OSeMOSYS — UTOPIA¶

What to build and how hard to run it, 1990–2010, to meet three end-use demands at least discounted cost.

✔ Verified against OSeMOSYS — objective 29446.86269, matched to rtol=1e-09. Asserted upstream in tests/test_gnu_mathprog.py, and re-run here directly on GLPK.

The only optimum in this corpus that comes from outside Python. UTOPIA is the reference system bundled with MARKAL and the case OSeMOSYS validates itself against. Its model is GNU MathProg and its solver is GLPK. Neither shares a line of code, a data model or a language family with anything here.

The model¶

The same model, as math

OSeMOSYS's UTOPIA: what to build and how hard to run it, 1990-2010, to meet three end-use demands at least discounted cost. The reference system bundled with MARKAL. Discounting, the annuity, salvage value and the operational-life window are arithmetic over years, so they are folded into coefficients before the model is built; what is left is the decision. Optimum 29446.86269, from OSeMOSYS itself under GLPK.

Sets¶

Symbol	Meaning
\(\mathcal{T}\)	index \(t\) — `technology` — the plants and processes that may be built and run
\(\mathcal{F}\)	index \(f\) — `fuel` — energy carriers, produced by one technology and consumed by another
\(\mathcal{I}\)	index \(i\) — `timeslice` — the slices a year is dispatched over
\(\mathcal{M}\)	index \(m\) — `mode` — the way a technology is being operated
\(\mathcal{Y}\)	index \(y\) — `year` — the years the pathway covers
\(\mathcal{V}\)	index \(v\) — `vintage` — a second axis over the same years — capacity standing in a year was built in some vintage

Parameters¶

Symbol	Meaning
\(\mathrm{still\_live}\)	`still_live` over \(\mathcal{T} \times \mathcal{Y} \times \mathcal{V}\) — 1 where a vintage is still inside its technology's operational life in that year
\(\mathrm{residual\_capacity}\)	`residual_capacity` over \(\mathcal{T} \times \mathcal{Y}\) — capacity that already stood in 1990 and has not yet retired
\(\mathrm{build}^{\mathrm{cost}}\)	`build_cost` over \(\mathcal{T} \times \mathcal{V}\) — discounted cost of building a unit of capacity in a vintage
\(\mathrm{holding\_cost}\)	`holding_cost` over \(\mathcal{T} \times \mathcal{Y}\) — discounted fixed cost of holding a unit of capacity through a year
\(\mathrm{running\_cost}\)	`running_cost` over \(\mathcal{I} \times \mathcal{T} \times \mathcal{M} \times \mathcal{Y}\) — discounted variable cost of a unit of activity
\(\mathrm{year\_split}\)	`year_split` over \(\mathcal{I} \times \mathcal{Y}\) — share of the year a timeslice stands for
\(\mathrm{capacity}^{\mathrm{available}}\)	`capacity_available` over \(\mathcal{T} \times \mathcal{I} \times \mathcal{Y}\) — share of its capacity a technology can offer in a timeslice
\(\mathrm{input\_ratio}\)	`input_ratio` over \(\mathcal{T} \times \mathcal{F} \times \mathcal{M} \times \mathcal{Y}\) — fuel a technology consumes per unit of activity
\(\mathrm{output\_ratio}\)	`output_ratio` over \(\mathcal{T} \times \mathcal{F} \times \mathcal{M} \times \mathcal{Y}\) — fuel a technology produces per unit of activity
\(\mathrm{sliced\_demand}\)	`sliced_demand` over \(\mathcal{F} \times \mathcal{I} \times \mathcal{Y}\) — demand for a fuel placed on one timeslice
\(\mathrm{annual\_demand}\)	`annual_demand` over \(\mathcal{F} \times \mathcal{Y}\) — demand for a fuel placed on the year as a whole
\(\mathrm{max\_capacity}\)	`max_capacity` over \(\mathcal{T} \times \mathcal{Y}\) — most capacity a technology may stand at
\(\mathrm{min\_capacity}\)	`min_capacity` over \(\mathcal{T} \times \mathcal{Y}\) — least capacity a technology must stand at
\(\mathrm{reserve\_margin}\)	`reserve_margin` over \(\mathcal{Y}\) — how far firm capacity must exceed the demand of the moment
\(\mathrm{reserve\_tagged}\)	`reserve_tagged` over \(\mathcal{T} \times \mathcal{Y}\) — 1 where a technology's capacity counts towards the reserve
\(\mathrm{reserve\_demand}\)	`reserve_demand` over \(\mathcal{T} \times \mathcal{F} \times \mathcal{M} \times \mathcal{Y}\) — the activity the reserve margin is measured against
\(\mathrm{residual\_holding}\)	`residual_holding` (scalar) — fixed operating cost owed on the capacity that already stood in 1990

Variables¶

Symbol	Meaning
\(\mathit{activity}\)	`activity` over \(\mathcal{I} \times \mathcal{T} \times \mathcal{M} \times \mathcal{Y}\) — how hard a technology runs, per timeslice and mode
\(\mathit{build}\)	`build` over \(\mathcal{T} \times \mathcal{V}\) — how much capacity is built, and when

Definitions¶

Symbol	Meaning
\(\mathit{built\_capacity}\)	`built_capacity` over \(\mathcal{T} \times \mathcal{Y}\) — capacity standing in a year from every vintage still inside its life. A plant's life is read from data and differs by technology, so the window cannot be a fixed shift — it is an incidence table, the shape the KVL port uses for a cycle basis.
\(\mathit{capacity}\)	`capacity` over \(\mathcal{T} \times \mathcal{Y}\) — all the capacity standing in a year, including what was already there in 1990

Upright is what the data supplies — a parameter such as \(\mathrm{still\_live}\), a coordinate map, a label — and italic is what the solver chooses, such as \(\mathit{activity}\). An index is italic too, being what a quantifier chooses, and a set is script.

Objective¶

\[ \min \sum_{t \in \mathcal{T},\ v \in \mathcal{V}} \mathit{build}_{t,v} \cdot \mathrm{build}^{\mathrm{cost}}_{t,v} + \sum_{t \in \mathcal{T},\ y \in \mathcal{Y}} \mathit{built\_capacity}_{t,y} \cdot \mathrm{holding\_cost}_{t,y} + \sum_{t \in \mathcal{T},\ i \in \mathcal{I},\ m \in \mathcal{M},\ y \in \mathcal{Y}} \mathit{activity}_{i,t,m,y} \cdot \mathrm{running\_cost}_{i,t,m,y} + \mathrm{residual\_holding} \]

Subject to¶

within_capacity

\[ \sum_{m \in \mathcal{M}} \mathit{activity}_{i,t,m,y} \le \mathit{capacity}_{t,y} \cdot \mathrm{capacity}^{\mathrm{available}}_{t,i,y} \qquad \forall\, i \in \mathcal{I},\ t \in \mathcal{T},\ y \in \mathcal{Y} \]

fuel_balance

\[ \left( \sum_{t \in \mathcal{T}} \sum_{m \in \mathcal{M}} \mathit{activity}_{i,t,m,y} \cdot \mathrm{output\_ratio}_{t,f,m,y} \right) \cdot \mathrm{year\_split}_{i,y} \ge \mathrm{sliced\_demand}_{f,i,y} + \left( \sum_{t \in \mathcal{T}} \sum_{m \in \mathcal{M}} \mathit{activity}_{i,t,m,y} \cdot \mathrm{input\_ratio}_{t,f,m,y} \right) \cdot \mathrm{year\_split}_{i,y} \qquad \forall\, i \in \mathcal{I},\ f \in \mathcal{F},\ y \in \mathcal{Y} \]

annual_balance

\[ \sum_{i \in \mathcal{I}} \sum_{t \in \mathcal{T}} \sum_{m \in \mathcal{M}} \mathit{activity}_{i,t,m,y} \cdot \mathrm{output\_ratio}_{t,f,m,y} \cdot \mathrm{year\_split}_{i,y} \ge \mathrm{annual\_demand}_{f,y} + \sum_{i \in \mathcal{I}} \sum_{t \in \mathcal{T}} \sum_{m \in \mathcal{M}} \mathit{activity}_{i,t,m,y} \cdot \mathrm{input\_ratio}_{t,f,m,y} \cdot \mathrm{year\_split}_{i,y} \qquad \forall\, f \in \mathcal{F},\ y \in \mathcal{Y} \]

capacity_ceiling

\[ \mathit{capacity}_{t,y} \le \mathrm{max\_capacity}_{t,y} \qquad \forall\, t \in \mathcal{T},\ y \in \mathcal{Y} \]

capacity_floor

\[ \mathit{capacity}_{t,y} \ge \mathrm{min\_capacity}_{t,y} \qquad \forall\, t \in \mathcal{T},\ y \in \mathcal{Y} \]

reserve

\[ \left( \sum_{t \in \mathcal{T}} \sum_{f \in \mathcal{F}} \sum_{m \in \mathcal{M}} \mathit{activity}_{i,t,m,y} \cdot \mathrm{reserve\_demand}_{t,f,m,y} \right) \cdot \mathrm{reserve\_margin}_{y} \le \sum_{t \in \mathcal{T}} \mathit{capacity}_{t,y} \cdot \mathrm{reserve\_tagged}_{t,y} \qquad \forall\, i \in \mathcal{I},\ y \in \mathcal{Y} \]

Definitions¶

built_capacity

\[ \mathit{built\_capacity}_{t,y} = \sum_{v \in \mathcal{V}} \mathit{build}_{t,v} \cdot \mathrm{still\_live}_{t,y,v} \qquad \forall\, t \in \mathcal{T},\ y \in \mathcal{Y} \]

capacity

\[ \mathit{capacity}_{t,y} = \mathit{built\_capacity}_{t,y} + \mathrm{residual\_capacity}_{t,y} \qquad \forall\, t \in \mathcal{T},\ y \in \mathcal{Y} \]

Variable domains¶

activity

\[ \mathit{activity}_{i,t,m,y} \ge 0 \qquad \forall\, i \in \mathcal{I},\ t \in \mathcal{T},\ m \in \mathcal{M},\ y \in \mathcal{Y} \]

build

\[ \mathit{build}_{t,v} \ge 0 \qquad \forall\, t \in \mathcal{T},\ v \in \mathcal{V} \]

The tabs start from the instance's tables — one frame per parameter.

specsolve

description: >-
  OSeMOSYS's UTOPIA: what to build and how hard to run it, 1990-2010, to meet
  three end-use demands at least discounted cost. The reference system bundled
  with MARKAL. Discounting, the annuity, salvage value and the operational-life
  window are arithmetic over years, so they are folded into coefficients before
  the model is built; what is left is the decision. Optimum 29446.86269, from
  OSeMOSYS itself under GLPK.

dimensions:
  technology:
    description: the plants and processes that may be built and run
    dtype: str
  fuel:
    description: energy carriers, produced by one technology and consumed by another
    dtype: str
  timeslice:
    description: the slices a year is dispatched over
    dtype: str
  mode:
    description: the way a technology is being operated
    dtype: int
  year:
    description: the years the pathway covers
    dtype: int
  vintage:
    description: >-
      a second axis over the same years — capacity standing in a year was built
      in some vintage
    dtype: int

parameters:
  still_live:
    description: 1 where a vintage is still inside its technology's operational life in that year
    dims: [technology, year, vintage]
  residual_capacity:
    description: capacity that already stood in 1990 and has not yet retired
    dims: [technology, year]

  build_cost:
    description: discounted cost of building a unit of capacity in a vintage
    dims: [technology, vintage]
  holding_cost:
    description: discounted fixed cost of holding a unit of capacity through a year
    dims: [technology, year]
  running_cost:
    description: discounted variable cost of a unit of activity
    dims: [timeslice, technology, mode, year]

  year_split:
    description: share of the year a timeslice stands for
    dims: [timeslice, year]
  capacity_available:
    description: share of its capacity a technology can offer in a timeslice
    dims: [technology, timeslice, year]
  input_ratio:
    description: fuel a technology consumes per unit of activity
    dims: [technology, fuel, mode, year]
  output_ratio:
    description: fuel a technology produces per unit of activity
    dims: [technology, fuel, mode, year]

  sliced_demand:
    description: demand for a fuel placed on one timeslice
    dims: [fuel, timeslice, year]
  annual_demand:
    description: demand for a fuel placed on the year as a whole
    dims: [fuel, year]

  max_capacity:
    description: most capacity a technology may stand at
    dims: [technology, year]
  min_capacity:
    description: least capacity a technology must stand at
    dims: [technology, year]

  reserve_margin:
    description: how far firm capacity must exceed the demand of the moment
    dims: [year]
  reserve_tagged:
    description: 1 where a technology's capacity counts towards the reserve
    dims: [technology, year]
  reserve_demand:
    description: the activity the reserve margin is measured against
    dims: [technology, fuel, mode, year]

  residual_holding:
    description: fixed operating cost owed on the capacity that already stood in 1990
    dims: []

variables:
  activity:
    description: how hard a technology runs, per timeslice and mode
    dims: [timeslice, technology, mode, year]
    bounds:
      lower: 0
  build:
    description: how much capacity is built, and when
    dims: [technology, vintage]
    bounds:
      lower: 0

expressions:
  built_capacity:
    expression: sum(build * still_live, over=vintage)
    description: >-
      capacity standing in a year from every vintage still inside its life. A
      plant's life is read from data and differs by technology, so the window
      cannot be a fixed shift — it is an incidence table, the shape the KVL port
      uses for a cycle basis.
  capacity:
    expression: built_capacity + residual_capacity
    description: all the capacity standing in a year, including what was already there in 1990

constraints:
  within_capacity:
    description: a technology cannot run beyond the capacity standing that year
    dims: [timeslice, technology, year]
    expression: sum(activity, over=mode) <= capacity * capacity_available

  fuel_balance:
    description: >-
      every fuel balances in every timeslice — what is produced covers the
      demand placed on it plus what other technologies consume
    dims: [timeslice, fuel, year]
    expression: >-
      sum(sum(activity * output_ratio, over=mode), over=technology) * year_split
      >= sliced_demand
      + sum(sum(activity * input_ratio, over=mode), over=technology) * year_split

  annual_balance:
    description: and balances again over the year, for demands that are not sliced
    dims: [fuel, year]
    expression: >-
      sum(sum(sum(activity * output_ratio * year_split, over=mode), over=technology), over=timeslice)
      >= annual_demand
      + sum(sum(sum(activity * input_ratio * year_split, over=mode), over=technology), over=timeslice)

  capacity_ceiling:
    dims: [technology, year]
    expression: capacity <= max_capacity

  capacity_floor:
    dims: [technology, year]
    expression: capacity >= min_capacity

  reserve:
    description: firm capacity exceeds the electricity demand of the moment by the reserve margin
    dims: [timeslice, year]
    expression: >-
      sum(sum(sum(activity * reserve_demand, over=mode), over=fuel), over=technology) * reserve_margin
      <= sum(capacity * reserve_tagged, over=technology)

objective:
  sense: minimize
  description: discounted cost of building, holding and running the system over the pathway
  expression: >-
    sum(build * build_cost)
    + sum(built_capacity * holding_cost)
    + sum(activity * running_cost)
    + residual_holding

What the port had to decide¶

An operational life is a window read from data, and that is an incidence table. Capacity standing in a year is every vintage still inside its technology's life. The lives differ, so this is not a fixed shift. The port carries still_live[technology, year, vintage], one row per pair that is still live, and the standing capacity is a contraction against it. That is the shape pypsa_kvl uses for a cycle basis. Building it is arithmetic over years, which the limits count as data preparation.

Discounting, the annuity and salvage value never reach the model. OSeMOSYS spends four parameters and four constraint families on them: CapitalRecoveryFactor, PvAnnuity, DiscountFactor, and SV1–SV4 with three depreciation cases. Every one is a function of the year and the technology, so all of it folds into a single coefficient per (technology, vintage). What survives into the model is the decision: how much to build, and when.

The one piece that cannot fold is the fixed cost owed on capacity that already stood in 1990. It is owed whatever the model chooses, so it enters the objective as a constant.

Same answer, a twenty-third of the model¶

	rows	columns
OSeMOSYS, as generated by GLPK	119,273	147,171
this port	5,124	5,733

OSeMOSYS's long formulation defines an intermediate variable for every accounting quantity and ties each to its definition with an equality. RateOfProductionByTechnologyByMode alone is one per region × timeslice × technology × mode × fuel × year. Those are names for expressions, not decisions, and writing the model as expressions substitutes them. Both reach 29446.86269, so the substitution is exact.

What this port does not carry¶

Storage. UTOPIA declares a reservoir and OSeMOSYS carries fifteen constraints for it, but the instance builds none. NewStorageCapacity is empty in the reference solution, as is Trade. Their constraints are satisfied at zero, so the port omits them. The optimum agreeing to ten digits says the omission was safe.

Conversionls, Conversionld and Conversionlh, the three maps from a timeslice to its season, day type and daily time bracket, appear only in those storage constraints. With storage inert they read nothing, so this port is no evidence about grouping one axis several ways.

What it exercises¶

A window read from data as an incidence table; a second axis over the same years joined to the first; and a cost chain that is entirely data preparation. No construct here is new.