🔬 physics

Modeling medium and low voltage grids using population density

This paper presents a scalable model that uses population density and the Kruskal algorithm to generate realistic medium and low voltage grid structures, revealing that global distribution network characteristics follow multivariate power laws driven by population scaling.

Original authors: Emile Emery, Joseph Le Bihan, José Halloy

Published 2026-02-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Emile Emery, Joseph Le Bihan, José Halloy

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to build a massive, invisible spiderweb that connects every single house, factory, and shop in a country to the electricity grid. You don't have a map of the wires, and you don't know exactly where every transformer (the little boxes on poles) is located. In fact, in many parts of the world, that data is a secret or simply doesn't exist.

So, how do you figure out where to build this web and how much copper wire you'll need?

This paper by Emery, Le Bihan, and Halloy proposes a clever solution: Just look at the people.

Here is the story of their discovery, broken down into simple concepts.

1. The Big Idea: People = Power Needs

The authors started with a simple hunch: Wherever people live, electricity needs to go.
If you have a dense city like Paris, you need a tight, complex web of wires. If you have a sparse village in the countryside, you need long, straight lines connecting a few scattered houses.

They asked: Can we predict the entire electrical grid just by looking at a map of population density?

2. The Recipe: How They Built the "Fake" Grid

To test this, they built a computer model using France as a practice run (because France has very good data). Here is their recipe:

Step 1: The Population Map. They took a high-resolution map of France that shows exactly how many people live in every tiny square kilometer. Think of this as a "heat map" of human activity.
Step 2: The Substation Guess. They created a rule: "In crowded areas, we need fewer transformers per person because one big one can serve many. In empty areas, we need one transformer for every few people." It's like a taxi service: in a busy city, one bus stops at many places; in the country, you might need a private car for every few passengers.
Step 3: The Connection. Once they randomly "placed" these imaginary transformers based on the population, they used a mathematical trick (called the Kruskal algorithm) to connect them. Imagine you have a bunch of dots on a piece of paper and you want to draw lines to connect them all using the absolute shortest amount of string possible without creating loops. That's what the computer did.

3. The "Aha!" Moment: The Model Works!

When they compared their computer-generated "fake" grid to the real French grid, the results were surprisingly accurate.

The Shape: The fake grid looked just like the real one. It branched out into cities and stretched thin into the countryside.
The Length: Even though the computer didn't know where the actual roads were (and real wires often follow roads), the total length of the wires in the model was within 6% of the real total length.

The Analogy: Imagine trying to guess the total length of all the roads in a city without ever seeing a map. If you just counted the number of houses and assumed roads connect them efficiently, you'd be surprisingly close. That's what they did with electricity.

4. The Global Test: Does it Work Everywhere?

They took this "French recipe" and applied it to about 35 other countries (like the USA, India, and China) where they didn't have detailed grid maps.

The Twist: It worked great for the total length of the wires. Whether it's the US or India, if you know the population and the size of the country, you can guess the total amount of wire needed with high accuracy.
The Glitch: It was less accurate for counting the number of transformers. Why? Because different countries build their grids differently.
- Europe (like France): Uses fewer, larger, powerful transformers.
- USA: Uses many, smaller transformers closer to homes.
- Result: The model guessed the amount of wire correctly, but it couldn't guess the style of the boxes on the poles because that depends on local engineering choices, not just population.

5. Why This Matters: The "Copper" Connection

Why do we care about guessing the length of wires? Because wires are made of copper (and aluminum).

Recycling: If we know how much copper is "locked up" in the grid, we know how much we can recycle in the future.
Building the Future: As the world electrifies (more electric cars, more heat pumps), we need to build more grid. This model tells governments and companies: "Hey, if you plan to add 1 million people to this region, you'll need roughly X tons of copper and Y kilometers of wire."

6. The "Secret Sauce" of the Discovery

The most fascinating part of the paper is a mathematical pattern they found. They discovered that the total length of the grid follows a simple "Power Law."

Think of it like this:

If you double the population, the grid doesn't double in length; it grows a bit less than double (because you can share wires more efficiently in cities).
If you double the size of the country (but keep the population the same), the grid gets longer because the wires have to stretch further to reach the same number of people.

They found that Population and Land Area are the two main ingredients that determine the size of the grid. Surprisingly, how much electricity people actually use didn't matter much for the length of the wires. Whether a house uses a lot or a little power, the wire connecting it is roughly the same length.

Summary

This paper is like a crystal ball for infrastructure.
Instead of needing secret government blueprints to know where the power lines are, we can just look at where people live. By understanding the relationship between people and space, we can predict the material needs (like copper) for building and maintaining the electrical grid anywhere on Earth.

It turns a complex engineering problem into a simple geography problem: Follow the people, and the wires will follow.

1. Problem Statement

The global expansion of electricity distribution systems requires massive infrastructure deployment, significantly increasing the demand for materials like copper and aluminum. However, characterizing the spatial layout and total length of these networks is critical for:

Estimating future material requirements for electrification.
Assessing existing material stocks for recycling potential.

Challenges:

Data Scarcity: Unlike transmission networks (managed by few operators with open data), distribution networks (Medium Voltage - MV, and Low Voltage - LV) are fragmented among many local operators. Detailed data on cable locations and substation connectivity is often unavailable, sensitive, or non-harmonized.
Complexity: Existing modeling approaches using satellite imagery (e.g., night lights) often fail to capture rural connections or require heavy processing.
Gap: There is a lack of robust, scalable models that can estimate distribution network topology and length using only widely available demographic data.

2. Methodology

The authors propose a stochastic, bottom-up modeling framework that generates realistic MV network structures based solely on high-resolution population density data.

A. Data Inputs

Population Data: WorldPop database (30 arc-second resolution, ~1km²), providing granular population density ( $\rho$ ) for every grid cell.
Reference Data: Open data from ENEDIS (French DSO) covering >95% of France, including substation locations and line drawings, used for calibration and validation.

B. Core Algorithm

The model operates in two main stages:

Substation Sampling (Node Generation):
- The authors establish a non-linear relationship between substation density ( $n_{stat}$ ) and population density ( $\rho$ ).
- Regime 1 (Sparse): For $\rho < 8$ hab/km², $n_{stat} \propto \rho$ (linear).
- Regime 2 (Dense): For $\rho \ge 8$ hab/km², $n_{stat} \propto \rho^{0.5}$ (sublinear due to scale effects).
- Stochasticity: To account for randomness and "empty" cells, the number of substations in each grid cell is sampled from a Poisson distribution with the parameter derived from the density relationship. Substations are then placed uniformly within the cell.
Network Construction (Edge Generation):
- The generated nodes are connected using the Kruskal algorithm to form a Minimum Spanning Tree (MST).
- Rationale: Real distribution networks are primarily tree-like. While real networks contain loops (bridges) for resilience, the MST serves as a robust first-order proxy for total length and topology.
- Corrections: The raw MST length is adjusted using correction factors ( $\gamma \approx 1.537$ $γ \approx 1.537$ ) to account for:
  - Manhattan vs. Euclidean distance: Real cables follow roads, not straight lines.
  - Route vs. Cable length: Distinguishing between the physical path and the actual cable length (accounting for multi-cable corridors).
  - Loops: Adding a factor to approximate the extra length of non-tree bridges.

C. Validation and Scaling

Local Validation: The model is calibrated on France and compared against ENEDIS data for node counts and total cable length.
Global Application: The France-calibrated model is applied to ~35 other countries with high electrification rates.
Macro-Scaling: The authors derive multivariate power laws relating total network length to Population ($Pop$), Area ($Area$), and Electricity Consumption ($Conso$).

3. Key Results

A. Local Scale (France)

Topological Similarity: The model-generated network visually and topologically matches the real French grid, capturing the "tree-like" structure and the spatial clustering of nodes.
Quantitative Accuracy:
- Substation Count: The model estimates 88% of the actual number of substations.
- Total Cable Length: The model estimates 94% of the actual MV cable length.
- Insight: Local routing errors (e.g., cables following roads vs. straight lines) cancel out at the national scale, making the total length estimate highly accurate despite local discrepancies.

B. Global Scale (Cross-Country)

Substation Counts: The model shows significant bias when applied globally. It tends to overestimate substations in countries with "American-style" grids (many small, low-capacity transformers near customers) compared to "European-style" grids (fewer, larger transformers). This highlights that substation density is architecture-dependent.
Network Length: Despite substation count discrepancies, the total MV network length is estimated with high accuracy across diverse countries. The model captures the spatial distribution of demand effectively, regardless of the specific transformer architecture.
Power Laws:
- MV Length: Best described by $L_{MV} \propto Pop^{0.59} Area^{0.36}$ . Notably, electricity consumption has a negligible impact on this scaling, suggesting spatial distribution is the primary driver.
- LV Length: Depends more on consumption and less on total area ( $L_{LV} \propto Pop^{0.66} Area^{0.22}$ ), as LV networks are the "last mile" directly serving households.
- Ratio: A strong correlation exists between LV and MV lengths ( $L_{LV} \approx 1.63 \times L_{MV}$ ), consistent with literature.

C. Material Estimation

Using the estimated lengths and standard material intensities (copper for underground, aluminum for aerial), the model can estimate the total mass of materials invested in a country's grid.
This provides a tool for "Material Flow Analysis" (MFA) to identify copper stocks for recycling and future demand.

4. Key Contributions

Novel Modeling Approach: Demonstrates that a complex physical infrastructure (MV grid) can be reconstructed with high fidelity using only population density data, bypassing the need for sensitive or unavailable grid maps.
Multi-Scale Validation: Successfully bridges local (visual topology) and global (aggregated length) scales, proving that local population-network relationships hold at the national level.
Discovery of Scaling Invariants: Identifies that while substation counts vary by grid architecture, the total network length is a universal function of population and territory size, largely independent of consumption levels for MV networks.
Practical Tool for Planning: Provides a method to estimate infrastructure needs and material stocks in emerging economies where grid data is non-existent, supporting the UN Sustainable Development Goal 7 (Affordable and Clean Energy).

5. Significance and Implications

Resource Management: The model enables precise estimation of in-use copper stocks, crucial for circular economy strategies and recycling planning.
Electrification Planning: Offers a "what-if" tool for policymakers to forecast the material and spatial requirements of expanding grids in rural or developing areas.
Generalizability: The framework suggests that similar population-driven approaches could be applied to other infrastructure networks (water, gas, telecom) where demand is spatially correlated with population.
Limitations & Future Work: The model assumes a mature, optimized grid. Future work could incorporate topographic data (buildings/roads) to refine the "last mile" (LV) connections and adapt the model for decentralized renewable energy scenarios which may alter traditional tree topologies.

In summary, this paper establishes that population spatial distribution is the primary determinant of distribution grid extent, providing a powerful, data-light methodology for characterizing global electrical infrastructure.