4  Time resolution

Chapter 1 lived in a single year, a single slice, a single region — enough to see the energy balance and nothing to distract from it. This chapter adds time: a horizon of several years to invest across, a discount rate to weigh the future, hours and seasons within the year, ageing capacity (vintages), and the two things that only make sense once time is resolved — storage and intermittent renewables.

We keep one region (R1) throughout; regions arrive in Chapter 5.

source("R/workshop-model.R")
library(dplyr) # filter(), select(), joins used in the exercises

Outputs shown below are illustrative. With eval: false set for the book, nothing runs at render time — copy each chunk into your own session to see the real numbers, which is also how you check the ones in the prose.

4.1 A model that spans years

So far the “horizon” was one year. newHorizon() describes a study period and how it is divided into intervals, each solved as one representative milestone year. Investment decisions are made once per milestone and carried forward.

ELC <- newCommodity(
  name      = "ELC",
  desc      = "Electricity",
  unit      = "PJ",
  timeframe = "ANNUAL"
)

COA <- newCommodity(
  name      = "COA",
  desc      = "Coal",
  unit      = "PJ",
  timeframe = "ANNUAL",
  emis      = data.frame(comm = "CO2", unit = "kt/PJ", emis = 95)
)

GAS <- newCommodity(
  name      = "GAS",
  desc      = "Natural gas",
  unit      = "PJ",
  timeframe = "ANNUAL",
  emis      = data.frame(comm = "CO2", unit = "kt/PJ", emis = 56)
)

CO2 <- newCommodity(
  name      = "CO2",
  desc      = "Carbon dioxide",
  unit      = "kt",
  timeframe = "ANNUAL"
)

SUP_COA <- newSupply(
  name      = "SUP_COA",
  commodity = "COA",
  unit      = "PJ",
  supply    = data.frame(cost = 2.5)
)

SUP_GAS <- newSupply(
  name      = "SUP_GAS",
  commodity = "GAS",
  unit      = "PJ",
  supply    = data.frame(cost = 6.0)
)

# demand grows from 50 PJ (2025) to 80 PJ (2050); energyRt interpolates between
DEM_ELC <- newDemand(
  name      = "DEM_ELC",
  commodity = "ELC",
  unit      = "PJ",
  demand    = data.frame(year = c(2025, 2050), demand = c(50, 80))
)

ECOA <- newTechnology(
  name    = "ECOA",
  desc    = "Coal plant",
  input   = list(comm = "COA", unit = "PJ", combustion = 1),
  output  = list(comm = "ELC", unit = "PJ"),
  ceff    = data.frame(comm = "COA", cinp2use = 0.40),
  invcost = list(invcost = 2000),
  fixom   = 55,
  cap2act = 31.536,
  olife   = 30L
)

EGAS <- newTechnology(
  name    = "EGAS",
  desc    = "Gas plant",
  input   = list(comm = "GAS", unit = "PJ", combustion = 1),
  output  = list(comm = "ELC", unit = "PJ"),
  ceff    = data.frame(comm = "GAS", cinp2use = 0.58),
  invcost = list(invcost = 900),
  fixom   = 25,
  cap2act = 31.536,
  olife   = 25L
)

draw(ECOA) # each technology's reference-energy-system diagram: coal in, ELC out
draw(EGAS)

repo_YR <- newRepository(
  "repo_YR",
  ELC, COA, GAS, CO2,
  SUP_COA, SUP_GAS, DEM_ELC, ECOA, EGAS
)

The horizon below begins with a one-year base period and adds three ten-year intervals. autoplot() shows the milestone years it produces.

horizon_2025_2050_by10 <- newHorizon(
  period    = 2025:2055,
  intervals = c(1, 10, 10, 10)
)

autoplot(horizon_2025_2050_by10)

solve_model() does in one step what you did by hand in Chapter 1 — it expands the description over years and slices and sends it to the solver.

mod_YR <- newModel(
  name     = "YR",
  data     = repo_YR,
  region   = "R1",
  discount = WS_DISCOUNT,
  horizon  = horizon_2025_2050_by10
)

scen_YR <- solve_model(mod_YR)

getData(scen_YR, "vTechCap", merge = TRUE)   # capacity by technology and milestone
getData(scen_YR, "vObjective", merge = TRUE) # total discounted cost, MEUR

A handful of milestone years — not one per calendar year — keeps the model small while still spanning several decades of investment.

2.1 — Temporal aggregation and problem size. Re-solve the model with a fully annual horizon, newHorizon(period = 2025:2055), in which each calendar year constitutes its own period. Compare the resulting problem size (model_size()) and objective value with those of the milestone formulation. Assess whether the additional temporal detail materially alters the investment path, and whether it justifies the increase in problem dimension.

TipSolution 2.1

horizon_annual <- newHorizon(period = 2025:2055)

mod_annual <- newModel(
  name     = "YR_annual",
  data     = repo_YR,
  region   = "R1",
  discount = WS_DISCOUNT,
  horizon  = horizon_annual
)

scen_annual <- solve_model(mod_annual)

model_size(scen_YR)                             # four milestones
model_size(scen_annual)                         # one period per year

getData(scen_YR, "vObjective", merge = TRUE)
getData(scen_annual, "vObjective", merge = TRUE)

The annual formulation is several times larger, yet the investment path is nearly identical: with smooth, monotone demand there is little that a yearly grid captures which four milestones miss. Milestones are the default precisely because they recover most of the accuracy at a fraction of the size. The detail that does matter — hours and seasons — lies within the year, and occupies the remainder of this chapter.

4.2 Discounting: business vs social planner

Spanning years forces a question Chapter 1 could ignore: a euro spent in 2050 is not a euro spent today. energyRt uses two rates, and it is worth keeping them apart:

• SDR (social discount rate) — discounts the stream of system costs in the objective. This is the planner’s view of the future.
• WACC (weighted average cost of capital) — annuitises a lump investment into an equivalent annual cost (EAC). This is how expensive capital is to the entity that builds.

The discount = argument is a shorthand that sets both rates equal (discount = 0.05 ⇒ wacc = sdr = 0.05). A social planner adopts one low rate over each asset’s full life. A private investor applies a higher WACC and often recovers capital over a shorter payback than the asset operates.

# social planner: one low rate, capital recovered over the full technical life
mod_soc <- newModel(
  name     = "SOC",
  data     = repo_YR,
  region   = "R1",
  discount = 0.03,
  horizon  = horizon_2025_2050_by10
)

# investor: split rates, and the coal plant must repay its capital in 15 years
ECOA_inv <- update(
  ECOA,
  invcost = list(invcost = 2000, wacc = 0.10, payback = 15)
)

repo_inv <- newRepository(
  "repo_inv",
  ELC, COA, GAS, CO2,
  SUP_COA, SUP_GAS, DEM_ELC, ECOA_inv, EGAS
)

mod_inv <- newModel(
  name     = "INV",
  data     = repo_inv,
  region   = "R1",
  discount = data.frame(wacc = 0.10, sdr = 0.03),
  horizon  = horizon_2025_2050_by10
)

scen_soc <- solve_model(mod_soc)
scen_inv <- solve_model(mod_inv)

getData(scen_soc, "vTechEac", merge = TRUE)  # annualised capital charge
getData(scen_inv, "vTechEac", merge = TRUE)  # higher under investor terms

Same invcost, very different annual capital charge: a 10% WACC recovered over 15 years annuitises far more steeply than 3% over 30 years, so the investor’s coal plant appears dearer and the model leans harder on gas.

The wacc/sdr split works on every backend, but the finite payback feature is, as of energyRt 0.80.x, implemented only on GLPK for testing. Its support in other backends (GAMS, Pyomo, JuMP) is on the way.
Omitting discount entirely leaves both rates at 0 (an undiscounted model), so always set it.
There is no per-technology sdr — the social rate is a property of the model.

The same divergence can be shown without solving, using levcost() (whose discount argument is the WACC).

2.2 — The cost of capital and technology choice. A photovoltaic plant is almost entirely capital and carries no fuel cost; a gas peaker is the converse. Using levcost(), compute the levelised cost of a capital-intensive generator under (a) the social-planner assumption (3%, full technical life) and (b) the investor assumption (10%, 15-year payback). Determine which technology’s LCOE is more sensitive to the cost of capital, and explain how that sensitivity shifts the least-cost generation mix toward or away from renewables.

TipSolution 2.2

# a capital-heavy generator: high invcost, no fuel
SOLARish <- newTechnology(
  name    = "SOLARish",
  desc    = "Capital-heavy generator",
  input   = list(comm = "SOL", unit = "PJ"),
  output  = list(comm = "ELC", unit = "PJ"),
  ceff    = data.frame(comm = "SOL", cinp2use = 1),
  invcost = list(invcost = meur_gw(650)),
  fixom   = 12,
  cap2act = 31.536,
  olife   = 25L
)

lc_soc <- levcost(SOLARish, discount = 0.03) # planner

SOLAR_inv <- update(
  SOLARish,
  invcost = list(invcost = meur_gw(650), wacc = 0.10, payback = 15)
)

lc_inv <- levcost(SOLAR_inv, discount = 0.10) # investor

lc_soc$levcost_npv
lc_inv$levcost_npv

The capital-intensive plant’s LCOE rises sharply under investor assumptions — most of its lifetime cost is the annuity, and the annuity is precisely what the discount rate acts upon. A gas peaker scarcely moves, because most of its cost is fuel, which no discount rate touches. A high cost of capital is therefore a quiet thumb on the scale against renewables and in favour of fuel-burning plant: the business case and the social case can rank the same technologies differently from identical physics.

4.3 Within the year: seasons and hours

A commodity balanced on ANNUAL need only add up over the whole year — a solar panel could “supply” midnight demand. Real electricity balances every hour. A calendar divides the year into time slices; a commodity’s timeframe states which level it balances on.

WS_CAL # calendars$utopia_s4h24: 4 seasons x 24 h = 96 slices

head(WS_CAL@timeslice_share) # each slice's share of the year

autoplot(WS_CAL) # the calendar's nested season / hour structure

Calendars come from the timeslices package; its successor timescales generalises them to arbitrary nested timeframes, and its spatial twin geoscales does the same for regions — the time and space “bricks” of the optimal2050 ecosystem that Chapter 5 builds on.

To balance electricity hourly, set timeframe = "HOUR" on ELC and attach the calendar to the model.

ELC_h <- update(ELC, timeframe = "HOUR")

2.3 — Temporal resolution and the visibility of storage. Solve the electricity model on three calendars of increasing resolution — calendars$utopia_annual (1 slice), calendars$utopia_seasons (12), and calendars$utopia_s4h24 (96) — supplied through newModel(..., calendar = ). Observe how the problem size grows, and explain why the value of storage is invisible at annual resolution and becomes apparent only once the day acquires an internal shape.

TipSolution 2.3

mod_annual_cal <- newModel(
  name     = "RES_annual",
  data     = repo_YR,
  region   = "R1",
  discount = WS_DISCOUNT,
  calendar = calendars$utopia_annual,
  horizon  = horizon_2025_2050_by10
)

mod_seasons <- newModel(
  name     = "RES_seasons",
  data     = repo_YR,
  region   = "R1",
  discount = WS_DISCOUNT,
  calendar = calendars$utopia_seasons,
  horizon  = horizon_2025_2050_by10
)

mod_s4h24 <- newModel(
  name     = "RES_s4h24",
  data     = repo_YR,
  region   = "R1",
  discount = WS_DISCOUNT,
  calendar = calendars$utopia_s4h24,
  horizon  = horizon_2025_2050_by10
)

scen_annual_cal <- solve_model(mod_annual_cal)
scen_seasons    <- solve_model(mod_seasons)
scen_s4h24      <- solve_model(mod_s4h24)

model_size(scen_annual_cal)
model_size(scen_seasons)
model_size(scen_s4h24)

getData(scen_annual_cal, "vObjective", merge = TRUE)
getData(scen_seasons, "vObjective", merge = TRUE)
getData(scen_s4h24, "vObjective", merge = TRUE)

More slices, more rows — and a model able to distinguish noon from midnight. On an annual calendar a battery has no intraday gap to arbitrage, so the optimiser never builds one; the value it contributes in Section 4.5 cannot appear until the calendar carries sub-daily structure. Resolution is not decoration: it determines which technologies can even be seen to be useful.

4.4 Vintages: one technology, evolving parameters

A coal plant commissioned in 2040 is more efficient than one built in 2020 — yet it is still the coal-plant technology, not a new one. Vintages are how a single technology carries parameters that depend on the year it was built: efficiency, investment cost, fixed O&M, even operating life can each take a different value per build-year cohort, while the technology stays one object in the model. A vintage names that cohort — a build window (start/end) with its own numbers.

# one coal technology whose efficiency improves with each build-year cohort
ECOA_v <- newTechnology(
  name    = "ECOA",
  desc    = "Coal plant, improving by build year",
  input   = list(comm = "COA", unit = "PJ", combustion = 1),
  output  = list(comm = "ELC", unit = "PJ"),
  ceff    = data.frame(
    vintage  = c("2020", "2030", "2040"),
    comm     = "COA",
    cinp2use = c(0.40, 0.43, 0.46)   # newer builds burn coal more efficiently
  ),
  invcost = list(invcost = 2000),
  fixom   = 55,
  cap2act = 31.536,
  vintage = data.frame(
    vintage = c("2020", "2030", "2040"),
    start   = c(2020, 2030, 2040),
    end     = c(2029, 2039, 2050),
    olife   = 30
  )
)

draw(ECOA_v)

The electrolyser and vehicle technologies in Chapter 6 are built exactly this way — the FPEM100 electrolyser’s hydrogen efficiency climbs from 0.53 (2020) to 0.66 (2050) across its vintages, and the one spec still describes one machine.

A second, related use of the same machinery is an existing fleet that retires over time: give a technology a capacity stock declared at several years and energyRt reads it as a declining retirement path, optionally letting the optimiser retire units early.

# the same technology, now also carrying an existing fleet that retires by 2050
ECOA_fleet <- update(
  ECOA_v,
  capacity = data.frame(year = c(2025, 2050), stock = c(1.2, 0)),
  optimizeRetirement = TRUE
)

2.4 — Parameters by build year. Put ECOA_v in the model, solve, and read vTechCap / vTechNewCap by vintage. Confirm that capacity built in later years carries the more efficient coefficients, and use autoplot(mod, type = "windows") to visualise each vintage’s build window against its operating life.

TipSolution 2.4

repo_v <- newRepository(
  "repo_v",
  ELC, COA, GAS, CO2,
  SUP_COA, SUP_GAS, DEM_ELC, ECOA_v, EGAS
)

mod_v <- newModel(
  name     = "VINT",
  data     = repo_v,
  region   = "R1",
  discount = WS_DISCOUNT,
  horizon  = horizon_2025_2050_by10
)

autoplot(mod_v, type = "windows") + theme_ws()   # vintage build windows vs olife

scen_v <- solve_model(mod_v)

getData(scen_v, "vTechCap", merge = TRUE)    # installed capacity, by vintage and year
getData(scen_v, "vTechNewCap", merge = TRUE) # new build, by vintage

Each vintage is a distinct cohort: capacity built from 2040 carries cinp2use = 0.46 and burns less coal per unit of electricity than the 2020 cohort’s 0.40 — even though both are “the coal plant.” Vintaging is what lets one technology track a changing world (improving efficiency, falling costs, an ageing fleet) without multiplying into a dozen near-duplicate objects.

4.5 Storage and intermittent sources

Two technologies acquire meaning only once the day has a shape. Weather-driven generators (solar, wind) produce on a fixed profile rather than on command; storage moves energy between slices.

# free renewable resources (no fuel cost), balanced hourly
SOL <- newCommodity("SOL", unit = "PJ", timeframe = "HOUR")
WIN <- newCommodity("WIN", unit = "PJ", timeframe = "HOUR")

SUP_SOL <- newSupply(
  name      = "SUP_SOL",
  commodity = "SOL",
  unit      = "PJ",
  supply    = data.frame(cost = 0)
)

SUP_WIN <- newSupply(
  name      = "SUP_WIN",
  commodity = "WIN",
  unit      = "PJ",
  supply    = data.frame(cost = 0)
)

# capacity-factor profiles for R1 on this calendar (deterministic, shipped)
weather_WSOL <- prof$weather |>
  filter(resource == "WSOL") |>
  select(region, timeslice, wval)

weather_WWIN <- prof$weather |>
  filter(resource == "WWIN") |>
  select(region, timeslice, wval)

WSOL <- newWeather(
  name      = "WSOL",
  timeframe = "HOUR",
  weather   = weather_WSOL
)

WWIN <- newWeather(
  name      = "WWIN",
  timeframe = "HOUR",
  weather   = weather_WWIN
)

# waf.fx = 1 fixes output to the weather profile; because electricity is a
# free-disposal commodity (limtype "LO"), any surplus is dumped and appears in
# vBalance -- which is how we measure curtailment in @sec-curtailment
ESOL <- newTechnology(
  name    = "ESOL",
  desc    = "Solar PV",
  input   = list(comm = "SOL", unit = "PJ"),
  output  = list(comm = "ELC", unit = "PJ"),
  ceff    = data.frame(comm = "SOL", cinp2use = 1),
  weather = list(weather = "WSOL", comm = "SOL", waf.fx = 1),
  invcost = list(invcost = meur_gw(650)),
  fixom   = 12,
  cap2act = 31.536,
  olife   = 25L
)

EWIN <- update(
  ESOL,
  name    = "EWIN",
  desc    = "Onshore wind",
  input   = list(comm = "WIN", unit = "PJ"),
  ceff    = data.frame(comm = "WIN", cinp2use = 1),
  weather = list(weather = "WWIN", comm = "WIN", waf.fx = 1),
  invcost = list(invcost = meur_gw(1300)),
  fixom   = 35
)

# a 4-hour battery on electricity
STG_ELC <- newStorage(
  name      = "STG_ELC",
  desc      = "Grid battery, 4 h",
  commodity = "ELC",
  invcost   = list(invcost = meur_pj_kwh(200)),
  cap2stg   = 4,
  seff      = data.frame(inpeff = 0.95, outeff = 0.95),
  olife     = 20L
)

draw(ESOL)    # solar: the SOL resource in, electricity out
draw(STG_ELC) # the battery, charging and discharging electricity

Assemble the hourly model (electricity now timeframe = "HOUR") and solve.

repo_VRE <- newRepository(
  "repo_VRE",
  ELC_h, COA, GAS, CO2, SOL, WIN,
  SUP_COA, SUP_GAS, SUP_SOL, SUP_WIN, DEM_ELC,
  ECOA, EGAS, ESOL, EWIN, WSOL, WWIN, STG_ELC
)

mod_VRE <- newModel(
  name     = "VRE",
  data     = repo_VRE,
  region   = "R1",
  discount = WS_DISCOUNT,
  calendar = WS_CAL,
  horizon  = horizon_2025_2050_by10
)

scen_VRE <- solve_model(mod_VRE)

getData(scen_VRE, "vTechCap", merge = TRUE)                           # the mix
getData(scen_VRE, "vStorageInp", merge = TRUE, timeframe = "highest") # charging

autoplot(scen_VRE) + theme_ws() # the generation mix by technology and year

2.5 — Complementarity of wind and solar resources. Plot the capacity factors of WSOL and WWIN across the day and the seasons (autoplot(WSOL, calendar = WS_CAL)). Identify the periods in which one resource rises as the other falls, and explain how this complementarity alters the demand placed on storage.

TipSolution 2.5

autoplot(WSOL, calendar = WS_CAL) + theme_ws()
autoplot(WWIN, calendar = WS_CAL) + theme_ws()

Solar is a midday peak that collapses to zero at night; wind is flatter and often stronger overnight and in winter. Because the two peak at different times, a combination of them requires less storage than either alone — the battery need only bridge what the pair still leaves uncovered.

4.6 Curtailment

Because waf.fx = 1 fixes each renewable’s output to its weather profile, a panel or turbine produces its full potential every hour, whether or not it is needed. Electricity is a free-disposal commodity by default (limtype = "LO", so its balance is production ≥ consumption), so the model may simply dump any surplus. That dumped surplus — electricity generated but not consumed — is the balance variable vBalance, so curtailment reads straight off it, with no reconstruction:

vBalance(ELC) = Σ vOutTot − Σ vInpTot  ≥ 0    (the curtailed surplus)

Had we instead set waf.up = 1, the optimiser would under-produce rather than spill, vBalance would be identically zero (and so absent from the output, which records only non-zero balances), and curtailment would have to be inferred from potential minus output.

There is a catch: whether any surplus arises at all depends on how much renewable capacity is built. In the least-cost mix above, cheap dispatchable thermal is free to fill any gap, so the optimiser builds only modest solar and wind and backs the thermal fleet down when they are plentiful — almost nothing is spilled, and vBalance is essentially zero. Curtailment is a phenomenon of high renewable penetration, not of renewables as such.

2.6 — Quantifying curtailment. Confirm that the free least-cost mix curtails little, then force a large solar fleet — so that its midday output must exceed demand — re-solve, and read the curtailed surplus from vBalance. Explain how the surplus responds to storage and to further increases in solar capacity.

TipSolution 2.6

# the free least-cost mix spills little or nothing
getData(scen_VRE, "vBalance", comm = "ELC", merge = TRUE)

# force a high solar penetration: at least 8 GW installed
ESOL_big <- update(
  ESOL,
  capacity = data.frame(year = 2050, cap.lo = 8)
)

repo_curt <- newRepository(
  "repo_curt",
  ELC_h, COA, GAS, CO2, SOL, WIN,
  SUP_COA, SUP_GAS, SUP_SOL, SUP_WIN, DEM_ELC,
  ECOA, EGAS, ESOL_big, EWIN, WSOL, WWIN, STG_ELC
)

mod_curt <- newModel(
  name     = "CURT",
  data     = repo_curt,
  region   = "R1",
  discount = WS_DISCOUNT,
  calendar = WS_CAL,
  horizon  = horizon_2025_2050_by10
)

scen_curt <- solve_model(mod_curt)

# now the surplus is real: curtailed electricity by slice, then as an annual total
getData(scen_curt, "vBalance", comm = "ELC", merge = TRUE, timeframe = "highest")
getData(scen_curt, "vBalance", comm = "ELC", merge = TRUE)

With cheap thermal on hand the model has no reason to over-build renewables, so the free mix curtails almost nothing. Force a large solar fleet and the picture changes: on sunny midday slices its output exceeds what demand and the battery can absorb, and the surplus is dumped — a positive vBalance. Adding storage reduces it, as energy is shifted to the evening; adding still more solar increases it. Curtailment appears only when renewables are pushed beyond what demand and flexibility can take up — which is why the fully renewable system of Section 5.5 in the next chapter is where it truly bites.

4.7 What you built

Time turned a snapshot into a trajectory. You met the horizon and its milestones, the two discount rates and how the business and social cases diverge, sub-annual calendars and why resolution decides what is even visible, vintages as ageing fleets, and the pair — storage and intermittent generation — that exist only once the day has a shape, together with the curtailment that follows. Chapter 5 adds space: several regions, trade between them, and where a new load wants to be.