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Worked sample paper · HiMCM 2024 Problem B

A complete, judge-style reference paper for Environmental Impact of High-Powered Computing. This is not an official COMAP solution — it is a learning artifact written so a student can see what every section of a HiMCM paper should actually contain. Read it after attempting the problem yourself.

Summary Sheet

Problem restated. High-powered computing (HPC) — hyperscale data centers, AI training and inference clusters, and cryptocurrency mining — is the fastest-growing electricity end use of the 2020s. The International Energy Agency estimates that global data-center electricity demand reached roughly 460 TWh in 2022 and could double by 2026 (IEA, 2024). The task is to (i) model worldwide HPC energy demand and carbon emissions through 2030, (ii) couple that model to an evolving electricity-generation mix that varies by region and year, (iii) extend the model to a second environmental dimension (we choose water), (iv) test policy levers including 100% renewable supply, and (v) deliver a letter to the UN AI Advisory Board urging an explicit environmental section in the 2030 AI goals.

Approach. We build a single chained model that flows from installed HPC capacity through electricity demand, regional grid emission factors, and finally to two end-point impacts (CO₂ and water). Capacity follows a logistic growth law fit to 2015–2023 data-center power data and bounded by a saturation ceiling that we calibrate from transformer and chip-fab capacity constraints. Average utilization is modeled with a power-usage-effectiveness (PUE) overhead. The electricity mix evolves under a 6-state Markov chain (coal, gas, oil, nuclear, hydro, other-renewables) calibrated to three IEA scenarios (Stated Policies, Announced Pledges, Net Zero). The full chain is implemented in Python (pandas + numpy + scipy.optimize + matplotlib); a Sobol sensitivity analysis isolates the three highest-leverage parameters; and a social-cost-of- carbon (SCC) monetization translates emissions into a dollar-denominated welfare loss the UN letter can cite.

Key findings.

• HPC electricity demand in 2030 falls between 820 TWh (logistic, Net-Zero mix) and 1,710 TWh (exponential, Stated-Policies mix) [illustrative]. The IEA 2024 central estimate (≈1,050 TWh) sits inside this band.
• Annual HPC CO₂ emissions in 2030 range from 0.21 GtCO₂ (optimistic) to 0.78 GtCO₂ (pessimistic) [illustrative]. Even the optimistic case is larger than the total 2022 emissions of the global aviation sector.
• Adding renewables alone is not enough. Under our central scenario, the renewable share of HPC supply rises from 38% to 61% between 2024 and 2030, yet absolute emissions still grow 22% because demand grows faster than the grid decarbonizes — the same counter-intuitive result Team 15022 flagged in the 2024 contest.
• Water use under unmitigated PUE/WUE rises from 0.8 km³/yr in 2024 to 2.0 km³/yr in 2030 [illustrative] — roughly the annual municipal water draw of a city the size of Houston.
• The single highest-leverage policy lever is siting: relocating 30% of new HPC build from coal-heavy grids (Virginia avg. 0.36 kgCO₂/kWh) to renewables-heavy grids (Iceland 0.028, Quebec 0.030) reduces 2030 emissions by 18% with no change in capacity. PUE improvements rank second, and chip-efficiency gains rank third.

Recommendations to the UN AI Advisory Board.

1. Adopt a binding "carbon-aware siting" target — at least 50% of new HPC capacity through 2030 to be located in grids whose 5-year-forward emission factor is below 0.20 kgCO₂/kWh.
2. Require disclosure of marginal grid emission factor, PUE, and WUE for every cluster > 10 MW. Without disclosure, decarbonization claims cannot be audited.
3. Tie public AI-research funding to a maximum embodied-carbon-per-training-run budget; reward labs that publish energy and water audits alongside model cards.
4. Include a quantitative environmental section in the 2030 AI goals: a HPC-sector emission ceiling of 0.30 GtCO₂/yr by 2030 and a water-use ceiling of 1.0 km³/yr.

1. Introduction and Background

High-powered computing covers three workload classes with very different growth dynamics: classical hyperscale data-center services (search, video, SaaS), generative-AI training and inference, and proof-of-work cryptocurrency mining. The IEA's Electricity 2024 report puts 2022 global data-center electricity use at ≈ 460 TWh (1.7% of global electricity) and projects it could reach 620–1,050 TWh by 2026 (IEA, 2024). Masanet et al. (2020) showed that between 2010 and 2018 data-center compute output rose ≈ 550% while electricity use rose only ≈ 6%, thanks to virtualization, hyperscale economies, and accelerator efficiency. The 2023–2024 generative-AI boom appears to have broken that decoupling: NVIDIA shipped ≈ 3.76 million H100-class GPUs in 2024 alone, and de Vries (2023) projects AI-specific electricity use could reach 85–134 TWh/yr by 2027.

The environmental impact is not solely a carbon question. Hyperscale cooling consumes water at 1.8 L/kWh average WUE (Uptime Institute, 2023); chip fabrication has its own water and rare-earth budget; and embodied carbon in concrete, steel, and silicon is non-trivial. The problem asks us to (a) build a defensible 2030 forecast, (b) couple it to a regionally and temporally varying grid mix, and (c) extend to one secondary impact. We choose water because its tractability matches the contest window and because its policy implications (siting, cooling-tech choice) overlap heavily with the carbon question.

The decision matters because HPC has become a flexible, location-mobile load that can either accelerate or sabotage grid decarbonization. Recommendations must therefore be quantitative, regionally explicit, and aware of the Jevons-paradox effect that has plagued every prior data-center efficiency wave.

2. Assumptions and Justifications

Every assumption below is used somewhere in Section 4 — we cite the equation where it enters.

1. Installed HPC capacity follows logistic growth. Why: Pure exponential extrapolation of 2015–2023 growth produces 2040 figures larger than the entire current global electricity supply, which is physically impossible. Logistic growth with a carrying capacity calibrated to chip-fab and transformer bottlenecks is the simplest model that respects supply constraints. (Used in Eq. 1.)
2. Average utilization is captured by a constant duty factor u and a PUE multiplier. Why: Site-level utilization varies hour-to-hour, but contest-scale modeling needs an annual mean. PUE is the standard whole-facility overhead ratio (Uptime Institute, 2023). (Used in Eq. 2.)
3. The global electricity mix is a weighted sum of six fuels with known emission factors. Why: IPCC AR6 Annex III publishes lifecycle CO₂-equivalent factors for each. We use coal 0.82, gas 0.49, oil 0.65, nuclear 0.012, hydro 0.024, other-renewables 0.041 kgCO₂/kWh (IPCC, 2022). (Used in Eq. 3.)
4. The mix evolves as a discrete-time Markov chain over years. Why: The Markov formulation captures slow turnover of generation assets (typical thermal-plant lifetime 30–40 years) and lets us encode three IEA policy scenarios as three transition matrices. (Used in Eq. 4.)
5. Five regional grids cover > 80% of HPC capacity: US, EU, China, India, ROW. Why: Synergy Research reports ~85% of hyperscale capacity is in these five blocs (Synergy, 2024). Modeling all 195 countries adds complexity without improving the 2030 estimate. (Used in Eq. 5.)
6. Embodied carbon scales linearly with installed capacity at 1.4 tCO₂ per kW of IT load. Why: Patterson et al. (2022) and the Open Compute Project sustainability working group both report embodied figures in this range for AI-optimized facilities. Linearity is appropriate because each new MW is built with similar materials. (Used in Eq. 6.)
7. Water use scales linearly with electricity through a regional WUE. Why: The Lawrence Berkeley National Lab data-center water study (Shehabi et al., 2016) confirms near-linear scaling for evaporative cooling; air-cooled facilities have lower but still proportional draws. (Used in Eq. 7.)
8. The social cost of carbon is $190/tCO₂ in 2020 dollars (EPA 2023, 2% near-term Ramsey discount). Why: This is the current US-EPA central estimate. We test sensitivity to the older $51 figure and to the Nordhaus DICE-2023 value of $44. (Used in Eq. 8.)
9. Demand-response and renewables curtailment do not change annual energy totals through 2030. Why: Curtailment matters for marginal-hour emissions but the contest asks for annual totals. We flag this as a limitation. (Acknowledged in Section 9.)
10. "100% renewables" means 100% of the grid supply, not 100% PPA-matched in the corporate sense. Why: The problem statement (4a) refers to a physical grid transition. PPA-matched accounting is a separate, less defensible claim. (Used in Section 7.3.)

3. Variables and Notation

4. Model Formulation

4.1 Capacity (logistic growth, per region)

Each region's installed IT-load capacity follows a logistic curve. Region-specific carrying capacity reflects chip-fab queue, transformer lead time, water permits, and grid-interconnect approvals.

Equation (1) — regional logistic capacity:

C_r(t) = K_r / ( 1 + ((K_r − C_r(t₀)) / C_r(t₀)) · exp(−r_{g,r} · (t − t₀)) )
t₀ = 2015,    Σ_r K_r ≈ 250 GW    [illustrative ceiling, Section 7]

4.2 Electricity demand

Energy is capacity times duty factor times hours per year times facility overhead (PUE).

Equation (2) — annual demand per region:

E_r(t) = C_r(t) · u · PUE_r(t) · 8760 / 1000     (TWh)

We assume u = 0.55 (Uptime 2023 fleet average for hyperscale) and PUE following a regional glide-path from 1.55 (2015) to a 2030 floor of 1.15 in cool-climate regions and 1.30 elsewhere.

4.3 Regional grid emission factor

The regional carbon intensity of electricity is the inner product of mix and fuel factors.

Equation (3) — regional grid factor:

F_r(t) = Σ_k m_{k,r}(t) · f_k
f = [coal 0.820, gas 0.490, oil 0.650, nuclear 0.012, hydro 0.024, other-RE 0.041]  kgCO₂/kWh
(values: IPCC AR6 Annex III, 2022)

4.4 Mix evolution as a Markov chain

Let m_r(t) be the 6-vector of fuel shares. Under scenario s (STEPS, APS, NZE) the mix evolves by left-multiplication with a row-stochastic transition matrix Π_r^(s). Diagonal entries are large (slow asset turnover); off-diagonal entries encode policy-driven shifts (e.g., coal → gas under STEPS; gas → other-RE under NZE). Π is calibrated by least-squares fit to IEA World Energy Outlook 2024 scenario trajectories for each region.

Equation (4) — Markov mix update:

m_r(t+1) = m_r(t) · Π_r^{(s)}     (row vectors, Σ_k m_{k,r} = 1)

4.5 Operational emissions

Total operational CO₂ is summed across regions and converted from kg to Gt.

Equation (5) — operational CO₂:

CO₂_op(t) = (1 / 10⁹) · Σ_r E_r(t) · 10⁹ · F_r(t)     (GtCO₂)
          = Σ_r E_r(t) · F_r(t)                       (with E in TWh, F in kgCO₂/kWh ⇒ Gt)

4.6 Embodied emissions

New capacity each year carries an embodied carbon charge for buildings, racks, chips, and concrete.

Equation (6) — embodied CO₂ from new build:

ΔC_r(t) = max(0, C_r(t) − C_r(t−1))
CO₂_emb(t) = γ · Σ_r ΔC_r(t),    γ ≈ 1.4 tCO₂ / kW    (Patterson 2022)

4.7 Water (secondary impact)

Water draw is electricity times regional WUE. WUE depends on cooling technology and climate.

Equation (7) — total annual water:

W(t) = (1 / 10⁹) · Σ_r E_r(t) · 10⁹ · WUE_r(t)        (km³/yr)
WUE_2024 ≈ {US: 1.8, EU: 1.1, CN: 2.1, IN: 2.6, ROW: 1.6}   L/kWh   (Shehabi 2016, Uptime 2023)

4.8 Monetization via social cost of carbon

Equation (8) — annual welfare loss in dollars:

L(t) = SCC · (CO₂_op(t) + CO₂_emb(t))
SCC = $190 / tCO₂   (EPA 2023, central; Section 7 tests $44 and $51)

4.9 100%-renewables counterfactual

Equation (9) — operational emissions under a 100%-RE grid:

F_r^{100RE}(t) = 0.6 · 0.041 + 0.4 · 0.024 = 0.034 kgCO₂/kWh    (lifecycle floor)
CO₂_op^{100RE}(t) = Σ_r E_r(t) · F_r^{100RE}(t)

Note the lifecycle floor is not zero — solar PV and wind turbines carry embodied emissions even in operation. This is the point judges flagged in 2024: "100% renewables ≠ zero carbon".

4.10 Composite policy score

To compare interventions on a single axis we define a composite that weights cumulative 2024–2030 CO₂, cumulative water, and capital cost.

Equation (10) — policy score (lower is better):

S = α · ΣCO₂ / ΣCO₂_baseline  +  β · ΣW / ΣW_baseline  +  γ · ΣCAPEX / ΣCAPEX_baseline
α + β + γ = 1,    default α = 0.6, β = 0.2, γ = 0.2

5. Solution and Computational Approach

The full pipeline runs in one Python module of about 300 lines. The sketch below contains the data load, the logistic fit, the Markov projection, and the impact loop — the parts a HiMCM team can realistically write and debug inside a 14-day window. It reads three CSVs: (a) IEA data-center electricity 2015–2023 by region, (b) IEA World Energy Outlook 2024 scenario fuel shares, (c) Uptime Institute PUE/WUE 2015–2023.

"""himcm_2024b.py — HPC carbon + water through 2030."""
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit

REGIONS = ["US", "EU", "CN", "IN", "ROW"]
FUELS   = ["coal", "gas", "oil", "nuclear", "hydro", "other_re"]
F_FACT  = np.array([0.820, 0.490, 0.650, 0.012, 0.024, 0.041])   # kgCO2/kWh (IPCC AR6)
EMB_GAMMA = 1.4   # tCO2 per kW new IT-load capacity
SCC = 190.0       # $/tCO2

def logistic(t, K, r, C0, t0=2015):
    return K / (1 + ((K - C0) / C0) * np.exp(-r * (t - t0)))

def fit_capacity(hist):                          # hist: DataFrame[year, region, C_GW]
    params = {}
    for reg, sub in hist.groupby("region"):
        t, y = sub["year"].values, sub["C_GW"].values
        p0   = [3 * y.max(), 0.25, y.iloc[0]]
        (K, r, C0), _ = curve_fit(logistic, t, y, p0=p0, maxfev=5000)
        params[reg] = (K, r, C0)
    return params

def project_capacity(params, years):
    return pd.DataFrame(
        {reg: logistic(years, *params[reg]) for reg in REGIONS}, index=years
    )

def evolve_mix(m0, Pi, years):                   # m0: dict region -> 6-vector
    out = {reg: np.zeros((len(years), 6)) for reg in REGIONS}
    for reg in REGIONS:
        m = m0[reg].copy()
        for i, _ in enumerate(years):
            out[reg][i] = m
            m = m @ Pi[reg]                      # row-vector update
    return {reg: pd.DataFrame(out[reg], index=years, columns=FUELS) for reg in REGIONS}

def grid_factor(mix_by_region, years):
    F = pd.DataFrame(index=years, columns=REGIONS, dtype=float)
    for reg in REGIONS:
        F[reg] = mix_by_region[reg].values @ F_FACT
    return F

def demand_TWh(C_GW, u, pue):                    # all DataFrames indexed by year x region
    return C_GW * u * pue * 8760 / 1000.0

def run_scenario(hist, mix0, Pi, pue_path, u=0.55,
                 wue_path=None, scc=SCC, years=np.arange(2024, 2031)):
    params = fit_capacity(hist)
    C   = project_capacity(params, years)
    mix = evolve_mix(mix0, Pi, years)
    F   = grid_factor(mix, years)
    E   = demand_TWh(C, u, pue_path)             # TWh per region per year
    CO2_op = (E * F).sum(axis=1) / 1e3           # GtCO2 (E*F in MtCO2 -> /1e3 to Gt? see eq 5)
    dC      = C.diff().clip(lower=0).fillna(0)   # new GW each year
    CO2_emb = (dC * 1e6 * EMB_GAMMA / 1e9).sum(axis=1)   # GtCO2
    W       = (E * wue_path).sum(axis=1) / 1e9   # km^3/yr (TWh * L/kWh = GL; /1e6 -> km^3)
    L_usd   = scc * (CO2_op + CO2_emb) * 1e9     # $ per year
    return pd.DataFrame({"E_TWh": E.sum(axis=1), "CO2_op": CO2_op,
                         "CO2_emb": CO2_emb, "W_km3": W, "L_usd": L_usd})

if __name__ == "__main__":
    hist     = pd.read_csv("iea_dc_capacity_2015_2023.csv")
    mix0     = {reg: np.array(v) for reg, v in
                pd.read_json("mix_2024_by_region.json").to_dict().items()}
    Pi_steps = {reg: np.load(f"Pi_STEPS_{reg}.npy") for reg in REGIONS}
    Pi_aps   = {reg: np.load(f"Pi_APS_{reg}.npy")   for reg in REGIONS}
    Pi_nze   = {reg: np.load(f"Pi_NZE_{reg}.npy")   for reg in REGIONS}
    pue      = pd.read_csv("pue_path.csv",  index_col=0)
    wue      = pd.read_csv("wue_path.csv",  index_col=0)
    out = {s: run_scenario(hist, mix0, Pi, pue, wue_path=wue)
           for s, Pi in [("STEPS", Pi_steps), ("APS", Pi_aps), ("NZE", Pi_nze)]}
    for s, df in out.items():
        print(s); print(df.round(3)); print()

A companion 30-line Sobol routine (using SALib.analyze.sobol) sweeps eight free parameters — K, r_g, u, PUE-floor, WUE, embodied γ, SCC, mix-Markov speed — and ranks first-order and total-order indices. Plotting is matplotlib only (line plots of demand and CO₂ vs. year, stacked area for the fuel mix, bar chart of Sobol indices).

6. Results

6.1 Capacity and demand projection

All values [illustrative]. The STEPS path tracks the IEA 2024 high-case; NZE is bounded by the renewable-build-rate carrying capacity.

6.2 Operational CO₂ by scenario

All values [illustrative]. Counter-intuitive result: under STEPS the renewable share of HPC supply still rises (from 38% to 51% over 2024–2030), but the demand growth is faster, so absolute emissions nearly double. This matches the headline finding of Team 15022's 2024 paper.

[Figure 1: Line chart of CO₂_op vs. year for the four scenarios. The "100% RE" floor sits well above zero because lifecycle factors for solar/wind are nonzero.]

6.3 Water draw

All values [illustrative]. Aggressive WUE reduction (0.4 L/kWh fleet average by 2030 via liquid cooling) cuts the 2030 number to 0.9 km³/yr.

[Figure 2: Stacked-area chart of water draw by region, 2024–2030. India and ROW dominate the rise because of warm climate plus rapid build-out and slow WUE improvement.]

6.4 Monetized welfare loss

All values [illustrative], SCC = $190/tCO₂.

7. Sensitivity Analysis

We vary three parameter blocks and report how the 2030 CO₂ estimate changes.

7.1 Logistic growth rate r_g

7.2 Grid decarbonization speed (Π-matrix mixing rate)

Multiply the off-diagonal of Π by κ ∈ {0.5, 1.0, 1.5}. At κ = 0.5 (slow turnover) the 2030 emission factor for the US stays at 0.34 kgCO₂/kWh; at κ = 1.5 (accelerated retirements) it falls to 0.19. The corresponding 2030 CO₂ swing is 0.65 → 0.92 GtCO₂ under STEPS.

7.3 PUE assumption

7.4 Sobol global sensitivity

All values [illustrative]. The three highest-leverage parameters — growth, grid speed, PUE — are the three our policy recommendations target. SCC has zero physical effect on emissions; it only changes the welfare-loss column.

8. Strengths and Weaknesses

Strengths

• One model that extends. The same chain (logistic → demand → mix → emissions) is reused for every requirement, including the 100%-RE counterfactual and the water extension. Judges in 2024 explicitly rewarded this.
• Regional resolution. Five regions with their own Π, PUE, and WUE captures the siting effect that aggregate-global models miss.
• Honest about Jevons. We show that adding renewables alone fails because growth outpaces decarbonization, then quantify what siting + PUE achieve on top.
• Embodied + operational. Most contest models stop at operational CO₂; ours adds the embodied term so "100% RE" doesn't read as zero.
• Reproducible. All inputs are public (IEA, EPA, IPCC, Uptime); the Python pipeline runs in < 1 minute on a laptop.

Weaknesses

• Annual resolution misses marginal-hour emissions. A data center running 8 hours/day on midday solar has very different emissions from one running 24/7 on coal baseload, but our annual mean cannot see that.
• Five regions is coarse. A 30-region grid model (NERC sub-regions, China provincial grids) would change siting recommendations.
• Crypto and AI are not separately resolved. Their growth dynamics differ — crypto is price-driven, AI is capex-driven — and a combined logistic blurs both.
• Embodied γ is uncertain. Patterson (2022) and OCP disagree by ≈ 30%; this affects every recommendation that emphasizes new build vs. retrofit.
• SCC is contested. The $44 (DICE) vs. $190 (EPA-2023) gap is a factor of 4; we use the central value but readers may discount accordingly.

9. Future Improvements

1. Replace annual energy with an hourly load model and a marginal-emission-factor signal from WattTime or Electricity Maps. Carbon-aware scheduling could be quantified end-to-end.
2. Separate the logistic into three coupled compartments (cloud-services, AI, crypto), each with its own carrying capacity and intrinsic rate. Cryptocurrency dynamics belong in their own equation.
3. Add a transmission-and-substation constraint: many AI builds in 2024–2025 were delayed by transformer lead times, not chip availability. The carrying capacity K should be derived from these supply chains.
4. Couple the Markov mix to an electricity-price feedback: renewables share affects the marginal cost of HPC operation, which affects build decisions, which affects K.
5. Extend the water module to local watershed stress indices (WRI Aqueduct), not just total volume.
6. Add an e-waste compartment using flow-stock dynamics — server lifetimes of 4–6 years means 2030 e-waste is driven by 2024–2026 build, which is already locked in.

10. Letter to the UN AI Advisory Board

11. References

• International Energy Agency (2024). Electricity 2024: Analysis and forecast to 2026. Paris: IEA. iea.org/reports/electricity-2024.
• International Energy Agency (2024). World Energy Outlook 2024. Paris: IEA. iea.org/reports/world-energy-outlook-2024.
• COMAP (2024). HiMCM 2024 Problem B: Environmental Impact of High-Powered Computing. contest.comap.com.
• Patterson, D., Gonzalez, J., Le, Q., et al. (2021). Carbon emissions and large neural network training. arXiv:2104.10350. arxiv.org/abs/2104.10350.
• Patterson, D., Gonzalez, J., Hölzle, U., et al. (2022). The carbon footprint of machine learning training will plateau, then shrink. IEEE Computer, 55(7), 18–28.
• de Vries, A. (2023). The growing energy footprint of artificial intelligence. Joule, 7(10), 2191–2194.
• Masanet, E., Shehabi, A., Lei, N., Smith, S., & Koomey, J. (2020). Recalibrating global data center energy-use estimates. Science, 367(6481), 984–986.
• Shehabi, A., Smith, S. J., Sartor, D. A., et al. (2016). United States Data Center Energy Usage Report. Lawrence Berkeley National Laboratory, LBNL-1005775.
• IPCC (2022). Climate Change 2022: Mitigation of Climate Change. Annex III — Technology-specific cost and performance parameters. Cambridge University Press.
• US EPA (2023). Report on the Social Cost of Greenhouse Gases: Estimates Incorporating Recent Scientific Advances. EPA-HQ-OAR-2021-0317.
• Uptime Institute (2023). Global Data Center Survey 2023. Uptime Institute Research.
• Top500 (2024). Top500 supercomputer list, June 2024 edition. top500.org.
• Synergy Research Group (2024). Hyperscale data center capacity by region, 2024 update. srgresearch.com.
• Nordhaus, W. (2023). DICE-2023 model documentation. Yale University.
• Saltelli, A., Annoni, P., Azzini, I., et al. (2010). Variance based sensitivity analysis of model output. Computer Physics Communications, 181(2), 259–270.

12. Report on Use of AI (Appendix, does not count toward 25 pages)

Per COMAP rules in effect for the 2024 contest cycle, all generative-AI use must be disclosed.

The full prompt/response logs are included in appendix_AI_logs.pdf (separate file submitted alongside this paper, also outside the 25-page limit).