Compute Crunch

Act 1: Route the world

The locations are real, but the demand is made up for these puzzles. Latency is estimated using great-circle distance, a signal speed of two thirds the speed of light, and a 1.4× multiplier to account for indirect fiber routes. Regional power prices and grid carbon intensities are rough examples.

The linear program

The solver chooses how many megawatts of compute each data center allocates to each city. It excludes routes that exceed the latency limit and keeps each site within its capacity. Unmet city demand adds a $1,400 per megawatt-hour penalty to the cost. Training can run at any site; postponing it adds a $500 per megawatt-hour penalty.

minimize

• Σ price · x + 1400 · unserved + 500 · unserved training

subject to

• Σ x into each city + unserved = demand
• Σ x out of each site ≤ capacity
• Σ carbon · x ≤ cap chapter 2
• x ≥ 0, only on routes within the latency limit

The solver

A Python service uses GLOP, the linear solver in Google’s OR-Tools, to find the lowest-cost allocation. It also returns dual values, which describe how the minimum cost changes as constraints are relaxed. The savings shown on the page are calculated by running the solver again with one extra megawatt of capacity at a site, or one extra tonne allowed under the carbon cap, and comparing the costs.

Adding capacity

In chapter 3, new capacity comes in 25 MW blocks. SCIP solves the integer program to choose where to build, then GLOP allocates the compute. Each block has a cost, so adding capacity only helps if the savings justify it.

The model treats every megawatt of compute as interchangeable. It doesn’t account for GPU types, network capacity, queueing, or fairness.

Act 2: Data Center Alley

The substations, their neighborhoods, and the shape of the network come from OpenStreetMap’s map of the 230 kV and 500 kV lines in Loudoun County, simplified to two 500 kV hubs and seven substations. Line lengths are real. Line limits, loads, prices, and land costs are illustrative; utilities don’t publish substation headroom.

Power flow uses the standard DC approximation: the flow on a line is the difference in voltage angle across it divided by its reactance, which here is proportional to its length. So power can’t be sent down one line. Every injection spreads across every path at once, and one full line limits the whole network.

minimize

• Σ price · supply + 1400 · dark + Σ land · campus + Σ cost · new line

subject to

• supply − load + dark = flow out − flow in, at every bus
• flow = (θa − θb) / km, on every line in service
• −limit ≤ flow ≤ limit
• each campus on exactly one substation chapter 1
• each line in or out of service, at most one new chapter 2

The solvers

Placing campuses is a mixed-integer program, solved with SCIP. Choosing which lines to build or switch out is another, solved with SCIP: a line out of service drops its flow equation, written with a big-M constraint. Both return a decision, and GLOP then re-solves the power flow exactly. The key move is measured with the same counterfactual approach as Act 1: force the tempting choice, or close the opened breaker, and solve again.

Chapter 2 is Braess’s paradox on a power grid. Adding a line adds a new loop, and the physics of that loop can push more power onto a line that is already full. Removing a line can do the opposite. Grid operators use this on purpose; it is called optimal transmission switching.

Watching the solver think

Where a chapter needs integer decisions, SCIP searches with branch and bound. The replay is real: the service reruns SCIP’s deterministic search, stopping after its first, second, third solution and so on, and records each plan with the lower bound SCIP had proved at that moment. The times shown are SCIP’s own.

Act 3: Campuses keep coming

The grid is Act 2’s, solved once per year from 2027 to 2030 as one mixed-integer program. Each project gets a start year; it counts only once its lead time has passed, and one crew can start one project a year. Turbines are rented per year, and in the last chapter breakers can be opened per year too. The score is the average hour across the four years. Lead times, costs, and campus sizes are illustrative.

Act 4: Who signs?

This is a two-stage stochastic program. Turbines ordered today are the first stage and cost the same in every future; rush orders are the second, chosen after the signings are known, and they are scarce. Chapter 1 measures the value of the stochastic solution: how much worse the plan built for the average signing does across the real futures. Chapter 2 minimizes the most expensive future instead, which is robust optimization. Chapter 3 is the expected value of perfect information: the best blind plan against the best plan for each future, weighted by the odds.

Sources

• Dietrich Braess, “Über ein Paradoxon aus der Verkehrsplanung,” Unternehmensforschung 12 (1968): 258–268. The original statement of Braess’s paradox, for road traffic.
• Dirk Witthaut and Marc Timme, “Braess’s paradox in oscillator networks, desynchronization and power outage,” New Journal of Physics 14 (2012): 083036. Braess’s paradox in power grids.
• Emily B. Fisher, Richard P. O’Neill, and Michael C. Ferris, “Optimal transmission switching,” IEEE Transactions on Power Systems 23, no. 3 (2008): 1346–1355.
• John R. Birge, “The value of the stochastic solution in stochastic linear programs with fixed recourse,” Mathematical Programming 24 (1982): 314–325.
• John R. Birge and François Louveaux, Introduction to Stochastic Programming, 2nd ed. (Springer, 2011).
• Aharon Ben-Tal, Laurent El Ghaoui, and Arkadi Nemirovski, Robust Optimization (Princeton University Press, 2009).
• Map data © OpenStreetMap contributors, available under the ODbL.