobi setting

Model Scale¶

Typical model dimensions (for the default configuration with 400 regions):

Regions: 400 sub-national optimization regions
Crops: ~70 crop types
Resource classes: 3-4 yield quality classes per region
Variables: ~1-5 million decision variables
Constraints: ~1-10 million constraints
Solve time: Minutes to hours depending on region count and solver

The model scales roughly linearly with the number of regions. Reducing aggregation.regions.target_count in the configuration will decrease solve time at the cost of spatial resolution.