Build a nation’s pathway: a multi-region, multi-decade model

Write Prairieland’s five regions to 2050 as one linear program with vintages, residual value, links, a cumulative carbon budget and an import cap, then test it against foresight, shocks and full years of hours

Modeler lesson M13 of Energy Systems, Step by Step: a fictional nation’s five regions to 2050 as one linear program with plant vintages, residual value, transmission, a cumulative carbon budget and an import cap; the budget’s dual as a carbon value; perfect foresight against decade-by-decade decisions; shocks; twelve time slices checked against full years of hours.
  • Modeler · L3
  • 45 min guided · 60 min lab · about 2 h practice
  • After M3 and M8 and A20 and A21

The lesson in one table: Prairieland's pathway, run as one linear program (2025 dollars, illustrative)

Pathway Cost to 2050 (present value) CO₂ 2026–2050 Imports, highest year Full years of hours (2030, 2050; 2025 and 2024 weather)
Central, perfect foresight $401.6 billion 3,600 of 3,600 Mt 32% all 4 checks survive
Central, decade by decade $412.0 billion 3,600 of 3,600 Mt 36% all 4 checks survive
Deep cuts (3,000 Mt), perfect foresight $394.6 billion 3,000 of 3,000 Mt 35% 2 of 4 checks fail (CO₂ more than 10% above the slices)
Central without the peak-hour rule $391.9 billion 3,600 of 3,600 Mt 35% 4 of 4 checks fail (demand unserved, up to 240 hours)

Precomputed with this engine on 7 November 2026. Cost: the supply side and every fuel bought, 2026–2050 in 2025 dollars; end-use equipment left out.

With JavaScript on, you solve each pathway with a real solver in your browser, move the budget and read its dual, compare perfect foresight with decade-by-decade decisions, inject a shock and map near-optimal plans. Prairieland is fictional; its numbers are illustrative.

  • Illustrative: Prairieland, its regions, fleet, links, demand and shocks (prairieland.json); the existing plants' operating costs, the penalties and the peak-hour rule are set for teaching (e4-pathways.json).
  • Based on public data: new plants' costs from EIA's 2026 assumptions, fuel prices from EIA's 2025 data, CO₂ factors from the IPCC, hourly shapes from SPP's 2025 and 2024 grid (EIA-930).
  • Left out of the cost: end-use equipment (heat pumps, cars, boilers): electrification is a choice you set, not one the model makes.
  • Solver: HiGHS (open source), loaded on your first run (about 3.5 MB, once). The full-year checks come from a library precomputed with this engine.
  • Against lesson A20: no pathway here gets below about 2,700 Mt (straight-line counting, central demand). A20 meets 2,600 Mt because its fastest path electrifies buildings and industry years sooner (step 7).

Build it in Python · notebook lab, about 60 minutes, then 120 minutes of practice

The engine's pathway with scipy.optimize.linprog and HiGHS.

  • The LP: vintages and residual value, the slices and their stores, links with losses, fuels bought domestic first, the budget and the import cap.
  • Reading it: the budget's dual as a carbon value, a tighter budget, the run without the peak-hour rule.
  • Practice: the straight line to zero, decade by decade, a shock's re-plan, near-optimal plans, two ways of counting the years.

What it checks: assert cells match the browser's cost, CO₂ by model year, imports, what is built and the budget's dual.

  1. Download the notebook and the three data files; put them in one folder.
  2. Upload all four to Google Colab (File → Upload notebook, then the folder icon), or open the notebook locally with Python 3, NumPy and SciPy 1.9 or later.
  3. Run all cells, then try the practice tasks at the end.

Notebook (.ipynb) Prairieland (JSON) Engine settings (JSON) Precomputed runs (JSON)

What you just did, as diagrams

Vintages
A plant runs in its model year and the later ones its life covers; it pays only for the years up to 2050, the rest is its residual value.
The budget
CO₂ summed over the model years, each with its weight, stays within the budget; its dual is what one more tonne allowed would save.
The import cap
Crude imports and LNG stay within a share of primary energy in every model year.
Foresight
Perfect foresight plans 2030–2050 at once; decade by decade sees two model years at a time and costs more.
Time
Twelve slices stand for each year; a full year of hours checks which conclusions survive.
Go deeper · ModelerThe pathway in symbols

Regions \(r\) (5), model years \(p\) (5; each stands for the five years ending in it), slices \(s\) (12, with hours \(h_s\)), technologies \(g\), links \(\ell\) (7). The rows below are selected and schematic; money is in millions of 2025 dollars ($M).

Symbol What it is Unit Decided or given
\(n_{r,g,q}\) built in region \(r\), technology \(g\), model year \(q\) (the firm option: one national \(n_{q}\)) MW decided, within the build limits
\(k_{u,p}\) capacity of existing fossil unit \(u\) kept open in \(p\) (never rises again) MW decided
\(x_{\cdot,s,p}\) output, battery charge and discharge, link flows in slice \(s\) MW (slice average) decided
\(a_{\ell,q}\) transfer limit added to link \(\ell\) MW decided
\(f_p\) fuels bought: domestic and imported crude, domestic gas and LNG, coal PJ a year decided
\(y_p\), \(i\) the calendar year of model year \(p\) (2030, …, 2050); the real discount rate, as a decimal (5% is 0.05) year; per year given
\(D_p\) \(\sum_{t=y_p-4}^{y_p} (1+i)^{-(t-2025)}\), the discount sum of a model year years given
\(A_g\) capital × the capital recovery factor over the life, plus fixed O&M $ per kW a year given (fixed ranges)
\(L_{r,p}\), \(d_s\) average busbar load (the region’s use plus 6% grid losses, at the plants’ outlets); the slice’s demand factor MW; ratio given (the plan)
\(c_f\) the price of fuel \(f\) (domestic gas, LNG, coal, crude oil), on net calorific values $ per GJ given (fixed ranges)
\(\alpha_{g,s}\) availability of plant type \(g\) in slice \(s\) (wind and solar: that slice’s output per MW) share given
\(\lambda_\ell\) the share of a link’s flow lost on the way (1% per 100 km) share given
\(x^{\text{dis}}\), \(x^{\text{ch}}\), \(u_{r,s,p}\) battery discharge and charge; electricity not served MW (slice average) decided
\(w_p\) the budget’s weight of each model year (2, 5, 5, 5, 5, 3 with straight lines between) years given

\[\min \sum_p D_p \Big(\sum_f c_f f_p + \text{O\&M}_p\Big) + \sum_{r,g,q} \frac{A_g}{1000}\, n_{r,g,q} \sum_{p \ge q,\; y_p - y_q < \text{life}_g} D_p + \text{links} + \text{penalties} + \text{ties}\]

Fuel \(f_p\) in PJ times \(c_f\) in $ per GJ is $M (1 PJ is a million GJ); \(A_g\) in $ per kW-year times MW, divided by 1,000, is $M a year. Penalties and the tie rule are never part of a reported cost.

\[\sum_g x_{g,r,s,p} + x^{\text{dis}}_{r,s,p} - x^{\text{ch}}_{r,s,p} + \sum_{\ell \to r} (1 - \lambda_\ell)\, x_{\ell,s,p} - \sum_{r \to \ell} x_{\ell,s,p} + u_{r,s,p} = L_{r,p}\, d_s\]

\[x_{g,r,s,p} \le \alpha_{g,s}\Big(K^{\text{exist}}_{g,r,p} + \sum_{q \le p,\; y_p - y_q < \text{life}_g} n_{r,g,q}\Big)\]

\[\sum_p w_p\, \text{CO}_2(f_p) + w_{2025}\cdot 214.9 \le B \qquad \text{imports}(f_p) \le \text{cap} \cdot \text{primary}_p \quad \forall p\]

  • The dual as a price. The budget’s dual \(\mu\) is in present-value dollars a tonne. As a price in model year \(p\)’s own dollars it is \(\mu\, w_p / D_p\); only in a model of every year, each counted once, is it \(\mu (1+i)^{t-2025}\).
  • Batteries. They cycle within each representative day; the scarce-wind week’s store may enter full and run down over seven days.
  • Stress hours. At two stress hours of 2025, each region’s load plus 10% must be covered by available capacity and the links.
Go deeper · ResearcherWhere this model stops
  • Electrification is set by you, not chosen: without end-use equipment costs, the model would electrify as fast as allowed, and a plan that electrifies more looks cheaper than it is.
  • No clean fuels for the oil and gas that stay: deep budgets are out of reach here, not necessarily in the world.
  • Twelve slices smooth the sunniest, windiest and tightest hours; the full-year check shows how much (spilled energy several times higher).
  • Perfect foresight is a benchmark, not a forecast: real decisions are made decade by decade, by many actors.
  • Plans within a millionth of the least cost can build a gigawatt more or less: report what they share, or say which plan you show (near-optimal methods map them).

Check yourself

A wind farm built in 2045 should be charged its whole cost in a model that ends in 2050.

  • Myth
  • Fact
Answer

Myth. It serves for decades after 2050; charging only its payments to 2050 leaves the rest as residual value, so late builds are not penalised.

Under a cumulative budget, one tonne cut in any year of 2026–2050 has the same value in present-value dollars.

  • Myth
  • Fact
Answer

Fact. Every tonne uses the same budget. In a model of every year its price in each year's dollars then rises at the discount rate; here the model years' weights bend that path (step 6).

If twelve slices serve every MWh, so will the full year.

  • Myth
  • Fact
Answer

Myth. Without the peak-hour rule, the central plan leaves demand unserved for up to 240 hours in 2024's weather: slice averages hide the tightest hours.

Numbers, sources and what’s next

Inputs the model uses

Input (2025 dollars) Default Range you can set (set for teaching) Source, and the source's own range
Wind farm, capital $1,712/kW (EIA Table 3) $1,500/kW to $2,400/kW EIA AEO2026 Table 4, its regions: $1,478/kW to $4,335/kW
Solar farm, capital $1,484/kW (EIA Table 3) $1,350/kW to $1,900/kW EIA AEO2026 Table 4, its regions: $1,342/kW to $1,949/kW
Battery (4 hours, 85% round trip), capital $1,521/kW (EIA Table 3) $1,380/kW to $1,690/kW NREL's 2025 battery update (low) and EIA AEO2026 Table 4 (high): $1,371/kW to $1,698/kW
Gas combined cycle, capital $1,032/kW (EIA Table 3) $850/kW to $1,400/kW EIA AEO2026 Table 4, its regions: $830/kW to $1,445/kW
Gas turbine, capital $1,158/kW (EIA Table 3) $950/kW to $1,580/kW EIA AEO2026 Table 4, its regions: $946/kW to $1,585/kW
Generic firm clean plant, capital $7,000/kW (set for teaching) $3,000/kW to $11,000/kW EIA AEO2026 Table 4, firm low-carbon plants that burn no fuel: $2,685/kW to $11,568/kW
New transmission $1,500 per MW-km (set for teaching) $800 per MW-km to $3,000 per MW-km set for teaching
Domestic gas $4.05/GJ (EIA, 2025: $3.85 per MMBtu, on net calorific values) $2.00/GJ to $9.00/GJ EIA Electric Power Annual, Table 7.4
LNG imports $10.00/GJ (set for teaching) $6.00/GJ to $30.00/GJ set for teaching
Coal (Ridge County) $2.41/GJ (EIA, 2025: $2.42 per MMBtu, on net calorific values) $1.50/GJ to $4.00/GJ EIA Electric Power Annual, Table 7.4
Crude oil $12.00/GJ (set for teaching: about $70 a barrel) $7.00/GJ to $22.00/GJ set for teaching
CO₂ factors: crude oil, gas, coal 73.3, 56.1, 94.6 t per TJ fixed IPCC 2006 Guidelines, Volume 2, net calorific values
Build limits, a region a period wind 6, solar 8, batteries 6, gas 4 + 4, firm 2 GW fixed set for teaching: one generic rule, no technology's own timeline
Discount rate (real) 5% (set for teaching) 2% to 8% set for teaching
Hourly shapes SPP's 2025 grid (slices); 2024 for the independent year fixed EIA Hourly Electric Grid Monitor (Form EIA-930)

Keep going

Back to the course map

Related lab: a SciPy/HiGHS notebook that rebuilds engine E4's pathway LP and asserts the browser's numbers, about 60 minutes, then 120 minutes of practice.

Built on and used in

  • M3 (capacity expansion on representative weeks) and M8 (deciding before uncertainty is known).
  • The same engine runs the Pathway Explorer and the capstone C4.
  • Open tools for national pathway models: PyPSA, GenX, JuMP and Pyomo.

Sources and further reading

Spotted an error, or stuck? Write to hlli@ksu.edu. Corrections are made in place and dated here.

Text licensed CC BY 4.0, code MIT: teachers and students may reuse and adapt it, with credit to Honglin Li.