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, its solver is GLPK, and neither shares a line of code, a data model or a language family with anything here — which is the strongest independence a port can have.

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

Objective

\[\min \sum_{t \in \mathcal{T},\enspace v \in \mathcal{V}} \mathit{build}_{t,v} \cdot \mathit{build}^{\mathrm{cost}}_{t,v} + \sum_{t \in \mathcal{T},\enspace y \in \mathcal{Y}} \left( \sum_{v \in \mathcal{V}} \mathit{build}_{t,v} \cdot \mathit{still\_live}_{t,y,v} \right) \cdot \mathit{holding\_cost}_{t,y} + \sum_{t \in \mathcal{T},\enspace i \in \mathcal{I},\enspace m \in \mathcal{M},\enspace y \in \mathcal{Y}} \mathit{activity}_{i,t,m,y} \cdot \mathit{running\_cost}_{i,t,m,y} + \mathit{residual\_holding}\]

Subject to

within_capacity

\[\sum_{m \in \mathcal{M}} \mathit{activity}_{i,t,m,y} \le \left( \sum_{v \in \mathcal{V}} \mathit{build}_{t,v} \cdot \mathit{still\_live}_{t,y,v} + \mathit{residual\_capacity}_{t,y} \right) \cdot \mathit{capacity\_available}_{t,i,y} \qquad \forall\thinspace i \in \mathcal{I},\enspace t \in \mathcal{T},\enspace y \in \mathcal{Y}\]

fuel_balance

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

annual_balance

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

capacity_ceiling

\[\sum_{v \in \mathcal{V}} \mathit{build}_{t,v} \cdot \mathit{still\_live}_{t,y,v} + \mathit{residual\_capacity}_{t,y} \le \mathit{max\_capacity}_{t,y} \qquad \forall\thinspace t \in \mathcal{T},\enspace y \in \mathcal{Y}\]

capacity_floor

\[\sum_{v \in \mathcal{V}} \mathit{build}_{t,v} \cdot \mathit{still\_live}_{t,y,v} + \mathit{residual\_capacity}_{t,y} \ge \mathit{min\_capacity}_{t,y} \qquad \forall\thinspace t \in \mathcal{T},\enspace y \in \mathcal{Y}\]

reserve

\[\left( \sum_{t \in \mathcal{T}} \sum_{f \in \mathcal{F}} \sum_{m \in \mathcal{M}} \mathit{activity}_{i,t,m,y} \cdot \mathit{reserve\_demand}_{t,f,m,y} \right) \cdot \mathit{reserve\_margin}_{y} \le \sum_{t \in \mathcal{T}} \left( \sum_{v \in \mathcal{V}} \mathit{build}_{t,v} \cdot \mathit{still\_live}_{t,y,v} + \mathit{residual\_capacity}_{t,y} \right) \cdot \mathit{reserve\_tagged}_{t,y} \qquad \forall\thinspace i \in \mathcal{I},\enspace y \in \mathcal{Y}\]

Variable domains

activity

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

build

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

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

lpspec

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
    foreach: [timeslice, technology, mode, year]
    bounds:
      lower: 0
  build:
    description: how much capacity is built, and when
    foreach: [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
    foreach: [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
    foreach: [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
    foreach: [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:
    foreach: [technology, year]
    expression: capacity <= max_capacity

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

  reserve:
    description: firm capacity exceeds the electricity demand of the moment by the reserve margin
    foreach: [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: >-
    build * build_cost
    + built_capacity * holding_cost
    + 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 — and 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 same shape pypsa_kvl uses for a cycle basis, and building it is arithmetic over years, which is where the ceiling puts it.

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

Not a fair fight, and worth being precise about why: OSeMOSYS's long formulation defines an intermediate variable for every accounting quantity — RateOfProductionByTechnologyByMode alone is one per region × timeslice × technology × mode × fuel × year — and ties each to its definition with an equality. Those are not decisions; they are names for expressions. Substituting them is what any modeller does by hand when the formulation is not being generated, and it is what writing the model as expressions does automatically.

The point is not that 5,124 beats 119,273. It is that both reach 29446.86269, so the substitution is exact, and the smaller model is the one a reader can hold.

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 — and the optimum agreeing to ten digits is what says the omission was safe.

That has a consequence worth recording, because it was the reason this model was first proposed for the corpus. 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 cannot be evidence about grouping one axis several ways. A model that needs it is still wanted.

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 — which, for a model of this size from a stack this far away, is the result.