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# Power grid network growth model based on rural-urban differentiation

Emile Emery, Sébastien Aumaître, Hervé Bercegol

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Emile Emery, Sébastien Aumaître, Hervé Bercegol. Power grid network growth model based on rural-urban differentiation. PRX Energy, 2025, 4, pp.033014. 10.1103/kysp-hgxt.cea-05282360

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# Power Grid Network Growth Model Based on Rural-Urban Differentiation

Emile Emery,*, Sébastien Aumaître, and Hervé Bercegol

Université Paris-Saclay, CEA, CNRS, SPEC, 91191 Gif-sur-Yvette, France

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(Received 11 February 2025; revised 28 May 2025; accepted 7 August 2025; published 9 September 2025)

The development of clean energy solutions requires an extension of electrical grids in size and power. Network evolution modeling is crucial for revealing critical insights into transmission grid growth dynamics. This study proposes an efficient analytical framework based on spatial network physics to enhance previous models and deliver better predictions. Utilizing the French transmission grid as a case study, our methodology demonstrates its aptitude to estimate energy invested and consumed, as well as topological and spatial characteristics. This new approach balances computational efficiency with predictive accuracy, making it particularly valuable for building datasets for research and contributing to well-informed infrastructure development forecast science.

DOI: 10.1103/kysp-hgxt

## I. INTRODUCTION

Despite often being treated as secondary in investment strategies, the development of electrical networks is a fundamental component of any successful energy transition strategy [1]. However, it is not an easy and automatic consequence of production investment. To help decision-makers, there is a need for tools to model energy networks and estimate their global cost and opportunities for optimization and gains. To efficiently test the many evolution scenarios, such a model must be simple yet able to capture the properties of the real network. In this context, network theory that provides powerful tools for understanding complex systems across disciplines [2] is very helpful.

Pagani and Aiello used this tool to develop evolution models in the energy transition context, but with a focus on the distribution layer only [3]. Some generative models are based on multiple topological characteristics taken as the input [4]. In contrast, our approach seeks to minimize the number of required inputs. Sequential growth models, i.e., graph models where nodes are added one by one, seem relevant to represent the evolution of power grids. Numerous graph theoretical studies built those models. In the early years of power grid network modeling, Roy et al. proposed a sequential growth where the connections were selected according to spatial distance and connectivity of the target node [5]. Unfortunately, the output networks are forced to be trees, failing to reproduce the loop phenomenon, which

is crucial for realistic robustness. Rosas-Casals et al. [6] developed a model that simulated well the history of the 400 kV layer of the French transmission grid (Réseau de transport d'électricité, RTE for short). They used exponentially decreasing statistics for the length of newly added edges and a cost function for the temporal part. It shows good results but only for the 400 kV layer. Two other power grid generation procedures are presented in a book by Mei et al. [7] and in a publication by Schultz et al. [8]. Both models show a sequential addition of nodes supplemented with at least one minimal-length edge to generate a small-world structure intuitively. Nevertheless, they do not consider how the introduction of a partial preferential attachment could improve the accuracy of degree distribution prediction, as demonstrated by the work of Irfan et al. in a nonspatial case [9].

Based on these ideas, the growth model implemented previously [10] relies on the hypothesis of the uniform spread of substations. Here, we aim to refine that model to consider the territorial inhomogeneities in power demand and the associated topology. The paper is organized as follows. In the first part, we recall our previous uniform growth model and propose a method to mimic dense urban areas and their specific topology. The second part underlines improvements to the refined model when we compare it to an actual electrical network.

## II. HOMOGENEOUS GROWTH MODEL

In our previous homogeneous model, we first defined an area $S$, then at each step $t$, we added a node ($n_t$) to a uniformly random location; connected it to its spatially nearest neighbor; and, with a probability $q$, connected it to its second-nearest neighbor. On average, the final number of edges after $N$ steps ($N$ added nodes) was $M$ =

<footnote>*Contact author: emile.emery@etik.com</footnote>

<footnote>Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.</footnote>
$N(1 + q)$, so we selected $q$ according to the objective numbers $M$ and $N$. This ensures the inequalities $N < M < 2N$ and the connectedness of the final graph, which are standard properties among power grids [11,12]. The growth dynamics are used here for the fingerprints they add to the final structure of the graph [13]; these are associated with the characteristics of an electrical network. On average, the components of the adjacency matrix of the network at step $t$ yielded

$$A_{ij}(q) = \begin{cases} 1 & \text{if } (i,j) = (1,2) \text{ or } (2,1), \\ 0 & \text{if } i = j, \\ \frac{1+q}{\max(i,j)-1} & \text{otherwise,} \end{cases} \quad (1)$$

with $i,j \le t$. This is useful for the analytical computation of various measures and metrics. The birth process was a sequential Poisson sprinkling. Consequently, the probability of finding $k$ nodes on a surface $s$, knowing the density $\lambda$, was

$$Q(k, s, \lambda) = \frac{(\lambda s)^k e^{-\lambda s}}{k!} \quad (2)$$

Thus, when we added node $n_t$ at step $t$ to the space of density $\lambda_t = (t-1)/S$, the probability that $n_t$ had its nearest neighbor at a distance between $r$ and $r + dr$ was the following [14]:

$$P_t(r)dr = 2\pi r \lambda_t Q(0, \pi r^2, \lambda_t)dr. \quad (3)$$

And for the next-nearest neighbor, it was

$$P'_{t}(r)dr = 2\pi r \lambda_{t} Q(1, \pi r^{2}, \lambda_{t})dr. \quad (4)$$

Knowing this, the distance between $n_t$ and its nearest neighbor at step $t > 1$ was

$$\ell_t = \int_0^\infty r P_t(r) dr = \frac{1}{2} \sqrt{\frac{S}{t-1}}. \quad (5)$$

And for the second-nearest neighbor,

$$\ell'_{t} = \int_{0}^{\infty} r P'_{t}(r) dr = \frac{3}{2} \ell_{t}, \quad (6)$$

with, by definition, $\ell_1 = \ell'_1 = \ell'_2 = 0$. Taking the large $N$ limit, the average total spatial length of the graph was

$$L_{\text{tot}} = \sum_{t=1}^{N} \ell_t + q \ell'_t \approx \left(1 + \frac{3}{2}q\right) \sqrt{NS}. \quad (7)$$

When this process was applied to a square territory with the area of France, we recovered a long-length distribution close to the one of the national transmission grid (see

<figure><img src="image_4.png" /><figcaption>FIG. 1. Length distribution for growth models versus French power grid data. Outputs from growth models are averaged over 50 iterations, and vertical lines represent standard errors.</figcaption></figure>

Fig. 1) and the same small-world behavior thanks to the long-distance shortcut lines linking low $t$ nodes [10]. The degree distribution presented an exponential decay with an already known but still unexplained offset between $1/\langle k \rangle$ and the exponent of the exponential fit [8]. Below, we propose adjustments to upgrade the model, to remove this offset, and to explain the short-length distribution.

### III. URBAN INHOMOGENEITY

The aforementioned procedure [10] simplified the previously established models [7,8] to explore different types of complexity from this basis. A fruitful way to refine this type of procedure is to go beyond the homogeneity hypothesis [15]. This is done by increasing the probability of hosting new nodes in urban areas, which are defined as follows. Let $N_u$ be the number of circular urban areas. To represent the additional need for substations to feed the city, $c \in [1, N_u]$, we define an urbanization function, $U_c(r)$, that decreases with $r$, the distance from the center of the urban area, $c$, emulating the densification of infrastructures in urban centers. This function is continuous to follow the need to go beyond the rural-urban dichotomy [16] and is proportional to the birth probability gain of this specific urban location. As this increase in probability is proportional to the number of supply units for the area, it is a function of the population according to a power law [17]:

$$U_c(r) \propto d_c(r)^\gamma, \quad (8)$$

where $d_c(r)$ is the population density at a distance $r$ from the center of the urban area, $c$. It presumably follows
Clark's law [18]:

$$d_c(r) \propto e^{-r/r_0}, \qquad (9)$$

with $r_0$ being of the order of the characteristic size of a city. So, we could write

$$U_c(r) = K_c e^{-\gamma r/r_0}, \qquad (10)$$

where $K_c$ is proportional to the city's total energy consumption and has an inverse length dimension. These definitions replicate the known fact that the number of elements in the grid increases with the power demand by the city and decreases with the distance from the center [19]. The probability of adding an urban node at a distance between $r$ and $r + dr$ from the center of the urban area, $c$ (or, in other words, the probability gain for this location during the growth), is thus given by

$$p_c(r)dr = U_c(r)dr. \qquad (11)$$

The probability of adding a rural node to this exact location is

$$p_c^{\text{rur}}(r)dr = (1 - \tau) \frac{2\pi r dr}{S}, \qquad (12)$$

with the proportion of urban nodes being

$$\tau = \frac{r_0}{\gamma} \sum_c K_c. \qquad (13)$$

More details of the framework are given in the Supplemental Material [20].

## IV. PREFERENTIAL ATTACHMENT REFINING

Breaking down the birth process into rural dynamics, producing low-density regions, and urban dynamics, producing high-density areas, enables the definition of different wiring mechanisms. As shown in Fig. 1, the homogeneous growth process already gives a reasonable approximation of the length distribution. The main inaccuracy of this model lies in the overestimation of the decay of the degree distribution (see Fig. 2). A fat-tail degree distribution is generally obtained with preferential attachment, which relies on the assumption of a connection probability increasing with the degree [21]. Changing all connection procedures to preferential attachment would significantly impact the length distribution since the edge attribution would favor connectivity over local length minimization. However, when implementing that change for dense urban areas only, the additional length is negligible. After adding a node $n_t$ to an urban area, the algorithm looks at the $N_{NN}$ nearest neighbors and adds a link between $n_t$ and its neighbor of highest degree among those $N_{NN}$ (if there is a tie, the spatially closest is chosen). This reflects the assumed economy of scale occurring when the degree increases on a given substation. Figure 3 illustrates the change in the final structure with a focus on an urban area.

## V. RESULTS

We take the French transmission grid as an applied case, following Ref. [10]. Thus, we select order $N = 4626$, size $M = 7206$, and area $S \simeq 540 000 \text{ km}^2$. The following parameters seemed reasonable for inhomogeneous growth. The exponent $\gamma = 0.85$ from West [23] is a standard sub-linear scaling that reflects the economy of scale in urban

<figure><img src="image_5.png" /><figcaption>FIG. 2. Degree distribution for the two growth models versus French power grid data (RTE). Figure on the left represents the initial homogeneous model [10], and the one on the right is the upgraded inhomogeneous model. Outputs are averages over a hundred iterations, and standard error bars are represented by black vertical lines. Exponents of the exponential and power law fits are given. Insets, visualization of the output of the model, taking parameter values, as discussed in the results section.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>FIG. 3. Comparison, using GEPHI [22], of one urban area to the surrounding rural territory. Typical radius, $r_0$, of the cities is shown with a dashed circle. Size and color of the nodes represent the degree.</figcaption></figure>

infrastructure. The Clark exponent for France is taken as $r_0^{-1} = 0.28$, following statistical works on the urban area of Lyon [24]. For the urban rate, we take $\tau = 0.2$, which is the product of the proportion of lowest voltage nodes with the proportion of total electrical power that is consumed in urban areas [25]. Comparing the connectivity of the model with the real-world topology on a restricted urban area, we choose to fix $N_{NN} = 2$ (see the Supplemental Material [20] for details). We select the population values of the $N_u = 11$ cities with more than 200 000 inhabitants to write

the urbanization functions with

$$K_c \propto \frac{g_c}{\sum_k g_k}, \qquad (14)$$

where $g_c$ represents the overall population of city c and $1 \le c \le N_u$. A higher value of $N_u$ will induce no significant change in the results developed in the following. For the simple square models, the territory is a square of side length $\sqrt{s}$. The algorithms are also compatible with the geographic format, and we use the continental French shape as a territory for geographic models. We found no fundamental differences between those two types of territory in the numerical experiments, as shown in Figs. 1 and 4 (as well as the Supplemental Material [20]). Figure 2 and Table I show that the inhomogeneous version of the growth process fits the French transmission degree data better than the homogeneous one. The existence of high degrees in nested areas shifts the exponential slope to an exponent close to that of the French grid: $1/(\langle k \rangle) \approx 0.35$ [10]. Even if the exponential is still the best fit of the inhomogeneous model, an interesting result is that this model's slope is closer to a power law than the homogeneous one. In Fig. 1, we find that the distribution of long-distance shortcuts is close to that from the RTE network for all types of growth. As expected, the inhomogeneous growth increases the proportion of short-distance edges, making it a better approximation of the actual network. Figure 4 shows that Eq. (7) gives a reasonable estimate for the total cable length of the grid model in the homogeneous growth case, slightly above the inhomogeneous growth. This gives access to the amount of matter and energy invested in the infrastructure. Table I proves the length is nearing that

<figure><img src="image_7.png" /><figcaption>FIG. 4. Total length in meters (see factor $10^8$) of the edges of the graph for varying N at $q = q_{RTE}$ (left) or varying q at $N = N_{RTE}$ (right). Outputs from growth models are averaged over 15 iterations for each data point, and error bars are too small to be visible with an average amplitude of $2.8 \times 10^5$ m. Minimum spanning tree cases estimate the values of the lower bounds for the total length of a spatially connected network computed according to Refs. [26,27].</figcaption></figure>
<table><caption>TABLE I. The Kullback-Leibler (KL) divergence is a measure of the difference between two distributions [28]. The higher the KL divergence, the higher the difference between the degree distribution of the growth model (GM) on average and the degree distribution of real-world data. The small-world propensity (SWP) measure is close to one when the network shows small-world behavior [10,29]. All the averages here are made on 50 samples. The average total length, $L_{tot}$, for various models must be compared to the two values 103 541 and 87 913 km for RTE, depending on whether or not we count the parallel edges multiple times.</caption><thead><tr><th></th><th>Homogeneous GM</th><th>Inhomogeneous GM</th><th>Homogeneous GM in France</th><th>Inhomogeneous GM in France</th></tr></thead><tbody><tr><td>KL divergence</td><td>0.1301</td><td>0.0957</td><td>0.1289</td><td>0.0962</td></tr><tr><td>SWP φ</td><td>0.999 ± 1.4 × 10<sup>-4</sup></td><td>0.999 ± 1.8 × 10<sup>-4</sup></td><td>0.999 ± 1.6 × 10<sup>-4</sup></td><td>0.999 ± 1.6 × 10<sup>-4</sup></td></tr><tr><td>weighted SWP φ<sub>w</sub></td><td>0.973 ± 4.2 × 10<sup>-4</sup></td><td>0.966 ± 1.1 × 10<sup>-3</sup></td><td>0.972 ± 3.9 × 10<sup>-4</sup></td><td>0.999 ± 1.1 × 10<sup>-3</sup></td></tr><tr><td>L<sub>tot</sub> (km)</td><td>92 981 ± 1031</td><td>87 048 ± 1150</td><td>93 510 ± 1195</td><td>89 560 ± 1118 km</td></tr></tbody></table>

of RTE and the small-world property is satisfied, which is related to efficiency [10] and optimization in terms of stability and synchronizability [30].

The redundancy of the model can be measured through the Fiedler eigenvalue (also called algebraic connectivity), as detailed in Ref. [8]. Figure 5 shows that this metric exhibits a linear growth with $q$ and that the real-world redundancy is obtained when giving the corresponding input in the growth model. However, contrary to the energetic analysis, we need to consider only the simple edges from the data to obtain the alignment with $\lambda_2^{RTE}$. This leads to a lower value, $q'_{RTE} = (M' - N)/N$, with $M' = 6272$.

The evolution of the energy invested in Joule losses and infrastructure, depending on $q$, is given in Fig. 6 for the inhomogeneous model (see the Supplemental Material [20] for details on the method). In particular, we retrieve findings by Jorge et al. [32] that Joule losses dominate the energy costs of networks. This fact is of prime importance when new devices, such as storage facilities, have to be connected to the grid. One interested in energy investment

and losses only can use the simpler homogeneous model, which is quicker and shows similar results (see the Supplemental Material [20]). In the real development process of the grid, one should avoid Braess's paradox, which is when poorer performance results from network reinforcement [33]. However, any such local effect is averaged out by the method used here, which deals with mean behaviors.

## VI. CONCLUSION AND OUTLOOK

We developed a methodology to build network models from first principles with high-voltage transmission power grid characteristics. The presented model aims to estimate global-scale energy and material investments given a few inputs. It does not capture aspects like the precise dynamics of power flows or synchronization stability that may have an influence on the effective layout of the grid [34].

<figure><img src="image_8.png" /><figcaption>FIG. 5. Evolution of the Fiedler eigenvalue of the output graph with $q$. Colored vertical lines represent standard error. Value $\lambda_2^{RTE}$ corresponding to the data network is obtained for the input $q'_{RTE}$ given by the number of nodes and simple edges in the data.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>FIG. 6. Joule losses for 15 different values of $q$; average and standard errors are computed over 10 different iterations. 100% Joule loss value corresponds to a power loss of 133.93 GW. For comparison, the dotted horizontal line refers to the percentage of Joule losses in the real French transmission power grid [31]. Estimate of energy invested in infrastructure is also shown (see the Supplemental Material [20] for details).</figcaption></figure>
However, this infrastructure topology prediction tool is relevant to the energy transition perspective in a context of resource scarcity, and its flexibility makes it valuable for various types of land areas, energy mixes, or development scenarios. A simple formula, Eq. (7), is derived to estimate the total length of electric cables needed to equip a country of area S with N substations. Inhomogeneity was added by dividing the birth process into a rural one that generated a relevant length distribution and an urban one with a light preferential attachment feature that could refine the degree distribution. This reduces the total network length by a very short amount when parameters are taken in the relevant range. The topological properties do not depend on the shape of the territory, which is a promising feature for international usage. A natural extension of this research could include the growth of the low-voltage electrical network. Scaling down the process, more details on the differentiation between rural and urban structures may be added, following works on the consumption spatial distribution [35] or the interconnection with socioecological systems [36]. Scaling up, further investigation is needed to test the model and its generalizations to other regions, once other high-quality weighted spatial topology databases are available. Leveraging these results, future research might explore the coupling between multiple national grids, paying particular attention to the internetwork bottlenecks. On top of these models, other criteria could be added to introduce storage systems, electric vehicles, and smart grid aspects.

The code developed for this paper is available at [37].

## ACKNOWLEDGMENTS

We would like to acknowledge interesting discussions with Olivier Vidal, François Verzier, Nicolas Jonqueres, and Baptiste Andrieu. On top of a well-documented and open-access website [38], we also benefited from information on RTE networks, notably through exchanges with Nicolas Lhuillier, Guillaume Busato, and Jean-Yves Bourmaud. E.E. wants to thank Francesco Zigliotto and Nikita Deniskin for helpful discussions and Celestin Zimmerlin for his feedback. This research was funded via FOCUS TIME CEA.

H.B. and S.A. supervised this work. E.E. performed the investigation, formal analysis, software development, and initial draft writing. All authors contributed to the study's conceptualization, validation, final writing, review, and editing.

The authors have no conflicts of interest to declare.

## DATA AVAILABILITY

The data that support the findings of this article are openly available [38–40].

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